## 2 Numerical Method and Simulation Conditions  
Every mathematical model for the simulation of microwave ablation consists of three fundamental components. The first component is the model of the antenna probe (or applicator) that generates a microwave field in the tissue. The second component describes the heat distribution in the tissue including sources and sinks and the phase changes. In our case, the microwave field is the source of heat, and the heat sinks are represented by the blood perfusion term in the heat transfer equation. The third part deals with the effect of heat on tumor cells and their destruction. All these components of the ablation model depend on a variety of material parameters, which themselves depend on the various states of the tissue. For small tumors or tumors adjacent to vital organs, MWA with antennas operating at 2.45 GHz is recommended due to its more localized ablation zone [26; 73]. The antennas are usually mechanically and geometrically complex, and the simulation relies on having accurate electromagnetic material and tissue properties. In this study, we use a compact 10-slot microwave antenna with an impedance pi-matching network that creates near-spherical ablation zones, schematically represented in Figure 1[53; 54]. The finely tuned, impedance matching eliminates the damaging of the surrounding healthy tissues. The multislot radiating probe is composed of several periodic elements equal to a linear uniform antenna array. Each periodic element individually comprises a slot with a width of 0.6 mm and a spacing conductor of 0.8 mm between adjacent slots. The required ablation shape is achieved by optimizing the distance between the adjacent slots and the number of the slot. In comparison with previously developed antennas [74; 75; 76; 77], this multislot coaxial antenna produces a more localized heating pattern and a smaller overtreatment region for spherical liver tumors. To define as realistic a simulation model as possible, we did not use a spherical tumor geometry, as is usually the case. In our analysis, we shall use the data from the 3D-IRCADb-01 database [70] that includes several sets of CT scans of patients manually segmented by clinical experts. We perform a simulation on the data reported for patient 16 in the database. The shapes of the tumor (and the surrounding liver) are shown in Figure 2a. Figure 2b illustrates a 10-slot microwave antenna inserted into a liver tumor with four test points (A, B, C, and D) marked in red, blue, green and purple, and located at different sites. Points A and D are positioned close to the heating center along the antenna shaft. Points B and C were placed along with the radial direction corresponding to the maximum transverse of each antenna. At these points, we shall follow the time dependence of temperature and tissue damage during the ablation process. They correspond to the usual positions of thermocouples in experimental studies [54]. For this study, the 3D finite elements method (FEM) is used to solve coupled electromagnetic-field and heat-transfer equations, including all details of antenna design and properties of healthy and tumoral tissue. Our 3D model is created within the COMSOL Multiphysics FEM-based simulation platform [68; 71]. Equations that describe the propagation of microwaves in tissue by an antenna have forms [71; 72]:  
\[\
abla^{2}\mathbf{E}-\mu_{r}\mathbf{k}_{0}^{2}\bigg{(}\varepsilon_{r}-\frac{ \mathrm{j}\sigma}{\omega\varepsilon_{0}}\bigg{)}\mathbf{E}=0, \tag{1}\]  
with the angular frequency \(\omega\), the vacuum propagation constant \(k_{0}=\omega/c_{0}\), the electric field vector \(\mathbf{E}\), and the tissue electrical conductivity \(\sigma\). \(\varepsilon_{0}\), \(\varepsilon_{r}\), and \(\mu_{r}\) are the vacuum dielectric constant, relative permittivity, and permeability of the tissue, respectively. Figure 1: Schematic view of the 10-slot microwave antenna with an impedance match network.