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MICROWAVE
ENERGY ABSORPTION
in TISSUE
ENVIRONMENTAL PROTECTION AGENCY
Office of Research and Monitoring
-------
ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND MONITORING
TWINBROOK RESEARCH LABORATORY
The technical publications issued by the Twinbrook Research
Laboratory report results of intramural and contractor pro-
jects. A limited number of the reports are distributed to
regional, state, and local radiological health agencies, to
universities and libraries, to other government and private
agencies, and to interested individuals.
The Twinbrook Research Laboratory technical reports listed
below are available from the National Technical Information
Service, Springfield, Virginia 22151. Microfiche copies are
$0.95 and paper copies are $3.00 unless otherwise noted.
Please refer to the PB number when ordering the reports.
PB 207 079 Krypton 85: A Review of the Literature and an
Analysis of Radiation Hazards (January 1972)
PB Microwave Energy Absorption in Tissue
(February 1972)
-------
MICROWAVE
ENERGY ABSORPTION
in TISSUE
Richard A. Tell
Twfnbrook Research Laboratory
12709 Twinbrook Parkway
Rockville, Maryland 20852
FEBRUARY 1972
&
\
ui
(9
T
K&h
ENVIRONMENTAL PROTECTION AGENCY
Office of Rue«di and Monitoring
VCkahington, D.C. 20460
-------
DISCLAIMER
The mention of conmercial preparations or
products in no way constitutes endorsement
by the U.S. Environmental Protection Agency
or its affiliates.
ii
-------
FOREWORD
The IWinbrook Research Laboratory plans and conducts
research studies related to the impact of environmental
radiation on the health and well-being of mankind.
Through investigations of the biological effects of
both ionizing and nonionizing radiation, the Laboratory
seeks to define potentially hazardous levels of radiation
on which to base criteria for the development of environ-
mental exposure standards.
These studies rely heavily on an ability to physically
specify the extent of irradiation in experimental subjects.
This report concerns technical aspects of the dosimetry
problem associated with biological subjects exposed to
energy in the microwave region of the electromagnetic
spectrum.
The author discusses the dielectric properties of various
tissues and explains the approaches used to calculate
wave impedance throughout multilayered tissue systems. The
coupling factor between exposure and absorbed energy is
derived; energy distribution, its development as heat, and
its relation to temperature rise in tissues is discussed.
Additional information of this type is sought on a continu-
ing basis and the interest and comments of individuals
concerned with various aspects of radiation protection of
man and his environment are solicited.
William A. Mills, Ph.D.
Director
Twiribrook Research Laboratory
iii
-------
CONTENTS
Foreword iii
Abstract vii
Acknowledgments viii
Introduction 1
Dielectric Properties of Tissues 4
Impedance Calculations in Multilayered Tissue Systems 9
Total Dose Absorbed by System 15
Dose Distribution within System 23
Comments on Temperature Rise in Tissues 39
Summary and Conclusions 45
References ..... 47
Appendixes
A. Verification of Identity: tan 6 = ^ 51
B. Derivation of Wavelength Reduction in a Dielectric Material . 52
C. Verification that ag = 53
Tables
1. Relative Dielectric Constant and Resistivity
of Body Tissues at 37° C 6
2. Temperature Coefficient of Relative Dielectric Constant and
Specific Resistance of Body Tissues in Percent per Degree. . 9
3. Dielectric Properties of Water at Selected
Frequencies and Temperatures 10
4. Parameters for Tissues in Figure 7 17
5. Densities and Heat Capacities of Various Tissues 41
Figures
1. Field Strength and Power Density in Free Space 2
2. Plane Slab Model Tissue System 3
3. Complex Dielectric Constant of Skin, Fat, and Muscle 7
4. Vector Representation of Z* 13
5. Vector Representation of K* 13
6. Propagation Constant of Skin, Fat, and Muscle 14
7. Model Configuration for Smith Chart Solution 18
8. Smith Chart Solution 19
CO
oc
a,
«=*:
-------
9. 915 MHz Power Absorption in Tissues 21
10. 2450 MHz Power Absorption in Tissues 21
11. 8500 MHz Power Absorption in Tissues 22
12. Relative Heat Development Rate in Two-Layer Model 28
13. Relative Heat Development Rate in Three-Layer Model .... 30
14. Thermal Dose Distribution at 915 MHz 33
15. Thermal Dose Distribution at 2450 MHz 34
16. Thermal Dose Distribution at 8500 MHz 37
17. Effect of Skin Thickness on Fraction of Total Heat
Developed in Skin and Fat 38
18. Maximum Tissue Temperature Elevation in Two-Component
System 43
19. Maximum Tissue Temperature Elevation in Three-Component
vi
-------
ABSTRACT
Various approaches may be used in describing microwave interactions
with animal systems. This paper is intended as a guide to several
dosimetric techniques useful with a slab model configuration of
biological tissues. A detailed account is given of the calculational
concepts, gathered from the literature, which are used to determine
the degree of power absorption within such tissue systems as well as
the spatial distribution of this absorbed dose as heat and consequently,
the tissue temperature elevations which may be experienced in the model.
Both a graphic-analytic technique using the Smith chart and a mathematical
derivation of the appropriate computing formulas are given. Adequate
reference is made to direct the reader to pertinent literature concerning
other models and mathematical methods involved in microwave dosimetry.
vii
-------
ACKNOWLEDGMENTS
The author acknowledges the efforts of several persons
who helped to assemble this report: Miss Judy Kuhn and
Mr. Jerry Gaskill of the Data Systems and Mathematics
Staff for their assistance in developing the computer
applications; Mrs. Dorothy Punga and Miss Rona Fox for
their help in typing the manuscript; all of the personnel
of the Audio Visual Support Branch for preparing the
figures; and finally Mr. Donald M. Hodge, for his editorial
comments and effort in composing the final report.
viii
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INTRODUCTION
Interaction of electromagnetic (EM) energy with a material may-
be characterized by the various mechanisms that are active during
the interaction process. Somecf these mechanisms convert radio
frequency (RF) and microwave energy to heat within the target
material; others are so-called field evoked force effects that
are not necessarily associated with a temperature rise in the inter-
acting medium. Although field evoked force effects, which include
pearl chain formation, particle orientation, molecular resonance,
and membrane excitation, have been studied in some detail (1-4),
experiments aimed at finding biophysical responses to these
phenomena without the presence of significant thermal effects have
proved difficult. Consequently, most biological effects research in the
United States has been based on the assumption that the primary
effective mechanism involved is heating induced by EM energy
throughout the tissue. This approach is based on the practicality
of the situation; subtle nonthermal effects have been observed both
here and abroad (5-7), principally in the U.S.S.R. where a rather
complete compilation of this type of information has recently been
made (8). However, the theories developed to explain these effects
(primarily information theory) have not been substantially subjected
to experiment to ensure widespread acceptance of their application
in this case. Accordingly, RF and microwave dosage measurements
(absorbed energy) have been tied to the Joule heating effect and
attempts to measure and predict the degree of local heat developed
throughout the irradiated object. This is not to say that other
parameters, such as electric field strength within the tissue or
surface current density might not provide a better measurement of
absorbed dose, but other reasons such as lack of instrumentation
and the subjective nature of biological interpretation lead one,
at least presently, generally back to the heat concept. To this
end this report discusses the relationship between exposure and'
absorbed energy and the conversion of this energy to heat in various
tissue systems.
Because of the nature of thermal regulation in animals, dose
rate is the most useful term for relating biophysical responses to
RF and microwave energy exposure. That is, at low energy input
rates, below the intrinsic heat dissipation capability of the animal,
negligible responses may occur. At greater energy deposition rates,
biological responses may be indicative of stress on the animal's thermal
regulatory system. Fortunately, at microwave frequencies a conmon
expression for radiation intensity, because of technical reasons, is
in terms of free space exposure dose rate or power density, i.e.
niW/cm2. Other measurements such as electric field strength (volts/,
meter), usually (see Figure 1) used at radio frequencies, are
-------
2
convertible to power density. Thus, if an expression describing the
absorption efficiency and power absorption pattern for a particular
biological system can be obtained, then the resultant heating rate
at a specific point may be directly established.
1 1 F 1 1 1 1 1
1—r i i i m i
1 1—r t i i 11
l I i i I i n
1 1 1 1 1 1 1 L
-
-
>
II 1 II | 1 "1
<
i i i i i 111
it
L- -J
1 111 1J_1_L
0.01 0.1 1 10 100 1000
POWER DENSITY ( milliwatts centimeters2)
Figure 1. Field Strength and Power Density in Free Space.
The problem of micrwave dosimetry in humans and laboratory
animals is severely couplicated by the critical and highly variable
relationship of energy absorption upon tissue parameters. Unlike
high-energy photons such as x rays, microwaves may be strongly
reflected by an interface between two different tissues. The total
energy or power extracted from an incident beam of microwaves may
vary from a small amount to essentially 100 percent depending
greatly on the particular dielectric properties and the physical
configuration of the tissues. Consequently, unlike ionizing
radiation dosimetry, no fixed conversion factors can be developed
-------
3
which relate, in general, exposure measurements in the free field to
absorbed dose. On the contrary, every animal or specific absorber,
because of its own particular makeup, has its own conversion factor.
At first sight, because of this highly variable coupling of
animal with electromagnetic field, the problem of dosimetry appears
unmanageable, and one is inclined to adhere solely to specifying
free field exposures. However, if the interactions between EM
energy and simplified tissue systems are examined carefully, one
can gain an insight into the heat development process and spatial
distribution of this heat. From here one may examine more coup lex
models of systems in an attempt to describe the actual energy
deposited in the system.
For the sake of simplicity in analysis and explanation the system
described in this paper consists of a series of planar slabs that
are arranged to simulate the configuration of typical human tissues.
These layers are depicted by Figure 2 and are assumed to be infinite
AIR
Figure 2. A Section of Plane Slab Model Tissue System.
in extent. Also, the base layer of muscle tissue is assumed to be
semi-infinite in depth, i.e., there are no underlying reflective
surfaces to which a wave might penetrate. This constraint is ful-
filled in most practical cases by the relatively high absorption
characteristics of the muscle tissue. A further supposition is
that the test system is exposed to plane electromagnetic waves.
This report details the calculations used in describing EM
energy absorption and distribution within plane slab model tissue
systems. The application of these procedures has been reported
-------
4
previously in numerous other publications (9-12). In this report
the calculations are extended to different frequencies and combina-
tions of tissue configurations with sufficient explanation to enable
one to carry out computations for specific conditions. First, the
dielectric properties of various tissues are discussed, including
factors that influence their values and techniques used in their
measurement. Secondly, the approach used to calculate the wave
impedance throughout multilayered tissue systems is explained. Once
this is accomplished, the coupling factor between exposure and absorbed
energy is derived. After the total dose is found, the determination
of its distribution as heat is discussed with emphasis on ultimately
relating heat development to temperature rise within the tissues, the
report is concluded with comnents on application of the techniques to
experimental studies, a description of other types of models and
theoretical approaches, and a summary of the results obtained in this
paper.
DIELECRTRIC PROPERTIES OF TISSUES
Dielectric response of a material placed in a sinusoidal field
may be described by two parameters, the conplex permittivity, (e*), and
complex permeability, (y*). These two parameters relate to the
dielectric's ability to store electric and magnetic energy. Custom-
arily, the conplex permittivity is expressed in the form,
e* = e' - je"
where e', or the dielectric constant, represents the dielectric's
storage capacity for electric field energy and e", the loss factor,
represents the loss of electric field energy due to frictional losses
by charge flow and dipolar rotation. In the MKSC system of units,
the dielectric constant for vacuum is designated e and has a value
of 8.854 x 10 12 farads/meter. Generally one work? in terms of
relative measures of a materials permittivity, i.e., the ratio of
the actual permittivity of the material to that of vacuum,
where K* = K' - jK". K' is the relative dielectric constant and K"
is the relative loss factor, both quantities being dimensionless.
A similar relation exists for the ccmplex relative permeability,
^mag ~ y0 = ^"ntag " j^mag
-------
5
_ 6
where y = 1.257 x 10 henry/meter. However, because biological
tissues represent essentially no magnetic perturbation to EM fields,
u* is generally set equal to yQ; hence, the complex relative
permittivity is the parameter of interest in most dosimetric
calculations.
The relative permittivities of biological materials usually are
characterized as being frequency dependent. This dependency is
characterized, for most biological tissue, by the evidence of three
regions of dispersion. In the first of these regions, below 300-400 Miz,
the cell meirbranes, which act as layers of high capacity and resistance,
become short circuited as the frequency is increased and peimit the
cell's contents to participate in current conduction. Thus, the
capacitance and specific resistance of the tissue decrease as frequency
increases in this region. Dispersion beyond 3000-6000 NHz is due to
rotation of water molecules that are contained in most tissues.
Electrical polarity of the water molecule causes it to rotate in
unison with the applied electrical field until the inertia of the
molecule becomes prohibitively large. At this critical frequency,
approximately 20,000 NHz for water, both the resistance and dielectric
constant decrease. The remaining intermediate dispersion region
represents a frequency range (~400 to 4,000 NHz) in which cell
membranes no longer affect tissue properties, yet the water molecule
is not strongly frequency dependent. Detailed accounts of the theory
of dielectrics can be found in the literature (13-15).
A collection of dielectric data by Schwan (16) is reproduced in
Table 1 where values of relative dielectric constant and the resistivity
of various tissues are entered. Several other sources of dielectric
data for biological tissues are. available (17, 18). Resistivity,
p, or its reciprocal, the dielectric conductivity, a = 1/p, is sometimes
given in lieu of the relative loss factor K". The units of p in
Table 1 are ohm-meters allowing ready conversion of p to the
relative loss factor through the relation
K"
pOJEo (l)£o
where: K" = relative loss factor (dimensianless)
p = resistivity (sometimes called specific resistance) in
units of ohm-meters
<*> = angular frequency = 2tt f (f in units of Hz)
-12
e = permittivity of vacuum - 8.854 x 10 F/m.
0 _ l
o = conductivity (mhos/meter) (ohm-meters)
-------
CT\
TABLE 1. RELATIVE DIELECTRIC CONSTANT AND
RESISTIVITY OF BODY TISSUES AT 37° Ca
Tissue
25
SO
Frequency, MHz
100
200
400
700
1000
3000
8500
Relative Dielectric Constant, K'
Muscle
Heart Muscle
Liver
Spleen
Kidney
Lung
Skin
Brain
Fat
Bone Marrow
Muscle
Heart Muscle
Liver
Spleen
Kidney
Lung
Skin
Brain
Fat
Bone Marrow
103-115
136-138
200
200
160
1.85-2.10
2.20
85-97
88-93
135-140
119-132
110-114
11-13
6.8-7.7
1.13-1.47
1.73-1.95
1.28-1.51
0.90-1.45
2.60-4.50
1.90-2.10
17.00-25.QQ
28.00-50.00
71-76
76-79
100-101
87-92
65
81-83
1.54-1.79
1.20-1.40
1.80-1.95
56
59-63
50-56
62
35
4.5-7.5
52-64
52-56
44-51
53-55
35
46-48
4-7
Resistivity p, ohm-meters
0.95-1.05
0.95-1.15
1.10-1.50
0.90
1.60
10.50-35.00
0.85-0.90
0.85-1.00
1.05-1.30
0.85
1.40
1.10-1.30
9.00-28.00
52-53
50-55
42-51
50-53
34
0.73-0.79
0.78-0.95
0.85-1.15
Q. 76-0.77
1.30
49-52
46-47
43-46
5.3-7.5
4.3-7.3
0.75-0.79
0.98-1.06
Q.90^1.10
6.70-12.00
10.00-23.00
45-48
42-43
40-45
3.9-7.2
4.2-5.8
0.43-0.46
0.49-0.50
Q.37'0.50
4.40-9.00
4.45-8.60
40-42
34-38
36
3.5-4.5
4.4-5.4
0.12
0.15-0.17
0.14
2.40-3.70
2.10-6.00
aData taken from reference 16
-------
7
Mean values of K' and K" (derived from Table 1) for skin, fat, and
muscle tissues are plotted in Figure 3, over a frequency range of
400 to 8500 Nflz. The range of p, and consequently K", for fat is
indicative of the highly variable water content in fatty tissue.
I i i 1 i i I _J I 1 1 1—I—I—1—1 io
0.4 0.7 1 2 3 5 7 9
FREQUENCY (GHZ)
Figure 3A. Complex Dielectric Constant of Skin.
Dielectric parameters vary also with temperature of the material.
Table 2 gives temperature coefficients of dielectric constants and
resistivities for various tissues at several different frequencies.
In general the dielectric constant is affected statistically
less by temperature change than is the resistivity (19). The temperature
dependence of K' and K" for water is shown in Table 3.
Generally, the dielectric constant is essentially independent
of tissue temperature and the loss factor K" shows small (about
2 percent/°C) changes with temperature over the region of interest
-------
I I I I ¦ I I I I I I I I I I
0.4 0.7 1 2 3 5 7 9
FREQUENCY ( GHZ)
Figure 3B. Complex Dielectric Constant of Fat.
-I 1 1 L—L.
J 1 1 L_J_
60
SO
0.4
0.7 1
2 3
5 7 9
FREQUENCY (GHZ)
Figure 3C. Complex Dielectric Constant of Muscle.
-------
9
TABLE 2. TEMPERATURE COEFFICIENT OF RELATIVE DIELECTRIC CONSTANT AND
SPECIFIC RESISTANCE OF BODY TISSUES IN PERCENT PER DEGREE Ca
Frequency, MHz
Tissue 33 200 lOffi)
AK1
100 °C"
Tissues with
high H90 content
0.5
0.2
-0.4
Fatty tissue
L
-
1.3
1.1
0.91 NaCl
-0.4
-0.4
-0.4
Ap
100 — oc
Tissues with
high H2O content
t-2
-1.8
-*•3
Fatty tissue
-(1.7 - 4.3)
-4.9
-4.2
0.91 NaCl
-2.0
-1.7
-1.3
^aken from reference 16.
in living animals; therefore, in recognition of the normal variance
in the measured dielectric paramenter values, temperature effects
have been given no consideration here. Schwan discusses this
particular aspect in more detail (11).
Various techniques for determining dielectric properties have
been utilized; DC and low frequency AC fields are usually measured
using bridge techniques while transmission line and waveguide
approaches are used at ultrahigh frequencies. Descriptions of the
application of these methods are abundant in the literature (20-24).
IMPEDANCE CALCULATIONS IN MJLTI- LAYERED TISSUE SYSTEMS
Solutions to the field distributions within a multilayered
tissue system are based on transmission line concepts. This approach
requires knowledge of the electrical impedance of the various
media involved.
In vacuum, or free space,
zx - 1 pr - (377 + jO) ohms (Eq. 1)
o
-------
10
TABLE 3. DIELECTRIC PROPERTIES OF WATER AT
SELECTED FREQUENCIES AND TEMPERATURES3
Frequency, MHz
°C Parameter 300 3,000 10,000 25,000
1.5
K'
K"
86.5
2.77
80.5
25.0
38.0
39.1
15.0
6.38
5
K'
KM
85.2
2.33
80.2
22.1
41.0
39.0
17.5
6.91
15
K'
K"
81.0
1.70
78.8
16.2
49.0
34.3
25.0
8.25
25
K'
KM
77.5
1.24
76.7
12.0
55.0
29.7
34.0
9.01
35
K'
K"
74.0
0.925
74.0
9.40
58.0
25.5
41.0
8.82
45
K'
K"
71.0
0.746
70.7
7.49
59.0
23.6
46.0
12.7
55
K'
KM
68.0
0.626
6.75
0.601
60.0
21.6
49.0
12.0
65
K'
K"
64.5
0.542
64.0
4.90
59.0
18.9
50.5
6.31
75
K»
K"
61.0
0.470
60.5
3.99
57.0
16.0
51.5
5.41
85
K'
KM
57.0
0.416
56.5
3.09
54.0
14.0
-
95
K'
KM
52.0
0.364
52.0
2.44
-
-
aData taken from reference 35
-------
11
where Z* is the characteristic wave impedance of free space and
y* and e* are as previously defined. In a material
* =
(Eq. 2)
where Z* is the characteristic wave impedance of the given material
and y* Hind e* are the complex permeability and peimittivity,
respectively? If Eq. 2 is divided by Eq. 1,
-As pointed out earlier, y^ is set equal to y* in biological materials, and
¦z* V?= z* /v%" 7% (Eq'3)
» • *m
If the modulus or magnitude of the wave jf,e^i5ru^r°^®rial
is needed, the modulSs I Iff I is used for the quantity under the
radical sign, i.e.,
Zo
,Z*' "v + Kir "
where Z* is written simply as Z since, jj^gdance. in^
xe.1; tfte asterisk is used mere?y to «££«*£ ^ce^xn
general, is a complex quantity .Itisc crim re£ers t0
intrinsic wave impedance of a medium v.tne ^
a given medium) in the familiar form
z/S - "m + % (Etl' 4)
-------
12
where R represents the resistive conponent of Z*, and X represents
the reactive component of Z* as illustrated in Fig. 4. Equation
3 may be manipulated as follows:
^m
Zq z„ ^ * j^~ z„ j k; + jiig-
J K " J K - 'W + K1
Z° .-'J •/1
-------
13
a complex factor composed of the linear attenuation factor a (units
of reciprocal length) and the phase factor g (measured in radians
per unit length) where
y* = a + jB = jto/ e* y *
j Xm = j 12hi* I sin 0 <
Rm = |^*| cos»
Figure 4. Vector Representation of Zm.
Figure 5. Vector Representation of K^.
-------
14
Expressions for a and 6 will not be derived here but only the
formulas stated:
2tt
a = —
^ {/TTi^J - 1 ) (Eq. 6)
6 = [/TT tan2 6 + l)
(Eq. 7)
where X0 is the free space (vacuum) wavelength of the wave and
tan 6 = Kjn/Km. The unit of length used for X0 determines the units
for a and 3. Values of a and B have been calculated, based on the
dielectric data in Table 1, and are plotted as a function of
frequency in Fig. 6.
15.0
10.0
6.0
3.0
1.0
S .6
o ,
«r .3
a.
o
cc
a.
.1
.06
.03
J
,/
/
£ MUSCLE^
A
B SKIN
/
/
' B FAT
/
/
/
a
MUSC
-ly/
SKIN
a FAT .
300 400 600 1,000 3,000 6,000 10,000
FREQUENCY (MHZ)
Figure 6. Propagation Constant of Skin, Fat, and Muscle.
-------
15
Occasionally, a slightly different foim for a is used when
dielectric data are given in terms of K' and p (resistivity)
rather than K' and K". This expression takes the form
2^
a =
J1
r 60*o '
ro|
I?i +
^ P
- 1
(Eq. 8)
Here p must be in units of ohm-cm and XQ in units of cm. Appendix
A verifies the identity
tan 5
60AC
From the expression for the phase factor 3, the wavelength of an
EM wave (AjJ with freespace wavelength X , in a given material can
be computed as
2n
¦^m = 0
/ 1 + tan2 6 + 1
Appendix B gives the derivation of another, equivalent expression
for wavelength reduction in a dielectric material.
TOTAL DOSE ABSORBED BY SYSTEM
To calculate the total energy deposited in any infinite slab
dielectric system by a plane wave, the reflection coefficient r*
must be deteimined for the exterior surface of the system. |r*| is
defined as the ratio of the amplitude of the reflected EM wave to
the amplitude of the incident wave; specifically, |r*| is related
to the amplitudes of the electrical field (voltage) components of the
EM waves and consequently, when squared, |r*|2 refers to the power
reflected at a given interface. Once r* is determined, the total
energy or power extracted from an inpinging EM wave may be computed
for the total system. At this point only total deposited energy
or power is computed; its distribution throughout the system is
not specified.
For the slab model system considered here, assuming the base
layer to be infinite in depth, the surface area for impinging
waves semi-infinite in extent, and plane wave propagation, it can
-------
16
be shown that the impedance at any point p through the tissue
system is given by
Z? cosh Yd + Z£ sinh Yd
ZP = Zc Z* cosh yd + Z* sinh yd ^ ^
Z* refers to the complex characteristic, or intrinsic impedance
o§ the material containing the point p, and Z£ is the complex
effective load impedance resulting from the discontinuity with
underlying layers for the propagating wave, y is the prop-
agation constant for the material containing point p and d is the
distance of point p from the interface defining Z£. A stepwise
process is used for systems composed of more than one material
starting at the interface lying deepest and working outward toward
the surface; each layer of material successively becomes the load,
with a corresponding effective load impedance, for the next adjacent
medium. In this method, the surface impedance at the air-tissue
interface is finally determined from which r* may be computed as
Z* Csurface) ~ Z
r* Z* (surface) + Z0 fEq* 101
Subsequently, the percentage power absorbed by the system, 1 - |r*|2
can be obtained.
Because of the inherent complexity of confutation for this
procedure, a simplified technique utilizing the Smith chart can
be used to evaluate surface reflection coefficients. A brief
description of this method, by way of an example, is useful before
details are given concerning the computer generated solutions of
Equations 9 and 10 as applied to some tissue systems.
As an exanple of the Smith chart technique of determining
surface reflection,consider a model system composed of 3 cm of
fat tissue and 0.2 cm of skin tissue arranged over a semi-infinite
layer of muscle tissue exposed to EM energy at a frequency of
2450 mz. The model configuration is shown in Fig. 7 with material
parameters indicated.
The first step involves making a table of derived parameters
for the various tissues. Table 4 gives this information for the
model under consideration.
-------
17
TABLE 4. PARAMETERS FOR TISSUES IN FIGURE 7
Tissue K* K" |Z*[ R X e Am
(ohm) (ohm) (deg.) (cm) (cm'1)
Skin 42.9 14.0 56.1 55.4 8.82 9.05 1.85 0.541
Fat 5.83 1.01 155.0 154.4 13.3 4.92 5.05 0.107
Muscle 47.6 13.7 53.6 53.1 7.51 8.05 1.76 0.504
Values for normalized inpedance will be plotted for each numbered
point which is indicated in Fig. 7. For the first point, point 2,
the load inpedance at the interface is represented by the intrinsic
wave inpedance for nuscle since it is assumed that no waves entering
the nuscle are reflected. This load inpedance, Z£, must be normalized
with respect to the wave inpedance of fat in order to be plotted on
the Smith chart and is determined as
Z*l =
$
= . S3,6 B 0#346
155.0
Z* is $2 =
X. iS |Z
4.92° = 3.13°.
- e* - 8605
sin 3.13 . Z| may now be plotted
R then
The phase angle of Z5
is |2*| cos 3.13 ana ^
at Z* = 0.345 + j 0.0169 (see Fig. 8). Now, the thickness of fat,
3.0 on, is expressed in terms of the wavelength for 2450 Miz waves
in fat tissue. 3.0 cm is found to be 0.594 and point 2 on the
chart is rotated in a clockwise manner until 0.594 wavelengths have
been passed. Since one full revolution on the Smith chart represents
one-half wavelength, the new point is located at an angular displacement
equivalent to a reading of 0.597 wavelengths toward the generator,
allowing for an initial displacement of approximately 0.003 for
point 2. A circle of constant voltage standing wave ratio (VSWR)
is scribed about the origin for the appropriate degree of shift.
-------
18
AIR
K ^ = 1.00
Ka = 0.00
©
£0= 377 ohms
SKIN
Kj = 42.9
K$ = 14.0
©
@
0.2 cm.-
FAT
Kp = 5.83
Kp = 1.01
©
3.0 cm.-
©
NOT DRAWN TO SCALE
MUSCLE
kM = 476
KM =13.7
Semi-infinite
in extent
Figure 7. Model Configuration for Smith Chart Solution.
The radius is reduced by an amount equal to
e"2apXp _ e(-2)(0.107)(3.0 cm) _ 0.526
of its initial length. This completes the plotting of point 3
on the chart as Z* = 0.75 + j 0.38. To go from point 3 to point 4,
the modulus and pnase angle for Z| are computed again separately.
A notation is made that the phase angle for Zt is = tan -1(0.38/0.75)=
26.9° 3 5
The modulus |ZJ|, normalized with respect to skin, is confuted as
, , iz*,i |z?i
IZ?I = = 2.32 ohms
-------
19
Figure 8. Smith Chart Solution. Chart grid reproduced by
permission of Phillip H. Smith. Copyrighted 1949 and 1966
by Kay Electric Company, Pine Brook, N. J.
-------
20
The phase angle of Z| is ^4 = 3 ~ 0 S = 17.8 . As before the
resistive and reactive components of Z| are determined as = 2.21
and X4 = 0.71. Again, point 4 mist be rotated about the origin
equal to a displacement of 0.2 ay 1.85 A and reduced in radius
because of attenuation in the skin, by an emount e "O.^lo _ Qggg
This yields point 5 on the diagram located at
Z* = 1.15 - j 0.77
Notice the negative sign of the imaginary part due to the side of
the Smith chart upon which the point falls. Note consequently that
the phase angle associated with Z? is now negative, 4>r = tan
(-.670) = -33.8°.
With the modulus of the impedance at point 5, |Z*| = 1.38, and
Z* may now be computed as,
|Z*| [Z*|
~ — — tcos ("33.8°) + j sin (-33.8°)]
|r*|, the voltage reflection coefficient modulus, may now be determined
as the fractional length of the radius of the outer circle on the
Smith chart. In this case, |r*| = 0.71 and |r*|2, the power reflection
coefficient is found to be .50 yielding a value of 0.50 for 1- |r*|2,
the percentage of incident power that is absorbed by the slab tissue
sys tem.
The reflection coefficient and its respective phase angle may
be determined directly from the diagram by reading the appropriate
scale at the perimenter of the circle. This completes the detailed
description of the graphic-analytic analysis of iirpedance transformations.
A more complete and exact analysis of the problem can be performed
by programming the mathematical approach on a computer. " Such a
program designed to compute values of 1 - |r*| 2 for an N layer system
of tissue slsibs, each with a characteristic K1 and K", and thickness,
at a given frequency, was written. The output format as plots in
Figures 9, 10 and 11 illustrate how the percentage of incident
power absorbed in the system varies as a function of skin and fat
thickness at 915, 2450, and 8500 Mta respectively. The first two
frequencies are of particular interest from a hunan exposure viewpoint
because they are designated as medical and industrial frequencies which
include the use of diathermy and microwave ovens.
-------
0.5 CM SKIN
_L
-L
X
X
o.o
Figure 9.
W THICKNESS (CK.i
915 MHz Power Absorption in Tissue.
4 * *
THJCW&5S <(M.)
Figure 10. 2450 MHz Power Absorption in Tissue,
-------
22
FAT THICKNESS (CM)
Figure 11. 8500 MHz Power Absorption in Tissue.
At 915 Miz, with to 0.4 ail skin, the presence of fat tissue
always increases the total amount of power which is absorbed by the
system. The addition of skin to a fat muscle complex with fat dimensions
greater than 3 cm generally decreases the percentage power absorbed,
while at fat thicknesses less than 3.0 cm, the inclusion of skin to
the model may enhance power absorption. This relationship is critically
dependent on the particular tissue dimensions.
Over the entire range of tissue dimensions considered, which, in
general, span the maximum range over which these dimensions would
ever take in humans, at 915 Mlz the absorbed power may vary between
35 and 97 percent, or by a factor of approximately three.
At 2450 MIz, the presence of skin is more effective in reducing
the extent of total power absorption. In this case, the presence of
fat tissue also increases absorption as long as the skin thickness
is less than 0.1 cm, compared with 0.4 cm at 915 NHz. This occurs
because skin tissue becomes markedly less transparent as the frequency
is increased. The possible range in percentage power absorption by
the model at 2450 NHz extends from about 21 to 100 percent, or a
-------
23
factor of about five. As skin thickness exceeds 0.1 cm, power
absorption at any given fat thickness greater than about 0.5 cm is
generally reduced.
As the frequency is increased to 8500 Miz, the presence of skin
is strikingly apparent. So long as the model is restricted to skin
layers less than 0.2 cm, the presence of fat generally increases the
amount of power absorbed. This time, for the range of possible
dimensions considered, the power absorbed by the system may range from
about 26 to 99 percent, or a factor of about four. Thus, 2450 Mlz
represents the frequency of EM energy, ol the three frequencies con-
sidered, which provides the most variable degree of possible coupling
between field and model.
All three figures are characterized by the appearance of periodic
relative peaks in power absorption. These peaks occur at intervals of
AF/2 with the first peak of any curve for no skin appearing at a fat
thickness equal to a quarter wavelength for the wave in fat at that
frequency. In this respect it may be assumed that the fat is acting
as a quarter-wave impedance transformer, optimizing the coupling of
energy, or power, between air and the tissue slabs. Increasing
the fat thickness illustrates the damping effect due to loss in
the fat.
From these data and the given incident power density of a
microwave exposure field, the coupling between a particular model
configuration and the field may be established. The exact distribution
of this absorbed power, as manifested in heat development, will next
be traced through selected systems.
DOSE DISTRIBUTION WITHIN SYSTEM
In deriving the spatial distribution of heating in the slab
model, the deepest lying interface is taken as the origin of
coordinates upon which all calculations will be referenced. An
effective incident wave, E , will be assumed. It is a composite
of the sun of the initial wave transmitted directly from the
radiation source and all positively directed reflections arising
within the medium adjacent to the interface. As Schwan points out,
the reflection coefficient for such a sun of waves can be shewn to
be identical with that for a single wave (9). All distances measured
to the right of an interface are positive and distances to the left
are negative.
For a skin-fat-muscle system, consider the electric field at airy
point within the fat material. Here the electrical field may be
-------
24
expressed as the resultant of the incident and reflected waves.
%at ^incident + ^reflected
-ije-v.rfo.vj x<
0 (Eq. 11)
To arrive at an expression for the magnetic field, Maxwell's equation
from Faraday's law,
n X! 9 §
ExE=--=-=-y
3t
at
is expanded in rectangular coordinates
if-
8Ez
I ay
3Ey
3z
3 Ex
3Ez
3z
ax
+ k
3Ey
ax
9 Ex
3y
= -y
at
(iHx + ^Hy + kHz)
(Eq. 12)
For a plane wave traveling in the x direction of the slab model,
the only components of Eq. 12 which contribute are
~ 3x
9Hz
at
or,
H = -
3E
9x
dt
and when E varies in time according to E = EQe^WT,
H
fat
o^F
fe^F* - r«£YF*l
FM
x < 0
(Eq. 13)
-------
25
r*
where: FM = complex reflection coefficient at fat-muscle interface
yF = complex propagation constant for fat
yF = permeability of fat medium
= angular frequency of wave = 2 nf
Ui
The value of the E field transmitted into the muscle, at the
interface is
^transmitted ^o ^ rFNp
from equation 11 when x is zero. With this value of E just inside
the muscle tissue, the description of the E and H fields anywhere
in the muscle are obtained through a similar procedure as
W ¦ C1 + r*> v"w I)0 (a»-
H = CI + r*l e'^M* CVn I ci
muscle ^ x > 0
£or other layers of tissue, such as skin, which lie on top of the
fat tissue, equations similar to Eq. 11 and Eq. 13 are obtained with
appropriate substitution of dielectric parameters and a factor which
corrects for the change in the E field which occurs as the origin of
coordinates is moved to the left. The appropriate forms of these equations
are: x < 0
*skm-v"rpvYsxs~<*•
b
Hskin ~ Eo
"TFdF YS (e-Vs - r« eYSxS) X < 0 (Eq. 17)
jygd) ^ dp < 0
The intensity of the wave configuration in the tissyes, or energy
flow, is given by the complex Poynting vector, I - / n11flntitv
above the H signifies that the complex conjugate for the quantity
is used? This^lation is arrived at after considering the expression
( S • n da = [ (EX H) • n da
Jq" jS ~
-------
26
2
where S = the Poynting vector = E X H (watts/m ). Though this
relation can be interpreted as the total flow of energy (intensity)
through a closed surface at a point in the field, it cannot definitely
be concluded that the intensity of energy flow at a point is
S = E X H; there could be added to this quantity any vector integrating
to zero over a closed surface without affecting the total flow.
Gomplete details of this expression and how the complex Poynting
vector is related to the actual physical entities involved in heat
production are given by Stratton (25), pages 132-137.
When y = a + j g and r* = |r*| e^, the intensities in fat
and muscle are found to be
fat
l¥%-
eF + 3»F_ p e
2ypU)
0 0
x < 0
(Eq. 18)
e"2°FX " lrFM|2 + j2 lrFMl sin(2SPx + +PH)
muscle
1
2
+ j«M
2V
E0E0
x > 0
(Eq. 19)
1 +
2 Irft^l cos
I r*
EM
2)
Heat development rate per unit volume is computed by taking the
negative space rate of change of the real part of the expressions
for intensity (25).
3 n T _ 1 E2 re-2«FX | * |2 2apx
" Re fat - J ypo) bo [_e I'fm1
+ 2 |%| cos (2gpx + M)]
-------
27
Since
(see Appendix C), this expression reduces to
"""fat = I °fe! [e"2c,FX + lrml2 e2o,FX
x < 0
+ 21 rFNll cos (26Fx + ^FM^l
(Eq. 20)
Taking the negative derivative of the intensity in muscle, and
recalling that the electrical field must be continuous across the
interface, yields
®^muscle = 2 °M^o (* + lrFM^ + 2lr?Ml cos ^fm] e ^
where the term HDR refers to heat development rate.
Since, as pointed out, expressions for the E and H fields in any
other tissue layers exterior to the two base layers (muscle and fat
in this case) have forms such as Eqs. 16 and 17, foimulas similar
to Eq. 20 giving the heat development rate in other overlying tissues
are obtained. In the case of a layer of skin tissue over the fat
layer,
x > 0
HDRglHn = | osE2e"2<,[!dF [e"2t,sX + |rgp|2 e^S* (Eq. 22)
+ 21 r|F | cos(2Bsx t ^SF)1 x*°
d < 0
F
-------
28
Recall that as the distribution is worked out for each layer of
tissue, starting with the deepest lying interface, the origin of
coordinates is shifted to the left (in the model configuration) so
that it defines the next considered interface. The sign convention
for positive distances to the right and negative distances to the
left of the interface is maintained. In this fashion, any number
of tissue layers may be modeled and analyzed.
When only two layers of tissue are being considered, the heat
development rates obtained are directly comparable. This is because
a reference field, EQ, was established at the fat muscle interface
and all expressions for intensity or HDR, in muscle or fat, were
derived in terms of distance from this reference field. However,
as additional layers are added to the model, and the origin of
coordinates for deriving intensity and HDR in those layers is moved
to the left, i.e., toward the source of radiation, the electrical
field at this new interface must be written in terms of the reference
field at the muscle-fat interface. This requires compensation for
2 3 * 5 S 7
TISSUE THICKNESS (CM.)
Figure 12. Relative Heat Development Rate in Two-Layer Model.
-------
29
attenuation of the effective incident wave in the fat layer or multi-
plication by a factor of e'YF^F, d < 0. In the heat development
formula, the correction term appears as e'^YF^F, having been squared
in the process. This correction must be performed for all layers
other than the two base layers, taking into account the attenuation
occurring through all layers separating the new origin from the
initial interface.
Figure 12 shows the relative heat development rate in the
fat-muscle model at three different frequencies for a fixed fat
thickness of 3 cm. The abrupt rise just inside the muscle layer is
characteristic of the radical change in conductivity of the medium
compared to that of fat. Relative peaks in the heating rate may be
seen for frequencies high enough for standing wave formation to
occur within the fat layer; the first peak in such cases always
occurs a quarter wavelength in front of the reflective interface.
This corresponds, in fat, to approximately 0.5 cm and 1.3 cm at
8500 and 2450 Mta, respectively. At 915 MHz, 3.0 cm of fat is not
quite equal to a quarter wavelength so that the turning point of the
first peak is not visible. At 2450 MHz surface heating of the fat
layer is about 68 percent of the maximum heating which occurs deeper
within the fat; the greatest rate of heat development in the entire
model at this frequency occurs on the exterior-most surface of the
muscle. At 915 NHz and 8500 NUz, surface heating of the fat layer
predominates over the heating rate anywhere else in the medium.
Figure 13 illustrates the spatial distribution of the heat
development rate for a three component system (0.2 cm skin, 3.0 cm
fat and semi-infinite muscle) at 2450 MHz. In this diagram the scale
for the skin thickness has been expanded by a factor of 10 to permit
careful visual inspection of the heating distribution. Because
0.2 cm represents substantially less than a quarter wavelength in
skin at even 8500 Nfiz, no oscillation of the heating rate from the
standing wave pattern is seen. Skin heating is the obvious manis-
festation of microwave exposure for this model configuration at the
higher frequencies. This follows from the fact that skin tissue
becomes more lossy as the frequency is increased.
In summary, in slab tissue models with muscle as a base, tissue
heating, on the average, tends to increase as the model is transversed
axially from the deepest lying interface toward the exterior.
It is valuable to know the total quantity of heat developed
within each of the layers of the model so that comparisons can be
made as to which of the tissues receives the greatest thermal burden
from exposure. This may be accomplished by integrating each of the
expressions for heat development rates over the dimensions for the
respective tissue slabs. The percentages of total heat developed
in each tissue may then be plotted as a function of any tissue
-------
30
dimension. In the case here, the subcutaneous fat layer thickness
has been selected as the parameter to plot on the absicissa.
Expressions for the particular integrals of the HDR's (Eqs. 20,
21, 22) over total thicknesses di are obtained upon integration as:'
r^M
J0
,0
= gMEo
2oiM
1 - e
2aMdM
HDRp = <
1 + IffmI2 + 21Ffm' cos ^FM
+ e"2apdp + | 12 . 2 e2apdF
rM FM
2<*
p I r*"
+ LSln ^FM " sin(26pdp +
V
< 0
1.0
SKIN
FAT
MUSCLE
FT2n^' Rel^r"eat Devel°Pment Rate in Three-Layer Model.
Frequency is 2450 MHz. *
-------
Q'SEq g~2oipdp
2as
1
-1 + e"2aSdS + |r* I2 - |r* |2 e2aSdS
i SFi i1SFt
sin cf>SF - sin(23sdg + <()SF) > dg < 0, df < 0
The sign convention for the thickness d is the same as before,
muscle thickness being positive and fat and skin thicknesses (or
any other tissue layers exterior to the deepest interface) being
negative. Since the integral of heat development in skin has been
normalized with respect the remainder of the layers, because of the
correction factor e~2YFdF} the percentages of total heat developed
may be displayed by plotting
as a function of dp for a two component system. When skin, or a
three or more component system is used,
is plotted on the same graph against dp. Figures 14, 15, and 16
illustrate the results of such computations for two and three
component systems. Where skin tissue is present, its thickness is
varied to show the change in partition of heat development between
the three tissues. By observing the various ordinate values of the
curves at any particular fat thickness, the breakdown in total heat
developed within the specified model may be determined. Total heat
developed within the system is assigned a value of 100 percent.
In general, the curves illustrate that at the lower frequency,
deep heating of the muscular tissue is more predominate while at
the higher frequency, the heating of surface tissue is greater.
Also, as Schwan discusses (11) the frequency range of approximately
1000 to 3000 MHz represents a range in which both the distribution
of total incident energy absorbed by the model, and the development
of heat in the various tissues are highly variable. That is, there
is a transition zone, in terms of frequency, which more or less
connects the regions showing deep heating at low frequencies and
surface heating at higher frequencies. It is of interest from the
/HDRm
/HDR^j + /HDRp
/HDRm
/HDRn + /HDRp + /HDRg
is first plotted against dp and then
/HDRm + /HDRp
/m^ + /HDRp + /HDRg
-------
32
standpoint of public exposure to microwaves that the industrial-
medical frequency of 2450 MHz lies in this complicated interaction
region. The exact fraction of total heat developed within the various
tissues is a function of an infinite set of possible tissue dimension
combinations. Maximum values of fractional heating may be determined,
however, for every possible combination within specified limits of
the dimensions. If, for practical purposes, one considers skin
thickness to vary from 0 to 3 mm, and the fat layer to vary between
0 and 4 cm, the following itiaximura. values are obtained:
915 MHz
Maximum possible fractional skin heating = 60 percent
Maximum possible fractional fat heating = 29 percent
2450 MHz
Maximum possible fractional skin heating = 74 percent
Maximum possible fractional fat heating = 25 percent
8500 MHz
Maximum possible fractional skin heating = 93 percent
Maximum possible fractional fat heating = 9 percent
-------
100
MUSCLE
1 I I I I 1 I J | |
0 1 2 3 4 5 . 6 7 8 9 10
FAT THICKNESS ( CM.)
Figure 14A. Thermal Dose Distribution in
Two-Component System at 915 MHz.
FAT THICKNESS (CM.)
Figure 14B. Thermal Dose Distribution in Three-Component
System at 915 MHz; Skin Thickness,0.2 cm.
-------
34
Figure 15B. Thermal Dose Distribution in Three-Component
System at 2450 MHz; Skin Thickness, 0.1 cm.
-------
35
Figure 15C. Thermal Dose Distribution in Three-Component
System at 2450 MHz; Skin Thickness, 0.2 cm.
Figure 15D. Thermal Dose Distribution in Three-Component
System at 2450 Wz; Skin Thickness, 0.3 cm.
-------
SKIN
4 5 6
FAT THICKNESS (CM.)
Figure 15E. Thermal Dose Distribution in Three-Component
System at 24S0 MHz; Skin Thickness, 0.4 cm.
s w
MUSCLE
4 5 6
FAT THICKNESS (CM.)
Figure 15F. Thermal Dose Distribution in Three-Component
System at 2450 MHz; Skin Thickness, 0.5 cm.
-------
37
loo r
KUSCLE
4 5 6
FAT THICKNESS (CM.)
Figure 16A. Thermal Dose Distribution in
Two-Component System at 8500 MHz.
70 -
Figure 16B. Thermal Dose Distribution in Three-Component
System at 8500 fcWz; Skin Thickness 0.2 cm.
-------
38
As skin thickness is increased the amount, percentage wise, of total
heat developed within the skin layer becomes larger. Consequently,
for any given fat layer dimension, the fractional heat developed
within fat decreases. This general relationship is graphically
illustrated in. Figure 17 for 2450 MHz and 2.0 cm of fat.
SKIN THICKNESS ( mm.)
Figure 17. Effect of Skin Thickness on Fraction of
Total Heat Developed in Skin and Fat.
-------
39
COMMENTS ON TEMPERATURE RISE IN TISSUES
To evaluate the extent of temperature rise in the individual tissues
it is necessary first to determine the percentage of incident power
absorbed by the model, as from Figure 9, 10, or 11 (page 21). Once this
quantity is known, the location of this power as heat generated
can be determined as from Figures 14, 15, 16 (page 33). From the
amount of power (watts) delivered to each layer of tissue, the period of
exposure, the initial temperature of the particular tissue, and the
specific heat capacity of the tissue, one can estimate the final
tenperature of the medium. This estimation does not take into
account blood circulation in the model; radiation, convection, or
conduction of heat to the ambient environment; nor does it hold
when such low intensities of exposure are used that heat flow within
the model is significant (i.e., the time of exposure is long compared
with the time constants which characterize heat exchange in the model).
The effects of radiation, evaporation of perspiration, or conduction
such as would occur in an animal are also not considered. Thus,
under these restricted conditions, where for a transient period of
time the tissue temperature rise is linear with time of exposure,
an attenpt can be made to estimate what might be called a total
microwave thermal dose delivered to the model (a product of exposure
time and absorbed energy flux).
The use of such information to propose standards or limits for
human exposure to microwaves, however, should be made carefully
because of the obvious variables and assumptions involved. Specifying
the maximum tolerance dose (watt-min/air) for humans must incorporate
biological studies to determine what tissue teniperatures are
allowable for any exposure time. A corresponding study must be done
to find what tissue temperature elevation rates are associated with
these maximum allowable tissue or body tenperatures.
Several studies have been reported that deal with the effects
of heat stress in humans (26, 27). These data, as applied to the
case of microwave heating, have been used to illustrate the necessity
of lowering exposure limits in other than ideal environmental
circumstances (28). However, in light of the restrictions inposed,
it is still possible to examine the thermal load placed on various
tissues in the model system. These estimates provide a means of
visualizing, in an approximate manner, the heating that will occur
in living tissue systems of humans.
Thermal loads in various layers of the stratified model tissue
system may be evaluated in terms of maxinum temperature excursions
during the irradiation time. In general the thermal energy imparted
-------
40
to the muscle tissue is related to a corresponding temperature rise
by
= m. c AT
M M M
where: = energy input to muscle during specified time
m^ = mass of muscle tissue absorbing
c^ = specific heat capacity of muscle tissue
ATM = temperature excursion of muscle in degrees C
during time of energy input
This relation holds for times which are short compared with the thermal
diffusion time constants of the model; it assumes (1) that adiabatic
conditions prevail, and (2) the energy distribution and temperature
rise in the mass m^ is uniform. These conditions, of course, do not
typify the situation in reality; however, for estimation purposes only
the equation is reasonably accurate.
Since, generally, heat development per unit volume is the quantity
computed, the above energy relation can be written for fat and muscle
tissues as:* n _ „ u ^ AT
QF - PpVpCpATp
Q = p V c AT
M M M M M
Here the p's and V's are the densities and volumes for the two tissues.
The fractional temperature rise, then, of the fat with respect to muscle
is
ATF = Qf # PMVM°M
atm ppVpcF *
Since, strictly, heat development per unit volume is a function of the
spatial coordinates, here Q(x),
Q(x) = p c AT(x)
where T(x) is the temperature accurately expressed as a depth or
coordinate function. The total energy in the fat or muscle is then
obtained by integration, not simple multiplication as implied above.
Here, only a very simplified estimation process is discussed.
-------
41
for the time during which the Q's are delivered. If the unit of
volume is chosen the same in both cases and the Qp delivered in a
given time is related to the QM delivered in the same time as
Qp = F ^4
where F is the fraction of that Qp represents, then
AT.
F ^ PM°M
atm F pfcf
Thus, if the tissues are initially at the same temperature, the
maximum temperature elevation attainable in the fat medium can be
expressed as some part of the maximum temperature elevation attainable
in the underlying muscle tissue.
pnFm
— MAX
ATf MAX = F ~J~ AT 1 ]
fat PpCp muscle
If it is desired to find the same relation for a model with skin as
well, then
4W« - G 4We™*
Here G is the factor relating the ratio of maximum heat development
rate within skin to that in muscle. p~ and c^ are the density and
specific heat capacity, respectively, tor skin. Values given in
Table 5 for p and c for the various tissues are taken from the
literature (29).
TABLE 5. DENSITIES AND HEAT CAPACITIES OF VARIOUS TISSUES
Tissue
p Cfijn/cm3)
c (cal/gvPC)
Skin
1.20
0.81
Fat
0.92
0.55
Muscle
1.27
0.91
-------
42
Computations for tissue volume heat development rates were
programmed on a computer for a variety of tissue dimensions. The
maximum values of the heat development rates in the fat and skin
layers, normalized with respect to those in the muscle, F and G
respectively, were determined, multiplied by the appropriate ratios
of p and c, and plotted in Figures 18 and 19 as functions of fat
thickness. Figure 18 refers to a two layer system and figure 19
gives the results for a three layer system in which the skin thickness
is set at 0.2 cm. From these curves the relationship between the
various maximum tissue temperatures can be determined in terms of
the greatest temperature excursion occuring in the muscle layer.
The ordinate value in these figures is simply the ratio of the
maximum temperature excursion in either skin or fat to that in
muscle. From Figure 18 it is determined that in a fat muscle system,
the greatest temperature elevation in the fat layer will be the same
as that in the underlying muscle, provided the fat is about 1.8 cm.
thick and about twice as great with a fat thickness of 4.6 cm.
These figures are for an idealized case which does not incorporate
thermal loss or conduction within the tissue; in light of these
simplications, the temperatures implied will be high. In the case
of three-layer systems including skin, where the emissivity will
introduce significant infrared radiation, the estimate may be very
high compared to reality. This approximate estimation technique
agrees reasonably well with the experimental maximum fat and muscle
temperatures attained in work reported by Guy (30). His investigation
involved the recording of thermographic images, indicating maximum
tissue temperature, for various combinations of fat-muscle models
exposed to several frequencies of microwave energy. Surface heating
effects are very obvious in these graphs where Figure 19 c shows
the impractical task of deep muscular heating with very short
wavelength microwaves. Heating here is almost solely on the surface.
These computations imply a rather difficult, and sometimes impractical
measurement of the maximum base layer temperature in humans or
laboratory animals. However, the results for the model systems should
prove useful for qualitatively evaluating certain exposure situations.
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5.0
3.5
al 2.0
FAT THICKNESS! CM.)
Figure 18. Maximum Tissue Temperature Elevation in
Two-Component System at 2450 MHz; No Skin.
RATIO OF SKIN TO MUSCLE
RATIO Of FAT TO MUSCLE
4 9 6
FAT THJCKME35 f CM.)
Figure 19A. Maximum Tissue Temperature Elevation in
Hiree-Component System at 915 MHz; 0.2 cm Skin.
-------
"
70 -
' RATIO OF SKIN TO MUSCLE
RATIO OF FAT TO MUSCLE
*T I I I L
-I I L
3 4 5 6
FAT THICKNESS ( CM.)
S 10
Figure 19B. Maximum Tissue Temperature Elevation in
Three-Component System at 2450 MHz; 0.2 cm Skin.
ratio of skin TO MUSCLE j
RATIO OF FAT TO MUSCLE
FAT THICKNESS (CM.)
Figure 19C. Maximum Tissue Temperature Elevation in
Three-Component System at 8500 MHz; 0.2 cm Skin.
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45
SUMMARY AND CONCLUSIONS
The planar slab model for tissues is a very simplified model_
that is subject to a large number of constraints. The application
of this model to actual human and laboratory animal dosimetric
studies involves, consequently, an element of approximation when
precise dose estimates are desired. Generally, actual temperature
excursions experienced by the various tissues which comprise the
exterior part of the anatomy will be typically lower than predicted
here due to thermal exchange with blood and environment. Also, the
semi-infinite slab model should not be used when dealing with structures
of the anatomy characterized by significant curvatures, e.g., the
head. In these cases, the curvature plays an important role in
refracting the incident microwave beam with the possible development
of local relative hot spots which would not have been predicted with
the semi-infinite assumption.
A spherical model approximating a monkey head has recently been
used to predict heating distributions within the cranial cavity (31).
The approach used consisted of calculating both the scattered and
interior fields for a series of concentric spherical shells.
Analytical details of scattering from dielectric spheres and spherical
shells are available in the literature (32 - 34). Similar
investigations have also been carried out to determine effective cross
sections for a range of tissue equivalent sphere sizes (33). The
primary observation indicated that the cross section was critically
dependent on sphere radius, again suggesting the refractive role of
curved surfaces in microwave interactions. Thus, the total power
extracted from a plane wave by a dielectric sphere cannot, in
general, be computed simply by finding the product of incident
power density in the field and shadow cross section of the sphere.
These special considerations are given to reveal the complicated
nature of microwave dosimetry in irregularly shaped objects. Con-
servative dosimetric estimates based on the semi-infinite slab model
results should, therefore, take into account the greatest possible
degree of power absorption.
In this report we have mathematically described a planar slab
tissue model in terms of its microwave absorption characteristics.
Both the total power absorption and the development of heat spatially
have been examined for a wide range of tissue dimensions. The partition
of this heat, fractionally, in the various tissues has been obtained,
and finally the relation between maximum tissue temperature elevations
in the various tissues has been determined. The following conclusions
can be drawn:
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46
1. Beam power extraction, or coupling, may vary as much as a
factor of 5 over the frequency range of 915 to 8500 MHz and the
possible range of tissue dimensions considered.
2. Peak heating within the base layer is at the interface
between that layer and the next outermost layer. Peak heating
within other overlying tissues may occur at various positions
throughout the tissue, depending upon the particular combination
of tissue dimensions.
3. The fractional integrated heat development within skin
tissue increases with increase in frequency and increase in skin
thickness.
4. Maximum temperature rise in skin and fat tissues are
generally always higher than the greatest temperature excursion
in the deepest lying tissue. This is due to the enhanced surface
heating of a relatively lossy tissue and the ratios of the tissue
specific heats.
5. Biological tissues may be broadly classified, dielectrjcally,
either as high or low loss. Skin, muscle, blood, and similar non-
fatty tissues are high loss; fat and bone are low loss. Thus, for
practical purposes most biological systems may be modeled on the
basis of two types of material for microwave absorption analysis.
6. The semi-infinite model is not satisfactory for objects
with radii of curvature on the order of the wavelength or smaller
than that of the incident energy.
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47
References
1. Baldwin, B. R., P. C. Constant, Jr., R. W. Fetter, B. L. Jones,
V. W. Klein, E. J. Martin, Jr., L. Runge, D. L. Waidelich. 14
June 1960. Survey of radiofrequency radiation hazards, Summary
Report #2 (20 May 1960-19 May 1961). Midwest Research Institute,
Kansas City, Missouri. Contract No. Nobs-77142, Navy, Bureau of
Ships.
2. Saito, M., and H. P. Schwan. 1961. Biological effects of microwave
radiation, Vol. 1, pp. 85-97, Plenum Press, New York.
3. Schwarz, G., M. Saito, and H. P. Schwan. 1965. On the orientation
of non-spherical particles in an alternating electrical field. J.
Chem. Phys. 43:3562-3569.
4. Saito, M., L. D. Sher, and H. P. Schwan. July 1961. RF field-
induced forces on microscopic particles. Digest 4th International
Conference on Medical Electronics.
5. Frey, A. 1962. Human auditory system response to modulated electro-
magnetic energy. J. Appl. Physiol. 17:689.
6. Marha, K., J. Mjsil, and H. Tuha. 1968. Electromagnetic fields and
the living environment. State Health Publishing House, Prague,
Czechoslovakia.
7. Letanet, A. A., and Z. V. Gordon (ed.). 1960. The biological
action of ultrahigh frequencies, Moscow. Translated by U.S. Joint
Publications Research Service, JRPS:12471, February 1962, Washington,
D.C.
8. Presman, A. S. 1970. Electromagnetic fields and life, Plenum Press,
N.Y. - London.
9. Schwan, H. P., E. L. Carstensesi, and K. Li. Septenfrer 1953. Heating
of fat-muscle layers by electromagnetic and ultrasonic diathermy.
Transactions of the AIEE, p. 483.
10. Livenson, A. R. 1966. Determination of the coefficient of reflection
for multilayered systems of biological tissues in the microwave range.
Translated from Russian in 1968, Translation Division, Foreign
Technology Division, Wright Patterson AFB, Ohio. AD681254.
11. Schwan, H. P., and K. Li. Noveniber 1956. Hazards due to total
body irradiation by radar. Proc. I.R.E. 44:1572-1581.
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48
12. Schwan, H. P., and G. M. Piersol. 1954. The absorption of electro-
magnetic energy in body tissues, Part I. Am. J. Phys. Med. 33:371-
404.
13. von Hippel, A. R. 1954. Dielectrics and waves. The M.I.T. Press,
Cairbridge, Massachusetts.
14. Frohlich, H. 1949. Theory of dielectrics. Clarendon Press,
Oxford.
15. Schwan, H. P. 1957. Electrical properties of tissue and cell
suspensions, p. 147-209. In Advances in biological and medical
physics, Tobias and Lawrance (eds.), Vol. 5, Academic Press,
New York.
16. Schwan, H. P. 1958. Biophysics of diathermy, p. 55-115. In
Therapeutic heat, S. Licht (ed.). Elizabeth Licht, Publisher,
New Haven, Connecticutt.
17. Cook, H. F. October 1951. The dielectric behaviour of some types
of human tissues at microwave frequencies. Brit. J. Appl. Phys.
2:295.
18. England, T. S., and N. A. Sharpies. 1949. Dielectric properties of
the human body in the microwave region of the spectrin. Nature
163:487-488.
19. Schwan, H. P., and K. Li. 1953. Capacity and conductivity of body
tissues at ultrahigh frequencies. Proc. I.R.E. 41:1735-1740.
20. Sharpe, C. B. 1960. A graphical method for measuring dielectric
constants at microwave frequencies. Trans. I.R.E., MTT-8, p. 155.
21. Roberts, S., and A. von Hippel. 1946. A new method of measuring
dielectric constant and loss in the range of centimetre waves. J.
i^ppl. Phys. 17:610.
22. Harvey, A. F. 1963. Microwave engineering. Academic Press, London-
New York, p. 233-270.
23. GiLoaore, J. F. May 1966. Measurements of dielectric materials
with the precision slotted line. General Radio Experimenter,
published by General Radio Coup any, West Concord, Massachusetts.
24. Livenson, A. R. 1964. Electrical parameters of biological
tissue in the microwave range, Report II. Methods of gauging
electrical parameters of biological tissue. Translated from
Russian by Joint Publications Research Service, JPRS:26424,
16 September 1964, Washington, D. C.
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49
25. Stratton, J. A. 1941. Electromagnetic theory. McGraw-Hill Book
Company, New York.
26. Belding, H. A., and T. R. Hatch. August 1955. Index for evaluating
heat stress in terms of resulting physiological strains. Heating,
Piping, Air Conditioning 26:129-136.
27. Haines, G. F., and T. R. Hatch. Noventier 1952. Industrial heat
exposures - Evaluation and control. Heating and Ventilating,
p. 94-104.
28. Mumford, W. W. 1969. Heat stress due to RF radiation. Proc.
I.E.E.E. 57:172-177.
29. Lipkin, M., and J. D. Hardy. 1954. Measurement of some theimal
properties of hunan tissues. J. Appl. Physiol. 7:212-217.
30. Guy, A. W., J. F. Lehmaim, J. A. McDougall, and Carrol C. Sorenson.
1968. Studies on therapeutic heating by electromagnetic energy,
p. 26-45. In Thermal problems in biotechnology, American Society
of Mechanical Engineers, Heat Transfer Division, New York.
31. Shapiro, A. R., R. F. Lutomirski, and H. T. Yura. Septenber 1970.
Induced fields and heating within a cranial structure irradiated
by an electromagnetic plane wave, Technical Report P-4458-1.
The Rand Corporation, 1700 Main Street, Santa Monica, California.
32. Anne, A., M. Saito, 0. M. Salati, and H. P. Schwan. 1961. Relative
microwave absorption cross sections of biological interest. In
Proceedings of the fourth annual tri-service conference on the
biological effects of microwave radiation, Vol. 1, Plenum Press,
New York, p. 153-176.
33. Aden, A. L., and M. Kerker. 1951. Scattering of electromagnetic
waves from two concentric spheres. J. Appl. Phys. 22:601-605.
34. Chu, G., D. G. Dudley, and T. W. Bristol. 1969. Interaction
between an electromagnetic plane wave and a spherical shell.
J. Appl. Phys. 40:3904-3914.
35. van Hippel, A. 1954. Dielectric materials and applications,
John Wiley and Sons, New York.
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SI
APPENDIX A
Verification that tan <5 =
60A
£L
K'P
Given;
p = resistivity
A0 «¦ free space wavelength
a) » angular frequency
f * frequency
e0 = permittivity of free space
= permeability of free space
c = speed of light
Z« = = 120tt ohms = 377 ohms
u y en
£o
K* = relative permittivity
of medium
KM = relative loss factor
of medium
c2^oeo = 1
From:
KM =
P we.
P =
K"a)e,
Thus:
60X
TV
K'
60X K^_ ^ 60Xo2ufeo = 120irXofeo
K
ICW,
" T" Vo "f — = ~W~ zoC£o " "T" zo—
/
VoeO
K"
m.-jr-
K"
o v.
T-zo-TaT-
Uius:
60Xn K"
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52
APPENDIX B
Wavelength Reduction in a Dielectric Material
The index of refraction, n, is related to the velocity of electromagnetic
waves in vacuum and a dielectric material by
c
n = —
where:
c = velocity of light
v = phase velocity of the wave in a dielectric medium.
Now:
n = -SL. = l/yo£o = J ym£m
v 1 V ^o
If tijn/iio is 1 as in the case of biological materials, then
= v1C*T or, v =
v m A*~
m
Since v = f-2^,
c
Xm" f
If f is in units of MHz, then
, _ 30,000
xm Ccm) « —
t t Qth) ]
The magnitude of K* is used under the radical sign. For materials with
lew loss, such as fat, direct use of K' rather than the magnitude of K*
will yield fairly accurate results.
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APPENDIX C
Verification that ag = HSp.
Given:
Y * joj /e*y* (1)
Y = a + je (2)
for biological tissues y* = = po
thus
Y - ju Ae' - je"Kv£l (3)
Square (2) and (3) and set equal. Equate imaginary and real parts
to get:
2 ae * (du^ct (Imaginary part) (4)
e2 - a2 - 0)2v^e' (Real part)
From (4)
~ ^
Op - 7
s. GOVERNMENT PRINTING OFFICE : 1972-484-484/143
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