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SKELETAL DOSIMETRY: A HYPERBOLOID REPRESENTATION OF THE BONEMARROW INTERFACE TO REDUCE VOXEL EFFECTS IN 3D IMAGES OF TRABECULAR BONE By DIDIER A. RAJON 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 2002 ACKNOWLEDGMENTS Several people contributed greatly to this work. First, I would like to express my sincere thanks to Dr. Wesley E. Bolch for his support, encouragement, and guidance throughout my graduate work at the University of Florida. A mere thank you cannot express my gratitude for his tolerance and patience over the last two years. I appreciate the opportunity that he has provided me to pursue my interest in health physics. I also thank Dr. David Hintenlang, Dr. Samim Anghaie, Dr. Stephen Blackband, Dr. Anthony Ladd, and Dr. Lionel Bouchet for their suggestions and for being members of my committee. I would also like to thank all of the students who contribute to the Bone Imaging Dosimetry project, for providing the bonesample images that I used in this research. I also thank Dr. Derek Jokisch and Dr. Phillip Patton for their contributions and support. I also thank all of the faculty and graduate students in the department of Nuclear and Radiological Engineering for their help and support during my five years of graduate research. They all contributed to my knowledge and made the work enjoyable. I also thank the staff of the department of Nuclear and Radiological Engineering for their devotion and comprehension in resolving various problems. I also thank the department of Nuclear and Radiological Engineering for its financial support over these years. Finally I would like to thank some people very close to me. These include my parents and my sister for having supported my decisions and accepted a long separation during these five years, and also all the friends I met since I arrived in Gainesville. They all contributed to the success of my experience by being present when I needed them. TABLE OF CONTENTS page A C K N O W L E D G M EN T S .................................................................................... ii TABLE OF CON TEN TS ............................. .... .......................................................... iv L IST O F FIG U R E S ....................................................... ............................ viii LIST OF TABLES ................................................... xii ABSTRACT ..................................... ................ xiv CHAPTER 1 IN T R O D U C T IO N ..................................................... .............................. 1 2 B A C K G R O U N D ............................................................................................ 7 Bone Structure and Physiology .......................... ........ ..................... 7 B one C ell Types .......................... ....................... ..... 8 Variation of the Trabecular Bone Microstructure .................................... 9 Internal D osim etry ........................................ ............. ......... 11 Past Skeletal Dosimetry Works and Models ....................................... ...... 13 Use of Nuclear M agnetic Resonance ............................. ....................... 16 Bone Sample Imaging and Image Processing .............. ........................... 18 3 VOXELSIZE EFFECTS IN 3D NMR MICROSCOPY PERFORMED FOR TRABECULAR BONE DOSIMETRY ............................ ................ 25 Introduction ........................... ........ .. .. ........ ........ ........ 25 Previous Studies in Trabecular Bone Dosimetry ...................................... 25 Use of NMR Microscopy .................................. .................... 26 VoxelSize Effects in Skeletal Dosimetry Calculations ............................ 27 M material and M methods .......................................................... ......... .......... 28 Construction of the Mathematical Model of Trabecular Bone .................. 28 Construction of the Segmented Images of the Model .............................. 33 ElectronTransport Sim ulations ............................. ..... .................. 34 SV alue Calculations .................................................................... ...... 35 R results and D discussion ................................. ............................................... ........ 36 M them atical Sam ple ............................... ... ............................ 36 Segmented Images ........................................ ........ 37 Absorbed Fractions and their Absolute Errors ........................................ 39 Absorbed Doses and their Relative Errors .............. ......................... 44 Conclusion ... .................................................. .... ........ 46 4 SURFACEAREA OVERESTIMATION WITHIN 3D DIGITAL IMAGES AND ITS CONSEQUENCES FOR SKELETAL DOSIMETRY ................. 65 Introduction ..... ........................... ................. ........ 65 BoneM arrow D osim etry ............ ............................ ................. .... 66 Voxelization Effects in BoneMarrow Dosimetry ...................................... 67 M material and M ethods .......................... .................................... ...... 69 Construction of the 3D Segmented Images of the SingleSphere Model .... 69 Volume Fraction Occupied by the Spheres within the Segmented Images .. 71 Theoretical Surface Area of a Voxelized Sphere ....................................... 71 Measurement of the Surface Area of a Voxelized Sphere .......................... 74 Consequences of an Error in Surface Area on the Absorbed Fraction ........ 74 ElectronTransport Sim ulations ............................. ..... .................. 76 Statistical Analysis .................................................... 78 Results and Discussion ........................ ........... ...................... 80 Volume Fraction Occupied by Spheres within the Segmented Images ....... 80 Surface Area of the 3D Segmented Images .............. ....................... 80 Absorbed Fractions within the 3D Segmented Images ............................... 81 Conclusion ... .................................................. .... ........ 83 5 INTERACTION WITH 3D ISOTROPIC AND HOMOGENEOUS RADIATION FIELDS: A MONTECARLO SIMULATION ALGORITHM 99 Introduction ........................ ............... ....... ................. .. ........ 99 ThreeDimensional Homogeneous and Isotropic Fields ............................ 100 MonteCarlo Methods ..................................... .......... 101 Random Number Generators .................................. ......... 101 M material and M ethods ............................................................. 102 General Description of the Algorithm ..................... .................. 102 Choosing a Random Point P on the Surface of the Sphere ...................... 104 Choosing a Random Point P' on the Surface of the Disk ......................... 105 Local Coordinate System of the Plane ...................................... 105 Point P' in the Local Coordinate System ........................................ 107 Coordinates of P' in the Initial Coordinate System ............................. 108 D direction of the P article ............................................................................ 108 Im plem entation of the Algorithm ......................................... ......... 109 Application to ChordLength Distributions ........................................ 111 Results and Discussion ........................................ ................... 112 General Algorithms ......................................... ......... 112 ChordLength Distribution Examples ................................. 113 Conclusion ..... ............................. ... ................ 114 6 VOXEL EFFECTS WITHIN DIGITAL IMAGES OF TRABECULAR BONE AND THEIR CONSEQUENCES ON CHORDLENGTH DISTRIBUTION MEASUREMENTS .............. ...................... 121 Introduction ...................................................... 121 Background ............................. ....... ....... ... ............. 122 ChordLength Distributions within Trabecular Bone Samples ................. 122 Voxel Effects on ChordLength Distribution Measurements .................... 124 Minimum Acceptable Chord (MAC) Selection Criteria ............................ 125 Mathematical Model of Trabecular Bone ........................................ 126 M material and M ethods ...................... ............ .... ... .. ........ .... 128 ChordLength Distributions through the Mathematical Model ............... 128 ChordLength Distributions through SingleSphere Models .................... 133 R results and D iscu ssion ............. .................................................................. 134 Mathematical Model of Trabecular Bone ........................................ 134 SingleSphere Models .............................................. 137 Conclusion ..... ............................. ... ................ 139 7 MARCHINGCUBE ALGORITHM: REVIEW AND TRILINEAR INTERPOLATION ADAPTATION FOR IMAGEBASED DOSIMETRIC M O D E L S .............................. ........................................ 15 6 Introduction ........ ................ ............................ ............... 156 MarchingCube Algorithm ..................................... .......... 159 Original MC Algorithm .............................................. 160 Am biguity Problem ............................. ..... .......................... 163 H ole Problem ..................................................... ......... 165 Im age Size and Processing Tim e ......................................... .......... 166 Marching Tetrahedrons .............................................. 167 Extended MC Algorithm ..................................... ......... 168 Facial D eciders ............................................................................ .......... 169 Optimizing the Image Size .......................... .......... ......... 171 Material and Methods ............................. .......... 172 M them atical B one Sam ple ...................................................................... 173 Direct and Reverse Extended MC Algorithm ........................................ 175 TrilinearInterpolation Isosurface .............................................................. 177 Intersection of a Straight Line with a Hyperboloid ................................... 181 Results and Discussion ............................ ............ ........... ......... 184 SurfaceArea Measurement ............................ ......... 184 ChordLength Distributions ........................................... 185 Conclusion .... ......................................................... 188 8 HYPERBOLOID REPRESENTATION OF THE BONEMARROW INTERFACE WITHIN 3D NMR IMAGES OF TRABECULAR BONE: APPLICATIONS TO SKELETAL DOSIMETRY ...................................... 213 Introduction ........................................ .. .......................... 213 Hyperboloid MarchingCube Algorithm .................................................. 214 M material and M ethods ................................... ............. .... 216 Revised Mathematical Model of Human Trabecular Bone ...................... 216 AF Calculations within the Mathematical Bone Model ............................. 219 SValue Calculations within the Mathematical Bone Model .................... 221 Application to NMR Microscopy Images of Human Trabecular Bone ....... 221 Results and Discussion .................................... .............. 223 Volume Fraction and Surface Area ........................... .......... 223 Absorbed Fractions of Energy .... .......................... ......... 224 Radionuclide S Values ................... ........ ....... ................... 226 Applications to NMR Microscopy Images of Human Trabecular Bone ...... 230 Conclusion ..... ............................. ... ................ 231 9 CONCLUSIONS AND FUTURE WORK ............................................... 249 C o n clu sion s .................................................................... 2 4 9 Future w ork ...... ....................... ..................................................................... 251 APPENDIX A LOOKUP TABLES FOR MARCHINGCUBE ALGORITHM ..................... 255 B GENERAL TOOLS (C++ PROGRAM S) ....................................................... 276 C LOOKUP TABLE TOOLS (C++ PROGRAMS) ........................................ 291 D MARCHINGCUBE IMAGE TOOLS (C++ PROGRAMS) .......................... 311 E MC TRIANGLE GENERATION (C++ PROGRAMS) .................................. 338 F MC CHORDLENGTH DISTRIBUTION (C++ PROGRAMS) ..................... 356 G HMC CHORDLENGTH DISTRIBUTION (C++ PROGRAMS) ............ 369 H HMC VOLUME FRACTION (C++ PROGRAMS) ........................................ 380 I HMC TRANSPORT CODE (EGSNRC USER CODE) ................................ 383 REFERENCES .......................... 411 BIOGRAPHICAL SKETCH ...................................... ............ 426 LIST OF FIGURES Figure page 21 Femur head showing the different constituents of the bone structure ......... 20 22 Microstructure of compact bone and trabecular bone ................................ 21 23 Microstructure of trabecular bone showing the types of bone cells and their location ............................... ............................................................. 22 24 Differing paths of two electrons emitted within anisotropic regions of trabecular bone ...... ................................................. ........ 23 25 Trabecular bone 3D image obtained from a 4.7 Tesla NMR system ........... 24 31 Electron micrograph of the trabecular latticework within a lumbar vertebra 49 32 Voxelsize effects on the dose calculation for a single electron track traveling within a bone trabecula ....................................................... 50 33 Edge effect when trying to position circles within a squared field ............. 51 34 Transverse slice, 1.6 cm x 1.6 cm, through the ROI of the mathematical sample of trabecular bone ................................. ...................... 52 35 Chordlength distributions: Spiers cervical vertebra compared to the m them atical m odel ............................. .. ...................................................... 53 36 Segmentation of the mathematical sample ............... ........................ 54 37 Geometrical parameters as a function of image resolution for the mathematical sample ................................. ........................... 55 38 Absorbed fraction differences between the segmented images and the reference values ................................................................................ 56 39 Relative error in the absorbed fraction for the segmented images and using results from the mathematical bone sample as reference values ................. 57 310 Relative error in the absorbed dose for monoenergetic electron sources in the marrow cavities ...................................................................................... 58 311 Relative error in the S value calculated for five radionuclides of interest in skeletal dosimetry .. ...................................................... 59 41 Cubical sample of trabecular bone reconstructed from a 3D NMR image obtained at 4.7 tesla ....................................... ..................... 86 42 Perimeter of a circle as represented by a digital image ............................... 87 43 Analytical derivation of the surface area of a voxelized sphere ................... 88 44 Consequence of the surfacearea error on the absorbedfraction calculation 89 45 Expected evolution of the crossabsorbed fraction overestimation as a function of the voxel size ................................. ....................... 90 46 Volume fraction occupied by the sphere within the segmented image as a function of the voxel size ................................. ....................... 91 47 Surface area of the segmented spheres, as a function of the voxel size ....... 92 48 Relative error on absorbed fractions inside the sphere ............................... 93 49 Relative error on absorbed fractions outside the sphere ............................. 94 51 Technique used to simulate a homogeneous and isotropic infinite 3D field of radiation when it interacts with a fixed object ....................................... 115 52 Geometric construction used to derive the trajectory of the particles ......... 116 53 C program that implements Algorithm 1 ......................................... 117 54 C program that implements Algorithm 2 .......................................... 118 55 Chordlength distributions measured through a sphere of unit diameter ..... 119 56 Chordlength distributions measured through a cube of unit width ............ 120 61 Cubical sample of trabecular bone reconstructed from a 3D NMR image obtained at 4.7 tesla ....................................... ............ ......... 144 62 Chordlength distributions measured by Spiers and colleagues within a thin slice of a human cervical vertebra ............. ....................... 145 63 Bone trabecula chordlength distribution measured through a NMR image of a hum an sam ple of trabecular bone ....................................................... 146 64 Segmentation of the mathematical model of trabecular bone .................... 147 65 Comparison between the theoretical and the measured distribution through the mathematical model .................................... .......... 148 66 Threedimensional chordlength distributions through segmented images of the mathematical model .................................... .......... 149 67 Distributions measured through the mathematical model using different techniques ............................................................. 150 68 Distributions through a single sphere of radius 500 tm using different techniques .................................................... 151 71 MC algorithm principle for a simple 2D image ................................... 191 72 MC algorithm configurations and patterns in 2D .................................. 192 73 MC patterns proposed by Lorensen for 3D images ................................. 193 74 Ambiguity problem in 2D .............................................. 194 75 Hole problem as a consequence of the ambiguity problem ................... 195 76 Hole problem for a 22 x 22 x 22 image of a trabecular bone sample .......... 196 77 MarchingTetrahedron method .................................... 197 78 MC patterns proposed to solve the hole problem using the direct design for ambiguous faces .. ............................. ......... .......... 198 79 MC patterns proposed to solve the hole problem using the reverse design for am biguous faces ........................................................ .................... 199 710 The four triangulations that solve the ambiguity problem for pattern 10 .... 200 711 Transverse slice through the mathematical sample of trabecular bone ........ 201 712 Synopsis of the database created by the MC algorithm .............................. 202 713 Localization of the eight vertices and twelve edges of a marching cube ..... 203 714 Trilinearinterpolated isosurface examples within unit cubes .................... 204 715 Trilinear interpolated isosurfaces within unit cubes ................................... 205 716 Trilinear interpolated isosurfaces within unit cubes of pattern 10 ............... 206 717 Trilinear interpolated isosurfaces within several adjacent cubes, showing how the surface connects ..................................... .......... 207 718 Surface area as a function of image resolution for the mathematical sample 208 719 Chordlength distributions measured through the mathematical bone sample for four different voxel sizes .................................................. 209 720 Corrected chordlength distributions measured with the trilinear technique 210 81 Cubical sample of a human lumbar vertebra ....................................... 234 82 3D representation of the triangulated bonemarrow interface obtained from the MC algorithm ............................. ......... 235 83 Transverse slice, 1.2 x 1.2 cm2, through the revised mathematical sample of trabecular bone ........................................ 236 84 Transverse slice, 0.896 x 2.12 cm2, through a NMR microscopy image of human trabecular bone .............................................. 237 85 Geometrical parameters as a function of image resolution for the revised m them atical sam ple ........................................................ ................ 238 86 Relative error of the absorbed fractions calculated for simulated images of the mathematical sample as a function of the voxel size ....................... 239 87 Relative error of the absorbed fractions calculated for simulated images of the mathematical sample as a function of the voxel size ....................... 240 88 Relative error of the S values calculated for four radionuclides of interest in skeletal dosimetry as a function of the voxel size .................................. 241 89 Relative error of the S values calculated for four radionuclides of interest for skeletal dosimetry as a function of the voxel size ................................. 242 810 Absorbed fraction: voxel code compared to HMC code ...................... 243 811 Relative error of the AF calculated with the voxelcode ...................... 244 LIST OF TABLES Table page 31 Characteristics of the 19 segmented images for a voxelsize range from 1000 [tm to 16 [tm ....... .................................................. 60 32 Tissue compositions used for bone and marrow in the mathematical bone sam ple .... .................................................................. 61 33 Radiation characteristics of the radionuclides used for Svalue calculations 62 34 Absorbed fractions calculated within the mathematical bone sample for the marrow region, the bone region, and the region beyond the dosimetric region of interest .................................. .................................... 63 35 S values calculated within the mathematical bone sample for the five cho sen radionu clides ............ ................................................................ 64 41 Characteristics of the eight singlesphere models covering a typical range of bonem arrow cavity sizes. ..................................................................... 95 42 Voxel sizes for the different 3D images of the singlesphere models .......... 96 43 List of electron energies used in the study ....................................... 97 44 Electron ranges in bone marrow and cortical bone .................................... 98 61 Characteristics of the 4 segmented images and of the mathematical trabecular bone model used in the chordlength distribution study ............ 152 62 Characteristics of the 4 segmented images of the singlesphere marrow cavity model used in the chordlength distribution study ...................... 153 63 Mean chord lengths for the 4 segmented images and the mathematical bone model using the different techniques to record the chordlength distributions ......... ..................... .................................... 154 64 Mean chord lengths for the 4 segmented images of the singlesphere marrow cavity model using the different techniques to record the chordlength distributions ................................... .......... 155 71 Characteristics of the 17 segmented images of the mathematical bone sam ple ... ...................................................... .... ......... 211 72 Graylevels used for the isosurface equation of the cubes presented in Figures 714, 715, 716, and 717 ........................... .......... 212 81 Characteristics of the 19 segmented images of the mathematical bone sam ple ................................ ............................................................. 245 82 Energies used for AF calculations ........................................ ......... 246 83 Radiation characteristics of the radionuclides used for the Svalue calculations ......... ................................................ .......... 247 84 S values calculated within the real bone sample for both the voxel and the HMC representations of the bonemarrow interface ................................. 248 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 SKELETAL DOSIMETRY: A HYPERBOLOID REPRESENTATION OF THE BONEMARROW INTERFACE TO REDUCE VOXEL EFFECTS IN 3D IMAGES OF TRABECULAR BONE By Didier A. Raj on December 2002 Chairman: Dr. Wesley E. Bolch Major Department: Nuclear and Radiological Engineering Radiation damage to the hematopoietic bone marrow is clearly defined as the limiting factor to the development of internal emitter therapies. Current dosimetry models rely on chordlength distributions measured through the complex microstructure of the trabecular bone regions of the skeleton in which most of the active marrow is located. Recently, Nuclear Magnetic Resonance (NMR) has been used to obtain highresolution threedimensional (3D) images of small trabecular bone samples. These images have been coupled with computer programs to estimate dosimetric parameters such as chordlength distributions, and energy depositions by monoenergetic electrons. This new technique is based on the assumption that each voxel of the image is assigned either to bone tissue or to marrow tissue after application of a threshold value. Previous studies showed that this assumption had important consequences on the outcome of the computer calculations. Both the chordlength distribution measurements and the energy deposition calculations are subject to voxel effects that are responsible for large discrepancies when applied to mathematical models of trabecular bone. The work presented in this dissertation proposes first a quantitative study of the voxel effects. Consensus is that the voxelized representation of surfaces should not be used as direct input to dosimetry computer programs. Instead we need a new technique to transform the interfaces into smooth surfaces. The Marching Cube (MC) algorithm was used and adapted to do this transformation. The initial image was used to generate a continuous graylevel field throughout the image. The interface between bone and marrow was then simulated by the isograylevel surface that corresponds to a predetermined threshold value. Calculations were then performed using this new representation. Excellent results were obtained for both the chordlength distribution and the energy deposition measurements. Voxel effects were reduced to an acceptable level and the discrepancies found when using the voxelized representation of the interface were reduced to a few percent. We conclude that this new model should be used every time one performs dosimetry estimates using NMR images of trabecular bone samples. CHAPTER 1 INTRODUCTION Dosimetry assessment within the trabecular bone regions of the skeleton has been a challenging investigation over the past four decades. The importance of these studies is that bones serve as the "housing" for hematopoietic marrow, the tissue responsible for producing a variety of different blood cells. Therefore, irradiation of bone regions has the potential to cause severe damage to the bone marrow and, as a consequence, can be life threatening for the individual exposed to the irradiation. Several situations may result in internal irradiation of trabecular bone regions. These include * Occupational exposures to boneseeking radionuclides (Brodsky 1996) * Therapy procedures using injected radiopharmaceuticals that transit through the skeletal system (Rubin and Scarantino 1978; Sgouros 1993; Siegel et al. 1990) * Therapeutic procedures for palliation of bone pain associated with bone cancers (Samaratunga et al. 1995). In most of these situations, the radionuclide is a beta or an alpha emitter and the amount of energy deposited in healthy marrow can be extremely important. Therefore, the active bone marrow is firmly established as the doselimiting organ in radionuclide therapies (Rubin and Scarantino 1978; Sgouros 1993; Siegel et al. 1990; Zanzonico and Sgouros 1997). Occupational exposures of boneseeking radionuclides will be a challenge to internal dosimetry cleanup and waste management programs within the U. S. Department of Energy (DOE). Currently, numerous sites need to be safely decontaminated or decommissioned. Potential inhalation or ingestion of boneseeking radionuclides (by cleanup workers or by offsite populations) is a concern in most of these sites (DOE 1995). Six such radionuclides are 90Y, 91Y, 95Zr, 95Nb, 90Sr and 226Ra. Over the past 50 years both 90Sr and 226Ra have been studied extensively in bone dosimetry (Hindmarsh et al. 1958; Spiers 1966a; Spiers 1967; Vaughan 1960; Vaughan 1973). Radiopharmaceuticals have been used increasingly as therapy agents in recent decades. Therefore a more accurate trabecular bone dosimetry is needed to limit risk to patients. The risk is present in bone therapy using boneseeking radiopharmaceuticals, and also in any therapy in which the radionuclide transits through the blood system. The bone marrow is continuously irrigated by blood vessels and is thus exposed to the radiation. More 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 nuclearmedicine techniques by maximizing the administration of therapeutic doses of radiopharmaceuticals. Radiopharmaceuticals are also used for palliation of bone pain. This treatment is accomplished with boneseeking betaemitting radionuclides. Phosphorus32, 89Sr, 131, and 186Re are four such radionuclides considered for this treatment (Samaratunga et al. 1995). Radiopharmaceutical treatments of bone pain cause marrow to receive a significant amount of the deposited energy. More accurate trabecular bone dosimetry has the potential to improve our understanding of the dangers associated with the scenarios previously mentioned. Therapeutic applications of radiation and radioactive materials would benefit from better dosimetry of these regions, and health risks associated with boneseeking radiopharmaceuticals could be calculated more accurately. Within the trabecular bone regions, the bone trabeculae have a thickness on the order of 300 tm and the marrow cavities have widths on the order of 1 mm. The smallest geometrical features can be on the order of 100 tm for both the bone and the cavities. Because the features of such bone microstructure are so small, energetic electrons will be able to traverse several different marrow cavities while continuously depositing energy. Therefore, a trabecular bone dosimetry can be done by either a study of the path length of the particles through the different regions, or by coupling a trabecular bone model with a transport code that can perform a calculation in such a microscopic structure. Instead of a model that could be difficult to build, bone samples can be imaged and the digital image coupled with the transport code. Nuclear magnetic resonance (NMR) and quantitative computed tomography (QCT) are two nondestructive techniques investigated during the last decade to provide bone sample images. The last studies use very high resolution invitro NMR imaging (Chung et al. 1996; Chung et al. 1995a; Hwang et al. 1997; Link et al. 1998b; Majumdar et al. 1996; Wessels et al. 1997) as well as QCT imaging (Ciarelli et al. 1991; Cody et al. 1989; Cody et al. 1991; Cody et al. 1996; Engelke et al. 1993; Goulet et al. 1994; Kinney et al. 1995; Kuhn et al. 1990; Link et al. 1997; Link et al. 1998a; Link et al. 1998b; McCubbrey et al. 1995; Muller et al. 1994; Muller and Ruegsegger 1996; Ruegsegger et al. 1996). A resolution of 50 tm can be achieved with both techniques. In 1995, an investigation of the feasibility of the use of NMR images to transport electrons trough threedimensional (3D) digitized representation of the trabecular bone microstructure was initiated at the University of Florida (Jokisch et al. 2001a; Jokisch et al. 1998; Jokisch et al. 2001b; Patton et al. 2002a; Patton et al. 2002b) and satisfactory results were achieved. Chordlength distributions through both the bone trabeculae and the marrow cavities were also acquired using these 3D images. These distributions are of great interest for bonemarrow dosimetry since they can be compared with electron ranges to deduce energy deposition through the bone and the marrow regions. Furthermore, all current models of bonemarrow dosimetry are based on these distributions. A few years ago, a study was initiated in an effort to assess the minimum voxel size required for NMR imaging in order to obtain reliable results for energy deposition calculation using MonteCarlo codes to transport the particles through the images (Rajon 1999). Our aim was to reduce imaging time by choosing the largest image resolution possible without altering results, so that in vivo imaging could become a reality. This study reached two conclusions. First, for highenergy electrons, there is no need to obtain highresolution images. With a voxel size as large as 300 tm the computer simulations are already in good agreement with the expected results. Second, for lowenergy electrons the calculation could lead to overestimation of the crossdose (marrow cavities irradiating the bone trabeculae or viceversa) by up to 40 % for 50 keV electrons. This second conclusion was unexpected and stimulated further investigation of what caused the overestimation of the crossdose and how the problem could be solved. The work presented in this dissertation proposes to answer these two questions. Chapter 2 gives the important background required to understand the whole development that follows. Chapter 3 summarizes the previous study that led to the two conclusions sited above. It describes the development of a mathematical bone sample and how it was used to benchmark the accuracy of the energy deposition calculation within 3D images. It shows that the digitization of a 3D image into voxels produces an overestimation of the surface area of the boundary between bone and marrow if measured along this voxelized representation of the boundary. As a consequence, the energy deposition by shortrange electrons initiated from one side of the boundary and irradiating the other side is also overestimated. Chapter 4 proposes to explain the origin of the surfacearea overestimation in studying single sphere models. It also discusses how this overestimation of the surface area will lead to an overestimation of the crossenergy deposition calculated within the voxelized representation. Then, computer simulations of electron transport are conducted within these models to verify the prediction. The experimental results are shown to be in perfect agreement with the predicted results. Therefore, it is concluded that a better representation of the interface between bone and marrow is required. Chapter 5 develops a new MonteCarlo technique to simulate an isotropic and homogeneous radiation field that surrounds any object. This technique is then used in Chapter 6 to measure chordlength distributions within representations of trabecular bone samples. In Chapter 6, the mathematical models are used to assess the accuracy of the chordlength distribution measurements through the voxelized representation of the boundary. Older studies showed that these measurements were affected by voxel effects and some techniques were developed to reduce these effects (Jokisch et al. 2001b). A reevaluation of these techniques is proposed and the overall results showed that their consequences depend on the voxel size. The conclusion is, again, that a better representation of the interface between bone and marrow is required. Chapter 7 proposes a new technique to model the bonemarrow interface by a smooth surface. The technique is an adaptation of the MarchingCube (MC) algorithm and produces an isosurface within the gray level field of the image. This isosurface representation is shown to preserve the surface area of the interface and is expected to solve the problem of the energydeposition calculation. It also gives excellent results when applied to chordlength distribution measurements and definitely solves the voxel effect problems. Finally, Chapter 8 uses the same adaptation of the MC algorithm to couple the bone sample geometry with the MonteCarlo transport code and to calculate the energy deposition by electrons through this representation of the bonemarrow interface. The technique is shown to eliminate the systematic overestimation found with the voxelized representation. CHAPTER 2 BACKGROUND Bone Structure and Physiology Bone has two main types of histological structure: cortical bone (also referred to as hard compact bone) and trabecular bone (also referred to as spongy bone or cancellous bone). Figure 21 shows the difference between compact bone and trabecular bone. Cortical bone comprises 80% of the skeletal mass (Berne et al. 1993). One of the few structures that penetrate the compactness of the cortical regions is the haversian canal system. Figure 22 is a more detailed illustration of the compact bone structure. The Haversian canals are pathways used by the circulatory system to supply the living bone cells with nutrients. The dosimetry of cortical bone has been studied (Akabani 1993; Beddoe 1976a; Beddoe 1977). Recently Bouchet and Bolch (1999) developed a 3D transport model of cortical bone at the University of Florida. Trabecular bone consists of a complex network of bone spicules (also called trabeculae) that surround cavities of marrow. Figure 23 shows the complexity of the trabecular bone microstructure. The marrow cavities contain both active marrow (also called red marrow) and inactive marrow (also called yellow marrow). The active marrow is responsible for the production of blood cells, whereas inactive marrow is just marrow in which the hematopoietic cells have been replaced with fat cells. To allow the blood vessels to irrigate the bone marrow, the cavities are connected to one another thus also forming a complex network. As a consequence trabeculae and marrow cavities are two separate lattices that interlace to each other (Figure 23). Trabecular bone exists in the inner regions of the vertebra, ribs, skull, pelvis and the end of the long bones. The presence of trabecular bone in the central part of long bones primarily occurs in newborns and very young children. But the trabecular regions gradually decay with age and rapidly disappear from these regions. Where the bone surface interfaces either a Haversian canal (in cortical bone) or a marrow cavity (in trabecular bone), there is a thin layer of osteogenic cells called the endosteum. A similar layer exists on the exterior of the cortical bone called the periosteum. Because of the presence of red marrow the trabecular bone regions are of great importance for dosimetry studies. They constitute the area of interest in this research. Bone Cell Types The cellular components of cortical and trabecular bone are the same. Three types of bone cells control the production and destruction of osteoid matrix. Osteoblasts are responsible for ossification (the formation of bone). They are responsible for the formation of a collagenbased osteoid matrix and the deposition of calcium in the form of calcium phosphate into this matrix. Osteoblasts exist on the endosteal and periosteal surfaces, and are the most radiosensitive of the three bone cells (Vaughan 1960). The stem cells that produce the osteoblasts also exist on the endosteal and periosteal surfaces. These preosteoblasts are very radiosensitive. Osteoclasts counteract the work of the osteoblasts by destroying or resorbing bone. Osteoclasts exist on the endosteal bone surfaces and are capable of removing calcium phosphate and destroying the osteoid matrix. They do this by producing hydrolytic enzymes that digest minerals and bone matrix (Vaughan 1975). Osteocytes, the third type of bone cells, are simply osteoblasts that bury themselves within the osteoid matrix over time. Obviously, osteocytes are not on the endosteal surface. However, osteocytes are believed to aid in the osteogenic processes by staying in contact with osteoblasts through channels called canaliculi. Osteocytes may become active osteoblasts again if they are uncovered by bone resorption later in life. For these reasons, the degree to which the bone volume is dosimetrically important is not fully understood. Figure 23 also shows the location of these different bone cells. Variation of the Trabecular Bone Microstructure The creation and resorption rates result in a process called bone remodeling that represents approximately a 15% bone mass turnover every year for an adult human (Berne et al. 1993). Bone mass peaks in humans between 20 and 30 years of age. Remodeling reaches equilibrium around 35 to 40 years of age, and then decreases for the remainder of life (Berne et al. 1993). Because women have smaller overall mass than men, this natural loss of bone, especially when coupled with hormonal losses due to menopause, can create bone structural problems such as osteoporosis. Trabecular structure varies with age (Atkinson 1965; Atkinson 1967; Snyder et al. 1974), gender (Atkinson 1967; Mosekilde 1989), skeletal site (Eckerman 1985), and skeletal orientation (Atkinson 1967; Atkinson and Woodhead 1973; Hahn et al. 1992; Mosekilde 1989). Because of these variations, trabecular microstructure data is more properly characterized by distributions as opposed to mean values. The ossification and resorption rates mentioned above also vary with skeletal site and orientation. Obviously, any change in the size of trabeculae corresponds to a change in marrow cavity size. Additionally, the percentage of active marrow that fills the cavities changes with age (Custer and Ahlfeldt 1932; Ellis 1961; Mechanik 1926). For these reasons, any study on the microstructure of trabecular regions of bone must pay attention to age and genderrelated changes in that microstructure. Trabecular bone structure can also be highly anisotropic in its structure (Cowin 1989; Turner 1992; Williams and Lewis 1982). Several studies found a difference between horizontal and vertical trabecular structure in bones such as vertebrae (Atkinson 1967; Atkinson and Woodhead 1973; Hahn et al. 1992; Mosekilde 1989). These studies suggest that the ossification and resorption processes in bone respond to compression and stress. In the case of a vertebra, the horizontal struts are not as structurally important as the vertical segments and they thin considerably faster than the vertical segments with subject age. At least one study did not find a horizontal resorption preference (Snyder et al. 1993). Trabecular and cavity sizes also vary with skeletal location. The fractional mass of active marrow also varies with skeletal location. Some skeletal sites, such as the adult vertebrae, are more important to marrow dosimetry than others because they contain a larger portion of active marrow. Since yellow marrow does not contain appreciable populations of hematopoietic stem cells, skeletal regions containing yellow marrow are only of dosimetric importance in endosteal tissues. The geometry and composition of the trabecular regions of the skeleton create several dosimetry problems. Since bonemarrow cavities are located within the trabecular bone structure, the dimensions of the two interlacing regions must be accurately known in order to calculate the absorbed dose to these sites. The anisotropic structure of these regions further complicates any dosimetry studies because it is difficult to apply any sort of uniform modeling technique to such a complex geometry. Furthermore, the small sizes of trabecular and cavity regions, relative to the ranges of typical beta particles emitted from boneseeking radionuclides, imply that an electron may traverse several cavities while continuously depositing kinetic energy. Two electrons with the same energy and starting point may take completely different paths, traversing differing amounts of marrow and bone as shown in Figure 24. Internal Dosimetry Energy absorption by body tissue while exposed to internal ionizing radiation has been studied intensively and numerous assessment techniques have been proposed. In 1968, a new technique was introduced by the Society of Nuclear Medicine's Medical Internal Radiation Dose (MIRD) Committee (Loevinger and Berman 1968a; Loevinger and Berman 1968b). This technique was improved over time and is now known as MIRD Primer (Loevinger et al. 1991). The purpose of this section is to briefly describe this technique. Each organ or tissue of the body can be seen as a source of radiation if it has been contaminated by a radionuclide or as a target organ when it absorbs energy from one or several source organs. Of course, a source organ always irradiates itself and is also seen as a target organ. As a consequence of a contamination by a radionuclide, the dose D received by a target organ is the sum of the doses received from the different source organs and can be defined by Dk = D(k+h) (21) h In Equation (21) k represent the target organ and h the different source organs. The MIRD Primer clearly divides the assessment of D(k+h) into two separate problems: * The cumulated activity Ah represents the total number of disintegrations of the radionuclide that occur during the contamination time. * The S value S(kh) represents the average dose received by the target organ k per disintegration within the source h. Using these definitions, Equation (21) can be rewritten as Dk =AhS(kh). (22) h The cumulated activity is calculated by integrating the activity within the source organ over time. Calculation of the activity is usually complex and must consider * Build up of activity in the organ (may occur through various biological path ways) * Physical decay of the radionuclide * Several biological decays (such as transition to another organ or natural elimination of the radionuclide from the body). This calculation is simplified by introducing a residence time (Loevinger et al. 1991) that relates the cumulated activity to the activity introduced to the body. The purpose of this dissertation is not the assessment of cumulated activity. Therefore, I do not discuss determination of residence time here. The S value represents the average dose received by a target organ when a single disintegration occurs in a source organ. Depending on the radionuclide, several particles can be emitted per disintegration, or a particle can be associated to a probability of being emitted: its yield. To account for these different situations, the S value is decomposed into S(kh) (23) mk In Equation (23), i represents the different particles that can be emitted by the radionuclide. Each particle is defined by its type (photon, electron, alpha, etc) and its energy. The parameter n, is the yield of this specific particle, E, is its initial energy, and ,,(kh) is the average fraction of initial energy that is absorbed by the target organ. The parameter mk is the mass of the target organ. The yield n, and the initial energy E, are characteristics of the radionuclide. They are provided by the radionuclide spectrum. As an example, for a beta particle, i refers to a fraction of the beta spectrum that corresponds to an energy from E, to E, + dE,. In this case, n, represents the probability for the initial energy to be between E, and E, + dE,. The absorbed fraction (AF) ,(kh) depends on the geometry of the two organs, the tissue composition of the two organs, as well as the tissue composition and the geometry of the organs that lie between the source and the target. The AF can be determined by analytical methods such as Point Kernel, or by using MonteCarlo transport codes. The purpose of this dissertation is to provide a more accurate calculation of the AF of energy within bone marrow. Past Skeletal Dosimetry Works and Models F. W. Spiers is responsible for most of the early work on skeletal dosimetry. While at the University of Leeds, Spiers (1949; 1951) began with relatively simple studies on bone and soft tissue interface, and the unique dosimetry associated with that region. Spiers (1963) later looked at influences of the percentage of active marrow on trabecular dosimetry. He used active marrow distribution data derived by Ellis (1961) from the work of Mechanik (1926) and Custer and Ahlfeldt (1932). Spiers (1966b; 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 betaemitters. This characterization was originally performed using microscopes and visual inspection. At this point, Spiers' group began working on a method to describe the geometry in terms of frequency distributions of linear path lengths through the trabecular and marrow regions. These distributions are called chordlength distributions. A variety of methods for obtaining these distributions exist. They depend on the origin and direction of the rays relative to the object. As pointed out by Eckerman et al. (1985), failure to account for the distinct nature of these distributions can result in misunderstanding some aspects of the radiation transport processes. Three fundamental methods of randomly obtaining these frequency distributions are relevant in trabecular dosimetry. * Meanfreepath randomness (or [[randomness). A chord of a convex body is defined by a point in space and a direction. The point and the direction are chosen randomly from independent uniform distributions. This kind of randomness results, for example, if the convex body is exposed to a uniform, isotropic field of straight lines. * Interior radiator randomness (or Irandomness). A chord is defined by a point in the interior of the convex body and a direction. The point and the direction are chosen randomly from independent uniform distributions. This kind of randomness results, for example, if the convex body contains a uniform distribution of point sources, each of which emits radiation isotropically. * Surface radiator randomness (or Srandomness). A chord is defined by a point on the surface of the convex body and a direction. The point and the direction are chosen randomly from independent uniform distributions. This kind of randomness results, for example, if the surface of a convex body contains a uniform distribution of point sources, each of which emits radiation isotropically (Kellerer 1971). When an electron starts from the bonemarrow interface (surfaceseeking radionuclide) its first path length is better characterized by an Srandomness. When it starts from the volume of a bone trabecula or a marrow cavity, its first path length is better characterized by an Irandomness. After this electron has left the first medium, its next path lengths are better characterized by a jrandomness. As seen previously (Figure 24), a single electron is likely to cross several bone and marrow regions before it loses its entire energy. Therefore, the overall electron transport is better characterized by a [randomness. As a consequence, [randomness is the method most investigated and thus used for our study. Eventually, the Leeds group was able to automate the process of obtaining chord distributions (Beddoe 1976b; Beddoe et al. 1976; Darley 1972; Spiers 1969). They physically sectioned trabecular bone regions into thin slices and took contact radiographs of the slices. In their techniques, the radiograph is mounted on a turntable below a light microscope. Above the turntable is a photomultiplier tube. As the turntable rotates, the radiograph is also moved radially, creating scan lines of minimal arc. The duration of a light pulse seen by the photomultiplier corresponds to a marrow chord length. The opposite is true of trabecular chords. Darley (1972) developed the original apparatus. Later, Beddoe (1976a) used the system and improved on the radiography and preparation of the bone sections. The distance between consecutive scan lines in the Leeds optical scanner is approximately 8 tm. The Leeds scanner had a dead time of 39 lts after the registration of a pulse length. This corresponds to a lost path length of 100 tm as the turntable continued to rotate. They justified this loss by concluding that the lost path length occurred almost entirely over the features which were not measured (Beddoe 1976a). The bone and marrow chord lengths are measured in separate scans. Thus, if marrow chords were being measured and the next bone chord was less than 100 itm, the next marrow chord would be registered as being smaller than actual size. The opposite would be true for the bone chord scan. The effective resolution of their scanning system including film noise was reported as 11.5 tm (Beddoe 1976a). To obtain omnidirectional distributions, Spiers and his colleagues had to make some symmetry assumptions. In the case of human vertebrae, the Beddoe (1976a) study found symmetry in one direction. The conclusion was that a scanning measurement in any set of parallel planes cut parallel to the symmetry axis was sufficient to generate an omnidirectional distribution. Use of Nuclear Magnetic Resonance Interest in bone micromorphology extends well beyond the radiation dosimetry community. Many studies focused on trabecular microstructure in an attempt to measure various structural parameters that might prove to be statistically reliable predictors of bone fracture risk in diseases such as osteoporosis. These studies have traditionally been performed using optical microscopy (Odgaard et al. 1990; Parfitt et al. 1983) or scanning electron microscopy (Boyde et al. 1986; Whitehouse and Dyson 1974; Whitehouse et al. 1971). These techniques require substantial sample preparation and are inherently destructive to the specimen. Furthermore, physical sectioning does not permit viewing the sample in multiple planes. Considering the anisotropic nature of trabecular bone, multiplane viewing is a highly desirable feature. Over the past decade, several groups have investigated the use of two nondestructive techniques for highresolution imaging of trabecular bone: QCT and NMR microscopy. QCT has been used to measure regional bone mineral density (Bone et al. 1994; Cody et al. 1989; Cody et al. 1991; Flynn and Cody 1993; Grampp et al. 1996; Kleerekoper et al. 1994a; Kleerekoper et al. 1994b; Link et al. 1997; Link et al. 1998a; Link et al. 1998b; Majumdar et al. 1997) and bone structural parameters (Chevalier et al. 1992; Durand and Ruegsegger 1992; Kuhn et al. 1990; Link et al. 1997; Link et al. 1998a; Link et al. 1998b; Majumdar et al. 1997; Muller and Ruegsegger 1996) at various skeletal sites in vivo. Several studies have used QCT for in vitro analyses of trabecular bone (Ciarelli et al. 1991; Cody et al. 1989; Cody et al. 1991; Cody et al. 1996; Engelke et al. 1993; Goulet et al. 1994; Kinney et al. 1995; Kuhn et al. 1990; Link et al. 1997; Link et al. 1998a; Link et al. 1998b; McCubbrey et al. 1995; Muller et al. 1994; Muller and Ruegsegger 1996; Ruegsegger et al. 1996). NMR microscopy represents an alternative to QCT for analyzing trabecular bone microstructure. This imaging technique is ideally suited for this purpose because bone marrow is composed primarily of water and lipids, thus providing an abundant source of proton MR signal. Conversely, bone does not contain hydrogen in the abundance or chemical form needed to produce a sufficient signal. The bonemarrow interface thus ensures a high intrinsic contrast in the resulting image. Several investigators have used NMR microscopy to quantify trabecular microarchitecture both in vivo (Chung et al. 1994; Foo et al. 1992; Gordon et al. 1997; Grampp et al. 1996; Guglielmi et al. 1996; Jara et al. 1993; Majumdar et al. 1997; Ouyang et al. 1997; Sebag and Moore 1990; Song et al. 1997; Wong et al. 1991) and in vitro (Chung et al. 1996; Chung et al. 1995a; Hwang et al. 1997; Majumdar et al. 1996; Wessels et al. 1997). Isotropic resolutions for these in vitro studies have been reported in the range of 50 to 150 tm. NMR microscopy of human subjects in vivo have yielded images with voxels ranging from 78 to 200 tm in plane and with slice thickness of 500 to 700 tm using clinical MR units at a field strength of 1.5 T (Jara et al. 1993; Majumdar and Genant 1995). Improved resolutions for in vivo NMR imaging would be possible using newer 3 Tesla clinical machines. NMR microscopy presents three distinct advantages over QCT for studying the microarchitecture of trabecular bone for improved radiation dosimetry modeling. First, NMR does not use ionizing radiation and thus applications to in vivo imaging are not dose limited. Second, NMR images may be acquired in any arbitrary plane without sample or subject repositioning. QCT would require sample positioning to acquire a specific orientation. Finally, NMR microscopy can be used to assess the fat and lipid mass fractions of the marrow spaces (Ballon et al. 1991; Ballon et al. 1996; Dixon 1984; Glover and Schneider 1991). NMR microscopy allows one to assess the trabecular and marrow chordlength distributions through the sample, and also to determine the fraction of marrow space occupied by active marrow (i.e., the regional marrow cellularity). Bone Sample Imaging and Image Processing Since Spiers' study, very little has been done in trabecular bone dosimetry. The chordlength distributions measured at Leeds during the 1970s became the reference and are still used today. While the research of Spiers and his students was superb in its level of detail and completeness, future improvements in skeletal dosimetry, particularly for medical applications, require an expanded database of marrow and trabecular chord distributions. These databases should more fully encompass variations in trabecular microstructure with both subject age and gender. Newer techniques in medical imaging can now be applied to this particular task. A few years ago, studies were initiated at the University of Florida (Jokisch 1997; Jokisch 1999; Jokisch et al. 2001a; Jokisch et al. 1998; Jokisch et al. 2001b; Patton 1998; Patton 2000; Patton et al. 2002a; Patton et al. 2002b; Rajon 1999) to look at techniques that could be used to improve on, or expand, the microstructure database gathered at Leeds. This work resulted in choosing nuclear magnetic resonance (NMR) microscopy coupled with imageprocessing techniques to obtain chord distributions and MonteCarlo transport codes to perform dose calculations. Figure 25 shows an example of 3D image obtained with an NMR system. This raw image is a digital image in which each voxel is a gray level number between 0 and 255. If the image on Figure 25 shows perfectly the location of the bone regions and the marrow regions, it's because your eyes constitute a powerful tool that can analyze the image. A computer program does not have this ability and the image needs to be processed in order to separate the bone and marrow voxels. This was done by Jokisch et al. (1998) using an imageprocessing technique that can be divided into three steps. * Image extraction in which a region of interest is extracted from the entire image (already done in Figure 25) * Image thresholding and segmentation using a mathematical process to determine the gray level limit between bone and marrow * Image filtering to eliminate abnormal high or low intensity voxels arising from signal noise. Articular Cartil Epiphy Periosteum Epiphyseal Linel Trabecular Bone. Medullary Canal Figure 21. Femur head showing the different constituents of the bone structure. Adapted from a study by Takahashi (1994). Haversian System Marrow Cavities Vessel I Haversian Canal Periosteum olkmann s Canal Blood Vessel 0 1 mm 2 mm Figure 22. Microstructure of compact bone and trabecular bone. Adapted from a study by Marieb (1998). 4 Trabeculae Interstital Lamellae Osteoclast Osteocyte SOsteoblasts  Marrow Cavities 0 0.5 1 mm I I I Figure 23. Microstructure of trabecular bone showing the types of bone cells and their location. Adapted from a study by Tortora (1992). Figure 24. Differing paths of two electrons emitted within anisotropic regions of trabecular bone. Trabecular bone 3D image obtained from a 4.7 Tesla NMR system. The image was taken over an 11 hour and 10 minute acquisition time (TR = 600 ms, TE =9.1 ms, spectral width: 123,457 Hz, 2 averages, matrix: 512 x 256 x 256, field of view: 4.5 x 2.25 x 2.25 cm3). The sample comes from the right femur head of a 51yearold man. The sample size is 1.81 x 1.06 x 0.80 cm3. The resolution is 88 x 88 x 88 jm3. Figure 25. CHAPTER 3 VOXELSIZE EFFECTS IN 3D NMR MICROSCOPY PERFORMED FOR TRABECULAR BONE DOSIMETRY1 Introduction Assessment of radiation absorbed dose within trabecular regions of the skeleton continues to be an important challenge in medical dosimetry. Skeletal dosimetry is important in that trabecular bone serves as the "housing" for hematopoietic marrow, the tissue responsible for the production of a variety of different blood cells. Several situations may result in internal irradiation of bone marrow. These include occupational exposures to boneseeking radionuclides (Brodsky 1996), radionuclide therapy of tumors (Rubin and Scarantino 1978; Sgouros 1993; Siegel et al. 1990), and bone pain palliation treatments (Samaratunga et al. 1995). Previous Studies in Trabecular Bone Dosimetry Active bone marrow (also referred to as red marrow) is located within the trabecular regions of the skeleton (Clayman 1995; Marieb 1998). The difficulty in assessing marrow dose is due to the complex structure of the trabecular regions. Figure 31 shows how the trabecular lattice interlaces with the marrow cavities. In Figure 31, the thickness of the trabeculae is 300 Ltm and the sizes of the marrow cavities are on the order of 1000 Ltm. Furthermore, several studies have shown that the bone microstructure varies as a function 1 This chapter was published by Medical Physics in November 2000: Rajon DA, Jokisch DW, Patton PW, Shah AP, and Bolch WE. 2000. Voxel size effects in threedimensional nuclear magnetic resonance microscopy performed for trabecular bone dosimetry. Med Phys 27: 26242635. of age (Atkinson 1965; Ouyang et al. 1997; Snyder et al. 1993) and between the sexes at a given age (Atkinson 1967; Mosekilde 1989). F. W. Spiers and colleagues conducted the first comprehensive studies on bone dosimetry at the University of Leeds during the 1960's. Their work demonstrated that a trabecular bone sample could be characterized by its chordlength distributions acquired through both the bone trabeculae and the marrow cavities. Consequently, one of their main accomplishments was to measure these two chordlength distributions within thin physical sections of trabecular bone using an optical scanning system (Beddoe 1976b; Spiers 1966b; Spiers 1967). His results were the first attempt to characterize the trabecular microstructure, and they have since been used as the basis for almost all subsequent skeletal dosimetry models (Bouchet and Bolch 1999; Bouchet et al. 2000; Bouchet et al. 1999b; Eckerman 1985; Spiers et al. 1978a; Spiers et al. 1978b; Stabin 1996). Today, Spiers' research still serves as a reference data source in trabecular bone dosimetry and his distributions are used to compare the results of this present study. Use of NMR Microscopy During the 1990s, nuclear magnetic resonance (NMR) was shown to be an important tool for quantifying the trabecular microstructure. Several investigators have used NMR microscopy to obtain 3D images of invitro samples with image resolutions below 100 tm (Chung et al. 1996; Chung et al. 1995b; Hwang et al. 1997; Link et al. 1998b; Wessels et al. 1997). Recently, Jokisch et al. (1998) investigated a technique to calculate the chordlength distribution in these highresolution images. Using a 600MHz NMR spectrometer they obtained a 59 x 59 x 78 itm3resolution image of a thoracic vertebra. After image filtration and image segmentation of the raw data, Jokisch et al. showed that the chordlength distributions calculated from this image were consistent with the Spiers' distributions obtained in both the cervical and lumbar vertebrae for a similarly aged male subject. This study showed that NMR is an excellent tool for acquiring 3D images of trabecular bone. Coupling these images with a particle transport code thus allows one to assess the absorbed dose in bone marrow using particletransport techniques within the full realistic 3D structure of the skeletal region. At the University of Florida, we have acquired images of trabecular bone samples for subsequent dosimetry analysis at both high and moderate magnetic field strengths (14.1 T and 4.7 T) with acquisition times of several hours. One of the many parameters to be optimized in these images is the voxel resolution of the final 3D image as controlled by the image matrix size and physical size of the sample. A decreasing voxel size necessarily increases the required acquisition time to maintain a reasonable signaltonoise ratio (SNR). Consequently, it is important to consider the dosimetric consequences of image voxel size in order to optimize our NMR imageacquisition techniques. VoxelSize Effects in Skeletal Dosimetry Calculations Voxelsize effects are demonstrated in Figure 32. This figure shows a schematic slice of a voxelized image, as in those obtained by Jokisch et al. (1998) via NMR microscopy, after image segmentation and filtration. The two curved lines represent the true boundaries between a bone trabecula and its two adjacent marrow cavities. The grid represents the pixels of the image. Each pixel may be assigned to marrow (the white pixels) if more than 50% of its volume is within marrow cavities. The pixel is assigned to bone (the dark pixels) if less than 50% of its volume is composed of marrow. Consider an electron traveling within a bone trabecula as indicated by the black arrow. In the real bone sample, the electron will deposit its entire energy in bone. In the digital image composed of pixels, the particle travels across both marrow (white pixels) and bone (dark pixels). As a result, the electron traveling in the digital image deposits its energy in both marrow and bone overestimating the energy deposition within marrow. Obviously, this error will be reduced if the voxel size is reduced, but what is not known is the minimum voxel size needed to reduce the error to an acceptable value. In some cases, the overestimate of absorbed dose to the marrow (situation described above) may be compensated by an overestimate of dose to the bone for a particle traveling on the other side of the boundary. A few other effects can be expected from the voxelization of the sample. Among them, one can image the variation of the volume of each media when the voxel size is increased and also the variation of the surface area of the boundary between bone and marrow when a curved interface is approximated by a series of orthogonal rectangular surfaces. These two effects are discussed in detail when analyzing the results of this chapter. Material and Methods Construction of the Mathematical Model of Trabecular Bone A mathematical bone model was created in this study to evaluate the effects of the voxel size on electron transport calculations. The model had to be constructed so that charged particles deposit their energy within the different media as they would if traveling within a real bone sample. From the studies of Spiers et al., it is known that this condition can be reasonably met if both the trabecular and marrow cavity chordlength distributions within the mathematical sample match their corresponding distributions within a real skeletal sample. In this study, the Spiers chord distributions for the cervical vertebrae were thus selected for initial investigation. Figure 31 shows that the boundaries between bone and marrow in the trabecular regions are smooth curved surfaces. Consequently, the mathematical sample had to be built with smooth surfaces and without sharp angles. Surface discontinuities are responsible for voxel effects and must be avoided within the mathematical sample. To satisfy these characteristics, a cubical bone volume was filled with spheres each representing a marrow cavity. The bone trabeculae are represented by the spaces between the spheres. The size of the sample cube had to be large enough so that it could represent a contiguous piece of trabecular bone, yet small enough so that the computer simulations of particle transport could treat the segmented images in a reasonable amount of time. The sample size that satisfies these two conditions was found to be 1.6 x 1.6 x 1.6 cm3. The sample had to be uniform in its microstructure over its entire volume and for this one needed to deal with "the edge effect". To understand "the edge effect", consider the case of a 2D image. Imagine a square in which one tries to fit circles (Figure 33). If the centers of the circles are randomly chosen inside the exterior square shown in Figure 33, at least one quarter of the area of each circle overlaps the exterior square (this fraction becomes one eighth the volume of a sphere for a 3D model). In Figure 33, this situation applies for the exterior square where one does not see spherical overlaps smaller than a quartercircle. Much smaller fractions of marrow cavities, however, are found along the cut edges of a real trabecular bone sample. Next, consider the interior square shown in Figure 33. With the same distribution of spheres, and if the distance between the two squares is greater than the maximum radius of the circles, the distribution of spheres within the interior square is uniform over its entire volume and no "edge effects" are present. For the current study, the maximum radius of the spheres is less than 3 mm, and thus the exterior cube of our mathematical sample must be 6 mm wider than the interior cube. As a result, the mathematical trabecular bone sample was built by positioning the centers of the marrow spheres randomly throughout a 2.2 x 2.2 x 2.2 cm3 cubical region, but only the central 1.6 x 1.6 x 1.6 cm3 cube was selected as the region of interest (ROI) for dosimetry studies. The next decision in building the mathematical sample was to choose the size of the spheres. Their radii had to follow a probability distribution that provides, for the whole sample, a marrow chordlength distribution similar to that of the Spiers cervicalvertebra distribution. To avoid surface discontinuities, the spheres needed to be located so that they do not overlap with one another. After several trials using various distributions of radii, it was found that an exponential distribution worked best: P(r)= Pme P, (31) where r is the radius of the spheres and Pm is a parameter that allows changes to the average value of the distribution. The value of Pm that gives the best fit to the Spiers' marrow chordlength distribution for the cervical vertebrae was found to be 44 cm1. An advantage of this mathematical sample is that the number of spheres does not affect the marrow chordlength distribution. While keeping the same radius distribution, an increase in the number of spheres within the sample will increase the volume fraction of marrow but not its chordlength distribution. Increasing the number of spheres inside the cube will also reduce the bone volume and will obviously reduce the mean trabecula chord length. Therefore, once the marrow radius distribution is defined, the only concern is to try to fit as many spheres as needed within the mathematical sample cube so that the mean trabecula chordlength distribution is reduced to a value comparable to the Spiers' value. A Cprogram was written to build the sample by fitting the spheres inside the sample cube. The filling of the sample cube was a long process and the program was not able to fit more than 28,200 spheres in the sample cube, giving a bone chordlength distribution slightly skewed from that of Spiers. The final mathematical sample thus contains 28,200 spherical marrow cavities. This large number is a major concern if one thinks about a transport code having to deal with such a large number of regions. With the sample cube occupying a volume of 2.2 x 2.2 x 2.2 cm3, and as only the central 1.6 x 1.6 x 1.6 cm3 is considered the dosimetric region of interest, more than 60% of the volume of the original sample cube is contained outside the central ROI. Consequently, many spheres located along the edge will never overlap the region of interest. Therefore, a last step before the mathematical sample is complete is to remove all spheres located in the 2.2 x 2.2 x 2.2 cm3 bone sample that do not in some way intersect the central ROI. The total number of remaining marrow spheres within the ROI is thus reduced to 11,605. Using the radius distribution of Equation (31), it is possible to calculate both the total volume of marrow and the total surface area of the bonemarrow interface. The analytical expressions for both the marrow volume and the interface area are: 8rcN V = and (32) Pm' S = (33) where N is the number of spheres. With N = 11,605 and Pm = 44 cm1, the total marrow volume and interface surface area are 3.424 cm3 and 150.7 cm2, respectively. The mean marrow chord throughout the mathematical bone sample may be estimated through Cauchy's theorem (4V/S) as 909 tm. Note that these values are inclusive of all marrow spheres whereas some spheres only partially overlap the ROI boundary. Consequently, the true values of the marrow volume fraction and the surface area of the bonemarrow interface within the dosimetric ROI are expected to be smaller than predicted by Equations (32) and (33). A Cprogram was also created to calculate the chordlength distribution for both the marrow cavities and bone trabeculae within the mathematical sample. As the sample is made of spheres, chord lengths are isotropic, and the calculation can be performed using a uniform field of parallel rays crossing the entire sample along one of the three axes. Another program calculates the volume fraction of marrow within the sample. As it is difficult to find an analytical expression for the volume fraction of a sphere that overlaps a cube, a MonteCarlo calculation was performed. The accuracy of this method can be measured by the standard deviation of the result. When a point is selected randomly inside the cubical sample, it can be either in bone or in marrow. The probability of each event follows a binomial distribution and the standard deviation of a sample of N points is given by S=P ) (34) N In Equation (34), p is the probability for the point to be in marrow. This value of will be maximal for p = 0.5. With N = 1,000,000 the standard deviation is then less than 0.05%. Construction of the Segmented Images of the Model Once the mathematically defined bone model is created representing a cubical specimen of trabecular bone, a grid structure is placed over the sample representing the voxel dimensions of an NMR image acquisition. Segmentation of the grid structure yields a binary image similar to that obtained by Jokisch et al. Segmentation of the mathematical sample is achieved by assigning each voxel to either bone or marrow according to the percentage of marrow (volume inside the spheres) and bone (volume outside the spheres) it contains. A total of 19 samples were created with cubical voxel sizes ranging from 16 ltm to 1000 tm (Table 31). A simplified segmentation algorithm was adopted in which, for each voxel of the image, the volume fraction of marrow it contains is determined. If this fraction is greater than 50%, the voxel is assigned to marrow and if not, it is assigned to bone. The complexity comes from the calculation of the volume of intersection between a cubical voxel and a marrow sphere. Many different situations exist and few have analytical solutions. Consequently, a combination of analytical and MonteCarlo methods were used for image segmentation. Three different Cprograms were written to perform the segmentation process. The first creates a file that contains one byte per voxel within setting the tissue medium of each voxel. The second program calculates the location of all voxels that are fully inside each marrow sphere of the sample and assigns these voxels to marrow. The third program sweeps through all voxels of the image previously undefined and for each it checks how many spheres overlap the voxel. If no overlapping regions are found, the voxel is assigned to bone. If at least one sphere overlaps the voxel, the program uses a MonteCarlo technique to calculate the percentage of voxel volume intersected by one or more spheres. If this cumulative percentage exceeds 50%, the voxel is assigned to marrow; otherwise, it is assigned to bone. The decision to segment at a volume fraction of 50% is somewhat arbitrary, but sufficient for the present study as it was more important to be consistent across all voxelized images. Other percentages or even a mass fraction (considering the different tissue densities) could have also been employed. Once the segmentation is performed, another program is used to compress the data file so that each byte of the file can store eight voxels. Finally, two programs calculate, for each segmented image, the marrow volume fraction within the image and the surface area of the bonemarrow interface. The two series of results are in Table 31. ElectronTransport Simulations The next step in the study is to couple the segmented voxelized images with a MonteCarlo transport code to study the evolution of absorbedfraction results while the voxel size is changed. The goal is to calculate the absorbed fractions (0) for a source of monoenergetic electrons born within the marrow cavities and for the bone trabeculae and marrow cavities as target regions. For both sets of absorbed fractions, simulations of electron transport are made within both the mathematical bone sample and within each of its simulated, segmented voxel images. Particle transport calculations were performed using the EGS4PRESTA code. Transport simulations were performed only within the central 1.6 x 1.6 x 1.6 cm3 ROI of the bone sample. Every particle leaving this volume was discarded. A homogeneous source of monoenergetic electrons was used with particle emissions beginning within those marrow spheres completed enclosed in the ROI. For both the bone and the marrow as target regions, the absorbed fraction deposited in that region was calculated. A total of six electrons energies were selected ranging 0.05 MeV to 2 MeV. Two different EGS4 user codes were developed: one for the mathematical sample ("reference code") and a second for the 19 segmented images ("test codes"). For both programs, the same calculation was performed. The only difference was the way the geometry was defined. For the "reference code" the geometry is defined by equations of spheres, whereas the "test code" is defined by equations of planes delineating the voxel slices of the particular image. The parameters used by the EGS4 transport code were also kept constant between both programs. The media used for both bone and marrow were those defined by the ICRU (1992) Report 46 for an adult. The bone regions were composed of pure skeletoncortical bone and the marrow regions were made of pure skeletonred marrow. Descriptions of both tissues are given in Table 32. The densities are 1.92 g/cm3 for bone and 1.03 g/cm3 for marrow. In both bone and marrow regions, the energy cut off for both electron and photon transport was set to 10 keV. For the PRESTA extension, the ESTEPE parameter was set at 0.05. Each calculation (6 times the "reference code" and 114 times the "test code") was performed for 10 runs of 100,000 initial electrons. The results were averaged over the 10 runs and a standard deviation was calculated to estimate the standard error. SValue Calculations The absorbed fractions calculated with EGS4 were then used to calculate the S values (dose per unit cumulated activity) as defined by the Medical Internal Radiation Dose (MIRD) Committee for several radionuclides (Loevinger et al. 1991). Five nuclides were chosen for their interest in trabecular bone dosimetry (131I, 32P, 33P, 153Sm, and 117mSn) and because they cover a large range of beta particle energies. Table 33 shows the radiological characteristics of the five radionuclides selected. S values were calculated using the decay scheme tables of Eckerman et al. (1993). In this study, only marrow sources of electrons were considered, even though two of these radionuclides (153Sm and 117mSn) are associated with radiochemicals that exclusively localize in or on the surface of the bone trabeculae. As discussed later, however, conclusions drawn regarding voxelization errors in the crossdose to bone will hold equally true for the crossdose to marrow when the sourcetarget designations are reversed. Results and Discussion Mathematical Sample The mathematical sample of trabecular bone was created as a cube (2.2 cm on edge) that contains 28,200 marrow spheres. Only 11,605 spheres remained in the sample after exclusion of those spheres fully outside the central 1.6 x 1.6 x 1.6 cm3 ROI. The average radius for all spheres is 227 im. The volume fraction of marrow within the ROI is 0.6979 + 0.0005. The total volume of marrow strictly found within the ROI is thus 2.859 cm3, a value less than that predicted by Equation (32) as discussed earlier. The true surface area of the bonemarrow interface found strictly within the ROI may be estimated as (150.7 cm2) x (2.859 cm3 / 3.424 cm3) or 125.8 cm2. Figure 34 shows a 1.6 x 1.6 cm2 cross section of the final sample within the ROI. The image is perpendicular to the Xaxis, and is located at X=0 (the center of the cube). Black regions represent the lattice of trabeculae while white circles represent the marrow cavities. In Figure 35, the chordlength distributions for both the marrow cavities (Figure 35A) and the bone trabeculae (Figure 35B) are compared with those for the cervical vertebra in the data of Spiers and colleagues (Whitwell 1973). Within the mathematical sample, the mean chord lengths are 871 tm and 377 tm for the marrow cavities and the bone regions, respectively. The former approaches that given by Equations (32) and (33) and Cauchy's theorem (909 [tm). The corresponding values from Spiers data are 909 tm and 280 tm, respectively. As noted earlier, the difference between the values of 871 and 909 tm are attributed to those spherical marrow cavities that only partially overlap the ROI boundary. Figure 35B shows a large difference in shape between the chordlength distributions for the bone trabeculae. This difference is due to the difficulty in fitting a sufficient number of marrow spheres within the bone cube and is reflected in the difference between the mean chord lengths. Nevertheless, our goal was not to have an exact representation of trabecular bone, but to create a mathematical model representative of trabecular bone. The shapes of both distributions are similar, however. One may also interpret our mathematical sample as a specimen of trabecular bone with trabeculae slightly thicker than those measured by Spiers for his cervicalvertebra specimen. Segmented Images Segmented images were created for 19 different voxel sizes listed in Table 31. The fifth column of Table 31 gives the volume fraction of marrow in each segmented image. As the voxel size increases, the volume fraction increases reaching 100% at very poor resolution. As there is more than 50 percent marrow in the mathematical sample (69.79 %), the larger the voxel size, the more likely each is to contain more than 50% marrow. Therefore, as the voxel size increases, there is a higher probability for each voxel to be assigned to marrow, and the percentage of marrow voxels within the entire sample thus increases. Figure 36 shows the result of segmentation for four different image resolutions taken for the same slice through the mathematical bone sample. The resolutions are 667 [tm (24 voxels per dimension) for Figure 36A, 286 tm (56 voxels per dimension) for Figure 36B, 80 tm (200 voxels per dimension) for Figure 36C, and 25 tm (640 voxels per dimension) for Figure 36D. For a voxel size of 286 tm, the image appears blocky and visually inaccurate, yet the volume fraction of marrow approaches the true value (70.84 % compared to 69.79 %). For a voxel size smaller than 300 tm, the error on the volume fraction is only about one percent. Because of poor resolution, some voxels overestimate their true volume fraction of marrow and some overestimate their volume fraction of bone, with both errors canceling one another yielding an acceptable marrow fraction over the entire image. Figure 37A shows how the marrow volume fraction converges toward the reference value (the horizontal doted straight line) as the resolution is improved. The last column of Table 31 shows the total surface area of the bonemarrow interface within the segmented sample. The results are compared with the reference value (the horizontal doted straight line) in Figure 37B. This area increases almost linearly from almost zero at 1000 tm to 191 cm2 at very high resolution. This is partly because of the volume fraction of marrow. For a large voxel size, only a few voxels are assigned to bone. As a consequence, the surface area between bone and marrow is reduced (see Figure 36A). Between Figures 36B and 36D the volume fraction of marrow is almost the same, but the surface area in Figure 36D is still larger than that in Figure 36B (Table 31). In Figure 36B one can see that, because of the lower image resolution, marrow spheres are connected to each other through marrow voxels, thus reducing the interface surface area. This problem does not appear in Figure 36D (or it appears very infrequently). This explains why the surface area continues to increase whereas the volume fraction remains constant. At high resolution, the area converges just above 191 cm2, which is 50 % larger than the surface area of marrow spheres within the true mathematical sample (125.8 cm2). Voxelization of the images thus introduces an overestimation of the interface surface area that is not reduced with further reductions in voxel size. Absorbed Fractions and their Absolute Errors EGS4 transport simulations were made for monoenergetic electron sources located within the trabecular marrow space of the 19 segmented images as well as the reference mathematical trabecular bone sample. For each simulation, absorbed fractions of energy were estimated for targets in the marrow cavities, the bone trabeculae, and regions outside the dosimetric ROI. The latter was used to check the energy deposition balance within the system. In no case did the standard deviation on the absorbed fraction exceed 0.0015. Reference absorbed fractions for the mathematical sample are listed in Table 34. They are shown to decrease with increasing electron energy, as more electron kinetic energy is lost to the skeletal regions outside the dosimetric ROI. The convergence of the absorbed fraction to marrow is shown in Figure 38A as a function of voxel size and electron energy, while Figure 38B shows the corresponding values for bone as the target tissue. The abscissa in Figures 38A and 38B is the voxel size ranging from high resolution (16 [tm) to lowresolution (500 [tm) images. The ordinate is the difference between the reference absorbed fraction and the value found within each of the 19 segmented images: A = ,eg ref (35) A positive value corresponds to an overestimate of the absorbed fraction within the segmented image and a negative value corresponds to an underestimate. Six general points can be made in viewing the data from Figure 38. They are 1) Both diagrams are fairly symmetric. For all electron energies and voxel sizes, an overestimate of the marrow absorbed fraction leads to an underestimate of the bone absorbed fraction and vice versa. 2) The absorbed fraction for large voxel sizes is overestimated in marrow and underestimated in bone. 3) For highenergy electrons (>400 keV), the error decreases as the voxel size is reduced. Figures 38A and 38B show that the absorbed fractions calculated within the segmented images converge to the reference value (the difference converges to zero) as the resolution is improved. 4) For electrons >400 keV, convergence of the results to the reference value is improved with increasing initial electron energy. 5) At low electron energies (<400 keV), the data of Figure 38 show that for large voxel sizes (> 350 [tm), smaller absorbedfraction differences are seen at lower electron energies. 6) At low energy electrons and at high image resolution, the results do not converge to the reference value. To further understand these trends, it is helpful to revisit the errors introduced via voxelization in both the marrow volume fraction ("volumeerror effect") and the surface area of the bonemarrow interface ("surfaceerror effect") (Figure 37). The influence of errors in the marrow volume fraction at large voxel sizes is easily understood. At low image resolution, the overestimation of the total marrow volume results in an increase in the absorbed fraction to marrow and a corresponding decrease in the absorbed fraction to the bone trabeculae. The surfaceerror effect is more complex. As shown in Figure 37B, the surface area of the bonemarrow interface increases as the voxel size decreases. This error was shown to overestimate the true surface area at small voxel sizes and to underestimate the area at large voxel sizes. As the surface area is artificially increased in highresolution images, particle escape at the interface surface is increased and the absorbed fraction to bone trabeculae is overestimated. Within lowresolution images, the surface area is underestimated and thus the absorbed fraction to bone is underestimated. For low energy electrons born within the marrow cavities, energy loss to bone is contributed by only those electrons emitted from a marrow layer immediately adjacent to the bonemarrow boundary. The thickness of this effective source layer is approximately equal to the electron range in marrow. If the electron range is much smaller than the voxel size, the effective source volume is roughly equal to the total surface area times the electron range in marrow. The right angles between the squared surfaces of the voxels do not introduce a significant contribution to this volume, and thus the variation of the effective source volume is roughly proportional to the corresponding variation in interface surface area. Consequently, an increase in surface area increases the effective source volume and the absorbed fraction to bone by the same proportion. On the other hand, if the electron range exceeds the voxel size, the effective source volume is more complex and becomes smaller than that predicted by the product of the surface area and the electron range. At higher electron energies, an increase in surface area does not necessarily increase the effective source volume by the same proportion, and thus overestimates in the absorbed fraction to bone are not as significant as they are at lower energies. A general conclusion is thus that overestimates in the absorbed fraction to bone are only significant if the voxel size is large compared to the electron range. Note that identical but converse arguments would hold true if one were to consider instead a bone source of lowenergy electrons irradiating the marrow in a highresolution image. Let us now return to the six observations noted in Figures 38A and 38B. Point 1): The total size of the bone sample and its interior ROI is the same whether it is taken as the pure mathematical sample or one of its 3D segmented images. The loss of energy due to particle escape, therefore, is constant and only depends on the initial electron energy. Point 2) is a direct consequence of the errors in the marrow volume fraction. At large voxel sizes, the volume of bone is small compared to that within the mathematical sample and thus the bone absorbed fraction is underestimated. This error increases as the voxel size increases. Point 3) is also a consequence of the errors in the marrow volume fraction. As the voxel size is reduced, the marrow volume approaches the reference volume and the absorbed fractions converge toward the reference values. For electrons exceeding 400 keV, their range exceeds the voxel dimensions (1300 tm for 400 keV electrons in water). Consequently, the surfaceerror effect only appears at large voxel sizes where it is completely masked by the volumeerror effect that acts in the opposite direction. Absorbed fractions converge to reference values around 200 tm where the surfaceerror effect is too weak to have a noticeable consequence. Point 4). For highenergy electrons, the deposition of energy is expected to be more uniform throughout the whole sample than for lowenergy electrons. Consequently, the absorbed fractions to both marrow and bone would be those given by the ratio of masses for the two media. One can therefore expect that the dependence of the results on the voxel size has its origin in the variation of the marrow volume fraction within the sample. Furthermore, convergence is also improved at high electron energies where the surfaceerror effect is smaller. Point 5). For lowenergy, shortranged electrons below 400 keV, the absorbed fraction to marrow approaches unity. Consequently, the dependence of the absorbed fraction on the volumeerror effect is weak for this energy range. A consequence of points 4) and 5) is that for large voxel sizes (> 350 [tm) the absorbedfraction error increases with electron energy below 400 keV, and decreases with electron energy above 400 keV. The calculation performed for 400 keV electrons seems to give the largest absorbedfraction error in images of poor resolution. Point 6). For small voxel sizes (below 400 [tm) the volumeerror effect is no longer strong and only the surfaceerror effect is dominant. For low electron energies, the electron range becomes smaller than the voxel size (e.g., 43 tm in water for 50 keV electrons). The surfaceerror effect thus leads to an overestimate of the bone absorbed fraction. When the voxel size is very small and becomes smaller than the electron range, the absorbed fraction converges again toward the reference value. Absorbedfraction curves in Figure 38B for both 50 and 100 keV electrons show an overestimate of the bone absorbed fraction that increases to a voxel size approximately equal to their range (43 tm and 143 jtm, respectively). At higher voxel resolutions, the absorbedfraction errors decrease again as they approach their reference values. The surfaceerror effect no longer influences the calculation if the voxel size becomes significantly smaller than the electron range. Absorbed Doses and their Relative Errors The discussion above reflects the absolute error made on the absorbed fraction with changes in voxel resolution. An alternative evaluation of the voxelsize effects on dose calculations to skeletal tissues using NMR microscopy is to look at the relative error of the absorbed fractions. Figure 39 shows the relative error (in percentage) for the absorbed fraction to both marrow (Figure 39A) and bone (Figure 39B) expressed as: Ar = eg ref .100%. (36) Oref For the marrow results, Figures 38A and 39A are almost identical since the absorbed fraction is on the order of unity for an electron source located within the marrow cavities. For the bone trabecula results, Figure 39B shows that the relative error is very important for low energy electrons. The absorbed fractions to the bone trabeculae are small because of the short ranges of the particles in the marrow source region. Here the relative error can reach 65 % for 50 keV electrons. This error is directly attributable to the 50% overestimate in the surface area of the bonemarrow interface (Figure 37B). By dividing each absorbed fraction by the mass of the corresponding target region (segmented image or reference sample), the corresponding relative error on the absorbed dose to either marrow (Figure 310A) or the bone (Figure 310B) from monoenergetic electrons sources in marrow may be calculated: r seg Oref A = mseg mref .100%. (37) mref Smref ) Figures 39 and 310 are similar for small voxel sizes, but differ at large voxel sizes. Since the mass of the target is proportional to its volume, the absorbedfraction error at large voxel sizes is attenuated by the mass error when the corresponding dose is calculated. Consequently, the error on the dose is smaller than that for the absorbed fraction. For small voxel sizes where the volume fraction of marrow is almost constant and equal to its true value, the relative error in the absorbed dose is approximately similar to the relative error in the absorbed fraction. The results of Figures 38 to 310 are shown for monoenergetic electron sources in marrow. To investigate potential errors in marrow and bone dose due to radionuclides localized in marrow, radionuclide S values were then determined in which absorbed fractions for monoenergetic electrons were weighting across their betaparticle energy spectra. These S values are given in Table 35 for 32p, 33P, 131I, 153Sm, and 117mSn. Figure 311 shows the relative error on the S value as a function of voxel size. For 32P, the mean electron energy is high and Figure 311 shows good convergence of the S value across all voxel sizes. There is less than 1% error for both the bone and the marrow dose below 300 tm. Iodine131 and 153Sm are intermediateenergy electron emitters. For these radionuclides, the surfacearea effect begins to become important and the relative error for S(bonemarrow) cannot be reduced to less than 5% even at high voxel resolution. For the very lowenergy emitters ll7mSn and 33P, the relative error on the S value remains as high as 25% for the crossdose to bone, but is less than 4% for the selfdose to marrow. At voxel resolutions of 450 tm, the result is exact but no conclusions can be drawn. This value of 450 tm is a consequence of the reduction of the surface area of the bonemarrow interface as the voxel size is increased. This surfacearea reduction depends on the marrow volume fraction within the mathematical sample and it would be incorrect to suggest that 450 rtm is an optimal value for either 117mSn or 33P. It is only an optimal value for the specific geometry used in this study. For a different trabecular bone sample with a different marrow volume fraction, the optimal value would most likely be different. Conclusion The voxelsize impact on dose calculations within a voxelized and segmented 3D NMR image of trabecular bone may be summarized according to three different aspects. First, the geometry of the segmented image is obviously different from that of the true bone sample. Figure 32 shows that cubical voxels may lead to overestimates and underestimates of energy deposition within the media if a particle is traveling close to a boundary. These errors tend to cancel one another on average and have probably no consequence on the dose calculation. Second, the volume fraction of marrow is overestimated at large voxel sizes, but as shown in Figure 37A this overestimate is small below 300 tm and there is no consequence to the dose calculation for image resolutions below 300 tm. Third, the surface area of the bonemarrow interface is overestimated at small voxel sizes (below 300 [tm) and underestimated at large voxel sizes as shown in Figure 37B. For highenergy electron emitters, this effect is without consequence in that the electron range exceeds the voxel dimensions. The surfaceerror effect occurs over a range of voxel sizes for which reductions in bone volume fraction with improved resolution attenuate the effect of increases in estimated surface area. The surfaceerror effect is thus without consequence for highenergy electrons. For lowenergy electrons emitted within the marrow cavities, the surfaceerror effect has little consequence on the selfdose to the marrow cavities (maximum error <5% at 200 keV in Figure 39A). Nevertheless, for lowenergy electrons, the surfaceerror effect does lead to an overestimate of the crossdose to bone trabeculae for small voxel sizes and an underestimate of the bone dose at large voxel sizes. The overestimate in bone crossdose approaches 25% for radionuclides with a mean beta energy of 100 keV and is maximal for 100 tm voxels. This error approaches 5% for radionuclides with a mean beta energy of 200 keV and is maximal also around 100 itm. No conclusion can be drawn for the optimal voxel size for lowenergy beta emitters in that the voxelsize range for overestimates and underestimates in crossregion dose depends on the variation of the surfaceerror effect with voxel size which further depends upon the specific geometry of the sample. At voxel resolutions below 100 tm, the error in crossdose is shown to turn over and again converge to the reference value as the voxel size becomes smaller than the electron range. This can be seen in both Figure 310 and Figure 311, but the convergence occurs at voxel sizes on the order of a few micrometers, well beyond the current capabilities of NMR on samples of this nature within reasonable acquisition time. The largest error encountered in the dosimetry of voxelized images was shown to be the influence of the surfaceerror effect on the crossregion dose for lowenergy electrons. Fortunately, in these instances, the selfdose will dominate because of the short range of the particles. In this study, the marrow cavities were selected as the source region of electron emissions. Because of the symmetry of the voxelized image, a similar overestimate in marrow dose would be expected for lowenergy electron sources in the bone trabeculae. Again, the selfdose to bone would dominate and the uncertainty in the average crossdose to the marrow cavity may be unimportant. Nevertheless, the surfaceerror effect warrants further investigation considering that studies by Lord (1990) indicate that the majority of marrow stem cells are located near the endosteal surface where the dose would be higher than the average marrow cavity dose. In summary, the present study found that an NMR image of trabecular bone composed of voxels <300400 tm provides accurate dosimetry for highenergy electron emitters and for both the selfdose to marrow and the crossdose to the bone trabeculae. For radionuclides with a mean beta energy of 300 keV or higher, the error is reduced to a few percent at this resolution. This finding is significant in that this voxelsize range approaches that obtainable for trabecular bone imaging via clinical MRI units at 1.5 T under conditions of minimal patient motion. While there is no impact of the surfaceerror effect on the selfdose to marrow for lowenergy beta emitters, overestimates in the interface surface area must be taken into account when considering the crossdose to bone at low electron energies. Further investigations are warranted including the use of additional mathematically defined bone samples of differing trabecular microstructure (e.g., cranium, ribs, etc.). In these studies, systematic dosimetry errors introduced by surface area overestimates with lowenergy electrons and within highresolution images can be evaluated. Correction factors might then be developed and applied to dosimetry results. 0 0.5 1 mm I I I Figure 31. Electron micrograph of the trabecular latticework within a lumbar vertebra. Adapted from a study by Clayman (1995). Voxelsize effects on the dose calculation for a single electron track traveling within a bone trabecula. The curved lines indicate the true surfaces of the bonemarrow interface. The gridlines indicate pixels within a corresponding digital image of the sample. Figure 32. Figure 33. Edge effect when trying to position circles within a squared field. Figure 34. Transverse slice, 1.6 cm x 1.6 cm, through the ROI of the mathematical sample of trabecular bone. 0.0010 . 9000 r 01)0.1 C 0 0002j 0 0l i0 il i11 _______ r'trLl iii' .111 ~1I 'III~i Marrow Cavities \. 0 500 l lr(io 1500 211ro 2500 3000i 3501f 4000 Ciord Length (Tim) S ill 1 I. ,,l r . Bone Traboculae 0 200 400 G00 UO0 1000 1200 1400 C'iltr i L rNmltl (inil Figure 35. Chordlength distributions: Spiers cervical vertebra compared to the mathematical model. A) Marrow cavities. B) Bone trabeculae. G 004 E 0 0 S0.001 0..nn Segmentation of the mathematical sample. The four images have a different resolution, but represent the same slice through the mathematical sample. A) 667 itm. B) 286 itm. C) 80 im. D) 25 itm. .*. I m I* U U m mU U U U El5 Figure 36. 1000 Rrp C. l llriori mi ri 0 200 400 ,00 800 Geometrical parameters as a function of image resolution for the mathematical sample. A) Marrow volume fraction (reference value is 69.79%). B) Surface area of the bonemarrow interface (reference value is 125.8 cm2). Figure 37. 1000 H r i' t1',1 ii ( rn .i 0 06 0.04 , o 2 0 00 0.02 004 0 50 100 1 5. 200 250 3C00 '350 400 150 500 0 0.1 0.02 0.00 0 02 0 04 0 IL) Bone Trabcc LIJC 100 1 250 300 350 00 50 500 0 ".,, '\ C 0 50 100 150 200 250 300 350 400 450 500 Ri Sohlir inr iiii) Absorbedfraction differences between the segmented images and the reference values. The reference is obtained within the mathematical sample. The source of radiation is monoenergetic electrons emitted within the marrow cavities. A) Target regions considered are the marrow cavities. B) Target regions considered are the bone trabeculae. Figure 38. R i, ,i iri n ( mIi) .1 0. ( 75 *OC ':.''v 5 UI 2.5 DO 2 5 xr If  .. VA n ^ I *^ ~_  ::^ Marrow Cavities 0 50 100 150 200 250 300 350 400 450 500 R': u1ltion (, rr) Bone Trabeculae c ou r,1: ,,,' .............. 0.100 MeV . .. .. I.. ,. .,.,   0 U.) FAL'' P l. r;. Iil v L: 0 S0 5 10io 1io 2oo 250 300 350 400 450 ro00 Riusioluticr.i filii Relative error in the absorbed fraction for the segmented images and using results from the mathematical bone sample as reference values. The source of radiation is monoenergetic electrons emitted within the marrow cavities. A) Target regions considered are the marrow cavities. B) Target regions considered are the bone trabeculae. 20 Figure 39. c 1PM L '' r ... Marrow Cait es 20 i. i M:.  3U  'f 00 Pr.solLtirfln ( m'! fl . T . L .i Sonile I raberLt I iae '. rI i I 20 4 *'0 M ,0*1 20 4 0 :.":i r,,10',..1' ' n SL, 1"J j r..i,. 5 200 400 600 800 100U R s. ,r:.II itirii ( inm ) Figure 310. Relative error in the absorbed dose for monoenergetic electron sources in the marrow cavities. A) Target regions considered are the marrow cavities. B) Target regions considered are the bone trabeculae. 0 200 Marrow Cavities SP33 "I 200 400 600 800 1000 /  Bon Trabrc lae 1 ,00 GO11 7r,' S 200 00 .00 (00 1000 Figure 311. Relative error in the S value calculated for five radionuclides of interest in skeletal dosimetry. A) Target regions considered are the marrow cavities. B) Target regions considered are the bone trabeculae. 4 0 4 8  12 R ('ul litluli fii 1) R oILl [lull1 f lmi Table 31. Characteristics of the 19 segmented images for a voxelsize range from 1000 [tm to 16 tm. Voxel size Voxels per Number of File size ([tm) dimension voxels (million of (million) bytes) 1000.00 666.67 500.00 400.00 333.33 285.70 250.00 200.00 153.85 125.00 100.00 80.00 62.50 50.00 40.00 31.25 25.00 20.00 16.00 16 24 32 40 48 56 64 80 104 128 160 200 256 320 400 512 640 800 1000 0.004 0.014 0.033 0.064 0.111 0.176 0.262 0.512 1.124 2.097 4.096 8.000 16.777 32.768 64.000 134.217 262.144 512.000 1,000.000 0.000 0.002 0.004 0.008 0.014 0.022 0.033 0.064 0.141 0.262 0.512 1.000 2.097 4.096 8.000 16.777 32.768 64.000 125.000 Fraction of Surface Area marrow voxels (%) 99.56 87.60 78.31 73.75 71.64 70.84 70.28 69.78 69.60 69.56 69.56 69.60 69.63 69.67 69.70 69.72 69.74 69.75 69.76 of Interface (cm2) 1.04 38.99 77.01 100.36 115.24 125.82 133.37 144.84 156.10 163.90 170.84 176.51 181.30 184.56 186.91 188.83 190.00 190.84 191.42 Table 32. Tissue compositions sample. used for bone and marrow in the mathematical bone Element Symbol Atomic Mass fraction Mass fraction in number in bone marrow (%) (%) Hydrogen H 1 3.4 10.5 Carbon C 6 15.5 41.4 Nitrogen N 7 4.2 3.4 Oxygen O 8 43.5 43.9 Sodium Na 11 0.1  Magnesium Mg 12 0.2  Phosphorus P 15 10.3 0.1 Sulfur S 16 0.3 0.2 Chlorine C1 17  0.2 Potassium K 19  0.2 Calcium Ca 20 22.5  Iron Fe 26  0.1 62 Table 33. Radiation characteristics of the radionuclides used for Svalue calculations. Mode of decay 1311 P 32p p 33p p 153Sm  117mS T.T Avg. Energy (MeV) 0.191 0.695 0.077 0.225 0.135 Max. Energy (MeV) 0.606 1.710 0.249 0.809 0.159 Halflife (days) 8.04 14.3 25.3 1.95 13.6 63 Table 34. Absorbed fractions calculated within the mathematical bone sample for the marrow region, the bone region, and the region beyond the dosimetric region of interest. Electron energy ((marrow<marrow) ()(bone+marrow) )(beyond ROI+marrow) (MeV) 0.05 0.9745 0.0232 0.0023 0.10 0.9200 0.0726 0.0074 0.20 0.7859 0.1926 0.0215 0.40 0.6222 0.3232 0.0546 1.00 0.5079 0.3292 0.1629 2.00 0.4034 0.2702 0.3264 64 Table 35. S values calculated within the mathematical bone sample for the five chosen radionuclides. The target masses used for the calculation correspond to the volume of tissue enclosed in the limit of the mathematical bone model. Radionuclide S(marrowsmarrow) (mGy/MBqs) S(bone+marrow) (mGy/MBqs) 131I 7.83 x 103 3.20 x 103 32P 2.08 x 102 1.56 x 102 33P 3.72 x 103 5.56 x 104 153Sm 1.07 x 102 4.35 x 103 117mSn 7.39 x 103 1.22 x 103 CHAPTER 4 SURFACEAREA OVERESTIMATION WITHIN 3D DIGITAL IMAGES AND ITS CONSEQUENCES FOR SKELETAL DOSIMETRY1 Introduction Trabecular bone is that portion of the adult human skeleton that houses the hematopoietic marrow, the tissue responsible for the production of various blood cells. The high mitotic activity of active bone marrow thus makes this organ one of high radiation sensitivity, and is thus of high interest in the dosimetry of radionuclide therapies where marrow toxicity is generally doselimiting (Sgouros 1993; Sgouros et al. 2000). Internal irradiation of bone marrow may result from occupational exposures to boneseeking radionuclides (Brodsky 1996), radionuclide therapy of tumors (Siegel et al. 1990), and bone pain palliation treatments (Samaratunga et al. 1995). In these different situations, the absorbed dose to the bone marrow results from both the physical aspects of the radiation transport and energy deposition, as well as the biological aspects that determine source location and the cumulative number of nuclear decays. In regard to the physical aspects of skeletal dosimetry, the microstructural complexity of trabecular bone (Clayman 1995; Marieb 1998) imposes challenges to the construction of dosimetry models of this skeletal region, particularly as needed to assess absorbed fractions of energy for electron sources. 1 This chapter was published by Medical Physics in May 2002: Rajon DA, Patton PW, Shah AP, Watchman CJ, and Bolch WE. 2002. Surface area overestimation within threedimensional digital images and its consequences for skeletal dosimetry. Med Phys 29: 682693. BoneMarrow Dosimetry Bonemarrow dosimetry has been investigated extensively during the past forty years. During the 1960s to early 1970s, Spiers and colleagues conducted the first comprehensive studies in skeletal dosimetry at the University of Leeds (Beddoe 1976b; Beddoe et al. 1976; Spiers 1966b; Spiers et al. 1978a; Spiers et al. 1978b; Whitwell 1973; Whitwell and Spiers 1976). After establishment of a referenceman skeletal model, other investigators have expanded these techniques to permit dose estimation specific to medical internal dosimetry (Bouchet and Bolch 1999; Bouchet et al. 2000; Eckerman 1985; Eckerman and Stabin 2000; Snyder et al. 1974; Snyder et al. 1975; Stabin 1996). Nevertheless, the complexity of the bone microstructure and its variations with patient age (Mosekilde 1986; Ouyang et al. 1997; Snyder et al. 1993) have made it difficult to define models for bonemarrow dosimetry that provide improved patient specificity over the single reference model for the adult male. During the 1990s, several groups have investigated the use of threedimensional imaging techniques such as microcomputed tomography (microCT) and nuclear magnetic resonance (NMR) imaging to acquire 3D digital images of trabecular bone samples. These techniques have been used to measure regional bone mineral density and structural parameters (Chung et al. 1996; Chung et al. 1995b; Hwang et al. 1997; Link et al. 1998b; Wessels et al. 1997). Recently, NMR microscopy has been applied to the study of skeletal dosimetry (Jokisch et al. 2001a; Jokisch et al. 1998; Jokisch et al. 2001b). This technique uses highfield NMR spectrometers to acquire highresolution (6090 [tm) 3D images of trabecular bone. The images are then directly coupled to MonteCarlo radiation transport codes to calculate the deposition of energy by monoenergetic electrons within bone marrow. The code used is EGS4PRESTA as it allows voxelized geometry containing millions of voxels. Figure 41 shows a reconstruction of a 3D NMR image of a trabecular bone sample. On this image, the average thickness of the bone trabeculae is 300 tm and the average size of the marrow cavities is 1000 tm. Voxelization Effects in BoneMarrow Dosimetry A 3D image is a digital representation of a real object. In the case of trabecular bone, two media are present within the image: the bone trabeculae and the marrow cavities. In the digital image, each voxel is assigned to bone or to marrow by image thresholding and segmentation (Jokisch et al. 1998). Consequently, the interface between the two media, which is most likely a curved surface in the real bone sample, appears as a jagged surface in the 3D digital image because of the rectangular shape of the voxels. As a consequence, a 3D digital image is not a totally faithful representation of the real sample. This problem has already been studied for chordlength distributions within NMR images of trabecular bone (Jokisch et al. 2001b) and has been defined by what the author calls pixel (or voxel) effects A and B. Both pixel effects A and B are a consequence of the rectangular shape of the voxels (or pixels in 2D geometry). Pixel effect A creates some sharp spikes over the entire chordlength distribution, whereas pixel effect B overestimates the amount of short chord lengths within the geometry. Pixel effect B thus contributes to an underestimation of the mean of the chordlength distribution. For calculations of energy deposition, the 3D digital image is then coupled to a MonteCarlo radiation transport code. Consequently, voxel effects are also expected to result in dosimetry errors when radiation particles are transported close to the bonemarrow interface. To evaluate the magnitude of these voxelization effects, a theoretical study was conducted previously using a mathematical model of trabecular bone as explained in Chapter 3. The model was constructed of nonuniformly sized spheres representing marrow cavities, with the intervening spaces representing the bone trabeculae. The experiment involved * Coupling the mathematical bone model with the radiation transport code EGS4 * Simulating 3D imaging of the bone model through its voxelization at various degrees of image resolution * Coupling these voxelized images within EGS4 * Comparing the dosimetry results with and without image voxelization. The study permitted the identification of three voxel effects. First, a geometry effect occurs when a particle travels parallel to the bonemarrow interface and deposits its energy alternatively within bone and marrow as a direct result of the jagged representation of the interface. This effect overestimates or underestimates the absorbed fraction to marrow tissues depending upon which side of the true interface the particle is traveling (marrow or bone). These errors were shown to cancel when the particle transport is averaged across the entire image. Second, a volume effect is seen to overestimate the volume fraction of marrow at large voxel sizes. Below 300 tm, however, the error in the marrow volume fraction becomes insignificant and without consequence for the dose calculation. The third effect is a surfacearea effect. Because of the voxelization of the image, the surface area of the bonemarrow interface is not respected. Measurements within the mathematical bone model show that the total surface area throughout the entire image increases continuously from near zero at poor resolutions to a convergence value at high resolution. Unfortunately, the convergence value is 50% higher than the true surface area within the mathematical model of trabecular bone. This overestimation has important consequences when the electron range is small compared to the voxel size. In this situation, the crossabsorbed fraction (e.g., a marrow source irradiating bone, or a bone source irradiating marrow) can also be overestimated by up to 50%. According to the study (see Chapter 3), for some lowenergy beta emitters used for skeletal dosimetry, this effect can lead to a 25% overestimation of the crossregion absorbed dose. The purpose of the present chapter is to give a scientific explanation for the bonemarrow interface overestimation, and its consequence on absorbedfraction calculations. The mathematical bone model developed for the previous study and detailed in Chapter 3 was made of thousands of spheres uniformly distributed throughout a cube representing a sectioned piece of trabecular bone. To understand how it is possible that the surface area of the bonemarrow interface, when measured through a voxelized image, converges to a value different from its true value, a separate study is presented here with models representing isolated marrowcavity spheres. In the following text, the previous model will be referred as the mathematical trabecularbone model, whereas the models developed in the present study will be referred as singlesphere models. Material and Methods Construction of the 3D Segmented Images of the SingleSphere Models A set of eight models was created in which a single sphere was located at the center of a cube representing a small trabecular bone sample encompassing a single marrow cavity. The radius of the sphere and the size of the cube were selected according to the following criteria: * The entire set should cover the range of typical marrow cavity sizes found in trabecular bone * The series of cube sizes and sphere radii should follow approximately a geometric progression * In each model, the sphere should be surrounded by a typical trabecular thickness * The sphere radius should be chosen so that it does not represent an integer number of voxels. The last condition ensures that the bonemarrow interface is rarely adjacent to a voxel side. Such an undesirable situation could introduce geometrical effects that could alter the results since it is never realized within a real bone sample where the marrow cavities are randomly located. Table 41 contains the characteristics of the series of singlesphere models. The ten digits of the sphere radius guarantees the fourth condition defined above. Once the singlesphere models are defined, a grid structure is placed over the cubes, representing the voxel dimensions of a 3D image acquisition. Different voxel sizes are used for each model to cover the largest possible range of image resolution. The minimum voxel size is set to 2.5 itm, which corresponds to the range of a 10 keV electron in bone marrow and of a 15 keV electron in cortical bone. The maximum voxel size is set so that each image contains at least one voxel of each media. Table 42 lists the voxel sizes used for the study. A geometric progression is also used for the choice of the number of voxels per dimension. The segmentation of the different images into bone and marrow voxels is based upon a technique similar to that developed for the mathematical bone model and explained in Chapter 3. The principle is to calculate the volume fraction of each voxel inside the sphere. If this volume fraction exceeds 0.5, the voxel is assigned to marrow; otherwise, it is assigned to bone. When the volume fraction is not easy to calculate analytically, a MonteCarlo sampling technique is used. Once the segmentation is complete, a compression technique similar to that developed for the mathematical bone model is used. This technique permits images as large as two billion voxels to be handled by the EGS4 transport code. Volume Fraction Occupied by the Spheres within the Segmented Images The study detailed in Chapter 3 showed that the volume fraction of marrow is not represented well for voxel sizes larger than 300 tm. To verify this result, and to include its consequences in the data interpretation of the present study, a computer program was created to calculate the marrow volume fraction in the voxelized singlesphere models. Theoretical Surface Area of a Voxelized Sphere As the resolution of a 3D image is improved, it would appear that the voxelized surface of the image would approach the exact surface of the real object. Nevertheless, the voxelized surface still follows rectangular shapes and the area measured within the 3D image does not converge to the real surface area, even though the voxel size is reduced to an infinitely small value. The problem is demonstrated in 2D as follows. Consider a circle of radius R as shown in Figure 42. If a 2D image of the circle is made with a digital imaging system, the result will be the grid structure shown in that same figure. For this example, let us assume that the imaging system assigns each pixel as dark or white according to the fraction of its perimeter that is outside or inside the circle. The white pixels will then represent the interior of the circle (marrow) and the dark pixels will represent the exterior (bone). Therefore, the perimeter of the circle, if measured within the image, is equal to the length of the cumulative interface between the white pixels and the dark pixels. Each pixel side that belongs to this perimeter can be moved as shown by the arrows to form the large dashed square. Therefore, the perimeter of the voxelized circle is equal to the perimeter of the dashed square. The size of this square depends on the pixel size and can be slightly smaller or slightly larger than the diameter of the circle. When the pixel size is reduced to an infinitely small value, the size of the square converges to the diameter of the circle. The total perimeter of the image of the circle is therefore equal to 8 times the radius of the circle. This, if compared with the exact perimeter of the circle, corresponds to an overestimate equal to 8R 2nrR 8R = 0.2732 27.32 %. (41) 27rR This overestimate is the limit of convergence when the pixel size is reduced to an infinitely small value. For a larger pixel size, the overestimate will be slightly smaller or slightly larger than the convergence value. This variation about the convergence value results from the fact that, depending on the ratio between the circle radius and the pixel size, the circle will not exactly fit within the larger dashed square. The 3D problem is more complex and can be solved analytically. A sphere of radius R can be considered as a set of slices along the zaxis. Each slice is a flat disk with radius r varying with the distance z. This is represented in Figure 43 where a 2D projection of the sliced sphere is shown. The vertical rectangles represent the disks seen on edge. The thickness dz of the disks is assumed to be the voxel size of the 3D image. The surface area of the edge of each voxelized disk can be derived from the previous result for the circle by multiplying the perimeter of the disk (within the voxelized image) by its thickness (the voxel size). That is ds = 8rdz. (42) When the voxel size is reduced to an infinitely small value, the total surface area can be calculated by the integration R S= 2 8rdz. (43) However, this expression only accounts for the surface area of the edges of the voxelized disks. These edges are perpendicular to the xaxis or the yaxis but never perpendicular to the zaxis; therefore, and because of the symmetry of the problem, they account for only two thirds of the total surface area of the voxelized sphere. The total surface is then obtained by S =3 8rdz. (44) Using a variable change: r = Rsin, and z= RcosO dz =Rsin d. (45) The integration thus becomes: S = 24R22 (sin 0)2d (46) The calculation gives the result: S = 6n7R2. (47) Equation (47) shows that reducing the voxel size will make the surface area of a voxelized sphere converge to a value that is 50% higher than the real surface area of the sphere (41cR2). For larger voxel sizes the surface area will be slightly larger or slightly smaller than the convergence value according to the ratio of the voxel size and the sphere radius. Measurement of the Surface Area of a Voxelized Sphere To check the theoretical result given by Equation (47), the surface area of the spheres was measured within each simulated image. A computer program was created to perform this task. Its principle is to sweep every voxel of the image and to check if the six adjacent voxels (or less if the voxel is on the edge of the cube) are composed of a different medium. Each time one adjacent voxel is found of a different medium, the surface area of the interface between the current and the adjacent voxel is added to the total surface area. By doing this, each surface is counted twice, and therefore the final result is reduced by a factor of two to obtain the true surface area of the voxelized sphere. Consequences of an Error in Surface Area on the Absorbed Fraction The overestimation of the surface area will affect the result of the MonteCarlo transport code when the particles are emitted close to the boundary of the two media, as each electron will have a greater than normal chance of crossing that boundary. This surfacearea effect will vary for different voxel sizes as demonstrated in Figure 44. Figure 44A demonstrates the situation of a voxel size small compared to the electron range. The diagram is in 2D but it can be easily transposed to the 3D situation. The thick solid straight line represents the real boundary between the medium of the radiation source (lower right side of the line), and the medium of the irradiated target (upper left side). The thin solid jagged line represents the same boundary, but as seen within the corresponding digital image. This boundary follows the shape of the rectangular image pixels. The arrows represent electrons (assumed monoenergetic) crossing the boundary. In both situations (the real boundary and the jagged one), electrons crossing the boundary must leave the source at a distance shorter than their range in the source medium. Therefore, in the real situation, the effective source area from which the electrons can reach the other medium extends from the boundary to the thick dashed straight line, the distance between the two lines being the electron range. In the case of the digital image, the effective source area is more complex, as the boundary is no longer straight. Each dashed circle represents the limit of the effective source area from which the electrons can reach each corer of the image boundary located on the source side of the real interface. The radius R is equal to the electron range. These circles extend slightly over the thick dashed straight line since their radius is equal to the distance between the two thick lines. The overall limit of the effective source area is the envelope of these circles. One can see in Figure 44A that the source area enclosed between the thin jagged line and the envelope of the circles is nearly identical to the source area enclosed between the two thick lines. Therefore, if the voxel size is small compared to the electron range, the error in interface length introduces little error on the crossregion absorbedfraction calculation since approximately the same number of electrons cross the boundary in the real case as within the digital image. Figure 44B represents exactly the same geometrical features as in Figure 44A, but for a voxel size that is larger than the electron range. Only the circles centered at the summits of the thin solid line have been represented, but one can easily understand that the envelope of the effective source area from which the electrons can reach the other medium extends from the solid thin jagged line to the dashed thin line (envelope of the circles). In this case, the two source areas (between the thick lines and between the thin lines) are different. Since the electron range is smaller than the voxel size, the shape of the interface affects the source area from which the electrons can reach the other side. This source area converges to the length of the thin solid line multiplied by the electron range when the voxel size becomes very large compared to the electron range. The 3D situation is similar and the effective source volume from which the electrons can reach the other side of the boundary is proportional to its surface area. Therefore, a 50 percent overestimation of the surface area overestimates the effective source volume for cross irradiation by 50 percent as well. Correspondingly, the crossregion absorbed fraction is also overestimated by as much as 50%. The real boundary in the example problem of Figure 44 is a straight line; this represents a geometry large in comparison to the electron range. In a smaller relative geometry, the shapes would be more complex but the principle would remain the same. For the singlesphere models, let us assume that monoenergetic electrons start from the outside of the sphere (the bone trabeculae) and that the absorbed fraction within the sphere (the marrow cavity) is to be calculated. Taking into account the surfacearea effect alone, and for a very large voxel size, the resulting absorbed fraction is expected to overestimate its exact value by 50% since the electron range is much smaller than the voxel size. By reducing the voxel size, the voxel dimensions will approach the electron range and thus the absorbed fraction would become closer to its exact value. If the voxel size is chosen very small, the results should converge. This scenario is shown in Figure 45. The voxel size at which the convergence appears should be around, or at least function of, the electron range in the source region. ElectronTransport Simulations The next step in the study is to calculate the absorbed fractions of energy and to analyze their evolution with the voxel size. For this, particle transport calculations were performed using the EGS4PRESTA code. Two transport simulations were performed. One for a uniform monoenergetic electron source located within the sphere (e.g., marrow source) and a second for a uniform monoenergetic electron source located outside the sphere (e.g., bone source) but within the limits of the bone cube. For each simulation, six electron energies were used. As the surfacearea effect only influences the calculation for lowenergy electrons (see Chapter 3), the maximum energy was set to 320 keV. The minimum energy was set to 10 keV. For each electron energy and for both source regions, one set of EGS4 runs was performed for each singlesphere model. Each set represents one simulation within the exact singlesphere model and one simulation within each 3D image (each voxel size shown in Table 42). Therefore, two different EGS4 user codes were developed: a "reference code" for the exact sphere model, using the equation of the sphere delineating the marrow cavity boundary, and an "image code" for the different segmented images, using equations of planes separating the voxels. The "reference code" was executed for each model and for each electron energy. The "image code" was executed for each model, for each electron energy, and for each image resolution. The parameters used by the EGS4 transport were the same for both codes. The characteristics of the bone and marrow media were identical to those used for the study of the mathematical trabecular bone model (see Chapter 3). For the PRESTA extension, the ESTEPE parameter was set to 0.05. For each execution, absorbed fractions of energy were estimated for targets inside the sphere, outside the sphere but inside the cube, and outside the cube (energy lost for this study). Only the cross absorbedfraction results are shown and analyzed in this work. The absorbed fractions calculated with the "image code" were compared to the absorbed fractions calculated with the "reference code". The absorbedfraction relative error Aro produced by the overestimation of the boundary surface area was then evaluated using the expression: Ar = " e x100%, (48) where ,ma, is the absorbed fraction calculated using the "image code" and ,ref is the absorbed fraction calculated using the "reference code". Statistical Analysis Combining all parameters described above, a total of 1,764 EGS4 simulations were performed. To minimize the calculation time, a statistical analysis was performed prior to the EGS4 simulations. Instead of one execution of N histories, 100 runs were performed. For each run, only one hundredth of Nhistories were used. This allows calculating a mean and a standard deviation within the sample of 100 runs. The standard deviation of the mean can also be estimated, as well as a 95 percent confidence interval assuming a normal distribution of the mean. As it is not known whether or not the distribution of the absorbed fraction follows a normal distribution, the later assumption is verified through application of the central limit theorem. This explains the large sample (100 runs in this case) used for this analysis. The number of histories was chosen so that the 95 percent confidence interval for the mean was reduced to about 2% for each absorbed fraction calculated. As the accuracy of the result varies mostly with the electron energy and the source region, a different number of histories was set for each situation to reach the 2% goal and to minimize the calculation time. Table 43 shows the electron energies used and the number of histories per run for both types of sources. Note that this 95 percent confidence interval is representative of the statistical fluctuation of the results because of the randomness of the MonteCarlo method. It does not represent the accuracy of the technique used. It signifies that another experiment using the same MonteCarlo transport code and the same experimental conditions would have a 95 percent chance to find a result within the confidence interval. The purpose of this work is not to assess the accuracy of the dosimetry technique, but to quantify the error due to the surfacearea overestimate. To calculate the 95 percent confidence interval for the relative error, the errorpropagation technique was used. According to the theory (Knoll 2000; Turner 1995), the standard deviation of function f(x,y) is given by the equation < 7j2 f (7J2 (49) In our situation, using Equation (48) to calculate the partial derivatives, the expression becomes 2 2 2(70_2 +26,.(To..s (Ar = "ef 2 (410) In Equation (410), (of and oe, are the standard deviations of the mean calculated with N = 100 runs of respectively the "reference code" and the "image code" using the relation = N )2 c,= (411) N(N 1) The 95 percent confidence interval is therefore obtained by: Err950 = +1.96 'ref 2 (412) Oref Results and Discussion The results from the eight singlesphere models are quite consistent. Therefore, only 4 models are shown below. The models chosen are 1400, 0728, 0350, and 0069. Volume Fraction Occupied by the Spheres within the Segmented Images The volume fraction occupied by the spheres within the segmented images is compared with the exact value in Figure 46 for each model. In this figure, the horizontal solid lines represent the exact volume fraction of the spheres, with values given in column 6 of Table 41. For each model, the volume fraction converges toward its exact value as the voxel size is reduced. The convergence occurs around a voxel size that decreases as the radius of the marrow sphere decreases. The study of the mathematical trabecular bone model (Chapter 3) showed that the error on the volume fraction at large voxel sizes, referred to as the volumefraction effect, was not a concern for dosimetry calculations since it occurs only above 300 tm in a typical bone sample. Therefore, in the remainder of this study, we focus only on voxel sizes small enough so that the surfacearea effect can be studied in isolation. Surface Area of the 3D Segmented Images The surface areas calculated within the segmented images are compared with their exact values in Figure 47. In this figure, the horizontal solid lines represent the exact surface areas of the marrow spheres whose values are given in column 4 of Table 41. The dashed lines are at 1.5 times the exact values. These results are in excellent agreement with predicted values calculated in the materials and methods section of this chapter. Furthermore, the surface area is expected to oscillate around a convergence value. It can be larger or smaller than the convergence value according to the ratio between the voxel size and the sphere radius. These oscillations are particularly evident for the small spheres; for the larger ones, they are barely visible, but still present. At large voxel sizes, the oscillations disappear for all spheres since the surfacearea effect is overwhelmed by the volume fraction effect. For very large voxel sizes, the surface area drops to zero since the volume fraction also decreases toward zero. Absorbed Fractions within the 3D Segmented Images The next step was to calculate the absorbed fraction of energy deposited by electrons within the different models. In this section, only the crossabsorbed fractions are considered since the selfabsorbed fractions are reasonably accurate even at small voxel sizes (according to the results of Chapter 3). The results are presented in Figure 48 for electron sources in bone irradiating marrow, and in Figure 49 for electron sources in the marrow irradiating bone. In each figure, the four selected models are shown: Figure 48A and 49A are for model 1400, Figure 48B and 49B are for model 0728, 48C and 49C are for model 0350, 48D and 49D are for model 0069. For each graph, the curves represent the relative error in the crossabsorbed fraction given by Equation (48) as a function of the voxel size and electron energy. The error bars show the 95 percent confidence intervals calculated by Equation (412). These curves are to be compared with the expected results shown schematically in Figure 45. First, for voxel sizes larger than the resolution at which the volume fraction converges, the results cannot be interpreted by the surfacearea error alone since it is overwhelmed and largely influenced by the volume fraction. This is why, at these voxel sizes, the absorbed fraction can be overestimated or underestimated, depending upon the volume fraction of the segmented sphere. At this resolution, one can see that the shapes of the absorbedfraction curves follow more or less the shapes of the volume fraction curves. Second, for a voxel size small enough so as to minimize the volumefraction effect, the surfacearea effect is visible in each model as predicted in the material and methods section of this chapter. The 50% overestimate of the surface area leads to a 50% overestimate of the crossabsorbed fraction down to a voxel size that depends on the electron energy. Below this voxel size, the absorbed fraction converges to its exact value. This convergence is only visible at high energies; at very lowenergy electron sources (e.g., 10 and 20 keV), the shape of the absorbedfraction curve indicates that a similar effect would occur at a voxel size below the range used in the study. These results are in strong agreement with Figure 45. Third, the resolution at which the absorbed fraction starts to converge toward the exact value can be compared with the electron range as shown in Table 44 (data from ICRU (1984) Publication 37). At electron energies from 20 keV to 80 keV, the electron range is equal to the voxel size that corresponds to a relative error of 25% (i.e., the middle of the convergence slope). For the 10 keV curves, the voxel size is not extended to a value small enough to show the same consequence, but an extrapolation of these curves can be easily made. At higher electron energies (160 and 320 keV curves), the convergence slope falls above the volumefraction convergence value. Once again, the convergence slope can be deduced from the lower part of the curves, and the extrapolated shapes would show a convergence slope at a voxel size close to the electron range. Finally, one can see that, at voxel sizes that are not affected by the volume fraction effect, the sphere radius does not change the shape of the curves. The surfacearea effect depends only on the ratio between the electron range and the voxel size, as predicted in the material and methods section of this chapter. Conclusion The surfacearea effect resulting from the segmentation of an object within its 3D digital image has been investigated using singlesphere models of trabecular bonemarrow cavities. It has been shown, both analytically and experimentally, that a consequence of the image segmentation is an overestimation of the interface surface area. This overestimate, identified as a surfacearea effect, is a result of the rectangular shape of the voxels that constitute the image and cannot be reduced by improved resolution. In the case of single spheres, the overestimate is as high as 50%. This finding is consistent with the studies of presented in Chapter 3 where a similar overestimate of bonemarrow interface area was found in a mathematical model of trabecular bone. The dosimetry consequence of the surfacearea effect is an overestimate of the absorbed fraction by electrons crossirradiating a region adjacent to the source region (bone source irradiating marrow, for example). At a voxel size larger than the electron range, this overestimate is equal to the overestimate of the surface area (50% for spherical objects). As the voxel size is reduced, the error decreases and becomes insignificant when the voxel size falls below the electron range. For highenergy electrons, this effect is inconsequential since the electron range is significantly larger than the voxel size used to image trabecular bone samples. For electron energies below 100 keV, however, the effect is significant to a voxel size on the order of a few micrometers. With typical resolutions used in NMR microscopy (from 60 to 100 [tm), and for very lowenergy beta emitters (e.g., 117mSm and 33P), the dose calculation using these absorbed fractions leads to a 25% overestimate within the mathematical model of trabecular bone, as shown in Chapter 3. This mathematical model was designed with dosimetry features as close as possible to those of a real trabecular bone sample, and thus the same consequence is expected within NMR images of real samples. As shown in Figure 45, the problem can be incrementally solved by improving the image resolution. Figures 48 and 49 shows that for lowenergy electrons (let us say 40 keV) the resolution needs to be pushed downward to 5 tm in order to achieve a close estimate of the crossdose. This resolution is 10 times better than what is currently performed in NMR microscopy of trabecular bone (Patton et al. 2002a) and 6 times better than available with microCT systems (Ruegsegger et al. 1996). Increased resolution of the NMR microscopy images are possible via improvements in the signaltonoise ratio of the NMR system acquisition via higher field strengths (e.g., 20 T) and/or longer imaging times. Compromises occur, however, in that one must contend with susceptibility artifacts at the bonemarrow interface at higher fields, and longer imaging sessions can become cost prohibitive. In addition, as the images become larger with decreasing voxel size, computer storage can potentially be an issue. Nevertheless, other solutions to the problem may be sought. One such solution would be to perform an interpolation of the graylevel values of the original NMR image to calculate a polygonal isosurface that would represent the bonemarrow interface. In this manner, image voxels at the interface would encompass one or more polygons separating the marrow tissues from the osseous bone tissue. The exact positions and relative angles between polygons would be a function of both the relative gray level of the interface voxels and the resolution of the image (i.e., voxel size). The resulting collection of polygons would thus permit a more smooth representation of the true bonemarrow interface surface and would thus better represent the true surface area. Such a polygonal isosurface has already been used with trabecular bone microstructure (Muller et al. 1994) and its construction is based on the MarchingCube algorithm developed during the 1980's by Lorensen and Cline (1987). In this algorithm, the resulting isosurface is only made of triangles that insure that each piece of the surface can be modeled with equations of planes. Through the application of this technique, previous problems identified in either the assessment of bone and marrow chordlength distributions (Jokisch et al. 2001b), as well as absorbed fractions for crossirradiation by lowenergy electrons (see Chapter 3) would be significantly reduced from those found using voxelized geometries. The storage capacity of the computer would still remain a major concern, since the number of triangles would be on the same order of the number of voxels, but the image resolution no longer needs to be tremendously improved, and may even be reduced if the voxel effects are shown to be significantly reduced. 