Range 

A. Onedimensional models 

Infinite plane tissue layers (isotropic)  Multilayers  Planewave  Transmissionline 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  4332450 MHz  b, c  Used to calculate relative heating patterns.  
(Anisotropic)  Multilayers  Planewave  Transmissionline model (together with Thevenin's theorem)  0.001100 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). 
Semiinfinite homogeneous slab  One layer  Nonperiodic electric field with TE and TM components  Planewave spectrum analysis  All calculations at 2450 MHz 
e  Nearfield analysis of aperture sources. 
Six layers  Epolarized planewave  Optical refractionSnell's law  Up to 3.0 GHz  f  Layering increases average SAR; an enhancement factor needed to correct results from homogeneous models.  
Homogeneous semiinfinite slab and multilayered slab    Near field of a source leaking radiation  Planewavespectrum approach  2450 MHz  g, h  For fields nearly constant over at least a freespace 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. Twodimensional models  
Infinite circular cylinder  Length l= 1.75 m, radius a = 11.3 cm; l = 14 cm, a = 1.58 cm  Planewave  Modematching technique and geometrical optics 
3006000 MHz  i, j  In good agreement with geometrical optics at the highfrequency limit. 
Radius such that (formula) k_{o}a < 2.4  Coaxial loop antenna  Longwavelength approximation method  Depends on (formula) k_{o}a < 2.4  k  Formulation simple to use, but valid results are in lowfrequency range. Power absorption coefficient is plotted against loop dimensions and cylinder radius.  
5 and 10 cmradius cylinders in presence of a reflector  Planewave TE and TH with k vector normal to plane of a reflector  Pointmatching method combined with imaging  Up to 1 GHz  l  SAR calculated as function of frequency and distance from reflector.  
Radius: 2.2424 cm 
Planewave E and H polarization  a. Geometrical optics approximation b. Modematching technique 
200100 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  Modematching technique and moment method  4332450 MHz  m, n  Microwave heating calculated in simulated human thigh. 
Three layers  Direct con tact aperture source (TE_{10} mode) 
Field expansion in terms of threedimensional cylindrical waves and matching the boundary conditions  4332450 MHz  o  Microwave heating calculated in phantom models of human limbs exposed to a directcontact aperture source.  
Multilayered cylinders of arbitrary cross section  a = 1.5 b, a = 0.239 m, b = 0.159 m for ellipticalcal cylinder  Planewave  Surface integral equations derived via vector Green's theorem and boundary condition  All calculations at 300 MHz  p  Calculations made for oneand twolayered circular and elliptical cylinders. 
17.5 x 19 cm triplelayered 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 MHz10 GHz  r  Absorption effects due to clothing are negligible below 2 GHz. Layering changes average SAR values in the 0.4 to 8GHz frequency range.  
Homogeneous infinite cylinder  62.6 x 62.6 cm  Planewave E polarization  Stacked twodimensional 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 convolutiontype 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  Modematching technique  27, 100, and 300 MHz  t  With proper choice of geometry and polarization 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 solenoidal field  Galerkin's method with linear basis function  Lowfrequency 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 twodimensional models of cross sections of the human body. Numerical results agree with measured values in central region of the cross section.  
C. Threedimensional models 

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/r_{o} = 12.68  Epolarized planewave  a. Antenna theory b. Curve fitting used along with circuit theory 
160 MHz  x  For human model in direct contact with a ground plane, an orderofmagnitude 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 freespace absorption characteristics. 
Spherical model  Radius of the sphere r = 10 cm  Planewave  Mie theory  1010,000 Mhz  y, z  Examined distribution of generated heat. 
r = 25.57 cm  Planewave  Mie theory  120 MHz  aa,bb  Power deposition in a spherical model of man70 kg.  
r = 5 cm  Planewave  Mie theory  1012,000 MHz  cc  Distribution of heating potential.  
4r = 10 cm  Planewave  Mie theory  10010,000 MHz  Type of nonuniformity described in a radiusfrequency 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 experimental 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 (7cm radius) and infrahuman  Epolarized 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 = 710 cm  Planewave  Mie theory  0.13 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 = 3.310 cm  Planewave  Mie theory  0.110 GHz  hh  Energy distribution examined in three spheres3.3, 6, and 10cm radiiwith 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.  
Sixlayered cranial structure of maximum radius 3.3 cm  a. Current loop p  z b. Electric dipole m  z, both axially above the model 
Statespace 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 m^{3} 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 firstorder 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 = 103000 MHz for a = 3 cm and b = 2 cm.  
Up to mansize 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 mansize model)  oo, pp  Good results up to resonance for averageman 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  20100 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  Epolarized planewave  Empirical curvefittingprocedure  10 Mhz10GHz  rr  Provides simple empirical formula for calculating average SAR over broadfrequency band. Formula is only for Epolarized 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 m^{3}  Planewave E, H, and K polarization  Pointmatching technique  Up to resonance of 130 MHz  tt  Absorbed power density plotted for sphericallike model. Solution does not converge for a/b > 1.5.  
a/b = 6.34, a = 0.875 m, volume = 0.7 m^{3}  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  Longwavelength approximation method  Results at 27 MHz  vv  Analysis useful where longwavelength approximation is valid but wave impedances are not 377 ?, and for nearfield irradiation in which incident fields are quasiuniform.  
Near field of a short electric dipole  Longwavelength 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  10600 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 farfield value. 
Prolate spheroid  a/b = 6.34, a = 0.875 m, weight = 70 kg  Planewave E and K polarizations  Iterative extended boundary condition method  27300 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 cm^{3}  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 m^{3}, and b/c = 2  Planewave  Pertgurbation technique  130 MHz for man model  ag  Firstorder analysis valid for longwavelength 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 electricfield 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  Threedimensional and densitographic pictures of electromagneticfield distribution with locations of hot spots shown. 
Bodyofrevolution 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  Momentmethod solution of electricfield 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  Epolarized 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 orderofmagnitude enhancement in SAR values is predicted at frequencies below 30 MHz. 
Moment method  10600 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  Wholebody and partbody 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  Nearfield exposure  An empirical relationship  Less than 350 MHz  aq  Empirical formula for average SAR in man under a twodimensional nearfield exposure. Average SAR is lower for n earfield exposure than for planewave irradiation conditions. 
Block model and cylinddrical model of man  Height = 1.75 m, weight = 70 kg  Near field of resonant thinwire 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 nearzone 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  10750 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 antennabody spacing is calculated at 27 MHz. Calculated SARdistribution 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 twodimensional leakage electric field. Average SAR under nearfield conditions is always less than or equal to the farfield 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 1801132 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 