## 4 Discussion  
An Electric-Potential Estimation (EPE) method for accurate prediction of the potential distribution in the vicinity of cylindrical electrodes was developed. This study demonstrated that the use of EPE can significantly improve the accuracy of both the electric-potential distribution and the electrical resistance for any grid resolution and electrodes configuration (diameter, orientation, and position). The improvement is particularly large at low-resolution, which can be attributed to the fact that EPE creates a grid-independent rendition of the electrode in the grid. Specifically, the EPE considers the circular surface of the applied electrodes by reshaping the electric-potential distributions in the direct vicinity and between the electrodes, resulting in a more accurate prediction of the electric-field distribution, particularly, within low-resolution models with a voxel size larger than the electrode diameter. The improvement of the accuracy means the low-resolution model becomes reliable, which is associated with a significant gain in the calculation time of, e.g., 15,000\(\times\) as was presented in Section 3.3. Application in Heterogeneous Tumor Models, which makes it attractive to be implemented into the numerical treatment planning of electrical energy-based ablation modalities, such as IRE. Next to IRE, the EPE can also be utilized in numerical calculations of other electrical energy-based ablation modalities such as reversible electroporation, and thermal ablation, where no local adaptive mesh refinement is utilized. For example, [PERSON] and [PERSON] investigated the effects of realistic geometry and blood vessels on the treatment outcome  
Figure 13: Electric-field distributions of Case 2 within the x,z-plane of the tumor model in Figure 5. Electric-field distribution of the high-resolution model (black line), and low-resolution models excluding EPE (+) and including EPE (*) along the lines O\({}_{1}\) (**A**) and O\({}_{2}\) (**B**). The grey rectangles represent the electrodes. of radiofrequency ablation using the Finite Difference Method by modeling a single cylindrical electrode with a diameter of \(1.5\times 10^{-3}\) m within an uniform grid (voxel size of \(0.5\times 10^{-3}\) m in the plane normal to the electrode) [21; 34]. As illustrated in Figure 1, a slight shift in the electrode position can alter the electrode discretization resulting in inaccurate electric-field distribution. In such cases, the EPE can be applied to improve the accuracy of the electric-potential distribution in the vicinity of the electrode. In this example, the coaxial cable model is recommended to validate the modeling of a single electrode, though one may also use different analytical models. For example, in a study performed by [PERSON] et al. [15], in which numerical modeling of electroporation was provided in the liver, the authors designated a fixed electric-potential amplitude of \(\Phi_{1}/5\) (V) to the estimation voxels to avoid a large increase in electric conductivities in the vicinity of the needle electrodes due to the overestimation of the electric field. For a low-resolution grid, a fixed electric-potential can distort the electric-field distribution in between the needle pairs, and therefore, one must be cautious with the selection of the appropriate electric-potential values near the electrode. In this study the presented homogeneous and heterogeneous examples, e.g., in Figures 4 and 5, are less complex than the real anatomies, since the real anatomies display far more spatial heterogeneity in tissue properties (different tissue types). Still, reconstruction of these real anatomies is based on 1-2 mm resolution CT or MR scans, thus we can expect that our low-resolution simulations will yield reliable results in realistic clinical heterogeneous tumor models. In addition to the simplification of the tissues, the electrical conductivities in this study were assumed to be constant, as we focused on evaluating the mechanism of EPE.