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Multiphysics Design Optimization of Microwave Ablation Antennas

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M ULTIPHYSICS D ESIGN O PTIMIZATION OF
M ICROWAVE A BLATION A NTENNAS
by
S HASHWAT S HARMA
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
c Copyright 2016 by Shashwat Sharma
ProQuest Number: 10194483
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Abstract
Multiphysics Design Optimization of Microwave Ablation Antennas
Shashwat Sharma
Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
2016
Microwave ablation is based on localized heating of biological tissues, enabled by an electromagnetic field. Antennas for ablation are commonly designed in a forward approach to generate
a particular Specific Absorption Rate profile within the target. However, little attention has
been dedicated to designing antennas inversely, to allow controllable synthesis of temperature
profiles customized for the application. Also, most existing designs do not account for thermal inhomogeneities in tissue. An inverse, multiphysics methodology for microwave ablation
antenna design is presented herein, that involves optimizing the antenna’s current distribution
to synthesize a desired temperature profile, while accounting for heat diffusion. The results
indicate and quantify the clear advantages of this multiphysics approach. This methodology
is successfully applied towards designing an easily configurable printed dipole microwave ablation antenna that addresses several limitations of existing designs. This design provides the
functionality of a phased array, and allows controlled synthesis of temperature profiles.
ii
To the shoulders on which I have stood.
iii
Acknowledgements
I would like to thank several people who have played an immense role in my work and personal
development over the last two years.
Firstly, I want to express my gratitude to Professor Costas D. Sarris, my supervisor, for
entrusting me with this project, for his patience and constant guidance and insight, and for
supporting me in a way that not only culminated in this work, but has also greatly enhanced
my personal growth as a researcher. Professor Sarris’ supervision was the ideal combination
of providing support, yet allowing for independence, and I appreciate that very much. I also
want to thank Professors Triverio and Eleftheriades for their invaluable courses, and for their
willingness to be on my defense committee; as well I’d like to thank the chair of my committee,
Professor Herman, for his time and interest in my research.
I would also like to thank my colleagues in the Electromagnetics Group, particularly Trevor
Cameron, Hans-Dieter Lang, Tony Liang, Neeraj Sood and Xingqi Zhang (in alphabetical
order), for their time, advice and patience in various matters ranging from typesetting to
antenna design. In particular, I’d like to acknowledge their sense of humour, their tolerance for
my sense of humour, and their excellent taste in coffee. I would also like to acknowledge Alex
Wong for helping me understand impedance surfaces.
I’d like to thank my parents and sister for being able to radiate love, support and encouragement from a distance of 11000 km. Finally, I want to thank Adriana, for inspiring and
encouraging me constantly, and for putting up with my odd work hours.
iv
Contents
1 Introduction
1.1
1.2
1
A Review of Literature in Microwave Ablation Technology . . . . . . . . . . . . .
2
1.1.1
MWA and Other Ablation Techniques . . . . . . . . . . . . . . . . . . . .
3
1.1.2
Monopole MWA Antennas in Literature . . . . . . . . . . . . . . . . . . .
4
1.1.3
Dipole MWA Antennas in Literature . . . . . . . . . . . . . . . . . . . . .
5
1.1.4
Other MWA Antennas in Literature . . . . . . . . . . . . . . . . . . . . .
6
1.1.5
Gaps in Literature and Motivation for our Work . . . . . . . . . . . . . .
7
Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Background and Motivation
9
2.1
The Physics of Microwave Ablation . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2
Effects of Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3
Electric Field Focusing v. Temperature Focusing . . . . . . . . . . . . . . . . . . 12
2.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 The Multiphysics Optimization Design Methodology
3.1
3.2
3.3
16
Optimization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.1
Standard Approach: SAR-Based Optimization . . . . . . . . . . . . . . . 18
3.1.2
Multiphysics Approach: Temperature-Based Optimization . . . . . . . . . 18
3.1.3
Optimization Method Comparison . . . . . . . . . . . . . . . . . . . . . . 19
Optimization and Computation Details
. . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1
Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.2
Simulation Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.3
Target Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Results and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.1
Confinement of the Ablation Zone . . . . . . . . . . . . . . . . . . . . . . 24
3.3.2
Temperature Gradient at Target Boundary . . . . . . . . . . . . . . . . . 25
3.3.3
Optimized Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4
Physical Implementation of Optimized Current Elements – A Proof of Concept . 26
3.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
v
4 A Printed Antenna Concept for Microwave Ablation
30
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2
Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3
Design Description and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1
Effects of Varying Capacitance Values . . . . . . . . . . . . . . . . . . . . 34
4.3.2
Effects of Varying the Feed Position . . . . . . . . . . . . . . . . . . . . . 34
4.3.3
Design Optimization for a Test Target . . . . . . . . . . . . . . . . . . . . 35
4.3.4
Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4
Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Conclusions
5.1
5.2
41
Significance of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1.1
Clinical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.1.2
Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.1
More Efficient Multiphysics Optimization . . . . . . . . . . . . . . . . . . 43
5.2.2
Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.3
Antenna Reconfigurability . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.4
Further Antenna Design Optimization . . . . . . . . . . . . . . . . . . . . 44
5.2.5
Fabrication and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Appendices
45
A Finite Differences Approximation of the Bioheat Equation
45
A.1 Electric Field Computation in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.1.1 Discretization of Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . 46
A.2 Computation of the Bioheat Equation in 2D . . . . . . . . . . . . . . . . . . . . . 50
A.3 Validation of FDFD Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
B Material Property Estimation by Inverse Scattering
52
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
B.2 The Inverse Scattering Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
B.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
B.4 Relevance to Microwave Ablation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Bibliography
55
vi
List of Tables
3.1
Material properties used in simulations. . . . . . . . . . . . . . . . . . . . . . . . 22
3.2
Relative, Euclidean and infinity norm errors for comparing optimization methods. 24
4.1
Material properties used in simulations. . . . . . . . . . . . . . . . . . . . . . . . 34
4.2
Input parameters - variable capacitance test cases. Capacitance values are in pF;
reactances are in Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3
Port parameters - variable capacitance test cases. . . . . . . . . . . . . . . . . . . 36
4.4
Port parameters - variable feed position test cases. . . . . . . . . . . . . . . . . . 36
4.5
Optimized port parameters for each test case. . . . . . . . . . . . . . . . . . . . . 39
vii
List of Figures
1.1
Cross-section of the monopole MWA antenna concept. . . . . . . . . . . . . . . .
4
1.2
Cross-section of the triaxial MWA antenna concept. . . . . . . . . . . . . . . . .
4
1.3
Cross-section of the monopole MWA antenna concept with a choke.
. . . . . . .
5
1.4
Cross-section of the dipole MWA antenna concept. . . . . . . . . . . . . . . . . .
6
1.5
Cross-section of the floating-sleeve dipole MWA antenna concept. . . . . . . . . .
7
2.1
Diros RFA probes with three tines. . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2
Diros RFA probe temperature profiles in ◦ C, with the 50◦ C contour in black. . . 10
2.3
(a) 2D SAR profile (in W/kg) and (b) corresponding temperature profile (in ◦ C)
with the 50◦ C contour. kth values in W/m/K. . . . . . . . . . . . . . . . . . . . . 12
2.4
Geometry for the field and temperature focusing study.
. . . . . . . . . . . . . . 13
2.5
Temperature confinement as a function of electric field beam width and thermal
conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6
(a) Electric field (at −6 dB) and (b) corresponding temperature (at 50◦ C) contours on a 2D slice through the domain, for various beam widths at a particular
value of thermal conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1
Overview of the optimization-based methodology. . . . . . . . . . . . . . . . . . . 17
3.2
Temperature as a function of time at four probe points. . . . . . . . . . . . . . . 21
3.3
Source geometry for optimization with indexing. . . . . . . . . . . . . . . . . . . 21
3.4
Depiction of the simulation domain and discretization into slices for optimization
and visualization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5
Left: One xy-slice of target profiles for (a) E-based and (c) T -based optimization.
Right: Points over which optimization is performed (in black) for selected xyslices for (b) E-based and (d) T -based optimization. Red curves represent target
boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6
Temperature profiles at selected xz-slices of the 3D domain. (a) SAR-based
optimization, and (b) T -based optimization. Target region boundary shown in
red; 50◦ C contour shown in black.
. . . . . . . . . . . . . . . . . . . . . . . . . . 25
viii
3.7
Surface plot of T for an xz-slice at y ≈ 0 cm. (a) SAR-based optimization, and
(b) T -based optimization. Target region boundary shown in red. Empty region
in the centre represents position of source array. . . . . . . . . . . . . . . . . . . . 26
3.8
Surface plot of normalized |E| (in dB) for an xz-slice at y ≈ 0 cm. (a) SAR-based
optimization, and (b) T -based optimization. Target region boundary shown in
red. Empty region in the centre represents position of source array. . . . . . . . . 26
3.9
Top row: Optimized current source element magnitudes. (a) SAR-based optimization, and (b) T -based optimization. Bottom row: Optimized current source
element phases. (d) SAR-based optimization, and (e) T -based optimization. . . . 27
3.10 Proposed proof of concept design and its surface current distribution, compared
to target currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.11 (a) Temperature profile generated by the proposed design at selected xz-slices
of the 3D domain. Target region boundary shown in red; 50◦ C contour shown
in black. (b) Surface plot of T generated by the proposed design for an xz-slice
at y ≈ 0 cm. Target region boundary shown in red. Empty region in the centre
represents position of source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1
Proposed antenna design. (a) Perspective view with dimensions and blow-up of
metallization layers. (b) Front and side views including outer teflon casing. . . . 32
4.2
Simulation geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3
Temperature as a function of time at four probe points. . . . . . . . . . . . . . . 36
4.4
(a) and (c): Surface current distributions. (b) and (d): Corresponding 50 ◦ C
contours at one central slice. (a) and (b): variable capacitance test cases. (c)
and (d): variable feed position test cases. . . . . . . . . . . . . . . . . . . . . . . 37
4.5
Optimized current distributions on one dipole. . . . . . . . . . . . . . . . . . . . 38
4.6
Optimized temperature profiles in ◦ C. Black: 50◦ C contour. Red: target boundary. 40
A.1 2D FDTD Yee cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2 Equivalent 2D FDTD Yee cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
A.3 (a) Ez computed via homemade FDFD. (b) Ez computed in COMSOL. (c) Ez
along the vertical test line shown in (a) and (b). Blue line: computed in COMSOL. Red crosses: computed via FDFD. (d) Ez along the horizontal test line
shown in (a) and (b). Blue line: computed in COMSOL. Red crosses: computed
via FDFD. Ez values are in V/m. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
B.1 Geometry for 2D FDTD-based inverse scattering algorithm. Small black squares:
transmitter/receiver pairs. Yellow shaded square: unknown domain being imaged. 53
B.2 (a) Actual electrical permittivity profile to be imaged. (b) Imaged permittivity profile obtained via inverse scattering algorithm. Actual profile in semitransparent red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
ix
B.3 (a) Actual electrical conductivity profile to be imaged. (b) Imaged conductivity profile obtained via inverse scattering algorithm. Actual profile in semitransparent red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
x
Chapter 1
Introduction
There are several malignant medical conditions that are caused by, or result in, the unwanted
presence of cells in localized regions of the body; in such cases, often the only promising type
of treatment is to destroy these cells selectively. This form of treatment is called ablation.
There are several ways in which tissue destruction can be attained; one popular method is to
selectively heat unwanted tissue to deleterious temperatures. This is referred to as thermal
ablation therapy. For example, cancerous tumours are mutations in healthy cells, leading to
the uncontrolled growth of malignant masses of tissue. In many cases, mutated cells develop
resistance to traditional treatment methods such as radiation and chemotherapy, or are not
easy to surgically remove from the body. In these cases, one clinically successful alternative
has been to heat the tumours to temperatures high enough to destroy the malignant cells [1].
Another example is irregular heart rhythms (arrhythmia) caused by areas of abnormal heart
tissue. In order to restore normal heart rhythms, it may be necessary to destroy the abnormal
tissue via heat [2]. Yet another example is nerve endings in areas of the body inflamed due to
osteoarthritis, causing the patient chronic pain. In this case, if the osteoarthritis itself cannot
be cured, the patient can be spared chronic pain by heating, or ablating, the affected areas
to kill the nerve cells [3]. In all of these cases, it is important to develop and improve upon
technologies that are able to deliver heat to target regions accurately and quickly, without
damaging surrounding healthy tissue and organs.
Ablation therapy involves rapid delivery of heat to a targeted region in tissue, inducing temperatures in excess of 50◦ C [1], and is generally carried out over the span of 5−10 minutes. Heat
can be delivered to the target cells in several ways, including ultrasound and electromagnetic
radiation. In microwave ablation (MWA), heat delivery is enabled by electric fields radiated
by interstitially-applied antennas operating at microwave frequencies. MWA has clinical applications in all of the medical conditions mentioned above, most popularly for the treatment of
malignant tumours [1, 4, 5].
One major challenge with MWA is designing microwave sources to selectively apply heat to a
targeted region without damaging surrounding healthy tissue. Several existing ablation antenna
designs attempt to achieve this confinement by focusing an electric field within the target [4,6].
1
Chapter 1. Introduction
2
This assumes that a focused electric field or specific absorption rate (SAR) profile will lead
to a focused temperature profile. However, the relationship between an applied electric field
and the consequent temperature profile is generally more complex, especially when the local
electrical and thermal properties of tissue are inhomogeneous. Thus, focusing or confining the
electric field does not guarantee an optimal temperature profile for ablation of a thermally
inhomogeneous target region.
Moreover, microwave ablation antennas are generally designed in a forward approach, and
are able to generate radiation patterns, and thus temperature profiles, that are specific to a
particular design [2, 7–10]. However, different applications of ablation may require different
shapes and sizes of temperature profiles. One major limitation of most designs in literature
is their inability to generate controllable temperature profiles customized to the shape of the
region being targeted. The design of a particular ablation antenna must be informed by the
application for which it is needed. Thus, a single antenna design that can be easily configured
for several different applications would provide a cost-effective and versatile contribution to
ablation therapy in a clinical setting.
The aim of this work is to tackle the shortcomings of existing MWA source design procedures,
and the limitations of existing MWA antennas, by proposing a new, more intelligent inverse
methodology for designing these antennas, that accounts for thermal inhomogeneities and heat
diffusion effects. Subsequently, the advantages of this methodology are demonstrated by using
it to design an easily configurable printed MWA antenna that addresses several limitations of
existing designs. The proposed design allows the user to synthesize a source current distribution
based on a desired target to ablate, offering the functionality of a phased array implemented
on the tip of an interstitial probe.
In the following section, a review of existing MWA technologies is presented. Advantages
of MWA compared to other ablation technologies such as High-Intensity Focused Ultrasound
(HIFU) and Radio Frequency Ablation (RFA) are discussed. Various MWA designs, their
results, and their limitations are discussed and used to motivate the need for a new inverse
methodology and more easily configurable MWA antennas.
1.1
A Review of Literature in Microwave Ablation Technology
MWA is a relatively new form of therapy. The use of an interstitial microwave-based device
for liver tumour therapy was first proposed by Tubuse [11] in 1979. The technique was subsequently applied successfully in a clinical setting by Tubuse et al. in 1985 [12]. Although
initially this technique was mostly popular in Japan and China, as of 2003, MWA antennas
have been commercially sold in the United States courtesy of Vivant Medical, Inc (Mountain
View, Calif., USA). There are now several commercial outlets for MWA technology, and various
advancements in MWA antenna design continue to be made. Most MWA antennas are thin,
needle-like coaxial-based devices, and the most popular ones are either monopole, dipole or slot
Chapter 1. Introduction
3
antennas [7–10, 13], with some helical and loop-based antennas also emerging recently [14]. In
the following subsections, we first present a comparison of MWA versus other popular ablation techniques, and subsequently summarize popular MWA antenna types and discuss their
advantages and disadvantages. This analysis is then used to motivate our work.
1.1.1
MWA and Other Ablation Techniques
There are several alternatives to microwave ablation in terms of destroying malignant tissue
in a percutaneous or noninvasive way. One of the earliest methods developed for this purpose
is High-Intensity Focused Ultrasound (HIFU), first introduced by Lynn et al. [15] in 1942. In
this method, high intensity acoustic waves at frequencies between 0.5–10 MHz are focused using
acoustic lenses into a tight target spot, resulting in the deposition of heat energy in that spot.
This is completely non-invasive technique that has gained significant popularity and garnered
many improvements over the years. HIFU has been successful in accurately targeting tumours
in clinical trials [16], however its success rate is not statistically different from MWA [17], and
modern HIFU applicators generally require the optimization of hundreds or even thousands of
acoustic source elements to properly focus the energy, making it a more expensive technology.
Also, successful HIFU ablation requires significantly longer treatment time than MWA [17].
Another popular ablation technique that is closely related to MWA is Radio Frequency
Ablation (RFA). RFA is similar to MWA in that thin needle-like applicators are fed with
alternating currents to generate heat interstitially. However, in RFA, lower frequencies are
used as compared to MWA (300–500 kHz), and the mechanism by which tissue is heated is
quite different. An RFA applicator consists of a needle-like structure with a metallic section
to which an alternating voltage is applied, and a ground that is either placed interstitially on
the same needle, or is placed on the interface between skin and air. This forms a current path
through the tissue near the applicator, which generates heat in the immediate vicinity of the
needle. The heat then diffuses through the nearby target tissue to create an ablation zone [16].
Although RFA has been successfully applied in a clinical setting and is commercially available,
due to its heating mechanism it is unable to generate ablation zones as large as in MWA. It
also requires a conducting path through tissue to be formed, which may create unnecessary
heating in non-targeted regions, since the conducting path is difficult to control. Also, certain
types of tissue are not conducive to the passage of electric current, and thus are not suitable for
RFA. This limits its applications and makes resulting ablation zones unpredictable. In order to
manipulate the shapes of temperature profiles, RFA applicators are often designed with sharp
tips or multiple tines, which is quite invasive [3,18]. These limitations are quite easily overcome
by MWA [19–22]; MWA is able to produce larger ablation zones more quickly, without relying
on any conduction paths through tissue. Although each of these ablation methods has its
advantages and have been in use longer than MWA, the use of microwaves offers many of the
same benefits while overcoming several of the disadvantages of HIFU and RFA.
4
Chapter 1. Introduction
1.1.2
Monopole MWA Antennas in Literature
These antennas are constructed from thin, semirigid coaxial cable, with the inner conductor
made to extend beyond the tip of the outer conductor by stripping the outer conductor [23],
as shown in Fig. 1.1. In [23], three types of monopole antennas are considered; each one
is a variation in how far the dielectric extends beyond the tip, and the metallization on the
tip. Simulation and experimental results were presented for each case, and in general, the
temperature patterns obtained in each case were either nonuniform, or not large enough for
practical application. Moreover, the temperature profiles were elongated along the tips of the
antennas due to backwards current flow. Although the size of the patterns can be somewhat
modified by varying the length of the extended tips or the input power, the poor return loss
and nonuniform temperature profiles are major drawbacks. These designs do not allow the user
control over the shape of the temperature profiles that are produced, and are not tunable to
local electrical or thermal properties of tissue.
Coaxial feed, outer conductor
Monopole antenna
Coaxial feed, inner conductor
Teflon
Figure 1.1 Cross-section of the monopole MWA antenna concept.
More promising results were presented by Brace et al. [7], where a coaxial monopole antenna
is placed inside a biopsy needle, to effectively yield a triaxial antenna, as shown in Fig. 1.2.
This design greatly reduces the return loss and is able to generate much larger ablation zones
(≈ 3 cm in diameter) at moderate input powers (≈ 50 W). However, this design also offers no
control on the shapes of generated temperature profiles; the profiles are elliptical, whereas in
the treatment of tumours, spherical profiles are more sought after. It cannot be configured to
suit a particular application, since there are very few design parameters that can be varied,
that would have an appreciable effect on the shape of the temperature profiles.
Coaxial feed, outer conductor
Monopole antenna
Biopsy needle
Coaxial feed, inner conductor
Teflon
Figure 1.2 Cross-section of the triaxial MWA antenna concept.
A modification of the above was presented by Prakash et al. [24] in 2009, where rather than
5
Chapter 1. Introduction
a biopsy needle, the outer conductor of a coaxial monopole antenna was fitted with a quarterwavelength metallic choke, as shown in Fig. 1.3. The quarter-wavelength nature of the choke
prevents flow of current along the outer surface of the outer conductor of the coax feed, thus
reducing the backward flow of current and subsequent unwanted heading along the feed line.
This leads to larger and more spherical ablation zones. The authors optimize the parameters of
the choke-based antenna to obtain near-spherical ablation zones. However, their simulation and
experimental results both indicate somewhat elongated ablation zones, and significant unwanted
heating along the feed, although less so than previous monopole designs. This design also does
not offer control of the shape of the ablation zone.
Coaxial feed, outer conductor
Monopole antenna
λ/4 Choke
Coaxial feed, inner conductor
Teflon
Figure 1.3 Cross-section of the monopole MWA antenna concept with a choke.
1.1.3
Dipole MWA Antennas in Literature
One of the earliest types of MWA antenna was a coaxial-based dipole, also built with a thin
semirigid coaxial cable, where a coaxial transmission line ends into a gap, with the inner conductor connected across the gap to a metallic extension on the other side of the gap, as shown
in Fig. 1.4. Jones et al. [25] presented this classical dipole antenna for MWA in 1989; they also
investigated the use of multiple dipoles inserted to form an array. They were able to generate
large (over 4 cm diameter) ablation zones using up to 8 or 9 antennas, and concluded that the
complex SAR profiles generated did not seem immediately useful in the clinic, and may require
further analysis with even more complex arrangements and variations in phase. These results
are discouraging, since inserting more than one antenna is quite invasive, and since then, various
designs have been proposed that achieve better results without having to use multiple antennas.
These designs also suffer from serious drawbacks: no temperature profiles are presented and
thermal properties are ignored, and the SAR or temperature profile shapes are not controllable
in any way.
Hürter et al. [26] proposed a modification of this type of antenna in 1991. The authors
propose a sleeve along the coax feed line connected to the outer conductor that prevents the
backward flow of current, thus ensuring that most of the power is deposited at the tip of the
antenna (near the gap) rather than along the feed line. The authors compare this design to the
more standard dipoles without a sleeve, and are successful in showing that the electric field is
indeed more concentrated near the dipole gap, rather than along the feed line, leading to tighter
6
Chapter 1. Introduction
Coaxial feed, outer conductor
Coaxial feed, inner conductor
Extension
Dipole gap
Teflon
Figure 1.4 Cross-section of the dipole MWA antenna concept.
ablation zones. Unlike other designs without such a sleeve, the radiation and temperature
patterns of this design do not depend on the insertion depth of the antenna, which is a major
advantage. However, once again all of the analysis provided by the authors is in the electrical
domain; only SAR patterns (which depend on the applied electric field, electric conductivity
and tissue density) are presented, with no mention of corresponding temperature profiles. Thus,
thermal properties and heat diffusion effects are not accounted for. Also, as in the cases above,
these designs produce a fixed SAR pattern, with little control over the shape of the generated
fields or temperature profiles.
More recently, a modification of Hürter’s sleeve-based dipole antenna has been presented,
where the sleeve is floating rather than fixed. This was first presented by Yang et al. [9] in
2006. A coaxial dipole antenna, similar to the ones presented above, is encased in a teflon
coating. A metallic sleeve is slid onto the teflon cylinder at a variable position, such that it is
electrically isolated from the coax cable, unlike the design presented by Hürter. This is depicted
in Fig. 1.5 The floating nature of the sleeve allows an added degree of freedom in terms of design
parameters that can be tuned to generate a suitable temperature profile. Prakash et al. [27]
subsequently presented an optimization procedure for the various design parameters of the
floating sleeve MWA antenna to produce the largest ablation zones. Both works yielded very
encouraging results, with significant reduction in the unwanted heating of the feed line, and the
ablation zones were quite successfully localized near the antenna tips. However, in both studies,
once again only SAR patterns are discussed. The fact that SAR patterns are well localized to
antenna tips does not imply that the temperature profile will also be thus confined. Also, the
various design parameters that can be adjusted in this case only modify the size and extent
of the SAR patterns, not their shapes. Although these designs yield large ablation zones with
reflection coefficients well below −15 dB, they do not offer control and specificity to particular
applications. The generic SAR profiles may be useful in some applications, but not in others.
1.1.4
Other MWA Antennas in Literature
A loop antenna was presented by Shock et al. [14], where a probe is inserted into the target
region, and a metallic loop is deployed with the help of electrocautery (i.e. the heating up of the
microwave probe) to cut through the tissue. The authors also investigated the use of multiple
7
Chapter 1. Introduction
Coaxial feed, outer conductor
Extension
Floating sleeve
Coaxial feed, inner conductor
Dipole gap
Teflon
Figure 1.5 Cross-section of the floating-sleeve dipole MWA antenna concept.
loops; experimental results indicate that these antennas are able to produce high temperatures
within the ablation zones, and generate fairly spherical temperature profiles. However, this
method suffers from several disadvantages; the loops are only effective if they are accurately
placed such that they perfectly encircle the target region. Thus, a particular loop can only be
applied to a specific target type and size. Also, deployment of the loop requires electrocautery
surgery which is very invasive, and creates the risk of harming healthy, important nerve cells
or surrounding tissue. Also, much like aforementioned designs in literature, there is no control
over the shapes of generated temperature profiles.
Luyen et al. [28] very recently presented a coaxial-based helical antenna, which avoids the
use of a balun or choke of any kind to prevent back flow of current. However, the results
indicate elongated, uncontrollable temperature profiles with significant tail-end heating of the
feed line. The advantages of this design include a return loss of −30 dB and the fact that the
lack of a balun allows the antenna to have a lower diameter than other designs; however, the
disadvantages in ablation zone shapes do not sufficiently justify the absence of a balun.
1.1.5
Gaps in Literature and Motivation for our Work
The various designs that have been proposed in literature have shown great improvement over
the years, and have collectively proved that MWA is a promising and viable clinical technique
for various applications. These designs provide several advantages over other ablation methods
such as RF and HIFU; however, several issues exist among these designs, and there is significant
room of improvement in the field.
As we claimed earlier, one common lacking feature among these designs is that they produce
generic temperature profiles whose shapes cannot be controlled or modified; this is a fundamental limitation of the fact that these antennas are designed in a forward approach with no
specific target profile in mind. Another common flaw is that many of these designs focus on the
electric field or SAR profiles generated, but do not consider heat diffusion around the antennas,
or the local thermal properties of tissue. These are the two primary areas of improvement that
are targeted in this thesis.
Chapter 1. Introduction
1.2
8
Overview of the Thesis
The thesis is organized as follows: first, we provide background regarding our initial work in
electromagnetic ablation. The physics of MWA is discussed, and the relations between the
thermal and electrical domains are studied through simulations to motivate our investigation
into multiphysics methods and the need for an inverse antenna design approach.
A methodology for the optimization of MWA sources is then presented, and the standard
electric field-based approach is compared with the proposed multiphysics approach. The results
obtained via each approach are then analyzed and discussed. A proof-of-concept MWA source
design is discussed.
The multiphysics methodology is then put to practice in order to design a printed microwave
ablation antenna, that can be configured to synthesize a range of temperature profile shapes
and sizes. The antenna provides the functionality of a phased array in allowing control over its
current distribution, which can be optimized through simple physical and electrical parameters
to generate a desired temperature profile. The performance of the antenna is analyzed to assess
its feasibility in a clinical setting.
The work is concluded with a discussion of the significance and relevance of the proposed
ideas, as well as future directions that may be taken to bring these ideas into a clinical setting.
Chapter 2
Background and Motivation
Our exploration into electromagnetic ablation methods began with performing multiphysics
simulations of RFA probes designed by Diros Technology Inc., Markham, ON, Canada. During
the collaboration we simulated the electric field and temperature profiles of new RF probes
that were experimentally validated at Diros, and are now commercially available [29]. The
simulations were set up in the commercial finite element method solver, COMSOL Multiphysics
(COMSOL Inc., Burlington, MA). The relation between the electric and thermal domains is
modeled by the Pennes Bioheat Equation [30] (see below for details), and the probes were
modeled in the time domain to evaluate thermal profiles after 10 minutes of application. Fig. 2.1
shows one of the new RFA probes proposed by Diros. The reasoning behind using three sharpedged tines is based on the conventional wisdom that sharp corners lead to higher local electric
field intensities, leading to higher temperatures around the probes.
The simulated temperature profiles, along with the 50◦ C contours, are shown in Fig. 2.2.
The temperature profile is extended near the tips due to the three tines. The validity of the
simulations was confirmed by experimental results provided by Diros. The probes were inserted
into bovine liver, which was sliced post-therapy to study the ablation zones.
The fundamental design ideology behind these, and several other RFA probes is that the
physical shapes of the probe tips are manipulated in order to control the shape and extent of
the ablation zone. However, the sharp-edged nature of these designs makes them quite invasive.
Additionally, if the ablation zone shapes and sizes were to be manipulated via the electrical
properties of the probe rather than its physical characteristics, it could allow for more versatile,
simpler, smaller and less invasive designs. This was one of our initial motivators for looking
into a more intelligent, inverse way to design ablation devices.
Upon our investigation into electromagnetic ablation methods, the advantages of microwavebased techniques over RF probes (outlined in the previous chapter) led us to apply the idea of an
inverse design procedure to MWA antennas, rather than RFA probes. In order to develop a more
intelligent methodology for the design of MWA antennas, it is necessary to study the physics
of MWA, and the factors that relate the applied electric field to the consequent temperature
profile. In the following sections, through simulations, we study the relations between the
9
10
Chapter 2. Background and Motivation
(a)
(b)
(c)
Figure 2.1 Diros RFA probes with three tines.
(a)
(b)
Figure 2.2 Diros RFA probe temperature profiles in ◦ C, with the 50◦ C contour in black.
Chapter 2. Background and Motivation
11
electrical and thermal domains, and understand the influence of thermal properties and heat
diffusion.
2.1
The Physics of Microwave Ablation
In MWA, the ablation zone is generated via a combination of heat energy deposited by the
applied electric field, and the diffusion of that deposited energy through tissue [31]. The electric
field, oscillating at microwave frequencies, causes the water molecules to constantly rotate to
align their polarity with the field. This increases the local kinetic energy, thus generating heat
and resulting in ablation if the temperature is high enough [1,19]. Since the relative permeability
of biological tissue is generally 1, the complex dielectric permittivity determines the transmission
of microwave energy into tissue. The complex permittivity is treated as a relative permittivity
that represents the real part, and an electric conductivity that represents the imaginary part.
In microwave ablation, electrical energy is primarily transmitted into tissue via displacement
currents generated by the applied field. This is different from RF ablation, where the energy is
primarily deposited in an ohmic fashion via a conductive path formed through tissue between
the source and ground. Thus, in RF ablation, the applicator is essentially a probe, while in
microwave ablation, the applicator is an antenna [19].
The mathematical relation between an applied electric field E and resulting temperature
profile T in biological tissue is well approximated by the Pennes Bioheat Equation [30].
ρCp
∂T
1
= σ|E|2 + ∇ · (k∇T ) + Wb Cb (Tb − T )
∂t
2
(2.1)
where ρ, Cp , k and σ are the density, heat capacity, thermal conductivity and electrical conductivity of tissue, and Wb and Cb are the perfusion rate and heat capacity of blood. The
first term on the right represents electrical energy deposition in tissue; the second term on the
right represents heat diffusion through tissue; and the third term represents the cooling effect
of blood perfusion through tissue. Another quantity relevant to MWA is the specific absorption
rate (SAR), which is defined as follows:
SAR =
σ |E|2
2ρ
(2.2)
The SAR is an indication of energy deposition in tissue due to an applied electric field, that
takes into account tissue density.
It is clear that the temperature profile depends on both, the applied electric field, and the
diffusion of heat through the tissue, which in turn depends on the local thermal conductivity.
Higher values of k imply increased diffusion of heat around the region of deposition. Thus, it
is clear that the shape and size of the ablation zone depend on two factors: heat deposition
(via the strength and pattern of the applied electric field); and heat diffusion (via the thermal
properties and inhomogeneities of the target region).
12
Chapter 2. Background and Motivation
0.01
kth = 0.9
60
kth = 0.9
kth = 0.1
1500
0
y (m)
2000
0.005
y (m)
0.01
2500
kth = 0.1
0.005
55
0
50
1000
45
-0.005
-0.005
th
th
-0.01
-0.01
-0.005
500
k = 0.7
k = 0.3
0
0.005
0.01
k = 0.7
k = 0.3
th
th
-0.01
-0.01
-0.005
x (m)
(a)
0
0.005
40
0.01
x (m)
(b)
Figure 2.3 (a) 2D SAR profile (in W/kg) and (b) corresponding temperature profile (in ◦ C)
with the 50◦ C contour. kth values in W/m/K.
2.2
Effects of Thermal Conductivity
The influence of thermal conductivity on a temperature profile resulting out of an applied
electric field can be demonstrated by a simple simulation of (2.1) in a domain with varying
thermal conductivities. A 2D square domain with four quadrants of exactly the same electrical
and thermal properties, except thermal conductivities that range from 0.1 to 0.9 W/m/K, is
simulated. The source is assumed to be a circular surface current in the centre of the domain,
flowing in the direction perpendicular to the 2D plane (ẑ). The resulting SAR and temperature
profiles are shown in Fig. 2.3. As expected, the SAR profile in Fig. 2.3a is symmetric about
the source current, since the electrical properties and densities of each of the four materials
is the same. However, it is clear from Fig. 2.3b that the corresponding temperature profile is
significantly distorted due to varying thermal conductivities. This occurs, as expected, because
the rates of thermal conduction through each of the four materials is significantly different. This
clearly indicates the need to design and study MWA antennas in the context of the temperature
profiles and not just the SAR profiles they generate. This is important to allow accurate ablation
of the targets, and to minimize damage to healthy surrounding tissue.
2.3
Electric Field Focusing v. Temperature Focusing
In order to develop a more intelligent design methodology for MWA antennas, it is necessary
not only to study and quantify the roles of heat deposition and heat diffusion, but also to study
the fundamental limits on the ability to control and shape temperature profiles. To this end,
and to motivate our investigation into multiphysics methods, a study was conducted to analyze
13
Chapter 2. Background and Motivation
the extent to which focusing of an electric field allows focusing of the temperature profile, and
the limits beyond which heat diffusion effects take over.
The purpose of this initial study was to, through simulations, understand and quantify the
extent to which each of these factors (heat deposition and heat diffusion) contributes towards
the temperature distribution.
In this study, a dipole antenna array with 5 × 5 elements placed near human liver tissue
was simulated to focus electric fields within the tissue to varying degrees, and the resulting
temperature profiles were computed using (2.1). The computations were carried out using
COMSOL Multiphysics. The geometry is depicted in Fig. 2.4. The excitation of each element
was chosen via the Dolph-Tschebyscheff method [32] to control and vary the maximum beam
width of the generated electric field profile. The beamwidth is controlled by specifying the
desired side lobe level, RdB , in dB. The excitations an are chosen using the following equations
[33]:
an =
M
+1
X
(−1)M −q+1 (z0 )2(q−1)
q=n
1
z0 =
2
"
(q + M − 2)!(2M )
εn (q − n)!(q + n − 2)!(M − q + 1)!
#
q
1/P
q
2
R0 + R0 − 1
+ R0 −
(2.3a)
1/P
R02 − 1
R0 = 10RdB /20
(2.3b)
(2.3c)
If N is the number of array elements along one direction, then M = (N − 1) / (2), n =
1, 2, . . . M + 1, ε equals 2 if n = 1 and 1 otherwise, and P = N − 1. In this way we can find the
excitations required along each dimension. The excitation vectors can then be matrix-multiplied
to obtain the excitation of each element in the 2D array.
Liver
Air
Dipole array
Figure 2.4 Geometry for the field and temperature focusing study.
14
Chapter 2. Background and Motivation
T profile (kth = 0.10)
Diameter of 500C isotherm (cm)
3.6
T profile (kth = 0.30)
T profile (kth = 0.57)
3.55
T profile (kth = 0.75)
T profile (kth = 0.95)
3.5
3.45
3.4
3.35
3.3
3.25
2.95
3
3.05
3.1
3.15
3.2
3.25
Diameter of -6 dB electric field contour (cm)
2
2
1.5
1.5
z (cm)
z (cm)
Figure 2.5 Temperature confinement as a function of electric field beam width and thermal
conductivity.
1
0.5
1
0.5
0
0
-2
-1
0
1
2
-2
-1
0
x (cm)
x (cm)
Sources
Sources
(a)
1
2
(b)
Figure 2.6 (a) Electric field (at −6 dB) and (b) corresponding temperature (at 50◦ C) contours
on a 2D slice through the domain, for various beam widths at a particular value of thermal
conductivity.
The maximum diameter of the 50◦ C isotherm at steady state is reported in Fig. 2.5 as a
function of the maximum electric field beam width at −6 dB, for different thermal conductivities
of tissue, in order to understand the contributions of deposited electric energy and heat diffusion
Chapter 2. Background and Motivation
15
to the obtained temperature profiles. Fig. 2.6a and Fig. 2.6b are examples of a slice of the electric
field and corresponding temperature contours (at −6 dB and 50◦ C) obtained for a particular
value of thermal conductivity, for different beam widths. Two main conclusions can be drawn
from Fig. 2.5 and Fig. 2.6:
• There is a fundamental limit to which the temperature profile can be focused or confined,
regardless of how focused the electric field is, for a given amount of energy deposited.
This limit arises due to heat diffusion around the area where the electric field is confined;
the energy deposited via an electric field will diffuse around the region even if the electric
field itself is confined to a small region. This must be taken into account when attempting
to ablate a given target region.
• The 50◦ C isotherm diameter is inversely related to the thermal conductivity. This can be
understood as follows: although a lower thermal conductivity implies less heat diffusion,
this reduced diffusion causes an accumulation of heat near the source, thus raising the
overall temperature around the source. Given that biological tissue may be thermally
inhomogeneous, and that tumours can exhibit thermal conductivities ranging from 0.3
to 0.7 Wm−1 K−1 [34], [35], the significant relation between temperature and thermal
conductivity must be taken into account in designing antennas for ablation.
The observations made above clearly demonstrate the complex relation between applied
fields and consequent temperature profiles, and indicate the necessity for taking these complexities into account when designing MWA antennas.
2.4
Summary
The purpose of this chapter was to establish the need for an inverse multiphysics approach
to designing MWA antennas. Through simulations, the relation between the electric field and
temperature profiles were studied, as well as the effect of thermal conductivity. The results of
these studied showed that the thermal profile depends significantly on both heat deposition in
the tissue, and heat diffusion through tissue. It was also established that simply focusing or
shaping the electric field or SAR profile is not sufficient to guarantee optimal generation of a
desired ablation zone. In the next chapter, an inverse multiphysics design approach is described
and analyzed.
Chapter 3
The Multiphysics Optimization Design
Methodology
With the results of the aforementioned studies and the understanding of heat deposition and
heat diffusion, we are now in a position to propose an inverse, optimization-based methodology
for MWA antenna design.
One heat-based therapeutic technique that has been used successfully in clinical applications
is hyperthermia, where an antenna array placed outside the body is used to focus the electric
field at a target within the body [36]. This is achieved by appropriately picking the excitation
phase and magnitude of each element of the array. Thus, it is an inverse technique, where
the source is optimized in order to confine the electric field or specific absorption rate (SAR)
at the target. In [37], an antenna array was proposed that allows sub-wavelength focusing
and shaping of the magnetic field in the near-field of the array via source optimization. This
technique can also be applied to shaping the electric field in the near-field. However, little
attention has been given to applying this inverse approach of source optimization in designing
interstitial antennas for microwave ablation. The purpose of this part of the work is to apply the
concept of source optimization to the design of interstitial MWA antennas, and subsequently
to develop a methodology for the design of these antennas to synthesize a desired temperature
profile based on some target.
Since the generated temperature profile depends on the applied electric field, it also indirectly depends on the current distribution on the antenna that produces the electric field. Thus,
the approach taken in this work is to optimize the current distribution on the antenna based
on a desired electric field or temperature profile.
Fig. 3.1 outlines the methodology of this approach: given a target to ablate, a corresponding
target temperature profile, Ttarget , and target electric field profile, Etarget , are defined. The
ablation antenna is modeled as an array of surface current elements placed inside the target
region, such that the current distribution on the elements generates an electric field E and
corresponding temperature profile T . The current element magnitudes and phases are then
16
Chapter 3. The Multiphysics Optimization Design Methodology
17
optimized such that the generated field or temperature profile best approximates the target
profile. Rather than manipulating the physical shape of the antenna, we use the properties
of antenna arrays to, through their near fields, control the generated temperature profiles.
This allows for the possibility of designing antennas that can generate customized temperature
profiles for different applications.
As mentioned previously, the standard approach in designing MWA antennas or optimizing
arrays for hyperthermia involves focusing or confining the electric field or SAR within the target, which does not account for the dynamics of heat diffusion and thermal inhomogeneities in
the target region. Thus, to optimize the current elements, a multiphysics approach is employed
that accounts for thermal inhomogeneities in tissue. The results are compared to the standard
approach of simply generating strong and confined electric fields or SAR profiles, to confirm
the claim that a purely field-based optimization procedure does not guarantee an optimal temperature distribution for a given target in biological tissue. Although it is intuitively apparent
that a temperature-based multiphysics optimization approach would work better than a purely
field- or SAR-based one, it is useful to quantify the differences between the two approaches and
to ascertain whether the multiphysics approach is worth the added computational costs.
In many cases where ablation is necessary, particularly for the treatment of tumours, it is
important for MWA antennas to generate symmetric, spherical ablation zones [24]. Thus, for
the purpose of this work, we focus on designs that are symmetric about the axis of the antenna.
The current distribution on the ablation antenna can be thought of as a linear array of
discrete current elements with magnitudes I (i) and phases β (i) , where i is an index to each
element. Our aim is to optimize I (i) and β (i) such that they produce an electric field profile E,
and consequently a temperature profile T , that best approximates a given target temperature
profile, Ttarget . In optimizing the current elements, we compare two optimization approaches:
the standard electric field-based approach where the elements are optimized such that the SAR
is confined to within the target (SAR-based optimization), and a multiphysics approach based
on synthesizing Ttarget directly (T -based optimization).
Study the
relation
between E and
T in tissue
Identify Ttarget ,
and corresponding Etarget
Model microwave
ablation antenna
as discrete surface
current elements
Study and compare E-based and
T -based optimization methods
I
Optimize surface
current elements
to generate
Etarget and Ttarget
Pick an
optimization
method based
on results
Realize optimized
current distribution
by designing an
appropriate antenna
Figure 3.1 Overview of the optimization-based methodology.
Chapter 3. The Multiphysics Optimization Design Methodology
3.1
3.1.1
18
Optimization Theory
Standard Approach: SAR-Based Optimization
In this approach, a locally high electric field and thus SAR is generated within the target, such
that energy deposition is minimized outside the target. The SAR is defined as follows:
SAR =
σ |E|2
2ρ
(3.1)
Since we assume no prior information about the actual field intensities required to induce
ablative temperatures, the optimization is performed in two steps.
• First, we consider normalized SAR only, and attempt to shape the pattern such that the
−6 dB SAR isoline lies within the target region, and decays below −10 dB near the target
boundary. Based on these constraints, we define a normalized target SAR distribution,
SARtarget . I (i) and β (i) are then optimized by minimizing the following objective function:
fE =
m,n
SARm,n
norm − SARtarget
X
2
(3.2)
m,n∈K
where K is the set of indices over which the optimization is performed and SARm,n
norm is
the normalized SAR distribution generated by the source that consists of I (i) and β (i) .
• At this point, an appropriate scaling of the optimized current magnitudes is required to
induce ablative temperatures within the target. Thus, we introduce a scaling factor b that
is multiplied with the optimized current magnitudes. The scaling factor itself is optimized
such that the resulting temperature at the boundary of the target approaches 50◦ C. This
is achieved by minimizing the following cost function:
fb =
X
(T m,n − 50)2
(3.3)
m,n∈B
where B is the boundary of the target region. The updated optimized source current
magnitudes are bI (i) , and the optimized phases are still β (i) .
3.1.2
Multiphysics Approach: Temperature-Based Optimization
In this approach, a target temperature profile, Ttarget , is defined such that the temperature is
above 50◦ C within the target region, close to 50◦ C at the target boundary, and decays to normal
body temperature (37◦ C) elsewhere. The goal is to optimize the source currents to generate
a temperature profile that approaches Ttarget . To achieve this, I (i) and β (i) are optimized by
Chapter 3. The Multiphysics Optimization Design Methodology
19
minimizing the following objective function:
fT =
X
m,n
T m,n − Ttarget
2
(3.4)
m,n∈K
where K is the set of indices over which the optimization is performed. No assumptions or
conditions are placed on the corresponding electric field required to generate Ttarget . This
method involves computing first the field profile, and then the consequent temperature profile
via (2.1) at each iteration of the optimization.
3.1.3
Optimization Method Comparison
The results obtained via the two optimization methods described above are to be compared on
the basis of two sets of metrics:
Confinement of the Ablation Zone
To quantify how closely the 50◦ C contour matches the target boundary, we define three comparison metrics: the L1 norm f1 , the Euclidean norm f2 and the infinity norm f∞ with respect
to the achieved and target temperatures at the target boundary:
f1 =
X T m,n − T m,n target
(3.5a)
m,n∈B
f2 =
X
m,n
T m,n − Ttarget
2
(3.5b)
m,n∈B
m,n f∞ = max T m,n − Ttarget
m,n∈B
(3.5c)
where B refers to the target boundary. Lower values of f1 , f2 and f∞ indicate better optimization results. Since the errors are computed only on the target boundaries, the number of points
over which the errors are calculated is the same for each optimization method, thus making for
a meaningful comparison.
Temperature Gradient at Target Boundary
An important consideration in ablation is to confine the heat damage to the target region, and
minimize the damage to surrounding tissue. Thus, it is important to quantify the temperature
gradient at the target boundary and the rate at which it decays to normal body temperature.
To quantify the rate at which T drops below ablative levels outside the target, we define gB
as the x-component of the gradient of the optimized temperature profile, evaluated at a single
point on the target boundary. We also compare gmax , the maximum of the x-component of the
20
Chapter 3. The Multiphysics Optimization Design Methodology
gradient of the optimized temperature profile on a line along x̂ in the center of the target.
gB = ∇T · x̂|(xb ,zb )
(3.6a)
gmax = max (∇T · x̂)
(3.6b)
line
where (xb , yb ) ∈ B and B is the set of points corresponding to the target boundary.
3.2
Optimization and Computation Details
COMSOL Multiphysics was used to solve the forward problem of computing electric fields and
temperature profiles. This was interfaced with Matlab (The MathWorks Inc., Natick, MA) for
the optimization procedure and for visualization of results. In both optimization approaches,
a sequential quadratic programming (SQP) algorithm was employed. The function space is
assumed to be convex, to allow the use of a constrained, convex optimizer. However, since
the function space has 2i dimensions, convexity is not guaranteed, and it may be better to use
a global optimization method. However, since the purpose here is to compare two different
objective functions for optimization, a convex optimizer is used to conserve computational
resources. A frequency of 2.45 GHz was used in all simulations; this is one of the clinical
standards for microwave ablation [4].
Electric fields are computed in the frequency domain. In order to reduce computational cost,
it is beneficial to solve (2.1) at steady state using a stationary solver. To establish the validity
of this approach, the sources are simulated to compute temperatures in the time domain. The
temperatures were probed at four points at varying distances (r) from the antenna, and plotted
as a function of time (t) to study the time domain characteristics. The results are shown in
Fig. 3.2.
It is clear that steady state in the thermal domain is achieved at t ≈ 5 minutes, and most
ablation treatments last at least that long. Thus, computing temperatures at steady state is
justified for the purposes of source optimization. Although tissue properties exhibit nonlinear
changes at high temperatures, especially above of 100◦ C, these effects are not considered in this
work since regions with such high temperatures are relatively small and confined only to the
core of the target. Thus the accuracy of the results obtained are sufficient for the purposes of
demonstrating the inverse design procedure and for comparing optimization methods.
3.2.1
Sources
The ablation antenna is divided into six independent surface current source elements. Each
surface current element is a cylindrical surface of radius a = 1 mm and height 2 mm, and all
six elements are arranged in a linear array as shown in Fig. 3.3. The current elements are not
included as part of the domain in which fields and temperatures are computed. The surface
Chapter 3. The Multiphysics Optimization Design Methodology
21
160
r = 0.4 cm
r = 0.7 cm
r = 1.1 cm
r = 2.1 cm
140
T ( °C)
120
100
80
60
40
2
4
6
8
10
t (min)
Figure 3.2 Temperature as a function of time at four probe points.
(i)
current density of the ith element, Js , is defined as
Js(i) =
I (i) jβ (i)
e
ẑ
2πa
(3.7)
where I (i) and β (i) are the magnitudes and phases of each current element as defined earlier.
The current elements are encapsulated by an outer layer of teflon (PTFE), a biocompatible
material commonly used as the outer sheath in microwave ablation antennas [8], [9].
i=6
i=5
i=4
2.0 mm
i=3
2.6 mm
i=2
3 mm
i=1
Figure 3.3 Source geometry for optimization with indexing.
22
Chapter 3. The Multiphysics Optimization Design Methodology
3.2.2
Simulation Domain
A cylindrical three-dimensional inhomogeneous domain was used for all simulations, as shown
in Fig. 3.4a. The cylinder has a radius of 3.7 cm, and a total height of 5.6 cm. The target
region to be ablated is a tumour in human liver, which is approximately a distorted sphere
of radius 1 cm. The electrical and thermal material parameters used for all simulations are
given in Table 3.1 [3, 38]. The domain is terminated on all sides with a perfectly matched layer
(PML) [39].
Table 3.1 Material properties used in simulations.
Liver
Tumour
Teflon
Blood
Relative
Relative
Electrical
Thermal
permittivity permeability conductivity conductivity
εr
µr
σ (S/m)
kth (W/m/K)
Heat
capacity
C
(J/kg/K)
Density
ρ
(kg/m3 )
43.0
54.9
2.1
3540
3540
1143
3617
1079
1079
1806
1050
1
1
1
1.69
1.99
1 × 10−25
0.52
0.54
0.27
Since a 3D domain is used, the visualization of data is done on a slice-by-slice basis. The
electric fields and temperature profiles are interpolated into a 3D grid of 150 × 150 × 150 cells.
In other words, we consider 150 slices of 150 × 150 cells in the xz-plane. The size of each grid
Liver
Target 4 cm
..
.
..
.
Sources
(a)
(b)
Figure 3.4 Depiction of the simulation domain and discretization into slices for optimization
and visualization.
23
Chapter 3. The Multiphysics Optimization Design Methodology
2
y (cm)
normalized |E|
1
0.5
1
0
-1
-2
0
2
1
0
-1
-2
5
4
3
2
2
x (cm)
z (cm)
x (cm)
0
(a)
5
4
3
5
4
3
2
1
z (cm)
(b)
2
y (cm)
55
50
T ( °C)
-2
45
1
0
-1
40
-2
2
1
0
-1
x (cm)
(c)
-2
5
4
3
2
z (cm)
2
0
x (cm)
-2
2
1
z (cm)
(d)
Figure 3.5 Left: One xy-slice of target profiles for (a) E-based and (c) T -based optimization.
Right: Points over which optimization is performed (in black) for selected xy-slices for (b)
E-based and (d) T -based optimization. Red curves represent target boundary.
cell is approximately 0.35 × 0.35 × 0.37 mm. This is depicted in Fig. 3.4b.
3.2.3
Target Profiles
The target field profile for E-based optimization, Etarget , and the target temperature profile
for T -based optimization, Ttarget , are defined over the entire domain. However, optimization is
only performed over a set of coordinates in the vicinity of the target boundary, as described
below.
E-based Optimization
We define a normalized Etarget such that it has a value of 1 within the target, and 0 outside.
Fig. 3.5a shows Etarget and Fig. 3.5b shows the points over which optimization is performed.
These points are picked concentric to the target, both inside and outside of it.
Chapter 3. The Multiphysics Optimization Design Methodology
24
T -based Optimization
In the ideal case, the optimized temperature profile would be at or above 50◦ C within the
target, and would decay to 37◦ C just outside. The rate at which temperature decays outside
the target is a function of diffusion as well as energy deposition via the electric field. In the
ideal case, there would be no electromagnetic energy deposition outside the target, and so the
temperature profile at and outside the target boundary would be dictated purely by diffusion.
Thus, Ttarget is defined such that it has a value of 50◦ C at the target boundary, and decays to
37◦ C outside due to diffusion only, in accordance with (2.1) with |E| set to 0. Fig. 3.5c shows
Ttarget and Fig. 3.5d shows the points over which optimization is performed. These points are
picked concentric to the target, at the boundary and outside of it.
3.3
Results and Comparisons
The results are presented and discussed here from three points of view: the confinement of the
50◦ C contour to the target boundary, the T gradients achieved at the target boundary, and the
optimized current distributions obtained via each optimization method.
3.3.1
Confinement of the Ablation Zone
Fig. 3.6 is a 3D depiction of the optimized temperature profiles for selected slices on the xzplane, for both optimization methods. In each case, the 50◦ C contour is well confined to the
boundary of the target at all slices. However, in the T -based case the 50◦ C contour is closer
to the desired profile. Table 3.2 lists the comparison metrics f1 , f2 and f∞ for each method.
As expected, the T -based multiphysics method performs better than the purely SAR-based
method. In terms of the relative error, Euclidean norm and infinity norm, the T -based method
provides a 39.3%, 70.1% and 67.9% improvement over the SAR-based method.
Table 3.2 Relative, Euclidean and infinity norm
errors for comparing optimization methods.
f1
f2
f∞
gB
gmax
SAR-based
optimization
T -based
optimization
4.96 × 104
2.30 × 105
18.2
30.8 ◦ C/cm
8.06 × 103 ◦ C/cm
3.01 × 104
6.87 × 104
5.85
44.8 ◦ C/cm
2.22 × 104 ◦ C/cm
Chapter 3. The Multiphysics Optimization Design Methodology
(a)
25
(b)
Figure 3.6 Temperature profiles at selected xz-slices of the 3D domain. (a) SAR-based optimization, and (b) T -based optimization. Target region boundary shown in red; 50◦ C contour
shown in black.
3.3.2
Temperature Gradient at Target Boundary
Fig. 3.7 shows a surface plot of the optimized T profiles for each method for one xz slice at
y ≈ 0 cm. It is clear that the temperature falls off with a much sharper gradient in the T -based
case than in the SAR-based case, and thus is less likely to cause damage to surrounding healthy
tissue. Sharper T gradients allow for more precise targeting. Table 3.2 lists the comparison
metrics gB and gmax to quantify these gradients. At the boundary of the target, along the x̂
direction, T -based optimization provides a 45.5% sharper gradient. The maximum slope along
the the x̂ direction is 2.75 times steeper for the T -based case.
Fig. 3.8 shows the normalized SAR profile in dB for the same slice as above. It is interesting
to note that the electric field is more strongly confined within the target in the T -based case
than the SAR-based case. This contributes to the fact that sharper T gradients are achieved
in the former, since the T profile outside the target is influenced more by diffusion of heat than
deposition of energy via the electric field. This is a direct result of the target profiles that
were chosen in each optimization method. In fact, the importance of heat diffusion is part of
the reason that thermal inhomogeneities in tissue play an important role in the temperature
distribution, and must be taken into account in addition to the electric field distribution.
3.3.3
Optimized Currents
Fig. 3.9 shows the optimized current source magnitudes (top row) and corresponding phases
(bottom row) for each element, for each optimization method. The optimized phases are fairly
constant along the antenna, which indicates that controlling the current distribution magnitude
alone provides sufficient degrees of freedom in synthesizing the target temperature profiles. The
current magnitudes obtained are realistic, and follow a distribution that can be implemented
26
Chapter 3. The Multiphysics Optimization Design Methodology
150
T ( °C)
T ( °C)
150
90
70
100
130
110
90
100
70
50
50
50
2
1
0
-1
-2
5
3
4
50
2
2
1
z (cm)
x (cm)
0
-1
-2
3
4
5
z (cm)
x (cm)
(a)
2
(b)
Figure 3.7 Surface plot of T for an xz-slice at y ≈ 0 cm. (a) SAR-based optimization, and
(b) T -based optimization. Target region boundary shown in red. Empty region in the centre
represents position of source array.
-3
-6
-10
-10
-20
-30
2
1
0
-1
-2
x (cm)
(a)
-3
-6
-10
0
|E| (dB)
|E| (dB)
0
5
4
3
-10
-20
2
-30
2
z (cm)
1
0
-1
-2
x (cm)
5
4
3
2
z (cm)
(b)
Figure 3.8 Surface plot of normalized |E| (in dB) for an xz-slice at y ≈ 0 cm. (a) SAR-based
optimization, and (b) T -based optimization. Target region boundary shown in red. Empty
region in the centre represents position of source array.
in practice, as shown in the following section, where their realization is discussed.
In terms of computational cost, the SAR-based method took approximately six hours to
converge, while the T -based approach took approximately eight hours. The added cost is
justified by the significant advantages provided by the T -based method.
3.4
Physical Implementation of Optimized Current Elements – A
Proof of Concept
In this section, a possible implementation of the optimized current distributions is presented.
This design is a proof of concept demonstrating that ablation antennas can be designed in
an inverse way by synthesizing desired current distributions. A more comprehensive antenna
design is presented in Chapter 3; the purpose of this section in simply to establish that the
27
Chapter 3. The Multiphysics Optimization Design Methodology
2
1
Surface current magnitude (A)
Surface current magnitude (A)
1.2
0.8
0.6
0.4
0.2
0
1.5
1
0.5
0
1
2
3
4
5
6
1
2
Source current element
(a)
4
5
6
(b)
:
Surface current phase (rad)
:
Surface current phase (rad)
3
Source current element
3:/4
:/2
:/4
0
3:/4
:/2
:/4
0
1
2
3
4
5
6
Source element
(c)
1
2
3
4
5
6
Source element
(d)
Figure 3.9 Top row: Optimized current source element magnitudes. (a) SAR-based optimization, and (b) T -based optimization. Bottom row: Optimized current source element phases.
(d) SAR-based optimization, and (e) T -based optimization.
optimized current distributions are realistic.
Since the optimized surface current magnitudes have a central peak and a smooth decay
on each side, we can implement the distributions via a single cylindrical dipole antenna with
a varying radius along its axis. Since the optimized phases are nearly constant, we need only
deal with the current magnitudes. The cylinder can be divided into six sections, each section
corresponding to one of the sources in the optimization procedure described above. Based on
the continuity of the normal component of the current density at an interface, a simple method
to vary the current magnitude is to modulate the cross-sectional area of the antenna. Hence,
the radius of each section is chosen to be proportional to the surface current magnitude required
in that section.
The antenna was simulated inside the same tissue structure that was used for the optimization procedure, to enable direct comparison of the resulting temperature profiles. The
antenna cylinder radii are chosen such that the resulting current in each section corresponds
to the T -based optimized currents obtained earlier. The length of each section is equal to the
Chapter 3. The Multiphysics Optimization Design Methodology
2.50 mm
1.25 mm
0.41 mm
Feed point
1.26 mm
0.44 mm0.09 mm
2
Surface current (A)
28
Physical implementation
Target
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
Distance along probe (cm)
Figure 3.10 Proposed proof of concept design and its surface current distribution, compared
to target currents.
length of each source that was used for optimization (2 mm). Copper was used as the antenna
material. The feed point was placed in the middle of the largest section, with a 250 V applied
voltage, which provides the correct scaling of currents to match the target currents. The surface
currents are sampled at the midpoint of each cylindrical section for comparison with the target
currents that we are trying to synthesize.
Fig. 3.10 shows the structure that was simulated and the resulting surface current distribution along its length. The distribution is a fairly accurate replication of the optimized currents
desired in each section. Fig. 3.11a and Fig. 3.11b show the temperature profile generated by this
design. It is clear that the temperature profile is in good agreement with the profile generated
via T -based optimization of six sources as in Fig. 3.6b and Fig. 3.7b.
Although it is important to design a suitable feeding network for the proposed dipole antenna, this proof of concept demonstrates that current distributions obtained via optimization
can be implemented physically in principle, to generate controllable temperature profiles. The
presented design is fairly straightforward, reconfigurable (cylinders of different radii can be
assembled in any configuration), and resembles commercial antennas while achieving targeted
ablation in a more intelligent and controllable way. This design is presented as a straightfor-
29
Chapter 3. The Multiphysics Optimization Design Methodology
T ( °C)
150
130
110
90
100
70
50
50
2
1
0
-1
x (cm)
(a)
-2
5
4
3
2
z (cm)
(b)
Figure 3.11 (a) Temperature profile generated by the proposed design at selected xz-slices
of the 3D domain. Target region boundary shown in red; 50◦ C contour shown in black. (b)
Surface plot of T generated by the proposed design for an xz-slice at y ≈ 0 cm. Target region
boundary shown in red. Empty region in the centre represents position of source.
ward way of achieving desired current distributions on the surface of an antenna. However,
there several ways of manipulating source currents, and a more detailed and complete antenna
design is presented in the next chapter.
3.5
Summary
A multiphysics inverse design methodology for the design of MWA sources was presented in this
chapter. The multiphysics approach was compared with the standard approach of considering
only the electrical domain without accounting for thermal properties of tissue. The results
clearly indicated the advantages of the multiphysics approach in terms of generating more
accurate ablation zones based on a given target. The multiphysics approach provides steeper
temperature gradients at the target boundary, thus reducing damage to surrounding healthy
tissue. A proof-of-concept design consisting of a dipole with varying radii along its axis was
presented as a physical implementation of the optimized current distributions yielded by the
optimization methodology. A more detailed, printed antenna design is presented in the next
chapter.
Chapter 4
A Printed Antenna Concept for
Microwave Ablation
4.1
Introduction
The purpose of this chapter is to design a realistic MWA antenna concept that can be optimized
using the multiphysics procedure described above. The antenna is designed such that its current
distribution can be controlled and optimized based on a desired target temperature profile. This
is achieved by specifying simple design parameters to manipulate the current distribution on
the antenna. The antenna is easily configurable to allow ablating targets of different shapes and
sizes, and allows the possibility of reconfigurability. This antenna concept has not previously
been applied to microwave ablation, to the best of our knowledge.
4.2
Design Considerations
There are several constraints within which such an antenna must operate. The relative electrical
permittivity in biological tissue commonly lies in the range 40 − 65 [40]. At a frequency of
2.45 GHz this implies that the wavelength in tissue is approximately in the range 1.5 – 2.0 cm.
In general, from a clinical perspective, the diameter of an interstitial device such as an ablation
antenna or probe of any sort must not exceed 3 mm, for safe percutaneous insertion. Also, since
MWA is generally applied to ailments lying closer the skin, and for target regions no smaller
than ≈ 1 cm, the total length of the antenna should typically not exceed ≈ 2 cm. Since the
antenna must be placed within a biocompatible casing, the entire antenna must fit within a
cylinder of diameter 2.5 mm and height 2 cm for it to comply with the above constraints and
be comparable to existing clinically acceptable designs. In electrical lengths, this corresponds
to a maximum diameter on the order of ≈ λ/7, and maximum height on the order of ≈ 2λ. In
addition, another design consideration was to attempt a printed antenna design, as it would be
easier to manufacture, potentially cheaper and has not been attempted before in the context of
30
Chapter 4. A Printed Antenna Concept for Microwave Ablation
31
MWA, to the best of our knowledge.
Although the optimization of a set of current element magnitudes and phases brings to mind
the properties of phased arrays, the space and manufacturing constraints make it difficult to
implement phased arrays here. Even if four to six antenna elements were able to fit, it would
be very challenging to design a feed network that satisfies these space constraints while allowing sufficient control over individual element excitations without excessive coupling between
elements and the feed lines.
With the above considerations in mind, it is more reasonable to attempt to manipulate the
current distribution on a single antenna with a simple feed, rather than attempt to fit a phased
array with a complex feed network. The long but narrow nature of a typical dipole antenna
lends itself quite well to the space constraints described above. Also, simple manipulations to
the arms and feed of a dipole antenna can allow the current distribution to be manipulated to
some extent. Two straightforward ways of achieving this are loading the arms of the dipole
reactively, and varying the position of the feed point along the dipole. Thus, in this case,
the reactive loading on the arms, and the feed position along the dipole, are the optimization
parameters, rather than current magnitudes and phases.
Given that the reactive loading of the dipole arms and position of the feed are to be picked
based on an optimization procedure for a particular target temperature profile, there is no
fixed input impedance to which a feed can be matched. During the optimization procedure, the
entire antenna structure must be simulated at each iteration. Thus the widths of the lines that
feed the dipole cannot be picked beforehand to have a particular impedance; instead, it is more
meaningful to study the input impedance at the port that connects to the entire structure,
rather than at the dipole gap. The widths of the feed line are thus chosen based on space
constraints, and not their impedance. These geometric characteristics can also be included
as optimization variables (in addition to the reactive loading and feed position), however this
would greatly increase the computational cost.
For the same reasons cited earlier, we focus on designing the antenna such that it is capable
of generating symmetric, spherical ablation zones, while allowing the possibility of generating
non-symmetric profiles as well.
4.3
Design Description and Optimization
The geometry of the proposed antenna is shown in Fig. 4.1. It consists of two back-to-back
dipoles printed on either side of a PCB strip. This configuration is necessary such that the
structure is symmetric about the plane that cuts through the PCB strip, to enable the generation
of symmetric ablation zones. The dipoles are loaded along the arms with lumped capacitors,
which can be surface-mounts or printed. The printed arms of the dipole have gaps to allow for
this loading. There are two capacitors per arm; this number can be increased or decreased as
per specific requirements. Including more capacitors would allow finer control of the current
32
Chapter 4. A Printed Antenna Concept for Microwave Ablation
Port 1
Port 2
λ = 18.67 mm
C1
C2
C3
1.270 mm
1.500 mm
C4
C1
C2
C3
0.381 mm
C4
0.127 mm
λ = 18.67 mm
0.254 mm
0.127 mm
0.381 mm
Ground
(a)
Copper Shield
Teflon
Water
FR4
(b)
Figure 4.1 Proposed antenna design. (a) Perspective view with dimensions and blow-up of
metallization layers. (b) Front and side views including outer teflon casing.
distribution on the arms, but would increase the number of variables to optimize for, and thus
increase computational complexity.
The two dipoles are each one wavelength in length, where the wavelength is calculated
with respect to the permittivity of the biological tissue being targeted. Although the input
impedance of a single-wavelength dipole in infinite, since the arms are to be reactively loaded
Chapter 4. A Printed Antenna Concept for Microwave Ablation
33
5.0 cm
Liver
Ablation Antenna
Target
4.5 cm
Figure 4.2 Simulation geometry.
and the feed position is not fixed at the centre, this limitation does not apply. The PCB is
encased in teflon (PTFE) in accordance with existing ablation antenna designs [9, 10], and the
space in between the PCB strip and the teflon casing is assumed to be filled with flowing water
to maintain the inner temperature at 20◦ C [41]. Most surface mount capacitors available on
the market are rated for temperatures up to ≈ 150◦ C, and FR4, which is used as the PCB
material, is rated for temperatures up to ≈ 140◦ C; thus, the high temperatures reached during
ablation (≈ 100◦ C) are not expected to be of concern. Additionally, since cool flowing water
is pumped through the device, the temperatures near the PCB will be significantly lower than
100◦ C.
The dipoles are fed by a dual stripline, where the signal lines and ground planes are embedded within the multilayer board, and connected to the dipole arms through vias. The ground
planes are tapered gradually (over 3λ/4) to provide a balanced signal to the dipoles. Each
dipole and its feed are treated as a separate (but coupled) single-port network. Each port is
treated as a lumped port with a prescribed input voltage. It is assumed that the ports are fed
by a 50 Ω transmission line.
The region to be ablated is assumed to be in human liver, and the simulation domain is a
cylinder as shown in Fig. 4.2. The antenna is designed for an operating frequency of 2.45 GHz.
The electrical and thermal material properties used for simulating the antenna are given in
Table 4.1 [3, 38]. The resulting physical dimensions of the antenna are shown in Fig. 4.1. The
width of each arm is chosen to be 1 mm, the feed line width is 0.2 mm, and the total PCB
length is 3.25 cm.
To control the shape of the generated temperature profile, there are several sets of parameters that can be varied. The values of the capacitors, the positions of the capacitors along the
arms, and the feed point along the length of the dipole can all be picked to modulate the current
distribution on the arms. These parameters can also be optimized to obtain a desired temperature profile. We first study the effects of varying the capacitance values and the feed positions
independently, to establish that the current distribution on the dipole can be controlled to an
34
Chapter 4. A Printed Antenna Concept for Microwave Ablation
extent. We then present three optimization case studies to demonstrate the performance of the
design.
4.3.1
Effects of Varying Capacitance Values
The current through each capacitor should be inversely related to the reactance of that element.
Thus, by varying the values of the capacitors, and reactance and thus the current flow can be
controlled. Five test cases were simulated. The capacitance and reactance values, as well as the
input voltage Vin applied to each port, are outlined in Table 4.2. The two dipoles are identical
in each case. For these tests, the capacitors are placed at 1/6, 2/6, 4/6 and 5/6 of the way
along the length of the dipoles. In each case, the feed is at the center of each dipole.
4.3.2
Effects of Varying the Feed Position
In this case, the lumped capacitor positions are fixed at 0.15d, 0.25d, 0.75d and 0.85d along the
dipoles, where d is the total dipole length measured from the end of the dipole closer to the
feed. In this case, d = λ. The value of each capacitor is fixed at 65 pF. Three feed positions (as
measured from the end of the dipole closer to the feed) were studied: in case 1, the feed is at
0.5d; in case 2, it is at 0.33d; and in case 3, it is at 0.67d. In cases 1 and 2, an input voltage of
10 V was applied to each port. In case 3, an input voltage of 14 V was used. In all cases, the
Table 4.1 Material properties used in simulations.
Liver
Water
Teflon
FR4
Relative
Relative
Electrical
Thermal
permittivity permeability conductivity conductivity
εr
µr
σ (S/m)
kth (W/m/K)
Heat
capacity
C (J/kg/K)
Density
ρ
(kg/m3 )
43.0
80.0
2.1
4.5
3540
4187
1143
1369
1079
1000
1806
1900
1
1
1
1
1.69
5.5 × 10−6
1.0 × 10−25
0.004
0.52
0.59
0.27
0.3
Table 4.2 Input parameters - variable capacitance test cases. Capacitance
values are in pF; reactances are in Ω.
Case
Case
Case
Case
Case
1
2
3
4
5
Vin (V)
C1 (X1 )
C2 (X2 )
C3 (X3 )
C4 (X4 )
14
5
12
4
15
65.0
0.11
65.0
65.0
65.0
1.30
0.13
65.0
65.0
65.0
1.30
65.0
65.0
65.0
0.13
65.0
65.0
65.0
0.11
0.11
(−1)
(−600)
(−1)
(−1)
(−1)
(−50)
(−500)
(−1)
(−1)
(−1)
(−50)
(−1)
(−1)
(−1)
(−500)
(−1)
(−1)
(−1)
(−600)
(−600)
Chapter 4. A Printed Antenna Concept for Microwave Ablation
35
two dipoles are identical.
4.3.3
Design Optimization for a Test Target
In order to demonstrate the optimization of the design, we take three test cases, each consisting
of a target with a different shape. The multiphysics temperature-based optimization procedure
described earlier is employed here. Each target is a closed surface resembling an ellipsoid in
human liver tissue, as depicted in Fig. 4.2. A corresponding target temperature profile, Ttarget ,
is defined in each case such that the temperature is above 50◦ C within the target region, at
50◦ C at the target boundary, and decays to normal body temperature (37◦ C) elsewhere. The
goal is to optimize the capacitor values, input voltage at each port, and the feed position, to
generate a temperature profile T that approaches Ttarget . To achieve this, the parameters stated
above are optimized to minimize the following objective function:
fT =
X
m,n
T m,n − Ttarget
2
(4.1)
m,n∈B
where B is the set of coordinates corresponding to the target boundary. The capacitor positions
are fixed at 0.15d, 0.25d, 0.75d and 0.85d along the dipoles. These positions can also be set as
optimization variables, which may lead to more accurate results but a greater computational
cost. The capacitance values are constrained between 0.1 pF and 6.5 nF; the input voltage is
constrained within 0.1 V and 1000 V; and the feed position is constrained within 0.3d and 0.7d,
measured from the end of the dipole closer to the feed.
4.3.4
Simulation Details
The forward problems of computing electric fields and temperature profiles were solved using
COMSOL Multiphysics. This was interfaced with Matlab for the optimization procedure and
for visualization of results.
As described previously, in order to reduce computational cost, it is beneficial to solve (2.1)
at steady state using a stationary solver. To establish the validity of this approach, the antenna
was simulated to compute temperatures in the time domain. The temperatures were probed at
four points at varying distances (r) from the antenna, and plotted as a function of time (t) to
study the time domain characteristics. The results are shown in Fig. 4.3.
It is clear that steady state temperatures were achieved after approximately 7 minutes;
since microwave ablation is generally applied for 5 − 10 minutes, it is confirmed that results can
be computed at steady state to reduce computational cost. For optimization, the sequential
quadratic programming (SQP) algorithm was used for objective minimization. The simulation
domain is terminated on all sides with a scattering boundary condition: the COMSOL equivalent of a radiation boundary condition. For each test case, including optimization, the port
impedance, input power, delivered power and reflection coefficient are reported.
Chapter 4. A Printed Antenna Concept for Microwave Ablation
36
120
r = 0.4 cm
r = 0.7 cm
r = 1.1 cm
r = 2.1 cm
T ( °C)
100
80
60
40
2
4
6
8
10
t (min)
Figure 4.3 Temperature as a function of time at four probe points.
4.4
Results and Analysis
The dipole current distributions, temperature profiles and lumped port parameters for each of
the test cases are discussed in this section. The following port parameters are reported: input
power, Pin , total power delivered, Pdel , input impedance, Zin and reflection coefficient S11 .
Fig. 4.4a and Fig. 4.4b show the current distribution and temperature profiles for each
Table 4.3 Port parameters - variable capacitance test cases.
Case
Case
Case
Case
Case
1
2
3
4
5
Pin (W)
Pdel (W)
Zin (Ω)
S11 (dB)
13.9
10.0
29.7
17.0
17.6
12.0
9.70
28.3
16.8
15.4
42.3 − 36.4j
54.2 − 16.3j
51.9 + 22.9j
48.0 − 9.35j
60.7 − 40.8j
−8.51
−16.0
−13.2
−20.3
−8.94
Table 4.4 Port parameters - variable feed position test cases.
Case 1
Case 2
Case 3
Pin (W)
Pdel (W)
Zin (Ω)
S11 (dB)
25.6
16.8
15.6
24.6
15.8
13.7
52.8 + 20.6j
70.2 + 21.9j
84.4 + 35.2j
−14.1
−12.3
−9.02
37
Chapter 4. A Printed Antenna Concept for Microwave Ablation
test case with a varying capacitance. Results are only reported for one of the two ports, as
the two dipoles are identical in this case. The correspondence between reactance values, the
current peaks and temperature profiles can clearly be seen. For example, it is clear that a
higher reactance on the lower arm of each dipole causes the current peak to be pushed towards
the upper arm, and causes the temperature to flatten towards the tip of the antenna. Thus,
manipulating the arm reactance at various points allows controlling the current and temperature
distributions to an extent. Table 4.3 shows the resulting port parameters. The range of Zin
values are fairly easy to match, and the input powers required are realistic compared to existing
ablation antennas. Despite being unmatched at the ports, the S11 in each case is fairly low.
Case 1
Case 2
Case 3
Case 4
Case 5
30
1
Case 1
Case 2
Case 3
Case 4
Case 5
I (A)
0.6
20
z (mm)
0.8
10
0.4
0
0.2
-10
0
-5
-20
0
5
10
15
-20
20
-10
0
10
20
x (mm)
z (mm)
(a)
(b)
30
1
Case 1
Case 2
Case 3
20
z (mm)
0.8
I (A)
0.6
10
0.4
0
0.2
-10
0
-5
Case 1
Case 2
Case 3
-20
0
5
10
z (mm)
(c)
15
20
-20
-10
0
10
20
x (mm)
(d)
Figure 4.4 (a) and (c): Surface current distributions. (b) and (d): Corresponding 50 ◦ C
contours at one central slice. (a) and (b): variable capacitance test cases. (c) and (d): variable
feed position test cases.
Chapter 4. A Printed Antenna Concept for Microwave Ablation
38
Fig. 4.4c and Fig. 4.4d show results for varying feed positions. Results are only reported for
one of the two ports, as the two dipoles are identical in this case. It is clear that the current
distribution and shape of the temperature profile depend on the feed position. Pushing the feed
higher or lower on the dipole allows one to create different distortions in the ellipsoidal profiles.
The port parameters are once again realistic in terms of input powers required, and the ports
are fairly straightforward to match to an input line.
Fig. 4.6 shows the resulting temperature profiles in the three optimization test cases. The
plots on the left show selected slices through the 3D domain, while the plots on the right show
a corresponding slice in the centre of the domain. The target boundary is represented by red
dots; the 50◦ C contour is a black line. It is clear that in each case, the optimized temperature
profile very closely approximates the desired temperature profile (i.e. the 50◦ C contour closely
follows the target boundary). The three target regions differ in their shapes, and the antenna
is able to accurately ablate each type of target.
Case 1
Case 2
Case 3
0.8
I (A)
0.6
0.4
0.2
0
0
5
10
15
z (mm)
Figure 4.5 Optimized current distributions on one dipole.
Fig. 4.5 shows the current distribution on one of the dipoles for each optimization test
case. It is clear that the current distribution is successfully manipulated over the three cases to
synthesize the required target temperatures, in accordance with the initial goal of this work.
Table 4.5 lists the post-optimization port parameters for each port, for each optimization
test case: required input power (Pin ), the input impedance (Zin ), reflection coefficient (S11 ), and
feed point. The required input powers are realistic from a laboratory or clinical point of view.
Despite the fact that the dipoles are not matched to the feed, and the physical dimensions of the
design have not been optimized, the S11 is at most −13.3 dB, which indicates a fairly efficient
and realistic design. The range of input impedances can quite easily be matched by a matching
39
Chapter 4. A Printed Antenna Concept for Microwave Ablation
Table 4.5 Optimized port parameters for each test case.
Case I
Case II
Case III
Pin (W)
Port 1
Zin (Ω)
S11 (dB)
7.33
16.8
14.5
43.3 − 4.33j
59.5 + 16.1j
58.5 + 22.3j
−21.4
−15.5
−13.3
Pin (W)
Port 2
Zin (Ω)
S11 (dB)
Feed point
19.8
17.2
15.7
45.2 − 0.83j
60.0 + 15.5j
59.5 + 21.0j
−25.7
−15.6
−13.7
0.39d
0.40d
0.40d
network placed before the ports. The optimized capacitances lie in the range [1.07, 272] pF,
which are physically realizable within the space constraints.
4.5
Summary
The concept of multiphysics optimization was applied to the design of a printed dipole antenna
for MWA. By appropriately loading the dipole arms with capacitors, and varying the feed
position, the current distribution on the antenna can be manipulated and optimized to generate
a desired target temperature profile. Simulations of the antenna demonstrate that it is able to
accurately ablate targets with realistic power requirements and low return losses. Thus, the
antenna is a low-profile and simple design that can be incorporated into a surgical teflon needle,
and provides control over its current distribution without the need for a phased array.
40
Chapter 4. A Printed Antenna Concept for Microwave Ablation
90
2
z (cm)
80
70
1
60
0
50
40
-1
30
-2
-1
0
1
2
x (cm)
(a)
(b)
100
z (cm)
2
80
1
0
60
-1
40
-2
-1
0
1
2
x (cm)
(c)
(d)
100
2
90
z (cm)
80
1
70
60
0
50
40
-1
30
-2
-1
0
1
2
x (cm)
(e)
(f )
Figure 4.6 Optimized temperature profiles in ◦ C. Black: 50◦ C contour. Red: target boundary.
Chapter 5
Conclusions
It has been shown qualitatively and quantitatively that in order to achieve greater accuracy
in the treatment of malignant tissue via MWA, it is important to account for the thermal
properties and inhomogeneities of the target tissue, and thus the diffusion of deposited heat
energy around the region of deposition. The need for a multiphysics approach was established
through simulations, and a novel multiphysics optimization technique for the design of MWA
probes was presented, and compared with the standard electric field-based approach.
The multiphysics methodology was applied to design a novel printed MWA antenna concept. Simulation results of the antenna indicate that it is feasible and advantageous to design
ablation antennas in an inverse way to generate controllable, customizable temperature profiles
by manipulating the current distribution on the antenna.
5.1
Significance of this Work
To the best of our knowledge, the principle of source optimization to generate desired temperature profiles has not previously been applied to MWA. The shapes, sizes, and electrical
and thermal properties of regions that are to be ablated can vary significantly from patient to
patient; the inverse multiphysics design methodology can be used to design sources that can
target ablation zones more accurately while causing less damage to healthy tissue than existing
SAR-based methods.
The printed dipole MWA antenna presented is a manifestation of the multiphysics methodology. One of the most significant advantages of this design is that it achieves, through just
two dipoles, the temperature profile patterning that one would normally obtain using a phased
array. The antenna is a low-profile and simple design that can be incorporated into a surgical
teflon needle, and provides control over its current distribution without the need for a phased
array. Additionally, the optimization of the antenna parameters presented here is significantly
simpler than the optimization of tens, hundreds or thousands of elements that is required for
HIFU or phased arrays that are used for hyperthermia treatment [36]. Yet this design achieves
more accurate, customizable and easily configurable ablation zones than existing MWA designs.
41
Chapter 5. Conclusions
42
In addition, another advantage of this design is that several existing MWA antennas operate at
powers as high as 120 W [9, 24]. However, the proposed design can generate comparably large
ablation zones at a total power in the range approximately between 30–60 W, which allows for
significant power savings.
Since this design is printed and configured quite easily, one can envision an “ablation toolkit”
consisting of a series of printed MWA antennas, each one configured differently to generate a
different canonical temperature profile based on the typical shapes of ablation zones generally
required for a certain application. For example, the toolkit could consist of an antenna that
generates spherical profiles for ablating tumours, and another antenna that generates elliptical profiles for arrhythmia. The relative ease of manufacturing several antennas on a single
PCB, which can then be cut into strips, would allow these toolkits to be more cost-effective
than manufacturing an equal number of separate non-printed antennas that currently exist in
literature.
Additionally, this design methodology can quite easily be integrated into a clinical setting,
as described below.
5.1.1
Clinical Integration
From a clinical perspective, an ablation antenna with easily configurable current element magnitudes and phases can be incorporated into treatment planning as follows:
• A stack of images of the target region can be obtained via MRI, PET, or other imaging
techniques, and the different tissues and structures are identified by contouring techniques.
• The electrical and thermal properties of the tissues and structures are obtained as documented in literature.
• The image stack is taken as the geometry to be used in the optimization procedure
described above, with the MWA antenna positioned appropriately inside the target region.
• A target temperature profile is created with respect to the image stack, such that ablative
temperatures are achieved within the target.
• The parameters of the antenna are optimized using the multiphysics techniques described
above. If an ablation toolkit consisting of several such printed antennas is available, the
antenna configuration that most closely matches the required configuration can be picked
to perform the therapy.
Ablation antennas designed in this way would allow for more predictable, controllable and
customizable ablation of targets while reducing damage to surrounding healthy tissue.
Chapter 5. Conclusions
5.1.2
43
Publications
Parts of this work have been presented at one conference and been submitted to two journals,
currently being written and under review.
• S. Sharma and C. D. Sarris, “Novel Antenna Design Concept for Temperature Profile
Synthesis in Microwave Ablation,” Physical Review Letters, 2016 (in development).
• S. Sharma and C. D. Sarris, “A Novel Multiphysics Optimization-Driven Methodology
for the Design of Microwave Ablation Antennas,” IEEE Journal on Multiscale and Multiphysics Computational Techniques, 2016 (under review).
• S. Sharma, A. Ludwig and C. D. Sarris, “Design and Optimization of Microwave Ablation
Probes for Tumour and Pain Therapy,” 2015 IEEE International Symposium on Antennas
and Propagation and North American Radio Science Meeting, July 2015.
5.2
Future Work
The work presented above should be seen only as the foundations of an exciting new way to
think about the design of MWA antennas. There is much room for improvement in the presented
methodology and design, and we believe that there is immense scope for the adaptation of this
work in a clinical context. In order to achieve this, there is a significant amount of work yet to
be done. This is discussed in the following subsections.
5.2.1
More Efficient Multiphysics Optimization
The focus of the multiphysics optimization methodology presented here was not the optimizer
itself, but rather the idea of synthesizing a source current distribution based on a target temperature profile. However, the optimization procedure can be improved significantly by making
use of impulse response functions of each source element to allow faster field computations.
This would require homemade field computation code, as well as a method to couple the field
computation with either a commercial solver of the Bioheat equation, or a homemade computation of the same. A finite differences in frequency domain (FDFD) solver was developed for 2D
electric field impulse response computation and was validated with comparisons against COMSOL; as well, a 2D Bioheat equation solver was developed to be coupled with the FDFD solver
(see Appendix A for details). However, sufficient computational resources were not available
to allow solving for the electric field and temperature profiles in 3D in a similar non-iterative
way; instead, a commercial solver had to be used, which was unable to store and reuse impulse
response functions to speed up computation.
Chapter 5. Conclusions
5.2.2
44
Uncertainty Analysis
The local tissue geometry realistically consists of uncertainties in its dimensions (given that the
geometry is like obtained via an imaging method); as well, the material properties associated
with various parts of the tissue consist of uncertainties, as tissue properties are only known to
a limited extent. The multiphysics optimization methodology can be enhanced by accounting
for these uncertainties when optimizing for the current distributions.
5.2.3
Antenna Reconfigurability
The antenna design presented above was proposed with the possibility of reconfigurability in
mind. Since the lumped reactive elements on the dipole arms consist of capacitors, it should be
possible to instead use variable capacitors (varactors) and bias them appropriately to control
their capacitance. The biasing network would consist of DC signals, and thus would not cause
interference with the microwave frequency currents through the feed lines. However, the biasing
will have to be designed carefully such that it does not distort the local electric fields significantly. However, minor distortions can simply be accounted for in the optimization procedure.
This would likely require an additional layer of metallization in the PCB strip.
Additionally, reconfigurability may also be possible in the feed postion by designing a switching network. The feed can be connected via switches to several discrete points along the dipole
arms, and the actual via that feeds the dipole can be controlled by way of biasing the switches.
Although this would be a challenging component to incorporate on an already crowded board,
it is an avenue worth exploring for the significant amount of flexibility it would provide.
5.2.4
Further Antenna Design Optimization
Several of the design parameters, such as the line widths, the dipole widths, the number of capacitors used, and the positions of the capacitors, were chosen empirically. However, the design
can be further optimized by studying the effects of varying these parameters and optimizing
them to minimize the return loss.
5.2.5
Fabrication and Testing
The permormance of the proposed antenna must be tested experimentally by fabricating it and
applying it to a test mass of tissue. Several fabrication challenges need to be overcome to do
this: a water pump must be attached to the teflon casing as well as a water reservoir, to allow
water to flow in and out of the casing during operation. The PCB must be attached to the
teflon casing to keep it in place; this can likely be achieved with the use of epoxy. A matching
network must be designed to allow feeding the antenna ports with a standard 50 Ω transmission
line. The ablation zones generated by the antenna must be studied to validate the simulation
results.
Appendix A
Finite Differences Approximation of
the Bioheat Equation
A.1
Electric Field Computation in 2D
The electric field can be computed via finite differences in the frequency domain [42]. The
advantage of a homemade electric field solver in the context of the optimization methodology
presented above, is that one can compute and store the electric field impulse response (i.e. the
electric field due to a source current of magnitude 1 A and phase of 0 rad) for each source element.
These impulse responses can then be reused algebraically via the principle of superposition,
during each iteration of the optimization procedure, rather than having a commercial solver
evaluate the fields during each iteration. This approach is described below for the case of a 2D
domain. A similar approach can be followed for 3D, given sufficient computational resources.
To evaluate the electric field due to a current source in 2D, the equations to be discretized,
with current sources, are Maxwell’s equations in the frequency domain, in an anisotropic
medium (to account for the use of a perfectly matched layer (PML, [39]):
∇ × E = −jωµs̄¯H + Jm
(A.1a)
∇ × H = jωεs̄¯E + Je
(A.1b)
where E is the electric field intensity, and H is the magnetic field intensity. Je and Jm are
electric and magnetic current sources respectively. Also,

sy sz
 sx

s̄¯ =  0

0
0
sx sz
sy
0
45
0



0 

sx sy
sz
(A.2)
Appendix A. Finite Differences Approximation of the Bioheat Equation
46
where
σx
jωε
σy
sy = 1 +
jωε
σz
sz = 1 +
jωε
sx = 1 +
(A.3a)
(A.3b)
(A.3c)
σx , σy and σz are the conductivities in the x̂, ŷ and ẑ directions respectively and ε is the
permittivity of the medium.
If we now split (A.1a) and (A.1b) up into components, we obtain:
∂Ey
∂Ez
−
= −jωµsxx Hx + Jm,x
∂y
∂z
∂Ex ∂Ez
−
= −jωµsyy Hy + Jm,y
∂z
∂x
∂Ey
∂Ex
−
= −jωµszz Hz + Jm,z
∂x
∂y
∂Hy
∂Hz
−
= jωεsxx Ex + Je,x
∂y
∂z
∂Hx ∂Hz
−
= jωεsyy Ey + Je,y
∂z
∂x
∂Hy
∂Hx
−
= jωεszz Ez + Je,z
∂x
∂y
A.1.1
(A.4a)
(A.4b)
(A.4c)
(A.5a)
(A.5b)
(A.5c)
Discretization of Maxwell’s Equations
(A.4a) - (A.5c) can be discretized in space using the 2D Yee cell [42] shown in Fig. A.1. Since the
Ez
(i, j)
Ex , Hy
(i, j +
Ey , Hx
Hz
1
2 , j)
1
2, j
(i +
(i +
Ez
1
2)
(i, j + 1)
Ey , Hx
+
1
2)
(i + 12 , j + 1)
ŷ
Ez
(i + 1, j)
x̂
Ex , Hy
(i + 1, j + 12 )
Ez
(i + 1, j + 1)
Figure A.1 2D FDTD Yee cell.
Appendix A. Finite Differences Approximation of the Bioheat Equation
47
computation will take place in matrix form in Matlab, we can make the following transformation
to get integer indices to reference each field point.
1
i→i+ k
2
1
j→j+ k
2
(A.6a)
(A.6b)
where k = [0, 1, 2 . . .] and the length of k is equal to the number of nodes. This would result
in the equivalent Yee cell shown in Fig. 2. Now we can discretize (A.4a) - (A.5c), keeping in
Ez
(i, j)
Ey , Hx
(i + 1, j)
Ex , Hy
(i, j + 1)
Hz
(i + 1, j + 1)
Ez
(i, j + 2)
Ey , Hx
(i + 1, j + 2)
ŷ
Ez
(i + 2, j)
x̂
Ex , Hy
(i + 2, j + 1)
Ez
(i + 2, j + 2)
Figure A.2 Equivalent 2D FDTD Yee cell.
mind that derivatives in z vanish, since we’re working in 2D.
Ezi+2,j − Ezi,j
i+1,j
i+1,j
= −jωµi+1,j si+1,j
+ Jm,x
xx Hx
∆y
Ezi,j+2 − Ezi,j
i,j+1
i,j+1
= −jωµi,j+1 si,j+1
+ Jm,y
yy Hy
∆x
Exi+2,j+1 − Exi,j+1
i+1,j+1
−
= −jωµi+1,j+1 si+1,j+1
Hzi+1,j+1 + Jm,z
zz
∆y
−
Eyi+1,j+2 − Eyi+1,j
∆x
(A.7a)
(A.7b)
(A.7c)
Hzi+3,j+1 − Hzi+1,j+1
i+2,j+1
= jωεi+2,j+1 si+2,j+1
Exi+2,j+1 + Je,x
(A.8a)
xx
∆y
Hzi+1,j+3 − Hzi+1,j+1
i+1,j+2
= jωεi+1,j+2 si+1,j+2
Eyi+1,j+2 + Je,y
(A.8b)
yy
∆x
Hxi+3,j+2 − Hxi+1,j+2
i+2,j+2
−
= jωεi+2,j+2 si+2,j+2
Ezi+2,j+2 + Je,z
(A.8c)
zz
∆y
−
Hyi+2,j+3 − Hyi+2,j+1
∆x
The fields above have been defined at grid points shown in Fig. A.2. However, to reduce memory
usage and redundancy, we can define separate indices for each of the field components. If the
Appendix A. Finite Differences Approximation of the Bioheat Equation
48
total number of Yee cells is (Nx × Ny ), then the field components have the following indices:
Ex :
i = [1, Nx + 1]
j = [1, Ny ]
(A.9a)
Ey :
i = [1, Nx ]
j = [1, Ny + 1]
(A.9b)
Ez :
i = [1, Nx + 1]
j = [1, Ny + 1]
(A.9c)
Hx :
i = [1, Nx ]
j = [1, Ny + 1]
(A.9d)
Hy :
i = [1, Nx + 1]
j = [1, Ny ]
(A.9e)
Hz :
i = [1, Nx ]
j = [1, Ny ]
(A.9f)
The grid points for material properties such as permittivities and permeabilities should still be
defined with reference to Fig. A.2 and are hereafter referred to with the subscript ‘m’. (A.7a)
- (A.8c) now become:
Ezi+1,j − Ezi,j
im +1,jm
m +1,jm
Hxi,j + Jm,x
= −jωµim +1,jm sixx
∆y
Ezi,j+1 − Ezi,j
im ,jm +1
m ,jm +1
= −jωµim ,jm +1 siyy
Hyi,j + Jm,y
∆x
Exi+1,j − Exi,j
im +1,jm +1
−
= −jωµim +1,jm +1 sizzm +1,jm +1 Hzi,j + Jm,z
∆y
−
Eyi,j+1 − Eyi,j
∆x
(A.10a)
(A.10b)
(A.10c)
Hzi+1,j − Hzi,j
im +2,jm +1
m +2,jm +1
= jωεim +2,jm +1 sixx
Exi+1,j + Je,x
∆y
(A.11a)
Hzi,j+1 − Hzi,j
im +1,jm +2
m +1,jm +2
= jωεim +1,jm +2 siyy
Eyi,j+1 + Je,y
−
∆x
(A.11b)
Hyi+1,j+1 − Hyi+1,j
Hxi+1,j+1 − Hxi,j+1
im +2,jm +2
−
= jωεim +2,jm +2 sizzm +2,jm +2 Ezi+1,j+1 + Je,z
∆x
∆y
(A.11c)
∂Je,x ∂Je,y
1
1
2
2 Lx,2D +
2 Ly,2D + γ I Hz = ∂y − ∂x
(∆x)
(∆y)
where
2
2
γ = ω εµ
Lx,2D
sx sy
sz
(A.12)
2


L 0 0 ... 0 0




0 L 0 . . . 0 0


= .
.. 
..
 ..
.
.


0 0 0 ... 0 L
(A.13)
(A.14)
Appendix A. Finite Differences Approximation of the Bioheat Equation
49
where 0 is a nx × nx matrix of zeros, and the size of Lx,2D is nx ny × nx ny . Also, L is a matrix
of size nx × nx defined as:


−2 1
0
0 ...




0 . . .
 1 −2 1




1 −2 1 . . . 
0


L= .

 ..





. . . 0
1 −2 1 


... 0
0
1 −2
(A.15)
Also,
Ly,2D

−2I
I
0 ...


−2I I
 I

 ..
..
= .
.


 0
0
0

0
0
0 ...
0
0
0









I −2I
I 

0
I
−2I
0
0
0
..
.
(A.16)
where I is an nx × nx identity matrix and there are ny matrices along the diagonal. Now, if
we have a y-directed source Jy defined at a point (i, j), we can treat it as two sources for Hz
as in Inan, Marshall (2011) page 350 [42], and write the equation for Hz as follows:
1
1
2
Lx,2D +
Ly,2D + γ I Hz = − (∇ × J)z
(∆x)2
(∆y)2
(A.17)
with
(∇ × J)z |i,j+1 = −
Jy |i,j
2∆x
(∇ × J)z |i,j = 0
(∇ × J)z |i,j−1 =
Jy |i,j
2∆x
(A.18a)
(A.18b)
(A.18c)
(A.18d)
Now we can calculate Ex and Ey using Maxwell’s equations:
Ex =
1
∂Hz
jωεS̄¯(1, 1) ∂y
Ey = −
1
∂Hz
¯
jωεS̄ (2, 2) ∂x
(A.19a)
(A.19b)
50
Appendix A. Finite Differences Approximation of the Bioheat Equation
A.2
Computation of the Bioheat Equation in 2D
Given the electric fields computed above, the temperature profile is computed by discretizing
the Bioheat equation [30]:
ρCp
∂T
σ
= |E|2 + ∇ · (k∇T ) + Wb Cb (Tb − T )
∂t
2
(A.20)
where Cp is the heat capacity per unit mass, ρ is the mass density, k is the thermal conductivity,
Wb is the mass flow density for blood, Cb is the specific heat for blood, and Tb is the normal
blood temperature, 37◦ C. In the frequency domain, equation 1 becomes:
jωρCp T (ω) =
σ
|E(ω)|2 + ∇ · (k∇T (ω)) + Wb Cb (Tb − T (ω))
2
(A.21)
This can be discretized in 2D space as done before. First we define the following variables:
Ti,j+1 − 2Ti,j + Ti,j−1
∆x2
Ti+1,j − 2Ti,j + Ti−1,j
Ỹ =
∆y 2
X̃ =
Z̃ = 0
(A.22a)
(A.22b)
(A.22c)
Then,
−k X̃ + Ỹ + (jωρCp + Wb Cb ) Ti,j = 0.5σ (Ei,j )2 + Wb Cb Tb
(A.23)
Now, following the same discretization approach as in Inan, Marshall (2011), page 349 [42], we
can write this as:
−k
−k
2
2 Lx,2D +
2 Ly,2D + γI T = 0.5σ (E) + Wb Cb Tb
(∆x)
(∆y)
(A.24)
where T and E are vectorized, and
γ = jωρCp + Wb Cb
(A.25)
Now T can be calculated via matrix inversion.
A.3
Validation of FDFD Results
The electric field computation via FDFD was compared with COMSOL to confirm the correct
functioning of the PML. The z component of the electric field, Ez , computed by the homemade
FDFD code and by COMSOL are shown respectively in Fig. A.3a and Fig. A.3b. For a better
comparison, Fig. A.3c and Fig. A.3d show Ez along the lines drawn in Fig. A.3a and Fig. A.3b.
Fig. A.3c shows Ez along the vertical lines. Fig. A.3d shows Ez along the horizontal lines. It is
clear that the finite differences solver computes the electric fields accurately, and if implemented
Appendix A. Finite Differences Approximation of the Bioheat Equation
51
in 3D, it is a suitable method for calculating the electric field impulse responses for each source
element, and can be stored and used in the optimization procedure to speed up the process.
(a)
(b)
(c)
(d)
Figure A.3 (a) Ez computed via homemade FDFD. (b) Ez computed in COMSOL. (c) Ez
along the vertical test line shown in (a) and (b). Blue line: computed in COMSOL. Red crosses:
computed via FDFD. (d) Ez along the horizontal test line shown in (a) and (b). Blue line:
computed in COMSOL. Red crosses: computed via FDFD. Ez values are in V/m.
Appendix B
Material Property Estimation by
Inverse Scattering
B.1
Introduction
In this section, a separate but related project is discussed. It is often necessary to determine
the electrical, thermal or other material properties of an unknown object or domain. This is
particularly necessary for the optimization methodology discussed in this work; it is important
to estimate as accurately as possible the electrical and thermal properties of the target region
in tissue in order to optimize the MWA antenna. This is particularly difficult if the electrical
and thermal discontinuities are to be accounted for, which is important for accurate targeting,
as shown in Chapter 2.
A method to estimate and reconstruct the electrical properties of an unknown 2D domain
was presented by Rekanos in 2002 [43] (subsequently published in further detail in 2012 [44]),
and is presented and discussed here in the context of MWA. If a set of electromagnetic transmitters and receivers is placed outside the region of the interest to reconstruct, the scattered fields
can be analyzed to reconstruct the electrical properties of the target region. Rekanos presents
a finite-differences in time-domain algorithm to perform this reconstruction, and the algorithm,
referred to as an inverse scattering algorithm, is tested via simulations. We implement the
algorithm in Matlab.
B.2
The Inverse Scattering Algorithm
We assume that the unknown domain of interest is in 2D, and is surrounded by eight transmitter/receiver pairs evenly spaced around the target domain in a circle. This setup is illustrated
in Fig. B.1. The entire domain is divided into 70 × 70 Yee cells for the FDTD procedure [45],
and electromagnetic fields are evaluated in the entire domain. This includes an outer padding
of 6 PML cells [39] along the boundary. The target domain, in the centre of the whole domain,
52
53
Appendix B. Material Property Estimation by Inverse Scattering
consists of 26 × 26 cells. The size of each cell is 6.25 mm. Each transmitter/receiver is a point
current source in the z direction, which is normal to the plane of the page, facing outwards.
70
60
50
40
30
20
10
0
0
20
40
60
80
Figure B.1 Geometry for 2D FDTD-based inverse scattering algorithm. Small black squares:
transmitter/receiver pairs. Yellow shaded square: unknown domain being imaged.
The scattered fields are sampled at the receiver points. The transmitters are iteratively
excited one-by-one, and the scattered fields are sampled at each of the other seven receiver
points, other than the one that corresponds to the transmitter for a particular iteration. An
objective functional is defined based on the differences between the measured field and the
calculated field based on the current estimate of the properties of the unknown domain. This
functional is minimized using a conjugate gradients algorithm. The algorithm is presented in
detail in [43] and [44]; the purpose of this section is to outline the computational differences in
our implementation.
In our implementation, each transmitter provides a modulated Gaussian excitation as follows:
Jz = exp
− (t − t0 )2
t2w
!
sin (ωt)
(B.1)
where Jz is the source current density, t0 is the time point corresponding to the peak of the
modulated Gaussian, tw is related to the temporal standard deviation of the pulse, and ω is
the angular frequency of the modulating sinusoid. In this case, the frequency is chosen to be
2.31 GHz. 301 time steps are used, ranging from t = 0 to t = 2.17 ns, with a time step of
Appendix B. Material Property Estimation by Inverse Scattering
54
7.22 ps. This excitation is chosen such that it is temporally and spatially smooth enough to
prevent numerical errors in the FDTD implementation.
The electric fields generated by each source are calculated via homemade FDTD code based
on [45], accounting for the PML based on [46]. For the line-search minimization step in the
optimization algorithm described in [43] and [44], we used Matlab’s patternsearch function
to evaluate the step length that minimizes the described functional for each iteration. The
optimization procedure attempts to image the permittivity, permeability and conductivity of
the target region. For applications in MWA where targeted tissues are generally non-magnetic,
it is sufficient to image only for the permittivity and conductivity.
B.3
Results
To test the algorithm, we assume that the target region contains two objects, with a relative
permittivity profile as shown in Fig. B.2a, and an electrical conductivity profile as shown in
Fig. B.3a. The optimization procedure was run for 40 iterations. The resulting imaged relative
permittivity profile is shown in Fig. B.2b; the imaged conductivity profile is shown in Fig. B.3b.
It is clear that the imaged profiles are fairly good approximations of the targets. The infinity norm (maximum deviation) in relative permittivity is 0.59; the infinity norm in electrical
conductivity is 0.12 S/m.
(a)
(b)
Figure B.2 (a) Actual electrical permittivity profile to be imaged. (b) Imaged permittivity
profile obtained via inverse scattering algorithm. Actual profile in semi-transparent red.
B.4
Relevance to Microwave Ablation
Given an algorithm that is able to reconstruct the electrical properties of a target domain,
a set of transmitter/receiver pairs can be used to image these properties in the target region
to be ablated. Knowledge of these properties is necessary in order to implement the MWA
Appendix B. Material Property Estimation by Inverse Scattering
(a)
55
(b)
Figure B.3 (a) Actual electrical conductivity profile to be imaged. (b) Imaged conductivity
profile obtained via inverse scattering algorithm. Actual profile in semi-transparent red.
antenna optimization methodology described in this thesis. In addition, there are further future
directions that must be taken in order to make full use of the inverse scattering algorithm:
• The algorithm must be extended to 3D in order to allow full compatibility with 3D
ablation targets.
• An equivalent algorithm must be developed to reconstruct the thermal properties of the
target region to enable the use of the multiphysics optimization methodology.
• Uncertainty quantification should be incorporated into the inverse scattering procedure to
account for the finite resolution of medical imaging methods, and allow for more accurate
targeting in designing MWA antennas.
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