Acquisition and analysis of megavoltage linac beam transmission data for direct verification of photon spectra models in a treatment planning system

Transcription

1 The University of Toledo The University of Toledo Digital Repository Theses and Dissertations 2014 Acquisition and analysis of megavoltage linac beam transmission data for direct verification of photon spectra models in a treatment planning system Dimitra Leheta University of Toledo Follow this and additional works at: Recommended Citation Leheta, Dimitra, "Acquisition and analysis of megavoltage linac beam transmission data for direct verification of photon spectra models in a treatment planning system" (2014). Theses and Dissertations This Thesis is brought to you for free and open access by The University of Toledo Digital Repository. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of The University of Toledo Digital Repository. For more information, please see the repository's About page.

2 A Thesis entitled Acquisition and Analysis of Megavoltage Linac Beam Transmission Data for Direct Verification of Photon Spectra Models in a Treatment Planning System by Dimitra S. Leheta Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science in Biomedical Sciences (Medical Physics) Dr. Dianna Shvydka, Committee Chair Dr. E. Parsai, Committee Member Dr. David Pearson, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo December 2014

3 Copyright 2014, Dimitra S. Leheta This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author.

4 An Abstract of Acquisition and Analysis of Megavoltage Linac Beam Transmission Data for Direct Verification of Photon Spectra Models in a Treatment Planning System by Dimitra S. Leheta Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Science in Biomedical Sciences (Medical Physics) The University of Toledo December 2014 The dose calculation in Philips Pinnacle Treatment Planning System (TPS) uses collapsed cone convolution algorithm, which relies on beam spectrum information for calculation of scatter contribution. Typically beam spectra are derived from a set of measurements collected during commissioning of a linear accelerator. The purpose of this project is to measure beam spectra independently using transmission methodology and compare with those modeled by the Pinnacle TPS. Three photon beam energies were measured and analyzed, 6MV, 6MV flattening-filter-free (FFF), and 10MV. Transmission measurements were conducted for two materials, having high and low atomic numbers (Z), using a standard Farmer ionization chamber fitted with high-z and low-z buildup caps. The combination of two materials and two caps with different Z served to enhance discrimination between absorption of low and high-energy portions of the spectrum, thus improving the accuracy of the results. We chose commonly found attenuating materials, lead and polycarbonate, and standard build-up caps, made of polymethyl methacrylate (PMMA) and brass, with a goal of adaptation of an ideal setup to that achievable in a typical clinical setting. The data was analyzed using a iii

5 regularization technique implemented through spreadsheet-based calculations, enabling a potential user to restore spectra from transmission measurements without elaborate programming efforts. We successfully unfolded the spectra using transmission measurements and found the resultant spectra to deviate from those derived in the TPS beam models. The effect of such deviations on treatment planning was evaluated for 6MV-FFF beam through sideby-side comparison of representative plans, calculated with both the commissioned model spectrum and that obtained from the measurements. The differences between the model and unfolded spectra were reviewed through isodose distributions, and quantified in terms of maximum dose values for critical structures. While in most cases switching the model spectrum with the measured one did not result in drastic differences in the calculated dose, plans with deviations of 4 to 8% in the maximum dose values for critical structures were discovered. Since spectrum information is used by TPS in evaluation of the scatter dose, the anatomical sites with large scatter contributions are the most vulnerable to inaccuracies in the modeled spectrum. The developed approach of using readily available attenuators and build-up caps in combination with streamlined data processing confirmed feasibility of acquisition of megavoltage linac beam spectra in a typical radiation oncology clinic, offering an independent check of the TPS model. iv

6 Table of Contents Abstract... iii Table of Contents...v List of Tables... vii List of Figures... viii List of Abbreviations...x List of Symbols... xi 1 Introduction Literature Review Methods & Materials to Chapter Approach Set Up Equipment Attenuating Materials Response Function Regularization Technique Results & Discussion Experimental Data Response Function Restored Spectra...34 v

7 4.4 Comparison of Model and Restored Spectra Conclusions...46 References...49 A Regularization of Transmission Data...52 vi

8 List of Tables 4.1 Patient 1 Values Patient 2 Values Patient 3 Values Patient 4 Values Patient 5 Values Patient 6 Values Difference between TPS Model and Restored Spectrum PDD...45 vii

9 List of Figures 2.1 Digitized Response Functions for Tungsten and PMMA Build Up Caps Response of Attenuating Materials per NIST Values for Lead and Carbon Experimental Set Up Farmer Ionization Chamber Mass Attenuation Coefficient Comparison Ion Chamber Response Function with PMMA Build Up Cap Ion Chamber Response Function for Brass Build Up Cap MV Beam through Lead Attenuator using Brass Build Up Cap Response Function and Mass Attenuation Coefficient Comparison Response Function and Mass Attenuation Coefficient Comparison Response Function and Mass Attenuation Coefficient Comparison Build Up Cap Comparison through Lead Attenuator Build Up Cap Comparison through Polycarbonate Attenuator Build Up Cap Comparison through Lead Attenuator Build Up Cap Comparison through Polycarbonate Attenuator Build Up Cap Comparison through Lead Attenuator Build Up Cap Comparison through Polycarbonate Attenuator Restored 6MV Spectrum Compared with TPS Model...34 viii

10 4.12 Restored 6MV-FFF Spectrum Compared with TPS Model Restored 10MV Spectrum Compared with TPS Model Patient 1 Isodose Distribution Comparison Patient 2Isodose Distribution Comparison Patient 3 Isodose Distribution Comparison Patient 4 Isodose Distribution Comparison Patient 5 Isodose Distribution Comparison Patient 6 Isodose Distribution Comparison PDD Curve Comparison...45 ix

11 List of Abbreviations CT...Computed tomography DVH...Dose volume histogram EBRT...External beam radiation therapy EDW...Enhanced dynamic wedge FFF...Flattening filter free Ge...Germanium IMRT...Intensity modulated radiation therapy MC...Monte Carlo MCNP...Monte Carlo N-Particle Transit Code MLC...Multi leaf collimator MU...Monitor unit NaI...Sodium Iodine PMMA...Polymethylmethacrylate QA...Quality assurance SBRT...Stereotactic body radiation therapy SRS...Stereotactic Radiosurgery TPS...Treatment planning system VMAT...Volumetric modulated radiation therapy x

12 List of Symbols E α...discrete set of energies µ en...absorption attenuation coefficient µ...linear attenuation coefficient Ψ(E)...Energy fluence Ψ o...energy fluence - Trial function R(d,x i,e)...ion chamber response function including exponential relationship R(d,0,E)...Ion chamber response function T i...measured transmission data point x...attenuator thickness Z...Atomic number Z iα...square matrix xi

13 Chapter 1 Introduction Radiation oncology has benefitted from technology advances over the years, allowing for more precise and effective treatments. Equipment used in external beam radiation therapy (EBRT) has exponentially evolved since the first use of Radium for treatments of cancers and skin lesions in the early 1900 s. The development of microwave technology after World War II revolutionized radiation therapy leading to construction and use of the first medical linear accelerator, which are still prevalent in today s clinics across North America and Europe. Linear accelerators provide a much higher energy alternative to Cobalt 60 (Co-60) units, originally developed at about the same time. The use of linear accelerators has improved the quality of life of patients and the safety of hospital personnel. Linacs are capable of producing electron and x-ray beams, with energies typically in the range of 2-20MeV. Over the years treatment field shaping approaches developed from combining simple open and wedged fields to the streamlined features of enhanced dynamic wedges (EDW) and multileaf collimators (MLC s) to aid in modulation of the beam for more complex plans, such as intensity modulated radiation 1

14 therapy (IMRT), and recently to volumetric-modulated arc therapy (VMAT), allowing precise targeting of radiation dose delivery. As the equipment in radiation oncology evolves, the delivery of treatment advances and in turn adds complexity to dose calculations. At its infancy radiation therapy relied on hand calculations of the delivered dose, which is almost impossible to imagine today. With developments of computerized dose computation algorithms we became fully dependent on treatment planning systems (TPS) for calculations of radiation dose distributions and treatment delivery. With a TPS one can evaluate the doses deposited to the healthy tissues surrounding the target volume which allows the physician to confidently determine the patient s course of treatment. The planning starts with a three dimensional computed tomography (CT) scan of patient anatomy. The dose calculations are based on commissioning measurements, acquired before the institute s linear accelerator is declared clinic ready. A set of data required for TPS includes beam profiles at several depths in a water phantom, percent depth dose curves, and output factors, measured for all clinical beams for the full range of available field sizes. These measurements are additionally used as the baseline for quality assurance (QA) tests throughout the machine s lifetime. The TPS processes the data set and develops a comprehensive model for each beam, with the beam spectrum serving as an integral component of the model. While this model is used for all dose distribution calculations, there is no data available to validate the modeled beam spectrum. While there are many commercially available treatment planning systems, Philip s commercially available TPS will be discussed here. Philip s Pinnacle TPS uses 2

15 collapsed cone convolution superposition algorithm to calculate the dose distribution (Ramsey 1999). Many factors contribute to the calculated dose distribution. The TPS algorithm can account for the effects of beam modifiers, the surface of the patient, and tissue heterogeneities on the dose distribution (Philips Medical Systems 2010). For each point in phantom the dose calculation algorithm evaluates contributions from primary fluence and scatter separately. The scatter component changes as the beam goes through tissue interfaces, especially for small field sizes. Scatter is calculated using the energy spectra modeled within the TPS (Philips Medical Systems 2010), therefore these spectra will have direct influence on resulting dose distributions in plans with significant heterogeneities. At the University of Toledo Dana Cancer Center (UTMC) the majority of the patients are undergoing stereotactic radiosurgery (SRS) or stereotactic body radiation therapy (SBRT). Both treatment regimens incorporate a prescription of high dose per fraction in a small number of fractions to control small tumors or metastatic lesions. SRS and SBRT are specific to treatment of brain lesions and lesions within the patients body respectively, in both cases lesions are typically small. Stereotactic therapy and surgery enable good local control and low morbidity (Flickinger & Kondziolka 2014). The contribution of scatter from small field sizes and tissue interfaces can have a significant effect on the dose distribution if the energy spectra of the commissioned linear accelerator are not modeled accurately. To assure the accuracy of a TPS s modeled energy spectra there must be a way to validate it. This research had the objective of measuring and evaluating the bremsstrahlung beam spectra of a Varian TrueBeam linear accelerator for the 6MV, 6MV 3

16 flattening filter free (FFF), and 10MV energies. A comparison of the TPS modeled energy spectra with the measured data provides an independent check of the TPS model quality. 4

17 Chapter 2 Literature Review Measurements of megavoltage bremsstrahlung energy spectra are difficult to obtain; it often requires expensive equipment and thorough mathematical analysis. Literature has shown different methods of collecting energy spectra of a linear accelerator. The four major methods will be discussed in this review; they are: direct spectroscopy, Compton spectroscopy, Monte Carlo (MC) simulation, and transmission or depth dose measurements. Direct spectroscopy is mostly available to large research institutions due to the prohibitively high cost of detection equipment, which includes a high-quality radiation detector (often coupled to a photomultiplier tube) and a pulse-height analyzer. It is not common for medical physicists to measure the primary beam spectra of their clinic s linac and most clinic locations spend their money on equipment used on the regular basis for the management and quality control of their radiation devices. Among popular choices of detectors for direct spectroscopy are scintillation crystal detectors, which convert the kinetic energy of charged particles into detectable light. The conversion from charged particles to light holds a proportional relationship to 5

18 the energy deposited (Knoll 2010). This process must have high efficiency so to not loose information about the beam energy. In the experiment conducted by Sandifer and Taherzahed a large sodium iodine (NaI) scintillator was coupled with a RCA-8055 photomultiplier tube to measure thick-target bremsstrahlung for electron energy range of 5.3 to 20.9MeV. The spectrometer outputs connected directly into a 400-channel pulseheight analyzer and a pulse pile-up rejection system where the data was analyzed and created absolute bremsstrahlung spectra (Sandifer & Taherzahed 1968). The advantage of direct spectroscopy is obtaining the spectrum immediately after data collection, although the materials are costly and not easily obtained by a clinical medical physicists. Compton spectroscopy is another method in measuring the primary beam spectra. A Compton spectrometer is dependent on the Compton scatter due to the photons of the primary beam interacting through attenuating material of a defined length. The attenuating material is referred to as the second target. The detector is placed at a fixed angle to the incident beam. An example of coincidence gamma ray detection Valentine and Rooney in 1994 collect data on the prospect that the two scattered photons are simultaneously detected by two detectors. Landry and Anderson in 1994 published a paper that used a high efficiency germanium (Ge) detector to measure an accelerators bremsstrahlung spectra. They successfully extracted a spectrum and compared it to previously published results using Compton spectroscopy. Specifically comparing to the works of Levy and Waggener in 1974 who successfully measure the bremsstrahlung spectra of a 25MeV linear accelerator and 19MeV betatron with NaI(Tl) spectrometer system. The mentioned papers achieved the goal of measuring the bremsstrahlung spectra of their megavoltage linear accelerators but at the expense of using costly equipment. 6

19 While there is a variety of medical linacs available on the market, their manufacturers do not typically provide their customers with measured beam spectra. For some linacs the spectra become available through Monte Carlo simulations of the accelerator heads and a known function as the input for the energy spectrum. Since this process requires detailed knowledge of treatment head geometry (mostly proprietary) and is very labor intensive, only a small set of linacs where modeled in this manner. Often MC modeled spectra are available for research linear accelerators, not used clinically, with the assumption that the spectra for clinical linacs are similar. Potentially, MC models offer very high accuracy in calculations of beam properties, however, they have to be validated against measurements. Vlamynck et al in 1999 compared dose measurements with Monte Carlo s system BEAM/EGS4, modeling the details of the head of the accelerator including the rounded leaf edges of the multi leaf collimator (MLC). With the accurate MC model of their research accelerator they compared the simulation results with experimental dose measurements taken in a water phantom with a highresolution diamond detector. Diamond detectors are near tissue equivalent, small in size, and have good spatial resolution, making them near-ideal dosimeters. Their high cost leads to difficulty in acquiring it in a typical clinical setting. In 1991 Faddegon, Ross, and Rogers used MCNP to recreate their experimental measurements of a 15MeV bremsstrahlung spectrum. The experimental setup consisted of thick targets of various high atomic numbers and data points at angles ranging from zero to ninety degrees with a cylindrical NaI detector. They successfully extracted an energy spectra for their 15MeV beam and found that the bremsstrahlung spectrum was comparable to their experimental measurements. 7

20 The majority of publications on linac energy spectra come from research facilities. They use Monte Carlo simulations because they have access to their linac s detailed specifications. It is assumed that all energy spectra, from machine to machine, will follow a similar pattern. Although this may very well be generally true, the details of specific spectra belonging to the radiation oncology clinic s medical linear accelerator remain largely unknown. The most clinically useful approach is the spectrum unfolding technique based on transmission or depth dose measurements. While this approach is an indirect measurement, due to its simple setup and relatively low demand on measurement equipment, it is the most attractive one for medical linear accelerators. Analysis of these measurements, however, is less straightforward, owing to a fairly weak energy dependence of photon attenuation at radiation therapy energy range. The problem statement involves a system of linear equations, typically written in a matrix form. The solution to unfold the energy spectrum from the measurements is through inversion of the matrix, this mathematical solution causes transmission and depth dose measurements to be classified as ill-conditioned problems. An in-depth summary of publications devoted to restoring linac spectra based on transmission measurements is given in Functional forms for photon spectra of clinical lincas (Ali and Rogers 2011), where an method proposed by the authors results in the highest level of robustness and lowest error. The research presented in this paper uses the experimental approach of Ali and Rogers, coupled with the regularization technique developed for solution of ill-conditioned problems. The complete procedure can be adapted in any radiation oncology clinic. 8

21 The idea of the paper by Ali and Rogers An improved physics-based approach for unfolding megavoltage bremsstrahlung spectra using transmission analysis (Ali and Rogers 2012a) is to use a combination of attenuating materials and detectors providing different response in different parts of energy spectrum. The latter is facilitated by use of an ionization chamber fitted with different build-up caps. Instead of acquiring one data set with one attenuating material and one build up cap; they investigate the use of multiple datasets with significantly different atomic number attenuating materials paired with differing build up cap materials. The Exradin A19 Farmer chamber was used for the measurements. The effect of energy response was modeled by Monte Carlo simulation, resulting in a set of energy response functions for different build up caps. The calculations were done for energies from 100keV to 30MeV. An increase in Z of the build-up cap increased the sensitivity of the ion chamber in the higher energy ranges. Similarly, the low Z build up cap had a higher response at the lower end of the energy range of the beam, depicted in Figure 2.1. This response function can be used as a spectral weighting function helping to solve ill-conditioned problem of unfolding the energy spectrum from the transmission measurements. The values plotted in Figure 2.1 are digitized from Ali and Rogers 2012a publication. Their materials included the use of tungsten and PMMA for their high Z and low Z build up caps respectively. In this research project the materials used differ from the ones proposed in Ali and Rogers 2012a publication since the intension was to use materials conveniently available in the clinic. 9

22 R(d,E) (cm2/g) Tungsten PMMA Energy (MeV) Figure 2.1 Digitized Response Functions for Tungsten and PMMA Build Up Caps (Ali and Rogers 2012a) A slow variation of attenuation coefficient with energy in MeV range necessitates use of multiple attenuators with different Z. Lead was chosen because the mass attenuation coefficient changes with energy rapidly. Carbon was the second choice attenuator because the mass attenuating coefficient s opposite response as energy increases (Ali and Rogers 2012a). These two materials add spectral information eliminating degeneracy due to variation in the rate of change of their linear attenuation coefficients, as shown in Figure 2.2. A more common in clinical setting low Z attenuator, polycarbonate, was used to substitute carbon. Lead was kept as the high Z attenuating material because of its availability in the clinic. 10

23 10 1st Derivative of Linear Attenuation Coeff (cm-1) Lead 1 Carbon Energy (MeV) Figure 2.2 Response of Attenuating Materials Based on the Rate of Change of Linear Attenuation Coefficients, Calculated from NIST Values for Attenuation Coefficients of Lead and Carbon Analysis of the data collected used Equation 2.1 to extract the function of the energy spectrum, energy fluence Ψ(E). Ali and Rogers proposed a functional form for Ψ(E) which was claimed to be the best function for reliable extraction of the spectra from the transmission measurements. They were able to show that the use of two materials as free parameters that were necessary for robustness for the form when used for spectral unfolding from measure dosimetric data (Ali and Rogers 2011). Since the analytic approach taken in Ali and Rogers publications included solution of transmission equations through custom computer program, which requires high level computer skills, the same equation, was taken in this research and solved using a regularization technique in extracting the spectrum from the measured data. 11

24 Tmeasd,xi=ElEmRd,xi,EΨEdEElEmRd,0,EΨEdE (2.1) Avenues that Ali and Rogers did not explore were how the unfolded spectra compared to a modeled spectrum in a treatment planning system. Therefore this can be easily accomplished by a clinical medical physicist. This research will be a guide in helping clinical medical physicists in validating their treatment planning system energy spectra models, if applicable, with the measured spectra of their linac. This can be done with fairly easy measurement set up and accessible materials found in the clinic. Analysis of the data does not require programming skills, it only requires to have access to a spreadsheet program for data analysis. 12

25 Chapter 3 Methods & Materials 3.0 Approach The objective of this study is to unfold the energy spectrum of the primary beam of a Varian TrueBeam linear accelerator using transmission measurements. The method relies on exploiting the differences in transmission between photons of different energies as the photon beam goes through an attenuator. While the experimental setup for this type of measurement is fairly straightforward, the data analysis is not a trivial task due to the ill-conditioned nature of the posed inverse problem. Because attenuation coefficients vary relatively slow with energy in the radiation therapy energy range, it is virtually impossible to recover a spectrum by the direct solution of the attenuation system of equations. One of the recently proposed approaches by Ali and Rogers makes use of difference in attenuation properties of materials with different atomic numbers, resulting in greatly improved spectral sensitivity of the combined transmission data. Here the proposed approach of combining the data for two attenuators and two ion chamber build-up caps is further developed to allow for relatively simplified data processing. The data analysis developed here does not require programing for solving a 13

26 system of equations, but rather fully relies on regularization approach, which was implemented through setting up several spreadsheets. The analysis procedure could be repeated in a clinical setting following the prescription of this paper. 3.1 Set Up Transmission measurements were set up to approach narrow beam geometry, reducing scatter contribution to the detector signal. This was accomplished by setting the gantry and couch angles to ninety degrees; the gantry rotation decreases the scatter from the floor and walls compared to the source being at zero degrees, while the couch rotation allows for the materials to be set up along the length of the couch. The setup is schematically shown in Figure 3.1, where the distance between the primary barrier and the ion chamber was kept at least 150cm away from the wall for every set of measurements. 150cm between the ion chamber and wall minimized wall scatter contribution to the signal and allowed for enough space upstream for the third collimator and attenuating materials set up. A standard Farmer ion chamber of 0.6cc volume was used for the transmission measurements. The specifications of this chamber are shown in Figure 3.2 below. 14

27 Figure 3.1 Experimental Set Up The chamber was fixed to a stand, using vinyl set screws to hold it in place, while allowing for adjustment of its vertical position. At 222cm distance from the source using the field light and cross hairs, denoting the central axis of the primary beam, the ion chamber s sensitive volume was fixed to be visually bisected by the cross hairs. This positioning set the height of the table and height of the ion chamber. Once the ion chamber was set at a reasonable location centered on the central axis of the beam the other materials were placed on the couch upstream of the chamber. In the gantry of the linear accelerator the beam passes through a primary and secondary collimator. The secondary collimator incorporates the jaws and multi leaf collimator (MLC), which is a standard in all modern medical linear accelerators. The secondary collimator determines the field size of the beam. The combination of these components is referred to in Figure 3.1 as the first collimator. A third collimator was added to further decrease scatter and ensure only the primary beam reaches the ion chamber s sensitive volume. The third collimator comprised of a solid lead block of 5cm thickness had a 3cm diameter hole drilled through the long path length of the block. This 15

28 was placed 7cm upstream from the ion chamber. Blocks of similar dimensions of the third collimator were used to make a shield wall around the third collimator. To assure that setup is consistent between data sets, verification that the ion chamber is still on the beam s central axis and the third collimator s circumference is equidistant from the central axis was done before each measurement. The final step to the experimental set up was placing the attenuating material upstream of the ion chamber and the third collimator. The first piece of the attenuating material is set to 100cm source to surface distance (SSD), as shown in Figure 3.1. The first piece of attenuating material had a cross marking the center of the material and was used to align the material at the central axis of the beam. The field size was chosen to give maximum signal output based on the attenuating material pieces. Though the maximum output of the machine would be at 40x40cm 2 field size it is not realistic to find big enough attenuating materials to encompass the field size, furthermore the experimental set up is to achieve narrow beam geometry which requires minimal scatter photons to reach the detector. Therefore the field size was chosen to give maximum output but be small enough so the beam passed through the reasonably sized attenuating material. The lead pieces were cut into larger than 3x3cm 2 sizes allowing for a 3x3cm 2 field size to be encompassed by the attenuator. For the low attenuating material, it was more cost effective to purchase cylindrical pieces of polycarbonate. The 3cm diameter of the cylindrical pieces did not inscribe the square field size, therefore 2x2cm 2 field sizes were used for the lower attenuating material. This did not have an effect on the data, the larger field size intended to maximize the signal output. 16

29 Figure 3.2 Farmer Ionization Chamber 3.2 Equipment The transmission measurements were taken with a standard Farmer ion chamber having a 0.6cc volume. At the University of Toledo - Dana Cancer Center, this chamber is used for the monthly output quality checks. It is cross calibrated with our primary Farmer ion chamber, used for TG-51. The ion chamber connected to a CNMC Instruments Inc. Model 206 Dosimetry Electrometer, supplied with 300V. The electrometer was set to the most sensitive output and read data points in units of nanocoulombs (nc). Four different atomic number materials were used, as discussed in section 3.3. Version 7.0 of Microsoft Excel was used to analyze and unfold the spectra from the measurements. Lastly, Philip s Pinnacle Treatment Planning System s versions 9.6 and 9.8 energy spectra models for the Varian TrueBeam were being evaluated. 3.3 Attenuating Materials Following the methodology of Ali and Rogers the experimental measurements were conducted, substituting some of the materials for those easily found in standard 17

30 radiation oncology clinics. This is to develop a methodology implementable by clinical medical physicists at any radiation oncology clinic, therefore simplifying validations of treatment planning systems energy spectra models of clinical medical accelerator. The essence of Ali and Rogers proposed approach is that materials of varying atomic number (Z) respond differently at different energy ranges, as shown in Figure 3.3. The values for lead were directly taken from NIST (Hubbell & Seltzer 2009). Polycarbonate was scaled by density from polymethylmethacrylate (PMMA) material values obtained also from NIST. The graph clearly depicts the reasoning behind using a low and high Z attenuators. Lead s, the high Z material, mass attenuation coefficient varies more with energy than the lower Z attenuating material, polycarbonate, allowing to avoid degeneracies in the restored energy spectra. In deciding to use the method of two different attenuating materials, Ali and Rogers noticed that the two materials should reach a similar transmission output. To achieve the same amount of transmission it was necessary to have a longer final length of the polycarbonate relative to the lead. The data presented in the Results section below does not use the exact process as Ali and Rogers and will be further discussed in the Results section. The dataset used in this research includes twenty-four 0.317cm thick pieces of lead and ten polycarbonate pieces of 8.3cm thickness as the high attenuating and low attenuating materials, respectively. 18

31 10000 Mass Attenuation Coefficient (cm2/g) 1000 Lead 100 Polycarbonate Energy (MeV) 0.01 Figure 3.3 Mass Attenuation Coefficient Comparison between Lead and Polycarbonate (Hubbell & Seltzer 2009) 3.4 Response Function To add spectral information to the transmission measurements build up caps of different atomic number were used, in addition to the two differing attenuating materials. Theoretically a higher atomic number material should have a larger response in the higher energy ranges and the opposite is true for the lower atomic number build up cap, a larger response in the lower energy range. The buildup cap materials were PMMA and brass for the low Z and high Z material, respectively, the two materials commonly used for build-up caps in clinical settings. While a response function for PMMA was available through Ali and Rogers publication, the data for brass was obtained using Monte Carlo N-Particle Transit Code (MCNP).The response functions for both PMMA and brass used 19

32 for the 6MV beams are shown in Figures 3.4 and 3.5 below. The data is extrapolated from the MCNP simulation to values for energy bin sizes of 0.23MeV for the 6MV. While for the 10MV beam the data points were extrapolated from the simulation using 0.4MeV energy bin sizes. The findings with MCNP and the response function used for 10MV will be further discussed in the Results chapter Response Function (cm2/g) PMMA Cap Energy (MeV) Figure 3.4 Ion Chamber Response Function with PMMA Build Up (Ali & Rogers 2012) The response function depicts how the ion chamber responds at different energies when enclosed in different Z build up caps. The methodology of Ali and Rogers allows achieving better spectral information in the transmission measurements using a combination of four different materials, two cap materials and two attenuator materials. They also showed that adding more materials differing in atomic number does not 20

33 improve spectral information obtained from the measurements. Therefore this paper concentrates on using four different materials Response Function (cm2/g) Brass Cap Energy (MeV) Figure 3.5 Ion Chamber Response Function for Brass Build Up Cap (REU Program 2014, University of Toledo) 3.5 Regularization Technique Different techniques can be used in unfolding a spectrum from transmission measurements. Ali and Rogers offer a spectral functional form to tame the unfolding problem from transmission measurements, using numerical solution accomplished through a custom program. Since this is typically beyond the capabilities in terms of time and required effort for a typical clinical physicist, this paper explores a regularization technique in restoring the spectrum from experimental measurements. The regularization 21

34 technique allows to solve a system of equations for ill-conditioned problems using minimal knowledge of programing and the use of a spreadsheet. Using cells to organize the values obtained for the response function, energy bin sizes, attenuating material thickness, and measured data. Many spreadsheets allow for denoting the rows and columns of the matrix to create the system of equations which can be easily set up. Once the spreadsheet is organized in a familiar way the equations discussed below can be implemented using simple spreadsheet program equation commands. The problem setup begins with integral equation 3.1, which can be discretized as shown in Eq. 3.2, where Rα REα, Ψα ΨEα and E α is the discrete set of energies, and T i are experimental transmission values for corresponding attenuator thicknesses x i. The subscripts i and α represent the rows and columns in the matrix, ie the rows and columns of the excel spreadsheet, respectively shown in equation ElEmREe μexi TiΨ(E)dE=0 (3.1) αrαe µαxi TiΨαΔEα=0 (3.2) The steps to solving the system of linear equations through a regularization technique are shown in equations 3.3 and 3.4. These system of linear equations set up to find the spectrum from the experimental measurements. R α represents the value of the ion chamber response for different energies, µ α is the linear attenuation coefficient of the attenuating material, x i is the various attenuation thicknesses, and T i is the measured data point. The regularization technique starts with equation 3.5 when it is said that Ψαo Ψowhich is the trial function. The trial function is to have a shape that is expected 22

35 to be restored through this regularization technique. In this research the phase space files from Varian for the 6MV and 10MV beams were used to restore the spectra. A least squares approach dictates that equation 3.7 be a minimum and therefore using a smoothing technique of Phillips. The smoothing technique of Phillips serves to minimize the sum of the second derivative of the function, Ψα, and is achieved through minimization of equation 3.7 where S is the second difference vector and α is a parameter. The minimization discussed above yields equation 3.9 where C is the nxn matrix defined by equation 3.10 shown below. In summary a square matrix was created denoted by Z iα, a trial function (Ψ α ) o was chosen to compute (g i ) o. Followed by computing the vector δ from a system of linear equations. The solution to unfolding the spectrum from transmission measurements through a regularization technique is then given by equation The matrix Z iα shown in equation 3.10 was created and in doing so it had to be considered that all of these variables are functions of energy. Therefore an energy bin size had to be chosen to keep consistent throughout the analysis process, so each data point related to an energy within the range of the spectrum being restored. Since the lead attenuator allowed for 24 data points to be collected the bin size was calculated to have equal spacing of 0.23MeV to incorporate the entire energy range in a 6MV spectrum. The same had to be calculated for the 10MV spectrum, since obviously the bin sizes will be much larger in order to reach 10MeV with only 24 data points. Therefore bin sizes of 0.4MeV were used to restore the 10MV spectrum. Rαe µαxi TiΔEα Ziα (3.3) 23

36 αziαψα=0 or ZΨ=0 (3.4) ZΨo=go (3.5) Set Ψ=Ψo+δ;δ=correction yielding ZΨ=go+Zδ (3.6) x go+zδtgo+zδ (3.7) xα=go+zδtgo+zδ+αsts (3.8) ZTZ+αCδ= ZTgo+αCΨo (3.9) (3.10) Ψ=Ψo+δ (3.11) 24

37 Chapter 4 Results & Discussion 4.1 Experimental Data Transmission measurements were successfully collected for 6MV, 6MV-FFF, and 10MV photon beam energies on the Varian TrueBeam. A large amount of monitor units (MU s) were delivered to obtain a reliable reading of charge collected by the ion chamber. The MU s varied between energies; the 6MV beams only necessitated 1500MU where the 10MV beam needed 2500MU delivered to achieve a reasonable charge reading through the largest attenuator thickness. Assuming a monoenergetic beam of 6MeV and 10MeV for the 6MV, 6MV-FFF, and 10MV beams respectively, the linear attenuation coefficients µ and linear energy absorption coefficient µ en values provided by NIST were found and the broad and narrow beam attenuation curves were calculated using equation 4.1 and 4.2. Figure 4.1 below show the data collected for lead attenuator in combination with the brass and PMMA build up caps for 6MV. For comparison attenuation curves for 25

38 the broad and narrow beam attenuation are also shown. This exercise was meant to show that there is spectral information within the charge measurements that would be eventually restored into an energy spectrum. As the beam passes through the high Z material the beam is hardened filtering out the lower energies, therefore these datasets hold significant information about the beam s spectrum. The same relationship was found for all three energies for both build up caps attenuated by both attenuators. I=I0e µx (4.1) I=I0e µenx (4.2) Charge (nc) Measured Charge Broad Beam Narrow Beam Lead Thickness (cm) Figure 4.1 6MV Beam through Lead Attenuator Using Brass Build Up Cap 4.2 Response Function 26

39 The approach of Ali and Rogers requires knowledge of energy response function for ionization chamber with the buildup caps used in transmission measurement. Roughly it can be approximated by mass attenuation coefficient of the build-up cap material, but knowing the exact function, which can be obtained through Monte Carlo modeling, improves the accuracy of the restored spectra. While the response function for PMMA cap is available in Ali and Rogers publication, the brass cap response had to be obtained through MC modeling with MCNP5 package (ref. to Julianna s abstract). Different energy bin sizes had to be determined depending on the spectrum being restored in order to relate the number of measured data points to the range of energies within a spectrum. For the restoration of the 6MV beams a bin size of 0.23 MeV was used. An energy bin size of 0.4MeV was used to restore the 10MV spectrum. The result of the simulation for the brass build up cap is shown in Figure 4.2. It can be concluded that the brass build up cap did not have the expected similar response function of a tungsten build up. The high Z build up cap was to enhance the response in the higher energy ranges, but as shown in the graphs below that is not the case for brass. The brass build up cap had a higher response at the lower energy ranges. This result still added spectral differentiation to the transmission measurements. The graphs in Figure 4.3 and 4.4 show that it cannot be expected to have the same effect if only using the mass attenuation coefficient of the build up cap material instead of a response function. The use of a response function adds spectral information to the transmission measurements and is a useful and important value to have when restoring a spectrum from transmission measurements. Further discussion of possible developments will be discussed in Chapter 5, Conclusions. 27

40 Response Function (cm2/g) PMMA Response Function Brass Response Function Energy (MeV) Figure 4.2 Response Functions of Brass and PMMA Build Up Caps Response Function (cm2/g) Response Function Mass Attenuation Coefficient Energy (MeV) Figure 4.3 Response Function and Mass Attenuation Coefficient Comparison of PMMA Cap 28

41 0.6 (cm2/g) Response Function Mass Attenuation Coefficient Energy (MeV) Figure 4.4 Response Function and Mass Attenuation Coefficient Comparison of Brass Cap The resultant difference in attenuation is demonstrated in Figures For combination of two build-up caps, PMMA and brass, and two attenuators, lead and polycarbonate. Data points of the charge collected are plotted as a function of the corresponding attenuator thickness on a semi-logarithmic scale. As evident from these graphs, the effect of the combination of different buildup cap and attenuating material has different degree of influence on the measured transmission. Figures 4.11 and 4.12 show the comparison between the different build up caps used for the different attenuating materials for the 6MV beam. As shown in graph of Figure 4.11 there is a rather small response difference between the two build up caps as the thickness of lead increases. At the same time for the polycarbonate attenuator the response between different build up caps is visible for all thicknesses. This confirms that using a low Z attenuating material 29

42 adds spectral information to the transmission measurements. Figures 4.13 and 4.14 show a trend similar to that of 4.11 and 4.12, confirming the response difference between the brass and PMMA build up caps is apparent for the 6MV flattening filter free beam. Figures 4.15 and 4.16 show the 10MV beam attenuated through the lead and polycarbonate attenuators, respectively, with the different buildup caps. Figure 4.15 breaks a pattern that was seen for the lower energies, the lead attenuator does not show a response difference between the two buildup caps where the polycarbonate attenuator, shown in Figure 4.16, has a significant response difference between buildup caps. These graphs confirm that there is a difference between ion chamber response when using different atomic number build up caps, although having a build up cap made of tungsten alloy would add to the spectral information, as shown in Ali and Rogers publication. This type of cap is not typically found in clinical setting since tungsten is very difficult to machine, thus making this cap expensive. The brass cap, on the other hand, is typically used as a high-density build-up cap in small field in-air dosimetry. 30

43 Charge (nc) 0.1 Brass Cap PMMA Cap 0.01 Lead Thickness (cm) Figure 4.5 Build Up Cap Comparison through Lead Attenuator for 6MV Beam Brass Cap PMMA Cap Charge (nc) Polycarbonate Thickness (cm) Figure 4.6 Build Up Cap Comparison through Polycarbonate Attenuator for 6MV Beam 31

44 Charge (nc) 0.1 Brass Cap PMMA Cap 0.01 Lead Thickness (cm) Figure 4.7 Build Up Cap Comparison through Lead Attenuator for 6MV FFF Beam Charge (nc) 0.1 Brass Cap PMMA Cap 0.01 Polycarbonate Thickness (cm) Figure 4.8 Build Up Cap Comparison through Polycarbonate Attenuator for 6MV FFF Beam 32

45 Charge (nc) 0.1 Brass Cap PMMA Cap 0.01 Lead Thickness (cm) Figure 4.9 Build Up Cap Comparison through Lead Attenuator for 10MV Beam Charge (nc) 0.1 Brass Cap PMMA Cap 0.01 Polycarbonate Thickness (cm) Figure 4.10 Build Up Cap Comparison through Polycarbonate Attenuator for 10MV Beam 33

46 4.3 Restored Spectra The successfully unfolded spectra for 6MV, 6MV FFF, and 10MV are shown in Figures 4.11, 4.12, and 4.13 respectively. Through the regularization technique the spectrum for each energy was extracted from the transmission measurements. The spectra were obtained separately for the lead attenuator and two build up caps, which then were averaged to get Averaged Lead spectrum; similarly, the spectra extracted from polycarbonate attenuator data with both build up caps, were also averaged, resulting in Polycarbonate averaged spectrum. For comparison the Pinnacle TPS model spectrum is presented in each graph. Figure 4.11 displays the data for the 6MV energy beam. 25 Normalized Energy Fluence (MeV - 1 ) TPS Model Average of Lead Averaged Polycarbonate Energy (MeV) Figure 4.11 Restored 6MV Spectrum Compared with TPS Model 34

47 As shown in Figure 4.11 there is a noticeable difference between the restored spectra and the treatment planning system s model. Graphs in Figure 4.12 show the unfolded spectra for 6MV flattening filter free beam. This graph displays the graphs represented by the data collected from the lead attenuating material with both build up caps and the polycarbonate attenuating material with the two build up caps. The graph in Figure 4.19 shows the restored spectrum for the 10MV beam. The regularized data was combined in the same method as for the 6MV beams where the lead attenuating data with both build up caps were averaged as well as the polycarbonate attenuator Normalized Energy Fluence (MeV - 1 ) TPS Model Averaged Lead Averaged Polycarbonate Energy (MeV) Figure 4.12 Restored 6MV-FFF Spectrum Compared with TPS Model 35

48 Energy Fluence Normalized (MeV-1) Avereaged Polycarbonate TPS Model Averaged Lead Energy (MeV) Figure 4.13 Restored 10MV Spectrum Compared with TPS Model 4.4 Comparison of Model and Restored Spectra The purpose of this research was to validate Pinnacle s models of the energy spectra for the Varian TrueBeam. Pinnacle uses a polyenergetic kernel spectrum for each energy, the flaw in this is that it determines the energy bin size and therefore there is a limited amount of kernels used per spectrum. Pinnacle s set kernels forced that the unfolded data be extrapolated to these specific kernels to input the restored spectrum into the TPS. This was successfully completed. The unfolded spectrum for 6MV FFF for the averaged polycarbonate attenuator data was used to recalculate the beams for a SRS patient that received treatment of brain metastases. This was a good case to choose because of the small target volume size and multiple interfaces involved in the 36

49 attenuation of the beam before delivery of the prescription. Figure 4.13 show respectively the isodose lines of the commissioned treatment machine and the model based on spectra obtained experimentally. A quantified display of this case is shown in Table 4.1. The values were obtained from the dose volume histogram (DVH), which is calculated within the TPS and used to evaluate the quality of the plan before delivery. In addition to the brain lesion case, five lung lesion plans were investigated due to the drastic changes between tissue interfaces within the human thorax and the lung cavity itself. The largest discrepancy between the TPS model and the restored spectrum model is the max dose to the left lung for an IMRT plan treating a lesion in the right lung where the TPS model calculated the dose by 8% high. When EBRT is planned for lung lesions physicians look at the volume of the combined lungs getting maximum 20Gy, intending to keep the volume under 20cc. This value shows the necessity of a good model for calculating doses through tissue interfaces and low dose regions expanding over critical structure. With regards to coverage of the target volume, the different spectra did not have a significant effect on the coverage. And therefore it can be determined that the different energy spectra does not have a significant effect on whether the lesion is sufficiently covered by the prescription dose. 37

50 Figure 4.14 Patient 1 - Brain SRS Isodose Distribution Using Restored Spectrum Volumes Modeled spectrum Max Dose (cgy) Unfolded Spectrum Max Dose (cgy) % Diff Brainstem Lt Lens Lt Optic Nerve Optic Chiasm Rt Lens Rt Optic Nerve Lt Frontal (PTV1) Lt Frontal (PTV2) Table 4.1 Patient 1 Values Comparing Modeled Spectra versus Unfolded Spectra 38

51 Figure 4.15 Patient 2 - Right Lung Lesion Isodose Distribution Comparison between Restored and Modeled Spectra Volumes Modeled spectrum Max Dose (cgy) Unfolded Spectrum Max Dose (cgy) % Diff Bronchus Lt Lung Rt Lung Heart Spinal Cord Trachea ITV 60Gy Esophagus Table 4.2 Patient 2 Values Comparing Modeled Spectra versus Unfolded Spectra 39

52 Figure 4.16 Patient 3 -Right Lung Lesion Isodose Distribution Comparison between Restored and Modeled Spectra Volumes Modeled spectrum Max Dose (cgy) Unfolded Spectrum Max Dose (cgy) % Diff Bronchus Lt Lung Rt Lung Combined Lungs Spinal Cord LUL- ITV LUL- PTV Table 4.3 Patient 3 Values Comparing Modeled Spectra versus Unfolded Spectra 40

53 Figure 4.17 Patient 4 - Left Lung Lesion Isodose Distribution Comparison between Restored and Modeled Spectra Volumes Modeled spectrum Max Dose (cgy) Unfolded Spectrum Max Dose (cgy) % Diff Lt Lung Rt Lung Spinal Cord Combined Lung PTV PTV Table 4.4 Patient 4 Values Comparing Modeled Spectra versus Unfolded Spectra 41

54 Figure 4.18 Patient 5 Left Lung Lesion Isodose Distribution Comparison between Restored and Modeled Spectra Volumes Modeled spectrum Max Dose (cgy) Unfolded Spectrum Max Dose (cgy) % Diff Lt Lung Rt Lung Spinal Cord Combined Lung Lt Lung PTV PET ITV Table 4.5 Patient 5 Values Comparing Modeled Spectra versus Unfolded Spectra 42

55 Figure 4.19 Patient 6 Left Lung Lesion Isodose Distribution Comparison between Restored and Modeled Spectra Volumes Modeled spectrum Max Dose (cgy) Unfolded Spectrum Max Dose (cgy) % Diff Lt Lung Rt Lung Spinal Cord Combined Lung ITV PTV Table 4.6 Patient 6 Values Comparing Modeled Spectra versus Unfolded Spectra 43

56 While we did not find a profound difference in dose calculation results despite fairly significant difference between TPS modeled and experimentally measured spectra, case by case evaluation is warranted for plans involving high heterogeneities next to small critical structures. Scatter contribution to calculated dose would be the most significant under the stated circumstances. Pinnacle treatment planning system uses the model created from commissioning measurements to calculate the percent depth dose (PDD) curves for each energy beam. With this in mind the PDD curves of the 6MV-FFF beam between the TPS model and restored spectrum were evaluated. As shown in Figure 4.20 and Table 4.7 there are not significant difference between the two PDD s, but show differences at depths at which the beam quality is determined. In Figure 4.19 the difference curve is plotted alongside the two PDD s and shows that the restore spectrum is harder at higher energies while the TPS model based PDD is harder at the lower energies. 44

57 Percent Depth Dose TPS Model Restored Difference Percent Difference Depth (cm) Figure 4.20 Percent Depth Dose Curve Comparison between TPS Model and Restored Spectrum for 6MV-FFF beam Depth (cm) TPS Model PDD Restored Model PDD % Diff Table 4.7 Differences between TPS Model PDD and Restored Model PDD for 6MV-FFF beam 45

58 Chapter 5 Conclusions In conclusion, the 6MV, 6MV-FFF, and 10MV energy spectra of the Varian TrueBeam linac were obtained through a physics-based approach for unfolding megavoltage bremsstrahlung spectra using transmission analysis. We followed the general methodology proposed by Ali and Rogers (Ali & Rogers 2012a), combining transmission data through two attenuators with two ionization chamber build-up caps of substantially different atomic numbers. The acquired transmission data were successfully processed with a regularization technique, applied to solving an ill-conditioned inverse problem of spectrum restoration. In fulfilling the objective of adapting the general methodology to a realistic clinical setting, we made the following changes to the ideal measurement setup. Low Z attenuating material, carbon, was replaced with PMMA having similar Z but lower density. Additionally, one of the build-up cap materials, tungsten alloy, which is not typically found in a standard radiation oncology clinic, was replaced with a commonly used brass alloy, available to most clinical medical physicist. While such a substitution does not result in as strong change of ion chamber energy response function as that due to the tungsten alloy, this was compensated using an alternative data processing algorithm 46

59 of regularization. The data analysis with regularization technique developed here has an further advantage of using only a spreadsheet software such as Microsoft Excel, and does not require advanced programming skills. The experimentally obtained spectra were directly compared to those of Pinnacle Treatment Planning System models. The spectra were visually different for all three energies evaluated here. To check the sensitivity of measured data to changes in spectrum, we evaluated differences between the PDD curves based on the TPS model (corresponding to commissioning PDD) and the restored 6FFF beam spectrum. These differences were small for the most depths, reaching up to 6-8% only in the regions close to the surface and at largest depth of 35 cm. To assess the effect of spectrum on the treatment planning, an additional validation test was performed for the 6FFF energy beam. The spectrum modeled by TPS was substituted by the restored spectrum with an objective of calculating dose distribution in several representative treatment plans. The differences between the model and unfolded spectra were evaluated through isodose distributions, and quantified in terms of maximum dose values for critical structures. The resultant differences between the scatter doses delivered to critical structures near the target volume were mostly negligibly small. In several cases, however, isodose distributions were visually different and maximum dose to critical structures differed by ~4 to 8%. Since spectrum information is used by TPS in calculation of the scatter dose, the anatomical sites getting large scatter contributions are the most vulnerable to inaccuracies in the modeled spectrum. 47

60 A complete transmission measurement and data processing approach developed here was adapted for easy implementation by clinical medical physicists aiming at validation of their treatment planning system beam models. With the use of in house equipment, and following description of spectra unfolding through the regularization technique, clinics across the country can measure and restore the energy beam spectra of their treatment linac. This research can be further extended by improving the accuracy of the data analysis through finding the response functions of the ion chamber more accurately. This could be achieved by Monte Carlo modeling of the exact geometry of the ion chamber with build-up caps. In our study response functions were obtained by MC simulations of a simple cylindrical geometry of an ion chamber with dimensions of PMMA and brass caps used for transmission measurements. Ultimately, utilization of materials readily available in clinical settings, such as polycarbonate in place of pure carbon attenuator, and brass instead of tungsten alloy for a build-up cap, resulted in some compromise between the accuracy and ease of implementation of the approach used here: polycarbonate does not have as high density as carbon, and use of brass build-up cap did not result in as drastic change in ion chamber response function. In a situation when more advanced research equipment or additional funds for purchasing specialized materials are available, the restored spectra can be made more accurate. 48

61 References A. Ahnesjo: Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media. Medical Physics 16, 577 (1989). A. Ahnesjo and P. Andreo: Determination of effective bremsstrahlung spectra and electron contamination for photon dose calculations (1989). Armburster, Russell, and Kuehl: Spectrum Reconstruction from Dose Measurements as a Linear Inverse Problem. B.A. Faddegon, C.K. Ross, and D. W. O. Rogers: Angular distribution of bremsstrahlung from 15MeV electrons incident on thick targets of Be, Al, and Pb. Medical Physics 18, 727 (1991). Bushberg, J. (2002). The Essential Physics of Medical Imaging. Philadephia: Lippincott Williams & Wilkins. B. sheikh-bagheri and D.W. O. Rogers: Monte Carlo calculation of nine megavoltage photon beam spectra using the BEAM cod, Medical Physics 29, 391 (2002). D. J. Landry and D. W. Anderson: Measurement of accelerator bremsstrahlung spectra with a high-efficiency Ge detector, Medical Physics 18, 527 (1991). E. S. M. Ali and D. W. O. Rogers: An improved physics-based approach for unfolding megavoltage bremsstrahlung spectra using transmission analysis, (2012a). E.S.M. Ali, M. R. McEwen, and D. W. O. Rogers: Detailed high-accuracy megavoltage transmission measurements: A sensitive experimental benchmark of EGSnrc, Medical Physics, 39(10), (2012b). 49

62 E. S. M. Ali and D. W. O Rogers: Functional forms for photon spectra of clinical linacs, Physics in Medicine and Biology 57, (2011). E.S.M. Ali, M. R. McEwen, and D. W. O. Rogers: Report CLRP Data for an accurate transmission measurement benchmark, (2012c).Flickinger, J., & Kondziolka, D. A Multi-Institutional Experience with Stereotactic Radiosurgery for Solitary Brain Metastasis. International Journal of Radiation Oncology, Biology, and Physics, 28, Retrieved July 22, 2014, from Hasenbalg, Neuenshwander, Mini, and Born. Collapsed cone covolution and analytical anisotropic algorithm dose calculations compared to VMC++ Monte Carlo simulations in clinical cases. Physics in Medicine and Biology, 52, (2007). Hubbell, J., & Seltzer, S. (2009, September 17). Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 kev to 20 MeV for Elements Z = 1 to 92 and 48 Additional Substances of Dosimetric Interest. Retrieved September 22, Schulz, M. The Supervoltage Store. The American Journal of Roentgenology Radium Therapy and Nuclear Medicine, 124, Sandifer, C., & Taherzadeh, M. (1968). Measurement of Linac Thick-Target Bremsstrahlung Spectra Using a Large NaI Scintillation Spectrometer. Nuclear Science, 15(6), Retrieved August 20, 2014, from Starkschall, Steadham, Popple, Ahmad, and Rosen: Beam-commissioning methodology for a three-dimensional convolution/superposition photon dose algorithm. Journal of Applied Clinical Medical Physics, 1(2), 8-28 (1999). Thwaites, D., & Tuohy, J. (2006). Back to the future: The history and development of the clinical linear accelerator. Physics in Medicine and Biology, (51),

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64 Appendix A Regularization of Transmission Data This appendix will take the reader through step-by-step of how to set up an organized spreadsheet to easily use the regularization technique to unfold an energy spectrum from transmission measurements made using two attenuating materials and two build up cap materials of different atomic numbers. Figure A.1 To begin setting up the matrices that will be used in restoring the spectrum all of the data necessary to begin the initial setup will be most useful organized in a way that you, the user is comfortable with accessing the information. This the sample of the spreadsheet above the values inside the red box are the organized by energy bin size, linear attenuation coefficient of the attenuating material corresponding with the energy bin, the thickness of the attenuating material which also corresponds with specific energy 52

65 bin, the measured data points (normalized to the open field not attenuated charge measurement) that are corresponding to the attenuator thickness, and the response function of the build up cap which was used in measuring the transmission data. The linear attenuation coefficient is to denote the columns of the matrix and the thickness of attenuating material denotes the rows displayed in red in the figure above. The second step to creating the matrix is to take the values calculated with the exponential term and subtracting the measured data from the exponential calculated values shown in the figure below. Organize the measured data in one column in an orderly fashion where the equation can be easily executed to create the second matrix. Notice that when using the equation to subtract the measurement from the original matrix value a sample of what this would look like in a spread sheet is shown in a blue box in the top left hand corner of the figure above. These similar steps are taken until you reach your final matrix with the equation that represents matrix Z. This matrix can then be manipulated using the regularization technique discussed in this paper, and a trial 53