## 2 Materials and Methods
### Application in Heterogeneous Tumor Models  
The next step is to assess the performance of EPE in realistic tumor models with heterogeneous tissue properties and different electrode orientations, as would occur in clinical use. An example of EPE application to a 3D model was presented to demonstrate the performance of EPE in treatment planning. A spherical pancreatic tumor illustrated in Figure 5 was modeled with a pair of cylindrical electrodes at its extreme ends. The diameter of the tumor and the electrodes lengths were assumed to be 20 \(\times\) 10\({}^{-3}\) m, and the interdistance between the electrodes was assumed to be 19 \(\times\) 10\({}^{-3}\) m with the presumed voltage of 2850 V across the electrodes (voltage-to-distance ratio of 1500 V\(\cdot\)cm\({}^{-1}\)) [30]. The electric-field threshold of the pancreatic cancer was assumed to be 500 V\(\cdot\)cm\({}^{-1}\), based on pancreatic cell line experiments [32]. We also changed the electrode orientation: the electrode-tumor model was rotated around the y-axis by 45\({}^{\circ}\) (see Figure 5A) and placed in the center of a cube representing healthy pancreatic tissue, with dimensions that vary between 90 \(\times\) 10\({}^{-3}\) and 94 \(\times\) 10\({}^{-3}\) m depending on voxel size. Furthermore, a centralized grid was applied to the cubic model with an equilateral voxel dimension of 2 \(\times\) 10\({}^{-3}\) m (diameter of a clinical needle electrode is \(\sim\)1 \(\times\) 10\({}^{-3}\) m) to simulate a worst-case scenario regarding electrode diameter, position, and orientation. Next, two heterogeneous model cases were presented:  
* Case 1: the electrical conductivities of the human cancerous (sphere) and healthy (cube) tissues were assumed to be 0.20 and 0.35 S\(\cdot\)m\({}^{-1}\), respectively [32];
* Case 2: the assumed conductivity values of these tissue types were interchanged.  
Along the length of the electrodes the electric-potential values of EPE were assumed to be constant. Furthermore, a Dirichlet boundary condition (\(\Phi\) = 0 V) was applied to all BOS of the model. The effect of EPE was then evaluated by determining the electric-field distributions after the exclusion and inclusion of EPE, and compare the results to the simulation results of the same model along the lines O\({}_{1}\) and O\({}_{2}\) (see Figure 5) with (1) high-resolution (an equilateral voxel dimension of 0.25 \(\times\) 10\({}^{-3}\) m), and (2) with a model in which the electrode axes were positioned on the x-axis and was parallel to the z-axis Figure 5B. Outcomes were expressed as a mean of the deviations of the electric-field distributions of the low-resolution models ex/including EFEs relative to the electric-field values at the corresponding position within the high-resolution model within a radius of 15 \(\times\) 10\({}^{-3}\) m from the center. The radius 15 \(\times\) 10\({}^{-3}\) m was selected to include the whole tumor and a part of the surrounding healthy tissue, which mimics the clinical ablation zone that aims  
Figure 5: A spherical pancreatic tumor with a pair of cylindrical electrodes at its extreme ends inside a cube of healthy pancreatic tissue. In (**A**) the cancer model was rotated around the y-axis by 45\({}^{\circ}\) in the x,z-plane, while in (**B**) the electrodes were positioned parallel to the z-axis. Electric-field distributions from (**A**,**B**) were compared along the lines O\({}_{1}\) and O\({}_{2}\) in the x,z-plane.  
to include the tumor-free margin [33]. The electric field was calculated using the method described in Appendix A.