Radiofrequency Radiation Dosimetry Handbook - Fourth Edition

Table 5.5. Comparison of Theoretical Methods Used In Literature
To Calculate The Power Absorption By Biological Models






A. One-dimensional
Infinite plane tissue layers (isotropic) Multilayers Planewave Transmission-line model Unlimited a Good only at high frequencies where the body
curvature can be neglected. Can predict y/4
type resonances but not body resonance.
Two layers Rectangular aperature Fourier transform technique and field matching on common boundaries 433-2450 MHz b, c Used to calculate relative heating patterns.
(Anisotropic) Multilayers Planewave Transmission-line model (together with Thevenin's theorem) 0.001-100 MHz d Used to calculate effects of tissue anisotropy (at low frequencies) on microwave fields and power absorption. Small effects observed at higher frequencies (above 10 MHz).
Semi-infinite homogeneous slab One layer Nonperiodic electric field with TE and TM components Planewave spectrum analysis All calculations
at 2450 MHz
e Near-field analysis of aperture sources.
Six layers E-polarized planewave Optical refraction--Snell's law Up to 3.0 GHz f Layering increases average SAR; an enhancement factor needed to correct results from homogeneous models.
Homogeneous semi-infinite slab and multi-layered slab --- Near field of a source leaking radiation Planewave-spectrum approach 2450 MHz g, h For fields nearly constant over at least a free-space wavelength, energy deposition equal to or less than that resulting from planewave exposure. In the analysis, coupling of target to source not taken into account.
B. Two-dimensional models
Infinite circular cylinder Length l= 1.75 m, radius a = 11.3 cm; l = 14 cm, a = 1.58 cm Planewave Mode-matching technique
and geometrical optics
300-6000 MHz i, j In good agreement with geometrical optics at the high-frequency limit.
Radius such that (formula) koa < 2.4 Coaxial loop antenna Long-wavelength approximation method Depends on (formula) koa < 2.4 k Formulation simple to use, but valid results are in low-frequency range. Power absorption coefficient is plotted against loop dimensions and cylinder radius.
5- and 10- cm-radius cylinders in presence of a reflector Planewave TE and TH with k vector normal to plane of a reflector Point-matching method combined with imaging Up to 1 GHz l SAR calculated as function of frequency and distance from reflector.
2.24-24 cm
Planewave E and H polarization a. Geometrical optics approximation
b. Mode-matching technique
200-100 GHz j At higher frequencies the two methods give same result. Average SAR results tally with those from prolate spheroidal models at higher frequencies.
Multilayered infinite circular cylinder --- Planewave Mode-matching technique and moment method 433-2450 MHz m, n Microwave heating calculated in simulated human thigh.
Three layers Direct con-
tact aperture source (TE10 mode)
Field expansion in terms of three-dimensional cylindrical waves and matching the boundary conditions 433-2450 MHz o Microwave heating calculated in phantom models of human limbs exposed to a direct-contact aperture source.
Multilayered cylinders of arbitrary cross section a = 1.5 b, a = 0.239 m, b = 0.159 m for elliptical-cal cylinder Planewave Surface integral equations derived via vector Green's theorem and boundary condition All calculations at 300 MHz p Calculations made for one-and two-layered circular and elliptical cylinders.
17.5- x 19- cm triple-layered model Planewave TM Finite element 433 MHz q Finite element method and variational calculus used to approximately calculate internal fields. Results presented only for normalized inside field.
Multilayered infinite circular cylinders Outer radius 11.28 cm Planewave E and H polarization 10 MHz-10 GHz r Absorption effects due to clothing are negligible below 2 GHz. Layering changes average SAR values in the 0.4- to 8-GHz frequency range.
Homogeneous infinite cylinder 62.6 x 62.6 cm Planewave E polarization Stacked two-dimensional spectral iterative technique (SIT) 915 MHz, 2450 MHz s Method based on modeling the body by a set of planar parallel slabs and utilizing a convolution-type relationship between a current distribution on any slab and the field due to this current. Calculated data for the SAR distribution are given for two
Homogeneous infinite circular cylinder radius = 5 cm Near field of electric and magnetic line sources Mode-matching technique 27, 100, and 300 MHz t With proper choice of geometry and polariza-tion of the sources, it is possible to have deep penetration with maximum heating at the center and to move the maximum around.
Inhomogeneous infinite cylinders with arbitrary cross section Cross section of a human torso EM planewave and a sole-noidal field Galerkin's method with linear basis function Low-frequency region u Arbitrarily shaped polygonal cells are used to allow more accurate modeling of complex objects without excessive matrix sizes.
Annular phased array system Moment method with pulse basis function 70 MHz v SAR distribution calculated in two-dimensional models of cross sections of the human body. Numerical results agree with measured values in central region of the cross section.
C. Three-dimensional
Finnite planar model 16 x 12 x 4 cm for a fat muscle tissue block and 3 x 0.5 x 0.5 y for a muscle Planewave Tensor integral equation for the electric field inside the body Results presented up to 2.45 GHz w Method applicable for heterogeneous biological bodies.
Finite circular cylinder Height/radius l/ro = 12.68 E-polarized planewave a. Antenna theory
b. Curve fitting used along with circuit theory
1-60 MHz x For human model in direct contact with a ground plane, an order-of-magnitude enhancement tin SAR value may occur at frequencies below resonance. Enhancement rapidly decreases as model moves away from ground plane. Separation distance of about 7.5 cm from ground plane is sufficient to restore free-space absorption characteristics.
Spherical model Radius of the sphere r = 10 cm Planewave Mie theory 10-10,000 Mhz y, z Examined distribution of generated heat.
r = 25.57 cm Planewave Mie theory 1-20 MHz aa,bb Power deposition in a spherical model of man--70 kg.
r = 5 cm Planewave Mie theory 10-12,000 MHz cc Distribution of heating potential.
4r = 10 cm Planewave Mie theory 100-10,000 MHz Type of nonuniformity described in a radius-frequency diagram. Localized heating for 8 cm < a < 0.1 cm in the frequency range 300 MHz < f < 12 GHz is described.
2 < r < 50 cm
results for r
= 3 and 7 cm
Planewave Mic theory 915 and 2450 MHz dd Used to calculate heating patterns in mammalian brains. Selective absorption also indicated.
Human head (7-cm radius) and infra-human E-polarized planewave Tensor integral equation method 918 mHz for infrahuman head; 2450 MHz for human head ee Comparison made between heating of spherical and realistic models of humans and infrahuman heads. Lower EM heating induced in brain of realistic model than spherical model. Skull's bony structure tends to attenuate heating of the brain, including the eyes.
Multilayered spherical model Primate cranial structure of rhesus macaque monkey Planewave Mie theory Most of the results, 3 GHz ff results given of average SAR and mean square electric field in equatorial plane
Human skull, r = 7-10 cm Planewave Mie theory 0.1-3 GHz gg Additional SAR peak recognized at about 2.1 GHz. This is due to the y/r impedance matching effect which was also recognized in planar model (Schwan, 1968;;).
Human and animal heads, r = 3.3-10 cm Planewave Mie theory 0.1-10 GHz hh Energy distribution examined in three spheres--3.3-, 6-, and 10-cm radii--with emphasis on strong localized heating.
Outer radius maximum value = 6.6 cm Planewave Mie theory Results presented, 433 < f < 6000 MHz ii Examined SAR distribution in different layers.
Six-layered cranial structure of maximum radius 3.3 cm a. Current loop p | z
b. Electric dipole m | z, both axially above the model
State-space formulation Rresults only at 3 GHz jj, kk Model allows for an idealized continuously inhomogeneous structure. Heat potential distributions are calculated
Prolate spheroid /td> a = 1 m and a/b = 7.73, volume = 0.07 m3 and a/b up to 10 Planewave magnetic, electric, and cross polarization Perturbation theory, based on expanding all fields in a power series of -jk Up to 30 MHz ll Used to calculate the first-order internal electric field and SAR
0.02 < a < 0.0875 m, 2 < a/b < 6.34 Planewave magnetic, electric, and cross polarization Perturbation analysis Up to 30 MHz for man model; up to 1 GHz for mouse model d Average SAR presented as function of angle of incidence.
3 < a < 10 cm, a/b = 3.6 Planewave cross polarization Using the vector spheroidal wave function expansion 100 and 2450 MHz mm, nn Results presented for 915 and 2450 MHz. In Lin and Wu (1977),, peak SAR was plotted for f = 10-3000 MHz for a = 3 cm and b = 2 cm.
Up to man-size model a = 0.875, a/b = 6.34 Planewave Extended boundary condition method At and slightly pastg resonance (-60 MHz for 2/3 muscle tissue of man-size model) oo, pp Good results up to resonance for average-man model. Frequency limited because of ill- conditioned matrix. For lower dielectric constants, the method can be used up to higher frequency limits.
Man and animal sizes Planewave Geometrical optics 20-100 GHz for man model, f > 80 GHz for rat model i, qq Based on dividing surface of prolate spheroid into small planar subareas, all power transmitted into the spheroid is assumed to be absorbed and secondary internal reflections are neglected. Lower frequency limit is based on convergence within 20% of Mie solution for sphere with a radius = b of the spheroid.
a/b = 6.34, a = 0.875 m, weight = 70 kg E-polarized planewave Empirical curve-fittingprocedure 10 Mhz-10GHz rr Provides simple empirical formula for calculating average SAR over broad-frequency band. Formula is only for E-polarized incident planewaves.
Planewave E, H, and K polarization Geometrical optics approximation 6 GHz and beyond ss Method valid only in the frequency range where the body dimensions >>
1.25 < a/b < 1.5, 0.297 < a < 0.335 m, volume = 0.07 m3 Planewave E, H, and K polarization Point-matching technique Up to resonance of 130 MHz tt Absorbed power density plotted for spherical-like model. Solution does not converge for a/b > 1.5.
a/b = 6.34, a = 0.875 m, volume = 0.7 m3 Near field of electric dipole, E and K polarization Extended boundary condition method Results at 27 MHz uu For E polarization, average SAR oscillatges around its planewave value. For K polarization, SRA distribution suggests possible enhancement at regions of small radius of curvature.
a/b = 6.34, a = 0.875 m, weight = 70 kg Electric dipole located paralle to major axis of spheroid Long-wavelength approximation method Results at 27 MHz vv Analysis useful where long-wavelength approximation is valid but wave impedances are not 377 ?, and for near-field irradiation in which incident fields are quasi-uniform.
Near field of a short electric dipole Long-wavelength analysis 27.12 MHz ww Average SARs in a prolate spheroidal model of man are essentially the same as those for a block model of man at 27.12 MHz, even in near fields. For purposes of average SAR, this allows use of the simpler and less expensive prolate spheroidal calculations.
Near fields of aperture sources Extended boundary condition method 27 MHz xx, yy Average SAR and the SAR distribution due to near fields of large and small aperture sources are given. Calculated results conform to the understanding previously obtained from studying irradiation of the spheroidal models by EM planewave and by n ear fields of various elementary radiation sources.
Spherically capped cylinder The average man model and the small rat model EM planewave Surface integral equation 80 MHz to 2.45 GHz zz Average SAR curves for E, H, and K polarizations intersect at a frequency just above resonance, about 800 MHz for man models. This may be useful in cases where the average SAR must be independent of animal position.
Prolate spheroid · a/b = 6.34, a = 0.875 m, weight = 70 kg
· a/b = 3.1, a = 20 cm, weight = 3.5 kg
· a/b = 3, a = 3.5 cm, weight = 20 g
Small coaxial loop antenna Extended boundary condition method 10-600 MHz ab SAR distribution and average SAR are plotted as a function of separation distance from the loop. For distances less than 5 , average SAR values oscillate about the far-field value.
Prolate spheroid a/b = 6.34, a = 0.875 m, weight = 70 kg Planewave E and K polarizations Iterative extended boundary condition method 27-300 MHz ad, ae An iterative procedure for improving stability and extending frequency range of the extended boundary condition method (EBCM). Calculated data for SAR distribution and average SARs in the resonance and postresonance frequency range are presented.
Axisymmetric cranial structure height = 22.6 cm, volume = 4189 cm3 EM planewave Finite element method 1 and 3 GHz af Model composed of upper concentric spheres and lower concentric spheroids. Curves for SAR distribution in brain region are presented for detached model of the human cranial structure.
Ellipsoidal model Man model a = 0.875 m, volume = 0.07 m3, and b/c = 2 Planewave Pertgurbation technique 1-30 MHz for man model ag First-order analysis valid for long-wave-length a/ < 0.1. Curves of SAR vs. frequency show SAR to be strong function of size and orientation of the ellipsoid in the incident field. Strongest absorption was found when electric-field vector of the incident planewave was along the longest dimension of the ellipsoid.
Man and animal model 0.05 < 2a < 1.8, 1.7 < a/b < 4.5, and 1.3 < b/c < 2 Planewave Perturbation technique Up to 30 MHz for man model and to 1 GHz for the mouse ah Data used to extrapolate results of observed irradiation effects in animals to those expected to be observed in humans.
Ellipsoid Model of breast carcinoma embedded in nonabsorbing dielectric Planewave Boundary value solution in spheroidal coordinates Results at 2450 MHz, 5.8 GHz, 10 GHz ai Three-dimensional and densitographic pictures of electromagnetic-field distribution with locations of hot spots shown.
Body-of-revolution model Sphere resting on base of conical body; total height = 1.78 m Vertically and horizontally polarized planewave Surface integral equation method Results at 30, 80, and 300 MHz aj Strongest power deposition is for field polarized along longest dimension and for frequencies near the first resonance (i.e., 80 MHz); hot spots predicted in neck region.
Block model of man Height = 1.7 m, 120 cells; cell size was kept smaller than o/4 Planewave Tensor integral equation method Up to 500 MHz ak, al Integral equation solved by dividing the body into N cells, assuming a constant field inside each cell, and solving for the 3N unknowns using point matching. Also, hot spots are illustrated.
Height = 1.7 m, 180 cells; cell size < 10 cm Planewave Moment-method solution of electric-field integral equation Up to 200 MHz am Chen and Guru's work (1977) extended by
a. Using interpolant between field values at cell centers before carrying out the volume integral.
b. Choosing cell sizes and locations for realistic model of man.
Block model Height = 1.75 m, weight = 70 kg E-polarized planewave Image theory and moment method Less than 100 MHz an Experimental data support numerical results. Resonant frequency shifts from 77 MHz in free space to 47 MHz when standing on a ground plane. An order-of-magnitude enhancement in SAR values is predicted at frequencies below 30 MHz.
Moment method 10-600 MHz am Numerical calculations of absorbed energy deposition made for human model constructed with careful attention to both biometric and anatomical diagrams.
Planewave, vertical, and horizontal polarization Tensor integral equation method Up to 500 MHz ao Results for average SAR are compared with existing experimental results. Resonance and the effect of body heterogeneity on the induced field are studied.
Inhomogeneous block model of man Height=- 1.75 m, weight = 70 kg Planewave Moment method with pulse basis function 27.12 MHz and 77 MHz ap Whole-body and part-body average SAR for man in free space and under grounded conditions are given as function of angle of incident. In general for frequencies considered, average SAR varies smoothly with angle between the extrema.
Block model of man Height = 1.75 m, weight = 70 kg Near-field exposure An empirical relationship Less than 350 MHz aq Empirical formula for average SAR in man under a two-dimensional near-field exposure. Average SAR is lower for n ear-field exposure than for planewave irradiation conditions.
Block model and cylinddrical model of man Height = 1.75 m, weight = 70 kg Near field of resonant thin-wire antenna Moment method and finite element method 45 MHz, 80 MHz, and 200 MHz ar, as Temperature distribution in cylindrical model of man is calculated by a finite element solution of the transient heat conduction equation in which the internal heat generation is due to metabolism and absorption of EM energy. At least 50 W incident power is required before the body experiences any significant thermal effect from the near-zone antenna fields.
Block model Height = 1.68 m, max diameter = 0.36 m; height = 2.22 cm, max diameter = 3.8 m Uniform RF magnetic field Solution of vector potential by moment method 10-750 MHz at Electric fields induced by RF magnetic field inside a sphere, finite circular cylinder, and phantom models of humans are calculated. Calculated data are verified by experimental values and existing theoretical results.
Height = 1.7 m, weight = 68 kg Near field of a dipole antenna Moment method with pulse basis function 27, 80, and 90 MHz au Average SAR in the body as a function of antenna-body spacing is calculated at 27 MHz. Calculated SAR-distribution data agree qualitatively with the experiment values.
Inhomogeneous block model Height = 1.75 m, weight = 70 kg Near field of an RF sealer Moment method with pulse basis function Less than 350 MHz av Planewave spectrum approach used to calculate average SAR and SAR distribution in an inhomogeneous block model of man for a prescribed two-dimensional leakage electric field. Average SAR under near-field conditions is always less than or equal to the far-field planewave value.
Inhomogeneous block model of man Height = 1.75 m, weight = 70 kg Planewave E polarization Moment method with pulse basis functions 27.12 MHz aw Average SAR and SAR distributions are obtained for man models with 180-1132 cells by the moment method with pulse basis function. Calculated values of average SAR increase with the number of cells used.

a. Schwan, 1968 aa. Johnson and Guy, 1972 ab. Lakhtakia et al., 1982b
b. Guy, 1971b bb. Lin et al., 1973b ac. Lakhtakia et al., 1981
c. Guy and Lehmann, 1966 cc. Kritikos and Schwan, 1975 ad. Iskander et al., 1983
d. Johnson et al., 1975 dd. Lin et al., 1973a ae. Iskander et al., 1982b
e. Chatterjee, 1979 ee. Rukspollmuang and Chen, 1979 af. Morgan, 1981
f. Barber et al., 1979 ff. Shapiro et al., 1971 ag. Massoudi et al., 1977a
g. Chatterjee et al., 1980a gg. Joines and Spiegel, 1974 ah. Massoudi et al., 1977b
h. Chatterjee et al., 1980b hh. Weil, 1975 ai. Zimmer and Gros, 1979
i. Durney et al., 1976 ii. Neuder et al., 1976 aj. Wu, 1979
j. Massoudi et al, 1979a jj. Hizal and Tosun, 1973 ak. Chen and Guru, 1977b
k. Yoneyama et al., 1979 kk. Hizal and Baykal, 1978 al. Chen et al., 1976
l. Ruppin, 1979 ll. Durney et al., 1975 am. Hagmann et al., 1979a
m. Ho, 1975a mm. Lin and Wu, 1977 an. Hagmann and Gandhi, 1979
n. Ho et al., 1969 nn. Wu and Lin, 1977 ao. Chen and Guru, 1977c
o. Ho et al., 1971 oo. Barber, 1977a ap. Hagmann et al., 1981
p. Wu and Tsai, 1977 pp. Barber, 1977b aq. Chatterjee et al., 1982a
q. Neuder and Meijer, 1076 qq. Rowlandson and Barber, 1977 ar. Spiegel, 1982
r. Massoudi et al., 1979b rr. Durney et al., 1979 as. Spiegel et al., 1980
s. Kastner and Mittra, 1983 ss. Rowlandson and Barber, 1979 at. Lee and Chen, 1982
t. Morita and Andersen, 1982 tt. Ruppin, 1978 au. Karimullah et al., 1980
u. Hill et al., 1983 uu. Iskander et al., 1980 av. Chatterjee et al., 1980c
v. Iskander et al., 1982a vv. Massoudi et al., 1980 aq. Deford et al., 1983