# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors
## 2 Materials and Methods
### Bio-Heat Transfer Analysis  
Bio-heat transfer between the ellipsoidal tumor and the surrounding healthy tissue is expressed by the thermal energy balance for perfused tissues described by the Pennes bio-heat equation [93; 94]:  
\[\rho_{n}c_{n}\frac{\partial T(x,y,t)}{\partial t}=k_{n}\
abla^{2}T(x,y,t)-\rho_{ \text{b}}c_{\text{b}}w_{\text{b},n}[T(x,y,t)-T_{\text{b}}]+Q_{\text{met},n}+Q_ {\text{s}} \tag{5}\]  
where the subscript \(n\) stands for the tissue under consideration (\(n=1\) for tumor and \(n=2\) for healthy tissue) and the subscript b corresponds to blood properties. Also, \(\rho_{n}\) and \(\rho_{\text{b}}\) denote the densities of the tissues and the blood respectively, \(c_{n}\) and \(c_{\text{b}}\) are the corresponding heat capacities, \(T(x,y,t)\) is the local tissue temperature, \(k_{n}\) is the tissue thermal conductivity, \(w_{\text{b}}\) is the blood perfusion rate, and \(T_{\text{b}}=37\)\({}^{\circ}\)C is the blood temperature. The left-hand side term in Equation (5) expresses the time rate of change of internal energy per unit volume. The first term on the right-hand side of Equation (5) represents the heat conduction in the tissue. The second term represents an additional change in the internal energy per unit volume associated with blood perfusion in tissue, assuming that the  
Figure 3: Two representative computational meshes used in the study focused at the tumor region and the close area around it. Magnified views close to the tumor/healthy tissue boundary are also shown. Both meshes correspond to tumors with aspect ratio AR = 2. Figure 2: Schematic representation of the axisymmetric model, where \(y\)-axis is the revolution axis and \(x\)-axis is a symmetry axis (figure not to scale). The ellipsoidal tumor is assumed to be surrounded by a significantly larger spherical healthy tissue (\(R_{\text{h}}>>a\) or \(b\)). \(T_{\text{s}}\) corresponds to the temperature of the outer surface of the healthy tissue. rate of heat transfer between tissue and blood is proportional to the blood perfusion rate and the difference between the local tissue temperature and the blood temperature, as suggested in [65]. In addition, \(Q_{\text{met,n}}\) is the internal heat generation rate per unit volume associated with the metabolic heat production. Finally, \(Q_{\text{s}}\) is the power dissipation density by the MNPs. It is assumed no leakage of MNPs to the surrounding healthy tissue. Therefore, \(Q_{\text{s}}\) is only applied to the cancerous region filled with the MNPs, similarly to several earlier investigations, e.g., [15; 67; 69; 92]. Following the assumptions by [42; 44; 92; 95; 101], a uniform distribution of nanoparticles is assumed. The temperature of the surface of the healthy tissue region (see Figure 2), which is far away the tumor surface as explained in the previous section, is set to \(T_{\text{s}}=T_{\text{b}}\)[69; 75]. For the initial temperature condition we set \(T(x,y,t=0)=T_{\text{b}}\). A symmetry (zero heat flux) boundary condition is applied on the \(x\)- and \(y\)-axes (line OA and line OB respectively in Figure 2). Also, heat flux and temperature continuity are considered on the tumor-healthy tissue interface as follows [12; 93]:  
\[k_{n}\bigg{(}\frac{\partial T}{\partial\mathbf{n}}\bigg{)}_{n}=k_{n+1}\bigg{(} \frac{\partial T}{\partial\mathbf{n}}\bigg{)}_{n+1} \tag{6}\]  
\[T_{n}=T_{n+1} \tag{7}\]  
where \(\mathbf{n}\) is the direction vector perpendicular to the tumor surface. To carry out numerical simulations for the different AR cases, typical tissue thermophysical values (for example prostate tissue) are chosen from earlier works [36; 63; 67; 71; 94] and shown in Table 2. Note that we have assumed that the perfusion rate is independent from region and temperature [31; 58].