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16070658.1985.11720308

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Journal of Microwave Power
ISSN: 0022-2739 (Print) (Online) Journal homepage: http://www.tandfonline.com/loi/tpee19
Application of a Finite-Difference Technique to the
Human Radiofrequency Dosimetry Problem
R.J. Spiegel, M.B.E. Fatmi & K.S. Kunz
To cite this article: R.J. Spiegel, M.B.E. Fatmi & K.S. Kunz (1985) Application of a FiniteDifference Technique to the Human Radiofrequency Dosimetry Problem, Journal of Microwave
Power, 20:4, 241-254, DOI: 10.1080/16070658.1985.11720308
To link to this article: http://dx.doi.org/10.1080/16070658.1985.11720308
Published online: 17 Jun 2016.
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Date: 26 October 2017, At: 10:39
Application of a Finite-Difference Technique to
the Human Radiofrequency Dosimetry Problem
Downloaded by [UNSW Library] at 10:39 26 October 2017
R.J. Spiegel, M.B.E. Fatmi and K.S. Kunz
A powerful finite-difference numerical technique has
been applied to the human radiofrequency dosimetry
problem. The method possesses inherent advantages
over the method-of-moments approach in that its
implementation requires much less computer memory.
Consequently, it has the capability to calculate specific
absorption rates (SARs) at higher frequencies and
provides greater spatial resolution. The method is
illustrated by the calculation of the time-domain and
frequency-domain SAR responses at selected locations
in the chest. The model for the human body is
comprised of rectangular cells with dimensions of
4x4x6 em and dielectric properties that simulate
average tissue (2/3 muscle). Additionally, the upper
torso (chest) is configured by both homogeneous and
inhomogeneous models in which this region is
subdivided into 20,736 cells with dimensions of
1x 1 x 1 em. The homogeneous model of the chest
consists of cells with average tissue properties, and
the calculated results are compared with measurements acquired from a homogeneous phantom model
when the exposure frequency is 350 MHz. For the
inhomogeneous chest model the lungs and surrounding
region (ribs, spine, sternum, fat, and muscle) are
modeled with as much spatial resolution as allowed by
the 1 x 1 xl cm cells. Computed results from the
inhomogeneous chest model are compared with the
homogeneous model.
ABOUT THE AUTHORS
R.J. Spiegel is with the U.S. Environmental Protection
Agency, MD-74C, Research Triangle Park, NC 27711; M.B.E.
Fatmi is with Northrop Services Inc., Research Triangle Park,
NC 27711 and K.S. Kunz is with the Lawrence Livermore
Laboratory, Uvermore, CA 94550. Address al/ correspondence to R.J. Spiegel at the above address. Disclaimer: The
research described in this article has been reviewed by the
Health Effects Research Laboratory, US Environmental Protection Agency, and approved for publication. Approval does not
signify that the contents necessarily reflect the view and policies of the Agency nor does mention of trade names or commercial products constitute endorsement or recommendation
for use.
Copyright Е1985 by IMPI, Victoria, Canada: Manuscript
received September 24, 1985; in revised form December 3,
1985.
Dosimetry-Finite Difference Techniques
his paper describes a three-dimensional
T
(3-D) finite-difference (FD) technique for
calculating the specific absorption rate (SAR)
distribution in a human model exposed to radiofrequency (RF) radiation. The numerical
method is based on a FD solution of Maxwell's
curl equations which were recast into a
scattered-field formulation [1]. The method is
very general and can be used to predict instantaneous, as well as continuous wave (CW),
SARs for complex, inhomogeneous models located over realistic surfaces (grounds). In addition, it can be used to investigate the interaction with many different electromagnetic
(EM) source distributions, including both
plane-\yave (far-field) and near-field exposure
conditions. It is possible to obtain spectral information about the body's SAR over a wide
frequency range.
The FD approach possesses an important
inherent advantage over the more extensively
utilized method-of-moments (MOM); namely,
it requires much less computer memory. The
MOM [2, 3] is based on formulating Maxwell's
equations into an integral equation(s). This
integral equation(s) is then solved by approximation in which it is replaced by a linear system of equations, and the entire set of equations must be stored in the computer memory.
This requires large amounts of memory because an object divided into N cells or blocks
must be represented on the computer by an
array of 9N2 elements (the factor of 9 is a result of the vector nature of the internal fields).
In addition, the solution of the large system of
equations is slow since the number of computer operations is proportional to the square
of the number of elements in the matrix. With
large mainframe computers, this means the
MOM can handle models comprised of only a
few hundred blocks (if symmetry conditions
are applicable, then models comprised of approximately 1100 cells are possible [4]), whereas the FD method can analyze models that are
divided into many thousands of blocks. Con-
J.
MICROWAVE POWER 1985
241
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sequently, FD techniques allow more realistic
models to be constructed, which can be analyzed over higher frequency ranges.
One disadvantage of the FD method, however, is that its implementation necessitates
modeling a certain region of the space surrounding the object, as well as the object itself.
Thus, the computer input demands may be
more troublesome than the MOM, where only
the object need be considered.
A previous study [1] has investigated the
accuracy of this FD technique as it applies to
elementary shapes such as dielectric spheres.
In this paper, calculations are provided to determine the capabilities and limitations of the
technique as it applies to the human dosimetry
problem. The human model consists of several
thousand rectangular blocks that are arranged
to fit the body shape, and the SAR is calculated for each block. Results obtained for a
homogeneous model comprised of average tissue characteristics are compared with measurements at 350 MHz. Additionally, results
are presented for an inhomogeneous model
in which the chest (lungs, ribs, spine, and
sternum) are modeled in greater detail using
20,736 cells with dimensions of 1x1x1 cm.
FINITE-DIFFERENCE METHOD
This numerical method is based on a FD solution of Maxwell's curl equations in the timedomain. The EM fields are partitioned into incident and scattered fields, and the source of
EM energy is turned on at t = O. Hence, it is
called a time-domain, scattered-field formulation. In as much as the technique can be
mathematically involved, only a rather brief
discourse is presented here; more detailed information can be found elsewhere [5].
PartitiE!lil$ the total fields @, ij) into incident (E I , H) and scattered (E S , H S ) fields,
allows Maxwell's curl equations to be written
as [1]
aAs
=
at
-f.Lo -
oE
S
~
e - ' + aE s
at
--+
V
x E
=
where E = E i + E S;
242
J.
S
(1)
~~.
aE
Vx H
S
I
-
- (e - eo)
aE i
at
(2)
A = Ai + As; e and a are
MICROWAVE POWER 1985
the dielectric constant and conductivity of the
object, respectively; and eo = 8.85 X 10- 12 F/m.
Note that the total field is a sum of the incident
and scattered fields. Also observe that the
terms aE i and (e - eo) aEi/at in Equation (2)
represent the source or driving terms and they
are distributed over the object's (scatterer's)
volume.
The propagation, scattering, and absorption
of the fields emitted by an EM source are simulated on a grid of cells by solving Equations
(1) and (2) in which spatial and time derivatives are differenced linearly and exponentially [1], respectively. The positions of
the vector components are positioned about a
cell according to the so-called Yee grid [6]. Using this approach, and expressing Equations
(1) and (2) in scalar form, yields six equations
which are solved simultaneously for the six
field components (E~, E;, E~, H~, H;, and
H~) in which the spatial evaluation points for
E and As occur alternately along each axis of
the cell. The time evaluation also occurs at
alternate half-time steps. For example, with At
the cycle time and n the it~ation number, H
is evaluated at ndt, and ES is computed at
(n -~) dt. With this scheme, the new value
of any component of the field at each grid
point only depends on its previous value (in
time) and the previous values of the adjacent
components of the other field. Consequently,
a solution is obtained by simultaneously solving the six scalar equations by time-stepping
through the entire grid.
The space in which the field must be computed is, in theory, unbounded. In reality,
however, this is impossible because a computer can store and compute only a finite amount
of data. To circumvent this problem, it is
necessary to surround the object by a volume
large enough to contain the object and impose
some conditions which must be satisfied at
the boundary walls. The idea is to create the
numerical illusion of an infinite space. One
approach is to impose a radiation boundary
condition [5] on the exterior surfaces of the
boundary; that is, on the boundary require
the scattered field to behave as an outpropagating spherical wave of the form
E = Б(6, <1╗ g(t - r/c)/r
(3)
where r is the measure of distance and c is the
S
S
S
Dosimetry-Finite Difference Techniques
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speed of light. Because the field radiated (scattered) by the object must behave as a far-zone
field for this condition to be valid, the boundary volume must be significantly larger than
the object to avoid excessive nonphysical reflections off the outer boundary. It should be
pointed out that the ability to impose a simple
radiation boundary condition in this form is an
inherent advantage of the scattered-field formulation. Other FD versions [7,8] based on a
total-field formulation require a much more
contrived approach to reduce reflections off
the outer boundary.
Presently, the memory limitations of midsize computers restrict problem spaces (object
plus surrounding volume) to typically around
25,000 or 30,000 cells, with an upper limit
of 100,000 cells if more sophisticated programming is applied. Even with these upper
limits, spatial resolution available for modeling the interior detail of a biological body is
restricted. This restraint can be circumvented
by the application of a so-called expansion
technique [9]. Basically, the expansion approach uses an initial computer run with a
fairly coarse division of the problem space
with the computed data being stored on disk.
A certain portion of the body is then subdivided into a much finer division of cells and
is called a subvolume. This becomes the problem space for a second computer run. Now
only this subvolume is treated by imposing the
same incident field conditions as the first run.
In addition, interpolated tangential electric
fields generated by the first run are imposed
on the outer boundary of the subvolume.
These tangential fields numerically mimic the
response of the rest of the body, thereby ensuring that the subvolume response behaves
as if the remaining portions of the body are
still present. Obviously this process could be
carried out several times. For example, the
whole body of a human could be considered
for the first run in which the head is coarsely
modeled. The second run would replace the
crude approximation of the head with a much
finer rendition of the head including detail
such as the nose, eyes, brain, and skull.
A third run might even be employed that
focused on the intricate detail for the eyes.
Although this FD technique yields a transient solution to Maxwell's equations, it is also
Dosimetry-Finite Difference Techniques
possible to obtain CW steady-state solutions.
This can be accomplished by merely letting the
time-domain solution run for a period of time
long enough to achieve steady-state conditions for a CW source turned on at t = O.
On the other hand, since a pulsed waveform
contains a spectrum of frequencies, it is possible to obtain spectral information about the
body's SAR over a large frequency range with
only one execution of the computer program.
This is achieved by the application of the
Fourier transform in which the time-domain
response is transformed to the frequency
domain according to
00
~
J
f (t)eio>tdt.
(4)
21T
Because Equation (4) cannot, in general, be
analyzed analytically, it is necessary to numerically evaluate the integral by employing standard fast Fourier transform (FFT) routines.
These procedures, however, are well documented, and FFT subroutines are available for
most large computer systems.
To determine the body's true CW response
from transformed data requires careful consideration. From linear system theory, it is well
known that the output of a system o(t) is related to the input i(t) by the system impulse
response h(t) via
F (w)
=
-00
00
o(t)
=
J i(t -
T) h(T)dT
(5)
with the Fourier transform given by
O(w) = I(w) "(w).
(6)
If I(w) represents the incident field and O(w)
the observed response at a point in the body,
then it is clear from Equation (6) that "(w) is
the desired CW response because it contains
no spectral components of the source. In
theory, "(w) could be determined by simply
using a unit impulse for the incident field.
However, this type of waveform poses severe
problems when evaluating the FFT of Equation (4) since "(w) does not, in general, go to
zero as w approaches infinity. Consequently,
the numerical solution would probably not
converge. This problem can be alleviated by
using an incident field that contains a finite
spectral content, such as a damped sine waveform. Then I(w) will approach zero with in-
J.
MICROWAVE POWER 1985
243
creasing w, and the FFT technique will yield
accurate information. Of course, Equation (6)
must be solved for H(w) after the FFT has been
applied.
An upper frequency limitation of FO procedures can be based on the following physical
argument. It is assumed that the waveform is
spatially sampled no more coarsely than 'A/4 in
a time step. This means, that at the highest
frequency, the signal from one cell arrives at
an adjoining cell one time step later shifted by
no more than 'A/(4c) when the maximum dimension of the cell is traversed. Thus, the upper frequency, fmax , is given by
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c
f max = - 'A min
c
4 x maximum cell dimension'
(7)
Note that because both c and 'A are reduced by
the dielectric constant of the object, these effects cancel out in Equation (7) and the net
result is both c and 'A can be considered as
free-space quantities.
NUMERICAL RESULTS
A: Homogeneous Human Model
To determine the capabilities and limitations
of this FO technique as it applies to the human
dosimetry problem, a coarse homogeneous
model comprised of rectangular cells with
dimensions of 4 x 4 x 6 em and average tissue
properties (2/3 muscle tissue) was constructed. Figure 1 illustrates the model (both
front and side views), where the (i, j, k) indices define the location of each spatial grid
point (x, y, z). Note that the figure contains a
volume which surrounds the object; the reasons for this volume were previously discussed (see previous Section). The entire problem space contains 20,736 cells and the body is
comprised of 1172 cells. The cells within the
broken line are of constant size (4x4x6 em).
Those outside this boundary are allowed to
increase in size by a factor of 1.2 and 1.3 in the
y, z and x directions, respectively, for each
cell away from the dotted line. By assigning
appropriate permittivity values to the ground
beneath the model's feet, it is possible to
evaluate the effects of objects standing on the
earth's surface. A perfectly conducting surface
244
J.
MICROWAVE POWER 1985
can also be utilized, as well as completely eliminating it for free space calculations.
Figure 1 also contains views of the subvolume in which the right side of the chest
region is modeled using 20,736 cubical cells
with dimensions of 1x 1x 1 em, and defined by
the indices (i', j', k'). A side view of this subvolume is further delineated by Figure 2. The
coarse (1172 cell) model is used to compute the
scattered tangential electric fields on the subboundary. The incident electric field utilized
for the first calculation, as well as the tangential electric fields obtained from the first
calculation, are then imposed on this subvolume for a second calculation. The finer grid
of the second calculation, and consequently
finer time steps, imply that the values for both
the spatial and time points for the tangential
fields on the subboundary must be determined by interpolation.
The incident electric field is a damped sine
wave of the form
E~
= Eosin(27l'at')e-at'U(t')
(8)
where t' = retarded time = t - (y - y')/c, y'
= source point, y = observation point, c =
speed of light, U(t') = unit step function, Eo =
1 V1m, a = oscillation frequency. Observe that
in Equation (8), the field produced by the
source at point y at time t is due to an earlier
disturbance which originated at t' and point
y'. The incident field is a plane-wave which is
propagating in the y direction with its vector
orientation in the x direction. This corresponds to the electric-field vector being parallel to the major length of the body and propagating from the front to the back of the object.
This field is turned on at t = 0, is oscillating
at a frequency designated by the quantity a.
Figure 3 shows the Fourier transform of the
waveform, along with an insert which contains a plot of Equation (8), when a = 500
MHz. Note the wide frequency content of the
waveform.
The instantaneous SARin the body is calculated for each cell according to
SAR(t) = ~ I Ei(t) + ES(t) 12
(9)
P
where p is the density of the tissue associated
with the cell. Frequency-domain information
is determined by applying the FFT to each vec-
Dosimetry-Finite Difference Techniques
tor component of Ei and E separately. Then
the SAR(w) is formed from squaring the magnitude of the resultant total field in conjunction with Equation (6).
Figure 4 shows the calculated instantaneous
SARs at a central point in the chest (i = 12,
j = 12, k = 15) for both the coarse (run 1) and
the refined (run 2) homogeneous (E. = 38 and
(J = 0.95 5/m) models located in free space.
The'damped sine wave incident field is oscillating at 100 MHz. To satisfy the Courant
stability criterion [8], the time increment, dt,
was 0.085 and 0.017 ns for run 1 and run 2,
respectively. There are several important
points which should be gleaned from the figure. First, observe that for times prior to approximately 4 ns, the run 2 waveform is essentially equal to zero. On the other hand, the run
1 curve shows field buildup in this time interval. This clearly illustrates that the results of
run 2 are superior because the field should be
identically equal to zero. This is a result of the
fact that a finite amount of time is required for
the incident wave to propagate from the outer
boundary, where it is launched at t = 0, to the
observation point. Obviously, no field should
exist at a point in the body until the incident
field reaches that point. After 4 ns, they are
closer in shape, but the peak values of the
curves are different and there is a slight time
shift of one curve relative to the other. Note
also that the internal field oscillates only for a
few cycles and then is rapidly damped out by
the lossy tissue.
The reasons for the larger errors for the run
1 case are more apparent after a consideration
of the data contained in Figure 5, in which
the individual waveforms (E~, E~, and Ex) are
plotted. The approximate 2 ns lag time, before
the incident and scattered fields begin to
buildup, represents the time it takes for the
incident field to propagate from the outer
boundary to the observation point (i = 12,
j = 12, k = IS), when the medium is freespace. It is evident that the total field, Ex, is
formed from a superposition of E~ and E~
which involves taking the difference of two
very nearly equal waveforms. This is further
evidenced by the fact that the peak values for
Ex are approximately one order of magnitude
smaller than those for either E~ or E~. Thus, E~
must be computed with extreme accuracy in
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S
Dosimetry-Finite Difference Techniques
order that Ex be reasonably accurate. Obviously, for t < 4 ns the total field was not computed accurately enough because the magnitude of the error is, in fact, almost equal to
the signal which occurs after t = 4 ns.
The above errors increase for both run 1 and
run 2, assuming constant cell sizes for both
cases, as the frequency of the damped sine
wave increases. Therefore, in light of these
difficulties, the data presented hereafter have
been "time filtered". That is, for times when
the waveform should be zero, as based on
propagation times, the total f~ld isjorced
to be identically zero by setting E = - E'. This
procedure aids in reducing cumulative error
buildup as time advances, because initial
values are forced to be correct.
Figure 6 shows normalized SARs (W/kg per
mW/cm 2 ) versus frequency when an FFT routine is applied to the time-domain responses
(run 1 and run 2). The damped sine wave incident field is oscillating at 500 MHz. To apply
the FFT routine, the time-domain waveform
for run 1 was digitized at 0.085 ns intervals by
sampling the signal 4096(2 12 ) times over a time
interval of 348 ns. The run 2 waveform was
sampled 4096 times at 0.017 ns intervals for a
total duration period 69.6 ns. For both waveforms, after the first 20.4 ns, zero values were
assigned to the sampled waveform because
the damped sine wave responses exhibit very
small values for times greater than 20.4 ns.
The FD calculation of the waveforms for all
4096 points would be prohibitive in terms of
computer time. The frequency responses were
plotted only up to the limits dictated by Equation (7). However, the frequency points
(0.5/sampling interval) at which aliasing occur
are well above those limits. The lowerfrequency limits are determined by the duration of the sampled waveform; these values
are 1/(348 ns) = 2.87 MHz and 1/(69.6 ns) =
14.4 MHz for run 1 and run 2, respectively. As
observed from Figure 6 there is reasonable
agreement in the two curves; the large dip in
the run 1 curve at 102 MHz is most likely a
numerical artifact. The rippling in spectral
curves is a result of truncation of the waveforms in the time domain. Finally it should
be pointed out that the flattening of the curves
for frequencies below approximately 60 or
70 MHz is a result of a constant value (E. = 38)
S
J. MICROWAVE POWER 1985
245
K
1 25
REGION WHERE LOSSY DIELECTRIC
CHARACTERIZED BY u AND [
,30X16X16 CELLS
r--
I
I
I
5
I
I
10
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-,
-
I
I
OUTER BOUNDARY
I
36x24x24 CELLS
1
-t
~10
I
-t
~20
~
IREGION TO
BE EXPANDED
I
36x24x24 CELLS
PROBLEM SPACE
r"""'l_.......... -I-'~',;....--....
I
SUB-VOLUME
~
25
~~
L. -4 -J _
20
10
L.
"
j::
1
K'
25
30
Figure 1, Human model in a
20,736 cell problem space
with the subvolume indicated. The location of the
mesh points are given by the
indices (i, j, k) and (i', j',
k') for the problem space and
subvolume, respectively.
(a) Front view
35
246
J.
MICROWAVE POWER 1985
Dosimetry-Finite Difference Techniques
J
1
5
10
15
20
25
1r--r--r-,~--r-,-r-r--r-,-r-r-r-,-r-r--r-'-I'"""T-r-~-r-..,
f
REGION WHERE LOSSY DIELECTRIC
CHARACTERIZED BY u AND
30x'6x"
E
CEllS
---,
r- -
I
5
I
I
r-r
20
~
30
~
1
10
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10
I'
15
37
I
I
I
~
PROBLEM SPACE
OUTER BOUNDARY
36x24x24 CELLS
j~ I
SUB-VOLUME
1
...l..J.~
10
20 25
J'
I
REGION TO BE
EXPANDED
I
36x24x24 CELLS
I
20
I
I
-
I
r-
25
r-
I
I
I
I
I
30
I
I
L_
J
(b) Side view,
Dosimetry-Finite Difference Techniques
J,
MICROWAVE POWER 1985
247
.,,
volume as delineated in Figure 2. Because the
dielectric discontinuity (air/tissue interface) at
the corners and edges of the surface cells
strongly affects the numerical results, calculated results are presented only for interior
grid points at least one cell length from the
surface. Additionally, those interior grid
points on the sub-volume border were not
considered because their values are found by
interpolation of run 1 values. Figure 7 contains
the results of the comparison for a scan taken
along the center of the chest, from front-toback. Very good agreement is obtained. Note
the fairly rapid drop in SAR values as a function of penetration depth. Figure 8 shows plots
for scans across the chest at different depths
from the front surface; reasonable agreement
is obtained. As might be expected, the comparison is better for the scans nearer the center
of the chest. As mentioned earlier, the larger
errors near the surface are a result of the corners and edges associated with the numerical
model.
r---------- j
1
1
I
I
I
I
I
RUN 1
BOUNDARY~
10
1
L __
15
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I'
20
25
30
35
37,
1
I
I
I
10
15
J'
1
20 :
25
1
1
1
I
I
I
I
1
I
1
I
Figure 2. A side view enlargement of the
subvolume showing the refinement of the
exterior surface of the chest.
utilized for the tissue relative dielectric constant. In reality, for frequencies below approximately 100 MHz, the average tissue dielectric constant increases as the frequency decreases [10].
To indicate the accuracy of this FD method,
the calculated (run 2 case) SAR distributions in
the upper torso (chest) are compared with
measured local values [11] at 350 MHz. No
attempt was made to configure the entire numerical model of the human to be identical
with the experimental phantom model. However, the geometry of the chest region was
modeled reasonably accurately by the sub-
248
J.
MICROWAVE POWER 1985
B: Inhomogeneous Human Model
To indicate further this technique's capabilities, the internal organs and bones contained
within the subvolume were modeled with as
much resolution as could be obtained from
20,736 cells with dimensions of 1 x 1 x 1 em.
Figure 9 illustrates views of this subvolume
showing how the lung, ribs, sternum, and
spine were modeled. Dielectric properties for
the various materials are contained in Table 1.
As previously discussed, the coarse model
(Figure 1) was used to compute the scattered
tangential electric fields on the subvolume
border. These tangential fields are then imposed on the subvolume as boundary conditions for run 2 calculations. Again, the
incident field is a damped sine wave oscillating at 500 MHz.
Figures 10 and 11 contain results at 350 MHz,
which are also compared with data generated
by the homogeneous chest model. From figure 10 it is noted that for the front-to-back scan
in the center of the chest, the inhomogeneous
model yields lower SAR values, especially
near the front surface. This reduction near the
front surface is most likely due to bone material (sternum) being present in the inhomogeneous model. However, the side-to-side
Dosimetry-Finite Difference Techniques
14
Figure 3. Fourier transform (FFT) of a damped
sine wave oscillating at
500 MHz.
1 .?
13
~
'2
E
;.r
"
10
8
9
-0.6
и1.0
::
~
ии?
:r
7
~
~
6
w
5
..?
>
0
10 1
10 2
10 3
10 4
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FREQUENCY. MHz
1.2
1.0
Figure 4. Time-domain
SAR responses at a central point in the chest.
Observation points for
run 1 and run 2 are (12,
13, 15) and (19, 13, 13)
respectively. The incident
field is a normalized (Eo
= 1 Vim) damped sine
wave oscillating at 100
MHz.
...
"..
~
;:
0.8
0.6
ri
Ф
lI>
0.4
0.2
0.0
4
0
6
8
10
12
14
16
18
20
22
TIME. ns
-3 '--_----'-_ _..L..._----L_ _..L-_---'_ _....L_---lI...-_-'-_ _.L-_----'-_ _..L..._--'---l_3
o
'0
12
TIME. ns
Dosimetry-Finite Difference Techniques
'4
16
'6
20
22
24
Figure 5. Run 1 electric
field waveforms for the
incident (E~), scattered
(E~), and total (ExY fields
at location (12, 13, 15)
versus time. The incident
field is a normalized
(Eo = 1V1m) damped
sine wave oscillating at
100 MHz.
J. MICROWAVE POWER 1985
249
10-
1
r--------~--------,_-------...,
\
~
\
\
\
RUN 2
\/ '\
:
\
\
\ I
\
N
u
"-
~
E
:D
D.
..."-
Downloaded by [UNSW Library] at 10:39 26 October 2017
Cl
1'1 \1 \ 1\
,-.. \I \ 1\
I
\ I \
I \
V \
\
,
E
~
I \ 1\
I \ I \
I \I
\I
\1
10-3
Figure 6. Frequency-domain
SAR in the chest normalized
with respect to incident
power density. The observation points are (12, 13,
15) for run 1 and (19, 13,
13) for run 2.
I
\
;-r'
~
,
,
\
\
\
I ,
RUN1
\ I
Ii
c(
I I
III
I ,
II
II
V
10- 5 '-:10 1
--I..::-10 2
..J-
......
10 4
10 3
.1
FREQUENCY, MHz
[jON'
I
N
"-
FRONT
3:
E
MEASUREMENT
~
"c.
..."-'"
BACK
(I' = 20. K' = 5)
E
u
.05
RUN 2
3:
rx'
Ф
til
Figure 7. Comparison of the calculated and measured SAR distribution in the center of the chest (20,
j', 5) as a function of penetration
depth. The frequency is 350 MHz
and the incident power density is
1 mW/cm 2 ?
250
J.
MICROWAVE POWER 1985
BACK
o L..L.-....L---L.--..L:s==2:::~:::I::=:I:::.J
o
.1
.2
.3
.4
.5
.6
.7
.8
.9
CHEST DEPTH FRACTION
Dosimetry-Finite Difference Techniques
0.15
Table 1 Relative dielectric constant and conductivity of
various tissues for 350 MHz
A' \
\
- - MEASURED
- - - - CALCULATED
\
Relative dielectric constant
\
Ae:r--
\
N
FRONT
\
\
\
\
0.10
E
B' _________ B
C' - - - - - - - - -
E
Q;
Q,
l
1
Muscle
Lung
Fat
Bone
:
I
I
a:
et
VI
0.05
",
\
-...--. A
_--OCY-~
0.00 '----'--_"----'---=-=:..:...J..._'------'--_'------'------'
o
.1
.2
38,0
35,0
4.6
8.0
Conductivity
(<T,8/m)
0,95
0,73
0,06
0,05
BACK
.~
'"
"~
.><
Downloaded by [UNSW Library] at 10:39 26 October 2017
c
_____1I
\
\
\
\
"~"
(e)
,3
,4
.5
,6
.7
,8
.9
CHEST WIDTH, fraction
Figure 8, Comparison of the calculated and
measured SAR distribution across the chest
at different distances from the front surface, The top, middle, and bottom scans
are at locations (20, 21, k'), (20, 19, k')
and (20, 16, k'), respectively, The frequency is 350 MHz and the incident power
density is 1 mW/cm2 ,
scan through the lung, as shown in Figure 11,
exhibits a considerable SAR increase in the
lung as compared to the homogeneous case.
This inhanced SAR in the lung may be of significant importance, especially for RF induced
hyperthermia in cancer treatment.
DISCUSSION AND CONCLUSIONS
The capabilities and limitations of a FD technique, as it applies to the human dosimetry
problem, have been studied. It was found that
reasonable agreement could be obtained for
calculated SARs in the chest, when the irradiation frequency is 350 MHz, with measured
local SARs acquired from a phantom model of
a human. The model tended to yield more ac-
Dosimetry-Finite Difference Techniques
curate results near the center. The larger errors
at the surface are a result of the edges and
corners of the cells producing unrealistically
high local field strengths. Of course, these
larger errors near the surface could be reduced
by simply using smaller cells that are carefully
arranged to minimize the effects of edges and
corners.
It was also found that a scattered field formulation might not be the best way to cope
with this type of problem. This is a result of
the total field being formed by taking the difference of two very--->nearly equal waveforms.
Thus, even though E can be computed accurately by the numerical scheme, E tends to be
considerably less accurate. A total field formulation of the problem, therefore, may be a
more attractive alternate approach. However,
a total field formulation will require a much
more intricate approach [12] to deal with the
artificial boundaries.
Some of the problems in the implementation of the model reported here are a
result of having to translate the parameters
associated with the model into an acceptable
format for the computer code. Currently, the
input data have to be manually entered into
the computer. The 3-D spatial quantification of
all the cells associated with the model is very
difficult when the data is generated by hand.
Graphics systems are available for automating
this process for more complicated models, but
substantial software development 'may be required to tailor a graphics system for this task.
Finally, some mention should be made relative to the computer memory and run time
requirements for this code. Currently, an IBM
3081 mainframe computer is used for the calculations, and approximately 90 minutes are
required to generate the time-domain data
for the run 2 case. The run 1 situation takes
S
J. MICROWAVE POWER 1985
251
!
:
"---- -1
\-----'
:
r--- -
,
,
'
I
1
2
1i'2':"-ro-r,░-r-,-ro_'r'-,-r-r-,-',0+'.-r-r"".-r-r.,'
I
MUSCLE
I
AIB1
I
I
~--
MUSCLE
__ I
RIB2
'0
MUSCLE
RIB3
,.
MUSCLE
RlB4
RIB5
MUSCLE
20
*,_J1~Ti--STEANUM
MUSCLE
FAT
RIB6
2'
AlB7
r-~;L--
RIB6
STERNUM
---t+-.,::::;~~------:",
Downloaded by [UNSW Library] at 10:39 26 October 2017
30
MUSCLE
AIBS
,
I FAT
3'
LUNG
I
,
,'
,,
,
MUSCLE I RIB 9
:,
,,
,
I
LUNG
MUSCLE
Museu;:
,
I
.......-'-tL.J.-'-.J.tL....l.-'-""'"
37 ~+,--'-L.J.+,--'-L.J.y...-,-w.
I
SPINE
,,
STERNUM
AIR
K'
25
15
10
25
....-FAT
...-MUSCLE
+ - RlBS5T09
...- RIB4
20
....- MUSCLE
....- RIB3
+-- MUSCLE
....- RIB2
. - MUSCLE
-+-- RIB 1
15
J'
MUSCLE
10
....- RIB1
5
~RIB2
. . - RIBS 3 TO 9
....- MUSCLE
+-- FAT
AIR
LUNG
SPINE
Figure 9. Subvolume inhomogeneous model
of the chest. (a) Frontal view (b) Side view
(c) Top view.
252
J.
MICROWAVE POWER 1985
Dosimetry-Finite Difference Techniques
0.06
s--'
FRONT
-
FRONT
HOMOGENEOUS
1._
- - - INHOMOGENEOUS
0'
0.05
err
0.04
E
u
"-
3:
Q.
Ol
~
I'
,I
0.03
oct
0.02
(1'=20, K'=5)
N
E
u
"-
3:
E
l;
0.06
Q.
Ol
FRONT\.."/
'""-
3:
.... -
I
0.04
I
a:oct
III
BACK
I
4
8
12
16
20
DEPTH, em
30 minutes. Since IT and E are nonuniform
throughout the body, it is necessary to store
eight quantities (including six fields) for each
cell. Thus, for single precision calculations it
takes approximately 320 K bytes to store an
array associated with 20,000 cells. Naturally
for double-precision calculations this value
REFERENCES
1 Holland, R, Simpson, L., and Kunz, K.S. (1980).
Finite-difference analysis of EMP coupling to lossy
dielectric structures. IEEE Trans. EMC-22, (3):
203-209.
2 Chen, K.M. and Guru, B.S. (1977). Internal EM
field and absorbed power density in human torsos
induced by 1-500-MHz EM waves. IEEE Trans. MTT25: 746-755.
3 Hagmann, M.J., Gandhi, O.P., and Durney, C.H.
(1979) Numerical calculation of electromagnetjc
energy deposition for a realistic model of man. IEEE
Trans. MTI-27, (9): 804-809.
4 DeFord, J.F., Gandhi, O.P., and Hagmann, M.J.
(1983). Moment-method solutions and SAR calculations for inhomogeneous models of man with large
Dosimetry-Finite Difference Techniques
I
\
I
\
I
,,
\
I
I
\
I
I
I
I
\
I
I
\
\
\
\
, _.......
I
I
I
0
,/'\
\
0.00
0
Figure 10. Comparison of the homogeneous
and inhomogeneous SAR distribution in
the center of the chest (20, j', 5) as a function of penetration depth. The frequency is
350 MHz and the incident power density
is 1 mW/cm2 ?
I
I
I
I
II "
24
-I
,
'\/
0.02
0.00
0
HOMOGENEOUS
LUNG
/
I ,
I \
I
I
I
I
0.08
I \
I \
0.01
Downloaded by [UNSW Library] at 10:39 26 October 2017
II r,
0.10
,,,\,,
Ii
-
- - - INHOMOGENEOUS
I
'""-3:
III
BACK
: BACK
E
l;
(1'=20, J'=161
_____ ....J
I
N
i
--------- 0
4
8
12
16
20
24
SIDE-TO-SIDE SCAN, em
Figure 11. Comparison of the homogeneous
and inhomogeneous SAR distribution for a
scan across the chest (20, 16, k') through
the center of the lung, The frequency is
350 MHz and the incident power density
is 1 mW/cm2 ?
would be multiplied by a factor of two. Because the IBM 3081 computer has 14 M bytes
of fast (electronic) memory available, models
with well over 200,000 cells are possible, albeit
run times will increase substantially.
number of cells. IEEE Trans. MTT-31, (10): 848-851.
5 Holland, R (1976). THREE, a free-field EMP coupling
and scattering code. Mission Research Corporation,
Albuquerque, NM, AMRC-R-85.
6 Yee, K.S. (1966). Numerical solution of initial boundary value problems involving Maxwell's equations in
isotropic media. IEEE Trans AP-14, (3): 302-307.
7 Taylor, C.D., Lam, D.H., and Shumpert, T.H. (1969).
Electromagnetic pulse scattering in time-varying
inhomogeneous media. IEEE Trans. AP-17, (5):
585-589.
8 Taflove, A. and Brodwin, M.E. (1976). Numerical
solution of steady-state electromagnetic scattering
problems using the time-dependent Maxwell's equations. IEEE Trans. MTT-23, (8): 623-630.
9 Kunz, K.S. and Simpson, L. (1981). A technique
J. MICROWAVE POWER 1985
253
11 Kraszewski, A., Stuchly, M.A., Stuchly, 5.5., Hartsgrove, G., and Adamski, D. (1984). Specific absorption rate distribution in a full-scale model of man at
350 MHz. IEEE Trans. MTT-32, (8): 779-783.
12 Mur, G., (1981). Absorbing boundary conditions for
the finite-difference approximation of the timedomain electromagnetic-field equations. IEEE Trans.
EMC-23, (4): 377-382.
Downloaded by [UNSW Library] at 10:39 26 October 2017
for increasing the resolution of finite-difference solutions of Maxwell equations. IEEE Trans. EMC-23, (4):
419-422.
10 Durney, e.H., Johnson, e.e., Barber, P.W.,
Massoudi, H., Iskander, M.F., Lords, J.L., Ryser,
D.K., Allen, S.J., and Mitchell, J.e. (1978). Radiofrequency Radiation Dosimetry Handbook, second ed.,
Report SAM-TR-78-22, USAF School of Aerospace
Medicine, Brooks AFB, TX.
254
J. MICROWAVE POWER 1985
Dosimetry-Finite Difference Techniques
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