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Monte Carlo simulation of radiation transport in inhomogeneous phantoms irradiated using sterotactic… Cho, Jongmin 2004

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Monte Carlo Simulation of Radiation Transport in Inhomogeneous Phantoms Irradiated using Stereotactic Radiosurgery Beams. by Jongmin Cho B.Eng. (Electronic Material and Devices Engineering), INHA University, 1996 M.Sc. (High Energy Physics), Seoul National University, 1998 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science In the Faculty of Graduate Studies Department of Physics and Astronomy  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA FEBRUARY 2004 © Jongmin Cho, 2004  Library Authorization  In presenting this thesis in partial fulfillment 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.  Name of Author (please print)  The University of British Columbia Vancouver, BC Canada  Date (dd/mm/yyyy)  Abstract 6 MV stereotactic beams, with collimators of size 25 mm and 40 mm, were used to irradiate many homogeneous phantoms of different densities to quantitatively study charged particle disequilibrium. In each phantom, the magnitudes of the collisional KERMA and dose were compared. When the collisional KERMA has almost the same magnitude as dose, lateral charged particle equilibrium is relatively well established, which is the case for homogeneous water, aluminum and bone phantoms. However, when the collisional KERMA has significantly higher magnitude than the dose, there is serious charged particle disequilibrium, which is the case for homogeneous balsa and cedar phantoms. The same beams were used to irradiate many composite phantoms to study the origins of over-dosage and under-dosage in inhomogeneities. In the composite phantoms that contain balsa and cedar wood slabs, a reduction of dose in balsa and cedar was observed. This dose reduction is attributed to the longer lateral range of electrons in low density materials and therefore rapid increase of lateral charged particle disequilibrium. In the composite phantom that contains a bone slab, there was no significant change of dose in the bone slab. This is because there is no change of lateral charged particle equilibriumfrompolystyrene slabs to the bone slab. These observations have immediate relevance to small beam Stereotactic Radiosurgery in the head and neck region since this region has high density (bones) and low density (air cavities) material between water equivalent tissue.  ii  Acknowledgements My earnest appreciation goes to God who guided and helped me throughout my life, especially during the difficult time of the final stage of my M.Sc. thesis. I give thanks to Dr. James Robar and Dr. Ermias Gete, both of whom have given me much good advice throughout my research and thesis. Special thanks goes to Dr. Robar who provided me with a BEAM file. I express my deepest gratitude to the external readers, Dr. Paul Mobit, Dr. Jan Seuntjens and Dr. Jacob Haider for their time, effort and impartial, independent reviews of my thesis. Without their help, the completion of my M.Sc. thesis would have undoubtedly been impossible. My sincere thanks to Dr. Ding for his criticism and to Dr. Alex Mackay, Dr. Lloyd Skarsgard, Dr. Brenda Clark and Dean Ann Rose who have supported me through my most difficult times. I thank my colleagues: Peter Petric, Karl Otto, Alanah Bergman, Regan Sibbald, Gary Lim, and Moira Schmuland for their help with various aspects of this thesis including proof-reading and assistance with the experimental measurements. My fondest appreciation goes to Peter, Karl, Alanah and Vicky who have given me much help and advice. I thank the BC Cancer Agency and its staff for their funding and data. Dr. Brenda Clark provided diode measurement data, and Dr. Ellen Grein and Moira Schmuland provided ion chamber measurement data. My loving appreciation goes to all of my friends at St. Andrew's Hall as well as all the friends who prayed for me, and my family in Korea. Completing this work without the emotional supportfrommy friends and the financial support from my family would have been difficult. Lastly, I would like to thank Dr. David Measday for his encouragement and above all, his unwavering support, which went beyond the expectation a student can have of his supervisors.  iii  Table of Contents Abstract  "  Acknowledgements  Hi  Table of Contents  iv  List of Tables  vii  List of Figures  viii  1  Introduction  1  1.1 Stereotactic Radiosurgery (SRS)  1  1.2 Linear Accelerator  3  1.3 Motivation for Research  6  2 Radiation Physics  7  2.1 Photon Interactions  7  2.2 Electron Interactions  9  2.3 Stopping Powers  11  2.3.1 Mass Stopping Powers  11  2.3.2 Restricted Stopping Powers  12  3 Radiation Dosimetry  15  3.1 Particle Fluences  •  15  3.2 Linear Attenuation Coefficients  16  3.3 Mass Attenuation Coefficients  17  iv  3.4 K E R M A  18  3.5 Charged Particle Equilibrium  21  3.6 Dose...  24  3.7 Lateral Charged Particle Disequilibrium  26  3.8 Dosimetry in Inhomogeneous Media (Bragg-Gray Cavity Theory and Spencer-Attix Cavity Theory) 4 Monte Carlo Simulation in Radiation Therapy 4.1 Monte Carlo Simulation  31 34 34  4.2 Physical and Statistical Principles of Monte Carlo Simulation for Dose Calculation. 35 4.2.1 Example of Photon Transport - Compton Scattering  35  4.2.2 Example of Electron Transport - Condensed History (CH) Technique  41  5 Methods and Materials  42  5.1 EGSnrc User Codes  42  5.2 Beam Generation Using the BEAMnrc User Code...  44  5.3 Material Files (PEGS4 Files)  49  5.4 Dose Calculation Using the DOSXYZnrc User Code and Ion Chamber and Diode Measurement  52  5.5 Dose Calculation Using DOSRZnrc and Fluence and Spectrum Calculation Using FLURZnrc  57  5.6 Methodology of Collisional K E R M A Calculation from Photon Fluence  61  6 Results and Discussions  62  6.1 Comparisons of Monte Carlo Simulations with Ion Chamber and Diode Measurements  62  6.2 Collisional KERMA Calculation from the Photon Fluence in the Homogeneous Water Phantom  72  6.3 Collisional KERMA Calculations in Non-Unit Density Homogeneous Phantoms  78  6.4 Comparison of Dose with Collisional KERMA in the Composite Phantoms  84  7 Conclusions and Future Directions  91  7.1 Conclusions  91  7.2 Future directions  92  References  94  vi  List of Tables  Table 3.1 The ranges of electrons in various materials at several electron energies. _ 29 Table 5.1 Several physical properties of a number of materials used for phantoms. _ 50 Table 5.2 Materials of slabs used to construct composite phantoms.  vn  56  List of Figures  Figure 1.1 The head of a linear accelerator.  3  Figure 1.2 Linear accelerator.  5  Figure 2.1 Relative components of the photon cross section in water.  7  36  Figure 2.2 The energy transfer from a photon beam to a medium in a Compton scattering interaction. The photon interacts at point (a) with a free electron, transferring kinetic energy E to an electron and leaving the atom with reduced tr  energy in the form of a scattered photon. The transfer of energy at (a) is called KERMA (defined in the section 3.4). The electron with kinetic energy E  t r  collides with many atoms in the medium while losing energy ((b) and (c)). E at b r  (d) is the energy carried away by bremsstrahlung radiation resulting from an electron-nucleus collision. The delta ray at (c) is another electron resulting from a relatively violent electron-electron collision. The absorbed dose equals the KERMA less the energy carried away as bremsstrahlung radiation (called the collisional KERMA, which is defined in the section 3.4) and this energy is absorbed in the medium along the electron tracks at (b) and (c).  10  Figure 3.2 a) Lateral dose profile (perpendicular to the beam direction) of a pencil photon beam: the dose profile is spread laterally with a decreased magnitude because of the lateral electron dispersion caused by random electron scattering, b) lateral dose profile of a large field photon beam: Point A - dose is greater than the collisional KERMA because the decreased dose from lateral electron dispersion is compensated by dose contribution from adjacent electrons and dose is deposited by electrons generated further upstream, Point B - the dose is smaller than the collisional KERMA because of the lack of scattered electrons from the left, Point C - the dose is non-zero even though it is outside the field because of the scattered electrons from the right. Lateral charged particle equilibrium exists at point A, but not at points B and C.  39  27  Figure 3.3 a) Lateral dose profile of smallfieldstereotactic radiosurgical beams of megavolt energy irradiating phantoms made of low density media: Electron  viii  lateral dispersion is more pronounced in phantoms made of low density media and results in wide lateral dose spread. The combination of a lack of lateral electron equilibrium due to the small field, and more lateral dose spread caused by the low density media makes the lateral dose profile significantly lower than the lateral collisional KERMA profile, b) The depth dose curve has notably lower values than the depth collisional KERMA curve when the field size is small and the density of the phantom is low.  30  42  Figure 4.1 The selection of the kind of interaction through random number generation for a 10 MeV photon beam incident on a water phantom. The photo 36  electric probability is negligible. The proportion of Compton interaction is 37  and that of pair and triplet production is Figure 4.2 Schematic diagram of Compton scattering.  38  33  Figure 4.3 Klein-Nishina cross section for 6 MeV photon beam.  37  39  Figure 5.1 Component modules of the head of a Clinac 2100 C/D with 40 mm tertiary collimator.  46  72  Figure 5.2 The 3D diagram of the component modules of the head of a Clinac 2100 C/D.  49  72  Figure 5.3 Geometry of homogeneous phantoms used for DOSXYZnrc simulations. Water, aluminum, bone, balsa and cedar wood were used as materials of each homogeneous phantom.  54  Figure 5.4 Geometry of composite phantoms used for DOSXYZnrc simulations. Three phantoms were constructed. In each phantom, a bone slab, balsa wood slab, or cedar wood slab is sandwiched between water equivalent slabs made of polystyrene. The thickness and material of each slab is shown in Table 5.2.  56  Figure 5.5 Geometry of homogeneous phantoms used for DOSRZnrc and FLURZnrc simulations. Water, aluminum, bone, balsa and cedar wood were used as materials for each homogeneous phantom.  59  Figure 5.6 Geometry of composite phantoms used for DOSRZnrc and FLURZnrc simulations. Three phantoms were constructed. In each phantom, either a bone  IX  slab, a balsa wood slab, or a cedar wood slab is sandwiched between polystyrene slabs. Slab thicknesses are given in Table 5.2.  60  Figure 6.1 PDD curves comparing the MC simulation and the ion chamber measurement in the water phantom using a) 25 mm and b) 40 mm diameter beams, SSD = 98.5 cm.  65  Figure 6.2 Lateral dose profile comparison of the Monte Carlo simulation and the diode measurement in the water phantom at 7.5 cm depth using: a) 25 mm and b) 40 mm diameter beams, SSD = 92.5 cm.  67  Figure 6.3 Comparison of PDD curves for the Monte Carlo simulation and the ion chamber measurement using a 25 mm diameter beam for a) balsa and cedar wood phantoms and for b) aluminum and bone phantoms.  69  Figure 6.4 Comparison of PDD curves for the Monte Carlo simulation and the ion chamber measurement using a 40 mm diameter beam for a) balsa and cedar wood phantoms and for b) aluminum and bone phantoms.  70  Figure 6.5 Depth dose curves calculated from DOSXYZnrc and DOSRZnrc in the water phantom using the 40 mm diameter collimated radiation beam.  71  Figure 6.6 a) Variation of the photon fluence with depth, b) photon spectrum at a depth of 1.5 cm and c) photon energy fluence with depth in a water phantom irradiated by 25 mm and 40 mm diameter beams at SSD = 98.5 cm.  73  Figure 6.7 a) The collisional KERMA simulated from DOSXYZnrc was compared with the collisional KERMA calculated from equation [5-1] in the water phantom irradiated by 25 mm diameter beam. They are identical within statistical uncertainty, b) Comparison of the Monte Carlo depth dose curve with the depth collisional KERMA curve in the water phantom irradiated by 25 mm and 40 mm diameter beams, c) the region of depth 0 - 5 cm is magnified for better visibility.  76  Figure 6.8 Comparison of the lateral profile of the Monte Carlo dose with the lateral profile of the collisional KERMA at a depth of 5 cm in the water phantom irradiated by 25 mm and 40 mm diameter beams.  X  78  Figure 6.9 Comparison of Monte Carlo depth dose curves with the collisional KERMA in aluminum and bone phantoms irradiated by the 6 MV 25 mm diameter beam.  79  Figure 6.10 Comparison of the Monte Carlo depth dose curves with the collisional KERMA calculated from the photon fluence in balsa and cedar wood phantoms irradiated by the 6 MV 25 mm diameter beam.  80  Figure 6.11 Lateral profiles of the collisional KERMA and the Monte Carlo dose profile at a depth of 5 cm and b) the Monte Carlo dose profiles at various depths for the 25 mm diameter 6 MV beam in the balsa phantom.  83  Figure 6.12 Variation of the electron average energy with depth for various materials.  84  Figure 6.13 Comparison of the depth dose and the depth collisional KERMA in a polystyrene-balsa-polystyrene composite phantom down the centre line. The composite phantom is composed of a 2.54 cm thick balsa wood slab sandwiched between two polystyrene slabs. The borders of the inhomogeneity are indicated in the graph. This phantom is irradiated by a 6 MV beam with a 25 mm diameter collimator. The depth dose in a unit density homogeneous phantom (polystyrene) is shown for comparison.  86  Figure 6.14 Comparison of the depth dose curve and the collisional KERMA in a polystyrene-cedar-polystyrene composite phantom. The composite phantom is composed of a 3 cm thick cedar wood slab sandwiched between two polystyrene slabs. The borders of the inhomogeneity are indicated in the graph. This phantom is irradiated by a 6 MV beam with a 25 mm diameter collimator. The depth dose of a unit density homogeneous phantom (polystyrene) is shown for comparison.  87  Figure 6.15 Comparison of the depth dose curve and the collisional KERMA in a polystyrene-bone-polystyrene composite phantom. The composite phantom is composed of a 1 cm thick bone slab sandwiched between two polystyrene slabs. The position of the bone slab is indicated. This phantom is irradiated by a 6 MV beam with a 25 mm diameter collimator. The depth dose of a unit density homogeneous phantom (polystyrene) is shown for comparison.  XI  89  1 Introduction  1.1 Stereotactic Radiosurgery (SRS)  Stereotactic radiosurgery is the precise delivery of radiation to inoperable small lesions in the head and neck; the objective is to spare the surrounding normal regions using small radiation beams. ' 1  2  Stereotactic radiosurgery differs from conventional  radiotherapy in the following ways: the volume of tissue is usually smaller, localization is very precise, and the number of fractions delivered is much smaller, usually a single fraction, and thus the dose per fraction is much larger. ' Although stereotactic radiation 3 4  is also given as a fractionated treatment, in which case it is called Stereotactic Radiation Therapy (SRT), the issues of small photon field radiation dosimetry equally apply to 4  SRS and SRT. The most important feature of stereotactic radiosurgery is the localization of dose to a well-defined lesion to minimize the damage to neighboring normal brain structures. ' A steep dose fall-off outside the lesion is the ideal dose distribution; this is 2 5  achieved by using a large number of ports or arcs for photon radiosurgery and by improving depth dose characteristics for charged particle radiosurgery. Localization and dose in stereotactic radiosurgery need to be accomplished with a positional accuracy of ± 1 mm and a dose accuracy of 5 %.  6  Because of the small size of the stereotactic radiosurgical beam, which is usually 1 to 4 cm in diameter, it is difficult to accurately measure dose using conventional detectors. For ion chamber measurements, the measurement gives rise to volume averaging effects over the steep dose gradients.  1  7—11  *  The volume averaging effect occurs  when there is a gradient of dose over the area that a detector measures. Large field beams have fairly large flat lateral dose profiles, which make accurate dose measurements possible. The volume averaging effect is not critical for conventional ion chamber measurements until the beam size is reduced beyond approximately 4 cm in diameter. However, stereotactic radiosurgical beams, which are usually smaller than 4cm in diameter have quite uneven lateral dose profiles. Therefore the volume averaging effects of large sized conventional detectors is inevitable. For this reason, using small size detectors such as diamond detectors , diodes and films is desirable because they have a 12  better spatial resolution and therefore a minimal volume averaging effect. The principles of stereotactic radiosurgery as well as the characteristics of stereotactic  radiosurgical beams  references. ' 1-6  13-16  inhomogeneities 17  are discussed  in many  Surendra showed the reduction of dose in the presence of high density due to  inhomogeneities.  and their applications  the  enhanced  photon attenuation  in the  high  density  Timothy showed the reduction of dose following a low density  inhomogeneity due to the absence of lateral electron equilibrium.  Indrin showed the  drop of dose in low density lung equivalent material sandwiched between two solid waters quantitatively.  19  2  1.2 Linear Accelerator  x-rays Patient  Figure 1.1 The head of a linear accelerator.  3  The linear accelerator or linac is a machine that generates high energy x-rays or electrons for radiation therapy.  20-22  In the linac, electrons are injected into an  accelerating waveguide and then accelerated by electromagnetic fields. These accelerated electrons are bent 90° or 270° using bending magnets and then strike either a high Z target to produce photons or a scattering foil to produce a spread out electron beam. When the electrons hit the target, electron energy is released mainly as bremsstrahlung xrays. These photons are emitted primarily in the forward direction relative to the incident electrons at megavolt energies. Photons first pass through the conical hole of the tungsten primary collimator. Since photons are emitted primarily in the forward direction, the beam intensity is highest at the centre of the beam and the beam intensity decreases laterally from the centre. For the ideal radiation therapy, the beam intensity should be as uniform as possible. To make this non-uniform x-ray beam uniform, a flattening filter, usually made of copper, is placed in the beam. The thickness at the centre of the flattening filter is larger than the thickness at the edge. The higher intensity region at the beam centre therefore passes through the thicker part of the flattening filter thus, decreasing the intensity and making the beam intensity uniform across the beam. Below the flattening filter, several monitor ion chambers measure the radiation output as well as beam flatness and symmetry. A mirror for the light localizer is placed below the flattening filter. This mirror reflects a light source onto the same area that will be irradiated by the x-ray beam. This light field is used to monitor the position of the beam along with the external lasers that indicate the centre of the beam. The secondary  collimator is located below the mirror. It usually consists of two pairs of orthogonal blocks of heavy metal with the divergent opening at the centre. For stereotactic linac based radiosurgery a tertiary collimator is placed below the secondary collimator. This tertiary collimator may be a micro-multileaf collimator or a circular fixed collimator. Since the field size of the beam is defined at isocentre, the size of the collimator is designed to produce the desired field size taking into account beam divergence and collimator to isocentre distance. The linac gantry can rotate 360° about the isocentre and the distance from the radiation source to the isocentre is always the same. Additionally, the rotatable couch can translate and rotate to position the target in the appropriate spot for treatment.  Linac head j  Vertical  Stereotactic/  Lateral Isocentre . ijj£0  Longitudinal - into page .90'  T  /  y  Linac rotational angle  Couch rotational angle  Figure 1.2 Linear accelerator.  5  1.3 Motivation for Research  Dose calculations and measurements in inhomogeneous composite phantoms irradiated by small field stereotactic radiosurgical beams are difficult problems for two reasons. First, there is a lack of lateral charged particle equilibrium for the small field stereotactic beam.  12,  2 3 - 2 7  Second, there is a lack of longitudinal charged particle  equilibrium when the beam passes through different materials, as is the case of the headand-neck or lung treatment sites. ' The first part of this thesis studies the difference between the collisional KERMA and dose when lateral charged equilibrium exists or does not exist. For this study, both high and low density phantoms irradiated by small field stereotactic beams are designed to create the condition that lateral charged particle equilibrium exists and does not exist, respectively. The second part of this thesis investigates the origins of the overdosage and underdosage common in head-and-neck treatments using small field stereotactic beams. For this research, inhomogeneous composite phantoms were designed and irradiated by small field stereotactic beams to create conditions such that longitudinal and lateral charged particle equilibrium does not exist.  6  2 Radiation Physics 2.1 Photon Interactions When a photon beam irradiates a medium, electrons are produced by the photoelectric interaction, Compton scattering, and pair and triplet production.  For  water, the photoelectric interaction is dominant at a lower photon energy (kiloelectron voltage range), Compton scattering is dominant at an intermediate photon energy (0.1 MeV - few MeV), and pair and triplet production is dominant at a high photon energy (larger than few MeV). Rayleigh scattering is the coherent scattering of a photon with the bound electrons in an atom, as a result no energy is transferred to the electrons (in other words, no electrons are produced). Figure 2.1 shows the relative components of the photon cross section as a function of incident photon energy in water. Photons: Relative components 1 .  ,  ,  ,  ,—  i z n ^ . = . . » . >  T  '  •  •—  '  •  •  I  Photon energy (MeV)  Figure 2.1 Relative components of the photon cross section in water.  7  The photoelectric interaction, Compton scattering, pair and triplet production all create vacancies in atomic shells and as a result, characteristic x-rays or Auger electrons are produced.  Pair and triplet production create electron and positron pairs. The  37  positron created annihilates into two gamma rays, each with 0.511 MeV, when it collides at rest with an electron. From the combination of the above interactions (Photoelectric interaction, Compton scattering, pair and triplet production), the incident photon energy is eventually transferred to electrons in the form of electron excitation and ionization. The energy transferred at this stage can be represented by the average kinetic energy transferred from photons to electrons  (Etr).  Compton scattering is the dominant interaction in most radiation therapy that uses a few megavolt x-ray beam on the human body, most which is water equivalent. In Compton scattering, the energy transferfroma photon beam to the medium takes place in two steps. The first step entails the interaction of an incident photon of energy hv with an orbital electron, assumed to be free and stationary. The incident photon energy is much higher than the electron binding energy, so the electron binding energy is ignored and thus the electron is considered as afreeelectron. The incident photon with energy hv causes an electron to be set in motion with kinetic energy E and the rest of the energy, hv - E , leaves as a scattered photon. The tr  tr  Compton scattering typically occurs with electrons in the outermost shells where the binding energy of the electron is considered insignificant.  8  2.2 Electron Interactions  After the photon energy is transferred to an electron, the second step entails the transfer of energy from this high-energy electron to the medium through ionization and excitation, causing numerous electrons to be set into motion. The kinetic energy of the initial electron is continuously lost through many collisions along the electron track and eventually all of the energy is absorbed in the medium in the form of chemical changes and heat. The energy transfer of the initial electron to free or orbital electrons can be categorized into two different collisions, a soft collision and a hard collision. In a soft collision, the initial electron transfers only a small amount of energy. When a significant fraction of the energy of the initial electron is transferred, it is called a hard collision. An electron produced from a hard collision creates its own track and loses its energy while colliding with other electrons and this electron is called a delta ray. The combined energy transfer from soft and hard collisions is referred to as the average energy absorbed in the medium and is represented as Eab. In this second step, however, a small portion of energy is carried away as bremsstrahlung radiation without being absorbed in the medium. Bremsstrahlung radiation is emitted when an electron decelerates rapidly. As an example, for a 6 MeV photon beam interacting with a water phantom, the average kinetic energy that photons transfer to electrons (Etr) is 3.99 MeV and the average energy absorbed in the medium along the electron track (Eab)  is 3.91 MeV.  33  The difference between the incident photon energy,  6 MeV and the photon energy transferred to the primary electron, 3.99 MeV is 2.01 MeV. This energy is carried away by scattered photons.  9  Scattered  Bremsstrahlung radiation (d)  Figure 2.2 The energy transfer from a photon beam to a medium in a Compton scattering interaction. The photon interacts at point (a) with a free electron, transferring kinetic energy E  tr  to an electron and leaving the atom with reduced energy in the form of a  scattered photon. The transfer of energy at (a) is called KERMA (defined in the section 3.4). The electron with kinetic energy E collides with many atoms in the medium while tr  losing energy ((b) and (c)). Etr at (d) is the energy carried away by bremsstrahlung radiation resulting from an electron-nucleus collision. The delta ray at (c) is another electron resulting from a relatively violent electron-electron collision. The absorbed dose equals the KERMA  less the energy carried away as bremsstrahlung radiation  (called the collisional KERMA, which is defined in the section 3.4) and this energy is absorbed in the medium along the electron tracks at (b) and (c).  10  The difference between the photon energy transfer Etr and the average energy absorbed in the medium Eab is 0.08 MeV. This energy is carried away as bremsstrahlung radiation (Esr).  Etr — Eab +  Thus,  .33  [2-1]  ElSr  While the energy absorption ( Eab) occurs along the electron track over a distance equal to the range of the electron, the energy transfer (Etr) occurs at the point where the photon interacts with a free electron as illustrated in Figure 1.2.  2.3 Stopping Powers 2.3.1 Mass Stopping Powers  Charged particles (such as electrons and positrons) set in motion by the incident photon beam lose energy while transversing the medium. ' 33  34  This loss of energy is  dE described by the stopping power, which is expressed as — (i.e. the energy loss per unit dx thickness of medium in units of MeV/cm). The mass stopping power is a more useful quantity. This quantity is obtained by dividing the stopping power by the density of the medium, thus giving the energy loss per unit thickness of medium measured in g/cm . 2  s=-  \(dE\  .33  [2-2]  p\dx)  11  The mass stopping power, or the total mass stopping power, consists of the ionizational stopping power,  Si „ 0  and the radiative stopping power  (or it is also called the collision stopping power,  S ii) co  (Srad)-  S — Sic, + S d ~ Stot ra  [2-3]  33  [2-4]  33  where _l(dE^ \dxj  , kjrad —  P \ dx )  rai  The ionizational stopping power represents the rate of energy loss per unit path length of a charged particle through atomic excitations and ionizations in the absorbing medium and the radiative stopping power represents the rate of energy loss of a charged particle by the emission of bremsstrahlung radiation. The mass stopping power will simply be called the stopping power (in units of MeV cm /g) throughout this thesis, in 2  other words all the stopping powers used throughout this thesis will be the mass stopping powers. The stopping power is also called the unrestricted stopping power in contradistinction to the restricted stopping power that will be explained in the next section.  2.3.2 Restricted Stopping Powers  .  The unrestricted collision stopping power is the rate of energy loss per unit path length of a charged particle in both soft and hard collisions.  12  The separation of soft and  hard collisions is determined by the energy A (delta) or energy exchange value. If an electron collides with another electron while transferring energy of equal or greater than A to the secondary electron, this secondary electron forms its own track (delta ray in Figure 2.2). This collision is called a hard collision. On the other hand, if the energy transfer between electrons in the collision is smaller than A, it is called a soft collision. The restricted collision stopping power is the rate of energy loss per unit path length of a charged particle only in soft collisions. The restricted stopping power is the sum of the restricted collision stopping power and the restricted radiative stopping power. The restricted radiative stopping power is the rate of energy loss of a charged particle by the emission of bremsstrahlung radiation of energy smaller than A. For example, — , A = P 0.01 MeV is the restricted stopping power where energy exchanges less than 0.01 MeV are counted as energy deposition and energy exchanges greater than 0.01 MeV are excluded and treated as delta rays. In reality, however, the true reason for using the restricted stopping power is that any measuring instrument or any region in which we wish to calculate the energy deposited has a finite size. Using the unrestricted stopping power for dose calculation will include the whole energy deposited from delta rays that could have left a small region without depositing the whole energy in the region. This can lead to incorrect doses in small regions. Using the restricted stopping power excludes electrons that have left the region in the form of delta rays (whose energy is higher than A) but includes electrons that have entered into the region from other regions as delta rays. As the size of the region of interest decreases, the restricted stopping power with lower A needs to be used to prevent  13  the error caused by delta rays. The restricted stopping power decreases as A decreases because delta ray contribution decreased as A decreases. The restricted collisional and radiative stopping powers will simply be called the restricted stopping power throughout this thesis.  14  3 Radiation Dosimetry 3.1 Particle Fluences When a photon beam irradiates a medium, electrons are set in motion in the medium according to the photon and electron interactions mentioned above. Thus, there is a stream of photons and electrons in the medium. The particle fluence can be defined as dN/da, where dN is the number of particles incident on a small sampling sphere of a cross sectional area da. A n area da is considered as perpendicular to the direction of each particle, therefore particle fluence is independent of the incident angle of radiation.  37  Another definition of the particle fluence is the total length of track segments that are contained within any sampling volume where the incident fluence is calculated. This definition is identical to the above definition.  38  a given unit area is called the photon fluence  The number of photons that pass through ) and the number of electrons that passes  through a given area is called the electron fluence  Photon Fluence:  =—  (<f>e).  [3-1 f  5  da  Electron Fluence: <pe =  [3-2f  5  da where N = the number of photons, N = the number of electrons, and a - a unit area. p  e  It is useful to introduce the concept of energy fluence along with particle fluence. The amount of energy that passes through a given unit area is called the energy fluence or  15  the energy fluence of a photon beam (%). When a monoenergetic photon beam with energy hv passes through a unit area, the energy fluence is given as :  dN • hv  .35  [3-3]  P  da  When a photon beam with energy spectrum consisting of energies from zero to hv ax passes through a unit area, the energy fluence is given as : m  .35  [3-4]  3.2 Linear Attenuation Coefficients For N monoenergetic photons that are incident on a medium, the incident 0  photons are attenuated as they pass through the medium. The number of transmitted photons, N decreases exponentially as the thickness of the medium x increases and is given as :  N =  .33  Ne'^  [3-5]  where ju is called the linear attenuation coefficient. This equation holds only when the photon beam is monoenergetic and the beam is narrow. As was explained by Johns and Cunningham, broad beams attenuate less rapidly because more scattered photons are found along the beam direction compared to narrow beams.  16  33  An energy transfer coefficient, /Mr can also be defined as the fraction of the linear attenuation coefficient, which represents the fraction of the energy transferred by the interacting photons :  Etr  r  /Mr=/l—  .33 [3-6]  hv  In a similar way, the energy absorption coefficient, /Ub is a fraction of the linear attenuation coefficient, which represents the fraction of the energy absorbed from the total photon energy :  33  Eab  /Ub^/i—  [3-7]  hv  3.3 Mass Attenuation Coefficients  The attenuation produced in a layer of thickness Ax depends on the number of atoms and electrons present in the layer. Even if the layer is compressed to half the thickness, the same number of photons is attenuated because the layer has the same amount of atoms and electrons. In this case, the linear attenuation coefficient becomes twice while the thickness becomes a half. In order to eliminate this dependency on the material density, we can define the mass attenuation coefficient  17  (A  by dividing the  linear attenuation coefficient by the density of the material. Thus, — is not dependent on P  the density of material.  -pH N  = Noe  {  p  [3-8]  )  33  where px is the mass thickness of the layer measured in units of mass per unit area In a similar way, we can define the mass energy transfer coefficient  /Mr  v P  mass absorption coefficient  Pab  v P  and )  by dividing the energy transfer coefficient and energy J.  absorption coefficient by the density, respectively.  Err  /Mr  P  \PJ  /lab P  hv  lab KP;  hv  [3-9]  33  [3-10]  33  3.4 K E R M A  KERMA stands for Kinetic Energy Released in the Medium. KERMA is the sum of all initial energies  dEtr  transferred by incident photons at each point of interaction in a  volume dV of mass dm.  18  [3-11]  K E R M A can be expressed- as the product of the energy fluence of the photon beam and the mass energy transfer coefficient of the particular photon energy in the medium. For monoenergetic photon beams, the energy fluence of the photon beam can be expressed as the product of the photon fluence and its energy.  \  (  K =%  fj.tr  ( /Utr .. \  [3-12] 35  — <pphv  For a photon beam with a spectrum of energies, K E R M A can be expressed as the product of the energy fluence of the photon beam and the mass energy transfer coefficient averaged over the spectrum of the photon energies in the medium.  fi,tr  K =¥ X  P  P  d>"  d(/> (hv) fltrjhv)^ • hv • d(hv) d(hv) P J f  P  [3-13] 35  Using equation [3-9], K E R M A can be expressed as the product of the photon fluence, the mass attenuation coefficient and the average kinetic energy that photons transfer to electrons for a monoenergetic photon beam. For a photon beam with a spectrum of energy, the mass attenuation coefficient and the photon fluence should be averaged over the spectrum of the photon energies.  19  V  K = fo-hv  = (/>p-hv  \P;  Etr  hv  [3-14] 35  pj  According to the equation [3-14], K E R M A is the average kinetic energy that photons transfer to electrons. The photon energy is released in the medium in two ways, through collisions and through bremsstrahlung radiations. The photon energy released in the medium through collisions is called the collisional K E R M A . The collisional K E R M A ends up being absorbed in the medium and given as the following for a monoenergetic photon beam :  Kcoi  •hv (A  = Sp-hv • P  v  KP;  J  V  sab  hv  Eab  [3-15] 35  \pj  According to the equation [3-15], the collisional K E R M A is the average kinetic energy absorbed in the medium through collisions. For a photon beam with a spectrum of energies, the collisional K E R M A can be expressed as the product of the energy fluence of the photon beam and the mass absorption coefficient averaged over the spectrum of the photon energies in the medium.  Jilab  I  I*  1  Kco, ~P)  ^  •""•((^(hv) hv • d(hv) I f  jUab{ltV)'  p  J  d(hv)  Ksrem  is the K E R M A released through bremsstrahlung radiation and the sum of  KBrem  is K E R M A :  K  =Kcoi  [3-16] 35  Kco{  and  [3-17] 37  + KiBrent  20  3.5 Charged Particle Equilibrium  It is relatively simple to calculate the K E R M A using equations [3-13] and [3-14] when either the photon fluence with its energy spectrum or photon energy fluence is known. However, the absorbed dose cannot as easily be calculated if a state of charged particle equilibrium does not exist. Charged particle equilibrium occurs when each particle of certain energy leaving a given volume is replaced by an identical particle with the same energy entering the volume, in expectation values.  =  (E)  35  [3-18f  m  For megavolt photon beam irradiation, charged particle equilibrium does not occur at the surface because of the gradual build-up of charged particles. This region is called the build-up region. Beyond the build-up region, transient charged particle equilibrium is established. This concept is illustrated in Figure 3.1. The phantom is considered as being composed of several slabs of equal thickness and density along the beam direction. The incident photon beam sets 30 electron energy units into motion in the middle of each slab and these electrons travel a distance equivalent to 3 times the slab thickness depositing energy along the way. As these electrons travel across slabs, they deposit the dose value of 5 when they go through half of the slab, and 10 when they go through the entire slab. As can be seen in Figure 3.1 a), dose gradually builds up in the build-up region and saturates in the charged particle equilibrium region. In the charged particle equilibrium region  E^(2?)  ( i i  = E^(JE') and the collisional K E R M A is the same as the absorbed dose iw  21  provided there is no photon attenuation. If we assume that either bremsstrahlung losses do not occur at all, or all the bremsstrahlung losses are reabsorbed i n the phantom as ionizational or excitational energy, then the absorbed dose is the same as the total KERMA.  3 3  This can be achieved at the scoring regions located at around the centre o f  the beam when the phantom is irradiated by a very broad beam  since most  bremsstrahlung photons w i l l be reabsorbed in the phantom. In reality, strict charged particle equilibrium never occurs because o f photon attenuation, as illustrated in Figure 3.1 b). A s the photon beam passes through the phantom, it is attenuated exponentially, and at depth sets fewer electrons into motion than it does at the surface o f the phantom. In the example shown in Figure 3.1 b) the photon beam decreases by 10% when it passes the length o f a slab thickness. It generates electrons equivalent to the energy value o f 30 in the first slab. In the second slab this value drops to 27 (a decrease o f 10%), while in the third slab it drops to 24. In this case, the collisional K E R M A decreases continually in the entire region. O n the other hand, the absorbed dose increases gradually in the build-up region, and decreases continually after the build-up region. The location where both the collisional K E R M A and the absorbed dose decrease is the transient charged particle equilibrium region, which exists where the absorbed dose is proportional to the collisional K E R M A . In this region the absorbed dose curve is slightly higher than the collisional K E R M A curve because recoil electrons primarily travel i n the forward direction when they interact with megavolt photon beams,  33  and dose is deposited by electrons generated further upstream.  22  Phantom 3030Photon Beam  30303030-  25 •^deposited  25  40  45  45  45  40  45  45  45  45  15  25  30  30  30  Collisionai KERMA  Equilibrium Region •  dose  a)  Depth Phantom 30 27'  Photo Beam  24' 22' 20' 18'  25 -'deposited  25  38  39  35  32  38  39  35  32  27  15  28  26  23  21  Collisional KERMA  Transient Equilibrium Region  Depth  Figure 3.1 a) Illustration of charged particle equilibrium - no photon attenuation b) Illustration of transient charged particle equilibrium -photon attenuation  23  33  3.6 Dose  Here, we introduce the concept of the absorbed dose. The absorbed dose (D) is the energy absorbed per unit mass.  D =  UEab  [3-19]35  dm  Let us define the absorbed dose when the charged particle equilibrium is established. For a monoenergetic photon beam, the absorbed dose can be expressed as the product of the energy fluence of the photon beam and the mass absorption coefficient of the particular photon energy in the medium.  [3-20]35  = (jh>-hv  D =¥ X  P  V  P  V  P  For a photon beam with a spectrum of energies, the absorbed dose can be expressed as the product of the energy fluence of the photon beam and the mass absorption coefficient averaged over the spectrum of photon energies in the medium.  jUab |  D =¥ X  j**  P  K~P)  ^  dhv  • hv •  dhv  [3-21]37  According to equation [3-21], the dose can be expressed as the product of the photon fluence and the mass attenuation coefficient for the particular photon energy and  24  the average energy absorbed in the medium for a monoenergetic photon beam. For a photon beam with a spectrum of energies, the mass attenuation coefficient and the photon fluence should be averaged over the spectrum of photon energies.  jilab  D =¥ X  [3-22].35  P  VP  )  The absorbed dose is the sum of the energy lost through collisions by each electron along the electron track in a volume dV with a mass dm. At this moment, equation [3-15] is found to be the same as equation [3-21] and equation [3-16] is the same as equation [3-22]. In other words, the dose is found to be the same as the collisional KERMA according to the equations. However, the dose is not the same as the collisional KERMA for the reason that the collisional KERMA and the dose do not occur at the same spot. The collisional KERMA occurs at the point where a photon transfers energy to an atom or an electron while the dose occurs along electron tracks created by photon interactions further upstream. As a result, there is a shift of the position of the dose from the position of the collisional KERMA. Besides the problem of position shift, the longitudinal (transient) charged particle equilibrium and lateral charged particle equilibrium are required for the dose to be the same as the collisional KERMA. Under the condition of charged particle equilibriums (longitudinal and lateral), /3 is slightly greater than 1.  .37  [3-23}  D = Kcoi • p  25  3.7 L a t e r a l C h a r g e d P a r t i c l e D i s e q u i l i b r i u m  Lateral charged particle disequilibrium or the lack of lateral charged particle equilibrium usually occurs in small radiation fields like those encountered in Stereotactic Radiosurgery. Electrons generated in the phantom by Compton interaction, which is dominant interaction in this research primarily move in the forward direction however, with some angular distribution. Besides that, electrons created undergo random scattering in the phantom. As a result, there is lateral dispersion along with forward movement. Because of this lateral transport of electrons as shown in Figure 3.2a, the electron energy absorption in the medium (dose) occurs over a region wider than the geometric extent of the photon beam. On the other hand, the initial energy transferfromphotons to electrons (KERMA) occurs over a relatively well defined lateral region. Therefore the dose becomes smaller than the collisional KERMA in the lateral region for a pencil photon beam (Figure 3.2a). Consider a condition in which the photon field size is large enough so that there are many photons over a large area. Then the decrease of the dose at one lateral position caused by lateral electron dispersion is compensated by the dose contributed from neighboring lateral electron dispersions (point A in Figure 3.2b). At this region, lateral electron equilibrium is established. Point B in Figure 3.2b has a lower dose than the collisional KERMA even though it is within the photon beam field because of lack of scattered electrons from the left. There is an energy deposition at point C even though it is located outside the photon beam field due to electron transport reaching therefromthe right.  26  Lateral axis Centre of beam  Centre of beam  a)  b)  Figure 3.2 a) Lateral dose profile (perpendicular to the beam direction) of a pencil photon beam: the dose profile is spread laterally with a decreased magnitude because of the lateral electron dispersion caused by random electron scattering, b) lateral dose profile of a large field photon beam: Point A - dose is greater than the collisional KERMA because the decreased dose from lateral electron dispersion is compensated by dose contribution from adjacent electrons and dose is deposited by electrons generated further upstream, Point B - the dose is smaller than the collisional KERMA because of the lack of scattered electrons from the left, Point C - the dose is non-zero even though it is outside the field because of the scattered electrons from the right. Lateral charged 39  particle equilibrium exists at point A, but not at points B and C.  27  Lateral charged particle disequilibrium or the lack of lateral charged particle equilibrium usually occurs in a small field stereotactic beam of megavolt energy. Consider a small scoring region v where the dose is measured or calculated. For electronic equilibrium to exist in the small scoring region v, the following conditions must be satisfied.  (a) The photon beam boundary must be further away from the boundary of v than the maximum lateral range of the secondary electrons. (b) The photon beam fluence must be uniform. (c) The atomic composition and the density of the medium are homogeneous.  The condition (a) fails if the distance between the photon beam boundary and the boundary of v is closer than the maximum lateral range of secondary electrons. The condition (b) fails if there is significant irregularity of photon attenuation within the range of the secondary electrons. The condition (c) fail if there is fluctuation in composition such as bone or lung. Dose measurement using an ion chamber is not an easy task in the region where lateral charged particle equilibrium does not exist. The charged particle disequilibrium causes the rapid lateral gradient change of the dose. Because the ion chamber itself has a finite size and there is rapid lateral gradient change of the dose, the ion chamber measurement does not give the true lateral dose profile. In this case a correction would 11  7  be required to extract the profile since the electron fluence is not constant over the entire region of the ion chamber. For phantoms made of low density media, such as lung or balsa wood, lateral electron dispersion is more pronounced because electrons can travel farther in those  28  media, as a result these have longer lateral range of secondary electrons. In fact electrons created in low density materials have a smaller angular distribution than electrons created in high density materials.  40  However, electrons created in low density material have  larger lateral distribution or penumbra due to their longer electron range in low density material. When these phantoms made of low density media are irradiated by small field stereotactic radiosurgical beams, the lateral electron dispersion range is bigger than the beam field radius and this results in the dose being lower than the collisional K E R M A over the entire lateral region (Figure 3.3a). This makes the depth collisional K E R M A curve noticeably higher than the depth dose curve as can be seen in Figure 3.3b. Table 3.1 shows the ranges of electrons in various materials. The range of electrons increases as the density of material decreases, although the elemental composition matters as well. The secondary electrons created by a high energy photon beam are forwardpeaked along the direction of the photon beam and the average energy of secondary electrons have significantly lower energy than the maximum energy. Therefore, the average lateral range is considerably smaller than the range of electrons with average energy. The details will be discussed in the following section.  Table 3.1 The ranges of electrons in various materials at several electron energies.  Electron energy (MeV) 1 1.1 2 6.5 Density (g/cm ) 3  Aluminum 2.04 mm 2.29 mm 4.52 mm 14.76 mm 2.70  Bone, Cortical 2.63 mm 2.95 mm 5.86 mm 19.45 mm 1.85  29  41  Water 4.36 mm 5.01 mm 9.72 mm 32.63 mm 1.00 33  Air (Dry at N T P ) 4.076 m 4.573 m 9.004 m 29.10 m 0.00129  41  Figure 3.3 a) Lateral dose profile of small field stereotactic radiosurgical beams of megavolt energy irradiating phantoms made of low density media: Electron lateral dispersion is more pronounced in phantoms made of low density media and results in wide lateral dose spread. The combination of a lack of lateral electron equilibrium due to the smallfield,and more lateral dose spread caused by the low density media makes the lateral dose profile significantly lower than the lateral collisional KERMA profile, b) The depth dose curve has notably lower values than the depth collisional KERMA curve when the field size is small and the density of the phantom is low.  42  30  3.8 Dosimetry in Inhomogeneous Media (Bragg-Gray Cavity Theory and SpencerAttix Cavity Theory)  The presence of a dosimeter in a medium perturbs the charged particle fluence in the medium unless the dosimeter and the medium are identical with respect to atomic composition and density. The response of a dosimeter, therefore, depends on the geometry, construction and the surrounding medium. A cavity theory is used to relate the dose deposited in the cavity (sensitive volume of the detector) to that in the surrounding medium, which may be of different atomic number or composition. The various cavity theories are briefly reviewed in this section.  Bragg-Gray cavity theory  The simplest cavity theory is the Bragg-Gray cavity theory (Bragg, 1910, Gray, 1929, 1936). Under the following two conditions, the dose in the cavity is related to the dose in the surrounding medium according to equation [3-24].  (1) The cavity is small enough compared to the range of charged particles incident on it and the cavity does not perturb the fluence of the charged particles. (2) The photon interaction in the cavity is negligible so that the dose in the cavity is deposited only by charged particles crossing the cavity from the medium.  31  / —  \medium  where — I— I P  is the ratio of the average unrestricted mass collision stopping powers  cavity  of the medium to cavity. However, Bragg-Gray cavity theory (B-G theory) is not in agreement with experimental measurements, especially with cavity walls of high Z materials. The stopping power in B - G theory is evaluated under the assumption of the continuous slowing down approximation (CSDA).  However, delta rays produced by knock-on electron-electron  collisions enhance the equilibrium spectrum of lower energy electrons crossing the cavity. The resulting electron spectrum is further enhanced at low energies because differential cross section for delta ray production is inversely proportional to the square of the energy according to Moller (1931). Since the stopping power increases with decreasing electron energy, the presence of additional low energy electrons in the equilibrium spectrum would give larger stopping powers for both low and high Z materials than stopping powers evaluated using the CSDA assumption.  Spencer-Attix cavity theory  Spencer and Attix (1955) rederived the Bragg-Gray (B-G) theory to include the effect of the production of delta rays. The Spencer-Attix theory assumes charged particle equilibrium (CPE) and the absence of bremsstrahlung generation in addition to the two B - G  32  conditions in the above. The equilibrium electron spectrum is divided into two groups in the Spencer-Attix theory.  (1) Electrons that have kinetic energy T larger than or equal to a user-defined threshold energy A are able to cross the cavity. These electrons can therefore transport energy into the cavity. (2) Electrons that have kinetic energy T less than the threshold energy A are not able to cross the cavity. So these electrons deposit their energy locally and are not able to transport energy into the cavity.  According to the Spencer-Attix theory, the ratio of dose in the medium and the detector (or cavity) is given by : Dmedium I Lcoll  P pmax  dtpe(E)  \medium  gas  Lcoll  dE • dtfk(E) dE  VP  where, TE  medium  (E)  nedium ' dE TEmedium dE + TE  (E)  g  [3-25]  = 0e(A)  — ( A ) medium • A , and TE P J  gas  V  = <f>e(A)-(A)  37  gas '  imedium  and  sCOll  P  is the ratio of the mean restricted mass collision stopping powers of the  gas  medium to the detector (or cavity). The Bragg-Gray cavity theory as well as the SpencerAttix cavity theory fails as the cavity size increases.  33  4 Monte Carlo Simulation in Radiation Therapy 4.1 Monte Carlo Simulation  The name "Monte Carlo" was coined during the Manhattan Project of World War II, because of the similarity of statistical simulation to games of chance, and because, Monte Carlo, the capital of Monaco, was a centre for gambling and similar pursuits. Monte Carlo simulations are now used routinely in many diverse fields where analytic calculation is difficult. Such applications range from the simulation of complex physical phenomena such as radiation transport, and the simulation of the subnuclear processes in high energy physics experiments, to the simulation of chaotic problems such as weather forecasts and the simulation of nuclear explosions. The Monte Carlo simulation of radiation transport consists of the use of knowledge of probability distributions governing the individual interactions of electrons and photons to simulate their transport through matter.  43-45  The resultant distributions of  physical quantities of interest from a large number of simulated particles ("histories") provide a description of the average transport properties and associated distributions such as the radiation absorbed dose. Since the Monte Carlo method uses well-established principles of physics describing the nature of the interactions, even though some algorithms use approximation such as the electron algorithm, the technique assures accurate results if the code is implemented correctly, and enough histories are run.  34  44  4.2 Physical and Statistical Principles of Monte Carlo Simulation for Dose Calculation 4.2.1 Example of Photon Transport - Compton Scattering  As incident x-rays interact with matter in the phantom, they generate electrons through photoelectric interactions, Compton scattering, and pair and triplet production. Electron-positron pairs are produced through pair and triplet production, however, each positron, when it stops, annihilates to produce two 0.511 MeV photons. The electrons lose energy as they traverse matter through two basic processes: collision and radiation.  45  Monte Carlo simulation generates random numbers that are used to determine how far each particle travels before an interaction and which interaction take place. As the photon enters a medium, first it should be determined how far the photon will travel before an interaction takes place. This distance is determined from the following :  x  =  _±l  [4-lf  n ( j R 1 )  M  where Rl is a random number between 0 and 1 and /* is the photon attenuation cross section. The distance x is distributed from 0 to infinity. The type of interaction that takes place after the photon travels x cm is determined by sampling the cross section for each interaction. Let us take as an example, a 10 MeV photon incident on a water phantom. Figure 4.1 shows the variation of the fractional cross section of each interaction in water as the photon energy changes. For a high-energy photon the cross section of the photoelectric interaction is negligibly small (see Figure  35  2.1), therefore there are only two significant interactions, Compton scattering and pair and triplet production. Because the cross section of the photoelectric effect is very small for a high energy photon, the total interaction cross section (/u.) is the sum of the Compton scattering cross section (cr) and the pair and triplet production cross section (K),  that is ju = a + K . The relative probability for Compton scattering is given by the  ratio of the Compton scattering cross section to the total cross section (—) and the M probability for pair and triplet production is given by the ratio of the pair and triplet  production cross section to the total cross section (— ). M A random number R2 is again generated between 0 and 1. If R2 is between 0 and  — , Compton scattering occurs and if R2 is between — and M M  a  +  K  (  or  i) p i ;  a  r a n  d triplet  M  production occurs. For example, assume that R2 was between 0 and — , this means that  Compton scattering takes place.  36  20  30 40 Photon energy (MeV)  Figure 4.1 The selection of the kind of interaction through random number generation for a 10 MeV photon beam incident on a water phantom/ The photo electric probability is 16  negligible. The proportion of Compton interaction is  and that of pair and triplet  production is  Since Compton scattering occurs, the angles of the scattered photon and electron should be chosen. Figure 4.2 shows the schematic diagram of Compton scattering. The differential cross section for Compton scattering (Klein-Nishina cross section) for a photon scattering at an angle (p is given by  37  :  f  /iv' (hv  hv'  v  yhv'  .  2  sin  h  ^ <p  [4-2/  hv  where hv is the energy of an incident photon and ro = — — = 2.818 x 10"' cm is called 3  ntoc  the "classical electron radius". hv  and hv =  r  1+  hv ^  \nioC  [4-3]  is the energy of the scattered photon.  35  (l -cos^>)  )  Recoil  Scatterec photon hv'  33  Figure 4.2 Schematic diagram of Compton scattering.  The Klein-Nishina differential cross section for a 6 MeV incident photon beam is shown in Figure 4.3. The Klein-Nishina cross section varies with the angle of the scattered photon cp. A method is shown to determine the angle of the scattered photon q> . This method is called the "rejection method" and it works as follows and is shown in Figure 4.3 :  38  1. A random number cpn between 0 and 180 is generated (along the x-axis in Figure 0  0  4.3). 2. Another random number "Lit is generated that is between 0 and the maximum of d<j  c  3.  (<p-0) (between 0 and 7.94 x 10  along the y-axis in Figure 4.3).  ~ZR is compared with dQ  4. If ZR<  -((p ) then "ZR is under the Klein-Nishina cross section curve, and <pit is dQ c  R  adopted as the scattering angle. 5. If E«>  -(qm) then 2,R is above the Klein-Nishina cross section curve, and (pn is dQ c  discarded and steps 1 - 3 are repeated until the condition in 4 is satisfied.  K-N differential cross section 6 MeV photon , Discarded since dQ  y  '  Adopted since  90 135 Angle  of scattered  photon  180  (<p)  37  Figure 4.3 Klein-Nishina cross section for 6 MeV photon beam  39  From steps 1 through 5, scattering angles are produced according to the KleinNishina differential cross section. Once the angle of the scattered photon cp is determined, the energy of the scattered photon is determined according to equation [4-3]. The energy and the angle of the Compton recoil electron are determined as follows : Kinetic energy of the scattered electron is :  E = hv-hv'  [4-4f  5  and the angle of the recoil electron (6) is :  cot<9 =  hv 1  V  +  2  tan  (9\  [4-5J  25  ntoc J  The next step is to determine how far the scattered photon and recoil electron will travel before the next interaction occurs. For the scattered photon, equation [4-1] is again used to determine the distance the photon travels before it interacts. For the recoil electron, a different method is used to track it and to determine the kind of interaction it has. These features are also determined using random number generation and will be discussed in the following section.  40  4.2.2 Example of Electron Transport - Condensed History (CH) Technique  The transport of electrons in matter can be solved accurately using the analogue Monte Carlo technique. The analogue Monte Carlo technique uses the elementary collision theory (cross sections for electron interactions) to determine the distance each electron travels before a collision and to select the type of interaction. Bremsstrahlung interaction will create a photon and an inelastic collision will create another secondary electron. This process continues until the energy of all the particles falls below the cut-off energies and is absorbed in the geometry, or the particles leave the geometry under consideration. However, this process requires an extremely long calculation time considering that a typical megaelectron electron undergoes collisions of the order of one million with surrounding matter. To avoid this inconvenience, the condensed history technique was 46  developed by Berger. This technique condenses large numbers of transport and 47  collisions into a single electron step. The cumulative effect of individual transport and collision is condensed and applied to a single electron step by appropriately changing the energy and the direction of motion of the electron. This technique is induced from the fact that single collisions with the atoms cause only minor changes of the particle's energy and direction of motion. Besides that, Larsen has proved that the condensed 48  history technique is a solution of the Boltzmann transport equation in the limit of small step size.  41  5 Methods and Materials  5.1 EGSnrc User Codes  In the Monte Carlo simulations in this research, I have employed Electron Gamma Shower NRC (EGSnrc) user codes BEAMnrc , DOSXYZnrc , DOSRZnrc, 49  50  and FLURZnrc. EGSnrc is an improved version of E G S 4 ' 5 2  51  5 3  and doses calculated using  EGSnrc codes have been shown to give better agreement with measurements compared to the EGS4 codes.  54  The user code BEAMnrc simulates the head of the linear accelerator by constructing a series of component modules and following each particle starting at the exit windows of the accelerator. Each component module is built according to the geometric shape and material information that the manufacturer of the linear accelerator provides. The BEAM user code is used extensively to simulate various kinds of linacs and generate their phase space files.  55-60  The DOSXYZnrc and DOSRZnrc user codes were used to simulate radiation dose deposited in various rectangular and cylindrical phantoms respectively. The user code DOSXYZnrc is used for dose calculations in phantoms of rectangular shape. The user code DOSXYZnrc creates a rectangular phantom composed of user-defined voxels and materials and calculates the average dose deposited in each voxel from the external beam (in many cases this is a phase space file). The shape and size of the phantom are defined by the outer X, Y and Z dimensions. The phantom can be subdivided into many rectangular voxels according to the need of the user. Each voxel can be assigned to  42  contain various materials of different sizes with given X, Y, and Z values. The size of each voxel should be carefully designed to get the dose profiles such as lateral dose profile and depth dose curve. Many different materials can be used to build a phantom, and the outside region of the phantom is usually modelled as air. The beam (or incident particles) can be incident on the phantom from any direction. Various particles (electrons, photons and/or positrons) and various types of sources (parallel beams, point sources, phase space files, etc.) can be used as the incident beam. The user code DOSXYZnrc sums the energy deposited in each voxel by following all the particles' trajectories and calculates the dose in each voxel by dividing the energy deposit by the mass of the voxel. To get a statistically acceptable result in the scoring region, a minimum number of particles or histories (usually more than several millions) is necessary. The smaller the voxel size is, the more particles or histories are required to get statistically acceptable results. The number of particles or histories is linearly proportional to the computing time. Using the range rejection technique a carefully designed phantom can save much computing time. The range rejection technique terminates the history of a particle and deposits the energy of the particle in the current voxel when the range of the particle is less than the perpendicular distance of the closest boundary. The user code DOSXYZnrc can also calculate the dose in the phantoms created from dicom CT data to compare with results from commercial planning systems. It is widely used to compare actual dose measurements with a Monte Carlo simulation.  61-63  The DOSRZnrc user code is used for the dose calculation of cylindrical phantoms. The size and shape of phantoms in the DOSRZnrc user code are defined  43  through radius (R) and height (Z) values. The phantoms can be subdivided into 61 planar regions in the Z direction and 60 cylindrical regions in the radial direction (total 3660 voxels). The size of each voxel is thus determined by the thickness of each plane and cylinder. Except for the shape of the phantom, all other characteristics of the DOSRZnrc user code are identical to the DOSXYZnrc code. The DOSRZnrc user code is widely used in research involving dose calculation in cylindrical phantoms.  64-66  The FLURZnrc user code is used for particle fluence and spectrum calculations in cylindrical phantoms. The FLURZnrc user code has identical features to the DOSRZnrc user code, except for the fact that it calculates particle fluence instead of dose. The FLURZnrc user code is widely used for calculation of the photon and electron fluences and spectra. ' ' ' 60  63  64  67  5.2 Beam Generation Using the BEAMnrc User Code  Figure 5.1 shows each component module in the head of the Clinac 2100 C/D that was simulated. A 6 MV beam is produced by a 6.45 MeV monoenergetic electron beam of radius 0.5 mm hitting the target normally. The target is made up of very thin tungsten and copper overlapped. The flattening filter is made of copper and the shape is designed to make the photon beam as uniform as possible. The primary and secondary collimators are both made of tungsten and the secondary collimator is designed to make the beam 5 cm x 5 cm at 100 cm from the source. The stereotactic collimator is made of lead and yellow brass. Two stereotactic collimators are designed to make a 25 mm diameter beam  44  and a 40 mm diameter beam at 100 cm from the source. This was also the isocentre position. Originally, the simulation of the head of the Clinac 2100 C/D with a 25 mm diameter stereotactic collimator was designed by Dr. J. L. Robar who was at the BC 68  Cancer Agency, where he used the specifications from Varian and the EGS4 BEAM user code.  69  The author of this thesis simulated the same linac head using the newer version 70  BEAMnrc user code from the input file made by Dr. J. L. Robar.  The simulation of the  linac head that has the 40 mm diameter beam was designed by the author of this thesis by changing the size of the stereotactic collimator with the help of Dr. E. A. Gete who was at the BC Cancer Agency at that time.  71  The above beams are used throughout this thesis,  and each is called the 25 mm diameter beam and the 40 mm diameter beam, respectively. The BEAMnrc user code records the information of each particle that is generated by the linear accelerator in a phase space file. A phase space file contains information on charge, energy, position, angle, weight, and LATCH of each scored particle.  43  45  Z-coordinate (cm)  Target  X-coordinate (cm)  0 * Primary collimator  Flattening filter Ion chamber  Mirror  Secondary collimator (jaws)  First scoring plane (phase space file 1)  57-  — Stereotactic collimator mount  Tertiary collimator (stereotactic radiosurgical collimator) Second scoring plane (phase space file 2)  75 -  MKP*€MKS  Figure 5.1 Component modules of the head of a Clinac 2100 C/D with 40 mm tertiary collimator.  72  46  The 25 mm diameter beam at the isocentre was scored at two different planes: Z = 57 cm from the source (above the stereotactic collimator) and Z = 75 cm from the source (below the stereotactic collimator), each of which is named phase space file 1 and phase space file 2, respectively. The number of electrons incident on the target (the number of histories) is 1.35 x 10 . The total number of particles in phase space file 1 is 6,640,045 9  (the number of photons is 6,635,995 and the rest are electrons/positrons) and the total number of particles in phase space file 2 is 1,257,688 (the number of photons is 1,257,507 and the rest are electrons/positrons). The number of particles in phase space file 2 is smaller than the number of particles in phase space file 1 because only particles that went through the tertiary collimator (stereotactic radiosurgical collimator) are found in phase space file 2. The 40 mm beam at the isocentre was scored at one plane: Z = 75 cm from the source. The number of electrons incident on the target (the number of histories) was 3.5 x 10 and the total number of particles in the phase space file was 820,723 (the number 8  of photons is 820,546 and the rest are electrons/positrons). The cluster for the simulation consists of 9 computers, each of which has a Pentium II 333 MHz CPU. The computation time to produce phase space file 1 and phase space file 2 for the 25 mm diameter beam at the isocentre is 6.07 x 10 histories/hour on 5  a single Pentium II 333 MHz processor and 5.46 x 10 histories/hour on the whole 6  cluster. The computation time to produce the phase space file for the 40 mm diameter collimator was 6.14 x 10 histories/hour on a single Pentium II 333 MHz processor and 5  5.53 x 10 histories/hour on the whole cluster. The number of histories here is the 6  number of electrons incident on the target.  47  In all beam simulations, the global electron cut-off energy (ECUT) was set to 0.7 MeV, and the global photon cut-off energy (PCUT) is set to 0.01 MeV. When the energy of the particle falls below the cut-off energy, the history of the particle is terminated and its energy is deposited in the current region. Since the electron energy is defined as the total relativistic energy and the mass energy of an electron is 0.511 MeV, the kinetic energy deposited is equal to 0.7 MeV less this mass energy (0.7 MeV - 0.511 MeV = 0.189 MeV). Photon forcing or IFORCE was turned off. Bremsstrahlung photon splitting and Russian Roulette were used with the minimum number of bremsstrahlung photons set to 5 and the maximum number of bremsstrahlung photons set to 50. In bremsstrahlung photon splitting each bremsstrahlung interaction produces N photons greater than 1, each with a weight of 1/N. Bremsstrahlung photon splitting improves the statistics of bremsstrahlung photons at the expense of additional CPU time. Russian roulette eliminates all the secondary electrons except one and makes this electron carry the weight of the eliminated electrons. This feature is used to reduce CPU time. The boundary crossing algorithm of PRESTA-I (The Parameter Reduced Electron-Step Transport Algorithm I) was used and the electron step algorithm of PRESTA-II  73  '  7 4  was used. PRESTA is a special algorithm for electron transport.  PRESTA shortens the electron steps in the vicinity of an interface and as an electron moves away from an interface the electron steps are gradually lengthened. If the step size is large, it is necessary to consider path length corrections and lateral displacements after each step to convert true path length to geometrical path length. The choice of the electron step size significantly affects the accuracy. Problems such as the violation of  48  multiple scattering theory and the ambiguity of energy deposition occur when the step is too large as it crosses an interface. Shortening the step size near interfaces does not necessitate path length correction and lateral displacement and removes the problems mentioned above. PRESTA reduces computing time significantly compared to the previous EGS algorithm.  74  Figure 5.2 shows the 3D diagram of the linear accelerator head that was generated in the simulation and a few sample photons. This is for a small number of histories. For typical numbers of histories there would be photons exiting the tertiary collimator.  Figure 5.2 The 3D diagram of the component modules of the head of a Clinac 2100 C/D.  12  5.3 Material Files (PEGS4 Files)  49  Material files (PEGS4 files) were used to define the materials of components in the linac head and phantoms for dose and fluence calculations. PEGS4 code generates a material file (either a pure element or a composite material) from each element's atomic number and proportion and the density of the material. Material files are comprised of the data that is necessary to calculate photon and electron transport in each material. This includes mean free paths to calculate the photon attenuation, relative components to calculate photon-electron interaction cross sections, and restricted stopping powers to calculate the energy loss of electrons. The transport of photons and electrons are determined by these data. For BEAMnrc simulations, materials such as tungsten, lead, and copper were used to simulate the linac head. For DOSXYZnrc, DOSRZnrc, and FLURZnrc simulations, five different materials (water, bone, aluminum, balsa and cedar wood) were used for homogeneous phantoms and polystyrene was used additionally for composite phantoms. Table 5.1 shows atomic numbers, physical densities, and electron densities of various materials used for phantom materials, p is actual measurement for balsa and cedar wood.  Table 5.1 Several physical properties of a number of materials used for phantoms. Atomic number (Z)  Material  Physical density p (g/cm ) 2.70 1.85 1.00 0.32 0.16 1.18 1.044 3  Aluminum Bone'* Water Cedar wood' Balsa wood' Lucite Polystyrene 33  33  3  3  33  33  13.00 12.31 7.51 6.73 6.73 6.56 5.74  50  Electron density # of electrons/cm  3  7.83 xl0 5.91 xl0 3.34xlO 1.07xlO 0.54xlO 3.83xlO 3.38xlO  Z3  Z3  Z3  Z3  Z3  Z3  Z3  Balsa and cedar wood material files were created for this research according to their chemical composition and densities using PEGS4. Balsa and cedar wood consist of 6 % of hydrogen, 44 % of oxygen, and 50 % of carbon with a density of 0.16 g/cm and 3  0.32 g/cm respectively. 3  75  Two PEGS4 files (700icru and 521icru) were used and they  were created by the National Research Council Canada (NRC) based on the density effect correction described in ICRU report 37.  41  Two cut-off energies, AE and AP characterize  these two material files. AE and AP are the lowest energies in the data of the material files created by PEGS4. For this reason, A E is the lower energy (mass energy + kinetic energy) of knock-on electrons (or delta rays) that can be generated in the slowing down of electrons and AP is the secondary photon production energy threshold. Hence restricted stopping powers in the material file uses AE value as the lower energy threshold. 700icru has an AE of 0.7 MeV and a AP of 0.01 MeV while 521icru has an AE of 0.521 MeV and a AP of 0.01 MeV. If the mass energy of an electron is considered, 700icru has 0.7 MeV 0.511 MeV = 0.189 MeV as the lowest possible electron kinetic energy that can create secondary electrons. 521icru has an electron kinetic energy of 0.521 MeV - 0.511 MeV = 0.01 MeV as the lowest possible electron kinetic energy. In a dose calculation, 700icru significantly saves computing time because the energy of all electrons with kinetic energy smaller than 0.189 MeV are deposited locally. This yields very accurate results when voxel sizes are larger than the range of an electron with the kinetic energy of 0.189 MeV. As an example for water, electrons have a range of about 0.3 mm at this kinetic energy. In a spectrum calculation, however, using a material  51  file with too high an AE creates errors by ignoring the low energy electrons. It is thus advisable to use 521icru for spectrum calculations. In this research 700icru was used for BEAMnrc calculations and dose calculations with larger voxel sizes (more than 2 mm one side), and 521icru was used for dose calculations with smaller voxel sizes (0.2 mm one side) and fluence, spectrum calculations.  5.4 Dose Calculation Using the DOSXYZnrc User Code and Ion Chamber and Diode Measurement.  Five homogeneous phantoms (see Figure 5.3) and three composite phantoms (see Figure 5.4) were built for DOSXYZnrc simulations. The size of the scoring regions in the phantom in Figure 5.3 was made to approximate sizes of the ion chamber to compare with ion chamber measurement. The size of the scoring regions in the phantom in Figure 5.4 was made to approximate sizes of the radiographic film measurement. In this research, MC simulations in Figure 5.4 were not compared with radiographic film measurement. The size of the scoring regions and phantoms are shown in Figures 5.3, 5.4, 5.5, and 5.6. Water, aluminum, bone, balsa or cedar wood was used to build each homogeneous phantom. Aluminum is used to model high density tissues such as bone. Balsa and cedar wood models low density tissue such as lung or air cavities. All non-unit density homogeneous phantoms made of aluminum, bone, balsa and cedar wood are of rectangular shape with X = 7 cm, Y = 7 cm and Z = 15 cm as shown in Figure 5.3. Only  52  the water phantom has a Z = 24 cm because a 24 cm long water phantom was used for the ion-chamber measurements. The dose-scoring region has a rectangular area of 5 mm x 5 mm located at the centre of the XY surface along the Z-axis. Phase space files created using BEAMnrc are used as input files for DOSXYZnrc dose calculations. The phantoms are irradiated in the direction parallel to the positive Z direction normal to the X and Y directions along the centre of the XY plane. The distance between the source and the surface (SSD) of all phantoms was set to 100 cm. Ionizational measurements were done and provided by Dr. E. Grein who was at the BC Cancer Agency at that time using the PTW 0.6 cc Farmer ion chamber, the Markus chamber and Keithley 530 electrometer to calculate PDD curves in all homogenous phantoms.  Diode measurements were done at the BC Cancer Agency to  76  measure lateral dose profile in a water phantom using a p-type silicon photon diode detector (Scanditronix Medical AB, Stalgatan 14 Uppsala, S-75 450 Sweden) and provided by Dr. B. Clark.  77  53  6 M V photon stereotactic 25 mm and 40 mm diameter beams, perpendicular to and centred in the X Y plane and parallel to the Z axis of the phantom Dose scoring region: Rectangular column at the centre of the phantom with dimensions of (x, y)=(5, 5) mm along the Z-axis Z-thickness of scoring region in non-unit density homogeneous phantoms 0 - 7 cm 7-15 cm Z-axis thickness 2 mm 5 mm Z-thickness of scoring region in the water phantom 12-24 cm Z-axis 0-6 cm 6-12 cm thickness 2 mm 5 mm 10 mm Z = 15 cm for non-unit density phantoms Z = 24 cm for the water phantom  Figure 5.3 Geometry of homogeneous phantoms used for DOSXYZnrc simulations. Water, aluminum, bone, balsa and cedar wood were used as materials of each homogeneous phantom.  The composite phantom in Figure 5.4 was constructed for Monte Carlo DOSXYZnrc simulations and the material of each slab is shown in Table 5.2. Polystyrene was additionally used to construct three composite phantoms. Each of three composite phantoms was constructed with a non-unit density slab sandwiched between two almost 54  unit density slabs (polystyrene slabs) to measure the change of depth dose in the non-unit density slab. The first slab is a 3.61 cm thick polystyrene slab, the second slab is a non-, unit density material with a thickness given in Table 5.2 and the third slab is polystyrene that makes the total length of the phantom 13.61 cm. In DOSXYZnrc simulations in Figure 5.3, PEG4 file 700icru was used and the global electron cut-off energy (ECUT) was set to 0.7 MeV and the global photon cut-off energy (PCUT) was set to 0.01 MeV. ECUT should be equal to or greater than AE and PCUT should be equal to or greater than AP because AE and AP values determine the lowest energy for the data in the material file of electrons and photons respectively. As a very conservative requirement, ECUT is chosen so that the electron range at ECUT is less than 1/3 of the smallest dimension of the scoring region of the phantom. As an example, electrons have the range of 0.3 mm when the electron total energy (kinetic energy + mass energy) is 0.7 MeV. Since 1/3 of 0.3 mm is 0.1 mm and the thickness of the thinnest scoring region is 2 mm in Figure 5.3, this ECUT value is reasonable enough. Electron range rejection was turned off to enable a more accurate calculation. PRESTA-I was used in Figure 5.3 and EXACT was used in Figure 5.4 as a boundary crossing algorithm and PRESTA-II was used as an electron step algorithm in both Figures 5.3 and 5.5.  55  6 M V photon stereotactic 25 mm diameter beam, perpendicular to and centred in the X Y plane and parallel to the Z axis of the phantom.  Dose is scored only at a column which is located at the centre of the X Y plane with dimensions of (X,Y) = (1mm, 0.2mm) along the Z-axis.  Figure 5.4 Geometry of composite phantoms used for DOSXYZnrc simulations. Three phantoms were constructed. In each phantom, a bone slab, balsa wood slab, or cedar wood slab is sandwiched between water equivalent slabs made of polystyrene. The thickness and material of each slab is shown in Table 5.2.  Table 5.2 Materials of slabs used to construct composite phantoms. Composite  Polystyrene-balsa-  Polystyrene-cedar-  Polystyrene-bone-  phantoms  polystyrene  polystyrene  polystyrene  Slab 1  3.61 cm polystyrene  3.61 cm polystyrene  3.61 cm polystyrene  Slab 2  2.54 cm balsa  3.00 cm cedar  1.00 cm bone  Slab 3  7.46 cm polystyrene  7.00 cm polystyrene  9.00 cm polystyrene  56  5.5 Dose Calculation Using DOSRZnrc and Fluence and Spectrum Calculation Using FLURZnrc  Five homogeneous phantoms were built, each of which is made of water, aluminum, bone, balsa and cedar wood. Figure 5.5 shows the homogeneous cylindrical phantom that was designed for DOSRZnrc simulations as well as FLURZnrc simulations. Three composite phantoms are constructed for DOSRZnrc and FLURZnrc simulations. In each phantom, a bone slab, a balsa wood slab, and a cedar wood slab is sandwiched between two polystyrene slabs to study the effect of non-unit density materials on the dose and the collisional KERMA within and beyond the non-unit density materials. The composite phantoms constructed are shown in Figure 5.6 and the materials and slab thicknesses are the same as in Table 5.2. In this research, it is necessary to compare the photon and electron fluences and spectra results obtainedfromthe cylindrical scoring regionfromthe cylindrical phantoms with the dose results obtained in the rectangular scoring region from the rectangular phantoms. This is necessary because FLURZnrc calculates the photon and electron fluences and spectra in the cylindrical scoring region from the cylindrical phantoms only, while ion chamber measurements are done in the rectangular scoring region from the rectangular phantoms. To compare two different results from two different geometries, it is necessary to make the following checks. First, ion chamber measurements are compared with simulations from the DOSXYZnrc  user code. Secondly, simulations from the  DOSXYZnrc user code are compared with simulations from the DOSRZnrc user code to make sure the two results are the same within statistical uncertainties. Thirdly, the dose  57  results from the DOSRZnrc user code are compared with the results from the photon and electron fluences obtained from the FLURZnrc user code. Cylindrical composite phantoms in Figure 5.6 were designed to compare their results directly with the results from the rectangular composite phantoms in Figure 5.4. The principal idea of finding an equivalent cylindrical phantom to a rectangular phantom is to make the areas of the two phantoms and the scoring regions the same (thus X Y=7V  R , x y ^ ' f , where X, Y, x, and y are from Figure 5.3, R and r are from Figure 2  2  5.5). Since the X and Y dimensions of the phantoms used in the DOSXYZnrc simulation are 7cm and 7cm, the equivalent radius of the cylindrical phantom is 3.92 cm. The equivalent radius of the 5 mm x 5 mm scoring region is 2.8 mm. Figure 5.5 shows the equivalent cylindrical phantom of the rectangular phantom shown in Figure 5.3. The dose-scoring region is located at the centre of the phantom with a radius of 2.8 mm along the Z-axis. Dose calculations using the equivalent cylindrical phantom instead of a rectangular phantom are well benchmarked for depth dose calculation using large field beams. The PRESTA-I was used as a boundary crossing algorithm and PRESTA-II was used as an electron step algorithm in both DOSRZnrc and FLURZnrc simulations. The material file 700icru (AE = 0.7 MeV, AP = 0.01 MeV) was used in DOSRZnrc simulations and the material file 521icru (AE = 0.521 MeV, AP = 0.01 MeV) was used in FLURZnrc simulations. ECUT was set to AE, and PCUT was set to AP in both DOSRZnrc and FLURZnrc simulations.  58  6 MV photon stereotactic 25 mm and 40 mm diameter beams, perpendicular to and centred in the top plane and parallel to the Z axis of the phantom.  Scoring region: Cylindrical column at the centre of the phantom with dimensions of r = 2.8 mm  Z-thickness of scoring region of non-unit density homogeneous phantoms Z-axis 0-7 cm 7-15 cm thickness 2 mm 5 mm Z-thickness of scoring region in the water phantom Z-axis 0-6 cm 6-12 cm 12-24 cm thickness 2 mm 5 mm 10 mm  Z = 15 cm for non-unit density phantoms Z = 24 cm for the water phantom  Figure 5.5 Geometry of homogeneous phantoms used for DOSRZnrc and FLURZnrc simulations. Water, aluminum, bone, balsa and cedar wood were used as materials for each homogeneous phantom.  59  6 MV photon stereotactic 25 mm and 40 mm diameter beams, perpendicular to and centred in the top plane and parallel to the Z axis of the phantom. Scoring region: Cylindrical column at the centre of the phantom with dimensions of r = 2.8 mm  Figure 5.6 Geometry of composite phantoms used for DOSRZnrc and FLURZnrc simulations. Three phantoms were constructed. In each phantom, either a bone slab, balsa wood slab, or a cedar wood slab is sandwiched between polystyrene slabs. Slab thicknesses are given in Table 5.2.  60  5.6 Methodology of Collisional K E R M A Calculation from Photon Fluence  Since the cut-off energy of a photon spectrum is 0.01 MeV due to the parameters in the FLURZnrc simulations (AP = PCUT = 0.01 MeV), equation [3-16] is changed to the following equation.  .37  [5-1]-  Where, A = 0.01 MeV and  /M,b(hv)  P  is calculated from the mean free path of the  PEG4 data file using equation [3-10], since the linear attenuation coefficient is the inverse of the mean free path. The collisional KERMA curves calculated from equation [5-1] were compared with the KERMA curves calculatedfromthe DOSXYZnrc user code with ECUT > E  m a x  (the maximum energy of electrons). When the dose is calculated in DOSXYZnrc or DOSRZnrc when the ECUT value is higher than the maximum photon and electron energy, it gives the KERMA.  61  6 Results and Discussions 6.1 Comparisons of Monte Carlo Simulations with Ion Chamber and Diode Measurements  Figure 6.1 compares the percentage depth dose (PDD) curves simulated with Monte Carlo (MC) using the DOSXYZnrc user code and ion chamber measurements in the water phantom (see Figure 5.3). The ion chamber measurements were taken from the stereotactic radiosurgery commissioning data and provided by Dr. E. Grein.  76  Each  Monte Carlo simulation and ion chamber measurement was performed using 25 mm and 40 mm diameter stereotactic beams. The SSD was 98.5 cm and the medium surrounding the phantom was air. All of the simulations in this study have been done to achieve better than 1 % statistical uncertainty. The PDD curves for the Monte Carlo simulation and ion chamber measurements for both 25 mm and 40 mm diameter beams agree to within 1 % over the entire region. The maximum dose is located at approximately a depth of 1.5 cm in the phantom for the 25 mm diameter beam and 1.6 cm for the 40 mm diameter beam. The depth of the maximum dose increases as the field size increases in small field stereotactic beams due to the increased phantom scatter.  78  In the Monte Carlo calculation of Figure 6.1b, a fluctuation of dose with depth appears right after the depth of maximum dose. Considering that the depth dose has to decrease smoothly beyond the depth of maximum dose, it should be possible to draw a smooth depth dose curve within the error bars. However, the error bars are somewhat too  62  small to adequately draw a smooth depth dose curve. This is caused from the over recycling of the phase space file. The number of particles in the phase space file is 820,723, and the number of particles used for the dose calculation is 10 million. As a result the phase space file was recycled 11 times. It is ideal to recycle the phase space file up to 3 times because the Monte Carlo code rotates the phase space file by 90° after all particles are used in the phase space file. After the rotation of 90°, the particles in the phase space file look like new particles generated with different random numbers. In the same way, the phase space file can be recycled after rotated to 180° and 270°. However, when it is rotated to 360°, the phase space file uses exactly the same particles as the first use. The particles in the phase space file are generated using the random number generator, so the distribution of particles is not perfectly symmetrical as long as the number of particles is finite. Large error bars could compensate for the fluctuation of the depth dose curve and the fluctuation should be able to be modified to a smooth curve within the error bars. However, if the particles in the phase space file are over recycled, these particles have a particular asymmetric distribution, which accumulate dose in the phantom with a particular asymmetric dose distribution. On the other hand, the Monte Carlo code reduces the size of the error bars as the number of particles incident on the phantom increases. This generated a rather odd shaped depth dose curve as seen in Figure 6.1. This asymmetric problem can be improved by generating more particles in the phase space file; however, it is not an easy task since only a minute percentage of the particles are coming out of the stereotactic collimator relative to the number of the  63  original particles hitting the target. The yield rate of the 4 cm diameter beam is 0.2 %, which means that only 2 particles are recorded below the 4cm stereotactic collimator out of 1000 electrons hitting the target. The yield rate of the 2.5 cm diameter beam is less than 0.1 %. Approximately 4 weeks of calculation time is necessary, using 9 computers used for this research to obtain 2.5 millions of particles, which will limit the recycling of the particles to three. This long calculation time is a limiting factor. Over recycling of phase space files does not result in adverse effects in small field stereotactic beams as it does in large field beams. Since so many particles are distributed in a small area in stereotactic beams, the asymmetry of the particle distribution in stereotactic beams is not enhanced as greatly as large field beams. Besides this, for the 2.5 cm diameter phase space file, it is recycled 7 times which means it is over recycled only 4 times (the number of particles in the phase space file is 1,257,688,723 and the number of particles used for the dose calculation is 10 million).  64  0  2  4  6  8  10  12  14  16  18  20  22  24  14 "16  18  20  22  24  depth (cm)  a)  0  2  4  6  8  10  12  depth (cm)  b) Figure 6.1 PDD curves comparing the MC simulation and the ion chamber measurement in the water phantom using a) 25 mm and b) 40 mm diameter beams, SSD = 98.5 cm. 65  Figure 6.2 compares the lateral dose profiles simulated with the DOSXYZnrc user code to the commissioning data from the BC Cancer Agency measured with a diode (ptype silicon photon diode detector). The SSD is 92.5 cm and the dose is measured at a 77  depth of 7.5 cm. The dose at the centre of the beam is normalized to 1.0. Since the 25 mm and 40 mm diameter collimators are designed to form a beam of size 25 mm and 40 mm diameter respectively at 100 cm from the source, the dose drops rapidly at a radius of 12.5 mm and 20 mm respectively. The Monte Carlo simulation using a 25 mm diameter beam was performed with a 1 % statistical error, and with a 2 % statistical error for the 40 mm diameter beam. The curves for the 25 mm and 40 mm diameter beams agree within 1 % in the central region and within 2 - 3 % after the dose falloff outside the beam diameter. Photon diodes are shielded to reduce the effect of low photons on their responses. This could have caused the under-response of the photon diode after the dose falloff. The penumbral region demonstrates relatively good agreement. The dose profile in the penumbral region is very sensitive to the position and the size of the source. Even the minor positional error and size difference of the simulated target compared to the real target can cause noticeable errors.  61  The FWHM values for the 25 mm and the 40 mm diameter beams are 24.2 mm and 39.6 mm, respectively. The 80 % - 20 % penumbra for the 25 mm and the 40 mm diameter beams are 2.54 mm and 2.61 mm, respectively.  66  diode MC  1  Io  0.8  T3 0)  SS  0.6 0.4 0.2  -1  0 1 lateral distance (cm)  a)  - 1 0 1 lateral distance (cm)  2  b) Figure 6.2 Lateral dose profile comparison of the Monte Carlo simulation and the diode measurement in the water phantom at 7.5 cm depth using: a) 25 mm and b) 40 mm diameter beams, SSD = 92.5 cm.  67  Figures 6.3 - 6.4 compares the percentage depth dose curves on the central beam axis obtained from ion chamber measurements (PTW 0.6 cc Farmer ion chamber, the Markus chamber and Keithley 530 electrometer) in four non-unit density homogeneous phantoms. All the ion chamber measurements were measured and provided by Dr. E. Grein.  76  Each homogeneous phantom is made of aluminum, bone, balsa wood and cedar  wood, with properties as listed in Table 5.1 and irradiated by the 6 MV 25 mm and 40 mm diameter beams. It is recognized that for the Farmer chamber, charged particle equilibrium may not have been established for the low density phantoms. In cases of homogeneous aluminum, bone and cedar phantoms, the PDD curves from the Monte Carlo simulation agreed with the ion chamber measurements within statistical uncertainty for both the 25 mm and 40 mm diameter beams. In the case of the homogeneous balsa phantom, the doses from the ion chamber measurements are always higher than the doses from the Monte Carlo calculations for both 25 mm and 40 mm beams in the build-up regions. It is assumed that the more rapid electron build-up is established in the high density material in the ion-chamber wall compared to balsa wood, as a result, a higher dose is obtained in the ion chamber measurement compared to the Monte Carlo calculation. The large mechanical support of the Markus chamber accounts for significant dose perturbation due to lateral and longitudinal electron scattering according to general cavity theory. For example, the Markus chamber works well for unit density water 79  equivalent phantoms. In a balsa or bone phantom, the chamber behaves in a complex fashion because its back support acrylic has very different back scattering property than balsa or bone.  68  20 L  0l 0  i  i  2  4  i  6 depth (cm)  i 8  i  i—  10  12  a)  20 -  Ol 0  i  i  2  4  i  6 depth (cm)  i  i  i  8  10  12  b) Figure 6.3 Comparison of PDD curves for the Monte Carlo simulation and the ion chamber measurement using a 25 mm diameter beam for a) balsa and cedar wood phantoms and for b) aluminum and bone phantoms. 69  100  „  80  B  60  I  40  balsa, balsa, cedar, cedar,  ion chamber Monte Carlo ion chamber Monte Carlo  20  10  12  depth (cm)  a) aluminum, ion chamber aluminum, Monte Carlo bone, ion chamber bone, Monte Carlo  100  80 tn o  •o  60  S  -a  0 O)  ra  1° 4  20  4  6 depth (cm)  10  12  b) Figure 6.4 Comparison of PDD curves for the Monte Carlo simulation and the ion chamber measurement using a 40 mm diameter beam for a) balsa and cedar wood phantoms and for b) aluminum and bone phantoms.  70  The Monte Carlo simulation dose calculations shown in Figure 6.5 were done to verify the validity of the equivalent cylindrical phantom corresponding to a rectangular phantom as was explained above. A rectangular water phantom and a cylindrical water phantom shown in Figures 5.3 and 5.6 were irradiated by the 40 mm collimator beam and simulated using DOSXYZnrc and DOSRZnrc respectively. The two curves overlapped within their statistical uncertainties and the same results were obtained in all non-unit density homogeneous phantoms (each of which is made of aluminum, bone, balsa and cedar wood). This shows the validity of using DOSXYZnrc and DOSRZnrc interchangeably, which applies only to depth dose calculation. Therefore, for equivalent radiation field sizes, dose results from either DOSXYZnrc or DOSRZnrc will simply be specified as the dose from a Monte Carlo simulation.  Ol  0  I  ,  I  ,  I  2  4  6  8  10  ,  I  12 14 depth (cm)  ,  ,  I  ,  16  18  20  22  L _ J  24  Figure 6.5 Depth dose curves calculated from DOSXYZnrc and DOSRZnrc in the water phantom using the 40 mm diameter collimated radiation beam. 71  6.2 Collisional KERMA Calculation from the Photon Fluence in the Homogeneous Water Phantom  Figure 6.6(a) shows the variation of the photon fluence with depth on the central beam axis from which the depth collisional KERMA is calculated. The photon fluence in the 40 mm diameter beam is always higher than that in the 25 mm diameter beam. The bigger collimator allows more primary photons from the target and flattening filter, and more scattered photons are found in the 40 mm diameter beam in the water phantom, since the number of scattered photons increases as the field size increases. Figure 6.6(b) compares the spectrum of the photon fluences in the water phantom at a depth of 1.5 cm irradiated by the 25 mm and 40 mm diameter beams of 6 M V photons. The areas of two photon spectra were normalized to the same size to compare the magnitude of the low energy photons. The figure shows that a greater number of low energy photons, that are believed to be scattered photons, are found in the water phantom irradiated by the 40 mm diameter beam, in comparison to the 25 mm diameter beam. Figure 6.6(c) shows the photon energy fluence with depth in the water phantom irradiated by the 25 mm and 40 mm diameter beams. The difference between the photon energy fluence for the 40 mm and the 25 mm diameter beams is not as significant as the difference of the photon fluence because the water phantom irradiated by the 40 mm diameter beam contains more low energy photons as can be seen in Figure 6.6(b). The photon fluence includes both the incident photons and the secondary scattered photons.  72  .  40 mm diametef beam 25 mm diameter beam  \  o  i i  40 mm diameter beam — - 25 mm diameter beam  1  0.1  £ 0.4  02  1  a)  2 3 photon energy (MeV)  b )  x10' i .  40 mm diameter beam 25 mm diamter beam  cC~** 3.5  > CD  I  3  \  CJ CO  a. 12-5  + •+  •g O c  Ico  +  2  •+  >, 03  I  1.5  CD  10  15  20  25  depth (cm)  c) Figure 6.6 a) Variation of the photon fluence with depth, b) photon spectrum at a depth of 1.5 cm and c) photon energy fluence with depth in a water phantom irradiated by 25 mm and 40 mm diameter beams at SSD = 98.5 cm.  73  The collisional KERMA was calculated from the photon fluence above, using equation [5-1] in a water phantom irradiated by 25 mm and 40 mm diameter beams, and is shown in Figure 6.7. It was compared with the KERMA curve calculated using DOSXYZnrc with ECUT =10 MeV for a 25 mm diameter beam in Figure 6.7a. The two curves are indistinguishable within their statistical uncertainty, since the bremsstrahlung fraction is very small. For a 1 MeV photon, the bremsstrahlung fraction is less than 0.4 %. In principle, the KERMA is different from the collisional KERMA and should be always higher. However, for this particular photon energy, the two values are almost the same. It can be seen from Figures 6.7(b) and 6.7(c) that the photon fluence, as well as the collisional KERMA, decreases with depth due to the photon attenuation. In the transient longitudinal charged particle equilibrium region, the depth collisional KERMA curve has the same pattern as the depth dose curve, but with a slightly lower value. This means that for the 25 mm and 40 mm diameter beams in water, lateral electron equilibrium is established on central axis. The depth collisional KERMA from the 40 mm diameter beam is always higher than that from the 25 mm diameter beam because of the higher photon fluence from the 40 mm diameter beam. However, the difference between the collisional KERMA for the 40 mm and the 25 mm diameter beams is not as significant as the difference of the photon fluence as shown in Figure 6.6(a). This is attributed to the many low energy scattered photons, which contribute significantly to increase the photon fluence in the 40 mm diameter beam, but these low energy scattered photons do not increase the collisional KERMA to the same extent.  74  On the other hand, the difference between the collisional KERMA for the 40 mm and the 25 mm diameter beams is similar to the difference of the photon energy fluence (see Figure 6.6(c)) between two beams. It is attributed to the fact that the collisional KERMA is proportional to the photon energy fluence. Both depth the collisional KERMA curves are slightly smaller than their depth dose curves after the build-up region as was expected from equation [3-22]. However, the difference between the collisional KERMA and the depth dose curve is greater in the 40 mm diameter beam than the 25 mm diameter beam. This is attributed to the improvement of lateral electron equilibrium as the beam size increases. Even though there is a small difference between the depth collisional KERMA and the depth dose, the depth dose curve is very similar to the collisional KERMA after the build-up region in the case where lateral charged particle equilibrium is relatively well established.  75  collisional K E R M A from D O S X Y Z n r c . E C U T = 10 M e V collisional K E R M A from equation [1-42]  M C dose 40 mm collisional K E R M A 40 n M C dose 25 mm collisional K E R M A 25 n  ? 0.8 c 5 •  £ 0  a  10  15  depth (cm)  depth (cm)  a)  b)  MC dose 40 mm collisional K E R M A 40 mm M C dose 25 mm collisional K E R M A 25 mm  2  3 depth (cm)  0 Figure 6.7 a) The collisional KERMA simulated from DOSXYZnrc was compared with the collisional KERMA calculated from equation [5-1J in the water phantom irradiated by 25 mm diameter beam. They are identical within statistical uncertainty, b) Comparison of the Monte Carlo depth dose curve with the depth collisional KERMA curve in the water phantom irradiated by 25 mm and 40 mm diameter beams, c) the region of depth 0 - 5 cm is magnified for better visibility.  76  The lateral profiles of the collisional KERMA and the dose at a depth of 5 cm in the water phantom irradiated by both the 25 mm and 40 mm diameter beams are shown in Figure 6.8. The dose drops relatively gradually compared to the rapid drop of the collisional KERMA outside the field because of lateral electron scattering. The doses are slightly higher than the collisional KERMAs at the centre of the profiles in both the 25 mm and 40 mm diameter beams. The Monte Carlo simulation produces the lateral profile with positive distance starting from zero. The lateral profile with negative distance has the same pattern as the positive part since reflecting the positive part produced them. The FWHM values for the 25 mm and the 40 mm diameter beams are 27.04 mm and 43.10 mm, respectively. The 80 % - 20 % penumbra for both the 25 mm and the 40 mm diameter beams is 2.82 mm.  77  1.4  lateral distance (cm)  Figure 6.8 Comparison of the lateral profile of the Monte Carlo dose with the lateral profile of the collisional KERMA at a depth of 5 cm in the water phantom irradiated by 25 mm and 40 mm diameter beams.  6.3 Collisional K E R M A Calculations in Non-Unit Density Homogeneous Phantoms  Figure 6.9 shows the comparison of the Monte Carlo depth dose curves with the depth collisional K E R M A curves calculated from the photon fluence in aluminum and bone phantoms, similar to the results in water. After the build-up regions, the depth collisional K E R M A is very close to the depth dose curve in both the aluminum and bone phantoms. This is attributed to the improved lateral charged particle equilibrium in both the aluminum and bone phantoms. Electrons have shorter average ranges in aluminum  78  than bone, since aluminum is a denser material (see Tables 3.1). This shorter electron range in aluminum contributes to build better lateral charged particle equilibrium and, as a result, the dose is slightly greater than the collisional K E R M A in the aluminum phantom compared to the bone phantom.  16  14  — 12  >.  O  o  S 10 te ro c CL  CD  X10"  17  \  . i  dose from collisional dose from collisional  Monte Carlo aluminum K E R M A aluminum Monte Carlo bone K E R M A bone  f\ s %s  -  -  •g o c "S3 to o TD  i  i  10  i  12  i  14  depth (cm)  Figure 6.9 Comparison of Monte Carlo depth dose curves with the collisional KERMA in aluminum and bone phantoms irradiated by the 6 MV 25 mm diameter beam.  Figure 6.10 shows a comparison of the Monte Carlo depth dose curves with the depth collisional K E R M A for balsa and cedar wood phantoms. The depth collisional K E R M A curves are significantly higher than the depth dose curves in both phantoms because balsa and cedar wood are lower density materials and the electrons can travel farther and the lateral electron dispersion is more pronounced. In other words, severe  79  lateral charged particle disequilibrium as well as a severe drop of dose occurs because of wide lateral dose spread in phantoms made of low density materials.  16  x10  80  •17  dose from Monte Carlo, balsa — —e_ o  14  _  >^  collisional K E R M A balsa dose from Monte Carlo, cedar collisional K E R M A , cedar  12  O  _cu  £  ro a.  10  -+—'  Io  8  "35 t/3  6  8 depth (cm)  10  Figure 6.10 Comparison of the Monte Carlo depth dose curves with the collisional KERMA  calculated from the photon fluence in balsa and cedar wood phantoms  irradiated by the 6 MV 25 mm diameter beam.  The comparisons of lateral profiles of the collisional KERMA and the Monte Carlo dose at a depth of 5 cm in the balsa phantom irradiated by the 25 mm and the 40 mm diameter beams are shown in Figure 6.11(a). In this figure, the dose was also calculated from the electron fluence shown in Figure 6.11(b). The collisional KERMA curve falls very steeply outside the beam field, however, the dose drops very smoothly  80  and spreads laterally for each photon beam. This makes the dose even at the centre of the phantom, which is the scoring region, to be significantly lower than the collisional KERMA. This result agrees with the findings of Kornelsen and Young who found the drop of dose in low density materials. '  80 81  In the balsa phantom, when the beam size increases from 25 mm to 40 mm, the dose increases drastically. It is attributed to the rapid improvement of lateral charged particle equilibrium in the balsa phantom when the beam size changes from 25 mm to 40 mm even though the beam size 40 mm is not large enough to establish lateral electron equilibrium in the balsa phantom (It is because the collisional KERMA is still greater than the dose for the beam size 40 mm). On the other hand, the lateral charged particle equilibrium in the water phantom does not improve rapidly because the 25 mm diameter beam already has established lateral charged particle equilibrium at the centre of the beam. As shown in Figure 6.12, electrons have approximately an average energy of 1.1 MeV in all homogeneous phantoms. The range of electrons of 1.1 MeV in balsa wood will be much larger than 5.01 mm, which is the range of such electrons in water (see Table 3.1 and Figure 6.12). This causes severe lateral charged particle disequilibrium in the phantom, which was irradiated by the 25 mm diameter beam. On the other hand, the range of electrons in cedar wood will be shorter than the range in balsa wood. This shorter electron range causes less severe lateral charged particle disequilibrium in the cedar wood. As a result, the difference between the collisional KERMA and the depth dose curve in cedar wood is not as great as in the balsa wood.  81  The variation of the lateral dose profiles with depth in the balsa wood phantom is shown in Figure 6.11(b). There is severe lateral charged particle disequilibrium, thus there is no flat area even at the centre. These profiles are different than the lateral dose profiles in the water phantom (shown in Figure 6.8). In the water phantom, the lateral dose profile has a relatively flat area around the central area. This means that it has established relatively good lateral charged particle equilibrium around the centre. Electrons in water have an average electron range of 5.01 mm that is relatively small compared to the diameter of the beam, 25 mm, whereas for cedar wood the electron range is almost 15 mm, and for balsa wood it is 30 mm.  The FWHM values for the 25 mm and the 40 mm diameter beams are 25.63 mm and 42.81 mm, respectively. The 80 % - 20 % penumbra for the 25 mm and the 40 mm diameter beams are 6.48 mm and 6.20 mm, respectively. The 80 % - 20 % penumbra increased in the balsa phantom compared with the water phantom as the result of lateral charged particle disequilibrium.  82  lateral distance (cm)  a) 1  x 10" 2.5 cm depth 5.1 cm depth —»_ 10.2 cm depth —f— 14.7 cm depth  0.9  f^*^%  0.8  f ^ * ^ f \  0.7  -  0.6  c -a> o  I  0.4  I  1  S 0.3 0.2  0.1 0 -4  -1 0 1 lateral distance(cm)  b) Figure 6.11 Lateral profiles of the collisional KERMA and the Monte Carlo dose profile at a depth of 5 cm and b) the Monte Carlo dose profiles at various depths for the 25 mm diameter 6 MV beam in the balsa phantom.  S3  1.25 1.2-  depth (cm)  Figure 6.12 Variation of the electron average energy with depth for various materials.  6.4 Comparison of Dose with Collisional KERMA in the Composite Phantoms  In this section, the collisional K E R M A curves were overlaid on M C dose curves in the three composite phantoms to explain the change of dose in inhomogeneous slabs. In Figure 6.13, the M C depth-dose curve of the composite phantom follows the depth dose curve of the homogeneous polystyrene phantom in the first slab. In the polystyrene near the first interface of polystyrene and balsa wood, the dose drops slightly because of the lack of back scattering from low density balsa wood. ' 33  4 0  Then the dose drops  abruptly in the second slab (balsa wood), and builds up again in the third slab (polystyrene). In the balsa wood near the second interface of balsa wood and polystyrene, the dose increases slightly because of the back scattering from polystyrene, which has higher density than balsa w o o d . ' There is underdosage at the surface of the third slab 33  40  84  because of another build-up. '  '  After the build-up, the dose drops as the depth  increases but it stays at a higher value than the depth dose in the homogeneous polystyrene phantom.  19  The depth collisional KERMA curve in this composite phantom follows the depth dose curve of the homogeneous polystyrene phantom in the first slab beyond the buildup region. And the collisional KERMA drops less rapidly in the second slab as a result of less photon attenuation in the balsa wood and indeed the collisional KERMA is significantly higher than the dose. In the third slab, the depth collisional KERMA curve follows the depth dose curve beyond another build-up region. In the first slab, the depth dose curve and the depth collisional KERMA curve are almost indistinguishable after the build-up region. This indicates that longitudinal charged particle equilibrium as well as lateral charged particle equilibrium is relatively well established in this region. In the second slab, the depth dose curve drops significantly relative to the collisional KERMA as well as the depth dose in the homogeneous polystyrene phantom. This is a result of severe lateral charged particle disequilibrium in this region. A significant number of electrons are laterally dispersed 81  due to the fact that the density of balsa wood is very low. The magnitude of underdosage in the second slab will increase as the field size of the photon beam decreases and as the 31  photon energy increases. Since photons are not attenuated as much in the second slab, the depth collisional KERMA and the depth dose in the third slab are higher than the depth dose curve of the homogeneous polystyrene phantom. The depth dose follows the depth collisional KERMA beyond the build-up region in the third slab because longitudinal charged  85  particle equilibrium as well as lateral charged particle equilibrium is established again in the region. A study on a lung equivalent slab sandwiched between two solid polystyrene slabs was done by Chetty et a l . and agrees with the results in this thesis. 19  1.6  x10  -16  d o s e from M C in phantom materials collisional K E R M A d o s e from h o m o g e n e o u s polystyrene  1.4  1.2 O  Io  rr  1  5 " 0.8 c CD  CD JO  |  0.6  "CD  Polystyrene  Balsa  Polystyrene  o  "O  0.4  0.2  0  3.61  6.15 8 depth (cm)  10  12  14  Figure 6.13 Comparison of the depth dose and the depth collisional KERMA in a polystyrene-balsa-polystyrene composite phantom down the centre line. The composite phantom is composed of a 2.54 cm thick balsa wood slab sandwiched between two polystyrene slabs. The borders of the inhomogeneity are indicated in the graph. This phantom is irradiated by a 6 MV beam with a 25 mm diameter collimator. The depth dose in a unit density homogeneous phantom (polystyrene) is shown for comparison.  The second composite phantom consists of a 3 cm thick cedar wood slab sandwiched between two polystyrene slabs. Cedar wood is also a low density material  86  and the depth dose and the depth collisional K E R M A in this composite phantom follow a similar pattern as they did in the composite phantom that contains a balsa wood slab. The depth collisional K E R M A is higher than the depth dose in the second slab (cedar wood), but the difference is not as great as what occurred in the balsa wood slab. This difference is attributed to the fact that cedar wood is greater in density than balsa wood, so the lateral charged particle disequilibrium is not as severe (Figure 6.14)  x10 dose from M C in phantom materials collisional K E R M A d o s e from h o m o g e n e o u s polystyrene:  0.2 3.61  6.61  8  depth (cm)  10  12  14  Figure 6.14 Comparison of the depth dose curve and the collisional KERMA  in a  polystyrene-cedar-polystyrene composite phantom. The composite phantom is composed of a 3 cm thick cedar wood slab sandwiched between two polystyrene slabs. The borders of the inhomogeneity are indicated in the graph. This phantom is irradiated by a 6 MV beam with a 25 mm diameter collimator. The depth dose of a unit density homogeneous phantom (polystyrene) is shown for comparison.  87  The third composite phantom consisted o f a 1 cm thick bone slab sandwiched between two polystyrene slabs. Its depth dose and depth collisional K E R M A curves are shown in Figure 6.15. In the first slab, the depth dose curve in the composite phantom follows the depth dose curve o f the homogeneous polystyrene phantom with the exception o f an abrupt increase at the end o f the first slab. This increase is caused by back scattering from the second slab (bone) since bone is appreciably denser than polystyrene. The depth dose is then observed to drop more rapidly in the second slab than the depth dose o f the homogeneous polystyrene phantom because photons are attenuated more rapidly i n bone. The depth dose i n the third slab drops with depth and it stays at a lower value than the depth dose o f the homogeneous polystyrene phantom throughout this slab. The  depth collisional K E R M A  curve follows the depth dose curve i n the  composite phantom over the entire depth beyond the build-up region. This indicates that the lateral charged particle equilibrium is relatively well established over the entire phantom. In the third slab, the depth collisional K E R M A as well as the depth dose is lower than the depth dose curve o f the homogeneous polystyrene phantom because more photons are attenuated in the bone slab.  17  The overdosage and underdosage are observed  before and after the bone slab because o f the electron back scatterting and the lack o f electron back scattering respectively. ' 33  82  88  x10 d o s e from M C in phantom materials collisional K E R M A d o s e from h o m o g e n e o u s polystyrene  ®  I  1  o CO  ~ l 0,8  §5  |  Polystyrene Polystyrene  0.6  CO  o 0.4  0.2 0  3.61 4.61  0  6 8 depth (cm)  10  12  14  Figure 6.15 Comparison of the depth dose curve and the collisional KERMA in a polystyrene-bone-polystyrene composite phantom. The composite phantom is composed of a 1 cm thick bone slab sandwiched between two polystyrene slabs. The position of the bone slab is indicated. This phantom is irradiated by a 6 MV beam with a 25 mm diameter collimator. The depth dose of a unit density homogeneous phantom (polystyrene) is shown for comparison.  Johns and Cunningham discussed the K E R M A and the dose in a composite phantom consisting of a bone slab sandwiched between two muscle slabs irradiated using Co-60. They explained the drop in K E R M A and dose in the bone slab is because the mass absorption coefficient ratio of bone to muscle is smaller than 1.0 in the photon energy of Co-60.  33  In the same way there was not much change in the collisional  89  KERMA in the bone slab in Figure 6.15 since the ratio of mass absorption coefficients of bone to polystyrene is close to 1.0. For this reason, the dose in the bone slab did not change much except over and under responses at interfaces. In three composite phantoms in this section, the calculation of photon transport (collisional KERMA calculation from the photon fluence) approximately estimates dose when longitudinal and lateral charged particle equilibrium is relatively well established.  90  7 Conclusions and Future Directions  7.1 Conclusions  Major underdosage is observed in the entire region of homogeneous balsa and cedar phantoms irradiated by small field stereotactic beams because of serious lateral charged particle disequilibrium. The lateral profile of the collisional KERMA shows a flat area within the field and a sharp decrease outside the field. On the other hand, the lateral profile of dose shows a relatively moderate decrease due to the lateral charged particle disequilibrium. When lateral charged particle equilibrium is relatively well established, the dose has about the same or slightly greater value than the collisional KERMA and the dose has a relatively flat area at the centre of the lateral profile. This is the case for homogeneous water, aluminum, and bone phantoms irradiated by 25 mm and 40 mm diameter stereotactic beams. When there is severe lateral charged particle disequilibrium, the dose is significantly lower than the collisional KERMA and the dose does not have a flat area even at the centre of the lateral profile. This was the case for homogeneous balsa and cedar wood phantoms irradiated by 25 mm and 40 mm diameter stereotactic beam. The 6 MV photon stereotactic beam with the 25 mm diameter collimator irradiated three inhomogeneous composite phantoms. There was an abrupt drop of dose in the entire area of the balsa slab sandwiched between two polystyrene slabs. This dramatic drop of dose is attributed to the great extent of lateral charged particle disequilibrium in the low density balsa medium. Underdosage and overdosage are  91  observed in interfaces of polystyrene/balsa and balsa/polystyrene slabs because of the change of the back scattering characteristics at boundaries.  7.2 Future directions  The future research will be on the validity of radiographic film measurements for composite phantoms studied in this thesis. The dose to the radiographic film is obtained by matching the optical density of the radiographic film to the dose measured by an ionization chamber at each depth in a phantom. This matched curve between the optical density and the dose is called a sensitometric curve. Sensitometric curves on homogeneous phantoms have been studied widely, ' however, not much work has been 83  84  done with composite phantoms. From unpublished research done by the author of this thesis, the radiographic film measurement in inhomogeneous composite phantoms did not accurately calculate dose when the sensitometric curve obtained from a homogeneous water phantom was used to calculate the dose. The sensitometric curve from water is obtained by radiographic film measurement, with the calibration using a reliable dosimeter such as an ion chamber. An important characteristic of this sensitometric curve is that it is obtained under the conditions that there are no abrupt changes in the number of low energy photons with depth, as a result radiographic film response is relatively steady. However, an inconsistency of the radiographic film response arises when thefractionof low energy photons changes abruptly when a photon beam is going from a unit density phantom to a non-unit density phantom. There was a decrease of low energy photons in low density  92  material and an increase in high density material when a photon beam is going from unit density material. The sensitometric curve obtained using water, which is the condition of the steady radiographic film response with depth, cannot be used to calculate dose from the radiographic film measurement in inhomogeneous composite phantoms. Future studies should focus on the calculation of sensitometric curves for the use in composite phantoms. The perturbations caused by composite phantoms in radiographic film should be studied in detail.  93  References [1] D. A. Bosch, "Stereotactic Technique in Clinical  Neurosurgery," Springer Verlag,  Vienna, New York (1986). [2] K. D. Foote, W. A. Friedman, J. M. Buatti, F. J. Bova, and S. A. Meeks, "Linear accelerator radiosurgery in brain tumor management" Neurosurg. Clin. N. Am.  10(2), p. 203-242 (1999). [3] E. B. Podgorsak, G. B. Pike, M. Pla, A. Olivier, and L. Souhami, "Radiosurgery with photon beams: physical aspects and adequacy of linear accelerators," Radiotherapy and Oncology, 17, p. 349-358 (1990). [4] Y. C. Ahn, K. C. Lee, D. Y. Kim, S. J. Huh, I. H, Yeo, D. H. Lim, M. K. Kim, K. H. Shin, S. Park, and S. H. Chang, "Fractionated stereotactic radiation therapy for extracranial head and neck tumors" Int. J. Radiation Oncology Biol. Phys. 48(2),  p. 501-505 (2000). [5] H. M. Kooy, L. A. Nedzi, J. S. Loeffler, E. Alexander III, C-W. Cheng, E. G. Mannarino, E. J. Holupka, and R. L. Siddon, "Treatment planning for stereotactic radiosurgery of intra-cranial lesions" Int. J. Radiation Oncology Biol. Phys. 21, p. 638-693 (1991). [6] Task Group 42 Radiation Therapy Committee, American Association of Physicists in Medicine, "AAPMReport No. 54: Stereotactic Radiosurgery," American Institute of Physics, Woodbury, NY (1995). [7] P. Charland, E. El-Khatib, and J. Wolters, "The use of deconvolution and total least squares in recovering a radiation detector line spread function," Med. Phys. 25(2), p. 152-160(1998). [8] J. Dutreix, A. Dutreix, and M. Tubiana, "Chargedparticle stages," Phys. Med. Biol. 10, p. 177-190 (1965).  equilibrium and transition  [9] C. H. Sibata, H. C. Mota, A. S. Beddar, P. D. Higgins, and K. H. Shin, "Influence of detector size in photon beam profile measurements," Phys. Med. Biol. 36(5), p.  621-631 (1991). [10] M . Heydarian, P. W. Hoban, and A. H. Beddoc, "A comparison of dosimetry techniques in stereotactic radiosurgery," Phys. Med. Biol. 41(1), p. 93-110 (1996).  94  [11] K-S. Chang, F-F. Yin and K - W Nie, "The effect of detector size to the broadening of the penumbra - a computer simulated study," Med. Phys. 23(8), p. 1407-1411  (1996). [12] S. N. Rustgi and D. M. Frye, "Dosimetric characterization of radiosurgical beams with a diamond detector" Med. Phys. 22(12), p. 2117-2121 (1995). [13] J. L. Robar and B. G. Clark, "The use of radiographic film for linear accelerator stereotactic radiosurgical dosimetry,'" Med. Phys. 26(10), p. 2144-2150 (1999).  [14] T. Greitz, I. Lax, M. Bergstrom, J. Arndt, B-M. Berggren, H. Blomgren, J. Boethius, M. Lindquist, T. Ribbe, and L. Steiner, "Stereotactic radiation therapy of intracranal lesions. Methodologic aspects," Acta Radiologica, 2 5 , p. 81-89  (1986). [15] M . P. Heilbrun and T. S. Roberts, "CT stereotactic guidance systems," In R H  Wilkins and S Rengachary, editors, Neurosurgery, McGraw-Hill Book Company, New York, NY, p. 2481-2489 (1985). [16] M . H. Phillips, "Physical Aspects of Stereotactic Radiosurgery" Plenum Publishing  Corporation 233 Spring Street, New York, NY. (1993). [17] S. N. Rustgi, A. K. Rustgi, S. B. Jiang, and K. M. Ayyangar, "Dose perturbation caused by high-density inhomogeneities in small beams in stereotactic radiosurgery" Phys. Med. Biol. 43, p. 3509-3518 (1998).  [18] T. D. Solberg, F. E. Holly, A. A. F. De Salles, R. E. Wallace, and J. B. Smathers, "Implications of tissue heterogeneity for radiosurgery in head and neck tumors,"  Int. J. Radiat. Oncol. Biol. Phys. 32(1), p. 235-239 (1995). [19] I. J. Chetty, P. M . Charland, N. Tyagi, and D. L. McShan, "Photon beam relative dose validation of the DPM Monte Carlo code in lung-equivalent media," Med. Phys. 30(4), p. 563-573, April (2003). [20] A. L. Boyer, M. Goitein, A. J. Lomax, and E. S. Pedroni, "Radiation in the treatment of cancer," Physics Today, p. 34-36, September (2002). [21] C. J. Karzmark, "AdVances in linear accelerator design for radiotherapy," Med.  Phys. 11(2), p. 105-128 (1984). [22] P. Metcalfe, T. Kron, and P. Hoban, "The Physics of Radiotherapy X-Rays from Linear Accelerators, " Medical Physics Publishing Madison, WI, p. 1-38 (1997). [23] J. L. Robar, "Film Dosimetry and Three-Dimensional Verification of Conformal Dose Distributions in Stereotactic Radiosurgery," Ph.D. Thesis, University of  British Columbia (2000).  95  [24] B. E. Bjarngard, J. S. Tsai, and R. K. Rice, "Doses on the central axes of narrow 6MVx-ray beams," Med. Phys. 17(5), p. 794-799 (1990). [25] D. Letourneau, J. Pouliot, and R. Roy, "Miniature scintillating detector for small field radiation therapy" Med. Phys. 26(12), p. 2555-2561 (1999). [26] P. J. Biggs and C. C. Ling, "Electrons as the cause of the observed D shift with field size in high energy photon beams," Med. Phys. 6(4), p. 291-295 (1979). max  [27] K. De Vlamynck, H. Palmans, F. Verhaegen, C. De Wagter, W. De Neve, and H. Thierens, "Dose measurements compared with Monte Carlo simulations of narrow 6 MV multileaf collimator shaped photon beams," Med. Phys. 26(9), p.  1874-1882 (1999). [28] H. Saitoh, T. Fujisaki, R. Sakai, and E. Kunieda, "Dose distribution of narrow beam irradiation for small lung tumor," Int. J. Radiation Oncology Biol. Phys. 53(5), p. 1380-1387 (2002).  [29] C. Martens, N. Reynaert, C. De Wagter, P. Nilsson, M. Coghe, H, Palmans, H. Thierens, and W. De Neve, "Underdosage of the upper-airway mucosa for small fields as used in intensity—modulated radiation therapy: A comparison between radiochromic film measurements, Monte Carlo simulations, and collapsed cone convolution calculations," Med. Phys. 29(7), p. 1528-1535 (2002). [30] L. Wang, E. Yorke, and C. Chui, "Monte Carlo evaluation of tissue inhomogeneity effects in the treatment of the head and neck," Int. J. Radiat. Oncol. Biol. Phys.  50(5), p. 1339-1349 (2001). [31] J. L. Beach, M. S. Mendiondo, and O. A. Mendiondo, "A comparison of air-cavity inhomogeneity effects for cobalt-60, 6-, and 10-MV x-ray beams," Med. Phys. 14(1), p. 140-144(1987). [32] A. K. Rustgi, M. A. Samuels and S. N. Rustgi, "Influence of air inhomogeneities in radiosurgical beams," Med. Dosim. 22(2), p. 96-100 (1997). [33] H. E. Johns and J. R. Cunningham, "The Physics of Radiology," 4 Edition, Thomas, Springfield, IL (1983). th  [34] F. M. Khan, "77ze Physics of Radiation Therapy," 2 Edition, Williams and Wilkins, Baltimore MD (1994). nd  [35] F. H. Attix, "Introduction to Radiological Physics and Radiation Dosimetry," John  Wiley and Sons, USA (1986). [36] Obtained in water from Monte Carlo PEGS4 user code.  96  [37] E. B. Podgorsak, "Review of Radiation Oncology Physics: A Handbook for Teachers and Students," International Atomic Energy Agency, Vienna, Austria. April, (2003). [38] L . Papiez and J. J. Battista, "Radiance and Particle Fluence," Phys. Med. Biol. 39, 1053- 1062, June (1994). [39] Obtained in a water phantom from Monte Carlo DOSRZnrc and FLURZnrc user codes. [40] B. L . Werner, I. J. Das, F. M. Khan, A. S. Meigooni, "Dose perturbations at interfaces in photon beams," Med. Phys. 14(4), p. 585-595 (1987). [41] ICRU Report 37, "Stopping Powers for Electrons and Positions," International Commission on Radiation Units and Measurements, Inc. (1989). [42] Obtained in low density phantoms (balsa and cedar wood) using Monte Carlo DOSRZnrc and FLURZnrc user codes. [43] National Research Council of Canada "OMEGA-BEAM October (2000).  Workshop Manual"  [44] D. W. O. Rogers, "Monte Carlo techniques in radiotherapy" Physics in Canada, 58(2), p. 63-70, March (2002). [45] W. R. Nelson, H. Hirayama and D. W. O. Rogers, "The EGS4 Code System," December (1985). [46] I. Kawrakow and A. F. Bielajew, "On the condensed history technique for electron transport"Nucl. Instr. Meth. B142,p. 253-280 (1998). [47] M. J. Berger, "Monte Carlo calculation of the penetration and diffusion of fast charged particles," In B. Alder, S. Fernbach, and M. Rotenberg, editors, Methods in Computational Physics, volume 1, p. 135-215. Academic, New York, (1963). [48] E. W. Larsen, "A theoretical derivation of the condensed history algorithm," Ann. Nucl. Energy, 19: p. 701-714 (1992). [49] I. Kawrakow and D. W. O. Rogers, "The EGSnrc System, a status report MC2000 Conference Proceedings ," p. 135-140. (2000). [50] D. W. O. Rogers, B. A. Faddegon, G. X. Ding, C.-M. Ma, J. Wei, and T. R. Mackie, "BEAM: A Monte Carlo code to simulate radiotherapy treatment units," Med. Phys. 22, p. 503-524 (1995). [51] C. M. Ma, P. Reckwerdt, M. Holmes, D. W. O. Rogers, and B. Geiser, "DOSXYZ Users Manual," NRC Report PIRS 509b (1995).  97  [52] A. F. Bielajew, H. Hirayama, W. R. Nelson and D. W. O. Rogers, "History, overview and recent improvements of EGS4" Report NRCC/PIRS-0436, June (1994). [53] A. F. Bielajew and D. W. O. Rogers, "A standard timing benchmark for EGS4 Monte Carlo calculations;' Med. Phys. 19, p. 303-304 (1992) PIRS 509b (1995). [54] I. Kawrakow, "Accurate condensed history Monte Carlo simulation of electron transport. I. EGSnrc, the new EGS4 version" Med. Phys. 27(3), p. 485-498 (2000). [55] D. Sheikh-Bagheri and D. W. O. Rogers, "Monte Carlo calculation of nine megavoltagephoton beam spectra using the BEAM code" Med. Phys. 29(3), p. 391-402 (2002). [56] G. X. Ding, "Energy spectra, angular spread, fluence profiles and dose distributions of 6 and 18 MV photon beams: results of Monte Carlo simulations for a Varian 2100EXaccelerator," Phys. Med. Biol. 47, p. 1025-1046 (2002). [57] D. Sheikh-Bagheri, D. W. O. Rogers, C. K. Ross and J. P. Seuntjens, "Comparison of measured and Monte Carlo calculated dose distributions from the NRC linear accelerator" Med. Phys. 27, p. 2256-2266 (2000). [58] D. W. O. Rogers, B. A. Faddegon, G. X. Ding, C. M. Ma, J. Wei and T. R. Mackie, "BEAM: A Monte Carlo code to simulate radiotherapy treatment units," Med. Phys. 22, p. 503-524(1995). [59] W. van der Zee and J. Wekkeweerd, "Calculating photon beam characteristics Monte Carlo techniques" Med. Phys. 26(9) (1999).  with  [60] F. Verhaegen, I. J. Das and H. Palmans, "Monte Carlo dosimetry study of a 6 MV stereotactic radiosurgery unit" Phys. Med. Biol. 43, 2755-2768 (1998). [61] G. X. Ding, "Dose discrepancies between Monte Carlo calculations and measurements in the buildup region for a high-energy photon beam" Med. Phys. 29(11), p. 2459-2463 (2002). [62] G. X. Ding, C. Duzenli and N. I. Kalach, "Are neutrons responsible for the dose discrepancies between Monte Carlo calculations and measurements in the buildup region for a high-energy photon beam?" Phys. Med. Biol. 47(17), p. 32513261 (2002). [63] S. Y. Lin, T. C. Chu, J. P. Lin, "Monte Carlo simulation accelerator," Appl. Radiat. Isot. 55(6), p. 759-765 (2001).  98  of a clinical  linear  [64] D. W. O. Rogers and A. F. Bielajew, "Monte Carlo techniques of electron and photon transport for radiation dosimetry" in The Dosimetry of Ionizing Radiation, Vol III, Academic Press, p. 427-539 (1990). [65] B. Nilsson, A. Montelius, and P. Andreo, "A study of interface effects in Co beams using a thin-walled parallel plate ionization chamber" Med. Phys. 19(6), p. 1413-1421 (1992). 60  [66] B. Nilsson, A. Montelius, and P. Andreo, "Wall effects in plane-parallel ionization chambers," Phys. Med. Biol. 41(4), p. 609-623 (1996). [67] K. E. Sixel and B. A. Faddegon, "Calculation of x-ray spectra for radiosurgical beams," Med. Phys. 22(10), p. 1657-1661 (1995). [68] J. L. Robar, S. A. Riccio, and M. A. Martin, "Tumour dose enhancement using modified megavoltagephoton beams and contrast media, " Phys. Med. Biol. 47, p. 2433-2449 (2002). [69] P. M. Ostwald, T. Kron, and C. S. Hamilton, "Assessment of mucosal underdosing in larynx irradiation" Int. J. Radiat. Oncol. Biol. Phys. 36(1), p. 181-187 (1996). [70] Personal Communication with J. Robar at the BC Cancer Agency, Vancouver, BC, Canada, December (2001). [71] Personal Communication with E. Gete at the BC Cancer Agency, Vancouver, BC, Canada, February (2002). [72] Obtained from BEAMnrc user code. [73] I. Kawrakow, "Accurate condensed history Monte Carlo simulation of electron transport. II. Application to ion chamber response simulations" Med. Phys. 27(3), p. 499-513 (2000). [74] A. Bielajew and D. W. O. Rogers, "PRESTA: the parameter reduced electron-step transport algorithm for electron Monte Carlo transport" Nucl. Instrum. Methods. 18, p. 165-181 (1987). [75] D. Fengel, G. Wegener, "Wood: chemistry, ultrastructure, reactions" Walter de Gruyter,NY. (2003). [76] Personal Communication with E. Grein at the BC Cancer Agency, Vancouver, BC, Canada, January (2002). [77] Personal Communication with B. Clark at the BC Cancer Agency, Vancouver, BC, Canada, December (2002).  99  [78] K. E. Sixel and E. B. Podgorsak, "Buildup region of high-energy x-ray beams in radiosurgery," Med. Phys. 20(3), p. 761-764, (2003). [79] E. R. Epp, A. L. Boyer, and K. P. Doppke, "Underdosing of lesions resulting from lack of charged particle equilibrium in upper respiratory air cavities irradiated by 10MV x-ray beams," Int. J. Radiat. Oncol. Biol. Phys. 2(7-8), p. 613-619 (1977). [80] E. E. Klein, L. M.Chin, R. K. Rice, B. J. Mijnheer, "The influence of air cavities on interface doses for photon beams," Int. J. Radiat. Oncol. Biol. Phys. 27(2), p. 419-427(1993). [81] P.J. White, R.D. Zwicker, and D.T. Huang, "Comparison of dose homogeneity effects due to electron equilibrium loss in lung for 6 MV and 18 MV photons," Int. J. Radiat. Oncol. Biol. Phys. 34(5), p. 1141-1146 (1996). [82] G.X. Ding, and C W . Yu, "A study on beams passing through hip prosthesis for pelvic radiation treatment," Int. J. Radiat. Oncol. Biol. Phys. 51(4), p. 1167-1175 (1991). [83] C. Danciu, S. P. Basil, J. C. Rosenwald, and B. J. Mijnheer, "Variation of sensitometric curves of radiographic films in high energy photon beams," Med. Phys. 28, p. 966-974 (2001). [84] L. J. Bos, C. Danciu, Chee-Wai Cheng, Marco J. P. Brugmans, Astrid van der Horst, A. Minken, and B. J. Mijnheer, "Interinstitutional variations of sensitometric curves of radiographic dosimetric fdms" Med. Phys. 29, p. 1772-1780 (2002).  100  

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