# Finite difference time domain (FDTD) method for microwave imaging of inhomogeneous media for breast cancer detection

код для вставкиСкачатьFINITE DIFFERENCE TIME DOMAIN (FDTD) METHOD FOR MICROWAVE IMAGING OF INHOMOGENEOUS MEDIA FOR BREAST CANCER DETECTION by Xiaoye Chen B.S., Tianjin Polytechnic University, 2009 A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 2012 UMI Number: 1516598 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent on the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 1516598 Copyright 2012 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 This thesis for the Master of Science degree by Xiaoye Chen has been approved by Yiming Deng Tim Lei Mark Golkowski Date 07/23/2012 Chen, Xiaoye (M.S., Electrical Engineering) Finite Difference Time Domain (FDTD) Method for Microwave Imaging of Inhomogeneous Media for Breast Cancer Detection Thesis directed by Assistant Professor Yiming Deng ABSTRACT This thesis presented a parametric study of microwave breast cancer detection and imaging with finite-difference time-domain (FDTD) method. FDTD is one of the most widely used numerical modeling techniques to simulate electromagnetic wave propagation and interactions in biological tissues. The inhomogeneous breast tissue model with various microwave pulse widths/frequencies, tumor locations, materials dielectric properties and tissue compositions were systematically investigated in this work. The different scattering signal contrast due to the property differences among fatty, normal glandular, malignant and benign tissues in our study not only confirms the feasibility of detecting the breast cancer using microwave energy that has been extensively studied, but also serves as a valuable model-based tool for microwave sensing optimization and system development. iii This abstract accurately represents the content of the candidate’s thesis. I recommend its publication. Yiming Deng iv DEDICATION I dedicate this thesis to my parents, who gave me an appreciation of learning and taught me the value of perseverance and resolve. I also dedicate this thesis to my aunt for her support and understanding while I was completing this thesis. CONTENTS Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Breast Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Adipose Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Glandular Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Breast Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Mammography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Magnetic Resonance Imaging (MRI) . . . . . . . . . . . . . . . . 4 1.2.4 Breast Computed Tomography (CT) . . . . . . . . . . . . . . . . 5 1.2.5 Microwave Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.6 Thermoacoustic Imaging . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Motivation and Research Objectives . . . . . . . . . . . . . . . . . 8 2. Numerical Modeling for EM problems . . . . . . . . . . . . . . . . . . 11 2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 FDTD Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Yee’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 20 vi 2.4.1 Perfect Match Layer(PML) Boundary Conditions . . . . . . . . . 21 2.4.2 Absorbing Boundary Conditions(ABCs) . . . . . . . . . . . . . . 22 2.4.3 Perfect Electric Conductor(PEC) Boundary Conditions . . . . . . 22 2.4.4 Perfect Magnetic Conductor(PMC) Boundary Conditions . . . . . 23 3. FDTD Method for Microwave Breast Imaging . . . . . . . . . . . . . 24 3.1 Breast Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Breast Imaging using FDTD Measurement at Various Frequency . . 26 3.3 Breast Imaging Using FDTD Method at Various Frequencies with Different Dielectric Properties . . . . . . . . . . . . . . . . . . . . . 29 4. Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 39 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Appendix vii FIGURES Figure 1.1 The Normal Breast . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Mammogram of Breast Cancer . . . . . . . . . . . . . . . . . . . . 4 1.3 Targeted Breast Ultrasound of A Benign Mass . . . . . . . . . . . . 5 1.4 A Mass Showed on A Breast MRI . . . . . . . . . . . . . . . . . . . 6 1.5 Breast CT Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Relative Permittivity and Conductivity of Normal and Malignant Breast Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Thermoacoustic Imaging Process . . . . . . . . . . . . . . . . . . . 9 2.1 Time Step Leap-Frogging . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 The Position of Field Components On the Grid . . . . . . . . . . . 15 2.3 Location of the TE Fields in the Computational Domain . . . . . . 17 2.4 PEC Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 PMC Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 The Homogenous Cylindrical Breast Model . . . . . . . . . . . . . 24 3.2 The Breast Model with Tumor on the Left . . . . . . . . . . . . . . 25 3.3 The Breast Model with Tumor on the Right . . . . . . . . . . . . . 26 3.4 The Breast Model with Tumor Between the Normal and Fatty Tissue 27 3.5 The Breast Model with Tumor Outside of the Normal Tissue . . . . 3.6 Electric Field for the Breast Model with Tumor on the Left Frequency Range 400MHz-1GHz . . . . . . . . . . . . . . . . . . . . . 28 29 viii 3.7 Electric Field for the Breast Model with Tumor on the Right Frequency Range 400MHz-1GHz . . . . . . . . . . . . . . . . . . . . . 3.8 Electric Field for the Breast Model with Tumor on the Edge between Normal and Fatty Tissue Frequency Range 400MHz-1GHz . . . . . 3.9 30 31 Electric Field for the Breast Model with Tumor on the Outside of Fatty Tissue Frequency Range 400MHz-1GHz . . . . . . . . . . . . 31 3.10 Electric Field Difference Between the Model with and without Tumor (Tumor on the Left) at Various Frequencies . . . . . . . . . . . . . 32 3.11 Electric Field Difference Between the Model with and without Tumor (Tumor on the Right) at Various Frequencies . . . . . . . . . . . . 32 3.12 Electric Field Difference Between the Model with and without Tumor (Tumor on the Edge between Normal and Fatty Tissue) at Various Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.13 Electric Field for the Breast Model - Tumor on the left - Frequency Range 0.4GHz-1GHz with Different Dielectric Properties . . . . . . 33 3.14 Electric Field for the Breast Model - Tumor on the Right - Frequency Range 0.4GHz-1GHz with Different Dielectric Properties . . . . . . 35 3.15 Electric Field for the Breast Model - Tumor on the Edge between Normal and Fatty Tissue - Frequency Range 0.4GHz-1GHz with Different Dielectric Properties . . . . . . . . . . . . . . . . . . . . . . 35 3.16 Electric Field Difference Between the Model with and without Tumor (Tumor on the Left) at Various Frequencies with Different Dielectric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ix 3.17 Electric Field Difference Between the Model with and without Tumor (Tumor on the Right) at Various Frequencies with Different Dielectric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.18 Electric Field Difference Between the Model with and without Tumor (Tumor on the Edge between Normal and Fatty Tissue) at Various Frequencies with Different Dielectric Properties . . . . . . . . . . . 38 x TABLES Table 3.1 Dielectric properties of female breast tissue at 3.2GHz[17] . . . . . 3.2 The variation of the relative permittivity and the conductivity of normal and cancer tissue between 400MHz-1GHz[11] . . . . . . . . 27 34 xi 1. Introduction Breast cancer is the most common type of cancer among women and also results in the most prevalent form of malignant tumors. The mortality rate is currently very high[9], but early detection of breast cancer can help reduce its victims’ death rates. There are several breast cancer screening methods available for early detection. X-ray mammography is currently the most widely used imaging technique. Other established imaging techniques such as ultrasound and Magnetic Resonance Imaging (MRI) have been pursued with some success to supplement mammography in certain specific cases. Even with the combined use of mammography, ultrasound, and MRI techniques, current methods of screening for breast cancer still do not meet the ideal requirement, which has lower radiation, more sensitive, more accurate and less uncomfortable feeling for patient. 1.1 Breast Cancer More and more people have cancer these days, cancer of the prostate, liver, pancreas, colorectal, and so forth, and especially of the breast. Breast cancer is a malignant tumor that starts at the cellular level. There are two broad classifications of breast cancer: ductal carcinoma and lobular carcinoma. The most common type of breast cancer is ductal carcinoma, occurring in slightly over 90% of all cases, where the cancer cells are inside the ducts but have not spread through the walls of the ducts into the surrounding breast tissue. Ductal carcinoma is further divided into two types: ductal carcinoma in situ 1 and invasive ductal carcinoma. Lobular carcinoma in situ, however, is a unique lesion that is manifested by proliferation in the terminal ducts or acini. These lesions have loosely spaced large cells. This type of cancer does not usually penetrate through the wall of the lobule, thus its name[19]. Figure 1.1 shows the normal breast of female. Figure 1.1: The Normal Breast[25] 1.1.1 Adipose Tissue The adipose tissue in the breast can be divided into three main groups: subcutaneous, retromammary, and intraglandular.The majority of cancers develop 2 in the region of glandular tissue no greater than 1cm of fat layers. Unfortunately, the layers of fat do not completely isolate the glandular tissue from either the surrounding skin or muscle, which enables the cancer to spread[27]. 1.1.2 Glandular Tissue The glandular tissue in the breast consists of a number of discrete lobes, which are made up of lobules and ducts. Toward the nipple, each duct widens to form a sac (ampulla). During lactation, the bulbs on the ends of the lobules produce milk. Once milk is produced, it is transferred through the ducts to the nipple. 1.2 Breast Screening 1.2.1 Mammography In the US, mammography is currently the most effective method used for early detection and diagnosis of breast diseases in women. All major US medical organizations recommend mammography for women 40 years and older. Screening mammography reduces breast cancer mortality from about 35% to 20% in women aged 50 to 69 years and slightly less in women aged 40 to 49 years as seen at a follow-up 14 years after these procedures[13]. Mammography uses a low-dose x-ray system to examine breast tissue and is more affordable for patients. Unfortunately, the procedure also has large limitations. First, mammography is less sensitive in women with dense breast tissue, because the amount of fibroglandular tissue may represent an independent risk factor for developing breast cancer. Also, undergoing the mammography procedure exposes the patient to relative high level of ionizing radiation and causes discomfort due to breast compression[14]. 1.2.2 Ultrasound 3 Figure 1.2: Mammogram of Breast Cancer[35] Ultrasound imaging relies on high frequency sound waves that reflect with varying intensity from different tissues. In the breast, ultrasound is able to differentiate skin, fat,glandular tissue, and muscle. Breast ultrasound is frequently used as a targeted diagnostic examination technique. The technique is good because it does not require radiation, is non-invasive, is cheap, and is portable for patients. However, the breast fat and most cancer cells have similar acoustic properties, so an ultrasound cannot always detect tumors correctly. Since ultrasound procedures are performed using hand-held devices, the imaging results maybe highly operator-dependent. 1.2.3 Magnetic Resonance Imaging (MRI) 4 Figure 1.3: Targeted Breast Ultrasound of A Benign Mass[21] Magnetic resonance imaging relies on the interaction of RF energy and strong magnetec fields with the magnetic properties of certain atoms to produce high resolution images. Breast MRI is more sensitive than mammography, which is especially useful for young women at a substantially increased risk for breast cancer[13]. It can take good images around breast implants and dense breast tissue. The high cost of MRI and its relatively low specificity prohibit its routine use for screening in the general population. The results from MRI screenings of high-risk women also may not be suitable, MRI is more useful when apply to women at an average risk[13]. 1.2.4 Breast Computed Tomography (CT) Breast CT Scan techniques are able to detect tumors with diameters down to 2mm making them better than x-ray techniques, which have minimum tumor 5 Figure 1.4: A Mass Showed on A Breast MRI[6] detection diameters of 5mm. However, CT techniques pose an additional risk of radiation damage to the human body, so they cannot be used as a normal method of diagnosis. 1.2.5 Microwave Imaging In order to minimize the limitation of the methods mentioned before, new imaging modalities should be developed as replacement or incremental techniques to improve both sensitivity and specificity of the current imaging systems[3]. Among the emerging breast cancer imaging technologies, microwave imaging is one of the most promising and attractive methods[33]. Microwave imaging uses antennas to measure the interaction of electromagnetic waves (generally between 1 and 5GHz) with tissue of varying electrical properties. It is non-ionizing, sensitive enough for tumor detection, and specific to malignant tumors. Microwave imaging avoids breast compression and has faster exam times[32]. More specifically, microwave imaging spectroscopy for biomedical ap- 6 Figure 1.5: Breast CT Image[31] plications reconstructs the electrical properties of tissues over the microwave frequency range[3]. The microwave exam involves the propagation of very low levels of microwave energy through breast tissue to measure electrical properties[34]. Previous studies have shown that there is a significant dielectric property contrast between normal and malignant breast tissue[17]. Fig 1.6 shows graphs of conductivity and relative permittivity of normal and malignant breast tissue over the microwave frequency range 50MHz to 1GHz. The graphed data shows that the greatest electrical property contrast between normal and malignant breast tissue occurs between 600MHz and 1GHz. Therefore, microwave imaging can provide substantial functional information about the breast tissue health and can also be used as a detection and treatment response monitoring tool for breast cancer. 7 Figure 1.6: Relative Permittivity and Conductivity of Normal and Malignant Breast Tissue[17] 1.2.6 Thermoacoustic Imaging Thermoacoustic imaging is a very promising hybrid technique that combines elements of microwave and ultrasound imaging methods. Breast tissues characterization occurs based on thermoelastic wave generation by taking advantage of high microwave absorption coefficients contrast of biological tissues while retaining the superior spatial resolution of ultrasonic waves. Figure 1.7 shows the process of the generation of thermoacoustic waves[33]. 1.3 Motivation and Research Objectives There are mainly two methods used for modeling breast imaging: FEM and FDTD. Paulsen et al investigated four modalities of breast imaging by Finite element method(FEM) of specific material properties throughout some portion of the breast, which include Magnetic Resonance Elastography(MRE), Electrical Impedance Spectroscopy(EIS), Microwave Imaging Spectroscopy(MIS), and 8 Figure 1.7: Thermoacoustic Imaging Process[27] Near Infrared Spectroscopic Imaging(NIS)[14]. There are several FDTD models of the breast that have already been developed by different research groups. Hagness et al.[17] developed a 2-D FDTD model of the breast based on a planar configuration, which is defined by the orientation of the patient and the position of the antenna array elements. Another FDTD model of the breast based on actual MRI images was developed by Li et al.[17]. Fear et al.[17] developed a FDTD model based on the configuration of the patient positioned in the prone position, with the breast extending naturally through a hole in the examination table. Fear’s model is the homogeneous and heterogeneous FDTD tissue model we are using in this paper. 9 In this thesis, I present a Finite-Difference Time-Domain (FDTD) method for performing simulations of microwave propagation in a two-dimensional (2-D) model of the human breast in order to simulate a microwave propagation into a simple model of the breast. The breast model is comprised of skin and an annular layer of fatty tissue, interior to the fatty tissue is normal tissue, and the tumor is set inside of normal tissue. The remainder of this thesis is organized as follows: Chapter 2 gives a brief introduction of the FDTD(Finite-Difference Time-Domain) Method. Chapter 3 shows results of the electric field and the electric field differences between two models on breast imaging by using FDTD method. Finally, Chapter 4 draws a summary of the work accomplished in this thesis. 10 2. Numerical Modeling for EM problems The finite-difference time-domain (FDTD) is one of the primary available computational electrodynamics modeling techniques. It is used in solving electromagnetic scattering problems, because it can model an inhomogeneous object of arbitrary shape[10]. FDTD avoids the difficulties with linear algebra that limit the size of frequency-domain integral-equation and finite-element electromagnetics models to generally fewer than 109 electromagnetic field unknowns[2]. The FDTD method is arguably the simplest, both conceptually and in terms of implementation, of the full-wave techniques used to solve problems in electromagnetics. It can accurately tackle a wide range of problems, scaling with high efficiency on parallel-processing CPU-based computers, and extremely well on recently developed GPU-based accelerator technology, the parallel-processing computer architectures that have come to dominate supercomputing. Many researchers have contributed immensely to extend the method to many areas of science and engineering areas, such as scattering, radar cross-section, antennas, medical applications. However, the FDTD method is generally computationally expensive. Solutions may require a large amount of memory and computation time. 2.1 Maxwell’s Equations Since the FDTD method is one of the primary modeling technique for electromagnetic problems, so the stating point for the construction of an FDTD 11 algorithm are Maxwell’s time-domain equations. Maxwells equations in differential form relate the time derivative of the E-field to the curl of the H-field. The time-domain form of Maxwell’s equations are included below as a reference. → − → − − → ∂B = − ▽ ×E − M ∂t → − → − − → ∂D = ▽×H − J ∂t → − ▽· D = ρ F araday ′sLaw (2.1) Ampere′ sLaw (2.2) Gauss′ sLawf orElectricF ields (2.3) Gauss′ sLawf orMagneticF ields (2.4) ContinuityEquationf orElectricF ields (2.5) ContinuityEquationf orMagneticF ields (2.6) → − ▽ · B = ρ∗ with definitions → − ∂ ▽· J =− ρ ∂t − → ∂ ▽ · M = − ρ∗ ∂t where − → B: → − D: → − E: → − H: → − J: Magnetic flux density (W B/m2 ) Electric flux density (C/m2 ) Electric field intensity (V/m) Magnetic field intensity (A/m) Electric current density (A/m2 ) 12 − → M : Magnetic current density (V /m2 ) ρ: Electric charge density (C/m3 ) ρ∗ : Magnetic charge density (Wb/m3 ) Constitutive relations are necessary to supplement Maxwell’s equations and characterize the material media. Constitutive relations for linear, isotropic, and nondispersive materials can be written as:[12] − → → − → − B = µ H = µ0 µr H (2.7) − → → − → − D = ǫ E = ǫ0 ǫr E (2.8) − → → − → − J = J s ource + σ E (2.9) − → − → → − M = M s ource + σ ∗ H (2.10) where ǫ0 : Free space permittivity (8.854 × 10−12 F/m) ǫr : Relative permittivity ǫ: Permittivity (F/m) of the media. µ0 : Free-space permeability(4π × 10−7 H/m) µr : Relative permeability µ: Permeability (H/m) of the media. σ: Electric Conductivity(S/m) σ ∗ : Equivalent Magnetic Loss(ohms/m) 2.2 FDTD Methods To indicate time, the superscript n is related to time t by the following equation. The variable ∆t is referred to as the time-step size. t = n∆t (2.11) 13 In the FDTD method, the electric field and magnetic field values which are relative to time step n can be expressed as: E n [i, j, k], E n+1 [i, j, k] and H n+1/2 [i, j, k], H n−1/2 [i, j, k] Electric field values are first calculated at each time step, then the magnetic field values are calculated at half time steps, as shown in Fig 2.1, such an approach is known as leap frogging time step. Figure 2.1: Time Step Leap-Frogging[28] 2.3 Yee’s Algorithm A well-known, efficient implementation of the FDTD method is Yee’s algorithm. In 1966, Kane S.Yee derived an elegant, yet simple, time-dependent solution of Maxwells equations based on their differential form using central difference approximations of both the space and the time-derivatives. In the Yee 14 algorithm, the electric and magnetic fields are defined on an intertwined double mesh, where electric field components are circulated by four magnetic field components and magnetic field components are circulated by four electric field components[18]. The Yee cell is shown in Fig 2.2. Figure 2.2: The Position of Field Components On the Grid[29] The Yee’s method approximates the differentiation operators of the Maxwell equations with finite-difference operators in time and space. It can be seen from the Yee’s that the time derivative of the E field is dependent on the curl of the H field. This can be simplified to state that the change in the E field (the time 15 derivative) is dependent on the change in the H field across space (the curl). This results in the basic FDTD equation that the new value of the E field is dependent on the old value of the E field (hence the difference in time) and the difference in the old value of the H field on either side of the E field point in space. Naturally this is a simplified description, which has omitted constants, etc. But the overall effect is as described. The H field is found in the same manner. The new value of the H field is dependent on the old value of the H field (hence the difference in time), and also dependent on the difference in the E field on either side of the H field point. From Fig 2.3, each E field component is surrounded by four H field components, similarly each H field component is surrounded by four E field components. By using the FDTD method to replace the spatial and the time domain derivatives of Maxwell’s equations with finite difference approximations. Taylor’s theorem in addition to the central difference approximation is employed into Maxwell’s equations, one can obtain the FDTD equations[30]. For a 2D FDTD case in lossless media, under Cartesian coordinate system, the Maxwell’s equation can be expanded as: TE Waves: 1 ∂Ey ∂Ez ∂Hx =− ( − − (Msourcex + σ ∗ Hx )) ∂t µ ∂z ∂y (2.12) 1 ∂Ez ∂Ey ∂Hz =− ( − − (Msourcey + σ ∗ Hy )) ∂t µ ∂y ∂x (2.13) ∂Ey 1 ∂Hx ∂Hz = ( − − (Jsourcey + σEy )) ∂t ǫ ∂z ∂x (2.14) 16 Figure 2.3: Location of the TE Fields in the Computational Domain[26] TM Waves: ∂Ex 1 ∂Hz ∂Hy = ( − − (Jsourcex + σEx )) ∂t ǫ ∂y ∂z (2.15) 1 ∂Hy ∂Hx ∂Ez = ( − − (Jsourcez + σEz )) ∂t ǫ ∂x ∂y (2.16) 1 ∂Ez ∂Ex ∂Hy =− ( − − (Msourcez + σ ∗ Hz )) ∂t µ ∂x ∂z (2.17) The equations (2.12-2.17) can also write into a typical substitution of central differences for the time and space derivatives, 17 n+1/2 Ex |i+1/2,j,k = σi+1/2,j,k ∆t ∆t ǫi+1/2,j,k 2ǫi +1/2,j,k n−1/2 ( )Ex |i+1/2,j,k + ( ) σ ∆t σi+1/2,j,k ∆t 1 + 2ǫi+1/2,j,k 1 + 2ǫi +1/2,j,k i +1/2,j,k n n H | − Hz |i+1/2,j−1/2,k Hy |ni+1/2,j,k+1/2 ˙ z i+1/2,j+1/2,k 1− ( − ∆y − Hy |ni+1/2,j,k−1/2 ∆z −Jsourcex |ni+1/2,j,k ) n+1/2 Ey |i,j+1/2,k = (2.18) σi,j+1/2,k ∆t ∆t ǫi,j+1/2,k 2ǫi ,j+1/2,k n−1/2 ( )Ey |i,j+1/2,k + ( ) σ ∆t σi,j+1/2,k ∆t 1 + 2ǫi,j+1/2,k 1 + 2ǫi ,j+1/2,k i ,j+1/2,k n n H | − Hx |i,j+1/2,k−1/2 Hz |ni+1/2,j+1/2,k ˙ x i,j+1/2,k+1/2 1− ( − ∆z − Hz |ni−1/2,j+1/2,k ∆x −Jsourcey |ni,j+1/2,k ) n+1/2 Ez |i,j,k+1/2 = (2.19) σi,j,k+1/2 ∆t ∆t ǫi,j,k+1/2 2ǫi ,j,k+1/2 n−1/2 ( )Ez |i,j,k+1/2 + ( ) σ ∆t σi,j,k+1/2 ∆t 1 + 2ǫi,j,k+1/2 1 + 2ǫi ,j,k+1/2 i ,j,k+1/2 n n H | − Hy |i−1/2,j,k+1/2 Hx |ni,j+1/2,k+1/2 ˙ y i+1/2,j,k+1/2 1− ( − ∆x − Hx |ni,j−1/2,k+1/2 ∆y −Jsourcez |ni,j+1/2,k ) Hx |n+1 i,j+1/2,k+1/2 =( 1− 1+ (2.20) ∗ σi,j+1/2,k+1/2 ∆t 2µi ,j+1/2,k+1/2 )Hx |ni,j+1/2,k+1/2 ∗ σi,j+1/2,k+1/2 ∆t 2µi ,j+1/2,k+1/2 n+1/2 Ey |i,j+1/2,k+1 ˙ ( − n+1/2 Ey |i,j+1/2,k ∆z −Msourcex |ni,j+1/2,k+1/2 + 1) − +( ∆t µi,j+1/2,k+1/2 1+ ∗ σi,j+1/2,k+1/2 ∆t ) 2µi ,j+1/2,k+1/2 n+1/2 Ez |i,j+1,k+1/2 n+1/2 − Ez |i,j,k+1/2 ∆y (2.21) 18 Hy |n+1 i+1/2,j,k+1/2 =( 1− 1+ ∗ σi+1/2,j,k+1/2 ∆t 2µi +1/2,j,k+1/2 )Hy |ni+1/2,j,k+1/2 ∗ σi+1/2,j,k+1/2 ∆t +( 1+ 2µi +1/2,j,k+1/2 n+1/2 E | ˙ z i+1,j+1/2,k ( − n+1/2 Ez |i,j+1/2,k ∆x − ∆t µi+1/2,j,k+1/2 ∗ σi+1/2,j,k+1/2 ∆t 2µi +1/2,j,k+1/2 n+1/2 Ex |i+1/2,j,k+1 n+1/2 − Ex |i+1/2,j,k ∆z −Msourcey |ni+1/2,j,k+1/2 + 1) Hz |n+1 i+1/2,j+1/2,k =( 1− 1+ (2.22) ∗ σi+1/2,j+1/2,k ∆t 2µi +1/2,j+1/2,k ∗ σi+1/2,j+1/2,k ∆t )Hz |ni+1/2,j+1/2,k 2µi +1/2,j+1/2,k n+1/2 n+1/2 Ex |i+1/2,j+1,k − Ex |i+1/2,j,k ˙ ( ∆y −Msourcez |ni+1/2,j+1/2,k ) − +( ∆t µi+1/2,j+1/2,k 1+ ∗ σi+1/2,j+1/2,k ∆t ) 2µi +1/2,j+1/2,k n+1/2 n+1/2 Ey |i+1,j+1/2,k − Ey |i,j+1/2,k ∆x + 1) (2.23) In the 2D case, each field is represented by a 2D array, Ez (i, j),Hx (i, j), and Hy (i, j). The indices i and j account for the number of space steps in the x and y direction. The location of the fields in the mesh is shown in Fig 2.3. In order to solve these coupled continuous differential equations on a computer, they must be transformed into difference equations. For example. consider the following equation: 1 ∂Hx ∂Hz ∂Ey = ( − ) ∂t ǫ ∂z ∂x (2.24) which can change into: Eyn (i, j) − Eyn1 (i, j) 1 Hxn−1/2 (i, j + 1/2) − Hxn−1/2 (i, j + 1/2) = ( ∆t ǫ ∆z n−1/2 n−1/2 Hz (i + 1/2, j) − Hz (i − 1/2, j) − ) (2.25) ∆x 19 By manipulation of the resulting difference equation (number), we can generate an update equation for the field quantity Ey : Eyn (i, j) = Eyn−1 (i, j) + − ∆t [H n−1/2 (i, j + 1/2) − Hxn−1/2 (i, j − 1/2)] ǫ∆z x ∆t [H n−1/2 (i + 1/2, j) − Hzn−1/2 (i − 1/2, j)] ǫ∆x z (2.26) So by using the same method, the TE Waves write into: Hxn+1/2 (i, j + 1/2) = Hxn−1/2 (i, j + 1/2) + ∆t [Eyn (i, j + 1) − Eyn (i, j)] (2.27) µ0 ∆z Hzn+1/2 (i + 1/2, j) = Hxn−1/2 (i + 1/2, j) − ∆t [E n (i + 1, j) − Eyn (i, j)] (2.28) µ0 ∆x y The superscript n labels the time steps while the indices i and j label the space steps and ∆x and ∆z along the x and z directions. Normally, the time step is determined by the Courant limit: ∆t ≤ 1 q (c 1/(∆x)2 + 1/(∆z)2 ) (2.29) For the TM Waves, according to Fig 2.2, the electric field components Ex and Ez will move to the cell edges, while the magnetic field Hy will be located at the cell center. And the TM algorithm can be derived in a way similar to equation(2.27-2.28). 2.4 Boundary Conditions The domain size of an FDTD simulation is limited primarily by memory constraints. The update of all field quantities relies on storage of the field 20 quantities in the entire domain. As long as the interactions of interest are sufficiently distant from the edge of the computation domain, the boundaries of the domain can be ignored. In the following sections we will introduce several methods of boundary conditions. 2.4.1 Perfect Match Layer(PML) Boundary Conditions PML is a finite-thickness special medium surrounding the computational space based on fictitious constitutive parameters to create a wave-impedance matching condition, which is independent of the angles and frequencies of the wave incident on this boundary[7]. This is analogous to matching the impedance two transmission lines in order to prevent reflections. Basically, if a wave is propagating in medium A and it impinges upon medium B, the amount of reflection is dictated by the intrinsic impedances of the two media[28] Γ= ηA − ηB ηA + ηB (2.30) which are determined by the dielectric constants ǫ and permeabilities µ of the two media η= r µ ǫ (2.31) If we made µ and ǫ into complex number, we will get a medium lossy so the pulse will die out before it hits the boundary. There are two conditions to form a PML[5]: 1. The impedance going from the background medium to the PML must beconstant. 2. In the direction perpendicular to the boundary (the x direction, for instance), the relative dielectric constant and relative permeability must be inverse of those in the other directions. 2.4.2 Absorbing Boundary Conditions(ABCs) 21 Absorbing boundary conditions are necessary to keep outgoing E and H fields from being reflected back into the problem space, by using some function to estimate what the electric field components should be on the surface of the mesh boundary, electromagnetic fields should appear to be ”absorbed” into the boundary surface. Since there are no sources outside the problem space, and the fields at the edge must be propagating outward. So we can estimate the value at the end by using the value next to it[20]. Suppose we are looking for a boundary condition at k = 0. If a wave is going toward a boundary in free space at the speed c0 . So in one time step of the FDTD algorithm, it travels distance = c0 · ∆t = c0 · ∆x ∆x = 2 · c0 2 (2.32) This equation explains that it takes two time steps for a wave front to cross one cell. So the boundary condition might be Exn (0) = Exn−2 (1) (2.33) 2.4.3 Perfect Electric Conductor(PEC) Boundary Conditions Perfect Electric Conductor is used to model a perfectly conductive metal surface, so the PEC boundary conditions are the electric field components tangential to the surface must be zero. → − − → n ×E =0 (2.34) So to use the Yee Cell in the finite difference time domain scheme, we just need to set the electric field components equal to zero at every time step to satisfy the PEC boundary conditions. 2.4.4 Perfect Magnetic Conductor(PMC) Boundary Conditions 22 Figure 2.4: PEC Boundary Condition[4] PMC Boundary Conditions are the magnetic field components tangential to the surface must be zero. It produces a reflected wave where the electric field is not inverted while the magnetic field is inverted. → − − → n ×H =0 (2.35) Figure 2.5: PMC Boundary Condition[4] 23 3. FDTD Method for Microwave Breast Imaging 3.1 Breast Model The adult female human breast consists of two main tissue types, which are normal breast tissue and glandular tissue. Our breast model duplicates the model developed by Fear et al[17] in Fig 3.1. Figure 3.1: The Homogenous Cylindrical Breast Model[17] The patient lies in a prone position. An array of antennas is placed around and offset from the breast. For data acquisition, the source to the left of the antenna transmits a plane wave and the scattered returns are recorded at the antenna. The breast is modeled as a finite cylinder of breast tissue surrounded by an outer layer of skin. This cylindrical model is not realistically shaped. 24 The breast models have diameter of 8cm, the skin is modeled as a 2-mm-thicker layer. Tumors are modeled as sphere of diameters of 2mm, which located at the different locations in the breast model, as shown in Fig 3.2 - 3.5. Breast Cancer Model Tumor on the Left Antennas skin Fatty tissue 800 Normal tissue Tumor 1000 600 400 200 0 0 200 400 600 800 1000 Figure 3.2: The Breast Model with Tumor on the Left The domain size that is used with FDTD is 1001x1001 square cells of 1 grid p size and a time step of dt = 1/(c ∗ ((1/dx)2 + (1/dy)2)), where c = 3 ∗ 108 , dx = dy = 0.025cm. The thickness of the PML layer is 60 cells, and the number of iterations is 1500. The breast model is a heterogenous cylindrical model, as show in Fig 3.2-3.5, which the thickness of skin is 8dx, the normal tissue has a 320dx diameter and the fatty tissue has a 160dx diameter. The cancer tissue assigned on the left and right of the normal tissue with 24dx diameter. The electromagnetic wave assumed to be a sinusoid wave propagation through the 25 Breast Cancer Model Tumor on the Right Antennas skin Fatty tissue 800 Normal tissue Tumor 1000 600 400 200 0 0 200 400 600 800 1000 Figure 3.3: The Breast Model with Tumor on the Right breast model. An antenna receiver with 410dx diameter reads the electric field data. 3.2 Breast Imaging using FDTD Measurement at Various Frequency Initially, the breast model assumed to have the same dielectric properties as show in Table 3.1. The incident plane wave is assumed to be propagating in the direction normal to the z-axis. The electric field values shown in Fig 3.6-3.9. The electric field was calculate by using FDTD method with different models. The frequencies chose from 400MHz to 1GHz with same dielectric properties. The different lines in the figures represent the different frequencies. From the figures, we try to find out the optimized frequency for the breast cancer detection. 26 Breast Cancer Model Tumor on the Edge between Fatty and Normal Tissue 1000 Antennas skin Fatty tissue 800 Normal tissue Tumor 600 400 200 0 0 200 400 600 800 1000 Figure 3.4: The Breast Model with Tumor Between the Normal and Fatty Tissue Table 3.1: Dielectric properties of female breast tissue at 3.2GHz[17] Tissue Type Relative Permittivity Conductivity (S/m) Fatty Tissue 45 0.65 Normal Tissue 55 1.2 skin 36 1.1 Malignant Tissue 15 0.1 Since the dielectric properties of the tumor and the normal tissue are significantly different, to get the optimal frequency for the simulation, more calculations were employed. We calculate the difference between the model with and without tumor. In the calculation, since the electric field value is too small, 27 Breast Cancer Model Tumor on the Outside of Normal Tissue 1000 Antennas skin Fatty tissue 800 Normal tissue Tumor 600 400 200 0 0 200 400 600 800 1000 Figure 3.5: The Breast Model with Tumor Outside of the Normal Tissue we first calculate the intensity, then add all the points together for both model with and without tumor, at the end, subtract from each other to get the electric field difference between two models. Ez tumor = 360 X Ez tumor (3.1) 360 X Ez notumor (3.2) n=1 Ez notumor = n=1 Dif f Ez = |Ez tumor − Ez notumor| (3.3) Fig 3.10-3.12 show the electric field differences between the model with and without tumor, over frequency range from 400MHz to 1GHz. The electric field 28 5 8 Electric Field 0.4GHz − 1GHz x 10 400MHz 500MHz 600MHz 700MHz 800MHz 900MHz 1GHz 7 Ez value 6 5 4 3 2 1 0 0 50 100 150 200 250 Angle [Degree] 300 350 400 Figure 3.6: Electric Field for the Breast Model with Tumor on the Left Frequency Range 400MHz-1GHz value was evaluated as different frequencies respectively. From the figures above, we can get at 500MHz point for the model with tumor on the left, tumor on the edge between normal and fatty tissue, and tumor on the outside of the fatty tissue has the maximum value of the difference of electric field, and for the model with tumor on the right, we can get the maximum value at 800MHz point. So we can assume 500MHz would be an optimized frequency for breast cancer detection by now. To get more accurate results, the parametric study with different dielectric properties at various frequencies will be introduced below. 3.3 Breast Imaging Using FDTD Method at Various Frequencies with Different Dielectric Properties 29 5 8 Electric Field 0.4GHz − 1GHz x 10 400MHz 500MHz 600MHz 700MHz 800MHz 900MHz 1GHz 7 Ez value 6 5 4 3 2 1 0 0 50 100 150 200 250 Angle [Degree] 300 350 400 Figure 3.7: Electric Field for the Breast Model with Tumor on the Right Frequency Range 400MHz-1GHz The breast model assumed to have the different dielectric properties at different frequencies as showed in table 3.2. Same simulation was ran to find out if 0.7GHz is the optimal frequency. The same simulation method was implement for the data with different dielectric properties at various frequencies. By comparing all the results we got from the calculation above, we can easily find the maximum value of the difference of electric field value located at 500MHz. Therefore, 500MHz could be the optimized frequency for breast cancer detection by using microwave imaging in my research. 30 5 8 Electric Field 0.4GHz − 1GHz x 10 400MHz 500MHz 600MHz 700MHz 800MHz 900MHz 1GHz 7 Ez value 6 5 4 3 2 1 0 0 50 100 150 200 250 Angle [Degree] 300 350 400 Figure 3.8: Electric Field for the Breast Model with Tumor on the Edge between Normal and Fatty Tissue Frequency Range 400MHz-1GHz 5 8 Electric Field 0.4GHz − 1GHz x 10 400MHz 500MHz 600MHz 700MHz 800MHz 900MHz 1GHz 7 Ez value 6 5 4 3 2 1 0 0 50 100 150 200 250 Angle [Degree] 300 350 400 Figure 3.9: Electric Field for the Breast Model with Tumor on the Outside of Fatty Tissue Frequency Range 400MHz-1GHz 31 Electric Field Difference Between the Model With and Without Tumor at Vary Frequencies 7 7 x 10 6 Diff Ez 5 4 3 2 1 0 0.4 0.5 0.6 0.7 Frequency [GHz] 0.8 0.9 1 Figure 3.10: Electric Field Difference Between the Model with and without Tumor (Tumor on the Left) at Various Frequencies Electric Field Difference Between the Model With and Without Tumor at Vary Frequencies −9 8 x 10 7 6 Diff Ez 5 4 3 2 1 0 0.4 0.5 0.6 0.7 Frequency [GHz] 0.8 0.9 1 Figure 3.11: Electric Field Difference Between the Model with and without Tumor (Tumor on the Right) at Various Frequencies 32 Electric Field Difference Between the Model With and Without Tumor at Vary Frequencies 7 3.5 x 10 3 Diff Ez 2.5 2 1.5 1 0.5 0 0.4 0.5 0.6 0.7 Frequency [GHz] 0.8 0.9 1 Figure 3.12: Electric Field Difference Between the Model with and without Tumor (Tumor on the Edge between Normal and Fatty Tissue) at Various Frequencies 5 8 Electric Field 0.4GHz − 1GHz x 10 400MHz 500MHz 600MHz 700MHz 800MHz 900MHz 1GHz 7 Ez value 6 5 4 3 2 1 0 0 50 100 150 200 250 Angle [Degree] 300 350 400 Figure 3.13: Electric Field for the Breast Model - Tumor on the left - Frequency Range 0.4GHz-1GHz with Different Dielectric Properties 33 Table 3.2: The variation of the relative permittivity and the conductivity of normal and cancer tissue between 400MHz-1GHz[11] f(Hz) Normal Tissue Malignant Tissue Conductivity Relative Conductivity Relative (S/m) Permittivity (S/m) Permittivity 400MHz 0.85 59 0.1 19 500MHz 0.91 59 0.11 18 600MHz 0.95 57 0.13 17 700MHz 1.03 57 0.15 17 800MHz 1.1 57 0.17 16 900MHz 1.13 57 0.19 15 1GHz 1.18 57 0.2 15 34 5 8 Electric Field 0.4GHz − 1GHz x 10 400MHz 500MHz 600MHz 700MHz 800MHz 900MHz 1GHz 7 Ez value 6 5 4 3 2 1 0 0 50 100 150 200 250 Angle [Degree] 300 350 400 Figure 3.14: Electric Field for the Breast Model - Tumor on the Right Frequency Range 0.4GHz-1GHz with Different Dielectric Properties 5 15 Electric Field 0.4GHz − 1GHz x 10 400MHz 500MHz 600MHz 700MHz 800MHz 900MHz 1GHz Ez value 10 5 0 0 50 100 150 200 250 Angle [Degree] 300 350 400 Figure 3.15: Electric Field for the Breast Model - Tumor on the Edge between Normal and Fatty Tissue - Frequency Range 0.4GHz-1GHz with Different Dielectric Properties 35 Electric Field Difference Between the Model With and Without Tumor at Vary Frequencies 7 7 x 10 6 Diff Ez 5 4 3 2 1 0 0.4 0.5 0.6 0.7 Frequency [GHz] 0.8 0.9 1 Figure 3.16: Electric Field Difference Between the Model with and without Tumor (Tumor on the Left) at Various Frequencies with Different Dielectric Properties 36 Electric Field Difference Between the Model With and Without Tumor at Vary Frequencies 5 5 x 10 Diff Ez 4 3 2 1 0 0.4 0.5 0.6 0.7 Frequency [GHz] 0.8 0.9 1 Figure 3.17: Electric Field Difference Between the Model with and without Tumor (Tumor on the Right) at Various Frequencies with Different Dielectric Properties 37 Electric Field Difference Between the Model With and Without Tumor at Vary Frequencies 7 7 x 10 6 Diff Ez 5 4 3 2 1 0 0.4 0.5 0.6 0.7 Frequency [GHz] 0.8 0.9 1 Figure 3.18: Electric Field Difference Between the Model with and without Tumor (Tumor on the Edge between Normal and Fatty Tissue) at Various Frequencies with Different Dielectric Properties 38 4. Summary and conclusion This thesis studies the FDTD method for breast cancer detection. By comparing several different kinds of detection modalities, Microwave Imaging becomes a potential alternative detection modality for early stage breast cancer due to the high contrast of the dielectric properties between cancerous and healthy tissues, also the microwave imaging use safer dosage of radiation and non-invasive. Microwave detection is an efficient diagnostic modality for noninvasively visualizing dielectric properties of non-metallic bodies. Many application areas in biomedicine have been explored recently. Simulating the microwave via different frequencies and into different tissue dielectric properties to find out the optimized frequency for microwave imaging breast cancer detection is the most important part of this study and also a promising avenue for breast cancer detection. The FDTD method was presented in chapter 2 as an algorithm. Some important concepts of the FDTD method were mentioned: the Yee Cell, time stepping, analysis stability and accuracy, and boundary conditions. The forms of the finite difference equations used to perform the FDTD method were introduced. I have designed and implemented the FDTD algorithm in MATLAB to build the breast models. First we build up two different kinds of breast models, which are the breast model without tumor and the breast model with tumor at different locations. Then we send a plane wave source which is big enough to cover the breast model to read the data from the antenna. After getting 39 the raw data, some simple calculations were implemented to find the optimized frequency for breast cancer detection. According to the simulation, there is one condition where the optimized frequency does not match with others, but 500MHz-600MHz would be the optimized frequency for my parametric study of microwave breast cancer detection. 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Dec 2011. 43 APPENDIX 1 % 2D FDTD TM s i m u l a t i o n 2 % Boundary Co ndi t i o n : PML 3 % FDTD 2D PML 4 5 clear ; 6 clc ; 7 %%%%%%%%%%% i n i t i a l i z a t i o n%%%%%%%%%%% 8 T=3000; 9 IE = 1 0 0 1 ; 10 JE = 1 0 0 1 ; 11 npml =60; % PML c e l l s 12 cc =3∗10ˆ8; % speed o f l i g h t 13 f =0.8 e9 ; % frequency 14 lambda=cc / f ; % time s t e p s 15 16 mu 0 =4.0∗ p i ∗ 1 . 0 e −7; %p e r m e a b i l i t y o f f r e e s p a c e 17 e p s 0 =8.8 e −12; %p e r m i t t i v i t y o f f r e e s p a c e 18 19 e p s r 1 =36; % permittivity of skin 20 e p s r 2 =3; % permittivity of fatty tissue 21 e p s r 3 =10; % p e r m i t t i v i t y o f normal t i s s u e 44 e p s r 4 =15; % p e r m i t t i v i t y o f tumor 24 sigma1 = 1 . 1 ; % conductivity of skin 25 sigma2 =1; % conductivity of fatty tissue 26 sigma3 =4; % c o n d u c t i v i t y o f normal t i s s u e 27 sigma4 =3; % c o n d u c t i v i t y o f tumor 29 dx =0.25 e −3; % t he c e l l s i z e 30 dy=dx ; 31 dt =1/( cc ∗ s q r t ( ( 1 / dx )ˆ2+(1/ dy ) ˆ 2 ) ) ; 22 23 28 % time s t e p p i n g 32 33 t 0 =40; 34 spread = 15; 35 36 i c=round ( IE / 2 ) ; % source x po sit io n 37 j c=round ( JE / 2 ) ; % source y po sit io n 38 i c t i s s u e = 450; % fatty tissue x position 39 j c t i s s u e = 450; % fatty tissue y position 40 i c t umo r l ef t = 600; % tumor x p o s i t i o n 41 j c t umo r l ef t = 390; % tumor y p o s i t i o n 42 % i c t umo r l ef t = 390; % tumor x p o s i t i o n 43 % j c t umo r l ef t = 390; % tumor y p o s i t i o n 44 % ic tumor right = 510; 45 % jc tumor right = 390; 45 46 % ic tumor edge = 600; 47 % jc tumor edge = 390; 48 % ic tumor outside = 750; 49 % jc tumor outside = 390; 50 51 dz = z e r o s ( IE , JE ) ; % D ensi t y i n z d i r e c t i o n 52 ez = z e r o s ( IE , JE ) ; % E l e c t r i c f i e l d in z direction 53 hx = z e r o s ( IE , JE ) ; % Magnetic f i e l d i n x d i r e c t i o n 54 hy = z e r o s ( IE , JE ) ; % Magnetic f i e l d i n y d i r e c t i o n 55 i z = z e r o s ( IE , JE ) ; 56 i h x = z e r o s ( IE , JE ) ; 57 i h y = z e r o s ( IE , JE ) ; 58 ga1 = o nes ( IE , JE ) ; 59 60 e z n e w t=z e r o s ( 1 , 3 6 0 ) ; 61 R1=i n p u t ( ' s i z e o f t he s k i n ( r a d i u s ) : ' ) ; 62 R2=i n p u t ( ' s i z e o f t he f a t t y t i s s u e ( r a d i u s ) : ' ) ; 63 R3=i n p u t ( ' s i z e o f t he normal t i s s u e ( r a d i u s ) : ' ) ; 64 R4=i n p u t ( ' s i z e o f t he tumor ( r a d i u s ) : ' ) ; 65 66 67 68 69 f o r i =1: IE f o r j =1:JE %% s i z e o f s k i n x d i s t =( i c −i ) ; 46 70 y d i s t =( j c −j ) ; 71 d i s t =( x d i s t ˆ2+ y d i s t ˆ 2 ) ˆ 0 . 5 ; 72 i f d i s t <=R1 ga1 ( i , j ) = 1 . / ( e p s r 1 +(sigma1 ∗ dt / e p s 0 ) ) ; 73 74 75 end %% s i z e o f f a t t y t i s s u e 76 x d i s t =( i c −i ) ; 77 y d i s t =( j c −j ) ; 78 d i s t =( x d i s t ˆ2+ y d i s t ˆ 2 ) ˆ 0 . 5 ; 79 i f d i s t <=R2 ga1 ( i , j ) = 1 . / ( e p s r 2 +(sigma2 ∗ dt / e p s 0 ) ) ; 80 81 end 82 83 %% s i z e o f normal t i s s u e 84 % tumor on t he l e f t ( f a t t y t i s s u e ) 85 x d i s t 1 =( i c t i s s u e −i ) ; 86 y d i s t 1 =( j c t i s s u e −j ) ; 87 88 89 d i s t 1 =( x d i s t 1 ˆ2+ y d i s t ˆ 2 ) ˆ 0 . 5 ; 90 i f d i s t 1 <=R3 ga1 ( i , j ) = 1 . / ( e p s r 3 +(sigma3 ∗ dt / e p s 0 ) ) ; 91 92 end 93 47 94 % s i z e o f tumor t i s s u e 95 %tumor on t he l e f t ( f a t t y t i s s u e ) 96 x d i s t 2 =( i c t u m o r l e f t −i ) ; 97 y d i s t 2 =( j c t u m o r l e f t −j ) ; 98 d i s t 2 =( x d i s t 2 ˆ2+ y d i s t 2 ˆ 2 ) ˆ 0 . 5 ; 99 i f d i s t 2 <=R4 100 ga1 ( i , j ) = 1 . / ( e p s r 4 +(sigma4 ∗ dt / e p s 0 ) ) ; 101 end 102 end 103 104 end 105 106 %%%%%%%%%%%%%%%%% PML %%%%%%%%%%%%%%%%% 107 % i n i t i a l i z e t he PML parameter 108 g i 2 = o nes ( IE ) ; 109 g i 3 = o nes ( IE ) ; 110 f i 1 = o nes ( IE ) ; 111 f i 2 = o nes ( IE ) ; 112 f i 3 = o nes ( IE ) ; 113 114 g j 2 = o nes ( IE ) ; 115 g j 3 = o nes ( IE ) ; 116 f j 1 = o nes ( IE ) ; 117 f j 2 = o nes ( IE ) ; 48 118 f j 3 = o nes ( IE ) ; 119 120 121 f o r i =1:npml 122 xnum=npml−i ; 123 xn = 0 . 3 3 ∗ ( ( xnum/npml ) ˆ 3 ) ; 124 g i 2 ( i )=1.0/(1+xn ) ; 125 g i 2 ( IE−1− i )=1/(1+ xn ) ; 126 g i 3 ( i )=(1−xn )/(1+ xn ) ; 127 g i 3 ( IE−i −1)=(1−xn )/(1+ xn ) ; 128 xn = 0 . 2 5 ∗ ( ( ( xnum−0.5)/ npml ) ˆ 3 ) ; 129 f i 1 ( i )=xn ; 130 f i 1 ( IE−2− i )=xn ; 131 f i 2 ( i )=1.0/(1+xn ) ; 132 f i 2 ( IE−2− i )=1/(1+ xn ) ; 133 f i 3 ( i )=(1−xn )/(1+ xn ) ; 134 f i 3 ( IE−2− i )=(1−xn )/(1+ xn ) ; 135 end 136 137 f o r j =1:npml 138 xnum=npml−j ; 139 xn = 0 . 3 3 ∗ ( ( xnum/npml ) ˆ 3 ) ; 140 g j 2 ( j )=1.0/(1+xn ) ; 141 g j 2 ( JE−j −1)=1/(1+xn ) ; 49 142 g j 3 ( j )=(1−xn )/(1+ xn ) ; 143 g j 3 ( JE−1−j )=(1−xn )/(1+ xn ) ; 144 xn = 0 . 2 5 ∗ ( ( ( xnum−0.5)/ npml ) ˆ 3 ) ; 145 f j 1 ( j )=xn ; 146 f j 1 ( JE−2−j )=xn ; 147 f j 2 ( j )=1.0/(1+xn ) ; 148 f j 2 ( JE−2−j )=1/(1+ xn ) ; 149 f j 3 ( j )=(1−xn )/(1+ xn ) ; 150 f j 3 ( JE−2−j )=(1−xn )/(1+ xn ) ; 151 end 152 153 %%%%%%%%%%%%%% FDTD Main Loop%%%%%%%%%%%%%%%%% 154 R=i n p u t ( ' s i z e o f t he obseved c i r c l e ( r a d i u e ) : ' ) ; 155 f o r theta =1:1:360 156 x ( t h e t a )=R∗ co sd ( t h e t a )+ i c ; 157 y ( t h e t a )=R∗ s i n d ( t h e t a )+ j c ; 158 end 159 [ X,Y]= meshgrid ( 1 : 1 : 1 0 0 1 ) ; 160 161 162 163 f o r t =1:T f o r i =2:IE−1 f o r j =2:JE 164 dz ( i , j )= g i 3 ( i ) ∗ g j 3 ( j ) ∗ dz ( i , j ) + . . . 165 g i 2 ( i ) ∗ g j 2 ( j ) ∗ 0 . 5 ∗ ( hy ( i , j )−hy ( i −1, j ) . . . 50 −hx ( i , j )+hx ( i , j − 1 ) ) ; 166 end 167 168 end 169 170 f o r i =1: IE f o r j =1:JE 171 ez ( i , j )=ga1 ( i , j ) ∗ dz ( i , j ) ; 172 end 173 174 end 175 176 p u l s e= s i n ( 2 ∗ p i ∗ f ∗ dt ∗ t ) ; 177 ez ( 1 0 0 : 9 0 0 , 8 0 ) = p u l s e ; 178 179 f o r j = 1 : JE 180 ez ( 1 , j ) = 0 ; 181 ez ( IE , j ) = 0 ; 182 end 183 184 f o r i = 1 : IE 185 ez ( i , 1 ) = 0 ; 186 ez ( i , JE ) = 0 ; 187 end 188 189 f o r i =1: IE 51 f o r j =1:JE−1 190 191 c u r l e=ez ( i , j )−ez ( i , j +1); 192 hx ( i , j )= f j 3 ( j ) ∗ hx ( i , j )+ f j 2 ( j ) ∗ 0 . 5 ∗ c u r l e ; end 193 end 194 195 f o r i =1:IE−1 196 f o r j =1:JE 197 198 c u r l e=ez ( i +1 , j )−ez ( i , j ) ; 199 hy ( i , j )= f i 3 ( i ) ∗ hy ( i , j )+ f i 2 ( i ) ∗ 0 . 5 ∗ c u r l e ; end 200 end 201 202 203 ez new = i n t e r p 2 (X, Y, ez , x , y ) ; 204 e z n e w t=e z n e w t+ez new ; 205 206 i ma g esc ( ez ) ; ho l d on 207 t i t l e ( [ ' t= ' num2str ( t ) ' d e l t a ' ] ) 208 ho l d o f f ; 209 colorbar ; 210 pause ( 0 . 1 ) 211 212 end ; 213 52 214 figure ; 215 f o r theta =1:1:360 216 x1 ( t h e t a )=R1∗ co sd ( t h e t a )+ i c ; 217 y1 ( t h e t a )=R1∗ s i n d ( t h e t a )+ j c ; 218 219 x2 ( t h e t a )=R2∗ co sd ( t h e t a )+ i c ; 220 y2 ( t h e t a )=R2∗ s i n d ( t h e t a )+ j c ; 221 222 % tumor on t he l e f t ( f a t t y ) 223 x3 ( t h e t a )=R3∗ co sd ( t h e t a )+ i c t i s s u e ; 224 y3 ( t h e t a )=R3∗ s i n d ( t h e t a )+ j c t i s s u e ; 225 %tumor on t he l e f t 226 227 x4 ( t h e t a )=R4∗ co sd ( t h e t a )+ i c t u m o r l e f t ; 228 y4 ( t h e t a )=R4∗ s i n d ( t h e t a )+ j c t u m o r l e f t ; 229 230 end 231 232 233 234 p l o t ( x , y , x1 , y1 , x2 , y2 , x3 , y3 , x4 , y4 ) % p l o t ( x , y , x1 , y1 , x2 , y2 , x3 , y3 ) ho l d on 235 f i l l ( x1 , y1 , ' g ' ) 236 f i l l ( x2 , y2 , ' r ' ) 237 f i l l ( x3 , y3 , 'm' ) 53 238 f i l l ( x4 , y4 , ' c ' ) 239 a xi s equal 240 a xi s square 241 l e g e n d ( ' Antennas ' , ' Fatty t i s s u e ' , ' Normal t i s s u e ' , ' Tumor ' ) 242 t i t l e ( ' Breat Cancer Model Tumor on t he Right ' ) 243 244 figure ; 245 theta =1:1:360; 246 p l o t ( t het a , e z n e w t , ' b− ' , ' l i n e w i d t h ' , 3 ) 247 title ( ' Electric field ' ) 248 xlabel ( ' theta ' ) 249 y l a b e l ( ' Ez v a l u e ' ) 54

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