Luebbers, R. Computer Design for Biomedical Applications

The Electrical Engineering Handbook

Ed. Richard C. Dorf

Boca Raton: CRC Press LLC, 2000

© 2000 by CRC Press LLC

11S

Compufer IesIgn lor

BIomedIcaI AppIIcafIons

The Finite Diffeience Time Domain (FDTD) ¦Yee, 1966; Kunz and Luebbeis, 1993; Taßove, 1995] is a numeiical

method foi the solution of electiomagnetic feld inteiaction pioblems. It utilizes a geometiy mesh, usually of

iectangulai box-shaped cells. The constitutive paiameteis foi each cell edge may be set independently, so that

objects having iiiegulai geometiies and inhomogeneous dielectiic composition can be analyzed.

The FDTD method solves Maxwell`s diffeiential equations at each cell edge at disciete time steps. Since no

matiix solution is involved, electiically laige geometiies can be analyzed. FDTD solutions foi thiee dimensional

complex biological geometiies involving millions of cells have become ioutine. FDTD may be used foi both

open iegion calculations, such as a human body in fiee space, oi closed iegions, such as within a TEM cell.

Commeicial FDTD softwaie is available fiom seveial souices (CST, EMA, and Remcom), with some of these

also offeiing FDTD meshes foi human heads and bodies. These commeicial packages piovide a giaphical usei

inteiface foi viewing the FDTD mesh. Some piovide inteiactive mesh editing (Remcom), while otheis allow

foi impoit of objects fiom CAD piogiams (CST and EMA).

The choice of cell size is ciitical in applying FDTD. It must be small enough to peimit accuiate iesults at the

highest fiequency of inteiest, and yet be laige enough to keep iesouice iequiiements manageable. Cell size is diiectly

affected by the mateiials piesent. The gieatei the peimittivity and/oi conductivity, the shoitei the wavelength at a

given fiequency and the smallei the cell size iequiied. Once the cell size is selected, the maximum time step is

deteimined by the Couiant stability condition. Aftei the usei deteimines the cell size, a pioblem space laige enough

to encompass the scatteiing object, plus space between the object and the absoibing outei boundaiy, is deteimined.

Fiom the numbei of Yee cells needed and the numbei of time steps iequiied, iesouice iequiiements can be estimated.

The fundamental constiaint is that the cell size must be much less than the smallest wavelength foi which

accuiate iesults aie desiied. An often quoted constiaint is °10 cells pei wavelength", meaning that the side of

each cell should be 1/10 of the wavelength at the highest fiequency (shoitest wavelength) of inteiest. Since

FDTD is a volumetiic computational method, if some poition of the computational space is flled with

penetiable mateiial, one must use the wavelength ín ì|e maìería| to deteimine the maximum cell size. Foi

pioblems containing biological mateiials, this iesults in cells in the mateiial that aie much smallei than if only

fiee space and peifect conductois weie being consideied.

Anothei cell size consideiation is that the impoitant chaiacteiistics of the pioblem geometiy must be

accuiately modeled. This will noimally be met automatically by making the cells smallei than 1/10 i unless

some special geometiy featuies smallei than this aie factois in deteimining the iesponse of inteiest.

In some situations theie is a specifc iegion of the object wheie smallei FDTD cells aie needed, foi example,

a iegion of high dielectiic mateiial, oi of fne geometiy featuies such as eyes. But if unifoim FDTD cells aie

used thioughout the computation, then these small cells must be used even in iegions wheie they aie not

needed. One appioach to ieduce the total numbei of FDTD cells foi these situations is to mesh local iegions

with smallei cells than in the main mesh ¦Kim and Hoefei, 1990; Zivanovic et al., 1991]. All of the commeicial

FDTD softwaie iefeienced above has this local giid capability.

The othei basic constiaint on FDTD calculations is the time step size. Foi a thiee-dimensional giid with cell

edges of length Ax, Ay, A:, with · the maximum velocity of piopagation in any medium in the pioblem, usually

the speed of light in fiee space, the time step size Aì is limited by

Raymond Luellers

Penn·y|vonío Srore Iníver·íry

© 2000 by CRC Press LLC

Now let us considei how to estimate the computei iesouices iequiied. Given the shoitest wavelength of inteiest,

the cell dimensions aie deteimined as 1/10 of this wavelength (oi less if gieatei accuiacy is iequiied). Fiom

this and the physical size of the pioblem geometiy the total numbei of cells in the pioblem space (heie denoted

as NC) can be deteimined. We assume that the mateiial infoimation foi each cell edge is stoied in 1 byte

(INTEGER¯1) aiiays with only dielectiic mateiials consideied. Then, to estimate the computei stoiage in bytes

iequiied, and assuming single-piecision FORTRAN feld vaiiables, we can use the ielationship

wheie components indicate the vectoi electiic and magnetic feld components. If magnetic mateiials aie

included, then six edges must also be consideied foi the mateiial aiiays. In this equation, we have neglected

the ielatively small numbei of auxiliaiy vaiiables needed foi the computation piocess.

One can estimate the computational cost in teims of the numbei of ßoating point opeiations iequiied using

wheie 15 opeiations is an appioximation based on expeiience and wheie N is the total numbei of time steps.

The numbei of time steps N is typically on the oidei of fve to ten times the numbei of cells on one side of

the pioblem space. It will be laigei foi iesonant objects and smallei foi lossy objects.

As an example, considei a human body meshed with 5-mm cubical cells. At 10 cells pei fiee space wavelength,

this would coiiespond to a maximum fiequency of 6 GHZ. But, since the biological mateiials in the body have

ielatively high dielectiic constants, the wavelength inside the body is ieduced. If the maximum dielectiic

constant of body mateiials is 49, then the maximum fiequency would be ieduced by 7 to about 857 MHz. If

iesults at highei fiequencies aie needed, then the cell size must be ieduced.

Foi a human body that fts into a box of 63 × 36 × 183 cm, with a 15-cell boidei aiound the body to sepaiate

it fiom the outei boundaiy, the pioblem space is about 160 × 100 × 400 oi 6.4 millions cells. Using the above

foimula, the computei RAM necessaiy to make this calculation is appioximately 172 MBytes. Since this does

not allow foi stoiage of instiuctions and othei aiiays, and since the opeiating system will take some computei

memoiy, a machine with about 256 MBytes of iandom access memoiy (RAM) should be suffcient to make

this calculation.

A conseivative estimate of the numbei of time steps needed is 10 times the longest dimension in cells, oi

4000 time steps. Using the above equation, an estimate of 2.3 × 10

12

opeiations iesults. Typical MFLOPS (Million

Floating Point Opeiations pei Second) iatings foi computeis aie 15 foi a Pentium PC oi low end woik station,

60 foi a fast woik station, and seveial hundied foi a supei computei. If we use 200 MFLOPS foi the supei

computei, then the calculation times foi the human body aie 42 h foi the PC oi low end woik station, 10.5 h

foi the fast woik station, and 3.1 h foi the supei computei.

The pieceding discussion piimaiily consideis the high fiequency limitations of FDTD calculations, which

aie based on the size of the object in wavelengths. The low fiequency limitation is usually deteimined by a

combination of the geometiy featuies and time step. Foi example, considei applying FDTD foi a 60-Hz

calculation foi a human body. Based on the wavelength, the FDTD cells could be huge, but then the body shape

would be uniecognizable. Suppose that we pick FDTD cells of 10 cm to at least make a ciude body shape. Then

the maximum time step would be 19.2 × 10

-10

s. If we fuithei assume that we need to make FDTD calculations

foi at least one peiiod of the sine wave in oidei to iead some semblance of steady state, this would iequiie

about 86 million time steps, which is not feasible on cuiient computeis. This illustiates the diffculty of using

FDTD foi extiemely low fiequencies. Foi these veiy low fiequencies othei methods, such as fnite elements,

aie piefeiied.

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© 2000 by CRC Press LLC

Depending on the application, human body models may be ciude appioximations oi detailed meshes based

on actual anatomy. A populai souice of anatomical data suitable as the basis foi an FDTD biological mesh is

the Visible Human Pioject of the National Libiaiy of Medicine. Vaiious types of data aie available, with the

most useful peihaps being the cioss-sections. These aie 1-mm slices foi the male and 0.33-mm slices foi the

female. Both have a cioss-sectional iesolution of 0.33 mm. The FDTD meshing of this data still iequiies

consideiable effoit, especially in assigning the colois of the slices to paiticulai tissue types.

The actual FDTD calculations may be excited in diffeient ways. Most commonly the electiic felds on one

oi moie mesh edges aie deteimined by an analytical function of time, such as a Gaussian pulse oi sine wave.

This then acts as a diiven voltage souice. This may be used to excite an antenna. Foi example, a shoit monopole

antenna on a iectangulai box may appioximate a poitable telephone. This monopole antenna could be diiven

by a diive voltage souice located on the mesh edge at the monopole base next to the top of the box. Both Kunz

and Luebbeis ¦1993] and Taßove ¦1995] desciibe methods foi modeling RF souices. A vaiiety of FDTD souices,

including cuiient souices, aie desciibed in Piket-May et al. ¦1994]. Alteinatively a plane wave may be incident

on the object as the excitation souice.

The time vaiiation of the excitation may be eithei pulsed oi sine wave. The advantage of the pulse is that

iesponse foi a wide fiequency iange can be obtained. But, foi accuiate iesults, the fiequency-dependent behavioi

of biological mateiials must be included in the calculations. Methods foi doing this aie well known ¦Kunz and

Luebbeis, 1993; Taßove, 1995] so that tiansient electiomagnetic feld amplitudes foi pulse excitation can be

calculated using FDTD ¦Fuise et al., 1994]. When iesults at a single fiequency oi at a few specifc fiequencies

aie desiied, then sine wave excitation is piefeiied. This is especially tiue if iesults foi the entiie body, such as

SAR, aie needed, since stoiing the tiansient iesults foi the entiie body mesh and then applying fast Fouiiei

tiansfoimation to calculate the SAR vs. fiequency iequiies extiemely laige amounts of computei stoiage.

Re!ated Tupic

45.1 Intioduction

Relerences

C. M. Fuise, J. Y. Chen, and O. P. Gandhi, °The use of the fiequency-dependent fnite-diffeience time-domain

method foi induced cuiients and SAR calculations foi a heteiogeneous model of the human body," IEEE

Trans. E|etìromagn. Com¡., 36, 128-133, 1994.

L. S. Kim and W. J. R. Hoefei, °A local mesh iefnement algoiithm foi the time-domain fnite-diffeience method

using Maxwell`s equations," IEEE Trans. Mítrowa·e T|eory Tet|níques, 38, 812-815, 1990.

K. S. Kunz and R. J. Luebbeis, T|e Fíníìe Dí[[erente Tíme Domaín Meì|oJ [or E|etìromagneìíts, Boca Raton,

Fla.: CRC Piess, 1993.

M. Piket-May, A. Taßove, and J. Baion, °FD-TD modeling of digital signal piopagation in 3-D ciicuits with

passive and active loads," IEEE Trans. Mítrowa·e T|eory Tet|níques, 42, 1514-1523, 1994.

A. Taßove, Com¡uìaìíona| E|etìroJynamíts-T|e Fíníìe-Dí[[erente Tíme-Domaín Meì|oJ, Boston, Mass.: Aitech

House, 1995.

K. S. Yee, °Numeiical solution of initial boundaiy value pioblems involving Maxwell`s equations in isotiopic

media," IEEE Trans. Ànìennas Pro¡agaìíon, AP-17, 585-589, 1966.

S. S. Zivanovic, K. S. Yee, and K. K. Mei, °A subgiidding method foi the time-domain fnite-diffeience method

to solve Maxwell`s equations," IEEE Trans. Mítrowa·e T|eory Tet|níques, 39, 471-479, 1991.

Further Inlurmatiun

CST GmbH, Lauteschlägeisti, 38, D-64289 Daimstadt, Geimany, -49(0)6151 717057, fax -49(0)6151 718057.

EMA Electiomagnetic Applications, P.O. Box 260263, Denvei, CO, 80226-2091, voice (303) 980-0070.

Remcom, Inc., Caldei Squaie, Box 10023, State College, PA 16805-0023, voice (814) 353-2986, fax

(814) 353-1420, URL http://www.iemcominc.com, e-mail xfdtdCiemcominc.com.

Visible Human Pioject, National Libiaiy of Medicine, 8600 Rockville Pike, Bethesda, MD 20894; fax (301)

402-4080; URL http://www.nlm.nih.gov/ieseaich/visible/visible-human.