United States
Environmental Protection
Agency
Health Effects Research
Laboratory
Research Triangle Park NC 27711
Research and Development
EPA/600/S1-85/016 Sept. 1985
Project Summary
Dosimetry for In-Vitro Chick
Brain Calcium-Ion Efflux
Experiments by Numerical
Methods Based on Zonal
Harmonic Expansions
Guillermo Gonzalez and James C. Nearing
The full report discusses the calcula-
tion of the electric field and power den-
sity distribution in chick brain tissue
inside a test tube, using an off-center
spherical model. The off-center spheri-
cal model overcomes many of the limita-
tions of the concentric spherical model
and permits a more realistic modeling of
the brain tissue as it sits in the bottom of
the test tube surrounded by buffer solu-
tion. The effect of the unequal amount
of buffer solution above the upper and
below the lower surface of the brain is
evaluated.
The field distribution is obtained in
terms of a rapidly converging series of
zonal harmonics. A method that per-
mits the expansion of spherical harmon-
ics about an off-center origin in terms of
spherical harmonics at the origin is
developed to calculate in closed form
the electric field distribution.
Numerical results are presented for
the adsorbed power density distribution
and scaling ratios at carrier frequencies
of 50 MHz. 100 MHz, and 147 MHz. It
is shown that the off-center spherical
model yields scaling ratios in the brain
tissue that lie between the extreme
values predicted by the concentric and
isolated spherical models.
This Project Summary was developed
by EPA's Health Effects Research Lab-
oratory, Research Triangle Park, NC, to
announce key findings of the research
project that is fully documented in a
separate report of the same title (see
Project Report ordering information at
back).
Introduction
A problem of considerable importance
to the understanding of biological phe-
nomena produced by electromagnetic
radiation is the mechanism of interaction
of amplitude-modulated radio-frequency
[RF] waves on calcium-ion efflux from in
vitro chick brain. Experiments have shown
that when amplitude is modulated with
extremely low frequency [ELF] sinusoidal
waves, RF radiation can enhance the
calcium-ion efflux from in vitro chick
brain tissue.
In order to better understand the effects
of amplitude-modulated RF radiation on
calcium-ion efflux, the electromagnetic
fields and absorbed power density distri-
bution in the chick brain must be accu-
rately determined. Concentric spherical
models have been used to calculate the
field and power density distribution in the
chick brain. These models have provided
much valuable information about the
dosimetric aspects of the calcium-ion
efflux experiments. The models have also
been used to develop scaling ratios to
determine the required incident power
density that produces calcium-ion efflux
at other carrier frequencies.
This study overcomes the limitations of
the concentric spherical models. Concen-
-------
trie spherical models do not adequately
represent the chick brain half immersed
in the buffer medium, because the brain
as it sits in the bottom of the test tube is
surrounded by a layer of buffer solution
with a thickness that differs above and
below the brain. Concentric spherical
models use a uniform layer of buffer solu-
tion to surround the brain. To determine
the dosimetric effects of the non-uniform
layer, an off-center spherical model is
used to approximate the physical geom-
etry of the brain/buffer combination. The
off-center spherical model represents a
significant refinement to the concentric
spherical model, since the brain can be
modeled as a continuous, smooth, spher-
ical surface surrounded by unequal
amounts of buffer solution on its upper
and lower surfaces. The electric field and
absorbed power density distributions, and
scaling ratios for carrier waves of 50,
100, and 147 MHz are calculated using a
quasi-static approximation. This approx-
imation leads to a field representation in
terms of a convergent series involving
spherical harmonics.
Procedure
The schematic of the experimental set-
up is shown in Figure 1. The chick brain
tissue sits at the bottom of the test tube
and is surrounded by a non-uniform layer
of buffer solution.
The analytical model for the off-center
spherical model of the brain inside a test
tube is shown in Figure 2, along with the
direction of the incident electric field. This
model represents the geometry in Figure
1.1 n fact, except for the flat-topped buffer
region, the model is reasonably accurate.
The radii a and b, and the off center dis-
tance d, are selected so that the brain
volume occupies 0.3 ml, and the buffer
volume, 1 ml. Also, the limiting case as h
approaches zero is analyzed.
The mathematical formulation that fol-
lows is by no means a simple extension of
the concentric spherical case. To the best
of the authors' knowledge, exact solu-
tions to off-center spherical geometries
of the type shown in Figure 2 in terms of
zonal harmonic expansions have not
appeared in the literature.
The mathematical formulation of the
problem follows. The electric field, as-
suming quasi-static conditions, can be
expressed as
E = -V0
where is the electrostatic potential. The
quasi-static approximation is valid be-
cause of the dimensions, frequencies,
and electrical properties of the materials
involved. In our calculations, the relative
complex permittivity of the brain tissue at
the typical carrier frequencies used in
-1.42 cm -
Buffer
Solution
0.24 cm
calcium-ion studies are 100-J175 at 50
MHz, 72-J94.9 at 100 MHz, and 66-J69.2
at 147 MHz; for the buffer, the values are
71-J580 at 50 MHz, 71-J290 at 100 MHz,
and71-j200at147MHz.
The externally applied field, which
makes an angle of 90° with the z-axis
(i.e., JE = E0Uy), produces a potential
which can be expressed in spherical
coordinates in the form
V.It = -
Test Tube
or
where
V.x, = -
Brain
Hemisphere
Figure 1. Schematic of the chick brain
inside a test tube containing a
buffer solution.
Y5m (ft 0) = cos m0 P'S (cos 0)
Pi! (cosfl) are associated Legendre poly-
nomials.
The potential in regions I, II, and III sat-
isfies Poisson's equation, subject to the
appropriate boundary conditions. The
charge distributions at r. = a and n> = b
produce potentials V« and Vb of the form
Figure 2. Off center spherical model for the chick brain.
-------
V. =
r° rn
Va,,n =\ \
()nYnm(0a, 0) ( ra <
(D
Va out =
> 3
. and
Vb =
b ,n
Vbou, =
)" Y£m ( ft,, 0 ) ( rb < b
rb
V, = V., in + Vb, out + V.,,
V,, = V., out + Vb, out + V.,»
Vln = V., ln + Vb, in + V.,«
/a, in
•r r
Ln=Qt.m=(
I
and
b, ou, =
n
s=m
oo
n
s=m
r
»-s=
(2)
where anm and ftnm are constants to be
determined from the boundary conditions.
In terms of (1) and (2), the potentials in the
three regions are given by
(3)
(4)
(5)
The form selected for the potentials in
(1)and(2) satisfy the continuity conditions
at r, = a and rb = b. Therefore, the
boundary conditions that remain to be
satisfied at the interface are the continuity
of the normal component of the current
density at r, = a and rt> = b.
In order to apply the boundary condi-
tions at the surface of the off-center
sphere (i.e., at rb = b), we need to expand
the solution V., m in terms of the coord-
inates rb, 0b and 0, and the solution Vb, out
in terms of the coordinates ra, 0a, and 0.
The necessary expansions are
(6)
, 0)
(7)
where
and
(n-s) I (s+m) I
d'"n(s-m)l
(n-m) I (s-n) I
Applying the boundary conditions to
(3), (4), and(5), and using (6) and(7), a set
of linear equations for trie n=j (j=0,1,...)
and m=k (k=0,1,..,n) coefficients (i.e., ctjk
and /3|k) are obtained.
Fixing the maximum n to be Nmax, the
set of linear equations is solved for a(k and
/3|k. The results are then inserted into (1)
to (5) to calculate the potential, electric
field, and absorbed power density distri-
bution in the three regions.
Results and Discussion
The absorbed power density distribu-
tions, assuming E0=1 V/m, were calcu-
lated for several cases; the most impor-
tant was for a brain volume of 0.3 cm3,
buffer volume of 1 cm3, and h=0.05 cm.
Therefore, a=0.677 cm, b=0.415 cm,
d=0.212 cm, and t=0.474 cm. Also, the
limiting cases as h and d approach zero
are considered. For h — 0 (i.e., when the
spheres touch at the bottom) the dimen-
sions are a=0.677 cm, b=0.415 cm,
d=0.262 cm, and t=0.524 cm.
Figures 3 and 4 graphically illustrate
the absorbed power density distribution
produced by an incident carrier wave,
with E0=1 V/m, at 147 MHz for the cases
where h=0.05 cm andd — 0, respectively.
Inthe figures, each line represents a con-
stant absorbed power density level. The
Figure 3. Absorbed power density distri-
bution at 147 MHz. The dimen-
sions are 8=0.677 cm, b=0.415
cm. d=O.212 cm. h=O.OS cm and
t=0.474 cm. The number scale
(0 to 9) is linear, where 9 repre-
sents an absorbed power density
of 0.5 mW/m3 for an incident
field of Eo=7 V/m.
Figure 4. Absorbed power density distri-
bution at 147 MHz for a cen-
tered spherical model (d=O). The
dimensions are a=0.677 cm,
b=O.415 cm. andt=h=0.262 cm.
The number scale (0 to 9) is
linear, where 9 represents an
absorbed power density of 0.5
mW/m3 for an incident field of
Ea=1 V/m.
number scale is linear, ranging from 0 to
9, where 9 represents an absorbed power
density of 0.5 mW/m3. For example, the
value of 2 represents an absorbed power
density of (2/9)0.5=0.111 mW/m3. In
-------
Figure 4, the absorbed power density dis-
tribution in the brain region is constant at
a value of 0.109 mW/m3.
Absorbed power density patterns sim-
ilar to those shown in Figures 3 and 4 are
produced at 50 and 100 MHz, except that
the absorbed power levels are smaller.
It is significant to observe from Figure 3
that the absorbed power density distribu-
tion in the brain region is not uniform; the
bottom of the chick brain has a higher
absorbed power density than the top.
Also, in Figures 3 and 4, the surface fields
around the chick brain exhibit a signifi-
cant variation from the bottom to the top.
Scaling relations were calculated for
the off-center spherical model. Figure 5
gives the values of the scaling factor, P,
(50)/Pi(147), for points along the z-axis
inside the spherical brain between the
top (i.e., z=0.415 cm) and bottom (i.e.,
z= -0.415 cm). Figure 6 gives the values of
the scaling factors for points on the out-
side surface of the brain between the top
(i.e., £=0) and bottom (i.e., 8=ir) from 0 <
0< it. The scaling ratios at a given fre-
quency are continuous at the top and bot-
tom of the brain-buffer interface. Thisfol-
lows from the fact that the electric field is
tangential and continuous at the top and
bottom brain-buffer interface.
Conclusions and
R ecommendations
The off-center spherical model was
used to calculate the electric field and
absorbed power density distribution in
chick brain tissue inside a test tube. The
calculations were made with typical di-
electric data used in previous analytical
studies for carrier frequencies of 50 MHz,
100 MHz, and 147 MHz. In the experi-
mentally realistic case where the chick
brain is surrounded by lesser amounts of
buffer solution at the bottom, the pattern
of absorbed power densities showed a
marked concentration of absorbed power
near the bottom. The absorbed power
density within the brain showed only
gradual variations with position. Outside,
at the surface of the brain, the absorbed
power density distribution varied by fac-
tors of 6 to 10.
The scaling ratios were calculated as a
function of position for points inside, and
on the outside surface of the brain. The
off-center spherical model revealed that
the scaling ratio inside the brain tissue is
not constant; the maximum change (h=0
case) from the top of the brain to the bot-
tom was approximately 11% for P45Q)/
PrflOO), and 19% for Pi(50)/Pi(147). This
contrasts with the concentric spherical
7,-
£ 6
d=O
h=.OS cm
h=O
0.415 0.249 0.083 0 -0.083
z-axis (cm)
-0.249
-0.415
Figure 5. Scaling ratio values for P> (50)/P\ (147) along the z-axis of the brain.
0 18 36 54 72 90 108 126 144 162 180
Theta (degrees)
Figure 6. Scaling ratio values for P\ (50)/P\ (147) along the outside surface of the brain as a
function of Q.
model, which predicts no change. As
expected, the scaling ratios near the top
are essentially identical to the values
predicted by the concentric spherical
model; at the bottom, the values become
closer to those predicted by an isolated
spherical model. Thus, the off-center
spherical model yields internal scaling
ratios that lie between the extreme values
predicted by the concentric and isolated
spherical models.
Recent experimental studies have
shown that the penetration of radioactive
calcium-ions in chick brain tissue was no
greater than 1 mm from the surface.
Thus, it may be concluded that surface
field values are of paramount importance.
The profiles for the surface scaling ratios
are symmetric for the concentric spheri-
cal case, with the peak value occurring at
a point on the surface (0=90°), where the
external electric field is normal to the sur-
face. However, the off-center spherical
model yielded somewhat lower peak
values whose position shifted toward the
bottom (0=*99°). In addition, the surface
ratios were similar at the top of the brain
for all the models, but the values at the
bottom varied considerably, with the h=0
case approaching the value given for an
isolated sphere.
The scaling ratio results have provided
new insights into the interpretation of the
calcium-ion efflux phenomena. The exper-
imentalists can use the information pre-
sented in this study to interpret their
results and to formulate theories about
calcium-ion efflux phenomena.
S. GOVERNMENT PRINTING OFFICE:1985/559-l 11/20680
-------
Guillermo Gonzalez and James C. Nearing are with University of Miami. Coral
Gables. FL 33124).
Ronald J. Spiegel is the EPA Project Officer (see below).
The complete report, entitled "Dosimetry for In-Vitro Chick Brain Calcium-Ion
Efflux Experiments by Numerical Methods Based on Zonal Harmonic Expan-
sions, " (Order No. PB 85-227 528/AS; Cost: $8.50, subject to change) will be
available only from:
National Technical Information Service
5285 Port Royal Road
Springfield. VA 22161
Telephone: 703-487-4650
The EPA Project Officer can be contacted at:
Health Effects Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park. NC 27711
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
Official Business
Penalty for Private Use $300
EPA/600/S1-85/016
0000329 PS
ENVIR PROTECTION AGENCt
------- |