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This is to certify that the

dissertation entitled

LOCAL HEATING OF BIOLOGICAL BODIES
WITH HF ELECTRIC AND MAGNETIC FIELDS

presented by

Manochehr Kamyab Hessary

has been accepted towards fulfillment
of the requirements for

 

 

Ph. D. degree in E] ec. Engl".

Major professor

Date 6/7/22—

 

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

IVIESI_J RETURNING MATERIALS:
Place in book drop to
LJBRARJES remove this checkout from
w your record. FINES will

 

 

be charged if book is
returned after the date
stamped below.

 

 

 

LOCAL HEATING OF BIOLOGICAL BODIES
WITH HF ELECTRIC AND MAGNETIC
FIELDS

By

Manochehr Kamyab Hessary

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Electrical Engineering and Systems Science

1982

ABSTRACT

LOCAL HEATING OF BIOLOGICAL BODIES
WITH HF ELECTRIC AND MAGNETIC FIELDS

BY

Manochehr Kamyab Hessary

In this research the schemes of utilizing HF electric and
magnetic fields to locally heat a biological body are investigated,
with the application to hyperthermia cancer therapy or other medical
purposes. The HF electric field maintained by a capacitor-plate
applicator and the HF magnetic field produced by a current disk are
used to heat biological bodies locally.

Rigorous theoretical analysis for such applicators are presented
in this thesis. First the problem of a capacitor consisting of a
pair of flat-plate electrodes of arbitrary dimensions in free space is
studied. The distributions of the electric charges on the plates are
obtained numerically for variety of cases and the electric fields at
various points in free space are calculated. Following this study,
the heating pattern induced by a capacitor-plate applicator inside a
body is analyzed theoretically. Numerical results obtained on the basis
of the solutions of two coupled integral equations are presented for
several cases. After this a current disk applicator (pancake applicator)
is studied. The electric field and the heating pattern induced in a

body by a current disk placed on the body surface are obtained numerically.

Theoretical schemes are developed tosynthesize the voltage distribution
on a capacitor-plate applicator and the current distribution on a current
disk to obtain a desired heating pattern inside a body. The electric
fields inside a simulated body induced by different applicators are

measured for several cases and are compared with the theoretical values.

ACKNOWLEDGMENTS

I would like to express my gratitude to Dr. K.M. Chen, my major
Professor, for his assistance, encouragement and support throughout the
course of this research.

Special thanks are extended to Dr. D.P. Nyquist, a member of my
guidance committee, for his many helpful suggestions, especially during
the experimental part of this work.

I also appreciate the help I received from the other members of
my guidance committee, Dr. B. Ho, Dr. D. Reinhard and Dr. B. Drachman.

Finally, I am grateful to my brother, Iradj, whose help and support
played a big part in completion of my Ph.D. degree.

ii

TABLE OF CONTENTS

List of Tables .............................................. v
List of Figures ............................................. _ vi
Chapter Page
I INTRODUCTION .................................... 1

II ANALYSIS OF FLAT-PLATE CAPACITORS OF ARBITRARY
DIMENSIONS AND ARRANGEMENTS IN FREE SPACE ....... 5
2.1 Description of Problem ..................... 5
2.2 Moment Solution of Integral

Equation ................................... 7
2.3 Calculation of Matrix Elements ............. 10
2.3.1 Grounded Case ............................ 10
2.3.2 Floating potential Case .................. l4
2.4 Numerical Results .......................... 18
2.5 Computation of the Electric Field .......... 23
2.6 Comparison of Numerical Results with
Experimental Results ....................... 36
III LOCAL HEATING WITH HF ELECTRIC FIELD .......... 41
3.1 Problem Descriptions ....................... 41
3.2 Integral Equation fbr the Total Electric
Field in the Body .......................... 43
3.3 Integral Equation for the Induced
Charge on the Electrodes ................... 47
3.3.1 Moment Solution of Coupled Integral
Equations ................................ 49
3.4 Calculation of the Matrices Elements ....... 54
3.4.1 G Matrix ................................. 54
3.4.2 A Matrix ................................. 55
3.4.3 GS Matrix ................................ 56
3.4.4 C Matrix ................................. 57
3.5 Numerical Results for SAR's and Electric
Fields .................................... 59
3.6 Synthesis of the Potential Distribution
for Selective Heating ...................... 69
3.7 Numerical Results for Synthesized
Voltage Distributions ...................... 71

iii

Chapter

IV

VI Part I

Part II

VII

BIBLIOGRAPHY .................................................

TABLE OF CONTENTS (continued)

3.8 Comparison of Numerical Results and
Experimental Results .......................

LOCAL HEATING WITH HF MAGNETIC FIELD ............

4.1 Introduction ...............................
4.2 Theoretical Analysis .......................

4.2.1 Impressed Electric Field .................
4.2.2 Scattered Electric Field .................
4.3 Numerical Results ..........................
4.4 Synthesis of the Current Distributions
for Selective Heating ......................
4.5 Comparison of Theoretical Results with
Experimental Results .......................

EXPERIMENTAL SETUP ..............................

5.l Construction of an Implantable Probe .......
5.2 Construction of a Balun ....................
5.3 Experimental Setup for the Measurement
of the Electric Field in a Conducting
Medium Maintained by a Capacitor-Plate

Applicator and Probe-ElectrodeInteraction..

5.4 Experimental Setup for the Measurement
of the Electric Field in a Conducting

Medium Maintained by a Current Disk ........

A USER'S GUIDE TO COMPUTER PROGRAM USED TO
CALCULATE THE ELECTRIC FIELD INSIDE A
BIOLOGICAL BODY INDUCED BY A PAIR OF CAPACITOR-

PLATE APPLICATOR ................................

6.1 Description of data files ..................
6.2 Numerical Example ..........................

A USER'S GUIDE TO COMPUTER PROGRAM USED TO
DETERMINE THE ELECTRIC FIELD INSIDE A
BIOLOGICAL BODY INDUCED BY A CURRENT DISK

APPLICATOR ......................................

6.3 Formulation of the Problem .................
6.4 Description of the Computer Program ........
6.5 Structure of the Input Data Files ..........
6.6 An Example to use the Program ..............

SUMMARY .........................................

iv

Page

81
86

86
88

9O
96
103
107
111

111
113

115

122

126

128
130

141

141
143
143
146

155
158

LIST OF TABLES

Table Page
6.1 The Symbolic Names for the Input Variables

and Format Specifications Used in Program

"FIELD" ......................................... 129
6.2 The Symbolic Names for the Input Variables

and Format Specifications Used in Program

"EDDY" .......... , ................... . ........... 145

Figure

2.1

LIST OF FIGURES

A Pair of Flat, Parallel Electrodes Partitioned

into Subareas ...................................

A Cubical Volume of Free Space is Partitioned
into Four Symmetrical Quadrants (a), and (b)
the Geometry of Two Parallel Electrodes with
$2 Partitioned and Charge Densities on the

Subareas ........................ . ...............

A Square Cell is Approximated by a Circular One
for the Calculation of the Diagonal Elements

of Matrix Gs ...................................

Relation Between the Charge on and the Current
Flowing into the Electrodes. (a) Grounded

Potential Case (b) Floating Potential Case ......

The Capacitance vs Spacing of Two Parallel Plates

of Equal Size (a) Using the Method of Subareas
(b) Using

S2 for Two Cases of D = 2. (Solid Curves),
and D = 4. (Dashed Curves). (a) along Z (6)

along Zd ................... . ..................

The Distributions of theCharge Densities on S1
and 52 (Same Dimension) (a) along Z (b)

along Zd ................................... ...

The Distribution of Charge Densities on S1 and
52 (Different Dimensions) for Grounded (Solid
Lines) and Floating Potential Cases: (A) along
2, (8) along 2

Three Components of the Electric Field at the
Center of Subvolumes in Different Layers in
k of Free Space Between Two Electrodes of

Equal Dimension for Floating Potential Case .....

vi

C = 60 D" ............................
The Distributions of Charge Densities on S1 and

d ...............................

Page

11

15

17

19

21

22

24

27

LIST OF FIGURES (continued)
Figure Page

2.10 The X-Component of the Electric Field at
the Center of Subvolumes in Different Layers
in k of Free Space Between Two Electrodes
of Equal Size with S1 Grounded ................ 28

2.11 Three Components of the Electric Field at
the Center of Subvolumes in Different Layers
in k of Free Space Between Two Electrodes
of Different Sizes for Floating Potential
Case ............................................ 29

2.12 The X-Component of the Electric Field at the
Center of Subvolumes in Different Layers
in k of Free Space Between Two Electrodes of
Different Sizes, with S1 Grounded ............. 31

2.13 Distributions of the X-Component of the
Electric Field along the X Axis Between Two
Electrodes of Equal Dimension for Various
Ratios of D/a, where D is the Spacing
Between the Electrodes and a is the Dimension
of the Electrode S1 is Grounded ................ 32

2.14 Distributions of theX-Component of the
Electric Field along the X-axis Between
Electrodes of Different Dimensions for
Various Ratios of 51/52. The Potentials

of the Electrodes are left Floating ............. 34

2.15 Distributions of the X-Component of the
Electric Field along the X-axis Between Two
Electrodes of Different Dimensions for
Various Rations of 31/52 where S1 and

52 are the Surface Areas of the Two
Electrodes. (S1 is Grounded) .................. 35

2.16 Theoretical and Experimental Results for the
Electric Field at the Inner and Outer Surfaces
of one Electrode for Three Capacitors with
Various Electrode Dimensions and Seperations.
Solid Lines are the Theoretical Results and
Discrete Points Represent the Experimental
Results ......................................... 37

vii

LIST OF FIGURES (continued)
Figure Page

2.17 Theoretical and Experimental Results for
the Distribution of them-component of the
Electric Field along X axis Between Electrodes
of Equal Dimension. Solid lines Show the
Theoretical Results and the Discrete Points
Represent the Experimental Results .............. 38

2.18 Theoretical and Experimental Results for
the Distribution of the X-Component of
Electric Field along the X Axis for

Various Ratios of 51/5 . A-Floating

Potential 8 -S1 is Grgunded ................... 40
3.1 Different Arrangements of Capacitor Plates

Placed on the Biological Body for the

Purpose of Local Heating ........................ 42
3.2 A Pair of Electrodes Energized by a HF-Voltage

Placed across the Biological Body for Local

Heating ......................................... 44
3.3 The Geometry of a Body Placed Between Two

Electrodes (a). (b) the Side View of the

Body and the Electrodes ......................... 50
3.4 TheDistributions of the Electric Charge along

the Y Axis on the Electrod for the Free Space

Case and the Case with a Body Between the

Electrodes. In Figure (b) the Charge

Distributions are Normalized by their Maximum

Values to Show the Relative Variation ........... 60

3.5 Distributions of SAR and Induced Electric Field
in one Quarter of a Body Maintained by a
Capacitor-Plate Applicator with Electrodes of
the Same Size ................................... 62

3.6 Distributions of SAR and Induced Electric Field
in one Quarter of a Body Maintained by a
Capacitor-Plate Applicator with Electrodes of
the Same Size and One Electrode Grounded ........ 63

3.7 Distributions of SAR and Induced Electric Field
in one Quarter of a Body Maintained by a
Capacitor-Plate Applicator with Electrodes of
Different Sizes ................................. 64

viii

LIST OF FIGURES (continued)
Figure Page

3.8 Distributions of SAR and Induced Electric
Field in one Quarter of a Body Maintained by
a Capacitor-Plate Applicator with Electrodes
of Different Sizes and one Electrode Grounded... 66

3.9 Distributions of SAR and Induced Electric Field
in one Quarter of a Heterogeneous Body, a
Lower Conductivity Region at the Center. Main-
tained by a Capacitor-Plate Applicator with
Electrodes of the Same Size ..................... 67

3.10 Distributions of SAR and Induced Electric
Field in one Quarter of a Heterogeneous
Body, a Higher Conductivity Region at the
Center, Maintained by a Capacitor-Plate
Applicator with Electrodes of the Same Size ..... 68

3.11 Geometry of a Body and a Capacitor-Plate
Applicator with Subsectioned Electrodes for
Localized Heating at the Center of the Body.
The Distribution of Required Voltages is

Shown in the next Figure ........................ 72
3.12 Distributions of the Required Voltages on the

Electrodes to Obtain a Localized Heating at

the Center of the Body .......................... 73
3.13 4 Geometry of a Body and a Capacitor-Plate

Applicator with Subsectioned Electrodes for
Localized Heating at the Center of the Body
Surface. The Distribution of the Required

Voltages on the Electrodes is Shown in Next

Figure .......................................... 75
3.14 Distributions of the Required Voltages on the

Electrodes to Obtain a Localized Heating at

the Center of the Body Surface .................. 76
3.15 Geometry of a Body and Capacitor-Plate

Applicator with Subsectioned Electrodes for

Localized Heating at the Central Column of

The Body. The Distribution of Required

Voltages on the Electrodes is Shown in Next

Figures ......................................... 78

ix

Figure
3.16

LIST OF FIGURES (continued)

Distributions of the Amplitude of the
Required Voltage on the Electrode to Obtain
A Localized Heating at the Central Column

of the Body .....................................

Distributions of the Phase Angle of the
Required Voltage on the Electrodes to Obtain
a Localized Heating at the Central Column of

the Body ........................................

Distributions of the Theoretical and
Experimental Values of the X-Component of
Electric Field along Y Axis Maintained in
the Body Between Two Electrodes of Equal

Dimension .......................................

Distribution of the Theoretical and
Experimental Values of the X-Component of
Electric Field along the X Axis Maintained
in the Body Between Two Electrodes of equal

Dimension .......................................

Distributions of the Theoretical and
Experimental Values of the X-Component of
Electric Field along the X Axis Maintained
in the Body Between Two Electrodes of

Different Dimensions ............................

A Biological Body Consisted of Skin, Fat and
Muscle Layers Placed Between a Pair of

Electrodes for the Purpose of Local Heating .....

Geometry of a Circular Disk Carrying a Cir-
culatory Current Placed on a Body for Local

Heating .........................................

Distributions of Amplitude and Phase of the
Electric Field in Different Layers of a

Body Induced by a Disk of Uniform Current .......
Distributions of the Amplitude of the Electric

Field in Different Layers of a Body Induced
by a Disk of Uniform Current at 30 MHz and

100 MHz .........................................

Distribution of the Amplitude of Electric
Field in Different Layers of a Body Induced

by a Disk of Uniform Current ....................

Page

79

80

82

83

85

87

89

97

98

100

Figure

4.6

4.7

4.8

4.9

4.10

4.12

5.1

5.2

5.3

LIST OF FIGURES (continued)

Distributions of the Phase Angle of the
Electric Field in Different Layers of a
Body Induced by a Disk of Uniform Current .......

Distributions of the Electric Field in the
First Two Layers of a Body Induced by Three
Kinds of Loop Currents: a Single Loop Current,
a Uniform Surface Current and a Triangular
Surface Current .................................

Distribution of Phase and Amplitude of the
Required Current Density on a Disk to Maintain
a Localized Heating at the Center of a Body .....

Distributions of the Phase and Amplitude of

the Required Current on a Disk to Maintain

a Uniform Heating in the First Layer of the
Body ............................................

Distributions of the Theoretical (Solid Lines)
and Experimental (Discrete Points) Values

for the Electric Field in Different Layers of

a Body Induced by a Single Current Loop .........

Distributions of the Theoretical (Solid Lines)
and Experimental Values (Discrete Points) for
the Electric Field in Different Layers of a
Body Induced by a Disk of Uniform Current .......

Distributions of the Theoretical (Solid Lines)
and Experimental Values (Discrete Points) for
the Electric Field in Different Layers of a Body
Induced by a Triangular Type of Current
Distribution ....................................

A Non-Interferring, Electric Field Probe for
Measuring the Induced Electric Field in a
Biological Body .................................

(a) Direct Connection of a Coaxial Line to a
Two wire Line (b) Decomposition of the

Current on the Two-wire Line into Symmetric

and Antisymmetric Modes .........................

A Balun for Converting a Coaxial Line to a
Balanced Two-wire Line ..........................

xi

Page

101

102

105

106

108

109

110

112

114

116

Figure

5.4

5.5

5.6

5.7

5.8

6.1

6.2

LIST OF FIGURES (continued)

Experimental Setup for the Measurement of the
Electric Field in a Conducting Medium
Maintained by a Pair of Capacitor-Plate

Electrodes ......................................

Equivalent Circuit for a Probe. (a) an
Isolated Probe (b) a Probe Located Close to a

Ground Plane ....................................

Distribution of the Currents on a Short
Electric Dipole and its Image Caused by the

Ground Plane ....................................

Experimental Setup for the Measurment of the
Electric Field in a Conducting Medium

Maintained by a Current Disk ....................

Experimental Models for Three Kinds of
Current Distributions (a) Single Current
Loop (b) Uniform Current Distribution (c)

Triangular Current Distribution .................

Geometry of a Boyd Located Between Tow Energized

Electrodes for the Purpose of Local Heating.
The Numbering Order of k of the Body and

Electrodes are Shown ............................

A Circular Current Disk is Placed on a Body

for the Purpose of Local Heating ................

xii

Page

117

118

120

123

124

127

142

CHAPTER I
INTRODUCTION

Since its early days of discovery, electromagnetic radiation
and propagation has been utilized to benefit human societies and the
everincreasing impact of the EM waves on different aspects of human
life has been enormous.

Constant efforts by scientists and engineers to utilize the
potential usefulness of EM‘wavesto enhance the quality of life and their
incredible achievements has made it possible for many man's dreams to
become reality. Intercontinental satellite comunication, radar detection
systems, microwave technology, and using EM energy for medical purposes
in recent years are only few examples to mention.

The idea of using EM energy to induce hyperthermia in bio-
logical bodies for the purpose of cancer therapy has become the center
of attention of many medical researchers over the past decade. It has
been found that when the temperture of a tumor is raised a few degrees
above that of the surrounding tissues, accompanying chemo or radio-
therapy becomes more effective in treating tumors [1.49. In the combined
therapy of malignancies, the objective is to find a noninvasive method
by which to heat the tumor without overheating other parts of the body.
EM radiation has been found by many researchers to be a convenient

agent to heat a tumor locally.

Significant progress was made in the hyperthermia cancer
therapy when Leveen, et al. C 4 3 used 13.56 MHz EM radiation to
eradicate tumors in cancer patients. Holt I 5 J has used 433 MHz EM
radiation in combination with X-rays to cure many cancer patients.
Antich, et al. I 5 I used 27.12 MHz EM radiation to heat cutaneous
human tumors. Jaines, et a1. E 7 1 combined microwave with X-rays to
treat tumors in the bodies of terminal cancer patients. Many other
researchers Ui-13] have used EM radiation of 2450 MHz, 918 MHz or HF
range to heat tumors in animal bodies and reported significant tumor
regressions.

However, in order to improve the efficiency of applicators,
in-depth theoretical study of the EM fields inside biological bodies
induced by the applicators, as well as the heating pattern or power
deposition in the tumor and other parts of the body is needed.

In the present research the methods of local heating of a
biological body with HF electric field (capacitor-plate applicator) and
HF magnetic field (current disk applicator) are studied theoretically
and experimentally.

In Chapter II a theoretical analysis of flat-plate capacitors
of arbitrary dimenisons and arrangements in free space is presented.
The distribution of the electric charge on the plates is found numerically,
and the three components of the electric field at various points in
free space are calculated for variety of cases.

In Chapter III a capacitor-plate applicator is analyzed. Two
flat plate electrodes located across a biological body with properly applied

voltage distributions are used to heat the body locally. Based on the

tensor integral equation method (TIEM) developed by Chen, et al.[les 1,
two coupled integral equations are established from which the induced
electric field inside the body and the density of the electric charge

on the electrodes are obtained numerically. Then, the specific ab-
sorption rate (SAR) of the EM energy in the body is calculated for both
homogeneous and heterogeneous bodies. In this chapter, a theoretical
scheme is also developed for synthesizing the voltage distribution on
the plates in order to obtain a desired heating pattern in the body,

and some numerical examples for such a synthesis are given.

In Chapter IV the shortcoming of the capacitor-plate applicators
relating to the overheating of the fat layer in biological bodies is
explained. The main subject of this chapter is the theoretical study
of a current disk applicator (or a pancake applicator). The electric
field inside a biological body induced by a current disk applicator
placed on the surface of the body is calculated numerically for different
current distributions, and various results are presented. A theoretical
scheme for synthesizing the current distribution on the disk to obtain
a selective heating pattern is developed and some numerical examples
are given.

In order to verify the theoretical results, a series of experi-
ments was conducted where the electric fields inside a biological body
and in free space induced by applicators were measured. The experimental
results are compared with theoretical results at the end of each chapter.

The decriptions of the experimental setup and related problems are

given in Chapter V.

Chapter'VI contains a brief description of the computer programs
used to obtain the numberical results. The lists of programs and numer-

ical examples for each program are also included in this chapter.

CHAPTER II

ANALYSIS OF FLAT-PLATE CAPACITORS OF ARBITRARY
DIMENSIONS AND ARRANGEMENTS IN FREE-SPACE

In this chapter,a numerical method is developed to analyze the
problem of a capacitor which consists of two flat parallel electrodes
of arbitrary dimensions and seperated by a distance, and with two dif-
ferent potentials applied to the electrodes. The distribution of induced
electric charges on the surfaces of the electrodes are determined first.
After that,the electric field maintained between the electrodes, and

the capacitance between the electrodes are determined.

2.1 Discription of Problem

 

Consider a capacitor with two conducting parallel electrodes,

S1 and 52, as shown in Figure 2.1. Two a.c. potentials of VS and
1

Vs are applied to the electrodes, and a harmonic time variation of
2

ert is assumed for those potentials.
At any point F in free space the vector and scalar potentials,

+ _) .
A(r) and ¢(F), maintained by the current and charge on S1 and 52

can be shown to satisfy the following differential equations.

 

2 + 2 + _ "$6: )
v ¢(r) + 80¢(r) - - 60 (2.1.1)
vzlim + 333m = -110 30? ) (2.1.2)

Equations (2.1.1) and (2.1.2) can be solved easily by the Green's

function technique resulting in the following integral equations in

4N2 = Number of subareas
on $2

4N1 = Number of subareas
on S1

N‘NTTNz

rm =(Xm,Ym,Zm) field p01nt

s rn=(Xn,Yn,Zn) source point

 

Figure 2.1. A pair of f1at,parallel electrodes partitioned into
subareas.

 

we = EL] nS(F')e(F,F'>ds' (2.1.3)
o S +5
1 2
A(F) = “o ( JS(F')G(F,F')ds' (2.1.4)
S]+S2
. + +'
-jBO|r-r I
where G(F,F') = e + + is the free space Greens' function.
4n|r-r'|

Equation (2.1.3) may be solved for n5 subject to the condition of
vanishing of the tangential component of electric field, i.e.
+ -+ .->->
Et = -vto(r) - ijt(r) = 0
+ —> .+ +
at low frequencies lijtl << | vt¢(r)|, thus, Et “ -vto(r) = O. For

F's S1 and 52, which implies that o(F 6 51,52) = constant.

2.2. Moment Solution of Integral Equation

 

The induced electric charge nS(F) is difficult if not im-
possible to determine in closed form. However, there are numerical tech-
niques available by means of which the solution can be found. One such
method is the well known "Method of Moment" which is briefly discussed
here. More detailed descriptions can be found in other sources [14].

The moment method is one of the convenient ways of solving
integral equations by converting them into a set of simultaneous linear
equations in terms of the discretized values of the unknown. The latter
can be solved by digital computers with desired accuracy.

To solve equation (2.1.3), S1 and S2 are divided into a number
of subareas which hereafter are referred to as cells. The charge

th

density on the n cell is represented by ”n’ and the potential

at an arbitrary point P(F) in free space due to the charge on the

nth cell is given as

,F'Ms' (2.2.1)

3

- ;L_ *1
en(r) - 60 (A5" n (r )G(

where Asn represents the surface of nth

cell. If Asn is chosen
small enough, nn(F') varies insignificantly over Sn’ thus (2.2.1)

becomes

J G(F,F')ds' (2.2.2)
AS

The total potential at point F is
4N

mm = z and) =g—
0

n=1 n

where 4N is the total number of surface cells.
Next we require that (2.2.3) be satisfied at the center of

each cell with position vector Th

4N
+ 1 + +
(p = SP” ) = — Z n I (3(1" ,r')ds' (2.2.4)
m m 60 n=1 n Asn m
m = 1,2, ,4N

+ +
when m f n, we can assume that ‘F' = rn, where rn represents the

th

position vector for the center of n cell, thus (2.2.3) becomes

1 4N + + 4N
8h = §_'[ I nnG(rm’rn)ASn + I 5mnnn J G(rm,r')ds']
0 3:1 n=1 ‘ Asn
m = l,2,...,4N (2.2.5)
0 m i n
mn =

The second term on the right is the contribution to the potential
at the center of a cell due to the charge on the same cell.
Equation (2.2.5) comprises a set of 4N linear equations in

terms of n's and may be written in the matrix form as follows

 

 

 

 

 

 

F W F ‘ ' '
Ti "1
‘P2 n2
. = G
(2.2.5)
I cf’4N j . _ - “4N-
4NX1 4NX4N 4NX1
1

. = _ -)- +
w1th em" 60 Asne(rm.rn)

As explained earlier, at low frequencies the vanishing of the

tangential component of electric field on the electrodes requires that '

the potential on the surface of each electrode should be constant.

Thus, we have.

em = constant = i

 

where N1 and N2 are the number of cells on 8 of S1 and S

respectively.

10

2.3. Calculation of Matrix Elements

In this section the expressions for the matrix elements Gmn
will be developed. Two different cases of grounded and floating

potentials are treated seperately.

2.3.l Grounded Case: This case is commonly used in practice where one
electrode is grounded (S1 for example) and an a.c. potential is applied

to the other electrode. For this case VS = 0 and VS = V,where \Iis the
1 2

amplitude of the applied voltage. Substituting these values for em

into equation (2.2.5), we have

-1

 

 

 

 

 

 

- 71] 1 - 1 - 0 -[
n2
, 4N1
= V G 0
1
4N2 (2.3.1)
.“4N- L . L l -
4le 4Nx4N 4le

To make the problem more general, and at the same time to keep
the cost of computation down, we assume a four quadrant symmetry for the
problem. In other words, the geometry of electrodes is symmetric about
the planes 2 = O and y = 0 (Figure 2.2). The planes of symmetry

divide the space and two electrodes into four quadrants labled by

 

 

 

 

/

 

 

 

 

 

Y

(b)
Figure 2.2. A cubical volume of free space is partitioned into four symme-

trical quadrants.(a),and (b) the geometry of two parallel electrodes
with S2 partitioned and the charge densities on the subareas.

12

Roman numerals I through IV. It is noted that the cases where no
symmetry is present in the geometry the problem can be handled as well,
but they are of little value in practical applications.

Under the stated conditions we need only to compute the
charge density,and later in this chapter the electric field, only
in one quadrant. The desired quantities in the other quadrants can be
obtained by utilizing the symmetry. The charge densities on various

cells on the first quadrant of $2 and that of their counterparts on

other quadrants are shown in Figure 2.2.b.
Now we proceed with the computation of the matrix elements.
The off-diagonal elements will be computed first.

If m f n, by symmetry and refer to Figure 2.2.b. we have

I = II = III = IV
nil T1" nn nn

which reduces the size of G matrix in (2.3.1) to %-of that when there

is no symmetry present. With this symmetry, (2.3.l) reduces to

-1

 

 

 

 

 

r n] l T l ' 0
n2
. N1
= v GS 0
N2
I nN . L . _ 1 _

 

NX1 NXN NX1

13

 

where
GS=£QEG(F?I)+G(++II)+G++III ++IV
mn so m’ n rm’rn (rm’rn ) + G(r‘m’rn )]
(2.3.3)
. + +
. -> + e-JBolrm-rn
with G(r ,r ) = , and
m n 4n]? -? I
m n
—> _ c A A
rm - me + me + Zmz
+I_ “ " " +11 " A c
rn - an + Yny + an n - an - Yny + an
+ III _ c e c + IV _ c ‘ ‘
rn - an - Yny - an rn - an + Yny - an

where (X ,Y ,Z ) and (X ,Y ,Z ) are the cartesian coordinates of

 

 

m m m n n n
the centers of the mth cell (field point), and the nth cell (source
point), respectively. Thus, Gin can be written as
. I . II . III . IV
S as" e JBoRmn e'JBORmn e-JBoRmn e'JBoRmn
Gmn ‘ 411€OE RI T T, T R111 1 —IV—R 3
mn mn mn mn
(2.3.4)
where
I _ 2 2 2 8
Rmn - [(Xm ' Xn) + (Ym - Yn) + (Zm - Zn) 3
II 2 2 2 %
Rmn [(Xm - Xn) + (Ym + Y”) + (2m - 2n) 3
III _ _ 2 2 2 1x.
Rmn - [(xm X ) + (Ym + Y“) + (2m + Z ) J
IV 2 2 2 %
Rm" - [(Xm - X ) + (Ym - Y ) + (Zm + Zn) 1

14

and

D for m > N1 0 for n > N1

X = , X =
111 T1

0 otherwise 0 otherwise
0 = The distance between the electrodes.

For diagonal elements, m = n and as a result FE = Fh, which
makes the first term on the right hand side of (2.3.3) become infinite.
Since this term represents the contribuiton to the potential at the
center of a cell due to the charge on the same cell, it is easy to
avoid the singularity by evaluating this term analytically. For this
purpose, we approximate the square cell by a cicular one of the same

area as shown in Figure 2.3. The potential at the center of the circular

disk with uniform charge density n is.

 

“.18 r __ . Pi
e ='_fl__ I e o rdrdo = —wfl—--(e 38° 1' - l) (2 3 5)

c
Therefore, the diagonal elements can be obtained by substituting (2.3.5)

into (2.3.3) for ASn G(Ffi,Fn) and setting m = n:

 

 

D . II . III . IV
S 1 deal/e;- ASm e'JsoRmn e-JBoRmn e'JBoRmn
Gmm ' 238060 ' T) T 41reoE RII T “Tl—R I T ‘rv—R 3
mn mn mn
(2.3.6)

This concludes the evaluation of the matrix elements for the grounded

potential case.

2.3-2 Floating Potential Case: When the potentials of the electrodes

are floating, VS and VS are not known, except for the special case
1 2

15

It?
® a = 05/“

Figure 2.3. A square cell is approximated by a circular one for
the calculation of the diagonal elements of matrix GS

16

of S = $2, for which VS = 'VS = %-. However, it is noted that

2 1
once the voltage on one of the electrodes is obtained, that of the other

1

is determined from
- V = V (2.3.7)

This means that one extra unknown is introduced into the equations
(2.2.5). Therefore an additional equation is required in order that
the equation system (2.2.5) can be uniquely solved. This extra equation

can be derived from the continuity equation.

v-Zi = -jwp (2.3.8)

If we integrate (2.3.8) over the volume enclosed by surface S surrounding
electrode S1 as illustrated in Figure 2.4 , and apply the divergence

theorem we obtain

ql T 57.} (2.3.9)

where q1 is the total induced charge on S1 and I is the current

flowing out from electrode. Similarly for 52 we have

(2.3.10)

I
q2 3":

with q2 being the total charge on 52. Adding (2.3.9) and (2.3.10) and

N
noting that q] + q2 = z "n ASn , leads to the following relation .
n:

1

N
Zlnn AS” = 0 (2.3.11)
n:

17

 

 

 

 

 

 

 

 

 

 

 

 

 

52 v 12
q2
+
P-V
51 Q1
l 11
q1+q2= 0
(a)
u I
s
2 (12
+
,. a.» v
n -
,, -----._--..--t.5---.
51 1‘ .......... .---------./'
q1=-q2 V
(b)

Figure 2.4. Relation between the charge on and the current flowing into
the electrodes. (a) grounded potential case (b) floating
potential case.

18

Equation (2.3.11) in conjunction with the equation (2.2.5) gives the
following matrix representation of N + 1 equations in terms of charge

densities and the potential of S1 .

I
_J
1
l
I
J
1

I

m
M
11
<
o o o o
A
N
00
1..
N
v

I

—J

J o
2
N

 

.umu----------------,---- l.u__- ...] ..... 1

: V
_ AS1 . . ASn : O . 1.51 ‘ . O

 

 

 

 

 

 

2.4 Numerical Results

A computer program has been developed for solving (2.2.5) and
(2.3.12), and the induced charge density has been obtained for several
cases. The capacitance between the electrodes of the same dimension
was calculated as the ratio of the total induced charge and the applied
voltage for several values of the seperation between electrodes, and the
results are depicted in Figure 2.4. (the solid curve). The dashed curve

on Figure 2.4. is obtained from the formula for the electrostatic

capacitance of

c=e % ’ (2.4.1)

19

 

 

 

 

X
I
D 9:
c l
A
10 T
Q;
2 8 r
'5
u.
m
"E:
:2
T” 6
G)
U
C
.3
'5
(U 1-
E; 4
2 n
0.0 T 2:0 ' 4T0 T 4' P

spacing between the electrodes(cm)

Figure 2.5. The capacitance Vs spacing of two parallel plates of equal
size. (a) using the methode of subareas,(b) using c= 60 A/D.

20

The closeness of the two curves for small values of the seperation
between electrodes suggests that the value for capacitance obtained
from (2.4.1) is close to the exact value of capacitance obtained numer-
ically. For a large seperation between the electrodes the expression
(2.4.1) does not give accurate results for the capacitance, the numerical
method should be employed instead for exact evaluation of the capacitance
between the electrodes.

Figure 2.6. shows the distribution of the induced electric
charge on the upper electrode S1 and lower electrode 52 along the
edge (Z~axis), and the diagonal (Zd axis) of the electrodes. The
electrodes are of equal dimension (6 x 6 cm). The solid curves are
obtained for D = 2. cm, while the dashed curves are for the case when
0 = 4. cm. A floating voltage of 2. volts at 15 MHz is applied between
the electrodes. It is noted that the charges on the upper and lower
electrodes have equal magnitude, but they are 180° out of phase. It
is observed that the charge is distributed almost uniformly on the middle
section of each electrode and increases rapidly towards the edges,and at
the corners of electrodes the magnitude of charge is maximum.

In Figure 2.7. the lower electrode 51 of Figure 2.6. is
grounded and 0 is kept constant at 4. cm. The distributions of the
electric charge on the electrodes are shown along Z, and Zd (solid
curves). The charge distributions for floating potential case are also
included for the purpose of comparison (dashed curves). It is noted
that for the grounded potential case the magnitude of the charge on

the upper electrode is noticeably larger than that on the lower grounded

electrode.

21

$1=Sz=6x6cm

V= 2. volts

f=15MHz I) o
I

‘-——- D= 2.0 cm

 

 

 

 

 

 

- ----- D= 4.0 cm
717x109 coulomb/m2
2
\ T J 52
\, ,/
\.\.\ ....... . _ -- -- fi"./.
k L I a n .1 > Z
'2'0 _.’_]-.:..Q.....-._._ .1..- 2.0
.’.—-- ‘\.\.
- -2 S1
(a)

A ‘nx109 coulomb/m2
L4

 

 

 

 

: -4

 

(b)
Figure 2.6. The distributions of the charge densities on S and S for two
cases of D=2.(solid curves),and D=4.(dashed cu ves);(§)along Z

(b) along Zd

f= 15 MHz
Sl= 2= 6x6 cm
V= 2. volts

 

--—- S1 grounded
— -u__ floating potential

nxlo9 coulomb/m2

 

 

 

 

 

' 4

 

 

 

 

 

. -4
(b)

Figure 2.7. The distributions of the charge densities on S1 and 52 (same
dimension) (a) along Z. (b) along Zd

 

23

Figure 2.8. shows the distribution of the electric charge on
the surface of two electrodes with different dimensions. The solid
curves represent the case when the larger lower electrode is grounded,

and the dashed curves are obtained for the case of floating potentials.

2.5. Computation of the Electric Field

 

This section is devoted to the calculation of the electric field
in free space maintained by the charge and current on the surfaces of
the parallel electrodes. If o(F) and A(F) are the scalar and vector
potentials at point F in free space, the electric field at that point
is

Em = - ch(F) 4.216?) (25.1)
At low frequencies and in the near zone, the first term on the right
hand side of (2.5.1) predominates. In other words, in the near-zone
the electric field is mainly due to the charge on the surfaces of the

electrodes. Thus, the electric field can be expressed as;

 

-ieo|F-f'l
5(a) = a] ns(?~*') ‘3 + + dS' (2.5.2)
51+S2 |r - r'l

Taking the differential operator inside the integral, and after some
straightforward manipulation we get
++ ++ ++
+ + (1+jB lr-r'|)(r-r') -j8 lr-r'l
EM T7473? [ ”5““) +0 -> 3 e 0 ds'
0 S1+S2 lr - r'l .
(2.5.3)

 

f=15 MHz

Sl=6x6cm 4.cm

$2=4x4cm
V=2. volts

 

—— S1 grounded
_. ...... floating potential 9
t‘nxlo coulomb/m2

 

 

 

 

 

 

 

 

 

S2
2:0 4311 iii) 230 7‘2
.’”_"_“._“_._.u_a._.__.....n;::::::7‘, S
./-/ ‘ r-l \.‘ 1
(A)
A. 9 2
0x10 coulomb/m
'5
.3 I,
,/
b ./
,/
0’./
-JIU* -1.0 110' T 310 ' " Zd
/f/— - - - ‘\\
./ ' \.
’///’ ‘\\\l s
(B) “-2

Figure 2. 8. The distribution of charge densities on S1 and S (different
dimensions) for grounded(solid lines) and 1floatiFlg potential
cases:(A)along Z,(B)along Zd

25

The electric field at point F in free space due to the charge

on the nth subarea is approximately equal to

 

. + + + + . + +
E (F) g ASmnn (1+JBOIr-rnl)(r-rn) e-Jsolr-rnl
n 4n€ + ‘ + 3
0 Ir - rn|

The total electric field is then

4N ASnnn (1+je |F—F | (F4F ) -js |F-F

 

 

.. 4N .. ) I
1:(F)= z s (F) = 3 g" " e 0 " (2.5.4)
n=1 n n=1 41'60 |r - r |3

+
The three scalar components of E in the first quadrant can

be written as

 

N
I _ I II III IV
Ex - n2] nn (cn + (3n + cn + (2n ) (x-xn) (2.5.5)
E1 = i (cI + CIV) (Y-Y ) + (cII + c111) (v+v ) (2 5 6)
Y n=1 nn n n n n n n ' ‘
5‘ = % (cI + CII) (z-z ) + (cIII + CIV) (2+2 ) (2 5 7)
Z n=1 nn n n n n n n ‘ '
where
. k . k
AS 1+JB R -38 R
ck " ° " e 0 " k = I, II, III, IV

and (X,Y,Z) are the coordinates of the field point F, and

k _ + +k
Rn - [r - rnl

26

where F: is the position vector for the source point in the kth

quadrant. The numerical results for the electric field components are
shown for various cases in Figures 2.9 to 2.12.

Figure 2.9. shows the distirubiton of the three components of
the electric fields, maintained by the charges on the electrodes, at
the centers of the partitioned subvolumes (1 cm3) between the electrodes
of equal dimension (6 x 6 cm) when the potentials of the electrodes
are left floating. The applied voltage between the electrodes is 2.
volt and the frequency is 15 MHz. Due to symmetry only the fields in
one quarter of the free space between the electrodes are shown in the
Figure. It is seen that the X-component of the electric field pre-
dominates as expected. Also the field is rather uniform with a slight
decrease in the middle portion between electrodes.

Figure 2.10. shows the distribution of the X-component (dominant
component) of the electric field in free space between two electrodes
of equal diminsions (6 x 6 cm) seperated by a distance of 4. cm. A
voltage of 2. volts at 15 MHz is applied between the electrodes
while the lower electrode is grounded. For this case weaker electric
fields are maintained near the grounded electrode, even though the
electric field in the free space between electrodes is still rather
uniform. Figure 2.11. shows the similar distribution of electric fields
maintained in the free space between two electrodes of different dimen-
sions, the upper electrode is 4 x 4 cm, and the lower one is 6 x 6 cm.
The potentials of the electrodes are left floating and the applied voltage

is 2. volts at 15 MHz. The most noteworthy phenomenon in Figure 2.11

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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o—lcm»

Three components of the electric field at the centers of subvolumes in different

Figure 2.9.

1ng

for float

1mension

layers in k of free space between two electrodes of equal d

potential case.

.umucaocm
Pm saw: m~_m Peace $0 monocpom_m oz“ cmmzpmn muwam mmLO $0 x :_ mcmxeP

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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15 MHz

f =

\IA
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E
U
k0
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\D

S1

  
   

 

A
v

4x4cm

$2:

 

 

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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*Icn1-

layers in k of free space between two electrodes of different sizes for floating

Three components of the electric field at the center of subvolumes in different
potential case.

Figure 2.11.

30

is the distribution of the high electric field near the smaller upper
electrode. This pehnomenon is expected from physical intuition, and
has been used in practical applications to focus the electric field.

Figure 2.12 shows the similar results as that given in Figure
2.11. with two different features: The larger lower electrode is grounded,
and only the X-components of electric fields are shown. The interest-
ing point we observe from Figure 2.12. is that when the larger electrode
is grounded, the concentration of higher electric fields near the
smaller electrode becomes most outstanding. This phenomenon should
have practical applications.

Figure 2.13. shows the distribution of the X-component (dominant
component) of the electric field along the X-axis maintained between
two electrodes of equal dimension. The spacing 0 between electrodes
is kept at 4. cm and the electrode dimension (a) is varied to give
four cases of 3" 1.16, 1.75, 2.4, and 3.5. A voltage of 2. volts
at 15 MHz is applied to each of the four cases. The lower electrode
S1 is grounded, and the distribution of X-component of the electric
. field along the X-axis is shown in the Figure for the four cases. It
is noted that when the dimension of electrodes is about the same as
the spacing between them, the electric field between electrodes is
nearly uniform. However when the dimension of electrodes become con-
siderably smaller than the spacing, the distribution of the electric
field along the X-axis can be nonuniform; with the intensity of the
electric field decreasing quite rapidly from the upper electrode (un-

grounded) towards the larger electrode (grounded).

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32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x
41
s
f = 15 MHz 42 a :
a = D/a I : BI- V
D IV
V = 2. volts 1 g 9 -
= 1
S1 S2 51 l
4.-
Ex(v/m)
100‘» a= 3-51t00
60 '
20 .
. . . :_ x
0.0 2.0 4.0

distance from S1 (cm)

Figure 2.13. Distributions of the X-component of the electric field along
the X axis between two electrodes of equal dimension for va-
rious ratios of D/a , where D is the spacing between the el-
ectrodes and a is the dimension of the electrode. S1 is

grounded.

33

Figure 2.14. shows the distribution of the X-component of
electric field along the X-axis in the free space between two electrodes
of different dimensions. A voltage of 2. volts at 15 MHz is applied
between electrodes while their potential are left floating. Four
cases of different electrode dimensions are considered for different
values of 8, which is defined as the ratio of the surface areas of the
tow electrodes. The spacing between the electrodes, 0 is kept at
4. cm, it is observed that for B = l, the electric field is uniform.

For a > 1, however, the electric field intensity is higher near the
smaller electrode 52, and decreases towards the larger electrodes.

It is noted that as the ratio 8 increases, the electric field intensity
decreases more rapidly along the X-axis towards the larger electrode.
Similar phenomena are observed for the case when the larger electrode

is grounded. The results for this case are shown for four different
values of s, in Figure 2.15. comparing Figures 2.14 and 2.15, we

observe that for the same value of e if S1 is grounded the intensity
of the electric field increases near the smaller upper electrode S2

and reduces near the larger lower electrode S1 as compared with the
case of floating potential case. For this reason the grounded potential
case has advantage over the floating potential case for the purpose

of focusing the electric field.

'34

 

 

 

 

 

 

 

 

 

 

 

X
A
f = 15 MHz
8 = area of Sl/area of 32
S
D = 4. C111 2 :_ T
+
v = 2. volts E D 6') V
. l '-
I
S1
Ex(v/m)
h
100 '
60 -
20 '
. . . ,, X
0.0 2.0 4.0
distance from S (cm)

1

Figure 2.14. Distributions of the X-component of the electric field along
the X axis between electrodes of different dimensions for
various ratios of 51/52 .The potentials of the electrodes
are left floating.

f
B =area of Sl/area of 52
V = 2. volts

0

= 4. cm

 

 

 

 

 

 

0.0 2.0 4.0

distance from Sl(cm)

Figure 2.15. Distributions of the X-component 0f the electric field along
the X axis between two electrodes of different dimensions
for various ratios of S /S where S1 and S are the surface
areas of the two electrddeé. ( S1 is groundbd )

36

2.6. Comparison of Numerical Results with Experimental Results

 

In order to verify the theory and numerical results presented
in the preceding sections, a series of experiments was conducted to
measure the electric field between the electrodes for various cases of
capacitor dimensions and applied voltages. The details of the exper-
imental setup will be given in Chapter V, and only the experimental
results will be used to compare with theoretical results in this section.

Figure 2.16. shows the theoretical and experimental results
of the electric fields maintained at the inner and outer surfaces of
one electrode for three capacitors with various electrode dimensions
(a), and seperations D (i) D = 4. cm, and a = 4cm, (ii) 0 = 8
cm, and a = 6. cm, and (iii) 0= 4 cm and a = 6 cm. The electric
fields at the inner and at the outer surfaces of one electrode tends
to increase towards the edge of the electrode, somewhat proportional
to the induced surface charge on the electrode. The electric field at
the inner surface of electrode is in general, larger than that at the
outer surface. The ratio of the former to the latter becomes larger
if the electrode dimension (a) is increased, IN“ the seperation between
electrodes 0 is decreased. In the limit of an infinite a and a
finite 0, there is zero electric field at the outer surface of the
electrodes. The agreement between theory and experiment is considered
to be good.

In Figure 2.17. the variation of the normalized X-components
of the electric field along the X-axis maintained between two electrodes
of equal dimension (7 x 7 cm) has been shown for the cases of

D = 4 cm (A), and D = 8. cm (B). The solid lines show the theoretical

 

 

 

 

 

 

 

 

 

 

 

 

 

I
E. 9 o
Eout 1"
T ‘\\T“~$l._______s——r””’ = 4.0 cm
" 52 = 4.0 cm
E1" | E . o J
O Y OUt\.__/
le—a—r '
S
1
\ P h l 7 Y
:2.0 -1.0 O 1.0 2.0
f = 15 MHz
. Ex(j(m) .
E = 8.0 cm
in -
\ 0 - - 6.0 cm
0 o
E J
out 0 ' o
o 0
L s n n L A : Y
-3 O -2.0 -1.0 0.0 1.0 2.0 3.0
EX( /m)
E ' OT
in"~‘--~.:L-.-.--‘-—- . = 4.0 cm
.0 = 6.0 cm
0 .1
E0” 0 o
o O
1 l a 0.5 n . A g Y
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

distance along Y axis(cm)

Figure 2.16. Theoretical and experimental results of the electric field at
the inner and outer surfaces of one electrode for three capa-

citors with variouseflectrode dimensions and seperations

. So-

lid lines are the theoretical results and discrete points re-
present the experimental results.

 

 

 

38

 

 

 

 

 

 

 

Ex(v/m) -
4 i/theory )
X 1.0 , "I‘jr—— -;—'1FT ‘ 1'0
O 0
51 I experiment
1 i
: 0.6 I ‘ 0.6
1” i 0=4.o cm «
"' a1 ’
1 a=7.0 cm
52 0.2 r , a 0.2
51 =‘S2 = 6x6 cm T I L _"X
0.0 1.0 2.0 3.0 4.0
f = 15 MHZ distance from S](Cm)
A
EX(VIm)
‘ IL
1 0 _ - theory - 1.0
h --Ni-$“‘_~_____~ ¢// #__._,,.—'——v1rrzzrzze
o ‘ '
0.6 I- .\ . .T 0.6
experiment
D=8.0 cm
0.2 _ a=7.0 cm - 0.2
L I L J H L L A :
0.0 2.0 4.0 5-0 8'0
(cm)

distance from S1

B

Figure 2Ll7. Theoretical and experimental results for the distribution of
the X-component of the electric field along X axis between

electrodes of equal dimension. Solid lines show the
theoretical results and the discrete points represent the
experimental results.

39

results, while the discrete points represent the experimental values.
Again a good agreement between theory and experiment is obtained. In
Figure 2.18. the theoretical and experimental results for the variation
of the X-component of electric field along the X-axis between two
electrodes of different dimensions are shown for the floating potential
case (A), and for the grounded potential case (B). In each case

the results for 3 different values of B are shown. It is observed

that the agreement between theory and experiment is satisfactory.

Figure 2.18.

40

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

X
= 15 MHz
= 4.0 cm
15 {3 = area of S /area of S
I 2
Ex(v/m) D 2 Tv
“ I - 8=16 “
100 T - 100
80 T 80
s=4
6O . 6O
=1
20 . - 20
n g L ; x
0.0 1.0 2.0 3.0’ 410
distance from S (cm)
A
T s
+
E (v/m) 2
x h D 6? V I
100 . l ' B=9 (100
S
- 1
80 . é- . 80
B=4
60 I' _ 1+ 60
9'73 =1
40 ' ' 40
20 - ‘ 20
0.0 120 2.0 3.0 4.0T X
distance from S](cm )
B

Theoretical and experimental results for the distribution of
the x-component of electric field along the x axis for various

ratios of 51/52. A floating potential 8 S1 is grounded.

CHAPTER III
LOCAL HEATING WITH 1”: ELECTRIC FIELD

The heating effect of nonradiating electromagnetic fields has
been utilized over the past decade in order to eradicate the cancerous
tumor embedded inside a biological body. A great deal of research has
been conducted and successful results have been obtained on tumorhbearing
labratory animals and even the human body.

There exist several methods by means of which a biological body
can be locally heated. One such method is the local heating with
electric field, where the body is placed between two parallel electrodes
connected to a RF generator. Although this method has been proven
efficient in cancer therapy, very few engineering studies have
been conducted to analyze the distribution of the induced electric
field and the specific absorption rate of energy (SAR) in the body.

In this chapter we aim to analyze the theoretical aspect of
the problem and find the induced electric field in the body for various
configurations of the electrodes as shown in Figure 3.1. Also a scheme
will be developed to synthesize the potential distribution or the charge
distribution on the electrodes in order to obtain a desired heating

pattern in the body.

3.1 Problem Discriptions

 

In the preceding chapter a capacitor consisting of two electrodes

41

 

 

 

 

Electrodes of the same size
floating voltage

 

 

 

v1 =-v2 = v/2 and q1 = q2

 

   

f----- -- -7
I
-----q
1’ V ’
I /

 

 

 

 

.‘oflq '
I. * - up}. -

I L- --J' .

I __q2

 

 

 

J
— I
I
I

..--‘12-----."

 

 

heterogeneous body

 

. .25

 

/ .q
V]+

 

 

 

---------q

I

I, qz 4

----’ c - I,
I

’1 V ’I

b-- -.---.-.

I

 

\

 

 

Electrodes of different sizes

floating voltage

v1+v2 = v and q.l = q2

 

 

 

 

EL—

’.
’V I

‘-- -------J

 

 

 

Electrodes of different sizes
one electrode grounded

v2= 0.0, v1= v
q] *q2

Figure 3.1. Different arrangements of capacitor plates placed on the
biological body for the purpose of local heating.

43

of arbitrary dimensions in free space was studied in some detail,
and the distributions of the electric charges on the electrodes, as
well as the electric field in free space, were obtained.

When a biological body of conductivity 0, permittivity E
and permeability “o is placed between the electrodes, the problem
becomes more complicated in the sense that the charge induced on the
electrodes and the induced charge on and the induced current in the
body are coupled. This necessitates the establishment of more equations
from which more unknown quantities are to be solved as compared with
the problem treated in the preceding chapter.

The geometry of the problem is shown in Figure 3.2 where a
pair of electrodes are placed across a biological body. The density
of charge on the electrodes is denoted by n(F'), and the incident elec-
tric field maintained by n(F') is represented by ET(F). The total
electric field at any point F inside the body is the sum of the in-
cident electric field E1 and the electric field maintained by the
induced current and charge inside the body, or the scattered electric
field represented by ES.

By using the Maxwells' equations we will obtain two coupled
integral equations in terms of the induced charge density on the elec-
trodes and the unknown total electric field inside the body. Then the

moment method is employed to solve the integral equations numerically.

3.2 Integral Equation for the Total Electric Field in the Body

 

The incident EM fields must satisfy the source-free Maxwells’

equations in the free space between the electrodes:

44

 

 

 

   

0 (F)

nan.

 
 
 
 
 

biological body

 

 

 

 

Figure 3.2. A pair of electrodes energized by a HF voltage placed
across the biological body for local heating

vxETm =4...“ H (F) (3 2 1)
VXHT(F) = 3.. 60 ER?) (3 2 2)
V-E1(F) = 0 (3 2 3)

<
.
14'
—lo
A
"51'
V
ll
0

(3.2.4)

where a time variation factor of ert is assumed for the incident
fields.
The total EM fields at any point inside the body can be

expressed as the sum of the incident and scattered fields

15(2) = Elm + 2%?) (3.2.5)
A(r) = Film + fism (3.2.6)
E(F) and H(F) must satisfy the following set of Maxwells' equations.
vxE‘(F) = ganja?) (3.2.7)
'* + + . + + +
VXH(r) = [0(r) + Jw €(r)lE(r) (3.2.8)
v-[o(F) + jm e(F)iE(F) = 0 (3.2.9)
v.fi(F) = 0 (3.2.10)

substituting (3.2.5) and (3.2.6) into (3.2.7) and (3.2.8) leads to the

following equations
+i+ +S+ _ . *i-> . +s+ ,
vXE (r) + vXE (r) - -JmuoH (r) -JmuOH(r) (3.2.11)

vxfi‘(F) + vxfi$(F) = 5(F)E(F) + wae(F)-GOJE(F) + iweot

46

If we subtract (3.2.1) and (3.2.2) from the above two equations, the

result is
vxESCF) = 4.300115%) (3.2.13)
-> . '* + . 'T' -)-
vxfis(r) = (G(r) + “(am-sonar) + 3.20550) (3.2.14)
We define an equivalent current density jéq(F) by
+ + -> + +
Jeq(r) = 1(r)E(r) (3.2.15)

where 1(F) = 0(F) + jw (6(F)-eo) is called the complex conductivity.

Thus, the equation (3.2.14) reduces to

vxlism = data!) + meow) (3.2.16)
It is noted that the equivalent current density, J consists of

eq
the conduction current 0E, and the polarization current jm(6(F)-€O)E.

The appropriate Maxwells' equations for the scattered fields

can be summarized as follows

vxES(F) = 041101153) (3.2.17)
vxfism = Teqm + jweoES(F) (3.2.18)
v-ES(F) = SL0 peqm (3.2.19)
v-Fsm = 0 (3.2.20)

+
o is the equivalent charge density and is related to Je by the

eq
equation of continuity

q

V'jeqm - -Japeq('F) (3.2.21)
.VHJ (+)
oeq(F) = g—fq—t— (3.2.22)

The scattered electric field can be thought of as the field
maintained by Uéq(F), which flows within the conductive medium. It

can be shown that they are related by the following expression E 15]

 

 

+5 -)- = + eh. + 4'. _?5(?‘Ffl I
E (r) IV Jeq(r ) [P.V.G r,r ) 3jo€o J dv (3.2.23)
where G F,F') = jquCTF+ Zgfl G(F,F') is the free space dyadic Greens'
80
-iBO|F-F'|
function and G(F.F') = e 1+ + I ' is the scalar Greens' function for
' 4n r-r'

free space, and

AA AA AA

H
I ‘ xx + yy + 22

is the unit dyadic:

P.V. stands for the principle value and it means that the source point
should be excluded while evaluating the integral. With (3.2.23),
(3.2.5) becomes

(1 + LEI {E(F) - p.v. f 3(F')E(F').t*(t,t')dv = Elm (3.2.24)
(0 o V

.).

which is an integral equation for the total electric field E(F) inside

the body.

3.3 Integral Equation for the Induced Charge on the Electrodes
An integral equation can be derived for the induced charge

density on the electrodes by a proceduce similar to that employed for

48

the free space case in Chapter II. The difference however, is that in
the present case the charge induced on the surface of the body must be
taken into account for evaluating the potentials on the electrodes.
Thus, at any point ‘F on the surfaces of the electrodes the potential
V(F) can be expressed in terms of the charge induced on the surfaces

of the electrodes and that on the body surface.

V(F) = g—f n(F')G(F,F')ds + ELI nb(F")G(F,F")ds" (3.3.1)

5 b

+S S

1 2

Where S1 and S2 represent the surfaces of the electrodes, Sb
represents the body surface, n(F'), and nb(F") represent the charge
densities on the electrodes and the body surface, respectively. An
expression for "b in terms of jéq may be deduced from (3.2.21).

For a homogeneous biological body v-Oeq(F) is nonzero only
at the body-free space interface, where the induced charge is present.
If (3.2.21) is integrated over an infinitesimal volume enclosed by a
samll pillbox as shown in Figure 3.3.b, and the divergence theorem is
applied, ”b can be obtained as

.. 1006-? 7‘ (3.3.2)
.100

+. =
nb(r )
Where 6 is the outward unit vector normal to the surface of the body.
With (3.3.2), (3.3.1) can be rewritten as

V(F) = ELI n(F')G(F,F')ds' + J—f 1(F")ri-E(F")G(F,F")ds" (3.3.3)

601
o 51+52 0 Sb

 

where as before G(F,F') =

49

Equation (3.3.3) and (3.2.23) constitute a pair of coupled integral
equations for the unknowns n(F') and E(F).

The incident electric field TET(F) in (3.2.24) is maintained
by the charges distributed on S1 and S2 only, and can be written
in terms of the charge density n(F') as

Elm = - 51-] n(r')vr G(F,F')ds' (3.3.5)
o 51+52

For convenience, (3.2.24) and (3.3.3) are rewritten together.

J—ngieo {E(F) - P.V. [1(F')E(F')~TGT(F.F')dv EMF) (3.2.24)

(1 +
g—( f n(F')G(F,F')ds' + i (S 40")(652'61')marinas") = w?)

1 2 b (3.3.3)

n(‘F' )vr0(‘F,‘F' )ds'

*i + 1
where E (r) = - g—-f
° 2

S]+S

3. Moment Solution of Coupled Integral Equations

 

The two coupled integral equations may be converted into a
system of linear equations in terms of the unknowns by using the method
of moments. For this purpose, the body and the electrodes are assumed
to be symmetric with respect to the Y = 0 and Z = 0 planes as shown
in Figure 3.3. The first quadrant of the body is divided into N
subvolume cells, throughout each cell 1(F) and E(F) are assumed
to be constant. Similarly, the first quadrants of S1 and 52 are
partitioned into a total number of N' subareas, over each subarea

the charge density n is assumed to be constant.

 

 

 

 

 

 

 

LO

 

'50

 

/ I
” I~Ipf [4-6!‘
I .
A- l I
I 7
’1’ 0 .4“ I,
: $1 [’7"
“‘~.~ I
s‘ I
‘~J
(a)

 

ET I IFS

 

+ 1- +4-324-

 

 

3L;

 

4

 

111

center of the m

r:(center of the nth)
///’/’ electrode surface

cell

..'b
r] (center of the 1th)

\7; (

body surface cell

th)
body cell

Figure 3.3. The geometry of a body placed between two electrodes(a).

(b) The side view of the body and the electrodes.

51

From (3.3.3), the voltage at the center of the th subarea

located in the first quadrant can be written as

 

N A ++b +b
N' 4 - B n-E(r )r(r )
+ - l 1 +5 +5 k . t 2
V(rs) = vs = I n X e (r ,r )AS + J 2
111 Ill 60 k=1 k 1:] 111 k e 1:] OJ
4 i +5 +b t
1:

where i represents the quadrant number and ”k is the charge density

on the kth subarea on the electrodes. F3, and ‘F: are the position
vectors for the centers of the mth cell (field point) and kth cell
(source point) on the electrodes.

FE represents the position vector of the center of the 2th body
surface cell, and finally ASE, and 05% are the areas of the kth

th

cell on the electrode and the 1 cell on the body surface, respectively.

The induced electric field inside the body is mainly in the
IE
X

TX-E _ ____
Jw Jw '

 

x direction, this implies that nb =

Therefore, the equation (3.3.4) may be transformed into the following

matrix form.

 

 

 

 

' ' n
N NB 1 . -
l ' V
N' : nN' 2 N' (3.3.5)
S I C ---.n. = °
G : E1 .
i x .
L E a . ' VN 'T
N
B
. Ex (I

 

 

Where

_52

4 - jt 4 .
S , _l_ k 1 = i 2
i ,g '1 +5 +5 i _-_ 1 +5 +b ; b
Gmk - G (rm.rk). Gm, - G (rm.r£). Tl - r(r£ )

and NB is the total number of the subareas on the body surface. The

S

expressions for the elements of the matrices G and C will be

presented in the next section.

The matrix representation of (3.2.24) is given elsewhere [16 l.

and the result is of the following form:

 

 

 

 

 

l ' i
T Gxx 3 va ; ze Ex Ex T
T """" : """" 1 """" ""' "T"

Gxx 5 va i sz Ev 5y (3.3.6)

-.----.4------J----.-~1 --.-.-. --.1--.

: : 1

sz g GZY : Gzz J Ez LE2

’ 1. .

 

Equation (3.3.6) can be written in a more compact form as

l .11.] [E‘l

where [G] is a 3NX3N matrix and [E], and [ET] are column metrices

each with 3N elements.
The incident electric field matrix [E1] can be related to the
induced charge densities on the subareas of S1 and S2 by the

following matrix relation.

NI

1' 1

Ex AX 1 ”1
i .

3N .-EY“ = "fl"-.. . (3. 3.7)

. ll I

i N

E A

L z . . z l

 

 

 

 

53

With (3.3.7) substituted into (3.3.6), and after some rearrangements,

we obtain

 

 

 

l l
3N N
P 1 EX
I g : .----..
Gxx : va i ze : Ax Ev 3“
3N GYX g GYY i GYZ 5 AY E2 = 0 (3.3.8)
i“"‘i""'r°""i“"'“’ -
sz E sz 3 G22 : A2 "1
. I N
)“N' .

 

 

For the case when one electrode (5]) is grounded, equations (3.3.5)

and (3.3.8) may be combined into the following matrix representation.

 

 

 

 

 

 

_ 3N N'
Gxx : va : ze i Ax l Ex 0 l
------ r -
I I
3" va : GYY : sz : Av Ev .
r ----- f ------ t ------ 1 ------------- 4 .
sz E GZY i 622 5 AZ E2 = v 0 (3.3.9)
""" ?"""?"."'T_"""" '"51 o
: i E . ‘ ”l
N' c' g o 3 o 1 GS . 9 N
. 5 3 ‘ LnN' . . 1 . 2
N1+ N2= N

where, N1 and N2 are the numbers of subareas on 51 and $2, and

V is the applied voltage.

The matrix C' is related to the matrix C as
N N N-ng
N' c' = N' c

O

54

3.4 Calculation of the Matrices Elements

 

In this section the expressions for the elements of the matrices

G, A, as

, and C will be developed. The elements of the matrix G
have been given by Livesay and Chen [16 J, and only the results will

be presented here.

3.4.) G Matrix

For a four quadrant symmetry it is shown that

mn mn mn mn
mn _ l 2 3 4
prx - prx + GXPX + GXPX + GXPX (3.4.1)
mn mn mn mn
mn l 2 3 4
G = G - G + G - G (3.4.2)
XPY XPY XPY XPY XPY
mn mn mn mn
mn l 2 3 4
G = G G - G + G (3.4.3)
P l,2,3
X1 = X, X2 = Y, X3 = Z

In the above expressions the numbers l to 4 in the superscripts
represent the quadrant number.

For the diagonal elements we have m = n, and
nn1 = quOGP -JB a

'* . + O n .
xqu 38 {3[T(rn) + JwEOJ-Zt(rn)e (1+Jgoan)} (3.4.4)
o

G

where
q = l,2,3

l for P = q

Pq'-
0 for P f q

55

and

th

where AVn is the volume of the n volume cell.

For m f n, we have
i . i

em" = -‘ a r(? )AV e-Jam”{[( 1 )2-1-' 1 15
X X quo o n n. amn JO‘mn Pq
P q l 1
+ Cos em"i Cos em"1 [3-( i )2 + 3 i 1 (3 4 5)
XP Xq amn amn ’ '
where
i _ 1 i _ + +
amn ' BoRmn ’ Rmn " Irin] ' rnii
i 1
. m n m n
1 X - X 1 X - X
mn _ P P mn _
COS exp - T— , COS 9X - "QErJ—
mn q mn
i = l,2,3,4 (The quadrant number)
and
i i 1
+ _ m m m + ___ n n n
rm “ (X1: X2: X3), rn (X1 3 X2 9 X3 )

3.4.2 A Matrix
From (3.3.7) the three components of the incident electric field
at the center of the mth body subvolume located in the first quadrant

are given as

N
Em = Z Amnn
X n=l X n
Em = g. Amnn m = 1 2 N
Y n=1 y n
NI
m mn
Ez ’ 2 AZ nn

3
...:

56

By comparing the above equations with (2.5.5),(2.5.6), and (2.5.7) the

following expressions for the elements of the matrix A can be obtained

 

A?" = (c1 + c2 + c3 + c4 n) (xm x?) (3.4.6)
A?" = (c; + cfi) (x? xg)+ +(c2 + c 3) (x3 + x2) (3.4.7)
A?=<cM+c>< -xW)+<§+c1u +Q)
where
cfi = ::g 1+::§RE3 e-jB°R:" . k = 1.2.3.4
O
and m"
Rmn - l$111 ’ ?:|
Ph = (xT. 2. x ). rn - (x? . xgk. xgk)

again k represents the quadrant number.

3.4.3 GS Matrix

In Chapter II the expressions for the elements of the matrix

S

G were developed and the results are as follows.

For m f k (off-diagonal elements).
AS

GS = —€9- [G (F5.'FS) + 62 (rm . PS) + 63(PS. F5) + 64(FS, PS)J
mk o k k

(3.4.8)

(3.4.9)

where

 

i
-js WEI-F: |
1+5 +s e °
G (r , ) = #44
m k 1
4nI?S-F I
m k
m = l,2.. .N'
i = l,2,3,4
$3 = (X2, Y3. 2:) is the cartesian coordinates of the mth subarea
on the electrode.
+si 5" si 5‘ th
rk = (Xk . Yk . Z ) is the cartesian coordinates of the k sub-
th

area in the ‘i quadrant. It should be noted that all surface cells

are assumed to have the same area Ase. and

'Oifng]. r0 ifmgN

II
(I)
I

 

 

LDifN1<ng1+N2 o ifN1<m<N

L

1+N

2”;
where D is the distance between the electrodes.
For m = k (diagonal elements) we have.
A
S 1 ‘jB "2' ASe

_ o w 2 +5 +5 33+S +5 4+5 +5
Gmm ' ZJBOGO (e '1) +_ T: [G (rm m) + G (r m) + G (r m)J

(3.4.10)

3.4.4 C Matrix

From (3.3.5.a) we have

l,2,...,NB

° 1 m = l.2.....N'

where

 

 

‘i =1- i+S+b
Gmg G (rm,r£)
It is straightforward to show that
c = ‘5-53 [G'GS f”) + GZFS ‘5'”) + G3(+S *b) + G4(*S "b )3 (3 4 ll)
me J 60“) 1:2 m) 2 rm, 2 rm,r2, rmfl‘ 9. . .
i
'isoliifi I
where G‘(‘F5,?f)= e 1 , i = k,2,3,4
m 4 |‘f5-‘Fb |
" m
and $3 = (Y3, $3, $3) is the cartesian coordinates for the center of
the mth subarea on the electrodes with
r 0 for m i N1
5..
Xm —{
L D for N.I < m <_ N1+N2
4:" si s1 51 th
r1 = (X2 , Y2 , Z2 ) is the coordinates for the center of the z

subarea on the body surface located in the quadrant

r NB
9 for 2 5'7?

 

where g is the seperation between the body and the electrodes.
When the voltage between the electrodes is left floating, by
the same argument given in Chapter II for the free space case, (3.3.9)

can be modified into the following form:

 

 

 

 

 

 

3N N l
' E : ‘ . 1
o . ' '
Gxx : va :ze Ax 2° Ex 0
....... ' -----.L-.---.E--------.'---. .----.- F“"'
3N GYX f GYY :GYZ 5 AY E0 EY o 3N
,.---..-:---.--:-.---.;-------.L-.-- ....... J F ------
' 5 : (3 4 12)
sz f sz 5622 5 A2 5° Ez “'V 0 ' °
, i i E i n] 0 N.I
N c i o i o 5 GS 5" ° ' 1 N N1+N2=N'
3 : E : ”'N 2
..--.--!.-----5 ..... I. "°"'t"" ..----- ..--.-“
1 o o i o i AS »0 V o 1
L, ' : e ' J c S] .J L ..
where VS is the potential of the electrode Sland V is the applied
voltage.1

l

3.5 Numerical Results for SAR's and Electric Fields

 

Equations (3.3.9) and (3.4.l2) can be solved easily by the
computer for the total electric field inside a conducting body and
the density of induced charge on the electrodes. In this section the
numerical results for various geometries shown in Figure 3.l will be
presented. The dimension of the body is chosen to be 6X6X3 cm, and
a potential of 2 volts at 15 MHZ is applied between the electrodes.
The gap between the electrode and the body in each case is assumed to
be 2.5 mm.

Figure 3.4 shows the distributions of the electric charge density
along the edge (Y-axis) on the upper electrode for the free space
case and the case when a conducting body is introduced between the
electrodes. The electrodes have the same dimension (4X4 cm), and a
potential difference of 2 volts at l5 MHZ is applied between the

electrodes. From Figure 3.4.a it is seen that in the presence of a

f = l5 MHz
v = 2. volts
0’= 0.5 S/m
S1= S2 = 4x4 cm
D = 3.0 cm
9 = 2.5 mm
free space
------ - with body
\-

60

 

 

 

 

 

 

 

 

 

$2__
l +
D -—> Y 6.) v
51—7——
77x109 coulomb/m2
h
j
*5 ,/
,x
,/
'—- ..... Ar- ....... /
'3

 

 

 

 

. A . . t. Y
-2.0 -l.0 0.0 l.0 2,0
(a)
2

leO coulomb/m

2
.. - 4A . ’.
-2.0 -l.0 0 0 1.0 2 0

Figure 3.4. The distributions of the electric charge along the Y axis on
the electrod for the free space case and the case with a body
between the electrodes.In figure (b) the charge distributions
are normalized by their maximum values to show the relative

variation.

61

conducting body the induced charge on the electrodes increases greatly
as compared to that of the free space case. In Figure 3.4.b, the
charge distributions for the cases are hormalized by their maximum
values. It is noted that for the case when a conducting body is placed
between the electrodes, the distribution of the charge on the electrodes
is more uniform than that of the free space case.

In Figure 3.5 the electrodes of equal dimension (4X4 cm) with
floating potentials are used to heat the body. One quarter of the
body is divided into three layers and each layer contains 9 cubic cells

3 volume. The specific absorption rate of energy (SAR), and

of l cm
the components of the total electric field at the center of each cell
are shown. It is noted that the electric field, as is expected, is
mainly in the X-direction and is uniformly distributed in the body
between the electrodes. The magnitude of the electric field drops
drastically in the region of body outside the electrodes. This phenom-
enon is important in practical applications where it is required to
focus the EM energy in a desired region of a biological body for the
purpose of local heating.

Figure 3.6 shows the distributions of SAR's, and electric fields
in various body cells when the lower of the two identical electrodes
(4X4 cm) is grounded. It is seen that unlike the floating potential
case the shown quantities are not equal in the first and the third layers.
Instead, the SAR's are maximum near the upper electrode and decrease
along the X-axis towards the lower electrode.

In Figure 3.7 the distributions of SAR's and electric fields

in various cells are shown for the case when the electrodes of different

dimensions are placed across the body. The upper electrode is 2 x 2 cm

3rd layer
Electrode-body gap = 2.5 mm fix / / / /
e _.. 52 / f[ “""2nd layer
' 8° / /
a: 0.5 S/m " 2‘ \lst layer
f = 15 MHz V , ,—-§-—;’- -
_. - I
V " 2o VOItS If I, I,
Sl= $2= 4x4 cm -------77' ------ I ----- Y
’s I]
1.1 ...... I
*---4 cm --*
1+---- 6 cm ----'
lst layer 2nd layer 3rd layer
4.4 3.5 0.6 4.6 3.8 6.7 4.4 3.8 6.7
122.2 121.4 3.5 120.0 119.0 3.8 122.2 121.4 3.5
121.7 122.2 4.4 121.0 120.0 4.6 121.7 122.2 4.4
Absorbed power density (Ilwatt/Kgm)
13.0 11.0 4.0 13.0 12.0 5.0 13.0 11.0 4.0
0.0 0.1 0.2 0.0 0.0 0.0 0.0 0.1 0.2
0.8 0.9 0.2 0.0 0.0 0.0 0.8 0.9 0.2
69.0 69.0 11.0 69.0 69.0 12.0 69.0 69.0 11.0
0.1 0.5 0.9 0.0 0.0 0.0 0.1 0.5 0.9
0.4 0.5 0.1 0.0 0.0 0.0_, 0.4 0.5 0.1
69.0 69.0 13.0 69.0 69.0 13.0 69.0 69.0 13.0
0.1 0.4' 0.8 0.0 0.0 0.0 0.1 0.4 0.8
0.1 0.1 0.0 0.0 0.0 0.0 0.1 0.1 0.0
components of the-induced electric field ( Ex . Ey , Ez )(10mv/m) ‘

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

body maintained by a

the same size.

 

Figure.3.5. Distributions of SAR and induced electric field in one quarter of a
capacitor plate applicator with electrodes of-

63

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3rd layer
Electrode-body gap = 2.5 mm x s/Idj7fi // "”’
6': 80 ~ ///r52 ““‘“2nd layer
0’ = 0.5 S/m \
+ 1'. 2'. lst layer
V = 2. volts ' - I! I,
- - ' . 7'1 ..... 4....
51- 52 - 4x4 cm =------;/ I Y
‘1'- / I
, S1 /
L. ________
*---4 cm “"
+—-————- 6 cm ---—*
lst layer 2nd layer 3rd layer
5.4 5.5 3.9 5.2 4.8 1.9 5.5 5.4 2.2
98.0 95.0 5.5 121.4 120.8 4.8 150.2 154.6 5.4
101.0 98.0 5.4 121.1 121.4 5.2 144.1 1150.2 5.5
Absorbed power density (IIWatt/Kgm)
13.0 12.0 7.0 13.0 12.0 5.0 12.0 11.0 1.0 Ex
1.0 5.0 7.0 1.0 4.0 5.0 1.0 5.0 6.0 Ey
5.0 6.0 7.0 4.0 5.0 5.0 7.0 8.0 6.0 52
62.0 61.0 12.0 69.0 69.0 12.0 77.0 78.0 11.0
1.0 5.0 6.0 1.0 5.0 5.0 2.0 6.0 8.0
4.0 5.0 5.0 4.0 5.0 4.0 .0 6.0 1.0
63.0 62.0 13.0 69.0 69.0 13.0 75.0 77.0 12.0
1.0 4.0 5.0 1.0 4.0 4.0 1.0 5.0 7.0
1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0
components of the induced electric field ( Ex , Ey , EZ )(10mv/m) '

Figure 3.6. Distributions of SAR and induced electric field in one quarter of a

body maintained by a capacitor-plate applicator with electrodes of
the same size and one electrode grounded.

64

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3rd layer
Electrode-body gap = 2.5 mm x //'.// // 'ill’
cf: 80 “~an layer
a = 0,5 S/m 2: ‘1 \lst layer
f = 15 MHZ V+ /—-1-—""'-—-
= _ / '
V 2. volts / g}, [I
52 - 2x2 cm ......... 1‘ c» ...... I ..... y
s = 4x4 cm / A
1 S /
2..“4 ...... l
*---4 cm “"
4—-—-—-—- 6 cm ----'
lst layer 2nd layer 3rd layer
0.6 0.4 0.0 0.4 0.2 0.0 0.28 0.16 0.0
16.0 9.8 0.4 12.4 5.2 0.2 9.96 2.16 0.16
68.3 16.0 0.6 116.9 12.4 0.4 186.6 9.96 0.28
Absorbed power density (IJWatt/Kgm)
4.0 3.0 1.0 3.0 3.0 1 0 1.0 2.0 1.0.
0.8 0.9 0.2 0.8 0.9 0 2 1.0 1.0 0.1
1.0 1.0 0.2 2.0 0.7 0 2 2.0 1.0 0.1
24.0 19.0 3.0 21.0 14.0 3.0 17.0 9.0 2.0
0.6 0.0 1.0 0.6 0.1 0.7 0.9 0.5 1.0
5.0 0.0 0.9 5.0 0.1 0.9 8.0 0.5 1.0
51.0 24.0 4.0 67.0 21.0 3 0 85.0 17.0 1.0
3.0 5.0 1.0 4.0 5.0 2 0 4.0 8.0 2.0
3.0 0.6 0.8 4.0 0.6 0 8 4.0 0.9 1.0
‘ components of the induced electric field ( Ex , E.y , Ez )(10mv/m) ’

mmrn

Figure 3.7. Distributions of SAR and induced electric field in one quarter of a

body maintained by a capacitor-plate applicator with electrodes of
different sizes.

65

and the lower one is 4 X 4 cm and the potentials of the electrodes
are left floating. The interesting observation is that like in the
free space case, the intensity of the electric field is maximum near
the smaller electrode and decreases rather rapidly along the X-axis
towards the lower larger electrode. As a result the absorbed power is
maximum near the smaller electrode. Similar phenomenon occurs when the
lower larger electrode is grounded (Figure 3.8). In this case the
concentration of the power density near the smaller electrode becomes
even more significant compared to the floating potential case.

In Figure 3.9 a heterogeneous body with an embedded tumor is
placed between two electrodes of equal dimension (4 X 4 cm). The
conductivity of the tumor is assumed to be 0.35 S/m, and that of the
surrounding cells is 0.5 S/m. It is seen that the absorbed power in
the tumor is greater than that of the neighboring cells. On the other
hand, if the conductivity of the tumor is higher than that of the
surrounding cells, less power will be absorbed by the tumor. The
numerical results for this case are shown in Figure 3.10 where the

conductivity of the tumor is assumed to be 0.65 S/m.

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 3rd layer
Electrode-body gap = 2.5 mm x4 [I _Z/ // ‘lf”
6': 80 / / /// ““an layer
0': 0.5 S/m 4. 1:521 , 2 -c~.lst layer
f = 15 MHz V p /—-1--,'-—"'
= _ / ' l
V _2. volts .’ Ell //
S = 4x4 cm --------/-I F ------ I ----- Y
S = 2x2 cm ‘-' / l
_2 S /
1__L _____ /
*---4 cm'———’
+————-—- 6 cm ---—'
lst layer 2nd layer 3rd layer
0.9 0.4 0.4 0.6 0.3 0.2 0.6 0.2 0.2
10.4 4.4 0.4 9.2 2.6 0.3 10.0 1.5 0.2
60.1 10.4 0.9 ‘ 116.8 9.2 0.6 201.6 10.0 0.6
Absorbed power density (llwatt/Kgm)
4.0 3.0 2.0 2.0 2.0 1.0 0.9 0.9 0.1 Ex
1.0 2.0 2.0 1.0 2.0 2.0 1.0 3.0 2.0 Ey
4.0 1.0 2.0 3.0 0.8 2.0 4.0 1.0 2.0 E2
18.0 13.0 3.0 17.0 10.0 2.0 16.0 6.0 0.9
1.0 1.0 1.0 1.0 1.0 0.8 1.0 2.0‘ 1.0
7.0 1.0 2.0 7.0 1.0 2.0 10.0 2.0 3.0
48.0 18.0 4.0 68.0 17.0 2.0 89.0 16.0 0.9
4.0 7.0 4.0 4.0 7.0 3.0 5.0 10.0 4.0
4.0 1.0 1.0 4.0 1.0 1.0 5.0 1.0 1.0
components of the induced electric field ( Ex , Ey , EZ )(10mv/m) '

Figure 3.8. Distributions of SAR and induced electric field in one quarter of a
body maintained by a capacitor-plate applicator with electrodes of

different sizes and one electrode grounded.

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3rd layer
x
Electrode-body gap = 2.5 mm l/fi //«// "’/’
_ S 17' 77» // “—"’2nd layer
fl - 80 2 [/1 //
o = 0.5 S/m "/ 'iz ““-1st layer
= __ _:=‘/ll
f 15 MHz C5 / / {A ,l/ /, //V tumor
V = 2 V01tS /' ’7‘ I
at: 0.35 S/m ---"---j ...... ,{--.-L— Y
’ /
S = S = 4x4 cm / /
1 2 z’.__§.i_____/
*---4 cm ———'
+—-—-——- 6 cm ----'
Ist layer 2nd layer 3rd layer.

4.4 3.5 0.6 4.6 3.8 0.7 4.4 3.5 0.6
122.2 121.4 3.5 120.8 119.5 3.8 122.2 121.4 3.5
121.4 122.2 4.4 169.1 120.8 4.6 121.4 1122.2 4.4

Absorbed power density ( Watt/Kgm)
13.0 11.0 4.0 13.0 12.0 5.0 13.0 11.0 4.0

0.0 0.1 0.2 0.0 0.0 0.0 0.0 0.1 0.2

0.8 0.9 0.2 0.0 0.0 0.0 0.8 0.9 0.2

69.0 69.0 11.0 69.0 65.0 12.0 69.0 69.0 11.0
0.1 0.5 0.9 0.0 0.0 0.0 0.1 0.5 0.9
0.4 0.5 0.1 0.0 0.0 0.0 0.4 0-5 0.1

69.0 69.0 13.0 98.0 69.0 13.0 69.0 69.0 13.0

0.1 0.4 0.8 0.0 0.0 0.0 0.1 0.4 0.8

0.1 0.1 0.0 0.0 0.0 0.0 0.1 0.1 0.
components of the induced electric field ( Ex , Ey , EZ )(10mv/m) '

Figure 3.9. Distributions of SAR and induced electric field in one quarter of a
heterogeneous body, alower conductivity region at the center .

tained by a capacitor-plate applicator with electrodes of the same

size.

main-

68

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3rd layer
Electrode- body gap= 2. 5 mm ’3 l/zll/Z/Z/ "/”
e}: 80 ///52 L/ 2nd layer
0' = 0.5 S/m 1.". {/32 / \1st 101/EV
f = 15 MHz —-"--°~" -
l/ ; ’:‘/ i / fl tumor
V = 2. V01tS I. -; | // /
°t= 0.55 S/m --------/-/ =------;—----L- Y
I
S = S = 4x4 an l/ 51 I
1 2 z_ ________ /
*---4 cm “"
+———-——- 6 cm ----'
1st layer 2nd layer 3rd layer
4.4 3.5 0.6 4.6 3.8 0.7 4.4 3.5 0.6
122.2 121.4 3.5 120.8 119.4 3.8 122.2 121.4 3.5
121.8 122_2 4.4 93.8 120.8 4.5 121.8 122.2 4.4
Absorbed power density (n Watt/Kgm)
13.0 11.0 4.0 13.0 12.0 5. o 13.0 11.0 4.0 Ex
0.0 0.1 0.2 0.0 0.0 O. 0 0.0 0.1 0.2 Ey
0.8 0.9 0.2 0.0 0.0 0. 0 0.8 0.9 0.2 2
69.0 69.0 11.0 69.0 69.0 12. 0 69.0 69.0 12.0
0.1 0.5 0.9 0.0 0.0 0. O 0.1 0.5 0.9
0.4 0.5 0.1 0.0 0.0 0. 0 0.4 0.5 0.1
69.0 69.0 13.0 53.0 69.0 13. 0 69.0 69.0 13.0
0.1 0.4 0.8 0.0 0.0 0. 0 0.1 0.4 0.8
0.1 0.1 0.0 0.0 0.0 0. 0 0.1 0.1 0.0
components of the induced electric field ( Ex , Ey . Ez )(10mv/m) ’

Figure 3.10. Distributions of SAR and induced electric field in one quarter of a
heterogeneous body,a higher conductivity region at the center, mai-
ntained by a capacitor-plate applicator with electrodes of the same

size.

69

3.6 Synthesis of the Potential Distribution for Selective Heating
Theoretically it is possible to sythesize the voltage distri-
bution on the electrodes in such a way that the induced electric field
vanishes everywhere inside the body except in a specific cell. In
this section we attempt to find the scheme for such a synthesis.
A system of linear equations for the induced electric field
inside the body and the induced charge density on the electrodes was

developed in section 3.4. This is rewritten here for convenience:

 

 

 

 

 

 

( g 1 Ex 1 ' o
: ------
e = A EY .
a I": .
L ______ ,,,,, 1 ,,,,,,,, , ___z = (3.6.1)
5 s E 1 “1
c' 3 o 2 o 5 GS I ‘.’2
i i 5 . .“N'. IVN'I
s

The elements of the matrices G, G , A, C' were all defined earlier

in this chapter.

The equation (3.6.1) can be decomposed into the following two systems.

 

 

 

 

 

 

 

 

1' i F - ,. l
E _
X n] 1
G 0
EV + A ; = o (3.6.1.a)
Inl
l . E2. 1 1

'70

Where [G] is a 3NX3N matrix, and [A] is 3NXN' and

 

 

 

 

 

 

 

r A j,__A j Ex
' l ”1'5"" 3N Pv
: l 3 ”.1.“ } 1
N. c' 01 0; G's EZ J = Y2 N' (3.5.1.5)
E E 3 “,{m' i
. - = - .1 l W.
. 1N1
LnN'-

 

We demand that the induced electric field E(rh) = x1 V/m in the

mth body cell and zero elsewhere. Thus, (3.6.l.a) reduces to

 

 

NI
f 1 ..
. n] . G(m,l)
. G(m.2)
3N A z = . (3.5.2)
nN. .
L d“ ' Lem“)

 

 

 

 

In order that the equation (3.6.2) can be solved for n uniquely,
the number of equations must be equal to the number of unknowns. In
other words, the matrix [A] must be a square matrix. This is possible

only when
3N = N'

or the number of surface subareas is three times the number of body

cells. Satisfying this condition, we find from (3.6.2)

71

 

 

 

 

 

 

BM -1
71"1 r 1 '- G(m91) I}
I G(m,2)
3” ' = A ° (3.6.3)
“NI .
L _ L _ L(5(M,3N) .

once n's are determined from (3.6.3), the potential at any subarea

on the electrodes can be calculated from (3.6.l.b).

3.7 Numberical Results for Synthesized Voltage Distributions

Based on the discussion in the preceding section, the distri-
butions of potentials on the electrodes required to produce a localized
induced electric field inside the body is calculated for three cases,
and the results are presented here.

Figure 3.11 shows the geometry of a body with dimension
(6X6X9 cm) placed in between two electrodes of equal dimension
(4.8X4.8 cm). 0ne eighth of the body is divided into three layers with
four cells (1 cm3) in each layer. One quarter of the upper electrode
is divided into 6 rows each containing 6 subarea cells. The desired
patterns for the induced electric field in different layers of the body
are shown in the figure as well; where E = 1; in the cell located
at the center of the body and zero elsewhere. The distribution of the
potential on one quarter of the upper electrode which is required to
produce the desired pattern of the induced electric field is shown in

Figure 3.12. This figure indicates that the required voltages of

-72

 

7;— 3rd layer

/.<-— 2nd layer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
 

/‘— lst layer
f = 15 MHz //
a = 0.5 S/m : Y
€;= 80
E
U
0" ’P—"-.-—-7
/ I
I, I’
I, //
A. ...... z
*—"'6 cm ""
lst layer , 2nd layer 3rd layer
0 0 0 0 0 0
E in(v/m)
321 o o o o o .
row # 6
_row # 5
‘,r””’ row # 4
1/4 of electrode -.———— row # 3
row # 2
\ POW # 1

Figure 3.11.Geometry of body and a capacitor- plate applicator with
subsectioned electrodes for localized heating at the center

of the body. The distribution of required voltages is shown
in the next figure.

.xuoa 6:» co emacou msu pm mcwgmm;
umNFquo_ m :wmuno op mmuocpumpm ms“ :0 mommppo> cmcwacmc asp co mcovpznvcpmwo .NH.m mczmwd

 

73

 

 

 

 

 

 

 

 

Aguvmwxm _mcucmo mga soc» wucmamwu Aguvmwxm Pmcwcmu 65p Eocm mucmpmwu
¢.N ~.H o.o ¢.N N.H 0.0
a d d d H 4 a 1 u a 4 I 1 HI
- 111 ll. m in. . “In" .11" o
o .5591 .l u . 2x30.» 1 . ._ u .1 FILIIII
. [
.lllllk . H
m n m m 11111: . N-
. _ . u . n
. a u u n u .
u _ u m WIIII. filllla “ _ _ m
.1 .1 n . . _ t h m .. t" H .
:32) ..II. n . u .. N321 . a. M . ,. a o
. . . . . .
. n u . . n n m
n . n u u n . n 4
111.. "IL. .111. m n . N
- .
\ ..IIIIL L
. _
_ z 2
..Ill... 1111.... u .
. . . . . . . .
. . u _ . u . u .. I
n n n u n u _ n N
m m m u m m u u .
1 _ m u u m u _ m n . _ .
«fizocl _ m . M ” mfizocni.“ . m1 m m . o
u u u " rllll rlIIIL “ n . m .
m u n u m u u m
.IIIIIL . . . _ . .
. _ . . . _ 1 N
. . n n . n
. . . . .
. . _ - . . n
. u “ .IIII. L
. v

 

 

 

(salon AOL) 3691(0A pa1lddv

’74

neighboring sub-electrodes alternate their signs, and the amplitudes
of the required voltages are very high. It is also found that this
distribution of the required voltages is extremely unstable in that a
small error in the required voltages will lead to a quite different
pattern of induced electric field than that specified in Figure 3.11.
Figure 3.13 shows the case where a localized electric field
of 1% V/m is induced in the cell located at the centers of the body
surfaces which face the electrodes. The required potential distribution
on different rows of sub-electrodes is shown in Figures 3.l4(a) and
3.l4(b). Again, a voltage distribution of alternating signs between
neighboring sub-electrodes is required. However, the magnitude of the
required voltages is considerably smaller than the case of Figure 3.12.
Finally, Figure 3.15 shows the case when the induced electric field is
to be concentrated in the central column of the body along the X axis.

The amplitude and the phase distributions of the required potential

are shown in Figures3.16 and 3.17. The required potential distribution
for this case is considerably smaller in magnitude than the two previous
cases where a localized electric field is to be induced in an isolated
cell.

The theoretical results obtained in this section may not have
easy practical applications because the required voltage distribution
is of very high magnitude and with a rapid phase variation. However,
with the advent of computer technology, the implementation of such a

voltage distributions may not be a very difficult task.

 

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f..,zL 7Z—-3rd layer
~p————-2nd layer
x" +——~ lst layer
f = 15 MHZ 1 /’
a = 0.5 S/m 1:11-..- : Y
6,= 80
E
U
cn ,P"""‘7
I
I, I’
x’ //
4. ______ A
‘—“' 6 cm “"
lst layer ' 2nd layer 3rd layer
0 0 0 0 0 0
A E in (v/m)
0 0 0 0 1x 0 .
rOw # 6
.row # 5

1/4 of electrode

 

‘,,””’ row # 4

..—_————— row # 3
row # 2
. row # 1

Figure 3.13. Geometry ofiabody and a capacitor-plate applicator with
subsectioned electrodes for localized heating at the -

center 0f the body surface. The distribution of the
requ1red voltages on the electrodes is shown in next figures.

76

.mumwczm znoa 65p mo coucmu ms“ um
mcwumm; umNFquop a :Pmuno ou mmuocpumFm ms“ co mommpFo> umcwscmc mcp we mcowaznwcgmwo m.¢~.m m;:m_d

 

 

 

(salon SOL) 369310A paglddv

AEuVm_xm ngpcmu msu Eocm mucoumwu Asuvmwxm pmcucwu mnu soc» mucmpmwv
¢.N N.~ o.o ¢.N N.H o.o
. on
_ _ .
. . . u 1
u m .u u
.1... m u m m
u u _ n . . _llullu
" u n . u .u u u .
u u . u . . m u N
. . . . . .. I
m u n u . m m m u. a
u u m m u u u u u .
u _ . . . . m _ "
232.1 1.11. I. _ u .r :32.) m u u m (u o
n . u . . . . . .
..IIL. _ u u m n W n u .
. . u n n u u n
_ n u “ _IL " u A
u u . u u u
n u .. m “IIIL
" _ m .3
m n m .
m m ..L ..
. -

 

 

 

 

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nmNFquoP m :wmpao op monocpumpm on“ :o mommu_o> umcwscmc esp 4o mcowuzowcumwo .n.¢H.m «gnaw;

 

77

 

 

 

 

 

Asuvmwxm _mcucmu on“ soc» wucmpmwu AEuvmwxm Foggcmu on“ Eocm mocmum_w
¢.N ~.H o.o ¢.N N.H 0.0
m u .
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z n m m .1... . .-
n m n u u H
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u u u u m . N-
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m .
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omzoc r11111

 

 

 

 

($110A S01) 3691(0A pangdV

78

 

, Z rd layer

 

 

AH” 2nd la
yer

/

/

 

 

 

 

 

x" //‘"—“'1st layer
f = 15 MHZ & /’
rr = 0.5 S/m ng___ . Y
6?: 80
E
U
01 I,’-"""'-;’7
I I
l l
l’ //
l 4 ------ I

 

 

 

lst layer 1 2nd layer 3rd layer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0 0 0 _ 0‘ 0
£1 0 £1 0 £1 0
row # 6
,row # 5
‘,””” POW # 4
1/4 of electrode ..__————— row # 3
row # 2
“~“~‘N TOW # 1

Figure 3.15. Geometry of a body and capacitor-plate applicator with
subsectioned electrodes fOr localized heating at the -
central column of the body. The distribution of required
voltages on the electrode is shown in next figures.

-79

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A Eu Vmwxm _ccucmu ms“ soc; mucoumwc

A Eu vmwxm chpcmu map soc» mucmumwn

 

 

 

 

 

 

 

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u u
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oH

OH

OH

Ca

Ca

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“Euvmwxm —mcgcmu ms» soc» mucmumwn
¢.N

Aacvmwxo ngucmu ecu Eocw mucmumwu
~.F o e.~ N.p o

 

80

 

«*zoc

mfizog

mfizos

-----.--“

-.. - -.- .- -q
--------- --

1
l

J
l

-¢-------d
------o 01)]

.....-...1r

1

------P
------.

E

1
--.---..--1

-..---- - 0*

---~----(

1'

...-0...... 4

[

 

mwzog

Nfizoc

pkzoc

l
l

----- -- --J

l

l

----.--- - --dt

1

------- --1p

J

.-----------‘
-- -- -- --- -

l

1

(-----.--—

.----.---.
---—-----1

l.

l

l

.--------4

I

b--------

l

A

j

 

-----.- -d

[-

-.J-----.--

 

 

 

ompu

omp

ompu

omp

ompu

om—

(°Baa) atfiue aseud

81

3.8 Comparison of Numerical Results and Experimental Results

 

To confirm the theoretical results presented in section 3.6
a series of expreiments was conducted to measure the induced electric
field inside a simulated biological body, a volume of saline solution,
with an implantable electric field probe. Since the experimental set
up will be described in Chapter V, only the experimental results will
be used here to compare with the numerical results.

Figure 3.18 depicts a body of (8X8X4 cm) dimensions with a
conductivity of 0.5 S/m placed between two electrodes of equal dimen-
sion (4X4 cm). The spacing between the electrode and the body is
2.5 mm. A potential difference of 2. volts at 20 MHz is applied
between the electrodes. The distributions of the theoretical (solid
lines) and experimental (discrete points) values of the X-component of
the induced electric field inside the body along the Y-axis are shown
in Figure 3.18. On the same figure, the corresponding distributions
of the X-component of the electric field in free space along the Y-axis
are also included. It is seen that there is a good agreement between
theory and experiment.

In Figure 3.19 the electrodes of equal dimension (6X6 cm)
are placed across a body of dimensions (8X8X8 cm), and a conductivity
of 0.5 S/m. The applied voltage and frequency are the same as that in
Figure 3.18. The distributions of the theoretical and experimental
values of the X-component of the electric fields inside the body along

the X-axis are shown. The corresponding results for the free space

82

 

 

 

 

 

 

 

 

.53.]—
f = 20 MHz
4- 4cm "’
v = 2. volts
0 = 0.5 S/m DI XL. 65+ V
$1=S2 = 4x4 cm 1 y _
g _ 2.5 11111 .____81C111:f:g—+
o = 4.0 cm S1
theory Ex(v/m)
0 0 experiment

     

free space

 

inside the _
body
0 O

0.0

 

0
—4l
0

f(
(

 

-4.0

l
N
O
N
O
.b
0

Figure 3.18. Distributions of the theoretical and experimental values
of the X-component of electric field along Y axis maintained
in the body between two electrodesof equal dimension.

83

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f = 20 MHz ‘_—8 cm—_"
v = 2. volts x
6 = 0.5 S/m a) I (5+
n v
51 = $2 = 6x6 cm 3 -
g = 2.5 mm
theory ,
o 0 experiment Sl [ g
Ex(v/m)
free space 1
40- . \ o ‘40
O O
P \ . . . / ‘
10 . .10
I.
6 - ‘ 6
_ inside the
body
.- . . .1
2 . . 2. . 0 ’__ 2
0.0 2.0 8-0

4.0 6.
distance from S1 (cm

Figure 3.19. Distributions of the theoretical and experimental values of
the X-component of electric field along the X axis maintained

in the body between two electrodes of equal dimension.

84

case are also included in the same figures. The distributions of the
same quantities for'the case when the electrodes are of different
dimensions are shown in Figure 3.20. In both cases, Figures 3.19

and 3.20, a good agreement between theory and experiment is observed.

85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SL1—
f = 20 MHz
V = 2. volts 1 4 cm
0 = 0.5 S/m
S=6x6cm °° X +
1 g I C>_V
$2= 4x4 cm
9 = 2.5 mm 1
,
S1 9
theory

0 0 experiment

Ex(v/m)

80( 180

40 1 free space ‘ 40

10 4 10

8‘ d 8

4) . . ‘ 4
1n51de the
1 body
___0 o .42, o __£__————""”J"”’
. . . . . . . 2., x
0.0 2.0 4.0 6.0 8.0

distance from 51 (cm)

Figure3.20. Distributions of the theoretical and experimental values of
the X-component of electeric field along the X axis maintained
in the body between two electrodes<rfdifferent dimensions.

CHAPTER IV
LOCAL HEATING WITH HF MAGNETIC FIELD

Electromagnetic local heating of a biological body can be
accomplished with the application of an HF magnetic field. In fact
the utilization of an HF magnetic field provides an important advantage
over the use of an HF electric field for not overheating the region
under the skin. In this chapter, a pancake, or a current disk, which
produces an HF magnetic field for the purpose of FM local heating is

studied.

4.1. Introduction

 

The capacitor plate applicator used for local heating of a
biological body was discussed in detail in the preceding chapter.
This scheme suffers a major drawback of overheating the fat region
under the skin. This difficulty can be explained by considering a
biological body consisted of three different layers as shown in Figure
4.1. The layers 1 and 2 represent the skin and the fat layers, and
the layer 3 represnets the muscle.

From Maxwell's equations we have
v- (0 +1.. em?) = 0 (4.1.1)

where E(?) is the total electric field inside the body. Integrating
(4.1.1) over the volume V enclosed by the surface S as shown in

Figure 4.1, and applying the divergence theorem gives.

86

87

 

 

 

 

Electrode
0m. rm 1 Skin
Ef
. Fat
Of’erf 2 r---LQ-1
l l
: I
1 1
1. ...... a
E
m
Om’ 6rm 3 Muscle

 

 

+ + + + + + Electrode

Figure 4.1. A biological body consisted of skin, fat and muscle layers
placed between a pair of electrodes for the purpose of local
heating.

88

Em(om + jw Eoérm) = Ef(0f + jw eoerf) (4.1.2)

where the subscripts m and f stand for the muscle and the fat layers.
Thus, the ratio of the absorbed power density in the muscle layer to that

in the fat layer is given by

 

P E' 2 + ' E 6 2

_m.= Omlm| = Omlof 3” o rfl (4 1 2)

Pf 0115’)? 0) +j€€|2 "
f f f 0m w o rm

For example, at 27.12 MHz we have I 17]

Q
II

0.61 S/m, erm = 113

of 0.11 ~ 0.43 S/m, Erf = 20
therefore (4.1.2) gives

P

1.5 5 EE-
m

f 5.6

which implies that the density of the power absorbed in the fat layer is
between 1.5 to 5.6 times higher than that absorbed in the muscle layer.
Therefore, for the cases when the tumor is located inside the muscle
layer, a sever burn may occur in the fat layer before adequate heat can

be produced at the tumor.

4.2. Theoretical analysis

 

The geometry of the problem is shown in Figure 4.2. A circular
disk of radius 0 carrying a circulatory current is placed on the sur-
face of a body with conductivity 0(E), permittivity E(E), and per-
meability no. The density of the current on the disk is represented

by 4 K¢(p'). It is assumed that the geometry of the body and disk is

.89

0th ring on the disk

 
 
 

Fflfl”flfly,..— circular current disk

“‘"‘ bOdy (O: E ’11 )

 

 

 

 

4—— r 0
-‘ mth body ring
\ \ N- .l_,,9§§£1\
\ \‘-- _-.’l ’I -\ pr"
\“ - 12')” b
",::1::~¢.m

 

 

I I ...-it... g
'4‘ 1‘1” ( *N‘gu.‘ fl nth body ring
I\ s 4", ..I

I

 

 

 

Figure 4.2. Geometry of a circular disk carrying a ciculatory current
placed on a body for local heating.

90

rotationally symmetric, this implies that all quantities are independent

of the azimuthal angle 9.

4.2.1 Impressed electric field
The vector potential at any point E inside the body maintained
by the current is the absence of the body is
-.is R(‘F,’F')

 

“ x
11m = 41113 (P' 199(5)e ds' (4.2.1)

1+ 0

R( Jr")

where 'F and 'F' are the position vectors for the field and source
points, respectively, and Sd represents the surface of the disk. In

the cylindrical coordinate system, R(?,?') can be expressed as follows.
R(EQE') = [0'2 + 02 - 200' 005 0 + (z-L)2]L5 (4.2.2)

where p and p' are the radial distances of the field and source points
from the Z axis and L is the height of the body. Due to the symmetry
the field point is chosen at 4 = 0. The unit vector 4' can be written

in terms of the unit vectors 2 and y as
4' = - i sin 0‘ + 9 cos 0' (4.2.3)

with (4.2.3), (4.2.1) becomes

 

 

b 2." ‘jBoR(?,-F')
'A(p 2) = EQ'C-i I K (o')p'do' J sin 9' e d¢'
4" 0 'P 0 R(Ffr')
b 2 ‘JBORG3—F')
+ 9 I K¢(p')o'dp' I cos 4' e +.+ dw']
0 0 R(r,r'

91

The first term on the right hand side of (4.2.4) integrates to zero,
since the integrand of the inner integral is an odd function of 9'.
Thus, the impressed electric field 'E1 at any point is
. | . 1, 'e-JBORG‘IF') '
KW(0 )0 do J0 cos 9 R(F,F') d9
(4.2.5)

b

 

+' + .. jun
E1(p,Z) = 'jU-lA(D,Z) = " (P F—o‘ [0

4.2.2 Scattered electric field

 

As it is shown by (4.2.5), the impressed electric field is of the
circulatory type which induces an eddy current inside the body. The
induced current becomes a source for a secondary or the scattered electric
field (ES. To find 'E5 at any point inside the body the volume of the

body is divided into N rings. and the eddy current density inside the

nth ring is represented by 0". The vector potential at the center of
th . ‘+ .
the m ring due to an 15
A111" - a? [th 1]" T dV (4.2.6)
n ring
where

_ m 2 n 2 m n . m n 2 8
Rmn - [(ob) + (0b) - Zobpb cos 9 + (lb-lb) 1

so = 2 nf 70060, f is the operating frequency, and p: and 93 are

th and nth rings from the Z axis, re-

the radial distances of the m
spectively.

If the cross sections of the rings are small enough, the density
of the current Jn can be assumed to be constant over the cross section

and (4.2.6) reduces to

-A* _Auod 11
mn ‘ T'7F nsnprmn (4'2'7)
where
n e-Jso mn
H = J cos 9' d9'
mn 0 Rmn

and Sn is the cross section area of the nth ring.

The total vector potential at the center of the mth ring is
N N
"£1 "21°2fl J "S nprmn (4.2.8)

The vector potential at the center of a ring due to the current
in the same ring should be evaluated more carefully, since for this case
the integrand in (4.2.6) becomes infinite. It has been shown that
Afim can be expressed as [18 l

+ U

= “ _2.
Amm 9 2 J mS mprmm (4.2.9)
where
- 2“ W . -jBoRmm(0) I
Hmm - -S— [Rmm(a) - Rmm(0)] J cos 9 e d9
m 0
with a =/§?~, and a is the length of the side of the square cross

section of the ring, and
m c2 %
Rmm(c) = pb[2(1-COS cp)+ -(—p'b—)2—]
The induced current 3n is related to the total electric field

En by the following relation;

93

where Tn = (on + Jw(€rm - 60)) 15 the complex conduct1v1ty. Thus,

 

the scattered electric field 'E; at the center of the mth ring may be
written as

+5 - . + - A jwllo N n

Em - -JwAm - - 9 2 n2] SnprmnTnEn (4.2.10)

The total electric field at any point inside the body is the sum

of the impressed and scattered electric fields at that point.

+

_+i+s
Em - Em + Em (4.2.11)
substituting (4.2.10) into (4.2.11) gives
N n i
Em + n21 quOSnprmntnEn = Em (4.2.12)

m = l,2,...,N

Equation (4.2.12) can be transformed into the following matrix re-

 

 

 

 

 

 

presentation
)- N 1 p q
' 1
E1 E1 )
E 5‘
2 2
N M . = . (4.2.13)
1
L L E" . . EN -
where
.. - '1
Mnn _ 1 + quosnprnnTn
and

.- ' . n ..
an - Jf”osnprmn‘n

94

The value for the impressed electric field E1 maintained by

 

 

the current disk at the center of the mth body ring can be obtained from
(4.25) as.
E; = 51(95’25)
= jwuo )b K (0')o'dp' )fl Cos 9' e-j80[(pg)2+p'2'ngp' cos 2' + (Zg-L)ZJ% d9:
2" 0 9 0 [(pg)2+p'2-2pgp' cos 9' + (lg-HZ];E
(4.2.14)

To evaluate the integral in (4.2.14), the area of the disk is
divided into N' concentric surface rings, over the area of each ring

the density of the surface current is assumed to be constant. The im-

 

 

pressed electric field at the center of the mth body ring due to the
current flowing on the 0th surface ring can be written as
+1 _ . quo P
Emv - 9 2" KV v Avamv (4.2.15)
where
n ‘350[(RE)2 + oi-Zpgpv COS 9' + (lg-Hz];5
A = J cos 9' e d9'
2 2 . 2*
mv 0 [(03) + DV‘ZDEDv cos 9 + (ZS-L) 335
(4.2.16)

and Kv is the density of the current on the 0th ring of the disk.

th

pv and ApV are the radial distance and the width of the 0 surface

ring, respectively.
th

 

The total impressed electric field at the center of the m body
ring is
. N' . jwu N'
*1 = "’1 = A o
Em Z] Emv 9 2” 021 Kv pv Apv Amv (4.2.17)

95

Equation (4.1.17) can be written into the following matrix form.

 

 

 

 

 

 

 

' 51° 1 r 1
' 1
K1
N ° = P I (4.2.18)
°. K 1
1 1 N'
LEN f 1. -
where
quo
mv - 2n Kv pv Apv Amv

If the surface current K on the current disk is given, the
+° -
impressed electric field E1 is determined by (4.2.18). once E1 is

determined the induced electric field 'E can be obtained from (4.2.13).

96

4.3. Numerical results

 

The total induced electric field induced in a body for several
different cases is calculated numerically and the results are persented
in this section. Unless otherwise specified, in each case the operating
frequency is fixed at 100 MHz and the conductivity and relative per-
mittivity of the body are assumed to be 0.5 S/m, and 80, respectively.

In Figure 4.3 a disk of radius 2.cml carrying a uniform circu-
latory current with a density of 200 5%9- is placed on the surface of a
cylindrical body of 5 cm radius and 4. cm height. The body is divided
into 4 layers and each layer is divided into 5 rings. The distributions
for the amplitude and phase of the total electric field induced inside
each layer of the body are shown. The quasi-static approximations for
the phase and for the amplitude of the total electric field in the first
layer of the body are also included in the figure. It is seen that the
intensity of the electric field is zero at the center of the body as
expected. The total electric field obtained form the present numerical
method is close to the quasi-static solution, however, when the frequency
increases the discr pancy between the numerical solution and the quasi-
static solution becomes more significant. It is also noted that the
intensity of the electric field is high in the region of the body close
to the disk and drops significantly in both vertical and radial directions
as we move away from the disk.

In Figure 4.4 the distributions of the amplitude of the electric
field in different layers of a body are shown at 100, and 30 MHz. The
radius of the current disk is 3. cm and the body has a radius of 5 cm.
The current density on the disk is 200 Amp/m. At 100 MHz the intensity

of the field is considerably higher thanthat at 30 MHz. Comparing

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97

 

 

 

 

 

 

    

    
 

 

    

 

  

 

   

 

 

 

  
   

 

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99

Figures 4.3, and 4.4 we notice that for the same frequency and body
dimension, the amplitude of the induced electric field inside the body
increases considerably as the radius of the current disk increases.

In Figure 4.5 a current disk of radius 5 cm is located on the
surface of a body of l2 cm radius and l5. cm height. The body is divided
into 10 layers each containing 4 rings. The distributions of the
amplitude of the induced electric field in different layers are shown
in the same figure. The distributions of the phase angle of the induced
electric field in each layer is depicted in Figure 4.6.

Figure 4.7 shows the distributions of the amplitude of the total
electric field inside a body induced by three different kinds of current
distributions (l) a single current loop, (2) a uniform current distribution,
and (3) a triangular type of current distribution (zero at the edge of
the disk and increases linearly towards the center). The radius of the
body is 5 cm and its height is 4 cm. The distributions of the relative
amplitude of the electric field in the first two layers of the body are
depicted for each case. It is observed that for a single current loop
of radius 2.5 cm the maximum electric field occurs at a radial distance
equal to the radius of the current loop and the electric field decreases
rather rapidly on either side. When the current is distributed uniformly
over the surface of the disk the point of the maximum electric field
moves closer to the center of the body and the distribution of the elec-
ric field becomes more uniform. If the current on the disk has a tri-
angular form of distribution the peak of the electric field distribution
moves even closer to the center of the body as compared with the pre-

vious cases.

100

 

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102

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103

b

4.4. Synthesis of the current distributions for selective heating
In this section we attempt to develope a theoretical scheme for
synthesizing the distribution of the current on a disk in order to obtain
a desired heating pattern inside a biological body.
Suppose that it is desired to have an indicued electric field

th

of l v/m at the center of the m ring and zero electric field in the

rest of the body. From (4.2.13) and (4.2.18) we obtain

 

 

N' , .
r W M(m,1)
I K] " M(M,2)
N P I = ° (4.4.1)
K . .
. N . M(m.N)

 

 

 

 

It is required that the matrix P be a square matrix if K's are to

be determined uniquely, that is,

N = N' (4.4.2)

which means that the number of the rings in the body must be equal to
the number of the surface rings on the current disk.

Some numerical example are given here to illustrate this scheme.
In the first example a current disk of 3 cm radius is placed on the
surface of a body of l0 cm radius and 5 cm height. The body is divided
into a total of 25 rings. It is desired to produce an electric field
with an amplitude of l0 V/m in a ring of radius 0.5 cm which is located

at the center of the body, and zero in the remaining part of the body.

104

The distributions for the phase angle and the amplitude of the current
required to produce the desired pattern for the electric field are shown
in Figure 4.8. It is noted that the amplitude of the required current
density is extremely high at the center of the disk and it decreases
rapidly towards the edge of the disk. The phase angle for the required
current density undergoes rapid fluctuations from one surface ring to
another.

In the second example we consider a body of 15 cm radius and
4.5 cm height, and the radius of the current disk is chosen to be the
same as that of the body. The objective is to produce an induced elec-
tric field of amplitude 10 V/m which is uniformly distributed over the
volume of a ring in the first layer having an outer radius of 4.5 cm. The
distributions of the phase angle and the amplitude of the required current
density to produce the prescribed pattern are shown in Figure 4.9. It
is noted that the amplitude of the current is very much smaller as com-
pared to that obtained in the previous example.

It is found that like the case of voltage synthesis in a cap-
acitor plate applicator, the theoretical results are unstable and a
small error in the phase or the amplitude of the required current may
produce a distribution of electric field which is quite different from

the desired one.

-~l

)3] = 10 v/m
and zero elsewhere

II-6 cm-ol
8

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105
= 0.5 S/m

= lOO MHz
H

w m Hum-...“...H...“ H m. :H .......H......... m.

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E
r
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l :
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my! mm...H.-..H..H...:- .................. _

Ls..- . ”-...-d

Amplitude

 

 

P I I n P r

radial distance from the center of the disk (cm)
Distributions of phase and amplitude of the required current

density on a disk to maintain a localized heating at the

center of a body.

 

 

)

m

(

00000

105
105 .

0

1|. 9 8 7
0 0 0 0
1 9| 1. HI

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000
0.45..

(Deg)
8
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107

4.5 Comparison of theoretical results with experimental-results

The electric field induced inside a simulated biological body
(saline solution) by three kinds of current distributions was measured
by an implantable electric field probe. The measured values along with
the theoretical results in relative amplitude are shown for each case
in Figures 4.l0 to 4.l2. It is noted that due to the finite length
of the prob in each case the measured values deviate from the theoretical
values near the center, but for the region away from the origin the
agreement between experiment and theory is considered to be good. The
details of the experimental set up and the simulations for different

current distributions will be explained in Chapter V.

108

 

 

 

f = 15 MH
a = 3 cm
0 = 0.5 S/m
er = lOO
KCP ~ G(r-a)
E(v/m)
Relative
l .
0.8 ,
O O
0.6 -
O
0 4 b . . ' o
. . . . r’/’ lst layer
0.2 . o *,,.v 2nd layer
0
*-——-3rd layer
0 / n 1 1 n 1 _‘; r
l 2 3 4 5

radial distance from the center (cm)

Figure 4.l0. Distributions of the theoretical (solid lines) and experimental
values (discrete points) for the electric field in different
layers of a body induced by a single current loop.

 

109

m
(19

 

15 MHz
3 cm
hh-6 cm-* 0 = 0.5 S/m
Er = lOD
K ~ l
E(v/m) f
Relative
1. F
O
0.8 .. o
O
O
0.6 ' o
0.4 . o o o
O O
O
0.2 . , -' ' . ' ,,z’lst layer
0 ' ._——-2nd layer
+- - 3rd layer
0 l 2 3 4 5

Figure 4.ll.

radial distance from the center (cm)

Distributions of the theoretical (solid lines) and experimental
values (discrete points) for the electric field in different
layers of a body induced by a disk of uniform current.

 

llO

 

 

a
f = l5 MHz
a = 3 cm
a = 0.5 S/m
r = lOO r
Ke ~ (1 - 39
E(v/m)
Relative
l. .
O
0.8 . '
O
o
0.6 '
O
0.4 » ‘ '
O O .
o 2. ' . o . . o e”” lst layer
. ' ,___— 2nd layer
+————-3rd layer
0 l 2 3 4 5 r

Figure 4.l2.

radial distance from the center (cm)

Distributions of the theoretical (solid lines) and experimental
values (discrete points) for the electric field in different
layers of a body induced by a triangular type of current
distribution.

CHAPTER V
EXPERIMENTAL SETUP

In order to verify the theoretical results obtained in the
preceding chapters, a series of experiments was conducted where the
induced electric field inside a body maintained by a capacitor plate
applicator or a current disk was measured. In this chapter the descrip-

tions of the setups for such measurements are given.

5.l Construction of an Implantable Probe

 

The procedure involving the construction of an inexpensive
implantable probe has been given by K.M. Chen, et al. [19 J. ‘This probe
has been tested successfully and proven efficient in measuring the in-
duced electric field inside a phantom model of a biological body. A
brief description of the probe is presented in this section.

In Figure 5.l the schematic diagram of the probe is shown.

This probe consists of a conventional short electric dipole loaded with
a zero bias microwave diode (Microwave Associates, MA 40234). The probe
output is connected to the measuring device (d.c. voltmeter or SNRM)
with a pair of very thin resistive parallel wires. The resistive wires
are loaded with two series of resistors in the sections adjacent to the
diode. This scheme reduces the induced current in the lead wires, and
consequently, minimizes the noise in the measurment. The probe and the

lead wires are encased in a plexiglass stick with the help of epoxy glue.

lll

112

zero bias microwave diode
(microwave Associates)

MA 40234

receiving probe

   

._ resistor (3 K m

 

plexigleee___..
stick

 

 

 

 

 

Figurefil. A non-interferring. electric field probe for
ueeeuring the induced electric field in a biological
body.

34

113

The electric field in any direction at a given point can be

measured by orientating the probe parallel to that direction.

5.2. Construction of a balun.

 

A difficulty encountered in the measurment of the electric field
is associated with the direct connection of a coaxial line to a balanced
two-wire line. The problem can be explained by considering the situation
depicted in Figure 5.2.

In Figure 5.2 a coaxial transmission line is directly connected
to a two-wire line. Like in most practical applications the outer con-
ductor of the coaxial line is assumed to be grounded. Since the wires land
2 of the two-wire are line at different potentials with respect to ground,
the capacitances with respect to ground of the wires are different and
as a result the currents i1 and i2 in the wires are different. This
implies that a current id = iI-i2 flows on the outer surface of the
coaxial line.

The current on the two wire line in Figure 5.2a may be decomposed
into symmetric and antisymmetric modes as shown in Figure 5.2b. For the
symmetric mode, equal currents flow in opposite directions on the two
wires, while for antisymmetric mode, two currents of equal magnitude
flow in the same direction. The anitsymmetric component of the line
current does not energize the applicator but it produces a strong electric
field around the applicator and will lead to an ambiguous probe measur-
ment.

One way to eliminate the antisymmetric component of the current

is by making the impedance seen by id in Figure 5.2d very high and thus,

‘ll4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)
. . is
1 =11+12
s 2 ZL
-15
. . la
1 = 11-12 2
a 2 L
..).
1a

 

 

Figure 5.2. (a) direct connection of a coaxial line to a two wire line.
(b) decomposition of the current on the two-wire line into
symmetric and antisymmetric modes.

115

prevent the current from flowing on the outer surface of the coaxial
line. This can be accomplished by using a balance to unbalance convertor
or a balun for short. One such balun is shown in Figure 5.3 where a
coaxial line is forked into a pair, one of which is a dummy. The center
conductor of the coaxial line is connected to the shield of the dummy
coaxial line and the shields are joined to the paralle-wire line. The
coaxial line and the dummy are wound on a ferrite toroid to prevent a
short circuit at the input terminal of the paralled-wire and at the same
time to suppress the current flowing on the outer surface of the coaxial
line. The detailed theory and the structure for different kinds of

baluns are reported in many sources [20,21]-

5.3. Experimental setup for the measurement of the electric field in

 

a conducting medium maintained by a capacitor-plate applicator and probe-

 

electrode interaction.

 

The setup for the measurment of the electric field inside a
body maintained by a capacitor-plate applicator is shown in Figure 5.4.
An HF voltage, form an HF signal generator processed through an ampli-
fier is applied to a pair of capacitor plate placed across a box of
plexiglass filled with saline solution. A variable inductor is used
for the tunning purpose.

The equivalent circuit for an isolated probe is shown in Figure
5.5a where Zin is the input impedance of the short electric dipole

and Z is the impedance of the diode, V is the equivalent driving

L
voltage induced by the impressed electric field, and V0 is the output

116

balanced two-wire line

 
 
   

ferrite
toroid

 

. dummy coaxial
coax1al cable

cable

 

 

 

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-

 
  

 

 

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A balun for converting a coaxial line to a balanced

Figure 5.3.
two-wire line.

'117

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118

 

 

 

 

Z.
in fi+
+
v_ 2L vO
(a)
Zin
*r=+
+ I
V_ ZL V0
Zm

Equivalent curcuits for a probe. (a) an isolated probe

(b) a probe located close to a grounded plane.

.119

voltage of the probe. When the probe is located close to the surface

of the conducting electrode, the image effect due to the electrode should
be considered. The image effect of the electrode can be taken account
for by introducing an additional impedance Zm in the equivalent circuit
(Figure 5.5b). Therefore, in measuring the electric field between two
electrodes the effect of conducting electrodes on the performance of the
probe should be investigated and corrections should be made in the mea-
sured values of the electric field if necessary.

It is possible to obtain a rough estimate for Zm. To do this we
assume that the electrode is of infinite extent and held at zero potential.
We also neglect the effect of the lead wires. With these assumptions,
the conducting plane in Figure 5.5b can be replaced by an image dipole
as shown in Figure 5.6. The mutual impedance between the dipole (l)
and its image (2) can be obtained from the follwoing expression [22 J:

- 1 -h-d
zm - - 1.9-mam [M EZZ(z)I](z)dz (5.3.1)

 

where d is the distance from the center of the dipole to the conducting
plane and h is half of the dipole length. I](-d) and 12(d) are the
magnitudes of the currents at the centers of the dipole and its image,
respectively. E22 represents the 2 component of the electric field

at the surface of the dipole maintained by the current on the image dipole
(2), and finally 11(2) is the current distribution on the dipole.

Since h << A the currents 11(2) and 12(2) on the dipole and its

image can be approximated with triangular distributions of

(-leflL

C,( h ) (5.3.1)

120

Figure 5.6.

s
‘4

 

z
2
Image dipole
d
- --1----- - - - — ground
plane
The original
dipole
1

Distribution of the currents on a short electric dipole

and its image caused by the ground plane.

121

12(2) = 10(1- lad ) (5.3.2)

where

I](-d) = 12(d) = IC

The Z component of the electric field at the surface of the
dipole due to the current flowing on the image dipole can be obtained
from the following relation,

2

 

 

522(2) = - :43 3—2 [A2Z(z)] - jw A22(z)1 (5.3.3)
0 32
where
u I d+h ._
A2Z(z) = gfl° I (1 - 'Zh dl) 21-2 dz' (5.3.4)
d-h
-jBOIZ'-Z)
withthe assumption that e = l

Equation (5.3.4), after evaluating the integral, gives

_ o o _ g_ 5_ g__ z_ d+h-z
A22(1)" 47- E“ h + hW‘d-z-h + (1 + h 11”” d—z 3 (5°35)

 

 

with (5.3.5), (5.3.3) becomes

jwuolo h l

 

- ___ 1.
522(2) ‘ ‘ 25 [32 (d-z)(d+h-z)(d-h-z) + 2 A22] (5°3°6)
0
Then from (5.3.1) we obtain
2 - -h'd E ( )(1 lfill-)3 (5 3 7)
m "' h-d 22 Z ’ h 2 e o

It is traight forward but very tedious to evaluate the integral in

(5.3.7). However, it can be done numerically with computer. For

l22
h = 0.5 cm, d = 0.7 cm and f = l5 MHz we obtain
Zm = j 2800 Q

The input impedance of a short dipole receiving antenna is

well known [23] and is given by

 

coffin (2;) ~11
Z- = "j h
1n Bo 11
where to = 120n. For h = 0.5 cm, g.. 10 and f = 15 MHZ,
z. = - j 100 x 103a.

1n

Thus, it is obvious that Zm << Z.

1n' Consequenlty, the effect of the

ground plane on the performance of the probe can be neglected.

5.4. Experimental setup for the measurment of the electric field in

 

a conducting medium maintained by a current disk.

 

The experimental setup for this case is shown in Figure 5.7.
The capacitor C is connected in series with the current disk to create
a resonance in the circut. This allows the maximum current to flow in
the current disk. Two 259 resistors in the circuit are used to prevent
short circuit in” the output terminals of the amplifier.

In Figure 5.8 the experimental models for three kinds of current
distributions are shown. In Figure 5.8.a, a single current loop
consists of several turns of enameled copper wire. The total length
of the wire is chosen to be shorter than a wavelength to ensure a con-
stant current distribution along the loop. In Figure 5.8.b, a single

piece of enameled copper wire is wound into a number of circular loops

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Figure 5.8. Experimental models for three kinds of current distributions:
(a) single current loop (b) uniform current distribution
(c) triangular current distribution.

125

with equal spacing to approximate a uniform current distribution.
Finally the experimental model for a triangular current distribution

is shown in Figure 5.8.c where the spacing between two adjacent loops

is varied accordingly.

CHAPTER VI
A USER'S GUIDE TO COMPUTER PROGRAM USED TO
CALCULATE THE ELECTRIC FIELD INSIDE A BIOLOGICAL
BODY INDUCED BY A PAIR OF CAPACITOR PLATE
APPLICATOR

Part I of this chapter briefly explains the computer program used
to determine the electric field inside a biological body induced by a pair
of capacitor-plate applicator placed across that body. This program en-
ables one to evaluate the density of electric charge on the electrodes
as well.

The geometry of the problem is shown in Figure 6.1. One quarter
of the body is divided into a number of cubic cells. Similarly, one quarter
of each electrode is partitioned into subareas. The matrix representation
for a set of linear algebraic equations in terms of the unknowns induced
electric field at the center of each body cell, and the charge densities

on different subareas of electrodes was presented in Chapter III and is

rewritten here

 

 

 

 

 

' . -‘ ' ' ' 1
G : A E o
I = v (5.1)
---.----:----- ...--.-- .--..6 .....
c' ; GS n
L ' - - - L 1 .1)

 

Given the necessary data the program solves equation (6.l) for E and

n, for both floating and grounded potential cuses.

126

127

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.l. Geometry of a body located between two energized electrodes
for the purpose of local heating. The numbering order of k
of the body and electrodes are shown.

128

6.1 Description of input data files.

The symbolic name for the program is "FIELD" and the sequential
structure of the data files, the format specifications and the symbolic
names of the input variables used in the program are outlined in Table

6.l, and the information on each data file are explained below.

First data file - contains only one card which defines the following

variables.

"D" - The thickness of the body between electrodes in meter.

"DL"-Defines the electrode-body spacing in meter.

"V" - Shows the amplitude of the applied voltage in volts.

"SLP,SUP“ - Specify g of the dimensions of the lower and upper electrodes,
respectively in meter.

"NSL,NSU" - Define the number of partitions along the x-axis on k of the
lower and upper electrodes, respectively.

"V mode" - Shows the mode of applied voltage (floating or grounded).

Second data file - with only one card containing the following informations
"Comp" - Specifies the components of the induced electric field and may
have one of the following forms.
l'X--only" For x-component of the induced electric field. This code is
used when the other components of electric field are neglegible.
"X,Y,Z9 - Used when it is desired to compute all three components of induced
electric field.
"Q(j), j = l,4"-Is the symbolic name for quadrants.

"FMEG"-Reads the frequency of applied voltage in MHz.

129

Table 6.1. The symbolic names for the input variables and format speci-
cations used in program “FIELD".

File No. Card No. Columns Variable Name Format
1 1 l-lO D FlO.4
ll-20 DL FlO.4
21-30 V FlO.4
31-40 SLP FlO.4
41-50 SUP FlO.4
51-53 NSL I3
54-56 NSU 13
59-61 V mode A3
2 l 1-6 COMP A6
ll-13 O1 to 04 4A1
18-27 FMEG Fl0.0
3 1 1-2 NX 12
5-7 NY 12
11-12 N2 12
4 l-N l-lO AMX FlO.3
11-20 AMY FlO.3
21-30 AMZ FlO.3
31-40 RELPI FlO.3
41-50 SIGI FlO.3
51-60 DXCM FlO.3
61-70 DYCM FlO.3
71-80 DZCM FlO.3

130

Third data file - contains one card which specifies the number of cells

in X,Y,Z directions (NX, NY, NZ).

Fourth data file - contains as many cards as there are number of cells in
k of biological body, and on each card the following informations are

punched.

"AMX”, "AMY", ”AMZ" which correspond to the maximum boundaries of a
cell in the X,Y and Z directions in centimeter with reference to
the origin.

"RLEPI" and "SIGI" are the codes for relative dielectric constant and
conductivity (mho/m) of each cell.

"DXCM", "DYCM", "DZCM" are the dimension of the cell in X,Y and Z,
in centimeter.
This concludes the description of input data files. An example

is worked out in the next section to supplement this user's guide.

6.2 Numerical example In this example the electrodes are considered to
be of different sizes with lower electrode grounded. The lower electrode
3x3 cm and the upper electrode is 2.0x2.0 cm. A voltage of 2. volts

at 15 MHz is applied between electrodes. The numbering order of
electrodes subareas is presented in Figure 6.1. Based on the given in-

formations, the input data files have the following form.

131

 

01 003002 GRD*

File No. Informations on the file.
1 0.02 0.0025 2.0 0.0l5 0.
2 X,Y,Z. 1234 15.0
3 02 02 02
4.1 1.0 1.0 1.0 80.0 0.5 1
4.2 1.0 2.0 1.0 80.0 0.5 l
4.3 1.0 1.0 2.0 80.0 0.5 l
4.4 1.0 2.0 2.0 80.0 0.5 1
4.5 2.0 1.0 1.0 80.0 0.5 l
4.6 2.0 2.0 1.0 80.0 0.5 l
4.7 2.0 1.0 2.0 80.0 0.5 1
4.8 2.0 2.0 2.0 80.0 0.5 l
*GRD is the code for grounded potential case, for floating

case "FLT" should be used instead.

OOOOOOOO

1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
potential

It is noted that NX * NY * NZ is the number of cells in k of the body.

The numerical results and program listing are presented in the

following pages.

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gum—u: U¢<nomxatoaaou~ uccapuudu no wcwz¢czm u!» 2o mu_h~mzdc wozcru

PART II
A USER'S GUIDE TO COMPUTER PROGRAM
USED TO DETERMINE THE ELECTRIC FIELD

INSIDE A BIOLOGICAL BODY INDUCED BY A
CURRENT DISK APPLICATOR

Part II of Chapter VI describes the computer program used for
calculating the electric field inside a biological body, maintained by a
current disk placed on the surface of the body for the purpose of local
heating. This program is a modified version of the computer program

developed by Lee [l8].

6.3 Formulation of the problem

 

The geometry of problem is shown in Figure 6.2 where a current
disk of radius a is placed on the surface of a rotationally symmetric
biological body. In order to calculate the induced electric field inside
the body, the body is divided into a number of rings. Similary the area
of the current disk is divided into a number of surface rings. The induced
electric field at the center of each body ring can be obtained from the

following equation which was derived in Chapter IV.

_ N

 

 

 

 

 

 

141

142

a ‘ 1 current
disk

 

 

lst la er
\. y

 

\ I
1 \\.__ ______ ...—s", bOdy

e-—"'—""—'

KM / (0(F).6(F) .uo)

 

 

Figure 6.2. A circular current disk is placed on a body for the purpose
of local heating.

143

where N is the nubmer of body rings, and

I

 

 

 

 

 

 

N
- 11 '- ..
E-I -
[K]
= P
° LK'_
-é;- L .J N

with N' being the number of surface rings on the current disk. The

elements of M and P matrices were defined in Chapter IV.

6.4 Description of the Computer Program.
The program "EDDY" uses the following complex functions and

subroutines.

"LEEMAT" - Calculates the elements of [M] matrix
"HMAT" - Calculates the elements of [P] matrix
"FMMC" "FMMS" "FMNC" "FMNS" "F1" "F2" - are functions which determine

H A

mm’ mv°
"DCADRE" - is a subroutine for numerical integration called from the

the integrands of integrals Hmn’
computer library.
"CMATP" - subroutine which solves linear algebric equations by using

Gauss - Seidel method.

6.5 Structure of the input data files.
The structure of the data files and relating formats is presented

in Table 6.2, and the informations on each data file are explained here.

144

First data file - contains one card with the following informations.

 

"N"- defines the total number of body rings.
"NPLAT"— specifies the number of rings on the surface of current disk.
"DIA"-the diameter of the disk in meter.

"FREINM"-specifies the operating frequency in MHz.

Second data files - reads the density of the currents on different rings

 

on the disk.

Third data file - contains only one card which specifies the following

 

variables.

”NPAR" - for the integrals H H Amv’ the integration interval 00

mm’ mn’
and 180° will be divided into "NPAR" subinterval in order to
save the computational cost.

"AERR” - desired minimum absolute error for the numerical integration.

"RERR" - desired munimum relative error for the numerical integration.

Fourth data file - contains N cards each with the following information.

 

"XEND" "ZEND" correspond to the maximum boundaries (in cm) of a ring

corss - section in the X- and Z-direction with reference to the origin.

"XAA" "ZBB" - are the codes for the dimensions of the-ring cross-section
in X- and Z- direction (in cm).

"REP” "SIG" - define the dielectric constant and conductivity (S/m)
of each circular ring.
This concludes the specifications of theinput data for the program.

A numerical example is presented in the next section.

Table 6.2.

File No.
l

145

The symbolic names for the input variables and format

specifications used in program "EDDY"/

Card No.
l

1-N

Column

1-3
4-6
11-20
21-30

1-5

6-15
16-25

1-12
13-24
25-36
37-48
49-50
51-63

Variable Name

N

NPLAT

DIA

FREINM

CD(I)

1, NPLAT

NPAR
AERR
RERR
XEND
ZEND
XAA
ZBB
REP
SIG

Format

13

I3
F10.4
FlO.4

NPLAT (FlO.4)
15

F10.

F10.

F12.

F12.

F12.

F12.

F12.

F13.

mmmmmmmm

146

6.6 An example to use the program

As an example, we consider a body of 2.cm height and 4 cm radius.
The body is divided into two layers each containing 4 rings. The current
disk is considered to be of 5. cm diameter and is divided into 5 concentric

rings each having a current density of 200 Amp/m.
The sequential order of the input data files is as follows

File No. Information on the file

l 008005 0.05 100.0

2 200.0 200.0 200.0 200.0 200.0

3 18. 0.0 0.01
4.l 1.0 1.0 1.0 1.0 80.0 0.5
4.2 2.0 1.0 1.0 l.O 80.0 0.5
4.3 3.0 l.0 1.0 1.0 80.0 0.5
4.4 4.0 1.0 1.0 1.0 80.0 0.5
4.5 l.0 2.0 1.0 l.0 80.0 0.5
4.6 2.0 2.0 l.O 1.0 80.0 0.5
4.7 3.0 2.0 1.0 1.0 80.0 0.5
4.8 4.0 2.0 1.0 l.0 80.0 0.5

The following pages are devoted to numerical results and computer program

listing.

ollolSoDR ’QST

07/00/02

FTN 9.8055?

FNIIH”

"PT

TIHIS

PROGRAM EDDY

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CHAPTER VII
SUMMARY

A numerical method is developed in Chapter II for determining
the electric charge densities on the surfaces of two squares, parallel
electrodes of arbitrary dimensions and spacing, energized by a HF
voltage. The denisty of electric charge is calculated numerically by
the moment-method technique for several cases, and the accurate value
for capacitance between two electrodes is evaluated. The induced charge
is calculated for both floating and grounded excitations, and significantly
different distributions are observed. As expected, the charge densities
are nearly uniform near the center of the electrodes with their increase
with the well-known singularity near the edges.

Based on the calculated charge density the components of electric
field at various points in free space are calculated for a variety of
cases. It is noted that for two electrodes of equal size, separated
by a distance small compared to the dimensions of the electrodes, the
electric field between electrodes is uniform. In the case of two electrodes
of different sizes, the electric field is mostly concentrated near the
smaller electrode.

In the next two chapters the problem of local heating of a bio-
logical body with HF electric and magnetic fields is discussed. This

study was motivated by the fact that HF electromagnetic radiation has

155

156

been utilized to heat tumor-bearing biological bodies for the purpose
of hyperthermia cancer therapy.

A pair of energized electrodes can be used to induce heating in
biological bodies. This problem is analysed theoretically in Chapter III.
Two coupled integral equations for the unknowns total electric field
inside body and charge density on the electrodes are established and
solved numerically for different cases. The distribution of specific
absorption rate (SAR) of energy in the body is calculated for both
homogeneous and heterogeneous bodies. From numerical results it is
observed that power is absorbed mainly in that part of the body located
between electrodes. For a body with an embedded tumor, the magnitude
of absorbed power at the location Of tumor is a function of tumor con-
ductivity; less conductive tumors absorb more power than those with
higher conductivity.

In Chapter IV the shortcoming of a capacitor plate applicator,
relating to overheating of the fat layer in biological bodies, is discussed.
This difficulty can be eliminated by using a current disk applicator.

The problem of a current disk applicator is analyzed theoretically
in Chapter IV. The electric field induced inside a biological body,
having rotational symmetry, by a current disk placed on the surface of
that body is calculated. Numerical results are presented for several
cases.

It is found that the induced electric field is strong near the
surface of the body, in the vicinity of current disk, with approximately
equal amplitudes in fat and muscle layers near the interface. The

intensity of induced electric field decreases rapidly in the direction

157

away from the disk. This suggests that the current disk applicators are
more effective on tumors located near the surface of the body. For
tumors located deep inside the body capacitor plate applicators are
perferable.

It is possible to synthesize the voltage distribution on a
capacitor plate applicator and the current distribution on a current
disk applicator in order to obtain a localized heating pattern inside
a biological body. The theoretical procedures are presented in Chapters
III and IV, respectively.

To confirm the accuracy of the theory, a series of experiments
was conducted where the electric field induced inside a simulated bio-
logical body by different applicators was measured and was compared with
the theoretical values. In each case a good agreement between theory

and experiment was observed.

BIBLIOGRAPHY

10.

BIBLIOGRAPHY

H.D. Suit and M. Schayder "Hyerthermia: Potentia] as an antitumor
agent," Cancer, 34, 1, 122-129, 1974.

J. Overgaad, "Combined adriamycin and hyperthermia treatment of a
murine mammary carcinoma havivo,fl Cancer Research 36, 3077-3081,
1976.

 

J.E. Robinson, M.J. Nizenberg and H.A. McGready, "Radiation and
hypertherma1 response of norma1 tissue in situ," Radio1ogx,
113, 1, 195-198, 1974.

H.H. Leveen, S. Napnick, V. Piccone, G. Fa1k and N. Ahmed, "Tumor
eradication by radiofrequency therapy," JAMA, 230, 29, 2198-2200,
May 17, 1976.

J.A.G. Ho1t, "Effects of 434 MHz e1ectromagnetic waves on human
cancer," Presented at 1977 Internationa1 symposium on bio1ogica1
effects of e1ectromagnetic waves, Air1ie, Virginia, October 30 -
November 4, 1977.

P. Antich, N. Tokita, J. Kim and E. Hahn, "Se1ective heating of
cutaneous human tumors," Presented at 1977 Internationa1 Micro-
wave symposioum, San Diego, Ca1ifornia, June 21-23, 1977.

N.T. Jaines, R.U. Thomas, K. Noe11, L.S. Mi11er and K.T. Noodward,
"Techniques and resu1ts of using microwaves and X-rays for the
treatment of tumors in man," Presented at 1977 Internationa1
Symposioum on bio1ogica1 effects of e1ectromagnetic waves,
Air1ie, Virginia, October 30 - November 4, 1977.

J.A. Dickson "Destruction of so1id tumors by heating with radio-
frequency energy," presented at 1977 Internationa1 symposioum
on bio1ogica1 effects of e1ectromagnetic waves, Air1ie, Virginia,
October 30 - November 4, 1977.

0.8. Marmor, N. Hahn and 6.”. Hahn "Tumor cure and ce11 surviva1
after 1oca1ized radio-frequency heating" Cancer Research, 37,
879-883, March 1977.

 

R. Baker, V. Smith, L. Kobe, T. Phi11ips, "A system for deveioping
microwave induced hyperthermia in sma11 anima1s" Presented at
1977 Internationa1 Microwave Symposioum, San Diego, Ca1ifornia,
June 21-23, 1977.

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12.

13.

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159

A. Cheung, D.McCu11och, J. Robinson and G. Samaras, "Bo1using

C.C.

S.B.

techniques for batch microwave irradiation of tumors in far
fie1d," Presented at 1977 Internationa1 Microwave Symosiqum,
San Diego, Ca1ifornia, June 21-23, 1977.

Johnson, H. P1enk and C.H. Durney, "A 915 MHz exposure system
for hyperthermia treatment of cancer," Presented at 1977
Internationa1 Symposioum on bio1ogica1 effects of e1ectromagnetic
waves, Air1ie, Virginia, October 30 - Novermber 4, 1977.

Fie1d and J.w. Hand "The response of pig skin to combined
X-irradiation and microwave heating," Presented at 1977 Inter-
nationa1 symposioum on bio1ogica1 effects of e1ectromagnetic
waves, Air1ie, Virginia, October 30 - November 4, 1977.

Harrington, R.F. Fie1d Computation by Moment Methods, Ch. 1, Macmi11an,

Van

K.M.

R.S.

J.H.

K.M.

1968.

B1ade1, J., "Some remarks on Green's dyadic for infinite space,"
IRE Trans. Antennas Propagat., V01. AP—9, PP. 563—566 November
1961.

 

Chen, "Interaction of e1ectromagnetic fie1ds with bio1ogica1
bodies" Research Topics in E1ectomagnetic Nave Theory, Edited
by J.A. Kong, Chapter 13, pp. 290-347, John Ni1ey and Sons, 1981.

 

E11iott, H.H. Harrison and F.K. Storm, "Hyperthermia:
E1ectromagnetic Heating of Deep-Seated Tumors," IEEE Trans-
action on biomedica1 engineering, V01. BME-29, No. 1, pp. 61-64
January 1982.

 

 

Lee, "Induced e1ectromagnetic fie1ds in conducting bodies by
magnetic fie1ds and microwave," Ph.D. Dissertation, Mich.
State Univ. East Lansing, MI 1977.

 

Chen, S. Rukspo11muang, D.P. Nyquist, "Measurment of Induced
E1ectric Fie1ds in a Phantom Mode1 of Man" To appear in Radio
Science, 1982.

”The Radio Amateurs Handbook" pub1ished by the American Radio

 

W.L.
E.C.

G.S.

Re1ay League, 1981.
Weeks, "Antenna Engineering”, pp. 167-179, McGraw-Ha11, 1968.

 

Jordan, "E1ectromagnetic Waves and Radiating Systems”, Ch. 11,
Prentice-Ha11, Inc. 1950.

 

Smith "Ana1ysis of Miniture E1ectric Fie1d Probes with Resistive
Transmission Lines", IEEE Transaction on Microwave theory and
techniques, V01. MTT-29, No. 11, pp. 1213-1224, November 1981.

          
 
      

     
 

     

 

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