Luebbers, R. “Computer Design for Biomedical Applications”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
118
Computer Design for
Biomedical Applications
The Finite Difference Time Domain (FDTD) [Yee, 1966; Kunz and Luebbers, 1993; Taflove, 1995] is a numerical
method for the solution of electromagnetic field interaction problems. It utilizes a geometry mesh, usually of
rectangular box-shaped cells. The constitutive parameters for each cell edge may be set independently, so that
objects having irregular geometries and inhomogeneous dielectric composition can be analyzed.
The FDTD method solves Maxwell’s differential equations at each cell edge at discrete time steps. Since no
matrix solution is involved, electrically large geometries can be analyzed. FDTD solutions for three dimensional
complex biological geometries involving millions of cells have become routine. FDTD may be used for both
open region calculations, such as a human body in free space, or closed regions, such as within a TEM cell.
Commercial FDTD software is available from several sources (CST, EMA, and Remcom), with some of these
also offering FDTD meshes for human heads and bodies. These commercial packages provide a graphical user
interface for viewing the FDTD mesh. Some provide interactive mesh editing (Remcom), while others allow
for import of objects from CAD programs (CST and EMA).
The choice of cell size is critical in applying FDTD. It must be small enough to permit accurate results at the
highest frequency of interest, and yet be large enough to keep resource requirements manageable. Cell size is directly
affected by the materials present. The greater the permittivity and/or conductivity, the shorter the wavelength at a
given frequency and the smaller the cell size required. Once the cell size is selected, the maximum time step is
determined by the Courant stability condition. After the user determines the cell size, a problem space large enough
to encompass the scattering object, plus space between the object and the absorbing outer boundary, is determined.
From the number of Yee cells needed and the number of time steps required, resource requirements can be estimated.
The fundamental constraint is that the cell size must be much less than the smallest wavelength for which
accurate results are desired. An often quoted constraint is “10 cells per wavelength”, meaning that the side of
each cell should be 1/10 of the wavelength at the highest frequency (shortest wavelength) of interest. Since
FDTD is a volumetric computational method, if some portion of the computational space is filled with
penetrable material, one must use the wavelength in the material to determine the maximum cell size. For
problems containing biological materials, this results in cells in the material that are much smaller than if only
free space and perfect conductors were being considered.
Another cell size consideration is that the important characteristics of the problem geometry must be
accurately modeled. This will normally be met automatically by making the cells smaller than 1/10 l unless
some special geometry features smaller than this are factors in determining the response of interest.
In some situations there is a specific region of the object where smaller FDTD cells are needed, for example,
a region of high dielectric material, or of fine geometry features such as eyes. But if uniform FDTD cells are
used throughout the computation, then these small cells must be used even in regions where they are not
needed. One approach to reduce the total number of FDTD cells for these situations is to mesh local regions
with smaller cells than in the main mesh [Kim and Hoefer, 1990; Zivanovic et al., 1991]. All of the commercial
FDTD software referenced above has this local grid capability.
The other basic constraint on FDTD calculations is the time step size. For a three-dimensional grid with cell
edges of length Dx, Dy, Dz, with v the maximum velocity of propagation in any medium in the problem, usually
the speed of light in free space, the time step size Dt is limited by
Raymond Luebbers
Pennsylvania State University
? 2000 by CRC Press LLC
Now let us consider how to estimate the computer resources required. Given the shortest wavelength of interest,
the cell dimensions are determined as 1/10 of this wavelength (or less if greater accuracy is required). From
this and the physical size of the problem geometry the total number of cells in the problem space (here denoted
as NC) can be determined. We assume that the material information for each cell edge is stored in 1 byte
(INTEGER*1) arrays with only dielectric materials considered. Then, to estimate the computer storage in bytes
required, and assuming single-precision FORTRAN field variables, we can use the relationship
where components indicate the vector electric and magnetic field components. If magnetic materials are
included, then six edges must also be considered for the material arrays. In this equation, we have neglected
the relatively small number of auxiliary variables needed for the computation process.
One can estimate the computational cost in terms of the number of floating point operations required using
where 15 operations is an approximation based on experience and where N is the total number of time steps.
The number of time steps N is typically on the order of five to ten times the number of cells on one side of
the problem space. It will be larger for resonant objects and smaller for lossy objects.
As an example, consider a human body meshed with 5-mm cubical cells. At 10 cells per free space wavelength,
this would correspond to a maximum frequency of 6 GHZ. But, since the biological materials in the body have
relatively high dielectric constants, the wavelength inside the body is reduced. If the maximum dielectric
constant of body materials is 49, then the maximum frequency would be reduced by 7 to about 857 MHz. If
results at higher frequencies are needed, then the cell size must be reduced.
For a human body that fits into a box of 63 ′ 36 ′ 183 cm, with a 15-cell border around the body to separate
it from the outer boundary, the problem space is about 160 ′ 100 ′ 400 or 6.4 millions cells. Using the above
formula, the computer RAM necessary to make this calculation is approximately 172 MBytes. Since this does
not allow for storage of instructions and other arrays, and since the operating system will take some computer
memory, a machine with about 256 MBytes of random access memory (RAM) should be sufficient to make
this calculation.
A conservative estimate of the number of time steps needed is 10 times the longest dimension in cells, or
4000 time steps. Using the above equation, an estimate of 2.3 ′ 10
12
operations results. Typical MFLOPS (Million
Floating Point Operations per Second) ratings for computers are 15 for a Pentium PC or low end work station,
60 for a fast work station, and several hundred for a super computer. If we use 200 MFLOPS for the super
computer, then the calculation times for the human body are 42 h for the PC or low end work station, 10.5 h
for the fast work station, and 3.1 h for the super computer.
The preceding discussion primarily considers the high frequency limitations of FDTD calculations, which
are based on the size of the object in wavelengths. The low frequency limitation is usually determined by a
combination of the geometry features and time step. For example, consider applying FDTD for a 60-Hz
calculation for a human body. Based on the wavelength, the FDTD cells could be huge, but then the body shape
would be unrecognizable. Suppose that we pick FDTD cells of 10 cm to at least make a crude body shape. Then
the maximum time step would be 19.2 ′ 10
–10
s. If we further assume that we need to make FDTD calculations
for at least one period of the sine wave in order to read some semblance of steady state, this would require
about 86 million time steps, which is not feasible on current computers. This illustrates the difficulty of using
FDTD for extremely low frequencies. For these very low frequencies other methods, such as finite elements,
are preferred.
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? 2000 by CRC Press LLC
Depending on the application, human body models may be crude approximations or detailed meshes based
on actual anatomy. A popular source of anatomical data suitable as the basis for an FDTD biological mesh is
the Visible Human Project of the National Library of Medicine. Various types of data are available, with the
most useful perhaps being the cross-sections. These are 1-mm slices for the male and 0.33-mm slices for the
female. Both have a cross-sectional resolution of 0.33 mm. The FDTD meshing of this data still requires
considerable effort, especially in assigning the colors of the slices to particular tissue types.
The actual FDTD calculations may be excited in different ways. Most commonly the electric fields on one
or more mesh edges are determined by an analytical function of time, such as a Gaussian pulse or sine wave.
This then acts as a driven voltage source. This may be used to excite an antenna. For example, a short monopole
antenna on a rectangular box may approximate a portable telephone. This monopole antenna could be driven
by a drive voltage source located on the mesh edge at the monopole base next to the top of the box. Both Kunz
and Luebbers [1993] and Taflove [1995] describe methods for modeling RF sources. A variety of FDTD sources,
including current sources, are described in Piket-May et al. [1994]. Alternatively a plane wave may be incident
on the object as the excitation source.
The time variation of the excitation may be either pulsed or sine wave. The advantage of the pulse is that
response for a wide frequency range can be obtained. But, for accurate results, the frequency-dependent behavior
of biological materials must be included in the calculations. Methods for doing this are well known [Kunz and
Luebbers, 1993; Taflove, 1995] so that transient electromagnetic field amplitudes for pulse excitation can be
calculated using FDTD [Furse et al., 1994]. When results at a single frequency or at a few specific frequencies
are desired, then sine wave excitation is preferred. This is especially true if results for the entire body, such as
SAR, are needed, since storing the transient results for the entire body mesh and then applying fast Fourier
transformation to calculate the SAR vs. frequency requires extremely large amounts of computer storage.
Related Topic
45.1 Introduction
References
C. M. Furse, J. Y. Chen, and O. P. Gandhi, “The use of the frequency-dependent finite-difference time-domain
method for induced currents and SAR calculations for a heterogeneous model of the human body,” IEEE
Trans. Electromagn. Comp., 36, 128–133, 1994.
L. S. Kim and W. J. R. Hoefer, “A local mesh refinement algorithm for the time-domain finite-difference method
using Maxwell’s equations,” IEEE Trans. Microwave Theory Techniques, 38, 812–815, 1990.
K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, Boca Raton,
Fla.: CRC Press, 1993.
M. Piket-May, A. Taflove, and J. Baron, “FD-TD modeling of digital signal propagation in 3-D circuits with
passive and active loads,” IEEE Trans. Microwave Theory Techniques, 42, 1514–1523, 1994.
A. Taflove, Computational Electrodynamics—The Finite-Difference Time-Domain Method, Boston, Mass.: Artech
House, 1995.
K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic
media,” IEEE Trans. Antennas Propagation, AP-17, 585–589, 1966.
S. S. Zivanovic, K. S. Yee, and K. K. Mei, “A subgridding method for the time-domain finite-difference method
to solve Maxwell’s equations,” IEEE Trans. Microwave Theory Techniques, 39, 471–479, 1991.
Further Information
CST GmbH, Lauteschl?gerstr, 38, D-64289 Darmstadt, Germany, +49(0)6151 717057, fax +49(0)6151 718057.
EMA Electromagnetic Applications, P.O. Box 260263, Denver, CO, 80226-2091, voice (303) 980-0070.
Remcom, Inc., Calder Square, Box 10023, State College, PA 16805-0023, voice (814) 353-2986, fax
(814) 353-1420, URL http://www.remcominc.com, e-mail xfdtd@remcominc.com.
Visible Human Project, National Library of Medicine, 8600 Rockville Pike, Bethesda, MD 20894; fax (301)
402-4080; URL http://www.nlm.nih.gov/research/visible/visible-human.
? 2000 by CRC Press LLC
