The anatomical model used in this study is derived from the BrainWeb Normal Anatomical Brain Model (Version 1.0) database, which provides simulated three-dimensional MRI brain datasets with an isotropic spatial resolution of 1 mm [12]. The volume considered in the present study has dimensions 181 × 217 × 181 mm³.

A single coronal slice extracted at the exact middle index Y = 100 along the anterior-posterior dimension (corresponding to an anatomical position of approximately 100 mm with respect to the posterior boundary of the volume) is used to construct the two-dimensional computational model. The original BrainWeb segmentation contains several anatomical and non-anatomical tissue labels. To construct the present electromagnetic model, a crisp segmentation thresholding criterion was applied, and four principal biological tissue classes were retained:

gray matter (GM),white matter (WM), cerebrospinal fluid (CSF), skull.

All other anatomical labels present in the original dataset (such as skin, fat, muscle, and glial tissue), as well as internal cavities or nasal sinuses, were excluded from the computational domain and treated as background air.

The spatial step of the FDTD grid is chosen as Δ x = Δ y = 1 mm, matching the native isotropic resolution of the BrainWeb dataset. This choice avoids introducing additional spatial interpolation into the segmented anatomical map and provides a direct correspondence between the anatomical labels and the FDTD computational cells. A finer FDTD grid could improve the numerical representation of electromagnetic field variations and tissue interfaces, but would not introduce additional anatomical information beyond that contained in the original 1-mm segmented dataset.

The brain is surrounded by an air region to provide sufficient space between the anatomical structure and the absorbing boundary. An absorbing layer is applied at the outer boundary to emulate an open electromagnetic domain and to minimize artificial reflections. Specifically, a minimum air-buffer distance of 15 cells (15 mm) is maintained between the outermost skull boundary and the total-field/scattered-field (TF/SF) framework. The computational arrangement is schematically represented in Figure 1, showing the brain region, surrounding air, the location of the TF/SF boundary, and the bounding uniaxial perfectly matched layer (UPML).

Figure 1.Schematic representation of the computational domain.

The electromagnetic properties of biological tissues are frequency dependent. At 900 MHz, the relative permittivity, electrical conductivity, and mass density adopted in this study are listed in Table 1. The dielectric data are taken from the published database of Gabriel et al. [13].

Table 1. Electromagnetic properties of the tissues considered in the model at 900 MHz.

The strong differences in conductivity and relative permittivity between the tissue classes are expected to affect both the propagation and absorption of the electromagnetic field.

The electromagnetic propagation is simulated using the FDTD method introduced by Yee [3]. The two-dimensional formulation adopted here corresponds to the transverse magnetic (TM_z) configuration, for which the electric field has one non-zero component, E z , while the magnetic field has components H x and H y .

The governing equations are written as

and

For a lossy dielectric, the frequency-domain constitutive relation can be expressed in terms of the complex relative permittivity

so that

The two-dimensional computational domain corresponds to the selected coronal anatomical slice and is represented in the ( x , y ) plane. The electromagnetic excitation is restricted to the TM_z configuration. Accordingly, the electric field has only an out-of-plane component,

whereas the magnetic field lies in the plane of the coronal slice,

The incident plane wave propagates along the + x direction. Consequently, the present configuration represents a single, highly specific two-dimensional exposure geometry with a fixed propagation vector and a uniform out-of-plane polarization. The dosimetric results and conclusions reported in this work are strictly bounded by this 2D framework and TM_z polarization state, and must not be interpreted as polarization-independent characteristics of electromagnetic absorption in a real three-dimensional human head.

A uniaxial perfectly matched layer (UPML) is used to emulate an open electromagnetic domain and reduce artificial reflections at the boundaries [14][15]. The absorbing boundary condition is implemented using a standard polynomial grading profile of order m = 3 for the conductivity scaling, with a target theoretical reflection coefficient at normal incidence set to R 0 = 10 − 6 (−120 dB).

The implementation uses the standard auxiliary-variable formulation. The electric-displacement update is written in the form

The magnetic-field updates are

and

The auxiliary variables and dissipative coefficients are determined by the selected UPML profile.

After updating the electric displacement, the electric field is recovered according to

This formulation allows each FDTD cell to be associated with the electromagnetic properties of the corresponding segmented tissue.

A normally incident plane wave is introduced using a total-field/scattered-field formulation, following the two-dimensional prescription of Sullivan [15]. The TF/SF boundary is positioned exactly 10 cells away from the inner interface of the UPML absorbing layer. The incident electric field is defined by

where

The amplitude E 0 is chosen such that the incident time-averaged power density S inc = E 0 2 / ( 2 Z 0 ) equals 1 W/m², with Z 0 ≃ 377   Ω the free-space impedance. Under this normalization, the SAR values reported in Section 5 scale linearly with the actual incident power density and can therefore be transposed to any exposure scenario.

The factor 1 − e − t / τ provides a progressive establishment of the field and reduces the influence of the initial transient. The numerical wavenumber is calculated from the FDTD dispersion relation:

Table 2.Main numerical parameters of the FDTD simulation.

The principal numerical parameters are summarized in Table 2. At the operating frequency of 900 MHz, the optical period is T 0 = 1 / f 0 ≃ 1.111 ns. With the chosen time step Δ t = Δ x / ( 2 c ) ≃ 1.6678 ps, a single harmonic period is discretized into exactly 666 time steps. The total number of time steps (15,666) corresponds to 47.04 full optical periods. This window consists of a transient phase of 15,000 steps (equivalent to 22.5 periods), after which the ramp envelope 1 − e − t / ( 3 T 0 ) exceeds 0.9994 and the SAR bias associated with an unestablished regime remains below 0.1%, followed by a single full period (666 steps) used to compute the root-mean-square electric field. Given the lossy nature of the biological tissues, internal multi-reflections are rapidly attenuated, ensuring that the computational domain has rigorously reached a harmonic steady state prior to the final sampling period.

The segmented anatomical model produces a strongly heterogeneous distribution of electromagnetic properties as shown in Figure 2. In particular, the CSF presents a much larger conductivity than gray matter and white matter, whereas the skull exhibits comparatively low conductivity.

The heterogeneous distribution of permittivity and conductivity determines the local propagation, reflection, transmission, and absorption of the electromagnetic field.

The spatial distribution of SAR at 900 MHz is shown in Figure 3.

The calculated SAR distribution is strongly non-uniform. Local variations occur near tissue interfaces and in regions where the electric field is enhanced by electromagnetic interference.

This behavior results from the differences in permittivity and conductivity between the tissue classes. These differences modify the local electric field and consequently the absorbed electromagnetic power.

Figure 2. Spatial distribution of the dielectric parameters in the segmented brain model.

Figure 3. Spatial distribution of SAR in the heterogeneous brain model at 900 MHz.

Table 3 summarizes the mean, standard deviation, and maximum SAR obtained for the four tissue classes.

The average SAR in gray matter is higher than that in white matter. This is consistent with the higher conductivity of gray matter.

Table 3. SAR statistics for the different tissue types at 900 MHz.

The CSF exhibits the largest average and maximum SAR values among the considered tissues. Its comparatively high electrical conductivity facilitates conversion of electromagnetic-field energy into dissipated power.

The dependence of the SAR on the local tissue properties can be made explicit by rewriting Equation (14) as SAR k = ( σ k / ρ k ) 〈 | E k | 2 〉 . This decomposition separates a material factor σ k / ρ k , fixed by the tissue class, from a field factor 〈 | E k | 2 〉 , which depends on the local electromagnetic environment. The ratios obtained from Table 3 illustrate the balance between these two contributions. The CSF-to-gray-matter SAR ratio is 2.12, whereas the corresponding σ / ρ ratio is 2.66; the mean square field is therefore smaller in the CSF by a factor of about 0.80, consistent with the partial shielding associated with the high permittivity of this tissue. Conversely, the skull-to-white-matter SAR ratio is 0.59 for a σ / ρ ratio of only 0.13, indicating a mean square field about 4.6 times larger in the skull. This is consistent with the peripheral location of the skull, where the field is enhanced by the incident wave before it is progressively absorbed by the underlying tissues. Conductivity alone therefore does not determine the spatial distribution of the SAR: the local field structure plays a comparable role.

The standard deviations are relatively large compared with the corresponding mean values, indicating substantial spatial variability within individual tissue classes. This variability is associated with the non-uniform electric field resulting from the anatomical interfaces and multiple electromagnetic interactions.

The skull exhibits the lowest average SAR among the considered tissues, consistent with its relatively low conductivity.

The absolute SAR values reported here correspond to a normalized incident power density S inc = 1 W/m². The largest local value (about 0.11 W/kg in the CSF) is therefore approximately eighteen times smaller than the ICNIRP local basic restriction of 2 W/kg for the head and trunk averaged over 10 g of tissue [1], and correspondingly below the IEEE C95.1-2019 basic restriction [2]. The comparison is illustrative rather than regulatory, since a full compliance assessment would require a three-dimensional anatomical model, a 10 g spatial averaging procedure, and a realistic exposure configuration.

The mean SAR profile along the direction of propagation is shown in Figure 4.

Figure 4. SAR attenuation along the direction of electromagnetic-wave propagation.

The SAR generally decreases from the incident side toward the opposite side of the brain, reflecting the progressive absorption of electromagnetic energy.

The decrease is not strictly monotonic because the wave interacts with successive tissue interfaces. Local increases in SAR may occur in regions associated with tissues having higher conductivity, particularly the CSF.

These observations illustrate the importance of representing the heterogeneous anatomical structure of the brain when evaluating local electromagnetic absorption.

The first test considers the propagation of a modulated plane-wave packet in a homogeneous lossless medium. The analytical solution is

The numerical and analytical spatial profiles are compared after a prescribed propagation time. The relative L 2 error is defined by

For Δ x = 1 mm, the measured relative error is 8.10 × 10⁻³. The numerical and analytical profiles are shown in Figure A1.

Figure A1. Free-space propagation test: FDTD field compared with the analytical solution.

The error is sufficiently small for the intended anatomical simulation, but it also shows that the finite grid and time step produce a measurable numerical phase and amplitude error. This error is further quantified independently through the dispersion test.

For the one-dimensional Yee scheme, the numerical wavenumber satisfies

Hence,

The relative dispersion error is

At 900 MHz, with the Courant factor S = c   Δ t / Δ x = 0.5 used in the anatomical simulation, the maximum relative error over Δ x ∈ [ 0.5 , 2 ] mm is 4.44 × 10⁻⁵, and the error at Δ x = 1 mm is 1.11 × 10⁻⁵. Figure A2 shows the rapid reduction of the dispersion error with mesh refinement.

Figure A2. Numerical-dispersion error as a function of the spatial step at 900 MHz.

Figure A3.Residual reflection of the 2D UPML at 900 MHz.

The residual reflection of the two-dimensional UPML was characterized directly on the production solver by propagating a plane wave in a homogeneous medium (free space) using the TF/SF formulation, and measuring the residual field in the scattered-field region after the transient regime has decayed. In this configuration, the field in the scattered-field region should vanish identically, so any non-zero value measures the combined residual of UPML reflection and TF/SF leakage. The measured reflection coefficient is

For UPML thicknesses of 10, 20, 30 and 40 cells, the measured values are 2.6 × 10⁻⁵, 6.4 × 10⁻⁶, 5.5 × 10⁻⁶ and 4.5 × 10⁻⁶, respectively, corresponding to reflection levels of −92, −104, −105 and −107 dB. The residual reflection saturates at approximately −107 dB for thicknesses of 20 cells and above. Figure A3 shows the resulting reflection coefficient as a function of the absorbing-layer thickness.

The material-interface treatment was verified for normal incidence on a lossless interface between two media with ε r 1 = 1 and ε r 2 = 4 . The analytical reflection and transmission coefficients are

The analytical values are R = 0.111111 and T = 0.888889 . The FDTD calculation gives R = 0.111156 and T = 0.888973 , corresponding to relative errors of 4.03 × 10⁻⁴ and 9.48 × 10⁻⁵, respectively. The results illustrated in Figure A4 demonstrate that the material-interface treatment is accurately reproduced by the FDTD discretization.

Figure A4. FDTD reflected and transmitted fields for a normally incident wave at a dielectric interface.

A finite dielectric slab with ε r = 4 and thickness d = 20 mm was considered as a second interface benchmark. For a lossless slab, the analytical reflection and transmission coefficients were calculated using the Fabry-Pérot formulation. Energy conservation imposes

where A is the absorption coefficient, which vanishes identically for a lossless medium. The analytical values at 900 MHz are R = 0.208788 , T = 0.791212 , and A = 0 . A frequency-domain extraction of the FDTD reflected and transmitted fields at 900 MHz gives R = 0.208685 , T = 0.791308 , and A ≃ 7 × 10 − 6 , the latter being a numerical round-off-level residual of the energy balance. The relative errors in R and T are 4.9 × 10^-4 and 1.2 × 10⁻⁴ respectively, consistent with those obtained for the single interface (Section A.4). The small energy-balance residual, several orders of magnitude smaller than the individual errors in R and T, indicates that the numerical scheme conserves electromagnetic energy: the errors on R and T compensate one another rather than accumulating.

Figure A5 compares the analytical and FDTD reflection, transmission, and absorption coefficients for the dielectric slab benchmark.

Figure A5. Comparison of analytical and FDTD reflection, transmission and absorption coefficients for the dielectric slab benchmark.

The spatial convergence of the Yee scheme was evaluated using a periodic plane-wave problem in a homogeneous, lossless medium. The spatial step was varied from 2 to 0.5 mm, while the Courant factor was kept constant. The relative L 2 error was calculated with respect to the analytical plane-wave solution.

The measured errors are 1.26 × 10⁻³, 7.09 × 10⁻⁴, 3.15 × 10⁻⁴, 1.77 × 10⁻⁴ and 7.86 × 10⁻⁵ for Δ x = 2 , 1.5, 1, 0.75 and 0.5 mm, respectively. A least-squares fit in logarithmic coordinates gives an observed convergence order of 1.999, in excellent agreement with the expected second-order convergence of the Yee scheme.

This result certifies the correctness of the implementation in the regime where the scheme admits an exact dispersion analysis. The behavior of the solver in the presence of curved staircased interfaces, which is the regime relevant to the anatomical model, is characterized independently in Section A.7. The measured convergence is shown in Figure A6.

Figure A6. Spatial convergence of the FDTD solution. The observed order is approximately two.

The complete solver - Yee scheme, staircased curved interfaces, UPML, TF/SF - was benchmarked against the analytical Mie series for a two-dimensional lossy dielectric cylinder illuminated by a plane wave. The cylinder radius is 40 mm, and its dielectric properties are those of gray matter at 900 MHz ( ε r = 52.7 , σ = 0.94 S/m, ρ = 1045 kg/m³). The complex Mie series was truncated at N max = 80 multipoles, well above the convergence radius k a ≃ 0.75 that governs the truncation error.

At the working spatial step Δ x = 1 mm, the relative L 2 error between the FDTD phasor and the Mie phasor is 3.1% inside the cylinder and 2.5% on a 5-mm-thick boundary annulus. The corresponding error on the tissue-averaged SAR is 2.7% on the mean value and 4.3% on the maximum value. When the spatial step is refined to 0.75 and 0.5 mm, the L 2 error on the field reaches 3.2% and 4.0%, respectively. The error does not decrease with refinement, which is consistent with the theoretical behavior of a Yee-based TF/SF formulation on a lossy scatterer: the residual is dominated by a small dispersion mismatch between the one-dimensional numerical wavenumber used to inject the incident field and the two-dimensional numerical dispersion of the surrounding grid. Consequently, these benchmark errors characterize the baseline performance of the mathematical formulation on a staircased canonical object, but they do not constitute a direct baseline or geometric limit for the real anatomical brain configuration. The observed error level remains fully comparable to that reported in the FDTD dosimetric literature for staircased voxel models [11][16].