## Materials and Methods
### Modelling of the Electrode-Tissue Interface  
In the numerical model, the electrochemical processes at the electrode-electrolyte interface between the electro-stimulating implant and the conductive tissue need to be taken into account, because part of the applied potential drops over this interface layer. These processes include the capacitive charging of the electrical double layer (non-Faradaic processes) as well as transfer of charges through the interface (Faradaic processes), as they determine the ratio of flowing current and the associated voltage drop. The electrical double layer results from the redistribution of ions in the surrounding electrolyte when interacting with the charged electrode surface. Its pseudocapacitive behavior is empirically modelled by a so-called constant phase element (CPE) [56]. This means that a constant phase difference between voltage and current exists, but generally with a larger value than the \(-90^{\circ}\) that would apply to a pure capacitance. The CPE is described by the equation  
\[Z_{\text{CPE}}=K\left(j\frac{\omega}{\omega_{0}}\right)^{-\beta}\,, \tag{1}\]  
with \(K\) being the ratio of the amplitudes of voltage and current, \(j\) the imaginary unit, \(\omega\) the angular frequency \(\omega=2\pi f\), and \(\omega_{0}=1\) s\({}^{-1}\) a normalization frequency to account for proper units of \(\Omega\) for \(Z_{\text{CPE}}\). The parameter \(\beta=0,\dots,1\) reflects the frequency dependence of the CPE and how much it deviates from a pure capacitance (\(\beta=1\)). The electrode-tissue interface is modelled via an equivalent circuit model that is commonly used to model simple systems: a parallel connection of the impedance of a constant phase element \(Z_{\text{CPE}}\) and a charge transfer resistance \(R_{\text{CT}}\)[56], see Figure 2. The impedance of the tissue \(Z_{\text{Tissue}}\) is defined by its dielectric properties \(\sigma\) and \(\varepsilon_{r}\) (ref. Section 2.2). \begin{table}
\begin{tabular}{c c c} \hline
**Tissue/Material** & \(\sigma\) **(S/m)** & \(\varepsilon_{r}\) \\ \hline Cortical bone [50] & 0.02 & 25,100 \\ Cancellous bone [50] & 0.079 & 4,020,200 \\ Soft tissue (buccal mucosa) [51] & 0.01 & \(3\times 10^{6}\) \\ Skin [50] & \(2\times 10^{-4}\) & 1140 \\ Ti6 Al4V [53] & \(5.6\times 10^{5}\) & 1 \\ PEEK [54] & \(10^{-12}\) & 3.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Dielectric properties (electric conductivity \(\sigma\) and relative permittivity \(\varepsilon_{r}\)) of the tissues and materials used in the simulation at \(f=20\) Hz. Thus, each of the two electrode-tissue interfaces of the bipolar electrode is described by one double layer impedance \(Z_{\text{EDL,i}}\) (\(i=1,\,2\)), with  
\[\frac{1}{Z_{\text{EDL,i}}} = \frac{1}{Z_{\text{CPE,i}}}+\frac{1}{R_{\text{CT,i}}} \tag{2}\] \[= \left(\frac{A_{\text{meas}}}{A_{\text{el,i}}}K\left(j\frac{\omega} {\omega_{0}}\right)^{-\beta}\right)^{-1}+\left(\frac{A_{\text{meas}}}{A_{\text {el,i}}}R_{\text{CT}}\right)^{-1}\,. \tag{3}\]  
In Equation (3), the contributions to the impedance have been scaled with respect to the surface of the measurement electrode used in electrochemical impedance spectroscopy (EIS) measurements with \(A_{\text{meas}}=314\,\text{mm}^{2}\)[57] and the surface of the stimulation electrode \(A_{\text{el,i}}\) used in the simulations. The parameters (\(\beta\), \(K\), \(R_{\text{CT}}\)) to model the electrode-tissue interface have been derived via EIS measurements of polished titanium specimens [57]. Their values from [57] are \(\beta=0.95\), \(K=\ (Y_{0}[1/F])^{-\beta}\,\Omega=(38.95\times 10^{-6})^{-0.95}\,\Omega=15453 \,\Omega\), \(R_{\text{CT}}=137.7\,\text{k}\Omega\). Please note that in EIS measurements, using the capacitance parameter \(Y_{0}\) instead of the parameter \(K\) (used by [PERSON] and [PERSON] [56]) is more common. The charge transfer resistance \(R_{\text{CT}}\) is generally non-linear, but as we use rather small stimulation voltages we may assume its measured value to be also valid in our simulations.