The specific absorption rate of tissues in rats exposed to electromagnetic plane waves in the frequency range of 0.05–5 GHz and SARwb in free-moving rats

Abstract

As electromagnetic exposure experiments can only be performed on small animals, usually rats, research on the characteristics of specific absorption rate (SAR) distribution in the rat has received increasing interest. A series of calculations, which simulated the SAR in a male rat anatomical model exposed to electromagnetic plane waves ranging from 0.05 to 5 GHz with different incidence and polarization, were conducted. The whole-body-averaged SAR (SARwb) and the tissue-averaged SAR (SARavg) in 20 major tissues were determined. Results revealed that incidence has great impact on SAR in the rat at higher frequencies owing to the skin effect and the effect on SARavg in tissues is much more apparent than that on SARwb; while polarization plays an important role under lower frequencies. Not only the incidence, but also the polarization in the rat keeps changing when the rat is in free movement. Thus, this article discussed a convenient way to obtain relatively accurate SARwb in a free-moving rat.

Appendix

Taking back irradiation for example, providing that the angle between the polarization and the long axis of the body is $\uptheta$, the effect caused by the external electric field $\left| \overrightarrow{E} \right|$ could be seen as the synergistic effect caused by the two fields, i.e. $\left| \overrightarrow{E} \right|\cos \theta$ along the long axis and $\left| \overrightarrow{E} \right|\sin \theta$ perpendicular to the long axis. As the electric field penetrates into the rat body, the value of these two fields in the body become $\alpha \left| \overrightarrow{E} \right|\cos \theta$ and $\beta \left| \overrightarrow{E} \right|\sin \theta$, respectively. α is the ratio of the electric field strength along the long axis inside to that outside the body. β is the ratio of the sselectric field strength perpendicular to the long axis inside to that outside the body.

$$SAR_{\theta } = \frac{\sigma }{\rho } \times \left[ {\alpha ^{2} E^{2} \left( {\cos \theta } \right)^{2} } \right] + \frac{\sigma }{\rho } \times \left[ {\beta ^{2} E^{2} \left( {\sin \theta } \right)^{2} } \right] = \frac{{\sigma \times E^{2} }}{\rho } \times \left[ {\alpha ^{2} \left( {\cos \theta } \right)^{2} + \beta ^{2} \left( {\sin \theta } \right)^{2} } \right]$$

$$\begin{gathered} \therefore {\text{ }}\overline{{SAR_{\theta } }} = \frac{{\mathop \sum \nolimits_{i} SAR_{{\theta _{i} }} }}{i} = \frac{{\mathop \sum \nolimits_{i} \frac{{\sigma \times E^{2} }}{\rho } \times \left[ {\alpha ^{2} \left( {\cos \theta _{i} } \right)^{2} + \beta ^{2} \left( {\sin \theta _{i} } \right)^{2} } \right]}}{i} \hfill \\ = \frac{{\sigma \times E^{2} }}{\rho } \times \left[ {\alpha ^{2} \overline{{\left( {\cos \theta } \right)^{2} }} + \beta ^{2} \overline{{\left( {\sin \theta } \right)^{2} }} } \right] \hfill \\ \end{gathered}$$

As the random distribution of $\theta$ in the interval $\left[0,\frac{\pi }{2}\right]$, probability density function (PDF) is

$$\text{f}\left( \theta \right)=\frac{1}{\frac{\pi }{2}}=\frac{2}{\pi }$$

$$\left( \theta \in \left[ 0,\frac{\pi }{2} \right] \right)$$

$$\therefore {\text{ }}\overline{{\left( {\cos \theta } \right)^{2} }} = {\text{EX}}\left[ {\left( {\cos \theta } \right)^{2} } \right] = \mathop \smallint \limits_{0}^{{\frac{\pi }{2}}} \left( {\cos \theta } \right)^{2} \times {\text{f}}\left( \theta \right)d\theta = \mathop \smallint \limits_{0}^{{\frac{\pi }{2}}} \left( {\cos \theta } \right)^{2} \times \frac{2}{\pi }d\theta = \frac{1}{2}$$

$$\overline{{{\left( \sin \theta \right)}^{2}}}=\text{EX}\left[ {{\left( \sin \theta \right)}^{2}} \right]=\underset{0}{\overset{\frac{\pi }{2}}{\mathop \int }}\,{{\left( \sin \theta \right)}^{2}}\times \text{f}\left( \theta \right)d\theta =\underset{0}{\overset{\frac{\pi }{2}}{\mathop \int }}\,{{\left( \sin \theta \right)}^{2}}\times \frac{2}{\pi }d\theta =\frac{1}{2}$$

$$\therefore {\text{ }}\overline{{SAR_{\theta } }} = \frac{{\sigma \times E^{2} }}{\rho } \times \left[ {\alpha ^{2} \overline{{\left( {\cos \theta } \right)^{2} }} + \beta ^{2} \overline{{\left( {\sin \theta } \right)^{2} }} } \right] = \frac{{\sigma \times E^{2} }}{\rho } \times \left[ {\alpha ^{2} \times \frac{1}{2} + \beta ^{2} \times \frac{1}{2}} \right]$$

$$\text{SA}{{\text{R}}_{45}}=\frac{\sigma }{\rho }\times \left[ {{\alpha }^{2}}{{E}^{2}}{{\left( \cos 45 \right)}^{2}} \right]+\frac{\sigma }{\rho }\times \left[ {{\beta }^{2}}{{E}^{2}}{{\left( \sin 45 \right)}^{2}} \right]=\frac{\sigma \times {{E}^{2}}}{\rho }\times \left[ {{\alpha }^{2}}\times \frac{1}{2}+{{\beta }^{2}}\times \frac{1}{2} \right]$$

$$\therefore {\text{ }}\overline{{SAR_{\theta } }} = {\text{SAR}}_{{45}}$$

In a word, the average SARwb under one incidence is equivalent to the SARwb when the angle between the body axis and the polarization is ${45}^{\circ}$.