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Physics and Computational Methods for X-ray Scatter Estimation and Correction in Cone-Beam Computed
Tomography
by
Gregory J. Bootsma
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Medical BiophysicsUniversity of Toronto
© Copyright by Gregory Bootsma 2013
Physics and Computational Methods for X-ray Scatter
Correction in Cone-Beam Computed Tomography
Gregory J. Bootsma
Doctor of Philosophy
Medical BiophysicsUniversity of Toronto
2013
Abstract
X-ray scatter in cone-beam computed tomography (CBCT) is known to reduce image quality by
introducing image artifacts, reducing contrast, and limiting computed tomography (CT) number
accuracy. The extent of the effect of x-ray scatter on CBCT image quality is determined by the
shape and magnitude of the scatter distribution in the projections. A method to allay the effects
of scatter is imperative to enable application of CBCT to solve a wider domain of clinical
problems. The work contained herein proposes such a method.
A characterization of the scatter distribution through the use of a validated Monte Carlo (MC)
model is carried out. The effects of imaging parameters and compensators on the scatter
distribution are investigated. The spectral frequency components of the scatter distribution in
CBCT projection sets are analyzed using Fourier analysis and found to reside predominately in
the low frequency domain. The exact frequency extents of the scatter distribution are explored
for different imaging configurations and patient geometries.
Based on the Fourier analysis it is hypothesized the scatter distribution can be represented by a
finite sum of sine and cosine functions. The fitting of MC scatter distribution estimates enables
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the reduction of the MC computation time by diminishing the number of photon tracks required
by over three orders of magnitude.
The fitting method is incorporated into a novel scatter correction method using an algorithm that
simultaneously combines multiple MC scatter simulations. Running concurrent MC simulations
while simultaneously fitting the results allows for the physical accuracy and flexibility of MC
methods to be maintained while enhancing the overall efficiency. CBCT projection set scatter
estimates, using the algorithm, are computed on the order of 1-2 minutes instead of hours or
days. Resulting scatter corrected reconstructions show a reduction in artifacts and improvement
in tissue contrast and voxel value accuracy.
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Dedicated to those who have suffered and are suffering from mental illness.
In loving memory of Elliott Glass (1976-2011).
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Acknowledgments
Many people have made this work possible both on a scientific and personal level and I would
like to thank you all for your time and support. Specifically, many thanks to my supervisor, Dr.
David Jaffray for providing me with the opportunity to work on this challenging and complex
problem. Without his guidance and support the discoveries and advancements created herein
would not exist. My sincere appreciation goes to my unofficial co-supervisor Dr. Frank
Verhaegan for all his assistance in using the EGS Monte Carlo code as well as his invaluable
feedback on journal and conference submissions. Thank you to my supervisory committee
members, Dr. Cynthia Ménard and Dr. Mike Rauth for their participation in this learning
process, you both always made me feel at ease and left your doors open to answer any questions
and concerns I may have had.
To all my fellow students and lab members, both past and present, my thanks goes to you for
making the lab an enriching and friendly work environment. I am much obliged to Drs. Noor
Mail, Douglas Moseley, and Jeff Siewerdsen for sharing their experience and knowledge with a
young scientist. Fanny Sie, thank you for your friendship, lunches and dish sessions. Steve
Bartolac, Mike Daly, Sam Richard, Sami Siddique, Nick Shkumat, Shawn Stapleton, and James
Stewart, here's to friendship, conferences, drinks, and getting to the other side of being a grad
student. For those in there who haven't reached the light at the end of that seemingly endless
tunnel: You will make it, just keep on keeping on!
A special thanks goes to my close friends for always being there for me, even after the cops show
up: Marty Rozee, Thomas Flood, Sandro Camilli, Mark Bergshoeff, Matt Bergshoeff, and Pat
Rijd (who garners special thanks for the daunting task of actually reading, editing, and making
suggestions on this thesis).
To my parents, brother, and sister I am forever grateful for all the love and support over the
years. George and Elizabeth, I could not have asked for a kinder and more supportive mother and
father-in-law. Last but not least, my gratitude goes to my beautiful ladies Magdalene and Kleo
who were always there for me, I love you both so much! Thanks for making me laugh, smile,
cry, and strive to be an all-around better person.
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Table of Contents
Abstract..............................................................................................................................ii
Acknowledgments .............................................................................................................v
List of Tables......................................................................................................................x
List of Figures....................................................................................................................xi
List of Abbreviations and Symbols..................................................................................xix
Chapter 1 Introduction ..................................................................................................................1
1 Motivation .......................................................................................................................1
2 Background.....................................................................................................................2
2.1 X-ray Interactions in Medical Imaging.....................................................................2
2.2 Effects of Scattering on X-ray Imaging....................................................................7
2.3 Computed Tomography...........................................................................................8
2.4 Cone-Beam Computed Tomography......................................................................9
3 Outline of Thesis...........................................................................................................11
3.1 Hypothesis.............................................................................................................12
3.2 Specific Aims.........................................................................................................12
3.3 Publication Status..................................................................................................14
Chapter 2 The Effects of Compensator and Imaging Geometry on the Distribution of X-ray Scatter in CBCT ..............................................................15
1 Introduction...................................................................................................................15
2 Methods and Materials..................................................................................................17
2.1 Monte Carlo Simulation System............................................................................17
2.2 Cone-Beam CT Imaging System...........................................................................19
2.3 Validation of Monte Carlo Model...........................................................................20
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2.3.1 Open-Field Detector Response Validation..................................................20
2.3.2 Object Projection Validation.........................................................................21
2.3.3 Scatter Signal Validation..............................................................................21
2.4 MC Scatter Distribution Simulations......................................................................25
2.4.1 Water Cylinder Scatter Distribution..............................................................25
2.4.2 Bowtie Filtration...........................................................................................26
3 Results..........................................................................................................................27
3.1 Monte Carlo Validation..........................................................................................28
3.1.1 Open-field Detector Response Validation...................................................28
3.1.2 Object Projection Validation.........................................................................30
3.1.3 Scatter Component Validation.....................................................................30
3.2 MC Scatter Distribution Simulations......................................................................33
4 Discussion and Conclusions.........................................................................................40
Chapter 3 The Spectrum of the X-ray Scatter Distributionin CBCT Projection Images.........................................................................................45
1 Introduction...................................................................................................................45
2 Materials and Methods..................................................................................................46
2.1 Monte Carlo Simulation System............................................................................46
2.1.1 X-ray Sources and Energy ..........................................................................46
2.1.2 Compensators..............................................................................................46
2.1.3 Phantoms.....................................................................................................47
2.1.4 Imaging Geometry.......................................................................................47
2.2 Scatter Spatial Frequency.....................................................................................49
2.3 Scatter Distribution Estimation from Limited Photon Simulations........................49
3 Results and Discussion.................................................................................................51
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3.1 Scatter Spatial Frequency Spectrum.....................................................................51
3.1.1 Cylinder........................................................................................................51
3.1.2 Anthropomorphic Phantoms........................................................................53
3.2 Scatter Distribution Estimation using Limited Photons.........................................59
4 Conclusions...................................................................................................................61
Chapter 4 Efficient Scatter Distribution Estimation and Correction in CBCT usingConcurrent Monte Carlo Fitting...................................................................................65
1 Introduction...................................................................................................................65
2 Materials and Methods..................................................................................................67
2.1 Concurrent Monte Carlo Fitting ............................................................................67
2.1.1 Monte Carlo Simulation Systems.................................................................68
2.1.2 Shared Memory...........................................................................................68
2.1.3 Concurrent Scatter Fitting............................................................................69
2.2 Scatter Correction .................................................................................................71
2.2.1 Concurrent Monte Carlo Fitting Scatter Correction.....................................71
2.2.2 Constant Scatter Correction........................................................................71
2.3 Experiments...........................................................................................................72
2.3.1 Simulated Phantom Data.............................................................................72
2.3.2 Measured Phantom and Patient Data..........................................................72
2.3.3 Concurrent Monte Carlo Fitting Parameters................................................73
2.3.4 Scatter Estimate Error..................................................................................75
2.3.5 Image Quality Metrics..................................................................................75
3 Results and Discussion.................................................................................................76
3.1 Simulated Phantoms.............................................................................................76
3.1.1 Scatter Estimate Error .................................................................................76
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3.1.2 Goodness of Fit Metrics...............................................................................79
3.1.3 Reconstruction Image Quality......................................................................79
3.2 Measured Pelvis Phantom and Patient ................................................................83
3.2.1 Pelvis Phantom............................................................................................83
3.2.2 Pelvis Patient .............................................................................................85
3.3 Efficiency ..............................................................................................................86
4 Conclusion....................................................................................................................87
Chapter 5 Retrospective and Prospective Deliberations and Contemplations............................89
1 Introduction...................................................................................................................89
2 Demonstrating Clinical Value........................................................................................90
3 Potential for Clinical Implementation............................................................................90
4 Improvements in Efficiency...........................................................................................91
5 Scatter Estimate Accuracy............................................................................................93
6 Noise Reduction Techniques........................................................................................94
7 Final Considerations.....................................................................................................95
References.......................................................................................................................98
Appendix A : Bowtie Filter Modeling..............................................................................104
Appendix B : Fourier Interpolation.................................................................................106
Copyright Acknowledgments.........................................................................................108
ix
List of Tables
Table 1: Weight fractions for elements composing water, soft tissue, and bone used in
computing the mass attenuation coefficients. The soft tissue and compact bone values come
from the International Commission on Radiation Units & Measurements (ICRU). .......................6
Table 2: Source and detector configurations used in simulations...............................................26
Table 3: Fitting parameters and associated coefficient of determination, R2, value for SPR and
SOCR data fit to Eq. (34) for the 16.4 and 30.6 cm diameter cylinders......................................34
Table 4: Spatial frequency width (SFW) values (in cm-1) along the horizontal and vertical (u,v)
frequency directions for the 30.6 cm diameter cylinder for various ADD, compensator, and
detector configurations. .............................................................................................................52
Table 5: SFW values for the pelvis and head phantom with and without the use of the AL16S. A
decrease in the horizontal frequencies is seen for both phantoms when a compensator is
employed...................................................................................................................................59
Table 6: The optimal ucut and vcut values and corresponding RMSE. The RMSE for using no filter
and using a filter with cutoffs selected from the SFW values are also shown for the case with the
F1 and AL16S compensators and without the use of a compensator. The error reduction for
using the optimal filter cutoffs is also presented.........................................................................59
Table 7: Spatial and angular frequency cutoffs values used in the Butterworth low-pass filter for
simulated phantoms and measured pelvis patient and phantom data........................................73
Table 8: Configuration parameters for CMCF algorithm for the simulated and measured
projection sets............................................................................................................................75
Table 9: Correlation between the two GOF metrics (Pearson correlation, r, and coefficient of
determination, R2) and scatter estimate error, Serr, computed for the interpolated scatter, SI, for
each of the simulated phantom configurations...........................................................................79
Table 10: Image quality metrics for the uncorrected, corrected, and primary only reconstructions
of the four imaging configurations. The time to compute the CMCF scatter estimate is also
shown for each phantom configuration......................................................................................82
x
List of Figures
Figure 1: Photon interactions with an atom relevant to x-ray imaging: (a) photoelectric
absorption, (b) incoherent (Compton) scattering, and (c) coherent (Rayleigh) scattering. The
photons initial energy, E0, is given by hν, where h is Planck's constant and ν is the photon's
frequency. In (a), photoelectric absorption, the photon's energy is absorbed in ejecting a
photoelectron, with the resulting photoelectron's energy, Ee, being equal to the photon's initial
energy, hν, minus the electron binding energy, Eb. In (b), incoherent scattering, the photon is
scattered with a change of energy specified as hν'. The recoil electron's energy is given as
Ee=hν-hν'. In (c), coherent scattering, the photon is scattered without a loss of energy...............3
Figure 2: The contribution percentage for each of the three interactions relevant to x-rays in
medical imaging for (a) water, (b) soft tissue, and (c) compact bone. The attenuation coefficient,
μ, as a function of energy is shown in (d) for each material. .......................................................6
Figure 3: Illustration showing photon intensities at a detector for the case with (right) and without
(left) scatter. Primary photons are indicated with solid lines and scattered photons are indicated
with dashed lines. The scattered photons add additional signal to areas of the detector causing
the attenuation of the primary beam to be underestimated estimated at those locations.............8
Figure 4: Scatter (green lines) and primary (red lines) x-ray contribution to the imaging signal in
a single detector row for a projection image of a water cylinder using a small (a) and large (b)
field of view (FOV). The x-ray coverage for the small and large FOV are indicated with
semitransparent yellow coloring and have approximately 5 cm and 20 cm coverage in the
vertical direction at a distance of 100 cm from the source, respectively. The increase in scatter
contribution to the horizontal line in the detector comes mainly from scatter photons outside the
plane of the primary photons contributing to the image signal....................................................10
Figure 5: CBCT reconstructions of a pelvis patient using three different field-of-views (FOV) with
dimensions of FOVs indicated on (a) coronal slice of pelvis patient. 2 cm FOV indicated with
light blue dashed lines, 10 cm FOV with white dashed lines, and top and bottom of image
indicate full FOV of ~26 cm. Axial slices from reconstruction using: (a) 2 cm FOV, (b) 10 cm
FOV, and (c) 26 cm FOV projections. A decrease in the image quality is clearly demonstrated
as the FOV increases.................................................................................................................11
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Figure 6: System diagram of the cone-beam CT Monte Carlo (CBCT MC) model. The system
consists of three main components each used to model a portion of the actual CBCT system.
(1) consists of a model of the x-ray tube that generates a phase space file containing the output
photons position, direction, energy, and statistical weights using BEAMnrc. The phase space
from (1) is then used as input to (2) a voxelized model of the compensator which results in a
phase space file. The phase space from (2) is then used as input to (3) a voxelized model
simulating the geometric configuration of the source and detector, the object to be imaged, and
rotation angle (θ) using modified DOSXYZnrc code which keeps track of primary and scattered
photons. The simulation in (3) generates another phase space file at the detector plane which is
input to (4) which models the detector response using a look up table based on previous MC
simulations. ...............................................................................................................................17
Figure 7: Top and side view diagram of CBCT bench-top system. The CBCT geometric
terminology and variables are illustrated: cylinder diameter (dcyl), source-to-axis distance (SAD),
axis-to-detector distance (ADD), source-to-detector distance (SDD), air gap (xgap), cone angle
(cone), fan angle (fan), and angle of rotation (θ). ........................................................................19
Figure 8: (a,b) Scatter measurement devices and (c) experimental setup on CBCT bench-top
system showing lead line beam stop. (a) Schematic front and side view illustration of lead disc
beam stop. The height, hdisc, and width, wdisc, of the acrylic plates into which the lead discs
inserted was 50 mm. The diameter of the lead discs, ddisc, were 5, 10, 15, and 20 mm. The
thickness of the disc and the plate, tdisc, was 4.7mm. (b) Schematic of lead bar beam stop held in
an acrylic plate. The dimensions of the beam stop are hplate=245 mm, wplate=225 mm, tplate=4 mm,
hline=tline=3 mm............................................................................................................................24
Figure 9: XZ profiles of the simulated F1 (a) and custom (b) bowtie filters. The thickness of the
filters in the y dimension was 8.5 cm. Equations (49), (50), (53), and (54) from Appendix A were
used to generate the surface curvature of the bowtie filters.......................................................27
Figure 10: Rows 1 and 2 show 2D images of the scatter-to-open-field center ratio (SOCR) for
the 16.4 and 30.6 cm diameter water cylinders at three different ADD settings of 18, 30, and 56
cm all with an SAD of 100 cm and a cone angle of 11.3°. Row 3 shows the same configuration
as row 2 except with the F1 bowtie filter in place. The images represent a detector size of
120(w) × 60(h) cm. The decrease in the scatter with increasing air gap is clearly evident for both
cylinders. A large degree of symmetry can also be seen in all the SOCR distributions.
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Additionally the effect of the bowtie filter on the scatter distribution shows not only a decrease in
the amount of scatter but also a flattening out of the scatter distribution....................................28
Figure 11: Open-Field validation for horizontal (a) and vertical (b) profiles of the measured and
simulated open-field projection images normalized by the center pixel value. The measured
open-field projection profiles have been corrected using the maximum normalized fixed input
fluence (FIF) profile shown in (e) and (f). The need for normalizing by the FIF is clearly seen by
the non-linear response across the vertical portion of the detector (f). The local percent
discrepancy (LPD) between the measured and simulated profiles is plotted in (c) and (d).........29
Figure 12: Cylindrical 16.4 cm diameter water phantom validation; (a) horizontal profile of a
open-field normalized x-ray projection taken with a SAD of 100 cm and ADD of 56 cm for
simulated and measured data. (b) The local percent discrepancy (LPD) between the measured
and simulated horizontal profiles................................................................................................30
Figure 13: Scatter signal validation for 16.4 cm diameter water cylinder using (a) scatter-to-
open-field ratio (SOR) and (b) scatter-to-primary ratio (SPR) for measured and MC simulated
data generated with a SAD of 100 cm and ADD of 56 cm. The measured data was calculated
using the 15 mm lead disc beam stop device.............................................................................31
Figure 14: SOR and SPR for measured and simulated data of a 16.4 cm diameter cylinder
imaged with an ADD of 30 (a,c) and 56 cm (b,d), both with a SAD of 100 cm. The measured
data (crosses) was estimated using the lead strip beam stop device. The simulated scatter data
(triangles) is found by keeping track of photons that scatter inside the object during the CBCT
MC simulation, whereas the simulated beam stop data (squares) comes from simulating the
lead strip beam stop approach using the CBCT MC system......................................................32
Figure 15: The percent of contaminating photons, Xu, in the open-field signal for a source-to-
detector distance (SDD) of 156 cm. The “Measured Pb Line” and “Simulated Pb Line” profiles
are for experimentally measured and MC simulated estimates using the lead bar beam stop,
respectively. The “Simulated” data is determined by separating photons originating from the
focal spot from those outside the focal spot region with a tolerance of +/-0.1 mm.....................33
Figure 16: Horizontal profiles of SOCR for varying ADD configurations (1-5 in Table 2 on p. 26),
for the 16.4 cm (a, c, e, and g) and 30.6 cm (b, d, f, and h) diameter cylinders at each of the four
cone angle configurations (1.4, 2.8, 5.7, and 11.3°, corresponding to a field of view at 100 cm
from the source of 2.4, 4.9, 10.0, and 19.8 cm, respectively).....................................................35
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Figure 17: Horizontal profiles of scatter distributions for varying SAD configurations (5-7 inTable
2 on p. 26), for the 16.4 cm (a, c, e, and g) and 30.6 cm (b, d, f, and h) diameter cylinders at
each of the four cone angle configurations (1.4, 2.8, 5.7, and 11.3°, corresponding to a field of
view at 100 cm from the source of 2.4, 4.9, 10.0, and 19.8 cm, respectively)............................36
Figure 18: Contour plots of fit of SPR (a,b) and SOCR (c,d) center pixel values using Eq. (17)
from the 16.4 and 30.6 cm diameter water cylinder MC simulations. The contours lines are
spaced equally with (a) 0.25, (b) 1, (c) 0.01, and (d) 0.002 increments. The fitting parameters
and resulting R2 values are found in Table 3. Each plot also displays three data points (+) drawn
(randomly) from the MC simulations to illustrate goodness of fit................................................37
Figure 19: Horizontal profiles of the SOCR signal for the 16.4 (a, c, e) and 30.6 cm (b, d, f)
diameter water cylinders broken into different interaction contributions for a geometric
configuration with a SAD of 100 cm, an ADD of 56 cm, and a cone angle of 11.3°; without any
bowtie filtration (a, b), with the F1 bowtie (c, d) and with the custom bowtie (e, f). The percent
decrease for the total SOCR signal at the center for the 16.4 cm cylinder was 6 and 34% for the
F1 and custom bowtie, respectively; for the 30.6 cm cylinder the percent decrease was 19 and
55% for the F1 and custom bowtie, respectively. In all cases except (c) using a bowtie filter
results not only in a reduction in the magnitude of the scatter but also in a reduction to the
structure in the SOCR profile predominately caused by the coherent scatter............................38
Figure 20: The the horizontal SPR signal profile measured at the center of the detector for the
16.4 (a) and 30.6 cm (b) diameter water cylinders with and without bowtie filtration. The percent
decrease for the 16.4 cm diameter cylinder when using the F1 and custom bowtie filters
measured at the center was 11 and 30%, respectively. The percent decrease for the 30.6 cm
diameter cylinder when using the F1 and custom bowtie filters measured at the center was 27
and 56%, respectively................................................................................................................39
Figure 21: The percentage of contamination photons, Xu, in the open-field signal with and
without the bowtie filter in place measured at a SDD of 156 cm. The contribution of the Xu signal
significantly increased as a percentage of the total open-field signal when either of the bowtie
filters are in place.......................................................................................................................40
Figure 22: Profile for compensator, AL16S, designed to compensate for a 16.4 cm diameter
cylinder. AL16S is composed of aluminum with a center thickness of 1 mm and a modulation
factor of 7.9................................................................................................................................46
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Figure 23: Axial (a) and sagittal (b) slices showing density values for voxelized head phantom
used in the MC simulations........................................................................................................47
Figure 24: (a) Axial and (B) coronal slices of the density values for the voxelized pelvis phantom
used in the MC simulations........................................................................................................48
Figure 25: (a-d) The normalized detector scatter distribution, Sn, and (e-h) the corresponding
logarithm of the spatial frequency, FS, for the 30.6 cm diameter water cylinder at ADD values of
18, 30, 44, and 56 cm................................................................................................................51
Figure 26: The normalized scatter distribution (a-c) and the corresponding logarithm of the FS
(d-f) for different bowtie filter implementations: (a,d) none, (b,e) F1, (c,f) AL16S.......................53
Figure 27: (a) Horizontal profiles along u axis (v=0) and (b) vertical profiles along v axis (u=0) for
the spatial frequencies of Sn for the 30.6 cm diameter water cylinder with different compensator
configurations (none, F1, and AL16S)........................................................................................53
Figure 28: Scatter distribution projections, Sn, for frontal views (θ=0°) of the pelvis (a,c) and
head (b,d) phantom. Images (a) and (b) are without the use of a compensator, whereas images
(c) and (d) are with a compensator. An increase in the signal intensity of Sn can clearly be seen
at the edges of the pelvis (a) and head (b) phantom when a compensator is not used due to the
increased coherent scattering contribution, when a compensator is used [(c)and (d)] these edge
effects are significantly diminished.............................................................................................54
Figure 29: Scatter sinograms for the center row (a,c) and center column (b,d) of Sn for the pelvis
phantom. The first row of images (a,b) is without a compensator and the second row (c,d) is
with the AL16S compensator. Periodic signals can clearly be seen in the angular direction due
to the ellipsoidal shape of the pelvis phantom. ..........................................................................55
Figure 30: Scatter sinograms for the center row (a,c) and column (b,d) of Sn for the head
phantom. The first row of images (a,b) is without a compensator and the second row (c,d) is
with the AL16S compensator.....................................................................................................56
Figure 31: (a) Central horizontal profile, (b) central vertical profile, and (c) central angular profile
of Sn for both head and pelvis phantoms with (dashed lines) and without (solid lines) the use of
the AL16S compensator.............................................................................................................57
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Figure 32: Logarithm images of FS for the pelvis phantom with (a-c) and without (d-f) the use of
the AL16S compensator for the three central planes (u-v, v-, and u-). A strong off axis signal
with a slope of -1 cm/turn is seen in the image of the u-ω plane shown (c) and (f), resulting from
the rotationally variant shape of the phantom.............................................................................58
Figure 33: Logarithm images of FS for the head phantom with (a-c) and without (d-f) the use of
the AL16S compensator for the three central planes (u-v, v-ω, and u-ω). Similar to the pelvis
phantom an off axis frequency component is seen in the u-ω plane shown in (c) and (f)...........58
Figure 34: (a) Contour plot of the resulting RMSE values between the gold standard and the low
pass filtered limited photon simulation (LPS) Sn signals for the 30.6 cm diameter water cylinder
with no compensator for a range of ucut and vcut values. The optimal cutoff values are found when
ucut and vcut are 0.05 and 0.045 cm-1, respectively resulting in an RMSE value of 6.1. The optimal
value is marked with a '+' on the contour plot. (b) The resulting shape of the optimal low pass
Butterworth filter in the frequency domain..................................................................................60
Figure 35: LPS Sn projection for 30.6 cm diameter water cylinder (no compensator) (a) without
and (b) with low pass filtering. The low pass filter cutoffs used in (b) are 0.065 and 0.045 cm-1
for u and v, respectively. (c) Gold standard scatter simulation Sn result. (d) The percent absolute
error between the filtered and gold standard Sn signal. (e) The central horizontal profile of the
gold standard, LPS, and filtered LPS Sn signals.........................................................................61
Figure 36: Top row shows 0° Sn projection for the pelvis phantom for the limited photon
simulation (LPS) using (a) 106 photons, (b) low-pass filtered LPS (using optimal cutoff values),
and (c) the gold standard (> 109 photons) Sn data. The LPS Sn signal uses an angular sampling
rate of 1°. The second row shows the same data but in the form of a sinogram composed of the
center horizontal row of Sn at each projection angle, θ...............................................................62
Figure 37: RMSE as a function of the angular sampling rate (ASR) for each of the four phantom
imaging conditions. The results using the SFW and optimal low-pass filter cutoff values are
shown as dashed lines with squares (□) and solid lines with crosses (+), respectively..............63
Figure 38: Optimal low-pass filter cutoff values, (a) ucut, (b) vcut, and (c) ωcut, for the different
angular sampling rates used in each of the four phantom imaging configurations. Two outliers at
dθ=72° and 90° were removed from the ωcut data for the pelvis phantom with the AL16S
compensator. The optimization for these two points resulted in a selection of the highest value
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of ωcut searched (35.28 turns-1) indicating that no filtering in the angular direction is optimal for
these cases. ..............................................................................................................................64
Figure 39: System diagram of the components involved in the concurrent MC fitting (CMCF)
algorithm. The two main systems: (1) MC simulation and (2) concurrent scatter fitting. These
two systems communicate through a shared memory space. The MC simulation system is made
up of a MC simulation thread manager which launches and manages NS MC simulation threads.
The shared memory consists of set of particle buffers which store the particles being generated
from the NS MC simulations. The concurrent scatter fitting consists of an analysis manager
which has sub-components responsible for reading particles from the buffer, creating a detector
response signal from the particles, fitting the detector response, and evaluating the goodness of
fit. ..............................................................................................................................................67
Figure 40: Simplified flow diagram for the CMCF algorithm showing the processing of photons
through the scatter fitting system. The end result is a interpolated estimate of the scatter
distribution, SI, using the scatter distribution fit, SF, of the Monte Carlo scatter data, SMC...........69
Figure 41: The scatter estimate error, Serr, for the raw Monte Carlo, SMC, (dashed lines) and the
low-pass Fourier fit, SF, (solid lines) scatter estimates as a function of computation time for the
two phantoms (head and pelvis) with and without a bowtie (BT)................................................77
Figure 42: The Serr for the interpolated scatter data, SI, for each of the of the phantom
configurations plotted as a function of (a) computation time and (b) the Pearson correlation
coefficient, r................................................................................................................................78
Figure 43: The scatter signal for the original data used in the fitting model, the interpolated fit,
and the gold standard simulation for the pelvis phantom with the AL16S compensator. The
original and fit data being shown is after 53.2 secs of run time. The fit has a Pearson correlation
coefficient of 0.61 and a Serr value of 0.07. The top row shows the scatter signal for the detector
position at the 0° projection angle. The bottom row shows the scatter sinogram for the horizontal
row at center of the detector at each angular position, θ............................................................78
Figure 44: CBCT reconstruction of the pelvis (top 2 rows) and head (bottom 2 rows) using
primary only (column 1), primary and scatter (column 2), and primary and scatter corrected
using CMCF algorithm. The CMCF corrections are shown for a GOF of r≈0.6. The time to
compute the scatter estimate was under 2 minutes for all four scenarios..................................80
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Figure 45: Voxel values plotted in arbitrary units (A.U.) for horizontal profiles in an axial slice of
reconstructions of the head and pelvis phantoms with and without the use of a compensator
from primary only, uncorrected, and corrected projection images. The profile locations are
shown in Figure 44.....................................................................................................................83
Figure 46: Axial slices from reconstructions of the anthropomorphic phantom taken on the
Elekta Synergy for the 26 cm FOV (a) uncorrected , (b) with CMCF scatter correction, (c)
constant scatter correction, and (d) the 2 cm FOV. The display window values were [0.09,0.3]
for all images. The horizontal and vertical profile lines plotted in Figure 47 are shown in (d).....85
Figure 47: (a) Horizontal and (b) vertical profiles through axial slices in reconstructions of the
anthropomorphic phantom showing voxel intensity differences for reconstructing with a 2 cm
FOV, and 25 cm FOV with no scatter correction, CMCF scatter correction, and constant scatter
correction...................................................................................................................................85
Figure 48: Axial slices from reconstructions of the pelvis patient data: (a) 26 cm FOV
uncorrected, display window: [0.12,0.23], (b) 26 cm FOV CMCF scatter corrected, display
window: [0.11,0.30] (c) 26 cm FOV constant scatter correction, display window: [0.13,0.25], and
(d) 2 cm FOV uncorrected, display window: [0.11,0.30]. The vertical and horizontal profile
locations for Figure 49 are shown as dashed lines in (d)...........................................................86
Figure 49: (a) Horizontal and (b) vertical profiles through axial slices in reconstructions of the
pelvis patient showing voxel intensity differences for reconstructing with a 2 cm FOV, and 25 cm
FOV with no scatter correction, CMCF scatter correction, and constant scatter correction........87
xviii
List of Abbreviations and Symbols
β User specified threshold for lowest signal after scatter correction
Δp User specified value for CMCF algorithm indicating number of new photons per
pixel required before a new fit is generated
ξ Scatter factor
θ Projection angle or rotation angle
λ Wavelength
μ Attenuation coefficient
μE Attenuation coefficient for x-rays with an energy of E of an absorbing medium
ν Frequency
ρ Density
σ Standard deviation
eσ Scattering cross section per electron
σC Compton (incoherent) scattering coefficient
σR Rayleigh (coherent) scattering coefficient
τ Attenuation coefficient for the photoelectric effect
φ Photon scattering angle
cone Cone angle
fan Fan angle
ωcut Angular cutoff frequency used in a low-pass filter
Ω Solid angle
A Molecular weight (grams per mole)
ADD Axis-to-detector distance
xix
A.U. Arbitrary units
BEAMnrc A Monte Carlo simulation system built on EGSnrc for modelling radiotherapy
sources
c Speed of light (299792.4 km/s)
C++ Programming language developed by Bjarne Stroustrup
CBCT Cone-beam computed tomography
CMCF Concurrent Monte Carlo fitting
CNR Contrast-to-noise ratio
CPU Central processing unit
CT Computed Tomography
dcyl Cylinder diameter
DC Direct current, the DC signal/value/term in a Fourier transform is the zero
frequency term (constant),
DBS Directional bremsstrahlung splitting
DOSXYZnrc A Monte Carlo code based on EGSnrc for calculating dose distributions in
voxelized phantoms
E Energy
E0 Initial energy
ECUT Electron cutoff energy used in any EGSnrc based Monte Carlo code
EGS Electron gamma shower
EGSnrc A Monte Carlo simulation system which extending EGS4 and is maintained by
the National Research Council of Canada (NRC)
F Fourier transform
FS Frequency spectrum of the scatter distribution, generated using the Fourier
xx
transform
FIF Fixed input fluence
FFT Fast Fourier transform
FFTW Open source library for computing the discrete Fourier transform
g User specified goodness of fit value
GEANT4 A Monte Carlo simulation toolkit for the passage of particles through matter
GOF Goodness of fit
GPU Graphics processing unit
h Panck's constant (4.135668 ×10-15 eV s)
HVL Half-value layer
I Transmitted intensities of an x-ray beam measured with a detector
I0 Input x-ray intensities of an x-ray beam measured with a detector
Iblock Detector signal in the shadow of the beam block with no object present
Icalib Calibrated and normalized projection image
Ifif Maximum normalized detector response to the fixed input fluence
Iopen Detector response to open-field signal
Iobj Detector signal of x-rays transmitted through an object
Iobj+block Detector signal in the shadow of the beam block with the object present
I0open Value of the open-field signal at the center pixel of the detector
I'open Iopen normalized by fixed input fluence signal, Ifif
Ip Projection image representing line integrals of the attenuation coefficients
Iprim Measured intensities resulting from primary x-rays at the detector
xxi
Iscat Measured intensities resulting from scattered x-rays at the detector
ICRU International Commission on Radiation Units & Measurements
Kair Air kerma
kerma Kinectic energy released per unit mass
K-N Klein-Nishina
LPD Local percent discrepancy
LPS Limited photon simulation
m0 Electron rest mass (510.9989 keV/c2)
Mθ Number of zeros required to pad Fourier transform for a given Fourier
interpolation
MC Monte Carlo
MTF Modulation transfer function
N Number of output x-rays
N0 Number of input x-rays
NA Avogadro's constant (6.022 × 1023 mole-1)
NB The order of the Butterworth filter
Np Maximum number of photons per projection
NIST National Institute of Standards and Technology
NRC National Research Council of Canada
P Projection set
Pcorr Scatter corrected projection set
Pobj Primary signal, Pu, transmitted through the object
Pu Focal component of the bremsstrahlung x-ray source
xxii
P'u Off-focal component of the bremsstrahlung x-ray source
PBS Portable batch system
PCUT Photon cutoff energy used in any EGSnrc based Monte Carlo code
PEGS4 Program for generating coss section data used by EGSnrc
PHSP Phase space
PMMA Poly(methyle methacrylate), often referred to as acrylic glass
r Pearson correlation coefficient
R Radon transform
r0 Classical electron radius
R2 Coefficient of determination
RMSE Root-mean-square error
S Scatter distribution created by the imaged object
Sblock Portion of scatter signal lost due to using a beam blocker
Serr Scatter estimate error metric
SF Fit of the scatter distribution
SI Interpolation of a fit of the scatter distribution
SMC Scatter distribution detector signal produced by MC simulation
Sn Scatter detector signal normalized by the open-field center pixel value and
multiplied by 104
Su Secondary source of scattered x-rays
SAD Source-to-axis distance
SDD Source-to-detector distance
SFW Scatter frequency width
xxiii
SFWu Scatter frequency width for horizontal frequencies
SFWv Scatter frequency width for vertical frequencies
SFWω Scatter frequency width for angular frequencies
SOR Scatter-to-open-field ratio
SOCR Scatter-to-open-field center pixel ratio
SPR Scatter-to-primary ratio
t thickness
tcup A metric measuring the “cupping” artifact in a reconstruction
ucut Horizontal cutoff frequency used in a low-pass filter
V Reconstructed volume
vcut Vertical cutoff frequency used in a low-pass filter
Verr Reconstructed volume error metric
Vsf Scatter free reconstructed volume
xgap The distance between the object and the detector, often referred to as the air
gap
Xobj Contaminating x-rays, Xu, transmitted through the object
Xu Contaminating x-rays, consisting of secondary source and off-focal component
XCOM Photon cross sections database
Z Atomic number
xxiv
Chapter 1 Introduction
1 Motivation
Cone-beam computed tomography (CBCT) is a common imaging modality in image guided
surgery [1]-[3], image guided radiation therapy [4]-[6], small animal imaging [7], [8], dental
imaging [9], [10], and industrial applications [11]. CBCT offers volumetric isotropic imaging,
high spatial resolution, good soft tissue contrast, scalable field-of-view (FOV), and fast
acquisition times. Despite all it's aforementioned positive attributes CBCT suffers from
substantial degrading effects due to scatter on the overall image quality [12]. Specifically,
increased scatter causes image artifacts, a reduction in image contrast and contrast-to-noise
(CNR), and a loss of computed tomography (CT) number accuracy. The overall cost of the loss
in image quality caused by scatter is far reaching in both diagnostic and therapeutic uses of
CBCT, and can:
● impede a radiologists diagnostic ability
● make it challenging to define the boundaries of anatomy and disease (e.g. contouring) which is essential to planning radiotherapy and surgery
● confound automated algorithms for segmentation/contouring, multimodal image registration, and computer aided diagnosis
● limit the ability to perform accurate radiotherapy dose planning on CBCT and
1
subsequently adaptive radiation therapy due to lack of CT number accuracy [13]-[15]
A way to correct or compensate for the effects of scatter would have a significant impact on the
clinical usefulness of CBCT and other x-ray based imaging modalities. The body of work
contained in this thesis looks at characterizing the scatter distribution in CBCT projection images
and implementing a novel algorithm to correct for its detrimental effects. In the following
sections a brief overview of the physics and mathematics behind both x-ray image formation and
scattering will be given, as well as an introduction to both CT and CBCT. In the final section the
hypothesis of this work will be stated along with an outline of the work contained herein.
2 Background
The signal differences or contrast in x-ray projection images is a result of the attenuation of x-
rays which is dependent on the x-ray energy, absorbing medium, and absorbing medium
thickness as described by the Beer-Lambert law. For a x-ray beam consisting of N0 x-rays with
an energy E, entering a uniform medium of thickness t, the number of output x-rays, N, is given
as
N = N 0e−μ E t (1)
where E is the attenuation coefficient of the absorbing medium for x-rays of an energy E. The
transmitted 2D intensities, I(x,y), from a parallel polyenergetic x-ray beam through an object
with the given volumetric linear attenuation coefficient distribution, E(x,y,z), where z is the axis
perpendicular to the detector can be stated as [16]:
I ( x , y)=∫ I 0( x , y , E)e−∫μ E (x , y , z) dz
dE (2)
where I0(x,y,E) is the unattenuated incident x-ray source spectrum. If we assume the incident x-
ray beam can be approximated as a monoenergetic beam, Eq. (2) can be simplified to
I p(x , y )=−ln( I ( x , y)
I 0(x , y ))=∫μ (x , y , z )dz
(3)
where Ip is the projection image representing the line integrals of the attenuation coefficients.
2.1 X-ray Interactions in Medical Imaging
In observation, x-ray photons of energy levels typical of those used in x-ray imaging (40-150
keV) undergo one of three possible interactions that contribute to the linear attenuation
2
coefficient, μ. The relevant interactions are photoelectric absorption, incoherent (Compton)
scattering, and coherent (Rayleigh) scattering (see Figure 1).
(a) (b) (c)
Figure 1: Photon interactions with an atom relevant to x-ray imaging: (a) photoelectric absorption, (b) incoherent (Compton) scattering, and (c) coherent (Rayleigh) scattering. The photons initial energy, E0, is given by hν, where h is Planck's constant and ν is the photon's frequency. In (a), photoelectric absorption, the photon's energy is absorbed in ejecting a photoelectron, with the resulting photoelectron's energy, Ee, being equal to the photon's initial energy, hν, minus the electron binding energy, Eb. In (b), incoherent scattering, the photon is scattered with a change of energy specified as hν'. The recoil electron's energy is given as Ee=hν-hν'. In (c), coherent scattering, the photon is scattered without a loss of energy.
Photoelectric absorption [see Figure 1(a)] occurs when a photon interacts with an orbital electron
and in the process transfers all of it's energy into freeing the electron, which is often referred to
as a photoelectron. For x-ray energies around 100 keV and below, the mass attenuation
coefficient for the photoelectric effect is defined by the following proportional relationship [17]:
τρ ∝( Z
E0)3
(4)
where τ is the attenuation coefficient for the photoelectric effect, ρ the density, Z the atomic
number, and E0 the incoming photons energy.
Incoherent (Compton) scattering occurs when a photon interacts with an orbital electron and
imparts some, but not all, of its energy to an orbital electron [see Figure 1(b)]. The orbital
electron with which the photon interacts, often referred to as a Compton electron or
photoelectron, is either freed from the atom or moved to excited state. The photon, with its
decreased energy and increased wavelength, has its path diverted from its prior direction. The
change in wavelength can be determined by the Compton scattering equation as
λ ' −λ=h
m0 c(1−cosφ)
(5)
where is the wavelength of the photon before scattering, ' is the wavelength of the photon
3
e-
e-
e- e-
e-
photone-
e-
e-
e- e-
e-photon
e-
photoelectronE
e=hν-E
b
e-
e-
e-
e-
e-
photon e-
recoil electron
scattered photon hν'
φnucleus
hν
hν
hν
E0=hν
Ee=hv-hν'
after scattering, m0 is the electron rest mass, φ is the photon scattering angle, h is Planck's
constant and c is the speed of light. Furthermore, the Klein-Nishina (K-N) formula can be used to
provide an approximate prediction of the angular distribution of photons interacting with a free
electron based upon the impinging photons energy. The K-N formula for unpolarized radiation
gives the differential cross section for a photon scattering at an angle φ per unit solid angle ( )
and per electron as [17],[18]:
(d σed Ω )
K −N
=12
r02(hν '
hν )2
(hν 'hν
+hνhν '
−sin2φ)
(6)
where eσ is the cross section per electron, r0 is the classical electron radius, ν the photon's initial
frequency, ν' the scattered photon's frequency, and hν'/hν is the ratio of photon energy before and
after the collision. The photon energy ration can be defined as
hν 'hν
=1
1+hν
m0 c2 (1−cosφ )
(7)
which results from combining the Compton scattering equation in Eq. (5) with the relationship
λ=c /ν (8)
Since this differential cross-section is for a free electron and electrons in atoms are bound by a
given energy, the K-N formula is generally modified by additional factors, such as the
incoherent-scattering function [18], to account for the binding energies. Integrating the K-N
differential cross section over all scatter angles results in total K-N cross section per electron ( eσ)
which can be related to Compton mass attenuation coefficient as [17]:
σ Cρ =
N A Z
Aσe
(9)
given in (cm2/g) where NA is Avogadro's constant, Z the atomic number, A the molecular weight,
and ρ the density of attenuating medium.
Coherent (Rayleigh) scattering occurs when a photon impinging upon an atom interacts with an
electron and the photon's direction is diverged by some angle without loss of energy (Figure 1.c).
The cause of the scattering is due the photon interacting with the electric field of the electron.
The Thomson formula can be used to approximate the angular distribution of coherent scattered
photons. The Thomson cross section for unpolarized x-rays is [17],[18]:
4
(d σed Ω )
Th
=12
r02(1+cos2 φ) .
(10)
This formula can be viewed simply as a special case of the K-N formula when the scattered
photons energy equals the incoming photon energy (hν'=hν). To account for the energy binding
an electron to an atom additional factors are used for a better approximation, such as the atomic
form factor [18]. The mass attenuation coefficient for Rayliegh scattering, σR/ρ, is approximated
by the following proportional relationship [17] of
σ Rρ ∝
Z2
E02
(11)
where σR is the Rayleigh scattering coefficient.
The total mass attenuation coefficient, μ/ρ, for photon interactions in the energy ranges relevant
for medical x-ray imaging (1-200 keV) is given by the sum of the contributing interactions as
μρ = τ
ρ +σ Cρ +
σ Rρ
(12)
Looking at the mass attenuation coefficients for each of the different interactions, the percentage
of x-ray photons scattered or absorbed depends on two things: the atomic number, Z, of the
matter the photon is passing through, and the energy of the photon. Values for each of the
different mass attenuation coefficients are available from the National Institute of Standards and
Technology (NIST) in the XCOM database for energies ranging from 1 keV to 100 GeV for any
element, compound or mixture. The mass attenuation coefficient for either a compound or
mixture can be computed as a sum of the weight fractions of the separate elements as
μρ =∑
m
f m(μρ )m
(13)
where fm is the fractional weight of the mth atomic element.
The percent contribution to the total attenuation coefficient for photoelectric absorption, coherent
scattering, and incoherent scattering is plotted Figure 2 as a function of initial photon energy for
water, soft tissue, and compact bone along with the total mass attenuation coefficient for these
materials as a function of energy. The mass attenuation values for each material were computed
using the XCOM database with the fractional weights given in Table 1. The fractional weights
for soft tissue and compact bone come from the International Commission on Radiation Units &
5
0 50 100 1500
20
40
60
80
100
Energy (keV)
Pe
rce
nt I
nte
ract
ion
Co
ntr
ibu
tion
Water
Coherent Scattering
Incoherent Scattering
Photoelectric Absorption
0 50 100 1500
20
40
60
80
100
Energy (keV)
Pe
rce
nt I
nte
ract
ion
Co
ntr
ibu
tion
Soft Tissue
0 50 100 1500
20
40
60
80
100
Energy (keV)
Pe
rce
nt I
nte
ract
ion
Co
ntr
ibu
tion
Compact Bone
0 50 100 15010
-2
100
102
104
Energy (keV)
(1
/cm
)
Water
Soft Tissue
Compact Bone
Measurements (ICRU) definitions. For all three materials the photoelectric effect is the dominant
interaction at lower energies. Water and soft tissue have similar attenuation contribution
distributions and mass attenuation coefficients. For water and soft tissue the photoelectric effect
remains dominant until about 30 keV, whereas for compact bone the photoelectric effect is
6
Table 1: Weight fractions for elements composing water, soft tissue, and bone used in computing the mass attenuation coefficients. The soft tissue and compact bone values come from the International Commission on Radiation Units & Measurements (ICRU).
Weight Fractions
H C N O Mg P S Ca
Water 0.1119 / / 0.8881 / / / /
ICRU Soft Tissue 0.1012 0.1110 0.0260 0.7618 / / / /
ICRU Compact Bone 0.0640 0.2780 0.0270 0.4100 0.0020 0.0700 0.0020 0.1470
(a) (b)
(c) (d)
Figure 2: The contribution percentage for each of the three interactions relevant to x-rays in medical imaging for (a) water, (b) soft tissue, and (c) compact bone. The attenuation coefficient, μ, as a function of energy is shown in (d) for each material.
dominant until around 50 keV. For all three materials in the higher energy range (> ~50 keV)
incoherent scattering is the most probable interaction. Coherent scattering has less than a 15%
contribution at it's maximum for all the materials and energies investigated, it contributes slightly
more than incoherent scattering at very low energies and slightly more than the photoelectric
effect at the higher end of the energy spectrum. The mean energy of photons from a x-ray tube
(tungsten anode) operating at input kilovoltage potentials (kVp) between 80-120 kVp (which are
common energies for both CT and CBCT) is approximately between 40-60 keV depending on
the applied filtration (computed using Spektr [19]). Within this energy range, incoherent
scattering and photoelectric absorption will be the dominant interactions.
2.2 Effects of Scattering on X-ray Imaging
Equations (1)-(3) make an assumption that all interactions contributing to the attenuation
coefficient, μ, do not contribute to the image intensity signal I. This assumption is valid if either
none of the scattered photons reach the detector or all the interactions consist of photoelectric
absorption. Since the only situation consisting of almost entirely photoelectric interactions are at
very low energy the only case where scattered photons don't contribute to the signal is under a
condition known as “narrow-beam geometry” (e.g. small x-ray beam and small detector placed
far from object) [17]. In real imaging conditions this is generally far from reality. The signal I in
most imaging conditions is actually a product of the primary (unattenuated) photons, Iprim and
scattered photons, Iscat:
I =I prim I scat (14)
as illustrated in Figure 3. Taking Eq. (3) and simplifying it for the photons contributing to a
single pixel through a single absorbing medium with a thickness t and substituting in Eq. (14) we
can solve for μ as:
μ=1t
(ln( I 0)−ln ( I prim+I scat))
(15)
From Eq. (15) it can be seen that as the Iscat contribution increases the value of μ is decreased and
thus underestimated.
7
Figure 3: Illustration showing photon intensities at a detector for the case with (right) and without (left) scatter. Primary photons are indicated with solid lines and scattered photons are indicated with dashed lines. The scattered photons add additional signal to areas of the detector causing the attenuation of the primary beam to be underestimated estimated at those locations.
2.3 Computed Tomography
The projection image, Ip, representing the line integral of the attenuation coefficient [see Eq. (3)],
forms the basis from which computed tomography (CT) reconstructs volumetric images. The
general idea of CT is that the volumetric details of an object (e.g. attenuation coefficient
distribution) can be reconstructed by taking a series of x-ray images at different projection
angles, θ, around the object of interest. The mathematical basis of this reconstruction is known as
the Radon transform, R, which is stated as
I p(x ' , y ' ,θ )=R {μ (x , y , z)} (16)
where x' and y' are the image coordinates, and θ is the projection angle and x, y, and z are the
object coordinates. Given an infinite number of projection angles the objects attenuation
coefficients and thus the volumetric representation of the imaged object, V, can be determined by
taking the inverse Radon transform:
V ( x , y , z )=μ (x , y , z)=R−1{I p(x ' , y ' ,θ )} (17)
In practice the reconstruction of the estimated linear attenuation coefficients is computed using
either filtered back-projection, iterative algebraic reconstruction, or iterative statistical
reconstruction methods [16]-[21].
8
Ideal (Primary only)
X-Ray Source
Object
Detector
Primary and Scatter
There are two fundamental assumptions that are made in reconstructions based on Eq. (3) and
(17) that can lead to reconstruction errors and artifacts. The first is the monoenergetic
approximation made in Eq. (3). In reality most x-ray sources generate a distribution of x-rays
that are polyenergetic. In general, the approximation of the polyenergetic beam as a
monoenergetic beam with a mean energy EM gives very reasonable results but it can lead to what
is known as beam hardening artifacts, especially when the object contains materials with high
atomic numbers. Beam hardening occurs due to changes in the x-ray energy spectrum. These
changes in the spectrum occur because x-rays of varying energies have different attenuation
coefficients so as the x-rays travel through an absorbing medium the mean energy tends to
increase, as lower energy x-rays are more likely to be absorbed. This shift in x-ray spectrum
causes the reconstructed μ values to be underestimated. Visually beam artifacts can appear as
cupping artifacts, streaks, and dark bands. The streaks and dark bands occur around objects with
high atomic numbers, such as bone, due to the increased attenuation. Several techniques have
been proposed to correct for the use of polyenergetic x-ray sources in both CT and CBCT and
the problem is an active area of research [22]-[25].
The second assumption has to do with the interactions through which the x-rays are attenuated,
as discussed in section 2.1. The reconstruction algorithms based on the Radon transform assume
the interactions resulting in the attenuation of photons prevent the x-rays from reaching the
imaging detector and contributing to the intensity signal, I. This can also be stated as the
assumption that all x-rays arriving at the imaging detector traveled in a straight line from their
point of origin to the imaging plane, allowing for the image signal to be a result of linear line
integrals. In Eq. (15) it was shown how the two scattering interactions (coherent and incoherent)
result in photons reaching the detector causing an underestimation of μ which results in visible
errors in the reconstructed volume which will be demonstrated subsequently in the next section
describing CBCT.
2.4 Cone-Beam Computed Tomography
Cone-beam computed tomography is an extension of traditional CT. Traditional CT uses a small
vertical FOV (a fan-beam essentially imaging a single axial slice) in conjunction with a moving
table to create a volumetric image using a spiral (helical) scanning technique involving multiple
rotations of the tube and detector invented by Willi Kalender. CBCT on the other hand uses a
large vertical FOV (cone-beam) along with a large (often flat) detector allowing for a isotropic
9
volume to be reconstructed from only a single rotation of the tube and detector. The ability of
CBCT to acquire large volumes in a single rotation makes it suitable to applications with limited
operating space, mechanical constraints, and/or require fast acquisition times such as image-
guided therapies (e.g. surgical [1] and radiation therapy [5]). Like CT the reconstruction
algorithms for CBCT are based on the Radon transform and a filter back-projection technique for
CBCT is outlined by Feldkamp, Davis, and Kress [26]. The use of a large “cone-beam” allowing
for a larger imaging volume and a more efficient image acquisition has the undesired effect of
increasing the scatter contribution to the image signal as shown in Figure 4 as it is almost the
exact opposite of a “narrow-beam geometry”. The result of this increase in scatter on the
10
(a) Small FOV
(b) Large FOV
Figure 4: Scatter (green lines) and primary (red lines) x-ray contribution to the imaging signal in a single detector row for a projection image of a water cylinder using a small (a) and large (b) field of view (FOV). The x-ray coverage for the small and large FOV are indicated with semitransparent yellow coloring and have approximately 5 cm and 20 cm coverage in the vertical direction at a distance of 100 cm from the source, respectively. The increase in scatter contribution to the horizontal line in the detector comes mainly from scatter photons outside the plane of the primary photons contributing to the image signal.
X-ray Source Water Cylinder Detector
Primary Photons
Scattered Photons
1FOVFullPatient.header
reconstruction is clearly illustrated in Figure 5 where axial slices of CBCT reconstructions
created with increasingly larger FOVs are shown. Looking at the axial image for the smallest
[see Figure 5(b)] and largest [see Figure 5(d)] FOV there is a substantial loss of contrast in both
the soft tissue and bone, as well as an increase in shading artifacts.
(a) (b)
(c) (d)
Figure 5: CBCT reconstructions of a pelvis patient using three different field-of-views (FOV) with dimensions of FOVs indicated on (a) coronal slice of pelvis patient. 2 cm FOV indicated with light blue dashed lines, 10 cm FOV with white dashed lines, and top and bottom of image indicate full FOV of ~26 cm. Axial slices from reconstruction using: (a) 2 cm FOV, (b) 10 cm FOV, and (c) 26 cm FOV projections. A decrease in the image quality is clearly demonstrated as the FOV increases.
3 Outline of Thesis
The aim of this thesis is to use physics and computational models to characterize and correct
scatter in CBCT imaging in an efficient and accurate manner. In order to obtain an efficient
estimate of the scatter distribution an understanding of the parameters that control the
distribution of scatter in CBCT is needed. The underlying physics of photon interactions can be
precisely characterized using Monte Carlo (MC) simulations which allow the generation of
scatter distributions across all projections in a CBCT image acquisition. The resulting scatter
distribution from the MC simulation is a finite three dimensional image representing the scatter
at each pixel, projection, and angle. The MC representation is a finite sampling of the continuous
11
three dimensional scatter function, S(x,y,θ), that occurs during the acquisition of a CBCT
projection set, where x and y are horizontal and vertical detector locations and θ is the projection
rotation angle. Knowing the scatter function, S(x,y,θ), for a given CBCT geometry and object
would allow for the correction of the degrading image quality effects that are associated with
scatter in a CBCT reconstruction. A thorough understanding of the nature of the scatter
distribution in CBCT, acquired using MC simulations, will make it possible to exploit various
priors (e.g. symmetry, shape, complexity) and computational methods (e.g. variance reduction
techniques, parallel processing, data fitting) to produce an accurate and efficient scatter
correction mechanism in CBCT.
3.1 Hypothesis
The central hypothesis to this thesis is:
Through advanced computational physics models it is possible to rapidly and accurately
estimate and subtract the underlying scatter distribution, S(x,y,θ), from CBCT projection
images of an object to remove artifacts arising from scatter.
3.2 Specific Aims
Specific Aim 1: Development and validation of a flexible Monte Carlo model for simulation of
CBCT x-ray projections by which the nature of the underlying scatter distribution at the detector
can be investigated for various objects and imaging geometries.
In order for a thorough understanding of the scatter distribution to be attained a method for
examining the scatter distribution in the projection images under different imaging scenarios is
required. The ability to measure scatter is an arduous task that is not suited for making estimates
across the imaging plane but best suited for point based measurements as will be discussed in
Chapter 2. On the other hand computational models using MC methods provide a robust method
for estimating the entire scatter distribution from a complex object contributing to the signal in a
projection image [27]-[36].
The CBCT MC model used herein extends work done by Jarry [27], [28] incorporating recent
advances in variance-reduction techniques applied to x-ray production from bremsstrahlung
targets [37]-[39] and allowing for a separation of each type of scatter interaction (coherent and
incoherent). The model is verified as a whole by comparing simulated projections of a known
12
object to measured projections of the same object. Validation of the x-ray tube model are
performed by comparing the open-field projections of simulated and measured data. Finally the
scatter simulation portion of the model is evaluated by taking scatter measurements using beam
stop techniques[35], [40], [41]. An overview of the MC system used along with the validation
against measurements is given in Chapter 2. which has been adapted from the paper published in
Medical Physics [42] of which I was the first author and conducted all experiments and analysis.
Specific Aim 2: Characterization of the scatter distribution through the use of the validated
Monte Carlo model.
The creation of a validated CBCT MC that can accurately estimate the scatter distribution in the
projection images allows for the characterization of the effect different imaging configurations
(object size, compensator, FOV, etc) has on the resulting scatter distribution. More importantly
the potential for finding ways of simplifying the scatter distribution through the use of fitting
functions becomes possible. In Chapter 2 the effects of imaging geometry and compensator on
the resulting scatter distribution is explored for head (diameter=16.5 cm) and body
(diameter=30.6 cm) cylinders. In particular the ability of a compensator (to not only diminish the
magnitude of the scatter distribution but also limit it's spatial complexity is noted. In Chapter 3
the spectrum of the scatter distribution in CBCT projection images is analyzed using Fourier
analysis showing the scatter distribution is contained within in the low frequency domain. This
work builds on the initial work published in the SPIE proceedings [43] of which I was the first
author and conducted all experiments and analysis.
Specific Aim 3: Create a system that estimates the scatter distribution function, S(x,y,θr), in an
efficient and accurate manner for any CBCT geometry and object and integrate it into a CBCT
scatter correction technique.
MC simulations provide an excellent method for estimating scatter in CBCT images but
currently, the extensive computational times involved make this a clinically irrelevant scatter
correction method [27]. Various techniques can be used to speed up MC simulations such as
parallel processing and variance reduction techniques [44]-[47], but these techniques have still
been unsuccessful in achieving near real-time performance. Recent research has shown that a
significant reduction in the number of particles required to generate an accurate MC simulation
can be achieved by using image processing fitting and filtering techniques [44], [47]-[51]. The
13
fact that the scatter distribution is limited to the low frequency domain suggests the efficacy of
fitting the scatter distribution to a limited sum of sine and cosine functions using Fourier based
low-pass filtering and interpolation techniques. In Chapter 3 the low-pass filter cutoff
frequencies for a head and pelvis phantom are computed and used in estimating the scatter
distribution from limited photon simulations. The results are compared to gold standard MC
simulation results to estimate the error. In Chapter 4 an efficient algorithm is outlined using
concurrent MC simulations combined with Fourier interpolation to significantly reduce scatter
simulation times.
Specific Aim 4: Evaluation of the scatter correction system to correct for scatter induced
artifacts in CBCT images.
In Chapter 4 a set of image quality metrics are outlined to evaluate the proposed scatter
correction technique. Simulated head and pelvis projection sets are corrected using the scatter
correction algorithm and evaluated using the proposed metrics. The error is also computed
against the simulated scatter free (primary photons only) reconstructions. Finally the algorithm is
tested on two projection sets (pelvis patient and phantom) obtained from the Elekta Synergy
Platform (Elekta, Crawley, West Sussex, UK).
3.3 Publication Status
Chapter 2 is an adapted version of the paper published in Medical Physics entitled “The effects
of compensator and imaging geometry on the distribution of x-ray scatter in CBCT” [42].
Chapter 3 is adapted from the paper tentatively accepted for publication in Medical Physics
entitled “Spatial frequency spectrum of the x-ray scatter distribution in CBCT projections”.
Finally, Chapter 4 and 5 are to be submitted to Medical Physics in a paper entitled “Efficient
scatter distribution estimation and correction in CBCT using concurrent Monte Carlo fitting”.
For all aforementioned publications I am the first author and conducted all experiments and
analysis.
14
Chapter 2 The Effects of Compensator and Imaging Geometry
on the Distribution of X-ray Scatter in CBCT
1 Introduction
There has been a large body of research aimed at better understanding and correcting the scatter
distribution in x-ray imaging techniques (CT, CBCT, and Radiography). Research has shown
that Monte Carlo (MC) simulations are a powerful tool in exploring and understanding the
characteristics and contributions of scatter to the image signal [27], [29]-[36], evaluating new
hardware and software based scatter correction techniques [52]-[56], and as a potential method
for reducing scatter induced image degradation. [28], [44], [46]-[48], [57].
This chapter details the investigation of the scattered radiation distribution in CBCT for various
imaging parameters and objects through the use of MC models. The MC simulations are
performed by building on the existing MC model previously described by Jarry et al. [27], [28].
The results of the MC scatter simulation from the system had not previously been validated
against measured scatter estimates and a good portion of the chapter details results from scatter
measurements taken using methods similar to those outlined by Chen et al. [35] and Fahrig et
al. [41]. The direct comparison between MC simulated and experimentally measured scatter
profiles is limited in the literature with the only example the authors are aware of being in work
done by Chen et al. [35]. In both the work done by Chen et al. [35] and ourselves discrepancies
15
between the simulated and measured scatter profiles can clearly be seen near the boundary of the
object being imaged. In order to better understand the nature of these differences we have
performed additional simulations to show the discrepancies are due to limitations in the
measurement technique and not the simulations.
The CBCT MC system is used to probe the effects of various imaging parameters (air gap, cone
angle, object size, and compensation) on the resulting scatter distribution. Previous work has
clearly shown, through both MC simulations and measurements, that there exists a relationship
between the amount of scatter in the projection image and both the air gap [34], [35], [58] and
cone angle [12], [33]. A decrease in scatter can be achieved through either increasing the air gap
or decreasing the cone angle. This relationship is further validated over an extensive range of air
gap and cone angle values using MC simulations and a functional relationship between the
amount of scatter found in the image and these two parameters is formulated.
Compensators were originally designed to “compensate” for the x-ray fluence changes created
by the object being imaged to create a more uniform x-ray flux at the detector to help cope with
the detectors limited dynamic range. Compensators are generally made out of either aluminum or
copper and often have a profile similar to the shape of a bowtie and are thus commonly referred
to as bowtie filters. Using compensators in CBCT offers another imaging means for improving
image quality by not only creating a more uniform fluence at the detector but also minimizing
scatter, reducing patient dose, and creating a more uniform fluence at the detector [59], [60].
Experiments by Graham et al. [59] showed that through the use of a compensator scatter could
be reduced by more than 40%. Bowtie filtration has also been included in previous CBCT MC
simulation studies [34], [36], [61] investigating patient dose and scatter. There is however, a lack
of data in regards to how compensators affect the shape of the scatter distribution created by the
object being imaged and the amount of secondary scattered radiation generated by the
compensator itself. We investigate two different bowtie filters for their impact on the scatter
emanating from the object and for potentially increasing the contamination scatter radiation.
This chapter represents a comprehensive study of the effects of various imaging parameters on
the resulting scatter distribution in CBCT imaging through the use of MC methods. Not only can
the scatter be reduced through the careful selection of imaging parameters and filtration, but the
structure of the scatter distribution can also be diminished. The decrease in the structure found in
16
the scatter signal may prove to be beneficial in scatter correction techniques that use primary
modulation, [55], [62], where the scatter is assumed to be contained primarily in the low-
frequency portion of the Fourier domain, and MC methods using fitting functions to improve
computational efficiency [44], [47], [48].
2 Methods and Materials2.1 Monte Carlo Simulation System
The CBCT imaging system was modeled using the EGSnrc MC code [63] extending simulation
work previously done by Jarry et al. [27], [28]. The system consists of four different components
as illustrated in Figure 6. The system allows for separation of the x-ray fluence at the detector
into scatter and primary components. The MC model simulates the x-ray source (including
housing, inherent and added filtration, collimation), bowtie filtration, object and imaging
17
Figure 6: System diagram of the cone-beam CT Monte Carlo (CBCT MC) model. The system consists of three main components each used to model a portion of the actual CBCT system. (1) consists of a model of the x-ray tube that generates a phase space file containing the output photons position, direction, energy, and statistical weights using BEAMnrc. The phase space from (1) is then used as input to (2) a voxelized model of the compensator which results in a phase space file. The phase space from (2) is then used as input to (3) a voxelized model simulating the geometric configuration of the source and detector, the object to be imaged, and rotation angle (θ) using modified DOSXYZnrc code which keeps track of primary and scattered photons. The simulation in (3) generates another phase space file at the detector plane which is input to (4) which models the detector response using a look up table based on previous MC simulations.
Target
Inherent Filtration
Added Filtration
Collimators
Detector
1. X-Ray TubeSoftware: BEAMnrcInput: Energy, # of Electrons, tube geometry,
and collimator settingsOutput: Phase space file at exit plane
collimators
Voxelized Phantom
3. Object and GeometrySoftware: dosxyznrc_phsp_laracoInput: Phase space from bowtie simulation,
rotation angle, SAD, ADD, # of photons, voxelized phantom
Output: Phase space file at detector position
4. DetectorSoftware: Program FlatDetect.cppInput: Phase space, energy response look
up tableOutput: Image representing energy response
of detector
Voxelized Bowtie
2. CompensatorSoftware: dosxyznrc_phsp_laracoInput: Phase space from x-ray tube
simulation, voxelized compensatorOutput: Phase space file after compensator
X
Z
geometry. The x-ray sources were modeled using BEAMnrc [64]. The compensator and object
are modeled using voxelized geometries simulated in a modified version of the DOSXYZnrc
program [65] called dosxyznrc_phsp_laraco which keeps track of scattering particles (coherent
and incohorent) and outputs a phase space file containing all the particle data (weight, direction,
energy, position) for particles passing through the specified plane. The detector response is
computed in a custom software program that uses a lookup table that relates the photons energy
and direction to the resulting detector signal. The detector response is separated into the resulting
contributions from coherent, incoherent, and primary x-rays allowing for analysis of the scatter
distribution in projection images. The structure of the MC model remains similar to that outlined
by Jarry et al. [27] with the major modifications outlined subsequently.
Efficiency improvements to the BEAMnrc x-ray tube simulation have been made by
incorporating a recent variance reduction technique known as directional bremsstrahlung
splitting (DBS) [38]. The optimal splitting number can be calculated according to techniques
outlined by Mainegra-Hing and Kawrakow [39]. The optimal splitting number depends on many
factors such as kVp, quantity of interest (e.g. fluence, dose), field size, scoring zone size, and
distance from the source to the scoring zone [39], [66], [67]. Since the simulations involve
different source-to-detector distances (SDD) and scoring zone sizes simulated with the same
phase space file output at the exit of the x-ray tube a splitting number of 2000 with a splitting
field radius of 10 cm located 30 cm from the focal spot was chosen. A lower splitting number
decreases the chance of having photons originating from the same bremsstrahlung event arrive at
the same pixel which adversely effects the history-by-history calculated statistical uncertainty of
the fluence. The optimal directional splitting number for our x-ray tube model with a source-to-
detector distance (SDD) of 156 cm and a pixel size of 10x10 mm2 was found to be around 14000,
providing a relative efficiency gain of three orders of magnitude. Additional changes to the
CBCT MC simulation include the use of bound Compton scattering, which simulates the binding
effects and Doppler broadening using the impulse approximation [65] and Rayleigh scattering.
The simulation code has also been modified such that radiation arriving at the detector can not
only be divided into primary and scatter but also into the types of scattering events the photons
underwent. The x-ray tube model for all simulations was run with ~40109 input electrons with a
kinetic energy of 100 keV (E0=611 keV). The Monte Carlo parameters for the electron cutoff
energy (ECUT) and photon cut off energy (PCUT) were set to 523 keV and 10 keV, respectively
18
for the x-ray tube. In the compensator, object and geometry portion of the simulation (modified
DOSXYZnrc code) the ECUT is raised to 10 MeV, ignoring electron transport and increasing
simulation efficiency.
2.2 Cone-Beam CT Imaging System
Measurements for all validation experiments were taken using a CBCT bench-top system [68].
The system consists of an x-ray tube (Rad-94 with Sapphire housing; tungsten-rhenium-
molybdenum-graphite target; 0.4-0.8mm focal spot; 14º anode angle; Varian Medical Systems)
powered by a constant potential generator (CPX 380, EMD Inc.), and a flat panel detector
(Paxscan 4030A; 2048(w) × 1536(h) pixel matrix; 0.194 mm pixel pitch; 397 mm(w) × 298
mm(h) pixel area; CsI scintillator; Varian Medical Systems). The flat panel detector, rotation
stage, and x-ray tube are all mounted on precision linear positioners (Parker-Daedel). The x-ray
tube and detector each have 3 linear positioners and the rotation stage has a single linear
positioner, including the rotation a total of 8 degrees of freedom is provided by the system. The
entire system (linear positioners, rotation stage, and flat panel detector) is under computer
control allowing for precise definition of image acquisition geometries.
19
2.3 Validation of Monte Carlo Model
An initial validation of the CBCT MC simulation system was performed by Jarry et al. [27].
Comparisons between measurements and simulations were done for the first and second half-
value layers (HVL), open-field profile measurements and projection images of a cylindrical
water and anthropomorphic head phantom with all comparisons showing agreement within 10%.
The separated scatter component of the CBCT MC model was not validated as an independent
signal in Jarry's previous work [27] and a good portion of the experiments conducted in this
research focus on completing this task. A similar validation of the scatter profiles simulated
using the GEANT4-based GATE MC simulation package for CBCT breast imaging has been
conducted by Chen et al. [35]. In the validation experiments comparing the simulated and
measured detector response the pixel values have been averaged into 64x64 pixel groups
resulting in a pixel pitch of 1.24 cm in both directions.
2.3.1 Open-Field Detector Response Validation
To ensure that the modifications to the CBCT MC system did not affect the accuracy of the
simulation a set of experiments were conducted to validate the system in its entirety. The first set
of measurements compared the simulated and measured detector response to an in-air
measurement (open-field) of the detector response. Validation was limited to the horizontal and
vertical axis intersecting the center of the panel. Due to the fact that the actual detector had a
variable response at different pixels to the same input fluence a calibration was required. To
create a functional relationship characterizing the variation in different pixel response to the
same input fluence a set of images were created by exposing a group of pixels to the same fixed
input fluence (FIF). The FIF was generated by drilling a 4.75 mm diameter hole in a 3 mm sheet
of lead and placing it between the source and the detector while exposing the panel at 100 kvp
and 0.4 mAs. The variation in pixel response was determined by fixing the position of the source
and lead sheet while translating the detector between exposures. An orthogonal set of 64
horizontal and 32 vertical, equally spaced, measurements were acquired. For each location 5
exposures were averaged together over a 10×10 region of pixels. A correction was then applied
to the open field response of the panel as
I open'
x , y=I openx , y
I fif x , y
(18)
20
where x and y are locations on the panel, Ifif is the detector response to the fixed input fluence
normalized by the maximum value, and Iopen is the panel response to an open-field signal. It
should be noted that this normalization is only applied for open-field validation purposes. All
other investigations were simply normalized using an open-field exposure.
The differences between the measured and simulated results were quantified by looking at the
local percent discrepancy between the signals. The local percent discrepancy was calculated as
LPD x , y =I simulation x , y − I measurement x , y
I measurement x , y
(19)
2.3.2 Object Projection Validation
To validate the MC imaging and object geometry a set of measured and simulated projection
images were made of the 16.4 cm diameter water cylinder. The projection image of the cylinder
was normalized, both in the simulation and measurement, as:
I calib x , y=I proj x , y −I dark x , y
I open x , y− I dark x , y
(20)
where Iproj is the projection image and Idark is the detector image without any input fluence. In the
case of the MC simulation Idark is zero. The horizontal profile at the center of the panel for the
measured and simulated data was compared using the LPD.
2.3.3 Scatter Signal Validation
The approach taken to get an estimate of the scatter component in a projection image is similar to
the beam stop methods outlined previously [12], [35] ,[40], [41], [69]. The beam stop method
relies on the collection of four images that can be mathematically manipulated to extract an
estimate of the scatter signal. Specifically these images are an open-field projection, Iopen, a
projection of the object whose scatter distribution is sought, Iobj, the portion of projection of the
object with the primary blocked (usually using a lead object), Iobj+block, and an image of the
primary blocker, Iblock. Iopen contains the bremsstrahlung source which consists of photons
emanating from the focal spot and those generated outside the focal spot (off-focal radiation),
along with a secondary source of photons resulting from bremsstrahlung source photons being
scattered by the tube housing, filtration, compensator, and the collimators. This can be written as,
I open=PuP ' uS u (21)
21
where Pu is the focal component of the bremsstrahlung source, P'u is the off-focal component of
the bremsstrahlung source and Su the secondary source of scattered photons. The secondary and
off-focal bremsstrahlung sources are grouped together as the contamination component,
X u=P ' uSu . (22)
The projection of the object is defined as
I obj=Pobj X objS (23)
where Pobj is the primary transmitted through the object, Xobj the contamination fluence
transmitted through the object and S the scatter generated in the object, which contains scattered
photons originating both from the Pu and Xu. The projection image with the beam blocker in front
of the object is given as
I objblock= X objS−S block≈ X objS (24)
and contains the attenuated contamination, Xobj, the scatter from the object, S, and a loss of
scattered photons due to the presence of the beam blocker, Sblock. In our work we found Sblock to be
negligible due to the small size of the blocker and thus it was set to zero. Finally, the image of
just the beam blocker is given as
I block= X u. (25)
The most common approach [35], [41], [69] taken in attempting to estimate either the scatter or
the SPR is to assume that the contamination component undergoes an attenuation similar to that
of the focal component of the primary, where the attenuated primary is calculated using Beer's
law as
Pobj=Pu e− t . (26)
An estimate of the attenuation of Pu by the object can be obtained from the given measurements
by combining equations (21)-(26) and solving for the attenuation,
e− t
=Pobj
Pu
≈I obj− I objblock
I open−I block
. (27)
Assuming the contamination photons travel a similar pathlength as the primary the attenuated
contamination component can be estimated, using the estimate of the attenutation of the primary
by the object given in Eq. (27), as
22
X obj≈ X u e− t
=I block I obj−I objblock
I open−I block. (28)
This estimate of the attenuated contamination component can then be used to estimate the object
scatter,
S≈ I objblock− I block I obj− I objblock
I open−I block= I open I objblock− I obj I block
I open− I block
(29)
and in a similar manner we can also obtain an estimate of the scatter-to-primary ratio (SPR),
where
SPR=
I objblock
I obj−I objblock
−I block
I open−I block
. (30)
In our work we have also created two additional scatter ratio relationships, the scatter-to-open-
field ratio (SOR),
SOR=
SI open
(31)
and the scatter-to-open-field center pixel ratio (SOCR)
SOCR=
S
I open0 (32)
where I0open is the value of the open-field detector response at the center pixel of the panel. These
two new ratios highlight the nuances in the scatter signal that might otherwise be hidden by the
primary signal in the SPR.
It is important at this point to reiterate that the estimate of Xobj obtained in Eq. (28) and used in
the subsequent equations for getting an estimate of the scatter makes the assumption that the
pathlength the contamination photons travel is the same as the pathlength of the focal source
photons. This assumption may not always be accurate and measurements by Johns and Yaffe
[40] found the ratio of attenuation between Pu and Xu varied from 0.81 to 0.89. We will delve
deeper into the validity of this assumption in the results and discussion sections of this chapter.
23
We experimentally estimated the scatter with both a lead disc and a lead bar. Lead discs of
varying size (0.5, 1, 1.5 and 2 cm) with a thickness of 0.5 cm (over 18 times the half-value layer
at 100 kVp) were held in a custom mechanism manufactured of acrylic for reproducible
placement in the imaging geometry (Figure 8). Measurements for Iobj+block and Iblock were collected
with 5 frames averaged for each lead disc size at 18 different displacements ~1 cm apart in the
horizontal dimension. The Icyl and Iopen images were estimated from 20 and 50 frame averages,
24
respectively. All four images were collected with a fixed geometry with an SAD of 100 cm and
ADD of 56 cm. The object used was the 16.4 cm diameter water cylinder.
The second beam stop device consisted of 0.3×0.3 cm lead bar, 22.5 cm in length fixed in acrylic
(Figure 8). This linear beam stop allowed for a horizontal profile of the scatter distribution to be
estimated in the shadow of the lead bar. As with the lead disc experiment, frame averaging (50
frames) was employed and the object of interest was the 16.4 cm diameter water cylinder. Two
imaging geometries were investigated with an ADD of 30 and 56 cm and the SAD fixed at 100
cm.
The SPR and SOR measurements from the lead disc and line experimental measurements were
compared to the simulated SOR and SPR measurements using the scatter signal determined from
the CBCT MC. In addition a voxelized model of the linear beam stop was created and the four
projection images required to estimate the SOR and SPR using the beam stop technique (Iopen,
Iblock, Iobj+bloc, and Iobj) were MC simulated for both ADD geometries. The simulated images were
then used to calculate the same quantities allowing differences between the beam stop estimated
scatter signal and the MC estimate to be quantified.
In addition to examining the scatter signal generated in the object a comparison of the
contamination scatter, Xu, was performed. The Xu signal can be obtained from both the Iblock
projection image and can be separated out of the MC model. In our study the x-ray tube portion
of the simulation doesn't keep track of scattered and primary particles. The Xu signal was
estimated from the MC phase space file by backprojecting the primary particle trajectories to the
focal spot. If the photons did not intersect within 0.1 mm of the electron impact region defining
the focal spot then they were considered to be “contamination”.
2.4 MC Scatter Distribution Simulations
2.4.1 Water Cylinder Scatter Distribution
The CBCT MC scatter simulation was employed to examine the effect of the object and imaging
geometry on the resulting spatial scatter distribution at the detector. Specifically, simulations
were carried out to determine the effect of the cone angle (cone), SAD, ADD and the cylinder
diameter (dcyl) on the resulting scatter distribution. The source and detector configurations are
listed in Table 2. Each of the seven configuration were repeated with four different cone angles,
25
{1.4, 2.8, 5.7, and 11.3°}, and two different cylinder diameters {16.4 and 30.6cm} resulting in
52 simulated configurations (Note: the ADD of 9 cm was removed from all 30.6cm
configurations). All configurations use a fan angle of 21.3º, a tube input potential of 100 kVp,
and a MC detector size of 120(w) cm 60(h) cm. The resulting scatter distributions database
was used to derive relationships between the scatter and two of the more dominant parameters,
the cone angle and ADD. The relationship between the scatter and the aforementioned imaging
parameter was evaluated for the two ratios, SPR and SOCR, looking only at the center pixel.
Here, the primary used in computing the SPR includes the contamination photons unlike in the
measurements using Eq. (30).
2.4.2 Bowtie Filtration
The effects of bowtie filtration on the spatial distribution of scatter was examined. Two bowtie
filters were modeled in the dosxyznrc_noscat_phsp portion of CBCT MC model after the
collimators. A voxelized model of a bowtie similar to the F1 filter used in the CBCT system on
the Elekta Synergy Platform (Elekta, Crawley, West Sussex, UK) and a custom bowtie designed
to compensate for a 16.4 cm diameter cylinder were created out of aluminum material (PEGS4).
A profile of both compensators can be seen in Figure 9 and details of the modeling process can
be found in the Appendix A.
The bowtie simulations were run with a cone angle of 11.3°, a fan angle of 21.3º, a SAD of 100
cm and an ADD of 56 cm for both the 16.4 and 30.6 cm diameter water cylinders. Two
additional ADD measurements of 18 and 30 cm were made for the F1 bowtie with the 30.6 cm
diameter cylinder and the remaining parameters the same as in the case of the 56 cm ADD.
26
Table 2: Source and detector configurations used in simulations.
Configuration SAD (cm) ADD (cm) Magnification
1 100 9 1.09
2 100 18 1.18
3 100 30 1.30
4 100 44 1.44
5 100 56 1.56
6* 75 56 1.75
7* 50 56 2.12*These configurations truncate the image of the 30.6 cm diameter cylinder due to the fixed size of the fan angle.
3 Results
A sample of the 2D images representing the SOCR for both cylinders (16.4 and 30.6 cm) without
bowtie filtration and the 30.6 cm cylinder with the F1 bowtie filter at three different ADD
configurations (18, 30, and 56 cm) for an ADD of 100 cm and a cone angle of 11.3º are shown in
Figure 10. The images represent a small portion of the data generated during this work and for
simplification purposes the rest of the data will be displayed in the shape of horizontal and
vertical profiles centered on the detector. The simulation times for the fluence arriving at the
detector for the 16.4 and 30.6 diameter water cylinders with a cone angle of 11.3º and a SDD of
156 cm were 6.6 and 28.8 hours, respectively. The large cylinder was simulated with 2×109
photons and 3082×3 voxels, compared to 5×108 photons and 1662×3 voxels for the smaller
cylinder, resulting in the significantly longer simulation times (> 300%). The aforementioned
simulation times do not include the time required for simulating the phase space file at the exit of
the x-ray tube which was only simulated once for each cone angle. The simulation of the x-ray
tube with a 11.3º cone angle with 40×109 input electrons took 24.5 days . The simulations were
run on an Intel Core 2 Quad processor (Q6600) with a 2.4 GHz clock speed.
27
(a) (b)
Figure 9: XZ profiles of the simulated F1 (a) and custom (b) bowtie filters. The thickness of the filters in the y dimension was 8.5 cm. Equations (49), (50), (53), and (54) from Appendix A were used to generate the surface curvature of the bowtie filters.
-6 -4 -2 0 2 4 6-2
-1
0
1
2
x [cm]
z [c
m]
-6 -4 -2 0 2 4 6
0
1
2
3
x [cm]
z [c
m]
ADD
18 cm 30 cm 56 cmC
ylin
de
r D
iam
ete
r
16.4
cm
30.
6 c
m
30.6
cm
w/
F1
Figure 10: Rows 1 and 2 show 2D images of the scatter-to-open-field center ratio (SOCR) for the 16.4 and 30.6 cm diameter water cylinders at three different ADD settings of 18, 30, and 56 cm all with an SAD of 100 cm and a cone angle of 11.3°. Row 3 shows the same configuration as row 2 except with the F1 bowtie filter in place. The images represent a detector size of 120(w) × 60(h) cm. The decrease in the scatter with increasing air gap is clearly evident for both cylinders. A large degree of symmetry can also be seen in all the SOCR distributions. Additionally the effect of the bowtie filter on the scatter distribution shows not only a decrease in the amount of scatter but also a flattening out of the scatter distribution.
3.1 Monte Carlo Validation
3.1.1 Open-field Detector Response Validation
The normalized measured open-field is compared to the simulated data in Figure 11 (a-d). The
maximum normalized detector response profiles to the FIF, used in the correction given in Eq.
(19), are shown in Figure 11 (e,f). The mean absolute percent discrepancy between the simulated
and FIF corrected measured data for the horizontal and vertical profiles was 1.10.7% and
0.70.9%, respectively. The normalized cross-correlation (NCC) of the measured and simulated
profile was also computed before and after the FIF correction to illustrate the impact of local
variation in detector response. The horizontal signal went from a NCC value of 0.93 to 0.98 after
the correction, showing a slight improvement. On the other hand the vertical signal showed a
dramatic improvement in the NCC value, going from -0.11 before the correction to 0.97 after the
correction. This is due to a variation in the gain across the detector in the vertical direction.
28
0
0.01
0.02
0.03
0.04
0.05
29
(a) (b)
(c) (d)
(e) (f)
Figure 11: Open-Field validation for horizontal (a) and vertical (b) profiles of the measured and simulated open-field projection images normalized by the center pixel value. The measured open-field projection profiles have been corrected using the maximum normalized fixed input fluence (FIF) profile shown in (e) and (f). The need for normalizing by the FIF is clearly seen by the non-linear response across the vertical portion of the detector (f). The local percent discrepancy (LPD) between the measured and simulated profiles is plotted in (c) and (d).
-20 -15 -10 -5 0 5 10 15 200.6
0.7
0.8
0.9
1
1.1
Horizontal Position [cm]
Nor
mal
ized
Det
ecto
r S
igna
l
Measured
Simulated
-15 -10 -5 0 5 10 150.6
0.7
0.8
0.9
1
1.1
Vertical Position [cm]
Nor
mal
ized
Det
ecto
r S
igna
l
-20 -15 -10 -5 0 5 10 15 20-2
0
2
4
6
8
Horizontal Position [cm]
Loca
l % D
iscr
epan
cy
-15 -10 -5 0 5 10 15-2
0
2
4
6
8
Vertical Position [cm]
Loca
l % D
iscr
epan
cy
-20 -15 -10 -5 0 5 10 15 200.8
0.9
1
1.1
Horizontal Position [cm]
Nor
mal
ized
Det
ecto
r S
igna
l
-15 -10 -5 0 5 10 150.8
0.9
1
1.1
Vertical Position [cm]
Nor
mal
ized
Det
ecto
r S
igna
l
3.1.2 Object Projection Validation
The horizontal profile of the calibrated detector signal for the measured and simulated projection
image of the 16.4 cm diameter cylinder is shown in Figure 12 along with the LPD profile. The
largest discrepancies can be found close to the edge of the water cylinder and at the center of the
cylinder, all of which have an absolute discrepancy less than 6%. The discrepancies at the edges
are believed to be due to misalignment of the phantom in the measurements and the lack of off-
focal radiation in the MC simulation. As for the center the small differences in the signal are
inflated due to the relative nature of the LPD measurement. Overall the comparison between the
simulation and measurement shows excellent agreement with a NCC value of 0.99.
3.1.3 Scatter Component Validation
The resulting SOR and SPR measurements for the 1.5 cm lead disc along with the CBCT MC
estimate are shown in Figure 13. A large discrepancy between the measured and simulated
scatter signal in the region surrounding the edge of the cylinder, located around 12.8 cm in the
projection image, can clearly be seen in the SOR signal. The discrepancy is also present in the
SPR data but it is difficult to see because the primary signal, Pobj, dominates the signal at these
locations. In both the SPR and SOR signals measured using the beam stop technique we get
invalid negative values of the functions. This discrepancy arises due to the assumptions made in
calculating the scatter from the object, S, in Eq. (29). For this estimate of the scatter to hold, the
30
Figure 12: Cylindrical 16.4 cm diameter water phantom validation; (a) horizontal profile of a open-field normalized x-ray projection taken with a SAD of 100 cm and ADD of 56 cm for simulated and measured data. (b) The local percent discrepancy (LPD) between the measured and simulated horizontal profiles.
(b)(a)
-20 -10 0 10 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Horizontal Position [cm]
Flo
od N
orm
aliz
ed D
etec
tor
Sig
nal
Simulated
Measured
-20 -10 0 10 20-8
-6
-4
-2
0
2
4
6
8
Horizontal Position [cm]
Loca
l % D
iscr
epan
cy
off-focal and secondary source photons, Xu, must travel a pathlength through the object similar to
the photons emanating from the focal spot. The assumption is more realistic near the center of
the cylinder, which lies on the central part of beam and is the thickest part of the object, but as
we move to the edge of the object the validity of the assumption deteriorates. The reason for this
is that the pathlength through the cylinder for primary photons traveling near the edge of the
cylinder is changing rapidly between photons arriving at adjacent horizontal pixels. It is thus
more likely that at the edge of cylinder the pathlength of the focal photons would differ from the
contamination photons.
(a) (b)
Figure 13: Scatter signal validation for 16.4 cm diameter water cylinder using (a) scatter-to-open-field ratio (SOR) and (b) scatter-to-primary ratio (SPR) for measured and MC simulated data generated with a SAD of 100 cm and ADD of 56 cm. The measured data was calculated using the 15 mm lead disc beam stop device.
The results from the lead bar beam stop experiments (Figure 8) for the two ADD imaging
configurations are shown in Figure 14. The three profiles confirm that the discrepancies between
the MC scatter estimates and the beam stop measurements are a result of the measurement
method (see MC modeled beam stop) and not from a simulation error. It can also be seen in the
results for the SPR data that the beam stop measurement technique tends to underestimate the
SPR. In comparing the simulated SPR to the simulated beam stop measurement the beam stop
SPR result is underestimated by a factor of 14% and 3% for the ADD of 30 and 56 cm,
respectively.
31
-20 -10 0 10 20-0.01
0
0.01
0.02
0.03
Horizontal Position [cm]
SO
R
Measured
Simulated
-20 -10 0 10 20-0.1
0
0.1
0.2
0.3
0.4
0.5
Horizontal Position [cm]
SP
R
There is still some noticeable difference between the simulated and measured lead bar SOR and
SPR results at the edges of the cylinder. The most likely source of this difference lies in the
contamination component of the measured and simulated signal. As previously mentioned, the
contamination component consists of both the off-focal bremsstrahlung photons and the
secondary source consisting of photons scattered inside the x-ray tube (e.g. collimators, filters,
housing). Figure 15 shows a plot of the percent contamination in the open-field exposure
determined by the lead bar measurement and corresponding simulation. In addition the percent
32
ADD
30 cm 56 cm
(a) (b)
(c) (d)
Figure 14: SOR and SPR for measured and simulated data of a 16.4 cm diameter cylinder imaged with an ADD of 30 (a,c) and 56 cm (b,d), both with a SAD of 100 cm. The measured data (crosses) was estimated using the lead strip beam stop device. The simulated scatter data (triangles) is found by keeping track of photons that scatter inside the object during the CBCT MC simulation, whereas the simulated beam stop data (squares) comes from simulating the lead strip beam stop approach using the CBCT MC system.
-20 -10 0 10 20-0.01
0
0.01
0.02
0.03
0.04
Horizontal Position [cm]
SO
R
Simulated
Measured Beam StopSimulated Beam Stop
-20 -10 0 10 20-0.01
0
0.01
0.02
0.03
0.04
Horizontal Position [cm]
SO
R
-20 -10 0 10 20-0.2
0
0.2
0.4
0.6
0.8
1
Horizontal Position [cm]
SP
R
-20 -10 0 10 20-0.2
0
0.2
0.4
0.6
0.8
1
Horizontal Position [cm]
SP
R
contamination profile, defined by photons originating outside the focal spot region, is shown for
simulated contamination photons determined through back-projection. Looking at the
contamination profiles two things are quite clear, first the simulation is significantly different
from the measured profile (57% less on average) and second the beam stop estimate of the
simulated contamination is less than the actual contamination in the simulation, underestimating
the contamination by 15% on average.
3.2 MC Scatter Distribution Simulations
Figure 16 plots the horizontal SOCR profiles for the two different cylinder diameters (16.4 and
30.6 cm) at various ADD configurations [geometry 1 through 5 listed in Table 2 on p. 26] and
the four different cone angles (1.4, 2.8, 5.7, and 11.3). The effect of varying the ADD is quite
pronounced for all cone angles and cylinder diameters. As a reminder, the ADD is related to the
air gap, xgap, through the size of the cylinder diameter as:
x gap=ADD−
d cyl
2. (33)
The plots in Figure 16 clearly show a decrease in scatter as the air gap is increased. Neitzel
showed, that under low and medium scatter conditions, air gaps of 20 cm are more effective than
using conventional grids in digital radiography [58].
33
Figure 15: The percent of contaminating photons, Xu, in the open-field signal for a source-to-detector distance (SDD) of 156 cm. The “Measured Pb Line” and “Simulated Pb Line” profiles are for experimentally measured and MC simulated estimates using the lead bar beam stop, respectively. The “Simulated” data is determined by separating photons originating from the focal spot from those outside the focal spot region with a tolerance of +/-0.1 mm.
-20 -10 0 10 200
5
10
15
20
Horizontal Position [cm]
% C
onta
min
atio
n
Measured Pb Line
Simulated
Simulated Pb Line
Figure 17 plots various SAD configurations (geometry 5 through 7 listed in Table 2 on p. 26) for
the same set of cone angles and cylinder diameter dimensions as in Figure 16. The effect of
varying the SAD distance is less pronounced than the effect of changing the ADD, specifically in
the smaller cylinder. Looking at the data for a cone angle of 11.3 and the 16.4 cm diameter
cylinder a change in the SAD from 100 to 50 cm results in a percentage decrease in the SOCR at
the center by a factor of 11.5%. Comparatively a change in the ADD from 56 to 9 cm results in a
623% increase in the SOCR. A more substantial change in the SOCR value between the 50 and
75 cm SAD for the 30.6 cm diameter cylinder is believed to be due to the increase in the exposed
portion of cylinder in the 75 cm SAD. As can be seen in Figure 16 and 17, both the air gap and
the cone angle play a defining role in the resulting scatter distribution at the detector. Specifically
we have found that the combined effect of the cone angle and air gap on the resulting SPR and
SOCR center pixel values can be fit to the generalized model
f x gap ,cone=c1 1
c2 x gap11−1
c3 cone1. (34)
The resulting fitting parameters (c1, c2, and c3) and the coefficient of determination, R2, values for
the two different cylinder diameters are listed in Table 3. A contour plot for the SPR and SOCR
fitted functions, along with a random sample of actual data points to demonstrate the goodness of
fit, is shown in Figure 18.
34
Table 3: Fitting parameters and associated coefficient of determination, R2, value for SPR and SOCR data fit to Eq. (34) for the 16.4 and 30.6 cm diameter cylinders. The units of xgap and Φcone in Eq. (34) are centimeters and degrees, respectively.
16.4 cm 30.6 cm
SPR SOCR SPR SOCR
c1 6.91 0.21 36.64 0.08
c2 0.20 0.20 0.07 0.07
c3 0.11 0.12 0.04 0.03
R2 0.99 0.99 1.00 1.00
dcyl=16.4cm dcyl=30.6cm
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 16: Horizontal profiles of SOCR for varying ADD configurations (1-5 in Table 2 on p. 26), for the 16.4 cm (a, c, e, and g) and 30.6 cm (b, d, f, and h) diameter cylinders at each of the four cone angle configurations (1.4, 2.8, 5.7, and 11.3°, corresponding to a field of view at 100 cm from the source of 2.4, 4.9, 10.0, and 19.8 cm, respectively).
35
-60 -40 -20 0 20 40 600
0.02
0.04
0.06
0.08
0.1
Horizontal Position [cm]
SO
CR
cone=1.4o
ADD=9cm
ADD=18cm
ADD=30cmADD=44cm
ADD=56cm
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
0.02
0.025
0.03
Horizontal Position [cm]
SO
CR
cone=1.4o
-60 -40 -20 0 20 40 600
0.02
0.04
0.06
0.08
0.1
Horizontal Position [cm]
SO
CR
cone
=2.8o
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
0.02
0.025
0.03
Horizontal Position [cm]
SO
CR
cone
=2.8o
-60 -40 -20 0 20 40 600
0.02
0.04
0.06
0.08
0.1
Horizontal Position [cm]
SO
CR
cone=5.7o
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
0.02
0.025
0.03
Horizontal Position [cm]
SO
CR
cone=5.7o
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
0.02
0.025
0.03
Horizontal Position [cm]
SO
CR
cone
=11.3o
-60 -40 -20 0 20 40 600
0.02
0.04
0.06
0.08
0.1
Horizontal Position [cm]
SO
CR
cone
=11.3o
dcyl=16.4cm dcyl=30.6cm
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 17: Horizontal profiles of scatter distributions for varying SAD configurations (5-7 inTable 2 on p. 26), for the 16.4 cm (a, c, e, and g) and 30.6 cm (b, d, f, and h) diameter cylinders at each of the four cone angle configurations (1.4, 2.8, 5.7, and 11.3°, corresponding to a field of view at 100 cm from the source of 2.4, 4.9, 10.0, and 19.8 cm, respectively).
36
-60 -40 -20 0 20 40 600
0.004
0.008
0.012
0.016
0.02
Horizontal Position [cm]
SO
CR
cone
=1.4o
SAD=100cm
SAD=75cm
SAD=50cm
-60 -40 -20 0 20 40 600
0.002
0.004
0.006
0.008
0.01
Horizontal Position [cm]
SO
CR
cone
=1.4o
-60 -40 -20 0 20 40 600
0.004
0.008
0.012
0.016
0.02
Horizontal Position [cm]
SO
CR
cone
=2.8o
-60 -40 -20 0 20 40 600
0.002
0.004
0.006
0.008
0.01
Horizontal Position [cm]
SO
CR
cone
=2.8o
-60 -40 -20 0 20 40 600
0.004
0.008
0.012
0.016
0.02
Horizontal Position [cm]
SO
CR
cone
=5.7o
-60 -40 -20 0 20 40 600
0.002
0.004
0.006
0.008
0.01
Horizontal Position [cm]
SO
CR
cone
=5.7o
-60 -40 -20 0 20 40 600
0.004
0.008
0.012
0.016
0.02
Horizontal Position [cm]
SO
CR
cone
=11.3o
-60 -40 -20 0 20 40 600
0.002
0.004
0.006
0.008
0.01
Horizontal Position [cm]
SO
CR
cone
=11.3o
dcyl=16.4cm dcyl=30.6cmS
PR
(a) (b)
SO
CR
(c) (d)
Figure 18: Contour plots of fit of SPR (a,b) and SOCR (c,d) center pixel values using Eq. (17) from the 16.4 and 30.6 cm diameter water cylinder MC simulations. The contours lines are spaced equally with (a) 0.25, (b) 1, (c) 0.01, and (d) 0.002 increments. The fitting parameters and resulting R2 values are found in Table 3. Each plot also displays three data points (+) drawn (randomly) from the MC simulations to illustrate goodness of fit.
The effects of the different bowtie filters on the resulting scatter distribution are shown in Figure
19 and 20. The simulated contributions to the total scatter distribution of photons that just
undergo either Compton or Rayleigh scattering and those that underwent both interaction types
for a 16.4 and 30.6 cm diameter cylinder with and without bowtie filtration are plotted as
horizontal profiles in Figure 19. The resulting scatter contributions for Compton and Rayleigh
scattering are similar in shape to those published by Kyriakou et al. [70]. In the case without a
bowtie filter in place, see Figure 19 (a) and (b), Compton scattering can be seen to contribute to a
37
Air Gap [cm]
Con
e A
ngle
[de
gree
s]
0.91 0.33
0.100.25
0.5
0.75
1
1.5
2
1.25
0 10 20 30 40 50 600
5
10
15
Air Gap [cm]
Con
e A
ngle
[de
gree
s]
5.49 2.00
0.411
2
3
45
67
810
0 10 20 30 40 50 600
5
10
15
Air Gap [cm]
Con
e A
ngle
[de
gree
s]
0.005
0.026
0.0250.01
0.02
0.03
0.040.
050.07
0 10 20 30 40 50 600
5
10
15
Air Gap [cm]
Con
e A
ngle
[de
gree
s]
0.005
0.006
0.0010.002
0.004
0.00
6
0.00
80.010.01
20.01
6
0 10 20 30 40 50 600
5
10
15
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
Horizontal Position [cm]
SO
CR
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
Horizontal Position [cm]
SO
CR
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
Horizontal Position [cm]
SO
CR
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
Horizontal Position [cm]
SO
CR
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
Horizontal Position [cm]
SO
CR
Total
ComptonRayleigh
Both
-60 -40 -20 0 20 40 600
0.005
0.01
0.015
Horizontal Position [cm]
SO
CR
larger portion of the total signal but it is the Rayleigh scattering that creates the signal with more
structure due to it's forward peaked angular distribution. The structure is located towards the
edges of the cylinder where photons are less likely to undergo multiple interactions including
photoelectric absorption, which is more likely in the thicker center of the cylinder.
16.4 cm 30.6 cm
With
out B
owtie
(a) (b)
F1
Bow
tie
(c) (d)
Cus
tom
Bow
tie
(e) (f)
Figure 19: Horizontal profiles of the SOCR signal for the 16.4 (a, c, e) and 30.6 cm (b, d, f) diameter water cylinders broken into different interaction contributions for a geometric configuration with a SAD of 100 cm, an ADD of 56 cm, and a cone angle of 11.3°; without any bowtie filtration (a, b), with the F1 bowtie (c, d) and with the custom bowtie (e, f). The percent decrease for the total SOCR signal at the center for the 16.4 cm cylinder was 6 and 34% for the F1 and custom bowtie, respectively; for the 30.6 cm cylinder the percent decrease was 19 and 55% for the F1 and custom bowtie, respectively. In all cases except (c) using a bowtie filter results not only in a reduction in the magnitude of the scatter but also in a reduction to the structure in the SOCR profile predominately caused by the coherent scatter.
A decrease in the total SOCR profiles for both cylinders is seen when either bowtie filter is in
place. The average percent decrease of the total SOCR, across the entire field of view of the
38
projection image, when using the F1 filter for the 16.4 and 30.6 cm cylinder was 9 and 27%,
respectively. The average percent decrease for the custom bowtie filter was 35 and 62% for the
16.4 and 30.6 cm cylinder, respectively. The custom bowtie filter is more effective for both
cylinders in decreasing the SOCR and minimizing the structure that is predominantly created by
the coherent portion of the signal.
(a) (b)
Figure 20: The the horizontal SPR signal profile measured at the center of the detector for the 16.4 (a) and 30.6 cm (b) diameter water cylinders with and without bowtie filtration. The percent decrease for the 16.4 cm diameter cylinder when using the F1 and custom bowtie filters measured at the center was 11 and 30%, respectively. The percent decrease for the 30.6 cm diameter cylinder when using the F1 and custom bowtie filters measured at the center was 27 and 56%, respectively.
A decrease in the SPR profile is also found for both cylinders (Figure 20) when either bowtie is
used, with the large cylinder again seeing a greater decrease. The SPR , unlike the SOCR, the
decrease is located primarily within the central portion of the cylinder with the SPR actually
increasing near the edge of the cylinder and in the open air portion of the signal. The largest
increase in the SPR can be seen in the case of the large cylinder with the custom bowtie, with an
increase greater than 100% in portions of the signal. The increase in the SPR signal is related to
the decrease in the primary caused by the use of the bowtie filter and not an actual increase in the
scatter as can be seen in the SOCR profiles. The bowtie itself acts as an additional secondary
source of scattered photons in the x-ray beam. Figure 21 shows the horizontal profile of the
percent contamination in the open-field with and without the bowtie filters in place. As expected
there is an increase in the contamination signal which increases with distance from the central
axis. The lack of symmetry in the signal is due to the heel effect present in the open-field signal
used in the denominator of this ratio.
39
-30 -20 -10 0 10 20 300
0.25
0.5
0.75
1
Horizontal Position [cm]
SP
R
Custom Bowtie
F1 BowtieWithout Bowtie
-30 -20 -10 0 10 20 300
1
2
3
Horizontal Position [cm]
SP
R
4 Discussion and Conclusions
We have shown through experiments that the CBCT MC model presented accurately estimates
primary and scatter fluences in the CBCT imaging components and geometry. These
experimental measurements and their simulated counterparts have provided insight into
techniques for measuring object dependent scatter and the open-field response of the detector. In
addition the measurements allowed us to recognize limitations in the simulations due to the
contamination photons in the fluence emitted from the x-ray source.
In order to properly compare the measured and simulated data it is important to correct the
measured detector response to account for this variation. Variations in the detector response
between pixels exposed to the same fluence were found when examining the open-field detector
response. In our measurements, using the Varian 4030A detector, up to a 6% signal difference
between pixels was found.
Discrepancies between simulated and beam stop measured scatter profiles led us to investigate
the accuracy of the beam stop measurement technique along with potential shortcomings of the
MC model. The measurement technique itself was examined by using simulations. Limitations in
the beam stop measurement appear to arise due to the difficulty in estimating the attenuation of
the contamination photons by the object itself. It is our conclusion that the beam stop technique
works best for estimating the scatter for objects that are of uniform thickness. This allows the
40
Figure 21: The percentage of contamination photons, Xu, in the open-field signal with and without the bowtie filter in place measured at a SDD of 156 cm. The contribution of the Xu signal significantly increased as a percentage of the total open-field signal when either of the bowtie filters are in place.
-30 -20 -10 0 10 20 300
5
10
15
20
25
30
Horizontal Position [cm]
% C
onta
min
atio
n
Custom Bowtie
F1 BowtieWithout Bowtie
assumption that contamination photons travel a similar pathlength as the primary photons to be
closer to the truth. It would be worthwhile to develop alternative measurement techniques to
quantify scatter profiles from objects with varying thickness, but it is beyond the scope of this
research.
The CBCT model shows a significant discrepancy (5% absolute difference) between the
simulated and measured contamination signal (see Figure 15). This difference has two potential
sources: (i) secondary scatter sources that were not modeled, such as the CsI(Tl) scintillator and
the detector cover and (ii) off-focal radiation. Secondary scatter from the detector components
should have minimal effect of image quality and measurements as their path should not be
diverted greatly due to the short distance these scatter photons can travel before detection and
thus were not included in our simulations. This is verified in Figure 8 from Chen et al. [35]
where the horizontal profile of the SPR signal from the CsI crystals and detector cover result in a
fairly straight line. We believe that the majority of the differences seen between the measurement
and simulation can be attributed to the off-focal component. The off-focal bremsstrahlung source
is partially caused by electrons striking the anode outside of the focal spot due to the design of
the focusing cup and electron field distortions, but primarily it is caused by electrons
backscattered at the anode that return to the anode outside the focal spot [71]. A recent
modification of the BEAMnrc code by Ali and Rogers [71] allows for proper simulation of
electrons backscattering and re-enter the anode and we plan to incorporate this modification into
the CBCT MC model in the future. Looking at the values reported by Ali and Rogers in Figure
11 [71], an estimate of the percent increase in the air kerma (Kair) at the patient plane resulting
from the inclusion of off-focal effects would be around 5-6% for our x-ray tube. Since Kair is
proportional to the fluence we can estimate that if the off-focal effects were included in the MC
model the simulated percent contamination shown in Figure 15 would increase to around 10-
11%, which would be in much closer agreement to the measured result.
The validated CBCT MC model was used to explore the effect of varying imaging and object
parameters (cylinder diameter, cone angle, source-to-axis distance, air gap, filtration) on the
resulting scatter distribution. It was shown that the air gap and the cone angle are two parameters
that have a dominant effect on the scatter produced by an object. A fitting model was generated
that accurately relates the center pixel value of either the SPR or SOCR to the cone angle and the
air gap. This model provides a useful and quick tool for estimating the scatter contribution for
41
different imaging scenarios.
In the last section the effect of bowtie filtration on the resulting scatter distribution was
investigated. The results show that the scatter signal is significantly diminished (up to 62% on
average) by the use of the bowtie filter. Not only is the scatter reduced in magnitude, the shape of
the horizontal scatter profile is flattened out, reducing the structure created by the coherent
portion of the signal when the bowtie filter is used. The flattening effect of the bowtie filter
suggests that combining the use of the bowtie filter with a simple scatter subtraction algorithm
would be an effective scatter reduction method. In the case where a scatter subtraction algorithm
is combined with the use of a bowtie the custom bowtie is the better choice for both cylinders. If
the bowtie filter is used without additional scatter correction the more effective choice is to use
the F1 bowtie for the larger cylinder and the custom filter for the smaller cylinder. The custom
filter is more effective in reducing the overall magnitude and structure in the scatter distribution
for the larger cylinder, but it also substantially diminishes the primary in the outer region of the
cylinder so that the SPR actually increases in this region (Figure 20).
Despite the bowtie's ability to reduce and flatten out the scatter distribution, it has the detrimental
effect of acting as a secondary source of photons increasing the contamination component in the
flood field. Because of this secondary source effect the benefits of the bowtie filter may decrease
to a point where it's use is actually detrimental for imaging configurations with small SPR
signals (e.g. when the cone angle is small). The contamination component will act to reduce the
high frequency components in the projection images reducing the contrast in sharp edges. This
effect can be seen in the modulation transfer function (MTF) measurements in previous work
done by Mail et al. [60] where the MTF is diminished for the line pairs per cm greater than 7.
The increase in percent contamination caused by the bowtie filter and the effect it will have on
the reconstructed CBCT images is thus worth further investigation. It would also be useful to
investigate the effect the bowtie filter will have on the off-focal portion of the contamination as it
may serve to diminish this aspect of the signal. Furthermore, the effects of varying the shape,
size, and material of the bowtie and the results this has on both the scatter and contamination is
also an area for further study. We conducted an initial investigation into the effects of different
bowtie materials which was published in the SPIE proceedings [43]. Also included in that study
are the effect of the F1 of the custom bowtie on anthropomorphic phantoms, which shall be
discussed more in the next chapter.
42
Qualitatively the results provided by our MC simulation appear to give similar results to recent
MC analysis and measurement found in the literature [34], [35], [70]. The shape of the scatter
profiles determined by our simulation for the 16.4 cm cylinders contain similar structural
elements as in MC simulations of a 14 cm diameter cylinder with an imaging magnification
factor of 1.5 shown in Figure 11 of the article by Chen et al. [35] using a GEANT4 based system.
The deviations between the simulated and the beam stop measured scatter also agree with our
results shown in Figure 11.b, with the largest deviations found around the edges of the cylinder.
The SPR results shown in Figure 12 by Chen et al. [35] also show a similar shape and magnitude
(SPR=0.55 at the center) when compared to our results in Figure 14(d) with a similar
magnification factor and object (SPR=0.45). Our results show a slightly lower SPR magnitude
because of the larger air gap and higher kVp used in our simulations. The results in Figure 12 by
Chen et al. [35] also show a similar underestimate of the SPR when comparing the beam stop
measurements to the simulated results.
Our findings that the structure found in the scatter distribution is largely a result of the coherent
scatter is confirmed in work done by Kyriakou et al. [70] Similar structural patterns are shown in
the both the total scatter signal and the separated coherent scattering components, determined by
a hybrid MC simulation, shown in Figure 3 of their article [70]. In another article published by
Kyriakou and Kalender [34] the effects of varying the cone angle and air gap on the SPR are
investigated for a flat panel detector CT system again using a hybrid MC system. Their results
shown in Figures 4 and 6 [34] appear to agree with the relationships determined by our work.
The results shown in their Figures 4 and 6 [34] have slightly lower SPR values than those
predicted using Eq. (34) which is probably a result of the fact that the SAD is larger and a bowtie
is not used in the simulations to predict the coefficients found in Table 3.
The nature of the scatter distributions shown in this work suggests the possibility of reducing the
magnitude and structure found in the scatter distribution by the selection of imaging parameters.
There is definite structure in the scatter distribution that is largely a result of the coherent scatter.
The structure and magnitude of the scatter distribution can be diminished not only through the
reduction of the cone angle and increasing the air gap but also with the proper selection of
bowtie filtration. A simplified scatter distribution reduces the complexity of the function needed
to correct for the scatter distribution in x-ray images. If the structure is significantly diminished it
may result in sufficient scatter reduction through the application of a simple scatter subtraction
43
algorithm, such as subtracting a constant [52]. The database of scatter distributions generated in
this chapter provides a starting point for determining the types of basis functions required for a
model driven approach to reduce the number of particles used in scatter distribution estimation.
In the next chapter the Fourier analysis of the scatter distribution will be applied in an attempt to
characterize the extent of the structure seen in the scatter distribution. Additionally, the potential
of using a sum of sines and cosines as a basis function for the scatter distribution is proposed
and investigated using the information gleaned from the Fourier analysis of the scatter
distribution.
44
Chapter 3 The Spectrum of the X-ray Scatter Distribution
in CBCT Projection Images
1 Introduction
The previous chapter looked at the effect of various imaging parameters (e.g. air gap,
compensator, cone angle) on the scatter distribution [42]. A key finding from this work was that
the use of a compensator not only reduced the magnitude of the scatter, which had been
previously reported by two other publications [59], [60], but also modulated the shape of the
scatter distribution. We conducted further studies using anthropomorphic phantoms with
different compensator shapes showing that the shape of the compensator played a significant role
in both the magnitude and shape of the resulting scatter distribution [43]. It is the goal of this
chapter to characterize the structure and complexity of the scatter distribution by examining the
spectrum of the spatial and angular frequencies of scatter distribution for different imaging
conditions.
It is a common perception that the scatter distribution in CBCT projection images is
predominately contained in the lower spatial frequencies of the projection image. This
assumption is central to a recent scatter correction algorithm using primary modulation [55],
[62], [72], but its validity has yet to be thoroughly explored and quantified. The domain of the
spatial and angular frequencies of the scatter distribution are quantified through the use of a
45
Fourier analysis of scatter distributions created using the validated CBCT MC model [27], [42],
described in Chapter 2. The effect of imaging geometry, object shape, and compensators on the
scatter spectrum was also investigated. It is also shown how knowledge of the scatter spectrum
can be applied to reduce the statistical noise in MC simulations of the scatter distribution using a
reduced number of photons, decreasing the computational cost of generating accurate scatter
estimates using MC methods.
2 Materials and Methods2.1 Monte Carlo Simulation System
The MC models were generated using the CBCT MC system outlined in Chapter 2 (see Figure 6
p. 17) . MC simulations were done for a water cylinder and two anthropomorphic phantoms with
and without the use of a compensator, the details of which are outlined in the following
subsections.
2.1.1 X-ray Sources and Energy
The x-ray source used in the simulations is identical to the one described in Chapter 2 in sections
2.1 and 2.2.
2.1.2 Compensators
Two different compensators were modeled for use in the MC CBCT system. Both compensators
were composed of aluminum with a density of 2.699 g/cm3. The first compensator was a model
of the Elekta F1 filter (Elekta, Crawley, West Sussex, UK), identical to the one from Chapter 2
[see Figure 9(a) p. 27]. The mathematical representation of the profile can be found in Appendix
A. The second compensator, AL16S, was designed to compensate for a cylinder with a radius of
8.2 cm using the equations outlined
Appendix A. The AL16S filter had a height
restriction limited to 3 cm which resulted in
a modulation factor of 7.9 (see Figure 22).
The modulation factor is defined as the ratio
of the maximum attenuation provided by the
filter to that at the center of the filter. The
design of AL16S is similar to that of the
46
Figure 22: Profile for compensator, AL16S, designed to compensate for a 16.4 cm diameter cylinder. AL16S is composed of aluminum with a center thickness of 1 mm and a modulation factor of 7.9.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6-1
0123
4
x [cm]
z [c
m]
custom filter outlined in Chapter 2 but it hasbeen smoothed with a zero-phase forward and
reverse average filter to remove any sharp edges that can contribute to image artifacts in the
reconstructions. In addition the AL16S is placed in the opposite direction (curved surface
towards the source) of the custom bowtie filter. Both the F1 and AL16S filters were placed 28.5
cm from the source, spanning a distance from the source of 28.5 cm to 31.5 cm.
2.1.3 Phantoms
A water cylinder with a diameter of 30.6 cm and two anthropomorphic phantoms were used to
evaluate spatial frequencies of the scatter distribution. The digital MC head (see Figure 23) and
pelvis (see Figure 24) phantoms were created from CT data of actual anthropomorphic phantoms
[73] using the ctcreate executable included in the BEAMnrc distribution [64]. The resulting
anthropormorphic phantoms had a voxel pitch of ~0.2 cm in each dimension. Each voxel in the
head and pelvis phantom was also assigned PEGS4 cross-sectional data of either air, soft tissue,
lung tissue, or cortical bone based upon the correlated CT number.
(a) (b)
Figure 23: Axial (a) and sagittal (b) slices showing density values for voxelized head phantom used in the MC simulations.
2.1.4 Imaging Geometry
The imaging geometry for the simulations is a sub-sample of the geometry in our previous study
[42]. We used a single cone and fan angle of 11.3 and 21.3 degrees, respectively resulting in a
field of view at 100 cm of 18.8 cm × 9.9 cm (width × height) for all the phantoms. The source-
to-axis distance (SAD) was 100 cm and the axis-to-detector distance (ADD) was 56 cm for the
47
x [cm]
z [c
m]
-10 -5 0 5 10
-10
-5
0
5
10
z [cm]
y [c
m]
-10 -5 0 5 10
-10
-5
0
5
10
g/cm
30
0.5
1
1.5
2
head and pelvis phantom and simulations were done with and without the AL16S compensator.
(a) (b)
Figure 24: (a) Axial and (B) coronal slices of the density values for the voxelized pelvis phantom used in the MC simulations.
For the water cylinder we used the varying ADD data {18, 30, 44, and 56 cm} with a SAD of
100 cm data from Chapter 2. Additional simulations with the water cylinder were done with both
the compensators for the ADD of 18 cm. The ADD of 18 cm was chosen because previous
research (see Figure 16 on p. 35) showed this configuration to have the most complex scatter
distribution.
For the water cylinder only a single projection image is needed to capture all the frequencies of
scatter distribution as the object is rotationally invariant. As the head and pelvis phantom are not
rotationally invariant projection images were collected at angular increments of 1 degree over
360 degrees around the angle of rotation, .
The projection images were computed for a virtual detector having a height and width of 80 and
120 cm, respectively except in the case of the data taken from previous work (30.6 cm cylinder
with varying ADDs) where the height of the detector was limited to 60 cm. The detector
response is based on the Paxscan 4030 (Varian Medical Systems, Inc., Palo Alto, California,
USA) which uses a cesium iodide scintillator. The simulated detector pixel pitch was 1.24 cm.
The scatter detector response was also normalized as
Sn=10000S
I open0
(35)
where S is the detector signal for the scattered photon distribution and I0open is the total detector
signal at the center pixel of an in-air measurement.
48
x [cm]
y [c
m]
-15 -10 -5 0 5 10 15
-10
-5
0
5
10
g/cm
3
0
0.5
1
1.5
2
x [cm]
z [c
m]
-15 -10 -5 0 5 10 15
-10
-5
0
5
10
2.2 Scatter Spatial Frequency
The spatial frequency for the scatter distribution, FS, was computed by first multiplying the Sn
projection data by a two-dimensional Tukey [74], also known as a tapered cosine, window then
taking the absolute magnitude of the Fourier transform. Mathematically this can be written as
F S (u , v ,ω )=∣F {T ( x , y)S n( x , y ,θ )}∣ (36)
where F is the Fourier transform, T is the Tukey window, x the horizontal pixel position, y the
vertical pixel position, θ is the projection angle, u the horizontal frequency, v the vertical
frequency, and the angular frequency.
To determine an estimate of the frequency span of the scatter distribution we devised the scatter
frequency width (SFW) metric. The SFW is defined as the highest absolute frequency, along a
given frequency axis (e.g. u, v, or ) at which FS is greater than 1% of the DC value of FS. The
SFW in the u direction (v=0, =0), corresponding to the horizontal frequencies is denoted SFWu.
Likewise the SFW for the vertical and angular frequencies are denoted SFWv and SFW,
respectively. The SFW is computed using interpolated values of FS to obtain a more accurate
estimate of the frequency width.
2.3 Scatter Distribution Estimation from Limited Photon Simulations
A gold standard (>109 input photons) and limited photon (106 input photons) scatter projection
simulation of the 30.6 cm diameter water cylinder with an SAD of 100 cm and ADD of 56 cm
was created with and without the AL16S and F1 compensators to evaluate the ability to recover
the 2D scatter signal from a noisy estimate. Each of the limited photon simulation (LPS) S n
signals were low-pass filtered in the frequency domain using a third-order 2D Butterworth filter
defined as
h u , v =1
1 2 uucut
2
2 vvcut
2
N B
(37)
where ucut and vcut are the cutoff frequencies, and NB is the order of the Butterworth filter.
To determine the optimal filter cutoff values brute force optimization was employed. The root-
49
mean-square error (RMSE) between the filtered LPS and the gold standard Sn signal for a range
of ucut and vcut values equal spaced at increments of 0.005 between 0.01 and 0.4 cm-1 was
calculated. The ucut and vcut combination generating the minimum RMSE was selected as the
optimum. The error from the optimal result was compared to the result using the SFW values as
the cutoffs in the low-pass filter.
The ability to estimate the 3D scatter distribution from a LPS was evaluated for the head and
pelvis phantoms with and without the use of the AL16S compensator. The gold standard data
used over 109 input photons per projection and the LPS data used 106 input photons per
projection. The LPS data was low-pass filtered in the frequency domain using a third-order 3D
Butterworth filter defined as:
h u , v ,=
1
1 2uucut
2
2vvcut
2
2
cut 2
N B
(38)
where cut is the cutoff frequency for the angular frequencies. The filtered LPS Sn signal was then
compared to the gold standard using the RMSE.
The effect of the angular sampling rate was investigated by using a set of LPS scatter projections
sampled at different angular increments, dθ. The LPS data was then compared to the gold
standard data (dθ=1°) by interpolating the data using Fourier interpolation after the low-pass
filter was applied. Fourier interpolation is a useful interpolation method for estimating a
continuous signal from a set of discrete samples [75], [76]. Fourier interpolation can be
computed efficiently by appropriately zero-padding the fast Fourier transform of a signal and
then taking the inverse fast Fourier transform of the padded data (see Appendix B). The number
of zeros, Mθ, to add to the spectrum data required to interpolate a 360° projection set sampled
at intervals of dθ to a sampling interval of 1 is given as
M θ =360−
360d θ
. (39)
The optimal low-pass filter cutoffs (ucut, vcut, and cut) were computed for each phantom
configuration and sampling interval, dθ, using a brute force search minimizing the RMSE
between the low-pass Fourier interpolated LPS and gold standard Sn data. The brute force search
was performed over values of ucut and vcut spanning 0.005 to 0.095 cm-1 at increments of 0.005
50
cm-1 and cut spanning 0.72 to 35.28 turn-1 at increments of 0.72 turn-1 [turn=degree/360=rad/
(2π)].
3 Results and Discussion3.1 Scatter Spatial Frequency Spectrum
3.1.1 Cylinder
The normalized detector scatter distribution signal, Sn, for the 30.6 cm diameter water cylinder
and the logarithm of the corresponding spatial frequency, FS, is shown in Figure 25 for each of
the four ADD values. As previously reported [42], [58] a decrease in the magnitude of the scatter
distribution when increasing the ADD (air gap) is clearly seen. In addition to the decrease in
magnitude a reduction in the structure in the scatter distribution (particularly in the horizontal
direction) is also noted with increasing ADD values. The magnitude of the two vertical peaks in
Sn at a horizontal position of approximately ±20 cm are clearly diminished with increasing
ADD. The spatial frequencies of the scatter distribution is, as suspected, largely contained in the
lower frequencies. As the ADD is increased the amount of signal in the higher frequencies is also
reduced. The SFW values can be found for each ADD value in Table 4. When comparing the
SFW values at the end points of the ADD values (18 and 56 cm), without a compensator, it was
found that the SFWu value decreases by 33%, whereas the SFWv value increased by 14%.
ADD=18 cm ADD=30 cm ADD=44 cm ADD=56 cm
Sn
(a) (b) (c) (d)
log
10(
FS)
(e) (f) (g) (h)
Figure 25: (a-d) The normalized detector scatter distribution, Sn, and (e-h) the corresponding logarithm of the spatial frequency, FS, for the 30.6 cm diameter water cylinder at ADD values of 18, 30, 44, and 56 cm.
51
x [cm]
y [c
m]
-40 -20 0 20 40
-20
0
20
20 40 60
20
400
100
200
u [1/cm]-0.4 -0.2 0 0.2 0.4
u [1/cm]-0.4 -0.2 0 0.2 0.4
10 20 30
10
20
30
40
0
2
4
6
u [1/cm]
v [1
/cm
]
-0.4 -0.2 0 0.2 0.4-0.4
-0.2
0
0.2
0.4
x [cm]
y [c
m]
-40 -20 0 20 40
-20
0
20
u [1/cm]-0.4 -0.2 0 0.2 0.4
x [cm]
y [c
m]
-40 -20 0 20 40
-20
0
20
x [cm]
y [c
m]
-40 -20 0 20 40
-20
0
20
Table 4: Spatial frequency width (SFW) values (in cm-1) along the horizontal and vertical (u,v) frequency directions for the 30.6 cm diameter cylinder for various ADD, compensator, and detector configurations.
ADD [cm] 18 30 44 56
Compensator AL16S F1 None None None None None
Detector Size [cm×cm] 120×80 120×80 120×80 120×60 120×60 120×60 120×60
SFWu [cm-1] 0.039 0.041 0.059 0.060 0.058 0.054 0.039
SFWv [cm-1] 0.048 0.048 0.046 0.065 0.079 0.080 0.080
The effect of using a compensator on the spatial frequencies of the scatter distribution for the
30.6 cm diameter cylinder with an ADD of 18 cm can be seen in Figure 26. Both compensators
are effective in diminishing both the magnitude and structure of Sn. Qualitatively it is seen that
the AL16S compensator is more effective than the F1 at minimizing the magnitude and structure
of the scatter distribution. The effect of the compensators on Sn is also translated into a reduction
in the magnitude of the higher spatial frequencies in FS. The FS spectrum for the AL16S
compensator sees a drop in the magnitude for all frequencies (see Figure 27). For both
compensators the frequencies in the v direction remain largely unchanged due to the fact that the
compensators are uniform in the y direction. In the u direction a qualitative decrease in the
magnitude of the higher frequencies (>0.5 cm-1) can be seen for both compensators (see Figure
26). The SFW is fairly adept at quantifying these changes (see Table 4) as the SFWv remains
constant with and without the compensators at an ADD of 18 cm. It is worth noting the
difference in the SFWv values (0.05 and 0.07 cm-1) for the two similar cases without a
compensator at an ADD of 18 cm in Table 4. The reason for the difference is a result of the
detector size difference in the y-direction between the two data sets. The data set with the smaller
detector height (60 cm) truncates the scatter distribution [see Figure 25(a)] resulting in a
spectrum with higher frequencies in v dimension of FS. When either compensator is used the
SFWu decreases by a factor of 33% when compared to the case without a compensator. The use
of the AL16S compensator at an ADD of 18 cm is equivalent to an SFWu value for an ADD of
56 cm (air gap ~40 cm) without a compensator. One limitation of the SFW metric, due to the fact
the SFW is calculated relative to the DC signal of FS, is it does not capture the overall decrease in
the magnitude of FS that is seen for the AL16S compensator but not as much for the F1 filter.
52
No Compensator F1 AL16S
S
n
(a) (b) (c)
log
10(
FS
)
(d) (e) (f)
Figure 26: The normalized scatter distribution (a-c) and the corresponding logarithm of the FS (d-f) for different bowtie filter implementations: (a,d) none, (b,e) F1, (c,f) AL16S.
(a)
(b)
Figure 27: (a) Horizontal profiles along u axis (v=0) and (b) vertical profiles along v axis (u=0) for the spatial frequencies of Sn for the 30.6 cm diameter water cylinder with different compensator configurations (none, F1, and AL16S).
3.1.2 Anthropomorphic Phantoms
Sample Sn projections for the head and pelvis phantom with and without the AL16S compensator
are shown in Figure 28. The images clearly show a decrease in the magnitude and structure of
the scatter when using a compensator. A marked decrease is seen in the high signal intensity
structures at the edges of the phantoms (located around x equals ±25 cm for the pelvis and x
equals ±10 cm for the head). These high intensity peaks are due to the increase in coherent
scattering that is allowed to escape at the air interface of the phantom. The compensator
53
x [cm]
y [c
m]
-40 -20 0 20 40
-20
0
20
x [cm]
y [c
m]
-40 -20 0 20 40
-20
0
20
u [1/cm]
v [1
/cm
]
-0.4 -0.2 0 0.2-0.4
-0.2
0
0.2
0.4
u [1/cm]-0.4 -0.2 0 0.2
u [1/cm]-0.4 -0.2 0 0.2
x [cm]
y [c
m]
-40 -20 0 20 40
-20
0
20
0 0.02 0.04 0.06 0.08 0.10
1
2
3
4x 10
5
u [1/cm]
FS
No Compensator
F1 Compensator
AL16S Compensator
0 0.02 0.04 0.06 0.08 0.10
1
2
3
4x 10
5
FS
v [1/cm]
asfda
10 20 30
10
20
30
40
50
600
2
4
6
asfda
20 40 60
20
40
600
100
200
significantly reduces the fluence in these regions, resulting in the decrease in the scatter signal.
In the pelvis projection data [Figure 28 (a) and (c)] the scatter intensity peak, located
approximately at (x,y)=(0,-10) cm, that results from the air gap in the phantom's legs is not
diminished by the use of a compensator, as the fluence in this region is not significantly
attenuated by the compensator.
Pelvis Head
With
out
Com
pens
ator
(a) (b)
AL1
6S C
ompe
nsat
or
(c)
\
(d)
Figure 28: Scatter distribution projections, Sn, for frontal views (θ=0°) of the pelvis (a,c) and head (b,d) phantom. Images (a) and (b) are without the use of a compensator, whereas images (c) and (d) are with a compensator. An increase in the signal intensity of Sn can clearly be seen at the edges of the pelvis (a) and head (b) phantom when a compensator is not used due to the increased coherent scattering contribution, when a compensator is used [(c)and (d)] these edge effects are significantly diminished.
Sinograms composed of the center column and row from the Sn projection data for the head and
pelvis data sets are shown in Figure 29 and Figure 30, respectively. Profiles along the central
axes of Sn are plotted in Figure 31. The sinograms clearly illustrate the periodic nature of Sn with
respect to the projection angle. Looking at the the angular axis central profiles in Figure 31(c) an
underlying function in Sn with a period of 180° can clearly be seen for both phantoms. In the
same figure it also seen that the pelvis and head phantom Sn signal are out of phase by 90°. Both
the periodicity and phase of the Sn functions are a result of the ellipsoidal shape of the phantoms.
If the pelvis and head phantoms are modeled as ellipses then at θ=0° and 180° the major axis in
54
bar
10 20
10
20
30
40
50
600
50
100
150
x [cm]
y [c
m]
-45 -30 -15 0 15 30 45
-30
-15
0
15
30
bar
10 20
10
20
30
40
50
600
50
100
150
x [cm]
y [c
m]
-45 -30 -15 0 15 30 45
-30
-15
0
15
30
x [cm]
y [c
m]
-45 -30 -15 0 15 30 45
-30
-15
0
15
30
x [cm]
y [c
m]
-45 -30 -15 0 15 30 45
-30
-15
0
15
30
the head phantom is parallel to the beam direction, whereas the minor axis of the pelvis is
parallel to the beam direction and vice versa for the phantoms at θ=90° and 270°. When the
major axis (thicker portion) of the phantom is parallel to the beam fewer scattering photons are
allowed to escape the phantom compared to when the minor axis is parallel resulting in the
“valleys” of the Sn signal shown in Figure 31(c).
The spectral analysis of the Sn signal for the pelvis and head phantoms with and without the use
of a compensator are found in Figure 32 and Figure 33, respectively. For all configurations the
majority of the signal is found in the low spatial and angular frequencies. The use of the
compensator decreases the high frequencies, that are largely due to noise, for both phantoms.
55
With
out
Com
pens
ator
(a) (b)
Al1
6S C
ompe
nsat
or
(c) (d)
Figure 29: Scatter sinograms for the center row (a,c) and center column (b,d) of Sn for the pelvis phantom. The first row of images (a,b) is without a compensator and the second row (c,d) is with the AL16S compensator. Periodic signals can clearly be seen in the angular direction due to the ellipsoidal shape of the pelvis phantom.
test
10 20
10
20
30
40
50
600
50
100
150test
10 20
10
20
30
40
50
600
50
100
150
x [cm]
[d
egre
es]
-45 -30 -15 0 15 30 450
50
100
150
200
250
300
350
y [cm]
[d
egre
es]
-30 -15 0 15 300
50
100
150
200
250
300
350
x [cm]
[d
egre
es]
-45 -30 -15 0 15 30 450
50
100
150
200
250
300
350
y [cm]
[d
egre
es]
-30 -15 0 15 300
50
100
150
200
250
300
350
The SFW values for each of the three axes are found in Table 5. For the pelvis phantom the
SFWu metric increases with the use of the compensator. This can be explained by two factors the
first of which is the fairly large peak in scatter caused by the air gap between the phantoms legs
[see Figure 24(b) and Figure 28(a) and 28(c)] which is not significantly diminished by the use of
the compensator. The second factor has to do with the fact that the SFW does not capture the
overall magnitude decrease between two different spectrums due to the DC normalization. The
head phantom sees a decrease of ~26% in the SFWu when the compensator is employed. For
both phantoms with and without the use of the AL16S the horizontal frequencies of the scatter
distribution are below 0.05 cm-1 suggesting a pixel pitch a large as 10 cm in the x dimension
could be used to capture horizontal frequencies of Sn.
With
out
Com
pens
ator
(a) (b)
Al1
6S C
ompe
nsat
or
(c) (d)
Figure 30: Scatter sinograms for the center row (a,c) and column (b,d) of Sn for the head phantom. The first row of images (a,b) is without a compensator and the second row (c,d) is with the AL16S compensator.
56
test
10 20
10
20
30
40
50
600
50
100
150
test
10 20
10
20
30
40
50
600
50
100
150x [cm]
[d
egre
es]
-45 -30 -15 0 15 30 450
50
100
150
200
250
300
350
y [cm]
[d
egre
es]
-30 -15 0 15 300
50
100
150
200
250
300
350
x [cm]
[d
egre
es]
-45 -30 -15 0 15 30 450
50
100
150
200
250
300
350
y [cm]
[d
egre
es]
-30 -15 0 15 300
50
100
150
200
250
300
350
(a) (b) (c)
Figure 31: (a) Central horizontal profile, (b) central vertical profile, and (c) central angular profile of Sn for both head and pelvis phantoms with (dashed lines) and without (solid lines) the use of the AL16S compensator.
The SFWv values remain largely unchanged when comparing the cases with and without the use
of the compensator for both the head and pelvis phantom. This corresponds to the fact that the
compensator does not modulate the fluence in the y direction. The SFWv values are slightly
higher for the head phantom when compared to the pelvis phantom, but both are below 0.07 cm -1
suggesting a minimum pixel pitch of ~7 cm in the y dimension could be used to properly sample
the vertical component of the scatter distribution.
The SFW value is decreased when using the compensator with the pelvis phantom by 31%,
whereas for the head phantom the value remains constant but less then the value for the pelvis
phantom. The angular frequency for all configurations is below 6.5 turn -1 corresponding to a
minimum angular sampling of every 25 degrees to properly capture the angular frequencies in Sn.
There is also a dominant frequency signal located off the central axes which can be seen in the
images of the u-plane shown in Figures 32 and 33. The slope of the off-axis component is
approximately -360 cm/turn or -1 cm/°. Looking at the x-θ sinograms in Figures 29 and 30 an off
axis pattern can be seen corresponding to the off axis spectrum signal in FS. The most prominent
example is seen in Figure 29(a) with the structures found at linear locations defined by θ=-x+b,
for b=90° and 270°. These structures are a result of the rotationally variant elliptical shape of the
head and pelvis and the resulting peaks and valleys caused by the loss and addition of coherent
and incoherent scattering during rotation.
57
-40 -20 0 20 400
50
100
150
200
y [cm]S
n
0 90 180 270 3600
50
100
150
200
Sn
[degree]-60 -40 -20 0 20 40 600
50
100
150
200
x [cm]
Sn
Pelvis
Pelvis with AL16SHead
Head with AL16S
With
out
Com
pens
ator
(a) (b) (c)
AL1
6S C
ompe
nsat
or
(d) (e) (f)
Figure 32: Logarithm images of FS for the pelvis phantom with (a-c) and without (d-f) the use of the AL16S compensator for the three central planes (u-v, v-, and u-). A strong off axis signal with a slope of -1 cm/turn is seen in the image of the u-ω plane shown (c) and (f), resulting from the rotationally variant shape of the phantom.
With
out
Com
pens
ator
(a) (b) (c)
Al1
6S C
ompe
nsat
or
(d) (e) (f)
Figure 33: Logarithm images of FS for the head phantom with (a-c) and without (d-f) the use of the AL16S compensator for the three central planes (u-v, v-ω, and u-ω). Similar to the pelvis phantom an off axis frequency component is seen in the u-ω plane shown in (c) and (f).
58
test
10 20
10
20
30
40
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600
2
4
6
8
10
test
10 20
10
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2
4
6
8
10
test
10 20
10
20
30
40
50
600
2
4
6
8
10
test
10 20
10
20
30
40
50
600
2
4
6
8
10
u [1/cm]
v [1
/cm
]
-0.2 0 0.2
-0.2
0
0.2
v [1/cm]
[
1/tu
rn]
-0.2 0 0.2-50
-25
0
25
50
u [1/cm]
[
1/tu
rn]
-0.2 0 0.2-50
-25
0
25
50
u [1/cm]
v [1
/cm
]
-0.2 0 0.2
-0.2
0
0.2
v [1/cm]
[
1/tu
rn]
-0.2 0 0.2-50
-25
0
25
50
u [1/cm]
[
1/tu
rn]
-0.2 0 0.2-50
-25
0
25
50
u [1/cm]
v [1
/cm
]
-0.2 0 0.2
-0.2
0
0.2
v [1/cm]
[
1/tu
rn]
-0.2 0 0.2-50
-25
0
25
50
u [1/cm]
[
1/tu
rn]
-0.2 0 0.2-50
-25
0
25
50
u [1/cm]
v [1
/cm
]
-0.2 0 0.2
-0.2
0
0.2
v [1/cm]
[
1/tu
rn]
-0.2 0 0.2-50
-25
0
25
50
u [1/cm]
[
1/tu
rn]
-0.2 0 0.2-50
-25
0
25
50
Table 5: SFW values for the pelvis and head phantom with and without the use of the AL16S. A decrease in the horizontal frequencies is seen for both phantoms when a compensator is employed.
Pelvis Head
No Compensator AL16S No Compensator AL16S
SFWu [cm-1] 0.024 0.035 0.049 0.036
SFWv [cm-1] 0.043 0.043 0.063 0.064
SFWω [turn-1] 6.299 4.699 2.900 2.900
3.2 Scatter Distribution Estimation using Limited Photons
A contour plot of the RMSE values between the gold standard and filtered LPS scatter
distributions for each evaluated u and v low-pass cutoff is shown in Figure 34 along with the
shape of the corresponding optimal low-pass filter for the 30.6 cm diameter cylinder 2D scatter
data without a compensator at an ADD of 18 cm. The optimal u and v low-pass filter cutoffs and
corresponding RMSE values are given in Table 6 with and without the use of the different
compensators. A significant reduction in error (greater than 80%) is accomplished through the
use of the optimal low-pass filter when compared to the RMSE when no low-pass filter is used
(see Table 6). The mean percent absolute error was found to be below 3% for all three filtered
scatter estimates when compared to the gold standard estimates. The qualitative and quantitative
effects of the low-pass filtering approach are demonstrated in Figure 35 which shows the
unfiltered and filtered LPS Sn projection data along with the gold standard projection and percent
absolute error for the filtered data. The increase in computational efficiencies is quite significant
as the reduction in the number of photons used to calculate the low-pass filtered LPS Sn is greater
than 3 orders of magnitude.
Table 6: The optimal ucut and vcut values and corresponding RMSE. The RMSE for using no filter and using a filter with cutoffs selected from the SFW values are also shown for the case with the F1 and AL16S compensators and without the use of a compensator. The error reduction for using the optimal filter cutoffs is also presented.
No Compensator F1 AL16S
Optimal ucut [cm-1] 0.050 0.035 0.030
Optimal vcut [cm-1] 0.045 0.055 0.055
RMSE (Optimal u,v) 3.97 2.33 1.27
RMSE (SFWu,v) 4.14 2.34 1.31
RMSE (No filter) 34.88 24.24 12.74
Optimal % Error Reduction 88.63 90.38 90.04
The cutoff values determined through the optimization are fairly similar to the SFWu and SFWv
values given in Table 4 suggesting the potential of using these values in selecting filter cutoff
59
values. The RMSE values when using the SFWu and SFWv as the filter cutoffs are shown in
Table 6. The LPS low-pass filtered results for the no compensator and AL16S case using the
SFW selected cutoffs had RMSE values within 4% of the optimal value. It is also worth noting
that without any low pass filtering the LPS using the AL16S compensator resulted in an RMSE
that is almost half that of the other two configurations. This suggests that the use of this
compensator in conjunction with a low-pass filter could result in even fewer photons needed to
estimate Sn.
(a) (b)
Figure 34: (a) Contour plot of the resulting RMSE values between the gold standard and the low pass filtered limited photon simulation (LPS) Sn signals for the 30.6 cm diameter water cylinder with no compensator for a range of ucut
and vcut values. The optimal cutoff values are found when ucut and vcut are 0.05 and 0.045 cm-1, respectively resulting in an RMSE value of 6.1. The optimal value is marked with a '+' on the contour plot. (b) The resulting shape of the optimal low pass Butterworth filter in the frequency domain
A 3D example case of estimating Sn from the LPS data can be found in Figure 36 for the pelvis
phantom without a compensator. The original unfiltered LPS, filtered LPS, and gold standard Sn
estimates are shown as 2D projections and sinograms. Both qualitatively and quantitatively it can
be seen that the low-pass filtered LPS data is a good estimate of the gold standard Sn. The filtered
LPS Sn signal [Figure 36(b)] has an RMSE of only 1.8 compared to an RMSE of 30.5 for the
case without low-pass filtering [Figure 36(a)]. A plot of the RMSE values between the low-pass
filtered and gold standard Sn data as a function of the angular sampling rate, dθ, for the pelvis and
head phantom with and without bowtie filtration is shown in Figure 37. The plot shows data
computed using the brute-force optimized and SFW (see Table 5) selected low-pass cutoffs with
similar RMSE values resulting for both. The resulting RMSE for all four configurations and both
filter cutoff selections remains under an RMSE of 5 for dθ less then 45° after which the RMSE
60
u [1/cm]
v [1
/cm
]
2422
20
2018
16
1614
12
12
10
8
864.0
0.1 0.2 0.3 0.4
0.1
0.2
0.3
0.4
u [1/cm]
v [1
/cm
]
-0.4 -0.2 0 0.2 0.4-0.4
-0.2
0
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0.2
0.4
0.6
0.8
1
begins to rise, especially in the case of the pelvis
phantom without the compensator. The use of the
compensator decreases the error for both phantoms
by more than 37% in all cases. In some cases the
compensator caused reductions in error up to 69% for
the pelvis and 53% for the head. The compensator
caused a larger reduction in error for all pelvis
phantom cases when compared to similar cases for
the head phantom. The optimal low-pass cutoff
values (ucut, vcut, and cut) for each angular sampling
rate is plotted in Figure 38. Two outlier data points
at dθ=72° and 90° were removed from the optimal
cut data for the pelvis phantom without a
compensator. The points removed resulted in a
selection of the highest cutoff value evaluated
suggesting the optimal solution was to have no
angular filtering. The cutoff values generally
decrease with decreased sampling (increasing dθ)
except in a few cases were there are slight upward
fluctuations. For both the pelvis and head phantom
when a compensator is used the optimal ucut value is
either less than or equal to the case without a
compensator. In the case of the vcut values the use of a
compensator tends to increase the optimal vcut value.
The optimal cut value is largely unchanged by the
use of the compensator. The use of a compensator
which modulates the fluence in the vertical direction
maybe useful in reducing the scatter distribution's
vertical frequencies and is left for future research.
4 Conclusions
For all objects and imaging configurations
61
(a)
(b)
(c)
(d)
(e)
Figure 35: LPS Sn projection for 30.6 cm diameter water cylinder (no compensator) (a) without and (b) with low pass filtering. The low pass filter cutoffs used in (b) are 0.065 and 0.045 cm-1 for u and v, respectively. (c) Gold standard scatter simulation Sn result. (d) The percent absolute error between the filtered and gold standard Sn
signal. (e) The central horizontal profile of the gold standard, LPS, and filtered LPS Sn signals.
Sn
0
100
200
300
400
Sn
0
100
200
300
400
Sn
0
100
200
300
400
% A
bsol
ute
Err
or
0
10
20
30
40
50
-60 -40 -20 0 20 40 600
200
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600
x [cm]
Sn
Gold Standard
LPS
Filtered LPS
investigated the scatter distribution in the CBCT projection images was predominately contained
within the low-frequency domain, both in the spatial and angular frequencies. Our Fourier
analysis of the MC simulated scatter distribution data, using the SFW metric, for a body size
cylinder (30.5 cm diameter) and two anthropomorphic (head and pelvis) phantoms show that the
spatial frequencies of the scatter distributions are contained below 0.1 cm -1 and the angular
frequencies below 7 turn-1. These values suggest a global minimum spatial pitch and an angular
sampling of every 5 cm and 25º, respectively to properly sample the scatter distribution of CBCT
projections. These spatial and angular sampling values are within a reasonable range of other
reported values by Ning et al. [77] used to accurately sample and estimate the scatter distribution
from a beam stop array, where the spatial sampling pitch was ~2 cm and angular projection
intervals between 11.5º and 90º depending on the thickness variations of the object being
investigated. The slightly higher spatial and angular sampling rates used by Ning et al. may have
been a result of the spatial and angular interpolation functions (cubic splines applied
independently to spatial then angular dimensions) used to estimate the scatter distribution and the
imaging and object configuration.
LPS Filtered LPS Gold Standard
(a) (b) (c)
(d) (e) (f)
Figure 36: Top row shows 0° Sn projection for the pelvis phantom for the limited photon simulation (LPS) using (a) 106
photons, (b) low-pass filtered LPS (using optimal cutoff values), and (c) the gold standard (> 109 photons) Sn data. The LPS Sn signal uses an angular sampling rate of 1°. The second row shows the same data but in the form of a sinogram composed of the center horizontal row of Sn at each projection angle, θ.
We found the frequency content of the scatter distribution to be dependent on the imaging
62
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configuration (air gap, compensator) and shape of the object of interest. Both the use of a
compensator and increasing the air gap were shown to decrease the horizontal frequencies of the
scatter distribution effectively leaving the vertical frequencies relatively unchanged in the water
cylinder's scatter distribution. The horizontal frequencies, u, were found to be usually contained
within sightly lower frequencies than that of the vertical frequencies, v. The head phantom had a
scatter distribution contained in similar horizontal frequencies, higher vertical frequencies, and
lower angular frequencies compared to those of the pelvis phantom. The potential for decreasing
both the horizontal and vertical frequencies through the use of a compensator maybe possible by
employing a compensator that modulates the fluence field in both directions. The inclusion of
vertical fluence modulation maybe especially relevant to the imaging of head anatomy due to the
increase in higher vertical frequencies (Table 5). The vertical frequency increase in the head
phantom is likely a result of the air-object interfaces at the neck and jawline with create edges
predominately located horizontally (horizontal edge in image results in higher vertical
frequencies in spectrum). The vertical frequency contribution to FS is also likely to increase
further if the top portion of the head becomes visible in the projection image. The crown of the
head would create a largely horizontal air-object interface that will result in an increase of
coherent scattering.
Along with providing an estimate of the spatial and angular sampling rates required to estimate
the scatter distribution, the Fourier analysis provides a basis for more efficiently estimating the
63
Figure 37: RMSE as a function of the angular sampling rate (ASR) for each of the four phantom imaging conditions. The results using the SFW and optimal low-pass filter cutoff values are shown as dashed lines with squares (□) and solid lines with crosses (+), respectively.
0 10 20 30 40 50 60 70 80 90 100 110 1200
5
10
15
20
RM
SE
d [degrees]
Pelvis
Pelvis with AL16SHead
Head with AL16S
scatter. MC estimates of the scatter in projection images can be sped up by limiting the number
of photons used, as the computational time is linearly correlated with the number of photons run.
Limiting the number of photons has the adverse effect of increasing the statistical noise in the
MC simulated results [see Figure 36(a)]. Through our frequency analysis of the scatter
distribution in CBCT it has been shown that the scatter distribution lies in the low frequency and
the statistical noise is generally contained in the high frequencies. The scatter estimate can thus
be recovered from a limited photon simulation through the use of a low-pass filter. Our results
show that using a low-pass filter allows both 2D and 3D estimates of the scatter distribution to be
recovered using only 106 input photons per projections, which translates into a computation
savings of 3 orders of magnitude per projection. Additionally we have shown that through the
use of low-pass filtered Fourier interpolation the scatter distribution can be recovered without a
substantial increase in error using only 15 projections, reducing computational costs by an
additional factor of 24. The combined computational savings afforded by the use of low-pass
filtering and interpolation through the application of the Fast Fourier transform provides a
potential mechanism by which MC based scatter correction could be performed for a clinical
system in a reasonable time frame. In the next chapter such a system is constructed by
integrating the Fourier fitting and interpolation into a MC based scatter correction algorithm. The
scatter estimation and correction system is tested on both simulated and measured CBCT data to
examine the image quality improvements of the method.
(a) (b) (c)
Figure 38: Optimal low-pass filter cutoff values, (a) ucut, (b) vcut, and (c) ωcut, for the different angular sampling rates used in each of the four phantom imaging configurations. Two outliers at dθ=72° and 90° were removed from the ωcut
data for the pelvis phantom with the AL16S compensator. The optimization for these two points resulted in a selection of the highest value of ωcut searched (35.28 turns-1) indicating that no filtering in the angular direction is optimal for these cases.
64
0 30 60 90 1200
0.02
0.04
0.06
0.08
0.1
u cut [
1/cm
]
d [degrees]
Pelvis
Pelvis with AL16S
Head
Head with AL16S
0 30 60 90 1200
0.02
0.04
0.06
0.08
0.1
v cut [
1/cm
]
d [degrees]0 30 60 90 120
0
5
10
15
cu
t [1/
turn
s]
d [degrees]
Chapter 4 Efficient Scatter Distribution Estimation and
Correction in CBCT usingConcurrent Monte Carlo Fitting
1 Introduction
A method for correcting scatter using Monte Carlo (MC) simulations was previously outlined by
Jarry et al. [27], [28] and involves estimating the scatter in each projection using a MC
simulation consisting of a phantom of the object being imaged and model of the imaging
geometry. The MC phantom can either have the density and material properties derived from a
prior CT scan of the patient aligned to CBCT reconstruction being corrected (a scenario quite
possible in image guided radiation therapy) or from the uncorrected CBCT reconstruction. The
scatter estimations are then subtracted from the original projection images to form a set of scatter
corrected projection images used to reconstruct a scatter free estimate of the CBCT
reconstruction. This method in its original configuration required a significant computational
time (430 h on a single CPU), making it largely clinically irrelevant.
In Chapter 3 the spectrum of the scatter distribution was investigated for a cylinder and two
anthropomorphic phantoms. The results showed the spatial and angular frequencies of the scatter
distribution vary depending on the imaging parameters (e.g. air gap, compensator), but in general
are contained in the lower end of the frequency domain (spatial frequencies < 0.1 cm -1, angular
65
frequencies < 4/ rad-1). The results also included an attempt at recovering the scatter distribution
signal from a limited photon simulation (LPS), consisting of 106 photons, using an optimized
low-pass filter. The results showed the scatter distribution signal could be recovered with
minimal deviation from a gold standard scatter estimate created using over 109 photons. This
approach presents a significant reduction in computation of three orders of magnitude. This type
of approach (using denoising to reduce computational time) in MC simulations is not new and
has been used in dose calculations with various techniques such as Savitzky-Golay curve-fitting
[78], digital filters [79], and wavelet thresholding [80]. A comparison of denoising techniques
used to reduce the noise in several MC dose calculations scenarios can be found in the
publications by El Naqa et al. [81]. Work has also been done to reduce the computational cost in
estimating the scatter distribution in cone-beam micro-CT projection images using Richardson-
Lucy (RL) methods to reduce the noise in MC simulations [44], [48]. Zbijewski and Beekman
[48] show that the MC computation time could be diminished by up to four orders of magnitude
with their RL fitting technique. Unlike the aforementioned approaches, our scatter estimation is
for CBCT projection data with much larger objects which have been shown to have significant
structure in the scatter, due largely to coherent scattering [34], [42], [70]. The most unique aspect
of our algorithm lies not in the application of the denoising technique, but in the fact that
multiple MC simulations are run concurrently with a denoising technique that is composed of a
fitting and interpolating function. The incorporation of interpolation into the denoising process
allows for a reduction in the number of projection angles that have to be simulated.
Additionally, the fact that the fitting is run simultaneously with the MC simulations allows the
number of photons used to be determined dynamically by testing the goodness of fit (GOF) of
each scatter fit estimate. Our approach has been written in an object-oriented manner to allow
different MC code systems incorporating the latest computational methods, such as variance
reduction techniques (e.g. forced detection [45], [46], interaction splitting with Russian Roulette
[82]) and GPU implementations [83], to be easily integrated with any desired fitting function.
In this chapter we will describe the implementation of our novel scatter estimation method that
simultaneously combines multiple MC CBCT scatter projection simulations through the use of a
fitting function. We quantify the performance of using this scatter estimation method in
correcting CBCT reconstructions from both simulated and measured projection data. The
potential for computational efficiency increases in estimating the scatter while using a
66
compensator is also examined. It is hypothesized that the the reduction in both the magnitude and
structure in the scatter distribution caused by the use of a compensator [42], [43], [59], [60] will
reduce the number of photons required to get an accurate estimate. The performance of the
scatter correction will be evaluated both in terms of efficiency and improvements in image
quality.
2 Materials and Methods2.1 Concurrent Monte Carlo Fitting
The basis of the concurrent MC fitting (CMCF) algorithm is to simultaneously combine scatter
distribution estimates from MC simulations with a fitting function in real-time to reduce the
number of photons required to get an accurate estimate of the scatter distribution. The algorithm
consists of three components: (1) MC simulation, (2) shared memory, and (3) concurrent scatter
fitting (see Figure 39). The MC simulation consists of a MC thread manager which launches
concurrently run MC CBCT scatter distribution simulation threads that simulate a sub-sample of
the projection angles used in the reconstruction set, P(x,y,θ), to be corrected, where x and y are
67
Figure 39: System diagram of the components involved in the concurrent MC fitting (CMCF) algorithm. The two main systems: (1) MC simulation and (2) concurrent scatter fitting. These two systems communicate through a shared memory space. The MC simulation system is made up of a MC simulation thread manager which launches and manages NS MC simulation threads. The shared memory consists of set of particle buffers which store the particles being generated from the NS MC simulations. The concurrent scatter fitting consists of an analysis manager which has sub-components responsible for reading particles from the buffer, creating a detector response signal from the particles, fitting the detector response, and evaluating the goodness of fit.
Shared MemoryMC Simulation
MC Thread Manager
MC CBCT Simulation Thread 1
MC CBCT Simulation Thread 1
MC Scatter DistributionSimulationThreads
ParticleBuffer(s)
Concurrent Scatter Fitting
Analysis Manager
Particle Reader
Scatter DetectorResponse Model
Fitting and InterpolationFunction
Goodness of Fit (GOF)Analysis
the horizontal and vertical detector positions, respectively and θ is the projection angle. The
particles generated by each MC simulation are stored in a particle buffer which is accessed by
the concurrent scatter fitting component and turned into a down-sampled scatter detector signal,
SMC(x,y,θ). At specific intervals, based on the number of photons processed, SMC is fit to a
function, SF(x,y,θ). If SF is evaluated to have met a user specified GOF criteria the MC
simulations are terminated by sending a signal to the MC thread manager and an interpolation, SI,
is created to match the sampling rate and region used in the projection set P. Each of the
components are described in more detail in the following subsections. The CMCF has been
written in an object oriented manner such that it would be easy to change various parts of the
system such as the MC system used to generated the scattered photons or the type of fitting
function used to estimate the scatter distribution.
2.1.1 Monte Carlo Simulation Systems
The MC projection simulation threads launched in our CMCF algorithm are based on a modified
version of the EGSnrc MC code [63] and a thorough description and validation of the MC
system can be found in Chapter 2. The thread manager used to launch and terminate the MC
threads is the Portable Batch System (PBS) running on a cluster with 50 nodes. Each node in the
cluster has two 3.0 GHz Intel X5472 Xeon quad-cores (Intel Corp., Santa Clara, California,
USA) with a 3 GHz clock speed. The tested configuration of the algorithm had each projection
angle simulated run on a single thread. The CMCF algorithm is configurable to allow multiple
simulation threads to be run for each projection angle to get further efficiencies from unused
cores when the projection angle sampling is less than the number of cores. The number of
projection angles simulated is configurable through a user supplied parameter.
2.1.2 Shared Memory
The shared memory exists as a mechanism for communicating the current information generated
from the MC simulation threads simultaneously with the scatter fitting components. In the
current configuration the shared memory consists of a set of particle buffers, one for each
projection angle. The buffers used in conjunction with our EGS based code are a set of PHSP
files, written to the systems hard drive, which store particle information (e.g. position, direction,
energy) at the plane of the detector for each projection angle. The buffers used in the code are
only dependent on the particle reader component in the concurrent scatter fitting system and can
68
easily be extended to incorporate buffers used to store synchronous photon data from other MC
systems.
2.1.3 Concurrent Scatter Fitting
The concurrent scatter fitting system has been written in C++ with an easily extendable object
oriented class hierarchy. The three main components of the concurrent scatter fitting system
(particle reader, scatter detector response model, and fitting function) are each modeled in a
separate virtual base class which can be extended to incorporate different MC systems, fitting
functions, and detector models. The generalized data processing of photons coming from the MC
simulations is outlined in flow diagram seen in Figure 40. For simplicity the flow of a single
photon through the concurrent scatter fitting system will be described, whereas in actuality
multiple photons are read in and processed from the particle buffer(s) being populated by the MC
simulations. The particle read in is first examined to its type, with only photons being processed.
69
Figure 40: Simplified flow diagram for the CMCF algorithm showing the processing of photons through the scatter fitting system. The end result is a interpolated estimate of the scatter distribution, SI, using the scatter distribution fit, SF, of the Monte Carlo scatter data, SMC.
Read in photon(s) from buffer
Compute response and add to SMC
Fit detector signal, SMC, to specified
function SF
Particle buffer(s)
Correct photontype?
Inside detector?
Enough photons?
Good fit?
MC simulationthread(s)
NO
NO
NO
NO
Terminate simulations
YES
YESYES
YES
Create SI, an interpolation of
SF over the entire projection
space
Upon finding a photon in the particle buffer (PHSP file) it is processed by a detector response
component which determines if the photon is the correct photon type. If the photon is a scattered
photon inside the specified detector region the detector response, based on it's energy, statistical
weight and direction, is computed and it is added to the scatter signal, SMC. If a sufficient number
of photons have been added to SMC it is fit to a user specified function SF. The number of photons,
∆p, deemed to be sufficient is a user specified criteria supplied as an average number of new
photons per pixel calculated across all detector pixel elements and photons. The fitting function
GOF is subsequently computed and if the user specified level of GOF, g, is met the MC
simulations are terminated and SF is used to interpolate the detector scatter signal, SI, at every
pixel and projection angle in the projection set, P, to be corrected. If the current GOF is less than
g the cycle begins again until either g is met or the MC simulations each run through the
maximum number of input photons allowed per projection, Np.
The fitting function implemented, and used in the test cases described subsequently, was a
limited sum of sines and cosines. The fitting and interpolation was accomplished through low-
pass Fourier filtering and interpolation. The forward and inverse Fourier transform were
accomplished using the fast Fourier transform (FFT) implemented in the free FFTW subroutine
library [84]. The low-pass filter is implemented in the frequency domain as a three dimensional
3rd order Butterworth filter, h(u,v,), defined by Eq. (38) in Chapter 3.
The fit of SMC is defined as
S F x , y ,=F −1 { ˚S MCu , v ,⋅h u , v ,} (40)
where F-1 is the inverse Fourier transform and ˚S MC is the Fourier transform of SMC. Subsequently
any Fourier transform of a given function will be similarly noted with the superscript circle. The
interpolation of the fit, SF, is denoted SI and is computed using Fourier interpolation by taking the
inverse Fourier transform of the appropriately zero-padded S̊ F (as described in the Appendix B).
In order for the algorithm to dynamically determine when an accurate estimate of the scatter has
been achieved a GOF metric is required that correlates with the actual error in the current scatter
estimate. The use of a GOF metric allows the user to specify the desired level of fitness without
having to determine the actual number of photons to be run. The Pearson correlation coefficient,
70
r, and the coefficient of determination, R2, were tested as potential GOF metrics. The Pearson
correlation coefficient was computed as:
r=
∑x , y ,
S F x , y ,−S F S MC x , y ,−S MC
∑x , y ,
S F x , y ,−S F ∑x , y ,
S MC x , y ,−S MC (41)
where S F and S MC are the mean values of SF and SMC, respectively. The coefficient of
determination was computed as
R2=1−
∑x , y ,
S F x , y ,−S MC x , y ,2
∑x , y ,
S MC x , y , −S MC .
(42)
2.2 Scatter Correction
2.2.1 Concurrent Monte Carlo Fitting Scatter Correction
The CMCF scatter corrected projection set, Pcorr, is created as
P corr x , y ,=P x , y ,−S I x , y , . (43)
If any projection angle, Φ, has a pixel with a value less than the specified threshold, β, the
following correction
P ' corri , j ,= P corri , j ,−min Pcorr i , j , (44)
is applied to all pixel locations in that projection, where min is the minimum pixel value of the
projection for angle Φ.
2.2.2 Constant Scatter Correction
A simple constant scatter correction was also used for the patient data in this study as a
comparison against the CMCF scatter correction. The algorithm works by subtracting a constant
scatter estimate from each of the projections. The value of the constant C(θ) for a given
projection angle, θ, is computed as the average pixel value of the non-air pixels of that projection
multiplied by a scatter factor ξ. If any of the corrected projections has pixels with values less
than the threshold, β, the correction outlined in Eq. (44) is used.
71
2.3 Experiments
2.3.1 Simulated Phantom Data
Four different simulated MC data sets were created for two different anatomical imaging sites
(head and pelvis) with and without the use of a compensator. These simulated data sets are
identical to those of the head and pelvis used in Chapter 3, with the only difference being that the
pixel pitch for the 360 degree projection set used in the reconstruction is 0.31 cm.
Reconstructions, using a Feldkamp filtered back-projection algorithm [26], of the projection data
inside the source FOV (detector size: 188×98 pixels) were done for both the primary and total
(scatter and primary) signal to serve as references to the scatter corrected reconstructions. The
reconstructions were done with a voxel size of 2×2×2 mm3 resulting in a 3D volume with
18818898 voxels.
2.3.2 Measured Phantom and Patient Data
A set of projections imaging the pelvis of a patient and an anthropomorphic phantom [73] were
used to test the CMCF algorithm on real data. The data was acquired using the Elekta Synergy
Platform (Elekta, Crawley, West Sussex, UK) with an offset detector geometry (detector offset
11.5 cm laterally). The x-ray tube was the DX9-30/50-150 (COMET Technologies, Stamford,
CT, USA) with the input potential set to 120 kVp. The source-to-axis distance (SAD) was 100
cm for both phantoms. The patient data had an axis-to-detector distance (ADD) of 60 cm and the
phantom an ADD of 53 cm. The detector used was the RID1640 (PerkinElmer, Waltham,
Massachusetts, USA) with a 10241024 array of pixels with a symmetric pitch of 0.4 mm. The
projection images were averaged down by a factor of 4 (256256 pixels, pitch of 1.6 mm) before
being reconstructed. The projections were collected with two different cone angles resulting in a
vertical field of view (FOV) of approximately 2 cm and 26 cm at a distance of 100 cm from the
source. The 2 cm FOV projection set was used to get an estimate of a scatter free reconstruction.
Using Eq. (34) from Chapter 2 the estimated scatter-to-primary ratio (SPR) for a 30.6 cm
diameter (body) cylinder at a FOV of 2 cm (cone angle ~1.1°) and FOV of 26 cm (cone angle
~14.8°) is 0.44 and 3.7, respectively for the 53 cm ADD and 0.39 and 3.3, respectively for the
60 cm ADD. The use of a 2 cm FOV results in a 88% reduction in the SPR and is used to
approximate a scatter free reconstruction.
72
The patient data used 329 and 328 projections for the 2 cm and 26 cm FOV, respectively. The
phantom data used 321 and 322 projections for the 2 cm and 26 cm FOV, respectively. Both the
patient and phantom data were reconstructed using a isotropic voxel size of 1 mm3, resulting in a
3D volume with 400×400×256 voxels. An additional set of projections of the 26 cm FOV data,
down sampled by a factor of 8 (pitch of 3.6 mm), was reconstructed for use in making the MC
phantom used in CMCF algorithm. The reconstruction used in the MC phantom creation had a
voxel size of 2×2×2 mm3. The phantom was created using the ctcreate application included in
the BEAM MC code distribution [85].
2.3.3 Concurrent Monte Carlo Fitting Parameters
The low pass filter cutoffs in the horizontal, vertical, and angular frequency cutoffs used for the
four simulated imaging scenarios and the measured pelvis (patient and phantom) data are given
in Table 7. The cutoff values for the simulated phantoms were calculated by rounding up values
from the SFW data given in Table 5 from Chapter 3. The values for the measured pelvis phantom
and patient projection data were selected from the optimal filter cutoff for the pelvis phantom
using a angular sampling rate of 24˚ per projections (see Figure 38 in Chapter 3).
Table 7: Spatial and angular frequency cutoffs values used in the Butterworth low-pass filter for simulated phantoms and measured pelvis patient and phantom data.
Spatial and Angular Frequency Cutoffs
Compensator ucut (cm-1) vcut (cm-1) cut (rad-1)
Pelvis Phantom None 0.03 0.05 7/(2π)
AL16S 0.04 0.05 5/(2π)
Head Phantom None 0.06 0.07 3/(2π)
AL16S 0.04 0.07 3/(2π)
Measured Data None 0.05 0.05 6/(2π)
The CMCF system requires the user to select the number of MC projection angles to be
simulated along with the size of the simulated detector and the horizontal and vertical pixel pitch
used to aggregate the photons from the MC simulations. The detector size for the simulated
phantom data remains the same as in Chapter 3 with a width of 120 cm and height of 80 cm. A
larger detector is used for the experimental data with a width and height of 164 cm. The large
detectors are used to capture the full extent of the scatter distribution to obtian a near zero scatter
signal at the edges of the simulated detector. Using data from Chapter 3 on the spatial and
angular frequencies of the scatter distribution, minimum values for the sampling rate parameters
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can be estimated by applying the Nyquist sampling theory to the scatter frequency width (SFW)
data. The minimum angular sampling rate (dθ) was found to be between 29 and 62° per
projection. The minimum horizontal (dx) and vertical sampling rate (dy) were to be between 10
cm to 20 cm and 7 cm to 11 cm per pixel, respectively. The range of sampling rate values are
attributed to variations in the phantoms being imaged and whether or not a compensator is used
(see Table 5 in Chapter 3). The sampling rates chosen in the processing of the MC data for the
simulated phantom data sets were 1.24 cm per pixel for both x and y and 24° per projection.
Similar settings were used for the Elekta Synergy patient and phantom data sets with a x and y
pitch of 1.28 cm and an angular sampling of 24° per projection. The samplings rates were chosen
to maximize efficiency well ensuring the full spectrum of the scatter frequencies were captured.
The spatial sampling rates only effect the computational time required for a fit and the amount of
memory required. The computation time on fitting (e.g. taking FFT and inverse FFT) data of
this size (3D matrices for simulated: 64×96×15 and measured: 128×128×15) is sufficiently under
the amount of time required to generate the Δp new photons required for the next fit so a higher
spatial sampling was used to get a better result from the low-pass filtering. The angular sampling
rate on the other hand has a much higher impact on the computational load, as an increase in the
number angular samples results in an increase to the number of MC simulations. The current
value was chosen to be close to the minimum while properly sampling an entire period of the
signal so that minimal errors are produced in the Fourier interpolation. The chosen value for the
angular sampling rate result in a total of 15 MC simulation threads.
An upper bound on the number of photons used per projection simulation, Np, was used as a
safeguard against non-converging simulations. For both the simulated and measured projection
Np was set to 107 photons per projection. The number of new photons per pixel parameter, Δp,
was set to a minimum average of 10 new photons added per pixel before a fit of the SMC data is
performed. For testing purposes no GOF stopping value, g, was supplied for the simulated
phantom data. Instead each fit, SF, along with the corresponding interpolation, SI, was output
with the input SMC data and corresponding GOF metric. This was done to test how well the
different GOF metrics corresponded to the actual error in SI, as well as determine the GOF
function and value to use as a stopping constraint. For the measured data g was set to 0.6 with
the reasoning outlined in the GOF results.
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Finally, the minimum signal value, β, allowed in a projection after scatter correction has been
performed was 1 and 20 for the simulated and measured projection sets, respectively. The large
discrepancy in the β value results from higher detector signal range in the measured data, and the
large signal fluctuations due to statistical noise in the simulated data resulting in numerous pixels
close to zero. A summary of all the aforementioned parameters (excluding the filter cutoff
values found in Table 7) can be found in Table 8 or the simulated and measured projection data.
Table 8: Configuration parameters for CMCF algorithm for the simulated and measured projection sets.
Detector Size Sampling Rates Stopping Values Minimum Signal
Height(cm)
Width (cm)
dx (cm/pixel)
dy (cm/pixel)
dθ(°/proj.)
g Np ∆p β
Simulated 80 120 1.24 1.24 24 n/a 107 10 1
Measured 164 164 1.28 1.28 24 0.6 107 10 20
2.3.4 Scatter Estimate Error
For the simulated phantom data the possibility of calculating an estimate of the error for any one
of the scatter estimates (e.g. SMC, SF, SI), is possible because a gold standard estimate, SG, exists
in the form the projection data simulated with > 109 photons. The measure of error in the scatter
is calculated as follows:
Serr S ,SG =mean[ S x , y ,−SG x , y ,
SG x , y , 2
]x , y , ∈{∣x∣29cm ,∣y∣15cm∣∣360 ˚ }
(45)
where the limits define the detector area and projection angles used in the reconstruction, S is
any scatter estimate (e.g. SMC, SI, or SF) and SG is the corresponding gold standard estimate of the
scatter. The two GOF metrics were evaluated by how well they correlated with Serr(SI,SG).
2.3.5 Image Quality Metrics
The resulting scattered corrected reconstructions were evaluated for the simulated data sets by
looking at a set of metrics measuring the reduction in the cupping/shading artifact, increase in the
contrast-to-noise ratio (CNR), and reconstruction voxel accuracy. The tcup metric from
Siewerdsen and Jaffray [12] was used to quantified the amount of “cupping” or nonunifomity in
an image, as
75
t cup=100×V skin−V center
V skin (46)
where V skin and V center are the average voxel value in regions at the skinline (periphery)
and center of the phantom, respectively. A square region in a single axial slice from the volume,
consisting of 4×4 voxels (8×8 mm2), was used in the computation of the averages.
The CNR was computed as
CNR=2∣V A−V B∣
A B (47)
where V A and V B are the mean voxel values in neighboring regions A and B, respectively
and σA and σB are the standard deviation in the voxel values in A and B. The regions for A and B
were selected as 4x4 voxel regions in a axial slice representing bone and surrounding tissue,
respectively in the phantoms.
A measure of the volume voxel error, Verr, based on the ERRE metric from Gao et al. [62] was
made by comparing corresponding voxels in a reconstructed volume, V, against the
corresponding voxels in a scatter free volume, Vsf. The Vsf volume was created by reconstructing
projections using only the primary photons as input. The error metric was defined as
V err=100 mean[V x , y , z −V sf x , y , z
V sf x , y , z 2
]x , y , z ∈ ROI
(48)
where the ROI of interest consisted of all voxels that were part of the phantom (a.k.a not air)
based on a thresholding of the volume.
3 Results and Discussion3.1 Simulated Phantoms
3.1.1 Scatter Estimate Error
The scatter estimate error for SMC and SF are plotted as a function of time for the head and pelvis
phantom with and without the use of a bowtie filter in Figure 41. There is a clear reduction in the
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amount of error in the filtered data compared to the unfiltered MC data. The error in the fit data
quickly (< 100 sec.) reaches a steady state for all four imaging configurations. Looking at the
data at the one minute mark we see the Serr has a value between 0.04 and 0.07 which translates to
a reduction in error between 85 and 90% for each of the configurations. The phantom
configurations in which a bowtie filter was employed result in the least amount of and largest
reduction in error when compared to the case without a bowtie filter. The pelvis SMC data with the
bowtie filter had an error larger than the no bowtie case, but the error values for the SF data is
reversed, clearly showing how the use of a compensator increases the effectiveness of the low-
pass filtering. Similar error reductions can be found for the interpolated scatter function, SI, with
the Serr values plotted in Figure 42(a) as a function of time. The largest reduction of error occurs
early on, with the error values of SI for each of the simulated configurations dropping to values
between 0.06 and 0.09 after only a minute of computation time. Figure 43 demonstrates
qualitatively the reduction in error by showing the scatter estimates of the original MC data, SMC,
the interpolated fit, SI, and the gold standard estimate, SG, for the pelvis phantom with a bowtie
filter after 53 seconds of run time. The similarity between SI and SG is noteworthy, especially
when looking at the sinogram data in Figure 43. The Serr values for SI are slightly larger then
those for SF across the entire time interval. This is to be expected due the large amount of angular
interpolation in SI. This error could be reduced slightly by increasing dθ (see Figure 37 in
Chapter 3) but at a cost of increasing the computational load. As in the SF case, when the bowtie
77
Figure 41: The scatter estimate error, Serr, for the raw Monte Carlo, SMC, (dashed lines) and the low-pass Fourier fit, SF, (solid lines) scatter estimates as a function of computation time for the two phantoms (head and pelvis) with and without a bowtie (BT).
0 100 200 300 400 500 60010
-2
10-1
100
Se
rr
Time [sec]
Pelvis
Pelvis w/ BTHead
Head w/ BT
filter is used the Serr values for the interpolated scatter estimates are reduced for both the pelvis
and head phantoms. In the pelvis and head phantom data the reduction in error when using the
bowtie filter is 13% and 9%, respectively at the one minute mark.
(a) (b)
Figure 42: The Serr for the interpolated scatter data, SI, for each of the of the phantom configurations plotted as a function of (a) computation time and (b) the Pearson correlation coefficient, r.
78
0 100 200 300 400 500 6000.05
0.06
0.07
0.08
0.09
0.1
Se
rr
Time [sec]0.4 0.5 0.6 0.7 0.8 0.9
0.05
0.06
0.07
0.08
0.09
0.1
Ser
rPearson Correlation (r)
Pelvis
Pelvis w/ BT
Head
Head w/ BT
Figure 43: The scatter signal for the original data used in the fitting model, the interpolated fit, and the gold standard simulation for the pelvis phantom with the AL16S compensator. The original and fit data being shown is after 53.2 secs of run time. The fit has a Pearson correlation coefficient of 0.61 and a Serr value of 0.07. The top row shows the scatter signal for the detector position at the 0° projection angle. The bottom row shows the scatter sinogram for the horizontal row at center of the detector at each angular position, θ.
y [c
m]
Original Data - SMC
-40 -20 0 20 40
-20
0
20
Interpolated Fit - SI
-40 -20 0 20 40
-20
0
20
Gold Standard - SG
-40 -20 0 20 40
-20
0
20
[d
egre
es]
x [cm]-40 -20 0 20 40
0
90
180
270
x [cm]-40 -20 0 20 40
0
90
180
270
x [cm]-40 -20 0 20 40
0
90
180
270
3.1.2 Goodness of Fit Metrics
The resulting Pearson correlation between the GOF functions (R2 and r) and the interpolated
scatter error, Serr(SI, SG), was computed for the simulated phantoms and is shown in Table 9. Both
functions show an adequate correlation with Serr. Both have a negative correlation value due to
the fact that a decrease in error results in an increase in the GOF value. In all cases r shows a
better correlation to Serr than R2 does and for both GOF metrics the data using the compensator is
better correlated than the case without the compensator. A plot of r versus Serr is shown in Figure
42(b). The majority of the decrease in Serr is seen to occur at an r value between 0.6 and 0.8
depending on the phantom and imaging configuration.
Table 9: Correlation between the two GOF metrics (Pearson correlation, r, and coefficient of determination, R2) and scatter estimate error, Serr, computed for the interpolated scatter, SI, for each of the simulated phantom configurations.
Compensator r R2
Pelvis None -0.74 -0.65
AL16S -0.89 -0.84
Head None -0.87 -0.82
AL16S -0.95 -0.93
3.1.3 Reconstruction Image Quality
Axial slices of the reconstructed head and pelvis phantoms with and without the use of the
AL16S compensator are shown in Figure 44. The axial slices reconstructed from the simulated
projections using only primary photons, all photons (uncorrected), and all photons corrected by
CMCF estimate of the scatter are displayed in each of the columns. The CMCF scatter estimate
used to correct the projections had an r value of ~0.6. The location of the horizontal profiles for
the head and pelvis data plotted in Figure 45 are shown in the primary only axial images without
the use a compensator as dashed lines. A clear increase in the image quality can be seen between
the uncorrected and CMCF corrected images with a decrease in shading artifacts and an increase
in contrast. Quantification of the image quality improvements in terms of the CNR improvement
for bone, the decrease in cupping (tcup), and the decrease in the reconstruction error (Verr) are
found in Table 10 along with the time to compute the CMCF estimate of the scatter.
Qualitatively, the presence of “cupping” and shading artifacts associated with scatter are much
more prominent in the larger pelvis phantom, which can be explained by the higher scatter-to-
primary ratios (SPR) reported for thicker objects [32], [34], [42]. The tcup metric decreases with
79
the use of the CMCF scatter correction for all imaging configurations. The tcup achieves a value
equal to the scatter free case for all imaging configurations except for the pelvis phantom without
a compensator. The remaining “cupping” can be seen in the axial images of the pelvis phantom
between the two femur bones. A slight decrease of the center voxel values versus those at the
edge of the phantom can also be seen in the primary only results when looking at the profiles
shown in Figure 45 for the pelvis phantom without a compensator. The corresponding decrease
in the primary only reconstructed image suggests that the decrease maybe partially related to
80
Primary Only Uncorrected CMCF Corrected
Pel
vis
With
out C
ompe
nsat
or
AL1
6S C
ompe
nsat
or
Hea
d
With
out C
ompe
nsat
orA
L16S
Com
pens
ator
Figure 44: CBCT reconstruction of the pelvis (top 2 rows) and head (bottom 2 rows) using primary only (column 1), primary and scatter (column 2), and primary and scatter corrected using CMCF algorithm. The CMCF corrections are shown for a GOF of r≈0.6. The time to compute the scatter estimate was under 2 minutes for all four scenarios.
prim MCFHaroldPZTOTAL.headerMCFHaroldPPSCR61xShift1.header
MCFHaroldPAlRSBTZPRIMARY.headerMCFHaroldPAlRSBTZTOTAL.headerMCFHaroldPAlRSBTPSCR61.header
prim MCFJackTOTAL.header MCFJackA15R61.header
MCFJackAlRSBTPRIMARY.headerMCFJackAlRSBTTOTAL.headerMCFJackAlRSBTA15R60.header
beam hardening. The photons traveling laterally through the phantom would encounter an
increased amount of bony anatomy (pelvis and both femurs) at the center potentially shifting the
mean energy of the x-rays higher due to the increased photoelectric absorption. The other
potential source of this error, at least in the case of the CMCF corrected images, is the method
used to correct projections with pixels that fall below the threshold β. The images with pixel
values below β are corrected with a constant signal shift. This shift occurs more frequently in the
lateral x-ray views due to the low photon count and increased statistical noise. Again the
decreased photon signal is caused by the significant photon attenuation by the pelvis and
femur(s) at these angles. This effect is seen less in the pelvis reconstruction when the
compensator is used because the photon distribution is enhanced for the central portion of the
beam relative to edge due to the fluence modulation profile of the compensator. The use of a
constant shift for values below β to account for scatter estimation can also lead to an increase in
streak like artifacts seen in the pelvis correction cases. These streak artifacts could potentially be
reduced by using a soft cutoff function proposed by Zhu et al. [55].
Table 10: Image quality metrics for the uncorrected, corrected, and primary only reconstructions of the four imaging configurations. The time to compute the CMCF scatter estimate is also shown for each phantom configuration.
Time (s) CNR Contrast Noise tcup Verr
Pelvis Uncorrected / 9.7 0.03 0.003 32% 17%
CMCF(r≈0.6) 92.67 14.6 0.13 0.009 19% 3%
Primary Only / 24.2 0.15 0.006 8% /
Pelvis w/ AL16S Uncorrected / 13.5 0.04 0.003 -20% 16%
CMCF(r≈0.6) 53.20 14.8 0.14 0.009 -8% 2%
Primary Only / 23.8 0.13 0.006 -8% /
Head Uncorrected / 11.1 0.19 0.017 16% 10%
CMCF(r≈0.6) 54.09 13.7 0.21 0.016 2% 1%
Primary Only / 13.8 0.21 0.016 2% /
Head w/ AL16S Uncorrected / 10.4 0.12 0.012 -19% 8%
CMCF(r≈0.6) 34.92 14.0 0.16 0.012 -13% 1%
Primary Only / 14.0 0.15 0.012 -13% /
The increase in bone contrast relative to the background tissue signal is clearly seen in both the
axial slices (Figure 44) and profile plots (Figure 45) of the CMCF corrected data compared to the
uncorrected data. The increase in contrast is much more substantial for the pelvis phantom
compared to the head phantom due to the larger SPR in the pelvis projections. When looking at
the CNR this is less apparent due to an increase in noise in the pelvis reconstruction when the
81
CMCF algorithm is applied. The increased noise caused by scatter correction is related to the fact
that the increase in signal caused by the scattered photons decreases the noise in the projection
image and when the CMCF scatter estimate is subtracted the signal is lost but the noise from
scatter remains. The noise increase caused by the scatter correction is a function of the SPR, as
shown in Zhu et al. [86]. The estimated SPR for a head cylinder (16.4 cm diameter) and body
cylinder (30.6 cm diameter) using the simulated image geometry (ADD=56 cm and 11.3°) is
0.43 and 4.1, respectively using Eq. (34). The effect of the large SPR difference is seen in the
noise results for the two phantoms with and without a compensator. The noise values remain
relatively constant for the CMCF corrected head phantom but triples in value for the CMCF
corrected pelvis phantom. The CNR bone measurements have an increase of 10% and 50% for
82
Without Compensator AL16S Compensator
Pe
lvis
He
ad
Figure 45: Voxel values plotted in arbitrary units (A.U.) for horizontal profiles in an axial slice of reconstructions of the head and pelvis phantoms with and without the use of a compensator from primary only, uncorrected, and corrected projection images. The profile locations are shown in Figure 44.
-20 -10 0 10 200
0.1
0.2
0.3
0.4
Position [cm]
Vox
el V
alue
(A
.U.)
-20 -10 0 10 200
0.1
0.2
0.3
0.4
Position [cm]
Vox
el V
alue
(A
.U.)
CMCF Corrected
Primary Only
Uncorrected
-10 -5 0 5 100
0.1
0.2
0.3
0.4
Position [cm]
Vox
el V
alue
(A
.U.)
-10 -5 0 5 100
0.1
0.2
0.3
0.4
Position [cm]
Vox
el V
alue
(A
.U.)
the pelvis phantom with and without a compensator, respectively and 34% and 23% for the head
phantom with and without a compensator, respectively. The reconstructed voxel error, Verr, was
significantly improved for all four imaging scenarios when the CMCF scatter correction was
used. The decrease in error was greater than 80% for all four imaging scenarios, resulting in a
reconstruction error below 4% when the CMCF algorithm was used.
3.2 Measured Pelvis Phantom and Patient
3.2.1 Pelvis Phantom
Axial slices taken from reconstructions of the 2 cm FOV, 26 cm FOV uncorrected, 26 cm FOV
CMCF scatter corrected, and 26 cm FOV constant scatter corrected anthropomorphic phantom
projection data are shown in Figure 46. The CMCF scatter estimates used in the correction took
122 seconds to compute. The increase in time compared to the simulated phantom studies has to
do with the larger MC detector size and smaller pixel pitch used. The increased projection data
size results in an increased FFT computation time. Qualitatively both the constant and CMCF
scatter corrected reconstructions [Figure 46(b) and 46(c)] show improvements to the overall
image quality (reduction in shading and an increase in contrast) compared to the uncorrected
reconstruction [Figure 46(a)]. Looking at the horizontal and vertical profiles shown in Figure 47
the CMCF correction provides a better reduction in the shading and cupping artifacts than the
constant scatter correction method. The contrast in the bone signal for the CMCF data is also
markedly improved over the simple constant correction. In general the CMCF data comes closer
to the reduced scatter reconstruction created from the 2 cm FOV projection data. Streak artifacts
can be seen in all the axial slices as a result of the under sampling (< 1 projection per degree) of
the projection space. If the imaging geometry did not include an offset detector a sampling of 1
projection per degree would have been sufficient to eliminate the streaking artifact. Generally a
angular sampling rate closer to 0.5 degrees is used in an offset geometry but the measured data
sets (patient and phantom) were taken from a database containing images acquired using an early
version of the Elekta Synergy, when a higher angular sampling rate was not used.
83
(a) (b)
(c) (d)
Figure 46: Axial slices from reconstructions of the anthropomorphic phantom taken on the Elekta Synergy for the 26 cm FOV (a) uncorrected , (b) with CMCF scatter correction, (c) constant scatter correction, and (d) the 2 cm FOV. The display window values were [0.09,0.3] for all images. The horizontal and vertical profile lines plotted in Figure 47 are shown in (d).
(a) (b)
Figure 47: (a) Horizontal and (b) vertical profiles through axial slices in reconstructions of the anthropomorphic phantom showing voxel intensity differences for reconstructing with a 2 cm FOV, and 25 cm FOV with no scatter correction, CMCF scatter correction, and constant scatter correction.
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-15 -10 -5 0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
Vertical Position [cm]
Vox
el I
nten
sity
(A
.U.)
-20 -15 -10 -5 0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
Horizontal Position [cm]
Vox
el I
nten
sity
(A
.U.)
2 cm FOV
25 cm FOV
25 cm FOV CMCF
25 cm FOV Constant
3.2.2 Pelvis Patient
Images of the reconstructed patient data are shown for 2 cm FOV, 26 cm FOV uncorrected, 26
cm FOV CMCF scatter corrected, and 26 cm FOV constant scatter corrected projection data in
Figure 48. The CMCF scatter estimate took 114 seconds to compute for the patient data. The
improvement in image quality is striking for the CMCF corrected reconstruction with large
improvements in the reduction of the shading artifacts. The CMCF algorithm reduces shading
artifacts in the posterior of the patient that remain when using the constant scatter correction.
Both soft tissue and boney anatomy contrast have also been improved. The voxel intensity
values for the CMCF corrected results shown in the horizontal and vertical profiles in Figure 49
are nearly identical to those of the 2 cm FOV. There is still some signal loss near the skinline on
the left and right side of the patient, even the 2 cm FOV data has a loss in the skinline seen in the
lower left and right side of the axial image. This loss can be associated with image lag effects
resulting from signal remaining in the detector after previous x-ray exposures [87] and maybe
improved by using a lag correction algorithm, such as the method outlined by Noor et al. [88].
(a)
(b)
(c) (d)
Figure 48: Axial slices from reconstructions of the pelvis patient data: (a) 26 cm FOV uncorrected, display window: [0.12,0.23], (b) 26 cm FOV CMCF scatter corrected, display window: [0.11,0.30] (c) 26 cm FOV constant scatter correction, display window: [0.13,0.25], and (d) 2 cm FOV uncorrected, display window: [0.11,0.30]. The vertical and horizontal profile locations for Figure 49 are shown as dashed lines in (d).
85
-20 -15 -10 -5 0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Horizontal Position [cm]
Vox
el I
nten
sity
(A
.U.)
2 cm FOV
25 cm FOV25 cm FOV CMCF
25 cm FOV Constant
(a) (b)
Figure 49: (a) Horizontal and (b) vertical profiles through axial slices in reconstructions of the pelvis patient showing voxel intensity differences for reconstructing with a 2 cm FOV, and 25 cm FOV with no scatter correction, CMCF scatter correction, and constant scatter correction.
3.3 Efficiency
The scatter estimates for both the simulated and measured data were estimated in half a minute to
two minutes, compared to the 200-400 hours required if the estimates were created using >109
photons per projection on the same platform used for the CMCF algorithm (16 cores from two
3.0 GHz Xeon X5472 Processors Intel Corp., Santa Clara, California, USA). The speed up can
be explained by three major sources of computational savings. The first is the reduction in the
number input photons per projection required to accurately estimate the scatter distribution. Due
to the use of Fourier fitting the number of photons per projection drops from ~109 to ~106, a
savings of 3 orders of magnitude. The second computational cost reduction comes from the
Fourier interpolation of the projection data which allows for a reduction in the sampling
frequency of x, y, and θ. Reducing the angular sampling, dθ, gives the largest computational
savings as it directly reduces the number of MC projection angle simulations required, whereas x
and y simply translate into computational savings for the CMCF analysis. The savings from the
use of interpolation in our experiments was a factor 24, as we went from having to simulate up to
360 projections to only 15. Finally, it was found in the simulated experiments that the use of a
compensator reduced the CMCF computational time by a factor between 1.5 and 1.75. This
reduction is believed to be due to the smoother scatter distribution created when using a
compensator. Combining all three reduction methods results in a maximum saving of over 4
orders of magnitude in computational savings, which matches the savings seen in our results.
86
-10 -5 0 5 100
0.05
0.1
0.15
0.2
0.25
0.3
Vertical Position [cm]
Vox
el I
nten
sity
(A
.U.)
There are some additional computational costs associated with using the CMCF method, the
largest of which is computing the interpolation using the inverse Fourier transform. For the
measured data the time used for computing the inverse transform was ~40 seconds (3D data
matrix size: 1024×1024×360), comparatively the time on the simulated data was ~3 seconds (3D
data matrix size: 256×384×360). For the measured data this accounts for approximately 30% of
the total computation time. Unlike the rest of the analysis (collecting and fitting MC data) which
is done simultaneously with the MC simulations, the interpolation is done after the simulations
are complete and is thus an additive time. The MC simulations themselves comprise about 60%
of the total computational time. The other 10% of the time is spent initializing the simulations
and outputting the scatter estimate to a file.
4 Conclusion
We have described a novel CMCF scatter correction method that uses a sparse number of MC
projection simulations run concurrently with a fitting algorithm. The fitting function decreases
the statistical noise in the MC scatter distribution estimates allowing for a reduction in the
number of photon tracks required to compute an accurate estimate of the scatter distribution. The
ability to interpolate the fit of the scatter distribution over detector pixel locations (x and y) and
angles (θ) allows for an additional reduction in both memory and computational requirements.
In this study the fitting function was chosen as a limited sum of sine and cosine functions
implemented using a low-pass filtered Fourier transform of the scatter data often referred to as
Fourier fitting and interpolation. The selection of the frequency truncation of the sum of sine and
cosine functions was determined based on our previous investigation into the spectrum of the
scatter distribution in CBCT projection images. The denoising provided by the low-pass filter
resulted in a >103 reduction in the number of photon tracks required. The interpolation allowed
the number of projections simulated to be reduced by a factor of 24. Using the CMCF algorithm
to estimate the scatter for simulated data for 360 projections with a detector size of 188×98
pixels took between 35-93 seconds . Estimates of the scatter distribution for the measured pelvis
(phantom and patient) data consisting of 322-329 projections with a resolution of 256×256 pixels
took between 114-122 seconds. The use of a compensator was also shown to decrease the CMCF
scatter estimate computation time by a factor of 1.5 to 1.75. This reduction is mainly believed to
be due to the smoother scatter distribution provided by using a compensator.
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The CMCF correction was shown to improve the image quality both qualitatively and
quantitatively in the simulated pelvis and head phantom projection cases. Improvements in the
contrast were clearly seen for both the bony anatomy. Measurements made in the simulated
phantoms showed a 10% to 50% increase in the CNR for bone when the CMCF correction was
used. The CNR measurements of the CMCF corrected reconstructions showed slight
improvements when a compensator was used. The cupping was measured to be reduced by 32%
to 88% in the simulated head and pelvis phantoms. When the compensator was employed a
reverse cupping effect was seen to be produced in the reconstructions. The voxel error
measurement, Verr, decreased between 82% and 90% resulting in a maximum Verr value of 3%.
The image quality in the measured phantom and patient data was also seen to improve.
Qualitatively the contrast in both the bony and soft tissue anatomy was improved, especially in
the patient data case. Due to a lack of ground truth a voxel error measurement could not be
computed, but the signals were visually compared to a reduced scatter reconstruction made using
a reduced field of view of 2 cm. The profiles of the CMCF reconstruction show a much better
signal agreement with the 2 cm FOV then both the uncorrected and constant scatter corrected
results. There are still some image artifacts seen to remain in the CMCF corrected images. Some
of these artifacts are related to the under sampling of the projection space (streaks) and others to
image lag (skin line signal loss). There is potential that the skin line loss in CMCF images is
exacerbated be the correction employed to projections whose signal drops below a specified
threshold and better solutions to this issue are to be investigated. The next chapter will look at
the clinical feasibility of the CMCF scatter correction approach as well as suggest possible
methods and mechanism to improve the algorithms accuracy and efficiency.
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Chapter 5 Retrospective and Prospective Deliberations and
Contemplations
1 Introduction
The CMCF scatter correction system proposed and implemented in this thesis provides a method
to subtract the object and imaging geometry specific scatter distribution to significantly diminish
the deleterious influences of scattered photons in CBCT reconstructions. The improvements were
shown to lessen the shading and cupping artifact, increase the CNR, and improve overall
reconstruction accuracy in a clinically viable time frame. The importance of a scatter correction
method for CBCT is self evident from the pelvis patient reconstruction shown in Figure 5(d) and
again in Figure 48(a), with scatter induced diminished contrast, shading artifacts, and voxel
value inaccuracy. The correction of these effects would increase the accuracy and efficacy of
CBCT in it's current applications, as well as facilitate the expansion of CBCT in other domains
such as adaptive radiation therapy [89]. Specifically, the correction method was applied to the
aforementioned patient data [Figure 48(b)] with prominent improvements to the reconstructed
image quality. In this final chapter the clinical feasibility and value of our approach will be
discussed along with the potential areas for improvements in it's design, with comparisons to
similar approaches to scatter estimation and correction.
89
2 Demonstrating Clinical Value
Besides being clinically feasible there must be enough added value to justify the cost of
implementing a scatter correction system. As demonstrated in Chapter 4 the algorithm clearly
improved the contrast of both bone and soft-tissue, as well as increased the overall reconstruction
accuracy, but the question remains as to how these improvements translate into improvements in
clinical outcomes. Measuring clinically benefit is not as simple as measuring improvements in
CNR, but is important in gaining acceptance by clinical staff. We propose two ways in which
the clinical improvements generated by the scatter correction algorithm could be quantified in
the future. Both methods involve generating several volumetric data sets of either patients
and/or phantoms scanned both using a traditional (small cone angle) CT and CBCT system, with
the resulting volume coordinate systems co-registered. For the CBCT system the volume would
be reconstructed with and without the scatter correction mechanism. In the first test a specific
item (e.g. tissue, organ, contrast insert) would be contoured on all three volumes (CT, CBCT,
scatter corrected CBCT) either manually by a clinician or using an automatic segmentation
algorithm [90], [91]. The location and volumetric differences of the contoured regions in the
CBCT volumes could then be analyzed against the co-registered CT volume. The second test
would consist of registering a known coordinate shifted and/or rotated version of the uncorrected
and corrected CBCT volumes to the CT volume using an automatic image registration algorithm
and measuring the error in the resulting registration. These tests provide a starting point for
evaluating the clinical value added by using the CMCF algorithm. Subsequent tests could also
be devised to evaluate the potential reduction in dose calculation errors on CBCT data from
using the CMCF algorithm, similar to the approach taken by Bazalova et al. [14] to evaluate the
effects of CT artifact corrections on dose calculations.
3 Potential for Clinical Implementation
There are several factors affecting the feasibility of implementing a scatter correction method.
These include: the additional time it takes to create the scatter free reconstruction, the financial
cost of implementing the system that performs the correction, and the degree of integration with
which the scatter corrections are made (e.g. is there additional input or work flow changes
required of the physician, patient, or therapist). Currently the scatter correction proposed has
been shown to correct patient images (~320 projections, 256×256 pixels) in about 2 minutes
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using 16 processors (a computer cluster was employed). The requirement of a cluster would
make this implementation clinically unfeasible due to the financial burden of installing and
operating a cluster. As an alternative we have recently constructed a workstation using the ASUS
KGPE-D16 server motherboard (ASUSTeK Computer Inc., Beitou District, Taipei, Taiwan) with
dual AMD Opteron 6200 series CPU (Advanced Micro Devices, Inc., Sunnyvale, California,
U.S.A.). This motherboard was configured with 32 cores, each with a clock speed of 2.2 GHz,
and assembled into a workstation costing under $4000 (including: a monitor, 2×120 GB solid
stated hard drives, 2×1 TB hard drive, 4×8 GB DDR3 Memory, and a RAID controller). The
system has a reasonable financial cost and provides a suitable computational platform that will
allow for many of the possible efficiency increases, which shall be discussed later. The algorithm
would be hidden from the end user, much as the reconstruction algorithm is, and would not
require any changes to work flow once configured. The only additional hardware or work flow
change that would be beneficial to the performance of the CMCF scatter correction system (but
not required) is the use of a compensator in the CBCT imaging system.
4 Improvements in Efficiency
The current version of the scatter correction algorithm completes an estimate of the scatter
distribution for a projection data set size of 256×256×360 (pixel height×pixel width×number of
projections) in approximately 2 minutes. This time is appropriate for some but not all imaging
scenarios and a faster computational time is always more desirable. There are several options
available to the CMCF method to decrease processing time.
The largest computational cost in the algorithm is the generation of photons from the MC
simulation. The simplest option to decrease this time is to increase the number of cores or the
clock speed of the cores used to run the MC simulations. In the current environment of processor
manufacturing the shift has been away from higher clock speeds (due to thermal, current leakage,
and power consumption issues) and instead towards multicore chips. Due to the “embarrassingly
parallel” nature of MC simulations (each photon is an independent interaction) the shift towards
multiple cores is advantageous. Instead of using a single core per MC projection angle
simulation, the simulation of a single angle could be divided up over NC multiple cores.,
decreasing the MC simulation time (which accounts for ~60% of the computation time) by a
factor of NC. The current workstation built for testing clinical applications has a total of 32 cores
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so it is feasible to have Nc=2, while still leaving two cores dedicated to the analysis process and
any operating system threads. Alternative hardware speedups exist, such as MC GPU
implementations, and this is something that could be easily integrated into the existing CMCF
algorithm [83]. Besides hardware speedups there exists numerous other methods to decrease the
MC photon generation time. One method is to simplify the MC phantom by increasing the voxel
size and limiting the number of materials used in the composition of the phantom. When using a
isotropic voxels the largest computational savings come from reducing the voxel size of the
phantom used, with little to no savings seen in reducing the number of materials (e.g. soft tissue,
water, air, lung) used to represent the object. For an isotropic voxel with a pitch of Vp (volume of
voxel: Vp×Vp×Vp) our initial experiments show the computational savings of increasing Vp by a
factor of NV to be roughly a factor of NV. The savings are due to a decrease in the number of
boundary condition checks in the MC photon transport code..The factor NV must be chosen such
that accuracy of the scatter simulations can be maintained. It is our hypothesis, from our previous
experiments [42], [43], that one of the most important boundary locations to maintain is the
location of the skin or object air interface, as this dominates in determining the shape of the
coherent scattering distribution. Maintaining the air-object boundaries while using uniform voxel
size is not an optimal solution for increasing the MC simulation efficiency. One way to maintain
the shape of the objects boundaries while minimizing the number of voxel boundary crossings is
to use an octree representation of the MC phantom [92], [93]. Octree volume representations are
the 3D equivalent of quadtree representations of 2D images and allow for the regional grouping
of like voxels into larger voxels while maintaining the differentiated nature of smaller voxels.
The grouping of similar voxels reduces the memory storage requirements as well as limits the
number of voxel boundaries used. The reduction in the number of a materials assigned to the
phantom would be more relevant to the photon transport efficiency when using an octree versus
a isotropic voxel representation. Using fewer materials increases the potential of large
“groupings” of voxels in the octree. Finally, the MC simulation time can be decreased by using
variance reduction techniques such as forced detection [44]-[47] and interaction splitting
combined with Russian Roulette [82].
The second largest potential area for efficiency improvement is in the inverse fast Fourier
transform (FFT-1) used to interpolate the scatter data. In correcting the data collected on the
Elekta Synergy system the FFT-1 accounted for over 30% of the total computational time. The
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current implementation uses a sequential version of the FFT-1 from the FFTW library [84]. Since
the interpolation using the FFT-1 is done after all the necessary MC simulation data has been
collected the cores used by the MC threads could be reassigned to computing a parallel version
of the FFT-1. The FFTW library includes a few different parallel implementations of the forward
and inverse FFT. The time speed up when using a parallel version of the FFT isn't necessarily
linear with the number of cores, and the speed up is dependent on the size of the problem.
Parallel benchmark data from the FFTW website (http://www.fftw.org) shows that for a
multidimensional FFT on data of a size 256×256×256 running on 8 processors had a speed up of
over 4 times that of using a single processor. We estimate that by implementing the most
straight forward efficiency reductions of doubling the voxel size of the MC phantom (NV=2),
simulating each projection using 2 cores (Nc=2), and using a parallel implementation of the FFT
the total computation time would be reduced to under a minute for a scatter estimate that can be
used in correcting clinical projection images.
5 Scatter Estimate Accuracy
The accuracy of the MC scatter estimate is obviously tied to the accuracy of the physics
modeling the particle interaction, but also to the accuracy of the physical object/patient and
imaging geometry model used in the simulation. In the simulated head and pelvis phantoms
shown in Chapter 4 the MC object model is identical to the model used to create the simulated x-
ray images. Under these conditions the system was able to estimate the scatter with an error
value less than 10% in under 2 minutes. When using these scatter estimates to correct CBCT
projections the resulting reconstruction error was less than 3%. In the measured pelvis phantom
and patient data from Chapter 4 the object model is created from the original CBCT projection
data and inherently has errors due to scatter and other imaging artifacts (e.g. beam hardening,
lag). A potential way to circumvent this problem is to use a prior CT scan of the patient, such as
a planning CT in radiation therapy, to create the MC object model. Alternatively, some MC
scatter correction techniques [47] propose to use an iterative correction technique, wherein the
object/patient model is updated by successive reconstructions which have been corrected by the
previous scatter estimate. Such approaches could significantly increase the computation time, as
both multiple scatter estimates and reconstructions must be computed.
The required exactness of the object/patient model for an accurate estimate of the scatter
93
distribution to be determined is still an open question. Our earlier experiments examining the
nature of the scatter distribution [42], [43] show that the need for an accurate definition of the
objects-air boundary is definitely important in properly representing the coherent portion of the
scatter distribution. We have performed a preliminary investigation showing that the definition
of the object models internal anatomy may be less important for estimating the scatter
distribution. In the investigation MC phantoms assigned a single material (e.g. tissue) and
density were compared to those with multiple materials and densities with the single material and
density phantoms resulting in similar scatter distributions. Similarly, scatter correction results
shown in Figure 48(b) using an object model created with inherent error from the scatter affected
reconstruction seen in Figure 48(a) suggest that an exact model may not be necessary. The
required geometric precision of the MC phantom used is a promising area for future research that
could determine the necessary internal and external phantom material delineations required for
an accurate simulation of the scatter distribution.
6 Noise Reduction Techniques
The use of a denoising technique to reduce the statistical noise in MC simulations is not in itself
unique, several other groups have applied these methods in accelerating MC estimates of dose
[78]-[81] and scatter [44], [47], [48]. The denoising methods employed in the other examples of
scatter estimation are the locally-adaptive Savitzky-Goley (LASG) filter [47] and the
Richardson-Lucy method [44], [48]. A Fourier fitting technique implemented using a low-pass
filter has a few advantages in reducing computation times in MC scatter estimation. It is an
efficient non-iterative method that can be easily parallelized [84]. The computational complexity
for a multidimensional FFT on a data set with a total element size N is O(Nlog2N). The low-pass
filter is tuned specifically to take advantage of the low-frequency scatter distribution for common
patient imaging sites [94], [43] and can be easily tuned offline to other shapes and sizes [43].
Additionally, since the coherent, incoherent, and multiple scatter signal are already separated in
the MC simulations different frequency cutoffs could be used to process each of these
components contribution to the detector signal, similar to the approach taken by Sisniega et al.
[95] In particular, it might be advantageous to have a higher cutoff for the coherent scatter signal
due to its more forward directed scattering distribution which results in a greater degree of
structure. The most desirable trait of using a Fourier fitting technique is the inherent ease in
which an interpolation can be computed by taking a zero-padded FFT-1 (see Appendix B) of the
94
frequency domain filtered data. The other aforementioned fitting methods do not currently
incorporate a mechanism for interpolation, which has allowed our algorithm to reduce the
computational load by a factor of 24. One draw back to using the FFT implementation is that
both the original and interpolated data sets must have a constant sampling frequency. This is
most detrimental in the sampling of the original data as the change in the scatter distribution with
respect to θ is nonuniform, especially when a compensator is not used, as seen in the scatter
distribution sinograms in Figure 29 and Figure 30 of Chapter 3. We are investigating the
potential of integrating alternative regression models such as locally weighted scatterplot
smoothing (LOWESS or LOESS ) [96], [97] and Gaussian regression (a.k.a. kriging) [98] into
our CMCF algorithm to provide denoising and interpolation without the fixed sampling rate
constraint.
7 Final Considerations
The choice of a scatter correction method is not an easy task, as there exists a plethora of
correction techniques that have been developed in which x-ray based imaging has continued to
become an increasing useful imaging modality. A comprehensive comparison of our method to
all other techniques is out of the scope of this study and the reader is referred to a recent two part
review by Rührnschopf and Klingenbeck [99], [100] in which a substantial number of the scatter
correction methods are reviewed through a generalized framework. For the sake of simplicity we
limit our discussion to other scatter estimation techniques and make a note that scatter rejections
methods (e.g. scatter grids, air gaps, collimation, and compensators) are complementary to our
approach by simply incorporating them into the MC model. In general MC based scatter
estimation methods provide a explicitly object based scatter estimate without the need for
additional imaging hardware, increased patient x-ray exposure, or introduction of additional
imaging artifacts often associated with measurement based estimation techniques using beam
stops [77], [101], [102] or primary modulation [55], [62]. MC based approaches use the most
complete modeling of the physics involved in the image formation and thus potentially provide
the most accurate scatter estimate. The high level of accuracy in MC approaches traditionally
comes at an increased computational time [28] compared to other analytical estimation methods
such as 2D scatter kernels which rely on a variety of approximations to increase their
computational efficiency presenting a tradeoff in scatter estimate accuracy [100]. The largest,
and possibly only, drawback in a MC approach compared to any other method is the time to
95
scatter estimate (TTSE), which is extremely important in implementing an on-line correction for
clinical procedures (see Potential for Clinical Implementation, section 3). Previous applications
of denoising techniques to MC scatter distribution estimates [44], [48], [82] have shown that a
speedup on the order of 10 to 104 is possible. Our own results show decreases on the order of 103,
with further reductions possible with the use of a compensator. The benefit of our denoising
method is that the CMCF method uses a fitting function that combines noise reduction with
interpolation allowing for a further reduction in computation time by limiting the number of MC
projections simulations needed. The addition of interpolation reduced the number of projection
simulations necessary by a factor of 24. The current TTSE is approximately 2 minutes, and
further efficiency improvements seem quite plausible. Another important difference between our
method and other MC based techniques is that in the CMCF method the MC simulations are run
concurrently with a fitting function allowing for real-time feedback in the quality of the scatter
estimate, allowing the number of photon tracks used to be determined dynamically based on a
desired level of GOF.
Using MC based approach may provide benefits beyond just estimating the scatter. We feel there
is potential to use the same MC system to estimate the scatter distribution, compute an estimate
of the imaging dose delivered to the patient, and compute a correction for beam hardening.
Currently within our own MC system the patient dose is being computed for each of the
projection angles simulated in the CMCF method. It has previously been shown that MC dose
estimates, like the MC scatter distribution estimate, can be denoised [78]-[81] to reduce the
number of particle tracks required while maintaining accuracy. Work done on computing doses
for intensity modulated arc therapy, which continuously delivers varying radiation beams while
rotating around the patient, found that the dose from the continuous beam can often be
approximated by calculating the dose from static beams equally spaced every 10º [103]-[105].
Beam hardening artifacts occur due to the monoenergetic assumption made in most
reconstruction algorithms, whereas in reality x-ray sources used in medical imaging devices are
polyenergetic (see Chapter 1 Section 2.3 Computed Tomography). In our simulation the full
spectrum and spatial distribution of the polychromatic photons of the input source beam is
created through a separate simulation of the entire x-ray tube (e.g. input electron beam, tungsten
target, filtration, etc), but can also be generated through the use of an analytical tool such as
Spektr [19]. A beam hardening correction factor could potentially be computed using the energy
96
spectrum information from the primary photons transported through the MC object model.
We therefore believe a MC based approach to scatter correction is the most accurate and useful
approach with the greatest potential for clinical viability and further improvements in efficiency.
Indeed, with the reduction of computational times achieved in this study the barrier of MC being
clinically relevant has been reduced and in some situations entirely eliminated. The MC
approach uses the most accurate physical models of fundamental x-ray imaging processes and
may provide solutions to problems outside of just scatter induced artifacts. Though the exact
clinical value of implementing a scatter correction is hard to quantify the impact could be quite
large even if the scenarios in which a clinical out come is improved is small. There are
approximately 2000 CBCT systems used in image-guidance on linacs (e.g. Elekta Synergy,
Varian OBI) world wide and if we estimate that on average these units are used to collect
treatment images for 20 patients a day, 250 days a year then approximately 10 million images are
collected. If the scatter correction algorithm improves the outcome for only 0.1% of these patient
treatments that would still be 10,000 cases with an improvement.
97
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Appendix A : Bowtie Filter Modeling
Both bowtie filter models were 12 cm in length (x-dimension), 8.5 cm in height (y-dimension).
The F1 and custom bowtie filter had a center thickness of 0.3 and 0.1 cm respectively. The
curved surface of the F1 bowtie was modeled as two polynomials given as
z top=0.036 x20.15 (49)
and
zbottom=−0.036 x2−0.15 (50)
where the center of the bowtie is set as the origin (0,0) and the units are in centimeters. The
curved surfaces were sampled with x and z voxel pitches of 0.25 mm. A profile of the F1 filter is
shown in Figure 9(a). In the CBCT MC simulation the center of the F1 bowtie filter was
positioned at a distance of 30 cm from the focal spot.
The custom bowtie filter was designed to compensate for a 16.4 cm diameter water cylinder. The
path length, lc, of a ray from a point source through a cylinder with a radius rcyl that also passes
through a point x positioned at a distance df from the center of the cylinder can be shown to be
l c x =2 rcyl
2 x2d f −SAD
2 rcyl2
−SAD2 x2
d f −SAD 2x2 . (51)
Thus the pathlength, lf, the same ray must travel through a filter, made of a material with an
attuenation of µf, designed to compensate for the attenuation of the cylinder, consisting of a
material with an attenuation coefficient uc,can be determined as
l f x =
f
c
2 rcyl−lc x . (52)
We can then determine the thickness of material, t, required for the filter, in the z direction, at a
position x' located at a distance df from the cylinder as
t x ' =l f x
SAD−d f
SAD−d f 2x2 (53)
where x' is computed as
x '=x
xl f x
SAD−d f 2x2
. (54)
106
The simulated model of our custom bowtie is designed for a water cylinder with a radius of 8.2
cm positioned such that the SAD is 100 cm and located 28.5 cm from the focal spot (df=71.5
cm). The average linear attenuation for water and aluminum are estimated to be µc=0.211 cm-1
and µf=0.7079 cm-1, respectively. The thickness of compensator, t(x'), was scaled by a factor of
0.6 to limit the maximum thickness of the filter to 3 cm, including the center filtration thickness
of 0.1 cm. The curved surfaces were sampled using a voxel pitch of 0.2 mm in the x and z
directions. A profile of the custom bowtie filter can be found in Figure 9(b). The flat surface of
the bowtie was placed closest to the focal spot at a distance of 28.5cm.
107
Appendix B : Fourier Interpolation
For the sake of simplicity the 1D case of Fourier interpolation is described which can be easily
extended to any multidimensional case. We define a 1D signal f(x) sampled at N discrete points
at intervals of dx (sampling frequency of us=1/dx) resulting in the discrete function fn defined as
f n= f x0ndx , n=0,1,2,... , N −1 (55)
where x0 is the starting sampling position. The Fourier transform f(x) is
f̊ u =F { f x}=∫−∞
∞
f xe− j2 ux dx (56)
where the variable u represents frequency. The discrete Fourier transform of fn is given as
f̊ k =F { f n}=∑n=0
N −1
f n e− j2 kn
N
(57)
where k is the the kth element in the frequency domain. Just as element n in fn corresponds to a
position x, given as x=x0+ndx, k likewise corresponds to the frequency u, where
u=kN
−1
2 dx, k=0,1,2,... , N −1. (58)
The inverse discrete Fourier transform of f̊ k is defined as
f n=F −1{ f̊ k }=
1N
∑k=0
N −1
f̊ k ej 2 k n
N (59)
A Fourier interpolation of fn, with an increased sampling frequency of a factor of b can be
accomplished by zero-padding f̊ k and taking the inverse Fourier transform. The number of zeros
to pad f̊ k with is given as M=N(b-1), where b is limited to values that result in integers values of
M. The zero-padding of f̊ k is specified as
g̊ l={f̊ l , 0≤l≤
N −12
f̊ l −M , Nb−N2
≤l≤Nb−1
0,N −1
2lNb−
N2
(60)
108
where l=1,2,3,...,Nb-1. The resulting interpolated function gl is sampled at a frequency of b/dx.
To minimize the interpolation error between the interpolated function, gl, and f(x) it is important
to have used a sampling rate, us, greater than twice the maximum frequency of f(x) when creating
fn as stated in the Nyquist sampling theorem, though this condition is not sufficient to eliminate
any errors. It is also important that f(x) is periodic and the sampling length N is properly chosen
to minimize errors. A more thorough analysis of the errors associated with Fourier interpolation
of band-limited signal, including upper-bounds on interpolation errors, can be found in Xu et al.
[76].
109
Copyright Acknowledgments
Chapter 2 is published and Chapter 3 has been accepted for publication in the International
Journal of Medical Physics. Permission has been granted by the journal to republish the
aforementioned articles and/or excerpts from them.
110