# Assessment and Categorization of Biological Effects and
## 3 RF-EMF Exposure
### Governing Mathematical Equations  
The governing equations are the Maxwell EMF equations and the [PERSON]'s bio-heat tissue equation [46]. The external source in the bio-heat equation corresponds to the SAR, which is determined from the EMF equations. These governing equations can be given by:  
\[\mathbf{\
abla}\times\mathbf{H}=\mathbf{J} \tag{3}\]  
\[\mathbf{J}=\mathbf{\sigma}\;\mathbf{E}+\mathbf{j}\;\omega\;\mathbf{D}+\mathbf{J} \mathbf{e} \tag{4}\]  
\[\mathbf{E}=-\mathbf{\
abla}\;\mathbf{V}-\mathbf{j}\;\omega\;\mathbf{A} \tag{5}\]  
\[\mathbf{B}=\mathbf{\
abla}\times\mathbf{A} \tag{6}\]  
\[\mathbf{c}\;\rho\;\partial\mathrm{T}/\partial\mathbf{t}=\mathbf{\
abla}\;(\mathbf{ k}\;\mathbf{\
abla}\mathrm{T})+\rho\;(\mathrm{SAR})+\mathrm{q}^{\mathrm{met}}-c_{b}\;\rho\;_{b}\; \omega_{\mathrm{b}}\;(\mathrm{T}-\mathrm{T}_{\mathrm{b}}) \tag{7}\]  
In the EM Equations (3)-(6), \(\mathbf{H}\) and \(\mathbf{E}\) are the magnetic and electric fields, \(\mathbf{B}\) and \(\mathbf{D}\) are the magnetic and electric inductions, \(\mathbf{A}\) and \(\mathbf{V}\) are the magnetic vector and electric scalar potentials. \(\mathbf{J}\) and \(\mathbf{J}\mathbf{e}\) are the total and source current densities, \(\mathbf{\sigma}\) is the electric conductivity and \(\omega\) is the frequency pulsation. The symbol \(\mathbf{\
abla}\) is a vector of partial derivative operators, and its three possible implications are gradient (product with a scalar field), divergence and curl (dot and cross products, respectively, with a vector field). The magnetic and electric comportment laws between \(\mathbf{B}\)/\(\mathbf{H}\) and \(\mathbf{D}\)/\(\mathbf{E}\) are represented by the permeability \(\upmu\) and the permittivity \(\varepsilon\), respectively.  
As well in the [PERSON]'s bio-heat Equation (7), \(\mathbf{k}\) is tissue thermal conductivity, \(\mathrm{T}\) is local temperature of tissue, qnet is the basal metabolic heat source in \(\mathrm{W}/\mathrm{m}^{3}\), \(\mathrm{c}_{\mathrm{b}}\) is blood specific heat in \(\mathrm{J}/(\mathrm{kg.^{\circ}C})\), \(\rho_{b}\) is blood density in \(\mathrm{kg}/\mathrm{m}^{3}\), \(\omega_{\mathrm{b}}\) is blood perfusion rate (1/s), \(\mathrm{T}_{\mathrm{b}}\) blood temperature. \(\mathbf{\
abla}\cdot(\mathbf{k}\;\mathbf{\
abla}\mathrm{T})\) represents simple heat equation in differential form and \(\rho\;(\mathrm{SAR})\) represents the influence of electromagnetic energy absorbed in the human tissues.  
The solution of Equations (3)-(6) allows us to determine the induced EMF, for a given frequency pulsation, in the body tissues. As well, the SAR (1) can be computed using the fields resulting from such solution. Additionally, the EMC analysis checking the perturbations due to EMF exposers of instruments (embedded or not) can be verified from such solution. [PERSON]'s bio-heat equation [46] given by (7) is usually used to determine heat transfer in living tissues. Thermal behavior in tissues due to EMF exposure through the SAR is governed by Equations (1), (3)-(7). The EMF equations and the heat transfer equation must be solved in a coupled fashion. Due to slow thermal time behavior compared to fast EM time behavior, weak coupling with successive solution can be used.