Luebbers, R. Computer Design for Biomedical Applications The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
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 pennsylvania State University 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 omplex 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 lso 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 mport 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 directhy 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 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 1/102 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 dges of length Ax, Ay, Az, 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 At is limited by c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 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
v△t≤l (△x2( 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 nis 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 4 byte stora cell cell 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 Operations= NC X 6 components/ cell x 15 operations/component x N 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 x 36x 183 cm, with a 15-cell border around the body to separate it from the outer boundary, the problem space is about 160 x 100 x 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 x 102operations 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 ne maximum time step would be 19.2 x 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 element ferred e 2000 by CRC Press LLC
© 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 ¥ 1012 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. v t xy z D DDD £ ( ) + ( ) + ( ) 1 111 222 storage NC components cell bytes component edges cell byte edge = ¥ +¥ Ê Ë Á ˆ ¯ ˜ *6 4 3 1 Operations NC components cell operations component N =¥ ¥ ¥ 6 15
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 lost useful perhaps being the cross-sections. These are l-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 monopo 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, cluding 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 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 re desired, then sine wave excitation is preferred. This is especially true if results for the entire body, such 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 Topi 45.1 Introduction References C. M. Furse, J. Y Chen, and O. P. Gandhi, The use of the frequency-dependent finite-difference time-domain nethod 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 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 KK Mei, A subgridding method for the time-domain finite-difference method to solve Maxwells equations, " IEEE Trans. Microwave Theory Techniques, 39, 471-479, 1991 Further information CST GmbH, Lauteschlagerstr, 38, D-64289 Darmstadt, Germany, +49(0)6151717057, fax +49(0)6151718057 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,Urlhttp://www.remcominc.com,e-mailxfdtd@remcominc.com. Visible Human Project, National Library of Medicine, 8600 Rockville Pike, Bethesda, MD 20894; fax(301) 402-4080;urLhttp://www.nlm.nihgov/research/visible/visible-human e 2000 by CRC Press LLC
© 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, Lauteschlägerstr, 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