developing a tri-hybrid algorithm to predict patient x-ray

Developing a tri-hybrid algorithm to predict patient x-ray scatter

into planar imaging detectors for therapeutic photon beams

by

Kaiming Guo

A Thesis submitted to the Faculty of Graduate Studies of

The University of Manitoba

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

University of Manitoba

Winnipeg, Manitoba

Copyright © 2020 by Kaiming Guo

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ABSTRACT

In vivo dosimetry via transmission imaging of the therapy beam can verify the

intended treatment plan was delivered to the patient. However, the EPID (electronic

portal imaging device) transmission images are contaminated with patient-generated

scattered photons. If this component can be accurately estimated, its effect can be

removed and therefore the resulting in vivo patient dose estimate will be more accurate.

This thesis presents the development of a ‘tri-hybrid’ (TH) algorithm to provide accurate

estimates of patient-generated photon scatter at the EPID.

The TH method combines three approaches: 1) Analytical methods to solve exactly

for singly-scattered photon fluence. 2) For multiply scattered photon fluence, a modified

hybrid Monte Carlo (MC) simulation method was applied, using only a few thousand

histories. From each second and higher-order interaction site in the MC simulation,

energy fluence entering all pixels of the EPID scoring plane was calculated using

analytical methods. 3) For the bremsstrahlung and positron annihilation component, a

convolution/superposition approach was employed using pre-generated pencil beam

scatter kernels superposed on the incident fluence.

Since no experimental measurement method is available to directly confirm the

separate subcomponents of photon scatter, a Monte Carlo simulation tool was developed

to separately score them. This tool was used as the ‘gold standard’ for the development

and validation of the TH method.

The TH-predicted total patient-scattered photon fluence entering the EPID, as well its

energy spectra, are compared with full Monte Carlo simulation (EGSnrc) for validation.

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A variety of phantoms are tested, including simple slab and anthropomorphic CT, as well

as monoenergetic and polyenergetic beams with different field sizes.

For these tests, the proposed TH method was demonstrated to be in good agreement

with full Monte Carlo simulation, generally within 1%. Parameters of the TH method

were optimized to maintain an accuracy of <2% while improving execution speed. The

optimized TH method takes as little as ~70 seconds to execute on a single (non-parallel)

CPU, while full MC simulations took over 30 hours. It is concluded that this patient-

generated scattered photon fluence prediction algorithm is relatively fast and accurate and

is suitable for implementation into clinical in vivo dosimetry approaches.

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CONTRIBUTIONS

This thesis presents an algorithm to accurately estimate patient scatter into the MV

imager of a modern linear accelerator (LINAC), developed at CancerCare Manitoba and

the University of Manitoba. The original project was proposed by my supervisor, Dr.

Boyd McCurdy. The thesis author, as the lead investigator in this work, has solely

accomplished the contributions listed below:

• Developed and validated a custom version of the DOSXYZnrc usercode (EGSnrc

Monte Carlo simulation package) for use as a reference tool for first, second, third

and higher order scattered fluence.

• Developed and validated an analytical approach (AnA) for estimating the x-ray

Compton and Rayleigh singly scattered fluence for simple phantoms in

therapeutic and diagnostic energy ranges.

• Developed and validated a custom (‘hybrid’) Monte Carlo algorithm to generate a

second and higher order scattering estimate utilizing a singly scattered source

under initially ideal conditions (i.e. monoenergetic beam energy and parallel

geometry), and then extending to divergent geometry and polyenergetic beam

energies.

• Customized an existing pencil beam scatter kernel algorithm to predict electron

interaction generated photons entering the EPID.

• Combined and validated the AnA, hybrid Monte Carlo, and pencil beam scatter

kernel methods mentioned above, to produce a ‘tri-hybrid’ algorithm to calculate

patient scatter entering the EPID.

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Timothy Van Beek has kindly provided his ray tracing code. Dr. Harry Ingleby, and

Dr. Eric Van Uytven also have provided some insights on technical issues and good

suggestions.

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ACKNOWLEDGEMENTS

I would like to take this opportunity to thank the people who have contributed to this

project in many unique ways.

First, I would like to thank my supervisor and mentor, Dr. Boyd McCurdy for taking

me as his student. Boyd is the who got me feet wet in the field of medical physics since

my second year of undergraduate. During the past seven years, Boyd has been committed

to providing me with the best learning experience including training on the linac and its

software, attending conferences, publishing manuscripts, applying for a medical physics

residency job and so on. His insightful guidance and advice have made substantial

differences in my work, as well as in my life. I believe this dissertation would not be

possible without his guidance, input, patience, dedication and encouragement.

I would like to thank Dr. Boyd McCurdy, Dr. Eric Van Uytven, Dr. Idris Elbakri, Dr.

Francis Lin, and Dr. Yang Wang for serving on my committee and for their time, support,

and helpful advice during my research and academic development.

I would also like to thank the entire Division of Medical Physics, CancerCare

Manitoba, with special thanks to Alana Dahlin, Jovanka Halilovic, Tracy Tyefisher

Luanne Scott for their assistance in my graduate study; Timothy Van Beek has kindly

provided his ray tracing code; James Beck and Dr. Ryan Rivest for providing me CT

images; Dr. Eric Van Uytven, Dr. Idris Elbakri, Dr. Jorge Alpuche and Dr. Harry Ingleby

for great discussion on EGSnrc.

I would also like to thank the Department of Physics and Astronomy, University of

Manitoba, especially to Robyn Beaulieu, Susan Beshta, Aymsley Bishop Mahon, Wanda

http://umanitoba.ca/faculties/science/people/staff_pages/1835.html

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Klassen, Maiko Langelaar, Andriy Yamchuk, Dr. Ruth Cameron and other professors for

their support in my graduate study. In particular I would like to thank Dr. Shelly Page and

Dr. Stephen Pistorius for their encouragement to let me pursue the physcis research.

I would also like to express my appreciation to all present and former fellow students

in Medical Physics at University of Manitoba for their constant suggestions and

friendship, including Troy Teo, Hongyan Sun, Peter McCowan, Bryan McIntosh,

Mohammadreza Teimoorisichani, Geng Zhang, Hongwei Sun, Azeez Omotayo, Pawel

Siciarz, Parandoush Abbasian, Princess Anusionwu, Suliman Barhoum, Sajjad Aftabi,

Fatimah Eashour, Adnan Hafeez and Sawyer Rhae Badiuk.

I would like to gratefully acknowledge sources of funding that I had over these years

from CancerCare Manitoba Foundation, the University of Manitoba.

I am very grateful to Geri McDonlod and Ellis Shippam for their generous assistance

as host family, which made my new life in Winnipeg much easier.

I would like to extend my deepest gratitude to my grandparents on my father side

Longfeng Lu and Yongping Guo, grandparents on my mother side Youmei Li and Leyi Li,

my parents Lihang Li, Xiurui Mi, and Runhong Guo, my wife Yanlin Dan, my daughter

Ningpei (Danica) Guo, other family members and personal friends for their

encouragement, inspiration and support through the years. Words are limited to describe

how important they are in all aspects of my life.

When I am writing this part, lots of memories, friends, and good moments had

emerged in front of my eyes, about things happened in the past 9 years in Winnipeg

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where I worked for my B.Sc. and PhD degrees. I appreciated to all the memorable

moments I had. Thank you!

At this tense moment of spread of COVID-19, best wishes to everyone to stay safe

and healthy, with hopes that life can be back to normal in the near future.

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To my beloved grandmother, parents, wife, and daughter

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LIST OF ABBREVIATIONS

ANA Analytical approach

ANN Artificial neural network

BR Bremsstrahlung Radiation

CBCT Cone-beam computed tomography

CS Compton Scattering

CT Computed tomography

EBRT External beam radiation therapy

EGS Electron Gamma Shower

EIG Electron-interaction-generated photons

EPID Electronic portal imaging devices

GPU Graphics processing unit

HB Hybrid method

HU Hounsfield unit

IGRT Image Guided Radiation Therapy

IMRT Intensity Modulated Radiation Therapy

KV Kilovoltage

LINAC Linear accelerator

MC Monte Carlo

MCHHB The number of MC simulation histories of the hybrid method

MS Multiply-scattered photons

MV Megavoltage

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NBND Narrow beam and narrow detector

NBWD Narrow beam and wide detector

NEF Normalized energy fluence

PA Positron Annihilation

PBSK Pencil beam patient-scatter kernel

PBSKL Pencil beam patient-scatter kernel library

PDI Percentage difference image

PE Photoelectric Effect

PP Pair Production

QA Quality assurance

rRMSE Relative root mean square error

RS Rayleigh Scattering

RT Radiation therapy

SDD Source–to-detector distance

SF Scatter factor

SID Source-to-interaction site distance

SNR Signal-to-noise ratio

SS Singly-scattered photons

SSD Source-to-surface distance

SSF Single scatter fraction

STD Standard deviation

TH Tri-hybrid method

TPS Treatment planning system

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VMAT Volumetric modulated arc therapy

WBND Wide beam and narrow detector

WBWD Wide beam and wide detector

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TABLE OF CONTENTS

ABSTRACT .............................................................................................................................................. ii

CONTRIBUTIONS ................................................................................................................................... iv

ACKNOWLEDGEMENTS ...................................................................................................................... vi

LIST OF ABBREVIATIONS ..................................................................................................................... x

TABLE OF CONTENTS ........................................................................................................................ xiv

LIST OF FIGURES ................................................................................................................................. xvii

LIST OF TABLES ............................................................................................................................... xxvii

Chapter 1: Introduction ..................................................................................................................... 1

1.1 General introduction ....................................................................................... 1

1.2 Motivation ..................................................................................................... 13

1.3 Outline of the thesis ...................................................................................... 16

Chapter 2: Physics Background .................................................................................................. 19

2.1 Overview of photon interactions in matter ................................................... 19

2.2 Overview of electron interactions in matter: energy loss and generation of

secondary photons ........................................................................................ 30

2.3 Discrete sampling of analytical solutions for photon scattering ................... 37

2.4 Patient generated photon scatter in linac medical imaging .......................... 40

2.5 Methods to limit photon scatter in medical imaging .................................... 48

2.6 Overview of Monte Carlo technique ............................................................ 58

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2.7 Overview of EPID image for treatment verification .................................... 68

Chapter 3: Development of Monte Carlo Based Validation Tool Box ............................ 71

3.1 Introduction .................................................................................................. 71

3.2 Methods and Materials ................................................................................. 75

3.3 Results .......................................................................................................... 87

3.4 Discussion .................................................................................................... 93

3.5 Conclusion ................................................................................................... 94

Chapter 4: A Tri-Hybrid Method to Estimate the Patient-Generated Scattered

Photon Fluence Components to the EPID Image Plane ................................ 95

4.1 Introduction .................................................................................................. 95

4.2 Methods and Materials ................................................................................. 98

4.3 Results ........................................................................................................ 113

4.4 Discussion .................................................................................................. 125

4.5 Conclusion ................................................................................................. 126

4.6 Appendix .................................................................................................... 127

Chapter 5: Performance Optimization of a Tri-Hybrid Method for estimation of

patient scatter into the EPID.................................................................................130

5.1 Introduction ................................................................................................ 130

5.2 Methods and Materials ............................................................................... 132

5.3 Results ........................................................................................................ 142

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5.4 Discussion ................................................................................................... 163

5.5 Conclusion .................................................................................................. 165

5.6 Appendix ..................................................................................................... 166

Chapter 6: Summary and Future work ................................................................................... 169

6.1 Summary ..................................................................................................... 169

6.2 Future work ................................................................................................. 174

Appendix A: Basic Monte Carlo Simulation for Photon Radiation Transport ............ 177

A.1 Basic Concept of Monte Carlo simulation in Radiation Transport............. 177

A.2 Photon-Specific Variance Reduction Techniques ...................................... 189

Appendix B: Sensitivity to Phantom Sampling of Analytical Modeling of Singly-

Scattered Fluence into an EPID ........................................................................... 201

B.1 Summary ..................................................................................................... 201

B.2 Introduction ................................................................................................. 201

B.3 Methods and Materials ................................................................................ 202

B.4 Results ......................................................................................................... 203

B.5 Conclusion................................................................................................... 204

Appendix C: Publications and Communications .................................................................... 206

C.1 List of Publications ..................................................................................... 206

C.2 List of Conference Publications .................................................................. 206

References: ......................................................................................................................................... 209

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LIST OF FIGURES

Figure 1.1 (a) Varian TrueBeam® radiotherapy system; (b) CBCT kilovoltage imaging

system (left) and corresponding 3D thorax CBCT for a moving lung tumour

target, where the red arrows highlight motion blurring in 3D; (c) megavoltage

imaging system shown with the treatment couch at two different angles. ....... 4

Figure 1.2 Beam arrangement, isodose distribution of target volume and organs–at-risk

for a left-sided breast cancer case using tangential (a) IMRT and (b) VMAT

plans, with DVHs shown in (c) [3]. For the patient with medium prostate

volume (48.4 cm3), DVHs of the (d) PTV and (e) rectum in prostate IMRT

(solid lines) and VMAT (broken lines) plans. Depths of the body contours

were reduced by 0, 1 and 2 cm [4]. .................................................................. 8

Figure 1.3 Diagram of 2D beam-detector setup and qualitative assessment of the received

scatter signal (dashed red lines) under (a) narrow beam and narrow detector,

(b) narrow beam and wide detector, (c) wide beam and narrow detector, and

(d) wide beam and wide detector situations. .................................................. 10

Figure 1.4 (a) Basic concept of contrast reduction by scattered radiation; An example of

scatter effect using two x-ray images of a knee phantom without (b) and with

(c) blocks of Plexiglas adjacent to the knee phantom. The blocks produce a

large amount of scatter, and degrade the image contrast. .............................. 12

Figure 2.1 Mass attenuation coefficients for carbon over the energy range 0.01 to 100

MeV. 𝝉/𝝆, 𝝈_𝑹/𝝆, 𝝈/𝝆, and 𝒌/𝝆 indicate the contribution of PE, Rayleigh

scattering, Compton scattering, and PP, respectively. 𝝁/𝝆 is their sum. [31] 21

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Figure 2.2 Kinematics of Rayleigh scattering: the incident photon is scattered at the angle

θ, without losing energy. ................................................................................. 22

Figure 2.3 (a) Kinematics of Compton scatter: the incident photon is scattered at angle φ,

and the Compton electron is ejected at angle θ with kinetic energy KE. E0 and

E’ are the energies of the incident and scattered photons respectively. (b)

Graph of the relationship of energy of incident (hv) and scattered photons

(hv’). ................................................................................................................ 24

Figure 2.4 Graph of the relationship of energy of incident and scattered photons [30]. ... 25

Figure 2.5 (a) Kinematics of the Photoelectric Effect, involving the collision of a photon

with energy 𝐸 and a tightly bound electron with binding energy EB. The

photon is completely absorbed and the electron is ejected with kinetic energy

EK. (b) Photoelectric atomic cross sections for various absorber media (i.e.

elements). Note the discontinuous K, L, and M shell absorption edges [30]. 28

Figure 2.6 The schematic of pair production in which a photon passes in the vicinity of a

nucleus and spontaneously forms a positron and an electron. ........................ 30

Figure 2.7 The mass radiation stopping power (thick lines) and collision stopping power

(thin lines) versus electron kinetic energy for aluminum, water, and lead.[30]

......................................................................................................................... 32

Figure 2.8 Bremsstrahlung radiation: electromagnetic radiation produced by

the deceleration of a charged particle when deflected by another charged

particle, typically a high energy electron incident on an atomic nucleus. The

process follows the conservation of energy and momentum, as shown two

examples of emission of low-energy and high-energy bremsstrahlung x-ray.

......................................................................................................................... 34

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Figure 2.9 Kinematics of at-rest (left) and in-flight (right) positron annihilation. ............ 36

Figure 2.10 Incident photons interact with a phantom/patient voxel, and scattered photons

are detected by an ideal detector at angle θ relative to the incident beam. .... 38

Figure 2.11 X-rays reaching the imaging plane without interacting in the patient/phantom

are primary photons. Some x-rays interact and are entirely absorbed within

the patient, while others are scattered either once or multiple times before

reaching the imaging plane. ............................................................................ 41

Figure 2.12 Comparison of radiographic image with patient-generated x-ray scatter

suppressed (left hand side) vs. not suppressed (right hand side).................... 41

Figure 2.13 (a) Illustration of the reduction of patient-generated scatter entering the

imager through reducing the x-ray beam field size. (b) Illustration of the

divergence of patient-generated scatter into the imager for various air gaps.

The scattered photons diverge more quickly compared with the primary beam

for any given air gap (three example air gaps shown). .................................. 43

Figure 2.14 (a) Example of CBCT cupping artifact for a homogeneous water cylinder

with a 10% scatter-to-primary (SPR) ratio, and (b) with a 120% scatter-to-

primary (SPR) ratio. (c) CBCT streaking artifact illustrated for a

homogeneous water cylinder with two dense material inserts with a 10% SPR

ratio, and (d) with 120% SPR ratio [47]. ........................................................ 46

Figure 2.15 The dependence of scatter fluence on air gap (17 cm thick slab with 30x30

cm2 field size) for polyenergetic beams of energy 6 MV (closed circles) and

24 MV (open circles) [57]. ............................................................................. 49

Figure 2.16 (a) The design of an anti-scatter grid for KV imaging, with lead strips

oriented along one dimension separated by a low attenuating interspace

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material such as carbon fiber or plastic. (b) Two images of the AP projection

of a pelvis phantom were obtained at 75 kV without using (left) and with

using an anti-scatter grid (right). ..................................................................... 52

Figure 2.17 (a) A deep learning method applied for automatic segmentation of anatomical

images of a nasopharynx patient, from [67]. (b) Training, validation and

testing processes of the CNN require three different datasets. The model is

trained using a training dataset. During the training, the validation dataset is

used to monitor and minimize bias in the model. Finally, independent test

datasets are used to test the generalization capability of the model for

completely new data. ....................................................................................... 57

Figure 2.18 The structure of the EGSnrc code system and how it interfaces to a user code

[73]. ................................................................................................................. 63

Figure 2.19 Profile dose curve along Y-axis comparing Geant4 (black), EGSnrc (red), and

measurement data (blue) for a 4x4 cm2 field in a homogeneous (water)

phantom. [80] .................................................................................................. 66

Figure 2.20 Comparison of PDD curves in a lung-slab phantom measured with

thermoluminescent dosimeters (solid line) and simulated using EGSnrc Monte

Carlo code (solid dark line) for field sizes of (a) 10×10 cm2, (b) 5×5 cm2, (c)

2×2 cm2, and (d) 1×1 cm2 [81]. ....................................................................... 67

Figure 3.1 The simulation was performed under (a) parallel and (b) divergent beam

geometry. Three phantoms, (c) Water, (d) LWRL, and (e) Thorax, using

monoenergetic and polyenergetic beams, are used to test the Monte Carlo

validation tool against the analytical calculations. .......................................... 86

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Figure 3.2 The (a) primary and (c) singly scattered NEF map, with the 0.06Mev incident

beam and field size of 10 x 10 cm2 incident on the thorax phantom, including

the percentage difference map and its histogram, as well as corresponding

central horizontal and vertical profiles in (b) and (d). .................................... 91

Figure 3.3 The central horizontal (left) / vertical (right) profiles of SSF for the thorax

phantom when the incident beam energy is (a) 60 keV, (b)100 keV, and (c) 6

MV. ................................................................................................................. 92

Figure 4.1 Schematic describing the analytical algorithm to calculate the single and

multiple scatter component into the imaging plane, where the physics process

are detailed in equation 4-3 and 4-7. ............................................................ 102

Figure 4.2 Schematic describing and contrasting the methods of calculating multiply

scattered photons entering the imager plane generated using (a) full Monte

Carlo simulation (i.e. DOSXYZnrc based patient scatter validation tool) with

one billion photon histories and (b) developed hybrid method logic flow

which generated an estimation of multiply scattered photons at the imager

plane. ............................................................................................................ 106

Figure 4.3 Testing was performed using divergent beam geometry (a), and using three test

phantoms including (b) water, (c) LWRL (left-half water, right-half lung), and

(d) thorax, with monoenergetic and polyenergetic beams............................ 111

Figure 4.4 The comparison between the MC simulation and TH methods, for total

scattered and individual scattered NEF components, for a 6MV photon beam,

10x10 cm2 field size irradiating the thorax phantom. .................................. 119

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Figure 4.5 The comparison of central horizontal (left-hand column) and vertical (right-

hand column) profiles between the TH method and Monte Carlo simulation,

when the incident energy is (a) 1.5 MeV , (b) 5.5 MeV, and (c) 12.5 MeV. 120

Figure 4.6 The comparison of scattered energy spectrum (with 10 bins) at the center pixel

of the imaging plane between the MC simulation and TH method for 6 MV

(left) and 18 MV (right) treatment beam irradiating a (a) water phantom (b)

thorax CT phantom with the field size of 10x10 cm2. .................................. 121

Figure 4.7 The comparison of mean energy spectrum across the imaging plane between

the MC simulation and TH method for 6 MV (left) and 18 MV (right)

treatment beam irradiating a (a) water phantom (b) thorax CT phantom with

the field size of 10x10 cm2. ........................................................................... 122

Figure 4.8 The comparison of central horizontal (left) and vertical (right) profiles between

the TH method and full Monte Carlo simulation, with a 6MV photon beam

irradiating the thorax phantom with field sizes of (a) 4x4 cm2, (b) 10x10 cm2,

and (c) 20x20 cm2. ........................................................................................ 123

Figure 4.9 The central horizontal (left-hand panel)/vertical (right-hand panel) profile of

SF for thorax phantoms when the energy of the incident beam is at 1.5 MeV,

5.5 MeV, and 12.5MeV with field size of 10x10 cm2. The symbol ‘*’

represents the percentage difference of the scatter factor between TH

calculation and full MC simulation. .............................................................. 124

Figure 4.10 The logic flow of the PBSK calculation for the EIG component into the

scoring plane: the radiological path length (RPL) and corresponding air gap

(AG) are calculated for each ray line from the x-ray source to the imaging

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plane pixels. Based on the given RPL and AG, bi-linear interpolation is used

on the patient scatter kernel library to generate the required patient scatter

kernel for the given ray line. The patient EIG scattered energy fluence kernel

is applied at the point of intersection in the imaging plane of each discretely

sampled ray line and summed over all rayline contributions to yield an

estimate of the EIG scatter fluence entering the imager. ............................. 128

Figure 5.1 (a) The workflow of the TH method (i.e. the combination of ANA, HB, and

PBSK methods) to estimate the total patient-generated scatter into the

imaging plane. (b) The resultant NEF compared with the full Monte Carlo

simulation fluence result (i.e. using the ’dosxyznrc_K’ validation tool) with 1

billion photon histories. ................................................................................ 135

Figure 5.2 The tests were performed with (a) divergent beam geometry. Three phantoms,

(b) water, (c) pelvis, and (d) thorax were used to investigate the effect of

various sampling issues in the implementation of the tri-hybrid method. ... 139

Figure 5.3 Comparison of the ANA method to full MC simulation for the single scatter

component from the water phantom at different spatial resolution (x-axis).

The accuracy (i.e. dots) and precision (i.e. error bars) for 6 and 18 MV

polyenergetic beams (top row and bottom row, respectively) with different

energy bin sampling (indicated by symbols, 0.25, 0.5 and 1 MeV correspond

to 24,12,6 bins for 6 MV, and 72, 36,18 for 18MV) with field sizes of 4x4,

10x10, and 20x20 cm2 (left, middle and right columns, respectively). ........ 143


component from the CT pelvis phantom at different spatial resolution (x-axis).


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component from the CT thorax phantom at different spatial resolution (x-axis).






Figure 5.6 Distribution of singly scattered centers (colour varying with z coordinate) with

various voxel sampling sizes with field sizes of (a) 4x4 cm2, (b) 10x10 cm2,

and (c) 20x20cm2. ......................................................................................... 148

Figure 5.7 Distribution of multiply scattered centers with a range of MC simulation

histories (i.e. 2K, 6K, 10K, 20K, 60K and 100K histories) inside the CT

pelvis phantom when it is irradiated by a 6 MV polyenergetic beam with field

sizes of (a) 4x4 cm2 and (b) 20x20cm

2. ...................................................... 151

Figure 5.8 Comparing HB method against full MC simulation for multiple scatter

component. The accuracy (i.e. symbol) and precision (i.e. error bar) are

indicators of performance for different numbers of Monte Carlo histories used

for the HB method, for 6 and 18 MV beams, irradiating the CT thorax

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phantom with field sizes of 4x4 (squares), 10x10 (circles), and 20x20

cm2(triangles). .............................................................................................. 152

Figure 5.9 The histogram of the multiple scatter centers (‘Counts’) per order of multiple

scatter for (a) 6 MV and (b) 18 MV incident beams and field size 20x20 cm2

irradiating on the pelvis phantom. ................................................................ 153

Figure 5.10 Calculation efficiency of the TH method when the (a) water, (b) pelvis, (c)

thorax phantoms are irradiated by 6 and 18 MV treatment beams with

different field sizes (4x4 cm2, 10x10 cm2, and 20x20 cm2) versus the number

of histories used in the HB MC simulation, using the recommended sampling

settings for the single scatter calculation. ..................................................... 159

Figure 5.11 The comparison of central horizontal (left) and vertical (right) profiles

between the TH method and full Monte Carlo simulation, with the 18 MV

photon beam irradiating the pelvis phantom with field sizes of (a) 4x4 cm2, (b)

10x10 cm2, and (c) 20x20 cm

2 using the optimal sampling settings of the TH

method. ......................................................................................................... 160

Figure 5.12 The comparison of total and individual scattered NEF component between

the full MC simulation against the TH method, for a 6MV photon beam with

a field size of 4 x 4 cm2 irradiating the pelvis phantom with the recommended

sampling settings. ......................................................................................... 161

Figure 5.13 Comparing the mean energy distribution from the TH method against full MC

simulation for the total patient-generated scatter component for the water

phantom irradiated by the 6MV beam. (a) Using 0.5, 1, and 1 cm3 voxel size

sampling with respect to the field sizes of 4x4, 10x10, and 20x20 cm2 and

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20K MCHHB, and (b) using a 0.2 cm3 voxel resolution and 100K MCHHB

for all field sizes. ........................................................................................... 162

Figure A. 1 (a) The histogram with bin size of 0.05 shows a PDF of pseudo-random

number sequence with 10 million elements; (b) a part of the scatterplot of the

random number sequence (vertical axis, (0, 0.1)) versus same sequence

lagging 10 elements (horizontal axis, (0, 0.1)); (c) the graph of autocorrelation

coefficients (blue bar) for a given series through lagging 1-10 elements with

the 95% confidence interval (blue line). ....................................................... 181

Figure A.2 (a) the normalized energy spectrum and (b) the cumulative probability

function of a typical 6MV treatment beam. ................................................. 183

Figure A.3 (a) the probability density and (b) cumulative probability of the photon

interaction based on the ratio of the mass attenuation coefficients for

individual interaction types to the total mass attenuation coefficient. .......... 185

Figure A.4 the logic scheme of the Monte Carlo simulation on radiation transport........ 189

Figure A.5 Example of a stretched ( 𝑐 = 12 ) and shortened ( 𝑐 = −1 ) distribution

compared to an unbiased (𝑐 = 0) distribution. In all three cases, 𝑐𝑜𝑠𝜃 = 1.

The horizontal axis is in units of mean free paths. [124] .............................. 198

Figure B.1 Geometry of the singly-scattered fluence entering a portal imaging device; (b)

the physics process of equation B-1& fluence map, and the validation with

Monte Carlo simulation at the phantom sampling resolution of 1 cm. ......... 205

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LIST OF TABLES

Table 2-1 The purpose of general subroutines in the EGS code system [73] .................. 63

Table 3-1 Output phase-space file of DOSXYZnrc-based patient scatter validation tool 78

Table 3-2 EGSnrc Monte Carlo transport parameters used .............................................. 85

Table 3-3 The mean and standard deviation (STD) of percentage differences between the

MC simulation (109 histories) and analytical calculated NEF for various

monoenergetic beams and phantoms under parallel beam geometry. ............ 89

Table 3-4 The mean and standard deviation (STD) of percentage differences between the

MC simulation (109 histories) and analytical calculated NEF for various

monoenergetic beams and phantoms under divergent beam geometry. ......... 90

Table 4-1 Comparison of patient-scattered photon fluence entering an EPID, calculated

with full MC simulation and ANA, HB, and PBSK methods. Results are

divided into single, multiple, EIG, and total scatter fluence for the three

phantoms tested here, using incident beam energies of 1.5, 5.5, and 12.5 MeV.

‘Accuracy’ and ‘Precision’ are indicators of the average and standard

deviation of percentage differences across the entire image plane respectively.

...................................................................................................................... 117

Table 4-2 Comparison of patient-scattered photon fluence entering an EPID, calculated

with full MC simulation and ANA, HB, and PBSK methods. Results are

divided into single, multiple, and EIG scattered fluence components, as well

as total scattered fluence for the three phantoms tested here, using incident

beam energies of 6 MV and 18 MV. ‘Accuracy’ and ‘Precision’ are indicators

xxviii

of the average and standard deviation of percentage differences across the

entire image plane respectively. .................................................................... 118

Table 4-3 Output phase-space file of DOSXYZnrc-based scatter scoring tool box ....... 127

Table 4-4 EGSnrc Monte Carlo transport parameters used in DOSXYZnrc .................. 129

Table 5-1 Comparison of patient-scattered photon entering an EPID calculated with full

MC simulation and the TH method using an incident beam energy of 6 and 18

MV for the water phantom. For the ANA method, the 0.5, 1, and 2 cm3 voxel

sampling sizes with respect to the three field sizes 4x4, 10x10, and 20x20 cm2

are used. ‘Accuracy’ and ‘Precision’ are indicators of the average and

standard deviation, respectively, of percentage differences across pixels in the

entire image plane. ........................................................................................ 156



MV for the pelvis phantom. For the ANA method part, the 0.5, 1, and 2 cm3

voxel sampling sizes with respect to the field sizes 4x4, 10x10, and 20x20

cm2 are used. ‘Accuracy’ and ‘Precision’ are indicators of the average and

standard deviation, respectively, of percentage differences across the entire

image plane. .................................................................................................. 157



MV for the thorax phantom. For the ANA method, the 0.5, 1, and 2 cm3 voxel

sampling sizes with respect to three field sizes 4x4, 10x10, and 20x20 cm2 are

used. ‘Accuracy’ and ‘Precision’ are indicators of the average and standard

xxix

deviation, respectively, of percentage differences across the entire image

plane. ............................................................................................................ 158

Table 5-4 EGSnrc Monte Carlo transport parameters used in DOSXYZnrc ................. 166

Table 5-5 List of the total number of multiply scattered interaction centers generated

within the phantoms (i.e. water, pelvis, and thorax), when irradiated by a 6

MV beam with various field sizes (4x4, 10x10, and 20x20 cm2). ............... 167

Table 5-6 List of total number of multiply scattered interaction centers generated within

the phantoms (i.e. water, pelvis, and thorax), when irradiated by an 18 MV

beam with various field sizes (4x4, 10x10, and 20x20 cm2). ....................... 168

Table A-1 Relative efficiency versus the parameter C of exponential transformation

biasing for calculation of the dose at various depths in water irradiated by 7-

MeV photons [118]. ..................................................................................... 199

Table B-1 The average and maximum difference in singly-scattered fluence for various

phantom resolutions compared to the highest resolution. ............................ 204

1

Chapter 1: Introduction

1.1 General introduction

Since x-rays were discovered in 1895 by Wilhelm Conrad Roentgen, they have found

a wide range of applications in fields such as medicine, crystallography, food inspection,

as well as airport security. The majority of x-ray use today is in medical applications,

such as cancer treatment with high energy radiation therapy (RT), and diagnostic imaging

applications such as planar x-ray imaging and computed tomography.

External beam therapy (EBT) using high energy x-rays (i.e. photons) was introduced

into RT in the 1950’s and 1960’s, initially using highly radioactive sources (i.e. Cobalt-

60), and then with the modern medical linear accelerator (‘linac’). Today, high energy

photon therapy is the cornerstone modality for radiation treatment throughout the world.

However, improved effectiveness and accuracy of RT continues to be a significant goal

today [1].

Electronic portal imaging devices (EPID) were developed in the 1980’s and 1990’s

to form an image with the high-energy (i.e. MV) therapeutic beam using camera-based

systems or liquid ionization arrays. In the early 2000’s, flat-panel detector (FPD)

technology was developed for commercial implementation in this role and is still the

dominant option today. Initially, the MV imager was used for anatomical imaging to

improve the accuracy of patient setup during radiotherapy sessions. However, in the late

2000’s, KV x-ray imaging systems (i.e. diagnostic systems) physically mounted onto the

radiotherapy treatment units became commercially available. These low-energy x-ray

2

imaging systems provide superior anatomical images for patient setup, but MV imaging

systems are still widely used as backup anatomical imaging systems and also now for the

additional application of treatment verification. A brief description of the linac will be

presented in Section 1.1-a.

Simultaneous with the imaging developments, other technological developments in

radiotherapy occurred, such as fluence modulation and inverse plan optimization, which

provided an increase in the quality of the dose distributions delivered to patients but with

increased complexity of the radiation therapy process. These developments are briefly

described in Section 1.1-b. Since radiation delivery and patient dosimetry have become

significantly more complicated over the last two decades [2], there has been (and

continues to be) strong interest in utilizing the EPID during treatment delivery to acquire

transmission images of the therapeutic beam, which can then be used for patient dose

verification including in vivo dose verification. In vivo dose verification is the

confirmation of the radiation dose delivered to the patient through a direct measurement

of the therapeutic beam.

The EPID images obtained from the transmitted treatment beam suffer from poor

image contrast due to the smaller attenuation differences of tissues at higher photon

energies, and also due to the patient-generated scatter reaching the EPID that blurs the

image. In contrast, using the kilovoltage x-ray source will better visualize the patient’s

anatomy due to the larger attenuation differences between tissues at lower energies (i.e.

providing better image contrast compared to MV), but will suffer from even more

blurring due to increased patient-generated scatter (which is only partially offset by anti-

scatter grids). The patient scatter issue is described in Section 1.1-c.

3

Once the patient-generated scatter is estimated, its effect can be removed. Thus,

accurate estimation of patient-generated scatter that reaches the imaging plane (in either

MV or KV applications) is of strong interest in modern radiation therapy.

1.1-a. A brief description of the medical linear accelerator (or ólinacô)

In modern radiation treatment units (Figure 1.1), a medical linear accelerator

generates therapeutic x-rays with megavoltage (MV) energies. Very commonly, a

diagnostic x-ray tube and associated dedicated imager are mounted on the same treatment

unit, providing kilovoltage (KV) x-rays to obtain reasonably good quality anatomical

imaging, such as projection radiographs or cone-beam computed tomography (CBCT)

volumetric images.

In a typical treatment, KV images are acquired just prior to the custom MV beam

treatment delivery, to provide anatomical localization of the patient in their treatment

position. The patient position can then be adjusted in three-dimensions to match the

planning CT, ensuring accurate geometrical targeting of the tumor. The therapeutic MV

x-ray beam is delivered immediately afterwards, taking several minutes (2-10 minutes) to

deliver the prescribed energy pattern to the patient. The MV imager (i.e. EPID) can

capture the portion of the therapy beam exiting the patient, which can then be used to

verify the complex intensity pattern of the MV beam (i.e. through dose estimation in the

patient, known as in vivo dosimetry).

4

Figure 1.1 (a) Varian TrueBeam® radiotherapy system; (b) CBCT kilovoltage imaging

system (left) and corresponding 3D thorax CBCT for a moving lung tumour target, where

the red arrows highlight motion blurring in 3D; (c) megavoltage imaging system shown

with the treatment couch at two different angles.1

1 Screenshot from “UAB first to use new HyperArc high definition radiotherapy in the U.S.”

https://youtu.be/a8__X1v5JsY posted by University of Alabama at Birmingham, posted on Oct. 24, 2017

https://youtu.be/a8__X1v5JsY

5

1.1-b. A brief description of modern RT Techniques

Sophisticated RT techniques are being increasingly employed clinically, taking

advantage of improvements of on-linac x-ray imaging systems, together with

improvements in both the ability to deliver modulated radiation fluence patterns, as well

as the ability to mathematically optimize and generate the individually customized,

complex fluence patterns. These developments help to improve the conformality of the

radiation dose to the tumor and improve the tumour control probability (TCP), while

reducing normal tissue complication probability (NTCP). The development and

introduction of x-ray imaging tools and techniques in RT has become inseparable as the

need for accurate patient anatomical and set-up information is ever more critical for

adaptation of the patient’s treatment to their disease. In traditional radiation therapy,

doctors diagnose, stage, and image the disease at the beginning of treatment. They use

this data to plan the whole course of cancer treatment up-front.

Intensity Modulated Radiation Therapy (IMRT) is a radiation delivery method where

the beam fluence is modulated before entering the patient. By doing this over several

static gantry angles (e.g. 5-9), with the modulated field applied at each static gantry angle,

the delivered energy pattern to the patient can be complexly shaped to match the 3D

shape of the tumour. This method required advances in patient imaging technology (the

advent of 3D image sets, i.e. visualize the tumour in 3D), fluence modulation, and

computerized inverse planning.

Rotational IMRT, known commercially as either VMAT (Elekta) or RapidArc™

(Varian Medical Systems), is an even more complex radiation dose delivery technique

6

than IMRT. For this technique, the gantry rotates while the therapeutic radiation beam

remains continuously on, and involves simultaneous modulation of radiation aperture,

dose rate, gantry speed, and potentially collimator and couch speeds. This technique

increases the complexity of radiation delivery by utilizing several additional degrees of

freedom, and is able to further maximize dose coverage of the tumour while also reducing

the damage to surrounding normal tissues.

Comparing IMRT and VMAT for an example of a left-sided breast cancer treatment,

Figure 1.2 (a-c) shows the beam arrangement, isodose distribution of the target volume

and organs–at-risk, as well as Dose-Volume Histograms (DVHs) of tangential IMRT and

VMAT plans, respectively [3]. VMAT gives more lung and heart dose while maintaining

target coverage similar to IMRT, so in this case IMRT may be preferred to VMAT .

Reviewing another example comparing IMRT and VMAT in a prostate treatment plan,

the VMAT plan shows a significant reduction in dose to organs-at-risk while maintaining

similar target coverage and conformality, as illustrated in Figure 1.2 (d-e) [4].

Another example of using advanced imaging technology is Image Guided Radiation

Therapy (IGRT). This encompasses the methods (imaging type, frequency, anatomy

matching, and applied tolerances) used to image and adjust the patient at setup for each

therapeutic fraction delivery. Considering intrafractional tumor motion due to organ

filling and patient breathing during the RT process [5], [6], as well as patient anatomical

changes during treatment (i.e. weight loss or weight gain) and tumour growth or

shrinkage, the original treatment plan may need to be modified to ensure the intended

prescription dose is realized. This is accomplished through a method called Adaptive

Radiation Therapy (ART). ART requires imaging, so changing anatomical parameters of

7

the patient, as well as treatment dose received by the anatomy, can be estimated and fed

back into the treatment planning system (TPS) for adaptive re-planning.

8

Figure 1.2 Beam arrangement, isodose distribution of target volume and organs–at-

risk for a left-sided breast cancer case using tangential (a) IMRT and (b) VMAT plans,

with DVHs shown in (c) [3]. For the patient with medium prostate volume (48.4 cm3),

DVHs of the (d) PTV and (e) rectum in prostate IMRT (solid lines) and VMAT

(broken lines) plans. Depths of the body contours were reduced by 0, 1 and 2 cm [4].

(c)

(d) (e)

9

1.1-c. A brief description of the impact of patient-generated x-ray scatter

As mentioned previously, it is well-known that x-ray scatter is a major effect that

degrades x-ray image quality. In this section, we briefly and qualitatively illustrate the

effect of scatter by examining four distinct beam-detector configurations.

For purposes of illustration we will consider photons with a (1) narrow or a (2) wide

field size irradiating a thin slab; the transmitted signal will be received by only a (1)

narrow or (2) wide detector array. Figure 1.3 illustrates the four physical combinations of

the example source/detectors: (a) narrow beam and narrow detector (NBND), (b) narrow

beam and wide detector (NBWD), (c) wide beam and narrow detector (WBND), and (d)

wide beam and wide detector (WBWD).

Beginning with the NBND situation, the small beam size generates a small amount of

scattered photons in the patient, some of which reach the detector. If the detector array is

enlarged (i.e. NBWD), the detector will receive more of these scattered photons.

Meanwhile, increasing the beam size (i.e. WBWD) leads to more scattered photons

generated within the patient, and thus more reach the detector. For the broad beam

configuration, if the size of detector array can be small (i.e. WBND), the effect of scatter

on the detected signal will be less significant compared to the WDWD case.

10

Figure 1.3 Diagram of 2D beam-detector setup and qualitative assessment of the received

scatter signal (dashed red lines) under (a) narrow beam and narrow detector, (b) narrow

beam and wide detector, (c) wide beam and narrow detector, and (d) wide beam and wide

detector situations.

11

These idealized 2D examples are applicable to the clinical applications relevant to

this thesis. The situation of IMRT or VMAT fields being imaged by a planar MV detector

sometime closely correspond to NBWD, while cone-beam computed tomography systems

most closely correspond to WBWD. The conventional computed tomography systems

one would encounter in CT simulation correspond to WBND, and even the least common

scenario of NBND corresponds to the experimental setup needed to measure narrow beam

attenuation coefficients (e.g. to be sampled by Monte Carlo simulation).

The impact of scattered radiation on x-ray image quality is shown in Figure 1.4,

where the image has become blurred, and both the contrast and the signal-to-noise ratio

(SNR) are reduced. Since the cone-beam CT data reconstruction requires taking a set of

x-ray projection images, the impact of patient-generated x-ray scatter will be seen in a

significant reduction in the quality of the reconstructed CBCT images, and the CT index

(used to infer electron density) would not be accurately calculated. More detailed

information about photon scatter in medical imaging will be described in Section 2.4.

12

Figure 1.4 (a) Basic concept of contrast reduction by scattered radiation2; An example of

scatter effect using two x-ray images of a knee phantom without (b) and with (c) blocks

of Plexiglas adjacent to the knee phantom. The blocks produce a large amount of scatter,

and degrade the image contrast3.

2 http://www.sprawls.org/ppmi2/SCATRAD/#CHAPTER%20CONTENTS 3 http://www.upstate.edu/radiology/education/rsna/radiography/scatter.php

(a)

(b) (c)

13

1.2 Motivation

Many different approaches have been investigated to minimize the effect of scattered

radiation on the detected image. Most are applicable to KV imaging while some are also

useful for MV imaging.

Various physical approaches have been used to suppress the impact of scatter, such

as using a small field of radiation, increasing the distance between the patient and

imaging plane, and adding an anti-scatter grid. However, each physical method has

associated drawbacks, the details of which are described in Section 2.5-a.

In contrast to physical approaches, numerous hardware- and/or software-based

techniques have been proposed to estimate and correct for the scatter signal and therefore

improve image quality. Even with advances in computer hardware, these calculation

techniques often take too long to be practical, so some groups have examined variance

reduction techniques incorporating deterministic methods [7], [8] to further reduce

calculation times. Using pencil beam scatter methods (a simplification of the radiation

transport in the patient), the calculation can be finished in a much shorter time, but at the

cost of limited accuracy for both KV and MV energies [9]–[11]. One group investigated

combining exact first order x-ray scatter calculations with a 2D pencil beam integration in

heterogeneous medium [12], but the results still demonstrated limited accuracy. Other

approaches that incorporate a measured estimate of scatter into a calculational correction

have been proposed. These include the collimator-shadow continuation method [13], the

beam stop technique [14], and the primary modulation method [15], but these are

14

generally inconvenient or not clinically usable. More detail on these methods is provided

in Section 2.5-b.

Full Monte Carlo approaches are generally considered the most accurate (i.e. the

‘gold standard’) of all the purely calculation-based approaches to x-ray scatter estimation,

but the associated calculation times are typically much too long to achieve the required

statistical accuracy for clinical applications [16], [17]. In the Monte Carlo (MC)

simulation of an experimental setup, all available information (geometric and physical

descriptions of the materials as well as radiation source) can be included to create a

realistic model. All possible photon interactions (coherent scatter, incoherent scatter,

photoelectric absorption, and pair/triplet production) in the kilovoltage or megavoltage

energy ranges are considered in the simulation, accounting for attenuation and scatter in

the different components of the experimental setup. Electron and positron interactions in

the materials are also modeled, including electron-electron and electron-nucleus

interactions with energy and range straggling, as well as photon creation due to

bremsstrahlung and positron annihilation events (important for higher photon energies).

This ensures the full MC method is an excellent tool for the accurate characterization of

x-ray systems, and further details are presented in Section 2.6.

In addition to being used as the gold standard for radiation transport, certain ‘partial’

implementations of MC modeling (i.e. where only thousands of histories are used instead

of billions) can be used to help overcome the accuracy limitations of empirical and semi-

empirical models while maintaining reasonably fast execution times. This is an example

of a ‘hybrid’ method. This term describes calculational solutions that combine the best

features of analytical methods and Monte Carlo methods to quickly but accurately

15

estimate scattered photon fluence. Generally, in radiation transport research, hybrid

methods feature a partial Monte Carlo method to track scatter sites of multiply scattered

x-rays, and an analytical method to estimate scattered x-rays generated from those

interaction sites into the imaging plane. A relatively recent review of x-ray scatter

estimation techniques [18] suggests that hybrid approaches represent the best hope for a

fast yet accurate solution to this problem. In this thesis, we develop a unique hybrid

method that combines three calculational approaches including analytical, partial Monte

Carlo, and pencil beam scatter methods, as discussed in more detail in Chapter 4.

Our research group has a strong background investigating scattered MV radiation

[19]–[26] over the last 20 years. Monte Carlo techniques, the transport of scattered

radiation, as well as analytical methods for singly-scattered photons have been

investigated in the past. The physics of x-ray scatter in MV beams was studied and we also

successfully developed a rudimentary x-ray scatter prediction algorithm [20]–[23] for patient

and phantom situations. Building on that work, we have spent much effort in the

development of a patient dose verification system for x-ray radiation therapy, including

creating and validating our own detailed x-ray source model to predict fluence exiting the

linear accelerator [24], [25].

Recently, we have developed a patient dose reconstruction approach that takes the

measured therapy transmission EPID images, converts them to an estimate of primary

fluence (by removing a calculated estimate of the patient-generated photon scattered

fluence entering the detector), and then back-projects this through the patient model to

find the 3D dose delivered to the patient by the treatment beam [26], [27]. Our work in

the area of x-ray image prediction has also been incorporated into a real-time patient

16

treatment monitoring software (research only) developed by our collaborators at the

University of Newcastle (Newcastle, Australia), led by Dr. Peter Greer [28], [29].

However, we recognize that the patient scatter component of our predictive model is

the least accurate step in our modeling, and ultimately limits the degree to which we can

verify delivered treatments. This motivates us to develop a more accurate method of

estimating x-ray scatter for MV scatter entering the EPID imager.

1.3 Outline of the thesis

Chapter 2 introduces the physical principles behind x-ray propagation in matter,

including photon interactions, secondary photon production associated with electron

interactions, and an overview of the physics of photon scattering. The effects of patient

generated photon scatter in linac medical imaging are described, as well as a summary of

methods to suppress/remove photon scatter. Finally, a brief overview of the EGSnrc

Monte Carlo simulation software package, which is used as a tool throughout this thesis,

is provided.

Chapter 3 describes a Monte Carlo based photon-scatter validation tool, which is

developed with EGSnrc by modifying the user code DOSXYZnrc. The tool provides the

ability for users to separately track individual components of photon scatter fluence in

simulations. A detailed description of the modifications, as well as the validation, is

presented. The content is published in the peer-reviewed journal Physics in Medicine and

Biology (May 2020).

In Chapter 4, we detail the development and validation of a tri-hybrid method to

accurately predict the scatter fluence entering the EPID imager. The method combines

17

three separate techniques which are applied to the different components of the patient-

generated scatter – an analytical model for singly scattered photons, a hybrid Monte Carlo

model for multiply scattered photons, and a pencil beam scatter kernel model for photons

arising from electron interactions (i.e. bremsstrahlung and positron annihilation). The

content of this chapter has been published in the peer-reviewed journal Physics in

Medicine and Biology (Sept 2020).

In Chapter 5, the impact of the sampling resolution of a variety of algorithm

parameters used in the previously developed tri-hybrid method are studied (e.g. voxel size,

energy spectra bin width, and number of histories used in the hybrid method). The

sampling is optimized for speed while maintaining an accuracy of at least 2% in predicted

scattered photon fluence and at least 5% in predicted mean energy spectra, for various

clinical beam energies, sizes, and three phantom configurations. This optimization is

important to allow the new method to be efficiently implemented in the clinical setting.

The content of this chapter has been submitted to the peer-reviewed journal Biomedical

Physics & Engineering Express, and is currently under review.

General conclusions are presented in Chapter 6 as well as a discussion of future work

that should be carried out based on the work presented in this thesis.

19

Chapter 2: Physics Background

This chapter introduces the physical principles underlying x-ray propagation in

matter, including photon interactions, secondary photon production (from electron

interactions), the physics of photon scattering, the effect of patient-generated photon

scatter in linac medical imaging, methods to suppress/remove photon scatter in medical

imaging, and a brief overview of the EGSnrc Monte Carlo simulation radiation transport

software package.

2.1 Overview of photon interactions in matter

When x-rays travel through a medium, the photons may interact with the medium

through four different interaction processes: Photoelectric Effect (PE), Rayleigh

Scattering (RS), Compton Scattering (CS), and Pair Production (PP). RS needs to be

taken into account at the KV energy range, and PP becomes increasingly important when

the photon energy is over 1.022 MeV [30], [31].

As a result of these interactions, for a narrow beam of mono-energetic photons, the

fractional reduction in the number of photons, 𝑑𝑁/𝑁 , is proportional to the travel

distance 𝑑𝑥 and the linear attenuation coefficient, 𝜇 .This constant is defined as the

probability per unit path length that a photon interacts within the absorber, and is a

function of both the energy of the photon, and the atomic number Z of the material. The

change in the number of photons in the beam at a particular distance traveled in the

medium is:

𝑁𝑡 = 𝑁0𝑒−∫ 𝜇(𝑙)𝑑𝑙

𝑙0 (Eq. II-1)

20

where 𝑁𝑡 is the number of transmitted photons, 𝑁0 is the number of photons incident on

the surface of the medium, and l is the depth in the medium. For heterogeneous media,

since is a function of the various materials (i.e. atomic numbers), and is also a

function of the depth of the beam in the medium.

There are three other common representations of the attenuation coefficient for

photon interactions. The ‘mass attenuation coefficient’ (𝜇/𝜌) is defined as the probability

(in units of area per mass) that a photon will interact with an absorbing medium:

𝜇

ρ(𝑙) =

𝜇(𝑙)

𝜌(𝑙) (Eq. II-2)

The ‘atomic cross section’ (𝜎𝐴) is 𝜇 divided by the number of atoms per volume of

the absorber (𝑁𝑎𝜌/𝑀), where 𝑁𝑎 is Avogadro’s constant, and 𝑀 is the molecular weight

(g/mole). 𝜎𝐴 defines the probability in units of area that a photon will interact with an

atom (i.e. the nucleus, or a tightly bound electron). The ‘electronic cross section’ (𝜎𝑒) is

defined as μ divided by the number of electrons per volume of the absorber (𝑁𝑎𝜌𝐶𝑀/𝑀),

where 𝐶𝑀 is the molecular charge (electrons /molecule). 𝜎𝑒 defines the probability in

units of area that a photon will interact with an electron (i.e. free or loosely bound).

The mass attenuation coefficient (𝜇/𝜌) of the medium can be decomposed into the

individual contributions from each specific interaction process, as in Equation (II-3),

including photoelectric effect (PE), Rayleigh scattering (R), Compton scattering (C), and

pair production (PP – which includes triplet production). As an example, the

corresponding contributions with respect to incident x-ray energy when the medium is

carbon, are shown in Figure 2.1. Similar equations can be found for the atomic and

electronic cross sections. The kinematics and cross section of each process will be

discussed in the following section.

21

𝜇

𝜌(𝑙) =

𝜇

𝜌𝑃𝐸(𝑙) +

𝜇

𝜌𝑅(𝑙) +

𝜇

𝜌𝐶(𝑙) +

𝜇

𝜌𝑃𝑃(𝑙) (Eq. II-3)

Figure 2.1 Mass attenuation coefficients for carbon over the energy range 0.01 to 100

MeV. 𝝉/𝝆, 𝝈_𝑹/𝝆, 𝝈/𝝆, and 𝒌/𝝆 indicate the contribution of PE, Rayleigh scattering,

Compton scattering, and PP, respectively. 𝝁/𝝆 is their sum. [31]

2.1-a. Rayleigh (coherent) scattering

Rayleigh scattering is the elastic scattering of electromagnetic radiation by a bound

atomic electron instead of a ‘free’ electron (Figure 2.2). The prerequisite is that the size of

the atom is smaller than the wavelength of the radiation, λ, and the atom is neither ionized

nor excited. This process is more probable only at very low energies (15 to 30 keV) and

in high Z materials [32], [33]. The electric field of the incident photon’s electromagnetic

wave will cause all of the electrons in the scattering atom to oscillate in phase. Then, the

atom’s electron cloud immediately reemits a photon of the same energy but slightly

22

different direction. The Rayleigh atomic cross section (in unit of 𝑐𝑚2/𝑎𝑡𝑜𝑚 ) is

proportional to 𝑍2/(ℎv)2.

The differential cross-section for Rayleigh (coherent) scattering is the product of the

Thomson differential cross section and the molecular coherent form factor 𝐹𝑀2(𝑥) as

follows:

𝑑𝜎

𝑑𝛺(𝜃, 𝑥) =

𝑟𝑜2

2(1 + 𝑐𝑜𝑠2 𝜃)𝐹𝑀

2(𝑥) (Eq. II-4)

where 𝑟0 is the classical electron radius (i.e. 2.8179 × 10−13 ) and 𝜃 is the scattering

angle. The scattering angle depends on both 𝑍 and ℎ𝑣. This angle decreases further with

increasing photon energy, and the scattering becomes more forward peaked. Also, the

scattering angle distribution becomes broader when lower energy photons interact with

high Z materials [31].

𝐹𝑀(𝑥) carries information about the molecular structure. Under this circumstance,

the scattering event is considered as being due to a free atom. Thus, it can be calculated

by the sum rule, adding the atomic scattering factor 𝐹2(𝑥, 𝑍𝑖) for i different elements,

Nucleus

Figure 2.2 Kinematics of Rayleigh scattering: the incident photon is scattered at

the angle θ, without losing energy.

𝐸𝑖 θ

23

weighted by the atomic abundance 𝑤𝑖/𝑀𝑖 as shown in equation II-5 below. This is known

as the independent atomic model or the ‘free-gas’ model.

𝐹𝑀2(𝑥) = 𝑊 ∗ ∑

𝑤𝑖

𝑀𝑖𝐹2(𝑥, 𝑍𝑖) (Eq. II-5)

where 𝑊 is the molecular weight of the material, 𝑤𝑖 and 𝑀𝑖 are the mass fraction and

atomic mass of element 𝑖, and 𝐹2(𝑥, 𝑍𝑖) is the atomic coherent form factor [32], [34]. The

value of the transferred momentum 𝑥 (Å−1) depends on the scattering angle:

𝑥 =ℎ𝑣

12.398 𝑘𝑒𝑉𝑠𝑖𝑛 (

𝜃

2) (Eq. II-6)

The contribution of Rayleigh scattering to the total mass attenuation coefficient is

observed to be fairly small. Rayleigh scattering is typically not important for radiation

dosimetry since energy transfer does not occur, and is considered negligible for MV

energies as the cross section value is extremely low compared to other interaction types.

However, it may be more significant in low energy medical imaging applications.

2.1-b. Compton (Incoherent) scattering

Compton scattering occurs when an incident photon with incident energy 𝐸0 ,

interacts with a free electron (Figure 2.3 (a)). The x-ray is scattered through an angle 𝜑

relative to the incident photon’s direction, and a lower energy, 𝐸’. At the same time, the

electron (known as the Compton electron) departs with kinetic energy KE, at angle 𝜃.

Based on the conservation of energy and momentum, the following equations provide a

complete kinematic solution:

{𝐸′ =

𝐸0

1+(𝐸0 𝑚𝑒𝑐2⁄ )(1−𝑐𝑜𝑠𝜑)

𝑐𝑜𝑠𝜃 = (1 + 𝐸0 𝑚𝑒𝑐2⁄ )𝑡𝑎𝑛 (

𝜑

2) (Eq. II-7)

24

where 𝑚𝑒𝑐2 is the rest mass energy of an electron.

Based on equation (II-7), it can be seen that for low energy incident photons (i.e.

𝐸0 ≪ 𝑚0𝑐2) or straight-ahead scattering, (i.e. 𝜑 = 0) the 𝐸’ and 𝐸0 are approximately the

same. This means that the Compton electron does not acquire any kinetic energy, and the

scattering process is nearly elastic at low photon energies (Figure 2.3 (b)).

The Klein-Nishina differential cross-section for Compton scattering at angle 𝜑 for an

incident photon of energy 𝐸0 is:

𝑑𝜎

𝑑𝛺(𝜃, 𝐸0)𝐾𝑁 =

𝑟02

2(𝐸′

𝐸0)2

(𝐸′

𝐸0+

𝐸0

𝐸′ − 𝑠𝑖𝑛2 𝜑) (Eq. II-8)

where 𝑟0 is the classical electron radius and 𝐸′ is the energy of the scattered photon.

Nucleus

Figure 2.3 (a) Kinematics of Compton scatter: the incident photon is scattered at angle

φ, and the Compton electron is ejected at angle θ with kinetic energy KE. E0 and E’ are

the energies of the incident and scattered photons respectively. (b) Graph of the

relationship of energy of incident (hv) and scattered photons (hv’).

ɗ

ű

𝐸0 = ℎ𝑣

𝐸′ = ℎ𝑣’

𝐾𝐸

(a) (b)

25

Figure 2.4 shows the angular distribution of the Klein-Nishina differential cross-

section, where higher photon energies result in more forward-directed photon scatter [30].

It is important to note that this derivation of the electronic cross section for Compton

scattering is independent of the atomic number Z of the absorber because the electron is

considered to be free. The subsequent radiation transport of the created Compton electron

and its interactions with matter will be discussed in Section 2.2. Furthermore, for this

work the bound Compton effect was not considered, and thus the binding energies of the

atomic electrons participating in the Compton interactions are ignored. However, this

effect is only significant for photon energies below 100 KeV [34], [35].

Figure 2.4 Graph of the relationship of energy of incident and scattered photons [30].

26

2.1-c. Photoelectric effect (PE)

The photoelectric effect (Figure 2.5 (a)) occurs when a photon collides with a tightly

bound electron (e.g. electrons in the inner shell of an atom of high Z materials). The

photon is completely absorbed, and the electron is ejected (known as a ‘photoelectron’)

with kinetic energy 𝐸𝐾, independent of its scattering angle 𝜃. For the photoelectric effect

to occur, the incident photon energy must be equal to or greater than the binding energy

EB of the electron it interacts with, i.e., 𝐸 ≥ 𝐸𝐵 . The kinetic energy imparted to the

photoelectron is the difference between these energies, 𝐸𝑘 =𝐸 − 𝐸𝐵. The scattering angle

of the photoelectron decreases with increasing incident photon energy. Sometimes for

excited or ionized atoms, the vacancy will be filled by outer-shell electrons, with the

simultaneous emission of a characteristic x-ray or an Auger electron4 will be emitted.

In general, the atomic cross section for the photoelectric effect, 𝜎𝑃𝐸 (in unit of

𝑐𝑚2/𝑎𝑡𝑜𝑚), is proportional to roughly 𝑍5 for relativistic photons and as low as 𝑍4for

low energy photons [30], and 𝜎𝑃𝐸 integrated over all angles of photoelectron emission is:

𝜎𝑃𝐸 ≅ 𝑘𝑍𝑛

(ℎ𝑣)𝑚 (Eq. II-9)

where 𝑘 is a constant for a given energy and material, [𝑛,𝑚] ≅ [4, 3] at ℎ𝑣 = 0.1 MeV

and below, and 𝑛 and 𝑚 gradually increase or decrease, respectively, with the increase of

incident photon energy (e.g. 𝑛 increases to 4.6 at 𝐸 = 3 𝑀𝑒𝑉; m decreases to 1 at 𝐸 =

5 𝑀𝑒𝑉).

Figure 2.5 (b) further shows the atomic cross sections of various materials over a

range of photon energies from 0.001 MeV to 1000 MeV. The ‘saw-tooth-like’

4 An Auger electron is a low energy electron sometimes emitted when higher shell electrons fill lower shell

vacancies and carries the difference in energy between the two valence bands as kinetic energy.

27

discontinuities observed are due to absorption edges. The incident photon cannot interact

with electrons whose binding energy exceeds the incident photon energy. Therefore once

the incident photon energy exceeds the binding energy of a particular valence shell, it is

able to interact with electrons in that valence, and the interaction coefficient increases.

This effect is more apparent at the larger binding energies of inner electron shells (e.g. K,

L, and M shells) for the higher atomic number media.

An electron orbital vacancy created in a lower shell may be filled through numerous

transitions, and a cascading combination of characteristic x-rays and Auger electron

emissions can occur. This means that a single photon undergoing a photoelectric effect

can create more than one emitted electron or x-ray from the atom. The Auger electrons

will have kinetic energies equal to the differences in their respective binding energies.

Dosimetrically, the approximation of the combined mean energy transferred to all

photoelectrons is of interest because this energy will be deposited as dose in the medium.

28

Incident

Photon (𝐸)

Photoelectron

(𝐸𝑘 = 𝐸 − 𝐸𝑏)

Bound Energy (𝐸𝑏)

(

a)

(b)

(a)

Figure 2.5 (a) Kinematics of the Photoelectric Effect, involving the collision of a

photon with energy 𝐸 and a tightly bound electron with binding energy EB. The

photon is completely absorbed and the electron is ejected with kinetic energy EK. (b)

Photoelectric atomic cross sections for various absorber media (i.e. elements). Note

the discontinuous K, L, and M shell absorption edges [30].

29

2.1-d. Pair Production (PP)

Pair production can only occur when the energy of the incident photon exceeds the

threshold energy, ℎ𝑣 ≥ 2𝑚𝑒𝑐2 (i.e. 1.022 MeV), for pair production.

Pair production occurs when the incident photon interacts with the Coulomb field of

the atomic nucleus. The incident photon will spontaneously convert to an electron, e−,

and a positron, e+ (i.e. the positron is the antiparticle to the electron - the two particles

have identical rest masses and rest mass energies, and charges that are equal in magnitude

but opposite in sign), as shown in Figure 2.6. During the interaction process the energy,

charge, and momentum must be conserved [30], [31]. Due to the large mass of the

absorbing nucleus, its recoil velocity is negligible.

The derivation of the atomic cross section 𝜎𝑃𝑃 for pair production is complicated and

based on several approximations [36]. It has the form:

𝜎𝑃𝑃 = 𝛼𝑟𝑒2𝑍2𝑃(𝐸, 𝑍) (Eq. II-10)

where P(𝐸,Z) is a complex function of photon energy and absorber atomic number. The

atomic cross section for pair production is proportional to the atomic number squared (i.e.

𝑍2). Above the threshold of 1.022 MeV, the probability of pair production increases as

the incident photon energy increases.

The interactions of the resultant electron and positron can generate secondary

photons as discussed in section 2.2.

30

2.2 Overview of electron interactions in matter: energy loss

and generation of secondary photons

X-rays interacting throughout the patient will generate high energy charged particles

through Compton scatter (Compton electron) and pair production (electron and positron

pair). In contrast to the neutral photon, the electron is a charged particle and therefore can

interact at a distance with other charged atomic entities such as orbital electrons and the

atomic nucleus, through its Coulomb electric field. Typically, each electron gradually

loses its kinetic energy through many thousands of interactions with the orbital electrons

(higher number of interactions for higher energy electrons) [31]. When an electron

collides with a target atom, the energy transfer varies depending on how far away the

incident electron is from the atom.

Collisional losses, in general, refer to electron-electron collisions. In addition to

transferring some energy, they can result in ejecting a valence electron in the absorber

producing ionization or excitation. Collisions can be either hard (when the incident

electron trajectory is within the radius of the atom), or soft (when the incident electron

Figure 2.6 The schematic of pair

production in which a photon passes

in the vicinity of a nucleus and

spontaneously forms a positron and

an electron.

Eγ

e-

e+

31

trajectory passes far outside the radius of the atom). Hard collisions involve larger energy

transfers compared to soft ones. Soft collisions account for roughly 50% of the electron’s

total energy loss into the medium. The energy lost by the electron depends on

characteristics of both the target (atomic composition of the medium) and the energy of

the electron.

Less frequently, the electron will interact with the atomic nucleus, resulting in a

larger energy loss but also associated with the creation of a photon. These are termed

radiative losses and the produced photons are known as ‘bremsstrahlung’ photons, and

will be discussed in detail in Section 2.2-a.

The rate of energy loss per unit length a charged particle travels in an absorbing

medium is called the linear stopping power. The linear stopping power divided by the

density of the absorber is called the ‘mass stopping power’ or just the ‘total stopping

power’, 𝑆𝑡𝑜𝑡 . The total stopping power of an absorber is the sum of the collisional

stopping power, 𝑆𝑐𝑜𝑙, and the radiative stopping power, 𝑆𝑟𝑎𝑑.

𝑆𝑡𝑜𝑡𝑎𝑙 = 𝑆𝑐𝑜𝑙 + 𝑆𝑟𝑎𝑑 (Eq. II-11)

The radiative and collisional stopping powers for aluminum, water, and lead are

shown in Figure 2.7. Collisional stopping power does not depend heavily on Z and

remains fairly constant for varying absorber densities at relativistic energies (>1 MeV),

while radiative losses are more strongly dependent on Z.

In context of design of the treatment head for a medical linac, a suitable target

material must be selected, accounting for both the collisional and radiative stopping

power characteristics of the absorbing material. The linac produces a beam of electrons

32

which is directed at the target, with the purpose of the target to convert those electrons to

photons through radiative interactions. Tungsten is a popular choice due to its high atomic

number and thus higher bremsstrahlung radiation yield for relativistic electron energies,

and also possesses a high thermal capacity. In some cases, a layer of copper may be

attached to the base of the tungsten target in order to help conduct heat away to a water-

cooling system.

At megavoltage energies, many x-rays interact through the Compton scatter process,

and some of those Compton-scattered electrons will have enough energy to generate

bremsstrahlung radiation within the patient. If the incident x-ray beam energy is over

1.022 MeV, pair production will also occur. The electron and position pair that are

generated through this process may go on to produce secondary photons. The electron

Figure 2.7 The mass radiation stopping power (thick lines) and collision stopping

power (thin lines) versus electron kinetic energy for aluminum, water, and lead.[30]

33

may produce a bremsstrahlung photon while the positron will eventually suffer

annihilation with an electron (termed positron annihilation) creating a pair of annihilation

photons. These processes will be discussed in the following two sections.

2.2-a. Bremsstrahlung Radiation (BR)

Bremsstrahlung radiation is produced when an electron (or positron) decelerates

during an inelastic collision with an absorbing nucleus’ Coulomb field (Figure 2.8) 5.

During this change in acceleration, a fraction of the kinetic energy (KE) of the electron is

lost and emitted as an x-ray. This can be as large as the entire initial KE of the electron or

very small, e.g., a 6MV electron beam bombarding a tungsten target produces a

continuous x-ray spectrum in the range of 0-6 MeV. In general, the average x-ray energy

of a continuous bremsstrahlung spectrum is roughly one third of the maximum kinetic

energy of the incident electrons.

5 http://physicsopenlab.org/2017/08/02/bremsstrahlung-radiation/

34

Bremsstrahlung radiation is mainly used for generating the photon treatment beam in

medical linacs. The radiative stopping power is given as the product of the cross section

for the emission of bremsstrahlung radiation 𝜎𝑟𝑎𝑑, the atomic density 𝑁𝑎 (Avogadro’s

number/atomic mass number, 𝑁𝐴/𝐴), and the initial total energy of the electron, 𝐸𝑖, which

is a sum of the electron’s rest mass and initial kinetic energy:

𝑆𝑟𝑎𝑑 = 𝑁𝑎𝜎𝑟𝑎𝑑𝐸𝑖 (Eq. II-12)

The cross section, 𝜎𝑟𝑎𝑑, is proportional to the square of the atomic number of the

absorbing nucleus Z. Combining equation II-12 with the well-established Bethe and

Heitler result [30] for 𝜎𝑟𝑎𝑑 gives:

Figure 2.8 Bremsstrahlung radiation: electromagnetic radiation produced by

the deceleration of a charged particle when deflected by another charged particle,

typically a high energy electron incident on an atomic nucleus. The process follows

the conservation of energy and momentum, as shown two examples of emission of

low-energy and high-energy bremsstrahlung x-ray.

35

𝑆𝑟𝑎𝑑 = 𝛼𝑟02𝑍2 (

𝑁𝐴

𝐴)𝐸𝑖𝐵𝑟𝑎𝑑 (Eq. II-13)

where 𝛼 is the fine structure constant (1/137), 𝑟0 is the classical electron radius (~2.8 fm),

and 𝐵𝑟𝑎𝑑 is a slowly varying function of 𝑍 and 𝐸𝑖.

For most stable elements 𝑍/𝐴 = 0.5 and approaches 0.4 for higher atomic number

elements like tungsten. Therefore, the radiative stopping power is roughly a function of

the absorber’s Z, and the larger the Z the greater the radiation yield. The radiation yield,

which defines the fraction of initial kinetic energy emitted as radiation, will increase with

higher atomic number absorbers and with higher initial kinetic energies of the electron.

For interactions in low atomic number tissues found in patients, radiation yield is smaller

than for heavier elements but can still be significant at the commonly used photon

treatment energies of 6 and 18 MV.

2.2-b. Positron Annihilation (PA)

The positron is an antiparticle to the electron and is generated through pair

production interactions (as described in Section 2.1-d). The two particles have identical

rest masses and rest mass energies, and charges that are equal in magnitude but opposite

in sign.

Positrons generated through pair production interaction will annihilate when

encountering an electron, meaning that both the positron and electron disappear and give

rise to two photons. The positron may annihilate when it is at rest (i.e. when it has no

remaining kinetic energy) or in flight (when it is has some remaining kinetic energy) as

illustrated in Figure 2.9.

36

In both cases, the electron interacted with is considered stationary and free. The most

common annihilation occurs after the positron has lost its entire kinetic energy and

annihilates with an orbital electron of the absorber. This annihilation creates two photons

each with energy of 0.511 MeV (the rest mass of the annihilating positron and electron),

and the photons travel at 180° away from each other, obeying conservation of total charge,

total energy, and total momentum. An ‘in-flight’ positron with non-zero kinetic energy

can also annihilate with a free or orbital electron and will also obey conservation of

energy, momentum, and charge. The in-flight annihilation will create two photons of

varying energy and outgoing scattering angles that depend on the impact parameters of a

two-body elastic collision. In contrast to the annihilation at rest, the two generated

photons will not move in exact opposite directions to each other due to the conservation

of momentum [30], [31].

𝑒−

𝑒+

𝛾1

𝛾2

𝑒+

𝑒−

Figure 2.9 Kinematics of at-rest (left) and in-flight (right) positron annihilation.

37

2.3 Discrete sampling of analytical solutions for photon

scattering

A critical concept needed for Chapters 3 and 4 of this thesis is the general equation

for estimating the number of scattered photons at an imaging pixel, due to scatter

originating in a patient voxel [37]. Considering a uniform monoenergetic photon beam

incident on a homogeneous target volume/voxel 𝑑𝑉 , the number of scattered photons

which are received by a detector pixel (Figure 2.10) is estimated as:

𝑁𝑠(𝜃, 𝐸) = 𝑁0𝑛𝑠.𝑐𝑑𝜎

𝑑Ω(𝜃, 𝐸)ΔΩ (Eq. II-14)

Where 𝑁𝑠 and 𝑁0 are the number of scattered and incident photons respectively, 𝑛𝑠.𝑐.

is the number of scattering centers in the target volume, and 𝑑𝜎

𝑑Ω is the differential cross

section per scatter center and accounts for the probability of an incident particle being

scattered through an angle 𝜃. In addition, 𝛥𝛺 is the solid angle subtended by a discrete

imaging detector element with area of 𝑑𝐴 and accounts for the inverse square relationship

from the scatter centre to the finite area defined by that detector element, with the

following expression:

𝛥𝛺 = 𝑑𝐴 ∙𝑐𝑜𝑠(𝜃)

𝑟2 (Eq. II-15)

38

Figure 2.10 Incident photons interact with a phantom/patient voxel, and scattered photons

are detected by an ideal detector at angle θ relative to the incident beam.

Equation II-14 has a few assumptions that should be noted. First, all scatter events

for a given phantom voxel are assumed to occur at the center of the voxel. Second, all

photons scattered from a given phantom voxel to a given detector pixel are assumed to

have a common trajectory, along a rayline connecting the voxel center to the pixel center.

In Compton scattering, the density of scattering centers is given by the electron

density of the phantom voxel material, and the differential cross-section is calculated

using the Klein-Nishina approximation (see section 2.1-b). For Rayleigh scattering, the

density of scattering centers is given by the density of atoms (for elements) or molecules

(for compounds) in the target, and the differential cross-section is computed using a ‘form

factor’ for molecules and compounds (see Section 2.1-a).

The most common use of the general scatter equation is to estimate the singly

scattered x-ray photon fluence [14], [38]–[43], which represents a large fraction of the

total scatter for both diagnostic and therapeutic energy ranges, although the exact fraction

depends on photon energy, field size, and the geometry of the irradiated object. For

example, Kyriakou et al. found that singly scattered photons represented approximately

39

70% of total scatter fluence for a 12 cm diameter water phantom at 40 keV [42]. For

megavoltage therapeutic beams (e.g. 6 and 18 MV), the contribution from singly scattered

photons to total scatter can increase up to 90% depending on the field size and air gap.

40

2.4 Patient generated photon scatter in linac medical imaging

As shown in Figure 2.11, x-rays from the incident radiation beam that reach the

imaging plane without interacting in the patient/phantom are considered primary photons.

However, many x-rays interact within the patient/phantom, either being fully absorbed by

the medium (photoelectric or pair production interaction), or are scattered once or

multiple times (i.e. Compton or Rayleigh interactions), and some of those scattered

photons will reach the imaging plane. As mentioned in Section 2.1, the ideal radiographic

image would be composed of transmitted primary photons only. Contrast (i.e. differences

between neighboring regions on an imaging plane) is important measures of medical

image quality. However, the scattered photons generated in the patient are an additional

contaminating component of the radiographic image, and will reduce contrast and

contrast resolution as shown in the Figure 2.126.

6 https://www.upstate.edu/radiology/education/rsna/radiography/scattergrid.php

41

Figure 2.11 X-rays reaching the imaging plane without interacting in the

patient/phantom are primary photons. Some x-rays interact and are entirely absorbed

within the patient, while others are scattered either once or multiple times before

reaching the imaging plane.

Figure 2.12 Comparison of radiographic image with patient-generated x-ray scatter

suppressed (left hand side) vs. not suppressed (right hand side).

42

Commonly both KV and MV imaging systems use a flat panel detector and will be

susceptible to the effects of patient-generated scattered photons (as mentioned in Section

1.1-c). There are four major physical factors that impact the characteristics of the patient-

generated scattered photons reaching the imaging plane: incident beam energy, x-ray field

size and fluence distribution, patient geometry (i.e. thickness and composition), and the

air gap between the patient and the imaging plane. For both KV and MV imaging systems,

the effect of the incident beam energy will be addressed in Sections 2.4-a and b. The x-

ray field size (as shown in Figure 2.13(a)7) and patient geometry (mainly thickness)

determine the irradiated volume, and the amount of scattered radiation produced is

generally proportional to the irradiated patient volume [44], [45]. In addition, increasing

the distance (i.e. air gap) between the patient's body and the imaging detector will reduce

the amount of patient-generated scattered radiation reaching the detector since the

scattered radiation leaving a patient's body is more divergent than the primary x-ray beam

at the imaging plane, as illustrated in Figure 2.13 (b).

7 http://www.sprawls.org/ppmi2/SCATRAD/

43

Figure 2.13 (a) Illustration of the reduction of patient-generated scatter entering

the imager through reducing the x-ray beam field size. (b) Illustration of the

divergence of patient-generated scatter into the imager for various air gaps. The

scattered photons diverge more quickly compared with the primary beam for any

given air gap (three example air gaps shown).

Vario

us air g

aps

(a)

(b)

44

2.4-a. Photon scatter effect on CBCT KV imaging

Since its commercial availability on modern medical linear accelerators beginning

in 2008, diagnostic x-ray imaging has been used to help set up the patient before the

therapeutic x-ray beam is delivered. These systems allow high quality anatomical

imaging of the patient at the time of treatment, which provides critical positioning

verification just prior to delivering the radiation treatment. They can provide ‘cone-

beam’ computed tomography (CBCT) image sets (volumetric 3D image data sets) that

are similar to conventional computed tomography (CT) data sets, as well as simple

planar projection radiographs.

The hybrid Monte Carlo technique used for fast but accurate photon scatter

estimates, was originally developed for KV applications such as CBCT imaging. Since

this technique is modified and incorporated into MV imaging applications in this thesis,

a brief review of the effect of photon scatter in CBCT imaging is justified.

For decreasing x-ray energies through the KV energy range, Rayleigh scattering

starts to become more significant relative to Compton scattering. The Rayleigh

component is strongly forward peaked, and at lower energies the angular distribution of

Compton scattered photons spreads significantly (i.e. the Klein–Nishan cross section

becomes ‘peanut shaped’). Since lower energy photons have a shorter mean free path, the

multiply scattered and singly scattered components become similarly important.

In addition, since only broad conical beams of x-rays are available for use in the

linac-mounted KV imaging systems, instead of the narrow fan-beams which are

employed in conventional diagnostic CT units, many more patient scattered x-rays are

produced and reach the detector during CBCT imaging versus conventional CT. This

45

causes severe contamination of the projection images by the scattered x-rays, resulting

in significant image artifacts (such as cupping and other shading artifacts) and relatively

poor contrast of the CBCT data sets. This hinders their clinical usefulness because

image contrast is degraded (such that anatomy can’t be seen as easily) [46], and also the

conversion of the CBCT data to a physical or electron density map is not very accurate,

so they also can’t be used reliably for calculating patient dose distributions for adaptive

radiation treatment planning [47] (Figure 2.14). Removal of the effect of scatter from

the projection images results in CBCT being much more effective for both anatomy

segmentation and patient dose calculation.

46

2.4-b. Photon scatter in EPID MV imaging

It has been well established that patient dose verification can be accomplished

with the EPID, either using the 2D planar dose (with the measured transmission EPID

image compared to a pre-calculated or ‘predicted’ transmission image), or estimating the

3D dose distribution in the patient (which can be compared to that intended by the

treatment planning system) [25], [26], [48]–[51]. If a difference is found between the

Figure 2.14 (a) Example of CBCT cupping artifact for a homogeneous water cylinder

with a 10% scatter-to-primary (SPR) ratio, and (b) with a 120% scatter-to-primary

(SPR) ratio. (c) CBCT streaking artifact illustrated for a homogeneous water cylinder

with two dense material inserts with a 10% SPR ratio, and (d) with 120% SPR ratio

[47].

47

measured and expected radiation delivery, the source of the difference could be corrected

in a following treatment fraction, thereby potentially improving the patient outcome.

For 3D in vivo patient dose estimates, typically the measured transmitted fluence

is corrected for patient-generated scatter, whose effects are estimated and removed from

the measured transmission image. The resulting estimate of ‘primary only’ fluence

transmission can then be backprojected to calculate delivered dose to the patient. Even

though published EPID-based in vivo dosimetry methods have achieved better than 90%

agreement in dose comparisons, it is still recognized that the accurate estimate of patient-

generated scatter fluence entering the EPID is still a significant challenge to using EPIDs

for patient in vivo dosimetry [52]–[55]. The sensitivity and specificity of this comparison

method is dependent on the accuracy achievable in estimating the patient-generated

scatter fluence component from the 2D images. Therefore, this issue ultimately limits the

accuracy of patient dose verification applications, including in vivo patient dose methods.

Currently for patient-specific quality assurance measurements, an accuracy of 3%/3mm

[56] is desired, therefore this level of accuracy would also be a reasonable target for in

vivo dosimetry applications.

When energy is increased through the MV energy range, Rayleigh scatter is

negligible, and the distribution of Compton scattered photons becomes more forward-

peaked. Also, since high energy photons have longer mean free paths, singly scattered

photons become more dominant than the multiply scattered photons. Distinctly for MV

energy beams, the electron-interaction-generated (EIG) scattered photons fluence (i.e.

bremsstrahlung radiation and positron annihilation) becomes more important due to its

proportionality with increasing energy. Overall, the patient-generated scattered x-ray

48

fluence entering MV images can be significant, making up as much as 30% of the MV

image signal [21], [57]. The presence of this scatter reduces image contrast and reduces

the ability to confidently verify the treatment delivery in dose verification applications.

2.5 Methods to limit photon scatter in medical imaging

There has been much work done to improve image quality through various means

of suppressing or removing the estimated patient scatter component of the individual

projection images, mainly focusing on the diagnostic energy range, although some

approaches may be used at the therapeutic energy range. Clearly for planar projection

imaging, reducing or removing the scatter component directly improves image contrast

and contrast resolution. For CBCT volume reconstruction, by limiting the effect of

scatter in the projection image before volumetric image reconstruction takes place, the

associated artifacts are removed from the CBCT image and the dataset is improved for

patient dose calculation and anatomical identification/contouring. Both hardware

(equipment/measurement-based) and software (i.e. calculation-based) methods have

been explored in the past couple of decades, and are described in the following

subsections.

2.5-a. Hardware approaches to supress scatter impact

As shown in Figure 2.13, simply increasing the distance (i.e. air gap) between the

patient's body and the image detector will reduce the amount of scattered radiation

reaching the image relative to the primary fluence, since the scattered radiation leaving

the patient's body is more divergent than the primary x-ray beam. This is illustrated in

Figure 2.15 for MV energies, and the scatter faction slowly decreased with the increasing

49

air gap [57]. For example for a 70 kV x-ray beam, increasing the air gap from 10 to 100

cm, will decrease the scatter-to-primary ratio by 83% [58]. However, the greater the air

gap, the stronger the magnification a projection image has, which limits the maximum

field size that can be imaged by a given sized imaging detector.

Another effective approach for removing at least some of the patient-generated

scatter is through use of an anti-scatter grid (Figure 2.168), which is typically placed on

top of the imaging panel for diagnostic image systems [18]. The grid strips are made of a

material which highly attenuates x-rays (e.g. lead) and is oriented divergently in the

direction of the primary x-ray beam, although typically only in one dimension. Since the

x-ray beam direction is aligned with the grid, much of the primary radiation passes

8 https://www.upstate.edu/radiology/education/rsna/radiography/scattergrid.php

Figure 2.15 The dependence of scatter fluence on air gap (17 cm thick slab with 30x30

cm2 field size) for polyenergetic beams of energy 6 MV (closed circles) and 24 MV

(open circles) [57].

50

through the interspaces without encountering the lead strips. In contrast, most scattered

radiation leaves the patient's body in a direction different from that of the primary beam.

Since the scattered radiation is not generally aligned with the lead strips, and the scattered

photon energies are generally less than the primary photon energy, the scattered radiation

is more readily absorbed. The ideal grid would absorb all scattered radiation and allow all

primary x-rays to penetrate to the image receptor. Unfortunately, there is no ideal grid,

because all such devices absorb some primary radiation and allow some scattered

radiation to pass through. Even when such scatter reduction approaches are employed, x-

ray projection data will still contain significant scattered photon signal, which will cause

significant artifacts if uncorrected, for example in CBCT applications [59].

For megavoltage (MV) x-ray imaging applications, such as MV fluoroscopic imaging

and MV cone beam computed tomography (MV-CBCT), the use of a MV anti-scatter grid

is not practical since a MV grid would need to be extremely thick to reduce patient scatter

relative to primary (many cm), resulting in a bulky and costly device which would also

itself serve as a scatter source, reducing its effectiveness [60]. Another experimental

approach to account for patient scatter in MV photon beams involved a beam-stop array

(BSA), which was utilized to estimate patient scatter by taking two sets of measurements,

with and without the beam-stop array. This was then used to correct for the cupping

artifacts in MV computed tomography; however, since two sets of images are required,

the BSA approach increased patient dose during the image acquisition [61]. Another

experimental method applied to MV energies was the use of a Cerenkov radiation based

electronic portal imaging device (CPID), which was designed and evaluated for its ability

to suppress scattered x-rays. Due to its larger bulk and mass compared to current

51

commercial EPIDs, it required a re-engineering of the linac gantry [62]. Thus,

experimental methods for scatter reduction are currently not feasible for routine

application to MV imaging.

52

Figure 2.16 (a) The design of an anti-scatter grid for KV imaging, with lead strips

oriented along one dimension separated by a low attenuating interspace material such

as carbon fiber or plastic. (b) Two images of the AP projection of a pelvis phantom

were obtained at 75 kV without using (left) and with using an anti-scatter grid (right).

(a)

(b)

53

2.5-b. Software/calculation approaches to supress scatter impact

This group of methods typically consists of various calculational techniques to

estimate the amount of scatter in an image, which can then be subtracted from the

measured image to produce an estimate of a primary-only image. Calculational

techniques include full Monte Carlo simulation, analytical methods, pencil beam

convolution/superposition methods, and most recently ‘hybrid’ approaches that combine

analytical and Monte Carlo techniques, which have shown much promise [18].

Monte Carlo simulation has been demonstrated as the benchmark approach in

accuracy when dealing with radiation transport problems. It predicts macroscopic

behaviour of radiation transport based on randomly sampling the probability density

functions associated with the underlying physical processes for individual radiation

particles. By repeatedly sampling these over a very large number of particles, a

statistically accurate macroscopic solution is eventually determined. Since Monte Carlo

simulation is a powerful research tool used throughout this thesis, an overview of the

Monte Carlo simulation method and the specific software package used in this thesis

(EGS/BEAM) are provided in Section 2.6.

As mentioned, due to the stochastic nature of Monte Carlo simulation, a large

number of particle histories are required to achieve the desired stochastic accuracy in a

solution, which means a heavy computational cost. Therefore, many variance reduction

techniques have been examined to accelerate the Monte Carlo simulation process (details

are described in Appendix A.2).

54

Since singly scattered photons are the most significant component of total scattered

signal for both KV and MV imaging, many researchers have investigated using analytical

modeling for patient scatter estimates for different imaging modalities [15], [32], [41],

[43], [63], [64]. The method is based on the numerical integration of the analytical

equations governing coherent (Rayleigh) and incoherent (Compton) scatter kinematics.

By sampling over a lattice of ‘scatter source’ points within the volume defined by the

intersection of the beam with the patient, singly scattered fluence contributions to a point

on the imaging plane may be exactly calculated. Repeating this process for the grid of

pixels on the image plane provides a map of singly scattered fluence for the entire imager.

The polyenergetic energy spectrum of the incident x-ray beam may be divided into

discrete bins and by repeating the analytical process for each energy bin, the singly

scattered fluence of a polyenergetic beam may be predicted. The attenuation along each

ray line may be calculated exactly using a ray tracing algorithm to find the exact

radiological pathlength through the remaining tissue. The probability of interaction is

found using the corresponding (Klein-Nishina or Rayleigh) differential cross section.

‘Hybrid’ methods combine the best features of analytical and Monte Carlo

techniques to quickly but accurately estimate scattered photon fluence [41]. Generally,

hybrid methods feature a Monte Carlo method to track scatter centers along the path of

several thousand radiation particles (not the billions typically required in a full Monte

Carlo simulation), and then analytically ray-trace scattered photon fluence from each

interaction centre to all pixels over the entire detector. Hybrid method shows the highest

potential for a fast and accurate solution to estimate x-ray scattering based on recent

review articles [18].

55

Pencil beam scatter kernel superposition methods (PBSKs) are widely applied

commercially because the method has the advantage of being computationally efficient.

The pencil beam scatter kernels themselves are generated using Monte Carlo simulation

with a finite set of configurations. For image scatter prediction applications, the scatter

produced at the imager by each pencil beam is given by the corresponding kernel (i.e.

predetermined point-spread functions). A fast adaptive scatter kernel method (fASKS)

used object-dependent asymmetric scatter kernels to estimate and subtract scatter from

KV projection images [10].

Recently a novel method was proposed that deterministically solves the linear

Boltzmann transport equation using iterative applications of the analytical technique [40],

[65]. Three main steps are involved: (a) Photons are ray-traced from the x-ray beam

source into voxels of the phantom/patient where they experience their first scattering

event and form scattering sources. (b) Photons are propagated from their first scattering

sources across the object in all directions, to form second scattering sources; this process

is repeated until a defined maximum-order scattering is reached (i.e. iteratively). (c)

Photons are ray-traced from all scattering sources within the object to all pixels in the

detector. This approach is computationally expensive but is accelerated by the use of

graphics processing units (GPUs) and is commercially implemented as Acuros® CTS

(Varian Medical Systems) to estimate and remove patient-generated scatter in KV

projection images. However, the approach is limited to low energy diagnostic x-ray

imaging due to the lack of electron interaction modeling.

Within the last three years, there have been significant advances in the application of

‘deep learning’ algorithms to image processing applications. Deep learning methods

56

essentially train an artificial neural network (ANN) with known inputs and outputs that

are paired together, then introduce new (i.e. previously unseen) inputs and obtain new

outputs predicted by the network. In the radiation oncology realm, the most common

application of these methods to date has been in automated segmentation of patient

anatomy [66]. In this application, an input data set is a CT dataset and the paired output

are the physician-segmented anatomical structures on the CT data. Hundreds of paired

training data sets are used to train the ANN, then a new input dataset (unpaired) is

introduced and the ANN outputs the segmented anatomy (Figure 2.17 (a) and (b)) [67].

There has been some interest in applying these techniques to the CBCT scatter-

contamination problem, although most work has focused on a ‘slice-by-slice’ technique,

where the input CBCT data set and the paired output data set (a corresponding CT data

set) are used to train the ANN, fed in slice-by-slice [68]–[71]. However, the effectiveness

of this technique is limited by the differences in anatomy (and therefore scatter effects)

between the paired CBCT and CT scans, which also requires 3D image registration.

Furthermore, the approach does not make use of available prior knowledge, which some

researchers are trying to exploit by applying the deep learning tools to the 2D projection

images (instead of the 3D reconstructed images), essentially working directly in the

domain of the scatter fluence. In this approach the input data are the CBCT projection

images and the paired outputs are the corresponding CBCT projection images with the

scatter removed. The prior knowledge here then is the accurate estimate of the scatter

signal in each of the projection images, which can then be removed from all the measured

2D projection images, prior to 3D image reconstruction. Nomura et al. have recently

demonstrated feasibility of this approach using only a few simple geometric phantoms as

57

training data sets [72]. The challenge of this technique is that the user requires expertise

to generate the accurate scatter estimates that are needed for the paired training data sets.

Figure 2.17 (a) A deep learning method applied for automatic segmentation of

anatomical images of a nasopharynx patient, from [67]. (b) Training, validation and

testing processes of the CNN require three different datasets. The model is trained

using a training dataset. During the training, the validation dataset is used to monitor

and minimize bias in the model. Finally, independent test datasets are used to test the

generalization capability of the model for completely new data.

(a)

(b)

58

2.6 Overview of Monte Carlo technique

2.6-a. Monte Carlo technique

‘Monte Carlo’ is the name of a European city with an international reputation as a

gambling destination. This inspired the naming of a group of mathematical techniques

that are based on random sampling, since randomness underlies gambling. Specifically,

for solving radiation transport problems, the underlying probability distributions of the

various physical particle-interaction events, are randomly sampled. The Monte Carlo

method is considered the gold standard in terms of accuracy for solving radiation

transport problems, but is associated with a heavy computational cost. Furthermore, the

Monte Carlo approach allows one to investigate parameters which may not be physically

measurable.

A random trajectory for an x-ray photon is simulated through the knowledge of the

probability distributions governing the individual interactions of the particle in the

various materials involved. The probability distributions must be sampled in a truly

random fashion to reduce systematic errors. The physical quantity of interest can be

determined by summing over a large number of particles. The more particles that are used,

the more accurate the solution that is obtained. There are approaches available that can

make more efficient use of each photon history in order to achieve output of a given

statistical accuracy but with fewer histories, and these are termed ‘variance reduction

techniques’. The rationale of Monte Carlo simulation in radiation transport and some

common variance reduction techniques are discussed in Appendix A.

59

The Monte Carlo technique is utilized throughout this thesis as an energy fluence

validation tool, but is also used in generating scatter energy fluence kernels, and also as

part of a custom hybrid MC method. Each of these will be described in following chapters.

With the increase in cost effectiveness of computing power combined with the

widespread availability of well-developed and extensively validated computer software

packages (i.e. EGSnrc9, Geant410, Penelope11, and MCNP12), the Monte Carlo technique

has been increasingly relied upon as a powerful tool for radiation transport in the field of

medical physics (radiotherapy, diagnostic imaging, and nuclear medicine).

Among the different available radiation transport Monte Carlo simulation packages,

EGSnrc has been thoroughly established to be in good agreement with measurement,

within experimental uncertainty for both KV and MV energy ranges [73]–[75]. A detailed

description of EGSnrc can be found in Sections 2.6-b and 2.6-c.

2.6-b. Overview of EGSnrc

The first version of the EGS (Electron Gamma Shower) Monte Carlo code was

written in the early 1960s, and later developed into EGS4 in the 1980’s, which was

further developed into EGSnrc in the 2000’s by the Ionizing Radiation Standards group of

the National Research Council (NRC) of Canada [73]. EGSnrc can be used to solve

radiation transport problems involving coupled transport of electrons (including positrons)

and photons through matter in arbitrary geometries, for the electron energies ranging from

10 eV to 100 GeV and for photon energies of 1 eV to 100 GeV.

9 https://nrc-cnrc.github.io/EGSnrc/ 10 https://geant4.web.cern.ch/ 11 https://www.oecd-nea.org/tools/abstract/detail/nea-1525 12 https://mcnp.lanl.gov/

60

Compared with the previous EGS4 version, EGSnrc incorporates significant

advances in several aspects of electron transport, including: a new electron transport

algorithm PRESTA-II, a more accurate boundary crossing algorithm, and improved

sampling algorithms for a variety of energy and angular distributions. The detailed

description of updates of each sub-version of the EGSnrc system can be found in the

EGSnrc manual, which is available online13 for the interested reader.

The EGSnrc code system is written in Mortran, an extended Fortran language. To use

EGS, a ‘user code’ is mandatory. As implied in the name, the ‘user code’ is written by the

user and describes the geometry of the radiation transport problem including materials

and radiation source. The user code interfaces to the underlying physics routines, thereby

allowing the user to customize the simulation while minimizing the risk of introducing

errors into the physics of the radiation transport steps, which are insulated from

modification by the user. Of the many pre-written user codes available, there are two

main user codes for the purpose of modelling external beam radiotherapy systems:

BEAMnrc (simulation of the linear accelerator head), and DOSXYZnrc (simulation of the

patient/phantom). Users can also make their own user code or modify an existing user

code to fulfill their research interests.

Each user code must contain calls to two main EGSnrc subroutines HATCH and

SHOWER that are associated with setup of the radiation transport problem, and

incorporate other subroutines (HOWFAR, HOWNEAR, and AUSGAB) to determine the

geometry and output scoring. The general workflow is shown in Figure 2.18, and the

specific communication with EGS by means of basic subroutines is listed in Table 2.1.

13 https://nrc-cnrc.github.io/EGSnrc/doc/pirs701-egsnrc.pdf

61

The user code used throughout this thesis is a modified version of the DOSXYZnrc

user code [76]. The original purpose of the DOSXYZnrc user code (i.e.

dosxyznrc.mortran) is to calculate the 3D dose deposited in a rectilinear phantom. We

modified its AUSGAB subroutine to score the phase-space information (including

interaction history) of photons exiting from the phantom, and then project these photons

to the defined imaging plane to obtain the fluence. These modifications were needed to

create the validation toolbox that was then used to test our proposed tri-hybrid method.

The detailed description and testing of the validation toolbox can be found in the Chapter

3.

During the development of the tri-hybrid method, the photon scatter interaction

centres and scatter order need to be tracked as part of the hybrid MC method used for

calculating multiply scattered photon fluence. We therefore modified AUSGAB again to

obtain the phase space of the photon prior to scattering to fulfill our needs. The detailed

description can be found in Chapter 4.

An input (*.egsinp) file provides specific details to define the simulation geometry,

media, and the radiation source. To simplify our input files, the *.egsphant files are used

to provide the number of media, name of each medium, and 3D volumetric position,

density and medium index. For CT based phantoms, the stand-alone code, ctcreate,

allows one to create the CT phantom in the format of *.egsphant from CT data in DICOM

(Digital Imaging and Communications in Medicine)14 format.

Finally, a core step of the Monte Carlo simulation method in EGS is a ‘pre-

processing’ step that, ahead of the actual simulation, defines the material data that

14 https://www.dicomstandard.org/

62

includes all required cross section data for each of the media involved in the simulation.

A stand-alone utility program, PEGS4, generates the material data files containing the

cross-section information for the materials of interest in the calculations. When a material

data set is generated using PEGS4, lower energy bounds for the production of secondary

electrons and photons, AE and AP respectively, are defined. These parameters represent

the lowest energy for which the material data are generated. The corresponding upper

bounds are UE and UP. Of note, the EGSnrc user code can be run in a ‘pegs-less’ mode

(i.e. without using a *.peg4dat file), where the photon cross sections can be generated on-

the-fly. However, throughout this thesis, the Monte Carlo simulations are run with

*.pegs4dat files.

63

Table 2-1 The purpose of general subroutines in the EGS code system [73]

HATCH Establish media data

SHOWER Initiate the radiation transport

HOWFAR & HOWNEAR Specify the geometry

AUSGAB Score and output the results

Control variance reduction

Figure 2.18 The structure of the EGSnrc code system and how it interfaces to a user

code [73].

64

2.6-c. Accuracy of EGSnrc based Monte Carlo simulation

Since the physical processes are simulated by randomly sampling particle interaction

cross sections, the results are subject to statistical uncertainty. This uncertainty is

estimated by splitting the simulation into ten independent batches, each processing the

same number of incident particles, for each scored quantity of interest. The final value of

the quantity of interest will be the average of the results of ten batches, while the

uncertainty estimate on this quantity is given by the standard deviation of the mean

(across the ten batches). The number of batches can be specified by the user, but ten is

commonly used.

Random sampling is an essential feature of any Monte Carlo simulation to help

ensure a high accuracy by preventing or reducing systematic errors. Essentially the

“pseudo” random number generator (RNG) is the engine of any Monte Carlo simulation.

As it imitates the true stochastic nature of particle interactions. Since a random number is

generated every time an interaction distribution needs to be sampled, this is an extremely

important component of any Monte Carlo method.

EGSnrc is supplied with two random number generators: RANLUX and RANMAR.

The generator used with EGS4 is RANMAR, while the default generator for EGSnrc is

RANLUX, which comes with a variety of “luxury levels” that select the quality (at the

cost of speed) for the random number sequences [77]. The important features of these

random number generators are a) that they produce a deterministic sequence of numbers

whose properties approximate the properties of sequences of random numbers with the

same seed on different machines, and b) that they can be initialized to guarantee

independent random number sequences when doing parallel computing.

65

The residual uncertainties in photon/electron interaction cross section data are

currently the limiting factor for the accuracy in the simulation result for the quantity of

interest. Any error in the cross-section data will be directly transferred to the final results

via the random sampling process [78], [79].

With a major advantage in the widespread use of EGSnrc throughout the medical

physics research community, and also the significant contribution of the NRC

development group providing support, the accuracy of EGSnrc has been continuously

improved, tested, and validated. For example, Faddegon et al. investigated the accuracy

of the EGSnrc Monte Carlo simulation software package, comparing results to measured

fluence profiles where EGSnrc and PENELOPE results agreed with measurement within

one standard deviation of experimental uncertainty [74]. Yani et al. compared the MC

code systems EGSnrc and Geant4 against experimental measurement for a 10 MV photon

beam with a field size of 4x4 cm2 irradiating a homogeneous phantom. Agreement

between the dose distribution from EGSnrc and the experimental data in a homogenous

water phantom was observed (Figure 2.19) [80]. For a 15 MV photon beam and field

sizes of 1×1, 2×2, 5×5, and 10×10 cm2 irradiating a soft tissue-lung phantom, EGSnrc

calculations agreed with the experimental results for all field sizes (Figure 2.20) [81].

66

EGSnrc was also used to compare the response of an aluminum-walled thimble

chamber to that of a graphite-walled thimble chamber for a Co-60 beam. The Monte

Carlo calculated values of the chamber response differ from the expected by only 0.15%

and 0.01% for the graphite and aluminum chambers, respectively [78]. Also, detailed

transmission measurements were performed and used to benchmark the EGSnrc system.

Comparisons imply that EGSnrc is accurate within 0.2% for relative ion chamber

response calculations over a wide range of spectral variations with transmission [79].

Figure 2.19 Profile dose curve along Y-axis comparing Geant4 (black), EGSnrc (red),

and measurement data (blue) for a 4x4 cm2 field in a homogeneous (water) phantom.

[80]

67

Figure 2.20 Comparison of PDD curves in a lung-slab phantom measured with

thermoluminescent dosimeters (solid line) and simulated using EGSnrc Monte Carlo

code (solid dark line) for field sizes of (a) 10×10 cm2, (b) 5×5 cm2, (c) 2×2 cm2, and

(d) 1×1 cm2 [81].

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2.7 Overview of EPID image for treatment verification

The current generation of amorphous-silicon EPIDs have been demonstrated to be

useful as dosimeters rather than only providing anatomical images as EPIDs were

originally developed for in the 1980’s and 1990’s. They have been shown to possess

good dosimetric characteristics such as linearity with dose and dose rate, high spatial

resolution, good reproducibility and stability, as well as the conveniences of digital output

and being directly mounted on the linear accelerator [82]–[84]. A comprehensive

literature review of EPID dosimetry has been present by van Elmpt et al [52].

The measured transmitted EPID images can be used to determine the energy fluence

exiting the patient or phantom. This energy fluence can then be backprojected through the

patient/phantom data set and used to calculate the patient dose that has been delivered.

Various methods presented in the literature have studied algorithms that provide 0D (ie.

point), 2D (ie. planar), or 3D (ie. volumetric) dose estimates in the patient. These are then

compared with the original planned dose distribution obtained from the treatment

planning system (TPS) [85], [86].

An alternative method is to generate a two-dimensional predicted portal dose image,

which is created by simulating the portal image as determined by beam, patient, and EPID

characteristics. This is a simpler approach than the previously described methods that

backproject dose into the patient since only the dose to the EPID image itself need be

predicted. Once this is done, then the predicted EPID image can be compared to the

actual measured treatment image captured by the EPID. Assuming the patient geometrical

set up has been verified as is standard practice, if the predicted EPID dose pattern

69

matches with measurement, the correct dose is assumed to be delivered within a desired

tolerance inside the patient [87]–[89].

There are many proposed methods for portal dose image prediction. One approach is

to apply the full Monte Carlo technique, where radiation transport through the treatment

unit head, through the patient, and dose deposition in the image detector are fully

simulated [90] for each incident treatment beam. Some groups have used pre-calculated

Monte Carlo dose kernels [87], [88], [91] or analytical dose kernels [89], [92] to represent

the dose delivered to the detector system. Usually, these dose kernels are convolved with

a photon fluence map incident on the EPID.

Our research group initially developed a two-step approach to predict EPID

transmission images. This approach convolves Monte Carlo generated EPID dose kernels

with a model estimate of incident EPID photon fluence for static fields [24]. This work

has been incorporated into a real-time patient treatment monitoring software (research

only) developed by our collaborators at the University of Newcastle (Newcastle,

Australia), led by Dr. Peter Greer [28], [29].

More recently our research group has developed a 3D patient dose reconstruction

approach that takes the measured therapy transmission EPID images, converts them to an

estimate of incident primary fluence (by removing a calculated estimate of the patient-

generated scattered photon fluence entering the detector), and then back-projects this

through the patient model to find the 3D dose delivered to the patient by the treatment

beam [26], [27]. The patient-generated scattered photon fluence in that model is a pencil-

beam method where the fluence incident on the patient is convolved/superposed with

pencil-beam scatter fluence kernels valid for the radiological thickness and air gap of

70

each discretely sampled rayline. These pencil-beam scatter fluence kernels are selected

from a library of kernels pre-generated using Monte Carlo simulation techniques. While

execution speed is very fast, the pencil-beam nature of the model inherently limits the

accuracy of this method for estimating patient scatter fluence into the EPID. Only patient

density variations along the rayline are accounted for, and even these effects are further

simplified by applying a center-of-mass assumption. Furthermore, divergence of the

rayline is not accounted for in the current pencil beam model. Early accuracy estimates of

this method [22] show differences as high as ~7% of total signal (even for simple

geometric phantoms), which inherently limits the accuracy of any resulting reconstructed

patient dose. The tri-hybrid method of estimating patient scatter, as developed in this

thesis, will provide an estimate of patient scatter fluence at the EPID accurate to within

<2% of total signal, with minimal impact to overall execution time.

71

Chapter 3: Development of Monte Carlo Based Validation Tool Box

This chapter details a Monte Carlo based photon-scatter validation tool, which is

developed with EGSnrc by modifying the user code DOSXYZnrc. The tool allows the

separate scoring of various sub-components of patient/phantom photon scatter, and is

used as a validation tool in Chapter 4. The material in this chapter has been reprinted and

adapted from Physics in Medicine and Biology, Volume 65, Number 9, Kaiming Guo,

Harry Ingleby; Idris Elbakri, Timothy Van Beek, Boyd McCurdy “Development and

validation of a Monte Carlo tool for analysis of patient-generated photon scatter”,

Copyright (2020), with permission from IOP Publishing Corporation.

3.1 Introduction

Radiation therapy (RT) is extensively used in cancer treatment. Modern radiation

treatment units use a medical linear accelerator to generate therapeutic x-rays at

megavoltage (MV) energies. Mounted on the same treatment unit, a diagnostic x-ray tube

provides kilovoltage (KV) x-rays for anatomical imaging of the patient. Each of the MV

and KV sources employ a dedicated planar imaging system.

The KV imaging units can also be operated as a cone-beam computed tomography

(CBCT) system to reconstruct a volumetric image set of the patient by using a set of

multi-angular x-rays projections. However, these CBCT systems suffer from the presence

of scattered x-rays generated in the patient, which show up as background signal in the

projected images. This unwanted signal reduces contrast and adds artifacts to the

72

reconstructed volumetric image sets. This limits the usefulness of CBCT since diseased

tissue cannot be seen as easily on CBCT images as compared with diagnostic computed

tomography (CT) images, as well as making the CBCT density conversion less accurate

which in turn limits its usefulness for dose calculation. Accurate prediction of x-ray

scatter present in KV image acquisition will allow for its removal and will result in

improved CBCT image quality. This will provide improved anatomical guidance to target

diseased tissue and a more accurate estimate of the patient’s physical density map. Patient

radiation dose calculations, necessary for customized planning of the radiation treatments,

require a 3D density representation of the patient (either physical or electron density) and

this is typically provided by a diagnostic CT scan of the patient taken before treatment

begins. Ideally, CBCT data sets could be used for daily patient dose calculation, which

then can be used to assess and adapt the patient treatment. However, it is well known that

the unwanted scatter contaminating the CBCT data sets reduces the accuracy of the 3D

density map, increasing error in patient dose calculation applications [93]. So far this has

been handled by cumbersome techniques involving multiple calibration geometries [94],

[95] but would be ideally solved by removal of the scatter before image reconstruction,

which is an area of active investigation[25], [40], [65], [96].

The therapeutic MV x-ray beam is delivered shortly after anatomical verification

imaging, taking several minutes to deliver the prescribed energy pattern. The MV

imaging system, termed the electronic portal imaging device (EPID), can be utilized for

treatment validation to verify that the patient receives the planned dose. This comparison

can be done in many ways. Some researchers compare the measured transmission dose

distribution with a precalculated portal dose [25], [51]. Some others convert the 2D

73

images to fluence estimates and backproject this to reconstruct the 3D radiation dose

distribution delivered to the patient [26], [48], [49]. If a difference is found between the

measured and expected radiation delivery, the source of the difference can be corrected in

following treatments, thereby improving the patient outcome. Even though these EPID-

based methods achieved more than 90% agreement using dose-difference evaluations, the

sensitivity and specificity of this comparison are dependent on the accuracy achievable in

estimating the 2D images. Scattered x-rays generated in the patient are a significant

component, making up as much as 30% of the MV image signal, and therefore dose

verification applications will improve when this scatter is accurately removed. The

presence of scatter also reduces image contrast and reduces the ability to confidently

verify the treatment delivery (i.e. forces increased tolerances in the acceptability criteria

of dosimetric evaluations).

When studying the effect of patient-scattered radiation in imaging applications, it is

necessary to quantify the physical characteristics (including the nature of scattering event)

of the scattered radiation. Monte Carlo (MC) techniques are considered the gold standard

in terms of accuracy for solving radiation transport problems but are associated with a

heavy computational cost. Amongst different radiation transport Monte Carlo simulation

packages, EGSnrc has been established to be in good agreement with measurement,

within experimental uncertainty for KV and MV energy ranges [73], [74]. The accuracy

of EGSnrc for both KV and MV energies in terms of additional measurable physics

variables (i.e. energy spectrum, half-value layer, and ion chamber dose) has been shown

[97], [98]. Experimentally, the energy fluence can not be directly measured but only be

calculated. The kerma (i.e. kinetic energy released in medium) has to be determined

74

ahead by measuring the total dose under the (assumed) condition of electronic

equilibrium.

In this paper, we aim to separately score the subcomponents of photon scatter by

modifying an existing EGSnrc user code (DOSXYZnrc), which has been demonstrated as

the benchmark in radiation transport over the KV and MV energy ranges [76], [99]. This

modification requires scientific validation to ensure it has been correctly implemented.

However, to the best of our knowledge no experimental technique is available to directly

confirm by measurement these separate subcomponents of photon scatter, hence our

comparison to exact analytical calculation whenever possible.

MC simulation can be used to study the relative importance of various interaction

and to test alternative radiation transport solutions, including analytical and hybrid

approaches. For example, MC scatter separation has been used to improve image contrast

by differentiating the primary and scattered signal entering into a flat panel MV imager

[16]. Acuros CTS (Varian Medical Systems, Palo Alto, CA), a linear Boltzmann solver

for CT scatter, is validated for primary and scattered energy fluence by utilizing a scatter

separation tool within the Monte Carlo simulation code Geant4 [40]. Single and multiple

scatter of KV-energy CBCT images based on a mathematical model have been

investigated by implementing additional tracking of the fluence contributions from

different scatter orders [15], [32], [41], [43], [63]. Megavoltage x-ray beams used for

cancer treatment will also lead to pair production and subsequent positron annihilation as

well as creation of bremsstrahlung photons within the patient. Custom MC simulation has

also been applied in those situations to provide insight regarding their contribution to the

transmission image formation [21], [57].

75

Our group and others are developing hybrid methods (i.e. combining pure analytical

approaches with modified MC approaches) as an efficient yet accurate solution to

calculate the patient-generated scattered photon contribution for both KV and MV images.

Using the hybrid method, singly scattered photon fluence is calculated using an analytical

technique, while multiply scattered photon fluence is calculated using a MC technique

that has been modified to significantly improve sampling efficiency. The work presented

in this paper will be a very useful instrument to help researchers evaluate the accuracy of

hybrid methods. As part of our work on this topic, we have developed a Monte Carlo

simulation tool for investigating the individual components of patient-scattered photon

fluence. In this work we present the development of a critical tool based on EGSnrc and

validation against analytical solutions using various homogeneous and heterogeneous

phantoms and several photon beam energies.

3.2 Methods and Materials

3.2-a. Monte Carlo toolbox generation

The original DOSXYZnrc code is mainly used for three-dimensional absorbed dose

calculations in a Cartesian coordinate system. We modified the DOSXYZnrc user code to

develop a tool to investigate patient-generated scattered photons. The subroutine

AUSGAB in DOSXYZnrc was originally designed to score the dose distribution in

phantom voxels, but this routine can also be customized to register various physical

characteristics of an individual particle interaction. By modifying the AUSGAB routine,

the user can extract critical information about the individual particle histories, without

making any changes to the EGSnrc core radiation transport code. One of the features of

76

EGSnrc allows the user to study specific physics interactions and record the phase-space

information of the particle before and/or after the interaction.

The argument IARG is utilized to guide calling of AUSGAB for various physical

situations. A total of 26 IARG conditions15 can be tracked by turning on or off flags in the

specialized flag array IAUSFL. For example, IAUSFL can be turned on when the current

particle experiences a Compton scatter event, and the AUSGAB routine will record this.

In order to quantitatively record the number of scatter events, it requires the user to add

their own tracking algorithm associated with the IARG argument to the AUSGAB

subroutine. Overall, by turning on the corresponding IAUSFL flags and modifying the

subroutine AUSGAB, we track various physical interactions of the particle with the

media. In this work, the IAUSFL flags 8, 14 & 15, 19, and 25 are turned on when events

of interest to photon scattering applications occurred: after a bremsstrahlung event, at-rest

positron annihilation events, in-flight positron annihilation events, Compton scatter, and

Rayleigh scatter, respectively.

The LATCH parameter is also used to record additional scatter-related information

from each particle history on the “stack” during its transport. We defined the

corresponding value is stored at the LATCH digit positions of 1, 103106, 109 for

Compton scatter ( #𝑐𝑠 ), Rayleigh scatter ( #𝑟𝑠 ), positron annihilation ( #𝑝𝑎 ), and

Bremsstrahlung (#𝐵𝑟𝑒𝑚), respectively. The use of the LATCH parameter allows us to

track and record all interaction types for one particle history, according to:

𝐿𝐴𝑇𝐶𝐻 = #𝑐𝑠 + 103 ∗ #𝑟𝑠 + 106 ∗ #𝑝𝑎 + 109 ∗ #𝐵𝑟𝑒𝑚

15 The number of IAGR conditions is 26 for the EGSnrc verion (released in 2015), and is larger for the most

current EGSnrc version.

77

The next step is differentiating the scatter order, whereby the LATCH parameter is

disassembled to determine the number of Compton scatter, Rayleigh scatter, positron

annulation, and Bremsstrahlung events. If #𝐵𝑟𝑒𝑚 and #𝑝𝑎 are not zero, the secondary

photons are identified and filtered out. The rest will be due to the scatter of the primary

photon beam. If the number of Compton or Rayleigh scatter is equal to 1, the components

are treated as singly scattered. Then, the remaining component is the multiple scatter with

the individual counts of number of Rayleigh or Compton scatters, which allows

differentiating of the scatter orders.

Upon completion of a simulation, we grouped scored photons into six different

categories including: (1) the primary photon (i.e. photons that have experienced no

interaction while traveling through matter and have the same energy and direction of

travel as the incident photons), the photons that only scattered once through (2) Compton

or (3) Rayleigh scattering (i.e. 1st Compton scatter and 1st Rayleigh scatter, respectively),

(4) the photons which scattered more than once (i.e. multiply scattered photons), (5) the

secondary photons resulting from bremsstrahlung (i.e. radiation resulting from rapid

deceleration of electrons traveling in high speed), and (6) the secondary photons resulting

from position annihilation.

We take advantage of the geometric boundary check built into DOSXYZnrc to

record information for the photons crossing a surface boundary. All the current photon

particle’s phase-space information (i.e. position, direction, and energy) is recorded in

customized phase-space files of 6 different groups, if it is at a user-specified x, y and z

boundary location, together with a positive z direction motion.

78

The corresponding output files are unformatted (i.e. binary) with length of 30 bytes

for each record, which contains nine basic variables (Table 3-1), so that it can be easily

read for data analysis. After the simulation, users can specify according to their own

research interests to generate images of primary photons, single Compton scatter, single

Rayleigh scatter, multiple scatter, and the contribution from secondary photons, reaching

a predefined imaging plane. The imaging plane is a user-defined pixelated “virtual

detector” or “scoring plane” which is located underneath the phantom or patient.

After DOSXYZnrc photon-tracking stops (at the exit surface of the phantom), the

photons are simply projected to the defined scoring plane where the particle fluence and

energy are recorded and binned into pixels according to their intersection location on the

scoring plane. In this note, we only compare the received energy fluence to the scoring

plane from the contribution of each category. By normalizing to the incident energy

fluence (i.e. at the entrance to the phantom), the corresponding normalized energy fluence

(NEF) at each detector element is obtained as 𝐹𝑖,𝑗.

Table 3-1 Output phase-space file of DOSXYZnrc-based patient scatter validation tool

Variable Definition Data type

X, Y, Z Position of particle in coordinate system (cm) real*4

U, V, W Direction cosine of particle with respect to x, y, z axes real*4

E Total energy of particle (Mev) real*4

n_r_byte The number of Rayleigh Scatter Events uint8

n_c_byte The number of Compton Scatter Events uint8

79

3.2-b. Validation with EGSnrc stack parameter

The first verification was performed on exit primary NEF. For the monoenergetic

incident photon case, the exit primary photons’ energy should be identical to the incident

energy. Also, taking advantage of the EGSnrc code system [99], the stack pointer (NP)

for primary, singly scattered and multiply scattered photons should equal to 1, since they

all originate from the parent particle. For positron annihilation and bremsstrahlung

components, the NP of the scored photon should be larger than 1 since they are generated

by the daughter particle.

3.2-c. Validation with analytical approach (ANA)

Based on first principles, we can exactly calculate primary, singly Rayleigh scattered,

and singly Compton scattered photon fluence at the scoring plane. The MC tool will be

validated with these two main categories (i.e. primary and singly scattered photons) under

both parallel and divergent beam geometry for several test phantoms.

Primary NEF

The exit primary NEF map is obtained based on Eq. 3-1, for a single ray line

transported from source to each detector element:

𝐹𝑖,𝑗𝑃 = 𝑒−𝐼𝑖,𝑗 ∗ 𝐺𝑝 (Eq. 3-1)

where 𝐼𝑖,𝑗 is the attenuation term (expressed discretely in Eq. 3-2) from source to detector

elements (i, j). 𝐺𝑝 is the inverse-square term for the primary beam, and 𝐺𝑝 =

1 𝑜𝑟 (𝑆𝑆𝐷

𝑆𝐷𝐷)2

for parallel and divergent beam geometry respectively, where SSD is source-

80

to-surface distance, SDD is source-to-detector distance and unit fluence is assumed at the

entrance to the phantom.

A modified exact 3D ray tracing algorithm [100] was applied to account for phantom

inhomogeneity, and to calculate geometrical path length (𝐺𝑃𝐿𝑖,𝑗𝑚 ) through the phantom, as

well as the geometric path length fraction 𝑤𝑖,𝑗𝑚 of each material:

𝐼𝑖,𝑗 = ∑𝜇

𝜌(𝐸)𝑖,𝑗

𝑚 ∗ 𝑤𝑖,𝑗𝑚 ∗ 𝜌𝑚 ∗ 𝐺𝑃𝐿𝑖,𝑗

𝑚𝛼𝑚=1 (Eq. 3-2)

where 𝛼 is the total number of media, and 𝜇

𝜌(𝐸)𝑖,𝑗

𝑚 is the mass attenuation coefficient of a

given material at energy E. This approach allows exact accounting of the attenuation

through various media composing the phantom or patient, although it requires a media-

mapping algorithm (with inherent assumptions) to assign CT Hounsfield density data to

specific media.

Analytical Singly Scattered NEF

For coherent (i.e. Rayleigh) and incoherent (i.e. Compton) singly-scattered photons,

voxels inside the irradiated volume were sampled as interaction sites. We assumed the

interaction site is located at the center of each voxel, with the scattering angle determined

as the direction from the interaction site to the center of each pixel in the scoring plane.

Coherent scatter is the elastic scattering of electromagnetic radiation by a bound

atomic electron instead of a ‘free’ electron and occurs only at low energies (15 to 30 keV)

and in high Z materials [33].

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The probability of interaction is found using the Rayleigh differential cross section,

which is the product of the Thomson differential cross section and the molecular coherent

form factor 𝐹𝑀2(𝑥):

𝑑𝜎

𝑑Ω

𝑟(𝜃, 𝑥) =

𝑟𝑜2

2(1 + cos2 𝜃)𝐹𝑀

2(𝑥) (Eq. 3-3)

where 𝑟0 is the classical electron radius, 𝜃 is the scattering angle, and 𝐹𝑀(𝑥) carries

information not only about the molecular structure, but also about the environment. For

this validation, we have considered the scattering event as being due to a free atom. Thus,

it can be calculated by the sum rule, adding the atomic scattering factor 𝐹𝑀2(𝑥, 𝑍𝑖) for i

different elements, weighted by the atomic abundance. This is known as the independent

atomic model or the free-gas model [32], [34].

𝐹𝑀2(𝑥) = 𝑊 ∗ ∑

𝑤𝑖

𝑀𝑖𝐹2(𝑥, 𝑍𝑖) (Eq. 3-4)

where W is the molecular weight of the material, wi and Mi are the mass fraction and

atomic mass of element i, and F2(x, Zi) is the atomic coherent form factor. The value of

the transferred momentum x (Å-1), is ℎ𝑣

12.398 𝑘𝑒𝑉𝑠𝑖𝑛 (

𝜃

2) . The generation of form factor has

been described in the Section 2.1-a.

For incoherent scatter, an incident x-ray photon with incident energy 𝐸 interacts with

a free electron, and is scattered through an angle φ relative to the incident photon’s

direction and possesses a lower energy, E’ . The electron receives some energy in the

interaction and based on the conservation of energy and momentum, the energy of the

scattered photon is:

𝐸′ = 𝐸

1+(𝐸0 𝑚0𝑐2⁄ )(1−𝑐𝑜𝑠𝜑) (Eq. 3-5)

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The probability of interaction is found using the Klein-Nishina differential cross

section, while the energy of the scattered photon is established using Compton kinematics

based on a given scattered angle (i.e. between the interaction voxel and the scoring plane

pixel).

𝑑𝜎

𝑑𝛺

𝑐(𝜃, 𝐸) =

𝑟02

2(𝐸′

𝐸)2

(𝐸′

𝐸+

𝐸

𝐸′ − 𝑠𝑖𝑛2 𝜑) (Eq. 3-6)

The singly scattered NEF is the sum of singly Rayleigh scattered NEF (i.e. 𝐹𝑖,𝑗𝑐 ) and

singly Compton scattered NEF (i.e. 𝐹𝑖,𝑗𝑟 ). Based on the scattered angle from the voxel to

the scoring plane pixel, and the 𝐺𝑃𝐿 associated with its own combination of medium

fraction, the scattered x-ray fluence at the scoring plane from each interaction site is

determined. After integrating over the beam-covered-volume, the net NEF at the scoring

plane will have different expressions for Rayleigh and Compton scatter, which are shown

as deterministic solutions in Eq. 3-7 and 3-8.

𝐹𝑖,𝑗𝑟 (𝐸) = ∫ 𝑒−𝐼0 ∗ 𝐺𝑠 ∗ 𝑛𝑠𝑐

𝑟 ∗𝑑𝜎

𝑑Ω𝑖,𝑗

𝑟(𝜃, 𝐸) ∗ 𝛥𝛺𝑖,𝑗 ∗ 𝑒−𝐼𝑖,𝑗

𝑟 𝑑𝑣 (Eq. 3-7)

𝐹𝑖,𝑗𝑐 (𝐸) = ∫ 𝑒−𝐼0 ∗ 𝐺𝑠 ∗ 𝑛𝑠𝑐

𝑐 ∗𝑑𝜎

𝑑Ω𝑖,𝑗

𝑐(𝜃, 𝐸) ∗ 𝛥𝛺𝑖,𝑗 ∗ 𝑒−𝐼𝑖,𝑗

𝑐 ∗𝐸𝑖,𝑗

′

𝐸𝑑𝑣 (Eq. 3-8)

where 𝐺𝑠 is the inverse square correction term for singly scattered photons, 𝐺𝑠 =

1 𝑜𝑟 (𝑆𝑆𝐷

𝑆𝐼𝐷)2

with respect to parallel and divergent beam geometry to account for the

inverse-square effect, where SSD is source-to-surface distance, SID is the source-to-

interaction site distance; 𝐼0 is the attenuation term from source to a interaction site, 𝐼𝑖,𝑗 is

the attenuation term from interaction site to a pixel within the scoring plane; 𝛥𝛺𝑖,𝑗 is the

solid angle subtended by a discrete detector element (i, j) with respect to the interaction

83

site; 𝑛𝑠𝑐𝑟 and 𝑛𝑠𝑐

𝑐 are the number of scattering centres in an interaction site/voxel for

Rayleigh and Compton Scatter respectively. 𝑛𝑠𝑐𝑟 is simplified as the number of atoms

within a voxel, and 𝑛𝑠𝑐𝑐 is the number of electrons within this voxel.

3.2-d. Validation testing

The simulation setup of the imaging system is illustrated in Figure 3.1(a) and (b)

under the divergent and parallel beam geometries respectively, using a 100 cm source–

surface distance (SSD), a 150 cm source–detector distance (SDD), and with the isotropic

x-ray point source collimated to the detector under the divergent beam geometry.

Incident beams used for testing included several monoenergetic beams (0.06, 0.1, 1.5,

5.5 MeV) and a 6 MV polyenergetic treatment beam (1 MeV wide energy bins), with

field size of 10 10 cm2 at 100 cm from source) to irradiate a 40 40 20 cm3 geometric

phantom. The phantom was composed of isotropic 2 mm voxels. Some preliminary work

was performed using 1cm3 voxel resolution, but we found using 2mm3 voxels improved

accuracy from approximately 1% to 0.1% when comparing with the Monte Carlo

simulation (on average over phantoms used in this study) for divergent beams, and

therefore used 2mm3 voxels for all comparisons presented here (both divergent and

parallel beam geometries). Three geometric phantoms shown in Figure 3.1 were used to

test the validation tool, including a homogeneous water phantom (40 40 20 cm3, ρ =

1.0 g/cm3), an ‘LWRL’ phantom which is composed of the left-half as water (40 20

20 cm3, ρ = 1.0 g/cm3), and the right-half as lung (40 20 20 cm3, ρ = 0.26 g/cm3), and

finally a thorax CT phantom for testing with increased heterogeneity and asymmetry

(composed of air ρ = 0.0012 g/cm3, lung ρ = 0.26 g/cm3, soft tissue ρ = 1 g/cm3, and bone

84

ρ = 1.85 g/cm3). The exiting photons are then projected to the scoring plane, which has

dimensions of 40 40 cm2 with 1 cm2 pixel size, placed 30 cm underneath of the

phantom’s exit surface. As mentioned in Section II.A, each category of exit photons from

the Monte Carlo tool are spatially binned to the scoring plane pixels (i.e. imaging plane).

Both MC simulation and the analytical calculation (ANA) were executed on a laptop

with Intel Core (i7)-6600U 2.60 GHz processors and 8 GB of RAM. The EGSnrc MC

simulation parameters used are listed in Table 3-2. The simulations reported here were

not parallelized and were performed on a single core during the simulation, which takes

about 32 hours for the Monte Carlo simulation include the projection the exiting photon

to scoring plane.

Validation is performed by quantitatively comparing exit primary NEF and 1st-

scattered NEF (i.e. sum of 1st Compton and Rayleigh scatter) to the corresponding

analytical calculations. We calculated the percentage difference image (PDI) between the

full Monte Carlo and ANA prediction for Primary, Compton and Rayleigh single scatter,

then obtained the histogram of the PDI. The mean and standard deviation (STD) of the

pixels in each PDI is treated as an indicator of accuracy and precision, respectively.

The single scatter fraction (SSF) was calculated as the ratio of the single scatter to the

sum of primary and single scatter over the central horizontal and vertical profiles on the

scoring plane; both profiles and percentage differences are compared.

85

Table 3-2 EGSnrc Monte Carlo transport parameters used

Transport parameter Value

Global ECUT 0.521

Global PCUT 0.01

Global SMAX 1e10

ESTEPE 0.25

XIMAX 0.5

Boundary Cross Algorithm PRESTA-I

Skin depth for BCA 0

Electron-step algorithm PRESTA-II

Spin effects On

Brems angular sampling Simple

Brems cross sections BH

Bound Compton scattering Off

Compton cross sections default

Pair angular sampling Simple

Pair cross sections BH

Photoelectron angular sampling Off

Rayleigh scattering On

Atomic relaxations Off

Electron impact ionization Off

Photon cross sections XCOM

Photon cross-sections output Off

86

Figure 3.1 The simulation was performed under (a) parallel and (b) divergent beam

geometry. Three phantoms, (c) Water, (d) LWRL, and (e) Thorax, using

monoenergetic and polyenergetic beams, are used to test the Monte Carlo validation

tool against the analytical calculations.

87

3.3 Results

The custom Monte Carlo tool was able to track and separately score the primary,

scattered, and other photon fluence components as expected. As expected, NPs of the

primary, singly and multiply scattered photons are consistently equal to 1, and NPs of the

positron annihilation and bremsstrahlung component are greater than 1. For quantitative

comparison, Tables 3-3 and 3-4 detail the comparison of MC and ANA predicted primary

and 1st scattering NEF for Water, LWRL, and Thorax phantoms at incident monoenergies

of 0.06, 0.1, 1.5, and 5.5 MeV under parallel and divergent beam geometry, respectively.

Figure 3.2 shows primary and single scatter NEF maps, for the 0.06 Mev beam and field

size of 10 x 10 cm2 incident on the thorax phantom, the percentage difference map and its

histogram; a comparison of vertical and horizontal profiles through the central axis are

also shown in the last column.

3.3-a. Exit Primary NEF

For the monoenergetic beam cases, as expected the energy of scored primary photons

is the same as the incident energy. The mean and standard deviation (STD) of the

percentage errors are correspondingly limited to 0.1% and 0.3% for parallel beam

geometry. Under divergent beam geometry, due to the inverse-square law relatively fewer

primary photons will reach the scoring plane, and therefore slight increases in the

standard deviation of percentage errors for Water and LWRL configurations are expected,

but they are still within 0.5%. For the thorax phantom, considering greater attenuation

from bony tissue, fewer primary photons will be able to reach the scoring plane, and the

average standard deviations are slightly increased.

88

3.3-b. Exit Singly Scattered NEF

Under either divergent or parallel beam geometry (Table 3-3 and 3-4), regardless of

phantom type, the accuracy of singly scattered NEF is under 0.2% and overall STDs are

within 2%.

The precision improves with increasing incident energy. This is due to the chance of

singly scattered coincidences increasing, so more scatter will reach the scoring plane thus

reducing the stochastic noise of the MC simulation. For the 6MV incident beam example

used in this situation, the mean and STD of PDI are 0.06% and 0.61% for parallel

geometry, and the mean and STD of PDI are -0.1% and 0.69% for divergent geometry.

3.3-c. Single scatter fraction (SSF)

Average percentage differences of SSF are within 1% among all tested

configurations. In Figure 3.3, the central horizontal and vertical profiles of SSF for the

thorax phantom shows the degree of agreement between MC and ANA results when the

energy of the incident beam is at 60 keV, 100 keV, and 6 MV.

89

Table 3-3 The mean and standard deviation (STD) of percentage differences between the MC simulation (109 histories) and analytical

calculated NEF for various monoenergetic beams and phantoms under parallel beam geometry.

Phantom Water LWRL Thorax

Energy incident beam

(MeV) 0.06 0.1 1.5 5.5 0.06 0.1 1.5 5.5 0.06 0.1 1.5 5.5

Primary

Mean (%) 0.07 -0.02 -0.06 -0.01 0.02 0.01 -0.02 -0.01 0.01 0.00 -0.05 0.00

STD (%) 0.27 0.17 0.17 0.12 0.18 0.14 0.14 0.12 0.25 0.16 0.05 0.13

1st Compton

scattering

Mean (%) 0.11 0.03 -0.07 -0.09 -0.08 -0.14 -0.12 -0.15 -0.13 -0.11 -0.08 -0.11

STD (%) 1.80 1.23 0.47 0.49 1.14 0.97 0.56 0.59 1.49 1.02 0.48 0.54

1st Rayleigh

scattering

Mean (%) 0.85 1.00 NA NA -0.58 0.64 NA NA 0.48 0.65 NA NA

STD (%) 0.87 1.01 NA NA 1.15 1.12 NA NA 0.92 0.96 NA NA

1st scattering

Mean (%) 0.05 0.03 -0.07 -0.09 -0.10 -0.14 -0.12 -0.15 -0.15 -0.10 -0.08 -0.11

STD (%) 1.58 1.18 0.47 0.49 1.06 0.95 0.56 0.59 1.36 0.98 0.48 0.54

90

Table 3-4 The mean and standard deviation (STD) of percentage differences between the MC simulation (109 histories) and analytical

calculated NEF for various monoenergetic beams and phantoms under divergent beam geometry.


Energy incident beam

(MeV) 0.06 0.1 1.5 5.5 0.06 0.1 1.5 5.5 0.06 0.1 1.5 5.5

Primary

Mean (%) -0.01 0.05 -0.02 0.03 0.04 0.04 0.01 0.03 0.36 0.4 0.32 0.38

STD (%) 0.46 0.33 0.19 0.27 0.34 0.3 0.31 0.29 0.84 0.73 0.61 0.52

1st Compton

scattering

Mean (%) 0.12 0.02 -0.1 -0.1 0.03 0 -0.11 -0.11 -0.19 -0.12 0.19 -0.17

STD (%) 1.82 1.19 0.46 0.5 1.03 0.82 0.5 0.57 1.46 1 0.47 0.51

1st Rayleigh

scattering

Mean (%) 0.32 0.3 NA NA 0.1 0.2 NA NA -0.15 0.04 NA NA

STD (%) 1.27 1.2 NA NA 0.88 0.8 NA NA 1.11 1.16 NA NA

1st scattering

Mean (%) 0.08 0.03 -0.1 -0.1 0.01 0.01 -0.11 -0.11 -0.16 -0.08 0.19 -0.17

STD (%) 1.5 1.11 0.46 0.5 0.89 0.75 0.5 0.57 1.2 0.9 0.47 0.51

91

Figure 3.2 The (a) primary and (c) singly scattered NEF map, with the 0.06Mev

incident beam and field size of 10 x 10 cm2 incident on the thorax phantom,

including the percentage difference map and its histogram, as well as corresponding

central horizontal and vertical profiles in (b) and (d).

92

Figure 3.3 The central horizontal (left) / vertical (right) profiles of SSF for the thorax

phantom when the incident beam energy is (a) 60 keV, (b)100 keV, and (c) 6 MV.

(a)

(b)

(c)

93

3.4 Discussion

Comparing singly scattered NEF, the average and standard deviation of accuracies

across all phantom tests are -0.09% and 0.06%, respectively for parallel beam geometry,

and are -0.04 % and 0.1%, respectively for divergent beam geometry. This level of

agreement between analytical calculation and Monte Carlo simulation demonstrates the

user code modification have been implemented correctly. The small increase in the

standard deviation is due to partial volume effects in the analytical calculation, since

some voxels (at the beam edge) are not considered if their voxel center is not contained

within the divergent beam. This issue could be resolved if finer resolution was applied to

phantom voxel sampling.

Our group is investigating the development of a hybrid method (combining Monte

Carlo simulation and analytical calculation) to estimate patient scatter for various x-ray

beam applications. Thus, the presented, validated customized MC user code is a critical

tool that allows users to separate, track, and score all different components of patient

scattered photon fluence entering the imaging plane. In addition to the phantom sampling,

the energy spectrum sampling of the polyenergetic beam could be another factor affecting

the agreement between MC and ANA. In the future, we could investigate the significance

of the sampling resolution in the phantom (spatial) or beam energy spectrum on the

accuracy of the ANA method, so we can optimize the settings for the analytical

simulation for accuracy versus speed.

94

3.5 Conclusion

In this chapter, A DOSXYZnrc-based MC tool was developed to estimate

contributions of primary and several separate components of patient-generated scattered

photon fluence into a user-defined imaging plane for arbitrary incident beam energies.

This model is successfully validated based on comparison between primary, 1st Compton

and Rayleigh scattered fractional energy fluence maps generated using first principle

analytical techniques. The NEF comparison results are within 0.2% for the various

phantom configurations and beam energies tested here. In the future, this MC tool will be

a critical test instrument for the future development of imaging applications requiring

patient-generated scatter fluence prediction or applications requiring separation of

patient-generated scatter fluence components.

This tool was extensively used for the development of a new ‘tri-hybrid’ method

presented in detail in Chapter 4 and 5 of this thesis. As another example, the tool will also

be valuable when training newly available artificial intelligence applications to predict

patient-generated scatter fluence. These applications require separate, accurate

calculations of patient scatter as described in more detail in Section 6.2 of this thesis.

Note that this validation tool is freely available from the authors for research

purposes upon request.

95

Chapter 4: A Tri-Hybrid Method to Estimate the Patient-Generated Scattered Photon Fluence Components to the EPID Image Plane

This chapter details the development and validation of a tri-hybrid method to

accurately predict the patient-generated scatter fluence entering the EPID imager. The

material in this chapter has been reprinted and adapted from the peer-reviewed journal

Physics in Medicine and Biology, Volume 65, Kaiming Guo, Harry Ingleby; Eric Van

Uytven, Idris Elbakri, Timothy Van Beek, Boyd McCurdy “A Tri-Hybrid Method to

Estimate the Patient-Generated Scattered Photon Fluence Components to the EPID Image

Plane”, Copyright (2020), with permission from IOP publishing Corporation.

4.1 Introduction

External beam radiation therapy (EBRT) is used extensively in cancer treatment,

delivering ionizing radiation to the cancerous region while attempting to spare the normal

tissues. With the increased development and use of advanced RT techniques including

higher dose prescriptions and lower fractions, the need for patient-specific dose

verification has increased. The increasing complexity of treatment plans makes it more

difficult to discover possible errors, and conventional pre-treatment quality assurance

(QA) approaches might not be adequate to ensure patient safety [49], [54], [55], [101]–

[103].

After a number of incorrect radiation delivery incidents in various countries over the

last several years [25], [55], the importance of monitoring the actual dose delivered to the

96

patient has become more evident, and therefore in vivo dose measurement has been

receiving increasing attention as an additional and very effective QA approach [101],

[103], [104]. The megavoltage beam imaging system, termed the ‘electronic portal

imaging device’ (EPID), was originally developed for anatomical position verification,

but has been shown to be useful for in vivo dosimetry applications [52], [105], [106].

Differences between measured and intended doses could be due to, for example, changes

in tumor size, patient weight, organ motion, mechanical failure of the MLC (multileaf

collimator) preventing delivery of the intended fluence, or linac output variations.

Patient in vivo dose verification can be accomplished in many ways, commonly

including single point-dose or 3D dose distributions in the patient (which can be

compared to the treatment planning system), or 2D planar dose at the EPID (with the

measured transmission image compared to a pre-calculated or ‘predicted’ transmission

image) [25], [26], [48]–[51]. If a difference is found between the measured and expected

delivery, the source of the difference could be identified and corrected in a following

treatment fraction, thereby potentially improving the patient outcome.

However, there are still some challenges to use EPIDs for patient in vivo dosimetry

[52]–[55]. One of those challenges is that the photon fluence entering the EPID is

contaminated with patient-generated scattered photons, which limits the accuracy of in

vivo patient dose calculations. This scattered x-ray component can be significant, making

up as much as 30% of the MV image signal [21], [57]. To improve the accuracy of in vivo

dosimetry methods, many researchers try to eliminate the patient scatter signal

contribution from the measured EPID image by estimating it and then subtracting it from

the measured image (whose signal is due to total incident fluence) [25], [26], [49], [53],

97

[107], [108]. Then, the remaining transmitted primary fluence is backprojected to a plane

above the patient as an estimate of the incident primary fluence, which can finally be used

by a patient dose calculation algorithm to estimate 3D dose to a CT (computed

tomography) or CBCT (cone beam computed tomography) representation of the patient.

Therefore, the performance of dose verification applications will improve when this

patient scatter component is more accurately removed. If uncorrected, the presence of

scatter also reduces image contrast and reduces the ability to confidently verify the

treatment delivery in dose verification applications (since it forces increased tolerances in

the acceptability criteria).

Patient-generated scatter entering the planar detector can be classified into three

components: singly-scattered photons (SS), multiply-scattered photons (MS), and

electron-interaction-generated (EIG) scattered photons (i.e. bremsstrahlung and positron

annihilation). Several groups have used analytical methods to estimate the singly

scattered energy fluence [9], [32], [42], [43], [109]. The MS component is known to be a

smooth, broad function and has been treated as proportional to the singly scattered photon

distribution [12]. ‘Hybrid methods’, which combine Monte Carlo simulation with

analytical methods, have been shown to accurately estimate the multiply-scattered

component for CBCT images [41]. A 2011 review article of x-ray scatter estimation

techniques [18] suggests that hybrid approaches represent the best hope for a fast yet

accurate solution to this problem. More recently, a different method to estimate patient-

generated KV photon scatter for CBCT was developed using a linear Boltzmann transport

equation solver [40].

98

Some researchers have examined estimating the EIG scatter contribution to the EPID

image [21], [57], but so far no analytical calculations have been developed to the best of

our knowledge. This scatter component becomes significant for higher energy radiation,

where pair production and bremsstrahlung interactions become more frequent.

In the current work, analytical (ANA) calculations are used to estimate SS photons, a

hybrid (HB) algorithm is implemented to estimate the MS photon component, and the

EIG component is estimated by using a convolution/superposition pencil beam patient-

scatter kernel (PBSK) method. Combining these three different scatter prediction methods,

termed tri-hybrid (TH) method, we investigate its feasibility and accuracy for estimating

total patient-generated scattered energy fluence entering an EPID. To our knowledge, this

is the first work reporting the combination of three different patient-generated scatter

fluence calculation methods. We developed all the original code for the three methods of

estimating the singly scattered, multiply scattered, and EIG scattered components of

patient generated scatter fluence. Especially for such hybrid methods, there has been little

work investigating their application to MV energies.


4.2-a. Singly Scattered Photon Fluence

Based on first principles, analytical techniques (ANA), can be used to calculate

primary fluence, singly-Rayleigh scattered fluence, and singly-Compton scattered fluence.

Since in megavoltage energy range of therapeutic beams, the incoherent (i.e. Compton)

scatter dominates the coherent (i.e. Rayleigh) scatter, contributions of Rayleigh scatter are

not considered here. To practically implement the ANA technique, the phantom or patient

99

volume is voxelized in three-dimensions along a regular Cartesian grid, while the imaging

plane (i.e. scoring plane) is pixelized in two-dimensions also along a regular Cartesian

grid (independent of the phantom/patient grid). Voxels in the irradiated volume were

sampled as scatter-source sites (at the center of each voxel) for incoherent singly-

scattered photons, with outgoing fluence calculated to the center of all imaging plane

pixels.

For incoherent scatter, the energy of the scattered photon is:

𝐸′ = 𝐸

1+(𝐸 𝑚0𝑐2⁄ )(1−𝑐𝑜𝑠𝜃) (Eq. 4-1)

The probability of an incoherent interaction occurring is governed by the Klein-Nishina

differential cross section:

𝑑𝜎

𝑑𝛺

𝑐(𝜃, 𝐸) =

𝑟02

2(𝐸′

𝐸)2

(𝐸′

𝐸+

𝐸

𝐸′ − 𝑠𝑖𝑛2 𝜃) (Eq. 4-2)

where 𝐸=energy of incident photon, 𝐸’=energy of scattered photon, 𝜃=angle of scatter,

𝑚0=mass of electron, 𝑟0=classical electron radius.

Referring to the geometry illustrated in Figure 4.1, the Compton singly-scattered

photon energy fluence contribution, 𝛹𝑐𝑃2, to an imaging plane pixel 𝑃3(𝑥3, 𝑦3, 𝑧3) from a

single interaction site 𝑃2 in the patient/phantom (at 𝑥2, 𝑦2, 𝑧2), and for a single energy E in

the incident spectrum, is calculated analytically as:

𝛹𝑐𝑃2(𝑥3, 𝑦3, 𝑧3, 𝐸) = 𝛷𝑖𝑛𝑐(𝑥1, 𝑦1, 𝑧1, 𝐸) ⋅ exp (−𝐼𝑃2−𝑃1 ) ⋅ (

𝑧1𝑧2

)2∙[𝜌𝑒(𝑥2, 𝑦2, 𝑧2) ⋅

𝑉(𝑥2, 𝑦2, 𝑧2)] ⋅𝑑𝜎

𝑑𝛺𝑃3−𝑃2

𝑐(𝜃, 𝐸) ⋅ 𝛥𝛺𝑃3−𝑃2 ⋅ exp (−𝐼𝑃3−𝑃2 ) ∙ 𝐸′𝑃3−𝑃2

(Eq. 4-3)

100

where 𝛷𝑖𝑛𝑐 is the incident fluence distribution at a plane (𝑧1) between the patient and the

linac source, point (𝑥1, 𝑦1, 𝑧1) lies on the rayline between the linac source (0, 0, 0) and

the scattering voxel (𝑥2, 𝑦2, 𝑧2), (𝑧1

𝑧2)2

is the inverse square law between points 𝑃1 and 𝑃2,

(𝑒∙ 𝑉) is the number of scattering centers per voxel (electron density multiplied by

voxel volume), 𝑑𝜎

𝑑𝛺𝑃3−𝑃2

𝑐(𝜃, 𝐸) is the Klein-Nishina cross section for scatter between a

scattering voxel 𝑃2 and a pixel 𝑃3, 𝛥𝛺𝑃3−𝑃2 is the solid angle defined by the pixel at P3

from scattering voxel at 𝑃2. 𝐸′𝑃3−𝑃2 is the energy of scattered photon along the direction

from P2 to P3. 𝐼𝑃2−𝑃1 and 𝐼𝑃3−𝑃2

are the attenuation terms between point P1 in the incident

fluence distribution and point P2 (the scattering voxel), and between point P2 (the

scattering voxel) and point P3 (the scoring pixel), respectively. In general, the attenuation

term, 𝐼𝐵−𝐴 is the radiological pathlength between point A and point B weighted by the

corresponding attenuation coefficient for each voxel on the path (i.e. the exponential of

this term gives the attenuation between the two points).

𝐼𝐵−𝐴 = ∑𝜇

𝜌(𝐸)𝑙 ∗ 𝜌𝑙 ∗ 𝑤𝑙 ∗ 𝐺𝑃𝐿𝛼

𝑙=1 (Eq. 4-4)

where 𝛼 is the total number of media involved in phantom/patient from source to

interaction site, 𝜇

𝜌(𝐸)𝑙 is mass attenuation coefficient of material l at energy E. An exact

3D ray tracing algorithm [100] was applied to account for phantom inhomogeneity, which

will calculate geometric path length (𝐺𝑃𝐿), as well as the portion of the pathlength 𝑤𝑙

composed of the various media materials.

Assuming there are 𝑛 scattering voxels (i.e. like 𝑃2 ) within the primary beam

coverage, summing the singly-scattered photon fluence over the entire irradiated volume

101

gives the total singly-scattered photon energy fluence at an imaging plane pixel (at

𝑥3, 𝑦3, 𝑧3) as:

𝛹𝑐 (𝑥3, 𝑦3, 𝑧3, 𝐸) = ∑ 𝛹𝑐𝑃2

𝑖𝑛 𝑖=1 (Eq. 4-5)

To make a consistent comparison with the conventional Monte Carlo simulation, the

total singly-scattered photon energy fluence is normalized to the incident energy fluence

at the phantom top surface, so we define the normalized energy fluence (NEF) at arbitrary

pixel (i, j) of the imaging plane as:

𝐹𝑖,𝑗𝑐 =

𝛹𝑐

𝛹𝑖𝑛𝑐=

𝛹𝑐

𝛷𝑖𝑛𝑐∙𝐸 (Eq. 4-6)

102

Figure 4.1 Schematic describing the analytical algorithm to calculate the single and

multiple scatter component into the imaging plane, where the physics process are

detailed in equation 4-3 and 4-7.

103

4.2-b. Multiply-Scattered Photon Fluence

To estimate the multiply scattered photon signal (i.e. a photon that experiences two

or more scattering events in the patient/phantom before entering the imager), our group

modified a hybrid method (HB) [18], [41], with the logic flow shown in Figure 4.2. The

proposed hybrid method consists of a Monte Carlo phase followed by an analytical

calculation phase, taking advantage of the strengths of each of these two methods. We

modified the well-benchmarked DOSXYZnrc user code [76], [99] to track and output

individual photon scatter interaction information for each incident particle history. After

completing the Monte Carlo simulation with only a few histories (thousands instead of

billions), the location of each interaction site, energy and direction of photons prior to

each scatter event are tracked and stored in phase-space format. This file is then used in

the second stage, where the analytical step estimates multiple scatter energy fluence into

the imaging plane from all MC interactions sites. The analytical step uses the cross-

section probability for the discrete direction exiting the second (or higher) order

interaction site, and accounts for the attenuation and inverse square effect from the

interaction site to each pixel of the detector.

The modifications made to DOSXYZnrc 16 are summarized here. The subroutine

AUSGAB (the dedicated scoring code) provides the ability to extract the necessary

detailed information about an EGSnrc simulation without making any change to the EGS

code itself. The argument IARG is utilized to guide calling AUSGAB for specific

situations. A total of 26 IARG situations can be turned on or off via turning on/off the

flags in the associated specialized array IAUSFL. Using the IAUSFL flag option, we

16 DOSXYZnrc used in this chapter is user code from EGSnrc released in 2015.

104

modified the subroutine AUSGAB to allow tracking of the various physical interactions

within the media. Specifically, extra IAUSFL flags (i.e. 18) are turned on when Compton

interaction events are about to occur. We take advantage of the ‘stack’ parameters of

EGSnrc to access all scatter interaction information, and each photon’s pre-scattering

phase-space information (i.e. direction with respect to coordinate system, particle weight,

energy, and scatter order) is recorded in an output file. Each record in the corresponding

output binary file has a length of 40 bytes, which contains 10 basic variables (listed in

Appendix 4.6 Table 4-3). By filtering out photon record information with stack pointers

(NP) greater than 1, we remove secondary charged particles and only select photon

scattering events.

This set of Monte Carlo simulation generated scattering centers is then used to

compute the multiply scattered energy fluence imparted to the detector. Since almost

none of the simulated histories actually result in a photon incident on the detector, the

location of each scattering event in the MC history is assumed to release a scattered

photon to every pixel in the detector, as shown in Figure 4.2 (b). The analytical (ANA)

method, as used for the singly scattered fluence calculation, is then applied to each

scattering center to account for attenuation, inverse square effect, and Klein-Nishina cross

section, between the scattering site and each pixel in the imaging plane. However, for the

hybrid method, since the radiation transport from the source to the interaction site has

been taken into account in the Monte Carlo simulation stage, this means the probability of

interaction has already been considered during the Monte Carlo stage, so the weight of

fluence calculated to the imaging plane from each interaction site is unity (as long as

other variance reduction techniques are not applied during the Monte Carlo stage). Using

105

the P2 and P3 geometry illustrated in Figure 4.1, the multiply-scattered photon energy

fluence contribution, 𝛹𝑚𝑃2 , to an arbitrary imaging plane pixel 𝑃3 (𝑥3, 𝑦3, 𝑧3 ) from an

arbitrary multiple-scatter interaction site 𝑃2(𝑥2, 𝑦2, 𝑧2) in the patient/phantom, is

calculated as:

𝛹𝑚𝑃2(𝑥3, 𝑦3, 𝑧3, 𝐸) =

1

𝜎𝑙(𝐸)∗ (

𝑑𝜎

𝑑𝛺)𝑃3−𝑃2

(𝜃, 𝐸) ∗ 𝛥𝛺𝑃3−𝑃2 ∗ 𝑒𝑥𝑝 (−𝐼𝑃3−𝑃2 ) ∗ 𝐸′𝑃3−𝑃2

(Eq. 4-7)

where 𝐸 is the energy of the photon before the interaction occurs at the 𝑃2 interaction site;

𝜎𝑙 is the total cross section at the 𝑃2 interaction site which is labeled with material index 𝑙;

(𝑑𝜎

𝑑𝛺)𝑃3−𝑃2

(𝜃, 𝐸) is the Klein-Nishina cross section for scatter between a scattering voxel

𝑃2 and an imaging plane pixel 𝑃3 , 𝛥𝛺𝑃3−𝑃2 is the solid angle defined by the pixel at

𝑃3 from scattering voxel at 𝑃2; 𝐼𝑃3−𝑃2 is attenuation term between point 𝑃2 (the scattering

voxel) and point 𝑃3 (the scoring pixel); 𝐸′𝑃3−𝑃2 is the energy of scattered photon along the

direction from P2 to P3;

Similar to the singly scattered NEF, in order to compare with the conventional Monte

Carlo simulation, the multiply-scattered photon energy fluence is normalized to the

incident energy fluence at the top surface of the phantom, which can be calculated with

the Monte Carlo simulation input parameters. Therefore, the multiply scattered NEF

received at arbitrary pixel (i, j) of scoring plane is given as:

𝐹𝑖,𝑗𝑚 =

1

𝛹𝑖𝑛𝑐⋅ ∑ 𝛹𝑚

𝑃2𝑘

𝑛𝑘=1 (Eq.4-8)

106

Figure 4.2 Schematic describing and contrasting the methods of calculating multiply scattered photons entering the imager plane

generated using (a) full Monte Carlo simulation (i.e. DOSXYZnrc based patient scatter validation tool) with one billion photon

histories and (b) developed hybrid method logic flow which generated an estimation of multiply scattered photons at the imager

plane.

107

4.2-c. Electron-Interaction-Generated Photon Fluence

There is no exact analytical method available to estimate photon fluence due to

secondary electron interactions in the phantom/patient to the best of our knowledge. This

component of fluence includes bremsstrahlung photons and positron annihilation photons

(indirectly due to pair production interactions). To estimate this component, we use a

convolution/superposition pencil beam technique. A pencil beam patient-scatter kernel

(PBSK) approach to calculate patient scatter fluence at the imager plane has been shown

to be of reasonable but limited accuracy compared to full Monte Carlo simulation [21].

However, in the current hybrid model work, we seek to improve overall accuracy of the

patient scatter fluence estimate and therefore limit the PBSK application to only the

electron interaction generated fluence component, which allows us to add it to the results

of the analytical (singly scattered fluence) and hybrid (multiply scattered fluence) model

estimates.

To implement this approach, a ‘patient scatter fluence kernel’ (sometimes referred to

as a ‘water scatter fluence kernel’) is created using standard Monte Carlo methods.

Initially a photon scatter fluence map (as a phase-space file) is generated at the exit

surface of a uniformly thick (thickness ‘t’), homogenous water phantom due to an

infinitesimal pencil beam of photons, perpendicularly incident on the phantom surface.

The distribution of scattered fluence at a range of air gaps beyond the exit surface is

determined by projecting the photons scored in the phase-space file to virtual imaging

planes at several user-defined discrete air gap distances (the default used here is 5 cm

intervals). A library of patient scatter fluence kernels is generated by repeating this

approach over various thicknesses, t, of homogeneous water phantom. By only counting

108

the EIG component in the patient scatter fluence kernels, only that fluence component

will be predicted when implementing the pencil beam convolution/superposition fluence

calculation.

The patient scatter kernels are used together with a priori information regarding the

treatment setup to calculate the EIG portion of the patient-scattered photon fluence at the

imaging plane. This a priori information includes the incident beam energy spectrum and

incident relative fluence distribution, as well as the phantom/patient density information

which is obtained from computed tomography data.

The PBSK method works as follows (shown in Appendix 4.6 Figure 4.10): The

phantom/patient density information and imaging plane orientation is input to the 3D ray

tracing algorithm, and the radiological path length (RPL) and corresponding air gap (AG)

are calculated for each ray line from the x-ray source to the imaging plane pixels. Based

on the given RPL and AG, a bi-linear interpolation is used on the patient scatter kernel

library (PBSKL) to generate the required patient scatter kernel for the given ray line and

air gap combination. The patient scatter fluence kernel (for the EIG component) is applied

at the point of intersection in the imaging plane and integrated over every ray line. For

application to divergent beam geometries, the radiological pathlength of the tilted raylines

are exactly calculated, although the fluence kernel distribution is not corrected for this tilt.

109

4.2-d. Validation Testing

The simulation geometry is illustrated in Figure 4.3(a) under divergent beam

conditions (ideal point source), using a 100 cm source–surface distance (SSD), and a 150

cm source–detector distance (SDD).

When measuring x-ray transmission images with the EPID, it is impossible to

distinguish the various components of phantom/patient generated x-ray scatter (i.e. the

detector only measures the total signal -- primary plus all scatter). Therefore,

experimental validation is not possible, so in order to validate our tri-hybrid scatter

prediction method, we compare it against our previously developed and tested EGSnrc-

based photon scatter research tool (named ‘Dosxyznrc_K’) [110], which uses full Monte

Carlo simulation techniques and can separately track and score a variety of types of

scattered photons.

Incident photon fields used for testing included monoenergetic beams at 1.5, 5.5, and

12.5 MeV as well as 6 and 18 MV polyenergetic treatment beams (sampled at 1 MeV

wide energy bins), with a field size of 1010 cm2 (at 100 cm from source) to irradiate a

404020 cm3 geometric phantom with isotropic 1 cm3 voxels. Three phantom

configurations, as shown in Figure 4.3, were used for testing, including a homogeneous

water phantom (404020 cm3, ρ = 1.0 g/cm3), an ‘LWRL’ phantom which is defined as

the left-half water (402020 cm3, ρ = 1.0 g/cm3) and right half lung (402020 cm3, ρ =

0.26 g/cm3), and finally a thorax CT phantom for testing with increased heterogeneity

(composed of air ρ = 0.0012 g/cm3, lung ρ = 0.26 g/cm3, soft tissue ρ = 1 g/cm3, and bone

110

ρ = 1.85 g/cm3). The imaging plane was defined with dimensions of 40 40 cm2 and a 1

cm2 pixel size, located 30 cm underneath of the phantom’s exit surface.

111

Figure 4.3 Testing was performed using divergent beam geometry (a), and using three

test phantoms including (b) water, (c) LWRL (left-half water, right-half lung), and (d)

thorax, with monoenergetic and polyenergetic beams.

112

Full MC simulations and the TH calculations (i.e. combining ANA, HB and PBSK

methods as programmed in MATLAB) were executed on a laptop with an Intel Core (i7)-

6600U 2.60 GHz processor and 8 GB of RAM (i.e. single core not parallelized). The

EGSnrc MC simulation parameters used are listed in Appendix 4.6 Table 4-4.

The validation is performed by quantitatively comparing singly-scattered NEF,

multiply-scattered NEF, and EIG NEF to their corresponding components obtained from

full Monte Carlo simulation. A percentage difference image (PDI) was calculated

between the full Monte Carlo and the predictions for each individual component, and a

histogram of the PDI was calculated. The mean and standard deviation (STD) of the PDI

were treated as an indicator of accuracy and precision, respectively. Regarding the energy

spectrum of the total scattered fluence, the predicted mean energy spectrum (MES) across

the imaging plane, as well as energy spectra at the image plane center (i.e. on the central

axis for these examples) were compared using overlapped histograms to corresponding

full Monte Carlo simulation. The performance of the TH method is also evaluated by

using the thorax CT phantom irradiated by a 6 MV treatment beam when varying the field

sizes from 4x4, 10x10, and 20x20 cm2. The scatter factor (SF), calculated as the ratio of

the single scatter fluence to the primary plus scatter fluence, was also compared to Monte

Carlo using the central horizontal and vertical profiles on the scoring plane, and

percentage difference.

113

4.3 Results

By implementing the ANA, HB, and PBSK methods, the singly, multiply, and EIG

scattered NEFs were calculated, respectively. The individual scatter fluence components

and the total patient-scattered NEF (i.e. summation of singly, multiply, and EIG scattered

contributions) were compared to the full MC simulation results. The accuracy and

precision for the fluence results for the different phantoms and beam energies are listed in

Table 4-1 (monoenergetic) and Table 4-2 (polyenergetic). Overall, accuracy for the total

scatter fluence calculations using the tri-hybrid method are within 0.4% of full Monte

Carlo simulation, with precision within 1%. Figure 4.4 shows the comparison between the

full MC simulation and the TH method using a 6 MV beam, for the individual scattered

fluence components and the total scattered fluence. The corresponding PDIs and

histograms are also shown for each category. Figure 4.5 illustrates central horizontal and

vertical profiles of NEF and corresponding percentage differences, which are obtained by

using the TH method and full Monte Carlo simulation, with incident monoenergetic

beams of 1.5, 5.5 and 12.5 MeV. Figure 4.6 and 4.7 illustrates energy spectrum of the

total scattered photon fluence at the center pixel of the imaging plane and the comparison

of MES across the imaging plane, respectively, between the full MC simulation and TH

method for both 6 MV and 18 MV treatment beams and a field size of 10x10 cm2

irradiating the water phantom and thorax CT phantoms. For the mean energy spectra, the

overlapped areas between Th and MC are at least 98% for the different cases. The

performance of the TH method is studied when varying the field size, as illustrated in

Figure 4.8. It shows the percentage differences of totally-scattered NEF for central

horizontal and vertical profile are within ± 1 %.

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4.3-a. Singly scattered NEF

For the MV energy range, singly scattered fluence is the dominant component of the

total scattered NEF, as observed in Figure 4.5. As expected, the distribution of singly

scattered fluence is increasingly forward directed as beam energy increases. This

dominates the multiply scattered fluence especially in the central regions of the imager

under the primary beam, and in turn causes the precision of total scatter fluence to worsen

slightly (i.e. increase) with increasing energy, as well as towards the edges of the imaging

plane, as the stochastic noise present in the multiply scattered photon fluence becomes

more noticeable since multiply scattered photons becomes a relatively larger contribution

of the total fluence.

Overall, accuracy and precision of the singly scattered photon fluence are within 0.5%

and 1%, respectively, regardless of phantom type or incident energy. Regarding the

performance of ANA method with the change of heterogeneity, a slight loss of accuracy

is observed when introducing more media, but the ANA method still remains well within

1% accuracy for the calculation of the singly scattered fluence for the phantoms tested

here.

4.3-b. Multiply Scattered NEF

In Figure 4.5, the proportion of multiply scattered photon fluence to total scattered

fluence decreases with increasing incident photon energy. This component is about 10%

of total scattered signal for the case of the 6MV beam on the thorax phantom, and reduces

to approximately 4% at 18 MV. For 12.5 MeV, the ratio drops below 2%.

115

The precision of the HB method worsens (i.e. increases) for multiply scattered

photon fluence when compared to full MC simulation for higher energy incident beams.

This is due to a decrease in the relative fluence of multiply scattered photons at the

scoring plane and therefore an increase in stochastic noise for this component.

Overall, the accuracy and precision of the HB method are within 1.2% and 2.5%,

respectively, when the proportion of multiply scattered photons is 4% or more. The

accuracy and precision increase to 1.5% and 4%, respectively, when the proportion is less

than 2%.

Regarding the performance of the HB method with a change of heterogeneity, a 1%

worsening (i.e. increase) in precision at 12.5 MeV is observed when introducing different

media (as compared to smaller fluctuations at lower energies). However, the accuracies

amongst all phantom configurations and energies examined here are within 1%.

4.3-c. Electron interaction generated scattered NEF

In contrast to multiply scattered photons, the ratio of EIG scattered fluence to total

scattered fluence increases as incident photon energy increases. The PBSK method can

provide reasonably accurate prediction of the EIG component, with overall accuracy

within 1% for all energy and configurations tested here. The precision varies depending

on the significance of the EIG component. When the proportion of EIG to total is above 5%

(i.e. for high energy beams), the precision is within 4%. However, the precision can

worsen to 10% when the EIG component is less than 2%, for example at low incident

energy.

116

Regarding the performance of the PBSK method with the change of heterogeneity,

the accuracy amongst all tested phantoms and energies are within 1%. However,

worsening of precision (i.e. increasing) to 2% are observed when introducing

heterogeneous media.

4.3-d. Scatter Fraction (SF)

Figure 4.9 illustrates the central horizontal and vertical profiles of the SF comparing

the TH method and full MC simulation for the thorax phantom, for beam energies of 1.5

MeV, 5.5 MeV, and 12.5 MeV, as well as their percentage differences. Average

percentage differences of SF are within 1% among all tested configurations. The average

SF is about 10% at 1.5 MeV and 6 MV for the thorax phantom.

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Table 4-1 Comparison of patient-scattered photon fluence entering an EPID, calculated with full MC simulation and ANA, HB,

and PBSK methods. Results are divided into single, multiple, EIG, and total scatter fluence for the three phantoms tested here,

using incident beam energies of 1.5, 5.5, and 12.5 MeV. ‘Accuracy’ and ‘Precision’ are indicators of the average and standard

deviation of percentage differences across the entire image plane respectively.


Energy incident beam 1.5

MeV

5.5

MeV

12.5

MeV

1.5

MeV

5.5

MeV

12.5

MeV

1.5

MeV

5.5

MeV

12.5

MeV

Single Scatter Accuracy (%) -0.01 0.26 0.33 -0.12 0.06 -0.27 -0.01 -0.23 -0.41

Precision (%) 0.57 0.61 0.56 0.56 0.79 0.94 0.57 0.53 0.86

Multiple Scatter Accuracy (%) -1.08 0.05 -0.78 -0.62 0.65 0.56 -0.14 -0.15 -0.38

Precision (%) 1.02 2.05 1.22 1.13 2.1 3.33 1.04 2.50 4.08

EIG Scatter Accuracy (%) 0.24 -0.42 -0.18 -1.23 -1.09 -0.03 0.99 -0.43 0.67

Precision (%) 4.62 1.48 3.13 9.39 7.16 5.60 6.98 3.85 3.17

Total Scatter Accuracy (%) 0.33 0.22 0.29 0.32 0.05 -0.17 -0.03 -0.16 - 0.21

Precision (%) 0.49 0.59 0.68 0.65 0.90 1.33 0.55 0.70 1.04

118

Table 4-2 Comparison of patient-scattered photon fluence entering an EPID, calculated with full MC simulation and ANA, HB,

and PBSK methods. Results are divided into single, multiple, and EIG scattered fluence components, as well as total scattered

fluence for the three phantoms tested here, using incident beam energies of 6 MV and 18 MV. ‘Accuracy’ and ‘Precision’ are

indicators of the average and standard deviation of percentage differences across the entire image plane respectively.


Energy incident beam 6 MV 18 MV 6 MV 18 MV 6 MV 18MV

Single Scatter Accuracy (%) 0.33 0.35 0.47 0.24 -0.06 -0.03

Precision (%) 0.56 0.62 0.68 0.75 0.56 0.67

Multiple Scatter Accuracy (%) -0.48 -1.09 0.31 -0.73 0.34 -0.60

Precision (%) 1.22 1.53 1.77 2.14 1.21 2.48

EIG Scatter Accuracy (%) -0.18 -0.02 -0.38 -0.76 -0.18 -0.66

Precision (%) 3.13 2.00 9.63 5.17 5.19 3.84

Total Scatter Accuracy (%) 0.15 0.10 0.41 -0.06 0.05 -0.17

Precision (%) 0.50 0.58 0.61 0.80 0.55 0.68

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Figure 4.4 The comparison between the MC simulation and TH methods, for total scattered and individual scattered NEF

components, for a 6MV photon beam, 10x10 cm2 field size irradiating the thorax phantom.

120

Figure 4.5 The comparison of central horizontal (left-hand column) and vertical (right-

hand column) profiles between the TH method and Monte Carlo simulation, when the

incident energy is (a) 1.5 MeV , (b) 5.5 MeV, and (c) 12.5 MeV.

121

(a)

(b)

6 MV 18 MV

18 MV 6 MV

Figure 4.6 The comparison of scattered energy spectrum (with 10 bins) at the center

pixel of the imaging plane between the MC simulation and TH method for 6 MV (left)

and 18 MV (right) treatment beam irradiating a (a) water phantom (b) thorax CT

phantom with the field size of 10x10 cm2.

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(a)

(b)

6 MV 18 MV

18 MV 6 MV

Figure 4.7 The comparison of mean energy spectrum across the imaging plane

between the MC simulation and TH method for 6 MV (left) and 18 MV (right)

treatment beam irradiating a (a) water phantom (b) thorax CT phantom with the field

size of 10x10 cm2.

123

Figure 4.8 The comparison of central horizontal

(left) and vertical (right) profiles between the TH

method and full Monte Carlo simulation, with a

6MV photon beam irradiating the thorax phantom

with field sizes of (a) 4x4 cm2, (b) 10x10 cm2, and

(c) 20x20 cm2.

(a) (b)

(c)

124

Figure 4.9 The central horizontal (left-hand panel)/vertical (right-hand panel) profile of SF for thorax phantoms when the energy

of the incident beam is at 1.5 MeV, 5.5 MeV, and 12.5MeV with field size of 10x10 cm2. The symbol ‘*’ represents the

percentage difference of the scatter factor between TH calculation and full MC simulation.

125

4.4 Discussion

Comparing the total scattered fluence between the TH method and full MC

simulation, the average accuracy and precision across all phantom tests and beam

energies were 0.08% and 0.2%, respectively. This level of accuracy between the TH and

full Monte Carlo simulation provides strong validation that the TH method has been

implemented correctly.

Examining calculation times, for the 6MV beam incident on the thorax phantom, the

full Monte Carlo simulation using 1 billion histories takes about 32h, while the TH

method takes <80 seconds (without using parallel computing). Breaking down the

calculation time for each method within the TH approach, the ANA method using ~ 2800

interaction centers and six energy bins to compute singly scattered fluence takes 31.8

seconds. The HB method calculated the contribution from multiply scattered fluence

using ~38,000 interaction centers (generated by MC simulation with 20,000 incident

histories), takes about 46.7 seconds. For the EIG component, the PBSK calculation is

completed within 0.6 seconds. Since the phase-space information of all interaction centers

(about 55,000 interaction sites) is known for the singly and multiply scattered fluence

calculations, therefore these calculations can potentially be completed much more quickly

with parallel computing (i.e. graphics processing units parallelism). For example, GPU

parallelism with a single NVIDIA 9800 GX2 (circa 2009) was applied for analytical

photon scatter calculations in KV imaging, and completed a 323 voxel calculation in 4.3

seconds [43].

126

The TH predicted energy spectra of the patient generated photon scatter compared

well to those of the full Monte Carlo simulation. This is expected to be an important

aspect of maintaining accuracy of predicting the energy response of the EPID imaging

system, which is a characteristic of the locally implemented EPID in vivo dosimetry

algorithms [26], [27]; however, this has not been studied in detail in this work.

Even though the current results show good agreement in patient-generated photon

scatter fluence between the TH method and full MC simulation, there are still some

aspects that need to be explored, such as sampling issues (e.g. the phantom voxel

sampling and beam energy spectrum bin size), the significance of multiply scattered order

(e.g. some higher-order scatter contributions might be negligible), as well as the

dependence of accuracy of the HB method on the number of histories.

As part of our ongoing EPID in vivo dosimetry research, the TH method aims to

estimate the patient scatter fluence component incident on the MV imager, which in the

local algorithm implementation is currently calculated only using the pencil beam scatter

kernel approach (for all patient scatter fluence components), which limits its achievable

accuracy [22]. Therefore, future work will investigate the effect on our in vivo dosimetry

program when implementing the newly developed TH method.

4.5 Conclusion

In this work, we propose and implement a tri-hybrid method to estimate total patient-

generated scattered photon fluence and have demonstrated it to be in good agreement

with full Monte Carlo simulation for several phantoms and beam energies. The TH

method as implemented with a single CPU here takes a relatively short time (~80 seconds)

127

to execute without the use of parallel computing. However, the present validation is

limited since it does not include complex beam field tests (i.e. IMRT or VMAT fields),

and both spatial sampling and energy sampling aspects of the TH method could be

optimized for further improved performance.

4.6 Appendix

Table 4-3 Output phase-space file of DOSXYZnrc-based scatter scoring tool box

Variable Definition Data type

X, Y, Z Position of particle in coordinate system (cm) real*4

U, V, W Direction cosine of particle with respect to x, y, z axes real*4

E Total energy of particle (MeV) real*4

Weight The weight of particle real*4

N_hist The index of history number real*4

SC_order The order for each interaction site real*4

128

Figure 4.10 The logic flow of the PBSK calculation for the EIG component into the scoring plane: the radiological path length

(RPL) and corresponding air gap (AG) are calculated for each ray line from the x-ray source to the imaging plane pixels. Based

on the given RPL and AG, bi-linear interpolation is used on the patient scatter kernel library to generate the required patient

scatter kernel for the given ray line. The patient EIG scattered energy fluence kernel is applied at the point of intersection in the

imaging plane of each discretely sampled ray line and summed over all rayline contributions to yield an estimate of the EIG

scatter fluence entering the imager.

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Table 4-4 EGSnrc Monte Carlo transport parameters used in DOSXYZnrc


Global ECUT 0.521

Global PCUT 0.01

Global SMAX 1e10

ESTEPE 0.25

XIMAX 0.5




Spin effects On








Rayleigh scattering Off





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Chapter 5: Performance Optimization of a Tri-Hybrid Method for estimation of patient scatter into the EPID

This chapter details the impact of the sampling resolution of a variety of algorithm

parameters used in the previously developed tri-hybrid method. The content of this

chapter has been submitted to the peer-reviewed journal Physics in Medicine and Biology,

and is currently under review. Kaiming Guo, Harry Ingleby; Eric Van Uytven, Idris

Elbakri, Timothy Van Beek, Boyd McCurdy “Performance Optimization of a Tri-Hybrid

Method for estimation of patient scatter into the EPID”.

5.1 Introduction

In previous work we pointed to the necessity for a fast yet accurate method for scatter

estimation in EPID images acquisitions for portal in vivo dosimetry. Patient scatter

remains a challenge for accurate reconstruction of the 3D dose delivered to the patient

[52]–[55], and an accurate scatter estimation technique that can be executed in a clinically

acceptable timeframe is of interest.

Previously [111], we reported the development of a tri-hybrid (TH) method that

estimates the patient-generated photon scatter energy fluence image based on three

categories of scatter (i.e. single scatter, multiple scatter, and electron interaction generated

scatter). The combination of three distinct predictive methods (analytical calculation,

Monte Carlo simulation, and superposition/convolution of a pencil beam scatter kernel)

customized to each category of scatter, ensures a highly accurate solution overall.

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An analytical approach (ANA) is used to estimate the single scatter component to the

imaging plane, based on the first principles of Compton scatter kinematics. For multiply

scattered photons, a hybrid method (HB) utilizes only a small number of histories of MC

simulation to extract the phase space information of photons prior to individual scattering

events, and then follows with an analytical calculation on the (weighted) outgoing scatter

fluence projected to the entire imaging plane. The secondary photons resulting from

bremsstrahlung and also from positron annihilation are categorized as ‘electron

interaction generated’ (EIG) scatter, and this scattered photon component is predicted

using a convolution/superposition approach employing pencil beam scatter kernels which

are superposed on the incident fluence distribution.

Comparison against full Monte Carlo simulation results using various test

configurations (i.e. different phantoms, incident beam energies and field sizes) showed

average and standard deviation of percent difference of patient scatter estimates at the

EPID imaging plane to be within 0.5% and 1%, respectively, with high spatial and energy

resolution. Executing on a single CPU, run times for accurate results with high resolution

sampling will take more than 5 hours for an 18 MV, 10x10 cm2 field, although this will

vary depending on the size of the scattering volume (i.e. phantom/patient size, field size).

The nature of the solution allows implementing GPU parallelism, which would

accelerate the computing process; however, sampling (e.g. of the phantom, of the multiple

scatter, and of the beam energy spectrum) is still a critical issue that requires thorough

investigation to optimize the trade-off between the desired accuracy and the required

computing time, as the ultimate goal is for real-time calculation speeds. Thus, in this

work, we explore the tradeoff between the sampling settings and the achieved accuracy,

132

to find optimal operating settings for future clinical implementation, with results

demonstrated on geometric phantoms and clinical examples.


Figure 5.1 shows a schematic of the workflow for (a) the TH method and (b) full

Monte Carlo simulation, to estimate patient generated scattered normalized energy

fluence (NEF), which is defined as the energy fluence entering the imager normalized to

the incident energy fluence entering the phantom/patient. There are three components

involved in the TH approach: an analytical (ANA) method for singly scattered energy

fluence, a hybrid (HB) method for multiply scattered energy fluence, and a

convolution/superposition of pencil beam scatter kernel (PBSK) method for electron-

interaction-generated photon energy fluence. All three methods are forms of numerical

integration and were developed based on sampling of a voxelized phantom/patient and

pixelized imaging plane in Cartesian coordinates, while the beam energy spectrum was

also sampled as discrete energy bins.

ANA method --- Voxels inside the irradiated volume were sampled as Compton

scatter interaction sites. Scattered x-rays from each site are assumed to travel along

straight lines to each pixel within the scoring plane at the EPID. Based on an exact ray-

tracing algorithm [100] and the 3D phantom/patient density map, the angle, physical

distance and radiological path length of each ray-line can be determined, and the

phantom/patient inhomogeneity can be taken into account. The probability of interaction

is found using the Klein-Nishina differential cross section, while the energy of the

scattered photon is established using Compton kinematics. The incident photon beam

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energy spectrum is divided into discrete energy bins and the entire fluence calculation is

repeated for each bin. Integrating the calculation over all energy bins and over all

irradiated phantom/patient voxels provides the total singly scattered photon fluence

entering the imaging plane.

HB method --- To estimate the higher order patient scatter fluence (i.e. two or

more scattering events), a hybrid method is applied which combines two different

techniques (i.e. Monte Carlo simulation followed by analytical calculation). In the Monte

Carlo stage, a modified DOSXYZnrc user code is used to track the interaction history of

multiply scattered x-rays. Using a Monte Carlo simulation with only a few histories

(thousands instead of billions), the location of each interaction site is tracked, as well as

the direction and energy of the photon prior to reaching each interaction site. All this

information is input to the second stage --- an analytical calculation. Each MC interaction

site is assumed to produce scatter fluence that enters each pixel in the imaging plane, with

the energy fluence at each pixel calculated using the corresponding cross section

probability for the discrete direction exiting the second (or higher) order scatter

interaction site, and accounting for the attenuation through the patient/phantom from the

interaction site to each pixel of the detector.

PBSK method --- a convolution/superposition approach was employed using

pencil beam scatter kernels (PBSK) superposed on the incident fluence to calculate the

bremsstrahlung and positron annihilation (positrons produced due to pair production)

component. The kernel library is pre-generated using Monte Carlo simulation techniques

for a variety of patient water-equivalent thicknesses and air gaps (i.e. distance between

the patient exit surface and the imager surface). The appropriate PBSK to apply for each

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sampled ray-line is chosen from the precalculated library by using bilinear interpolation

based on the radiological pathlength and air gap. Discretely summing this product over all

incident raylines yields the distribution of the patient-generated EIG scatter fluence

entering the imager.

Within the TH method, there are several crucial sampling settings that trade off

calculation time against accuracy in the predicted fluence, and these are especially

important for the relatively more time-consuming ANA and HB methods (vs the PBSK

method). Note that the EIG NEF settings are not studied in the current work. Instead we

employ the previous optimized recommendation of 0.5 cm2 sampling resolution for the

convolution/superposition PBSK method for all tests [22].

135

Figure 5.1 (a) The workflow of the TH method (i.e. the combination of ANA, HB, and PBSK methods) to estimate the total patient-

generated scatter into the imaging plane. (b) The resultant NEF compared with the full Monte Carlo simulation fluence result (i.e.

using the ’dosxyznrc_K’ validation tool) with 1 billion photon histories.

(b)

(a)

136

5.2-a. Significance of sampling issues (phantom and energy spectrum) on single

scatter

Since the scatter distribution is broad and smoothly varying over the scoring plane,

some researchers suggest using a coarse phantom sampling resolution. For example for

cone beam computed tomography (CBCT), an isotropic 8 mm voxel was utilized to

accurately estimate 120 KV x-ray scatter contamination with a large incident field size of

261x196 mm2 [41]. Similarly, the Acuros CTS algorithm is able to provide accurate

scatter estimation for a 125 kVp energy spectrum with isotropic 1.25 cm3 voxels [40].

However, those works did not focus on optimized sampling, and in general little previous

work has been done to examine the impact of sampling the energy spectra in particular.

For our TH method, we investigate voxel sampling issues for several phantom/patient

geometries and also sampling of two clinically realistic polyenergetic beam spectra.

Specifically, phantom/patient isotropic voxel resolution is varied as 0.2, 0.25, 0.5, 1, 2,

and 4 cm, while the polyenergetic spectra sampling is varied over energy bin sizes of 0.25

MeV, 0.5 MeV, and 1MeV (while not significantly changing the mean energy of the

spectrum). The accuracy of the resulting calculations of fluence are compared to

corresponding full Monte Carlo simulation results in terms of percentage differences as

explained in Section 5.2-C below.

5.2-b. Significance of Monte Carlo history number and scattered order sampling for

multiple scatter

The hybrid method has several sampling considerations. Utilizing more Monte Carlo

histories will result in more scattering centers being sampled, which is expected to

137

increase the HB method accuracy at the cost of a longer calculation time. This effect is

studied by varying the number of simulation histories for the HB method (i.e. 2K, 4K, 6K,

8K, 10K, 20K, 40K, 60K, 80K, and 100K) and then examining the resulting accuracy for

various test configurations (i.e. phantom/patient, field size, and beam energy) by

comparing to full Monte Carlo simulation results in terms of percentage differences in

scatter fluence at the imaging plane, as explained in Section 5.2-C below.

For a typical 6 MV therapeutic beam, the maximum number of Compton scattering

events (or ‘order) in one photon history can approach 30 (although the average is 2-3),

before exiting a 20 cm thick patient. This is highly dependent on the size of the phantom

and the incident beam energy. The hybrid method can be sped up if one truncates at a

fixed maximum order of scatter, at the cost of decreased accuracy. The effect of

truncating at a range of different scatter orders (𝑛 ∈ [2, 15], [2, 20], [2,∞)) is examined

by again comparing the TH scatter fluence to full Monte Carlo simulation in terms of

percentage differences.

5.2-c. Validation Testing

The simulation setup of the imaging system is illustrated in Figure 5.2(a) for

divergent beam geometry using an ideal point source, a 90 cm source–surface distance

(SSD) and a 140 cm source–detector distance (SDD).

When measuring transmission EPID images experimentally, it is impossible to

distinguish the various components of phantom/patient generated x-ray scatter, i.e. the

detector only measures the total signal of primary plus all scattered photons. Therefore, in

order to validate our scatter prediction model, we have to compare it against full Monte

138

Carlo simulation. Previously we developed and tested an EGSnrc-based validation tool

for photon scatter (named ‘Dosxyznrc_K’) [110], which uses full Monte Carlo simulation

techniques and can separately track a variety of types of scattered photons. We use this

tool here as the ‘gold standard’ for the accuracy assessment of the TH model scatter

fluence predictions.

For this work, three different phantoms are used (illustrated in Figure 5.2) including a

homogeneous water phantom (404020 cm3, ρ = 1.0 g/cm3)a pelvis CT phantom

(composed of air ρ = 0.0012 g/cm3, soft tissue ρ = 1 g/cm3, and bone ρ = 1.85 g/cm3), and

a thorax CT phantom (composed of air ρ = 0.0012 g/cm3, lung ρ = 0.26 g/cm3, soft tissue

ρ = 1 g/cm3, and bone ρ = 1.85 g/cm3), for testing with increased heterogeneity

approaching realistic patient situations. The phantoms are irradiated with two

polyenergetic beams (6 MV and 18MV) [112], and with three different field sizes (4x4

cm2, 10x10 cm2, and 20x20 cm2). The EPID imaging plane was defined with dimensions

of 40 40 cm2 and a 1 cm2 pixel size, located 30 cm underneath the phantom’s exit

surface (i.e. air gap of 30 cm). The sampling resolution of the imaging plane is fixed at 1

cm2 for all studies performed here. This was selected based on the approach taken in prior

work [14], [113], where frequency analysis of patient-scattered fluence entering an

imager was performed in order to set the imaging plane sampling resolution at 5 cm and 2

cm, respectively, for KV applications. An analysis of MV scatter in test situations in the

current study (not shown here) indicate that a 1 cm2 sampling resolution will be a

conservative setting. Furthermore, a 30 cm air gap was chosen for use here for all test

cases since this is typically the closest the EPID imager is to the patient during routine

clinical use. Therefore, the investigations performed here represent a conservative

139

estimate of sampling requirements (i.e. if the imager is further away, sampling resolutions

will be relaxed compared to those required at 30 cm air gap, thus ensuring accuracy will

not decrease).

Full MC simulations and the tri-hybrid (TH) calculations (i.e. combined ANA, HB

and PBSK methods as programmed in MATLAB) were executed on a laptop with an Intel

Core (i7)-6600U 2.60 GHz processor and 8 GB of RAM (i.e. single core, not parallelized).

Figure 5.2 The tests were performed with (a) divergent beam geometry. Three

phantoms, (b) water, (c) pelvis, and (d) thorax were used to investigate the effect of

various sampling issues in the implementation of the tri-hybrid method.

140

The EGSnrc MC simulation parameters used in this work are listed in Appendix 5.6,

Table 5-4.

The validation is performed by quantitatively comparing singly-scattered NEF and

multiply-scattered NEF calculated over the entire imaging plane to their corresponding

values obtained from full Monte Carlo simulation. A percentage difference image (PDI)

is calculated between the full Monte Carlo and the predictions for each component, and a

histogram of the PDI is calculated. The mean and standard deviation (STD) of the PDI is

treated as an indicator of accuracy and precision, respectively, for singly-scattered NEF,

multiply-scattered NEF and total scattered NEF.

The relative root mean square error (rRMSE) of total-scatter NEF is calculated as

another measure of the performance of the TH method:

𝑟𝑅𝑀𝑆𝐸 = (1

𝑁∑

(𝑥𝑖𝑇𝐻−𝑥𝑖

𝑀𝐶)2

𝑥𝑖𝑀𝐶

𝑁𝑖=1 )

1

2 (Eq. 5-1)

where 𝑁 is the number of pixels in the imaging plane, and 𝑥𝑖𝑇𝐻 and 𝑥𝑖

𝑀𝐶 are the estimated

value of the NEF signal from TH method and MC simulation in the imaging plane

correspondingly.

Based on calculated rRMSE and the CPU time of calculation (𝑡𝐶𝑃𝑈), the efficiency

can be estimated using the following expression [8], which helps identify the optimal

sampling settings of the TH method for total-scattered NEF:

𝜀 =1

𝑡𝐶𝑃𝑈∙𝑟𝑅𝑀𝑆𝐸2 (Eq. 5-2)

It is well-known that the indirect a-Si EPID detector designs (used with almost all

modern linacs) have a unique energy response that is different from that of water [114],

141

[115], and which is important to consider for accurate conversion of fluence entering the

EPID to signal/dose generated in the EPID. Therefore, the mean energy distributions

across the entire imaging scoring plane are compared between the TH method and full

Monte Carlo simulation using the optimized settings.

The required accuracy of any scatter fluence prediction algorithm will be determined

by the application it is being used for. In the current work we choose an objective of +/-2%

accuracy in total scatter fluence at the imaging plane. While the imaging plane

contribution of the three scatter components considered here varies based on the phantom

geometry, field size, and beam energy, we can make some reasonable assumptions to help

set accuracy objectives for each scatter component. Since it is known that singly scattered

photon fluence will dominate, we expect to have more relaxed accuracy requirements for

the multiply scattered photon component and the EIG photon component, relative to the

singly scattered component. To estimate these accuracy requirements, we assume a ratio

of 70% singly scattered fluence, 20% multiply scattered fluence, and 10% EIG fluence.

This is considered conservative since typically singly scattered fluence is >70% for

therapeutic beams. Assuming the individual component error contributions are

independent, we can add them in quadrature and require that the total cannot exceed the

target of 2%. Thus, we have an estimate of the error in the calculation of the total scatter

fluence as:

𝜎𝑡𝑜𝑡𝑎𝑙 = √(𝜎𝑠𝑠)2 + (𝜎𝑚𝑠)2 + (𝜎𝑒𝑖𝑔)2 (Eq. 5-3)

A simple approach to achieve a 2% maximum uncertainty target is to limit each

component of scatter to contribute 1% or less of the total scatter error, or 𝜎𝑡𝑜𝑡𝑎𝑙 =

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√1 + 12 + 12 ≅ 1.7% . Thus the estimate for an acceptable error on the individual scatter

components is √1

0.72= 1.4% for the singly scattered component, √

1

0.22= 5% for the

multiply scattered component, and √1

0.12 = 10% for the EIG component. These set the

accuracy targets needed in order to select the optimal sampling settings.

5.3 Results

5.3-a. ANA method - Singly scattered NEF

Comparing the ANA method to full MC simulation for the single scatter component,

Figures 5.3 – 5.5 illustrate the changes in accuracy and precision of the ANA method for

different field sizes with different spatial voxel size sampling and different energy bin

sampling, for the phantoms examined here (i.e. water, CT pelvis, and CT thorax

phantoms, respectively). In Figures 5.3 – 5.5 it is evident that the change in energy bin

resolution from 0.25 to 1 MeV (per bin) has much less of an effect on the scatter fluence

accuracy compared with the changes in the phantom voxel sampling resolution. In fact,

errors larger than 2% (of total patient scatter fluence) were observed only when the

sampling of either energy spectrum (6 or 18 MV) increased beyond 1 MeV (it is not

shown in the figure). Therefore, the optimal energy spectrum sampling is considered to be

1 MeV per bin. As expected, as either the voxel sampling size or the energy bin sampling

size increases, the predicted fluence accuracy slightly decreases for all testing

configurations up until a threshold.

14

3

Figure 5.3 Comparison of the ANA method to full MC simulation for the single scatter component from the water phantom at

different spatial resolution (x-axis). The accuracy (i.e. dots) and precision (i.e. error bars) for 6 and 18 MV polyenergetic beams (top

row and bottom row, respectively) with different energy bin sampling (indicated by symbols, 0.25, 0.5 and 1 MeV correspond to

24,12,6 bins for 6 MV, and 72, 36,18 for 18MV) with field sizes of 4x4, 10x10, and 20x20 cm2 (left, middle and right columns,

respectively).

144

Figure 5.4 Comparison of the ANA method to full MC simulation for the single scatter component from the CT pelvis phantom at

different spatial resolution (x-axis). The accuracy (i.e. dots) and precision (i.e. error bars) for 6 and 18 MV polyenergetic beams (top

row and bottom row, respectively) with different energy bin sampling (indicated by symbols, 0.25, 0.5 and 1 MeV correspond to

24,12,6 bins for 6 MV, and 72, 36,18 for 18MV) with field sizes of 4x4, 10x10, and 20x20 cm2 (left, middle and right columns,

respectively).

14

5

Figure 5.5 Comparison of the ANA method to full MC simulation for the single scatter component from the CT thorax phantom at

different spatial resolution (x-axis). The accuracy (i.e. dots) and precision (i.e. error bars) for 6 and 18 MV polyenergetic beams

(top row and bottom row, respectively) with different energy bin sampling (indicated by symbols, 0.25, 0.5 and 1 MeV correspond

to 24,12,6 bins for 6 MV, and 72, 36,18 for 18MV) with field sizes of 4x4, 10x10, and 20x20 cm2 (left, middle and right columns,

respectively).

146

For either the 6 MV or 18 MV beam energies with the small field size of 4x4 cm2,

using a fine resolution (i.e. 0.2 and 0.25 cm3) maintains accuracy within 0.8%. When

voxel sampling resolution increased to 0.5 cm3, the accuracy decreases but is still within

1%. Increasing to 1 cm3 resolution, the accuracy is strongly impacted (maximum of 18%)

for all the tested phantoms. For either the 6 MV or 18 MV beam with the field size of

10x10 cm2 accuracy better than 1% is maintained with voxel resolution at 1 cm3, but

decreases significantly (maximum of 16%) when voxel sampling is increased to 2 cm3 for

all the tested phantoms. With the large field size of 20x20 cm2, accuracy better than 1% is

maintained even at voxel sampling of 2 cm3, but drops significantly (up to maximum of

12%) when increasing voxel size to 4 cm3.

This drastic change is due to larger spatial sampling, which will lead to increasing

partial volume effects at the edge of the beam (i.e. regions of steep dose gradient). In the

extreme case of an idealized binary fluence incident beam, some voxels at the beam edge

would not be considered if their voxel center happened to lie just outside the divergent

beam. Figure 5.6 illustrates this effect and shows the singly scattered center distribution

for 4x4, 10x10, and 20x20 cm2 field sizes with various voxel sampling sizes. While this

issue is minimized by using a finer resolution of phantom voxel sampling, using a fine

resolution increases the calculation time geometrically. For example, for an 18 MV beam

with energy bin sampling of 1 Mev and the 4x4 cm2 field, changing the voxel size from

0.5 cm3 to 0.2 cm3 leads to a calculation time increase by a factor of ~181.

For the ANA method using a 1 MeV energy bin resolution, a voxel sampling

resolution of 0.5, 1, and 2 cm3 is able to maintain the desired accuracy for 4x4, 10x10,

and 20x20 cm2 field sizes, respectively for the homogeneous water and CT pelvis

147

phantoms. However, when dealing with the heterogeneous thorax phantom at a field size

of 20x20 cm2, the 1 cm voxel sampling resolution was needed to maintain desired

accuracy. Therefore, it is recommended to use 0.5 cm3 voxel resolution at field sizes

below 10x10 cm2, and 1.0 cm3 voxel resolution at field sizes equal to or larger than 10x10

cm2.

148

(a)

(b)

(c)

Figure 5.6 Distribution of singly scattered centers (colour varying with z coordinate)

with various voxel sampling sizes with field sizes of (a) 4x4 cm2, (b) 10x10 cm2, and

(c) 20x20cm2.

149

5.3-b. HB method - Multiply Scattered NEF

Figure 5.7 illustrates the distribution of multiply scattered centers (MSC) for the 4x4

and 20x20 cm2 fields, using the 6 MV beam on the CT pelvis phantom with an increasing

number of tracked histories. The broad, distributed nature of these center locations is

demonstrated when varying the number of MC simulation histories of the hybrid method

(MCHHB) between 2K to 100K.

Figure 5.8 shows the accuracy of the HB method versus full MC simulation as a

function of the number of MCHHB, for all combinations of beam energy and field size

irradiating the CT pelvis phantom. As the number of tracked histories is increased, the

accuracy converges, as expected. At 100K of tracked histories, accuracies for all tested

situations are within 1%. The selection of 20K tracked histories ensures the accuracy of

the HB method to be within the target accuracy of 5% for this scatter component (over all

test configurations examined here).

The number of multiply scatter interaction sites generated within the phantoms (i.e.

water, pelvis, and thorax), for all tested combinations of beam energies and field sizes

ranged from around 1500 to nearly 200,000 (see Tables 5-5 and 5-6 in Appendix 5.6), as

expected, the number of sites scales basically proportionally with the number of MCHHB.

As incident energy increases, the number of scattering sites is generally reduced for the

water and pelvis phantoms due to the longer mean free path of the high energy photons,

but are more similar for the thorax phantom between 6 and 18 MV beam energies since a

large portion of lung tissue inside the thorax phantom will lead to longer mean free paths

for both energies. The average calculation time per scatter center is about 0.0015 sec for

the analytical stage.

150

Regarding the multiple scatter order sampling, the histograms in Figure 5.9 illustrate

the counts of multiply scattered centers versus the scatter order. The counts decrease

exponentially with the increase of the scatter order. For the 20 cm thick water phantom

test, the scatter order varies between 19 – 34 depending on the incident energy and the

field size. However, if the multiply scattered centers used are limited to between scatter

order 2 and 15, then the overall time of HB calculation drops only very modestly (5% of

the HB time) while the accuracy and precision is reduced by 4%. This is due to the rapid

falloff of higher order scatter interactions. Therefore, we conclude that truncating the

sampling of scatter order is not critical to improve efficiency of the HB method, and

recommend leaving it unchanged.

151

Figure 5.7 Distribution of multiply scattered centers with a range of MC simulation

histories (i.e. 2K, 6K, 10K, 20K, 60K and 100K histories) inside the CT pelvis

phantom when it is irradiated by a 6 MV polyenergetic beam with field sizes of (a)

4x4 cm2 and (b) 20x20cm

2.

(b)

(a)

152

Figure 5.8 Comparing HB method against full MC simulation for multiple scatter component. The accuracy (i.e. symbol) and

precision (i.e. error bar) are indicators of performance for different numbers of Monte Carlo histories used for the HB method, for 6

and 18 MV beams, irradiating the CT thorax phantom with field sizes of 4x4 (squares), 10x10 (circles), and 20x20 cm2(triangles).

153

(a)

(b)

Figure 5.9 The histogram of the multiple scatter centers (‘Counts’) per order of

multiple scatter for (a) 6 MV and (b) 18 MV incident beams and field size 20x20 cm2

irradiating on the pelvis phantom.

154

5.3-c. TH method - Total scattered NEF

By using the tri-hybrid method (i.e. combining ANA, HB, and PBSK methods), the

singly, multiply, EIG scattered NEF were calculated, respectively. Summing these

together yields the total patient-scattered NEF. Implementing the sampling settings

determined in Sections 5.3-a and b, the impact on the accuracy of the total scattered NEF

is assessed.

Tables 5-1 – 5-3 detail the comparison of patient-scatter calculated with full MC

simulation and the TH method using incident beam energies of 6 and 18 MV for the water,

pelvis, and thorax phantoms at different field sizes, and different settings of MCHHB.

The TH calculation times are in the range of ~15 seconds to ~5 minutes. All accuracies lie

within the target of ± 2%, and precision estimates are also under 2%. The rRMSE

decreases from 0.42% to 0.06% when increasing the number of MCHHB from 2K to

100K, and the precision improves by about 50% while the computing time increases by

up to 9.5 times.

Figure 5.10 illustrates the calculation efficiencies of the TH method when the water,

pelvis, thorax phantoms are irradiated by the 6 and 18 MV polyenergetic treatment beams

with different field sizes (4x4 cm2, 10x10 cm2, and 20x20 cm2), versus the histories used

in the HB MC simulation, using the recommended settings for single scatter. The patterns

showed the 20K MCHHB yields the optimal efficiency for most test cases while 10K

MCHHB occasionally showed a bit higher efficiency. Therefore, the optimal number of

MCHHB to estimate total-scattered NEF is recommended as 20K.

155

These recommended settings are applied to two clinically realistic examples. Figure

5.11 shows the cross-plane and in-plane profile comparisons for the total scatter, and each

sub-component of scatter for each tested field size, for an 18 MV treatment beam incident

on the pelvis phantom. Figure 5.12 shows the comparison of the total and individual

scattered NEF components for the thorax phantom using a 6MV beam and 10 x 10 cm2

field size.

In terms of the mean energy of scattered fluence incident on the scoring plane, the

overlapping histograms shown in Figure 5.13 illustrate the comparison of the mean

energy distributions across the scoring plan pixels between the TH and full MC method,

for the 6MV beam irradiating the water phantom with three different field sizes. Using

the recommended settings (figure 5.13 (a)) for the TH method ensures differences in the

mean energies are less than 5% compared to full MC simulation. As field size increases,

the differences in the mean energy distributions decrease. The overlapped areas of the

mean energy spectra histograms are at least 95% of mean energy distribution from Monte

Carlo simulation. Using a 0.2 cm3 voxel resolution and 100K MCHHB for all field sizes

(figure 5.13 (b)), not many differences are observed comparing with recommended

setting for 20 x 20 cm2 field sizes, but it will increase overlapped areas to over 98% for

the field sizes of 4x4 and 10x10 cm2 .

156

Table 5-1 Comparison of patient-scattered photon entering an EPID calculated with full MC simulation and the TH method using

an incident beam energy of 6 and 18 MV for the water phantom. For the ANA method, the 0.5, 1, and 2 cm3 voxel sampling sizes

with respect to the three field sizes 4x4, 10x10, and 20x20 cm2 are used. ‘Accuracy’ and ‘Precision’ are indicators of the average

and standard deviation, respectively, of percentage differences across pixels in the entire image plane.

Field Size 4x4 cm2 10x10 cm2 20x20 cm2

MCHHB 2K 6K 10K 20K 100K 2K 6K 10K 20K 100K 2K 6K 10K 20K 100K

6

MV

Accuracy 1.90% 0.46% -0.84% -0.75% -0.18% 2.29% 0.56% -0.44% -0.35% 0.21% -1.11% 0.30% -0.12% -0.04% 0.23%

Precision 1.00% 0.85% 0.73% 0.66% 0.60% 1.18% 0.99% 0.88% 0.75% 0.69% 1.20% 1.13% 0.92% 0.63% 0.56%

rRMSE 0.32% 0.19% 0.15% 0.12% 0.09% 0.38% 0.24% 0.13% 0.11% 0.11% 0.29% 0.19% 0.14% 0.10% 0.09%

tCPU (sec) 31.5 42.2 52.9 78.7 289.7 25.8 35.9 47.1 73.3 280.5 87.0 97.5 107.1 131.6 329.6

18

MV

Accuracy 0.16% 0.31% 0.10% -0.01% -0.04% 0.42% 0.57% 0.36% 0.25% 0.22% 1.67% 0.71% 0.32% 0.48% 0.29%

Precision 1.58% 1.25% 0.87% 0.77% 0.51% 1.61% 1.29% 0.93% 0.82% 0.58% 1.34% 1.10% 0.76% 0.65% 0.52%

rRMSE 0.19% 0.13% 0.10% 0.09% 0.06% 0.21% 0.16% 0.14% 0.12% 0.09% 0.31% 0.17% 0.13% 0.11% 0.11%

tCPU (sec) 82.0 88.5 96.9 114.3 254.7 65.4 72.2 79.0 96.5 235.5 249.2 255.6 262.6 279.4 412.1

157


an incident beam energy of 6 and 18 MV for the pelvis phantom. For the ANA method part, the 0.5, 1, and 2 cm3 voxel sampling

sizes with respect to the field sizes 4x4, 10x10, and 20x20 cm2 are used. ‘Accuracy’ and ‘Precision’ are indicators of the average

and standard deviation, respectively, of percentage differences across the entire image plane.



6

MV

Accuracy 2.00% 1.14% -0.72% -0.25% -0.16% 2.34% 1.16% 0.31% 0.13% -0.29% -2.25% -0.85% -0.51% -0.16% 0.21%

Precision 1.44% 1.20% 1.11% 1.11% 1.03% 1.25% 1.06% 0.85% 0.81% 0.71% 1.39% 1.27% 0.80% 0.79% 0.64%

rRMSE 0.40% 0.27% 0.21% 0.17% 0.17% 0.42% 0.24% 0.16% 0.13% 0.10% 0.40% 0.23% 0.15% 0.12% 0.10%

tCPU (sec) 31.4 42.0 50.9 77.1 284.3 25.9 36.0 45.9 71.4 277.0 86.9 96.4 105.0 129.2 323.0

18

MV

Accuracy 1.21% -1.65% -0.89% -0.87% -0.55% -2.60% -1.18% -0.89% -0.46% -0.62% -0.30% 0.48% -0.22% -0.46% -0.02%

Precision 1.61% 1.38% 1.18% 1.10% 1.06% 1.19% 0.90% 0.83% 0.64% 0.52% 1.89% 1.22% 1.19% 1.07% 0.88%

rRMSE 0.27% 0.23% 0.21% 0.18% 0.17% 0.33% 0.17% 0.14% 0.11% 0.09% 0.26% 0.23% 0.19% 0.16% 0.13%

tCPU (sec) 82.1 88.8 95.5 113.2 250.5 65.1 71.9 79.2 97.2 235.3 249.3 255.9 262.5 279.3 410.9

158


an incident beam energy of 6 and 18 MV for the thorax phantom. For the ANA method, the 0.5, 1, and 2 cm3 voxel sampling sizes

with respect to three field sizes 4x4, 10x10, and 20x20 cm2 are used. ‘Accuracy’ and ‘Precision’ are indicators of the average and

standard deviation, respectively, of percentage differences across the entire image plane.



6

MV

Accuracy -1.86% -1.05% -0.85% -0.73% -0.77% -0.85% -0.43% -0.11% -0.31% -0.28% -0.65% 0.55% 0.25% 0.26% -0.21%

Precision 1.23% 1.26% 1.08% 1.04% 1.03% 1.15% 0.96% 0.87% 0.75% 0.72% 1.31% 1.17% 0.91% 0.78% 0.61%

rRMSE 0.28% 0.21% 0.18% 0.16% 0.16% 0.20% 0.16% 0.14% 0.11% 0.10% 0.19% 0.17% 0.14% 0.12% 0.09%

tCPU (sec) 29.3 35.7 42.1 58.5 188.7 24.2 31.3 37.8 55.0 192.8 85.4 92.0 98.4 115.4 246.6

18

MV

Accuracy -0.61% -1.07% -0.77% -0.26% -0.58% 1.73% 0.89% -0.80% -0.50% -0.27% -1.03% -1.15% -0.83% -0.68% -0.75%

Precision 1.47% 1.38% 0.95% 0.93% 0.89% 1.40% 1.18% 0.81% 0.88% 0.75% 1.59% 0.92% 0.83% 0.77% 0.65%

rRMSE 0.23% 0.21% 0.15% 0.14% 0.14% 0.31% 0.19% 0.13% 0.11% 0.09% 0.27% 0.22% 0.18% 0.15% 0.14%

tCPU (sec) 80.5 84.7 88.9 99.5 184.0 63.8 68.4 73.0 83.9 174.3 248.1 252.3 256.2 266.7 354.5

159

Figure 5.10 Calculation efficiency of the TH method

when the (a) water, (b) pelvis, (c) thorax phantoms

are irradiated by 6 and 18 MV treatment beams with

different field sizes (4x4 cm2, 10x10 cm2, and 20x20

cm2) versus the number of histories used in the HB

MC simulation, using the recommended sampling

settings for the single scatter calculation.

(a) (b)

(c)

160

Figure 5.11 The comparison of central horizontal

(left) and vertical (right) profiles between the TH

method and full Monte Carlo simulation, with the 18

MV photon beam irradiating the pelvis phantom with

field sizes of (a) 4x4 cm2, (b) 10x10 cm

2, and (c)

20x20 cm2 using the optimal sampling settings of the

TH method.

(a) (b)

(c)

161

Figure 5.12 The comparison of total and individual scattered NEF component between the full MC simulation against the TH

method, for a 6MV photon beam with a field size of 4 x 4 cm2 irradiating the pelvis phantom with the recommended sampling

settings.

162

Figure 5.13 Comparing the mean energy distribution from the TH method against full MC simulation for the total patient-generated

scatter component for the water phantom irradiated by the 6MV beam. (a) Using 0.5, 1, and 1 cm3 voxel size sampling with respect

to the field sizes of 4x4, 10x10, and 20x20 cm2 and 20K MCHHB, and (b) using a 0.2 cm

3 voxel resolution and 100K MCHHB for

all field sizes.

(a)

(b)

163

5.4 Discussion

For the MV energy range, the single scatter fluence is the dominant component of

total scattered NEF, especially for smaller fields and higher beam energy. The accuracy

of estimating the singly scattered component is mainly dependent on the voxel sampling

resolution. At the imaging plane, the multiple scatter is of less magnitude than single

scatter and is also a broader distribution, for all field sizes. It was found that limiting the

scatter order sampling was not a significant factor in reducing calculation time. As

expected, a larger number of MCHHB yield a more accurate and precise estimation of

multiple scatter fluence and reduce its rRMSE contribution to the total-scattered NEF. As

the incident beam energy is increased, the contribution of the EIG fluence component

increases, and the shape of the EIG fluence varies noticeably with field size (e.g. Figure

5.11), since this component is very forward directed. The mean energy spectra predicted

by the TH method overlapped within 5% area of the full MC simulation energy spectra.

Based on the maximum slope of the a-Si detector energy response curve above 0.5 MeV

[23], a 5% error in the TH predicted energy spectra would result in a maximum ~0.6%

error in a subsequent dose calculation using the predicted energy fluence image. While

this is a very rough estimate, it indicates that the level of agreement observed in the TH

predicted energy spectra here is more than adequate for the purposes of accurate

conversion of incident energy fluence to dose in the EPID.

Based on the estimated required patient-generated scatter fluence accuracy, the

recommended sampling settings are determined in Sections 5.3-a, b and c. By using these

sampling settings, the accuracy (and precision) in the total-scattered NEF of the TH

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patient scatter prediction method across all tests are within 0.8% (and 1.2%) of full Monte

Carlo simulation for all test cases, which is within our target accuracies.

In terms of total calculation time of the TH method, we examine the 18MV

therapeutic beam with field size of 4x4 cm2 irradiated on the pelvis phantom as an

example. Using the recommended sampling settings, the TH method calculation time can

be analyzed by scatter component/algorithm. The ANA method uses a voxel sampling

resolution of 0.5 cm3 (generates ~ 3K scatter source centers) and 18 energy bins to

compute the singly scattered NEF, taking about 77 seconds. The HB method calculated

the contribution from multiple scatter using approximately 26K multiply scattered

interaction centers, which is generated by MC simulation using 20K histories. The MC

simulation part takes about 0.8 seconds, and then about 36 seconds to accomplish the

remaining analytical calculation step. For the EIG component, the PBSK method

completes the calculation within 0.6 seconds. Therefore, for this example, the full TH

method takes ~ 113 seconds, without using parallel computing. In contrast, TH method

with high spatial, energy resolution and 100K MCHHB takes more than 5 hours to

complete, and the full Monte Carlo simulation using 1 billion histories, including scoring

fluence entering the detector, takes about 32h.

While with optimized sampling, the TH method takes a relatively short time

compared to the full Monte Carlo simulation, there is still another technique to speed up

the calculation. Since the majority of calculation time is spent estimating the singly and

multiply scattered components based on the large number of scatter centers, and the

phase-space information of all the interaction centers is known, such a calculation can be

potentially completed by parallel computing using, for example, GPU (graphics

165

processing units) parallelism. The GPU parallelism with a single NVIDIA 9800 GX2 was

applied for fast analytical calculation for a singly scattered fluence map in low energy KV

imaging, and it accomplished a 323 voxel calculation in 4.3 second [43]. The GPU in that

earlier work had only 128 cores. In the current market, GPUs with over 4000 cores are

available at low cost, and therefore we expect reprogramming the TH method to take

advantage of GPU processing will significantly accelerate the calculation (to about one

second with 4000 cores).

As part of our EPID in vivo dosimetry research program, the development of the TH

method aims to more accurately estimate the patient scatter fluence component into the

EPID imager, which in our local implementation is currently calculated only using a

simple pencil beam scatter kernel approach, which limits its accuracy. Therefore, the

future work will implement the TH method into our clinical in vivo EPID dosimetry

program, using the sampling resolutions recommended here.

5.5 Conclusion

In this paper, we investigate the sampling issues of a recently developed tri-hybrid

method to estimate the total patient-generated scattered photon energy fluence entering an

imaging detector. Using the recommended sampling resolutions, the TH method with

optimal sampling setting takes a significantly shorter calculation time compared to the

high-resolution sampling setting and full Monte Carlo simulation, while showing

quantitative agreement with full Monte Carlo simulation results within the target accuracy

of 2% for energy fluence and 5% for mean energy spectra. Optimized sampling as

implemented here on a single CPU is faster than full Monte Carlo simulation by a factor

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of roughly 1000. In the future, we are interested in translating the TH method to a GPU

platform and implementing it into a clinically used in vivo EPID dosimetry program.

5.6 Appendix

Table 5-4 EGSnrc Monte Carlo transport parameters used in DOSXYZnrc


Global ECUT 0.521

Global PCUT 0.01

Global SMAX 1e10

ESTEPE 0.25

XIMAX 0.5




Spin effects On








Rayleigh scattering Off





167

Table 5-5 List of the total number of multiply scattered interaction centers generated within the phantoms (i.e. water, pelvis,

and thorax), when irradiated by a 6 MV beam with various field sizes (4x4, 10x10, and 20x20 cm2).

Phantom Water Pelvis Thorax

Field Size

(cm2) 4x4 10x10 20x20 4x4 10x10 20x20 4x4 10x10 20x20

2K Histories 3823 3717 3599 3721 3782 3516 2216 2599 2425

4K Histories 7381 7346 7216 7432 7202 6917 4439 5160 4790

6K Histories 11397 10925 11059 11280 10948 10264 6795 7642 7159

8K Histories 15107 14595 14634 14924 14302 13659 9186 9969 9472

10K Histories 19019 18869 17890 17580 18010 16392 11325 12238 11722

20K Histories 37427 37505 35289 36249 36147 33612 23010 24486 23817

40K Histories 74793 75208 70209 73209 72613 68723 45779 48281 46857

60K Histories 112574 111817 105230 109667 109032 103316 68962 73296 69856

80K Histories 150258 148871 140797 146822 145572 136842 92405 98306 93217

100K Histories 187505 184935 176206 183706 182449 171472 115693 122527 117139

168

Table 5-6 List of total number of multiply scattered interaction centers generated within the phantoms (i.e. water, pelvis, and

thorax), when irradiated by an 18 MV beam with various field sizes (4x4, 10x10, and 20x20 cm2).

Phantom Water Pelvis Thorax

Field Size (cm2) 4x4 10x10 20x20 4x4 10x10 20x20 4x4 10x10 20x20

2K Histories 2533 2581 2326 2625 2401 2449 1498 1490 1588

4K Histories 4916 5088 4607 5014 4761 4737 3026 3140 3048

6K Histories 7185 7427 6912 7413 7250 7120 4477 4723 4570

8K Histories 9669 9850 9291 9808 9761 9362 6140 6351 6175

10K Histories 13110 12309 11883 12162 12438 11793 7430 7996 7304

20K Histories 25559 24750 23846 24763 25246 23796 15022 15797 14783

40K Histories 50933 49302 47465 49699 49056 47202 29940 31626 30710

60K Histories 75346 74190 70936 73977 73824 70729 44957 47963 45803

80K Histories 100306 99267 94735 97788 98589 93654 59930 63887 61501

100K Histories 125454 123650 118252 122428 123501 117386 75103 80077 77282

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Chapter 6: Summary and Future work

6.1 Summary

Advanced RT techniques are continuously developed and are often utilized to deliver

higher dose prescriptions or treatment with fewer fractions. The increasing complexity of

treatment plans makes it more difficult to discover possible errors, and conventional pre-

treatment quality assurance (QA) approaches are not adequate to ensure patent safety [49],

[54], [55], [101]–[103]. Thus, the need for improved patient-specific dose verification has

increased. Therefore in vivo dose measurement has been receiving increasing attention as

an additional and very effective QA approach [101], [103], [104]. The megavoltage beam

imaging system (EPID) has been shown to be useful for in vivo dosimetry applications.

Patient dose verification can be accomplished in many ways, commonly including

single point-dose or 3D dose distributions in the patient (compared to the treatment

planning system), or 2D planar dose at the EPID (with the measured transmission image

compared to a pre-calculated or ‘predicted’ transmission image). Some researchers

compare the measured EPID transmission dose distribution with a pre-calculated portal

dose [25], [51]. Some others convert the 2D images to fluence estimates and backproject

this to reconstruct the 3D radiation dose distribution delivered to the patient [26], [48],

[49]. If a difference is found between the measured and expected radiation delivery, the

source of the difference can be corrected in following treatments, thereby improving the

patient outcome.

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However, there are still some challenges to use of EPIDs for routine patient in vivo

dosimetry [52]–[55]. One of those challenges is that the photon fluence entering the EPID

is contaminated with patient-generated scattered photons, which limits the accuracy of in

vivo patient dose calculations. This scattered x-ray component can be significant, making

up as much as 30% of the MV image signal [21], [57]. To improve the accuracy of in vivo

dosimetry methods, many researchers try to eliminate the patient scatter signal

contribution from the measured EPID image by estimating it and then subtracting it from

the measured image [25], [26], [49], [53], [107], [108]. Then, the remaining transmitted

primary fluence is backprojected to a plane above the patient as an estimate of the

incident primary fluence, which can finally be used by a patient dose calculation

algorithm to estimate 3D dose to a CT or CBCT representation of the patient. The

performance of dose verification applications will improve when this patient scatter

component is more accurately accounted for, since uncorrected scatter reduces image

contrast and reduces the ability to confidently verify the treatment delivery by forcing

increased tolerances in acceptability criteria.

Patient-generated scatter entering the MV planar detector can be classified into three

components: singly-scattered photons, multiply-scattered photons, and electron-

interaction-generated (EIG) photons (i.e. due to bremsstrahlung and positron annihilation).

Several groups have used analytical methods to estimate the singly scattered energy

fluence [9], [32], [42], [43], [109]. The multiply-scattered component is known to be a

smooth, broad function and has been simply approximated as proportional to the singly

scattered photon distribution [12]. However ‘hybrid methods’, which combine Monte

Carlo simulation with analytical methods, have been shown to accurately estimate the

171

multiply-scattered component of CBCT projection images [41]. A 2011 review article of

x-ray scatter estimation techniques [18] suggests that hybrid approaches represent the best

hope for a fast yet accurate solution to this problem.

Our research group has a strong background investigating scattered MV radiation

[19]–[26]. Recently, we developed a patient dose reconstruction approach that removes

the patient-scatter component from the measured therapy transmission images (i.e. on-

treatment EPID images), to estimate the 3D dose delivered to the patient by the treatment

beam [26], [27]. However, we recognize that the patient scatter component of our

predictive model is the least accurate step in our modeling, and may ultimately limit the

degree to which we can verify delivered treatments. The objective of this thesis was to

develop a much more accurate method (i.e. the tri-hybrid method) to estimate patient-

generated x-ray scatter entering into the MV image detectors, while maintaining a high

execution speed.

Before we could confidently validate our proposed tri-hybrid method, a customized

Monte Carlo (MC) simulation user code was developed for investigating the individual

components of patient-scattered photon fluence, as described in Chapter 3. This MC tool

is based on the EGSnrc/ DOSXYZnrc user code. The IAUSFL flag options associated

with subroutine AUSGAB, combined with LATCH tracking, are used to classify the

various interactions of particles with the media. Photons are grouped into six different

categories: primary, 1st Compton scatter, 1st Rayleigh scatter, multiply scattered,

bremsstrahlung, and positron annihilation. We take advantage of the geometric boundary

check in DOSXYZnrc to write exiting photon particle information to a phase-space file.

The tool is validated using homogeneous and heterogeneous phantom configurations with

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monoenergetic and polyenergetic beams under parallel and divergent beam geometry,

comparing MC simulated exit primary fluence and singly-scattered fluence to

corresponding analytical calculations.

This Monte Carlo tool has been validated to separately score the primary and scatter

fluence components for energies relevant to both KV and MV radiotherapy imaging

applications. The results are acceptable for the various configurations and beam energies

tested here. Overall, the mean percentage differences are less than 0.2% and standard

deviations less than 1.6%. This will be a critical test instrument for research in photon

scatter applications, particularly for the development of hybrid methods, and is freely

available from the authors for research purposes.

Chapter 4 details the work in developing and validating an algorithm to provide

accurate estimates of the total patient-generated scattered photon fluence entering the MV

imager. In this work, analytical calculations (ANA) are used to estimate the singly-

scattered photon fluence component, a hybrid (HB) algorithm is implemented to estimate

the multiply-scattered photon fluence component, and the EIG component is estimated by

using a convolution/superposition pencil beam scatter kernel (PBSK) method. Combining

these three different scatter prediction methods, termed the tri-hybrid method (TH), we

investigate its feasibility and accuracy for estimating total patient-generated scattered

energy fluence entering an EPID.

The total patient-scattered photon fluence entering the imager was compared with a

corresponding full MC simulation (EGSnrc) for several homogeneous and heterogeneous

test cases. The proposed tri-hybrid method is demonstrated to agree well with full MC

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simulation, with the average fluence differences and standard deviations found to be

within 0.5% and 1% respectively, for the test cases examined here.

Chapter 5 investigates the most significant sampling issues of the TH method, to

optimize the efficiency for a target accuracy of 2% in the predicted energy fluence and 5%

in the predicted mean energy spectrum. The sampling parameters examined included: a

range of patient voxel size, the number of Monte Carlo histories used in the modified MC

method (i.e. hybrid method), the scatter order in hybrid method, and also the energy bin

sizes of the incident energy spectrum. Three phantoms (homogeneous water, CT pelvis,

CT thorax) were tested with 6 and 18 MV polyenergetic treatment beams at field sizes of

4x4, 10x10 and 20x20 cm2.

For the different sampling setting, the tri-hybrid method was compared to full Monte

Carlo simulation. With the optimized sampling, accuracy and precision of the total-

scattered NEF of the TH patient scatter prediction method are within 0.9% and 1.2%,

respectively, comparing with full Monte Carlo simulation results for all test cases. For the

mean energy distribution across the imaging plane, the overlapped predicted histogram

coincides with 95% of the mean energy distribution from the Monte Carlo simulation.

The method takes as little as ~73 seconds to execute on a single (non-parallel) CPU,

while with non-optimized sampling the TH method took as long as 5 hours, and full MC

simulations took over 30 hours.

In summary, a new tri-hybrid method was developed and validated for predicting

patient-generated scatter entering an MV imager. The method applies a custom algorithm

to each of the three main components of scatter fluence to optimize the accuracy-speed

tradeoff. Since experimental techniques cannot separate various subcomponents of MV

174

photon fluence, a special Monte Carlo simulation reference tool was developed to allow

direct comparison of the accuracy of the three components of scatter fluence, and

therefore facilitated validation of the tri-hybrid method. The sampling of the tri-hybrid

method was optimized to ensure desired accuracy was achieved while maintaining

superior execution speed.

6.2 Future work

Although the TH method as implemented on a single CPU here takes a relatively

short time compared with conventional full Monte Carlo simulation, even with optimal

sampling settings it still takes far too long for routine clinical use (i.e. target speed will be

less than 1 second per projection). Therefore, translating the TH method onto a GPU

platform will be an essential step to gain the efficiency required for routine clinical

application. Since the majority of calculation time is spent estimating the singly and

multiply scattered component using a large number of scatter (interaction) centers, and

the phase-space information of all interaction centers is known, these calculations can be

potentially completed by parallel computing (e.g. GPU - graphics processing units). The

GPU parallelism with a single NVIDIA 9800 GX2 was utilized for similar fast analytical

calculations for a KV imaging application [43], and it accomplished a 323 voxel

calculation in 4.3 seconds. The GPU used in that work was released in 2008, and featured

only 128 cores. In the current market, for example, a GeForce RTX 2080Ti has 4352

cores which is 32 times more powerful than the GPU used in [43]. GPU implementation

of the method, which involves reprogramming the TH method to take advantage of the

available parallel processing, could significantly accelerate the entire calculation to near

175

real-time execution speeds. This level of performance will allow the TH method to be

implemented into the clinical in vivo dosimetry program in our local clinic. Currently, our

in vivo dosimetry software uses only the pencil beam scatter kernel method to estimate

patient generated scatter fluence at the EPID, which limits its accuracy. The ultimate goal

is to have an efficient yet highly accurate patient-scatter removal method that will

improve the accuracy of the in vivo dosimetry results, which will improve our ability to

ensure the patient received the intended dose.

Furthermore, there is an opportunity to translate our experience with MV scatter and

image formation into the KV energy domain, where it could provide improvement to

cone-beam CT applications. To accomplish this, the TH method would have to be

significantly reworked, since the bremsstrahlung and positron annihilation photon

component becomes negligible at KV energies, while the importance of Rayleigh

scattering increases. This implies that the method at the KV energy range would be more

similar to a hybrid approach, which has been explored by several researchers over the

past 10 years [41], [96], [116], [117]. However, there may be room for additional novel

contributions, for example forced discrete sampling of MC interaction sites, iterative

analytical scatter, or the use of pre-calculated ray-trace information.

One more possible future work opportunity could be the investigation of artificial

intelligence techniques for patient-generated scatter prediction (for both MV and KV

applications). Such applications, including machine learning and neural network methods,

have been receiving significant attention from medical physics researchers over the last 2-

3 years. Potentially a large amount of known combinations of patient, incident beam, and

total patient scatter fluence maps could be used to train an artificial neural network. Then

176

a previously unseen patient and beam could be input and the new output (i.e. scattered

fluence) would be predicted by the network. A key feature of this technique is to generate

the accurate scatter estimates that are needed for the paired training data sets, which could

benefit from techniques developed in this work. If successful, this approach may exceed

the performance of the hybrid method (i.e. accuracy and speed).

177

Appendix A: Basic Monte Carlo Simulation for Photon Radiation Transport

A.1 Basic Concept of Monte Carlo simulation in Radiation

Transport

Monte Carlo (MC) simulation is used in a wide range of scientific applications, e.g.

astrophysics, environmental engineering, cell biology and so on. It can be generally

described as any technique that provides an approximate solution through statistical

sampling. This method is useful for dealing with problems having a probabilistic

interpretation. This technique can provide a solution to a macroscopic system through a

simulation of its microscopic interactions, and can handle complex and multidimensional

problems [118], [119].

Based on the law of large numbers in probability theory [120], the average of the

results obtained from a large number of individual trials should be close to the expected

value. By repeatedly running microscopic ‘experiments’ that measure the full history of

an individual particle, and tracking the results, the average over many thousands (or

millions, or billions) of individual particle histories will yield a macroscopic solution. The

use of random numbers for sampling the probabilities describing the underlying physical

processes is required to obtain a high quality stochastic solution.

The rationale of Monte Carlo simulation and its application in radiation transport will

be discussed in the following sections.

178

a. Probability Theory, Sampling Methods, and Random Number Generators

A normalized probability distribution function (PDF), 𝑓(𝑥) , for each physical

process is defined over a range [a, b] and is normalized to have unit area (under the curve).

A PDF must have properties such that it is both integrable and non-negative so that its

cumulative probability function (CPF) 𝐹(𝑥) is constructed to range between 0 and 1 (i.e.

𝐹(𝑏) = 1).

𝐹(𝑥) = ∫ 𝑓(𝑥)𝑑𝑥𝑥

𝑎 (Eq. A-1)

The usefulness of mapping 𝐹(𝑥) onto the range of a random variable, 𝜀, where 0 <

𝜀 < 1 is that the equation can be easily inverted to solve for the value 𝑥 = 𝐹−1(𝜀). In

general, there are two approaches to determine 𝑥: the “direct method” and the “discrete

method”, for continuously and discretely distributed PDFs, respectively [121]. These will

be discussed in more detail in the following Section A.1-b.

Random numbers are a key requirement in the Monte Carlo method. Truly random

number sequences come from physical events displaying true randomness (e.g. the decay

of a radioisotope), but the process of gathering such physical random number sequences

can be time consuming, and the length of the sequence is limited by practicality.

As a convenient alternative, there exist mathematical algorithms that can generate

nearly random number sequences. A random number is generated successively by using a

recurrence formula in the form of 𝑅𝑛+1 = ℜ(𝑅𝑛), and is called a pseudo-random number.

Generating pseudo-random numbers has been intensively studied and is a subfield within

mathematics. A common method to calculate a pseudo-random number sequence is the

linear congruence approach [122]:

179

𝑅𝑛+1 ≡ 𝑚𝑜𝑑(𝑎𝑅𝑛 + 𝑏,𝑚 ) (Eq. A-2)

where a, b, and m are positive integers and the divider m is the length of the integer value

allowed in the computer’s compiler (e.g. 𝑚 = 232 for 32 bit). A random number

generator (RNG) should be fast, able to create long sequences before repetition is

encountered, and display good statistical characteristics (i.e. be truly random).

The randomness of the RNG should be well understood. There are many approaches

available to test the randomness of a given sequence.

1. Examining the probability distribution function (PDF) as shown in Figure A.1

(a): by observing the probability distribution function of a random number

sequence, the relative frequency should be the same for each bin regardless of bin

size.

2. A lag plot as shown in Figure A.1 (b), plots the random number sequence against

a version of itself that has been shifted (or ‘lagged’) by a specific number of

elements. For purely random data, there should not be any identifiable structure in

the lag plot. If the lag plot exhibits some distinct patterns, it indicates that the

underlying data are not completely random.

3. The autocorrelation function can be used to detect non-randomness in data [123].

For given measurements, 𝑌1 , 𝑌2 , …, 𝑌𝑁 , the lag k autocorrelation function is

defined as

𝑟𝑘 =∑ (𝑌𝑖 − ��)𝑁−𝑘

𝑖=1 (𝑌𝑖+𝑘 − ��)

∑ (𝑌𝑖 − ��)2𝑁𝑖=1

180

The randomness is evaluated by computing autocorrelations for random number

sequences at varying lags. If random, such autocorrelation coefficients should be

nearly zero for all lag values. If non-random, then one or more of the

autocorrelation coefficients will be significantly non-zero. As illustrated in Figure

A.1(c), autocorrelation coefficients (blue bar) for a given series through lagging 1-

10 elements are nearly zero.

4. Statistical tests for uniformity and independence17 : these algorithms test the

randomness of an RNG based on hypothesis testing. There are typically two

hypotheses statements:

H0 (null hypothesis) the sequence was produced uniformly and randomly

distributed

Ha (Alternative hypothesis) the sequence was not produced uniformly and

randomly distributed

The probability P-value is calculated by assuming the null hypothesis is true

for a specific statistical test. The P-value is compared with the required level of

statistical significance 𝛼 (usually set to be 0.05). If the P-value > 𝛼, the test does

not reject the null hypothesis, essentially concluding that there is not enough

evidence of randomness.

17 www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/freqtest.htm

www.eg.bucknell.edu/~xmeng/Course/CS6337/Note/master/ node42.html

181

b. Direct and Discrete Probability Method on Radiation Transport

Probability distribution functions can be used to sample the emission location,

direction, and energy of particles from a radiation source, the location of the interaction

points of the radiation particle in the medium, the interaction type, and the physical

properties of the resulting particles (i.e. any particles created by an interaction with the

medium).

Figure A. 1 (a) The histogram with bin

size of 0.05 shows a PDF of pseudo-

random number sequence with 10 million

elements; (b) a part of the scatterplot of

the random number sequence (vertical

axis, (0, 0.1)) versus same sequence

lagging 10 elements (horizontal axis, (0,

0.1)); (c) the graph of autocorrelation

coefficients (blue bar) for a given series

through lagging 1-10 elements with the

95% confidence interval (blue line).

(a) (b)

(c)

182

All the physical variables are described by their own PDFs. Some of them (e.g.

emitting direction and location of interaction) are continuously distributed and can be

represented by an analytical equation, and others (e.g. interaction types) are discretely

distributed and represented by a lookup data table. Since the PDFs are integrable and non-

negative, they are mapped onto CPFs in order to allow the use of uniformly distributed

random numbers to sample the direction, energy, and spatial location of the particles

[118], [119], [121]. The detailed sampling techniques used in Monte Carlo simulation of

radiation transport will be discussed in the following sections.

Source Sampling

Suppose that a polyenergetic divergent beam strikes on a homogeneous water

phantom. The energy spectrum of the beam provides the relative fraction of photon

fluence with respect to the individual energy bins, which is considered a discrete PDF.

Figure A.2 (a) shows the normalized energy spectrum (i.e. normalized to ensure the area

under the curve is unity) of a typical 6 MV treatment beam with an energy bin size of 1

MeV. By integrating this PDF, the CPF is obtained (Figure A.2 (b)). A random number 𝜀

is sampled and transformed by the CPF, providing the energy of the source photon in the

MC simulation.

183

If we assume the source emits photons isotropically, the same number of the photons

is emitted per unit solid angle in any direction (on average). Thus, the random isotropic

direction of an emitted source photon will need to be sampled. If the solid angle is 𝑑Ω =

𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙 , where 𝜃 and 𝜙 are zenith and azimuthal angle, respectively, then the

normalized probability distribution function is found by the definition of the solid angle:

𝑝(𝜃) =𝑠𝑖𝑛𝜃

∫ 𝑠𝑖𝑛𝜃𝑑𝜃𝜋0

=1

2sin(𝜃); 𝑝(𝜙) =

𝑑𝜙

∫ 𝑑𝜙 2𝜋0

=1

2𝜋

Based on the random sampling from their own CPF, the angle 𝜃 and 𝜙 can be

defined for a source photon. Applying the fundamental principle to the cumulative

probability function, the value of 𝜃 and 𝜙 can be resolved.

{𝜀1 = ∫ 𝑝(𝜃)𝑑𝜃 =

1

2(1 − 𝑐𝑜𝑠𝜃)

𝜃

0

𝜀2 = ∫ 𝑝(𝜙)𝑑𝜙 =1

2𝜋

𝜙

0

{𝜃 = cos−1(2𝜀1 − 1) ;

𝜙 = 2𝜋𝜀2

(A-3)

Travel Length Sampling

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40P

rob

ab

ilit

y D

istr

ibu

tio

n F

un

cti

on

Energy (Mev)

Probability Distribution Function

Figure A.2 (a) the normalized energy spectrum and (b) the cumulative probability

function of a typical 6MV treatment beam.

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

0.0

0.2

0.4

0.6

0.8

1.0

Cu

mu

lati

ve

Pro

ba

bil

ity

Fu

nc

tio

n

Energy (Mev)

Cumulative Probability Function

184

Once the energy and the travel direction of the incident photon are known, the

location of the next interaction site (i.e. particle travel length) needs to be determined.

Based on Equation (II-1), at a distance 𝑙 along the direction of the beam, the fraction of

photons not having undergone any interactions is 𝑃𝑛𝑐(𝑙) = 𝑒−𝜇𝑙 . Thus, the fraction of

photons that interacted with the medium before reaching a depth l is a cumulative

probability function (CPF), written as:

𝑃𝑐(𝑙) = 1 − 𝑒−𝑢𝑙 (Eq. A-4)

When 𝑙 approaches infinity, 𝑃𝑐(𝑙) will approach 1 which means that all the photons

are attenuated. Similarly, when 𝑙 approaches 0, 𝑃𝑐(𝑙) will approach 0, meaning no

photons are attenuated. The 𝑃𝑐(𝑙) is in the range 0 to 1.

When simulating the paths of individual photons, one needs to determine the length

from the current position to the next interaction site. The free path 𝜆 is defined as:

𝜆 = −ln(1−𝑃𝑐(𝜆))

𝜇 (Eq. A-5)

and represents the distance traveled by a moving particle between successive interactions,

which will modify its direction and/or energy. A uniformly distributed random number 𝜀

in the range of 0 to 1, is used to sample the cumulative distribution function 𝑃𝑐(𝜆). The

length to the next interaction point can then be determined as:

𝜆 = −ln(1−𝜀)

𝜇 (Eq. A-6)

Since simulations are based on large number theory and 1 − 𝜀 goes from 1 to 0, (1 −

𝜀) and 𝜀 are randomly selected, uniformly distributed numbers, and they are thus

considered to be equivalent. The path length can be further reduced to:

185

𝜆 = −𝑙𝑛 (𝜀)/𝜇 (Eq. A-7)

Interaction Sampling

Similarly, the discrete PDF concept is used to select the interaction type during

individual photon simulation. When the photon ‘arrives’ at the next interaction site, the

next step is to decide which interaction will happen. For the diagnostic and therapeutic

energy ranges, there are mainly four interactions: Photoelectric effect, Rayleigh scattering,

Compton scattering and pair production. The PDF of interactions is determined by the

ratio of the specific interaction cross sections over the total cross section. The interaction

will be chosen by a uniformly distributed random number 𝜀 applied to the generated CPF.

As shown in Figure A.3, when the normalized probabilities are given for all possible

reactions, the current interaction can be determined by mapping random number 𝜀 on the

corresponding CPF. Again, referring to Figure A.3, when the photoelectric effect occurs

Figure A.3 (a) the probability density and (b) cumulative probability of the photon

interaction based on the ratio of the mass attenuation coefficients for individual

interaction types to the total mass attenuation coefficient.

186

( 0 < 𝜀 < 𝑃𝑃𝐸 ), the photon is absorbed (i.e. photon history ends). When Rayleigh

scattering (𝑃𝑃𝐸 ≤ 𝜀 < 𝑃𝑃𝐸 + 𝑃𝑅𝑆) or Compton scattering (𝑃𝑃𝐸 + 𝑃𝑅𝑆 ≤ 𝜀 < 𝑃𝑃𝐸 + 𝑃𝑅𝑆 +

𝑃𝐶𝑆 ) occurs, the photon moves in a new direction and with a new energy. If pair

production (𝑃𝑃𝐸 + 𝑃𝑅𝑆 + 𝑃𝐶𝑆 ≤ 𝜀 < 1) occurs, the photon history will terminate, but a

positron and electron pair will be produced.

Direction/Energy sampling for the resultant particles

If photoelectric absorption occurred, the participating shell must be determined.

Assume the photoelectron only emits from the K or L shell, the random number 𝜀 is

chosen to map onto the discrete CPF to determine the interacting shell (e.g. interaction

with the K shell if 𝜀1 ≤ 𝑃𝑘 , where 𝑃𝑘 is the probability of all photoelectric interactions

that occur with the K shell; otherwise, the interaction will occur with the L shell).

If the K-shell photoelectron is ejected, the vacancy may be filled by an electron from

a higher shell (e.g. L, M, N, etc.), along with emitting fluorescence photon(s) or Auger

electron(s). Thus, another random number 𝜀2 is introduced to determine whether

fluorescence occurred (e.g. if 𝜀2 ≤ 𝑃𝑘𝑙, where 𝑃𝑘𝑙 is the fraction of the fluorescence yield

from filling the vacancy in a K-shell by an L-shell electron, the fluorescence photon is

produced with energy of the difference between the K and L’s binding energy; if 𝑃𝑘𝑙 <

𝜀2 < 𝑃𝑙∗ , where 𝑃𝑙∗ is the fraction of the total fluorescence photon yield from a shell

higher than L, then the fluorescence photon is ejected with the average energy from

filling from all shells higher than L; when 𝜀2 > 𝑃𝑙∗ , an Auger electron is produced).

187

The low kinetic energy of photoelectrons and Auger electrons is typically assumed to

be deposited locally. The direction of any fluorescence photon is chosen from an isotropic

distribution in the same way as Equation (A-3).

If the interaction is Rayleigh or Compton Scattering, the first step is to sample from

the corresponding differential cross section so that the zenith angle 𝜙 with respect to the

incident photon direction and the energy of either scattered electron or photon can be

determined. Numerical inversion is commonly used to provide the differential probability,

Δ𝑃𝑖,𝑗(𝜙𝑖 , 𝐸𝑗) =𝑑𝑒𝜎(𝜙𝑖,𝐸𝑗)

𝑑ΩΔΩ, where the differential cross section has been evaluated at an

energy 𝐸𝑗 at the midpoint of the 𝑖𝑡ℎ angular interval that is 2𝜋 sin(𝜙𝑖) Δ𝜙𝑖 wide. The

discrete normalized CPF is generated in a lookup table:

𝑃𝑚,𝑗 = (∑ Δ𝑃𝑖,𝑗(𝜙𝑖, 𝐸𝑗)𝑚𝑖=1 )/(∑ Δ𝑃𝑖,𝑗(𝜙𝑖 , 𝐸𝑗)

𝑛𝑖=1 ) (Eq. A-8)

where 𝑚 is an interval number corresponding to the angle 𝜙𝑚 and 𝑛 is the total number of

intervals. The random number 𝜀 is applied to map onto the CPF lookup table. To be

accurate for a rapidly changing distribution, a large number of intervals must be used, but

the intervals do not necessarily need to be equally spaced.

If 𝑃𝑚,𝑗 < 𝜀1 < 𝑃𝑚+1,𝑗, the angle 𝜙 can be found by interpolating between 𝜙𝑚 and

𝜙𝑚+1. Once sampled, the zenith angle determines the energy of the outgoing photon by

solving the appropriate kinematics equations. The azimuthal angle 𝜃 of the scattered

photon with respect to the incoming photon direction is equally distributed between 0 and

2𝜋, so the other random number 𝜀2 is used to determine the angle 𝜃 as shown in the

Equation (A-3). The direction of the scattered photon will be transformed back to the

phantom coordinate system to transport it [118].

188

When pair production occurs, the incoming photon disappears, and an electron and

a positron are created. Charged particle transport is not considered here, however, if in-

flight positron annihilation is negligible, then positron annihilation can be assumed to

take place at the location in which the pair production occurred. Both annihilation

photons with energy 0.511 MeV are then transported. The direction (𝜃1, 𝜙1) of one of the

annihilation photons is chosen from an isotropic distribution in the same way as Equation

(A-3). The other annihilation photon is given the opposite direction (i.e. 𝜃2 = 𝜃1 + 𝜋,

𝜙2 = 𝜙1 + 𝜋).

c. Overview of logic flow for MC Simulation of Radiation Transport

Figure A.4 illustrates a logic flow diagram for Monte Carlo simulation of photon

transport. As mentioned in the previous section, the detailed knowledge of the physical

processes involved in photon transport are used to determine the parameters of the event

by sampling from an appropriate probability distribution [118], [124]. Since the future of

a photon or an electron is independent from its previous history at any point in the

simulation, it is possible to process phase-space data after the simulation.

For this discussion, consider a source photon placed on the ‘stack’ (i.e. a queue for

particles that need to be simulated) – the stack holds all required physical information

about the impending particles, including the current location, direction, energy, etc. The

concept of a “stack” of particles is essential for data management during the simulation

because with each photon interaction it is possible to create one or more additional

resultant particles (electrons, positrons, fluorescent X-rays, etc.), and the necessary

physical parameters for each particle are stored by the stack during simulation.

189

Photon histories are terminated and removed from the stack because: (i) the photon

has been absorbed (and its energy is deposited locally), (ii) the energy of the photon falls

below a cutoff value (also resulting in energy being deposited locally), or (iii) the photon

leaves the geometric volume of interest. The details of when the history is terminated and

how the energy cutoff is chosen depends on what quantities are of interest, which can be

monitored or tracked (using a “scoring” component). For example, if the absorbed dose to

the medium is the parameter of interest for the simulation, the energy deposited by

interactions in a particular geometric region will be recorded throughout the simulation.

A.2 Photon-Specific Variance Reduction Techniques

In MC simulation, each initial particle is given equal ‘statistical weight’ to the

outcome, and it is attempted to faithfully duplicate the microscopic radiation transport

Figure A.4 the logic scheme of the Monte Carlo simulation on radiation transport.

190

process, achieving a macroscopic solution by averaging over millions or billions of

individual particle histories. Even through identical source particles may be initiated with

equal statistical weights, their individual microscopic behavior will not be the same.

The accuracy of a Monte Carlo calculated quantity [118], [120] is mainly restricted

by the statistical noise, because the influence of Monte Carlo method approximations is

much smaller. For example, systematic errors are typically <1%. The statistical variation

of the quantity of interest is expected and this statistical noise can be decreased by using a

larger number of histories, at the cost of longer calculation times. Variance reduction

concepts were developed to increase sampling efficiency and therefore also improve

simulation efficiency.

There are a variety of techniques to decrease the statistical fluctuations of Monte

Carlo calculations without increasing the number of particle histories. These techniques

are known as variance reduction techniques (VRTs). Efficiency in the context of Monte

Carlo simulation can be described as [118], [120]:

𝜖 =1

𝜎2𝑇 (Eq. A-9)

where 𝜎2 is variance of the quantity of interest [𝜎2 =1

𝑐∑ (

Δ𝐷𝑖

𝐷𝑖)2

𝑛𝑖=1 ] , and T is the

computing time to obtain the variance 𝜎2.

When applying a variance reduction technique, the statistical uncertainty of the

simulation is improved using the same number of particle histories, thus increasing

efficiency of the Monte Carlo simulation. In general, these techniques are considered to

be application-specific. In some cases, different techniques can interfere with each other

so choosing VRT methods strongly depends on the application of interest.

191

Monte Carlo Variance Reduction techniques can be generally divided into four

categories [124], [125]:

1. The truncation method (e.g. geometry truncation and energy cut off);

2. The population control method (e.g. Russian roulette and particle splitting);

3. The modified sampling method (e.g. interaction forcing)

4. Sectioned problems (e.g. pre-calculated results).

If variance reduction techniques are employed, the statistical weight of the particle

must be monitored and adjusted to avoid potential bias which would otherwise provide

erroneous results. The following section will cover how the main variance reduction

techniques are applied to Monte Carlo simulation of photon transport.

a. Interaction Forcing

For some applications only focused on the location of where the photon interacted,

efficiency of MC simulation will be reduced if some photons leave the geometry without

interacting [124].

Equation (A-4) gives the cumulative probability distribution of the path length in a

homogeneous medium. Thus, the probability that a particle is attenuated between distance

𝑙 and 𝑙 + 𝑑𝑙 is given by

𝑑𝑃𝑐(𝑙) = 𝜇𝑒−𝜇𝑙𝑑𝑙 (Eq. A-10)

The mean free path length �� is defined as the average distance traveled before an

interaction occurs. It represents the penetrating ability of the x-ray beam. The

corresponding expected value 𝐸(𝜆) or �� is found:

192

𝐸(𝜆) = �� =∫ 𝑙∗𝜇 𝑒−𝜇𝑙𝑑𝑙∞0

∫ 𝜇 𝑒−𝜇𝑙𝑑𝑙∞0

=1

𝜇 (Eq. A-11)

Commonly, the free path length will be expressed by the number of mean free path

lengths, which is labeled 𝑘 =𝜆

��= 𝜇𝜆. The Equation (A-10) can be written as

𝑑𝑃𝑐(𝑘) = 𝑝(𝑘)𝑑𝑘 = 𝑒−𝑘𝑑𝑘

The number of mean free path lengths 𝑘 varies from 0 to infinity and the

corresponding cumulative probability18 will be:

𝑃𝑐(𝑘) = ∫ 𝑒−𝑘𝑑𝑘∞

0= 1 − 𝑒−𝑘 (Eq. A-12)

Similarly to the derivation of Equation (A-4), 𝑘 = −ln (1 − 𝜀), where 𝜀 is a random

number uniformly distributed in the range between 0 and 1.

The length a photon travels through the geometry may be finite, but there is a non-

zero and sometimes large probability that photons leave the geometry of interest without

interacting so that computational time is wasted tracking these photons.

This waste can be prevented if the photons are forced to interact in the geometry of

interest. This is achieved by constructing a new probability distribution by

renormalization, as

𝑝𝑛𝑒𝑤(𝑘) =𝑒−𝑘

∫ 𝑒−𝑘′𝑑𝑘′𝛼

0

=𝑒−𝑘

1−𝑒−𝛼 (Eq. A-13)

where 𝛼 is the total number of mean free paths along the direction of the photon to the

edge of the geometry. The corresponding new cumulative probability function will be

18 lim

𝑘→∞𝑃𝑐(𝑘) = 1

193

𝑃𝑐𝑛𝑒𝑤(𝑘) = ∫ 𝑝𝑛𝑒𝑤(𝑘)𝑑𝑘

𝛼

0=

1−𝑒−𝑘

1−𝑒−𝛼 (Eq. A-14)

Thus, 𝑘 is restricted to the range between 0 and 𝛼, and 𝑘 is selected as:

𝑘 = − ln(1 − 𝜀(1 − 𝑒−𝛼)) (Eq. A-15)

Since the photon has been forced to interact within the geometry of the simulation, its

weighting factor needs to be changed in order to avoid bias. This entails making the ‘new’

value of weighting factor 𝐸(𝑤𝑛𝑒𝑤) the same as before performing the VRT.

𝑝𝑛𝑒𝑤(𝑘) ∗ 𝑤𝑛𝑒𝑤 = 𝑝𝑜𝑙𝑑(𝑘) ∗ 𝑤𝑜𝑙𝑑 (Eq. A-16)

where 𝑤𝑜𝑙𝑑 is the previous weighting factor. 𝑝𝑜𝑙𝑑(𝑘) is equal to 𝑝(𝑘) in this case.

Therefore, the resulting weighting factor (𝑤𝑛𝑒𝑤) is calculated as

𝑤𝑛𝑒𝑤 = 𝑤𝑜𝑙𝑑(1 − 𝑒−𝛼) (Eq. A-17)

When the interaction is forced, (1 − 𝑒−𝛼) simply multiplies the old weighting factor.

This approach can easily be used repeatedly to force the interaction of descendants of

scattered photons. It also may be used in conjunction with other techniques, depending on

the application.

The increase in efficiency can be dramatic when applying this VRT, especially for

applications in small cavity theory (i.e. ion chamber problems). For example in reference

[124], only 6% of photons would have interacted in the ion chamber, but when using

interaction forcing, the efficiency improved by a factor of 2.3.

b. Russian Roulette & Particle Splitting

Two situations are encountered in photon transport simulations where a photon may

generate a large number of secondary particles, or a particle remains within the

194

considered geometry without escape, leading to long particle histories [121], [124]. In

either case, a method for early termination of the particle history may be beneficial to

improve efficiency, but the process needs to be kept unbiased. Russian Roulette (RR) and

Particle Splitting (PS) are two complementary importance-sampling methods which can

achieve that goal.

Consider a particle weight of 𝑤𝑜𝑙𝑑 , the Russian Roulette random variable can be

described in equation form as the particle is ‘killed’ (i.e. removed from the simulation)

with probability 1 − 𝑝, and the particle will survive with probability 𝑝.

{𝐾𝑒𝑒𝑝 𝑃𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑝 𝑤𝑛𝑒𝑤 = 𝑤𝑜𝑙𝑑/𝑝 𝐾𝑖𝑙𝑙 𝑃𝑎𝑟𝑡𝑖𝑐𝑙𝑒 1 − 𝑝 𝑤𝑛𝑒𝑤 = 0

(Eq. A-18)

The expected weight of the particle is conserved before and after the decision. The

expectation value of the statistical weight is 𝐸(𝑤𝑛𝑒𝑤) = (1 − 𝑝) ∗ 0 + 𝑝 ∗ 𝑤𝑜𝑙𝑑 ∗1

𝑝=

𝑤𝑜𝑙𝑑. The killed particles are removed from the simulation and the remaining particles

with new weight are sampled. The process is repeated with different values of the kill

probability P until the number of particles is reduced to a manageable size to avoid long

histories. By terminating the particle history in this manner, the statistical weight is

preserved.

In contrast to RR, the PS method increases the sample size. It relies on a particle

being split into numerous similar particles. To avoid bias, the statistical weight of the

original particle is distributed amongst its replacement particles. This is done by: 𝑤𝑖 =

𝑤𝑜𝑙𝑑

𝑛, 𝑖 = 1,2, … , 𝑛, where n is the number of split particles. The expected value of the

new particle weights is their sum, 𝐸(𝑤𝑛𝑒𝑤) = 𝐸(𝛴𝑤𝑖) = 𝑛 ∗𝑤𝑜𝑙𝑑

𝑛= 𝑤𝑜𝑙𝑑, and conserved

195

with the original photon’s statistical weight (i.e. before the split). The PS method controls

the total number of tracks and the relative number of tracks in various regions of phase

space by assigning a different value 𝑛. If particle histories not contributing to final results

are avoided, the number of paths would ideally be proportional to their contribution to the

final results.

When RR and PS are combined as an importance-sampling method, not only the

number of scoring particles increases but also the particles’ weights to the scored

parameter (i.e. the parameter of interest in the simulation) tend to be nearly constant. It

leads to a significant reduction of the variance of the scored quantity [7]. Considering a

medium subdivided into regions (𝑟1, 𝑟2 ,…, 𝑟𝑛), each of them assigned an importance

(𝐼1, 𝐼2 ,…, 𝐼𝑛). When a particle with weight of 𝑤𝑜𝑙𝑑 enters a region (𝑟𝑗) from a region (𝑟𝑖),

the importance ratio (𝑛) between regions is calculated as (𝐼𝑗/𝐼𝑖). If the new region has

greater importance than the previous one (𝐼𝑗 > 𝐼𝑖 ), the particle is split into 𝑛 identical

particles of weight, 𝑤𝑛𝑒𝑤 =𝑤𝑜𝑙𝑑

𝑛 , where n is not necessary to be an integer as long as

splitting is done in a probabilistic manner so that the expected number of splits is equal to

the importance ratio.

On the other hand, when the importance ratio 𝑛 is less than 1, the RR will be used.

With the surviving probability (𝑝𝑠𝑢𝑟𝑣𝑖𝑣𝑎𝑙 ) of 𝑛 , the particle is killed with probability

𝑝𝑘𝑖𝑙𝑙 = 1 − 𝑛 . The surviving particle is kept in the simulation with a new statistical

weight 𝑤𝑛𝑒𝑤 = 𝑤𝑜𝑙𝑑/𝑛. If the regions have equal importance, neither RR nor PS is used.

However, the parameters for both PS and RR techniques are difficult to determine,

which is a matter of experience with a given type of application. For example, in the

196

EGSnrc package, couple splitting functions can be chosen and significant gains in the

photon scatter calculation efficiency are observed for Cone-Beam Computed

Tomography (CBCT) applications when the splitting parameters are optimized [7].

c. Exponential Transform

The exponential transformation of photon path lengths is a VRT designed to enhance

efficiency of deep penetration problems (e.g. shielding design) or surface problems (e.g.

build-up region in photon beams) [118], [119], [124]. Consider photon transport in a

simple slab geometry with a normally incident photon beam. By stretching or shortening

the photon path length, the technique biases the sampling procedure to give more photon

interactions in either deep or shallow regions and therefore improve the efficiency for

those applications.

To implement this method, define 𝑘 to be the distance measured in the number of

mean free paths to the next photon interaction. The corresponding scaled distance 𝑘′ is

calculated as:

𝑘′ = 𝑘(1 − 𝑐 ∗ 𝑐𝑜𝑠𝜃) (Eq. A-19)

Where 𝜃 is the angle that the photon makes with the z-axis, and 𝑐 is a stretching

parameter that adjusts the magnitude of the scaling.

Substituting Equation (A-19) into Equation (A-12), the biased cumulative probability

function 𝑃𝑐𝑛𝑒𝑤(𝑘′) and probability distribution function 𝑝𝑛𝑒𝑤(𝑘′) will be:

𝑃𝑐𝑛𝑒𝑤(𝑘′) = 1 − 𝑒−𝑘(1−𝑐∗cos𝜃) (Eq. A-20)

𝑝𝑛𝑒𝑤(𝑘′) = (1 − 𝑐 ∗ cos 𝜃)𝑒−𝑘(1−𝑐∗cos𝜃) (Eq. A-21)

197

With the purpose of sampling the stretched or shortened number of mean free paths 𝑘

to the next interaction point from selection of a random number 𝜀.

𝑘 = −𝑙𝑛𝜀/(1 − 𝑐 ∗ 𝑐𝑜𝑠𝜃) (Eq. A-22)

The average number of mean free path lengths will be:

𝑘 =∫ 𝑘′∗𝑝𝑛𝑒𝑤(𝑘)𝑑𝑘∞0

∫ 𝑝𝑛𝑒𝑤(𝑘)𝑑𝑘∞0

=1

1−𝑐∗𝑐𝑜𝑠𝜃 (Eq. A-23)

Based on Equation (A-23), since 𝑘′ is a positive number, then c must be less than one (i.e.

c < 1).

When 𝑐 = 0, we will have an unbiased probability distribution. When 𝑐 is in the

range 0 to 1, this corresponds to “path-length stretching”, meaning that the average

distance to an interaction is stretched in the forward direction, which is useful in shielding

design problems. When 𝑐 is less than zero, for forward-going photons the average

distance to the next interaction site will be shortened, and the point of interaction moved

closer to the surface of the medium which will be beneficial to applications examining the

photon beam’s build-up region. Figure A.5 shows the effect on the interaction probability

when changing the stretching parameter from -1 to ½ [124].

The sampled distance and weighting factor can even become negative when c is

below -1 (i.e. 𝑐 ≤ −1), of interest in surface dose applications. If one restricts the biasing

to the incident photons which are directed along the axis of interest (i.e. u> 0), then 𝑐 <=

−1 may be used.

198

To keep the Monte Carlo simulation unbiased, similarly to Section III-a, the expected

value of the weighting factor 𝐸(𝑤𝑛𝑒𝑤) is kept the same as before performing the VRT.

𝑝𝑛𝑒𝑤(𝑘) ∗ 𝑤𝑛𝑒𝑤 = 𝑝𝑜𝑙𝑑(𝑘) ∗ 𝑤𝑜𝑙𝑑 (Eq. A-24)

where 𝑤𝑜𝑙𝑑 is the previous weighting factor. 𝑝𝑜𝑙𝑑(𝑘) is equal to 𝑝(𝑘) in this case. Solving

Equation (A-24) for 𝑤𝑛𝑒𝑤 yields,

𝑤𝑛𝑒𝑤 =𝑒−𝑐∗𝑐𝑜𝑠𝜃

1−𝑐∗𝑐𝑜𝑠𝜃𝑤𝑜𝑙𝑑 (Eq. A-25)

Table A-1 shows the relative efficiency when calculating the dose in different depth

bins for a 7-Mev photon beam incident on a water slab. As |𝑐| increases, the efficiency

for calculating the dose near the surface improves by a factor of 3 compared with the

unbiased case, even while the computing time per history goes up by a factor of 2.

However, the efficiency for calculating the dose at deep depths gets worse at larger values

of |𝑐| because fewer photons now get to those depths [124].

The optimal choice of parameter 𝑐 is problem-dependent (e.g. 𝑐 = −6 for studies of

surface regions in dose build-up curve). In general, the stretching parameter should be

chosen carefully to prevent particles from having too large a weight, since these rare

Figure A.5 Example of a

stretched (𝑐 =1

2 ) and shortened

(𝑐 = −1 ) distribution compared

to an unbiased ( 𝑐 = 0 )

distribution. In all three cases,

𝑐𝑜𝑠𝜃 = 1. The horizontal axis is

in units of mean free paths. [124]

199

cases can occasionally lead to an increase in the variance [118]. If severe biasing is

applied, then as seen in Equation A-25, the weighting factors for the occasional photon

that penetrates very deeply can get very large. If this photon is backscattered and interacts

in the surface region where one is interested in gaining efficiency, then the calculated

variance may not be increased.

Table A-1 Relative efficiency versus the parameter C of exponential transformation

biasing for calculation of the dose at various depths in water irradiated by 7-MeV photons

[118].

Relative efficiency of calculated dose History

C 0 - 0.25 cm 6 – 7 cm 10 – 30 cm 103

0 1 1 1 100

-1 1 1 4 70

-3 1.4 1.2 0.6 55

-6 2.7 2.8 0.07 50

Interaction forcing and weight windowing techniques (PS & RR) are recommended

for use associated with the exponential transformation so that computing time can be

reduced and the unwanted increase in variance associated with large weighted particles

can be avoided [124], [126].

d. Sectioned Problems (Use of Pre-computed Results)

A ‘sectioned problem’ is one that can be split into separate, manageable parts that

can be separately simulated. In some applications, the output of Monte Carlo simulations

of different subsections of the problem can be used as input to a new section(s) of the

problem, and solved through another simulation. These applications tend to be very

specialized, but the sectioned problem approach can be very effective at improving

overall simulation efficiency [124].

200

An example of this approach is the study of the effects of photon scatter in a

radiation therapy unit. In reference [124], the simulation of a Cobalt-60 external beam

treatment unit was divided into three parts. Firstly, the radioactive Cobalt-60 source was

modeled in detail and a phase space file was generated containing energy, direction, and

position of those particles leaving the source and entering the collimator system. These

data were then used repeatedly as source particles in modelling the radiation transport

through the collimators and filters of the therapy head (different collimator settings and

filters could be studied using the same pre-generated phase-space source file). Then the

effects of photon scatter and the contaminant electrons downstream from the therapy head

were studied by separating the simulation again to just focus on the fluence at the patient

location.

By splitting the problem into three parts, the total amount of time used to simulate

the radiation transport from a 60Co therapy head to dose deposition at the patient location

was reduced by 10-100 times. Indeed some Monte Carlo radiation transport simulation

software has been specifically developed to handle ‘sectioned problems’. A good example

of this is the BEAMnrc and DOSxyznrc codes in the EGSnrc package [76], [127].

BEAMnrc is used to obtain the characteristics of the radiation unit’s treatment head (i.e. a

phase-space file of the exiting photons). DOSxyznrc is capable of transporting the

BEAMnrc resultant phase-space file(s) or using the characteristics of the resultant

treatment head fluence to calculate the patient dose downstream of the treatment head

geometry.

201

Appendix B: Sensitivity to Phantom Sampling of Analytical Modeling of Singly-Scattered Fluence into an EPID

B.1 Summary

As the dominant photon scattering source within the patient, singly-scattered

Compton fluence was evaluated based on a first principles technique which takes into

account patient heterogeneity. At a high sampling frequency, the predicted fluence results

are in good agreement with Monte Carlo simulation for a variety of phantom

configurations. Decreasing the phantom sampling frequency (i.e. increasing the voxel size)

had a small impact on predicted fluence at the EPID, up to 0.92% averaged over the EPID

surface in this heterogeneous phantom setup.

B.2 Introduction

Radiation therapy (RT) is widely applied in cancer treatment. Improved effectiveness

and accuracy of RT continues to be a significant goal. However, scattered radiation

(Figure B.1(a)) unavoidably generated in the patient will negatively impact both the KV

and MV imaging applications, resulting in image artifacts, reducing image contrast, and

also reducing accuracy of treatment delivery verification. Therefore, a fast and accurate

model to predict patient x-ray scatter is required to remove this scattered component in

both MV and KV energy ranges. At an earlier stage, our group focused on modeling the

transport of photons from source to EPID for MV x-ray dosimetry. Since the first scatter

202

fluence has been demonstrated to dominate scatter reaching the EPID over the therapeutic

range of high energy photons [41], [57], an overview of the implemented method will be

given, and the predicted fluence results will be compared to those of Monte Carlo

simulation for a variety of sampling sizes (i.e. voxel sizes) for LWRL phantom

configurations.

B.3 Methods and Materials

Voxels inside the irradiated volume were sampled as Compton interaction sites. The

scattered x-rays from each site will travel along a specific angle to each pixel within the

pixelized scoring plane at the EPID. Based on a ray-tracing algorithm [100] and the 3D

phantom/patient density map, the angle, physical distance and radiological path length

can be determined to take phantom/patient inhomogeneity into account. The probability

of interaction is found using the Klein-Nishina differential cross section, while the energy

of the scattered photon is established using Compton kinematics. The incident photon

beam energy spectrum is divided into six discrete energy bins and the entire calculation is

repeated for each bin. Integrating the calculation over all energy bins and phantom/patient

voxels will provide the total fluence. The mathematical descriptions are provided below

and illustrated in Figure B.1:

Φ(dA) = ∰∫ Φ0𝑒−𝐼(𝐸, 𝑟1 , 𝑟2 ) ∙ 𝑑𝜎(𝐸0, θ) ∙ 𝑒−𝐼(𝐸1 , 𝑟2 , 𝑟3 )𝑑𝐸𝑑𝑉

𝐸𝑚𝑎𝑥

𝐸𝑚𝑖𝑛; (B-1)

where 𝐼(𝐸, 𝑟𝑎 , 𝑟𝑏 ) = ∫𝜇

𝜌(𝐸) ∙ 𝜌(𝑟 − 𝑟𝑎 ) ∙

𝑑(𝑟 −𝑟𝑎 )

|𝑟 −𝑟𝑎 |

|𝑟𝑏 −𝑟𝑎 |

0; and 𝐸1 =

𝐸0

1+(𝐸0

0.511)∙(1−cos(θ))

;

and 𝑑𝜎(𝐸0, θ) = 𝑟0

2

2∙ (

𝐸1

𝐸0)2

∙ (𝐸0

𝐸1+

𝐸1

𝐸0− 𝑠𝑖𝑛2θ) ∙

𝑑𝐴∙𝑐𝑜𝑠θ

| 𝑟3 − 𝑟2 |2∙ 𝜌𝑒(𝑟 − 𝑟1 )𝜌(𝑟 − 𝑟1 );

203

I(E, ra , rb ) is defined as the integrated attenuation coefficient for energy E along the path

|rb − ra |; E is the incident photon energy for a particular energy bin; E1 is the scattered

photon energy; r0 is the ‘classical electron radius’, and dσ is the corrected Klein-Nishina

differential cross section (cm2) accounting for the inverse-square law and conversion to

planar fluence. In this work, calculations were performed for a phantom of (404020)

cm3, irradiated with an incident energy spectrum typical for a 6 MV photon beam, using a

field size of 1010 cm2, with the EPID scoring plane of 40 x 40 cm2 placed 30 cm

underneath the phantom. The sampling throughout the scoring plane was performed at 1

cm2 resolution. The impact on the total fluence at the EPID plane is calculated for a

variety of phantom sampling resolutions (0.2, 0.25, 0.5, 1.0, 2.5, 5 cm), and the results are

compared to Monte Carlo simulations (EGSnrc) which directly scored the first scatter

fluence under the same geometry.

B.4 Results

The analytical model simulates singly-scattered photons in a simple inhomogeneous

phantom configuration with comparison with Monte Carlo simulations. The uncertainty

estimated on the Monte Carlo scored fluence accounts for most of the observed

differences. Table B-1 shows a decreased fluence with increasing phantom sampling size

by comparing with the fluence map of 0.2 cm. A significant increase in error is observed

when sample resolution is larger than 1 cm. Reducing the sampling size leads to dramatic

simulation time increases for each decrease of resolution.

204

B.5 Conclusion

Based on a first principles technique, taking into account patient heterogeneity, this

approach can accurately predict patient-generated singly-scattered fluence entering a

portal imaging device. The output fluence can be coupled to a convolution style dose

prediction algorithm. We suggest limiting sample size to 1cm, which offer acceptable

trade-off of accuracy versus time. In future, we need to examine more complex phantom

geometries and patient CT data.

Table B-1 The average and maximum difference in singly-scattered fluence

for various phantom resolutions compared to the highest resolution.

Phantom

Resolution (cm)

Average Max Time

0.2 0 % 0 % 20.7 h

0.25 -0.003 % -0.06 % 5.4 h

0.5 -0.02 % -0.39 % 462.9 sec

1 -0.08 % -0.89 % 40.9 sec

2.5 -0.35 % -2.88 % 2.8 sec

5 -0.92 % -5.91 % 0.3 sec

205

Figure B.1 Geometry of the singly-scattered fluence entering a portal imaging device;

(b) the physics process of equation B-1& fluence map, and the validation with Monte

Carlo simulation at the phantom sampling resolution of 1 cm.

206

Appendix C: Publications and Communications

C.1 List of Publications

Guo, K., Ingleby,H., and McCurdy, B., et al. (2020) " Performance Optimization of a Tri-

Hybrid Method for estimation of patient scatter into the EPID", Physics in Medicine &

Biology (Under Review)

Guo, K., Ingleby,H., and McCurdy, B., et al. (2020) "A Tri-Hybrid Method to Estimate

the Patient-Generated Scattered Photon Fluence Components to the EPID Image

Plane", Physics in Medicine & Biology (Published on 14-September-2020)

Guo, K., Ingleby,H., and McCurdy, B., et al. (2020) " Technical note: Development and

validation of a Monte Carlo tool for analysis of patient-generated photon scatter",

Physics in Medicine & Biology (Published on 04-May-2020)

Teo P. T., Guo, K., Ahmed B. S., Pistorius S., (2019) "Evaluating a potential technique

with local optical flow vectors for automatic organ-at-risk (OAR) intrusion detection

and avoidance during radiotherapy", Physics in Medicine & Biology (Published on 1-

July-2019)

Teo, P.T., Guo, K., Pistorius S., et al. (2019), “Reducing the tracking drift of an

uncontoured tumor for a portal-image based dynamically adapted conformal

treatment”, Medical & Biological Engineering & Computing (Published on 14-May-

2019)

C.2 List of Conference Publications

Guo K., McCurdy B. (2020), “Development of Tri-Hybrid Method to Estimate Patient

Scattered Photon Fluence into the EPID Image Plane”, Joint AAPM/COMP Annual

Scientific Meeting (accepted).

207

Guo K., McCurdy B. (2019), “Hybrid Approach to Estimate Patient Scattered Energy

Fluence into a MV Imager from a Therapy Beam”, 65rd COMP Annual Scientific

Meeting, Kelowna.

Guo K., McCurdy B. (2019), “Hybrid Method for Estimating Multiply X-ray Patient

Scatter into MV imager”, International Conference on Monte Carlo Techniques for

Medical Applications, Montreal.

Guo K., McCurdy B. (2019), “Hybrid Approach to Estimate Patient Multiply Scattered

Energy Fluence into an Electronic Portal Imaging Device (EPID)”, WESCAN2.0

conference, Edmonton.

Guo K., Zhao Y., Beek, T., McCurdy B. (2018), “Analytical Modeling for the Singly-

Rayleigh-Scattered Fluence in CBCT Application”, 2018 CARO-COMP-CAMRT

Joint Scientific Meeting, Montreal.

Guo K., Ingleby H., McCurdy B. (2017), “Development and validation of a Monte Carlo

tool for analysis of patient-generated photon scatter”, COMP 63rd Annual Scientific

Meeting, Ottawa.

Guo K., Teo P. T., Wang Y., Pistorius S. (2017), “Detection and tracking of multiple

targets on portal images using feature-based learning and weighted optical flow

algorithm”, 63rd COMP Annual Scientific Meeting, Ottawa.

Teo P.T., Guo K., Bruce N., Pistorius S., et al. (2016), “Incorporating tracking and

prediction of tumor motion in a motion-compensating system for adaptive

radiotherapy”, IEEE Imaging Systems & Techniques Proceedings, Chania, Greece,

pp.77

Guo K., Teo P.T., Kawalec, P., and Pistorius S., (2016), “Poster - 51: A tumor motion-

208

compensating system with tracking and prediction – a proof-of-concept study”. 62nd

COMP Annual Scientific meeting, St. John's, Canada. Medical Physics, 43(8), 4948-

4949

Guo K., Beek T., McCurdy B. (2016), “Sensitivity to Phantom Sampling of Analytical

Modeling of Singly-Scattered Fluence into an EPID”. Electronic Patient Imaging

(EPI) Conference, St. Louis.

Teo P. T., Guo K., Pistorius S., et al. (2016), “Detection and tracking of intrusions at the

edges of a treatment field using local Optical Flow”. Electronic Patient Imaging (EPI)

Conference, St. Louis.

Teo P. T., Guo K., Pistorius S., et al. (2015). “SU-E-J-58: Comparison of Conformal

Tracking Methods Using Initial, Adaptive and Preceding Image Frames for Image

Registration”. 57th AAPM preceding, Med. Phys. 42(6): 3277

Teo P. T., Guo K., Pistorius S., et al. (2015), “Drift correction techniques in the tracking

of lung tumor motion”, World Congress on Biomedical Engineering and Medical

Physics, Toronto, Canada.

209

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