## 2 Materials and Methods  
### Plan2 Heat: In-House Developed Software Package  
#### 2.1.1 Summary of the Applied Finite Difference Method in Plan2 Heat  
In this study the software package Plan2 Heat was used to simulate the electric-potential distribution and electrical resistance for IRE applications. Plan2 Heat is an in-house developed treatment planning package for a variety of hyperthermia applications [24; 25; 26], also supporting IRE simulations. Since the energy density of the electric field resulting from the IRE pulses is much larger than the energy density of the magnetic field, the electro-quasi-static approximation can be applied [12]. The quasi-static solver of Plan2 Heat calculates the electric-potential distribution using the [PERSON]'s equations by combining [PERSON]'s law of induction, and [PERSON]'s law  
\[\
abla\times\mathbf{E}=-\mathrm{j}\omega\mu_{x}\mu_{0}\mathbf{H}\,, \tag{1}\]  
\[\
abla\times\mathbf{H}=(\sigma+\mathrm{j}\omega\varepsilon_{x}\varepsilon_{0}) \mathbf{E}\,, \tag{2}\]  
Figure 1: Examples of the discretization of cylindrical electrodes within a uniform rectangular (**A**) centralized grid, and (**B**) shifted grid. This illustration assumes a \(1\times 10^{-3}\) m diameter electrode and a \(0.5\times 10^{-3}\) m voxel size. The voxel size is varied during this study.  
where:  
* \(\mathbf{\
abla}\times\) (m\({}^{-1}\)) is the curl;
* \(\mathbf{E}\) (V\(\cdot\)m\({}^{-1}\)) is the electric field;
* \(\omega\) (rad\(\cdot\)s\({}^{-1}\)) is the angular frequency;
* \(\upmu_{\mathrm{r}}\) is the dimensionless relative permeability;
* \(\upmu_{0}\) (H\(\cdot\)m\({}^{-1}\)) is the permeability of free space;
* \(\mathbf{H}\) (A\(\cdot\)m\({}^{-1}\)) is the magnetic field;
* \(\sigma\) (S\(\cdot\)m\({}^{-1}\)) is the electrical conductivity;
* \(\varepsilon_{\mathrm{r}}\) is the dimensionless relative permittivity;
* \(\varepsilon_{0}\) (F\(\cdot\)m\({}^{-1}\)) is the permittivity of free space.  
Here, the vector quantities are expressed in bold and italic. Assuming that the magnetic field is negligible, we can transform Equation (1) into:  
\[\
abla\times\mathbf{E}=0\, \tag{3}\]  
allowing the electric-field component to be expressed as:  
\[\mathbf{E}=-\
abla\Phi\, \tag{4}\]  
with \(\Phi\) (V) as in the scalar electric potential. Using the electro-quasi-static assumption, the continuity equation:  
\[\
abla\cdot\mathbf{J}=0\, \tag{5}\]  
can be turned into:  
\[-\
abla\cdot\kappa\
abla\Phi=0\, \tag{6}\]  
where \(\mathbf{J}\) (A\(\cdot\)m\({}^{-2}\)) is the electrical current density, and \(\kappa=\sigma+\mathrm{j}\omega\varepsilon_{\mathrm{r}}\varepsilon_{0}\) (S\(\cdot\)m\({}^{-1}\)) is the complex admittance.  
To calculate the electric-potential distribution, Equation (6) was converted to a parabolic partial differential equation by introducing a pseudo-time \(\tau\) (s), and a spatial-dependent relaxation function F(x, y, z) to smooth large differences in admittance, such that Equation (6) was turned into:  
\[\frac{1}{\mathrm{F}(\mathrm{x},\mathrm{y},\mathrm{z})}\frac{\partial\Phi}{ \partial\tau}=\
abla\cdot\kappa\
abla\Phi\, \tag{7}\]  
with (x, y, z) as Cartesian coordinates expressed in (m) [19]. Subsequently, Equation (7) was solved by marching on in pseudo-time, allowing the potential \(\Phi\) to diffuse in space starting from an initial distribution, until a stationary state (\(\partial\Phi/\partial\tau=0\) V\(\cdot\)s\({}^{-1}\)) was reached.