Journal of Microwave Power ISSN: 0022-2739 (Print) (Online) Journal homepage: http://www.tandfonline.com/loi/tpee19 Application of a Finite-Difference Technique to the Human Radiofrequency Dosimetry Problem R.J. Spiegel, M.B.E. Fatmi & K.S. Kunz To cite this article: R.J. Spiegel, M.B.E. Fatmi & K.S. Kunz (1985) Application of a FiniteDifference Technique to the Human Radiofrequency Dosimetry Problem, Journal of Microwave Power, 20:4, 241-254, DOI: 10.1080/16070658.1985.11720308 To link to this article: http://dx.doi.org/10.1080/16070658.1985.11720308 Published online: 17 Jun 2016. Submit your article to this journal View related articles Citing articles: 2 View citing articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tpee19 Download by: [UNSW Library] Date: 26 October 2017, At: 10:39 Application of a Finite-Difference Technique to the Human Radiofrequency Dosimetry Problem Downloaded by [UNSW Library] at 10:39 26 October 2017 R.J. Spiegel, M.B.E. Fatmi and K.S. Kunz A powerful finite-difference numerical technique has been applied to the human radiofrequency dosimetry problem. The method possesses inherent advantages over the method-of-moments approach in that its implementation requires much less computer memory. Consequently, it has the capability to calculate specific absorption rates (SARs) at higher frequencies and provides greater spatial resolution. The method is illustrated by the calculation of the time-domain and frequency-domain SAR responses at selected locations in the chest. The model for the human body is comprised of rectangular cells with dimensions of 4x4x6 em and dielectric properties that simulate average tissue (2/3 muscle). Additionally, the upper torso (chest) is configured by both homogeneous and inhomogeneous models in which this region is subdivided into 20,736 cells with dimensions of 1x 1 x 1 em. The homogeneous model of the chest consists of cells with average tissue properties, and the calculated results are compared with measurements acquired from a homogeneous phantom model when the exposure frequency is 350 MHz. For the inhomogeneous chest model the lungs and surrounding region (ribs, spine, sternum, fat, and muscle) are modeled with as much spatial resolution as allowed by the 1 x 1 xl cm cells. Computed results from the inhomogeneous chest model are compared with the homogeneous model. ABOUT THE AUTHORS R.J. Spiegel is with the U.S. Environmental Protection Agency, MD-74C, Research Triangle Park, NC 27711; M.B.E. Fatmi is with Northrop Services Inc., Research Triangle Park, NC 27711 and K.S. Kunz is with the Lawrence Livermore Laboratory, Uvermore, CA 94550. Address al/ correspondence to R.J. Spiegel at the above address. Disclaimer: The research described in this article has been reviewed by the Health Effects Research Laboratory, US Environmental Protection Agency, and approved for publication. Approval does not signify that the contents necessarily reflect the view and policies of the Agency nor does mention of trade names or commercial products constitute endorsement or recommendation for use. Copyright Е1985 by IMPI, Victoria, Canada: Manuscript received September 24, 1985; in revised form December 3, 1985. Dosimetry-Finite Difference Techniques his paper describes a three-dimensional T (3-D) finite-difference (FD) technique for calculating the specific absorption rate (SAR) distribution in a human model exposed to radiofrequency (RF) radiation. The numerical method is based on a FD solution of Maxwell's curl equations which were recast into a scattered-field formulation [1]. The method is very general and can be used to predict instantaneous, as well as continuous wave (CW), SARs for complex, inhomogeneous models located over realistic surfaces (grounds). In addition, it can be used to investigate the interaction with many different electromagnetic (EM) source distributions, including both plane-\yave (far-field) and near-field exposure conditions. It is possible to obtain spectral information about the body's SAR over a wide frequency range. The FD approach possesses an important inherent advantage over the more extensively utilized method-of-moments (MOM); namely, it requires much less computer memory. The MOM [2, 3] is based on formulating Maxwell's equations into an integral equation(s). This integral equation(s) is then solved by approximation in which it is replaced by a linear system of equations, and the entire set of equations must be stored in the computer memory. This requires large amounts of memory because an object divided into N cells or blocks must be represented on the computer by an array of 9N2 elements (the factor of 9 is a result of the vector nature of the internal fields). In addition, the solution of the large system of equations is slow since the number of computer operations is proportional to the square of the number of elements in the matrix. With large mainframe computers, this means the MOM can handle models comprised of only a few hundred blocks (if symmetry conditions are applicable, then models comprised of approximately 1100 cells are possible [4]), whereas the FD method can analyze models that are divided into many thousands of blocks. Con- J. MICROWAVE POWER 1985 241 Downloaded by [UNSW Library] at 10:39 26 October 2017 sequently, FD techniques allow more realistic models to be constructed, which can be analyzed over higher frequency ranges. One disadvantage of the FD method, however, is that its implementation necessitates modeling a certain region of the space surrounding the object, as well as the object itself. Thus, the computer input demands may be more troublesome than the MOM, where only the object need be considered. A previous study [1] has investigated the accuracy of this FD technique as it applies to elementary shapes such as dielectric spheres. In this paper, calculations are provided to determine the capabilities and limitations of the technique as it applies to the human dosimetry problem. The human model consists of several thousand rectangular blocks that are arranged to fit the body shape, and the SAR is calculated for each block. Results obtained for a homogeneous model comprised of average tissue characteristics are compared with measurements at 350 MHz. Additionally, results are presented for an inhomogeneous model in which the chest (lungs, ribs, spine, and sternum) are modeled in greater detail using 20,736 cells with dimensions of 1x1x1 cm. FINITE-DIFFERENCE METHOD This numerical method is based on a FD solution of Maxwell's curl equations in the timedomain. The EM fields are partitioned into incident and scattered fields, and the source of EM energy is turned on at t = O. Hence, it is called a time-domain, scattered-field formulation. In as much as the technique can be mathematically involved, only a rather brief discourse is presented here; more detailed information can be found elsewhere [5]. PartitiE!lil$ the total fields @, ij) into incident (E I , H) and scattered (E S , H S ) fields, allows Maxwell's curl equations to be written as [1] aAs = at -f.Lo - oE S ~ e - ' + aE s at --+ V x E = where E = E i + E S; 242 J. S (1) ~~. aE Vx H S I - - (e - eo) aE i at (2) A = Ai + As; e and a are MICROWAVE POWER 1985 the dielectric constant and conductivity of the object, respectively; and eo = 8.85 X 10- 12 F/m. Note that the total field is a sum of the incident and scattered fields. Also observe that the terms aE i and (e - eo) aEi/at in Equation (2) represent the source or driving terms and they are distributed over the object's (scatterer's) volume. The propagation, scattering, and absorption of the fields emitted by an EM source are simulated on a grid of cells by solving Equations (1) and (2) in which spatial and time derivatives are differenced linearly and exponentially [1], respectively. The positions of the vector components are positioned about a cell according to the so-called Yee grid [6]. Using this approach, and expressing Equations (1) and (2) in scalar form, yields six equations which are solved simultaneously for the six field components (E~, E;, E~, H~, H;, and H~) in which the spatial evaluation points for E and As occur alternately along each axis of the cell. The time evaluation also occurs at alternate half-time steps. For example, with At the cycle time and n the it~ation number, H is evaluated at ndt, and ES is computed at (n -~) dt. With this scheme, the new value of any component of the field at each grid point only depends on its previous value (in time) and the previous values of the adjacent components of the other field. Consequently, a solution is obtained by simultaneously solving the six scalar equations by time-stepping through the entire grid. The space in which the field must be computed is, in theory, unbounded. In reality, however, this is impossible because a computer can store and compute only a finite amount of data. To circumvent this problem, it is necessary to surround the object by a volume large enough to contain the object and impose some conditions which must be satisfied at the boundary walls. The idea is to create the numerical illusion of an infinite space. One approach is to impose a radiation boundary condition [5] on the exterior surfaces of the boundary; that is, on the boundary require the scattered field to behave as an outpropagating spherical wave of the form E = Б(6, <1╗ g(t - r/c)/r (3) where r is the measure of distance and c is the S S S Dosimetry-Finite Difference Techniques Downloaded by [UNSW Library] at 10:39 26 October 2017 speed of light. Because the field radiated (scattered) by the object must behave as a far-zone field for this condition to be valid, the boundary volume must be significantly larger than the object to avoid excessive nonphysical reflections off the outer boundary. It should be pointed out that the ability to impose a simple radiation boundary condition in this form is an inherent advantage of the scattered-field formulation. Other FD versions [7,8] based on a total-field formulation require a much more contrived approach to reduce reflections off the outer boundary. Presently, the memory limitations of midsize computers restrict problem spaces (object plus surrounding volume) to typically around 25,000 or 30,000 cells, with an upper limit of 100,000 cells if more sophisticated programming is applied. Even with these upper limits, spatial resolution available for modeling the interior detail of a biological body is restricted. This restraint can be circumvented by the application of a so-called expansion technique [9]. Basically, the expansion approach uses an initial computer run with a fairly coarse division of the problem space with the computed data being stored on disk. A certain portion of the body is then subdivided into a much finer division of cells and is called a subvolume. This becomes the problem space for a second computer run. Now only this subvolume is treated by imposing the same incident field conditions as the first run. In addition, interpolated tangential electric fields generated by the first run are imposed on the outer boundary of the subvolume. These tangential fields numerically mimic the response of the rest of the body, thereby ensuring that the subvolume response behaves as if the remaining portions of the body are still present. Obviously this process could be carried out several times. For example, the whole body of a human could be considered for the first run in which the head is coarsely modeled. The second run would replace the crude approximation of the head with a much finer rendition of the head including detail such as the nose, eyes, brain, and skull. A third run might even be employed that focused on the intricate detail for the eyes. Although this FD technique yields a transient solution to Maxwell's equations, it is also Dosimetry-Finite Difference Techniques possible to obtain CW steady-state solutions. This can be accomplished by merely letting the time-domain solution run for a period of time long enough to achieve steady-state conditions for a CW source turned on at t = O. On the other hand, since a pulsed waveform contains a spectrum of frequencies, it is possible to obtain spectral information about the body's SAR over a large frequency range with only one execution of the computer program. This is achieved by the application of the Fourier transform in which the time-domain response is transformed to the frequency domain according to 00 ~ J f (t)eio>tdt. (4) 21T Because Equation (4) cannot, in general, be analyzed analytically, it is necessary to numerically evaluate the integral by employing standard fast Fourier transform (FFT) routines. These procedures, however, are well documented, and FFT subroutines are available for most large computer systems. To determine the body's true CW response from transformed data requires careful consideration. From linear system theory, it is well known that the output of a system o(t) is related to the input i(t) by the system impulse response h(t) via F (w) = -00 00 o(t) = J i(t - T) h(T)dT (5) with the Fourier transform given by O(w) = I(w) "(w). (6) If I(w) represents the incident field and O(w) the observed response at a point in the body, then it is clear from Equation (6) that "(w) is the desired CW response because it contains no spectral components of the source. In theory, "(w) could be determined by simply using a unit impulse for the incident field. However, this type of waveform poses severe problems when evaluating the FFT of Equation (4) since "(w) does not, in general, go to zero as w approaches infinity. Consequently, the numerical solution would probably not converge. This problem can be alleviated by using an incident field that contains a finite spectral content, such as a damped sine waveform. Then I(w) will approach zero with in- J. MICROWAVE POWER 1985 243 creasing w, and the FFT technique will yield accurate information. Of course, Equation (6) must be solved for H(w) after the FFT has been applied. An upper frequency limitation of FO procedures can be based on the following physical argument. It is assumed that the waveform is spatially sampled no more coarsely than 'A/4 in a time step. This means, that at the highest frequency, the signal from one cell arrives at an adjoining cell one time step later shifted by no more than 'A/(4c) when the maximum dimension of the cell is traversed. Thus, the upper frequency, fmax , is given by Downloaded by [UNSW Library] at 10:39 26 October 2017 c f max = - 'A min c 4 x maximum cell dimension' (7) Note that because both c and 'A are reduced by the dielectric constant of the object, these effects cancel out in Equation (7) and the net result is both c and 'A can be considered as free-space quantities. NUMERICAL RESULTS A: Homogeneous Human Model To determine the capabilities and limitations of this FO technique as it applies to the human dosimetry problem, a coarse homogeneous model comprised of rectangular cells with dimensions of 4 x 4 x 6 em and average tissue properties (2/3 muscle tissue) was constructed. Figure 1 illustrates the model (both front and side views), where the (i, j, k) indices define the location of each spatial grid point (x, y, z). Note that the figure contains a volume which surrounds the object; the reasons for this volume were previously discussed (see previous Section). The entire problem space contains 20,736 cells and the body is comprised of 1172 cells. The cells within the broken line are of constant size (4x4x6 em). Those outside this boundary are allowed to increase in size by a factor of 1.2 and 1.3 in the y, z and x directions, respectively, for each cell away from the dotted line. By assigning appropriate permittivity values to the ground beneath the model's feet, it is possible to evaluate the effects of objects standing on the earth's surface. A perfectly conducting surface 244 J. MICROWAVE POWER 1985 can also be utilized, as well as completely eliminating it for free space calculations. Figure 1 also contains views of the subvolume in which the right side of the chest region is modeled using 20,736 cubical cells with dimensions of 1x 1x 1 em, and defined by the indices (i', j', k'). A side view of this subvolume is further delineated by Figure 2. The coarse (1172 cell) model is used to compute the scattered tangential electric fields on the subboundary. The incident electric field utilized for the first calculation, as well as the tangential electric fields obtained from the first calculation, are then imposed on this subvolume for a second calculation. The finer grid of the second calculation, and consequently finer time steps, imply that the values for both the spatial and time points for the tangential fields on the subboundary must be determined by interpolation. The incident electric field is a damped sine wave of the form E~ = Eosin(27l'at')e-at'U(t') (8) where t' = retarded time = t - (y - y')/c, y' = source point, y = observation point, c = speed of light, U(t') = unit step function, Eo = 1 V1m, a = oscillation frequency. Observe that in Equation (8), the field produced by the source at point y at time t is due to an earlier disturbance which originated at t' and point y'. The incident field is a plane-wave which is propagating in the y direction with its vector orientation in the x direction. This corresponds to the electric-field vector being parallel to the major length of the body and propagating from the front to the back of the object. This field is turned on at t = 0, is oscillating at a frequency designated by the quantity a. Figure 3 shows the Fourier transform of the waveform, along with an insert which contains a plot of Equation (8), when a = 500 MHz. Note the wide frequency content of the waveform. The instantaneous SARin the body is calculated for each cell according to SAR(t) = ~ I Ei(t) + ES(t) 12 (9) P where p is the density of the tissue associated with the cell. Frequency-domain information is determined by applying the FFT to each vec- Dosimetry-Finite Difference Techniques tor component of Ei and E separately. Then the SAR(w) is formed from squaring the magnitude of the resultant total field in conjunction with Equation (6). Figure 4 shows the calculated instantaneous SARs at a central point in the chest (i = 12, j = 12, k = 15) for both the coarse (run 1) and the refined (run 2) homogeneous (E. = 38 and (J = 0.95 5/m) models located in free space. The'damped sine wave incident field is oscillating at 100 MHz. To satisfy the Courant stability criterion [8], the time increment, dt, was 0.085 and 0.017 ns for run 1 and run 2, respectively. There are several important points which should be gleaned from the figure. First, observe that for times prior to approximately 4 ns, the run 2 waveform is essentially equal to zero. On the other hand, the run 1 curve shows field buildup in this time interval. This clearly illustrates that the results of run 2 are superior because the field should be identically equal to zero. This is a result of the fact that a finite amount of time is required for the incident wave to propagate from the outer boundary, where it is launched at t = 0, to the observation point. Obviously, no field should exist at a point in the body until the incident field reaches that point. After 4 ns, they are closer in shape, but the peak values of the curves are different and there is a slight time shift of one curve relative to the other. Note also that the internal field oscillates only for a few cycles and then is rapidly damped out by the lossy tissue. The reasons for the larger errors for the run 1 case are more apparent after a consideration of the data contained in Figure 5, in which the individual waveforms (E~, E~, and Ex) are plotted. The approximate 2 ns lag time, before the incident and scattered fields begin to buildup, represents the time it takes for the incident field to propagate from the outer boundary to the observation point (i = 12, j = 12, k = IS), when the medium is freespace. It is evident that the total field, Ex, is formed from a superposition of E~ and E~ which involves taking the difference of two very nearly equal waveforms. This is further evidenced by the fact that the peak values for Ex are approximately one order of magnitude smaller than those for either E~ or E~. Thus, E~ must be computed with extreme accuracy in Downloaded by [UNSW Library] at 10:39 26 October 2017 S Dosimetry-Finite Difference Techniques order that Ex be reasonably accurate. Obviously, for t < 4 ns the total field was not computed accurately enough because the magnitude of the error is, in fact, almost equal to the signal which occurs after t = 4 ns. The above errors increase for both run 1 and run 2, assuming constant cell sizes for both cases, as the frequency of the damped sine wave increases. Therefore, in light of these difficulties, the data presented hereafter have been "time filtered". That is, for times when the waveform should be zero, as based on propagation times, the total f~ld isjorced to be identically zero by setting E = - E'. This procedure aids in reducing cumulative error buildup as time advances, because initial values are forced to be correct. Figure 6 shows normalized SARs (W/kg per mW/cm 2 ) versus frequency when an FFT routine is applied to the time-domain responses (run 1 and run 2). The damped sine wave incident field is oscillating at 500 MHz. To apply the FFT routine, the time-domain waveform for run 1 was digitized at 0.085 ns intervals by sampling the signal 4096(2 12 ) times over a time interval of 348 ns. The run 2 waveform was sampled 4096 times at 0.017 ns intervals for a total duration period 69.6 ns. For both waveforms, after the first 20.4 ns, zero values were assigned to the sampled waveform because the damped sine wave responses exhibit very small values for times greater than 20.4 ns. The FD calculation of the waveforms for all 4096 points would be prohibitive in terms of computer time. The frequency responses were plotted only up to the limits dictated by Equation (7). However, the frequency points (0.5/sampling interval) at which aliasing occur are well above those limits. The lowerfrequency limits are determined by the duration of the sampled waveform; these values are 1/(348 ns) = 2.87 MHz and 1/(69.6 ns) = 14.4 MHz for run 1 and run 2, respectively. As observed from Figure 6 there is reasonable agreement in the two curves; the large dip in the run 1 curve at 102 MHz is most likely a numerical artifact. The rippling in spectral curves is a result of truncation of the waveforms in the time domain. Finally it should be pointed out that the flattening of the curves for frequencies below approximately 60 or 70 MHz is a result of a constant value (E. = 38) S J. MICROWAVE POWER 1985 245 K 1 25 REGION WHERE LOSSY DIELECTRIC CHARACTERIZED BY u AND [ ,30X16X16 CELLS r-- I I I 5 I I 10 Downloaded by [UNSW Library] at 10:39 26 October 2017 -, - I I OUTER BOUNDARY I 36x24x24 CELLS 1 -t ~10 I -t ~20 ~ IREGION TO BE EXPANDED I 36x24x24 CELLS PROBLEM SPACE r"""'l_.......... -I-'~',;....--.... I SUB-VOLUME ~ 25 ~~ L. -4 -J _ 20 10 L. " j:: 1 K' 25 30 Figure 1, Human model in a 20,736 cell problem space with the subvolume indicated. The location of the mesh points are given by the indices (i, j, k) and (i', j', k') for the problem space and subvolume, respectively. (a) Front view 35 246 J. MICROWAVE POWER 1985 Dosimetry-Finite Difference Techniques J 1 5 10 15 20 25 1r--r--r-,~--r-,-r-r--r-,-r-r-r-,-r-r--r-'-I'"""T-r-~-r-.., f REGION WHERE LOSSY DIELECTRIC CHARACTERIZED BY u AND 30x'6x" E CEllS ---, r- - I 5 I I r-r 20 ~ 30 ~ 1 10 Downloaded by [UNSW Library] at 10:39 26 October 2017 10 I' 15 37 I I I ~ PROBLEM SPACE OUTER BOUNDARY 36x24x24 CELLS j~ I SUB-VOLUME 1 ...l..J.~ 10 20 25 J' I REGION TO BE EXPANDED I 36x24x24 CELLS I 20 I I - I r- 25 r- I I I I I 30 I I L_ J (b) Side view, Dosimetry-Finite Difference Techniques J, MICROWAVE POWER 1985 247 .,, volume as delineated in Figure 2. Because the dielectric discontinuity (air/tissue interface) at the corners and edges of the surface cells strongly affects the numerical results, calculated results are presented only for interior grid points at least one cell length from the surface. Additionally, those interior grid points on the sub-volume border were not considered because their values are found by interpolation of run 1 values. Figure 7 contains the results of the comparison for a scan taken along the center of the chest, from front-toback. Very good agreement is obtained. Note the fairly rapid drop in SAR values as a function of penetration depth. Figure 8 shows plots for scans across the chest at different depths from the front surface; reasonable agreement is obtained. As might be expected, the comparison is better for the scans nearer the center of the chest. As mentioned earlier, the larger errors near the surface are a result of the corners and edges associated with the numerical model. r---------- j 1 1 I I I I I RUN 1 BOUNDARY~ 10 1 L __ 15 Downloaded by [UNSW Library] at 10:39 26 October 2017 I' 20 25 30 35 37, 1 I I I 10 15 J' 1 20 : 25 1 1 1 I I I I 1 I 1 I Figure 2. A side view enlargement of the subvolume showing the refinement of the exterior surface of the chest. utilized for the tissue relative dielectric constant. In reality, for frequencies below approximately 100 MHz, the average tissue dielectric constant increases as the frequency decreases [10]. To indicate the accuracy of this FD method, the calculated (run 2 case) SAR distributions in the upper torso (chest) are compared with measured local values [11] at 350 MHz. No attempt was made to configure the entire numerical model of the human to be identical with the experimental phantom model. However, the geometry of the chest region was modeled reasonably accurately by the sub- 248 J. MICROWAVE POWER 1985 B: Inhomogeneous Human Model To indicate further this technique's capabilities, the internal organs and bones contained within the subvolume were modeled with as much resolution as could be obtained from 20,736 cells with dimensions of 1 x 1 x 1 em. Figure 9 illustrates views of this subvolume showing how the lung, ribs, sternum, and spine were modeled. Dielectric properties for the various materials are contained in Table 1. As previously discussed, the coarse model (Figure 1) was used to compute the scattered tangential electric fields on the subvolume border. These tangential fields are then imposed on the subvolume as boundary conditions for run 2 calculations. Again, the incident field is a damped sine wave oscillating at 500 MHz. Figures 10 and 11 contain results at 350 MHz, which are also compared with data generated by the homogeneous chest model. From figure 10 it is noted that for the front-to-back scan in the center of the chest, the inhomogeneous model yields lower SAR values, especially near the front surface. This reduction near the front surface is most likely due to bone material (sternum) being present in the inhomogeneous model. However, the side-to-side Dosimetry-Finite Difference Techniques 14 Figure 3. Fourier transform (FFT) of a damped sine wave oscillating at 500 MHz. 1 .? 13 ~ '2 E ;.r " 10 8 9 -0.6 и1.0 :: ~ ии? :r 7 ~ ~ 6 w 5 ..? > 0 10 1 10 2 10 3 10 4 Downloaded by [UNSW Library] at 10:39 26 October 2017 FREQUENCY. MHz 1.2 1.0 Figure 4. Time-domain SAR responses at a central point in the chest. Observation points for run 1 and run 2 are (12, 13, 15) and (19, 13, 13) respectively. The incident field is a normalized (Eo = 1 Vim) damped sine wave oscillating at 100 MHz. ... ".. ~ ;: 0.8 0.6 ri Ф lI> 0.4 0.2 0.0 4 0 6 8 10 12 14 16 18 20 22 TIME. ns -3 '--_----'-_ _..L..._----L_ _..L-_---'_ _....L_---lI...-_-'-_ _.L-_----'-_ _..L..._--'---l_3 o '0 12 TIME. ns Dosimetry-Finite Difference Techniques '4 16 '6 20 22 24 Figure 5. Run 1 electric field waveforms for the incident (E~), scattered (E~), and total (ExY fields at location (12, 13, 15) versus time. The incident field is a normalized (Eo = 1V1m) damped sine wave oscillating at 100 MHz. J. MICROWAVE POWER 1985 249 10- 1 r--------~--------,_-------..., \ ~ \ \ \ RUN 2 \/ '\ : \ \ \ I \ N u "- ~ E :D D. ..."- Downloaded by [UNSW Library] at 10:39 26 October 2017 Cl 1'1 \1 \ 1\ ,-.. \I \ 1\ I \ I \ I \ V \ \ , E ~ I \ 1\ I \ I \ I \I \I \1 10-3 Figure 6. Frequency-domain SAR in the chest normalized with respect to incident power density. The observation points are (12, 13, 15) for run 1 and (19, 13, 13) for run 2. I \ ;-r' ~ , , \ \ \ I , RUN1 \ I Ii c( I I III I , II II V 10- 5 '-:10 1 --I..::-10 2 ..J- ...... 10 4 10 3 .1 FREQUENCY, MHz [jON' I N "- FRONT 3: E MEASUREMENT ~ "c. ..."-'" BACK (I' = 20. K' = 5) E u .05 RUN 2 3: rx' Ф til Figure 7. Comparison of the calculated and measured SAR distribution in the center of the chest (20, j', 5) as a function of penetration depth. The frequency is 350 MHz and the incident power density is 1 mW/cm 2 ? 250 J. MICROWAVE POWER 1985 BACK o L..L.-....L---L.--..L:s==2:::~:::I::=:I:::.J o .1 .2 .3 .4 .5 .6 .7 .8 .9 CHEST DEPTH FRACTION Dosimetry-Finite Difference Techniques 0.15 Table 1 Relative dielectric constant and conductivity of various tissues for 350 MHz A' \ \ - - MEASURED - - - - CALCULATED \ Relative dielectric constant \ Ae:r-- \ N FRONT \ \ \ \ 0.10 E B' _________ B C' - - - - - - - - - E Q; Q, l 1 Muscle Lung Fat Bone : I I a: et VI 0.05 ", \ -...--. A _--OCY-~ 0.00 '----'--_"----'---=-=:..:...J..._'------'--_'------'------' o .1 .2 38,0 35,0 4.6 8.0 Conductivity (<T,8/m) 0,95 0,73 0,06 0,05 BACK .~ '" "~ .>< Downloaded by [UNSW Library] at 10:39 26 October 2017 c _____1I \ \ \ \ "~" (e) ,3 ,4 .5 ,6 .7 ,8 .9 CHEST WIDTH, fraction Figure 8, Comparison of the calculated and measured SAR distribution across the chest at different distances from the front surface, The top, middle, and bottom scans are at locations (20, 21, k'), (20, 19, k') and (20, 16, k'), respectively, The frequency is 350 MHz and the incident power density is 1 mW/cm2 , scan through the lung, as shown in Figure 11, exhibits a considerable SAR increase in the lung as compared to the homogeneous case. This inhanced SAR in the lung may be of significant importance, especially for RF induced hyperthermia in cancer treatment. DISCUSSION AND CONCLUSIONS The capabilities and limitations of a FD technique, as it applies to the human dosimetry problem, have been studied. It was found that reasonable agreement could be obtained for calculated SARs in the chest, when the irradiation frequency is 350 MHz, with measured local SARs acquired from a phantom model of a human. The model tended to yield more ac- Dosimetry-Finite Difference Techniques curate results near the center. The larger errors at the surface are a result of the edges and corners of the cells producing unrealistically high local field strengths. Of course, these larger errors near the surface could be reduced by simply using smaller cells that are carefully arranged to minimize the effects of edges and corners. It was also found that a scattered field formulation might not be the best way to cope with this type of problem. This is a result of the total field being formed by taking the difference of two very--->nearly equal waveforms. Thus, even though E can be computed accurately by the numerical scheme, E tends to be considerably less accurate. A total field formulation of the problem, therefore, may be a more attractive alternate approach. However, a total field formulation will require a much more intricate approach [12] to deal with the artificial boundaries. Some of the problems in the implementation of the model reported here are a result of having to translate the parameters associated with the model into an acceptable format for the computer code. Currently, the input data have to be manually entered into the computer. The 3-D spatial quantification of all the cells associated with the model is very difficult when the data is generated by hand. Graphics systems are available for automating this process for more complicated models, but substantial software development 'may be required to tailor a graphics system for this task. Finally, some mention should be made relative to the computer memory and run time requirements for this code. Currently, an IBM 3081 mainframe computer is used for the calculations, and approximately 90 minutes are required to generate the time-domain data for the run 2 case. The run 1 situation takes S J. MICROWAVE POWER 1985 251 ! : "---- -1 \-----' : r--- - , , ' I 1 2 1i'2':"-ro-r,░-r-,-ro_'r'-,-r-r-,-',0+'.-r-r"".-r-r.,' I MUSCLE I AIB1 I I ~-- MUSCLE __ I RIB2 '0 MUSCLE RIB3 ,. MUSCLE RlB4 RIB5 MUSCLE 20 *,_J1~Ti--STEANUM MUSCLE FAT RIB6 2' AlB7 r-~;L-- RIB6 STERNUM ---t+-.,::::;~~------:", Downloaded by [UNSW Library] at 10:39 26 October 2017 30 MUSCLE AIBS , I FAT 3' LUNG I , ,' ,, , MUSCLE I RIB 9 :, ,, , I LUNG MUSCLE Museu;: , I .......-'-tL.J.-'-.J.tL....l.-'-""'" 37 ~+,--'-L.J.+,--'-L.J.y...-,-w. I SPINE ,, STERNUM AIR K' 25 15 10 25 ....-FAT ...-MUSCLE + - RlBS5T09 ...- RIB4 20 ....- MUSCLE ....- RIB3 +-- MUSCLE ....- RIB2 . - MUSCLE -+-- RIB 1 15 J' MUSCLE 10 ....- RIB1 5 ~RIB2 . . - RIBS 3 TO 9 ....- MUSCLE +-- FAT AIR LUNG SPINE Figure 9. Subvolume inhomogeneous model of the chest. (a) Frontal view (b) Side view (c) Top view. 252 J. MICROWAVE POWER 1985 Dosimetry-Finite Difference Techniques 0.06 s--' FRONT - FRONT HOMOGENEOUS 1._ - - - INHOMOGENEOUS 0' 0.05 err 0.04 E u "- 3: Q. Ol ~ I' ,I 0.03 oct 0.02 (1'=20, K'=5) N E u "- 3: E l; 0.06 Q. Ol FRONT\.."/ '""- 3: .... - I 0.04 I a:oct III BACK I 4 8 12 16 20 DEPTH, em 30 minutes. Since IT and E are nonuniform throughout the body, it is necessary to store eight quantities (including six fields) for each cell. Thus, for single precision calculations it takes approximately 320 K bytes to store an array associated with 20,000 cells. Naturally for double-precision calculations this value REFERENCES 1 Holland, R, Simpson, L., and Kunz, K.S. (1980). Finite-difference analysis of EMP coupling to lossy dielectric structures. IEEE Trans. EMC-22, (3): 203-209. 2 Chen, K.M. and Guru, B.S. (1977). Internal EM field and absorbed power density in human torsos induced by 1-500-MHz EM waves. IEEE Trans. MTT25: 746-755. 3 Hagmann, M.J., Gandhi, O.P., and Durney, C.H. (1979) Numerical calculation of electromagnetjc energy deposition for a realistic model of man. IEEE Trans. MTI-27, (9): 804-809. 4 DeFord, J.F., Gandhi, O.P., and Hagmann, M.J. (1983). Moment-method solutions and SAR calculations for inhomogeneous models of man with large Dosimetry-Finite Difference Techniques I \ I \ I ,, \ I I \ I I I I \ I I \ \ \ \ , _....... I I I 0 ,/'\ \ 0.00 0 Figure 10. Comparison of the homogeneous and inhomogeneous SAR distribution in the center of the chest (20, j', 5) as a function of penetration depth. The frequency is 350 MHz and the incident power density is 1 mW/cm2 ? I I I I II " 24 -I , '\/ 0.02 0.00 0 HOMOGENEOUS LUNG / I , I \ I I I I 0.08 I \ I \ 0.01 Downloaded by [UNSW Library] at 10:39 26 October 2017 II r, 0.10 ,,,\,, Ii - - - - INHOMOGENEOUS I '""-3: III BACK : BACK E l; (1'=20, J'=161 _____ ....J I N i --------- 0 4 8 12 16 20 24 SIDE-TO-SIDE SCAN, em Figure 11. Comparison of the homogeneous and inhomogeneous SAR distribution for a scan across the chest (20, 16, k') through the center of the lung, The frequency is 350 MHz and the incident power density is 1 mW/cm2 ? would be multiplied by a factor of two. Because the IBM 3081 computer has 14 M bytes of fast (electronic) memory available, models with well over 200,000 cells are possible, albeit run times will increase substantially. number of cells. IEEE Trans. MTT-31, (10): 848-851. 5 Holland, R (1976). THREE, a free-field EMP coupling and scattering code. Mission Research Corporation, Albuquerque, NM, AMRC-R-85. 6 Yee, K.S. (1966). Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans AP-14, (3): 302-307. 7 Taylor, C.D., Lam, D.H., and Shumpert, T.H. (1969). Electromagnetic pulse scattering in time-varying inhomogeneous media. IEEE Trans. AP-17, (5): 585-589. 8 Taflove, A. and Brodwin, M.E. (1976). Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations. IEEE Trans. MTT-23, (8): 623-630. 9 Kunz, K.S. and Simpson, L. (1981). A technique J. MICROWAVE POWER 1985 253 11 Kraszewski, A., Stuchly, M.A., Stuchly, 5.5., Hartsgrove, G., and Adamski, D. (1984). Specific absorption rate distribution in a full-scale model of man at 350 MHz. IEEE Trans. MTT-32, (8): 779-783. 12 Mur, G., (1981). Absorbing boundary conditions for the finite-difference approximation of the timedomain electromagnetic-field equations. IEEE Trans. EMC-23, (4): 377-382. Downloaded by [UNSW Library] at 10:39 26 October 2017 for increasing the resolution of finite-difference solutions of Maxwell equations. IEEE Trans. EMC-23, (4): 419-422. 10 Durney, e.H., Johnson, e.e., Barber, P.W., Massoudi, H., Iskander, M.F., Lords, J.L., Ryser, D.K., Allen, S.J., and Mitchell, J.e. (1978). Radiofrequency Radiation Dosimetry Handbook, second ed., Report SAM-TR-78-22, USAF School of Aerospace Medicine, Brooks AFB, TX. 254 J. MICROWAVE POWER 1985 Dosimetry-Finite Difference Techniques

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