## 4 Materials and Methods
### Field from Annular Dipole Antenna Array and Optimization Procedure
In a general case where the amplitude and the phase of the \(n\)-th dipole antenna are \(A_{n}\) and \(\phi_{n}\), respectively, then the effective \(E\)-field at the current point \(\mathbf{r}\) (see Equation (5)) affected by the element matrix of \(N=8\) dipole antennas is established by  
\[\mathbf{E}\left(A_{1},\ \ldots\,A_{N},\ \phi_{1},\ \ldots\,\phi_{N}, \mathbf{r}\right)=\sum\limits_{n=1}^{N}A_{n}\exp(\mathrm{j}\phi_{n})\ \cdot\sum\limits_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[\mathrm{j}\Phi_{n,i}( \mathbf{r})]\ \mathbf{e}_{i} \tag{6}\]  
In the following optimization procedure, it is necessary to determine the specific absorption rate SAR (W/kg) used in the modified [PERSON] bioheat transfer model (Equation (1)). By combining Equations (2) and (6), one can obtain the resultant value for the SAR parameter at the given location derived from the \(N\)-element dipole antenna array, i.e.,  
\[\text{SAR }(A_{1},\ \ldots\ \mathcal{A}_{N},\ \phi_{1},\ \ldots\ \phi_{N}, \mathbf{r})=\sum\limits_{n=-1}^{N}\sum\limits_{m=-1}^{N}A_{n}\frac{x,y,z}{z_{ 0}}\sum\limits_{i}^{x,y,z}E_{n,i}(\mathbf{r})E_{n,i}(\mathbf{r})\cos\left[ \Phi_{n,i}(\mathbf{r})\!-\!\Phi_{m,i}(\mathbf{r})\!+\!\phi_{n}\!-\!\phi_{m}\right] \tag{7}\]  
Therefore, the SAR coefficient obtained from the \(N=8\) dipole antenna array is the sum of \(N^{2}=64\) terms for each of the spatial components of the \(\mathbf{E}\)-field vector (\(E_{x}\), \(Ey\), \(Ez\)). Then, using the following substitution,  
\[\sum\limits_{i}\alpha_{i}\cos(\omega_{i}+\phi)=\beta\cos(\Omega+\phi) \tag{8}\]  
where  
\[\beta=\sqrt{\left(\sum\limits_{i}\alpha_{i}\cos(\omega_{i})\right)^{2}+\left( \sum\limits_{i}\alpha_{i}\sin(\omega_{i})\right)^{2}} \tag{9}\]  
and  
\[\Omega=\arctan\left[\frac{\sum\limits_{i}\alpha_{i}\sin(\omega_{i})}{\sum \limits_{i}\alpha_{i}\cos(\omega_{i})}\right] \tag{10}\]  
yields the SAR definition that depends from two spatial-dependent parameters but which at the same time is independent of the dipole antennas' amplitudes and phases, namely,  
\[\text{SAR }(A_{1},\ \ldots\,A_{N},\ \phi_{1},\ \ldots\,\phi_{N}, \mathbf{r})=\sum\limits_{m=-1}^{N}\sum\limits_{n=1}^{N}A_{m}A_{n}\ \beta_{mn}(\mathbf{r})\cos[\Omega_{mn}(\mathbf{r})+\phi_{n}-\phi_{m}] \tag{11}\]  
with  
\[\mathbf{E}_{n}(\mathbf{r})=\sum_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[i\Phi_{n,i} (\mathbf{r})]\mathbf{e}_{i} \tag{5}\]0  
and  
\[\mathbf{E}_{n}(\mathbf{r})=\sum_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[i\Phi_{n,i} (\mathbf{r})]\mathbf{e}_{i} \tag{5}\]1  
In the optimization procedure for amplitude- and phase-dependent power deposition inside the volume of interest, it is necessary to establish the average SAR value for such volume, defined as  
\[\mathbf{E}_{n}(\mathbf{r})=\sum_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[i\Phi_{n,i} (\mathbf{r})]\mathbf{e}_{i} \tag{5}\]2which after discretization and once Equation (11) is taken into account gives the following relation:  
\[\mathbf{E}_{n}(\mathbf{r})=\sum_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[i\Phi_{n,i} (\mathbf{r})]\mathbf{e}_{i} \tag{5}\]3  
where \(M\) is the number of volumetric cells into which the selected volume \(V\) was discretized. \(M\) can vary from several to several thousand cells, depending on the volume of \(V\). The equivalent equation with parameters independent of the coordinates take the following form:  
\[\mathbf{E}_{n}(\mathbf{r})=\sum_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[i\Phi_{n,i} (\mathbf{r})]\mathbf{e}_{i} \tag{5}\]4  
where  
\[\mathbf{E}_{n}(\mathbf{r})=\sum_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[i\Phi_{n,i} (\mathbf{r})]\mathbf{e}_{i} \tag{5}\]5  
and  
\[\mathbf{E}_{n}(\mathbf{r})=\sum_{i}^{x,y,z}E_{n,i}(\mathbf{r})\exp[i\Phi_{n,i} (\mathbf{r})]\mathbf{e}_{i} \tag{5}\]6  
It is necessary to emphasize that Equation (16) does not depend on the location (spatial-dependent parameters) or the amplitude and the phase of individual antennas in the matrix, but only on the specific excitation parameters of the dipole antennas, namely, \(\beta^{\prime}_{\ mm}\) and \(\Omega^{\prime}_{\ mm}\).