## 2 Materials and Methods
### Estimation of the RF Deposited Power at the Tips of a Pair of Generic Implants  
Initially, it was observed that when a wire was excited by a plane wave along its length, there was induced power at the tip of the proximal unexcited wire because of the coupling between them, as shown in Figure 1b. Therefore, the total absorbed power at the tip of each wire was considered as the summation of the scattered power caused by the incident electric field and the induced power caused by the coupling with the proximal wire. The total deposited power at the pair of wires was determined as the summation of the absorbed power at the tip of each individual wire. Similarly, the incident electric field along the trajectory of each wire was assumed as the summation of the incident field at its position and the scattered electric field due to the proximal wire. Moreover, as the presence of an implant influences the currents in a nearby implant, we followed the approach of [31], where the temperature rise was estimated with higher precision using the measured TF of each wire separately, instead of the consideration of both wires as a whole. In particular, we considered the TF of the excited lead by a current source, as well as the TF of the proximal unexcited wire. Taking the above into consideration, we modified the Tier 3 approach and proposed Equation (2) to calculate the deposited power at the tip of lead \(i\), when lead \(j\) is present. Table 1 defines the variables of the equation. \[P_{i}=\ A_{ii}\bigg{(}\int_{L}E_{tan,ji}(z)S_{ii}(z)dz\ \bigg{)}^{2}+A_{ij}\bigg{(} \int_{L}E_{tan,ij}(z)S_{ij}(z)dz\ \bigg{)}^{2} \tag{2}\]  
Hence, to evaluate the absorbed power around the tips of two leads (lead 1 and 2), we assessed the summation of the deposited power at the tip of each lead with Equation (3):  
\[ \begin{array}{l}P=\ A_{11}\big{(}\int_{L}E_{tan,21}(z)S_{11}(z)dz\ \big{)}^{2}+A_{12}\big{(}\int_{L}E_{tan,12}(z)S_{12}(z)dz\ \big{)}^{2}\\ +A_{22}\big{(}\int_{L}E_{tan,12}(z)S_{22}(z)dz\ \big{)}^{2}+A_{21}\big{(}\int_{L}E_{ tan,21}(z)S_{21}(z)dz\ \big{)}^{2}\end{array}  \tag{3}\]  
The electric field \(\mathrm{E_{tan,ij}}\) (z) was calculated from [19]:  
\[E_{tan,ij}(z)=E_{inc,j}(z)+E_{Si}(z)\int_{L}TF_{i}(z)E_{inc,i}(z)dz, \tag{4}\]  
where \(\mathrm{E_{inc,j}}\) (z) is the incident electric field along the path of the lead \(j\) without the presence of any proximal lead, whereas the rest of the equation defines the scattered electric field \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Variables Relative to the Induced Power**} & \multicolumn{2}{c}{**Variables Relative to the Induced Power**} \\ \multicolumn{2}{c}{**Cause the Tangential Electric Field**} & \multicolumn{2}{c}{**Cause the Coupling with the Proximal Lead**} \\ \hline \multirow{3}{*}{\(\mathrm{S_{ii}}\) (z)} & TF of the excited lead & \multirow{3}{*}{\(\mathrm{S_{ij}}\) (z)} & TF of the proximal \\  & \(i\), in the presence of lead \(j\) & & \\ \hline \multirow{6}{*}{\(\mathrm{A_{ii}}\)} & Deposited power at & \multirow{3}{*}{\(\mathrm{A_{ij}}\)} & Deposited power at \\  & the tip of lead \(i\) when & & the tip of lead \(i\), when \\  & excited along its & & lead \(j\) is excited along \\  & length with & & its length with \\  & iso-electric field of & & \\  & 1 V m\({}^{-1}\) rms, in the & & iso-electric field of \\  & presence of lead \(j\) & & 1 V m\({}^{-1}\) rms \\ \hline \multirow{6}{*}{\(\mathrm{E_{tan,ij}}\) (z)} & The incident & \multirow{3}{*}{\(\mathrm{E_{tan,ij}}\) (z)} & The incident \\  & tangential electric & & tangential electric \\  & field along the & & field along the \\  & trajectory of lead \(i\), & & trajectory of lead \(j\), \\ \cline{1-1}  & taking into account & & taking into account \\ \cline{1-1}  & also the scattered & & also the scattered \\ \cline{1-1}  & electric field caused & & electric field caused \\ \cline{1-1}  & by lead \(j\). & & by lead \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the variables of Equation (2).