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

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FINITE 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
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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.
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P.O. Box 1346
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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.
The core of the FDTD method is a set of equations that are repeatedly
performed in a very predictable way, but it took a long time to run the program.
For future work it would be beneficial to examine how to reduce memory taking
space to allow larger FDTD models, like the 3D breast model to be studied.
40
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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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