## 2 Materials and Methodology  
Recently, [PERSON] et al. [19] focused on evaluating the extent of tumor damage considering the tumor margins for irregular tumors, while sparing the healthy tissue. Their analysis shows that a thermal damage of 10 is achieved within the entire tumor region. Additionally, a thermal damage of 4-10 is attained beyond the tumor margins, ensuring adherence to the protocol which states that the volume of damaged healthy tissue should not exceed the tumor volume. Both the conditions (i.e., temperatures above 55 \({}^{\circ}\)C near tumor boundaries and less than 5% thermal damage to healthy tissue) are achieved for a 3 mm distance beyond the tumor periphery. Corresponding SAR values and exposure duration are reported to achieve the ablation of the considered tumor while minimizing the damage beyond 3 mm (margin) of the tumor boundary. In summary, the iterative computational experiments suggest that margins less than 5 mm are sufficient for the given patient-specific tumor and the typical clinical criterion that requires sacrificing the 10 mm of excessive healthy tissue fringes does not need to be met. To predict the temperature behavior of the tissue, the porcine liver has been chosen as the material for this simulation model. A modified version of [PERSON] bioheat model has been developed for this purpose [3; 20]:  
\[\rho c\frac{2\Omega(\mathbf{r},z,t,P)}{dt} = \
abla\cdot[k\
abla T(\mathbf{r},z,t,P)]+Q_{laser}(\mathbf{r},z,t,P)\] \[+Q_{bio}+Q_{H}(\mathbf{r},z,t,P)\]where \(\rho\), \(c\), and \(k\) represent the density, specific heat, and thermal conductivity of the tissue, respectively. \(T\) denotes the tissue temperature. \(Q_{laser}\), defined as the irradiated laser power [20], is given by [21]:  
\[Q_{laser}(\mathbf{r},z,t)=\mu_{a}I(\mathbf{r},z,t) \tag{2}\]  
where \(\mu_{a}\) represents the absorption coefficient and \(I\) denotes the photon distribution in the tissue. The last term on the right-hand side in Equation (1) represents the energy of the tissue evaporation, denoted as \(Q_{H}\),  
\[Q_{H}=-\rho\alpha\cdot\frac{\partial P(\%)}{\partial t} \tag{3}\]  
The symbol \(\alpha\) denotes the latent heat of tissue, which is replaced by the latent heat of water. The parameter \(P\) (\(\%\)) represents the percentage of necrotic tissue, serving as a descriptor of the damage incurred [22; 23; 24]:  
\[P(\%)=[1-\exp(-\Omega)]\cdot 100\% \tag{4}\]  
where \(\Omega\) is defined by the Arrhenius rate process model to analyze the thermal damage of the tissue:  
\[\Omega=\int_{0}^{t}A\cdot\exp\biggl{(}-\frac{E_{a}}{RT(t)}\biggr{)}dt \tag{5}\]  
where \(A\) is the frequency factor, \(E_{a}\) is the activation energy of the denaturation reaction, and \(R\) is the universal gas constant. Therefore, the energy of the tissue evaporation is given by:  
\[Q_{H}(r,z,t,P) =-\rho\alpha \tag{6}\] \[\times\frac{\partial\left\{1-\exp\left[-\int\limits_{0}^{t}A \cdot\exp\left(-\frac{E_{a}}{RT(t)\cdot\epsilon)P}dt\right]\right\}\cdot 100\%}{ \partial t}\]  
According to the theory of diffusion approximation, \(I\) comprises collimated part \(I_{c}\) and the diffuse part \(I_{d}\)[1; 25]. The collimated intensity \(I_{c}\) is given by [PERSON]'s law [21]:  
\[I_{c}(\mathbf{r},z,t)=I_{0}\exp\left(-2\frac{r^{2}}{w_{0}^{2}}\right)\exp \left(-\mu_{t}^{\prime}\cdot z\right)\cdot round\left[0.1321+\exp\left(-\frac {t}{\tau}\right)\right] \tag{7}\]  
where \(I_{0}\) represents the initial laser intensity at the surface, \(w_{0}\) denotes the radius of the laser beam on the surface, and \(\mu_{t}\) = \(\mu_{a}+\mu_{s}\)' stands for the effective attenuation coefficient, with \(\mu_{a}\) being the absorption coefficient. \(\mu_{s}\) = \(\mu_{s}\) (\(1-g\)) represents the effective scattering coefficient, where \(g\) signifies the anisotropy factor, and \(\mu_{s}\) represents the scattering coefficient. The diffuse part \(I_{d}\) is calculated by the diffusion approximation equation [26]:  
\[\left\{