# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors
## 3 Computational Results and Discussion  
Magnetic (Fe\({}_{3}\)O\({}_{4}\)) nanoparticles are selected as heat mediators assuming typical magnetic properties, as shown in Table 4. The magnetic field properties are also presented in Table 4. Note that for the selected \(H_{0}\) and \(f\) values we find \(H_{0}\)\(f\) = 1.496 \(\times\) 10\({}^{9}\) A\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\), which falls between the limits of Atkinson-Brezovich (4.85 \(\times\) 10\({}^{8}\) A\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\)) and Dutz-Hergt (5 \(\times\) 10\({}^{9}\) A\(\cdot\)m\({}^{-1}\cdot\)s\({}^{-1}\)) criterions [29; 30]. Also, the nanoparticles volume fraction  
Figure 4: Comparison of the present computational results for various dimensionless treatment times with the closed-form solution by [PERSON] et al. [67]. \(T^{*}\) is the dimensionless temperature, \(r^{*}\) the dimensionless distance from the tumor center and \(t^{*}\) is the dimensionless time, as defined in [PERSON] et al. [67]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mesh Number** & **Number of Cells** & **Temperature Location 2 mm above Tumor Center (\({}^{\circ}\)C)** \\ \hline
1 & 9500 & 41.581 \\
2 & 15,740 & 41.852 \\
3 & 32,781 & 41.911 \\
4 & 57,468 & 41.915 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mesh sensitivity analysis results. we used is low. Therefore, the effective tumor parameters of MNPs-saturated tissue are nearly identical to tumor parameters without nanoparticles that are used in the model. By substituting these parameters in Equation (8) we find \(Q_{\text{s}}=1.91\times 10^{5}\) W/m\({}^{3}\), which is within the range of earlier publications [63; 65; 68]. The computational results are carried out for a 30 min treatment, since in magnetic hyperthermia it is desirable to have a treatment duration as short as possible for safety purposes [76; 109]. The duration the AMF is switched on and heats the nanoparticles is assumed to be 22 min [76]. After that time and for the remaining eight minutes of the treatment, the magnetic field is off and stops heating. To gain an initial understanding of the tissue temperature distribution, in Figure 5 the temperature field is presented for a region near oblate tumors after 22 min of treatment at various AR values. Note that all tumor shapes have the same volume. In all cases the maximum treatment temperature is observed at the tumor center. As the aspect ratio AR increases, this maximum temperature decreases. This is also the case for the temperature on other regions inside the tumor. A similar behavior is observed for the prolate spheroidal tumors as shown in Figure 6. This behavior is consistent with the results of our earlier preliminary work [99]. Note that the tissue and nanoparticle parameters used in [99] are different than the ones used in the present work. Moreover, the bio-heat equation in [99] was solved under a steady state condition. In the current investigation we have used the more realistic temperature time dependent approach which further allows us to determine the extent of the tissue thermal damage with the Arrhenius thermal damage model. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(M_{\text{d}}\) (kA\(\cdot\)m\({}^{-1}\)) & 446 \\ \(\mathcal{K}\) (k\(\cdot\)m\({}^{-3}\)) & 41 \\ \(\rho_{\text{nano}}\) (kg\(\cdot\)m\({}^{-3}\)) & 5180 \\ \(R\) (nm) & 9.5 \\ \(\eta\) (Pa\(\cdot\)s) & \(6.53\times 10^{-4}\) \\ \(\varphi\) & \(4.8\times 10^{-4}\) \\ \(f\) (kHz) & 220 \\ \(H_{0}\) (A\(\cdot\)m\({}^{-1}\)) & 6800 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Magnetic nanoparticles and magnetic field parameters [33; 36; 47; 49; 63]. Figure 5: Treatment temperature field after 22 min of heating for oblate spheroidal tumor shapes with different aspect ratios. (**a**) AR = 1, (**b**) AR = 2, (**c**) AR = 4 and (**d**) AR = 8. Figure 7 shows time-dependent temperature profiles at the tumor center for all the considered shapes. The AR value appears to have a significant effect on the tumor temperature evolution.