457

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-22, NO. 6, NOVEMBER 1975

The Distribution of Heating Potential Inside Lossy Spheres H. N. KRITIKOS, MEMBER,

IEEE, AND

Abstract-The distribution of the heating potential inside a lossy sphere having the same electrical characteristics as those of brain tissue was investigated in the frequency region of 10 MHz to 12 GHz. The conditions under which a potential hot spot appears inside the sphere and its shape were determined and were displayed in a radius vs. frequency diagram. The results show that hot spots appear inside only for spheres with radii 8 cm > R > 0.1 cm and only in the frequency range of 300 < f < 12,000 MHz. It was found that the heating potential is always non-unifonnly distributed, and in cases where it was maximum inside the sphere it was larger by a factor of approximately ten to the values calculated by the plane slab or the average cross section model.

1. INTRODUCTION IN ATTEMPTING TO assess the effects of electromagnetic fields on biological tissues it is important to isolate and clarify the various separate mechanisms which contribute to the total phenomenon. One such mechanism is the process by which the heating potential is distributed inside the test objects. A non-uniform distribution implies the possibility of non-uniform temperature rise or the creation of hot spots. It is therefore important to determine the conditions under which this non-uniform distribution takes place. Non-uniform distributions have been reported by Guy [11 who investigated both theoretically and experimentally spherical and, cylindrical phantoms. Shapiro et al. [2] have shown the existence of potential hot spots in multilayered spheres. An excellent summary of these contributions and other related work up to 1972 has been provided by Johnson and Guy [3]. Kritikos and Schwan [4] have demonstrated that size is an important parameter in the potential formation of hot spots, while for larger size spheres (10 cm in radius) the maximum heating potential was always at the front surface of the sphere; for smaller sizes (5 cm in radius) the heating potential was sharply peaked inside the sphere. While the reported investigations have established the fact that for certain sizes and frequencies the heating potential peaks inside the test object, the precise conditions for this phenomenon to occur are not well known. The reported work here is a contribution in establishing those conditions and in determining the exact size and shape of these potential hot spots.

HERRMAN P. SCHWAN, FELLOW,

IEEE

fields is a complex one involving many different physical mechanisms. Basically these phenomena can be divided into the mechanism of heat development caused by absorption of electromagnetic energy and heat dissipation and the resulting temperature rise. While recognizing that the heat dissipation is a very important part of the total process, we will focus our attention to the problem of heating potential. The problem of heating potential can be studied by recognizing that the biological tissues are lossy dielectrics, thus providing Joules heat when they are subjected to electromagnetic fields.1 Schwan [5], [6] has made extensive studies of the dielectric and conductive properties over a wide range of frequencies. For the purposes of this investigation the dielectric properties of brain tissue will be chosen in order to use the result to discuss potential overheating of the brain. The conductivity a and relative dielectric constant er are somewhat less than those of muscle and are taken to be [5]

er = 5 + 1

35

+(flf0)2

a=7+ 550 ecoo

I

(1)

+(f/f0)2 (flfO

)2

nmho

cm

(2)

where o

10-11 in = 20 X 109 Hz. =36-a F/cm f0

(3)

The next question to consider is what is the best possible geometrical configuration to perform our modeling. Taking into account the complexity of the human skull and the lengthy computer calculations involved in direct numerical methods, it was decided as the first step to study the simple problem of the homogeneous sphere. This model is very effective in demonstrating the mechanism by which the heating is non-uniformly distributed.

3. THE HEATING POTENTIAL AND NORMALIZED ABSORPrION CROSS SECTION The heating potential 4J is the rate by which heat is generated in the tissues. It is defined as 1 = 4 a lEl2 (E is the 2. THE HEATING OF BIOLOGICAL MATERIAL AND electric field). In presenting the results, it is convenient to MATHEMATICAL MODELING calculate the normalized differential absorption cross section The problem of determining the temperature rise of living (NDAC) animals or human beings which are exposed to electromagnetic E 2 AS=u

Manuscript received March 12, 1974; revised December 16, 1974. This work was supported by the Office of Naval Research under Contract N00014-67-A-0216.0015, the National Institutes of Health under Grant 5 ROI HL01253, and the National Science Fbundation under Grant GK 40119. The authors are with the Department of Bioengineering, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pa. 19174.

1.20rrin

cm-1(4

(4)

which is the heat generated per unit volume (cc) per incident power density (watts/cm2). The electric field is calculated from the well-known expressions [4] 1 The heating potential is defmed in Section 3.

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, NOVEMBER 1975

458

_ 00 E= E

2n+1I

i(n I) +

[an-moin- ibtniein]

REGION I MAXIMUM HEATING OCCURS AT FRONT SURFACE

(5)

III

where t

an

=-

i/p

hn (p) [NPjn (Np)] '- in (Np) [phn (P)'

(6)

i/Np

(7) [Npjn (Np)] 'hn (p) - N2jn (Np) [ph,n (P)]' The spherical vector harmonics mo1, and iTem, are given by n

(8)

Moln = V X RIoln 1

feln =ki

VX VX

_1

RIein

(9)

where R is the radius vector

To!e =sisinmP, (cos 0) j,(kR) Cos

k2 = CO/c, X = 2rrf c= 3 X 1010 cm/sec where f= frequency in Hz

(10)

Fig. 1. The radius vs. frequency diagram.

(1 1)

k1 = W)2pE + iCua

N=

k2

p

= kjA

(A = radius of sphere).

(12)

The above series was evaluated by using an IBM-360 computer. The approximate number of terms for convergence was found to be 2 INPI. 4. THE DISTRIBUTION OF THE HEATING POTENTIAL In all cases the distribution of the heating potential is not uniform. The type of distribution depends on the size of the sphere and the frequency. The k2A product alone is not sufficient to adequately describe the phenomenon because biological tissues are dispersive and scaling is not useful. One is therefore forced into a classification of the mechanisms in terms of a radius of sphere A vs. frequency f diagram. In the A-f diagram one recognizes two regions which are defined by the position of the maximum heating potential. I. Maximum heating potential occurs at front surface. II. Maximum heating potential occurs inside the sphere. Regions I and II are shown in Fig. 1. The boundary line dividing the two regions has been obtained by a numerical searching technique. A discussion of the regions is given below. I. Maximum Heating Potential Occur at Front Surface In this region the heating potential is always at the front surface (q5 = 0, 0 = ir). In trying to understand the physical mechanisms involved it is convenient to recognize two asymptotic cases where some excellent approximations can be used. These are the high frequency case (p >> 1) or Geometrical Optics region, and the low frequency case (p > 1: In this region very little energy penetrates into the sphere. The heating is a skin phenomenon (see Figs. 2 and 3). The maximum heating occurs at the front surface where one can very effectively use the Kirchoff approximation. In the Kirchoff approxima-

k AXIMM HEATING

3

2

NDACz

4 M5cm

.2

.4

E PLANE

.3

FREQUENCY= 3000 MHz

Fig. 2. Distribution of the normalized absorption differential cross section (N.A.D.C. in 1/cm) E plane. Frequency = 3000 MHz.

k22

MAXIMUI

2

3

4

MAIU

H PLANE

5cm

FREOUENCY= 3000MHz

Fig. 3. Distribution of the normalized absorption differential cross section (N.A.D.C. in 1/cm) H plane. Frequency = 3000 MHz.

459

KRITIKOS AND SCHWAN: DISTRIBUTION OF HEATING POTENTIAL

tion the heating potential at the front end is equal to that obtained for an infinite slab. One has then differential absorption at the surface. 2k1

AS=a

2

k1 +k2

(13)

1.20ir.

The heating decays exponentially into the sphere with a penetration length L 1 (14) L= 21mk1 The exact solutions obtained with spherical harmonic series agree very closely to the above approximation. IB. The Rayleigh Region p X1 the refracted field converges and becomes focused in a region which is roughly X1 /2 by X1 /2. This is a region which can be characterized as a quasi-physical optics region. Alternatively when A1 > A the magnetic and electric modes resonate inside the sphere. This region will be characterized as the resonance region. These two regions can be described by asymptotic expressions which greatly simplify the understanding of the heating mechanism. A description of these regions follows. IIA. The Quasi-Optics Region (See Fig. 1): In this region and specifically only in the vicinity of segment a to b, the flow of the energy inside can be described by the laws of the physical optics. Assuming the simplest possible model, one can consider the diffracted energy to illuminate an area half the surface area of the sphere 4irA2 and be focused down to an area of the order of magnitude of (X1/2)2. The gain in power is then the ratio of the two areas which is 87r((A/Xl ))2. At the same time the energy is attenuated because of the losses by an amount e 1) then the heating potential peaks inside the sphere. Exact calculation has shown that this is the case. In fact part of the boundary which separates region I from II (1500 MHz