Developing a tri-hybrid algorithm to predict patient x-ray ...
Transcript of 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
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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
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
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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 5.5 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). ........ 145
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
Table 5-2 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 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
Table 5-3 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 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
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.
18
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
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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.
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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].
81
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
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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.
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 (%) -0.01 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.
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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
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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],
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[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].
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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 Methods and Materials
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
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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)
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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
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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)
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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.
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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.
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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
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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)
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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.
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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
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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.
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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.
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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%.
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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.
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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.
Phantom Water LWRL Thorax
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
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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.
Phantom Water LWRL Thorax
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.
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(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.
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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)
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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].
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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
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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
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 Off
Atomic relaxations Off
Electron impact ionization Off
Photon cross sections XCOM
Photon cross-sections output 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.
5.2 Methods and Materials
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)
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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
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Table 5-2 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 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.
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 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
Table 5-3 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 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.
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 -1.86% -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)
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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
164
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
166
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
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 Off
Atomic relaxations Off
Electron impact ionization Off
Photon cross sections XCOM
Photon cross-sections output Off
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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.
170
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
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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
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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).
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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.
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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
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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)
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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.
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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
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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
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