IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MIT-26, NO. 8, AUGUST 1978 !507

of using it in regions with low dielectric constants and the

difficulty of designing an interference-free lead wire sys-

tem for the probe. An implantable electric field probe

with an interference-free lead-wire system has subse-

quently been developed by our laboratory. This informa-

tion

[1]

[2]

[3]

[4]

[5]

[6]

will be published in a future paper.

I?EFERENCES

C. C. Johnson and A. W. Guy, “Nonionizing electromagnetic wave

effects in biological materials and systems,” Proc. IEEE, vol. 60,pp.692–718, June 1972.C. C. Johnson, “Research needs for establishing a radio frequencyelectromagnetic radiation safety standard: J. Microwaoe Power,vol. 8, pp. 367–388, Nov. 1973.S. M Michelson, “Review of a program to assess the effects onman from exposure to microwaves,” J. ,$ficrowaoe Power, vol. 9,June 1974.H. Bassen, M. Swicord, and J. Abita, “A miniature broad-bandelectric field probe,” Ann. N. Y. Acad. Sci., vol. 247, pp. 481–493,1975.A. Cheung, H. Bassen, M. Swicord, and D. Witters, “Experimental

calibration of a miniature electric field probe within simulatedmuscular tissues,” in Proc. 1975 fJRSI Bioeffects Symp., to be

published.G. Smith, “A comparison of electrically-short bare and insulatedprobes for measuring the local radio frequency electric field in

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

biological systems,” IEEE Trans. Biomed. Eng., vol, ME-2!2, PP.477–783, Nov. 1975.B. S. Guru and K. M. Chen, “Experimental and theoretical :studieson electromagnetic fields induced inside finite biological bodies,”IEEE Tran.r. Microwave Theory Tech., vol. M’IT-24, pp. 433-440,

July 1976.H. Bassen, W. Herman, and R. Ross, “EM probe with fiber optictelemetry system; Microwave J., pp. 35–47, Apr. 1977.

H. Bassen, A. Cheung, and K. M. Chen, “Comments on ‘Experi-mental and theoretical studies on electromagnetic fields induced

inside finite biological bodies,’” IEEE Trans. Microwave Theory

Tech., vol. M’IT-25, pp. 623-624, July 1977.

R. W. P. King, “The many faces of the insulated antennafl Proc.

IEEE, vol. 64, 228-238, Feb. 1976.J. A. Stratton and L. J. Chu, “Forced oscillation of a conducting

sphere,” J. Appl. Phys., vol. 12, pp. 230–248, Mar. 19’41.M. Abramowitz and I. A. Stegun, Handbook of A4athe,maticalFunctions. New York: Dover, 1970.

L. Infeld, “The influence of the width of the gap upon the thmxy

of antennas; Quart. Appl. Math., vol. V, pp. 113-132, Jul~y !1947.S. H. Mousavinezhad, “Implantable electromagnetic field probes

in finite biological bodies,” Michigan State Uuiv., East Laming,Ph.D. dissertation, 1977.

H. Mousavinezhad, K. M. Chen, and D. P. Nyquist, “Implimtablefield probes in finite biolo~cal bodiesfl presented at the 1S176!ht.

IEEE/AP-S Symp., Univ. of Massachusetts, Amherst, Ott. 1976.

D. Livesay and K. M. Chen, “EM field induced inside arbitrarily-

shaped biological bodies,” IEEE Trans Microwaoe Theoiy Tech.,vol. MTT-22, pp. 1273–1280,Dec. 1974.

Heat Potential Distribution in anInhomogeneous Spherical Model of a Cra~~ia~ll

Structure Exposedl to Microwaves Due toLoop or IDipole Antennas

ALTUNKAN HIZAL AND YAHYA KEMAL BAYKAL

Abstract—An inhomogeneoaa sphericafmodel of a 3.3-cm radhw crankf

structure is assured to be placed symmetrkafly in the near field of’ a smaffloop antenna or an electrical dipole antenna at 3 GHz. The transitions

between the layers are taken to be sharp but sinusoidal. Cslcnlations of theheat potentiaf are performed using a sphericaf wave expansion technique in

which finear differential equations are solved for the rmknowm ❑rrltipole

coefficients. The results are afso compared with the plane-wave! excita-

Manuscript received July 8, 1977; revised February 21, 1978.A Hizal is with the Electrical Engineering Department, Middle East

Technical University, Ankara, Turkey.Y. K. Baykal was with the Electrical Engineering Department, Middle

East Technical University, Ankara, Turkey. He is now at NorthwesternUniversity, Evanston, IL 60201.

tions. It is seen that a more uniform distribution of tbe heat poteMiaJoccurs for the dipole aatemra excitation which is afso similar to theE-plane distribution in the case of plane-wave excitation. For the loop

excitation, a significant hot spot occurs near tbe center of the stractwe.

1. INTRODUCTION

T HE PREDICTION of the heat potential distribution

in a cranial structure excited by a microwave radiat-

ion is of interest for the purposes of medical treatment

and searching out the radiation hazards. For this purpose,multilayered spherical models [1], [2] irradiated by a pllane

wave have been analyzed. It is found that a nonuniform

0018-9480/78 /0800-0607$00.75 @ 1978 IEEE

608 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MIT-26, NO. 8, AUGUST 1978

heat potential distribution, which gives rise to the “hot-

spots” in the structure, occurs mostly in the resonance

region where the radius of the structure ranges from about

6 to 16 percent of the free-space wavelength [2].

Previous investigations are based mostly on the assump-

tion of a plane-wave incidence. It might be of interest also

to study the heat potential distribution for sources other

than a plane wave. In this connection, Guy [3] has

analyzed layered biological tissues exposed to radiation

from a rectangular aperture. The results indicated that the

heating of the tissues exceeds that produced by a plane

wave. Ho [4] has presented results for the dose rate

distributions with the unit mass density, which is equiv-

alent to the heat potential, for five layered monkey heads

of 3.3 cm at 2.45 GHz for a TEIO contact aperture source.

The results indicate that the aperture source predomi-

nantly yields absorption on the surface, and microwave

energy penetration into the central region is weak. In the

present investigation, small loop or dipole antennas are

considered as the sources of radiation which excite an

inhomogeneous spherical model of a cranial structure

situated symmetrically in the near zone of these antennas.

The idealized model chosen is the six-layered spherical

model with a maximum radius of 3.3 cm which might

correspond to a monkey’s or an infant’s heat. The outer

five layers make up the inhomogeneous medium, the brain

being the homogeneous core. The frequency is chosen to

be 3 GHz at which peak internal heat potential makes a

maximum for the plane-wave excitation [2]. The permittiv-

ity and the conductivity of the layers are taken at this

specific frequency with sharp sinusoidal transitions be-

tween the layers. The choice of a continuously inhomoge-

neous model allows one to study the effect of these

boundaries on the heat potential distribution. The current

distribution on the infinitesimally thin loop and dipole

antennas are assumed to be constant and sinusoidal, re-

spectively.

11. FORMULATION OF THE PROBLEM

The scattering problem cited in Fig. 1 is solved by the

state-space method [5] after having it extended to the case

with a homogeneous dielectric inner core.

1) For the electrical loop excitation, the incident elec-

tric field for r< (a*+ d2)1i2 may be expressed by the m = O

terms of the spherical vector harmonics expansion as

E.$(r, d ) = Z. ~ Cnfljn(kr) -$ Pn (1)~=]

where c.= {(2n + 1)/[4nn(n + l)]} ‘/2,j~(kr) is the spherical

Bessel function with k= 21T/A being the wavenumber in

free space, P.= Pn(cos f?) is the Legendre function, and r,

O, @ are the spherical coordinates. ZO = ( ~/qJlf2 = 1207r

ohms is the intrinsic impedance of the free space. The

multipole coefficient ~~ is given by

%= [ 1–ak21&i!m.h~l) k(a2+ d2)1’2 -$ P. (2)s

L

t

/(1. homogeneous core )

CSF, Duro,Bore, Fat, Skm i’+’”” -+

Fig. 1. Geometry of the problem for loop or dipole antenna excita-tions.

where 10 is the current in amps, h$l)(kr) is the spherical

Hankel function, and 6,= tan- ‘(a/d). The scattered elec-

tric field in the inhomogeneous dielectric layer for rl < r <

r2 for exp ( –jtif) time dependence is expressed by

where the multipole coefficients satisfy the first-order cou-

pled linear differential equations (state equations) for

p~(kr) + 11 and P;(kr) given in [5]. The unknown multi-pole coefficients are found by solving these differential

equations subject to the boundary conditions

~J(kr2) = O. (5)

Here, F~(lclr-l) is the Mie scattering function [6] given by

F~(k,rl) =jkrlj~(krl) Tn /[jn(k,rl) –jkrlh~’)(krl) Tn] (6)

Tn= [r,j.(krJ ]’j.(klrJ –j.(krl)[ r&(krJ ] (7)

where the prime indicates the derivative with respect to r,,

kl = k(~J1/2, and c1= e,, + jul/(a~O) is the complex dielec-

tric constant of the homogeneous inner core.

The total field in the homogeneous inner core for 0< r

< r, is given by

HIZAL AND BAYKAL.: HSAT POTENTIAL DISTRIBUTION

where the multipole coefficient ~~c is obtained from

E~(rl,@) = E$(rl, O) + E~(rl, O) and the ortbogonality ofthe Legendre’s functions on a spherical surface. It can

klr,). (9)

The heat potential is given by Pk = (1 /2)cl E@\2,where

E* is the total electric field in the dielectric and is given

by E@= Et + E:, (1) and (3) for the inhomogeneous layerand E+= E;, and (8) for the homogeneous inner core.

2) For the electric dipole excitation, the incident elec-

tric field for r< (d– h) may be expressed by the m = O

terms of the vector spherical harmonics expansion as

Ej(r, tl)= –20 ~ n(n+ l)c~aJ~(kr)&P~~=1

E~(r, O)= –.230 ~ c~a~[rj~(kr)]’~~P~~=1

where the multipole coefficient a; is given by

a~=kIOc~{k(d– h)h:l)[k(d– fi)]+k(d+h)h:lJ

(lo)

(11)

.[k(d+ h)] -2kdfi$1)(kd) COS kh} (12)

for the sinusoidal current distribution l(r) =10 sin [k(h – Ir

– do]. The scattered electric field in a vacuum gap of

vanishingly small thickness at r in the inhomogeneous

dielectric layer for r,< r < r2 is expressed by

E:(r, r3)= –ZO ~ n(n+l)cn~=]

o[a~(kr)j.(kr)+ai(kr)hil) (kr)] ~F’n (13)

E~(r,t9)= –ZOn~l c~{a~(kr)[rj~(kr)]’

+ a~(kr)[rk$])(kr) ]’} ~ -$P~ (14)

where the multipole coefficients a; (kr) and a~(kr) satisfy

the state equations for a~(kr) + a; and a~(kr) given in [5]

which should be solved subject to the boundary condi-

tions

al(krl) = ~.(klrl)a~(krl) + a; (15)

a~(krz) = O (16)

where the Mie scattering function F~(klrJ is given by (6),

but now

T.= [~&(krl)]’jn(klrl)-.L(krJ~ [rl~n(klrl)]’. (17)

6(19

The total field in the homogeneous inner core for Cls; r

< rl is given by

E~(r,6J )= – 2?.(61) -1/2~$, ~(”+ W%J’.(JW);;P.I

(18)

E~(r, O)= – ZO(tJ- 1/2 5 %a;[tin(wl’-+:l-’n~=1

where the multipole coefficient a: is obtained from

E~(r,, 0) = E~(rl, O) i- E~(rl, 19) and the orthogonality of

the Legendre’s functions as

-t a#[rh~lJ(krl)]’} /[rl j.(kr)]’. (20)

The heating potential for the dipole excitation is given

by P~ = (1/2) a(lE,12+ IE@12), where E, and E@ are the

spherical components of the total field and are given “by

E,= (c)- l(E; -1-E:), EO= i?; -!- E~ for the inhornogeneous

layer and E,= Es, E@= E; for the homogeneous inner

core.

HI. NUMERICAL RESULTS

The profiles shown in Fig. 2 correspond to those givem

by Shapiro ei al. [1] at 3 GHz. The sinusoidal transitions

between the layers can be made sharper or smoother by

adjusting the values el, ez, es and dl, d2, d~.

A computer program is developed which calculates the

multipole coefficients of the scattered field for the prolb-

Iem illustrated in Fig. 1 and Fig. 2. The state equations a,re

integrated by Hamming’s modified predictor corrector

method [8] for each n, and linear interpolation is applied

to the scattering coefficients to find a close estimate of

these coefficients at points where the heat potential is to

be found. To make sure that the incident field,, which is

the near field of the loop or dipole antenna, is found

correctly, these fields are also calculated by numerical

integration for the loop antenna and from the analytical

expression [7] for the dipole antenna. For the cc~mparison

ka = 0.3, kh = n/2, and d (6.3 and 13 cm) are taken for the

antennas and the electric field is computed at (9 (30 and

90°) and r (0.159 and 3.3 cm). In the expansions (l), (10),

and (1 1), n~a,= N =11 terms are used. It is found that the

multipole expansions calculate the incident fields correctly

up to four significant digits. The computer program

calculating the multipole coefficients and, hence, the

scattered fields is also checked by solving a homogeneous

sphere as an inhomogeneously coated sphere by the pr-

esent formulation and comparing the results with those

obtained from the analytical Mie formulation [6] for both

loop and dipole antenna excitations. A very good agree-

ment is obtained between the two results. Table I presents

the results of the comparison for the loop antenna excita-

tion. The convergence of the spherical wave expansions

used slows down as the complex dielectric constant, and

610 133EETRANSACTIONS ON M2CROWAVE THEORY ANO ‘IECHN3QUES, VOL. MTT-26, NO. 8, AUGUST 1978

Cr

6

777?

-----/

i: ‘––– ‘-––II I

[ I

1 I

I I

I III I

I5 --––..[–––--––l–-l-

1 , 1 +26 \=.6 8 28 288 29.3 30 31.3 32 5133

2500

213ao1900

200

CfImtiml

II III e,+II

I

1I

I I II I!

I I II I /

1 II 4+ d 4

1 1121I

, ,; ~1( id--- , - /_ -,-b

-l\,,

I1,

~!

I

I

1

t1I!

-d3 :

r [mm)

, , , I 1 ! br[mml26r1= 268 28 28$ 29.3 30 31 3 32 r2= 33

Fig. 2. The variation of the dielectric constant and conductivity withradius in Fig. 1.

TABLE I

HEAT POTENTIAL DISTRIBUTION (W/m3) m A HOMOGE~OUS

SPHERS EXCITED BY A SMALL LooP ANTENNA (r] = 2.68 cm,

rz=3.3cm, 0=900, ka=0.3, ~=3Gfi, IO=lpA, c,=42, u=2

r(m)

O 0:18

1 3559

2 6830

2 9300

313001J

0.0J18

1,3559

2 6802

2 9!300

3 3000

mho/m, N= 11)

As coated sphere by As homogeneotis sphere;,resent fot-m”lat, or, by MIe formuldtlon

O 12206 #10-”

o 16W8X1O-9

I.43820x10”’U

(l,46984x10-’o

o 50617 X1O-”3

0 80916 X1O-’3

U.12627X10-13

O S9832 ,10-”

J 92661 x10-11

u ~7570x111-11

,.12217x10-11

o 167OOX1O-9

o 43849 Y.1O-’O

0.47184,10-10

0 51208 K10-’0

o 80939 X1O-’3

O 12632,10-10

0 80867 x10-11

U 93026 x10-11

0. Y8183x10-’1

d(cnl)

63

12,3

the optical radius (krJ increase and the radial separation

of the antenna and the field point decrease. In the present

applications it is found that, with N = 9 and 11

coefficients, the expansions converged satisfactorily. The

computer time required increases with N, the thickness of

the inhomogeneous layer, and the slope of the variation of

Ph (watt s/m3)

5 A iL

3

2

10-9

7I

5L

3

2

-i o10 /

&

.1l.= lpA

T

ka=O.3., ~=1 Ocm

e6.3 Cm

L

/ 30

-J3.3:iL

-11

10 ~/1 I t I I I I I I i

36912151821 2L 27 :~mm~

Fig. 3. Distributions of heat potential due to loop at d= 6.3 cm.

; (wotts/m3)

I 4tlz-10

10

1?

T

,0 l.. l~A

7 ka. O.36 12.3c

51 A.10cm

1

L u //3 .—

‘J33cm~ ~O._ .. .2 -

10

-12

:/,,,,,,,,,

I

10

:

3 6 9 12 15 18 21 2L 27 30 33

r(mml

Fig. 4. Distributions of heat potential due to loop at d= 12.3 cm.

the complex dielectric constant. Table I is produced in 20

min using double precision arithmetic in an IBM-370/145

system.

The heat potentials (PJ are shown in Figs. 3–5 for the

loop excitations and in Figs. 6–8 for the dipole excita-

tions.

HIZAL AND isAYXAL: HSAT POTENTIAL DISTR2BOTION

~ (wcitts/m3 )

-lo7X1O

5 -

L -

3 -

10

7 -

5

Ll.= lpA

3

zxld’ -I I , 1 1 1 1 1 I369121518 2! 2L 27 30 ’33

r(mm)

Fig. 5. Comparison of distributions of heat potential for two different

loop radii.

~ (watts/m3 ~

Pzt ““T:m 10=JIA

T---”!..L

A=IO cm

, , I I ) t 8 1 I ,

10-’

765

L

3

2

10-9

~3 6 9 12 15 18 21 2L 27 30 33

r [mm]

Fig. 6. Distributions of heat potential due to a half-wave dipole atd= 10 cm.

For the loop excitations, heating the exact center of the

sphere is not possible, as the loop does not radiate in the

direction of its axis. However, a peak is observed close tothe origin in the O= 90° plane. Increasing the distance of

the loop to the sphere decreases the peak. The dip in the

heat potential in the vicinity of the 3 l-mm radius is

obviously due to the small conductivity of the bone and

fat layers. It is seen that this dip shifts slightly with the

radius of the loop. The results shown are for el = 2.0 mm,

d,= .2 =2.5 mm, dz= d~ = 1.35 mm, and e~=0.97 mm.

When the values el = 0.955 mm, dl = e2= 0.192 mm, dz = d~

=0.675 mm, and e3= 0.478 mm are used for sharper

transitions between the layers, the results did not change

significantly except that the dip around the 31-mm radius

raised up slightly.

For the dipole excitation, the heat potential shows un-

dulations with a more uniform distribution compared to

611

$(watts/m3)

r —1

9.109 -

7 -

5 -L -

3 -

2 -

10-9 z7 -6 -5 -L -

3 -

2,10’0 .

~z10=l~A--

r-”--l

[

5cnl ~.10 cm

--

30 I

~-~36912 15 18 21 2.4

r (mm)

Fig. 7. Distributions of heat potential due to a half-wave di~oh~ at~= 13 cm.

P~ [watts/m31

108

f, IQ.lyA

h.se:”

;-w

klocm

+

~-..2.5 cm 1..

T

5cm

~=loc:. z+; F$M- F-l--...’=’fire....‘;irn---;:[----‘,1

d=10cm,2h=A

~~L.J3 6 9 12 15 18 21 24 27 30 33

r(mm)

Fig. 8. Comparison of distributions of heat potential due to variousdipole excitations.

the loop excitation. The effect of the distance of the di]pole

to the sphere and the length of the dipole (Fig,s. 7 and 8)

on the heat potential seems to be only to change the level

612 IEEE TRANSACTIONS ON MICROWAVE TlU30RY AN33 TECHNIQUES, VOL. MTT-26, NO. 8, AUGUST 1978

Ph /lE’12

.9

I

+/+-+,.7

/ ‘\.5 ‘\+.L

\+.3

\

J_-_J3691215 18 21 24 27 30 33

r (mm]

Fig. 9. Cornpanson of incident field normalized heat potentials at

8= 90° for loop antenna, dipole antenna, and plane-wave excitations.

of the variations. The smoother and the sharper transi-

tions did not make any significant change in the results.

The results for plane-wave excitations are obtained

from the literature [1] for comparison and given in Fig. 9.

These results correspond to profiles with abrupt transi-

tions, and the heat potential is normalized with the mean-

squared incident power. For comparison, the curves at

9= 90° for loop and dipole excitations are normalized by

the near-field mean-squared incident powers (Fig. 10) and

also given in Fig. 9. It is seen that a similarity existsbetween the normalized heat potential for the plane-wave

excitation in the direction of the incident electric field and

the dipole excitation at 9= 90°. For the case of the loop,

both the incident power and the normalized heat potential

show a large variation towards the center of the brain.

IV. CONCLUSION

Multilayered and inhomogeneous spherical models of acranial structure of 33-mm radius at 3 GHz are rather an

idealized model of a real physical structure. Nevertheless,

the numerical results could have some relevance to the

actual case.

It is seen that no considerable hot spots occur for the

dipole excitation and a rather uniform heat potential

distribution occurs. On the other hand, a hot spot is

7

[

,/

6 ,/

5 ,/

L ,/

3 —— +—–––––~n:-~e– —–

2/!

/

10-8 :/

: :/ Dipole

L -x —~ —x-x-x_x_-x —x—x —x— x

3 -X—x

3 6 9 12 15 18 21 2k 27 30 33r [mm)

Fig. 10. Comparison of mean-squared incident powers at O= 90°.

observed close to the origin for the loop antenna excita-

tion at 6 = 90°. When compared with the plane-wave

excitation, it is seen that the exact center of the head can

best be heated by the plane-wave incidence. Also, for all

excitations, a concentration of heat at this frequency and

size can be observed at the skin because of the high

conductivity of this outermost layer.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

I@FE~NCES

A. R. Shapiro, R. F. Lutomirski, and H. T. Yura, “Induced fieldsand heating within a craniaf structure irradiated by an electromag-netic plane wave,” IEEE Trans. Microwave Theory Tech., vol.MTT- 19, pp. 187-196, Feb. 1971.C. M. Weit, “Absorption characteristics of multilayered spheremodels exposed to UHF/microwave radiation,” IEEE Trans. Bio-

med. Eng., vol. BME-22, pp. 48–476, Nov. 1975.

A. W. Guy, “Electromagnetic fields and relative heating patternsdue to a rectangular aperture source in direct contact with bilayered

biological tissue: IEEE Trans. Microwave Theory Tech. (SuecialIssue on Biological Efiects of Microwaves), vol. MIT’-~~, pp.214–223,Feb. 1971.H. S. Ho, “Contrast of dose distributions in phantom heads due toaperture and plane wave sources.” Ann. N. Y. A cad. Sci.. vol. 247.pi. 454-472, i975.A. Hizal and H. Tosun, “State-space formulation of scattering with

application to spherically symmetrical objects,” Can. J. l%ys., vol.51, no. 5, pp. 549-558, 1973.

~qfi Stratton, Electromagnetic Z7teory. New York: McGraw-Hill,. . . .R. Mittra, Computer Techniques for Electromagnetic. New York:Pergamon. 1973.R. ~. Hamming, Numerical Methoa3 for Scientists and Engineers.New York: McGraw-Hifl, 1962, pp. 393-426.