Relationship between peak spatial-averaged specific absorption rate and peak temperature elevation in human head in frequency range of 1–30 GHz

In this study, both simplified (multi-layer cubes) and realistic (MRI-based) anatomical models were considered. Specifically, a seven-layer model, consisting of the skin (1.5 mm), fat (1.5 mm), muscle (2.5 mm), skull (4.5 mm), dura (1.0 mm), cerebrospinal fluid (1.0 mm), and brain (sufficiently thick), was considered as the model of the head (Drossos et al 2000).

MRI-based human voxel models for different ages, genders, and races were considered as realistic anatomical models. This study employed the Japanese male model TARO and female model HANAKO (Nagaoka et al 2004). In addition, the European models of Virtual Family (Duke, Billie) (Christ et al 2010) and NORMAN (Dimbylow 1997) were considered. These models have a resolution of a few millimetres and are segmented into 51–80 anatomical tissues and organs. All the models were truncated at the bottom of the neck because the rest of the body does not affect the human interaction with the dipole antenna.

The finite-difference time-domain (FDTD) method (Taflove and Hagness 2003) was used to conduct electromagnetic dosimetry on a human body exposed to microwaves emitted from a dipole antenna. The SAR is defined as

$$\begin{eqnarray}&&\text{SAR}\left(\mathbf{r}\right)=\frac{\sigma \left(\mathbf{r}\right)}{2\rho \left(\mathbf{r}\right)}|\mathbf{E}\left(\mathbf{r}\right){{|}^{2}}\end{eqnarray} \tag{ 1 }$$

where |E| is the peak value of the electric field at position r and σ and ρ are the conductivity and mass density, respectively, of the tissue. The dielectric properties of the tissues were determined with a 4-Cole–Cole dispersion model (Gabriel et al 1996), where the upper frequency at which the measured data were considered is 20 GHz. As the frequency increased, the penetration depth of the electromagnetic wave decreased, indicating that most of the energy is absorbed in the skin layer at high frequencies. At a frequency of 20 GHz, the difference between the skin dielectric properties of the two research groups is a few percent, as summarised in Sasaki et al (2014). Note that the conductivity of the child tissue was assumed to be identical to that of the adult because the variation in human tissues is within a few percent across age (Wang et al 2006).

A major source of the computational error in the FDTD method is the discretization error, which depends on the ratio between the cell size and the wavelength in biological tissue. A well-known rule to suppress the numerical dispersion error in FDTD simulations is that the maximum cell size should be smaller than one tenth of the wavelength; the numerical error is less than 10% for the local 10 g averaged and the whole-body averaged SAR under this condition (Pickwell et al 2004, Dimbylow and Bolch 2007, Kühn et al 2009, Laakso 2009). The original resolution of TARO, HANAKO, and NORMAN is 2 mm. Thus, each voxel was further divided into 64 voxels with a resolution of 0.5 mm up to 10 GHz and into 512 voxels with a resolution of 0.25 mm above 10 GHz to keep the computational accuracy as high as possible. An in-house smoothing algorithm with manual editing was applied in order to maintain the reliability of the models (Nagaoka et al 2005). Similarly, the finest resolution of Duke and Billie is 0.5 mm and thus one cell was divided into 64 with a resolution of 0.25 mm for frequencies higher than 10 GHz. Even if we changed the resolution of the model resolution higher than 0.25 mm, almost identical results were obtained even at 30 GHz. Another factor affecting the numerical error is the reflection error of the absorbing boundary, but it is negligibly small as reported in previous studies (Lazzi et al 2000, Findlay and Dimbylow 2006, Laakso et al 2007, Kühn et al 2009). Our computational results are shown to be in good agreement with those by other research groups via inter-comparison (Dimbylow et al 2008).

The temperature in the human model is computed by solving the bioheat equation (Pennes 1948). This equation, which takes into account various heat exchange mechanisms such as heat conduction, blood perfusion, and electromagnetically induced heating, is represented by the following expression:

$$\begin{eqnarray}&&C\left(\mathbf{r}\right)\rho \left(\mathbf{r}\right)\frac{\partial T\left(\mathbf{r},t\right)}{\partial t}=\nabla \cdot \,\left(K\left(\mathbf{r}\right)\nabla T\left(\mathbf{r},t\right)\right)+\rho \left(\mathbf{r}\right)\text{SAR}\left(\mathbf{r}\right)+Q\left(\mathbf{r},t\right)-B\left(\mathbf{r},t\right)\left(T\left(\mathbf{r},t\right)-{{T}_{\text{B}}}\left(\mathbf{r},t\right)\right)\end{eqnarray} \tag{ 2 }$$

where T is the temperature of the tissue, T_B is the blood temperature, C is the specific heat of the tissue, K is the thermal conductivity of the tissue, Q is the metabolic heat generation, and B is the term associated with blood perfusion. Furthermore, r and t are the position vector and time, respectively.

The temperature elevation due to handset antennas can be considered to be small enough that it does not activate the thermoregulatory response, because the temperature elevation at the SAR limit (2 W kg⁻¹ for the general public) is less than 0.5 °C (<3 GHz) (Hirata and Fujiwara 2009). Additionally, the blood temperature is assumed to be spatiotemporally constant because the power absorption due to antennas (the typical output power of handset antennas is less than a few hundred milliwatts) is much smaller than the metabolic heat generation of a male adult (~100 W), resulting in a marginal body-core temperature elevation. Then, the blood temperature in (2) is simplified as constant (${{T}_{\text{B}}}\left(\mathbf{r},t\right)$  =  37 °C). The boundary condition for (2) is given by

$$\begin{eqnarray}&&-K(r)\frac{\partial T\left(\mathbf{r},t\right)}{\partial n}=H\cdot \left({{T}_{\text{s}}}\left(\mathbf{r},t\right)-{{T}_{\text{e}}}(t)\right)\end{eqnarray} \tag{ 3 }$$

where H, T_s, and T_e respectively denote the heat transfer coefficient, the surface temperature of the tissue, and the temperature of air (independent of the position). Samaras et al (2006) pointed out the weakness of the boundary condition (3) for the bioheat equation when applied to the human model with the stair-casing approximation. Improved algorithm for reducing computational error has been proposed (Neufeld et al 2007, Laakso 2009). Such an algorithm would be essential when calculating the temperature itself. However, that error marginally influences temperature elevation, as shown in Wang and Fujiwara (1999). Thus, we used the conventional formula.

The bioheat equation subjected to the boundary condition was solved to obtain the thermally steady-state temperature elevation. Thus, the left-hand-side term of (2) was assumed to be zero and all the right-hand-side terms were independent of t. The equation was discretised using a finite difference method and solved by applying the geometric multi-grid method (Laakso 2009). The following were the parameters of the multi-grid method: the number of grid levels was 6, with the resolution halved for each level; the thermal properties of each lower-resolution grid were the arithmetic averages of those of the higher-resolution grid; the pre- and post-smoothing operations consisted of four and one iterations, respectively, of the Gauss–Seidel method; and finally, the prolongation and restriction operations were full weighting and linear interpolation. The iteration was continued until the relative residual of the solution was less than 10⁻⁷.

The thermal parameters used in this study are the same as those in Hirata and Shiozawa (2003). These parameters were borrowed primarily from Duck (1990). Dominant factors affecting the temperature elevation for localized exposure is the blood flow rate (Hirata et al 2006a, Laakso and Hirata 2011). According to our recent study, the variation of the temperature elevation due to variability in the measured blood perfusion rate is  ±20% for frequencies 1–3 GHz, gradually decreases with the increase of the frequency, and is  ±10% at frequencies above 10 GHz (Ota et al 2014). The heat transfer coefficient between the skin and air and that between the eye surface and air were set to 5.0 and 15 W (m² · °C)⁻¹, respectively, which are the typical values at a room temperature of 23 °C (Fiala et al 2001).

Our computational scheme for thermal computation was in good agreement with measured results for human exposure to ambient heat (Hirata et al 2015) as well as small animals exposed to microwaves (Hirata et al 2006b, Hirata et al 2011). In addition, our computational results for handset antennas were in good agreement with those by other groups for frequencies below 3 GHz, as summarized in IEEE-C95.1 (2006) and plane wave exposure (Hirata et al 2013) in comparison with Bakker et al (2011).

Figure 1 shows the exposure scenarios considered in this study. As shown in this figure, a separation distance of 15 mm was chosen from the closest model surface excluding the pinna and the half-wave dipole antenna. The reason the dipole antenna was chosen is that commercial communications systems for frequencies in the 6 GHz band or higher are currently not well established. In addition, the dipole antenna is often considered in human exposure standards; for example, WHO (2010) cites only one paper that derived the correlation between SAR and temperature elevation (Hirata and Fujiwara 2009). The lengths of the antenna were adjusted to 151, 76, 51, 38, 31, 25, 21, 17, 14, 13, 11, 8.5, 6.5, and 4.3 mm for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, and 30 GHz, respectively, so that it would resonate at the corresponding frequencies. The positions of the centre of the dipole antenna relative to the pinna are shown in figure 1. For comparison, the output power of the antenna is taken as 1 W. Note that the typical output powers of the mobile terminals in a few GHz band are less than a few hundred milliwatts.

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As a metric for the dosimetric evaluation, the peak value of the SAR averaged over 10 g of tissue was considered in the international guidelines and standards (ICNIRP 1998, IEEE-C95.1 2006). The SAR averaging schemes in the standards and guidelines differ in three aspects: the number of tissues allowed in the averaging process, the shape of the averaging volume, and the inclusion or exclusion of the pinna in the averaging volume. Based on these differences, six algorithms for SAR averaging, labelled algorithms (I)–(VI), were then considered, and their properties are listed in table 1. When considering the cubic averaging shape (algorithms (I) and (II)), the algorithm specified in the IEEE C95.3 standard (2002) was applied to clarify the treatment of the air region and the pinna. The pinna is treated separately from the head (IEEE-C95.1 2006) with a basic restriction of 4 W kg⁻¹, which is twice as large as the head and trunk and identical to that of the limbs. In contrast, no clear definition of the tissue number (stated to be contiguous) is provided in the ICNIRP guidelines (1998). Therefore, averaging algorithm (I) may correspond to that prescribed in the IEEE standard, whereas algorithms (IV) and (VI) correspond to the scheme in the ICNIRP guidelines. The peak temperature elevation in the pinna is not considered when algorithms (I), (III), and (V) are employed, because the SAR in the pinna is not considered.

In this study, we mainly focused on the heating factor, which is defined as the ratio of the peak temperature elevation to the peak 10 g-averaged SAR.

When the peak temperature evaluation is considered at the thermal steady state, it is appropriate to express the approximate peak temperature elevation in the head or brain as

$$\begin{eqnarray}&&\Delta {{T}_{\text{peak}}}=\alpha \times \text{SA}{{\text{R}}_{10\,\text{g}}},\end{eqnarray} \tag{ 4 }$$

where SAR, ΔT, and α are the SAR averaged over 10 g of tissue (W kg⁻¹), the peak temperature elevation estimated by the regression line (°C), and the slope of the regression line (°C · kg W⁻¹) which corresponds to the heating factor, respectively, and are determined by using the method of least squares (Sachs 2012). Note that the intercept of the regression line is set to zero because there is no temperature elevation without microwave power absorption. To evaluate the effectiveness of the estimation scheme for the peak temperature elevation, the coefficient of determination R² is introduced as

$$\begin{eqnarray}&&{{R}^{2}}=1-\frac{{\sum}^{}{{(\Delta {{T}_{i}}-\Delta {{{\widehat{T}}}_{i}})}^{2}}}{{\sum}^{}{{(\Delta {{T}_{i}}-\Delta {{{\overline{T}}}_{i}})}^{2}}},\end{eqnarray} \tag{ 5 }$$

where ΔT_i is the peak temperature elevation for the ith scenario, $\Delta {{\overline{T}}_{i}}$ is the mean value of ΔT_i, and $\Delta {{\widehat{T}}_{i}}$ is the value of ΔT_i estimated from the regression line.

In addition to the coefficient of determination, the coefficient of variation (CV) is introduced to discuss the variability of the heating factor at various frequencies. The coefficient of variation is defined as the ratio of the standard deviation of $\alpha$ to $\overline{\alpha}$.

Figure 2 shows the distributions of the SAR and temperature elevation at frequencies of 2 and 8 GHz. The dipole antenna is located at the reference position (black circle in figure 1) and oriented in the up–down (superior–inferior) direction. As shown in figure 2, the distribution of the temperature elevation is smoother than that of the SAR. The penetration depth of the temperature elevation is larger than that in the SAR. These differences are due to the fact that the heat diffusion length is larger than the penetration depth of the electromagnetic wave (Hirata et al 2006a, Samaras et al 2007), which is on the order of a few millimetres at these frequencies.

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Figure 3 shows the frequency dependence of the peak value of the SAR averaged over 10 g of tissue in TARO obtained using different averaging algorithms when the dipole antenna is located at position (e) and oriented in the up–down (superior–inferior) direction. As shown in figure 3, the different averaging algorithms yield different SAR tendencies, with the highest average SAR values appearing above 15 GHz for the cases in which the pinna is considered. This is because more than 60% of the power is absorbed around the pinna at frequencies higher than 7 GHz (see figure S1 (stacks.iop.org/PMB/61/5406/mmedia)). As the frequency increases, the penetration depth of the electromagnetic field decreases. The SARs computed with algorithms (III) and (V) and those with algorithms (IV) and (VI) are different at lower frequencies, but almost coincide with each other at frequencies higher than 12 GHz.

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In this section, the SAR and temperature elevation in five head models are computed for the dipole antenna located at the reference position in figure 1 and oriented in the up–down (superior–inferior) and front–back (anterior–posterior) directions; thus, 10 scenarios were considered for each frequency.

As shown in figures 4(a), (c) and (e), a comparison of the results of the anatomically based and multi-layer cube models using averaging algorithms (I), (III), and (V) reveals that the heating factors of the cube models are lower. The difference between the anatomically based and multi-layer models is more obvious for algorithms (III) and (V) than for algorithm (I). Figure 4 also demonstrates that the mean and median values of the heating factor peak at approximately 12 GHz for averaging algorithms (I) and (II) and approximately 6–10 GHz for the remaining algorithms. This broader frequency range may be attributable to the averaging shape of the SAR.

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As shown in figure 4(a), the median and mean values of the heating factor for averaging algorithm (I), which is similar to the averaging algorithm in the IEEE standard (IEEE-C95.1 2006), was rather flat (0.20–0.27 °C · kg W⁻¹) for frequencies up to 4 GHz, and then it increased gradually with a further increase in frequency but became unstable at frequencies higher than 12 GHz. As can be confirmed from figure 5(a), a similar CV is observed for algorithm (II).

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Figures 4(c) and (d) show the heating factors for averaging algorithms (III) and (IV), which permit any averaging volume shape but only a single type of tissue. One notable difference is the variability at 4 and 6 GHz in figure 4(d). It can be confirmed from figure 5(a) that the CV is largest at frequencies of 4 and 6 GHz. The CVs for algorithms (III) and (IV) are comparable to each other at frequencies higher than 10 GHz.

Figures 4(e) and (f) show the heating factor when any number of tissues is allowed in the averaging volume. As shown in figures 4(e) and (f), the tendencies of the heating factors are similar to those in the case of averaging algorithms (III) and (IV). However, the CVs of algorithms (V) and (VI) are smaller than those of algorithms (III) and (IV) at most frequencies.

At frequencies higher than 10 GHz, the CVs of algorithms (I) and (II) are smaller than those of algorithms (III)–(VI). The heating factors of algorithms (I) and (II) increase with increasing frequency for frequencies higher than 4 GHz.

Figure 6 demonstrates the effect of varying the antenna position (see figure 1) on the heating factor by using the head models Duke, TARO and NORMAN for averaging algorithms (I)–(VI). The reason for choosing these three models is that they are developed with different tissue classification algorithms. Comparing figure 6 with figure 4, similar mean and median values of the heating factors are observed in the two figures (see also figure S2). The antenna feeding point is a larger cause of variability in the heating factor than the model anatomy, which can be confirmed by comparing the respective CVs in figures 5(a) and (b). The variability becomes greater with increasing frequency. As with the case of model variability in section 3.2, the CVs of algorithms (I) and (II) are smaller than those of algorithms (III)–(VI) at frequencies higher than 10 GHz.

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The statistical variation of the heating factor is discussed to give insight into the international standards and guidelines. At each frequency, we considered 40 cases: two polarizations in five head models at the reference point and 11 feeding positions (including the reference point) in three head models for vertical polarization. The upper frequency where the SAR is applicable is 3 GHz in the IEEE standard and 10 GHz in the ICNIRP guidelines. Thus, a total of 120 cases (40 cases at 1, 2, and 3 GHz) were considered for the former, whereas 280 cases (40 cases at 1, 2, 3, 4, 6, 8, and 10 GHz) were considered for the latter. As mentioned above, the intercept of the regression line was set as non-existent. All the data (up to 30 GHz, 400 cases) were considered when discussing the upper applicable frequency of the SAR. Similarly, in figure S4, the data of 200 cases (40 cases at 1, 2, 3, 4, and 6 GHz) were also plotted for frequencies up to 6 GHz, which is the upper transition frequency of IEEE.

As shown in figure 7, the R² values for algorithms (I) and (V) are comparable to each other and show modest correlations (0.25  <  R²  <  0.64) between the SAR and temperature elevation. On comparison, algorithm (V) shows a better R² value than algorithm (III), suggesting that the inclusion of multiple tissues in the averaging volume provide better correlation. It can be expected from figure 5 that the CV for algorithm (V) is smaller than that for algorithm (III).

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As shown in figure 8, a modest correlation was observed between the SAR and temperature elevation for frequencies up to 10 GHz in comparison with algorithms (I), (III), and (V) where the SAR and temperature in the pinna are not considered. Strong correlations (R²  >  0.64) were observed between peak temperature elevation including the pinna and peak SAR with algorithms (II), (IV), and (VI), in which the SAR and temperature elevation in the pinna is considered, suggesting that the peak temperature elevation in the pinna can be estimated from the peak SAR when considering the pinna. Conversely, the SAR computed with algorithms (II), (IV), and (VI) may not be reasonably used to estimate the peak temperature elevation in the head excluding the pinna (see figure S3,); R² was in the range between 0.0801–0.354.

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As shown in figure 9, the values of R², except for those of algorithms (III) and (V), became small when the data for frequencies from 1 to 30 GHz were considered. The decrease in R² values in comparison with the case where the target frequency range was 1–10 GHz, except for the algorithms (III) and (V), is expected from figure 5 that the CVs at frequencies higher than 10 GHz are larger than those at lower frequencies. The increase in the R² values in algorithms (III) and (V) may be coincidental, but the mean and median values of the heating factor are rather uniform in the frequency range considered herein.

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The 95th percentile values of the heating factor are listed in table 2 for the above three frequency ranges. By multiplying the values presented in this table and the SAR averaged over 10 g of tissue, the temperature elevation in the head can be conservatively estimated. The 95th percentile value of the heating factor at frequencies under 3 GHz was the lowest and increased when the target frequencies extended to 10 and 30 GHz. This is because the SAR concentrates around the antenna due to both the shallower penetration depth of electromagnetic waves and smaller antenna dimensions, and thus the SAR averaged over the volume of 10 g of tissue may not be sufficient.

Figure 10 shows the heating factor in the brain of the different head models. Note that the peak 10 g-averaged SAR was calculated using averaging algorithms (I) and (IV). The same scenarios as in figure 4 were considered; the SAR and temperature elevation in five head models were computed for the dipole antenna located at the reference position and oriented in the up–down and front–back directions; 10 scenarios were considered for each frequency.

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As shown in figure 10, the brain temperature reduced with an increase in the frequency, which is attributable to the decrease in the penetration depth of the electromagnetic fields. However, the brain temperature did not converge to zero, because the heat generated around the surface may diffuse up to a few centimetres into the head. A similar tendency was also observed for the other averaging algorithms, but to avoid repetition, these results are not shown here.