## 2 Materials and Methods
### Analytical Validation  
The analytical solutions for the electric-potential distribution and electrical resistances were compared to the numerical solutions with and without using EPE for both centralized and shifted grids (see Figure 1). The comparison was performed for equilateral voxel  
Figure 4: (**A**) The transverse plane of cylindrical electrode model. The boundaries at the outer surface of the domain model (BOS) were assumed to be \(\Phi_{\text{BOS}}=0\) V. The modeling domain has an equilateral dimension in the x,y-plane. (**B**) The coronal plane of the model with the electrode of length L interposed between two isolation layers (colored aqua blue), through which J\({}_{\text{BOS}}=0\) A\(\cdot\)m\({}^{-2}\). (**C**) Discretization of the electrodes (represented by the circular lines). Electrode voxels are grey and their corresponding estimation voxels are represented by the orange voxels. The red contour represents the surfaces of the voxels that were used to calculate the electrical resistance. dimensions that varied between 0.5 \(\times\) 10\({}^{-3}\) m and 3.0 \(\times\) 10\({}^{-3}\) m in the (x, y)-plane, which are comparable with the pixel size range (0.5 \(\times\) 10\({}^{-3}\)-1.0 \(\times\) 10\({}^{-3}\) m) and slice thickness size range (2.0 \(\times\) 10\({}^{-3}\)-6.0 \(\times\) 10\({}^{-3}\) m) used for CT scans [20]. Choosing 3.0 \(\times\) 10\({}^{-3}\) m as the maximal voxel dimension yields sufficient voxels between the electrodes to provide an adequate overview of the electric-field distribution. Furthermore, the Dirichlet boundary condition (\(\Phi\) = 0 V) was applied to the remaining BOS of the model. As mentioned before, the electric-potential values of the estimation voxels are fixed. To calculate the electrical resistance considering EPE, the total electrical current was calculated using:  
\[\mathrm{I_{total}}=\mathbf{J}\cdot\mathbf{n}\mathrm{dA}\;, \tag{14}\]  
with \(\mathbf{n}\) as a unity vector normal to the red contour shown in Figures 3C and 4C. In a discrete manner [19; 27], the electric current density between two adjacent voxel nodes i and j can be calculated by:  
\[\mathrm{I_{ij}}=\mathrm{J_{ji}}\mathrm{A_{ij}}=\frac{\kappa_{ij}}{\mathrm{h_{ ij}}}(\Phi_{i}-\Phi_{j})\mathrm{A_{ij}}\;, \tag{15}\]  
with:  
\[\kappa_{ij}=\frac{2\kappa_{i}\kappa_{j}}{\kappa_{i}+\kappa_{j}}\;, \tag{16}\]  
where:  
* i is the node (the center) of an estimation voxel;
* j is the voxel node neighboring the voxel node i across the red contour;
* \(\mathrm{I_{ij}}\) (A) is the current flow between voxel nodes i and j;
* \(\mathrm{J_{ji}}\) (A\(\cdot\)m\({}^{-2}\)) is the current density between voxel nodes i and j;
* \(\mathrm{A_{ij}}\) (m\({}^{2}\)) is the area of the surface between voxel nodes i and j;
* \(\kappa_{i}\) (S\(\cdot\)m\({}^{-1}\)) is the admittance of the voxel with voxel node i, it is assumed that each voxel has a uniform admittance;
* \(\kappa_{j}\) (S\(\cdot\)m\({}^{-1}\)) is the admittance of the voxel with voxel node j;
* \(\kappa_{ij}\) (S\(\cdot\)m\({}^{-1}\)) is the effective admittance between the voxel nodes i and j;
* \(\mathrm{h_{ij}}\) (m) is the distance between the voxel nodes i and j;
* \(\Phi_{i}\) (V) is the electric-potential value at the voxel node i. Then, along the red contour (or for all estimation voxels) \(\mathrm{I_{total}}\) is calculated using:  
\[\mathrm{I_{total}}=\sum_{i}\sum_{j}\mathrm{I_{ij}} \tag{17}\]  
The analytical models described in the previous section were used to validate EPE, since their cylindrical shape is similar to the shape of IRE needle electrodes applied in the clinic. In the next paragraph, the choices for the model parameters are presumed based on parameters used in the clinic: For the coaxial cable model, the inner electrode was assumed to have a diameter of 1 \(\times\) 10\({}^{-3}\) m, which is comparable with the clinical IRE needle electrodes [30]. The outer electrode diameter was assumed to be 40 \(\times\) 10\({}^{-3}\) m to attain a distance of 20 \(\times\) 10\({}^{-3}\) m between the inner and outer electrode.