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Computational aspect of tomographic microwave imaging for biomedical applications

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COMPUTATIONAL ASPECT OF TOMOGRAPHIC
MICROWAVE IMAGING FOR BIOMEDICAL APPLICATIONS
A Thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
by
AMIR H. GOLNABI
Thayer School of Engineering
Dartmouth College
Hanover, New Hampshire
MAY 2012
Examining Committee:
Chairman _______________________
Paul M. Meaney, Ph.D.
Member ________________________
Keith D. Paulsen, Ph.D.
Member ________________________
Andrea Borsic, Ph.D.
Member ________________________
Elise C. Fear, Ph.D.
_____________________
Brian W. Pogue
Dean of Graduate Studies
Copyright 2012 Amir H. Golnabi
UMI Number: 3544495
All rights reserved
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Abstract
Microwave imaging (MI) is based on recovering dielectric properties (permittivity and
conductivity) of materials. Over the last two decades, MI has attracted increasing interests in
biomedical applications, in particular, for breast cancer screening and therapy monitoring,
mainly due to the significant dielectric property contrast between normal and malignant breast
tissue. Compared to other conventional imaging modalities, such as X–ray mammography, the
non–ionizing and non–compressive nature of MI makes it very attractive from a patient’s
perspective. In addition, MI has a relatively high sensitivity to detect small tumors, and
potentially high specificity to determine whether a suspicious area is malignant or benign at a
significantly lower cost level compared to methods such as magnetic resonance imaging (MRI)
and nuclear medicine.
Several years ago, a clinical microwave imaging system was developed at Dartmouth
College. During microwave data acquisition, electromagnetic fields propagate through and
scatter from the tissue in a three–dimensional (3D) fashion. However, in order to reduce the
computational complexity and to speed up the image reconstruction process, it is often assumed
that the behavior of electromagnetic waves in 3D space can be represented as a simplified 2D
model. While it benefits from less intensive computational demands, such assumption can lead
into increased level of artifacts in the recovered dielectric properties. Therefore, in order to
improve the accuracy and quality of reconstructed images, we have developed a viable, fast, and
user-friendly 3D microwave image reconstruction.
We have previously demonstrated that we can recover clinically useful microwave
tomographic images for breast cancer detection without using any prior spatial information about
the tissue. However, in order to enhance the quality of our previous results and recover more
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accurate dielectric property distributions, we have implemented a new approach to combine the
functional information of the MI with the high spatial resolution of other imaging modalities
such as MRI. This approach is based on developing a new image reconstruction strategy that
exploits structural information about the object being imaged through some constraints in the
microwave property contrast.
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Acknowledgements
This dissertation would not have been possible without the guidance and the help of several
individuals who in one way or another contributed and extended their valuable assistance in the
preparation and completion of this study. First and foremost, I would like to show my sincerest
gratitude to my advisor, Prof. Paul Meaney, who has supported me throughout my doctoral
studies with his patience and knowledge whilst allowing me the room to grow and expand my
professional and personal experiences.
I should also express my gratitude to my other advisor, Prof. Keith D. Paulsen, who has
constantly supported me with encouragement and enthusiasm. Despite his extremely busy
schedule, Keith has always made himself available to me for any questions or concerns. In
addition, his valuable comments and questions have helped me become a better scientist who is
more competent in the field.
In addition to my advisors, I would like to thank the rest of my thesis committee, Prof.
Andrea Borsic and Prof. Elise Fear for their insightful comments and hard questions. I would
also like to thank Prof. Brian Pogue, Prof. Dan Lynch, Prof. George Cybenko, Prof. Alex Hartov,
Dr. Ron Lasky, and Dean Joe Helble for their advice and support during my time at Thayer.
I am grateful to Amin Plaisted for his invaluable help with this dissertation, and to Dr.
Hamid Ghadyani for providing me with the 2D and 3D mesh generating codes. His help has been
a key to completing some of the most important parts of my research. Although Margaret
Fanning and Sherri Geimer no longer work in the MIS research group, they both deserve my
sincere thanks. Margaret and Sherri have been excellent role models for me as very organized
and detailed oriented researchers. It was a pleasure working with them.
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I am indebted to many of my current and former colleagues who have supported me during
the last five years at Dartmouth: Matt Pallone for his continuous positive attitudes, Neil Epstein
and Alex Bijamov for useful discussions, and Tian Zhuo for his xcritical comments and inputs. I
would like to thank Dr. Tomasz Grzegorczyk, Preston Manwaring, Dr. Qianqian Fang, and Dr.
Elia Attardo for their assistance and thoughtful suggestions.
Xiaoyao, Parisa, Lamia, AJ, Ashley, Alex, and Lyubomir, I am grateful for having had
such a wonderful friends at Thayer with whom to share the good times working, studying,
playing, or just taking a break. Also my special thanks goes to my dearest friend Niusha who has
been there for me whenever I needed her.
Finally, I give my deepest gratitude to my parents and my wife Marzie for supporting me
unconditionally and for believing in my capabilities. Marzie has been very understanding and
patient with me during my very long hours in the office including nights and weekends. Without
her encouragement, I would not be where I am today without her heartening.
v
Dedications
I dedicate this work to the loves of my life, my wife Marzie, my daughter Niusha, my
mother Shahnaz, my father Jalil, my sister Elham, and my brother Mahdiar.
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Table of Contents
1. Introduction ..................................................................................................................................1
1.1. Motivation ............................................................................................................................1
1.2. Overview of Different Microwave Imaging Methods .........................................................4
1.3. Tomographic Microwave Imaging .......................................................................................8
1.4. Project Aims .......................................................................................................................10
1.5. Proposed Work ...................................................................................................................12
2. 3D Microwave Image Reconstruction .......................................................................................15
2.1. 2D vs. 3D Modeling of the Electromagnetic Field ............................................................15
2.2. Challenges With Respect to 2D Reconstruction Algorithm ..............................................19
2.3. Proposed method ................................................................................................................21
2.3.1. User Interface ...........................................................................................................21
2.3.2. 3D Data Acquisition ................................................................................................22
2.3.3. Input–File Format ....................................................................................................23
2.3.4. Optimizing the Reconstruction Algorithm...............................................................29
2.4. Results – Simulations .........................................................................................................30
2.4.1. FDTD Grid Density .................................................................................................31
2.4.2. Number of Reconstruction Nodes ............................................................................35
2.4.3. 3D images from in-plane data vs. stack of 2D images ............................................40
2.4.4. Data Selection for Reconstruction ...........................................................................43
2.4.5. Number of Iterations ................................................................................................45
2.5. Results – Phantom Experiments.........................................................................................49
2.5.1. Two Cylindrical Inclusions ......................................................................................49
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2.5.2. Breast-shaped Phantom Experiment with an Spherical Inclusion ...........................54
2.6. Results – Initial Clinical Data ............................................................................................63
2.6.1. Breast Imaging .........................................................................................................63
2.6.2. Bone Imaging ...........................................................................................................72
3. Incorporation of a Priori Structural Information into the Microwave Image
Reconstruction Algorithm .........................................................................................................77
3.1. Introduction ........................................................................................................................77
3.2. A Mathematical Framework of Microwave Image Reconstruction Algorithm .................79
3.2.1. Overview ..................................................................................................................79
3.2.2. Soft prior Regularization..........................................................................................82
3.2.3. Hard priors – Parameter Reduction..........................................................................85
3.2.4. Error Analysis ..........................................................................................................86
3.3. Integration of Microwave Tomographic Imaging System with MRI ................................88
3.3.1. Motivation ................................................................................................................88
3.3.2. Integration of MI System into MRI .........................................................................89
3.3.3. MR Image Artifact Reduction Strategies .................................................................91
3.3.4. Multipath Signal Assessment ...................................................................................93
3.3.5. MR RF Gradient Signal Disruption of Microwave Signals .....................................95
3.4. Results – 2D .......................................................................................................................99
3.4.1. Simulation Experiments ...........................................................................................99
3.4.2. Phantom Experiments ............................................................................................131
3.4.3. Bone Imaging: Monitoring Changes in the Bone Dielectric Property
Distributions............................................................................................................178
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3.4.4. Clinical Breast Microwave Imaging in MR ...........................................................186
3.5. Results – 3D .....................................................................................................................191
3.5.1. Simulation Experiments .........................................................................................191
3.5.2. Phantom Experiments ............................................................................................220
4. 3D Microwave Image Reconstruction GUI .............................................................................235
5. Summary and Conclusion ........................................................................................................244
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List of Figures
Figure 1.1. Tree diagram of the current MI methods for breast cancer detection ...........................4
Figure 1.2. MIST system: (a) antenna configuration on two independently-moving plates
A (pink) and B (blue), (b) imaging tank, (c) exam platform, and (d) cabling and
fluid reservoir underneath the table .....................................................................................9
Figure 2.1. Simulation experiment setup with a cylindrical inclusion or 1.0 cm radius ................17
Figure 2.2. Calibrated amplitude/power (top) and phase (bottom) values of the
synthesized data using the 2D and 3D models as a function of the relative
receivers for a single transmitter (tx = 1) ...........................................................................18
Figure 2.3. RMS difference values of the amplitude and phase ....................................................19
Figure 2.4. Schematic antenna configuration: Two interleaved arrays of 8 antennas with
each sub-array (A and B) being able to move independently from the other ....................23
Figure 2.5. Schematic summary of steps to create 3D image reconstruction input files
from the MIST system output data ....................................................................................24
Figure 2.6. Input File–1 .................................................................................................................25
Figure 2.7. Input File–2 .................................................................................................................25
Figure 2.8. Input File–3 .................................................................................................................26
Figure 2.9. Input File–3’ ................................................................................................................27
Figure 2.10. Input data selection GUI in MATLAB ......................................................................28
Figure 2.11. Schematic summary of reconstruction algorithm ......................................................29
Figure 2.12. Schematic of the imaging domains evaluated: The background diameter was
14 cm (antennas are positioned on a 15.2 cm diameter). The spherical inclusion of
1.5 cm radius was centered at (3, 0, 0 cm). ........................................................................30
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Figure 2.13. 1300 MHz contour slice images extracted from the reconstructed permittivity
(top) and conductivity (bottom) profiles for the simulation experiment using the
finite difference grid spacing cases (a) to (f) in Table 2.1. ................................................34
Figure 2.14. Reconstructed permittivity (left) and conductivity (right) profiles along with
the true property values, as a function of the number of cells per wavelength (D)
in the forward solution .......................................................................................................35
Figure 2.15. 1300 MHz reconstructed permittivity (top) and conductivity (bottom)
profiles for the simulation experiment on (a) 600, (b) 1587, (c) 2611, (d) 3767, (e)
4608, (f) 6095, (g) 7669, and (h) 9321 node meshes. ........................................................39
Figure 2.16. Reconstructed permittivity (left) and conductivity (right) profiles along with
the true property values, as a function of the number of reconstruction nodes .................39
Figure 2.17. Total reconstruction times (min) per iteration as a function of the number of
reconstruction nodes ..........................................................................................................40
Figure 2.18. (a) 2D and (b) 3D reconstructed permittivity (top row) and conductivity
(bottom row) slices (in z-direction) of the simulation experiment with -100 dBm
added noise at 1300 MHz, using only in-plane data ..........................................................42
Figure 2.19. 3D reconstructed permittivity (top row) and conductivity (bottom row) slices
(in z-direction) of the simulation experiment with -100 dBm added noise at 1300
MHz, using (a) the full-data (all 7 planes of data, including both in-planes and
cross-planes), and (b) multi-plane data with two consecutive planes ...............................44
Figure 2.20. 3D Relative errors as a function of the number of iterations from a
simulation experiment........................................................................................................46
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Figure 2.21. Relative permittivity (top) and conductivity (bottom) difference values for
iterations (a) 40, (b) 60, (c) 80, and (d) 100 (i.e. for D20,40, D20,60, D20,80, and
D20,100, respectively), along with the mean of the difference values ± 2 standard
deviations (i.e. 95% of all difference data) ........................................................................48
Figure 2.22. Setup for the phantom experiment with two cylindrical inclusions: (a) The
square-based inclusion tilted toward the right side of the imaging domain, (b)
Both inclusions immersed in a coupling medium and surrounded by the antenna
array ...................................................................................................................................49
Figure 2.23. 1100 MHz reconstructed permittivity values of the 3D phantom experiment
with two cylindrical inclusions. The iso-surface thresholds are εr,CircInc = 10.0 and
εr,SqInc = 33.0. .....................................................................................................................51
Figure 2.24. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) slices
of the phantom experiment with two cylindrical inclusions using 3D
reconstruction algorithm with (a) full-data set, (b) 5 multi-plane data sets with
two consecutive planes, (c) 5 in-plane data sets, (d) 5 individual in-plane data
sets, and (e) using 2D reconstruction algorithm with 5 individual in-plane data
sets......................................................................................................................................54
Figure 2.25. Breast-shaped phantom experiment setup: (a) The spherical saline gel target
inclusion suspended in the plastic breast model, (b) Rapid–prototyped plastic
breast model submerged in the imaging tank ....................................................................55
Figure 2.26. 3D reconstructed (a) permittivity and (b) conductivity images, using the
multi-plane data set at 1300 MHz. The iso-surface thresholds are εr,BR = 15.0, σBR
= 1.0 S/m, and εr,Inc = 65.0 and σInc = 2.3 S/m. ..................................................................56
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Figure 2.27. Extracted slices from the 3D reconstructed images in Figure 2.26:
Permittivity (top row) and conductivity (bottom row) ......................................................58
Figure 2.28. 2D reconstructed dielectric properties: Permittivity (top row) and
conductivity (bottom row) .................................................................................................59
Figure 2.29. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) slices
of the breast-shaped phantom experiment at z = 3 cm using the 3D reconstruction
algorithm with (a) full-data set, (b) 9 multi-plane data sets with two consecutive
planes, (c) 5 in-plane data sets, and (d) using the 2D reconstruction algorithm
with 9 individual in-plane data sets ...................................................................................61
Figure 2.30. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) slices
of the breast-shaped phantom experiment at z = 0 cm using the 3D reconstruction
algorithm with (a) full-data set, (b) 9 multi-plane data sets with two consecutive
planes, (c) 5 in-plane data sets, and (d) using the 2D reconstruction algorithm
with 9 individual in-plane data sets ...................................................................................61
Figure 2.31. 3D microwave imaging of a normal subject in the clinical MIST system ................64
Figure 2.32. Extracted slices from the 1100 MHz 3D reconstructed images using the fulldata set: Permittivity (top row) and conductivity (bottom row) ........................................65
Figure 2.33. 1100 MHz 2D reconstructed dielectric properties: Permittivity (top row) and
conductivity (bottom row) .................................................................................................65
Figure 2.34. 1100 MHz reconstructed permittivity (top) and conductivity (bottom) slices
of the breast-shaped phantom experiment at z = 1.5 cm using the 3D
reconstruction algorithm with (a) the full-data (FD) set, (b) 6 multi-plane data sets
with four consecutive planes (4CS) (two planes above and two planes below), (c)
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6 multi-plane data sets with two consecutive planes (2CS) (one plane above and
one plane below), (d) 6 in-plane (IP) data sets, and (e) using the 2D reconstruction
algorithm with 6 individual in-plane data sets. ..................................................................67
Figure 2.35. Relative permittivity (top) and conductivity (bottom) difference images with
respect to the recovered dielectric property values using the full-data (FD) set: (a)
full-data – multi-plane data sets with four consecutive planes (i.e. FD – 4CS) (b)
full-data – multi-plane data sets with two consecutive planes (i.e. FD – 2CS), and
(c) full-data – in-plane data sets (i.e. FD – IP) ..................................................................69
Figure 2.36. Relative permittivity (top) and conductivity (bottom) difference values in
Figure 2.35 as a function of the reconstruction nodes, along with the mean of the
difference values ± 2 standard deviations (i.e. 95% of all difference data): (a) fulldata – multi-plane data sets with four consecutive planes (i.e. FD – 4CS) (b) fulldata – multi-plane data sets with two consecutive planes (i.e. FD – 2CS), and (c)
full-data – in-plane data sets (i.e. FD – IP)........................................................................71
Figure 2.37. Heel imaging in the microwave imaging system: (a) The heel extended
through an aperture in the mounting plate, (b) The heel surrounded by the antenna
array ...................................................................................................................................72
Figure 2.38. 2D and 3D reconstruction meshes: (a) 6.9 cm radius circular mesh: 559
nodes and 1044 triangular elements, (b) 6.9 cm radius, 5 cm height cylindrical
mesh: 3132 nodes and 12707 tetrahedral elements............................................................73
Figure 2.39. Several views of the 3D reconstructed permittivity image of the heel at 1300
MHz, with an iso-surface threshold of εr = 26 ...................................................................74
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Figure 2.40. 1300 MHz, (a) 2D and (b) slices of the 3D reconstructed permittivity (top)
and conductivity (bottom) images at 4 planes of data separated by 1 cm, starting
from the ankle (P1) and moving down towards the heel apex (P4). ..................................75
Figure 3.1. Soft prior regularization matrix L with two distinct regions .......................................83
Figure 3.2. MI-MR imaging: (a) the portable data acquisition system in the control room
with the auxiliary cables attached to the bulkhead connectors in the wall, (b) the
imaging tank outside of the MR bore showing the semi-rigid cables extending
from the tank ......................................................................................................................90
Figure 3.3. MI tall tank with a large cylinder breast phantom (green) and smaller
inclusions surrounded by the monopole antenna array: (a) side view and (d) topdown view ..........................................................................................................................92
Figure 3.4. T2–weighted MR images of the phantom experiment: (a) at the center plane
of the active part of the antennas and (b) at a plane 3 cm below the interface
between the semi-rigid coaxial line and the antenna. For (a) and (b) the phantom
included a 5.4 cm radius outer cylinder and the 2.1 and 1.0 cm radius inclusions,
and the coaxial line center conductors were silver-plated copper. (c) Image at the
center plane of the active part of the antennas for the 5.4 cm radius outer cylinder
and just the 1.0 cm radius inclusion. In this case the coaxial cable center
conductor was steel. ...........................................................................................................93
Figure 3.5. Monopole antenna array: (a) in the MI shortened tank with a breast phantom
(green), and straight 5.0 cm long feedlines (short), (b) with 9.0 cm long serpentine
feedlines (tank wall removed) ............................................................................................95
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Figure 3.6. Plots of the 1300 MHz amplitudes as a function of receiver number for all 16
transmitting antennas when the imaging tank is operating in the MR bore with
MR data being acquired simultaneously for the cases where (a) no low pass
filters, and (b) two low pass filters installed, respectively. ................................................97
Figure 3.7. Schematic of the imaging domains evaluated. The background diameter was
14 cm (antennas are positioned on a 15.2 cm diameter). The circular inclusion’s
center with radius r = 1.40 cm was located at (0, –3 cm). ...............................................100
Figure 3.8. Reconstruction parameter meshes (a) uniformly distributed 473 node mesh,
(b) uniformly distributed 961 node mesh, and (c) 915 node mesh with preferential
node deployment in the inclusion in Figure 3.7 ...............................................................100
Figure 3.9. Simulated 1300 MHz reconstructed permittivity (top) and conductivity
(bottom) images (a) without priors on the 473 node mesh, (b) without priors on
the 961 node mesh, (c) without priors on the 915 node mesh with preferential
node deployment in the inclusion, and (d) with the soft prior regularization on the
915 node mesh. ................................................................................................................101
Figure 3.10. Comparison of the 1300 MHz reconstructed permittivity (top) and
conductivity (bottom) values using the no priors and soft prior regularization with
different levels of added noise: (a) –110, (b) –100, (c) –90, (d) –80 dBm ......................103
Figure 3.11. Comparison of 1300 MHz reconstructed permittivity (top row) and
conductivity (bottom row) profiles for the no priors and the soft prior
regularization: (a) no permittivity contrast (left column), (b) no conductivity
contrast (right column). ....................................................................................................105
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Figure 3.12. (a) Schematic of the imaging domains evaluated. The arbitrarily shaped
inclusion was located in the upper part of the imaging domain. The reconstructed
values of the simulation experiment with arbitrarily shaped inclusion were
extracted at 30 points evenly distributed along the line x = –2 cm. (b) Customized
soft prior reconstruction mesh comprised of 1725 nodes and 3248 triangular
elements. ..........................................................................................................................106
Figure 3.13. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) images
from a phantom experiment with arbitrarily-shaped target inclusion for (a) no
priors on the 473 node mesh, and (b) soft priors on the customized 1725 node
mesh in Figure 3.12(b) .....................................................................................................107
Figure 3.14. Comparison of the 1300 MHz reconstructed (a) permittivity and (b)
conductivity profiles (along the line x = –2 cm) in Figure 3.13 phantom
experiment with arbitrarily-shaped target inclusion for the no priors and soft prior
regularization ...................................................................................................................108
Figure 3.15. Customized soft prior reconstruction meshes used in 1 and 3–inclusion
simulation experiments: (a) 402 node mesh with one inclusion and two distinct
regions (RBK and RI1), (b) 1203 node mesh with one inclusion and two distinct
regions (RBK and RI1), (c) 407 node mesh with three inclusions and two distinct
regions (RBK and RI1), (d) 1203 node mesh with three inclusions and two distinct
regions (RBK and RI1), (e) 407 node mesh with three inclusions and four distinct
regions (RBK, RI1, RI2, and RI3), and (f) 1203 node mesh with three inclusions and
four distinct regions (RBK, RInc1, RInc2, and RInc3) .............................................................109
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Figure 3.16. Soft prior reconstructed (a) permittivity and (b) conductivity values using
two meshes with different number of nodes, along with the exact properties of
background and inclusion at 1300 MHz...........................................................................110
Figure 3.17. Maximum soft prior recovered (a) permittivity and (b) conductivity values
along with the exact properties of the background and target inclusions at 1300
MHz using the customized reconstruction meshes in Figures 3.15(c) – (f) .....................112
Figure 3.18. Maximum soft prior recovered (a) permittivity and (b) conductivity values
along with the exact properties of the background and target inclusions at 1300
MHz using the customized reconstruction meshes in Figures 3.15(e) and (f) .................115
Figure 3.19. Comparison between the soft prior reconstructed (a) permittivity and (b)
conductivity values of I1 with and without additional targets (three inclusions and
one inclusion cases, respectively) ....................................................................................117
Figure 3.20. Customized reconstruction mesh comprised of 1196 nodes and 2215
triangular elements, accounting for two inclusion regions: a false circular region
of interest (in red) with radius of 1.4 cm centered at (0, 3 cm), and the actual
circular target zone (in green) with radius of 1.4 cm centered at (0, -3 cm) ....................119
Figure 3.21. Transect plots of the 1300 MHz reconstructed soft prior permittivity (top)
and conductivity (bottom) profiles along the y-axis for three simulation cases with
a false region of interest. Dielectric properties of the actual inclusion were set to
those of the (a) 40:60, (b) 55:45 and (c) 70:30 mixtures of glycerin and water,
reported in Table 3.3 ........................................................................................................121
Figure 3.22. Relative errors between the maximum soft prior recovered properties and the
actual values in Figure 3.21 .............................................................................................121
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Figure 3.23. (a) Patient’s breast MR image (b) The customized soft prior mesh composed
of 1465 nodes and 2706 triangular elements containing only internal structure of
the breast – adipose (orange) and fibroglandular (blue) tissue, in (a).
Reconstructed dielectric property values were extracted along the dark blue line
across the breast and fibroglandular tissues. ....................................................................123
Figure 3.24. Simulated 1300 MHz reconstructed permittivity (top) and conductivity
(bottom) images with -100 dBm added noise, using the soft priors (left) and no
priors (right) for cases (a) – (d) in Table 3.4....................................................................125
Figure 3.25. Comparison of the 1300 MHz recovered permittivity (top) and conductivity
(bottom) values along an arbitrary line (illustrated in Figure 3.23(b)) in Figure
2.24: (a) – (d) correspond to different target values in case (a) – (d) in Table 3.4 ..........127
Figure 3.26. Comparison of the 1300 MHz soft prior reconstructed permittivity (top) and
conductivity (bottom) values with the standard (Stnd w) and new (w*0.1) scaling
factor along the same arbitrary line shown in Figure 3.23(b): (a) – (d) correspond
to different target values in case (a) – case (d) in Table 3.4. ...........................................130
Figure 3.27. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) images
from a phantom experiment with a single circular inclusion for (a) no priors on
the 473 node mesh and (b) soft priors on the 915 node mesh in Figure 3.8 ....................132
Figure 3.28. Comparison of the 1300 MHz reconstructed permittivity (left) and
conductivity (right) profiles extracted along the y-axis in Figure 3.26 phantom
experiment using the no priors and soft prior regularizations. ........................................133
Figure 3.29. Comparison of the reconstructed permittivity (left), and conductivity (right)
values from a phantom experiment using different spatial prior coefficient λ
xix
values at (a) 900, (b) 1100, (c) 1300, (d) 1500, (e) 1700, (f) 1900, (g) 2100, (h)
2300, and (i) 2500 MHz ...................................................................................................138
Figure 3.30. Weighted (a) εr and (b) σ errors for a phantom experiment over a range of
frequencies from 900 to 2500 MHz using six different soft prior coefficients: λ =
0.01, 0.1, 1, 10, 100, and 1000 .........................................................................................139
Figure 3.31. Customized soft prior reconstruction meshes used for the phantom
experiments with a circular inclusion of radius: (a) 0.65 cm – 1329 nodes and
2530 triangular elements, (b) 1.4 cm – 1037 nodes and 1943 triangular elements,
and (c) 2.1 cm – 813 nodes and 1498 triangular elements ...............................................141
Figure 3.32. Comparison of the reconstructed permittivity (left), and conductivity (right)
values from three phantom experiments with different sized inclusions (of radius
0.65, 1.4, and 2.1 cm) using the no prior (LM) and soft prior (SP) regularizations
at (a) 900, (b) 1100, (c) 1300, (d) 1500, and (e) 1700 MHz ............................................143
Figure 3.33. Customized soft prior reconstruction meshes used for the phantom
experiments with: (a) circular inclusion – 1329 nodes and 2530 triangular
elements, (b) square-shaped inclusion – 1096 nodes and 2096 triangular elements,
and (c) diamond-shaped inclusion – 1040 nodes and 1984 triangular elements .............144
Figure 3.34. Comparison of the reconstructed permittivity (left), and conductivity (right)
values from a phantom experiment with three different shaped but similar sized
inclusions (1.3 cm diameter circular, 1.3 cm side length square, and 1.3 cm side
length diamond-shaped cylinders) using the no prior (LM) and soft prior (SP)
regularizations at (a) 900, (b) 1100, (c) 1300, (d) 1500, and (e) 1700 MHz ....................147
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Figure 3.35. Comparison of the soft prior reconstructed permittivity (left) and
conductivity (right) values of five different contrast-level target-inclusions (50:50,
60:40, 70:30, 90:10 and 100:0 mixture of glycerin:water) at (a) 900, (b) 1100, (c)
1300, (d) 1500, and (d) 1700 MHz, along with linear least-square fits ...........................150
Figure 3.36. Comparison of the soft prior reconstructed permittivity (left), and
conductivity (right) values of a target inclusion using several soft prior
customized meshes with different node densities: case (a) number of nodes in the
inclusion region (NInc) changes, case (b) number of nodes in the background
region (NBK) changes, and case (c) number of nodes in both regions (NTotal)
changes proportionally .....................................................................................................154
Figure 3.37. 1300 MHz soft prior reconstructed permittivity (top) and conductivity
(bottom) images of a phantom experiment using the reconstruction mesh with a
false region of interest in Figure 3.20 ..............................................................................156
Figure 3.38. Comparison of the 1300 MHz soft prior reconstructed permittivity (left), and
conductivity (right) profiles from a phantom experiment using the reconstruction
mesh with a false region of interest in Figure 3.20 ..........................................................157
Figure 3.39. 1100 MHz reconstructed permittivity (top) and conductivity (bottom) images
of a phantom experiment using different prior inclusion-sized meshes: (a) 0.8 cm,
(b) 1.0 cm, (c) 1.2 cm, (d) 1.4 cm (the true size), (e) 1.6 cm, (f) 1.8 cm, and (g) 2.0
cm were considered as the radius of the inclusion in the reconstruction mesh. The
actual size (radius of 1.4 cm) and location (0, – 3 cm) of the inclusion is outlined
by a circle on the reconstructed images. ..........................................................................159
xxi
Figure 3.40. Comparison of the extracted 1100 MHz reconstructed permittivity (left), and
conductivity (right) values in Figure 3.39 along with the exact property
distributions for a phantom experiment using different inclusion–sized soft prior
meshes ..............................................................................................................................160
Figure 3.41. Comparison of the extracted (a) 1300, (b) 1500, and (c) 1700 MHz
reconstructed permittivity (left), and conductivity (right) profiles along with the
exact property distributions for a phantom experiment using different inclusion–
sized meshes.....................................................................................................................162
Figure 3.42. 1100 MHz reconstructed permittivity (top) and conductivity (bottom) images
of a phantom experiment using different prior inclusion–located meshes: x = 0
and y–coordinates of (a) – 3.9, (b) – 3.0 (the true location), (c) – 2.1, (d) – 1.2, (e)
– 0.3, (f) 0.6, and (g) 1.5 cm were considered as the center of the inclusion in the
reconstruction mesh. The actual size (radius of 1.4 cm) and location (0, – 3 cm) of
the inclusion is outlined by a circle on the reconstructed images. ...................................164
Figure 3.43. Comparison of the extracted (a) 1100, (b) 1300, (c) 1500, and (d) 1700 MHz
reconstructed permittivity (left), and conductivity (right) values along with the
exact property distributions for a phantom experiment using different inclusion–
located meshes .................................................................................................................166
Figure 3.44. Real breast–shaped model: (a) breast mold, (b) rapid–prototyped breast
model................................................................................................................................168
Figure 3.45. 3D optical scanning of the model ............................................................................169
Figure 3.46. Exported 3D model (a) 3D surface mesh, (b) Formation of a solid object and
extraction of 2D slices in the SolidWorks software package ..........................................170
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Figure 3.47. 2D customized soft prior meshes used in a breast–shaped phantom
experiment with circular target inclusions of (a) 0.65 cm and (b) 1.0 cm radius, as
well as a polyline along which the actual and reconstructed dielectric property
distributions were extracted. The mesh in (a) is comprised of 1778 nodes and
3468 triangular elements, whereas the one in (b) contains 1788 nodes and 3488
triangular elements. ..........................................................................................................171
Figure 3.48. Comparison of the 1300 MHz reconstructed permittivity (left) and
conductivity (right) profiles along an arbitrary polyline for the breast–shaped
phantom experiments with (a) small inclusion of 0.65 cm radius and (b) medium–
sized inclusion of 1 cm radius, using the no priors and soft prior regularization ............172
Figure 3.49. T2–weighted MR images of the phantom experiment at the center plane of
the active part of the antennas ..........................................................................................174
Figure 3.50. 2D reconstruction meshes used for the phantom experiment in MR: (a) No
priors, with 559 uniformly distributed nodes and 1044 triangular elements, and (b)
soft priors, with 1034 nodes and 1821 triangular elements .............................................175
Figure 3.51. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) images
of the multi–region phantom case using the (a) no priors (circles indicate the exact
object locations) and (b) soft prior regularization............................................................176
Figure 3.52. Experiment flow diagram for evaluating the correlation between bone
mineral and dielectric properties of porcine trabecular bone specimens .........................179
Figure 3.53. Microwave data acquisition of the bone sample .....................................................180
Figure 3.54. Soft prior microwave image reconstruction mesh comprised of 1107 nodes
and 2092 triangular elements ...........................................................................................181
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Figure 3.55. The 1100 MHz reconstructed permittivity (top) and conductivity (bottom)
images of the (a) first, (b) second, and (c) final (5th) microwave scans...........................181
Figure 3.56. Relationship between the dielectric properties of saline–saturated bones at
1100 MHz and BVF of dry bone specimens ....................................................................182
Figure 3.57. X–ray CT images of the patient’s heels for the (a) normal weight–bearing
and (b) injured leg (White zones within the calcaneous bones are the radiologists’
labels.) ..............................................................................................................................184
Figure 3.58. 2D customized reconstruction meshes of the (a) normal weight–bearing and
(b) affected leg comprised of 1196 nodes and 2264 triangular elements, and 815
nodes and 1519 triangular elements, respectively ...........................................................185
Figure 3.59. 1300 MHz soft prior permittivity (top) and conductivity (bottom) images for
the heel of a (a) normal and an (b) injured leg utilizing the CT images as prior
information during the reconstruction process ................................................................186
Figure 3.60. Photograph of a patient in the combined MI-MRI system (a) before entering
the MR bore and (b) inside the bore ................................................................................187
Figure 3.61 (a) MR T1-weighted image of the patient’s left breast at the plane of the
active part of the monopole antennas, and (b) the corresponding segmented soft
prior mesh ........................................................................................................................188
Figure 3.62. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) images
of the patient’s left breast utilizing the (a) no priors and (b) soft prior
regularizations, respectively ............................................................................................189
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Figure 3.63. 1300 MHz 3D reconstructed permittivity (top) and conductivity (bottom)
images of the simulation experiment with – 100 dBm added noise using (a) the
soft prior regularization and (b) no prior spatial information ..........................................192
Figure 3.64. Extracted 3D recovered permittivity (left) and conductivity (right) values
along with the exact property values, as a function of the number of nodes in the
soft prior reconstruction mesh .........................................................................................195
Figure 3.65. 1300 MHz 3D reconstructed permittivity (top) and conductivity (bottom)
profiles for the simulation experiment using different soft prior coefficients λ = (a)
0.01, (b) 0.1, (c) 1.0, (d) 10, and (e) 100..........................................................................198
Figure 3.66. Extracted 3D recovered permittivity (left) and conductivity (right) values
along with the exact property distributions, as a function of the base – 10
logarithm of the soft prior coefficient λ ...........................................................................198
Figure 3.67. Relative RMS errors of the recovered dielectric property distributions as a
function of the base – 10 logarithm of the soft prior regularization coefficient λ ...........199
Figure 3.68. Extracted soft prior 3D recovered permittivity (left) and conductivity (right)
values along with the exact property distributions, using different inclusion–sized
meshes as prior information .............................................................................................201
Figure 3.69. Extracted soft prior 3D recovered permittivity (left) and conductivity (right)
values along with the exact property distributions, using different inclusion–
located meshes as prior information ................................................................................204
Figure 3.70. 1300 MHz 3D reconstructed images using the corresponding soft prior
meshes with a false region of interest in Table 3.18 ........................................................208
xxv
Figure 3.71. 3D soft prior recovered (a) permittivity and (b) conductivity values extracted
at the center of each region, along with the exact properties, using customized
reconstruction meshes in Table 3.20 ................................................................................211
Figure 3.72. Soft prior 3D recovered (a) permittivity and (b) conductivity values
extracted at the center of each region, along with the exact properties, using 4region customized reconstruction meshes in Table 3.20 .................................................214
Figure 3.73. Comparison between the soft prior reconstructed (a) permittivity and (b)
conductivity values of I1 with and without additional targets (i.e. multi-inclusion
and single inclusion cases, respectively) .........................................................................216
Figure 3.74. Recovered permittivity (top) and conductivity values at the center of the (a)
cylindrical and (b) spherical inclusions as a function of the inclusion size, using
different reconstruction approaches (2D and 3D, with no priors, soft priors, and
hard priors) .......................................................................................................................219
Figure 3.75. Complex-shaped phantom experiment with a cylindrical and a cup-shaped
gelatin inclusion: (a) cylindrical inclusion (CylInc) inserted into the cup-shaped
gelatin inclusion (GelInc), (b) both inclusions submerged in the imaging tank, and
(c) top view of the phantom setup in the microwave imaging tank .................................221
Figure 3.76. MRI Imaging of the phantom experiment with a cylindrical and a cupshaped gelatin inclusions: (a – b) pictures of the phantom placed in an empty
microwave imaging tank inside the MR bore, and (c) top view of the schematics
of the phantom in the MRI scanner..................................................................................223
xxvi
Figure 3.77. (a) One of the corresponding MR images of the inclusions, (b) the
corresponding 3D soft prior mesh composed of 3393 nodes and 17027 tetrahedral
elements ...........................................................................................................................224
Figure 3.78. A vertical slice through (a) soft prior and (b) no prior reconstructed
permittivity (top) and conductivity (bottom) images of the phantom experiment
with a cylindrical and a cup-shaped gelatin inclusions at 1300 MHz ..............................225
Figure 3.79. Breast–shaped phantom experiment with two arbitrarily shaped target
inclusions: (a) Two arbitrary-shaped gelatin target inclusions suspended in the
plastic breast model, (b) rapid–prototyped plastic breast model submerged in the
imaging tank, and (c) schematic configuration of the breast phantom in the
microwave imaging tank. .................................................................................................227
Figure 3.80. MRI Imaging of the breast phantom experiment with two inclusions: (a – b)
pictures of the phantom placed in an empty microwave imaging tank inside the
MR bore, and (c) top view of the phantom configuration in the MRI scanner................229
Figure 3.81. (a) Stack of binary-segmented MRI images of the breast phantom
experiment with two target inclusions, (b) the corresponding 3D soft prior mesh
composed of 7540 nodes and 34591 tetrahedral elements...............................................230
Figure 3.82. A vertical slice through the (a) soft prior and (b) no prior reconstructed
permittivity (top) and conductivity (bottom) images of the breast phantom
experiment with two target inclusions at 1100 MHz .......................................................231
Figure 4.1. Input file selection: (a) file format and (b) raw measurement file selection
windows ...........................................................................................................................235
xxvii
Figure 4.2. Sample antenna configuration plot where 6 planes of data with a slice
thickness of 1 cm have been acquired ..............................................................................236
Figure 4.3. Output file format selection window .........................................................................237
Figure 4.4. 3D data selection window .........................................................................................238
Figure 4.5. Example of a multi-plane data selection with 2 consecutive planes .........................239
xxviii
List of Tables
Table 2.1. Finite difference grid spacing (in mm) used to compute the forward solution
for the simulation experiment, along with the corresponding reconstruction times
per iteration ........................................................................................................................32
Table 2.2. Parameter reconstruction meshes used to solve the inverse problem for the
simulation experiment........................................................................................................36
Table 2.3. Independently measured dielectric properties of the phantom experiment with
two cylindrical inclusions ..................................................................................................50
Table 2.4. Independently measured dielectric properties of the background coupling
medium (BK), breast model (BR), and target inclusion (Inc) at 1300 MHz.......................55
Table 3.1. Relative (a) permittivity and (b) conductivity errors for the multiple inclusions
simulation experiment using different reconstruction meshes with two and four
distinct regions .................................................................................................................113
Table 3.2. Relative (a) permittivity and (b) conductivity errors for the multiple inclusion
simulation experiment with different property contrasts using the soft prior
reconstruction meshes with four distinct regions in Figures 3.15(e) and (f) ...................116
Table 3.3. Dielectric properties of different mixtures of glycerin and water used for the
simulation experiments with a false region of interest ....................................................118
Table 3.4. Dielectric property values of the background medium, breast, and
fibroglandular region, in four breast simulation experiments, cases (a) – (d), at
1300 MHz .........................................................................................................................124
Table 3.5. Soft prior relative dielectric property errors in the fibroglandular region,
breast-shaped simulation experiments with two different scaling factors .......................130
xxix
Table 3.6. Independently measured dielectric properties of the background medium (BK)
and target inclusion (Inc) over the range of frequencies evaluated .................................134
Table 3.7. Properties of the soft prior customized meshes used for the phantom
experiment with a single target inclusion of 1.4 cm radius..............................................152
Table 3.8. Weighted soft prior εr and σ errors for a phantom experiment using a
reconstruction mesh with a false region of interest for different soft prior
coefficients: λ = 0.01, 0.1, and 1.0 at 1300 MHz .............................................................158
Table 3.9. Characteristics of the soft prior reconstruction meshes used for a phantom
experiment with imperfect prior size of the target inclusion ...........................................159
Table 3.10. Characteristics of the soft prior customized meshes used for a phantom
experiment with imperfect prior location of the target inclusion ....................................163
Table 3.11. Independently measured dielectric properties of the background medium
(BK), breast (BR), and inclusions (Inc) at 1300 MHz ......................................................169
Table 3.12. (a) The actual and averaged recovered property values of the fibroglandular
and tumor inclusion regions in the phantom experiment in the MR bore using the
no priors, soft priors, and hard priors, (b) The corresponding relative dielectric
property errors ..................................................................................................................177
Table 3.13. Pearson coefficient for the recovered dielectric properties at various
frequencies .......................................................................................................................183
Table 3.14. The permittivity and conductivity RMS errors associated with the inclusion
region in Figure 3.63 ........................................................................................................193
Table 3.15. Customized reconstruction meshes used for the simulation experiment with
soft prior regularization....................................................................................................194
xxx
Table 3.16. Characteristics of the 3D soft prior customized meshes used for a simulation
experiment with imperfect prior size of the target inclusion ...........................................200
Table 3.17. Characteristics of the 3D soft prior customized meshes used for a simulation
experiment with imperfect prior location of the target inclusion ....................................203
Table 3.18. Characteristics of the 3D soft prior customized meshes used for a simulation
experiment with a false target inclusion (Bk = background, TI = True Inclusion,
and FI = False Inclusion) .................................................................................................206
Table 3.19. Recovered dielectric properties (εr and σ) of the true (TI) and false inclusion
(FI) extracted at the center of each region .......................................................................209
Table 3.20. Soft prior reconstruction meshes used for the simulation experiment with
three spherical target inclusions .......................................................................................210
Table 3.21. Relative (a) permittivity and (b) conductivity errors for the multi-inclusion
simulation experiment using different reconstruction meshes with two and four
distinct regions. The negative sign indicates that the reconstructed property value
is underestimated. (i.e. less than the exact property values) ...........................................212
Table 3.22. Relative (a) permittivity and (b) conductivity errors for the multi-inclusion
simulation experiment with different property contrasts using two soft prior
reconstruction meshes with 4 distinct regions .................................................................215
Table 3.23. Characteristics of the customized 2D and 3D meshes used for the simulation
experiments with different sized and shaped target inclusions ........................................217
Table 3.24. Independently measured dielectric properties of the background medium
(Bk), cylindrical inclusion (CylInc), and cup-shaped gelatin inclusion (GelInc) at
1300 MHz .........................................................................................................................222
xxxi
Table 3.25. Soft-prior and no prior RRMS errors of the recovered properties in each
region of the breast phantom experiment with two target inclusions ..............................226
Table 3.26. Independently measured dielectric properties of the background medium
(Bk), breast model (Br), and the target inclusions (Inc1 and Inc2) at 1100 MHz ............228
Table 3.27. Soft prior and no prior RRMS errors of the recovered properties in each
region of the breast phantom experiment with two target inclusions ..............................232
Table 3.28. The computed contrast enhancements of the target inclusions with respect to
the breast region, when the prior structural information of the phantom is used
during the reconstruction .................................................................................................233
Table 4.1. A list of inputs/3D reconstruction parameters along with a brief description of
them..................................................................................................................................240
xxxii
List of Acronyms
2D
Two-dimensional
3D
Three-dimensional
ACS
American Cancer Society
Adp
Adipose
Bk
Background
Br
Breast
BVF
Bone Volume Fraction
CS
Consecutive planes
CT
Computed Tomography
DHMC
Dartmouth Hitchcock Medical Center
EIS
Electrical Impedance Spectroscopy
Err
Error
FD
full-data
FDTD
Finite Difference Time Domain
Fg
Fibroglandular
GUI
Graphical User Interface
Inc
Inclusion
IP
In-plane
LM
Levenberg Marquardt
MI
Microwave Imaging
MIS
Microwave Imaging Spectroscopy
MRE
Magnetic Resonance Elastography
xxxiii
MRI
Magnetic Resonance Imaging
NIR
Near Infrared
NP
Number of Planes
RF
Radiofrequency Field
RX
Receiver
SA
Spectrum Analyzer
SP
Soft Prior
TSAR
Tissue Sensing Adaptive Radar
TK
Tikhonov
TM
Transverse Magnetic
Tu
Tumor
TX
Transmitter
UWB
Ultrawide Band
VHF
Very High Frequency
xxxiv
1. Introduction
1.1. Motivation
Microwave imaging (MI) is based on recovering dielectric properties (permittivity and
conductivity) of materials. Over the last two decades MI has attracted increasing interest in
biomedical applications. In particular, interest in breast cancer screening and therapy monitoring.
Excluding skin cancer, breast cancer is the most commonly diagnosed cancer in the United
States, and the second leading cause of cancer death in women, exceeded only by lung cancer
[1]. In fact, it was estimated that over 230,000 new cases of invasive breast cancer would be
diagnosed among women in the U.S. in 2011 [1]. The American Cancer Society (ACS) estimates
that a woman in the United States has a 1 in 8 chance of developing invasive breast cancer
during her lifetime [1]. Despite the overall decline in breast cancer mortality over the past
decade, about 40,420 deaths were expected in 2011 alone [1]. It has been shown that early
detection of breast cancer is among the most effective ways to improve patient’s long-term
survival [2].
The most common method for detection in clinical practice is X-ray mammography, which
is generally effective for the broad population of women over 50 years of age. However,
screening mammography has substantial limitations, primarily a high false-positive rate (up to
29%), which can result in unnecessary and costly surgical interventions [3]. Breast cancer
detection is a particularly challenging problem in younger women and those with
radiographically dense breasts. In these cases, the increased levels of fibroglandular tissue can
easily obscure small tumors or masquerade as an abnormality because of the tissue overlap on
plane film, and as a result, the overall diagnostic performance of mammography can be
1
significantly degraded [4]. Mammography has other drawbacks from the patient’s perspective,
including uncomfortable and painful breast compression and exposure to ionizing radiation [5].
Other clinical standards, such as ultrasound and magnetic resonance imaging (MRI), have also
been used to detect breast cancer. While both can achieve high spatial resolution, neither can
provide information about the molecular-level changes occurring in breast tissue at the present
time [6, 7].
In response to these shortcomings, alternative and/or complementary medical imaging
modalities are being developed to improve both the sensitivity and specificity of current imaging
tools, and to supply more functional information about the tissue properties comprising the
breast. These properties can include electrical or optical characteristics, temperature, or tissue
elasticity [5]. At Dartmouth, four different modalities, namely, microwave imaging spectroscopy
(MIS), electrical impedance spectroscopy (EIS), near infrared imaging (NIR), and magnetic
resonance elastrography (MRE), are being developed for breast cancer detection [8]. These
alternative imaging modalities may prove to be reliable detection tools because they provide
significant and consistent contrast between normal and malignant breast tissue [9-12].
Early microwave studies for biomedical applications showed a significant dielectric
property contrast between normal and malignant breast tissues [13-15]; however, more recent
data reported by Lazbnik et al. [16] has indicated that the properties of the normal breast are
more variable than originally thought and that the contrast may not be as great for some types of
breast tissue. This is particularly true for radiographically denser breast with higher
concentrations of fibroglandular tissue [16]. Notwithstanding, early clinical microwave imaging
studies on patients with suspected tumors has demonstrated significant discrimination between
those with malignant cancers versus those with benign lesions and other normal tissues [17, 18].
2
Compared to other conventional imaging modalities, such as X–ray mammography, the
non–ionizing and non–compressive nature of MI makes it very attractive from the patient’s
perspective [19]. In addition, MI has a relatively high sensitivity for detecting small tumors, and
potentially high specificity to determine whether a suspicious area is malignant or benign at a
significantly lower cost level compared to methods such as MRI and nuclear medicine [19].
3
1.2. Overview of Different Microwave Imaging Methods
At radio and microwave frequencies, several methods have been investigated for MI,
particularly for breast cancer detection. These methods include: passive, hybrid, and active MI.
These techniques are summarized in a tree diagram illustrated in Figure 1.1.
Figure 1.1. Tree diagram of the current MI methods for breast cancer detection
Passive MI is based on using microwave radiometry to measure temperature differences in
the tissue. The principle of operation of this method is that there is a measurably greater
temperature in tumor region compared with healthy breast tissue, which can be potentially
detected using radiometers [20-25]. In hybrid methods, also called microwave–acoustic imaging,
the principle of operation is to use microwave energy to illuminate the tissue and measure the
pressure waves generated by the higher conductivity tumor regions using an ultrasound
transducer [26-28]. In this case, the dielectric contrast selectively chooses the tumor for enhance
absorption and vibration, which is subsequently detected by the ultrasound. Finally, the basic
idea in the active MI is to illuminate the tissue with microwaves and measure the scattered or
reflected signals around the tissue [29-33].
4
Due to the dielectric property contrasts between normal and malignant tissue, in the
presence of a tumor, microwave signals traveling through the tissue encounter a change in
electrical properties, which causes the incident wave to scatter and create some variation in phase
and the amount of detected energy at the receivers [19]. This information can be ultimately used
to obtain the associated dielectric property maps of the tissue at microwave frequencies.
Active MI approach may be classified as tomographic and confocal. Tomographic MI is a
transmission–reflection imaging technique where the complete spatial dielectric property
distributions of the object are obtained from the transmitted (incident) and scattered (received)
electromagnetic fields [34]. Confocal MI, on the other hand, is a time-domain imaging technique
that exploits the principles of radar synthetic focusing, and instead of recovering the complete
dielectric property profiles of the object, only locations of significant scatterers are reconstructed
[35].
The main idea behind the confocal MI is to create a map of microwave scattering arising
from the contrast in dielectric properties within the imaging domain. This approach is very
closely related to optical confocal systems; however, electromagnetic signals at microwave
frequencies benefit from a greater penetrability in tissue [36]. After transmitting an ultra-short
pulse into the tissue, the backscatter waveforms are collected, filtered, and time–shifted to create
synthetic focal points. Returning waveforms from the scattering tissue add coherently at the focal
points, while those surrounding the confocal points add incoherently [30, 37, 38].
Confocal microwave imaging was first presented by Hagness et al. in 1998 [37].
Traditionally, radar imaging is only a reflection technique and utilizes monostatic radar, where
the signal is transmitted and received by the same antenna. In addition, due to the structural
5
complexity of the living tissue, more measurements are desired and as a result, sensor arrays
have been introduced to increase the amount of measurement data [39, 40].
Radar-based imaging systems include microwave imaging via space time beamforming
[41, 42], and tissue sensing adaptive radar (TSAR) [40, 43, 44]. In the former, the patient lies
supine with the antennas scanned over the naturally flattened breast, and some advanced clutter
reduction algorithms are used to calculate the location of the scatterer(s) within the imaging
domain. On the other hand, in the TSAR systems the patient lies prone and the antennas are
scanned around the breast, which is suspended through a hole in the examination table. In this
case, simpler clutter reduction algorithms are used to create the corresponding TSAR images [40,
43, 44].
Transmission-based confocal imaging is a relatively new radar imaging technique where
the transmitted signals are collected and used to exploit the varying attenuation induced by the
electrical property differences within the tissue. Craddock et al at the University of Bristol has
successfully developed a clinical radar–based UWB microwave system for breast cancer
detection [45]. Their system uses a real aperture array of UWB antennas positioned on a section
of a hemi-sphere (conforming to the breast’s curvature) and operates in a multi-static mode [45].
Microwave tomography commonly refers to imaging by cross-sectioning or slicing through
an inhomogeneous object/tissue. The word tomography is derived from the Greek tomē ("a
cutting") or tomos ("a cut" or "section") and graphein ("to write") [46]. This method aims at the
complete reconstruction of the dielectric property distributions (permittivity and conductivity) of
the tissue being imaged. It should be noted that despite recovering the complete 3D map of the
actual dielectric properties of the tissue at once (instead of 2D slices), the three-dimensional MI
is frequently referred to as 3D tomography.
6
Several research groups have been investigating the 2D and 3D tomographic MI in
biomedical applications. Researchers at Duke have developed a fixed 3D array of antennas
positioned throughout the surface of the imaging chamber and connected to the source and
measurement devices through an RF switching network [47]. They use a fast volume integral
equation method (diagonal tensor approximation, DTA) to accelerate the computation of the
forward solution [48-50]. The 3D imaging systems and methods at Duke have been tested on
several phantoms from experimental data, and their results indicate that the detection and
imaging of inclusions with dielectric contrast is possible [51].
Similarly, other researchers have been studying microwave tomography as a diagnostic
modality for non-invasive assessment of functional and pathological conditions of biological
tissues. They have developed both 2D and 3D imaging systems [52-54], in addition to the image
reconstruction methods [55-57] for experimental imaging of animals [53, 54, 58] and functional
imaging of extremity soft tissues [59]. However, the prototypical imaging system developed at
Dartmouth College [9] is the only MI system thus far which has been successfully progressed to
clinical trial stage. Details of this system are provided in the following section.
7
1.3. Tomographic Microwave Imaging
Several research groups have been investigating tomographic MI for breast cancer
detection [60, 61], but to the best of our knowledge, we at Dartmouth College, are the only group
to this date, who has developed an actual clinical system [9, 29]. Figure 1.2 shows our latest
clinical microwave imaging system (MIST), located at Dartmouth Hitchcock Medical Center
(DHMC). The imaging array consists of 16 monopole antennas positioned on a 15.2 cm diameter
circle. They are placed on one of the two independently–moving plates, A or B, as shown in
Figure 1.2(a).
During data acquisition, the antenna array scans through a number of vertical positions in
0.5, 0.75, or 1.0 cm increments. This configuration enables us to collect both in-plane (i.e. when
all antennas are at the same height) and cross-plane (i.e. when the two sets of antennas are at
different heights) data. However, due to a time limitation and the lack of a viable, fast, and user–
friendly three–dimensional (3D) image reconstruction procedure, only in-plane patient data is
currently collected and the corresponding images are only reconstructed in two dimensions (2D).
Sequentially, at each antenna position, that antenna acts as a transmitter while the response is
measured at each of the remaining 15 antennas. The system is designed to operate over the
frequency range from 500 MHz to 3 GHz.
The patient lies prone on the bed with the breast to be examined suspended through an
aperture in the top of the tank, as shown in Figure 1.2(b). The tank is filled with a coupling
liquid, closely mimicking the average constitutive parameters of the breast [62], maximizing the
amount of microwave energy coupled into the tissue. The imaging platform (c) and the space
underneath (d) are shown in Figure 1.2. This space is used to house the data acquisition
instrumentation, illumination tank modules, and the coupling medium reservoir.
8
(a)
(b)
(c)
(d)
Figure 1.2. MIST system: (a) antenna configuration on two independently
moving plates A
independently-moving
(pink) and B (blue), (b) imaging tank, ((c) exam platform, and (d)) cabling and fluid reservoir
underneath the table
9
1.4. Project Aims
Question 1: Can a viable, fast, and user–friendly image reconstruction procedure be
developed for 3D microwave imaging?
To recover representative electrical property profiles of tissue from measurement data,
matching the numerical model to the physical situation is one of the most important factors to
ensure the correct interpretation of the data (by the model) and algorithmic convergence.
Modeling of 3D electromagnetic wave propagation in a complex medium is generally
computationally expensive, even with modern computational resources. Using 2D methods to
model this inherently 3D phenomenon may save significant computational time; however, it
likely imposes simplifications, which may introduce image artifacts. Our goal is to develop a
practical 3D image reconstruction procedure, which is computationally feasible and balances the
tradeoff between the accuracy and efficiency of the model. Other issues such as compatibility
with previously developed reconstruction techniques should also be taken into account. In order
to evaluate the performance of the reconstruction algorithm, a number of simple and complex–
shaped 3D simulation and phantom experiments will be performed.
Question 2: Will a priori structural information from other imaging modalities, such as MRI,
improve the quality of microwave images and enrich the accuracy of the reconstructed
property distributions?
This question will be answered through a series of 2D and 3D simulations and phantom
experiments. Moreover, different analyses including: frequency, noise level, shape, size and
number of regions of interest, imperfect or false priors, and contrast levels will be performed to
evaluate how prior spatial information of the object being imaged can enhance the quality and
accuracy of the recovered property distributions. We will also extend the technique of using prior
10
structural information to 2D patient data. Our ultimate goal is to collect simultaneously 3D
microwave data and the corresponding anatomical information from MRI. We believe this
additional information will improve the accuracy of the reconstructed property distributions
significantly, enabling us to characterize smaller regions of interest more precisely.
11
1.5. Proposed Work
Currently our imaging system operates primarily in the 2D mode, where multiple vertical
planes of data (in-plane configuration) are collected, and the corresponding 2D slice images are
reconstructed. In this mode, the image reconstruction is based on the assumption that the
scattering problem can be reasonably represented as a 2D problem [63]. Using 2D methods to
model a 3D phenomenon significantly reduces the computation time; however, it may impose
excessive simplifications, which can introduce image artifacts. This has been certainly used as a
rationale for why other 2D imaging efforts by other researchers failed [64]. In the past, we have
performed studies showing that the match between the 2D model and 3D actual measurements is
quite good and is not a deterrent for reconstructing 2D images, especially for the phase [63, 65].
Certainly, this has been part a key reason we have been able to recover useful 2D images to this
point.
Previously, a 3D microwave image reconstruction procedure based on a finite–difference
time–domain (FDTD) algorithm was developed at Dartmouth College [66]. The original
implementation of this reconstruction algorithm was written in FORTRAN and a few simple–
geometry simulations and phantom experiments were performed to evaluate the efficiency and
accuracy of the results [67]. One of the aims of this work is to modify and redesign the existing
algorithm and develop a not only viable and fast, but also a user–friendly 3D image
reconstruction module which is computationally feasible and balances the tradeoff between the
accuracy and efficiency of the model. In chapter 2 a full discussion about the 3D microwave
image reconstruction, including the proposed redesigning and optimization methods, along with
the results of several simulations, phantom experiments, and preliminary patient data are
presented.
12
At the core of active MI is the inverse reconstruction algorithm. Since the inverse
electromagnetic problem is inherently non–linear, iterative schemes are usually used in the
image reconstruction process [68, 69]. Moreover, due to the ill–posed nature of the problem,
regularization procedures are required to impose additional constraints on the reconstructed
images [69, 70]. This is often accomplished by introducing prior information about the object
being imaged, which is helpful to ensure convergence of the reconstruction algorithm to the
correct electromagnetic property distribution [71].
We have previously demonstrated that we could recover clinically useful microwave
tomographic images for breast cancer detection without using any prior spatial information about
the tissue [9]. However, in order to enhance the quality of our previous results and recover more
accurate dielectric property distributions, we propose a new approach combining the functional
information of the MI (property contrast) with the high spatial resolution of other imaging
modalities such as MRI and X–ray CT. This approach is based on developing new image
reconstruction/regularization strategies that exploit structural information about the object being
imaged through some constraints in the microwave property contrast [72]. In chapter 3 the
mathematical framework of our microwave imaging reconstruction algorithm along with the new
regularization schemes are presented. Then, our prototyped MI-MRI multi-modality imaging
system is described in detail, and finally, the new regularization approaches are evaluated
through several 2D and 3D simulations, phantom experiments, and initial clinical data.
The final outcome of this project is two graphical user interfaces (GUI) for the 2D and 3D
FDTD microwave image reconstructions. These GUIs are designed to enable the user to control
and adjust some of the important parameters in the reconstruction procedure, and to visualize the
13
final reconstructed images more conveniently. A detailed description of the developed GUIs is
presented in chapter 4.
14
2. 3D Microwave Image Reconstruction
2.1. 2D vs. 3D Modeling of the Electromagnetic Field
In our image reconstruction algorithm, the numerical modeling of the propagation of
electromagnetic (EM) waves is governed by Maxwell's equations. Solutions to Maxwell’s
equations in 2D cylindrical waveguides can be approximated as linear combinations of the
Bessel functions, known as Hankel functions of the first and second kind of nth order with an
argument hr, Hn(hr), where h2 = γ2 + k2, γ is the propagation constant, k is the wave number, and
r is the distance from the source to the observation point in the cylindrical coordinate system
[73]. For the far field solution:

π nπ 
2 ± j hr− 4 − 2 
lim ( H n (hr)) =
e
r→∞
π hr
(2.1)
and as a result, the field E is proportional to [73]:
()
E r ∝
1 jhr
e
πr
(2.2)
On the other hand, in antenna and radiation problems in 3D space, solutions to Maxwell’s
equations can be approximated using dyadic Green’s function G [74], which in turn can be
expressed in terms of the scalar Green’s function g as:


1
G r, r′ =  I + 2 ∇∇ g r, r′
 k

( )
( )
(2.3)
where r and r′ are the Cartesian coordinates of the observation point and the source,
respectively. Using the radiation field approximation, the electric field E is proportional to [74]:
15
()
E r ∝
1 jhr
e
πr
(2.4)
Based on the Poynting’s theorem, the magnitude of the electromagnetic radiation is
2
2
proportional to E [73]. From (2.2) and (2.4), E ∝
2
E ∝
1
in the 2D case, whereas for the 3D case,
r
1
. In both cases, the phase varies as a function of r, the distance from the source to the
r2
observation point, i.e. receiver location.
Based on simulation and phantom experimentation, we have noticed a strong correlation
between the magnitude of the electric field and the recovered conductivity values. We have
observed a similar relationship between the phase and the recovered permittivity values. As
discussed above, the phase varies as a function of the distance r in both 2D and 3D models,
whereas the magnitude varies differently in the two models (1/r in 2D vs. 1/r2 in 3D). Therefore,
we hypothesize that there exists a greater mismatch between the magnitudes of electric field
modeled in 2D and 3D, as compared to the corresponding phase values. In addition, since the
electromagnetic wave propagation is a 3D phenomenon, this may explain why we have been able
to recover more accurate permittivity values using our 2D model. On the other hand, due to the
greater mismatch between the actual and 2D modeled magnitudes of the electric field, overall the
2D reconstructed conductivity images contain more artifacts.
In order to validate our hypothesis, we performed a simulation experiment as shown in
Figure 2.1. A cylindrical inclusion of 1.0 cm radius with dielectric properties of εr = 40 and σ = 2
S/m was centered at (3, 0 cm). The permittivity and conductivity values of the surrounding
background medium was set to εr = 24.0 and σ = 1.13 S/m, respectively. For the 2D simulated
data, a cross section of the setup in Figure 2.1 was used, whereas for the 3D case, the entire
16
cylinder of 3.0 cm height was considered. In both cases, only one plane of data (16 transmitters ×
15 receivers) was synthesized at 1100 MHz. In addition, the data was calibrated with respect to
the homogeneous background.
Figure 2.1. Simulation experiment setup with a cylindrical inclusion or 1.0 cm radius
Figure 2.2 shows the calibrated amplitude/power (top) and phase (bottom) values of the
synthesized data using the 2D and 3D models as a function of the relative receivers for a single
transmitter (tx = 1). In both power and phase plots, the mismatch between the two models
appears in the regions where the projection of the actual object (cylinder/circle) is recorded (i.e.
between the receiver # 5 and 13).
17
Figure 2.2. Calibrated amplitude/power (top) and phase (bottom) values of the synthesized data
using the 2D and 3D models as a function of the relative receivers for a single transmitter (tx = 1)
In order to quantify the mismatch for all receivers and compare the 2D and 3D models, the
RMS difference values of the amplitude and phase were calculated at each transmitter tx as:
15
D =
V
tx
(V
2D
rx
− Vrx3D )
∑ max(V ) − min(V )
15
(2.5)
rx=1
where V is the amplitude or phase, rx is the receiver number, and the superscript 2D and 3D
correspond to the amplitude or phase values synthesized in the 2D and 3D model, respectively.
The calculated RMS difference values are plotted in Figure 2.3.
18
Figure 2.3. RMS difference values of the amplitude and phase
The results in Figure 2.3 confirm our hypothesis that there is a greater mismatch between
the magnitudes of electric field modeled in 2D and 3D, as compared to the corresponding phase
values. In fact, the amplitude RMS differences are between 2-3 times larger than the counterpart
phase values.
2.2. Challenges With Respect to 2D Reconstruction Algorithm
During microwave data acquisition, electromagnetic fields propagate through and scatter
from the tissue in a three–dimensional (3D) fashion [75]. However, in order to reduce the
computational complexity and to speed up the image reconstruction process, it is often assumed
that the behavior of electromagnetic waves in 3D space can be represented by a simplified 2D
model. While it benefits from less intensive computational demands, such assumptions,
including the field confinement to only the transverse magnetic (TM) mode, can lead to an
increased level of artifacts in the recovered dielectric properties. Moreover, since only in-plane
data is collected in 2D imaging, if the region of interest is small enough to fall between two
19
consecutive imaging slices, the 2D reconstruction algorithm may not detect the target accurately.
Therefore, in order to improve the accuracy and quality of reconstructed images, a viable 3D
microwave image reconstruction scheme is desired. Several years ago, a 3D microwave
reconstruction algorithm was developed by Fang et al [66]. The algorithm was based on the
FDTD modeling of electromagnetic wave propagation and it was implemented in FORTRAN. In
order to demonstrate its feasibility, the algorithm was tested on several simulated data and
phantom experiments with simple–geometry configurations [66, 67]. Despite the numerical
efficiency of the reconstruction scheme, the lack of transparent and flexible input–data selection
process and the lack of a user–friendly interface prevented further development of a practicable
3D image reconstruction procedure.
20
2.3. Proposed method
2.3.1. User Interface
MATLAB (The MathWorks, Inc., Natick, MA) is a high–performance language for
technical computing, which integrates computation, visualization, and programming in a user–
friendly environment. The MATLAB graphical user interface (GUI) is also a powerful tool
exploiting advanced computational and graphical functions of MATLAB. However, for
computationally complex problems such as our 3D microwave image reconstruction, MATLAB
is inherently less efficient than other prominent programming languages, such as FORTRAN, C,
or C++. Nonetheless, MATLAB enables us to evaluate our codes using “performance profiling”
and take advantage of MEX–files to improve speed of computationally intensive portions of our
program. MEX–files are MATLAB EXecutables, which provide an interface between MATLAB
and subroutines written in FORTRAN, C, or C++. When compiled, MEX–files can be run from
within MATLAB in the same way as any other built–in functions. To support the development of
MEX–files, MATLAB offers external interface functions to facilitate data transfer between
MEX–files and MATLAB, and the ability to call MATLAB functions from FORTRAN, C, or
C++ code [76]. Finally, MATLAB offers the Parallel Computing Toolbox to parallelize high–
level constructs, such as for–loops, using multi–core processors or computer clusters. Parallel
computing is very appealing in 3D microwave image reconstruction, especially to solve the
forward model, where the electromagnetic field has to be calculated for each source (in our case,
for each transmitter), as well as calculating the Jacobian matrix [77]. The forward problem is
certainly the most time consuming portion of the image reconstruction algorithm, but since it can
be solved for each source independently, the field calculation corresponding to each antenna can
21
be parallelized. In an optimal situation, the forward model for all sources can be calculated
concurrently by using as many core–processors/cluster–nodes as the number of sources.
2.3.2. 3D Data Acquisition
Based on the current design of our imaging system as described in Section 1.3, the 16
monopole antennas are grouped into two interleaved arrays of 8 antennas, and they are placed on
two independently–moving plates, which in turn are attached to separate pairs of opposing
computer controlled motors positioned underneath the tank. Since all antennas on the same
mounting plate move together and have the same height, the collected measurement data
contains some redundancies. For example, as is illustrated in Figure 2.4, when antennas 2 and 4
on plate A act as transmitter–receiver pairs, redundant data is collected at all possible imaging
planes of plate B. In theory, all redundant measured amplitudes and phases are supposed to be
equal, simply because the same antennas at the same locations transmit and receive the signal.
However, there are minor differences between these measurements in practice, mainly due to the
contribution of multipath signals related to other neighboring antennas.
In order to handle redundant data, several approaches can be considered. One of them is
that each redundant transmitter–receiver pair can be considered a new pair by assigning a
different Source ID. The main advantage of this approach is that no measured data is discarded
and the contribution of external sources is also taken into account. However, this assumption
could make the number of forward solutions larger, and as a result, the computational time could
increase significantly. In addition, due to the fact that the difference between redundant data is
minimal (usually less than 0.1 dBm in amplitude and 1 degree in phase,) this approach does not
appear to be beneficial.
22
Figure 2.4. Schematic antenna configuration: Two interleaved arrays of 8 antennas with each
sub-array (A and B) being able to move independently from the other
In order to account for all measured data and also minimize the number of unnecessary
forward solutions, we use a more efficient approach. The average of redundant amplitudes and
phases are calculated and used for each repeated transmitter–receiver pair.
2.3.3. Input–File Format
In order to develop an efficient, flexible, and transparent image reconstruction procedure,
we designed a new input file format. Figure 2.5 shows a schematic summary of steps to create
3D image reconstruction input files from the MIST system output data.
23
Figure 2.5. Schematic summary of steps to create 3D image reconstruction input files from the
MIST system output data
The first step is data subtraction and phase unwrapping, where calibrated data is formed by
subtracting calibration data from raw measurement data, and then, the phase is unwrapped. In
the second step, MIST system output files are converted into a single and flexible format: Files
1, 2, and 3. Based on the measurement configuration, a unique Source ID is assigned to each
possible antenna position. These IDs along with their corresponding antenna positions are
recorded in File–1. (Figure 2.6)
File–2 records the order in which measurements were acquired at each frequency. As
illustrated in Figure 2.7, transmitter (TX) and receiver (RX) sequences corresponding to
measurements are listed in a table, in which a unique Measurement ID is assigned to each row. It
should be noted that there are redundancies in (TX, RX) pairs (circled in Figure 2.7), which need
to be identified and handled as described in the previous section.
24
Figure 2.6. Input File–1
Figure 2.7. Input File–2
File–3 consists of a table containing calibrated and phase unwrapped data for each
frequency. As shown in Figure 2.8, each row of this table includes a measurement ID, TX and RX
IDs, along with their corresponding data in the last two columns. Similar to File–2, File–3
contains redundant measurements (circled pairs in Figure 2.8). In order to compare these
redundancies, the second and third columns in File–3 are sorted in ascending order, thus a list of
all data corresponding to each (TX, RX) pair appears in a block.
25
Figure 2.8. Input File–3
Files 1, 2, and 3 are a basis for all possible subsequent reconstructions. The next step is to
generate input files that are free of redundancies and are ready to be used in the image
reconstruction algorithm. These input files include File–1’, 3’ and 4. As illustrated in Figure 2.5
and Figure 2.9, File–3’ consists of two 2–dimensional matrices (one for amplitude and one for
phase), which are formed based on File–2 and File–3. The size of each matrix is n_Sources ×
n_Sources, where n_Sources is the number of selected sources in File–1’. Entry ampi,j or phasei,j
of each matrix indicates a unique amplitude or phase transmitted by Source ID Si and received at
Source ID Sj. At this stage, data redundancy is eliminated, as described in Section 2.3.2.
Moreover, if no data has been collected for a given (TX, RX) pair – an impossible pair, the
corresponding entry of matrices in File–3’ are empty. For example, since each antenna cannot
act as both transmitter and receiver simultaneously, the diagonals of the matrices are always
empty.
26
Figure 2.9. Input File–3’
File–4 consists of a n_Sources × n_Sources matrix of flags (0’s or 1’s), which are linked to
matrices in File–3’ (Figure 2.9). RX’s associated with 1 are used for the inversion in the
reconstruction algorithm, whereas those associated with 0 are not. The flagged matrix entries
corresponding to empty cells in File–3’ (diagonal and impossible pairs) always remain zero,
simply because no data has been collected at those (TX, RX) pairs.
Prior to image reconstruction, the user can define what data will be used in the
reconstruction algorithm by activating (switching the flags on) or deactivating (switching the
flags off) receivers. This process is done through a MATLAB GUI, where each checked box
represents a flag corresponding to (TX, RX) pair (i, j), as shown in Figure 2.10. Diagonal and
impossible pairs of the matrix are disabled so the user cannot switch them on. A few buttons,
including SELECT ALL, CLEAR ALL, SELECT IN–PLANE, and MULTI PLANES are also
designed to help the user make selections faster and more efficiently. Based on the number of
27
planes used to acquire measurements, the user can select a specific set of in-plane data (for
example P1, P2, …) by checking/un–checking the corresponding checkboxes, in Figure 2.10.
Once the transmitter/receiver selection process is done, the user presses CONTINUE and the flag
file (File–4) will be generated. This file is used in the reconstruction algorithm to select specific
transmitter/receiver pairs for the inversion process.
Based on the selected transmitter/receiver pairs in File–4, some antennas may not be
selected to act as either transmitter or receiver. Therefore, in order to reduce the size of input
data and avoid unnecessary forward solution calculation, the unused antennas (sources) in File–4
are removed from the antenna list in File–1 and the new list of sources with their spatial
coordinates is saved into File–1’. Accordingly, Files–3’ and 4 are updated with the new source
IDs.
Figure 2.10. Input data selection GUI in MATLAB
28
2.3.4. Optimizing the Reconstruction Algorithm
As described in Section 2.3.1, the forward solution for each source can be computed
independently. In order to speed up this process, they are calculated in parallel. Once forward
solutions are obtained, the Jacobian matrix is formed. The components of the Jacobian matrix are
partial derivatives (or the sensitivities) of the computed electric field with respect to the complex
contrast in the object. Depending upon the number of sources and the resolution of
reconstruction mesh, the computation of the Jacobian matrix can be intensive in MATLAB. In
order to speed up this process, we use MEX–files to populate the Jacobian matrix. Figure 2.11
illustrates a schematic summary of steps in the image reconstruction algorithm.
In the current study, we evaluate several important aspects of 3D microwave imaging in
simulations, phantom experiments, and some preliminary patient data.
Figure 2.11. Schematic summary of reconstruction algorithm
29
2.4. Results – Simulations
Simulated 3D measurement data is generated by our FDTD forward solver for different
shaped and different sized target inclusions. In order to simulate the system noise, –100 dBm
synthetic noise is also introduced to the data. In addition, in order to avoid so called “inverse
crime,” different meshes are used for the forward and inverse solvers [78]. More specifically, the
finite difference grid used for generating the simulated data is 50% denser than the one used for
the reconstruction.
Figure 2.12. Schematic off the imaging domains evaluated: The background diameter was 14 cm
(antennas are positioned on a 15.2 cm diameter). The spherical inclusion of 1.5 cm radius was
centered at (3, 0, 0 cm).
Figure 2.12 shows a schematic setup of the simulation experiment
experiments studied in this section.
112 monopole antennas are configured in 7 evenly–spaced circles of 15.2 cm diameter and 1 cm
separated from each other. A spherical inclusion of 1.5 cm radius with dielectric properties of εr,
30
Inc
= 40.0 and σInc = 2.0 S/m and centered at (x, y, z) = (3, 0, 0 cm) is embedded in a background
medium with dielectric properties of εr, BK = 22.4 and σBK = 1.23 S/m.
2.4.1. FDTD Grid Density
We use a dual-mesh approach for the image reconstruction procedure, where the forward
solution is calculated on a rectangular uniform FDTD grid, while the distribution of the
constitutive parameters are reconstructed on a tetrahedral-element mesh [67]. It is generally
assumed that a resolution of 10-20 grid points per wavelength is sufficient to guarantee
acceptable convergence for forward solution [79]. In this section, we study the effects of the
finite difference (FD) grid density on the recovered dielectric property distributions.
The permittivity (ε) and permeability (µ) determine the speed of propagation of a wave (c)
through a given medium by
c=
1
(2.6)
εµ
The wavelength (λ) is determined by
λ=
2π c
ω
=
2π
(2.7)
ω 2εµ
where ω is the angular frequency. To determine the size of the FD grid for the forward solution,
we use the already known background medium permittivity value to calculate the wavelength.
The resolution of the FD grid (∆x, ∆y, and ∆z) is then calculated by
31
∆x = ∆y = ∆z =
λ
(2.8)
D
where D is the number of cells per wavelength. Table 2.1 contains a list of FD grid spacing used
to compute the forward solution for the simulation experiment described in Section 2.4.
Table 2.1. Finite difference grid spacing (in mm) used to compute the forward solution for the
simulation experiment, along with the corresponding reconstruction times per iteration
Case
D
∆x = ∆y = ∆z
(mm)
Reconstruction Time
(min)/ iteration
(a)
(b)
(c)
(d)
(e)
(f)
30
25
20
15
10
5
1.52
1.83
2.29
3.05
4.58
9.16
16
9
4.5
2.5
1.7
0.8
Figure 2.13 shows the corresponding reconstructed images at 1300 MHz on a mesh
composed of 3594 uniformly distributed nodes and 15095 tetrahedral elements.
32
(a) D = 30
(b) D = 25
(c) D = 20
(d) D = 15
33
(e) D = 10
(f) D = 5
Figure 2.13. 1300 MHz contour slice images extracted from the reconstructed permittivity (top)
and conductivity (bottom) profiles for the simulation experiment using the finite difference grid
spacing cases (a) to (f) in Table 2.1.
In all 6 cases above, both permittivity and conductivity profiles of the target inclusion are
successfully recovered from those of the background medium. In order to compare the recovered
properties with their true values, the reconstructed dielectric profiles in Figure 2.13 were
extracted at the center of the inclusion (3, 0, 0 cm), as well as the center of the imaging domain
(0, 0, 0 cm) – i.e. background region, and they are plotted as a function of the number of cells per
wavelength (D in equation 2.8) in Figure 2.14.
34
Figure 2.14. Reconstructed permittivity (left) and conductivity (right) profiles along with the true
property values, as a function of the number of cells per wavelength (D) in the forward solution
For D = 10, the conductivity values of the target in Figure 2.14 does not seem to have
reached an asymptote – i.e. it is still getting better, albeit slowly, for even larger D. However,
neither reconstructed permittivity nor conductivity values of the background medium and the
target inclusion appear to be significantly sensitive to the size of the FDTD grid spacing when
over 10 cells per wavelength (D > 10) are used to calculate the forward solution. Therefore, in
order to guarantee that the FDTD grid size does not affect the recovered property distributions,
without significantly increasing the total reconstruction time, we use 20 cells per wavelength in
the forward solver.
2.4.2. Number of Reconstruction Nodes
The total reconstruction process time is directly proportional to the number of nodes (N) in
the property reconstruction mesh. More specifically, N is the size of the solution of the inverse
problem, where dielectric property distributions are calculated. Therefore, the larger N is, the
more computationally intensive the image reconstruction becomes. On the other hand, as N
increases, dielectric property values are calculated at more nodes within the imaging domain. As
35
a result, the reconstructed image has a higher quality, and the property distributions are
potentially more accurate. In order to quantitatively study the sensitivity of the recovered
property profiles to the parameter reconstruction mesh density, we reconstructed 3D dielectric
property profiles of the simulation experiment described in the previous section on different
reconstruction meshes. A list of these meshes (including the number of nodes and elements) is
shown in Table 2.2.
Table 2.2. Parameter reconstruction meshes used to solve the inverse problem for the simulation
experiment
Case
Number of Nodes
Number of Elements
(a)
600
2275
(b)
1587
7545
(c)
2611
11378
(d)
3767
17761
(e)
4608
22184
(f)
6095
29796
(g)
7669
39372
(h)
9321
47978
In order to exclude the effects of other factors, all parameters except the reconstruction
mesh were kept the same in this experiment. Figure 2.15 shows the reconstructed property
profiles for cases (a) through (h) in Table 2.2.
36
(a) 600 Nodes
(b) 1587 nodes
(c) 2611 Nodes
(d) 3767 nodes
37
(e) 4608 Nodes
(f) 6095 nodes
(g) 7669 Nodes
(h) 9321 nodes
38
Figure 2.15. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) profiles for the
simulation experiment on (a) 600, (b) 1587, (c) 2611, (d) 3767, (e) 4608, (f) 6095, (g) 7669, and
(h) 9321 node meshes.
Although both permittivity and conductivity profiles of the target inclusion are effectively
detected from the background medium in all cases, the recovered dielectric properties appear to
be improved as the number of nodes in the reconstruction mesh increases. In order to verify this
observation, the reconstructed dielectric profiles in Figure 2.15 were extracted at the center of the
inclusion (3, 0, 0 cm), as well as at the center of the imaging domain (0, 0, 0 cm) – i.e.
background region. Figure 2.16 shows the extracted property values along with the exact
solutions as a function of the number of reconstruction nodes in Table 2.1. Furthermore, for each
case, the total reconstruction time per iteration was recorded and they are plotted as a function of
the number of reconstruction nodes in Figure 2.17.
Figure 2.16. Reconstructed permittivity (left) and conductivity (right) profiles along with the true
property values, as a function of the number of reconstruction nodes
39
Figure 2.17. Total reconstruction times (min) per iteration as a function of the number of
reconstruction nodes
Results in Figure 2.16 confirm that the recovered dielectric properties are more accurate as
the number of reconstruction nodes increases. However, the changes in the property values are
not significant when reconstruction meshes with over 3000 nodes are used. Moreover, the
reconstruction times in Figure 2.17 increase quadratically with the number of reconstruction
nodes. Therefore, for a configuration of this size, it appears to be sufficient to use a mesh
composed of approximately 3000-4000 nodes.
2.4.3. 3D images from in-plane data vs. stack of 2D images
During the data acquisition, when the transmitting antenna is at the same height as the 15
receiving antennas – i.e. both plates A and B in Figure 1.2(a) are at the same level, in-plane data
is collected. On the contrary, when they are at different levels, cross-plane data is collected. A
complete set of data (full-data) includes both in-plane and cross-plane data at multiple vertical
positions. In 3D imaging, data can be acquired in either in-plane or a combination of in-plane
and cross-plane forms.
40
On the other hand, only in-plane data is collected and reconstructed for 2D imaging.
Therefore, the only difference between a 3D reconstructed image from in-plane data and a stack
of 2D images reconstructed from the same set of data is the forward model and the size of the
respective inversion problems. Propagation of electromagnetic waves in a complex medium is a
3D phenomenon, and using 2D methods to model such a phenomenon is under the assumption
that the scattering problem can be reasonably represented as a 2D problem. Nonetheless, this
assumption may impose over simplifications on the reconstruction process that consequently
introduce image artifacts. In order to study the effects of using a 2D model for a 3D problem, we
have used the same set of in-plane synthetic data in Section 2.4 to reconstruct images in 2D and
3D on uniformly distributed triangular and tetrahedral element meshes composed of 473 and
4608 nodes, respectively.
(a) 2D
41
(b) 3D
Figure 2.18. (a) 2D and (b) 3D reconstructed permittivity (top row) and conductivity (bottom
row) slices (in z-direction) of the simulation experiment with -100 dBm added noise at 1300
MHz, using only in-plane data
Figure 2.18(a) and (b) show 7 slices of the corresponding 2D and 3D reconstructed images,
respectively. In terms of the location and recovered dielectric properties of the target inclusion,
both sets of images are very similar and they closely match to the exact values. However, the
correct size of the spherical inclusion is recovered more precisely in the 3D case. More
specifically, since the radius of the inclusion was 1.5 cm and it was centered at z = 0 cm, the
target should only appear in slices between z = –1.5 and z = 1.5 cm. While only a small trace of
the inclusion shows on the subsequent 3D slices of z = –2 and z = 2 cm, a significantly higher
mismatch appears on the corresponding 2D slices. This effect is even more prominent in 2D
conductivity images, where higher property values arise on the first and last slices too. This
confirms what Fang et al previously showed that due to the data-model mismatch, the 2D
reconstructed microwave images may show traces of the object in planes that extend beyond its
actual locations [67]. In addition to the size, dielectric properties of the target inclusion are also
42
slightly higher and closer to the exact values when the 3D model is used to reconstruct the
images.
Based on the observations above, the 3D FDTD model seems to be superior to its 2D
counterpart. Nonetheless, the recovered 2D images are quite accurate, which makes our 2D
microwave imaging a compelling alternative to 3D imaging when limited computational
resources are available.
2.4.4. Data Selection for Reconstruction
As described in Section 2.3.3, during the data selection process in the 3D GUI, several predefined data selection options, including full-data, in-plane, and multi-plane are available. In the
current configuration of our imaging system, if data is acquired in NP planes, NP×16×15 2D inplane measurements are collected, whereas the equivalent number of collected in-plane and
cross-plane measurements is NP×16×((NP-1)×8). In 2D, the number of measurements increases
linearly with NP, however, in 3D the increase is exponential. If each antenna had the flexibility
to move independent of other antennas, (i.e. if instead of only two independent moving plates,
they were mounted on 16 independent moving plates) the number of 3D measured data would be
about 1.8 times larger (NP×16×NP×15). The increased number of 3D measurements amplifies
the computational complexity of the forward problem, which is the most time-consuming part of
the reconstruction algorithm. However, using cross-plane data may potentially enable us to
better characterize smaller targets that can fall between two consecutive in-planes (2D slices).
Nonetheless, cross-plane data for receivers close to the transmitter might not be very useful,
especially when separated by multiple levels.
In order to study the effects of different data selections on the 3D reconstructed dielectric
property distributions, images of the synthetic data described in Section 2.4 were reconstructed
43
using full-data and multi-plane data with two consecutive planes. Figure 2.19 shows 7 slices of
the corresponding reconstructed images.
(a) full-data
(b) multi-plane data with two consecutive planes
Figure 2.19. 3D reconstructed permittivity (top row) and conductivity (bottom row) slices (in zdirection) of the simulation experiment with -100 dBm added noise at 1300 MHz, using (a) the
full-data (all 7 planes of data, including both in-planes and cross-planes), and (b) multi-plane
data with two consecutive planes
In terms of the size and location of the target, both sets of images are identical to Figure
2.18(b), where only in-plane data was used for reconstruction. However, the recovered properties
44
of the inclusion are slightly more accurate as the number of measured data increases from inplane to full-data set selection. More specifically, when full-data set is used for the
reconstruction, the elevated inclusion region in the middle slice (z = 0 cm) in Figure 2.19(a)
reaches its maximum value in both permittivity and conductivity images. As less data is
progressively used in multi-plane and in-plane data selections, the corresponding inclusion
values decrease accordingly. Nonetheless, the increased amount of data used in full-data set
selection adds significantly to the computational complexity of the forward problem, which
might not be fully compensated for by the subtle differences in the reconstructed property
profiles. Moreover, the amount of background artifacts increases as more data is selected for the
reconstruction. This effect is even more prominent in the conductivity images. Therefore, using
multi-plane data seems to be an effective selection to tradeoff between the accuracy of the
recovered properties and computational demand of the reconstruction process. In addition, when
limited data or scarce computational resources are available, using only in-plane data can be
sufficient to obtain reliable 3D dielectric property distributions.
2.4.5. Number of Iterations
Due to the iterative nature of the reconstruction procedure, a stopping criterion is needed to
terminate the algorithm. In our 2D reconstruction algorithm, convergence is typically achieved
within less than 15 iterations. However, we allow all reconstructions to execute for 20 iterations
[80] as a simple way to ensure convergence in each case and verify that the relative error is
below a threshold of 0.2 (in most cases). In order to establish an analogous norm to terminate the
3D reconstruction algorithm, the relative error values of the simulation experiment in the
previous section are plotted as a function of the number of iterations in Figure 2.20.
45
Figure 2.20. 3D Relative errors as a function of the number of iterations from a simulation
experiment
The relative errors reach the threshold of 0.2 after 17 iterations, and they do not decrease
significantly after about iteration number 20. In order to verify this observation with the
reconstructed property values, the relative permittivity and conductivity difference values with
respect to the 20th iteration were calculated at each node using the following formula:
D20,N = V20 − VN
(2.9)
where V20 and VN are the reconstructed property values at iteration number 20 and N,
respectively. Figure 2.21 shows the relative permittivity (top) and conductivity (bottom)
difference values for iterations 40, 60, 80, and 100 (i.e. D20,40, D20,60, D20,80, and D20,100) at all
reconstructed nodes. For statistical purposes, the mean of the difference values, along with ± 2
standard deviations are also plotted in Figure 2.21.
46
(a) D20,40
(b) D20,60
47
(c) D20,80
(d) D20,100
Figure 2.21. Relative permittivity (top) and conductivity (bottom) difference values for iterations
(a) 40, (b) 60, (c) 80, and (d) 100 (i.e. for D20,40, D20,60, D20,80, and D20,100, respectively), along
with the mean of the difference values ± 2 standard deviations (i.e. 95% of all difference data)
Overall, all difference values are very close to zero, which indicates that there are minor
variations in the reconstructed property values after 20 iterations. More specifically, 95% of all
difference data points fall in a range of 1 unit for relative permittivity and 0.05 S/m for
conductivity profiles at all subsequent iterations, suggesting that similar to the 2D algorithm, the
stopping criterion of 20 iterations is generally a reliable means of ensuring convergence in the
3D reconstruction algorithm.
48
2.5. Results – Phantom Experiments
In order to study the behavior of our new 3D reconstruction process with experimental
data, our clinical MIST system was used to perform several phantom experiments with different
geometries and setups.
2.5.1. Two Cylindrical Inclusions
In the first phantom experiment, two cylindrical inclusions (4.2 cm radius circular-based
and 1.0 cm side-length square-based cylinders) were filled with 90:10 and 50:50 mixtures of
glycerin:water, respectively. As illustrated in Figure 2.22, the square-based cylinder was tilted
toward the right side of the imaging domain, and both inclusions were immersed in a coupling
medium composed of 80:20 glycerin:water mixture.
(a)
(b)
Figure 2.22. Setup for the phantom experiment with two cylindrical inclusions: (a) The squarebased inclusion tilted toward the right side of the imaging domain, (b) Both inclusions immersed
in a coupling medium and surrounded by the antenna array
49
The corresponding permittivity and conductivity values of the phantom were measured
using a dielectric probe (Agilent, Santa Clara CA, 85070E Dielectric Probe Kit – High
temperature probe,) and they are reported in Table 2.3.
Table 2.3. Independently measured dielectric properties of the phantom experiment with two
cylindrical inclusions
Frequency
(MHz)
εr, BK
σBK
εr, CircInc
σCircInc
εr, SqInc
σSqInc
1100
24.8
1.12
9.7
0.78
53.7
1.15
1300
21.0
1.2
8.3
0.83
51.2
1.48
The phantom was then imaged in our clinical system (MIST) with the antenna array set to
collect 3D data in 5 vertical steps of length 1 cm. 3D images were reconstructed at 1100 MHz on
a cylindrical mesh composed of 3132 uniformly distributed nodes and 12707 tetrahedral
elements.
The reconstructed permittivity profile using multi-plane data (two consecutive planes),
along with the iso-surfaced inclusion regions are shown in Figure 2.23. It should be noted that
due to the small contrast between conductivity values of the inclusions and the background
medium, the corresponding reconstructed conductivity image is not shown here.
50
Figure 2.23. 1100 MHz reconstructed permittivity values of the 3D phantom experiment with
two cylindrical inclusions. The iso-surface thresholds are εr,CircInc = 10.0 and εr,SqInc = 33.0.
Both circular-based (with lower permittivity values than those of the background) and
square-based (with higher permittivity values than those of the background) cylindrical
inclusions are detected successfully. Moreover, the location of the inclusions and the tilting of
the square-based cylinder are reconstructed correctly. However, in terms of the property values,
the recovered permittivity values of the square-based cylindrical inclusion are underestimated,
which might be due to the small size of the cylinder and the large contrast between the
permittivity values of the inclusion and those of the surrounding background medium.
In order to assess the effects of different data selection schemes on the recovered property
profiles, four data selection scenarios were used for 3D reconstructing the phantom experiment:
(a) full-data set (i.e. all acquired data), (b) 5 multi-plane data sets with two consecutive planes,
(c) 5 in-plane data sets (one reconstruction using 5 in-planes), and (d) 5 individual in-plane data
sets (5 separate reconstructions, each using only one in-plane data set). In addition, for
comparison purposes, the 5 individual in-plane data sets were used to reconstruct images in 2D
(i.e. 5 separate 2D reconstructions at the matching vertical positions). Figure 2.24(a) – (d) show
the extracted 2D slices of the 3D reconstructed images at 1300 MHz corresponding to the data
51
selection scenarios (a) – (d) described above. The equivalent 2D reconstructed slices are shown
in Figure 2.24(e).
(a) 3D – full-data
(b) 3D – 5 multi-plane
52
(c) 3D – 5 in-plane
(d) 3D – 5 individual in-plane
(e) 2D – 5 individual in-plane
53
Figure 2.24. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) slices of the
phantom experiment with two cylindrical inclusions using 3D reconstruction algorithm with (a)
full-data set, (b) 5 multi-plane data sets with two consecutive planes, (c) 5 in-plane data sets, (d)
5 individual in-plane data sets, and (e) using 2D reconstruction algorithm with 5 individual inplane data sets
In terms of the location and property values of the phantom, the reconstructed 3D images
(a) – (d) are not significantly different, and they are all superior to the equivalent 2D images in
(e). Overall, the more data that is used for the reconstruction, the more features show on the
reconstructed images. The additional features do not all necessarily add to the accuracy of the
recovered property values. In fact, the increased level of artifacts in (a), for example, may be due
to the larger amount of measured data used in the full-data scheme.
In Figure 2.24(c) and (d), the main difference between the selected data is that in (c) all 5
in-plane data sets are used at once to reconstruct a single 3D image on a cylindrical mesh;
whereas in (d), each of the 5 in-plane data sets is used separately to reconstruct five 3D images
on the same cylindrical mesh. Therefore, the amount of measured data used in (d) is five times
smaller than that used in (c), which may explain the lower detail-level observed in Figure
2.24(d).
2.5.2. Breast-shaped Phantom Experiment with an Spherical Inclusion
In order to perform a more anatomically-shaped phantom experiment, a spherical target
inclusion of 2.0 cm radius made from a saline gel was suspended in a prototyped plastic breast
model, as shown in Figure 2.25(a). The breast model was filled with a 88:12 glycerin:water
mixture, and it was then submerged in the imaging tank filled with a matching liquid composed
54
of a 80:20 mixture of glycerin:water, as illustrated in Figure 2.25(b). The independently
measured dielectric property values of the coupling medium, breast model, and the target
inclusion at 1300 MHz are reported in Table 2.4.
(a)
(b)
Figure 2.25. Breast-shaped phantom experiment setup: (a) The spherical saline gel target
inclusion suspended in the plastic breast model, (b) Rapid–prototyped plastic breast model
submerged in the imaging tank
Table 2.4. Independently measured dielectric properties of the background coupling medium
(BK), breast model (BR), and target inclusion (Inc) at 1300 MHz
Frequency
(MHz)
εr,BK
σBK
εr,BR
σBR
εr,Inc
σInc
1300
22.3
1.32
15.1
1.01
67.8
1.80
In the present phantom experiment, 3D microwave data was acquired in multiple planes.
More specifically, the antenna array transmitted and received the signal at 9 equally–spaced (1
cm) vertical positions, and as a result, 9 × 9 × 240 measurements were collected. For comparison
purposes, selective data was used to reconstruct images both in 2D and 3D. In the 2D
55
reconstruction, only in-plane data (with all 16 antennas at the same height collecting a total of 9
× 240 measurements) was used, whereas in the 3D reconstruction, several data selection
scenarios including full-data, in-plane, and multi-plane (two consecutive planes) were
considered. Figure 2.26 shows the 1300 MHz reconstructed images using the multi-plane data set
on a 3D mesh composed of 4849 uniformly distributed nodes and 21381 tetrahedral elements.
For illustration purposes, the 3D iso–surface of the breast and target inclusion are displayed in
Figure 2.26.
(a)
(b)
Figure 2.26. 3D reconstructed (a) permittivity and (b) conductivity images, using the multi-plane
data set at 1300 MHz. The iso-surface thresholds are
εr,BR = 15.0, σBR = 1.0 S/m, and εr,Inc =
65.0 and σInc = 2.3 S/m.
In terms of the location, the target inclusion as well as the breast region is accurately
detected in both permittivity and conductivity images. However, the recovered properties of the
target inclusion are overestimated. More specifically, the recovered permittivity and conductivity
of the target at the center of the inclusion are 91.5 and 2.46 S/m, respectively, which indicates
56
about 35% overestimation of dielectric properties. Moreover, the size of the reconstructed
inclusion permittivity appears to be smaller than that of the conductivity.
In order to compare the 3D reconstructed images with those reconstructed in 2D, 9 slices
from the 3D volume in Figure 2.26 were extracted and are shown in Figure 2.27. Similarly,
Figure 2.28 shows the 2D reconstructed dielectric properties at 9 equally–spaced (1 cm) vertical
positions. In both figures, the images in the top row correspond to permittivity, whereas those in
the bottom row represent the recovered conductivity values.
57
Figure 2.27. Extracted slices from the 3D reconstructed images in Figure 2.26:: Permittivity (top
row) and conductivity (bottom row)
58
Figure 2.28. 2D reconstructed
econstructed dielectric properties: Permittivity (top row) and conductivity
(bottom row)
Similar to the 3D reconstructed images iin Figure 2.27,, the breast as well as the target
inclusion is successfully detected in the 2D images in Figure 2.28. In addition, as the
reconstructed planes move toward the nipple (i.e. z decreases), the outline of the breast model
becomes smaller and finally vanishes at about z = –2 cm in both sets of images. Nonetheless, the
object seems to appear at different heights in the permittivity and conductivity images (at z = 4
cm in
εr and at z = 3 cm in σ). In addition, the level of artifacts, especially in the background
59
region, is significantly increased in the 2D reconstructed images. This effect is even more
prominent in the corresponding conductivity profiles. In terms of the location of the target, since
the radius of the spherical inclusion was 2.0 cm, it should only appear in the first three
reconstructed images from the top of the breast model (z = 4, 3, and 2 cm). While the 3D
reconstructed images capture the correct location of the inclusion, the 2D reconstructed
conductivity images show traces of the target at lower planes (as low as z = 1 cm), which
indicates that the 3D reconstructed conductivity images are superior to those reconstructed in 2D,
and confirms that due to the data-model mismatch, the 2D reconstructed microwave images may
show traces of the object in planes that extend beyond its actual locations.
Similar to the analysis performed in Section 2.5.1, three different data selection schemes
including (a) full-data set (i.e. all acquired data), (b) 5 multi-plane data sets with two consecutive
planes, and (c) 5 in-plane data sets were used for reconstructing the breast-shaped phantom
experiment at 1300 MHz. Figure 2.29(a) – (c) show the extracted 2D slices of the 3D
reconstructed images (permittivity on top and conductivity on bottom) at z = 3 cm, corresponding
to the data selection scenarios (a) – (c), respectively. The equivalent 2D reconstructed slices are
shown in Figure 2.29(d). Likewise, Figure 2.30 shows the counterpart reconstructed images at z
= 0 cm, where only the distal parts of the breast are captured.
60
(a)
(b)
(c)
(d)
Figure 2.29. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) slices of the
breast-shaped
shaped phantom experiment at z = 3 cm using the 3D reconstruction algorithm with (a)
full-data set, (b) 9 multi-plane data sets with two consecutive planes, (c) 5 in-plane
plane data sets, and
(d) using the 2D reconstruction algorithm with 9 individual in-plane data sets
(a)
(b)
(c)
(d)
Figure 2.30. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) slices of the
breast-shaped
shaped phantom experiment at z = 0 cm using the 3D reconstruction algorithm with (a)
61
full-data set, (b) 9 multi-plane data sets with two consecutive planes, (c) 5 in-plane data sets, and
(d) using the 2D reconstruction algorithm with 9 individual in-plane data sets
Overall, the 3D reconstructed images are superior to those reconstructed in 2D, largely
because the outline of the breast is more visible and the target inclusion is characterized more
accurately. The property profiles in multi-plane and in-plane cases, (b) and (c), are very similar
and they both present less background artifacts than the full-data case in (a).
Despite the fact that the target inclusion is correctly detected in both permittivity and
conductivity images in Figure 2.29, the size mismatch between the reconstructed inclusion
permittivity and conductivity profiles can be seen in all cases, including the 2D reconstructions.
This mismatch is even more prominent in (a) where full-data set was used for reconstruction,
which may be due to the larger amount of measured data used in the full-data case.
62
2.6. Results – Initial Clinical Data
Until very recently, all of our clinical breast data had been collected and reconstructed only
in the 2D format (in-plane 2D slices). Since we have developed a viable and user-friendly
interface to select and reconstruct 3D measurement data, we are planning to extend our data
acquisition from 2D (only in-plane) to 3D (both in-plane and cross-plane). In Section 2.6.1, we
will evaluate the performance of the reconstruction algorithms by comparing the 2D and
preliminary 3D reconstructed breast images of a normal subject. In Section 2.6.2, we will present
the 3D results for the initial clinical bone imaging, using a simple adaptation of our existing
microwave imaging system.
2.6.1. Breast Imaging
In this study, a normal subject (patient# 551, a volunteer with no known breast
abnormalities) of 45 years old was imaged in our clinical system (MIST) using the 3D data
acquisition, as shown in Figure 2.31. Based on the patient’s size and age, a matching liquid
composed of a 80:20 mixture of glycerin:water was used, and six planes of microwave data (both
in-plane and cross-plane, separated by 1 cm) were collected. The images were reconstructed
both in 2D and 3D on triangular- and tetrahedral-element meshes, respectively. The 2D circular
reconstruction mesh was comprised of 559 nodes and 1044 triangular elements with a 6.9 cm
radius, whereas the 3D cylindrical reconstruction mesh had a radius of 6.9 cm, a height of 6 cm,
and consisted of 6095 nodes and 29796 tetrahedral elements, respectively.
63
Figure 2.31. 3D microwave imaging of a normal subject in the clinical MIST system
For comparison purposes, selective data was used to reconstruct images both in 2D and 3D.
In the 2D reconstruction, only in-plane data (with all 16 antennas at the same height collecting a
total of 6 × 240 measurements) was used, whereas in the 3D reconstruction, several data
selection scenarios including full-data, multi-plane (four and two consecutive planes), and inplane were considered. Figure 2.32 shows 6 slices from the 3D reconstructed volume at 1100
MHz using the full-data set. Similarly, Figure 2.33 shows the 2D reconstructed dielectric
properties at 6 equally–spaced (1 cm) vertical positions. In both figures, the images in the top
row correspond to permittivity, whereas those in the bottom row represent the recovered
conductivity values.
64
Figure 2.32. Extracted slices from the 1100 MHz 3D reconstructed images using the full-data set:
Permittivity (top row) and conductivity (bottom row)
Figure 2.33. 1100 MHz 2D reconstructed
econstructed dielectric properties: Permittivity (top row) and
conductivity (bottom row)
Overall, the permittivity images appear to be superior to the conductivity counterparts,
counterpar and
despite the low contrasts between the dielectric properties of the coupling medium and the
heterogeneously dense breast tissue, the outline of the breast is roughly detected in both 3D and
2D reconstructed images in Figure 2.32 and Figure 2.33.. In addition, as the reconstructed planes
65
move toward the nipple (i.e. z decreases), the contour of the breast becomes smaller and finally
vanishes at about z = –1.5 cm in both sets of images. Based on the patient’s age and her breast
density type, the elevated dielectric properties (especially the permittivity) at the right center of
the reconstructed images closer to the chest wall may correspond to the fibroglandular tissue.
While the 2D and 3D reconstructed permittivity images are comparable, the conductivity
counterparts are more different. In fact, the higher property values of the fibroglandular region
only appear in the first three planes (i.e. z = 2.5 to 0.5 cm) in the reconstructed 3D images, as
well as in the 2D reconstructed permittivity profiles. However, this region arises in the lower
planes in the 2D reconstructed conductivity images, which confirms one more time that due to
the greater mismatch between the actual and 2D modeled magnitudes of the electric field, the 2D
reconstructed conductivity images contain more artifacts, and may show traces of the tissue (in
this case fibroglandular) in planes that extend beyond its actual location.
In order to study the effects of different data selection schemes on the 3D recovered
dielectric properties, four different cases were used for reconstructing the patient data described
above: (a) the full-data set (i.e. all acquired data), (b) 6 multi-plane data sets with four
consecutive planes (two planes above and two planes below), (c) 6 multi-plane data sets with two
consecutive planes (one plane above and one plane below), and (d) 6 in-plane data sets. Figure
2.34(a) – (d) show the extracted 2D slices of the 3D reconstructed images (permittivity on top
and conductivity on bottom) at z = 1.5 cm, corresponding to the data selection scenarios (a) – (d),
respectively. The equivalent 2D reconstructed slices (using only one in-plane data set at z = 1.5
cm) are shown in Figure 2.34(e). It should be noted that for comparison purposes, the range of
the recovered permittivity values in Figure 2.34 are different from those in Figure 2.33.
66
(a)
(b)
(c)
(d)
(e)
Figure 2.34. 1100 MHz reconstructed pe
permittivity
rmittivity (top) and conductivity (bottom) slices of the
breast-shaped
shaped phantom experiment at z = 1.5 cm using the 3D reconstruction algorithm with (a)
the full-data (FD) set, (b) 6 multi
multi-plane data sets with four consecutive planes (4CS) (two planes
above and two planes below), (c) 6 multi-plane data sets with two consecutive planes (2CS) (one
plane above and one plane below), (d) 6 in-plane (IP) data sets,, and (e) using the 2D
reconstruction algorithm with 6 individual in-plane data sets.
At a first glance, all 3D reconstructed images in Figure 2.34 seem very similar and there
are minor differences between them and their 2D counterparts
counterparts. The outline off the breast appears
to be more distinguishable in the 3D reconstructed permittivity images, compared to that
reconstructed in 2D. In general, the 2D reconstructed images look sharper and appear to have
higher quality compared to the 2D slices of the 3D re
reconstructed images.. This is due to the fact
that the 2D slices of the 3D reconstructed images transect the 3D mesh tetrahedral elements at
oblique angles, and as a result, the breast structures in Figure 2.34(a) – (d) appear more blurred
compared to those in Figure 2.34
34(e).
67
In order to evaluate the differences between the 3D reconstructed images using the data
selection cases in Figure 2.34,, the recovered dielectric properties of the multi-plane
plane (b and c) and
in-plane (d) cases were subtracted from the full-data case in (a). The difference images are
plotted in Figure 2.35
(a) FD – 4CS
(b) FD – 2CS
68
(c) FD – IP
Figure 2.35. Relative permittivity (top) and conductivity (bottom) difference images with respect
to the recovered dielectric property values using the full-data (FD) set: (a) full-data
data – multi-plane
data sets with four consecutive planes (i.e. FD – 4CS) (b) full-data – multi-plane
lane data sets with
two consecutive planes (i.e. FD – 2CS), and (c) full-data – in-plane data sets (i.e. FD – IP)
The amount of measured data decreases from the full-data set to multi-plane
plane and then to inplane data selections.. Accordingly, the variations in the difference images increase from case (a)
to (c) in Figure 2.35.. However, most of the variations appear on the exterior part of the
reconstructed images and thee data selection scheme seems to have no major effects on the
recovered dielectric properties of the breast. In order to quantify the variations, the relative
permittivity (top) and conductivity (bottom) difference values in Figure 2.35 were extracted and
they are plotted in Figure 2.36
36 as a function of the reconstruction nodes. In addition, for
69
statistical analysis purposes, the mean of the difference values, along with ± 2 standard
deviations are also plotted in Figure 2.36.
(a) FD – 4CS
(b) FD – 2CS
70
(c) FD – IP
Figure 2.36. Relative permittivity (top) and conductivity (bottom) difference values in Figure
2.35 as a function of the reconstruction nodes, along with the mean of the difference values ± 2
standard deviations (i.e. 95% of all difference data): (a) full-data – multi-plane data sets with
four consecutive planes (i.e. FD – 4CS) (b) full-data – multi-plane data sets with two consecutive
planes (i.e. FD – 2CS), and (c) full-data – in-plane data sets (i.e. FD – IP)
Overall, all difference values are relatively close to zero, which indicates that there are
minor variations in the 3D reconstructed property values using different data selection schemes.
In case (c) where the largest deviations appear, 95% of all difference data points fall in a range of
± 2.5 units for the relative permittivity and ± 0.1 S/m for the conductivity profiles. These
71
variations may increase if less regularization is used during the reconstruction. For the present
result, the Tikhonov regularization coefficient value of 50 was used.
2.6.2. Bone Imaging
Microwave imaging can be used for detecting dielectric property changes in bone tissue
with age or due to injury–related bone loss [81, 82]. Microwave tomography for heel imaging is
particularly interesting, mainly because there is a significant dielectric property contrast between
the bone and surrounding soft tissue. Moreover, like breast tissue, the heel is relatively easy
accessible and it can fit into our microwave antenna array for imaging. In this section, we
evaluate the feasibility of microwave heel imaging using a simple adaptation of our existing
breast imaging system, as shown in Figure 2.37.
(a)
(b)
Figure 2.37. Heel imaging in the microwave imaging system: (a) The heel extended through an
aperture in the mounting plate, (b) The heel surrounded by the antenna array
This was used to acquire four planes of microwave data (both in-plane and cross-plane,
separated by 1 cm) of a normal heel (pt# 901, R) at 1300 MHz. The images were reconstructed
72
both in 2D and 3D on triangular
triangular- and tetrahedral-element meshes, respectively.. The 2D circular
reconstruction mesh was comprised of 559 nodes and 1044 triangular elements with a 6.9 cm
radius, whereas the 3D cylindrical reconstruction mesh had a radiu
radius of 6.9 cm,
cm height of 5 cm,
and consisted of 3132 nodes and 12707 tetrahedral ele
elements respectively, as shown in Figure
2.38(a) and (b).
(b)
(a)
Figure 2.38. 2D and 3D reconstruction meshes: (a) 6.9 cm radius circular mesh: 559 nodes and
1044 triangular elements, (b) 6.9 cm radius, 5 cm height cylindrical mesh: 3132 nodes and 12707
tetrahedral elements
Figure 2.39 shows several views of the 3D reconstructed permittivity image of the heel
with an iso-surface
surface threshold of
εr = 26, which captures the bone/soft tissue and the
heel/coupling liquid interfaces.. The light blue shaded area surrounding the heel corresponds to
the cylindrical reconstruction zone
zone. The
he underside of the surface exterior appears to conform
nicely to the overall heel geometry. Moreover, several important features including, the large
high-water-content
content zones corresponding to the soft tissue nearer to the front of the foot,
foot the
73
centrally located vertical calcane
calcaneous
us bone, and the narrowing of the heel towards the posterior of
the foot, can be observed in the 3D reconstructed images in Figure 2.39.
Figure 2.39. Several
everal views of the 3D reconstructed permittivity image of the heel at 1300 MHz,
with an iso-surface threshold of εr = 26
Figure 2.40(a)
(a) and (b) show the corresponding 2D and horizontal slices of the 3D
reconstructed permittivity (top) and conductivity (bottom) images, respectively. The four image
pairs in Figure 2.40(a)
(a) correspond only to the in-plane data acquired from the ankle (P1) down
towards the heel apex (P4), whereas those in Figure 2.40(b) correspond to both in-plane and
cross-plane data (two consecutive
ive planes selection).
In both cases, as the images progress towards the heel apex (P4), the outline of the heel
shrinks until it completely disappears in the lower planes. Moreover, the calcaneous bone located
at the center of the heel has lower properti
properties
es compared to the surrounding soft tissue, which
74
shows that the recovered property distributions are representative of the actual dielectric
properties of the tissues.
(a) 2D
(b) 3D
Figure 2.40. 1300 MHz,, (a) 2D and (b) slices of the 3D reconstructed permittivity (top) and
conductivity (bottom) images at 4 planes of data separated by 1 cm,, starting from the ankle (P1)
and moving down towards the heel apex (P4).
75
In terms of the size, location, feature shapes, and overall property distributions, the 3D
reconstructed images are similar to their 2D counterparts. However, since the 2D slices of the 3D
reconstructed images transect the 3D mesh tetrahedral elements at oblique angles, the heel
structures in Figure 2.40(b) appear more blurred compared to those in Figure 2.40(a). Moreover,
the calcaneous appears in the first two planes of the permittivity and the first three planes of the
conductivity images in Figure 2.40(a), whereas in Figure 2.40(b), the bone is only visible in the
first plane of the permittivity and the first two planes of the conductivity images. Based on the
relative vertical position of the antennas to the heel during the exam, the recovered location of
the calcaneous bone seems to be more accurate in the 3D reconstructed images. This observation
confirms again that due to the data-model mismatch, the 2D reconstructed microwave images
may show traces of the object in planes that extend beyond its actual locations.
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3. Incorporation of a Priori Structural Information into the
Microwave Image Reconstruction Algorithm
3.1. Introduction
In prior studies, we have shown that MI has substantial potential for diagnosing breast
lesions greater than 1cm in size [9, 17] – a result that was driven largely by the excellent
specificity of the technique (rather than sensitivity as in the case of MR). We have also shown
that MI can be a useful tool to monitor breast cancer treatment response to neoadjuvant
chemotherapy [18, 83, 84]. However, MI suffers from a relatively poor spatial resolution
compared to other conventional imaging modalities such as MRI or X-ray CT. Moreover, due to
the use of the regularization method to stabilize the MI image reconstruction process, detecting
boundaries of the regions of interest in the recovered microwave properties is a difficult task, and
as a result, the diagnostic potential of MI is often limited.
In response to these shortcomings, we have developed a method to combine the functional
information available through MI (i.e. property contrast) with the high spatial resolution
anatomical information available from other imaging modalities, such as MRI, to recover more
accurate dielectric properties and potentially better characterize smaller masses [72]. This
approach has been developed successfully in a related imaging system that utilizes a combination
of NIR tomography and MRI methods for breast cancer detection [85-89].
A mathematical framework of microwave imaging reconstruction algorithm is presented in
Section 3.2. Several regularization techniques, including the ones that incorporate the spatial
priors, are also discussed in that section. In order to combine the functional information of the
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MI with the spatial information from the MRI, we have developed a prototype MR-MI system,
which is described in details in Section 3.3. Finally, the results of the 2D and 3D simulations,
phantom experiments, and initial clinical data are presented in Sections 3.4 and 3.5, respectively.
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3.2. A Mathematical Framework of Microwave Image Reconstruction
Algorithm
3.2.1. Overview
The reconstruction process in microwave imaging is based on determining the distribution
of the constitutive parameters within tissue where the dielectric properties are embedded in the
squared complex-valued wave number, which can be written as
k 2 ( r ) = ω 2µ0ε ( r ) − jωµ 0σ ( r )
(3.1)
where r is the position vector in the imaging domain, ω is the angular frequency, j is the
imaginary unit, µ0 is the free-space permeability, ε is the permittivity, and σ is the conductivity
[90].
The reconstruction algorithm mainly consists of solving two problems: the forward
problem and the inverse (or optimization) problem. The forward problem involves computing the
output (scattered field) from known inputs (microwave excitation) and system properties
(dielectric property distribution of the tissue being imaged). In our algorithm, the calculation of
the forward solution is based on Maxwell’s equations and it is computed using a finite difference
time domain (FDTD) algorithm [79]. In the FDTD method, a frequency domain forward field
response is produced for each transmitter and the individual field values are extracted at each
receiver location [66]. The length of the vector k2 is N, the number of reconstruction parameters.
The inverse problem, on the other hand, estimates the properties of an unknown volume
(dielectric properties of the tissue) from a known input (microwave excitation) and measured
field values. Since the inverse electromagnetic problem is non-linear, iterative Gauss-Newton
schemes are well suited for the application [68, 69]. In our case, the solution of the inverse
79
problem is based on an improved Gauss-Newton (GN) iterative approach, called LevenbergMarquardt (LM), with a variance stabilizing transformation in which, the measured electric field
vector Em is matched iteratively with the computed electric field vector Ec(k2) calculated using
the forward model for a given distribution of the constitutive parameters stored in the vector k
2
[65, 91]. Although regularized Gauss-Newton methods can be susceptible to convergence to
local minima [92, 93], we have found that log transformation serves to mitigate these effects [65,
94]. Indeed, recent studies of convergence indicate that our Gauss-Newton algorithm reaches the
same image (i.e., solution) for widely different initial estimates suggesting that it is not easily
trapped by local minima [95]. Completely different approaches to inverse problem solutions such
as genetic algorithms and stochastic processes are also possible and have been applied
successfully to MI as well [96-98].
In addition, due to the ill-posed nature of the inverse problem, some form of regularization
method must be utilized in order to impose additional constraints [69, 70]. Regularization is
often accomplished by introducing a priori information about the tissue being imaged, which
may be necessary to ensure convergence of the reconstruction algorithm to the correct
electromagnetic property distribution [71]. In the field of microwave imaging, a number of
studies have investigated the incorporation of different types of priors ranging from the internal
and/or external shape of the body to information about tissue dielectric properties including their
upper and lower bounds [99-108]. For example, Crocco et al assumed that the object was
homogeneous and that the permittivity value of the target was known [109]. In their shape
reconstruction simulation study, El-Shenawee et al, used estimates of inclusion properties and
location, and exact knowledge of the number of targets to aid convergence to a viable solution
[110].
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In our algorithm, the Tikhonov regularization [69] is used to stabilize the reconstruction
procedure, albeit with added smoothing. The objective function is
2
2
2
2
2
2
Ω = Γ m − Γ c (k 2 ) + Φ m − Φc (k 2 ) + λ L(k 2 − k02 )
m
c
m
(3.2)
c
where Γ and Γ are the log magnitudes and Φ and Φ are the phases of the measured and
computed field values, respectively [65, 66, 91, 94], λ is the weighting coefficient, also known as
the Tikhonov regularization parameter, and L is a positive definite, dimensionless regularization
2
matrix. k02 is a prior estimate of k , and the two-norm V2 of a vector of length M (in this case
M is the number of measurements) is the square root of the sum of the squares of the complex
modulus of its elements:
M
v
2
=
∑v
2
(3.3)
i
i=1
In our previous and current studies, the choice of λ is derived empirically. After some
manipulation, equation 3.2 can be solved for the iterative property update, ∆kη2

∆ kη =  J J + λ L L   J T


2
T
T
−1

 Γ m − Γc (kη2 )  T
2
2
−
−
L
L
(
k
k
)
 m
η
0 
c
2 

Φ − Φ (kη ) 

(3.4)
where J is the Jacobian matrix, which has dimensions 2M × 2N, and it consists of derivatives of
the log magnitude and phases of the computed field values with respect to the property values at
each of the N reconstruction parameter mesh nodes. kη2 is the vector k2 at iteration η and is
updated as
∆ kη2 = kη2+1 − kη2
(3.5)
In addition, we use a dual-mesh approach where the forward solution is computed on a
uniform FDTD grid (rectangular lattice in 2D and rectangular cuboid in 3D), while the
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electromagnetic property parameters are reconstructed on a triangular (2D) or tetrahedral (3D)
element mesh, placed concentrically within the antenna array [67].
In our original reconstruction algorithm, the regularization matrix L in equation 3.4 was set
to the identity matrix, which applied the same weight to the values at all nodes within the
imaging domain. In addition, for the right hand term in equation 3.4, k02 was set to kη2 as in the
case for the Levenberg-Marquardt (LM) algorithm [68], leading to a simplified version of the
update equation
 Γ m − Γ c ( kη2 ) 
∆ kη =  J J + λ I  J  m
c
2 
 Φ − Φ ( kη ) 
2
T
−1
T
(3.6)
Throughout this chapter, The LM method, also referred as the “no priors”, corresponds to
the update equation 3.6.
3.2.2. Soft prior Regularization
In order to incorporate prior spatial information of the tissue (or phantom) being imaged
into the microwave imaging, we have modified our reconstruction algorithm similarly to that
reported in [85, 87-89, 111, 112]. In the new algorithm, called soft prior regularization, the
spatial prior is considered “soft” because it does not force the property estimates inside an
identified region to be constant. Instead, the known boundary data is used to adjust the
regularization to smooth the property estimates within pre-identified regions, while limiting the
smoothing across the boundaries (to preserve property changes at the interface with other
tissues/regions). The basic idea behind the soft prior regularization is to more heavily weight the
uniformity within regions that are assumed to have the same or similar dielectric properties. In
addition, when two different regions share the same boundary, the smoothing across their
common interface is penalized [111]. In the current implementation, we incorporate a priori
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information about the anatomical structure of the tissue through the regularization matrix L in
equation 3.4. According to known information about the tissue’s structure (derived from any
high spatial resolution source, such as MR images), each node in the reconstruction mesh is
labeled with an associated region number. Given two nodes, i and j, in the reconstruction mesh,
with their associated matching regions, Ri and Rj, the corresponding entry in the L matrix is
defined as:



lij = 



0
if Ri ≠ R j
−1
N Ri
if Ri = R j
1
if i = j
(3.7)
where is the number of nodes in region Ri – 1. Figure 3.1 illustrates how the regularization
matrix L is constructed in a simplified case, where there are just two regions, R1 and R2,
containing N1 + 1 and N2 + 1 nodes, respectively.
Figure 3.1. Soft prior regularization matrix L with two distinct regions
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T
Based on the above construction of L, L L in equation 3.4 is an approximation of a second
order Laplacian smoothing operator inside each region, which limits the smoothing across the
boundary of distinct regions [113, 114]. Since the structure of the tissue being imaged does not
change during the iterative image reconstruction algorithm, both the regularization matrix L and
T
Laplacian smoothing operator L L, can be calculated once and stored at the beginning of the
reconstruction process. In this way, redundant calculations are avoided and the algorithm
becomes more efficient.
Effective encoding of spatial priors in microwave image reconstruction requires software
tools for region identification and segmentation, in addition to mesh generation. We have used
both commercial software (e.g. Mimics, SolidWorks) as well as our own algorithms to postprocess the structural information. In the current study, we apply a collection of user-selected
options, such as region growing methods in Mimics, to segment regions of interest in the MR
images. When prior spatial structures are available, a more specific and customized
reconstruction parameter mesh is desired. In fact, sufficient nodes need to be deployed on the
boundary of adjacent regions, so that the interfaces in the reconstructed image are smooth. We
will use methods developed in-house for surface (for 2D) and volume (for 3D) mesh generation,
which control the nodal sampling density to capture the irregular object geometry. Finally, the
subsequent user-defined mesh needs to be co-registered with the acquired microwave data – This
step usually involves shifting, rotating, and/or flipping the mesh. Due to the different coordinate
systems, co-registration of the microwave and MR images can be a very challenging task.
Therefore, we utilize additional markers during both imaging sessions (i.e. MI and MRI) to
ensure accuracy in the co-registration process.
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Further on in this chapter, we evaluate the performance of the MI with the soft prior
regularization relative to no priors in a number of simulations, phantom experiments, and some
preliminary patient data. The corresponding 2D and 3D results are presented in Sections 3.4 and
3.5, respectively.
3.2.3. Hard priors – Parameter Reduction
Unlike the soft prior regularization technique, the hard priors approach uses a priori
constraints through parameter reduction. The basic idea behind the hard priors is to reduce the
size of the reconstruction parameters by assuming that there are homogeneous dielectric property
distributions within each of the pre-identified regions [88]. As a result, given structural
information with R distinct regions, single values for permittivity and conductivity are
reconstructed within each region. The region identification process is identical to that in the soft
priors. However, the parameter reduction is implemented during the property update in equation
3.6, by defining a new Jacobian matrix:
Jɶ = JKɶ
(3.8)
, and K is the a priori parameter reduction matrix defined as:
where 1
kij = 
0
if node(i ) ∈ R j
(3.9)
if node(i ) ∉ R j
where node(i) is the ith node in the parameter reconstruction mesh, and Rj is jth region identified
from the prior structural information. The dimensions of K are the number of nodes by the
number of regions (N×R), and since the dimensions of the original Jacobian matrix J are two
times the number of measurements by two times the number of reconstruction nodes (2M × 2N),
has the dimensions of 2M × R. In the new sensitivity matrix, all the columns corresponding to
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the same region are summed, and as a result, the number of reconstruction parameters drops
from N (number of nodes in the reconstruction mesh) to R (number of pre-identified regions of
interest). Therefore, no matter how many nodes are placed in each region, the number of
unknowns is set to the number of regions in the prior structural data R. As long as R < M, no
regularization is required during the image reconstruction, and consequently equation 3.6 can be
simplified to
 Γ m − Γ c ( kη2 ) 
T ɶ −1 ɶ T
ɶ
∆ kη =  J J  J  m
c
2 
 Φ − Φ ( kη ) 
2
(3.10)
We compare the performance of the hard priors with the soft priors as well as no priors in a
simulation study case in Section 3.5.1.7.
3.2.4. Error Analysis
Throughout the remainder of this chapter, we will use an error analysis to compare
quantitatively the soft prior and no prior results. Since the true values of the dielectric property
distributions are known in simulations and phantom experiments, we can calculate the relative
error between the true/independently-measured property distributions and the reconstructed
values as:
N
Errw = ∑ wn
n=1
recon
exact
V(n)
−V(n)
(3.11)
exact
V(n)
where V(n)recon is the reconstructed dielectric property value (either permittivity or conductivity)
at node n in the reconstruction mesh, whereas V(n)exact is the true value of the selected dielectric
property at that location. To account for the fact that nodes in the reconstruction meshes may not
86
be uniformly distributed, a weighting factor is added at each location computed as wn = An /A,
where An is the area of the elements surrounding node n and A is the total imaging area.
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3.3. Integration of Microwave Tomographic Imaging System with MRI
3.3.1. Motivation
Unlike microwave tomographic imaging, MRI has an excellent spatial resolution (down to
0.5 mm) and it is broadly used in clinical practice. In fact, MRI is the most sensitive medical
exam for breast cancer surveillance, especially in the dense breast [115]. The high spatial
resolution of internal anatomical structures that MRI provides is based on the natural tissue
contrast in terms of the relaxation times [116]. Breast MRI has been used successfully to monitor
treatment response to neoadjuvant chemotherapy and is deployed routinely for cancer staging
and surgical treatment planning [117-119].
Despite the high cost of the imaging system and the relative expense of an exam, the
number of clinical breast MR imaging studies is likely to continue to grow in the future, mainly
due to the extraordinarily detailed 3D anatomical information that can be obtained [120, 121].
Nonetheless, breast MRI has substantial limitations, including a high false positive rate and the
fact that it cannot provide information about a tissue’s physiological state [121]. As a result, MRI
has been combined with other imaging modalities to improve the specificity of diagnosis and
therapy [122-125]. For instance, in the combined NIR-MRI imaging, MRI contributes excellent
spatial resolution, while the NIR tomography provides the complementary functional
information about the tissue [126].
Similar to the NIR-MRI concept mentioned above, we proposed to combine the high
spatial resolution of MRI with the high specificity of the microwave dielectric properties. One of
the most apparent challenges for combining microwave tomography with MRI is the physical
conflict between the small MR bore size and our requirement to utilize a dielectric coupling bath.
Based on our experience from earlier studies, we have been able to strategically reduce the size
88
of the imaging tank enough so that it can accommodate many women even within an MR bore.
In addition, the non-ferrous monopole antennas and their unique low profile design make them
ideal for operating in the MR environment while inducing only minimal and acceptable MR
image disruptions. Along with the microwave antennas affecting the MR signals, the RF
gradients used in the MR gradient sequences produce high power signal bursts that can saturate
and desensitize receivers and subsequently disrupt the desired electromagnetic signals. In the
following four Sections, 3.3.2 – 3.3.5, we will discuss these issues in details and present
strategies to overcome barriers in developing a multi-modality MI-MR imaging system with
minimal unwanted effects.
3.3.2. Integration of MI System into MRI
As with all research performed in an MR environment, the electronics are generally
positioned outside of the examination area. In our case, this is achieved by using long, flexible
coaxial cables which connect our imaging tank in the bore to a set of 16 SMA flange connectors
positioned on a bulkhead panel between the exam room and the external control room. Figure
3.2(a) shows our clinical microwave imaging system located just outside and below the
connector panel with short coaxial cables running from the connectors down to the transceiver
modules inside the bed.
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(a)
(b)
Figure 3.2. MI-MR imaging: (a) the portable data acquisition system in the control room with the
auxiliary cables attached to the bulkhead connectors in the wall, (b) the imaging tank outside of
the MR bore showing the semi-rigid cables extending from the tank
While the extended cable lengths (7.9 dB return path attenuation at 1.5 GHz for the 6.5 m
cables – LMR200 from Times Microwave Systems, Wallingford, CT) adds significant signal
loss, it does allow us to utilize the MR–based tank while locating all electronics out of the MR
room. Since signal losses in the tank increase with frequency, for future development, we intend
to implement lower loss cables (LMR600 – 2.0 dB loss at 1.5 GHz for the return path loss over
6.5 m long cables), which will ultimately allow us to operate at higher frequencies.
An overriding principle for the MI-MRI multi–modality integration is to minimize the
amount of metal near the MR bore, especially removing anything that may be ferromagnetic. As
can be seen in Figure 3.2(b), the cables leading directly away from the illumination tank are 3.58
mm diameter, semi–rigid coaxial cables made from a copper outer conductor and a silver–plated,
copper center conductor. Once these have extended at least 1 meter from the tank, the cables are
connected to the flexible coaxial lines described above, which are more convenient for the long
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span across the room to the bulkhead interface connectors. The microwave antenna array
configuration in this new system is identical to that of our original clinical imaging system, and
each antenna is capable of operating as both transmitter and receiver in the frequency range of
700 MHz to 3 GHz. However, in our actual prototype system, the antennas are not movable and
do not have the capability of collecting data at multiple planes, and as a result, only 240
measurements (16 transmitters × 15 receivers) can be collected for each image.
3.3.3. MR Image Artifact Reduction Strategies
The most fundamental challenge in combining microwave imaging with an MR system is
the need to bring metallic objects into the bore. Ferromagnetic materials are clearly prohibited,
but many other types of metals such as titanium, aluminum, copper and silver are safe to position
within the magnet. However, these metals can still disrupt the images.
Figure 3.3 shows a 19 cm height, 13 cm radius microwave imaging tank used for a phantom
experiment in the MR. The phantom was comprised of a 5.4 cm radius cylindrical container
(84:16 glycerin:water mixture) with 2.1 cm and 1.0 cm radius cylindrical inclusions filled with
72% and 55% glycerin mixtures, respectively. The associated copper–clad, semi–rigid coaxial
feed lines and the stainless steel flange connectors were placed on the under surface of the tank
(the coaxial cables consisted of a 1.16 mm diameter center conductor, a 3.05 mm insulator
diameter and 3.58 mm outer conductor diameter).
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(a)
(b)
Figure 3.3. MI tall tank with a large cylinder breast phantom (green) and smaller inclusions
surrounded by the monopole antenna array: (a) side view and (d) top-down view
Figure 3.4 shows the corresponding T2–weighted MR images of the phantom experiment.
Both Figure 3.4(a) and (b) are for silver–plated copper center conductors. The first imaging plane
in (a) transects the monopole antennas, while the one in (b) intersects the coaxial feedlines 3 cm
below the antenna/coaxial line interface plane. For the imaging plane in the active part of the
antennas in Figure 3.4(a), there are minor artifacts surrounding each antenna, but these
perturbations do not impact the images of the phantom. In fact, for co-registration purposes, the
positions of the antennas can be accurately marked as a reference with respect to the microwave
imaging array geometry. At the plane transecting the feedline in (b), the artifacts surrounding the
antenna array have increased and there are minor perturbations to the shapes of the interrogated
phantom and inclusions. For Figure 3.4(c), each monopole antenna and coaxial line used a steel
center conductor, and the phantom only had the 2 cm diameter inclusion. The MR image in (c) is
completely disrupted due to the presence of ferromagnetic materials in the magnet. This
sequence of images emphasizes that the ferromagnetic materials cannot be used in the combined
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MI-MRI system. In contrast, it also demonstrates that other metals can still disrupt the image
quality, and that reducing the metal footprint clearly reduces artifacts. In this regard, the
monopoles are optimal in that they employ the least amount of metal possible of any antenna.
(a)
(b)
(c)
Figure 3.4. T2–weighted MR images of the phantom experiment: (a) at the center plane of the
active part of the antennas and (b) at a plane 3 cm below the interface between the semi-rigid
coaxial line and the antenna. For (a) and (b) the phantom included a 5.4 cm radius outer cylinder
and the 2.1 and 1.0 cm radius inclusions, and the coaxial line center conductors were silverplated copper. (c) Image at the center plane of the active part of the antennas for the 5.4 cm
radius outer cylinder and just the 1.0 cm radius inclusion. In this case the coaxial cable center
conductor was steel.
3.3.4. Multipath Signal Assessment
Multipath signals are problematic for any communication or radar system. They are
especially troublesome for near field systems where there is less opportunity for natural
attenuation from simple physical isolation of the transmitters and receivers [127, 128]. One of
the principle ways we have overcome this issue with our system has been the adoption of a lossy
coupling bath.
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In earlier work, we determined that the tank side walls could be placed as close as 7 cm
from the antennas without signal reflections from the wall interfering with desired transmitting
signals [129]. In addition, we found that we could operate with the tips of the monopole antennas
as close as 2 mm from the top air/liquid interface, while only inducing minor artifacts in the
microwave images [29]. However, our earliest systems outside of the MR used relatively deep
tanks (usually 24 to 30 cm deep), which worked well and did not appear to create unwanted
multipath signals.
As we proceeded to translate this technology to integration with the MR, space was a
premium and we initially reduced the internal tank height towards 8.0 cm with shorter, straight
antenna feedlines shown in Figure 3.5(a). However, the reduced-height configuration induced
multiple unwanted signals to our microwave data acquisition system. After performing a
thorough analysis, we concluded that the offending multipath signals were the result of surface
waves propagating along the outside coaxial feedlines and along the tank base/liquid interface
[130]. Given that the majority of the previous loss derived primarily from the external coaxial
feed path, we determined that it was essential to maintain a minimum feed length of 9.0 cm for
sufficient attenuation. Utilizing straight feedlines, this would have required a minimum internal
tank height of 11.7 cm (including 2.5 cm for the length of the monopole antenna and 2 mm
between the antenna tips and liquid/air interface). However, by precision bending the semi-rigid
coaxial lines into a serpentine shape as shown in Figure 3.5(b), we were able to retain the
required shorter, 8.0 cm height MR-based tank.
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(a)
(b)
Figure 3.5. Monopole antenna array: (a) in the MI shortened tank with a breast phantom (green),
and straight 5.0 cm long feedlines (short), (b) with 9.0 cm long serpentine feedlines (tank wall
removed)
3.3.5. MR RF Gradient Signal Disruption of Microwave Signals
The RF gradient pulses from the MR coils are primarily at the Larmor frequency (128 MHz
for the 3T system) and radiate freely into the MR bore. However, through our own internal
measurements using the same configuration as in Section 3.3.3, except for replacing the custom
data acquisition system with an Agilent E4402B Spectrum Analyzer (SA), we detected quite
high signal levels for the Larmor signal and its harmonics. For these measurements, the signals
were measured after being coupled to the monopole antennas submerged in the 80:20
glycerin:water coupling bath which then transit through the coaxial cables and through the
bulkhead to the SA. In this case the MR-based signals had to propagate through the bath and
couple to the antennas – the coupling bath attenuation decreases rapidly at lower frequencies,
especially as low as the Larmor fundamental frequency. In addition, the coupling efficiency to
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the antenna is suboptimal, especially given that the polarization of the signal is uncertain.
However, this is the same configuration for when the data acquisition system is attached to the
channel cables and is representative of the experimental conditions for both phantom tests and
patient exams.
The power levels measured for the Larmor fundamental and the first six harmonics (128,
255, 383, 510, 638, 766, and 893 MHz, respectively) were nominally 0, – 40, –50, – 67, –72, –
77, and –79 dBm, respectively. Because of the intermittent nature of the pulses, these power
levels were estimates based on visual observations of the SA during continuous operation of the
MR system. The signal strength roll off demonstrates the normal decrease expected for
increasing harmonic numbers. In addition, the wave attenuation increases substantially in the
glycerin bath with frequency, which also adds to the signal degradation. Other losses included
the coaxial cable attenuation as mentioned in Section 3.3.3, which decreased somewhat at lower
frequencies.
Using our custom data acquisition system, Figure 3.6(a) shows plots of the raw measured
signal amplitudes (uncalibrated) at 1300 MHz for 15 receivers collecting data from all 16
transmitters in a homogeneous bath due to the 1 mW (0 dBm) transmitted signals. Most of
measured data exhibit the typical pattern for received signals, i.e. a symmetric parabolic plot.
However, the power levels for the four corrupted sets (transmitters 2, 5, 13, and 16) are typically
20 to 40 dB lower than for the other receivers. Because the MR RF gradients are intermittent and
not synchronized with the microwave data acquisition, the distortions did not affect all data sets.
Had the signal distortion been from simple constructive or destructive addition of one of the
Larmor harmonics with the desired signals, we would have expected positive and negative
distortions around the predicted pattern with the disruptions visible mostly at the lower signal
96
levels. Instead, the distortions are most likely the result of amplifier de-sentization, whereby an
unwanted signal is processed simultaneously with the desired one and has a substantially higher
amplitude, such that it effectively turns off the small signal amplification process for all other
frequencies.
(a) No low pass filters
(b) With two low pass filters
Figure 3.6. Plots of the 1300 MHz amplitudes as a function of receiver number for all 16
transmitting antennas when the imaging tank is operating in the MR bore with MR data being
97
acquired simultaneously for the cases where (a) no low pass filters, and (b) two low pass filters
installed, respectively.
Given that the Larmor fundamental is roughly 0 dBm after being received by an antenna,
after the first stage of 20 dB receiver amplification it would have easily saturated that amplifier
and even more so in the subsequent cascaded element. It was more than apparent that this was
the offending mechanism. After installing two VHF-740 Mini-Circuits high pass filters
(combined attenuation of 122 dB at 128 MHz), the unwanted distortions completely disappeared,
as shown in Figure 3.6(b).
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3.4. Results – 2D
In this section, we evaluate the performance of the MI with the soft prior regularization
relative to no priors in 2D simulations, phantom experiments, and some preliminary patient data
where the boundaries of different regions of interest were known.
3.4.1. Simulation Experiments
2D simulated measurement data was generated by the hybrid boundary element/finite
element approach [63] for different target shapes. The images were reconstructed in 2D using the
update equation (3.6) with no spatial information and the update equation (3.4) with the soft
prior regularization algorithm described in Section 3.2.2.
3.4.1.1. Mesh Resolution and Noise Level
Figure 3.7 illustrates a circular inclusion centered at (x, y) = (0, –3 cm) with a radius of 1.4
cm and dielectric properties of
εr,Inc = 51.16 and σInc = 1.44 S/m embedded in a background
medium with dielectric properties of εr,BK = 15.60 and σBK = 0.90 S/m.
Figure 3.8 shows different parameter reconstruction meshes used for this experiment.
Noise ranging from -110 dBm to -80 dBm was added to the synthetic measurements. Figure 3.9
shows reconstructed images at 1300 MHz (noise level of -100 dBm) using the no priors (on 473,
961, and 915 node meshes) and soft prior (on the 915 node mesh) regularizations, respectively.
99
Figure 3.7. Schematic of the imaging domains evaluated. The background diameter was 14 cm
(antennas are positioned on a 15.2 cm diameter). The circular inclusion’s center with radius r =
1.40 cm was located at (0, –3 cm).
(a)
(b)
(c)
Figure 3.8. Reconstruction parameter meshes (a) uniformly distributed 473 node mesh, (b)
uniformly distributed 961 node mesh, and (c) 915 node mesh with preferential node deployment
in the inclusion in Figure 3.7
100
(a)
(b)
(c)
(d)
Figure 3.9. Simulated 1300 MHz reconstructed permittivity (top) and conductivity (bottom)
images (a) without priors on thee 473 node mesh, (b) without priors on the 961 node mesh, (c)
without priors on the 915 node mesh with preferential node deployment in the inclusion, and (d)
with the soft prior regularization
rization on the 915 node mesh
mesh.
Without spatial priors, the reconstructed images are very similar, which indicates that
increasing the number of nodes or changing their distribution to be preferentially greater within
the inclusion in the reconstruction mesh does not improve the quality of the recovered images.
Both regularizations
ions recovered the inclusion in the permittivity and conductivity images, but
those from the soft prior technique are more accurate in terms of the location of the inclusion and
its property values. In addition, the recovered background permittivity and co
conductivity
nductivity values
are more uniform in the soft prior case. Figure 3.9 confirms that these improvements are not due
to the reconstruction mesh, but to the regularizat
regularization matrix, which incorporates structural priors
into the reconstruction procedure. Therefore, from this point on, only the 473 node mesh will be
101
used for the 2D reconstructions with no priors. The no priors weighted permittivity and
conductivity errors (described in 3.2.4) were 0.172 and 0.128, respectively, while those with the
soft prior regularization were reduced by 83% to 0.028 for εr and by 87% to 0.016 for σ.
In order to compare the reconstructed and the true dielectric properties, vertical transects of
the permittivity and conductivity profiles along the y-axis are shown in Figure 3.10 for added
noise levels of -110, -100, -90, and -80 dBm.
(a) –110 dBm
(b) –100 dBm
102
(c) –90 dBm
(d) –80 dBm
Figure 3.10. Comparison of the 1300 MHz reconstructed permittivity (top) and conductivity
(bottom) values using the no priors and soft prior regularization with different levels of added
noise: (a) –110, (b) –100, (c) –90, (d) –80 dBm
As expected, artifacts increased in both the permittivity and conductivity images without
priors as the noise level rose, especially in the -80 dBm conductivity images where the
fluctuations are significant. The soft prior regularization tolerates the added noise significantly
better with relatively minor decreases in the recovered inclusion permittivity and only slightly
greater reductions in conductivity at the highest noise level. Using the soft priors, the
reconstructed permittivity values were underestimated (~10-15%) in the inclusion at all noise
levels. Notwithstanding, the method successfully characterized the inclusion given the large
103
property contrast with the background. In addition, even when considerable noise was added to
the measured data (-90 dBm), the algorithm with the soft prior regularization recovered the
inclusion conductivity very accurately, and started to underestimate (~15%) its property values
only at higher noise levels (-80 dBm). Based on the results shown above, -100 dBm noise is the
level where we can see a degradation effect on the reconstructed images. In practice, the noise
floor of our system is closer to -135 dBm, and as a result, we generally have a very large signal to
noise ratio (SNR) even for the weakest signals transmitted from the opposing antennas [31].
3.4.1.2. Estimation of the Parameter Coupling
In order to study the effects of the no prior and the soft prior regularization on
reconstruction parameter coupling, two simulation experiments with the same setup and the same
background medium (εr, BK = 15.60 and σBK = 0.90 S/m) as the previous experiment (Figure 3.7)
were performed. In the first case, no permittivity contrast existed in the inclusion region
(dielectric properties were
εr,
Inc
= 15.60 and
σInc = 1.44 S/m). The second case had no
conductivity contrast in the inclusion (dielectric properties were
εr, Inc = 51.16 and σInc = 0.90
S/m). Figure 3.11 shows transects of the 1300 MHz reconstructed permittivity (top) and
conductivity (bottom) profiles along the y-axis for the first and second experiments, using the
473 and 915 node meshes, respectively (with -100 dBm of added noise).
104
(a)
(b)
Figure 3.11. Comparison of 1300 MHz reconstructed permittivity (top row) and conductivity
(bottom row) profiles for the no priors and the soft prior regularization: (a) no permittivity
contrast (left column), (b) no conductivity contrast (right column).
While both methods (no priors and the soft priors) handle the no-permittivity contrast case
(a) effectively, the soft prior regularization is clearly superior when no conductivity contrast
exists in the inclusion (b). The soft prior regularization profile almost overlaps the exact solution,
whereas the no priors curve has significant differences with the exact solution. Quantitatively,
the permittivity – conductivity parameter coupling is only about 5% with the soft prior
regularization in both cases in Figure 3.11.
105
3.4.1.3. Arbitrarily-Shaped Target
An arbitrarily-shaped target inclusion, as shown in Figure 3.12(a), with dielectric properties
of
εr,
Inc
= 40.0 and
properties of
σInc = 1.30 S/m was embedded in a coupling medium with dielectric
εr, BK = 15.60 and σBK = 0.90 S/m. Based on the exact location of the target, the
customized mesh in Figure 3.12(b) was generated and used for the soft prior regularization
algorithm.
(a)
(b)
Figure 3.12. (a) Schematic of the imaging domains evaluated. The arbitrarily shaped inclusion
was located in the upper part of the imaging domain. The reconstructed values of the simulation
experiment with arbitrarily shaped inclusion were extracted at 30 points evenly distributed along
the line x = –2 cm. (b) Customized soft prior reconstruction mesh comprised of 1725 nodes and
3248 triangular elements.
106
Figure 3.13 shows the 1300 MHz reconstructed images with -100 dBm added noise both
with and without priors, on the 1725 ((Figure 3.12(b))) and 473 node meshes, respectively.
(a)
(b)
Figure 3.13. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) images from a
phantom
tom experiment with arbitrarily
arbitrarily-shaped target inclusion for (a) no priorss on the 473 node
mesh, and (b) soft priors on the customized 1725 node mesh in Figure 3.12(b)
While the no prior permittivit
permittivity image shows the target at roughly the correct location, the
recovered target is considerably shifted towards the top part of the counterpart conductivity
image in Figure 3.13(a),
(a), and the level of artifacts is significantly higher. In contrast, the
reconstructed soft prior images not only recover the complex shape of the target and its exact
location, but they also appear to recover the dielectric property distributio
distributions
ns more accurately.
accurately In
order to confirm this observation, transects of the reconstructed
onstructed permittivity and conductivity
profiles in Figure 3.13 were extracted along the line x = -2 cm,, and they are plotted along with
the exact solution in Figure 3.14(a)
(a) and (b), respectively
respectively.
107
(a)
(b)
Figure 3.14. Comparison of the 1300 MHz reconstructed (a) permittivity and (b) conductivity
profiles (along the line x = –2 cm
cm) in Figure 3.13 phantom
tom experiment with arbitrarily-shaped
arbitrarily
target inclusion for the no priors and soft prior regularization
The transected profiles in Figure 3.14 confirm that the soft prior recovered property
distributions are more accurate. In fact, tthe no priors weighted permittivity and conductivity
errors (described in 3.2.4) were 0.
0.398 and 0.411,, respectively, while those with the soft prior
regularization were
ere reduced by 94% to 0.022 for εr and by 87% to 0.051 for σ.
3.4.1.4. Multiple Inclusions and Number of R
Regions
In order to study the effects of the number of inclusions, as well as the number of nodes
and regions in the reconstruction mesh on the recovered property values when the soft prior
regularization is used, three simulation experiments with one and three inclusions were
performed. In all cases, -100 dBm noise was added to data.
a) Single Inclusion: In the first experiment, a single inclusion of 1.5 cm radius centered at
(x, y) = (3, 0.0 cm) with dielectric properties of εr, I1 = 40.0 and σI1 = 2.0 S/m was placed in a 7
108
cm radius circular region (mimicking an 80:20 mixture of glycerin:water background medium)
located at the center of the imaging domain (x = y = 0) with dielectric properties of εr, BK = 22.4
and σBK = 1.23 S/m. Two customized reconstruction meshes with 402 and 1203 nodes, as shown
in Figure 3.15(a) and (b), respectively, were then used to reconstruct the dielectric property
values of the inclusion and the background region at 1300 MHz.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.15. Customized soft prior reconstruction meshes used in 1 and 3–inclusion simulation
experiments: (a) 402 node mesh with one inclusion and two distinct regions (RBK and RI1), (b)
1203 node mesh with one inclusion and two distinct regions (RBK and RI1), (c) 407 node mesh
with three inclusions and two distinct regions (RBK and RI1), (d) 1203 node mesh with three
109
inclusions and two distinct regions (RBK and RI1), (e) 407 node mesh with three inclusions and
four distinct regions (RBK, RI1, RI2, and RI3), and (f) 1203 node mesh with three inclusions and
four distinct regions (RBK, RInc1, RInc2, and RInc3)
Since the recovered properties within each region were very similar, the maximum
reconstructed values for both meshes along with the exact properties in each region were selected
and are displayed in Figure 3.16..
(a)
(b)
Figure 3.16. Soft prior reconstructed (a) permittivity and (b) conductivity values using two
meshes with different number of nodes, along with the exact properties of background and
inclusion at 1300 MHz
Both permittivity and conductivity values of the background medium are reconstructed
reco
very closely to the exact properties. Moreover, the inclusion is successfully characterized with
both meshes. However, the recovered property values of the target-inclusion
inclusion, especially the
permittivity values, are closer to exact
exact, when the finer mesh (1203 nodes) is used for
reconstruction.
b) Multiple Inclusions with Identical Dielectric Properties:
Similar to the first
experiment, in the second experiment a circular background region with a radius of 7 cm was
110
placed at the center of the imaging domain. In this case, three target inclusions (I1, I2, and I3)
with the same radius of 1.5 cm were centered at (x, y)I1 = (3, 0 cm), (x, y)I2 = (–2, 3 cm), and (x,
y)I3 = (–2, –3 cm), respectively. The dielectric properties of all inclusions were set to εr, I1 = εr, I2
=
εr, I3 = 40.0 and σI1 = σI2 = σI3 = 2.0 S/m; however, two different cases were considered for
the reconstruction:
b1) All reconstruction nodes in the three inclusions were assigned the same region number
(i.e. RI1 = RI2 = RI3), and as a result, only two distinct regions were considered: one for the
background (RBK) and one for all inclusions (RI). In this case, two customized meshes with
different numbers of nodes (407 and 1203 nodes) were used for the reconstruction, as shown in
Figure 3.15(c) and (d), respectively.
b2) The reconstruction nodes in each inclusion were assigned a different region number,
assuming different inclusions and permitting their property distributions to be recovered
independent of each other. In this case, four distinct regions were considered for the
reconstruction: one for the background (RBK) and three for inclusions (RI1, RI2, and RI3). Similar
to (b2), two customized meshes with different number of nodes (407 and 1203 nodes) but with
four distinct regions were used for the reconstruction, as shown in Figure 3.15(e) and (f),
respectively.
Figure 3.17 shows the maximum soft prior reconstructed (a) permittivity and (b)
conductivity values of each region, along with their exact property values at 1300 MHz, using the
customized reconstruction meshes in Figure 3.15(c) – (f)
111
(a)
(b)
Figure 3.17. Maximum soft
oft prior recovered (a) permittivity and (b) conductivity values along
with the exact properties of the background and target inclusions at 1300 MHz using the
customized reconstruction
struction meshes in Figure 3.15(c) – (f)
Similar to the results of the first experiment with only one inclusion, both permittivity and
conductivity values of the background region are recovered very accurately, and all inclusions
are successfully characterized.. When two-region meshes (Figure 3.15(c)
(c) and (d))
(d) are used, the
recovered values of all targets are the same and they are all very close to the exact property
distributions. However, when four
four-region meshes (Figure 3.15(e) and (f))) are used, some
variations appear between the recove
recovered properties of inclusions, though all inclusions were
112
assigned the same property values. In these cases, the deviation from the exact solution is more
prominent in the recovered permittivity values. A complete summary of the relative dielectric
property errors using different reconstruction meshes with two and four distinct regions is
presented in Table 3.1. The negative sign indicates that the reconstructed property value is
underestimated (i.e. less than the exact property value).
Table 3.1. Relative (a) permittivity and (b) conductivity errors for the multiple inclusions
simulation experiment using different reconstruction meshes with two and four distinct regions
Rel. Err.
Rel. Err.
Rel. Err.
Rel. Err.
εr, BK
εr, I1
εr, I2
εr, I3
Mesh 407 – 2r
Mesh 407 – 4r
Mesh 1203 – 2r
-0.4%
-0.4%
0.0%
-11.5%
-10.8%
-7.5%
-11.5%
-11.3%
-7.5%
-11.5%
-12.0%
-7.5%
Mesh 1203 – 4r
0.0%
-7.5%
-8.0%
-8.2%
Rel. Err.
Rel. Err.
Rel. Err.
Rel. Err.
σ BK
σ I1
σ I2
σ I3
0.0%
0.0%
0.8%
0.0%
-2.0%
-5.0%
1.0%
-2.5%
-2.0%
-4.0%
1.0%
-0.5%
-2.0%
2.0%
1.0%
5.0%
Reconstruction Mesh
(a)
Reconstruction Mesh
Mesh 407 – 2r
Mesh 407 – 4r
Mesh 1203 – 2r
Mesh 1203 – 4r
(b)
When a finer soft prior mesh (1203 node) is used, the reconstructed properties are overall
closer to the exact solution, and both relative permittivity and conductivity errors decrease in all
regions. Moreover, restricting variations between different regions that have the same properties
(i.e. using meshes with two distinct regions instead of four) can improve the accuracy of the soft
prior reconstructed dielectric properties.
113
c) Multiple Inclusions with Different Dielectric Properties: This experiment was
identical to the second experiment in (b), except for the property values of the target inclusions.
In this case, dielectric properties of
S/m, and
εr, I1 = 27.0 and σI1 = 1.45 S/m, εr, I2 = 35.0 and σI2 = 1.66
εr, I3 = 40.0 and σI3 = 2.0 S/m were assigned to I1, I2, and I3, respectively. Since
dielectric properties of the inclusions were different, only customized meshes with four distinct
regions (BK, I1, I2, and I3) shown in Figure 3.15(e) and (f) were used for the soft prior
reconstructions. Figure 3.18 shows the maximum soft prior recovered dielectric property values
of all regions along with the exact property values at 1300 MHz.
114
(a)
(b)
Figure 3.18. Maximum soft
oft prior recovered (a) permittivity and (b) conductivity values along
with the exact properties of the background and target inclusions at 1300 MHz using the
customized reconstruction meshes in Figure 3.15(e) and (f)
Similar to the results of the first and second experiments, both permittivity and
conductivity values of the background region are recovered very accurately, and all target
inclusions are successfully characterized
characterized. Moreover, the correct pattern of property increase from
I1 to I3 can be seen in the soft prior reconstructed permittivity and conductivity profiles. A
complete error analysis of the recovered soft prior property values is presented in Table 3.2. The
negative
egative sign indicates that the reconstructed pproperty
roperty value is underestimated (i.e. less than the
exact property value).
115
Table 3.2. Relative (a) permittivity and (b) conductivity errors for the multiple inclusion
simulation experiment with different property contrasts using the soft prior reconstruction
meshes with four distinct regions in Figure 3.15(e) and (f)
Reconstruction Mesh
Mesh 407 – 4r
Mesh 1203 – 4r
Rel. Err. Rel. Err. Rel. Err. Rel. Err.
εr, BK
εr, I1
εr, I2
εr, I3
-0.9%
-0.4%
-3.7%
-3.0%
-8.3%
-5.4%
-12.5%
-7.8%
(a)
Rel. Err.
Rel. Err.
σ BK
σ I1
σ I2
σ I3
Mesh 407 – 4r
1.6%
0.0%
-3.6%
-4.0%
Mesh 1203 – 4r
1.6%
0.0%
-1.8%
-1.0%
Reconstruction Mesh
Rel. Err. Rel. Err.
(b)
Similar to the results of cases (a) and (b), when a finer soft prior mesh (1203 node) is used,
the relative permittivity and conductivity errors decrease. This effect is more prominent on larger
relative errors as the dielectric property contrast between the background medium and target
inclusions increases from I1 to I3.
In the first and second experiments in cases (a) and (b), the true dielectric properties of I1
centered at (x, y)I1 = (– 3, 0 cm) were identical and equal to
εr, I1 = 40.0 and σ I1 = 2.0 S/m.
Therefore, in order to study the effects of additional regions of interest (in this case I2 and I3) on
the recovered dielectric properties of I1, the maximum recovered soft prior property values of I1
in the first and second simulation experiments (on 1203 node mesh with one and three target
inclusions shown in Figure 3.15(b) and (f), respectively) are plotted in Figure 3.19.
116
(a)
(b)
Figure 3.19. Comparison between the soft prior reconstructed (a) permittivity and (b)
conductivity values of I1 with and without additional targets (three inclusions and one inclusion
cases, respectively)
The difference in recovered dielectric property values in Figure 3.19 is minimal (2.75%
(2.75 in
permittivity and 2.5% in conductivity
conductivity), which suggests that additional targets do not have a
major effect on the 2D reconstructed soft prior dielectric property profiles.
3.4.1.5. Sensitivity to a False Region
egion of Interest
The behavior of the soft prior regularization in regions which are identified prior to the MI
property estimation, but do not actually have contrast, is of interest because of the potential for
an approach to identifying false inclusions based solely on the presumed structural information.
In order to study how the dielectric property contrast between the background and
a target zone
would affect the sensitivity of the soft prior regularization algorithm to a false region of interest,
three simulation experiments were performed. Dielectric properties of the background coupling
medium and the actual target inclusion in th
thee experiments were set to those of the 86:14, 40:60,
55:45, and 70:30 mixtures of glycerin and water, respectively. The corresponding property
values are notes in Table 3.3.
117
Table 3.3. Dielectric properties of different mixtures of glycerin and water used for the
simulation experiments with a false region of interest
Glycerin:water Mixture
(case)
Relative Permittivity
Conductivity (S/m)
40:60 Inc – Case (a)
55:45 Inc – Case (b)
70:30 Inc – Case (c)
86:14 BK – (all cases)
62.6
51.2
40
15.6
1.12
1.44
1.64
0.90
Figure 3.20 shows the customized reconstruction mesh with a false inclusion region (in
red) of 1.4 cm radius centered at (0, 3 cm) along with the actual circular target zone (in green)
centered at (0, – 3 cm) with a radius of 1.4 cm.
118
Figure 3.20. Customized reconstruction mesh comprised of 1196 nodes and 2215 triangular
elements, accounting for two inclusion regions: a false circular region of interest (in red) with
radius of 1.4 cm centered at (0, 3 cm), and the actual circular target zone (in green) with radius of
1.4 cm centered at (0, -3 cm)
In all three cases ((a), (b), and (c) in Table 3.3, -100 dBm of noise was added to the
synthetic measurements and images were reconstructed at 1300 MHz using the soft prior
regularization algorithm. The corresponding transect plots of the reconstructed permittivity (top)
and conductivity (bottom) along the y-axis are shown in Figure 3.21.
119
(a)
(b)
(c)
120
Figure 3.21. Transect plots of the 1300 MHz reconstructed soft prior permittivity (top) and
conductivity (bottom) profiles along the y-axis for three simulation cases with a false region of
interest. Dielectric properties of the actual inclusion were set to those of the (a) 40:60, (b) 55:45
and (c) 70:30 mixtures of glycerin and water, reported in Table 3.3
The false region appears as a weak decrease in both permittivity and conductivity images
when the contrast between the true target-inclusion and the background medium (86:14 mixture
of glycerin and water) is higher, as in case (a) 40:60 glycerin:water inclusion. The presence of
the false region declines as the concentration of glycerin and water in the actual target-inclusion
gets closer to that of the background medium (86% glycerin). In order to quantify the effect of
inclusion property contrast on the sensitivity to the false region of interest, the relative errors
between the maximum soft prior reconstructed properties and the actual values are plotted for all
three cases in Figure 3.22.
8%
% False Inclusion
Permittivity Error
6%
4%
% False Inclusion
Conductivity Error
2%
0%
40% glyrecin
55% glycerin
70% glycerin
Figure 3.22. Relative errors between the maximum soft prior recovered properties and the actual
values in Figure 3.21
In both permittivity and conductivity images, the relative error in the false inclusion region
is always under 7%. In fact, this error decreases as the concentration of glycerin and water in the
actual target inclusion gets closer to that of the background medium. In general, conductivity
121
images are less affected by the presence of the false prior structural information. In particular,
when dielectric properties of the inclusion are set to those of 55% and 70% glycerin, the relative
errors of the reconstructed conductivity values in the false inclusion region with respect to the
true background conductivity values are very close to zero. These results show that the soft prior
regularization algorithm can effectively handle misleading situations where an incorrect region
of interest is identified. The reconstructed dielectric property distributions in an erroneously
identified region of interest are clearly different from those in the true region of interest, and they
tend to be very close to those of the background medium.
3.4.1.6. Simulation Based On the MR Images of a Normal Patient
In order to study the behavior of the soft prior regularization approach in a more realistic
scenario where prior clinical structural information of a subject is available, an MR image of a
patient’s breast, as illustrated in Figure 3.23(a), was used to create several breast-shaped
simulation experiments. The same MR image was also used to generate a two-region conformal
soft prior mesh in Figure 3.23(b), composed of 1465 nodes and 2706 triangular elements.
122
(a)
(b)
Figure 3.23. (a) Patient’s breast MR image (b) The customized soft prior mesh composed of
1465 nodes and 2706 triangular elements containing only internal structure of the breast –
adipose (orange) and fibroglandular (blue) tissue, in (a). Reconstructed dielectric property values
were extracted along the dark blue line across the breast and fibroglandular tissues.
Four sets of dielectric property values were assigned to the identified region of interest
(fibroglandular region), resulting in four different breast simulation experiments. These values,
along with the dielectric properties of the breast and the background medium at 1300 MHz are
presented in Table 3.4.
123
Table 3.4. Dielectric property values of the background medium, breast, and fibroglandular
region, in four breast simulation experiments, cases (a) – (d), at 1300 MHz
εr
σ
Background
Medium
All cases
Breast
All
Cases
Region of
interest
Case (a)
Region of
interest
Case (b)
Region of
interest
Case (c)
Region of
interest
Case (d)
15.3
0.98
11.0
0.90
27.0
1.14
35.0
1.45
40.0
1.66
48.0
2.00
Figure 3.24(a) – (d) show the 1300 MHz reconstructed permittivity (top) and conductivity
(bottom) images of the breast-shaped simulation experiments with -100 dBm added noise using
the soft priors (left) and no priors (right) for cases (a) – (d) (as described in Table 3.4),
respectively. The soft prior images were reconstructed on the two-region conformal mesh in
Figure 3.23(b), whereas the no prior images were reconstructed on a uniform 1193 node mesh
within the imaging domain (containing the breast and part of the background medium around it.)
(a)
(b)
124
(c)
(d)
Figure 3.24. Simulated 1300 MHz reconstructed permittivity (top) and conductivity (bottom)
images with -100 dBm added noise, using the soft priors (left) and no priors (right) for cases (a) –
(d) in Table 3.4
Comparing the two regularization techniques in terms of finding the correct shape and
location of the region of interest, the soft prior regularization technique is significantly superior
to the no priors, mainly due to the fact that the prior structural information is used in the soft
prior algorithm. In the no prior images on the right, the shape of the region of interest is different
in the reconstructed permittivity and conductivity images. More specifically, the ‘C’ shape
appeared in the conductivity images (especially when the dielectric property contrast is higher, as
in case (c) and case (d)) and is closer to the exact shape of the fibroglandular region. In
accordance with the true property contrast increase from cases (a) to (d), both the soft prior and
no prior recovered property distributions increase from cases (a) to (d). In order to analyze the
results more quantitatively, transects of the reconstructed permittivity (top) and conductivity
125
(bottom) profiles in Figure 2.24 were extracted along an arbitrary line (across the breast and
fibroglandular region, as shown in Figure 3.23(b)) and are plotted in Figure 3.25.
(a)
(b)
126
(c)
(d)
Figure 3.25. Comparison of the 1300 MHz recovered permittivity (top) and conductivity
(bottom) values along an arbitrary line (illustrated in Figure 3.23(b)) in Figure 2.24: (a) – (d)
correspond to different target values in case (a) – (d) in Table 3.4
While the no prior images show the fibroglandular region at a roughly correct location,
both permittivity and conductivity components of the soft prior images contain the complex
shape and exact location of this region. In terms of the recovered dielectric property
distributions, the soft prior permittivity values nearly overlap the exact solution in all 4 cases (a)
– (b), while the conductivity counterparts are underestimated in the breast region and
overestimated in the fibroglandular region. The soft prior conductivity mismatch in the
fibroglandular region becomes more prominent as the dielectric property contrast between
regions increases from case (a) toward case (d). In particular, the relative soft prior conductivity
127
error in the fibroglandular region increases from 4% in case (a) by nearly a factor of 8 to 31% in
case (d).
Gauss–Newton image reconstruction in microwave imaging is formulated in terms of the
complex wave number squared (k2), from which the relative permittivity and conductivity values
2
are extracted. The magnitude of the average real and imaginary components of k can be
significantly out of balance depending on the operating frequency and material characteristics of
the object being imaged, and as a result, the reconstructed property distribution values can be
imbalanced toward one parameter (relative permittivity or conductivity) or the other. Therefore,
in order to balance the property recovery, a pre-scaling procedure is required at the property
update stage of the reconstruction [131]. Since the soft prior recovered conductivity values in
Figure 3.25 are significantly overestimated, the soft prior images were re-reconstructed with
exactly the same reconstruction parameters, but with a 10 times smaller scaling factor applied to
the conductivity values at the nodes inside the fibroglandular region. Figure 3.26 shows the
transects of the new reconstructed permittivity (top) and conductivity (bottom) profiles with the
original and new scaling factor along the same arbitrary line in Figure 3.25 using the soft prior
regularization.
128
(a)
(b)
(c)
(d)
129
Figure 3.26. Comparison of the 1300 MHz soft prior reconstructed permittivity (top) and
conductivity (bottom) values with the standard (Stnd w) and new (w*0.1) scaling factor along the
same arbitrary line shown in Figure 3.23(b): (a) – (d) correspond to different target values in case
(a) – case (d) in Table 3.4.
Since the scaling factor applied to the permittivity values was kept the same, the
reconstructed permittivity values in Figure 3.25 and Figure 3.26 are very similar. On the
contrary, due to the using of a smaller scaling factor, the conductivity mismatch in Figure 3.26 is
significantly reduced. In fact, the relative soft prior conductivity errors in the fibroglandular
region in cases (a), (b), (c), and (d) decrease to 4%, 6%, 9%, and 15%, respectively. A complete
summary of the relative dielectric property errors in all four cases with two different scaling
factors is shown in Table 3.5. The negative sign indicates that the reconstructed property value is
underestimated (i.e. less than the exact property values).
Table 3.5. Soft prior relative dielectric property errors in the fibroglandular region, breast-shaped
simulation experiments with two different scaling factors
Case (a) Case (b) Case (c) Case (d)
Relative Error εr, Default Scaling Factor
1%
-0.3%
-0.5%
-1%
Relative Error εr, Smaller Scaling Factor
1%
-1%
-2%
-3%
Relative Error σ, Default Scaling Factor
4%
17%
27%
31%
Relative Error σ, Smaller Scaling factor
4%
6%
9%
15%
130
3.4.2. Phantom Experiments
In order to study the behavior of the soft prior regularization with experimental data,
several phantom experiments with different geometries and setups are performed at multiple
frequencies. The results are presented in the following sections.
3.4.2.1. No priors Versus the Soft prior Regularization
The geometry used in this study was the same as the simulations with the circular inclusion
described in 3.4.1.1. Specifically, a 1.4 cm radius, thin-walled plastic cylinder filled with a
mixture of 55% glycerin and 45% water was offset by 3 cm along the y-axis in the background
medium (86:14 glycerin:water mixture).
Figure 3.27 shows the 1300 MHz reconstructed images using the no priors and soft prior
regularization.
131
(a)
(b)
Figure 3.27. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) images from a
phantom experiment with a single circular inclusion for (a) no priors on the 473 node mesh and
(b) soft priors on the 915 node mesh in Figure 3.8
In both instances, the inclusion is evident. Several background artifacts appear in the no
prior images, which are more prominent in the conductivity parameter. Incorporating the spatial
priors substantially improves the quality of both the permittivity and conductivity images.
Weighted permittivity and conductivity errors decrease from 0.329 and 0.302 to 0.015 and 0.045,
respectively, when the spatial structure of the phantom is incorporated through the soft prior
regularization. Figure 3.28 shows the extracted reconstructed dielectric properties along the yaxis from the two approaches relative to the exact values.
132
Figure 3.28. Comparison of the 1300 MHz reconstructed permittivity (left) and conductivity
(right) profiles extracted along the y-axis in Figure 3.26 phantom experiment using the no priors
and soft prior regularizations.
Clearly, the soft prior regularization dramatically reduces the spatial oscillations within the
background, but it also recovers the dielectric properties in the inclusion more accurately.
Improvements are even more prominent in the conductivity images in which case the recovered
inclusion values are over-estimated and displaced (toward the boundary) without priors. These
artifacts are eliminated when the spatial structure of the phantom is incorporated through the soft
prior regularization.
3.4.2.2. Choice of the Soft Prior Coefficient
In all previous results, the soft prior coefficient, λ in equation 3.4, was set to unity as a
default, but in this section, a more detailed study of the effects of λ as a function of frequency is
presented. Data from the same experimental setup described in 3.4.2.1 was used and images were
reconstructed at 900, 1100, 1300, 1500, 1700, 1900, 2100, 2300, and 2500 MHz. The
133
independently measured dielectric properties of the coupling medium and the inclusion are
reported in Table 3.6 as a function of frequency.
Table 3.6. Independently measured dielectric properties of the background medium (BK) and
target inclusion (Inc) over the range of frequencies evaluated
Frequency
(MHz)
εr, BK
σBK
εr, Inc
σInc
900
1100
1300
1500
1700
1900
19.3
17.3
15.6
14.4
13.6
12.8
0.72
0.82
0.90
0.97
1.03
1.10
56.1
53.5
51.2
48.9
46.4
44.5
0.80
1.12
1.44
1.77
2.09
2.42
2100
2300
2500
12.1
11.5
11.1
1.14
1.20
1.24
42.3
40.4
38.8
2.74
3.04
3.35
Based on testing over a wide range of values in simulation and phantom experiments, a
spectrum of the soft prior coefficients (λ = 0.01, 0.1, 1, 10, and 100) was used for the
reconstruction. Lower values, such as 0.001 for λ, allowed the solution to diverge in some cases,
while higher values tended to suppress the recovered inclusion properties.
Transects along the y-axis of the reconstructed images for
εr and σ at 900, 1100, 1300,
1500, 1700, 1900, 2100, 2300, and 2500 MHz for the range of soft prior coefficients are shown
in Figure 3.29.
134
(a) 900 MHz
(b) 1100 MHz
(c) 1300 MHz
135
(d) 1500 MHz
(e) 1700 MHz
(f) 1900 MHz
136
(g) 2100 MHz
(h) 2300 MHz
(i) 2500 MHz
137
Figure 3.29. Comparison of the reconstructed permittivity (left), and conductivity (right) values
from a phantom experiment using different spatial prior coefficient λ values at (a) 900, (b) 1100,
(c) 1300, (d) 1500, (e) 1700, (f) 1900, (g) 2100, (h) 2300, and (i) 2500 MHz
In general, at lower frequencies up to 1700 MHz (e), no significant difference occurs
between the reconstructed values when λ = 0.1, λ = 1.0 and λ = 10. When λ = 0.01, the recovered
permittivity values in the inclusion region are close to the exact values, while the conductivity
counterparts are slightly underestimated. In addition, the recovered properties of the background
region begin to deviate from the true levels as the frequency increases. At lower frequencies, the
λ = 100 reconstructions appear to estimate the inclusion conductivity values closer to the true
levels than the corresponding permittivity values, which are noticeably underestimated across all
frequencies. This increasing mismatch may be due to the excessive smoothing of the soft prior
coefficient.
In order to analyze the results quantitatively, the weighted permittivity and conductivity
errors associated with each image were computed and are plotted as a function of the soft prior
coefficient λ in Figure 3.30.
138
(a)
(b)
Figure 3.30. Weighted (a) εr and (b) σ errors for a phantom experiment over a range of
frequencies from 900 to 2500 MHz using six different soft prior coefficients: λ = 0.01, 0.1, 1, 10,
100, and 1000
139
The weighted property errors for λ = 100 and λ = 1000 are elevated at all frequencies, when
compared to the λ = 1 and λ = 10 cases. Similar behavior is observed for λ = 0.01 and λ = 0.1, but
reducing the soft prior weighting coefficient λ appears to be more sensitive at higher frequencies,
specifically over 1700 MHz. These results suggest that values in the range of λ=1.0 and λ=10
appear to be appropriate soft prior weighting coefficients over our reconstruction frequency
range (usually from 1100 to 1700 MHz). However, the stability of the weighting parameter λ
investigated here is based on the present set of experiments, and may need to be adjusted in other
configurations.
3.4.2.3. Size of the Target
Three thin-walled plastic cylinders of 0.65, 1.4, and 2.1 cm radii filled with a mixture of
50% glycerin and 50% water were centered at (x, y)Inc = (0, 3) cm in a background medium
comprised of 80:20 glycerin:water mixture. Microwave data was then collected at a frequency
range from 900 to 1700 MHz in steps of 200 MHz using our experimental imaging Metrikos
System (Metrikos, Inc., Hopkinton, MA), and the images were reconstructed using the no priors
on the 473 node mesh in Figure 3.8(a), and the soft prior regularization on the customized
reconstruction meshes illustrated in Figure 3.31.
140
(a)
(b)
(c)
Figure 3.31. Customized soft prior reconstruction meshes used for the phantom experiments with
a circular inclusion of radius: (a) 0.65 cm – 1329 nodes and 2530 triangular elements, (b) 1.4 cm
– 1037 nodes and 1943 triangular elements, and (c) 2.1 cm – 813 nodes and 1498 triangular
elements
Figure 3.32 shows transects of the reconstructed permittivity (left) and conductivity (right)
images along the y-axis at a frequency range from 900 to 1700 MHz in steps of 200 MHz, using
the no priors (LM) and soft prior (SP) along with the exact properties of the inclusion.
(a) 900 MHz
141
(b) 1100 MHz
(c) 1300 MHz
(d) 1500 MHz
142
(e) 1700 MHz
Figure 3.32. Comparison of the reconstructed permittivity (left), and conductivity (right) values
from three phantom experiments with different sized inclusions (of radius 0.65, 1.4, and 2.1 cm)
using the no prior (LM) and soft prior (SP) regularizations at (a) 900, (b) 1100, (c) 1300, (d)
1500, and (e) 1700 MHz
Clearly, the soft prior regularization not only reduces the spatial oscillations within the
background in all cases and at all frequencies, but also recovers the dielectric properties in the
smallest inclusion (of 0.65 cm radius) much more accurately. The soft prior improvements are
even more prominent in the conductivity images at higher frequencies in which case the
recovered inclusion values are over-estimated, and in some cases the inclusion is displaced
(toward the boundary) without priors. These artifacts are eliminated when the spatial structure of
the phantom is incorporated through the soft prior regularization. Except for the last frequency
(1700 MHz), the recovered soft prior permittivity values of the mid-sized inclusion (of 1.4 cm
radius) are generally closer to the exact solution, while the most accurate reconstructed soft prior
conductivity values appear in the smallest inclusion.
143
3.4.2.4. Shape of the Target
Three different base shaped thin-walled plastic tubes including a 1.3 cm diameter circleshaped, a 1.35 cm side length square-shaped and a 1.35 cm side length diamond-shaped cylinders
were filled with a mixture of 50% glycerin and 50% water, and then were centered at (x, y)Inc =
(0, 3) cm in a background medium comprised of 80:20 glycerin:water mixture. Similar to the
previous section, microwave data was collected at a frequency range from 900 to 1700 MHz in
steps of 200 MHz using the Metrikos System, and the images were reconstructed with no priors
on the 473 node mesh in Figure 3.8(a) and with the soft prior regularization on the customized
reconstruction meshes illustrated in Figure 3.33.
(a)
(b)
(c)
Figure 3.33. Customized soft prior reconstruction meshes used for the phantom experiments
with: (a) circular inclusion – 1329 nodes and 2530 triangular elements, (b) square-shaped
inclusion – 1096 nodes and 2096 triangular elements, and (c) diamond-shaped inclusion – 1040
nodes and 1984 triangular elements
144
Figure 3.34 shows transects of the reconstructed permittivity (left) and conductivity (right)
images along the y-axis at a frequency range from 900 to 1700 MHz in steps of 200 MHz, using
the no priors (LM) and soft prior (SP) along with the exact properties of the inclusion.
(a) 900 MHz
(b) 1100 MHz
145
(c) 1300 MHz
(d) 1500 MHz
(e) 1700 MHz
146
Figure 3.34. Comparison of the reconstructed permittivity (left), and conductivity (right) values
from a phantom experiment with three different shaped but similar sized inclusions (1.3 cm
diameter circular, 1.3 cm side length square, and 1.3 cm side length diamond-shaped cylinders)
using the no prior (LM) and soft prior (SP) regularizations at (a) 900, (b) 1100, (c) 1300, (d)
1500, and (e) 1700 MHz
Similar to the previous phantom experiment results presented in 3.4.2.3, when the soft prior
regularization is used, the spatial oscillations within the background are significantly damped in
all cases and at all frequencies. Overall, the no prior (LM) reconstructed dielectric properties are
very similar in all three inclusions, which indicates that the Levenberg-Marquardt regularization
is not very sensitive to the shape of the region of interest. Alternatively, the soft prior
regularization appears to respond quite differently to different inclusion shapes. In particular,
over all frequencies the recovered dielectric properties in the circular inclusion are more accurate
than those in the square and diamond-shaped inclusions in which case the recovered soft prior
permittivity values are underestimated, whereas their conductivity counterparts are overestimated. This behavior may be due to the sharp corners in the square and diamond-shaped
cylinders.
3.4.2.5. Dielectric Property Contrast of the Target
In order to study the effects of dielectric property contrast between the target and
background medium on the soft prior reconstructed property distributions, five phantom
experiments with a relatively small target-inclusion were performed. The schematic setups in all
cases were alike: A circular thin-walled plastic tube of 0.65 cm radius filled with different
mixtures of glycerin and water (50:50, 60:40, 70:30, 90:10, and 100:0 glycerin:water solutions)
147
was centered at (x, y)Inc = (0, 3) cm in a background medium comprised of 80:20 glycerin:water
mixture. Microwave data was collected at a frequency range from 900 to 1700 MHz in steps of
200 MHz using the Metrikos System, and the soft prior images were reconstructed on the
customized 1329 node mesh, illustrated in Figure 3.33(a).
The maximum recovered soft prior permittivity (left) and conductivity (right) of the five
target-inclusions with different contrast-levels against the independently measured properties
(i.e. the exact values) are plotted in Figure 3.35. For analytical purposes, the linear least-square
fits, along with the linear regression slopes and their corresponding R2 values were calculated
and they are shown in Figure 3.35.
(a) 900 MHz
148
(b) 1100 MHz
(c) 1300 MHz
(d) 1500 MHz
149
(e) 1700 MHz
Figure 3.35. Comparison of the soft prior reconstructed permittivity (left) and conductivity
(right) values of five different contrast-level target-inclusions (50:50, 60:40, 70:30, 90:10 and
100:0 mixture of glycerin:water) at (a) 900, (b) 1100, (c) 1300, (d) 1500, and (d) 1700 MHz,
along with linear least-square fits
In Figure 3.35, concentration of glycerin (%g) in the target-inclusion is labeled next to each
data point. Two metrics are considered in this study: the slope of the linear least-square fit shows
the pattern of the soft prior reconstructed properties at different contrast-levels, whereas the R2
value is a measure of goodness-of-fit. In an ideal case, the linear fit would have both slope and
R2 value of unity – i.e. all data points would fall on the line y = x. Data points above y = x
indicate that the recovered dielectric properties are overestimated (reconstructed value > exact
value), whereas those underneath represent underestimated property values. In general, the
permittivity R2 values across all frequencies and the conductivity R2 values at 1100, 1300 and
1500 MHz are very close to unity (> 0.93), which confirms a strong linear correlation between
the soft prior reconstructed property values of different contrast-level inclusions. In addition, the
slopes of the linear fits at 1100, 1300 and 1500 MHz are also close to unity (1), indicating the
150
correct pattern of the soft prior reconstructed properties at different contrast-levels. Overall, the
soft prior recovered properties of the inclusions are more degraded at the lowest and the highest
frequencies (900 and 1700 MHz). In fact, the mismatch between the reconstructed and actual
conductivity values is significantly more prominent. Nonetheless, it should be noted that the
range of actual conductivity values of different mixture of glycerin and water used in this
experiment is constricted at 900 MHz (varying only from 0.33 to 0.88 S/m).
3.4.2.6. Number of the Reconstruction Nodes
In order to study how the soft prior recovered dielectric properties are affected by the
number of nodes in the customized reconstruction mesh, measurement data from the singletarget phantom experiment described in 3.4.2.1 was used to reconstruct soft prior images at 1300
MHz, using a set of customized meshes with different number of nodes in each region. In this
analysis, three cases were considered: (a) increasing number of nodes in the target-inclusion
region (NInc), while keeping number of nodes in the background region (NBK) in the same bulk
part, (b) increasing number of nodes in the background region (NBK), while number of nodes in
the target-inclusion region (NInc) stays in the same range, and (c) increasing number of nodes in
both regions (NBK and NInc) proportionally. Table 3.7 contains all details about the corresponding
soft prior meshes in each case, including the total number of nodes (NTotal), number of nodes on
the perimeter of the background (NΩBK) and target inclusion (NΩInc) regions, maximum area of
triangular elements (maxA) in each region, and number of nodes in each region (N).
151
Table 3.7. Properties of the soft prior customized meshes used for the phantom experiment with
a single target inclusion of 1.4 cm radius
Case (a) more nodes in the target-inclusion region
NTotal
255
NΩBK
30
maxABK
0.5
NΩInc
6
maxAInc
3
NBK
247
NInc
8
251
269
288
337
30
30
30
30
0.5
0.5
0.5
0.5
10
15
20
25
0.5
0.3
0.1
0.05
235
243
233
233
16
26
55
104
754
30
0.5
25
0.01
267
487
1304
1955
5224
30
0.5
25
0.005
315
30
0.5
25
0.003
327
30
0.5
25
0.001
410
Case (b) more nodes in the background region
989
1628
4814
NTotal
405
1159
2303
NΩBK
30
40
50
maxABK
0.3
0.1
0.05
NΩInc
10
10
15
maxAInc
0.5
0.5
0.3
NBK
389
1137
2267
11327
60
NTotal
251
NΩBK
30
maxABK
0.5
NΩInc
10
maxAInc
0.5
NBK
235
NInc
16
414
1204
2366
11686
30
40
50
60
0.3
0.1
0.05
0.01
15
25
30
30
0.3
0.1
0.05
0.01
388
1141
2260
11167
26
63
106
519
0.01
15
0.3
11238
Case (c) more nodes in both regions
NInc
16
22
36
89
For each of the three cases above, the maximum soft prior recovered property values in the
inclusion region were extracted, and they are plotted as a function of the number of
reconstruction nodes (in logarithmic scale) in Figure 3.36. It should be noted that since the
152
recovered dielectric properties of the background medium nearly overlap the exact properties in
all cases, only reconstructed values of the target are presented here.
(a) NInc
(b) NBK
153
(c) NTotal
Figure 3.36. Comparison of the soft prior reconstructed permittivity (left), and conductivity
(right) values of a target inclusion using several soft prior customized meshes with different node
densities: case (a) number of nodes in the inclusion region (NInc) changes, case (b) number of
nodes in the background region (NBK) changes, and case (c) number of nodes in both regions
(NTotal) changes proportionally
In all three cases, the soft prior recovered inclusion conductivity values (plots on the right)
appear to be less sensitive to the number of reconstruction nodes. In fact, all reconstructed
conductivity values are within roughly 6% of the exact values. Nonetheless, the optimal
inclusion conductivity value is reached when approximately 25-100 nodes are placed in the
inclusion region. That is when the 269 node mesh is used in case (a) (i.e. 3rd point from the left in
Figure 3.36(a)), the 1159 node mesh is used in case (b) (i.e. 2nd point from the left in Figure
3.36(b)), and the 414 node mesh is used in case (c) (i.e. 2nd point from the left in Figure 3.36(c)).
On the other hand, changing the number of reconstruction nodes has a more significant
effect on the recovered inclusion permittivity values. More specifically, as the number of nodes
increases, generally the soft prior permittivity values of the target inclusion are recovered more
154
accurately in all three cases (as much as %20). This observation suggests that the denser
reconstruction mesh can improve the accuracy of the soft prior reconstructed permittivity values.
However, no obvious reason is found for the slight decrease in the permittivity values of the 4th
data point in Figure 3.36(a).
Intuitively, when additional nodes are inserted in the non-target region (here, the
background) as in case (b), the recovered dielectric properties of the target inclusion should not
change. However, the reconstructed inclusion permittivity values in Figure 3.36(b) are improved
as the number of nodes in the background region increases. This may be due to the fact that the
number of nodes in the inclusion region also increases (although at a much lower rate). In effect,
as shown in Table 3.7, the number of inclusion nodes increases from 16 to 89 as the number of
background nodes rises from 389 to 11238. It would be desirable to keep the number of inclusion
nodes constant, but in order to comply with certain mesh quality criteria, additional nodes ought
to be inserted in the inclusion region as the number of background nodes was increased.
3.4.2.7. Sensitivity to a False Region of Interest
In order to study the sensitivity of the soft prior algorithm to false regions of interest in
measurement data, we used the data generated in the phantom experiment in 3.4.2.1, where only
a single inclusion at the lower location (0, –3 cm) actually existed. The customized
reconstruction mesh in Figure 3.20 containing a false inclusion region of 1.4 cm radius centered
at (0, 3 cm), and the actual target zone was used to reconstruct the soft prior images at 1300
MHz.
Figure 3.37 shows the 1300 MHz reconstructed images using the soft prior regularization
coefficients of λ = 0.01, λ = 0.1, λ = 1.0 and λ = 10, respectively. The false region appears as a
155
weak increase (~ 3 to 6%) in the permittivity images, but with a more prominent decrease (~ –
4% to – 20%) in the conductivity images.
(a) λ = 0.01
(b) λ = 0.1
(c) λ = 1.0
(d) λ = 10
Figure 3.37. 1300 MHz soft prior reconstructed permittivity (top) and conductivity (bottom)
images of a phantom experiment using the reconstruction mesh with a false region of interest in
Figure 3.20
Transect plots along the y-axis in the reconstructed permittivity and conductivity images
are presented in Figure 3.38.
156
Figure 3.38. Comparison of the 1300 MHz soft prior reconstructed permittivity (left), and
conductivity (right) profiles from a phantom experiment using the reconstruction mesh with a
false region of interest in Figure 3.20
Consistent with the images in Figure 3.37, the permittivity values within the false inclusion
region in Figure 3.38 are close to the true background, while the counterpart conductivity
profiles are more noticeably affected by the misleading information and they exhibit lower
properties than those of the background medium, especially when larger values of λ are used.
Since the contribution from the soft prior regularization is reduced when λ = 0.01, the dielectric
properties are not significantly influenced by the presence of the false inclusion region (~ 3% and
~ – 4% error in the permittivity and conductivity images, respectively). Nonetheless, for smaller
values of λ, more artifacts are observed in the background region. However, as the soft prior
weighting coefficient λ increases to 1.0 and 10, the level of the error increases in the false region
of interest (~ 6% and ~ – 20% in the permittivity and conductivity images, respectively). A
summary of the weighted soft prior property errors is presented in Table 3.8.
157
Table 3.8. Weighted soft prior εr and σ errors for a phantom experiment using a reconstruction
mesh with a false region of interest for different soft prior coefficients: λ = 0.01, 0.1, and 1.0 at
1300 MHz
Frequency Errw, ε r
(MHz)
λ =0.01
1300
0.232
Errw, σ
λ =0.01
Errw, ε r
λ =0.1
Errw, σ
λ =0.1
Errw, ε r
λ =1
Errw, σ
λ =1
Errw, ε r
λ =10
Errw, σ
λ =10
0.246
0.143
0.159
0.131
0.162
0.127
0.157
For λ = 1, Errw, ε r = 0.131 and Errw, σ = 0.162, which are larger relative to the case when the
exact spatial structure of the phantom was used in Section 3.4.2.1 ( Errw, ε r = 0.015 and Errw, σ =
0.045), but significantly lower than those obtained without priors ( Errw, ε r = 0.329 and Errw, σ =
0.302).
3.4.2.8. Sensitivity to Imperfect Spatial Priors
Obtaining perfect a priori structural information of the tissue being imaged may not be
feasible in practice, and therefore, evaluating the sensitivity of the soft prior technique to
imperfect priors is essential. In this section, two types of priors’ imperfections, namely the size
and location of the target, are studied:
a) Imperfect prior size of the target inclusion: Table 3.9 presents some attributes of a set of
different inclusion-sized soft prior meshes that was used to reconstruct the same data generated
in the phantom experiment described in 3.4.2.1, with a target inclusion of 1.4 cm radius centered
at (0, –3 cm).
158
Table 3.9. Characteristics of the soft prior reconstruction meshes used for a phantom experiment
with imperfect prior size of the target inclusion
Inclusion Location
Actual Inclusion Prior Inclusion
Number of
(In the actual
Total
Size
Size (In the
Nodes in the
experiment and in
Number of
(In the experiment)
mesh)
Inclusion
the mesh)
Nodes
Radius (cm)
Radius (cm)
Region
Center (cm)
(0, –3 cm)
(0, –3 cm)
(0, –3 cm)
(0, –3 cm)
(0, –3 cm)
(0, –3 cm)
(0, –3 cm)
1.4
1.4
1.4
1.4
1.4
1.4
1.4
0.8
1.0
1.2
1.4
1.6
1.8
2.0
873
1054
969
1036
978
954
874
217
288
288
318
318
326
318
Figure 3.39 shows the 1100 MHz reconstructed images using different soft prior meshes in
Table 3.9.
(a) 0.8
(b) 1.0
(c) 1.2
(d) 1.4
(e) 1.6
(f) 1.8
(g) 2.0 cm
Figure 3.39. 1100 MHz reconstructed permittivity (top) and conductivity (bottom) images of a
phantom experiment using different prior inclusion-sized meshes: (a) 0.8 cm, (b) 1.0 cm, (c) 1.2
cm, (d) 1.4 cm (the true size), (e) 1.6 cm, (f) 1.8 cm, and (g) 2.0 cm were considered as the radius
159
of the inclusion in the reconstruction mesh. The actual size (radius of 1.4 cm) and location (0, – 3
cm) of the inclusion is outlined by a circle on the reconstructed images.
As the inclusion radius in the reconstruction mesh increases from Figure 3.39(a) – (g) the
recovered inclusion permittivity values decrease, while the counterpart conductivity values
increase. In order to compare these values with the actual property distributions, the
reconstructed permittivity and conductivity values of the background and target inclusion in
Figure 3.39 were extracted and they are plotted in Figure 3.40 as a function of the inclusion
radius in the reconstruction mesh.
Figure 3.40. Comparison of the extracted 1100 MHz reconstructed permittivity (left), and
conductivity (right) values in Figure 3.39 along with the exact property distributions for a
phantom experiment using different inclusion–sized soft prior meshes
Consistent with the reconstructed images in Figure 3.39, as the radius of the inclusion in
the soft prior mesh increases from 0.8 to 2 cm, the inclusion recovered permittivity values in
Figure 3.40 decrease, while the counterpart conductivity values increase. In addition, the
recovered and exact background values overlap very closely. As expected, when the exact size of
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the inclusion is used in the reconstruction mesh, the most accurate inclusion permittivity value is
obtained. Using prior structural information with smaller–sized inclusions misleads the soft prior
algorithm to overestimate the inclusion permittivity values, while using larger–sized inclusions
results in underestimation of properties. However, imperfect prior information of the size of the
inclusion has a very different effect on the recovered conductivity values. As the inclusion radius
in the soft prior mesh increases, the recovered conductivity values approach the exact solution,
and as a result, the most accurate values are not obtained with the exact–sized target inclusion.
At 1100 MHz, there is about 30% change in recovered permittivity and 27% change in recovered
conductivity values when approximately 78% change is applied to the area of the target region
(changing the inclusion radius from 1.2 to 1.6 cm).
In order to analyze the sensitivity to imperfect size of the target at other operating
frequencies, the reconstructed permittivity and conductivity values of the background and target–
inclusion at a frequency range from 1300 to 1700 MHz in steps of 200 MHz were extracted and
plotted versus the inclusion radius in the reconstruction mesh in Figure 3.41.
(a) 1300 MHz
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(b) 1500 MHz
(c) 1700 MHz
Figure 3.41. Comparison of the extracted (a) 1300, (b) 1500, and (c) 1700 MHz reconstructed
permittivity (left), and conductivity (right) profiles along with the exact property distributions for
a phantom experiment using different inclusion–sized meshes
Overall, at higher frequencies in Figure 3.41 the same pattern as the 1100 MHz
reconstructed property values in Figure 3.40 is seen: As the inclusion radius in the reconstruction
mesh increases from 0.8 to 2 cm, the recovered inclusion permittivity values decrease, while the
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counterpart conductivity values increase. In terms of the relative property variations, when
approximately 78% change is applied to the area of the target region, the recovered permittivity
values at 1300, 1500 and 1700 MHz vary about 28%, 30%, and 28% respectively, while the
counterpart conductivity changes decrease from 13% at 1300 MHz to only 3% at 1700 MHz.
b) Imperfect prior location of the target inclusion: Table 3.10 presents some attributes of a
set of different inclusion–located soft prior meshes that was used to reconstruct the same data
generated in the phantom experiment described in 3.4.2.1, with a target inclusion of 1.4 cm
radius centered at (0, – 3 cm).
Table 3.10. Characteristics of the soft prior customized meshes used for a phantom experiment
with imperfect prior location of the target inclusion
Inclusion Size
Inclusion Location Prior Inclusion
Number of
(In the actual
Total
(In the
Location
Nodes in the
experiment and in
Number of
experiment)
(In the mesh)
Inclusion
the mesh)
Nodes
Center (x, y) cm Center (x, y) cm
Region
Radius (cm)
1.4
1.4
(0, –3)
(0, –3)
(0, –4.5)
(0, –4.2)
882
895
288
288
1.4
1.4
1.4
1.4
1.4
(0, –3)
(0, –3)
(0, –3)
(0, –3)
(0, –3)
(0, –3.9)
(0, –3.6)
(0, –3.3)
(0, –3)
(0, –2.7)
897
906
914
912
911
288
288
288
288
288
1.4
1.4
1.4
(0, –3)
(0, –3)
(0, –3)
(0, –2.4)
(0, –2.1)
(0, –1.8)
909
903
904
288
288
288
1.4
1.4
1.4
1.4
1.4
(0, –3)
(0, –3)
(0, –3)
(0, –3)
(0, –3)
(0, –1.5)
(0, –1.2)
(0, –0.9)
(0, –0.6)
(0, –0.3)
894
913
924
899
920
288
288
288
280
288
163
1.4
1.4
(0, –3)
(0, –3)
(0, 0)
(0, 0.3)
899
916
288
288
1.4
1.4
1.4
1.4
(0, –3)
(0, –3)
(0, –3)
(0, –3)
(0, 0.6)
(0, 0.9)
(0, 1.2)
(0, 1.5)
899
909
916
869
288
288
288
280
Figure 3.42 shows the 1100 MHz reconstructed images using some of the soft prior meshes
in Table 3.10.
(a) -3.9
(b) -3.0
(c) -2.1
(d) -1.2
(e) -0.3
(f) 0.6
(g) 1.5 cm
Figure 3.42. 1100 MHz reconstructed permittivity (top) and conductivity (bottom) images of a
phantom experiment using different prior inclusion–located meshes: x = 0 and y–coordinates of
(a) – 3.9, (b) – 3.0 (the true location), (c) – 2.1, (d) – 1.2, (e) – 0.3, (f) 0.6, and (g) 1.5 cm were
considered as the center of the inclusion in the reconstruction mesh. The actual size (radius of 1.4
cm) and location (0, – 3 cm) of the inclusion is outlined by a circle on the reconstructed images.
As the inclusion in the reconstruction mesh moves away from its actual location in the
phantom experiment setup (black circle in Figure 3.42), the recovered permittivity profiles
decrease toward the background, while the corresponding conductivity profiles slightly increase
and then decrease lower than the background conductivity values.
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In order to analyze the results more quantitatively, the reconstructed permittivity and
conductivity values of the background and target inclusion in Figure 3.42 were extracted along
the y–axis and they are plotted in Figure 3.43(a) versus the inclusion center in the reconstruction
mesh. The same set of property values reconstructed at 1300, 1500, and 1700 MHz are presented
in Figure 3.43(b), (c), and (d), respectively.
(a) 1100 MHz
(b) 1300 MHz
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(c) 1500 MHz
(d) 1700 MHz
Figure 3.43. Comparison of the extracted (a) 1100, (b) 1300, (c) 1500, and (d) 1700 MHz
reconstructed permittivity (left), and conductivity (right) values along with the exact property
distributions for a phantom experiment using different inclusion–located meshes
Overall, when the prior and actual target locations match, the reconstructed inclusion
permittivity values approach the exact profiles. However, the most accurate values are not
recovered when the mismatch is zero, but when the inclusion is offset along the y–axis about 0.3
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cm. In fact, this can be due to the slight misplacement of the actual inclusion tube during the
phantom experiment. When the partial overlap between the actual and prior target area
decreases, the recovered inclusion permittivity profiles approach the background values. The
imperfect prior location of the target seems to have a dissimilar effect on the recovered
conductivity profiles. In general, as the center of the inclusion in the reconstruction mesh shifts
down from its actual location, the reconstructed inclusion conductivity values are overestimated.
Similar behavior is seen when the prior center of the inclusion moves up from y = – 3 cm to
about y = – 1.5 cm. The conductivity profiles then start to decrease significantly to the extent that
they even go below the background values.
In terms of the recovered background properties, the most accurate values at all frequencies
are obtained around the exact location of the inclusion. As the mismatch between the prior and
actual target locations increases – i.e. the y–coordinate of the inclusion center in the
reconstruction mesh moves away from – 3 cm, both reconstructed background permittivity and
conductivity values increase, which may be an indication of averaged property distribution
between the background and actual target regions.
The results presented in this section verified that the soft prior algorithm is not overly
sensitive to the exact size and location of the target region and it can tolerate priors’ imperfection
to some extent.
3.4.2.9. Breast–Shaped Phantom Experiments
In order to study the behavior of the soft prior regularization technique in more practical
circumstances, several phantom experiments with different sized target inclusions were
performed. In order to conduct real breast–shaped experiments, MR scans of a real breast of cup
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size B (425 ml) was used to create a breast mold, as shown in Figure 3.44(a). This mold was then
used to make the rapid–prototyped plastic breast model in Figure 3.44(b).
(a)
(b)
Figure 3.44. Real breast–shaped model: (a) breast mold, (b) rapid–prototyped breast model
The plastic used in the model was chosen to be thin enough so that the microwave signals
would not be significantly distorted as they propagated through it. Two cylinders – of 0.65 and
1.0 cm radius – were inserted into the breast model as tumor inclusions. The dielectric properties
of the breast were chosen to mimic those of the scattered breast by using a 88:12 mixture of
glycerin and water, whereas the inclusions were set to be similar to a tumor with a mixture of
50:50 glycerin:water. The breast phantom was also submerged in a coupling liquid composed of
a 80:20 mixture of glycerin and water. The 1300 MHz relative permittivity and conductivity
values of the actual phantom components are presented in Table 3.11.
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Table 3.11. Independently measured dielectric properties of the background medium (BK), breast
(BR), and inclusions (Inc) at 1300 MHz
Frequency
(MHz)
εr, BK
σBK
εr, BR
σBR
εr, Inc
σInc
1300
15.3
0.98
13.0
0.83
58.6
1.02
The required a priori structural information of the phantom setup for the soft prior
regularization was obtained before microwave data acquisition by using our 3D optical scanner
shown in Figure 3.45 (VIUscan color laser scanner, CreaForm Inc. Quebec, Canada).
Figure 3.45. 3D optical scanning of the model
First, the breast model as well as the two tumor inclusions was scanned by the laser
scanner. Next, the resulting 3D surface mesh, illustrated in Figure 3.46(a), was imported into the
SolidWorks (SolidWorks Corp., Concord, MA) software package, where the solid object in
Figure 3.46(b) was formed and 2D slices were extracted.
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(a)
(b)
Figure 3.46. Exported 3D model (a) 3D surface mesh, (b) Formation of a solid object and
extraction of 2D slices in the SolidWorks software package
Finally, different 2D customized meshes containing the exact location of the tumor
inclusions with respect to the breast region were obtained and used in the reconstruction
algorithm with soft prior regularization. The soft prior meshes used in the present experiment are
illustrated in Figure 3.47.
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(a)
(b)
Figure 3.47. 2D customized soft prior meshes used in a breast–shaped phantom experiment with
circular target inclusions of (a) 0.65 cm and (b) 1.0 cm radius, as well as a polyline along which
the actual and reconstructed dielectric property distributions were extracted. The mesh in (a) is
comprised of 1778 nodes and 3468 triangular elements, whereas the one in (b) contains 1788
nodes and 3488 triangular elements.
Figure 3.48 shows transects of the 1300 MHz reconstructed permittivity (top) and
conductivity (bottom) profiles along the arbitrary polyline across the imaging domain in Figure
3.47, using the no priors (on 473 node mesh) and soft prior regularization (on the 1778 and 1788
node meshes).
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(a)
(b)
Figure 3.48. Comparison of the 1300 MHz reconstructed permittivity (left) and conductivity
(right) profiles along an arbitrary polyline for the breast–shaped phantom experiments with (a)
small inclusion of 0.65 cm radius and (b) medium–sized inclusion of 1 cm radius, using the no
priors and soft prior regularization
The tumor inclusion is detected both with and without using prior structure of the phantom
setup in (a) and (b). Nonetheless, the soft prior images are clearly superior in terms of the
recovered dielectric properties of the different regions. The reconstructed soft prior permittivity
values of the background medium and the breast nearly overlap the exact solution in Figure
3.48(a) and (b). However, the corresponding tumor inclusion values are not as precise as
expected, possibly due to the significant contrast between the breast and tumor inclusion. The
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mismatch between the recovered soft prior properties and the exact permittivity values of the
inclusion deteriorates as its radius decreases from 1 to 0.65 cm.
The recovered soft prior conductivity profiles are overall more accurate than the
permittivity counterparts; however, similar to the permittivity profile, as the size of the target
inclusion becomes smaller, the mismatch between the reconstructed soft prior properties and the
exact conductivity values increases.
3.4.2.10. Phantom Experiment in the MR Bore
Accurate a priori structural information of the object being imaged can be obtained from
different sources. In this section we show the results of combining microwave data with actual
MR images obtained simultaneously in the MR bore, as described in 3.3.3.
In this experiment, the MR system was acquiring data during the microwave acquisition.
The tank was filled with an 80% glycerin solution with dielectric properties of εr, BK = 20.9 and
σBK = 1.22 S/m, respectively. The imaging target was a 5.4 cm radius tube filled with an 84%
solution of glycerin with 2.1 cm and 1.0 cm radius offset cylindrical inclusions having 72% and
55% glycerin:water mixtures. The properties of the three regions were:
1.02 S/m,
εr, Br = 16.1 and σBr =
εr, Fg = 30.2 and σFg = 1.51 S/m, and εr, Tu = 51.2 and σTu = 1.44 S/m, respectively.
While the three regions are nominally meant to mimic the properties of a fatty breast (Br),
fibroglandular tissue (Fg) and tumor (Tu), in actuality they only achieve this for the permittivity.
Because of the relaxation behavior of the glycerin:water solutions, their conductivity values in
the 1300 MHz region are only fractionally different. However, for the purposes of testing the
robustness of the hardware and software under challenging, high contrast situations, this
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configuration is more than adequate. Figure 3.49 shows the corresponding MR T2–weighted
image of the horizontal plane through the target and the center of the active part of the antennas.
Figure 3.49. T2–weighted MR images of the phantom experiment at the center plane of the
active part of the antennas
The location of the target and associated inclusions with respect to the antenna positions is
clearly evident. Figure 3.50(a) shows the no prior reconstruction mesh comprised of 559
uniformly distributed nodes and 1044 triangular elements, while Figure 3.50(b) shows the soft
prior reconstruction mesh containing 1034 nodes and 1821 elements discretized from the MR
image. (Note that the MR image and mesh are horizontal mirror images of each other since the
MR system produces images viewed from top down while the microwave system is configured
for en face view images – i.e. from underneath – during breast patient exams.)
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(a)
(b)
Figure 3.50. 2D reconstruction meshes used for the phantom experiment in MR: (a) No priors,
with 559 uniformly distributed nodes and 1044 triangular elements, and (b) soft priors, with 1034
nodes and 1821 triangular elements
Figure 3.51(a) and (b) show the recovered 1300 MHz permittivity (top) and conductivity
(bottom) images of the target using the no priors and the soft prior regularization exploiting the
MR–based prior information.
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(a)
(b)
Figure 3.51. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) images of the
multi–region
region phantom case using the (a) no priors (circles indicate the exact
ct object locations)
loca
and
(b) soft prior regularization.
In Figure 3.51(a), the breast region and both target inclusions are visible in the permittivity
and conductivity images, but with a fair number of artifacts within the breast phantom and the
surrounding bath. Circles have been drawn over the conventional images to indicate the exact
locations of the objects. Overall, tthe soft prior images in Figure 3.51(b)
(b) appear to recover the
property distributions more accurately
accurately,, especially the permittivity profile of the tumor inclusion.
inclusion
In order to evaluate whether using tthe hard priors would improve the characterization of the
tumor inclusion, the MR–based
based prior information was used to reduce the size of the
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reconstruction parameters, as described in 3.2.3. Then, the images were reconstructed using the
hard priors.
The actual and maximum recovered property values of the fibroglandular and tumor
inclusion regions both with and without priors are reported in Table 3.12(a). While the
reconstructed properties with no priors are significantly different from the actual values, the soft
prior and hard prior values are generally closer to the exact properties. In order to quantify the
improvement, the corresponding relative dielectric property errors were calculated and they are
presented in Table 3.12(b). The negative sign indicates that the reconstructed property value is
underestimated (i.e. less than the exact property values).
Table 3.12. (a) The actual and averaged recovered property values of the fibroglandular and
tumor inclusion regions in the phantom experiment in the MR bore using the no priors, soft
priors, and hard priors, (b) The corresponding relative dielectric property errors
εr, Fg
σFg
εr, Tu
σTu
Actual values
30.2
1.51
51.2
1.44
Avg. Reconstructed – No priors
25.1
1.28
29.1
1.25
Avg. Reconstructed – Soft priors
28.6
1.41
47.3
1.27
Avg. Reconstructed – Hard priors
28.7
1.41
50.7
1.12
(a)
εr, Fg
σFg
εr, Tu
σTu
% Relative Error – No priors
-16.9%
-15.2%
-43.2%
-13.2%
% Relative Error – Soft priors
-5.3%
-6.6%
-7.6%
-11.8%
% Relative Error – Hard priors
-5.0%
-6.6%
-1.0%
-22.2%
(b)
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Both soft and hard priors appear to have a similar impact on the recovered property values
of the fibroglandular region (~70% and ~55% less errors in the reconstructed permittivity and
conductivity profiles, respectively). However, they improve the reconstructed properties of the
target inclusion differently. Specifically, the hard priors significantly improve the permittivity
profile of the tumor region (decreasing the error to only 1%), while they have a negative effect
on the counterpart conductivity values (the error increases to 22%). The soft priors, on the other
hand, have a more stable effect on both permittivity and conductivity images, improving them by
42% and 10%, respectively.
3.4.3. Bone Imaging: Monitoring Changes in the Bone Dielectric Property
Distributions
Bone quality assessment is one of the areas where microwave imaging can be used to
detect and monitor dielectric property changes. Bone dielectric properties have been studied
extensively up to 5 MHz [132, 133]. These investigations have demonstrated strong correlations
between the electrical and clinically important mechanical properties of bone. For instance, a
recent study by Peyman et al showed that dramatic changes in bone dielectric properties occur in
vivo with age (not typical in other tissues) and the age–related dynamics of bone physiology may
be evident through interrogation of dielectric properties as a means of bone health monitoring
[134].
In the current study, we will investigate the correlation between dielectric properties in the
microwave frequency range and the bone volume fraction, which is an established measure of
overall bone health. The results are supported by a series of ex–vivo bone imaging, as well as
some initial clinical patient data from a recent pilot study for microwave imaging of the heel
using a simple adaptation of our existing breast imaging system.
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3.4.3.1. Ex–vivo Bone Imaging
We performed a series of ex
ex–vivo imaging on several de–marrowed
marrowed porcine trabecular
bone specimens. Figure 3.52 illustrates a flow diagram used for testing and processing the bone
samples.
Figure 3.52. Experiment flow diagram for evaluating the correlation between bone mineral and
dielectric properties of porcine
rcine trabecular bone specimens
In each cycle, first a desiccator was used to remove the water and moist
moisture
ure from the bone
pores. Then, a micro–CT
CT scan was used to calculate the bone volume fraction (BVF) of each
sample. Next, the specimens were immersed in 18 mm diameter test tubes containing a 0.9%
saline solution to remove any air traces during the microwave measurements. As shown in
Figure 3.53,, the test tubes were then placed at a position 3 cm offset from the center of the
microwave imaging tank filled with a matching liquid comprised of an 80:20 mixture of
glycerin:water solution.. 2D microwave data was acquired at a frequency range from 500 to 2500
MHz at a step size of 200 MHz.. Finally, the bone samples were de
de–calcified
calcified by being put into an
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acid treatment procedure. At the end of each cycle, all samples were rinsed in tap water for about
10 minutes.
Figure 3.53. Microwave data acquisition of the bone sample
Since prior structural information of the bone samples (i.e. size and location of the test
tubes) was available from the X–ray micro–CT, the soft prior regularization technique was used
to obtain higher quality images and recovering more accurate dielectric property distributions.
Figure 3.54 shows the customized soft prior reconstruction mesh used for recovering the
dielectric properties of all bone samples. The mesh consists of 1107 nodes and 2092 triangular
elements.
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Figure 3.54. Soft prior microwave image reconstruction mesh comprised of 1107 nodes and 2092
triangular elements
Figure 3.55 shows the representative 1100 MHz soft prior permittivity (top) and
conductivity (bottom) images of the (a) first, (b) second, and (c) final (5th) microwave scans.
(a)
(b)
(c)
Figure 3.55. The 1100 MHz reconstructed permittivity (top) and conductivity (bottom) images of
the (a) first, (b) second, and (c) final (5th) microwave scans
181
It is interesting to note that while the dry diameters for all samples decreased about 12.2%
due to the acid treatments, when
en the samples were immersed in the saline, they expanded slightly
to snuggly fit up against the test tube walls. The properties of the recovered targets demonstrated
a clear increasing trend from the first to the last imaging session.
In order to evaluate the correlation between microwave dielectric properties and BVF, the
peak recovered permittivity and conductivity values of the saline–saturated bone specimens in
the test tube were extracted and are plotted as a function of bone volumee fraction in Figure
3.56(a) and (b) respectively.
(a)
(b)
Figure 3.56. Relationship between the dielectric properties of saline–saturated
saturated bones at 1100
MHz and BVF of dry bone specimens
In both Figure 3.56(a) and (b)
(b),, as the BVF increases, the dielectric property values decrease
and the corresponding R2 correlation coefficients are 0.351 and 0.544, respectively. In terms of
testing for statistical significance, we applied the Pearson correlation test [135],, which produced
coefficient values of – 0.592 for permitti
permittivity and -0.738
0.738 for conductivity. For the case of N = 30,
absolute values of the Pearson coefficient greater than 0.373 are statistically significant to a p–
p
182
value of 0.05 or better. As reported in Table 3.13, the results from a similar analysis performed
on the 900 and 1300 MHz recovered dielectric properties were also statistically significant.
Table 3.13. Pearson coefficient for the recovered dielectric properties at various frequencies
Frequency
(MHz)
Permittivity
Conductivity
Pearson r–value Pearson r–value
900
1100
– 0.526
– 0.592
– 0.679
– 0.738
1300
– 0.603
– 0.753
These results suggest a significant correlation between the tissue dielectric properties
(especially conductivity) and the bone volume fraction at our microwave frequency range, which
agree with the strong relationship between bone content and the dielectric properties at lower
frequencies, previously found by Sierpowska et al [132].
3.4.3.2. Initial Clinical Heel Imaging
We recruited patients who had required external support of one leg for approximately 4 – 6
weeks. Significant trabecular bone loss can be expected in the heel of a non–weight bearing leg,
especially for the immediate period after cast or support removal. Studies have shown that
trabecular-rich bones such as those in the ankle and calcaneus undergo dramatic changes in
mineralization levels – decreases as much as 35% – when patients reduce weight-bearing for six
weeks or more [136-138]. Figure 3.57 shows the CT cross sections through the calcaneous of
both heels of a patient (#902) who wore a boot on one leg for at least 6 weeks (while using
crutches) and recovered for several weeks (2 – 4) without support prior to imaging. According to
the radiologist, the nominal Hounsfield units for the affected leg were 36.5 while those for the
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normal weight–bearing leg were 49.2, indicating a 25.8% drop in bone density during the non–
or limited weight–bearing time.
(a)
(b)
Figure 3.57. X–ray CT images of the patient’s heels for the (a) normal weight–bearing and (b)
injured leg (White zones within the calcaneous bones are the radiologists’ labels.)
Since prior anatomical structure of the heels was available from X–ray CT images,
customized reconstruction meshes were created (Figure 3.58) and used in the soft prior
regularization technique.
184
(a)
(b)
Figure 3.58. 2D customized reconstruction meshes of the (a) normal weight–bearing and (b)
affected leg comprised of 1196 nodes and 2264 triangular elements, and 815 nodes and 1519
triangular elements, respectively
Figure 3.59(a) and (b) show the 2D recovered soft prior permittivity (top) and conductivity
(bottom) images for the normal (fully weight–bearing) and injured (non– or limited weight–
bearing) heels, respectively, suggesting slightly elevated values for the affected calcaneous. In
this case, the recovered dielectric properties in the normal calcaneous were εr = 13.5 and σ =
0.81 S/m; while the values in the injured calcaneous were εr = 16.7 and σ = 0.92 S/m. Since bone
has significantly lower dielectric properties comparing to the surrounding soft tissue, major
trabecular bone loss in the heel of a non–weight bearing leg (i.e. injured heel) can result in higher
bone dielectric properties. The findings are consistent with expectations in that the permittivity
and conductivity values were about 24% and 14% higher for the injured calcaneous,
respectively.
185
(a)
(b)
Figure 3.59. 1300 MHz soft prior permittivity (top) and conductivity (bottom) images for the heel
of a (a) normal and an (b) injured leg utilizing the CT images as prior information during the
reconstruction process
3.4.4. Clinical Breast Microwave Imaging in MR
In order to assess the performance of our combined MI-MRI system, we imaged a 43 year
old woman who had heterogeneously dense breasts, a body mass index of 25 and did not have
any known cancers. Figure 3.60(a) shows the patient lying prone with her left breast suspended
through an aperture in the top of the illumination tank right before entering the MR bore. Figure
3.60(b) shows the patient inside the bore.
186
(a)
(b)
Figure 3.60. Photograph of a patient in the combined MI-MRI system (a) before entering the MR
bore and (b) inside the bore
For this pilot test case, we took care to choose a woman with fairly low body mass index
while also having modestly sized breasts to allow for sufficient tissue to be pendant down to the
plane of the antennas. In addition, while previously categorized as having dense breasts, we
deliberately reviewed earlier normal patient MR images and selected this subject based on there
being a fair amount of distinct fibroglandular regions which were useful in testing the soft prior
regularization scheme. It should be noted that we took particular care to not inadvertently create
any conductive loops (i.e. from the various feed cables under the patient) that might have caused
localized heating.
Figure 3.61(a) shows the anatomically coronal MR T1-weighted image at the plane of the
active part of the monopole antennas. The antennas are clearly visible and can be accurately
registered with respect to the breast tissue. As discussed in Section 3.3.3, there is no noticeable
distortion in the breast portion of the images due to the proximity of metal objects. After
187
segmenting the adipose and fibroglandular tissues in the MR image, the data was analyzed and
co-registered with the corresponding microwave images. Then, our in-house developed codes
were used to create a customized reconstruction mesh for use in the soft prior regularization
algorithm. The resulting mesh shown in Figure 3.61(a) was comprised of 2473 nodes and 4647
triangular elements. For the fibroglandular tissue, there are three discrete zones (one larger and
two smaller), which we tied together as a single zone for the purposes of the soft prior
reconstruction.
(a)
(b)
Figure 3.61 (a) MR T1-weighted image of the patient’s left breast at the plane of the active part
of the monopole antennas, and (b) the corresponding segmented soft prior mesh
Figure 3.62 shows the 1300 MHz reconstructed permittivity (top) and conductivity
(bottom) images with the no priors and soft prior regularization, respectively. For the former, the
breast perimeter is roughly detected in both permittivity and conductivity (largely based on the
fact that the breast properties are generally lower than those of the background), but with
188
considerable artifacts in and outside of the breast. The soft prior images are essentially
homogeneous for the two zone types in both property images.
(a)
(b)
Figure 3.62. 1300 MHz reconstructed permittivity (top) and conductivity (bottom) images of the
patient’s left breast utilizing the (a) no priors and (b) soft prior regularizations, respectively
While we do not have exact validation of the tissue properties for this patient, the recovered
values are reasonable compared with published ex-vivo values for the associated tissue types
[139]. In this instance, the recovered adipose property values were εr, Adp = 8.8 and
σAdp = 0.09
S/m, while those for the fibroglandular tissue were εr, Fg = 40.7 and σFg = 0.96 S/m. Comparable
published values for these tissue types were: εr, Adp = 4.8 and σAdp = 0.05 S/m; εr, Fg = 39.4 and
189
σFg = 0.75 S/m, respectively (assuming roughly 50% adipose content in the fibroglandular
range) [139].
The patient results from our combined MI-MRI are encouraging, because they demonstrate
that it is possible to collect both MR and microwave data on a patient even in such a confined
space along with the various integration challenges. These first images are promising in that they
demonstrate that the scattered signals contain information about the target that indicates which
subzones have lower and higher properties. This will be particularly valuable when imaging
patients in later studies where this approach may add valuable diagnostic information concerning
the specific nature of suspicious lesions.
190
3.5. Results – 3D
As discussed in Section 3.4, incorporating structural information of the object being
imaged into our 2D image reconstruction algorithm through the soft prior regularization
approach significantly improved the quality of the recovered images, and as a result, a more
accurate reconstruction of dielectric property distributions was obtained. In this section, we will
extend the idea of using prior spatial information with microwave imaging to our 3D image
reconstruction algorithm. In addition to a complete analysis of the behavior of the soft prior
technique in 3D, we will perform a comprehensive comparison between the reconstructed 2D
and 3D images using the prior spatial information.
3.5.1. Simulation Experiments
In order to study the effects of different factors such as the soft prior coefficient,
reconstruction mesh density, imperfect spatial priors, and size, shape, and number of target
inclusions on the 3D soft prior technique, several 3D simulation experiments will be performed.
The schematic setup of the simulation experiments studied in this section is the same as the one
described in Section 2.4.
3.5.1.1. No Priors versus Soft Prior Regularization
Figure 3.63(a) and (b) show the 1300 MHz 3D reconstructed images of the synthetic data
with and without incorporating prior spatial information, respectively. For visualization
purposes, horizontal slices normal to the XY plane at z = 0, along with iso-surfaces of the target
inclusion are presented in Figure 3.63.
191
(a)
(b)
Figure 3.63. 1300 MHz 3D reconstructed permittivity (top) and conductivity (bottom) images of
the simulation experiment with – 100 dBm added noise using (a) the soft prior regularization and
(b) no prior spatial information
While the target inclusion is successfully detected in both cases, the soft prior reconstructed
dielectric properties are much closer to the actual property values in both permittivity and
conductivity images. Moreover, the level of background artifacts is significantly reduced in the
images with incorporated structural information. In order to compare quantitatively the no priors
and soft prior images, the relative root mean square error (RRMSE) for the inclusion region is
calculated as
 Vnrecon −Vnexact 
RRMSE = ∑

exact
V


n
n=1
N
2
N
(3.12)
192
where N is the total number of nodes in the inclusion region, Vn
recon
is the reconstructed
dielectric property value (either permittivity or conductivity) at node n (in the reconstruction
mesh), and Vn
exact
is the true value of the selected dielectric property at that location. The
permittivity and conductivity RMS errors associated with the inclusion region for each image in
Figure 3.63 are summarized in Table 3.14.
Table 3.14. The permittivity and conductivity RMS errors associated with the inclusion region in
Figure 3.63
Soft priors
No priors
Permittivity
RRMS Error
0.8%
22%
Conductivity
RRMS Error
8%
18%
The soft prior regularization significantly improves the recovered dielectric property
distributions, especially the permittivity values. In fact the permittivity RRMS errors are about
27 times smaller when the soft prior regularization is used. This improvement is less significant
in the counterpart conductivity profiles, as the soft prior RRMS errors are only about two times
smaller than that of the no prior case.
3.5.1.2. Number of the Reconstruction Nodes
In order to study how the number of nodes in the customized reconstruction mesh affects
the soft prior recovered dielectric properties, a set of customized meshes with different node
densities was used to reconstruct the synthetic data in Section 2.4 with the soft prior
regularization at 1300 MHz. A list of these meshes (including the number of nodes and elements)
193
is reported in Table 3.15. In order to minimize the effects of other factors, all parameters except
the reconstruction mesh were kept the same in this experiment.
Table 3.15. Customized reconstruction meshes used for the simulation experiment with soft prior
regularization
Case
Number of Nodes
Number of Elements
(a)
(b)
(c)
2180
3083
4509
11033
16257
21330
(d)
7972
40686
In all cases, both permittivity and conductivity profiles of the target inclusion were
effectively characterized, however, they did not appear to be significantly different as the
number of reconstruction nodes increased. In order to analyze this observation quantitatively, the
reconstructed dielectric properties of the inclusion and the background region were extracted at
the center of the inclusion (3, 0, 0 cm) and at the center of the imaging domain (0, 0, 0 cm),
respectively. Figure 3.64 shows the extracted property values along with the exact solutions, as a
function of the number of reconstruction nodes in Table 3.15.
194
Figure 3.64. Extracted 3D recovered permittivity (left) and conductivity (right) values along with
the exact property values, as a function of the number of nodes in the soft prior reconstruction
mesh
Results in Figure 3.64 confirm that the recovered soft prior dielectric properties are not
highly sensitive to the reconstruction mesh density, especially in the permittivity images, as the
recovered properties overlap with the exact values in all four cases. Nonetheless, the recovered
conductivity values improve marginally, as the reconstruction mesh density increases from 2000
to over 4000 nodes, suggesting that for a configuration of this size, it appears to be sufficient to
use a mesh composed of approximately 3000 – 4000 nodes.
3.5.1.3. Choice of the Soft-Prior Coefficient
In the previous 3D soft prior reconstructions, the soft prior coefficient, λ in equation 3.4,
was set to 10 as a default. However, in this section a more detailed study of the effects of λ on
the recovered dielectric properties is presented. A spectrum of the soft prior coefficients (λ =
0.01, 0.1, 1, 10, and 100) was used for reconstructing the synthetic data described in Section 2.4.
Lower values, such as 0.001 for λ, allowed the solution to diverge in some cases, while higher
195
values tended to suppress the recovered inclusion properties. Figure 3.65
3
shows the
corresponding soft prior reconstructed images at 1300 MHz.
(a) λ = 0.01
(b) λ = 0.1
196
(c) λ = 1.0
(d) λ = 10
(e) λ = 100
197
Figure 3.65. 1300 MHz 3D reconstructed permittivity (top) and conductivity (bottom) profiles for
the simulation experiment using different soft prior coefficients λ = (a) 0.01, (b) 0.1, (c) 1.0, (d)
10, and (e) 100
In order to analyze the results quantitatively, the reconstructed dielectric properties of the
inclusion and the background region were extracted at the center of the inclusion (3, 0, 0 cm),
and at the center of the imaging domain (0, 0, 0 cm), respectively. Figure 3.66 shows the
extracted property values along with the exact solutions, as a function of the soft prior coefficient
λ.
Figure 3.66. Extracted 3D recovered permittivity (left) and conductivity (right) values along with
the exact property distributions, as a function of the base – 10 logarithm of the soft prior
coefficient λ
As the soft prior coefficient decreases, less regularization is applied and as a result, the
overall quality of the reconstructed images degrades. This effect is even more prominent in the
conductivity images, where the recovered properties of the target inclusion are significantly
lower than the exact solution, and also closer to those of the coupling medium. On the other end,
higher regularization values may cause excessive smoothing effects on the reconstructed images,
198
and as a result, the recovered properties of the inclusion are underestimated. For the intermediate
values, no observable difference can be seen between the reconstructed values when λ = 1.0 to λ
= 100 are used. When λ = 10, both recovered permittivity and conductivity values in the
inclusion region are the closest to the exact solution, and those of the background match the true
levels. The corresponding permittivity and conductivity RRMS errors as a function of the soft
prior regularization coefficient λ are plotted in Figure 3.67.
Figure 3.67. Relative RMS errors of the recovered dielectric property distributions as a function
of the base – 10 logarithm of the soft prior regularization coefficient λ
The relative RMS error plot confirms that the recovered soft prior permittivity values are in
general more accurate than the counterpart conductivity values. Moreover, the minimum error is
achieved when the soft prior coefficient of λ = 10 is used.
199
3.5.1.4. Sensitivity to Imperfect Spatial Priors
Obtaining perfect a priori structural information of the object being imaged may not be
possible in practice, therefore, it is essential to evaluate the sensitivity of the soft prior technique
to imperfect priors. In this section, we extend the analysis performed in Section 3.4.2.8 to the 3D
soft prior algorithm, and study two types of priors’ imperfections (the size and location of the
target inclusion) in the simulation experiment described in Section 2.4.
a) Imperfect prior size of the target inclusion: Table 3.16 presents some attributes of a set
of different inclusion-sized 3D soft prior meshes that was used to reconstruct the synthetic data
with a spherical target inclusion of 1.5 cm radius centered at (3, 0, 0 cm). In all reconstruction
meshes, the total number of nodes, as well as the number of nodes in the inclusion region, was
kept within approximately the same order of magnitude.
Table 3.16. Characteristics of the 3D soft prior customized meshes used for a simulation
experiment with imperfect prior size of the target inclusion
Inclusion Location
(In the actual
experiment and in the
soft prior mesh)
Center (cm)
Actual
Prior Inclusion Size
Inclusion Size
Total
Number of
(In the soft prior
(In the
number of nodes in the
mesh)
experiment)
nodes
inclusion
Radius (cm)
Radius (cm)
(3, 0, 0 cm)
(3, 0, 0 cm)
(3, 0, 0 cm)
1.5
1.5
1.5
0.7
0.8
0.9
3255
3209
3431
252
284
333
(3, 0, 0 cm)
(3, 0, 0 cm)
(3, 0, 0 cm)
1.5
1.5
1.5
1.0
1.1
1.2
3078
3358
3257
332
149
170
(3, 0, 0 cm)
(3, 0, 0 cm)
(3, 0, 0 cm)
(3, 0, 0 cm)
1.5
1.5
1.5
1.5
1.3
1.4
1.5
1.6
3323
3249
3083
3007
182
198
240
241
200
(3, 0, 0 cm)
(3, 0, 0 cm)
1.5
1.5
1.7
1.8
3693
2808
276
339
(3, 0, 0 cm)
(3, 0, 0 cm)
(3, 0, 0 cm)
(3, 0, 0 cm)
1.5
1.5
1.5
1.5
1.9
2.0
2.1
2.2
2932
2945
3059
3016
358
438
504
546
(3, 0, 0 cm)
(3, 0, 0 cm)
(3, 0, 0 cm)
1.5
1.5
1.5
2.3
2.4
2.5
3213
2898
2815
608
613
627
In order to compare the recovered dielectric properties with the exact solution, the
reconstructed permittivity and conductivity values of the background and the target inclusion
were extracted at the center of each region, and are plotted in Figure 3.68, as a function of the
spherical inclusion radius in the reconstruction mesh (i.e. the radius of the inclusion in the prior
information).
Figure 3.68. Extracted soft prior 3D recovered permittivity (left) and conductivity (right) values
along with the exact property distributions, using different inclusion–sized meshes as prior
information
201
In Figure 3.68, the exact size of the target inclusion is marked with a green vertical line,
which indicates that when the exact prior information is used, the most accurate property
distributions are recovered. More specifically, as the radius of the inclusion in the reconstruction
mesh increases, both reconstructed permittivity and conductivity values of the target inclusion
gradually approach those of the background medium (Bk). On the other end, as the prior size of
the inclusion decreases, the reconstructed conductivity values also approach those of the
background medium, while the permittivity counterparts first increase and then decreased toward
the background values. In all cases, the recovered and exact background values overlap very
closely. In terms of the sensitivity of the algorithm to the imperfect prior size of the target, there
is about a 31% change in the recovered permittivity and 15% change in the recovered
conductivity values, when approximately 126% change is applied to the volume of the target
region (changing the inclusion radius from 1.3 to 1.7 cm). Comparing these numbers with those
reported in 2D soft prior case in Section 3.4.2.8(a), reveals that the 3D soft prior algorithm is less
sensitive to the imperfect prior size of the target.
b) Imperfect prior location of the target inclusion: Table 3.17 presents some attributes of a
set of different inclusion–located soft prior meshes that was used to reconstruct the synthetic data
with a spherical target inclusion of 1.5 cm radius centered at (3, 0, 0 cm). In all reconstruction
meshes, the total number of nodes, as well as the number of nodes in the inclusion region, was
kept approximately the same.
202
Table 3.17. Characteristics of the 3D soft prior customized meshes used for a simulation
experiment with imperfect prior location of the target inclusion
Inclusion Size
Inclusion Location
(In the actual
(In the
experiment and in
experiment)
the soft prior mesh)
Center (cm)
Radius (cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
1.5
(3, 0, 0 cm)
Prior Inclusion
Location
(In the soft prior
mesh)
Center (cm)
(0, 0, 0 cm)
(0.2, 0, 0 cm)
(0.4, 0, 0 cm)
(0.6, 0, 0 cm)
(0.8, 0, 0 cm)
(1.0, 0, 0 cm)
(1.2, 0, 0 cm)
(1.4, 0, 0 cm)
(1.6, 0, 0 cm)
(1.8, 0, 0 cm)
(2.0, 0, 0 cm)
(2.2, 0, 0 cm)
(2.4, 0, 0 cm)
(2.6, 0, 0 cm)
(2.8, 0, 0 cm)
(3.0, 0, 0 cm)
(3.2, 0, 0 cm)
(3.4, 0, 0 cm)
(3.6, 0, 0 cm)
(3.8, 0, 0 cm)
(4.0, 0, 0 cm)
(4.2, 0, 0 cm)
(4.4, 0, 0 cm)
(4.6, 0, 0 cm)
(4.8, 0, 0 cm)
(5.0, 0, 0 cm)
(5.2, 0, 0 cm)
(5.4, 0, 0 cm)
203
Number of
Total
Nodes in
Number of
the
Nodes
Inclusion
Region
2925
226
3041
238
3053
233
3024
241
3036
243
3005
239
3016
239
3087
235
3060
235
3007
241
3120
233
3157
241
3118
233
3094
240
3068
237
3083
240
2936
230
3044
230
2925
231
3046
233
2954
227
3267
235
3274
263
3204
253
3134
267
3047
253
3033
263
2969
260
In order to compare the recovered dielectric properties with the exact solution, the
reconstructed permittivity and conductivity values of the background and the target inclusion
were extracted at the center of each region. They are plotted in Figure 3.69, as a function of the
X-coordinates of the center of the prior spherical inclusion in the reconstruction mesh (i.e. the
center of the inclusion in the prior information).
Figure 3.69. Extracted soft prior 3D recovered permittivity (left) and conductivity (right) values
along with the exact property distributions, using different inclusion–located meshes as prior
information
In Figure 3.69, the exact location of the target inclusion is marked with a green vertical
line. Overall, the imperfect prior location of the target seems to have a similar effect (hill-shaped
curve) on both recovered permittivity and conductivity profiles. When the prior and actual target
locations match, the reconstructed permittivity and conductivity values of the inclusion approach
their exact profiles. However, the most accurate values are not recovered when the mismatch is
zero, but when the prior inclusion is slightly offset along the X–axis. Nonetheless, the offset
directions for the permittivity and conductivity profiles are the opposite of each other, which
makes the recovered values at the actual target location (green line) the most accurate
204
simultaneous solutions for both permittivity and conductivity. When the partial overlap between
the actual and prior target volumes decreases, the recovered property distributions of the
inclusion approach those of the background medium.
In all cases, the recovered and exact background values overlap very closely and only
minor variations occur when imperfect prior information is used. However, the most accurate
values are obtained at the exact prior location of the inclusion. In terms of the sensitivity of the
algorithm to the imperfect prior location of the target, there is only about a 4% change in the
recovered permittivity and 15% change in the recovered conductivity values when the prior
target region is shifted approximately 87% of its radius (i.e. shifting the X-coordinates of the
center of the inclusion from 2.6 to 3.4 cm).
The results presented in this section verify that the soft prior algorithm is not overly
sensitive to the exact size and location of the target region, and that it can tolerate priors’
imperfection to some extent.
3.5.1.5. Sensitivity to a False Region of Interest
In order to study the sensitivity of the 3D soft prior algorithm to false regions of interest,
the synthetic data described in 2.4 was used. In the actual experiment, there was only one
spherical inclusion of 1.5 cm radius centered at (3, 0, 0 cm). However, as a part of prior
information, an additional false region of interest of the same size was identified in the soft prior
reconstruction mesh. The center of the false inclusion was set to 5 random locations. Some
attributes of these meshes are presented in Table 3.18. In all reconstruction meshes, the total
number of nodes, as well as the number of nodes in the true and false inclusion regions, was kept
approximately the same.
205
Table 3.18. Characteristics of the 3D soft prior customized meshes used for a simulation
experiment with a false target inclusion (Bk = background, TI = True Inclusion, and FI = False
Inclusion)
Case
Number
of
Elements
Number
of
Nodes
Number
of Bk
Nodes
Number
of TI
Nodes
Number
of FI
Nodes
TI
Center
(cm)
FI
Center
(cm)
(a)
(b)
(c)
19496
19494
18834
3687
3687
3583
3293
3292
3173
204
205
204
190
190
206
(3, 0, 0)
(3, 0, 0)
(3, 0, 0)
(-1, -1, 0)
(-2, 2, 1.2)
(1, -4.5, -1)
(d)
18353
3490
3081
206
203
(3, 0, 0)
(1, 4.5, -1.2)
(e)
18480
3514
3117
206
191
(3, 0, 0)
(-5, 0, -1)
Figure 3.70(a) – (e) show the 1300 MHz 3D reconstructed images using the corresponding
soft prior meshes with a false region of interest in Table 3.18. For illustration purposes, the
reconstructed images in each case are sliced in such a way that both the true and false inclusion
regions can be seen. Moreover, iso-surface thresholds of εr = 39 and σ = 1.85 S/m are applied to
the reconstructed permittivity and conductivity images, respectively.
206
(a) FI centered at (-1, -1, 0)
(b) FI centered at (-2, 2, 1.2)
(c) FI centered at (1, -4.5,
4.5, -1)
(d) FI centered at (1, 4.5, -1.2)
207
(e) FI centered at (-5, 0, -1)
Figure 3.70. 1300 MHz 3D reconstructed images using the corresponding soft prior meshes with
a false region of interest in Table 3.18
The false region of interest does not appear prominently in any of the images in Figure
3.70. It onlyy presents as a subtle variation from the background properties. In order to study the
sensitivity of the 3D soft prior algorithm to a false region of interest, the recovered dielectric
properties of all 5 cases were extracted at the center of the true and false inclusions, and they are
presented in Table 3.19,, along with their corresponding error values.
208
Table 3.19. Recovered dielectric properties (εr and
σ) of the true (TI) and false inclusion (FI)
extracted at the center of each region
Case
εr, TI
%Error
σTI
%Error
(a)
39.40
2%
1.89
(b)
(c)
(d)
(e)
39.44
39.51
39.32
39.53
1%
1%
2%
1%
1.90
1.89
1.89
1.89
εr, TI
εr, FI
%Error
5%
22.51
5%
5%
6%
6%
22.38
22.48
22.46
22.29
σTI
σFI
%Error
0.5%
1.28
4%
0.1%
0.4%
0.3%
0.5%
1.25
1.26
1.26
1.25
2%
3%
3%
1%
εr, FI
σFI
From Table 3.19, for all cases the false region appears with a very small increase or
decrease (less than 1%) in the permittivity images, but appears with a slightly larger increase
(from 1% to 4%) in the conductivity images. In the true inclusion region, while the permittivity
errors are less than 2%, the counterpart conductivity errors are somewhat larger, but still less
than 6%.
The results in this section show that the 3D soft prior algorithm is able to manage situations
where a false region of interest has been identified in the prior structural information. In such
situations the false region of interest only shows a weak variation of the background properties.
3.5.1.6. Multiple Inclusions and Number of Regions
In order to study the effects of the number of inclusions as well as the number of nodes and
regions in the reconstruction mesh on the recovered property values, when the 3D soft prior
algorithm is used, the simulation experiment described in Section 2.4 was repeated, but this time
there were three spherical inclusions of 1.5 cm radius. The target inclusions I1, I2, and I3 were
centered at (x, y, z)I1 = (3, 0, 0 cm), (x, y, z)I2 = (– 2, 3, 0 cm), and (x, y, z)I3 = (– 2, – 3, 0 cm),
209
respectively. Similar to the study performed on the 2D soft prior algorithm in Section 3.4.1.4, the
following cases were considered:
a) Multiple Inclusions with Identical Dielectric Properties: In the synthetic data, the
dielectric properties of all inclusions were set to εr, I1 = εr, I2 = εr, I3 = 40.0 and σI1 = σI2 = σI3 =
2.0 S/m. The 3D soft prior images were then reconstructed using four different customized
meshes. The first two meshes were composed of 5545 nodes and 27389 tetrahedral elements,
whereas the other two meshes were composed of 2459 nodes and 12656 tetrahedral elements. In
the first and third mesh, all inclusion regions were assigned the same region number (i.e. RI1 =
RI2 = RI3); therefore, only two distinct regions were considered: one for the background (Bk) and
one for all inclusions (I). In the second and forth meshes, each inclusion region was assigned a
different region number (i.e. RI1, RI2, and RI3 had independent properties); therefore, four distinct
regions were considered in this case: one for the background (Bk) and three for each inclusion
(I1, I2, and I3). A summary of the soft prior reconstruction meshes used for this simulation
experiment is presented in Table 3.20.
Table 3.20. Soft prior reconstruction meshes used for the simulation experiment with three
spherical target inclusions
Mesh5545_2r
Mesh5545_4r
Mesh2459_2r
Number of
Nodes
5545
5545
2459
Number of
Elements
27389
27389
12656
Number of
Distinct Regions
2
4
2
Mesh2459_4r
2459
12656
4
3D Mesh ID
210
Figure 3.71 shows the 3D soft prior reconstructed dielectric properties extracted at the
center of each region, along with their exact property values
values, using four customized
reconstruction meshes in Table 33.20.
(a) Permittivity
(b) Conductivity
Figure 3.71. 3D soft prior recovered (a) permittivity and (b) conductivity values extracted at the
center of each region, along with the exact properties
properties, using customized reconstruction meshes in
Table 3.20
Overall, all recovered property distributions are fairly close to the exact solution. When 2region meshes (Mesh5545_2r and Mesh2459_2r)) are used, the recovered values of all targets are
211
the same. On the other hand, when 4-region meshes (Mesh5545_4r and Mesh2459_4r) are used,
some variations appear between the recovered properties of the inclusions, although they were
assigned the same property values. In these cases, the deviation from the exact solution is more
prominent in the recovered conductivity profiles. A complete summary of the relative dielectric
property errors using different reconstruction meshes with two and four distinct regions is shown
in Table 3.21.
Table 3.21. Relative (a) permittivity and (b) conductivity errors for the multi-inclusion
simulation experiment using different reconstruction meshes with two and four distinct regions.
The negative sign indicates that the reconstructed property value is underestimated. (i.e. less
than the exact property values)
Rel. Err.
Rel. Err.
Rel. Err.
Rel. Err.
εr, BK
εr, I1
εr, I2
εr, I3
Mesh5545_2r
Mesh5545_4r
Mesh2459_2r
-0.6%
-0.6%
-0.9%
0.1%
-0.2%
-2.2%
0.1%
-0.2%
-2.2%
0.1%
0.2%
-2.2%
Mesh2459_4r
-0.9%
-1.7%
-2.5%
-3.0%
Rel. Err.
Rel. Err.
Rel. Err.
Rel. Err.
σ BK
σ I1
σ I2
σ I3
2.5%
2.5%
2.4%
2.4%
-2.1%
-4.0%
-4.7%
-6.3%
-2.1%
-2.2%
-4.7%
-4.8%
-2.2%
-0.8%
-4.8%
-4.4%
Reconstruction Mesh
(a)
Reconstruction Mesh
Mesh5545_2r
Mesh5545_4r
Mesh2459_2r
Mesh2459_4r
(b)
When a finer soft prior mesh (5542 nodes) is used, the reconstructed properties are overall
closer to the exact solution and both relative permittivity and conductivity errors decrease in
212
almost all regions. Moreover, it seems that restricting variations between different regions that
have the same properties (i.e. using meshes with two distinct regions instead of four) can
improve the accuracy of the soft prior reconstructed dielectric properties.
b) Multiple Inclusions with Different Dielectric Properties: This case was identical to
the previous experiment, except for the property values of the target inclusions. In this case,
dielectric properties of εr, I1 = 27.0 and σI1 = 1.45 S/m, εr, I2 = 35.0 and σI2 = 1.66 S/m, and εr, I3
= 40.0 and
σI3 = 2.0 S/m were assigned to I1, I2, and I3, respectively. Since the dielectric
properties of the inclusions were different, only customized meshes with four distinct regions
(Mesh5545_4r and Mesh2459_4r
Mesh2459_4r) were used for the soft prior reconstruction. Figure 3.72 shows
the 3D soft prior reconstructed dielectric properties extracted at the center of each region,
region along
with their exact property values.
(a)
213
(b)
Figure 3.72. Soft prior 3D recovered (a) permittivity and (b) conductivity values extracted at the
center of each region, along with the exact properties
properties, using 4-region customized reconstruction
meshes in Table 3.20
Similar to the previous experiment’s results, both permittivity and conductivity values of
the background region are recovered very accurately, and all inclusions are characterized
successfully. Moreover, the correct pattern of property increase from I1 to I3 can be seen in the
soft prior reconstructed permittivity and conductivity profiles. A complete error analysis of the
reconstructed soft prior property values is reported in Table 3.22. The negative
egative sign indicates that
the reconstructed property
roperty value is underestimated. (i.e. less than the exact property values)
214
Table 3.22. Relative (a) permittivity and (b) conductivity errors for the multi-inclusion
simulation experiment with different property contrasts using two soft prior reconstruction
meshes with 4 distinct regions
Rel. Err.
Rel. Err.
Rel. Err.
Rel. Err.
εr, BK
εr, I1
εr, I2
εr, I3
– 0.5%
– 0.6%
0.6%
0.3%
0.1%
– 1.6%
– 0.1%
– 3.4%
Rel. Err.
Rel. Err.
Rel. Err.
Rel. Err.
σ BK
σ I1
σ I2
σ I3
Mesh5545_4r
2.5%
0.2%
– 0.7%
– 1.2%
Mesh2459_4r
2.4%
– 1.0%
– 2.4%
– 4.6%
Reconstruction Meshes
Mesh5545_4r
Mesh2459_4r
(a)
Reconstruction Mesh
(b)
In general, when a finer soft prior mesh (5545 nodes) is used, the relative permittivity and
conductivity errors decrease. This effect is more prominent on larger errors as the dielectric
property contrasts between the background medium and target inclusions increase from I1 to I3.
In the simulation experiments described in Section 2.4, the target inclusion was centered at
the same location as I1 – i.e. (x, y, z)I1 = (3, 0. 0 cm), and the dielectric properties were identical
and equal to εr, I1 = 40.0 and σI1 = 2.0 S/m. Therefore, in order to study the effects of additional
regions of interest (in this case I2 and I3) on the recovered dielectric properties of I1, the
synthetic data in 2.4 was reconstructed, using the 5545 node mesh with only one inclusion region
I1 (I2 and I3 regions in the reconstruction mesh were assigned the same region number as that of
the background medium). The recovered soft prior property values of I1 along with the exact
solution for the single and multi-inclusion cases are plotted in Figure 3.73.
215
(a)
(b)
Figure 3.73. Comparison between the soft prior reconstructed (a) permittivity and (b)
conductivity values of I1 with and witho
without additional targets (i.e. multi-inclusion
inclusion and single
inclusion cases, respectively)
The difference in recovered dielectric property values in Figure 3.73 is minimal (0.5%
(0.5 in
permittivity and 3.5% in conductivity
conductivity), which suggests that additional targets do not have a
major effect on the 3D reconstructed soft prior dielectric property profiles.
3.5.1.7. Size and Shape of the Target
In this section, we study the effects ooff the size and shape of the target inclusion on the
recovered dielectric properties both with and without using prior structural information. We also
compare the corresponding reconstructed images in 2D and 3D.
Similar to the previous simulation experiments, synthetic 3D measurement data with –100
dBm of added noise was generated by the FDTD forward solver for different shaped and
different sized target inclusions. More specifically, 48 monopole antennas were configured in
three evenly–spaced circles with a 15.2 cm diameter and a 0.5 cm separation from each other.
Five different sized spherical and cylindrical shaped inclusionss of 0.25, 0.5, 0.75, 1.0, and 1.25
cm radius, centered at (x, y, z)) = (3
(3, 0, 0 cm), with dielectric properties of εr, Inc = 40.0 and σInc =
216
2.0 S/m were embedded in a background medium with dielectric properties of
εr, Bk = 22.4 and
σBk = 1.23 S/m.
For 3D reconstructions, the full-data selection containing both in-plane and cross-plane
data was used, whereas for 2D reconstructions, only the middle in-plane data was selected. All
no prior 2D and 3D images were reconstructed on a circular mesh composed of 559 uniformly
distributed nodes and 1044 triangular elements, and on a cylindrical mesh comprised of 1298
uniformly distributed nodes and 5259 tetrahedral elements, respectively. On the other hand, the
hard prior and soft prior images were reconstructed on customized 2D and 3D meshes
summarized in Table 3.23.
Table 3.23. Characteristics of the customized 2D and 3D meshes used for the simulation
experiments with different sized and shaped target inclusions
Inclusion
Shape
Cylinder
Cylinder
Inclusion Size 2D/3D Number of Number of
Radius (cm) Mesh
Nodes
Elements
0.25
2D
803
1537
0.25
3D
3000
13863
Cylinder
Cylinder
Cylinder
Cylinder
Cylinder
Cylinder
0.5
0.5
0.75
0.75
1.0
1.0
2D
3D
2D
3D
2D
3D
512
2160
421
1891
418
1726
962
9220
788
7850
780
7029
Cylinder
Cylinder
Sphere
Sphere
1.25
1.25
0.25
0.25
2D
3D
2D
3D
429
1628
803
6382
800
6562
1537
35131
Sphere
Sphere
Sphere
0.5
0.5
0.75
2D
3D
2D
512
3493
421
962
18534
788
217
Sphere
Sphere
Sphere
Sphere
0.75
1.0
1.0
1.25
3D
2D
3D
2D
1990
418
1556
429
9846
780
7428
800
Sphere
1.25
3D
1175
5305
For each case, the recovered dielectric property distributions were extracted at the center of
the inclusion. Figure 3.74(a) and (b) show the reconstructed values of the cylindrical and
spherical inclusions as a function of the inclusion size, using different reconstruction algorithms
(i.e. 2D and 3D, with no priors, soft priors, and hard priors).
218
(a)
(b)
Figure 3.74. Recovered permittivity (top) and conductivity values at the center of the (a)
cylindrical and (b) spherical inclusions as a function of the inclusion size, using different
reconstruction approaches (2D and 3D, with no priors, soft priors, and hard priors)
Overall, as the size of the inclusion decreases, the recovered dielectric properties become
less accurate. In addition, the properties of the cylindrical inclusion are superior to those of the
spherical inclusion, especially for the larger size targets. This improvement is even more
prominent when prior structural information is incorporated into the reconstruction algorithm.
219
The no prior 3D reconstructed permittivity values (red dotted lines in the top parts of Figure
3.74) are, overall, similar to those reconstructed in 2D with and without prior information (all
blue lines). However, the corresponding counterpart conductivity values (the bottom graphs in
Figure 3.74) are significantly superior when the radius of the inclusion is greater than 0.5 cm,
which confirms that the 3D reconstruction algorithm has a greater improvement on the recovered
conductivity profiles.
Comparing the two priors methods (i.e. hard priors and soft priors), it appears that the hard
prior reconstructed properties are slightly superior to those reconstructed with the soft prior
regularization. This effect is significantly more enhanced in the 3D reconstructed profiles (red
dotted lines) of smaller sized spherical inclusions in Figure 3.74(b). This may be due to the fact
that with the hard priors, the number of unknowns – i.e. the size of the reconstruction parameters
– is drastically reduced from 418 – 803 in 2D and 1175 – 3000 in 3D to only two parameters for
each of the two identified regions (BK and Target), and as a result, no regularization was
required during the reconstruction. On the other hand, in the soft prior technique, the number of
unknowns (i.e. the number of nodes in the customized reconstruction mesh) is significantly
greater than the number of measurements, which requires additional regularizations to stabilize
the image reconstruction.
3.5.2. Phantom Experiments
In order to study the robustness of the 3D soft prior technique in more complex-shaped
scenarios, and to evaluate the accuracy of the reconstructed dielectric property distributions in
real measured data, two phantom experiments are performed. The MR images of the
corresponding experiments are acquired separately, and the prior spatial information is used for
the 3D soft prior algorithm.
220
3.5.2.1. Cylindrical and Cup-shaped Gelatin Inclusions
In the first phantom experiment, a thin-walled plastic tube of 1.0 cm radius, filled with a
60:40 mixture of glycerin:water was inserted into a cup-shaped gelatin inclusion, as illustrated in
Figure 3.75(a). The inclusions were then submerged in the imaging tank filled with an 86:14
mixture of glycerin:water, as illustrated in Figure 3.75(b). Figure 3.75(c) shows the top view of
the phantom setup in the microwave imaging system. The independently measured dielectric
properties of the coupling medium (Bk), cylindrical inclusion (CylInc), and the gelatin inclusion
(GelInc) at 1300 MHz are reported in Table 3.24.
(a)
(b)
(c)
Figure 3.75. Complex-shaped phantom experiment with a cylindrical and a cup-shaped gelatin
inclusion: (a) cylindrical inclusion (CylInc) inserted into the cup-shaped gelatin inclusion
(GelInc), (b) both inclusions submerged in the imaging tank, and (c) top view of the phantom
setup in the microwave imaging tank
221
Table 3.24. Independently measured dielectric properties of the background medium (Bk),
cylindrical inclusion (CylInc), and cup-shaped gelatin inclusion (GelInc) at 1300 MHz
Frequency (MHz)
εr, Bk
σBk
1300
10.8
0.75
εr, CylInc σCylInc εr, GelInc σGelInc
50.60
1.29
42.45
2.36
In the present phantom experiment, 3D microwave data was acquired at multiple planes.
More specifically, the antenna array transmitted and received the signal at 8 equally–spaced (1
cm) vertical positions, and as a result, 8 × 8 × 240 measurements were collected. For the 3D
reconstructions, the multi-plane (two consecutive planes) data selection was used.
In order to obtain the corresponding structural information, the phantom inclusions were
also imaged with the MRI. For co-registration purposes, the inclusions were placed in exactly the
same location in an empty tank with an equal-sized antenna array, as shown in Figure 3.76(a)
and (b). Figure 3.76(c) shows the top view of the phantom configuration in the MRI scanner.
(a)
(b)
222
(c)
Figure 3.76. MRI Imaging of the phantom experiment with a cylindrical and a cup-shaped
gelatin inclusions: (a – b) pictures of the phantom placed in an empty microwave imaging tank
inside the MR bore, and (c) top view of the schematics of the phantom in the MRI scanner
Figure 3.77(a) shows one of the corresponding MR images of the inclusions, which were
post-processed and segmented to create the resultant 3D soft prior mesh in Figure 3.77(b).
223
(a)
(b)
Figure 3.77. (a) One of the corresponding MR images of the inclusions, (b) the corresponding 3D
soft prior mesh composed of 3393 nodes and 17027 tetrahedral elements
The customized soft prior mesh in Figure 3.77(b) is composed of 3393 nodes and 17027
tetrahedral elements. While this mesh was used to reconstruct microwave images with prior
spatial information, the no prior images were reconstructed on a 6.9 cm radius, 10 cm height
cylindrical mesh composed of 5301 uniformly distributed nodes and 23714 tetrahedral elements.
Figure 3.78(a) and (b) show the 1300 MHz reconstructed permittivity (top) and
conductivity (bottom) with and without prior spatial information, respectively. In order to view
the recovered dielectric properties of the regions of interest, the reconstructed volumes in Figure
3.78 were sliced vertically through the inclusion regions.
224
(a)
(b)
Figure 3.78. A vertical slice through (a) soft prior and (b) no prior reconstructed permittivity
(top) and conductivity (bottom) images of the phantom experiment with a cylindrical and a cupshaped gelatin inclusions at 1300 MHz
When the soft prior regularization is used, the contrast between the inclusions is
successfully detected in both permittivity and conductivity images. In fact, the higher
permittivity values of the cylindrical inclusion versus the higher conductivity values of the
gelatin inclusion are depicted in the recovered dielectric property profiles in Figure 3.78(a).
However, when no prior structural information is used during the reconstruction, the recovered
target inclusions are not distinguishable from each other. Moreover, while the cylindrical
225
inclusion appears indistinctly on the top-center of the reconstructed conductivity image, there are
no indications of such region in the counterpart permittivity image in Figure 3.78(b).
In order to analyze the recovered dielectric properties quantitatively, the soft prior and no
prior RRMS errors of the recovered properties are calculated in both inclusion regions (i.e.
Cylindrical – CylInc and Gelatin – GelInc inclusions), and the results are summarized in Table
3.25.
Table 3.25. Soft-prior and no prior RRMS errors of the recovered properties in each region of the
breast phantom experiment with two target inclusions
Soft priors
No priors
εr,CylInc
σCylInc
εr,GelInc
σGelInc
εr,Total
σTotal
RRMS
Error
RRMS
Error
RRMS
Error
RRMS
Error
RRMS
Error
RRMS
Error
0.07
0.55
0.41
0.22
0.15
0.41
0.05
0.37
0.14
0.43
0.15
0.35
The soft prior regularization significantly improves the recovered dielectric property
distributions. In fact, the RRMS errors are overall about 3 to 8 times smaller when the soft prior
regularization is used. This improvement is less significant in the conductivity profile of the
cylindrical inclusion, as the soft prior RRMS error is only about 2 times smaller than that of the
no prior case.
3.5.2.2. Breast Phantom with Two Target Inclusions
In this experiment we assessed the 3D soft prior algorithm in a more realistic configuration,
where two arbitrarily–shaped target inclusions were placed inside a prototyped plastic breast
model, as shown in Figure 3.79(a). The breast model was filled with a 88:12 glycerin:water
mixture (mimicking the dielectric properties of a scattered breast), and it was then submerged in
226
the imaging tank filled with a ma
matching
tching liquid composed of a 86:14 mixture of glycerin:water, as
illustrated in Figure 3.79(b). Figure 3.79(c) shows the schematic configuration of the phantom.
The independently measured dielectric properties of the coupling medium, breast model, and two
target inclusions at 1100 MHz are reported in Table 3.26.
(a)
(b)
(c)
Figure 3.79. Breast–shaped
shaped phantom experiment with two arbitrarily shaped target inclusions: (a)
Two arbitrary-shaped gelatin target inclusion
inclusions suspended in the plastic breast model,
model (b) rapid–
prototyped plastic breast model submerged in the imaging tank, and (c) schematic configuration
of the breast phantom in the microwave im
imaging tank.
227
Table 3.26. Independently measured dielectric properties of the background medium (Bk), breast
model (Br), and the target inclusions (Inc1 and Inc2) at 1100 MHz
Frequency
(MHz)
εr, Bk
σBk
εr, Br
σBr
1100
12.2
0.71
14.2
0.77
εr, Inc1 σInc1
35.0
1.17
εr, Inc2
σInc2
35.0
1.17
In the present phantom experiment, 3D microwave data was acquired at multiple planes.
More specifically, the antenna array transmitted and received the signal at 11 equally–spaced (1
cm) vertical positions, and as a result, 11 × 11 × 240 measurements were collected. For the 3D
reconstructions, the multi-plane (two consecutive planes) data selection was used.
In order to obtain the corresponding structural information, the breast model was also
imaged with the MRI. For co-registration purposes, the breast model with the target inclusions
was placed in exactly the same location in an empty tank with an equal-sized antenna array, as
shown in Figure 3.80(a) and (b). Figure 3.80(c) shows the top view of the phantom configuration
in the MRI scanner. For illustration purposes, an MRI coronal slice of the phantom containing
both target inclusions is placed at the center of the imaging domain.
(a)
(b)
228
(c)
Figure 3.80. MRI Imaging of the breast phantom experiment with two inclusions: (a – b) pictures
of the phantom placed in an empty microwave imaging tank inside the MR bore, and (c) top
view of the phantom configuration in the MRI scanner
After the experiment was done, the MR images were post-processed and segmented. Figure
3.81(a) shows a stack of binary-segmented images used to create the corresponding 3D soft prior
mesh in Figure 3.81(b).
229
(a)
(b)
Figure 3.81. (a) Stack of binary-segmented MRI images of the breast phantom experiment with
two target inclusions, (b) the corresponding 3D soft prior mesh composed of 7540 nodes and
34591 tetrahedral elements
The customized soft prior mesh in Figure 3.81(b) is composed of 7540 nodes and 34591
tetrahedral elements. While this mesh was used to reconstruct microwave images with prior
spatial information, the no prior images were reconstructed on a 6.9 cm radius, 11 cm height
cylindrical mesh composed of 5720 uniformly distributed nodes and 25800 tetrahedral elements.
Figure 3.82(a) and (b) show the 1100 MHz reconstructed permittivity (top) and
conductivity (bottom) with and without prior spatial information, respectively. In order to view
the recovered dielectric properties of the regions of interest, the reconstructed volumes in Figure
3.82 were sliced vertically through the breast region.
230
(a)
(b)
Figure 3.82. A vertical slice through the (a) soft prior and (b) no prior reconstructed permittivity
(top) and conductivity (bottom) images of the breast phantom experiment with two target
inclusions at 1100 MHz
When the soft prior regularization is used, the target inclusions are successfully
characterized in both permittivity and conductivity images, although they appear weaker in the
conductivity counterpart. When no prior spatial information is incorporated into the microwave
reconstruction algorithm, the target inclusions are only successfully detected in the permittivity
images. In the counterpart conductivity images, only the lower target (Inc2) appears to be
detected. In addition, its size is significantly larger and its location is drastically shifted from the
231
actual location. In all cases, the recovered property distributions of the breast are closely matched
to the exact values.
In order to analyze the recovered dielectric properties of the phantom experiment
quantitatively, the soft prior and no prior RRMS errors of the recovered properties were
calculated in each region (i.e. Br, Inc1 and Inc2), and the results are summarized in Table 3.27.
Table 3.27. Soft prior and no prior RRMS errors of the recovered properties in each region of the
breast phantom experiment with two target inclusions
εr, Br
σBr
εr, Inc1
σInc1
εr, Inc2
σInc2
εr, Total
σTotal
RRMS
Error
RRMS
Error
RRMS
Error
RRMS
Error
RRMS
Error
RRMS
Error
RRMS
Error
RRMS
Error
Soft
priors
0.06
0.11
0.09
0.28
0.09
0.11
0.06
0.12
No
priors
0.29
0.25
0.44
0.44
0.35
0.23
0.30
0.25
The soft prior regularization significantly improves the recovered dielectric property
distributions, especially the permittivity values. In fact the permittivity RRMS errors are about 5
times smaller when the soft prior regularization is used. This improvement is less significant in
the counterpart conductivity profiles, as the soft prior RRMS errors are only about 2 times
smaller than that of the no prior case.
In order to evaluate how the incorporated prior structural information improves the contrast
between the target inclusions and the background breast region, the percentage of the contrast
enhancement (CE) of the targets is calculated as:
CEti =
SP
−100 × (CtNP
−
C
t
i
i )
(3.13)
CtNP
i
232
where and are the no priors and soft priors contrasts of the ith target inclusion with
respect to the true properties of the breast region, respectively, and they are calculated as
NP
ti
C
∑
=
N NP
VnNP N
n=1
ExactBR
V
and
SP
ti
C
∑
=
N SP
V SP N
n=1 n
ExactBR
(3.14)
V
where VnNP and VnSP are the no priors and soft priors reconstructed dielectric property values
(either permittivity or conductivity) at node n (in the corresponding reconstruction meshes with
ExactBR
NNP and NSP nodes), respectively, whereas V
is the true dielectric property value of the
breast region.
The computed contrast enhancements of the target inclusions with the incorporated prior
structural information are presented in Table 3.28.
Table 3.28. The computed contrast enhancements of the target inclusions with respect to the
breast region, when the prior structural information of the phantom is used during the
reconstruction
% Contrast Enhancement (CE)
εr
σ
Upper target (Inc1)
Lower target (Inc2)
44.3
28.6
25.2
8.3
Positive values in Table 3.28 indicate that incorporating prior spatial information of the
phantom enhances the contrast between the target inclusions and background breast region.
Overall, this improvement is more prominent in the permittivity images, as the contrast
enhancement is as high as 45% in Inc1. The corresponding contrast enhancements in the
counterpart conductivity images are relatively smaller, especially in the lower inclusion (Inc2),
233
which shows that using prior structural information can improve the contrast between the lower
target and its background by only about 8%.
234
4. 3D Microwave Image Reconstruction GUI
In order to enable the user to interactively select the input data and the desired
reconstruction parameters, we have developed a MATLAB Graphical User Interface (GUI) for
3D microwave image reconstruction.
Prior to image reconstruction, the user selects the raw measurement and calibration (only
the homogeneous bath) data acquired by our clinical microwave imaging system (MIST), as
shown in Figure 4.1. This data can be in either 2D or 3D format.
(a)
(b)
Figure 4.1. Input file selection: (a) file format and (b) raw measurement file selection windows
235
The information about the acquired data (i.e. patient ID, side, exam date, used frequencies,
and antenna configuration including the number of planes for plates A and B in Figure 1.2(a) and
the corresponding slice thicknesses) is automatically extracted from the header of the measured
data file. Based on this information, the location of each antenna (either transmitting or
receiving) in the 3D space is calculated and a plot containing the antenna configuration is
displayed to the user. An example of such a plot is shown in Figure 4.2.
Figure 4.2. Sample antenna configuration plot where 6 planes of data with a slice thickness of 1
cm have been acquired
Once the measured and calibration data are subtracted from each other, the calibrated data
is phased unwrapped and stored in two multi-dimensional matrices, one for the amplitude and
one for the phase. At this stage, the data redundancy is handled as described in Section 2.3.2.
Next, as illustrated in Figure 4.3, the user can select the output file format. It should be noted that
the output file here is the input file for the reconstruction algorithm. That is to say, if the “2D”
236
output file format is selected, the calibrated measured data at each plane is stored in a file with
the right format to be used in the 2D reconstruction algorithm. Similarly, the “3D” output file
format selection corresponds to input data files for the 3D reconstruction algorithm.
Figure 4.3. Output file format selection window
Beside the file format, the antenna numbering system is different in the input files to the
2D and 3D reconstruction algorithms. More specifically, in the 2D case, the in-plane antennas on
a circle are numbered sequentially (1 – 16), whereas in the 3D case, first the antennas on plate B
(in Figure 1.2(a)) are numbered sequentially for each plane (plane 1: 1 – 8, plane 2: 9 – 16, etc.),
and then, the same process is applied to those on plate A. The reason we have chosen this
antenna numbering system is that, unlike the 2D case where all antennas on plates A and B move
together, during the 3D data acquisition there could be a different number of planes with
different slice thicknesses for plates A and B. Therefore, sequentially numbering the antennas
based on the plates on which they are mounted is the most consistent way for the 3D antenna
numbering system.
If the “3D” output file format is selected, the user can decide what portion of the acquired
data should be used for the reconstruction. As described in Section 2.3.3, this data selection is
done through a matrix of checkboxes, where each checked box represents a flag corresponding to
(TX, RX) pair (i, j), as shown in Figure 4.4. Diagonal and impossible pairs of the matrix are
disabled so the user cannot switch them on. A few buttons, including SELECT ALL, CLEAR
237
ALL, SELECT IN–PLANE, and MULTI PLANES are also designed to help the user make
selections faster and more efficiently. Based on the number of planes used to acquire
measurements, the user can select a specific set of in-plane data (for example P1, P2, …) by
checking/un–checking the corresponding checkboxes, in Figure 4.4.
Figure 4.4. 3D data selection window
The MULTI PLANES option allows the user to select multiple planes of data with a specific
number of consecutive planes. For example, as shown in Figure 4.4, if 2 consecutive planes
(2CS) are selected, for each transmitting antenna, all those in the same plane in addition to those
on the other plate located one plane above and one plane below are automatically checked as
receivers. Figure 4.5 shows an example of a multi-plane data selection with 2 consecutive planes.
238
Figure 4.5. Example of a multi-plane data selection with 2 consecutive planes
Once the transmitter/receiver selection process is done, the user presses CONTINUE and
proceeds to the next step, which is the 3D reconstruction input/parameter selection. A list of
these inputs/reconstruction parameters along with a brief description of them is reported in Table
4.1.
239
Table 4.1. A list of inputs/3D reconstruction parameters along with a brief description of them
Input/parameter
ptID
exam_date
side
Description
Patient ID
Exam (data acquisition) date
‘L’ or ‘R’
frequency
Frequency list
n_iteration
Number of iterations
fbgmedia
meshID
IsCoformalMesh
IsProfiling
IsParallel
ParallelConfig
IsSoftPrior
SP_coef
n_region
IsHardPrior
IsSimu
SimuNoise
PMLLayerNo
Background medium property filename
Mesh ID (the corresponding nodes and element files have the same ID)
‘y’ if a conformal mesh is used, ‘n’ otherwise: The mesh will be centered
with respect to the center of the imaging domain (z = 0 is always the midplane
‘y’ if the user wants to use MATLAB profiling, ‘n’ otherwise
‘y’ if the user wants to use MATLAB parallel toolbox for the forward
solution, ‘n’ otherwise
If IsParallel = ‘y’, the user needs to select one of the preset parallel
configurations (‘2nodes’, ‘4nodes’, etc.)
‘y’ if the user wants to use the soft prior regularization technique. In that
case, the soft prior reconstruction mesh (with different regions) is required.
‘n’ otherwise
If IsSoftPrior = ‘y’, the user needs to select a weighting factor as a soft
prior coefficient.
If IsSoftPrior = ‘y’, the user needs to specify how many different regions
are in the soft prior reconstruction mesh
‘y’ if the user wants to use the hard prior technique. In that case, the soft
prior reconstruction mesh (with different regions) is required. ‘n’ otherwise
‘y’ if reconstructing a simulation experiment, ‘n’ otherwise
If IsSimu = ‘y’, the user needs to specify how much noise has been added
to the synthetic data (the format is the |power level|). Example: 120, 100, or
0 for no added noise
Number of PML layers (by default = 5)
WaveLengthDiv
Number of cells per wavelength (D in equation 2.8)
TikhonovAlpha
Tikhonov Regularization coefficient (α)
Once the input parameters are selected, the 3D reconstruction algorithm begins to run in
the background. The user can return later to access the 3D reconstructed images, which are
conveniently in .vtk format. The open source scientific visualization software, ParaView
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(Kitware, Inc., Clifton Park, New York), can be used to display the reconstructed images in 3D.
In addition, the results are stored in a “Tecplot-ready” file for viewing or manipulation, using the
Tecplot 360 (Tecplot, Inc., Bellevue, WA).
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5. Summary and Conclusion
This project focused on the computational aspect of tomographic microwave imaging for
biomedical applications. Throughout this work, three main tasks were completed:
1) Our existing microwave 3D image reconstruction algorithm was optimized and
redesigned to a faster, more practical, flexible, and also user–friendlier 3D image reconstruction
module which is computationally feasible and balances the tradeoff between the accuracy and
efficiency of the model. Through several simulation, phantom experiments, and clinical exams it
was shown that using 3D microwave imaging can be beneficial in a number of ways including:
a) The increased amount of data used during the reconstruction procedure can reveal more details
about the internal structure of the imaged object/tissue, b) Despite the fact that our 2D
reconstructed images have been proved to be very informative and clinically useful, the 3D
reconstruction algorithm can help improve the recovered dielectric property distributions even
further, and c) due to the greater data–model mismatch, the 2D reconstructed images (especially
the conductivity) may show traces of the object in planes that extend beyond its actual location.
Using a 3D model during the reconstruction process can improve the accuracy of the images in
terms of both the location of the target and its recovered dielectric properties.
In addition, it was shown that using different data selections (full-data, multi-plane, and inplane) in the 3D reconstruction algorithm can impact the quality of the reconstructed images.
Specifically, the results suggested that using the multi-plane scheme with some consecutive
planes of data enables us to utilize only a subset of data during the reconstruction and to get rid
of the unwanted or potentially corrupted measurements.
242
The redesigned input–file system, as well as the optimization of the reconstruction
algorithm has made it possible to develop a not only viable and fast, but also a user–friendly 3D
reconstruction module, which is computationally feasible and balances the tradeoff between the
accuracy of efficiency of the model. Taking advantage of the parallel computing and MATLAB
executable files (MEX) has enabled us to reconstruct a 3D image in minutes, instead of
previously days or even weeks.
2) We implemented a new approach combining the functional information of the MI with
the high spatial resolution of other imaging modalities such as MRI and X–ray CT. This
approach was based on developing new image reconstruction/regularization strategies (called
soft and hard priors) that exploit structural information about the object being imaged through
additional constraints to the microwave property contrast.
We evaluated the performance of both 2D and 3D image reconstruction algorithms with the
inclusion of prior structural information through a series of simulations, phantom experiments,
and initial clinical data. Different factors such as: frequency, noise level, shape, size and number
of regions of interest, imperfect or false priors, and contrast levels were studies in depth to assess
the robustness of the new image technique.
The results showed that incorporating prior high–resolution spatial information into the MI
can significantly enhance the quality and accuracy of the recovered dielectric property
distributions. In some cases, the weighted permittivity and conductivity errors were reduced by
over 80% when the soft prior regularization was used. The hard priors had a similar effect,
reducing the relative errors by up to 95%.
In addition, it was shown that additional target regions do not have a major effect on the
reconstructed soft prior dielectric property profiles. Moreover, the soft prior regularization
243
algorithm is not overly sensitive to the exact size and location of the target, and it can tolerate
priors’ imperfection to some extend. For example, in 3D phantom experiments, changing the
priors’ volume of the target region over 125% only produced 31% change in the recovered
permittivity and 15% change in the counterpart conductivity values. Similarly, shifting the prior
target region by over 85% only changed the recovered permittivity and conductivity values by
4% and 15%, respectively. Additionally, it was shown that the soft prior technique can
effectively handle misleading situations where even an incorrect region of interest is identified.
In those cases, the relative error in the false target region was always under 7%, and in fact, this
error became even smaller (approaching zero) as the contrast between the actual target and the
background medium was decreased.
3) The final outcome of this project was two graphical user interfaces (GUI) for the 2D and
3D FDTD microwave image reconstructions. These GUIs were designed to enable the user to
control and adjust important parameters in the reconstruction procedure, and to visualize the final
reconstructed images more conveniently.
244
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