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 ░ 2 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. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 CHAPTER 1 2 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1 1.2 1.3 1.4 1.5 . . . . . 12 14 16 17 19 ADAPTIVE MICROWAVE IMAGING FOR BREAST CANCER DETECTION 25 2.1 2.2 25 26 26 26 27 28 29 29 31 34 34 35 37 2.3 2.4 2.5 2.6 3 Background . . . . . . . . . . . . . . Microwave Imaging . . . . . . . . . . Microwave Induced Thermal Acoustic Microwave Hyperthermia Treatment Outline of This Dissertation . . . . . . . . . . . . . . . Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . 58 60 61 62 63 63 65 67 TIME REVERSAL BASED MICROWAVE HYPERTHERMIA TREATMENT OF BREAST CANCER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1 4.2 4.3 5 . . . . . . . . . Hyperthermia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 78 81 81 82 82 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . 86 5.1 5.2 . . . . 86 87 . . 87 . . 88 4.4 Introduction . . . . . . . . . . . Time-Reversal Based Microwave Model and Numerical Results . 4.3.1 Model and Simulation . . 4.3.2 Numerical Results . . . . Conclusions . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 92 95 96 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: ? ? ? ? ? ? ? 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. 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Hamilton and D. T. Blackstock, Nonlinear Acoustics. San Diego, CA: Academic Press, 1997. [114] D. T. Blackstock, Fundamentals of Physical Acoustics. New York, NY: John Wiley and Sons, 2000. 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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