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INTERACTION OF ELECTROMAGNETIC FIELDS

WITH HETEROGENEOUS BIOLOGICAL SYSTEMS

by

Sutus Rukspollmuang

A DISSERTATION

Submitted to
Michigan State University
in partial fu1fillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering and Systems Science

1979

ABSTRACT

INTERACTION OF ELECTROMAGNETIC FIELDS

WITH HETEROGENEOUS BIOLOGICAL SYSTEMS

by

Sutus Rukspollmuang

This thesis presents the theoretical and experimental results
of the induced electric field inside a biological system when it is
irradiated by a non-ionized electromagnetic radiation. This study
was conducted because of the need of quantifying the induced EM
field in a biological body in the study of potential EM radiation
hazards and in the biomedical applications involving EM radiation.

A numerical method based on a tensor integral equation is
briefly outlined. The accuracy of this numerical method is verified
by the exact solution of Mie theory for the induced EM heating
inside the homogeneous spherical models of human and animal heads.
The numerical method is also used to determine the induced EM
heating in a realistic model of human or animal head that consists
of a brain of realistic shape and eyes surrounded by a bony
structure.

The induced electric fields in irradiated, electrically small
cubes filled with phantom material were measured by an electric
field probe. The measured results were in good agreement with

theoretical results obtained from the tensor integral equation

method. An implantable electric field probe with an interference-
free wire system was constructed at a nominal cost for the purpose
of measuring the induced electric fields in a phantomnmodel when it
is irradiated by EM waves of various frequencies. A phantom model
of man which was constructed with thin plexiglass filled with
phantom material, was irradiated by 500 to 3000 MHz EM waves in a
microwave anechoic chamber. The distribution of the measured
electric field was compared with the distribution of theoretical
results obtained numerically from the tensor integral equation
method. A quanlitative agreement was obtained between experiment
and theory.

A study has been conducted to investigate effective methods of
inducing hyperthermia in the tumors embedded in animal and human
bodies by ultilizing EM fields. The distributions of SARs in
biological bodies with embedded tumors induced by various EM fields
are theoretically quantified to assess the effectiveness of various
local EM heating schemes.

The tensor integral equation method is combined with an
iteration process to provide a scheme that extends the tensor
integral equation method to handle a body consisting of'a very large
number of cells, while sidestepping the problem of computer storage
limitation. In some medical applications, a local part of a
biological body is magnetized and irradiated by an EM field. To
analyze such a body the existing tensor integral equation method is
generalized to handle a body with an arbitrary permeability in
addition to arbitrary conductivity and permittivity. In addition,
three computer programs used in this study are described along with

their instructions and the program listings.

DEDICATION

"To my parents, Dr. Natee and Suparb Rukspollmuang,

my wife Chanita, and my son Davin."

ii

ACKNOWLEDGEMENTS

I wish to express my appreciation to my major professor, Dr.
K.M. Chen, for his guidance, encouragement and invaluable advice
throughout the course of this work. I also wish to thank the other
members of my guidance committee, Dr. D.P. Nyquist, Dr. J. Asmussen,
Dr. B. Ho and Dr. J. Shapiro, for their time and assistance.

This research was supported by the National Science Foundation
under Grant ENG 74-12603, and in part, by U.S. Army Research Office
under Grant DAAG 29-76-6-0201.

I would like to express my deep appreciation to my parents
and to my wife, Chanita, for their support, understanding and
encouragement during my graduate study. I sincerely thank Joanna

Gruber who has done a great job typing this thesis.

iii

TABLE OF CONTENTS

LIST OF TABLES O O O O O O O O O I O I O O I O O O O 0

LIST OF FIGURES ' I I O O O O O O O O O I O O O O I

INTRODUCT I ON I O O O O O O O O O O O O O O O O O O 0

CHAPTER 2 REVIEW OF TENSOR INTEGRAL EQUATION METHOD

2.1

2.2

2.3
2.4

CHAPTER 3

3.1
3.2
3.3

3.4

CHAPTER 4

4.1
4.2
4.3
4.4
CHAPTER 5
5.1
5.2

5.3

Description of problem . . . . . . . . . . .

Tensor Integral Equation for the Induced
Electric Field . . . . . . . . . . . . . . .

Moment Solution of Tensor Integral Equation
Calculation of Matrix Elements . . . . . . .

INDUCED EM FIELDS IN SPHERICAL BODIES...
HUMAN AND ANIMAL HEADS . . . . . . . . . .

The Mie Theory 0 I O O O O O O O O O O O O O
Formulation of the Problem . . . . . . . .
Induced EM Field in Homogeneous Spheres . .

Induced EM Heating in Realistic Models of
Human and Animal Heads . . . . . . . . .

INDUCED EM FIELDS INSIDE THE CUBICAL BODIES
EXPERIMENTAL VERIFICATION . . . . . . . . .

Experimental Set Up . . . . . . . .
Construction of Probe . . . . . . . . . . .
Theoretical and Experimental Results . . . .
Summary . . . . . . . . . . . . . . . . . .
INDUCED EM FIELDS INSIDE HUMAN BODIES . . .
Experimental Set Up . . . . . . . . . . . .
Theoretical and Experimental Results . .

DISCUSSIOH e e e e e e o o o o e o e e 0

iv

vii

viii

10

l3
l3
14

17

33

44
44
45
49
56
60
61
61

67

CHAPTER 6

CHAPTER 7

CHAPTER 8

CHAPTER 9

INDUCED EM FIELDS IN HETEROGENEOUS BIOLOGICAL
SYSTEM AND APPLICATION TO HYPERTHERMIA CANCER
THERAPY ’ C C I O C O I O O O O O O O O C C O C

6.1 Comparison of Experiment and Theory in a
Heterogeneous Biological System . . . . .

6.2 Hyperthermia in Animal and Human Bodies
Induced by EM Fields . . . . . . . . . .

16.3 Theoretical Model of a Biological Body

W1 th Tumor O O O O O O O O O O O O I I I
6.4 Part-Body Irradiation with HF Electric Field
6.5 Hyperthermia with Microwave or UHF Irradiation
TENSOR INTEGRAL EQUATION METHOD COMBINED WITH
ITERATION TECHNIQUE FOR QUANTIFYING INDUCED
EM FIELD IN BIOLOGICAL SYSTEM . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . .
7.2 Theoretical Development . . . . . . . . .

7.3 Example . . . . . . . . . . . . . . . .

GENERALIZED TENSOR INTEGRAL EQUATION METHOD FOR
BODIES WITH ARBITRARY ELECTRICAL PARAMETERS

8.1 Introduction . . . . . . . . . . . . . .
8.2 Theoretical Development . . . . . . . . .
8.3 Example . . . . . . . . . . . . . . .
A USER'S GUIDE TO COMPUTER PROGRAM FOR INDUCED
ELECTRIC FIELD INSIDE AN ARBITRARILY SHAPED,
FINITELY CONDUCTING BIOLOGICAL BODY . . . . .
Part I PROGRAM FIELDS . . . . . . . . . . . .
9.1 Description of the program . . . . .
9.2 Data Structure and Input Variables

9.3 Description.of the Input Variables .

9.4 How to Use the Program . . . . . . .

76

76

81

83
83

91

99
99
99
106

110
110
110

124

132
132
132
134
134

138

Part II PROGRAM ITERATE

9.5
9.6
9.7
9.8

9.9

Formulation of the Problem

Description of Computer Program .

Structure of the Data File

An Example to use the Program .

Printed Output

Part III PROGRAM EMFIELD

BIBLIOGRAPHY

9.10 Formulation of the Problem
9.11 Description of the Computer Program .

9.12 Structure of the Data File and Input

Variables

9.13 An Example to Use the Program .

9.14 Printed Output

vi

151
151
152
152
157
159
170
170

170

171
175
175

189

3.1

9.1

9.2

9.3

LIST OF TABLES

Comparisons of numerical results on the average and
maximum beatings calculated from the numerical method
and that obtained from the exact solution of Mie theory
(incident power density = l mW/cmz) . . . . . . . . .

The symbolic names of input variables and corresponding
specifications for the data files used in the data
structure for the program "FIELDS" . . . . . . . . .

The symbolic names of input variables and corresponding
specifications for the data files used in the data
structure for the program "ITERATE" . . . . . . . . .

The symbolic names of input variables and corresponding

specifications for the data files used in the data
structure for the program "EMFIELD" . . . . . . . .

vii

32

136

155

173

3.4(a)

3.4(b)

Distributions of heating along the X, Y and Z
axes of a "cubic spherical" brain of 7 cm radius
induced by a plane EM wave of 918 MHz propagat-
ing in the + Z direction with a power density

of 1 mW/cm . Electrical properties of the brain,
the average and maximum beatings are shown. One
eighth of the "cubic sphere" is constructed with
40 cubic cells.

Distributions of heating along the X, Y and Z
axes of a "cubic spherical" brain of 7 cm radius
induced by a plane EM wave of 918 MHz propagat—
ing in he + Z direction with a power density of
l mW/cm . Electrical properties of the brain,
the average and maximum beatings are shown. One
eighth of the "cubic sphere" is constructed with
73 cubic cells.

3.4(c) Distributions of heating along the X, Y and Z axes

3.5(a)

3.5(b)

of a spherical brain of 7 cm radius induced by a
plane EM wave of 918 MHz propagating in t e + Z
direction with a power density of 1 mW/cm .
Electrical properties of the brain, the average
and maximum beatings are shown. Numerical results
are obtained from the exact solution of Mie theory.

Distributions of heating along the X, Y and Z axes
of a "cubic spherical” brain of 7 cm radius in-
duced by a plane EM wave of 2450 MHz propagating
in the 5 Z direction with a power density of

1 mW/cm . Electrical properties of the brain,

the average and maximum beatings are shown. One
eighth of the "cubic sphere" is constructed with
40 cubic cells.

Distributions of heating along the X, Y and Z axes
of a "cubic spherical" brain of 7 cm radius, in-
duced by a plane EM wave of 2450 MHz propagating

in tb + Z direction with a power density of l
mW/cm.. Electrical properties of the brain, the
average and maximum beatings are shown. One eighth
of the "cubic sphere" is constructed with 73

cubic cells.

ix

24

25

26

28

29

3.5(c)

3.6(a)

3.6(b)

3.7(a)

3.7(b)

3.8(a)

3.8(b)

4.1

4.2

4.3

Distribution of heating along the X, Y and Z axes
of a spherical brain of 7 cm radius induced by a
plane EM wave of 2450 MHz propagating in Ehe +'Z
direction with a power density of l mW/cm .
Electrical properties of the brain, the average
and maximum beatings are shown. Numerical results
are obtained from the exact solution of Mie theory.

Distribution of SARs inside a human head induced
by a plane EM wave of 918 MHz with a vertically
polarized electric field of l V/m.

Distribution of SARs inside a human brain, without
the surrounding bony structure, induced by a plane
EM wave of 918 MHz with a vertically polarized
electric field of 1 V/m.

Distribution of SARs inside a human head induced
by a plane EM wave of 2450 MHz with a vertically
polarized electric field of 1 V/m.

Distribution of SARs inside a human brain, without
the surrounding bony structure, induced by a plane
EM wave of 2450 MHz with a vertically polarized
electric field of 1 V/m.

Distribution of SARs inside an animal bead induced
by a plane EM wave of 2450 MHz with a vertically
polarized electric field of l V/m.

Distribution of SARs inside an animal brain, with—
out tbe surrounding bony structure, induced by a
plane EM wave of 2450 MHz with a vertically polarized
electric field of 1 V/m.

The schematic diagram of the experimental setup
for the measurement of induced electric field
inside the phantom model.

(a) An implantable electric field probe immersed
inside a conducting body.

(b) An implantable electric field probe with
interference—free lead wires.

A cubical phantom model is illuminated by an EM
wave. One eighth of the cube is divided into
27 cubic cells.

30

36

37

38

40

41

42

47

48

51

4.4

4.5

4.6

4.7

4.8

5.1

5.2

5.3

Theoretical and experimental results of the
chomponents of the induced electric field, E ,
along the Z axes of the 2 cm cube, placed at

the locations of a maximum electric field and a
maximum magnetic field of a 750 MHz, EM standing
wave.

Theoretical and experimental results of the
X-components of the induced electric field, E ,
along the Z axes of the 4 cm cube, placed at the
locations of a maximum electric field and a
maximum magnetic field of a 750 MHz, EM standing
wave.

Theoretical and experimental results of the
X-components oftflmzinduced electric fields, Ex’
along the Z axes of the 2 cm cube, placed at the
locations of a maximum electric field and a max-
imum magnetic field of a 750 MHz, EM standing
wave.

Theoretical and experimental results of the
X-components of the induced electric fields, E ,
along the Z axes of the 2 cm cube, placed at
the location of a maximum electric field and a
maximum magnetic field of a 1 GHz, EM standing
wave.

Theoretical and experimental results of the
chomponents of the induced electric field, E ,
along the Z axes of the 4 cm cube, placed at

the locations of a maximum electric field and

a maximum magnetic field of a 1 GHz, EM standing
wave.

Experimental setup for measuring the induced
electric field inside a phantom model of human
body.

Geometry and dimensions of the phantom model of
man.

Relative distribution of the X—components of
the measured induced electric fields inside the
phantom model of man, excited by a vertically
polarized, travelling EM wave of 2500 MHz at
normal incidence.

xi

52

54

55

57

58

62

63

65

5.4

5.5

5.6

5.7

5.8

5.9

5.10

6.1

Relative distribution of the chomponents of
theoretical induced electric fields inside the
phantom model of man, excited by a vertically
polarized, travelling EM wave of 2500 MHz at
normal incidence. (lO4-cell model)

Relative distribution of the X-components of the
measured induced electric fields inside the phan-

tom model of man, excited by a vertically polarized,

travelling EM wave of 2000 MHz at normal incidence.

Relative distribution of the X-components of
theoretical induced electric fields inside the
phantom model of man, excited by a vertically
polarized, travelling EM wave of 2000 MHz at
normal incidence (104-ce11 model).

Relative distribution of the X—components of the
measured induced electric fields inside the
phantom model of man, excited by a vertically
polarized, travelling EM wave of 500 MHz at
normal incidence.

Relative distribution of the X-components of
theoretical induced electric fields inside the
phantom model of man, excited by a vertically
polarized, travelling EM wave of 500 MHz at
normal incidence (246-ce11 model).

Distribution of the X—component of the induced

electric fields (the electric mode) excited by
a symmetrically impressed electric field in a

4 cm phantom cube, and the comparison of numer—
ical results based on the 216 cell subdivision
and the 512 cell subdivision.

Distribution of the X-component of the induced
electric fields (magnetic mode) excited by an
antisymmetrically impressed electric field in a

4 cm phantom cube, and the comparison of numerical
results based on the 216 cell subdivision and the
512 cell subdivision.

Geometry and dimensions of the phantom model.

xii

66

68

69

70

71

73

74

77

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

‘ 6.10

Theoretical and experimental results of the
X—components of the induced electric field, Ex,

at the tumor as a function of the tumor conduc—
tivity, excited by a vertically polarized,
travelling EM wave of 600 MHz at normal incidence.

Theoretical and experimental results of the
X-components of the induced electric field, E ,

at the tumor as a function of the tumor conducfivity,
excited by a vertically polarized, travelling EM
wave of 600 MHz at normal incidence.

A simulated biological body (6x6x12 cm) with an
embedded tumor ( x2x4 cm) irradiated by a uniform
electric field (E1) of l V/m (max. value) at
frequency (f) across the top and bottom of the
body and over the area of the tumor. The uniform
electric field is maintained by two capacitor-
plate electrodes.

Distribution of SARs inside a simulated biological
body of Fig. 6.4 when f = 15 MHz, E1 = l V/m x,
o =.O.62 S/m, 0t = 0.31 S/m, and cr = 150.

SAR in the tumor and in the neighboring cells
varying as a function of the tumor conductivity
(0 ) for the case of f = 15 MHz, ii = 1 V/m x,
o = 0.62 S/m and cr = 150.

Distribution of SARs inside a human body with an
embedded tumor when f = 15 MHz, E1 = 1 V/m 2,

o = 0.62 S/m, o = 0.31 S/m and e = 150. The
SARs in the tumor for the cases of o = 0.62

S/m and at = 1.24 S/m are also given.

Distribution of SARs inside the simulated body of
Fig. 6.4, but with the tumor located at the body
surface. Parameters are: f = 15 MHz, E1 = 1 Vm 2,
o = 0.62 S/m, at = 0.31 S/m and at = 150.

SARs in the surface tumor and in the neighboring
cells varying as functions of the tumor conductiv-
ity (at). Other parameters are: f = 15 MHz, E1 =
1 V/m a, o = 0.62 S/m, and Er = 150.

The simulated body of Fig. 6.4 with a surface
tumor is irradiated by a microwave in a waveguide.

xiii

79

8O

85

87

88

90

92

93

94

6.11

6.12

7.1

7.2

7.3

8.1

8.2

8.3

Distribution of SARs inside the simulated body
of Fig. 6.10. Parameters are: f = 2.45 GHz,

E a 1 vlmy, o = 2.21 S/m, ot a 1.1 S/m and

er = 47. The SARs in the tumor for the cases of
(It = 2.21 S/m and ot = 4.42 S/m are also given.

Distribution of SARs inside the simulated body of
Fig. 6.4 with a surface tumor under the whole-
body irradiation. Parameters are: f = 600 MHz,

E1 = 1 e-jkz V/m 2, o = 1.48 S/m, o = 0.74 S/m
and 5r = 53. The SARs in the tumor for the cases
of ct = 1.48 S/m and at = 2.96 S/m are also given.
An irradiated body is subdivided into N cells, and
each of the N cells is then subdivided again into
8 subcells.

X components of induced electric fields inside a
muscle layer of 4x2x0.5 cm irradiated by an 1 GHz
EM wave, numerically computed when 1/4 of the body
is subdivided into 8 cells, 64 cells, and 8 cells
with iteration process.

Distributions of the X components of induced
electric fields inside a muscle layer of 4x2x0.5

cm irradiated by an 1 GHz EM wave, numerically
determined when 1/4 of the body is subdivided into

8 cells, 64 cells and 8 cells with iteration process.

A simulated muscle layer (12x2x12 cm) with a
magnetized central part irradiated by a uniform
field (Hi) of 1 A/m (max. value) at 30 MHz in
Y—direction. The body is divided into 36 of

2 cm cubic cells.

The absorbed power density in cell A varying as
a function of relative permeability of the mag—
netized region inside a muscle layer (8x2x8 cm)
for the case of frequency = 30 MHz, H1 = 1 A/m
9, o = 0.62 S/m and Er = 150.

Distribution of induced electric fields (or currents)
inside a muscle layer (8x2x8 cm) with a magnetized
region (shaded region, u = 10) when frequency =

30 MHz, Hi = 1 A/m y, o E 0.62 S/m and er = 150.

xiv

95

97

101

107

108

125

126

127

8.4

8.5

9.1

9.2

9.3

The absorbed power densities in cells A and B
varying as functions of the relative permeability
of magnetized region (shaded region) inside a
muscle layer (12x2x12 cm) for the case of frequency
= 30 MHz, Hi = 1 A/m y, o = 0.62 S/m and er = 150.

Distribution of induced electric fields (or currents)
inside a muscle layer (12x2x12 cm) with magnetized
region (s aded region, u = 10) when frequency =

30 MHz, H = 1 A/m y, o E 0.62 S/m and er = 150.

A two layer biological body illuminated by an EM
wave at normal incidence is shown divided into 4
quadrants under symmetry conditions.

A two layer biological body illuminated by EM wave
at normal incident is shown divided into 8 cells
in the lst quadrant.

A layer of biological body illuminated by electro-
magnetic wave at normal incidence is shown divided
into 4 cells.

129

130

135

154

172

CHAPTER 1

INTRODUCTION

In recent years, many researchers have studied the induced
electromagnetic heating inside biological systems because of the
controversy of potential health hazards due to non-ionizing EM
radiation and the applications of EM radiation in biomedical area.
Electromagnetic waves in the frequency range of HF to UHF may cause
adverse effects in biological systems. Some of these effects can
be harmful at high intensities, causing cancer, burns, cataracts,
etc. However, if it is under the controlled condition at lower
intensities, electromagnetic radiation can be used for therapeutic
purpose and to make useful diagnostic measurements. In order to
predict the effects of EM radiation on biological systems, and for
the applications of EM radiation, induced EM heating inside biological
body needs to be determined. The determination of the induced EM
fields inside biological system can be approached from both the
theoretical and experimental viewpoints.

Theoretically, biological systems have been approximated by

simple mathematical models such as planeslabs, spheres or cylinders.
These simple models, however, can provide only very approximate
results. In reality, a biological system is usually a heterogeneous
finite body with an irregular shape, and it is necessary to use a

realistic model for the biological system if accurate results on

the induced EM fields inside the system are needed.

To handle such an irregular geometry, the only potent method is
the numerical method with the help of a high-speed computer. This
research deals with the theoretical and experimental studies of the
induced EM fields inside biological systems irradiated by various
types of EM radiation. The numerical technique called the tensor
integral equation method has been used in this research, and this
method is outlined in Chapter 2.

The accuracy of tensor integral equation method is verified by the
exact solution of Mie theory and the experimental results in Chapter
3 and 4, respectively. In Chapter 3, we compare the numerical results
with the exact solutions of Mie theory for the induced EM beatings
inside homogeneous spherical models of human and animal heads at 918
and 2450 MHz. The induced EM beatings inside realistic models of
human and animal heads are also included. Chapter 4 contains numerical
and experimental results for the induced electric fields inside the
cubical phantom models with different sizes.

Chapter 5 is devoted to the theoretical and experimental studies
on the induced electric field inside human body irradiated by EM
waves of various frequencies. A phantom model of man was constructed
with thin plexiglass filled with phantom material. The model was
irradiated by 500 to 3000 MHz EM wave in a microwave anechoic chamber.
Induced electric fields were probed over 28 locations in one side of
the model. The distribution of the measured electric fields was
compared with the distribution of theoretical results obtained from

the tensor integral equation method.

Chapter 6 is devoted to the theoretical study on local EM heating
of tumors in biological bodies. One of the therapies for cancer is
that of hyperthermia in combination with chemotherapy. When the
temperature of a tumor is raised a few degrees above that of surrounding
tissue, accompanying chemotherapy has been found to be effective in
treating the tumor. The purpose of this study is to find a noninvasive
method by which to heat the tumor without overheating other parts of
the body. In order to find such a method, we have theoretically
studied the heating pattern induced inside the body with tumor when
it is irradiated by various EM fields with certain schemes.

The combination of the tensor integral equation method (TIEM)
with an iteration technique is discussed in Chapter 7. When the TIEM
is applied to quantify the induced EM field in an electrically large
body, it is necessary to divide the body into a large number of cells
to obtain accurate results. This will lead to a large number of
unknowns in the numerical calculation and overloading the computer
storage. This method of combining an iteration technique with TIEM
is designed to overcome these problems.

In some biological applications it may be feasible to introduce
nontoxic magnetic powder into a local region so that the absorbed
power at the local region is enhanced when it is irradiated by an EM
field. The generalized TIEM which is designed to handle the bodywith
arbitrary permeability in addition to arbitrary conductivity and
permittivity is discussed in Chapter 8.

Chapter 9 includes a description and listing of the computer

programswuuxiin this study. Part 1, program FIELDS is used to quantify

the induced electric field inside an arbitrarily shaped biological
body. Part 2, program ITERATE, is the extension of program FIELDS
with an addition of iteration process. This program is useful for
a large body with a large number of cells. Part 3, program EMFIELD,
is used to quantify the induced electric and magnetic fields inside
an arbitrarily shaped biological body with arbitrary permeability,
permittivity and conductivity. Definitions of input variables, the
construction of data files and the useage instruction are given in

each part.

CHAPTER 2

REVIEW OF TENSOR INTEGRAL EQUATION METHOD

In this chapter the tensor integral equation method is briefly
outlined. Induced electric field inside the irradiated,arbitrary-
shape biological body or system was obtained by this tensor integral
equation method which surved as our theory in this study. The accuracy
of our theory (numerical result) has been checked by comparing with the
exact solution of Mie theory in Chapter 3 and comparing with the

experimental result in Chapter 4.

2.1 Description of problem

The theoretical method used in this study is based on a tensor
integral equation develOped by Livesay and Chen (1). When a bio-
logical system is illuminated by an electromagnetic wave, an electro—
magnetic field is induced inside the body and an electromagnetic wave
is scattered by the body in the region exterior to the body. In
general the biological system is an irregularly shaped heterogeneous
conducting medium and its electrical parameters are dependent on fre-
quency of incident electromagnetic wave and locations within the body.

Conventionally they are assumed to be

+

o = o (w,r)

e = a (an?)
ue’
LlO

The induced electromagnetic field inside the body, in general,

depends on the body's physiological parameters and geometry, as well as

5

the frequercy and polarization of the incident wave.

Based on Maxwell's equatirnm we can obtain a tensor integral
equation which relates unknown induced electric field inside the system
to the incident electric field. After using a pulse-function expansion
of the unknown induced electric field and point-matching, we employ the

method of moments to solve the integral equation numerically.

2.2 Tensor Integral Equation for the Induced Electric Field

Consider a finite biological body of arbitrary shape with permit-
tivity 6(f), conductivity of CC?) and permeability p0, illuminated in
free space by an incident electronagnetic wave with an electric field
91+ 41+
E (r) and magnetic field H (r). We can write Maxwell's equations for

this incident EM field in free space as

v x $36) = muff?) (2.1)

v x iiim = j weo’fiiG) (2.2)
+1 ->

V ° E (r) = 0 (2.3)
+1 +

V ° H (r) = 0 (2.4)

where no and so are the permeability and permitiivity of free space.
When a biological body is illuminated by the incident electromagnetic
field, it creates a distribution of induced charges and currents
throughout the body. These charges and currents produce a scattered
field. Thus, the total electromagnetic field inside the body is the

sum of the incident field and the scattered field:

E (r) = E (E’) + E (r) (2.5)
H (r) = Ei(¥) + fis(¥) (2.6)

Combining ea. (21), (2.2), (2.3), (2.4) with eq. (2.5), (2.6)
and Maxwell's equations for the total electromagnetic field we obtain

Maxwell's equations for the scattered field as
¥ ) (2.7)
V x 35(3) = m?) + jw [e(?) - 2.01} E ('r’) + jeeOESG) (2.8)
Defining an equivalent volume current density jeq (f) as
Equ) = r G) EG’) (2.9)
where T (f) = O (f) + jw [€(f)-€O] is the equivalent complex conductivity.
Eq. (2.8) can be rewritten as
v x is (P) = Squ) + jeeo‘e’sé’) (2.10)

+
The equation of continuity for Jeq(f) defines an equivalent volume

charge density peq(¥) as

v - 3' (¥) + jwp ($) = o (2.11)

01'
p (3?) = 31v - leqd’) (2.12)

Taking the divergence of eq. (2.10) and using eq. (2.12) gives

+ S + be (P)
v - E (r) = + (2.13)
O

+s+ +S+
Finally, Maxwell's equations for E (r) and H (r) can be written

38

v x E (r) = - j wuofiSG) (2.14)

v x ES(¥) = 38q(¥) + j weoES(¥) (2.15)
+3 —) _ _ -+

v . E (r) — so peq(r) (2.16)

v - 1336?) = o (2.17)

The scattered electric field ES(?) within the body can be deter-

mined from the following equation(2,3):

 

+ +
, J (r)
E36?) = emfl’qu) -‘é(¥,¥') dv' -3—J‘?§€— (2.18)
O
V
where
EEG?) = -jw€ fi+5731] 1:62?)
o 2
k
0
—> +'
‘i’+ +' e'jkolr"r I
(rsr ) 4fl1r_rl_li
H .‘uA AA AA
I = xx + yy + 22

 

P.V. symbol means the principle value of the integral, and GY;,E')

is the free space tensor green's function.

By substituding eq. (2.18) in eq. (2.5), rearranging terms,
-> -+ +++
and recalling that Jeq(r) = T(r) E(r), we can obtain a tensor integral

equation as

+ .
[1 . Egg] a?) - m. .(Ev)E(:v).8(-;,:v)dv' = ER?) (2.19)
0
+1 ->- -> + +
E (r) and T(r') are known quantities and E(r) is the total induced

field inside the body.

2.3 Moment Solution of Tensor Integral Equation

It is very difficult to solve the tensor integral equation by
performing integral which involved unknown E(f) inside the integral.
One simple possibility is to solve the tensor integral equation

numerically by using the method of moments.

If the body is partitioned into N subvolumes or cells and E(f)
and T(;) are assumed to be constant within each cell, tensor integral
equation (eq. 2.19) can be transformed into 3N simultaneous equations

for Ex’ E , and E2 at the center of N cells by the point matching

Y

method. These simultaneous equations can be written into a matrix

form as
r '1' 1 r1 ‘
G ' G ' G E E
_-EX-L-XZ_I_’_‘Z_-_’_‘- -1‘-
I I
I I i
G G E = E (2.20)
--X"-1-YZ_'-ZZ___Z_ -Y-
I I
G 'G 'G E E1
zx I zy I 22 z 2
L ' ' - L J L a

 

 

 

 

 

 

10

The [G ] matrix is a 3N x 3N matrix, while I: E] and[E1] are
3N column matrices expressing the total electric field and the
incident electric field at the centers of N cells. The elements
of [G ]matrix have been evaluated in the next section. Therefore,
with the known incident electric field EYE) the total induced electric
field E(f) inside the body can be obtained from eq. (2.20) by

inverting the [ G ] matrix.

2.4 Calculation of Matrix Elements

The expressions for the elements of each NxN submatrix [Gx x ],

 

P q
p, q = 1,2,3 are given in this section.
Let
x1 = X, x2 = y and x3 = z
The (m,n)th off diagonal element of the [Gx x:] matrix is given :1)
P q

by

_ 'J'O‘

mn jwuokoT(rn)Avne mu 2
C = (o -1 - jo ) 6
x x 3 mm mm pq
P q 47rd
mn
mn mu 2
+ cos 6x cos 6 (3 - omn + BjomUI], m # n (2.21)
P q
where
+ —>
o = k R ; R = Ir - r I
mm 0 mn mn m n
a m
mn xp — xn Xm - xn
COS 0 =———-P— , cos an = —-————9—q
R x
p R

11

n n
- (X1 9 x2 9 x3)

The (n,n)th diagonal element of the [Gx x 1 matrix is given by (l)
p q ‘

iju T(¥n)

 

p q ‘ 3ko
T(rn)]
+ [1 + mg;- (2.22)

where

3Av 1/3

H 4n

 

After all the elements of [Gj] matrix are determined, the total

induced electric field E(¥) inside the body can be obtained by in-

 

 

r 1
verting the G matrix as
I. .I
I- 1 F - -
I I
E C I G ' G -1 Y E1
__x_ __X’_‘.'_’.£Y_'__XE_ .13.
I I
= I I 1
E G l G I G E
y yx I yy I yz y
I
- _ _ - - _ 1 - - -I. _ _ _ _ i _
E G I G " G E
z zx I zy I 22 z
L . .. . . I J. l

 

 

 

 

12

After E(E) field is determined, the absorbed power density is
determined from P = 0/2 IEIZ. To verify the accuracy of the tensor
integral equation method, the numerical results generated by this
method were compared with the existing exact solution and experimen-

tal results in the following chapters.

CHAPTER 3

INDUCED EM FIELDS IN SPEERICAL BODIES...

HUMAN AND ANIMAL HEADS

In this chapter the accuracy of the tensor integral equation
method which serves as our theoretical tool is checked. The induced
heating patterns inside a homogeneous spherical brain obtained by
the tensor integral equation method are compared with the correspond-
ing results obtained from the exact solution of Mie theory (4,5).
After the theory was verified it was used to predict the induced

heating patterns inside human and animal heads.

3.1 The Mic Theory (4,5)

Consider a sphere of radius a which is illuminated by a plane
wave, whose electric field is linearly polarized in the x-direction
and propagates in the + z direction. The expression of this incident
field in terms of vector spherical wave functions is:

(X)
E1 e-jwt 2n+1 +

= E0 n=1 E(EiITI (Moln - ) (3'1)

where E0 is the amplitude of the incident electric field.

When EM wave is incident upon a sphere, it will give rise to a
forced oscillation of free and bound charges synchronous with the
applied field. This oscillation of charges will set up a secondary

field both inside and outside the sphere.

13

14

According to Mie theory, the electric field E induced inside a
conducting sphere by a plane EM wave with an incident electric field
E1 can be calculated from the general vector spherical wave solution

of the wave equation as

00
+ +1 . n 2n+1 + +
E E nil (J) n(n+l) (anMoln j bn Nolm) (3'2)
where the vector functions M and N are defined, and the co-
oln olm

efficients an and bn are obtained in Stratton (5). Numerical results
computed from eq. (3.2) were considered to be the exact solution to

the problem.

3.2 Formulation of the problem

To check the accuracy of the tensor integral equation method
(outlined in Chapter 2), the method was employed to determine the
distributions of the absorbed power density or the EM heating induced
by plane EM waves of 918 MHz and 2450 MHz in the spherical models
of animal and human brains having radii of 3 cm and 7 cm, respectively.
Numerical results obtained from this method were then compared with
the exact solution of Mie theory.

In order to apply the numerical method, a sphere is first
approximated by a "cubic sphere" which is constructed with a number
of small cubic cells. Figure 3.1 shows an example that one eighth
of a sphere is approximated by one eighth of a "cubic sphere" which is
censtructed with 73 small cubic cells. It is evident that a better approxi—
mation can be achieved by a larger number of smaller cubic cells. However,
to economize the computing time, the number of cubic cells or subdivisions

should be compromised.

15

In the present study, we subdivide one eighth of the sphere either
into 40 or 73 cubic cells.

A plane EM wave propagating in the + z direction is assumed to
be incident upon the sphere with a vertically polarized electric
field E1 in the x-direction and the associated magnetic field in the
y-direction. The E1 field can be expressed as

-jk z
o

E = x E0 2 v/m (3.3)

where E0 is the amplitude of the incident electric field and is equal

to

E0 = ZCOPi v/m (3.4)

In eq. (3.4), P is the incident power density in W/m2 and

i
Co is the impedance of free space having a value of 377 ohms. In the

following examples, we used P = l mW/cm2 and E0 = 66.83 V/m. K0 in

1
eq. (3.3) is the propagation constant of the EM wave in free space.

With eq. (3.3), E1 at the center of each cubic cell of the "cubic
sphere" can be specified. With this information on E1, electrical
properties of cubic cells and given geometry of the "cubic sphere",
the induced electric field E in each cubic cell is numerically com-
puted based on the tensor integral equation method. After E is
determined, the absorbed power density or the specific absorption rate
(SAR) of the EM energy is obtained from P = O/ZIEIZ. The average
heating is obtained by averaging out P inside the sphere.

The maximum heating is identified by the maximum value of P at a

certain location inside the sphere. The curve showing the relative

16

F I
I ‘1‘ I’
\’

I” ’

--- —--J---

/ ,I x / \
4L -’

 

 

 

\ I 1/8 "cubic sphere"

 

\

\
\
’

 

\

—--—-

 

\
\

z~---1r

 

\
\
\
\
\
\
\
+\
\
\
V
\
\
\
\
\
_-P

N
--q
\
\
\
\
\

pppppp

l/I

 

 

‘
‘
O
I
I
l
I
I
I
I
‘
\ \
\
\
X
‘s
\
\
\
T7
I
n
x 1
\
\
..<

 

 

~
|—---

 

 

 

   
  
 

1/8 sphere

‘2

Figure 3.1 One eighth of a sphere is approximated by one eighth

of a "cubic sphere" which is constructed with 73 small

cubic cells.

17

heating as a function of location is obtained by normalizing the
distribution of P with respect to the maximum heating. The dielectric
constant Er and conductivity 0 of the brain at 918 MHz and 2450 MHz

are obtained from values reported by Schwan (6).

3.3 Induced EM Field in Homogeneous Spheres

As the first example, we consider the case of a spherical
model of animal brain of 3 cm radius exposed to a plane EM wave of
918 MHz propagating in the + z direction and with a power density of
1 mW/cmz. At this frequency, the dielectric constant er of the sphere
(brain) is assumed to be 35 and the conductivity 0:0.7 s/m. The brain
is approximated by a "cubic sphere" and one eighth of it is construct-
ed by 40 cubic cells. The numerical results are shown in Figure 3.23
where relative heatings along the x, y and z axes inside the "cubic
sphere" are plotted, and the average and maximum heatings are indicated.
The three curves marked X, Y and Z show the distributions of the
relative heatings or the relative SARs along the X, Y and Z axes,
respectively. These curves show strong standing wave patterns with
a peak heating located somewhere in the front half of the brain. The
average and maximum heatings are found to be 0.3202 mW/cm3 and 0.885
mW/cm3, respectively.

To check the accuracy of the numerical results presented in
Figure 3.2a, the induced EM heating was computed in a spherical brain
of the same radius based on the exact solution of Mie theory. The
corresponding results are shown in Figure 3.2b. The relative heating
curves along the X, Y and Z axes based on the exact solution resemble

closely with that shown in Figure 3.2a. It is noted that the curves

18

(40 Subdivisions)

 

 

 

 

 

. . , 2
Brain (3 cm radius) P1 = 1 mW/cm
Freq. = 918 MHz Ave. heating = 0.3202 mW/cm3
8r = 35, o = 0.7 S/m Max. heating = 0.885 mW/cm3
1.0 .
Z
)—
008- .
_ X
m
:
“30.6b
m
m
n I- I
2
"-4 0.4?-
‘5 . Y
H
a; r '
0.2,
y.
0 j l 1 1 l l l L 1 1 1
-3.0 -2.0 —1.0 0 1.0 2.0 3.0
cm

Figure 3.2a. Distributions of heating along the x, y and z axes
of a "cubic spherical" brain of 3 cm radius induced
by a plane EM wave of 918 MHZ propagating in the +2
direction with a power density of 1 mW/cmz. Electrical
properties of the brain, the average and maximum
heatings are shown. One eighth of the "cubic sphere"
is constructed with 40 cubic cells.

19

(Exact Solution)

. . 2
Brain (3 cm radius) P1 - l mW/cm

0.295‘mW/cm3
3

Freq. = 918 MHz Ave. heating

a = 35, o = 0.7 S/m Max. heating 0.814 mW/cm

r

 

0.6"-

0.4?

Relative heating

 

 

l J_ L L l l 1 AJ 1
-3.0 -2.4 -1.8 -l.2 -.6 0 .6 1.2 1.8 2.4 3.0

 

0

cm

Figure 3.2b. Distributions of heating along the x, y and z axes of a
spherical brain of 3 cm radius induced by a plane EM
wave of 918 MHz prOpagating in the +2 direction with a
power density of 1 mW/cm . Electrical properties of the
brain, the average and maximum heatings are shown.
Numerical results are obtained from the exact solution
of Mie theory.

20

shown in Figure 3.2b are the distributions of SARs along the X, Y and
Z axes while those curves shown in Figure 3.2a are, strickly speaking,
the distributions of SARs along the centers of cubic cells which line
adjacent to the X, Y and Z axes. Therefore, a perfect agreement
between those two sets of curves is not expected. The average and
maximum heatings based on the exact solution are found to be 0.295
mW/cm3 and 0.814 mW/cm3, respectively. These values are in agreement
with the corresponding numerical results shown in Figure 3.2a with a
deviation of less than 9%. The comparison of Figure 3.2a and 3.2b
confirms the accuracy of our numerical method.

In the second example, we consider the same spherical model of
animal brain of the first example exposed to a plamaEM wave of 2450
MHz propagating in the + z-direction with a power density of 1 mW/cmz.
At this frequency, Er and o of the brain are assumed to be 30.9 and
1.1 S/m, respectively. Figure 3.33 shows the numerical results on
the distributions of relative heating, the average and maximum heating
in the spherical brain that is approximated by a "cubic sphere" with
one eighth of it constructed with 40 cubic cells. The distributions
of heating along the X, Y and Z axes show a strong resonant peak in
the center of the brain. The maximum heating near the center of the
brain is found to be 1.576 mW/cm3 which is about twice the value for
the case of 918 MHz. The average heating is found to be 0.235 mW/cm3
which is about the same as the case of 918 MHz.

The corresponding results for this example based on the exact
solution of Mie theory are given in Figure 3.3b. The distributions of

relative heating along the X, Y and Z axes show similar but somewhat

sharper resonant peaks than that shown in Figure 3.33. The average

21

(40 Subdivisions)

 

 

 

 

 

 

Brain (3 cm radius) Pi = l mW/cm2
Freq. = 2450 MHz Ave. heating = 0.235 mW/cm3
Er = 30.9, o = 1.1 S/m Max. heating = 1.571 mW/cm3
1.0
L-
0.8 " X
P
2‘
mi 0.6 ’
4)
m .
m -
.c
g
”4 0.4 r
4.)
m
H +-
m
m
0.2 ' ' - -
l L L L L 1 L l 1 l I
0 -3.0 -2.0 -1.0 o 1.0 2.0 3.0

Figure 3.3a. Distributions of heating along the x, y and z axes of a
"cubic spherical" brain of 3 cm radius induced by a plane
EM wave of 2450 MHz propagating in the +2 direction with
a power density of l mW/cm . Electrical properties of
the brain, the average and maximum heatings are shown.
One eighth of the "cubic sphere" is constructed with 40
cubic cells.

22

and maximum heatings are 0.278 mW/cm3 and 1.698 mW/cm3, respectively.
These values based on the exact solution are quite close to the
numerical results given in Figure 3.3a.

In the third example, we consider a spherical model of human brain
of 7 cm radius exposed to a plane EM wave of 918 MHz propagating in
the + z direction with a power density of l mW/cmz. At this frequency,
er and O of the brain are assumed to be 35 and 0.7 s/m, respectively.
Figure 3.4a shows the numerical results on the induced EM heatings
calculated with the model of a "cubic sphere" of 7 cm radius, with one
eighth of it constructed with 40 cubic cells. We observe that a
resonance is induced in the brain and the peak heating occurs in the
central part of the brain. The average and maximum heatings are found
to be 0.1065 mW/cm3 and 0.5937 mW/cm3, respectively.

‘When one eighth of the same "cubic sphere" of 7 cm radius is
constructed with 73 smaller cubic cells, numerical results on the
induced EM heating are somewhat modified as shown in Figure 3.4b. The
resonant peak of heating become sharper compared with that of Figure
3.4a and the average and maximum heating become 0.115 mW/cm3 and 0.619
mW/cm3, respectively.

The corresponding numerical results in a spherical brain of 7 cm
radius based on the exact solution of Mie theory are shown in Figure
3.4c. The resonant peaks of heating in Figure 3.4c are somewhat
sharper than that of Figure 3.4b, and the average and maximum heating
are found to be 0.117 mW/cm3 and 0.458 mW/cm3, respectively.

If we compare the results of Figures3.4a and 3.4b with that of

Figure 3.4c, it is clear that numerical results of our numerical

23

(Exact Solution)

 

 

 

 

 

Brain (3 cm radius) Pi = 1 mW/cm2
Freq. = 2450 MHz Ave. heating = 0.278 mW/cm3
Er = 30.9, o = 1.1 S/m Max. heating = 1.698 mW/cm3
1.0 t
L, X
0.8
+—
o\
.5
u 0.6 "
m
o
S ._
m
.i’.
u 0.4 h-
m
H
o
“‘ l
1’
0.2
0 L, l 1 L J l l l l
-3.0 -2.4 -1.8 -l.2 -.6 0 .6 1.2 1.8 2.4 3.0
cm

Fibure 3.3b.

DistributionSof heating along the x, y and z axes of a
spherical brain of 3 cm radius induced by a plane EM
wave of 2450 MHz propagating in the +2 direction with
a power density of 1 mW/cm . Electrical properties of
the brain, the average and maximum heatings are shown.
Numerical results are obtained from the exact solution
of Mie theory.

Relative heating

24

(4O Subdivisions)

Brain (7 cm radius) Pi = 1 mW/cm2
Freq. = 918 MHz Ave. heating = 0.1065 mW/cm3
er = 35, o = 0.7 S/m Max. heating = 0.5937 mW/cm3

 

 

1-° V v
I

 

 

 

   

 

Y
Z
41 1 1 1 .1 1 1 1 1 1 1 1 l
O -6.0 -4.0 -2.0 0 2.0 4.0 6.0
cm

Figure 3.4a. Distributions of heating along the x, y and z axes of a
"cubic spherical" brain of 7 cm radius induced by a plane
EM wave of 918 MHz propagaaing in the +2 direction with
a power density of 1 mW/cm . Electrical properties of the
brain, the average and maximum heatings are shown. One
eighth of the "cubic sphere" is constructed with 40 cubic

cells.

Relative heating

25

(73 Subdivisions)

Brain (7 cm radius) Pi = l mW/cm2

Freq. = 918 MHz Ave. heating = 0.115 mW/cm3

er = 35, o = 0.7 S/m Max. heating = 0.619 mW/cm3
1.0

 

 

 

 

 

 

 

-6.0

Figure 3.4b.

-4.0 -2.0 0 2.0 4.0 6.0
cm

Distributions of heating along the x, y and z axes of a

"cubic spherical" brain of 7 cm radius induced by a plane

EM wave of 918 MHz propa ating in the +2 direction with a

power density of 1 mW/cm . Electrical properties of the

brain, the average and maximum heatings are shown. One

eighth of the "cubic sphere" is constructed with 73 cubic cells.

Brain (7 cm radius) Pi = l mW/cm

Freq. = 918 MHz Ave. heating

Er = 35,

26

(Exact Solution)

2

0.117 mW/cm3

o = 0.7 S/m Max. heating 0.458 mW/cm3

 

1.0

0.6 r

I

Relative heating

 

/

‘ 1 1"

1 ' 1 1 1 1 1 1 1

 

 

0

-7.0 -5.6 -4.2 -2.8 -1.4 0 1.4 2.8 4.2 5.6 7.0

Figure 3.4c.

cm

Distributionsof heating along the x, y and z axes of a
spherical brain of 7 cm radius induced by a plane EM

wave of 918 MHz propagating in theufiz direction with a power
density of 1 mW/cmz. Electrical properties of the brain,
the average and maximum heatings are shown. Numerical
results are obtained from the exact solution of Mie theory.

27

method can be improved significantly by increasing the number of
subdivision on the sphere. It appears that when one eighth of a
spherical brain of 7 cm radius is subdivided into 73 cubic cells,
our numerical method is capable of producing satisfactory results
at 918 MHz.

The last example is for the case of the same spherical model
of human brain exposed to a plane EM wave of 2450 MHz propagating
in the + z direction with a power density of 1 mW/cmz. er and O
for the brain are assumed to be 30.9 and 1.1 s/m respectively.
Numerical results on the induced EM heatings calculated in a "cubic
spherical" brain of 7 cm radius, and with one eighth of it construct-
ed with 40 cubic cells, are shown in Figure 3.5a. The corresponding
numerical results calculated in the same "cubic spherical" brain
but with one eighth of it constructed with 73 cubic cells are shown
in Figure 3.5b. Numerical results for a spherical brain of 7 cm
radius obtained from the exact solution of Mie theory are shown in
Figure 3.5c. Comparing the results of Figure33.Sa and 3.5b with
that of Figure 3.5c, it is observed that when one eighth of the brain
of 7 cm radius is subdivided into 40 cubic cells, our numerical
method produced poor results at 2450 MHz. However, if the subdivision
is increased from 40 to 73, much improved results are obtained. The
main difficulty our numerical method encounted in this case was the
failure in predicting the rapidly attenuating nature of the induced
heating in the front surface of the brain. The reason for this
difficulty is that the skin depth of a 2450 MHz EM wave is quite

shallow in a spherical brain tissue of 7 cm radius, and the numerical

Relative heating

28

(40 Subdivisions)

 

 

 

 

Brain (7 cm radius) Pi = l mW/cm2
Freq. = 2450 MHz Ave. heating = 0.068 mW/cm3
6r = 30.9, o = 1.1 S/m Max. heating = 0.141 mW/cm3
1.0
0.8 P
006 ’-
z ' A
0.4 ’ O «\
0.2 ‘ ‘ /‘T
Y v
0 1 1 1 ' 1 1 1 1 1 .1 1 1 1
-6.0 -4.0 -2.0 0 2.0 4.0 6.0
cm

Figure 3.5a.

Distributions of heating along the x, y and z axes of a
"cubic spherical" brain of 7 cm radius induced by a plane
EM wave of 2450 MHz propagating in the +2 direction with
a power density of l mW/cm . Electrical properties of the
brain, the average and maximum heatings are shown. One

eighth of the "cubic sphere" is constructed with 40 cubic
cells.

\

Relative heating

29

(73 Subdivisions)

Brain (7 cm radius) Pi = 1 mW/cm
Freq. = 2450 MHz Ave. heating = 0.094 mW/cm
er = 30.9, o = 1.1 S/m Max. heating = 1.1 mW/cm3

 

 

 

 

 

 

 

-600 -400 -200 0 2.0 4.0 6.0

Figure 3.5b. Distributions of heating along the x, y and z axes of a
"cubic spherical" brain of 7 cm radius induced by a plane
EM wave of 2450 MHz propagating in the +2 direction with a
power density of l mW/cmz. Electrical properties of the
brain, the average and maximum heatings are shown. One
eighth of the "cubic sphere" is constructed with 73 cubic

cells.

3

Brain (
Freq. =
er = 30

1.0

30

(Exact Solution)

7 cm radius) Pi = l mW/cm2
2450 MHz Ave. heating - 0.092 mW/cm3

.9, o = 1.1 S/m Max. heating = 0.396 mW/cm3

 

0.8-

0.6

0.4"

Relative heating

0.2"

‘Y

 

0

\“" A

l l l l l 11, 1 1

 

 

-7.0

Figure 3.5c.

-5.6 -4.2 -2.8 -l.4 0 1.4 2.8 4.2 5.6 7.0

cm

Distributions of heating along the x, y and z axes of a
spherical brain of 7 cm radius induced by a plane EM wave
of 2450 MHz propagating in the +2 direction with a power
density of 1 mW/cmz. Electrical properties of the brain,
the average and maximum heatings are shown. Numerical
results are obtained from the exact solution of Mie theory.

31

result on the induced EM heating in each cubic cell is the average
heating within the cell, therefore unless the size of the cubic cell
is small or comparable with the skin depth, the rapidly attenuating
nature of the induced EM heating cannot be predicted by our numerical
method. In Figure 3.5a, the rapidly attenuating nature of the induced
EM heating in the front surface is missing because the cubic cell in
this case is relatively large electrically. However, as the size of
the cubic cell is reduced as in Figure 3.5b, the rapidly attenuating
nature of the induced EM heating is recovered in the front surface of
the sphertal brain. The average heatings predicted by the numerical
method are quite good, especially, for the case of 73 subdivision.
For the maximum heating, numerical results at this frequency are poor.
From this example, it appears that to calculate the internal EM
field induced by a 2450 MHz EM wave inside a typical spherical model
of human brain with our numerical method, the subdivision of one eighth
of the spherical brain into 73 cubic cells can only yield fair results.
More accurate results will necessitate the subdivision of the spherical
brain into a larger number of smaller cubic cells. Fortunately, the
case of 2450 MHz is not as important as the case of 918 MHz, because
the latter induces a strong resonance in a human brain. For the fre-
quency of 918 MHz, our numerical method yields satisfactory results
when one eighth of the sphere brain is subdivided into 73 cubic cells.
Table 3.1 shows comparisons of numerical results on the average
and maximum heatings produced by our numerical method with that obtained
from the exact solution of Mie theory. From this table, we can conclude

that our numerical method predicts the average heating very well even

32

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33

though the brain is subdivided into a relatively small number of cubic
cells. For the maximum heating, our method can also produce satis-
factory results if the dimension of cubic cell is kept to be electri-
cally small, for example, one tenth of the free space wavelength or

smaller.

3.4 Induced EM heating in Realistic Models of Human and Animal Heads.

The controversy of potential health hazards due to non-ionizing
electromagnetic radiation has led to many studies on the induced EM
heating in the brain and eyes of humans and animals. Existing theore-
tical studies by many researchers (4, 7-9) on the induced EM heating
in the brain were either based on a homogeneous spherical model or a
multilayer spherical model. It was reported in these studies that a
UHF EM wave can excite an EM resonance and create a hot spot inside
the human brain, and a microwave can cause a similar phenomenon
inside an animal brain.

In reality, the brain is not spherical in shape and it is
surrounded by other tissues of irregular geometries. Therefore, it
seems important to quantify the induced EM heating inside a realistic
model of a human head or an animal head that consists of a brain of
realistic shape and eyes surrounded by a bony structure. To handle
such an irregular geometry, the only potent method is the numerical
method.

After the accuracy of the numerical method was verified in the
case of spherical brains as explained in the preceeding section, the
method was employed to quantify the induced EM heating in a realistic

model of human or animal head that consists of a brain of realistic

34

shape and two eyes surrounded by a bony structure as shown in Figure
3.6a, where the brain and eyes are indicated by shaded regions. The
head was subdivided into 180 cubic cells of various sizes in the
numerical calculation. We have also calculated the EM heating induced
in the bare brain without the surrounding bony structure. This case
was considered for the purpose of assessing the effect of the surround-
ing bony structure on the induced EM heating in the brain.

Figure 3.6a shows the distribution of the EM heating or SARs in
mW/m3 inside a human head, with dimensions of 18x18x24 cm, induced by
a plane EM wave of 918 MHz with a vertically polarized electric field
of 1 V/m, incident upon the head from its front surface. The dielectric
constant Er and conductivity 0 for the brain and eyes are assumed to
be 51.0 and 1.6 S/m, respectively, at this frequency. Er and o for
the surrounding bony structure are assumed to be 5.6 and 0.101 S/m,
respectively. Since the human head is in near resonance at the
frequency of 918 MHz, strong induced SARs inside the head are expected.
The distribution of SARs in Figure 3.6a shows that the induced SARs
are generally strong inside the head, with the maximum SAR located
at the central part of the head. The SARs in the brain and eyes are
relatively low compared with that in the surrounding bony structure.
However, the maximum SAR in the brain can reach a value of 21.5 mW/m3.
The total power dissipated in the brain is 3.202x10"6 W and that in
the whole head is 4.522x10-5 W. It is noted that if the incident EM
wave has a power density of l mW/cmz, the induced SARs in Figure 3.6a

should be multiplied by a factor of 4.466x103. Thus, the maximum

heating inside the brain is estimated to be 0.096 mW/cm3. This value

35

is about one fifth of the value predicted with the model of a spherical
brain as shown in Figures3.4a, 3.4b and 3.4c.

Figure 3.6b shows the distribution of SARs inside a bare human
brain, without the surrounding bony structure, induced by a plane EM
wave of 918 MHz with a vertically polarized electric field of l V/m.

Er and O of the brain and eyes are assumed to be 51.0 and 1.6 S/m,
respectively. The induced SARs in the central part of the bare brain
are found to be considerably higher than the SARs induced inside the
brain surrounded by the bony structure as shown in Figure 3.6a. The
total dissipated power in the bare brain is found to be 4.591x10-6 W
which is about 43% higher than the case of the brain surrounded by
the bony structure. This example implies that the surrounding bony
structure around the brain tends to reduce the induced EM heating
inside the brain.

Figure 3.7a shows the distribution of SARs in the same human head
of Figure 3.6a induced by a plane EM wave of 2450 MHz with a vertically
polarized electric field of l V/m. Er and G for the brain and eyes are
assumed to be 47 and 2.21 S/m, respectively, while Er and o for the
surrounding bony structure are assumed to be 5.5 and 0.15 S/m, respect-
ively. At this frequency, the induced field is mainly concentrated
near the front surface of the head, and thus, induced SARs in the
brain are generally very low. The interesting point to observe is
the SAR induced inside the eyes. Even though the eyes are located in
the front surface of the head and the EM wave is directly incident
upon them, the induced SAR inside the eyes is very small compared with

that in the surrounding bony structure. The total dissipated power in

36

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Total power in head

 

Distribution of SARs inside a human head induced by a plane
EM wave of 2450 MHz with a vertically polarized electric

field of 1 V/m.

Figure 3.7a.

39

7

the brain at this frequency is 1.404x10- W and that in the head is

2.179x10-“5W3 These values are much smaller compared with the case
of 918 MHz.

When the bare brain without the surrounding bony structure is
immersed in the same EM wave of 2450 MHz, the induced SARs inside the
brain are shown in Figure 3.7b. These values are somewhat greater
than the case of Figure 3.73, but they are still very low compared with
the case of 918 MHz. The total dissipated power in the bare brain is
found to be 4.20x10'7 w.

For the last example, we consider an animal head exposed to a
plane EM wave of 2450 MHz because this frequency is known to excite
a resonance in a brain with a radius of about 3 cm. Figure 3.8a shows
the distribution of SARs inside an animal head, with dimensions of
9x9le cm, induced by a plane EM wave of 2450 MHz with a vertically
polarized electric field of 1 V/m. Er and O for the brain, eyes and
the surrounding bony structure are assumed to be the same as the case
of Figure 3.7a. From Figure 3.8a, it is observed that the induced
SARs inside the animal brain are quite high and are in the same order
of magnitude as the induced SARs inside the human brain when it is
exposed to an EM wave of 918 MHz. The SARs in the surrounding bony
structure are even higher than the SARs in the brain. The SAR in the
animal eye is 9.6 mW/m3, which is about 10 times higher than that in
the human eyes when they are exposed to a 918 MHz EM wave of the same
power density. The total power dissipated in the animal brain is

7 6

6.814x10- w and that in the animal head is 9.411x10’ w.

When the bare animal brain without the surrounding bony structure

40

 

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41

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Total power in Brain = 6.814 x 10‘7 W

9.411 x 10'6 w

Total power in head

 

Distribution of SARs inside an animal head induced by a
plane EM wave of 2450 MHz with a vertically polarized

electric field of 1 V/m.

Figure 3.8a.

42

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43

is exposed to the same EM wave of 2450 MHz, the induced SARs inside
the brain are shown in Figure 3.8b. It is observed that the induced
SARs in the bare brain is much higher than the case of Figure 3.8a.
The total power dissipated in the bare animal brain is found to be
1.702x10-6 W which is more than twice the value of the case shown in
Figure 3.8a. From this example, it appears that the surrounding bony
structure has a significant effect on the induced EM heating in the
animal brain when the animal head is exposed to 2450 MHz EM wave.

To summarize the findings in this section, it appears that the
bony structure surrounding the brain tends to reduce the induced EM
heating in the brain. The induced EM heating in the brain calculated
on the realistic model of a brain within a head is significantly
different from the results obtained with an idealistic model of a
spherical brain. The induced EM heating in eyes is found to be

relatively low compared with that in the surrounding bony structure.

CHAPTER 4

INDUCED EM FIELDS INSIDE THE CUBICAL BODIES

EXPERIMENTAL VERIFICATION

To confirm the accuracy of the tensor integral equation, a series
of experiments have been conducted to measure the induced electric
field inside the cubic boxes with different size. The phantom materials
of varying conductivity were used to model the biological body (phantom
model). The phantom models were exposed to a maximum electric field and a
maximum magnetic field of a standing EM wave which was created in front of
the reflector.

The accuracy of thetheory has been checked by comparing the induced elec-

tric field inside the phantom cubic boxes with the corresponding experimental

results.

4.1 Experimental Set Up

The set up for this experiment is shown in Fig. 4.1. The experi-
ment was conducted inside a large microwave anechoic chamber in which
a standing EM wave was created by radiating an EM wave upon a metallic
reflector. Standing waves of the electric field E1 and the magnetic
field of H1 set up in front of the reflector are depicted in Fig. 4.1.
The electric field of the wave is polarized horizontally and the
magnetic field is vertically polarized.

When the phantom cube is placed at the location of a maximum

electric field, the impressed electric field on the cube is

Ei=

N)

E1 cos kz (4.1)
max

44

45

where z = 0 corresponds to the center of the cube. This symmetrically
impressed electric field will excite a linear electric mode of induced
electric field in the cube.

On the other hand, when the cube is placed at the location of a
minimum electric field, or a maximum magnetic field, the impressed

electric field on the cube is

E1 = x E1 sin kz (4.2)
max

This impressed electric field is antisymmetrical with respect to
the center of the cube, 2 = 0, and it will excite a circulartory
magnetic mode of induced electric field in the cube. It is noted that
this magnetic mode can be considered as excited by the magnetic field

of the EM wave, from a different point of view.

4.2 Construction of Probe

The main difficulty in the direct measurement of the induced
electric field in a phantom model or biological body is the availability
of a workable, implantable electric field probe. Although a miniature
electric field probe, capable of measuring the electric field from
0.915 to 10 GHz., has been reported by Bassen e£_al, (1975), the
frabication of this probe requires thin-film technique and it is not
commercially available. There is a need for an implantable electric
field probe which can be constructed inexpensively and handled

ruggedly for researchers in the bioelectromagnetic area. In this

study such a probe has been developed and used in experiments of

measuring induced electric field inside a biological body.

46

A conventional electric field probe for measuring the electric
fields of EM waves consists of a short dipole loaded with a microwave
diode and connected with a pair of very thin, high resistive wires.
This probe can be used to measure the electric field of an EM wave in
space by orientating its lead wires perpendicular to the electric
field vector to minimize the induced current in the lead wires, or to
minimize the interference caused by the lead wires. However, when
implantable electric field probe loaded with a microwave diode is
inserted into a finite conducting body to measure the internal electric
field, a great difficulty is usually encountered. The situation is
depicted in Fig. 4.2a which shows that the electric field on the body
surface is much higher than the internal electric field and is mainly
perpendicular to the surface, and in parallel with the lead wires.
Thus, whilethe probe is excited by a weak internal electric field,
the strong electric field on the body surface can induce a large
current on the lead wires over the section adjacent to the probe. Unless
the probe system is perfectly symmetrical, the antisymmetrical component
of the induced current on the lead wires will be detected by the diode
and adds a very large noise to the probe output. It is also found that
this noise can not be minimized sufficiently with a pair of very thin,
high resistance wires placed very close or twisted around.

A scheme to overcome this difficulty is shown in Fig. 4.2b. In
this scheme, the section of lead wires adjacent to the probe is con-
structed with two series of lumped resistors of 3 K9. This large
resistance minimizes the current induced in the lead wires by the

strong surface electric field and, consequently, minimizes the noise

47

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An implantable electric field probe
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zero-bias microwave diode
(Microwave Associates, MA 40234)
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An implantable electric field probe
with interference-free lead wires,

49

component of the probe output signal. The probe itself is made from a
zero-bias microwave diode (Microwave Associates, MA 40234). The probe
and the lead wire system are encased in a plexiglass stick with the
help of epoxy glue. The probe is very rugged and inexpensive and its

dimension is about 1 cm.

4.3 Theoretical and Experimental Results

The experimental results on the induced electric fields inside
the cubical phantom models are compared with the theoretical values of
the induced electric fields obtained from the tensor integral equation
method.

The cubical phantom model used in the study is depicted in Fig.
4.3. For the theoretical analysis, one eighth of the cube is divided
into 27 cubic cells.

Figure 4.4a shows the case of a cubical phantom model of 2x2x2 cm
placed at the location of a maximum electric field of a 750 MHz standing
wave in front of the reflector. The dielectric constant and conductivity
are assumed to be 50 and 4.5 S/m, respectively. The numerical results
on the X-component of the induced electric fields along the Z-axis
inside the cube are plotted in this figure. The results show a linear
electric mode of induced electric field which is rather uniform in the
cube.

In the corresponding experiment, a phantom model of a cubic box
with dimension 2x2x2 cm was constructed with thin plexiglass filled
with phantom material of the same dielectric constant and conductivity
as above (er = 50, 0 = 4.5 S/m). The X-component of the induced

electric field has been probed at a 2 mm interval along the z-axis.

50

By inserting the probe from the back surface along z-axis and taking
the data every increment of’2 mm until the probe reaches the front
surface. The experimental results of the x-components of the induced
electric fields along the z—axis inside the cube were also plotted in
Fig. 4.4a. The experimental results compare very well with the theory
for this linear electric mode of induced electric field inside the
cube.

In Fig. 4.4b, we consider the same cube but now the cube is
located at the location of the minimum electric field. In the theore-
tical calculation, the dielectric constant and conductivity are
assumed to be the same as before and again one eighth of the cube is
divided into 27 cubic cells. The impressed field for this case is
E1 = x Emax sin kz with the minimum electric field located at the
center of the cube. The same experimental procedure was used to
measure the induced electric field. The theoretical and experimental
results are plotted in the same figure for comparison. A good agree-
ment is obtained between theory and experiment and these results show
a circulatory magnetic mode of induced electric field excited in the
cube.

In Fig. 4.5a, we consider a phantom cube with dimensions of
4x4x4 cm, placed at the location of a maximum electric field of 750 MHz
standing wave. In the numerical calculation, the same dielectric
constant and conductivity (er = 50, O = 4.5 S/m) are assumed and one
eighth of the cube is divided into 27 cubic cells. In this experiment,
a phantom material with the same dielectric constant and conductivity

was packed in the cubic plexiglass box of size 4x4x4 cm. The x-component

Figure 4.3.

51

4x

 

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an EM wave.

divided into 27 cubic cells.

One eighth of the cube is

 

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52

Electric mode

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

5 Emax
' A
E I I x
3 _, ' T 1‘ .1?
E 1 E
x _ . :
2 —0— experiment ‘
1 .. —-a— theory '— Zcm“
O 1 I 1
-1.0 "Os5 O 005 1.0
(a) location along the center line of the cube
Magnetic mode f = 750 MHZ
E =
7 r 50
, 6 = 11.5 S/m
6 t. —._exper1ment
5 +theory
9c 47
1" I’L“ q
EX 1 /‘/E .' a
3 / "J
2 fin. ,. 20mg
max
1
O I 1
"1s0 "0s5 0 005 1.0
(b) location along the center line of the cube
Figure 4.4. Theoretical and experimental results of the x-components

of the induced electric field, E , along the z-axis of
the 2-cm cube, placed at the locations of a maximum
electric field and a maximum magnetic field of a 750 MHz,
EM standing wave.

53

of the induced electric field was probed along the z-axis. Theoretical
and experimental results show a good agreement for this case. In

Fig. 4.5b, the same phantom cube is placed at the location of a minimum
electric field. For this case a good agreement is again obtained
between experiment and theory.

From these two examples of 2 cm cube and 4 cm tube, it is found
that the magnetic mode becomes much greater than the electric mode in
the larger cube.

Next we will study the effect of the conductivity of the biolo-
gical system on the induced electric field inside the body. We consider
a 2x2x2 cm phantom cubic with the dielectric constant and conductivity
of 50 and 3.0 S/m, respectively. In the numerical calculation, one
eighth of the cube was divided into 27 cubic cells and the induced
electric field inside the cube was calculated. In the experiment, the
field probe was inserted into the body along the z-axis to measure
the x-component of the induced electric field along the z-axis. Fig.
4.6a shows the experimental and theoretical results when the cube was
placed at the location of a maximum electric field of a 750 MHz standing
wave. Fig. 4.6b shows the experimental and theoretical results when
the cube was placed at the location of a maximum magnetic field of the
same standing wave. Both results show a good agreement between theory
and experiment. From these results we observe that when the conduc-
tivity decreases the linear electric mode of the induced electric field
becomesgreater than the circulatory magnetic mode of the induced
electric field in the cube.

After three examples have been checked at 750 MHz, we proceeded

Fig;

54

Electric mode

 

 

 

 

 

 

 

 

 

 

6
5 .
6 _
3 P
Ex 2 _ —_4y_. experiment
1 _ -ib- theory
0 1 I 1
—2.0 -1.0 0 _ 1.0 2.0
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~ I theory a I 4.5 Slm
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Figure 4.5 Theoretical and experimental results of the x—components

of the induced electric field, , along the z-axis of
the 4-cm cube, placed at the locations of a maximum
electric field and a maximum magnetic field of a 750 MHz,
EM standing wave.

55

Electric mode

 

 

 

 

 

 

 

 

 

 

 

 

5
r“ .‘
L; g/ V
3 r
E
x 2 _
—-e-— experiment 5'23")“
1 ‘ -+- theory
0 7 I I L
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- +theory
3
Ex
2
1
0 1 1
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(1’) location along center line of the cube
Figure 4.6 Theoretical and experimental results of the x-components

of the induced electric fields, E , along the z-axis of
the 2-cm cube, placed at the loca ions of a maximum
electric field and a maximum.magnetic field of a 750 MHz,
EM standing wave.

56

to check the validity of the theory at a high frequency of 1 GHz. The
phantom cube was exposed to 1 GHz standing wave in the experiment.

Figs. 4.7a and 4.7b show the theoretical and experimental results
on the x-components of the induced electric fields along the z-axis of
the phantom cube of 2x2x2 cm when the cube is placed at the maximum
electric field location and the maximum magnetic field location of the
1 GHz standing wave, respectively. The dielectric constant and conduc-
tivity are assumed to be 50 and 4.5 S/m, respectively. The results show
that the theory agree very well with the experiment.

Figures 4.8a and 4.8b show the theoretical and experimental results
on the x-components of the induced electric fields along the z-axis of
the phantom cube of 4x4x4 cm when the cube is placed at the maximum
electric field location and the maximum magnetic field location of 1 GHz
standing wave, respectively. The dielectric constant and conductivity
are assumed to be 50 and 1.62 S/m, respectively. The agreement between
experiment and theory at this frequency is only fair because the
electrical dimension of the cube becomes larger in this case.

From these two examples we also observe that when the cube
becomes larger the magnetic mode tends to dominate the electric mode,
and when the conductivity decreases the linear electric mode tends to

increase in magnitude.

4.4 Summary

After the accuracy of our theory has been checked with Mie theory
in Chapter 3, it was reconfirmed in this chapter by comparing it with
a series of experimental results on the induced electric field

measured in phantom cubes exposed to EM standing waves. Up to this

(a)

(b)

Figure 4.7

57

Electric mode

 

 

 

 

 

5

h r

:3.LaAa:j::::::::=t=it::tijh<:f

2 _ , - .
"—experiment

1' - _‘_theory

0 .4 1 1

“100 -005 O 005 100

location along the center line of the cube

= 1 GHz
Magnetic mode

 

—0— experiment
+ theory

 

 

 

 

 

0 I l
-1.0 -0.5 0 0.5 1.0

location along the center line of the cube

Theoretical and experimental results of the x-components
of the induced electric fields, E , along the z-axis of
the 2-cm cube, placed at the loca ions of a maximum
electric field and a maximum magnetic field of a 1 GHZ:
EM standing wave.

(a)

(b)

Figure 4.8.

58

Electric mode

 

 

  
  

   

6
5 -
u
3
2
+ experiment
1 + theory
0 J l l
“'2 o O '1 o O O 1 o O

 

 

 

 

location along the center line of the cube

Magnetic mode

 

 

 

 

f 1 GHz
Er: 5O
5 = 1.62 S/m

  

_.+N>
—P«
tn

 

’I
/"'L"C1n _‘
41
Ignax:

10

9 _ + experiment

8 + theory

r

7 ..

6 ..

5 .-

a _

3 ..

2 ..

1 L

o ‘ '
'200 -100 O 100 200

location along the center line of the cube

Theoretical and experimental results of the x—components
of the induced electric field, E , along the z-axis of
the 4 cm cube, placed at the locations of a maximum
electric field and a maximum magnetic field of a 1 GHz,

EM standing wave.

59

point the accuracy of our theory has been established. We will use our
theory to predict the induced electric field inside a human body with
a realistic model in the next chapter. Also in this chapter we observe
that the circulatory magnetic mode of the induced electric field has a

significant effect in a large body at a low frequency.

CHAPTER 5
INDUCED EM FIELDS INSIDE HUMAN BODIES

The theoretical quantification of the induced electric field and
the specific absorption rate (SAR) of EM energy inside a realistic
model of human body irradiated by EM waves has been reported by Chen
and Guru (10,11) and by Gandhi gt a1. (12). Experimentally, the in-
duced SARs was determined indirectly by measuring the temperature
distribution in an irradiated phantom model of man with a thermo-
graphical method by Guy_gg.al. (14) or by a liquid-crystal temperature
probe by Gandhi gt a1. (13). The temperature distribution in a
phantom model may not correspond to the true distribution of the
induced SARs because of the heat dissipation. A direct measurement
of the induced electric field inside a phantom model of man should
provide a more accurate distribution of the induced SARs.

The main difficulty in the direct measurement of the induced
electric field in a phantom model of man or in a biological body is
the availability of a workable, implantable electric field probe.

We have described an implantable electric field probe which can be
constructed inexpensively and handled ruggedly in Section 4.2. We
have used this field probe to measure the induced electric field
inside a phantom model of human body. Characteristics of the probe
were checked by measuring the induced electric fields in small phantom
cubes in Chapter 4. The measured induced electric fields with this
probe were confirmed by theoretical results, and, thus, a good working

condition for this probe was assured.

60

61

5.1 Experimental set Up

The schematic diagram of the experimental set up for measuring
the induced electric field in the phantom model of human body is shown
in Fig. 5.1. The model was placed in a large microwave anechoic chamber
and was irradiated by a travelling EM wave of 500 to 3000 MHz incident
normally from front to back. An array antenna was used as a radiating
source for the range of 500 to 1000 MHz and a horn antenna was used
as a radiating source for the range of 1000 to 3000 MHz. A phantom
model of man with 1/5 dimension of a typical man was constructed with
thin plexiglass and filled with phantom material of appropriate con-
ductivity and permitdvity. Induced electric fields were probed over
28 locations in one side of the model. Detailed dimensions of the
model are depicted in Fig. 5.2. Since the scaling factor of 5 was
used in the model, thus, to simulate the actual human body the conduc-
tivity of the phantom model needs to be five times that of the human
body and, at the same time, the model should have the same permitdvity

as the human body.

5.2 Theoretical and Experimental results

The theoretical and experimental results on the induced electro-
magnetic fields inside the phantom model of human body are shown in
Figs.5.3 to 5.8.

Figure 5.3 shows the relative distribution of measured induced
electric field inside the phantom model of the human body for the
case of 2500 MHz., conductivity and dielectric constant are 7.4 S/m
and 50, respectively. In this experiment, a field probe was inserted

intd the body through the hole in the back surface and measured the

62

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of the phantom.model of man.

ions

Geometry and dimens

Figure 5.2.

64

induced electric field in the back layer first and then front layer
by moving the probe further inside the body. The measurement was
repeated at each of the 28 locations in the body. Figure 5.4 shows
the distribution of corresponding theoretical results obtained from
the tensor integral equation method. The theoretical results were
obtained when the geometry of the phantom model was subdivided into
104 cubic cells of various sizes in the numerical calculation.
Incident EM wave was assumed to be a plane wave travelling in the
+2 direction with a vertically polarized electric field.

Comparing Figs. 5.3 and 5.4, a quanlitative agreement is
obtained between experiment and theory: the maximum field is found
in the neck, and other high field regions in the arms, the legs, and
the front part of the head. It is noted that this case simulates the
case of a typical man with o = 1.48 S/m and er = 50 irradiated by an
EM wave of 500 MHz.

Figures 5.5 and 5.6 show the relative distributions of measured
induced electric field and theoretical induced electric field inside
the phantom model of human body with o = 7.0 S/m and er a 50
irradiated by a travelling EM wave of 2000 MHz. Theoretical results
were obtained with the 104—cell model as the previous case. A quan-
litative agreement is again obtained between experiment and theory:
the maximum field is found in the lower part of the arm, and other
high field regions are found in the neCk, the front part of the head
and the legs.

Figures 5.7 and 5.8 show the relative distributions of experimental

and theoretical results on the induced electric fields in a phantom

 

Figure 5.3

65
Relative distribution of IEx

(Experiment)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

52 13
k4 é?

v V '
15526 @0599
,.---L---_ ,---J .....
13540 ¢1E13
L---J---- '.....l.......1
----1---- L----,----.
23:45 4J§16
L----L—--- _----..---.

. l
imT’" """
2 1 g 3 6_ 5 1 .1 6

. ll
L
l
p----<
4 o f ' 2500 MHZ
+ Vertical polarization
3 6 er... 50
0'~ 7.4 Slm
X n".
10
.---4
.9!

 

 

 

Relative distribution of the x—components of the
measured induced electric fields inside the phantom
model of man, excited by a vertically polarized,
travelling EM wave of 2500 MHz at normal incidence,

 

Figure 5.4.

66
Relative distribution of IEx

 

 

 

 

 

 

 

 

 

 

 

 

fiheory)
(I04 - Cell Model)
_+__. _
.. T31
7; 3'; '
..|/

H i ‘ _,/
10535 29:21
lr----L---_‘ . r"'J"“"
----1 ..... L----r .....
11:45 11:16
3T5}? '1'}???
EST}? [33? '5'

1.----. " '
l
L .....
4 o f ' 2500 MHZ
_ ..... Vertical polarization
3.: 6r ' 50
a' - 7.4 Slm
x
5:!
-Y

 

 

 

Relative distribution of the x-components of theoretical
induced electric fields inside the phantom model of man,
excited by a vertically polarized, travelling EM wave of
2500 MHz at normal incidence (104-cell model).

 

67

model of human body with o = 4.5 S/m and cr = 50 irradiated by a
travelling EM wave of 500 MHz. It is noted that theoretical results
shown in Fig. 5.8 are obtained when the body is subdivided into 246
cubic cells of various sizes. A larger number of cells, or a finer
cell subdivision, was needed to produce a set of theoretical results
which agreed quanlitatively with the experimental results at 500 MHz.
This point will be discussed again in the next section. We observe
a quanlitative agreement between experiment and theory in Figs. 5.7
and 5.8: at 500 MHz., the maximum field point moves to the thigh, and
high field regions are found in the arm and the back side of the
torso. A disagreement between theoretical and experimental results

is found in the neck.

5.3 Discussion

In the course of our study on the induced electric field in a
phantom model of man, it was found that the agreement between
experiment and thory tended to deteriorate as the frequency is lowered.
This phenomenon seemed to contradict the thinking that at a lower
frequency, theoretical results should be more accurate because the
cell size would be electrically smaller. After a careful examination
of this phenomenon we have found the possible reason. When a phantom
body is irradiated by a travelling EM wave with an impressed electric
field,

A

i e-sz = x E; cos kz - xj E; sin kz (5.1)

an electric mode and a magnetic mode of induced electric field are

Figure 5.5

68

Relative distribution of Ex

( Experhnani)

 

 

 

 

 

 

 

 

 

 

 

 

_fl‘ —*
6" l 212
i----a -_--
7() 1‘9
3;] ioal
I 7 I
15353 63:13
1----LT--_ ,---J .....
-----
17 g 62 2.8: 6.6
L-_.'.J ......... i .....
i
2i§43 63:12
.----L---- _----L----
I
r-~~-r---- «a '-'-“1 -----
21:50 62:30
...... L---- .----J----.
i u
4.9
_____J f: 2000 MHz
...... Vertical polarization
Gib
..... l E r~ so ‘
}(
A l----d
di2
3,)’

 

 

 

 

Relative distribution of the x-Components of the measured
induced electric fields inside the phantom model of man,
excited by a vertically polarized, travelling EM wave of
2000 MHz at normal incidence.

Figure 5.6.

69

Relative distribution of Ex
( Theory )
( lO4-cell model )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 ‘ l I.
l ..... r----
7’2 <3]
‘I
III 7:1
, l ,4r
.23;SI 35E26
r----LT--~ r...J .....
313:27!) I 9 lill
L-_-J---- ;""L .....
2i;e6 12530
,---_L---- t----L---_
l
r----r---- '- ---------
I I
48:36 l5l44
l' ..... L.--..- .-..-...' .....
l I
4.4.
r---- F = 2000 MHz
...... Vertical polarization
6‘!
p---" 5350
o’~7.0 S/m
5C)

 

 

Relative distribution of the x-components of theoretical
induced electric fields inside the phantom model of man,
excited by a vertically polarized, travelling EM wave of
2000 MHz at normal incidence (104-cell model).

Figure 5.7.

70

Relative distribution of Ex

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(Experiment)
I 15

24
92'

i

25 3 24
._..-L7--a

22 £5;

L---J----

40 5 6a 60 l 50
.----L---- ---.L---.
l’--'r--- . ........ q

45 360 ‘62: 56
. ..... L ........ J----.
l i

4 J
' l
273;? f - 500 MHz
WM- Vertical Polarization
...... ' 6 r~ 50
"7:4 0' ~ 4.5 S/m
X
l) i""
50
*- Y

Relative distribution of the x-components of the measured
induced electric fields inside the phantom model of man,

excited by a vertically polarized, travelling EM wave of

500 MHz at normal incidence.

Figure 5.8.

71

Relative distribution of Ex

 

 

 

 

 

 

 

 

 

 

 

 

(iheo )
(246 - Ceril Model)
I 5 ‘—1 It)
| 6' ”(4"
H 5r
l .. g n a! ..
' -..-l...--. ,---.l----..
as: far..."
L---J---- ....i......1
ET}: Xi}:
----L--- ---.L----
[337. '3'???
..... L---- .----.i----1
I i
1
l
2’33} f - 500 MHz
““4 Vertical Polarization
e - 50
(I3 r
0'-4.5 S/m
x b-co-i
H 54»
----+
: Y

 

 

 

 

Relative distribution of the x-components of theoretical
induced electric fields inside the phantom model of man,
excited by a vertically polarized, travelling EM wave of
500 MHz at normal incidence (246-cell model).

 

72

both excited in the body. For the phantom model of man we used in
the experiment, the magnetic mode of induced electric field starts to
dominate the electric mode as the frequency becomes lower than 1000
MHz. The electric mode of induced elctric field is linear in nature,
and its numerical results converge well. On the other hand, the
magnetic mode of induced field is circulartory, and we often encounter
the difficulty of numerical convergence. Thus, for a specific cell
subdivision, the accuracy of the numerical results on the electric
mode of induced electric field is higher than that of the magnetic
mode. If the magnetic mode becomes dominant in the total induced
electric field, a finer cell subdivision is needed to produce accurate
results on the induced electric field. This is the possible reason
for the phenomenon we have observed in the 500 MHz case, in which a
246-cell model is needed to yield a set of theoretical results which
agreed with the experimental results.

To demonstrate this phenomenon further, two examples are given
in Figs. 5.9 and 5.10. Figure 5.9 shows the x-components of the in-
duced electric fields, IEXI, inside a 4-cm phantom cube with o = 4.5
S/m and Er = 50, excited by a symmetrically impressed electric field
of-E.i = x cos kz at 750 MHz. With this E1, an electric mode of electric
field is induced inside the cube.' In the upper part of Fig. 5.9 is
shown the distributions of IEXI within l/8 of the cube, obtained
when the cube is subdivided into 216 cubic cells. In the lower part
of Fig. 5.9, the distribution of [Ex] within 1/8 of the cube, obtained
with the 512-cell subdivision, is shown. Comparing these two sets of

numerical results, we observe an excellent convergence for the

 

 

750 MHz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- so
r .
8 4.5 S/m
- it cos kz
3
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"“44- i“‘~l- "“~+-
24.3) ' ~...---- 22.) Ha..." 23 i “in-«4
.2 I o I 1 5:
24" 32% «3360 M2"; 7.0
L____ “-L____. ____d
(216-cell subdivision)
"5*Iin (inV94nl)
ital- “ Mi .' - .
‘~l|~-i'l"?6.c L-L"5:I+.4§ 5
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3.0:22 Punt--. 2L3: :‘~L-. ---
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”may agpwlt--. 30:22.6: ““4...

Figure 5.9.

 

 

 

 

 

 

 

 

 

 

 

 

 

(512-cell subdivision)

Distribution of the x-component of the induced electric fields
(the electric mode) excited by a symmetrically impressed
electric field in a 4-cm phantom cube, and the comparison of
numberical results based on the 216-cell subdivision and the
512-cell subdivision.

74

 

 

 

 

'”" "‘~~qt_.Eg 8 2 sin kz
x \‘u ________ .
31“\\\\\\ . f - 750 MHz
ll: 'i'i‘i 8r ' 50
I“ a... .3333? a '. 4.5 s/m
E1 - i sin kz

 

 

 

 

 

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' .0 I I 2 '
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“" s “ . e is“‘ . o
28 ' "‘xL--." I "“4“-.. f“4~-- b
‘ film. lZ‘l IS", :l53 l‘ I 5-3 :52: ' ""1
‘~‘:“~. 42 i‘-‘4.‘~ :5.8 _~“:~. :5.3
I “ . ‘4‘. sq I “I~~.
:67 i
I
I
——-t

 

 

 

 

 

 

 

(216-cell subdivision)

IE.| In (mV/m)

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‘ I

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1
I
I
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 72'“ I ~I‘~ _-_J 215: P‘W-a IO . I.“ '\.
.‘. ' ‘1 I28. I N --q .OI I 1 s..-“
I '4‘.6:43.° I I :2707526.4 l. .9.8 :9.‘ !&q
L— ___N ' _,

(512-cell subdivision)

Figure 5.10. Distribution of the x-component of the induced electric fields
(magnetic mode) excited by an antisymmetrically impressed
electric field in a 4-cm phantom cube, and the comparison of
numerical results based on the 216-cell subdivision and the
512-cell subdivision.

75

numerical results for the electric mode of induced electric field.
Figure 5.10 shows the distribution of the x-components of the
induced electric fields, IEXI, inside the same 4-cm cube excited by
an antisymmetrically impressed electric field of'E.1 8 x sin kz at
750 MHz. Two sets of numerical results for [Exl are given in Fig.
5.10 for the 216-cell subdivision and the 512-cell subdivision. It
is observed that the results for the 512-cell subdivision deviate
significantly from that of the 216-cell subdivision, especially at
the outer layer of the cube. This implies a poor numerical conver-
gence, thus, to produce accurate results for the magnetic mode of

induced electric field the cell size should be smaller compared with

the case of the electric mode shown in Fig. 5.9.

CHAPTER 6

INDUCED EM FIELDS IN HETEROGENEOUS BIOLOGICAL

SYSTEM AND APPLICATION TO HYPERTHERMIA CANCER THERAPY

In Chapter 3-5, the induced electric field inside an irradiated
homogeneous biological system has been studied. In this chapter, we
proceede to check the validity of the theory when applied to a hetero-
geneous biological system. We have conducted experiments to verify
the theory, and then applied the theory to the hyperthermia cancer

therapy.

6.1 Comparison of experiment and theory in a heterogeneous biological

system

Two sets of experimental and theoretical results on the induced
electric fields in two different heterogeneous biological systems are
compared. We used the same experimental set up as we did for measuring
the induced electric field in the phantom model of human body. An
array antenna was used as a radiating source at 600 MHz. A phantom
model of rectangular box with dimensions of 6x12x2 cm was constructed
with thin plexiglass and filled with phantom material of appropriate
conductivity and permitdyity. A tumor (heterogeneity) was assumed to
be a part of the body (phantom material) that has a conductivity
differs ' from that of the body. As this experimental model was
irradiated, induced electric fields were probed over 18 locations in
one side of the body. Detailed dimensions of the model are depicted

in Fig. 6.1.

76

77

I A /
1‘7 W6
.1 \

)4.

Figure 6.1. Geometry and dimensions of the phantom model.

 

 

 

 

\/

78

Figures 6.2 (a) and (b) show the theoretical and experimental'
results of the induced electric field at the tumor inside the phantom
model varying as a function of the tumor conductivity. The tumor
with dimensions of 2x4x2 cm is embedded in the center of the body as
shown in the upper right hand corner of Fig. 6.2. At this frequency,
conductivity and dielectric constant of the body are assumed to be
4.5 S/m and 70, respectively. Four different tumor conductivities
have been considered while the dielectric constant of the tumor is
assumed to be equal to that of the body. In the experiment, each
time a phantom material of 2x4x2 cm dimensions with appropriate tumor
conductivity was packed at the center of the body (box) and then the
rest of the body was filled with a phantom material with conductivity
and dielectric constant equal to that of the body. The induced
electric fields were probed over 18 locations in one side of the body,
nine locations in the front and the back layer, respectively. Only
the induced electric fields at the tumor were plotted as a function of
the tumor conductivity because the induced electric fields elsewhere
were only slightly changed when the tumor conductivity was changed.
Figures 6.2 (a) and (b) show that the induced electric fields at the
tumor in the front and the back layer decrease as the tumor conductiv-
ity is increased from 2.0 S/m to 6.5 S/m. Corresponding theoretical
results on the induced electric fields at the tumor and other parts of
the body were generated. A good agreement between theory and experiment
was obtained.

In Figures 6.3 (a) and (b), the same phantom model is considered
but the location and size of the tumor are changed. Two tumors each

with dimensions of 2x2x2 cm are embedded inside each half of the body

79

 

 

 

 

 

 

 

 

 

 

 

 

 

Freq. = 600 MHz 0 = 4.5 S/m
g1 a 9 e-jkoz 8r = 70
(a) Front layer at x'O cm, y=l cm, z=0.5 cm “3
Theory
80 .“'
L ‘r- Exp.
E 60 F.
x
("W/m) 50 -
4O )-
30 l l 1 l l l
0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Tumor Conductivity 0t (S/m)
(b) Back layer at x80 cm, y=l cm, z=l.5 cm
80 L_
70 P
60 __
EX
(mV/m) 50 r
40 __
30 1 L_ I, 1 L, 14

 

 

 

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Tumor Conductivity ot (S/m)

Figure 6.2. Theoretical and experimental results of the
x-components of the induced electric field, E ,
at the tumor as a function of the tumor conductivity,
excited by a vertically polarized, travelling EM
wave of 600 MHz at normal incidence.

(a)

80

7O
6O
(mV/m) 50

4O

30

(b)

80

7O

(mV/m) 50

4O

30

80

 

 

 

 

 

 

 

 

 

 

Freq. = 600 MHz 0 = 4.5 S/m
E1 e-jkoz e = 70
r
Front layer at x=0 cm, y=3 cm, z=0.5 cm fit
_ O-— Theory
A- - - Exp .

I I I I I I

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Tumor Conductivity ot (S/m)

Back layer at x=0 cm, y=3 cm, z=l.5 cm

 

 

 

 

 

.— Theory
‘ A— - .EXP-
1 l i l J I
0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Figure 6.3.

Tumor Conductivity ot (S/m)

Theoretical and experimental results of the
x-components of the induced electric field, E ,

at the tumor as a function of the tumor conductivity,
excited by a vertically polarized, travelling EM
wave of 600 MHz at normal incidence.

81

as shown in the upper right hand corner of Figure 6.3. Conductivity
and dielectric constant of the body are assumed to be 4.5 S/m and 70,
respectively. Again four different conductivities of the tumor have
been considered. Figures 6.3 (a) and (b) show the theoretical and
experimental results of the induced electric fields at the tumor in
the front and the back layer, respectively,as a function of the tumor
conductivity. A good agreement is again obtained between experiment

and theory for this case.

6.2 Hyperthermia in animal and human bodies induced by EM fields

One of the promising therapies for cancer is that of hyperthermia
in combination with chemotherapy or ionizing radiations. When the
temperature of a tumor is raised a few degrees above that of surrounding
tissues, accompanying chemo- or radio- therapy has been found to be
effective in treating the tumor (15,16,17). In the combined cancer
therapy, the objective is to find a noninvasive method by which to heat
the tumor without overheating other parts of the body.

A convenient means of heating embedded tumors in a biological
body noninvasively is to utilize the EM radiation. When an EM field
of a certain frequency is applied in a particular manner to a biological
body with an embedded tumor, it is difficult to predict the distribution
of the induced field inside the body because the body with the tumor
represents electrically a finite heterogeneous body. Thus, it is a
non-trivial engineering problem to construct an effective scheme for
local EM heating. In general, an effective local EM heating of embedded
tumor depends on the following factors: (1) the type of EM irradiation,

part-body or whole-body; (ii) the frequency of the EM field; (iii) the

82

type of applied field, electric, magnetic or electromagnetic; (iv) the

location of the tumor inlfluybody; (v) the conductivity and permittivity
of the tumor relative to that of the surrounding tissues; and (vi) the

heat diffusion from the tumor.

The purpose of this study is to theoretically predict the heating
pattern induced inside the body with the tumor when it is irradiated
by various EM fields under certain schemes, some of which are commonly
used for the hyperthermia purpose. From these theoretical results, the
effectiveness of various EM heating schemes can be assessed. We will
theoretically quantify the induced electric field and the specific
absorption rate (SAR) of the EM energy in the theoretical model of a
biological body with an embedded tumor under various schemes of EM
irradiation. The scheme which can induce a localized high SAR in the
tumor while maintaining low SARs in the surrounding tissues is considered
to be an effective scheme. In the present study, the heat diffusion
from the tumor will not be considered. It is well known that the heat
diffusion from the tumoris poor because of a sluggish blood supply to
the tumor. This phenomenon is advantageous from the viewpoint of
maintaining hyperthermia at the tumor.

We consider, first, the part-body irradiation with HF electric
fields. -We found that this scheme is effective for internal tumors,
especially those with lower conductivities. We also found that this
scheme of irradiation cannot selectively heat surface tumor. Secondly,
we consider the irradiation with microwave or UHF EM fields. At these
frequency ranges, application of a localized EM radiation at the tumor
may create hot spots at various locations away fromthe tumor, instead

of heating the tumor.

83

6.3 Theoretical model of a biological body with tumor

In this study, we use biological bodies of simple geometries to
simulate animal and human bodies with embedded tumors. The body is
considered to be homogeneous with certain electrical properties and
the tumor is assumed to be a local region with a conductivity that
differs from that of the surrounding tissue. The permittivity of the
tumor is assumed to be the same as that of the surrounding tissue.
These assumptions are supported by recent in 2139 measurements of
electrical properties of tumors in mice by Bordette 25 g1. (18).
Another reason for the assumption of the same permittivity for the
body and for the tumor is that effect of the permittivity on the
induced electric field in the body is insignificant, especially in
the lower frequency ranges.

The body is assumed to be partially or wholly irradiated by EM
energy of various frequency ranges including HF, VHF, UHF, and
microwave.

The first step of our study is to determine the induced electric
field inside a heterogeneous body, consisting of a homogeneous body
with an embedded tumor of different electrical properties, as induced
by the applied EM field. After this quantity is obtained, the SAR of
EM energy in the tumor and at any other'pointhf the body are determined.
An effective EM heating should induce a localized, high SAR in the

tumor and low SARs in the surrounding tissues.

6.4 Part-Body Irradiation with HF Electric Field
The electric fields of the HF range (3 to 30 MHz) maintained

between two capacitor-plate electrodes have been used to heat embedded

84

tumor in animals (19) and in human bodies (20). We will analytically
show that this scheme of EM heating is very effective for internal
tumors embedded in the central part of the body, however, this scheme
cannot provide a selective heating for surface tumors. We will also
show that the tumor conductivity relative to that of the surrounding
tissue plays an important role in this type of EM heating.

The first example is a simulated animal body with the dimensions
of 6x6le cm having a tumor of 2x2x4 cm located at the center of the
body and under the partial irradiation of a uniform electric field
as shown in Fig. 6.4. A uniform electric field (E1) of 1 V/m
(max. value) at 15 MHz is applied across the top and bottom of the
body and only over the area of tumor (2x4 cm).

For the numerical calculation of the absorbed power density, the
body is divided into 27 2-cm cubic cells as shown in Fig. 6.4. The
tumor occupies the 13th cell and its image and has a conductivity
(at) of 0.31 S/m while the conductivity of the body (0) is 0.62 S/m.
The dielectric constant (er) of the tumor and the body is assumed to
be 150.

The distribution of SARs in this simulated biological body is
shown in Fig. 6.5. The SAR in the tumor reaches a maximum value of
150.9uW/m3 while the immediate neighboring cells, the 10th and the
16th cell, only have a value of 44.8 uW/m3. If the tumor conductivity
is increased from 0.31 S/m to 0.496, 0.62, 0.744, and 1.24 S/m, the
SAR in the tumor will decrease from 150.0 uW/m3 to 103.7, 84.9, 71.7
and 43.8 uW/ms, respectively. The results for this case are summarized
in Fig. 6.6. The absorbed power density at other parts of the body

is only altered slightly by the change in the tumor conductivity.

85

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mo mane use um>o was atom one «o sauuon tam aou one mmouum va hoaaavoum
um A03Ha> .xmav a\> H mo namv vHon cauuoaam showed: a he amumavmuuw

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O-.. C‘.-. ”b C..-

         
 

> a
s as

 

 

 

 

 

 

 

69-h, .
.

V----?---‘

 

 

 

     

 

 

--J---.

 

 

 

-"J--

 

86

When 0t 3 o, the SAR in fluatumor is about twice that of the surround-
ing tissue. If at = 0.50, the SAR in the tumor can be about four
times that of the surrounding tissue. The important point not to
overlook is that even the tumor conductivity is considerably higher
than that of the surrounding tissue, the SAR in the tumor is still
higher than that in the surrounding tissues. This result implies

that an electric field of the HF range maintained by a capacitorhplate
applicator can be used to selectively heat the internal tumors of
various kinds as long as the tumor is located in the central part

of the body.

The examples show in Figs. 6.4 to 6.6 are for a simulated animal
body with an internal tumor. The case of a human body with an internal
tumor is considered next. Figure 6.7 depicts a human body with an
internal tumor of 5x5x5 cm embedded inside the upper torso. A uniform
electric field of 15 MHz with the intensity of l V/m maintained by a
capacitor-plate applicator is applied over the tumor area across the
body as shown. For the numerical calculation ofthe induced electric
field, only a volume of the body with dimensions of 15x15x20 cm
centered around the tumor is considered because the fringe field is
small in other parts of the body. This volume of the body is divided
into 36 5-cm cubic cells stacked in four layers, and with the tumor
occupying the center cell of the second layer. For this particular

example, we assume that o = 0.5 o = 0.31 S/m, and at = 150. From

(2
the distribution of SARs shown in Fig. 6.7, it is observed that the
SAR in the tumor reaches a maximum value of 417.8 uW/m3 while that

in the cells immediately above and below the tumor are 98.2 uW/m3

87

   

 

 

 

i «15MHz
Ii -1V/m§

 

9
4’.

  

 

SARs (uw/m3)

 

 

 

 

 

 

    

 

 

  

    

mm mm
I I
S S AV“;
0.. n .m m.
. . fl /
0 0 l as
C2
I I I “.06
U I r be
T 8 O a
.1
.00
d e.
e m
t/
av
1
m1
.1 =
S
.1
a+E
. /J a...
a
.1
var rev mm
0. ... I «.1 J .1 U
v.0...v0 m
IIII‘IOIOJ'UIOO =
4.. e Sf
f n
.e
n W
O
.1/4
t 0
U6
b
.1 00
fl...
EF
3
if
D 0
5
6
e
r
U
Go
.1
F

= 150.

0.31 S/m, and Er

0'
t

88

 

 

 

i = 15 MHz
'E'- lV/m 32
160 t.
a’- 0. 62 S/m
' SARs in tumor 8". 150
120 _ ’ 0? variable
a?
E
3
3 so -
O:
3‘. SARs In neitlyshboring
«40 .. -------------------
O"
I II J I I
o, 0.3 0.6 0.9 1.2 1.5
crt (Slm)

Figure 6.6. SAR in the tumor and in the neighboring cells varying as

a function of the tumor conductivity (at) for the case

of f = 15 MHz, ‘3” = 1 V/m x, o = 0.62 S/m and er = 150..

89

and 234.1 uW/m3, respectively. The fringe field in the vicinity of
the tumor is insignificantly small. We have also calculated the SAR
in the tumor for different tumor conductivities: when 0t - o - 0.62
S/m, the SAR in the tumor is 235.9 uW/m3, and if ot - 20 - 1.24 S/m,
the SAR in the tumor becomes 121.9 uW/ma. From these results, one
observes that an embedded tumor inside a human body can also be sel-
ectively heated by a HF electric field produced by a simple capacitor-
plate applicator.

Up to this point, we have demonstrated that a uniform electric
field in the HF range can be used to selectively heat an internal
tumor embedded inside an animal or a human body. We will now show
that this type of electric field can notixautilized to selectively
heat surface tumors because it heatstimatissue immediately below the
tumor excessively.

Figure 6.8 shows the distribution of SARs in the simulated
animal body depicted in Fig. 6.4, but with the tumor located in the
middle of the upper body surface, when the same uniform electric
field of 15 MHz with the intensity of 1 V/m is applied over the tumor
area and across the body. For this example, we assume that at - 0.50

= 0.31 S/m and Er = 150. It is observed in Fig. 6.8 that the SAR
in the tumor is 80.3 uW/m3 while that in the cell immediately below
the tumor is 84.7 uW/m3. If the tumor conductivity is higher than
0.5 o, the SAR in the tumor decreases further while the SAR in the
cell immediately below the tumor remains about 85 uW/m3. This
phenomenon is summarized in Fig. 6.9. It is observed in Fig. 6.9 that

the SAR in the tumor is about the same as that in the neighboring

90

I I Body

 

SI
4—— I layer

/'
/ 4— 2ndlayer
/

4— 3rd layer

 

     

/ *— 4Ih'°,.'

 

 

 

 

 

 

 

 

 

 

 

 
  

235.9 )LW/ma if (€0.62 S/m
121.9 [.w/m3 if 6.51.24 S/m

SAR at Tumor =

Figure 6.7. Distribution of SARs inside a human body with an
embedded tumor when f = 15 MHz, E1 = lV/m x,
O = 0.62 S/m, at = 0.31 S/m, and Er = 150. The
SARs in the tumor for the cases of O = 0.62 S/m

t
and at = 1.24 S/m are also given.

91

cell if 0t 3 0.5 o, the SAR in the tumor is reduced to about one
half of that in the neighboring cell if 0t 5 o, and if %:> o , the
SAR in the tumor decreases further while the SAR in the neighboring
cell remains relatively unchanged.

From these results shown in Figs 6.8 and 6.9, it is evident that
a uniform electric field in the HF range maintained by a capacitor-
plate applicator cannot be utilized to selectively heat a surface
tumor without severely heating the tissue immediately below the tumor.
This situation may be somewhat improved by increasing the area of
the lower electrode of the applicator in such a way that the-induced
current starting from the upper electrode flows through the tumor
and then diffuses into the tissue below the tumor before it reaches

the lower electrode. The reduction of the current density in the

neighboring tissue will cause a decrease in the SAR and the heating.

6.5 Hyperthermia with Microwave or UHF Irradiation

In this section, we aim to show that EM fields of the UHF to
microwave range (e.g. 500 toIMXNiMHz) should be carefully applied to
a biological body to induce a local heating at the tumor. An improper
scheme of irradiation may cause severe heating at locations away
from the tumor. This problem is essentially caused by the fact that
electrical dimensions of experimental animal such as rats or mice
are in the "resonance region" of this frequency range. Thus, hot
spots may be induced at unintended locations inside the body even
though only the tumor region is irradiated. Figure 6.10 shows the
simulated animal body with a surface tumor as considered in Fig. 6.8

being irradiated by a microwave of 2.45 GHz in a waveguide. Assuming

SAR: (uwIm3I

 

Figure 6.8.

92

 
    

 

   

C---'--...

 

'7
o
J
I
O

 

 

 

 

 

f - 15 MHz
1 Vlm I
0. 62 Slm
0. 31 Slm
5r - 150

"it.
I

q

 

.3

 

Distribution of SARs inside the simulated body of
Fig. 6.4, but with the tumor located at the body
surface. Parameters are: f = 15 MHz, E1 = 1 V/m x,

G 8 0.62 Slm, at = 0.31 S/m and Er = 150.

93

 

 

f = 15 MHz
15' = lV/m SE
160 - cr= 0. 62 Slm
6r= 15o
(Tt = variable
120 -
IE SARs in neighboring cells
E
3 80 -
CE
5.
40 -
o 1.5

 

o-t (Slm)

Figure 6.9. SARs in the surface tumor and in the neighboring cells
varying as functions of the tumor conductivity (at).
Other parameters are: f = 15 MHz, E1 = l V/m x,

O = 0.62 S/m and er = 150.

94

Biological body

 

 

 

 

 

 

--‘

  

 

 

   

 

 

4------

 

-.(...

I

i
~4-f--

l

i
T..-

i

i
Waveguide

L.)—

I

i
fiPOLC-J- )-

incident , ~.
wave \\l" .- ' " .
_ i 1

Figure 6.10. The simulated body of Fig. 6.4 with a surface tumor

is irradiated by a microwave in a waveguide.

95

SARs (mWImB)

Pa rtiai-Body
Irradiation

   

 

 

 

 

 

 

 

 

 

f = 2.45 GHz
is“ I Wm?
6=zmsm
0’t -- l.| Slm
er = 47

3 . _
SAR at Tumor . 0.2 lem3 ti 0'. - 2. 2| Slm
0.3 lem IfO’t =4.42 Slm

Figure 6.11. Distribution of SARs inside the simulated body of Fig. 6.10.
Parameters are: f = 2.45 GHz, E1 =1V/m y, o 8 2.21 S/m,
ot a 1.1 S/m and er = 47. The SARs in the tumor for the

cases of at = 2.21 S/m and at = 4.42 S/m are also given.

96

that the tumor is placed in the center of the waveguide so that

the maximum electric field of TE10 mode is incident upon the tumor.
For this arrangement only one third of the middle section of the
body is irradiated by the microwave. The cells being irradiated are
the lst, the 4th, the 7th, the 10th, the 12th, the 16th, the 19th,
the 22nd and the 25th, and their image in the other half of the body
(see Fig. 6.4). The incident electric fields to these cells are

assumed to be that of TE mode with an intensity of l V/m in the

10
y-direction. We also assume that o = 0.5 o = 1.1 S/m and er 8 47

t
at 2.45 GHz. Under this irradiation, the distribution of SARs in
the body is shown in Fig. 6.11. It is surprising to observe that
the SAR in the tumor is, in effect, a minimum value of 0.1 mW/m3
instead of an expected maximum. The highest SARs reaching a value
of 5.8 mW/m3 are induced in the regions not irradiated. Other high
SARs are also induced in various point of the body. We have also
calculated for the cases of ot = o = 2.21 S/m and 0t = 2 o = 4.42
S/m. The SAR in the tumor for these two cases are still very small
at 0.2 mW/m3 and 0.3 mW/ma, respectively, while high SARs are observ-
ed at regions away from the tumor. This unexpected heating pattern
is due to the resonance phenomenon induced in a 6x6x12 cm biological
body by a 2.45 GHz microwave. This example implies that to irradiate
experimental animals such as rats with a microwave of 2.45 GHz, the
potential resonance phenomenon should be taken into account. To avoid
this phenomenon, the surface tumor may be drawn through a slot on a

shielded animal strainer and the microwave is then applied exclusively

to the tumor (21). It is also noted that mice irradiated by the

SA Rs (mW/m3)

Whole; Body
I rradIatIon

 

97

   
  
  

   

 

 

I
unr- ----'J--

 

 

 

 

 

f = 600MHz
Ii - l 9'ij Vlm 32
'O’= l.48 Slm
q=dmwm

er-sa

 

 

SAR at Tumor- 3-6 mW/m3 if 0? =I.48 Slm

Figure 6.12.

3.5 mm3 if 0; =2.96 Slm

Distribution of SARs inside the simulated body of Fig. 6.4

with a surface tumor under the whole-body irradiation.
Parameters are: f = 600 MHz, E1 - l e-jkz V/m x, o - 1.48 S/m,
0t - 0.74 S/m and Er = 53. The SARs in the tumor for the cases

of 0t - 1.48 S/m and 0t = 2.96 S/m are also given.

98

scheme of Fig. 6.10 at 2.45 CHz (22) will not exhibit an unexpected
heating pattern because mice are electrically much smaller and a
resonance is not possible at this frequency.

One example is given to show the distribution of SARs in the
same simulated body with a surface tumor induced by a UHF field under
the whole-body irradiation. This example is depicted in Fig. 6.12
where a plane EM wave of 600 MHz with an electric field of l V/m in
the x-direction is incident upon the front surface of the body where
the tumor is located. For this example, we assume that ot = 0.50
= - 0.74 S/m and 8r = 53 at 600 MHz. From Fig. 6.12, one observes a
rather uniform distribution of SARs throughout the body with higher
SARs induced in the rearside of the body. No peaking of the SAR in
the tumor is observed for this case. The SAR in the tumor is in-
creased to 3.6 mW/m3 if ot = o = 1.48 S/m. However, when 0t ' 2 o =
2.96 S/m, the SAR in the tumor decreases to 3.5 mW/m3. The SARs in
other part of the body are only affected slightly by the change in
the tumor conductivity. From this example, it seems that the maximum
SAR in the tumor is obtained when ot = 0. However, this scheme of
irradiation cannot selectively produce a peak SAR in the tumor even
though the electrical properties of the tumor may be significantly

different from that of the surrounding tissue.

CHAPTER 7

TENSOR INTEGRAL EQUATION METHOD COMBINED WITH ITERATION
TECHNIQUE FOR QUANTIFYING INDUCED EM

FIELD IN BIOLOGICAL SYSTEM

7.1 Introduction

The tensor integral equation method has been applied to solve
many problems involving the interaction of EM fields with I
biological system. Although this method has been powerful in many
problems, it has some difficulties. The major difficulty is on the
numerical convergence when it is applied to an electrically large
body. In order to generate accurate numerical results, it is necessary
to divide the body into a large number of electrically small volume
cells. This, in turn, leads to an unmanageably large number of
unknowns in the numerical calculation. Since a conventional computer
may have difficulty in inverting a matrix larger than 300 x 300 due
to storage limitation, it is desirable to devise schemes to extend
the tensor integral equation method to handle a body consisting of a
very large number of cells, while sidestepping the problem of computer

storage limitation.

7.2 Theoretical Development

We have develoPed a scheme which combines an iteration process
with the tensor integral equation method. This method is not a simple

numerical average process; it is a process consistant with Maxwell's

99

100

equations. This method is explained here.

1. As the first step, we subdivide an irradiated body as shown
in Fig. 7.1 into N cells,where N is the maximum number of cells which
can be handled by the computer with a reasonable cost. We then quantify
the induced electric field in the body with this N-cell model based on the
tensor integral equation method. The numerical results of this N-cell
model will be considered as the first-order solution for the induced
electric field.

2. Each of N cells will be subdivided, one at a time, further into

th

8 subcells as shown in Fig. 7.1. Let us exclude the m cell from the

body temporarily and consider its 8 subcells, m1,m ,.. ..... ,m

2 8'

3. Next, in the absence of the mth cell, calculate the equivalent

incident electric fields at the centers of the 8 subcells, located at

f , f , ......., f . The equivalent incident electric field at the

ml m2 m8

center of m1 subcell, or at ffii, is equal to the sum of the original
incident electric field at ¥fi and the scattered electric field maintained

1
by the first-order induced currents in the N-l cells (the mth cell

excluded) at ffi . That is

i
+inc + +inc + +5 +
Eeq (rm ) = E (rm ) + E (r ) (7.1)
i 1 “'1
and
—> + + 0
ES (r ) = j J (39).“? , 'r") dv’, i = 1,2,...8
m eq m
1 i
V Avm
where

Ziqu’I) = u?) E(ii') (7.3)

101

 

 

 

I ’ I
1’ 2’ . mth cell
F--"‘f
I I "/
a- .--” ' ...... A
’1 I. m. ,’I
I ’ I
’I I ' -(’ I
r ----- ’ . : total cell no. - N
I ‘5 '. l J total subcell no. - 8N
*w : a t I I”
. -_-- m6 ___J.-.V’
f . ' I
i l i 1 i I
l I I"
: l i/ I
I ___,_-|r '
Inc—«D-OT- : :
I I . 2'
I l ’
L ..... I/
-inr.

Fig.7.1. An irradiated body is subdivided into N cells, and each
of the N cells is then subdivided again into 8 subcells.

102

4+
G(¥§i, f') is the tensor green function. Scattered electric

field at f; can be written as

1

+3 -+

E (rmi) =

V-AVm

1(9) E(I').‘é(¥m , 15) dv'

1

++ 9+ +
We may represent the inner product E(r') - C(rmi, r') as

 

 

(7.4)

 

 

9+ 1' ++'
where C(rm ,r ) is the 3Nx3N matrix and E(r ) is a 3N column matrix

T (r

+ +
m .r')

i
with zero values for matrix elements corresponding to the mth
Let
x1 = x, x2 = y, x3 =
then Gx is given by
P q
l 2
+ + ’3
G (r .r') = -M [5 +‘2—"'—]
k x x
xpxq Eli 0 Pq 07¢]? p

Peq

1

1,2,3

cell.

(7.6)

103

+s+
Each scalar component of E (rm ) may be written as
i

ES (3? ) = f u?) [ >33 0 (3? fr") E (15)] dv'
X m X X m X
p i q=1 p q i q

V—AVm
p = 1,2,3 (7.7)

We can transform eq. (7.4) into a matrix equation by using the method

of moments. The scattered electric field maintained by the first-order

induced currents in the N-l cells (the mth cell excluded) at :5 become
1
3 N
E: (3? ) = 2: z [T(¥n) (2.xx (r ,f') dv'] E (r)
p mi q=1 n=1 p q i q
11*111 V
(7.8)
Let
m n
l _ '* "*I I
Gx x - T(rn) Gx x (rm ,r ) dv ’ m i n (7.9)
pq pq i
V
n
As a first approximation, we have
min
G - G (+ I) [IV (710)
x x - T<rn) x x rm ’rn n ' '

104

where

vn = f dv' (7.11)
V

n
By using eq. (7.6) to evaluate G (I ,f ) gives
x x m n
p q i
m n
i _ + 3
x x - -jmuoko T(rn) Avn exp(-onm n) / 4flqm n
P q i i
2 min m n
. [(qm n -1-jam n) 6 + cos 6x cos 6
1 1 Pq p xq
. (3-0I2 + 3ja )] m 1‘ n (7 12)
m n m n '
i i
where
+ +
am n a koR'm n ’ Rm n = 1‘In - rnI
i i i 1
m1 “I n
n (x -x ) m n (x - x )
x R n x R.In n
P mi q 1
+ m1 m1 m1 -+ n n n
rmi = (x1 . x2 . x3 ). rn - (x1. x2. x3)
then
3 N m n
+3 +, 1
Ex (rm ) - X 2 Gx x Ex (Ifi) (7.13)
p 1 q=1 n=1 p q q

“*1“ i = 192,044.,8

105

After the first-order solution of the.induced electric field is

+

obtained in the first step. Then the scattered electric field at rm
. i 1

can be computed by eq. (7.13). And, equivalent incident electric

field at rm can be written as
i

N m n
+inc (13m) g Einc (gm) + Z ‘5 1 (gm’;n) . E(irn) (7.14)
eq 1 1 n=1 1
min

4. After equivalent incident electric fields at the centers of
8 subcells are calculated based on the first-order solution of the
induced current, we can consider the mth cell as an isolated body
being irradiated by these equivalent incident electric fields at 8
points inside the mth cell. We have subdivided the mth cell into 8
subcells and the incident electric fields at the centers of these 8
subcells are known, therefore, the induced electric fields intthese 8 sub-
cells can now be readily determined based on the tensor integral
equation method.

5. After each of the N cells is treated separately following
the above steps, the second-order solutions for the induced electric
fields in the body are determined.

6. If more acculate results are needed, each of 8 subcells can
be subdivided further into 8 subcells again and the iteration process
can be repeated.

The important advantage of this technique is that since the

iteration of the induced electric field of each cell is carried out

seperately, it does not require the inversion of a large matrix and,

106

then, computer storage problem is sidestepped.

7.3 Example

An example is given here to show interesting results. Figure 7.2
shows the numerical results on the x-components of the induced electric
fields in a muscle layer of 4x2x0.5 cm irradiated by a 1 GHz plane wave.

Due to symmetry, only 1/4 of the body is considered, and it is
subdivided into 8 cells, 64 cells or 8 cells with iteration process in
the numerical calculation. The incident electric field is l V/m (max.
value) and the body conductivity and dielectric constant are assumed
to be 1.62 S/m and 50, respectively. It is observed in Figure 7.2
that a significant improvement in numerical results is obtained with
the 64-cell model or the 8-cell model with iteration process over the
8-cell model. It is also noted that numerical results based on the
64-ce11 model and 8-cell model with iteration process are quite close.
The numerical results based on these three models are graphically
compared in Figure 7.3, in which the x—components of the induced
electric fields along four lines in the body are shown. From Figure
7.3, it is evident that numerical results based on the 64-cell model
and the 8—cell model with iteration process are quite comparable. The
most significant difference, however, is in the computation cost: with
the 64-ce11 model the computation cost was $40, but it was only $14
with the 8-cell model with iteration process. It is to be emphasized
that saving computation cost is one big advantage with the iteration
technique, but the more important point is that with the iteration
technique the body can now be divided into many more cells without

encountering the computer storage problem.

107

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

X
JI
0.5cm I =1 2
J J J --.2.-.Ir"/
IO.|3I Io.Iso f I ,.
I ' r.--L-.Jr
5 2
+1 .. 00‘"
Jo.I91:o.2lo B “° 4cm 2 '.
'r......I.....1 'L——J"""Y
' ->
_ I
(8 cell m0d21):o,234:0,243 IEXI
I
E .... z.--“ fiinc
20.25! :0.259
2----2----2 Fm...
(front layer) (back layer) 2 . 1 GHz
7 r7 ' a - 1.62 S/m
2'1“.” Ill..." $912.12: .I.l_:._l_4 c - SO
:.I;IJ. I1..I7J.I9 .362 ISL-29' 2! ’
IExI L23, 23921211223 (MI-cell model) :3. -L- .Ei.-'-- IEXI
;.23..232. 23:22 2. 23;. .2222:
g.. --L -- -- u.-- --
2 .-.2'2.2' 3222.54.22. 5:2.51.“2..-31J24.
---.2I' 21129 3.222222332122222
Leia; 225.21 5.2.222-!.-2.2.'-.2.2.
Lax-2.2222. Iii-2.2.22.2.E-32.
(front layer) (back layer)
129229129221 !.II_;. IoKII 2 .I4
'I_L_'.l.6 .IB‘ .l9 [n.lsrlms
2 —-r 1'“
13.52293 22 lth' .I_BII22 2.I
IQ .2I_I.2I .23J.24IB—ce11 with iteration) Fl. .22i.23_§._ .24 IEXI
.33J-- 2’2 233.2235. £2523? 232' 2112':
”L252. 25%? l. 25?. 25 .26I.21
£322.29 2.13.3.3. iisJ-fi .25.J: ?§J. 39

 

Fig.7222 X components of induced electric fields inside a muscle layer of
6x2x0.5 cm irradiated by an 1 GHz EM wave, numerically computed
when 1/6 of the body is subdivided into 8 cells, 64 cells, and 8
cells with interation process.

108

 

0.5
s
--..... B-cell with iteration /
I f - 16R:
-——6b-cell model a _ 1.62 S]-
-----8-cell model Einc ‘ " 5°
' nc - 31 VIII
a fly
flint

 

 

+....2cm._‘.'

1. Along y-0.25cII, s-0.125cn

 

 

 

 

 

 

 

   

4‘ Along z-OJScIII z-0.125cm
:- P
3 h 3 b
P P
x 2 b 2 ’
(cl) - 2-
l P 1 "
o 1 o l L l
0 0 0.1
+ +
ls‘l (V/m) lsxl (V/m)

 

 

X
(cm)

 

 

 

 

 

 

 

Ifixl (v/n) Ifixl (V/In)

Fig,‘7.3.Distributions of the x components of induced electric fields inside
a muscle layer of 432:0.5 cm irradiated by an 1 CH: EH wave, numeri-
cally determined when 1/4 of the body is subdivided into 8 cells, 64
cells and 8 cells with iteration process.

109

It is noted that the method described in this chapter is
particularly useful in the EM local heating of a body where EM energy
is concentrated at a local region and detailed distribution of the
absorbed power density in that local region is needed. For this
problem, only the induced electric fields in that region need to be

iterated and, thus, very accurate results can be obtained.

CHAPTER 8

GENERALIZED TENSOR INTEGRAL EQUATION METHOD

FOR BODIES WITH ARBITRARY ELECTRICAL PARAMETERS

8.1 Introduction

The existing tensor integral equation was formulated for nondmagnetic
conducting bodies such as usual biological bodies. In some biological
applications, it may be feasible to introduce notoxic magnetic powder
into a local region of the body so that when the body is irradiated
by an EM field or a magnetic field, the absorbed power density at the
local region is enhanced. To analyze such a system the existing tensor
integral equation can be generalized to handle a body with an arbitrary
permeability in addition to arbitrary conductivity and permittivity.
This generalized method will also be useful in the study of the inter-
action of EM fields with magnetic materials in solid—state electronic

area or in other related fields.

8.2 Theoretical development

Consider a finite biological body of arbitrary shape with arbitrary
electrical parameters characterized by 0(f), C(f) and u(f), illuminated
in a free space by an incident EM wave with an electric field Ei(;)
and a magnetic field Ri(¥). When a biological body is illuminated by
the incident EM field, it creates a distribution of induced charges and
currents throughout the body. The induced current in the body includes
three types of currents; the conduction current, the polarization

current and the magnetization current. These charges and currents

llO

111

produce a scattered field. The total EM field inside the body is the

sum of incident field and the scattered field:

EEG?) = E165) +ES(¥) (8.1)
EEG) = $316?) +fis(?) (8.2)

As developed in Chapter 2, the scattered electric field maintained by
the conduction and polarization currents can be determined from the

following equation (2,3):

+3 + + +' 4+ + +' ' 3936?)
E (r) = PV Jeq(r ) ' G(r,r ) dv - 3jwe¢ (8.3)
V

where

 

jqu) = [o + jw(€-eo)] E(f)
C(rfi') = -jwu0[‘I + 171—2— VV] d) (r,r')
o
+ + -jk|r-—r'
¢(r.r') = e _, _,
4nlr—r'l

+
Let's introduce new notations for the equivalent current Jeq and the

tensor green function relating to the electric field as:

386?) = [mm (e—eo)] E(E’)
'Ce(?.?') = -jwu [‘1 + —-l— W] ¢(? 3?)
e O k 2 ’

112

red’) = {om +jm[e(‘r’)-eo]}

*S

The scattered electric field E (f) in equation 8.3 can then be written

as

Te (BEG?)

3jw€o (8'4)

E36?) = pv f red?) E(f') ‘5: (1%?) dv' -
V

The scattered electric field maintained by the magnetization current

can be determined from the magnetic vector potential as

E36) - viz—MM?)

E

_ if v x [2:3 (P) ¢(?,’r")] dv' (8.5)
V :11

By using a vector identity V x (OK) = ¢VXK + vwa and let w = ¢(;,¥')

+ ++
and A = J (r'),equation 8.5 becomes
m

"1556?) = -] [¢(¥,¥') \7 x 3111‘?) + v¢(¥,'r") x 3m(‘r")] dv' (8.6)
v
+ +
Since V x J(r') = 0, equation 8.6 reduces to
ES(?) = -/ V¢(¥,‘r") x 3m('r") dv' (8.7)
v

9 ++
From a tensor identity (I x A) - E = X x D, we let A = V¢(r,r') and

113

B = 3fi(;'), equation 8.7 then becomes

ESG) = -f [*f x V¢(¥,?')]- 3* d") dv' (8.8)
V m
Defining
836,?) = -‘ix val-’3')
3m(r') = rm(1~”)fi(¥')
where

Trudi) = jw[u(?) - “0]

Equation 8.8 can be rewritten as

836’) = f rmG') fiG') ”6:630 dv' (8.9)
V
By substituting equations 8.4 and 8.9 in equation 8.1, we obtain

an integral equation for the induced electric field as

 

] EEG) - PVf T G") ‘86:”) .‘59 (1",?) dv'
O V e e

4+
- f “r 6') 135') ~ Ge ($.15) dv' = EH?) (8.10)
m m
V .
By induction, the scattered magnetic field maintained by magnetization
current can be obtained as

Imam)
3jwu

0

i536?) = PVf 11nd") in?) - 6:6,?) dv' - (8.11)
V

114

where
0)“; + + . 4+ 1
Gm(r,r') = -JwEO [I + :7 VV] ¢<¥,¥')

0

And the scattered magnetic field maintained by the conduction and

polarization currents can be obtained as

fisé’) =fv 166:") ‘86:") 35:6,?) dv' (8.12)

where

‘1’ x val-’3')

C)
A
H

U
H
\:
II

By substituting equations 8.11 and 8.12 in equation 8.2, we obtain an

integral equation for the induced magnetic field as

 

T G") + -> —> + + + +
[1 + gljwuo] H(r) - PVfV Tm(r') H(r') - E:(Lr') dv'
-j r ('9) ER?) WE‘RE?) dv' = 13%?) (8.13)
V e e

1(f) and a

If an incident EM field with an electric field E
magnetic field fii(;) are defined, the total induced electric field
E(;) and the total induced magnetic field E(f) inside the body can

be determined from the following two coupled tensor integral equations.

115

T6?) + + + + +
[1 + —e———]E(r) - PVfTeG') E(r') . Ee(r,r
O

3jw€ e
V
— [I (’E') 156-") -"ée('r’,¥') dv' = ER?
111 m
V
+

 

Tm(r) + + -> + + + +
[1 + 333on] H(r) - PV f1m(r') H(r') ' fihfi') dv'
V

-fre(‘r") E(r") WEEK?) dv' = 1516?)
V
where
Te(¥) = o(¥) + jw [8(f) - so]
rmd’) = jw [11(2) - no]
%:(+,r') = ~3mu [I + —;—x7v ] ¢(r,r')
k
0
afiéfi') = -jw€o [I+-};—%—VV] 8G,?)
0
+5:(r,r') = I x V 0G,?)
33%?) = 3 x v 86.5,?)
-jko|?-?'|
”(r,r') = e

 

(ml-11?]

(8.14)

(8. 15)

116

+
I - identity tensor

k I (1.1/us
00

0

PV symbols mean the principle values of the integrals.

These two coupled equations are more complicated than the tensor
integral equation treated in the previous chapter. It is evident that
they can only be solved numerically for a finite body with an irregular
shape. By using the method of momennswe can transform these coupled
integral equations into a matrix equation. After that the induced
electric and induced magnetic fields are obtained by inverting the
matrix. Computer program for this problem is explained in part 3 of

Chapter 9.

Tranformation to matrix equation

We may represent the inner product of EC: in Eq. 8.14 as

 

 

 

 

’ . . ‘ F
Ge (EH?) Ge ("£3") Ge (1?) E (9)]
e e e X
xx xy xz
-+ 4+ + .
E(r)-ce(r,}") = Ge ($.15) Ge ($.15) Ge (12?) E (9)1
e e e e y
yx yy yz
6: (r2?) 0: (m c: 63') 828')
_ zx zy 22 d L J
(8.16)
Let
x1 = x, x2 = y, x3 = 2

Then G: (;,;') is given by

x x
P q

117

 

 

 

 

 

e 1 ’32 1
t a _ ~ - 1
Ge (r.r> quo[5pq 2 0x 3x ¢(r.r)
x x k
99 0
P.q = 1.2.3 (8-17)
+He
We also represent the inner product H-Gm as
' e e e '+'+ T P -+ T
G (r,r') Gm (r,r') Gm (r,r') H(r')
xx xy xz
fi(¥').‘c*;(‘r’,?') = Ge (11") a: (1?) (:8 (r,r') HyGF')
yx W Y?-
a; (r,r') a; (r,r') c; <‘r’.r') 828')
L zx zy 22 J L
(8.18)

The scalar components of Eq. 8.14 may be written as

 

+
T (r) 3
[1 + 3:008 JEX (“r’) - PV freG') 2 c: ($.15) Ex (15) dv'
o p q=1 x x q
v p q
+ 3 e + -> -) 1 ->
- Tm(r') Z Gm (r,r') Hx (r') dv' = Ex (r)
=1 .
q xpyq q P
V
p = 1,2,3 (8.19)

We can transform eq. 8;uiinto a matrix equation by using the method of
momencL ‘We partition the body into N subvolumes and assume that E(f),
E(f), Te(;)’ Tm(;) are conStant in each subvolume.

Requiring that eq. 8.19 is satisfied at each 3%, the center of the mth

118

subvolume Vm, we have

 

 

P q=1n=
V P q
n
3 N +
.E (r)- 2 Z [T (r) [G (fm,r') dv']
=1 =1 x
q q n v qu
n
+ 1 a,
.Hx (rn) — Ex (rm) (8.20)
q P
Defining
mn
“3 = I " +' ' 8. 1
Gex x Te(rn) PV ‘Jr C x x (rm,r ) dv ( 2 )
P q V P q
n
and
mn
e - + e + +' '
Gm - Tm(rn) .jr Gm (rm,r ) dv (8.22)
x x x x
P q V P q
n
We can write eq 8 20 as
3 N Inn T (r)
2‘. z [6: - 8 m (1 + 3:“): )] . Ex (itn)
q=1 n=1 x x pq o q
P q
mn
e . + a _ i +
+ Gmx x qu(rn) Exp(rm)
P q
n = 1,2, . N

1,2,3 (8.23)

'6
ll

119

mn

Let G: be an N‘x N matrix given by
x x
P q
mn mn T (r )
e '— e e m
= - —-————- .24
Ge Ge apq 6m [1 + 3wa ] (8 )
x x x x o
P q P q

Let [Ex ], [Hx ], and [Bi ] be N—dimensional vectors. As m and p

P P P
range over all possible values in eq. 8.23, we obtain a matrix

equation approximation for eq. 8.14 as

 

 

 

 

 

Ge Ge Ge E 1
e e e X
xx xy xz
Ge Ge Ge E
e e e y
yx yy yz
Ge Ge Ge E
ezx ezy ezz z
- J L. J
- Ge Ge Ge F 11 FE
m m m x X
xx xy xz
+ Ge Ge Ge H E1
m m m y B- y (8.25)
yx yy yz
ce Ge Ge H 81
zx ”2y mzz z 2 J
k 4L J 11

 

 

 

 

 

120

Symbolically, we can write eq. 8.25 as

(8.26)

Similarly, for equation 8.15 we may represent the inner products

fi-Emand-E°‘€nas
m e

and

P m
G... < . 0
xx
m + ,
Gm (r. )
yx
m +
Gm (r. ')

 

 

m +

Gm (1', ')
xy

6‘“ ( . ')
W

m +

Gm (r. ')
2?

GIn (;,r')
e

X?

+

a: (r. '>
yy

6‘“ ( . ')
2y

The scalar components of equation 8.15 may be written as

 

 

 

 

 

 

121

 

+
T (r) 3
[1 + 3‘“ ]H ('2’) - pv Tm(-1:') 2 8: (“Hr") 11x (3?) dv'
j(“no xp q=1 x x q

V P q

3
- fr (2?) 2 cm (32?) E (13') dv' = H1 (1?)
e e X x
q=1 x x q p

P q
v

p = 1.2.3 (8.29)

By using the method of moments,we partition the body into N subvolumes
and E(f), Te(;), fi(;), Tm(;) are assumed to be constant in each sub-

volume. Requiring that equation 8.29 is satisfied at each :5, we have

+
T (r ) 3 N
m m * _ + m + +1 1
[l + 3:1on ]Hx (rm) 2 £1[Tm(rn) PV / Gmx x (rm,r ) dv]

 

p q=1 n=
Vn P q
H (+) g 1; -)- Gm + +' ' +

x rn - Te(rn) e (rm,r ) dv Ex (rn)

q q=1 n=1 x x q

. P q
‘IH
.. 1 '*
- Hx (rm)
P
p = 1,2,3
(8.30)
Defining
..mmn _ + m + +' '
Gm - Tm(rn) PV f Gm (rm,r ) dv (8.31)
x x . x x
P q V P q
n
and
m mn
m
Ge = Te(?h) .jr Ge (? ,f')‘dv' (8.32)
x x x x m
P ‘1 v p q

n

122

We can write equation 8.30 as

3 N mn .T (r )
2 z [6: -6 6mn(1+-§§‘—@1‘9—)] H G“)
q=1 n=1 x x pq o q
P q
mn
m + i +
+ Gex x - Exq(rn) - pr(rm) (8.33)
P q
m = l’zgoooooooN
p = 1,2,3
m mn
Let Gm be an N x N matrix given by
x x
P q
m mn -m mn Tm(rm) 34)
G = G - 6 6 (1-+-—————) (8.
mx x mx x pq mm 3jumo
P q P q

As m and p range over all possible valuesin equation 8.33 we obtain

another matrix equation.

123

 

 

 

 

P p 1
GIn Gm cm 1 H
m m m X
xx xy xz
G“ G” Gm H
m m m y
yx W Y?-
Gm am cm H
m m m 2
_ zx zy 22 L
Gm Gm Gm f E H1
e X X
xx xy xz
+ Gm cm 8‘“ E = - H1 (8.35)
e e e y y
yx yy yz
Gm cm 0‘“ E H1
e e e Z Z
ZX 2y 22 J L. J L J

 

 

 

 

 

 

Symbolically equation 8.35 can be represented as

m m i
[Gm][H] + [Ge][E] = - [H ] (8.36)
and combining equations 8.25, 8.26, 8.35, 8.36, we have
. , 1
MM + GZHH] “FEE
[Gm][E] + FGmHH] = - r111]
‘3 L m .

Where [Ge], [Ge], [cm], [cm] are 3N x 3N matrices and [ E] , [ H],
e m m e

[BI], [H1] are 3N dimensional vectors. Finally we can write

 

 

 

 

124

 

 

 

 

 

 

. . . .
G: G: E 1 E11
a: a: H H1

1 . L . L 1

This matrix equation represents 6N simultaneous equations for 6N
unknowns. If the incident electric field Ei(;) and the incident magnetic
+1+ ++
field H (r) are specified, the total induced electric field E(r) and the
total induced magnetic field E(f) inside the body can be determined from

equation 8.37 by inverting the [ G] matrix.

8.3 Example

Two examples are given here to show theoretical results of the
magnetic heating inside biological systems which are injected with
magnetic materials and when they are irradiated by a uniform magnetic
field. Figure 8.1 shows the theoretical model of a muscle layer of
12x2x12 cm with a magnetized central part (shaded region) where the
magnetic property was modified to possess an arbitrary permeability.
The body is divided into 36 cells and the incident field is assumed
to be a 30 MHz uniform magnetic field in y-direction.

First example is the case of a muscle layer of 8x2x8 cm irradiated
by a 30 MHz uniform magnetic field. The body is divided into 16 of 2
cm-cubic cells and the permeability of the magnetized, central 4 cells
has been modified by a magnetic powder injection. Figure 8.2 shows
the absorbed power density in cell A located at x = 3, y = 1, z - 3 cm
as a function of relative permeability of the magnetized part. At this
frequency the conductivity and dielectric constant are assumed to be

0.62 S/m and 150, respectively, and the relative permeability of the

125

 

 

 

 

 

 

 

ll,"
H
K
)(
A
LICW‘
2:
.IL ‘,__.__.
H /
3 _.__J L
(a
<5?
4 Y

 

 

 

Figure 8.1 A simulated muscle layer (12x2x12 cm) with a
magnetized central part irradiated by a uniform
field (81) of 1 A/m (max. value) at 30 MHz in
y-direction. The body is divided into 36 of 2 cm-

cubic cells.

126

 

 

 

 

 

 

 

 

 

 

3 X
P ( VW/ m ) A
I
/////
.0016 "' :[A/uy/
.5!
.001le h
.0012 "'

I

.0010

    

= 30 MHz

 

 

   

-.0008 81 = 1 A/m y
0 = 0.62 S/m
.0006 E = 150
r
u = 1.0 (unshaded region)
.0004 r
“r = varied (shaded region)
.0002
1 2 3 4 5 10 50 100

Relative Permeability Ur of the Magnetized Region

Figure 8.2 The absorbed power density in cell A varying as a
function of relative permeability of the magnetized
region inside a muscle layer (8x2x8 cm) for the case
of frequency a 30 MHz, 81 = l A/my, o = 0.62 S/m and

at = 150.

\
7

127

 

Freq. = 30 MHz

Di = l A/m y

 

 

u = 1.0 (unshaded region)

u = 10 (shaded region)

 

 

/‘ \

7 /
f /) A
\ Z
\ ./

 

 

 

 

V?

 

Figure 8.3

E
2

Distribution of induced electric fields (or currents)
inside a muscle layer (8x2x8 cm) with a magnetized
region (shaded region, pr = 10) when frequency = 30 MHz,

ii = 1 A/m y, o = 0.62 S/m and er = 150.

128

unshaded region (unmagnetized region) is assumed to be unity. It is
found that the absorbed power is rapidly increased when l < ur < 10
and slowly increased when 10 < pr < 100, where pr is the relative
permeability of the magnetized region.

These results imply the effectiveness of magnetic heating induced
by a uniform magnetic field in a magnetized body. The distribution of
induced currents inside the body is shown in Figure 8.3. The relative
permeability of the magnetized region is assumed to be pr 8 10. As
expected the induced currents are circulatory on xz-plane around the
direction of the incident magnetic field.

Another example is the case of a muscle layer of 12x2x12 cm with
a magnetized region, excited by a 30 MHz uniform magnetic field as
depicted in Figure 8.1. The body is divided into 36 2 cm-cubic cells
with 16 cells consisting of the magnetized region (shaded region).
Figure 8.4 shows the absorbed power densities in cells A and B located
at x - 5, y = 1, z = 3 cm and x = 5, y = 1, z - 5 cm, respectively, as
functions of the relative permeability “r of the magnetized region.
This result shows that the absorbed power density increases rapidly
when l < ur < 10 and then start to saturate after “r > 50. Figure 8.5
shows the current distribution induced inside the muscle layer of
Figure 8.4 when “r = 10. It is clearly shown in this figure that
circulartory currents are induced inside the body with their magnitudes
increasing with the distance from the center of the body.

A study on this generalized TIEM at this point is not complete.
Further studies on the numerical convergence and the accuracy test

are needed in the future. One may be able to find new distribution

 

129

 

 

 

P (W/m3)
l-—lzcm-——1 A
.007;
Freq. = 30 MHz
.006_ 351 = 1 May
0 = 0.62 S/m
.00 =
5r er 150
Ur = 1.0 (unshaded region)
.0014—
Dr = varied (shaded region)
.003—
.002 ..
.001 _.
B
A
0 l l J l 1 l l
1 2 3 h 5 10 50 100
Relative Permeability Ur
Figure 8.4 The absorbed power densities in cells A and B varying

as functions of the relative permeability of magnetized

region (shaded region) inside a muscle layer (12x2x12 cm)

for the case of frequency = 30 MHz, 81 = l A/m y,

0 = 6.2 S/m, and Er = 150.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r W%f/ // \ E Z?“
I %//////:///):/ A l :
1%?%%1”

\ / // ///1% 1

131

functions beside the pulse functions that when combined with the moment
method, will lead to a more efficient solution to these coupled tensor

integral equations.

CHAPTER 9

A USER’S GUIDE TO COMPUTER PROGRAM FOR INDUCED
ELECTRIC FIELD INSIDE AN ARBITRARLLY SHAPED,

FINITELY CONDUCTING BIOLOGICAL BODY

This chapter explains the computer programs used to obtain the
numerical results on the induced EM fields in an irradiated biological
body, based on the tensor integral equation method. There are 3
computer programs used.

First, program "FIELDS" is used to quantify the induced electric
field at various locations of the biological system.

Second, program "ITERATE" is used to quantify the induced
electric fields at the centers of 8 subcells in each cell of the
biological system based on the first-order solutions of induced
electric fields from program "FIELDS".

Third, program "EMFIELD" is used to quantify both induced
electric field and induced magnetic field at various locations of
a biological system with arbitrary permittivity, permeability and
conductivity.

A listing of the program deck and instructions for their useage
are also provided.

PART I PROGRAM FIELDS

9.1 Description of the program

This program is the modification of the program "FIELDS" developed

132

133

by Guru (23). The program is exactly the same mathematically but was
modified to conform with a standard FORTRAN IV . The reason for this
modification is to increase the "capacity" of the program to be able
to handle a larger number of cells. By using the "MERIT" network,
this program can handle the maximum size of the matrix up to 300 x 300
(100 cells).

The "MERIT" network (MTS) is the computing facility which combines
the three host computing facilities located at Michigan State University,
University of Michigan and Wayne State University. The IBM G&H
compilers, waterloo Fortran and interactive FORTRAN (IF) are available
under MTS at the University of Michigan. At Wayne State University
the IBM FORTRAN G&H extended compilers, waterloo Fortran and IF
are available under MTS. CDC FORTRAN extended version 4(FTN4) and
Minnesota FORTRAN are available under scope/Hustler at Michigan State
University.

With this computing facility CPU is ten times faster than CDC
6500 with 250 K 32 bit word real memory and unlimited virtual storage.

A biological system is divided into N small cubic cells with the
side of each cubic cell not exceeding 1/4 where 1 is the wavelength
inside the body. The body is illuminated by an electromagnetic plane
wave. Only normal incidence will be considered and the incident field

is given by
Ei(?) = x e-jkoz

Program "FIELDS" will calculate the induced electric field and

power density at the center of each cell inside the body.

134

9.2 Data Structure and Input Variables

The data file for program FIELDS, showing the input variables,
their FORMAT specifications, and their locations within the file, is
outlined in Table 9.1. A detailed description of the variables is
given in the next section.

A sample partitioning scheme for an arbitrary biological system
is shown in Fig. 9.1. The biological body is divided into 4 quadrants
and the center of the body is assumed to be the reference point. The
body has two layers in z-direction. The quadrants are numbered in the
clockwise order. In this example we assume that all the cells have the
same physical dimension and electrical parameters. Under the symmetri-
cal conditions, it is always intended to solve for the induced field
in first quadrant and then interpret the result in other quadrant.

It is noted that the body can also be divided into 8 quadrants if
the symmetry conditions for the body and the incident electric field
exist for this division. For example,a plane EM wave in exponential

-jk 2

form, e o , can be divided into cos koz and —j sin koz,
The induced electric fields in the body due to cos koz and -j sin

koz can be determined separately using 8 quadrants symmetry. The total

induced electric fields in the body are then the sum of these two

modes.

9.3 Description of the Input Variables
In this section we describe the function of each input variable

and explain how it is used in PROGRAM FIELDS.

13S

 

 

 

 

 

X
Al
I
I I
I I
’----v---
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I I I
14 I ' I
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p ------ auf---L—--" %y

I
I
I
I

+ I

E1 l
l

 

 

 

Figure 9.1. A two-layer biological body illuminated by an EM wave
at normal incidence is shown divided into 4 quadrants

under symmetry conditions.

136

 

 

 

 

 

 

 

 

File No. Card No. Symbolic name Columns Format
1 1 NDIV 1 ll
2 l COMP 1-3 A3
Q(J), J = 1,8 ll-l8 8I1
FMEG 21-31 F10.0
SCAT 41-45 A5
3 1 NX 1-2 12
NY 6-7 12
NZ 11-12 12
4 1 N 1-3 I3
5 l—N AMX 1-10 F10.3
AMY 11-20 F10.3
AMZ 21-30 F10.3
RLEPl 31-40 F10.3
8101 41-50 F10.3
DXCM 51-60 F10.3
DYCM 61-70 F10. 3
DZCM 71-80 F10.3
Table 9.1 The symbolic names of input variables and corresponding

specifications for the data files used in the data

structure for the program "FIELDS".

 

First data file

 

NDIV

Second data file

 

COMP

Q(J), J=1.8

FMEG

SCAT

Third data file

 

NX,NY,NZ

Fourth data file

 

N

Fifth data file

 

137
This variable allows the user to control the accuracy
with which the elements of [G] are evaluated.

being the code name for the component of the induced

electric field which may have any one of the following

forms
"XXX" x-component only
"XAY" x- and y-components

"XYZ" all three components
is the symbolic name for the quadrants. Quadrant 1-8
corresponding to column 11—18. If any one of the
quadrants used then punch the quadrant number in
corresponding column otherwise is "0" (zero).
read frequency of incident wave in Mega Hz.
being a code name for the incident wave.
EXPKZ,COSKZ,SINKZ -—- for the exponential, cosine,

sine variation of the incident electric field.

defines the maximum number of cells in x-, y- and

z-directions.

Total number of cells being considered.

There are as many as "N" data cards which help to

AMX,AMY,AMZ

RLEPl,SIG1

138

simulate the biological body. Each card contains

maximum boundaries of a cell in x-, y- and 2-

directions in cm.

are the codes for relative dielectric constant and

conductivity (Slm).

DXCM,DYCM,DZCM are the dimensions of the cell in x-, y- and z-

9.4 How to Use the Program

directions in cm.

We will construct the data file for the example shown in Fig. 9.1.

Let us assume that the incident field is of exponential form, the

frequency of the incident wave is 2.45 GHz, and the electric parameters

are e = 50 80, O = 2.21 S/m with a cell volume of lxlxl cm. The

sequential order of the data files is as follows.

File No.

5.1
5.2
5.3
5.4
5.5
5.6

5.7

5.8

2

XYZ 12340000

02 02 02
008

1.0 1.0
1.0 2.0
2.0 1.0
2.0 2.0
1.0 1.0
1.0 2.0
2.0 1.0
2.0 2.0

Information on the File

 

2450.0 EXPKZ

1.0 50.0 2.21 1.0 1.0 1.0
1.0 50.0 2.21 1.0 1.0 1.0
1.0 50.0 2.21 1.0 1.0 1.0
1.0 50.0 2.21 1.0 1.0 1.0
2.0 50.0 2.21 1.0 1.0 1.0
2.0 50.0 2.21 1.0 1.0 1.0

2.0 50.0 2.21 1.0 1.0 1.0

2.0 50.0 2.21 1.0 1.0 1.0

139

The list of the control cards needed for execution of the program in

MERIT network is as follows:

 

Card No. It's purpose
1 Authorization to use the computer
2 Job card (MSU)
3 Pass word
4 Input to MERIT network
5.
6
7 Job card (UM)
8 Pass work (UM)
9

The deck card structure is as follows:
Control cards

Program

$ENDrILE

$SET DEBUG=ON

$SDS SET ERRORDUMP=ON

$RUN - LOAD SPUNCH=*PUNCH*

data cards

$ENDFILE
$SIGNOFF

6/7/8/9

Information on the card

 

PNC
B,CM30000,T100,RGl,JC100.
PW=SUTUS
DISPOSE,INPUT,IN=UM.

7/8/9

7/8/9

$310 xszq T=160 PRIo-D P=300
FIELDS

$RUN *FTN PAR=FORMAT-IBM

<02 HI
1100)
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(IF-OK?
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MAKES USE OF THE FOLLOWING SUBPROGRAMS

TO FIND PHASE ANGLES OF THE INDUCED FIELDS

RAH FIELDS

'2
.3

PRC

THE

ANGLES
CMATP
GMAT
RFN

MATRIX BY BACK SUBSTITUTIDN

OF G MATRIX
DISTANCE R FROM ONE CELL TC ANOTHER

UHCH OUT ON COMPUTER CARDS THE FESULT
r’

EIC FIELDS

N

N BY

SLLVES
CEFINES ELEMENTS

FINDS

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151

PART 2 PROGRAM ITERATE

9.5 Formulation of the Problem

The numerical method of program "FIELDS" sometimes encounters
difficulties in the numerical convergence when the body is electrically
large. It is evident that in order to generate accurate results and a
good numerical convergence the cell size has to be electrically small,
thus, a large number of cells is needed. This, in turn, leads to an
unmanageably large number of unknowns in the numerical calculation.
The program "ITERAIE" is designed to overcome this difficulty.

Program "ITERATE" needs the first-order solutions on the induced
electric fields from program "FIELDS". Starting with a biological
body with N cells (this N-cells model gives reasonably good results and
a reasonable computer cost), the induced electric fields in these N
cells are obtained based on the tensor integral equation method.

In the next step, program "ITERATE" considers any one of N cells
in the body (e.g. mth cell) and divide this cell into 8 subcells. It
then calculates the equivalent incident electric field at the center
of each of the 8 subcells, which is equal to the incident electric
field of the incident plane wave plus the scattered electric field
maintained by the induced currents and charges in the rest of the
cells in the body.

Finally, program "ITERATE" solves the induced electric fields at
the centers of 8 subcells. After repeating each of these N cells, we
determine the induced electric fields in the biological body with 8

N cells.

152

9.6 Description of Computer Program

The computer program is coded in standard FORTRAN IV and can be
compiled on any FTN compiler. The main program is symbolically named
as "ITERATE". Program "ITERATE" uses the following complex functions
and subroutines for the numerical evaluation of induced electric fields
inside a biological system.

"GMAT" is a complex function which calculates the elements of
G matrix.

"CMATP" is a subprogram which calculates the induced electric
field in each cell by solving a 24x24 matrix.

"ANGLES" This subprogram determines the phase angle between real
and imaginary parts of the induced electric field in
each cell.

A listing for the main program "ITERATE", the complex function
"GMAT", and subprograms "GMAT?" and "ANGLES" is given at the end of

this part.

9.7 Structure of the Data File

The data file for program "ITERATE" showing the input variables,
their FORMAT specifications and their locations within the file, is
outlined in Table 9.2. A detailed description of the variables is
given in this section.

A sample partitioning scheme for a biological body is shown in
Fig. 9.2 and we will use this sample to run the program. The body is
a cube of size 4x4x4 cm and it is divided into 8 subcells with the

size of 2x2x2 cm. The body has two layers in the z—direction and these

153

2 layers touch each other. The reference point is shown in the figure
which is located at the lower left corner of the body. The location
of each cell is read with respect to the reference point.

After defining the location and electrical properties of each cell,
we find the induced electric field at the center of each cell by using
program "FIELDS" with an incident field of E1(¥) = 2 e-jkoz. These
results will be considered as the first-order solutions and input
variables for program "ITERATE".

It is important to identify the order of these 8 subcells and
their number system. These subcells are numbered in the clockwise
order beginning with the back layer. In the program "ITERATE", the
center location of each subcell is calculated based on this number

system.

Next, the information in each data file is explained below.

First data file

 

This data file has only one data card which defines

a symbolic name "NDIV" under I-format and "FMEG" under F-format.

NDIV This variable allows the user to control the accuracy
with which the elements of [quare evaluated.
FMEG read frequency of the incident electromagnetic wave

in MHz under F-format.

Second data file

 

This data file has only one data card which defines a symbolic

name "NT" under I-format in the first three columns of the data card.

154

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--I.-
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NJ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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HI
———-'—-.—_I
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I
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Fig. 9.2 A two—layer biological body illuminated by an EM wave

at normal incident is shown divided into 8 cells in

the lst quadrant.

155

 

 

 

Table 9.2 The symbolic names of input variables and corresponding
specifications for the data files used in the data
structure for the program "ITERATE".

File No. Card No. Symbolic Name Columns Format

1 l NDIV 1 Il
FMEG 2-11 F10.0

2 1 NT 1-3 13
3 l-NT AMX 1-10 F10.3
AMY 11-20 F10.3
AMZ 21-30 F10.3
RLEPl 31-40 F10.3
8161 41-50 F10.3
DXCM 51-60 F10.3
DYCM 61-70 F10.3
DZCM 71-80 F10.3
4 l—NT E (K) ,K=l ,NT 1-24 2E12 . 5
E(KPN),KPN=NT,2NT 29-52 2E12.5
E(KNN),KNN=2NT, 3NT 57-80 2E12.5

5 1 NQ 1-2 12

 

 

 

 

 

 

156

It tells the computer about the total number of cells being considered.

"NT" is also the total number of cell in the body.

Third data file

 

This data file has as many as NT data cards. This set of data
cards helps simulate the biological system and each card contains the
following information.

AMX,AMY,AMZ These codes correspond to the maximum boundaries of
a cell in the x-, y- and z-directions in cm.

RLEPl,SIGl are the codes for relative dielectric constant and
conductivity of the cell.

DXCM,DYCM,DZCM are the symbolic names for the dimensions of the

cell in x-, y- and z-directions in cm.

Fourth data file

 

This data file has NT data cards and this set of data defines
the induced electric field at the center of each cell (first-order

solution). Each card contains the x-, y— and z- components of the

induced electric field.

Fifth data file

 

This data file has only one card which defines a symbolic name
"NQ" under I-format in the first two columns of data card. It tells
the computer how many quadrants have been used. If more than one
quadrant are used only the cells in the lst quadrant are considered.

To get better understanding of how these variables are used,an
example is worked out in the next section for induced electric fields

in a biological body as shown in Fig. 9.2.

157

9.8 An example to use the program
We will construct the data files for a sample problem in this
section, to illustrate how the input variables are used. Also we
will discuss some of the important features of the printed output.
A sample problem is a biological body as shown in Fig. 9.2.
The body has 8 cells with a cell volume of 2x2x2 cm. Let's assume
that the frequency of the incident wave is 1.0 GHz and the electrical
parameters of the biological body are E = 50 so, and O = 1.62 S/m.
The first-order solutions of the induced electric fields at the centers

of 8 cells are punched on the computer cards as

Card no. Ex Ey E2
1 6.318E-2 2.488E-2 8.373E-3 9.604E-4 8.185E-3 -1.055E-2
2 6.318E-2 2.488E-2 -8.373E-3-9.604E-4 8.185E-3 -1.055E-2
3 6.318E-2 2.488E—2 -8.373E-3-9.604E-4 -8.185E-3 1.055E-2
4 6.318E-2 2.488E-2 8.373E-3 9.604E-4 -8.185E-3 1.055E-2
5 6.513E-2-2.87OE-2 8.505E-3-l.819E-3 -8.693E-3 -9.694E—3
6 6.513E-2-2.870E-2 -8.505E-3 1.819E-3 -8.693E-3 -9.694E-3
7 6.513E-2-2.870E-2 -8.505E-3 1.819E-3 8.693E-3 9.694E-3
8 6.513E-2-2.87OE—2 8.505E-3—l.819E-3 8.603E-3 9.694E-3

After knowing all these induced fields, we can start setting the
data files with the aid of Section 9.7 and Table 9.2. The sequential

order of the data files is as follows:

File no.

-'::—‘—‘
2
3.1
3.2
3.3
3.4
3.5
3.6
3.7

3.8

158

Information on the file

 

2 1000.0

008

2.0 2.0 2.0 50.0
2.0 4.0 2.0 50.0
4.0 2.0 2.0 50.0
4.0 4.0 2.0 50.0
2.0 2.0 4.0 50.0
2.0 4.0 4.0 50.0
4.0 2.0 4.0 50.0

4.0 4.0 4.0 50.0

1.62

1.62

1.62

1.62

1.62

1.62

1.62

1.62

2.0 2.0 '2.0
2.0 2.0 2.0
2.0 2.0 2.0
2.0 2.0 2.0
2.0 2.0 2.0
2.0 2.0 2.0
2.0 2.0 2.0

2.0 2.0 2.0

First—order solution (see the previous page)

01

The list of the control cards needed for the execution of the

program is as follows:

Card no.

1

2

Its purpose

 

Authorization to use the computer

job card

Pass word

Identification name
Compile the program
Execute the program

End of control card

Information on the card
PNC
B,CM60000,T300,RGl,JCZOOO.
AUTORFL,PART.

Pw=SUTUS

HAL,BANNER,SR.

FTN(R=3)

LGO.

7/8/9

159

9.9 Printed Output
Although most of the items in the output are self-explaintory,
a few of them need to be explained. The printed output consist of
4 output files. First, second and third output files are the data
from the input files in order to check on the data used for the input

variables and for further future references.

First output file consists of the maximum boundaries limited of each

 

cell as read in through the symbolic code names "AMX', "ANN" and "AMZ'
and the dimensions of each cell "DXCM", "DYCM" and "DZCM" in centimeters

in x-, y- and z-direction, respectively.

Second output file, the internally calculated coordinates in the x-,

 

y- and z-directions for the central location of each call, its volume

and its permittivity and conductivity.

Third output file, the components of the induced electric field in

 

each cell (the first-order solution).

Fourth output file consists of as many sets as the number of cells

 

considered. Each set contains:

1. The coordinates in the x-, y- and z-directions for the
central location of each subcell, its volume, permittivity
and conductivity.

2. The equivalent incident electric field at the center of each
subcell.

3. The most needed results for the induced electric field and

the power density in each subcell in addition to the frequency

160

of incident wave and total power absorbed by the cell.
4. The real and imaginary parts of each component of the induced
electric field in each subcell along with absolute magnetude

and phase angle.

Listing of the program

 

A Fortran listing of the PROGRAM ITERATE and its subprogram
begins on the next page. The subprogram is listed in order of their
first appearance in the main program. The program required approxi-

mately 24600 octal words of storage.

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170

PART 3 PROGRAM EMFIELD

9.10 Formulation of the Problem

In this computer program, we consider the biological body-with
arbitrary permeability, permittivity and conductivity. When such a
body is irradiated by an EM wave, both the induced electric field
and the induced magnetic fields need to be determined. Furthermore
these induced electric fields and magnetic fields are coupled together.

We now have to solve the system with 6 unknowns, three components
each for the induced electric and magnetic fields, in each cell. Thus,
the size of[G]matrix increases twice to 6Nx6N, where N is the number of
cells in the body. The formulas used to compute the elements of[Q:]
matrix become more complicated because of the coupling term between
the electric and magnetic fields.

In this program, the incident electromagnetic wave is assumed to
be a simple plane wave with an electric field polarized in the x-
direction and a magnetic field polarized in the y-direction. The
arbitrarily shaped biological body is divided into N small cubic cells
with the side of the cube not exceeding A/4.

After defining the incident electric and magnetic fields,
electrical parameters and physical dimensions, program EMFIELD then
calculates the induced electric and magnetic fields, and the power

density at the center of each cell in the body.

9.11 Description of the Computer Program
The computer program is coded in standard FORTRAN IV. The main

program is symbolically named as "EMFIELD". Program EMFIELD uses the

171

following complex functions and subroutines for the numerical evaluation
of induced electric fields and induced magnetic fields inside the

biological body.

GMATl and GMATZ are 2 complex functions which calculate the
elements of G matrix.

GMATP is a subprogram which calculates the induced
electric and magnetic fields in each cell.

RFN is a subprogram which calculates the distance
between one cell and another.

ANGLES This subprogram determines the phase angle
between real and imaginary parts of the induced

electric and magnetic fields.

A listing for the main program "EMFIELD", the complex function
GMATl and GMATZ, subprogram CMATP, RFN and ANGLES is given at the

end of this part.

9.12 Structure of the Data File and Input Variables

The sequential structure of the data files, the format specifi-
cations and the symbolic names of the variables appearing in each
file are outlined in Table 9.3.

Figure 9.3 shows a sample biological body with one layer which
consists of 4 cells in the lst quadrant (there is no symmetry condition
which exists in general). The body is a block of size 20x20x10 cm
with each cell size of lelelO cm. The reference point is located
at the lower left corner of the body. We will show how to construct

the data file in the next section. The information in each data file

172

_>x

 

 

 

 

Figure 9.3 A layer of biological body illuminated by Electromagnetic

wave at normal incidence is shown divided into 4 cells.

173

 

 

 

Table 9.3 The symbolic names of input variables and corresponding
specifications for the data files used in the data
- structure for theprogram-EMFIELD.-
File No. Card No. Symbolic Name. Columns Format
1 l NDIV l 11
2 l COMP 1-3 A3
Q1 11 Il
FMEG 21-30 F10.0
SCAT 41-45 A5
3 l NX 1-2 12
NY 6-7 12
NZ 11—12 12
4 1 N 1—3 13
5 l-N AMX l-lO F10.3
AMY 11-20 F10.3
AMZ 21-30 F10.3
RLMUl 31-36 F6.3
RLEPl 37-42 F6.3
3101 43-50 F8.3
DXCM 51-60 F10.3
DYCM 61-70 F10.3
DZCM 71—80 F10.3

 

 

 

 

 

 

174

is as follows:

First data file This data file has one card which defines a symbolic
name NDIV. This variable allows the user to control the accuracy

with which the elements of [G lare evaluated.

Second data file consists of one data card which defines the compon-

 

ents of induced electric and magnetic fields, quadrant, frequency of

the incident wave and type of the incident field.

 

Third data file consists of one data card which defines the maximum
number of cells in the x-, y- and z-direction. The symbolic names for

these numbers are NX, NY, and NZ, respectively.

Fourth data file has only one data card which defines a symbolic

 

name "N" under I-format in the first three columns of the data card.

"N" is the total number of cells being considered.

Fifth data file This data file has as many as "N" data cards. This

 

set of data cards helps simulate the biological system and each card
contains the following information.
AMX,AMY,AMZ These codes correspond to the maximum boundaries
of a cell in the x-, y- and z- direction in cm.
RLMU1,RLEP1,SIG1 are the codes for permeability, dielectric constant
and conductivity of the cell.

DXCM,DYCM,DZCM are the symbolic names for the dimension of the cell

in x-, y- and z-directions in cm.

175

9.13 ‘An Example to Use the Program

Let us now try to determine the electric and magnetic fields
induced inside a biological system as shown in Fig. 9.3 by an incident
EM wave. Let's assume that the frequency of the incident wave is 500
MHz and the electrical parameters of the biological body are u - 1.2uo,
8 = 53E:o and O = 1.45 S/m with the cell volume of 10x10x10 cm. We use
"EXPKZ", an exponential variation for the incident EM wave. From

section 12 and Table 9.3, the sequential order of the data files is as

 

follows.

File No. Information on the File
1 2
2 XYZ 1 500.0 EXPKZ
3 02 02 01
4 004
5.1 10.0 10.0 10.0 1.2 53.0 1.45 10.0 10.0 10.0
5.2 10.0 20.0 10.0 1.2 53.0 1.45 10.0 10.0 10.0
5.3 20.0 10.0 10.0 1.2 53.0 1.45 10.0 10.0 10.0
5.4 20.0 20.0 10.0 1.2 53.0 1.45 10.0 10.0 10.0

9.14 Printed output

The printed output consists of 2 parts. First part is the echo
of the data from the input files in order to check on the data needed
for the input variables and for further references. Second part is
the required output. The information on each output file can be

explained as follows.

176

First output file consists of the maximum boundary limits of each

 

cell as read in through the symbolic code names AMX, AMY and AMZ and

the dimensions of each cell DXCM, DYCM and DZCM in cm.

Second output file consists of the central location of each cell in

 

the x-, y- and z-directions, its permeability, dielectric constant and

conductivity.

Third output file The components of the incident electric and magnetic

 

fields in each cell and the type of its variation.

Fourth output file The results for the induced electric field and the

 

power density in each cell in addition to the frequency of the incident
field and the total power absorbed in the biological system. Following
these results are the real and imaginary parts of each component of

the induced electric field in each cell along with its absolute magne«

tude and phase angle.

Fifth output file The results for the induced magnetic fields in each

 

cell in addition to the frequency of the incident wave in the body.
Also the real and imaginary parts of each component of the induced

magnetic field along with its absolute magnetude and phase angle.

Listing of the program

 

A Fortran listing of the program EMFIELD and its subprogram
starts on the next page. The program requires approximately 12000

octal words of storage.

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BIBLIOGRAPHY

 

10.

ll.

12.

BIBLIOGRAPHY

Livesay, D.E., and K.M. Chen, "Electromagnetic fields induced
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Chen, K.M., "A Simple Physical Picture of Tensor Green's
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Johnson, C., and A.W. Guy, "Non-ionizing EM effects in biological
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Stratton, J.A., Electromagnetic Theory, McGraw—Hill Co., New
York, 1941, pp. 563-567.

 

Schwan, H., "Survey of microwave absorption characteristics of
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Kritikos, H. and H. Schwan, "Hot spots generated in conducting
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Shapiro, A.R., et a1., "Induced fields and heating within the
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Kritikos, H. and H. Schwan, "Formation of hot spots in multilayer
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Chen, K.M., and B.S. Guru (1977a), "Induced EM fields inside
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Chen, K.M. and B.S. Guru (1977b), "Internal EM field and absorbed
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Baker, R.J., V. Smith, T.L. Phillips, L. Kobe and L. Kane, "A
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Guru, B.S., "An experimental and theoretical study of the inter-
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