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Reference Skeletal Dosimetry Model for an Adult Male Radionuclide Therapy Patient Based on 3D Imaging and Paired-Image R...


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REFERENCE SKELETAL DOSIMETR Y MODEL FOR AN ADULT MALE RADIONUCLIDE THERAPY PATIENT BASED ON 3D IMAGING AND PAIREDIMAGE RADIATION TRANSPORT By AMISH P. SHAH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004

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Copyright 2004 by Amish P. Shah

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iii ACKNOWLEDGMENTS Several people contributed a great deal to this work. First, I would like to thank Dr. Wesley E. Bolch for his support, guidan ce, and encouragement. I appreciate his patience over the past 5 years; but most importantly, I valu e his friendship and everything he has done to help me mature into a better student. I also thank Dr. David Hintenlang, Dr. Edward Dugan, Dr. Christopher Batich, and Dr. Didier Rajon for their suggestions and for being part of my committee. I would also like to thank my colleagues that reside in the halls throughout the Department of Nuclear and Radi ological Engineering for their friendship. I also thank all the current and former members of the Bone Imaging and Dosimetry project for the time we have spent working together and ponde ring over trabecular bone. Specifically, I thank Dr. Phillip Patton and Dr. Derek Jokisch for sharing their infinite wisdom in all aspects of bone dosimetry and life. I would also like to recognize the faculty and staff in the departments of Nuclear and Radiological Engineering and Biomedical Engineering for their assistance. As I am in the middle of the two departments, both we re instrumental in their help and support over the past 5 years. I also thank the Gr aduate School and Dr. Bolch, for the opportunity to pursue an interest in business manageme nt. Without their he lp, I could not have concurrently earned a Master of Science in Business Management. Finally, I would like to thank some people ve ry close to my heart. My parents and my brother, Angesh, have been right there beside me every step of the way. Their love

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iv and support are immeasurable and cannot be understood with words. I thank my wife, Priti, for always knowing that things are going to work out, and never worrying about making the wrong decisions in life. Among other things, I thank her fo r bringing that gift into my life. I also thank Priti for her love, in fectious laughter, and most of all, patience.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................ix LIST OF FIGURES.......................................................................................................xxiv ABSTRACT...................................................................................................................xxxi CHAPTER 1 INTRODUCTION........................................................................................................1 2 BACKGROUND..........................................................................................................6 Bone Structure and Physiology....................................................................................6 Radionuclide Therapies for Cancer..............................................................................8 Marrow Toxicity...........................................................................................................9 Previous Methods of Trabecular Bone Dosimetry.....................................................10 Internal Dosimetry Calculations.................................................................................13 3 PAIRED-IMAGE RADIATION TRANSPORT MODEL FOR SKELETAL DOSIMETRY.............................................................................................................17 Introduction.................................................................................................................17 Materials and Methods...............................................................................................21 Cadaver Selection................................................................................................21 In-Vivo Computed Tomography Scanning.........................................................21 Bone Harvesting and Ex-Vivo Computed Tomography Scanning.....................22 Image Segmentation of Spongiosa and Cortical Bone Regions..........................23 Microimaging of Trabecular Spongiosa..............................................................24 Voxel-Based Infinite Spongiosa Transport (VBIST) Model...............................25 Paired-Image Radiation Tran sport Model (L4 Vertebra)....................................26 Paired-Image Radiation Trans port Model (Proximal Femur).............................28 Results........................................................................................................................ .29 Absorbed Fractions to Active Marrow within the L4 Vertebra...........................29 Absorbed Fractions to Active Marrow within the Proximal Femur....................31 Absorbed Fractions to Endosteal Tissues............................................................32

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vi Discussion...................................................................................................................33 Conclusion..................................................................................................................35 4 BETA-PARTICLE ENERGY LOSS TO CORTICAL BONE VIA PAIRED-IMAGE RADIATION TRANSPORT: CORRECTI ONS TO CLINICAL MODELS OF SKELETAL TISSUE DOSE......................................................................................48 Introduction.................................................................................................................48 Materials and Methods...............................................................................................52 Cadaver Selection................................................................................................52 In-Vivo Computed Tomography Scanning.........................................................52 Bone Harvesting and Ex-Vivo Computed Tomography Scanning.....................53 Image Segmentation of Spongiosa and Cortical Bone Regions..........................54 Micro-Computed Tomography of Trabecular Spongiosa...................................55 Voxel-Based Infinite Spongiosa Transport (VBIST) Model...............................55 Paired-Image Radiation Tr ansport (PIRT) Model...............................................56 Results........................................................................................................................ .60 Absorbed Fractions to Active Marrow within the Pelvis....................................60 Absorbed Fractions to Active Marrow within the Cranium................................61 Absorbed Fractions to Active Marrow within the Rib Cage...............................62 Absorbed Fractions to Endosteal Tissues............................................................64 Discussion...................................................................................................................65 Conclusion..................................................................................................................66 5 SKELETAL CHORD-LENGTH DISTRI BUTIONS FOR ICRP REFERENCE MALE VERSUS THE UF REFERE NCE MALE CANCER PATIENT...................82 Introduction.................................................................................................................82 Current Reference Male Skeletal Model.............................................................82 University of Florida Reference Male Cancer Patient........................................84 Materials and Methods...............................................................................................85 Bone Specimen Selection....................................................................................85 Microimaging of Trabecular Spongiosa..............................................................86 Measurement of Chord-Length Distributions.....................................................87 Averaging of ChordLength Distributions..........................................................88 Reference Skeletal Sites......................................................................................88 Results........................................................................................................................ .89 Discussion...................................................................................................................91 Femoral Head and Neck......................................................................................91 Cervical and Lumbar Vertebrae..........................................................................92 Ribs......................................................................................................................93 Cranium...............................................................................................................94 Pelvis (Os Coxae)................................................................................................95 Remaining Marrow-Containing Bones of the Skeleton......................................95 Weighting Schemes for Non-Imaged Bone Sites in the Leeds Data...................96 Conclusion..................................................................................................................98

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vii 6 CHORD-BASED VERSUS VOXELBASED METHODS OF ELECTRON TRANSPORT IN SKELETAL DOSIMETRY........................................................122 Introduction...............................................................................................................122 Materials and Methods.............................................................................................124 Cadaver Selection..............................................................................................124 Trabecular Microstructure Acquisition.............................................................125 Voxel-Based Infinite Spongiosa Transport (VBIST) Model.............................126 Chord-Based Infinite Spongiosa Transport (CBIST) Model.............................127 Chord-Length Distributions for th e UF Reference Cancer Patient...................130 Convergence Limits for Absorbed Fractions under CBIST and VBIST...........131 Results and Discussion.............................................................................................133 Trabecular Microstructure of the Leeds and UF Reference Subjects................133 Electron Dosimetry Comparisons between the UF and Leeds Microstructures..............................................................................................135 Comparison of CBIST and VBIST for Marrow Space Targets........................138 Comparison of CBIST and VBIST for Bone Endosteum Targets....................140 Conclusion................................................................................................................141 7 REFERENCE SKELETAL DOSIMETR Y MODEL FOR AN ADULT MALE RADIONUCLIDE THERAPY PATIENT...............................................................166 Introduction...............................................................................................................166 Materials and Methods.............................................................................................170 Reference Adult Male Cadaver Selection.........................................................171 Skeletal-Image Database for the UF RMCP.....................................................171 Radiation Transport Modeling..........................................................................173 Paired-Image Radiation Tr ansport (PIRT) Model.............................................175 Mass Calculation for UF RMCP.......................................................................177 Skeletal Averaging of Absorbed Frac tions and S Values for the UF RMCP....178 Results.......................................................................................................................1 81 Discussion.................................................................................................................182 Comparison of UF and ICRP Reference Tissue Masses...................................182 PIRT Model Simulations – Sacrum...................................................................185 Energy Loss to Cortical Bone............................................................................186 Skeletal-Averaged Absorbed Fractions.............................................................187 Site-Specific Radionuclide S Values.................................................................190 Skeletal-Averaged S Values..............................................................................192 Scalability of the UF Reference Skeletal Model...............................................194 Conclusion................................................................................................................196 8 CONCLUSIONS AND FUTURE WORK...............................................................219 Conclusions...............................................................................................................219 Future Work..............................................................................................................222 Improvements in the use of Voxel Models........................................................222 Improvements in the Characterization of Active Marrow.................................223

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viii Improvements to the Skeletal Database............................................................224 Scaling of Reference S Values to a Patient.......................................................225 Clinical Application of Reference S Values......................................................226 APPENDIX A UNIVERSITY OF FLORIDA REFERENCE ADULT MALE RADIONUCLIDE PATIENT: CT IMAGE DETAILS...........................................228 B MICROIMAGE PARAMETERS FO R SKELETAL SPONGIOSA.......................236 C IMAGE PROCESSING (C PROGRAMS)..............................................................238 D IMAGE PROCESSING TECHNIQUE....................................................................251 E PAIRED-IMAGE RADIATION TRANSPORT (PIRT) MODEL (EGSNRC USER CODE)...........................................................................................................273 F PIRT MODEL FOR THE PROXIMAL FEMUR (EGSNRC USER CODE)..........316 G 66-YEAR UF REFERENCE MALE CANCER PATIENT CHORD-LENGTH DISTRIBUTIONS AND THE TR ILINEAR CHORD-LENGTH CALCULATION......................................................................................................360 H TABLES OF SITE-SPECIFIC AB SORBED FRACTIONS FOR THE UF REFERENCE MALE...............................................................................................387 I TABLES OF SITE-SPECIFIC S VALUES FOR THE UF REFERENCE MALE...............................................................................................425 REFERENCES................................................................................................................450 BIOGRAPHICAL SKETCH...........................................................................................459

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ix LIST OF TABLES Table page 3-1 Tissue compositions (% by mass) and ma ss densities used in either the IST and PIRT models of skeletal dosimetry..............................................................45 3-2 Tissues masses used in the paired-i mage radiation transport (PIRT) model (100% marrow cellularity)..................................................................................46 3-3 Ratio of the radionuclide S value for an active marrow (TAM) target as given by the infinite spongiosa transport (IST) model to that given by the pairedimage radiation transport (PIRT) model.............................................................47 4-1 Tissues masses used in the paired-ima ge radiation transport (PIRT) model......80 4-2 Ratio of the radionuclide S value for an active marrow (TAM) target as given by the voxel-based infinite spongiosa tran sport (VBIST) model to that given by the paired-image radiati on transport (PIRT) model.......................................81 5-1 Comparison of measured mean chord lengths with values published from the University of Leeds (Whitwell 1973).........................................................121 6-1 Mean values of trabecular and marro w cavity chord-lengths as given by the present UF study and those published from the University of Leeds...............163 6-2 Ratios of absorbed frac tions to active marrow (UF values to Leeds values under CBIST)....................................................................................................164 6-3 Ratios of absorbed fractions to bone endosteum (UF values to Leeds values under CBIST)....................................................................................................165 7-1 Comparison between reference masse s used at UF (pre sent study) and ICRP 89 for tissues within the marrow cavities................................................211 7-2 Pertinent values used in the calcula tion of data for the PIRT model of skeletal dosimetry.............................................................................................212 7-3 Measurements of spongiosa volume a nd cortical bone volume within each of the five (5) lumbar vertebrae of the UF Reference Male Cancer Patient.....213

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x 7-4 Surface-to-volume ratios within skel etal sites of trabecular spongiosa found within the UF 66-year male, the Leeds 44-year male.......................................214 7-5 Comparison of assigned ce llularity factors and fracti ons of different skeletal components between UF (present stu dy) and Eckerman and Stabin (2000)....215 7-6. Skeletal-averaged absorbed fractions for monoenergetic electrons for the UF Reference Male Cancer Patient and those of the ICRP Reference Man...........216 7-7 Radiation characteristic s of radionuclides used for calculation of S values.....217 7-8 Skeletal-averaged S values (mGy/MBq -s) for different combinations of source and target regions within the spongiosa and cortical bone....................218 A-1 Ex-vivo CT parameters used in the PIRT model for radiation transport in order to define the binary contours of the macroimage at each skeletal site....235 B-1 MicroCT image parameters used in th e PIRT model in order to define the microimage of spongiosa in radiation transport................................................237 D-1 Breakdown of dimensions necessary for opening images in Adobe................257 D-2 Example of the method for determin ation of the region of interest..................264 G-1 Normalized 3D chord-length distributi ons through the bone trabeculae of the left and right femur heads, left and right femur necks. First 50 bins...............361 G-2 Normalized 3D chord-length distribu tions through the bone trabeculae of the left and right femur heads, left and right femur necks. Second 50 bins.....362 G-3 Normalized 3D chord-length distributi ons through the bone trabeculae of the pubis, ilium, ischium, right and left s capula, sternum, and the averages for the os coxae and scapula. First 50 bins............................................................363 G-4 Normalized 3D chord-length distributi ons through the bone trabeculae of the pubis, ilium, ischium, right and left s capula, sternum, and the averages for the os coxae and scapula. Second 50 bins........................................................364 G-5 Normalized 3D chord-length distributi ons through the bone trabeculae of the right and left clavicle, ri ght and left humerus, C3 and C6 vertebra, and the averages for those respective bone sites First 50 bins......................................365 G-6 Normalized 3D chord-length distributi ons through the bone trabeculae of the right and left clavicle, ri ght and left humerus, C3 and C6 vertebra, and the averages for those respective bone sites. Second 50 bins................................366

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xi G-7 Normalized 3D chord-length distributi ons through the bone trabeculae of the T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages for the thoracic and lumbar vertebra. First 50 bins..........................................367 G-8 Normalized 3D chord-length distributi ons through the bone trabeculae of the T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages for the thoracic and lumbar vertebra. Second 50 bins.....................................368 G-9 Normalized 3D chord-length distributi ons through the bone trabeculae of the right and left upper rib, middle rib, and lowe r rib. Also shown is the average for the single rib. First 50 bins.........................................................................369 G-10 Normalized 3D chord-length distributi ons through the bone trabeculae of the right and left upper, middle, and lower ri bs. Also shown is the average for the single rib. Second 50 bins..........................................................................370 G-11 Normalized 3D chord-length distributi ons through the bone trabeculae of the mandible, frontal bone, occipital bone, a nd right and left parietal bones. Also shown is the average for the cranium. First 50 bins................................371 G-12 Normalized 3D chord-length distributi ons through the bone trabeculae of the mandible, frontal bone, occipital bone, a nd right and left parietal bones. Also shown is the average for the cranium. Second 50 bins...........................372 G-13 Normalized 3D chord-length distribu tions through the marrow cavities of the left and right femur heads, left and ri ght femur necks, and their respective averages. First 50 bins......................................................................................373 G-14 Normalized 3D chord-length distribu tions through the marrow cavities of the left and right femur heads, left and ri ght femur necks, and their respective averages. Second 50 bins.................................................................................374 G-15 Normalized 3D chord-length distribu tions through the marrow cavities of the pubis, ilium, ischium, right and left s capula, sternum, and the averages for the os coxae and scapula. First 50 bins............................................................375 G-16 Normalized 3D chord-length distribu tions through the marrow cavities of the pubis, ilium, ischium, right and left s capula, sternum, and the averages for the os coxae and scapula. Second 50 bins........................................................376 G-17 Normalized 3D chord-length distribu tions through the marrow cavities of the right and left clavicle, ri ght and left humerus, C3 and C6 vertebra, and the averages for those respective bone sites. First 50 bins....................................377 G-18 Normalized 3D chord-length distribu tions through the marrow cavities of the right and left clavicle, ri ght and left humerus, C3 and C6 vertebra, and the averages for those respective bone sites. Second 50 bins................................378

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xii G-19 Normalized 3D chord-length distribu tions through the marrow cavities of the T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages for the thoracic and lumbar vertebra. First 50 bins..........................................379 G-20 Normalized 3D chord-length distribu tions through the marrow cavities of the T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages for the thoracic and lumbar vertebra. Second 50 bins.....................................380 G-21 Normalized 3D chord-length distribu tions through the marrow cavities of the right and left upper rib, middle rib, and lowe r rib. Also shown is the average for the single rib. First 50 bins.........................................................................381 G-22 Normalized 3D chord-length distribu tions through the marrow cavities of the right and left upper, middle, and lower ri bs. Also shown is the average for the single rib. Second 50 bins..........................................................................382 G-23 Normalized 3D chord-length distribu tions through the marrow cavities of the mandible, frontal bone, occipital bone, a nd right and left parietal bones. Also shown is the average for the cranium. First 50 bins................................383 G-24 Normalized 3D chord-length distribu tions through the marrow cavities of the mandible, frontal bone, occipital bone, and right and left parietal bones. Second 50 bins......................................................................................384 H-1. Absorbed fractions in the right proxi mal femur for sources in the trabecular bone volume at varying marrow cellularity......................................................388 H-2 Absorbed fractions in the left proximal femur for sources in the trabecular bone volume at varying marrow cellularity......................................................388 H-3 Absorbed fractions in the right hum erus for sources in the trabecular bone volume at varying marrow cellularity...............................................................388 H-4 Absorbed fractions in the left humer us for sources in the trabecular bone volume at varying marrow cellularity...............................................................389 H-5 Absorbed fractions in the cervical ve rtebra for sources in the trabecular bone volume at varying marrow cellularity......................................................389 H-6 Absorbed fractions in the thoracic ve rtebra for sources in the trabecular bone volume at varying marrow cellularity......................................................389 H-7 Absorbed fractions in the lumbar vert ebra for sources in the trabecular bone volume at varying marrow cellularity...............................................................390 H-8 Absorbed fractions in the sacrum fo r sources in the trabecular bone volume at varying marrow cellularity............................................................................390

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xiii H-9 Absorbed fractions in the os coxae for sources in the trabecular bone volume at varying marrow cellularity............................................................................390 H-10 Absorbed fractions in the cranium fo r sources in the trabecular bone volume at varying marrow cellularity............................................................................391 H-11 Absorbed fractions in the mandible for sources in the trabecular bone volume at varying marrow cellularity...............................................................391 H-12 Absorbed fractions in the ribs for so urces in the trabecul ar bone volume at varying marrow cellularity................................................................................391 H-13 Absorbed fractions in the sternum for sources in the trabecular bone volume at varying marrow cellularity...............................................................392 H-14 Absorbed fractions in the right clavicle for sources in the trabecular bone volume at varying marrow cellularity...............................................................392 H-15 Absorbed fractions in the left clavic le for sources in the trabecular bone volume at varying marrow cellularity...............................................................392 H-16 Absorbed fractions in the right s capula for sources in the trabecular bone volume at varying marrow cellularity...............................................................393 H-17 Absorbed fractions in the left scapula for sources in the trabecular bone volume at varying marrow cellularity...............................................................393 H-18 Absorbed fractions in the femur h ead for sources in the trabecular bone volume at varying marrow cellularity...............................................................393 H-19 Absorbed fractions in the right proxi mal femur for sources in the trabecular active marrow at varying marrow cellularity....................................................394 H-20 Absorbed fractions in the left proximal femur for sources in the trabecular active marrow at varying marrow cellularity....................................................394 H-21 Absorbed fractions in the right humer us for sources in the trabecular active marrow at varying marrow cellularity..............................................................394 H-22 Absorbed fractions in the left humerus for sources in the trabecular active marrow at varying marrow cellularity..............................................................395 H-23 Absorbed fractions in the cervical ve rtebra for sources in the trabecular active marrow at varying marrow cellularity....................................................395 H-24 Absorbed fractions in the thoracic ve rtebra for sources in the trabecular active marrow at varying marrow cellularity....................................................395

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xiv H-25 Absorbed fractions in the lumbar ve rtebra for sources in the trabecular active marrow at varying marrow cellularity....................................................396 H-26 Absorbed fractions in the sacrum for sources in the trabecular active marrow at varying marrow cellularity............................................................................396 H-27 Absorbed fractions in the os coxae for sources in the trabecular active marrow at varying marrow cellularity..............................................................396 H-28 Absorbed fractions in the cranium for sources in the trabecular active marrow at varying marrow cellularity..............................................................397 H-29 Absorbed fractions in the mandible for sources in the trabecular active marrow at varying marrow cellularity..............................................................397 H-30 Absorbed fractions in the ribs for sources in the trabecular active marrow at varying marrow cellularity............................................................................397 H-31 Absorbed fractions in the sternum for sources in the trabecular active marrow at varying marrow cellularity..............................................................398 H-32 Absorbed fractions in the right clavicle for sources in the trabecular active marrow at varying marrow cellularity..............................................................398 H-33 Absorbed fractions in the left clavic le for sources in the trabecular active marrow at varying marrow cellularity..............................................................398 H-34 Absorbed fractions in the right scapula for sources in the trabecular active marrow at varying marrow cellularity..............................................................399 H-35 Absorbed fractions in the left scap ula for sources in the trabecular active marrow at varying marrow cellularity..............................................................399 H-36 Absorbed fractions in the femur head for sources in the trabecular active marrow at varying marrow cellularity..............................................................399 H-37 Absorbed fractions in the right proxi mal femur for sources in the trabecular bone surface at varyi ng marrow cellularity......................................................400 H-38 Absorbed fractions in the left proximal femur for sources in the trabecular bone surface at varyi ng marrow cellularity......................................................400 H-39 Absorbed fractions in the right hum erus for sources in the trabecular bone surface at varying marrow cellularity...............................................................400 H-40 Absorbed fractions in the left humer us for sources in the trabecular bone surface at varying marrow cellularity...............................................................401

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xv H-41 Absorbed fractions in the cervical vert ebra for sources in the trabecular bone surface at varying marrow cellularity...............................................................401 H-42 Absorbed fractions in the thoracic vert ebra for sources in the trabecular bone surface at varying marrow cellularity...............................................................401 H-43 Absorbed fractions in the lumbar vert ebra for sources in the trabecular bone surface at varying marrow cellularity...............................................................402 H-44 Absorbed fractions in the sacrum fo r sources in the trabecular bone surface at varying marrow cellularity............................................................................402 H-45 Absorbed fractions in the os coxae fo r sources in the trabecular bone surface at varying marrow cellularity............................................................................402 H-46 Absorbed fractions in the cranium fo r sources in the trabecular bone surface at varying marrow cellularity...........................................................................403 H-47 Absorbed fractions in the mandible for sources in the trabecular bone surface at varying marrow cellularity............................................................................403 H-48 Absorbed fractions in the ribs for so urces in the trabecular bone surface at varying marrow cellularity................................................................................403 H-49 Absorbed fractions in the sternum for sources in the trabecular bone surface at varying marrow cellularity............................................................................404 H-50 Absorbed fractions in the right clavicle for sources in the trabecular bone surface at varying marrow cellularity...............................................................404 H-51 Absorbed fractions in the left clavic le for sources in the trabecular bone surface at varying marrow cellularity...............................................................404 H-52 Absorbed fractions in the right s capula for sources in the trabecular bone surface at varying marrow cellularity...............................................................405 H-53 Absorbed fractions in the left scapula for sources in the trabecular bone surface at varying marrow cellularity...............................................................405 H-54 Absorbed fractions in the femur h ead for sources in the trabecular bone surface at varying marrow cellularity...............................................................405 H-55 Absorbed fractions in the right proxi mal femur for sources in the trabecular bone endosteum at varying marrow cellularity.................................................406 H-56 Absorbed fractions in the left proximal femur for sources in the trabecular bone endosteum at varying marrow cellularity.................................................406

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xvi H-57 Absorbed fractions in the right hum erus for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................406 H-58 Absorbed fractions in the left humer us for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................407 H-59 Absorbed fractions in the cervical vert ebra for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................407 H-60 Absorbed fractions in the thoracic ve rtebra for sources in the trabecular bone endosteum at varying marrow cellularity.................................................407 H-61 Absorbed fractions in the lumbar vert ebra for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................408 H-62 Absorbed fractions in the sacrum for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................408 H-63 Absorbed fractions in the os coxae for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................408 H-64 Absorbed fractions in the cranium for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................409 H-65 Absorbed fractions in the mandibl e for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................409 H-66 Absorbed fractions in the ribs for so urces in the trabecular bone endosteum at varying marrow cellularity............................................................................409 H-67 Absorbed fractions in the sternum for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................410 H-68 Absorbed fractions in the right clavicle for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................410 H-69 Absorbed fractions in the left clavicle for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................410 H-70 Absorbed fractions in the right s capula for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................411 H-71 Absorbed fractions in the left scapula for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................411 H-72 Absorbed fractions in the femur h ead for sources in the trabecular bone endosteum at varying marrow cellularity.........................................................411

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xvii H-73 Absorbed fractions in the right proximal femur for sources in the cortical bone volume at varying marrow cellularity......................................................412 H-74 Absorbed fractions in the left proxim al femur for sources in the cortical bone volume at varying marrow cellularity......................................................412 H-75 Absorbed fractions in the right humer us for sources in the cortical bone volume at varying marrow cellularity...............................................................412 H-76 Absorbed fractions in the left humerus for sources in the cortical bone volume at varying marrow cellularity...............................................................413 H-77 Absorbed fractions in the cervical vertebra for sources in the cortical bone volume at varying marrow cellularity...............................................................413 H-78 Absorbed fractions in the thoracic ve rtebra for sources in the cortical bone volume at varying marrow cellularity...............................................................413 H-79 Absorbed fractions in the lumbar ve rtebra for sources in the cortical bone volume at varying marrow cellularity...............................................................414 H-80 Absorbed fractions in the sacrum for sources in the cortical bone volume at varying marrow cellularity................................................................................414 H-81 Absorbed fractions in the os coxae for sources in the co rtical bone volume at varying marrow cellularity............................................................................414 H-82 Absorbed fractions in the cranium fo r sources in the co rtical bone volume at varying marrow cellularity............................................................................415 H-83 Absorbed fractions in the mandible fo r sources in the co rtical bone volume at varying marrow cellularity............................................................................415 H-84 Absorbed fractions in the ribs for sources in the cortic al bone volume at varying marrow cellularity................................................................................415 H-85 Absorbed fractions in the sternum fo r sources in the cor tical bone volume at varying marrow cellularity................................................................................416 H-86 Absorbed fractions in the right clav icle for sources in the cortical bone volume at varying marrow cellularity...............................................................416 H-87 Absorbed fractions in the left clav icle for sources in the cortical bone volume at varying marrow cellularity...............................................................416 H-88 Absorbed fractions in the right scap ula for sources in the cortical bone volume at varying marrow cellularity...............................................................417

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xviii H-89 Absorbed fractions in the left scap ula for sources in the cortical bone volume at varying marrow cellularity...............................................................417 H-90 Absorbed fractions in the femur h ead for sources in the cortical bone volume at varying marrow cellularity...............................................................417 H-91 Absorbed fractions in the right proxi mal femur for sources in the trabecular marrow cavity at varying marrow cellularity....................................................418 H-92 Absorbed fractions in the left proximal femur for sources in the trabecular marrow cavity at varying marrow cellularity....................................................418 H-93 Absorbed fractions in the right hum erus for sources in the trabecular marrow cavity at varying marrow cellularity....................................................418 H-94 Absorbed fractions in the left humerus for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................419 H-95 Absorbed fractions in the cervical ve rtebra for sources in the trabecular marrow cavity at varying marrow cellularity....................................................419 H-96 Absorbed fractions in the thoracic ve rtebra for sources in the trabecular marrow cavity at varying marrow cellularity....................................................419 H-97 Absorbed fractions in the lumbar ve rtebra for sources in the trabecular marrow cavity at varying marrow cellularity....................................................420 H-98 Absorbed fractions in the sacrum for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................420 H-99 Absorbed fractions in the os coxae for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................420 H-100 Absorbed fractions in the cranium for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................421 H-101 Absorbed fractions in the mandible for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................421 H-102 Absorbed fractions in the ribs for so urces in the trabecular marrow cavity at varying marrow cellularity................................................................................421 H-103 Absorbed fractions in the sternum for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................422 H-104 Absorbed fractions in the right clav icle for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................422

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xix H-105 Absorbed fractions in the left clavic le for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................422 H-106 Absorbed fractions in the right scap ula for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................423 H-107 Absorbed fractions in the left scap ula for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................423 H-108 Absorbed fractions in the femur h ead for sources in the trabecular marrow cavity at varying marrow cellularity.................................................................423 H-109 Absorbed fractions in all skeletal sites for any source in the spongiosa (TAM, TBV, TBS, TBE, or TMS) irradia ting the cortical bone volume (CBV)..........424 H-110 Absorbed fractions in all skeletal site s for sources and targ ets in the cortical bone volume (CBV self-irridiation)..................................................................424 I-1 S values in the right proximal fe mur for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides..............................426 I-2 S values in the left proximal fe mur for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides..............................426 I-3 S values in the right humerus for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides..............................................426 I-4 S values in the left humerus for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides..............................................427 I-5 S values in the cervical vertebra for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides...........................................427 I-6 S values in the thoracic vertebra for sources in the tr abecular bone volume at varying marrow cellularity for 10 radionuclides...........................................427 I-7 S values in the lumbar vertebra for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides...........................................428 I-8 S values in the sacrum for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides............................................................428 I-9 S values in the os coxae for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides............................................................428 I-10 S values in the cranium for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides............................................................429

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xx I-11 S values in the mandible for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides............................................................429 I-12 S values in the ribs for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides............................................................429 I-13 S values in the sternum for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides............................................................430 I-14 S values in the right clavicle for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides..............................................430 I-15 S values in the left clavicle for sources in the trabecu lar bone volume at varying marrow cellularity for 10 radionuclides..............................................430 I-16 S values in the right scapula for sources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides..............................................431 I-17 S values in the left scapula for s ources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides..............................................431 I-18 S values in the femur head for s ources in the trabecular bone volume at varying marrow cellularity for 10 radionuclides..............................................431 I-19 S values in the right proximal femu r for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides.............................432 I-20 S values in the left proximal femur for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides.............................432 I-21 S values in the right humerus for s ources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................432 I-22 S values in the left humerus for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................433 I-23 S values in the cervical vertebra for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides...........................................433 I-24 S values in the thoracic vertebra for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides...........................................433 I-25 S values in the lumbar vertebra for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides...........................................434 I-26 S values in the sacrum for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides............................................................434

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xxi I-27 S values in the os coxae for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................434 I-28 S values in the cranium for source s in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................435 I-29 S values in the mandible for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................435 I-30 S values in the ribs for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides............................................................435 I-31 S values in the sternum for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................436 I-32 S values in the right clavicle for so urces in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................436 I-33 S values in the left clavicle for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................436 I-34 S values in the right scapula for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................437 I-35 S values in the left scapula for s ources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................437 I-36 S values in the femur head for sources in the trabecular active marrow at varying marrow cellularity for 10 radionuclides..............................................437 I-37 S values in the right proximal fe mur for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides..............................438 I-38 S values in the left proximal fe mur for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides..............................438 I-39 S values in the right humerus for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides..............................................438 I-40 S values in the left humerus for s ources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides..............................................439 I-41 S values in the cervical vertebra for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides...........................................439 I-42 S values in the thoracic vertebra for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides...........................................439

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xxii I-43 S values in the lumbar vertebra for s ources in the trabecul ar bone surface at varying marrow cellularity for 10 radionuclides..............................................440 I-44 S values in the sacrum for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides............................................................440 I-45 S values in the os coxae for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides............................................................440 I-46 S values in the cranium for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides............................................................441 I-47 S values in the mandible for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides............................................................441 I-48 S values in the ribs for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides............................................................441 I-49 S values in the sternum for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides............................................................442 I-50 S values in the right clavicle for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides..............................................442 I-51 S values in the left clavicle for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides..............................................442 I-52 S values in the right scapula for sources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides..............................................443 I-53 S values in the left scapula for s ources in the trabecula r bone surface at varying marrow cellularity for 10 radionuclides..............................................443 I-54 S values in the femur head for s ources in the trabecular bone surface at varying marrow cellularity for 10 radionuclides..............................................443 I-55 S values in the right proximal femur for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides...........................................444 I-56 S values in the left proximal femur for sources in the co rtical bone volume at varying marrow cellularity for 10 radionuclides...........................................444 I-57 S values in the right humerus for s ources in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................444 I-58 S values in the left humerus for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................445

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xxiii I-59 S values in the cervical vertebra fo r sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................445 I-60 S values in the thoracic vertebra for sources in th e cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................445 I-61 S values in the lumbar vertebra fo r sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................446 I-62 S values in the sacrum for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides............................................................446 I-63 S values in the os coxae for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides............................................................446 I-64 S values in the cranium for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides............................................................447 I-65 S values in the mandible for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides............................................................447 I-66 S values in the ribs for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides............................................................447 I-67 S values in the sternum for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides............................................................448 I-68 S values in the right clavicle for so urces in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................448 I-69 S values in the left clavicle for s ources in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................448 I-70 S values in the right scapula for s ources in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................449 I-71 S values in the left scapula for s ources in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................449 I-72 S values in the femur head for sources in the cortical bone volume at varying marrow cellularity for 10 radionuclides..............................................449

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xxiv LIST OF FIGURES Figure page 2-1. Vertebral body showing the different t ypes of bone tissue in one particular skeletal site. Adapted from a st udy by Fagerburg and Lafferty (1998)..................15 2-2. Microstructure of compact and cancell ous bone. Illustrati on includes entire osteon known as the Haversian system....................................................................16 3-1. Schematic of the PIRT model constructed for the L4 vertebra.................................37 3-2. Schematic of the PIRT model cons tructed for the right proximal femur..................38 3-3. Electron absorbed fractions to active bone marrow within the L4 vertebrae for three source tissues correspond to 100% marrow cellularity...................................39 3-4. Electron absorbed fractions to activ e bone marrow within the L4 vertebrae at reference cellularity for three source tissues............................................................40 3-5. Electron absorbed fractions to activ e bone marrow within the proximal femur for three source tissues correspond to 100% marrow cellularity...................................41 3-6. Electron absorbed fractions to activ e bone marrow within the proximal femur at reference cellularity for three source tissues............................................................42 3-7. Electron absorbed fractions to the trabecular bone endosteum within the L4 vertebra for three source tissu es – TAM, TBV, and TBS........................................43 3-8. Electron absorbed fractions to the tr abecular bone endosteum within the proximal femur for three source tissues – TAM, TBV, and TBS............................................44 4-1. Schematic of the PIRT model cons tructed for the pelvis (os coxae)..........................68 4-2. Schematic of the PIRT mode l constructed for the cranium........................................69 4-3. Schematic of the PIRT model constructed for the ribs...............................................70 4-4. Electron absorbed fractions to activ e bone marrow within the os coxae at 100% marrow cellularity for three source tissues – TAM, TBV, and TBS.......................71 4-5. Electron absorbed fractions to active bon e marrow within the os coxae at reference cellularity for three source ti ssues – TAM, TBV, and TBS.....................................72

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xxv 4-6. Electron absorbed fractions to activ e bone marrow within the cranium at 100% marrow cellularity for three source tissues – TAM, TBV, and TBS.......................73 4-7. Electron absorbed fractions to active bone marrow within the cranium at reference cellularity for three source ti ssues – TAM, TBV, and TBS.....................................74 4-8. Electron absorbed fractions to active bone marrow within the ribs at 100% marrow cellularity for three source ti ssues – TAM, TBV, and TBS.....................................75 4-9. Electron absorbed fractions to active bone marrow within the ribs at reference cellularity for three source ti ssues – TAM, TBV, and TBS.....................................76 4-10. Electron absorbed fractions to the trab ecular bone endosteum within the os coxae for three source tissues – TAM, TBV, and TBS......................................................77 4-11. Electron absorbed fractions to the tr abecular bone endosteum within the cranium for three source tissues – TAM, TBV, and TBS......................................................78 4-12. Electron absorbed fractions to the trab ecular bone endosteum within the ribs for three source tissues – TAM, TBV, and TBS............................................................79 5-1. Schematic demonstrating the acquisition of chord-lengths across bone trabeculae and marrow cavities at scanning angle in a single transverse plane of a 3D microCT digital image...........................................................................................100 5-2. Normalized, omnidirectional chord-le ngth distributions through the marrow cavities of the femoral head and neck as measured with physical sectioning and automated light microscopy...................................................................................101 5-3. Normalized, omnidirectional chord-le ngth distributions through the bone trabeculae of the femoral head and neck as measured with physical sectioning and automated light microscopy............................................................................102 5-4. Chord-length distributions through marrow cavities of the cervical and lumbar vertebra...................................................................................................................103 5-5. Chord-length distributions through bone trabeculae of the cervical and lumbar vertebra...................................................................................................................104 5-6. Chord-length distributions thro ugh marrow cavities of the ribs...............................105 5-7. Chord-length distributions thro ugh bone trabeculae of the ribs...............................106 5-8. Chord-length distributions through marrow cavities of the cranium. Values for individual bones of the cranium in th e UF male subject as shown as well............107 5-9. Chord-length distributions through bone trabeculae of the cranium. Values for individual bones of the cranium in th e UF male subject as shown as well............108

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xxvi 5-10. Chord-length distributions through marro w cavities of the pelvis (os coxae).......109 5-11. Chord-length distributi ons through bone trabeculae of the pelvis (os coxae)........110 5-12. Chord-length distributions through marro w cavities of the scapula, clavicle, and humerus in the UF male subject......................................................................111 5-13. Chord-length distributi ons through bone trabeculae of the scapula, clavicle, and humerus in the UF male subject......................................................................112 5-14. Chord-length distribu tions through marrow cavities of the sacrum, sternum, and mandible in the UF male subject.....................................................................113 5-15. Chord-length distributions through bone trabeculae of the sacrum, sternum, and mandible in the UF male subject.....................................................................114 5-16. Chord-length distributi ons through marrow cavities of the thoracic vertebra........115 5-17. Chord-length distributions through bone trabeculae of the thoracic vertebra........116 5-18. Chord-length distributions th rough marrow cavities of the sacrum.......................117 5-19. Chord-length distributions th rough bone trabeculae of the sacrum.......................118 5-20. Chord-length distributions thro ugh marrow cavities of the humerus.....................119 5-21. Chord-length distributions th rough bone trabeculae of the humerus.....................120 6-1. Normalized, omnidirectional chord-le ngth distributions through the marrow cavities of the Leeds 44 -year reference male.........................................................144 6-2. Normalized, omnidirectional chord-le ngth distributions through the marrow cavities of the UF 66-year re ference male cancer patient......................................145 6-3. Normalized, omnidirectional chord-le ngth distributions through the bone trabeculae of the Leeds 44-year reference male.....................................................146 6-4. Normalized, omnidirectional chord-le ngth distributions through the bone trabeculae of the UF 66-year reference male cancer patient..................................147 6-5. Electron absorbed fractions to the activ e bone marrow within the parietal bone for three source tissues – TAM, TBV, and TBS..........................................................148 6-6. Electron absorbed fractions to the bone endosteum within the parietal bone for three source tissues – TAM, TBV, and TBS..........................................................149 6-7. Electron absorbed fractions to the act ive bone marrow within the femoral head for three source tissues – TAM, TBV, and TBS....................................................150

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xxvii 6-8. Electron absorbed fractions to the bone endosteum within the femoral head for three source tissues – TAM, TBV, and TBS..........................................................151 6-9. Electron absorbed fractions to the act ive bone marrow within the ribs for three source tissues – TAM, TBV, and TBS...................................................................152 6-10. Electron absorbed fractions to the bone endosteum within the ribs for three source tissues – TAM, TBV, and TBS...................................................................153 6-11. Electron absorbed fractions to the activ e bone marrow within the femoral head of the UF reference male cancer patient.................................................................154 6-12. Electron absorbed fractions to the activ e bone marrow within the femoral neck of the UF reference male cancer patient.................................................................155 6-13. Electron absorbed fractions to the active bone marrow within the L4 lumbar vertebra of the UF reference male cancer patient...................................................156 6-14. Electron absorbed fractions to the active bone marrow within the C6 cervical vertebra of the UF reference male cancer patient...................................................157 6-15. Electron absorbed fractions to the activ e bone marrow within the ilium of the UF reference male cancer patient...........................................................................158 6-16. Electron absorbed fractions to the activ e bone marrow within the parietal bone of the UF reference male cancer patient.................................................................159 6-17. Electron absorbed fractions to the trab ecular endosteum within the ilium of the UF reference male cancer patient...........................................................................160 6-18. Electron absorbed fractions to the trab ecular endosteum within the ribs of the UF reference male cancer patient...........................................................................161 6-19. Comparison of electron transport pa ths through the trabecular endosteum under either CBIST simulations or VBIST si mulations for two different initial trajectory angles 1 and 2......................................................................................162 7-1. Multiple generations of the radiation tr ansport codes used at the University of Florida....................................................................................................................199 7-2. Representative vertebral imag es used in the PIRT model........................................200 7-3. Electron absorbed fractions to activ e bone marrow within the sacrum (70% ICRP reference cellularity) for three source tissues – TAM, TBV, and TBS........201 7-4. Electron absorbed fractions to bone endosteum within the sacrum for three source tissues – TAM, TBV, and TBS...................................................................202

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xxviii 7-5. Electron absorbed fractions to the co rtical bone volume from electron sources in the spongiosa tissues (TAM, TBS, and TBV)....................................................203 7-6. Electron absorbed fractions to the co rtical bone volume from electron sources in the cortical bone cortex itself.............................................................................204 7-7. Skeletal averaged electron absorbed fr actions to active bone marrow within the entire skeleton for four source tissues in comparison to Eckerman.......................205 7-8. Skeletal averaged electron absorbed fractions to bone endosteum within the entire skeleton for five source tissues – TAM, TBV, TBS, TMC, and CBV........206 7-9. Skeletal averaged electron absorbed fr actions to active bone marrow within the entire skeleton for four source tissues – TAM, TBV, TBS, and CBV...................207 7-10. Variations in the S(TAM TAM ) with different radionuclides on the skeletal-site-specific radionuclide S values given by the PIRT model..................208 7-11. Variations in the S(TAM rS) for 90Y on the skeletal-site-s pecific radionuclide S values given by the PIRT model for th e UF reference male cancer patient.......209 7-12. Variations in the S(TAM TAM ) to the cranium based on varying marrow celluarity and 5 radionuclides given by the PIRT model.......................................210 A-1. In-vivo computed tomography scout scans............................................................228 A-2. Cranium images shown for vi sualization of skeletal site.......................................229 A-3. Mandible images shown for 3D vi sualization of the skeletal site..........................230 A-4. Clavicle images shown for 3D visualization of skeletal site..................................230 A-5. Scapulae images shown for 3D visualization of skeletal site.................................230 A-6. Cervical vertebra images shown fo r 3D visualization of skeletal site....................231 A-7. Thoracic vertebra images shown for 3D visualization of skeletal site...................231 A-8. Sacrum images shown for 3D visualization of skeletal site...................................231 A-9. Lumbar vertebrae images shown for 2D and 3D visualization of the skeletal site..........................................................................................................................2 32 A-10. Os coxae (pelvic) images shown for 2D and 3D visualization of the skeletal site..........................................................................................................................2 33 A-11. Proximal femur images shown for 2D a nd 3D visualization of the skeletal site..........................................................................................................................2 34

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xxix B-1. Example of a 3D reconstruction of spongiosa acquired from microCT imaging of a bone section from a skeletal site of interest....................................................236 D-1. Illustrative example of header and data files for the microCT data. In this case, C0000040 is the file name......................................................................................252 D-2. Pictorial example of the head er file from the C0000040 data set...........................254 D-3. Pictorial example of how to use the ConvertMicroCT.exe program......................255 D-4. Pictorial example for the execution of the Plane.exe program...............................256 D-5. Example of the opening window for “_Kslice.raw” images in Adobe Photoshop...................................................................................................257 D-6. Example image of after opening _Kslice.raw in Adobe Photoshop.......................258 D-7. Example image of Figure D-8 afte r “auto-leveling” in Adobe Photoshop.............259 D-8. Example of the opening window fo r “_Jslice.raw” images in Photoshop..............259 D-9. Example of the opening window fo r “_Islice.raw” images in Photoshop..............260 D-10. Example of ROI determinati on with the I Slice in Photoshop.............................261 D-11. Example of ROI determinati on with the J Slice in Photoshop.............................262 D-12. Example of ROI determinati on with the K Slice in Photoshop............................263 D-13. Pictorial example of the execu tion of the Histogram.exe program......................265 D-14. Pictorial example of gray-level hi stogram data plot in Microsoft Excel..............265 D-15. Example of how to import a text file into SigmaPlot...........................................266 D-16. Screen capture of the SigmaPlo t window after plotting the histogram................267 D-17. Regression wizard window displaying the list for the curve-fitting equation......267 D-18. Example window in the process for obtaining the curve-fitting parameters........268 D-19. Example of Regression Wizard wi ndow providing the four parameters..............269 D-20. Pictorial example of the executi on of the FindThresh. exe program and its corresponding output used to dete rmine the optimum threshold...........................270 D-21. Pictorial example of the executi on of the ResizeCTimage.exe program..............271 D-22. Pictorial example of the execu tion of the MedianFi lter.exe program..................272

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xxx D-23. Pictorial example of the execu tion of the ReadImage.exe program.....................272 G-1. Schematic of how chord lengths and el ectrons could travel through a section of trabecular spongiosa...............................................................................................360 G-2. Pictorial example for the Humerus_Left image execution of the Tri-Linear Chord-Length Distribution program......................................................................386

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xxxi Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REFERENCE SKELETAL DOSIMETR Y MODEL FOR AN ADULT MALE RADIONUCLIDE THERAPY PATIENT BASED ON 3D IMAGING AND PAIREDIMAGE RADIATION TRANSPORT By Amish P. Shah December 2004 Chair: Wesley E. Bolch Major Department: Biomedical Engineering The need for improved patient-specificity of skeletal dose estimates is widely recognized in radionuclide therapy. Current clinical models for marrow dose are based on skeletal mass estimates from a variety of sources and linear chord-length distributions that do not account for particle escape into cortic al bone. To predict marrow dose, these clinical models use a scheme that requires separate calculations of cumulated activity and radionuclide S values. Selection of an appropr iate S value is generally limited to one of only three sources, all of which use as input th e trabecular microstruc ture of an individual measured 25 years ago, and the tissue mass deri ved from different individuals measured 75 years ago. Our study proposed a new modeling appro ach to marrow dosimetry—the Paired Image Radiation Transport (PIRT) model—that properly accounts for both the trabecular microstructure and the cortical macrostructure of each skeletal site in a reference male

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xxxii radionuclide patient. The PIRT model, as ap plied within EGSnrc, requires two sets of input geometry: (1) an infinite voxel array of segmented microimages of the spongiosa acquired via microCT; and (2) a segmente d ex-vivo CT image of the bone site macrostructure defining both the spongiosa (m arrow, endosteum, and trabeculae) and the cortical bone cortex. Our study also propos ed revising reference skeletal dosimetry models for the adult male cancer patient. Sk eletal site-specific radionuclide S values were obtained for a 66-year-old male reference patient. The derivation for total skeletal S values were unique in that the necessa ry skeletal mass and electron dosimetry calculations were formulated from the same source bone site over the entire skeleton. We conclude that paired-image radiationtransport techniques provide an adoptable method by which the intricate, an isotropic trabecular microstruc ture of the skeletal site; and the physical size and shape of the bone can be handled together, for improved compilation of reference radionuclide S valu es. We also conclude that this comprehensive model for the adult male cancer patient should be implemented for use in patient-specific calculations for radi onuclide dosimetry of the skeleton.

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1 CHAPTER 1 INTRODUCTION Bone marrow, the highly organized tissue that comprises different blood-forming cells in the body, is considered the dose-limiting organ in many radiation therapy applications, namely radioimmunotherapy (Lim et al. 1997; Sgouros 1993; Siegel et al. 1990). Hematopoiesis, the development of red and white blood cells from the proliferation and differentiation of stem cells occurs within the bone marrow. Sites for hematopoiesis are located only within the ax ial skeleton of the human body. Within the axial skeleton are regions of trabecular bone Trabecular bone regi ons thus provide the “housing” for the hematopoietic element of bone marrow within the human body. Dosimetric assessment of trabecular bone regions is an important area within internal dosimetry, considering the role thes e bone sites play in both the skeletal and hematopoietic systems. Since bone marrow is located within trabecular bone regions, radiation incident on bone is li kely to also cause damage to the marrow. Radionuclides that localize in bone, especially charged-pa rticle emitters, have the potential to cause damage to both endosteal tissues and bone marrow. Several situations may result in internal irradiation of trabecu lar bone regions. These includ e therapy procedures that use injected radiopharmaceuticals that transit th rough the skeletal system, occupational exposures to bone-seeking radionuc lides, and therapeutic proce dures for the palliation of bone pain associated with bone cancers. The amount of energy deposited in healthy marrow from therapeutic radiopharmaceuti cals often limits the amount of activity prescribed in these procedures.

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2 Use of radiopharmaceuticals as therapy agen ts has increased during the last few decades. Therefore, more accurate trabecular-b one dosimetry is needed to minimize risk to patients. Risk arises from any treatment plan in which a radionuclide travels through the circulatory system. These radiopha rmaceuticals emit radiation particles while traveling through the blood stream of the patien t. Bone marrow is continuously irrigated by blood vessels and thus exposed to this radiation. Some of the energy is also deposited in trabecular bone regions. Accurate trabecul ar bone dosimetric models will allow one to calculate the dose to both the bone and the bone marrow with more precision. Improved skeletal dosimetry will allow physicians to better understand the bi ological effects of specific therapy procedures, which in turn wi ll help improve nuclear medicine techniques by optimizing the administration of therap eutic doses of radi opharmaceuticals. Radiopharmaceuticals are also used for bone -pain palliation. This treatment is accomplished with bone-seeking beta-emitting ra dionuclides. Iodine-131, 32P, 89Sr, and 186Re are four types of radionuclides consider ed for this treatment (Samaratunga et al. 1995). Nevertheless, marrow receives a sign ificant amount of the deposited energy from these radiopharmaceuticals in bone pain treatments. Increased accuracy in trabecular-bone dosim etry has the potential to improve our understanding of the consequences associated with the scenarios previously mentioned. Therapeutic applications of radiation and ra dioactive materials will benefit from better dosimetry within these regions. Health ri sks associated with bone-seeking radionuclides can also be more calculated more accurate ly. Thus, current methods to improve trabecular-bone dosimetry are directed at more correctly measuring th e microstructure of these skeletal regions.

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3 Previously at the University of Florida (UF), an investigation was initiated on the feasibility of using magnetic resonance (MR) imaging to transport electrons through 3D digital images of the trabecular-bone microstr ucture. Chord-length distributions through both the bone trabeculae and the marrow caviti es were also acquired using these 3D images. These distributions are important for skeletal dosimetry since they can be compared with electron ranges to deduce en ergy deposition through the bone and marrow regions. Furthermore, all current models of skeletal dosimetry are based on these distributions. More recently at UF, chord-length meas urements of voxelized images were found to be difficult to calculate and are highly dependent on the methods used to remove voxel effects (Jokisch et al. 1999; Rajon et al. 2000) Consequently, an approach of directly coupling 3D MR images to the EGSnrc radiat ion transport code was developed (Jokisch et al. 1999; Patton 2002a). This voxel-based spongiosa transport approach allowed for radiation transport in a real geometry, thus serving as a be nchmark set of calculations for all existing trabecular-dosimetry models. Our study aimed to build on these previous studies and develop a new model for skeletal dosimetry for an adult male radionuc lide patient. Chapter 2 gives the important background required to better understand the whol e development that follows. Chapter 3 proposes a new Monte-Carlo technique for sk eletal dosimetry, Paired-Image Radiation Transport (PIRT). This voxel-based techniqu e is an adaptation of previous voxel-based models at the UF, with major improvements to the approach for modeling the physical macrostructure of each skeletal site, as define d by the cortical bone cortex. The prototype Monte-Carlo (PIRT) codes presented were di rectly compared to outdated methods for

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4 modeling the trabecular micros tructure, by comparing two sk eletal sites previously investigated at UF with older voxel-based c odes. This new technique of modeling for skeletal dosimetry was then used (Chapter 4) to provide skeletal dosimetry in bone sites that have not been studied accurately in the past. The flat bones of the human skeleton have presented a formidable task in stylis tic modeling methods for Monte-Carlo codes to track electrons through a single skeletal site such as the cranium or iliac crest. Comparisons were made (Chapt er 4) to older methods for skeletal dosimetry that are currently accepted for use in clinical dosim etry. Chapter 4 introduces micro-computed tomography (microCT), as a better technique fo r imaging the regions of trabecular bone. Chapter 5 details a new reference individual for skeletal dosimetry. The trabecular microstructure of the UF adult male cancer patient chosen must be thoroughly examined for validity for use in skeletal-dosimetry mode ls. At present time, skeletal dose estimates in clinical dosimetry are fundamentally relian t on a single set of c hord-length distribution measurements performed at the University of Leeds for a single 44-year male subject. Presented here is an alternative set of chor d-length distribution data of a 66-year male subject (Chapter 5). In Chapter 6, the chordlength distributions we re used to assess the dosimetric difference between the UF adult male cancer patient and the 44-year male subject previously developed for radiation pr otection. Differences in dosimetry between two types of radiation transport models (c hord-based and voxel-based models) were investigated (Chapter 6). Chapter 7 provides the complete data set for the new reference adult male radionuclide patient. Skeletal mass estimat es and bone site-specific dosimetry was measured through the use of microCT imagi ng, ex-vivo CT scans, and the new PIRT

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5 methodology for radiation transpor t (Chapter 7). With the us e of all these tools, our study presents complete skeletal-site speci fic radionuclide S values tabulated for the reference male cancer patient. The S values presented will allow for better estimates of the absorbed dose to the skeletal system.

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6 CHAPTER 2 BACKGROUND Bone Structure and Physiology The skeleton, composed of bones, cartilage s, joints, and ligaments, accounts for 20% of the body mass. Bones make up the bul k of the skeleton. Th e bones in the human skeleton are grouped into two ma in categories: the axial a nd appendicular skeleton. The axial skeleton forms the long axis of th e body, and includes the bones of the skull, vertebrae, sternum, ribs, pelvis, and the proximal ends of the long bones (Gatter and Brown 1997). The appendicular skeleton cons ists of bones that make up the upper and lower limbs. In the normal adult, sites of hematopoiesis are restricted to the axial skeleton; the marrow in these sites is portrayed as red or active, because of the presence of erythroid elements. In terms of trabecu lar bone dosimetry, the axial skeleton and the hematopoietically active marrow are the primary regions of interest. Every bone in the skeleton is composed of two mains types of tissue: cortical bone (the hard compact bone that forms the dense but smooth external la yer); and trabecular bone (the spongy or cancellous bone that form s the honeycomb structure within the dense shell). The honeycomb structure of the cance llous bone actually is a lattice of small needle-like, flat pieces calle d trabeculae. Cortic al bone comprises 80% of the skeletal mass. Figure 2-1 shows the distincti on between compact and cancellous bone. Cortical bone is made up of canals and passageways that serve as regions for nerves, blood vessels, and lymphatic vessels. The building blocks of cortical bone are the osteon or Haversian system. Each osteon is an elongated cylinder orient ed parallel to the

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7 long axis of the bone. Running through the core of the osteon is the Haversian canal; this canal contains small blood vessels and nerv e fibers. As seen in Figure 2-2, each Haversian system is surrounded by concentric cylinder-shaped layers called lamellae. At the junctions of the lamellae are spider-s haped mature bone cells (also known as osteocytes). These osteocytes receive ti ssue fluid from the Haversian canals, through canals called the cana liculi. Together, the Havers ian canal, surrounding lamellae, osteocytes, and canaliculi make up the oste on or Haversian system (Figure 2-2). The thin lining of cells along the interf ace between the Haversian canals and the bone surfaces is known as the endosteum. The endosteum is a delicat e connective-tissue membrane that covers the trabeculae of the spongy bone in the marrow cavities (Marieb 1998). The endosteum is composed primarily of osteoblasts (bone-forming cells) and osteoclasts (bone-destroying cells). Ther e is also a periosteum, a double-layered membrane composed of osteoblasts and osteoc lasts, which covers the exterior of the cortical bone. At both the periosteal a nd endosteal surfaces, bone production occurs where added bone strength is needed, or at s ites of injury. The mechanism that regulates this process is the res ponse of bone to mechanical stress and gravity. In contrast, bone resorption is triggered by the parathyroid hormone. When ionic levels of calcium in the blood decline, the hormone is released from the parathyroid glands. By resorbing the bone, the calcif ied bone matrix is broken down, thereby releasing calcium, which is then releas ed to the bloodstream. In young individuals, osteoblasts are more actively dominant than osteoclasts, and thus form more bone. However, in adults, the process begins to e quilibrate, and soon the osteoclasts dominate.

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8 Nevertheless, for the remainder of adult life, bone resorption ou tlabors bone production, resulting in a net loss of bone mass (Berne and Levy 1993). Several factors create variations in trab ecular bone microstructure. Trabecular structure varies with age (Atkinson 1965; Snyder et al. 1993), gender (Mosekilde 1989; Patton 2000), skeletal site (Eckerman 1985a; Patton 2000), and skeletal orientation (Hahn et al. 1992; Mosekilde 1989; Patton 2000). Age-related ch anges, along with the natural progression of aging, include thi nning and loss of bone trabecul ae. In terms of skeletal site and orientation, mechanical stress and grav ity play a large role in the variability of bone trabeculae. For instan ce, weight-bearing tr abeculae should be thicker than nonweight-bearing trabeculae. Also, the rate of bone loss is greater for trabeculae in a horizontal rather than a vertical orie ntation (Mosekilde 1989; Parfitt 1983). The intricate geometry and composition of the trabecular bone regions of the skeleton create several dosimetry problems Since the bone marrow cavities are located within the trabecular bo ne structure, the dimensions of the “honeycomb” configuration must be known to accurately calculate any dose to this region. However, the anisotropic framework of these regions further complicat es any attempt to apply a uniform modeling technique to the trabecular bone geometry. Radionuclide Therapies for Cancer Absorbed dose to the bone marrow occurs in several ways. On e significant method is radioimmunotherapy. Radi oimmunotherapy (RIT) involve s tagging antibodies with radionuclides for use treating cancers outside of the hematopoietic system, such as osteosarcoma (bone cancer), liver cancer, a nd other tumor growths. However, the process of RIT seems to be dose limited by bone marrow, because of the accumulation of activity within the marrow (L im et al. 1997; Sgouros 1993; Siegel et al. 1990).

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9 Monoclonal antibodies (mAbs) of fer therapeutic choices for patients with hematological malignancies. These mAbs are used to deli ver radioisotopes selectively to malignant tissue and thus increase the speci ficity of toxicity effects. In RIT, tumor cells may be killed by the antibody effect and also by th e crossfire effect of irradiation. For radioisotopes commonly used (90Y and 131I), beta particle emission eliminates tumor cells within a range of 1000-5000 m of their deposition. Radiotherapy can also be localized w ithin the hematopoietic system through methods of bone marrow ablation, using ionizi ng radiation to destroy malignancies in bone marrow. Malignancies such as H odgkin's or non-Hodgkin's lymphoma, multiple myeloma, and leukemia all require marrow ablation before bone marrow transplants (Juweid et al. 1995). Bone marrow ablati on is the process before a bone marrow transplant. Bone marrow abla tion involves high levels of chemotherapy and/or external beam radiation. Ablation of marrow by irradiation for the purposes of marrow transplants is limited by the radiation damage to healthy osteogenic tissue. Bayouth and Macey (1993) define the prototypical radionuclid e, for ablation therapy, as being able to deliver an adequate dose to the marrow, allo w for reinfusion of the new marrow in a short period of time, and minimize the radiation dos e to other outside or gans. One method of marrow ablation is to administer bone-seek ing radiopharmaceuticals that accumulate in the skeleton, and deliver largely concentr ated radiation dose to the adjacent marrow cavity (Bayouth and Macey 1993). Marrow Toxicity Monoclonal antibodies conjugate d to radioisotopes or to low-energy Auger electron emitters have been studied for the targeted radiotherapy of cancer. Similar to clinical

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10 experience with immunotoxins only minor responses to radioimmunotherapy have been achieved in subjects with solid tumors. Because of the severe bone-marrow toxicity associated with these high amounts of radi oactivity, patients may receive bone-marrow stem-cell infusions after treatment. Poor tumor penetration of the antibodies and doselimiting bone-marrow toxicity has severely restricted the effectiveness of radioimmunotherapy for solid tumors such as breast, ovarian, or colorectal cancer. Myelosuppression after radioimmunotherapy wa s not due to direct targeting of bone marrow stem cells by the antibodies, but rath er to nonspecific irradiation of bone-marrow stem cells caused by high levels of circul ating radioactive antibodi es perfusing the bone marrow. The bone marrow is very sensit ive to ionizing radiation, with severe bone-marrow suppression developing at extrem ely low absorbed doses. Bone-marrow toxicity limits the dose of monoclonal anti bodies that could be safely administered. Previous Methods of Trabecular Bone Dosimetry Current bone-dosimetry models stem from early data collected by Spiers and colleagues at the University of Leeds in E ngland over 20 years ago. Spiers (1951) first investigated the effects of a bone surface interface on the dose to the adjacent soft tissue. He calculated that the presen ce of a bone interface increased the dose to the surrounding soft tissue by as much as a factor of four. In 1963, Spiers (1963) began investigating how active marrow is distribute d throughout the human body and the impact this had on trabecular bone dosimetry. Spiers (1966a ; 1967) was the first to recognize that the anisotropic structure of trab ecular bone required a unique method for characterizing the geometry, in order to perform accurate skeletal internal dosimetry of beta-emitters. He then developed the concept of linear path length distributions (chord lengths) to describe the physical dimensions of these regions By knowing the complete frequency

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11 distribution of marrow path lengths and trabeculae path lengths, the fraction of a particle’s kinetic energy de posited in each type of tissue could be calculated. Dose calculations were obtained by c oupling the frequency distribu tions to a one-dimensional transport model that used these distributions to follow electrons through trabecular bone. Spiers obtained chord-length di stributions for approximately 5 to 7 skeletal sites from three males of ages 1.7, 9, and 44 years (B eddoe 1976). Almost all trabecular-bone dosimetry calculations are still based on data obtained from the Spiers’ work. Several different methods for trabecular dos imetry were developed that used chord distributions. In terms of dos imetry calculations, several a ssumptions must be made in order to use chord distributions. First, beta particles are assumed to travel in approximately straight paths through a give n media (bone or marrow). Any attempt to use the chord distributions to create a geometry that allows a beta particle to travel a distance not equal to the sampled chord-le ngth distribution inco rrectly uses these distributions. Second, the chord-length dist ributions are assumed to be an accurate representation of a person’s tr abecular microstructure. Thir d, chord-length distributions are assumed to be independent of one anot her. A chord-length-based Monte Carlo radiation dosimetry model randomly samples from both bone and marrow distributions without considering any dependence one distribution may have on the other. Spiers and his students were the first to consider random sampling from the chord-length distributions us ing Monte Carlo techniques (S piers et al. 1978; Whitwell 1973; Whitwell and Spiers 1976). Experimental ly measured chordlength distributions were coupled with range-energy relationships to calculate dose conversion factors for seven radionuclides. These dose-conversion fact ors were used to determine the absorbed

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12 fraction values in ICRP P ublication 30 (1979). This report recommended absorbed fractions for beta particles fo r use in radiation protection of skeletal tissues. For beta particles originating in the bone volume, a single value of absorbed fraction is recommended. For beta particles originati ng on the bone surface, one absorbed fraction for low-energy beta particles (< 0.2 MeV), a nd one for high-energy be ta particles (> 0.2 MeV) are recommended. These absorbed frac tions values are roughly based on the dose conversion factors from Whitwell (1973; 1976) Subsequently, the ICRP’s relatively energy-independent absorbed fractions of energy were implemented in the MIRDOSE2 program (Stabin 1996) for use in nuclear medi cine dosimetry. In this same computer program, the self-absorbed fraction to the marrow was assumed to be unity at all energies, as suggested in Part 3 of ICRP Publication 30 (1979). Spiers data were used later to calcu late the dose to marrow cavities and the endosteal layer for monoenergetic electrons emitted uniformly and isotropically within trabeculae and marrow cavities (Eckerman 1985 a, 1985b, 2000). He noted that the range of electrons in marrow to that in bone is near ly constant for energies up to 4 MeV. This allowed reduction of the two media to a single homogeneous me dium, by extending the length of the sampled trabeculae chord lengths by this ratio. M onte Carlo techniques were then applied to this single medium. Another model using Spiers data was de veloped by Bouchet et al. (1999, 2000) at the University of Florida. This model assu mes that electrons do not deviate far from a straight-line path (similar to other models). However, it improves on previous models, because it is a 3D model that accounts for delta rays and bremsstrahlung radiation, and is able to consider electron backsc atter at bone-ma rrow interfaces.

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13 More recent work at UF shows that c hord-length measurements of voxelized images are difficult to calculate and are highly dependent on the methods used to remove voxel effects (Rajon et al. 2000). Thus, these UF studies have directly coupled threedimensional Nuclear Magnetic Resonan ce (NMR) images to the EGS4-PRESTA transport code, to acquire energy deposition wi thin the marrow cavities. This allows for radiation transport in the actual trabecular ge ometry, thus serving as a benchmark set of calculations for existing trabecu lar-dosimetry models (Patton et al. 2002a). This method, based on geometric models, showed improvement to localized skeletal dosimetry because of the addition of measured trabecular bone spongiosa (bone trabeculae and marrow) and cortical bone cortex. These additions furt her corrected any overestimation of energy deposition seen in models assuming an infi nite trabecular region (Patton et al. 2002b). Internal Dosimetry Calculations The Medical Internal Radiation Dose (M IRD) Committee established a method for calculating internal dose (Loevinger et al. 1991). According to this method, the key component is the S value, which is the aver age dose received by the target organ per disintegration within th e source organ. S values for spec ific radionuclides are calculated for specific source rs and target rt regions using the following formula: i T S T i i S Tm r r r r S (2-1) where i is the mean energy emitted per nuclear transition, i(rTrS) is the absorbed fraction (AF) of energy in the ta rget region for the ith radiation type that originated in the source region, and mT is the mass of the target region. The S value is then used in the calculation of dose. As a result of a contamination by a radionuclide, the dose D is defined by

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14 h h k h kS A D) ( ~ (2-2) where the cumulated activity ~ hA represents the total number of disintegrations of the radionuclide that occur during th e contamination time. In our study, the determining AF is the critical component for approximating the dose calculation. The AF depends on the geometry of two organs, on the tissue compos ition of two organs, and on what organs lie in between these two organs. The AF can be determined by analytical methods such as Point Kernel, or by using Monte-Carlo transp ort codes. The purpose of our study was to provide a better approximation of the AF of energy within the bone marrow for use in clinical dosimetry.

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15 Figure 2-1. Vertebral body show ing the different types of bone tissue in one particular skeletal site. Adapted from a study by Fagerburg and Lafferty (1998).

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16 Figure 2-2. Microstructure of compact and cancellous bone. I llustration includes entire osteon known as the Haversian system. Adapted from a study by (Marieb 1998).

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17 CHAPTER 3 PAIRED-IMAGE RADIATION TRANSPORT MODEL FOR SKELETAL DOSIMETRY1 Introduction The skeletal system represents one of the more complex challenges in internal dosimetry. This distributed organ, with its wi de variety of bone sizes and configurations, encompasses the hematopoietic tissues of th e active (red) bone marrow, as well as the osteogenic tissues of the endosteum, both of which are relevant targets for short-term deterministic and long-term probabilistic radiat ion effects. Of primary importance is the 3D microscopic architecture of the bone trabeculae which se parate and define the marrow cavities. For short-ranged ra diations (alpha particles and lower-energy beta particles), knowledge of this 3D microstr ucture is necessary and suffi cient for accurate computation of particle transport through these skeletal tissues. For longer-ranged radiations (such as intermediate to high-energy beta particles), further consider ation should be given to the 3D macrostructure of the skel etal site, including the locati on and extent of cortical bone into which escaping particles may penetrate. The vast majority of initial studies in skeletal dosimetry were conducted at the University of Leeds (Spiers 1951; Spiers 1966; Spiers 1966; Spie rs 1967; Spiers 1968; Spiers 1969). Spiers (1966; 1967) was the firs t to recognize that the anisotropic structure 1 This chapter was accepted for publication by The Jour nal of Nuclear Medicine and will be published in February 2005: Shah AP, Bolch WE, Rajon DA, Patton PW, and Jokisch DW. A Paried-Image Radiation Transport model for skeletal dosimetry J Nuc Med: accepted September 2004.

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18 of trabecular bone required a unique method for characterizi ng the trabecular geometry as needed for accurate skeletal dosimetry of beta-emitters. Consequently, he and his students constructed an optical bone scanning system which measured linear chord-length distributions across 2D radiogra phs of excised bone tissue sl ices. Using these frequency distributions of linear chor d lengths through both bone trab eculae and marrow cavities, the fraction of a particle’s kinetic energy de posited in each tissue type was estimated. Spiers and his students obtained chord length distributions in the lumbar vertebrae for several subjects, as well as at several skeletal sites of a 1.7-y ear child (5 sites), a 9-year child (5 sites), and a 44-year male (7 s ites) (Beddoe 1976; Beddoe 1976; Whitwell 1973; Whitwell 1976). In many ways, the chord-le ngth distribution data measured for the 44year male has served to define many of th e skeletal attributes of Reference Man as defined by the International Commission on Radiological Protection (ICRP) (1975; 2002). Furthermore, all skeletal dosimetry models published and presently used in clinical dose assessment are fundamentally reliant upon this single set of adult chordlength distributions (Eckerman 1985; Bouc het 2000; Bouchet 1999; Eckerman 2000; Stabin 2002). In the technique described above, radiati on particles are effectively transported within an infinite region of trabecular spongiosa (defined as the combined tissues of the bone trabeculae, endosteum, and marrow cavities). Models of skeletal dosimetry used in current clinical practice, su ch as the Eckerman and Stab in model (2000) of MIRDOSE3 and its successor codes (Stabi n 1996), belong to a class of models called CBIST or C hord-B ased I nfinite S pongiosa T ransport, and do not account for particle escape to

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19 cortical bone. Consequently, absorbed frac tions to skeletal tissues are potentially overestimated in CBIST models for higher-energy beta emitters. One of the first attempts to account for energy loss to cortical bone was made by Spiers’ doctoral student JR Whitwell ( 1976; 1973). She introduced a trabecular equilibrium factor, Qtrab, to account for the finite extent of the spongiosa. This correction factor was determined for several radionuclides of interest in radia tion protection and for each of the 7 skeletal sites for which chordlength distributions were obtained in the 44year male subject. For 90Y, the highest correction noted by Whitwell was for the parietal bone (Qtrab = 0.672) while the lowest was for the head of the femur (Qtrab = 0.980). Nevertheless, these values of Qtrab were determined using simplified geometries for both spongiosa and cortical bone (e.g., planes and spheres). In a more recent study by Patton et al. (2002), NMR microscopy was applied to the study of the 3D microstructure of bone trabeculae within the femoral and humeral heads of three subjects: a 51-year male, an 82-year female, and a 89-year female. To account for energy lost to cortical bone, an ex-vivo CT scan of the excised femoral or humeral head was obtained prior to spongiosa sectioni ng. From spatial measurements on the CT images, a spherical region of spongiosa wa s constructed surrounded by a spherical shell of cortical bone. Electrons of various initial energies were thus transported (via the ESG4 radiation transport code) simultaneously within the NMR microimage (constructed of voxels of bone and marrow), and within st ylized model of the femoral or humeral head. Comparisons were subsequently made between energy-weighted absorbed fractions to active marrow under pa rticle transport in either (1) an infinite extent of spongiosa, or (2) the stylized model of the bone site. Patton et al. demonstrated that,

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20 without explicit consideration of energy loss to cortical bone, radionuclide S values for 32P and 90Y could potentially over-estimate active marrow dose by 6% and 11%, respectively, in the femoral head – values that exceeded the 2% corrections predicted by Whitwell. This tendency to overestimate dose to active marrow under infinite spongiosa transport had also been demonstrated by Jokisc h et al. (2001) for the thoracic vertebra in which the physical extent of the vertebral spong iosa was delineated in a stylized model of the vertebral body (e.g., truncated circular cylin der). Due to their geometric complexity, however, no attempt was made to include the ve rtebral processes in the stylized vertebral model (which account for up to ~2 5% of vertebral spongiosa). In the present study, we significantly ex tend the skeletal modeling approach originally explored by Jokisc h et al. and Patton et al. to fully account for the 3D macrostructural dimensions of skeletal site s within which dose estimates are desired. A Paired-Image Radiation Transpor t or PIRT model for skeletal dosimetry is introduced in which radiation particles are tracked simu ltaneously within two different segmented digital images: (1) an ex-vivo CT image of the entire skeletal site outlining regions of trabecular spongiosa, cortical bone, and su rrounding tissues, and (2) an ex-vivo NMR microscopy image of the interior bone tr abeculae and marrow cavity microstructure representative of that found in spongiosa volumes of the larger CT image. The PIRT model is demonstrated within two skeletal sites obtained from a single male cadaver: the L4 vertebra and the right proximal femur. In addition, representa tive site-specific S values are calculated and compared to thos e obtained under partic le transport within infinite regions of spongiosa for a variety of radionuclides of interest in skeletal imaging and therapy.

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21 Materials and Methods Cadaver Selection Candidate subjects for study were obtained through the State of Florida Anatomical Board located on the University of Florida (UF) campus. Cadave r selection criteria included (1) an age between 50 – 75 years (rep resentative of typica l radionuclide therapy patients), (2) a body mass index of 18.5 – 25 kg m-2 (CDC recommended healthy range), and (3) a cause of death that would preclud e significant skeletal deterioration. The subject identified was a 66-year male approximately 68 kg in total mass and 173 cm in total height at the time of death (BMI of 22.7 kg m-2). The subject died suddenly of complications associated with cardiomyopathy. In-Vivo Computed Tomography Scanning Prior to bone harvesting, the male cadav er was subjected to whole-body imaging via multi-slice helical CT at a pitch nece ssary to reconstruct contiguous 1-mm axial slices. The images were acquired on a Siem ens Sensation 16 unit within the Department of Radiology at UF Shands Hospital. Imag e reconstruction was performed with a bone filter at an in-plane pixel resolution of 977 m x 977 m. The CT image sets were then transferred to workstations within the A dvanced Laboratory for Radiation Dosimetry Studies (ALRADS) in the UF Department of Nuclear & Radiological Engineering for image processing and data storage. The in -vivo CT scans provided image data for (1) selecting the anatomical region from which the bone site would be harvested, and (2) constructing 3D anatomic models of skelet al sites where bone harvesting (and thus exvivo CT scanning) might be incomplete (e.g., facial bones of the skull).

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22 Bone Harvesting and Ex-Vivo Computed Tomography Scanning Following detailed review of the whol e-body in-vivo CT images, bone harvesting was conducted. Thirteen major sk eletal sites were taken from the male cadaver including the entire vertebral column and both proximal femora. Once each skeletal site was excised, it was cleaned of excess tissue, bagge d, labeled, and stored frozen until ex-vivo CT imaging could be scheduled. Post-har vest, ex-vivo CT imaging was conducted at higher resolution (1.0 mm slice thickness with an in-plane resolution of 0.3 x 0.3 mm) than permitted for in-vivo scans. The ex-v ivo CT scans provided image data for (1) identifying the location and extent of trab ecular spongiosa to be sectioned for NMR microscopy, (2) quantifying volumes of trabecu lar spongiosa and cortical bone within the bone site, and (3) constructing 3D anatomic models of the bone site for subsequent paired-image radiation transport simulations. Following detailed review of the ex-vivo CT scans, physical sections of trabecular spongiosa were taken from each bone site. Sections representing as large a region of spongiosa as possible, given the constraint s of the bone shape and the NMR imaging system (e.g., cuboidal samples taken from a s pherically shaped femoral head). Marrowintact sections of spongiosa were ba gged, labeled, and kept frozen until NMR microimaging sessions could be arranged. Fo r the lumbar vertebra 2 cuboidal sections (roughly 1.25 cm x 1.25 cm x 2.5 cm on edge ) were cut from the vertebral body representing ~24% of the total vertebral body spongiosa. For the right proximal femur, 4 cuboidal sections were cut from the femoral head (~20% of total spongiosa within the head) and 4 sections were cut from the femo ral neck (~16% of the total spongiosa within the neck).

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23 Image Segmentation of Spongiosa and Cortical Bone Regions To create tomographic anatomic models fo r use in internal dosimetry, radiation transport codes must be able to decipher the boundaries of each tissue region for which an independent dose assessment is to be made Limitations of CT image acquisition can result in an overlap of grayscale values for tissues of interest, thus precluding the use of simple automated methods of boundary definiti on. In the present study, the program CT_Contours was adopted for use in segmenting spongiosa and cortical bone within each ex-vivo CT image set (Nipper 2002). This program is based upon Interactive Data Language (IDL) version 5.5 and can output labeled contour file s in a variety of formats including binary files for EGSnrc (K awrakow 2000) and ASCII text for MCNP (Briesmeister 1997). CT_Contours displays th e current CT information, as well as a color overlay of the contours being edited. Th e contours can be created using a variety of tools including: basic thresholding, pixel growing, voxel growing, region growing and manual segmentation. The voxels contained in the individual contours are filled with the desired segmentation value, generating volumes of voxels with identical tag values. In the present study, these volu mes represent individual regions of either trabecular spongiosa or cortical bone within the skeletal site. CT_Contours was written to have the option of displaying the images using 15 different fi lters including Gaussian smoothing (3 3, 5 5, or 7 7), median (3 3, 5 5, or 7 7), Roberts edge detecti on, Sobel edge detection, Prewitt edge detection, isot ropic edge detection, hist ogram equalization, adaptive histogram equalization, sharpening and Kuwahara (3 3 or 5 5) filtering. By altering regions of a separate contour dataset, the desired segm entation can be performed. CT_Contours was designed so that ROI crea tion or modification can be performed in either the transverse, sagi ttal, or coronal plane.

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24 Microimaging of Trabecular Spongiosa NMR microscopy of trabecular bone for th e purposes of skeletal dosimetry has been discussed previously (Jokisch 2001; Patton 2002; Shah 2003; Bolch 2002; Bolch 2002). NMR imaging requires physi cal sectioning of the excise d sample and digestion of the marrow tissues. Samples of trabecular bone sections are first immersed and suspended within a circulati ng solution of sodium hypochlorite for approximately three hours. The samples are then rinsed in wate r and re-immersed in a new solution. This process is repeated up to th ree times depending on the size of the sample. Visual inspection is used to determin e the number of repetitions need ed. To ensure that water completely fills all marrow cavities, each sample is placed in a container filled with Gddoped water under vacuum. While immersed, the sample is placed in a smaller container needed for insertion into the magnet. This im aging container is then sealed and taken to the Advanced Magnetic Resonance Imaging a nd Spectroscopy (AMRIS ) facility at the UF McKnight Brain Institute for NMR microscopy. NMR microscopy images in the presen t study were acquired on a Bruker 40-cm wide bore imaging spectrometer, operated at a 470-MHz proton resonance frequency (11 T magnetic field strength). The system is fitted with a small gradient set (for microimaging), consisting of 3-axes magnetic field gradients, with a 22-Gauss/cm maximum gradient amplitude in all three orthogonal directions. A 35-mm diameter quadrature birdcage coil of le ngth 45 mm is used in order to obtain the best signal-tonoise ratio (SNR). For all imaging sessi ons, a RARE encode 3D spin-echo pulse sequence is used to obtain fully three-dimensiona l images of the samples. Fields of view are typically 3.2 cm x 3.2 cm x 3.2 cm with matrix dimensions of 512 x 512 x 512. The resulting spatial resolution of the 3D images is thus 63 m x 63 m x 63 m. Smaller

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25 voxel dimensions can be achieved at UF (~ 58 m), but at the cost of smaller sample sizes and increased imaging time (to preserve signal-to-noise). Post-acquisition image processing, including gray-level threshol ding, voxel segmentation, and 3D median filtering, have been reported previously (Jok isch 1998; Patton 2002). For use in radiation transport simulations, interior regions-of-interest (ROIs) ar e taken to avoid both physical distortions (bone saw tearing) and imaging distortions (NMR a liasing) at the edges of the sectioned specimen. Voxel-Based Infinite Spongiosa Transport (VBIST) Model Following NMR microscopy of our skeletal samples, a series of Voxel-Based Infinite Spongiosa Transport, or VBIST, m odels were created to approximate (via 3D transport) the results of current CBIST mode ls. First, marrow voxels within the binary NMR microscopy image are further labeled into voxels of active (red) marrow and inactive (yellow) marrow at a pre-determined value of marrow cellularity. This process has been outlined previously by Shah et al. (2003), and is based upon microscopy measurements of the spatial distribution of adipocytes within normal bone marrow biopsies covering a broad range of marrow cellu larities. Skeletal endosteum is further defined as a 10m at the bone-marrow interface as prev iously described by Jokisch et al. (1999). The resulting 4-tissue 3D model of trabecular spongiosa is coupled to the EGSnrc radiation transport code (Kawrakow 2000) for el ectron and beta particle transport simulations. Source tissues include the trabecular active marrow (TAM), trabecular bone surfaces (TBS), and trabec ular bone volume (TBV). Bone surface sources are approximated as a 0.1m layer on the marrow side of the bone-marrow voxel interface. Target ti ssues include the TAM and th e trabecular bone endosteum

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26 (TBE). Once a given electron reaches the physical edge of the 3D NMR microscopy image, that particle is re-introduced to the image at a corre sponding location at its opposing edge. The processes of particle transport within the image of spongiosa and its re-introduction are continued unt il all initial kinetic energy is expended. Particle histories are continued (50,000 to 2,000,000) until coeffici ents of variation on the absorbed fraction are below 1%. It is noted, however, that results given here for our voxel-based IST model are only approximate to those gi ven from chord-based IST models. In a previous study by Jokisch et al. (2001), the authors quest ion the sampling independence of the marrow and bone chord length distri butions within existing CBIST models, and suggest a 3D joint distribution might be more appropriate to descri bing the full 3D microarchitecture of particle tr ansport within the spongiosa regions of trabecular bone. Paired-Image Radiation Transport Model (L4 Vertebra) In contrast to the VBIST model desc ribed above, the Paired-Image Radiation Transport or PIRT Model supplements the 3D microscopic histology provided by the NMR microscopy image with the 3D macroscopic histology given in the corresponding ex-vivo CT image. The latter provides additi onal data for particle transport including (1) the spatial extent of the tr abecular spongiosa (e.g ., vertebral processes and body) and (2) the spatial extent of the surrounding cortical bone (which laterally encompasses the vertebral body, forms the lamina separating the ve rtebral processes, and is absent at the superior and inferior body-disc interfaces). A schematic of the PIRT model for the L4 vertebra of the 66-y ear male is given in Figure 3-1. The ex-vivo CT image is shown in the upper left in which segmented regions of spongiosa and cortical bone surfaces are hi ghlighted in orange and white, respectively. Two representative transverse slices ar e shown (upper middle and upper right) where

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27 regions of spongiosa (orange) a nd cortical bone (light blue ) are again differentiated. Superimposed over the entire ex-vivo CT image is a 3D ar ray of the replicate cuboidal NMR microscopy images each representing the 3D microstructure of the individual bone trabeculae and marrow cavities. A 3D renderi ng of the NMR microimage is thus shown in the lower left of Figure 3-1. Finall y, a single transverse slice through the NMR microimage is shown in the lower right disp laying individual voxels of bone (black) and total marrow (white). In the EGSnrc implementation of the vert ebral PIRT model, individual electrons are tracked simultaneously within the coor dinates of the NMR microimage (indicating locations in TBV, TBE, TAM, or trabecula r inactive marrow – TIM), and the coordinates of the CT macroimage (indicat ing locations in either spongi osa, cortical bone volume – CBV, or surrounding tissues – muscle, soft tissue or vertebral discs). Elemental compositions and mass densities assumed within the PIRT mode l are shown in Table 3-1. When the particle is shown to leave the s pongiosa of the CT macroimage, tracking within the NMR microimage is halted and the partic le is transported within a homogeneous region of cortical bone defined only by the larg er voxels of the ex-vivo CT macroimage. Upon particle escape from outer surface of the b one site, particle tracking is terminated. In cases where the particle leaves cortical bone and re-enters th e interior spongiosa, particle tracking in the NMR microimage is resumed. The PIRT model is thus far more anatomically realistic than is the geometry provided by either CBIST or VBIST models, especially for higher-energy, longer-range d electrons and beta particles. The principle approximation inherent within the PIRT model is that the trabecular microstructure given by the physical secti on of spongiosa (as imaged via NMR) is

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28 uniform across all CT-segmented regions of spongiosa within the skeletal site. As a result, the trabecular microstruc tures of the various vertebral processes (spinal, superior articular, and transverse) ar e implicitly assumed to be approximated by that imaged within the vertebral body. In cases where more than one physical section of spongiosa has been imaged by NMR, the PIRT m odel can be re-run using different NMR microimages. The resulting microimage-speci fic absorbed fraction profiles can thus be averaged either uniformly or weighted by th e volume of spongiosa sectioned. Finally, it is noted that the PIRT model permits explic it consideration of a cortical bone volume (CBV) as a potential radioactivity source – a feature not permitted within the CBIST or VBIST models of skeletal dose. Paired-Image Radiation Transp ort Model (Proximal Femur) A corresponding schematic of the PIRT m odel for the right proximal femur of the 66-year male subject is shown in Figure 3-2. In adults, hematopoiesis occurs primarily within the proximal epiphysis of the femur, and thus the macr ostructural mode l (shown in the upper right of Figure 3-2 and given by the ex -vivo CT) is terminated inferiorly at the point where the lesser trochanter merges anat omically with the femoral diaphysis. As with the University of Leeds chord-length measurements for their 44-year male, the biomechanics and thus the trabecular microstruc ture are notably different within femoral head and femoral neck; consequently, 3D NMR microscopy images were taken separately from the head and neck regions of the proximal femur. Representative transverse NMR image slices are shown in th e lower middle and lower right of Fig. 3-2. For each tissue source region in the model (T AM, TBV, or TBS), two different transport simulations are performed – one in which electrons are starte d within the spongiosa of the femoral head (orange voxels of the ex-vi vo CT transverse slice), and one in which

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29 electrons are started within the spongiosa of the femoral neck (red voxels of the ex-vivo CT transverse slice). In each case, only the corresponding NMR microscopy image is used within the PIRT model (head or neck mi croimage). Final absorbed fraction results for the entire proximal femur are taken as mass weighted averages of results from the head-only and neck-only spongiosa source trans port calculations. Table 3-2 displays the various source and target tissues masses for both the proximal femur and lumbar vertebra PIRT dosimetry models (given as the product of their segmented volume and the reference densities of Table 3-1). The fina l row of Table 3-2 gives values of marrow volume fraction (MVF) defined as the fraction of all voxels within the NMR microimage that are assigned to marrow tissues following im age thresholding. Here it is noted that the MVF of the femoral head is 64.5%, while it is 75.5% within the femoral neck. The MVF within the L4 vertebral body, however, was measured at 87%. Results Absorbed Fractions to Acti ve Marrow within the L4 Vertebra Figures 3-3 and 3-4 display values of el ectron absorbed fraction to active (red) bone marrow within the L4 vertebra of the 66-year male subject. Figure 3-3 corresponds to an assumption of 100% marrow cellularity (no voxels of adipos e tissue are labeled within the NMR microimage), while Figure 3-4 corresponds to an assumed marrow cellularity of 70% (reference value in both ICRP Publications 70 and 89) (1995; 2002). In each graph, solid lines indicate energy-de pendent absorbed fractions obtained from PIRT model simulations, while dashed lines indicate those derived from VBIST model simulations. For either model and at bot h cellularities, three source tissues are considered: TAM (diamonds), TBS (tri angles), and TBV (circles).

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30 At source energies below ~100 keV, th e two model types yield essentially equivalent results, as boundary effects at the spongiosa-corti cal bone interface (within the PIRT model) play a negligible role in m odifying the pattern of energy deposition to active marrow voxels (as seen within the VBIS T model). Model equivalency is noted to extend to electrons of ~200 keV initial ener gy when emitted within the volume of the bone trabeculae (TBV sources). As the electron initial energy increases above 100-200 keV, energy deposition to active marrow as predicted under VBIST mode l simulations increasingly over-predicts that given by the more anatomically realis tic PIRT model. As previously noted for chord-based skeletal models under either CBIST or VBIST simulations, absorbed fractions asymptotically approach a lim ited value independent of the source tissue (Eckerman 1985; Bouchet 1999; Jokisch 2001b). At 100% cellularit y, the VBIST model absorbed fraction to active marrow approaches a value of 0.76 at hi gh electron energies, while it approaches a limiting value of 0.53 at 70% cellularity (70% of 0.76). Similarly, absorbed fractions to active marrow pred icted under PIRT model simulations also converge in a source-independent manner, but this convergence value is energy dependent as more and more electron energy is lost to the surr ounding cortical bone (and potentially surrounding tissues). With the PIRT model results serving as the local standard, percent errors in self-absorbed fraction to active marrow given by the VBIST model are 7% at 500 keV, 16% at 1 MeV, and 85% at 4 MeV. Corresponding percent errors are 7%, 16%, and 88% for TBS sources, and 7%, 18%, and 89% for TBV sources. These percent errors are roughly equi valent at both marrow cellularities.

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31 Absorbed Fractions to Active Ma rrow within the Proximal Femur Figures 3-5 and 3-6 display values of el ectron absorbed fraction to active marrow for TAM, TBS, and TBV sources located wi thin the spongiosa of the right proximal femur of the 66-year male subject. Figures 3-5 and 3-6 correspond to marrow cellularities of 100% and 25%, re spectively, where the latter is the default cellularity for the upper femur given in ICRP Publications 70 and 89. In each graph, the individual absorbed fraction profiles for electron sources in the femoral head and in the femoral neck have been averaged according to the tota l mass of source tissue in the head and neck regions of the proximal femur, respectively. In Figure 3-6, the ordinate has been expanded to better view differen ces in modeling results at high electron energies. At the lowest energy considered (10 keV), a value of (TAM TAM) = 0.98 is seen under both VBIST and PIRT simulations. Patterns of divergence between the two modeling approaches (VBIST versus PIRT) in the proximal femur are seen to occur at lo wer energies compared to those found within the L4 vertebra (~100 keV for TAM sources, ~5 0 keV for TBS sources, and ~100 keV for TBV sources). Furthermore, it is seen that at 4 MeV (the highest energy considered), full convergence of the absorbed fraction to active marrow under both VBIST and PIRT model simulations has not yet been reached for the three source regions. Nevertheless, the energy-independent (VBIST) and energydependent (PIRT) patterns of convergence are still evident at electron initial energi es exceeding 1 MeV. With the PIRT model results serving as the local standard, percen t errors in self-absor bed fraction to active marrow (100% cellularity) gi ven by the VBIST model are 6% at 500 keV, 12% at 1 MeV, and 31% at 4 MeV. Corresponding perc ent errors are 22%, 27%, and 44% for TBS

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32 sources, and 12%, 21%, and 44% for TBV sour ces. These percent errors are ~20-50% higher when the marrow cellularity of the proxi mal femur is reduced to 25% (fat fraction of ~75%). Absorbed Fractions to Endosteal Tissues Figures 3-7 and 3-8 display values of abso rbed fraction to the trabecular endosteal tissues defined as a 10m layer of soft tissue on the marrow-side of the bone-marrow interface within the NMR microimages. Fi gure 3-7 gives results for TBS, TBV, and TAM electron sources emitted within the L4 vertebra containing bone marrow at 70% cellularity. Figure 3-8 shows da ta for these same source tissu es within the right proximal femur at 25% marrow cellularity. In both gr aphs, the ordinate scale is expanded to a maximum value of 0.16 to facilitate viewing mo del differences at higher energies. At the lowest energy considered (10 keV), a value of (TBE TBS) = 0.5 is seen under both VBIST and PIRT simulations. At each energy for each model, higher ab sorbed fractions are noted for electron sources on the trabecular surfaces, while lower absorbed fractions are seen for electron sources emitted within the active bone marrow. Intermediate absorbed fractions are shown for bone volume sources which peak in value at a source energy of ~100 keV in both skeletal sites. As expected, VBIST m odel simulations approach energyand sourceindependent convergence values at high electron initial energies (0.032 in the L4 vertebra, and 0.045 in the proximal femur), while sour ce-independent convergence values for the PIRT model are shown to continually dec line with increasing s ource energy above 1 MeV. This decline is slightly more prominent in the L4 vertebra than seen in the proximal femur, and is accountable in part by cortical bone losses within the vertebral

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33 processes. In these anatomic regions of the vertebra (whi ch encompass ~25% of total vertebral spongiosa), the surface-to-volume ra tio of trabecular spongiosa is higher than that found in the vertebral body, and thus elec tron escape to cortical bone is greater for individual electron emissions. Discussion As a further means of comparing the VB IST and PIRT model results, radionuclide S values were calculated for a wide range of be ta-particle emitters of interest in skeletal tissue imaging and radionuclide therapy. Ab sorbed fractions to active bone marrow given in Figures 3-3 through 3-6, along with tissue mass data of Table 3-2 and betaparticle energy spectra from Eckerman et al. (1994), were used to calculate S values under the MIRD schema for nine different radi onuclides. Ratios of the S value based on VBIST-model absorbed fractions to that using PIRT-model absorbed fractions are displayed in Table 3-3 for both skeletal sites and at both 100% and ICRP-reference marrow cellularities. For low-energy beta-emitters such as 33P, 169Er, and 177Lu, absorbed fractions given by the VBIST model simulati ons overestimate radionuclide S values for TAM, TBS, and TBV sources by only 1% to 5% in the L4 vertebra. Higher errors are noted in the proximal femur, particularly for bone trabeculae volum e sources (ratios of 1.17 to 1.23). For radionuclides at intermediate beta energies (Eave of 225 keV to 583 keV), S value ratios range from 1.05 to 1.14 in the L4 vertebra and from 1.04 to 1.26 in the proximal femur. For radionuclides in the highest beta-energy range (Eave of 695 to 934 keV), S value ratios range from 1.15 to 1.24 in the L4 vertebra and from 1.11 to 1.30 in the proximal femur. It is reasonable to a ssume that similar errors are also present in radionuclide S values derived from chord-ba sed models (Eckerma n 2000; Bouchet 2000)

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34 which, as in the VBIST simulations of the present study, assume an infinite region of spongiosa during particle transport. Prior to the full development of the pa ired-image radiation transport methodology given here, the UF ALRADS research group ha d attempted to correct for energy loss to cortical bone by applying a stylized model of the skeletal site macrostructure. For example, in the study by Patton et al. (2002) a spherical region of spongiosa surrounded by a spherical shell of cortical bone was app lied to the femoral heads of three different individuals based upon CT image analysis. In that study, it was demonstrated that infinite spongiosa transport yi elded radionuclide S values for 32P that were ~5-8% higher than those in which cortical bone energy lo ss was accounted for via stylistic modeling of the femoral head. For the higher-energy 90Y, the infinite spongios a transport results gave S values 8% to 11% higher. In the pres ent study, however, the full 3D histological macrostructure of the proximal femur (head as well as neck and trochanter regions) is treated within the PIRT model simulations Corresponding corrections to infinite spongiosa transport are shown in the pres ent study (by the PIRT model) to be significantly higher (up to 1.26 for 32P and up to 1.30 for 90Y) than indicated previously by Patton et al. (2002) for the femoral head. These larger corrections are attributed to enhanced particle energy loss at three s pongiosa regions of the PIRT femur model: the femoral neck, the trochanters, and the bottom interface of the model (where particles are lost to inactive marrow of the femoral di aphysis – see Fig. 3-2). These regions of enhanced electron escape were not present with in the spherical femoral head model of the Patton et al. study.

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35 Improved macrostructural modeling of the skeleton via the PIRT model methodology will potentially lead to improve ments in correlations between marrow dose estimates and observed patient myelotoxicity. For example, clinical studies of the bone pain palliation agents 153Sm-EDTP (Turner 1991; Collins 1993; Farhanghi 1992) and 186Re-HEDP (Kucuk 2000; Giannakenas 2000; Br eitz 1998) have shown patient marrow toxicities that were lower than expected based on marrow dose estimates from standard CBIST skeletal dose models (e.g., MIRDOSE2 and MIRDOSE3). While various studies have been initiated to explain these disc repancies including improvements in activity uptake quantification (van Rensburg 1998; Bren ner 2001), the data of Table 3-3 indicates that perhaps values of marrow dose were simp ly overestimated in these studies, as the standard clinical models do not properly a ccount for particle escape from marrow-filled regions of spongiosa. For both bone surf ace and volume sources, infinite spongiosa transport is shown in Table 3-3 to overestimate the femo ral marrow self-dose by 10-22% for 153Sm and 13-24% for 186Re, while the vertebral marrow self-dose is overestimated by 6% for 153Sm and 9% for 186Re. Conclusion A paired-image radiation transport (PIR T) model for skeletal dosimetry is presented in which electrons and beta part icles are tracked simultaneously within two different segmented digital images: (1) an ex -vivo CT image of the skeletal site with segmented regions of trabecular spongiosa, cortical bone, and surrounding tissues, and (2) an in-vitro NMR microscopy image of th e interior bone trabeculae and marrow cavity microstructure representative of that found within spongiosa regions of the ex-vivo CT image. Example dose calculations under the PIRT methodology within the L4 vertebra and right proximal femur of an adult 66-year male subject demonstr ate a divergence from

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36 standard infinite spongiosa transport (VBIST) methods at energies as low as 50-200 keV depending upon the source tissue and skeletal site Calculations of radionuclide S values under both methodologies imply that current chordbased models used in clinical skeletal dosimetry may over-estimate dose to active bone marrow in these two skeletal sites by ~4% to 23% for low-energy beta emitters (33P, 169Er, and 177Lu), by ~4% to 25% for intermediate-energy beta emitters (153Sm, 186Re, and 89Sr), and by ~11% to 30% for highenergy beta emitters (32P, 188Re, and 90Y). Higher errors are noted for bone-volume seekers, while lower errors are seen for s ource emissions within the active bone marrow. While the proximal femur and lumbar verteb ra are investigated in the present study, potentially larger errors in skeletal dosimetry are presumed to exist in skeletal sites with disproportionately smaller volumes of s pongiosa (e.g., ribs, cranium, and sternum). The PIRT methodology supersedes previous stylized modeling attempts by the UF ALRADS research group to account for the fini te spatial extent of trabecular spongiosa and the presence of cortical bone. This a pproach thus renders obsolete any need for mathematical modeling of the either simple or complex bone site geometries. Furthermore, the technique increases the pros pects for expanded availa bility of reference skeletal dosimetry models for both genders and of individuals of varying stature and skeletal size for use in radionuclide ther apy treatment planning of cancer in which marrow toxicity is of concern.

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37 Figure 3-1. Schematic of the PIRT m odel constructed for the L4 vertebra.

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38 Figure 3-2. Schematic of the PIRT model constructed for the right proximal femur

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39 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TAM Source TBS Source TBV Source L4 Vertebra 100% Marrow Cellularity Figure 3-3. Electron absorbed fractions to ac tive bone marrow within the L4 vertebrae for three source tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from the IST model. Data for the figure correspond to 100% marrow cellularity.

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40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TAM Source TBS Source TBV Source L4 Vertebra 70% Marrow Cellularity Figure 3-4. Electron absorbed fractions to ac tive bone marrow within the L4 vertebrae at reference cellularity for three source ti ssues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from the IST model. Data correspond to the ICRP 70 reference cellularity of 70%.

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41 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) Proximal Femur 100% Marrow CellularityTAM Source TBS Source TBV Source Figure 3-5. Electron absorbed fractions to active bone marrow within the proximal femur for three source tissues – TAM, TBV, a nd TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from the IST model. Data correspond to 100% marrow cellularity.

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42 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) Proximal Femur 25% Marrow CellularityTAM Source TBS Source TBV Source Figure 3-6. Electron absorbed fractions to active bone marrow within the proximal femur at reference cellularity for three source tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from the IST model. Data correspond to the ICRP 70 reference cellularity of 25%.

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43 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TBS Source TBV Source TAM Source L4 Vertebra 70% Marrow Cellularity Figure 3-7. Electron absorbed fractions to th e trabecular bone endos teum within the L4 vertebra for three source tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while t hose given by dashed lines are from the IST model.

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44 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TBS Source TBV Source TAM Source Proximal Femur 25% Marrow Cellularity Figure 3-8. Electron absorbed fractions to the trabecular bone e ndosteum within the proximal femur for three source tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from the IST model.

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45Table 3-1. Tissue compositions (% by mass) and mass densities used in either the IST and PIRT m odels of skeletal dosimetry. Tissue or Region a H C N O Trace Mass Density (g cm-3) Trabecular Active Marrow (TAM) 10.5 41.4 3.4 43.9 0.1 P, 0.2 S, 0.2 Cl, 0.2 K, 0.1 Fe 1.03 Trabecular Inactive Marrow (TIM) 11.5 64.4 0.7 23.1 0.1 Na, 0.1 S, 0.1 Cl 0.98 Trabecular Bone Endosteum (TBE) 10.5 25.6 2.7 60.2 0.1 Na, 0.2 P, 0.3 S, 0.2 Cl, 0.2 K 1.03 Trabecular Bone Volume (TBV) 3.4 15.5 4.2 43.5 0.1 Na, 0.2 Mg, 10.3 P, 0.3 S, 22.5 Ca 1.92 Cortical Bone Volume (CBV) 3.4 15.5 4.2 43.5 0.1 Na, 0.2 Mg, 10.3 P, 0.3 S, 22.5 Ca 1.92 Surrounding Tissues 10.5 25.6 2.7 60.2 0.1 Na, 0.2 P, 0.3 S, 0.2 Cl, 0.2 K 1.03 a TAM – “adult red marrow”, TIM – “adult yellow marrow”, TBE – “adu lt ICRU-44 soft tissue (male)”, TBV – “adult cortical bone”, CBV – “adult cortical bone” (Appe ndix A of ICRU Report 46) (ICRU 1992).

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46Table 3-2. Tissues masses used in the paired -image radiation transport (PIRT) model (100% marrow cellularity). The marrow volu me fractions are taken from the 3D NMR microscopy images of excised cube of spongiosa. Tissue / Quantity L4 Vertebra Femoral Head Femoral Neck Proximal Femur Trabecular Active Marrow (TAM) 153.3 g 15.80 g 26.30 g 42.1 g Trabecular Bone Endosteum (TBE) 3.2 g 0.68 g 1.12 g 1.8 g Trabecular Bone Volume (TBV) 117.0 g 4.55 g 7.55 g 12.1 g Cortical Bone Volume (CBV) 74.4 g 26.6 g Marrow Volume Fraction (MVF)a 87% 64.5% 75.5% a Ratio of total marrow voxels to total voxels in the bina ry 3D NMR microscopy images of trabecular spongiosa.

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47Table 3-3. Ratio of the radionuclide S valu e for an active marrow (TAM) target as gi ven by the infinite s pongiosa transport (IS T) model to that given by the paired-image radiation transport (PIRT) model. Eave Emax L4 Vertebra – 100% Cellularity Proximal Femur – 100% Cellularity Radionuclide (keV) (keV) TAM Source TBS Source TB V Source TAM Source TBS Source TBV Source P-33 77 239 1.01 1.02 1.02 1.02 1.09 1.21 Er-169 100 351 1.02 1.04 1.03 1.02 1.08 1.23 Lu-177 133 498 1.03 1.05 1.04 1.03 1.09 1.23 Sm-153 225 809 1.05 1.06 1.06 1.04 1.10 1.23 Re-186 323 1075 1.08 1.09 1.09 1.06 1.13 1.24 Sr-89 583 1492 1.13 1.14 1.13 1.10 1.18 1.26 P-32 695 1854 1.15 1.16 1.15 1.11 1.19 1.26 Re-188 764 2000 1.17 1.19 1.18 1.12 1.21 1.28 Y-90 934 2282 1.21 1.23 1.22 1.14 1.23 1.30 Eave Emax L4 Vertebra – 70% Cellularity Proximal Femur – 25% Cellularity Radionuclide (keV) (keV) TAM Source TBS Source TB V Source TAM Source TBS Source TBV Source P-33 77 239 1.01 1.02 1.01 1.00 1.03 1.17 Er-169 100 351 1.02 1.04 1.02 1.02 1.05 1.19 Lu-177 133 498 1.03 1.05 1.04 1.04 1.07 1.21 Sm-153 225 809 1.05 1.06 1.06 1.07 1.10 1.22 Re-186 323 1075 1.08 1.09 1.09 1.09 1.13 1.24 Sr-89 583 1492 1.13 1.14 1.14 1.13 1.18 1.25 P-32 695 1854 1.15 1.16 1.16 1.15 1.19 1.26 Re-188 764 2000 1.18 1.19 1.20 1.16 1.21 1.27 Y-90 934 2282 1.22 1.23 1.24 1.19 1.24 1.28

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48 CHAPTER 4 BETA-PARTICLE ENERGY LOSS TO CORTICAL BONE VIA PAIRED-IMAGE RADIATION TRANSPORT: CORRECTI ONS TO CLINICAL MODELS OF SKELETAL TISSUE DOSE2 Introduction Accurate models of skeletal tissue dose are needed in both radiation protection (e.g., predicting risks for leukemia and bone cancer induction following inhalation of long-lived bone-seeking radionuclides) and in radionuclide therapy (e.g., correlations of marrow dose and toxicity for radiopharmaceutic als subject to either specific or nonspecific skeletal tissue uptake). Ideally th ese models must take into account both the microscopic structure of the bone trabecu lae and marrow cavities, as well as the macroscopic structure of the bone site its elf (shape and volume of the trabecular spongiosa and the exterior cortex of cortical bone). For alpha emitters and low-energy beta emitters, only the microscopic characterizati on of the bone site is needed in the dose model, as these particles typically expend their full emission energy within the trabecular spongiosa. For intermediateto higher-ene rgy beta emitters, however, energy loss to the exterior cortical bone is to be expected, especially at t hose skeletal sites with high spongiosa surface-to-volume ratios (flat bones such as the cranium and ribs). Current models of skeletal dosimetry us ed in both health physics and medical 2 This chapter has been submitted to Medical Physics: Shah AP, Bolch WE, Rajon DA, Patton PW, and Jokisch DW. Submitted. Beta-particle energy loss to co rtical bone via paried-image radiation transport: corrections to clinical models of skeletal tissue dose. Med Phys: submitted.

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49 physics track alpha and beta particles within the skeleton through an infinite region of trabecular spongiosa, thus ne glecting effects introduced by th e 3D macrostructure of the bone site. These IST, or infin ite spongiosa transport, models use as their input either (1) linear chord-length distributions measured across the trabeculae and marrow cavities (Beddoe et al. 1976; Whitwell and Spiers 1976 ), or (2) 3D digital images of that microstructure (Jokisch et al. 2001; Patton et al. 2002). Subsequentl y, we refer to these two modeling approaches as CBIST (chordbased IST) or VBIST (voxel-based IST) skeletal dose models, respectively. The skel etal dose model used in current clinical practice, the Eckerman and Stabin model (2000) of MIRDOSE3 (S tabin 1996) and its successor code, belongs to th e CBIST model classification. In the present study, we discuss a new appro ach to skeletal dosimetry using PairedImage Radiation Transport (PIRT). In the PI RT skeletal model, ra diation particles are tracked simultaneously within two different segmented digital images: (1) an ex-vivo CT image of the skeletal site outlining regions of trabecular spongiosa and cortical bone, and (2) an in-vitro microCT image of the s pongiosa microstructure (bone trabeculae and marrow cavities). In Shah et al. (2004), we compared dosimetry results between VBIST and PIRT model transport simulations for electron and beta-particle emitters within the proximal femur and lumbar vertebrae of a 66year adult male. In the current study, we extend this comparison to include three othe r skeletal sites with high percentages of active bone marrow: the pelv is, cranium, and ribs. Microimaging of trabecular spongi osa: NMR microscopy vs. microCT. Our research group has previously reported on the use of NMR microscopy to obtain 3D microimages of the trabecular micro-architect ure for skeletal dosimetry (Jokisch et al.

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50 1998; Jokisch et al. 2001; Bolc h et al. 2002; Bolch et al. 2002; Patton et al. 2002; Patton et al. 2002; Shah et al. 2003; Shah et al 2004). Optimal images from NMR microscopy require physical samples of spongiosa be s ubjected to marrow digestion. The marrow cavities of the sample are then filled with Gd-doped water for enhanced MR signal from voxels within the marrow cavities. Marrow diges tion is efficient for those skeletal sites with relatively large and ex ternally accessible marrow cavi ties (e.g, femur head/neck and vertebra). In contrast, ma rrow digestion can be incomplete for skeletal sites with inaccessible and relatively small marrow cav ities (e.g., cranium, sternum, etc.). Alternatively, sectioned pieces of trabecular spongiosa may be imaged directly via NMR as marrow-intact samples. Problems with th is approach, however, include poor signal-tonoise ratios (requiring longer and more costly imaging times) and corresponding difficulties in image segmentation and thresholding (less distinct peaks between marrow and bone voxels in the gray-level histogram). For marrow-digested samples, strong MR signals are emitted somewhat uniformly within the water-filled marrow cavities. In contrast, marrow-intact samples emit MR signals separately from the active (red) and inactive (yellow or fat) marrow regions of th e cavities, thus broadening the marrow signal peaks in the gray-level histogram and co mplicating image thresholding of the marrowbone interfaces. For both marrow-digested and marrow-intact samples, one must also contend with limitations in sample size considering th e small imaging bore of high-field NMR systems. Typically, only a relatively small physical section of the trabecular spongiosa can be utilized in NMR microscopy, and multiple samples must then be imaged to properly account for site-to-site variations in the trabecular microstructure across a given

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51 skeletal site (e.g., several spongiosa cubes within the vertebral body of the lumbar vertebrae). An attractive alternative to NMR microsc opy for skeletal dosimetry is the use of microCT imaging of physical samples of s pongiosa (Regsegger et al. 1996; Dufresne 1998; Muller et al. 1998). As with NMR microscopy, enhanced image contrast is achieved with microCT for marrow-digested sa mples, without a need to fill the empty marrow cavities with water to improve signal. However, microCT images of marrowintact sections of trabecul ar spongiosa yield approximate ly the same level of image contrast as seen for marrow-digested NMR microimages. MicroCT imaging of marrowintact samples is thus a very acceptable method for acquiring microimages for skeletal dosimetry modeling – an option that requires ve ry little sample preparation and thus is achievable at all skeletal sites regardless of our ability to fu lly digest the marrow tissues. Improved signal-to-noise ratios with microCT images also permit enhancements in image segmentation and thresholding. Typically, one finds a more clear distinction between the bone peak and marrow peak within the gray-l evel histogram of the microCT image, over that seen in comparable images obtained from NMR microscopy (especially for marrowintact NMR samples). The final disadvantage of NMR microscopy is its restriction on sample size. Currently, with use of the high-field magnets (4.7 T and 11 T) at the Advanced Magnetic Resonance Imaging and Spectroscopy (AMRIS ) facility at the UF McKnight Brain Institute, the resolution obtained for a spongiosa sample is approximately 63 m x 63 m x 63 m at a field of view of 3.2 cm x 3.2 cm x 3.2 cm. The corresponding maximum sample size is a cuboidal section approxi mately 1.25 cm x 1.25 cm x 2.5 cm on edge.

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52 With microCT, a slightly be tter voxel reso lution of 60 m x 60 m x 60 m can be achieved for a cubical physical section of spongiosa as large as 5.3 cm x 5.3 cm x 5.3 cm on edge. It is noted, however, that for many sk eletal sites, the physic al shape and size of the bone may not permit a physical sectioning of its spongiosa to this maximum sample size. Materials and Methods Cadaver Selection Candidate subjects for study were obtained through the State of Florida Anatomical Board located on the University of Florida (UF) campus. Cadave r selection criteria included (1) an age between 50 – 75 years (rep resentative of typica l radionuclide therapy patients), (2) a body mass index of 18.5 – 25 kg m-2 (CDC recommended healthy range), and (3) a cause of death that would preclud e significant skeletal deterioration. The subject identified was a 66-year male approximately 68 kg in total mass and 173 cm in total height at the time of death (BMI of 22.7 kg m-2). The subject died suddenly of complications associated with cardiomyopathy. In-Vivo Computed Tomography Scanning Prior to bone harvesting, the male cadav er was subjected to whole-body imaging via multi-slice helical CT at a pitch nece ssary to reconstruct contiguous 1-mm axial slices. The images were acquired on a Siem ens Sensation 16 unit within the Department of Radiology at UF Shands Hospital. Imag e reconstruction was performed with a bone filter at an in-plane pixel resolution of 977 m x 977 m. The CT image sets were then transferred to workstations within the A dvanced Laboratory for Radiation Dosimetry Studies (ALRADS) in the UF Department of Nuclear & Radiological Engineering for image processing and data storage. The in-vi vo CT scans provided image data in order to

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53 (1) select the anatomical region from which the bone site would be harvested, and (2) construct 3D anatomic models of skeletal sites where bone harvesting (and thus ex-vivo CT scanning) might be incomplete (e.g., rib cage). Bone Harvesting and Ex-Vivo Computed Tomography Scanning Following detailed review of the whol e-body in-vivo CT images, bone harvesting was conducted. Fourteen major skeletal sites were taken from the male cadaver including the pelvis (pelvis), the cranium (cranial cap), and several ribs from both the right and left side of the rib cage. Once each skeletal si te was excised, it was cleaned of excess tissue, bagged, labeled, and stored frozen until ex-vivo CT imaging could be scheduled. Postharvest, ex-vivo CT imaging was conducted at the highest resolution permitted based on sample size (1.0 mm slice thickness with an in-plane resolu tion of 0.65 mm x 0.65 mm for the pelvis, 0.23 mm x 0.23 mm for the ribs). The ex-vivo CT scans provided image data for (1) identifying the location and extent of trabecular spongiosa to be sectioned for microCT imaging; (2) quantifying both trab ecular spongiosa and cortical bone volumes within the bone site; and (3 ) constructing 3D anatomic models of the bone site for subsequent paired-image radia tion transport simulations. Following detailed review of the ex-vivo CT scans; physical sections of trabecular spongiosa were taken from each bone site. Sections representing as large a region of spongiosa as possible were taken, given th e constraints of th e bone shape and the microimaging system (e.g., cuboidal samples ta ken from a spherically shaped femoral head). Marrow-intact sections of spongiosa were bagged, la beled, and kept frozen until microimaging sessions were arranged. For th e left parietal bone, 2 cuboidal sections (roughly 4.9 cm x 2.8 cm x 1.3 cm on edge) were cut from the cranial section representing ~10% of the total spongiosa within the cranial ca p. For the left middle rib, 4

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54 cylindrical sections were cut (~12% of total spongiosa within the left side of the rib cage), and 6 sections were cut from different bones of the pelvis (~25% of the total spongiosa within the entire pelvis). These phys ical sections of trab ecular spongiosa were originally intended to be imaged via NMR mi croscopy, and thus they were cut at sizes less than the maximum sizes permitted by microCT imaging (given above). Image Segmentation of Spongiosa and Cortical Bone Regions To create tomographic anatomic models fo r use in internal dosimetry, radiation transport codes must be able to decipher the boundaries of each tissue region for which an independent dose assessment is to be made Limitations of CT image acquisition can result in an overlap of grayscale values for tissues of interest, thus precluding the use of simple automated methods of boundary defini tion. In the presen t study, the program CT_Contours was adopted for use in segmenting spongiosa and cortical bone within each ex-vivo CT image set (Nipper et al. 2002). This program is based upon Interactive Data Language (IDL) version 5.5 and can output labeled contour file s in a variety of formats including binary files for EGSnrc (K awrakow 2000) and ASCII text for MCNP (Briesmeister 1997). CT_Contours displays th e current CT information, as well as a color overlay of the contours being edited. Th e contours can be created using a variety of tools including: basic thresholding, pixel growing, voxel growing, region growing, and manual segmentation. The voxels contained in the individual contours are filled with the desired segmentation value, generating volumes of voxels with identi cal tag values. In the present study, these volumes represent individual regions of either trabecular spongiosa or cortical bone within the skeletal site. CT_Cont ours was written to have the option of displaying the images using 15 diffe rent filters including Gaussian smoothing (3 3, 5 5, or 7 7), median (3 3, 5 5, or 7 7), Roberts edge detection, Sobel edge

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55 detection, Prewitt edge detection, isotropi c edge detection, histogram equalization, adaptive histogram equalization, sharpening and Kuwahara (3 3 or 5 5) filtering. By altering regions of a separate contour dataset, the desired se gmentation can be performed. CT_Contours was designed so that ROI crea tion or modification can be performed in either the transverse, sagi ttal, or coronal plane. Micro-Computed Tomography of Trabecular Spongiosa Micro-tomographic imaging of cuboidal samples of spongiosa was performed on desktop cone-beam CT40 or CT80 scanners (Scanco Medical AG, Bassersdorf, Switzerland) yielding 3D image data sets at a voxel resolution of 60 m x 60 m x 60 m. Although a resolution of 30 m3 could be obtained at an equivalent sample size, the higher resolution images exceed the maximum allowable binary array size of both the image processing and radiation transport code s. Post-acquisition image processing steps included (1) selection of an ideal volume of interest, (2) gray-level th resholding, (3) voxel segmentation, and (4) 3D median filteri ng, all of which have been previously reported in Jokisch et al. (1998) and Patton et al.(2002). Voxel-Based Infinite Spongiosa Transport (VBIST) Model Following microCT imaging of our skeletal samples, a series of VBIST models were created to approximate (via 3D trans port) the results of current CBIST models. First, marrow voxels within the binary microC T image are further labeled into voxels of active (red) marrow and inactive (yellow) ma rrow at a pre-determined value of marrow cellularity. This process has been outlined previously by Shah et al. (2003), and is based upon microscopy measurements of the spatial distribution of adipocytes within normal bone marrow biopsies covering a broad range of marrow cellulariti es. The trabecular

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56 bone endosteum (TBE) is further defined as a 10m at the bone-marrow interface as previously described by Jokisc h (2001). The resulting 4-ti ssue 3D model of trabecular spongiosa is coupled to the EGSnrc radiation transport code for electron (beta particle) transport simulations. Source tissues include the trabecular active marrow (TAM), trabecular bone surfaces (TBS), and trabec ular bone volume (TBV). Trabecular bone surface (TBS) sources are approximated as a 0.1m layer on the marrow side of the bone-marrow voxel interface. Target tissues include both the active marrow and bone endosteum. Once a given electron reaches the ph ysical edge of the 3D micro-image, that particle is re-introduced to the image at a corresponding lo cation at its opposing edge. The processes of particle transport within th e image of spongiosa a nd its re-introduction are continued until all initial kinetic energy is expended. Particle histories are continued (50,000 to 2,000,000) until coefficients of vari ation on the absorbed fraction are below 1%. Paired-Image Radiation Transport (PIRT) Model The Paired-Image Radiation Transport or PIRT model supplements the 3D microscopic histology provided by the microCT image with the 3D macroscopic histology given in the correspond ing ex-vivo CT image. The latter provides additional data for particle transport including (1) the sp atial extent of the tr abecular spongiosa (e.g., ilium, pubis and ischium bones of the pelvis) a nd (2) the spatial extent of the surrounding cortical bone (which laterally encompasses the entire pelvis). A schematic of the PIRT model of the pelv is from the 66-year male is given in Figure 4-1, where the ex-vivo CT image is s hown in the upper left. A representative coronal slice is shown in the upper right wher e regions of spongiosa (o range) and cortical

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57 bone (blue) are differentiated. Superimposed over the entire ex-vivo CT image is a 3D array of replicate cuboidal microCT images each representing the 3D microstructure of the individual bone trabeculae and corresponding marrow caviti es. A 3D rendering of the microCT image is thus shown in the lower left of Figure 4-1. Finall y, a single transverse slice through the microCT image is shown in the lower right displa ying individual voxels of bone (black) and total marrow (white), a pattern inverted from that within the original microCT image. In the EGSnrc implementation of the PIRT model, individual el ectrons are tracked simultaneously within the coordinates of the microCT image (indicating locations in TBV, TBE, TAM, or trabecular inactive marro w – TIM), and the coordinates of the CT macroimage (indicating locations in either spongiosa, cortical bone volume – CBV, or surrounding tissues – muscle or soft tissue). Elemental compositions and mass densities assumed within the PIRT model are shown in Table 3-1 (refer to Chapter 3). When the particle is shown to leave the spongiosa of the CT macroimage, tracking within the microCT image is halted and the particle is transported within a homogeneous region of cortical bone defined only by the larger voxe ls of the ex-vivo CT macroimage. Upon particle escape from the outer surface of the bone si te, particle tracking is terminated. In cases where the particle leaves cortical bone and re-enters the interior spongiosa, particle tracking in the microCT image is resume d. The PIRT model is thus far more anatomically realistic than is the geomet ry provided by the VBIST model, especially when accounting for higher-ener gy, longer-ranged electrons. The principle approximation inherent within the PIRT model is that the trabecular microstructure given by the phys ical section of spongiosa (as imaged via microCT) is

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58 uniform across all CT-segmented regions of spongiosa within the skeletal site. As a result, the trabecular microstruc tures of the various other regions of the pelvis (pubis and ischium) are implicitly assumed to be approxima ted by that imaged within the ilium. In cases where more than one physical section of spongiosa has been imaged by microCT, the PIRT model can be re-run using different microimages re presentative of different spongiosa regions of the bone site. The resu lting microimage-specific absorbed fraction profiles can thus be averaged either uniform ly or weighted by the volume of spongiosa sectioned. In the case of the pelvis, the mi crostructure of the pubis and ischium can be sampled, utilized, and the resulting transport da ta can be averaged. Finally, it is noted that the PIRT model permits explicit consider ation of a cortical bone volume (CBV) as a potential radioactivity source – a feature not permitted within chord-based models of skeletal dose (CBIST). In this study, two other bone sites repr esentative of flat bones in the human body were subjected to electron transport within th e PIRT model: the ribs and the cranium. As with the pelvis, the cranium and ribs have several regions in which sampling of the trabecular structure can be performed. In th e case of the cranium, final dosimetry data can be averaged from sampling of the frontal, occipital, left pariet al and right parietal bones. In the present study, we focus on the mi crostructure of the left parietal bone as shown in Figure 4-2. The upper left corner of the Figure 4-2 shows the ex-vivo image of the cranial cap. Only the outer cortex of the cranium can be seen with the coronal suture, nearly transverse in direction, between the frontal and parietal bone s, and the sagittal sutures, medially placed, between the right a nd left parietal bones. Two representative transverse slices are shown (upper middle a nd upper right) where regions of spongiosa

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59 (occipital, frontal, right parietal left parietal) and cortical bone are again differentiated. A 3D rendering of the microCT image of the pari etal bone is thus shown in the lower left of Figure 4-2. Finally, one transverse and one coronal slice thr ough the microCT image are shown in the lower right displaying indi vidual voxels of bone (black) and total marrow (white). The need for multiple spongiosa sampling sites also occurs in the ribs, in which the left and right rib cages each contain 12 individual rib bones To accurately sample the trabecular microstructure of th e rib cage, 3 ribs were chosen from the right and left side. In the present study (Figure 4-3), we fo cus only on a single rib – the middle or 7th rib of the left rib cage. The upper left image in Figure 4-3 show s the spongiosa regions in the middle portions of both the left and right rib cage. Differentiation of spongiosa and cortical bone within the left middle rib are shown in the upper right. A 3D rendering of the microCT image for the middle left rib is shown in the lower left of Figure 4-3. A single transverse and coronal slice through the microCT imag e shown in the lower right displaying individual voxels of bone (b lack) and total marro w (white). Table 4-1 displays the various source a nd target tissues masses for the pelvis, cranium, and left rib cage of the 66-year male subject. Valu es for cortical bone mass are estimated as the product of the tissue density (1.92 g cm-3) and their cortical volumes from either the in-vivo CT image (left rib cage) or ex-vivo CT image (pelvis and cranium). Mass estimates for total marrow, bone endosteum, and bone trabeculae in each skeletal site are calculated as the product of (1) the total spongiosa volume from the CT macroimage (in-vivo or ex-vi vo), (2) the tissue volume fraction taken from the microCT image, and (3) the tissue density (values given in Chapter 3, Ta ble 3-1). As an example,

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60 values of the marrow volume fraction (MVF) the fraction of tissue volume assigned to marrow in the segmented microCT image are given at the bottom of Table 4-1 for the left ilium (85.3%), left parietal bone (60.0%), and left middle rib (88.8%), respectively. Results Absorbed Fractions to Active Marrow within the Pelvis Figure 4-4 and Figure 4-5 display values of electron absorbed fraction to active (red) bone marrow within the pelvis of the 66year male subject. Figure 4-4 corresponds to an assumption of 100% marrow cellularity (no voxels of adipos e tissue are labeled within the microCT image), while Figure 4-5 corresponds to an assumed marrow cellularity of 48% (reference adult value in both ICRP Publica tions 70 and 89) (ICRP 1995; ICRP 2002). In each graph, solid li nes indicate energy-dependent absorbed fractions obtained from PIRT model simula tions, while dashed lines indicate those derived from VBIST model simulations. For e ither model and at both cellularities, three source tissues are considered: active marrow (diamonds), bone surfaces (triangles), and bone trabeculae (circles). The two model types yield essentially equi valent results only at electron energies below ~50 keV where boundary effects at the spongiosa-cortical bone interface (within the PIRT model) play a negligible role in modifying the pattern of energy deposition to active marrow voxels (as seen within the VBIS T model). Model equivalency is noted to extend to electrons of ~80-100 keV initial ener gy when emitted along the surfaces of the bone trabeculae (TBS sources). As the electron initial en ergy increases above 50-100 keV, energy deposition to active marrow as predicted under VBIST mode l simulations increasingly over-predicts that given by the more anatomically realis tic PIRT model. As previously noted for

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61 skeletal models under either CBIST or VBIST simulations, absorbed fractions asymptotically approach a limited value i ndependent of the source tissue (Eckerman 1985; Bouchet et al. 1999; Joki sch et al. 2001b). At 100% cellularity, the VBIST model absorbed fraction to active marrow approaches a value of ~0.75 at high electron energies, while it approaches a limiting value of 0.36 at 48% cellularity (48% of 0.75). Similarly, absorbed fractions to active marrow pred icted under PIRT model simulations also converge in a source-independent manner, but this convergence value is noted to be energy dependent as more and more electron energy is lost to th e surrounding cortical bone (and potentially surrounding tissues). With the PIRT model results serving as the local standard, percent errors in self-absorbed fraction to active marrow given by the VBIST model are 17% at 500 keV, 34% at 2 MeV, and 70% at 4 MeV. Corresponding percent errors are 8%, 30%, and 68% for TBS sources, and 22%, 36%, and 72% for TBV sources. These percent erro rs are roughly equivalent at both marrow cellularities. Absorbed Fractions to Active Marrow within the Cranium Figure 4-6 and Figure 4-7 display values of electron absorbed fraction to active marrow for TAM, TBS, and TBV sources located within the spongiosa of the cranium of the 66-year male subject. Figures 4-6 and Figure 4-7 correspond to marrow cellularities of 100% and 38%, respectively, where the latter is the default cellularity for the cranium given in ICRP Publications 70 and 89. For e ither model and at bot h cellularities, three source tissues are considered: active marrow (diamonds), bone surfaces (triangles), and bone trabeculae volumes (circles). In Fig. 47, the ordinate has b een expanded to better view differences in modeling results at hi gh electron energies. At the lowest energy

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62 considered (10 keV), a value of unity for (TAM TAM) is seen under both VBIST and PIRT simulations. Patterns of divergence between the two modeling approaches (VBIST versus PIRT) in the cranium are seen to occur at higher energies compared to those found within the pelvis (~100 keV for TAM and TBS sources). Model equivalency is noted to extend to electrons of ~200 keV initial energy when emitted within the volume of the bone trabeculae (TBV sources). At 100% cellular ity, the VBIST model ab sorbed fraction to active marrow approaches a value of 0.44 at hi gh electron energies, while it approaches a limiting value of 0.17 at 38% cellularity (38% of 0.44). Similarly, absorbed fractions to active marrow predicted under PIRT model si mulations also converge in a sourceindependent manner, but again this converg ence value is energy dependent. With the PIRT model results serving as the local standa rd, percent errors in self-absorbed fraction to active marrow (100% cellula rity) given by the VBIST mode l are 18% at 500 keV, 88% at 2 MeV, and 200% at 4 MeV. Correspondi ng percent errors are 22%, 93%, and 208% for TBS sources, and 21%, 93%, and 208% for TBV sources. Similar to the pelvis, these percent errors are r oughly equivalent when the marrow cellularity of the cranium is reduced to 38% (fat fraction of ~62%). Absorbed Fractions to Active Marrow within the Rib Cage Figures 4-8 and 4-9 display values of el ectron absorbed fraction to active marrow for TAM, TBS, and TBV sources located within the spongiosa of the left rib cage of the 66-year male subject. Figure 4-8 and 4-9 co rrespond to marrow cellularities of 100% and 70%, respectively, where the latter is the defa ult cellularity for the ribs given in ICRP Publications 70 and 89. In each graph, solid lines indicate energy-dependent absorbed

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63 fractions obtained from PIRT model simula tions, while dashed lines indicate those derived from VBIST model simu lations. At the lowest en ergy considered (10 keV), a value of (TAM TAM) = 1.0 is seen under both VBIST and PIRT simulations, as expected. Patterns of divergence between the two modeling approaches (VBIST versus PIRT) in the ribs are seen to mirror those seen in the cranium (both flat bones of the axial skeleton). At 4 MeV (the hi ghest energy considered), full convergence of the absorbed fraction to active marrow under VBIST model simulations has not yet been reached for the three source regions. Nevertheless, the energy-independen t (VBIST) and energydependent (PIRT) patterns of convergence are still evident at elec tron initial energies exceeding 1 MeV. At 100% cellularity, the VBIST model absorbed fraction to active marrow approaches a value of ~0.82 at high electron energies. With the PIRT model results serving as the local standard, percen t errors in self-absor bed fraction to active marrow (100% cellularity) gi ven by the VBIST model are 21% at 500 keV, 124% at 2 MeV, and 313% at 4 MeV. Corresponding pe rcent errors are 16% 136%, and 327% for TBS sources, and 31%, 55%, and 337% for TB V sources. These percent errors are roughly equivalent when the marrow cellularity of the rib cage is reduced to 70% (fat fraction of ~30%). The higher errors in dos imetry for the ribs under VBIST simulations is not unexpected, considering that this bone site has both a high surface-to-volume ratio of spongiosa (higher chance for electron escape to cortical bone), as well as a high marrow volume fraction within its spongiosa (l esser chance for energy absorption within the bone trabeculae).

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64 Absorbed Fractions to Endosteal Tissues Figures 4-10 through 4-12 display values of absorbed fraction to the trabecular endosteal tissues defined as a 10m layer of soft tissue on th e marrow-side of the bonemarrow interface within the microCT images. Figure 4-10 gives results for TBS, TBV, and TAM electron sources emitted within th e pelvis containing bone marrow at 48% cellularity. Figures 4-11 and 4-12 show corresponding values within the cranium and left rib cage, respectively, also at reference ma rrow cellularities (38% for cranium and 70% for the ribs). In all three gr aphs, the ordinate scale is e xpanded to a maximum value of 0.10 to facilitate viewing model differences at higher energi es. At the lowest energy considered (10 keV), a value of (TBE TBS) = 0.5 is seen under both VBIST and PIRT simulations (half-space source-target geomet ry for all bone sites). Also, changes in the marrow cellularity at each bone site had no direct effect on the absorbed fraction to the endosteal tissues. Thus, reported in this investigati on are the only the absorbed fraction values at the reference ce llularity for each bone site. At each energy and for each model, higher absorbed fractions are noted for electron sources on the trabecular surfaces, while lower absorbed fractions are seen for electron sources emitted within the active bone marrow. Intermediate absorbed fractions are shown for bone volume sources which peak in value at a source energy of ~100 keV in the pelvis and rib skeletal sites. Within the cranium, values of (TBE TBV) peak in value at a source energy of ~200 keV. As expected, VBIST model simulations approach energyand source-independent convergence values at high electron initial energies (0.028 in the pelvis, 0.030 in the cranium, a nd 0.015 in the left rib cage), while sourceindependent convergence values are shown to continually decline wi th increasing source

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65 energy above 1 MeV. This decline is more prominent in the cranium and the ribs than seen in the pelvis (ratios of 3.0 versus 1.5 at high energies), and is accountable in part by cortical bone losses and particle escape from these two flat bones. In these anatomic regions of the skeleton, the surface-to-volume ratio of trabecular spongiosa is higher than that found in the pelvis, a nd thus electron escape to co rtical bone is greater. Discussion As a further means of comparing the VB IST and PIRT model results, radionuclide S values were calculated for a wide range of be ta-particle emitters of interest in skeletal tissue imaging and radionuclide therapy. Ab sorbed fractions to active bone marrow given in Figures 4-4 through 4-9, along with both the tissue mass data of Table 4-1 and beta-particle energy spectra from Eckerman et al. (1994), were used to calculate radionuclide S values under the MIRD schema for ten different radi onuclides. Ratios of the S value based on VBIST-model absorbed fractions to those using PIRT-model absorbed fractions are displayed in Table 4-2 for all three skeletal sites and at both 100% and ICRP-reference marrow cellularities. For low-energy beta-emitters such as 33P, 169Er, and 177Lu, absorbed fractions given by the VBIST model simulations overestimate radionuclide S values for TAM, TBS, and TBV sources by only 2% to 13% in the cranium. Higher errors are noted in the left rib cage, particularly for bone trabeculae volume sources (ratios of 1.18 to 1.24). In co mparison to both the cranium and the ribs, even higher errors were also noted for the tr abecular volume sources in the pelvis (ratios of 1.25 to 1.28). For radionuclides at intermediate beta energies (Eave of 192 keV to 583 keV), S value ratios range from 1.07 to 1.34 in the cranium, from 1.08 to 1.48 in the ribs, and from 1.04 to 1.24 in the pelvis. For radi onuclides in the highe st beta-energy range (Eave of 695 to 934 keV), S value ratios range from 1.30 to 1.54 in the cranium, from 1.37

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66 to 1.76 in the ribs, and from 1.14 to 1.29 in the pe lvis. It is reasonable to assume that similar errors are also present in radionuclid e S values derived from chord-based models (Bouchet et al. 2000; Eckerman and Stabin 2000 ) which, as in the VBIST simulations of the present study, assume an infinite regi on of spongiosa during pa rticle transport. Conclusion A paired-image radiation transport (PIR T) model for skeletal dosimetry is presented in which electrons (beta particle s) are tracked simultaneously within two different segmented digital images: (1) an ex -vivo CT image of the skeletal site with segmented regions of trabecular spongiosa, cortical bone, and surrounding tissues, and (2) an ex-vivo microCT image of the in terior bone trabeculae and marrow cavity microstructure representative of that found within spongiosa regions of the ex-vivo CT image. Example dose calculations under th e PIRT methodology within the cranium, ribs, and pelvis of an adult 66-year male subj ect demonstrate a divergence from standard infinite spongiosa transport (VBIST) me thods at energies as low as 50-200 keV depending upon the source tissue and skeletal site Calculations of radionuclide S values under both methodologies imply that current chordbased models used in clinical skeletal dosimetry may over-estimate dose to active bone marrow in these three skeletal sites by ~2% to 28% for low-energy beta emitters (33P, 169Er, and 177Lu), by ~4% to 48% for intermediate-energy beta emitters (131I, 186Re, and 89Sr), and by ~14% to 76% for highenergy beta emitters (32P, 188Re, and 90Y). Higher errors are noted for bone-volume seekers, while lower errors are seen for s ource emissions within the active bone marrow. These finding are consistent with those inves tigated previously in the proximal femur and lumbar vertebrae of the same 66-year male subject. The PIRT model thus supersedes previous stylized modeling attempts by the UF ALRADS research group to account for

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67 the infinite spatial extent of trabecula r spongiosa and cortical bone, and provides a method for expanding the availability of re ference models needed for clinical bone marrow dose estimates to radionuclide therapy patients.

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68 Figure 4-1. Schematic of the PIRT model c onstructed for the pelvis (os coxae).

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69 Figure 4-2. Schematic of the PIRT mode l constructed for the cranium.

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70 Figure 4-3. Schematic of the PIRT m odel constructed for the ribs.

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71 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TAM Source TBS Source TBV Source Pelvis (IIium) 100% Marrow CellularityA Figure 4-4. Electron absorbed fractions to active bone marrow within the os coxae at 100% marrow cellularity for three sour ce tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations.

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72 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TAM Source TBS Source TBV Source Pelvis (Ilium) 48% Marrow CellularityB Figure 4-5. Electron absorbed fractions to active bone marrow within the os coxae at reference cellularity for three source ti ssues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations. The ICRP reference cellularity is 48%.

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73 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TAM Source TBS Source TBV Source Cranium (Parietal) 100% Marrow CellularityA Figure 4-6. Electron absorbed fractions to active bone marrow within the cranium at 100% marrow cellularity for three sour ce tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations.

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74 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TAM Source TBS Source TBV Source Cranium (Parietal) 38% Marrow CellularityB Figure 4-7. Electron absorbed fractions to active bone marrow within the cranium at reference cellularity for three source ti ssues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations. The ICRP reference cellularity is 38%.

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75 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) Ribs (Left Middle)100% Marrow CellularityTAM Source TBS Source TBV Source A Figure 4-8. Electron absorbed fractions to active bone marrow within the ribs at 100% marrow cellularity for three source tissu es – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations.

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76 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) Ribs (Left Middle) 70% Marrow CellularityTAM Source TBS Source TBV Source B Figure 4-9. Electron absorbed fractions to activ e bone marrow within the ribs at reference cellularity for three source tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations. The ICRP reference cellularity is 70%.

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77 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TBS Source TBV Source TAM Source Pelvis (Ilium) 48% Marrow CellularityA Figure 4-10. Electron absorbed fractions to the trabecular bo ne endosteum within the os coxae for three source tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations.

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78 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TBS Source TBV Source TAM Source Cranium (Parietal) 38% Marrow CellularityB Figure 4-11. Electron absorbed fractions to the trabecular bone endosteum within the cranium for three source tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations.

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79 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TBS Source TBV Source TAM Source Ribs (Left Middle) 70% Marrow CellularityC Figure 4-12. Electron absorbed fr actions to the trab ecular bone endosteum within the ribs for three source tissues – TAM, TBV, a nd TBS. Data shown by solid lines are from the PIRT model, while those given by dashed lines are from VBIST simulations.

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80Table 4-1. Tissues masses used in the paired -image radiation transport (PIRT) model (100% marrow cellularity). The marrow volu me fractions are taken from the 3D microCT imag es of excised sections of spongiosa. Tissue / Quantity Os Coxae Cranium Left Rib Cage Trabecular Active Marrow (TAM) 471.40 g 67.80 g 87.42 g Trabecular Bone Endosteum (TBE) 19.30 g 4.91 g 1.49 g Trabecular Bone Volume (TBV) 157.50 g 90.48 g 25.76 g Cortical Bone Volume (CBV) 392.50 g 361.95 g 140.15 g Marrow Volume Fraction (MVF)a 85.3% (left ilium) 60.0% (lef t parietal) 88.8% (left 7th) a Ratio of total marrow voxels to total image voxels from the binary 3D microCT images of trabecular spongiosa

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81Table 4-2. Ratio of the radionuclide S valu e for an active marrow (TAM) target as gi ven by the voxel-based infinite spongiosa transport (VBIST) model to that given by the pa ired-image radiation tr ansport (PIRT) model Eave Emax Pelvis – 100% Cellularity Pelvis – 48% Cellularity Radionuclide (keV) (keV) TAM Source TBS Source TBV Source TAM Source TBS Source TBV Source P-33 77 239 1.04 1.04 1.28 1.03 1.01 1.27 Er-169 100 351 1.05 1.05 1.27 1.04 1.11 1.09 Lu-177 133 498 1.07 1.05 1.26 1.06 1.02 1.25 Re-186 323 1075 1.14 1.09 1.24 1.13 1.08 1.24 Sr-89 583 1492 1.19 1.13 1.25 1.19 1.12 1.25 P-32 695 1854 1.21 1.14 1.26 1.20 1.14 1.25 Re-188 764 2000 1.22 1.16 1.27 1.22 1.16 1.27 Y-90 934 2282 1.25 1.19 1.29 1.25 1.19 1.29 Eave Emax Cranium – 100% Cellularity Cranium – 38% Cellularity Radionuclide (keV) (keV) TAM Source TBS Source TBV Source TAM Source TBS Source TBV Source P-33 77 239 1.02 1.08 1.07 1.02 1.06 1.05 Er-169 100 351 1.04 1.11 1.09 1.03 1.09 1.08 Lu-177 133 498 1.05 1.13 1.12 1.05 1.11 1.11 Re-186 323 1075 1.15 1.23 1.23 1.13 1.19 1.22 Sr-89 583 1492 1.27 1.34 1.34 1.26 1.30 1.34 P-32 695 1854 1.31 1.38 1.38 1.30 1.33 1.38 Re-188 764 2000 1.36 1.44 1.44 1.34 1.39 1.44 Y-90 934 2282 1.46 1.54 1.53 1.45 1.49 1.53 Eave Emax Ribs – 100% Cellularity Ribs – 70% Cellularity Radionuclide (keV) (keV) TAM Source TBS Source TBV Source TAM Source TBS Source TBV Source P-33 77 239 1.03 1.04 1.18 1.03 1.00 1.18 Er-169 100 351 1.05 1.06 1.20 1.04 1.02 1.21 Lu-177 133 498 1.07 1.08 1.23 1.06 1.05 1.24 Re-186 323 1075 1.20 1.18 1.35 1.19 1.18 1.35 Sr-89 583 1492 1.35 1.32 1.48 1.35 1.33 1.49 P-32 695 1854 1.41 1.38 1.54 1.20 1.14 1.25 Re-188 764 2000 1.48 1.46 1.62 1.48 1.46 1.63 Y-90 934 2282 1.61 1.59 1.75 1.62 1.59 1.76

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82 CHAPTER 5 SKELETAL CHORD-LENGTH DISTRIBUTI ONS FOR ICRP REFERENCE MALE VERSUS THE UF REFERENC E MALE CANCER PATIENT3 Introduction The active (red) bone marrow of the trabecu lar regions of the adult skeleton, along with the endosteal tissues aligning the interior bone surfaces, is an important target tissue in radiation protection (e.g., induction of leukemia and bone cancer following long-term internal exposure). In radioimmunotherapy (RIT), the active marro w has been identified as the dose-limiting organ in these treatment s, thus placing incr eased importance on methods for patient-specific marrow dos imetry (Sgouros 1993; Sgouros et al 2000). The need to avoid myelotoxicity in radionuc lide therapy can, in many cases; result in suboptimal therapy of the targeted lesion by the radioimmunoconjugate. Current Reference Male Skeletal Model At present, modeling techniques for estimati ng skeletal tissue dose in both radiation protection and in therapy nuclear medicine are fundamentally based upon research conducted by F.W. Spiers (1966; 1978; 1981) an d his students at the University of Leeds in the late 1960s to late 1970s. The Leed s studies led to the development of a novel optical scanning system for thin sections of trabecular bone. Li ght transmission and 3 This chapter has been submitted to Health Phys ics: Shah AP, Bolch WE, Rajon DA, Jokisch DW, and Patton PW. submitted. A comparison of skeletal chord-length distributions between the ICRP reference male and the UF reference male cancer patient. Health Phys: submitted.

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83 absorption patterns measured through th e marrow cavities and bone trabeculae, respectively, were translated into linear, omni-directional chord-length distributions across these skeletal microstructures (Darle y 1972; Spiers 1968). The chord-length distribution thus provided a convenient and efficient means for describing the complex 3D geometry of the skeletal micro-archit ecture as needed for radiation transport simulations. The Leeds research group subse quently published chordlength distributions in the lumbar vertebrae for several subjects, as well as at several skeletal sites of a 1.7-year child (5 sites), a 9-year child (5 si tes), and a 44-year male (7 sites) (Beddoe 1976; Beddoe et al. 1976; Whitwell 1973; Whitwell 1976). These chord-length distributions were used by Leeds student Whitwell (1973; 1976) to derive dose factors (marrow and endosteal doses per unit skeletal activity burden) for several radionuclides of interest in radiation protection. Fi gure 5-1 demonstrates schematically the measurement of bone and marrow chords for one representative ac quisition direction across the bone section. The first tabulations of radi onuclide S values for medical dosimetry were published by Snyder et al. (1974) in Oak Ridge Nati onal Laboratory (ORNL) Report Number 5000 for combinations of source and target organs in an adult heterogeneous anatomic model. For the skeletal system, Snyder relied on th e previous work of Whitwell (1973) whereby the Leeds skeletal dose factors were converted to specific abso rbed fractions of energy as a function of the average beta particle en ergy. In 1975, modifications of the ORNL 5000 S values were published by the Medical Internal Radiation Dosimetry (MIRD) Committee of the Society of Nuclear Medici ne as MIRD Pamphlet No. 11 (MIRD 11) (Snyder et al. 1975). In 1979, th e International Commission on Radiological Protection

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84 (ICRP) published the bone dos imetry model of ICRP P ublication 30 (1979) where recommended values of electron absorbed fractions in the ICRP 30 bone model were derived primarily from data based on the L eeds chord-length distribut ions. As a result, the skeletal microstructure implicitly embodi ed within the current ICRP reference male (ICRP 1995; ICRP 2002) can be traced to the or iginal optical scanning measurements and chord-length distributions cons tructed at the University of Leeds in the early 1970s for their 44-year male subject. In 1985, the Spiers chord-length distribu tions were used once again by Eckerman (1985) to determine electron absorbed fractions as a function of pa rticle energy within each of the seven skeletal sites of the Leed s 44-year male subject. These values were then used to establish fluence-to-dose res ponse functions for use in photon dosimetry of the skeleton (Cristy and Eckerm an 1987). The Eckerman mo del of chord-based electron transport using the Leeds chord-length distri butions was subsequen tly updated in 2000 (Eckerman and Stabin 2000) and in 2002 (Stabin et al 2002) and is the basis for the skeletal tissue model in both the MIRDOS E (Stabin 1996) and OLINDA (Stabin 2004) computer codes. In a separate study by B ouchet et al. (1999), the Leeds chord-length distributions were used to create a 3D trans port geometry for electron s in trabecular bone. Values of electron absorbed fraction to active marrow and endosteum were shown to agree with those of the Eckerman and Stab in model by 10% for a majority of the source-target tissue combinations. University of Florida Reference Male Cancer Patient While the current ICRP Reference Male (R M) skeletal model provides tissue dose estimates adequate for use in prospectiv e radiation protectio n, the model can be considered limited in its ability to provide either individual-speci fic skeletal doses in

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85 retrospective dose-reconstruction studies, or patient-specific skeletal doses in radionuclide cancer therapy. These limitations include (1) lack of consideration of energy loss to cortical bone for interm ediate-to-high energy beta sources (Patton et al. 2002b; Shah et al. 2004), (2) reliance on fixed reference va lues of marro w cellularity (Cristy 1981; Custer and Ahlfeldt 1932), (3) use of multiple data sources for skeletal tissue masses (Mechanik 1926; Trotter and Hixon 1974), and (4) lack of bone sitespecific data on spongiosa volumes, cortical bone volumes, and marrow volume fractions (ICRP 1995). To address the need for a more comprehens ive and internally c onsistent model for skeletal tissue dose, we have performed a variety of in-vivo and ex-vivo CT imaging studies of the skeleton of a single 66-year ma le cadaver – an age representative of those considered for radionuclide cancer therapy. In addition, sections of trabecular spongiosa were imaged under micro-computed tomography revealing high-resolution details of the individual bone marrow cavities and bone trab eculae in 14 skeletal sites within the UF Reference Male Cancer Patient (RMCP). In th e present study, we evaluate differences in the trabecular microstructures of these two reference individuals through side-by-side comparisons of their marrow cavity and bone trabeculae chord distributions. A companion study of dosimetry results (e.g., electron absorbed fractions) between the Leeds RM and the UF RMCP at equivalent bone sites of the skeleton is reported separately in Chapter 6 of this dissertation. Materials and Methods Bone Specimen Selection Candidate subjects for study were obtained through the State of Florida Anatomical Board located on the UF campus. Cadaver sele ction criteria included (1) an age between

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86 50 – 75 years (representative of typical ra dionuclide therapy patients), (2) a body mass index of 18.5 – 25 kg m-2 (CDC recommended healthy ra nge) (Heyward and Stolarczyk 1996), and (3) a cause of death that would preclude significant skeletal deterioration. The subject identified was a 66-year male approximately 68 kg in total mass and 173 cm in total height at the time of death (BMI of 22.7 kg m-2). The subject died suddenly of complications associated with cardiomyopathy. Over forty bone samples were removed from the cadaver. After removal, the samples were stored frozen at -17 C until imaged. Microimaging of Trabecular Spongiosa Physical sectioning was performed on all excised skeletal sites. For example, cuboidal sections of spongios a were cut from the vertebra l body of C3, C6, T3, T6, T11, L2, and L4 to assess the trabecul ar microstructure of various regions of the spine. Microtomographic imaging of the samples wa s performed using desktop cone-beam CT40 or CT80 scanners (Scanco Medical AG, Bassersdo rf, Switzerland) yielding 3D image data sets at a voxel resolution of 60 m x 60 m x 60 m. Although a resolution of 30 m3 could be obtained at an equivalent sample si ze, the higher resolution images exceeded the maximum allowable binary array size of both our image processing and radiation transport codes. Previous studies by Rajon et al. (2002) using mathematical models of trabecular bone had indicated that accurate estimates of marrow dose can be achieved at this resolution over a broad range of electron energies. Post-acquisition image processing steps included (1) selection of the volume of interest, (2 ) gray-level thresholding, (3) voxel segmentation, and (4) 3D median filteri ng, all of which have been previously reported by Jokisch et al.(1998) and by Patton et al.(2002a)

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87 Measurement of Chord-Length Distributions The problems associated with acquiring c hord distributions across digital images were identified in Jokisch et al. (2001). The stair-steppe d representation of bone/marrow interfaces within digita l images give rise to voxel effects when measuring pathlengths across these regions. Accurate technique s for both the generation (Rajon and Bolch 2003a) and measurement (Rajon and Bolch 2003b) of -random chords through any 3D object were subsequently investigated by Ra jon and Bolch. These investigators further addressed issues relating to voxel effects imposed on the measured chord-length distributions. Rajon and Bolch showed that vox el effects increase the frequency of short chords and consequently reduce the mean c hord-length by ~30% at resolutions of ~60 m. These investigators further expressed th e need for a smoother representation of the bone-marrow interface within the 3D digital image. The method recommended was an extension of the Marching Cube algorithm in which a trilinear interpolation of the eight gray-levels of the neighboring voxe ls are used to create a scalar field of gray-level values at the point of interest (e.g., the marching cube ). This Trilinear Interpolation Marching Cube (or TLI-MC) algorithm offers a bone-marrow interface surface that is reasonably smooth and continuous. Furthermore, it elim inates image-output data size problems (storing millions of triangles that constitute the surface) and simplifies the computations that are present in other surface-smoothing methods (e.g., the Marching Tetrahedron or Marching Cube algorithms). By usi ng the TLI-MC algorithm and measuring distributions at an image resolution of 60 m, Rajon and Bolch (2003a) showed significant improvements to the true distribution found within a mathematical simulation model of trabecular bone. Rajon and Bolch furt her state the need to remove an artificial

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88 effect on the chord-length dist ributions (increased frequenc y at the short chord lengths below 90 microns) generated from the use of the TLI-MC technique. In the present study, all chord-length distributions through th e 33 spongiosa physical sections (as taken from the 14 major skeletal sites) were constructed using the TLI-MC technique and accordingly followed many of the methods as outlined by Rajon and Bolch (Rajon and Bolch 2003a). Averaging of Chord-Length Distributions In the present study, several bone regions were sampled in order to determine the average chord-length distributi on for a particular bone site. For example, four different bones were sampled within the cranium: left pa rietal, right parietal, frontal, and occipital bones. For each cranial bone, physical sec tions were imaged under microCT and the resulting images were used to generate chord-length distribu tions for that bone site. To report a single distribution for the cranium of the UF-RMCP, it is necessary to take into account each of the four separate distributions Probability densitie s within each bin of the chord distributions for the individua l cranium bones were averaged based on weighting schemes defined by the volume of each physical bone section at each particular skeletal site. This average di stribution remained normalized across the entire range of chord lengths. In this st udy, bone trabeculae and marrow cavity chord distributions were tabulated in 20m and 100m bin widths, respectively, and out to a maximum chord-length of 5000 m (bone trabeculae) and 10000 m (marrow cavities). Reference Skeletal Sites In the University of Leeds studies, optical scanning measurements were performed on contact radiographs of trabecular bone sectio ns taken from seven skeletal sites of a

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89 44-year male subject. These skeletal site s included: the lumbar vertebra (L3), the cervical vertebra (C4), the ribs, the iliac crest, the fe mur head, the femur neck, and the parietal bone of the cranium. In the presen t study, chord-length di stributions taken from 3D microCT images of spongiosa in these same seven skeletal site s within the UF-RMCP are shown for comparison. For the lumbar ve rtebra, chord distribu tions were obtained from vertebral body spongiosa microimages of L2 and L4, whereas chord distributions from C3 and C6 were averaged to report a si ngle distribution for the cervical vertebrae. The 2nd, 7th, and 11th ribs from the right and left rib cage (total of 6) were all averaged to generate chord-distributions for the ribs of the UF-RMCP. All thr ee pelvic bones of the os coxae (ilium, ischium, and pubis), and a ll four bones of the cranium (right and left parietal bones, frontal bone, and occipital bone) were averag ed to develop the os coxae and cranium chord distribution data, respectively. Lastly, both femoral heads and necks from the right and left proximal epiphysis of the femur were averaged to construct the femoral head and neck data. Additionally, se veral bones sites were sampled in this study from regions not present in the Leeds data se t. These include the scapulae (right/left), clavicles (right/left), humerus (right/left), sacrum, mandible, sternum and thoracic vertebra (T3, T6, and T11). Results In this study, microCT imaging has b een used to expand and improve upon the original University of Leeds chord-length di stribution data used presently in radiation dosimetry skeletal models. Differences in the chord distributions between the Leeds and UF studies draw attention to potential probl ems within the Leeds data and may result from two sources. First, ther e might exist real physical di fferences in the trabecular microstructure of these two individuals – one a 44-year male and one a 66-year male.

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90 Age-dependent bone thinning (ost eopenia) is just one exampl e of possible difference we might observe. As cancer patients treated wi th radionuclide therapy are generally in the age range of 50 to 75 years, th e trabecular microstructure of the 66-year UF RMCP might better resemble that of indi vidual patients requiring marrow dose estimates, particularly for male patients. Second, differences in the measured chor d-length distributions might exist which are attributable to the acquisition method. In the case of the L eeds data, the chordlengths were measured on 2D physical se ctions and by light ab sorption/transmission timing measurements. Careful selection of cut angles when prep aring the 2D physical sections provided justifica tion for reporting the Leeds chor d-length distributions as omni-directional (i.e., representing the 3D stru cture). For this pres ent study, 3D digital images were acquired directly via microCT, and chord-length measurements were then acquired using 3D ray-tracing techniques with in the images. Consequently, the method of chord measurement is very different in th e Leeds studies and in the present UF studies and these differences might further contribute to a divergence of distribution shapes. Figures 5-2 through 5-15 display normalized chord-length dist ributions across both bone trabeculae and marrow cavities in the UF samples (given in Appendix G), as measured through microCT imaging and image an alysis. In addition, these figures show the respective chord distributions for each bone site as measured by the Leeds’ optical bone scanner and reported in Appendix C of Whitwell’s thesis (1973) Values of mean chord lengths are given in Table 5-1 for both the Leeds 44-year male and the UF 66-year male subject.

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91 Discussion Femoral Head and Neck In Figure 5-2, chord-length distributions across the marrow cavities of the femoral head and neck are compared between the L eeds 44-year RM and the UF 66-year RMCP. For both individuals, marrow cavities are shown to be generally larger in the femoral neck than in the femoral head. For the femo ral head, the chord dist ributions or the two individuals have similar shapes, both p eak at a marrow chord-length of ~650 m, and both show very similar frequencies for chords exceeding 1300 m. In both regions of the femur, the UF RMCP distributions show a lo wer frequency of smaller marrow chords as compared to the Leeds RM. We further not e that the Leeds femoral head and neck distributions in Figure 5-2 show a disc ontinuous upturn within the first 100-300 m. This pattern of increasing sma ller chords is suggestive of pi xel effects as discussed by Jokisch et al. (2001). With the TLI-MC algor ithm applied to the 3D microCT images, a continuous and smooth distribution of marrow chord-lengths is seen for both femoral regions in the UF RMCP. As given in Table 5-1, the mean marrow chord is 1043 m in the femoral head and 1454 m in the femoral neck of the UF RMCP. Comparable averages in the Leeds RM are slightly higher at 1157 m and 1655 m, respectively. Corresponding distributions of bone chord-le ngths are shown in Figure 5-3 for the femoral head and neck of both individuals. Cl ose agreement is seen in the femoral neck for the two subjects and bot h distributions show a peak in frequency at ~200 m. However, a smaller and unexplained sec ond frequency peak is observed at ~30 m in the Leeds distribution. While the data of Figur e 5-3 indicate that bone chord-lengths are comparable in the femoral head and neck of the UF RMCP (mean bone chords of 347 m

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92 each), greater differences are noted for bone c hords in these femoral regions of the Leeds RM (mean bone chords of 232 m and 314 m, respectively). For the latter, a very prominent peak in chord frequency is noted in the Leeds femoral head distribution at ~140 m. Cervical and Lumbar Vertebrae Figure 5-4 compares marrow c hord distributions within the cervical and lumbar vertebrae of both the UF RMCP and L eeds RM. In addition, the marrow chord distribution of the thoracic ve rtebrae is shown for the UF s ubject (a site not reported in the Leeds data). In general, marrow chor d distributions seen in both bone sites are reasonably comparable between the two individuals. For ma rrow chords in the range of 0 1000 m, the UF distributions rise to a peak frequency (~550 m for the CV, ~700 m for the TV, and ~650 m for the LV) and then decline. In contrast, peak distributions are noted only at extremely lo w chord-lengths for the Leeds RM in both the cervical and lumbar vertebrae. This feature is suggestive, yet not conclusive, of pixel effects present in the Leeds distributions (Jokisch et al. 2001). Mean marrow chord-lengths are 1038 m (CV) and 1479 m (LV) for the UF RMCP, while they are 910 m (CV) and 1233 m (LV), respectively, in the Leeds RM. The mean marrow chord-length in the UF thoracic vertebrae is 1368 m. Figure 5-5 displays the corresponding c hord distributions across bone trabeculae within the spinal column of both referen ce individuals. Here, we note that very comparable distributions are shown in the cerv ical vertebra of the UF and Leeds subjects (mean bone chords of 282 m and 279 m, respectively). Larger differences are shown in the distribution of bone chords in the lu mbar vertebrae, where a higher frequency of

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93 smaller bone chords is seen in the Leeds subject (frequency peak at ~110 m). The mean LV bone chord-length in the UF RMCP is 316 m, but is 246 m in the Leeds RM. In the UF subject, microCT images of the trab ecular structure of the cervical (C3 and C6) and thoracic (T3, T6, and T11) vertebrae show ed relatively consistent shapes in their bone chord-length distributions Greater differences, however, were noted in the bone chord-length distributions of the lumbar region of the spine (L2 versus L4). Ribs Figures 5-6 and 5-7 display chord distri butions across the marrow cavities and bone trabeculae, respectively, in the ribs. For the UF data, lighter dashe d, dotted, and dash-dot lines are used to indicate the individual rib distributions wh ich were then averaged to create the composite rib distributions (ope n circles). For marrow chords exceeding ~900 m, close agreement is seen between the UF and Leeds reference individuals. While the UF marrow chords in the ribs peak in frequency around 400-500 m, the Leeds distribution is slightly depressed from 200 to 700 m, and then increases steeply at chords lengths below ~200 m. A more divergence in dist ribution shape is evident for the bone chord-lengths as shown in Figure 5-7. Here, the UF distribution shows a gradually increase in chord frequency peaking at ~200-210 m, and then falling gradually until it matches the frequency of th e Leeds data at bone chords exceeding ~600 m. The Leeds data, however, show a bimodal distribution of bone chords in a manner similar to that found in the Leeds femoral h ead data. Mean values of marrow and bone chord lengths in the ribs of the UF RMCP are 1630 m and 302 m, respectively. Corresponding values for th e Leeds subject are 1703 m and 266 m, as shown in Table 5-1.

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94 Cranium Figures 5-8 and 5-9 displa y chord-length distributions across the marrow cavities and bone trabeculae in the cran ial bones of the skeletal. For the UF data, individual distributions are given for the two parietal bones, the frontal bone, and the occipital bone, along with an averaged distribution representi ng the entire cranium. For the Leeds data, distributions are only given for the pariet al bone of their 44-year male subject. Dramatically different shapes are seen in the chord distributions between the UF and Leeds subjects for both spongiosa tissues. For the marrow cavities, the UF cranial averaged chords rise and peak in frequency at ~450 m, while the Leeds chords increase in frequency almost linearly for chords below ~600-700 m, with very low frequencies observed for chords exceeding ~1500 m. The UF average marrow chord-length for the cranium is 751 m, while the Leeds parietal aver age marrow chord-length is only 389 m. For the bone trabeculae, the UF chord di stributions again rise to a peak frequency at ~250 m, after which the frequency distribution declines and approaches that of the Leeds parietal distribution at bone chords exceeding ~400 m. In contrast, the Leeds bone chord distribution also rises to a peak (~300 m), but this peak is much broader. Also, a small spike in frequency is noted in the 2nd chord bin of the L eeds distribution at 40 m. Again, this pattern of increasing smalle r chords is suggestive of pixel effects as mentioned above for the femoral head and n eck, and can also be seen within the rib distribution (Figure 5-6). The mean bone chord-lengths in the UF cranial data and in the Leeds parietal bone data are 465 m and 511 m, respectively.

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95 Pelvis (Os Coxae) The final skeletal site for which direct comparisons can be made between the UF 66-year male and the Leeds 44-year male is in the pelvis (Figures 5-10 and 5-11). For the Leeds individual, marrow and bone chords dist ributions were acquired across samples of the iliac crest. For the UF individual, however, several cuboidal sections of spongiosa were taken and imaged from each of the th ree major bones comprising the pelvis – the ilium, the ischium, and the pubis. In Figure 5-10, frequencies in both the UF and Leeds chord distributions rise for decreasing marrow chords and peak in at ~600 – 700 m. However, the distribution of iliac crest marro w chords shown in the Leeds subject are shifted in favor of smaller marrow cavities in comparison to the UF individual. While the mean marrow chord-length in the iliac crest of the Leeds RM is 904 m, the UF individual shows mean marrow chord-lengths of 1508, 1593, and 1493 m in the ilium, ischium, and pubis (pelvic average of 1523 m). For the bone trabeculae, the UF data display a very similar distributional shape to the Leeds iliac crest distribution (except at small bone chords). The peak frequency for the Leeds data is located at ~150 m, while peak frequencies are shown in the UF individual at 170 m (ilium), 190 m (ischium), and 240 m (pubis). Mean chord lengths for bone trabeculae in the ilium within the UF and Leeds individuals are very similar (245 and 242 m, respectively). Remaining Marrow-Containing Bones of the Skeleton There are six remaining bone sites whic h contain active marrow in the adult skeleton for which chord-length distribution data for marrow cavitie s and bone trabeculae are not available for the Leeds 44-year ma le subject. These include the scapulae, clavicles, humeri, sacrum, sternum, and mandi ble. Figures 5-12 and 5-13 display chord

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96 distribution data in the UF reference individual for the marrow cavities and bone trabeculae, respectively, within the scapulae, clavicles, and humeri. Each curve in Figures 5-12 and 5-13 represents the average chord distribution for both the left and right skeletal sites of this individual. Marrow c hord distributions in Figure 5-12 for these three upper torso skeletal sites are remarkable similar, all peaking in frequency at chord-lengths 500 – 700 m. Bone trabeculae are shown in Figure 5-13 to be slightly larger in the scapula (mean bone chord of 417 m) and slightly smaller in the clavicles (mean bone chord of 315 m). Marrow and bone chord distributions for the three remaining bone sites are shown in Figures 5-14 and 5-15, respectively. The mean marrow chord-length in the sacrum (1116 m) is shown to fall intermediate to that in the cervical (1038 m) and thoracic and lumbar (1368 and 1479 m) vertebrae. The bone trabeculae of the sacrum are shown to thicker on average (mean bone chord of 330 m) than seen in the other regions of the spine (282, 282, and 316 m in the cervical, thoracic, and lumbar vertebrae, respectively). Furt hermore, the mandible (mean marrow chord of 1273 m) is shown to display larger marrow cavities than seen in the cranium (mean marrow chord of 751 m). In contrast, the bone trabeculae of the mandible (mean bone chord of 335 m) were seen to be slightly th inner than found throughout the cranium (mean bone chord of 465 m). Weighting Schemes for Non-Imaged Bone Sites in the Leeds Data As noted earlier, the Leeds chord-length distribution data for their 44-year male subject have been used extensively in both bone dosimetry in radiation protection and in nuclear medicine. However, since the Leeds data are limited to only measurements in seven skeletal sites (two of which are need ed for the proximal femur), weighting schemes

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97 have been proposed by which chord-length dist ributions and dosimetry data at other nonimaged skeletal sites can be approximated. The larger and more extensive set of chord distribution assembled for the UF reference individual thus provide s a unique opportunity to assess these weighting schemes. Table 8 of Bouchet et al. (2000) displays the various skeletal weighting schemes as originally proposed by Whitwell (1973). Three of these schemes are evaluated in the present st udy as shown in Figures 5-16 to 5-21. In the Leeds data, chord distributions we re acquired in the cervical and lumbar regions of the spine, but not in the thoracic regions (at least the data were not reported in Whitwell’s dissertation). Consequentl y, it was proposed that the spongiosa microstructure of the thoracic vertebrae coul d be estimated as a 50:50 weighting of the cervical and lumbar chord distributions for bot h the marrow cavities and bone trabeculae. This 50/50 weighting scheme is shown in Fi gures 5-16 and 5-17 as open circles. Data shown as closed circles are the microCT measur ed chord-length distributions as averaged across T3, T6, and T11. Relatively good agreement is shown in both tissue compartments, particular for the bone trabecul ae. For comparison, the estimated thoracic vertebrae distributions using the Leeds data ar e shown as dot-dashed lines in both figures. A second approximation involves a 60% il iac crest and 40% lumbar vertebra weighting to approximate the spongiosa micr ostructure of the sacrum. As shown in Figure 5-18, this scheme is fairly accurate in predicting the true s acral distribution of marrow chords in the UF individual. The sa me scheme slightly under-predicts the peak frequency of bone trabeculae chor d-lengths in the sacrum of th e UF individual (Fig 5-19). Finally, the weighting sche me proposed by Whitwell to approximate the spongiosa microstructure of the humerus involved an 80:20% weighting of the femoral head and

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98 femoral neck, respectively. As shown in Figures 5-20 and 5-21, this technique (as applied only to the UF data) provides an exce llent approximation of the true chord-length distributions measured across the marrow cav ities and bone trabecul ae of the proximal humerus. Discrepancies between the Leeds estimate and the UF measurements of the bone chord-length distribution of the proximal humerus shown in Figure 5-21 are directly attributed to the differences seen earlier in the bone chords of the femoral heads of these two individuals (see Figure 5-3). Conclusion In this study, we present 3D chord-le ngth probability distributions across the marrow cavities and bone trabeculae at multiple skeletal sites of a single 66-year male subject (UF reference male cancer patient). These distributions are then compared to those assembled at the University of Leed s in the early 1970s, and which form the microstructural basis for current ICRP refe rence male skeletal models used in both radiation protection and medical dosimetry. A review of me an chord lengths given in Table 5-1 indicate that, on average, th e marrow cavities with in the ribs (1630 m) of the UF 66-year male subject are fairly consistent with those seen in the Leeds 44-year male (mean marrow chord of 1703 m over the same size range of 10000 m). Larger marrow cavities are noted on average in the UF RMCP for the cervical vertebrae (1038 vs. 910 m), ilium (1508 vs. 904 m), femoral head (1043 vs. 1157 m), lumbar vertebrae (1479 vs. 1233 m), and parietal bone (812 vs. 389 m), while smaller marrow cavities are noted in the UF subj ect for the femoral neck (1454 m versus 1655 m in the Leeds subject). The mean chord-lengths fo r the bone trabeculae show close agreement for the two reference individuals in the iliu m and cervical vertebrae. Thicker trabeculae

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99 were seen on average in the UF individual for the femoral head (ratio of 1.50), femoral neck (ratio 1.11), ribs (ratio 1.13), and lumbar vertebrae (r atio of 1.28), while thinner trabeculae were seen on average in the UF i ndividual for the parietal bone of the cranium (ratio of 0.91). In three cases were prominen t discrepancies in chor d-distributional shape noted between the two reference individuals: the bone trabeculae in the ribs (Fig. 5-7) and the marrow cavities and bone trabeculae in th e cranium (Figs. 5-8 and 5-9). Overall, the distributional shapes for all UF measuremen ts were fairly consis tent across skeletal site where a moderate-to-steep rise in chord frequency is seen for small chords, followed by a peak in the chord frequency (300-700 m for marrow cavity chords and 100 – 300 m for the bone trabeculae) and gradual decline in chord frequency thereafter. Finally, it is noted that the UF chord distributio ns and the corresponding library of voxel microimages of trabecular spongiosa cover the fu ll range of skeletal sites for which active bone marrow is present in the adult. Thes e images thus provide the framework for a more comprehensive assessment of skeletal tissue dose in radionuclide therapy patients than afforded by the existing data fr om the University of Leeds studies.

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100 Figure 5-1. Schematic demonstrating the acquisition of chord-lengths across bone trabeculae and marrow cavities at scanning angle in a single transverse plane of a 3D microCT digital image. Two chord-lengths are shown for the bone trabeculae (white arrows) and ma rrow cavities (black arrows). Final distributions are assembled following ra y-tracing of chords across both tissues in fully 3D space within the image.

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101 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) UF Femur Head Leeds Femur Head UF Femur Neck Leeds Femur Neck (A) Marrow Cavities Figure 5-2. Normalized, omnidirectional chor d-length distributions through the marrow cavities of the femoral head and neck as measured with physical sectioning and automated light microscopy (Wh itwell 1973) and the corresponding 3D chord distributions as measured thro ugh microCT imaging, image processing, and the trilinear technique (present study).

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102 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0060 0100200300400500600700800Chord length ( m)Normalized frequency (per m) UF Femur Head Leeds Femur Head UF Femur Neck Leeds Femur Neck (B) Bone Trabeculae Figure 5-3. Normalized, omnidirectional c hord-length distributions through the bone trabeculae of the femoral head and neck as measured with physical sectioning and automated light microscopy (Wh itwell 1973) and the corresponding 3D chord distributions as measured thro ugh microCT imaging, image processing, and the trilinear technique (present study).

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103 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0011 0500100015002000250030003500Chord length ( m)Normalized frequency (per m) UF Cervical Leeds Cervical Vert UF Lumbar Leeds Lumbar Vert UF Thoracic (A) Marrow Cavities Figure 5-4. Chord-length distributions th rough marrow cavities of the cervical and lumbar vertebra.

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104 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0060 0100200300400500600700Chord length ( m)Normalized frequency (per m) UF Cervical Vert Leeds Cervical Vert UF Lumbar Vert Leeds Lumbar Vert UF Thoracic Vert (B) Bone Trabeculae Figure 5-5. Chord-length distri butions through bone trabeculae of the cer vical and lumbar vertebra.

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105 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) UF Rib Leeds Ribs UF Upper Lt Rib UF Middle Lt Rib UF Lower Lt Rib Marrow Cavities Figure 5-6. Chord-length distributions through marrow cavities of the ribs. Values for individual rib bones used to compute the UF average are shown for the upper left rib (dashed line), middl e left rib (dotted line), a nd lower left rib (dot-dash line).

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106 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0.0055 0.0060 0100200300400500600700Chord length ( m)Normalized frequency (per m) UF Ribs Leeds Ribs UF Upper Lt Rib UF Middle Lt Rib UF Lower Lt Rib Bone Trabeculae Figure 5-7. Chord-length distri butions through bone trabeculae of the ribs. Values for individual rib bones used to compute the UF average are shown for the upper left rib (dashed line), middl e left rib (dotted line), a nd lower left rib (dot-dash line).

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107 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 0.0020 05001000150020002500Chord length ( m)Normalized frequency (per m) UF Cranium Leeds Parietal Bone UF Left Parietal UF Right Parietal UF Frontal Bone UF Occipital Bone Marrow Cavities Figure 5-8. Chord-length distributions thr ough marrow cavities of the cranium. Values for individual bones of the cranium in th e UF male subject as shown as well.

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108 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0100200300400500600700800Chord length ( m)Normalized frequency (per m) UF Cranium Leeds Parietal Bone UF Left Parietal UF Right Parietal UF Frontal Bone UF Occipital Bone Bone Trabeculae Figure 5-9. Chord-length distri butions through bone trabeculae of the cranium. Values for individual bones of the cranium in th e UF male subject as shown as well.

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109 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) UF Os Coxae Leeds Iliac Crest UF Pelvis Ilium UF Pelvis Ischium UF Pelvis Pubis (A) Marrow Cavities Figure 5-10. Chord-length distributions through marrow cavities of the pelvis (os coxae). Values for individual bones of the pelvis in the UF male subject as shown as well.

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110 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0100200300400500600700800Chord length ( m)Normalized frequency (per m) UF Os Coxae Leeds Iliac Crest UF Ilium UF Ischium UF Pubis Bone Trabeculae Figure 5-11. Chord-length distri butions through bone trabeculae of the pelvis (os coxae). Values for individual bones of the pelvis in the UF male subject as shown as well.

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111 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) UF Scapula UF Clavicle UF Humerus (A) Marrow Cavities Figure 5-12. Chord-length distributions thr ough marrow cavities of th e scapula, clavicle, and humerus in the UF male subject.

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112 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0100200300400500600700800Chord length ( m)Normalized frequency (per m) UF Scapula UF Clavicle UF Humerus (B) Bone Trabeculae Figure 5-13. Chord-length dist ributions through bone trabecul ae of the scapula, clavicle, and humerus in the UF male subject.

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113 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) UF Sacrum UF Sternum UF Mandible (A) Marrow Cavities Figure 5-14. Chord-length distributions th rough marrow cavities of the sacrum, sternum, and mandible in the UF male subject.

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114 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0100200300400500600700800Chord length ( m)Normalized frequency (per m) UF Sacrum UF Sternum UF Mandible (B) Bone Trabeculae Figure 5-15. Chord-length dist ributions through bone trabecu lae of the sacrum, sternum, and mandible in the UF male subject.

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115 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) UF 50% CV : 50% LV UF Thoracic Leeds 50% CV : 50% LV (A) Marrow Cavities Figure 5-16. Chord-length distributions thr ough marrow cavities of the thoracic vertebra (present study) and as approximated using the weighted average of the cervical and lumbar vertebrae.

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116 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0100200300400500600700800Chord length ( m)Normalized frequency (per m) UF 50% CV : 50% LV UF Thoracic Vert Leeds 50% CV : 50% LV (B) Bone Trabeculae Figure 5-17. Chord-length dist ributions through bone trabecul ae of the thoracic vertebra (present study) and as approximated using the weighted average of the cervical and lumbar vertebrae.

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117 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) UF 60% IC : 40% LV UF Sacrum Leeds 60% IC : 40% LV (A) Marrow Cavities Figure 5-18. Chord-length distributions thr ough marrow cavities of th e sacrum (present study) and as approximated using a wei ghted average of the iliac crest and lumbar vertebrae.

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118 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0100200300400500600700800Chord length ( m)Normalized frequency (per m) UF 60% IC : 40% LV UF Sacrum Leeds 60% IC : 40% LV (B) Bone Trabeculae Figure 5-19. Chord-length dist ributions through bone trabecul ae of the sacrum (present study) and as approximated using a wei ghted average of the iliac crest and lumbar vertebrae.

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119 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) UF 80% FH : 20% FN UF Humerus Leeds 80% FH : 20% FN (A) Marrow Cavities Figure 5-20. Chord-length distributions thr ough marrow cavities of the humerus (present study) and as approximated using a weight ed average of the femoral head and neck.

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120 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 0100200300400500600700800Chord length ( m)Normalized frequency (per m) UF 80% FH : 20% FN UF Humerus Leeds 80% FH : 20% FN (B) Bone Trabeculae Figure 5-21. Chord-length dist ributions through bone trabecul ae of the humerus (present study) and as approximated using a weight ed average of the femoral head and neck.

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121Table 5-1. Comparison of measured mean chord lengths with valu es published from the Universi ty of Leeds (W hitwell 1973). Mean Bone Trabeculae Chord Length ( m) Description of UF Study UF (present study)aWhitwell (1973)aUF (present study)bWhitwell (1973)bFemoral Head average of left, right 10431157348232 Femoral Neckaverage of left, right14541655347314 Cervical Vertebraaverage of C3, C61038910282279 Thoracic Vertebraaverage of T3, T6, T111368*282* Lumbar Vertebraeaverage of L2, L414791233316246 Sacrumsingle bone site1116*330* Os coxae average of 3 pelvic bones1523*280* Iliumsingle iliac crest1508904245242 Ischiumsingle bone site1593*330* Pubic Bonesingle bone site1493*280* Craniumaverage of 4 cranial bones751*465* Frontal Bonesingle bone site676*470* Parietal Boneaverage left, right812389469511 Occipital Bonesingle bone site598*491* Mandiblesingle bone site1273*335* Ribsaverage of 6 ribs16301703302266 Humerusaverage of left, right1169*357* Scapulaaverage of left, right1179*417* Sternumsingle bone site1404*273* Clavicleaverage of left, right1535*315*Note: indicates values not reported in Whitwell (1973)a Normalized distribution out to 10,000 m.b Normalized distribution out to 2000 m.Mean Marrow Cavity Chord Length ( m)

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122 CHAPTER 6 CHORD-BASED VERSUS VOXELBASED METHODS OF ELECTRON TRANSPORT IN SK ELETAL DOSIMETRY4 Introduction Current models of skeletal dosimetry used in both radiation pr otection and medical dosimetry track alpha and beta particles within the skeleton through an infinite region of trabecular spongiosa, thus ne glecting effects introduced by th e 3D macrostructure of the bone site. These infinite spongiosa transport (o r IST) models use as their input either (1) linear chord-length distributions measur ed across the trabeculae and marrow cavities(Beddoe et al. 1976; Whitwell and Spiers 1976), or (2) 3D digital images of that microstructure (Jokisch et al. 2001a; Pa tton et al. 2002a). These two modeling approaches can thus be refe rred to as CBIST (chord-based IST) or VBIST (voxel-based IST) skeletal dose models, respectively. The two skeletal dose models currently used in clinical practice the Eckerm an and Stabin model (2000) of MIRDOSE3 (Stabin 1996) and its successor code OLINDA (Stabin 2004) belongs to the CBIST model classification. Chord-length di stributions used in the Ecke rman and Stabin model are taken from studies conducted at the Univers ity of Leeds in the early 1970’s on a single adult male, 44-year of age (Beddoe et al 1976; Whitwell 1973; Whitwell and Spiers 1976). Furthermore, the Leeds chord-leng th distribution data provide the basic 4 This chapter has been submitted to Medical Physics: Shah AP, Jokisch DW, Watchman CJ, Rajon DA, Patton PW, and Bolch WE,. submitted. Chord-based ve rsus voxel-based methods of electron transport in skeletal dosimetry. Med Phys: submitted.

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123 framework for many of the tissue and dosimetry features of the skeletal dose model published by the International Commissi on on Radiological Prot ection (ICRP) (1995; 2002). While the current ICRP Reference Male (R M) skeletal model provides tissue dose estimates adequate for use in prospectiv e radiation protectio n, the model can be considered limited in its ability to provide either individual-speci fic skeletal doses in retrospective dose-reconstruction studies, or patient-specific skeletal doses in radionuclide cancer therapy. These limitations include (1) lack of consideration of energy loss to cortical bone for intermedia te-to-high energy beta s ources (Patton et al. 2002b; Shah et al. 2004a), (2) reliance on fixe d reference values of marrow cellularity (Cristy 1981; Custer and Ahlfeldt 1932), (3) use of multiple data sources for skeletal tissue masses (Mechanik 1926; Trotter and Hixon 1974), and (4) lack of bone sitespecific data on spongiosa volumes, cortical bone volumes, and marrow volume fractions (ICRP 1995), quantities which can conceivably be measured directly or indirectly in individual patients. To a ddress the need for a more comprehensive and internally consistent model for skeletal tissue dose, we have performed a variety of in-vivo and exvivo CT imaging studies of the skeleton of a single 66-year male cadaver – an age representative of those considered for radi onuclide cancer therapy. In addition, sections of trabecular spongiosa were imaged under micro-computed tomography revealing highresolution details of the i ndividual bone marrow cavitie s and bone trabeculae at 14 skeletal sites within this UF Refere nce Male Cancer Patient (RMCP). In the present study, we pursue two funda mental objectives. The first objective involves direct comparisons of the elect ron dosimetry within the trabecular

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124 microstructure of the Leeds Reference Male to that within the UF Reference Male Cancer Patient. In this comparison, we fix the tr ansport method to CBIST model simulations, thus providing uniformity with current skelet al dosimetry practice (Eckerman and Stabin 2000). The second objective involves a dir ect comparison of th e CBIST and VBIST transport methodologies using id entical data on the trabecula r microstructure (in this case, the UF RMCP). In both portions of th e study, it is acknowle dged that values of absorbed fraction to skeletal tissues will not account for energy loss to cortical bone (both CBIST and VBIST follow particles through an infinite region of spongiosa), and thus they will lead to over-estimates of tissue dos e at high electron emi ssion energies. Energy loss to cortical bone has been previously documented by Patton et al. (2002b) and Shah et al. (2003) using VBRST ( voxel-based restricted transport) simulations and more recently by Shah et al. (2004a) using PIRT (pairedimage radiation transport) simulations. Nevertheless, this comparison of CBIST and VBIST electron dosimetry methodologies is of critical interest as both the health physics(Petoussi-Hen ss et al. 2002; Xu et al. 2000) and medical physics (Yoriyaz et al. 2000; Z ubal and Harell 1992) fields move toward more image-based approaches to internal dosimetry. Materials and Methods Cadaver Selection Candidate subjects for study were obtained through the State of Florida Anatomical Board located on the University of Florida (UF) campus. Cadave r selection criteria included (1) an age between 50 – 75 years (rep resentative of typica l radionuclide therapy patients), (2) a body mass index of 18.5 – 25 kg m-2 (CDC recommended healthy range), and (3) a cause of death that would preclud e significant skeletal deterioration. The subject identified was a 66-year male approximately 68 kg in total mass and 173 cm in

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125 total height at the time of death (BMI of 22.7 kg m-2). The subject died suddenly of complications associated with cardiomyopathy. Trabecular Microstructure Acquisition Following skeletal harvesting of the axial skeleton, physical secti ons of trabecular spongiosa were taken from each bone site. Sections representing as large a region of spongiosa as possible were taken, given th e constraints of th e bone shape and the microimaging system (e.g., cuboidal samples ta ken from a spherically shaped femoral head). Marrow-intact sections of spongiosa were bagged, la beled, and kept frozen until microimaging sessions were arranged. Samples taken for this investigation were matched to those of the Leeds study: femur head, fe mur neck, cervical verteb ra, lumbar vertebra, cranium, os coxae, and ribs. In addition, sa mples were images from the sites not covered in the Leeds study including the humerus, sca pula, clavicle, mandible, thoracic vertebra, and sacrum. Micro-tomographic imaging of cuboidal samples of spongiosa was performed on desktop cone-beam CT40 or CT80 scanners (Scanco Medical AG, Bassersdorf, Switzerland) yielding 3D image data sets at a voxel resolution of 60 m x 60 m x 60 m. Although a resolution of 30 m3 could be obtained at an equivalent sample size, in some cases the higher resolution images exceed the maximum allowable binary array size of both the image processing and radiation transport codes. Post-acquisition image processing steps included (1) selection of an ideal volume of interest, (2) gray-level thresholding, (3) voxel segmentation, and (4) 3D median filtering, all of which have been previously reported in Jokisch et al.(1998) and Patton et al.(2002a).

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126 Voxel-Based Infinite Spongiosa Transport (VBIST) Model Following microCT imaging of our skeletal samples, a series of VBIST models were created to approximate (via 3D trans port) the results of current CBIST models. First, marrow voxels within the binary microC T image are further labeled into voxels of active (red) marrow and inactive (yellow) ma rrow at a pre-determined value of marrow cellularity. This process has been outlined previously by Shah et al. (2003), and is based upon microscopy measurements of the spatial distribution of adipocytes within normal bone marrow biopsies covering a broad range of marrow cellularities In the present study, however, only a marrow cellularity of 100% was considered (no inclusion of adipocytes within the marrow cavities). The trabecular bone endosteum (TBE) is further defined as a 10m at the bone-marrow interface as prev iously described by Jokisch et al. (2001a). The resulting 3D model of trab ecular spongiosa is coupl ed to the EGSnrc radiation transport code (Kawrakow 2000) for electron (beta pa rticle) transport simulations. Source tissues in this study include the trabecular marrow space (TMS), the trabecular bone surfaces (TBS), and the tr abecular bone volume (TBV). TBS sources are approximated as a 0.1m layer on the marrow side of the bone-marrow voxel interface. Target tissues include both the marrow space and bone endosteum. Once a given electron reaches the physical edge of the 3D micr o-image, that particle is re-introduced to the image at a corresponding lo cation at its opposing edge. The processes of particle transport within the image of spongiosa and its re-introduction are continued until all initial kinetic energy is e xpended. Particle histories are continued (50,000 to 2,000,000) until coefficients of variation on the absorbed fraction are below 1%.

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127 Chord-Based Infinite Spongiosa Transport (CBIST) Model Calculations of electron absorbed frac tion were further performed using chordlength distributions published in Whitwell ( 1973) for the Leeds 44-year male, as well as digital acquired from imaging data on th e UF 66-year male. In general, three fundamental types of random chord distributions exist: mean free path, I – interior radiator, and S – surface radiator (Coleman 1969; Eckerman et al. 1985; Kellerer 1971). Coleman (1969) further explains that I-random distributions (used for volume source simulations) can be derived from -random distributions (used for tissue region traversals). However, for S-random di stributions (used for surface sources), corresponding mathematical correlation do not exist, and thus S-random distributions must be measured directly within th e sample. At the present time, only -random, and thus I-random, distributions are available for both the Leeds and UF reference subjects. In a CBIST simulation, the bone and ma rrow chord-length distributions are randomly and alternately sampled with each sampled chord representing the potential pathlength of an electron in the current tis sue. Range-energy data are then used to calculate residual energies at tissue-interfaces and thus energy loss within each traversed tissue region. In the presen t study, electron range-energy re lationships for active (red) marrow and endosteum are derived through ra nge-scaling from liquid water using the Bragg-Kleeman rule (Knoll 2000) and elem ental tissue compositions from ICRU Publication 46 (ICRU 1992). For the osseous ti ssues of the bone trabeculae, “compact bone” range-energy data from ICRU Publication 37 (ICRU 1984) is similarly adjusted via Bragg-Kleeman to match the cortical bone reference composition given in ICRU Publication 46. Three sources tissues were modeled in th is study: trabecular marrow

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128 space (TMS), the trabecular bone volume (TBV ), and trabecular bone surfaces (TBS). Target tissues included the trabecular marrow space (marrow at 100% cellularity) and the trabecular bone endosteum (TBE). The CBIST transport methodology is best described by first considering an electron-emitter uniformly distri buted within the ti ssues of the bone trabeculae (i.e., TBV source). The transport code first ra ndomly samples a bone chord-length dT max from the I-random cumulative density function CDFI (dT max) for the skeletal site of interest (e.g., cervical vertebra). This sa mpled chord length is treated as the maximum possible distance that an electron may travel within its bone trabecula prior to entering the endosteal layer. The transport distance actually taken, dT, is thus uniformly sampled across the interval [0, dT max]. The range-energy function fo r electrons in bone tissue is then used to determine the total energy expended by the particle within that bone trabecula. If residual kinetic energy remains, the particle is further transported into (and potentially across) the ad jacent endosteal layer. Chord lengths across the endosteal layer must be considered in tandem with random sampling of the Leeds or UF marrow-cavity chord length, dMC, as described in Bouchet et al. (1999). For an electro n emerging from a bone trabecula, a random marrow-cavity chord length dMC is sampled under -randomness ( CDF) for the same skeletal site. The value of dMC is at most composed on two endosteal chord lengths (near and far side of the marrow cavity) and an intervening chord length across the marrow space: dMC = dE1 + dMS + dE2. (6-1)

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129 Values for dE1 and dE2 are determined through uniform sa mpling of the cosine of the entry angle () across each endosteal la yer (half-space geometry): 1E11 2E22 [0:1] d=(10m) [0:1] d=(10m) withand with (6-2) If dE1 + dE2 > dMC, then dE1 = dE2 = dMC /2 and dMS = 0, (6-3) the electron travels fully within the endosteal layer roughly para llel to the surface of the bone trabecula, and does not enter the marrow space. If, however, dE1 + dE2 < dMC, then dMS = dMC – ( dE1 + dE2). (6-4) The electron range-energy func tion in endosteal tissues is then used to determine the kinetic energy lost within the first endosteal la yer. If residual kine tic energy still exists, and dMS > 0, the electron is further transported within the tissues of the marrow space. Emission sites within the marrow (TMS source) or on the bone surfaces (TBS source) are considered in a similar fashion. For the latter, -random chord distributions are employed as an approximation of S-random dist ributions. It is estimated that errors associated with this approach are only seen at low electron energi es (below 30 keV), and become increasingly smaller with increasing electron energy. Absorbed fractions were calculated with sufficient histories to reduce the coefficient of variance to below 1% for primar y targets and for most secondary (adjacent) targets. Coefficients of variation were le ss than 3% (exhibited at energies below ~20 keV) for tertiary targets (TMS target fo r a TBV source with TBE as the intervening tissue).

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130 Chord-Length Distributions for the UF Reference Cancer Patient Accurate techniques for the both the generation (Rajon and Bolch 2003b) and measurement (Rajon and Bolch 2003a) of -random chords through any 3D object were previously investigated by Raj on and Bolch. These investigat ors further addressed issues relating to voxel effects impos ed on the measured chord-leng th distributions resulting from the stair-stepped representation of ti ssue interfaces within the digital image (e.g., surfaces of the bone trabeculae). Rajon a nd Bolch (2003a) showed that voxel effects increase the frequency of short chords and consequently reduce the mean chord-length by ~30% at resolutions of ~60 m. These investigators furt her expressed the need for a smoother representation of the bone-marrow in terface within the 3D digital image. The method recommended was an extension of the Marching Cube algorithm in which a trilinear interpolation of the eight gray-levels of the ne ighboring voxels are used to create a scalar field of gray-level va lues at the point of interest (e.g., the marching cube). This Trilinear Interpolation Marching Cube (o r TLI-MC) algorithm offers a bone-marrow interface surface that is reasonably smooth and continuous. Furthermore, it eliminates image-output data size problems (storing millions of triangles that constitute the surface) and simplifies the computations that are present in other surf ace-smoothing methods (e.g., the Marching Tetrahedron or Marching Cube algorithms). By using the TLI-MC algorithm and measuring distributions at an image resolution of 60 m, Rajon and Bolch (2003a) showed significant improvement s to the true distribution found within a mathematical simulation model of trabecular bon e. In the present study, all chord-length distributions through the 33 s pongiosa physical sections (a s taken from the 14 major

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131 skeletal sites) were constructed using th e TLI-MC technique and the related methods outlined by Rajon and Bolch (2003a). Convergence Limits for Absorbed Fractions under CBIST and VBIST As previously described by Eckerman (1985) and Bouchet et al. (1999), values of absorbed fraction under CBIST c onverge to a single source -independent value at high electron emission energies. These convergen ce values correspond to the fractional track length in the target tissue and are given by the following expressions: MCE CBIST MCTdd TMSS dd 2 (6-5) E CBIST MCTd TBES dd 2 and (6-6) T CBIST MCTd TBVS dd (6-7) where dT dMC dE and are the mean chord lengths across the bone trabeculae, marrow cavities, and endosteal la yer, respectively, for the bone site of interest, and is the ratio of CSDA ranges in marrow and in bone tissue at high electron energy (Bouchet et al. 1999). Values of dT and dMC are shown in Table 6-1 for the seven Leeds skeletal sites as given by their published distributions (Whitwell 1973). Be low these values are mean trabecular and marrow cavity chord length s for seven of the corresponding skeletal sites in the UF 66-year-old male subject. The next two columns of Table 6-1 show tabulated mean chord lengths across the e ndosteal layer and marro w space—two tissue regions which in combination define the marrow cavity. Note that these means are not taken directly from the measured chord-leng th distributions, but are determined via the CBIST algorithm defined above. For example, while the endosteal layer is uniformly

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132 considered to be a 10m tissue layers on the bone surfaces at all skeletal sites, Equations 6-1 through 6-4 show that the maxi mum endosteal chord is directly limited by the sampled marrow cavity chord length. In the final three column s of Table 6-1, the convergence limits on the absorbed fraction at high particle energy are displayed, as given by Equations 6-5 thr ough 6-7. The variable rS denotes any of the various electron source regions within the skeletal site. Jokisch et al. (2001a) deri ved corresponding convergen ce limits for infinite spongiosa transport within voxel-based models of the trabecular microstructure. These expressions are given as TMS VBIST TMSTBETBVm TMSS mmm (6-8) TBE VBIST TMSTBETBVm TBES mmm and (6-9) TBV TBE TMS TBV VBISTm m m m S TBV ) ( (6-10) where mTMS, mTBE, and mTBV are the tissue masses for the marrow space, bone endosteum, and bone trabeculae within the physical sections of spongiosa used for microCT imaging, and is the ratio of mean mass co llisional stopping powers in bone and marrow tissues evaluated at hi gh electron energy. As derived by Jokisch et al. (2001a), Equations 6-5 through 6-7 yield equivalent convergence limits on the absorbed fraction as given by Equations 6-8 through 6-10 for similar targets provided that TTBV MC TMSdTdx N N dTdx / /, (6-11)

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133 where NT and NMC are the cumulative number of electron tracks through the bone trabeculae and marrow cavities, respectiv ely. Under the assumption that NT NMC and that for 4-MeV electrons, the ratio of mean linear stopping powers in bone and in marrow is ~1.67, the authors conc lude that CBIST and VBIST s hould (in principle) produce equivalent convergence results at high sour ce energies. This agreement would then follow only if the chord distribution data are consistent with the volume (or mass) fractions displayed within the 3D voxelized mo del of the skeletal sample. In the lower portion of Table 6-1, mass estimates are given for the three tissue regions for each of the physical sections of spongiosa taken from th e UF 66-year male. Values of absorbed fraction convergence under VBIST are then gi ven as defined in Equations 6-8 through 6-10 above. Results and Discussion Comparisons are made in this study between the Leeds trabecular microstructure, defined by -random bone and marrow chord-length di stributions, and th e UF trabecular microstructure, defined by voxel-based images and -random chord-length distributions generated within those images. For a co mparison between VBIST and CBIST transport models using UF skeletal data it is important to first compare current published skeletal data (used clinically) to the ne wly acquired UF skeletal data. Trabecular Microstructure of the Leeds and UF Reference Subjects Figures 6-1 and 6-2 display normalized -random chord-length distributions across the marrow cavities of the Leeds reference male and the UF reference male, respectively, at several skeletal sites. With the exception of the cranium, the marrow chord distributions show remarkable similarity in both overall distribution shape and frequency.

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134 At marrow chords exceeding 1500 m, both reference individuals show similar patterns of chord frequency. At marrow chords below 1500 m, the UF individual displays a consistent distributional shape among the different skeletal sites in which chord frequencies rise to a peak value within the range of 300 m (ribs) to 650 m (femoral head and humerus). In both individuals, marrow chords <1000 m are found to range in frequency from 0.0004 to 0.0008 m-1. In the Leeds individual, however, greater discontinuities are seen in distributional shape (as compared to the UF marrow distributions). Furthermore, large upward excursions in fr equency are seen for many of the Leeds skeletal sites within the first couple of frequency bins. This pattern is similar to that arising from pixel-type effects as described by both Joki sch et al. (2001b) and Rajon et al. (2000). A clear outlier in marrow chord frequency shape is the cranium as seen in both individuals. For the UF individual, marrow ch ords in the cranium (volumeweighted average from CT microimages of the left parietal, right pa rietal, occipital, and frontal bone spongiosa) rise to a maximum frequency of 0.00097 m-1 at 250 m. For the Leeds individual, however, marrow chords in the cranium increase in frequency with decreasing chord-length, reaching a maximum frequency of ~0.00192 m-1 only within the first bin of the distribution (0-100 m). A corresponding comparison of the chordlength distributions across the bone trabeculae of these two reference individuals is made in Figures 6-3 and 6-4. Several conclusions can be drawn in the comparison of distributional shapes in Figure 6-3 (Leeds subject) and in Figure 6-4 (UF s ubject). First, one sees greater skeletal site variability in the trabecular microstructure of the Leeds indi vidual than seen in the UF individual for all sites exclusive of the cranium. Second, for the Leeds subject, a few bone sites (ribs

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135 and femoral neck) display a smaller seconda ry peak in bone-chord frequency at small sizes, a pattern not seen in th e UF individual. Third, as shown in Table 6-1, mean trabeculae chord-lengths are shown to be larg er in the UF individual within the femoral head and neck, lumbar vertebrae, and ribs than seen in the Leeds individual, whereas roughly equivalent mean bone chord-lengths ar e seen in the cervical vertebrae and ilium of the os coxae. Fourth, the largest bone c hord-lengths are found in the cranium of both individuals, with the average bone chord being slightly lower in the UF individual (mean of 504 m) than seen in the Leeds individual (mean of 511 m). Also shown in Table 6-1 are values of the tissue masses for the UF skeletal sites as defined within the segmented microCT images of spongiosa. In voxel-based models, tissue masses are the product of the tissue volume with in the digital image and refe rence tissue densities from ICRU Publication 46 (1992). Electron Dosimetry Comparisons between the UF and Leeds Microstructures Energy-dependent electron absorbed fracti ons are shown in Figures 6-5 through 610 for the Leeds reference individual (dashe d lines) and the UF reference individual (solid lines), as determined using the CBIST transport model in the parietal bone, femoral head, and ribs, respectively. The absorbed fractions to TMS and TBE are given for 3 source regions (TMS, TBV, TBS) at each skelet al site. For simplicity, the present study only considers bone marrow at 100% cellular ity; consequently, any differences in electron CBIST dosimetry may be attributed solely to corresponding differences in the marrow and bone chord-length distributions for these tw o individuals (and not on methods to account for energy loss to inactive or yellow marrow). At the present time, it

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136 is difficult to explicitly consider marrow cel lularity during particle transport in chordbased models of particle transport, an issue previously highlighted in Bolch et al. (2002). In Figures 6-5 and 6-6, absorbed fractions under the Leeds and UF chord distributions in the parietal bone are shown for TMS (Figure 6-5) and TBE (Figure 6-6) targets. As the mean marrow chord-length for the parietal bone in the UF 66-year male is approximately twice that found in the Leeds 44-year male (806.5 m versus 388.9 m – see Table 6-1), one would thus e xpect a divergence of values of (TMS rS) at high energies in the two reference individu als. In Figure 6-5, values of (TMS rS)UF approach a convergence limit of ~0.43, while the microstructu re of the Leeds individual predicts that (TMS rS)Leeds approach a convergence limit of only ~0.25. These differences are a direct reflec tion of differences in chord-le ngth distribution as given in Figures 6-1 through 6-4 for the parietal bone. In contrast, the mean marrow chord-length in the femoral head of the UF 66-year male is only 75% of that found in the Leeds 44year male (867 and 1157 m, respectively), and thus a higher convergence of (TMS rS) is shown in Figure 6-7 for the Leeds indi vidual over that for the UF subject. Finally, Figure 6-9 demonstrates even closer agreement in the convergence of (TMS rS) at high electron energies in the ribs. At this skeletal site, mean marrow chord-lengths are very close (~1703 m each), while the mean bone chord-length in the ribs of the UF individual is ~24% la rger than that in the Leeds subject. For a TBE target, the figures (Figs. 6-6, 68, and 6-10) show differences in values of (TBE rS)UF and (TBE rS)Leeds at source energies below 1 MeV. For higherenergy particles, however little difference in (TBE rS) is noted between the Leeds and UF reference individuals as convergence values are shown to be essentially equivalent for

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137 electron sources irradiating th e trabecular endosteum (see Ta ble 6-1). As before, the closest agreement in values of (TBE rS) are seen for the ribs across all energies. In the parietal bone, values of (TBE TBV) are also noted to be ve ry similar for both subjects, while values of (TBE TMS) are notably higher in th e Leeds subject. The latter observation is again a reflection of the smalle r marrow cavities in the parietal bone of the Leeds subject where electrons emitted within the marrow space are ab le to approach and then irradiate the trabecular e ndosteum to a greater extent than in the UF individual. In Table 6-2, further comparisons are given between the energy-dependent electron absorbed fractions to active bone marrow with in the Leeds reference individual and their corresponding values in the UF reference individual, under CBIST simulations. Three sections of absorbed fraction ratios are give n in Table 6-2: TMS sources (upper third), TBS sources (middle third), and TBV sources (lower third). For each source tissue, two averages are taken of the absorbed fracti on ratio: one for elec tron energies below 100 keV, and one for electron energies greater than 100 keV. For TMS sources, the greatest differences in electron dosimetry between th e two reference individuals are found in the cranium, where at energies exceeding 100 keV, values of (TMS TMS)UF are shown to exceed values of (TMS TMS)Leeds by factors of 1.39 (200 keV) to 1.71 (4 MeV). Within the 6 remaining skeletal sites, howev er, electron dosimetry is shown to vary on average only 15% at high energies, with much closer agreement at low electron energies. Similar comparisons can be made for bone surface and bone trabeculae sources; where again, the largest difference are seen at high electron energies in the cranium (mean ratios of 1.61 in the cranium at energies 100 keV). Note that ratios are not given for electrons below 30 keV for su rface and bone volume sources, as values of

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138 absorbed fraction are zero in both models (ele ctron energy is fully absorbed in either the bone trabeculae and/or 10m endosteal layer). Comparable ratios of (TBE S)UF to (TBE S)Leeds are given in Table 6-3 for sources in the TMS, TBS, or TBV spongiosa ti ssues. For marrow sources irradiating the trabecular endosteum, absorbed fraction ra tios in CBIST dosimetry between the two reference individuals range from a low of 0.40 (cranium) to a high of 1.35 (femoral head) for electron energies below 100 keV, and from a low of 0.70 (os coxae) to a high of 1.12 (femoral head and lumbar vertebrae) at energies above 100 keV. For bone surface sources, the greatest difference in values of (TBE TBS) between the UF and Leeds individual is shown in Table 6-3 to occur in the os cox ae at higher electron energies (mean ratio of 0.83). For bone volume s ources, the greatest difference in CBIST dosimetry is shown to be in the femoral head at low electron energi es (mean ratio of 0.67 at energies 100 keV). Comparison of CBIST and VBIST for Marrow Space Targets In the second portion of the study, chord-length distribut ions from the UF 66-year male subject are used to compare chor d-based (CBIST) and voxel-based (VBIST) methods for infinite spongiosa transport in the skeletal tissu es. In Figures 6-11 to 6-16, absorbed fractions to the marrow space (TMS ) are shown for three source regions (TMS, TBS, and TBV) and six skeletal sites: the femoral head, femoral neck, lumbar vertebra, cervical vertebra, os coxae, and cranium, re spectively. At electron energies below ~50 keV, values of (TMS TMS) are approximately equiva lent under the two transport methodologies at all skeletal sites considered. At high energies (~4 MeV), values of (TMS TMS) approach their theoretically-der ived convergence values for both CBIST

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139 and VBIST simulations in femoral head, fe moral neck, lumbar vertebrae, cervical vertebrae, and cranium. As shown by Joki sch et al. (2001a), c onvergence values are expressed in terms of mass fractions for voxe l-based infinite spongiosa transport models (see Equations 6-8 through 6-10 and values given in Table 6-1). For each of the six bone sites, CBIST convergence falls slightly below that seen for VBIST convergence. The mean ratio of VBIST to CBIST convergen ce limits was found to be 0.945 0.35 for all skeletal sites. At electron energies exceeding 50 keV, Fi gures 6-11 to 6-16 indicate that, for the six bone sites considered, the chord-based IST model consistently pred icts smaller values of (TMS TMS) than given by the voxel-based IS T model. This difference can be attributed to the fact that, under a CBIST simulation, sampled chord-lengths in the marrow space are directly equated to electron path-lengths within the marrow tissues. As a result, non-linear trajectories resulting fr om elastic and inelastic collisions are not properly accounted for under CBIST, and the ma rrow space is thus traversed with lower energy loss than seen under full 3D voxelbased transport. As the electron energy increases, however, the assumption of a linear pathlength under CBIST become less and less in error, and both mode ling approaches (if applied to the same skeletal microstructure) yield similar values of (TMS TMS) within the agreement of their convergence limits. In contrast, excellent agreement between CBIST and VBIST radiation transport is seen across all bone sites for both TBS and TB V electron sources at energies up to ~800 keV to 1 MeV. Exceptions are noted, however, for values of (TMS TBV) in the femoral head (Figure 6-11) and the os coxae (Figure 6-15) at energies as low as ~200

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140 keV, where VBIST results exceed those given by CBIST simulations. These discrepancies warrant further i nvestigation. One possible ex planation is that in these particular skeletal sites, bone and marrow c hord-lengths are not properly characterized by separate and independent probability density functions. In the CBIST method, the bone chord distribution is sampled alternatel y and independently of the marrow chord distribution. As discussed in Jokisch et al. (2001b), this f undamental tenant of the CBIST method might be invalid especially for those bone sites which exhibi t anisotropy in their trabecular microstructure. These authors s uggest that a better implementation of chordbased transport would entail the use of a joint probability distribution in which the selection of the bone trabeculae chord would be dependent (not independent) of the value of the previously select ed marrow cavity chord. Comparison of CBIST and VBIST for Bone Endosteum Targets As a final comparison, we show in Figures 6-17 and 6-18, values of (TBE TMS), (TBE TBS), and (TBE TBV) for electron sour ces in the os coxae and ribs. Similar results are noted at othe r skeletal sites. As the electron energy increases, CBIST and VBIST values of (TBE rS) begin to increa singly diverge, and approach different convergence limits under th e two modeling methodologies. In both Figure 6-17 (os coxae) and 6-18 (ribs), a hi gher convergence value is given under chordbased transport than under voxel-based transpor t. In fact, ratios of CBIST to VBIST convergence limits for TBE targets range from a low of 1.86 in the femoral head to a high of 3.56 in the ribs (skeletal mean ratio of 2.2 0.6). This separation of convergence values at high electron energy was previous ly demonstrated by J okisch et al. (2001a), who further noted that CBIST and VBIST c onvergence values should, in theory, be

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141 identical (independent of the modeling tec hnique). One explanation for these observed discrepancies is the way in which pathlengt hs across trabecular endosteum are simulated under CBIST (infinite half-space) versus VB IST (transport geometry presented by the segmented 3D image). Consider for the moment, two electrons emerging from a bone trabecula and entering the adjacent endosteal layer as shown in Figure 6-19. For both the CBIST model (upper section) and VBIST model (lower se ction), the first electron traverses the endosteal layer at entry angle 1, while the second crosses at entry angle 2. As the endosteal pathlength for angle 1 is relatively short, energy deposition under CBIST or VBIST are expected to be similar. For a wider entry angle 2, however, a relatively long endosteal pathlength (assume d to be linear) would arise under CBIST (limited only by the sampled marrow cavity chor d). As shown in the lower section of Figure 6-19, however, this same electron unde r voxel-based transport will most likely scatter out of the endosteal layer, thus de positing some of its energy to other tissues regions (bone or marrow). Statistically th en, one would expect that 3D voxel-based transport techniques would result in a lowe r estimate of TBE energy deposition than predicted under a CBIST mode l of electron transport. Conclusion The present study was conducted (1) to co mpare chord-based electron dosimetry within two reference skeletal models – the 44year male from the University of Leeds and the 66-year male from the University of Flor ida, and (2) to compar e chord-based versus voxel-based methods of skeletal dosimetry usi ng imaging data from the UF subject. The following conclusions can be drawn from obj ective 1 of the study. First, minimal differences are seen in electron skeletal dos imetry for low-energy emitters irradiating the

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142 marrow tissues (where the marrow cavity si ze is not important) and for high-energy emitters irradiating the trabecular endost eum (where the CBIST algorithm assumes infinite half-space geometry independent of the trabecular microstructure). Second, significant but predictable differences are seen in the electron dosimetry of high-energy emitters irradiating the marrow space and (to a lesser extent) low-energy emitters irradiating the trabecular endosteum. Third, convergence limits for (TMS rS) are seen to (1) differ to a greater extent in bone sites with significantly different marrow cavity chord-length distributions (e.g., parietal bone), and (2) to approach a common convergence limit when equivalent marrow chorddistributions are noted (e.g., ribs). The following additional conclusions can be drawn from objective 2 of the study. First, chord-based IST models of electron tr ansport consistently yield lower values of (TMS TMS) than predicted under voxel-base d IST models, a feature directly attributed to the inability of chord-based models to account for non-linear electron trajectories in the marrow tissues. Second, excellent agreement is seen between CBIST and VBIST values of (TMS TBS), especially for electrons of energies below 500 keV. This finding is particularly significant in that the CBIST model in this study approximates the trajectories of su rface-source emissions by sampling -random chord distributions as an approximation to samp ling S-random distributions. The agreement seen here lends strong suppor t to the conclusion that random chord sampling is an approximate, yet reasonable substitute wh en S-random chord distributions are not available. Future efforts in image-based skel etal dosimetry studies should investigate this issue further through direct acquisition of S -random chord distributio ns within the 3D digital images of spongiosa. Third, excellent agreement is also seen between CBIST and

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143 VBIST values of (TMS TBV), especially for electrons of energies below 500 keV. In the cervical vertebra and cranium, the CBIST-VBIST agreement in (TMS TBV) exceeds to even higher energies (~1 MeV). In the other four bone sites considered, however, VBIST values of (TMS TBV) begin to rise above those predicted under CBIST simulations. This latter observation wa rrants further investigation as previously noted. Fourth, at high electron energies, values of (TMS rS) under CBIST consistently fall below the corresponding convergence li mits as seen under VBIST (~5.5% on average). Values of (TMS rS) at high energy shown in Figures 6-11 through 6-16 confirm that the transport calculations ar e valid and do indeed converge to values predicted under either Equations 6-5 thr ough 6-7 (CBIST) or Equations 6-8 through 6-10 (VBIST). Consequently, one may infer that this ~5.5% discrepancy in VBIST-CBIST convergence limits reflects the degree to wh ich sampled chord-leng th distributions are able (or not able) to fully ch aracterize the 3D microstructure of the skeletal spongiosa. Finally, the data of Figures 6-17 and 6-18 indicate that significant differences in dosimetry of the trabecular endosteum exist be tween CBIST or VBIST electron transport. Convergence limits on (TBE rS) given by CBIST are shown to exceed those given by VBIST by an average of a factor of 2.2. The overestimate of energy deposition to TBE is attributed to limitations in the CBIST algorithm used to simulate electron traversal of this tissue layer (e.g., half-space geometry with linear electron trajectories).

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144 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) Femur Head Femur Neck Parietal Bone Ribs Iliac Crest Cervical Vertebra Lumbar Vertebra Leeds Marrow Cavities Figure 6-1. Normalized, omnidirectional c hord-length distributions through the marrow cavities of the Leeds 44-year reference male.

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145 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 05001000150020002500300035004000Chord length ( m)Normalized frequency (per m) Femur Head Femur Neck Cranium Ribs Os Coxae Cervical Vert Thoracic Vert Lumbar Vert Humerus UF Marrow Cavities Figure 6-2. Normalized, omnidirectional c hord-length distributions through the marrow cavities of the UF 66-year reference male cancer patient.

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146 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 01002003004005006007008009001000Chord length (m)Normalized frequency (per m) Femur Head Femur Neck Parietal Bone Ribs Iliac Crest Cervical Vertebra Lumbar Vertebra Leeds Bone Trabeculae Figure 6-3. Normalized, omnidirectional c hord-length distributions through the bone trabeculae of the Leeds 44-year reference male.

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147 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 0.0050 01002003004005006007008009001000Chord length (m)Normalized frequency (per m) Femur Head Femur Neck Cranium Ribs Os Coxae Cervical Vert Thoracic Vert Lumbar Vert Humerus UF Bone Trabeculae Figure 6-4. Normalized, omnidirectional c hord-length distributions through the bone trabeculae of the UF 66-year reference male cancer patient.

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148 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST Leeds (TMS) CBIST Leeds (TBV) CBIST Leeds (TBS) Convergence Value (Leeds) CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (UF) Parietal Bone (Marrow Space Target) TMS TBV TBS Convergence Limits Figure 6-5. Electron absorbed fractions to the active bone marrow within the parietal bone for three source tissues – TAM, TBV, and TBS. Data shown by solid lines represent CBIST transport using th e Leeds’ chord length distributions, while those given by dashed lines are from the chord length distributions generated within the UF reference patient.

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149 0.00 0.05 0.10 0.15 0.20 0.25 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE CBIST Leeds (TMS) CBIST Leeds (TBV) CBIST Leeds (TBS) Convergence Value (Leeds) CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (UF) Parietal Bone (Endosteum Target) TBS Convergence Values Figure 6-6. Electron absorbed fractions to th e bone endosteum within the parietal bone for three source tissues – TAM, TBV, and TBS. Data shown by solid lines represent CBIST transport using the L eeds’ chord length distributions, while those given by dashed lines are from th e chord length distri butions generated within the UF reference patient.

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150 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST Leeds (TMS) CBIST Leeds (TBV) CBIST Leeds (TBS) Convergence Value (Leeds) CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (UF) Femoral Head (Marrow Space Target) TMS TBV TBS Convergence Values Figure 6-7. Electron absorbed fractions to the active bone marrow within the femoral head for three source tissues – TAM, TB V, and TBS. Data are shown for both the Leeds and UF chord-length dist ributions under CBIST simulations.

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151 0.00 0.05 0.10 0.15 0.20 0.25 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE CBIST Leeds (TMS) CBIST Leeds (TBV) CBIST Leeds (TBS) Convergence Value (Leeds) CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (UF) Femoral Head (Endosteum Target) TBS Convergence Values TBV TMS Figure 6-8. Electron absorbed fractions to th e bone endosteum with in the femoral head for three source tissues – TAM, TBV, and TBS. Data are shown for both the Leeds and UF chord-length distributions under CBIST simulations.

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152 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST Leeds (TMS) CBIST Leeds (TBV) CBIST Leeds (TBS) Convergence Value (Leeds) CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (UF) Ribs (Marrow Space Target) TMS TBV TBS Convergence Values Figure 6-9. Electron absorbed fractions to the active bone marrow within the ribs for three source tissues – TAM, TBV, and TBS. Data are shown for both the Leeds and UF chord-length distributions under CBIST simulations.

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153 0.00 0.05 0.10 0.15 0.20 0.25 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE CBIST Leeds (TMS) CBIST Leeds (TBV) CBIST Leeds (TBS) Convergence Value (Leeds) CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (UF) Ribs (Endosteum Target) TBS Convergence Values TBV TMS Figure 6-10. Electron absorbed fr actions to the bone endosteum within the ribs for three source tissues – TAM, TBV, and TBS. Data are shown for both the Leeds and UF chord-length distributi ons under CBIST simulations.

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154 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (CBIST) VBIST UF (TMS) VBIST UF (TBV) VBIST UF (TBS) Convergence Value (VBIST)TMS TBV TBS Femur Head (Marrow Space Target) Figure 6-11. Electron absorbed fractions to the active bone marrow within the femoral head of the UF reference male cancer patient. Solid lines indicate values under VBIST simulations, while dashed lines indicate values under CBIST simulations.

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155 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (CBIST) VBIST UF (TMS) VBIST UF (TBV) VBIST UF (TBS) Convergence Value (VBIST)TMS TBV TBS Femur Neck (Marrow Space Target) Figure 6-12. Electron absorbed fractions to the active bone marrow within the femoral neck of the UF reference male cancer patient. Solid lines indicate values under VBIST simulations, while dashed lines indicate values under CBIST simulations.

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156 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (CBIST) VBIST UF (TMS) VBIST UF (TBV) VBIST UF (TBS) Convergence Value (VBIST)TMS TBV TBS L4 Lumbar Vertebra (Marrow Space Target) Figure 6-13. Electron absorbed fractions to the active bone marrow within the L4 lumbar vertebra of the UF reference male cancer patient. Two simulation methods are compared: VBIST (solid lines) and CBIST (dashed lines) simulations.

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157 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (CBIST) VBIST UF (TMS) VBIST UF (TBV) VBIST UF (TBS) Convergence Value (VBIST)TMS TBV TBS C6 Cervical Vertebra (Marrow Space Target) Figure 6-14. Electron absorbed fractions to the active bone marrow within the C6 cervical vertebra of the UF reference male cancer patient. Two simulation methods are compared: VBIST (solid lines) and CBIST (dashed lines) simulations.

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158 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (CBIST) VBIST UF (TMS) VBIST UF (TBV) VBIST UF (TBS) Convergence Value (VBIST)TMS TB V TBS Os Coxae (ilium) (Marrow Space Target) Figure 6-15. Electron absorbed fractions to the active bone marrow within the ilium of the UF reference male cancer patient. Two simulation methods are compared: VBIST (solid lines) and CBIST (dashed lines) simulations.

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159 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron Energy (MeV)Absorbed Fraction to TMS CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (CBIST) VBIST UF (TMS) VBIST UF (TBV) VBIST UF (TBS) Convergence Value (VBIST)TMS TBV TBS Cranium (parietal bone) (Marrow Space Target) Figure 6-16. Electron absorbed fractions to the active bone marrow within the parietal bone of the UF reference male cancer patient. Two simulation methods are compared: VBIST (solid lines) and CBIST (dashed lines) simulations.

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160 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (CBIST) VBIST UF (TMS) VBIST UF (TBV) VBIST UF (TBS) Convergence Value (VBIST)TBS TAM TBV Os Coxae (ilium) Endosteum Target Convergence Values Figure 6-17. Electron absorbed fractions to the trabecular endosteu m within the ilium of the UF reference male cancer patient. Two simulation methods are compared: VBIST (solid lines) and CBIST (dashe d lines) simulations, along with their respective convergence values.

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161 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.010.1110Electron Energy (MeV)Absorbed Fraction to TBE CBIST UF (TMS) CBIST UF (TBV) CBIST UF (TBS) Convergence Value (CBIST) VBIST UF (TMS) VBIST UF (TBV) VBIST UF (TBS) Convergence Value (VBIST)TBS TAM TBV Ribs Endosteum Target Convergence Values Figure 6-18. Electron absorbed fractions to the tr abecular endosteum within the ribs of the UF reference male cancer patient. Two simulation methods are compared: VBIST (solid lines) and CBIST (dashe d lines) simulations, along with their respective convergence values.

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162 Figure 6-19. Comparison of electron transp ort paths through the trabecular endosteum under either CBIST simulations or VBIST simulations for two different initial trajectory angles 1 and 2.

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163Table 6-1. Mean values of tr abecular and marrow cavity chord-lengths as given by the present UF study and those published from the University of Leeds. Values of mean chord lengths across the endosteal layer and the marro w space (which in combination define the marrow cavity) are shown as well based upon the CBIST algorithm presente d in this study. Values for sourceindependent absorbed fractions at high electron energies (co nvergence limits) are given for TB E, TMS, and TBV targets in both the Leeds and UF bone sites. At the bottom of the ta ble, corresponding values of convergence limits under VBIST simulations are shown. Mean TrabecularMean MarrowMean EndostealMean Marrow (TMS S) (TBE S) (TBV S) SourceSkeletal Site Chord ( m)Cavity Chord ( m)Chord ( m)Space Chord ( m) Leeds Femur head231.71156.652.41051.80.6810.0680.251 Leeds Femur neck314.11655.455.11545.20.7090.0510.241 Leeds Parietal bone511.4388.940.4308.10.2480.0650.687 Leeds Ribs265.61703.354.31594.70.7430.0510.207 Leeds Illiac Crest242.1903.950.2803.50.6140.0770.309 Leeds Cervical Vertebra279.2910.749.0812.70.5900.0710.339 Leeds Lumbar Vertebra245.51233.251.61130.00.6880.0630.250 UFFemur head (right)348.5866.850.8765.20.5280.0700.402 UFFemur neck (right)354.41551.855.51440.80.6720.0520.276 UFParietal bone (left)503.6806.550.3705.90.4280.0610.510 UFRib (7th left)329.21702.655.21592.20.7070.0490.244 UFOs coxae (iliac crest)245.21507.955.31397.30.7290.0580.214 UFCervical vertebra (C6)279.21046.252.1942.00.6230.0690.308 UFLumbar vertebra (L4)288.31056.552.5951.50.6190.0680.313 Bone TrabeculaeEndosteal LayerMarrow Space Mass (g)Mass (g)Mass (g) UFFemur head (right)45.23.754.50.5520.0380.410 UFFemur neck (right)45.73.6108.60.7090.0230.268 UFParietal bone (left)90.54.967.80.4400.0320.528 UFRib (7th left)20.91.587.40.8120.0140.175 UFOs coxae (iliac crest)157.519.3471.40.7460.0310.224 UFCervical vertebra (C6)22.82.345.00.6640.0340.302 UFLumbar vertebra (L4)96.99.9182.60.6530.0350.311 Measured Chord Distributions Values from VBIST Convergence Limits Values from CBIST AlgorithmCBIST Convergence Limits

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164 Table 6-2. Ratios of absorbed fractions to active marrow (UF valu es to Leeds values under CBIST). Ratio of (TMS TMS)UF to (TMS<-TMS)LeedsFemur HeadFemur NeckRibsOs CoxaeCraniumCervical VertLumbar Vert Energy (MeV)(right)(right)(7th left)(iliac crest)(left parietal)(C6)(L4) 0.0101.001.001.001.001.011.001.00 0.0151.001.001.001.001.011.001.00 0.0201.001.001.001.001.021.001.00 0.0301.001.001.001.011.031.001.00 0.0400.991.001.001.011.051.011.00 0.0500.991.001.001.021.071.011.00 0.1000.960.991.001.051.171.010.98 0.2000.890.970.971.131.391.020.92 0.5000.800.950.941.171.581.040.89 1.0000.790.950.951.181.651.050.90 1.5000.790.950.951.181.681.050.90 2.0000.780.950.951.181.691.050.90 4.0000.780.950.951.191.711.050.90Mean ( 100 keV)0.991.001.001.011.051.001.00Mean ( 100 keV)0.830.960.961.151.551.040.91 Ratio of (TMS TBS)UF to (TMS<-TBS)LeedsFemur HeadFemur NeckRibsOs CoxaeCraniumCervical VertLumbar VertEnergy (MeV)(right)(right)(7th left)(iliac crest)(left parietal)(C6)(L4) 0.010 0.015 0.020 0.0301.001.001.011.021.061.021.02 0.0401.001.011.021.021.081.031.02 0.0501.011.011.021.031.141.041.03 0.1001.000.970.951.031.221.051.01 0.2000.870.940.841.071.471.080.94 0.5000.820.960.941.141.711.070.93 1.0000.800.950.951.161.721.060.91 1.5000.790.950.951.171.721.060.91 2.0000.790.950.951.181.721.060.91 4.0000.780.950.951.181.731.060.90Mean ( 100 keV)1.001.001.001.021.121.041.02Mean ( 100 keV)0.840.950.931.131.611.060.93 Ratio of (TMS TBV)UF to (TMS<-TBV)LeedsFemur HeadFemur NeckRibsOs CoxaeCraniumCervical VertLumbar VertEnergy (MeV)(right)(right)(7th left)(iliac crest)(left parietal)(C6)(L4) 0.010 0.015 0.020 0.0300.670.880.811.001.101.020.86 0.0400.670.900.821.011.081.020.87 0.0500.670.900.831.021.111.030.87 0.1000.670.940.891.061.231.080.92 0.2000.731.001.071.111.511.131.05 0.5000.790.961.041.151.701.080.97 1.0000.780.960.981.171.721.060.93 1.5000.780.950.971.181.721.060.92 2.0000.780.950.961.181.731.060.91 4.0000.780.950.961.181.731.060.91Mean ( 100 keV)0.670.900.841.021.131.040.88Mean ( 100 keV)0.760.960.981.151.621.080.94

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165 Table 6-3. Ratios of absorbed fractions to bone endosteum (UF values to Leeds values under CBIST). Ratio of (TBE TMS)UF to (TBE<-TMS)LeedsFemur HeadFemur NeckRibsOs CoxaeCraniumCervical VertLumbar VertEnergy (MeV)(right)(right)(7th left)(iliac crest)(left parietal)(C6)(L4) 0.0101.341.000.910.500.310.731.02 0.0151.341.000.920.510.330.751.04 0.0201.341.020.930.520.340.761.06 0.0301.351.030.960.540.370.791.09 0.0401.361.050.990.560.420.831.14 0.0501.371.071.030.580.460.871.19 0.1001.371.071.000.610.570.921.22 0.2001.201.020.850.640.730.941.12 0.5001.091.040.960.710.910.971.11 1.0001.061.030.980.740.930.971.10 1.5001.051.030.980.740.920.971.09 2.0001.051.030.970.750.930.971.09 4.0001.041.030.970.750.940.971.09Mean ( 100 keV)1.351.040.960.540.400.811.11Mean ( 100 keV)1.121.030.960.700.850.961.12 Ratio of (TBE TBS)UF to (TBE<-TBS)LeedsFemur HeadFemur NeckRibsOs CoxaeCraniumCervical VertLumbar VertEnergy (MeV)(right)(right)(7th left)(iliac crest)(left parietal)(C6)(L4) 0.0101.001.001.001.001.001.001.00 0.0151.001.001.001.001.001.001.00 0.0201.001.001.001.001.001.001.00 0.0301.001.000.990.990.990.990.99 0.0401.000.970.960.970.960.970.97 0.0501.000.970.940.950.890.940.95 0.1000.990.920.840.930.830.940.89 0.2000.890.940.850.920.880.960.91 0.5000.980.990.930.830.940.961.01 1.0001.011.000.950.790.940.971.05 1.5001.021.010.970.780.940.971.06 2.0001.021.020.970.770.940.971.07 4.0001.031.020.970.760.940.971.08Mean ( 100 keV)1.000.980.960.980.950.980.97Mean ( 100 keV)0.990.990.930.830.910.961.01 Ratio of (TBE TBV)UF to (TBE<-TBV)LeedsFemur HeadFemur NeckRibsOs CoxaeCraniumCervical VertLumbar VertEnergy (MeV)(right)(right)(7th left)(iliac crest)(left parietal)(C6)(L4) 0.0100.670.890.810.991.021.000.85 0.0150.670.890.810.991.021.000.85 0.0200.660.890.810.991.011.000.85 0.0300.660.890.810.991.011.000.85 0.0400.660.890.820.991.001.000.85 0.0500.660.890.810.990.980.990.85 0.1000.670.930.880.980.961.000.88 0.2000.800.991.110.970.971.001.02 0.5000.940.991.000.840.930.971.03 1.0000.991.010.980.790.940.971.06 1.5001.011.010.980.780.940.971.08 2.0001.011.020.980.770.940.971.07 4.0001.021.020.980.760.940.971.08Mean ( 100 keV)0.670.890.820.991.001.000.86Mean ( 100 keV)0.921.000.990.840.950.981.03

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166 CHAPTER 7 REFERENCE SKELETAL DOSIMETR Y MODEL FOR AN ADULT MALE RADIONUCLIDE THERAPY PATIENT5 Introduction Toxicity of the hematopoietically activ e bone marrow continues to be a primary limitation in radiolabeled antibody therapy (D ivgi et al. 1995; Kaminski et al. 1996; Vriesendorp et al. 1996). For both intravenous and non-intravenous (L arson et al. 1991; Rosenblum et al. 1991; Stewart et al. 1990) administrations in radioimmunotherapy, suppression of the active marrow has limited th e amount of radioactivity that may be given to the patient. Techniques must ther efore be sought by which marrow toxicity may be predicted in these patients (Sgouros et al. 2000; Shen et al. 2002; Si egel et al. 2003). Analyses of hematological toxicity vers us absorbed dose to active marrow have confirmed that absorbed dose is a fundamental indicator of marrow toxicity, but only if the dose estimate is made as specific as possi ble to the patient in question (Blumenthal et al. 1999; Juweid et al. 1999; Liu et al. 1997; O'D onoghue et al. 1998; Sgouros et al. 1997; Sgouros et al. 1996a). Through the MIRD schema D=AS the absorbed dose to active marrow must include separate assessmen ts of the cumulated activity within the source region (marrow or bone), as well as the radionuclide S value (absorbed dose per unit cumulated activity). While extensive e fforts have been made in patient-specific 5 This chapter has been submitted to The Journal of Nuclear Medicine: Shah AP, Bolch WE, Rajon DA, Patton PW, Brindle JM, and Jokisch DW. Submitted. A image-based reference skeletal model for the adult male radionuclide therapy patient. J Nuc Med: submitted.

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167 assessments of activity uptake in bone or ma rrow (Juweid et al. 1995; Macey et al. 1995; Sgouros 1993; Sgouros et al. 1996b; Siegel et al. 1990), very little research has been focused on improving the patient specificity of radionuclide S values for skeletal tissues. Indeed, the physics models used to evaluate electron and beta-particle absorbed fractions in the skeletal tissues have not been fundamentally updated since the mid-1970s. The skeleton is perhaps the most diff icult of all organ systems to model anatomically in a patient-specific manner. For many organs of the body (e.g., lungs or liver), regional CT scans can be used to crea te image contours of patient anatomy. When coupled with fused images of radiopharm aceutical uptake (SPECT-CT or PET-CT), all pertinent information is available for a pati ent-specific dose assessment of the tumor and surrounding normal tissues using point-kernel co nvolution, voxel S value convolution, or direct Monte Carlo radiation tran sport. In the skeleton, howe ver, the relevant tissues of the bone trabeculae, bone endosteum, and marrow cavities are exceedingly small and cannot be imaged directly through in-vivo CT or MR. Consequently, one must rely upon previously developed reference skeletal mode ls (RSM) in which ex-vivo microimages of the excised trabecular micro-architecture (acquired via contact radiography, NMR microscopy, or microCT) are used for radiation transport and/or tissue mass determination. To date, the only complete set of skeletal microstructural data, in a format sufficient for radiation transport simulations is that generated by FW Spiers at the University of Leeds in the late 1960s to mid-1970s. The Leeds group used optical scanning techniques to acquire linear c hord-length distributions across the bone

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168 trabeculae and marrow cavities of the lumbar ve rtebrae for several subjects, as well as at several skeletal sites in a 1.7-year child (5 sites), a 9-year child (5 sites), and a 44-year male (7 sites) (Beddoe 1976; Beddoe et al. 1976; Whitwell 1973; Whitwell and Spiers 1976). The bone and marrow chord-distributions for the Leeds 44-year male subject have been used as the fundamental basis for dete rmining electron and beta particle absorbed fractions embodied in essentially all published RSMs. These include the skeletal models of (1) MIRD Pamphlet No. 11 for nuclear medicine (Snyder et al. 1975), (2) ICRP Publication 30 for radiati on protection (1979), (3) ORNL TM-8381 for photon internal dosimetry (Cristy and Eckerman 1987), (4) Bouchet et al. (2000, 1999) for nuclear medicine, (5) the Eckerman & Stabin (2000) model used in the MIRDOSE 3.0 dosimetry code, and (6) the Stabin & Siegel (2003) m odel used within the OLINDA 1.0 dosimetry code. Of equal importance in the determinati on of radionuclide S values are skeletal tissue masses; unfortunately, tissue masses were not reported for the Leeds 44-year male subject, and thus the various RSMs mentione d above have had to rely on a variety of patient and/or cadaver studies as summari zed in ICRP Publications 70 and 89 (ICRP 1995; ICRP 2002). These studies include thos e by (1) Mechanik in 1926—total marrow masses and their relative distribution within the skeleton, (2) Custer in 1974—reference marrow cellularities, (3) Trotter and Hixon in 1974—bone tissue masses, and (4) Beddoe in 1976—bone trabeculae surface-to-volume ratio s for estimating endosteal tissue masses (Mechanik 1926; Woodard and Holodny 1960; Custer 1974; Trotter and Hixon 1974; Beddoe 1976). The ICRP Reference Man skeletal model is thus assembled from a wide range of subject populations and measuremen t techniques. More fundamentally, the

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169 anatomical source for absorbed fractions is not the same as the anatomical sources from which reference tissue masses are defined. Conceptionally, a reference sk eletal dosimetry model may be used under two very different applications: (1) prospective dose assessment in radiati on protection, and (2) retrospective (dose-response correlations) or prospective (treatment planning) dose assessment in medical dosimetry. Under the first application, the RSM should be designed to match as closely as possible the typical or average individual in the worker population. The skeletal tissu e doses determined by such a model are used solely to establish dose limits based upon an acceptable ri sk of radiation injury and/or stochastic cancer risk. The dose value itself is not intended to be rescaled to any specific individual. This is in contrast to a RSM for medical patient dosimetry. In this case, the model does not necessarily have to represent the “average” or “typical” patient (if such an individual can even be defined). Instead, the medica l RSM should be known in explicit detail, especially for those anatomical and physiologi cal parameters which ar e subject to patientto-patient variability and can be assess ed in individual patients through in-vivo imaging (skeletal volumes, marrow cellularity, etc.). In this respect, the ICRP Reference Man model, which utilizes the Leeds 44-year male chord-distribution data, is a perfectly acceptable model for ra diation protection. The ICRP model, however, can be considered limited in its ability to provide patientspecific doses in radionuclide therapy. These limitations include (1) lack of consideration of energy loss to cortical bone for intermediate-to-high energy beta sources (Patton et al. 2002b; Shah et al. 2004a), (2) reliance on fixed reference values of marrow cellularity (Cristy 1981; Custer and Ahlfeldt 1932), (3) use of multiple data sources for

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170 skeletal tissue masses (Mechanik 1926; Trotte r and Hixon 1974) separate from that used to determine values of abso rbed fractions, and (4) lack of bone site-specific data on spongiosa volumes, cortical bone volumes, and marrow volume fractions (ICRP 1995). In studies by Ballon et al. ( 1996), it has been shown that marrow cellularity (which can vary greatly from patient-to-patient) can be assessed non-invasively using MR techniques and can thus be used to refine patient-specifi c skeletal doses. Furthermore, Shen et al. (2002) found relatively strong correlations be tween marrow dose and toxicity provided the reference model marrow dose was rescal ed using CT-based estimates of spongiosa volumes within the patient’s lumbar vertebrae. A problem encountered by these authors was that comparable spongiosa volumes in the ICRP reference model were not available, and pooled CT volumetry data from severa l patients had to be used instead. To address the need for a more comprehens ive, detailed, and internally consistent model for skeletal tissue dose, we have perf ormed a variety of in-vivo and ex-vivo CT imaging studies of the entire skeleton of a 66-year male cadaver—an age more representative of radi onuclide cancer therapy patients th an the Leeds 44-year male. In addition, sections of tr abecular spongiosa were im aged under micro-computed tomography revealing high-resolution detail s of the individual bone marrow cavities and bone trabeculae in 14 sk eletal sites within this UF Reference Male Cancer Patient (RMCP). The resulting model directly addr esses each of the four limitations outlined above in currently available clinical models of skeletal tissue dose. Materials and Methods The experimental and computational methods for skeletal dosimetry developed at the University of Florida (UF) have been desc ribed previously (Shah et al. 2004a). In the

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171 present study, several additional steps have been taken to develop the comprehensive skeletal dosimetry model for electron sources within our reference male cancer patient. Reference Adult Male Cadaver Selection The UF reference male cancer patient was obtained through the State of Florida Anatomical Board located on the UF campus. Selection criteria included (1) an age between 50 – 75 years (representative of typical radionuclide therapy patients), (2) a body mass index of 18.5 – 25 kg m-2 (CDC reco mmended healthy range) (Heyward and Stolarczyk 1996), and (3) a cause of death th at would preclude significant skeletal deterioration. The subject identified was a 66-year male approximately 68 kg in total mass and 173 cm in total height at the time of death (BMI of 22.7 kg m-2). The subject died suddenly of complications a ssociated with cardiomyopathy. Skeletal-Image Database for the UF RMCP Construction of a complete database of the UF RMCP skeletal model involves several imaging modalities and image processing techniques. Initially, the UF RMCP was subjected to whole-body imaging via multislice helical CT at a pitch necessary to reconstruct contiguous 1-mm axial slices. The images were acquired on a Siemens Sensation 16 unit within the Department of Radiology at UF Shands Hospital. Image reconstruction was performed with a bone filte r at an in-plane pixel resolution of 977 m x 977 m. The CT image sets were then transfe rred to workstations within the Advanced Laboratory for Radiation Dosimetry Studies (ALRADS) in the UF Department of Nuclear & Radiological Engineering for image processing and data storage. The in-vivo CT scans provided image data for (1) selecting the anatomical region from which each skeletal site would be harves ted, and (2) constructing 3D an atomic models of skeletal

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172 sites where bone harvesting (and thus ex-vivo CT scanning) might be incomplete (e.g., facial bones of the skull). Skeletal harvesting was then performed following review of the whole-body invivo CT images. Thirteen major skeletal sites were taken from the RMCP cadaver. These included the entire vertebral column (cervical, lumbar, thoracic, and sacrum), proximal femora (2), proximal humerii (2), cranium, mandible, st ernum, ribs (6), os coxae, clavicles (2), and scapulae (2). Once each skeletal site was excised, it was cleaned of excess tissue, bagged, labeled, and stored frozen until ex-vivo CT imaging was scheduled. Post-harvest, ex-vivo CT imag ing was conducted at higher resolution (1.0 mm slice thickness) with an in-plane resoluti on as high as allowable by the field of view (FOV) for each skeletal site (higher than pe rmitted for in-vivo scans). The maximum and minimum in-plane resolutions ranged betw een 0.0977 mm x 0.0977 mm for the clavicles (FOV 5.0 cm) to 0.6563 mm x 0.6563 mm for the os coxae (FOV – 33.6 cm). Ex-vivo CT scans provided image data for (1) identif ying the location and extent of trabecular spongiosa to be sectioned for microCT imag ing, (2) quantifying volumes of trabecular spongiosa and cortical bone within the bone site, and (3) constructing 3D anatomic macrostructural models of the skeletal si te for subsequent paired-image radiation transport (PIRT) simulations (Shah et al. 2004a). Image segmentation was accomplished using the program CT_Contours based upon In teractive Data Language (IDL) version 6.0, the details of which have been pr eviously reported (Nipper et al. 2002). Following detailed spongiosa/cortical bone segmentation of all th e ex-vivo CT data and creation of contours for each skeletal site, physical sec tions of trabecular spongiosa were taken from each skeletal site of the UF RMCP. Limitations on the size of each

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173 section were two-fold, the given constraint s of the bone shape (e.g., cuboidal samples taken from a spherically shaped femoral head) and the microCT imaging system. Marrow-intact sections of spongiosa were furt her bagged, labeled, and kept frozen until microimaging sessions were arranged. MicroCT imaging of 33 cuboidal samples of spongiosa from the 13 major skeletal sites was performed on desktop cone-beam CT40 or CT80 scanners (Scanco Medical AG, Bassersdorf, Switzerland) yielding 3D im age data sets at a voxel resolution of 60 m x 60 m x 60 m. Post-acquisition image processing steps included (1) selection of an ideal volume of interest, (2) gray-level threshol ding, (3) voxel segmentation, and (4) 3Dweighted median filtering, all of which have been previously repor ted in Jokisch et al. (Jokisch et al. 1998, 2001b) and in Pa tton et al. (Patton et al. 2002a). Radiation Transport Modeling Currently, at the University of Florida, attempts to develop better skeletal dosimetry transport models to improve patien t specificity in bone marrow dosimetry have been ongoing for the past 5 years. Four ge nerations of improvements at UF have been developed over the earlier models adopted as the standard in skeletal dosimetry. Figure 7-1 shows the differences between the 4 ge nerations of transport modeling for an example skeletal – a single vertebra. The 1st generation model, and which is us ed currently in the MIRDOSE and OLINDA codes, is called CBIST —chord-based infinite spongios a transport. In a CBIST transport model, radiation particles are fo llowed through the trabecular microstructure through random and alternate sampling of bone and marrow cavity chord-length distributions. Range-energy re lationships in bone and marrow, as in the Eckerman and

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174 Stabin model (2000); or interaction cross sect ions and particle transport, as in the Bouchet et al. model (1999) are used to assess energy loss across the trabeculae and marrow tissues. No consideration is given, however, to the 3D macrostructure of the skeletal site. The 2nd generation model is voxel-based infinite spongiosa transport, or VBIST. Here, the 3D microstr ucture of the bone site, as acquired via microCT, is used directly in particle radiati on transport. These voxel-based 3D images properly account for the anisotropic structure seen in a bone site. Marrow voxels within the binary microimage are further labeled into voxels of active (red) marrow and inactive (yellow) marrow at a pre-determined value of marrow cel lularity. This process has been outlined previously by Shah et al. (2003), and is based upon microscopy measurements of the spatial distribution of adipocytes within normal bone marrow biopsies covering a broad range of marrow cellularities. The trabecular bone endosteum (TBE) is defined as a 10m layer at the bone-marrow interface as previous ly described by Jokisch et al (2001b). As with the CBIST model, no consideration is given to energy loss outside the spongiosa in VBIST. To account for this energy loss, one may invoke a 3rd generation model VBRST or voxel-based restricted s pongiosa transport. Here, a stylized mathematical description of the entire skeletal s ite is used to restrict particle transport to regions of spongiosa found in that particle bone site. VBRST has been well documented in previous work done at UF (Jokisch et al 2001b; Bolch et al. 2002; Patton et al. 2002b; Rajon et al. 2002; Shah et al. 2003). In Jokisch et al. ( 2001b), a downturn in the energy deposited to the marrow space (absorbed fraction) was seen due to the macrostructural boundaries of the skeletal site with VBRS T models in comparis on to earlier CBIST models. This downturn was only significan t at source energies exceeding ~2 MeV,

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175 where particles can escape the spongiosa regions and travel in to the cortical bone cortex or leave the bone site altogether. VBRS T was developed for the proximal femoral epiphysis, humeral epiphysis, and the vertebra l body. These bone sites were model as two concentric spheres for the femur a nd humerus; and two c oncentric cylinders intersected by a plane for the body of the vert ebra. For many of the irregularly shaped bones of the skeleton, mathematical descriptio ns of their cortical bone cortex modeling can be extremely problematic (e.g., processes of the vertebrae, bones of the os coxae, shape of the scapulae, etc.). The 4th generation model shown in Figure 7-1, PIRT or paired-image radiation trans port, overcomes this limita tion by directly using a highresolution ex-vivo image of the skeletal site to model all regions of both cortical bone and trabecular spongiosa. In all models excep t for CBIST, an NMR microscopy or microCT image of a physical section of spongiosa is used to follow radiation particles across individual bone trabeculae a nd marrow cavities. In the present study, the PIRT model is used to assess electron absorbed fractions acro ss a variety of skeletal tissues in the UF RMCP. Paired-Image Radiation Transport (PIRT) Model The paired-image radiation transport or PIRT model supplements the 3D microscopic histology provided by the mi croCT image with the 3D macroscopic histology given in the corres ponding contour of the ex-vi vo CT image (Shah et al. 2004a). Elemental compositions and mass densi ties for each skeletal region were taken from the data given in ICRU Publication 46 (19 92) and are presented in Table 3-1. In the PIRT model, five tissue regions are defined: trabecular ac tive marrow (TAM), trabecular inactive marrow (TIM), the tr abecular bone volume (TBV), trabecular bone endosteum (TBE), and cortical bone volume (CBV). We further define the trabecular bone surfaces

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176 (TBS) as the tissue boundary between the bone trabeculae (TBV) and the endosteal layer (TBE). Finally, the trabecular marrow cavity (TMC) is defined as the combined tissues of the TBE and TAM (all soft tissues excl usive of the TIM). The term spongiosa describes the combined tissues interior to the CBV (e.g., TBV, TBE, TAM, and TIM together). In Figure 7-2, we show several exampl es from the skeletal image database generated for the UF RMCP, all of which ar e necessary components of the PIRT model for radiation transport. Within Figure 7-2, the entire vertebral column has been partitioned into four separate transport regions: the cervical vertebrae, thoracic vertebrae, lumbar vertebrae, and sacrum. Each row of im ages (A, B, C, and D) represents the same image type for each vertebral section, as labe led in separate columns. In row A, 3D isosurface reconstructions of the 2D contours generated from the segmentation of the exvivo CT scans at each vertebral skeletal site ar e given. A single 2D transverse slice of the segmented contour for each vertebra is given in row B. The red regions represent skeletal spongiosa and the blue regions represent th e cortical bone cortex. In row C, 3D reconstructions of the microCT image data of physical sections of trabecular spongiosa are given for each vertebral region: 6th cervi cal vertebra, 3rd thor acic vertebra, 4th lumbar vertebra, and a single section from the sacrum. Finally, row D shows single 2D slices through the microCT image data after image processing techniques have been implemented for each vertebra. The PIRT m odel tracks electrons (beta-particles) through direct coupling of the macrostructural images of the bone sites (rows A and B) as well as the microstructural images of these same bone sites (rows C and D).

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177 Mass Calculation for UF RMCP The calculation of source and target tissue masses is an important step in defining the tissue structure of the UF RMCP. In th e present study, mass estimates require the use of (1) macrostructural volume informati on (segmented ex-vivo CT images), (2) microstructural volume information (segment ed microCT spongiosa images), and (3) the volume percentage of hematopoietically active versus inactive bone marrow at each skeletal site (marrow cellularity factor as measured or assumed). At skeletal site j within the UF RMCP, the mass of trabecular active marrow (TAM) is thus calculated as the product of four terms: TAMjjjTAM jmSVMVFCF (7-1) where SVj is the spongiosa volume, MVFj is the marrow volume fraction, and CFj is the cellularity factor for skeletal site j while TAM is the mass density of active marrow (1.03 g cm-3). In this investigation, marrow cellula rity is explicitly treated as a model parameter and allowed to vary from as low as 10% to as high as 100%. As a result, the UF RMCP does not have a single value of total active marrow mass, but a range of potential active marrow masses, each corresponding to a unique set of energy-dependent electron absorbed fractions. Since the exis ting ICRP Reference Man model is based on a fixed set of bone-site-specific marrow cellulari ties given in ICRP Publication 70 (ICRP 1995), we have adopted this same set of CF values for comparison purposes only. Calculations of tissue masses for the tr abecular bone endosteum (TBE) and the bone trabeculae (TBV) are reported using similar expressions and is given by TBEjjTBE jmSVEVF and (7-2) TBVjjTBV jmSVBVF (7-3)

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178 In Equations 7-2 and 7-3, EVFj and BVFj are the endosteal layer and bone trabeculae volume fractions, respectivel y, at skeletal site j ; while TBE and TBV are the mass densities of these tissues: 1.03 g cm-3 and 1.92 g cm-3, respectively (see Table 3-1). A unique feature of the PIRT skeletal dosimetry model, not found in existing clinical models, is the explicit treatment of the cortical bone cortex of each skeletal site as a possible source and/or target region. Th e measurement of cortical bone mass is done through the tabulation of the voxels designate d as cortical bone in a contour created through the segmentation of a corresponding ex-vivo CT scan and can be written for a particular bone site j : CBVjCBV jmCBV (7-4) In Equation (7-4), CBVj is the cortical bone volume (cm3) tabulated from the contour data of a particular skeletal site j and CBV is the density of cortical bone (1.92 g cm-3). All corresponding densities are taken from ICRU 46 (1992) and are gi ven in Table 3-1. Skeletal Averaging of Absorbed Frac tions and S Values for the UF RMCP Bone site-specific absorbed fractions and radionuclide S values were computed for the entire axial skeleton at 33 separate bone sites using the PIRT model for electron dosimetry and the image database of the UF RMCP. These absorbed fractions and corresponding S values were subsequently us ed to compile average values corresponding to 17 different skeletal regions of the UF RMCP. For example, microCT images were acquired of vertebral body spongiosa a nd physically sectioned from the T3, T6, and T11 vertebrae. The PIRT model was then run i ndividually three times using (1) the entire macroimage of the thoracic vertebral series (rows A and B, column 2 of Figure 7-2), and (2) the individual microCT images of either the T3, T6, or T11 microimages. The three

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179 resulting sets of energy-dependent absorbed fractions were then averaged (weighted by the relative mass of the microimage) to yiel d a single set of energy-dependent absorbed fractions for the thoracic vertebral series as a whole. The absorbed fraction profiles and radionuclide S values were tabulated for a range of 10 cellularitie s across two target regions (TAM and TBE) and for 4 sour ce regions (TAM, TBV, TBS, CBV). For comparison with existing clinical models of skeletal dosimetry, it is useful to additionally derive both skel etal-averaged absorbed frac tions and skeletal-averaged radionuclide S values for each source and target tissue in the skeleton of the UF RMCP. Skeletal-averaged absorbed fractions to TAM and TBE target regions were computed as follows. We define fS,j as the ratio of source tissue mass in bone site j ( mS,j) to the total mass of source tissue found throughout the skeleton ( mS)Skel ,, , SjSj Sj SjS Skel jmm f mm (7-5) Skeletal-averaged absorbed fractions, Skel(rT rS) are thus calculated as the sum of their values at each skeletal site j weighted by the fractional ma ss of source tissue at that skeletal site, ,;SkelTSSjTS j jrrfrrCF (7-6) In the UF RMCP skeletal dosimetry m odel, PIRT simulations were performed across a full range of marrow cellularities at ea ch skeletal site. C onsequently, application of Equation 7-6 implicitly involves an assumption of the marrow cellularity ( CFj) at each skeletal site in the averaging expression. These values of marrow cellularity must also correspond to those chosen in the calculation of the mass fraction of the source region (Equation 7-1), if that source region contains active bone marrow (TAM). We note here

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180 that, while skeletal-averaged absorbed frac tions are based upon reference values of marrow cellularity, the cellularity-dependent data of Appendix H gives the user total flexibility in deriving skelet al-averaged absorbed fracti ons (and thus radionuclide S values) that are uniquely speci fic to an individual patient provided that whole-body MR imaging is used to non-invasively measure skeletal-site-specific marrow cellularity, as per the techniques of Ballon et al. (1996). Skeletal-averaged radionuclide S values are calculated using the derivation of Bouchet et al. (2000). Starting assumptions for this derivation are (1 ) that the skeletal averaged absorbed dose is given as the ratio of the total energy deposited within all skeletal sites j and the total mass of skeletal target tissue mSkel,T; and (2) that the cumulated activity throughout the entire skeleton is pa rtitioned to individu al skeletal sites j according their fractional mass of source tissue rS, TS j j SkelTS SkelTrr Drr m and (7-7) ,, jS jSSkelS SkelSm AA m (7-8) The skeletal-averaged radionuclide S value is then computed as ,, SkelTSSjTjTS j jSrrffSrr (7-9) where fS,j and fT,j are the fractional masses of source and target tissue in skeletal site j, respectively, and S(rT rS)j is the radionuclide S value for skeletal site j As with Equation 7-6, one must be consistent in the selection or assignment of marrow cellularity across individual skeletal sites; consequently, values of fS,j, fT,j, and S(rT rS)j must all be taken from PIRT model simula tions run at various assigned values of marrow cellularity

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181 CFj. In this study, we have selectively applied Equati ons 7-6 and 7-9 using adult reference cellularities given in ICRP Public ation 70. However, it is again emphasized that the UF RMCP skeletal model can be applied to estimate active bone marrow or endosteal tissue dose at any skeletal site for any value of patient marrow cellularity. Results Skeletal mass estimates for the UF refe rence male cancer patient are given in Tables 7-1 and 7-2, for each bone site within the skeletal system. Table 7-1 includes mass estimates for the marrow space tissues, while Table 7-2 includes those for the osseous tissues. Table 7-2 also provides pe rtinent information from the UF RMCP, such as tissue volume fractions, needed to calculate each component of skeletal mass. Table 7-3 gives a breakdown of spongiosa and cortical bone volumes within each of the five lumbar vertebrae. These values are given in response to the study by Shen et al. (2002) where corresponding values in the ICRP Refe rence Man were not available for patientspecific corrections to the marrow absorbed dos e. Table 7-4 gives values of trabecular surface-to-volume ratios in comparison to t hose reported in the University of Leeds studies and others cited in I CRP Publication 70. Absorbed fraction data compiled from the UF RMCP model are given in Appendix H for the entire skeleton for both TAM and TBE target regions and for 6 electron sour ce regions (TAM, TBS, TBV, TBE, TMC, and CBV) over a range of marrow cellularities (10% to 100%). Mass fractions for each skeletal site, at ICRP 70 reference cellulari ties, are given in Table 7-5 for each tissue region. These values are necessary in the ca lculation of the skeletal-averaged absorbed fractions for each source-target combination, as reported in Table 7-6. The values given in Table 7-6 are based on the absorbed frac tion data generated in the PIRT model at reference cellularities as defi ned by ICRP 70 and the mass frac tions given in Table 7-5,

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182 also at reference cellularities. For comparis on, Table 7-8 also disp lays skeletal-averaged absorbed fractions reported in both Eckerm an and Stabin [MIRDOSE3] (2000) and in Stabin and Siegel [OLINDA] (2003). Appendix I provides site-sp ecific radionuclide S-values for TAM and TBE target tissues and TBV, TAM, TBS, and CBV source tissues over a range of marrow cellularity from 10% to 100%. Site-specific S values were calculated at varying cellularities, thus the masses used in the calculation of the S valu e for each skeletal site varied accordingly. For the tabulation of S values, 10 radionuclid es are considered with radiation decay schemes given in Table 7-7. In all but 3 of these radionuclides, th e photon component of the emission spectrum is either very small or totally absent (i.e. pure -emitters). For 131I, 177Lu, and 153Sm, the S values reported in this study represent only the electron ( particle) component of the total S value. Skeletal -averaged S values were subsequently calculated for all source and targ et combinations and ar e given in Table 7-8 for all 10 radionuclides. Values from th e OLINDA code are given for comparison. Discussion Comparison of UF and ICRP Reference Tissue Masses Tables 7-1 and 7-2 display a variety of imaged-based volumetric data for the UF Reference Male Cancer Patient including bone -site-specific values of (1) spongiosa and cortical bone volumes as seen in the segmented ex-vivo CT images, and (2) volume fractions for bone marrow (both active and in active), trabecular endosteum, and the bone trabeculae as seen in the segmented ex-vivo microCT images. With the application of ICRP reference cellularities as given in ICRP Publication 70 (see colu mn 2 of Table 7-1), the UF RMCP is noted to have 967.2 g of active marrow, 2446.6 g of inactive marrow,

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183 124.5 g of trabecular endosteum, 1414.7 g of trabecular bone, and 4069.6 g of cortical bone. Corresponding skeletal tissue masses can be assigned to the ICRP Reference Man using data given in ICRP Publication 70 (1995) and following the analysis of Bouchet et al. (2000). Here, we can see that the ICRP Reference Man has 21% higher active marrow mass (1170 g) than the UF RMCP, only 1% higher inactive marrow mass (2480 g), and 12% higher cortical bone mass (4547.2 g). In contrast, the ICRP Reference man is noted to have only 72% of the trabecular endos teal mass (89.7 g) and only 67% of the trabecular bone mass (952.8 g) of the UF RMCP. The total skeletal mass of the ICRP Reference Man is reported to be 10,500 g, which includes a total of 1350 g for cartila ge and miscellaneous tissues (endosteum, periosteum, and blood vessels). With the ex clusive of these latter tissues, a revised skeletal mass of the ICRP Reference Man w ould thus be 9150 g. The corresponding total skeletal mass of the UF RMCP (masses of TBV, CBV, TAM, and TIM) is 8898.1 g, a value which is 97% of the ICRP value. Again, it is emphasized that the active and inactive marrow masses reported above are not fi xed in the UF model, but are attained only for specific ICRP reference values of ma rrow cellularity and can thus be changed as needed to match a given patient’s marrow status. In Table 7-4, 3D imaged-based measurem ents of surface-to-vol ume ratios (S/V) of the trabecular microstructure are given for all skeletal sites in the UF RMCP. The mean values of S/V are 16.0 3.1 mm2/mm3 across the 17 skeletal regions of the model. This value compares favorably with that given for 5 skeletal sites within the University of Leeds 44-year reference individual (basis for the ICRP Reference Man model) – 16.1 4.8 mm2/mm3. Other studies cited in ICRP Pub lication 70 have indicated both smaller

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184 S/V ratios (~11 mm2/mm3) as well as larger S/V ratios (~19 mm2/mm3) for the lumbar vertebrae and the iliac crest. In its Publication 70, th e ICRP adopted a single value of 18 mm2/mm3 to be applied to all bones within th e ICRP Reference Man skeleton. This single value of the trabecular S/V ratio yields a total skeletal trabecular surface area of 10.5 m2 for the ICRP Reference Man, which when multiplied by a uniform thickness of 10m gives a total TBE mass of 108 g, a value 87% of that given for the UF RMCP. The discrepancy between 108 g of TBE and 89.7 g of TBE in the ICRP Reference Man is that the later estimate by Bouchet et al. (2000) was determined by allowing the trabecular S/V ratio to vary with skeletal s ite, and not remain fixed at 18 mm2/mm3. This example demonstrates the advantage of an image-based skeletal reference model, where tissue mass estimates are given explicitly via imag e segmentation analysis, and not through the application of model assumptions draw n from diverse literature studies. Fractional distributions of skeletal tissue mass in the two reference individuals are given in Table 7-5. Reference cellularities are taken from ICRP Publication 70 (1995) for the UF model, and from Eckerman & St abin (2000) for the I CRP model. The bone sites with the highest percentage of active marrow in the UF model (fTAM) are the os coxae (23%), thoracic vertebra (17.9%), lumb ar vertebrae (13.6%), and ribs (12.2%) for a total of 66.7% of total skeletal active marrow. These same skeletal sites comprise a total of 79.7% of all active marrow in the ICRP m odel: os coxae (33.3%), thoracic vertebra (17.4%), lumbar vertebrae (9.8 %), and ribs (19.2%). The 4 bone sites with the largest fraction of bone trabeculae mass in the UF model (fTBV) are the os coxae (12.4%), proximal femora (12.8%), distal femora (12. 8%), and lower leg bones (11.1%). In the ICRP model, roughly equi valent values of fTBV are found in the proximal (16.3%) and

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185 distal (16.5%) femora; however, much smaller contributions to TBV mass are assigned to the os coxae (only 1.8%) and the lower legs bones (only 4.0%) than seen via CT and microCT image analysis in the UF model. In the ICRP model, 45.2% of all TBV mass is localized in the spine, while the correspondi ng total in the UF model is only 18.0%. Finally, reasonable agreement is seen in th e fractional distribution of cortical bone volume (fCBV) at several skeletal sites: cranium (8.9% UF and 12.8% ICRP), ribs (7.1% UF and 12.0% ICRP), and os coxae (9.6% UF and 14.0% ICRP). These values of fCBV are slightly lower in the UF model since a hi gher fractional distributi on of cortical bone is assigned to the extremities of the UF model (total of 45.6%), than is assigned in the ICRP model (total of 38.0%). PIRT Model Simulations – Sacrum PIRT simulations, by definition, account fo r full electron energy loss to the cortex of each skeletal site where intermediate-t o-high energy electrons deposit energy outside the regions of trabecular spongiosa. This ability to fully account for the skeletal macrostructure is demonstrated in Figures 73 and 7-4 for target ti ssues of the active bone marrow and trabecular endosteum, respectively, within the sacrum of the UF RMCP. As an example, we chose to discuss only one bone site—the sacrum, a skeletal site not individually considered in the ICRP model. In each figure, the fraction of electron en ergy deposited to their respective target tissue is shown as a function of electr on emission energy for sources uniformly distributed with the sacral TAM, TBS, and TB V tissues. Solid lines indicate PIRT model results, while dashed lines indicate corres ponding VBIST simulations. A divergence in model results is shown to occur at electron en ergies as low as 200 keV in the sacrum, for both targets and all three source regions. At 1 MeV and 4 MeV, the VBIST model

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186 overestimates values of (TAM rS) given by the PIRT model by 21% and 100%, respectively. Furthermore, the VB IST model overestimates values of (TBE rS) given by PIRT by 17% at 1 MeV and by 94% at 4 MeV. Energy Loss to Cortical Bone The importance of accounting for electron energy lost to cortical bone can also be demonstrated by plotting values of (CBV rS) where rS may be any of the various source tissues within the trab ecular spongiosa. These abso rbed fractions are given in Figure 7-5 for several of the skeletal regions of the UF RMCP. Values of (CBV rS) are seen to be independent of both the marro w cellularity as well as the specific source region within the spongiosa of th e skeletal site (either TAM, TBS, or TBV). At electron energies below ~100 keV, very little energy lo ss to cortical bone is noted, and thus CBIST or VBIST codes give valid estimates of energy deposition. As the electron energy increases, however, more electron energy is deposited with the cortex of the skeletal site. For six of the skeletal sites, values (CBV rS) reach a maximum value at energies between 1 MeV and 2 MeV and decr ease thereafter (as en ergy is expended to soft tissues outside the skeletal site). The greatest amount of energy loss to cortical bone is noted for the mandible ( max 0.36 at 2 MeV), the cranium ( max 0.32 at 2 MeV), and the ribs ( max 0.26 at 1.5 MeV). The lowest amount of electron energy escape is noted for the right humerus and femur. Figure 7-6 shows corresponding values of (CBV CBV) for electron sources localized in the CBV cortex of these same 9 skeletal s ites. As the electron energy increases, the self-dose to cortical bone decrea ses. Self-dose to cortical bone is noted to be highest (less energy escape) for the mandi ble, humerus, and cranium, while lowest

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187 (more energy escape) for the cervical vertebrae, lumbar vertebrae, and ribs. These trends result directly from the three-dimensional shap e of the skeletal site, as well as the cortex thickness as seen by ex-vivo CT image segmen tation. It is envisioned that future extensions of the PIRT model could incor porate microimages of the haversian canals (obtained via microCT or electron microsc opy), whereby accounting for energy loss to radiobiologically sensitive tissu es of the cortical endosteum. Skeletal-Averaged Absorbed Fractions At the present time, only two comprehensiv e skeletal reference models are widely available for marrow dose estimates in nuclear medicine: (1) the Eckerman & Stabin model (2000) of the MIRDOSE 3 code (Sta bin 1994), and (2) the Stabin & Siegel model (2003) of the OLINDA 1.0 code (Sta bin 2003). Both models utilize CBIST simulations of electron transport based on th e 44-year male Leeds chord-distributions, and tissue mass estimates taken from the I CRP Reference Man. Electron absorbed fractions in these models are given, not by indi vidual skeletal site, but as averages across the entire skeleton. Consequently, we have applied Equation 7-6 to the absorbed fraction data of Appendix H to facilitate mode l comparisons. We further note that skeletal-averaged absorbed fractions given in Eckerman & Stabin (2000) were also calculated using Equation 7-6, and not by the weighting schemes indicated in Table 4 of their article (KF Eckerman, personal communication, November 2004). In both cases, ICRP reference marrow cellularities are app lied to each, the Eckerman & Stabin model (by CF scaling of the marrow space) and the UF model (by radiation transport modeling). In Figure 7-7, skeletal-averaged values of (TAM rS) are given for the UF RMCP (closed symbols) along with values of Skel(TAM rS) published by Eckerman & Stabin

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188 (open symbols) (see Table 7-6). For each source tissue region rS, differences between the two RSMs can be attributed to differences in (1) modeling approach (PIRT versus CBIST), (2) skeletal microstructure (66y ma le versus 44y male), (3) imaging technique (3D microCT versus 2D optical scanning), (4) tissue region definiti on (explained below), and/or (5) fractional mass distri butions of source a nd target tissues. Three discrepancies are noted in Figure 7-7. First, for TAM sources at low energy, values of Skel(TAM TAM) for the UF model are noted to c onverge to unity at low energy, while those of Eckerman & Stabin converge to ~0 .61 (their model’s skeletal-averaged marrow cellularity – see Table 7-5). In the Eckerman & Stabin model, explicit treatment of marrow cellularity is considered only after CBIST transport, and thus values of Skel(TMS TMS) for the trabecular marrow space (TMS) must be scaled uniformly by the reference cellularity following particle transport. This error has since been empirically corrected in th e OLINDA code (see below). A second discrepancy is noted for values of Skel(TAM TAM) at high energy where values in the Eckerman & Stabin model approach an energy-independent convergence limit at energies exceeding ~1 MeV. At high energies, the Eckerman & Stabin model predicates a constant value of ~0.44 for Skel(TAM TAM), while the UF RMCP predicts a continual decline in Skel(TAM TAM) that approaches 0.35 at 1 MeV and decreases to 0.19 at 4 MeV. The third discrepancy between these two mode ls occurs at low electron energies for sources localized uniformly across the bone su rfaces. In the UF m odel, these electrons must first penetrate the 10m endosteal layer before they can enter the marrow space and irradiate the active bone marrow. In the Eckerman & Stabin model, a roughly

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189 energy-independent value of Skel(TAM TBS) 0.17 is given for electron energies below ~100 keV, thus indicating that the TBE is implicitly included in their definition of active marrow. Values of (TAM TBV), however, are roughly comparable in the two models up to a couple of hundred keV, after which the Eckerman & Stabin model again overestimates the fraction of electron energy deposited to active marrow (as per their CBIST simulations). Finally, a small but additional contribution to TAM dose from electron sources in the cortical bone cortex is given by the UF model – a feature not considered in the CBIST model construction. A similar comparison is given in Figure 7-8 for energy deposition to the trabecular bone endosteum. As shown in Table 7-6, values of Skel(TBE TBS) are both ~0.5 for low-energy electrons in both the UF and Ec kerman & Stabin m odel (not shown in Figure 7-8). As the electr on energy increases beyond ~1 MeV, however, the models diverge in their estimat es of endosteal dose, with the Eckerman & Stabin model again predicting a greater amount of energy deposit ion to TBE (no account for energy loss to cortical bone). Values of Skel(TBE TBV), however, are similar for electrons below ~500 keV, above which a similar divergence is noted. For electron sources emitted within the active bone marrow, clear differences in Skel(TBE TAM) are noted for lowenergy electrons (< 100 keV) between the two m odels. In the Eckerman & Stabin model, an active marrow source is considered to be localized not only within the marrow space, but also within the endosteal layer itself. In the UF model, an active marrow source is localized outside of the endosteum layer. By invoking a TMC electron source in the UF model (where electron emissions are consid ered in both the TBE and TAM tissues),

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190 closer agreement between the UF and Eckerman & Stabin models is realized for marrow electron sources less than 100 keV (d ashed-dot-dot line in Figure 7-8). In Figure 7-9, we make additional comparisons betw een the UF model and the Stabin & Siegel model of OLINDA version 1.0. As shown for low-energy electron sources in the active marrow, the Stabin & Siegel model empirically adjusts previous estimates of Skel(TAM TAM) upward to converge at unity for very low electron energies. This adjustment is noted in Figure 7-9 to be slightly steeper than values given in the current UF model. At higher energies, values of Skel(TAM TAM) are equivalent to those given in Figure 7-7 under the Ecke rman and Stabin model, and again they converge to an energy-independent value of ~0.43. In contrast, values of Skel(TAM TBS) and Skel(TAM TBV) in the Stabin & Siegel model are noted to converge to ~0.3, a value higher than that given in the Eckerman & Stabin model of MIRDOSE3 (~0.24). As seen in Table 7-6, ve ry slight changes are noted in values of Skel(TBE rS) between the models of MIRDOSE3 and OLINDA. At very low electron energies, values of Skel(TAM TBS) in the Stabin & Siegel model converge to a value slightly higher than that give n in the Eckerman & Stabin mo del (0.21 versus 0.17). This newer value of 0.21 is now shown to be c onsistent with the marrow cellularities and fractional bone surface areas ( fTBS) given for the ICRP Reference Model in the Eckerman & Stabin model (see Table 7-5). Site-Specific Radionuclide S Values With changes in both bone shape, size and trabecular microstructure, large variations in the radionuclid e S value may be seen acro ss the skeleton of radionuclide therapy patients. As an example of this va riation, we plot in Figure 7-10 values of

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191 S(TAM TAM) for 7 skeletal regions of the UF RMCP for 5 differe nt radionuclides covering a range of beta-particle energi es (see Table 7-7). Values of S(TAM TAM) are noted to be highest within the cervical ve rtebra and lowest in the os coxae for each radionuclide considered. For very low-energy emitters such as 33P and 177Lu, the 2nd and 3rd highest contribution to active marrow dose is found in the cranium and ribs, respectively. For high-energy emitters, such as 32P and 90Y, however, the S value for self-dose to active marrow is noted to be higher in the ribs (2nd highest) than in the cranium (3rd highest). Figure 7-11 displays values of S(TAM rS) for 90Y across all skeletal sites in the UF RMCP in decreasing order of active marro w self-dose. The table insert of Figure 7-11 lists the skeletal sites by increasing TAM mass for the UF model. In Figure 7-11, we see that in only 9 of the 17 skeletal si tes is the relative magnitude of the S value predicted solely by the corres ponding inverse ratio of target tissue mass (larger the mass, smaller the S value). For the remaining skel etal sites, the magnitude of the absorbed fraction (TAM rS) plays an equally important role in the overall magnitude of the radionuclide S value, and thus the ordering of th ese skeletal sites differs from that in the mass table (showing mass is not the only major factor). The influence of marrow cellularity on values of S(TAM TAM) is shown in Figure 7-12 for the cranium. As was show n previously by Bolch et al. (2002), the sensitivity of S(TAM TAM) to changes in marrow cellularity increases as the mean energy of the beta emissions decrease. This sa me inverse relationship can be seen in this investigation as well. Figure 712 shows a ~5% decrease in S(TAM TAM)cranium for

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192 90Y as the cellularity decreases from 100% to 30%, however, there is an almost 30% decrease in S(TAM TAM)cranium for 177Lu with the same change in cellularity. Skeletal-Averaged S Values Skeletal-averaged S values are given in Table 7-8 for the UF RMCP for a broad range of radionuclides of interest in skelet al dosimetry and for several source-target combinations. These values are determined using the weighting scheme of Equation 7-9 and the fractional tissue masses given in Table 7-4 for ICRP 70/89 reference marrow cellularities. Corresponding S values are s hown in the bottom of Table 7-8 as given by the OLINDA 1.0 code. Direct comparisons to OLINDA 1.0 are complicated by severa l factors. First, with radionuclides such as 131I, OLINDA reports values that include the photon component of the marrow dose, while the UF values provide data only for the electron/beta component of the emission spectrum. Photon contributi ons require the ability to consider crossskeletal site irradiation, a nd thus a full-body anthropomor phic model of the individual must be utilized. A good example of these mode ls would be the ORNL series of stylized models in the OLINDA code (Cristy and Eckerman 1987). Photon contributions to skeletal dose using the UF model may be assessed through fluence-to-dose conversion coefficients as described by Eckerman (1985a), and then s ubsequently applied to any full-body anatomic model of the patient. This work is currently in progress using the UF RSM. Second, values of SSkel(TBE CBV)OLINDA are not provided since CBIST simulations of beta/electron transport treat cort ical bone independently from trabecular bone (no electron cross-fire between CBV a nd spongiosa tissues). However, non-zero entries are given for SSkel(TAM CBV)OLINDA and are attributed to the photon cross-dose

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193 from cortical regions of the skeleton. Values for SSkel(TAM CBV)UF, however, represent direct electron/beta contributions to active marrow dose from emissions within the cortex of cortical bone at each skeletal site (e.g, cranium CBV irradiating cranial TAM). Third, tissue regions are de fined differently within th ese two models. In the Stabin & Siegel model (2003), the endosteal la yer is included as pa rt of the active bone marrow (both source and target). While in the UF model, it is treated as a separate tissue layer. Also in the UF model, marrow cellulari ty is used to modify the volume fraction of active and inactive bone marrow within the marrow space exclusive of the endosteal layer. In contrast, the Stabin & Siegel mode l applies a cellularity scaling to the endosteal layer when it considers an active marrow ta rget, but does not apply this same scaling when considering dose only to the endosteal layer. Finally, direct comparisons of skeletal-a veraged S values are further complicated by the fact that different weighting schemes are applied in each model. In the UF model, bone-site-specific S values are doubly weight ed by the fractional masses of source and target tissue as given in Equation 7-9, thus following the starting assumptions of Equations 7-7 and 7-8. In the Stabin & Sieg el model, the following alternative weighting scheme is applied (MIRD schema applied to the skeletal-averaged absorbed fractions): , ,, iSjTS ij iSkelTS i ij i SkelTS SkelTSkelTfrr rr Srr mm (7-10) where mSkel,T is the total mass of target tissue across the entire skeleton. In comparing the weighting schemes of Equations 79 and 7-10, it is seen that the latter expression does not account for the fractional distribution of targ et tissue in the skeleton. As a result,

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194 Equation 7-10 is not consistent w ith the concept of an average skeletal tissue dose as it is defined in Equation 7-7. Use of the total ta rget mass in the denominator of Equation 7-10 tends to lower the estimate of SSkel(rT rS) in comparison to that given by Equation 7-9. In fact, by using the bone-site-specific S values of Appendix I for the UF model, Equation 7-10 is noted to yield values of SSkel(rT rS) that range from 95% to as low as 38% of values of SSkel(rT rS) given by Equation 7-9. The average ratio of SSkel(rT rS) given by Equation 7-10 to that given by Equation 7-9 is 0.74. Scalability of the UF Reference Skeletal Model One unique feature of the UF RSM is that it is imaged-based, and thus the model greatly facilitates patient-specific S value sca ling. Consider, for example, the sacrum a skeletal site frequently used in quant itative imaging of radiopharmaceutical marrow uptake. The mass of active marrow in the sa crum of the UF RSM may be determined using the data of Tables 7-1 and 7-2: 33159.50.7570.701.0387.0UF UFUFICRP TAMSacrumSacrumSacrumTAM SacrummSVMVFCF cmgcmg (7-11) With the capabilities to change the marrow cellularity term in Equation 7-11, the UF RSM is not fixed at a mass of 87.0 g for the sa cral active marrow. If instead the patient’s marrow cellularity within the sacrum is determ ined (via MR imaging) to be higher than normal (e.g, 90%) or lower than normal (e.g., 30%), then the UF model can be adjusted to report a sacral marrow mass of 111.9 g or 37. 3 g, respectively. Absorbed fractions and radionuclide S values at either 90% or 30% marrow cellularity can then be assigned from the data of Appendices H and I, respectively.

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195 Patient-specific scaling of the UF model S value can then be performed in the assessment of sacral marrow self-dose in the radionuclide therapy patient: UFUFUF PatientUF SacrumSacrumSacrumTAM SacrumSacrum PatientPatientPatient SacrumSacrumSacrumTAMSVMVFCF STAMTAMSTAMTAM SVMVFCF (7-12) UFUF UF SacrumSacrum Sacrum PatientPatient SacrumSacrumSVMVF STAMTAM SVMVF (7-13) UF UF Sacrum Sacrum Patient SacrumMVF STAMTAM MVF (7-14) Equation 7-12 reduces to Equation 7-13 direct ly as the UF reference skeletal model may be applied at any value of patient marro w cellularity. Further reduction of Equation 7-13 to Equation 7-14 can be made if CT-based estimates of sacral spongiosa volume are made in the patient as well. This estimate can be made simultaneously with marrow activity quantification if multi-modality imagi ng is applied (e.g., SPECT-CT or PET-CT). One problem with Equation 7-13 is that if the change in sacral SV is significant (e.g., patient’s skeletal size is much larger or mu ch smaller than that of UF 66-year male subject), the electron absorbed fractions of A ppendix H may need to be adjusted as well. Since the UF model is imaged-based, one can re-scale (upward or downward) the voxel dimensions of the ex-vivo CT images, and re-r un the PIRT model for each skeletal site of the model to better match the skeletal stature of the patient. In so doing, one may directly account for electron (beta particle) energy esca pe to cortical bone for different patient skeletal sizes. The final and most difficult adjustment of the reference skeletal S value would be to account for differences in marrow volume fraction between the UF 66-year model and that of the radionuclide ther apy patient. While voxel-rescal ing of the CT ex-vivo image

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196 can account for changes in SV, similar rescal ing of the microCT image of the trabecular microstructure is not warranted. Consequently, the present UF RSM implicitly assumes that the bone microstructure of the patient ex actly matches that of the patient for whom skeletal dose estimate is required. Expansion of the UF model to permit adjustments for changes in trabecular microstructure would entail a corresponding expansion of mi croCT images of spongiosa, all taken from cadavers and c overing a wide range of bone mineral densities (normal to osteopenic to osteoportic trabecu lar microstructures). This effort is presently ongoing. Matching of the patient to the skeletal re ference model might then entail additional measurements of volumetric bone mineral density, a process that can easily be incorporated into the SPECT-CT or PET-CT imaging session using a QCT (quantitative computed tomography) bone mineral density phantom Conclusion A comprehensive image-based reference skeletal model (RSM) is presented representing an adult male radionuclide ther apy patient. The model is based on ex-vivo CT images of individual skeletal sites ha rvested from the 66-year male cadaver (68 kg and 173 cm) in which regions of cortical bon e and trabecular spongiosa are segmented. Microscopic details of the bone trabeculae, marrow tissues, and trabecular endosteum are based on microCT images of physical sections of spongiosa. Paired-image radiation transport is used to determine electron absorb ed fractions for various source-target tissues and marrow cellularities. When evaluated at ICRP 70 reference cellula rities, the UF RSM is noted to have a smaller mass of active (red) bone marrow than defined in the ICRP Reference Man model (ratio of 0.827), roughly an equivalent mass of inactive marrow (ratio of 0.987), and a

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197 smaller mass of cortical bone (ratio of 0.895). In contrast, the image-based UF model has a higher mass of bone trabeculae (ratio of 1.49) and bone endosteum (ratio of 1.39) than the ICRP model. Trabecular surface-to-v olume ratios for the UF model are in good agreement with those reported for the Un iversity of Leeds 44-year RSM (~16 mm2/mm3), but are slightly lower than th e bone-independent value of 18 mm2/mm3 reported in ICRP Publication 70. The os coxae, thoracic vert ebrae, lumbar vertebrae, and ribs were the skeletal sites of highest active marrow conten t in both the UF and ICRP models (66.6% UF and 79.7% ICRP). Skeletal-averaged values of electron absorb ed fraction are noted to be significantly higher in the chord-based model of the OL INDA code, where electron energy escape to cortical bone is not considered. For example, values of Skel(TAM TAM)OLINDA are higher than those from the UF model by factors of 1. 13, 1.27, and 1.57 at electron energies of 500 keV, 1 MeV, and 2 MeV, resp ectively. When the electron emissions are localized on or within the bone trabeculae, corre sponding ratios of Skel(TAM TBS or TBV) are 1.59, 1.72, and 2.08, respectively. Furthermore, a non-tr ivial contribution to active marrow dose from cortical bone emissions is seen in the UF model. For example, the data of Figure 7-7 indicate that the sk eletal-averaged contri bution to active marrow dose from electron emissions in cortical bone under the UF model are 9.6%, 16.1%, and 21.1% of those seen for electron emissions in trabecular bone at 500 keV, 1 MeV, and 2 MeV, respectively. In the calculation of a skeletal-averaged S value, higher absorbed fractions in the Eckerman model are partia lly compensated for by use of a tissue mass weighting scheme in OLINDA (Equation 7-10) that generally yields lower radionuclide S values than given by the weighting sche me of the UF model (Equation 7-9).

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198 Finally, it is noted that image-based skel etal reference models offer significant advantages in patient-specific scaling of th e radionuclide S value. The UF model may be used as is to report skeletal tissues doses over a broad range of marrow cellularity. In addition, explicit tabulation of spongiosa and cortical bone volumes allow the user to adjust skeletal tissue masses in the model to better match CT-measured volumes seen in the patient, as per the study of Shen et al. (2002). Corresponding adjustments in the electron absorbed fraction are easily made through image rescaling and PIRT model simulations of electron transport. Future ex pansions of the model are also envisioned whereby measurements of volumetric bone mine ral density can be used to properly match the patient’s bone mineral density status to a reference library of microCT images covering a corresponding range of trabecular thicknesses and marrow volume fractions.

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199 Figure 7-1. Multiple generations of the radiatio n transport codes used at the University of Florida. This is an example of the tr eatment of a single vertebra within each radiation transport code. The upper most left is the 1st generation (CBIST) and the bottom right is th e 4th generation (PIRT).

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200 Figure 7-2. Representative vertebral images used in the PIRT model. Four types of images are shown, (A) 3D reconstructe d images of segmented 2D ex-vivo CT slices, (B) single 2D segmented ex-vi vo CT slices, (C) 3D reconstructed images of microCT data, and (D) 2D single slices through the microCT data set.

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201 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron energy (MeV)Absorbed fraction to TAM Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TAM source TBS source TBV source Sacrum 70% marrow cellularity Figure 7-3. Electron absorbed fractions to active bone marrow within the sacrum (70% ICRP reference cellularity) for three s ource tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those give by dashed lines are from VBIST simulations.

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202 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.010.1110Electron energy (MeV)Absorbed fraction to TBE Infinite Spongiosa (TAM) Infinite Spongiosa (TBV) Infinite Spongiosa (TBS) Paired Image (TAM) Paired Image (TBV) Paired Image (TBS) TBS source TBV source TAM source Sacrum 70% marrow cellularity Figure 7-4. Electron absorbed fractions to bone endosteum within the sacrum for three source tissues – TAM, TBV, and TBS. Data shown by solid lines are from the PIRT model, while those give by dash ed lines are from VBIST simulations.

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203 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.010.1110Electron energy (MeV)Absorbed fraction to CBV Cranium Mandible Right Humerus Right Femur Right Clavicle Ribs Cervical Vert Lumbar Vert Os Coxae Spongiosa tissue sources (TAM, TBS, TBV) Energy escape to the cortical bone cortex Figure 7-5. Electron absorbed fractions to th e cortical bone volume from electron sources in the spongiosa tissues (TAM, TBS, and TBV) The legend is listed in anatomical (cranial to caudal) order.

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204 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron energy (MeV)Absorbed fraction to CBV Mandible Right Humerus Cranium Right Femur Os Coxae Right Clavicle Cervical Vert Lumbar Vert Ribs CBV sources Self-irradiation of the cortical bone cortex Figure 7-6. Electron absorbed fractions to th e cortical bone volume from electron sources in the cortical bone cortex itself. The legend is given in decreasing value of the self-absorbed fraction at high electron energies.

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205 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron energy (MeV)Absorbed fraction to TAM Eckerman 2000 (TAM) Eckerman 2000 (TBV) Eckerman 2000 (TBS) UF (TAM) UF (TBV) UF (TBS) UF (CBV) TAM source TBS source TBV source CBV source Discrepancy: Uniform scaling vs explicit modeling of marrow cellularity Discrepancy: Energy loss to cortical bone is or is not considered in the model Discrepancy: Trabecular endosteum is either inclusive or exclusive of the marrow space Skeletal Averaged Absorbed Fractions at Reference Cellularities Figure 7-7. Skeletal averaged electron absorbed fractions to active bone marrow within the entire skeleton for four source tissu es – TAM, TBV, TBS, and CBV. Data shown by solid lines are from the PIRT model (present study), while those given by dashed lines are from the Eckerman and Stabin CBIST model.

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206 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.010.1110Electron energy (MeV)Absorbed fraction to TBE Eckerman 2000 (TAM) Eckerman 2000 (TBV) Eckerman 2000 (TBS) UF (TAM) UF (TBV) UF (TBS) UF (CBV) UF (TMC) TBS sourceTAM sourceTBV source CBV source Discrepancy: Energy loss to cortical bone is or is not considered in the model Discrepancy: Trabecular endosteum is either inclusive or exclusive of the marrow space Skeletal Averaged Absorbed Fractions at Reference CellularitiesTMC source Figure 7-8. Skeletal averaged electron absorb ed fractions to bone endosteum within the entire skeleton for five source tissues – TAM, TBV, TBS, TMC, and CBV. Data shown by solid lines are from th e PIRT model (present study), while those given by dashed lines are from th e Eckerman and Stabin CBIST model.

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207 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.010.1110Electron energy (MeV)Absorbed fraction to TAM Stabin 2003 (TAM) Stabin 2003 (TBV) Stabin 2003 (TBS) UF (TAM) UF (TBV) UF (TBS) TAM source TBS source TBV source Skeletal Averaged Absorbed Fractions at Reference Cellularities Figure 7-9. Skeletal averaged electron absorbed fractions to active bone marrow within the entire skeleton for four source tissu es – TAM, TBV, TBS, and CBV. Data shown by solid lines are from the PIRT model (present study), while those given by dashed lines are from the Stabin and Siegel CBIST model.

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208 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018 33P177Lu186Re32P90YRadionuclideS Value (mGy/MBq-s) Cranium Ribs Cervical Vertebra Thoracic Vertebra Lumbar Vertebra Sacrum Os CoxaeSite-Specific Values of S(TAM TAM) Figure 7-10. Variations in the S(TAM TAM ) with different radionuclides on the skeletal-site-specific radionuclide S valu es given by the PIRT model for the UF reference male cancer patient.

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209 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010Cl a vicle (Rig h t ) Cla v i c l e (L e f t ) Mandible S t ernum S cap u la (Ri g ht) Sc a pula (L e f t ) Ce rvical V e rteb r a Humerus ( Righ t ) Humeru s (Left ) Fe mu r (Rig h t ) Rib s Cra n iu m S a crum Femur (L e ft) L u mba r V er t ebra Thoracic Ve r tebra O s Co x a eSkeletal SiteS Value (mGy/MBq-s) S(TAM TAM) S(TAM TBS) S(TAM TBV) Radionuclide: 90Y mTAM (g) Clavicles (Right) 3.0 Clavicles (Left) 4.3 Manible 5.2 Humerus (Right) 15.0 Scapulae (Right) 15.2 Scapulae (Left) 16.3 Humerus (Left) 17.4 Sternum 19.6 Cranium 26.0 Cervical Vertebra e 32.1 Femur (Right) 40.8 Femur (Left) 42.1 Sacrum 87.0 Ribs 117.1 Lumbar Vertebrae 131.3 Thoracic Vertebrae 172.8 Os Coxa e 222.0 Figure 7-11. Variations in the S(TAM rS) for 90Y on the skeletal-site-specific radionuclide S values given by the PIRT model for the UF reference male cancer patient.

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210 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 33P177Lu186Re32P90YRadionuclideS Value (mGy/MBq-s) 100% 70% 30%Marrow CellularityS(TAM TAM) for the Cranium Figure 7-12. Variations in the S(TAM TAM ) to the cranium based on varying marrow celluarity and 5 radionuclides given by the PIRT model for the UF reference male cancer patient.

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211Table 7-1. Comparison between reference masses used at UF (pre sent study) and ICRP 89 for tissues within the marrow cavities. UF Reference Male Cancer Patient (present study) ReferenceActive MarrowActive MarrowTotal Marrow Active Marrow †Active Marrow †Endosteum§Inactive Marrow Cellularity Factor (%)Mass (g)% of tota l Mass (g)Mass (g)% of tota l Mass (g)Mass (g) Cranium‡38%88.97.6%68.526.02.5%5.540.4 Facial Bones 38%**4.41.70.2%0.22.6 Mandible 38%9.40.8%13.65.20.5%0.78.0 Ribs 70%188.416.1%167.3117.112.1%4.347.8 Sternum 70%36.33.1%28.019.62.0%1.18.0 Cervical Vertebra 70%45.63.9%45.832.13.3%2.213.1 Thoracic Vertebra 70%188.416.1%246.9172.817.9%10.370.5 Lumbar Vertebra 70%143.912.3%187.6131.313.6%9.053.5 Sacrum 70%115.89.9%124.387.09.0%6.535.5 Os Coxae 48%204.817.5%462.6222.023.0%18.3228.9 Femora, proximal 25%78.46.7%331.582.98.6%15.3236.5 Right 25%**163.140.84.2%7.8116.4 Left 25%**168.342.14.4%7.5120.1 Humeri, proximal 25%26.92.3%129.732.43.4%6.792.5 Right 25%**60.115.01.6%3.042.9 Left 25%**69.617.41.8%3.749.6 Clavicles 33%9.40.8%22.17.30.8%0.514.1 Right 33%**9.23.00.3%0.45.9 Left 33%**12.94.30.4%0.28.2 Scapulae 38%32.82.8%82.831.53.3%3.948.9 Right 38%**40.115.21.6%1.823.6 Left 38%**42.816.31.7%2.125.2 Skeletal Sites (w/o active marrow) Femora, lower 0%**576.70.00.0%14.6548.7 Fibula/Tibia/Feet 0%**600.20.00.0%12.8571.1 Humeri, lower 0%**182.70.00.0%6.7173.9 Ulna/Radius/Hands 0%**268.30.00.0%6.0255.3 TOTALS1170.099.9%3538.6967.2100%124.52446.6 indicates values not reported in ICRP 70 / 89 † indicates active marrow at ICRP 70 / 89 reference cellularity ‡ value includes facial bones (maxilla, na sal, zygomatic, lacrimal, palatine, vomer) § endosteum associated with trabecular regions of the skeletal (TBE) Approximate values for regions not containing active marrow in the adult Skeletal Sites ( w/ active marrow ) ICRP 70 / 89 Reference Man

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212Table 7-2.Pertinent values used in the calcu lation of data for the PIRT model of skelet al dosimetry. Values listed are generat ed from macroscopic segmentation of ex-vivo computed tomography (CT) of the each skeletal site as well as microscopic image processing of ex-vivo microCT data. Spongiosa* Marrow†Endosteum†Trabecular Bone†Trabecular BoneCortical*Cortical Bone Volume (cm 3 ) Volume FractionVolume FractionVolume FractionMass (g) Volume (cm 3 ) Mass (g) Cranium 117.70.5290.0430.42896.7188.5362.0 Facial Bones ‡ 5.80.7390.0370.2252.591.8176.3 Mandible 17.90.7390.0370.2257.733.764.7 Ribs 194.40.8350.0220.14353.4151.3290.5 Sternum 33.90.8020.0310.16710.913.926.6 Cervical Vertebra 57.30.7770.0370.18720.535.467.9 Thoracic Vertebra 292.10.8200.0340.14681.9105.4202.4 Lumbar Vertebra 237.30.7670.0370.19689.265.6125.9 Sacrum 159.50.7570.0400.20462.441.479.5 Os Coxae 558.40.8040.0320.164175.8204.4392.5 Femora, proximal 430.80.7380.0340.228181.480.8155.1 Humeri, proximal 173.10.7260.0380.23678.457.4110.2 Clavicles 25.90.8240.0210.1557.713.928.3 Scapulae 115.30.6970.0320.27059.869.3133.0 Skeletal Sites (w/o active marrow) Femora, lower§414.60.7380.0340.228181.4219.5421.5 Fibula/Tibia/Feet§593.20.7380.0340.228156.5373.2716.5 Humeri, lower§172.50.7260.0380.23678.1123.4236.9 Ulna/Radius/Hands§344.90.7260.0380.23670.4249.8479.8 TOTALS 1414.74069.6 indicates values determined from ex-vivo CT segmentation (macroimage) † indicates values determined from image processing of microCT data (microimage)‡ facial bones include maxilla, nasal, zygomatic, lacrimal, palatine, vomer § indicates estimated values for skeletal sites not containing active marrow in the adult Skeletal Sites (w/ active marrow)Approximate values for regions not containing active marrow in the adult Trabecular Spongiosa Regions Cortical Bone Regions

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213Table 7-3. Measurements of spongiosa volum e and cortical bone volume within each of the five (5) lumbar vertebrae of the UF Reference Male Cancer Patient. SpongiosaCortical Bone Skeletal Site Volume (cm 3 )Volume (cm 3 ) L1 Vertebra 42.5611.82 L2 Vertebra 45.5812.69 L3 Vertebra 49.1612.77 L4 Vertebra 51.4313.24 L5 Vertebra 48.6015.06 Totals: 237.365.6

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214Table 7-4. Surface-to-volume ratios within skeletal sites of trabecular spongiosa f ound within the UF 66-year male, the Leeds 44-year male, and within other individuals as adapted from in the lite rature (ICRP). The reference S/V ratio assumed within the ICRP 70 Reference Man m odel is 18 mm2 / mm3. Cranium10.27.84.1 Mandible16.4 Ribs15.218.511.5 Sternum18.3 Cervical Vertebra 19.611.5 Thoracic Vertebra20.212.0 Lumbar Vertebra18.919.711.412.014.6 23.0 23.9 14.0 21.0 A mstutz & Sission (1969) (1 adult) A rnold & Wei (1972) (39-50y) Beddoe et al. (1976) (39-55y) Bromley et al. (1966) (30-60y) Dyson et al. (1970) (1 adult) Sacrum19.4 Os Coxae19.617.218.017.4 20.2 21.4 18.6 21.6 Merk & Schenk (1970) (30-60y) Shulz & Delling (1976) (31-40y) Shulz & Delling (1976) (41-50y) Shulz & Delling (1976) (51-60y) Shulz & Delling (1976) (61-70y) Femora, proximal 17.39.510.0 Right14.9 Left15.2 Humeri, proximal Right14.1 Left18.1 Clavicles Right12.8 Left15.5 Scapulae Right11.3 Left12.7 Mean 1 SD 16.0 3.116.1 4.811.0 1.011.2 5.019.3 4.7 Lumbar Vertebrae 19.8 1.8 Os Coxae (Illiac Crest) ICRP Publication 70 18.0 ICRP Publication 70 References S/V Ratio (mm2 / mm3) Lloyd et al. (1968) (1 Adult) S/V Ratio (mm2 / mm3) Lloyd & Hodges (1971) (1 Adult) S/V Ratio (mm2 / mm3) Skeletal Sites S/V Ratio (mm2 / mm3) (UF Present Study) S/V Ratio (mm2 / mm3) Beddoe et al. (1976) (44y M)

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215Table 7-5. Comparison of assigne d cellularity factors a nd fractions of different skeletal components between UF (present study ) and Eckerman and Stabin (2000). Values are used in abso rbed fraction and S value skeletal average weighting Cellularity fTAMfTBVfTBSfCBVfTMCCellularity fTAMfTBVfTBSfCBVCranium 0.380.0250.0680.0420.0890.02710.420.0560.0260.0120.128 Facial bones 0.380.0020.0020.0020.0430.00170.420.0280.0130.0060.063 Mandible 0.380.0050.0050.0050.0160.0054* **** Ribs 0.700.1210.0380.0350.0710.11130.720.1920.0300.0310.120 Sternum 0.700.0200.0080.0090.0070.0189* **** Cervical Vertebra 0.700.0330.0150.0170.0170.03140.720.0270.1070.1100.009 Thoracic Vertebra 0.700.1790.0580.0830.0500.16790.720.1740.2760.2820.024 Lumbar Vertebra 0.700.1360.0630.0730.0310.12860.720.0980.0690.0710.008 Sacrum 0.700.0900.0440.0520.0200.0857* **** Os Coxae 0.480.2300.1240.1480.0960.22030.580.3330.0180.0180.014 Femora, proximal 0.250.0330.1630.1670.140 Right 0.250.0420.0640.0600.0180.0442 Left 0.250.0440.0640.0590.0200.0453 Humeri, proximal 0.250.0230.0300.0310.070 Right 0.250.0160.0280.0250.0140.0166 Left 0.250.0180.0270.0300.0130.0193 Clavicles 0.370.0080.0020.0020.009 Right 0.330.0030.0040.0030.0030.0031 Left 0.330.0040.0020.0020.0040.0041 Scapulae 0.420.0280.0090.0090.036 Right 0.380.0160.0210.0140.0170.0156 Left 0.380.0170.0220.0170.0160.0168 Femora, lower 0.000.0000.1280.1180.1040.01340.000.0000.1650.1690.207 Humeri, lower 0.000.0000.0550.0540.1760.00610.000.0000.0250.0260.047 Fibula/Tibia/Feet 0.000.0000.1110.1040.0580.01180.000.0000.0400.0400.082 Ulna/Radius/Hands 0.000.0000.0500.0490.1180.00550.000.0000.0260.0260.044 Skeletal Averaged Marrow Cellularity 0.570.61 indicates values not reported in Eckerman & Stabin (2000) indicates reference cellularity given in ICRP Publications 70 / 89 Adult Male Skeletal Sites UF (present study)Eckerman & Stabin (2000)

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216Table 7-6. Skeletal-averaged absorbed frac tions for monoenergetic electrons for the UF Reference Male Cancer Patient and those of the ICRP Reference Man as adapted from Eckerman & Stabin (2000) and in St abin & Siegel (2003). Energy (MeV)(TAM TAM)(TAM TBV)(TAM TBS)(TAM CBV)(TBE TAM)(TBE TBV)(TBE TBS)(TBE CBV)(TAM TBE)(TBE TBE)(TAM TMC)(TBE TMC) 0.010 0.98400.00000.00180.00000.00120.00310.50000.00000.01060.93780.94560.0401 0.015 0.96930.00010.00310.00010.00250.00630.50000.00000.02130.87240.93220.0386 0.020 0.95110.00020.00470.00010.00410.01040.50000.00000.03540.78970.91570.0368 0.030 0.90440.00060.02060.00030.00800.01980.47470.00000.06800.59560.87310.0325 0.040 0.84980.00250.05260.00040.01090.02730.39530.00010.09270.44110.82230.0288 0.050 0.79200.00580.07810.00070.01290.03230.31020.00010.11190.33800.76820.0264 0.100 0.59450.03170.13080.00210.01690.04050.13330.00030.14790.14090.58090.0221 0.200 0.49380.09470.16780.00580.02040.03820.06240.00070.17830.06510.48630.0223 0.500 0.39870.16040.19650.01710.02330.02980.03570.00230.20310.03630.39670.0239 1.000 0.34580.16040.18670.03080.02220.02590.02850.00440.19170.02870.34530.0225 1.500 0.30770.15010.17220.03640.02040.02340.02510.00540.17650.02530.30790.0207 2.000 0.27620.13910.15840.03730.01880.02140.02270.00570.16280.02280.27690.0190 4.000 0.19410.10320.11670.03130.01380.01590.01670.00520.11710.01670.19510.0140 Energy (MeV)(TAM TAM)(TAM TBV)(TAM TBS)(TAM CBV)(TBE TAM)(TBE TBV)(TBE TBS)(TBE CBV) 0.010 0.60900.00130.17300.04390.00360.5000 0.015 0.60900.00240.17100.04390.00680.4980 0.020 0.60700.00420.17200.04180.01210.4980 0.030 0.60400.00810.17300.03680.02100.4110 0.040 0.60000.01350.17200.03320.02860.3050 0.050 0.59500.01960.17000.03100.03350.2340 0.100 0.56300.06040.16400.02700.04260.0981 0.200 0.50000.14200.19000.02890.03640.0619 0.500 0.45200.20800.22700.02980.02950.0366 1.000 0.43800.22500.23400.02990.02870.0320 1.500 0.43550.22850.23600.02980.02870.0312 2.000 0.43300.23200.23800.02970.02870.0303 4.000 0.43100.23600.24000.02950.02870.0295 Energy (MeV)(TAM TAM)(TAM TBV)(TAM TBS)(TAM CBV)(TBE TAM)(TBE TBV)(TBE TBS)(TBE CBV) 0.010 0.99700.00160.21500.04400.00360.5020 0.015 0.96000.00290.21200.04400.00680.4970 0.020 0.92000.00510.21300.04190.01210.4980 0.030 0.84000.00990.21400.03680.02100.4120 0.040 0.77000.01650.21300.03330.02860.3040 0.050 0.73000.02400.21100.03100.03330.2350 0.100 0.59000.07380.20300.02700.04230.0980 0.200 0.50000.17400.23300.02890.03640.0620 0.500 0.45200.25500.28000.02980.02990.0371 1.000 0.43800.27600.28800.02990.02910.0325 1.500 0.43550.28050.29050.02980.02910.0317 2.000 0.43300.28500.29300.02970.02910.0308 4.000 0.43100.28900.29500.02960.02910.0299 Skeletal Averaged Electron Absorbed Fractions (UF present study) Skeletal Averaged Electron Absorbed Fractions (Eckerman & Stabin 2000 MIRDOSE3) Skeletal Averaged Electron Absorbed Fractions (Stabin & Siegel 2003 OLINDA)

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217Table 7-7. Radiation charac teristics of radionuclides used for calculation of S values. 33P -0.0770.24925.301.28E-020.00E+000%169E r -0.1000.3519.401.74E-021.52E-060%177Lu -0.1330.4986.712.46E-025.61E-0319%131I -0.1920.6068.023.11E-026.09E-0266%153Sm -0.2250.8091.954.50E-029.91E-0318%186Re -,EC 0.3231.0753.785.76E-023.29E-035%89S r -0.5831.49250.509.74E-021.35E-050%32P -0.6951.71014.301.16E-010.00E+000%188Re -0.7642.0000.711.28E-019.19E-037%90Y -0.9342.2822.671.56E-012.70E-070% (photon) (g-mGy/MBq-s) % Photon Contribution Half-life (days) (electron) (g-mGy/MBq-s) Emax (MeV) Radionuclide Decay Mode Eave (MeV)

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218Table 7-8. Skeletal-averaged S values (m Gy/MBq-s) for different combinations of so urce and target regions within the spongiosa and cortical bone of the UF Reference Male Cancer Patient (prese nt investigation) and the ICRP Reference Man adapted from the OLINDA computer code (2003). Skeletal Region33P169Er177Lu131I153Sm186Re89Sr32P188Re90Y Sskel(TAM TAM) 9.02E-061.14E-051.51E-052.46E-052.63E-052 .88E-054.26E-054.90E -055.38E-056.02E-05 Sskel(TAM TBV) 6.22E-071.17E-062.17E-063.69E-064.99E-068 .60E-061.63E-051.96E -052.15E-052.58E-05 Sskel(TAM TBS) 1.90E-062.80E-064.42E-066.36E-068.46E-061 .24E-052.13E-052.52E -052.76E-052.95E-05 Sskel(TAM CBV) 4.81E-089.46E-082.00E-073.81E-075.61E-071 .22E-063.13E-064.03E -064.80E-066.33E-06 Sskel(TBE TAM) 4.48E-065.76E-067.97E-061.34E-051.47E-051 .78E-052.87E-053.36E -053.71E-054.27E-05 Sskel(TBE TBV) 5.79E-067.62E-061.09E-051.38E-051.83E-052 .22E-053.34E-053.85E -054.23E-054.79E-05 Sskel(TBE TBS) 2.36E-052.65E-052.97E-052.94E-055.13E-053 .69E-054.45E-054.94E -055.65E-055.75E-05 Sskel(TBE CBV) 6.27E-081.25E-072.70E-075.22E-077.78E-072 .17E-065.69E-067.37E -068.80E-061.16E-05Skeletal Region33P169Er177Lu131I153Sm186Re89Sr32P188Re90Y Sskel(TAM TAM) 7.00E-068.84E-061.19E-051.55E-052.06E-052 .27E-053.75E-054.42E -054.99E-055.87E-05 Sskel(TAM TBV) 5.20E-071.05E-062.48E-065.08E-066.31E-061 .09E-052.18E-052.65E -052.98E-053.66E-05 Sskel(TAM TBS) 2.24E-062.98E-064.32E-066.91E-069.28E-061 .33E-052.35E-052.82E -053.18E-053.84E-05 Sskel(TAM CBV) 005.60E-086.11E-071.14E-073.34E-081.34E-1008.87E-080 Sskel(TBE TAM) 2.90E-063.78E-065.58E-068.20E-061.11E-051 .36E-052.31E-052.75E -053.11E-053.72E-05 Sskel(TBE TBV) 4.02E-065.61E-068.17E-061.05E-051.28E-051 .41E-052.34E-052.76E -053.11E-053.64E-05 Sskel(TBE TBS) 1.43E-051.52E-051.70E-051.83E-053.20E-052 .12E-052.80E-053.22E -053.70E-054.10E-05 Sskel(TBE CBV) ---------------------------------------S Values (mGy/MBq-s) UF Skeletal Averaged Electron / Beta Component Only Radionuclides S Values (mGy/MBq-s) OLINDA Skeletal Averaged Photon Component Included Radionuclides

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219 CHAPTER 8 CONCLUSIONS AND FUTURE WORK Conclusions Patient specificity in the absorbed dose to active marrow requires separate assessments of the amount of radioactivity taken up within skeletal tissues of that individual patient, and the absorbed dose per pa rticle emission (S value) considering the individual patient’s internal skeletal structure (normal, osteopenic, or osteoporotic trabecular bone), external skeletal structur e (size of the spongiosa and bone cortex), and marrow composition (cellularity). Extensive efforts have been made in patient-specific assessments of activity uptake in active marro w. These techniques include estimation of marrow activity concentration as a functi on of the peripheral blood concentration (Sgouros 1993). However, little research has been focused on improving the patient specificity of radionuclide S values. S values for skeletal dosimetry are assigned from a Reference Man model that is based on the 2D optical scans of seven bones within a single 44-year male subject measured some 30 years ago (Whitwell 1973; Whitwell 1976). Estimates of marrow tissue mass in Reference Man come from a variety of separate and independent data sources, with key studies dating back to 1926. The Reference Man skeletal model is thus limited in its ability to tailor S values for specific patients for a variety of reasons. Whitwell (1973) used experimentally measured chord length distributions along with range-e nergy relationships to calc ulate dose conversion factors for seven radionuclides of interest in hea lth physics. In thes e studies, only bone-and

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220 surface-seeking radionuclides were consider ed, and only the marrow and trabecular endosteum were chosen as target regions. Subsequent investigators have continued to redefine the skeletal dosimetry model of Reference Man, but all have relied extensively on this single chord-length data set obtained from a single adult male. While incremental improvements are noted, no fundamental revisi ons have been made to ensure improved patient-specificity of radionuc lide S values (Bouchet 2000). Chapters 3 and 4 have provided a new me thodology allowing for detailed modeling of the 3D macrostructure of individual marrow-containing bone s within the skeleton, thus permitting improved estimates of absorbed fractions and radionuclide S values for intermediate-to-high beta emitters. In Chapter 3, the use of Paired-Image Radiation Transport (PIRT) and magnetic resonance im aging have shown that for previously investigated skeletal sites th e dose is overestimated by up to ~30% for high-energy beta emitters (as compared to models based on infini te spongiosa transport). In Chapter 4, the PIRT methodology is applied to three different skeletal sites that have been difficult to model in skeletal dosimetry (i.e. flat bone s). With the use of microCT imaging and PIRT, calculations of radionucli de S values imply that curre nt chord-based models used in clinical skeletal dosimetry overestimate dose to the active marrow by up to ~75% for high-energy beta emitters. A new adult male reference model for skeletal dosimetry requires a detailed investigation into the currently adopted Univ ersity of Leeds model. Chapter 5 and 6 go into great detail to compare the University of Florida adult male cancer patient (66-year UF RMCP) to the current clinical refere nce model, using fixed modeling techniques (CBIST) to develop a fair comparison. In Ch apter 5, an alternative set of chord-length

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221 distribution data is presented for 14 skeletal sites of the 66-year UF RMCP. These chordlength distributions were obtained through the us e of trilinear interpol ation techniques to smooth the bone trabecular surfaces within di gital images of the trabecular spongiosa of the reference subject. Results have shown that the marrow cavities, on average, were similar with those seen in the Leeds 44-year male for the femoral head, lumbar vertebrae, and ribs. However, in several cases significan t differences in the c hord-distribution shape were noted between the two refe rence individuals (e.g. parietal bone of the cranium). In Chapter 6, these same chord-length distri butions were used to provide a dosimetry comparison between the two reference subjects. The chord-length dist ributions were also used to compare radiation transport methodol ogies. A version of the older chord-based transport model was directly compared to more recent voxel-based transport techniques using the same trabecular microstructure fr om the 66-year UF RMCP. Results have shown that a voxel-based transport approach better serves dosimetry models due to its ability to accurately model the 3D trabecula r microstructure and decrease significant limitations, as seen in older c hord-based transport models. Chapter 7 investigated the dosimetry and skeletal mass results for the 66-year UF RMCP. PIRT model techniques presented in Chapters 3 and 4 were used to obtain skeletal dosimetry data provided in Chapter 7. This work was uni que in that direct calculation of skeletal S values was internally consistent. Hence, site-specific absorbed fractions for electrons emitted within the trabecular structure and masses of the target and source tissues needed for dose assessment were measured in a single patient at each bone site of the axial skeleton. The radionuclide S values presented for the UF reference male cancer patient provides better estim ates of dose to the skeletal system. In turn, improved

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222 scaling of this data for variations in individual patients is possible due to the comprehensive data on the reference radionuclide patient provided in th is investigation. Future Work The development of a defined reference male cancer patient, using the PIRT methodology for the purposes of accurate skel etal dosimetry, has paved the way for several uses in patient-specific estimates in marrow dose models. The extent to which the ALRADS group at the University of Flor ida has progressed over the past 5 years has been enormous. Major obstacles in skeletal dose modeling include, but are not limited to, the following: (1) movement away from chor d-based radiation transport; (2) explicit consideration of adipose tissue within the bone marrow; (3) incorporat ions of variations in the trabecular microstructure of an indivi dual subject; and finall y, (4) addition of the cortical bone cortex or macrostructure of each sk eletal site in a patien t. The treatment of these 4 obstacles has resolved a major complication in skeletal dosimetry—the development of bone site specific radionuclide S values within a reference patient, in which absorbed fractions and target tissue masses are derived from the same individual and all as a function of marrow cellularity. Th ese advances and this reference data will allow other researchers to take steps in de veloping more patient-s pecific skeletal dose models. Improvements in the use of Voxel Models Although the 4 major obstacles in skeletal dosimetry have been handled in this investigation for the reference male cancer pa tient, improvements can always be made. For example, in regards to obstacle 1, Chapter 5 has shown that voxel-based dosimetry provides an easier and more accurate method of skeletal dosimetry over standard chordbased radiation transport models. With that said, it may be necessary to address “voxel

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223 effects” within the digital images of the tr abecular bone, as well as in the macrostructural model of each skeletal site. Rajon et al. (2003b) has shown th at when transport is made within a voxel image with voxel sizes on the same order as the electr on range large values of the absorbed fraction relative errors are seen. Image resolution in the present investigation is 60 m. At this resolution, Rajon et al. (2003b) show significant voxel effects will occur for energies below 150 keV. To resolve this issue, Rajon et al. suggest the use of the hyperboloid Marching Cube (HMC) algorithm coupled to EGSnrc. The HMC algorithm allows for a better represen tation of the bone-marrow interface, which will subsequently reduce the absorbed fracti on overestimation especially in cross-region situations (Rajon et al. 2003b). Another issue may be in the use of a voxel model for the macrostructural model of the skeletal site. A future step that can be taken is to quantify the error in the absorbed fraction, from e rrors found in the segmentation of the ex-vivo CT images used as the macrostructural data in PIRT. One can expect to find that with larger skeletal sites (containing a high per centage of spongiosa vol ume versus cortical bone volume), the absorbed fraction errors will be small. However, with skeletal sites that contain a small percentage of spongiosa volume (such as the flat bones), errors in segmentation may directly result in large differences in the absorbed fraction. Improvements in the Characterization of Active Marrow In regards to obstacle 2, marrow cellularity has been explicitly defined in the PIRT methodology and can be measured externally and scaled accordingly. Although the marrow cavity has been defined by adipose tissu e, it can further be partitioned to account for variations in the location of the hematopoietic tissues. Current investigations have only assessed the dose to active marrow unifo rmly across the marrow cavities. For some

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224 lower-energy emitters on the bone surfaces or in the bone volume, dose gradients can exist, which if quantified might improve the prediction capabilities of the model for correlations with marrow toxicity. Creation of models that partition the marrow voxels into bins, indexed according to their distance from the bone surfaces, could be done to account for this issue. This type of marrow dosimetry will be an improvement over the standard assessment of the mean dose to the marrow cavities and may show that there are shielding effects of surface-a dhered adipocytes, as well as so me dose fall-off due to finite particle ranges. Further goals of this work could be to establish marrow dose profiles (histograms) to assign weighted absorbed dos es to different cellular elements of the active marrow. Currently available data on the spatial distribution of marrow cell populations could be used to calculate the mean dose and dose distribution to specific populations of cells in the marrow. This may improve the correlation between marrow toxicity and absorbed dose. Improvements to the Skeletal Database Although this investigation has developed comprehensive data for a reference male cancer patient, the need for more subjects is necessary. As explained previously, in obstacles 3 and 4, the methodology to utili ze the trabecular micr ostructure and the skeletal site macrostructure has been devel oped. Future improvements can be made to the library of trabecular microstructures that will allow for use of different spongiosa data for varying patients. Variations in the trab ecular microstructure are seen between gender (Patton 2000), age-related change (Atkins on 1965), and most importantly, osteoporotic changes in the trabecular micr ostructure (Berne and Levy 1993). With more reference individuals, one will be able to “mix and ma tch” varying trabecular microstructures with different skeletal macrostructures. This dir ectly relates to the additions necessary for the

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225 skeletal database for the macrostructural data. The same situation occurs with the macrostructure, a single complete skeletal model has been developed for a 66-year-old reference male. Additions to the macrostructural database can be made for female subjects, younger reference patients (similar to the age of the ICRP reference male), and even pediatric patients. Subsequently, with an increase in the skeletal database for various types of subjects, one can report skeletal dosimetry results using the PIRT methodology. Difficulties in obtaining ex-vi vo CT data for these various types of patients, can be reduced by the ability to scale the macrostructure of a single individual to the composition of another based on simple vol umetric measurements within in-vivo CT scans. Scaling the CT macrostructural da ta from the 66-year UF RMCP in this investigation will allow for more reference do simetry models to be generated, provided that the issue with obtaining the trabecular micros tructure can be resolved. In the case of pediatric patients, one may be able to comb ine the University of Leeds chord-length distributions for pediatri c patients (Beddoe 1976) w ith the scaled-down voxel macrostructural model from the UF RMCP and obtain accurate skeletal dosimetry. Regardless, further data must be collected from subjects as mentioned above, beginning with a reference female cancer patient a nd a more representative ICRP reference individual. Scaling of Reference S Values to a Patient Currently studies at the University of Fl orida have focused on electron transport as related to beta particle dosimetry of the marrow. Another step for this project will be to additionally look at lowa nd high-energy photon self-dose to the marrow and endosteum. Cross dose to the marrow from sources outside the skeleton or from adjacent skeletal

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226 regions must continue to be assessed via standard met hods using whole-body computational models. Upon consideration for photon contributions to the radionuclid e S values, future work should utilize the reference skeletal dosimet ry data provided in this investigation for other patients. To do so, scaling approach es must be developed for patient-specific dosimetry. Shen et al. (2002) investigat ed methodologies in which the marrow dose could be determined without knowing the mass of the patient’s active marrow. However, they concluded that the best indicator fo r the estimation of marrow dose was by obtaining the required active marrow mass through imagi ng of the lumbar vertebra (Shen et al. 2002). Work by Bolch et al. (2002a) has atte mpted to scale S values by anthropometric factors (i.e. body mass index, lean body mass, etc.) and tissue volumes to determine more patient-specific marrow dose estimates. Th ey found that spongiosa volume provided a good parameter for scaling at higher energies while active marrow volume served as a better parameter at lower energies (<70 keV) ; therefore not requiring the use of active marrow mass, a difficult parameter to m easure in-vivo (Bolch et al. 2002a). Future utilization of the PI RT model and reference skeletal dosimetry data found in this investigation will be necessary to produce a more realistic S value for a given patient. We suggest that spongiosa volume ratios be taken between a single patient and the 66year UF RMCP (from this investigation) for a given skeletal site and used to scale the reference S values at a predetermined marrow cellularity. Clinical Application of Reference S Values Clinical applications to this work may be promising. With the possibility of scaling the 66-year UF RMCP S values by spongiosa vo lume, one can further scale to any given patient who enters a clinic. Scaling spongi osa volumes by all skeletal sites may be

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227 difficult, therefore it is sugge sted that one skeletal site (that can easily be measured via CT scans) be chosen in a patient. Deve lopment of factors would then be necessary to correlate the spongiosa volume of the chosen skel etal site to all others within the patient. Subsequently, a clinical scenario such as the following can be imagined: A patient enters the clinic, treatment pl anning for radionuclide therapy has begun, and patient specific S values are needed. Th e patient then undergoes a quick pelvic CT scan and the spongiosa volume of the sacrum is measured within the patient. With this new spongiosa volume quantity, the previously provided reference skeletal dosimetry data, and the predetermined rati os for spongiosa differences between the sacrum and all other bone sites, the clinic may now provide tailored S values at a given marrow cellularity for the patient. With the advent of better imaging modalities, measurements of marrow cellularity can be done in-vivo through improved techniques of magnetic resonance imaging (MRI). Additionally, with the capabilities of SPECT (Single Photon Emission Computed Tomography) or PET/CT dual-scanners (Positron Emission Tomography-Computed Tomography), a single scan may provide crucial imaging details and activity concentrations to help targeting and treat ment planning for radiation therapy. Future methods similar to that described in the previous clinic al scenario suggest that it may be possible to bette r predict S values in a give n radionuclide therapy patient, and thus better pred ict toxicity to the bone marrow. The methods provided in this investigation have opened the door to these possibilities.

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228 APPENDIX A UNIVERSITY OF FLORIDA REFERE NCE ADULT MALE RADIONUCLIDE PATIENT CT IMAGE DETAILS This appendix contains details regarding the subject chosen as the University of Florida Reference adult male cancer patient The appendix contains information on the in-vivo and ex-vivo computed tom ography CT imaging as well as image captures from both imaging techniques. Figure A-1. In-vivo computed tomography scout scans

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229 Figure A-2. Cranium images shown fo r visualization of skeletal site.

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230 Figure A-3. Mandible images shown for 3D visualization of the skeletal site. Figure A-4. Clavicle images shown fo r 3D visualization of skeletal site. Figure A-5. Scapulae images shown for 3D visualization of skeletal site.

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231 Figure A-6. Cervical vertebra images show n for 3D visualization of skeletal site. Figure A-7. Thoracic vertebra images show n for 3D visualization of skeletal site. Figure A-8. Sacrum images shown for 3D visualization of skeletal site.

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232 Figure A-9. Lumbar vertebrae images shown fo r 2D and 3D visualization of the skeletal site.

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233 Figure A-10. Os coxae (pelvic) images shown for 2D and 3D visualization of the skeletal site.

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234 Figure A-11. Proximal femur images shown for 2D and 3D visualization of the skeletal site.

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235 Table A-1. Ex-vivo CT parameters used in the PIRT model for radiation transport in order to define the binary contours of the macroimage at each skeletal site. Skeletal SiteNumber of SlicesSlice ThicknessIn-plane ResolutionField of View (z-direction)(mm)(microns)(cm) Left Femur1631.00203.1250010.4 Right Femur1561.00203.1250010.4 L4 Vertebra481.00322.6600016.5 Right Humerus1421.00128.906006.6 Left Humerus1471.00128.906006.6 Os Coxae2281.00656.2500033.6 Sternum2111.0097.656305.0 Sacrum1531.00322.6600016.5 Right Scapula2131.00322.6600016.5 Left Scapula2221.00322.6600016.5 Right Clavicle1701.0097.656305.0 Left Clavicle1571.0097.656305.0 Upper Right Rib2131.00289.0630014.8 Middle Right Ri b 2671.00289.0630014.8 Lower Right Rib2091.00289.0630014.8 Upper Left Rib2101.00234.3750012.0 Middle Left Rib2591.00234.3750012.0 Lower Left Rib1901.00234.3750012.0 Mandible1531.00443.3590022.7 Cranium1831.00443.3590022.7 Lumbar Vertebra1751.00322.6600016.5 Thoracic Vertebr a 3141.00322.6600016.5 Cervical Vertebr a 1291.00322.6600016.5

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236 APPENDIX B MICROIMAGE PARAMETERS FO R SKELETAL SPONGIOSA This appendix contains the pertinent mi croimage information for use within the Paired-Image Radiation Transport model. These parameters can be used for the input parameters for the microimage with in PIRT. All spongiosa sections for each skeletal site were acquired through mi croCT. Image processing and image preparation was performed via techniques outlined in Appendix C and Appendix D. Figure B-1. Example of a 3D reconstr uction of spongiosa acquired from microCT imaging of a bone section from a skeletal site of interest.

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237Table B-1. MicroCT image parameters used in the PIRT model in order to define the microimage of spongiosa in radiation transpo rt. ThresholdMarrow Volume Fraction CT Data IJKX (cm)Y (cm)Z (cm)(%)MarrowBone 1Right Femur Head1651003300.00600.00600.006015070.62864.8349.8 2Left Femur Head1801803600.00600.00600.006016076.971119.1358.7 3Left Femur Neck1851603200.00600.00600.006014179.211282.4368.5 4Right Femur Neck1851753900.00600.00600.006013882.061517.1360.5 5Left Parietal Bone751851800.00600.00600.006015859.97806.3508.9 6Right Parietal Bone902853800.00600.00600.006015960.97816.9471.0 7Frontal Bone702853800.00600.00600.006015856.48675.6488.5 8Occipital Bone65603600.00600.00600.006016051.49597.6521.4 9Mandible852252800.00600.00600.006015577.551224.8354.6 10Pelvis Pubis2601803100.00600.00600.006014882.411455.8282.9 11Pelvis Ischium2801003300.00600.00600.006014483.111548.4334.6 12Pelvis Ilium6101451500.00600.00600.006014885.311448.5245.3 13C3 Vertebra1251351900.00600.00600.006015982.501019.1289.1 14C6 Vertebra1502101450.00600.00600.006014280.161044.0279.4 15T3 Vertebra2251952750.00600.00600.006015284.401334.5290.3 16T6 Vertebra1803302050.00600.00600.006013582.001219.2316.3 17T11 Vertebra1602104050.00600.00600.006015483.281245.5249.3 18L2 Vertebra1703401500.00600.00600.006013682.111334.2340.5 19L4 Vertebra1452152000.00600.00600.006014878.741057.5288.8 20Sacrum2001752400.00600.00600.006014879.631093.9333.2 21Sternum4002101300.00600.00600.006014683.251355.6280.8 22Right Humerus4102301450.00600.00600.006014974.601123.7435.5 23Left Humerus2402653200.00600.00600.006015778.221180.4318.5 24Right Scapula400701400.00600.00600.006014872.841184.1439.8 25Left Scapula145752650.00600.00600.006014473.131154.5432.2 26Right Clavicle270701800.00600.00600.006015877.761100.8353.4 27Left Clavicle602351400.00600.00600.006014491.281681.1300.6 28Upper Right Rib38060500.00600.00600.006014979.671343.5287.2 29Middle Right Rib295801050.00600.00600.006014889.901901.3337.1 30Lower Right Rib39585350.00600.00600.006014786.011252.0248.5 31Upper Left Rib70652200.00600.00600.006014485.101345.8258.9 32Middle Left Rib100654600.00600.00600.006015088.781610.6330.4 33Lower Left Rib10075950.00600.00600.006014484.711238.4290.3 Average Chords Length Voxel DimensionsVoxel Resolution

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238 APPENDIX C IMAGE PROCESSING (C PROGRAMS) This appendix contains all the C programs used to prepare microimages from raw data to usable images for radiation transport within PIRT. They are ConvertMicroCT.c: Converts the gray le vels of the binary image from signed integer to unsigned integer. Plane.c: Creates 3 two-dimensional slices in each dimension I, J, K in order to determine the best region of interest within the microCT raw data. Histogram.c: Creates a gray-level histogram from the region of interest chosen in the previous step. FindThresh.c: Displays the threshold valu e that should be used in segmenting the microimage into two array number ResizeImage.c: Creates binary image that thresholds the gray-levels and resizes the raw microimage into the determined region of interest. MedianFilter.c: Filters the new image in a median filter, weighted to remove the noise in the image. ReadImage.c: Displays the pertinent values of the new image. Each program given in this appendix must be compiled at least once before use and must receive inputs when prompted. The compile command in the Linux environment is the following, “gcc –o program program.c –Wl, --stack, 0x400000” with “program” and “program.c” being the name of the actual program that one is compiling. ConvertMicroCT.c /***************************************************/ /* Amish P. Shah */ /* This Program Converts MicroCT Data into a usable*/ /* image for image processing */ /***************************************************/ #include main (int argc, char *argv[]) { int i, j, k, imax, jmax, kmax; char tmp; unsigned char tmp2; unsigned char output[30]; FILE *inp; FILE *out; if (argc!=3) printf("Usage: ConvertMicroCT input output\n"); else {

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239 printf("\n"); printf("Please convert the RAW MicroCT image DATA\n"); printf("\n"); printf("Enter I matrix size: "); scanf("%d", &imax); printf("Enter J matrix size: "); scanf("%d", &jmax); printf("Enter K matrix size: "); scanf("%d", &kmax); printf("Converting MicroCT IMAGE DATA file...\n "); inp = fopen(argv[1],"rb"); out = fopen(argv[2],"wb"); if (inp==NULL) printf("error opening file\n"); else { for (i=0; i #define Imx 1000 #define Jmx 1000 #define Kmx 1000 unsigned char image[Imx][Jmx][Kmx]; main (int argc, char *argv[]) { int imax, jmax, kmax; int i, j, k, min, max, smax; char output[30]; FILE *inp; FILE *out; if (argc!=2) printf("Usage: plane input\n"); else { printf("Enter I Matrix Size: "); scanf("%d", &imax); printf("Enter J Matrix Size: "); scanf("%d", &jmax);

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240 printf("Enter K Matrix Size: "); scanf("%d", &kmax); printf("Enter min Slice Number for image: "); scanf("%d", &min); printf("Enter max Slice Number for image: "); scanf("%d", &max); smax = max + 1; inp = fopen(argv[1],"rb"); if (inp==NULL) printf("error opening file\n"); else { for (i=0; i
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241 Histogram.c /******************************************************/ /* AMISH P. SHAH */ /* histogram.c */ /* This program gives the gray level histogram. */ /* Data requires knowledge of the region of interest */ /******************************************************/ #include main(int argc, char *argv[]) { int i, j, area,k, c,temp; float pdf[256]; int tmpim1; float red[256]; unsigned char im1; FILE *inp; FILE *otp; int width,K1,height,J1,depth,I1; char inputfile[30]; char outputfile[30]; int Imin,Jmin,Imax,Jmax,Kmin,Kmax; if(argc!=12) printf("Usage: histogram input output I J K Imin Imax Jmin Jmax Kmin Kmax\n"); else { inp=fopen(argv[1], "rb"); I1 = atoi(argv[3]); J1 = atoi(argv[4]); K1 = atoi(argv[5]); Imin = atoi(argv[6]); Imax = atoi(argv[7]); Jmin = atoi(argv[8]); Jmax = atoi(argv[9]); Kmin = atoi(argv[10]); Kmax = atoi(argv[11]); area = 0; depth = I1; width = K1; height = J1; for(i=0; i<256; i++) { red[i]=0; } /*Reading the file*/ for(i=0; i=Imin && i<=Imax && j>=Jmin && j<=Jmax && k>=Kmin && k<=Kmax) { red[tmpim1]=red[tmpim1]+1; area=area+1; } } fclose(inp); /*Compute and display Probability Density Function*/ otp=fopen(argv[2], "w");

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242 for(i=0; i<256; i++) { pdf[i]= red[i]/area; fprintf(otp, "%d %.8f\n", i, pdf[i]); /*fprintf(otp, "%d\n", red[i]);*/ /*fwrite(&pdf[i],1,1,otp);*/ } printf("\n"); printf("\n"); printf("Done Creating Histogram for Data Set: %s \n", argv[2]); fclose(otp); return 0; } } FindThresh.c /***************************************************************/ /* Amish Shah */ /* findthresh.c */ /* The threshold value can be figured from the data provided */ /* Parameters required are given in SigmaPlot. M,mu,SigM,SigB */ /***************************************************************/ #include #include main() { int i, j; float a,b,c,thresh; float M,SigM,SigB,mu; printf("Enter M: "); scanf("%f", &M); printf("Enter mu: "); scanf("%f", &mu); printf("Enter SigM: "); scanf("%f", &SigM); printf("Enter SigB: "); scanf("%f", &SigB); printf("THRESH\t\tDifference\n"); /*Reading the file*/ for(thresh=0; thresh<256; thresh=thresh+1) { a=((SigB*SigB)*(M))/(thresh*(1-M)*(sqrt(2*3.14159))*SigM); b=exp(((thresh-mu)*(thresh-mu))/(2*SigM*SigM)-((thresh*thresh)/(2*SigB*SigB))); c=a-b; if(c<256 && c> -256) printf("%f\t%f\n", thresh,c); } return 0; } ResizeImage.c /****************************************************/ /* Amish Shah */ /* ResizeImage.c */ /* Program that Converts a Particular Slice from */ /* gray level to black & white based on a threshold */ /* value. Also, the raw image is narrowed to a ROI.*/ /****************************************************/ #include main (int argc, char *argv[]) { int i, j, k;

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243 unsigned char tmp; char output[30]; FILE *inp; FILE *out; int depth, height, width, thresh; int Imin, Imax, Jmin, Jmax, Kmin, Kmax; if (argc!=3) printf("Usage: ResizeImage input output\n"); else { inp = fopen(argv[1],"rb"); out = fopen(argv[2],"wb"); printf("\nEnter Dimensions of Image Matrix\n"); printf("Enter I Dimension: "); scanf("%d", &depth); printf("Enter J Dimension: "); scanf("%d", &height); printf("Enter K Dimension: "); scanf("%d", &width); printf("\nEnter Threshold Value for new Binary Image: "); scanf("%d", &thresh); printf("\nEnter Dimensions for Region of Interest\n"); printf("Enter I minimum Dimension: "); scanf("%d", &Imin); printf("Enter I maximum Dimension: "); scanf("%d", &Imax); printf("Enter J minimum Dimension: "); scanf("%d", &Jmin); printf("Enter J maximum Dimension: "); scanf("%d", &Jmax); printf("Enter K minimum Dimension: "); scanf("%d", &Kmin); printf("Enter K maximum Dimension: "); scanf("%d", &Kmax); if (inp==NULL) printf("error opening file\n"); else { printf("Creating 3D Image w/ a Threshheld Region of Interest... \n"); for (i=0; i=Imin && i=Jmin && j=Kmin && k
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244 #include #include #include #define SIZE 27 #define Imx 800 #define Jmx 800 #define Kmx 800 unsigned char gray[Imx][Jmx][Kmx]; unsigned char tmp,smooth[Imx][Jmx][Kmx]; int i,j,k,g,total; FILE *inp; FILE *otp; main (int argc, char *argv[]) { int width, height, depth; int x,y,bound, omed, c, test; int hood[SIZE]; if (argc!=3) printf("Usage: MedianFilter input output\n"); else { inp = fopen(argv[1],"rb"); otp = fopen(argv[2],"wb"); printf("\nEnter Dimensions of New Image Matrix\n"); printf("Enter I Dimension: "); scanf("%d", &depth); printf("Enter J Dimension: "); scanf("%d", &height); printf("Enter K Dimension: "); scanf("%d", &width); printf("\nCreating 3D Filtered Image...\n"); printf("opening file\n"); for(i=0; i
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245 hood[19]=(int)gray[i][j][k-1]; hood[20]=(int)gray[i+1][j][k-1]; hood[21]=(int)gray[i-1][j+1][k-1]; hood[22]=(int)gray[i][j+1][k-1]; hood[23]=(int)gray[i+1][j+1][k-1]; hood[24]=(int)gray[i-1][j-1][k-1]; hood[25]=(int)gray[i][j-1][k-1]; hood[26]=(int)gray[i+1][j-1][k-1]; for(g=0; g<27; g++) { total = total + hood[g]; } test = (int)gray[i][j][k]; if (test==255) { if(total<1536) { smooth[i][j][k] = 0; } else { smooth[i][j][k] = 255; } } else { if (test == 0){ if(total>4585) { smooth[i][j][k] = 255; } else { smooth[i][j][k] = 0; } } else { printf("error in voxel\n"); } } } for(i=0; i
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246 ReadImage.c /*************************************************************/ /* Amish P. Shah */ /* ReadImage.c */ /* This program provides all relevant IMAGE information */ /*************************************************************/ #include #include #include #define Imx 1000 #define Jmx 1000 #define Kmx 1000 #define SEED 0 #define FALSE 0 #define TRUE 1 double FatPercentage, BoneCoverage, FatPercentage2; int FatVoxel, FatSurface, i, j, k, BoneVoxel, TotalVoxel, MarrowVoxel; int MarrowSurface, MarrowVoxel2, MarrowSurface2; int NeighborVoxel (int i, int j, int k); int HEIGHT, WIDTH, DEPTH; int sides, TotalSides; double MarrowVolFraction; unsigned char tmp[Imx][Jmx][Kmx]; FILE *inp; main (int argc, char *argv[]) { int NSV; FatVoxel=0; FatPercentage=0; TotalVoxel=0; MarrowVoxel=0; BoneVoxel=0; FatPercentage2=0; MarrowSurface=0; FatSurface=0; MarrowSurface2=0; MarrowVoxel2=0; sides=0; TotalSides=0; MarrowVolFraction=0; /* Reads in the file and assign it to tmp*/ if (argc!=2) printf("Usage: ReadImage input\n"); else { inp = fopen(argv[1],"r"); if (inp==NULL) printf("error opening file\n"); else { printf("\nEnter Dimensions of Image Matrix\n"); printf("Enter I Dimension: "); scanf("%d", &HEIGHT); printf("Enter J Dimension: "); scanf("%d", &WIDTH); printf("Enter K Dimension: "); scanf("%d", &DEPTH); for (i=0; i
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247 FatVoxel = FatVoxel + 1; } else { if (tmp[i][j][k] == 255) { MarrowVoxel = MarrowVoxel + 1; } } } TotalVoxel = TotalVoxel +1; } } } } fclose (inp); } for (i=0; i122){ if (tmp[i-1][j][k] == 0) sides = sides + 1; if (tmp[i+1][j][k] == 0) sides = sides + 1; if (tmp[i][j+1][k] == 0) sides = sides + 1; if (tmp[i][j-1][k] == 0) sides = sides + 1; if (tmp[i][j][k-1] == 0) sides = sides + 1; if (tmp[i][j][k+1] == 0) sides = sides + 1; } } } } for (i=1; i122){ if (tmp[i-1][j][0] == 0) sides = sides + 1; if (tmp[i+1][j][0] == 0) sides = sides + 1; if (tmp[i][j+1][0] == 0) sides = sides + 1; if (tmp[i][j-1][0] == 0) sides = sides + 1; } if (tmp[i][j][DEPTH-1]>122){ if (tmp[i][j+1][DEPTH-1] == 0)

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248 sides = sides + 1; if (tmp[i][j-1][DEPTH-1] == 0) sides = sides + 1; if (tmp[i+1][j][DEPTH-1] == 0) sides = sides + 1; if (tmp[i-1][j][DEPTH-1] == 0) sides = sides + 1; } } } for (k=1; k122){ if (tmp[i-1][0][k] == 0) sides = sides + 1; if (tmp[i+1][0][k] == 0) sides = sides + 1; if (tmp[i][0][k+1] == 0) sides = sides + 1; if (tmp[i][0][k-1] == 0) sides = sides + 1; } if (tmp[i][WIDTH-1][k]>122){ if (tmp[i][WIDTH-1][k+1] == 0) sides = sides + 1; if (tmp[i][WIDTH-1][k-1] == 0) sides = sides + 1; if (tmp[i+1][WIDTH-1][k] == 0) sides = sides + 1; if (tmp[i-1][WIDTH-1][k] == 0) sides = sides + 1; } } } for (j=1; j122){ if (tmp[0][j-1][k] == 0) sides = sides + 1; if (tmp[0][j+1][k] == 0) sides = sides + 1; if (tmp[0][j][k+1] == 0) sides = sides + 1; if (tmp[0][j][k-1] == 0) sides = sides + 1; } if (tmp[HEIGHT-1][j][k]>122){ if (tmp[HEIGHT-1][j][k+1] == 0) sides = sides + 1; if (tmp[HEIGHT-1][j][k-1] == 0) sides = sides + 1; if (tmp[HEIGHT-1][j+1][k] == 0) sides = sides + 1; if (tmp[HEIGHT-1][j-1][k] == 0) sides = sides + 1; } } } /****/ for (i=1; i121){ if (tmp[i-1][j][k] == 0) TotalSides = TotalSides + 1; if (tmp[i+1][j][k] == 0) TotalSides = TotalSides + 1; if (tmp[i][j+1][k] == 0) TotalSides = TotalSides + 1; if (tmp[i][j-1][k] == 0) TotalSides = TotalSides + 1;

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249 if (tmp[i][j][k-1] == 0) TotalSides = TotalSides + 1; if (tmp[i][j][k+1] == 0) TotalSides = TotalSides + 1; } } } } for (i=1; i121){ if (tmp[i-1][j][0] == 0) TotalSides = TotalSides + 1; if (tmp[i+1][j][0] == 0) TotalSides = TotalSides + 1; if (tmp[i][j+1][0] == 0) TotalSides = TotalSides + 1; if (tmp[i][j-1][0] == 0) TotalSides = TotalSides + 1; } if (tmp[i][j][DEPTH-1]>121){ if (tmp[i][j+1][DEPTH-1] == 0) TotalSides = TotalSides + 1; if (tmp[i][j-1][DEPTH-1] == 0) TotalSides = TotalSides + 1; if (tmp[i+1][j][DEPTH-1] == 0) TotalSides = TotalSides + 1; if (tmp[i-1][j][DEPTH-1] == 0) TotalSides = TotalSides + 1; } } } for (k=1; k121){ if (tmp[i-1][0][k] == 0) TotalSides = TotalSides + 1; if (tmp[i+1][0][k] == 0) TotalSides = TotalSides + 1; if (tmp[i][0][k+1] == 0) TotalSides = TotalSides + 1; if (tmp[i][0][k-1] == 0) TotalSides = TotalSides + 1; } if (tmp[i][WIDTH-1][k]>121){ if (tmp[i][WIDTH-1][k+1] == 0) TotalSides = TotalSides + 1; if (tmp[i][WIDTH-1][k-1] == 0) TotalSides = TotalSides + 1; if (tmp[i+1][WIDTH-1][k] == 0) TotalSides = TotalSides + 1; if (tmp[i-1][WIDTH-1][k] == 0) TotalSides = TotalSides + 1; } } } for (j=1; j121){ if (tmp[0][j-1][k] == 0) TotalSides = TotalSides + 1; if (tmp[0][j+1][k] == 0) TotalSides = TotalSides + 1; if (tmp[0][j][k+1] == 0) TotalSides = TotalSides + 1; if (tmp[0][j][k-1] == 0) TotalSides = TotalSides + 1; } if (tmp[HEIGHT-1][j][k]>121){ if (tmp[HEIGHT-1][j][k+1] == 0) TotalSides = TotalSides + 1;

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250 if (tmp[HEIGHT-1][j][k-1] == 0) TotalSides = TotalSides + 1; if (tmp[HEIGHT-1][j+1][k] == 0) TotalSides = TotalSides + 1; if (tmp[HEIGHT-1][j-1][k] == 0) TotalSides = TotalSides + 1; } } } /*End of reading in the file*/ MarrowVoxel2 = MarrowVoxel+FatVoxel; MarrowSurface2 = MarrowSurface + FatSurface; FatPercentage = (((double)FatVoxel/(MarrowVoxel2))*100); BoneCoverage = (((double)FatSurface/(MarrowSurface2))*100); MarrowVolFraction = (((double)MarrowVoxel2/(TotalVoxel))*100); printf("FatPercentage %f \n", FatPercentage); printf("TotalVoxels %d \n", TotalVoxel); printf("BoneVoxels %d \n", BoneVoxel); printf("FatVoxels %d \n", FatVoxel); printf("MarrowVoxelsB4Fat %d \n", MarrowVoxel2); printf("MarrowSurf#B4Fat %d \n", MarrowSurface2); printf("Number of Active Marrow Surface Sides is %d \n", sides); printf("Number of All Endosteum Surface Sides is %d \n", TotalSides); printf("FatSurf# %d \n", FatSurface); printf("BoneCoverage %f \n", BoneCoverage); printf("Marrow Space Vol Fraction %f \n", MarrowVolFraction); } } int NeighborVoxel(int i, int j, int k) { int IsNeighborBone; IsNeighborBone = FALSE; if (i != 0) { if (tmp[i-1][j][k] == 0) { IsNeighborBone = TRUE; } } if (i != (HEIGHT-1)) { if (tmp[i+1][j][k] == 0) { IsNeighborBone = TRUE; } } if (j != 0) { if (tmp[i][j-1][k] == 0) { IsNeighborBone = TRUE; } } if (j != (WIDTH-1)) { if (tmp[i][j+1][k] == 0) { IsNeighborBone = TRUE; } } if (k != 0) { if (tmp[i][j][k-1] == 0) { IsNeighborBone = TRUE; } } if (k != (DEPTH-1)) { if (tmp[i][j][k+1] == 0) { IsNeighborBone = TRUE; } } return(IsNeighborBone); }

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251 APPENDIX D IMAGE PROCESSING TECHNIQUE This appendix contains all the vital steps required to take in order to successfully convert the raw microCT data received from S canco Medical AG into digital images in a format necessary for use in radiation transpor t codes, specifically Paired Image Radiation Transport (PIRT). The associated C program s used to prepare microimages from raw data to usable images for radiation transpor t within PIRT are given in Appendix C. The PIRT models written in the EGSnrc platform are given in Appendix E and Appendix F. Necessary tools (software) for the image processing steps are Cygwin installation on the PC. Cygwin is a Linux-like environment for Windows. It consists of two parts: a DLL (cygwi n1.dll) which acts as a Linux API emulation layer providing substantial Linux API func tionality; and a coll ection of tools, which provide Linux look and feel. Fr ee downloadable software offered at www.cygwin.com. Adobe Photoshop, minimum Adobe P hotoshop 5.0. Copyright 1998 Adobe Systems Incorporated. SigmaPlot 2001 for Windows Version 7.0. Copyright 1986-2001 SPSS Inc. and the associated library to fit the gaussian-rayleigh distribution. Seven C Programs written or modified by Amish P. Shah (2004) and given in Appendix C. These programs are the following: ConvertMicroCT.c, Plane.c, Histogram.c, FindThresh.c, ResizeImage.c, MedianFilter.c, and ReadImage.c. Each program given in Appendix C must be compiled at least once before use and must receive inputs when prompted. Image Processing Steps (Cookbook) This guide will provide steps necessary for image processing for one skeletal site; however, these steps are similar for all data se ts obtained from microC T imaging. First, grab the data and the header file from th e CD and place into a specific folder of your choice (keeping in mind where the files have been placed). For ease, one may place these two files in the folder with the C programs used in the Linux environment. The two files

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252 should look similar to that seen in Figure D-1. One then must open and print the header file in Microsoft WordPad, as shown in Figure D-2, in order to obtain parameters necessary for image processing. In this exampl e, the files are associated to a cube from the humerus. Figure D-1. Illustrative example of header a nd data files for the microCT data. In this case, C0000040 is the file name. In Figure D-2, the important numbers to highl ight are the three at the top row. In this example, the numbers are 532, 324, and 300. These values define the image dimensions of the raw image data set obt ained from image acquisition. The MOST important aspect of these numbers will be what they represent throughout the rest of this process. The right-most number (300) is wh at we will term “I” as in the number of

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253 voxels in the “I” dimension. The middle numbe r (324) is what will be termed “J” as in the number of the voxels in the “J” dimension. The left most number (532) is “K” and defines the number of voxels in the “K” dimension. These numbers will vary with each cube or section from each skeletal site when received from microCT image acquisition. The third row of numbers la beled “voxel size in mm” repr esent the resolution of each voxel; in this example (and in most cases) it is 60 m or 0.0600 mm. Some other important lines in the header file are the ones that describe the data set labeled “Patient Name” and “Original File,” basically telli ng you what you are looking at in the file labeled C0000040_DATA.RAW. As one can see in this example (Figure D-2), the file is the left humerus acquired in May 2004.

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254 Figure D-2. Pictorial example of the h eader file from the C0000040 data set. The following six steps all re quire the use of a Linux envi ronment. One must start Cygwin on their respective PC and type “tcsh” to work in the environment required. It will be necessary to change directories into the one containing the image processing codes. This image processing cookbook does not go into details of working in a Linux environment; one must become familiar with Linux commands. The next step converts the raw image data set to an image data set that we would like to use within our C programs. The C program that will be used is ConvertMicroCT.exe. This program requires that you call the program as well as define where the input (image) is and the output (image) will go. In case you cannot remember what is required to run the

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255 program, all that is necessary is the entering of the program executable file (.exe). Figure D-3 provides an example for executing this step In this example, one can see that the input image was located in a folder unde r “…My Documents/MicroCT/May Samples” and the output image will be placed in the curr ent folder with all the C programs (labeled as Humerus_Left.IGL. As shown in Figur e D-3, the program requests three input numbers – the I, J, and K dimensions. In this example, that would be 300 for I dimension, 324 for J dimension, and 532 for the K dimension. Figure D-3. Pictorial ex ample of how to use the ConvertMicroCT.exe program The next step is to take three 2-dimensiona l slices of the image data set in order to determine the ideal region of interest. In order to do this, one must run the Plane.exe program within the Linux environment. Figur e D-4 shows an example of how to do so. An image input is required to run the executable (.exe) file; the one th at should be used is the one that was converted in the last step. In the example here (Figure D-4), the C0000040_DATA image was converted and rena med to Humerus_Left.IGL and used with the Plane.exe program. The Plane.exe pr ogram will provide a slice in each of the I, J, and K dimensions at a give n voxel range. In Figure D-4, th e given value range starts at 150 and ends at 150, thus only 1 slice is show n for each dimension. If the range is set

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256 from 100 to 101, then 2 slices for each dimens ion is outputted (6 total slices). Choosing this range is important. One must choose a range in the approximate middle of the bone section. Generally it is safe to choose a slice in the middle of the smallest dimension, thus in this example 150 was chosen. This may not always work out, but later steps will be taken in those cases. The 3 output images will be placed in the directory that the C program (Plane.exe) resides in and are name d “_Kslice150.raw” with the value changing based on the number range chosen and the dime nsion letter given (i.e _Islice200.raw). Figure D-4. Pictorial example for th e execution of the Plane.exe program One point to note, the number range chosen should never exceed the value in any 1 dimension. For example in this case, the number range should not exceed 300. If necessary, one must edit the C code (to exclude that particular I, J, or K dimension) and re-compile the program. The next step requires the use of Adobe Photoshop. In Photoshop, one must open the raw images from each dimension. When the open command is used, a window will pop up asking for the dimensions of each slice. An example of this is shown in Figures D-5, D-8, and D-9. It is necessary to unders tand the dimensions of the image in order to

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257 correctly open each 2-dimensional slice. In Table D-1, one can see the breakdown of each slice for each dimension. Figure D-5. Example of the opening wi ndow for “_Kslice.raw” images in Adobe Photoshop Table D-1. Breakdown of dimensions nece ssary for opening images in Adobe Photoshop Photoshop window I Slice J Slice K Slice Width K dimension K dimension J dimension Height J dimension I dimension I dimension With each opening, a new window will open showing a single 2-dimensional image similar to one that resembles trabecula r spongiosa as shown in Figure D-6. J I

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258 Figure D-6. Example image of after opening _Kslice.raw in Adobe Photoshop. After opening the image in Photoshop, one shoul d hit “Ctrl +Shift + L” to auto-level the image so it will be easier to view, as shown in Figure D-7. The same process should be conducted for the “_Islice.raw” and the “_ Jslice.raw.” Table D-1 shows the proper dimensions for opening images in Adobe Photoshop for the I and J dimensions. Figures D-8 and D-9 show examples for those two other dimensions.

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259 Figure D-7. Example image of Figure D-8 after “auto-leveling” in Adobe Photoshop. Figure D-8. Example of the opening window for “_Jslice.raw” images in Photoshop K I

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260 Figure D-9. Example of the opening window for “_Islice.raw” images in Photoshop. After opening the other two images from the remaining two dimensions, one must again “auto-level” the two images for a clearer view of the trabecular spongiosa. Finally, the Adobe Photoshop screen should look simila r to Figure D-10 through Figure D-12. In these figures, one must determine the optimum region of interest (ROI). Determination of the ROI is a difficult pr ocess; one must be able to understand the 3D image received from microCT acquisitio n. Table D-2 shows an example of the simplest way to record the values for the determination of the ROI. To start, one must choose a dimension to begin with. In this ex ample, we will begin with the I dimension, or the upper right image in Figure D-10. We will begin by boxing an ROI using the “box” tool given in the toolbar. By looki ng at the “info” box, gi ven in Photoshop, the dimensions of x (width) and y (height) are given. The goal of this is to determine the “best” minimum and maximum value in the x (width) direction and in the y (height) K J

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261 direction. By looking at the I slice image, we can determine the optimum ROI in the J and K dimensions. As shown in Figure D-10, we can move the cursor in Photoshop and record the x and y values for the image. Tabl e D-1 shows that the x value is equal to the range in the K dimension or width and the y va lue is equal to the J dimension or height (only with the I slice image) Table D-2 shows the values for the ROI in the J and K dimensions for the I slice image. You should be gin to fill in a table similar to Table D-2. Figure D-10. Example of ROI determina tion with the I Slice in Photoshop. 30 width (x) 500 300 height (y) 30 Info Box

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262 Figure D-11. Example of ROI determina tion with the J Slice in Photoshop. The same process should be taken with the other two images. With the J slice image, Table D-1 shows that the x value is e qual to the range in the K dimension or width and the y value is equal to the I dimension or height (only with the J slice image). Figure D-11 shows this example with the Jslice.raw image (bottom left image). Table D-2 shows the values for the ROI in the I and K dimensions for the J slice image. Again, the same process should be taken with the last image. With the K slice image, Table D-1 shows that the x value is equal to the range in the J dimension or width and the y value is equal to the I dimension or height (only with the K slice image). Figure D-12 shows this 15 height (y) 250 25 width (x) 500 Info Box

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263 example with the Kslice.raw image (top left im age). Table D-2 shows the values for the ROI in the I and J dimensions for the K slice image. Figure D-12. Example of ROI determina tion with the K Slice in Photoshop. An important aspect to note is that if you choose a valu e for the three slice images in the I, J, and K dimensions (in this case it was 150) and the min or max range in I, J, and K do not include that value then you must repeat the Plane.exe process and choose a slice that is within the max and min range you ha ve chosen for the ROI. In this example, that is not the case, therefor e you can continue and determin e the final ROI for the three dimensions. Your table should be similar to Table D-2 without the final two rows. The 1 height (y) 251 25 width (x) 310 Info Box

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264 third row “ROI Range” is filled by choosi ng the higher number in the minimum range and the lower number in the maximum range for each dimension. Just as an example, to determine the range in the K dimension, one must take the minimum K from the I dimension (30) and the maximum K from the J dimension (500). The same must be done for the other two dimensions. The last row in Table D-2 is filled by the difference in the range for each dimension. For example, in the K dimension, the total dimension of the ROI in the K dimension is 470 (500 – 30 = 470). Table D-2. Example of the method for de termination of the region of interest. HEIGHT (y) = JWIDTH (x) = KHEIGHT (y) = IW IDTH (x) = KHEIGHT (y) = IWIDTH (x) = JMinimum30301525125Maximum300500250500251310ROI Range ROI Dimensions235270470I Dimension SliceJ Dimension SliceK Dimension Slice15 25030 30030 500 The new ROI is a single set of numbers, that will be calculated for each microCT image. For this example the ROI of interest is 15 – 250 for the I dimension, 30 – 300 for the J dimension, and 30 – 500 for the K di mension. Consequently, the new image dimension for the ROI will be 235 for I, 270 for J, and 470 for the K dimension. These values will be used in the next step – obt aining the gray-level histogram of the ROI within the microCT image. Figure D-13 shows an example of how to run the histogram.exe program. For the histogram.exe program, it is necessary to input the image file location, the new output file (.txt), the original dimensions for the image (300 for I, 324 for J, and 532 for K dimension in this example), and then input th e new ROI range in I, J, and K dimensions.

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265 Figure D-13. Pictorial example of the execution of the Histogram.exe program. After computing the gray-lev el histogram and outputting in to the “.txt” file, one must open the histogram file in Microsoft Exce l and graph the data. An example of this is shown in Figure D-14. Figure D-14. Pictorial example of gray-level histogram data plot in Microsoft Excel

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266 An important number to record is the peak value of the first peak in the gray-level histogram. In the example given below, the peak value is 141 and can be found by looking at the graph in Excel. This va lue will be used in later steps. After opening the histogram file into Mi crosoft Excel, one must copy the first two columns of data and paste in to the SigmaPlot 2001 program. On the other hand, one can open the “.txt” directly into SigmaPlot by importing the text and choosing to delimit by white space (shown in Figure D-15). Figure D-15. Example of how to im port a text file into SigmaPlot. After opening the file in SigmaPlot 2001, one must hit “F3” to start the graph wizard. In order to graph properly, one must choose “scatter plot”, then “simple scatter”, then “XY pair”, then choose “column 1” fo r “X”, “column 2” for “Y”, and finally hit “finish” for a suitable graph. The Si gmaPlot 2001 window will then look similar to Figure D-16 and the SigmaPlot graph will l ook exactly the same as the plot in the Microsoft Excel window, if not try again in SigmaPlot.

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267 Figure D-16. Screen capture of the Sigm aPlot window after plotting the histogram. Figure D-17. Regression wizard window displa ying the list for the cu rve-fitting equation.

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268 In order to fit the curve, one must star t the regression wizard by hitting “F5” and choosing “Gaussian Rayleigh” from the list (as shown in Figure D-17). After highlighting the option for “Gaussian Raylei gh,” hit the “Edit Code…” tab and a window will open as shown in Figure D-18. In th e “Initial Parameters” window, edit the “mu” value. In this example, the value was change d to 141 – the same value as the peak in the gray-level histogram as seen in the Micr osoft Excel plot (shown in Figure D-14). Figure D-18. Example window in the process for obtaining the curve-fitting parameters After changing the value for “mu,” h it “Run” and a new window should pop up labeled “Regression Wizard.” An example of this window is shown in Figure D-19. Record the values given in this window for th e four parameters: M, mu, sigM, and sigB. In the example given below the M = 0.7520, the mu = 141.2, the sigM = 3.132, and the sigB = 56.72. These values will be used in later steps.

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269 Figure D-19. Example of Regression Wizar d window providing the four parameters In order to determine the threshold calcula tion, one must use these four parameters. As predicted under the tissue classification m odel of Chung et al. in 1993, the distribution is bimodal and follows the Gaussian Rayleigh distribution. In that model, a nonlinear least-squares technique is used to fit the histogram from which an optimal image threshold is selected by numerical solution. In this work, image thresholding is performed automatically by one of the C codes provided that incorporates the results of the application of the tissue classification m odel. This C code was developed by methods incorporated in Derek Jokisch’ s dissertation. Figure D-20 pr ovides an example of how to execute the program to obtain the threshold va lue necessary to segment the 3D image. After executing the FindThresh.exe program and inputting the four parameters as requested, a list of values will be printed on the screen. As shown in Figure D-20, the value that is considered the threshold is the one that comes after the last positive value (or the first negative number in the second set of negative numbers). In this example, the threshold value is 152, as seen below.

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270 Figure D-20. Pictorial example of the execu tion of the FindThresh.exe program and its corresponding output used to dete rmine the optimum threshold. After determining the optimum threshold, the next three steps are simple. The ResizeCTimage.exe program was written to th reshold the image, resize the image to the region of interest chosen above, and swap the image tag values – the black (tag value = 0) voxels are bone and the white voxels (tag value = 255) are marrow space. Figure D-21 provides an example of this program ex ecution. First, the execution of the ResizeCTimage.exe requires the input of th e converted image and the naming of the new output image (in this case, we call it Hu merus_Left_ROI IGL). Also for the ResizeCTimage.exe it is necessary to first input the old image dimensions from the raw data obtained from microCT image acquisiti on (in this case 300 for I, 324 for J, and 532 for K); then input the calculated threshold value (in this example, T = 152); and then

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271 finally input the maximum and minimum range fo r the ROI in the I, J, and K dimensions (as decided above and shown in the fourth row of Table D-2). Figure D-21. Pictorial example of the ex ecution of the ResizeCTimage.exe program. One should note that the new image should have a file size equivalent to the multiplication of the three new dimensions (235 x 270 x 470 = 29,821,500) and can be checked by listing the files and l ooking at the attributes. The ne xt step is to median filter the image to reduce the “noise” in the 3D im age. This step requires the use of the MedianFilter.exe, the image i nput (in this case, Humerus_Le ft_ROI IGL), and the new image output (in this example, Humerus_Left .gray). An example of this execution is shown in Figure D-22. It is important to us e the new dimensions of the image (235, 270, and 470), not the old raw image dimensions (300, 324, and 532). Just as a note, one

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272 should notice that the labeling of the new im age as “.gray” is th e tag that defines a finished image. Figure D-22. Pictorial example of the execution of the MedianFilter.exe program. The image is now processed and ready for use in Paired Image Radiation Transport and for mass calculations. The final step is to get the important parameters of the final image processed bone section. The program ReadImage.exe provides all the important aspects of the image, such as: marrow volume fraction, voxel sides that contain endosteum, total number of bone voxels, marrow voxels, and if necessary – fat voxels. Figure D-23 provides an example of the ex ecution of the ReadImage.exe program. Figure D-23. Pictorial example of the ex ecution of the ReadImage.exe program.

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273 APPENDIX E PAIRED-IMAGE RADIATION TRANSPORT (PIRT) MODEL (EGSNRC USER CODE) This appendix contains the MORTRAN user c ode of the EGSnrc radiation code that has been developed to transport electrons within a pair of 3D images that define a single skeletal site. It uses to input data from (1) a pre-pro cessed microimage of skeletal spongiosa (microCT) and (2) binary contour data generated through segmentation of an ex-vivo CT scan. This appendix also shows examples of the following files used within this code: Configuration file Input file Output file EGSnrc MORTRAN User Code !INDENT M 4; "INDENT EACH MORTRAN NESTING LEVEL BY 4" !INDENT F 2; "INDENT EACH FORTRAN NESTING LEVEL BY 2" "This line is 80 characters long, use it to set up the screen width" "23456789|123456789|123456789|123456789|123456789|123456789|123456789|123456789" "******************************************************************************" Amish P. Shah " *********************** " * " PairedImage.mortran " * " *********************** " " This program calculates the absorbed fraction of energy within the bone " trabeculae, the marrow space components (inactive/active marrow), the bone " endosteum, and the surrounding cortical bone of a given bone site. " The geometry is defined by two images. An microimage is used to simulate an " infinite field of trabecular bone. " A CT image (lower resolution) is place over the trabecular bone region to " define the limits of the trabecular bone region via a cortical bone shell " with surrounding soft tissue. " Particles that goes outside the cortical shell are discarded. " The source is defined in any one of the regions that a target exists: bone " volume, active marrow, inactive marrow, cortical bone, and bone endosteum. " " Several things must be defined for each run: both images' configurations, " the source, the output file, the location of the input file, etc.. " " The PEGS file is: microimage (so no one forgets!) " the type of particle: -1 for electrons, 0 for photons " the initial energy of the particles " the number of histries per configuration. " The results are in the file Output.dat. " "******************************************************************************" "----------------------------------------" Step 1: To override the EGSnrc macros

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274 "----------------------------------------" 1) so that all real variables are in double precision REPLACE {$REAL} WITH {DOUBLE PRECISION} 2) the size of the arrays used by EGSnrc. REPLACE {$MXMED} WITH {3} "3 medium in the problem (default 10)" REPLACE {$MXREG} WITH {6} "6 geometric regions (default 2000)" REPLACE {$MXSTACK} WITH {100} "less than 100 particles on stack at once" REPLACE {$MXMDSH} WITH {100} "max. nb of shells per medium for "incoherent scattering" REPLACE {IFIX} WITH {INT} REPLACE {0.,} WITH {0.0D0,} 3) for compatibility with the old EGS4. REPLACE {$CALL-HOWNEAR(#);} WITH {CALL HOWNEAR({P1},X(NP),Y(NP),Z(NP),IRL);} "-------------------------------------------" Step 1.a. To define user constant values "-------------------------------------------" REPLACE {$REG_TRAB} WITH {1} region within the bone trabeculae REPLACE {$REG_MARR} WITH {2} region within the marrow cavities REPLACE {$REG_CORT} WITH {3} region within the cortical bone REPLACE {$REG_OUTSIDE} WITH {4} region outside the study REPLACE {$REG_LOST} WITH {5} region for lost particles REPLACE {$REG_FAT} WITH {6} region within the FAT REPLACE {$MED_BONE} WITH {0} to represent bone in MICRO image REPLACE {$MED_MARR} WITH {255} to represent marrow MICRO image REPLACE {$MED_FAT} WITH {122} to represent FAT in MICRO image REPLACE {$MED_SPONG} WITH {5} to represent spongiosa in CT image" REPLACE {$MED_CORT} WITH {45} to represent cort. bone in CT image" REPLACE {$MED_TISS} WITH {25} to represent softtissue in CT image" REPLACE {$IMAGE_FILE_MICRO} WITH {23} file to read the image REPLACE {$IMAGE_FILE_CT} WITH {23} file to read the image REPLACE {$INPUT_FILE} WITH {25} file to get the parameters REPLACE {$OUTPUT_FILE} WITH {26} file to record the results REPLACE {$N_RUN} WITH {100} number of run for each configuration" REPLACE {$INFINITY} WITH {1.0D99} to simulate infinity long distance REPLACE {$PI} WITH {3.1415926535897932D+00} need Pi in Source " this is to solve the boundary crossing problem. The particle is " transported a little farther than the exact boundary REPLACE {$BOUNDARY_THICKNESS} WITH {1.0D-09} that's 0.1 Angstrom " for the geometrical model "************************Change These Parameters*******************************" 1) the MICRO image REPLACE {$MICRO_VOXEL_SIZE_X} WITH {0.00600D+00} "microimage voxel res (cm) REPLACE {$MICRO_VOXEL_SIZE_Y} WITH {0.00600D+00} in cm REPLACE {$MICRO_VOXEL_SIZE_Z} WITH {0.00600D+00} in cm REPLACE {$MICRO_IMAGE_NX} WITH {610} # of voxels along (O,x)" REPLACE {$MICRO_IMAGE_NY} WITH {145} # of voxels along (O,y)" REPLACE {$MICRO_IMAGE_NZ} WITH {150} # of voxels along (O,z)" 2) the CT image "macroimage voxel res (cm) REPLACE {$CT_VOXEL_SIZE_X} WITH {0.0656250D+00} REPLACE {$CT_VOXEL_SIZE_Y} WITH {0.0656250D+00} in cm REPLACE {$CT_VOXEL_SIZE_Z} WITH {0.1000000D+00} in cm REPLACE {$CT_IMAGE_NX} WITH {512} # of voxels along (O,x)" REPLACE {$CT_IMAGE_NY} WITH {512} # of voxels along (O,y)" REPLACE {$CT_IMAGE_NZ} WITH {228} # of voxels along (O,z)" "**************************Change the Above Parameters************************" "-------------------------------------------------" Step 1.b. To define the user common variables

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275 "-------------------------------------------------" a) for scoring the results REPLACE {COMIN/SCOR/;} WITH {COMMON/SCOR/ CumulEnergyTrabeculae,CumulEnergyMarrow,CumulEnergyCortical, CumulEnergyOutside,CumulEnergyLost, CumulEnergyFat,CumulEnergyEndo; $REAL CumulEnergyTrabeculae; $REAL CumulEnergyMarrow; $REAL CumulEnergyCortical; $REAL CumulEnergyOutside; $REAL CumulEnergyLost; $REAL CumulEnergyFat; $REAL CumulEnergyEndo;} b) for the geometry REPLACE {COMIN/GEOM/;} WITH {COMMON/GEOM/MICROBoneImage, MICROBoneImage2, CTBoneImage; BYTE MICROBoneImage($MICRO_IMAGE_NZ $MICRO_IMAGE_NY $MICRO_IMAGE_NX); CHARACTER MICROBoneImage2($MICRO_IMAGE_NX,$MICRO_IMAGE_NY,$MICRO_IMAGE_NZ); BYTE CTBoneImage($CT_IMAGE_NZ $CT_IMAGE_NY $CT_IMAGE_NX);} "--------------------------------------------------------" Step 1.c. To define the variables of the main program "--------------------------------------------------------" $IMPLICIT-NONE; to make sure that all variables are declared " 1) all the common that you need in the main programm COMIN/BOUNDS,MEDIA,MISC,USEFUL,RANDOM,GEOM,SCOR/; The above expands into COMMON statements " BOUNDS contains ECUT and PCUT " MEDIA contains NMED and the array concerning media " MISC contains the medium per region and Rayleigh parameters " USEFUL contains electron rest mass " RANDOM contains the RANMAR parameters " GEOM passes info to HOWFAR and HOWNEAR routines " SCOR passes info to AUSGAB routine " 2) local variables of the main program $REAL XIN, YIN, ZIN; particle location (to give to SHOWER) $REAL UIN, VIN, WIN; particle direction (to give to SHOWER) $REAL EIN; particle energy (to give to SHOWER) $REAL WTIN; particle weight (to give to SHOWER) $INTEGER IQIN; particle type (to give to SHOWER) $INTEGER IRIN; particle region (to give to SHOWER) $INTEGER PartNo; particle # to loop for each particle $INTEGER RunNo; run number to loop for each run $INTEGER ConfigNo; configuration number to loop for each one LOGICAL NoMoreConfig; to test the end of the input file $INTEGER ParticleType; particle type got from the input file $REAL KineticEnergy; kinetic energy got from the input file $INTEGER NumberOfHistories; number of histories got from the input file $INTEGER ParticlePerRun; number of particles per run " for statistical results: mean, standard deviation, standard deviation " of the mean, 95% confidence interval, and 95% confidence error $REAL AFTrabeculae; $REAL MeanAFTrabeculae; $REAL StdDevAFTrabeculae; $REAL StdDevOfMeanAFTrabeculae; $REAL ConfIntOfMeanAFTrabeculae; $REAL ConfErrOfMeanAFTrabeculae; $REAL AFMarrow; $REAL MeanAFMarrow; $REAL StdDevAFMarrow; $REAL StdDevOfMeanAFMarrow; $REAL ConfIntOfMeanAFMarrow; $REAL ConfErrOfMeanAFMarrow; $REAL AFFat; $REAL MeanAFFat; $REAL StdDevAFFat; $REAL StdDevOfMeanAFFat; $REAL ConfIntOfMeanAFFat; $REAL ConfErrOfMeanAFFat; $REAL AFEndo; $REAL MeanAFEndo; $REAL StdDevAFEndo; $REAL StdDevOfMeanAFEndo;

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276 $REAL ConfIntOfMeanAFEndo; $REAL ConfErrOfMeanAFEndo; $REAL AFCortical; $REAL MeanAFCortical; $REAL StdDevAFCortical; $REAL StdDevOfMeanAFCortical; $REAL ConfIntOfMeanAFCortical; $REAL ConfErrOfMeanAFCortical; $REAL AFOutside; $REAL MeanAFOutside; $REAL StdDevAFOutside; $REAL StdDevOfMeanAFOutside; $REAL ConfIntOfMeanAFOutside; $REAL ConfErrOfMeanAFOutside; $REAL AFLost; $REAL MeanAFLost; $REAL StdDevAFLost; $REAL StdDevOfMeanAFLost; $REAL ConfIntOfMeanAFLost; $REAL ConfErrOfMeanAFLost; $INTEGER NumByte, NumX, NumY, NumZ; CHARACTER tmp; 3) system functions invoked in the main program $REAL DSQRT; INTRINSIC DSQRT; "----------------------------------------" Step 2. To initialize the EGSnrc data "----------------------------------------" 1) to place medium names in an array. " $S is a MORTRAN macro to expand strings CHARACTER*4 MEDARR(24,$MXMED); $INTEGER I, J; DATA MEDARR /$S'Bone',20*' ',$S'Marrow',18*' ',$S'Fat',21*' '/; NMED = $MXMED; "Set number of media." DO J = 1,$MXMED [ DO I=1,24 [ MEDIA(I,J) = MEDARR(I,J); ] this is to avoid a DATA STATEMENT for a variable in COMMON" NMED and DUNIT default to 1, i.e. one medium and we work in cm ] 2) to initialize the medium in each region MED($REG_TRAB) = 1; "cortical bone in the bone trabeculae" MED($REG_MARR) = 2; "bone marrow in the marrow cavities" MED($REG_FAT) = 3; "fat marrow in the marrow cavities" MED($REG_CORT) = 1; "cortical bone in the cortical shell" MED($REG_OUTSIDE) = 0; "vacuum outside the study region MED($REG_LOST) = 0; "vacuum if particles are lost (does not matter)" 3) to initialize the cutoff energy for both electrons and " photons in each region ECUT($REG_TRAB) = 0.005 + PRM; 5 keV + rest mass for electrons PCUT($REG_TRAB) = 0.001; 1 keV for photons ECUT($REG_MARR) = 0.005 + PRM; PCUT($REG_MARR) = 0.001; ECUT($REG_FAT) = 0.005 + PRM; PCUT($REG_FAT) = 0.001; ECUT($REG_CORT) = 0.005 + PRM; PCUT($REG_CORT) = 0.001; ECUT($REG_OUTSIDE) = 0.005 + PRM; PCUT($REG_OUTSIDE) = 0.001; ECUT($REG_LOST) = 0.005 + PRM; PCUT($REG_LOST) = 0.001; 4) to ask EGSnrc to treat the Rayleigh scattering in each region IRAYLR($REG_TRAB) = 1; IRAYLR($REG_MARR) = 1; IRAYLR($REG_FAT) = 1; IRAYLR($REG_CORT) = 1;

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277 IRAYLR($REG_OUTSIDE) = 1; IRAYLR($REG_LOST) = 1; 5) to initialize the random number generator IXX = 1; JXX = 1; seed # to initialize the random number series $RNG-INITIALIZATION; "---------------------------------------------------------------" Step 3. To pick up the cross sections precalculated by pegs4 "---------------------------------------------------------------" CALL HATCH; data file must be assigned to unit 12 PRINT *, 'End of HATCH'; "------------------------------------------" Step 3.a. To initialize the output file "------------------------------------------" "********Make Sure to Change the Path of the OUTPUT File in new directory******" OPEN ( UNIT=$OUTPUT_FILE, FILE='/c/users/Amish/egsnrc/PIRT2/Output.dat', STATUS='unknown' ); "********Make Sure to Change the Path of the OUTPUT File in new directory******" WRITE($OUTPUT_FILE, '(A,A)') 'Absorbed fractions for irradiation ', 'from various sources.'; "*************** Remove Comments from the Source that you Choose***************" WRITE($OUTPUT_FILE, '(A,A)') 'Absorbed fractions for irradiation ', 'from bone trabeculae.'; 'from active bone marrow.'; " 'from inactive bone marrow.'; " 'from trabecular bone surface.'; " 'from cortical bone.'; "*************** Remove Comments from the Source that you Choose***************" "---------------------------------------------" Step 3.b. To open and read the image files "---------------------------------------------" "********************Change the Input MicroIMAGE File Path*********************" OPEN(25, FILE='/c/users/Amish/RITman/MicroCTimages/PelvisIliumCT.60.gray', ACCESS='DIRECT',ERR=95,FORM='FORMATTED',RECL=1); GOTO 101; 95 PRINT *, 'error opening'; 101 PRINT *, 'ok opening MICRO/MicroCT image file'; "********************Change the Input MicroIMAGE File Path*********************" NumByte = 1; DO NumX=1, $MICRO_IMAGE_NX [ DO NumY=1, $MICRO_IMAGE_NY [ DO NumZ=1, $MICRO_IMAGE_NZ [ READ(25, '(A1)', REC=NumByte) tmp; MICROBoneImage2(NumX,NumY,NumZ)=tmp; NumByte = NumByte + 1; ] ] ] CLOSE (25); PRINT *, 'ok reading MICRO image file';

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278 "********************Change the Input MAcroIMAGE File Path*********************" OPEN($IMAGE_FILE_CT, FILE='/c/users/Amish/RITman/CTimages/Pelvis.con', ACCESS='DIRECT', FORM='UNFORMATTED', RECL=$CT_IMAGE_NZ*$CT_IMAGE_NY*$CT_IMAGE_NX); PRINT *, 'ok opening CT image file'; READ($IMAGE_FILE_CT, REC=1) CTBoneImage; CLOSE ( $IMAGE_FILE_CT ); "********************Change the Input MAcroIMAGE File Path*********************" "*********Make Sure to Change the Path of the Input File in new directory******" "-----------------------------------------------------" Step 3.c. For each configuration in the input file "-----------------------------------------------------" One execution is performed for each line of the input file OPEN ( UNIT=$INPUT_FILE, FILE='/c/users/Amish/egsnrc/PIRT2/Input.dat', STATUS='old' ); READ ( $INPUT_FILE, ); to skip the first line "*********Make Sure to Change the Path of the Input File in new directory******" NoMoreConfig = .FALSE.; ConfigNo = 0; LOOP [" until no more line in the file "-------------------------------------------------" Step 3.d. To read a new line in the input file "-------------------------------------------------" READ ( $INPUT_FILE, *, END = :EndInput: ) ParticleType, KineticEnergy, NumberOfHistories; GO TO :NextInput:; :EndInput: NoMoreConfig = .TRUE.; :NextInput: CONTINUE; "-----------------------------------------------------------------------" Step 3.e. If a new line exists, initialize the data for this config. "-----------------------------------------------------------------------" IF (~NoMoreConfig) [ 1) to display the new configuration ConfigNo = ConfigNo + 1; PRINT *, 'Configuration no:', ConfigNo; 2) how many particles per run? ParticlePerRun = NumberOfHistories / $N_RUN; 3) to output the parameters of the configuration WRITE($OUTPUT_FILE, '(A)') '; WRITE($OUTPUT_FILE, '(A,I3)') 'Configuration No:', ConfigNo; WRITE($OUTPUT_FILE, '(A)') 'The calculation is performed for:'; WRITE($OUTPUT_FILE, '(A,I5,A)') ', $N_RUN, runs'; IF (ParticleType = 0) [ WRITE($OUTPUT_FILE, '(A,I6,A)') ', ParticlePerRun, photons per run'; ] ELSE [ WRITE($OUTPUT_FILE, '(A,I6,A)') ', ParticlePerRun, electrons per run'; ] WRITE($OUTPUT_FILE, '(A,I8,A)') Total: ', ParticlePerRun*$N_RUN, histories.'; WRITE($OUTPUT_FILE, '(A,F7.3,A)') Initial kinetic energy: ', KineticEnergy, MeV.'; 4) to initialize the statistical data

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279 MeanAFTrabeculae = 0.0; MeanAFMarrow = 0.0; MeanAFFat = 0.0; MeanAFEndo = 0.0; MeanAFCortical = 0.0; MeanAFOutside = 0.0; MeanAFLost = 0.0; StdDevAFTrabeculae = 0.0; StdDevAFMarrow = 0.0; StdDevAFFat = 0.0; StdDevAFEndo = 0.0; StdDevAFCortical = 0.0; StdDevAFOutside = 0.0; StdDevAFLost = 0.0; "-------------------------" Step 3.f. For each run "-------------------------" DO RunNo=1,$N_RUN [ PRINT *, Run no:', RunNo; "------------------------------------------------------------" Step 4. To initialize the geometry for HOWFAR and HOWNEAR "------------------------------------------------------------" done when reading the input file "---------------------------------------------------------" Step 5. To initialize the scoring variables for AUSGAB "---------------------------------------------------------" CumulEnergyTrabeculae = 0.0; CumulEnergyMarrow = 0.0; CumulEnergyFat = 0.0; CumulEnergyEndo = 0.0; CumulEnergyCortical = 0.0; CumulEnergyOutside = 0.0; CumulEnergyLost = 0.0; "------------------------------" Step 5.a. For each particle "------------------------------" DO PartNo=1, ParticlePerRun [ to have a display of the progression of the code IF (MOD(PartNo,100) = 0) [ "PRINT *, Particle: ', PartNo;" ] "--------------------------------------------" Step 6. To define the particle parameters "--------------------------------------------" IF (ParticleType = 0) [ EIN = KineticEnergy; initial kinetic energy" ] ELSE [ EIN = KineticEnergy + PRM; initial kinetic + rest mass energy" ] IQIN=ParticleType; WTIN=1.0; weight = 1 since no variance reduction used" to get the initial location and direction of the particle. "************Remove Comments from the Source that you Choose*******************" CALL SourceBoneVolume(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); CALL SourceActiveMarrow(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceFatMarrow(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceBoneEndosteum(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceBoneSurface(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceCorticalBone(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); "************Remove Comments from the Source that you Choose*******************" "------------------------------------"

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280 Step 7. To transport the particle "------------------------------------" CALL SHOWER(IQIN,EIN,XIN,YIN,ZIN,UIN,VIN,WIN,IRIN,WTIN); ] "-------------------------------------------------------------" Step 7.a. To calculate and display the result for this run "-------------------------------------------------------------" AFTrabeculae = CumulEnergyTrabeculae / (ParticlePerRun KineticEnergy); AFMarrow = CumulEnergyMarrow / (ParticlePerRun KineticEnergy); AFFat = CumulEnergyFat / (ParticlePerRun KineticEnergy); AFEndo = CumulEnergyEndo / (ParticlePerRun KineticEnergy); AFCortical = CumulEnergyCortical / (ParticlePerRun KineticEnergy); AFOutside = CumulEnergyOutside / (ParticlePerRun KineticEnergy); AFLost = CumulEnergyLost / (ParticlePerRun KineticEnergy); "PRINT *, Data for this run:';" "PRINT *, AF in bone trabeculae: ',AFTrabeculae;" "PRINT *, AF in MARROW: ',AFMarrow;" "PRINT *, AF in fat: ',AFFat;" "PRINT *, AF in endo: ',AFEndo;" "PRINT *, AF in cortical: ',AFCortical;" "PRINT *, AF in outside: ',AFOutside;" "PRINT *, AF in lost: ',AFLost;" "PRINT *, Total in AF: ',AFTrabeculae + AFMarrow +" AFEndo + AFFat + AFCortical + AFOutside + AFLost;" "---------------------------------------------" Step 7.b. To cumulate the statistical data "---------------------------------------------" MeanAFTrabeculae = MeanAFTrabeculae + AFTrabeculae; MeanAFMarrow = MeanAFMarrow + AFMarrow; MeanAFFat = MeanAFFat + AFFat; MeanAFEndo = MeanAFEndo + AFEndo; MeanAFCortical = MeanAFCortical + AFCortical; MeanAFOutside = MeanAFOutside + AFOutside; MeanAFLost = MeanAFLost + AFLost; StdDevAFTrabeculae = StdDevAFTrabeculae + AFTrabeculae*AFTrabeculae; StdDevAFMarrow = StdDevAFMarrow + AFMarrow*AFMarrow; StdDevAFFat = StdDevAFFat + AFFat*AFFat; StdDevAFEndo = StdDevAFEndo + AFEndo*AFEndo; StdDevAFCortical = StdDevAFCortical + AFCortical*AFCortical; StdDevAFOutside = StdDevAFOutside + AFOutside*AFOutside; StdDevAFLost = StdDevAFLost + AFLost*AFLost; ] End of this run "----------------------------------------------" Step 7.c. To calculate the statistical data "----------------------------------------------" a) the mean MeanAFTrabeculae = MeanAFTrabeculae / $N_RUN; MeanAFMarrow = MeanAFMarrow / $N_RUN; MeanAFFat = MeanAFFat / $N_RUN; MeanAFEndo = MeanAFEndo / $N_RUN; MeanAFCortical = MeanAFCortical / $N_RUN; MeanAFOutside = MeanAFOutside / $N_RUN; MeanAFLost = MeanAFLost / $N_RUN; b) the standard deviation of the sample StdDevAFTrabeculae = StdDevAFTrabeculae $N_RUN*MeanAFTrabeculae*MeanAFTrabeculae; StdDevAFMarrow = StdDevAFMarrow $N_RUN*MeanAFMarrow*MeanAFMarrow; StdDevAFFat = StdDevAFFat $N_RUN*MeanAFFat*MeanAFFat; StdDevAFEndo = StdDevAFEndo $N_RUN*MeanAFEndo*MeanAFEndo; StdDevAFCortical = StdDevAFCortical $N_RUN*MeanAFCortical*MeanAFCortical; StdDevAFOutside = StdDevAFOutside $N_RUN*MeanAFOutside*MeanAFOutside; StdDevAFLost = StdDevAFLost $N_RUN*MeanAFLost*MeanAFLost; StdDevAFTrabeculae = StdDevAFTrabeculae / ($N_RUN 1); StdDevAFMarrow = StdDevAFMarrow / ($N_RUN 1); StdDevAFFat = StdDevAFFat / ($N_RUN 1);

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281 StdDevAFEndo = StdDevAFEndo / ($N_RUN 1); StdDevAFCortical = StdDevAFCortical / ($N_RUN 1); StdDevAFOutside = StdDevAFOutside / ($N_RUN 1); StdDevAFLost = StdDevAFLost / ($N_RUN 1); StdDevAFTrabeculae = DSQRT(StdDevAFTrabeculae); StdDevAFMarrow = DSQRT(StdDevAFMarrow); StdDevAFFat = DSQRT(StdDevAFFat); StdDevAFEndo = DSQRT(StdDevAFEndo); StdDevAFCortical = DSQRT(StdDevAFCortical); StdDevAFOutside = DSQRT(StdDevAFOutside); StdDevAFLost = DSQRT(StdDevAFLost); c) the standard deviation of the mean */ StdDevOfMeanAFTrabeculae = StdDevAFTrabeculae / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFMarrow = StdDevAFMarrow / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFFat = StdDevAFFat / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFEndo = StdDevAFEndo / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFCortical = StdDevAFCortical / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFOutside = StdDevAFOutside / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFLost = StdDevAFLost / DSQRT(DBLE($N_RUN)); d) the 95% confidence interval of the mean */ ConfIntOfMeanAFTrabeculae = 1.96*StdDevOfMeanAFTrabeculae; ConfIntOfMeanAFMarrow = 1.96*StdDevOfMeanAFMarrow; ConfIntOfMeanAFFat = 1.96*StdDevOfMeanAFFat; ConfIntOfMeanAFEndo = 1.96*StdDevOfMeanAFEndo; ConfIntOfMeanAFCortical = 1.96*StdDevOfMeanAFCortical; ConfIntOfMeanAFOutside = 1.96*StdDevOfMeanAFOutside; ConfIntOfMeanAFLost = 1.96*StdDevOfMeanAFLost; e) the 95% confidence error of the mean */ ConfErrOfMeanAFTrabeculae = 100.0 ConfIntOfMeanAFTrabeculae / MeanAFTrabeculae; ConfErrOfMeanAFMarrow = 100.0 ConfIntOfMeanAFMarrow / MeanAFMarrow; ConfErrOfMeanAFFat = 100.0 ConfIntOfMeanAFFat / MeanAFFat; ConfErrOfMeanAFEndo = 100.0 ConfIntOfMeanAFEndo / MeanAFEndo; ConfErrOfMeanAFCortical = 100.0 ConfIntOfMeanAFCortical / MeanAFCortical; ConfErrOfMeanAFOutside = 100.0 ConfIntOfMeanAFOutside / MeanAFOutside; ConfErrOfMeanAFLost = 100.0 ConfIntOfMeanAFLost / MeanAFLost; "------------------------------------------------------" Step 8. To print out the results to the output file "------------------------------------------------------" WRITE($OUTPUT_FILE, '(A,A)') Absorbed fractions with 95%', confidence intervals:'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in Trabeculae: ', MeanAFTrabeculae, +/', ConfIntOfMeanAFTrabeculae,' (', ConfErrOfMeanAFTrabeculae, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in Fat: ', MeanAFFat, +/', ConfIntOfMeanAFFat,' (', ConfErrOfMeanAFFat, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in Endo: ', MeanAFEndo, +/', ConfIntOfMeanAFEndo,' (', ConfErrOfMeanAFEndo, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in MARROW: ', MeanAFMarrow, +/', ConfIntOfMeanAFMarrow,' (', ConfErrOfMeanAFMarrow, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in corticalshell: ', MeanAFCortical, +/', ConfIntOfMeanAFCortical,' (', ConfErrOfMeanAFCortical, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in outside: ', MeanAFOutside, +/', ConfIntOfMeanAFOutside,' (', ConfErrOfMeanAFOutside, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF lost: ', MeanAFLost, +/', ConfIntOfMeanAFLost,' (', ConfErrOfMeanAFLost, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14)')

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282 Total AF: ', MeanAFTrabeculae + MeanAFMarrow + MeanAFCortical+ MeanAFEndo + MeanAFFat + MeanAFOutside + MeanAFLost; ] ] End of this configuration UNTIL (NoMoreConfig); "--------------------------------------------" Step 8.a. Don't forget to close the files "--------------------------------------------" CLOSE($INPUT_FILE); CLOSE($OUTPUT_FILE); END; End of main program "******************************************************************************" SourceBoneVolume "******************************************************************************" " The SourceBoneVolume subroutine returns a particle starting within the bone " regions of the image. The source is isotropic and uniform within the BONE ." The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceBoneVolume(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneVolume; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( InsideBoneVolume(XSrc, YSrc, ZSrc)) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction

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283 "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_TRAB; END; End of subroutine SourceBoneVolume "******************************************************************************" SourceActiveMarrow "******************************************************************************" " The SourceActiveMarrow subroutine returns particles starting within the " marrow regions of the MICRO image. The source is isotropic and uniform " within the Active Marrow. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceActiveMarrow(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideActiveMarrow; LOGICAL InsideBoneEndosteum; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( (InsideActiveMarrow(XSrc, YSrc, ZSrc)) .AND. (~InsideBoneEndosteum(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ]

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284 ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_MARR; END; End of subroutine SourceActiveMarrow "******************************************************************************" SourceFatMarrow "******************************************************************************" " The SourceFatMarrow subroutine returns particles starting within the " marrow regions of the MICRO image. The source is isotropic and uniform " within the Fat Marrow or InActive Marrow. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceFatMarrow(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideFatMarrow; LOGICAL InsideBoneEndosteum; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3;

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285 XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( (InsideFatMarrow(XSrc, YSrc, ZSrc)) .AND. (~InsideBoneEndosteum(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_FAT; END; End of subroutine SourceFatMarrow "******************************************************************************" SourceBoneEndosteum "******************************************************************************" " The SourceBoneEndosteum subroutine returns particles starting within the " endosteum regions of the image. The source is isotropic and uniform within" the 10 micron layer of the endosteum. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceBoneEndosteum(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSourceMarr; LOGICAL InsideSourceFat; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneEndosteum; LOGICAL InsideActiveMarrow; LOGICAL InsideFatMarrow; "-------------------------------------"

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286 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( (InsideActiveMarrow(XSrc, YSrc, ZSrc)) .AND. (InsideBoneEndosteum(XSrc, YSrc, ZSrc))) [ InsideSourceMarr = .TRUE.; ] ELSE [ InsideSourceMarr = .FALSE.; ] IF ( (InsideFatMarrow(XSrc, YSrc, ZSrc)) .AND. (InsideBoneEndosteum(XSrc, YSrc, ZSrc))) [ InsideSourceFat = .TRUE.; ] ELSE [ InsideSourceFat = .FALSE.; ] ] UNTIL ( (InsideSourceFat) .OR. (InsideSourceMarr) ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" IF (InsideSourceMarr) [ RegSrc = $REG_MARR; ] IF (InsideSourceFat) [ RegSrc = $REG_FAT; ] END; End of subroutine SourceBoneEndosteum "******************************************************************************" SourceBoneSurface "******************************************************************************" " The SourceBoneSurface subroutine returns particles starting within the " endosteum regions of the image. The source is isotropic and uniform within" the 0.1 micron layer of the endosteum. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceBoneSurface(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc);

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287 $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSourceMarr; LOGICAL InsideSourceFat; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneSurface; LOGICAL InsideActiveMarrow; LOGICAL InsideFatMarrow; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( (InsideActiveMarrow(XSrc, YSrc, ZSrc)) .AND. (InsideBoneSurface(XSrc, YSrc, ZSrc))) [ InsideSourceMarr = .TRUE.; ] ELSE [ InsideSourceMarr = .FALSE.; ] IF ( (InsideFatMarrow(XSrc, YSrc, ZSrc)) .AND. (InsideBoneSurface(XSrc, YSrc, ZSrc))) [ InsideSourceFat = .TRUE.; ] ELSE [ InsideSourceFat = .FALSE.; ] ] UNTIL ( (InsideSourceFat) .OR. (InsideSourceMarr) ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" IF (InsideSourceMarr) [ RegSrc = $REG_MARR;

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288 ] IF (InsideSourceFat) [ RegSrc = $REG_FAT; ] END; End of subroutine SourceBoneSurface "******************************************************************************" SourceCorticalBone "******************************************************************************" " The SourceCorticalBone subroutine returns particles starting within the " marrow regions of the MICRO image. The source is isotropic and uniform " within the Cortical Bone of the CT Image. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceCorticalBone(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideCorticalBone; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( InsideCorticalBone(XSrc, YSrc, ZSrc) ) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1);

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289 Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_CORT; END; End of subroutine SourceCorticalBone "******************************************************************************" HOWFAR "******************************************************************************" " The HOWFAR subroutine measures the distance between the location of the " particle (X0, Y0, Z0) and the next boundary crossed by the particle when " traveling to the direction (Up, Vp, Wp). " The returned values are: " IDISC is set to 1 if we need to discard the particle " USTEP is shortened if the boundary is reached by the particle " IRNEW is set with the region number that lies beyond the boundary " "******************************************************************************" SUBROUTINE HOWFAR; $IMPLICIT-NONE; to make sure that all variables are declared " COMMON variables COMIN/STACK,EPCONT/; The above expands into COMMON statements " STACK contains IR(NP), X,Y,Z(NP), and U,V,W(NP) " EPCONT contains USTEP: the distance EGSnrc is to transport the part. " local variables $REAL X0, Y0, Z0; the position of the particle $REAL Up, Vp, Wp; the direction of the particle $INTEGER IReg; the region number" $REAL Distance; the distance to the boundary $REAL XNew, YNew, ZNew; location of particle after current step " user functions invoked in the subroutine LOGICAL InsideBoneVolume; LOGICAL InsideBoneEndosteum; LOGICAL InsideActiveMarrow; LOGICAL InsideFatMarrow; LOGICAL InsideCorticalBone; $REAL BoundaryDistance; "--------------------------------" 1) To get the data from EGSnrc "--------------------------------" X0 = X(NP); Y0 = Y(NP); Z0 = Z(NP); Up = U(NP); Vp = V(NP); Wp = W(NP); IReg = IR(NP); "-----------------------------------------" 2) To check the data returned by EGSnrc "-----------------------------------------" if a mismatch is detected, the particle is discarded (IDISC=1) " IR(NP) is set to the region $REG_LOST so that AUSGAB can detect the " problem (IRNEW is not used by EGS since it does not transport the " particle before it calls AUSGAB) " a) to check the region numbers IF ( (IReg ~= $REG_TRAB) & (IReg ~= $REG_MARR) & (IReg ~= $REG_FAT) & (IReg ~= $REG_CORT) & (IReg ~= $REG_OUTSIDE) ) [ PRINT *, 'Error in HOWFAR: wrong region number: ', IReg; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] b) to check if the region number matches the location IF (IReg = $REG_TRAB) [ IF (~InsideBoneVolume(X0, Y0, Z0)) [ PRINT *, 'Error in HOWFAR: particle is not in bone.'; IDISC = 1;

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290 IR(NP) = $REG_LOST; RETURN; ] ] ELSEIF (IReg = $REG_MARR) [ IF (~InsideActiveMarrow(X0, Y0, Z0)) [ PRINT *, 'Error in HOWFAR: particle is not in marrow.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] ELSEIF (IReg = $REG_FAT) [ IF (~InsideFatMarrow(X0, Y0, Z0)) [ PRINT *, 'Error in HOWFAR: particle is not in fat.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] ELSEIF (IReg = $REG_CORT) [ IF (~InsideCorticalBone(X0, Y0, Z0)) [ PRINT *, 'Error in HOWFAR: particle is not in cortical.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] ELSE [ IF (InsideBoneVolume(X0, Y0, Z0) | InsideFatMarrow(X0, Y0, Z0) | InsideActiveMarrow(X0, Y0, Z0) | InsideCorticalBone(X0, Y0, Z0) ) [ PRINT *, 'Error in HOWFAR: particle is not outside.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] "----------------------------------------------------------------" 3) To discard the particle if it goes outside the study region "----------------------------------------------------------------" IF (IReg = $REG_OUTSIDE) [ IDISC = 1; ] ELSE [ "----------------------------------------------" 4) To calculate the distance to the boundary "----------------------------------------------" Distance = BoundaryDistance(X0, Y0, Z0, Up, Vp, Wp); "----------------------------------------------------------------------" 5) To make sure the particle jumps on the other side of the boundary "----------------------------------------------------------------------" Distance = Distance + $BOUNDARY_THICKNESS; "---------------------------------------------------" 6) To check if the distance is shorter than USTEP "---------------------------------------------------" IF ( Distance < USTEP ) [ USTEP = Distance; ] "------------------------------------------------" 7) To calculate the region beyond the boundary "------------------------------------------------" a) to calculate the new position XNew = X0 + USTEP*Up; YNew = Y0 + USTEP*Vp; ZNew = Z0 + USTEP*Wp; b) to calculate the new region IF (InsideBoneVolume(XNew, YNew, ZNew)) [ IRNEW = $REG_TRAB; ] ELSEIF ((InsideActiveMarrow(XNew, YNew, ZNew))) [

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291 IRNEW = $REG_MARR; ] ELSEIF (InsideCorticalBone(XNew, YNew, ZNew)) [ IRNEW = $REG_CORT; ] ELSEIF ((InsideFatMarrow(XNew, YNew, ZNew))) [ IRNEW = $REG_FAT; ] ELSE [ IRNEW = $REG_OUTSIDE; ] ] END; End of subroutine HOWFAR "******************************************************************************" HOWNEAR "******************************************************************************" " The HOWNEAR subroutine measures the shortest distance between the location " of the particle (X0, Y0, Z0) and the boundary of the actual region IReg. " The returned values are: " TPerp is the shortest distance from the particle location to " the boundary of the region IReg " "******************************************************************************" SUBROUTINE HOWNEAR(TPerp, X0, Y0, Z0, IReg); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL TPerp; the shortest distance to return $REAL X0, Y0, Z0; the current location of the particle $INTEGER IReg; the current region of the particle " user functions invoked in the subroutine $REAL ClosestBoundary; "-----------------------------------------------------" 1) To check if the particle has become out of study "-----------------------------------------------------" IF (IReg = $REG_OUTSIDE) [ TPerp = 0.0; so that HOWFAR is called and discard the particle ] ELSE [ "---------------------------------------" 2) To calculate the shortest distance "---------------------------------------" TPerp = ClosestBoundary(X0, Y0, Z0); "--------------------------------------------------------------------" 3) To make sure the particle will not be too close to the boundary "--------------------------------------------------------------------" TPerp = TPerp $BOUNDARY_THICKNESS; IF (TPerp < 0.0) [ TPerp = 0.0; ] ] END; End of subroutine HOWNEAR "******************************************************************************" AUSGAB "******************************************************************************" " The AUSGAB subroutine cumulates the energy deposited within the regions. " The energy is stored in the 'CumulEnergy' variables. " " Input: " IARG : A flag (see EGSnrc documentation) which is set to 3 if the " particle is discarded by the HOWFAR subroutine, in our " situation, that means that the particle is going outside " the study region or that it has been lost. " "******************************************************************************"

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292 SUBROUTINE AUSGAB(IARG); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $INTEGER IARG; $REAL X0, Y0, Z0; COMMON variables COMIN/STACK,EPCONT,SCOR/; The above expands into COMMON statements " STACK contains IR(NP) " EPCONT contains EDEP: the energy deposited now " SCOR contains the variables to cumulate the energy deposited " local variables $INTEGER IReg; to store the region number" LOGICAL InsideBoneEndosteum; "--------------------------------" 1) To get the data from EGSnrc "--------------------------------" IReg = IR(NP); X0 = X(NP); Y0 = Y(NP); Z0 = Z(NP); "---------------------------------------------------------" 2) To test if the particle has been discarded by HOWFAR "---------------------------------------------------------" IF (IARG = 3) [ test why it has been discarded IF (IReg = $REG_OUTSIDE) [ CumulEnergyOutside = CumulEnergyOutside + EDEP; ] ELSEIF (IReg = $REG_LOST) [ CumulEnergyLost = CumulEnergyLost + EDEP; ] ELSE [ PRINT *, 'Error in AUSGAB: wrong region number after discard.'; RETURN; ] ] ELSE [ "-----------------------------------------------" 3) To cumulate the energy in the right region "-----------------------------------------------" IF (IReg = $REG_TRAB) [ CumulEnergyTrabeculae = CumulEnergyTrabeculae + EDEP; ] ELSEIF (IReg = $REG_MARR) [ IF (InsideBoneEndosteum(X0, Y0, Z0)) [ CumulEnergyEndo = CumulEnergyEndo + EDEP; ] ELSE [ CumulEnergyMarrow = CumulEnergyMarrow + EDEP; ] ] ELSEIF (IReg = $REG_CORT) [ CumulEnergyCortical = CumulEnergyCortical + EDEP; ] ELSEIF (IReg = $REG_FAT) [ IF (InsideBoneEndosteum(X0, Y0, Z0)) [ CumulEnergyEndo = CumulEnergyEndo + EDEP; ] ELSE [ CumulEnergyFat = CumulEnergyFat + EDEP; ] ] ELSE [ PRINT *, 'Error in AUSGAB: wrong region number after transport.'; RETURN; ] ] END; End of subroutine AUSGAB

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293 "******************************************************************************" Function InsideBoneVolume "******************************************************************************" " Test if a given position (X, Y, Z) is inside the trabeculae voxels of the " duplicated MICRO image and in the spongiosa region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideBoneVolume(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; the voxel itself " system functions invoked in the main program $INTEGER MOD; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideBoneVolume = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideBoneVolume = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the MICRO image "---------------------------------------------------" I = (X / $MICRO_VOXEL_SIZE_X); I = MOD(I, $MICRO_IMAGE_NX); to shift to the copy of the image J = (Y / $MICRO_VOXEL_SIZE_Y); J = MOD(J, $MICRO_IMAGE_NY); to shift to the copy of the image K = (Z / $MICRO_VOXEL_SIZE_Z); K = MOD(K, $MICRO_IMAGE_NZ); to shift to the copy of the image VoxelNo = (K*$MICRO_IMAGE_NY + J)*$MICRO_IMAGE_NX + I + 1; "--------------------------------"

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294 5) to get and test the medium "--------------------------------" VoxelValue2 = MICROBoneImage2(I,J,K); IF (VoxelValue2 = CHAR(0)) [ InsideBoneVolume = .TRUE.; ] ELSE [ InsideBoneVolume = .FALSE.; ] ] ] END; End of function InsideBoneVolume "******************************************************************************" Function InsideActiveMarrow "******************************************************************************" " Test if a given position (X, Y, Z) is inside the active marrow voxels of " the duplicated MICRO image and in the spongiosa region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideActiveMarrow(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image CHARACTER VoxelValue2; the voxel itself $INTEGER VoxelValue; the voxel itself " system functions invoked in the main program $INTEGER MOD; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; LOGICAL InsideBoneEndosteum; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideActiveMarrow = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [

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295 InsideActiveMarrow = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the MICRO image "---------------------------------------------------" I = (X / $MICRO_VOXEL_SIZE_X); I = MOD(I, $MICRO_IMAGE_NX); to shift to the copy of the image J = (Y / $MICRO_VOXEL_SIZE_Y); J = MOD(J, $MICRO_IMAGE_NY); to shift to the copy of the image K = (Z / $MICRO_VOXEL_SIZE_Z); K = MOD(K, $MICRO_IMAGE_NZ); to shift to the copy of the image VoxelNo = (K*$MICRO_IMAGE_NY + J)*$MICRO_IMAGE_NX + I + 1; "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = MICROBoneImage2(I,J,K); IF (VoxelValue2 = CHAR(255)) [ InsideActiveMarrow = .TRUE.; ] ELSE [ InsideActiveMarrow = .FALSE.; ] ] ] END; End of function InsideActiveMarrow "******************************************************************************" Function InsideBoneEndosteum "******************************************************************************" " Test if a given position (X, Y, Z) is inside the bone endosteum of the " marrow cavity voxels of the duplicated MICRO image and in the spongiosa " region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideBoneEndosteum(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER I2, J2, K2; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; the voxel itself " system functions invoked in the main program $INTEGER MOD; CHARACTER EDGEIPOS,EDGEINEG,EDGEJPOS,EDGEJNEG,EDGEKPOS,EDGEKNEG; $REAL P1, P2, P3, P4, P5, P6, PDIST; $REAL XMax, XMin, YMax, YMin, ZMax, ZMin; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [

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296 InsideBoneEndosteum = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideBoneEndosteum = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the MICRO image "---------------------------------------------------" I = (X / $MICRO_VOXEL_SIZE_X); I = MOD(I, $MICRO_IMAGE_NX); to shift to the copy of the image J = (Y / $MICRO_VOXEL_SIZE_Y); J = MOD(J, $MICRO_IMAGE_NY); to shift to the copy of the image K = (Z / $MICRO_VOXEL_SIZE_Z); K = MOD(K, $MICRO_IMAGE_NZ); to shift to the copy of the image VoxelNo = (K*$MICRO_IMAGE_NY + J)*$MICRO_IMAGE_NX + I + 1; "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = MICROBoneImage2(I,J,K); IF ((VoxelValue2 = CHAR($MED_MARR)) .OR. (VoxelValue2 = CHAR($MED_FAT)))[ P1 = 1.0; P2 = 1.0; P3 = 1.0; P4 = 1.0; P5 = 1.0; P6 = 1.0; EDGEIPOS = CHAR(255); EDGEINEG = CHAR(255); EDGEJPOS = CHAR(255); EDGEJNEG = CHAR(255); EDGEKPOS = CHAR(255); EDGEKNEG = CHAR(255); "CHECK FOR BONE VOXEL NEIGHBORS" "DETERMINE WHERE BONE SURFACES ARE(IF THEY ARE)" IF (I .EQ. ($MICRO_IMAGE_NX)) [ EDGEIPOS = MICROBoneImage2(1,J,K) ; ] ELSE [ EDGEIPOS = MICROBoneImage2(I+1,J,K); ] IF (I .EQ. (1)) [ EDGEINEG = MICROBoneImage2($MICRO_IMAGE_NX,J,K) ; ] ELSE [ EDGEINEG = MICROBoneImage2(I-1,J,K) ; ] IF (J .EQ. ($MICRO_IMAGE_NY)) [ EDGEJPOS = MICROBoneImage2(I,1,K) ; ] ELSE [

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297 EDGEJPOS = MICROBoneImage2(I,J+1,K) ; ] IF (J .EQ. (1)) [ EDGEJNEG = MICROBoneImage2(I,$MICRO_IMAGE_NY,K) ; ] ELSE [ EDGEJNEG = MICROBoneImage2(I,J-1,K) ; ] IF (K .EQ. ($MICRO_IMAGE_NZ)) [ EDGEKPOS = MICROBoneImage2(I,J,1) ; ] ELSE [ EDGEKPOS = MICROBoneImage2(I,J,K+1) ; ] IF (K .EQ. (1)) [ EDGEKNEG = MICROBoneImage2(I,J,$MICRO_IMAGE_NZ) ; ] ELSE [ EDGEKNEG = MICROBoneImage2(I,J,K-1) ; ] I2 = (X / $MICRO_VOXEL_SIZE_X); J2 = (Y / $MICRO_VOXEL_SIZE_Y); K2 = (Z / $MICRO_VOXEL_SIZE_Z); XMin = (I2) $MICRO_VOXEL_SIZE_X; XMax = XMin + $MICRO_VOXEL_SIZE_X; YMin = (J2) $MICRO_VOXEL_SIZE_Y; YMax = YMin + $MICRO_VOXEL_SIZE_Y; ZMin = (K2) $MICRO_VOXEL_SIZE_Z; ZMax = ZMin + $MICRO_VOXEL_SIZE_Z; IF(EDGEIPOS .EQ. CHAR(0)) [ P1= XMax X; ] IF(EDGEINEG .EQ. CHAR(0)) [ P2= X XMin; ] IF(EDGEJPOS .EQ. CHAR(0)) [ P3= YMax Y; ] IF(EDGEJNEG .EQ. CHAR(0)) [ P4= Y YMin; ] IF(EDGEKPOS .EQ. CHAR(0)) [ P5= ZMax Z; ] IF(EDGEKNEG .EQ. CHAR(0)) [ P6 = Z ZMin; ] PDIST=10.0; IF (P1 .LE. PDIST) [ PDIST = P1; ] IF (P2 .LE. PDIST) [ PDIST = P2; ] IF (P3 .LE. PDIST) [ PDIST = P3; ] IF (P4 .LE. PDIST) [ PDIST = P4; ] IF (P5 .LE. PDIST) [ PDIST = P5; ] IF (P6 .LE. PDIST) [ PDIST = P6;

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298 ] IF (PDIST .LT. 0.0) [ "PRINT *, ERROR IN PDIST'; "PRINT *, PDIST ', PDIST; ] InsideBoneEndosteum = .FALSE.; IF (PDIST .LE. 0.0010) [ InsideBoneEndosteum = .TRUE.; ] ] ELSE [ InsideBoneEndosteum = .FALSE.; ] ] ] END; End of function InsideBoneEndosteum "******************************************************************************" Function InsideBoneSurface "******************************************************************************" " Test if a given position (X, Y, Z) is inside the bone endosteum of the " marrow cavity voxels of the duplicated MICRO image and in the spongiosa " region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideBoneSurface(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER I2, J2, K2; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; the voxel itself " system functions invoked in the main program $INTEGER MOD; CHARACTER EDGEIPOS,EDGEINEG,EDGEJPOS,EDGEJNEG,EDGEKPOS,EDGEKNEG; $REAL P1, P2, P3, P4, P5, P6, PDIST; $REAL XMax, XMin, YMax, YMin, ZMax, ZMin; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideBoneSurface = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------"

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299 I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideBoneSurface = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the MICRO image "---------------------------------------------------" I = (X / $MICRO_VOXEL_SIZE_X); I = MOD(I, $MICRO_IMAGE_NX); to shift to the copy of the image J = (Y / $MICRO_VOXEL_SIZE_Y); J = MOD(J, $MICRO_IMAGE_NY); to shift to the copy of the image K = (Z / $MICRO_VOXEL_SIZE_Z); K = MOD(K, $MICRO_IMAGE_NZ); to shift to the copy of the image VoxelNo = (K*$MICRO_IMAGE_NY + J)*$MICRO_IMAGE_NX + I + 1; "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = MICROBoneImage2(I,J,K); IF ((VoxelValue2 = CHAR($MED_MARR)) .OR. (VoxelValue2 = CHAR($MED_FAT)))[ P1 = 1.0; P2 = 1.0; P3 = 1.0; P4 = 1.0; P5 = 1.0; P6 = 1.0; EDGEIPOS = CHAR(255); EDGEINEG = CHAR(255); EDGEJPOS = CHAR(255); EDGEJNEG = CHAR(255); EDGEKPOS = CHAR(255); EDGEKNEG = CHAR(255); "CHECK FOR BONE VOXEL NEIGHBORS" "DETERMINE WHERE BONE SURFACES ARE(IF THEY ARE)" IF (I .EQ. ($MICRO_IMAGE_NX)) [ EDGEIPOS = MICROBoneImage2(1,J,K) ; ] ELSE [ EDGEIPOS = MICROBoneImage2(I+1,J,K); ] IF (I .EQ. (1)) [ EDGEINEG = MICROBoneImage2($MICRO_IMAGE_NX,J,K) ; ] ELSE [ EDGEINEG = MICROBoneImage2(I-1,J,K) ; ] IF (J .EQ. ($MICRO_IMAGE_NY)) [ EDGEJPOS = MICROBoneImage2(I,1,K) ; ] ELSE [ EDGEJPOS = MICROBoneImage2(I,J+1,K) ; ] IF (J .EQ. (1)) [ EDGEJNEG = MICROBoneImage2(I,$MICRO_IMAGE_NY,K) ; ] ELSE [

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300 EDGEJNEG = MICROBoneImage2(I,J-1,K) ; ] IF (K .EQ. ($MICRO_IMAGE_NZ)) [ EDGEKPOS = MICROBoneImage2(I,J,1) ; ] ELSE [ EDGEKPOS = MICROBoneImage2(I,J,K+1) ; ] IF (K .EQ. (1)) [ EDGEKNEG = MICROBoneImage2(I,J,$MICRO_IMAGE_NZ) ; ] ELSE [ EDGEKNEG = MICROBoneImage2(I,J,K-1) ; ] I2 = (X / $MICRO_VOXEL_SIZE_X); J2 = (Y / $MICRO_VOXEL_SIZE_Y); K2 = (Z / $MICRO_VOXEL_SIZE_Z); XMin = (I2) $MICRO_VOXEL_SIZE_X; XMax = XMin + $MICRO_VOXEL_SIZE_X; YMin = (J2) $MICRO_VOXEL_SIZE_Y; YMax = YMin + $MICRO_VOXEL_SIZE_Y; ZMin = (K2) $MICRO_VOXEL_SIZE_Z; ZMax = ZMin + $MICRO_VOXEL_SIZE_Z; IF(EDGEIPOS .EQ. CHAR(0)) [ P1= XMax X; ] IF(EDGEINEG .EQ. CHAR(0)) [ P2= X XMin; ] IF(EDGEJPOS .EQ. CHAR(0)) [ P3= YMax Y; ] IF(EDGEJNEG .EQ. CHAR(0)) [ P4= Y YMin; ] IF(EDGEKPOS .EQ. CHAR(0)) [ P5= ZMax Z; ] IF(EDGEKNEG .EQ. CHAR(0)) [ P6 = Z ZMin; ] PDIST=10.0; IF (P1 .LE. PDIST) [ PDIST = P1; ] IF (P2 .LE. PDIST) [ PDIST = P2; ] IF (P3 .LE. PDIST) [ PDIST = P3; ] IF (P4 .LE. PDIST) [ PDIST = P4; ] IF (P5 .LE. PDIST) [ PDIST = P5; ] IF (P6 .LE. PDIST) [ PDIST = P6; ] IF (PDIST .LT. 0.0) [ "PRINT *, ERROR IN PDIST'; "PRINT *, PDIST ', PDIST; ]

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301 InsideBoneSurface = .FALSE.; "****** This is a 0.1 micron volume source for Surface Sources **************" IF (PDIST .LE. 0.000010) [ InsideBoneSurface = .TRUE.; ] ] ELSE [ InsideBoneSurface = .FALSE.; ] ] ] END; End of function InsideBoneSurface "******************************************************************************" Function InsideFatMarrow "******************************************************************************" " Test if a given position (X, Y, Z) is inside the Fat voxels of the " duplicated MICRO image and in the spongiosa region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideFatMarrow(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; the voxel itself " system functions invoked in the main program $INTEGER MOD; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; LOGICAL InsideBoneEndosteum; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideFatMarrow = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo);

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302 IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideFatMarrow = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the MICRO image "---------------------------------------------------" I = (X / $MICRO_VOXEL_SIZE_X); I = MOD(I, $MICRO_IMAGE_NX); to shift to the copy of the image J = (Y / $MICRO_VOXEL_SIZE_Y); J = MOD(J, $MICRO_IMAGE_NY); to shift to the copy of the image K = (Z / $MICRO_VOXEL_SIZE_Z); K = MOD(K, $MICRO_IMAGE_NZ); to shift to the copy of the image VoxelNo = (K*$MICRO_IMAGE_NY + J)*$MICRO_IMAGE_NX + I + 1; "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = MICROBoneImage2(I,J,K); IF ((VoxelValue2 = CHAR($MED_FAT))) [ InsideFatMarrow = .TRUE.; ] ELSE [ InsideFatMarrow = .FALSE.; ] ] ] END; End of function InsideFatMarrow "******************************************************************************" Function InsideCorticalBone "******************************************************************************" " Test if a given position (X, Y, Z) is inside the cortical region of the " image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideCorticalBone(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; the voxel itself " user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideCorticalBone = .FALSE.; ] ELSE [

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303 "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideCorticalBone = .TRUE.; ] ELSE [ InsideCorticalBone = .FALSE.; ] ] END; End of function InsideCorticalBone "******************************************************************************" Function InsideCT_CortShell "******************************************************************************" " Test if a given position (X, Y, Z) is inside the limits of the CT image " The outer limit of the CT image is 512 x 512 " Also, test if the given position is in the ROI within the CT image " ROI defined by everything within outside edge of CorticalBone (not tissue) " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the CT image. " .FALSE. if the position is not inside the CT image. " "******************************************************************************" LOGICAL FUNCTION InsideCT_CortShell(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself "-------------------------------------" 1) to check if outside the CT image "-------------------------------------" IF ( (X < 0.0) | (X >= $CT_IMAGE_NX $CT_VOXEL_SIZE_X) | (Y < 0.0) | (Y >= $CT_IMAGE_NY $CT_VOXEL_SIZE_Y) | (Z < 0.0) | (Z >= $CT_IMAGE_NZ $CT_VOXEL_SIZE_Z) ) [ InsideCT_CortShell = .FALSE.; ] ELSE [ "-----------------------------------------------------" 2) to check if in the tissue region of the CT image "-----------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; VoxelValue = CTBoneImage(VoxelNo);

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304 IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_TISS) [ InsideCT_CortShell = .FALSE.; ] ELSE [ InsideCT_CortShell = .TRUE.; ] ] END; End of function InsideCT_CortShell "******************************************************************************" Function BoundaryDistance "******************************************************************************" " Returns the distance from the position (X, Y, Z) to the nearest boundary " of the voxel when following the direction (U, V, W) " The two images are tested and the closest voxel limit is returned. " " Input: " X, Y, Z: the position to be tested. " U, V, W: the direction to follow. " " Return: " the distance to the boundary. " "******************************************************************************" $REAL FUNCTION BoundaryDistance(X, Y, Z, U, V, W); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; $REAL U, V, W; local variables $REAL Distance; $REAL ShortestDistance; $INTEGER I, J, K; to store the position of the voxel $REAL XMin, YMin, ZMin; for the boundary of the voxel $REAL XMax, YMax, ZMax; for the boundary of the voxel "-------------------------------------------------------------------" 1) to calculate the boundary of the current voxel in the CT image "-------------------------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); XMin = I $CT_VOXEL_SIZE_X; XMax = XMin + $CT_VOXEL_SIZE_X; YMin = J $CT_VOXEL_SIZE_Y; YMax = YMin + $CT_VOXEL_SIZE_Y; ZMin = K $CT_VOXEL_SIZE_Z; ZMax = ZMin + $CT_VOXEL_SIZE_Z; "---------------------------------------------------------" 2) to measure the distance to the boundary of the voxel "---------------------------------------------------------" ShortestDistance = $INFINITY; a) along the X axis IF ( U > 0.0 ) [ Distance = (XMax X) / U; ] ELSEIF ( U < 0.0 ) [ Distance = (XMin X) / U; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] b) along the Y axis

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305 IF ( V > 0.0 ) [ Distance = (YMax Y) / V; ] ELSEIF ( V < 0.0 ) [ Distance = (YMin Y) / V; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] c) along the Z axis IF ( W > 0.0 ) [ Distance = (ZMax Z) / W; ] ELSEIF ( W < 0.0 ) [ Distance = (ZMin Z) / W; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] "--------------------------------------------------------------------" 3) to calculate the boundary of the current voxel in the MICROimage" "--------------------------------------------------------------------" I = (X / $MICRO_VOXEL_SIZE_X); J = (Y / $MICRO_VOXEL_SIZE_Y); K = (Z / $MICRO_VOXEL_SIZE_Z); XMin = I $MICRO_VOXEL_SIZE_X; XMax = XMin + $MICRO_VOXEL_SIZE_X; YMin = J $MICRO_VOXEL_SIZE_Y; YMax = YMin + $MICRO_VOXEL_SIZE_Y; ZMin = K $MICRO_VOXEL_SIZE_Z; ZMax = ZMin + $MICRO_VOXEL_SIZE_Z; "---------------------------------------------------------" 4) to measure the distance to the boundary of the voxel "---------------------------------------------------------" a) along the X axis IF ( U > 0.0 ) [ Distance = (XMax X) / U; ] ELSEIF ( U < 0.0 ) [ Distance = (XMin X) / U; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] b) along the Y axis IF ( V > 0.0 ) [ Distance = (YMax Y) / V; ] ELSEIF ( V < 0.0 ) [ Distance = (YMin Y) / V; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] c) along the Z axis IF ( W > 0.0 ) [ Distance = (ZMax Z) / W; ] ELSEIF ( W < 0.0 ) [

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306 Distance = (ZMin Z) / W; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] "---------------------------" 5) to return the distance "---------------------------" BoundaryDistance = ShortestDistance; END; End of function BoundaryDistance "******************************************************************************" Function ClosestBoundary "******************************************************************************" " Returns the shortest distance from the position (X, Y, Z) to the nearest " boundary of the voxel. " The two images are tested and the closest voxel limit is returned. " " Input: " X, Y, Z: the position to be tested. " " Return: " the shortest distance to the boundary. " "******************************************************************************" $REAL FUNCTION ClosestBoundary(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables " local variables $REAL Distance; $REAL ShortestDistance; $INTEGER I, J, K; to store the position of the voxel $REAL XMin, YMin, ZMin; for the boundary of the voxel $REAL XMax, YMax, ZMax; for the boundary of the voxel "-------------------------------------------------------------------" 1) to calculate the boundary of the current voxel in the CT image "-------------------------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); XMin = I $CT_VOXEL_SIZE_X; XMax = XMin + $CT_VOXEL_SIZE_X; YMin = J $CT_VOXEL_SIZE_Y; YMax = YMin + $CT_VOXEL_SIZE_Y; ZMin = K $CT_VOXEL_SIZE_Z; ZMax = ZMin + $CT_VOXEL_SIZE_Z; "---------------------------------------------------------" 2) to measure the distance to the boundary of the voxel "---------------------------------------------------------" ShortestDistance = $INFINITY; Distance = X XMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = XMax X; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = Y YMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = YMax Y;

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307 IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = Z ZMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = ZMax Z; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] "--------------------------------------------------------------------" 3) to calculate the boundary of the current voxel in the MICROimage" "--------------------------------------------------------------------" I = (X / $MICRO_VOXEL_SIZE_X); J = (Y / $MICRO_VOXEL_SIZE_Y); K = (Z / $MICRO_VOXEL_SIZE_Z); XMin = I $MICRO_VOXEL_SIZE_X; XMax = XMin + $MICRO_VOXEL_SIZE_X; YMin = J $MICRO_VOXEL_SIZE_Y; YMax = YMin + $MICRO_VOXEL_SIZE_Y; ZMin = K $MICRO_VOXEL_SIZE_Z; ZMax = ZMin + $MICRO_VOXEL_SIZE_Z; "---------------------------------------------------------" 4) to measure the distance to the boundary of the voxel "---------------------------------------------------------" Distance = X XMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = XMax X; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = Y YMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = YMax Y; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = Z ZMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = ZMax Z; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] "---------------------------" 5) to return the distance "---------------------------" ClosestBoundary = ShortestDistance; END; End of function ClosestBoundary

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308 Configuration File #!/bin/csh # standard.configuration (SID 1.6 last edited 00/02/27) # # # When an EGSnrc code is compiled, the egs_compile script # calls this file to create the overall source code by # concatinating different files in a specified order. Generally # user codes will require extensions to this bare bones configuration # file. See the examples for dosrznrc etc. # If no configuration file is present on the user-code directory # (i.e. $HOME/egsnrc/user_code) then this file is used. # # Note that order IS IMPORTANT since the last definition of a macro # is the one that is used. # # catecho is a simple little script to concatonate the named file # and echo things to the terminal depending on whether EGS_PERT # is set or not. # # The EGSnrc system has been structured to work with either the # RANLUX or the RANMAR random number generators. To switch # which rng to use, comment out the one not wanted (in 2 places). # The only difference to the user is that RANLUX requires a # luxury level (0 to 4) plus an initial seed (any positive integer) # whereas RANMAR needs two initial seeds between 1 and roughly 30,000. # # This is part of the EGSnrc Code System # Copyright NRC 2000 echo "Entering $HEN_HOUSE/standard.configuration (SID 1.6) echo "---------------------------------------------------------------------" echo " echo Using machine: $my_machine" echo " echo "%L" > .mortjob.mortran # Mortran switch to turn listing on $HEN_HOUSE/catecho "$HEN_HOUSE/egsnrc.macros "egsnrc standard macros" $HEN_HOUSE/catecho "$HEN_HOUSE/lib/$my_machine/machine.mortran" "machine macros" #$HEN_HOUSE/catecho "$HEN_HOUSE/ranlux.macros "RNG macros" $HEN_HOUSE/catecho "$HEN_HOUSE/ranmar.macros "RNG macros" if ($?EGS_PERT != 1) echo "----------------------------------------------------" $HEN_HOUSE/catecho "$1.mortran "user-code source" if ($?EGS_PERT != 1) echo "----------------------------------------------------" #$HEN_HOUSE/catecho "$HEN_HOUSE/ranlux.mortran" "RNG initialization" $HEN_HOUSE/catecho "$HEN_HOUSE/ranmar.mortran" "RNG initialization" $HEN_HOUSE/catecho "$HEN_HOUSE/nrcaux.mortran "NRC auxilliary subs" $HEN_HOUSE/catecho "$HEN_HOUSE/egsnrc.mortran "egsnrc subroutines" echo " echo "-----------------------------------------------------------------" echo "end of standard.configuration(SID 1.6). .mortan.mortjob created" echo "-----------------------------------------------------------------" echo "

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309 Input File particle type particle energy histories -1 0.010 100000 -1 0.015 100000 -1 0.020 1000000 -1 0.030 1000000 -1 0.040 1000000 -1 0.050 1000000 -1 0.100 1000000 -1 0.200 1000000 -1 0.500 1000000 -1 1.000 500000 -1 1.500 500000 -1 2.000 100000 -1 4.000 100000 Output File Absorbed fractions for irradiation from bone trabeculae. Configuration No: 1 The calculation is performed for: 100 runs 1000 electrons per run Total: 100000 histories. Initial kinetic energy: 0.010 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.99617353893884 0.00031772128347 ( 0.03%) AF in Fat: 0.00005595111884 0.00003570754961 ( 63.82%) AF in Endo: 0.00367777859519 0.00031420133113 ( 8.54%) AF in MARROW: 0.00005100112030 0.00003450549248 ( 67.66%) AF in corticalshell: 0.00003052952721 0.00002453570522 ( 80.37%) AF in outside: 0.00001120069958 0.00001547189273 (138.13%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 0.99999999999997 Configuration No: 2 The calculation is performed for: 100 runs 1000 electrons per run Total: 100000 histories. Initial kinetic energy: 0.015 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.99209903186890 0.00053180028848 ( 0.05%) AF in Fat: 0.00009947270981 0.00005171961831 ( 51.99%) AF in Endo: 0.00750881097946 0.00052303260122 ( 6.97%) AF in MARROW: 0.00011895123804 0.00004891424324 ( 41.12%) AF in corticalshell: 0.00013475275254 0.00005903948523 ( 43.81%) AF in outside: 0.00003898045120 0.00003226687897 ( 82.78%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 0.99999999999996 Configuration No: 3 The calculation is performed for: 100 runs 10000 electrons per run

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310 Total: 1000000 histories. Initial kinetic energy: 0.020 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.98742322461241 0.00016908185556 ( 0.02%) AF in Fat: 0.00021620700975 0.00002403811356 ( 11.12%) AF in Endo: 0.01176466047470 0.00016706752133 ( 1.42%) AF in MARROW: 0.00029943168589 0.00002758913999 ( 9.21%) AF in corticalshell: 0.00022090488970 0.00002627266958 ( 11.89%) AF in outside: 0.00007557132767 0.00001506415865 ( 19.93%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000012 Configuration No: 4 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.030 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.97475149383534 0.00028246160757 ( 0.03%) AF in Fat: 0.00084859272887 0.00004113402027 ( 4.85%) AF in Endo: 0.02263377380447 0.00025591275325 ( 1.13%) AF in MARROW: 0.00116243204442 0.00005235533620 ( 4.50%) AF in corticalshell: 0.00041536637593 0.00003448894657 ( 8.30%) AF in outside: 0.00018834121093 0.00002212032010 ( 11.74%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 0.99999999999996 Configuration No: 5 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.040 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.95934609115757 0.00030790999339 ( 0.03%) AF in Fat: 0.00351528753560 0.00007422360526 ( 2.11%) AF in Endo: 0.03112602816434 0.00025266979779 ( 0.81%) AF in MARROW: 0.00506123281395 0.00010572062339 ( 2.09%) AF in corticalshell: 0.00067133728453 0.00004125307966 ( 6.14%) AF in outside: 0.00028002304413 0.00002595193813 ( 9.27%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000011 Configuration No: 6 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.050 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.94125860394055 0.00038733729654 ( 0.04%) AF in Fat: 0.00828091897413 0.00013139053638 ( 1.59%) AF in Endo: 0.03685645146887 0.00026702620801 ( 0.72%) AF in MARROW: 0.01200805223567 0.00016535787516 ( 1.38%) AF in corticalshell: 0.00108108113936 0.00005562528793 ( 5.15%) AF in outside: 0.00051489224145 0.00003605121182 ( 7.00%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000003 Configuration No: 7 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.100 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.83851505156197 0.00057145594886 ( 0.07%) AF in Fat: 0.04377118493170 0.00026376930578 ( 0.60%) AF in Endo: 0.04592893168121 0.00020207939131 ( 0.44%) AF in MARROW: 0.06648636179310 0.00035106811589 ( 0.53%)

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311 AF in corticalshell: 0.00363199694286 0.00010533122513 ( 2.90%) AF in outside: 0.00166647308916 0.00007024426533 ( 4.22%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 8 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.200 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.60499136666091 0.00066164753822 ( 0.11%) AF in Fat: 0.13187267655352 0.00034985137509 ( 0.27%) AF in Endo: 0.04323356229926 0.00011815015141 ( 0.27%) AF in MARROW: 0.20363761251115 0.00040599102717 ( 0.20%) AF in corticalshell: 0.01130968568044 0.00017798482735 ( 1.57%) AF in outside: 0.00495509629471 0.00011574505269 ( 2.34%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 9 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.500 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.37341691898257 0.00050685143751 ( 0.14%) AF in Fat: 0.20953948532408 0.00025445414057 ( 0.12%) AF in Endo: 0.03408028467036 0.00005898312722 ( 0.17%) AF in MARROW: 0.32399344932064 0.00033626672044 ( 0.10%) AF in corticalshell: 0.03959572864992 0.00030417579308 ( 0.77%) AF in outside: 0.01937413305243 0.00021632506990 ( 1.12%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 10 The calculation is performed for: 100 runs 5000 electrons per run Total: 500000 histories. Initial kinetic energy: 1.000 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.31189131116089 0.00049783426086 ( 0.16%) AF in Fat: 0.20758411360983 0.00029855712315 ( 0.14%) AF in Endo: 0.03018794121065 0.00005294149425 ( 0.18%) AF in MARROW: 0.32136423648797 0.00041654813518 ( 0.13%) AF in corticalshell: 0.07323142562073 0.00052705701795 ( 0.72%) AF in outside: 0.05574097190993 0.00050027869817 ( 0.90%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 11 The calculation is performed for: 100 runs 5000 electrons per run Total: 500000 histories. Initial kinetic energy: 1.500 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.27761245291727 0.00048261786030 ( 0.17%) AF in Fat: 0.19579437882970 0.00029603391134 ( 0.15%) AF in Endo: 0.02744342049639 0.00004761652634 ( 0.17%) AF in MARROW: 0.30365427572946 0.00044902268743 ( 0.15%) AF in corticalshell: 0.08630556625506 0.00049874573741 ( 0.58%) AF in outside: 0.10918990577212 0.00071295286715 ( 0.65%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 12 The calculation is performed for:

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312 100 runs 1000 electrons per run Total: 100000 histories. Initial kinetic energy: 2.000 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.25159376181906 0.00093031311605 ( 0.37%) AF in Fat: 0.18255607942573 0.00061360836090 ( 0.34%) AF in Endo: 0.02507531069563 0.00009225676750 ( 0.37%) AF in MARROW: 0.28346690550113 0.00089656129864 ( 0.32%) AF in corticalshell: 0.08928935509622 0.00100051990906 ( 1.12%) AF in outside: 0.16801858746223 0.00180230793927 ( 1.07%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 13 The calculation is performed for: 100 runs 1000 electrons per run Total: 100000 histories. Initial kinetic energy: 4.000 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.18573199795076 0.00073319557587 ( 0.39%) AF in Fat: 0.13923111972031 0.00058179064793 ( 0.42%) AF in Endo: 0.01870227382423 0.00008146119045 ( 0.44%) AF in MARROW: 0.21603629899094 0.00091484115811 ( 0.42%) AF in corticalshell: 0.07406424015795 0.00070312496887 ( 0.95%) AF in outside: 0.36623406935580 0.00198932225417 ( 0.54%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Absorbed fractions for irradiation from Active marrow. Configuration No: 1 The calculation is performed for: 100 runs 1000 electrons per run Total: 100000 histories. Initial kinetic energy: 0.010 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.00005228999984 0.00003732522374 ( 71.38%) AF in Fat: 0.01130912295027 0.00056039705694 ( 4.96%) AF in Endo: 0.00165973326093 0.00020282698987 ( 12.22%) AF in MARROW: 0.98685722762826 0.00059328238751 ( 0.06%) AF in corticalshell: 0.00008850902344 0.00004689484180 ( 52.98%) AF in outside: 0.00003311713730 0.00003214163963 ( 97.05%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000003 Configuration No: 2 The calculation is performed for: 100 runs 1000 electrons per run Total: 100000 histories. Initial kinetic energy: 0.015 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.00019205294737 0.00006613714024 ( 34.44%) AF in Fat: 0.02401494475502 0.00080846131952 ( 3.37%) AF in Endo: 0.00296376651321 0.00028014975056 ( 9.45%) AF in MARROW: 0.97254014730268 0.00084451706013 ( 0.09%) AF in corticalshell: 0.00023496016436 0.00007923188125 ( 33.72%) AF in outside: 0.00005412831733 0.00004023688840 ( 74.34%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 0.99999999999997 Configuration No: 3 The calculation is performed for: 100 runs 10000 electrons per run

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313 Total: 1000000 histories. Initial kinetic energy: 0.020 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.00025255265178 0.00002503177870 ( 9.91%) AF in Fat: 0.03935040650526 0.00028142938325 ( 0.72%) AF in Endo: 0.00535073632315 0.00009551083965 ( 1.79%) AF in MARROW: 0.95462341361033 0.00030141011232 ( 0.03%) AF in corticalshell: 0.00031652004467 0.00002970833437 ( 9.39%) AF in outside: 0.00010637086495 0.00001879712365 ( 17.67%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000015 Configuration No: 4 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.030 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.00095192162551 0.00004646414861 ( 4.88%) AF in Fat: 0.07769979017934 0.00045997065415 ( 0.59%) AF in Endo: 0.01006414639599 0.00014969419124 ( 1.49%) AF in MARROW: 0.91039357846353 0.00047168431639 ( 0.05%) AF in corticalshell: 0.00065309560671 0.00003981917289 ( 6.10%) AF in outside: 0.00023746772889 0.00002446619753 ( 10.30%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 0.99999999999997 Configuration No: 5 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.040 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.00411707478309 0.00009338193451 ( 2.27%) AF in Fat: 0.12283667497734 0.00050610573077 ( 0.41%) AF in Endo: 0.01360436910464 0.00017734133020 ( 1.30%) AF in MARROW: 0.85793455207185 0.00056024040982 ( 0.07%) AF in corticalshell: 0.00111201347958 0.00004856752275 ( 4.37%) AF in outside: 0.00039531558350 0.00003458853471 ( 8.75%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 6 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.050 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.00988377924049 0.00013769420316 ( 1.39%) AF in Fat: 0.16980256092716 0.00059553351346 ( 0.35%) AF in Endo: 0.01605839733911 0.00016873338723 ( 1.05%) AF in MARROW: 0.80201044583958 0.00063811998752 ( 0.08%) AF in corticalshell: 0.00166366922132 0.00006341742268 ( 3.81%) AF in outside: 0.00058114743236 0.00004744071214 ( 8.16%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 7 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.100 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.05320983373348 0.00035765268686 ( 0.67%) AF in Fat: 0.30883620899115 0.00056779475411 ( 0.18%) AF in Endo: 0.02113194736323 0.00013508574717 ( 0.64%) AF in MARROW: 0.60958121725542 0.00060282004694 ( 0.10%)

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314 AF in corticalshell: 0.00521108440111 0.00010929288615 ( 2.10%) AF in outside: 0.00202970825562 0.00008467587405 ( 4.17%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 8 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.200 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.16107655977185 0.00048073455120 ( 0.30%) AF in Fat: 0.29649162057482 0.00040445417575 ( 0.14%) AF in Endo: 0.02547957826139 0.00010910940954 ( 0.43%) AF in MARROW: 0.49643813032063 0.00048071705434 ( 0.10%) AF in corticalshell: 0.01458250560698 0.00020089649299 ( 1.38%) AF in outside: 0.00593160546433 0.00011885953679 ( 2.00%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 9 The calculation is performed for: 100 runs 10000 electrons per run Total: 1000000 histories. Initial kinetic energy: 0.500 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.25653571205504 0.00045361300210 ( 0.18%) AF in Fat: 0.25163928055888 0.00025820148899 ( 0.10%) AF in Endo: 0.02886261193152 0.00006782583134 ( 0.23%) AF in MARROW: 0.39945108916020 0.00035427559668 ( 0.09%) AF in corticalshell: 0.04274875555160 0.00031921050814 ( 0.75%) AF in outside: 0.02076255074276 0.00023690462903 ( 1.14%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 10 The calculation is performed for: 100 runs 5000 electrons per run Total: 500000 histories. Initial kinetic energy: 1.000 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.25484487639770 0.00046628914278 ( 0.18%) AF in Fat: 0.22713548335228 0.00025454323533 ( 0.11%) AF in Endo: 0.02761668643564 0.00006375160286 ( 0.23%) AF in MARROW: 0.35626832510805 0.00037654460401 ( 0.11%) AF in corticalshell: 0.07590703315697 0.00054877849328 ( 0.72%) AF in outside: 0.05822759554936 0.00040125011018 ( 0.69%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 11 The calculation is performed for: 100 runs 5000 electrons per run Total: 500000 histories. Initial kinetic energy: 1.500 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.24135807328668 0.00041835429899 ( 0.17%) AF in Fat: 0.20794818747639 0.00029330652152 ( 0.14%) AF in Endo: 0.02568890160106 0.00005180003174 ( 0.20%) AF in MARROW: 0.32510585613165 0.00043359451905 ( 0.13%) AF in corticalshell: 0.08802848426782 0.00052082692545 ( 0.59%) AF in outside: 0.11187049723640 0.00072347070453 ( 0.65%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 12 The calculation is performed for:

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315 100 runs 1000 electrons per run Total: 100000 histories. Initial kinetic energy: 2.000 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.22540153142851 0.00091145123494 ( 0.40%) AF in Fat: 0.19091189687058 0.00060475259560 ( 0.32%) AF in Endo: 0.02378261890204 0.00009612952338 ( 0.40%) AF in MARROW: 0.29810525743776 0.00089525871002 ( 0.30%) AF in corticalshell: 0.09018674428577 0.00108171519812 ( 1.20%) AF in outside: 0.17161195107533 0.00175822376683 ( 1.02%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000 Configuration No: 13 The calculation is performed for: 100 runs 1000 electrons per run Total: 100000 histories. Initial kinetic energy: 4.000 MeV. Absorbed fractions with 95% confidence intervals: AF in Trabeculae: 0.17303408269506 0.00077469705662 ( 0.45%) AF in Fat: 0.14293387758550 0.00058461966832 ( 0.41%) AF in Endo: 0.01806950365424 0.00009224561228 ( 0.51%) AF in MARROW: 0.22275740845251 0.00095664014699 ( 0.43%) AF in corticalshell: 0.07461395392928 0.00079885603241 ( 1.07%) AF in outside: 0.36859117368341 0.00218394295505 ( 0.59%) AF lost: 0.00000000000000 0.00000000000000 ( NaN%) Total AF: 1.00000000000000

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316 APPENDIX F PIRT MODEL FOR THE PROXIMAL FEMUR (EGSNRC USER CODE) This appendix contains the MORTRAN user code of the EGSnrc radiation code that has been developed to tr ansport electrons with in the varying spongi osa regions of the proximal femur (femur head and neck) with ac count for the macrostructure of a skeletal site (cortical bone cort ex). It uses an input data fro m a pre-processed microimage of skeletal spongiosa (microCT) for the femur head and neck. Appendix E shows examples of the associated files used w ithin this EGSnrc code, such as the input and output files. EGSnrc MORTRAN User Code !INDENT M 4; "INDENT EACH MORTRAN NESTING LEVEL BY 4" !INDENT F 2; "INDENT EACH FORTRAN NESTING LEVEL BY 2" "This line is 80 characters long, use it to set up the screen width" "23456789|123456789|123456789|123456789|123456789|123456789|123456789|123456789" "******************************************************************************" Amish P. Shah " ********************* " * " FemurPIRT.mortran " * " ********************* " " This program calculates the absorbed fraction of energy within the bone " trabeculae, the marrow space components (inactive/active marrow), the bone " endosteum, and the surrounding cortical bone of a given bone site. " The geometry is defined by two images. An NMR image is used to simulate an " infinite field of trabecular bone. " A CT image (lower resolution) is place over the trabecular bone region to " define the limits of the trabecular bone region via a cortical bone shell " with surrounding soft tissue. " Particles that goes outside the cortical shell are discarded. " The source is defined in any one of the regions that a target exists: bone " volume, active marrow, inactive marrow, cortical bone, and bone endosteum. " " Several things must be defined for each run: both images' configurations, " the source, the output file, the location of the input file, etc.. " " The PEGS file is: nmrcube (so no one forgets!) " the type of particle: -1 for electrons, 0 for photons " the initial energy of the particles " the number of histries per configuration. " The results are in the file Output.dat. " "******************************************************************************" "----------------------------------------" Step 1: To override the EGSnrc macros "----------------------------------------" 1) so that all real variables are in double precision REPLACE {$REAL} WITH {DOUBLE PRECISION} 2) the size of the arrays used by EGSnrc. REPLACE {$MXMED} WITH {3} "3 medium in the problem (default 10)" REPLACE {$MXREG} WITH {6} "6 geometric regions (default 2000)" REPLACE {$MXSTACK} WITH {100} "less than 100 particles on stack at once" REPLACE {$MXMDSH} WITH {100} "max. nb of shells per medium for "incoherent scattering"

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317 REPLACE {IFIX} WITH {INT} REPLACE {0.,} WITH {0.0D0,} 3) for compatibility with the old EGS4. REPLACE {$CALL-HOWNEAR(#);} WITH {CALL HOWNEAR({P1},X(NP),Y(NP),Z(NP),IRL);} "-------------------------------------------" Step 1.a. To define user constant values "-------------------------------------------" REPLACE {$REG_TRAB} WITH {1} region within the bone trabeculae REPLACE {$REG_MARR} WITH {2} region within the marrow cavities REPLACE {$REG_CORT} WITH {3} region within the cortical bone REPLACE {$REG_OUTSIDE} WITH {4} region outside the study REPLACE {$REG_LOST} WITH {5} region for lost particles REPLACE {$REG_FAT} WITH {6} region within the FAT REPLACE {$MED_BONE} WITH {0} to represent bone in NMR image REPLACE {$MED_MARR} WITH {255} to represent marrow in NMR image REPLACE {$MED_FAT} WITH {122} to represent FAT in NMR image REPLACE {$MED_NECKSPONG} WITH {5} to represent spongiosa in CT image REPLACE {$MED_HEADSPONG} WITH {65} to represent spongiosa in CT image REPLACE {$MED_CORT} WITH {45} to represent cort. bone in CT image REPLACE {$MED_TISS} WITH {25} to represent soft tissue in CT image REPLACE {$IMAGE_FILE_NMR} WITH {23} file to read the image REPLACE {$IMAGE_FILE_CT} WITH {23} file to read the image REPLACE {$INPUT_FILE} WITH {25} file to get the parameters REPLACE {$OUTPUT_FILE} WITH {27} file to record the results REPLACE {$N_RUN} WITH {100} number of run for each configuration" REPLACE {$INFINITY} WITH {1.0D99} to simulate infinity long distance REPLACE {$PI} WITH {3.1415926535897932D+00} need Pi in Source " this is to solve the boundary crossing problem. The particle is " transported a little farther than the exact boundary REPLACE {$BOUNDARY_THICKNESS} WITH {1.0D-09} that's 0.1 Angstrom " for the geometrical model " 1) the NMR image "************************Change These Parameters*******************************" REPLACE {$NMR_VOXEL_SIZE_X} WITH {0.00600D+00} in cm REPLACE {$NMR_VOXEL_SIZE_Y} WITH {0.00600D+00} in cm REPLACE {$NMR_VOXEL_SIZE_Z} WITH {0.00600D+00} in cm REPLACE {$NMR_IMAGE_NX} WITH {262} nb of voxels along (O,x) REPLACE {$NMR_IMAGE_NY} WITH {101} nb of voxels along (O,y) REPLACE {$NMR_IMAGE_NZ} WITH {203} nb of voxels along (O,z) " 2) the CT image REPLACE {$CT_VOXEL_SIZE_X} WITH {0.0203125D+00} in cm REPLACE {$CT_VOXEL_SIZE_Y} WITH {0.0203125D+00} in cm REPLACE {$CT_VOXEL_SIZE_Z} WITH {0.1000000D+00} in cm REPLACE {$CT_IMAGE_NX} WITH {512} nb of voxels along (O,x) REPLACE {$CT_IMAGE_NY} WITH {512} nb of voxels along (O,y) REPLACE {$CT_IMAGE_NZ} WITH {163} nb of voxels along (O,z) "************************Change These Parameters*******************************" "-------------------------------------------------" Step 1.b. To define the user common variables "-------------------------------------------------" a) for scoring the results REPLACE {COMIN/SCOR/;} WITH {COMMON/SCOR/ CumulEnergyTrabeculae,CumulEnergyMarrow,CumulEnergyCortical, CumulEnergyOutside,CumulEnergyLost, CumulEnergyFat,CumulEnergyEndo; $REAL CumulEnergyTrabeculae; $REAL CumulEnergyMarrow; $REAL CumulEnergyCortical; $REAL CumulEnergyOutside; $REAL CumulEnergyLost; $REAL CumulEnergyFat; $REAL CumulEnergyEndo;} b) for the geometry REPLACE {COMIN/GEOM/;} WITH {COMMON/GEOM/NMRBoneImage, CTBoneImage;

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318 CHARACTER NMRBoneImage($NMR_IMAGE_NX,$NMR_IMAGE_NY,$NMR_IMAGE_NZ); BYTE CTBoneImage($CT_IMAGE_NZ*$CT_IMAGE_NY*$CT_IMAGE_NX);} "--------------------------------------------------------" Step 1.c. To define the variables of the main program "--------------------------------------------------------" $IMPLICIT-NONE; to make sure that all variables are declared " 1) all the common that you need in the main programm COMIN/BOUNDS,MEDIA,MISC,USEFUL,RANDOM,GEOM,SCOR/; The above expands into COMMON statements " BOUNDS contains ECUT and PCUT " MEDIA contains NMED and the array concerning media " MISC contains the medium per region and Rayleigh parameters " USEFUL contains electron rest mass " RANDOM contains the RANMAR parameters " GEOM passes info to HOWFAR and HOWNEAR routines " SCOR passes info to AUSGAB routine " 2) local variables of the main program $REAL XIN, YIN, ZIN; particle location (to give to SHOWER) $REAL UIN, VIN, WIN; particle direction (to give to SHOWER) $REAL EIN; particle energy (to give to SHOWER) $REAL WTIN; particle weight (to give to SHOWER) $INTEGER IQIN; particle type (to give to SHOWER) $INTEGER IRIN; particle region (to give to SHOWER) $INTEGER PartNo; particle # to loop for each particle $INTEGER RunNo; run number to loop for each run $INTEGER ConfigNo; configuration number to loop for each one LOGICAL NoMoreConfig; to test the end of the input file $INTEGER ParticleType; particle type got from the input file $INTEGER SourceNo; source number to loop for each one LOGICAL NoMoreSource; to test the end of the input2 file $INTEGER SourceType; particle type got from the input file $REAL KineticEnergy; kinetic energy got from the input file $INTEGER NumberOfHistories; number of histories got from the input file $INTEGER ParticlePerRun; number of particles per run " for statistical results: mean, standard deviation, standard deviation " of the mean, 95% confidence interval, and 95% confidence error $REAL AFTrabeculae; $REAL MeanAFTrabeculae; $REAL StdDevAFTrabeculae; $REAL StdDevOfMeanAFTrabeculae; $REAL ConfIntOfMeanAFTrabeculae; $REAL ConfErrOfMeanAFTrabeculae; $REAL AFMarrow; $REAL MeanAFMarrow; $REAL StdDevAFMarrow; $REAL StdDevOfMeanAFMarrow; $REAL ConfIntOfMeanAFMarrow; $REAL ConfErrOfMeanAFMarrow; $REAL AFFat; $REAL MeanAFFat; $REAL StdDevAFFat; $REAL StdDevOfMeanAFFat; $REAL ConfIntOfMeanAFFat; $REAL ConfErrOfMeanAFFat; $REAL AFEndo; $REAL MeanAFEndo; $REAL StdDevAFEndo; $REAL StdDevOfMeanAFEndo; $REAL ConfIntOfMeanAFEndo; $REAL ConfErrOfMeanAFEndo; $REAL AFCortical; $REAL MeanAFCortical; $REAL StdDevAFCortical; $REAL StdDevOfMeanAFCortical; $REAL ConfIntOfMeanAFCortical; $REAL ConfErrOfMeanAFCortical;

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319 $REAL AFOutside; $REAL MeanAFOutside; $REAL StdDevAFOutside; $REAL StdDevOfMeanAFOutside; $REAL ConfIntOfMeanAFOutside; $REAL ConfErrOfMeanAFOutside; $REAL AFLost; $REAL MeanAFLost; $REAL StdDevAFLost; $REAL StdDevOfMeanAFLost; $REAL ConfIntOfMeanAFLost; $REAL ConfErrOfMeanAFLost; $INTEGER NumByte, NumX, NumY, NumZ; CHARACTER tmp; 3) system functions invoked in the main program $REAL DSQRT; INTRINSIC DSQRT; "----------------------------------------" Step 2. To initialize the EGSnrc data "----------------------------------------" 1) to place medium names in an array. " $S is a MORTRAN macro to expand strings CHARACTER*4 MEDARR(24,$MXMED); $INTEGER I, J; DATA MEDARR /$S'Bone',20*' ',$S'Marrow',18*' ',$S'Fat',21*' '/; NMED = $MXMED; "Set number of media." DO J = 1,$MXMED [ DO I=1,24 [ MEDIA(I,J) = MEDARR(I,J); ] this is to avoid a DATA STATEMENT for a variable in COMMON" NMED and DUNIT default to 1, i.e. one medium and we work in cm ] 2) to initialize the medium in each region MED($REG_TRAB) = 1; "cortical bone in the bone trabeculae" MED($REG_MARR) = 2; "bone marrow in the marrow cavities" MED($REG_FAT) = 3; "fat marrow in the marrow cavities" MED($REG_CORT) = 1; "cortical bone in the cortical shell" MED($REG_OUTSIDE) = 0; "vacuum outside the study region MED($REG_LOST) = 0; "vacuum if particles are lost (does not matter)" 3) to initialize the cutoff energy for both electrons and " photons in each region ECUT($REG_TRAB) = 0.005 + PRM; 5 keV + rest mass for electrons PCUT($REG_TRAB) = 0.001; 1 keV for photons ECUT($REG_MARR) = 0.005 + PRM; PCUT($REG_MARR) = 0.001; ECUT($REG_FAT) = 0.005 + PRM; PCUT($REG_FAT) = 0.001; ECUT($REG_CORT) = 0.005 + PRM; PCUT($REG_CORT) = 0.001; ECUT($REG_OUTSIDE) = 0.005 + PRM; PCUT($REG_OUTSIDE) = 0.001; ECUT($REG_LOST) = 0.005 + PRM; PCUT($REG_LOST) = 0.001; 4) to ask EGSnrc to treat the Rayleigh scattering in each region IRAYLR($REG_TRAB) = 1; IRAYLR($REG_MARR) = 1; IRAYLR($REG_FAT) = 1; IRAYLR($REG_CORT) = 1; IRAYLR($REG_OUTSIDE) = 1; IRAYLR($REG_LOST) = 1; 5) to initialize the random number generator IXX = 1; JXX = 1; seed # to initialize the random number series $RNG-INITIALIZATION; "---------------------------------------------------------------" Step 3. To pick up the cross sections precalculated by pegs4 "---------------------------------------------------------------" CALL HATCH; data file must be assigned to unit 12

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320 PRINT *, 'End of HATCH'; "------------------------------------------" Step 3.a. To initialize the output file "------------------------------------------" "********Make Sure to Change the Path of the OUTPUT File in new directory******" OPEN ( UNIT=$OUTPUT_FILE, FILE='/c/users/Amish/egsnrc/FemurPIRT/OUtput.dat', STATUS='unknown' ); "---------------------------------------------" Step 3.b. To open and read the image files "---------------------------------------------" OPEN($IMAGE_FILE_NMR, FILE='/c/users/Amish/RITman/NMRimages/RtFemurNeck.10.gray', "********Make Sure to Change the Path of the INPUT Image File *****************" ACCESS='DIRECT',ERR=95,FORM='FORMATTED',RECL=1); GOTO 101; 95 PRINT *, 'error opening'; 101 PRINT *, 'ok opening NMR image file'; NumByte = 1; DO NumX=1, $NMR_IMAGE_NX [ DO NumY=1, $NMR_IMAGE_NY [ DO NumZ=1, $NMR_IMAGE_NZ [ READ($IMAGE_FILE_NMR, '(A1)', REC=NumByte) tmp; NMRBoneImage(NumX,NumY,NumZ)=tmp; NumByte = NumByte + 1; ] ] ] CLOSE ($IMAGE_FILE_NMR); PRINT *, 'ok reading NMR image file'; "********Make Sure to Change the Path of the INPUT CT File *********************" OPEN($IMAGE_FILE_CT, FILE='/c/users/Amish/RITman/CTimages/LtFemurHDNK.con', "********Make Sure to Change the Path of the INPUT CT File *********************" ACCESS='DIRECT', FORM='UNFORMATTED', RECL=$CT_IMAGE_NZ*$CT_IMAGE_NY*$CT_IMAGE_NX); PRINT *, 'ok opening CT image file'; READ($IMAGE_FILE_CT, REC=1) CTBoneImage; CLOSE ( $IMAGE_FILE_CT ); PRINT *, 'ok reading CT image file'; "-----------------------------------------------------" Step 3.c. For each configuration in the input file "-----------------------------------------------------" One execution is performed for each line of the input file OPEN ( UNIT=$INPUT_FILE, "*********Make Sure to Change the Path of the Input File in new directory******" FILE='/c/users/Amish/egsnrc/FemurPIRT/Input.dat', STATUS='old' ); "*********Make Sure to Change the Path of the Input File in new directory******"

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321 READ ( $INPUT_FILE, ); to skip the first line NoMoreConfig = .FALSE.; ConfigNo = 0; LOOP [" until no more line in the file "-------------------------------------------------" Step 3.d. To read a new line in the input file "-------------------------------------------------" READ ( $INPUT_FILE, *, END = :EndInput: ) ParticleType, KineticEnergy, NumberOfHistories; GO TO :NextInput:; :EndInput: NoMoreConfig = .TRUE.; :NextInput: CONTINUE; "-----------------------------------------------------------------------" Step 3.e. If a new line exists, initialize the data for this config. "-----------------------------------------------------------------------" IF (~NoMoreConfig) [ 1) to display the new configuration ConfigNo = ConfigNo + 1; PRINT *, 'Configuration no:', ConfigNo; 2) how many particles per run? ParticlePerRun = NumberOfHistories / $N_RUN; 3) to output the parameters of the configuration WRITE($OUTPUT_FILE, '(A)') '; WRITE($OUTPUT_FILE, '(A,I3)') 'Configuration No:', ConfigNo; WRITE($OUTPUT_FILE, '(A)') 'The calculation is performed for:'; WRITE($OUTPUT_FILE, '(A,I5,A)') ', $N_RUN, runs'; IF (ParticleType = 0) [ WRITE($OUTPUT_FILE, '(A,I6,A)') ', ParticlePerRun, photons per run'; ] ELSE [ WRITE($OUTPUT_FILE, '(A,I6,A)') ', ParticlePerRun, electrons per run'; ] WRITE($OUTPUT_FILE, '(A,I8,A)') Total: ', ParticlePerRun*$N_RUN, histories.'; WRITE($OUTPUT_FILE, '(A,F7.3,A)') Initial kinetic energy: ', KineticEnergy, MeV.'; 4) to initialize the statistical data MeanAFTrabeculae = 0.0; MeanAFMarrow = 0.0; MeanAFFat = 0.0; MeanAFEndo = 0.0; MeanAFCortical = 0.0; MeanAFOutside = 0.0; MeanAFLost = 0.0; StdDevAFTrabeculae = 0.0; StdDevAFMarrow = 0.0; StdDevAFFat = 0.0; StdDevAFEndo = 0.0; StdDevAFCortical = 0.0; StdDevAFOutside = 0.0; StdDevAFLost = 0.0; "-------------------------" Step 3.f. For each run "-------------------------" DO RunNo=1,$N_RUN [ PRINT *, Run no:', RunNo; "------------------------------------------------------------" Step 4. To initialize the geometry for HOWFAR and HOWNEAR "------------------------------------------------------------" done when reading the input file "---------------------------------------------------------" Step 5. To initialize the scoring variables for AUSGAB "---------------------------------------------------------" CumulEnergyTrabeculae = 0.0; CumulEnergyMarrow = 0.0; CumulEnergyFat = 0.0; CumulEnergyEndo = 0.0;

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322 CumulEnergyCortical = 0.0; CumulEnergyOutside = 0.0; CumulEnergyLost = 0.0; "------------------------------" Step 5.a. For each particle "------------------------------" DO PartNo=1, ParticlePerRun [ to have a display of the progression of the code IF (MOD(PartNo,100) = 0) [ "PRINT *, Particle: ', PartNo;" ] "--------------------------------------------" Step 6. To define the particle parameters "--------------------------------------------" IF (ParticleType = 0) [ EIN = KineticEnergy; initial kinetic energy" ] ELSE [ EIN = KineticEnergy + PRM; initial kinetic + rest mass energy" ] IQIN=ParticleType; WTIN=1.0; weight = 1 since no variance reduction used" to get the initial location and direction of the particle. "************Remove Comments from the Source that you Choose*******************" CALL SourceHeadBoneVolume(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); CALL SourceNeckBoneVolume(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); CALL SourceHeadActiveMarrow(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceNeckActiveMarrow(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceHeadBoneSurface(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceNeckBoneSurface(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceCorticalBone(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceHeadBoneEndosteum(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceNeckBoneEndosteum(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceHeadFatMarrow(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); " CALL SourceNeckFatMarrow(XIN,YIN,ZIN,UIN,VIN,WIN,IRIN); "************Remove Comments from the Source that you Choose*******************" "------------------------------------" Step 7. To transport the particle "------------------------------------" CALL SHOWER(IQIN,EIN,XIN,YIN,ZIN,UIN,VIN,WIN,IRIN,WTIN); ] "-------------------------------------------------------------" Step 7.a. To calculate and display the result for this run "-------------------------------------------------------------" AFTrabeculae = CumulEnergyTrabeculae / (ParticlePerRun KineticEnergy); AFMarrow = CumulEnergyMarrow / (ParticlePerRun KineticEnergy); AFFat = CumulEnergyFat / (ParticlePerRun KineticEnergy); AFEndo = CumulEnergyEndo / (ParticlePerRun KineticEnergy); AFCortical = CumulEnergyCortical / (ParticlePerRun KineticEnergy); AFOutside = CumulEnergyOutside / (ParticlePerRun KineticEnergy); AFLost = CumulEnergyLost / (ParticlePerRun KineticEnergy); "PRINT *, Data for this run:';" "PRINT *, AF in bone trabeculae: ',AFTrabeculae;" "PRINT *, AF in MARROW: ',AFMarrow;" "PRINT *, AF in fat: ',AFFat;" "PRINT *, AF in endo: ',AFEndo;" "PRINT *, AF in cortical: ',AFCortical;" "PRINT *, AF in outside: ',AFOutside;" "PRINT *, AF in lost: ',AFLost;" "PRINT *, Total in AF: ',AFTrabeculae + AFMarrow +" AFEndo + AFFat + AFCortical + AFOutside + AFLost;" "---------------------------------------------"

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323 Step 7.b. To cumulate the statistical data "---------------------------------------------" MeanAFTrabeculae = MeanAFTrabeculae + AFTrabeculae; MeanAFMarrow = MeanAFMarrow + AFMarrow; MeanAFFat = MeanAFFat + AFFat; MeanAFEndo = MeanAFEndo + AFEndo; MeanAFCortical = MeanAFCortical + AFCortical; MeanAFOutside = MeanAFOutside + AFOutside; MeanAFLost = MeanAFLost + AFLost; StdDevAFTrabeculae = StdDevAFTrabeculae + AFTrabeculae*AFTrabeculae; StdDevAFMarrow = StdDevAFMarrow + AFMarrow*AFMarrow; StdDevAFFat = StdDevAFFat + AFFat*AFFat; StdDevAFEndo = StdDevAFEndo + AFEndo*AFEndo; StdDevAFCortical = StdDevAFCortical + AFCortical*AFCortical; StdDevAFOutside = StdDevAFOutside + AFOutside*AFOutside; StdDevAFLost = StdDevAFLost + AFLost*AFLost; ] End of this run "----------------------------------------------" Step 7.c. To calculate the statistical data "----------------------------------------------" a) the mean MeanAFTrabeculae = MeanAFTrabeculae / $N_RUN; MeanAFMarrow = MeanAFMarrow / $N_RUN; MeanAFFat = MeanAFFat / $N_RUN; MeanAFEndo = MeanAFEndo / $N_RUN; MeanAFCortical = MeanAFCortical / $N_RUN; MeanAFOutside = MeanAFOutside / $N_RUN; MeanAFLost = MeanAFLost / $N_RUN; b) the standard deviation of the sample StdDevAFTrabeculae = StdDevAFTrabeculae $N_RUN*MeanAFTrabeculae*MeanAFTrabeculae; StdDevAFMarrow = StdDevAFMarrow $N_RUN*MeanAFMarrow*MeanAFMarrow; StdDevAFFat = StdDevAFFat $N_RUN*MeanAFFat*MeanAFFat; StdDevAFEndo = StdDevAFEndo $N_RUN*MeanAFEndo*MeanAFEndo; StdDevAFCortical = StdDevAFCortical $N_RUN*MeanAFCortical*MeanAFCortical; StdDevAFOutside = StdDevAFOutside $N_RUN*MeanAFOutside*MeanAFOutside; StdDevAFLost = StdDevAFLost $N_RUN*MeanAFLost*MeanAFLost; StdDevAFTrabeculae = StdDevAFTrabeculae / ($N_RUN 1); StdDevAFMarrow = StdDevAFMarrow / ($N_RUN 1); StdDevAFFat = StdDevAFFat / ($N_RUN 1); StdDevAFEndo = StdDevAFEndo / ($N_RUN 1); StdDevAFCortical = StdDevAFCortical / ($N_RUN 1); StdDevAFOutside = StdDevAFOutside / ($N_RUN 1); StdDevAFLost = StdDevAFLost / ($N_RUN 1); StdDevAFTrabeculae = DSQRT(StdDevAFTrabeculae); StdDevAFMarrow = DSQRT(StdDevAFMarrow); StdDevAFFat = DSQRT(StdDevAFFat); StdDevAFEndo = DSQRT(StdDevAFEndo); StdDevAFCortical = DSQRT(StdDevAFCortical); StdDevAFOutside = DSQRT(StdDevAFOutside); StdDevAFLost = DSQRT(StdDevAFLost); c) the standard deviation of the mean */ StdDevOfMeanAFTrabeculae = StdDevAFTrabeculae / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFMarrow = StdDevAFMarrow / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFFat = StdDevAFFat / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFEndo = StdDevAFEndo / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFCortical = StdDevAFCortical / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFOutside = StdDevAFOutside / DSQRT(DBLE($N_RUN)); StdDevOfMeanAFLost = StdDevAFLost / DSQRT(DBLE($N_RUN)); d) the 95% confidence interval of the mean */ ConfIntOfMeanAFTrabeculae = 1.96*StdDevOfMeanAFTrabeculae; ConfIntOfMeanAFMarrow = 1.96*StdDevOfMeanAFMarrow;

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324 ConfIntOfMeanAFFat = 1.96*StdDevOfMeanAFFat; ConfIntOfMeanAFEndo = 1.96*StdDevOfMeanAFEndo; ConfIntOfMeanAFCortical = 1.96*StdDevOfMeanAFCortical; ConfIntOfMeanAFOutside = 1.96*StdDevOfMeanAFOutside; ConfIntOfMeanAFLost = 1.96*StdDevOfMeanAFLost; e) the 95% confidence error of the mean */ ConfErrOfMeanAFTrabeculae = 100.0 ConfIntOfMeanAFTrabeculae / MeanAFTrabeculae; ConfErrOfMeanAFMarrow = 100.0 ConfIntOfMeanAFMarrow / MeanAFMarrow; ConfErrOfMeanAFFat = 100.0 ConfIntOfMeanAFFat / MeanAFFat; ConfErrOfMeanAFEndo = 100.0 ConfIntOfMeanAFEndo / MeanAFEndo; ConfErrOfMeanAFCortical = 100.0 ConfIntOfMeanAFCortical / MeanAFCortical; ConfErrOfMeanAFOutside = 100.0 ConfIntOfMeanAFOutside / MeanAFOutside; ConfErrOfMeanAFLost = 100.0 ConfIntOfMeanAFLost / MeanAFLost; "------------------------------------------------------" Step 8. To print out the results to the output file "------------------------------------------------------" WRITE($OUTPUT_FILE, '(A,A)') Absorbed fractions with 95%', confidence intervals:'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in Trabeculae: ', MeanAFTrabeculae, +/', ConfIntOfMeanAFTrabeculae,' (', ConfErrOfMeanAFTrabeculae, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in Fat: ', MeanAFFat, +/', ConfIntOfMeanAFFat,' (', ConfErrOfMeanAFFat, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in Endo: ', MeanAFEndo, +/', ConfIntOfMeanAFEndo,' (', ConfErrOfMeanAFEndo, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in MARROW: ', MeanAFMarrow, +/', ConfIntOfMeanAFMarrow,' (', ConfErrOfMeanAFMarrow, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in corticalshell: ', MeanAFCortical, +/', ConfIntOfMeanAFCortical,' (', ConfErrOfMeanAFCortical, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF in outside: ', MeanAFOutside, +/', ConfIntOfMeanAFOutside,' (', ConfErrOfMeanAFOutside, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14,A,F16.14,A,F6.2,A)') AF lost: ', MeanAFLost, +/', ConfIntOfMeanAFLost,' (', ConfErrOfMeanAFLost, '%)'; WRITE($OUTPUT_FILE,'(A,F16.14)') Total AF: ', MeanAFTrabeculae + MeanAFMarrow + MeanAFCortical+ MeanAFEndo + MeanAFFat + MeanAFOutside + MeanAFLost; ] ] End of this configuration UNTIL (NoMoreConfig); "--------------------------------------------" Step 8.a. Don't forget to close the files "--------------------------------------------" CLOSE($INPUT_FILE); CLOSE($OUTPUT_FILE); END; End of main program "******************************************************************************" SourceHeadBoneVolume "******************************************************************************" " The SourceBoneVolume subroutine returns a particle starting within the bone " regions of the image. The source is isotropic and uniform within the BONE ." The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers)

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325 Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceHeadBoneVolume(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneVolume; LOGICAL InsideHeadSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ((InsideBoneVolume(XSrc, YSrc, ZSrc)) .AND. (InsideHeadSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_TRAB; END; End of subroutine SourceHeadBoneVolume "******************************************************************************" SourceNeckBoneVolume "******************************************************************************" " The SourceBoneVolume subroutine returns a particle starting within the bone " regions of the image. The source is isotropic and uniform within the BONE ."

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326 The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceNeckBoneVolume(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneVolume; LOGICAL InsideNeckSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ((InsideBoneVolume(XSrc, YSrc, ZSrc)) .AND. (InsideNeckSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_TRAB; END; End of subroutine SourceNeckBoneVolume

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327 "******************************************************************************" SourceHeadActiveMarrow "******************************************************************************" " The SourceActiveMarrow subroutine returns particles starting within the " marrow regions of the NMR image. The source is isotropic and uniform within" the Active Marrow. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceHeadActiveMarrow(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideActiveMarrow; LOGICAL InsideBoneEndosteum; LOGICAL InsideHeadSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ((InsideActiveMarrow(XSrc, YSrc, ZSrc)) .AND. (~InsideBoneEndosteum(XSrc, YSrc, ZSrc)) .AND. (InsideHeadSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2;

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328 USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_MARR; END; End of subroutine SourceHeadActiveMarrow "******************************************************************************" SourceNeckActiveMarrow "******************************************************************************" " The SourceActiveMarrow subroutine returns particles starting within the " marrow regions of the NMR image. The source is isotropic and uniform within" the Active Marrow. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceNeckActiveMarrow(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideActiveMarrow; LOGICAL InsideBoneEndosteum; LOGICAL InsideNeckSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ((InsideActiveMarrow(XSrc, YSrc, ZSrc)) .AND. (~InsideBoneEndosteum(XSrc, YSrc, ZSrc)) .AND. (InsideNeckSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction

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329 "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_MARR; END; End of subroutine SourceNeckActiveMarrow "******************************************************************************" SourceHeadFatMarrow "******************************************************************************" " The SourceFatMarrow subroutine returns particles starting within the " marrow regions of the NMR image. The source is isotropic and uniform within" the Fat Marrow or InActive Marrow. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceHeadFatMarrow(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideFatMarrow; LOGICAL InsideBoneEndosteum; LOGICAL InsideHeadSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( (InsideFatMarrow(XSrc, YSrc, ZSrc)) .AND. (~InsideBoneEndosteum(XSrc, YSrc, ZSrc)) .AND. (InsideHeadSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.;

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330 ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_FAT; END; End of subroutine SourceHeadFatMarrow "******************************************************************************" SourceNeckFatMarrow "******************************************************************************" " The SourceFatMarrow subroutine returns particles starting within the " marrow regions of the NMR image. The source is isotropic and uniform within" the Fat Marrow or InActive Marrow. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceNeckFatMarrow(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideFatMarrow; LOGICAL InsideBoneEndosteum; LOGICAL InsideNeckSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2;

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331 $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( (InsideFatMarrow(XSrc, YSrc, ZSrc)) .AND. (~InsideBoneEndosteum(XSrc, YSrc, ZSrc)) .AND. (InsideNeckSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_FAT; END; End of subroutine SourceNeckFatMarrow "******************************************************************************" SourceHeadBoneEndosteum "******************************************************************************" " The SourceBoneEndosteum subroutine returns particles starting within the " endosteum regions of the image. The source is isotropic and uniform within" the 10 micron layer of the endosteum. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceHeadBoneEndosteum(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneEndosteum; LOGICAL InsideActiveMarrow; LOGICAL InsideFatMarrow; LOGICAL InsideHeadSpongiosa;

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332 "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ((InsideBoneEndosteum(XSrc, YSrc, ZSrc)) .AND. (InsideHeadSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL (InsideSource); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" IF (InsideActiveMarrow(XSrc, YSrc, ZSrc)) [ RegSrc = $REG_MARR; ] ELSE [ RegSrc = $REG_FAT; ] END; End of subroutine SourceHeadBoneEndosteum "******************************************************************************" SourceNeckBoneEndosteum "******************************************************************************" " The SourceBoneEndosteum subroutine returns particles starting within the " endosteum regions of the image. The source is isotropic and uniform within" the 10 micron layer of the endosteum. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceNeckBoneEndosteum(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables

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333 COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneEndosteum; LOGICAL InsideActiveMarrow; LOGICAL InsideFatMarrow; LOGICAL InsideNeckSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ((InsideBoneEndosteum(XSrc, YSrc, ZSrc)) .AND. (InsideNeckSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL (InsideSource); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" IF (InsideActiveMarrow(XSrc, YSrc, ZSrc)) [ RegSrc = $REG_MARR; ] ELSE [ RegSrc = $REG_FAT; ] END; End of subroutine SourceNeckBoneEndosteum "******************************************************************************" SourceHeadBoneSurface "******************************************************************************" " The SourceBoneSurface subroutine returns particles starting within the " endosteum regions of the image. The source is isotropic and uniform within" the 1 micron layer of the endosteum. " The direction is equiprobable, that means that:

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334 Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceHeadBoneSurface(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneSurface; LOGICAL InsideActiveMarrow; LOGICAL InsideFatMarrow; LOGICAL InsideHeadSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ((InsideBoneSurface(XSrc, YSrc, ZSrc)) .AND. (InsideHeadSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL (InsideSource); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" IF (InsideActiveMarrow(XSrc, YSrc, ZSrc)) [

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335 RegSrc = $REG_MARR; ] ELSE [ RegSrc = $REG_FAT; ] END; End of subroutine SourceHeadBoneSurface "******************************************************************************" SourceNeckBoneSurface "******************************************************************************" " The SourceBoneSurface subroutine returns particles starting within the " endosteum regions of the image. The source is isotropic and uniform within" the 1 micron layer of the endosteum. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceNeckBoneSurface(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideBoneSurface; LOGICAL InsideActiveMarrow; LOGICAL InsideFatMarrow; LOGICAL InsideNeckSpongiosa; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ((InsideBoneSurface(XSrc, YSrc, ZSrc)) .AND. (InsideNeckSpongiosa(XSrc, YSrc, ZSrc))) [ InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL (InsideSource); "----------------------------" 2) to return the direction

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336 "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" IF (InsideActiveMarrow(XSrc, YSrc, ZSrc)) [ RegSrc = $REG_MARR; ] ELSE [ RegSrc = $REG_FAT; ] END; End of subroutine SourceNeckBoneSurface "******************************************************************************" SourceCorticalBone "******************************************************************************" " The SourceCorticalBone subroutine returns particles starting within the " marrow regions of the NMR image. The source is isotropic and uniform within" the Cortical Bone of the CT Image. " The direction is equiprobable, that means that: " Phi is equiprobable within the [0, 2Pi] interval, " Theta is not equiprobable within [0, Pi], but cos(Theta) is " equiprobable within the [-1, 1] interval. " Hence, the Phi and Theta values are (if Rn1 and Rn2 are two random numbers) " Phi = 2*Pi*Rn1 " Theta = arcos(1 2*Pi) " "******************************************************************************" SUBROUTINE SourceCorticalBone(XSrc,YSrc,ZSrc,USrc,VSrc,WSrc,RegSrc); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL XSrc; $REAL YSrc; $REAL ZSrc; $REAL USrc; $REAL VSrc; $REAL WSrc; $INTEGER RegSrc; COMMON variables COMIN/RANDOM,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $REAL Random1, Random2, Random3; $REAL Theta, Phi; LOGICAL InsideSource; system functions invoked in subroutine $REAL DACOS, DCOS, DSIN; INTRINSIC DACOS, DCOS, DSIN; user functions invoked in the subroutine LOGICAL InsideCorticalBone; "-------------------------------------" 1) to return the starting position "-------------------------------------" The three coordinates are first chosen within the image " Then a test checks if it is located within a bone voxel. LOOP [ until the position is inside bone $RANDOMSET Random1; $RANDOMSET Random2; $RANDOMSET Random3; XSrc = $CT_VOXEL_SIZE_X $CT_IMAGE_NX Random1; YSrc = $CT_VOXEL_SIZE_Y $CT_IMAGE_NY Random2; ZSrc = $CT_VOXEL_SIZE_Z $CT_IMAGE_NZ Random3; IF ( InsideCorticalBone(XSrc, YSrc, ZSrc) ) [

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337 InsideSource = .TRUE.; ] ELSE [ InsideSource = .FALSE.; ] ] UNTIL ( InsideSource ); "----------------------------" 2) to return the direction "----------------------------" To choose a random direction. In the spherical coordinate frame: " Phi is equiprobable within the [0, 2Pi] interval " cos(Theta) is equiprobable within the [-1, +1] interval $RANDOMSET Random1; $RANDOMSET Random2; Theta = DACOS(1 2.0*Random1); Phi = 2.0 $PI Random2; USrc = DSIN(Theta) DCOS(Phi); VSrc = DSIN(Theta) DSIN(Phi); WSrc = DCOS(Theta); "--------------------------------" 3) to return the region number "--------------------------------" RegSrc = $REG_CORT; END; End of subroutine SourceCorticalBone "******************************************************************************" HOWFAR "******************************************************************************" " The HOWFAR subroutine measures the distance between the location of the " particle (X0, Y0, Z0) and the next boundary crossed by the particle when " traveling to the direction (Up, Vp, Wp). " The returned values are: " IDISC is set to 1 if we need to discard the particle " USTEP is shortened if the boundary is reached by the particle " IRNEW is set with the region number that lies beyond the boundary " "******************************************************************************" SUBROUTINE HOWFAR; $IMPLICIT-NONE; to make sure that all variables are declared " COMMON variables COMIN/STACK,EPCONT/; The above expands into COMMON statements " STACK contains IR(NP), X,Y,Z(NP), and U,V,W(NP) " EPCONT contains USTEP: the distance EGSnrc is to transport the part. " local variables $REAL X0, Y0, Z0; the position of the particle $REAL Up, Vp, Wp; the direction of the particle $INTEGER IReg; the region number" $REAL Distance; the distance to the boundary $REAL XNew, YNew, ZNew; location of particle after current step " user functions invoked in the subroutine LOGICAL InsideBoneVolume; LOGICAL InsideActiveMarrow; LOGICAL InsideFatMarrow; LOGICAL InsideCorticalBone; $REAL BoundaryDistance; "--------------------------------" 1) To get the data from EGSnrc "--------------------------------" X0 = X(NP); Y0 = Y(NP); Z0 = Z(NP); Up = U(NP); Vp = V(NP); Wp = W(NP); IReg = IR(NP); "-----------------------------------------" 2) To check the data returned by EGSnrc "-----------------------------------------" if a mismatch is detected, the particle is discarded (IDISC=1) " IR(NP) is set to the region $REG_LOST so that AUSGAB can detect the

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338 problem (IRNEW is not used by EGS since it does not transport the " particle before it calls AUSGAB) " a) to check the region numbers IF ( (IReg ~= $REG_TRAB) & (IReg ~= $REG_MARR) & (IReg ~= $REG_FAT) & (IReg ~= $REG_CORT) & (IReg ~= $REG_OUTSIDE) ) [ PRINT *, 'Error in HOWFAR: wrong region number: ', IReg; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] b) to check if the region number matches the location IF (IReg = $REG_TRAB) [ IF (~InsideBoneVolume(X0, Y0, Z0)) [ PRINT *, 'Error in HOWFAR: particle is not in bone.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] ELSEIF (IReg = $REG_MARR) [ IF (~InsideActiveMarrow(X0, Y0, Z0)) [ PRINT *, 'Error in HOWFAR: particle is not in marrow.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] ELSEIF (IReg = $REG_FAT) [ IF (~InsideFatMarrow(X0, Y0, Z0)) [ PRINT *, 'Error in HOWFAR: particle is not in fat.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] ELSEIF (IReg = $REG_CORT) [ IF (~InsideCorticalBone(X0, Y0, Z0)) [ PRINT *, 'Error in HOWFAR: particle is not in cortical.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] ELSE [ IF (InsideBoneVolume(X0, Y0, Z0) | InsideFatMarrow(X0, Y0, Z0) | InsideActiveMarrow(X0, Y0, Z0) | InsideCorticalBone(X0, Y0, Z0) ) [ PRINT *, 'Error in HOWFAR: particle is not outside.'; IDISC = 1; IR(NP) = $REG_LOST; RETURN; ] ] "----------------------------------------------------------------" 3) To discard the particle if it goes outside the study region "----------------------------------------------------------------" IF (IReg = $REG_OUTSIDE) [ IDISC = 1; ] ELSE [ "----------------------------------------------" 4) To calculate the distance to the boundary "----------------------------------------------" Distance = BoundaryDistance(X0, Y0, Z0, Up, Vp, Wp); "----------------------------------------------------------------------" 5) To make sure the particle jumps on the other side of the boundary "----------------------------------------------------------------------" Distance = Distance + $BOUNDARY_THICKNESS; "---------------------------------------------------" 6) To check if the distance is shorter than USTEP "---------------------------------------------------"

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339 IF ( Distance < USTEP ) [ USTEP = Distance; ] "------------------------------------------------" 7) To calculate the region beyond the boundary "------------------------------------------------" a) to calculate the new position XNew = X0 + USTEP*Up; YNew = Y0 + USTEP*Vp; ZNew = Z0 + USTEP*Wp; b) to calculate the new region IF (InsideBoneVolume(XNew, YNew, ZNew)) [ IRNEW = $REG_TRAB; ] ELSEIF ((InsideActiveMarrow(XNew, YNew, ZNew))) [ IRNEW = $REG_MARR; ] ELSEIF (InsideCorticalBone(XNew, YNew, ZNew)) [ IRNEW = $REG_CORT; ] ELSEIF ((InsideFatMarrow(XNew, YNew, ZNew))) [ IRNEW = $REG_FAT; ] ELSE [ IRNEW = $REG_OUTSIDE; ] ] END; End of subroutine HOWFAR "******************************************************************************" HOWNEAR "******************************************************************************" " The HOWNEAR subroutine measures the shortest distance between the location " of the particle (X0, Y0, Z0) and the boundary of the actual region IReg. " The returned values are: " TPerp is the shortest distance from the particle location to " the boundary of the region IReg " "******************************************************************************" SUBROUTINE HOWNEAR(TPerp, X0, Y0, Z0, IReg); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL TPerp; the shortest distance to return $REAL X0, Y0, Z0; the current location of the particle $INTEGER IReg; the current region of the particle " user functions invoked in the subroutine $REAL ClosestBoundary; "-----------------------------------------------------" 1) To check if the particle has become out of study "-----------------------------------------------------" IF (IReg = $REG_OUTSIDE) [ TPerp = 0.0; so that HOWFAR is called and discard the particle ] ELSE [ "---------------------------------------" 2) To calculate the shortest distance "---------------------------------------" TPerp = ClosestBoundary(X0, Y0, Z0); "--------------------------------------------------------------------" 3) To make sure the particle will not be too close to the boundary "--------------------------------------------------------------------" TPerp = TPerp $BOUNDARY_THICKNESS; IF (TPerp < 0.0) [ TPerp = 0.0; ] ] END; End of subroutine HOWNEAR

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340 "******************************************************************************" AUSGAB "******************************************************************************" " The AUSGAB subroutine cumulates the energy deposited within the regions. " The energy is stored in the 'CumulEnergy' variables. " " Input: " IARG : A flag (see EGSnrc documentation) which is set to 3 if the " particle is discarded by the HOWFAR subroutine, in our " situation, that means that the particle is going outside " the study region or that it has been lost. " "******************************************************************************" SUBROUTINE AUSGAB(IARG); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $INTEGER IARG; $REAL X0, Y0, Z0; COMMON variables COMIN/STACK,EPCONT,SCOR/; The above expands into COMMON statements " STACK contains IR(NP) " EPCONT contains EDEP: the energy deposited now " SCOR contains the variables to cumulate the energy deposited " local variables $INTEGER IReg; to store the region number" "--------------------------------" 1) To get the data from EGSnrc "--------------------------------" IReg = IR(NP); X0 = X(NP); Y0 = Y(NP); Z0 = Z(NP); "---------------------------------------------------------" 2) To test if the particle has been discarded by HOWFAR "---------------------------------------------------------" IF (IARG = 3) [ test why it has been discarded IF (IReg = $REG_OUTSIDE) [ CumulEnergyOutside = CumulEnergyOutside + EDEP; ] ELSEIF (IReg = $REG_LOST) [ CumulEnergyLost = CumulEnergyLost + EDEP; ] ELSE [ PRINT *, 'Error in AUSGAB: wrong region number after discard.'; RETURN; ] ] ELSE [ "-----------------------------------------------" 3) To cumulate the energy in the right region "-----------------------------------------------" IF (IReg = $REG_TRAB) [ CumulEnergyTrabeculae = CumulEnergyTrabeculae + EDEP; ] ELSEIF (IReg = $REG_MARR) [ CALL BoneEndosteum; ] ELSEIF (IReg = $REG_CORT) [ CumulEnergyCortical = CumulEnergyCortical + EDEP; ] ELSEIF (IReg = $REG_FAT) [ CALL BoneEndosteum; ] ELSE [ PRINT *, 'Error in AUSGAB: wrong region number after transport.'; RETURN; ]

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341 ] END; End of subroutine AUSGAB "******************************************************************************" Function InsideBoneVolume "******************************************************************************" " Test if a given position (X, Y, Z) is inside the trabeculae voxels of the " duplicated NMR image and in the spongiosa region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideBoneVolume(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; system functions invoked in the main program $INTEGER MOD; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideBoneVolume = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideBoneVolume = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the NMR image "---------------------------------------------------" I = (X / $NMR_VOXEL_SIZE_X); I = MOD(I, $NMR_IMAGE_NX); to shift to the copy of the image J = (Y / $NMR_VOXEL_SIZE_Y); J = MOD(J, $NMR_IMAGE_NY); to shift to the copy of the image K = (Z / $NMR_VOXEL_SIZE_Z);

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342 K = MOD(K, $NMR_IMAGE_NZ); to shift to the copy of the image "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = NMRBoneImage(I,J,K); IF (VoxelValue2 = CHAR(0)) [ InsideBoneVolume = .TRUE.; ] ELSE [ InsideBoneVolume = .FALSE.; ] ] ] END; End of function InsideBoneVolume "******************************************************************************" Function InsideActiveMarrow "******************************************************************************" " Test if a given position (X, Y, Z) is inside the active marrow voxels of " the duplicated NMR image and in the spongiosa region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideActiveMarrow(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; system functions invoked in the main program $INTEGER MOD; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideActiveMarrow = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue;

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343 ] IF (VoxelValue = $MED_CORT) [ InsideActiveMarrow = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the NMR image "---------------------------------------------------" I = (X / $NMR_VOXEL_SIZE_X); I = MOD(I, $NMR_IMAGE_NX); to shift to the copy of the image J = (Y / $NMR_VOXEL_SIZE_Y); J = MOD(J, $NMR_IMAGE_NY); to shift to the copy of the image K = (Z / $NMR_VOXEL_SIZE_Z); K = MOD(K, $NMR_IMAGE_NZ); to shift to the copy of the image "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = NMRBoneImage(I,J,K); IF ((VoxelValue2 = CHAR(255))) [ InsideActiveMarrow = .TRUE.; ] ELSE [ InsideActiveMarrow = .FALSE.; ] ] ] END; End of function InsideActiveMarrow "******************************************************************************" Function InsideBoneEndosteum "******************************************************************************" " Test if a given position (X, Y, Z) is inside the bone endosteum of the " marrow cavity voxels of the duplicated NMR image and in the spongiosa " region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideBoneEndosteum(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER I2, J2, K2; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; system functions invoked in the main program $INTEGER MOD; CHARACTER EDGEIPOS,EDGEINEG,EDGEJPOS,EDGEJNEG,EDGEKPOS,EDGEKNEG; $REAL P1, P2, P3, P4, P5, P6, PDIST; $REAL XMax, XMin, YMax, YMin, ZMax, ZMin; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI

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344 "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideBoneEndosteum = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideBoneEndosteum = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the NMR image "---------------------------------------------------" I = (X / $NMR_VOXEL_SIZE_X); I = MOD(I, $NMR_IMAGE_NX); to shift to the copy of the image J = (Y / $NMR_VOXEL_SIZE_Y); J = MOD(J, $NMR_IMAGE_NY); to shift to the copy of the image K = (Z / $NMR_VOXEL_SIZE_Z); K = MOD(K, $NMR_IMAGE_NZ); to shift to the copy of the image "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = NMRBoneImage(I,J,K); IF ((VoxelValue2 = CHAR(255)) .OR. (VoxelValue2 = CHAR(122)))[ P1 = 1.0; P2 = 1.0; P3 = 1.0; P4 = 1.0; P5 = 1.0; P6 = 1.0; EDGEIPOS = CHAR(255); EDGEINEG = CHAR(255); EDGEJPOS = CHAR(255); EDGEJNEG = CHAR(255); EDGEKPOS = CHAR(255); EDGEKNEG = CHAR(255); "CHECK FOR BONE VOXEL NEIGHBORS" "DETERMINE WHERE BONE SURFACES ARE(IF THEY ARE)" IF (I .EQ. ($NMR_IMAGE_NX)) [ EDGEIPOS = NMRBoneImage(1,J,K);; ] ELSE [ EDGEIPOS = NMRBoneImage(I+1,J,K) ; ] IF (I .EQ. (1)) [ EDGEINEG = NMRBoneImage($NMR_IMAGE_NX,J,K); ] ELSE [ EDGEINEG = NMRBoneImage(I-1,J,K) ; ] IF (J .EQ. ($NMR_IMAGE_NY)) [ EDGEJPOS = NMRBoneImage(I,1,K);

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345 ] ELSE [ EDGEJPOS = NMRBoneImage(I,J+1,K) ; ] IF (J .EQ. (1)) [ EDGEJNEG = NMRBoneImage(I,$NMR_IMAGE_NY,K); ] ELSE [ EDGEJNEG = NMRBoneImage(I,J-1,K) ; ] IF (K .EQ. ($NMR_IMAGE_NZ)) [ EDGEKPOS = NMRBoneImage(I,J,1); ] ELSE [ EDGEKPOS = NMRBoneImage(I,J,K+1) ; ] IF (K .EQ. (1)) [ EDGEKNEG = NMRBoneImage(I,J,$NMR_IMAGE_NZ); ] ELSE [ EDGEKNEG = NMRBoneImage(I,J,K-1) ; ] I2 = (X / $NMR_VOXEL_SIZE_X); J2 = (Y / $NMR_VOXEL_SIZE_Y); K2 = (Z / $NMR_VOXEL_SIZE_Z); XMin = (I2) $NMR_VOXEL_SIZE_X; XMax = XMin + $NMR_VOXEL_SIZE_X; YMin = (J2) $NMR_VOXEL_SIZE_Y; YMax = YMin + $NMR_VOXEL_SIZE_Y; ZMin = (K2) $NMR_VOXEL_SIZE_Z; ZMax = ZMin + $NMR_VOXEL_SIZE_Z; IF(EDGEIPOS .EQ. CHAR(0)) [ P1= XMax X; ] IF(EDGEINEG .EQ. CHAR(0)) [ P2= X XMin; ] IF(EDGEJPOS .EQ. CHAR(0)) [ P3= YMax Y; ] IF(EDGEJNEG .EQ. CHAR(0)) [ P4= Y YMin; ] IF(EDGEKPOS .EQ. CHAR(0)) [ P5= ZMax Z; ] IF(EDGEKNEG .EQ. CHAR(0)) [ P6 = Z ZMin; ] PDIST=10.0; IF (P1 .LE. PDIST) [ PDIST = P1; ] IF (P2 .LE. PDIST) [ PDIST = P2; ] IF (P3 .LE. PDIST) [ PDIST = P3; ] IF (P4 .LE. PDIST) [ PDIST = P4; ] IF (P5 .LE. PDIST) [ PDIST = P5;

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346 ] IF (P6 .LE. PDIST) [ PDIST = P6; ] IF (PDIST .LT. 0.0) [ PRINT *, ERROR IN PDIST'; PRINT *, PDIST ', PDIST; ] IF (PDIST .LE. 0.0010) [ InsideBoneEndosteum = .TRUE.; ] ELSE [ InsideBoneEndosteum = .FALSE.; ] ] ELSE [ InsideBoneEndosteum = .FALSE.; ] ] ] END; End of function InsideBoneEndosteum "******************************************************************************" Function InsideBoneSurface "******************************************************************************" " Test if a given position (X, Y, Z) is inside the bone endosteum of the " marrow cavity voxels of the duplicated NMR image and in the spongiosa " region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideBoneSurface(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER I2, J2, K2; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; system functions invoked in the main program $INTEGER MOD; CHARACTER EDGEIPOS,EDGEINEG,EDGEJPOS,EDGEJNEG,EDGEKPOS,EDGEKNEG; $REAL P1, P2, P3, P4, P5, P6, PDIST; $REAL XMax, XMin, YMax, YMin, ZMax, ZMin; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideBoneSurface = .FALSE.; ] ELSE [

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347 "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideBoneSurface = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the NMR image "---------------------------------------------------" I = (X / $NMR_VOXEL_SIZE_X); I = MOD(I, $NMR_IMAGE_NX); to shift to the copy of the image J = (Y / $NMR_VOXEL_SIZE_Y); J = MOD(J, $NMR_IMAGE_NY); to shift to the copy of the image K = (Z / $NMR_VOXEL_SIZE_Z); K = MOD(K, $NMR_IMAGE_NZ); to shift to the copy of the image "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = NMRBoneImage(I,J,K); IF ((VoxelValue2 = CHAR($MED_MARR)) .OR. (VoxelValue2 = CHAR($MED_FAT)))[ P1 = 1.0; P2 = 1.0; P3 = 1.0; P4 = 1.0; P5 = 1.0; P6 = 1.0; EDGEIPOS = CHAR(255); EDGEINEG = CHAR(255); EDGEJPOS = CHAR(255); EDGEJNEG = CHAR(255); EDGEKPOS = CHAR(255); EDGEKNEG = CHAR(255); "CHECK FOR BONE VOXEL NEIGHBORS" "DETERMINE WHERE BONE SURFACES ARE(IF THEY ARE)" IF (I .EQ. ($NMR_IMAGE_NX)) [ EDGEIPOS = NMRBoneImage(1,J,K);; ] ELSE [ EDGEIPOS = NMRBoneImage(I+1,J,K) ; ] IF (I .EQ. (1)) [ EDGEINEG = NMRBoneImage($NMR_IMAGE_NX,J,K); ] ELSE [ EDGEINEG = NMRBoneImage(I-1,J,K) ; ] IF (J .EQ. ($NMR_IMAGE_NY)) [ EDGEJPOS = NMRBoneImage(I,1,K); ] ELSE [ EDGEJPOS = NMRBoneImage(I,J+1,K) ; ]

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348 IF (J .EQ. (1)) [ EDGEJNEG = NMRBoneImage(I,$NMR_IMAGE_NY,K); ] ELSE [ EDGEJNEG = NMRBoneImage(I,J-1,K) ; ] IF (K .EQ. ($NMR_IMAGE_NZ)) [ EDGEKPOS = NMRBoneImage(I,J,1); ] ELSE [ EDGEKPOS = NMRBoneImage(I,J,K+1) ; ] IF (K .EQ. (1)) [ EDGEKNEG = NMRBoneImage(I,J,$NMR_IMAGE_NZ); ] ELSE [ EDGEKNEG = NMRBoneImage(I,J,K-1) ; ] I2 = (X / $NMR_VOXEL_SIZE_X); J2 = (Y / $NMR_VOXEL_SIZE_Y); K2 = (Z / $NMR_VOXEL_SIZE_Z); XMin = (I2) $NMR_VOXEL_SIZE_X; XMax = XMin + $NMR_VOXEL_SIZE_X; YMin = (J2) $NMR_VOXEL_SIZE_Y; YMax = YMin + $NMR_VOXEL_SIZE_Y; ZMin = (K2) $NMR_VOXEL_SIZE_Z; ZMax = ZMin + $NMR_VOXEL_SIZE_Z; IF(EDGEIPOS .EQ. CHAR(0)) [ P1= XMax X; ] IF(EDGEINEG .EQ. CHAR(0)) [ P2= X XMin; ] IF(EDGEJPOS .EQ. CHAR(0)) [ P3= YMax Y; ] IF(EDGEJNEG .EQ. CHAR(0)) [ P4= Y YMin; ] IF(EDGEKPOS .EQ. CHAR(0)) [ P5= ZMax Z; ] IF(EDGEKNEG .EQ. CHAR(0)) [ P6 = Z ZMin; ] PDIST=10.0; IF (P1 .LE. PDIST) [ PDIST = P1; ] IF (P2 .LE. PDIST) [ PDIST = P2; ] IF (P3 .LE. PDIST) [ PDIST = P3; ] IF (P4 .LE. PDIST) [ PDIST = P4; ] IF (P5 .LE. PDIST) [ PDIST = P5; ] IF (P6 .LE. PDIST) [ PDIST = P6; ]

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349 IF (PDIST .LT. 0.0) [ PRINT *, ERROR IN PDIST'; PRINT *, PDIST ', PDIST; ] IF (PDIST .LE. 0.000010) [ InsideBoneSurface = .TRUE.; ] ELSE [ InsideBoneSurface = .FALSE.; ] ] ELSE [ InsideBoneSurface = .FALSE.; ] ] ] END; End of function InsideBoneSurface "******************************************************************************" Function InsideFatMarrow "******************************************************************************" " Test if a given position (X, Y, Z) is inside the Fat voxels of the " duplicated NMR image and in the spongiosa region of the CT image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideFatMarrow(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; system functions invoked in the main program $INTEGER MOD; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideFatMarrow = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------"

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350 VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideFatMarrow = .FALSE.; ] ELSE [ "---------------------------------------------------" 4) to calculate the voxel number in the NMR image "---------------------------------------------------" I = (X / $NMR_VOXEL_SIZE_X); I = MOD(I, $NMR_IMAGE_NX); to shift to the copy of the image J = (Y / $NMR_VOXEL_SIZE_Y); J = MOD(J, $NMR_IMAGE_NY); to shift to the copy of the image K = (Z / $NMR_VOXEL_SIZE_Z); K = MOD(K, $NMR_IMAGE_NZ); to shift to the copy of the image "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = NMRBoneImage(I,J,K); IF ((VoxelValue2 = CHAR($MED_FAT))) [ InsideFatMarrow = .TRUE.; ] ELSE [ InsideFatMarrow = .FALSE.; ] ] ] END; End of function InsideFatMarrow "******************************************************************************" Function InsideHeadSpongiosa "******************************************************************************" " Test if a given position (X, Y, Z) is inside the cortical region of the " image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideHeadSpongiosa(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideHeadSpongiosa = .FALSE.; ] ELSE [ "--------------------------------------------------"

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351 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] "PRINT *, 'Called InsideHeadSpongiosa ';" IF (VoxelValue = $MED_HEADSPONG) [ InsideHeadSpongiosa = .TRUE.; "PRINT *, 'Found HeadSpongiosa ';" ] ELSE [ InsideHeadSpongiosa = .FALSE.; ] ] END; End of function InsideHeadSpongiosa "******************************************************************************" Function InsideNeckSpongiosa "******************************************************************************" " Test if a given position (X, Y, Z) is inside the cortical region of the " image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideNeckSpongiosa(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself " user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideNeckSpongiosa = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1;

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352 "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_NECKSPONG) [ .OR. (VoxelValue = $MED_HEADSPONG)) [" InsideNeckSpongiosa = .TRUE.; ] ELSE [ InsideNeckSpongiosa = .FALSE.; ] ] END; End of function InsideNeckSpongiosa "******************************************************************************" Function InsideCorticalBone "******************************************************************************" " Test if a given position (X, Y, Z) is inside the cortical region of the " image. " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the region. " .FALSE. if the position is not inside the region. " "******************************************************************************" LOGICAL FUNCTION InsideCorticalBone(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself " user functions invoked in the subroutine LOGICAL InsideCT_CortShell; "--------------------------------------------" 1) to check if (X, Y, Z) is inside the ROI "--------------------------------------------" IF (~InsideCT_CortShell(X, Y, Z)) [ InsideCorticalBone = .FALSE.; ] ELSE [ "--------------------------------------------------" 2) to calculate the voxel number in the CT image "--------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; "--------------------------------" 3) to get and test the medium "--------------------------------" VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_CORT) [ InsideCorticalBone = .TRUE.; ]

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353 ELSE [ InsideCorticalBone = .FALSE.; ] ] END; End of function InsideCorticalBone "******************************************************************************" Function InsideCT_CortShell "******************************************************************************" " Test if a given position (X, Y, Z) is inside the limits of the CT image " The outer limit of the CT image is 512 x 512 " Also, test if the given position is in the ROI within the CT image " ROI defined by everything within outside edge of CorticalBone (not tissue) " " Input: " X, Y, Z: the position to be tested. " " Return: " .TRUE. if the position is inside the CT image. " .FALSE. if the position is not inside the CT image. " "******************************************************************************" LOGICAL FUNCTION InsideCT_CortShell(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables COMIN/GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself "-------------------------------------" 1) to check if outside the CT image "-------------------------------------" IF ( (X < 0.0) | (X >= $CT_IMAGE_NX $CT_VOXEL_SIZE_X) | (Y < 0.0) | (Y >= $CT_IMAGE_NY $CT_VOXEL_SIZE_Y) | (Z < 0.0) | (Z >= $CT_IMAGE_NZ $CT_VOXEL_SIZE_Z) ) [ InsideCT_CortShell = .FALSE.; ] ELSE [ "-----------------------------------------------------" 2) to check if in the tissue region of the CT image "-----------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); VoxelNo = (K*$CT_IMAGE_NY + J)*$CT_IMAGE_NX + I + 1; VoxelValue = CTBoneImage(VoxelNo); IF (VoxelValue < 0) [ VoxelValue = 256 + VoxelValue; ] IF (VoxelValue = $MED_TISS) [ InsideCT_CortShell = .FALSE.; ] ELSE [ InsideCT_CortShell = .TRUE.; ] ] END; End of function InsideCT_CortShell "******************************************************************************" Function BoundaryDistance "******************************************************************************" "

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354 Returns the distance from the position (X, Y, Z) to the nearest boundary " of the voxel when following the direction (U, V, W) " The two images are tested and the closest voxel limit is returned. " " Input: " X, Y, Z: the position to be tested. " U, V, W: the direction to follow. " " Return: " the distance to the boundary. " "******************************************************************************" $REAL FUNCTION BoundaryDistance(X, Y, Z, U, V, W); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; $REAL U, V, W; local variables $REAL Distance; $REAL ShortestDistance; $INTEGER I, J, K; to store the position of the voxel $REAL XMin, YMin, ZMin; for the boundary of the voxel $REAL XMax, YMax, ZMax; for the boundary of the voxel "-------------------------------------------------------------------" 1) to calculate the boundary of the current voxel in the CT image "-------------------------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); XMin = I $CT_VOXEL_SIZE_X; XMax = XMin + $CT_VOXEL_SIZE_X; YMin = J $CT_VOXEL_SIZE_Y; YMax = YMin + $CT_VOXEL_SIZE_Y; ZMin = K $CT_VOXEL_SIZE_Z; ZMax = ZMin + $CT_VOXEL_SIZE_Z; "---------------------------------------------------------" 2) to measure the distance to the boundary of the voxel "---------------------------------------------------------" ShortestDistance = $INFINITY; a) along the X axis IF ( U > 0.0 ) [ Distance = (XMax X) / U; ] ELSEIF ( U < 0.0 ) [ Distance = (XMin X) / U; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] b) along the Y axis IF ( V > 0.0 ) [ Distance = (YMax Y) / V; ] ELSEIF ( V < 0.0 ) [ Distance = (YMin Y) / V; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] c) along the Z axis IF ( W > 0.0 ) [ Distance = (ZMax Z) / W; ] ELSEIF ( W < 0.0 ) [

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355 Distance = (ZMin Z) / W; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] "--------------------------------------------------------------------" 3) to calculate the boundary of the current voxel in the NMR image "--------------------------------------------------------------------" I = (X / $NMR_VOXEL_SIZE_X); J = (Y / $NMR_VOXEL_SIZE_Y); K = (Z / $NMR_VOXEL_SIZE_Z); XMin = I $NMR_VOXEL_SIZE_X; XMax = XMin + $NMR_VOXEL_SIZE_X; YMin = J $NMR_VOXEL_SIZE_Y; YMax = YMin + $NMR_VOXEL_SIZE_Y; ZMin = K $NMR_VOXEL_SIZE_Z; ZMax = ZMin + $NMR_VOXEL_SIZE_Z; "---------------------------------------------------------" 4) to measure the distance to the boundary of the voxel "---------------------------------------------------------" a) along the X axis IF ( U > 0.0 ) [ Distance = (XMax X) / U; ] ELSEIF ( U < 0.0 ) [ Distance = (XMin X) / U; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] b) along the Y axis IF ( V > 0.0 ) [ Distance = (YMax Y) / V; ] ELSEIF ( V < 0.0 ) [ Distance = (YMin Y) / V; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] c) along the Z axis IF ( W > 0.0 ) [ Distance = (ZMax Z) / W; ] ELSEIF ( W < 0.0 ) [ Distance = (ZMin Z) / W; ] ELSE [ Distance = $INFINITY; ] IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] "---------------------------" 5) to return the distance "---------------------------" BoundaryDistance = ShortestDistance; END; End of function BoundaryDistance "******************************************************************************" Function ClosestBoundary

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356 "******************************************************************************" " Returns the shortest distance from the position (X, Y, Z) to the nearest " boundary of the voxel. " The two images are tested and the closest voxel limit is returned. " " Input: " X, Y, Z: the position to be tested. " " Return: " the shortest distance to the boundary. " "******************************************************************************" $REAL FUNCTION ClosestBoundary(X, Y, Z); $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X, Y, Z; COMMON variables " local variables $REAL Distance; $REAL ShortestDistance; $INTEGER I, J, K; to store the position of the voxel $REAL XMin, YMin, ZMin; for the boundary of the voxel $REAL XMax, YMax, ZMax; for the boundary of the voxel "-------------------------------------------------------------------" 1) to calculate the boundary of the current voxel in the CT image "-------------------------------------------------------------------" I = (X / $CT_VOXEL_SIZE_X); J = (Y / $CT_VOXEL_SIZE_Y); K = (Z / $CT_VOXEL_SIZE_Z); XMin = I $CT_VOXEL_SIZE_X; XMax = XMin + $CT_VOXEL_SIZE_X; YMin = J $CT_VOXEL_SIZE_Y; YMax = YMin + $CT_VOXEL_SIZE_Y; ZMin = K $CT_VOXEL_SIZE_Z; ZMax = ZMin + $CT_VOXEL_SIZE_Z; "---------------------------------------------------------" 2) to measure the distance to the boundary of the voxel "---------------------------------------------------------" ShortestDistance = $INFINITY; Distance = X XMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = XMax X; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = Y YMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = YMax Y; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = Z ZMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = ZMax Z; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] "--------------------------------------------------------------------" 3) to calculate the boundary of the current voxel in the NMR image "--------------------------------------------------------------------" I = (X / $NMR_VOXEL_SIZE_X); J = (Y / $NMR_VOXEL_SIZE_Y); K = (Z / $NMR_VOXEL_SIZE_Z);

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357 XMin = I $NMR_VOXEL_SIZE_X; XMax = XMin + $NMR_VOXEL_SIZE_X; YMin = J $NMR_VOXEL_SIZE_Y; YMax = YMin + $NMR_VOXEL_SIZE_Y; ZMin = K $NMR_VOXEL_SIZE_Z; ZMax = ZMin + $NMR_VOXEL_SIZE_Z; "---------------------------------------------------------" 4) to measure the distance to the boundary of the voxel "---------------------------------------------------------" Distance = X XMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = XMax X; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = Y YMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = YMax Y; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = Z ZMin; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] Distance = ZMax Z; IF ( Distance < ShortestDistance ) [ ShortestDistance = Distance; ] "---------------------------" 5) to return the distance "---------------------------" ClosestBoundary = ShortestDistance; END; End of function ClosestBoundary "******************************************************************************" BoneEndosteum "******************************************************************************" " " "******************************************************************************" SUBROUTINE BoneEndosteum; $IMPLICIT-NONE; to make sure that all variables are declared " parameters of the routine $REAL X0, Y0, Z0; COMMON variables COMIN/STACK,EPCONT,SCOR,GEOM/; The above expands into COMMON statements " GEOM contains the image " local variables $INTEGER I, J, K; to store the position of the voxel $INTEGER IReg; the region number" $INTEGER VoxelNo; the voxel number within the image $INTEGER VoxelValue; the voxel itself CHARACTER VoxelValue2; system functions invoked in the main program $INTEGER MOD; CHARACTER EDGEIPOS,EDGEINEG,EDGEJPOS,EDGEJNEG,EDGEKPOS,EDGEKNEG; $REAL P1, P2, P3, P4, P5, P6, PDIST; $REAL XMax, XMin, YMax, YMin, ZMax, ZMin; INTRINSIC MOD; user functions invoked in the subroutine LOGICAL InsideCT_CortShell; PDIST = 10.0; X0 = X(NP); Y0 = Y(NP);

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358 Z0 = Z(NP); IReg = IR(NP); VoxelNo = 0; "---------------------------------------------------" 4) to calculate the voxel number in the NMR image "---------------------------------------------------" I = (X0 / $NMR_VOXEL_SIZE_X); I = MOD(I, $NMR_IMAGE_NX); to shift to the copy of the image J = (Y0 / $NMR_VOXEL_SIZE_Y); J = MOD(J, $NMR_IMAGE_NY); to shift to the copy of the image K = (Z0 / $NMR_VOXEL_SIZE_Z); K = MOD(K, $NMR_IMAGE_NZ); to shift to the copy of the image "--------------------------------" 5) to get and test the medium "--------------------------------" VoxelValue2 = NMRBoneImage(I,J,K); IF ((VoxelValue2 = CHAR($MED_MARR)) .OR. (VoxelValue2 = CHAR($MED_FAT)))[ P1 = 1.0; P2 = 1.0; P3 = 1.0; P4 = 1.0; P5 = 1.0; P6 = 1.0; EDGEIPOS = CHAR(255); EDGEINEG = CHAR(255); EDGEJPOS = CHAR(255); EDGEJNEG = CHAR(255); EDGEKPOS = CHAR(255); EDGEKNEG = CHAR(255); "CHECK FOR BONE VOXEL NEIGHBORS" "DETERMINE WHERE BONE SURFACES ARE(IF THEY ARE)" IF (I .EQ. ($NMR_IMAGE_NX)) [ EDGEIPOS = NMRBoneImage(1,J,K);; ] ELSE [ EDGEIPOS = NMRBoneImage(I+1,J,K) ; ] IF (I .EQ. (1)) [ EDGEINEG = NMRBoneImage($NMR_IMAGE_NX,J,K); ] ELSE [ EDGEINEG = NMRBoneImage(I-1,J,K) ; ] IF (J .EQ. ($NMR_IMAGE_NY)) [ EDGEJPOS = NMRBoneImage(I,1,K); ] ELSE [ EDGEJPOS = NMRBoneImage(I,J+1,K) ; ] IF (J .EQ. (1)) [ EDGEJNEG = NMRBoneImage(I,$NMR_IMAGE_NY,K); ] ELSE [ EDGEJNEG = NMRBoneImage(I,J-1,K) ; ] IF (K .EQ. ($NMR_IMAGE_NZ)) [ EDGEKPOS = NMRBoneImage(I,J,1); ] ELSE [ EDGEKPOS = NMRBoneImage(I,J,K+1) ; ] IF (K .EQ. (1)) [ EDGEKNEG = NMRBoneImage(I,J,$NMR_IMAGE_NZ); ] ELSE [ EDGEKNEG = NMRBoneImage(I,J,K-1) ; ]

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359 I = (X0 / $NMR_VOXEL_SIZE_X); J = (Y0 / $NMR_VOXEL_SIZE_Y); K = (Z0 / $NMR_VOXEL_SIZE_Z); XMin = I $NMR_VOXEL_SIZE_X; XMax = XMin + $NMR_VOXEL_SIZE_X; YMin = J $NMR_VOXEL_SIZE_Y; YMax = YMin + $NMR_VOXEL_SIZE_Y; ZMin = K $NMR_VOXEL_SIZE_Z; ZMax = ZMin + $NMR_VOXEL_SIZE_Z; IF(EDGEIPOS .EQ. CHAR($MED_BONE)) [ P1 = XMax X0; ] IF(EDGEINEG .EQ. CHAR($MED_BONE)) [ P2 = X0 XMin; ] IF(EDGEJPOS .EQ. CHAR($MED_BONE)) [ P3 = YMax Y0; ] IF(EDGEJNEG .EQ. CHAR($MED_BONE)) [ P4 = Y0 YMin; ] IF(EDGEKPOS .EQ. CHAR($MED_BONE)) [ P5 = ZMax Z0; ] IF(EDGEKNEG .EQ. CHAR($MED_BONE)) [ P6 = Z0 ZMin; ] IF (P1 .LE. PDIST) [ PDIST = P1; ] IF (P2 .LE. PDIST) [ PDIST = P2; ] IF (P3 .LE. PDIST) [ PDIST = P3; ] IF (P4 .LE. PDIST) [ PDIST = P4; ] IF (P5 .LE. PDIST) [ PDIST = P5; ] IF (P6 .LE. PDIST) [ PDIST = P6; ] IF ((PDIST .LT. 0.0))[ PRINT *, ERROR IN PDIST'; PRINT *, PDIST ', PDIST; ] IF (PDIST .LE. 0.0010) [ CumulEnergyEndo = CumulEnergyEndo + EDEP; ] ELSE [ IF (IReg = $REG_MARR) [ CumulEnergyMarrow = CumulEnergyMarrow + EDEP; ] ELSE [ CumulEnergyFat = CumulEnergyFat + EDEP; ] ] ] ELSE [ PRINT *, ERROR IN BoneEndosteum'; ] RETURN; END; End of BoneEndosteum

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360 APPENDIX G 66-YEAR UF REFERENCE MALE CANCER PATIENT CHORD-LENGTH DISTRIBUTIONS AND THE TRILIN EAR CHORD-LENGTH CALCULATION This appendix contains the necessary soft ware to generate tables of chord-length values. The generation of chord-length distribution was done using the Trilinear Interpolation Marching Cube Algorithm as given by Rajon (2003). This appendix provides the chord-length distribution code using trilinear te chniques. This appendix also contains the table of chord-length distri butions within the bone trabeculae and marrow cavities for the 66-year UF reference male can cer patient. Chapter 5 provides a complete discussion of the data and methods provided within this appendix. Figure G-1. Schematic of how chord lengths and electrons could tr avel through a section of trabecular spongiosa.

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361 Chord-Length Distributions through the bo ne trabeculae of the 66-year UF RMCP Table G-1. Normalized 3D c hord-length distributions throug h the bone trabeculae of the left and right femur heads, left and ri ght femur necks, and their respective averages. Probability distributions are given through the first 50 bins with a maximum length of 1000 microns. bin width Left Femur Head Left Femur Neck Average Femur Head Right Femur Head Right Femur Neck Average Femur Neckm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 10 0.00059240.00055730.00061560.00066530.00053330.0005436 30 0.00068790.00079170.00071030.00075850.00077090.0007798 50 0.00081030.00093160.00083250.00088000.00090130.0009143 70 0.00101260.00104230.00104550.00111590.00104320.0010428 90 0.00142460.00124250.00146670.00155670.00130360.0012774 110 0.00180900.00148170.00186330.00197950.00159940.0015490 130 0.00213070.00178820.00217730.00227730.00193270.0018708 150 0.00243890.00213030.00248520.00258430.00230730.0022314 170 0.00263710.00244400.00266430.00272250.00264060.0025563 190 0.00274010.00268970.00274220.00274670.00283040.0027701 210 0.00272170.00283700.00269480.00263730.00292220.0028856 230 0.00261030.00282320.00257490.00249900.00286610.0028477 250 0.00245880.00266250.00241360.00231690.00271860.0026945 270 0.00226300.00245780.00221460.00211080.00249600.0024796 290 0.00205410.00221410.00201400.00192810.00226380.0022425 310 0.00184140.00196040.00180870.00173840.00198400.0019739 330 0.00163450.00173470.00160560.00154370.00175390.0017457 350 0.00146780.00152840.00144010.00138080.00151710.0015219 370 0.00131410.00134580.00129580.00125640.00132220.0013323 390 0.00116520.00119240.00115400.00113000.00116500.0011767 410 0.00105240.00106350.00104290.00102240.00101400.0010352 430 0.00092590.00094220.00092410.00092030.00090050.0009184 450 0.00083670.00085000.00084290.00085630.00079650.0008195 470 0.00075450.00076430.00076180.00077740.00072210.0007402 490 0.00068790.00069430.00069870.00072160.00064440.0006658 510 0.00063890.00062870.00064510.00065840.00058030.0006011 530 0.00057890.00058100.00058980.00061310.00053450.0005544 550 0.00053590.00052810.00054610.00056810.00048350.0005026 570 0.00049970.00048890.00050560.00051820.00044080.0004615 590 0.00045670.00044910.00046260.00047510.00041100.0004273 610 0.00042580.00041530.00043340.00044990.00037890.0003945 630 0.00039700.00038470.00040250.00041410.00034340.0003611 650 0.00036990.00036050.00037350.00038100.00033130.0003438 670 0.00035280.00032850.00035400.00035660.00030170.0003132 690 0.00033300.00031540.00033420.00033670.00028410.0002975 710 0.00030500.00028810.00030700.00031120.00026380.0002742 730 0.00029150.00027210.00029150.00029150.00024570.0002570 750 0.00027200.00025400.00027100.00026900.00023190.0002414 770 0.00025950.00023740.00025800.00025460.00021570.0002250 790 0.00023820.00022600.00023730.00023520.00020290.0002128 810 0.00021950.00020780.00022210.00022770.00018950.0001974 830 0.00021200.00019340.00021310.00021540.00017690.0001840 850 0.00020520.00018740.00020360.00020010.00016740.0001760 870 0.00018640.00017050.00018520.00018250.00015730.0001630 890 0.00017830.00016230.00017770.00017630.00014800.0001541 910 0.00016330.00015630.00016300.00016240.00014090.0001475 930 0.00015960.00014510.00015640.00014970.00012750.0001351 950 0.00014410.00013610.00014200.00013740.00012340.0001288 970 0.00013880.00012650.00013800.00013610.00011420.0001195 9900.00012850.00012210.00012700.00012370.00010720.0001136

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362 Table G-2. Normalized 3D c hord-length distributions throug h the bone trabeculae of the left and right femur heads, left and ri ght femur necks, and their respective averages. Probability distributions are given through the second 50 bins for a maximum length of 2000 microns. bin width Left Femur Head Left Femur Neck Average Femur Head Right Femur Head Right Femur Neck Average Femur Neckm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 1010 0.00011950.00011600.00011770.00011390.00010400.0001092 1030 0.00011440.00010770.00011260.00010870.00009670.0001014 1050 0.00010620.00010120.00010650.00010700.00009310.0000966 1070 0.00009910.00009830.00009670.00009160.00008430.0000903 1090 0.00009370.00008910.00009210.00008870.00008520.0000869 1110 0.00009100.00008690.00008960.00008660.00007840.0000820 1130 0.00008350.00008000.00008280.00008120.00007190.0000754 1150 0.00007980.00007720.00007890.00007680.00007250.0000745 1170 0.00007470.00007430.00007280.00006880.00006540.0000692 1190 0.00006750.00007150.00006630.00006380.00006330.0000668 1210 0.00006670.00006620.00006500.00006150.00005890.0000620 1230 0.00006150.00006300.00005940.00005500.00005850.0000605 1250 0.00005750.00006230.00005780.00005850.00005510.0000582 1270 0.00005710.00005770.00005520.00005110.00005100.0000539 1290 0.00005090.00005420.00005010.00004830.00004920.0000513 1310 0.00005420.00005280.00005090.00004370.00004810.0000501 1330 0.00004730.00004740.00004620.00004380.00004560.0000464 1350 0.00004610.00004720.00004410.00003990.00004240.0000445 1370 0.00004180.00004380.00004100.00003950.00004100.0000422 1390 0.00003920.00004260.00003880.00003790.00004050.0000414 1410 0.00003790.00004020.00003700.00003510.00003650.0000381 1430 0.00003530.00003770.00003370.00003030.00003460.0000359 1450 0.00003220.00003720.00003240.00003290.00003370.0000352 1470 0.00003100.00003600.00003040.00002930.00003180.0000336 1490 0.00003140.00003280.00002960.00002570.00003070.0000316 1510 0.00003040.00003230.00002860.00002480.00002920.0000306 1530 0.00002650.00003020.00002610.00002510.00002830.0000291 1550 0.00002560.00002800.00002530.00002470.00002730.0000276 1570 0.00002480.00002770.00002400.00002250.00002550.0000265 1590 0.00002430.00002620.00002320.00002070.00002540.0000258 1610 0.00002230.00002600.00002180.00002070.00002330.0000245 1630 0.00002110.00002420.00002020.00001840.00002160.0000227 1650 0.00002040.00002260.00001960.00001810.00002280.0000227 1670 0.00001930.00002120.00001880.00001760.00002090.0000210 1690 0.00001790.00002150.00001700.00001530.00001940.0000203 1710 0.00001750.00002030.00001690.00001570.00002040.0000204 1730 0.00001650.00002030.00001610.00001520.00001820.0000191 1750 0.00001510.00001880.00001410.00001190.00001750.0000180 1770 0.00001420.00001940.00001360.00001230.00001780.0000185 1790 0.00001480.00001830.00001380.00001150.00001760.0000179 1810 0.00001440.00001720.00001340.00001130.00001620.0000166 1830 0.00001230.00001610.00001190.00001100.00001690.0000165 1850 0.00001240.00001520.00001140.00000910.00001490.0000151 1870 0.00001220.00001520.00001130.00000940.00001540.0000153 1890 0.00001130.00001430.00001050.00000880.00001480.0000146 1910 0.00001150.00001490.00001070.00000900.00001500.0000149 1930 0.00001130.00001350.00001040.00000850.00001310.0000133 1950 0.00000840.00001350.00000840.00000830.00001380.0000137 1970 0.00000940.00001380.00000870.00000730.00001290.0000133 19900.00000960.00001290.00000850.00000610.00001260.0000127

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363 Table G-3. Normalized 3D c hord-length distributions throug h the bone trabeculae of the pubis, ilium, ischium, right and left sca pula, sternum, and the averages for the os coxae and scapula. Probability distributions are given through the first 50 bins with a maximum length of 1000 microns. bin width PubisIschiumIlium Average Os Coxae Right Scapula Left Scapula Average Scapula Sternumm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 10 0.00067420.00064190.00081990.00071840.00039060.00041430.00040060.0007814 30 0.00092980.00089670.00113790.00099610.00066230.00063150.00064920.0010903 50 0.00111760.00103820.00139890.00119860.00073710.00072630.00073250.0013592 70 0.00134780.00113490.00167030.00141030.00072660.00074580.00073470.0015760 90 0.00173250.00130650.00210580.00176000.00078790.00082510.00080360.0019694 110 0.00212990.00152280.00263730.00216020.00091600.00094830.00092970.0023923 130 0.00259450.00174480.00319080.00259620.00110850.00110020.00110500.0028461 150 0.00312370.00200940.00372290.00306030.00136190.00131580.00134240.0032786 170 0.00346100.00224730.00399680.00335010.00166330.00149860.00159350.0035704 190 0.00362010.00247440.00390520.00343630.00199300.00173280.00188270.0036287 210 0.00349080.00268450.00361280.00333330.00227620.00196780.00214550.0034690 230 0.00317630.00277180.00311360.00305280.00249090.00219580.00236590.0031086 250 0.00276890.00270660.00263210.00270430.00248130.00230400.00240620.0026367 270 0.00240030.00256900.00219620.00236920.00241320.00231510.00237170.0022424 290 0.00202690.00242360.00181980.00205170.00227390.00227760.00227550.0018558 310 0.00171950.00218590.00151520.00176270.00209760.00215510.00212190.0015426 330 0.00147380.00196650.00125710.00151910.00193020.00199450.00195750.0012945 350 0.00124440.00176910.00104750.00130480.00177700.00183320.00180080.0010984 370 0.00107450.00154810.00088380.00112440.00158480.00170020.00163370.0009370 390 0.00094180.00134230.00076840.00097960.00145120.00152730.00148340.0008212 410 0.00080860.00118860.00064800.00084590.00133000.00139810.00135890.0007091 430 0.00072230.00104230.00057870.00075070.00118360.00128100.00122490.0006418 450 0.00063960.00092590.00050590.00066320.00107710.00115050.00110820.0005704 470 0.00057540.00081920.00045590.00059340.00098840.00106000.00101880.0005044 490 0.00050860.00073280.00040480.00052730.00088600.00098380.00092750.0004573 510 0.00046660.00065490.00037050.00047910.00082970.00089050.00085550.0004163 530 0.00041820.00060470.00032350.00043080.00075750.00083170.00078890.0003812 550 0.00038060.00054370.00029420.00039040.00069710.00076140.00072430.0003452 570 0.00036080.00049340.00026810.00036070.00065050.00071100.00067610.0003159 590 0.00031840.00045420.00024250.00032510.00059170.00065720.00061950.0002914 610 0.00029190.00041340.00022320.00029760.00055940.00059420.00057420.0002636 630 0.00026300.00037000.00019920.00026680.00051480.00056460.00053590.0002449 650 0.00024360.00035130.00018100.00024800.00046990.00051270.00048810.0002206 670 0.00022600.00031860.00016210.00022620.00044130.00048650.00046050.0002051 690 0.00020020.00029060.00015270.00020570.00042390.00044900.00043450.0001826 710 0.00018600.00026900.00014290.00019130.00038630.00042070.00040090.0001700 730 0.00017490.00025060.00012460.00017580.00035760.00040040.00037570.0001577 750 0.00015490.00023170.00011280.00015900.00034480.00037750.00035870.0001431 770 0.00014610.00021600.00009470.00014510.00032430.00034570.00033340.0001315 790 0.00013540.00019830.00008940.00013460.00030170.00032290.00031070.0001275 810 0.00012140.00018380.00008210.00012290.00028570.00030200.00029260.0001131 830 0.00011040.00017280.00007640.00011380.00027130.00028970.00027910.0001015 850 0.00010590.00015780.00006760.00010510.00025160.00026410.00025690.0000948 870 0.00009260.00014440.00006170.00009440.00024160.00025830.00024870.0000933 890 0.00008430.00013140.00005650.00008610.00023330.00023800.00023530.0000803 910 0.00007900.00012520.00005490.00008190.00021690.00022990.00022240.0000828 930 0.00007300.00010840.00004860.00007310.00020870.00021060.00020950.0000777 950 0.00006610.00010460.00004250.00006720.00019250.00020500.00019780.0000698 970 0.00006070.00009190.00003890.00006070.00017950.00019570.00018640.0000637 9900.00005320.00008940.00003100.00005430.00016430.00017850.00017030.0000609

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364 Table G-4. Normalized 3D c hord-length distributions throug h the bone trabeculae of the pubis, ilium, ischium, right and left sca pula, sternum, and the averages for the os coxae and scapula. Probability distributions are given through the second 50 bins for a maximum length of 2000 microns. bin width PubisIschiumIlium Average Os Coxae Right Scapula Left Scapula Average Scapula Sternumm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 1010 0.00005140.00008490.00003620.00005430.00016060.00017210.00016550.0000584 1030 0.00005190.00007580.00002820.00004940.00015360.00016560.00015870.0000517 1050 0.00004480.00006740.00002900.00004480.00014190.00015260.00014650.0000527 1070 0.00004390.00006640.00002780.00004370.00013460.00014140.00013750.0000462 1090 0.00004140.00005950.00002490.00004000.00012610.00013510.00012990.0000457 1110 0.00003790.00005790.00002260.00003740.00012160.00013440.00012700.0000418 1130 0.00003670.00005620.00002350.00003680.00011730.00011760.00011740.0000399 1150 0.00003190.00005090.00001720.00003140.00011120.00010990.00011070.0000411 1170 0.00003210.00004400.00001870.00003030.00009770.00010690.00010160.0000349 1190 0.00002680.00004580.00001460.00002720.00009580.00010220.00009850.0000358 1210 0.00002730.00004130.00001320.00002570.00009260.00009780.00009480.0000360 1230 0.00002550.00004150.00001250.00002490.00008670.00008780.00008720.0000292 1250 0.00002180.00003450.00001200.00002140.00008290.00008400.00008330.0000288 1270 0.00002230.00003280.00000960.00002040.00007840.00007840.00007840.0000272 1290 0.00001950.00003590.00000880.00001970.00007200.00007710.00007420.0000281 1310 0.00001850.00002690.00000950.00001740.00007340.00007400.00007360.0000261 1330 0.00001810.00002550.00000880.00001660.00006350.00006930.00006600.0000225 1350 0.00001860.00002580.00000840.00001670.00006080.00006400.00006210.0000231 1370 0.00001650.00002440.00000780.00001540.00005860.00006270.00006030.0000195 1390 0.00001570.00002450.00000800.00001510.00005790.00005650.00005730.0000200 1410 0.00001340.00002050.00000650.00001270.00005770.00005320.00005580.0000205 1430 0.00001390.00002040.00000570.00001260.00005020.00005170.00005080.0000198 1450 0.00001260.00001950.00000530.00001170.00005270.00004680.00005020.0000182 1470 0.00001020.00001640.00000520.00000990.00005120.00004700.00004940.0000180 1490 0.00001090.00001630.00000380.00000970.00004650.00004400.00004540.0000155 1510 0.00000890.00001490.00000500.00000900.00004190.00004250.00004220.0000158 1530 0.00001040.00001280.00000390.00000870.00004410.00004140.00004300.0000143 1550 0.00000840.00001570.00000340.00000840.00004140.00003830.00004010.0000130 1570 0.00000790.00001290.00000390.00000770.00003930.00003560.00003770.0000116 1590 0.00000740.00001350.00000310.00000740.00003820.00003610.00003730.0000139 1610 0.00000840.00001100.00000250.00000690.00003410.00003210.00003320.0000116 1630 0.00000690.00001190.00000300.00000680.00003550.00002980.00003310.0000117 1650 0.00000770.00001210.00000290.00000710.00003250.00002640.00002990.0000100 1670 0.00000610.00001240.00000220.00000630.00003330.00002960.00003170.0000108 1690 0.00000750.00001000.00000160.00000600.00003100.00002640.00002900.0000121 1710 0.00000650.00000960.00000180.00000560.00003000.00002190.00002650.0000089 1730 0.00000590.00000990.00000250.00000570.00002890.00002370.00002670.0000084 1750 0.00000560.00000850.00000180.00000490.00002800.00002270.00002570.0000092 1770 0.00000640.00000720.00000140.00000480.00002650.00001970.00002360.0000074 1790 0.00000450.00000720.00000120.00000400.00002510.00002120.00002350.0000096 1810 0.00000420.00000650.00000110.00000370.00002780.00002160.00002510.0000082 1830 0.00000540.00000650.00000220.00000450.00002460.00001820.00002190.0000065 1850 0.00000470.00000540.00000180.00000390.00002200.00001760.00002010.0000068 1870 0.00000370.00000590.00000140.00000340.00002110.00001700.00001940.0000070 1890 0.00000360.00000580.00000100.00000320.00002220.00001650.00001980.0000069 1910 0.00000390.00000590.00000100.00000340.00001930.00001530.00001760.0000078 1930 0.00000390.00000550.00000140.00000340.00001940.00001260.00001650.0000062 1950 0.00000350.00000540.00000100.00000310.00001820.00001720.00001780.0000053 1970 0.00000340.00000390.00000080.00000260.00001750.00001350.00001580.0000054 19900.00000240.00000350.00000070.00000210.00001900.00001310.00001650.0000058

PAGE 397

365 Table G-5. Normalized 3D c hord-length distributions throug h the bone trabeculae of the right and left clavicle, ri ght and left humerus, C3 and C6 vertebra, and the averages for those respective bone sites. Probability distributions are given through the first 50 bins with a maximum length of 1000 microns. bin width Left Clavicle Right Clavicle Average Clavicle Right Humerus Left Humerus Average Humerus C3 VertebraC6 Vertebra Average Cervical V ertebram p(L)dLp(L)dLp(L)dLp(L)dLp(L) dLp(L)dLp(L)dLp(L)dLp(L)dL 10 0.00080120.00063630.00074070.00031530.00052040.00043800.00058530.00059810.0005929 30 0.00108350.00076790.00096760.00053230.00065890.00060800.00092520.00082610.0008670 50 0.00135360.00090130.00118760.00060910.00078360.00071350.00108820.00103370.0010562 70 0.00150920.00110280.00136000.00062050.00099860.00084670.00126970.00124960.0012579 90 0.00179480.00152910.00169730.00067610.00144490.00113590.00152800.00171020.0016351 110 0.00199020.00191980.00196430.00078440.00192710.00146790.00183320.00227420.0020923 130 0.00229250.00228390.00228940.00095940.00243540.00184230.00221960.00291400.0026276 150 0.00245690.00262680.00251930.00121620.00290680.00222740.00273070.00353050.0032006 170 0.00263070.00284700.00271010.00155340.00319340.00253430.00322780.00386960.0036049 190 0.00262400.00294650.00274240.00196400.00326300.00274100.00353890.00385760.0037262 210 0.00267960.00287570.00275160.00236770.00316960.00284730.00361560.00355190.0035782 230 0.00261800.00269920.00264780.00264040.00293180.00281470.00343520.00309910.0032377 250 0.00246240.00243670.00245300.00267280.00264770.00265780.00303060.00262990.0027951 270 0.00236610.00217960.00229760.00262130.00234500.00245600.00263010.00222550.0023924 290 0.00206830.00191840.00201320.00245370.00207670.00222820.00222030.00187960.0020201 310 0.00192460.00167180.00183180.00224130.00180440.00197990.00185380.00157270.0016886 330 0.00170400.00150110.00162950.00203400.00159680.00177250.00155340.00133810.0014269 350 0.00154660.00132620.00146570.00184600.00138470.00157010.00131360.00114190.0012127 370 0.00137670.00117530.00130280.00164680.00121150.00138640.00109990.00099170.0010363 390 0.00128620.00103580.00119420.00147080.00107450.00123380.00093640.00085130.0008864 410 0.00112820.00092810.00105470.00133160.00095090.00110390.00081980.00075180.0007799 430 0.00100880.00086200.00095490.00118740.00083920.00097920.00070570.00065820.0006778 450 0.00090640.00076840.00085570.00106570.00075600.00088050.00063680.00059290.0006110 470 0.00080120.00071160.00076830.00097790.00068350.00080180.00056820.00053590.0005492 490 0.00072160.00064350.00069300.00088170.00061220.00072050.00050850.00048470.0004946 510 0.00063870.00059690.00062330.00079490.00056770.00065900.00047030.00043810.0004513 530 0.00057580.00054520.00056450.00073520.00051510.00060360.00043830.00040110.0004165 550 0.00049980.00049820.00049920.00068020.00048390.00056280.00039190.00037230.0003804 570 0.00045240.00048020.00046260.00062560.00043580.00051210.00036740.00034420.0003538 590 0.00040750.00044540.00042140.00057140.00039970.00046870.00034130.00031230.0003242 610 0.00037070.00040490.00038320.00053550.00037410.00043900.00032080.00029100.0003033 630 0.00035370.00039530.00036900.00049010.00034480.00040320.00029760.00027250.0002828 650 0.00030110.00036610.00032500.00047280.00032030.00038160.00027930.00025260.0002636 670 0.00026770.00034050.00029440.00043490.00030040.00035440.00025910.00023130.0002428 690 0.00026310.00032020.00028410.00040980.00027870.00033140.00023470.00022070.0002265 710 0.00021270.00029630.00024340.00039410.00026410.00031640.00021370.00019950.0002054 730 0.00019720.00027120.00022440.00037240.00024380.00029550.00019930.00018220.0001893 750 0.00017020.00025930.00020290.00035530.00021980.00027420.00018480.00017240.0001775 770 0.00015920.00025430.00019410.00033640.00020960.00026060.00016530.00016100.0001628 790 0.00015220.00022990.00018070.00030300.00019720.00023970.00015630.00014500.0001496 810 0.00013790.00021270.00016540.00030220.00017800.00022790.00014250.00013830.0001401 830 0.00012000.00020550.00015140.00028030.00016370.00021060.00013190.00012330.0001268 850 0.00011640.00018530.00014170.00026970.00015630.00020190.00011870.00011630.0001173 870 0.00010970.00017270.00013280.00025390.00014330.00018780.00011270.00010330.0001072 890 0.00010150.00017200.00012740.00023970.00013240.00017550.00009790.00010390.0001014 910 0.00008930.00016880.00011850.00023230.00012340.00016710.00009360.00009160.0000924 930 0.00008720.00015230.00011110.00021470.00011450.00015480.00008520.00008490.0000850 950 0.00007720.00014500.00010210.00020290.00010940.00014700.00007860.00007480.0000764 970 0.00007870.00013520.00009940.00019320.00009510.00013450.00006760.00007380.0000712990 0.00006290.00012300.00008500.00017850.00009160.00012650.00006540.00006740.0000666

PAGE 398

366 Table G-6. Normalized 3D c hord-length distributions throug h the bone trabeculae of the right and left clavicle, ri ght and left humerus, C3 and C6 vertebra, and the averages for those respective bone sites. Probability distributions are given through the second 50 bins for a maximum length of 2000 microns. bin width Left Clavicle Right Clavicle Average Clavicle Right Humerus Left Humerus Average Humerus C3 VertebraC6 Vertebra Average Cervical V ertebram p(L)dLp(L)dLp(L)dLp(L)dLp(L) dLp(L)dLp(L)dLp(L)dLp(L)dL 1010 0.00006530.00011810.00008470.00017190.00008200.00011820.00006060.00006080.0000607 1030 0.00006140.00011620.00008150.00016280.00007820.00011220.00005540.00005800.0000570 1050 0.00004830.00010520.00006920.00015480.00007270.00010570.00005010.00005220.0000513 1070 0.00004560.00010110.00006600.00014410.00006770.00009840.00004750.00004890.0000483 1090 0.00004310.00009190.00006110.00013850.00006320.00009350.00004180.00004260.0000423 1110 0.00004100.00009000.00005900.00012610.00005930.00008620.00003880.00003960.0000393 1130 0.00003620.00007580.00005070.00012230.00005610.00008270.00003750.00003610.0000367 1150 0.00003430.00007610.00004970.00011230.00005100.00007570.00003370.00003580.0000349 1170 0.00003280.00007650.00004890.00011110.00004740.00007300.00003090.00003030.0000305 1190 0.00002730.00007220.00004380.00010600.00004490.00006940.00002980.00003140.0000307 1210 0.00002490.00006910.00004120.00009550.00004140.00006310.00002700.00002910.0000282 1230 0.00002800.00006750.00004250.00008980.00003780.00005870.00002590.00002610.0000260 1250 0.00002950.00006190.00004140.00008960.00003670.00005800.00002350.00002470.0000242 1270 0.00001760.00005740.00003220.00008330.00003410.00005390.00002090.00002320.0000222 1290 0.00001520.00005230.00002880.00007860.00003040.00004980.00002040.00002100.0000207 1310 0.00002100.00005290.00003270.00007730.00002990.00004900.00002070.00002000.0000203 1330 0.00002010.00004610.00002960.00006990.00002680.00004410.00001790.00001850.0000182 1350 0.00001790.00004470.00002780.00007130.00002590.00004410.00001640.00001770.0000172 1370 0.00001760.00004260.00002680.00006250.00002380.00003930.00001450.00001520.0000149 1390 0.00001550.00004010.00002450.00006450.00002220.00003920.00001420.00001520.0000148 1410 0.00001520.00003560.00002270.00005690.00002160.00003580.00001350.00001370.0000136 1430 0.00001190.00003940.00002200.00005480.00002080.00003450.00001230.00001310.0000128 1450 0.00001310.00003490.00002110.00005260.00001910.00003260.00001080.00001150.0000112 1470 0.00000880.00003100.00001700.00005160.00001790.00003140.00000970.00001030.0000100 1490 0.00001190.00003510.00002040.00004880.00001620.00002930.00000900.00000960.0000094 1510 0.00001030.00002840.00001700.00004360.00001670.00002750.00001010.00000990.0000100 1530 0.00000850.00002910.00001610.00004520.00001450.00002680.00000870.00000800.0000083 1550 0.00000880.00002830.00001600.00004280.00001430.00002580.00000720.00000800.0000077 1570 0.00001030.00002710.00001650.00004140.00001340.00002470.00000780.00000750.0000076 1590 0.00000910.00002350.00001440.00003570.00001140.00002110.00000640.00000710.0000068 1610 0.00000820.00002640.00001490.00003690.00001230.00002220.00000590.00000620.0000060 1630 0.00000670.00002330.00001280.00003300.00001120.00002000.00000550.00000660.0000061 1650 0.00000730.00002010.00001200.00003660.00001000.00002070.00000500.00000690.0000061 1670 0.00000400.00002040.00001000.00003230.00000920.00001850.00000430.00000540.0000050 1690 0.00000490.00002240.00001130.00002910.00000870.00001690.00000560.00000600.0000058 1710 0.00000670.00002160.00001210.00002930.00000820.00001670.00000430.00000490.0000047 1730 0.00000490.00001890.00001000.00002580.00000830.00001540.00000320.00000470.0000041 1750 0.00000400.00001640.00000850.00002630.00000740.00001500.00000320.00000500.0000042 1770 0.00000490.00001630.00000910.00002500.00000710.00001430.00000380.00000390.0000038 1790 0.00000430.00001590.00000850.00002430.00000690.00001390.00000290.00000390.0000035 1810 0.00000460.00001360.00000790.00002160.00000670.00001270.00000370.00000380.0000037 1830 0.00000270.00001330.00000660.00002050.00000570.00001160.00000240.00000300.0000028 1850 0.00000270.00001160.00000600.00001930.00000650.00001160.00000220.00000310.0000027 1870 0.00000240.00001170.00000580.00001890.00000540.00001080.00000240.00000330.0000030 1890 0.00000300.00001050.00000580.00001980.00000420.00001050.00000210.00000290.0000026 1910 0.00000240.00001290.00000630.00001800.00000530.00001040.00000190.00000230.0000021 1930 0.00000270.00001130.00000590.00001670.00000500.00000970.00000200.00000240.0000023 1950 0.00000240.00001090.00000550.00001670.00000500.00000970.00000180.00000250.0000022 1970 0.00000240.00001240.00000610.00001500.00000420.00000860.00000200.00000260.00000231990 0.00000090.00000930.00000400.00001550.00000400.00000860.00000110.00000240.0000018

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367 Table G-7. Normalized 3D c hord-length distributions throug h the bone trabeculae of the T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages for the thoracic and lumbar vertebra. Proba bility distributions are given through the first 50 bins with a maximum length of 1000 microns. bin width T3 VertebraT6 VertebraT11 Vertebra Average Thoracic V ertebra L2 VertebraL4Vertebra Average Lumbar V ertebra Sacrumm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 10 0.00054610.00083790.00103160.00081450.00051830.00060280.00055360.0004330 30 0.00087650.00119940.00124470.00111280.00084580.00075160.00080640.0007215 50 0.00106600.00133110.00142230.00127940.00098160.00090400.00094920.0008540 70 0.00120010.00138530.00179950.00147520.00105530.00114540.00109300.0009205 90 0.00141250.00144890.00247070.00180460.00118170.00161690.00136370.0010568 110 0.00168260.00155930.00302100.00212410.00133200.00212470.00166360.0012757 130 0.00203650.00175790.00339930.00243680.00157750.00267920.00203840.0016042 150 0.00250300.00208960.00364760.00278150.00186270.00323730.00243770.0020716 170 0.00303970.00241080.00369120.00307160.00219330.00363510.00279650.0026537 190 0.00346050.00263720.00349490.00320800.00253690.00371110.00302810.0031061 210 0.00373280.00287000.00317590.00325500.00287210.00353180.00314810.0034290 230 0.00366740.00299800.00277950.00313280.00310410.00318910.00313960.0034454 250 0.00322310.00279860.00239210.00278770.00297980.00277440.00289390.0031524 270 0.00277640.00257110.00202950.00244180.00280450.00238150.00262750.0028165 290 0.00236820.00233960.00172520.00212780.00255770.00201860.00233220.0024659 310 0.00196620.00203150.00144870.00180110.00222530.00170480.00200760.0021042 330 0.00164140.00175160.00124050.00153270.00195820.00144550.00174370.0018009 350 0.00137730.00154560.00105850.00131680.00167480.00123090.00148910.0015451 370 0.00116350.00134890.00089950.00112820.00144820.00104560.00127980.0013105 390 0.00100590.00117260.00078600.00098050.00126250.00090630.00111350.0011334 410 0.00086030.00105140.00068260.00085790.00112150.00079580.00098520.0009871 430 0.00075300.00091560.00060080.00075060.00098260.00070730.00086750.0008678 450 0.00067870.00083310.00053330.00067610.00087040.00063270.00077100.0007703 470 0.00059480.00074750.00047390.00060050.00078430.00057010.00069470.0006966 490 0.00053150.00068190.00043520.00054530.00069080.00051690.00061800.0006243 510 0.00047470.00060430.00038750.00048500.00063410.00046900.00056510.0005685 530 0.00043980.00055220.00034880.00044320.00056710.00042790.00050890.0005268 550 0.00039400.00049670.00032180.00040110.00052950.00039480.00047320.0004808 570 0.00036240.00046350.00029140.00036940.00048280.00036140.00043210.0004472 590 0.00033860.00041760.00026540.00033770.00044040.00033560.00039660.0004123 610 0.00030790.00038600.00024150.00030920.00041200.00031430.00037110.0003873 630 0.00028250.00035040.00022270.00028280.00037950.00028990.00034200.0003606 650 0.00025580.00032680.00020120.00025900.00035980.00026740.00032110.0003366 670 0.00023500.00030300.00018580.00023920.00032630.00024760.00029340.0003129 690 0.00021610.00027900.00016840.00021920.00030270.00022460.00027000.0002948 710 0.00020120.00025790.00015250.00020190.00028610.00020560.00025240.0002717 730 0.00018690.00023740.00013880.00018590.00025950.00019330.00023180.0002568 750 0.00017100.00021750.00012900.00017090.00024890.00017380.00021750.0002341 770 0.00015410.00020510.00011890.00015780.00022690.00016010.00019890.0002176 790 0.00014310.00018900.00010570.00014440.00020620.00014880.00018220.0002048 810 0.00013030.00017660.00009670.00013310.00019170.00013530.00016810.0001880 830 0.00012090.00015750.00008750.00012060.00018110.00012990.00015970.0001765 850 0.00010880.00015030.00008120.00011220.00016880.00012030.00014850.0001625 870 0.00010130.00013330.00007390.00010170.00015900.00010700.00013720.0001541 890 0.00009130.00012260.00006700.00009260.00014480.00009840.00012540.0001420 910 0.00008330.00011920.00006330.00008760.00013000.00009230.00011430.0001295 930 0.00007530.00010720.00005660.00007880.00012850.00008320.00010960.0001206 950 0.00007110.00010220.00005480.00007520.00011870.00007980.00010240.0001118 970 0.00006570.00009050.00004720.00006700.00010590.00007240.00009180.0001033 9900.00006040.00008580.00004500.00006300.00009690.00006690.00008430.0000979

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368 Table G-8. Normalized 3D c hord-length distributions throug h the bone trabeculae of the T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages for the thoracic and lumbar vertebra. Proba bility distributions are given through the second 50 bins for a maximum length of 2000 microns. bin width T3 VertebraT6 VertebraT11 Vertebra Average Thoracic V ertebra L2 VertebraL4Vertebra Average Lumbar V ertebra Sacrumm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 1010 0.00005430.00007840.00004120.00005730.00009070.00006440.00007970.0000924 1030 0.00005030.00007350.00003530.00005240.00008850.00005900.00007620.0000833 1050 0.00004730.00006780.00003570.00004970.00008150.00005300.00006960.0000805 1070 0.00004340.00006290.00003070.00004510.00007530.00004780.00006380.0000732 1090 0.00003940.00005650.00002970.00004140.00006840.00004500.00005860.0000692 1110 0.00003660.00005170.00002680.00003790.00006530.00004330.00005610.0000644 1130 0.00003350.00005040.00002520.00003600.00006400.00004000.00005400.0000624 1150 0.00003070.00004570.00002350.00003290.00005890.00003580.00004920.0000565 1170 0.00002920.00004450.00002110.00003120.00005660.00003340.00004690.0000538 1190 0.00002710.00004070.00001860.00002840.00004940.00003070.00004160.0000489 1210 0.00002580.00003810.00001940.00002750.00004640.00003000.00003950.0000466 1230 0.00002440.00003610.00001680.00002540.00004480.00002710.00003740.0000444 1250 0.00002120.00003150.00001500.00002230.00004300.00002560.00003570.0000401 1270 0.00001830.00002960.00001400.00002040.00003740.00002410.00003190.0000380 1290 0.00001690.00002810.00001420.00001950.00003530.00002200.00002970.0000360 1310 0.00001670.00002610.00001280.00001830.00003490.00002090.00002900.0000325 1330 0.00001610.00002340.00001160.00001680.00003360.00001920.00002760.0000319 1350 0.00001480.00002080.00001080.00001530.00002780.00001860.00002390.0000280 1370 0.00001390.00002040.00001010.00001460.00002810.00001680.00002340.0000268 1390 0.00001330.00001920.00000970.00001390.00002690.00001550.00002210.0000255 1410 0.00001190.00001850.00000910.00001300.00002250.00001590.00001970.0000227 1430 0.00001120.00001650.00000870.00001200.00002480.00001480.00002070.0000218 1450 0.00001140.00001600.00000740.00001140.00002030.00001360.00001750.0000206 1470 0.00000970.00001550.00000690.00001060.00002150.00001270.00001780.0000192 1490 0.00000960.00001570.00000650.00001040.00001830.00001180.00001560.0000200 1510 0.00000890.00001390.00000620.00000950.00001800.00001050.00001490.0000179 1530 0.00000760.00001330.00000660.00000910.00001590.00001050.00001370.0000168 1550 0.00000710.00001220.00000560.00000820.00001420.00000910.00001210.0000158 1570 0.00000700.00001170.00000510.00000790.00001510.00000940.00001270.0000149 1590 0.00000630.00001070.00000470.00000710.00001340.00000850.00001140.0000144 1610 0.00000580.00000860.00000460.00000630.00001130.00000780.00000980.0000126 1630 0.00000560.00000830.00000390.00000590.00001120.00000780.00000980.0000118 1650 0.00000530.00000870.00000410.00000600.00001110.00000770.00000970.0000111 1670 0.00000470.00000620.00000330.00000470.00001070.00000670.00000900.0000101 1690 0.00000490.00000580.00000350.00000470.00001100.00000640.00000910.0000091 1710 0.00000390.00000750.00000310.00000480.00000980.00000580.00000810.0000096 1730 0.00000440.00000660.00000300.00000460.00000890.00000590.00000760.0000097 1750 0.00000380.00000580.00000290.00000410.00000960.00000500.00000770.0000084 1770 0.00000360.00000600.00000290.00000410.00000840.00000510.00000700.0000079 1790 0.00000360.00000550.00000260.00000380.00000800.00000510.00000680.0000078 1810 0.00000380.00000530.00000230.00000370.00000730.00000480.00000620.0000069 1830 0.00000280.00000430.00000210.00000300.00000640.00000460.00000570.0000069 1850 0.00000290.00000420.00000200.00000300.00000690.00000380.00000560.0000065 1870 0.00000230.00000410.00000200.00000280.00000680.00000370.00000550.0000062 1890 0.00000230.00000360.00000180.00000250.00000590.00000370.00000500.0000056 1910 0.00000210.00000410.00000150.00000260.00000650.00000340.00000520.0000056 1930 0.00000210.00000380.00000140.00000240.00000590.00000340.00000490.0000049 1950 0.00000200.00000330.00000170.00000230.00000580.00000320.00000470.0000045 1970 0.00000200.00000320.00000130.00000210.00000510.00000280.00000420.0000047 19900.00000170.00000280.00000130.00000190.00000470.00000290.00000390.0000045

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369 Table G-9. Normalized 3D c hord-length distributions throug h the bone trabeculae of the right and left upper rib, middle rib, and lower rib. Also shown is the average for the single rib. Probability distri butions are given through the first 50 bins with a maximum length of 1000 microns. bin width Upper Rt Rib Middle Rt Rib Lower Rt RibUpper Lt Rib Middle Lt Rib Lower Lt RibAverage Ribm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 10 0.00066980.00044410.00106120.00083280.00044550.00088180.0006218 30 0.00086780.00064730.00146940.00118580.00061560.00116840.0008614 50 0.00100990.00070140.00175830.00144170.00071620.00141850.0010057 70 0.00118910.00071740.00203110.00173000.00076980.00168310.0011323 90 0.00142550.00080690.00247880.00209660.00103040.00205730.0013882 110 0.00191750.00100900.00290040.00242300.00127730.00246410.0016948 130 0.00246270.00127530.00332720.00275730.00166790.00286530.0020709 150 0.00301330.00169660.00351020.00302100.00208360.00308840.0024450 170 0.00340970.00227440.00362350.00315050.00265570.00329840.0028669 190 0.00358680.00289900.00359300.00321130.00304720.00330840.0031777 210 0.00357440.00351050.00331940.00323850.00331170.00319780.0033798 230 0.00327330.00362250.00302900.00316110.00345060.00287950.0033486 250 0.00286010.00342420.00245350.00285880.00324670.00261200.0030599 270 0.00243550.00316800.00202700.00257890.00288150.00222060.0027155 290 0.00212560.00279840.00164090.00220810.00258850.00187960.0023772 310 0.00187880.00242900.00136320.00185260.00231430.00161180.0020729 330 0.00153700.00210280.00117940.00155510.00191090.00131690.0017435 350 0.00138170.00179510.00096770.00129630.00167850.00112240.0015033 370 0.00120930.00153820.00080440.00109040.00145790.00097560.0012932 390 0.00099850.00130410.00070230.00091730.00122900.00080690.0010912 410 0.00087020.00111840.00059190.00078040.00105230.00071640.0009369 430 0.00079000.00098830.00053970.00067580.00092800.00063640.0008307 450 0.00070650.00083470.00048380.00057770.00082740.00056290.0007261 470 0.00062170.00076280.00041930.00049690.00071180.00048480.0006384 490 0.00054150.00067830.00037420.00044320.00063800.00045610.0005701 510 0.00047480.00059900.00033320.00040540.00057690.00040280.0005091 530 0.00043480.00053520.00030750.00034940.00052070.00037250.0004586 550 0.00040070.00047440.00026240.00032930.00049520.00034840.0004211 570 0.00036030.00043040.00024120.00029860.00044600.00032890.0003820 590 0.00031350.00038690.00023890.00027250.00037880.00030960.0003393 610 0.00029300.00034150.00021100.00024870.00035150.00027930.0003082 630 0.00025430.00034130.00019940.00021800.00031680.00026450.0002868 650 0.00024580.00030240.00017670.00020980.00029440.00023680.0002629 670 0.00022290.00027370.00018190.00019320.00026720.00022400.0002420 690 0.00020610.00025380.00014760.00016830.00025880.00021650.0002247 710 0.00017750.00023380.00013570.00014690.00022190.00019130.0001988 730 0.00016300.00020980.00012490.00013480.00021090.00018440.0001842 750 0.00016270.00019810.00010400.00012020.00020780.00016850.0001748 770 0.00015290.00018920.00011370.00010440.00019250.00015750.0001652 790 0.00013610.00017190.00010880.00010520.00016830.00015010.0001500 810 0.00010940.00015980.00009470.00009170.00016300.00014040.0001381 830 0.00010780.00014040.00008720.00008540.00015550.00013080.0001282 850 0.00009830.00013170.00008090.00008640.00014720.00012580.0001211 870 0.00008990.00012760.00006820.00007580.00011690.00011130.0001057 890 0.00008650.00011290.00006080.00006100.00012260.00010680.0001004 910 0.00007810.00010460.00006150.00005550.00012000.00009940.0000954 930 0.00006970.00010870.00006110.00005180.00010410.00008620.0000890 950 0.00006230.00009440.00005370.00004990.00009230.00008470.0000794 970 0.00006130.00008770.00004850.00005020.00008700.00008180.0000751 9900.00005090.00008360.00004550.00004120.00008660.00006970.0000704

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370 Table G-10. Normalized 3D c hord-length distributions throug h the bone trabeculae of the right and left upper, middle, and lower ri bs. Also shown is the average for the single rib. Probability distributions ar e given through the second 50 bins for a maximum length of 2000 microns. bin width Upper Rt Rib Middle Rt Rib Lower Rt RibUpper Lt Rib Middle Lt Rib Lower Lt RibAverage Ribm p(L)dLp(L)dLp(L)dLp(L) dLp(L)dLp(L)dLp(L)dL 1010 0.00005350.00007070.00004210.00002930.00008880.00007180.0000665 1030 0.00004950.00007260.00004700.00003510.00007380.00006460.0000625 1050 0.00005150.00006700.00003800.00002770.00006420.00006120.0000561 1070 0.00004710.00006540.00003620.00002620.00006990.00005930.0000564 1090 0.00003910.00005190.00003760.00002800.00006200.00005230.0000493 1110 0.00003400.00005740.00003060.00002220.00005270.00005460.0000459 1130 0.00002530.00005550.00002310.00002110.00005930.00004780.0000449 1150 0.00003370.00004610.00002870.00002110.00004260.00004640.0000387 1170 0.00002660.00004400.00002530.00001430.00003780.00004240.0000344 1190 0.00002660.00004160.00002310.00001720.00004830.00003590.0000366 1210 0.00002050.00004130.00002420.00001480.00004530.00003320.0000345 1230 0.00002630.00004060.00002310.00001080.00003430.00003350.0000310 1250 0.00002530.00003360.00001640.00001290.00003650.00003510.0000293 1270 0.00001850.00003540.00001710.00001080.00003210.00003410.0000274 1290 0.00001780.00002830.00001860.00001320.00002810.00003220.0000245 1310 0.00001950.00003170.00002120.00000920.00002720.00002930.0000250 1330 0.00001580.00002670.00001940.00000950.00002810.00002380.0000229 1350 0.00001550.00002670.00001450.00000900.00002280.00002660.0000208 1370 0.00001140.00002410.00001830.00000770.00002200.00002320.0000194 1390 0.00001310.00002350.00001080.00000900.00002370.00002460.0000193 1410 0.00001620.00002090.00001230.00000710.00002070.00002630.0000181 1430 0.00000940.00001940.00001120.00000450.00002370.00001920.0000169 1450 0.00000980.00001640.00001010.00000690.00001930.00002130.0000151 1470 0.00001520.00002170.00001450.00000710.00001930.00002270.0000178 1490 0.00000910.00001660.00000970.00000420.00001980.00002220.0000150 1510 0.00000810.00001750.00001040.00000580.00002370.00001930.0000164 1530 0.00000710.00001400.00001190.00000340.00001580.00001430.0000124 1550 0.00000810.00001510.00000670.00000290.00001050.00001560.0000105 1570 0.00000980.00001520.00001230.00000580.00001270.00001390.0000123 1590 0.00000300.00001280.00000890.00000340.00001760.00001610.0000119 1610 0.00000740.00001270.00000710.00000420.00001490.00001240.0000112 1630 0.00000570.00001250.00000860.00000500.00001050.00001370.0000099 1650 0.00000570.00001420.00000820.00000290.00000970.00001210.0000097 1670 0.00000540.00001030.00000890.00000320.00001100.00001320.0000092 1690 0.00000370.00000920.00000710.00000240.00001190.00001140.0000086 1710 0.00000810.00000870.00000860.00000320.00001140.00001110.0000091 1730 0.00000540.00000920.00000560.00000180.00000920.00001370.0000079 1750 0.00000540.00000860.00000600.00000160.00000700.00000970.0000067 1770 0.00000440.00000980.00000520.00000240.00000570.00001050.0000066 1790 0.00000440.00000940.00000480.00000130.00000750.00001110.0000069 1810 0.00000400.00000700.00000260.00000260.00000700.00001100.0000060 1830 0.00000440.00000650.00000630.00000340.00000480.00000980.0000056 1850 0.00000640.00000690.00000190.00000130.00000880.00000870.0000063 1870 0.00000440.00000620.00000220.00000110.00000920.00001000.0000062 1890 0.00000340.00000580.00000710.00000130.00000400.00000900.0000049 1910 0.00000200.00000800.00000410.00000080.00000400.00000710.0000047 1930 0.00000170.00000330.00000450.00000130.00000440.00000640.0000036 1950 0.00000200.00000580.00000260.00000180.00000570.00000710.0000046 1970 0.00000170.00000790.00000190.00000340.00000570.00001030.00000541990 0.00000400.00000620.00000340.00000080.00000750.00000840.0000056

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371 Table G-11. Normalized 3D c hord-length distributions throug h the bone trabeculae of the mandible, frontal bone, occipital bone, and right and left pari etal bones. Also shown is the average for the cranium. Probability distributions are given through the first 50 bins with a maximum length of 1000 microns. bin width Mandible Average Cranium Left Parietal Right Parietal Frontal Bone Occipital Bonem p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 10 0.00103950.00019040.00013560.00020110.00019100.0002107 30 0.00127820.00025560.00020800.00026690.00025550.0002613 50 0.00146290.00028090.00021820.00029170.00028480.0002957 70 0.00170500.00031630.00022370.00034670.00031190.0002946 90 0.00216490.00041370.00025930.00047130.00039800.0003734 110 0.00243440.00056660.00033070.00066880.00052820.0004842 130 0.00260720.00080110.00048760.00096290.00072040.0006714 150 0.00268080.00116330.00076330.00140310.00101720.0009984 170 0.00269320.00163710.00119860.00192830.00144190.0014488 190 0.00254780.00207870.00170770.00232810.00191830.0018737 210 0.00238910.00243620.00221750.00263530.00229440.0022086 230 0.00220170.00260190.00252260.00269530.00254620.0023955 250 0.00202460.00260870.00263720.00263510.00259920.0024265 270 0.00178600.00251420.00260640.00246310.00258150.0023418 290 0.00161350.00235580.00246910.00227980.00243490.0022543 310 0.00143840.00216330.00233980.00208140.00223170.0020482 330 0.00129880.00196440.00212680.00188740.00203290.0018408 350 0.00116470.00175370.00192510.00166960.00181810.0016841 370 0.00104130.00156030.00171970.00148130.00161700.0015197 390 0.00093920.00140010.00154120.00134160.00143230.0013822 410 0.00085580.00124930.00137310.00118440.00128930.0012635 430 0.00076430.00111420.00123610.00106170.00113850.0011306 450 0.00071440.00101230.00109380.00096190.00103950.0010698 470 0.00064160.00091550.00100300.00086750.00093470.0009893 490 0.00059930.00083450.00091200.00079310.00084900.0009058 510 0.00054600.00078000.00084650.00074540.00079570.0008182 530 0.00049880.00072150.00077660.00069940.00072260.0007704 550 0.00047950.00066890.00071490.00064240.00067480.0007394 570 0.00044550.00062190.00066200.00059840.00062950.0006728 590 0.00040810.00059410.00061700.00056700.00060980.0006569 610 0.00038470.00056060.00057450.00054450.00056760.0006101 630 0.00036990.00053640.00054490.00051680.00054990.0005839 650 0.00033480.00050500.00052690.00048780.00050810.0005690 670 0.00031650.00048360.00049280.00046040.00050240.0005268 690 0.00029770.00047190.00047550.00045000.00049020.0005186 710 0.00027790.00044740.00045100.00042360.00046850.0004929 730 0.00025630.00042050.00042680.00039570.00044490.0004501 750 0.00025130.00041190.00041020.00039470.00043310.0004197 770 0.00022810.00039270.00039700.00036970.00041260.0004377 790 0.00021430.00037410.00038240.00035470.00038900.0004135 810 0.00021450.00036280.00037390.00034500.00037940.0003765 830 0.00019960.00034710.00035870.00032830.00036160.0003786 850 0.00018690.00032920.00034740.00030840.00034360.0003625 870 0.00017950.00031310.00032990.00029520.00032310.0003540 890 0.00017240.00029860.00031240.00028300.00030840.0003287 910 0.00016320.00028110.00030210.00026310.00029090.0003163 930 0.00015450.00027010.00028540.00025480.00028000.0002950 950 0.00014960.00026260.00027240.00025170.00027150.0002729 970 0.00013880.00024710.00026570.00023010.00025650.0002808 9900.00012810.00023630.00025340.00022310.00024500.0002502

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372 Table G-12. Normalized 3D c hord-length distributions throug h the bone trabeculae of the mandible, frontal bone, occipital bone, and right and left pari etal bones. Also shown is the average for the cranium. Probability distributions are given through the second 50 bins with a maximum length of 2000 microns. bin width Mandible Average Cranium Left Parietal Right Parietal Frontal Bone Occipital Bonem p(L)dLp(L)dLp(L)dL p(L)dLp(L)dLp(L)dL 1010 0.00012440.00022350.00023700.00021260.00022870.0002467 1030 0.00011930.00021360.00022670.00020650.00021560.0002295 1050 0.00011030.00020700.00022140.00019730.00020960.0002345 1070 0.00011110.00019190.00020550.00018120.00019620.0002194 1090 0.00009810.00018720.00019660.00017610.00019390.0002113 1110 0.00009460.00017800.00019060.00017130.00018070.0001875 1130 0.00009050.00016950.00018510.00016290.00016990.0001851 1150 0.00009260.00015910.00017900.00015300.00015780.0001729 1170 0.00008400.00015220.00017190.00014720.00014860.0001714 1190 0.00007810.00014690.00016210.00014170.00014510.0001666 1210 0.00007570.00013910.00015140.00013180.00013980.0001638 1230 0.00007130.00013170.00014410.00012620.00013270.0001425 1250 0.00006880.00012680.00014040.00011920.00012930.0001416 1270 0.00006620.00011870.00013120.00011160.00012160.0001298 1290 0.00006350.00011430.00012820.00010680.00011540.0001367 1310 0.00006330.00011000.00012020.00010470.00011100.0001236 1330 0.00005600.00010580.00011110.00010220.00010600.0001205 1350 0.00005390.00010120.00010910.00009930.00010010.0001054 1370 0.00005230.00009560.00010320.00009260.00009490.0001069 1390 0.00005090.00009010.00010130.00008630.00008960.0000999 1410 0.00004790.00008590.00009540.00008090.00008700.0000968 1430 0.00004680.00008140.00009100.00007920.00007850.0000949 1450 0.00004460.00007870.00008870.00007660.00007500.0000955 1470 0.00004360.00007360.00008200.00006990.00007400.0000827 1490 0.00004090.00007060.00007500.00006940.00006840.0000827 1510 0.00004000.00006800.00007390.00006620.00006590.0000819 1530 0.00003660.00006460.00007020.00006050.00006580.0000765 1550 0.00003770.00006220.00006950.00005820.00006390.0000678 1570 0.00003540.00005870.00006230.00005580.00005800.0000763 1590 0.00003350.00005630.00006360.00005440.00005510.0000635 1610 0.00003180.00005500.00005850.00005420.00005300.0000655 1630 0.00003100.00005140.00006070.00004630.00005270.0000635 1650 0.00003170.00004950.00005320.00004790.00004850.0000604 1670 0.00002910.00004490.00005360.00004280.00004250.0000573 1690 0.00002690.00004600.00005270.00004470.00004390.0000542 1710 0.00002750.00004310.00004980.00004070.00004220.0000529 1730 0.00002750.00004050.00004490.00003750.00004190.0000459 1750 0.00002580.00003800.00004260.00003530.00003810.0000476 1770 0.00002180.00003730.00004260.00003660.00003550.0000428 1790 0.00002070.00003500.00003920.00003360.00003420.0000428 1810 0.00002150.00003450.00003900.00003310.00003350.0000426 1830 0.00001920.00003280.00003650.00003060.00003250.0000440 1850 0.00002090.00002910.00003330.00002790.00002820.0000345 1870 0.00001800.00002850.00003590.00002760.00002560.0000376 1890 0.00001550.00002740.00003300.00002660.00002490.0000372 1910 0.00001570.00002560.00002870.00002460.00002410.0000354 1930 0.00001710.00002540.00002930.00002450.00002400.0000325 1950 0.00001690.00002410.00002510.00002240.00002500.0000292 1970 0.00001730.00002340.00002580.00002060.00002460.00003121990 0.00001580.00002170.00002590.00001990.00002170.0000273

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373 Chord-Length Distributions through the marrow cavities of the 66-year UF RMCP Table G-13. Normalized 3D c hord-length distributions throug h the marrow cavities of the left and right femur heads, left and ri ght femur necks, and their respective averages. Probability distributions are given through the first 50 bins with a maximum length of 5000 microns. bin width Left Femur Head Left Femur Neck Average Femur Head Right Femur Head Right Femur Neck Average Femur Neckm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 50 0.00024680.00034210.00028750.00037470.00033140.0003360 150 0.00034110.00049840.00038760.00048740.00036830.0004241 250 0.00048880.00058830.00053150.00062310.00043850.0005027 350 0.00060280.00060370.00064290.00072890.00045760.0005203 450 0.00069620.00060320.00073290.00081160.00047700.0005311 550 0.00074830.00059540.00078350.00085870.00048700.0005335 650 0.00074540.00057090.00077830.00084890.00048620.0005225 750 0.00070190.00053020.00073170.00079550.00047530.0004988 850 0.00063890.00047950.00066060.00070710.00045900.0004678 950 0.00055850.00044190.00057340.00060550.00042960.0004349 1050 0.00048590.00039960.00049200.00050510.00040190.0004009 1150 0.00042160.00036090.00042130.00042080.00037300.0003678 1250 0.00036660.00032870.00036300.00035530.00034220.0003364 1350 0.00032160.00029890.00031340.00029570.00031760.0003096 1450 0.00028800.00027390.00027540.00024830.00029420.0002855 1550 0.00024950.00024810.00023710.00021050.00027380.0002628 1650 0.00022270.00022730.00020950.00018110.00025360.0002423 1750 0.00019390.00020700.00018070.00015260.00023210.0002214 1850 0.00017100.00018800.00015670.00012600.00021950.0002060 1950 0.00015250.00017280.00013730.00010460.00020020.0001884 2050 0.00013560.00015780.00011990.00008620.00018530.0001735 2150 0.00012290.00014560.00010710.00007320.00017280.0001611 2250 0.00010860.00013150.00009330.00006050.00016460.0001504 2350 0.00009730.00012170.00008290.00005200.00015010.0001379 2450 0.00008770.00011230.00007380.00004400.00014060.0001285 2550 0.00008010.00010260.00006640.00003690.00013030.0001184 2650 0.00007150.00009320.00005870.00003120.00012200.0001096 2750 0.00006470.00008600.00005240.00002620.00011290.0001014 2850 0.00005750.00007940.00004630.00002230.00010510.0000941 2950 0.00005170.00007170.00004130.00001890.00009950.0000876 3050 0.00004700.00006810.00003710.00001610.00009130.0000814 3150 0.00004210.00006330.00003300.00001330.00008410.0000752 3250 0.00003830.00005810.00002990.00001200.00007880.0000700 3350 0.00003470.00005370.00002680.00000980.00007250.0000644 3450 0.00003060.00004900.00002360.00000860.00006780.0000598 3550 0.00002830.00004590.00002150.00000700.00006430.0000564 3650 0.00002500.00004270.00001900.00000620.00005860.0000518 3750 0.00002270.00003880.00001710.00000500.00005450.0000478 3850 0.00002030.00003580.00001530.00000450.00005220.0000452 3950 0.00001810.00003440.00001350.00000370.00004740.0000419 4050 0.00001650.00003080.00001230.00000310.00004380.0000382 4150 0.00001520.00002940.00001130.00000280.00004170.0000364 4250 0.00001360.00002750.00001000.00000210.00003720.0000330 4350 0.00001190.00002550.00000870.00000200.00003500.0000309 4450 0.00001100.00002440.00000800.00000160.00003240.0000289 4550 0.00000980.00002200.00000710.00000140.00003080.0000270 4650 0.00000920.00002070.00000670.00000120.00002890.0000254 4750 0.00000880.00001920.00000640.00000100.00002670.0000235 4850 0.00000740.00001770.00000530.00000090.00002390.0000212 49500.00000650.00001690.00000460.00000070.00002270.0000202

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374 Table G-14. Normalized 3D c hord-length distributions throug h the marrow cavities of the left and right femur heads, left and ri ght femur necks, and their respective averages. Probability distributions ar e given through the second 51 bins with a maximum length of 10100 microns. bin width Left Femur Head Left Femur Neck Average Femur Head Right Femur Head Right Femur Neck Average Femur Neckm p(L)dLp(L)dLp(L)dL p(L)dLp(L)dLp(L)dL 5050 0.00000590.00001590.00000420.00000070.00002170.0000192 5150 0.00000560.00001480.00000400.00000050.00001940.0000174 5250 0.00000510.00001360.00000360.00000050.00001820.0000162 5350 0.00000450.00001300.00000320.00000040.00001690.0000152 5450 0.00000400.00001150.00000280.00000030.00001530.0000137 5550 0.00000390.00001110.00000270.00000030.00001460.0000131 5650 0.00000320.00000990.00000230.00000020.00001380.0000121 5750 0.00000290.00000940.00000200.00000020.00001290.0000114 5850 0.00000290.00000880.00000200.00000010.00001230.0000108 5950 0.00000260.00000790.00000180.00000010.00001130.0000099 6050 0.00000220.00000740.00000160.00000010.00001020.0000090 6150 0.00000200.00000720.00000140.00000010.00000990.0000088 6250 0.00000190.00000660.00000130.00000010.00000910.0000080 6350 0.00000160.00000600.00000110.00000010.00000850.0000074 6450 0.00000150.00000560.00000110.00000010.00000830.0000072 6550 0.00000150.00000550.00000100.00000010.00000740.0000066 6650 0.00000120.00000450.00000080.00000000.00000670.0000058 6750 0.00000110.00000450.00000080.00000000.00000650.0000056 6850 0.00000110.00000410.00000080.00000000.00000590.0000051 6950 0.00000090.00000360.00000070.00000000.00000570.0000048 7050 0.00000080.00000380.00000050.00000000.00000550.0000047 7150 0.00000070.00000330.00000050.00000000.00000460.0000041 7250 0.00000070.00000310.00000050.00000000.00000420.0000037 7350 0.00000060.00000270.00000040.00000000.00000400.0000035 7450 0.00000060.00000240.00000040.00000000.00000370.0000032 7550 0.00000050.00000230.00000040.00000000.00000350.0000030 7650 0.00000040.00000220.00000030.00000000.00000320.0000028 7750 0.00000040.00000210.00000030.00000000.00000310.0000027 7850 0.00000030.00000190.00000020.00000000.00000290.0000025 7950 0.00000030.00000180.00000020.00000000.00000290.0000024 8050 0.00000030.00000160.00000020.00000000.00000230.0000020 8150 0.00000020.00000140.00000020.00000000.00000220.0000019 8250 0.00000020.00000130.00000010.00000000.00000220.0000018 8350 0.00000020.00000120.00000010.00000000.00000190.0000016 8450 0.00000020.00000120.00000010.00000000.00000180.0000015 8550 0.00000020.00000100.00000010.00000000.00000180.0000015 8650 0.00000020.00000090.00000010.00000000.00000160.0000013 8750 0.00000010.00000090.00000010.00000000.00000160.0000013 8850 0.00000010.00000090.00000010.00000000.00000140.0000012 8950 0.00000010.00000080.00000010.00000000.00000140.0000011 9050 0.00000010.00000080.00000010.00000000.00000100.0000009 9150 0.00000010.00000070.00000010.00000000.00000120.0000010 9250 0.00000010.00000050.00000010.00000000.00000100.0000008 9350 0.00000010.00000050.00000000.00000000.00000090.0000007 9450 0.00000010.00000040.00000010.00000000.00000080.0000007 9550 0.00000000.00000050.00000000.00000000.00000060.0000006 9650 0.00000010.00000030.00000000.00000000.00000070.0000005 9750 0.00000000.00000040.00000000.00000000.00000060.0000005 9850 0.00000000.00000030.00000000.00000000.00000060.00000059950 0.00000000.00000030.00000000.00000000.00000040.0000004 10050 0.00000020.00000300.00000020.00000000.00000540.0000044

PAGE 407

375 Table G-15. Normalized 3D c hord-length distributions throug h the marrow cavities of the pubis, ilium, ischium, right and left sca pula, sternum, and the averages for the os coxae and scapula. Probability distributions are given through the first 50 bins with a maximum length of 5000 microns. bin width PubisIschiumIlium Average Os Coxae Right Scapula Left Scapula Average Scapula Sternumm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 50 0.00017930.00022530.00018750.00019370.00036340.00030910.00034040.0003673 150 0.00025900.00035340.00029360.00029500.00049310.00045190.00047560.0005533 250 0.00035280.00038940.00039900.00037850.00056850.00054860.00056010.0006773 350 0.00042680.00041120.00048480.00044370.00058310.00059170.00058670.0006890 450 0.00050230.00043680.00055340.00050430.00058290.00061050.00059460.0006597 550 0.00056470.00045030.00060810.00055170.00058140.00061610.00059610.0006071 650 0.00059930.00046580.00062010.00057340.00056380.00060650.00058190.0005371 750 0.00059180.00046790.00059220.00056100.00054670.00059710.00056810.0004879 850 0.00054840.00046260.00053890.00052360.00052300.00057730.00054600.0004377 950 0.00049480.00044110.00048930.00047940.00048840.00053220.00050700.0003933 1050 0.00044450.00041450.00043200.00043260.00045030.00047580.00046110.0003520 1150 0.00040230.00038400.00038170.00039030.00041640.00042500.00042000.0003153 1250 0.00036710.00036100.00034090.00035620.00037200.00037480.00037320.0002858 1350 0.00033780.00033790.00031030.00032800.00033930.00034160.00034030.0002562 1450 0.00030980.00031170.00028520.00030140.00030310.00030410.00030350.0002329 1550 0.00028450.00029220.00025760.00027680.00027890.00027370.00027670.0002136 1650 0.00026050.00027330.00024210.00025710.00024700.00024810.00024750.0001965 1750 0.00023850.00025830.00021880.00023640.00022750.00022350.00022580.0001806 1850 0.00021760.00023870.00019930.00021630.00020190.00020040.00020120.0001654 1950 0.00020010.00021990.00017960.00019770.00018180.00018020.00018110.0001531 2050 0.00018280.00020510.00016540.00018210.00016350.00016440.00016390.0001436 2150 0.00017110.00018850.00015210.00016870.00014860.00014680.00014780.0001305 2250 0.00015680.00017650.00014270.00015670.00013790.00013080.00013490.0001211 2350 0.00014310.00016530.00013220.00014470.00012210.00011440.00011880.0001110 2450 0.00013280.00015390.00012000.00013350.00010910.00010240.00010620.0001039 2550 0.00012200.00014160.00011050.00012280.00009850.00009000.00009490.0000978 2650 0.00011290.00012980.00010320.00011370.00008900.00008180.00008590.0000903 2750 0.00010460.00012190.00009470.00010540.00007950.00007350.00007700.0000862 2850 0.00009590.00011150.00008740.00009670.00007110.00006580.00006890.0000796 2950 0.00008890.00010340.00008170.00008990.00006540.00006030.00006330.0000732 3050 0.00008170.00009330.00007670.00008280.00005980.00005440.00005750.0000716 3150 0.00007590.00008820.00007060.00007700.00005280.00004710.00005040.0000644 3250 0.00007010.00008260.00006710.00007210.00004860.00004150.00004560.0000610 3350 0.00006480.00007710.00006060.00006630.00004340.00003830.00004130.0000575 3450 0.00005910.00007050.00005740.00006130.00003860.00003330.00003630.0000539 3550 0.00005620.00006550.00005350.00005750.00003710.00002980.00003400.0000505 3650 0.00005100.00006120.00004900.00005280.00003180.00002610.00002940.0000483 3750 0.00004750.00005580.00004550.00004890.00003010.00002340.00002720.0000453 3850 0.00004300.00005310.00004300.00004550.00002650.00002050.00002400.0000432 3950 0.00004090.00004970.00004130.00004330.00002380.00001860.00002160.0000411 4050 0.00003680.00004610.00003780.00003950.00002140.00001670.00001940.0000386 4150 0.00003560.00004300.00003630.00003770.00001920.00001500.00001740.0000352 4250 0.00003290.00003990.00003350.00003480.00001690.00001350.00001550.0000339 4350 0.00002950.00003540.00003160.00003170.00001520.00001120.00001350.0000317 4450 0.00002820.00003380.00002960.00003010.00001340.00001100.00001240.0000300 4550 0.00002590.00003190.00002770.00002800.00001240.00000970.00001120.0000288 4650 0.00002400.00002890.00002610.00002600.00001130.00000830.00001010.0000270 4750 0.00002230.00002720.00002440.00002420.00000960.00000680.00000840.0000255 4850 0.00002000.00002480.00002220.00002200.00000960.00000680.00000840.0000239 49500.00001930.00002310.00002130.00002090.00000780.00000620.00000710.0000235

PAGE 408

376 Table G-16. Normalized 3D c hord-length distributions throug h the marrow cavities of the pubis, ilium, ischium, right and left sca pula, sternum, and the averages for the os coxae and scapula. Probability distributions are given through the second 51 bins with a maximum length of 10100 microns. bin width PubisIschiumIlium Average Os Coxae Right Scapula Left Scapula Average Scapula Sternumm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 5050 0.00001820.00002110.00001950.00001940.00000700.00000540.00000630.0000216 5150 0.00001660.00001960.00001890.00001810.00000630.00000450.00000550.0000209 5250 0.00001500.00001860.00001900.00001740.00000580.00000380.00000490.0000193 5350 0.00001350.00001640.00001650.00001530.00000540.00000330.00000450.0000186 5450 0.00001350.00001580.00001610.00001500.00000490.00000310.00000410.0000175 5550 0.00001230.00001440.00001530.00001390.00000450.00000280.00000380.0000163 5650 0.00001090.00001320.00001440.00001270.00000340.00000240.00000300.0000159 5750 0.00001040.00001240.00001360.00001210.00000350.00000230.00000300.0000141 5850 0.00000970.00001110.00001250.00001110.00000310.00000160.00000250.0000140 5950 0.00000890.00000970.00001190.00001020.00000320.00000180.00000260.0000128 6050 0.00000830.00000910.00001120.00000950.00000220.00000160.00000190.0000120 6150 0.00000800.00000910.00001060.00000920.00000220.00000140.00000180.0000113 6250 0.00000740.00000760.00000940.00000820.00000210.00000120.00000170.0000113 6350 0.00000660.00000700.00000880.00000750.00000160.00000110.00000140.0000104 6450 0.00000610.00000720.00000820.00000710.00000200.00000080.00000150.0000092 6550 0.00000570.00000650.00000800.00000670.00000130.00000060.00000100.0000092 6650 0.00000540.00000590.00000810.00000650.00000160.00000060.00000120.0000087 6750 0.00000500.00000540.00000760.00000600.00000120.00000050.00000090.0000083 6850 0.00000460.00000470.00000720.00000550.00000120.00000050.00000090.0000079 6950 0.00000420.00000470.00000630.00000510.00000130.00000040.00000090.0000068 7050 0.00000380.00000460.00000610.00000480.00000110.00000040.00000080.0000066 7150 0.00000360.00000390.00000590.00000450.00000070.00000030.00000050.0000063 7250 0.00000340.00000390.00000540.00000420.00000090.00000030.00000060.0000058 7350 0.00000320.00000350.00000550.00000410.00000070.00000030.00000050.0000056 7450 0.00000310.00000300.00000460.00000360.00000070.00000020.00000050.0000056 7550 0.00000280.00000280.00000440.00000340.00000050.00000020.00000040.0000052 7650 0.00000270.00000260.00000430.00000330.00000060.00000020.00000040.0000047 7750 0.00000240.00000240.00000370.00000290.00000040.00000020.00000030.0000043 7850 0.00000210.00000230.00000370.00000270.00000050.00000020.00000040.0000036 7950 0.00000210.00000220.00000340.00000260.00000030.00000010.00000020.0000039 8050 0.00000180.00000190.00000340.00000240.00000040.00000010.00000030.0000034 8150 0.00000180.00000160.00000300.00000210.00000030.00000010.00000020.0000035 8250 0.00000150.00000150.00000310.00000210.00000030.00000010.00000020.0000029 8350 0.00000150.00000160.00000270.00000190.00000030.00000010.00000020.0000028 8450 0.00000140.00000160.00000300.00000200.00000020.00000010.00000020.0000025 8550 0.00000120.00000130.00000270.00000180.00000020.00000010.00000010.0000024 8650 0.00000110.00000110.00000220.00000150.00000010.00000010.00000010.0000022 8750 0.00000110.00000100.00000210.00000150.00000010.00000010.00000010.0000020 8850 0.00000090.00000100.00000200.00000130.00000010.00000000.00000010.0000022 8950 0.00000090.00000080.00000190.00000120.00000010.00000000.00000010.0000017 9050 0.00000080.00000080.00000190.00000120.00000010.00000000.00000010.0000017 9150 0.00000080.00000080.00000130.00000100.00000010.00000000.00000010.0000015 9250 0.00000080.00000070.00000140.00000100.00000020.00000000.00000010.0000015 9350 0.00000070.00000080.00000130.00000090.00000010.00000000.00000010.0000014 9450 0.00000050.00000060.00000120.00000080.00000010.00000000.00000010.0000012 9550 0.00000060.00000060.00000140.00000090.00000010.00000000.00000000.0000011 9650 0.00000050.00000060.00000100.00000070.00000010.00000000.00000010.0000011 9750 0.00000040.00000050.00000100.00000070.00000010.00000000.00000010.0000011 9850 0.00000040.00000050.00000090.00000060.00000010.00000000.00000000.0000010 99500.00000050.00000040.00000090.00000060.00000000.00000000.00000000.0000010 10050 0.00000410.00000600.00001190.00000740.00000040.00000000.00000030.0000112

PAGE 409

377 Table G-17. Normalized 3D c hord-length distributions throug h the marrow cavities of the right and left clavicle, ri ght and left humerus, C3 and C6 vertebra, and the averages for those respective bone sites. Probability distributions are given through the first 50 bins with a maximum length of 5000 microns. bin width Left Clavicle Right Clavicle Average Clavicle Right Humerus Left Humerus Average Humerus C3 VertebraC6 Vertebra Average Cervical V ertebram p(L)dLp(L)dLp(L)dLp(L)dLp(L) dLp(L)dLp(L)dLp(L)dLp(L)dL 50 0.00036120.00035220.00035790.00027830.00021790.00024220.00037360.00033590.0003515 150 0.00047140.00046550.00046920.00039250.00030860.00034230.00045330.00043330.0004416 250 0.00048930.00060940.00053340.00052810.00044870.00048060.00066170.00059230.0006209 350 0.00050240.00068460.00056930.00060010.00054750.00056870.00077710.00068920.0007254 450 0.00051350.00071780.00058850.00063470.00062780.00063060.00081860.00074260.0007740 550 0.00047330.00070170.00055720.00065200.00067510.00066580.00080500.00075570.0007760 650 0.00044250.00065390.00052010.00064850.00068460.00067010.00074070.00072860.0007336 750 0.00041700.00060410.00048570.00063170.00066710.00065290.00064990.00067710.0006659 850 0.00039100.00055480.00045110.00059770.00061810.00060990.00055530.00060510.0005845 950 0.00037680.00050800.00042500.00055130.00055980.00055640.00048160.00052740.0005085 1050 0.00036750.00045680.00040030.00049870.00049990.00049940.00042450.00045410.0004419 1150 0.00034290.00041050.00036770.00044660.00044550.00044590.00037260.00039470.0003856 1250 0.00029760.00036500.00032240.00039810.00039980.00039910.00032890.00034880.0003406 1350 0.00027290.00031570.00028860.00034770.00035210.00035030.00028820.00030670.0002990 1450 0.00025440.00027480.00026190.00030960.00031290.00031160.00025310.00027190.0002641 1550 0.00024150.00024240.00024180.00027350.00027910.00027690.00022140.00023940.0002320 1650 0.00022250.00021230.00021870.00023960.00024870.00024500.00019530.00021150.0002048 1750 0.00019800.00018670.00019380.00021330.00022120.00021800.00016960.00018560.0001790 1850 0.00017590.00016320.00017130.00018880.00019760.00019410.00014880.00016460.0001581 1950 0.00016000.00014440.00015430.00016570.00017620.00017200.00013320.00014640.0001409 2050 0.00015360.00012840.00014430.00014780.00015790.00015380.00011890.00012820.0001244 2150 0.00014090.00011490.00013130.00013070.00014160.00013720.00010350.00011380.0001095 2250 0.00013400.00010230.00012230.00011540.00012510.00012120.00008930.00010120.0000963 2350 0.00012860.00009350.00011570.00010180.00011150.00010760.00007940.00009080.0000861 2450 0.00012710.00008620.00011210.00009230.00009850.00009600.00007070.00008110.0000768 2550 0.00011770.00007680.00010270.00008350.00008900.00008680.00006340.00007340.0000693 2650 0.00011250.00007080.00009720.00007450.00007950.00007750.00005790.00006430.0000617 2750 0.00010280.00006400.00008860.00006590.00007100.00006890.00005170.00005590.0000542 2850 0.00010110.00005800.00008530.00006000.00006450.00006270.00004580.00005070.0000487 2950 0.00009380.00005390.00007920.00005240.00005790.00005560.00004130.00004480.0000434 3050 0.00008620.00004690.00007180.00004770.00005190.00005020.00003770.00003940.0000387 3150 0.00008170.00004220.00006720.00004270.00004660.00004500.00003350.00003600.0000350 3250 0.00007620.00003930.00006260.00003870.00004160.00004040.00003000.00003270.0000316 3350 0.00007860.00003480.00006250.00003450.00003760.00003640.00002660.00002860.0000278 3450 0.00007160.00003210.00005710.00003170.00003360.00003280.00002460.00002560.0000252 3550 0.00006810.00002970.00005400.00002810.00003050.00002950.00002280.00002310.0000229 3650 0.00006310.00002780.00005010.00002520.00002710.00002640.00002060.00002080.0000207 3750 0.00006070.00002540.00004780.00002250.00002460.00002380.00001910.00001900.0000191 3850 0.00005820.00002220.00004500.00002030.00002190.00002120.00001790.00001670.0000172 3950 0.00005250.00002010.00004060.00001830.00001960.00001910.00001600.00001510.0000155 4050 0.00005160.00001800.00003930.00001640.00001750.00001710.00001460.00001360.0000140 4150 0.00004750.00001710.00003630.00001520.00001590.00001560.00001360.00001220.0000127 4250 0.00004570.00001590.00003480.00001320.00001470.00001410.00001300.00001120.0000119 4350 0.00004360.00001460.00003290.00001180.00001260.00001230.00001140.00000960.0000103 4450 0.00003960.00001330.00003000.00001060.00001190.00001140.00001070.00000870.0000095 4550 0.00003750.00001120.00002780.00001000.00001060.00001040.00001030.00000750.0000086 4650 0.00003800.00001100.00002810.00000900.00000970.00000940.00000910.00000710.0000079 4750 0.00003620.00000870.00002610.00000790.00000870.00000840.00000800.00000620.0000069 4850 0.00003010.00000850.00002220.00000720.00000780.00000750.00000750.00000570.00000654950 0.00003230.00000740.00002320.00000660.00000690.00000680.00000710.00000500.0000058

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378 Table G-18. Normalized 3D c hord-length distributions throug h the marrow cavities of the right and left clavicle, ri ght and left humerus, C3 and C6 vertebra, and the averages for those respective bone sites. Probability distributions are given through the second 51 bins with a maximum length of 10100 microns. bin width Left Clavicle Right Clavicle Average Clavicle Right Humerus Left Humerus Average Humerus C3 VertebraC6 Vertebra Average Cervical V ertebram p(L)dLp(L)dLp(L)dLp(L)dLp(L) dLp(L)dLp(L)dLp(L)dLp(L)dL 5050 0.00002950.00000740.00002140.00000600.00000630.00000620.00000620.00000420.0000050 5150 0.00003140.00000630.00002220.00000550.00000560.00000560.00000580.00000420.0000049 5250 0.00003160.00000590.00002220.00000510.00000520.00000510.00000540.00000360.0000043 5350 0.00002940.00000550.00002060.00000440.00000450.00000450.00000500.00000340.0000040 5450 0.00002740.00000490.00001920.00000410.00000420.00000410.00000430.00000290.0000035 5550 0.00002580.00000440.00001790.00000340.00000380.00000360.00000420.00000260.0000033 5650 0.00002610.00000400.00001800.00000310.00000350.00000340.00000390.00000230.0000030 5750 0.00002380.00000400.00001650.00000300.00000310.00000310.00000360.00000190.0000026 5850 0.00002250.00000330.00001540.00000270.00000280.00000270.00000310.00000180.0000024 5950 0.00002430.00000290.00001640.00000220.00000240.00000230.00000300.00000160.0000021 6050 0.00002390.00000300.00001620.00000200.00000230.00000220.00000270.00000140.0000019 6150 0.00002270.00000260.00001540.00000200.00000200.00000200.00000240.00000130.0000018 6250 0.00002230.00000240.00001500.00000180.00000190.00000190.00000230.00000110.0000016 6350 0.00002140.00000220.00001440.00000150.00000160.00000160.00000210.00000110.0000015 6450 0.00002000.00000180.00001330.00000150.00000150.00000150.00000180.00000090.0000013 6550 0.00002010.00000160.00001330.00000120.00000130.00000130.00000170.00000080.0000012 6650 0.00001840.00000150.00001220.00000130.00000120.00000120.00000160.00000070.0000011 6750 0.00001640.00000130.00001080.00000110.00000110.00000110.00000170.00000060.0000010 6850 0.00001630.00000110.00001070.00000100.00000100.00000100.00000140.00000050.0000009 6950 0.00001570.00000110.00001030.00000090.00000090.00000090.00000120.00000050.0000008 7050 0.00001630.00000110.00001070.00000070.00000080.00000080.00000090.00000040.0000006 7150 0.00001440.00000100.00000950.00000080.00000080.00000080.00000110.00000040.0000007 7250 0.00001390.00000080.00000910.00000060.00000060.00000060.00000090.00000030.0000005 7350 0.00001110.00000080.00000730.00000050.00000060.00000060.00000080.00000030.0000005 7450 0.00001170.00000070.00000770.00000050.00000050.00000050.00000060.00000030.0000004 7550 0.00001100.00000070.00000720.00000050.00000040.00000050.00000060.00000030.0000004 7650 0.00001010.00000080.00000670.00000050.00000040.00000040.00000060.00000020.0000004 7750 0.00001040.00000060.00000680.00000030.00000040.00000040.00000050.00000020.0000004 7850 0.00000910.00000050.00000600.00000030.00000040.00000030.00000040.00000010.0000003 7950 0.00000850.00000050.00000560.00000030.00000030.00000030.00000040.00000020.0000003 8050 0.00000680.00000050.00000450.00000030.00000030.00000030.00000030.00000020.0000002 8150 0.00000680.00000040.00000440.00000030.00000020.00000020.00000020.00000010.0000002 8250 0.00000600.00000040.00000400.00000020.00000020.00000020.00000030.00000010.0000001 8350 0.00000640.00000050.00000420.00000020.00000020.00000020.00000020.00000010.0000001 8450 0.00000590.00000030.00000380.00000020.00000020.00000020.00000020.00000010.0000001 8550 0.00000560.00000030.00000370.00000020.00000010.00000020.00000010.00000010.0000001 8650 0.00000500.00000030.00000320.00000010.00000020.00000020.00000010.00000010.0000001 8750 0.00000510.00000030.00000330.00000020.00000010.00000010.00000010.00000000.0000000 8850 0.00000390.00000020.00000250.00000010.00000010.00000010.00000010.00000010.0000001 8950 0.00000340.00000020.00000220.00000010.00000010.00000010.00000010.00000000.0000001 9050 0.00000440.00000030.00000290.00000010.00000010.00000010.00000000.00000000.0000000 9150 0.00000410.00000030.00000270.00000010.00000010.00000010.00000000.00000000.0000000 9250 0.00000350.00000020.00000220.00000010.00000000.00000010.00000000.00000000.0000000 9350 0.00000380.00000020.00000240.00000010.00000010.00000010.00000000.00000000.0000000 9450 0.00000290.00000010.00000190.00000010.00000000.00000000.00000010.00000000.0000000 9550 0.00000300.00000010.00000200.00000000.00000010.00000000.00000000.00000000.0000000 9650 0.00000300.00000020.00000200.00000000.00000010.00000010.00000000.00000000.0000000 9750 0.00000320.00000010.00000210.00000010.00000000.00000010.00000000.00000000.0000000 9850 0.00000240.00000010.00000160.00000010.00000000.00000000.00000000.00000000.00000009950 0.00000250.00000010.00000160.00000000.00000000.00000000.00000000.00000000.0000000 10050 0.00004410.00000150.00002850.00000020.00000040.00000030.00000000.00000000.0000000

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379 Table G-19. Normalized 3D c hord-length distributions throug h the marrow cavities of the T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages for the thoracic and lumbar vertebra. Proba bility distributions are given through the first 50 bins with a maximum length of 5000 microns. bin width T3 VertebraT6 VertebraT11 Vertebra Average Thoracic V ertebra L2 VertebraL4Vertebra Average Lumbar V ertebra Sacrumm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 50 0.00022010.00023140.00023030.00022740.00018250.00023250.00020340.0002287 150 0.00030720.00036210.00030390.00032370.00025560.00037250.00030450.0003556 250 0.00041620.00044920.00046040.00044270.00033020.00055270.00042330.0005077 350 0.00048740.00048060.00058510.00052030.00036720.00068210.00049890.0006237 450 0.00054720.00050750.00067910.00058180.00039870.00077440.00055590.0007187 550 0.00059330.00052730.00072230.00061850.00042820.00081880.00059160.0007709 650 0.00061050.00053860.00070970.00062300.00044660.00079480.00059230.0007534 750 0.00059770.00052690.00064830.00059310.00045040.00072320.00056450.0006991 850 0.00055890.00049550.00057970.00054600.00044760.00063360.00052540.0006171 950 0.00050500.00045850.00051160.00049240.00042690.00054550.00047650.0005378 1050 0.00045110.00042120.00044770.00044030.00040550.00046380.00042990.0004686 1150 0.00040680.00038680.00039480.00039610.00037210.00039600.00038210.0004118 1250 0.00037180.00035600.00034850.00035840.00035020.00034540.00034820.0003625 1350 0.00033910.00032520.00031140.00032470.00032990.00030090.00031770.0003193 1450 0.00030800.00029980.00027990.00029520.00031150.00026540.00029220.0002787 1550 0.00028260.00027480.00024920.00026810.00028960.00023440.00026650.0002506 1650 0.00025850.00025460.00022590.00024560.00026860.00020740.00024300.0002200 1750 0.00023420.00023240.00020370.00022260.00024780.00018080.00021980.0001972 1850 0.00021310.00021570.00018390.00020350.00023170.00016020.00020180.0001764 1950 0.00019320.00020080.00016270.00018470.00021500.00013940.00018330.0001564 2050 0.00017450.00018640.00014830.00016890.00019880.00012340.00016720.0001402 2150 0.00016130.00017260.00013360.00015500.00019010.00011040.00015680.0001228 2250 0.00014530.00015830.00012070.00014060.00017670.00009730.00014350.0001095 2350 0.00013310.00014640.00011060.00012930.00016550.00008630.00013240.0000972 2450 0.00012170.00013620.00010030.00011870.00015670.00007700.00012340.0000881 2550 0.00011180.00012320.00009130.00010810.00014350.00006870.00011220.0000790 2650 0.00010210.00011340.00008320.00009900.00013520.00006170.00010440.0000707 2750 0.00009370.00010600.00007500.00009090.00012880.00005510.00009800.0000639 2850 0.00008580.00009940.00006930.00008420.00012110.00004980.00009120.0000575 2950 0.00007800.00009230.00006150.00007660.00011320.00004490.00008460.0000523 3050 0.00007360.00008440.00005860.00007170.00010820.00003980.00007960.0000456 3150 0.00006570.00007850.00005260.00006510.00010060.00003540.00007330.0000419 3250 0.00006140.00007210.00004820.00006010.00009590.00003220.00006920.0000365 3350 0.00005570.00006760.00004470.00005560.00008990.00002800.00006400.0000332 3450 0.00005230.00006140.00004070.00005100.00008400.00002630.00005990.0000300 3550 0.00004650.00005740.00003750.00004680.00007820.00002320.00005520.0000269 3650 0.00004380.00005230.00003490.00004330.00007320.00002140.00005150.0000241 3750 0.00003970.00004850.00003200.00003970.00007000.00001930.00004880.0000220 3850 0.00003630.00004500.00002970.00003670.00006570.00001710.00004530.0000200 3950 0.00003360.00004120.00002770.00003390.00006180.00001560.00004250.0000177 4050 0.00003170.00003840.00002530.00003160.00005830.00001410.00003980.0000167 4150 0.00002810.00003600.00002400.00002920.00005450.00001290.00003710.0000144 4250 0.00002640.00003340.00002080.00002670.00005090.00001170.00003450.0000127 4350 0.00002360.00003100.00001980.00002460.00004750.00001030.00003200.0000118 4450 0.00002200.00002830.00001860.00002280.00004410.00000960.00002970.0000105 4550 0.00002030.00002660.00001770.00002140.00004160.00000840.00002770.0000099 4650 0.00001860.00002390.00001630.00001950.00003960.00000780.00002630.0000090 4750 0.00001720.00002220.00001490.00001800.00003640.00000710.00002420.0000080 4850 0.00001550.00002080.00001390.00001660.00003410.00000610.00002240.0000070 49500.00001490.00001900.00001290.00001550.00003260.00000580.00002140.0000067

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380 Table G-20. Normalized 3D c hord-length distributions throug h the marrow cavities of the T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages for the thoracic and lumbar vertebra. Proba bility distributions are given through the second 51 bins with a ma ximum length of 10100 microns. bin width T3 VertebraT6 VertebraT11 Vertebra Average Thoracic V ertebra L2 VertebraL4Vertebra Average Lumbar V ertebra Sacrumm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 5050 0.00001340.00001770.00001190.00001420.00003050.00000500.00001980.0000060 5150 0.00001220.00001640.00001140.00001330.00002890.00000470.00001870.0000052 5250 0.00001140.00001480.00001050.00001210.00002720.00000410.00001760.0000049 5350 0.00001000.00001380.00000990.00001120.00002540.00000360.00001630.0000043 5450 0.00000930.00001300.00000920.00001050.00002370.00000330.00001520.0000037 5550 0.00000890.00001200.00000860.00000980.00002210.00000310.00001420.0000036 5650 0.00000800.00001140.00000800.00000910.00002070.00000270.00001320.0000031 5750 0.00000740.00001050.00000740.00000840.00001890.00000250.00001200.0000028 5850 0.00000680.00000970.00000670.00000770.00001750.00000230.00001110.0000026 5950 0.00000610.00000880.00000640.00000710.00001630.00000210.00001030.0000025 6050 0.00000590.00000810.00000580.00000650.00001470.00000170.00000930.0000022 6150 0.00000550.00000750.00000570.00000620.00001450.00000170.00000910.0000019 6250 0.00000470.00000690.00000510.00000560.00001320.00000140.00000830.0000017 6350 0.00000440.00000660.00000470.00000520.00001250.00000130.00000780.0000015 6450 0.00000410.00000600.00000480.00000490.00001170.00000100.00000720.0000014 6550 0.00000390.00000540.00000460.00000460.00001110.00000100.00000690.0000012 6650 0.00000340.00000510.00000390.00000410.00001080.00000090.00000670.0000011 6750 0.00000310.00000470.00000380.00000390.00000960.00000080.00000590.0000010 6850 0.00000290.00000430.00000360.00000360.00000890.00000070.00000550.0000010 6950 0.00000280.00000380.00000310.00000320.00000820.00000060.00000510.0000009 7050 0.00000250.00000390.00000310.00000320.00000800.00000050.00000490.0000007 7150 0.00000230.00000330.00000280.00000280.00000690.00000040.00000420.0000007 7250 0.00000200.00000330.00000270.00000270.00000700.00000040.00000420.0000006 7350 0.00000190.00000290.00000250.00000240.00000620.00000040.00000370.0000006 7450 0.00000180.00000260.00000230.00000230.00000570.00000040.00000350.0000005 7550 0.00000160.00000250.00000220.00000210.00000550.00000030.00000330.0000005 7650 0.00000140.00000220.00000180.00000180.00000520.00000030.00000320.0000004 7750 0.00000140.00000230.00000180.00000180.00000440.00000020.00000270.0000003 7850 0.00000130.00000200.00000170.00000170.00000410.00000020.00000250.0000003 7950 0.00000120.00000180.00000170.00000160.00000390.00000020.00000240.0000003 8050 0.00000100.00000170.00000150.00000140.00000340.00000020.00000210.0000002 8150 0.00000090.00000140.00000140.00000130.00000330.00000010.00000200.0000002 8250 0.00000080.00000150.00000130.00000120.00000310.00000010.00000190.0000002 8350 0.00000080.00000120.00000120.00000110.00000280.00000010.00000170.0000002 8450 0.00000080.00000120.00000130.00000110.00000230.00000010.00000140.0000002 8550 0.00000070.00000120.00000100.00000090.00000290.00000010.00000170.0000001 8650 0.00000060.00000100.00000080.00000080.00000210.00000010.00000130.0000001 8750 0.00000060.00000090.00000080.00000080.00000190.00000010.00000110.0000001 8850 0.00000050.00000080.00000080.00000070.00000190.00000010.00000110.0000001 8950 0.00000050.00000080.00000080.00000070.00000160.00000000.00000090.0000001 9050 0.00000040.00000080.00000070.00000060.00000150.00000000.00000090.0000001 9150 0.00000040.00000060.00000070.00000060.00000160.00000010.00000090.0000001 9250 0.00000040.00000070.00000060.00000050.00000130.00000000.00000080.0000001 9350 0.00000040.00000050.00000060.00000050.00000120.00000000.00000070.0000000 9450 0.00000030.00000050.00000050.00000040.00000120.00000000.00000070.0000000 9550 0.00000020.00000040.00000050.00000040.00000100.00000000.00000060.0000000 9650 0.00000030.00000050.00000050.00000040.00000090.00000000.00000050.0000001 9750 0.00000020.00000040.00000030.00000030.00000090.00000000.00000050.0000000 9850 0.00000020.00000040.00000040.00000030.00000070.00000000.00000040.0000000 99500.00000020.00000030.00000040.00000030.00000080.00000000.00000050.0000000 10050 0.00000220.00000280.00000350.00000290.00000790.00000010.00000460.0000002

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381 Table G-21. Normalized 3D c hord-length distributions throug h the marrow cavities of the right and left upper rib, middle rib, and lower rib. Also shown is the average for the single rib. Probability distri butions are given through the first 50 bins with a maximum length of 5000 microns. bin width Upper Rt Rib Middle Rt Rib Lower Rt RibUpper Lt Rib Middle Lt Rib Lower Lt RibAverage Ribm p(L)dLp(L)dLp( L)dLp(L)dLp(L) dLp(L)dLp(L)dL 50 0.00029060.00029370.00036410.00041010.00031440.00044910.0003325 150 0.00043350.00045270.00048390.00052580.00042360.00062910.0004661 250 0.00052830.00046840.00052340.00058740.00050240.00067060.0005208 350 0.00054930.00046420.00051720.00059550.00051510.00061000.0005218 450 0.00056540.00045420.00054940.00059240.00052800.00057530.0005262 550 0.00051270.00043920.00054280.00057940.00051330.00056410.0005083 650 0.00048340.00039420.00055660.00055840.00050320.00053630.0004873 750 0.00043020.00033840.00056640.00050960.00048150.00050750.0004534 850 0.00039990.00033170.00054410.00045430.00043160.00046810.0004208 950 0.00038650.00030630.00050890.00040010.00037070.00040780.0003788 1050 0.00039510.00028630.00044780.00035580.00033150.00037250.0003474 1150 0.00039050.00025640.00038430.00031350.00029610.00035130.0003140 1250 0.00035900.00024000.00035280.00029300.00026460.00031050.0002869 1350 0.00032660.00021410.00033330.00026840.00024830.00028500.0002642 1450 0.00030850.00019790.00030410.00024810.00022180.00026150.0002419 1550 0.00028970.00018740.00026750.00023330.00022010.00024840.0002293 1650 0.00026200.00017910.00024130.00021770.00020920.00023150.0002142 1750 0.00025430.00017270.00022580.00019800.00020230.00020870.0002038 1850 0.00024780.00017020.00019860.00018780.00018610.00019380.0001917 1950 0.00022900.00016280.00018890.00017400.00017830.00017600.0001810 2050 0.00020670.00015210.00016790.00015230.00017050.00016460.0001674 2150 0.00020610.00014350.00015230.00014640.00015080.00015510.0001556 2250 0.00018070.00012960.00013850.00013670.00014660.00014420.0001440 2350 0.00016040.00012370.00012850.00013050.00013720.00013220.0001343 2450 0.00015310.00011020.00011080.00012210.00012630.00011940.0001224 2550 0.00014180.00010220.00010630.00011660.00011570.00010870.0001137 2650 0.00013930.00010100.00009820.00010850.00010290.00010190.0001067 2750 0.00013480.00011000.00008350.00010330.00010390.00009130.0001057 2850 0.00012530.00011990.00007370.00009470.00010240.00008360.0001039 2950 0.00011380.00012270.00007120.00008860.00010570.00007990.0001031 3050 0.00009710.00013140.00006880.00008650.00010290.00007410.0001015 3150 0.00008420.00012520.00005770.00008520.00010250.00006820.0000963 3250 0.00007600.00012820.00005240.00007950.00009870.00006290.0000932 3350 0.00005870.00013420.00005190.00007440.00009120.00005710.0000893 3450 0.00005930.00012900.00004890.00007240.00008910.00005290.0000865 3550 0.00004670.00013500.00004620.00006750.00008140.00004850.0000830 3650 0.00003760.00013010.00004160.00006210.00007480.00004230.0000769 3750 0.00003450.00012420.00003140.00005870.00007360.00003940.0000727 3850 0.00002950.00011890.00003440.00005430.00006890.00003570.0000689 3950 0.00002300.00010820.00002300.00004980.00006390.00003270.0000616 4050 0.00002170.00010010.00002790.00004360.00006500.00003030.0000595 4150 0.00002050.00009140.00002320.00003780.00005430.00002670.0000523 4250 0.00001680.00008320.00002220.00003220.00005380.00002230.0000485 4350 0.00001700.00008060.00001980.00003000.00004990.00002160.0000460 4450 0.00001270.00007340.00001660.00002650.00004430.00001850.0000408 4550 0.00001290.00006680.00001780.00002400.00004350.00001660.0000386 4650 0.00001090.00006180.00001510.00002020.00004010.00001420.0000351 4750 0.00000800.00005880.00001180.00001800.00003650.00001360.0000322 48500.00001140.00005400.00001130.00001510.00003640.00001160.0000308 4950 0.00000980.00004990.00001380.00001420.00003140.00001210.0000282

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382 Table G-22. Normalized 3D c hord-length distributions throug h the marrow cavities of the right and left upper, middle, and lower ri bs. Also shown is the average for the single rib. Probability distributions ar e given through the second 51 bins with a maximum length of 10100 microns. bin width Upper Rt Rib Middle Rt Rib Lower Rt RibUpper Lt Rib Middle Lt Rib Lower Lt RibAverage Ribm p(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dLp(L)dL 5050 0.00000780.00005000.00001020.00001420.00003220.00001000.0000276 5150 0.00000730.00004490.00000930.00001180.00003090.00000780.0000253 5250 0.00000570.00003860.00000850.00001060.00002610.00000600.0000216 5350 0.00000480.00003990.00000650.00000940.00002480.00000560.0000210 5450 0.00000470.00004260.00000710.00000790.00002410.00000570.0000214 5550 0.00000550.00003920.00000550.00000790.00002190.00000420.0000196 5650 0.00000470.00003900.00000660.00000710.00002090.00000400.0000192 5750 0.00000450.00003590.00000560.00000710.00002300.00000360.0000189 5850 0.00000300.00003280.00000480.00000620.00001760.00000330.0000160 5950 0.00000480.00003200.00000500.00000470.00001770.00000230.0000158 6050 0.00000230.00003130.00000410.00000520.00001500.00000190.0000144 6150 0.00000310.00002930.00000250.00000390.00001550.00000130.0000137 6250 0.00000290.00002510.00000380.00000420.00001240.00000130.0000118 6350 0.00000270.00002340.00000400.00000370.00001040.00000090.0000107 6450 0.00000240.00002040.00000280.00000350.00001070.00000060.0000097 6550 0.00000200.00001850.00000340.00000330.00001130.00000050.0000094 6650 0.00000230.00001880.00000360.00000290.00000920.00000060.0000089 6750 0.00000150.00001520.00000240.00000240.00000850.00000040.0000074 6850 0.00000210.00001680.00000200.00000210.00000740.00000030.0000075 6950 0.00000200.00001270.00000180.00000220.00000760.00000020.0000064 7050 0.00000180.00001240.00000190.00000170.00000660.00000020.0000060 7150 0.00000130.00001070.00000200.00000180.00000510.00000010.0000050 7250 0.00000170.00001090.00000220.00000160.00000670.00000010.0000056 7350 0.00000140.00001000.00000160.00000130.00000540.00000000.0000048 7450 0.00000140.00000850.00000130.00000170.00000540.00000000.0000044 7550 0.00000090.00000790.00000170.00000150.00000490.00000000.0000041 7650 0.00000140.00000780.00000110.00000140.00000560.00000000.0000042 7750 0.00000130.00000740.00000170.00000100.00000400.00000000.0000037 7850 0.00000060.00000610.00000060.00000140.00000460.00000000.0000033 7950 0.00000070.00000520.00000060.00000100.00000330.00000000.0000027 8050 0.00000080.00000540.00000180.00000080.00000410.00000000.0000031 8150 0.00000090.00000490.00000100.00000090.00000400.00000000.0000029 8250 0.00000090.00000550.00000070.00000070.00000340.00000000.0000028 8350 0.00000090.00000440.00000070.00000080.00000260.00000000.0000023 8450 0.00000090.00000440.00000060.00000030.00000340.00000000.0000024 8550 0.00000040.00000390.00000040.00000050.00000300.00000000.0000021 8650 0.00000050.00000390.00000050.00000060.00000240.00000000.0000020 8750 0.00000090.00000300.00000010.00000060.00000270.00000000.0000018 8850 0.00000070.00000320.00000050.00000040.00000180.00000000.0000016 8950 0.00000080.00000250.00000060.00000040.00000260.00000000.0000017 9050 0.00000110.00000290.00000040.00000020.00000350.00000000.0000021 9150 0.00000070.00000310.00000010.00000030.00000210.00000000.0000016 9250 0.00000030.00000270.00000070.00000040.00000230.00000000.0000016 9350 0.00000040.00000200.00000080.00000020.00000210.00000000.0000014 9450 0.00000060.00000230.00000070.00000030.00000280.00000000.0000017 9550 0.00000050.00000260.00000080.00000040.00000180.00000000.0000014 9650 0.00000050.00000210.00000060.00000040.00000200.00000000.0000014 9750 0.00000020.00000160.00000010.00000020.00000210.00000000.0000011 9850 0.00000020.00000190.00000010.00000020.00000190.00000000.0000011 99500.00000030.00000220.00000070.00000040.00000110.00000000.0000011 10050 0.00000690.00003330.00000600.00000240.00004300.00000000.0000241

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383 Table G-23. Normalized 3D c hord-length distributions throug h the marrow cavities of the mandible, frontal bone, occipital bone, and right and left pari etal bones. Also shown is the average for the cranium. Probability distributions are given through the first 50 bins with a maximum length of 5000 microns. bin width Mandible Average Cranium Left Parietal Right Parietal Frontal Bone Occipital Bonem p(L)dLp(L)dLp(L)dL p(L)dLp(L)dLp(L)dL 50 0.00050730.00039190.00035550.00036560.00039970.0005983 150 0.00068650.00060010.00050840.00054550.00064110.0009203 250 0.00080340.00078950.00067090.00070380.00088220.0010949 350 0.00077520.00089290.00077240.00078450.00102680.0011363 450 0.00070450.00093930.00083580.00083030.00108050.0011177 550 0.00063030.00093740.00087090.00085000.00105480.0010284 650 0.00055650.00089040.00087460.00083110.00097240.0008877 750 0.00049580.00080630.00082100.00078250.00084590.0007309 850 0.00043650.00070080.00075300.00071010.00069400.0005796 950 0.00038440.00059200.00064970.00063170.00054880.0004475 1050 0.00033610.00048610.00054540.00054120.00042330.0003367 1150 0.00029610.00039450.00045380.00045400.00032420.0002548 1250 0.00026190.00031560.00037040.00037140.00024760.0001983 1350 0.00023040.00025250.00030220.00030310.00018930.0001533 1450 0.00020270.00020070.00024340.00024450.00014570.0001173 1550 0.00018180.00015850.00019310.00019590.00011160.0000910 1650 0.00016700.00012510.00015230.00015710.00008550.0000683 1750 0.00015140.00009970.00012230.00012620.00006670.0000535 1850 0.00013820.00007870.00009610.00010030.00005230.0000399 1950 0.00012540.00006350.00007770.00008230.00004060.0000313 2050 0.00011690.00005110.00006280.00006630.00003250.0000254 2150 0.00010800.00004150.00005060.00005380.00002640.0000208 2250 0.00009830.00003340.00004270.00004320.00002090.0000155 2350 0.00008980.00002730.00003400.00003620.00001640.0000125 2450 0.00008670.00002220.00002680.00003000.00001310.0000094 2550 0.00008040.00001800.00002140.00002460.00001050.0000073 2650 0.00007580.00001470.00001730.00002070.00000800.0000054 2750 0.00007140.00001220.00001330.00001730.00000660.0000041 2850 0.00006600.00000980.00001100.00001410.00000510.0000033 2950 0.00006290.00000820.00000910.00001170.00000440.0000026 3050 0.00006030.00000720.00000770.00001040.00000380.0000019 3150 0.00005720.00000610.00000600.00000890.00000340.0000013 3250 0.00005490.00000510.00000510.00000760.00000260.0000010 3350 0.00005120.00000430.00000390.00000660.00000210.0000006 3450 0.00004960.00000360.00000340.00000540.00000180.0000008 3550 0.00004750.00000280.00000280.00000430.00000130.0000004 3650 0.00004410.00000240.00000230.00000370.00000130.0000003 3750 0.00004110.00000200.00000200.00000310.00000100.0000002 3850 0.00003960.00000180.00000150.00000280.00000090.0000003 3950 0.00003850.00000150.00000110.00000240.00000080.0000001 4050 0.00003510.00000130.00000090.00000220.00000060.0000001 4150 0.00003400.00000100.00000080.00000170.00000040.0000001 4250 0.00003200.00000100.00000070.00000160.00000050.0000001 4350 0.00002980.00000080.00000060.00000130.00000040.0000000 4450 0.00002770.00000070.00000050.00000120.00000030.0000000 4550 0.00002590.00000060.00000050.00000100.00000030.0000000 4650 0.00002450.00000050.00000030.00000080.00000030.0000000 4750 0.00002280.00000040.00000030.00000070.00000020.0000000 4850 0.00002080.00000040.00000030.00000060.00000020.00000004950 0.00002040.00000030.00000020.00000050.00000010.0000000

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384 Table G-24. Normalized 3D c hord-length distributions throug h the marrow cavities of the mandible, frontal bone, occipital bone, and right and left pari etal bones. Also shown is the average for the cranium. Probability distributions are given through the second 51 bins with a maximum length of 10100 microns. bin width Mandible Average Cranium Left Parietal Right Parietal Frontal Bone Occipital Bonem p(L)dLp(L)dLp(L)dL p(L)dLp(L)dLp(L)dL 5050 0.00001930.00000030.00000020.00000050.00000010.0000000 5150 0.00001810.00000020.00000020.00000040.00000010.0000000 5250 0.00001710.00000020.00000020.00000030.00000010.0000000 5350 0.00001610.00000020.00000010.00000030.00000010.0000000 5450 0.00001490.00000020.00000010.00000030.00000010.0000000 5550 0.00001350.00000010.00000010.00000020.00000010.0000000 5650 0.00001280.00000010.00000010.00000020.00000010.0000000 5750 0.00001200.00000010.00000010.00000020.00000000.0000000 5850 0.00001170.00000010.00000010.00000010.00000000.0000000 5950 0.00001040.00000010.00000010.00000020.00000000.0000000 6050 0.00000970.00000010.00000000.00000010.00000000.0000000 6150 0.00000950.00000010.00000000.00000010.00000000.0000000 6250 0.00000910.00000000.00000000.00000010.00000000.0000000 6350 0.00000820.00000010.00000000.00000010.00000000.0000000 6450 0.00000780.00000000.00000000.00000010.00000000.0000000 6550 0.00000720.00000000.00000000.00000010.00000000.0000000 6650 0.00000760.00000000.00000000.00000000.00000000.0000000 6750 0.00000710.00000000.00000000.00000010.00000000.0000000 6850 0.00000640.00000000.00000000.00000010.00000000.0000000 6950 0.00000580.00000000.00000000.00000000.00000000.0000000 7050 0.00000530.00000000.00000000.00000000.00000000.0000000 7150 0.00000490.00000000.00000000.00000000.00000000.0000000 7250 0.00000500.00000000.00000000.00000000.00000000.0000000 7350 0.00000490.00000000.00000000.00000000.00000000.0000000 7450 0.00000400.00000000.00000000.00000000.00000000.0000000 7550 0.00000390.00000000.00000000.00000000.00000000.0000000 7650 0.00000350.00000000.00000000.00000000.00000000.0000000 7750 0.00000390.00000000.00000000.00000000.00000000.0000000 7850 0.00000330.00000000.00000000.00000000.00000000.0000000 7950 0.00000310.00000000.00000000.00000000.00000000.0000000 8050 0.00000280.00000000.00000000.00000000.00000000.0000000 8150 0.00000260.00000000.00000000.00000000.00000000.0000000 8250 0.00000260.00000000.00000000.00000000.00000000.0000000 8350 0.00000250.00000000.00000000.00000000.00000000.0000000 8450 0.00000240.00000000.00000000.00000000.00000000.0000000 8550 0.00000250.00000000.00000000.00000000.00000000.0000000 8650 0.00000200.00000000.00000000.00000000.00000000.0000000 8750 0.00000190.00000000.00000000.00000000.00000000.0000000 8850 0.00000210.00000000.00000000.00000000.00000000.0000000 8950 0.00000160.00000000.00000000.00000000.00000000.0000000 9050 0.00000170.00000000.00000000.00000000.00000000.0000000 9150 0.00000150.00000000.00000000.00000000.00000000.0000000 9250 0.00000150.00000000.00000000.00000000.00000000.0000000 9350 0.00000150.00000000.00000000.00000000.00000000.0000000 9450 0.00000120.00000000.00000000.00000000.00000000.0000000 9550 0.00000130.00000000.00000000.00000000.00000000.0000000 9650 0.00000100.00000000.00000000.00000000.00000000.0000000 9750 0.00000110.00000000.00000000.00000000.00000000.0000000 9850 0.00000100.00000000.00000000.00000000.00000000.00000009950 0.00000080.00000000.00000000.00000000.00000000.0000000 10050 0.00001360.00000000.00000000.00000000.00000000.0000000

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385 Chord-Length Distribution Software /*****************************************************************************/ /* TLChordLength.cpp */ /* */ /* This program calculates the chord length distribution of both the */ /* 'above the threshold' and the 'below the threshold' of a 3D object */ /* using the trilinear interpolation adaptation of the Marching Cubes */ /* algorithm. */ /* Input: */ /* The input image file: ImageName (no extension). */ /* The dimensions of the image: Nx, Ny, Nz in number of voxels. */ /* The dimensions of each voxel: Vx, Vy, Vz in centimeters. */ /* The threshold to separate the two media. */ /* The number of rays to fire */ /* The radius of the sphere that surrounds the image */ /* The number of histogram bins */ /* The step per bin (in cm) */ /* Output: */ /* The above chord length distribution in "ImageName_TL.CLM" */ /* The below chord length distribution in "ImageName_TL.CLB" */ /* */ /*****************************************************************************/ Instructions to properly execute the Trilinear Chord-Length Program To properly execute the Trilinear Chord-Le ngth distribution program one must first locate the microCT image of choice and make sure it is in the correct format. The format for this image is the raw image with no threshold value applied; however the region of interest (ROI) must be applied and the th reshold value must be known. Appendix D provides the necessary steps to determine the threshold value and with modifications to the ResizeCTimage.c code, one can resize the image into the ROI without thresholding. To execute the program one must enter the program name and the image to input (e.g. Humerus_Left_ROI .IGL without the “.IGL” extension. Figure G-1 shows a pictorial example of how to execute the code. Ot her inputs one must also provide are dimensions in K, J, and I directions. resolution of the cube 0.0060 = 60 microns. threshold value, ~152. number of chords to run 1,000,000. diagonal through the cube (i .e sqrt( (K/2)^2)+(J/2) ^2)+(I/2)^2) ) 0.0060). number of bins – 500 for trabeculae, 100 for marrow cavities. distance or thickness of each bin: 0.0020 cm for trabeculae, 0.0100 for cavities.

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386 Figure G-2. Pictorial exampl e for the Humerus_Left image execution of the Tri-Linear Chord-Length Distribution program. After the application of the TLChordLength.ex e program is run one must open the data and properly tabulate the probability distribution and the average chord length. A critical step in doing so is choosing a consistent manner to label the bin widths given in the distribution.

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387 APPENDIX H TABLES OF SITE-SPECIFIC ABSORBED FRACTIONS FOR THE UF REFERENCE MALE This appendix contains site-specific abso rbed fractions for the UF reference male cancer patient. The tables are separated by sk eletal site. Each bone site is labeled and given as volume-weighted averages of the sk eletal site, if necessary. The absorbed fraction is given as a function of increasing energy at 10 di fferent marrow cellularities. Six source regions are presented: trabecu lar active marrow (TAM ), trabecular bone volume (TBV), trabecular bone surface (T BS), trabecular bone endosteum (TBE), cortical bone volume (CBV), and trabecular marrow cavity (TMC). Data is given for only three target regions, TA M, TBE, and CBV. The 110 ab sorbed fraction tables (18 skeletal sites and 6 source regions plus 2 tables for CBV as a target region) given in this appendix were generated through the paired-ima ge radiation transpor t (PIRT) model and represent 33 microimages of spongiosa. The breakdown for each skeletal site is as follows. For the cranium (4), four regions we re averaged – the right and left parietal bones, the occipital bone, and the frontal bone. The mandible (1), sternum (1), and sacrum (1) were all single microstructure skeletal sites. The vertebra was separated into 3 regions, cervical, thoracic, and lumbar. E ach region had multiple vertebrae that were sectioned. For the cervical vertebra (2), the 3rd and 6th vertebrae were sectioned for microimaging. For the thoracic vertebra (3 ), the 3rd, 6th, and 11th vertebrae were sectioned for microimaging. For the lumbar vertebra (2), the 2nd and 4th lumbar vertebrae were sectioned for microimaging. The rib data was compiled through the acquisition and measurement through ribs (6), th e 2nd, 7th, and 11th ribs were taken from the right and left rib cage. For the os c oxae (3), the pelvic bones were sampled – the ilium, ischium and pubis. The right and left proximal humerii (2), clavicles (2), and scapulae (2) were all sectioned for microimagi ng. Lastly, both the right and left femora were sectioned for the microimaging and radi ation transport. However, each femur was subdivided into two regions, femur head and neck (4). The data presented for the reference adult male for the right proximal femur and the left proximal femur. The proximal femur model was developed through we ighting of the source tissue between the femur head and neck. This is outlined in ch apter 3. The volume-weighted average of the absorbed fraction data was done for the craniu m, ribs, lumbar vertebra, thoracic vertebra, and lumbar vertebra only.

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388 Table H-1. Absorbed fractions in the righ t proximal femur for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0000 0.00000.00000.0000 0.00000.00000.0027 0.0150.00020.00020.00010.00010.0001 0.00010.00010.0001 0.00010.00000.0056 0.020.00030.00030.00020.00020.0002 0.00010.00010.0001 0.00000.00000.0094 0.030.00140.00130.00110.00100.0008 0.00070.00060.0004 0.00030.00010.0177 0.040.00660.00590.00530.00460.0040 0.00330.00270.0020 0.00130.00070.0246 0.050.01560.01410.01260.01100.0094 0.00780.00620.0047 0.00310.00160.0294 0.10.08710.07850.06990.06130.05260. 04390.03520.02660.01800.00900.0371 0.20.26740.24220.21710.19050.16400. 13670.10950.08250.05540.02780.0364 0.50.50490.45470.40440.35430.30420. 25420.20410.15300.10190.05130.0299 10.54300.49040.43790.38320.32850. 27440.22030.16570.11110.05530.0277 1.50.54560.49280.44000.38590.33180. 27640.22100.16590.11080.05590.0265 20.53720.48390.43060.37780.32500. 27110.21720.16280.10850.05460.0254 40.47610.42920.38230.33480.28740. 23950.19160.14400.09630.04810.0220 (TAM TBV) Table H-2. Absorbed fractions in the left proximal femur for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0001 0.00010.00000.0000 0.00000.00000.0028 0.0150.00020.00020.00020.00010.0001 0.00010.00010.0000 0.00000.00000.0063 0.020.00040.00030.00030.00020.0002 0.00010.00010.0001 0.00010.00000.0097 0.030.00160.00140.00130.00120.0010 0.00090.00070.0006 0.00040.00020.0184 0.040.00660.00590.00520.00460.0040 0.00330.00260.0020 0.00130.00070.0249 0.050.01570.01410.01260.01100.0095 0.00790.00640.0048 0.00320.00170.0295 0.10.08740.07880.07030.06180.05330. 04450.03570.02670.01780.00880.0370 0.20.27030.24470.21910.19210.16510. 13790.11070.08310.05550.02770.0358 0.50.50520.45440.40350.35390.30440. 25400.20360.15300.10240.05100.0294 10.54470.49050.43640.38300.32960. 27520.22090.16590.11100.05550.0271 1.50.55110.49700.44290.38740.33190. 27710.22220.16720.11220.05650.0257 20.54590.49290.43980.38450.32920. 27550.22190.16620.11050.05560.0245 40.48670.43790.38910.34110.29310. 24450.19580.14720.09850.04930.0208 (TAM TBV) Table H-3. Absorbed fractions in the right humerus for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0001 0.00010.00010.0000 0.00000.00000.0020 0.0150.00050.00040.00030.00030.0003 0.00020.00010.0001 0.00010.00010.0059 0.020.00050.00040.00040.00030.0003 0.00020.00010.0001 0.00010.00000.0085 0.030.00170.00150.00130.00120.0010 0.00080.00070.0005 0.00030.00020.0166 0.040.00680.00600.00520.00450.0039 0.00320.00260.0020 0.00130.00060.0233 0.050.01510.01370.01220.01070.0092 0.00760.00590.0044 0.00300.00150.0274 0.10.08110.07340.06570.05810.05050. 04190.03320.02490.01660.00870.0353 0.20.25150.22740.20330.17870.15400. 12810.10210.07660.05120.02570.0345 0.50.49080.44260.39450.34560.29670. 24810.19940.14970.10010.05080.0300 10.51770.46670.41580.36400.31230. 26090.20960.15770.10580.05280.0277 1.50.50980.46040.41110.36040.30980. 25820.20670.15500.10320.05210.0265 20.49490.44550.39610.34660.29710. 24860.20010.15030.10050.05040.0249 40.42190.37960.33740.29590.25440. 21190.16940.12680.08410.04270.0207 (TAM TBV)

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389 Table H-4. Absorbed fractions in the left humerus for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0001 0.00010.00010.0001 0.00010.00010.0035 0.0150.00020.00020.00020.00020.0002 0.00020.00010.0001 0.00010.00000.0072 0.020.00040.00040.00030.00030.0002 0.00020.00010.0001 0.00010.00000.0124 0.030.00150.00140.00120.00110.0009 0.00080.00060.0005 0.00030.00010.0213 0.040.00770.00700.00620.00550.0048 0.00400.00320.0025 0.00170.00080.0303 0.050.01890.01700.01510.01300.0110 0.00920.00740.0055 0.00370.00170.0357 0.10.10200.09280.08360.07350.06340. 05280.04220.03170.02120.01100.0451 0.20.31970.28880.25800.22590.19380. 16280.13180.09900.06620.03270.0422 0.50.56110.50480.44860.39350.33830. 28210.22590.16980.11370.05660.0339 10.57600.51960.46320.40580.34840. 29010.23190.17460.11740.05870.0310 1.50.56320.50780.45230.39680.34130. 28460.22780.17070.11370.05710.0293 20.54500.49110.43720.38260.32790. 27410.22020.16530.11040.05510.0278 40.45830.41270.36700.32250.27810. 23160.18510.13890.09270.04660.0232 (TAM TBV) Table H-5. Absorbed fractions in the cervical vertebra for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00020.00020.00020.00020.0002 0.00020.00010.0001 0.00000.00000.0037 0.0150.00030.00030.00030.00020.0002 0.00020.00010.0001 0.00010.00000.0078 0.020.00070.00060.00060.00050.0004 0.00030.00030.0002 0.00010.00010.0118 0.030.00230.00210.00180.00160.0014 0.00110.00090.0007 0.00040.00020.0231 0.040.00890.00810.00720.00630.0055 0.00460.00370.0028 0.00190.00100.0318 0.050.02120.01920.01720.01510.0131 0.01100.00890.0067 0.00440.00210.0380 0.10.11460.10360.09250.08090.06920. 05780.04640.03500.02360.01180.0474 0.20.33740.30500.27260.23930.20600. 17180.13760.10390.07030.03470.0436 0.50.51030.46080.41130.36030.30940. 25770.20610.15500.10390.05210.0317 10.47290.42620.37940.33260.28580. 23840.19100.14340.09580.04790.0256 1.50.41460.37370.33280.29130.24990. 20860.16730.12570.08400.04210.0215 20.36320.32790.29260.25560.21860. 18210.14560.10900.07240.03650.0184 40.21340.19170.17010.14890.12770. 10640.08500.06390.04270.02140.0105 (TAM TBV) Table H-6. Absorbed fractions in the thoracic vertebra for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0001 0.00000.00000.0000 0.00000.00000.0036 0.0150.00030.00020.00020.00020.0002 0.00010.00010.0001 0.00010.00000.0071 0.020.00050.00050.00050.00040.0003 0.00030.00020.0002 0.00010.00000.0121 0.030.00200.00180.00160.00140.0012 0.00100.00080.0006 0.00040.00020.0237 0.040.00890.00810.00720.00630.0054 0.00450.00370.0027 0.00180.00090.0329 0.050.02130.01910.01700.01490.0128 0.01060.00850.0064 0.00420.00210.0389 0.10.11800.10630.09470.08300.07130. 05950.04770.03580.02380.01190.0483 0.20.34610.31320.28030.24640.21250. 17690.14130.10620.07100.03580.0436 0.50.57320.51650.45980.40350.34710. 29000.23290.17470.11650.05830.0312 10.55890.50410.44930.39320.33710. 28190.22660.17000.11340.05710.0263 1.50.51900.46660.41420.36380.31340. 26150.20960.15720.10490.05250.0232 20.47150.42590.38020.33260.28510. 23820.19140.14330.09520.04810.0208 40.33530.30150.26780.23470.20160. 16780.13410.10040.06680.03350.0143 (TAM TBV)

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390 Table H-7. Absorbed fractions in the lumbar vertebra for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0001 0.00010.00010.0001 0.00000.00000.0037 0.0150.00040.00030.00030.00020.0002 0.00020.00020.0002 0.00010.00010.0074 0.020.00060.00060.00060.00050.0004 0.00030.00030.0002 0.00010.00010.0119 0.030.00210.00190.00170.00150.0013 0.00100.00080.0006 0.00050.00020.0225 0.040.00860.00770.00680.00600.0053 0.00450.00370.0028 0.00190.00090.0306 0.050.02010.01810.01610.01410.0122 0.01010.00810.0061 0.00420.00200.0360 0.10.10970.09920.08860.07760.06660. 05530.04410.03320.02220.01090.0456 0.20.32660.29490.26310.23090.19860. 16630.13410.10070.06730.03370.0429 0.50.54850.49460.44060.38600.33130. 27660.22190.16700.11210.05600.0329 10.54870.49400.43920.38480.33030. 27590.22160.16650.11140.05590.0285 1.50.51830.46670.41500.36390.31280. 26100.20920.15710.10500.05290.0257 20.48240.43630.39010.34130.29250. 24390.19540.14670.09800.04930.0234 40.37020.33440.29860.26020.22170. 18540.14900.11190.07470.03730.0175 (TAM TBV) Table H-8. Absorbed fractions in the sacrum for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0000 0.00000.00000.0000 0.00000.00000.0041 0.0150.00030.00030.00030.00030.0002 0.00020.00020.0001 0.00010.00000.0079 0.020.00050.00030.00020.00020.0002 0.00020.00020.0001 0.00010.00000.0117 0.030.00200.00180.00150.00130.0012 0.00100.00080.0006 0.00040.00020.0233 0.040.00850.00770.00690.00600.0051 0.00430.00340.0026 0.00170.00080.0319 0.050.02040.01830.01620.01420.0122 0.01020.00820.0062 0.00410.00190.0370 0.10.11140.09990.08830.07740.06650. 05580.04520.03400.02270.01120.0465 0.20.33380.30200.27010.23680.20350. 16920.13490.10150.06810.03440.0440 0.50.53810.48390.42980.37740.32510. 27100.21690.16290.10880.05500.0333 10.51770.46770.41770.36660.31550. 26310.21070.15820.10570.05300.0286 1.50.48040.43270.38500.33790.29070. 24270.19460.14610.09750.04870.0257 20.44370.39970.35570.31190.26800. 22390.17980.13480.08970.04460.0232 40.32330.29050.25770.22520.19260. 16120.12980.09720.06460.03230.0162 (TAM TBV) Table H-9. Absorbed fractions in the os coxae for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00020.00020.00010.0001 0.00010.00010.0001 0.00010.00000.0037 0.0150.00030.00030.00020.00020.0002 0.00020.00010.0001 0.00000.00000.0070 0.020.00060.00050.00040.00030.0003 0.00030.00020.0002 0.00010.00010.0125 0.030.00210.00190.00170.00150.0013 0.00110.00090.0007 0.00040.00020.0232 0.040.00900.00810.00720.00630.0054 0.00460.00370.0028 0.00190.00090.0320 0.050.02110.01890.01670.01460.0125 0.01040.00840.0062 0.00410.00200.0381 0.10.11410.10320.09230.08050.06880. 05750.04620.03480.02340.01190.0468 0.20.33740.30460.27180.23850.20530. 17190.13850.10420.06980.03490.0416 0.50.57640.51970.46290.40610.34930. 29150.23370.17580.11790.05900.0301 10.59630.53740.47850.41930.36000. 30100.24210.18150.12100.06090.0264 1.50.57530.51880.46230.40570.34900. 29140.23370.17530.11690.05870.0243 20.54670.49360.44040.38580.33130. 27660.22190.16650.11110.05570.0224 40.42820.38680.34540.30210.25870. 21630.17390.13050.08720.04340.0170 (TAM TBV)

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391 Table H-10. Absorbed fractions in the cran ium for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00000.00000.00000.00000.0000 0.00000.00000.0000 0.00000.00000.0019 0.0150.00010.00010.00010.00010.0001 0.00000.00000.0000 0.00000.00000.0042 0.020.00020.00020.00020.00020.0001 0.00010.00010.0001 0.00000.00000.0065 0.030.00080.00070.00060.00060.0005 0.00040.00030.0003 0.00020.00010.0126 0.040.00430.00390.00350.00310.0027 0.00220.00180.0013 0.00090.00050.0175 0.050.01060.00950.00850.00740.0064 0.00530.00420.0032 0.00210.00110.0209 0.10.05720.05170.04620.04050.03480. 02910.02330.01750.01160.00570.0273 0.20.17560.15840.14110.12390.10670. 08900.07130.05370.03620.01800.0296 0.50.30790.27770.24740.21670.18600. 15520.12440.09340.06240.03130.0275 10.28210.25390.22580.19790.17010. 14180.11350.08520.05690.02840.0233 1.50.24330.21920.19500.17060.14620. 12190.09760.07330.04900.02450.0198 20.20950.18830.16710.14630.12540. 10480.08420.06310.04200.02090.0169 40.12960.11620.10280.09050.07820. 06510.05200.03900.02600.01310.0103 (TAM TBV) Table H-11. Absorbed fractions in the mandible for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00020.00020.00010.00010.0001 0.00010.00010.0001 0.00000.00000.0029 0.0150.00010.00010.00010.00010.0001 0.00000.00000.0000 0.00000.00000.0070 0.020.00030.00030.00030.00020.0002 0.00020.00010.0001 0.00010.00000.0102 0.030.00160.00140.00130.00110.0010 0.00080.00070.0005 0.00030.00020.0202 0.040.00750.00670.00600.00520.0045 0.00370.00300.0022 0.00150.00070.0284 0.050.01800.01620.01440.01270.0110 0.00920.00750.0055 0.00350.00170.0333 0.10.09640.08640.07640.06730.05830. 04860.03890.02920.01960.00940.0415 0.20.25470.22970.20470.17880.15280. 12850.10420.07830.05240.02590.0381 0.50.39730.35920.32120.28040.23960. 20030.16100.12090.08080.04100.0298 10.39510.35560.31610.27780.23940. 19960.15970.11980.07980.03990.0239 1.50.34890.31560.28220.24600.20990. 17580.14170.10660.07160.03510.0199 20.30330.27360.24390.21360.18330. 15290.12250.09170.06080.03060.0168 40.18300.16400.14510.12690.10870. 09090.07300.05460.03630.01840.0098 (TAM TBV) Table H-12. Absorbed fractions in the ribs for sources in the trab ecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00030.00030.00030.00020.0002 0.00020.00010.0001 0.00010.00000.0029 0.0150.00080.00070.00060.00050.0005 0.00040.00030.0002 0.00010.00010.0057 0.020.00130.00110.00100.00090.0008 0.00070.00050.0004 0.00030.00010.0095 0.030.00340.00310.00270.00240.0020 0.00170.00140.0010 0.00070.00030.0187 0.040.00990.00890.00800.00700.0060 0.00500.00390.0030 0.00200.00100.0258 0.050.02080.01870.01660.01460.0125 0.01050.00850.0063 0.00420.00210.0303 0.10.10410.09420.08420.07380.06330. 05290.04240.03180.02130.01080.0375 0.20.29170.26300.23440.20570.17690. 14780.11860.08910.05960.02990.0346 0.50.49580.44650.39720.34840.29960. 25000.20030.15060.10080.05040.0230 10.46040.41470.36900.32320.27740. 23150.18550.13930.09300.04660.0161 1.50.37520.33780.30050.26340.22620. 18840.15050.11290.07530.03760.0121 20.30140.27050.23960.21030.18100. 15080.12070.09050.06030.03010.0095 40.15350.13820.12300.10760.09210. 07670.06130.04600.03070.01530.0049 (TAM TBV)

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392 Table H-13. Absorbed fractions in the ster num for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0001 0.00000.00000.0000 0.00000.00000.0035 0.0150.00050.00050.00050.00040.0004 0.00030.00020.0002 0.00010.00010.0067 0.020.00080.00070.00060.00050.0004 0.00040.00030.0003 0.00020.00010.0110 0.030.00220.00200.00180.00160.0014 0.00120.00090.0007 0.00050.00030.0207 0.040.00890.00810.00720.00630.0055 0.00460.00370.0028 0.00190.00090.0296 0.050.02040.01820.01610.01410.0121 0.01010.00820.0060 0.00390.00190.0332 0.10.10680.09700.08710.07660.06600. 05490.04380.03270.02160.01110.0426 0.20.31010.28020.25020.21890.18760. 15700.12640.09560.06470.03270.0381 0.50.50600.45630.40660.35580.30510. 25460.20410.15370.10320.05150.0290 10.53030.47810.42600.37380.32160. 26750.21340.16030.10720.05400.0250 1.50.51010.45990.40970.35880.30800. 25640.20480.15420.10370.05140.0221 20.47000.42340.37690.32980.28270. 23580.18890.14220.09550.04770.0198 40.31340.28120.24890.21920.18960. 15790.12620.09470.06310.03150.0129 (TAM TBV) Table H-14. Absorbed fractions in the right clavicle for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00020.00020.00020.00010.0001 0.00010.00000.0000 0.00000.00000.0020 0.0150.00020.00020.00020.00010.0001 0.00010.00010.0001 0.00010.00010.0045 0.020.00050.00040.00040.00040.0004 0.00030.00020.0002 0.00010.00010.0081 0.030.00160.00150.00130.00120.0010 0.00080.00070.0005 0.00040.00020.0161 0.040.00630.00570.00510.00450.0038 0.00310.00240.0018 0.00120.00050.0223 0.050.01500.01350.01200.01050.0090 0.00770.00640.0049 0.00330.00160.0263 0.10.08110.07280.06450.05670.04890. 04100.03320.02510.01710.00810.0326 0.20.24740.22310.19890.17430.14970. 12500.10020.07520.05020.02530.0323 0.50.44500.39940.35390.31100.26810. 22390.17960.13440.08910.04570.0259 10.44940.40520.36110.31630.27150. 22560.17970.13550.09130.04550.0220 1.50.41500.37260.33020.28850.24680. 20610.16540.12430.08330.04160.0187 20.36580.33020.29450.25740.22030. 18350.14670.11010.07340.03730.0159 40.21700.19610.17510.15250.13000. 10810.08630.06480.04330.02150.0092 (TAM TBV) Table H-15. Absorbed fractions in the left clavicle for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00030.00030.00030.00030.0002 0.00010.00010.0001 0.00000.00000.0030 0.0150.00050.00050.00040.00030.0003 0.00020.00010.0001 0.00000.00000.0048 0.020.00120.00110.00100.00090.0008 0.00070.00060.0004 0.00030.00010.0085 0.030.00310.00280.00250.00220.0019 0.00160.00120.0009 0.00060.00030.0167 0.040.00920.00830.00740.00640.0054 0.00450.00360.0027 0.00190.00100.0227 0.050.01940.01750.01560.01370.0118 0.00980.00780.0059 0.00400.00190.0271 0.10.09550.08600.07650.06720.05780. 04840.03890.02920.01940.00990.0333 0.20.27570.24850.22130.19410.16700. 13950.11200.08380.05560.02900.0307 0.50.53360.48190.43030.37780.32530. 27030.21530.16200.10860.05440.0203 10.55980.50390.44800.39230.33660. 28070.22490.16970.11440.05620.0154 1.50.51190.46200.41210.36070.30940. 25870.20810.15630.10450.05160.0123 20.45470.40970.36480.31980.27490. 22990.18490.13820.09160.04620.0105 40.26540.24120.21700.18880.16070. 13370.10670.08030.05380.02680.0059 (TAM TBV)

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393 Table H-16. Absorbed fractions in the righ t scapula for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00010.00010.0001 0.00010.00010.0000 0.00000.00000.0019 0.0150.00030.00020.00010.00010.0001 0.00010.00010.0001 0.00010.00000.0041 0.020.00040.00030.00020.00020.0002 0.00020.00010.0001 0.00010.00000.0075 0.030.00140.00130.00120.00100.0009 0.00070.00060.0004 0.00030.00020.0141 0.040.00550.00490.00430.00390.0034 0.00280.00230.0017 0.00110.00050.0191 0.050.01270.01140.01000.00890.0077 0.00640.00510.0038 0.00240.00110.0216 0.10.06710.06140.05560.04910.04260. 03540.02830.02110.01390.00690.0286 0.20.19610.17660.15710.13900.12100. 10080.08070.06060.04050.01980.0292 0.50.38690.34900.31110.27230.23350. 19520.15700.11750.07810.03900.0239 10.40770.36630.32480.28540.24600. 20490.16380.12330.08290.04100.0208 1.50.37690.34020.30350.26660.22980. 19090.15200.11430.07660.03840.0185 20.34220.30750.27280.23880.20480. 17070.13670.10280.06890.03460.0163 40.22480.20190.17900.15720.13530. 11240.08950.06740.04520.02270.0104 (TAM TBV) Table H-17. Absorbed fractions in the left scapula for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00000.00000.00000.00000.0000 0.00000.00000.0000 0.00000.00000.0022 0.0150.00020.00010.00010.00010.0001 0.00010.00000.0000 0.00000.00000.0052 0.020.00030.00030.00020.00020.0001 0.00010.00010.0001 0.00010.00000.0079 0.030.00120.00110.00100.00080.0007 0.00060.00050.0004 0.00020.00010.0156 0.040.00550.00500.00440.00380.0033 0.00280.00230.0017 0.00120.00070.0215 0.050.01350.01220.01090.00950.0081 0.00680.00540.0040 0.00270.00130.0256 0.10.07340.06610.05890.05160.04430. 03720.03020.02270.01520.00740.0330 0.20.21860.19780.17700.15470.13230. 11090.08940.06680.04410.02270.0328 0.50.43350.39110.34870.30520.26180. 21870.17550.13160.08760.04420.0277 10.43720.39400.35080.30810.26540. 22190.17850.13400.08950.04490.0240 1.50.40510.36580.32650.28490.24320. 20250.16170.12150.08140.04070.0209 20.36060.32570.29090.25410.21740. 18170.14610.10940.07280.03620.0184 40.23480.21270.19050.16700.14360. 11910.09460.07120.04780.02390.0120 (TAM TBV) Table H-18. Absorbed fractions in the fe mur head for sources in the trabecular bone volume at varying marrow cellularity. (TBE TBV)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00010.00010.00000.00000.0000 0.00000.00000.0000 0.00000.00000.0026 0.0150.00020.00020.00020.00010.0001 0.00010.00000.0000 0.00000.00000.0061 0.020.00030.00020.00020.00020.0001 0.00010.00010.0000 0.00000.00000.0099 0.030.00150.00130.00120.00100.0009 0.00070.00060.0005 0.00030.00020.0192 0.040.00680.00600.00530.00470.0040 0.00340.00270.0020 0.00130.00070.0262 0.050.01590.01430.01270.01110.0096 0.00800.00640.0048 0.00320.00160.0307 0.10.08930.08030.07140.06270.05400. 04510.03630.02730.01840.00910.0390 0.20.27700.25110.22530.19700.16880. 14080.11290.08520.05750.02860.0390 0.50.50300.45360.40420.35380.30350. 25330.20300.15230.10150.05100.0337 10.52700.47510.42330.37110.31890. 26620.21350.16050.10750.05380.0319 1.50.52420.47260.42100.36900.31700. 26440.21170.15900.10620.05340.0305 20.51440.46400.41360.36250.31130. 26000.20870.15650.10430.05240.0293 40.45550.41110.36680.32120.27560. 22990.18420.13830.09250.04620.0254 (TAM TBV)

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394 Table H-19. Absorbed fractions in the righ t proximal femur for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99870.99570.99270.98980.9868 0.98390.98100.9781 0.97520.97090.0013 0.0150.99680.99050.98420.97850.9729 0.96650.96000.9542 0.94840.94160.0030 0.0200.99550.98560.97580.96580.9559 0.94540.93480.9251 0.91540.90550.0041 0.0300.99080.97080.95090.93150.9121 0.89200.87190.8529 0.83380.81370.0086 0.0400.98460.95350.92230.89180.8612 0.83030.79930.7682 0.73720.70510.0112 0.0500.97780.93500.89230.84920.8060 0.76320.72040.6771 0.63390.59100.0132 0.1000.93700.85850.78000.70060.6212 0.54060.46000.3796 0.29910.21790.0170 0.2000.83740.75960.68170.60190.5222 0.44220.36210.2818 0.20150.12010.0207 0.5000.70140.63260.56370.49620.4286 0.35890.28920.2199 0.15070.08120.0241 1.0000.65130.58790.52450.45980.3952 0.33070.26630.2011 0.13590.07000.0240 1.5000.61820.55810.49790.43650.3750 0.31380.25260.1904 0.12820.06540.0237 2.0000.59320.53500.47670.41790.3590 0.29990.24080.1817 0.12260.06250.0228 4.0000.50410.45470.40540.35600.3067 0.25600.20530.1544 0.10350.05230.0200 (TAM TAM) Table H-20. Absorbed fractions in the left proximal femur for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99890.99590.99300.99000.9869 0.98400.98100.9777 0.97430.97190.0010 0.0150.99740.99160.98580.97900.9723 0.96680.96140.9554 0.94930.94250.0026 0.0200.99550.98570.97590.96570.9556 0.94580.93600.9269 0.91770.90720.0040 0.0300.99040.97100.95170.93170.9118 0.89300.87420.8537 0.83320.81180.0080 0.0400.98480.95390.92300.89210.8612 0.82960.79800.7673 0.73660.70530.0110 0.0500.97830.93550.89270.84970.8066 0.76340.72010.6776 0.63510.59180.0130 0.1000.93730.85810.77900.70050.6219 0.54160.46130.3806 0.29990.21860.0171 0.2000.83980.76090.68200.60240.5228 0.44310.36340.2828 0.20210.12030.0205 0.5000.70700.63810.56910.49980.4304 0.36140.29230.2222 0.15220.08150.0241 1.0000.65510.59060.52610.46220.3982 0.33360.26890.2031 0.13720.07100.0242 1.5000.62330.56240.50140.43940.3774 0.31570.25390.1915 0.12900.06590.0237 2.0000.59570.53730.47900.41980.3605 0.30120.24180.1821 0.12240.06230.0232 4.0000.50630.45630.40620.35670.3073 0.25630.20540.1544 0.10350.05240.0201 (TAM TAM) Table H-21. Absorbed fractions in the right humerus for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99810.99580.99360.99000.9865 0.98310.97970.9769 0.97410.97190.0015 0.0150.99560.99070.98580.97980.9738 0.96660.95930.9543 0.94930.94190.0030 0.0200.99520.98510.97510.96510.9552 0.94590.93650.9276 0.91870.90640.0048 0.0300.99030.97010.95000.93040.9108 0.89150.87230.8525 0.83270.81210.0095 0.0400.98150.95130.92120.89040.8596 0.82840.79720.7661 0.73510.70530.0124 0.0500.97560.93240.88910.84650.8039 0.76080.71780.6750 0.63220.59120.0156 0.1000.93020.85130.77240.69340.6144 0.53530.45610.3766 0.29720.21700.0198 0.2000.81400.73930.66450.58640.5083 0.43040.35250.2743 0.19620.11760.0224 0.5000.64210.58050.51890.45670.3945 0.33040.26640.2026 0.13880.07500.0256 1.0000.58970.53220.47460.41700.3594 0.30050.24170.1827 0.12370.06350.0254 1.5000.55330.50160.44980.39460.3394 0.28340.22740.1717 0.11590.05900.0247 2.0000.53500.48100.42700.37450.3221 0.26870.21540.1625 0.10950.05620.0237 4.0000.43810.39460.35110.30730.2636 0.22000.17640.1328 0.08920.04530.0199 (TAM TAM)

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395 Table H-22. Absorbed fractions in the left humerus for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99850.99560.99270.98910.9856 0.98250.97930.9759 0.97250.97060.0016 0.0150.99660.99040.98420.97780.9714 0.96510.95870.9532 0.94780.94160.0033 0.0200.99410.98470.97540.96520.9550 0.94500.93500.9264 0.91780.90650.0047 0.0300.98920.96970.95010.93060.9112 0.89160.87190.8522 0.83250.81300.0096 0.0400.98200.95190.92190.89060.8593 0.82780.79630.7660 0.73560.70360.0134 0.0500.97360.93090.88820.84560.8031 0.76170.72030.6758 0.63140.58790.0159 0.1000.92570.84780.76990.69190.6140 0.53480.45550.3772 0.29880.21760.0201 0.2000.81160.73650.66150.58410.5066 0.42900.35140.2748 0.19820.11780.0244 0.5000.67240.60720.54210.47560.4091 0.34310.27710.2105 0.14400.07760.0290 1.0000.62410.56410.50400.44190.3797 0.31740.25520.1929 0.13070.06750.0290 1.5000.59580.53710.47850.41910.3598 0.30100.24230.1828 0.12330.06340.0280 2.0000.56640.50950.45260.39760.3427 0.28620.22960.1732 0.11680.05950.0266 4.0000.46820.42340.37850.33120.2838 0.23670.18950.1428 0.09620.04870.0225 (TAM TAM) Table H-23. Absorbed fractions in the cervi cal vertebra for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99830.99530.99230.98940.9865 0.98360.98070.9781 0.97550.97130.0014 0.0150.99630.99030.98420.97820.9722 0.96610.96000.9537 0.94740.94250.0027 0.0200.99410.98430.97450.96460.9546 0.94450.93430.9237 0.91320.90480.0046 0.0300.98840.96880.94930.92950.9096 0.89030.87090.8510 0.83110.81210.0088 0.0400.98130.95030.91930.88820.8571 0.82620.79530.7653 0.73540.70470.0120 0.0500.97190.92980.88760.84560.8035 0.76010.71660.6740 0.63140.58750.0143 0.1000.91970.84270.76580.68820.6105 0.53160.45270.3744 0.29620.21630.0185 0.2000.80050.72510.64960.57440.4992 0.42320.34710.2700 0.19280.11560.0222 0.5000.64230.58050.51870.45540.3920 0.32950.26690.2027 0.13850.07480.0242 1.0000.53590.48300.43010.37760.3251 0.27180.21840.1650 0.11170.05840.0217 1.5000.45210.40750.36280.31840.2740 0.22890.18390.1388 0.09380.04840.0188 2.0000.38970.35030.31100.27220.2334 0.19480.15610.1177 0.07930.04060.0163 4.0000.22390.20160.17940.15710.1348 0.11250.09010.0678 0.04550.02340.0095 (TAM TAM) Table H-24. Absorbed fractions in the thorac ic vertebra for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99860.99540.99210.98940.9867 0.98340.98010.9773 0.97450.97130.0013 0.0150.99700.99100.98490.97850.9720 0.96580.95960.9533 0.94700.94350.0024 0.0200.99510.98520.97530.96530.9552 0.94560.93600.9254 0.91490.90450.0040 0.0300.99000.97030.95060.93110.9116 0.89180.87200.8525 0.83300.81270.0080 0.0400.98400.95320.92240.89130.8602 0.82880.79740.7664 0.73550.70400.0107 0.0500.97670.93400.89120.84790.8046 0.76170.71890.6756 0.63220.58880.0128 0.1000.93310.85510.77710.69780.6185 0.53810.45760.3782 0.29870.21690.0171 0.2000.83180.75410.67650.59810.5196 0.44050.36140.2810 0.20060.11940.0209 0.5000.69470.62810.56150.49280.4240 0.35580.28760.2186 0.14950.08020.0240 1.0000.61500.55480.49470.43390.3732 0.31210.25110.1899 0.12880.06650.0227 1.5000.55030.49710.44390.38930.3347 0.28000.22530.1696 0.11390.05860.0207 2.0000.49520.44680.39840.34930.3003 0.25140.20250.1522 0.10190.05220.0189 4.0000.34300.30900.27490.24130.2077 0.17320.13870.1042 0.06960.03550.0132 (TAM TAM)

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396 Table H-25. Absorbed fractions in the lumb ar vertebra for sour ces in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99830.99560.99290.98960.9862 0.98330.98050.9773 0.97410.97280.0014 0.0150.99690.99080.98470.97830.9720 0.96630.96050.9543 0.94810.94280.0027 0.0200.99420.98490.97550.96530.9551 0.94520.93530.9251 0.91490.90630.0047 0.0300.98910.96970.95020.93070.9113 0.89150.87180.8521 0.83230.81200.0089 0.0400.98240.95150.92060.88970.8587 0.82800.79730.7659 0.73450.70240.0123 0.0500.97430.93170.88910.84710.8051 0.76190.71880.6758 0.63270.59040.0141 0.1000.92670.84890.77120.69220.6133 0.53530.45740.3770 0.29660.21650.0189 0.2000.81640.73970.66310.58620.5094 0.43100.35260.2750 0.19750.11860.0225 0.5000.67580.61030.54480.47860.4125 0.34590.27930.2121 0.14490.07810.0263 1.0000.60570.54610.48640.42680.3671 0.30730.24760.1873 0.12710.06540.0254 1.5000.55200.49900.44600.39050.3350 0.28030.22550.1701 0.11470.05880.0236 2.0000.50840.45870.40890.35880.3088 0.25780.20690.1559 0.10500.05370.0220 4.0000.38230.34400.30560.26770.2298 0.19200.15430.1157 0.07720.03940.0167 (TAM TAM) Table H-26. Absorbed fractions in the s acrum for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99870.99590.99320.99010.9871 0.98430.98140.9788 0.97630.97220.0014 0.0150.99600.99050.98510.97940.9737 0.96710.96060.9545 0.94840.94120.0029 0.0200.99400.98430.97460.96460.9545 0.94370.93280.9238 0.91470.90590.0047 0.0300.98790.96880.94960.93000.9103 0.89100.87170.8522 0.83280.81160.0095 0.0400.98150.95050.91950.88860.8576 0.82700.79640.7654 0.73450.70460.0133 0.0500.97210.92990.88770.84510.8025 0.76060.71870.6765 0.63430.58700.0152 0.1000.92350.84500.76640.68870.6111 0.53170.45240.3744 0.29640.21540.0195 0.2000.80040.72470.64900.57410.4993 0.42340.34750.2706 0.19370.11550.0241 0.5000.64810.58580.52350.45930.3951 0.33170.26830.2046 0.14090.07500.0273 1.0000.56870.51350.45840.40260.3467 0.29000.23330.1766 0.12000.06210.0257 1.5000.51300.46240.41170.36170.3117 0.26030.20890.1574 0.10580.05480.0235 2.0000.46540.41970.37400.32670.2793 0.23380.18830.1423 0.09630.04870.0214 4.0000.32850.29730.26620.23230.1984 0.16570.13290.1002 0.06740.03440.0154 (TAM TAM) Table H-27. Absorbed fractions in the os coxae for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99870.99580.99300.99030.9875 0.98370.97980.9772 0.97470.97260.0010 0.0150.99710.99100.98500.97890.9728 0.96670.96060.9542 0.94780.94260.0024 0.0200.99550.98600.97650.96640.9562 0.94580.93540.9257 0.91610.90500.0039 0.0300.99110.97140.95170.93190.9121 0.89240.87270.8530 0.83330.81280.0074 0.0400.98500.95450.92390.89260.8612 0.83000.79880.7675 0.73630.70510.0100 0.0500.97800.93600.89400.85070.8073 0.76430.72120.6778 0.63440.59070.0120 0.1000.94000.86080.78150.70210.6227 0.54300.46320.3817 0.30020.21760.0160 0.2000.84900.76950.68990.60980.5296 0.44870.36780.2854 0.20310.12140.0200 0.5000.72770.65780.58780.51730.4467 0.37410.30160.2292 0.15670.08370.0235 1.0000.67110.60460.53810.47230.4064 0.34010.27370.2068 0.14000.07230.0231 1.5000.62390.56200.50010.43890.3776 0.31580.25400.1914 0.12880.06590.0220 2.0000.58160.52440.46730.40980.3523 0.29450.23670.1780 0.11930.06130.0207 4.0000.44470.40010.35550.31200.2684 0.22390.17950.1351 0.09070.04590.0160 (TAM TAM)

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397 Table H-28. Absorbed fractions in the cr anium for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99730.99450.99160.98880.9860 0.98300.97990.9769 0.97390.97150.0022 0.0150.99420.98840.98260.97680.9709 0.96480.95860.9529 0.94710.94150.0049 0.0200.99080.98140.97200.96200.9520 0.94230.93260.9231 0.91350.90400.0077 0.0300.98190.96270.94360.92450.9053 0.88650.86760.8487 0.82970.81040.0147 0.0400.97060.94060.91060.88070.8508 0.82070.79060.7604 0.73020.70090.0200 0.0500.95700.91600.87500.83360.7921 0.75060.70900.6675 0.62590.58400.0231 0.1000.87770.80510.73260.65860.5845 0.50990.43520.3606 0.28600.21020.0287 0.2000.68580.62240.55910.49540.4317 0.36580.30000.2348 0.16960.10400.0294 0.5000.42430.38370.34300.30160.2603 0.21900.17770.1358 0.09390.05220.0269 1.0000.33170.29910.26650.23400.2015 0.16900.13660.1036 0.07060.03750.0229 1.5000.27380.24660.21940.19270.1660 0.13860.11130.0842 0.05720.03010.0193 2.0000.22960.20710.18450.16200.1395 0.11660.09360.0708 0.04800.02500.0165 4.0000.13990.12620.11260.09860.0847 0.07070.05670.0428 0.02890.01490.0102 (TAM TAM) Table H-29. Absorbed fractions in the ma ndible for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99820.99550.99280.99010.9874 0.98440.98140.9782 0.97500.97250.0012 0.0150.99590.99010.98440.97770.9710 0.96630.96160.9553 0.94910.94130.0034 0.0200.99420.98420.97420.96440.9547 0.94460.93450.9240 0.91360.90540.0050 0.0300.98750.96790.94820.92880.9094 0.89010.87090.8512 0.83150.81180.0095 0.0400.98060.94960.91860.88730.8561 0.82620.79640.7649 0.73330.70250.0124 0.0500.97080.92790.88500.84290.8008 0.75930.71770.6734 0.62910.58710.0154 0.1000.91680.83980.76270.68670.6107 0.53140.45200.3736 0.29530.21480.0199 0.2000.80520.73000.65470.57860.5024 0.42610.34980.2721 0.19440.11650.0235 0.5000.63630.57490.51340.45100.3885 0.32590.26330.2005 0.13760.07420.0244 1.0000.50250.45350.40460.35520.3058 0.25540.20500.1551 0.10530.05520.0213 1.5000.41200.37160.33130.29000.2487 0.20830.16790.1264 0.08500.04400.0181 2.0000.34550.31030.27520.24160.2080 0.17300.13810.1043 0.07060.03630.0152 4.0000.19740.17700.15670.13770.1187 0.09890.07920.0597 0.04030.02080.0088 (TAM TAM) Table H-30. Absorbed fractions in the ribs for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99880.99580.99290.98990.9869 0.98370.98060.9774 0.97430.97150.0008 0.0150.99730.99140.98550.97900.9726 0.96700.96150.9550 0.94860.94220.0017 0.0200.99590.98600.97610.96600.9560 0.94540.93480.9251 0.91540.90580.0028 0.0300.99120.97140.95160.93190.9121 0.89210.87210.8525 0.83280.81270.0051 0.0400.98590.95470.92350.89240.8612 0.82980.79840.7672 0.73600.70420.0070 0.0500.97990.93660.89330.85010.8069 0.76360.72040.6768 0.63330.58980.0083 0.1000.93980.86090.78200.70310.6242 0.54330.46230.3811 0.29990.21930.0108 0.2000.85150.77210.69280.61190.5310 0.44950.36800.2861 0.20430.12140.0128 0.5000.68880.62210.55540.48790.4204 0.35230.28430.2161 0.14790.07940.0143 1.0000.53030.47850.42670.37430.3218 0.26920.21650.1637 0.11080.05780.0121 1.5000.41380.37310.33230.29130.2504 0.20900.16760.1267 0.08570.04410.0096 2.0000.32910.29630.26350.23070.1980 0.16550.13300.1004 0.06770.03490.0076 4.0000.16700.15050.13410.11720.1004 0.08400.06760.0509 0.03410.01760.0039 (TAM TAM)

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398 Table H-31. Absorbed fractions in the st ernum for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99870.99580.99300.99030.9875 0.98470.98180.9787 0.97560.97140.0012 0.0150.99730.99090.98450.97840.9724 0.96650.96070.9556 0.95040.94160.0018 0.0200.99550.98540.97530.96550.9558 0.94590.93610.9257 0.91530.90730.0040 0.0300.99070.97090.95120.93110.9110 0.89150.87190.8523 0.83270.81210.0074 0.0400.98490.95370.92250.89180.8612 0.82950.79770.7662 0.73470.70470.0103 0.0500.97730.93440.89150.84850.8055 0.76320.72090.6767 0.63250.58930.0117 0.1000.93850.85980.78110.70140.6218 0.54130.46080.3804 0.30000.21930.0158 0.2000.84440.76600.68760.60730.5271 0.44610.36510.2839 0.20260.12080.0191 0.5000.72210.65260.58310.51300.4429 0.37100.29920.2274 0.15570.08220.0218 1.0000.62930.56890.50840.44640.3843 0.32130.25820.1949 0.13160.06830.0210 1.5000.55840.50280.44720.39300.3388 0.28320.22750.1716 0.11570.05950.0193 2.0000.49890.44890.39880.34890.2990 0.25070.20230.1523 0.10220.05260.0175 4.0000.32210.28930.25660.22510.1936 0.16120.12880.0969 0.06510.03300.0115 (TAM TAM) Table H-32. Absorbed fractions in the right clavicle for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99860.99560.99260.98960.9865 0.98340.98030.9773 0.97430.97120.0013 0.0150.99720.99100.98490.97920.9734 0.96810.96280.9551 0.94750.94210.0023 0.0200.99550.98530.97510.96520.9553 0.94420.93320.9244 0.91560.90600.0035 0.0300.99080.97110.95140.93200.9127 0.89240.87220.8527 0.83330.81220.0071 0.0400.98420.95230.92040.88960.8588 0.82850.79830.7664 0.73460.70520.0103 0.0500.97740.93470.89190.84930.8066 0.76330.72000.6758 0.63170.58970.0121 0.1000.93440.85490.77540.69770.6199 0.54050.46100.3795 0.29800.21800.0153 0.2000.83100.75290.67480.59690.5191 0.44000.36100.2809 0.20090.11900.0177 0.5000.67060.60640.54220.47540.4087 0.34280.27700.2103 0.14370.07730.0199 1.0000.56210.50700.45200.39560.3391 0.28390.22860.1726 0.11650.06090.0179 1.5000.47700.42890.38080.33550.2901 0.24180.19340.1461 0.09870.05100.0162 2.0000.40740.36640.32530.28600.2467 0.20570.16460.1242 0.08380.04280.0141 4.0000.23100.20910.18720.16340.1396 0.11680.09400.0707 0.04750.02420.0081 (TAM TAM) Table H-33. Absorbed fractions in the left clavicle for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99910.99640.99360.98970.9858 0.98370.98160.9779 0.97430.97170.0005 0.0150.99800.99200.98590.98030.9747 0.96880.96280.9565 0.95010.94070.0009 0.0200.99730.98720.97710.96670.9563 0.94670.93700.9273 0.91760.90430.0019 0.0300.99460.97460.95460.93460.9147 0.89470.87460.8547 0.83480.81550.0034 0.0400.99110.95950.92790.89650.8652 0.83320.80120.7693 0.73740.70620.0049 0.0500.98720.94360.89990.85510.8103 0.76750.72470.6804 0.63600.59400.0056 0.1000.96110.88070.80030.71790.6356 0.55310.47060.3879 0.30510.22040.0076 0.2000.90430.81910.73390.64890.5639 0.47670.38940.3022 0.21510.12680.0094 0.5000.78140.70610.63090.55420.4775 0.39990.32240.2449 0.16740.08890.0112 1.0000.67070.60480.53890.47280.4067 0.34040.27410.2065 0.13890.07210.0105 1.5000.57880.51980.46090.40570.3505 0.29290.23520.1771 0.11900.06070.0094 2.0000.50060.44900.39730.35000.3027 0.25210.20160.1520 0.10240.05210.0082 4.0000.28680.25810.22930.20020.1711 0.14300.11490.0866 0.05820.02950.0047 (TAM TAM)

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399 Table H-34. Absorbed fractions in the righ t scapula for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99860.99560.99260.98990.9871 0.98410.98110.9781 0.97510.97440.0018 0.0150.99650.99130.98620.98010.9739 0.96790.96180.9553 0.94880.93980.0024 0.0200.99420.98410.97400.96440.9547 0.94500.93520.9246 0.91410.90530.0044 0.0300.98910.96950.94990.93030.9107 0.89110.87160.8514 0.83130.81160.0086 0.0400.98200.95080.91960.88900.8584 0.82720.79600.7655 0.73510.70420.0112 0.0500.97440.93180.88920.84580.8025 0.76030.71820.6750 0.63190.58880.0132 0.1000.92660.84920.77180.69290.6140 0.53540.45680.3776 0.29850.21650.0171 0.2000.81710.74120.66530.58700.5087 0.43090.35320.2753 0.19740.11720.0193 0.5000.62340.56280.50220.44130.3804 0.31910.25790.1963 0.13470.07290.0212 1.0000.51210.46230.41260.36070.3089 0.25900.20920.1584 0.10760.05590.0194 1.5000.44070.39660.35250.30930.2662 0.22320.18030.1358 0.09130.04750.0173 2.0000.38610.34680.30740.26970.2321 0.19370.15530.1168 0.07830.04050.0153 4.0000.24290.21850.19410.17070.1474 0.12280.09830.0742 0.05010.02520.0098 (TAM TAM) Table H-35. Absorbed fractions in the left scapula for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99860.99560.99250.98970.9870 0.98350.98000.9782 0.97630.97270.0016 0.0150.99670.99010.98340.97780.9722 0.96640.96060.9550 0.94940.94270.0030 0.0200.99440.98450.97470.96480.9550 0.94510.93520.9249 0.91460.90430.0046 0.0300.98850.96920.94990.93030.9108 0.89090.87100.8512 0.83150.81190.0090 0.0400.98140.95050.91960.88880.8580 0.82620.79450.7641 0.73370.70310.0128 0.0500.97190.92920.88660.84580.8051 0.76190.71880.6749 0.63090.58950.0148 0.1000.92180.84570.76960.69080.6121 0.53300.45390.3759 0.29780.21630.0191 0.2000.80670.73100.65530.58010.5049 0.42680.34880.2716 0.19450.11710.0217 0.5000.61220.55200.49170.43100.3702 0.31120.25220.1921 0.13200.07130.0231 1.0000.51050.46030.41000.35960.3092 0.25920.20920.1579 0.10670.05540.0216 1.5000.44570.40180.35790.31300.2681 0.22440.18080.1364 0.09210.04740.0193 2.0000.39130.35300.31470.27600.2373 0.19800.15870.1197 0.08080.04170.0172 4.0000.25130.22620.20120.17660.1519 0.12630.10080.0760 0.05130.02610.0113 (TAM TAM) Table H-36. Absorbed fractions in the femu r head for sources in the trabecular active marrow at varying marrow cellularity. (TBE TAM)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.99830.99560.99290.98990.9868 0.98400.98120.9777 0.97430.97180.0015 0.0150.99630.99010.98380.97750.9713 0.96530.95940.9542 0.94900.94240.0035 0.0200.99420.98410.97400.96470.9554 0.94490.93430.9250 0.91560.90550.0054 0.0300.98770.96840.94910.92920.9093 0.89010.87090.8515 0.83210.81140.0107 0.0400.98030.94960.91900.88850.8581 0.82700.79590.7657 0.73550.70340.0146 0.0500.97180.92930.88680.84420.8016 0.75910.71660.6742 0.63190.59010.0172 0.1000.91950.84240.76520.68790.6106 0.53190.45330.3744 0.29560.21630.0219 0.2000.79300.71960.64620.57070.4951 0.41990.34460.2689 0.19310.11580.0262 0.5000.63570.57380.51190.44970.3876 0.32540.26320.2001 0.13700.07410.0297 1.0000.59140.53390.47640.41810.3599 0.30120.24250.1832 0.12390.06440.0297 1.5000.56760.51180.45600.40010.3442 0.28780.23140.1743 0.11720.06020.0291 2.0000.54590.49240.43900.38470.3305 0.27590.22140.1668 0.11220.05730.0283 4.0000.46970.42310.37660.33050.2844 0.23720.19010.1430 0.09590.04850.0248 (TAM TAM)

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400 Table H-37. Absorbed fractions in the righ t proximal femur for sources in the trabecular bone surface at varyi ng marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00440.00390.00340.00310.0027 0.00220.00170.0013 0.00090.00050.5000 0.0150.00800.00710.00620.00540.0046 0.00390.00320.0023 0.00150.00080.5000 0.020.01220.01110.01000.00880.0076 0.00640.00510.0038 0.00240.00130.5000 0.030.05440.04910.04380.03820.0326 0.02710.02160.0162 0.01090.00550.4747 0.040.14840.13350.11860.10400.0894 0.07460.05980.0450 0.03010.01510.3838 0.050.22400.20170.17940.15720.1349 0.11270.09050.0679 0.04520.02270.3026 0.10.37720.34050.30370.26670.22960. 19210.15470.11630.07790.03870.1312 0.20.48490.43750.39000.34230.29460. 24630.19790.14860.09920.05030.0615 0.50.58350.52650.46940.41150.35360. 29520.23670.17780.11900.05950.0360 10.58620.52910.47190.41330.35460. 29610.23760.17850.11940.06000.0303 1.50.57440.51810.46180.40470.34760. 29010.23270.17490.11710.05840.0282 20.55630.50160.44680.39180.33670. 28140.22600.16960.11320.05690.0266 40.48480.43730.38980.34160.29350. 24460.19580.14710.09840.04920.0224 (TAM TBS) Table H-38. Absorbed fractions in the left proximal femur for sources in the trabecular bone surface at varyi ng marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00480.00430.00380.00340.0029 0.00240.00200.0015 0.00090.00060.5000 0.0150.00800.00720.00640.00560.0047 0.00410.00350.0026 0.00180.00090.5000 0.020.01240.01120.00990.00870.0076 0.00630.00510.0038 0.00260.00120.5000 0.030.05500.04970.04450.03880.0332 0.02750.02180.0164 0.01110.00570.4747 0.040.14860.13390.11920.10420.0891 0.07440.05970.0448 0.02990.01480.3829 0.050.22400.20220.18030.15770.1351 0.11280.09050.0678 0.04520.02270.3018 0.10.37990.34260.30540.26820.23090. 19300.15500.11650.07800.03890.1308 0.20.48920.44120.39320.34470.29630. 24740.19850.14900.09960.05000.0613 0.50.58610.52850.47080.41280.35480. 29610.23750.17860.11970.05980.0358 10.59140.53380.47620.41610.35600. 29770.23930.17970.12000.06040.0298 1.50.57900.52220.46550.40770.34990. 29230.23470.17620.11780.05900.0277 20.56460.50910.45360.39730.34100. 28470.22840.17170.11510.05740.0262 40.49380.44400.39420.34590.29750. 24840.19920.14950.09980.05000.0219 (TAM TBS) Table H-39. Absorbed fractions in the righ t humerus for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00490.00480.00460.00390.0031 0.00260.00210.0015 0.00090.00070.5000 0.0150.00830.00760.00690.00580.0047 0.00400.00320.0024 0.00150.00070.5000 0.020.01260.01130.01010.00890.0078 0.00650.00530.0040 0.00260.00140.5000 0.030.06040.05430.04820.04230.0365 0.03040.02440.0183 0.01210.00620.4747 0.040.15400.13930.12470.10920.0936 0.07800.06230.0468 0.03130.01570.4013 0.050.22860.20660.18450.16120.1378 0.11500.09230.0694 0.04650.02310.3125 0.10.37170.33460.29760.26150.22540. 18810.15080.11270.07460.03780.1348 0.20.45790.41320.36840.32370.27900. 23320.18740.14100.09460.04670.0616 0.50.54310.48990.43680.38330.32970. 27490.22000.16520.11030.05540.0364 10.54230.48800.43370.38080.32780. 27340.21900.16460.11020.05490.0305 1.50.52550.47300.42060.36960.31870. 26560.21260.15950.10630.05380.0279 20.50510.45570.40630.35590.30550. 25550.20560.15440.10330.05140.0264 40.42640.38420.34190.29930.25670. 21370.17070.12840.08610.04300.0214 (TAM TBS)

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401 Table H-40. Absorbed fractions in the left humerus for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00540.00460.00380.00340.0029 0.00250.00200.0015 0.00100.00090.5000 0.0150.01010.00870.00740.00650.0056 0.00470.00380.0028 0.00180.00100.5000 0.020.01320.01190.01050.00920.0079 0.00660.00530.0039 0.00260.00150.5000 0.030.06090.05490.04890.04290.0369 0.03090.02480.0186 0.01240.00620.4747 0.040.15510.14030.12560.10930.0930 0.07770.06240.0466 0.03070.01600.3982 0.050.23050.20800.18560.16240.1392 0.11610.09310.0700 0.04700.02360.3127 0.10.38050.34440.30840.27010.23180. 19430.15670.11790.07910.03960.1330 0.20.48560.43990.39420.34580.29750. 24880.20010.15060.10100.05020.0631 0.50.59530.53570.47620.41820.36020. 29980.23940.18020.12100.06040.0392 10.59100.53270.47440.41630.35810. 29870.23930.17970.12000.06020.0334 1.50.57130.51590.46040.40300.34560. 28870.23190.17420.11650.05810.0309 20.55150.49700.44240.38790.33340. 27800.22260.16690.11120.05620.0291 40.46280.41640.37000.32480.27970. 23240.18520.13930.09350.04660.0235 (TAM TBS) Table H-41. Absorbed fractions in the cervi cal vertebra for sources in the trabecular bone surface at varyi ng marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00540.00480.00430.00390.0035 0.00280.00220.0017 0.00110.00060.5000 0.0150.00990.00890.00790.00710.0063 0.00510.00400.0029 0.00170.00110.5000 0.020.01370.01240.01110.00980.0085 0.00710.00570.0042 0.00280.00150.5000 0.030.06250.05640.05030.04410.0379 0.03170.02540.0190 0.01270.00640.4747 0.040.15850.14280.12720.11150.0958 0.08020.06450.0484 0.03230.01590.3998 0.050.23410.21070.18730.16430.1412 0.11800.09490.0714 0.04780.02380.3121 0.10.38700.34890.31070.27320.23580. 19720.15850.11940.08030.04040.1331 0.20.49250.44380.39520.34700.29880. 24960.20040.15070.10100.05050.0627 0.50.55210.49850.44500.38960.33420. 27910.22390.16800.11220.05610.0362 10.49040.44250.39460.34550.29630. 24750.19870.14890.09920.05010.0273 1.50.42630.38400.34170.29990.25820. 21500.17180.12900.08610.04310.0224 20.37020.33250.29480.25860.22250. 18560.14880.11140.07400.03740.0188 40.21780.19530.17280.15160.13040. 10840.08650.06500.04350.02170.0108 (TAM TBS) Table H-42. Absorbed fractions in the thorac ic vertebra for sources in the trabecular bone surface at varyi ng marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00530.00480.00440.00370.0030 0.00260.00220.0016 0.00100.00070.5000 0.0150.00960.00860.00760.00670.0058 0.00490.00390.0030 0.00210.00100.5000 0.020.01450.01310.01180.01020.0086 0.00710.00570.0044 0.00300.00150.5000 0.030.06370.05730.05090.04460.0383 0.03190.02560.0192 0.01290.00640.4747 0.040.15950.14370.12780.11200.0961 0.08010.06400.0482 0.03240.01640.3986 0.050.23570.21260.18950.16550.1416 0.11820.09480.0713 0.04770.02370.3127 0.10.39160.35360.31550.27740.23930. 20010.16080.12060.08050.04090.1340 0.20.51770.46680.41590.36520.31450. 26240.21030.15820.10610.05350.0630 0.50.61000.55050.49110.43070.37030. 30890.24750.18610.12460.06260.0358 10.57580.51990.46400.40650.34910. 29130.23350.17540.11720.05850.0281 1.50.52760.47520.42280.37060.31850. 26550.21260.15970.10690.05360.0243 20.48190.43300.38420.33700.28980. 24150.19310.14500.09680.04880.0215 40.33540.30170.26800.23520.20240. 16850.13450.10100.06740.03380.0146 (TAM TBS)

PAGE 434

402 Table H-43. Absorbed fractions in the lumbar vertebra for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00500.00450.00410.00340.0027 0.00230.00190.0014 0.00100.00050.5000 0.0150.00830.00760.00700.00600.0050 0.00410.00330.0026 0.00180.00090.5000 0.020.01270.01170.01070.00950.0083 0.00690.00550.0041 0.00270.00140.5000 0.030.05780.05230.04680.04100.0352 0.02950.02370.0178 0.01190.00600.4747 0.040.15330.13790.12240.10740.0924 0.07710.06170.0464 0.03110.01530.3907 0.050.22870.20610.18340.16120.1390 0.11580.09260.0694 0.04620.02260.3082 0.10.38320.34560.30790.27030.23280. 19440.15610.11760.07900.03910.1329 0.20.49240.44470.39690.34880.30070. 25070.20070.15080.10080.05050.0638 0.50.58560.52800.47030.41260.35480. 29640.23800.17860.11920.06010.0374 10.56500.50920.45330.39740.34140. 28490.22850.17160.11470.05730.0304 1.50.52710.47500.42290.37090.31900. 26640.21390.16040.10700.05340.0270 20.48900.44210.39530.34630.29720. 24800.19890.14940.10000.04990.0245 40.37320.33590.29860.26170.22490. 18780.15070.11310.07540.03770.0179 (TAM TBS) Table H-44. Absorbed fractions in the sacr um for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00490.00460.00420.00380.0033 0.00280.00230.0016 0.00100.00040.5000 0.0150.00900.00840.00770.00680.0058 0.00510.00430.0029 0.00150.00100.5000 0.020.01460.01280.01100.00960.0083 0.00720.00600.0045 0.00300.00130.5000 0.030.06050.05500.04940.04320.0369 0.03090.02500.0189 0.01280.00640.4747 0.040.15780.14110.12450.10910.0938 0.07840.06290.0474 0.03180.01560.3968 0.050.23130.20820.18520.16230.1394 0.11630.09310.0701 0.04710.02410.3130 0.10.37700.34230.30760.26990.23220. 19410.15610.11750.07890.03950.1345 0.20.48940.44040.39140.34320.29510. 24760.20000.14990.09970.05060.0630 0.50.57150.51530.45910.40340.34770. 28960.23160.17390.11630.05800.0378 10.53790.48380.42970.37800.32630. 27170.21710.16310.10910.05430.0306 1.50.49130.44270.39400.34520.29630. 24810.19990.14970.09960.05000.0268 20.45140.40760.36370.31700.27030. 22540.18050.13570.09090.04530.0237 40.32330.29150.25970.22630.19300. 16170.13040.09750.06470.03240.0166 (TAM TBS) Table H-45. Absorbed fractions in the os coxae for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00560.00500.00440.00390.0035 0.00290.00240.0017 0.00110.00060.5000 0.0150.00990.00890.00800.00700.0060 0.00490.00390.0028 0.00180.00090.5000 0.020.01510.01360.01200.01050.0089 0.00740.00580.0043 0.00290.00150.5000 0.030.06400.05780.05160.04510.0386 0.03230.02600.0195 0.01290.00640.4747 0.040.16030.14420.12810.11240.0966 0.08080.06490.0487 0.03240.01600.3992 0.050.23790.21410.19030.16670.1432 0.11920.09530.0715 0.04770.02410.3125 0.10.39800.35870.31940.28090.24240. 20260.16290.12230.08170.04110.1337 0.20.52600.47480.42370.37170.31970. 26700.21440.16150.10860.05490.0629 0.50.63940.57650.51350.45000.38650. 32290.25930.19500.13070.06530.0352 10.62660.56470.50290.44070.37860. 31630.25400.19080.12760.06380.0283 1.50.59620.53770.47910.41950.35990. 30030.24060.18070.12080.06090.0254 20.56270.50630.44990.39380.33770. 28210.22650.17030.11420.05720.0233 40.43570.39250.34930.30600.26260. 21920.17580.13200.08830.04400.0174 (TAM TBS)

PAGE 435

403 Table H-46. Absorbed fractions in the cran ium for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00450.00400.00350.00310.0027 0.00230.00180.0014 0.00090.00060.5000 0.0150.00780.00710.00630.00560.0048 0.00400.00320.0025 0.00180.00080.5000 0.020.01180.01060.00950.00820.0070 0.00580.00460.0034 0.00230.00120.5000 0.030.05680.05110.04550.03980.0342 0.02860.02310.0173 0.01150.00570.4747 0.040.14640.13190.11730.10270.0881 0.07340.05880.0441 0.02940.01490.3997 0.050.21720.19580.17450.15270.1310 0.10940.08780.0660 0.04410.02210.3128 0.10.34260.30940.27620.24240.20870. 17440.14010.10500.06990.03530.1345 0.20.37610.33880.30140.26420.22700. 18950.15190.11420.07650.03870.0608 0.50.35480.32020.28560.25000.21450. 17890.14320.10760.07190.03580.0352 10.30330.27280.24230.21270.18300. 15230.12160.09110.06060.03040.0267 1.50.25470.23000.20530.17940.15350. 12800.10250.07690.05120.02580.0218 20.21650.19530.17410.15240.13080. 10910.08750.06560.04380.02190.0184 40.13360.12030.10710.09370.08030. 06690.05360.04020.02690.01340.0111 (TAM TBS) Table H-47. Absorbed fractions in the mandible for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00600.00560.00510.00460.0041 0.00350.00280.0020 0.00110.00050.5000 0.0150.00970.00890.00810.00700.0058 0.00490.00410.0030 0.00190.00120.5000 0.020.01390.01260.01140.01000.0087 0.00710.00560.0041 0.00270.00130.5000 0.030.06270.05660.05060.04420.0379 0.03160.02540.0190 0.01270.00630.4747 0.040.15790.14250.12720.11100.0947 0.07920.06380.0478 0.03190.01580.3988 0.050.23330.20980.18630.16350.1408 0.11770.09470.0712 0.04760.02360.3141 0.10.38050.34340.30620.26890.23160. 19370.15580.11650.07710.03860.1364 0.20.48260.43500.38740.34040.29340. 24430.19530.14760.09980.04920.0679 0.50.51270.46310.41350.36220.31080. 25910.20730.15630.10540.05210.0375 10.45120.40620.36120.31640.27150. 22680.18220.13680.09130.04570.0268 1.50.38420.34450.30480.26760.23050. 19260.15470.11570.07670.03890.0216 20.32410.29180.25950.22620.19300. 16150.13010.09710.06420.03250.0177 40.18860.16920.14990.13130.11280. 09410.07550.05660.03760.01870.0100 (TAM TBS) Table H-48. Absorbed fractions in the ribs for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00590.00530.00470.00410.0036 0.00300.00240.0018 0.00120.00060.5000 0.0150.01000.00900.00800.00710.0061 0.00510.00410.0030 0.00190.00100.5000 0.020.01470.01340.01220.01070.0092 0.00770.00630.0047 0.00300.00140.5000 0.030.06360.05730.05100.04480.0385 0.03220.02580.0194 0.01290.00650.4747 0.040.15940.14400.12850.11260.0966 0.08070.06480.0487 0.03260.01600.4001 0.050.23570.21230.18890.16540.1420 0.11870.09540.0714 0.04750.02370.3115 0.10.38760.35000.31250.27410.23580. 19720.15870.11900.07930.04010.1328 0.20.49100.44300.39510.34660.29810. 24900.19990.15020.10050.05030.0606 0.50.56500.50950.45400.39760.34110. 28510.22910.17210.11520.05750.0285 10.48350.43570.38790.34000.29200. 24390.19570.14680.09780.04900.0183 1.50.38750.34880.31000.27140.23280. 19410.15540.11670.07800.03900.0135 20.30860.27800.24740.21680.18610. 15530.12440.09320.06200.03100.0105 40.15810.14220.12630.11080.09520. 07910.06300.04730.03160.01580.0053 (TAM TBS)

PAGE 436

404 Table H-49. Absorbed fractions in the ster num for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00530.00490.00450.00380.0031 0.00260.00220.0017 0.00110.00050.5000 0.0150.00980.00880.00780.00680.0058 0.00480.00380.0029 0.00200.00100.5000 0.020.01460.01320.01190.01030.0087 0.00730.00590.0045 0.00300.00150.5000 0.030.06390.05760.05130.04500.0387 0.03220.02570.0193 0.01280.00640.4747 0.040.16230.14650.13070.11460.0984 0.08170.06490.0490 0.03300.01600.3982 0.050.23920.21530.19150.16780.1442 0.12130.09840.0735 0.04860.02450.3087 0.10.39930.36150.32370.28390.24410. 20400.16390.12290.08180.04060.1324 0.20.51790.46720.41660.36540.31430. 26200.20980.15760.10540.05280.0620 0.50.60430.54560.48700.42600.36510. 30500.24490.18410.12320.06170.0351 10.57500.51870.46230.40510.34790. 29060.23330.17560.11790.05910.0272 1.50.53330.48060.42790.37540.32290. 26980.21670.16220.10780.05430.0236 20.48480.43630.38770.33900.29040. 24240.19440.14590.09750.04860.0209 40.31720.28600.25480.22240.19000. 15860.12730.09540.06350.03170.0131 (TAM TBS) Table H-50. Absorbed fractions in the right clavicle for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00490.00460.00430.00380.0033 0.00290.00250.0020 0.00140.00050.5000 0.0150.00840.00760.00670.00570.0047 0.00390.00320.0024 0.00160.00080.5000 0.020.01270.01160.01050.00910.0078 0.00630.00480.0036 0.00240.00130.5000 0.030.06030.05460.04880.04260.0365 0.03030.02420.0181 0.01210.00610.4747 0.040.15620.14080.12540.11010.0948 0.07880.06280.0473 0.03180.01610.4004 0.050.23150.20860.18560.16210.1386 0.11610.09350.0693 0.04510.02300.3108 0.10.37780.34100.30420.26680.22940. 19230.15530.11670.07820.03960.1313 0.20.46670.42200.37720.33100.28480. 23770.19060.14340.09620.04840.0608 0.50.53540.48190.42850.37620.32380. 27010.21630.16240.10850.05430.0325 10.49270.44290.39310.34470.29630. 24720.19820.14880.09930.05000.0245 1.50.43450.39210.34960.30680.26400. 21930.17470.13130.08790.04390.0203 20.37870.34280.30680.26930.23170. 19220.15260.11460.07670.03870.0171 40.22170.20040.17900.15650.13400. 11140.08880.06690.04510.02250.0097 (TAM TBS) Table H-51. Absorbed fractions in the left clavicle for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00750.00650.00560.00490.0042 0.00340.00260.0019 0.00120.00070.5000 0.0150.01210.01090.00980.00870.0077 0.00620.00480.0034 0.00210.00120.5000 0.020.01670.01500.01320.01160.0101 0.00840.00680.0051 0.00330.00150.5000 0.030.06680.06040.05390.04710.0402 0.03370.02730.0205 0.01370.00690.4747 0.040.16560.14920.13290.11630.0997 0.08330.06700.0501 0.03320.01670.3965 0.050.24190.21810.19430.17010.1459 0.12270.09960.0740 0.04830.02460.3093 0.10.39910.36040.32170.28240.24300. 20380.16460.12350.08250.04110.1293 0.20.52140.47200.42260.36960.31650. 26370.21100.15840.10590.05360.0572 0.50.63790.57520.51240.44960.38670. 32270.25870.19470.13070.06470.0255 10.60670.54820.48970.42800.36640. 30660.24680.18500.12330.06150.0170 1.50.53910.48530.43150.37800.32450. 27120.21800.16340.10880.05430.0136 20.47220.42450.37670.33000.28330. 23690.19050.14240.09420.04730.0113 40.27490.24770.22050.19290.16530. 13770.11010.08250.05500.02760.0063 (TAM TBS)

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405 Table H-52. Absorbed fractions in the righ t scapula for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00510.00460.00410.00350.0029 0.00240.00190.0015 0.00110.00030.5000 0.0150.00870.00770.00670.00590.0052 0.00430.00350.0028 0.00200.00090.5000 0.020.01300.01170.01040.00910.0077 0.00630.00480.0035 0.00220.00120.5000 0.030.06040.05450.04850.04260.0367 0.03070.02460.0184 0.01220.00600.4747 0.040.15530.13910.12290.10800.0931 0.07750.06200.0466 0.03130.01580.3995 0.050.22650.20480.18300.16060.1382 0.11510.09200.0688 0.04560.02210.3145 0.10.36720.33080.29440.25830.22210. 18530.14850.11170.07500.03700.1349 0.20.44100.39910.35720.31350.26970. 22550.18120.13560.09000.04500.0625 0.50.49880.44940.40010.35030.30050. 25110.20170.15140.10110.05060.0330 10.45880.41220.36560.32140.27710. 23030.18350.13790.09220.04660.0243 1.50.41000.36800.32590.28620.24640. 20540.16440.12310.08190.04160.0205 20.36270.32580.28900.25310.21730. 18110.14500.10890.07290.03640.0178 40.23070.20730.18390.16020.13650. 11460.09280.06950.04620.02310.0109 (TAM TBS) Table H-53. Absorbed fractions in the left scapula for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00580.00520.00460.00400.0035 0.00290.00220.0016 0.00090.00070.5000 0.0150.01010.00910.00820.00690.0057 0.00480.00380.0029 0.00200.00110.5000 0.020.01430.01270.01120.00990.0085 0.00730.00600.0046 0.00310.00150.5000 0.030.06180.05570.04960.04330.0371 0.03100.02490.0186 0.01230.00620.4747 0.040.15540.13970.12400.10830.0925 0.07750.06240.0470 0.03170.01590.3985 0.050.22720.20550.18390.16050.1372 0.11440.09150.0693 0.04700.02400.3123 0.10.36790.33160.29530.25920.22320. 18620.14920.11200.07480.03810.1353 0.20.44740.40390.36040.31710.27380. 22910.18450.13850.09250.04680.0621 0.50.49860.45010.40150.35200.30240. 25140.20030.15100.10180.05090.0341 10.46290.41710.37130.32510.27890. 23320.18760.14070.09380.04710.0263 1.50.41910.37730.33540.29380.25220. 21010.16800.12640.08480.04240.0227 20.37190.33510.29830.26180.22530. 18710.14880.11220.07550.03740.0198 40.24290.21840.19390.16990.14590. 12120.09650.07230.04800.02440.0125 (TAM TBS) Table H-54. Absorbed fractions in the fe mur head for sources in the trabecular bone surface at varying marrow cellularity. (TBE TBS)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.010.00520.00460.00400.00340.0029 0.00240.00200.0015 0.00090.00050.5000 0.0150.00800.00720.00650.00550.0046 0.00400.00340.0026 0.00180.00100.5000 0.020.01230.01110.00990.00890.0078 0.00650.00520.0038 0.00250.00120.5000 0.030.05420.04870.04330.03760.0318 0.02640.02100.0159 0.01070.00550.4747 0.040.14640.13190.11750.10270.0880 0.07360.05910.0444 0.02970.01480.3832 0.050.22120.19950.17780.15550.1333 0.11140.08950.0671 0.04480.02260.3024 0.10.37120.33450.29780.26150.22510. 18810.15100.11360.07630.03810.1316 0.20.47100.42460.37820.33160.28510. 23790.19080.14330.09580.04860.0629 0.50.55050.49600.44140.38730.33330. 27790.22250.16730.11200.05610.0394 10.55210.49770.44340.38770.33210. 27770.22330.16770.11210.05640.0339 1.50.54020.48720.43420.38020.32620. 27230.21840.16410.10990.05490.0319 20.52590.47400.42220.36950.31690. 26480.21270.15960.10660.05360.0304 40.46100.41500.36910.32400.27880. 23230.18570.13950.09340.04680.0258 (TAM TBS)

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406 Table H-55. Absorbed fractions in the righ t proximal femur for sources in the trabecular bone endosteum at varying marrow cellularity. (TBE TBE)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.03060.02760.02490.02190.0186 0.01540.01230.0093 0.00640.00320.9379 0.0150.06210.05590.04970.04350.0374 0.03110.02480.0187 0.01240.00640.8730 0.0200.10210.09200.08180.07170.0615 0.05120.04100.0309 0.02080.01060.7911 0.0300.19750.17830.15920.13940.1197 0.10020.07970.0607 0.04070.02090.5946 0.0400.27240.24410.21570.18920.1627 0.13510.10930.0811 0.05480.02750.4419 0.0500.32530.29330.26120.22880.1963 0.16360.13150.0989 0.06700.03310.3389 0.1000.42610.38450.34290.30090.2589 0.21600.17480.1300 0.08700.04330.1410 0.2000.50670.45770.40870.35810.3074 0.25770.20810.1567 0.10540.05210.0646 0.5000.58800.53000.47200.41370.3555 0.29650.23770.1786 0.11960.05970.0369 1.0000.59050.53270.47500.41510.3553 0.29710.24160.1791 0.11920.06000.0307 1.5000.57590.51820.46050.40410.3477 0.28930.23270.1736 0.11630.05850.0284 2.0000.55920.50300.44680.39220.3375 0.28270.22580.1704 0.11280.05670.0268 4.0000.48540.43470.38400.33690.2898 0.24240.19420.1459 0.09670.04860.0225 (TAM TBE) Table H-56. Absorbed fractions in the left proximal femur for sources in the trabecular bone endosteum at varying marrow cellularity. (TBE TBE)Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.03070.02760.02490.02190.0186 0.01550.01230.0094 0.00640.00320.9377 0.0150.06270.05640.05010.04390.0377 0.03140.02500.0188 0.01260.00640.8724 0.0200.10350.09320.08300.07270.0624 0.05200.04160.0313 0.02110.01070.7901 0.0300.19690.17780.15870.13900.1194 0.09990.07950.0605 0.04060.02080.5957 0.0400.27410.24550.21700.19030.1637 0.13590.11000.0816 0.05510.02760.4422 0.0500.32630.29410.26200.22940.1969 0.16400.13190.0992 0.06720.03320.3381 0.1000.42930.38730.34540.30310.2608 0.21760.17610.1310 0.08760.04360.1407 0.2000.51180.46230.41280.36170.3105 0.26030.21020.1583 0.10650.05260.0640 0.5000.59490.53630.47760.41870.3597 0.30010.24050.1807 0.12100.06040.0360 1.0000.59580.53750.47920.41880.3585 0.29980.24380.1807 0.12020.06050.0297 1.5000.58440.52580.46730.41010.3529 0.29360.23610.1761 0.11800.05940.0273 2.0000.56830.51120.45410.39850.3430 0.28730.22940.1732 0.11470.05770.0257 4.0000.49570.44390.39220.34410.2960 0.24760.19830.1490 0.09880.04960.0213 (TAM TBE) Table H-57. Absorbed fractions in the righ t humerus for sources in the trabecular bone endosteum at varying marrow cellularity. ( TBE TBE ) Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.03090.02780.02510.02210.0188 0.01560.01240.0094 0.00650.00330.9374 0.0150.06210.05590.04960.04350.0374 0.03110.02470.0186 0.01240.00640.8729 0.0200.10220.09210.08190.07180.0616 0.05130.04100.0309 0.02090.01060.7906 0.0300.19520.17630.15740.13790.1184 0.09910.07880.0600 0.04030.02060.5961 0.0400.27110.24290.21470.18830.1619 0.13440.10880.0807 0.05450.02730.4416 0.0500.32210.29040.25870.22650.1944 0.16200.13020.0980 0.06640.03280.3389 0.1000.41740.37660.33580.29470.2536 0.21150.17120.1273 0.08520.04240.1413 0.2000.47780.43150.38530.33760.2899 0.24300.19620.1477 0.09940.04910.0640 0.5000.54810.49410.44000.38570.3314 0.27640.22160.1665 0.11150.05570.0369 1.0000.54600.49260.43910.38380.3285 0.27470.22340.1656 0.11020.05550.0307 1.5000.52780.47490.42200.37040.3187 0.26510.21330.1591 0.10660.05360.0282 2.0000.50590.45510.40430.35480.3054 0.25580.20430.1542 0.10210.05130.0265 4.0000.42710.38250.33790.29650.2550 0.21330.17090.1284 0.08510.04270.0214 (TAM TBE)

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407 Table H-58. Absorbed fractions in the left humerus for sources in the trabecular bone endosteum at varying marrow cellularity. ( TBE TBE ) Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.03040.02740.02470.02170.0185 0.01530.01220.0093 0.00640.00320.9388 0.0150.06400.05760.05120.04480.0385 0.03210.02550.0192 0.01280.00660.8730 0.0200.10500.09460.08420.07370.0633 0.05270.04210.0318 0.02140.01090.7880 0.0300.19580.17680.15780.13830.1187 0.09940.07900.0602 0.04040.02070.5947 0.0400.27410.24560.21710.19040.1637 0.13590.11000.0816 0.05510.02760.4404 0.0500.32380.29190.26000.22770.1954 0.16280.13090.0985 0.06670.03290.3374 0.1000.42250.38120.34000.29830.2567 0.21410.17330.1289 0.08620.04290.1412 0.2000.50360.45490.40620.35590.3055 0.25610.20680.1557 0.10480.05170.0664 0.5000.60060.54140.48210.42260.3632 0.30290.24280.1824 0.12220.06100.0399 1.0000.59060.53280.47500.41520.3553 0.29720.24170.1791 0.11920.06000.0338 1.5000.57570.51800.46030.40400.3476 0.28920.23260.1735 0.11620.05850.0313 2.0000.54910.49390.43880.38510.3314 0.27760.22170.1673 0.11080.05570.0291 4.0000.45990.41190.36380.31920.2746 0.22970.18400.1382 0.09170.04600.0237 (TAM TBE) Table H-59. Absorbed fractions in the cervi cal vertebra for sources in the trabecular bone endosteum at varying marrow cellularity. ( TBE TBE ) Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.03300.02970.02680.02360.0200 0.01660.01330.0101 0.00690.00350.9362 0.0150.06540.05890.05230.04580.0394 0.03280.02610.0196 0.01310.00670.8711 0.0200.10400.09370.08340.07310.0627 0.05220.04180.0315 0.02120.01080.7890 0.0300.20200.18240.16280.14270.1225 0.10250.08150.0621 0.04170.02140.5923 0.0400.27970.25060.22150.19430.1670 0.13870.11230.0833 0.05630.02820.4385 0.0500.33460.30170.26870.23530.2019 0.16830.13530.1018 0.06890.03400.3364 0.1000.44960.40570.36180.31750.2731 0.22780.18440.1372 0.09180.04570.1410 0.2000.57920.52320.46720.40930.3514 0.29460.23790.1791 0.12050.05950.0689 0.5000.63750.57460.51170.44860.3855 0.32150.25770.1936 0.12970.06470.0373 1.0000.55980.50500.45020.39350.3368 0.28170.22910.1698 0.11300.05690.0273 1.5000.48130.43310.38480.33770.2906 0.24180.19450.1450 0.09720.04890.0223 2.0000.41230.37090.32950.28920.2488 0.20850.16650.1256 0.08320.04180.0184 4.0000.21390.19150.16920.14850.1277 0.10680.08560.0643 0.04260.02140.0104 (TAM TBE) Table H-60. Absorbed fractions in the thorac ic vertebra for sources in the trabecular bone endosteum at varying marrow cellularity. ( TBE TBE ) Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.03120.02810.02530.02230.0189 0.01570.01250.0095 0.00650.00330.9378 0.0150.06460.05810.05160.04520.0389 0.03230.02570.0194 0.01290.00660.8706 0.0200.10350.09330.08300.07270.0624 0.05200.04160.0313 0.02110.01080.7903 0.0300.19900.17970.16040.14060.1207 0.10100.08030.0612 0.04110.02100.5952 0.0400.27660.24780.21910.19210.1652 0.13720.11100.0824 0.05560.02790.4401 0.0500.32920.29680.26430.23150.1986 0.16550.13310.1001 0.06780.03350.3373 0.1000.43640.39380.35120.30820.2652 0.22120.17900.1332 0.08910.04430.1404 0.2000.53220.48070.42930.37610.3229 0.27070.21860.1646 0.11070.05470.0647 0.5000.61430.55370.49310.43220.3714 0.30980.24830.1866 0.12490.06240.0363 1.0000.57810.52150.46490.40640.3478 0.29090.23650.1753 0.11660.05870.0283 1.5000.52630.47360.42080.36930.3178 0.26440.21270.1586 0.10630.05350.0244 2.0000.48180.43340.38500.33790.2908 0.24360.19450.1468 0.09720.04890.0217 4.0000.33810.30280.26750.23470.2019 0.16890.13530.1016 0.06740.03380.0148 (TAM TBE)

PAGE 440

408 Table H-61. Absorbed fractions in the lumbar vertebra for sources in the trabecular bone endosteum at varying marrow cellularity. ( TBE TBE ) Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.03050.02740.02470.02180.0185 0.01540.01220.0093 0.00640.00320.9384 0.0150.06100.05490.04880.04270.0367 0.03060.02430.0183 0.01220.00630.8735 0.0200.10360.09340.08310.07280.0625 0.05200.04160.0314 0.02120.01080.7904 0.0300.19820.17900.15970.13990.1201 0.10060.08000.0609 0.04090.02090.5968 0.0400.27380.24540.21690.19020.1635 0.13580.10990.0816 0.05510.02760.4406 0.0500.32620.29410.26190.22940.1968 0.16400.13190.0992 0.06720.03320.3389 0.1000.42740.38570.34390.30180.2597 0.21660.17530.1304 0.08720.04340.1414 0.2000.51220.46270.41310.36190.3108 0.26050.21040.1584 0.10660.05260.0657 0.5000.59120.53290.47460.41610.3575 0.29820.23900.1796 0.12020.06010.0385 1.0000.56460.50930.45410.39690.3397 0.28410.23100.1712 0.11390.05740.0309 1.5000.52860.47560.42260.37090.3192 0.26550.21360.1593 0.10670.05370.0272 2.0000.49210.44270.39330.34510.2970 0.24880.19870.1500 0.09930.04990.0247 4.0000.37250.33370.29480.25860.2224 0.18610.14910.1120 0.07420.03730.0180 (TAM TBE) Table H-62. Absorbed fractions in the sacrum for sources in the trabecular bone endosteum at varying marrow cellularity. ( TBE TBE ) Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.02920.02630.02370.02080.0177 0.01470.01170.0089 0.00610.00310.9394 0.0150.06290.05660.05030.04400.0378 0.03150.02510.0189 0.01260.00650.8718 0.0200.10380.09350.08320.07290.0626 0.05210.04170.0314 0.02120.01080.7896 0.0300.19640.17730.15830.13870.1191 0.09970.07930.0604 0.04050.02080.5960 0.0400.27460.24600.21750.19070.1640 0.13610.11020.0818 0.05520.02770.4408 0.0500.32590.29380.26170.22920.1967 0.16390.13170.0991 0.06710.03310.3368 0.1000.42740.38570.34390.30180.2597 0.21660.17530.1304 0.08720.04340.1417 0.2000.50590.45690.40800.35750.3069 0.25730.20780.1564 0.10520.05200.0659 0.5000.57570.51890.46210.40510.3481 0.29030.23270.1748 0.11710.05850.0389 1.0000.53710.48460.43200.37760.3232 0.27030.21980.1629 0.10840.05460.0308 1.5000.49420.44470.39520.34680.2984 0.24830.19970.1489 0.09980.05020.0271 2.0000.45020.40500.35970.31570.2717 0.22760.18180.1372 0.09090.04570.0241 4.0000.32560.29160.25760.22600.1944 0.16260.13030.0979 0.06490.03260.0170 (TAM TBE) Table H-63. Absorbed fractions in the os coxae for sources in the trabecular bone endosteum at varying marrow cellularity. ( TBE TBE ) Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.03170.02850.02570.02260.0192 0.01600.01270.0097 0.00660.00330.9370 0.0150.06350.05720.05080.04450.0382 0.03180.02530.0191 0.01270.00650.8720 0.0200.10600.09550.08500.07440.0639 0.05320.04250.0321 0.02160.01100.7877 0.0300.20040.18090.16150.14150.1215 0.10170.08090.0616 0.04130.02120.5948 0.0400.27880.24980.22080.19370.1665 0.13830.11190.0830 0.05610.02810.4408 0.0500.33370.30080.26790.23460.2013 0.16780.13490.1015 0.06870.03390.3377 0.1000.44600.40240.35890.31490.2710 0.22600.18290.1361 0.09100.04530.1403 0.2000.55630.50250.44870.39310.3375 0.28290.22850.1720 0.11570.05720.0654 0.5000.66860.60260.53670.47050.4043 0.33720.27030.2031 0.13600.06790.0352 1.0000.65040.58670.52310.45720.3913 0.32730.26610.1973 0.13120.06610.0281 1.5000.61780.55590.49400.43350.3730 0.31030.24970.1862 0.12470.06280.0252 2.0000.58140.52300.46460.40770.3509 0.29400.23470.1772 0.11730.05900.0230 4.0000.43300.38780.34260.30060.2586 0.21630.17330.1302 0.08630.04330.0171 (TAM TBE)

PAGE 441

409 Table H-64. Absorbed fractions in the cranium for sources in the trabecular bone endosteum at varying marrow cellularity. ( TBE TBE ) Energy (MeV) 100%90%80%70%60%50%40%30%20%10% Celluarity Independent 0.0100.02940.02650.02390.02100.0179 0.01480.01180.0090 0.00620.00310.9383 0.0150.05950.05350.04760.04170.0358 0.02980.02370.0179 0.01190.00610.8744 0.0200.09920.08940.07950.06970.0598 0.04980.03980.0300 0.02030.01030.7918 0.0300.18900.17070.15230.13350.1146 0.09590.07630.0581 0.03900.02000.5992 0.0400.26140.23420.20700.18150.1561 0.12960.10490.0778 0.05260.02630.4451 0.0500.31000.27950.24900.21800.1871 0.15590.12530.0943 0.06390.03150.34
Permanent Link: http://ufdc.ufl.edu/UFE0008395/00001

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Title: Reference Skeletal Dosimetry Model for an Adult Male Radionuclide Therapy Patient Based on 3D Imaging and Paired-Image Radiation Transport
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
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Permanent Link: http://ufdc.ufl.edu/UFE0008395/00001

Material Information

Title: Reference Skeletal Dosimetry Model for an Adult Male Radionuclide Therapy Patient Based on 3D Imaging and Paired-Image Radiation Transport
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
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REFERENCE SKELETAL DOSIMETRY MODEL FOR AN ADULT MALE
RADIONUCLIDE THERAPY PATIENT BASED ON 3D IMAGING AND PAIRED-
IMAGE RADIATION TRANSPORT












By

AMISH P. SHAH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2004

































Copyright 2004

by

Amish P. Shah















ACKNOWLEDGMENTS

Several people contributed a great deal to this work. First, I would like to thank

Dr. Wesley E. Bolch for his support, guidance, and encouragement. I appreciate his

patience over the past 5 years; but most importantly, I value his friendship and everything

he has done to help me mature into a better student. I also thank Dr. David Hintenlang,

Dr. Edward Dugan, Dr. Christopher Batich, and Dr. Didier Rajon for their suggestions

and for being part of my committee.

I would also like to thank my colleagues that reside in the halls throughout the

Department of Nuclear and Radiological Engineering for their friendship. I also thank all

the current and former members of the Bone Imaging and Dosimetry project for the time

we have spent working together and pondering over trabecular bone. Specifically, I

thank Dr. Phillip Patton and Dr. Derek Jokisch for sharing their infinite wisdom in all

aspects of bone dosimetry and life.

I would also like to recognize the faculty and staff in the departments of Nuclear

and Radiological Engineering and Biomedical Engineering for their assistance. As I am

in the middle of the two departments, both were instrumental in their help and support

over the past 5 years. I also thank the Graduate School and Dr. Bolch, for the opportunity

to pursue an interest in business management. Without their help, I could not have

concurrently earned a Master of Science in Business Management.

Finally, I would like to thank some people very close to my heart. My parents and

my brother, Angesh, have been right there beside me every step of the way. Their love









and support are immeasurable and cannot be understood with words. I thank my wife,

Priti, for always knowing that things are going to work out, and never worrying about

making the wrong decisions in life. Among other things, I thank her for bringing that gift

into my life. I also thank Priti for her love, infectious laughter, and most of all, patience.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iii

LIST OF TABLES ..................................................... ix

LIST OF FIGURES ..................................... .............. xxiv

A B S T R A C T .................................................................................................................. x x x i

CHAPTER

1 IN T R O D U C T IO N ................................................. .............................................. .

2 B A C K G R O U N D .............. ......................... ................ ..6.... .....6

B one Structure and Physiology ...................................... ...................... ...............6...
R adionuclide Therapies for C ancer ......................................................... ...............8...
M arrow T oxicity .................................................. ............... .............. ... ............ 9
Previous Methods of Trabecular Bone Dosimetry ................................................10
Internal D osim etry C calculations ............................................................ ............... 13

3 PAIRED-IMAGE RADIATION TRANSPORT MODEL FOR SKELETAL
D O S IM E T R Y ............................................................................................................. 17

In tro d u ctio n ............................................................................................................... .. 1 7
M materials and M ethods .. ..................................................................... ................ 2 1
C adav er Selection ........................................... .. ...................... .. ...... ............ 2 1
In-Vivo Computed Tomography Scanning .................................... ................ 21
Bone Harvesting and Ex-Vivo Computed Tomography Scanning ..................22
Image Segmentation of Spongiosa and Cortical Bone Regions.......................23
M icroim aging of Trabecular Spongiosa................................ ............... 24
Voxel-Based Infinite Spongiosa Transport (VBIST) Model...............................25
Paired-Image Radiation Transport Model (L4 Vertebra)................................26
Paired-Image Radiation Transport Model (Proximal Femur) ..........................28
R e su lts.................. .......................... ... ........................................................ ........ 2 9
Absorbed Fractions to Active Marrow within the L4 Vertebra........................29
Absorbed Fractions to Active Marrow within the Proximal Femur.................31
Absorbed Fractions to Endosteal Tissues....................................... ................ 32









D isc u ssio n ................................................................................................................... 3 3
C o n c lu sio n .................................................................................................................. 3 5

4 BETA-PARTICLE ENERGY LOSS TO CORTICAL BONE VIA PAIRED-IMAGE
RADIATION TRANSPORT: CORRECTIONS TO CLINICAL MODELS OF
SK E L E TA L T ISSU E D O SE ......................................................................................48

Introduction ................................................................................... ...................... 48
M materials and M ethods .. ..................................................................... ................ 52
C adav er Selection ........................................... .. ...................... .. ...... ............ 52
In-Vivo Computed Tomography Scanning ....................................................52
Bone Harvesting and Ex-Vivo Computed Tomography Scanning .................. 53
Image Segmentation of Spongiosa and Cortical Bone Regions.......................54
Micro-Computed Tomography of Trabecular Spongiosa ...............................55
Voxel-Based Infinite Spongiosa Transport (VBIST) Model.............................. 55
Paired-Image Radiation Transport (PIRT) Model.........................................56
R e su lts.................. .......................... .... ....................................................... ........ 6 0
Absorbed Fractions to Active Marrow within the Pelvis...............................60
Absorbed Fractions to Active Marrow within the Cranium...............................61
Absorbed Fractions to Active Marrow within the Rib Cage...............................62
Absorbed Fractions to Endosteal Tissues....................................... ................ 64
D isc u ssio n ............................................................................................................... ... 6 5
C o n c lu sio n ............................................................................................................... .. 6 6

5 SKELETAL CHORD-LENGTH DISTRIBUTIONS FOR ICRP REFERENCE
MALE VERSUS THE UF REFERENCE MALE CANCER PATIENT................ 82

In tro d u ctio n ............................................................................................................... .. 8 2
Current Reference M ale Skeletal M odel ........................................ ................ 82
University of Florida Reference Male Cancer Patient .................................... 84
M materials and M ethods .. ..................................................................... ................ 85
B one Specim en Selection ............................................................... ................ 85
M icroim aging of Trabecular Spongiosa......................................... ................ 86
Measurement of Chord-Length Distributions ...............................................87
Averaging of Chord-Length Distributions .............. ....................................88
R reference Skeletal Sites ....................................... ....................... ................ 88
R e su lts....................................................................................................... ....... .. 8 9
D isc u ssio n ............................................................................................................... ... 9 1
Fem oral H ead and N eck ................. .......................................................... 91
Cervical and Lum bar V ertebrae ....................... .......................................... 92
R ib s ...................................................................................................... ........ .. 9 3
C ra n iu m ............................................................................................................... 9 4
Pelvis (O s Coxae) ............................................................. ..... ........................ ... 95
Remaining Marrow-Containing Bones of the Skeleton ................................. 95
Weighting Schemes for Non-Imaged Bone Sites in the Leeds Data................96
C o n c lu sio n ............................................................................................................... .. 9 8









6 CHORD-BASED VERSUS VOXEL-BASED METHODS OF ELECTRON
TRANSPORT IN SKELETAL DOSIMETRY ............................. ..................... 122

In tro d u ctio n ............................................................................................................... 12 2
M materials and M ethods ................... .............................................................. 124
Cadaver Selection .......................................................................... 124
Trabecular M icrostructure A acquisition ........................................ ................ 125
Voxel-Based Infinite Spongiosa Transport (VBIST) Model.......................... 126
Chord-Based Infinite Spongiosa Transport (CBIST) Model.......................... 127
Chord-Length Distributions for the UF Reference Cancer Patient .................130
Convergence Limits for Absorbed Fractions under CBIST and VBIST........ 131
R results and D discussion .... .............. .. ... ............................ .. .. .. ..... .. .. .... ............ .. 133
Trabecular Microstructure of the Leeds and UF Reference Subjects ..............133
Electron Dosimetry Comparisons between the UF and Leeds
M icrostructures .................................................. .. .... .. ........ .. ............. .. 135
Comparison of CBIST and VBIST for Marrow Space Targets ......................138
Comparison of CBIST and VBIST for Bone Endosteum Targets ..................140
C o n c lu sio n ............................................................................................................... 14 1

7 REFERENCE SKELETAL DOSIMETRY MODEL FOR AN ADULT MALE
RADIONUCLIDE THERAPY PATIENT......................................166

In tro d u ctio n .............................................................................................................. 16 6
M materials and M methods .............. ..................................................... 170
Reference Adult M ale Cadaver Selection ....... ... .................................... 171
Skeletal-Image Database for the UF RMCP ......................... .................. 171
R radiation Transport M odeling ..................................................... ................ 173
Paired-Image Radiation Transport (PIRT) Model.................. .................. 175
M ass C alculation for U F R M CP .......................................................................177
Skeletal Averaging of Absorbed Fractions and S Values for the UF RMCP.... 178
R e su lts..................................................................................................... .......... 18 1
D discussion ................................................ .... ... ...... ...... ............ ............... 182
Comparison of UF and ICRP Reference Tissue Masses.................................182
PIRT Model Simulations Sacrum....................................... 185
Energy Loss to Cortical Bone...... ........ ...... ..................... 186
Skeletal-Averaged Absorbed Fractions ....................................... ............... 187
Site-Specific Radionuclide S Values........................................ 190
Skeletal-A averaged S V alues ......................................................... ............... 192
Scalability of the UF Reference Skeletal Model .................... ...................194
C o n c lu sio n ............................................................................................................... 1 9 6

8 CONCLUSIONS AND FUTURE WORK........................................219

C o n c lu sio n s ............................................................................................................... 2 1 9
Future Work.................. ..... ............................... 222
Improvements in the use of Voxel M odels...... .... ................................... 222
Improvements in the Characterization of Active Marrow...............................223









Im provem ents to the Skeletal D database ....................................... ................ 224
Scaling of Reference S Values to a Patient ....... ... ................................... 225
Clinical Application of Reference S Values...... ................... ................... 226

APPENDIX

A UNIVERSITY OF FLORIDA REFERENCE ADULT MALE
RADIONUCLIDE PATIENT: CT IMAGE DETAILS ................ ...................228

B MICROIMAGE PARAMETERS FOR SKELETAL SPONGIOSA.....................236

C IM AGE PROCESSING (C PROGRAM S) ................................... ..................... 238

D IMAGE PROCESSING TECHNIQUE........................................251

E PAIRED-IMAGE RADIATION TRANSPORT (PIRT) MODEL (EGSNRC
U SE R C O D E ) ........................................................................................................... 2 7 3

F PIRT MODEL FOR THE PROXIMAL FEMUR (EGSNRC USER CODE) ..........316

G 66-YEAR UF REFERENCE MALE CANCER PATIENT CHORD-LENGTH
DISTRIBUTIONS AND THE TRILINEAR CHORD-LENGTH
C A L C U L A T IO N ...................................................................................................... 3 6 0

H TABLES OF SITE-SPECIFIC ABSORBED FRACTIONS FOR THE UF
REFEREN CE M ALE .................... ................................................................ 387

I TABLES OF SITE-SPECIFIC S VALUES FOR THE UF
REFEREN CE M ALE .................... ................................................................ 425

REFERENCES ............ ................... .. ........... .......................................450

BIOGRAPHICAL SKETCH ..................................................... 459















LIST OF TABLES


Table page

3-1 Tissue compositions (% by mass) and mass densities used in either the IST
and PIRT m odels of skeletal dosim etry ........................................ ................ 45

3-2 Tissues masses used in the paired-image radiation transport (PIRT) model
(100% m arrow cellularity) ............................................................. ................ 46

3-3 Ratio of the radionuclide S value for an active marrow (TAM) target as given
by the infinite spongiosa transport (IST) model to that given by the paired-
im age radiation transport (PIRT) m odel ....................................... ................ 47

4-1 Tissues masses used in the paired-image radiation transport (PIRT) model ......80

4-2 Ratio of the radionuclide S value for an active marrow (TAM) target as given
by the voxel-based infinite spongiosa transport (VBIST) model to that given
by the paired-image radiation transport (PIRT) model..................................81

5-1 Comparison of measured mean chord lengths with values published from
the University of Leeds (W hitwell 1973). ..................................... 121

6-1 Mean values of trabecular and marrow cavity chord-lengths as given by the
present UF study and those published from the University of Leeds............ 163

6-2 Ratios of absorbed fractions to active marrow (UF values to Leeds values
u n d er C B IS T ).................................................................................................... 164

6-3 Ratios of absorbed fractions to bone endosteum (UF values to Leeds values
u n d er C B IS T ).................................................................................................... 16 5

7-1 Comparison between reference masses used at UF (present study) and
ICRP 89 for tissues within the marrow cavities..................... ................... 211

7-2 Pertinent values used in the calculation of data for the PIRT model of
skeletal dosim etry ............. ............... ................................................ 2 12

7-3 Measurements of spongiosa volume and cortical bone volume within each
of the five (5) lumbar vertebrae of the UF Reference Male Cancer Patient.....213









7-4 Surface-to-volume ratios within skeletal sites of trabecular spongiosa found
within the UF 66-year male, the Leeds 44-year male..................................214

7-5 Comparison of assigned cellularity factors and fractions of different skeletal
components between UF (present study) and Eckerman and Stabin (2000) ....215

7-6. Skeletal-averaged absorbed fractions for monoenergetic electrons for the UF
Reference Male Cancer Patient and those of the ICRP Reference Man.........216

7-7 Radiation characteristics of radionuclides used for calculation of S values.....217

7-8 Skeletal-averaged S values (mGy/MBq-s) for different combinations of
source and target regions within the spongiosa and cortical bone ..................218

A-i Ex-vivo CT parameters used in the PIRT model for radiation transport in
order to define the binary contours of the macroimage at each skeletal site. ...235

B-i MicroCT image parameters used in the PIRT model in order to define the
microimage of spongiosa in radiation transport..................... ................... 237

D-1 Breakdown of dimensions necessary for opening images in Adobe ..............257

D-2 Example of the method for determination of the region of interest................264

G-1 Normalized 3D chord-length distributions through the bone trabeculae of the
left and right femur heads, left and right femur necks. First 50 bins.............361

G-2 Normalized 3D chord-length distributions through the bone trabeculae of
the left and right femur heads, left and right femur necks. Second 50 bins.....362

G-3 Normalized 3D chord-length distributions through the bone trabeculae of the
pubis, ilium, ischium, right and left scapula, sternum, and the averages for
the os coxae and scapula. First 50 bins....... ... ....................................... 363

G-4 Normalized 3D chord-length distributions through the bone trabeculae of the
pubis, ilium, ischium, right and left scapula, sternum, and the averages for
the os coxae and scapula. Second 50 bins....... ... .................................... 364

G-5 Normalized 3D chord-length distributions through the bone trabeculae of the
right and left clavicle, right and left humerus, C3 and C6 vertebra, and the
averages for those respective bone sites First 50 bins .............. ................... 365

G-6 Normalized 3D chord-length distributions through the bone trabeculae of the
right and left clavicle, right and left humerus, C3 and C6 vertebra, and the
averages for those respective bone sites. Second 50 bins..............................366









G-7 Normalized 3D chord-length distributions through the bone trabeculae of the
T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages
for the thoracic and lumbar vertebra. First 50 bins................ ................... 367

G-8 Normalized 3D chord-length distributions through the bone trabeculae of the
T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages
for the thoracic and lumbar vertebra. Second 50 bins. ...............................368

G-9 Normalized 3D chord-length distributions through the bone trabeculae of the
right and left upper rib, middle rib, and lower rib. Also shown is the average
for the single rib. First 50 bins. ...... ... ... ...................... 369

G-10 Normalized 3D chord-length distributions through the bone trabeculae of the
right and left upper, middle, and lower ribs. Also shown is the average for
the single rib. Second 50 bins. ...... ....... ...... ...................... 370

G- 11 Normalized 3D chord-length distributions through the bone trabeculae of the
mandible, frontal bone, occipital bone, and right and left parietal bones.
Also shown is the average for the cranium. First 50 bins..............................371

G-12 Normalized 3D chord-length distributions through the bone trabeculae of the
mandible, frontal bone, occipital bone, and right and left parietal bones.
Also shown is the average for the cranium. Second 50 bins. .........................372

G-13 Normalized 3D chord-length distributions through the marrow cavities of the
left and right femur heads, left and right femur necks, and their respective
av erages. F first 50 bin s..................................... .. ........ .................. ............... 373

G-14 Normalized 3D chord-length distributions through the marrow cavities of the
left and right femur heads, left and right femur necks, and their respective
averages. Second 50 bins ....... .......... ............ ...................... 374

G-15 Normalized 3D chord-length distributions through the marrow cavities of the
pubis, ilium, ischium, right and left scapula, sternum, and the averages for
the os coxae and scapula. First 50 bins. ...... ... ....................................... 375

G-16 Normalized 3D chord-length distributions through the marrow cavities of the
pubis, ilium, ischium, right and left scapula, sternum, and the averages for
the os coxae and scapula. Second 50 bins....... ... .................................... 376

G-17 Normalized 3D chord-length distributions through the marrow cavities of the
right and left clavicle, right and left humerus, C3 and C6 vertebra, and the
averages for those respective bone sites. First 50 bins ............................... 377

G-18 Normalized 3D chord-length distributions through the marrow cavities of the
right and left clavicle, right and left humerus, C3 and C6 vertebra, and the
averages for those respective bone sites. Second 50 bins..............................378









G-19 Normalized 3D chord-length distributions through the marrow cavities of the
T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages
for the thoracic and lumbar vertebra. First 50 bins................ ................... 379

G-20 Normalized 3D chord-length distributions through the marrow cavities of the
T3, T6, and T11 vertebrae; L2 and L4 vertebrae; sacrum; and the averages
for the thoracic and lumbar vertebra. Second 50 bins. ...............................380

G-21 Normalized 3D chord-length distributions through the marrow cavities of the
right and left upper rib, middle rib, and lower rib. Also shown is the average
for the single rib. First 50 bins. ...... ... ... ...................... 381

G-22 Normalized 3D chord-length distributions through the marrow cavities of the
right and left upper, middle, and lower ribs. Also shown is the average for
the single rib. Second 50 bins. ...... ....... ...... ...................... 382

G-23 Normalized 3D chord-length distributions through the marrow cavities of the
mandible, frontal bone, occipital bone, and right and left parietal bones.
Also shown is the average for the cranium. First 50 bins..............................383

G-24 Normalized 3D chord-length distributions through the marrow cavities of
the mandible, frontal bone, occipital bone, and right and left parietal
bones. Second 50 bins..................................... .. .......... ............ .. ........ .... .... 384

H-1. Absorbed fractions in the right proximal femur for sources in the trabecular
bone volume at varying marrow cellularity....... .................. ................... 388

H-2 Absorbed fractions in the left proximal femur for sources in the trabecular
bone volume at varying marrow cellularity....... .................. ................... 388

H-3 Absorbed fractions in the right humerus for sources in the trabecular bone
volum e at varying marrow cellularity....... ... ....................................... 388

H-4 Absorbed fractions in the left humerus for sources in the trabecular bone
volum e at varying m arrow cellularity....... ... ......................................3... 89

H-5 Absorbed fractions in the cervical vertebra for sources in the trabecular
bone volume at varying marrow cellularity....... .................. ................... 389

H-6 Absorbed fractions in the thoracic vertebra for sources in the trabecular
bone volume at varying marrow cellularity....... .................. ................... 389

H-7 Absorbed fractions in the lumbar vertebra for sources in the trabecular bone
volum e at varying marrow cellularity....... ... ....................................... 390

H-8 Absorbed fractions in the sacrum for sources in the trabecular bone volume
at varying m arrow cellularity ....... ...... ..... ...................... 390









H-9 Absorbed fractions in the os coxae for sources in the trabecular bone volume
at varying m arrow cellularity ...... ....... ..... ...................... 390

H-10 Absorbed fractions in the cranium for sources in the trabecular bone volume
at varying m arrow cellularity ...... .......... ....... ...................... 391

H-11 Absorbed fractions in the mandible for sources in the trabecular bone
volum e at varying marrow cellularity...... .... ....................................... 391

H-12 Absorbed fractions in the ribs for sources in the trabecular bone volume at
varying m arrow cellularity...... ............. ............ ...................... 391

H-13 Absorbed fractions in the sternum for sources in the trabecular bone
volum e at varying marrow cellularity...... .... ....................................... 392

H-14 Absorbed fractions in the right clavicle for sources in the trabecular bone
volum e at varying marrow cellularity....... ... ....................................... 392

H-15 Absorbed fractions in the left clavicle for sources in the trabecular bone
volum e at varying marrow cellularity....... ... ....................................... 392

H-16 Absorbed fractions in the right scapula for sources in the trabecular bone
volum e at varying marrow cellularity...... .... ....................................... 393

H-17 Absorbed fractions in the left scapula for sources in the trabecular bone
volum e at varying marrow cellularity...... .... ....................................... 393

H-18 Absorbed fractions in the femur head for sources in the trabecular bone
volum e at varying marrow cellularity...... .... ....................................... 393

H-19 Absorbed fractions in the right proximal femur for sources in the trabecular
active marrow at varying marrow cellularity...... ................... ................... 394

H-20 Absorbed fractions in the left proximal femur for sources in the trabecular
active marrow at varying marrow cellularity......................... ................... 394

H-21 Absorbed fractions in the right humerus for sources in the trabecular active
m arrow at varying m arrow cellularity. ........................ ................................. 394

H-22 Absorbed fractions in the left humerus for sources in the trabecular active
m arrow at varying m arrow cellularity. ........................ ................................. 395

H-23 Absorbed fractions in the cervical vertebra for sources in the trabecular
active marrow at varying marrow cellularity...... ................... ................... 395

H-24 Absorbed fractions in the thoracic vertebra for sources in the trabecular
active marrow at varying marrow cellularity...... ................... ................... 395









H-25 Absorbed fractions in the lumbar vertebra for sources in the trabecular
active marrow at varying marrow cellularity...... ................... ................... 396

H-26 Absorbed fractions in the sacrum for sources in the trabecular active marrow
at varying m arrow cellularity ...... ........ ..... ...................... 396

H-27 Absorbed fractions in the os coxae for sources in the trabecular active
m arrow at varying m arrow cellularity. ......................................... ................ 396

H-28 Absorbed fractions in the cranium for sources in the trabecular active
m arrow at varying m arrow cellularity. ......................................... ................ 397

H-29 Absorbed fractions in the mandible for sources in the trabecular active
m arrow at varying m arrow cellularity. ......................................... ................ 397

H-30 Absorbed fractions in the ribs for sources in the trabecular active marrow
at varying m arrow cellularity ...... ........ ..... ...................... 397

H-31 Absorbed fractions in the sternum for sources in the trabecular active
marrow at varying marrow cellularity .............................398

H-32 Absorbed fractions in the right clavicle for sources in the trabecular active
marrow at varying marrow cellularity .............................398

H-33 Absorbed fractions in the left clavicle for sources in the trabecular active
marrow at varying marrow cellularity .............................398

H-34 Absorbed fractions in the right scapula for sources in the trabecular active
marrow at varying marrow cellularity .............................399

H-35 Absorbed fractions in the left scapula for sources in the trabecular active
marrow at varying marrow cellularity .............................399

H-36 Absorbed fractions in the femur head for sources in the trabecular active
marrow at varying marrow cellularity .............................399

H-37 Absorbed fractions in the right proximal femur for sources in the trabecular
bone surface at varying marrow cellularity. ..................................400

H-38 Absorbed fractions in the left proximal femur for sources in the trabecular
bone surface at varying marrow cellularity. ..................................400

H-39 Absorbed fractions in the right humerus for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 400

H-40 Absorbed fractions in the left humerus for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 401









H-41 Absorbed fractions in the cervical vertebra for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 401

H-42 Absorbed fractions in the thoracic vertebra for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 401

H-43 Absorbed fractions in the lumbar vertebra for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 402

H-44 Absorbed fractions in the sacrum for sources in the trabecular bone surface
at varying m arrow cellularity ...... ........ ..... ...................... 402

H-45 Absorbed fractions in the os coxae for sources in the trabecular bone surface
at varying m arrow cellularity ...... ........ ..... ...................... 402

H-46 Absorbed fractions in the cranium for sources in the trabecular bone surface
at varying m arrow cellularity ...... ....... ..... ...................... 403

H-47 Absorbed fractions in the mandible for sources in the trabecular bone surface
at varying m arrow cellularity ...... ....... ..... ...................... 403

H-48 Absorbed fractions in the ribs for sources in the trabecular bone surface at
varying m arrow cellularity...... ............. ............ ...................... 403

H-49 Absorbed fractions in the sternum for sources in the trabecular bone surface
at varying m arrow cellularity ...... ........ ..... ...................... 404

H-50 Absorbed fractions in the right clavicle for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 404

H-51 Absorbed fractions in the left clavicle for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 404

H-52 Absorbed fractions in the right scapula for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 405

H-53 Absorbed fractions in the left scapula for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 405

H-54 Absorbed fractions in the femur head for sources in the trabecular bone
surface at varying marrow cellularity .............................................. 405

H-55 Absorbed fractions in the right proximal femur for sources in the trabecular
bone endosteum at varying marrow cellularity...................... ................... 406

H-56 Absorbed fractions in the left proximal femur for sources in the trabecular
bone endosteum at varying marrow cellularity...................... ................... 406









H-57 Absorbed fractions in the right humerus for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 406

H-58 Absorbed fractions in the left humerus for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 407

H-59 Absorbed fractions in the cervical vertebra for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 407

H-60 Absorbed fractions in the thoracic vertebra for sources in the trabecular
bone endosteum at varying marrow cellularity...................... ................... 407

H-61 Absorbed fractions in the lumbar vertebra for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 408

H-62 Absorbed fractions in the sacrum for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 408

H-63 Absorbed fractions in the os coxae for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 408

H-64 Absorbed fractions in the cranium for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 409

H-65 Absorbed fractions in the mandible for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 409

H-66 Absorbed fractions in the ribs for sources in the trabecular bone endosteum
at varying m arrow cellularity ...... ........ ..... ...................... 409

H-67 Absorbed fractions in the sternum for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 410

H-68 Absorbed fractions in the right clavicle for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 410

H-69 Absorbed fractions in the left clavicle for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 410

H-70 Absorbed fractions in the right scapula for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 411

H-71 Absorbed fractions in the left scapula for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 411

H-72 Absorbed fractions in the femur head for sources in the trabecular bone
endosteum at varying marrow cellularity. ..................................... 411









H-73 Absorbed fractions in the right proximal femur for sources in the cortical
bone volume at varying marrow cellularity....... .................. ................... 412

H-74 Absorbed fractions in the left proximal femur for sources in the cortical
bone volume at varying marrow cellularity....... .................. ................... 412

H-75 Absorbed fractions in the right humerus for sources in the cortical bone
volum e at varying marrow cellularity...... .... ....................................... 412

H-76 Absorbed fractions in the left humerus for sources in the cortical bone
volum e at varying m arrow cellularity...... .... ....................................... 413

H-77 Absorbed fractions in the cervical vertebra for sources in the cortical bone
volum e at varying m arrow cellularity...... .... ....................................... 413

H-78 Absorbed fractions in the thoracic vertebra for sources in the cortical bone
volum e at varying m arrow cellularity...... .... ....................................... 413

H-79 Absorbed fractions in the lumbar vertebra for sources in the cortical bone
volum e at varying marrow cellularity....... ... ....................................... 414

H-80 Absorbed fractions in the sacrum for sources in the cortical bone volume at
varying m arrow cellularity...... ............. ............ ...................... 414

H-81 Absorbed fractions in the os coxae for sources in the cortical bone volume
at varying m arrow cellularity ....... ...... ..... ...................... 414

H-82 Absorbed fractions in the cranium for sources in the cortical bone volume
at varying m arrow cellularity ...... ........ ..... ...................... 415

H-83 Absorbed fractions in the mandible for sources in the cortical bone volume
at varying m arrow cellularity ...... ........ ..... ...................... 415

H-84 Absorbed fractions in the ribs for sources in the cortical bone volume at
varying m arrow cellularity...... ............. ............ ...................... 415

H-85 Absorbed fractions in the sternum for sources in the cortical bone volume at
varying m arrow cellularity...... ............. ............ ...................... 416

H-86 Absorbed fractions in the right clavicle for sources in the cortical bone
volum e at varying marrow cellularity....... ... ....................................... 416

H-87 Absorbed fractions in the left clavicle for sources in the cortical bone
volum e at varying marrow cellularity....... ... ....................................... 416

H-88 Absorbed fractions in the right scapula for sources in the cortical bone
volum e at varying marrow cellularity....... ... ....................................... 417









H-89 Absorbed fractions in the left scapula for sources in the cortical bone
volum e at varying marrow cellularity....... ... ....................................... 417

H-90 Absorbed fractions in the femur head for sources in the cortical bone
volum e at varying marrow cellularity...... .... ....................................... 417

H-91 Absorbed fractions in the right proximal femur for sources in the trabecular
marrow cavity at varying marrow cellularity...... .................. ................... 418

H-92 Absorbed fractions in the left proximal femur for sources in the trabecular
marrow cavity at varying marrow cellularity...... .................. ................... 418

H-93 Absorbed fractions in the right humerus for sources in the trabecular
marrow cavity at varying marrow cellularity...... .................. ................... 418

H-94 Absorbed fractions in the left humerus for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 419

H-95 Absorbed fractions in the cervical vertebra for sources in the trabecular
marrow cavity at varying marrow cellularity...... .................. ................... 419

H-96 Absorbed fractions in the thoracic vertebra for sources in the trabecular
marrow cavity at varying marrow cellularity...... ................... ................... 419

H-97 Absorbed fractions in the lumbar vertebra for sources in the trabecular
marrow cavity at varying marrow cellularity...... .................. ...................420

H-98 Absorbed fractions in the sacrum for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 420

H-99 Absorbed fractions in the os coxae for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 420

H-100 Absorbed fractions in the cranium for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 421

H-101 Absorbed fractions in the mandible for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 421

H-102 Absorbed fractions in the ribs for sources in the trabecular marrow cavity at
varying m arrow cellularity...... ............. ............ ...................... 421

H-103 Absorbed fractions in the sternum for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 422

H-104 Absorbed fractions in the right clavicle for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 422


xviii









H-105 Absorbed fractions in the left clavicle for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 422

H-106 Absorbed fractions in the right scapula for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 423

H-107 Absorbed fractions in the left scapula for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 423

H-108 Absorbed fractions in the femur head for sources in the trabecular marrow
cavity at varying m arrow cellularity. ...... .......... ........................................ 423

H-109 Absorbed fractions in all skeletal sites for any source in the spongiosa (TAM,
TBV, TBS, TBE, or TMS) irradiating the cortical bone volume (CBV)..........424

H-1 10 Absorbed fractions in all skeletal sites for sources and targets in the cortical
bone volume (CBV self-irridiation) ............... .........................424

I-1 S values in the right proximal femur for sources in the trabecular bone
volume at varying marrow cellularity for 10 radionuclides............................426

1-2 S values in the left proximal femur for sources in the trabecular bone
volume at varying marrow cellularity for 10 radionuclides............................426

1-3 S values in the right humerus for sources in the trabecular bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 426

1-4 S values in the left humerus for sources in the trabecular bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 427

1-5 S values in the cervical vertebra for sources in the trabecular bone volume
at varying marrow cellularity for 10 radionuclides................ ...................427

1-6 S values in the thoracic vertebra for sources in the trabecular bone volume
at varying marrow cellularity for 10 radionuclides................ ...................427

1-7 S values in the lumbar vertebra for sources in the trabecular bone volume
at varying marrow cellularity for 10 radionuclides................ ...................428

1-8 S values in the sacrum for sources in the trabecular bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 428

1-9 S values in the os coxae for sources in the trabecular bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 428

1-10 S values in the cranium for sources in the trabecular bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 429









I-11 S values in the mandible for sources in the trabecular bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 429

1-12 S values in the ribs for sources in the trabecular bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 429

1-13 S values in the sternum for sources in the trabecular bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 430

1-14 S values in the right clavicle for sources in the trabecular bone volume at
varying marrow cellularity for 10 radionuclides. ........................ ................ 430

1-15 S values in the left clavicle for sources in the trabecular bone volume at
varying marrow cellularity for 10 radionuclides. ........................ ................ 430

1-16 S values in the right scapula for sources in the trabecular bone volume at
varying marrow cellularity for 10 radionuclides. ........................ ................ 431

1-17 S values in the left scapula for sources in the trabecular bone volume at
varying marrow cellularity for 10 radionuclides. ........................ ................ 431

1-18 S values in the femur head for sources in the trabecular bone volume at
varying marrow cellularity for 10 radionuclides. ........................ ................ 431

1-19 S values in the right proximal femur for sources in the trabecular active
marrow at varying marrow cellularity for 10 radionuclides .........................432

1-20 S values in the left proximal femur for sources in the trabecular active
marrow at varying marrow cellularity for 10 radionuclides .........................432

1-21 S values in the right humerus for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. ........................ ................ 432

1-22 S values in the left humerus for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. ........................ ................ 433

1-23 S values in the cervical vertebra for sources in the trabecular active marrow
at varying marrow cellularity for 10 radionuclides................ ...................433

1-24 S values in the thoracic vertebra for sources in the trabecular active marrow
at varying marrow cellularity for 10 radionuclides................ ...................433

1-25 S values in the lumbar vertebra for sources in the trabecular active marrow
at varying marrow cellularity for 10 radionuclides................ ...................434

1-26 S values in the sacrum for sources in the trabecular active marrow at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 434









1-27 S values in the os coxae for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 434

1-28 S values in the cranium for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 435

1-29 S values in the mandible for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 435

1-30 S values in the ribs for sources in the trabecular active marrow at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 435

1-31 S values in the sternum for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. ...................... ................ 436

1-32 S values in the right clavicle for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. ...................... ................ 436

1-33 S values in the left clavicle for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. ...................... ................ 436

1-34 S values in the right scapula for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. ...................... ................ 437

1-35 S values in the left scapula for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. ...................... ................ 437

1-36 S values in the femur head for sources in the trabecular active marrow at
varying marrow cellularity for 10 radionuclides. ...................... ................ 437

1-37 S values in the right proximal femur for sources in the trabecular bone
surface at varying marrow cellularity for 10 radionuclides............................438

1-38 S values in the left proximal femur for sources in the trabecular bone
surface at varying marrow cellularity for 10 radionuclides............................438

1-39 S values in the right humerus for sources in the trabecular bone surface at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 438

1-40 S values in the left humerus for sources in the trabecular bone surface at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 439

1-41 S values in the cervical vertebra for sources in the trabecular bone surface
at varying marrow cellularity for 10 radionuclides................ ...................439

1-42 S values in the thoracic vertebra for sources in the trabecular bone surface
at varying marrow cellularity for 10 radionuclides................ ...................439









1-43 S values in the lumbar vertebra for sources in the trabecular bone surface at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 440

1-44 S values in the sacrum for sources in the trabecular bone surface at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 440

1-45 S values in the os coxae for sources in the trabecular bone surface at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 440

1-46 S values in the cranium for sources in the trabecular bone surface at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 441

1-47 S values in the mandible for sources in the trabecular bone surface at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 441

1-48 S values in the ribs for sources in the trabecular bone surface at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 441

1-49 S values in the sternum for sources in the trabecular bone surface at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 442

1-50 S values in the right clavicle for sources in the trabecular bone surface at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 442

1-51 S values in the left clavicle for sources in the trabecular bone surface at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 442

1-52 S values in the right scapula for sources in the trabecular bone surface at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 443

1-53 S values in the left scapula for sources in the trabecular bone surface at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 443

1-54 S values in the femur head for sources in the trabecular bone surface at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 443

1-55 S values in the right proximal femur for sources in the cortical bone volume
at varying marrow cellularity for 10 radionuclides................ ...................444

1-56 S values in the left proximal femur for sources in the cortical bone volume
at varying marrow cellularity for 10 radionuclides................ ...................444

1-57 S values in the right humerus for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 444

1-58 S values in the left humerus for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 445









1-59 S values in the cervical vertebra for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 445

1-60 S values in the thoracic vertebra for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 445

1-61 S values in the lumbar vertebra for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 446

1-62 S values in the sacrum for sources in the cortical bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 446

1-63 S values in the os coxae for sources in the cortical bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 446

1-64 S values in the cranium for sources in the cortical bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 447

1-65 S values in the mandible for sources in the cortical bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 447

1-66 S values in the ribs for sources in the cortical bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 447

1-67 S values in the sternum for sources in the cortical bone volume at varying
m arrow cellularity for 10 radionuclides...... .... .................. ................... 448

1-68 S values in the right clavicle for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 448

1-69 S values in the left clavicle for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 448

1-70 S values in the right scapula for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 449

1-71 S values in the left scapula for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 449

1-72 S values in the femur head for sources in the cortical bone volume at
varying marrow cellularity for 10 radionuclides. .................. ...... ............ 449


xxiii















LIST OF FIGURES


Figure page

2-1. Vertebral body showing the different types of bone tissue in one particular
skeletal site. Adapted from a study by Fagerburg and Lafferty (1998) ............... 15

2-2. Microstructure of compact and cancellous bone. Illustration includes entire
osteon know n as the H aversian system ............................................... ............... 16

3-1. Schematic of the PIRT model constructed for the L4 vertebra..............................37

3-2. Schematic of the PIRT model constructed for the right proximal femur...............38

3-3. Electron absorbed fractions to active bone marrow within the L4 vertebrae for
three source tissues correspond to 100% marrow cellularity............................. 39

3-4. Electron absorbed fractions to active bone marrow within the L4 vertebrae at
reference cellularity for three source tissues....................................... ................ 40

3-5. Electron absorbed fractions to active bone marrow within the proximal femur for
three source tissues correspond to 100% marrow cellularity..............................41

3-6. Electron absorbed fractions to active bone marrow within the proximal femur at
reference cellularity for three source tissues....................................... ................ 42

3-7. Electron absorbed fractions to the trabecular bone endosteum within the L4
vertebra for three source tissues TAM, TBV, and TBS ........................................43

3-8. Electron absorbed fractions to the trabecular bone endosteum within the proximal
femur for three source tissues TAM, TBV, and TBS ................. ..................... 44

4-1. Schematic of the PIRT model constructed for the pelvis (os coxae).......................68

4-2. Schematic of the PIRT model constructed for the cranium...................................69

4-3. Schematic of the PIRT model constructed for the ribs..........................................70

4-4. Electron absorbed fractions to active bone marrow within the os coxae at 100%
marrow cellularity for three source tissues TAM, TBV, and TBS ....................71

4-5. Electron absorbed fractions to active bone marrow within the os coxae at reference
cellularity for three source tissues TAM, TBV, and TBS..............................72


xxiv









4-6. Electron absorbed fractions to active bone marrow within the cranium at 100%
marrow cellularity for three source tissues TAM, TBV, and TBS. ....................73

4-7. Electron absorbed fractions to active bone marrow within the cranium at reference
cellularity for three source tissues TAM, TBV, and TBS...............................74

4-8. Electron absorbed fractions to active bone marrow within the ribs at 100% marrow
cellularity for three source tissues TAM, TBV, and TBS...............................75

4-9. Electron absorbed fractions to active bone marrow within the ribs at reference
cellularity for three source tissues TAM, TBV, and TBS...............................76

4-10. Electron absorbed fractions to the trabecular bone endosteum within the os coxae
for three source tissues TAM TBV, and TBS ................................. ................ 77

4-11. Electron absorbed fractions to the trabecular bone endosteum within the cranium
for three source tissues TAM TBV, and TBS ................................. ................ 78

4-12. Electron absorbed fractions to the trabecular bone endosteum within the ribs for
three source tissues TAM TBV, and TBS....................................... ................ 79

5-1. Schematic demonstrating the acquisition of chord-lengths across bone trabeculae
and marrow cavities at scanning angle (p in a single transverse plane of a 3D
m icroC T digital im age .................. ............................................................. 100

5-2. Normalized, omnidirectional chord-length distributions through the marrow
cavities of the femoral head and neck as measured with physical sectioning and
autom ated light m icroscopy ........................................................ 101

5-3. Normalized, omnidirectional chord-length distributions through the bone
trabeculae of the femoral head and neck as measured with physical sectioning
and automated light microscopy ....... ....... ........ ..................... 102

5-4. Chord-length distributions through marrow cavities of the cervical and lumbar
v e rte b ra .............................................................................................................. ... 1 0 3

5-5. Chord-length distributions through bone trabeculae of the cervical and lumbar
v e rte b ra .............................................................................................................. ... 1 0 4

5-6. Chord-length distributions through marrow cavities of the ribs............................ 105

5-7. Chord-length distributions through bone trabeculae of the ribs............................ 106

5-8. Chord-length distributions through marrow cavities of the cranium. Values for
individual bones of the cranium in the UF male subject as shown as well..........107

5-9. Chord-length distributions through bone trabeculae of the cranium. Values for
individual bones of the cranium in the UF male subject as shown as well..........108









5-10. Chord-length distributions through marrow cavities of the pelvis (os coxae). ......109

5-11. Chord-length distributions through bone trabeculae of the pelvis (os coxae)........110

5-12. Chord-length distributions through marrow cavities of the scapula, clavicle,
and hum erus in the U F m ale subject .................................................................. 111

5-13. Chord-length distributions through bone trabeculae of the scapula, clavicle,
and humerus in the UF male subject. ...... ... .........................1... 12

5-14. Chord-length distributions through marrow cavities of the sacrum, sternum,
and m andible in the UF m ale subject. ...... ... ..........................................1... 13

5-15. Chord-length distributions through bone trabeculae of the sacrum, sternum,
and m andible in the UF m ale subject. ...... ... ..........................................1... 14

5-16. Chord-length distributions through marrow cavities of the thoracic vertebra........ 115

5-17. Chord-length distributions through bone trabeculae of the thoracic vertebra........ 116

5-18. Chord-length distributions through marrow cavities of the sacrum.................... 117

5-19. Chord-length distributions through bone trabeculae of the sacrum .....................118

5-20. Chord-length distributions through marrow cavities of the humerus...................119

5-21. Chord-length distributions through bone trabeculae of the humerus .................. 120

6-1. Normalized, omnidirectional chord-length distributions through the marrow
cavities of the Leeds 44-year reference male. .....................................144

6-2. Normalized, omnidirectional chord-length distributions through the marrow
cavities of the UF 66-year reference male cancer patient. ................................145

6-3. Normalized, omnidirectional chord-length distributions through the bone
trabeculae of the Leeds 44-year reference male.......................... ................... 146

6-4. Normalized, omnidirectional chord-length distributions through the bone
trabeculae of the UF 66-year reference male cancer patient...............................147

6-5. Electron absorbed fractions to the active bone marrow within the parietal bone for
three source tissues TAM TBV, and TBS....... ... ..................................... 148

6-6. Electron absorbed fractions to the bone endosteum within the parietal bone for
three source tissues TAM TBV, and TBS....... ... ...................................... 149

6-7. Electron absorbed fractions to the active bone marrow within the femoral head
for three source tissues TAM TBV, and TBS ......................... ................... 150


xxvi









6-8. Electron absorbed fractions to the bone endosteum within the femoral head for
three source tissues TAM TBV, and TBS....... ... ..................................... 151

6-9. Electron absorbed fractions to the active bone marrow within the ribs for three
source tissues TAM TBV, and TBS....... ... .......................................... 152

6-10. Electron absorbed fractions to the bone endosteum within the ribs for three
source tissues TAM TBV, and TBS....... ... .......................................... 153

6-11. Electron absorbed fractions to the active bone marrow within the femoral head
of the UF reference male cancer patient.............. ......................... 154

6-12. Electron absorbed fractions to the active bone marrow within the femoral neck
of the UF reference male cancer patient.............. ........................ 155

6-13. Electron absorbed fractions to the active bone marrow within the L4 lumbar
vertebra of the UF reference male cancer patient...... .................. .................. 156

6-14. Electron absorbed fractions to the active bone marrow within the C6 cervical
vertebra of the UF reference male cancer patient...... .................. ................... 157

6-15. Electron absorbed fractions to the active bone marrow within the ilium of the
UF reference m ale cancer patient...... .......... ....... ..................... 158

6-16. Electron absorbed fractions to the active bone marrow within the parietal bone
of the UF reference male cancer patient.............. ........................ 159

6-17. Electron absorbed fractions to the trabecular endosteum within the ilium of the
UF reference m ale cancer patient...... .......... ........ ..................... 160

6-18. Electron absorbed fractions to the trabecular endosteum within the ribs of the
UF reference m ale cancer patient ...... .......... ........ ..................... 161

6-19. Comparison of electron transport paths through the trabecular endosteum under
either CBIST simulations or VBIST simulations for two different initial
trajectory angles 0 1 and 02 .2 ........................................................................ 162

7-1. Multiple generations of the radiation transport codes used at the University of
F lo rid a ................................................................................................................. .. 1 9 9

7-2. Representative vertebral images used in the PIRT model..................................200

7-3. Electron absorbed fractions to active bone marrow within the sacrum (70%
ICRP reference cellularity) for three source tissues TAM, TBV, and TBS ........201

7-4. Electron absorbed fractions to bone endosteum within the sacrum for three
source tissues TAM TBV, and TBS....... ... .......................................... 202


xxvii









7-5. Electron absorbed fractions to the cortical bone volume from electron sources
in the spongiosa tissues (TAM, TBS, and TBV)........................ .................. 203

7-6. Electron absorbed fractions to the cortical bone volume from electron sources
in the cortical bone cortex itself. ......................... ......................... 204

7-7. Skeletal averaged electron absorbed fractions to active bone marrow within the
entire skeleton for four source tissues in comparison to Eckerman.....................205

7-8. Skeletal averaged electron absorbed fractions to bone endosteum within the
entire skeleton for five source tissues TAM, TBV, TBS, TMC, and CBV........206

7-9. Skeletal averaged electron absorbed fractions to active bone marrow within the
entire skeleton for four source tissues TAM, TBV, TBS, and CBV.................207

7-10. Variations in the S(TAM*-TAM ) with different radionuclides on the
skeletal-site-specific radionuclide S values given by the PIRT model ................208

7-11. Variations in the S(TAM*-rS) for 90Y on the skeletal-site-specific radionuclide
S values given by the PIRT model for the UF reference male cancer patient. ......209

7-12. Variations in the S(TAM*-TAM ) to the cranium based on varying marrow
celluarity and 5 radionuclides given by the PIRT model ..................................210

A-1. In-vivo computed tomography scout scans ....................................... ................ 228

A-2. Cranium images shown for visualization of skeletal site. ................................ 229

A-3. Mandible images shown for 3D visualization of the skeletal site ........................230

A-4. Clavicle images shown for 3D visualization of skeletal site..............................230

A-5. Scapulae images shown for 3D visualization of skeletal site...............................230

A-6. Cervical vertebra images shown for 3D visualization of skeletal site..................231

A-7. Thoracic vertebra images shown for 3D visualization of skeletal site .................231

A-8. Sacrum images shown for 3D visualization of skeletal site ..............................231

A-9. Lumbar vertebrae images shown for 2D and 3D visualization of the skeletal
site ....................................................................................................... ....... .. 2 3 2

A-10. Os coxae (pelvic) images shown for 2D and 3D visualization of the skeletal
site ....................................................................................................... ........ .. 2 3 3

A-11. Proximal femur images shown for 2D and 3D visualization of the skeletal
site ............................................................................................... ........ . ....... 2 3 4


xxviii









B-1. Example of a 3D reconstruction of spongiosa acquired from microCT imaging
of a bone section from a skeletal site of interest. ................................ 236

D-1. Illustrative example of header and data files for the microCT data. In this case,
C 0000040 is the file nam e ................................... ....................... ................ 252

D-2. Pictorial example of the header file from the C0000040 data set.........................254

D-3. Pictorial example of how to use the ConvertMicroCT.exe program....................255

D-4. Pictorial example for the execution of the Plane.exe program.............................256

D-5. Example of the opening window for "_Kslice.raw" images in
Adobe Photoshop ............... .............. .... ........ ...... ............... 257

D-6. Example image of after opening _Kslice.raw in Adobe Photoshop.....................258

D-7. Example image of Figure D-8 after "auto-leveling" in Adobe Photoshop...........259

D-8. Example of the opening window for "_Jslice.raw" images in Photoshop............259

D-9. Example of the opening window for "_Islice.raw" images in Photoshop............260

D-10. Example of ROI determination with the I Slice in Photoshop. ...........................261

D- 1. Example of ROI determination with the J Slice in Photoshop ...........................262

D-12. Example of ROI determination with the K Slice in Photoshop..........................263

D-13. Pictorial example of the execution of the Histogram.exe program ....................265

D-14. Pictorial example of gray-level histogram data plot in Microsoft Excel............265

D-15. Example of how to import a text file into SigmaPlot ................ ................266

D-16. Screen capture of the SigmaPlot window after plotting the histogram ..............267

D-17. Regression wizard window displaying the list for the curve-fitting equation .....267

D-18. Example window in the process for obtaining the curve-fitting parameters ........268

D-19. Example of Regression Wizard window providing the four parameters ..............269

D-20. Pictorial example of the execution of the FindThresh.exe program and its
corresponding output used to determine the optimum threshold .........................270

D-21. Pictorial example of the execution of the ResizeCTimage.exe program............271

D-22. Pictorial example of the execution of the MedianFilter.exe program ................272


xxix










D-23. Pictorial example of the execution of the Readlmage.exe program...................272

G-1. Schematic of how chord lengths and electrons could travel through a section of
trabecular spongiosa ....................................................................... 360

G-2. Pictorial example for the Humerus_Left image execution of the Tri-Linear
Chord-Length Distribution program. ...... ... .......................... 386


xxx















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

REFERENCE SKELETAL DOSIMETRY MODEL FOR AN ADULT MALE
RADIONUCLIDE THERAPY PATIENT BASED ON 3D IMAGING AND PAIRED-
IMAGE RADIATION TRANSPORT

By

Amish P. Shah

December 2004

Chair: Wesley E. Bolch
Major Department: Biomedical Engineering

The need for improved patient-specificity of skeletal dose estimates is widely

recognized in radionuclide therapy. Current clinical models for marrow dose are based

on skeletal mass estimates from a variety of sources and linear chord-length distributions

that do not account for particle escape into cortical bone. To predict marrow dose, these

clinical models use a scheme that requires separate calculations of cumulated activity and

radionuclide S values. Selection of an appropriate S value is generally limited to one of

only three sources, all of which use as input the trabecular microstructure of an individual

measured 25 years ago, and the tissue mass derived from different individuals measured

75 years ago.

Our study proposed a new modeling approach to marrow dosimetry-the Paired

Image Radiation Transport (PIRT) model-that properly accounts for both the trabecular

microstructure and the cortical macrostructure of each skeletal site in a reference male


xxxi









radionuclide patient. The PIRT model, as applied within EGSnrc, requires two sets of

input geometry: (1) an infinite voxel array of segmented microimages of the spongiosa

acquired via microCT; and (2) a segmented ex-vivo CT image of the bone site

macrostructure defining both the spongiosa (marrow, endosteum, and trabeculae) and the

cortical bone cortex. Our study also proposed revising reference skeletal dosimetry

models for the adult male cancer patient. Skeletal site-specific radionuclide S values

were obtained for a 66-year-old male reference patient. The derivation for total skeletal S

values were unique in that the necessary skeletal mass and electron dosimetry

calculations were formulated from the same source bone site over the entire skeleton.

We conclude that paired-image radiation-transport techniques provide an adoptable

method by which the intricate, anisotropic trabecular microstructure of the skeletal site;

and the physical size and shape of the bone can be handled together, for improved

compilation of reference radionuclide S values. We also conclude that this

comprehensive model for the adult male cancer patient should be implemented for use in

patient-specific calculations for radionuclide dosimetry of the skeleton.


xxxii














CHAPTER 1
INTRODUCTION

Bone marrow, the highly organized tissue that comprises different blood-forming

cells in the body, is considered the dose-limiting organ in many radiation therapy

applications, namely radioimmunotherapy (Lim et al. 1997; Sgouros 1993; Siegel et al.

1990). Hematopoiesis, the development of red and white blood cells from the

proliferation and differentiation of stem cells, occurs within the bone marrow. Sites for

hematopoiesis are located only within the axial skeleton of the human body. Within the

axial skeleton are regions of trabecular bone. Trabecular bone regions thus provide the

"housing" for the hematopoietic element of bone marrow within the human body.

Dosimetric assessment of trabecular bone regions is an important area within

internal dosimetry, considering the role these bone sites play in both the skeletal and

hematopoietic systems. Since bone marrow is located within trabecular bone regions,

radiation incident on bone is likely to also cause damage to the marrow. Radionuclides

that localize in bone, especially charged-particle emitters, have the potential to cause

damage to both endosteal tissues and bone marrow. Several situations may result in

internal irradiation of trabecular bone regions. These include therapy procedures that use

injected radiopharmaceuticals that transit through the skeletal system, occupational

exposures to bone-seeking radionuclides, and therapeutic procedures for the palliation of

bone pain associated with bone cancers. The amount of energy deposited in healthy

marrow from therapeutic radiopharmaceuticals often limits the amount of activity

prescribed in these procedures.









Use of radiopharmaceuticals as therapy agents has increased during the last few

decades. Therefore, more accurate trabecular-bone dosimetry is needed to minimize risk

to patients. Risk arises from any treatment plan in which a radionuclide travels through

the circulatory system. These radiopharmaceuticals emit radiation particles while

traveling through the blood stream of the patient. Bone marrow is continuously irrigated

by blood vessels and thus exposed to this radiation. Some of the energy is also deposited

in trabecular bone regions. Accurate trabecular bone dosimetric models will allow one to

calculate the dose to both the bone and the bone marrow with more precision. Improved

skeletal dosimetry will allow physicians to better understand the biological effects of

specific therapy procedures, which in turn will help improve nuclear medicine techniques

by optimizing the administration of therapeutic doses of radiopharmaceuticals.

Radiopharmaceuticals are also used for bone-pain palliation. This treatment is

accomplished with bone-seeking beta-emitting radionuclides. Iodine-131, 32P, 89Sr, and

186Re are four types of radionuclides considered for this treatment (Samaratunga et al.

1995). Nevertheless, marrow receives a significant amount of the deposited energy from

these radiopharmaceuticals in bone pain treatments.

Increased accuracy in trabecular-bone dosimetry has the potential to improve our

understanding of the consequences associated with the scenarios previously mentioned.

Therapeutic applications of radiation and radioactive materials will benefit from better

dosimetry within these regions. Health risks associated with bone-seeking radionuclides

can also be more calculated more accurately. Thus, current methods to improve

trabecular-bone dosimetry are directed at more correctly measuring the microstructure of

these skeletal regions.









Previously at the University of Florida (UF), an investigation was initiated on the

feasibility of using magnetic resonance (MR) imaging to transport electrons through 3D

digital images of the trabecular-bone microstructure. Chord-length distributions through

both the bone trabeculae and the marrow cavities were also acquired using these 3D

images. These distributions are important for skeletal dosimetry since they can be

compared with electron ranges to deduce energy deposition through the bone and marrow

regions. Furthermore, all current models of skeletal dosimetry are based on these

distributions.

More recently at UF, chord-length measurements of voxelized images were found

to be difficult to calculate and are highly dependent on the methods used to remove voxel

effects (Jokisch et al. 1999; Rajon et al. 2000). Consequently, an approach of directly

coupling 3D MR images to the EGSnrc radiation transport code was developed (Jokisch

et al. 1999; Patton 2002a). This voxel-based spongiosa transport approach allowed for

radiation transport in a real geometry, thus serving as a benchmark set of calculations for

all existing trabecular-dosimetry models.

Our study aimed to build on these previous studies and develop a new model for

skeletal dosimetry for an adult male radionuclide patient. Chapter 2 gives the important

background required to better understand the whole development that follows. Chapter 3

proposes a new Monte-Carlo technique for skeletal dosimetry, Paired-Image Radiation

Transport (PIRT). This voxel-based technique is an adaptation of previous voxel-based

models at the UF, with major improvements to the approach for modeling the physical

macrostructure of each skeletal site, as defined by the cortical bone cortex. The prototype

Monte-Carlo (PIRT) codes presented were directly compared to outdated methods for









modeling the trabecular microstructure, by comparing two skeletal sites previously

investigated at UF with older voxel-based codes. This new technique of modeling for

skeletal dosimetry was then used (Chapter 4) to provide skeletal dosimetry in bone sites

that have not been studied accurately in the past. The flat bones of the human skeleton

have presented a formidable task in stylistic modeling methods for Monte-Carlo codes to

track electrons through a single skeletal site such as the cranium or iliac crest.

Comparisons were made (Chapter 4) to older methods for skeletal dosimetry that are

currently accepted for use in clinical dosimetry. Chapter 4 introduces micro-computed

tomography (microCT), as a better technique for imaging the regions of trabecular bone.

Chapter 5 details a new reference individual for skeletal dosimetry. The trabecular

microstructure of the UF adult male cancer patient chosen must be thoroughly examined

for validity for use in skeletal-dosimetry models. At present time, skeletal dose estimates

in clinical dosimetry are fundamentally reliant on a single set of chord-length distribution

measurements performed at the University of Leeds for a single 44-year male subject.

Presented here is an alternative set of chord-length distribution data of a 66-year male

subject (Chapter 5). In Chapter 6, the chord-length distributions were used to assess the

dosimetric difference between the UF adult male cancer patient and the 44-year male

subject previously developed for radiation protection. Differences in dosimetry between

two types of radiation transport models (chord-based and voxel-based models) were

investigated (Chapter 6).

Chapter 7 provides the complete data set for the new reference adult male

radionuclide patient. Skeletal mass estimates and bone site-specific dosimetry was

measured through the use of microCT imaging, ex-vivo CT scans, and the new PIRT






5


methodology for radiation transport (Chapter 7). With the use of all these tools, our

study presents complete skeletal-site specific radionuclide S values tabulated for the

reference male cancer patient. The S values presented will allow for better estimates of

the absorbed dose to the skeletal system.














CHAPTER 2
BACKGROUND

Bone Structure and Physiology

The skeleton, composed of bones, cartilages, joints, and ligaments, accounts for

20% of the body mass. Bones make up the bulk of the skeleton. The bones in the human

skeleton are grouped into two main categories: the axial and appendicular skeleton. The

axial skeleton forms the long axis of the body, and includes the bones of the skull,

vertebrae, sternum, ribs, pelvis, and the proximal ends of the long bones (Gatter and

Brown 1997). The appendicular skeleton consists of bones that make up the upper and

lower limbs. In the normal adult, sites of hematopoiesis are restricted to the axial

skeleton; the marrow in these sites is portrayed as red or active, because of the presence

of erythroid elements. In terms of trabecular bone dosimetry, the axial skeleton and the

hematopoietically active marrow are the primary regions of interest.

Every bone in the skeleton is composed of two mains types of tissue: cortical bone

(the hard compact bone that forms the dense but smooth external layer); and trabecular

bone (the spongy or cancellous bone that forms the honeycomb structure within the dense

shell). The honeycomb structure of the cancellous bone actually is a lattice of small

needle-like, flat pieces called trabeculae. Cortical bone comprises 80% of the skeletal

mass. Figure 2-1 shows the distinction between compact and cancellous bone.

Cortical bone is made up of canals and passageways that serve as regions for

nerves, blood vessels, and lymphatic vessels. The building blocks of cortical bone are the

osteon or Haversian system. Each osteon is an elongated cylinder oriented parallel to the









long axis of the bone. Running through the core of the osteon is the Haversian canal; this

canal contains small blood vessels and nerve fibers. As seen in Figure 2-2, each

Haversian system is surrounded by concentric cylinder-shaped layers called lamellae. At

the junctions of the lamellae are spider-shaped mature bone cells (also known as

osteocytes). These osteocytes receive tissue fluid from the Haversian canals, through

canals called the canaliculi. Together, the Haversian canal, surrounding lamellae,

osteocytes, and canaliculi make up the osteon or Haversian system (Figure 2-2).

The thin lining of cells along the interface between the Haversian canals and the

bone surfaces is known as the endosteum. The endosteum is a delicate connective-tissue

membrane that covers the trabeculae of the spongy bone in the marrow cavities (Marieb

1998). The endosteum is composed primarily of osteoblasts (bone-forming cells) and

osteoclasts (bone-destroying cells). There is also a periosteum, a double-layered

membrane composed of osteoblasts and osteoclasts, which covers the exterior of the

cortical bone. At both the periosteal and endosteal surfaces, bone production occurs

where added bone strength is needed, or at sites of injury. The mechanism that regulates

this process is the response of bone to mechanical stress and gravity.

In contrast, bone resorption is triggered by the parathyroid hormone. When ionic

levels of calcium in the blood decline, the hormone is released from the parathyroid

glands. By resorbing the bone, the calcified bone matrix is broken down, thereby

releasing calcium, which is then released to the bloodstream. In young individuals,

osteoblasts are more actively dominant than osteoclasts, and thus form more bone.

However, in adults, the process begins to equilibrate, and soon the osteoclasts dominate.









Nevertheless, for the remainder of adult life, bone resorption outlabors bone production,

resulting in a net loss of bone mass (Berne and Levy 1993).

Several factors create variations in trabecular bone microstructure. Trabecular

structure varies with age (Atkinson 1965; Snyder et al. 1993), gender (Mosekilde 1989;

Patton 2000), skeletal site (Eckerman 1985a; Patton 2000), and skeletal orientation (Hahn

et al. 1992; Mosekilde 1989; Patton 2000). Age-related changes, along with the natural

progression of aging, include thinning and loss of bone trabeculae. In terms of skeletal

site and orientation, mechanical stress and gravity play a large role in the variability of

bone trabeculae. For instance, weight-bearing trabeculae should be thicker than non-

weight-bearing trabeculae. Also, the rate of bone loss is greater for trabeculae in a

horizontal rather than a vertical orientation (Mosekilde 1989; Parfitt 1983).

The intricate geometry and composition of the trabecular bone regions of the

skeleton create several dosimetry problems. Since the bone marrow cavities are located

within the trabecular bone structure, the dimensions of the "honeycomb" configuration

must be known to accurately calculate any dose to this region. However, the anisotropic

framework of these regions further complicates any attempt to apply a uniform modeling

technique to the trabecular bone geometry.

Radionuclide Therapies for Cancer

Absorbed dose to the bone marrow occurs in several ways. One significant method

is radioimmunotherapy. Radioimmunotherapy (RIT) involves tagging antibodies with

radionuclides for use treating cancers outside of the hematopoietic system, such as

osteosarcoma (bone cancer), liver cancer, and other tumor growths. However, the

process of RIT seems to be dose limited by bone marrow, because of the accumulation of

activity within the marrow (Lim et al. 1997; Sgouros 1993; Siegel et al. 1990).









Monoclonal antibodies (mAbs) offer therapeutic choices for patients with hematological

malignancies. These mAbs are used to deliver radioisotopes selectively to malignant

tissue and thus increase the specificity of toxicity effects. In RIT, tumor cells may be

killed by the antibody effect and also by the crossfire effect of irradiation. For

radioisotopes commonly used (90Y and 1311), beta particle emission eliminates tumor cells

within a range of 1000-5000 |tm of their deposition.

Radiotherapy can also be localized within the hematopoietic system through

methods of bone marrow ablation, using ionizing radiation to destroy malignancies in

bone marrow. Malignancies such as Hodgkin's or non-Hodgkin's lymphoma, multiple

myeloma, and leukemia all require marrow ablation before bone marrow transplants

(Juweid et al. 1995). Bone marrow ablation is the process before a bone marrow

transplant. Bone marrow ablation involves high levels of chemotherapy and/or external

beam radiation. Ablation of marrow by irradiation for the purposes of marrow

transplants is limited by the radiation damage to healthy osteogenic tissue. Bayouth and

Macey (1993) define the prototypical radionuclide, for ablation therapy, as being able to

deliver an adequate dose to the marrow, allow for reinfusion of the new marrow in a short

period of time, and minimize the radiation dose to other outside organs. One method of

marrow ablation is to administer bone-seeking radiopharmaceuticals that accumulate in

the skeleton, and deliver largely concentrated radiation dose to the adjacent marrow

cavity (Bayouth and Macey 1993).

Marrow Toxicity

Monoclonal antibodies conjugated to radioisotopes or to low-energy Auger electron

emitters have been studied for the targeted radiotherapy of cancer. Similar to clinical









experience with immunotoxins, only minor responses to radioimmunotherapy have been

achieved in subjects with solid tumors. Because of the severe bone-marrow toxicity

associated with these high amounts of radioactivity, patients may receive bone-marrow

stem-cell infusions after treatment. Poor tumor penetration of the antibodies and dose-

limiting bone-marrow toxicity has severely restricted the effectiveness of

radioimmunotherapy for solid tumors such as breast, ovarian, or colorectal cancer.

Myelosuppression after radioimmunotherapy was not due to direct targeting of bone

marrow stem cells by the antibodies, but rather to nonspecific irradiation of bone-marrow

stem cells caused by high levels of circulating radioactive antibodies perfusing the bone

marrow. The bone marrow is very sensitive to ionizing radiation, with severe

bone-marrow suppression developing at extremely low absorbed doses. Bone-marrow

toxicity limits the dose of monoclonal antibodies that could be safely administered.

Previous Methods of Trabecular Bone Dosimetry

Current bone-dosimetry models stem from early data collected by Spiers and

colleagues at the University of Leeds in England over 20 years ago. Spiers (1951) first

investigated the effects of a bone surface interface on the dose to the adjacent soft tissue.

He calculated that the presence of a bone interface increased the dose to the surrounding

soft tissue by as much as a factor of four. In 1963, Spiers (1963) began investigating how

active marrow is distributed throughout the human body and the impact this had on

trabecular bone dosimetry. Spiers (1966a; 1967) was the first to recognize that the

anisotropic structure of trabecular bone required a unique method for characterizing the

geometry, in order to perform accurate skeletal internal dosimetry of beta-emitters. He

then developed the concept of linear path length distributions (chord lengths) to describe

the physical dimensions of these regions. By knowing the complete frequency









distribution of marrow path lengths and trabeculae path lengths, the fraction of a

particle's kinetic energy deposited in each type of tissue could be calculated. Dose

calculations were obtained by coupling the frequency distributions to a one-dimensional

transport model that used these distributions to follow electrons through trabecular bone.

Spiers obtained chord-length distributions for approximately 5 to 7 skeletal sites from

three males of ages 1.7, 9, and 44 years (Beddoe 1976). Almost all trabecular-bone

dosimetry calculations are still based on data obtained from the Spiers' work.

Several different methods for trabecular dosimetry were developed that used chord

distributions. In terms of dosimetry calculations, several assumptions must be made in

order to use chord distributions. First, beta particles are assumed to travel in

approximately straight paths through a given media (bone or marrow). Any attempt to

use the chord distributions to create a geometry that allows a beta particle to travel a

distance not equal to the sampled chord-length distribution incorrectly uses these

distributions. Second, the chord-length distributions are assumed to be an accurate

representation of a person's trabecular microstructure. Third, chord-length distributions

are assumed to be independent of one another. A chord-length-based Monte Carlo

radiation dosimetry model randomly samples from both bone and marrow distributions

without considering any dependence one distribution may have on the other.

Spiers and his students were the first to consider random sampling from the

chord-length distributions using Monte Carlo techniques (Spiers et al. 1978; Whitwell

1973; Whitwell and Spiers 1976). Experimentally measured chord-length distributions

were coupled with range-energy relationships to calculate dose conversion factors for

seven radionuclides. These dose-conversion factors were used to determine the absorbed









fraction values in ICRP Publication 30 (1979). This report recommended absorbed

fractions for beta particles for use in radiation protection of skeletal tissues. For beta

particles originating in the bone volume, a single value of absorbed fraction is

recommended. For beta particles originating on the bone surface, one absorbed fraction

for low-energy beta particles (< 0.2 MeV), and one for high-energy beta particles (> 0.2

MeV) are recommended. These absorbed fractions values are roughly based on the dose

conversion factors from Whitwell (1973; 1976). Subsequently, the ICRP's relatively

energy-independent absorbed fractions of energy were implemented in the MIRDOSE2

program (Stabin 1996) for use in nuclear medicine dosimetry. In this same computer

program, the self-absorbed fraction to the marrow was assumed to be unity at all

energies, as suggested in Part 3 of ICRP Publication 30 (1979).

Spiers data were used later to calculate the dose to marrow cavities and the

endosteal layer for monoenergetic electrons emitted uniformly and isotropically within

trabeculae and marrow cavities (Eckerman 1985a, 1985b, 2000). He noted that the range

of electrons in marrow to that in bone is nearly constant for energies up to 4 MeV. This

allowed reduction of the two media to a single homogeneous medium, by extending the

length of the sampled trabeculae chord lengths by this ratio. Monte Carlo techniques

were then applied to this single medium.

Another model using Spiers data was developed by Bouchet et al. (1999, 2000) at

the University of Florida. This model assumes that electrons do not deviate far from a

straight-line path (similar to other models). However, it improves on previous models,

because it is a 3D model that accounts for delta rays and bremsstrahlung radiation, and is

able to consider electron backscatter at bone-marrow interfaces.









More recent work at UF shows that chord-length measurements of voxelized

images are difficult to calculate and are highly dependent on the methods used to remove

voxel effects (Rajon et al. 2000). Thus, these UF studies have directly coupled three-

dimensional Nuclear Magnetic Resonance (NMR) images to the EGS4-PRESTA

transport code, to acquire energy deposition within the marrow cavities. This allows for

radiation transport in the actual trabecular geometry, thus serving as a benchmark set of

calculations for existing trabecular-dosimetry models (Patton et al. 2002a). This method,

based on geometric models, showed improvement to localized skeletal dosimetry because

of the addition of measured trabecular bone spongiosa (bone trabeculae and marrow) and

cortical bone cortex. These additions further corrected any overestimation of energy

deposition seen in models assuming an infinite trabecular region (Patton et al. 2002b).

Internal Dosimetry Calculations

The Medical Internal Radiation Dose (MIRD) Committee established a method for

calculating internal dose (Loevinger et al. 1991). According to this method, the key

component is the S value, which is the average dose received by the target organ per

disintegration within the source organ. S values for specific radionuclides are calculated

for specific source rs and target rt regions using the following formula:


S(r, )- rs)= A 'A (r --rs (2-1)


where Ai is the mean energy emitted per nuclear transition, 0,#(rrT-rs) is the absorbed

fraction (AF) of energy in the target region for the ith radiation type that originated in the

source region, and mr is the mass of the target region. The S value is then used in the

calculation of dose. As a result of a contamination by a radionuclide, the dose D is

defined by










Dk I= Ah S(kh) (2-2)
h


where the cumulated activity Ah represents the total number of disintegrations of the

radionuclide that occur during the contamination time. In our study, the determining AF

is the critical component for approximating the dose calculation. The AF depends on the

geometry of two organs, on the tissue composition of two organs, and on what organs lie

in between these two organs. The AF can be determined by analytical methods such as

Point Kernel, or by using Monte-Carlo transport codes. The purpose of our study was to

provide a better approximation of the AF of energy within the bone marrow for use in

clinical dosimetry.

















q W -V
* 4


Trabecular
- P. [L'


Figure 2-1. Vertebral body showing the different types of bone tissue in one particular
skeletal site. Adapted from a study by Fagerburg and Lafferty (1998).


Compact
Bone


t' 1i,. it LLm













Haversian System


Periosteum K




| Blood
vessel
Volkmann's
Canal


Figure 2-2. Microstructure of compact and cancellous bone. Illustration includes
entire osteon known as the Haversian system. Adapted from a study by
(Marieb 1998).














CHAPTER 3
PAIRED-IMAGE RADIATION TRANSPORT MODEL FOR SKELETAL
DOSIMETRY1

Introduction

The skeletal system represents one of the more complex challenges in internal

dosimetry. This distributed organ, with its wide variety of bone sizes and configurations,

encompasses the hematopoietic tissues of the active (red) bone marrow, as well as the

osteogenic tissues of the endosteum, both of which are relevant targets for short-term

deterministic and long-term probabilistic radiation effects. Of primary importance is the

3D microscopic architecture of the bone trabeculae which separate and define the marrow

cavities. For short-ranged radiations (alpha particles and lower-energy beta particles),

knowledge of this 3D microstructure is necessary and sufficient for accurate computation

of particle transport through these skeletal tissues. For longer-ranged radiations (such as

intermediate to high-energy beta particles), further consideration should be given to the

3D macrostructure of the skeletal site, including the location and extent of cortical bone

into which escaping particles may penetrate.

The vast majority of initial studies in skeletal dosimetry were conducted at the

University of Leeds (Spiers 1951; Spiers 1966; Spiers 1966; Spiers 1967; Spiers 1968;

Spiers 1969). Spiers (1966; 1967) was the first to recognize that the anisotropic structure


1 This chapter was accepted for publication by The Journal of Nuclear Medicine and will be published in
February 2005: Shah AP, Bolch WE, Rajon DA, Patton PW, and Jokisch DW. A Paried-Image Radiation
Transport model for skeletal dosimetry. J Nuc Med: accepted September 2004.









of trabecular bone required a unique method for characterizing the trabecular geometry

as needed for accurate skeletal dosimetry of beta-emitters. Consequently, he and his

students constructed an optical bone scanning system which measured linear chord-length

distributions across 2D radiographs of excised bone tissue slices. Using these frequency

distributions of linear chord lengths through both bone trabeculae and marrow cavities,

the fraction of a particle's kinetic energy deposited in each tissue type was estimated.

Spiers and his students obtained chord length distributions in the lumbar vertebrae for

several subjects, as well as at several skeletal sites of a 1.7-year child (5 sites), a 9-year

child (5 sites), and a 44-year male (7 sites) (Beddoe 1976; Beddoe 1976; Whitwell 1973;

Whitwell 1976). In many ways, the chord-length distribution data measured for the 44-

year male has served to define many of the skeletal attributes of Reference Man as

defined by the International Commission on Radiological Protection (ICRP) (1975;

2002). Furthermore, all skeletal dosimetry models published and presently used in

clinical dose assessment are fundamentally reliant upon this single set of adult chord-

length distributions (Eckerman 1985; Bouchet 2000; Bouchet 1999; Eckerman 2000;

Stabin 2002).

In the technique described above, radiation particles are effectively transported

within an infinite region of trabecular spongiosa (defined as the combined tissues of the

bone trabeculae, endosteum, and marrow cavities). Models of skeletal dosimetry used in

current clinical practice, such as the Eckerman and Stabin model (2000) of MIRDOSE3

and its successor codes (Stabin 1996), belong to a class of models called CBIST or

Chord-Based Infinite Spongiosa Transport, and do not account for particle escape to









cortical bone. Consequently, absorbed fractions to skeletal tissues are potentially

overestimated in CBIST models for higher-energy beta emitters.

One of the first attempts to account for energy loss to cortical bone was made by

Spiers' doctoral student JR Whitwell (1976; 1973). She introduced a trabecular

equilibrium factor, Qtrab, to account for the finite extent of the spongiosa. This correction

factor was determined for several radionuclides of interest in radiation protection and for

each of the 7 skeletal sites for which chord-length distributions were obtained in the 44-

year male subject. For 90Y, the highest correction noted by Whitwell was for the parietal

bone (Qtrab = 0.672) while the lowest was for the head of the femur (Qtrab = 0.980).

Nevertheless, these values of Qtrab were determined using simplified geometries for both

spongiosa and cortical bone (e.g., planes and spheres).

In a more recent study by Patton et al. (2002), NMR microscopy was applied to the

study of the 3D microstructure of bone trabeculae within the femoral and humeral heads

of three subjects: a 51-year male, an 82-year female, and a 89-year female. To account

for energy lost to cortical bone, an ex-vivo CT scan of the excised femoral or humeral

head was obtained prior to spongiosa sectioning. From spatial measurements on the CT

images, a spherical region of spongiosa was constructed surrounded by a spherical shell

of cortical bone. Electrons of various initial energies were thus transported (via the

ESG4 radiation transport code) simultaneously within the NMR microimage (constructed

of voxels of bone and marrow), and within stylized model of the femoral or humeral

head. Comparisons were subsequently made between energy-weighted absorbed

fractions to active marrow under particle transport in either (1) an infinite extent of

spongiosa, or (2) the stylized model of the bone site. Patton et al. demonstrated that,









without explicit consideration of energy loss to cortical bone, radionuclide S values for

32P and 90Y could potentially over-estimate active marrow dose by 6% and 11%,

respectively, in the femoral head values that exceeded the 2% corrections predicted by

Whitwell. This tendency to overestimate dose to active marrow under infinite spongiosa

transport had also been demonstrated by Jokisch et al. (2001) for the thoracic vertebra in

which the physical extent of the vertebral spongiosa was delineated in a stylized model of

the vertebral body (e.g., truncated circular cylinder). Due to their geometric complexity,

however, no attempt was made to include the vertebral processes in the stylized vertebral

model (which account for up to -25% of vertebral spongiosa).

In the present study, we significantly extend the skeletal modeling approach

originally explored by Jokisch et al. and Patton et al. to fully account for the 3D

macrostructural dimensions of skeletal sites within which dose estimates are desired. A

Paired-Image Radiation Transport or PIRT model for skeletal dosimetry is introduced in

which radiation particles are tracked simultaneously within two different segmented

digital images: (1) an ex-vivo CT image of the entire skeletal site outlining regions of

trabecular spongiosa, cortical bone, and surrounding tissues, and (2) an ex-vivo NMR

microscopy image of the interior bone trabeculae and marrow cavity microstructure

representative of that found in spongiosa volumes of the larger CT image. The PIRT

model is demonstrated within two skeletal sites obtained from a single male cadaver: the

L4 vertebra and the right proximal femur. In addition, representative site-specific S

values are calculated and compared to those obtained under particle transport within

infinite regions of spongiosa for a variety of radionuclides of interest in skeletal imaging

and therapy.









Materials and Methods

Cadaver Selection

Candidate subjects for study were obtained through the State of Florida Anatomical

Board located on the University of Florida (UF) campus. Cadaver selection criteria

included (1) an age between 50 75 years (representative of typical radionuclide therapy

patients), (2) a body mass index of 18.5 25 kg m-2 (CDC recommended healthy range),

and (3) a cause of death that would preclude significant skeletal deterioration. The

subject identified was a 66-year male approximately 68 kg in total mass and 173 cm in

total height at the time of death (BMI of 22.7 kg m-2). The subject died suddenly of

complications associated with cardiomyopathy.

In-Vivo Computed Tomography Scanning

Prior to bone harvesting, the male cadaver was subjected to whole-body imaging

via multi-slice helical CT at a pitch necessary to reconstruct contiguous 1-mm axial

slices. The images were acquired on a Siemens Sensation 16 unit within the Department

of Radiology at UF Shands Hospital. Image reconstruction was performed with a bone

filter at an in-plane pixel resolution of 977 |tm x 977 |tm. The CT image sets were then

transferred to workstations within the Advanced Laboratory for Radiation Dosimetry

Studies (ALRADS) in the UF Department of Nuclear & Radiological Engineering for

image processing and data storage. The in-vivo CT scans provided image data for (1)

selecting the anatomical region from which the bone site would be harvested, and (2)

constructing 3D anatomic models of skeletal sites where bone harvesting (and thus ex-

vivo CT scanning) might be incomplete (e.g., facial bones of the skull).









Bone Harvesting and Ex-Vivo Computed Tomography Scanning

Following detailed review of the whole-body in-vivo CT images, bone harvesting

was conducted. Thirteen major skeletal sites were taken from the male cadaver including

the entire vertebral column and both proximal femora. Once each skeletal site was

excised, it was cleaned of excess tissue, bagged, labeled, and stored frozen until ex-vivo

CT imaging could be scheduled. Post-harvest, ex-vivo CT imaging was conducted at

higher resolution (1.0 mm slice thickness with an in-plane resolution of 0.3 x 0.3 mm)

than permitted for in-vivo scans. The ex-vivo CT scans provided image data for (1)

identifying the location and extent of trabecular spongiosa to be sectioned for NMR

microscopy, (2) quantifying volumes of trabecular spongiosa and cortical bone within the

bone site, and (3) constructing 3D anatomic models of the bone site for subsequent

paired-image radiation transport simulations.

Following detailed review of the ex-vivo CT scans, physical sections of trabecular

spongiosa were taken from each bone site. Sections representing as large a region of

spongiosa as possible, given the constraints of the bone shape and the NMR imaging

system (e.g., cuboidal samples taken from a spherically shaped femoral head). Marrow-

intact sections of spongiosa were bagged, labeled, and kept frozen until NMR

microimaging sessions could be arranged. For the lumbar vertebra, 2 cuboidal sections

(roughly 1.25 cm x 1.25 cm x 2.5 cm on edge) were cut from the vertebral body

representing -24% of the total vertebral body spongiosa. For the right proximal femur, 4

cuboidal sections were cut from the femoral head (-20% of total spongiosa within the

head) and 4 sections were cut from the femoral neck (-16% of the total spongiosa within

the neck).









Image Segmentation of Spongiosa and Cortical Bone Regions

To create tomographic anatomic models for use in internal dosimetry, radiation

transport codes must be able to decipher the boundaries of each tissue region for which

an independent dose assessment is to be made. Limitations of CT image acquisition can

result in an overlap of grayscale values for tissues of interest, thus precluding the use of

simple automated methods of boundary definition. In the present study, the program CT-

_Contours was adopted for use in segmenting spongiosa and cortical bone within each

ex-vivo CT image set (Nipper 2002). This program is based upon Interactive Data

Language (IDL) version 5.5 and can output labeled contour files in a variety of formats

including binary files for EGSnrc (Kawrakow 2000) and ASCII text for MCNP

(Briesmeister 1997). CT_Contours displays the current CT information, as well as a

color overlay of the contours being edited. The contours can be created using a variety of

tools including: basic thresholding, pixel growing, voxel growing, region growing and

manual segmentation. The voxels contained in the individual contours are filled with the

desired segmentation value, generating volumes of voxels with identical tag values. In the

present study, these volumes represent individual regions of either trabecular spongiosa

or cortical bone within the skeletal site. CT_Contours was written to have the option of

displaying the images using 15 different filters including Gaussian smoothing (3 x3, 5

x5, or 7 x7), median (3 x3, 5 x5, or 7 x7), Roberts edge detection, Sobel edge detection,

Prewitt edge detection, isotropic edge detection, histogram equalization, adaptive

histogram equalization, sharpening and Kuwahara (3 x3 or 5 x5) filtering. By altering

regions of a separate contour dataset, the desired segmentation can be performed.

CT_Contours was designed so that ROI creation or modification can be performed in

either the transverse, sagittal, or coronal plane.









Microimaging of Trabecular Spongiosa

NMR microscopy of trabecular bone for the purposes of skeletal dosimetry has

been discussed previously (Jokisch 2001; Patton 2002; Shah 2003; Bolch 2002; Bolch

2002). NMR imaging requires physical sectioning of the excised sample and digestion of

the marrow tissues. Samples oftrabecular bone sections are first immersed and

suspended within a circulating solution of sodium hypochlorite for approximately three

hours. The samples are then rinsed in water and re-immersed in a new solution. This

process is repeated up to three times depending on the size of the sample. Visual

inspection is used to determine the number of repetitions needed. To ensure that water

completely fills all marrow cavities, each sample is placed in a container filled with Gd-

doped water under vacuum. While immersed, the sample is placed in a smaller container

needed for insertion into the magnet. This imaging container is then sealed and taken to

the Advanced Magnetic Resonance Imaging and Spectroscopy (AMRIS) facility at the

UF McKnight Brain Institute for NMR microscopy.

NMR microscopy images in the present study were acquired on a Bruker 40-cm

wide bore imaging spectrometer, operated at a 470-MHz proton resonance frequency (11

T magnetic field strength). The system is fitted with a small gradient set (for

microimaging), consisting of 3-axes magnetic field gradients, with a 22-Gauss/cm

maximum gradient amplitude in all three orthogonal directions. A 35-mm diameter

quadrature birdcage coil of length 45 mm is used in order to obtain the best signal-to-

noise ratio (SNR). For all imaging sessions, a RARE encode 3D spin-echo pulse

sequence is used to obtain fully three-dimensional images of the samples. Fields of view

are typically 3.2 cm x 3.2 cm x 3.2 cm with matrix dimensions of 512 x 512 x 512. The

resulting spatial resolution of the 3D images is thus 63 |tm x 63 |tm x 63 |tm. Smaller









voxel dimensions can be achieved at UF (~ 58 [tm), but at the cost of smaller sample

sizes and increased imaging time (to preserve signal-to-noise). Post-acquisition image

processing, including gray-level thresholding, voxel segmentation, and 3D median

filtering, have been reported previously (Jokisch 1998; Patton 2002). For use in radiation

transport simulations, interior regions-of-interest (ROIs) are taken to avoid both physical

distortions (bone saw tearing) and imaging distortions (NMR aliasing) at the edges of the

sectioned specimen.

Voxel-Based Infinite Spongiosa Transport (VBIST) Model

Following NMR microscopy of our skeletal samples, a series of Voxel-Based

Infinite Spongiosa Transport, or VBIST, models were created to approximate (via 3D

transport) the results of current CBIST models. First, marrow voxels within the binary

NMR microscopy image are further labeled into voxels of active (red) marrow and

inactive (yellow) marrow at a pre-determined value of marrow cellularity. This process

has been outlined previously by Shah et al. (2003), and is based upon microscopy

measurements of the spatial distribution of adipocytes within normal bone marrow

biopsies covering a broad range of marrow cellularities. Skeletal endosteum is further

defined as a 10-[tm at the bone-marrow interface as previously described by Jokisch et al.

(1999). The resulting 4-tissue 3D model of trabecular spongiosa is coupled to the

EGSnrc radiation transport code (Kawrakow 2000) for electron and beta particle

transport simulations. Source tissues include the trabecular active marrow (TAM),

trabecular bone surfaces (TBS), and trabecular bone volume (TBV). Bone surface

sources are approximated as a 0. 1-[tm layer on the marrow side of the bone-marrow

voxel interface. Target tissues include the TAM and the trabecular bone endosteum









(TBE). Once a given electron reaches the physical edge of the 3D NMR microscopy

image, that particle is re-introduced to the image at a corresponding location at its

opposing edge. The processes of particle transport within the image of spongiosa and its

re-introduction are continued until all initial kinetic energy is expended. Particle histories

are continued (50,000 to 2,000,000) until coefficients of variation on the absorbed

fraction are below 1%. It is noted, however, that results given here for our voxel-based

IST model are only approximate to those given from chord-based IST models. In a

previous study by Jokisch et al. (2001), the authors question the sampling independence

of the marrow and bone chord length distributions within existing CBIST models, and

suggest a 3D joint distribution might be more appropriate to describing the full 3D micro-

architecture of particle transport within the spongiosa regions of trabecular bone.

Paired-Image Radiation Transport Model (L4 Vertebra)

In contrast to the VBIST model described above, the Paired-Image Radiation

Transport or PIRT Model supplements the 3D microscopic histology provided by the

NMR microscopy image with the 3D macroscopic histology given in the corresponding

ex-vivo CT image. The latter provides additional data for particle transport including (1)

the spatial extent of the trabecular spongiosa (e.g., vertebral processes and body) and (2)

the spatial extent of the surrounding cortical bone (which laterally encompasses the

vertebral body, forms the lamina separating the vertebral processes, and is absent at the

superior and inferior body-disc interfaces).

A schematic of the PIRT model for the L4 vertebra of the 66-year male is given in

Figure 3-1. The ex-vivo CT image is shown in the upper left in which segmented regions

of spongiosa and cortical bone surfaces are highlighted in orange and white, respectively.

Two representative transverse slices are shown (upper middle and upper right) where









regions of spongiosa (orange) and cortical bone (light blue) are again differentiated.

Superimposed over the entire ex-vivo CT image is a 3D array of the replicate cuboidal

NMR microscopy images each representing the 3D microstructure of the individual bone

trabeculae and marrow cavities. A 3D rendering of the NMR microimage is thus shown

in the lower left of Figure 3-1. Finally, a single transverse slice through the NMR

microimage is shown in the lower right displaying individual voxels of bone (black) and

total marrow (white).

In the EGSnrc implementation of the vertebral PIRT model, individual electrons

are tracked simultaneously within the coordinates of the NMR microimage (indicating

locations in TBV, TBE, TAM, or trabecular inactive marrow TIM), and the coordinates

of the CT macroimage (indicating locations in either spongiosa, cortical bone volume -

CBV, or surrounding tissues muscle, soft tissue or vertebral discs). Elemental

compositions and mass densities assumed within the PIRT model are shown in Table 3-1.

When the particle is shown to leave the spongiosa of the CT macroimage, tracking within

the NMR microimage is halted and the particle is transported within a homogeneous

region of cortical bone defined only by the larger voxels of the ex-vivo CT macroimage.

Upon particle escape from outer surface of the bone site, particle tracking is terminated.

In cases where the particle leaves cortical bone and re-enters the interior spongiosa,

particle tracking in the NMR microimage is resumed. The PIRT model is thus far more

anatomically realistic than is the geometry provided by either CBIST or VBIST models,

especially for higher-energy, longer-ranged electrons and beta particles.

The principle approximation inherent within the PIRT model is that the trabecular

microstructure given by the physical section of spongiosa (as imaged via NMR) is









uniform across all CT-segmented regions of spongiosa within the skeletal site. As a

result, the trabecular microstructures of the various vertebral processes (spinal, superior

articular, and transverse) are implicitly assumed to be approximated by that imaged

within the vertebral body. In cases where more than one physical section of spongiosa

has been imaged by NMR, the PIRT model can be re-run using different NMR

microimages. The resulting microimage-specific absorbed fraction profiles can thus be

averaged either uniformly or weighted by the volume of spongiosa sectioned. Finally, it

is noted that the PIRT model permits explicit consideration of a cortical bone volume

(CBV) as a potential radioactivity source a feature not permitted within the CBIST or

VBIST models of skeletal dose.

Paired-Image Radiation Transport Model (Proximal Femur)

A corresponding schematic of the PIRT model for the right proximal femur of the

66-year male subject is shown in Figure 3-2. In adults, hematopoiesis occurs primarily

within the proximal epiphysis of the femur, and thus the macrostructural model (shown in

the upper right of Figure 3-2 and given by the ex-vivo CT) is terminated inferiorly at the

point where the lesser trochanter merges anatomically with the femoral diaphysis. As

with the University of Leeds chord-length measurements for their 44-year male, the

biomechanics and thus the trabecular microstructure are notably different within femoral

head and femoral neck; consequently, 3D NMR microscopy images were taken

separately from the head and neck regions of the proximal femur. Representative

transverse NMR image slices are shown in the lower middle and lower right of Fig. 3-2.

For each tissue source region in the model (TAM, TBV, or TBS), two different transport

simulations are performed one in which electrons are started within the spongiosa of the

femoral head (orange voxels of the ex-vivo CT transverse slice), and one in which









electrons are started within the spongiosa of the femoral neck (red voxels of the ex-vivo

CT transverse slice). In each case, only the corresponding NMR microscopy image is

used within the PIRT model (head or neck microimage). Final absorbed fraction results

for the entire proximal femur are taken as mass weighted averages of results from the

head-only and neck-only spongiosa source transport calculations. Table 3-2 displays the

various source and target tissues masses for both the proximal femur and lumbar vertebra

PIRT dosimetry models (given as the product of their segmented volume and the

reference densities of Table 3-1). The final row of Table 3-2 gives values of marrow

volume fraction (MVF) defined as the fraction of all voxels within the NMR microimage

that are assigned to marrow tissues following image thresholding. Here it is noted that

the MVF of the femoral head is 64.5%, while it is 75.5% within the femoral neck. The

MVF within the L4 vertebral body, however, was measured at 87%.

Results

Absorbed Fractions to Active Marrow within the L4 Vertebra

Figures 3-3 and 3-4 display values of electron absorbed fraction to active (red)

bone marrow within the L4 vertebra of the 66-year male subject. Figure 3-3 corresponds

to an assumption of 100% marrow cellularity (no voxels of adipose tissue are labeled

within the NMR microimage), while Figure 3-4 corresponds to an assumed marrow

cellularity of 70% (reference value in both ICRP Publications 70 and 89) (1995; 2002).

In each graph, solid lines indicate energy-dependent absorbed fractions obtained from

PIRT model simulations, while dashed lines indicate those derived from VBIST model

simulations. For either model and at both cellularities, three source tissues are

considered: TAM (diamonds), TBS (triangles), and TBV (circles).









At source energies below -100 keV, the two model types yield essentially

equivalent results, as boundary effects at the spongiosa-cortical bone interface (within the

PIRT model) play a negligible role in modifying the pattern of energy deposition to

active marrow voxels (as seen within the VBIST model). Model equivalency is noted to

extend to electrons of -200 keV initial energy when emitted within the volume of the

bone trabeculae (TBV sources).

As the electron initial energy increases above 100-200 keV, energy deposition to

active marrow as predicted under VBIST model simulations increasingly over-predicts

that given by the more anatomically realistic PIRT model. As previously noted for

chord-based skeletal models under either CBIST or VBIST simulations, absorbed

fractions asymptotically approach a limited value independent of the source tissue

(Eckerman 1985; Bouchet 1999; Jokisch 2001b). At 100% cellularity, the VBIST model

absorbed fraction to active marrow approaches a value of 0.76 at high electron energies,

while it approaches a limiting value of 0.53 at 70% cellularity (70% of 0.76). Similarly,

absorbed fractions to active marrow predicted under PIRT model simulations also

converge in a source-independent manner, but this convergence value is energy

dependent as more and more electron energy is lost to the surrounding cortical bone (and

potentially surrounding tissues). With the PIRT model results serving as the local

standard, percent errors in self-absorbed fraction to active marrow given by the VBIST

model are 7% at 500 keV, 16% at 1 MeV, and 85% at 4 MeV. Corresponding percent

errors are 7%, 16%, and 88% for TBS sources, and 7%, 18%, and 89% for TBV sources.

These percent errors are roughly equivalent at both marrow cellularities.









Absorbed Fractions to Active Marrow within the Proximal Femur

Figures 3-5 and 3-6 display values of electron absorbed fraction to active marrow

for TAM, TBS, and TBV sources located within the spongiosa of the right proximal

femur of the 66-year male subject. Figures 3-5 and 3-6 correspond to marrow

cellularities of 100% and 25%, respectively, where the latter is the default cellularity for

the upper femur given in ICRP Publications 70 and 89. In each graph, the individual

absorbed fraction profiles for electron sources in the femoral head and in the femoral

neck have been averaged according to the total mass of source tissue in the head and neck

regions of the proximal femur, respectively. In Figure 3-6, the ordinate has been

expanded to better view differences in modeling results at high electron energies. At the

lowest energy considered (10 keV), a value of 4((TAM*-TAM) = 0.98 is seen under both

VBIST and PIRT simulations.

Patterns of divergence between the two modeling approaches (VBIST versus PIRT)

in the proximal femur are seen to occur at lower energies compared to those found within

the L4 vertebra (-100 keV for TAM sources, -50 keV for TBS sources, and -100 keV for

TBV sources). Furthermore, it is seen that at 4 MeV (the highest energy considered), full

convergence of the absorbed fraction to active marrow under both VBIST and PIRT

model simulations has not yet been reached for the three source regions. Nevertheless,

the energy-independent (VBIST) and energy-dependent (PIRT) patterns of convergence

are still evident at electron initial energies exceeding 1 MeV. With the PIRT model

results serving as the local standard, percent errors in self-absorbed fraction to active

marrow (100% cellularity) given by the VBIST model are 6% at 500 keV, 12% at 1

MeV, and 31% at 4 MeV. Corresponding percent errors are 22%, 27%, and 44% for TBS









sources, and 12%, 21%, and 44% for TBV sources. These percent errors are -20-50%

higher when the marrow cellularity of the proximal femur is reduced to 25% (fat fraction

of -75%).

Absorbed Fractions to Endosteal Tissues

Figures 3-7 and 3-8 display values of absorbed fraction to the trabecular endosteal

tissues defined as a 10-[tm layer of soft tissue on the marrow-side of the bone-marrow

interface within the NMR microimages. Figure 3-7 gives results for TBS, TBV, and

TAM electron sources emitted within the L4 vertebra containing bone marrow at 70%

cellularity. Figure 3-8 shows data for these same source tissues within the right proximal

femur at 25% marrow cellularity. In both graphs, the ordinate scale is expanded to a

maximum value of 0.16 to facilitate viewing model differences at higher energies. At the

lowest energy considered (10 keV), a value of 4 (TBE*-TBS) = 0.5 is seen under both

VBIST and PIRT simulations.

At each energy for each model, higher absorbed fractions are noted for electron

sources on the trabecular surfaces, while lower absorbed fractions are seen for electron

sources emitted within the active bone marrow. Intermediate absorbed fractions are

shown for bone volume sources which peak in value at a source energy of -100 keV in

both skeletal sites. As expected, VBIST model simulations approach energy- and source-

independent convergence values at high electron initial energies (0.032 in the L4 vertebra,

and 0.045 in the proximal femur), while source-independent convergence values for the

PIRT model are shown to continually decline with increasing source energy above 1

MeV. This decline is slightly more prominent in the L4 vertebra than seen in the

proximal femur, and is accountable in part by cortical bone losses within the vertebral









processes. In these anatomic regions of the vertebra (which encompass -25% of total

vertebral spongiosa), the surface-to-volume ratio of trabecular spongiosa is higher than

that found in the vertebral body, and thus electron escape to cortical bone is greater for

individual electron emissions.

Discussion

As a further means of comparing the VBIST and PIRT model results, radionuclide

S values were calculated for a wide range of beta-particle emitters of interest in skeletal

tissue imaging and radionuclide therapy. Absorbed fractions to active bone marrow

given in Figures 3-3 through 3-6, along with tissue mass data of Table 3-2 and beta-

particle energy spectra from Eckerman et al. (1994), were used to calculate S values

under the MIRD schema for nine different radionuclides. Ratios of the S value based on

VBIST-model absorbed fractions to that using PIRT-model absorbed fractions are

displayed in Table 3-3 for both skeletal sites and at both 100% and ICRP-reference

marrow cellularities. For low-energy beta-emitters such as 33P, 169Er, and 177Lu, absorbed

fractions given by the VBIST model simulations overestimate radionuclide S values for

TAM, TBS, and TBV sources by only 1% to 5% in the L4 vertebra. Higher errors are

noted in the proximal femur, particularly for bone trabeculae volume sources (ratios of

1.17 to 1.23). For radionuclides at intermediate beta energies (Eave of 225 keV to 583

keV), S value ratios range from 1.05 to 1.14 in the L4 vertebra and from 1.04 to 1.26 in

the proximal femur. For radionuclides in the highest beta-energy range (Eave of 695 to

934 keV), S value ratios range from 1.15 to 1.24 in the L4 vertebra and from 1.11 to 1.30

in the proximal femur. It is reasonable to assume that similar errors are also present in

radionuclide S values derived from chord-based models (Eckerman 2000; Bouchet 2000)









which, as in the VBIST simulations of the present study, assume an infinite region of

spongiosa during particle transport.

Prior to the full development of the paired-image radiation transport methodology

given here, the UF ALRADS research group had attempted to correct for energy loss to

cortical bone by applying a stylized model of the skeletal site macrostructure. For

example, in the study by Patton et al. (2002), a spherical region of spongiosa surrounded

by a spherical shell of cortical bone was applied to the femoral heads of three different

individuals based upon CT image analysis. In that study, it was demonstrated that

infinite spongiosa transport yielded radionuclide S values for 32P that were -5-8% higher

than those in which cortical bone energy loss was accounted for via stylistic modeling of

the femoral head. For the higher-energy 90Y, the infinite spongiosa transport results gave

S values 8% to 11% higher. In the present study, however, the full 3D histological

macrostructure of the proximal femur (head as well as neck and trochanter regions) is

treated within the PIRT model simulations. Corresponding corrections to infinite

spongiosa transport are shown in the present study (by the PIRT model) to be

significantly higher (up to 1.26 for 32P and up to 1.30 for 90Y) than indicated previously

by Patton et al. (2002) for the femoral head. These larger corrections are attributed to

enhanced particle energy loss at three spongiosa regions of the PIRT femur model: the

femoral neck, the trochanters, and the bottom interface of the model (where particles are

lost to inactive marrow of the femoral diaphysis see Fig. 3-2). These regions of

enhanced electron escape were not present within the spherical femoral head model of the

Patton et al. study.









Improved macrostructural modeling of the skeleton via the PIRT model

methodology will potentially lead to improvements in correlations between marrow dose

estimates and observed patient myelotoxicity. For example, clinical studies of the bone

pain palliation agents 153Sm-EDTP (Turner 1991; Collins 1993; Farhanghi 1992) and

186Re-HEDP (Kucuk 2000; Giannakenas 2000; Breitz 1998) have shown patient marrow

toxicities that were lower than expected based on marrow dose estimates from standard

CBIST skeletal dose models (e.g., MIRDOSE2 and MIRDOSE3). While various studies

have been initiated to explain these discrepancies including improvements in activity

uptake quantification (van Rensburg 1998; Brenner 2001), the data of Table 3-3 indicates

that perhaps values of marrow dose were simply overestimated in these studies, as the

standard clinical models do not properly account for particle escape from marrow-filled

regions of spongiosa. For both bone surface and volume sources, infinite spongiosa

transport is shown in Table 3-3 to overestimate the femoral marrow self-dose by 10-22%

for 153Sm and 13-24% for 186Re, while the vertebral marrow self-dose is overestimated by

6% for 153Sm and 9% for 186Re.

Conclusion

A paired-image radiation transport (PIRT) model for skeletal dosimetry is

presented in which electrons and beta particles are tracked simultaneously within two

different segmented digital images: (1) an ex-vivo CT image of the skeletal site with

segmented regions of trabecular spongiosa, cortical bone, and surrounding tissues, and

(2) an in-vitro NMR microscopy image of the interior bone trabeculae and marrow cavity

microstructure representative of that found within spongiosa regions of the ex-vivo CT

image. Example dose calculations under the PIRT methodology within the L4 vertebra

and right proximal femur of an adult 66-year male subject demonstrate a divergence from









standard infinite spongiosa transport (VBIST) methods at energies as low as 50-200 keV

depending upon the source tissue and skeletal site. Calculations of radionuclide S values

under both methodologies imply that current chord-based models used in clinical skeletal

dosimetry may over-estimate dose to active bone marrow in these two skeletal sites by

-4% to 23% for low-energy beta emitters (33P, 169Er, and 177Lu), by -4% to 25% for

intermediate-energy beta emitters (153Sm, 186Re, and 89Sr), and by ~11% to 30% for high-

energy beta emitters (32P, 8Re, and 90Y). Higher errors are noted for bone-volume

seekers, while lower errors are seen for source emissions within the active bone marrow.

While the proximal femur and lumbar vertebra are investigated in the present study,

potentially larger errors in skeletal dosimetry are presumed to exist in skeletal sites with

disproportionately smaller volumes of spongiosa (e.g., ribs, cranium, and sternum).

The PIRT methodology supersedes previous stylized modeling attempts by the UF

ALRADS research group to account for the finite spatial extent of trabecular spongiosa

and the presence of cortical bone. This approach thus renders obsolete any need for

mathematical modeling of the either simple or complex bone site geometries.

Furthermore, the technique increases the prospects for expanded availability of reference

skeletal dosimetry models for both genders and of individuals of varying stature and

skeletal size for use in radionuclide therapy treatment planning of cancer in which

marrow toxicity is of concern.






































Figure 3-1. Schematic of the PIRT model constructed for the L4 vertebra.


Ax




















~, U


Figure 3-2. Schematic of the PIRT model constructed for the right proximal femur


I "A
t4











1.0 4!
1.0 : -. ,. L4 Vertebra -
0.9 s100% Marrow Cellularity
0.9 :---------
TAM Source
0.8

0.7 _

C 0.6

m 0.5

S0.4
o TBS Source ___________________
0.3 1_T S --0- Infinite Spongiosa (TAM)
~ 0-- /A- Infinite Spongiosa (TBV)
0.2- -0- Infinite Spongiosa (TBS)
o. i ), TBV Source -*-Paired Image (TAM)
0.1 A ---Paired Image (TBV)
-*-Paired Image (TBS)
0.0 '
0.01 0.1 1 10
Electron Energy (MeV)


Figure 3-3. Electron absorbed fractions to active bone marrow within the L4 vertebrae for
three source tissues TAM, TBV, and TBS. Data shown by solid lines are
from the PIRT model, while those given by dashed lines are from the IST
model. Data for the figure correspond to 100% marrow cellularity.













1.0
-0- Infinite Spongiosa (TAM)
0.9 _- --A- Infinite Spongiosa (TBV)
0 --0- Infinite Spongiosa (TBS)
0.8 _- Paired Image (TAM)

0 TAM Source -- Paired Image (TBV)
-*-Paired Image (TBS)

C 0.6

U-

0.4






0.1 ourcL4 Vertebra -
70% Marrow Ce/llula/rity

0.01 0.1 1 10
Electron Energy (MeV)


Figure 3-4. Electron absorbed fractions to active bone marrow within the L4 vertebrae at
reference cellularity for three source tissues TAM, TBV, and TBS. Data
shown by solid lines are from the PIRT model, while those given by dashed
lines are from the IST model. Data correspond to the ICRP 70 reference
cellularity of 70%.












1.00

0.90

0.80

< 0.70
I-
o
C 0.60
o
m 0.50

. 0.40

0.30

0.20

0.10


0.00
0.


-0- Infinite Spongiosa (TAM)
Proximal Femur -- Infinite Spongiosa (TBV)
100% Marrow Cellularity -0-Infinite Spongiosa (TBS)
--4-- Paired Image (TAM)
-h-Paired Image (TBV)
TAM S e ---0*-Paired Image (TBS)
TAM Source / -






TBS Source f / A






TBV Source


01 0.1 1 1
Electron Energy (MeV)


Figure 3-5. Electron absorbed fractions to active bone marrow within the proximal femur
for three source tissues TAM, TBV, and TBS. Data shown by solid lines are
from the PIRT model, while those given by dashed lines are from the IST
model. Data correspond to 100% marrow cellularity.













0.40
Proximal Femur -
25% Marrow Cellularity -c- Infinite Spongiosa (TAM)
0.35 -A- Infinite Spongiosa (TBV)
-0- Infinite Spongiosa (TBS)
S4-- Paired Image (TAM)
T 0.30 A Paired Image (TBV)
STAM Source Paired Image (TBS)
0 0.25
C

0.20
J-


0 .1 5 T B S S o u r c e



0.05

TBV Source
0.00
0.01 0.1 1 10
Electron Energy (MeV)


Figure 3-6. Electron absorbed fractions to active bone marrow within the proximal femur
at reference cellularity for three source tissues TAM, TBV, and TBS. Data
shown by solid lines are from the PIRT model, while those given by dashed
lines are from the IST model. Data correspond to the ICRP 70 reference
cellularity of 25%.







43



0.16
-0- Infinite Spongiosa (TAM)
0.14 -A- Infinite Spongiosa (TBV)
-- -0- Infinite Spongiosa (TBS)
TBS Source Paired Image (TAM)
0.12 -- Paired Image (TBV)
-.1 -0- Paired Image (TBS)
o 0.10
C \ L4 Vertebra -
.2 0\ 70% Marrow Cellularity
M 0.08
LL..
-o TBV Source
0.06


0.04 ..


0.02
TAM Source
0.00 I , I
0.01 0.1 1 10
Electron Energy (MeV)


Figure 3-7. Electron absorbed fractions to the trabecular bone endosteum within the L4
vertebra for three source tissues TAM, TBV, and TBS. Data shown by solid
lines are from the PIRT model, while those given by dashed lines are from the
IST model.












0.16


0.14


w 0.12

o
I-
0
- 0.10
0

E 0.08
U-

S0.06
0
o

<* 0.04


0.02


0.00


0.01


Electron Energy (MeV)


Figure 3-8. Electron absorbed fractions to the trabecular bone endosteum within the
proximal femur for three source tissues TAM, TBV, and TBS. Data shown
by solid lines are from the PIRT model, while those given by dashed lines are
from the IST model.


-0- Infinite Spongiosa (TAM)
--A- Infinite Spongiosa (TBV)
-0- Infinite Spongiosa (TBS)
TBS Source -*-Paired Image (TAM)
-i Paired Image (TBV)
-- Paired Image (TBS)

\Proximal Femur -
;25% Marrow Cellularity


TBV Source .








TAM Source












Table 3-1. Tissue compositions (% by mass) and mass densities used in either the IST and PIRT models of skeletal dosimetry.


Tissue or Region a

Trabecular Active Marrow (TAM)
Trabecular Inactive Marrow (TIM)
Trabecular Bone Endosteum (TBE)
Trabecular Bone Volume (TBV)
Cortical Bone Volume (CBV)
Surrounding Tissues
a TAM "adult red marrow", TIM -


H C N

10.5 41.4 3.4
11.5 64.4 0.7
10.5 25.6 2.7
3.4 15.5 4.2
3.4 15.5 4.2
10.5 25.6 2.7
"adult yellow marrow", TBE -


0


Trace


Mass Density
(g cm-3)


43.9 0.1 P, 0.2 S, 0.2 Cl, 0.2 K, 0.1 Fe 1.03
23.1 0.1 Na, 0.1 S, 0.1 Cl 0.98
60.2 0.1 Na, 0.2 P, 0.3 S, 0.2 Cl, 0.2 K 1.03
43.5 0.1 Na, 0.2 Mg, 10.3 P, 0.3 S, 22.5 Ca 1.92
43.5 0.1 Na, 0.2 Mg, 10.3 P, 0.3 S, 22.5 Ca 1.92
60.2 0.1 Na, 0.2 P, 0.3 S, 0.2 Cl, 0.2 K 1.03
"adult ICRU-44 soft tissue (male)", TBV "adult cortical bone",


CBV "adult cortical bone" (Appendix A of ICRU Report 46) (ICRU 1992).












Table 3-2. Tissues masses used in the paired-image radiation transport (PIRT) model (100% marrow cellularity). The marrow volume
fractions are taken from the 3D NMR microscopy images of excised cube of spongiosa.


Tissue / Quantity
Trabecular Active Marrow (TAM)
Trabecular Bone Endosteum (TBE)
Trabecular Bone Volume (TBV)
Cortical Bone Volume (CBV)


L4 Vertebra
153.3 g
3.2 g
117.0g
74.4 g


Femoral Head
15.80 g
0.68 g
4.55 g


Femoral Neck
26.30 g
1.12 g
7.55 g


Proximal Femur
42.1 g
1.8 g
12.1 g
26.6 g


Marrow Volume Fraction (MVF)a 87% 64.5% 75.5%
a Ratio of total marrow voxels to total voxels in the binary 3D NMR microscopy images of trabecular spongiosa.













Table 3-3. Ratio of the radionuclide S value for an active marrow (TAM) target as given by the infinite spongiosa transport (IST)
model to that given by the paired-image radiation transport (PIRT) model.


Radionuclide
P-33
Er-169
Lu-177
Sm-153
Re-186
Sr-89
P-32
Re-188
Y-90


Radionuclide
P-33
Er-169
Lu-177
Sm-153
Re-186
Sr-89
P-32
Re-188
Y-90


Eave
(keV)
77
100
133
225
323
583
695
764
934

Eave
(keV)
77
100
133
225
323
583
695
764
934


Emax
(keV)
239
351
498
809
1075
1492
1854
2000
2282

Emax
(keV)
239
351
498
809
1075
1492
1854
2000
2282


L4 Vertebra 100% Cellularity
TAM Source TBS Source TBV Source
1.01 1.02 1.02
1.02 1.04 1.03
1.03 1.05 1.04
1.05 1.06 1.06
1.08 1.09 1.09
1.13 1.14 1.13
1.15 1.16 1.15
1.17 1.19 1.18
1.21 1.23 1.22

L4 Vertebra 70% Cellularity
TAM Source TBS Source TBV Source
1.01 1.02 1.01
1.02 1.04 1.02
1.03 1.05 1.04
1.05 1.06 1.06
1.08 1.09 1.09
1.13 1.14 1.14
1.15 1.16 1.16
1.18 1.19 1.20
1.22 1.23 1.24


Proximal Femur- 100% Cellularity
TAM Source TBS Source TBV Source
1.02 1.09 1.21
1.02 1.08 1.23
1.03 1.09 1.23
1.04 1.10 1.23
1.06 1.13 1.24
1.10 1.18 1.26
1.11 1.19 1.26
1.12 1.21 1.28
1.14 1.23 1.30

Proximal Femur 25% Cellularity
TAM Source TBS Source TBV Source
1.00 1.03 1.17
1.02 1.05 1.19
1.04 1.07 1.21
1.07 1.10 1.22
1.09 1.13 1.24
1.13 1.18 1.25
1.15 1.19 1.26
1.16 1.21 1.27
1.19 1.24 1.28
















CHAPTER 4
BETA-PARTICLE ENERGY LOSS TO CORTICAL BONE VIA PAIRED-IMAGE
RADIATION TRANSPORT: CORRECTIONS TO CLINICAL MODELS OF
SKELETAL TISSUE DOSE2

Introduction

Accurate models of skeletal tissue dose are needed in both radiation protection

(e.g., predicting risks for leukemia and bone cancer induction following inhalation of

long-lived bone-seeking radionuclides) and in radionuclide therapy (e.g., correlations of

marrow dose and toxicity for radiopharmaceuticals subject to either specific or non-

specific skeletal tissue uptake). Ideally these models must take into account both the

microscopic structure of the bone trabeculae and marrow cavities, as well as the

macroscopic structure of the bone site itself (shape and volume of the trabecular

spongiosa and the exterior cortex of cortical bone). For alpha emitters and low-energy

beta emitters, only the microscopic characterization of the bone site is needed in the dose

model, as these particles typically expend their full emission energy within the trabecular

spongiosa. For intermediate- to higher-energy beta emitters, however, energy loss to the

exterior cortical bone is to be expected, especially at those skeletal sites with high

spongiosa surface-to-volume ratios (flat bones such as the cranium and ribs).

Current models of skeletal dosimetry used in both health physics and medical


2 This chapter has been submitted to Medical Physics: Shah AP, Bolch WE, Rajon DA, Patton PW, and
Jokisch DW. Submitted. Beta-particle energy loss to cortical bone via paried-image radiation transport:
corrections to clinical models of skeletal tissue dose. Med Phys: submitted.









physics track alpha and beta particles within the skeleton through an infinite region of

trabecular spongiosa, thus neglecting effects introduced by the 3D macrostructure of the

bone site. These IST, or infinite spongiosa transport, models use as their input either (1)

linear chord-length distributions measured across the trabeculae and marrow cavities

(Beddoe et al. 1976; Whitwell and Spiers 1976), or (2) 3D digital images of that

microstructure (Jokisch et al. 2001; Patton et al. 2002). Subsequently, we refer to these

two modeling approaches as CBIST (chord-based IST) or VBIST (voxel-based IST)

skeletal dose models, respectively. The skeletal dose model used in current clinical

practice, the Eckerman and Stabin model (2000) of MIRDOSE3 (Stabin 1996) and its

successor code, belongs to the CBIST model classification.

In the present study, we discuss a new approach to skeletal dosimetry using Paired-

Image Radiation Transport (PIRT). In the PIRT skeletal model, radiation particles are

tracked simultaneously within two different segmented digital images: (1) an ex-vivo CT

image of the skeletal site outlining regions of trabecular spongiosa and cortical bone, and

(2) an in-vitro microCT image of the spongiosa microstructure (bone trabeculae and

marrow cavities). In Shah et al. (2004), we compared dosimetry results between VBIST

and PIRT model transport simulations for electron and beta-particle emitters within the

proximal femur and lumbar vertebrae of a 66-year adult male. In the current study, we

extend this comparison to include three other skeletal sites with high percentages of

active bone marrow: the pelvis, cranium, and ribs.

Microimaging of trabecular spongiosa: NMR microscopy vs. microCT. Our

research group has previously reported on the use of NMR microscopy to obtain 3D

microimages of the trabecular micro-architecture for skeletal dosimetry (Jokisch et al.









1998; Jokisch et al. 2001; Bolch et al. 2002; Bolch et al. 2002; Patton et al. 2002; Patton

et al. 2002; Shah et al. 2003; Shah et al. 2004). Optimal images from NMR microscopy

require physical samples of spongiosa be subjected to marrow digestion. The marrow

cavities of the sample are then filled with Gd-doped water for enhanced MR signal from

voxels within the marrow cavities. Marrow digestion is efficient for those skeletal sites

with relatively large and externally accessible marrow cavities (e.g, femur head/neck and

vertebra). In contrast, marrow digestion can be incomplete for skeletal sites with

inaccessible and relatively small marrow cavities (e.g., cranium, sternum, etc.).

Alternatively, sectioned pieces of trabecular spongiosa may be imaged directly via NMR

as marrow-intact samples. Problems with this approach, however, include poor signal-to-

noise ratios (requiring longer and more costly imaging times) and corresponding

difficulties in image segmentation and thresholding (less distinct peaks between marrow

and bone voxels in the gray-level histogram). For marrow-digested samples, strong MR

signals are emitted somewhat uniformly within the water-filled marrow cavities. In

contrast, marrow-intact samples emit MR signals separately from the active (red) and

inactive (yellow or fat) marrow regions of the cavities, thus broadening the marrow signal

peaks in the gray-level histogram and complicating image thresholding of the marrow-

bone interfaces.

For both marrow-digested and marrow-intact samples, one must also contend with

limitations in sample size considering the small imaging bore of high-field NMR

systems. Typically, only a relatively small physical section of the trabecular spongiosa

can be utilized in NMR microscopy, and multiple samples must then be imaged to

properly account for site-to-site variations in the trabecular microstructure across a given









skeletal site (e.g., several spongiosa cubes within the vertebral body of the lumbar

vertebrae).

An attractive alternative to NMR microscopy for skeletal dosimetry is the use of

microCT imaging of physical samples of spongiosa (Ruegsegger et al. 1996; Dufresne

1998; Muller et al. 1998). As with NMR microscopy, enhanced image contrast is

achieved with microCT for marrow-digested samples, without a need to fill the empty

marrow cavities with water to improve signal. However, microCT images of marrow-

intact sections of trabecular spongiosa yield approximately the same level of image

contrast as seen for marrow-digested NMR microimages. MicroCT imaging of marrow-

intact samples is thus a very acceptable method for acquiring microimages for skeletal

dosimetry modeling an option that requires very little sample preparation and thus is

achievable at all skeletal sites regardless of our ability to fully digest the marrow tissues.

Improved signal-to-noise ratios with microCT images also permit enhancements in image

segmentation and thresholding. Typically, one finds a more clear distinction between the

bone peak and marrow peak within the gray-level histogram of the microCT image, over

that seen in comparable images obtained from NMR microscopy (especially for marrow-

intact NMR samples).

The final disadvantage of NMR microscopy is its restriction on sample size.

Currently, with use of the high-field magnets (4.7 T and 11 T) at the Advanced Magnetic

Resonance Imaging and Spectroscopy (AMRIS) facility at the UF McKnight Brain

Institute, the resolution obtained for a spongiosa sample is approximately 63 rtm x 63 rtm

x 63 |tm at a field of view of 3.2 cm x 3.2 cm x 3.2 cm. The corresponding maximum

sample size is a cuboidal section approximately 1.25 cm x 1.25 cm x 2.5 cm on edge.









With microCT, a slightly better voxel resolution of 60 |tm x 60 |tm x 60 |tm can be

achieved for a cubical physical section of spongiosa as large as 5.3 cm x 5.3 cm x 5.3 cm

on edge. It is noted, however, that for many skeletal sites, the physical shape and size of

the bone may not permit a physical sectioning of its spongiosa to this maximum sample

size.

Materials and Methods

Cadaver Selection

Candidate subjects for study were obtained through the State of Florida Anatomical

Board located on the University of Florida (UF) campus. Cadaver selection criteria

included (1) an age between 50 75 years (representative of typical radionuclide therapy

patients), (2) a body mass index of 18.5 25 kg m-2 (CDC recommended healthy range),

and (3) a cause of death that would preclude significant skeletal deterioration. The

subject identified was a 66-year male approximately 68 kg in total mass and 173 cm in

total height at the time of death (BMI of 22.7 kg m-2). The subject died suddenly of

complications associated with cardiomyopathy.

In-Vivo Computed Tomography Scanning

Prior to bone harvesting, the male cadaver was subjected to whole-body imaging

via multi-slice helical CT at a pitch necessary to reconstruct contiguous 1-mm axial

slices. The images were acquired on a Siemens Sensation 16 unit within the Department

of Radiology at UF Shands Hospital. Image reconstruction was performed with a bone

filter at an in-plane pixel resolution of 977 |tm x 977 |tm. The CT image sets were then

transferred to workstations within the Advanced Laboratory for Radiation Dosimetry

Studies (ALRADS) in the UF Department of Nuclear & Radiological Engineering for

image processing and data storage. The in-vivo CT scans provided image data in order to









(1) select the anatomical region from which the bone site would be harvested, and (2)

construct 3D anatomic models of skeletal sites where bone harvesting (and thus ex-vivo

CT scanning) might be incomplete (e.g., rib cage).

Bone Harvesting and Ex-Vivo Computed Tomography Scanning

Following detailed review of the whole-body in-vivo CT images, bone harvesting

was conducted. Fourteen major skeletal sites were taken from the male cadaver including

the pelvis (pelvis), the cranium (cranial cap), and several ribs from both the right and left

side of the rib cage. Once each skeletal site was excised, it was cleaned of excess tissue,

bagged, labeled, and stored frozen until ex-vivo CT imaging could be scheduled. Post-

harvest, ex-vivo CT imaging was conducted at the highest resolution permitted based on

sample size (1.0 mm slice thickness with an in-plane resolution of 0.65 mm x 0.65 mm

for the pelvis, 0.23 mm x 0.23 mm for the ribs). The ex-vivo CT scans provided image

data for (1) identifying the location and extent of trabecular spongiosa to be sectioned for

microCT imaging; (2) quantifying both trabecular spongiosa and cortical bone volumes

within the bone site; and (3) constructing 3D anatomic models of the bone site for

subsequent paired-image radiation transport simulations.

Following detailed review of the ex-vivo CT scans; physical sections of trabecular

spongiosa were taken from each bone site. Sections representing as large a region of

spongiosa as possible were taken, given the constraints of the bone shape and the

microimaging system (e.g., cuboidal samples taken from a spherically shaped femoral

head). Marrow-intact sections of spongiosa were bagged, labeled, and kept frozen until

microimaging sessions were arranged. For the left parietal bone, 2 cuboidal sections

(roughly 4.9 cm x 2.8 cm x 1.3 cm on edge) were cut from the cranial section

representing -10% of the total spongiosa within the cranial cap. For the left middle rib, 4









cylindrical sections were cut (-12% of total spongiosa within the left side of the rib

cage), and 6 sections were cut from different bones of the pelvis (-25% of the total

spongiosa within the entire pelvis). These physical sections of trabecular spongiosa were

originally intended to be imaged via NMR microscopy, and thus they were cut at sizes

less than the maximum sizes permitted by microCT imaging (given above).

Image Segmentation of Spongiosa and Cortical Bone Regions

To create tomographic anatomic models for use in internal dosimetry, radiation

transport codes must be able to decipher the boundaries of each tissue region for which

an independent dose assessment is to be made. Limitations of CT image acquisition can

result in an overlap of grayscale values for tissues of interest, thus precluding the use of

simple automated methods of boundary definition. In the present study, the program

CT_Contours was adopted for use in segmenting spongiosa and cortical bone within each

ex-vivo CT image set (Nipper et al. 2002). This program is based upon Interactive Data

Language (IDL) version 5.5 and can output labeled contour files in a variety of formats

including binary files for EGSnrc (Kawrakow 2000) and ASCII text for MCNP

(Briesmeister 1997). CT_Contours displays the current CT information, as well as a

color overlay of the contours being edited. The contours can be created using a variety of

tools including: basic thresholding, pixel growing, voxel growing, region growing, and

manual segmentation. The voxels contained in the individual contours are filled with the

desired segmentation value, generating volumes of voxels with identical tag values. In

the present study, these volumes represent individual regions of either trabecular

spongiosa or cortical bone within the skeletal site. CT_Contours was written to have the

option of displaying the images using 15 different filters including Gaussian smoothing

(3 x3, 5 x5, or 7 x7), median (3 x3, 5 x5, or 7 x7), Roberts edge detection, Sobel edge









detection, Prewitt edge detection, isotropic edge detection, histogram equalization,

adaptive histogram equalization, sharpening and Kuwahara (3 x3 or 5 x5) filtering. By

altering regions of a separate contour dataset, the desired segmentation can be performed.

CT_Contours was designed so that ROI creation or modification can be performed in

either the transverse, sagittal, or coronal plane.

Micro-Computed Tomography of Trabecular Spongiosa

Micro-tomographic imaging of cuboidal samples of spongiosa was performed on

desktop cone-beam [tCT40 or [tCT80 scanners (Scanco Medical AG, Bassersdorf,

Switzerland) yielding 3D image data sets at a voxel resolution of 60 |tm x 60 |tm x 60

itm. Although a resolution of 30 |tm3 could be obtained at an equivalent sample size, the

higher resolution images exceed the maximum allowable binary array size of both the

image processing and radiation transport codes. Post-acquisition image processing steps

included (1) selection of an ideal volume of interest, (2) gray-level thresholding, (3)

voxel segmentation, and (4) 3D median filtering, all of which have been previously

reported in Jokisch et al. (1998) and Patton et al.(2002).

Voxel-Based Infinite Spongiosa Transport (VBIST) Model

Following microCT imaging of our skeletal samples, a series of VBIST models

were created to approximate (via 3D transport) the results of current CBIST models.

First, marrow voxels within the binary microCT image are further labeled into voxels of

active (red) marrow and inactive (yellow) marrow at a pre-determined value of marrow

cellularity. This process has been outlined previously by Shah et al. (2003), and is based

upon microscopy measurements of the spatial distribution of adipocytes within normal

bone marrow biopsies covering a broad range of marrow cellularities. The trabecular









bone endosteum (TBE) is further defined as a 10-jtm at the bone-marrow interface as

previously described by Jokisch (2001). The resulting 4-tissue 3D model of trabecular

spongiosa is coupled to the EGSnrc radiation transport code for electron (beta particle)

transport simulations. Source tissues include the trabecular active marrow (TAM),

trabecular bone surfaces (TBS), and trabecular bone volume (TBV). Trabecular bone

surface (TBS) sources are approximated as a 0.1-[tm layer on the marrow side of the

bone-marrow voxel interface. Target tissues include both the active marrow and bone

endosteum. Once a given electron reaches the physical edge of the 3D micro-image, that

particle is re-introduced to the image at a corresponding location at its opposing edge.

The processes of particle transport within the image of spongiosa and its re-introduction

are continued until all initial kinetic energy is expended. Particle histories are continued

(50,000 to 2,000,000) until coefficients of variation on the absorbed fraction are below

1%.

Paired-Image Radiation Transport (PIRT) Model

The Paired-Image Radiation Transport or PIRT model supplements the 3D

microscopic histology provided by the microCT image with the 3D macroscopic

histology given in the corresponding ex-vivo CT image. The latter provides additional

data for particle transport including (1) the spatial extent of the trabecular spongiosa (e.g.,

ilium, pubis and ischium bones of the pelvis) and (2) the spatial extent of the surrounding

cortical bone (which laterally encompasses the entire pelvis).

A schematic of the PIRT model of the pelvis from the 66-year male is given in

Figure 4-1, where the ex-vivo CT image is shown in the upper left. A representative

coronal slice is shown in the upper right where regions of spongiosa (orange) and cortical









bone (blue) are differentiated. Superimposed over the entire ex-vivo CT image is a 3D

array of replicate cuboidal microCT images each representing the 3D microstructure of

the individual bone trabeculae and corresponding marrow cavities. A 3D rendering of the

microCT image is thus shown in the lower left of Figure 4-1. Finally, a single transverse

slice through the microCT image is shown in the lower right displaying individual voxels

of bone (black) and total marrow (white), a pattern inverted from that within the original

microCT image.

In the EGSnrc implementation of the PIRT model, individual electrons are tracked

simultaneously within the coordinates of the microCT image (indicating locations in

TBV, TBE, TAM, or trabecular inactive marrow TIM), and the coordinates of the CT

macroimage (indicating locations in either spongiosa, cortical bone volume CBV, or

surrounding tissues muscle or soft tissue). Elemental compositions and mass densities

assumed within the PIRT model are shown in Table 3-1 (refer to Chapter 3). When the

particle is shown to leave the spongiosa of the CT macroimage, tracking within the

microCT image is halted and the particle is transported within a homogeneous region of

cortical bone defined only by the larger voxels of the ex-vivo CT macroimage. Upon

particle escape from the outer surface of the bone site, particle tracking is terminated. In

cases where the particle leaves cortical bone and re-enters the interior spongiosa, particle

tracking in the microCT image is resumed. The PIRT model is thus far more

anatomically realistic than is the geometry provided by the VBIST model, especially

when accounting for higher-energy, longer-ranged electrons.

The principle approximation inherent within the PIRT model is that the trabecular

microstructure given by the physical section of spongiosa (as imaged via microCT) is









uniform across all CT-segmented regions of spongiosa within the skeletal site. As a

result, the trabecular microstructures of the various other regions of the pelvis (pubis and

ischium) are implicitly assumed to be approximated by that imaged within the ilium. In

cases where more than one physical section of spongiosa has been imaged by microCT,

the PIRT model can be re-run using different microimages representative of different

spongiosa regions of the bone site. The resulting microimage-specific absorbed fraction

profiles can thus be averaged either uniformly or weighted by the volume of spongiosa

sectioned. In the case of the pelvis, the microstructure of the pubis and ischium can be

sampled, utilized, and the resulting transport data can be averaged. Finally, it is noted

that the PIRT model permits explicit consideration of a cortical bone volume (CBV) as a

potential radioactivity source a feature not permitted within chord-based models of

skeletal dose (CBIST).

In this study, two other bone sites representative of flat bones in the human body

were subjected to electron transport within the PIRT model: the ribs and the cranium. As

with the pelvis, the cranium and ribs have several regions in which sampling of the

trabecular structure can be performed. In the case of the cranium, final dosimetry data

can be averaged from sampling of the frontal, occipital, left parietal and right parietal

bones. In the present study, we focus on the microstructure of the left parietal bone as

shown in Figure 4-2. The upper left corner of the Figure 4-2 shows the ex-vivo image of

the cranial cap. Only the outer cortex of the cranium can be seen with the coronal suture,

nearly transverse in direction, between the frontal and parietal bones, and the sagittal

sutures, medially placed, between the right and left parietal bones. Two representative

transverse slices are shown (upper middle and upper right) where regions of spongiosa









(occipital, frontal, right parietal, left parietal) and cortical bone are again differentiated.

A 3D rendering of the microCT image of the parietal bone is thus shown in the lower left

of Figure 4-2. Finally, one transverse and one coronal slice through the microCT image

are shown in the lower right displaying individual voxels of bone (black) and total

marrow (white).

The need for multiple spongiosa sampling sites also occurs in the ribs, in which the

left and right rib cages each contain 12 individual rib bones. To accurately sample the

trabecular microstructure of the rib cage, 3 ribs were chosen from the right and left side.

In the present study (Figure 4-3), we focus only on a single rib the middle or 7th rib of

the left rib cage. The upper left image in Figure 4-3 shows the spongiosa regions in the

middle portions of both the left and right rib cage. Differentiation of spongiosa and

cortical bone within the left middle rib are shown in the upper right. A 3D rendering of

the microCT image for the middle left rib is shown in the lower left of Figure 4-3. A

single transverse and coronal slice through the microCT image shown in the lower right

displaying individual voxels of bone (black) and total marrow (white).

Table 4-1 displays the various source and target tissues masses for the pelvis,

cranium, and left rib cage of the 66-year male subject. Values for cortical bone mass are

estimated as the product of the tissue density (1.92 g cm-3) and their cortical volumes

from either the in-vivo CT image (left rib cage) or ex-vivo CT image (pelvis and

cranium). Mass estimates for total marrow, bone endosteum, and bone trabeculae in each

skeletal site are calculated as the product of (1) the total spongiosa volume from the CT

macroimage (in-vivo or ex-vivo), (2) the tissue volume fraction taken from the microCT

image, and (3) the tissue density (values given in Chapter 3, Table 3-1). As an example,









values of the marrow volume fraction (MVF) the fraction of tissue volume assigned to

marrow in the segmented microCT image are given at the bottom of Table 4-1 for the

left ilium (85.3%), left parietal bone (60.0%), and left middle rib (88.8%), respectively.

Results

Absorbed Fractions to Active Marrow within the Pelvis

Figure 4-4 and Figure 4-5 display values of electron absorbed fraction to active

(red) bone marrow within the pelvis of the 66-year male subject. Figure 4-4 corresponds

to an assumption of 100% marrow cellularity (no voxels of adipose tissue are labeled

within the microCT image), while Figure 4-5 corresponds to an assumed marrow

cellularity of 48% (reference adult value in both ICRP Publications 70 and 89) (ICRP

1995; ICRP 2002). In each graph, solid lines indicate energy-dependent absorbed

fractions obtained from PIRT model simulations, while dashed lines indicate those

derived from VBIST model simulations. For either model and at both cellularities, three

source tissues are considered: active marrow (diamonds), bone surfaces (triangles), and

bone trabeculae (circles).

The two model types yield essentially equivalent results only at electron energies

below -50 keV where boundary effects at the spongiosa-cortical bone interface (within

the PIRT model) play a negligible role in modifying the pattern of energy deposition to

active marrow voxels (as seen within the VBIST model). Model equivalency is noted to

extend to electrons of -80-100 keV initial energy when emitted along the surfaces of the

bone trabeculae (TBS sources).

As the electron initial energy increases above 50-100 keV, energy deposition to

active marrow as predicted under VBIST model simulations increasingly over-predicts

that given by the more anatomically realistic PIRT model. As previously noted for









skeletal models under either CBIST or VBIST simulations, absorbed fractions

asymptotically approach a limited value independent of the source tissue (Eckerman

1985; Bouchet et al. 1999; Jokisch et al. 2001b). At 100% cellularity, the VBIST model

absorbed fraction to active marrow approaches a value of -0.75 at high electron energies,

while it approaches a limiting value of 0.36 at 48% cellularity (48% of 0.75). Similarly,

absorbed fractions to active marrow predicted under PIRT model simulations also

converge in a source-independent manner, but this convergence value is noted to be

energy dependent as more and more electron energy is lost to the surrounding cortical

bone (and potentially surrounding tissues). With the PIRT model results serving as the

local standard, percent errors in self-absorbed fraction to active marrow given by the

VBIST model are 17% at 500 keV, 34% at 2 MeV, and 70% at 4 MeV. Corresponding

percent errors are 8%, 30%, and 68% for TBS sources, and 22%, 36%, and 72% for TBV

sources. These percent errors are roughly equivalent at both marrow cellularities.

Absorbed Fractions to Active Marrow within the Cranium

Figure 4-6 and Figure 4-7 display values of electron absorbed fraction to active

marrow for TAM, TBS, and TBV sources located within the spongiosa of the cranium of

the 66-year male subject. Figures 4-6 and Figure 4-7 correspond to marrow cellularities

of 100% and 38%, respectively, where the latter is the default cellularity for the cranium

given in ICRP Publications 70 and 89. For either model and at both cellularities, three

source tissues are considered: active marrow (diamonds), bone surfaces (triangles), and

bone trabeculae volumes (circles). In Fig. 4-7, the ordinate has been expanded to better

view differences in modeling results at high electron energies. At the lowest energy









considered (10 keV), a value of unity for 4((TAM*-TAM) is seen under both VBIST and

PIRT simulations.

Patterns of divergence between the two modeling approaches (VBIST versus PIRT)

in the cranium are seen to occur at higher energies compared to those found within the

pelvis (-100 keV for TAM and TBS sources). Model equivalency is noted to extend to

electrons of -200 keV initial energy when emitted within the volume of the bone

trabeculae (TBV sources). At 100% cellularity, the VBIST model absorbed fraction to

active marrow approaches a value of 0.44 at high electron energies, while it approaches a

limiting value of 0.17 at 38% cellularity (38% of 0.44). Similarly, absorbed fractions to

active marrow predicted under PIRT model simulations also converge in a source-

independent manner, but again this convergence value is energy dependent. With the

PIRT model results serving as the local standard, percent errors in self-absorbed fraction

to active marrow (100% cellularity) given by the VBIST model are 18% at 500 keV, 88%

at 2 MeV, and 200% at 4 MeV. Corresponding percent errors are 22%, 93%, and 208%

for TBS sources, and 21%, 93%, and 208% for TBV sources. Similar to the pelvis, these

percent errors are roughly equivalent when the marrow cellularity of the cranium is

reduced to 38% (fat fraction of -62%).

Absorbed Fractions to Active Marrow within the Rib Cage

Figures 4-8 and 4-9 display values of electron absorbed fraction to active marrow

for TAM, TBS, and TBV sources located within the spongiosa of the left rib cage of the

66-year male subject. Figure 4-8 and 4-9 correspond to marrow cellularities of 100% and

70%, respectively, where the latter is the default cellularity for the ribs given in ICRP

Publications 70 and 89. In each graph, solid lines indicate energy-dependent absorbed









fractions obtained from PIRT model simulations, while dashed lines indicate those

derived from VBIST model simulations. At the lowest energy considered (10 keV), a

value of 4((TAM*-TAM) = 1.0 is seen under both VBIST and PIRT simulations, as

expected.

Patterns of divergence between the two modeling approaches (VBIST versus PIRT)

in the ribs are seen to mirror those seen in the cranium (both flat bones of the axial

skeleton). At 4 MeV (the highest energy considered), full convergence of the absorbed

fraction to active marrow under VBIST model simulations has not yet been reached for

the three source regions. Nevertheless, the energy-independent (VBIST) and energy-

dependent (PIRT) patterns of convergence are still evident at electron initial energies

exceeding 1 MeV. At 100% cellularity, the VBIST model absorbed fraction to active

marrow approaches a value of -0.82 at high electron energies. With the PIRT model

results serving as the local standard, percent errors in self-absorbed fraction to active

marrow (100% cellularity) given by the VBIST model are 21% at 500 keV, 124% at 2

MeV, and 313% at 4 MeV. Corresponding percent errors are 16%, 136%, and 327% for

TBS sources, and 31%, 55%, and 337% for TBV sources. These percent errors are

roughly equivalent when the marrow cellularity of the rib cage is reduced to 70% (fat

fraction of -30%). The higher errors in dosimetry for the ribs under VBIST simulations

is not unexpected, considering that this bone site has both a high surface-to-volume ratio

of spongiosa (higher chance for electron escape to cortical bone), as well as a high

marrow volume fraction within its spongiosa (lesser chance for energy absorption within

the bone trabeculae).









Absorbed Fractions to Endosteal Tissues

Figures 4-10 through 4-12 display values of absorbed fraction to the trabecular

endosteal tissues defined as a 10-[tm layer of soft tissue on the marrow-side of the bone-

marrow interface within the microCT images. Figure 4-10 gives results for TBS, TBV,

and TAM electron sources emitted within the pelvis containing bone marrow at 48%

cellularity. Figures 4-11 and 4-12 show corresponding values within the cranium and left

rib cage, respectively, also at reference marrow cellularities (38% for cranium and 70%

for the ribs). In all three graphs, the ordinate scale is expanded to a maximum value of

0.10 to facilitate viewing model differences at higher energies. At the lowest energy

considered (10 keV), a value of 4(TBE-TBS) = 0.5 is seen under both VBIST and

PIRT simulations (half-space source-target geometry for all bone sites). Also, changes in

the marrow cellularity at each bone site had no direct effect on the absorbed fraction to

the endosteal tissues. Thus, reported in this investigation are the only the absorbed

fraction values at the reference cellularity for each bone site.

At each energy and for each model, higher absorbed fractions are noted for electron

sources on the trabecular surfaces, while lower absorbed fractions are seen for electron

sources emitted within the active bone marrow. Intermediate absorbed fractions are

shown for bone volume sources which peak in value at a source energy of -100 keV in

the pelvis and rib skeletal sites. Within the cranium, values of 4((TBE*-TBV) peak in

value at a source energy of -200 keV. As expected, VBIST model simulations approach

energy- and source-independent convergence values at high electron initial energies

(0.028 in the pelvis, 0.030 in the cranium, and 0.015 in the left rib cage), while source-

independent convergence values are shown to continually decline with increasing source









energy above 1 MeV. This decline is more prominent in the cranium and the ribs than

seen in the pelvis (ratios of 3.0 versus 1.5 at high energies), and is accountable in part by

cortical bone losses and particle escape from these two flat bones. In these anatomic

regions of the skeleton, the surface-to-volume ratio of trabecular spongiosa is higher than

that found in the pelvis, and thus electron escape to cortical bone is greater.

Discussion

As a further means of comparing the VBIST and PIRT model results, radionuclide

S values were calculated for a wide range of beta-particle emitters of interest in skeletal

tissue imaging and radionuclide therapy. Absorbed fractions to active bone marrow

given in Figures 4-4 through 4-9, along with both the tissue mass data of Table 4-1 and

beta-particle energy spectra from Eckerman et al. (1994), were used to calculate

radionuclide S values under the MIRD schema for ten different radionuclides. Ratios of

the S value based on VBIST-model absorbed fractions to those using PIRT-model

absorbed fractions are displayed in Table 4-2 for all three skeletal sites and at both 100%

and ICRP-reference marrow cellularities. For low-energy beta-emitters such as 33P, 169Er,

and 177Lu, absorbed fractions given by the VBIST model simulations overestimate

radionuclide S values for TAM, TBS, and TBV sources by only 2% to 13% in the

cranium. Higher errors are noted in the left rib cage, particularly for bone trabeculae

volume sources (ratios of 1.18 to 1.24). In comparison to both the cranium and the ribs,

even higher errors were also noted for the trabecular volume sources in the pelvis (ratios

of 1.25 to 1.28). For radionuclides at intermediate beta energies (Eave of 192 keV to 583

keV), S value ratios range from 1.07 to 1.34 in the cranium, from 1.08 to 1.48 in the ribs,

and from 1.04 to 1.24 in the pelvis. For radionuclides in the highest beta-energy range

(Eave of 695 to 934 keV), S value ratios range from 1.30 to 1.54 in the cranium, from 1.37









to 1.76 in the ribs, and from 1.14 to 1.29 in the pelvis. It is reasonable to assume that

similar errors are also present in radionuclide S values derived from chord-based models

(Bouchet et al. 2000; Eckerman and Stabin 2000) which, as in the VBIST simulations of

the present study, assume an infinite region of spongiosa during particle transport.

Conclusion

A paired-image radiation transport (PIRT) model for skeletal dosimetry is

presented in which electrons (beta particles) are tracked simultaneously within two

different segmented digital images: (1) an ex-vivo CT image of the skeletal site with

segmented regions of trabecular spongiosa, cortical bone, and surrounding tissues, and

(2) an ex-vivo microCT image of the interior bone trabeculae and marrow cavity

microstructure representative of that found within spongiosa regions of the ex-vivo CT

image. Example dose calculations under the PIRT methodology within the cranium, ribs,

and pelvis of an adult 66-year male subject demonstrate a divergence from standard

infinite spongiosa transport (VBIST) methods at energies as low as 50-200 keV

depending upon the source tissue and skeletal site. Calculations of radionuclide S values

under both methodologies imply that current chord-based models used in clinical skeletal

dosimetry may over-estimate dose to active bone marrow in these three skeletal sites by

-2% to 28% for low-energy beta emitters (33P, 169Er, and 177Lu), by -4% to 48% for

intermediate-energy beta emitters (131, 186Re, and 89Sr), and by -14% to 76% for high-

energy beta emitters (32P, 8Re, and 90Y). Higher errors are noted for bone-volume

seekers, while lower errors are seen for source emissions within the active bone marrow.

These finding are consistent with those investigated previously in the proximal femur and

lumbar vertebrae of the same 66-year male subject. The PIRT model thus supersedes

previous stylized modeling attempts by the UF ALRADS research group to account for






67


the infinite spatial extent of trabecular spongiosa and cortical bone, and provides a

method for expanding the availability of reference models needed for clinical bone

marrow dose estimates to radionuclide therapy patients.




































Figure 4-1. Schematic of the PIRT model constructed for the pelvis (os coxae).