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Proton Range Uncertainty as a Function of CT Image Noise

Permanent Link: http://ufdc.ufl.edu/UFE0025051/00001

Material Information

Title: Proton Range Uncertainty as a Function of CT Image Noise
Physical Description: 1 online resource (52 p.)
Language: english
Creator: Paige, Sandra
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: computed, proton, radiation, range, therapy, tomography, uncertainty
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Computed Tomography (CT) is now used routinely to acquire a patient image data set for use in radiation therapy treatment planning. This data set provides not only invaluable patient anatomical information, but also information relating characteristics of the image to proton physical properties. Specifically, the Hounsfield Units (HU), or CT numbers, composing the image are used for calibration of the relative proton stopping power. This calibration is then utilized in determining the necessary treatment plan. This process, however, incorporates certain inherent uncertainties. One such uncertainty is that the CT image has some intrinsic level of noise, which is then carried over into the calibration. In this paper CT image noise is taken to be a random variable. The proton stopping power is calculated based on the noise influenced CT numbers using random number generators. This leads to uncertainty in the subsequent range calculations. A Fortran program is used to calculate the proton range in water for one hundred thousand noise patterns. The resulting probability density function is graphed and analyzed. The full width at half maximum (FWHM) is evaluated for a variation of initial proton energies, noise amplitudes, and noise periods. Also considered is the effect of a local region of increased noise along the proton pathlength. Results indicate linear relationships between all variables and FWHM calculations. For energy variation, the FWHM relative to proton range is also evaluated. In this case, a power relationship is observed between initial proton energy and the resulting calculation. Conclusions from these numerical experiments signify the need to account for greater range uncertainties when using noisy CT images and larger grid sizes in treatment planning, especially with low energy proton beams.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sandra Paige.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Hintenlang, David E.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0025051:00001

Permanent Link: http://ufdc.ufl.edu/UFE0025051/00001

Material Information

Title: Proton Range Uncertainty as a Function of CT Image Noise
Physical Description: 1 online resource (52 p.)
Language: english
Creator: Paige, Sandra
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: computed, proton, radiation, range, therapy, tomography, uncertainty
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Computed Tomography (CT) is now used routinely to acquire a patient image data set for use in radiation therapy treatment planning. This data set provides not only invaluable patient anatomical information, but also information relating characteristics of the image to proton physical properties. Specifically, the Hounsfield Units (HU), or CT numbers, composing the image are used for calibration of the relative proton stopping power. This calibration is then utilized in determining the necessary treatment plan. This process, however, incorporates certain inherent uncertainties. One such uncertainty is that the CT image has some intrinsic level of noise, which is then carried over into the calibration. In this paper CT image noise is taken to be a random variable. The proton stopping power is calculated based on the noise influenced CT numbers using random number generators. This leads to uncertainty in the subsequent range calculations. A Fortran program is used to calculate the proton range in water for one hundred thousand noise patterns. The resulting probability density function is graphed and analyzed. The full width at half maximum (FWHM) is evaluated for a variation of initial proton energies, noise amplitudes, and noise periods. Also considered is the effect of a local region of increased noise along the proton pathlength. Results indicate linear relationships between all variables and FWHM calculations. For energy variation, the FWHM relative to proton range is also evaluated. In this case, a power relationship is observed between initial proton energy and the resulting calculation. Conclusions from these numerical experiments signify the need to account for greater range uncertainties when using noisy CT images and larger grid sizes in treatment planning, especially with low energy proton beams.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sandra Paige.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Hintenlang, David E.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0025051:00001


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1 PROTON RANGE UNCERTAINTY AS A FUNCTION OF CT IMAGE NO ISE By SANDRA L. PAIGE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009

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2 2009 Sandra L. Paige

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3 To Frankie

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4 ACKNOWLEDGMENTS I would like to thank a number of people who have been truly instrumental in my academic professional, and personal accomplishments. First, my greatest thanks go to my advisor, Dr. Alexei Chvetsov, for his patience and mentorship. I have learned so much from his unique perspective. I would li ke to thank Dr. Thomas Edwards, with whom I be gan my journey in radiologic physics, and Dr. Merton Hollister who helped me hang on at the most difficult part of my journey. I am forever grateful that these men chose to be educators. T heir support and encouragement has instilled a much appreciated l ove of l ifelong learning. I would also like to thank Dr. Marc Shapiro and Dr. Nik Wasudev for believing in me. The generosity of their time, knowledge, and professional guidance is sincerely treasured Their desire for excellence in their own careers h as inspired me to strive for it in my own. A special thanks to Dr. Kimberly Lee for her endless hours of counsel and for giving me a home away from home I am indebted to her humor, motivation, kindness and leadership. Most importantly, I would like to thank my family and friends for their love In one way or another they have all made great personal sacrifice in order to support me, without question or hesitation. I am so blessed to be surrounded by such amazing people I definitely would not be w here I am today without their contributions, and my successes are just as much their successes. I am especially thankful for my husband Steve. God has not given us an easy path, but he gave us each other. I love him with all my heart and could not imag ine sharing my life with anyone else. Heres looking forward to many good things as our journey continues.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................................... 4 LIST OF TABLES ................................................................................................................................ 7 LIST OF FIGURES .............................................................................................................................. 8 ABSTRACT .......................................................................................................................................... 9 CHAPTER 1 PROTON THERAPY ................................................................................................................. 11 Introduction ................................................................................................................................. 11 Importance of Range Uncertainty Analysis ............................................................................... 11 Previous work .............................................................................................................................. 12 2 THEORY ..................................................................................................................................... 16 Proton Interactions ...................................................................................................................... 1 6 Stopping Pow er ........................................................................................................................... 17 Bragg Peak ................................................................................................................................... 18 CT Calibration ............................................................................................................................. 18 3 EXPERIMENTS ......................................................................................................................... 23 Background .................................................................................................................................. 23 Program Details ........................................................................................................................... 25 Definition of Variables ........................................................................................................ 26 Stopping Power and Range Calculation ............................................................................. 27 4 METHODS .................................................................................................................................. 31 5 RESULTS .................................................................................................................................... 34 Initial Proton Energy ................................................................................................................... 34 Noise Period ................................................................................................................................ 35 Noise Amplitude ......................................................................................................................... 35 Local Noise .................................................................................................................................. 35 6 CONCLUSIONS ......................................................................................................................... 42 APPENDIX: FORTRAN PROGRAMS .......................................................................................... 44 Range Calculation ....................................................................................................................... 44 Clearing Routine ......................................................................................................................... 46

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6 Stopping Power Calculation ....................................................................................................... 47 Gaussian Random Number Generator ....................................................................................... 47 Uniform Random Number Generator ........................................................................................ 48 LIST OF REFERENCES ................................................................................................................... 50 BIOGRAPHICAL SKETCH ............................................................................................................. 52

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7 LIST OF TABLES Table page 5 1 FWHM calculations for energy ............................................................................................. 40 5 2 FWHM calculations for noise period .................................................................................... 40 5 3 FWHM calculations for noise amplitude .............................................................................. 41 5 4 FWHM calculations for local noise ...................................................................................... 41

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8 LIST OF FIGURES Figure page 1 1 Prostate treatment plan. .......................................................................................................... 15 2 1 Total stopping power for protons in water. .......................................................................... 22 3 1 Profile of CT Numbers in bone tissue for a 3mm image slice. ........................................... 28 3 2 Profile of CT Numbers in soft tissue for a 3mm image slice. ............................................. 28 3 3 Profile of CT Numbers in soft tissue for a 1mm image slice. ............................................. 29 3 4 Random number generation of 5% white Gaussian noise. .................................................. 29 3 5 Random number generation of 5% white uniform noise. .................................................... 30 3 6 Range calculation. .................................................................................................................. 30 4 1 PDF fo r reference conditions. ................................................................................................ 32 4 2 PDF for noise period variation. ............................................................................................. 32 4 3 PDF for noise amplitude variation. ....................................................................................... 33 5 1 Relationship of range uncertainty with respect to initial energy. ........................................ 37 5 2 Relationship of relative uncertainty with respect to initial energy. .................................... 37 5 3 Relationship of range uncertainty with respect to noise period. ......................................... 38 5 4 Relationship of range uncertainty with respect to noise amplitude. ................................... 38 5 5 Relationship of range uncertainty with respect to local noise. ............................................ 39

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9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science PROTON RANGE UNCERTAINTY AS A FUNCTION OF CT IMAGE NOISE By Sandra L. Paige August 2009 Chair: David Hintenlang Major: Nuclear Engineering Sciences Computed Tomography (CT) is now used routinely to acquire a patient image data set for use in radiation therapy treatment planning. T his data set provides not only invaluable patient anatomical information, but also information relating characteristics of the image to proton physical properties. Specifically, the Hounsfield Units (HU), or CT numbers, composing the image are used for ca libration of the relative proton stopping power. This calibration is then utilized in determining the necessary treatment plan. This process, however, incorporates certain inherent uncertainties. One such uncertainty is that the CT image has some intrin sic level of noise, which is then carried over into the calibration. In this paper CT image noise is taken to be a random variable. The proton stopping power is calculated based on the noise influenced CT numbers using random number generators. This lead s to uncertainty in the subsequent range calculations. A Fortran program is used to calculate the proton range in water for one hundred thousand noise patterns. The resulting probability density function is graphed and analyzed. The full width at half m aximum (FWHM) is evaluated for a variation of initial proton energies, noise amplitudes, and noise period s. Also considered is the effect of a local region of increased noise along the proton pathlength. Results indicate linear relationships between all variables and

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10 FWHM calculations. For energy variation, the FWHM relative to proton range is also evaluated. In this case, a power relationship is observed between initial proton energy and the resulting calculation. Conclusions from these numerical expe riments signify the need to account for greater range uncertainties when using noisy CT images and larger grid sizes in treatment planning, especially with low energy proton beams

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11 CHAPTER 1 PROTON THERAPY Introduction The ultimate goal of radiation therapy is to effectively deliver a treatment that affords both maximum injury on a precise target volume, and maximum sparing of normal tissue cells. Proton therapy is an attractive answer to this objective. As protons tra vel through matter, they release the majority of their energy at the end of their range. This inherent peak of energy loss, the Bragg peak, creates a steep dose gradient and is utilized to deliver a highly localized dose to diseased tissue while sparing s urrounding healthy tissue. For patients, t his translates in general to a reduced incidence of complications, a higher quality of life during treatment itself, and improved overall treatment outcomes. The realization of these benefits occurs only with exa ct localization of the finite range of the protons within the patient, and thus is highly dependent upon the accuracy of treatment planning and administration Although there exists notable potential for variation in localization during treatment administ ration, this paper evaluates range uncertainty arising during treatment planning. The treatment plan aims at accounting for uncertainties while maintaining conformal dose distribution. Range uncertainties in treatment planning occur as a result of two t hings. First is the systematic uncertainty due to the calibration process relating CT numbers to relative proton stopping power. Second is the statistical uncertainty due to random noise in the CT image. The noise manifests as nonphysical fluctuation of the CT numbers, which then affects the subsequent calibration Importance of Range Uncertainty Analysis Analysis of range uncertainties in proton therapy is important for several reasons. One reason is that minimizing range uncertainty will increase the effectiveness of treatment to the

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12 target volume. The foundation of the primary benefit of proton therapy lies in the compact dose distribution due to the Bragg peak at the end of the particles range, and the ability to localize the Bragg peak relative to the desired treatment volume. Considering a pristine Bragg peak, an underestimation of proton range ma y result in greater dose deposition in healthy tissue before the proton beam reaches the treatment volume. The tumor in this case, distal to the Bragg peak, would receive little if any dose. An overestimation of proton range may result in greater dose de position in healthy tissue beyond the treatment volume. The tumor in this case, proximal to the Bragg peak, would receive a smaller fraction of the prescribed dose. When considering a modulated proton beam with a spread out Bragg peak (SOBP), many pristi ne peaks are superimposed to achieve adequate target coverage. The goal of uniform dose over the target volume within the SOBP, however, comes with the cost of increased entrance dose. For that reason, this advantage can be somewhat negated if the SOBP m ust be increased due to uncertainty in proton range. Alternatively, range uncertainties may cause critical variations in dose distribution across the SOBP plateau. Any of these scenarios yield undesirable and unsatisfactory tumor control results. Minimizing range uncertainty can also help maximize normal tissue sparing. Figure 1 1 displays a prostate treatment plan with added safety margins to account for various types of uncertainty. Improving upon the degree of range uncertainty may permit the use of reduced margins around the target volume in treatment planning. For critical structure s adjacent to a target volume, this reduction could significantly affect viability and functionality post treatment. Previous work Range uncertainty analysis is a comm on research area in proton therapy. A great deal of work has been done in various areas. Mustafa and Jackson (1983) establish a calibration curve relating calculated CT numbers of tissue substitutes of known elemental composition to proton

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13 stopping power They note that by calculating linear stopping power of the medium relative to that of water (Equation 2 8) that it becomes primarily dependent on the relative electron density of the material (Equation 2 9) The calibration curve is then formed by com paring the calculated values of relative electron density to calculated values of relative linear attenuation coefficients, CT numbers, of the tissue substitutes. Schneider et al. (1996) continue this work in carrying out a stoichiometric calibration. T his included using measured CT numbers of tissue substitutes of known elemental composition to parameterize the response of the CT unit. Schaffner and Pedroni (1998) verify this stoichiometric calibration experimentally for various animal tissue samples. Noted in both these works is that a key source of uncertainty is the acquisition of the CT numbers themselves. Approaches to this problem include the use of alternative calibration methods, such as proton radiography (Schneider and Pedroni 1995 Schneid er et al. 2005), or proton computed tomography (Zygmanski et al. 2000) The goal of proton based imaging to obtain proton stopping powers directly is to limit errors in treatment planning due to ambiguity of information originating from CT image data (Zyg manski 1998). Another method is the use of positron emission tomography (PET) to verify proton range ( Nishio et al. 2006, Parodi et al. 2007). However, range verification using PET imaging must occur post treatment administration. With the advent of intensity modulated proton therapy (IMPT), optimization of treatment planning to include range uncertainties is even more important. Unkelbach et al. (2007) evaluate a probabilistic approach assuming range to be a random variable Fol lowing their work, Pflugfelder (2008) investigates a worst -case optimization which considers a minimum, nominal, and maximum range. In both of these papers, the range uncertainty is incorporated into the objective function of inverse treatment planning.

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14 Whether it is for these newer techniques, or for the addition of safety margins, knowledge of the probability density function (PDF) of proton range is necessary for optimizing the treatment plan.

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15 Figure 1 1. Prostate treatment plan.

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16 CH APTER 2 THEORY Proton Interactions Charged particles have distinctly different interactions with matter than uncharged particles, such as x ray photons. As p hoton s travel through tissue, they undergo interactions according to an exponential relationship and may pass through the medium with no interaction at all. Therefore it is impossible to predict an individual photons pathlength through an absorber. As p rotons travel through tissue, they experience Coulomb-force interactions with all electrons along their path, causing ionization and excitation of atoms Only a small amount of energy is lost during these types of interactions. max 24 mME Q Mm (2 1) Based on conservation of energy and momentum relationships, maxQ is equal to the maximum energy transfer where m is the rest mass of the electron, M is the rest mass of the proton, and E is the energy of the proton (Turner 1995) For proton energ y in the majority of the therapeutic range, this e q uals approximately 0.22%. This process allows a continuous slowing down approximation (CSDA) to be made in terms of the range and individual protons of a given energy can be characterized by a common pathlength. The CSDA Range is then represented by 010 ECSDAdER dE dx (2 2) the integration of the mean rate of energy loss dE dx w here 0E is the initial proton energy. Where energy loss is primarily through ionization, scattering of the proton beam is primarily through Multiple Coulomb Scattering (MCS) processes. In general both of these types

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17 of events cause little deviation from the proton path. The dose distribution then of the proton beam exhibits a sharp lateral fall -off when compared to that of photons (Zygmanski 1998). Stopping Power The CSDA range is an important measure of the proton beam. Treatment planning is based on the target location within the patient, and penetration depth of the protons must match this target location. Likewise the relationship governing energy loss is also important. The average rate of energy loss per unit pathl ength dE dx is represented by the Bethe -Bloch equation 224 22 2 0 22 24 2 ln 1ekzeN dE mc dxmc I (2 3) w here 0k is Coulombs constant, z is the atomic number of the proton, e is the magnitude of electron charge, 2mc is the electron rest mass, is the velocity of the proton equal to vc eN is the electron density of the medium, and I is the mean ionization energy of the medium. A eZN N A (2 4) T he electron density is based on the material density, the atomic number, Z Avogadros number, A N and the atomic mass, A T he mean ionization energy, I is taken from experimentally determined tabulated values (Schneider et al. 1996). Two important fea tures concerning the rate of energy loss are inferred from this relationship. First is the linear dependence on electron density In fact, this feature is utilized to correlate CT numbers to relative proton stopping power. S econd is the significant n onl inear

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18 dependence on energy, which is the feature that contributes to the Bragg peak and makes proton therapy so appealing. Bragg Peak As protons traverse matter, they very slowly lose energy until nearing the end of their range. At this point, the majority of their residual energy is released quickly. Mathematically, the leading term of Equation 23 sharply increases as goes to zero. Figure 2 1 displays this rise in total stopping power as energy decreases for protons in water. Therefore, as the depth of penetration increases and energy decreases, there is a resulting maximum energy deposition known as the Bragg peak It is this feature which is taken advantage of in proton therapy. The distal dose gradient falls from 90% to 10% with in several millimeters Therefore complications and side effects resulting from dose to surrounding normal tissue can be significantly reduced compared to photon radiotherapy. However, when considering a single Bragg peak, the width of th e peak at a 90% dose level may be only 0.5cm in the direction of the beam Thus t he depth dose distribution of a single pristine Bragg peak is not sufficient to treat a n entire target volume (Bussiere and Adams 2003) By modulating the incident proton beam energy, many Bragg peaks may be superimposed to create a plateau or SOBP, over which a relatively uniform dose distribution exists. By positioning the SOBP in coincidence with the target area, the healthy tissue sur rounding the tumor is spared much of the dose that it would receive in conventional photon radiotherapy. This results in fewer side effects and complications for the patient, and an overall improved treatment outcome. CT Calibration Conventional CT uses t he transmission of xrays to generate image data representative of differential attenuation within a patient. CT images are beneficial to treatment planning for

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19 several reasons. They provide visual anatomical and pathological information, as well as nume rical information regarding tissue heterogeneity. Each pixel comprising the CT image is assigned a CT number, or Hounsfield Unit (HU). The CT numbers indicate relative tissue density and are scaled from 1000 to 1000, with zero being equal to water. Ext reme negative values are representative of air, and extreme positive values are representative of dense materials such as bone. Often the concept of scaled HUs is applied so that the numerical scale is nonnegative (Schneider et al. 1996, Schaffner and P edroni 1998). In this case the relationship is expressed as 1000scHUHU (2 5 ) Specifically, the HU relationship is expressed as 1000xw wHU (2 6 ) where x is equal to the linear attenuation coefficient for the tissue, and w is equal to the linear attenuation coefficient of water. The linear attenuation coefficient is a measure of the fraction of x -ray photons that is either scattered or absorbed per unit thickness of the absorber. It may be broken down into the cross sections for the various photon interactions ,e incoh cohNZA (2 7 ) w here eN is the electron density, and incoh coh are the incoherent scattering photoelectric absorption, pair production, and coherent scattering electronic cross -section s respectively (Schneider et al. 1996). The common element then between the CT image information and proton physical characteristics is some dependence on electron density. Mustafa and Jackson (1983) explore this

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20 relationship by calculating H Us, elec tron density, and proton stopping power values for various materials By consid ering the stopping power of the material relative to that of water, the majority of the terms in Equation 2 3 cancel, leaving the relative electron density. 224 22 2 0 22 2 224 22 2 0 22 24 2 ln 1 () () 4 2 ln 1e tissue tissue water e waterkzeN mc mcI SE SE kzeN mc mc I (2 8 ) () ()e tissue tissue e water waterSEN SEN (2 9 ) E quation 2 8 assumes the material has a mean excitation energy equivalent to that of water equal to 75 eV For muscle tissue this assumption is valid with I equal to 74.7 eV. However the mean excitation energy for adipose tissue, 63.2 eV, and compact bone, 91.9 eV, introduce approximately a 2% or +2% error in stopping power. Since the proton path encompasses a variety of body tissues, the overall error fal ls below 2% (Schulte et al. 2005). Based on Equation 2 9, Mustafa and Jackson (1983) conclude a linear relationship when plotting relative stopping power versus relative electron density. However, there is not a linear relationship between CT numbers an d relative electron density. This is due to the fact that at energies employed for diagnostic CT imaging scattering and photoelectric absorption are the primary interactions. Schneider et al. (1996) take this into account in a stoichiometric calibration. Using tissue substitutes of known chemical composition, corresponding measurements of CT numbers are made. The response of the CT scanner is then parameterized 3.62 1.86(,)[ ]e ph coh incohNZAKZKZK (2 10) where K is a constant describing the photoelectric, coherent and incoherent scatter interaction cross -sections. Equation 2 10 is then used to calculate CT numbers for ICRP reference tissues.

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21 Relative electron density and proton stopping powers are also calculat ed for the ICRP tissues and a calibration curve is constructed by plotting the CT number versus relative stopping power. Schaffner and Pedroni (1998) perform animal tissue experiments to verify the validity of this stoichiometric calibration and report ra nge uncertainties of 1.1% in soft tissue, and 1.8% in bone tissue.

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22 Figure 2 1. Total stopping power for protons in water. 0 100 200 300 400 500 600 700 800 1.E 03 1.E 02 1.E 01 1.E+00 1.E+01 1.E+02 1.E+03Total Stopping Power, MeV/cmProton Energy, MeV

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23 CHAPTER 3 EXPERIMENTS Numerical experiments were performed using a Fortran program to evaluate the effect of CT ima ge noise on the probability density function ( PDF) of the proton range. Variations in the noise amplitude as well as noise period and incident proton energy were investigated. Image noise is considered to be a random variable. Random number generators were used to simulate white noise. Both a uniform distribution and a Gaussian distribution of noise were evaluated for each parameter. The resulting PDF s of all numerical experiments were compared with respect to their FWHM. Background It is important to determine any dependence of uncertainty on energy because the initial energy of the proton beam dictates the nominal range. The energy selected for a particular treatment then may depend on the amount of expected uncertainty. Proton therapy treatment energies range from tens of MeVs to hundreds of MeVs. For optic tumors such as retinoblastomas, the proton energy selected might be 70 MeV, whereas for prostate tumors the greater penetration depth of a 200 MeV beam might be necessary. By quantifying unc ertainty dependence on energy, it may be possible to identify certain anatomical locations that may require a more robust approach in treatment planning. The importance of uncertainty analysis is further emphasized when looking at uncertainty as a fraction of range. Noise contributions in the CT image arise from signal detection, registration, and processing. Therefore a CT image always has some background level of noise Schneider et al. (1996) report a variation of 1% to 2% even in homogeneous media. Furthermore they note a difference of 3% depending on the size and shape of the medium, and the location within the medium where the CT number is measured. Similarly Flampouri et al. (2007) investigated the

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24 dependence of CT nu mbers on the size of the object being imaged. Quoted range uncertainties for their experiments ranged from 1.5% to 2.4%. H i gh Z materials in the beam make the CT number measurements more unreliable. A commercial treatment planning system (Pinnacle 8.0m ) was used to collect CT image data. For example, a 3mm slice thickness CT image is used to measure profile s of CT numbers The first profile is taken over a 10cm section of bone and is shown in Figure 3 1. Comparing this profile to a profile of CT numb ers over a 16cm section of soft tissue, shown in Figure 3 2, the relative amplitude of noise in bone is greater than that of the soft tissue Examining Figure 3 1 from 3cm to 5cm, an average oscillation of +/ 200 CT number s is observed whereas in Figure 3 2 from 2cm to 6cm, average oscillation is approximately +/ 30. A similar finding is noted when examining the slice thickness effect on noise Figure 3 3 shows a 1mm slice thickness CT image and corresponding profile over a 16cm section of soft tissue Comparing this profile from 2cm to 6cm to Figure 3 2, the average oscillation of approximately +/ 100 CT numbers indicates greater noise amplitude in the thinner slice thickness image. The noise profile of Figure 3 3 also appears to exhibit a higher fr equency noise pattern. Noise may be described as fluctuation in the measurement of attenuation coefficients, and thus CT numbers displayed in the image pixels Since clinical images were used to analyze noise patterns, it was not possible to separate true subtle tissue inhomogeneities from background noise. Therefore the noise referred to includes signal variation due to physical and nonphysical components. Th e image noise typically has some characteristic distribution. It is assumed that CT image noise follows a Gaussian distribution, shown in Figure 3 4. H owever fo r comparison, t he numerical experiments performed consider both a Gaussian and a uniform distribution of noise shown in Figure 3 5 This distribution of CT numbers in turn affects th e

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25 calibration to relative proton stopping power, and ultimately the range calculated for treatment planning. The proton range itself now becomes a random variable wi th some probability distribution, where the FWHM may be used to quantify the resulting ran ge uncertainty. Program Details The main program is the range calculator, incorporating a clearing subroutine and functions to generate random numbers and calculate proton stopping power. Range is calculated for protons of a given energy in a homogeneous medium of unit density. Energy loss is assumed to follow the continuous slowing down approximation (CSDA) which is valid for the majority of the pathlength Calculations are based on dividing the proton pathlength into a grid, as diagramed in Figure 3 6 The initial energy of the pr oton 0E is known. As it traverses the grid, it loses energy at some rate, the stopping power. () dE SE dz (3 1) The stopping power within each grid unit is a function of initial energy. It is also a function of HU since the noise influenced CT numbers are used to determine the relative stopping power. Energy loss is then calculated as the product of the stopping power and the discrete grid unit length (,) dESEHUdz (3 2) Exit energy is calculated by integrating across the grid unit, and then becomes entrance energy for the next grid unit. 11 22 11 22(,)EZ ii EZ iidESEHUdz (3 3)

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26 1 11 1 2 22 2(,)i ii iEESEHUzz (3 4 ) The program iterates this calculation computing range when 1 2 iE is equal to zero Definition of Variables The main program reads the variables from an input file. The input file defines the initial energy of the proton, E0, the standard deviation of image noise (noise amplitude), SIG 1 and S IG2 the grid size (noise period), DZ1 and DZ2, the density of the medium, DENS1 and DENS2, the number of statistical simulations, IMAX, and the boundaries of the zones, N1, N2, N3, N4, and N5. The zones are used to divide the proton pathlength for assign ment of different variables, and are selected in the direction of travel and according to initial proton energy. For all numerical experiments performed in this work a homogeneous medium wa s assumed, thus density is equal to 1.00g/cm3 for all zones and D ENS1 is equal to DENS2. Hence there exists a potential for future investigation of density effects. Since the HU for water is equal to zero, this allows the random numbers generated to serve as the CT numbers in the numerical experiments. To account for statistical straggling, IMAX is equal to 99999 for all experiments. The noise period defined as DZ2 is also a constant in all numerical experiments The final zone in the path is selected to coincide with the end of the proton range. DZ2 is assigned to this zone to allow smaller calculation increments and sufficient data points for analysis. DZ1 then is a constant in the remaining zones, and is also the experimental parameter. Similarly, for all numerical experiments except those evalua ting local noise, the noise amplitude designated is considered a constant in all zones and SIG1 is equal to SIG2. For the local noise experiments a single zone is selected for SIG2 such that it is bordered proximally and distally by SIG1 zones

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27 Stopping Power and Range Calculation Once the variables have been assigned, the program first calculates the stopping power. The i nitial proton energy given in the input file dictates the stopping power obtained from tabulated PTRAN Monte Carlo mass stopping powe r values (Berger 1993). A subprogram is used to retrieve the tabulated values. Linear interpolation is used for stopping powers of energies not specified. Then random number generators are used to simulate either Gaussian or uniform CT image noise accor ding to the noise amplitude given in the input file Stopping power then becomes a random variable by multiplying the tabulated value by the random number and the density given in the input file. The new stopping power is multiplied by the noise period given in the input file and subtracted from the initial energy to give the exit energy. If the exit energy is less than 0.1MeV it is assumed that energy is deposited locally and the range is equal to the grid size, and the probability for traveling that distance is noted If the exit energy is greater than or equal to 0.1MeV, the tabulated stopping power is retrieved for that energy. In this case, the initial stopping power and the new stopping power are averaged and then multiplied by the random number The new exit energy is calculated in the same manner. The computation is iterated until either the relative difference in energies is less than or equal to 0.001, or exit energy meets the cut -off value of 0.1MeV. In which case, the exit energy becomes the entrance energy for the subsequent grid unit, or the range probability at that distance is noted, respectively.

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28 Figure 3 1 Profile of CT Numbers in bone tissue for a 3mm image slice. Figure 3 2 Profile of CT Numbers in soft tissue for a 3mm image slice.

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29 Figure 3 3 Profile of CT Numbers in soft tissue for a 1mm image slice. Figure 3 4. Random number generation of 5% white Gaussian noise. 0.85 0.90 0.95 1.00 1.05 1.10 1.15 0 20 40 60 80Normalized CT NumberDepth, cm Gaussian Image Noise Distribution

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30 Figure 3 5. Random number generation of 5% white uniform noise. Figure 3 6 Range calculation. 0.85 0.90 0.95 1.00 1.05 1.10 1.15 0 20 40 60 80Normalized CT NumberDepth, cm Uniform Image Noise Distribution

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31 CHAPTER 4 METHODS Proton range was computed using a Fortran program created by Sandison and Chvetsov (2000). The variables of initial proton energy, noise amplitude, and noise period were evaluated. Reference input value s were selected to be representative of average values encountered in prostate therapy, and include proton energy equal to 200 MeV, noise amplitude equal to 5%, noise period equal to 0.3 cm, and medium density of 1.00 g/cm3. The parameters were evaluated independently with all other variables held constant at reference conditions. All measurements were performed using random number generators to simulate first a Gaussian distribution of image noise, and then a uniform distr ibution of image noise along the proton pathlength. The proton range was computed for each set of variables. One hundred thousand noise patterns were utilized to model statistical straggling. Convergence of the range probability distribution for both forms of noise patterns approaches a Gaussian, as seen in Figure 4 1. This result is seen for all variables. The calculated FWHM of the PDF is utilized as a quantitative measure of range uncertainty. Initial proton energy was evaluated from 60 MeV up t o 260 MeV in 20 MeV increments. These energies incorporate the clinical range currently used for most treatments. Noise period, referenced as grid size or DZ 1 in the program, was evaluated from 0.05 cm to 0.5 cm in 0.05 cm increments The PDF of noise p eriod variation computed with a Gaussian distribution of image noise is shown in Figure 4 2. Noise amplitude, referenced as SIG1 was evaluated from 1% up to 15% in 1% increments The PDF of noise amplitude variation computed with a Gaussian distribution of image noise is shown in Figure 4 3 Areas of local noise referenced as SIG2, were also evaluated from 5% to 15%. R eference conditions were used with the exception of a designated zone of increased noise amplitude.

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32 Figure 4 1. PDF for reference conditions Figure 4 2. PDF for noise period variation. 0.0000E+00 5.0000E+03 1.0000E+04 1.5000E+04 2.0000E+04 2.5000E+04 25.0 25.5 26.0 26.5 27.0FrequencyRange, cm Gaussian Image Noise Distribution Uniform Image Noise Distribution 0 5000 10000 15000 20000 25000 25 25.5 26 26.5 27FrequencyRange, cm 0.10 grid 0.20 grid 0.30 grid 0.40 grid 0.50 grid

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33 Figure 4 3. PDF for noise amplitude variation. 0 10000 20000 30000 40000 50000 60000 24.5 25 25.5 26 26.5 27 27.5FrequencyRange, cm 0.01 sigma 0.05 sigma 0.10 sigma 0.15 sigma

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34 CHAPTER 5 RESULTS Output data was analyzed in Excel. A frequency plot of range values was generated, identifying range associated with maximum frequency. Data points bordering half -maximum frequency values were identified on both the ascending and descending slopes of the PDF. Linear interpolation was used to calculate range values at half -maximum frequency for each location, and the FWHM was determined from the difference. Uncertainty in proton range is quantified according to the FWHM of the range PDF. Calculations indicate range uncertainties that could affect placement and width of safety margins in proton therapy treatment planning. In itial Proton Energy First, to verify accuracy of range calculations in the Fortran code, the proton range with the resulting maximum probability was compared to the CSDA range given by PSTAR for protons in water. All range values computed were within 1.5mm of PSTAR tabulated values. This discrepancy is expected due to the simulation of noise in the numerical experiment. The FWHM v alues for proton energies from 60MeV to 260MeV are graphed in Figure 5 1. Regression fit shows a linear relat ionship for both noise patterns (r2=0.9971 for Gaussian, and r2=0.9905 for uniform). Table 5 1 lists the FWHM calculation results. For a Gaussia n distribution of noise, uncertainty ranges from 1.1mm (60MeV) to 4.1mm (260MeV), whereas for a uniform distribution of noise, uncertainty ranges from 0.7mm (60MeV) to 2.4mm (260MeV). The linear relationship is a reasonable result in that range itself is increasing with energy, so one would expect range straggling and therefore uncertainty to increase with energy as well. However, when the uncertainty is evaluated as a fraction of the total range in Figure 5 2, a power relationship is observed (r2=0.9985 for Gaussian, and r2=0.9923 for uniform). This corresponds to an uncertainty of 3.74% for 60MeV versus an uncertainty of 1.01% for 260MeV.

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35 Noise Period The FWHM values for noise periods from 0. 05cm to 0.5cm are graphed in Figure 5 3. Regression fit sho ws a linear relationship for both noise patterns (r2=0.9775 for Gaussian, and r2=0.9914 for uniform) Table 5 2 lists the FWHM calculation results. For a Gaussian distribution of noise, uncertainty ranges from 1.4mm (0.05cm) to 4.2mm (0.5cm), whereas for a uniform distribution of noise, uncertainty ranges from 0.9mm (0.05cm) to 2.4mm (0.5cm). The linear relationship is a reasonable result in that as the noise period increases the Fortran code computes the average stopping power over a larger grid increm ent, thus increasing uncertainty. Noise Amplitude The FWHM values for noise amplitudes from 1 % to 15% are graphed in Figure 5 4. Regression fit shows a linear relationship for both noise patterns (r2=0.9989 for Gaussian, and r2=0.9995 for uniform). Tab le 5 3 lists the FWHM calculation results. For a Gaussian distribution of noise, uncertainty ranges from 0.8mm (1%) to 9.5mm (15% ), whereas for a uniform distribution of noise, uncertainty ranges from 0. 6 mm ( 1 %) to 5.6mm (15%) The linear relationship is a reasonable result in that noise amplitude affects the range calculation most directly As the amplitude increases, the spread of possible CT numbers increases, the stopping power becomes more unreliable, and the resulting range calculation has a larger possible error. Local Noise Areas of increased local noise occur frequently in many treatment planning situations, i.e. prostate treatments where the beam passes through the hip bones and encounters bone/soft tissue interfaces Since the reference conditions for the numerical experiments include noise amplitude of 5%, that is considered the baseline and local noise is defined as an increase above 5%. The FWHM values for a region of increase local noise from 5% to 15% are gra phed in Figure 5 5. Regression fit shows a linear relationship for both noise patterns (r2=0.9986 for

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36 Gaussian, and r2=0.9984 for uniform). Table 5 4 lists the FWHM calculation results For a Gaussian distribution of noise, uncertainty ranges from 3.2mm (5%) to 7.0mm (15%), whereas for a uniform distribution of noise, uncertainty ranges from 1.9mm (5%) to 4.2mm (15%). Again, the linear relationship is a reasonable result, and the effect of noise amplitude on range uncertainty is emphasized.

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37 Figure 5 1 Relationship of range uncertainty with respect to initial energy. Figure 5 2 Relationship of relative uncertainty with respect to initial energy. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0 50 100 150 200 250 300FWHM, cmInitial Proton Energy, MeV Gaussian noise distribution Uniform noise distribution 0.00 0.01 0.02 0.03 0.04 0 50 100 150 200 250 300FWHM / RangeInitial Proton Energy, MeV Gaussian noise distribution Uniform noise distribution

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38 Figure 5 3 Relationship of range uncertainty with respect to noise period Figur e 5 4 Relationship of range uncertainty with respect to noise amplitude 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.00 0.10 0.20 0.30 0.40 0.50 0.60FWHM, cmNoise Period, cm Gaussian noise distribution Uniform noise distribution 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.05 0.10 0.15FWHM, cmNoise Amplitude Gaussian noise distribution Uniform noise distribution

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39 Figure 5 5 Relationship of range uncertainty with respect to local noise. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.00 0.05 0.10 0.15 0.20FWHM, cmLocal Noise Amplitude Gaussian noise distribution Uniform noise distribution

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40 Table 5 1. FWHM calculations for e nergy Variable Value FWHM cm (Gaussian noise) % of range FWHM cm (uniform noise) % of range Energy 60 MeV 80 MeV 100 MeV 120 MeV 140 MeV 160 MeV 180 MeV 200 MeV 220 MeV 240 MeV 260 MeV 0.11 0.14 0.18 0.21 0.23 0.27 0.30 0.32 0.34 0.38 0.41 3.74 2.82 2.39 1.97 1.68 1.54 1.39 1.26 1.13 1.07 1.01 0.07 0.08 0.12 0.13 0.14 0.16 0.18 0.19 0.20 0.22 0.24 2.45 1.67 1.58 1.22 1.04 0.93 0.82 0.73 0.66 0.63 0.60 Table 5 2. FWHM calculations for noise period Variable Value FWHM cm (Gaussian noise) % of range FWHM cm (uniform noise) % of range Noise period 0.05 cm 0.10 cm 0.15 cm 0.20 cm 0.25 cm 0.30 cm 0.35 cm 0.40 cm 0.45 cm 0.50 cm 0.14 0.19 0.23 0.27 0.30 0.32 0.36 0.37 0.39 0.42 0.55 0.75 0.91 1.03 1.17 1.24 1.39 1.42 1.53 1.61 0.09 0.12 0.14 0.15 0.17 0.19 0.20 0.21 0.23 0.24 0.37 0.45 0.53 0.59 0.66 0.73 0.78 0.82 0.90 0.93

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41 Table 5 3. FWHM calculations for noise amplitude Variable Value FWHM cm (Gaussian noise) % of range FWHM cm (uniform noise) % of range Noise amplitude 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.08 0.14 0.20 0.26 0.32 0.38 0.45 0.51 0.57 0.63 0.70 0.78 0.83 0.88 0.95 0.31 0.53 0.76 1.00 1.26 1.48 1.75 1.96 2.19 2.45 2.73 3.00 3.21 3.39 3.68 0.06 0.09 0.12 0.16 0.19 0.23 0.26 0.30 0.34 0.37 0.41 0.45 0.48 0.52 0.56 0.23 0.35 0.46 0.60 0.73 0.88 1.01 1.14 1.31 1.43 1.57 1.72 1.87 2.00 2.16 Table 5 4. FWHM calculations for l ocal noise Variable Value FWHM cm (Gaussian noise) % of range FWHM cm (uniform noise) % of range Local noise 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.32 0.36 0.39 0.43 0.47 0.50 0.54 0.59 0.64 0.66 0.70 1.24 1.39 1.51 1.66 1.80 1.94 2.10 2.27 2.45 2.53 2.72 0.19 0.21 0.23 0.25 0.27 0.29 0.32 0.34 0.37 0.39 0.42 0.74 0.81 0.88 0.96 1.04 1.13 1.24 1.33 1.41 1.50 1.61

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42 CHAPTER 6 CONCLUSIONS Proton therapy can offer unique benefits due to the dose deposition characteristics of the Bragg peak. However, these benefits are subject to range uncertainties from various sources. To name a few, p atient set up and geometry can affect the proton range, as well as the calibration performed to correlate CT numbers with relative proton stopping power. In this work, the uncertainty arising from CT image noise is investigated. A Fortran program wa s used to simulate white noise alon g the proton pathlength and compute a probability distribution of proton range. Parameters affecting the shape of the range PDF include initial proton energy, noise period, and noise amplitude. The FWHM of the PDF wa s calculated to quantify range uncerta inty for these parameters. Results show linear relationships between uncertainty and al l variables evaluated. Indicating larger possible error introduced into range calculation when calibrating relative stopping power from CT images exhibiting high amplit ude and or low frequency noise. The increased uncertainty may require adjustment of current methods of safety margin application. Furthermore, the relative uncertainty increases for lower energies due to their reduced range. This i ndicat es the possible need for more robust treatment planning for target volumes of small penetration depths often required for optic tumors or pediatric cases Further investigation could incorporate a multi -variable analysis to include density effects. This paper consider s only single variable effects on proton range PDF. However in a clinical environment, different combinations of these variables would be encountered. An evaluation of the Fourier transform of the noise profiles could also be performed. This could facil itate a more precise characterization of the noise distribution possibly assisting in distinguishing between true background noise and subtle tissue inhomogeneities Also, noise reduction techniques for

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43 CT imaging, such as smoothing and denoising algorithms, could reduce the standard deviation of the range PDF. Finally, adjusting radiographic technique used to acquire the CT image could affect the background noise. For example, increasing the mAs will i ncrease the number of photons in the beam and should reduce noise caused by signal detection and registration. A risk versus benefit analysis could be completed for the increased dose to the patient from higher image techniques versus improved proton rang e uncertainty. T he numerical experiments performed demonstrate that proton range uncertainty due to CT image noise can be clinically significant requiring adjustment of current guidelines for margins Average FWHM values, between 1% and 2%, were on the order of uncertainty arising from the relative stopping power calibration process itself.

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44 APPENDIX FORTRAN PROGRAM S Range Calculation The main program calculates the proton range and produces range probability distribution (Sandison and Chvetsov 2000) PROGRAM RANGECALC c IMPLICIT DOUBLE PRECISION (A H,O Z) INTEGER NMAX PARAMETER (NMAX=600) DOUBLE PRECISION FDE NS(NMAX) c COMMON/TAPEID/ M1,M2 ,M3 c c INPUT DATA M1=5 M2=6 OPEN(M1, FILE='INPUT .TXT' STATUS='OLD' ) OPEN(M2, FILE='OUTPU T.TXT', STATUS='UNKN OWN') c c E0 INITIAL ENERGY OF PR OTON c SIG 1 STANDARD DEVIATION O F NOISE c SIG 2 STANDARD DEVIATION O F INCREASED LOCAL NOISE c DZ1 AND DZ2 GRID SIZE IN ZONES 1 AND 2 c DENS1 AND DENS2 MEDIA DENSITY IN ZONES 1 AND 2 c IMAX NUMBER OF STATISTICA L SIMUL ATIONS c N1,N2,N3,N4,N5 BOUNDARIES OF ZONES c READ (M1,*) E0, S IG 1 SIG2, DZ1, DZ2, DENS1, DEN S2 WRITE(M2,2000) E0, S IG, DZ1, DZ2, DENS1, DENS2 c READ (M1,1001) IMAX, N1,N2,N3,N4,N5 WRITE(M2,1001) IMAX, N1,N2,N3,N4,N5 c WRITE(M2,9000) 9000 FORMAT(//' RN RANGE FDENS(N) RANGE R1') c CALL ZEROUT(NMAX,FDENS) c DO 300 I=1,IMAX c ISEED=I EIN=E0

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45 DO 100 N=1,NMAX IF (N.GE.N1 .AN D. N.L T .N2) THEN SIG=SIG1 DENS=DENS1 DZ=DZ1 ELSE IF (N.G E .N2 .AND. N.LT.N3) T HEN SIG=SIG2 DENS=DENS2 DZ=DZ1 ELSE IF (N.GE.N3 .AN D. N.L T .N4) THEN SIG=SIG1 DENS=DENS1 DZ=DZ1 ELSE IF (N.G E .N4 .AND. N.LE.N5) T HEN SIG=SIG1 DENS=DENS1 DZ=DZ2 END IF c SPIN =BERGER(EIN) R1 =1.D0+SIG*GRDM(ISEED) c R1 =1.D0+SIG*(2.D0 *URDM(ISEED) 1.D0) c SPRAN=R1*DENS*SPIN EOUT 0 =EIN SPRAN*DZ IF(EOUT0 .LT. 0.1D0) GOTO 200 c 101 SPOUT =BERGER(EO UT0) SPRAN =R1*DENS*(SPIN +SPOUT)*0.5D0 EOUT1 =EIN SPRAN*DZ IF(EOUT1 .LT. 0.1D0) GOTO 200 IF (ABS( (EOUT0 EOUT1)/EOUT0 ) .GT. 0.0001) THEN EOUT0 =EOUT1 GO TO 101 END IF RN=N EIN =EOUT1 100 CONTINUE c 200 FDENS(N 1)=FDENS(N 1) + 1.D0 c STOP 300 CONTINUE c RANGE=0.D0 ISEED=1 DO 600 N=1,NMAX

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46 IF (N.GE.N1 .AN D. N.L T .N2) THEN SIG=SIG1 DENS=DENS1 DZ=DZ1 ELSE IF (N.G E .N2 .AND. N.LT.N3) T HEN SIG=SIG2 DENS=DENS2 DZ=DZ1 ELSE IF (N.GE.N3 .AN D. N.L T .N4) THEN SIG=SIG1 DENS=DENS1 DZ=DZ1 ELSE IF (N.G E .N4 .AND. N.LE.N 5) THEN SIG=SIG1 DENS=DENS1 DZ=DZ2 END IF R1 =(1.D0+SIG*GRDM (ISEED))*DENS c R1 =1.D0+SIG*(2.D0 *URDM(ISEED) 1.D0) RANGE=RANGE+DZ RN=N c WRITE(M2,2000) RN,RA NGE, FDENS(N),RANGE, R1 600 CONTINUE c 2000 FORMAT(8D12.4) 1001 FORMAT(14I5) STOP END Clearing Routine Subprograms supporting the range calculator include a clearing routine, calculation of stopping power, and random number generators The stopping power values were taken from tabulated data ( Berger 1993). The random number generators were used in compliance with a free immediate license ( Press et al. 1992) SUBROUTINE ZEROUT(MAX,A) c***** REAL CLEARING ROUTINE ******** IMPLICIT DOUBLE PRECISION (A H,O Z) DIMENSION A(1) DO 1 I=1,MAX 1 A(I)=0.D0 RETURN END

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47 Stopping Power Calculation FUNCTION BERGER(E0) c c*** INTERPOLATES 1D ARRAY c IMPLICIT DOUBLE PRECISION (A H,O Z) COMMON / T APEID / M 1, M 2, M 3 DIMENSION D(41),E(41) DATA E A/2.6D2, 2.5D2, 2.4D2, 2.3D2, 2.2D2, B 2.1D2, 2.0D2, 1.9D2, 1.8D2, 1.7D2, C 1.6D2, 1.5D2, 1.4D2, 1.3D2, 1.2D2, D 1.1D2, 1.0D2, 9.0D1, 8.0D1, 7.0D1, E 6.0D1, 5.0D1, 4.0D1, 3.0D1, 2.0D1, F 1.5D1, 1.0D1, 8.0D0, 6.0D0, 5.0D0, G 4.0D0, 3.0D0, 2.0D0, 1.0D0, 0.8D0, H 0.6D0, 0.5D0, 0.4D0, 0.3D0, 0.2D0, E 0.1D0/ DATA D A/3.800D0, 3.911D0, 4.008D0, 4.114D0, 4.229D0, B 4.354D0, 4.492D0, 4.644D0, 4.812D0, 4.999D0, C 5.209D0, 5.445D0, 5.713D0, 6.021D0, 6.377D0, D 6.794D0, 7.289D0, 7.888D0, 8.625D0, 9.559D0, E 1.078D1, 1.245D1, 1.488D1, 1.876D1, 2.607D1, F 3.292D1, 4.567D1, 5.460D1, 6.858D1, 7.911D1, G 9.404D1, 1.172D2, 1.586D2, 2.608D2, 3.039D2, H 3.680D2, 4.132D2, 4.719D2, 5.504D2, 6.613D2, O 8.161D2/ c DO 120 II=2,41 IF(E(II).LT.E0) GOTO 220 120 CONTINUE BERGER=0.0 RETURN 220 DX = E(II) -E(II 1) A = (D (II) D(II 1)) / DX B = (D(II 1)*E(II) -D(II)*E(II 1)) / DX BERGER = A*E0 + B c 1000 FORMAT (10e12.5) RETURN END Gaussian Random Number Generator FUNCTION GRDM (IDUM ) INTEGER IDUM

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48 REAL GRDM c USES URDM c Returns a normally distributed deviate with zero mean and unit variance, using urdm(idum) c as the source of uniform deviates. INTEGER ISET REAL FAC,GSET,RSQ,V1,V2,R AN1 SAVE ISET,GSET DATA ISET /0/ c IF (IDUM.LT.0) IS ET=0 IF (ISET.EQ.0) THEN 1 V1=2.*URDM(IDUM ) 1. V2=2.*URDM(IDUM) 1. RSQ=V1**2+V2**2 IF(RSQ.GE.1..OR.RSQ.EQ.0.)GOTO 1 FA C=SQRT( 2.*LOG(RSQ)/RSQ) GSET=V1*FAC GRDM=V2*FAC ISET=1 ELSE GRDM=GSET ISET=0 ENDIF RETURN END Uniform Random Number Generator FUNCTION URDM(IDUM) INTEGER IDUM ,IA,IM,IQ,IR,NTAB,NDIV DOUBLE PRECISION URDM ,AM,EPS,RNMX PARAMETER (IA=16807,IM=2147483647,AM=1./IM,IQ=127773,IR=2836, NTAB=32,NDIV=1+(IM 1)/NTAB,EPS=1.2d7,RNMX=1. EPS) c c Returns a uniform random deviate between 0.0 and 1.0 (exclusive of the endpoint values) c INTEGER J,K,IV(NTAB),IY SAVE IV,IY DATA IV /NTAB*0/, IY /0/ c IF (IDUM.LE.0.OR.IY. EQ.0) THEN IDUM=MAX( IDUM,1) DO 11 J=NTAB+8,1, 1 K=IDUM/IQ IDUM=IA*(IDUM -K*IQ) IR*K IF (IDUM.LT.0) IDUM= IDUM+IM IF (J.LE.NTAB) IV(J)=IDUM 11 CONTINUE

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49 IY=IV(1) ENDIF c K=IDUM/IQ IDUM=IA*(IDUM -K*IQ) IR*K IF (IDUM.LT.0) IDUM= IDUM+IM J=1+IY/NDIV IY=IV(J) IV(J)=IDUM URDM=MIN(AM*IY,RNMX) RETURN END

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50 LIST OF REFERENCES Attix F H 1986 Introduction to Radiological Physics and Radiation Dosimetry (John Wiley & Sons, Inc.) Berger M J 1993 Proton Monte Carlo transport program PTRAN Technical Report NISTIR 5113 (Gaithersburg, MD: U.S. Department of Commerce) p27 Bussiere M and Adams J 2003 Treatment Planning for Conformal Proton Radiation Therapy Technology in Cancer Research & Treatment 2 389399 Chvetsov A and Paige S 2009 Influence of CT image noise on proton range uncertainty Med. Phys. 36 2734 Chvetsov A and Palta J 2008 Probability density distribution of proton range as a function of noise in CT images Med. Phys. 35 2754 Evans R 1982 The Atomic Nucleus (Malabar, FL: Krieger) Flampouri S, Slopsema R, Yeung D, Malyapa R, Keole S, Vergas C and Li Z 2007 Realistic estimation of proton range uncertainties and dosimetric implications Med. Phys. 34 2643 Li H, Liu R, Wu R, Sahoo N, Mohan R, Gillin M and Zhu X 2008 Effect of CT calibration phantom size on the range in proton radiation therapy Me d. Phys. 35 2955 Lomax A J 2008 Intensity modulated proton therapy and its sensitivity to treatment uncertainties 1: the potential effects of calculational uncertainties Phys. Med. Biol. 53 10271042 Lu H 2008 A potential method for in vivo range verification in proton therapy treatment Phys. Med. Biol. 53 14131424 Lu H 2008 A point dose method for in vivo range verification in proton therapy Phys. Med. Biol. 53 N415N422 Mustafa A and Jackson D 1983 The relation between xray CT numbers and charged particle stopping powers and its significance for radiotherapy treatment planning Phys. Med. Biol. 28 169176 Newhauser W, Giebeler A, Langen K, Mirkovic D and Mohan R 2008 Can megavoltage computed tomography reduce proton range uncertainties in treatment plans for patients with large metal implants? Phys. Med. Biol. 53 23272344 Nishio T, Ogino T, Nomura K and Uchida H 2006 Dose -volume delivery guided proton therapy using beam on-line PE T system Med. Phys. 33 41904197 Parodi K, Paganetti H, Cascio E, Flanz J, Bonab A, Alpert N, Lohmann K and Bortfeld T 2007 PET/CT imaging for treatment verification after proton therapy: A study with plastic phantoms and metallic implants Med. Phys. 34 419435

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51 Pflugfelder D, Wilkens J J and Oelfke U 2008 Worst cast optimization: a method to account for uncertainties in the optimization of intensity modulated proton therapy Phys. Med. Biol. 53 16891700 Press W H, Teukolsky S A, Vetterling W T and Fla nnery B P 1992 Numerical Recipes in Fortran 77 (Cambridge University Press) Rietzel E, Schardt D and Haberer T 2007 Range accuracy in carbon ion treatment planning based on CT -calibration with real tissue samples Radiation Oncology 2 1422 Sandison G and Chvetsov A 2000 Proton loss model for therapeutic beam dose calculations Med. Phys. 27 21332146 Schaffner B and Pedroni E 1998 The precision of proton range calculations in proton radiotherapy treatment planning: experimental verification of the relation between CT HU and proton stopping power Phys. Med. Biol. 43 15791592 Schneider U and Pedroni E 1995 Proton ra diography as a tool for quality control in proton therapy Med. Phys. 22 35363 Schneider U, Pedroni E and Lomax A 1996 The calibration of CT Hounsfield units for radiotherapy treatment planning Phys. Med. Biol. 41 111124 Schneider U, Pemler P, Besserer J, Pedroni E, Lomax A and Kaser Hotz B 2005 Patient specific optimization of the relation between CT Hounsfield units and proton stopping power with proton radiography Med. Phys. 32 195199 Schulte R W, Bashkirov V, Loss Klock M C, Li T, Wroe A J, Evseev I, Williams D C and Satogata T 2005 Density resolution of proton computed tomography Med. Phys. 32 10351046 Szymanowski H and Oelfke U 2003 CT calibration for twodimensional scaling of proton pencil beams Phys. Med. Biol. 48 861874 Turner J E 1995 Ato ms, Radiation, and Radiation Protection (New York: Wiley) Unkelbach J, Chan T C Y and Bortfeld T 2007 Accounting for range uncertainties in the optimization of intensity modulated proton therapy Phys. Med. Biol. 52 27552773 Zygmanski P 1998 Proton cone beam computed tomography PhD Dissertation University of Massachusetts, Amherst Zygmanski P, Gall K, Rabin M and Rosenthal S 2000 The measurement of proton stopping power using proton -cone -beam computed tomography Phys. Med. Bo il. 45 511528

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52 BIOGRAPHICAL SKETCH Sandra Paige was born and raised in Orlando, Florida. She graduated from Dr. Phillips High School in 1994, and continued her studies at Valencia Community College while working at Walt Disney World. She then earned a Bachelor of Science degree in radiologic science from the University of Central Florida in 2000. She spent seven years as a radiologic technologist, learning to love the many aspects of radiology. She then returned to school to pursue a graduat e degree in Medical Physics at the University of Florida graduating in 2009. She hopes to continue to contribute to the delivery of high quality patient care through the practice of Medical Physics.