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Microwave techniques for breast cancer detection and treatment

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MICROWAVE TECHNIQUES FOR
BREAST CANCER DETECTION AND TREATMENT
By
BIN GUO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2007
1
3311625
2008
3311625
c 2007 Bin Guo
░
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To my parents and my wife
3
ACKNOWLEDGMENTS
I would like to sincerely thank my advisor, Professor Jian Li, for her support,
encouragement, and guidance in this research. I am greatly indebted to her guidance
throughout the development of this dissertation. I also want to thank my co-advisor,
Professor Henry Zmuda. My research benefited greatly from his insightful suggestions on
electromagnetics. My special appreciation is due to Professor Liuqing Yang, and Professor
Rongling Wu for serving on my supervisory committee, and for their valuable guidance
and constructive comments. I am deeply grateful to Professor Mark Sheplak and Professor
Lou Cattafesta for their comments and suggestions.
I gratefully acknowledge Dr. Luzhou Xu, Yao Xie, and Dr. Yanwei Wang for their
help during this work. I wish to thank all the lab members, Dr. Jianhua Liu, Dr. Guoqing
Liu, Dr. Yijun Sun, Dr. Yi Jiang, Dr. Zhisong Wang, Dr. Hong Xiong, Xiayu Zheng,
Xumin Zhu, and William Roberts, in the Spectral Analysis Laboratory group with whom
I had the great pleasure of interacting. Finally, I would like to thank all of the people who
helped me during my Ph.D. study.
This dissertation is dedicated to my parents and my wife, for everything they did for
me.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CHAPTER
1
2
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.1
1.2
1.3
1.4
1.5
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ADAPTIVE MICROWAVE IMAGING FOR BREAST CANCER DETECTION
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2.1
2.2
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2.3
2.4
2.5
2.6
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Background . . . . . . . . . . . . . .
Microwave Imaging . . . . . . . . . .
Microwave Induced Thermal Acoustic
Microwave Hyperthermia Treatment
Outline of This Dissertation . . . . .
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Imaging
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Model and Problem Formulation . . . . . . . . . . . . .
2.2.1 Data Collection . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Early-Time Response Removal . . . . . . . . . . . . . .
2.2.3 Signal Time-Shifting, Windowing, and Compensation .
2.2.4 Data Model . . . . . . . . . . . . . . . . . . . . . . . .
Adaptive Microwave Imaging . . . . . . . . . . . . . . . . . .
2.3.1 Robust Capon Beamformer (RCB) . . . . . . . . . . . .
2.3.2 Amplitude and Phase Estimation (APES) . . . . . . . .
2.3.3 Comparison of Different Microwave Imaging Algorithms
Modeling and Simulation . . . . . . . . . . . . . . . . . . . . .
Numerical Examples . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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MULTI-FREQUENCY MICROWAVE INDUCED THERMAL ACOUSTIC IMAGING
FOR BREAST CANCER DETECTION . . . . . . . . . . . . . . . . . . . . . . 50
3.1
3.2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Microwave Properties of Human Breast . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Cutoff Frequency of Human Breast . . . . . . . . . . . . . . . . . . 52
3.2.2 Microwave Energy Absorption Properties of Breast Tissues and Tumor 53
3.3 Multi-frequency Adaptive and Robust Technology (MART) for Breast Cancer
Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1 Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.2 Multi-frequency Adaptive and Robust Technology (MART) . . . . . 58
5
3.3.2.1 Stage I . . . . . . . . . . . . .
3.3.2.2 Stage II . . . . . . . . . . . .
3.3.2.3 Stage III . . . . . . . . . . . .
3.4 Modeling and Simulations . . . . . . . . . . .
3.4.1 Electromagnetic Model and Simulation
3.4.2 Acoustic Model and Simulation . . . .
3.5 Numerical Examples . . . . . . . . . . . . . .
3.6 Conclusions . . . . . . . . . . . . . . . . . . .
4
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TIME REVERSAL BASED MICROWAVE HYPERTHERMIA TREATMENT
OF BREAST CANCER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.1
4.2
4.3
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Hyperthermia .
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77
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CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . .
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4.4
Introduction . . . . . . . . . . .
Time-Reversal Based Microwave
Model and Numerical Results .
4.3.1 Model and Simulation . .
4.3.2 Numerical Results . . . .
Conclusions . . . . . . . . . . .
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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Hardware Implementation for The Breast Cancer Detection and
Treatment Systems . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Waveform Diversity Based Ultrasound System for Hyperthermia
Treatment of Breast Cancer . . . . . . . . . . . . . . . . . . . .
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APPENDIX
A
A NEW FDTD FORMULATION FOR WAVE PROPAGATION IN BIOLOGICAL
MEDIA WITH COLE-COLE MODEL . . . . . . . . . . . . . . . . . . . . . . . 91
A.1
A.2
A.3
A.4
B
Introduction . . . . .
Formulations . . . .
Numerical Examples
Conclusions . . . . .
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91
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NONLINEAR ACOUSTIC EFFECT IN MICROWAVE INDUCED THERMAL
ACOUSTIC IMAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B.1 Shock Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B.2 Second Harmonic Content . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6
LIST OF TABLES
Table
page
1-1 Comparison of various microwave techniques . . . . . . . . . . . . . . . . . . . .
21
1-2 Typical dielectric properties of breast tissues . . . . . . . . . . . . . . . . . . . .
21
2-1 Merits and limitations of various microwave imaging algorithms . . . . . . . . .
38
3-1 Cole-Cole parameters for biological tissues . . . . . . . . . . . . . . . . . . . . .
68
3-2 Acoustic parameters for biological tissues . . . . . . . . . . . . . . . . . . . . . .
68
4-1 Typical thermal properties of breast tissues . . . . . . . . . . . . . . . . . . . .
83
A-1 Cole-Cole parameters for fatty tissue and muscle tissue . . . . . . . . . . . . . .
97
7
LIST OF FIGURES
Figure
page
1-1 Microwave techniques for breast cancer detection . . . . . . . . . . . . . . . . .
22
1-2 Dielectric properties of human tissues over a wide frequency band . . . . . . . .
23
1-3 Physical basis of the thermoacoustic effect . . . . . . . . . . . . . . . . . . . . .
23
1-4 Model of microwave induced TAI for breast cancer detection . . . . . . . . . . .
24
1-5 Physical basis of time-reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2-1 Cross Sections of a 3-D breast model . . . . . . . . . . . . . . . . . . . . . . . .
39
2-2 Data collection geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2-3 3D images obtained via four different imaging algorithms in the presence of a 6
mm-diameter tumor but in the absence of noise . . . . . . . . . . . . . . . . . .
41
2-4 Cross section images obtained via four different imaging algorithms in the presence
of a 6 mm-diameter tumor but in the absence of noise . . . . . . . . . . . . . . . 43
2-5 3D images obtained via four different imaging algorithms in the presence of a 6
mm-diameter tumor, and with SNR = ?8 dB . . . . . . . . . . . . . . . . . . .
44
2-6 Cross section images obtained via four different imaging algorithms in the presence
of a 6 mm-diameter tumor, and with SNR = ?8 dB . . . . . . . . . . . . . . . . 46
2-7 Comparison of 3D images obtained via four different imaging algorithms. A 4
mm-diameter tumor is present, and there is no noise. . . . . . . . . . . . . . . .
47
2-8 Comparison of cross section images obtained via four different imaging algorithms 49
3-1 Simplified breast model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3-2 Total conductivity of normal breast tissues and tumor as a function of frequency
69
3-3 Ratio of conductivity between tumor and normal breast tissue as a function of
frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3-4 Model of microwave induced TAI for breast cancer detection . . . . . . . . . . .
70
3-5 Data cube model. In Stage I, MART slices the data cube for each frequency
index. RCB is applied to each data slice to estimate the corresponding waveform. 71
3-6 Breast model for thermal acoustic simulation . . . . . . . . . . . . . . . . . . . .
72
3-7 Gaussian modulated microwave source . . . . . . . . . . . . . . . . . . . . . . .
73
3-8 Thermal acoustic signals at different stimulating frequencies f =200 MHz, 400
MHz, 600 MHz, and 800 MHz . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
8
3-9 Imaging results for the case of a single 1 mm-diameter tumor . . . . . . . . . . .
75
3-10 Imaging results for the two 1.5 mm-diameter tumors case . . . . . . . . . . . . .
76
4-1 Breast model and antenna array . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
4-2 Temperature distribution. The dish lines denote the location of the skin. . . . .
85
5-1 Ultrasound hyperthermia system . . . . . . . . . . . . . . . . . . . . . . . . . .
90
A-1 Time-domain waveform of a differential Gaussian pulse computed using our method
and the Debye model when compared with the exact solution . . . . . . . . . . 98
A-2 Reflection coefficient magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
99
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MICROWAVE TECHNIQUES FOR
BREAST CANCER DETECTION AND TREATMENT
By
Bin Guo
May 2007
Chair: Jian Li
CoChair: Henry Zmuda
Major: Electrical and Computer Engineering
Early detection and effective treatment of breast cancer play key roles in reducing the
breast cancer mortality rate of women. Microwave technique is a promising technology for
both early breast cancer detection and effective treatment.
The ultra-wideband (UWB) microwave imaging (MWI) method exploits the
significant contrast in dielectric properties between normal breast tissue and tumor.
Previously, data-independent methods, such as delay-and-sum (DAS) and space-time
beamforming, have been used for UWB microwave imaging. However, the low resolution
and the poor interference suppression capability associated with the data-independent
methods restrict their use in practice, especially when the noise is high and the backscattered
signals are weak. In this work, we focus on two data-adaptive methods for UWB
microwave imaging, which are referred to as the robust Capon beamforming (RCB)
method and the amplitude and phase estimation (APES) method. Due to their data-adaptive
nature, these methods outperform their data-independent counterparts in terms of
improved resolution and reduced sidelobe levels.
Another promising early breast cancer detection technique is microwave induced
thermal acoustic imaging (TAI), which combines the advantages of microwave stimulation
and ultrasound imaging and offers a high imaging contrast as well as high spatial
resolution at the same time. A new multi-frequency microwave induced thermal
10
acoustic imaging scheme for early breast cancer detection is proposed in this dissertation.
Significantly more information about the human breast can be gathered using multiple
frequency microwave stimulation. A multi-frequency adaptive and robust technique
(MART) is presented for image formation. Due to its data-adaptive nature, MART
can achieve better resolution and better interference rejection capability than its
data-independent counterparts, such as the DAS method.
The effectiveness of microwave induced local hyperthermia has been shown be
effective for the treatment of breast cancer. In this dissertation, a new time-reversal
(TR) based ultra-wideband microwave method for the hyperthermia treatment of breast
cancer is presented. Two high-resolution techniques, TR and RCB, are employed to
shape the transmitted signals both temporally and spatially. This method has better
electromagnetic energy focusing ability than the existing methods, and can provide the
necessary temperature gradients required for effective hyperthermia.
11
CHAPTER 1
INTRODUCTION
1.1
Background
Breast cancer is the most common nonskin malignancy in women and the second
leading cause of female cancer mortality [1, 2]. There are over 200,000 new cases of
invasive breast cancer diagnosed each year in the U. S., and one out of every seven women
in the U.S. will be diagnosed with breast cancer in their life time (the American Cancer
Society, 2006, URL: http://www.cancer.org). Early diagnosis and effective treatment
are the key to survive from breast cancer [3]. X-ray mammography, currently the most
common imaging method for early breast cancer detection, is not highly reliable, requires
painful breast compression, and exposes the patient to low levels of ionizing radiation [4].
Magnetic resonance imaging (MRI) is another option, but it is expensive and not suitable
for routine breast screening.
The limitations of existing breast cancer detection methods motivate the research for
new breast screening techniques. According to [5], an ideal breast cancer detection method
should have the following properties:
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has low health risk;
is sensitive to tumors and specific to malignancies;
detects breast cancer at a curable stage;
is noninvasive and simple to perform;
is cost effective and widely available;
involves minimal discomfort, so the procedure is acceptable to women;
provides easy to interpret, objective, and consistent results.
Microwave technique is attractive to patients for early breast cancer detection because
it has the potential to possess all these desirable properties. There are several microwave
techniques for breast cancer detection: narrow-band (NB) microwave tomography,
ultra-wideband (UWB) microwave imaging, and microwave induced thermal acoustic
imaging (TAI), as shown in Figure 1-1.
Microwave tomography uses narrow-band microwave signal, and reconstructs the
distribution of the dielectric properties of the breast by solving the inverse scattering
12
problem. The microwave tomography system is low cost and easy to build because it is a
narrow-band system. However, the inverse scattering problem is nonlinear and difficult to
solve. The sensitivity and resolution of this system is poor. Currently, a clinical system
based on this method has been developed [3]. UWB microwave imaging technique uses
broadband microwave pulse, and reconstructs the backscattered energy inside the breast.
The resolution of this technique is high, but the cost of the UWB microwave system is
higher than that of the narrow-band microwave system. Microwave induced TAI technique
uses a narrow-band microwave source to induce a broadband acoustic signal, and uses
the induced thermal acoustic signal for image formation. This technique reconstructs the
microwave energy absorption properties of the breast. The resolution of the microwave
induced TAI system is very high (sub-millimeter), and the cost of this system is between
the narrow-band microwave tomography system and the UWB microwave imaging system.
However, to make the thermal acoustic signal strong enough, the power of the microwave
source in the microwave induced TAI system is much higher than that in microwave only
systems (including NB microwave tomography and UWB microwave imaging). A detailed
comparison of the different microwave techniques is listed in Table 1-1.
Thermotherapy is usually used in cancer treatment to enhance the effects of
radiotherapy and chemotherapy [6]. Hyperthermia is often lethal to weakened malignant
cells. By increasing tumor blood flow, hyperthermia can also increase the efficacy of
chemotherapeutic drugs [7]. Not just for medical imaging, microwave techniques can also
be used for local hyperthermia, and can offer a number of advantages [8].
1.2
Microwave Imaging
Microwave imaging (MWI) is an active wave-based non-invasive imaging method. The
physical quantities imaged in MWI are the dielectric properties. The dielectric properties
of human tissues have been studied for more than 100 years [9]. The measurements of
more than twenty human tissues over the frequency band from 10 Hz to 20 GHz have been
13
published in [9?11]. The data is also available at the authors? website1 . Figure 1-2 shows
the dielectric properties of three human tissues, breast fat, dry skin, and muscle. The
figure shows that the dielectric properties of biological tissues are frequency dependent,
and the dispersion property is not a simple linear relationship with frequency. Another
observation from this figure is that different tissues have significantly different permittivity
and conductivity values.
Microwave detection of breast cancer exploits the significant contrast in dielectric
properties between normal tissue and cancerous tissue [12?15]. Moreover, these tumor
dielectric properties show no significant variation with tumor age [15]. The typical
permittivities and conductivities of tumor and different parts of the breast around 6 GHz
are listed in Table 1-2 [9?11, 16]. It shows that the permittivity and conductivity of tumor
are about 5 and 10 times greater than those of normal breast tissues, respectively. As a
result, the tumor microwave scattering cross-section is much larger than that of normal
breast tissue with the same size. Since the tumor backscattered microwave response is
much stronger than that of normal breast tissue, the tumor location can be accurately
determined from its backscattered wave field. During detection, antennas are located
on or near the skin of breast to measure the backscattered wave field and hence breast
compression can be avoided [3]. Safety standards, such as IEEE C95.1-1999, have been
defined based on specific absorption rate (SAR) of electromagnetic radiation. These
standards indicate that the acceptable device operating between 100 KHz and 6 GHz
should have a maximum SAR less than 1.6 W/kg over the body when averaged over 1 g
of tissue. Hence the microwave breast cancer detection system is expected to operate with
power levels one or two orders of magnitude lower than those of a cellular telephone, which
are much lower than the requirement of these standards [2].
1
Institute for Applied Physics Italian National Research Council. Dielectric Properties
of Body Tissues. Available online at http://niremf.ifac.cnr.it/tissprop
14
During the last decade, several research groups have been working on the active
microwave imaging system for breast cancer detection. There are two major classes of
active microwave imaging techniques: inverse scattering (tomography) [3, 17?21] and
UWB microwave imaging [22?26]. The goal of microwave tomography is to reconstruct
the distribution of the dielectric properties of the breast by solving the inverse scattering
problem using numerical techniques, such as the finite-element method (FEM) and the
method on moment (MoM). Because the inverse scattering problem is ill-conditioned and
nonlinear, this approach is inherently limited by its vulnerability to small experimental
uncertainties and noise [23].
Unlike the inverse scattering method, the UWB microwave imaging approach
transmits broadband microwave pulses from different locations on the breast surface
and records the backscattered responses from the breast. The backscattered responses
are processed coherently to form the image. This method previously has been used
predominantly in military radar applications, and was introduced in [22] for breast cancer
detection. Their pioneering work shows the promise of UWB microwave imaging for the
detection of breast cancer. However, only 2-D and simplified 3-D breast models have
been studied in the literature. The image formation methods considered so far, including
confocal microwave imaging [23, 24] and microwave imaging via space-time (MIST)
beamforming [25, 26], are data-independent.
In this dissertation, we focus on two data-adaptive algorithms for UWB microwave
imaging, referred to as the robust Capon beamformer (RCB) method [27, 28] and the
amplitude and phase estimation (APES) method [29, 30]. These data-adaptive methods
outperform their data-independent counterparts in terms of higher resolution and better
interference suppression capability.
1.3
Microwave Induced Thermal Acoustic Imaging
Microwave imaging exploits the significant contrast in dielectric properties between
normal and cancerous tissues [12?14]. However, the long wavelength of microwave limits
15
its resolution. Ultrasound is another option which offers a high spatial resolution because
of its short acoustic wavelength [31?33]. However, the contrast in acoustic properties
between normal and tumor tissues is very small due to both being soft tissues.
Microwave induced thermal acoustic imaging (TAI) combines the advantages of
microwave stimulation and ultrasound imaging [34], which offers a high imaging contrast
(due to the significantly different dielectric properties of tumor and normal breast tissues)
as well as high spatial resolution (due to the low propagation velocity or the short
wavelength of acoustic waves in biological tissues) at the same time. The physical basis
of the TAI is the thermoacoustic effect of biological tissues, as shown in Figure 1-3 [35].
When biological tissue absorbs energy and is heated, it expands and generates acoustic
signals. To use microwave induced TAI techniques for breast cancer imaging, a microwave
source with a short duration time is used to irradiate the breast uniformly, as shown
in Figure 1-4. The normal breast tissues, as well as tumors, absorb microwave energy
and emanate thermal acoustic waves by thermoelastic expansion. It is well-known that
malignant breast tissue has a higher water content [2, 12, 14, 36], with a much higher
conductivity than normal breast tissues (with low water content), as shown in Figure
1-2(b) and Table 1-2. As a result, the microwave energy absorbed by tumor and normal
breast tissues will be significantly different and a stronger acoustic wave will be produced
by the tumor. The acoustic waves generated in this manner carry the information about
the microwave energy absorption properties of the tissues under irradiation. The thermal
acoustic waves propagate out of the breast and are recorded by an acoustic sensor array
placed around the breast. Hence the tumor locations can be accurately determined from
the received signals since the malignant tumors are associated with larger acoustic signal
levels in the image construction.
During the last decade, several research groups have been working on the microwave
induced TAI of biological tissues [35, 37?41]. The microwave frequency used ranges from
400 MHz [35] to 3 GHz [34]. Image reconstruction algorithms include the widely used
16
delay-and-sum (DAS) method [39, 41], the frequency-domain inverse method [42, 43], and
the time-domain inverse method [34, 38].
In this dissertation, a multi-frequency microwave induced TAI system is proposed.
The multi-frequency microwave induced thermal acoustic signals will offer higher
signal-to-noise ratio (SNR) and higher imaging contrast than single-frequency microwave
induced thermal acoustic signals. Furthermore, the interference due to the inhomogeneous
breast tissues can be suppressed more effectively when multi-frequency microwave
induced thermal acoustic signals are used for image reconstruction. We also present a
multi-frequency adaptive and robust technique (MART) for multi-frequency microwave
induced TAI. This technique is based on the RCB [27, 44] method, and can achieve better
resolution and better interference rejection capability than the data-independent methods.
1.4
Microwave Hyperthermia Treatment
Many studies have been performed recently to show the effectiveness of the microwave
induced local hyperthermia in the treatment of breast cancer as well as other types of
cancer [6, 7, 45?49]. In hyperthermia treatment of cancer, malignant tumors are heated
to a temperature above 43? C for about thirty to sixty minutes. Because of the high water
content in malignant tumors, the conductivity of the tumor is significantly different from
other normal breast tissues over a wide frequency band, as shown in Table 1-2. As a
result, the heat absorption of the tumor is much greater than that of the host tissue over
an ultra wide band of microwave frequencies [50, 51]. However, major challenges still exist
for the wide use of microwave thermotherapy for breast cancer treatment:
?
Since the heat absorption properties of the breast skin are similar to the tumor but
quite different from the normal breast tissue, the breast skin can burn easily during
thermotherapy.
?
The heterogeneous properties of the breast tissue make it difficult to focus the
microwave energy onto the tumor.
Most of the recent studies concentrate on the narrow-band (NB) techniques, which
can focus the microwave energy at the desired location in the breast via adjusting the
17
amplitude and phase of the transmitted sinusoidal signal at each antenna [6, 46, 48].
However, due to the ineffective focusing ability of NB systems, complicated skin
temperature feedback is required to avoid burning the skin.
Recently, the feasibility of using the ultra-wideband (UWB) technique for microwave
hyperthermia treatment was investigated in [52]. The authors in [52] developed a UWB
space-time beamforming approach, which could provide better temperature selectivity
than the NB methods. However, the heterogeneous properties of the breast tissue make it
hard to align the super narrow pulses accurately.
To improve the microwave energy focusing ability and to provide the effective
temperature gradients required for effective hyperthermia for breast cancer treatment,
we consider a new UWB method, referred to as the time reversal based robust Capon
beamforming (TR-RCB) approach, for the hyperthermia treatment of breast cancer
in this dissertation. Time-reversal is a powerful method for focusing through complex
media. The physical basis behind this method is that the wave propagation is reciprocal
if we neglect the dispersion of the media, as shown in Figure 1-5. This method has
many applications in acoustics, such as medical imaging and therapy [53]. Recently,
the feasibility of the time-reversal of electromagnetic waves was demonstrated in [54].
Our application of time reversal is more challenging than the conventional ones such as
breaking up kidney stones into small pieces using ultrasound [55, 56]. In our application,
we must deal with the strong backscattering from the breast skin. Our method is to
process the backscattered signals to eliminate as much as possible undesired components
in them before time reversing and retransmitting them. Moreover, the state-of-the-art
robust Capon beamformer (RCB) [27, 28, 57] is used to further improve the energy focus
onto the tumor cells while placing nulls in the beam patterns to avoid potential burns in
other areas.
18
1.5
Outline of This Dissertation
In Chapter 2, two data-adaptive algorithms, RCB and APES, for UWB microwave
imaging are studied. Due to their data-adaptive nature, these methods outperform their
data-independent counterparts in terms of improved resolution and reduced sidelobe
levels. We compare these algorithms and illustrate their performance by using a complex
three dimensional (3-D) breast model with a small embedded tumor. The finite-difference
time-domain (FDTD) method is used to simulate the propagation of the microwave within
the breast.
In Chapter 3, a new multi-frequency microwave induced thermal acoustic imaging
scheme for early breast cancer detection is proposed. A multi-frequency adaptive
and robust technique (MART) is presented for image formation in this chapter also.
Significantly more information about the human breast can be gathered using multiple
microwave frequencies for stimulation. The effectiveness of this procedure is shown by
several numerical examples. The FDTD method is used to simulate the electromagnetic
field distribution, the absorbed microwave energy, and the acoustic field in the breast
model.
In Chapter 4, a new time-reversal (TR) based ultra-wideband (UWB) microwave
method for hyperthermia treatment of breast cancer is presented. Two high-resolution
techniques, time reversal (TR) and robust Capon beamformer (RCB), are employed
to shape the transmitted signals both temporally and spatially. As shown in the
two-dimensional (2D) numerical simulations, this method has better electromagnetic
(EM) energy focusing ability than the existing methods, and can provide the necessary
temperature gradients required for effective hyperthermia.
The conclusions and future work is provided in Chapter 5.
A new FDTD formulation for wave propagation in biological media with Cole-Cole
model is presented in Appendix A. The fractional order differentiators in the Cole-Cole
model is implemented in this new FDTD formulation.
19
Finally, the nonlinear acoustic effect in the microwave induced TAI system is analyzed
in Appendix B.
20
Table 1-1. Comparison of various microwave techniques.
Microwave Band
Signal Type
Signal Band
Cost
Complexity of the system
Source Power
Image Contrast
Resolution
NB Microwave UWB Microwave
Tomography
Imaging
Narrow
Wide
Microwave
Microwave
Narrow
Wide
Low
High
Low
Middle
Low
Low
High
High
Low
High
Microwave
Induced TAI
Narrow
Acoustic
Wide
Middle
High
High
High
Very High
Table 1-2. Typical dielectric properties of breast tissues.
Tissues
Immersion Liquid
Chest Wall
Skin
Fatty Breast Tissue
Nipple
Glandular Tissue
Tumor
Dielectric Properties
Permittivity Conductivity
9
0
50
7
36
4
9
0.4
45
5
11-15
0.4-0.5
50
4
21
Narrow-Band
Microwave
Narrow-Band
Antenna
Tumor
A
Wide-Band
Microwave
Wide-Band
Antenna
Tumor
Backscattered
Wave
B
Narrow-Band
Microwave
Ultrasonic
Transducer
Acoustic Wave
Tumor
C
Figure 1-1. Microwave techniques for breast cancer detection. A) Narrow-band microwave
tomography. B) Ultra-wideband microwave imaging. C) Microwave induced
thermal acoustic imaging.
22
BreastFat
SkinDry
Muscle
7
10
6
Relative Permittivity
10
5
10
4
10
3
10
2
10
1
10
2
10
4
10
6
10
8
10
10
10
Frequency (Hz)
A
1
10
0
Conductivity (S/m)
10
?1
10
?2
10
?3
10
BreastFat
SkinDry
Muscle
2
10
4
6
10
10
8
10
10
10
Frequency (Hz)
B
Figure 1-2. Dielectric properties of human tissues over a wide frequency band. A) Relative
permittivity. B) Conductivity.
Figure 1-3. Physical basis of the thermoacoustic effect.
23
Microwave
120
acoustic sensor
100
Y (mm)
80
breast
60
tumor
skin
40
20
chest wall
20
40
60
80
100
120
140
160
180
X (mm)
Antenna Array
Figure 1-4. Model of microwave induced TAI for breast cancer detection.
Sourec
Antenna Array
A
Focus point
B
Figure 1-5. Physical basis of time-reversal. A) Antenna array records the waveforms from
the point source. B) The received waveforms are time reversed and sent back
into the medium by the antenna array. The waveforms will refocus at the
point source location.
24
CHAPTER 2
ADAPTIVE MICROWAVE IMAGING FOR BREAST CANCER DETECTION
2.1
Introduction
In this chapter, we consider adaptive microwave imaging for breast cancer detection.
We present two data-adaptive algorithms for UWB microwave imaging, referred to as
the robust Capon beamformer (RCB) method and the amplitude and phase estimation
(APES) method. These data-adaptive methods outperform their data-independent
counterparts in terms of higher resolution and better interference suppression capability.
In the RCB method, the received microwave signals are passed through a robust
Capon beamformer to estimate the waveform [27, 28]. RCB is a robustified version of the
standard Capon beamformer (SCB), by allowing the array steering vector to be uncertain
within an uncertainty set.
APES is derived based on the least squares fitting of the beamformer output [29, 30]
under the assumption that the signal waveform is known. The drawback of this approach
is the requirement of the knowledge of the signal waveform, which is usually not satisfied.
However, APES has the potential of providing excellent imaging results if the signal
waveform can be properly estimated.
To validate these algorithms, we have developed a complex 3-D breast model. The
glandular tissues are set randomly inside the fatty breast tissue, and the dispersive and
inhomogeneous properties of the breast tissues are also considered in our model. A small
tumor is set at 2.7 cm below the skin. To obtain the backscattered waveform, the FDTD
method [58, 59] is used to simulate the propagation of the microwave within the breast.
The remainder of this chapter is organized as follows. In Section 2.2, we introduce the
data model and formulate the problem of interest. In Section 2.3, we present the RCB and
APES algorithms for microwave imaging. Comparisons between these algorithms, as well
as with DAS and MIST, are also presented in Section 2.3. Simulation results based on the
aforementioned 3-D breast model are provided in Section 2.4. Section 2.5 concludes this
chapter.
25
2.2
2.2.1
Data Model and Problem Formulation
Data Collection
We consider herein a bistatic radar model for the imaging system. A pair of
transmitter/receiver antennas are used to scan the breast at different positions. The
distance between the transmitter and the receiver is fixed. During each scan, the
antenna pair is located on the breast skin at a chosen position ri = [xi yi zi ]T . Here,
(и)T denotes the transpose. A broadband microwave pulse is sent by the transmitter, and
the backscattered signal is sampled by the receiver. Let Ei (t), i = 1, и и и , M , denote the
received signal by the ith channel at time instant t, and let riT and riR denote the positions
of the transmitter and receiver antennas for the ith channel, respectively, where M is the
number of channels or antenna pair positions.
Our goal is to detect the tumor by constructing 3-D images of the backscattered
energy p(r) as a function of imaging location r within the breast. For each specific imaging
location r, the backscattered energy p(r) is estimated from the received signals Ei (t),
i = 1, и и и , M , via adaptive beamforming.
Before the received signals are passed through an adaptive beamformer to estimate
the backscattered energy, they have to be preprocessed to remove the early-time response
and to compensate for the propagation loss.
2.2.2
Early-Time Response Removal
There are early-time and late-time contents in the received backscattered signals.
The early-time content is dominated by the incident pulse (direct propagation from the
transmitting antenna to the receiving antenna) and reflections from the breast skin. The
late-time content contains tumor backscattered signals and other backscattering due to the
inhomogeneous fatty tissue, glandular tissue, and chest wall. Due to the small distances
between the antennas and between the antenna pair and the skin, the magnitude of the
early-time content is much larger than that of the late-time content. We must remove the
early-time response to enhance the tumor response.
26
Because the distance between the transmitter and the receiver is fixed and the skin
tissues are similar at different positions, the signals recorded at various antenna locations
have similar direct propagations and skin reflections. Hence we can remove the early-time
content by subtracting a fixed signal out from all channels. This signal can be obtained
simply by averaging the recorded signals at all channels,
M
1 X
E?(t) =
Ei (t).
M i=1
(2?1)
After E?(t) is subtracted out from each channel, we have the preprocessed signal
Xi (t) = Ei (t) ? E?(t),
2.2.3
i = 1, и и и , M.
(2?2)
Signal Time-Shifting, Windowing, and Compensation
For the ith channel, we align the return from a specific imaging location r with the
returns from the same location for the other channels by time-shifting the signal Xi (t)
a number of samples ni (r). The discrete-time delay between the antennas and r can be
calculated as
$
ni (r) =
и
И%
1 kriT ? rk kriR ? rk
+
,
4t
C
C
(2?3)
where bxc stands for rounding to the greatest integer less than x, C is the velocity
of microwave propagating in breast tissues, and 4t is the sampling interval, which is
assumed to be sufficiently small. The time-shifted signal is denoted as
X?i (t, r) = Xi (t + ni (r)),
t = ?ni (r), и и и , T ? ni (r),
(2?4)
where T is the maximum time (rounded to the nearest multiples of the sampling interval)
needed by microwave pulse to propagate from the transmitter to the far side of the skin
or chest wall and back to the receiver. This T gives the maximum time interval we are
interested in.
After time-shifting, the backscattered signals from location r are aligned so that they
all start approximately from time t = 0 for all channels. Next the aligned signals are time
27
windowed by
?
?
? 1, 0 ? t ? N ? 1
Window(t) =
?
? 0,
otherwise
(2?5)
to isolate the backscattered signals from location r. The windowed signals are denoted by
X?i (t, r), t = 0, и и и , N ? 1, where N 4t is the approximate duration of the backscattered
signal from location r. Note that N can be determined approximately from the duration
of the transmitted signal waveform, which is known. Note also that only the center part
of X?i (t, r), t = 0, и и и , N ? 1, can be considered useful since the signal contents at the two
ends of the interval [0, N ? 1] are small.
Propagation attenuation occurs when the microwave propagates within the breast.
The attenuation of the tumor responses at various channels is different because the
distances from the transmitter to the imaging position r and back to the receiver are
different, with longer propagation indicating more severe attenuation, and vice versa.
Here we only compensate out the attenuation due to the propagation and ignore the lossy
medium effect because the propagation attenuation is the dominant factor. For the ith
channel, the compensation factor is given by
Ki (r) = kriT ? rk2 и kriR ? rk2 ,
(2?6)
and the compensated signal can be calculated as
yi (t, r) = Ki (r) и X?i (t, r),
2.2.4
t = 0, и и и , N ? 1.
(2?7)
Data Model
Without loss of generality, we consider imaging at the generic location r only. For
notational convenience, we drop the dependence of yi (t, r) on r, and simply denote it as
yi (t). Now we consider the signal
y(t) = [y1 (t) y2 (t) и и и yM (t)]T ,
28
t = 0, и и и , N ? 1.
(2?8)
After preprocessing, each snapshot y(t) can be modeled as
y(t) = a и s(t) + e(t),
(2?9)
where s(t) is the backscattered signal, a denotes the steering vector, and
e(t) = [e1 (t) e2 (t) и и и eM (t)]T ,
t = 0, и и и , N ? 1
(2?10)
is a term comprising both interference and noise. Since y(t) was properly time-shifted
and compensated for, the steering vector a is assumed to be [1 1 и и и 1]T . The problem of
interest then is to estimate the backscattered signal s(t) from y(t).
The energy of the estimated signal can be calculated as
p(r) =
N
X
s?2 (t)
(2?11)
t=1
which will be regarded as the backscattered energy from position r.
2.3
2.3.1
Adaptive Microwave Imaging
Robust Capon Beamformer (RCB)
The standard Capon beamformer (SCB) considers the following problem
min wT R?w
w
subject to wT a = 1,
(2?12)
where w is the beamformer?s vector, and
N ?1
1 X
R? =
y(t) и yT (t)
N t=0
4
(2?13)
is the sample covariance matrix. The solution to (2?12) is (see, e.g. [60])
w?SCB =
R??1 a
aT R??1 a
.
(2?14)
SCB has better resolution and much better interference rejection capability than
the data-independent beamformers. However, it suffers from severe performance
degradations when some of the underlying assumptions on the environment, sources,
29
propagation, or sensor array are violated. In a MWI system, various model errors occur.
For instance, there exist residues from the early-time signal removal step, inaccurate
signal compensation due to the non-homogeneous propagation media, non-stationary
interference and noise, and round-off errors in the signal time-shifting step. These errors
can be mitigated by allowing an error in the steering vector a [44].
To improve the performance of SCB in the presence of model errors, we assume
that the true steering vector is a?, which is a vector in the vicinity of a, and that the only
knowledge we have about a? is that
ka? ? ak2 ? ▓
(2?15)
where ▓ is a user parameter. To avoid the trivial solution a? = 0, we require that
▓ < kak2 .
(2?16)
We adopt the recently developed robust Capon beamforming (RCB) [27] approach to
make SCB robust against the errors in a.
Consider the theoretical covariance matrix used by SCB
where
R? = ? и aaT + Q
(2?17)
N ?1
1 X 2
?=
s (t)
N t=0
(2?18)
N ?1
ц
1 X Б
E e(t)eT (t) .
N t=0
(2?19)
4
and
4
Q=
Due to the possible errors described above, the signal term in R? is not well described by
? и aaT , but by ? и a?a?T [61].
First, we assume a? is given (the determination of a? will be discussed later on). Then
the RCB problem can be re-formulated as
min wT R?w
w
subject to wT a? = 1
30
(2?20)
which has the solution
w?RCB =
R??1 a?
a?T R??1 a?
.
(2?21)
Next, we determine a? via a covariance fitting approach [28, 60, 62]. Since a? is a vector
in the vicinity of a such that ? и a?a?T is a good fit to R?, we determine a? as the solution to
the following optimization problem
max ?
?,a?
subject to
R? ? ?a?a?T ? 0
ka? ? ak2 ? ▓.
(2?22)
Usually, ▓ is determined by the various errors discussed previously. In practice, ▓ can
be chosen experimentally by considering all the errors together. By using the Lagrange
multiplier method, the solution to (2?22) is given by [27]:
h
a? = a ? I + хR?
i?1
a,
(2?23)
where х ? 0 is the corresponding Lagrange multiplier that can be solved from the
following equation:
░│
┤?1 ░
░
░2
░ I + хR?
a░
░
░ = ▓.
(2?24)
The equation can be solved as described in [27]. After obtaining the value of х, the
estimate a? of the actual steering vector a is determined by (2?23).
Substituting the so-obtained a? into (2?21), we obtain w?RCB . The output of the RCB
beamformer is given by
T
s?RCB (t) = w?RCB
y(t),
(2?25)
and the backscattered energy can be calculated from (2?11) using s?RCB (t).
2.3.2
Amplitude and Phase Estimation (APES)
Previously, we have developed the RCB method based on the assumptions that the
signal waveform can be estimated. In this subsection we present the amplitude and phase
estimation (APES) method which explicitly assumes that the signal waveform is known.
31
Consider the following data model:
y(t) = a?s?(t) + e(t),
t = 0, и и и , N ? 1,
(2?26)
where ? is the unknown amplitude of the backscattered signal with waveform s?(t), t =
P ?1
2
0, и и и , N ? 1, assumed to be known. To avoid a scaling ambiguity, we let N
t=0 [s?(t)] = 1.
The APES method considers the following problem
N ?1
ц2
1 XБ T
min
w y(t) ? ?s?(t)
?,w N
t=0
subject to wT a = 1.
(2?27)
Here, the beamformer output wT y(t) is required to be as close as possible (up to a scaling
factor ?) to the known signal waveform s?(t). By design, the APES beamformer can
suppress the noise and interference, and at the same time, protect the signal of interest by
enforcing the equality constraint.
Let
N ?1
1 X
g=
y(t)s?(t).
N t=0
(2?28)
A straightforward calculation shows that the criterion function in (2?27) can be rewritten
as
N ?1
ц2
1 XБ T
w y(t) ? ?s?(t)
N t=0
?2
= wT R?w ? 2?wT g +
N
х
Х2
?
А
б2
?
= ? ? N wT g + wT R?w ? N wT g .
N
(2?29)
So the minimization of (2?29) with respect to ? is given by
?? = N и wT g.
32
(2?30)
Insertion of (2?30) into (2?29) yields the following minimization problem for the
determination of the APES beamformer
min wT Zw
w
subject to wT a = 1,
(2?31)
where we have defined
Z = R? ? N и ggT .
(2?32)
The solution to (2?31) is readily obtained as
w?APES =
Z?1 a
.
aT Z?1 a
(2?33)
aT Z?1 g
.
aT Z?1 a
(2?34)
Substituting (2?33) into (2?30), we have
?? = N и
Then the backscattered energy is ?? 2 .
Since we know the transmitted pulse and the dielectric properties of the tumor,
the waveform of the backscattered microwave from a small tumor can be calculated
theoretically. In practice, the size of the tumor is much smaller than the shortest
wavelength of the microwave pulse used in the MWI system. For example, the wavelength
of the electromagnetic wave at 5 GHz is 2 cm in the breast fatty tissue, but the size of the
tumor in its early stage is around 5 mm-diameter or less. Compared with the wavelength,
the tumor is relatively small, and can be approximated as a point target. Based on the
point target tumor model, we can calculate the theoretical tumor waveform. Although
the theoretical waveform is not exactly the same as the received backscattered waveform
due to the dispersive and non-homogeneous propagation medium, it can be argued that
the waveform of the backscattered signal is approximately known. For simplicity, in
the numerical experiments of the next section we just choose the normalized early-time
response as the backscattered signal waveform in our numerical examples.
33
2.3.3
Comparison of Different Microwave Imaging Algorithms
We have presented two adaptive imaging methods for the microwave image formation
problem. The RCB algorithm involves a user parameter ▓ which is used to make the
Capon method robust against model errors. The APES algorithm assumes that the
backscattered waveform is available, which has to be estimated before APES can be
applied. If we regard this waveform as some kind of ?user parameter,? this means that the
APES algorithm has N user parameters. The selection of these parameters will influence
the performance of APES significantly.
In general, data-adaptive methods outperform data-independent ones by providing
higher resolution and better interference suppression capability due to their data adaptive
nature. However, adaptive methods usually have higher computational complexities. A
qualitative comparison of these algorithms is given in Table 2-1.
2.4
Modeling and Simulation
For simulation purposes, a 3-D model of a cancerous breast is considered. The
3-D breast model includes fatty breast tissue, glandular tissue, skin, nipple, and chest
wall. To reduce reflections, the breast model is immersed in a liquid which has the
same permittivity as the breast fatty tissue and is lossless. The breast has a shape of a
hemi-sphere with a size of 10 cm in diameter. A 6 mm-diameter tumor is located at 2.7
cm below the skin (x = 70 mm, y = 90 mm, z = 60 mm). (Later, a 4 mm-diameter tumor
is used at the same location). The cross section images of the breast model at z = 60 mm
and y = 90 mm are shown in Figures 2-1(a) and 2-1(b), respectively.
The dielectric properties of the normal fatty breast tissue are assumed to be
random with variations of ▒10% around nominal values. Some glandular tissues with
dielectric constants between ?r = 11 and ?r = 15 are randomly distributed within the
breast. Because the transmitted signal is an UWB pulse, the dispersive properties of the
normal fatty breast tissue and tumor were also considered in the model. The frequency
dependence of the permittivity ?r (?) and conductivity ?r (?) can be modeled using the
34
following single-pole Debye model [24]:
??r (?) = ?r (?) +
?r (?)
?
?1
= ?r +
+
j??0
j??0 1 + j??
(2?35)
The parameters used in (2?35) are ?r = 7, ? = 0.15 S/m, ?1 = 3, ? = 7 ps for the normal
fatty breast tissue and ?r = 4, ? = 0.7 S/m, ?1 = 50, ? = 7 ps for tumor, respectively.
A pair of antennas is sequentially placed at seventy-two positions arranged in six
circles with twelve antenna pair positions each, as indicated in Figure 2-2, where each
circle is located on an X-Y plane. The offset between the transmitter and the receiver
is 1 cm. The UWB signal used to excite the transmitter antenna is an ultra-wideband
Gaussian pulse. The backscattered signals are collected by the receiver.
To obtain the backscattered signals, the finite-difference time-domain method
(FDTD) is used to simulate the propagation of the microwave in the breast tissues. The
grid cell size used by FDTD is 1 mm О 1 mm О 1 mm and the time step is 1.667 ps. The
model is terminated according to perfectly matched layer (PML) absorbing boundary
conditions [63?65]. The Z-Transform method [66?68] is used to implement the FDTD
method whenever frequency-dependent materials are involved.
2.5
Numerical Examples
In this subsection, we provide several examples to demonstrate the performances
of the data-adaptive RCB and APES methods. The length of the window in (2?5) is
N = 150. For comparison, two data-independent methods, MIST [25, 26] and DAS
[23, 24], are also applied to the same data sets. DAS is a simple method that uses the
following data-independent beamformer
w?DAS =
a
.
M
(2?36)
The goal of MIST is to design a beamformer that passes backscattered signals from r with
unit gain while attenuating signals from other locations [25]. This method designs the
35
beamformer using a theoretical homogeneous model, and the resulting spatial filter is also
data-independent.
Figures 2-3 and 2-4 show the imaging results obtained for the case where a 6
mm-diameter tumor is present in the breast. Figure 2-3 gives the 3-D images obtained
via 3-D RCB, APES, MIST, and DAS. The X-Y cross section images obtained via 3-D
RCB, APES, MIST, and DAS are shown in Figures 2-4(a), (c), (e), and (g), respectively.
Figures 2-4(b), (d), (f), and (h) are the corresponding X-Z cross section images. All of
the four methods can detect and locate the tumor correctly, while clutter shows up in
DAS. Due to the geometry of the virtual antenna array, the resolution in the Z-axis is
comparatively poorer. The data-adaptive methods perform similarly to one another,
and they demonstrate better resolution and lower sidelobes than their data-independent
counterparts.
In our next example, we present the imaging results when additive white Gaussian
noise is present. A white Gaussian noise ni (t) with zero-mean and variance ?02 is added to
the original received signals Ei (t),
Ei0 (t) = Ei (t) + ni (t)
and the signal-to-noise ratio (SNR) is defined as:
? PM h PT
1
1
SNR = 10 и log10 ?
M
i=1
(2?37)
i?
2
t=1 E?i (t)
T
?02
? dB,
(2?38)
where T has the same meaning as in (2?4). The E?i (t) is the idealized tumor backscattered
signal, which is not available in practice. However, we can obtain it in our simulations to
calculate SNR by doing the simulation twice, with and without the tumor, and calculating
their differences.
Figures 2-5 and 2-6 consider the noise influence (SNR = ?8 dB). All other parameters
as well as imaging tables are the same as in Figures 2-3 and 2-4, respectively. In this case,
more than two thirds of the seventy-two channels have tumor backscattered signals with
36
power lower than ?0 . Figures 2-5 and 2-6 show that the influence of the noise is reasonably
small when RCB and APES are used. MIST can also find the tumor, but its performance
is poor compared with RCB. The tumor is missed completely in the DAS images. These
imaging results indicate that RCB and APES are robust against the noise.
Our final example is similar to Figures 2-3 and 2-4 except that the tumor size is now
reduced to 4 mm-diameter. The microwave scattering cross-section of this tumor is much
smaller than that in the 6 mm case because the surface area of the 4 mm-diameter tumor
is less than half of that of the 6 mm-diameter tumor. So the backscattered energy from
the 4 mm-diameter tumor is much smaller, and the detection of the 4 mm-diameter tumor
is much more challenging than the 6 mm case. The imaging results of the four algorithms
are shown in Figures 2-7 and 2-8. The tumor can be detected by RCB, but clutter starts
to show up in the image. The APES images are slightly worse than those of RCB. The
tumor is completely buried in clutter in the MIST and DAS images.
2.6
Conclusions
In this chapter we have presented two data-adaptive microwave imaging (MWI)
methods for breast cancer detection, namely the RCB and the APES methods. A complex
3-D breast model was also developed to compare the performances of these adaptive
imaging algorithms. The proposed data-adaptive methods produce better imaging results
than their data-independent counterparts. Compared with MIST and DAS, RCB and
APES are more robust against noise. When the tumor is small (4 mm-diameter) in size,
only RCB and APES can still detect the tumor, while MIST and DAS cannot.
37
Table 2-1. Merits and limitations of various microwave imaging algorithms.
Robustness
Resolution
Sidelobe
Computational
complexity
User parameters
DAS
Yes
Low
High
MIST
Yes
Low
High
Low
Low
High
High
High
No
No
No
One
Many
38
SCB RCB APES
No
Yes
Yes
High High High
Low Low
Low
Model X?Y plane
Immersion
Liquid
160
140
Skin
Breast
Fat
Y(mm)
120
100
Tumor
80
Glandular
Tissue
60
40
20
0
0
20
40
60
80
100
120
140
160
X(mm)
A
Model X?Z plane
Immersion
Liquid
100
Nipple
Skin
80
Z(mm)
Breast
Fat
60
Tumor
Glandular Tissue
40
20
Chest Wall
0
0
20
40
60
80
100
120
140
160
X(mm)
B
Figure 2-1. Cross Sections of a 3-D breast model. A) z = 60 mm. B) y = 90 mm.
39
the ith channel
Tumor
z
ri
y
x
Figure 2-2. Data collection geometry.
40
A
B
C
D
Figure 2-3. 3D images obtained via four different imaging algorithms in the presence of
a 6 mm-diameter tumor but in the absence of noise. A) RCB with ▓ = 5. B)
APES with normalized early-time response as s?(t). C) MIST. D) DAS.
41
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
120
?4
140
?4
100
?6
?6
120
80
100
Z (mm)
Y (mm)
?8
?10
80
?8
?10
60
?12
?12
40
60
?14
?14
20
40
?16
20
?16
20
?18
20
40
60
80
100
120
140
160
180
40
60
80
100
120
140
160
180
X (mm)
?20
?18
?20
X (mm)
A
B
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
120
?4
140
?4
100
?6
?6
120
80
100
Z (mm)
Y (mm)
?8
?10
80
?8
?10
60
?12
?12
40
60
?14
?14
20
40
?16
20
?16
20
?18
20
40
60
80
100
120
140
160
180
40
60
80
100
X (mm)
?20
120
140
160
180
?18
?20
X (mm)
C
D
42
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
?4
140
120
?4
?6
120
100
?6
100
80
?8
?10
Z (mm)
Y (mm)
?8
80
?12
?10
60
60
?12
40
?14
?14
40
?16
20
?16
20
?18
20
20
40
60
80
100
120
140
160
180
40
60
80
100
120
140
160
180
X (mm)
?20
X (mm)
?18
?20
E
F
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
120
?4
140
?4
100
?6
?6
120
80
100
Z (mm)
Y (mm)
?8
?10
80
?8
?10
60
?12
?12
40
60
?14
?14
20
40
?16
20
?16
20
?18
20
40
60
80
100
120
140
160
180
40
60
80
100
X (mm)
?20
120
140
160
180
?18
?20
X (mm)
G
H
Figure 2-4. Cross section images obtained via four different imaging algorithms in the
presence of a 6 mm-diameter tumor but in the absence of noise. A) and B)
RCB with ▓ = 5. C) and D) APES with normalized early-time response as
s?(t). E) and F) MIST. G) and H) DAS.
43
A
B
C
D
Figure 2-5. 3D images obtained via four different imaging algorithms in the presence of a 6
mm-diameter tumor, and with SNR = ?8 dB. A) RCB with ▓ = 5. B) APES
with normalized early-time response as s?(t). C) MIST. D) DAS.
44
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
120
?4
140
?4
100
?6
?6
120
80
100
Z (mm)
Y (mm)
?8
?10
80
?8
?10
60
?12
?12
40
60
?14
?14
20
40
?16
20
?16
20
?18
20
40
60
80
100
120
140
160
180
40
60
80
100
120
140
160
180
X (mm)
?20
?18
?20
X (mm)
A
B
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
120
?4
140
?4
100
?6
?6
120
80
100
Z (mm)
Y (mm)
?8
?10
80
?8
?10
60
?12
?12
40
60
?14
?14
20
40
?16
20
?16
20
?18
20
40
60
80
100
120
140
160
180
40
60
80
100
X (mm)
?20
120
140
160
180
?18
?20
X (mm)
C
D
45
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
?4
140
120
?4
?6
120
100
?6
100
80
?8
?10
Z (mm)
Y (mm)
?8
80
?12
?10
60
60
?12
40
?14
?14
40
?16
20
?16
20
?18
20
20
40
60
80
100
120
140
160
180
40
60
80
100
120
140
160
180
X (mm)
?20
X (mm)
?18
?20
E
F
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
120
?4
140
?4
100
?6
?6
120
80
100
Z (mm)
Y (mm)
?8
?10
80
?8
?10
60
?12
?12
40
60
?14
?14
20
40
?16
20
?16
20
?18
20
40
60
80
100
120
140
160
180
40
60
80
100
X (mm)
?20
120
140
160
180
?18
?20
X (mm)
G
H
Figure 2-6. Cross section images obtained via four different imaging algorithms in the
presence of a 6 mm-diameter tumor, and with SNR
=
?8 dB. A) and B)
RWCB with ▓ = 5. C) and D) APES with normalized early-time response as
s?(t). E) and F) MIST. G) and H) DAS.
46
A
B
C
D
Figure 2-7. Comparison of 3D images obtained via four different imaging algorithms. A 4
mm-diameter tumor is present, and there is no noise. A) RCB with ▓ = 6. B)
APES with normalized early-time response as s?(t). C) MIST. D) DAS.
47
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
120
?4
140
?4
100
?6
?6
120
80
100
Z (mm)
Y (mm)
?8
?10
80
?8
?10
60
?12
?12
40
60
?14
?14
20
40
?16
20
?16
20
?18
20
40
60
80
100
120
140
160
180
40
60
80
100
120
140
160
180
X (mm)
?20
?18
?20
X (mm)
A
B
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?2
Image: X?Z plane at Y=9cm
120
?4
140
?4
100
?6
?6
120
80
100
Z (mm)
Y (mm)
?8
?10
80
?8
?10
60
?12
?12
40
60
?14
?14
20
40
?16
20
?16
20
?18
20
40
60
80
100
120
140
160
180
40
60
80
100
X (mm)
?20
120
140
160
180
?18
?20
X (mm)
C
D
48
Image: X?Y plane at Z=6cm
180
0
160
?2
0
?4
140
Image: X?Z plane at Y=9cm
?2
120
?6
?4
120
100
?6
100
80
?10
80
Z (mm)
Y (mm)
?8
?12
60
?14
40
?8
?10
60
?12
40
?14
?16
20
20
?16
?18
20
40
60
80
100
120
140
160
180
20
?20
40
60
80
100
120
140
160
180
?18
X (mm)
X (mm)
?20
E
F
Image: X?Y plane at Z=6cm
180
0
160
?2
0
Image: X?Z plane at Y=9cm
140
?4
120
?6
100
?2
?4
120
?6
80
100
Z (mm)
Y (mm)
?8
?10
80
?12
?8
?10
60
?12
40
60
?14
?14
20
40
?16
?16
20
?18
20
40
60
80
100
120
140
160
180
?18
X (mm)
20
40
60
80
100
120
140
160
180
?20
?20
X (mm)
G
H
Figure 2-8. Comparison of cross section images obtained via four different imaging
algorithms. A 4 mm-diameter tumor is present, and there is no noise. A) and
B) RCB with ▓ = 6. C) and D) APES with normalized early-time response as
s?(t). E) and F) MIST. G) and H) DAS.
49
CHAPTER 3
MULTI-FREQUENCY MICROWAVE INDUCED THERMAL ACOUSTIC IMAGING
FOR BREAST CANCER DETECTION
3.1
Introduction
Microwave induced TAI encounters several challenges. First, the human breast is
large in size, usually has an irregular shape if not compressed, and is covered with a 2 mm
thick skin with dielectric properties significantly different from the normal breast tissues.
Moreover the breast tissue is far from homogeneous. All these factors make it difficult to
approximate the back propagation properties of thermal acoustic signals inside the breast.
Due to the slow acoustic wave propagation speed or short wavelength in biological tissues,
the errors on the order of millimeters in determining the acoustic signal propagation path
lengths will severely degrade the image quality.
In this chapter, a multi-frequency microwave induced TAI system is proposed which
remedy the problems mentioned above. Instead of using a single frequency microwave
source, as generally done by other research groups in this field, here a multiple frequency
source is used, since the desired thermal acoustic signals can be induced by microwave
sources operating at a wide range of frequencies. We show in this dissertation that
the rich information collected from the multi-frequency stimulation can help mitigate
the challenges mentioned. The multi-frequency microwave induced thermal acoustic
signals will offer higher signal-to-noise ratio (SNR) and higher imaging contrast than
signal-frequency microwave induced thermal acoustic signals since much more information
about the human breast can be harvested from the multiple stimulating frequencies within
the microwave frequency band. Furthermore, the interference due to inhomogeneous breast
tissue can be suppressed more effectively when multi-frequency microwave induced thermal
acoustic signals are used for image reconstruction.
Another challenge encountered by microwave induced TAI is the need to develop
accurate and robust image reconstruction methods. DAS is a widely used reconstruction
algorithm in medical imaging. This method is data-independent and tends to suffer from
50
poor resolution and high sidelobe level problems. Data-adaptive approaches, such as
the recently introduced Robust Capon Beamforming (RCB) [27, 44] method, can have
much better resolution and much better interference rejection capabilities than their
data-independent counterpart. Several medical imaging algorithms [57, 69?71] based on
RCB have been developed and used for microwave imaging and thermal acoustic imaging.
Good performances of these algorithms have been reported.
We present a Multi-frequency Adaptive and Robust Technique (MART) based on
RCB for multi-frequency microwave induced TAI. There are three stages in our MART. In
Stage I, RCB is used to estimate the thermal acoustic responses from the grid locations
within the breast for each stimulating microwave frequency. Then, in Stage II, a scalar
acoustic waveform at each grid location is estimated based on the response estimates for
all stimulating frequencies from Stage I. Finally, in Stage III, the positive peak and the
negative peak of the estimated acoustic waveform at each grid location are determined,
and the peak-to-peak difference is computed and referred to as the image intensity.
To validate the effectiveness of the proposed algorithm, we develop a 2-D inhomogeneous
breast model, which includes skin, breast fatty tissues, glandular tissues, and the chest
wall. Small tumors are set below the skin. The finite-difference time-domain (FDTD)
method is used to simulate the electromagnetic field inside the breast tissues [58, 59].
The specific absorption rate (SAR) distribution is calculated based on the simulated
electromagnetic field [51, 72]. Then FDTD is used again to simulate the propagation of
the microwave induced thermal acoustic waves [73, 74].
The remainder of this chapter is organized as follows. In Section 3.2, the microwave
frequency properties of human breast are described. A proper microwave frequency
band for multi-frequency microwave induced TAI is also given in this section. MART
is proposed for image formation in Section 3.3. In Section 3.4, 2-D electromagnetic and
acoustic breast models are developed. The electromagnetic and acoustic simulation
51
methods are also presented in this section. Imaging results based on numerical examples
are provided in Section 3.5. Section 3.6 concludes the chapter.
3.2
3.2.1
Microwave Properties of Human Breast
Cutoff Frequency of Human Breast
In a microwave induced TAI system, the biological tissues should be heated by
microwave sources in a uniform manner, otherwise thermal acoustic signals will be induced
by a nonuniform microwave energy distribution, resulting in images difficult to interrupt.
It is well-known that high order electromagnetic field modes will be excited in a media if
the microwave works at a frequency higher than a cutoff frequency of the media [75], and
the microwave energy distribution is nonuniform at high order modes [76]. To minimize
the nonuniform microwave energy distribution inside the breast caused by the high order
electromagnetic modes, the microwave source should work at a frequency below a certain
cutoff frequency.
To estimate the cutoff frequency for the human breast, we consider the simplified
breast model shown in Figure 3-1(a) consisting of a semicircular dielectric waveguide with
a perfect magnetic conductor (PMC) at the bottom of the semicircle. Recall that the
tangential components of the magnetic field are zero on the surface of the PMC. The PMC
assumption is reasonable because the permittivity of the chest wall is much greater than
that of the normal breast tissues. In circular dielectric waveguide, if an electromagnetic
mode has a field distribution whose tangential magnetic field components are zero at
the center line of the circular waveguide, as shown in Figure 3-1(b), the introduction
of a PMC at the center line of the circular waveguide will not significantly change the
boundary conditions and hence will not significantly alter the mode distribution. The
modes in the semicircular dielectric waveguide can thus be estimated by determining the
modes in a corresponding circular dielectric waveguide.
The dominant mode of a circular dielectric waveguide is the HE11 mode, the cutoff
frequency of which is zero. The electromagnetic field distribution is near uniform at this
52
mode. The dominant mode is followed by the TE01, TM01, and HE21 modes. These
modes are degenerate, and have a cutoff frequency given by [75]
fc =
?01 C0
?
,
2?a ?r ? 1
(3?1)
where C0 is the speed of light in free space, ?01 = 2.405 is the first root of the Bessel
function of the first kind of order zero (J0 (?01 ) = 0), a and ?r are the radius and average
permittivity of the circular dielectric waveguide, respectively. TM01 and HE21, as well
as the interference between them, satisfy the zero tangential magnetic field component
condition at the center line of the circular waveguide. These modes can also exist in the
semicircular dielectric waveguide. By substituting the parameters of the breast model into
(3?1), we obtain the cutoff frequency of the semicircular breast model to be
fc =
2.405 и 3 О 108
?
= 812 MHz,
2? и 0.05 и 9 ? 1
(3?2)
where we have used a = 5 cm and ?r = 9 as typical values for human breast.
Consequently, the stimulating microwave frequency for the TAI system should be below
812 MHz.
3.2.2
Microwave Energy Absorption Properties of Breast Tissues and Tumor
It is well-known that the complex relative dielectric properties of a medium can be
expressed as
?r = ?r 0 ? j?r 00 ,
(3?3)
where ?0r is the relative permittivity and ?00r is the out-of phase loss factor which can be
written as
?00r =
?
,
?0 ?
(3?4)
with ? being the total conductivity, ?0 the free space permittivity, and ? the electromagnetic
frequency. The tissue absorption property of the electromagnetic wave energy is
1
Q(r) = ?|E(r)|2 ,
2
53
(3?5)
which is a function of the total conductivity and the electric field inside the tissue. If
we assume that the microwave energy distribution is uniform inside the breast in a TAI
system, the absorption of the microwave energy by the breast is characterized by the total
conductivity of the breast tissues:
?(?) = ?00r ?0 ?.
(3?6)
Hence, instead of using the attenuation coefficient ?, as used in [41], in this dissertation we
study the absorption properties of breast tissues using the total conductivity ?.
The dielectric properties of biological tissues can be accurately modeled by the
Cole-Cole equation [11]
?r (?) = ?? +
K
X
i=1
4?i
?0
+
,
1??
1 + (j??i ) i j??0
(3?7)
where K is the order of the Cole-Cole model, ?? is the high frequency permittivity, ?i
is the relaxation time, 4?i is the pole amplitude, ?i (0 ? ?i ? 1) is a measure of the
broadening of dispersion, and ?0 is the static ionic conductivity. The Cole-Cole parameters
for skin, breast fatty tissue, chest wall (mainly consisting of muscle), as well as tumor are
listed in Table 3-1 [77, 78]. Because we cannot find the values specific to the tumor, the
dielectric properties of the tumor is approximated using a Debye model [24, 25], which is a
special case of the Cole-Cole model.
Substituting (3?7) into (3?6), we obtain the total conductivity of the breast tissue as
follows:
├
?(?) = ?imag ?? +
K
X
i=1
?
4?i
+
1??
1 + (j??i ) i j??0
!
?0 ?,
(3?8)
which is a function of the stimulating microwave frequency, where imag(и) denotes the
imaginary part of the complex relative permittivity. Figure 3-2 gives the total conductivity
of breast fatty tissue and tumor over a frequency band from 100 MHz to 1000 MHz.
Note that the total conductivity increases with the microwave frequency, which means
that more microwave energy is absorbed and converted to heat by tissues at higher
54
microwave frequency region, or in other words, the SNR is higher in the received thermal
acoustic signals at higher stimulating microwave frequency region. On the other hand, the
penetration at higher microwave frequencies is smaller because the tissues are lossy. We
define the conductivity ratio between the tumor and the normal breast tissue as
r? (?) =
?tumor (?)
,
?breast (?)
(3?9)
and plot it as a function of frequency in Figure 3-3. A high conductivity ratio means that
more microwave energy is absorbed and converted to heat by tumor than by normal breast
tissues. In other words, the higher the conductivity ratio, the higher the imaging contract.
Figure 3-3 shows that the imaging contrast is higher at the lower microwave frequency
region because the conductivity ratio decreases with the frequency.
These microwave energy absorption properties of breast tissues and tumor motivate
us to consider inducing thermal acoustic signals with different microwave frequencies.
By taking into account the aforementioned cutoff frequency given in (3?2), we choose
a frequency range from 200 MHz to 800 MHz. The frequency step is 100 MHz, with a
total of 7 frequencies. Another advantage of using multiple frequencies for stimulation
is that more information about the inhomogeneous breast tissues will be harvested
from the multi-frequency microwave induced thermal acoustic signals. The microwave
energy distribution inside the breast model is not uniform because the human breast is
random media, and thermal acoustic signals will be induced by the inhomogeneous energy
distribution. These thermal acoustic signals will appear as clutter in the resulting images.
However, the inhomogeneous microwave energy distributions are different at different
stimulating frequencies because of the different microwave wavelengths in breast tissues.
When a multi-frequency microwave source is used for TAI, the thermal acoustic clutter
induced by the inhomogeneous breast tissues can be suppressed by our adaptive and
robust imaging algorithm.
55
3.3
Multi-frequency Adaptive and Robust Technology (MART) for Breast
Cancer Imaging
We consider a multi-frequency microwave induced TAI system as shown in Figure 3-4,
where an acoustic sensor array is arranged on a semicircle relatively close to the breast
skin. The location of each acoustic sensor is rj (j = 1, и и и , N ), where N is the number
of acoustic sensors. Assume that M = 7 microwave sources with different frequencies
are used to irradiate the breast model. Let pi,j (t) (i = 1, и и и , M ; j = 1, и и и , N ;
t = 0, и и и , T ? 1) denote the thermal acoustic signal induced by the ith frequency and
received by the j th acoustic sensor, where T is the recording time which is sufficiently long
to allow acoustic sensors to record all responses from the breast. Our goal is to detect
the tumor by reconstructing an image of the thermal acoustic response intensity I(r) as a
function of scan location r within the breast.
3.3.1
Data Preprocessing
Because breast skin, breast tissues, chest wall, and the tumor, absorb the microwave
energy and convert the energy to heat, all of them produce thermal acoustic signals. The
received thermal acoustic waveforms include the responses from tumor as well as from
other healthy breast tissues. The thermal acoustic signals generated by the skin are much
stronger than those by a small tumor because of the high conductivity of the skin and
the acoustic sensors being very close to the skin. We must remove the skin responses to
enhance the tumor responses. Because the distances between the acoustic sensors and the
nearest breast skin are similar to one another, the signals recorded by various sensors have
similar skin responses. Hence, we can remove the skin response by subtracting out a fixed
calibration signal from all received signals. This calibration signal can be obtained simply
by averaging the recorded signals from all channels.
Let xi,j (t) denote the signals after subtracting out the calibration signal. To process
the signals coherently for a focal point at r, we align the signals xi,j (t) by time-shifting
each signal a number of samples nj (r). The discrete time delay between r and the j th
56
acoustic sensor can be calculated as
$
nj (r) =
%
krj ? rk
,
4t c
(3?10)
where b?c stands for rounding to the greatest integer less than ?, krj ? rk is the distance
between rj and r, c is the velocity of the acoustic wave propagating in breast tissues, and
4t is the sampling interval, which is assumed to be sufficiently small. The time-shifted
signals are denoted as
x?i,j (t, r) = xi,j (t + nj (r)),
t = ?nj (r), и и и , T ? nj (r).
(3?11)
After time-shifting, the acoustic signals from the imaging location r are aligned so
that they all start approximately from time t = 0 for all channels. Now the aligned signals
are windowed by
?
?
? 1, 0 ? l ? L ? 1
Window(l) =
?
? 0,
otherwise
(3?12)
to isolate the signals from the focal point at r. The windowed signals are denoted as
x?i,j (l, r), l = 0, и и и , L ? 1, where L4t is the approximate duration of the thermal acoustic
pulse, which can be determined from the pulse duration of the pulsed microwave source.
Attenuation exists when acoustic waves propagate within the breast. This attenuation
has two parts: the attenuation due to the lossy media and the propagation attenuation.
Thus the attenuation of the tumor responses at various channels are different because of
the different distances between the imaging position r and the acoustic sensors. For the
2-D case considered here, the compensation factor for the j th channel is given by
Kj (r) = exp(? krj ? rk) и krj ? rk1/2 ,
(3?13)
where the first term of the right side of (3?13) compensates for the attenuation due to
the lossy media, and the second term compensates for the geometric attenuation. The
57
compensated signal can be calculated as
yi,j (l, r) = Kj (r) и x?i,j (l, r),
3.3.2
l = 0, и и и , L ? 1.
(3?14)
Multi-frequency Adaptive and Robust Technology (MART)
Without loss of generality, we consider imaging at a generic location r only. For
notational convenience, we drop the dependence of yi,j (l, r) on r, and simply denote it as
yi,j (l). Now we consider the data model
yi,j (l) = si,j (l) + ei,j (l),
(3?15)
where si,j (l) represents the tumor response and ei,j (l) represents the residual term, which
includes the noise and interference from breast skin, chest wall, and other responses. The
structure of the data model is a data cube as shown in Figure 3-5.
MART is a three-stage time-domain signal processing algorithm. In Stage I, MART
slices the data cube corresponding to each frequency index, and processes the each data
slice by the robust Capon beamformer (RCB) to obtain the thermal acoustic waveform
estimate for each stimulating frequency. Then, in Stage II, a scalar waveform is estimated
from all frequencies based on the waveform estimates from Stage I. Finally, the positive
peak and the negative peak of the estimated thermal acoustic waveform from Stage II are
found in Stage III. The peak-to-peak difference is calculated as the image intensity at the
focal point at r. The details of all three stages are given below.
3.3.2.1
Stage I
In Stage I, MART approximates the data model as
yi (l) = ai si (l) + ei (l),
(3?16)
where yi (l) = [yi,1 (l), и и и , yi,N (l)]T and ei (l) = [ei,1 (l), и и и , ei,N (l)]T . The scalar
waveform si (l) denotes the thermal acoustic signal generated at the focal location r
corresponding to the ith stimulating frequency. The vector ai is referred to as the array
58
steering vector, which is approximately equal to 1N О1 = [1, и и и , 1]T since all the signals
have been aligned temporally and their attenuation compensated for in the preprocessing
step. The residual ei (l) is the noise and interference term, which is assumed uncorrelated
with the signal.
There are two assumptions made to write the model given in (3?16). First, the
steering vector is assumed to vary with the microwave frequency (i) but nearly constant
with the time sample (l). Second, we assume that the thermal acoustic signal waveform
depends only on the microwave frequency (i) but not on the acoustic sensor (j). The
truth, however, is that the steering vector is not exactly known as it changes slightly
with both the stimulating frequency and time due to array calibration errors and other
factors. The signal waveform can also vary slightly with both the stimulating frequency
and acoustic sensor, due to the inhomogeneous and frequency-dependent medium within
the breast. The two aforementioned assumptions simplify the problem slightly. They cause
little performance degradations when used with our adaptive and robust algorithm.
In practice, the true steering vector in (3?16) is not 1N О1 . We assume that the true
steering vector ai lies in the vicinity of the assumed steering vector a? = 1N О1 , and the only
knowledge we have about ai is that
kai ? a?k2 ? ▓1 ,
(3?17)
where ▓1 is a user parameter, which may be determined depending on the various errors
discussed previously.
The true steering vector ai can be estimated via the following covariance fitting
approach of RCB [27, 44]
max ?i2
?i2 , ai
subject to
R?Yi ? ?i2 ai aTi ? 0
kai ? a?k2 ? ▓1 ,
59
(3?18)
where ?i2 is the power of the signal si (l) and
R?Yi
L?1
1X
=
yi (l)yiT (l)
L l=0
4
(3?19)
is the sample covariance matrix. The above optimization problem can be solved as
described in [27], and the estimated true steering vector is denoted here as a?i .
To obtain the signal waveform estimate, we pass the received signals through a Capon
beamformer [44, 60]. The weight vector of the beamformer is determined by using the
estimated steering vector a?i in the following expression:
wi =
R??1
Yi a?i
a?Ti R??1
Yi a?i
.
(3?20)
Then the estimated signal waveform corresponding to the ith stimulating frequency is
s?i (l) = wiT yi (l).
(3?21)
By repeating the aforementioned process for i = 1 through i = M , we obtain the
complete set of M waveform estimates:
s?(l) = [s?1 (l), и и и , s?M (l)]T .
3.3.2.2
(3?22)
Stage II
Since the stimulating microwave sources with various frequencies are assumed to
have the same power, we assume that the thermal acoustic responses from the tumor at
different stimulating frequencies have nearly identical waveforms. Note that the thermal
acoustic responses induced by the inhomogeneous microwave energy distribution (due to
the inhomogeneous breast tissues) are different at different stimulating frequencies. This
means that the elements of the vector s?(l) are all approximately equal to an unknown
scalar signal s(l), and the noise and interference term can be assumed uncorrelated with
60
this signal. In Stage II of MART, we adopt the data model
s?(l) = as s(l) + es (l),
(3?23)
where as is approximately equal to 1M О1 . However, the ?steering vector? may again be
imprecise, and hence RCB is needed again.
As we did in Stage I, we assume that the only knowledge about as is that
kas ? a?s k2 ? ▓2 ,
(3?24)
where a?s = 1M О1 is the assumed steering vector, and ▓2 is a user parameter. Again, the
true steering vector as can be estimated via the covariance fitting approach
max ? 2
? 2 , as
subject to
R?s ? ? 2 as aTs ? 0
kas ? a?s k2 ? ▓2 ,
(3?25)
where ? 2 is the power of the signal s(l), and
4
R?s =
L?1
1X
s?(l)s?T (l)
L l=0
(3?26)
is the sample covariance matrix.
After obtaining the estimated steering vector a?s , we obtain the adaptive weight vector
and the estimated signal waveform, respectively, as
w=
R??1
s a?s
a?Ts R??1
s a?s
(3?27)
and
s?(t) = wT s?(t).
3.3.2.3
(3?28)
Stage III
Because the thermal acoustic pulse is usually bipolar: a positive peak, corresponding
to the compression pulse, and a negative peak, corresponding to the rarefaction pulse [79],
we use the peak-to-peak difference as the response intensity for the imaging location r
61
in the third stage of MART. Compared with other energy or amplitude based response
intensity estimation methods, peak-to-peak difference can be used to improve imaging
quality with little additional computation costs.
The positive and negative peak values of the estimated waveform for the focal
location r will be searched based on the estimated waveform (3?28) obtained in Stage
II. Because of the nonuniform sound speed in biological tissues, the arrival time of the
acoustic pulse generated at location r cannot be calculated accurately. However, it was
reported in [37] that when the heterogeneity is weak, such as in breast tissues, amplitude
distortion caused by multi-path is not severe. We assume that the original peak remains a
peak in the estimated waveform, and the positive and negative peak values of the thermal
acoustic pulse can be searched as
й
P
+
P
?
= max
and
(3?29)
Й
min {s?(l)}, 0 ,
(3?30)
l?[41 ,42 ]
й
= min
Й
max {s?(l)}, 0
l?[41 ,42 ]
where [41 , 42 ] ? [0, L] is the searching range. Here 41 and 42 are user parameters, and
the details on how to choose them can be found in [71].
After the positive and negative peak values are found, the response intensity for the
focal point at location r is given as:
I(r) = P + ? P ? .
3.4
(3?31)
Modeling and Simulations
We consider 2-D breast models simulated in two steps. In the first step, the
electromagnetic field inside the breast model is simulated and the specific absorption
rate (SAR) distribution is calculated based on the simulated electromagnetic field. The
second step is for the acoustic wave simulation, where the SAR distribution obtained
62
in the first step is used as the acoustic pressure source through the thermal expansion
coefficient. In both steps, the FDTD method is used for the simulation examples.
3.4.1
Electromagnetic Model and Simulation
For simulation purposes, the 2-D electromagnetic breast model used is as shown in
Figure 3-6(a). The breast model is a 10 cm in diameter semicircle, which includes skin,
breast fatty tissue, glandular tissues, and chest wall (muscle). A 1 mm-diameter tumor is
embedded below the skin. The dielectric properties of the breast tissues as well as tumor
at the microwave frequency fi (i = 1, и и и , M ) were calculated based on the Cole-Cole
model in (3?7). The dielectric properties of the normal breast fatty tissue are assumed
random with a variation of ▒10% around the nominal values. The dielectric constants of
glandular tissues are between ?r = 11 and ?r = 15.
Figure 3-7 shows a Gaussian modulated electromagnetic wave used to irradiate the
breast from the top of the model, as shown in Figure 3-6(a). The time duration for the
Gaussian pulse is 1 хs. The electromagnetic field is simulated using the FDTD method
[58, 59]. The grid cell size used by FDTD is 0.5 mm О 0.5 mm and the computational
region is terminated by perfectly matched layer (PML) absorbing boundary conditions
[63, 65].
The SAR distribution is given as [51, 72]
SAR(r) =
?(r)E 2 (r)
,
2?(r)
(3?32)
where ?(r) is the conductivity of the biological tissues at location r, E(r) is the electric
field at location r, and ?(r) is the mass density of the biological tissues at location r.
3.4.2
Acoustic Model and Simulation
In the microwave induced TAI system, the microwave energy is small, and as a result,
the acoustic pressure field induced by microwave is also small. So the nonlinear acoustic
effect does not need to be considered in the TAI system.
63
The two basic linear acoustic wave generation equations are [34]
?
u(r, t) = ??p(r, t)
?t
(3?33)
1 ?
?
p(r,
t)
+
?p(r,
t)
+
?
T (r, t),
?c2 ?t
?t
(3?34)
?
and
? и u(r, t) = ?
where u(r, t) is the acoustic velocity vector, p(r, t) is the acoustic pressure field, ? is the
mass density, ? is the attenuation coefficient, ? is the thermal expansion coefficient, and
T (r, t) is the temperature. The values for these acoustic properties for different breast
tissues are listed in Table 3-2 [31, 52, 80?82]. The attenuation coefficient is calculated with
f = 0.15 MHz. The values for the tumor are approximated using the values for muscle
because we cannot find the values specific to the tumor.
Because the duration of the microwave pulse is much shorter than the thermal
diffusion time, thermal diffusion can be neglected [34], and the thermal equation is
Cp
?
T (r, t) = SAR(r, t),
?t
(3?35)
where Cp is the specific heat. Substituting (3?35) into (3?34) gives
? и u(r, t) = ?
1 ?
?
p(r, t) + ?p(r, t) +
SAR(r, t).
2
?c ?t
Cp
(3?36)
FDTD is used again to compute the thermal acoustic wave based on Equations
(3?33) and (3?36). More details about FDTD for acoustic simulations can be found in
[73, 74, 83?88].
The breast model for the acoustic simulation is constructed similarly to the model
for electromagnetic simulation. The velocities of the normal fatty breast tissue are also
assumed random with a variation of ▒5% around average values, as shown in Figure
3-6(b). An acoustic sensor array with 35 elements deployed uniformly around the breast
model is used to record the thermal acoustic signals. The distance between neighboring
acoustic sensors is 4 mm. The grid cell size used by the acoustic FDTD is 0.1 mm О
64
0.1 mm and the computational region is terminated by perfectly matched layer (PML)
absorbing boundary conditions [87, 88]. Note that the size of the FDTD cell for acoustic
simulation is much finer than that of the FDTD cell for electromagnetic simulation
because the wavelength of an acoustic wave is much smaller than that of a microwave.
The SAR distribution data is interpolated to achieve the designed grid resolution for the
acoustic breast model.
The typical microwave induced thermal acoustic responses from the tumor are plotted
in Figure 3-8(a) for stimulating frequencies of f =200 MHz, 400 MHz, 600 MHz, and 800
MHz. The signals are simulated based on the aforementioned 2-D breast model. To obtain
the signals, we perform the simulation twice at each stimulating frequency, with and
without the tumor, and record the acoustic signals in an acoustic sensor. The difference
of the two received signals is referred to as the thermal acoustic response only from the
tumor at the stimulating frequency. It can be seen that the thermal acoustic responses
from the tumor at different stimulating frequencies are similar to one another. The figure
also shows that the thermal acoustic signals are wide-band bipolar pulses, with a large
positive peak and a large negative peak. Figure 3-8(b) shows the normalized spectra of
the acoustic signals corresponding to the excitation in Figure 3-8(a). It is seen that the
frequency range of the acoustic signals is about from 1 KHz to 400 KHz. The dominant
band (3 dB band) of the signals ranges from 10 KHz to 180 KHz, and the corresponding
acoustic wavelength ranges from 150 mm to 8 mm in the breast tissues.
3.5
Numerical Examples
Several numerical examples are used in this section to demonstrate the effectiveness
of MART. For comparison purposes, the DAS method is applied to the same data set
also. We also present the imaging results for the single-frequency microwave induced TAI
at different stimulating frequencies. The corresponding image reconstruction method
is referred to as the single-frequency adaptive and robust technique (SART), which is
similar with MART but without Stage II of MART. In SART, RCB is used to estimate
65
the thermal acoustic waveform at a certain stimulating frequency just like in Stage I of
MART. Then the peak search method used in MART Stage III is applied to the estimated
waveform to determine the image intensity.
In the first example, a 1 mm-diameter tumor is embedded in the breast model at the
location (x = 70 mm, y = 60 mm). This is the challenging case of early breast cancer
detection because of the small tumor size. Figures 3-9(a) and 3-9(b) show the imaging
results for MART and DAS, respectively. The tumor is shown clearly in the MART
image (Figure 3-9(a)), and the size and location of the tumor is accurate. The tumor is
essentially missed by DAS as shown in Figure 3-9(b). Figures 3-9(c), 3-9(d), 3-9(e), and
3-9(f) are the imaging results for SART at the stimulating frequencies f =200 MHz, 400
MHz, 600 MHz, and 800 MHz, respectively. The figures show that SART can determine
the tumor correctly, but some clutter show up in the SART images. Note that the clutter
show up at different locations with different stimulating frequencies. By comparing the
images for MART and SART, it can be seen that the clutter are effectively suppressed by
MART when multiple stimulating frequencies are used.
In the second numerical example, two small 1.5 mm-diameter tumors are set inside
the breast model as shown in Figure 3-10(a). Their locations are at (x = 70 mm, y = 60
mm) and (x = 75 mm, y = 62.5 mm). The distance between the two tumors is 4 mm.
The imaging results using MART and DAS are shown in Figures 3-10(b) and 3-10(c),
respectively. The two tumors are seen clearly in the MART image. To show them clearly
we zoom in onto the tumor locations in Figure 3-10(d), where the two black circles mark
the actual sizes and locations of the two tumors. It is shown that MART can be used to
determine the locations and sizes of the two tumors accurately. The DAS image contains
much clutter. The two tumors cannot be separated clearly in the DAS image because of
the poor resolution of DAS. Figures 3-10(e) and 3-10(f) are the imaging results of SART
at stimulation frequencies f =300 MHz and 700 MHz, respectively. The tumors can be
seen in both of the SART images, but clutter show up between the two tumors in Figure
66
3-10(e) and 3-10(f), and the sizes of the two tumors in Figure 3-10(f) are larger than their
true sizes.
3.6
Conclusions
An investigation of using a multi-frequency microwave induced thermal acoustic
imaging (TAI) system for early breast cancer detection has been reported in this chapter.
The frequency band for this system has been given based on the cutoff frequency of
the human breast. A simplified semicircular dielectric waveguide mode was used to
calculate the cutoff frequency in this dissertation. By studying the microwave energy
absorption properties of breast tissue and tumor, we have shown that the multi-frequency
microwave induced TAI can offer higher SNR, higher imaging contrast, and more effective
clutter suppression capability than the traditional single-frequency microwave induced
TAI. A Multi-frequency Adaptive and Robust Technique (MART) has been presented
for image formation. This data-adaptive algorithm can achieve better resolution and
better interference rejection capability than its data-independent counterparts, such as
DAS. The feasibility of this multi-frequency microwave induced TAI system as well as
the performance of the proposed image reconstruction algorithm for early breast cancer
detection have been demonstrated by using 2-D numerical electromagnetic and acoustic
breast models. The absorbed microwave energy and the thermal acoustic field in the
breast models have been simulated using the FDTD method. Numerical examples have
been used to demonstrate the excellent performance of MART.
67
Table 3-1. Cole-Cole parameters for biological tissues.
Tissue
??
?
4?1
?1 (ps)
?1
4?2
?2 (ns)
?2
4?3
?3 (хs)
?3
4?4
?4 (ms)
?4
Breast
2.5
0.01
3.0
17.68
0.1
15
63.66
0.1
5.0E4
454.7
0.1
2.0E7
13.26
0.0
Skin
4.0
0.0002
32.0
7.23
0.0
1100
32.48
0.2
0
N/A
N/A
0
N/A
N/A
Muscle
4.0
0.2
50.0
7.23
0.1
7000
353.68
0.1
1.2E6
318.31
0.1
2.5E7
2.274
0.0
Tumor
4.0
0.2
50.0
7.0
0.0
0
N/A
N/A
0
N/A
N/A
0
N/A
N/A
Table 3-2. Acoustic parameters for biological tissues. (* f is the acoustic frequency, and
the unit is MHz.)
Tissue
Breast
Skin
Muscle
Tumor
? (kg/m3 ) c (m/s) ?? (dB/cm) ? (1/? C) Cp (J/(? C и kg))
1020
1100
1041
1041
1510
1537
1580
1580
0.75f 1.5
3.5
0.57f
0.57f
68
3E-4
3E-4
3E-4
3E-4
3550
3500
3510
3510
a=5 cm
Perfect Magnetic Conductor (PMC)
A
a=5 cm
a=5 cm
PMC
PMC
HE21
TM01
Electric field
Magnetic field
B
Figure 3-1. Simplified breast model. A) Semicircular dielectric waveguide with PMC. B)
Corresponding circular dielectric waveguide.
0.9
0.8
0.7
Total Conductivity
0.6
0.5
Tumor
Breast
0.4
0.3
0.2
0.1
0
100
200
300
400
500
600
700
800
900
1000
Frequency (MHz)
Figure 3-2. Total conductivity of normal breast tissues and tumor as a function of
frequency.
69
23
22
Ratio of Conductivity
21
20
19
18
17
16
100
200
300
400
500
600
700
800
900
1000
Frequency (MHz)
Figure 3-3. Ratio of conductivity between tumor and normal breast tissue as a function of
frequency.
Microwave
120
acoustic sensor
100
Y (mm)
80
breast
60
tumor
skin
40
20
chest wall
20
40
60
80
100
120
140
160
180
X (mm)
Figure 3-4. Model of microwave induced TAI for breast cancer detection.
70
MART Stage I
Acoustic
Sensor Index
N
Time
Index
1
f1
fi
fM
Frequency
Index
Figure 3-5. Data cube model. In Stage I, MART slices the data cube for each frequency
index. RCB is applied to each data slice to estimate the corresponding
waveform.
71
?r
60
55
Microwave
120
50
45
100
40
80
breast
Y (mm)
35
60
30
tumor
skin
glandular tissues
25
40
20
20
15
chest wall
10
20
40
60
80
100
120
140
160
180
X (mm)
5
A
c0 (m/s)
1650
120
1600
acoustic sensor
100
1550
Y (mm)
80
breast
1500
60
tumor
skin
40
1450
20
chest wall
1400
20
40
60
80
100
120
140
160
180
X (mm)
1350
B
Figure 3-6. Breast model for thermal acoustic simulation. A) Model for electromagnetic
simulation. B) Model for acoustic simulation.
72
1
Single frequency microwave
Gaussian pulse
0.8
0.6
0.4
0.2
0
?0.2
?0.4
?0.6
?0.8
0
1
2
3
4
5
6
time (хs)
Figure 3-7. Gaussian modulated microwave source.
73
Thermal Acoustic Signals from Tumor
f=200 MHz
f=400 MHz
f=600 MHz
f=800 MHz
8
Acoustic Prussure (Pa)
6
4
2
0
?2
?4
0
20
40
60
80
100
120
140
160
Time (хs)
A
Normalized Spectrums of The Thermal Acoustic Signals
1
f=200 MHz
f=400 MHz
f=600 MHz
f=800 MHz
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
Frequency (KHz)
B
Figure 3-8. Thermal acoustic signals at different stimulating frequencies f =200 MHz, 400
MHz, 600 MHz, and 800 MHz. A) Thermal acoustic responses from tumor
only. B) The normalized spectrums of the signals in A.
74
dB
dB
0
0
2D Image
2D Image
120
120
?5
?5
100
100
?10
?10
80
Y (mm)
Y (mm)
80
?15
60
40
?15
60
40
?20
?20
20
20
?25
20
40
60
80
100
120
140
160
?25
180
20
40
60
X (mm)
80
100
120
140
160
180
X (mm)
?30
?30
A
B
dB
dB
0
0
2D Image
2D Image
120
120
?5
?5
100
100
?10
?10
80
Y (mm)
Y (mm)
80
?15
60
40
?15
60
40
?20
?20
20
20
?25
20
40
60
80
100
120
140
160
?25
180
20
40
60
X (mm)
80
100
120
140
160
180
X (mm)
?30
?30
C
D
dB
dB
0
0
2D Image
2D Image
120
120
?5
?5
100
100
?10
?10
80
Y (mm)
Y (mm)
80
?15
60
40
?15
60
40
?20
?20
20
20
?25
20
40
60
80
100
120
140
160
?25
180
20
X (mm)
40
60
80
100
120
140
160
180
X (mm)
?30
?30
E
F
Figure 3-9. Imaging results for the case of a single 1 mm-diameter tumor. A) MART.
B) DAS. D) SART at stimulating frequency f =200 MHz. D) SART at
stimulating frequency f =400 MHz. E) SART at stimulating frequency f =600
MHz. F) SART at stimulating frequency f =800 MHz.
75
dB
0
120
2D Image
120
100
?10
100
breast
80
Y (mm)
Y (mm)
80
tumor
60
tumor
skin
glandular tissues
?20
60
40
40
?30
20
20
chest wall
20
20
40
60
80
100
120
140
160
40
60
80
100
120
140
160
180
?40
X (mm)
180
X (mm)
A
B
dB
dB
0
0
2D Image
70
2D Image
68
120
?10
?10
66
100
64
62
Y (mm)
Y (mm)
80
?20
60
?20
60
58
40
?30
56
?30
54
20
52
20
40
60
80
100
120
140
160
180
?40
?40
50
X (mm)
60
65
70
75
80
85
X (mm)
C
D
dB
dB
0
0
2D Image
2D Image
120
120
?10
?10
100
100
80
?20
Y (mm)
Y (mm)
80
60
40
?20
60
40
?30
?30
20
20
20
40
60
80
100
120
140
160
180
20
?40
X (mm)
40
60
80
100
120
140
160
180
?40
X (mm)
E
F
Figure 3-10. Imaging results for the two 1.5 mm-diameter tumors case. A) Breast model.
B) MART. C) DAS. D) zoom in of B. E) SART at stimulating frequency
f =300 MHz. F) SART at stimulating frequency f =700 MHz.
76
CHAPTER 4
TIME REVERSAL BASED MICROWAVE HYPERTHERMIA TREATMENT OF
BREAST CANCER
4.1
Introduction
In the last two decades, many studies have shown the effectiveness of the local
hyperthermia, induced by microwave, in the treatment of breast cancer [6, 45, 48]. Most
of the studies concentrate on the narrow-band (NB) techniques, which can focus the
microwave energy at the desired location in the breast via adjusting the amplitude and
phase of the transmitted sinusoidal signal at each antenna. Recently, the feasibility
of using the ultra-wideband (UWB) technique for microwave hyperthermia treatment
is investigated in [52]. The authors in [52] develop an UWB space-time beamforming
approach, which can provide better temperature selectivity than the NB methods.
We propose a new UWB method, referred to as the time reversal based robust Capon
beamformer (TR-RCB), for hyperthermia treatment of breast cancer. Two high-resolution
techniques, i.e., time reversal (TR) [55, 56] and robust Capon beamformer (RCB) [44, 57],
are employed to shape the transmitted signals both temporally and spatially.
Time-reversal is a powerful method for focusing through complex media, which
can turn the disadvantage of randomly inhomogeneous and/or multipath rich media
into an advantage. This method has many application in acoustics, such as ultrasound
imaging [89] and medical imaging and therapy [53]. Recently, the feasibility of the
time-reversal in electromagnetic wave had been proved in [54]. The physical basis behind
this method is that the wave propagation is unchanged when time is reversed if we neglect
the dispersion of the media. Robust Capon beamformer is a data-adaptive beamforming
method which has higher resolution and better interference suppression capability than the
data-independent ones, such as delay-and-sum (DAS) and space-time beamforming.
To validate our algorithm, we have developed two 2D breast models with a small
embedded tumor. The models include the fatty breast tissue, skin, chest wall, as
well as glandular tissues, which are set randomly inside the fatty breast tissue. The
77
finite-difference time-domain (FDTD) method [58, 59] is used to simulate the electromagnetic
(EM) distribution and the temperature distribution within the breast. As we will show
in the 2D numerical simulations, the proposed method has better electromagnetic energy
focusing ability than the existing methods and can provide the necessary temperature
gradients required for effective hyperthermia.
The remainder of this chapter is organized as follows. In Section 4.2, we present the
time reversal and RCB algorithms for microwave hyperthermia. Simulation results based
on two 2-D breast models are provided in Section 4.3. Section 4.4 concludes this chapter.
4.2
Time-Reversal Based Microwave Hyperthermia
We consider a multistatic microwave system with M antennas as shown in Figure
4-1. The location of the antennas are denoted as ri (i = 1, 2, и и и , M ). Let r0 denote
the location of the tumor, which is assumed to be estimated accurately a priori using, for
example, the microwave imaging method in [57]. To achieve the temperature selectivity
needed for effective hyperthermia treatment, we adopt two high-resolution techniques, i.e.,
time reversal and RCB, to shape the transmitted signal both temporally and spatially.
In this method, we first transmit a low-power pulse from one of antennas (assuming
the j th antenna). The backscattered signals are received by all antennas, then the
couplings among the antennas and the reflections from the skin are removed, and finally
the time gate is used to retain only the direct paths of the signals from the tumor location.
The time gate for the ith received signal is
(
gatei (t) =
1, ni ? t ? ni + N
0,
,
i = 1, 2, . . . , M,
(4?1)
otherwise
where N is the length of the time gate, which can be determined approximately from the
duration of the transmitted pulse. ni is the discrete-time delay between the transmitter
antenna (j th antenna) and the ith receiver antenna, which can be calculated as
$
и
И%
1 krj ? r0 k kri ? r0 k
+
, i = 1, 2, . . . , M,
ni =
4t
C
C
78
(4?2)
where bxc stands for rounding to the greatest integer less than x, C is the velocity of
microwave propagating in breast tissues, and 4t is the sampling interval. The time-gated
signal are denoted as xi (t) (i = 1, 2, и и и , M ), which are the backscattered signals from
tumor.
Then the time-gated signals xi (t) (i = 1, 2, и и и , M ) are time-reversed and
retransmitted into the breast simultaneously after being amplified and weighted as:
yi (t) = ? wi xi (T ? t),
i = 1, 2, . . . , M,
(4?3)
where yi (t) is the signal retransmitted by the ith antenna, ? is the amplifier gain
controlling the peak power of the retransmitted signals, wi is the weight will be discussed
later on, and T is the maximum propagation time of microwave signals within the breast.
Because of the time reversibility of the wave equation, the time reversed signals will
refocuse at the tumor location when they are retransmitted into the breast.
The beamforming weights can be calculated from the time-gated signals xi (t) (i =
1, 2, и и и , M ) using the RCB algorithm, which has been used for breast cancer detection
in [57]. We consider the following signal
X?(t) = [x?1 (t) x?2 (t) и и и x?M (t)]T ,
(4?4)
where
x?i (t) = xi (t + ni ),
i = 1, 2, и и и , M,
(4?5)
is the time delayed signal of xi (t). Since X?(t) is properly time-shifted, which can be
modelled as
X?(t) = a и s(t) + e(t),
(4?6)
where s(t) is the backscattered response of tumor, and
e(t) = [e1 (t) e2 (t) и и и eM (t)]T
79
(4?7)
is a term comprising both interference and noise. The steering vector a is assumed to be
и
1
1
1
a=
иии
kr1 ? r0 k kr2 ? r0 k
krM ? r0 k
ИT
,
(4?8)
which represents the propagation attenuation of the backscattered signal from tumor.
RCB algorithm considers the following problem
min wT R?w
subject to wT a? = 1,
w
(4?9)
where w is the beamformer?s vector, and
N ?1
1 X
R? =
X?(t) и X?T (t)
N t=0
4
(4?10)
is the sample covariance matrix. a? is the true steering vector, which is a vector in the
vicinity of a, and that the only knowledge we have about a? is that
ka? ? ak2 ? ▓
(4?11)
where ▓ is a user parameter. The goal of the weights in (4?9) is to suppress the interference
as much as possible while keeping the signal from the tumor location undistorted.
The a? can be solved as described in [44], and the final solution to (4?9) is
w=
R??1 a?
a?T R??1 a?
.
(4?12)
When the weights are used as a transmitting beamformer, it can approximately achieve a
unit gain at the tumor location while minimizing the gain at sensitive locations (such as
the breast skin and nipple).
The proposed TR-RCB method is summarized as follows:
Step 1: We transmit a lower-power pulse microwave signal from one of the antennas.
The backscattered signals are received by all antennas and then time-gated. Based
on the time-gated signals, xi (t) (i = 1, 2, . . . , M ), we perform the RCB algorithm
and get the beamforming weights, w = [w1 , w2 , и и и , wM ]T .
80
Step 2: The time-gated waveforms are time-reversed, weighted, amplified and then
retransmitted into the breast from all antenna simultaneously, as shown in (4?3).
The signals yi (t) in (4?3) can be transmitted repetitively with a certain repetition
rate, denoted as R, which can be used to control the average microwave power.
4.3
4.3.1
Model and Numerical Results
Model and Simulation
For simulation purposes, two 2D EM (TMz) models of the breast are established,
as shown in Figure 4-1. Figure 4-1(a) and 4-1(b) represent the vertical position and
horizontal position of the breast, respectively. A 6 mm in diameter tumor is embedded
2.7 cm below the skin in both models. There are 17 and 16 antennas deployed uniformly
around the vertical and the horizontal breast models, respectively. The typical dielectric
properties of the tumor and breast tissues around 6 GHz are listed in Table 1-2 [9?11,
16]. Because the transmitted signals are UWB pulse, the dispersive properties of the
normal fatty breast tissue and the tumor are also considered in the EM models [90]. The
frequency dependence of the permittivity ?r (?) and conductivity ?r (?) are modeled using
the single-pole Debye model as shown in (2?35). The electromagnetic field deposition
is simulated using the FDTD method. Then the electromagnetic power deposition at
location r, denoted as Q(r), is calculated as [91]
Q(r) = R
T
X
? Ez (r, t)
0
?Dz (r, t)
,
?t
(4?13)
where Ez (r, t) and Dz (r, t) are the electric field and flux density, respectively.
After obtaining the electromagnetic power deposition, the 2D thermal models,
corresponding to the 2D EM models, are used to calculate the temperature distribution in
the breast tissues. The thermal model is based on the well-known bio-heat equation [51]
? и (K(r) ?T (r)) + A(r) + Q(r) ? B(r)(T (r) ? TB ) = C(R)?(r)
81
?T (r)
.
?t
(4?14)
Where K(r) is the thermal conductivity, T (r) is the temperature, A(r) is metabolic
heat production, B(r) represents the heat exchange mechanism due to capillary blood
perfusion, and TB is the blood temperature, which can be assumed as the body temperature.
The thermal properties for our breast models are listed in Table 4-1. The more detail
discussion can be found in [51].
The thermal models are also simulated using the FDTD method [72, 92]. The body
temperature and the environmental temperature are set at 36.8 ? C and 20 ? C, respectively.
The convective boundary condition is used at the skin surface.
4.3.2
Numerical Results
In this subsection, the simulation results are used to demonstrate the performances of
our TR-RCB method. For comparison purposes, the space-time beamforming method is
also applied to the same models. Figure 4-2 shows the simulated temperature distribution
within the breast. Figure 4-2(a) and 4-2(b) give the results for the vertical and horizontal
breast models, respectively, when the TR-RCB is used. Figure 4-2(c) and 4-2(d) are the
temperature distribution within the vertical and horizontal breast model, respectively,
using space-time beamforming methods. As we can see, our TR-RCB method can
elevate the temperature of the target region greater than 43? C while maintaining the
temperatures of the healthy regions below 43? C. We also note that the proposed method
provides better temperature selectivity than the space-time beamforming method.
4.4
Conclusions
We have presented a new UWB method for microwave hyperthermia treatment
of breast cancer employing the time reversal and robust weighted Capon beamforming
techniques. As shown in the 2D numerical simulations, this method has better EM energy
focusing ability than the existing methods, and can provide the necessary temperature
gradients required for effective hyperthermia.
82
Table 4-1. Typical thermal properties of breast tissues.
Chest Wall
Skin
Fatty Breast Tissue
Glandular Tissue
Tumor
K ( mиW? C )
0.564
0.376
0.499
0.499
0.564
W
A (m
3)
480
1620
480
480
480
83
B ( m3Wи? C )
2700
9100
2700
2700
2700
C ( kgиJ? C )
3510
3500
3550
3550
3510
kg
? (m
3)
1020
1100
1020
1020
1020
120
antennas
100
breast
Y (mm)
80
60
tumor
glandular
tissue
skin
40
20
chest wall
20
40
60
80
100
120
140
160
180
160
180
X (mm)
A
180
160
antennas
140
breast
Y (mm)
120
100
glandular
tissue
tumor
80
60
skin
40
20
20
40
60
80
100
120
140
X (mm)
B
Figure 4-1. Breast model and antenna array. A) Vertical position. B) Horizontal position.
84
44
180
44
160
120
42
42
140
100
4412
38
37
80
37
37
100
39 37
38
43
39
43
Y (mm)
39
60
42
41
39
37
Y (mm)
40
120
40
37
80
40
36
36
60
20
40
34
20
40
60
80
100
120
140
160
34
180
20
X (mm)
32
20
40
60
80
100
120
140
160
180
32
X (mm)
A
B
44
180
44
160
120
42
42
140
100
38
43
80
39
37
20
Y (mm)
37
40
100
37
39
38
41
42
42
40
39
43
60
37
Y (mm)
37
39
41
37
120
40
80
36
37
60
36
37
40
34
20
40
60
80
100
120
140
160
34
180
20
X (mm)
32
20
40
60
80
100
120
140
160
180
32
X (mm)
C
D
Figure 4-2. Temperature distribution. The dish lines denote the location of the skin. A)
and B) TR-RCB. C) and D) Space-time beamforming.
85
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
5.1
Conclusions
In this dissertation, we have presented the microwave techniques for breast cancer
detection and treatment. The physical basis of microwave techniques is the significant
contrast in dielectric properties (permittivity/conductivity) between normal and
malignant breast tissues. The UWB microwave signals and the microwave induced
wide-band thermal acoustic signals have been used for detection and treatment. Several
data-adaptive beamforming algorithms have been used to form the image or to shape
the transmitted signals for the treatment. The detailed conclusions provided by our
investigations are as follows.
Two data-adaptive algorithms, RCB and APES, for UWB microwave imaging have
been studied. We compared these algorithms with other data-independent methods and
illustrated their performance by using a complex 3-D breast model. The propagation of
the UWB microwave within the breast model was simulated by using the FDTD method.
Due to their data-adaptive nature, RCB and APES outperform their data-independent
counterparts in terms of improved resolution and reduced sidelobe levels.
A new multi-frequency microwave induced thermal acoustic imaging system has
been proposed for early breast cancer detection. By studying the microwave energy
absorption properties of breast tissues and tumor, we have shown that the multi-frequency
microwave induced TAI can offer higher SNR, higher imaging contrast, and more effective
clutter suppression capability than traditional single-frequency microwave induced
TAI. A data-adaptive algorithm, MART, has been presented for image formation,
which can achieve better resolution and better interference rejection capability than its
data-independent counterparts. 2-D numerical EM and acoustic breast models have been
developed. The absorbed microwave energy and the thermal acoustic field in the breast
models have been simulated by using the FDTD method. The excellent performance of
MART has been demonstrated by numerical examples.
86
A new time-reversal based UWB microwave method for hyperthermia treatment
of breast cancer has been presented. Two high-resolution techniques, time-reversal and
robust Capon beamformor, have been employed to shape the transmitted signals both
temporally and spatially. The FDTD method has been used to simulate the absorbed
microwave energy deposition and the temperature distribution in the breast based on the
Maxwell equation and bio-heat equation, respectively. The numerical results show that
this method has better EM energy focusing ability than the existing methods, and can
provide the necessary temperature gradients required for effective hyperthermia.
We have also presented a new FDTD formulation for wave propagation in biological
media with Cole-Cole model. The fractional order differentiators in the Cole-Cole
model have been approximated by a polynomial whose coefficients were found using
a least-squares method. This new formulation can give more accurate simulation of
microwave propagation in biological media than the traditional FDTD formulation.
5.2
Future Work
It has been presented in this dissertation that the microwave techniques are promising
technologies for both early breast cancer detection and effective treatment. However, there
is much research to be done before these techniques can be used for clinical diagnosis and
treatment. Moreover, the development of other new techniques can also be considered to
use for the breast cancer detection and treatment.
Several possible directions for future work are as follows.
5.2.1
Hardware Implementation for The Breast Cancer Detection and Treatment Systems
In Chapters 2, 3, and 4, we have presented the theoretical feasibility study of the
microwave techniques for breast cancer detection and treatment by using numerical
examples. The next natural step is the hardware implementation of the system. Some
possible problems that need to be considered are listed below.
87
The first challenge of implementing the UWB microwave imaging/treatment system
for breast cancer is to design a high performance UWB antenna. Typical wideband
antennas include resistively loaded monopoles, dipoles, bow-tie antennas, and horn
antennas [93?97]. However, several special design requirements are important for the
UWB microwave imaging/treatment systems. For example, the antenna should have a low
voltage standing wave ratio (VSWR) over a wide frequency band; the dimension of the
antenna should be compact to fit on the breast.
The ideal electromagnetic (EM) field distribution for microwave induced TAI systems
is an uniform EM power distribution. Otherwise the biological tissues will be heated
nonuniformly and thermal acoustic signals will be induced by the nonuniform EM power
distribution. Moreover, these thermal acoustic signals will be shown as clutters in the
breast images. There are several challenges in realizing the uniform electromagnetic field
distribution inside human breast, such as the attenuation of the human tissues and the
nonhomogeneous electrical properties of the human breast. A potential electromagnetic
excitation structure is a wire array. By choosing the amplitude and phase of each wire
source, we may optimize the EM power distribution to achieve uniformity.
5.2.2
Waveform Diversity Based Ultrasound System for Hyperthermia
Treatment of Breast Cancer
The development of breast cancer imaging techniques, such as microwave imaging
[3, 57], ultrasound imaging [31, 32], thermal acoustic imaging [40], and MRI, has improved
the ability to visualize and accurately locate the breast tumor without the need for
surgery [98]. This has lead to the probability of noninvasive local hyperthermia treatment
of breast cancer. Many studies have been performed to demonstrate the effectiveness
of the local hyperthermia on the treatment of breast cancer [45, 48]. There are two
major classes of local hyperthermia techniques: microwave hyperthermia [47, 50] and
ultrasound hyperthermia [99]. The penetration of microwave is poor in biological tissues.
Moreover, the focal spot generated by microwave is undesirable at the normal/cancerous
88
tissues interface because of the long wavelength of the microwave. Ultrasound can
achieve much better penetration depths than microwave. However, because the acoustic
wavelength is very short, the focal spot generated by ultrasound is very small (millimeter
or submillimeter in diameter) compared to the large tumor region (centimeter in diameter
on average). Thus, many focal spots are required for complete tumor coverage, and this
can result in a long treatment time and missed cancer cells.
As shown in Figure 5-1(a), in a traditional phased-array ultrasound hyperthermia
system, the transmit beam pattern is achieved by adjusting the phase and amplitude
of the signal in each transmitter element. Waveform diversity is a new beampattern
design technique recently proposed for multi-input multi-output (MIMO) radar [100?104],
as shown in Figure 5-1(b). Unlike the standard phased-array technique, transmitting
multiple different waveforms via its transducers offers more flexibility for transmit
beampattern design. By designing the transmitted signal cross-correlation matrix under
the uniform elemental power constraint, the waveform diversity can be exploited to
maximize the power deposition at the entire tumor region while minimizing the impact on
the surrounding healthy tissue region.
We may adopt the waveform diversity technique for improved ultrasound hyperthermia
treatment of breast cancer. By choosing a proper covariance matrix of the transmitted
waveforms under the uniform elemental power constraint, the resulting ultrasound
system has the potential to provide a focal spot matched to the entire tumor region, and
meanwhile minimize the impact to the surrounding healthy breast tissues.
89
Tumor
W1s(t)
WM s(t)
Transmit Phased-Array
A
Tumor
Combinations of {sm(t)}
s M (t)
s1 (t)
MIMO Transmit Array
B
Figure 5-1. Ultrasound hyperthermia system. A) Phased-array ultrasound hyperthermia
system. B) Waveform diversity ultrasound hyperthermia system.
90
APPENDIX A
A NEW FDTD FORMULATION FOR WAVE PROPAGATION IN BIOLOGICAL
MEDIA WITH COLE-COLE MODEL
A.1
Introduction
The finite-difference time-domain (FDTD) method has been widely used to simulate
the electromagnetic wave propagation in biological tissues [25, 105]. An advantage to using
the FDTD method is that wide frequency band components of the electromagnetic field
can be computed simultaneously. The dielectric properties of biological tissues, however,
are dispersive and their variation with the frequency is very complex [9]. To incorporate
dispersion into the FDTD method, the frequency dependent dielectric properties have
often been described by a Debye model [106, 107] or single-pole conductivity model [108].
However, the Debye model and the single-pole conductivity model do not represent the
frequency variation of many biological tissues accurately over a wide frequency band. The
Cole-Cole model [109] offers an alternative approach which can be used to describe many
types of biological tissues accurately over a very wide frequency band [11].
The implementation of the Cole-Cole model in FDTD is difficult because of the
fractional order differentiators in the model [107, 110]. In [111, 112], the authors
transformed the Cole-Cole dispersion relation into the time domain which involves a
convolution integral, and approximated the convolution integral by a decaying exponential
series. The method is complicated because it considers the time domain convolution
integral directly.
In this appendix, a new FDTD formulation is presented for modeling of the
electromagnetic wave propagation in dispersive biological tissues. The frequency
dependent permittivity of the biological tissues is given by the Cole-Cole model [11].
The Z-transform [66?68] is used to represent the frequency dependent relationship
between the electric flux density and the electric field in Z domain, which leads to a
FDTD implementations directly. The fractional order differentiators in the Cole-Cole
model are approximated by a polynomial. The coefficients of the polynomial are found by
91
using a least-squares (LS) fitting method. To demonstrate the accuracy of the proposed
formulation, two numerical examples are given.
The remainder of this appendix is organized as follows. In Section A.2, we present the
new FDTD formulation for the modeling of electromagnetic wave propagation in dispersive
biological tissues with the Cole-Cole model. Several numerical examples are given in
Section A.3 to verify the accuracy of the proposed formulation. Section A.4 concludes this
appendix.
A.2
Formulations
We consider the Cole-Cole media whose frequency dependent relative permittivity is
given as
?r (?) = ?? +
M
X
i=1
?
4?i
+
,
?
i
1 + (j??i )
j??0
(A?1)
where M is the order of the Cole-Cole model, ?? is the high frequency permittivity, ?i is
the relaxation time, 4?i is the pole amplitude, and ? is the conductivity. To simply the
equation, the Cole-Cole model described in Equation (A?1) uses ?i (0 ? ?i ? 1) instead of
(1 ? ?i ) as the measure of the broadening of dispersion. The frequency domain relationship
between the electric flux density D and the electric field E is
D(?) = ?0 ?r (?)E(?).
(A?2)
To avoid the several orders of magnitude difference between the electric field and
magnetic field, we normalized D and E as
r
E? =
?0
E,
х0
D? = ?
1
D.
? 0 х0
(A?3)
Then we have
1
? D?
?ОH
=?
?t
? 0 х0
(A?4)
D?(?) = ?r (?)E?(?).
(A?5)
?H
1
= ??
? О E?.
?t
? 0 х0
(A?6)
92
The update equation for D? and H can be obtained from the discrete formula of
Equations (A?4) and (A?6) as usual [59]. To generate a discrete difference formula for
Equation (A?5), the Z-transform [66?68] is used to represent the relative permittivity in
the frequency domain in (A?5). By substituting the bilinear transform
j? ?
2 1 ? z ?1
4t 1 + z ?1
(A?7)
into (A?5), we obtain
?
D? = ??? +
?
M
X
i=1
│
1+
2?i
4t
4?
┤?i i
(1?z ?1 )?i
(1+z ?1 )?i
+
?
2?0 (1?z ?1 )
4t (1+z ?1 )
? E?,
(A?8)
where z ?1 is a time delay operator. By introducing the parameters
Si =
│
1+
2?i
4t
4?
┤?i i
(1?z ?1 )?i
(1+z ?1 )?i
E?
(A?9)
and
I=
?
2?0 (1?z ?1 )
4t (1+z ?1 )
E?,
(A?10)
(A?8) can be rewritten as
D? = ?? E? +
M
X
Si + I.
(A?11)
i=1
The update equation of I is given by expanding (A?10) as
I n = I n?1 +
?4t n
(E? + E? n?1 ),
2?0
(A?12)
where I n and E? n are one of the x, y, or z components of I and E? at time step n,
respectively.
The key point is how to obtain a recursive formula to discretize the fractional-order
differentiator in (A?9). Rewriting (A?9) as
Si =
│
1+
4?i
┤?i
2?i
4t
93
(z?1)?i
(z+1)?i
E?,
(A?13)
and then using a K-order polynomial to approximate the fractional-order differentiator as
?i
(z ▒ 1)
K?1
X
=
k
b▒
kz ,
(A?14)
k=0
?
where b+
k and bk (k = 1, и и и , K ? 1) are the coefficients of the polynomials corresponding
to (z + 1)?i and (z ? 1)?i , respectively. Because the time stable condition of the FDTD
approach guarantees that the time growth factor satisfying
» n+1 »
»E
»
»
»
» E n » ? 1,
(A?15)
the region of z is z ? [?1, 1].
To find the coefficients in (A?14), we define the least-squares error as follows:
Z
R2 =
1
"
(1 ▒ z)?i ?
?1
K?1
X
#2
k
b▒
kz
dz.
(A?16)
k=0
The condition for R2 to be a minimum is that
?R2
=0
?b▒
k
for k = 0, и и и , K ? 1, which gives
Z "
1
(1 ▒ z)?i ?
?1
K?1
X
(A?17)
#
j
b▒
(?z k ) dz = 0.
j z
(A?18)
j=0
The more concise matrix form of (A?18) is
Xb = a,
where
?
R1
R1
(A?19)
R1
?
K?1
1dz
zdz и и и
z
dz
?1
?1
?1
?
? R1
R1 2
R1 K
?
zdz
z dz и и и
z dz
?
?1
?1
?1
X=?
..
..
..
...
?
.
.
.
?
? R
R1
R1
1
z K?1 dz ?1 z K dz и и и ?1 z 2K?2 dz
?1
94
?
?
?
?
?
?
?
?
(A?20)
is a K О K matrix,
Б
цT
▒
▒
b = b▒
0 , b1 , и и и , bK?1
and
иZ
Z
1
a=
?i
(z + 1) dz,
?1
Z
1
?i
ИT
1
(z + 1) zdz, и и и ,
?1
(A?21)
?i K?1
(z + 1) z
dz
(A?22)
?1
are K О 1 vectors. The solution to (A?19) is
b = X?1 a,
(A?23)
which is the least-squares coefficients for (A?16).
Substituting (A?14) into (A?13) gives the relationship between Si and E?
Si =
│
1+
│
=
1+
2?i
4t
2?i
4t
4?
┤?i iPK?1
Pk=0
K?1
k=0
┤?i
k
b?
kz
k
b+
z
k
E?
4?i
PK?1 ? ?(K?1?k)
bk z
Pk=0
K?1 + ?(K?1?k)
k=0 bk z
E?
PK?1 + ?(K?1?k)
k=0 bk z
│ ┤?i P
E?.
= P
K?1 ? ?(K?1?k)
K?1 + ?(K?1?k)
2?i
b
z
b
z
+
k=0 k
k=0 k
4t
4?i
(A?24)
The update equation for Si is
" K?2 х
#
х Х? i Х
K?1
X
X
1
2?i
n?(K?1?k)
+ n?(K?1?k)
?
+
n
│ ┤?i
Si =
bk E?
?
bk Si
+
bk +
,
?
2?i
4t
b
b+
+
k=0
k=0
K?1
K?1
4t
(A?25)
where Sin is one of the x, y, or z components of Si (z) at time step n. Then the update
equation for E? can be obtained by substituting (A?12) and (A?25) into (A?11).
A.3
Numerical Examples
Two numerical examples are shown to verify the new formulation presented. In the
first example, a plane wave propagating in fat tissue is simulated. The waveform of the
plane wave is a differential Gaussian pulse with time duration 100 ps. The 4th-order
(M = 4) Cole-Cole models for fat tissue as well as muscle tissue are listed in Table
A-1 [11]. A 5th-order polynomial (K = 5) is used to approximate the fractional-order
95
differentiator. The waveform was recorded after it has propagated 10 cm inside the fatty
tissue, and is shown in Figure A-1 with the solid line. The exact solution, obtained using
a theoretical analytical method, is also plotted in Figure A-1 and shown by the dotted
line. It can be seen that the proposed result matches the exact solution very well. As
a comparison, a 5th-order Taylor series is also used to approximate the fractional-order
differentiator in (A?14). The Taylor result is much poorer than the proposed result
because of its low convergence speed. Plane wave propagation is also simulated in the
fatty tissue with an approximated Debye model [25] and plotted in Figure A-1 with the
dashed line. The waveform simulated using the proposed method agrees with the exact
solution much better than the result simulated using the Debye model.
In the second example, the reflection coefficients at an air/fat interface and a
fat/muscle interface are computed, and shown in Figure A-2(a) and A-2(b), respectively.
The incident wave used in the simulations is a Gaussian pulse with time duration 25 ps.
The order of the polynomial is the same as the first example. The solid lines are simulated
reflection coefficients using the new method, which agree with the exact solutions (dotted
lines) very well. As a comparison, the simulation results using the Debye model for fat and
muscle tissues are also plotted in Figure A-2 with dashed lines. The curves show that the
new method presented here greatly improves the accuracy over the Debye model.
A.4
Conclusions
In this chapter, a new FDTD formulation is applied to the solution of electromagnetic
wave propagation in biological tissues using the Cole-Cole model. The fractional-order
differentiator in the Cole-Cole model is approximated by a polynomial whose coefficients
are found using a least-squares method. Numerical results are provided to demonstrate the
accuracy of the new formulation.
96
Table A-1. Cole-Cole parameters for fatty tissue and muscle tissue.
Tissue
??
?
4?1
?1 (ps)
?1
4?2
?2 (ns)
?2
4?3
?3 (хs)
?3
4?4
?4 (ms)
?4
Fat
2.5
0.035
9.0
7.96
0.8
35
15.92
0.9
3.3E4
159.15
0.95
1.0E7
15.915
0.99
97
Muscle
4.0
0.2
50.0
7.23
0.9
7000
353.68
0.9
1.2E6
318.31
0.9
2.5E7
2.274
1.0
z=10cm
Exact
LS
Debye
Taylor
0.3
0.2
0.1
0
?0.1
?0.2
600
800
1000
1200
1400
1600
1800
2000
2200
2400
time (ps)
Figure A-1. Time-domain waveform of a differential Gaussian pulse computed using our
method and the Debye model when compared with the exact solution.
98
Reflection Coefficient
Exact
LS
Debye
0.8
0.75
Magnitude
0.7
0.65
0.6
0.55
0.5
0.45
0
1
2
3
4
5
6
7
8
9
10
Frequency (GHz)
A
Reflection Coefficient
0.55
Exact
LS
Debye
Magnitude
0.5
0.45
0.4
0.35
0
1
2
3
4
5
6
7
8
9
10
Frequency (GHz)
B
Figure A-2. Reflection coefficient magnitude. A) Air/fat interface. B) Fat/muslce
interface.
99
APPENDIX B
NONLINEAR ACOUSTIC EFFECT IN MICROWAVE INDUCED THERMAL
ACOUSTIC IMAGE
In microwave induced TAI system, because the frequencies of the thermal acoustic
signal are very high (several hundred KHz to several MHz), and because breast tissues
are highly nonlinear media, we need to verify whether the nonlinear acoustic effect is too
small to be ignored. It has been shown in [80] that the temperature rise is about 0.1 mK
and the corresponding acoustic pressure change is about 100 Pa in the microwave induced
TAI system. In this appendix, the shock distance and the second harmonic content for the
nonlinear acoustic signal are calculated. The results show that it is reasonable to ignore
the nonlinear effect in microwave induced TAI system.
B.1
Shock Distance
The shock distance in breast tissues is [113]
?s =
where
B
A
B
2A
АB
2 2A
+
B
?0 c 2
? c2
c
2A
би
би 0 и
и ?min = А B
,
p
p
fmax
1
2 2A + 1
(B?1)
(? 10) is the nonlinear factor of the breast tissues, ?0 (? 1000 kg/m3 ) is the
mass density of the breast tissues, and c (? 1500 m/s) is the sound speed inside the
breast tissues [31]. p is the acoustic pressure rise, and ?min and fmax are the minimal
acoustic wavelength and the maximal acoustic frequency of the thermal acoustic signal,
respectively. For our breast model, the acoustic pressure rise is p=100 Pa, and the
maximal acoustic frequency is fmax =500 KHz. By substituting the parameters into (B?1),
we obtain the shock distance in breast tissues to be
B
?0 c 2
c
5
1000 и 15002
1500
2A
А
б
и
?s =
и
=
и
и
= 2.8 О 104 m.
B
3
p
fmax
2(5 + 1)
100
500 О 10
2 2A + 1
(B?2)
Because the size of our breast model is only 10 cm, which is much smaller than the shock
distance, the nonlinear acoustic effect in the microwave induced TAI system can be
ignored.
100
B.2
Second Harmonic Content
In nonlinear acoustic, high order harmonic distortion components will be generated
during the acoustic wave propagation. For a given acoustic signal
u(t) = u0 sin(?t),
(B?3)
after it propagates a distance of x, the second harmonic content is [114]
1
B u0
u2 = u0 (1 +
) kx,
2
2A c
(B?4)
where u(t) is the acoustic velocity, ? is the frequency of the acoustic signal, u0 is the
amplitude of the basic frequency component, u2 is the amplitude of the second harmonic
component, and k = ?/c is wave number. The acoustic velocity u0 can be calculated from
the acoustic pressure
u0 =
p
,
Z0
(B?5)
where Z0 = ?0 c is the acoustic impedance. We define the second harmonic component and
basic frequency component ratio as
B
p
?
u2
1
)и
и и x.
= (1 +
2
u0
2
2A ?0 c c
(B?6)
By substituting the parameters into (B?6) and assuming the propagation distance x = 10
cm, we obtain the ratio of the second harmonic component and the basic frequency
component at ? = 2? и 500KHz as follows:
1
100
u2
2? и 500 О 103
= (1 + 5) и
и 0.1 = 2.79 О 10?5 .
и
u0
2
1000 и 15002
1500
(B?7)
The result shows that the second harmonic component is about -90 dB smaller than the
basic frequency component. Therefore, our analysis indicates that the high order harmonic
distortion can be ignored in the microwave induced TAI system.
101
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BIOGRAPHICAL SKETCH
Bin Guo received his B.E. and M.Sc. degree from Xian Jiaotong University, Xian,
China, in 1997 and 2000, respectively, and his Ph.D. degree from University of Florida in
2007. From April 2000 to April 2002, he was a lecturer with the Department of Electronic
and Information Engineering, Xian Jiaotong University, Xian, China. From April 2002
to July 2003, he was an Associate Research Scientist with the Temasek Laboratories,
National University of Singapore, Singapore. Since August 2003, he has been a research
assistant with the Department of Electrical and Computer Engineering, University
of Florida, Gainesville. His current research interests include medical imaging, signal
processing, electromagnetic theory, and computational electromagnetics.
112
imaging (TAI) system for early breast cancer detection has been reported in this chapter.
The frequency band for this system has been given based on the cutoff frequency of
the human breast. A simplified semicircular dielectric waveguide mode was used to
calculate the cutoff frequency in this dissertation. By studying the microwave energy
absorption properties of breast tissue and tumor, we have shown that the multi-frequency
microwave induced TAI can offer higher SNR, higher imaging contrast, and more effective
clutter suppression capability than the traditional single-frequency microwave induced
TAI. A Multi-frequency Adaptive and Robust Technique (MART) has been presented
for image formation. This data-adaptive algorithm can achieve better resolution and
better interference rejection capability than its data-independent counterparts, such as
DAS. The feasibility of this multi-frequency microwave induced TAI system as well as
the performance of the proposed image reconstruction algorithm for early breast cancer
detection have been demonstrated by using 2-D numerical electromagnetic and acoustic
breast models. The absorbed microwave energy and the thermal acoustic field in the
breast models have been simulated using the FDTD method. Numerical examples have
been used to demonstrate the excellent performance of MART.
67
Table 3-1. Cole-Cole parameters for biological tissues.
Tissue
??
?
4?1
?1 (ps)
?1
4?2
?2 (ns)
?2
4?3
?3 (хs)
?3
4?4
?4 (ms)
?4
Breast
2.5
0.01
3.0
17.68
0.1
15
63.66
0.1
5.0E4
454.7
0.1
2.0E7
13.26
0.0
Skin
4.0
0.0002
32.0
7.23
0.0
1100
32.48
0.2
0
N/A
N/A
0
N/A
N/A
Muscle
4.0
0.2
50.0
7.23
0.1
7000
353.68
0.1
1.2E6
318.31
0.1
2.5E7
2.274
0.0
Tumor
4.0
0.2
50.0
7.0
0.0
0
N/A
N/A
0
N/A
N/A
0
N/A
N/A
Table 3-2. Acoustic parameters for biological tissues. (* f is the acoustic frequency, and
the unit is MHz.)
Tissue
Breast
Skin
Muscle
Tumor
? (kg/m3 ) c (m/s) ?? (dB/cm) ? (1/? C) Cp (J/(? C и kg))
1020
1100
1041
1041
1510
1537
1580
1580
0.75f 1.5
3.5
0.57f
0.57f
68
3E-4
3E-4
3E-4
3E-4
3550
3500
3510
3510
a=5 cm
Perfect Magnetic Conductor (PMC)
A
a=5 cm
a=5 cm
PMC
PMC
HE21
TM01
Electric field
Magnetic field
B
Figure 3-1. Simplified breast model. A) Semicircular dielectric waveguide with PMC. B)
Corresponding circular dielectric waveguide.
0.9
0.8
0.7
Total Conductivity
0.6
0.5
Tumor
Breast
0.4
0.3
0.2
0.1
0
100
200
300
400
500
600
700
800
900
1000
Frequency (MHz)
Figure 3-2. Total conductivity of normal breast tissues and tumor as a function of
frequency.
69
23
22
Ratio of Conductivity
21
20
19
18
17
16
100
200
300
400
500
600
700
800
900
1000
Frequency (MHz)
Figure 3-3. Ratio of conductivity between tumor and normal breast tissue as a function of
frequency.
Microwave
120
acoustic sensor
100
Y (mm)
80
breast
60
tumor
skin
40
20
chest wall
20
40
60
80
100
120
140
160
180
X (mm)
Figure 3-4. Model of microwave induced TAI for breast cancer detection.
70
MART Stage I
Acoustic
Sensor Index
N
Time
Index
1
f1
fi
fM
Frequency
Index
Figure 3-5. Data cube model. In Stage I, MART slices the data cube for each frequency
index. RCB is applied to each data slice to estimate the corresponding
waveform.
71
?r
60
55
Microwave
120
50
45
100
40
80
breast
Y (mm)
35
60
30
tumor
skin
glandular tissues
25
40
20
20
15
chest wall
10
20
40
60
80
100
120
140
160
180
X (mm)
5
A
c0 (m/s)
1650
120
1600
acoustic sensor
100
1550
Y (mm)
80
breast
1500
60
tumor
skin
40
1450
20
chest wall
1400
20
40
60
80
100
120
140
160
180
X (mm)
1350
B
Figure 3-6. Breast model for thermal acoustic simulation. A) Model for electromagnetic
simulation. B) Model for acoustic simulation.
72
1
Single frequency microwave
Gaussian pulse
0.8
0.6
0.4
0.2
0
?0.2
?0.4
?0.6
?0.8
0
1
2
3
4
5
6
time (хs)
Figure 3-7. Gaussian modulated microwave source.
73
Thermal Acoustic Signals from Tumor
f=200 MHz
f=400 MHz
f=600 MHz
f=800 MHz
8
Acoustic Prussure (Pa)
6
4
2
0
?2
?4
0
20
40
60
80
100
120
140
160
Time (хs)
A
Normalized Spectrums of The Thermal Acoustic Signals
1
f=200 MHz
f=400 MHz
f=600 MHz
f=800 MHz
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
Frequency (KHz)
B
Figure 3-8. Thermal acoustic signals at different stimulating frequencies f =200 MHz, 400
MHz, 600 MHz, and 800 MHz. A) Thermal acoustic responses from tumor
only. B) The normalized spectrums of the signals in A.
74
dB
dB
0
0
2D Image
2D Image
120
120
?5
?5
100
100
?10
?10
80
Y (mm)
Y (mm)
80
?15
60
40
?15
60
40
?20
?20
20
20
?25
20
40
60
80
100
120
140
160
?25
180
20
40
60
X (mm)
80
100
120
140
160
180
X (mm)
?30
?30
A
B
dB
dB
0
0
2D Image
2D Image
120
120
?5
?5
100
100
?10
?10
80
Y (mm)
Y (mm)
80
?15
60
40
?15
60
40
?20
?20
20
20
?25
20
40
60
80
100
120
140
160
?25
180
20
40
60
X (mm)
80
100
120
140
160
180
X (mm)
?30
?30
C
D
dB
dB
0
0
2D Image
2D Image
120
120
?5
?5
100
100
?10
?10
80
Y (mm)
Y (mm)
80
?15
60
40
?15
60
40
?20
?20
20
20
?25
20
40
60
80
100
120
140
160
?25
180
20
X (mm)
40
60
80
100
120
140
160
180
X (mm)
?30
?30
E
F
Figure 3-9. Imaging results for the case of a single 1 mm-diameter tumor. A) MART.
B) DAS. D) SART at stimulating frequency f =200 MHz. D) SART at
stimulating frequency f =400 MHz. E) SART at stimulating frequency f =600
MHz. F) SART at stimulating frequency f =800 MHz.
75
dB
0
120
2D Image
120
100
?10
100
breast
80
Y (mm)
Y (mm)
80
tumor
60
tumor
skin
glandular tissues
?20
60
40
40
?30
20
20
chest wall
20
20
40
60
80
100
120
140
160
40
60
80
100
120
140
160
180
?40
X (mm)
180
X (mm)
A
B
dB
dB
0
0
2D Image
70
2D Image
68
120
?10
?10
66
100
64
62
Y (mm)
Y (mm)
80
?20
60
?20
60
58
40
?30
56
?30
54
20
52
20
40
60
80
100
120
140
160
180
?40
?40
50
X (mm)
60
65
70
75
80
85
X (mm)
C
D
dB
dB
0
0
2D Image
2D Image
120
120
?10
?10
100
100
80
?20
Y (mm)
Y (mm)
80
60
40
?20
60
40
?30
?30
20
20
20
40
60
80
100
120
140
160
180
20
?40
X (mm)
40
60
80
100
120
140
160
180
?40
X (mm)
E
F
Figure 3-10. Imaging results for the two 1.5 mm-diameter tumors case. A) Breast model.
B) MART. C) DAS. D) zoom in of B. E) SART at stimulating frequency
f =300 MHz. F) SART at stimulating frequency f =700 MHz.
76
CHAPTER 4
TIME REVERSAL BASED MICROWAVE HYPERTHERMIA TREATMENT OF
BREAST CANCER
4.1
Introduction
In the last two decades, many studies have shown the effectiveness of the local
hyperthermia, induced by microwave, in the treatment of breast cancer [6, 45, 48]. Most
of the studies concentrate on the narrow-band (NB) techniques, which can focus the
microwave energy at the desired location in the breast via adjusting the amplitude and
phase of the transmitted sinusoidal signal at each antenna. Recently, the feasibility
of using the ultra-wideband (UWB) technique for microwave hyperthermia treatment
is investigated in [52]. The authors in [52] develop an UWB space-time beamforming
approach, which can provide better temperature selectivity than the NB methods.
We propose a new UWB method, referred to as the time reversal based robust Capon
beamformer (TR-RCB), for hyperthermia treatment of breast cancer. Two high-resolution
techniques, i.e., time reversal (TR) [55, 56] and robust Capon beamformer (RCB) [44, 57],
are employed to shape the transmitted signals both temporally and spatially.
Time-reversal is a powerful method for focusing through complex media, which
can turn the disadvantage of randomly inhomogeneous and/or multipath rich media
into an advantage. This method has many application in acoustics, such as ultrasound
imaging [89] and medical imaging and therapy [53]. Recently, the feasibility of the
time-reversal in electromagnetic wave had been proved in [54]. The physical basis behind
this method is that the wave propagation is unchanged when time is reversed if we neglect
the dispersion of the media. Robust Capon beamformer is a data-adaptive beamforming
method which has higher resolution and better interference suppression capability than the
data-independent ones, such as delay-and-sum (DAS) and space-time beamforming.
To validate our algorithm, we have developed two 2D breast models with a small
embedded tumor. The models include the fatty breast tissue, skin, chest wall, as
well as glandular tissues, which are set randomly inside the fatty breast tissue. The
77
finite-difference time-domain (FDTD) method [58, 59] is used to simulate the electromagnetic
(EM) distribution and the temperature distribution within the breast. As we will show
in the 2D numerical simulations, the proposed method has better electromagnetic energy
focusing ability than the existing methods and can provide the necessary temperature
gradients required for effective hyperthermia.
The remainder of this chapter is organized as follows. In Section 4.2, we present the
time reversal and RCB algorithms for microwave hyperthermia. Simulation results based
on two 2-D breast models are provided in Section 4.3. Section 4.4 concludes this chapter.
4.2
Time-Reversal Based Microwave Hyperthermia
We consider a multistatic microwave system with M antennas as shown in Figure
4-1. The location of the antennas are denoted as ri (i = 1, 2, и и и , M ). Let r0 denote
the location of the tumor, which is assumed to be estimated accurately a priori using, for
example, the microwave imaging method in [57]. To achieve the temperature selectivity
needed for effective hyperthermia treatment, we adopt two high-resolution techniques, i.e.,
time reversal and RCB, to shape the transmitted signal both temporally and spatially.
In this method, we first transmit a low-power pulse from one of antennas (assuming
the j th antenna). The backscattered signals are received by all antennas, then the
couplings among the antennas and the reflections from the skin are removed, and finally
the time gate is used to retain only the direct paths of the signals from the tumor location.
The time gate for the ith received signal is
(
gatei (t) =
1, ni ? t ? ni + N
0,
,
i = 1, 2, . . . , M,
(4?1)
otherwise
where N is the length of the time gate, which can be determined approximately from the
duration of the transmitted pulse. ni is the discrete-time delay between the transmitter
antenna (j th antenna) and the ith receiver antenna, which can be calculated as
$
и
И%
1 krj ? r0 k kri ? r0 k
+
, i = 1, 2, . . . , M,
ni =
4t
C
C
78
(4?2)
where bxc stands for rounding to the greatest integer less than x, C is the velocity of
microwave propagating in breast tissues, and 4t is the sampling interval. The time-gated
signal are denoted as xi (t) (i = 1, 2, и и и , M ), which are the backscattered signals from
tumor.
Then the time-gated signals xi (t) (i = 1, 2, и и и , M ) are time-reversed and
retransmitted into the breast simultaneously after being amplified and weighted as:
yi (t) = ? wi xi (T ? t),
i = 1, 2, . . . , M,
(4?3)
where yi (t) is the signal retransmitted by the ith antenna, ? is the amplifier gain
controlling the peak power of the retransmitted signals, wi is the weight will be discussed
later on, and T is the maximum propagation time of microwave signals within the breast.
Because of the time reversibility of the wave equation, the time reversed signals will
refocuse at the tumor location when they are retransmitted into the breast.
The beamforming weights can be calculated from the time-gated signals xi (t) (i =
1, 2, и и и , M ) using the RCB algorithm, which has been used for breast cancer detection
in [57]. We consider the following signal
X?(t) = [x?1 (t) x?2 (t) и и и x?M (t)]T ,
(4?4)
where
x?i (t) = xi (t + ni ),
i = 1, 2, и и и , M,
(4?5)
is the time delayed signal of xi (t). Since X?(t) is properly time-shifted, which can be
modelled as
X?(t) = a и s(t) + e(t),
(4?6)
where s(t) is the backscattered response of tumor, and
e(t) = [e1 (t) e2 (t) и и и eM (t)]T
79
(4?7)
is a term comprising both interference and noise. The steering vector a is assumed to be
и
1
1
1
a=
иии
kr1 ? r0 k kr2 ? r0 k
krM ? r0 k
ИT
,
(4?8)
which represents the propagation attenuation of the backscattered signal from tumor.
RCB algorithm considers the following problem
min wT R?w
subject to wT a? = 1,
w
(4?9)
where w is the beamformer?s vector, and
N ?1
1 X
R? =
X?(t) и X?T (t)
N t=0
4
(4?10)
is the sample covariance matrix. a? is the true steering vector, which is a vector in the
vicinity of a, and that the only knowledge we have about a? is that
ka? ? ak2 ? ▓
(4?11)
where ▓ is a user parameter. The goal of the weights in (4?9) is to suppress the interference
as much as possible while keeping the signal from the tumor location undistorted.
The a? can be solved as described in [44], and the final solution to (4?9) is
w=
R??1 a?
a?T R??1 a?
.
(4?12)
When the weights are used as a transmitting beamformer, it can approximately achieve a
unit gain at the tumor location while minimizing the gain at sensitive locations (such as
the breast skin and nipple).
The proposed TR-RCB method is summarized as follows:
Step 1: We transmit a lower-power pulse microwave signal from one of the antennas.
The backscattered signals are received by all antennas and then time-gated. Based
on the time-gated signals, xi (t) (i = 1, 2, . . . , M ), we perform the RCB algorithm
and get the beamforming weights, w = [w1 , w2 , и и и , wM ]T .
80
Step 2: The time-gated waveforms are time-reversed, weighted, amplified and then
retransmitted into the breast from all antenna simultaneously, as shown in (4?3).
The signals yi (t) in (4?3) can be transmitted repetitively with a certain repetition
rate, denoted as R, which can be used to control the average microwave power.
4.3
4.3.1
Model and Numerical Results
Model and Simulation
For simulation purposes, two 2D EM (TMz) models of the breast are established,
as shown in Figure 4-1. Figure 4-1(a) and 4-1(b) represent the vertical position and
horizontal position of the breast, respectively. A 6 mm in diameter tumor is embedded
2.7 cm below the skin in both models. There are 17 and 16 antennas deployed uniformly
around the vertical and the horizontal breast models, respectively. The typical dielectric
properties of the tumor and breast tissues around 6 GHz are listed in Table 1-2 [9?11,
16]. Because the transmitted signals are UWB pulse, the dispersive properties of the
normal fatty breast tissue and the tumor are also considered in the EM models [90]. The
frequency dependence of the permittivity ?r (?) and conductivity ?r (?) are modeled using
the single-pole Debye model as shown in (2?35). The electromagnetic field deposition
is simulated using the FDTD method. Then the electromagnetic power deposition at
location r, denoted as Q(r), is calculated as [91]
Q(r) = R
T
X
? Ez (r, t)
0
?Dz (r, t)
,
?t
(4?13)
where Ez (r, t) and Dz (r, t) are the electric field and flux density, respectively.
After obtaining the electromagnetic power deposition, the 2D thermal models,
corresponding to the 2D EM models, are used to calculate the temperature distribution in
the breast tissues. The thermal model is based on the well-known bio-heat equation [51]
? и (K(r) ?T (r)) + A(r) + Q(r) ? B(r)(T (r) ? TB ) = C(R)?(r)
81
?T (r)
.
?t
(4?14)
Where K(r) is the thermal conductivity, T (r) is the temperature, A(r) is metabolic
heat production, B(r) represents the heat exchange mechanism due to capillary blood
perfusion, and TB is the blood temperature, which can be assumed as the body temperature.
The thermal properties for our breast models are listed in Table 4-1. The more detail
discussion can be found in [51].
The thermal models are also simulated using the FDTD method [72, 92]. The body
temperature and the environmental temperature are set at 36.8 ? C and 20 ? C, respectively.
The convective boundary condition is used at the skin surface.
4.3.2
Numerical Results
In this subsection, the simulation results are used to demonstrate the performances of
our TR-RCB method. For comparison purposes, the space-time beamforming method is
also applied to the same models. Figure 4-2 shows the simulated temperature distribution
within the breast. Figure 4-2(a) and 4-2(b) give the results for the vertical and horizontal
breast models, respectively, when the TR-RCB is used. Figure 4-2(c) and 4-2(d) are the
temperature distribution within the vertical and horizontal breast model, respectively,
using space-time beamforming methods. As we can see, our TR-RCB method can
elevate the temperature of the target region greater than 43? C while maintaining the
temperatures of the healthy regions below 43? C. We also note that the proposed method
provides better temperature selectivity than the space-time beamforming method.
4.4
Conclusions
We have presented a new UWB method for microwave hyperthermia treatment
of breast cancer employing the time reversal and robust weighted Capon beamforming
techniques. As shown in the 2D numerical simulations, this method has better EM energy
focusing ability than the existing methods, and can provide the necessary temperature
gradients required for effective hyperthermia.
82
Table 4-1. Typical thermal properties of breast tissues.
Chest Wall
Skin
Fatty Breast Tissue
Glandular Tissue
Tumor
K ( mиW? C )
0.564
0.376
0.499
0.499
0.564
W
A (m
3)
480
1620
480
480
480
83
B ( m3Wи? C )
2700
9100
2700
2700
2700
C ( kgиJ? C )
3510
3500
3550
3550
3510
kg
? (m
3)
1020
1100
1020
1020
1020
120
antennas
100
breast
Y (mm)
80
60
tumor
glandular
tissue
skin
40
20
chest wall
20
40
60
80
100
120
140
160
180
160
180
X (mm)
A
180
160
antennas
140
breast
Y (mm)
120
100
glandular
tissue
tumor
80
60
skin
40
20
20
40
60
80
100
120
140
X (mm)
B
Figure 4-1. Breast model and antenna array. A) Vertical position. B) Horizontal position.
84
44
180
44
160
120
42
42
140
100
4412
38
37
80
37
37
100
39 37
38
43
39
43
Y (mm)
39
60
42
41
39
37
Y (mm)
40
120
40
37
80
40
36
36
60
20
40
34
20
40
60
80
100
120
140
160
34
180
20
X (mm)
32
20
40
60
80
100
120
140
160
180
32
X (mm)
A
B
44
180
44
160
120
42
42
140
100
38
43
80
39
37
20
Y (mm)
37
40
100
37
39
38
41
42
42
40
39
43
60
37
Y (mm)
37
39
41
37
120
40
80
36
37
60
36
37
40
34
20
40
60
80
100
120
140
160
34
180
20
X (mm)
32
20
40
60
80
100
120
140
160
180
32
X (mm)
C
D
Figure 4-2. Temperature distribution. The dish lines denote the location of the skin. A)
and B) TR-RCB. C) and D) Space-time beamforming.
85
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
5.1
Conclusions
In this dissertation, we have presented the microwave techniques for breast cancer
detection and treatment. The physical basis of microwave techniques is the significant
contrast in dielectric properties (permittivity/conductivity) between normal and
malignant breast tissues. The UWB microwave signals and the microwave induced
wide-band thermal acoustic signals have been used for detection and treatment. Several
data-adaptive beamforming algorithms have been used to form the image or to shape
the transmitted signals for the treatment. The detailed conclusions provided by our
investigations are as follows.
Two data-adaptive algorithms, RCB and APES, for UWB microwave imaging have
been studied. We compared these algorithms with other data-independent methods and
illustrated their performance by using a complex 3-D breast model. The propagation of
the UWB microwave within the breast model was simulated by using the FDTD method.
Due to their data-adaptive nature, RCB and APES outperform their data-independent
counterparts in terms of improved resolution and reduced sidelobe levels.
A new multi-frequency microwave induced thermal acoustic imaging system has
been proposed for early breast cancer detection. By studying the microwave energy
absorption properties of breast tissues and tumor, we have shown that the multi-frequency
microwave induced TAI can offer higher SNR, higher imaging contrast, and more effective
clutter suppression capability than traditional single-frequency microwave induced
TAI. A data-adaptive algorithm, MART, has been presented for image formation,
which can achieve better resolution and better interference rejection capability than its
data-independent counterparts. 2-D numerical EM and acoustic breast models have been
developed. The absorbed microwave energy and the thermal acoustic field in the breast
models have been simulated by using the FDTD method. The excellent performance of
MART has been demonstrated by numerical examples.
86
A new time-reversal based UWB microwave method for hyperthermia treatment
of breast cancer has been presented. Two high-resolution techniques, time-reversal and
robust Capon beamformor, have been employed to shape the transmitted signals both
temporally and spatially. The FDTD method has been used to simulate the absorbed
microwave energy deposition and the temperature distribution in the breast based on the
Maxwell equation and bio-heat equation, respectively. The numerical results show that
this method has better EM energy focusing ability than the existing methods, and can
provide the necessary temperature gradients required for effective hyperthermia.
We have also presented a new FDTD formulation for wave propagation in biological
media with Cole-Cole model. The fractional order differentiators in the Cole-Cole
model have been approximated by a polynomial whose coefficients were found using
a least-squares method. This new formulation can give more accurate simulation of
microwave propagation in biological media than the traditional FDTD formulation.
5.2
Future Work
It has been presented in this dissertation that the microwave techniques are promising
technologies for both early breast cancer detection and effective treatment. However, there
is much research to be done before these techniques can be used for clinical diagnosis and
treatment. Moreover, the development of other new techniques can also be considered to
use for the breast cancer detection and treatment.
Several possible directions for future work are as follows.
5.2.1
Hardware Implementation for The Breast Cancer Detection and Treatment Systems
In Chapters 2, 3, and 4, we have presented the theoretical feasibility study of the
microwave techniques for breast cancer detection and treatment by using numerical
examples. The next natural step is the hardware implementation of the system. Some
possible problems that need to be considered are listed below.
87
The first challenge of implementing the UWB microwave imaging/treatment system
for breast cancer is to design a high performance UWB antenna. Typical wideband
antennas include resistively loaded monopoles, dipoles, bow-tie antennas, and horn
antennas [93?97]. However, several special design requirements are important for the
UWB microwave imaging/treatment systems. For example, the antenna should have a low
voltage standing wave ratio (VSWR) over a wide frequency band; the dimension of the
antenna should be compact to fit on the breast.
The ideal electromagnetic (EM) field distribution for microwave induced TAI systems
is an uniform EM power distribution. Otherwise the biological tissues will be heated
nonuniformly and thermal acoustic signals will be induced by the nonuniform EM power
distribution. Moreover, these thermal acoustic signals will be shown as clutters in the
breast images. There are several challenges in realizing the uniform electromagnetic field
distribution inside human breast, such as the attenuation of the human tissues and the
nonhomogeneous electrical properties of the human breast. A potential electromagnetic
excitation structure is a wire array. By choosing the amplitude and phase of each wire
source, we may optimize the EM power distribution to achieve uniformity.
5.2.2
Waveform Diversity Based Ultrasound System for Hyperthermia
Treatment of Breast Cancer
The development of breast cancer imaging techniques, such as microwave imaging
[3, 57], ultrasound imaging [31, 32], thermal acoustic imaging [40], and MRI, has improved
the ability to visualize and accurately locate the breast tumor without the need for
surgery [98]. This has lead to the probability of noninvasive local hyperthermia treatment
of breast cancer. Many studies have been performed to demonstrate the effectiveness
of the local hyperthermia on the treatment of breast cancer [45, 48]. There are two
major classes of local hyperthermia techniques: microwave hyperthermia [47, 50] and
ultrasound hyperthermia [99]. The penetration of microwave is poor in biological tissues.
Moreover, the focal spot generated by microwave is undesirable at the normal/cancerous
88
tissues interface because of the long wavelength of the microwave. Ultrasound can
achieve much better penetration depths than microwave. However, because the acoustic
wavelength is very short, the focal spot generated by ultrasound is very small (millimeter
or submillimeter in diameter) compared to the large tumor region (centimeter in diameter
on average). Thus, many focal spots are required for complete tumor coverage, and this
can result in a long treatment time and missed cancer cells.
As shown in Figure 5-1(a), in a traditional phased-array ultrasound hyperthermia
system, the transmit beam pattern is achieved by adjusting the phase and amplitude
of the signal in each transmitter element. Waveform diversity is a new beampattern
design technique recently proposed for multi-input multi-output (MIMO) radar [100?104],
as shown in Figure 5-1(b). Unlike the standard phased-array technique, transmitting
multiple different waveforms via its transducers offers more flexibility for transmit
beampattern design. By designing the transmitted signal cross-correlation matrix under
the uniform elemental power constraint, the waveform diversity can be exploited to
maximize the power deposition at the entire tumor region while minimizing the impact on
the surrounding healthy tissue region.
We may adopt the waveform diversity technique for improved ultrasound hyperthermia
treatment of breast cancer. By choosing a proper covariance matrix of the transmitted
waveforms under the uniform elemental power constraint, the resulting ultrasound
system has the potential to provide a focal spot matched to the entire tumor region, and
meanwhile minimize the impact to the surrounding healthy breast tissues.
89
Tumor
W1s(t)
WM s(t)
Transmit Phased-Array
A
Tumor
Combinations of {sm(t)}
s M (t)
s1 (t)
MIMO Transmit Array
B
Figure 5-1. Ultrasound hyperthermia system. A) Phased-array ultrasound hyperthermia
system. B) Waveform diversity ultrasound hyperthermia system.
90
APPENDIX A
A NEW FDTD FORMULATION FOR WAVE PROPAGATION IN BIOLOGICAL
MEDIA WITH COLE-COLE MODEL
A.1
Introduction
The finite-difference time-domain (FDTD) method has been widely used to simulate
the electromagnetic wave propagation in biological tissues [25, 105]. An advantage to using
the FDTD method is that wide frequency band components of the electromagnetic field
can be computed simultaneously. The dielectric properties of biological tissues, however,
are dispersive and their variation with the frequency is very complex [9]. To incorporate
dispersion into the FDTD method, the frequency dependent dielectric properties have
often been described by a Debye model [106, 107] or single-pole conductivity model [108].
However, the Debye model and the single-pole conductivity model do not represent the
frequency variation of many biological tissues accurately over a wide frequency band. The
Cole-Cole model [109] offers an alternative approach which can be used to describe many
types of biological tissues accurately over a very wide frequency band [11].
The implementation of the Cole-Cole model in FDTD is difficult because of the
fractional order differentiators in the model [107, 110]. In [111, 112], the authors
transformed the Cole-Cole dispersion relation into the time domain which involves a
convolution integral, and approximated the convolution integral by a decaying exponential
series. The method is complicated because it considers the time domain convolution
integral directly.
In this appendix, a new FDTD formulation is presented for modeling of the
electromagnetic wave propagation in dispersive biological tissues. The frequency
dependent permittivity of the biological tissues is given by the Cole-Cole model [11].
The Z-transform [66?68] is used to represent the frequency dependent relationship
between the electric flux density and the electric field in Z domain, which leads to a
FDTD implementations directly. The fractional order differentiators in the Cole-Cole
model are approximated by a polynomial. The coefficients of the polynomial are found by
91
using a least-squares (LS) fitting method. To demonstrate the accuracy of the proposed
formulation, two numerical examples are given.
The remainder of this appendix is organized as follows. In Section A.2, we present the
new FDTD formulation for the modeling of electromagnetic wave propagation in dispersive
biological tissues with the Cole-Cole model. Several numerical examples are given in
Section A.3 to verify the accuracy of the proposed formulation. Section A.4 concludes this
appendix.
A.2
Formulations
We consider the Cole-Cole media whose frequency dependent relative permittivity is
given as
?r (?) = ?? +
M
X
i=1
?
4?i
+
,
?
i
1 + (j??i )
j??0
(A?1)
where M is the order of the Cole-Cole model, ?? is the high frequency permittivity, ?i is
the relaxation time, 4?i is the pole amplitude, and ? is the conductivity. To simply the
equation, the Cole-Cole model described in Equation (A?1) uses ?i (0 ? ?i ? 1) instead of
(1 ? ?i ) as the measure of the broadening of dispersion. The frequency domain relationship
between the electric flux density D and the electric field E is
D(?) = ?0 ?r (?)E(?).
(A?2)
To avoid the several orders of magnitude difference between the electric field and
magnetic field, we normalized D and E as
r
E? =
?0
E,
х0
D? = ?
1
D.
? 0 х0
(A?3)
Then we have
1
? D?
?ОH
=?
?t
? 0 х0
(A?4)
D?(?) = ?r (?)E?(?).
(A?5)
?H
1
= ??
? О E?.
?t
? 0 х0
(A?6)
92
The update equation for D? and H can be obtained from the discrete formula of
Equations (A?4) and (A?6) as usual [59]. To generate a discrete difference formula for
Equation (A?5), the Z-transform [66?68] is used to represent the relative permittivity in
the frequency domain in (A?5). By substituting the bilinear transform
j? ?
2 1 ? z ?1
4t 1 + z ?1
(A?7)
into (A?5), we obtain
?
D? = ??? +
?
M
X
i=1
│
1+
2?i
4t
4?
┤?i i
(1?z ?1 )?i
(1+z ?1 )?i
+
?
2?0 (1?z ?1 )
4t (1+z ?1 )
? E?,
(A?8)
where z ?1 is a time delay operator. By introducing the parameters
Si =
│
1+
2?i
4t
4?
┤?i i
(1?z ?1 )?i
(1+z ?1 )?i
E?
(A?9)
and
I=
?
2?0 (1?z ?1 )
4t (1+z ?1 )
E?,
(A?10)
(A?8) can be rewritten as
D? = ?? E? +
M
X
Si + I.
(A?11)
i=1
The update equation of I is given by expanding (A?10) as
I n = I n?1 +
?4t n
(E? + E? n?1 ),
2?0
(A?12)
where I n and E? n are one of the x, y, or z components of I and E? at time step n,
respectively.
The key point is how to obtain a recursive formula to discretize the fractional-order
differentiator in (A?9). Rewriting (A?9) as
Si =
│
1+
4?i
┤?i
2?i
4t
93
(z?1)?i
(z+1)?i
E?,
(A?13)
and then using a K-order polynomial to approximate the fractional-order differentiator as
?i
(z ▒ 1)
K?1
X
=
k
b▒
kz ,
(A?14)
k=0
?
where b+
k and bk (k = 1, и и и , K ? 1) are the coefficients of the polynomials corresponding
to (z + 1)?i and (z ? 1)?i , respectively. Because the time stable condition of the FDTD
approach guarantees that the time growth factor satisfying
» n+1 »
»E
»
»
»
» E n » ? 1,
(A?15)
the region of z is z ? [?1, 1].
To find the coefficients in (A?14), we define the least-squares error as follows:
Z
R2 =
1
"
(1 ▒ z)?i ?
?1
K?1
X
#2
k
b▒
kz
dz.
(A?16)
k=0
The condition for R2 to be a minimum is that
?R2
=0
?b▒
k
for k = 0, и и и , K ? 1, which gives
Z "
1
(1 ▒ z)?i ?
?1
K?1
X
(A?17)
#
j
b▒
(?z k ) dz = 0.
j z
(A?18)
j=0
The more concise matrix form of (A?18) is
Xb = a,
where
?
R1
R1
(A?19)
R1
?
K?1
1dz
zdz и и и
z
dz
?1
?1
?1
?
? R1
R1 2
R1 K
?
zdz
z dz и и и
z dz
?
?1
?1
?1
X=?
..
..
..
...
?
.
.
.
?
? R
R1
R1
1
z K?1 dz ?1 z K dz и и и ?1 z 2K?2 dz
?1
94
?
?
?
?
?
?
?
?
(A?20)
is a K О K matrix,
Б
цT
▒
▒
b = b▒
0 , b1 , и и и , bK?1
and
иZ
Z
1
a=
?i
(z + 1) dz,
?1
Z
1
?i
ИT
1
(z + 1) zdz, и и и ,
?1
(A?21)
?i K?1
(z + 1) z
dz
(A?22)
?1
are K О 1 vectors. The solution to (A?19) is
b = X?1 a,
(A?23)
which is the least-squares coefficients for (A?16).
Substituting (A?14) into (A?13) gives the relationship between Si and E?
Si =
│
1+
│
=
1+
2?i
4t
2?i
4t
4?
┤?i iPK?1
Pk=0
K?1
k=0
┤?i
k
b?
kz
k
b+
z
k
E?
4?i
PK?1 ? ?(K?1?k)
bk z
Pk=0
K?1 + ?(K?1?k)
k=0 bk z
E?
PK?1 + ?(K?1?k)
k=0 bk z
│ ┤?i P
E?.
= P
K?1 ? ?(K?1?k)
K?1 + ?(K?1?k)
2?i
b
z
b
z
+
k=0 k
k=0 k
4t
4?i
(A?24)
The update equation for Si is
" K?2 х
#
х Х? i Х
K?1
X
X
1
2?i
n?(K?1?k)
+ n?(K?1?k)
?
+
n
│ ┤?i
Si =
bk E?
?
bk Si
+
bk +
,
?
2?i
4t
b
b+
+
k=0
k=0
K?1
K?1
4t
(A?25)
where Sin is one of the x, y, or z components of Si (z) at time step n. Then the update
equation for E? can be obtained by substituting (A?12) and (A?25) into (A?11).
A.3
Numerical Examples
Two numerical examples are shown to verify the new formulation presented. In the
first example, a plane wave propagating in fat tissue is simulated. The waveform of the
plane wave is a differential Gaussian pulse with time duration 100 ps. The 4th-order
(M = 4) Cole-Cole models for fat tissue as well as muscle tissue are listed in Table
A-1 [11]. A 5th-order polynomial (K = 5) is used to approximate the fractional-order
95
differentiator. The waveform was recorded after it has propagated 10 cm inside the fatty
tissue, and is shown in Figure A-1 with the solid line. The exact solution, obtained using
a theoretical analytical method, is also plotted in Figure A-1 and shown by the dotted
line. It can be seen that the proposed result matches the exact solution very well. As
a comparison, a 5th-order Taylor series is also used to approximate the fractional-order
differentiator in (A?14). The Taylor result is much poorer than the proposed result
because of its low convergence speed. Plane wave propagation is also simulated in the
fatty tissue with an approximated Debye model [25] and plotted in Figure A-1 with the
dashed line. The waveform simulated using the proposed method agrees with the exact
solution much better than the result simulated using the Debye model.
In the second example, the reflection coefficients at an air/fat interface and a
fat/muscle interface are computed, and shown in Figure A-2(a) and A-2(b), respectively.
The incident wave used in the simulations is a Gaussian pulse with time duration 25 ps.
The order of the polynomial is the same as the first example. The solid lines are simulated
reflection coefficients using the new method, which agree with the exact solutions (dotted
lines) very well. As a comparison, the simulation results using the Debye model for fat and
muscle tissues are also plotted in Figure A-2 with dashed lines. The curves show that the
new method presented here greatly improves the accuracy over the Debye model.
A.4
Conclusions
In this chapter, a new FDTD formulation is applied to the solution of electromagnetic
wave propagation in biological tissues using the Cole-Cole model. The fractional-order
differentiator in the Cole-Cole model is approximated by a polynomial whose coefficients
are found using a least-squares method. N
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