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Electromagnetic Fields in Biological Systems
of cubic cells (Yee cells), where the geometries under study are spatially approximated
(Taflove and Hagness 2000). The cell size must be small enough to permit accurate
results at the highest frequency of interest, taking into account that the materials present directly affect the wavelength. Once the cell size is selected, the maximum time step
is determined by the Courant stability condition (Taflove and Hagness 2000).
FDTD has been used in several EM problems, such as RF microwave antennas and
circuit design, and radar scattering from objects of various types, and it has been widely
applied to EM dosimetry in models of human beings and other animals. In applying the
FDTD method for numerical dosimetry calculation, the Yee cells correspond completely
to the voxels in biological models. This technique has the following main advantages
(Kuster 2002):
1. The ability to model complex geometries in a straightforward manner
2. Its direct derivation from Maxwell’s equations, which allows the modeling of
lossy and nonlinear materials without special treatment at interfaces
3. The possibility to obtain results for a broad frequency range
4. The linear increase of the computer memory requirement and the computation
time with the number of voxels
5. The lack of the requirement to invert large matrices, which allows the calculation
of models of the human body composed of million voxels (high spatial resolution)
6. Its suitability to evaluate the sensitivity of the results upon different parameters
(e.g., dependence of the absorption upon anatomy, posture, tissue parameters)
However, a disadvantage of the method has been that the EM fields in the space surrounding the body must also be calculated. This implies that the surrounding space must also
be modeled by mathematical cells. This greatly increases the number of cells for which
calculations must be made, thus requiring a large amount of computer memory. Another
drawback of the method is that there is no internal check for assessing the quality of the
solution, that is, the quantitative uncertainty of the resulting field distribution. Coarse
discretization or too small distances between the scatterer and absorbing boundaries
can cause errors that are difficult to detect. However, the use of a sufficiently wide margin between the scatterers and the boundaries results in a negligible error from boundary modeling and an improved accuracy of the far-field transformation. Furthermore,
major difficulties may be encountered in handling the sources. Describing small dimensions may lead to a compromise on the dimension of the space-discretization step, while
significant difficulties are encountered in modeling antenna structures not conforming to the used grid, such as helical antennas (Cavagnaro and Pisa 1996; Nikita et al.
2000a). Numerical artifacts or interface effects in the representation of inhomogeneous
body structures can be a significant source of error when modeling nonhomogeneous
space with the discrete FDTD scheme. In studies where the field interaction is highly
dependent on the shape of the boundary, this can lead to substantial error in the calculation. Improvements in the modeling process by implementing algorithms relying on
boundary-fitted grids should be used in order to minimize staircasing effects.
Nevertheless, since the 1990s, the use of FDTD has become the most powerful simulation technique in the area of dosimetry and has been widely used to assess human
exposure to spatially uniform or nonuniform (far-field or near-field) EM fields from
Electromagnetic Fields in Biological Systems
of cubic cells (Yee cells), where the geometries under study are spatially approximated
(Taflove and Hagness 2000). The cell size must be small enough to permit accurate
results at the highest frequency of interest, taking into account that the materials present directly affect the wavelength. Once the cell size is selected, the maximum time step
is determined by the Courant stability condition (Taflove and Hagness 2000).
FDTD has been used in several EM problems, such as RF microwave antennas and
circuit design, and radar scattering from objects of various types, and it has been widely
applied to EM dosimetry in models of human beings and other animals. In applying the
FDTD method for numerical dosimetry calculation, the Yee cells correspond completely
to the voxels in biological models. This technique has the following main advantages
(Kuster 2002):
1. The ability to model complex geometries in a straightforward manner
2. Its direct derivation from Maxwell’s equations, which allows the modeling of
lossy and nonlinear materials without special treatment at interfaces
3. The possibility to obtain results for a broad frequency range
4. The linear increase of the computer memory requirement and the computation
time with the number of voxels
5. The lack of the requirement to invert large matrices, which allows the calculation
of models of the human body composed of million voxels (high spatial resolution)
6. Its suitability to evaluate the sensitivity of the results upon different parameters
(e.g., dependence of the absorption upon anatomy, posture, tissue parameters)
However, a disadvantage of the method has been that the EM fields in the space surrounding the body must also be calculated. This implies that the surrounding space must also
be modeled by mathematical cells. This greatly increases the number of cells for which
calculations must be made, thus requiring a large amount of computer memory. Another
drawback of the method is that there is no internal check for assessing the quality of the
solution, that is, the quantitative uncertainty of the resulting field distribution. Coarse
discretization or too small distances between the scatterer and absorbing boundaries
can cause errors that are difficult to detect. However, the use of a sufficiently wide margin between the scatterers and the boundaries results in a negligible error from boundary modeling and an improved accuracy of the far-field transformation. Furthermore,
major difficulties may be encountered in handling the sources. Describing small dimensions may lead to a compromise on the dimension of the space-discretization step, while
significant difficulties are encountered in modeling antenna structures not conforming to the used grid, such as helical antennas (Cavagnaro and Pisa 1996; Nikita et al.
2000a). Numerical artifacts or interface effects in the representation of inhomogeneous
body structures can be a significant source of error when modeling nonhomogeneous
space with the discrete FDTD scheme. In studies where the field interaction is highly
dependent on the shape of the boundary, this can lead to substantial error in the calculation. Improvements in the modeling process by implementing algorithms relying on
boundary-fitted grids should be used in order to minimize staircasing effects.
Nevertheless, since the 1990s, the use of FDTD has become the most powerful simulation technique in the area of dosimetry and has been widely used to assess human
exposure to spatially uniform or nonuniform (far-field or near-field) EM fields from