FDTD method and dispersive algorithm

The FDTD method is one of the well-known numerical methods and has been widely used for EM dosimetry analysis of the human body due to its suitability to handle inhomogeneous tissues of the human body. For the numerical accuracy of the FDTD method, the CFL condition depending on the cell size of the computational domain must be satisfied. The cell size (${\Delta _i}$) must be at least one-tenth of the minimum wavelength of the tissues in the computational domain because the finer cell size provides solutions with better accuracy, especially at higher frequencies. Therefore, to ensure this condition and obtain accurate solutions over the frequency range up to 100 GHz, the cell size of the human body model is set to ${\Delta _i}$ = 0.0625 mm in this study.

The total-field scattered-field (TF/SF) formulation [Reference Elsherbeni AZ and Demir V30] integrated into the dispersive FDTD algorithm is used to generate an incident plane wave on the TFSF boundary. The incident plane wave considered as a far-field source is a Gaussian waveform including the frequency range up to 100 GHz. To terminate the FDTD computational domain, the convolution perfect matched layer boundary [Reference Elsherbeni AZ and Demir V30] condition is used in a dispersive algorithm proposed in [Reference Kaburcuk and Elsherbeni17, Reference Kaburcuk and Elsherbeni18]. The dispersive algorithm consists of three sub-calculations. In the first sub-calculation, EM analysis of the human body model with dispersive tissues is performed using the Debye models integrated into the FDTD method with the auxiliary differential equation to obtain solutions up to 100 GHz in a single simulation. Secondly, SAR, APD, and PTC recommended metrics by the ICNIRP guideline to evaluate EM wave exposure in the tissues are calculated at over the mmWave frequency range up to 100 GHz. Finally, calculation of the temperature rise due to the EM wave exposure is performed for all frequencies of interest.

The three-term Debye parameters (${\varepsilon _\infty }$: relative permittivity at infinite frequency, ${\varepsilon _{sk}}$: static relative permittivity, and ${\tau _k}$: relaxation time at kth term) of 54 different tissues in the human body are based on the values presented in [Reference Lumnitzer37] and provide EM solutions at frequencies up to 100 GHz. These parameters of selected tissues from 20 GHz to 100 GHz are tabulated in Table 2. The complex relative permittivity $\left( {\varepsilon _r^*\left( \omega \right)} \right)$ in equation (1.a) and conductivity $\left( {\sigma \left( \omega \right)} \right)$ in equation (1.b) for tissues is obtained from these parameters.

(1.a)\begin{align}\varepsilon _r^*\left( \omega \right) = \varepsilon _r^{\prime}\left( \omega \right) + j\varepsilon _r^{\prime\prime}\left( \omega \right) = {\varepsilon _\infty } + \mathop \sum \limits_{k = 1}^3 \frac{{{\varepsilon _{sk}} - {\varepsilon _\infty }}}{{1 + j\omega {\tau _k}}}\end{align}

(1.b)\begin{align}\sigma \left( \omega \right) = - \varepsilon _r^{\prime\prime}\left( \omega \right)*\omega *{\varepsilon _o}\end{align}

Table 2. Three-term Debye parameters of human tissues from 20 GHz to 100 GHz [Reference Lumnitzer37]

where ${\varepsilon _o}$ is the permittivity of free space.

Human body model and 1D multi-layered models

A 3D realistic human body voxel model, named TARO in [Reference Nagaoka, Watanabe, Sakurai, Kunieda, Watanabe, Taki and Yamanaka34] and shown in Fig. 1(a), is used in this study and consists of 51 different tissues such as skin, fat, muscle, bone cortical and marrow, blood, white and gray matter, CSF, cerebellum, lung, heart, kidney, liver, eye tissues, etc. with a resolution of ${\Delta _i}$ = 2 mm. The height and weight of the human body model used are 173 cm and 65 kg, respectively.

Figure 1. (a) 3D human body model [Reference Nagaoka, Watanabe, Sakurai, Kunieda, Watanabe, Taki and Yamanaka34], (b) 1D multi-layered models shown as dashed lines and derived from 2D cross sectional views (from top to bottom: forehead, eye, chest, forearm, and thigh models).

To satisfy the CFL conditions, the human body model with ${\Delta _i} =\ $2 mm is resampled with all tissues to achieve a new human body model with ${\Delta _i}$ = 0.0625 mm. 1D multi-layered models are then extracted from different regions of the human body model, including forehead, chest, forearm, thigh, and eye tissues with a high resolution. Therefore, effects of tissue arrangements of the models on the EM wave exposure metrics can be investigated. These 1D models presented in Fig. 1(b) and named forehead, eye, chest, forearm, and thigh models are marked as dashed lines on two-dimensional (2D) cross-sectional views of five different regions in the human body. The skin thickness and tissue arrangement of each model in the human body are different. Therefore, the forehead, chest, forearm, thigh, and eyelid models with different tissues arrangement, thickness of tissues and skin thickness are analyzed to have a more accurate assessment of EM dosimetry up to 100 GHz.

EM wave exposure metrics calculation

The SAR defined in [7, 8] is calculated to determine how much EM power absorbed per unit mass of tissues.

(2)\begin{align}SA{R_i} = \frac{{{\sigma _i}}}{{2{\rho _i}}}{\left| {{E_i}} \right|^2}\end{align}

where $\left| {{E_i}} \right|$ [V/m], ${\sigma _i}$ [S/m], and ${\rho _i}$ [kg/m³] are the total electric field strength, the electric conductivity, and mass density of the ith indexed cell of the tissues, respectively. The APD [Reference Diao, Rashed and Hirata29, Reference Sasaki, Mizuno, Wake and Watanabe32, Reference Li and Sasaki38, Reference Li, Hikage, Masuda, Ijima, Nagai and Taguchi39] crossing a unit area at ith indexed cell is calculated as follows:

(3)\begin{align}{\text{AP}}{{\text{D}}_i} = \frac{1}{2}Real\left[ {{E_i} \times H_i^*} \right]\end{align}

where ${E_i}$ and $H_i^*$ are the electric field and the complex conjugate of the magnetic field at ith indexed cell, respectively. For the assessment of the transmittance from air to skin tissue, the PTC [Reference Kodera, Taguchi, Diao, Kashiwa and Hirata9] is the ratio of TPD and IPD. The TPD is calculated in equation (4) for the 1D multi-layered model.

(4)\begin{align}{\text{TPD}} = \frac{1}{2}\mathop \int \limits_{i = 0}^{{i_{max}}} {\sigma _i}{\left| {{E_i}} \right|^2}{\text{d}}i\end{align}

where i_max is larger than penetration depth.

Thermal calculation

The resting state temperature distribution in the human body model is calculated based on the Pennes bioheat equation [Reference Li, Hikage, Masuda, Ijima, Nagai and Taguchi39] in the absence of a heating source (SAR = 0). The SAR distribution is considered as the heating source in the bioheat equation. Then, the final temperature rise distribution is computed by solving the bioheat equation again in the presence of the heating source (SAR $ \ne $ 0). Finally, the temperature rise distribution is obtained by taking difference of the resting state and final temperature distributions.

The Pennes bioheat equation [Reference Pennes40] is solved with the finite-difference approximation [Reference Sabbah, Dib and Al-Nimr21] at ith indexed cell of 1D multi-layered models with a resolution of ${\Delta _i}$ as follows:

(5)\begin{align}& T_i^{n + 1} = T_i^n + \frac{{{{{\Delta }}_{temp,i}}}}{{{C_i}}}\nonumber\\ &\quad \times\left[ {SA{R_i} {-} \frac{{{B_i}}}{{{\rho _i}}}\left[ {T_i^n {-} {T_b}} \right] {+} \frac{{{K_i}}}{{{\rho _i} \cdot {\Delta _i}^2}}\left[ {T_{i {+} 1}^n + T_{i {-} 1}^n {-} 2T_i^n} \right]} \right]\end{align}

where ${C_i}$ [J/(kg$ \cdot $°C)], [W/(m$ \cdot $°C)], and ${B_i}$ [W/(m³$ \cdot $°C)] are the heat capacity, the thermal conductivity, and the blood perfusion rate at ith indexed cell of 1D multi-layered models, respectively. $T_i^n$ is the temperature [°C] at time $n$ and at ith indexed cell, and ${T_b}$ is the blood temperature [°C] set to 37°C. The thermal time increment (${{{\Delta }}_{temp}}$) [Reference Wang and Fujiwara10], used in the iterative calculation of the temperature in equation (5), must satisfy the numerical stability.

(6)\begin{align}{{{\Delta }}_{temp,i}} \le \frac{{2 \cdot {\rho _i} \cdot {C_i} \cdot {\Delta _i}}}{{12 \cdot {K_i} + {B_i} \cdot {\Delta _i}^2}}\end{align}

At the interfaces of skin-air and internal cavity-air, the convective boundary condition is solved by the finite-difference approximation to model the heat exchange at the interfaces of skin-air and internal cavity-air. The convective boundary condition [Reference Wang and Fujiwara10] is defined as

(7)\begin{align}T_{{i_{min}}}^{n + 1} = \frac{{{K_i} \cdot T_{{i_{min}} + 1}^n + {T_{air}} \cdot H \cdot {{{\Delta }}_i}}}{{{K_i} + H \cdot {\Delta _i}}}\end{align}

where ${T_{air}}$ is the air temperature set to 20°C, $n$ is the unit normal vector to the interfaces, and $H$ is the convection heat transfer coefficient of 10.5 [W/(m²$ \cdot $°C)] from the skin to air [Reference Wang and Fujiwara10]. The mass density and thermal parameters of the human body tissues are based on the values presented in [Reference Hirata, Fujiwara and Shiozawa41].

The heating factor based on the APD, which is defined as a ratio of the temperature rise to APD, is calculated as a function of frequency up to 100 GHz. Above 6 GHz, the maximum heating factor should be at most 0.025 [°C$ \cdot $m²/W] based on [Reference Sasaki, Mizuno, Wake and Watanabe32].

EM analysis of 1D multi-layered models

The realistic 1D multi-layered models consisting of forehead, eye, chest, forearm, and thigh tissues with 2 mm skin thickness shown in Fig. 1(b) are analyzed using the dispersive algorithm for the frequency range from 1 GHz to 100 GHz. Therefore, effects of five different 1D models on the EM exposure metrics are investigated due to the EM far-field public exposure based on the ICNIRP guideline.

For the sake of comparison, the APDs of the 1D multi-layered models with 2 mm skin thickness presented in Fig. 1(b) and a simple four-layered 1D forearm model with 1 mm skin thickness presented in [Reference Sasaki, Mizuno, Wake and Watanabe32] are shown in Fig. 2 as a function of tissue depth at 30 GHz for the IPD of 1000 W/m². It can be noticed from Fig. 2 that APDs of the forearm models presented in Fig. 1(b) and a forearm model presented in [Reference Sasaki, Mizuno, Wake and Watanabe32] are in good agreement inside the first 1 mm thickness of the skin. Then, the thickness of the skin and other tissues makes a difference on the APDs of the forearm models. After 2 mm skin tissue depth, the APD of the closed-eye model decreases faster than the APDs of other models because tissues in the eye models are different from the tissues in other models, and the APDs of other models decreases at different rates toward deeper tissues because the arrangement and thickness of tissues in all models are different.

Figure 2. APDs of five different models and 1D forearm model presented in [Reference Sasaki, Mizuno, Wake and Watanabe32] when the IPD is 1000 W/m² for 30 GHz.

The heating factors of the 1D forehead, chest, and forearm models are calculated up to 100 GHz for the IPD of 10 W/m². In Fig. 3, the obtained heating factors of the 1D models are compared with those presented in [Reference Funahashi, Hirata, Kodera and Foster23] of a simple 1D planar model. They have good consistency. Above 6 GHz, the maximum heating factors of all 1D models in the skin are less than 0.025 [°C$ \cdot $m²/W], which is the maximum value of the heating factors recommended in [Reference Sasaki, Mizuno, Wake and Watanabe32]. Also, the heating factors of all models are in good consistence with those of a 2D multi-layered model presented in [Reference Diao, Rashed and Hirata29] which are around 0.02 [°C$ \cdot $m²/W] at frequencies above 10 GHz. As can be seen from Fig. 3, the heating factors of the models are different over the frequency range, and they are all frequency-dependent at frequencies below 60 GHz, but not frequency-dependent at frequencies above 60 GHz. The heating factors obtained in [Reference Funahashi, Hirata, Kodera and Foster23] for a simple 1D model and obtained here for different 1D multi-layered models are in acceptable agreement.

Figure 3. Heating factors of the simple 1D model in [Reference Funahashi, Hirata, Kodera and Foster23] and 1D models with 2 mm and realistic skin thickness.

Effect of skin thickness in the 1D multi-layered models on the EM exposure metrics

In Fig. 3, the heating factors of all 1D models having realistic skin thickness are compared with those having 2 mm skin thickness to show the effect of skin thickness. It can be seen from Fig. 3 that the heating factors of the models with realistic skin thickness are greater than those with 2 mm skin thickness above 30 GHz. Therefore, the heating factors of all models are significantly affected by the change in the skin thickness.

Figure 4 shows the APD distributions in the forehead, chest, and thigh models with the skin thickness of 2 mm and realistic values in Table 1 for the IPD of 1000 W/m² at 30 and 60 GHz. In Fig. 4, there is no significant difference in the APD distributions inside the skin tissues of the models, whereas different skin thickness of the models makes respectable difference on the APD distributions in the deeper tissues. It can be seen from Fig. 4 that APDs of the models with realistic skin thickness in the deeper tissues are higher than those of the models with 2 mm skin thickness.

Figure 4. APDs of different models with 2 mm and realistic skin thickness for the IPD of 1000 W/m² at (a) 30 and (b) 60 GHz.

PTCs of the forehead, chest, and thigh models with 2 mm and realistic skin thicknesses are shown in Fig. 5. The PTCs of different models are significantly affected by the change of skin thickness, especially at frequencies below 30 GHz. The PTC values of the models with 2 mm skin thickness are the same above 30 GHz due to their same skin thicknesses, whereas the change in the skin thickness makes a difference in the PTCs values of the models below 60 GHz. The fluctuations occurred in PTC depend on the frequency of EM exposure and the skin thickness of the models because the skin tissue of the models acts as an impedance matching layer.

Figure 5. PTCs of the forehead, chest, and thigh 1D models with 2 mm and realistic skin thickness.

EM analysis of eye layer with open and closed eyelid

Effects of the eyelid and its thickness in the eye model given in Fig. 1(b) on the EM wave exposure metrics are investigated up to 100 GHz. In the original voxel body model, the thickness of eyelid is 2 mm, whereas the realistic eyelid thickness is 0.57 mm, presented in Table 1.

Figure 6 shows the PTCs of the open and closed eye models with 2 mm and 0.57 mm eyelid thickness. The power transmission in the eye models is significantly affected by the presence and thickness of eyelid in the eye model. Above 10 GHz, the presence of the eyelid significantly increases the power transmission, while the eyelid thickness does not make a significant difference in PTC values. It can also be noticed at frequencies between 3 GHz and 10 GHz that the power transmission in the eye model with 2 mm eyelid thickness is higher than others. The heating factors of the open-eye model and closed-eye model with 2 mm and 0.57 eyelid thickness are shown in Fig. 7. The heating factors are affected by the presence and thickness of eyelid in the eye model. It is clearly seen that the heating factor values in the absence of the eyelid are higher than in the presence of the eyelid. The heating factors for all eye models are less than the recommended value of 0.025 [°C$ \cdot $m²/W] in [Reference Sasaki, Mizuno, Wake and Watanabe32]. It is also worth noting that EM absorption by the eye model may also be affected by the curvature of the eye and the presence of other curved tissues on the face, such as the nose.

Figure 6. PTCs of the open-eye and closed-eye models with 2 mm and 0.57 mm eyelid thicknesses.

Figure 7. Heating factors of the open-eye and closed-eye models with 2 mm and realistic eyelid thicknesses.