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Characterization of small high energy proton beams in homogenous and heterogenous media Charland, Paule 1999

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CHARACTERIZATION OF SMALL HIGH ENERGY PHOTON BEAMS IN HOMOGENEOUS AND HETEROGENEOUS MEDIA. By Paule Charland B. Sc. (Physics) Universite de Montreal M . Sc. (Medical Radiation Physics) McGi l l University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE' REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1999 © Paule Charland, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not. be allowed without my written permission. Physics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: // /WX 99 Abstract This thesis advances the study of small high energetic photon fields in radiotherapy. Small photon field irradiation is aimed at delivering a uniform dose to a well defined target while minimizing the dose to the surrounding normal tissue. The dosimetry of small x-ray fields is complicated by two factors: the relationship between detector size and field dimensions and the lack of equilibrium in lateral charged particles. Additionally, a longitudinal charged particle disequilibrium is present when materials with different atomic composition and density than water are introduced in a water-like phantom. , Small radiation dosimeters such as diamond, diodes film and a mini-ion chamber have a better spatial resolution to detect the steep dose fall-off at the edge of small photon fields than the large Markus chamber. The line spread function (LSF) of the film densitometer can be estimated by simple measurement of a slit image. Deconvolution of the measured, beam profile from a linear accelerator (linac) with the LSF of a detector yields an estimate of the true inherent beam profile of the linac. Conversely, the LSF of any detector can be estimated by deconvolution from measured data once the inherent profile is known. Similarly, a blurring function representing the finite source size effect of the head of the linac which is missing in a Monte Carlo simulation can be obtained. Because the deconvolution process is highly sensitive to noise, the Total Least Squares (TLS) approach offers a reasonable means to overcome this problem. To deal with inhomogeneous media, the density scaling theorem has been modified to incorporate the effect of a change in atomic number of a material. This modified scaling found an application in the convolution-superposition dose model and provided better agreement with the Monte Carlo generated data. ii The idea of electronic disequilibrium has been taken into account in our simple depth dose model. A prototype second order differential equation allowed energy to be carried away, analogous to the notion of electron range, and hence we were able to simulate a build-up region for the depth dose curve as well as inhomogeneities. iii Table of Contents Abstract List of Tables viii List of Figures ix Acknowledgement xvi Dedication xvii 1 Introduction: Small photon fields in radiotherapy 1 1.1 Stereotactic radiosurgery 1 1.1.1 History . 4 1.1.2 Requirements . .... •. 6 1.2 Dosimetry of narrow photon beams 6 1.3 Objectives 7 1.4 Overview 7 2 Theory 9 2.1 Dose and Kerma . 9 .2.1.1 Charged Particle Equilibrium 10 2.2 Beam Data . . . . 12 iv 2.2.1 Percentage depth dose . 12 2.2.2 Beam profile and off-axis ratio . . 14 2.2.3. Relative output factor ' 16 2.3 Dosimetry techniques . . . 17 2.3.1 Radiation detector size effects . • • • 18 2.4 Dose models • 21 2.4.1 Proposed simplistic depth dose model 26 2.5 Inhomogeneity corrections 27 2.5.1 Scaling theorem 28 2.5.2 Mass stopping power 29 2.5.3 Density effect 29 2.5.4 Continuous-slowing-down range 30 2.5.5 Mass scattering power • • • • • • • • • • 30 2.5.6 Inhomogeneity correction factors 31 2.5.7 Limitations of the scaling theorem 31 2.6 Summary 32 3 Materials and Methods 34 3.1 Materials 34 3.1.1 Linear accelerator • • • • 34 3.1.2 Dosimetry phantoms 37 3.1.3 Radiation detectors .38 . 3.2 Methods . . . . . . . . . . . . . . . . . . . . . . 39 v 3.2.1 Deconvolution 39 3.2.2 Computer simulation •. • • • 43 3.2.3 Densitometer LSF 44 3.2.4 Monte Carlo Simulation . . . 45 3.2.5 Scaling approach and radiological pathlength . 46 3.2.6 Solution of the depth dose model with oscillators 49 4 Results and Discussion 50 4.1 Part 1: Detector resolution 50 4.1.1 Beam profile and depth dose measurements 51 4.2 Part 2: Deconvolution 59 4.2.1 Simulation results . 59 4.2.2 Experimental implementation 64 4.3 Part 3: Scaling in inhomogeneous media : 75 4.3.1 Dose measurement in heterogeneous phantoms 75 4.3.2 Properties of different media 79 4.3.3 Dose calculations using various scalings 83 4.3.4 Discussion 90 4.4 Part 4: Simple dose model 92 • 4.4.1 Percentage Depth Dose in a homogeneous medium 92 4.4.2 Percentage Depth Dose in a heterogeneous medium 92 4.4.3 Comparison of calculations to measurements (Monte Carlo) . . . 93 4.4.4 Discussion . 96 vi 5 Conclusion 5.1 Conclusions 5.2 Future directions Bibliography Appendices A Historical Background B Spring model: solution L i s t o f T a b l e s 3.1 The properties of various phantom materials are given. The electron and physical densities (pe and p respectively) are relative to water and Z is the effective atomic number. The properties of bone are added for comparison. 37 3.2 The physical properties of detectors used in the measurement of dose for SRS fields are given • 38 4.1 Penumbra widths for a 3 x 3 cm2 6 MV EGS4 profile (raw data) convolved with different LSFs and profile widths measured with diode, film, Markus and IC10 ion chamber are given. The FWHM of the corresponding detec-tor LSF are also shown. R is the detector radius. . . . . . . . . . . . . . . 66 4.2 The properties of various phantom materials are given. The electron and physical densities (pe and p respectively) are relative to water and Zeff is the effective atomic number. The ratio of the average stopping powers relative to water are given for the mean energy of the 6 MV (1.91 MeV) • and 10 MV (2.91 MeV) photon spectra. 83 4.3 Depths of maximum dose (dmax) and depths in the fall-off regions for 80, '50 and 20% of the maximum dose are given for different selections of the parameters cn, UQ,. and f3. The number of oscillators •(TV + 1) is 50, the "time" of integration was 30 and the amplitude of the photon beam was 21. All the parameters are in arbitrary units in this table. 93 4.4 Fitting parameters. . 96 viii List of Figures 1.1 LinaC-based stereotactic radiosurgery. The stereotactic frame is attached to the patient's head (Adapted from [99], with permission.) 3 1.2 Overview of the thesis work . . . . . . . . 1 . . . 8 2.1 The concept of Charged Particle Equilibrium is illustrated. Electron ex enters the volume u with a kinetic energy Ek equal to that carried out by electron e2 • • 11 2.2 A typical photon beam depth dose curve is shown 13 2.3 A photon beam Tissue Maximum Ratio (TMR) measurement set up is shown. ; 14 2.4. A photon beam profile measurement set up is shown 15 2.5 A typical photon beam profile is shown for a given depth d •. . . 15 2.6 Electronic nonequilibrium at the edges of a high energy photon beam1. Point a receives electron tracks coming from all directions. Point b is inside the photon beam but receives fewer electron tracks than does a. Point c is outside of the photon beam but still receives some electron tracks. 16 2.7 A diagram of a point of dose calculation Q at a depth d for radiosurgerjf planning is shown 23 2.8 Illustration of the vectorial framework for the convolution/superposition method. The dose deposition occurs at a distance r — r' away from the radiation interaction site 24 ix 2.9 Chain of oscillators 27 3.1 A medical linear accelerator (linac) and its rotational and translational degrees of freedom are illustrated. The gantry rotates (6Q) around an axis located at the isbcenter. The couch rotates (6T) and has a translational motion in three dimensions. The collimator also rotates (6 c) about an axis centered at the isocenter (Adapted from [57], with permission.) . . . 35 3.2 The head assembly of a linear accelerator in photon beam mode equipped with a stereotactic collimator is shown. The electrons strike a target and produce bremsstrahlung photons which are forward peaked in intensity. These photons first go through a primary collimator and then pass through a flattening filter- to produce a uniform radiation distribution. Futher, . secondary and tertiary (stereotactic) collimators shape the photon beam (Adapted from [57], with permission.) . . 36 3.3 An illustration of the TLS and the LS methods for n — 1. The dashed LS lines are parallel to the b-axis while the solid TLS lines are perpendicular to the line az = b. . . . 42 3.4 The computer-simulated inherent profile generated from the convolution of a step function and a semi-elliptic distribution is shown along with these two generating functions 44 3.5 The electronic scattering power (r/p x A/Z).relative to Hydrogen is shown. 47 3.6 The energy deposition kernel is being modified to account for the change in scattering and stopping powers. 49 4.1 Mass collision stopping power ratios relative to water are shown for differ-ent media as a function of electron energy. 52 x 4.2 Variations of the mean energies of the photon and electron spectra with field sizes calculated in polystyrene at a depth of 2.5 cm with EGS4 for a 10 M V photon beam are shown 53 4.3 One half of the 6 M V beam profile is shown for a 3.0 cm diameter stereo-tactic applicator with 5 x 5 cm 2 primary collimator setting at 100 cm SSD and depth of maximum dose (1.6 cm). The dose profiles.are measured with the diamond, diode, film, and MINI chamber and are normalized to the 100% dose on central axis. 54 4.4 One half of the 10 M V beam profile is shown for a 1.0 cm diameter stereo-tactic applicator with 4 x 4 cm 2 primary collimator setting at 100 cm SSD and depth of maximum dose (1.92 cm). The dose profiles are mea-sured with the diamond, diode, film, Markus and MINI chambers and are normalized to the 100% dose on central axis 55 4.5 Percentage depth doses measure with diamond, diode, and mini chamber for 10 M V photon beams using a 1 cm diameter stereotactic applicator is shown for (a) a 4 x 4 cm 2 primary field size, and b) a 5 x 5 cm 2 field size. The source surface distance is 100 cm 56 4.6 One half of the 6 M V beam profile is shown for a 3 x 3 cm 2 field size at a 100 cm SSD and depth of maximum dose (1.5 cm). The dose profiles are measured with the diode, film, Markus and IC10 chambers and are normalized to the 100% dose on central axis. A n EGS4-calculated profile (raw data uncorrected for source size) is also shown for comparison. . . . 58 4.7 One half of the computer-simulated inherent and measured profiles are shown. The semi-elliptic distribution is normalized to 100 relative intensity on this graph 60 xi 4.8 The R M S error between the original semi-elliptic distribution of radius R = 6 units and the LSF recovered by deconvolution is plotted against the rank of the augmented matrix E for different levels of Gaussian noise ov : : . . . 4.9 The L S F (R = 6 units) recovered by deconvolution and T L S in the pres-ence of Gaussian noise av = 0.05 is shown for the ranks 512 and 350 (a and b respectively) of the augmented matrix E. In c) the original LSF is compared to the recovered one (Rank= 80) 4.10 6 M V beam profiles in water at a depth of 1.5 cm and for field size 3 x 3 cm 2 at 100 cm SSD are shown, a) Two profiles are generated by the convolution of the EGS4 simulated profile with the semi-elliptic distribution of radius R = 3.0 mm and 4.6 mm and are compared to the raw EGS4 profile and the one measured by the IC10 ion chamber, b.) Measured and EGS4-generated profiles are shown. The left side of the profiles is illustrated and these are normalized to 100% on central axis. (Note that the central axis position (0) is not displayed.) 4.11 The LSF as determined by direct measurement. of a slit image for the Wellhofer film densitometer and according to a step function of width 2R = 1.2 mm are shown 4.12 The realistic inherent profile as solved by deconvolving the film data with the L S F of the densitometer is presented along with the raw EGS4 pro-file, the film measured profile and the reconvolved profile generated by convolving the realistic inherent profile with the LSF of the densitometer. 4.13 The finite source size effect correction to the EGS4 data as solved with the deconvolution and T L S method is shown xii 4.14 The EGS4 profile corrected for finite source size is illustrated as well as the original raw EGS4 data and the realistic inherent profile of the Linac. 4.15 a) The LSF of the Markus chamber as solved with the deconvolution and . TLS method is shown for the. ranks 429 and 259 (a and b respectively) and in c) the peak of the LSF for a rank 101 of the augmented matrix E. 4.16 The profile generated by convolving the realistic inherent profile with the LSF of the Markus chamber as solved by deconvolution and TLS is com-pared with the profile as measured with the. Markus chamber and the realistic inherent profile. . 4.17 The experimental set up for dose measurements at the distal interface of a heterogeneous phantom is shown. The heterogeneous slab is located at a depth d and the SSD is 100 cm. The dosimeter is positioned at a depth D 4.18 Dose measurements at the distal interface of a heterogeneous phantom are shown. The heterogeneous slab is located at a depth of 3.5 cm and the SSD is 100 cm. The ratio of the dose in the heterogeneous phantom to the dose at the same position in the homogeneous phantom is taken (Inhomogeneity correction factor or ICF) 4.19 The 10 MV photon beam profile measured at a depth of 4.75 cm in a homogeneous polystyrene phantom is compared to that measured in an inhomogeneous phantom. In the inhomogeneous case, a 1.25 cm aluminum slab is located at a depth of 3.5 cm. The field diameter is 3.0 cm at a SSD of 100 cm. The doses are normalized to the dose on the central axis of the homogeneous phantom. . xiii 4.20 The 10 M V photon beam profile measured at a depth of 4.75 cm in a homogeneous polystyrene phantom is compared to that measured in an inhomogeneous phantom. In the inhomogeneous case, a 1.25 cm balsa slab is located at a depth of 3.5 cm. The field diameter is 3.0 cm at a SSD of 100 cm. The doses are normalized to the dose on the central axis of the homogeneous phantom • • • 80 4.21 The ratios of mass scattering powers ( ( r /p)w e ^ m ) a r e shown for various materials. 81 4.22 The ratio of the continuous slowing down approximation (CSDA) ranges (electrons/cm2) relative to water is shown for various materials 82 4.23 Depth dose curves for a 6 MV photon beam are calculated by Monte Carlo simulation and convolution/superposition in polystyrene. The ho-mogeneous case and the inhomogeneous case (1 cm aluminum slab at a depth of 4 cm) are illustrated in a). The field size is 3 x 3 cm2 at a SSD of 100 cm. In b) the rectilinear scaling is illustrated and compared to 3 other types of scaling and to measurements. . 85 4.24 Depth dose curves for a 6 M V photon beam are calculated by Monte Carlo simulation and convolution/superposition in polystyrene. The ho-mogeneous case and the inhomogeneous case (1.25 cm air gap at a depth of 3.5 cm) are illustrated in a). The field diameter is 3.0 cm at a SSD of 100 cm. In b) the rectilinear scaling is illustrated and compared to 3 other types of scaling . . 87 x i v 4.25 Depth dose curves for a 10 M V photon beam are calculated by Monte Carlo simulation and convolution/superposition in polystyrene. The ho-mogeneous case and the inhomogeneous case (0.6 cm air gap at a depth of 2.6 cm) are illustrated in a). The field diameter is 1.0 cm at a SSD of 100 cm. In b) the rectilinear scaling is illustrated and compared to 3 other types of scaling 88 4.26 Depth dose curves for a 10 M V photon beam are calculated by Monte Carlo simulation and convolution/superposition in a homogeneous polystyrene phantom. The heterogeneous phantom contains a 1.25 cm balsa slab at a depth of 3.5 cm. The field diameter is 3 cm at a SSD of 100 cm 89 4.27 "PDDs" in homogeneous media generated by the spring model. The total number of nodes (TV +1) is 50, the damping coefficient cn is 1 everywhere and the attenuation of the photon beam j3 is l/(N + 1). The units are arbitrary for these curves 94 4.28 "PDDs" are shown for an inhomogeneity generated by the spring model. The total number of nodes (JV + 1) is 50, the damping coefficient c„ is 1 everywhere except in the slab (n = 24 to 26) where it is 2 and the attenuation of the photon beam (3 is 1/(N.+ 1). The units are arbitrary for these curves 94 4.29 EGS4-simulated percent depth dose of a 10 M V photon beam of 1 cm diameter at SSD=100 cm in polystyrene is shown. The homogeneous case and cases with an air gap of thicknesses of 0.3, 0.6, 1.0, and 1.3 cm at a depth of 2.6 cm are illustrated. 95 4.30 The 10 M V photon beam in homogeneous media generated by the spring model and as calculated by Monte Carlo is shown 97 xv 4.31 Depth dose for a 10 MV photon as generated by the spring model mm air gap is present at a depth beam in heterogeneous media is shown and as calculated by Monte Carlo. A 3 of 2.6 cm xvi Acknowledgement I would like to acknowledge the continuous support in all matters given by Dr. Ellen El-Khatib whom I was privileged to have as a supervisor. I also would like to acknowledge the valuable insights provided by the members of my committee: Dr. Sherali Hussein, Dr. Edward Auld, Dr. Micheal McKenzie and Dr. Alex MacKay. A special thanks goes to my colleague and friend Siobhan Ozard for many helpful discussions. My final and special thanks goes to my parents for their continuous mental support and affection. xvii Dedication To my parents xviii Chapter 1 Introduction: Small photon fields in radiotherapy In this thesis the dosimetry of small photon field irradiation is investigated. The in-terest in and use of very small photon field irradiation to treat head lesions has grown. While conventional radiotherapy of malignant brain lesions has been performed with stan-dard radiotherapy equipment, relatively large field sizes and standard dose fractionation regimes, a recent technique termed stereotactic radiosurgery (SRS) has evolved [70, 99]. This technique typically consists of high dose brain irradiation in a single fraction using narrow photon beams. However there is increasing interest in its use to treat malignant brain tumors with a fractionated regimen (stereotactic radiotherapy or SRT). Another active area of research is in conformal therapy where the beams are conformed to the tar-get volume to minimize the treatment volume, and this often requires irregularly shaped small fields. A brief overview of the treatment phase for stereotactic radiosurgery along with the history of the procedure are presented in the following sections. A discussion of the dosimetry of small photon fields follows. And the remainder of this chapter consists of an overview of the research work. 1.1 Stereotactic radiosurgery Stereotactic radiosurgery is a treatment modality which is based on the ability to deliver high radiation doses to small intracranial lesions by means of external irradiation [99]. A beam of ionizing radiation is used, usually in a single fraction, as a surgical tool to destroy tissue. The entrance and exit doses are distributed so that tissue outside the 1 Chapter 1. Introduction: Small photon Gelds in radiotherapy 2 lesion is minimally irradiated. The root 'stereo' in stereotaxy implies a knowledge or perception of 3-D structure. Stereotactic techniques imply localization of intracranial targets with high accuracy [39, 40]. Treatment planning for stereotactic radiosurgery is fundamentally a 3-dimensional task, and requires accurate determination of the target volume and its spatial relationship to nearby critical structures in the brain [13, 121, 64]. A l l the radiosurgical techniques rely on stereotactic frames for target localization, treatment set-up and patient immobilization during the treatment (Figure 1.1). A popular stereotactic frame is the Brown-Roberts-Wells (BRW) 1 frame which is compatible with Computed Tomography (CT), Magnetic Resonance (MR) and radiographic imaging modalities. These systems employ fiducial markers which serve as identifiable landmarks in the acquired images. Localization is the process of finding the coordinates of the lesion in a stereotactic reference system. A 3-dimensional treatment planning system is used for the calculation of the 3-dimensional isodose distribution in conjunction with the stereotactic frame and its associated target localization software [82, 40]. The patient's anatomical information is taken into account by using C T and M R images prior to irradiation [86, 83]. Digital Subtraction Angiography (DSA) is generally used to localize the target in arteriovenous malformation (AVM) treatment. For linac-based stereotactic radiosurgery an isocen-trically mounted linear accelerator (linac) is used [37, 98, 35, 21, 89]. Target volumes treated by radiosurgery are usually assumed to be spherically symmetrical. Circular fields are obtained by using cylindrical collimators with circular openings placed onto the accessory tray holder of the linac (Figure 1.1). The small circular radiation beams have sizes of the order of 10 to 40 mm in diameter. Isocenter location, gantry arc rotation interval, couch angle, collimator field size, and dose are adjustable treatment parameters [119, 50, 19, 101]. 1 Brown-Roberts-Wells (BRW) frame, Radionics, Inc., Burlington MA. Chapter 1. Introduction: Small photon Gelds in radiotherapy 3 Figure 1.1: Linac-based stereotactic radiosurgery. The stereotactic frame is attached to the patient's head (Adapted from [99], with permission.) Stereotactic radiosurgery is an attractive therapy because it is a non-invasive proce-dure (in comparison with normal surgery)- and efficacious for treating lesions in inaccessi-ble regions of the brain. Its usefulness has been proven in the treatment of non-malignant brain diseases such as arteriovenous malformation (AVM) [63]. It can also be used for treatment of small tumors and is. being extended to treat lesions other than intracra-nial lesions [66]. For malignant tumors there is a preference to use fractionation (SRT) [25, 14]. Efforts are under way to design collimator systems which conform to the ir-regular shape of the lesions to be treated [12, 102]. Another possibility is to obtain the conformal collimation through dynamic field shaping [75, 59]. Chapter 1. Introduction: Small photon fields in radiotherapy 4 1.1.1 History Stereotaxy and radiotherapy were developed independently of the other. Stereotaxy has its roots in experimental research and clinical neurosurgery prior to radiosurgery. Robert Clarke in 1906 conceived of the notion of applying a 3-D coordinate system based on skull landmarks to the study of the brain [11]. He had realized the need for an apparatus that could be fixed to the cranium, permitting a greater certainty in the positioning of electrodes within deep brain structures. The first human stereotactic operation was reported in 1947 by Spiegel et al. [109]. The apparatus called a stereotactic frame has seen various modifications over the years. In 1949, the Swedish neurosurgeon Lars Leksell presented the first stereotactic apparatus that employed an arc attachment allowing angular adjustments to be made [61]. The discoveries of x-rays (1895) and radioactivity (1896) were rapidly followed by the application of ionizing radiations to the diagnosis and treatment of disease [8]. The first three-dimensional treatment of a brain lesion with a megavoltage unit took place in 1948 [49]. In 1951, Leksell introduced the idea of stereotactic radiosurgery that is to combine the use of an x-ray unit with a stereotactic frame [60]. Initially, the stereotactic frame that he had previously described and a special mobile collimator, attached to an x-ray tube, that circumscribed an arc over the head were used. Radiosurgery was initially performed with 200-300 kVp x-rays. In the late 1950's, high energy heavy charged particle beam therapy started. These beams are obtained from a cyclotron or synchrocyclotron. Leksell used protons. The advantages of using charged particle beams in therapy is that the physical dose distribution can be more sharply defined at depth thus sparing surrounding tissues. This much better distribution can be achieved because of the Bragg peak of dose absorption [85]. After protons, helium ions and heavier ion beams were also considered [99]. In 1968, Leksell introduced radiosurgery with multiple cobalt-60 sources [62]. The unit originally employed 179 cobalt-60 sources producing 179 Chapter 1. Introduction: Small photon Gelds in radiotherapy 5 converging beams. A similar unit is marketed today as a gamma knife [30, 29]. In 1972, computed tomography (CT) was introduced by Hounsfield et al. [43]. With the advent of CT it became possible to obtain tissue density and attenuation information in addition to accurate contours of internal organs. This then created an incentive for the development of more accurate dose calculation methods, in radiotherapy. In 1974, Larsson et al. proposed linacs as a new less expensive alternative for performing radio-surgery [58]. The recent development of linear accelerator-based radiosurgical techniques has increased the accessibility of radiosurgery. In 1979, Brown presented a prototype system in which a modified stereotactic frame served as the reference coordinate system during both imaging and surgery [15]. He was the first to develop a CT-compatible frame which allowed target localization to be determined using a computer. Once it was shown that computers could be used in image analysis and target localization, several groups concentrated their efforts on developing software systems and instrumentation that was compatible not only with radiographic techniques, but also with tomographic imaging modalities. These new systems have since been used in planning stereotactic procedures that include: radiosurgery, biopsies, electrode implantation and interstitial brachyther-apy [86, 16, 38, 81, 83]. With the evolution of three-dimensional medical imaging and the increasing complexity of the,radiosurgical treatments such as conformal techniques, treatment planning and dosimetric measurements have to keep up in achieving greater accuracy. This leads to the development of more sophisticated photon and electron dose calculation algorithms (see Chapter 2 for a review of existing dose models). The ideal dose model would allow fast accurate three-dimensional dose prediction in homogeneous as well as heterogeneous irradiated media. Moreover, dose models rely on beam charac-teristics which can be obtained empirically, analytically or numerically. Chapter 1. Introduction: Small photon fields in radiotherapy 6 1.1.2 Requirements Some factors to be considered for dose delivery in stereotactic radiosurgery are the follow-ing [88]: high spatial and numerical accuracy of dose delivery to the target, -knowledge of dose inside the target volume, steep dose fall-off outside the target volume, treat-ment accomplished in a reasonable amount of time, low skin dose and low eye lens dose, low or negligible scatter and leakage to radiosensitive organs and reasonable cost of the equipment. The spatial accuracy of dose delivery is affected by two components: the accuracy of the determination of the target coordinates and the accuracy of the subsequent delivery of the dose to the pre-determined target. The accuracy of target localization that can be achieved when DSA is used in conjunction with C T and M R I is within ± 1 mm. A n ideal radiosurgical technique would concentrate the dose (within ± 2 % of numerical accuracy) in the target volume and give a steep and isotropic dose fall-off outside the target volume (from 90% to 10% on the order of mm). These beams have dose gradients greater than 20% per millimeter in the target area and the isodose calculations require high spatial resolution (1-2 mm). Doses delivered in a single fraction are of the order of 10-30 Gray, so that small errors in dose calculation will be significant. [4, 87] 1.2 Dosimetry of narrow photon beams Very small collimated photon beams are used in radiosurgery. In narrow beams as well as at small depths the absorbed dose changes rapidly with beam radius and depth in phantom. This is attributed to the absence of both lateral and longitudinal electronic equilibrium in radiation fields of dimensions smaller than the maximum range of sec-ondary electrons2. These fields are referred to as small fields. For these very small fields 2These notions are described in Chapter 2. Chapter 1. Introduction: Small photon Gelds in radiotherapy 7 the size of the detector may perturb the signal and hence, the dosimetry of small x-ray fields is complicated by two factors: the relationship between detector size and field dimensions and the lack of equilibrium in lateral charged particles. Therefore the mea-surement and calculation of dose for very small photon fields used in radiotherapy is of great scientific and clinical relevance. 1.3 Objectives The main requirement of a system for SRS is a high efficiency in confining radiation to the lesion. The consequences of error can be disastrous. This work deals with find-ing accurate methods to characterize the radiosurgical beam in both homogeneous and heterogeneous media to allow a better calculation of dose distribution and consequently, better treatment planning. Since the large dose gradients in the typical SRS penumbra require dosimetry techniques with higher spatial resolution, the first part of the work involves the evaluation and selection of an adequate radiation detector and the formu-lation of a method for correcting for detector size effects. Another part of the project is to study and propose a method for handling the dose distribution in the presence of inhomogeneities. 1.4 Overview The theory pertaining to small field photon dosimetry is presented in Chapter 2. Mate-rials and Methods are discussed in Chapter 3. Results of the investigations are presented in four sections in Chaper 4 according to the schema shown in Figure 1.2: First, different detectors used, to resolve small fields are investigated. Second, an approach to obtain corrections for detector size effects is presented. Third, a new method for accounting for tissue inhomogeneity is described. Finally, an original simplistic depth dose model that includes the notion of electron range is proposed. Chapter 1. Introduction: Small photon fields in radiotherapy 8 Topic: Small High Energy Photon Beams Application: Stereotactic Radiosurgery or Radiotherapy Requirements: High Spatial Resolution & Dose Accuracy Problems or characteristics: 1) Sharp Penumbra (Detector Resolution) 2) Lateral Electronic Disequilibrium (Electron Range) 3) Inhomogeneities (Longitudinal Electronic Disequilibrium) Approach: Part 1. Comparison of the resolution of various detectors (probleml) Part 2. Deconvolution of detector size effects (problem 1) Part 3. Scaling approach (problems 2 & 3) Part 4. Simple depth dose model with notion of electron range (problem 2) and that can account for inhomogeneities (problem 3) Figure 1.2: Overview of the thesis work. C h a p t e r 2 T h e o r y The theoretical principles pertaining to the dosimetry of small high energy photon beams are presented in this chapter. The concept of radiation absorbed dose and other related concepts such as charged particle equilibrium are reviewed. A solution to the problem of the influence of radiation detector size on dosimetric data acquisition is proposed.. In addition, some of the existing dose models are discussed and we introduce a new simplistic depth dose model. Finally a review of the current status of inhomogeneity corrections for photon beams along with our proposed approach for properly scaling the dose in the presence of inhomogeneities are presented. 2.1 D o s e a n d K e r m a The absorbed dose from radiation is defined as the energy deposited in a medium per unit mass of material. High energy photon interactions with matter cause electrons to be set in motion. The transfer of photon energy to the charged particles, expressed as kinetic energy per unit mass at a point is called KERMA (iTmetic Energy .Released in the Medium, the 'a' has been added for phonetic reasons). These electrons lose their energy through interactions with the atoms and electrons of the medium through which they pass and have a range which depends on the starting energy spectrum. Because of this range of the electrons, the energy absorption does not occur at the point of interaction but all along their path at some distance downstream. In small high energy photon beams this leads to significant regions of charged particle nonequilibrium which is discussed in 9 Chapter 2. Theory 10 section 2.1.1 [1, 46].' 2.1.1 Charged Particle Equi l ibr ium The concept of equilibrium requires a closed system where the expectation value of a. macroscopic quantity describing the system is to'remain unchanged as a function of space and.time. The net energy e or energy deposited by photons or charged particles (CP) in a volume equals the difference in the energy entering and the energy exiting the volume [1, 70]: • -6 = ^(e™hotons + €CP) ~ Yl(ephotons + eCP) ": Charged Particle Equilibrium (CPE) is said to exist if the energy carried in and. out of the volume by charged particles is equal. That is: ... E(^)c/> - YS<"ui)c>> ' (2-2) Hence, for electronic equilibrium to exist in a small volume u, the following conditions must be satisfied throughout the volume V (Figure 2.1): 1) The photon beam boundary must be further away from the small volume u than the maximum range of the secondary electrons, 2) the photon fluence must be uniform, 3) and the atomic composition and the density of the medium must be homoge-neous. In small high energy photon fields, condition 1) fails since the photon beam bound-ary is closer to the central axis than the maximum lateral range of the electrons [77]. Chapter 2. Theory 11 Figure 2.1: The concept of Charged Particle Equilibrium is illustrated. Electron ex enters the volume u with a kinetic energy Ek equal to that carried out by electron e^. Condition 2) is not met if there is significant photon attenuation within the range of the secondary electrons while condition 3) would fail in the presence of bone, lung or air cavities. Under CPE, the volume u is small enough to allow the escape of radiative losses. The dose D is then the collision kerma K: D = K under CPE (2.3) In reality, CPE does not exist at any point. Transient charged-particle equlibrium (TCPE) exists within a region in which D is proportional to K D = (5K under TCPE, {(3 > 1) (2.4) A radiation detector generally consists of a sensitive volume filled with a certain medium. For measurements in air, the dosimeter should have a wall at least as thick as the maxi-mum range of the charged particles present to provide CPE or TCPE. Measurements in Chapter 2. Theory 12 disequilibrium situations generally are not interpretable or would require exotic compu-tations. 2.2 B e a m D a t a The dose calculation algorithms for treatment planning systems for SRS commonly rely on empirical data. Some of the small field data that must be determined for the dose calculation algorithms include [103]: 1) percentage depth doses, 2) dose profiles, and 3) relative output factors. These relative dose concepts (characteristics) of a beam will be defined and followed by a discussion of the dosimetric techniques (radiation detectors) used to obtain them. 2.2.1 P e r c e n t a g e d e p t h d o s e The percentage depth dose PDD is defined as the ratio of the radiation absorbed dose in a phantom (commonly water) at a depth d to the radiation absorbed.dose at the depth of maximum dose dmax for a given source-to-phantom-surface distance (SSD) and field size A defined at phantom surface [46] . This depth dmax is the depth at which a maximum dose is observed. Recall that for high energy photon beams, the photon interaction sets in motion electrons that deposit their energy at some distance away from the deposition site of the photon interaction and therefore there is a buildup of electrons up to a maximum. This maximum occurs at some depth beyond the irradiated surface and ranges from 0.5 cm for Cobalt-60 radiation to 5 cm for a 25 M V photon beam in water (for a 10 x 10 cm 2 field at SSD of 100 cm). Large fields used in radiotherapy have their depths of dose maxima decreased with increasing Chapter 2. Theory 13 1001 (/I o T3 <L> > Pi 0 d, max Depth Figure 2.2: A typical photon beam depth dose curve is shown. field size. In contrast, the depths of dose maxima increase with the field diameter in the range from 10-30 mm and this increase is proportional to the photon beam energy [106]. This means that electronic equilibrium can only be achieved at a greater depth because when the field size becomes smaller than the range of the electrons, the ratio of electrons scattering out to coming in at a given point is increased. A typical PDD is shown in A similar quantity which is more convenient to use with rotation geometry is known as Tissue Maximum Ratio TMR. The field size AQ is defined at the source to axis of gantry rotation distance (SAD). The ratio is as follows (Fig. 2.3): Figure 2.2. TMR(d, AQ) = 100 D(d,AQ) (2.6) D(d max: Chapter 2. Theory 14 Figure 2.3: A photon beam Tissue Maximum Ratio (TMR) measurement set up is shown. 2.2.2 B e a m p r o f i l e a n d o f f - a x i s r a t i o The off-axis ratio OAR is the ratio of radiation absorbed dose at a point a distance r away from the beam central axis to the absorbed dose at the same depth along the central axis for the same field size A and SSD (Figure 2.4) [46]. OAXr.4,A,SSm-m$ffi22 (2.7) When all the dose points at a particular depth are joined and normalized to 100% on the beam central axis, the resultant line is called a dose profile or beam profile. A typical beam profile is shown in figure 2.5. The penumbra region is characterized by unsharp edges. Lateral electronic equilibrium is never achieved at the edges of any radiation field. That is the 'lateral' range of the electrons is such that the amount of electrons scattering out of a volume located at the edge of the field is larger than the number of electrons coming in (Figure 2.6). In addition there are photons scattered out of the field. For small field sizes and high-energy beams, the entire beam profile may be characterized by Chapter 2. Theory 15 Figure 2.5: A typical photon beam profile is shown for a given depth d. Chapter 2. Theory 16 Photon beam Kerma Figure 2.6: Electronic nonequilibrium at the edges of a high energy photon beam. Point a receives electron tracks coming from all directions. Point b is inside the photon beam but receives fewer electron tracks than does a. Point c is outside of the photon beam but still receives some electron tracks. the penumbra region since the recoil electrons have a substantial lateral range in tissue, thereby increasing the penumbra beyond the geometrical beam. In a high energy inherent photon beam profile, the geometric penumbra width is determined by the source-size and the.collimator design. The width of the physical penumbra involves photons scattered out of the beam region and an additional contribution from secondary electron spread. The inherent beam profile (or primary profile) can be modeled as/a convolution, of a source function and a collimator profile function or any other beam transmission function that has been modified by blocks, wedges, or compensating filters [110]. 2.2.3 Relative output factor Since the radiation output from a linac and hence the dose delivered varies with the size of the radiation field used, the term relative output factor R O F is introduced. It is defined as the dose at dmax for a given field size AQ and S A D (Source to Axis of Chapter 2. Theory 17 gantry rotation Distance) to the dose at the same depth and SAD for a reference field size usually the 10 x 10 cm 2 field at which point linacs are typically calibrated to deliver a dose of 1 cGy per monitor unit (MU) [46]. Hence an output factor gives the dose in c G y / M U at isocenter and at dmax. . r>r,W(A \ - D(dmax,AQ,SAD) - R ° F ( A Q ) - D(dmax,10x 10, SAD) • . ( 2 : 8 ) For radiosurgical fields, AQ is the field size at SAD as produced by the tertiary cylindrical collimator 1. The combination of very small field size and high energy yields a smaller value for the relative output factor again due to electronic nonequilibrium conditions [103]. 2.3 Dosimetry techniques , Different types of radiation detectors are routinely used for the dosimetry of photon beams. One of the underlying assumptions for the accurate conversion of detector signal to radiation absorbed dose is that electron fluence across the detector is uniform which requires that the detector is situated in a region of electronic equilibrium. The dosime-try of small fields is more difficult because of the steep dose gradients and the lack of lateral electronic equilibrium conditions that complicate the interpretation of the dose measurements. The use of small volume detectors is necessary to achieve high spatial resolution [93]. Dose measurements for the small fields used in SRS have been reported using: regular film (Kodak X V ) , diode, thermoluminescent dosimeters (TLD) and ion-ization chambers [93, 5], and more recently, for the P T W diamond [95], scintillation detector [6, 7] and GafChromic film [67, 92]. The properties of these detectors are given in Chapter 3 Materials and Methods. 1See Chapter 3 Materials and Methods for a basic description of a linac and its collimation system. Chapter 2. Theory 18 2.3.1 Radiation detector size effects There are two competing processes by which a detector perturbs the true dose distri-bution. One is the change in electron transport if the material of the detector is not water equivalent and the other is the broadening of the measured penumbra due to the finite size of the detector. The scanning of a beam with a detector of finite size can be represented as a convolution (*) ofthe inherent beam profile (Pinherent) with the detector line spread function (LSF). The modified beam profile (Pmodified) after scanning can be represented mathematically by Pmodified{x) = Pinherent(x) * LSF(x) "(2.9) or n Pmodified(x) = Pinherent{x ~ x' + 1) LSF(x') (2.10) x'=l where x represents the lateral position along the profile. The modified profile is the profile as measured with a detector. The inherent dose profile of the beam is not obtained easily. It needs to be either modelled, Monte Carlo simulated2, or deduced by some other means. As for the LSF, a semi-elliptic distribution has often been used to represent the response of a cylindrical detector of effective sensing radius R such as the P T W (2R = 5.0 mm) and Farmer (2R = 6.1 mm) ion chambers [104, 42, 18]. LSF(x) = { ^(R*-Xi)h ii\\x\\<R 0 otherwise (2.11) where x is the lateral distance. Using the convolution theorem, equation 2.9 is equivalent to. 2See section 2.4 on dose models Chapter 2. Theory 19 JS[Pmodified(x)]=TS{Pinherent(x)}- F[LSF{x)} (2.12) where 7s, the Fourier transform is given by: 1 r°° ~ , r\f(x)\ = F(k) = -== \ f(x) e2mkxdx (2.13) where k is the phase number. The corresponding inverse Fourier transform is given by I roo F-l[F(k)\ = f{x) = -*= \ F(k) e-^dx (2.14) In the discrete case, the discrete Fourier transforms are computed, usually with the Fast Fourier Transform (FFT) algorithm. When the LSF is known, the inherent beam profile can be recovered by using equations 2.12 and 2.14 as follows Pinherent{x) = ^[^[Pmodifiedix)] / F[LSF(x)]] (2.15) Similarly, if the inherent profile is known the LSF of a detector can be found from LSF{x) = T-^iPmodiHedix)] I T[P%nherent{x)}} (2.16) However, because of sampling, error or noise in the measured data, deconvolution using the Fourier transform can fail to provide the solution. In fact, such a linear system (Eq. 2.10) is difficult to solve due to the smoothing action of the convolution operator [96]. This can be seen in the following example. Consider the continuous version of equation 2.10: fx I I I Pmodified(x) — / Pinherentix X ) LSF{x )dx JO (2.17) Chapter 2. Theory 20 Let us denote the exact solution to equation 2.17 by LSFx{x) = LSF(x)' ' (2.18) and a perturbed solution by LSF2(x) — LSF(x) + Csinux (2.19) The difference between the two corresponding outputs is Pmodified2(x) - Pmodifiediix) = Cj Pinherent(x - x') SlUUx'dx' (2.20) J o where the frequency u can be chosen high enough, even for large values of C such that the difference between the two modified profiles (equation 2.20) is made arbitrarily small. This can be shown as follows. If the input Pinherent is bounded, ||P<„herent(x) || < M (a constant) (2.21) It follows that Pmodified2{x) - Pmodifiedx{x) < CM f sinux'dx' = CM^~ C0SUX) (2.22) JO Ul From this we conclude that 2CM \\Pmodified2(x) Pmodifiedi (x)\\ < (2.23) U) From this last equation, we can see how by selecting u> to be sufficiently large, the difference between the two modified profiles can be made arbitrarily small. This means Chapter 2. Theory 21 that small differences in Pmodified(x) can map into large differences in LSF(x). This is a serious problem because in practice, the measured profile (Pmodified{x)) is always accompanied by a nonzero measurement error. In other words, if there are perturbations in the measured profile used as input in the deconvolution process to solve for the LSF, the resulting LSF will be erroneous. There have been several attempts to account and correct for the ionization chamber size effect by extrapolating measurements to zero detector size [104, 23, 24, 54] and by deconvolving measurements with the detector response function given by Equation 2.11 [104, 42]. The recovery (or deconvolution) of the LSF of detectors of any shape or the inherent profile from measured data would provide useful information and the details of the approach used to deal with this problem is presented in the next chapter (Chapter 3 Materials and Methods). 2.4 Dose models Radiation absorbed dose can be calculated by analytical models, by convolution tech-niques or by Monte Carlo simulations. Dose calculation algorithms for treatment plan-ning require a knowledge of the characteristics of the linac-generated photon beam and its interaction with tissues [26]. While an empirical determination of the characteristics with a detector implies a perturbation of the physical beam, analytical and numerical dose models (e.g. The convolution/superposition and the Monte Carlo methods) require specifications such as the effects of the components of the treatment head. Dose calcu-lations in regions of electronic nonequilibrium are usually handled by Monte Carlo or convolution techniques but not by more simple models. The models employed in most present photon dose planning systems are based on the concept of separating the ab-sorbed dose into two components 'primary' and 'scattered' dose. These models are valid provided that the electron range is smaller than the desired spatial resolution of the dose Chapter 2. Theory 22 distribution. The current existing dose models are: 1- Primary and scatter dose model [3] The total dose TJ at a depth d for a field size A is as follows: D{A,d) = Dp(0,d) + Ds(A,d) (2.24) The primary dose Dp is defined as the dose resulting from charged particles set in motion by photons interacting with the medium for the first time. The scatter dose Ds is the dose from charged particles set in motion by photons that have interacted within the medium more than once and from bremsstrahlung and annihilation photons created in the medium. For radiosurgery planning, the dose calculation3 for a point Q at depth d with a lateral displacement with respect to the isocenter is given by [70] . where TQ is the projected radial distance of the point Q (Figure 2.7). AQ is the field size at S A D . The total dose is a summation over the dose contribution from each arc. The primary dose component corresponds to using TMR(d, 0) and the scatter component is obtained from the subtraction, of this quantity from the total dose. This model alone does not account for the transport of electrons [72] but some attempts have been made to modify the primary and scattered dose model by incorporating the electron transport obtained from Monte Carlo simulations [122] and more recently, by including an empirical Electron Perturbation Factor [36]. 3See Beam data section 2.2 D(d,rQ) = ROF{AQ)TMR(d,AQ)OAR(rQ)( SAD (2.25) SAD-d Chapter 2. Theory 23 Source Central axis SAD Phantom surface Figure 2.7: A diagram of a point of dose calculation Q at a depth d for radiosurgery planning is shown. 2- Convolution models In the Convolution/Superposition (CVSP) method [65, 56], the photon-charged-particle transport is separated into primary photon interactions, charged-particle and secondary photon transport and energy deposition. The convolution model can take into account electrons and scattered photons by means of scattering kernels. The dose D(r) (J/Kg) is calculated as the convolution between the TERM A (Total Energy .Released in Medium, J /Kg) T(r) and the photon dose kernel K(r) ( cm - 3 ) , and is expressed as, in which the TERMA T(r) is computed from the product of the photon energy E (J), the photon fluence 0(r) (cm - 2 ) , and the mass attenuation coefficient p(E)/p (2.26) (cm 2/kg), Chapter 2. Theory 24 Source Figure 2.8: Illustration of the vectorial framework for the convolution/superposition method. The dose deposition occurs at a distance r — r' away from the radiation inter-action site. T(r) = ^-<j>(r)E (2.27) P It represents the total energy that is deposited in the medium for the initial photon interaction (Figure 2.8). The photon dose kernel K(r), describes the distribution of the relative energy deposition per unit volume following the initial photon inter-action at r, Kernels for polyenergetic photon beams are calculated by superposition of the mo-noenergetic data. The kernels can be obtained either by Monte Carlo simulation or extracted from measured data [20]. The Differential Pencil Beam (DPB) model [73] is another convolution approach. The DPB is defined as the dose distribu-tion relative to the position of the first collision per unit collision density for a Chapter 2. Theory 25 monoenergetic pencil beam of photons in an infinite unit density medium. 3- Monte Carlo simulation The Monte Carlo approach provides a bridge between measurements and analytic calculations. In the Monte Carlo approach, the transport of an incident particle, and of particles that are subsequently set in motion (a particle 'history') is simu-lated. The histories are determined from the probability distribution that controls each possible radiation interaction. A large number of histories is required to obtain a statistical uncertainty small enough such that the resulting dose distribution is re-alistic and reproducible. The EGS4 4 code [76] is a popular general purpose package for the Monte Carlo simulation of the coupled transport of electrons and photons in an arbitrary symmetric geometry. This technique is very time consuming since each interaction is modelled by computer and millions of histories are calculated to obtain good statistics. The code can be used to calculate dose in phantoms using an assumed photon beam spectrum at the phantom surface or alternatively the photon spectrum can be obtained from another code known as B E A M [94] which allows a simulation of the treatment head of the linac and asymmetric beams. 4- Analytical Models A full analytical solution to the complex electron-photon transport is unrealis-tic. The many variables affecting the dose distribution render equations almost impossible to solve. Reliable approximations must be made [26]. This cate-gory includes some simplistic dose models. Among them, some simple physical models have been presented to generate PDDs. A well-known dose deposition model for photons is based on coupled transport equations similar to the parent-daughter radioactive decay equations [28]. This model yields the dose fitting func-tion TJ (a:) = C(e~Xx — e~Ax) where all the constants are empirically determined. Electron Gamma Shower, version 4. Chapter 2. Theory 26 This analytical representation has also been used to fit radiosurgical depth dose data [120]. These first order differential equations treat electrons and photons alike, that is all particles are projected forward and there is no electron range. In fact the proposed analytical expressions for PDDs in the literature are either just a fitting function or are based on first order differential equations [32, 51, 113]. Second order differential equations seem more adequate to include energy being car-ried away by electrons. We propose a one-dimensional model for illustrating photon beam depth dose deposition which includes the notion of an electron range and which can be applied to heterogeneous media. This model is physically simplistic and cannot compete with Monte Carlo simulation but is rather a "toy model" that provides some insight into the dose deposition process. 2.4.1 Proposed simplistic depth dose model Consider the classical mechanics problem of one-dimensional lattices (chains) of particles with nearest neighbor interaction (Hooke's law is obeyed) [31]. Then the equation of motion in a damped system (damping coefficient c and spring constant k) is.given by mx'^t) + k{2xn{t) - xn+1(t) - xn_i(t)) = Cnx'Jt) (2.29) where x is a displacement. This chain involves a large number (N + 1) of coupled oscillators of equal "mass" m (Figure 2.9). The square of the frequency (UIQ) is defined as k/m. The following initial conditions are imposed: xn(0) = 0 (2.30) and x'n(0) = ae'P (2.31) Chapter 2. Theory 27 n = 0 n = N + l Figure.2.9: Chain of oscillators. These springs are set in motion by an exponentially decaying stimulus which is analo-gous to a photon beam. The oscillators can be seen as electrons allowed a certain "range" Of motion in a damping medium. These neighbouring vibrating; systems are able to trans-mit their energy to each other. The total energy deposited or "dose" in the system at a site "n" is related to the square of the velocity. • Dn = cnJXx'n(t))2dt ' , (2.32) The analytical solution to this integral is complex and given in the appendix. In-homogeneities for this macroscopic model can be introduced by changing the damping properties at some nodes. In the following section, a more general discussion about inhomogeneity correction approaches routinely used in current dose models is presented. 2.5 Inhomogeneity corrections The presence of inhomogeneities such as air cavities, bone and lung in a waterlike material perturb the dose distribution [107]. A lack of electronic equilibrium exists at. the interface Chapter 2. Theory 28 of any two different media. It has been a long-standing problem in radiation therapy to correct the dose distributions for these inhomogeneities. Most current inhomogeneity correction approaches are based on the "scaling theorem" which is explained below. 2.5.1 Scaling theorem According to O'Connor [79, 80], the dose deposited by photon beams in two media with different densities (but with the same atomic number) is the same provided all dimen-sions in the media are scaled inversely with the density of the media. The theorem was originally developed for photon transport but was later extended to electron transport. This scaling theorem is based on the linear dependence of the attenuation coefficient on density. For media of a given atomic number, both the stopping power and the angular scattering power5 are directly proportional to the physical density [45, 52, 124]. The necessary conditions for the theorem are: • the probability of an interaction of any specified kind bccuring in a small voxel must be directly proportional to the number of interaction centers in that voxel (the number of interaction centers per unit volume corresponds to the density in this context); • there must be no cooperative effects which depend on the relative spatial positions of the interaction centers; • ' . • there must be no change in any other parameter which influences the interaction probability. The theorem finds an application in the Compton effect, a process which has its mass attenuation coefficient independent of the atomic number and where the electrons are the interaction centers. 5 Both notions, stopping power and angular scattering power, are explicitly presented in sections 2.5.2 and 2.5.5 respectively of this chapter. Chapter 2. Theory 29 2.5.2 Mass stopping power The mass collision stopping power in a medium A of density p quantifies the energy dE lost by charged particles in a distance dx divided by the density and is given by [44] N IdE 27rr2mec2 A r Z r i , e2(e + 2) . _,. . f n o o N where I is the mean excitation energy, 8 is the density effect correction, and m = l - f + * / a - £ £ ) m • (2.34) where r e is the classical electron radius, e = E/mec2 the ratio of the kinetic energy E of the electrons to the rest energy, j3 the ratio of the velocity of the electron to the speed of light c, NA is Avogadro's constant, Z is the atomic number and finally MA is the molar mass of the substance A. 2.5.3 Density effect According to Sternheimer [111] the density effect correction is given by the following approximate analytic expression: 0 if x < XQ 2(lnl0)x + C + a{xx - x)b • if xe(rr0, a;i) (2-35) 2(lnl0)x + C if x > xx where p = momentum, m = mass of charged particle Chapter 2. Theory 30 •C = -2\n{I/hvp) - 1 up = p l a s m a f r e q u e n c y = y n r g C 2 / ^ n = e l e c t r o n d e n s i t y r e = c l a s s i c a l e l e c t r o n r a d i u s = e 2 / ( m 2 c 2 ) (2.36) and the fit parameters XQ, X x , a and b which must be chosen appropriately for each sub-stance to obtain an adequate fit for S are given in Sternheimer's paper [111]. When X\ is such that 5 has reached its asymptotic behavior then dE/dx no longer depends on the -ionization potentials but only on the electron density, (this argument, is being used in O'Connors' scaling theorem extended to electron transport). 2.5.4 Continuous-slowing-down range The range ro of an electron gives the mean path length which an electron would travel in the course of slowing down if its rate of energy loss along the entire track was always equal to the mean rate of energy loss [44]. It is given by the following integral: where EQ is the initial energy and Stot{E) is the total stopping power which includes the energy lost by radiative and collisional electron interactions. 2.5.5 Mass scattering power The mass scattering power r/p which characterizes the electrons passing through a medium is given by [44] (2.37) r/p \dfP_ p dl [ln(l + i f )2 ) - 1 + (1 + (f?)-1} (2.38) Chapter 2. Theory 31 where 6m and 0M are the cutoff angle due to the finite size of the nucleus and the screening angle respectively. 2.5.6 Inhomogeneity correction factors The inhomogeneity correction factors when used with the popular "Primary and Scatter" dose models usually take the form of multiplicative factors. They are based on the concept of effective pathlength (radiological length, de) which is de = pe • d where pe is the electron density relative to water and d is the physical pathlength [27]. In the Equivalent Tissue-air-ratio method (ETAR) [108], both the depth and the field size are scaled using weighting factors. This method however ignores secondary electron transport [69]. For small high energy photon beams, problems of electron transport are more • pronounced so dose modeling must take into account both electron longitudinal and lateral nonequilibrium effects. The rectilinear density scaling has also been applied to the convolution/superposition method. The dose at a point r in a medium of density pave is given by D(r)= [ T(r')p(r')K(r-r>,pave)/paved3r' (2.39) Jr' With the rectilinear scaling, the average density between the interaction and the deposi-tion site, Paw, is Pave = ~ — rP(r)dr' (2.40) / — I Jr 2.5.7 Limitations of the scaling theorem In the application of the scaling theorem to the ratio of the fluence of secondary elec-trons to the fluence of primary photons for high-energy x-ray beams certain limits are Chapter 2. Theory 32 encountered. One must take into account the density correction, which is related to the dielectric properties of the material, to the collision mass stopping power, for electrons. The density correction factor depends on the energy of the electrons and is a function of the number of atoms per unit volume. Hence the mass stopping power increases as the density diminishes. This effect diminishes with decreasing density and with decreasing electron energy. According to the work of Sternheimer [111], the correction amounts to 10% for 10 MeV electrons in water. In their analysis Woo and Cunningham [122] found that points immediately below an air gap were overestimated by the density scaling calculation. This discrepancy increases with the air gap thickness and with the field size reduction. And the density scaling will also fail to predict the perturbation effects near the inhomogeneities of different atomic composition. The effects of atomic number changes produce changes in electron multiple scattering and pair production. Attempts have been made to incorporate explicit electron transport in a photon-electron cascade model [126] and to describe the degree of electronic disequilibrium (lateral and longitudinal) [84]. An approach to account for both the lateral spread of the electrons and their stopping power is possible by using a different scaling method [47, 97]. Our scaling approach is detailed in Chapter 3 and is applied to the Convolution/Superposition method. 2.6 Summary We have introduced different concepts related to dose calculation for small photon fields. The properties of small photon beams can be summarized as follow: • Radiation beams of diameter 4.0 cm or less are commonly used in stereotactic radiosurgery (SRS); • Lateral electronic nonequilibrium (LEN) may exist even on the central axis and the Chapter 2. Theory 33 absorbed dose mainly stems from primary interactions; • The presence of inhomogeneities introduces a longitudinal electronic nonequilibrium to the dose distribution already modified by LEN; • The range of the electrons increases with photon energy (Compton process); • Steep dose gradients; , • The size and composition of a detector alter the dose measurement; • The depth of maximum dose increases with field size. In the next chapter, the procedures and materials we used for obtaining beam data, correcting for detector size effects and scaling the dose in the presence of inhomogeneities will be presented. Chapter 3 Materials and Methods Beam profiles and percentage depth doses, with and without inhomogeneities present in the phantom, can be obtained by one of the following means 1) measurement, 2) Monte Carlo simulation, 3) convolution/superposition, and 4) analytical modeling. This chapter describes the equipment used for the experimental work, the codes employed in our simulations and the mathematical approaches used in our studies. 3.1 Materials The experimental materials used are for: 1) the production of an external beam of radiation, 2) the use of a phantom during the irradiation to simulate human tissues, and 3) the dose measurement with the help of a radiation detector. 3.1.1 Linear accelerator 10 MV and 6 MV radiosurgical x-ray beams ranging from 0.5 to 4.0 cm in diameter at 100 cm from the source were produced with a linear accelerator (linac)1 using tertiary collimators consisting of compacted circular cerrobend2 (Fig. 3.1). In the treatment head of the linac in photon mode (Fig. 3.2), high energy electrons are focused on a target to produce bremsstrahlung photons. The beam then passes through a primary collimator. Just below this first collimator, the bremsstrahlung photons which are forward peaked in ^l inac 2100 C/D, Varian Associates, Palo Alto CA. 2Low melting temperature lead, bismuth, tin and cadmium. 34 Chapter 3. Materials and Methods 35 Figure 3.1: A medical linear accelerator (linac) and its rotational and translational de-grees of freedom are illustrated. The gantry rotates (6Q) around an axis located at the isocenter. The couch rotates (fly) and has a translational motion in three dimensions. The collimator also rotates (6 c) about an axis centered at the isocenter (Adapted from [57], with permission.) intensity, pass through a flattening filter to produce a uniformly intense radiation. Next comes a beam ion chamber to monitor the dose. Adjustable secondary pairs of collimators are located below the chamber and can be used to design any rectangular field shape. A mirror for the light localizer can slide in and out. The stereotactic collimator was of thickness 10 cm and was mounted on an aluminum plate which was inserted in a tray slot at 65.4 cm from the source. The secondary collimators of the linac were set to 5 x 5 cm2. Chapter 3. Materials and Methods 36 Electrons Target Beam monitor chamber Upper collimator - - * (Secondary) Primary collimator Flattening filter - - Mirror Tertiary collimator (Stereotactic) - Lower collimator (Secondary) Figure 3.2: The head assembly of a linear accelerator in photon beam mode equipped with a stereotactic collimator is shown. The electrons strike a target and produce bremsstrahlung photons which are forward peaked in intensity. These photons first go through a primary collimator and then pass through a flattening filter to produce a uniform radiation distribution. Futher, secondary and tertiary (stereotactic) collimators shape the photon beam (Adapted from [57], with permission.) Chapter 3. Materials and Methods 37 3.1.2 Dosimetry phantoms A tank3 made of an acrylic container and filled with water has been used for some of the beam data collection. Also solid polystyrene sheets were used in conjunction with film dosimetry. Two tissue heterogeneities that are present during SRS for the head and neck region were studied: air gaps and bones (we used aluminum as a bone substitute). The properties of the various phantom materials are shown in Table 3.1. The electron density, pe, is the number of electrons per unit volume and the physical density, p, is the mass per unit volume. •< . • Material Pe P ;• Z Water 1.000 1.000 •7.51 Polystyrene 1.011 1.044. 5.74 Air 0.00108 0.0012 7.78 Aluminum 2.343 2.699 13 Bone 1.575 1.650 12.31 Table 3.1: The properties of various phantom materials are given. The electron and physical densities (pe and p respectively) are relative to water and Z is the effective atomic number. The properties of bone are added for comparison. The photoelectric effect was considered to obtain the effective atomic number Z: Zm = axZ™ + a2Z™ + ... (3.41) where a; is the fraction of the electrons present in the mixture and m has a value of about 3.5. 3Wellhofer three dimensional watertank dosimetry system with the WP700C software package (Well-hofer Dosimetrie, Schwarzenbruck, Germany). Chapter 3. Materials and Methods 38 3.1.3 Radiation detectors Lateral dose profiles and percentage depth doses were obtained in the Wellhofer three dimensional watertank dosimetry system. The diode detector4, PTW diamond detector, Wellhofer IC10 ion chamber, NAC mini-chamber5 and Markus parallel plate chamber were all used in conjunction with the water tank. These detectors were connected to a digital electrometer6 for charge measurements. Dose profiles were also measured using film7 in solid polystyrene phantoms at an equivalent depth and scanned with the Well-hofer densitometer. The physical dimensions and properties of these detectors are listed in Table 3.2. Detector Material Sensitive area Sensitive volume Si diode p-type silicon 2.5 mm diameter 60pm thick + 0.45 mm silicon backing Kodak XV-2 Film Photographic emulsion Wellhofer densitometer 1.2 mm aperture PTW diamond diamond plate 2.3 mm diameter 0.33 mm thick Wellhofer IC10 air 6.0 mm diameter 0.14 cc3 Markus chamber air 5.4 mm diameter 2 mm thick NAC Mini-chamber air 1.9 mm diameter 0.0073 cc3 . Table 3.2: The physical properties of detectors used in the measurement of dose for SRS fields are given. 4Scanditronix p-type silicon semiconductor diode. 5The NAC mini chamber is constructed by Schreuder et al. [100]. 6Victoreen Model 500, Nuclear Associates, Cleveland, OH or. Keithley 614, Keithley Instruments, Cleveland, OH. ' 7Kodak X-Omat V Ready Pack. Chapter 3. Materials and Methods 39 3.2 Methods The methods presented involve the following topics: 1) the deconvolution approach to correct for detector size effects, 2) the direct measurement of the LSF of the film den-sitometer, 3) the pathlength scaling approach for phantoms containing inhomogeneities, and 4) the solution of our simple analytical depth dose model. 3.2.1 Deconvolution In principle, the following useful information can be extracted by deconvolution from measured broad lateral beam profiles: 1) Pencil beam kernels, 2) Detector LSF, 3) Ef-fective source size, and 4) Inherent beam profile of the linac [20, 42, 17]. Because it is an ill-posed problem, an attempt should be made to regularize the problem to offer a reasonable approximation. We have discussed in Chapter 2 (Eq. 2.10) that the convolution along the lateral position x of the inherent beam profile (Pinherent) with the detector line spread function (LSF) yields the modified beam profile (Pmodified)8'-Since the deconvolution process is an ill-posed problem that normally fails with the Fourier method, a better approach is to make use of the fact that the discrete deconvo-lution problem (equation 3.42) is essentially reduced to the problem of solving a set of linear equations. The relation, (equation 3.42), between the inherent profile, the modified profile and the LSF can be put into the following matrix form [90] n Prnodifiedipt)' ^ ] Pinherentix X -f- 1) LSF(x ) x'=l (3.42) Az = b (3.43) 8The inherent beam profile is the true beam profile prior to measurement while the modified beam profile is the profile "modified" by the detector during measurement. Chapter 3. Materials and Methods 40 where and A=. b = Pmodified (1) Pmodified^) •modified (m) J m x l PinherentiX) 0 Pinherent (2) Pinherent (1) Pinherentijl) Pinherentijl 1) Pinherentipi) PinherentijTl 1) LSF(1) LSF{2) LSF(n) J n x l 0 0 Pi inherent (1) PinherentijTl TI - f r 1) (3.44) The matrix A is highly ill-conditioned9 [34]. The conventional technique to find the solution is to use the least squares (LS) method [34] which can be formulated as an optimization problem to find the LS value of z, z^s as given by minimize ||Ab||E, subject to AZLS = b + Ab (3.45) where ||.||£ denotes the Euclidean norm and Ab represents the perturbation vector b. In the LS technique, there is an underlying assumption that the error is imposed only on the vector b while the matrix A is accurate. Since in our application both the matrix A and the vector b can be corrupted by noise and errors, a better optimal method is the 9See further 'Condition number'. A singular matrix has an infinite condition number and an i l l -conditioned matrix has its condition number too large. Chapter 3. Materials and Methods 41 total least squares (TLS) technique [90, 34]. The TLS value z, zTLs, is found from the following optimization problem . minimize | |AA : Ab||p, subject to (A + A A ) Z T L S = b + Ab (3.46) where \\.\\F denotes the Frobenius norm and A denotes the perturbation of the associated quantity. Thereby we bring Az = b into the form: A geometrical comparison of the goodness-of-fit between the TLS method and the LS method for the simplest case n = 1 is shown in Figure 3.3. In this illustration, a line is used to fit a set of data points. The LS method minimizes J2i<*i, which is the summation of the squares of the vertical distances oVs from the points to the line. On the other hand, the TLS technique is based on minimizing Yjifih where the /3j's are the perpendicular distances from the points to the line. " The TLS solution can be derived from the singular value decomposition [34] (SVD) of the augmented matrix E = [A : b]: where U = [ux, u2,.."., um] and V = [vi, v2,..., vn+i] are unitary matrices such that [A:b][zT\:-l]T = 0 (3.47) E = UWVT (3.48) UTU = Im and VTV = 7n + 1 and Im and In+X are identity matrices, and W is an m x (n + l) matrix expressed by S w = (3.49) 0 Chapter 3. Materials and Methods 42 ! \ p . » a Figure 3.3: A n illustration of the T L S and the LS methods for n = 1. The dashed LS lines are parallel, to the b-axis while the solid TLS lines are perpendicular to the line az — b. where S = diag(ax,...., an+i} is a diagonal matrix with ax > a2 > ... > an > an+x. The entries CTJ are called the "singular values" of E. The vector ZTLS 1 n T is parallel to the right singular vector vn+x. The TLS solution ZTLS is obtained from ZTLS - 1 Vn+1 (3.50) where ^ n + i ) n + i is the last component of v n + i . The TLS method has been shown to provide better performance than the LS method in different applications [91, 22]. However, the TLS method has not been as widely used as the LS method since the numerical computation of the TLS approach employing S V D is more complicated than that of the LS approach. The measured and inherent profiles are sent to our Fortran program D O S E (Decon-volution Of Size Effects) that executes the deconvolution and makes use of the T L S Chapter 3. Materials and Methods 43 SVD-based algorithm10 [117]. First, we construct the augmented matrix [A : b] and find its maximum and minimum singular values. For an overdetermined (mxn) matrix E, the condition number C^E) is defined by [34] °>& = 2m (3-51) where G\(E) is the largest singular value and o~n(E), the smallest. A large condition number implies an ill-conditioned matrix. Small errors in the data may lead to large errors in the computed LSF. Then we should somehow change the ill-conditioned problem to a well-conditioned problem [96]. To have a well-conditioned problem one can approximate the augmented matrix [A : b] with a lower rank augmented matrix [A : b]. This is achieved by zeroing the singular values that are smaller than a selected threshold. The threshold can be computed from the standard deviation of the errors ov on [A : b] and is y/2 • MAX(m, n + 1) • ov [117]. 3.2.2 Computer simulation A simulation has been done to provide us with guidelines in selecting a threshold for truncating the singular values and determining suitable rank approximations for our de-convolution approach. A computer simulated study of the effect of the size of a cylindrical detector on the measured penumbra was presented by Chang et al. [18]. We have used a similar procedure to generate our profiles. A computer simulated "inherent" beam profile has been generated by convolving a step function with a semi:elliptic distribution (Eq. 2.11) of radius R = 8 (arbitrary units) (Fig. 3.4). Furthermore, this inherent profile was convolved again with a semi-elliptic line spread function of radius R = 6 units simulating a cylindrical detector with such a radius to generate a "measured profile". Following the generation of these profiles of length 512, Gaussian noise with zero mean and standard 1 0Available on internet site http://netlib.bell-labs.com/netlib/vanhuffel/index.html Chapter 3. Materials and Methods 44 100 CL — — Step function Computer-simulated inherent profile Semi-elliptic distribution R=8 units 0 10 20 30 40 50 60 70 80 90 100 110 120 Off :axis distance (arbitrary units) Figure 3.4: The computer-simulated inherent profile generated from the convolution of a step function and a semi-elliptic distribution is shown along with these two generating functions. deviation ov (au =0.00, 0.05 and 0.25) was added to each of them. The noisy measured profile was then deconvolved with the noisy inherent profile using the SVD-based T L S approach to recover the LSF. In one example some preprocessing was applied to the data. We modified the mea-sured profiles so that they satisfy a theoretical symmetry condition about an inflection point [20]. The profile is "folded" horizontally about its central axis and also vertically about its mid-height. 3.2.3 Densitometer LSF Y i n [123] used a convolution method to assess the blurring of the penumbra due to film digitization. To estimate the blurring of the film produced by digitization a narrow line Chapter 3. Materials and Methods 45 on a film was scanned. A similar procedure was followed except that our method did not require any special equipment to print the narrow line. Our line was made by a very narrow and shallow cut with a fine sharp needle-like point on the film. The thickness of the line as measured with a microscope was 70pm and it darkened during normal film processing. The line was scanned and the resulting spread out profile was normalized to its area to obtain the LSF. The final LSF can be fitted to a division of a gaussian and exponentials by a second degree polynomial. 3.2.4 Monte Carlo Simulation Since an exact analytical solution to the complex transport problem of electrons is not possible, Monte Carlo simulation is the method of choice for studying dose distribution in nonequilibrium situations. We have used Monte Carlo calculations for generating data to be compared to 1) Beam profile measurements and radiation detector resolution and 2) Inhomogeneity effects and scaling approach using the Convolution/Superposition dose model. The Monte Carlo simulation was. performed using the-EGS4 code [76]. The P R E S T A algorithm was employed [9]. Each simulation required about 100 million initial photon histories to achieve a standard deviation smaller than 2% and was usually Completed in about lOOhrs on either a SUN Sparc 10 workstation or on a Pentium 133 MHz. For.the first study of beam profiles, the 6 M V published spectra from Mohan et al. [71] was used in the simulation. To assess the validity of the 6 M V energy spectrum used as input, the calculated depth doses were compared to the measured, ones for a 5 x 5 cm 2 field in water. EGS4 simulations were performed using cartesian geometry. The modeled beam was an isotropically radiating point source producing a field size 3 x 3 cm 2 at 100 cm SSD. This beam was incident on a water phantom of dimensions 5 x 5 cm 2 perpendicular to the beam and 7.2 cm deep. Scoring bins of 0.5 mm width were used to record the Chapter 3. Materials and Methods 46 dose profile at a depth of 1.5 cm along the central axis. Energy cut-off parameters used were ECUT = A E = 10 keV, PCUT = AP = 10 keV. To speed up the simulations the electron transport cutoff (ECUT) for voxels farther than 0.5 cm from the scoring region was increased to 1 MeV kinetic energy. This increase in ECUT away from the profile was tested and found to have no effect on the profile shape. Percentage depth doses have been generated with EGS4 in the second study to illus-trate the inhomogeneity effects. Some of these calculations, which were either in cartesian geometry or in cylindrical geometry, were done for 6 M V and 10 M V photon beam spec-tra [71]. The cut-off energy parameters were always 10 keV for this study. A phantom made of 0.5 mm width scoring bins was large enough to provide sufficient scattering. 3.2.5 Scaling approach and radiological pathlength The aim of this work was to improve the form of scaling involving relative electron densities currently used for dealing with inhomogeneities. The approach proposed was applied to the Convolution/Superposition code (CVSP) 1 1 and was compared to the more accurate Monte Carlo simulation method to handle inhomogeneities. An approximate solution to the problem of inhomogeneities by means of the convolu-tion method is to use radiological units (g/cm2) instead of length units (cm) to describe the kernels (the scaling factor reduces to O'Connors' density theorem). Changes in atomic numbers affect the interaction probabilities of processes such as pair production, radia-tive and collision mass stopping powers (Eq. 2.33), and mass angular scattering power (Eq. 2.38). There is no longer an exact scaling when the atomic number is changing. There is an increase in range with atomic number and hence in the angular spread. For low-density heterogeneities, the density distribution also needs to be accounted for. The scattering power P (in units of cm2/electron) can be set to be proportional to a power "Available on internet site http://www-madrad.radiology.wisc.edu/penbeam/index.html Chapter 3. Materials and Methods 47 a CO .> co 3^ 8 LU Figure 3.5: The electronic scattering power (r/p x A/Z) relative to Hydrogen is shown, of Z [53]: r A P=-'x---oc Z n (3.52) The electronic scattering power relative to Hydrogen is plotted against the atomic number Z in Figure 3.5. The values from ICRU report 35 [44] are used on this graph. Over a wide range of energy and atomic number, n e(0.837,0.955) and an average value 0.90 is chosen. Kornelson had found n=0.92 [53] using ICRU report 21 [45] for which the scattering power values were 10% higher than the ones from report 35. The effective atomic number for compounds can be calculated with the following formulae: (3.53) Chapter 3. Materials and Methods 48 where fi is the fraction by weight of electrons of the element Zi. We have accounted for the change in lateral scattering by scaling the convolution kernels K (as seen in Chapter 2 section 2.4) with a relative diffusion angle (Eq. 3.54). K%°* = KH20(P%r,pavej (3.54) where P^ is the ratio of the scattering power relative to water (W) as calculated using equations 3.52 and 3.53 of a material of effective atomic number Z.. The average electron density pave is scaled as follow Pave = P*Z Pave . (3.55) We have accounted for the difference in stopping power and range by scaling the kernels with the ratio of the stopping powers 5* between the inhomogeneity and water, as follows; Ksztop = S^KH20((S^)r,Pave). (3.56) where S^r is the ratio of the stopping power relative to water (W) of a material of effective atomic number Z at the energy deposition site and (Sw) is the mean mass stopping power ratio along the path.- The combination of Eqs. 3.54 and 3.56 ('Full scaling') yields the kernel to be used for the Convolution/Superposition (Figure 3.6). This scaling is an alternative to a Monte Carlo calculated electron track kernel [48] (or to scaling with "FET" [47]) to take full account of the stopping power and scattering characteristics of the medium). Chapter 3. Materials and Methods 49 K(r ,p) S z w K ( P z w < S z w > r , P w z Q ) Full scaling Figure 3.6: The energy deposition kernel is being modified to account for the change in scattering and stopping powers. 3.2.6 Solution of the depth dose model with oscillators The analytical solution to the depth dose model is given in the appendix and involves infinite summations. Hence, we have evaluated numerically the equations with their initial conditions using the mathematical package Mathematica12. 1 Wolfram Research Inc. Chapter 4 Results and Discussion Calculated and measured dose distributions are presented and discussed in this chap-ter. The following main themes are presented: the measurements obtained with different detectors are shown, followed by the corrections for detector size effects and for inhomo-geneities, and finally some examples of depth doses generated with our simple model are given. 4.1 Part 1: Detector resolution At first suitable radiation detectors for use in SRS were investigated. When measur-ing dose on the central axis of small fields, the absence of lateral electronic equilibrium complicates the interpretation of the measured value since the detector has a finite size and the dose may vary significantly from the centre to the periphery ofthe detector. Furthermore, if large dose gradients are present, the detector may not adequately resolve these when making measurements away from the central axis. To minimize these effects detectors of small diameter should be used [101]. Evaluations of the resolution of various detectors have been presented [41]. Different dosimetry techniques available at our in-stitution including the recently developed mini chamber [100] have been used for profile measurement in small fields and the detector resolution was compared. The properties of the detectors used are given in Chapter 3 Table 3.2. The material from which a detector is made can perturb the photon and electron 50 Chapter 4. Results and Discussion 51 fluence. Figure 4.1 illustrates the difference from unity of the stopping power ratios rela-tive to water for carbon (diamond), silicon (diode), air (ion chamber) and photographic emulsion (film). The diamond detector because of its near water equivalence (constant ratio near 1 with energy Fig. 4.1) and good spatial resolution has been found to be po-tentially ideal for small field measurements of photons as well as charged particles [118]. According to this figure (4.1), we see that a silicon diode.and an ion chamber filled with air would have different stopping powers than water. The diode was used with the stem parallel to the central axis to the beam to avoid an asymmetry in the profile caused by the shield [5]. The stopping power ratio ofthe photographic emulsion relative to water varies significantly with energy. This can potentially have some small effect for narrow photon beams since the mean energy of the spectrum varies slightly with the size of the field (beam hardening with field size reduction) as seen in figure 4.2. 4.1.1 B e a m profile and depth dose measurements 6 MV and 10 MV beam profiles taken at dmax in water for a 3.0 cm and a 1.0 cm diameter field respectively are shown in Figures 4.3 and 4.4. The radiation profiles measured with the diamond, diode and mini chamber consistently agree, the diode (sensitive area of 2.5 mm diameter) measuring a minimally steeper profile than both the diamond (sensitive area of 2.1 mm diameter) and mini chamber (sensitive area of 1.9 mm diameter), and that measured with film is less steep. The Markus chamber, because of its large size (sensitive area of 5.4 mm diameter), as expected produces the greatest blurring of the penumbra and hence exhibits the worst spatial resolution. Therefore the profiles measured with the mini chamber, diamond and diode detectors and also the film would most closely resemble the inherent radiation profile produced by the linac. These first three detectors also yield similar depth dose measurements (Fig. 4.5). Additional measured photon beam profiles and an EGS4-generated one are shown in Chapter 4. Results and Discussion 52 1.2 g "•8-DC 03 0.8 0.6 t c CL CL O CO s 0.4 0.2 0 . CarborVWater . AirAA/ater . SiliconAA/ater Film Emulsion/Water j i i i i i i i i i _i_ 0.1 0.3 0.5 1 3 5 10 15 Electron Energy (MeV) Figure 4.1: Mass collision stopping power ratios relative to water are shown for different media as a function of electron energy. Chapter 4. Results and Discussion 53 f S> 1.5 C 1 0.5 • • Photons Bectrons 1 . 2 3 4 Held diameter..(cm) Figure 4.2: Variations of the mean energies of the photon and electron.spectra with field sizes calculated in polystyrene at a depth of 2.5 cm with EGS4 for a 10 M V photon beam are shown. Chapter 4. Results and Discussion 54 so 0S-I DC 100 x^ x-x-x-x^ x-x-x-x^ x^ x,,. 80 P § 60 P 40 P 20 0 Film _x— Diode Diamond ____MINI ' l I I I L _ l I 1 1 1 1 "X~X rX =x =X EX =X =X fcX 0 0.4 0.8 1.2 1.6 2 2.4 2.8 Off-axis distance (cm) Figure 4.3: One half of the 6 M V beam profile is shown for a 3.0 cm diameter stereo-tactic applicator with 5 x 5 cm 2 primary collimator setting at 100 cm SSD and depth of maximum dose (1.6 cm). The dose profiles are measured with the diamond, diode, film, and MINI chamber and are normalized to the 100% dose on central axis. Chapter 4. Results and Discussion 55 Off-axis distance (cm) Figure 4.4: One half of the 10 MV beam profile is shown for a 1.0 cm diameter stereo-tactic applicator with 4x4 cm2 primary .collimator setting at 100 cm SSD and depth of maximum dose (1.92 cm). The dose profiles are measured with the diamond, diode, film, Markus and MINI chambers and are normalized to the 100% dose on central axis. Chapter 4. Results and Discussion 56 a) 0 L-J—' '—' '—' ' ' — I '—I 1 1 — I 1 — I " — I " 1 — I I l I I I I I I I I I I I I I I L 0.1 0.6 1.1 1.6 2.1 2.6 3.1 3.6 4.1 4.6 5.1 5.6 Depth (cm) 20 P 0 I ' 1 1 1 1 1 1 1 1 1 1 I 0 1 2 3 4 5 6 Depth (cm) Figure 4.5: Percentage depth doses measure with diamond, diode, and mini chamber for 10 M V photon beams using a 1 cm diameter stereotactic applicator is shown for (a) a 4 x 4 cm2 primary field size, and b) a 5 x 5 cm2 field size. The source surface distance is 100 cm. Chapter 4. Results and Discussion 57 Figure 4.6. Once again the diode and the film have similar resolution, while the Markus and IC10 ion chambers both lacked resolution. The EGS4 profile appears much steeper than any measured profile. The shape of this Monte Carlo simulated profile is further discussed in a subsequent section (4.2.2). A correction for the size of the detector can be applied by extrapolating to zero detector size or using deconvolution once the detector LSF is known [104]. We have studied this last approach and present it in the next section. Chapter 4. . Results and Discussion 58 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 Off-axis distance (cm) Figure 4.6: One half of the 6 M V beam profile is shown for a 3 x 3 cm 2 field size at a 100 cm SSD and depth of maximum dose (1.5 cm). The dose profiles are measured with the diode, film, Markus and IC10 chambers and are normalized to the 100% dose on central axis. A h EGS4-calculated profile (raw data uncorrected for source size) is also shown for comparison. Chapter 4. Results and Discussion 59 4.2 Part 2: Deconvolution We have developed a method to estimate the LSF of any detector and applied it to the Markus ionization chamber which has a complicated geometry. The proposed method does not assume any a priori function for the LSF but makes use of the measured data and the knowledge of an inherent beam profile. A computer simulation is used to demonstrate the approach. In addition, we present a simple method to estimate both the response function of a film densitometer by direct measurement of a slit image and the realistic inherent beam profile of a linac. A finite source size correction has been established for the Monte Carlo generated dose profile. 4.2.1 Simulation results The computer-simulated inherent and measured profiles are shown in Figure 4.7 with the corresponding LSF. The average distance or root mean square (RMS) error between the original LSF of radius R — 6 units (full length of the vector ZTLS) and the LSF recovered by deconvolution is plotted in Figure 4.8 against various ranks of the approximation matrix E. The RMS error values for the noisy situations (av = 0.05 and 0.25) for higher ranks would be too large and are set equal to 0.22 on this graph. When the noise is absent from the original data (ov = 0.0) we can see that the higher the rank of the matrix the better is the solution for the LSF (RMS error very small) and the increase in error for reduced ranks is relatively- slow. A better recovery process of the LSF for the noisy data implies achieving a significant reduction of the matrix E. The drop in RMS error with rank reduction is abrupt for noisy data with a small rise near lower ranks. The procedure of deconvolution and TLS will be illustrated for the case where ou = 0.05. We first construct an augmented matrix [A : b] for the noisy "inherent" profile and the noisy "measured" profile. The maximum ( c i ) and minimum ( 0 5 1 3 ) singular values for this augmented matrix would be Chapter 4. . Results and Discussion 60 Corrputer-simulated inherent profile Serri-elliptic distribution R=6 units (LSF) — Cornputer-sirrulated measured profile 10 20 30 40 50 60 70 80 90 100 110 120 Off-axis distance (arbitrary units) Figure 4.7: One half of the computer-simulated inherent and measured profiles are shown. The semi-elliptic distribution is normalized to 100 relative intensity on this graph. Chapter 4. Results and Discussion 61 0 100 200 300 400 500 Rank of matrix E Figure 4.8: The R M S error between the original semi-elliptic distribution of radius R = 6 units and the LSF recovered by deconvolution is plotted against the rank of the aug-mented matrix E for different levels of Gaussian noise ov. Chapter 4. Results and Discussion 62 cri = 18591.1 cr513 = 0.0 . By considering a rank of 513 for the augmented matrix [A : 6], the condition number would be infinite. The LSF which has an amplitude smaller than unity is hidden by a noise as high as an order of 108 in the TLS-solution vector for a rank of 512 (Figure 4.9a). For a second try, we have taken the first 350 singular values out of the 513 (o"350 = 0.72) and zeroed the rest (Figure 4.9b). In this case, the condition number of the approximated matrix is 25856.9 and there is still some major noise in the TLS solution vector. We approximate the highly ill-conditioned matrix with an even lower rank matrix and then find its LSF by using the TLS solution.. In this case, ai/ogo = 1878.1. This recovered LSF and the original one are illustrated in Figure 4.9c.. Some noise remains, in the tails of the vector ZTLS giving an average RMS error over the full length of the vector of 0.0027. The peaks of the LSF are within 5% relative error and a divergence appears mainly at the bottom edges of the peaks. No further improvement at this point was possible, and the solution is a best estimate in the total least squares sense. Not much improvement either was found when the measured profile was folded horizontally (mirrored) prior to deconvolution: we obtained 5.19% relative error for the peaks but the RMS error over full length ztis which was 0.00265 was slightly better and at a higher matrix rank of 104 (instead of 80). Ah additional fold in the vertical direction tends to modify the penumbra slightly and hence was avoided. 4 . 2 . 1 . a M a t r i x r a n k d e t e r m i n a t o r The proposed threshold which is obtained from the standard deviation of the errors is used as a starting point to find an appropriate rank estimation. However, we have found it difficult in general to rely only on this threshold and the situation becomes more complex when applied to real data. A more fundamental study is still needed to find a Chapter 4. Results and Discussion 63 1 .65 129 193 257 321 385 449 Spatial position (arbitrary units) 1 65 129 193 257 321 385 449 Spatial position (arbitrary units) c) 0.12 p 0.10 -0.08 -itud 0.06 -mpl 0.04 -< 0.02 -0.00 ( -0.02 I . Original LSF T — Recovered LSF I A. .A A A . -v. .A. -A . A 65 129 193 257 321 385 449 Spatial position (arbitrary units) Figure 4.9: The LSF (R = 6 units) recovered by deconvolution and TLS in the presence of Gaussian noise ov = 0.05 is shown for the ranks 512 and 350 (a and b respectively) of the augmented matrix E. In c) the original LSF is compared to the recovered one (Rank= 80). Chapter 4. Results and Discussion 64 robust strategy for finding an optimal rank determinator. Hence, a confirmation of the solution was obtained by reconvolving the recovered LSF with the inherent profile and comparing this with the measured data. The criteria was to have the higher portion of the profiles (dose levels higher than 10%) be equal. 4.2.2 Experimental implementation ~Y The effect of detector size on the intrinsic EGS4 penumbra has been studied by convolving the EGS4 profile with the semi-elliptic distribution (equation 2.11) for several detector radii R. The modelled profiles for a 3 x 3 cm2 6 MV photon beam at 1.5 cm in water are. shown in Figure 4.10a for R = 3.0 and 416 mm and are compared to the profile measured with the IC10 ion chamber. As expected the intrinsic EGS4 profile has the steepest falloff. We see that the IC10 ion chamber measurement is identical to the convolution of the EGS4 simulated profile with the semi-elliptic LSF of a detector of radius R = 4.6 mm. This provides confidence that the EGS4 does a good job of simulating the penumbra shape. However, the actual inner radius of the IC10 chamber is 3.0 mm. That is, convolution with an R = 3.0 mm LSF predicts a profile that is steeper than the measured IC10 profile. Our EGS4 profile might not be completely accurate in representing the inherent profile. The EGS4 simulated intrinsic profile is shown again in Figure, 4.10b as well as the profiles measured with film, diode and the Markus ion chamber. It is followed by two profiles that are very close to each other: the film and the diode which are believed to approach the true inherent profile of the linac. They show a better spatial resolution than the Markus chamber. The penumbra widths and the full width at half maximum (FWHM) of the corresponding detector LSF for measured and convolved profiles are presented in Table 4.1. As the detector radius is increased from 0.1 mm (essentially a perfect detector that reproduces the raw EGS4 profile) to 9.0 mm the effect of detector size in broadening the penumbra can be clearly seen. Chapter 4. Results and Discussion 65 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 Off-axis distance (cm) -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0, Off-axis distance (cm) Figure 4.10: 6 M V beam profiles in water at a depth of 1.5 cm and for field size 3 x 3 cm 2 at 100 cm SSD are shown, a) Two profiles are generated by the convolution of the EGS4 simulated profile with the semi-elliptic distribution of radius R = 3.0 mm and 4.6 mm and are compared to the raw EGS4 profile and the one measured by the IC10 ion chamber, b) Measured and EGS4-generated profiles are shown. The left side of the profiles is illustrated and these are normalized to 100% on central axis. (Note that the central axis position (0) is not displayed.) Chapter 4: Results and Discussion 66 FWHM |80 - 20%| |90 - 10% | (mm) (mm) (mm) Raw EGS4 - 1.7 3.7 R = 0.1mm 0.17 1.8 3.8 Wellhofer densitometer 0.997a 3.0 4.9 Diode - 3.1 5.1 R = 3.0mm 5.20 3.6 5.3 Markus 5.396 ' 5.1 7.7 ' IC10 - 5.1 7.7 R = 4.6mm 7.97 . 5.1 7.4 R = 7.0mm 12.12 7.3 10.3 R = 9.0mm. 15.59 9.1 12.9 a Direct measurement of a slit image 6 Deconvolution and TLS method », Table 4.1: Penumbra widths for a 3 x 3 cm2 6 MV EGS4 profile (raw data) convolved with different LSFs and profile widths measured with diode, film, Markus and IC10 ion chamber are given. The FWHM of the corresponding detector LSF are also shown. R is the detector radius. Although the EGS4 profile used in the. calculation might have the correct general shape, it is too steep in the penumbral region. This is not surprising when one considers the approximations made in the simulation. The most important approximation is the modelling of the source as an isotropically radiating point source of photons. This does not take into account the contaminant electrons and the scattered photons that originate from the flattening filter and other linac head components. Also the geometric penumbra is not accounted for. A blurring function to account for the finite source size effect needs to be determined. This correction to the EGS4-simulated profile will serve to broaden its penumbra and hence should produce a better agreement with the inherent profile of the linac. Chapter 4. Results and Discussion 67-4.2.2.a Realistic inherent profile The first step was to obtain a realistic inherent profile. We have used the LSF of the film densitometer obtained by scanning the narrow line on a film, this yields a F W H M of 0.997 mm, as well as the beam profile measured on film with the same densitometer to calculate a realistic inherent profile of the linac. The resulting LSF normalized to unit area is presented in Figure 4.11. A simple model where the LSF is assumed to be constant over the densitometer circular aperture of radius 0.6 mm is also shown for comparison [104]. The LSF according to this model, a step function, yields a FWHM of 1.2 mm and the shape of this LSF differs considerably from the other LSF. For this particular deconvolution problem, the LSF data composed the matrix A and the vector z that we are seeking was the inherent profile. In this task, the LSF is very much narrower than the profile as measured by the film where both were sampled at 0.1 mm. This better estimate of the inherent profile (final rank of the augmented matrix E 106/706) was used to solve for both the finite source size effect and for the LSF of the Markus chamber. An excellent match between this inherent profile convolved with the film LSF and the film measured profile confirmed the validity of the deconvolution solution (Figure 4.12). 4.2.2.b Effective source size for the EGS4-generated profile A blurring function for the finite source size effect which is not accounted for in the EGS4 profile has been determined using the realistic inherent profile along with deconvolution and TLS. The augmented matrix [A : b] was composed of the EGS4-generated inherent profile and the realistic inherent profile for the 3 x 3 cm2 field of 6 MV. A FWHM of 3.15 mm for the blurring function was found with the deconvolution technique (Figure 4.13). This solution was verified by convolving the raw EGS4 data with this blurring function and the resulting profile was in good agreement with the realistic inherent profile with small differences appearing only at dose levels lower than 10% (Figure 4.14). Chapter 4. Results and Discussion 68 1 . 0 0 . 9 I-0 . 8 0 . 7 •g 0.6 3 0.5 t o . 0.3 0.2 0.1 0.0 . Direct -measurement -Step function (2R = 1.2rrm) :3.0 -2.2 -1.4 -0.6 0.2 1.0 Spatial position (mm) 2.6 Figure 4.11: The LSF as determined by direct measurement of a slit image.for the Wellhofer film densitometer and according to a step function of width 2R — 1.2 mm are shown. .' Chapter 4. Results and Discussion 69 0 5 10 15 20 25 Off- axis distance (mm) Figure 4.12: The realistic inherent profile as solved by deconvolving the film data with the LSF of the densitometer is presented along with the raw EGS4 profile, the film measured profile and the reconvolved profile generated by convolving the realistic inherent profile with the LSF of the densitometer. Chapter 4. Results and Discussion 70 0.16 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 Spatial position (mm) Figure 4.13: The finite source size effect correction to the EGS4 data as solved with the deconvolution and TLS method is shown. Chapter 4. Results and Discussion 71 Off-axis distance (mm) Figure 4.14: The EGS4 profile corrected for finite source size is illustrated as well as the original raw EGS4 data and the realistic inherent profile of the Linac. Chapter 4. Results and Discussion 72 4.2.2.c L S F of the Markus chamber The LSF for the Markus chamber which has a 5.4 mm diameter sensitive volume and a large mechanical support (27 mm diameter) was obtained with the deconvolution and TLS technique. The augmented matrix [A : b] was constructed for the EGS4-generated inherent profile and the Markus measured profile for the 3x3 cm2 field of 6 MV. The maximum (01) and minimum (17434) singular values for this augmented matrix would be o~\ =6219.24 0-434 = 0.00 By considering a rank of 434 for the augmented matrix [A : 6], the condition number would be very large (infinite in practice). The vector ZTLS f° r a rank 429 (0429 = 1.15 x 10-13) is shown in Figure 4.15a. As we can see, the LSF is contaminated with a lot of noise (order ~ 1013) such that it is not possible to distinguish the real signal. An intermediate result where the first 259 singular values (0259 = 2.49) are retained is shown in Figure 4.15b. With a condition number <TI/<7259 = 2497.69, the problem becomes better conditioned. We can see a peak appearing corresponding to the LSF but that still needs some cleaning. Finally, we approximate the highly ill-conditioned matrix with an even lower rank matrix and then find the LSF from the TLS solution (Figure 4.15c). In this case, c T i / o i o i = 393.37 and the FWHM is 5.39 mm. An additional verification of the TLS solution was obtained by simply convolving the inherent profile with this LSF and comparing with the profile measured with the Markus chamber (Figure 4.16). The two profiles were found to be consistent. Chapter 4. Results and Discussion 73 a) •o 3 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 50 99 148 197 2 4 6 2 9 5 3 4 4 393 Spatial position (x 0.5 mm) b) d> •o a 1 50 99 148 197 246 295 344 393 Spatial position (x 0.5 mm) -6 -4 -2 0 2 4 6 Spatial position (mm) Figure 4.15; a) The LSF of the Markus chamber as solved with the deconvolution and TLS method is shown for the ranks 429 and 259 (a and b respectively) and in c) the peak of the LSF for a rank 101 of the augmented matrix E. Chapter 4. Results and Discussion 74 -32.0-24.0-16.0 -8.0 0.0 8.0 16.0 24.0 Off-axis distance (mm) Figure 4.16: The profile generated by convolving the realistic inherent profile with the LSF of the Markus chamber as solved by deconvolution and TLS is compared with the profile as measured with the Markus chamber and the realistic inherent profile. Chapter 4. Results and Discussion 75 4.3 Part 3: Scaling in inhomogeneous media In this study, it is shown that the density scaling approach for inhomogeneity correction in the convolution/superposition method fails to predict the dose in media of different atomic composition. An approach is proposed to improve the scaling. First, the per-turbations produced by the presence of inhomogeneities of different atomic numbers and different densities are illustrated in the following experiments. 4.3.1 Dose measurement in heterogeneous phantoms Dose measurements1 have been done in a heterogeneous phantom and also at the same position in a homogeneous polystyrene phantom (Figure 4.17). Slab type inhomogeneities have been investigated. An inhomogeneity correction factor (ICF) is defined as the ratio of the dose in a heterogeneous phantom Dhetero to the dose at the same distance in a homogeneous phantom Dhomo-j C F = D ^ t e r o ( 4 5 7 ) '•'homo The heterogeneous phantom consisted of a polystyrene phantom with a slab of a different material inserted horizontally at a depth d of 3.5 cm (perpendicular to the beam axis). The ratios of the dose as measured in the polystyrene directly beyond the slab of the heterogeneous medium (air or aluminum) relative to the dose at the same position in the homogeneous case (depth D) are plotted as a function of the field diameter and are shown in Figure 4.18. Two energies are compared: 6 MV (dotted line) and 10 MV (solid line). The decrease in the measured relative dose (ICF) is attributed to changes in electron transport and loss of electronic equilibrium. There is a loss in both longitudinal and lateral scattered electrons for the smaller field sizes. The perturbation at the distal 1 F i l m was used for off-axis measurements (profiles) and the Markus chamber was used for central axis measurements. Chapter 4. Results and Discussion 76 interface of the slab is more pronounced for the low density heterogeneity (air), for higher energy (10 MV) and smaller field size. The perturbation at the exit side caused by a 1.4 cm aluminum slab is smaller than the one due to the same thickness of air. The'dose reduction following the 1.4 cm aluminum slab is about 2% for a 10 MV beam and 9% for the 6 MV beam for the 1 cm diameter field. In comparison, the same thickness of air (1.4 cm) yields a dose reduction of about 40% for the 10 MV beam and 33% for the 6 MV beam and the 1 cm diameter field. For aluminum there is increased beam attenuation producing a decrease in dose, however, close to the interface there is an increase in electrons set in motion and this would produce an increase in dose. The beam attenuation is a more dominant effect in aluminum than the increase in electron fluence for the 6 MV photon beam compared to the 10 MV photon beam. But the two counter effects produce a minimal perturbation of the dose beyond a 0.1 cm aluminum slab whereas for air perturbations are observed for the smallest field sizes. Hence, air cavities such as the sinus and nasopharynx can produce uncertainties in dose distributions if neglected in dose calculations. For example, the diameter of a larynx is about 2 cm, so one would want to avoid having the treatment beam going through such . a cavity to reach the target. In the case of aluminum which has a high atomic number relative to water, the perturbation is less important at the higher energy (10 MV). The mass energy absorption coefficient of aluminum is close to the polystyrene one at higher energies. In reality, the electron density of compact bone and its effective atomic number are much less than that for aluminum. Therefore the perturbation is expected to be less in bone and in the skull which has a thickness somewhere between 0.5 cm and 1 cm. The ICF for a 10 x 1.0 cm2 field is also shown for comparison. At this larger field size, the dose reduction at the distal interface is not significant because lateral electronic equilibrium is established. This is true for the two types of heterogeneity studied. Beam profiles are also modified by the presence of inhomogeneities. In Figures 4.19 and 4.20 the profiles immediately after the heterogeneous slab (aluminum and balsa) is Chapter 4. Results and Discussion 77 Figure 4.17: The experimental set up for dose measurements at the distal interface of a heterogeneous phantom is shown. The heterogeneous slab is located at a depth d and the SSD is 100 cm. The dosimeter is positioned at a depth D. Chapter 4. Results and Discussion 78 o 10 xlO cm2 6 MV 1.4 cm air gap 10 MV 1.4 cm air gap -6MV1.4cmAI -10MV1.4cmAI -6 MV0.1 cm air gap 10 MV0.1 cm air gap - - -6 MV0.1 cmAI - ^ 1 0 M V 0 . 1 cmAI 1 1.5 2 2.5 3 3.5 Field diameter (cm) 4 . 4.5 5 5.5 Figure 4.18: Dose measurements at the distal interface of a heterogeneous phantom are shown. The heterogeneous slab is located at a depth of 3.5 cm and the SSD is 100 cm. The ratio of the dose in the heterogeneous phantom to the dose at the same position in the homogeneous phantom is taken (Inhomogeneity correction factor or ICF). Chapter 4. Results and Discussion 79 Off axis distance (cm) Figure 4.19: The 10 MV photon beam profile measured at a depth of 4.75 cm in a homogeneous polystyrene phantom is compared to that measured in an inhomogeneous phantom. In the inhomogeneous case, a 1.25 cm aluminum slab is located at a depth of 3.5 cm. The field diameter is 3.0 cm at a SSD of 100 cm. The doses are normalized to the dose on the central axis of the homogeneous phantom. sharper and more spread out respectively. The presence of inhomogeneities with different stopping and scattering powers perturb the transport of the secondary electrons differ-ently compared to inhomogeneities of only density variations. This is also observed by Yu [125]. 4.3.2 Properties of different media The mass scattering powers for different materials relative to water are shown in Figure 4.21. It can be seen that the scattering power varies very slowly with energy but is different for different materials. In Figure 4.22, the ranges (in electrons/cm2) are plotted Chapter 4. Results and Discussion 80 Off axis distance (cm) Figure 4.20: The 10 MV photon beam profile measured at a depth of 4.75 cm in a homogeneous polystyrene phantom is compared to that measured in an inhomogeneous phantom. In the inhomogeneous case, a 1.25 cm balsa slab is located at a depth of 3.5 cm. The field diameter is 3.0 cm at a SSD of 100 cm. The doses are normalized to the dose on the central axis of the homogeneous phantom. Chapter 4. Results and Discussion 81 2 cu I C L O ) c CD I CO CO E 05 g •Jo cc Figure 4.21: The ratios of mass scattering powers ((r/p)wai™m) are shown for various materials. for different media relative to water. The Z dependancy of the range and the angular spread (scattering) is such that an increase in these physical quantities is associated with a larger atomic number. The effective atomic numbers2 for different materials as calculated from equation 3.53 are given in Table 4.2. The 6 MV and 10 MV photon spectra used in the calculations in the next section have a mean energy of 1.91 MeV and 2.91 MeV respectively [71]. Balsa wood is often used to simulate lung tissue during measurements. Since balsa wood was used in our measurements, we have also used this wood in our EGS4 simula-tions. We generated the cross sections for balsa wood which are not available with the 2Note: For the calculation of Zef/, the scattering power was considered while for Z (Table 3.1), the photoelectric effect was used. Chapter 4. Results and Discussion 82 1.25 ct 0.95 0 5 10 E (MeV) 15 20 Figure 4.22: The ratio of the continuous slowing down approximation (CSDA) ranges (electrons/cm2) relative to water is shown for various materials. Chapter 4. Results and Discussion 83 Material Pe P Zeff SZterm 1-91 MeV SZterm 2-91 MeV Water 1.000 1.000 6.51 1.000 1.000 Polystyrene 1.011 1.044 5.24 0.978 0.976 Air 0.00108 0.0012 7.35 0.891 0.898 Aluminum 2.343 . 2.699 13.00 0.787 0.792 Balsa 0.16 0.17 6.56 1.000 1.000 Table 4.2: The properties of various phantom materials are given. The electron and physical densities (pe a n d p respectively) are relative to water and Zeff is the effective atomic number. The ratio of the average stopping powers relative to water are given for the mean energy of the 6 MV (1.91 MeV) and 10 MV (2.91 MeV) photon spectra. EGS4 distribution using the knowledge of its chemical composition and density3. The stopping power data for balsa were not available and were assumed to be identical to the ones for water since this is a biological material. 4.3.3 Dose calculations using various scalings EGS4 and convolution/superposition depth dose calculations along-with sonie measure-ments were made for a number of homogeneous and inhomogeneous phantoms as previ-ously shown in Figure 4.17. The point of calculation (dosimeter position on this figure) was varied to cover all the depths in the phantom (PDD). Polyenergetic photon beams of 10 MV and 6 MV were used in the calculations. Results were normalized to the maxi-mum dose value in the homogeneous water phantom and the source-surface distance was kept constant. In the inhomogeneous cases with the CVSP algorithm, the results were obtained for the so-called rectilinear scaling (O'Connor's scaling theorem, Ch. 2) and for the other proposed scalings as described in Chapter 3 and applied to the following heterogeneities: aluminum, air and balsa wood. The electron density only is used to scale the pathlength in the rectilinear scaling employed in superposition calculations [65]. Note 3 Chemical composition of balsa wood: C ( 4 9 % ) , H (6%), O (44%) and N (1%) [115]. Chapter 4. Results and Discussion 84 that the electron density is also present in the other scaling approaches as presented in Chapter 3. Hence when the expression 'scattering power' or 'stopping power' scalings are mentioned, these scalings are simply added to the previous rectilinear scaling. 'Full scaling' means that both 'scattering power' and 'stopping power' scalings are used in the algorithm along with the,rectilinear scaling. A 6 MV photon beam was used for the calculations of relative dose shown in Figure 4.23. The dose at depths in a homogeneous polystyrene phantom and one containing a 1-cm-thick aluminum slab placed at 4 cm from the surface are shown in this figure. There is a slight dose build-up at the proximal polystyrene-aluminum interface due to the electron backscattering into polystyrene from aluminum. There is also a small dose build-up in polystyrene at the distal aluminum-polystyrene interface due to increased electrons forward scattered. The dose following the aluminum slab remains lower than the dose in the homogeneous phantom- due to the decrease in radiation transmission. Generally there is good agreement between the EGS4 and CVSP calculated PDD in the homogeneous phantom. The deficiency of the rectilinear scaling is demonstrated in (a) where the dose perturbation effects are clearly underestimated inside and near the aluminum slab. However beyond the inhomogeneity there is agreement with EGS4. The CVSP algorithm based on density scaling simply treated aluminum as dense water. Aluminum has a smaller mass stopping power than water but a higher electron density. This will tend to increase the electron fluence and the amount of photon interaction respectively. The scattering of the electrons is also higher in aluminum than in water. In (b) when only either the 'scattering power' or the 'stopping power' scaling is used, the CVSP. dose in the aluminum deviates more from the Monte Carlo results. The result within the aluminum calculated using the 'full scaling' predicts more accurately the perturbation effect but some discrepancy still exists. Hence there is a better agreement between the CVSP curve with the full scaling and the one from EGS4 than there is with the rectilinear scaling for a high atomic number material such as aluminum. Chapter 4. Results and Discussion 85 a) 100 t 80 ON § 60 1 40 •35 20 0 0 — EGS4homo + CVSP homo EGS4 inhom CVSP inhom/rect scaling _i i i_i i_ j i_ 2 4 6 Depth (cm) 8 10 b) 100 80 60 d) w O T3 1 40 DC 20 . EGS4 inhom CVSP rectilinear scaling . CVSP Stop, power scaling . CVSP Scatt power scaling . CVSP Full scaling Measurements inhom 4 5 6 Depth (cm) 10 Figure 4.23:' Depth dose curves for a 6 MV photon beani are calculated by Monte Carlo simulation and convolution/superposition in polystyrene. The homogeneous case and the inhomogeneous case (1 cm aluminum slab at a depth of 4 cm) are illustrated in a). The field size is 3 x 3 cm2 at a SSD of 100 cm. In b) the rectilinear scaling is illustrated and compared to 3 other types of scaling and to measurements. Chapter 4. Results and Discussion 86 In Figure 4.24, the various scalings were applied to a low density inhomogeneity (air) which has an atomic number close to that of water and polystyrene. There is a significant dose reduction of about 5% at the proximal polystyrene-air interface that extends over about 3 mm and there is a high dose gradient in the distal build-up region. Backscattering from air is greatly reduced at the proximal interface. The second build-up occurs due to an increase in photon fluence following the air gap since the attenuation is reduced within it. Deeper in the phantom, the PDD is greater than the one for the homogeneous case. Wi th the heterogeneity, the 'scattering power' scaling does not affect the scaling much since only the Z dependance of the scattering power is used in the calculations. The -'stopping power' scaling is a slight improvement upon the rectilinear scaling. In all heterogeneous cases, the C V S P build-up and build-down curves each side Of the inhomogeneous slab diverge from the EGS4 ones. The disagreement between the C V S P and the Monte Carlo methods in those regions could be attributed to some lack in the C V S P code used. This is also observed for the PDDs in a 1 cm diameter 10 M V photon field shown in Figure 4.25. Again the 'stopping power' scaling yields a dose in air as calculated by C V S P closer to the EGS4-calculated one. . In Figure 4.26, the heterogeneous slab is made of balsa wood to simulate lung tissue. Even though the homogeneous PDDs are close to each other, there is a major divergence among the results from the C V S P algorithm and the EGS4 code in the inhomogeneous cases for all the various scalings. Balsa has an atomic number very similar to water and its stopping power is not accurately known. Hence only the information from the electron density of balsa is being used by the C V S P code. Our measurements seem to agree with the Monte Carlo data except inside the balsa and in the distal build-up region where the measured dose is above the EGS4-simulated dose. Chapter 4. Results and Discussion 87 a) CVSP homo ... CVSPirtomrect.scaling _EGS4homD ;_EGS4 inhom Measured homo ,_ Measured inhom 3 4 5 6 Depth (cm) 7 8 10 b) 0) W o T3 O > CD DC 100 c EGS4 homo EGS4 inhom CVSP inhom rect scaling CVSP inhom stop, scaling CVSP inhom scat, scaling CVSP inhom full scaling 4 5 6 Depth (cm) 8 10 Figure 4.24: Depth dose curves for a 6 M V photon beam are calculated by Monte Carlo simulation and convolution/superposition in polystyrene. The homogeneous case and the inhomogeneous case (1.25 cm air gap at a depth of 3.5 cm) are illustrated in a). The field diameter is 3.0 cm at a SSD of 100 cm. In b) the rectilinear scaling is illustrated and compared to 3 other types of scaling. Chapter 4. Results and Discussion 88 a) CD w. o T 3 CD > o cc 100 80 60 i 40 20 4 5 6 Depth (cm) 10 b) 100 t o •a _ra DC EGS4 homo - - - EGS4 inhom _ ^ C V S P rect. scaling fl C V S P scat, scaling -x—CVSP stop, scaling _ ^ - C V S P full scaling 4 5 6 Depth (cm) 8 10 Figure 4.25: Depth dose curves for a 10 M V photon beam are calculated by Monte Carlo simulation and convolution/superposition in polystyrene. The homogeneous case and the inhomogeneous case (0.6 cm air gap at a depth of 2.6 cm) are illustrated in a). The field diameter is 1.0 cm at a SSD pf 100 cm. In b) the rectilinear scaling is illustrated and compared to 3 other types of scaling. Chapter 4. Results and Discussion 89 <X> tn o o cc 0 EGS4 homo EGS4 inhom CVSP homo CVSP inhom rect. scaling CVSP inhom full scaling Measurements homo Measurements inhom 3 4 5 Depth (cm) Figure 4.26: Depth dose curves for a 10 M V photon beam are calculated by Monte Carlo simulation and convolution/superposition in a homogeneous polystyrene phantom. The heterogeneous phantom contains a 1.25 cm balsa slab at a depth of 3.5 cm. The field diameter is 3 cm at a SSD of 100 cm. Chapter 4. Results and Discussion 90 4.3.4 Discussion The introduction of a materialof .a different atomic composition changes the radiation transport of both photons and electrons in a. complex manner. In the case of a low density material with an atomic number close to water such as air and balsa, the primary transmitted radiation is increased because of the reduced attenuation inside the lower density material. This explains the higher dose (higher than the homogeneous case) in the distal region following the low density material slab. There are however less scattered photons inside the low density medium and also less charged particles set in motion. This explains the decrease in dose observed in the air gap and in the balsa slab. In the case of a high atomic number and high density medium such as aluminum, the primary transmitted radiation is decreased because of the increased attenuation inside the heterogeneous slab. This yields a lower dose in the distal region following the aluminum slab: Better agreement was obtained for dose calculations in heterogeneous phantoms with the Monte Carlo simulation when both the stopping and scattering properties of the material were included in the C V S P calculation (full scaling) for the high atomic number heterogeneity. There is only a minor improvement with the full scaling when the atomic number of the slab was close to water. The method is still applicable for media containing inhomogeneities of different electron densities and/or different atomic numbers. The proposed full scaling method is simple to apply since only a mean energy value was used to obtain the stopping power and only the Z dependence of the scattering power is used to scale the convolution kernel so that the 'distribution' of the energy deposition is accounted for. A n alternate approach would be to calculate the exact values of these two physical quantities according to the spectrum used. The convolution method did not resolve the build-up regions very well. The effects of the kernel and voxel resolutions as well as the ability to model the circular fields could require further modifications to the code. Measurements were not available inside the air cavity and inside aluminum. Chapter 4. Results and Discussion ' 91 There is generally a good agreement between measured data and Monte Carlo data. This gives confidence that no serious error occured in either method. However there is some discrepancy that remains unexplained in the case of balsa as a heterogeneity. The stopping power value of balsa is probably different than the value for water. Chapter 4. Results and Discussion 92 4.4 Part 4: Simple dose model Depth dose was modelled by the spring model. However, the analytical solution was . not simple so a direct numerical solution ofthe differential equations had to be applied. Homogeneous and slab-type heterogeneous phantoms have been considered. 4.4.1 Percentage Depth Dose in a homogeneous medium A homogeneous phantom would be modelled with a damping coefficient Cn that is con-stant for all sites n (depths). The effects of the various, parameters of the spring model (Eq. 2.29) on the depth dose curve are shown in Table 4.3. The amplitude a (Eq. 2.31) of the initial photon beam was set arbitrarily to 21 but does not affect the curve since it is normalized to 100% at the point of maximum dose afterward. As the damping constant of the medium Cn is increased from 1 to 10, the depth of maximum dose is reduced and the fall-off is more rapid. The damping constant has an analogy with the properties of the medium (stopping and scattering powers). A n increase in the "frequency" UJQ from 1 to 10 yields a more penetrating beam. This parameter could be related to the combined effect of the energy and field size of the photon beam. The effect of the variation of u>0 is also plotted in Figure 4.27. Still in Table 4.3, the effect of increasing (5 (Eq. 2.31.) is to reduce the depth of maximum dose and to cause a more abrupt fall-off. The parameter a has a behavior similar to the attenuation coefficient of a photon beam. The exact significance of these parameters should not be taken too rigorously as the separation of the primary and scattered dose is not explicit and therefore only analogies are possible, 4.4.2 Percentage Depth Dose in a heterogeneous medium Inhomogeneous cases are obtained by setting a different damping constant cn at some nodes (depths) (Fig. 4.28). In this figure, the nodes from 24 to 26 were set to twice the Chapter 4. Results and Discussion 93 dmax ^80 <^ 50 ^20 C n UJ0- /3-(iV + l ) 2.94 7.65 19.12 42.08 1 1 1 1.027 6.96 18.71 41.62 5 )> 1.00 6.68 18.43 41.34 10 j; )> 2.94 7.65 19.12 42.08 1 1 1 6.54 14.16 25.52 45.21 5 ii 11.00 22.75 33.32 45.28 » 10 ii 2.94 7.65 19.12 42.08 1 1 1 1.48 3.34 5.55 9.98 n » 5 1.00 2.46 3.66 5.89 ii 10 Table 4.3: Depths of maximum dose (dmax) and depths in the fall-off regions for 80, 50 and 20% of the maximum dose are given for different selections of the parameters Cn, uo, and f3. The number of oscillators (N + 1) is 50, the "time" of integration was 30 and the amplitude of the photon beam was 21. A l l the parameters are in arbitrary units in this table. damping constant of the remainder of the phantom. The resolution (smoothness) of the curve improves with the number of nodes (N + 1). A decrease in 'dose' is observed in the heterogeneous slab and is followed by another build-up. 4.4.3 Comparison of calculations to measurements (Monte Carlo) The effect of low-density tissue inhomogeneity on the dose in small high energy photon beams is illustrated in Figure 4.29. Ai r cavities which could be present in the head and neck region have been simulated by Monte Carlo. The dose drops dramatically by more than 20% in air and builds up again after the gap and exceeds by a few percent the dose in the homogeneous case. These findings agree with the ones from Solberg et al. [107]. This effect is accentuated as the gap is increased. We have attempted to reproduce some of these curves with our simple spring model. There were three parameters to be fitted which are shown in Table 4.4. These are nof best-fit parameters in a least-square sense but obtained by trial and errors. The homogeneous and heterogeneous 'PDDs' for this beam energy as calculated Chapter 4. Results and Discussion 94 0 10 20 • 30 40 50 Depth Figure 4.27: "PDDs" in homogeneous media generated by the spring model. The total number of nodes (N + 1) is 50, the damping coefficient Cn is 1 everywhere and the attenuation of the photon beam J3 is 1/(N + 1). The units are arbitrary for these curves. ICO 83 63 40 20 0 10 20 33 40 33 Dspth Figure 4.28: "PDDs" are shown for an inhomogeneity generated by the spring model. The total number of nodes (N+l) is 50, the damping coefficient c„ is 1 everj^where except in the slab (n = 24 to 26) where it is 2 and the attenuation of the photon beam j3 is l / ( i V + 1). The units are arbitrary for these curves. Chapter 4. Results and Discussion 95 100 80 § 60 •o i I 40 DC 20 V Homogeneous 3 mm air gap 6 mm air gap 10 mm air gap 13 mm air gap 6. 8 10 Depth (cm) 12 14 Figure 4.29: EGS4-simulated percent depth dose of a 10 M V photon beam of 1 cm diameter at SSD=100 cm in polystyrene is shown. The homogeneous case and cases with an air gap of thicknesses of 0.3, 0.6, 1.0, and 1.3 cm at a depth of 2.6 cm are illustrated. Chapter 4. Results and Discussion 96 Homogeneous 0.5 3 mm air gap 0.5 1.0 0.96/(iV + l) 0.5 0.96/(iV + l ) inside inhom 0.96/(N + 1) Table 4.4: Fitting parameters. with the spring model are also shown in Figures 4.30 and 4.31 respectively. The build-up and fall-off regions of this 10 M V beam can be represented with relatively good accuracy in the homogeneous case. The heterogeneous PDD as calculated with the spring model is a very rough approximation of the EGS4 data. 4.4.4 Discussion In conclusion, these equations provide a new simple approach to obtain PDDs including build-up and inhomogeneities but we are not able to determine a simple expression for the solution and this is a severe limitation. This model remains interesting as it is the first attempt in representing dose deposition with higher order differential equations. This model is at present not accurate enough for clinical use. Chapter 4. Results and Discussion 97 100 •a 1-1 o DC EGS4 homo Springs homo Depth (cm) Figure 4.30: The 10 M V photon beam in homogeneous media generated by the spring model and as calculated by Monte Carlo is shown. Chapter 4. Results and Discussion 98 r •••••••• i i i i i i i i i i i i i i i i i i i i i i i i i i 1 0 2 4 6 8 10 12 14 16 Depth (cm) Figure 4.31: Depth dose for a 10 M V photon beam in heterogeneous media is shown as generated by the spring model and as calculated by Monte Carlo. A 3 mm air gap is present at a depth of 2.6 cm. Chapter 5 Conclusion 5.1 Conclusions Careful collection of beam data and treatment planning must be carried out for linac-based radiosurgery to ensure accurate dose to the target volume. We have presented approaches to deal with three problems inherent to small photon beams in any media. Namely, the lack of detector resolution, the lateral electronic disequilibrium and the longitudinal electronic disequilibrium in heterogeneous media. We have presented a method which makes use of the deconvolution and SVD-based TLS approach to estimate the LSF of a bulky detector like the Markus chamber by using data provided by both an inherent profile estimation and measurements with the detector. The deconvolution process is highly noise sensitive and would normally yield an improper representation of the LSF. However, successful deconvolution in the presence of high levels of noise or error has been demonstrated with the TLS-SVD method. This was illustrated with computer simulated profiles. In the presence of Gaussian noise in the input data, the LSF in the vector solution was completely hidden under very high levels of noise. The T L S - S V D method allowed to approximate a solution. In a practical application, our technique is currently limited by the ability of the Monte Carlo data to simulate correctly the inherent data from the head of the linear accelerator (linac). To overcome this difficulty we have solved by deconvolution and T L S for a more realistic inherent beam profile of our linac using the information from both.profile data as measured with film and the film densitometer response function. The L S F of the densitometer was 99 Chapter 5. Conclusion 100 estimated with a simple method of direct measurement of a slit image and a full width at half maximum (FWHM) of 0.997 mm was recorded. Additionally, using the knowledge of this realistic inherent profile of the linac a blurring function representing the finite source size effect missing in our current Monte Carlo profile simulation was determined. Finally, with the realistic inherent beam profile we have applied the deconvolution and T L S method to find a LSF for the Markus chamber and found a resulting F W H M of 5.39 mm. Low density heterogeneities cause important dose reduction at the energies considered in the interface region. Significant dose reduction in balsa (lung) and air cavities as com-pared to water equivalent material (normal tissue) were observed. This would produce inaccuracies in dose distribution if not taken into account during treatment planning. The perturbation at the distal interface of a heterogeneous slab is less if 6 M V is used instead of 10 M V but about 4% of dose reduction was measured for the 6 M V beam following a 0.1 cm air cavity for the 1 cm diameter field. It is best to avoid going through an air cavity with the beam in an attempt to reach a target at the distal interface of that cavity. A drop in dose (about 5%) was also present at the proximal interface of a 1.25 cm air cavity due to less backscattering for the 6 M V beam of 3.0 cm diameter field. The perturbation was less if the heterogeneity consisted of a slab of higher atomic number material (aluminum) and this perturbation should be even less for bone. The skull is expected to cause only a negligible perturbation in SRS. The rectilinear scaling is not appropriate for high atomic number media but also does not account for the electron transport characteristics in low density media. The full scaling (when both the stopping and scattering properties of the material were included in the C V S P calculation) showed improvement over the rectilinear scaling for dose calculations in heterogeneous phantoms with the Monte Carlo simulation for the high atomic number heterogeneity. There is only a minor improvement with the full scaling when the atomic number of the slab was close to water. The full scaling method is still applicable for media containing inhomogeneities Chapter 5. Conclusion 101 of different electron densities and/or different atomic numbers. A prototype depth dose model with oscillators has been introduced in which some exchange of energy between the various sites in the media is possible. This is a rough approach to account for the notion of range of electrons set in motion by photons. A build-up region as well as the dose reduction in a heterogeneous slab can be simulated. 5.2 Future directions This section delineates areas of development which have yet to be explored. While the deconvolutions to account for detector size were all done on the raw data in this paper, other areas of investigation include preprocessing the data with smoothing, adding constraints to the Total Least Squares algorithm ("Constrained Total Least Squares" [117]), and finding a rigorous rank estimation approach. Once the inherent profile is, correctly determined, the deconvolution and T L S method is more flexible for finding the actual shape of the LSF of a detector. Moreover, it can be used on any possible shape of detector. Our Fortran program DOSE for recovering a detector LSF is at present limited by the ability of the EGS4 profile to model accurately the head of our linac. Future work can include using the B E A M code [94] to simulate the interactions in the linac head and produce a more accurate inherent dose profile. Profiles simulated using B E A M generated phase-space data should then allow the deconvolution method to more accurately solve for the different LSFs. This T L S SVD-based method could find various deconvolution applications in the medical physics field. The convolution/superposition code is a promising dose calculation method to account for electron transport and to achieve reasonable computation times. Studies and code modifications are needed to further improve the stereotactic field simulation along with dose calculations both inside and near heterogeneous media. Our scaling method is simple to implement but the level of sophistication can be increased and aimed at obtaining a Chapter 5. Conclusion 102 better match with measured and EGS4-simulated doses. Our investigation was restricted to slab geometry but other structures could be tested. In parallel to increasing the level of complexity in dose calculation there is also an interest in obtaining simpler and aesthetic models representing the complex radiation transport processes. The spring model is an attempt in this direction. The model is not a clinically realistic dose calculation tool. A simpler analytical solution is required to allow a handling of the parameters. The parameters could then be set semi-empirically instead of numerically extracted. Glossary Absorbed dose:' defined as the energy deposited by ionizing radiation per unit mass of material. The SI unit is Gray (J/kg). Bragg peak: Heavy charged particles penetrating a material in which nuclear interactions are negligible show a depth dose distribution in which most of the energy expenditure (peak) happens toward the end of the track. After this "Bragg peak" the dose decreases rapidly from its maximum as the particles run out of energy and stop. Charged particle equilibrium or electronic equilibrium: charged particle equilibrium or electronic equilibrium exists in a volume v if each charged particle of a given type and energy leaving v. is replaced by an identical particle of the same energy, in terms of expectation value. Fiducial marker: Marker employed as a standard of reference in imaging. Field size: defined as the projected collimator opening at the sour.ce-to-axis dis-tance. Full Width at Half Maximum (FWHM): Quantity to describe the narrowness of a distribution. Homogeneous medium: a medium that is comparable to unit density soft tissue in terms of electron density, physical density and atomic composition. Inhomogeneous media: a medium that is different from unit density soft tissue in terms of electron density,, physical density and atomic composition. 103 Chapter 5. Conclusion 104 • I s o c e n t e r : Point of intersection of the central beam axis and the gantry rotation axis. • L a t e r a l e l e c t r o n i c e q u i l i b r i u m : secondary charged particles are produced uni-formly throughout the volume V by an external source. Lateral charged particle' equilibrium is said to exist in a smaller volume v if the minimum distance separating the boundaries V and v is greater than the lateral range of the electrons. • L i n e S p r e a d F u n c t i o n ( L S F ) : Spatial distribution of the signal that results from an infinitely thin line stimulus. • M a s s S t o p p i n g P o w e r : defined as the expectation value of the rate of energy loss in a medium, per unit path length, per unit density, by a charged particle. • O f f - a x i s r a t i o ( O A R ) : defined as the ratio of the absorbed dose at a point away from the central axis to the absorbed dose at the same depth along the central axis of the beam. • P e n u m b r a : defined as the region at the edge of a radiation beam over which the dose rate changes rapidly as a function of distance from the beam central axis. The geometric penumbra is the portion of the penumbra produced by the finite radiation source size and the position of the collimators. Penumbra caused by the lateral scattering of the charged particles is called the radiation penumbra. • P e r c e n t a g e d e p t h d o s e ( P D D ) : defined as the quotient, expressed as a percent-age, of the absorbed dose at any depth to the absorbed dose at a fixed reference depth along the central axis of the beam. • P o l a r i z a t i o n e f f e c t : the nucleus and the orbital electrons of a medium are at-tracted in different directions by the electric field of a passing charged particle which results in the creation of electric dipoles with their own electric field. The Chapter 5. Conclusion 105 . atoms and the molecules become polarized developing a screening field against the influence of the passing particle which minimizes interaction probability reducing energy losses. • Radiological depth or pathlength (d e): defined as de = p e • d, where pe is the electron density of a phantom relative to water and d is the actual depth in the phantom. • Relative Output Factor ( R O F ) : defined as the ratio of the dose rate for a given field size at a reference depth to the dose rate at the same depth for the reference field size, • Source-to-axis distance ( S A D ) : defined as the distance from the source to the axis of gantry rotation. • Source-to-surface distance (SSD): defined as the distance from the source to the surface of a phantom. • Stereotactic radiosurgery (SRS): Precision radiation treatment of a particular (usually small) intracranial structure or lesion. 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Purdy, "Photon dose perturbations due to small inhomogeneities," Med. Phys. 14(1), 78-83 (1987). [126] C. X Yu, T. R. Mackie and J . W. Wong, "Photon dose calculation incorporating explicit electron transport," Med. Phys. 22(7), 1157-1165 (1995). Appendix A Historical Background 1895 Discovery of X-rays (C. Roentgen) 1896 1st X-ray treatment of a cancer patient 1906 Stereotaxy: Notion of 3-D coordinate system (R. Clarke) 1932 Cyclotron (E.O. Lawrence) 1947 1st human stereotactic operation 1949 Stereotactic apparatus of Swedish neurosurgeon Lars Leksell 1951 Stereotactic Radiosurgery (L. Leksell) late 1950s High energy heavy charged particle beam therapy 1968 Radiosurgery with Co-60 (L.Leksell) 1972 Computed tomography (Hounsfield) 1974 Use of Linac for radiosurgery 117 Appendix B Spring model: solution The equation of motion of coupled oscillators in a damped system from Chapter 2 (damp-ing coefficient Cn and spring constant k) is given by m<(t) + /c(2a;n(t)-xn+1(t)->n_1(t)) = cn<(t) (B.58) The solution has the following general form: xn{t) = Ane"L (B.59) We introduce the solution Eq. B.59 into equation B.58 and after some algebra we obtain the two following equalities: i V i + A f i _ 2 + JL[(2mq + cj)2 - 1] (B.60) g = - - [ l ± J l - ^ ( 2 - ^ 4 ^ ) " •• (B.61) m V Cn "n-We set An = c smn6 - (B.62) Using B.62 we can now rewrite Eq. B.59 as 118 Appendix B. Spring model: solution 119 xn(t) = Aneqt = c sinn8eqt : (B.63) To have the following boundaries x0 — xn+x — 0, weset 9 = mn/(N + 1) into Eq. B.63 with me J\f. We then get , -xn(t) = c s i n i ^ y t (B.64) This last expression for xn(t) along with Eq, B.61 yields: 9 = - 2 ^ ± 2 V ^ sin2m7r c 2 2(N + 1) 16mk which we insert into Eq. B.63 and set 7 = c/2m and u0 = ^k/m to get ) (B.65) 2 xn(t) = c s i n ( - ^ ^ - ) e 9 t - £ 7 3 ^ s i n ^ = - e x p [ ( - 7 ± 2zu;0 J - fyt) (B.66) One can verify that x n (t) is the solution to Eq. B.58. The following condition on the first derivative at t = 0 is also imposed: -<(0) = ae-W+V (B.67) We differentiate B.66, evaluate it at t = 0 and set it equal to B.67 _„ nvtVK n ae p n = }2 Brhsm——2iujc sih2m7r c 2 lN + l °\2(N + 1) 16mk which is (B.68) Appendix B. Spring model: solution 120 a J0N+l e - ^ s i n ( ^ ) d n dn • (B.69) r iV+i nm7r . nqn sin——-sin——-2zo>o = E ^ / 0 s m ^ S m > 7 ^ ^ 0 V 2 ( i V +1) 16m* The right hand side is V- D rN+l • nm-K • nqn Q- / s in 2 m7r Si-Wry = BQ—-—2lUQ< sin2g7r c2 (B.70) 2 ^ ° \ | . 2 ( i V + l) 16mA: and the left hand side of the equality B.69 after evaluation of the integral becomes:: af e - ^ s i n ( - ^ - ) d n = „ N + ' (1 - cosgyre-^^ 1)) . (B.71) and we can rewrite the coefficient Bq as ' 2g7ra x _ e - W W . 9 /?2 + ^-Aulrmsin(^) +~^2 We now get xn(t) as • ' ^(t) = £ ^ s i n ^ - e - ^ s i n h ^ / 7 2 - 4 ^ g r m y n 2 ( 2 ( ^ (B.73) Now x n is differentiated and squared to obtain the energy deposited. Dn= Cn(x'n(t))2dt (B.74) Jo The expression becomes rather large and there is no known simplification at this point. 

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