# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors
## 3 Computational Results and Discussion
At very initial times the temperature in the center increases rapidly and it is relatively independent of the aspect ratio and whether the tumor is an oblate or prolate spheroid. However, at intermediate times, the temperature becomes significantly dependent on the aspect ratio and it remains nearly insensitive to prolate or oblate spheroid shapes of the same AR. For instance, for AR = 1 after 22 min of treatment, the temperature is 46.62 \({}^{\circ}\)C, while for AR = 8 the maximum temperature is approximately 3.5 \({}^{\circ}\)C less for both prolate and oblate spheroidal tumors. The above treatment temperature trends could be explained in terms of the surface area of the tumors. From Equation (3), increasing the AR-value of an ellipsoid tumor leads to an increase of the surface area. For instance, for an oblate tumor with AR = 8 the surface area is approximately twice the surface area of a perfectly spherical tumor of the same volume and for a prolate tumor of AR = 8 it is approximately 1.5 times. Since the available surface area increases, the heat transfer from the nanoparticle-saturated tumor to the cooler healthy tissue increases, leading to a drop of the tumor temperature. Figure 8 shows the temperature values at two representative tumor surface probe locations as shown in Figure 8a. Observation point P1 is fixed on the tumor surface at the semi-minor axis and P2 fixed on the tumor surface at the semi-major axis. For a tumor with AR = 1, both probes give exactly the same temperature, as shown in Figure 8b. Of course,  
Figure 6: Treatment temperature field after 22 min of heating for prolate spheroidal tumor shapes with different aspect ratios. (**a**) AR = 2, (**b**) AR = 4 and (**c**) AR = 8. Figure 7: Behavior of temperature profiles at the tumor center over time for all analyzed models. this is due to the perfect spherical symmetry for this tumor and has been shown by several other works. However, as shown in Figure 8c for a prolate and an oblate tumor with AR = 2, the treatment temperature as a function of time, at P1 and P2, differs significantly. While at the start of the treatment the temperature rises rapidly in a similar manner at both probe locations, after a relatively short time the temperature history at P1 is substantially higher than the temperature at P2. For the oblate tumor, after 22 min of treatment, the temperature at P1 is approximately 2 \({}^{\circ}\)C higher than the temperature at P2. A similar temperature difference is observed at the corresponding locations for a prolate spheroidal tumor of the same aspect ratio, but with lower treatment temperature values than the oblate. It should be also noted that for the oblate spheroid the temperature at P1 is higher than the temperature at the corresponding location of a perfectly spherical tumor (AR = 1), as compared to Figure 8b. For tumors with AR = 4 (see Figure 8d) the temperature difference between the two probe locations, after 22 min of treatment, increases to approximately 3 \({}^{\circ}\)C (for both oblate and prolate shapes), which is also in the case of tumors with an AR of 8 (see Figure 8e). As expected, due to the larger available surface for heat transfer, the temperatures for the case of AR = 8 are lower than for tumors with an AR of 4. Figure 8: Treatment temperature on the tumor surface, at two tumor surface probe locations P1 and P2 shown in the scheme (**a**) for: (**b**) AR = 1, (**c**) AR = 2, (**d**) AR = 4 and (**e**) AR = 8. All ellipsoidal tumor shapes have the same volume. Once the treatment temperature of the proposed model is determined, the degree of thermal damage in the tumor and healthy tissue can be calculated using the Arrhenius kinetic model [21]. The maximum treatment temperature profiles as a function of treatment time presented in the previous figures can be used to evaluate the degree of thermal damage in the tumor center.