# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors
## 4 Comparison with Experiments
[92], as shown in Table 5.  
Figure 12: Two cases approximating the tumor shape from a histological cross-section by [PERSON] et al. [92] with a prolate spheroid. Note that the tumor histological cross-section has been redrawn from the original: (**a**) prolate spheroid shape, case A with AR \(\approx\) 1.29, on top of the redrawn tumor and (**b**) prolate spheroid shape, case B with AR \(\approx\) 1.57, on top of the redrawn tumor. Comparison of the present numerical model with the 3D numerical model and experiments by [PERSON] et al. [92] at the tumor center (probe center) for (**c**) Case A and (**d**) Case B and at the probe tip (approximately 3 mm from tumor center) for (**e**) Case A and (**f**) Case B. In Figure 13 the present model is compared with experimental measurements by [PERSON] et al. [91]. In their study, cancerous human breast cells were injected in nine mice. Hyperthermia was applied when the tumor reached a volume of 0.453 \(\pm\) 0.109 cm\({}^{3}\). The tumor volume in [PERSON] et al. [91] was calculated from Equation (4). Then 0.1 mL of a mixture that was injected in the tumor tissue containing 10 wt% Fe\({}_{3}\)O\({}_{4}\) powder, 36 wt% polymethylmethacrylate (PMMA) powder and 54 wt% MMA liquid. A magnetic field of frequency 626 kHz and strength 28.6 kA/m was applied for 180 s. [PERSON] et al. [91] report that the ferrous powder diameter was in the range of \(D\) = 20-50 nm. An ultrasound image section of the tumor from [PERSON] et al. [91] is shown at the top of Figure 13. The red dashed line corresponds to the tumor boundary as marked by [PERSON] et al. [91]. For the comparison, we recognize two cases. Given the tumor volume mentioned above for Case A (Figure 13a) we find AR \(\approx\) 2.5 with \(a\approx\) 3.5 mm and \(b\approx\) 8.8 mm, while for Case B, AR \(\approx\) 2.82 with \(a\approx\) 3.37 mm and \(b\approx\) 9.5 mm. Subsequently, we applied [PERSON]'s theory to estimate the amount of heat dissipated by the nanoparticles. We initially calculated the solid volume fraction of the nanoparticles in the tumor volume. The density of the injected mixture can be calculated from the following equation  
\[\rho_{\rm mixture}=w_{\rm FeSO_{4}}\rho_{\rm FeSO_{4}}+w_{\rm PMMA}\rho_{ \rm PMMA}+w_{\rm MMA}\rho_{\rm MMA} \tag{17}\]  
where \(\rho_{\rm FeSO4}\) = 5180 kg/m\({}^{3}\)[33], \(\rho_{\rm MMA}\) = 940 kg/m\({}^{3}\)[11], \(\rho_{\rm PMMA}\) = 1180 kg/m\({}^{3}\)[11] and the corresponding weight fractions are mentioned above. Substituting the values in the above equation we find \(\rho_{\rm mixture}\) = 1450 kg/m\({}^{3}\). The 0.1 mL mixture injected in the 453 mm\({}^{3}\) tumor volume has a mass of 0.1 mL \(\times\) 1450 mg/mL = 145 mg. Since 10 wt% of the mixture is the ferrous mass, the nanoparticles weight in the tumor is 14.5 mg. This corresponds a solid volume fraction in the tumor of \(\varphi\approx\) 6.16 \(\times\) 10\({}^{-3}\). Also, [PERSON] et al. [91] measured the magnetization saturation of the nanoparticles \(M_{\rm s}\) = 1560 A/m (or mass magnetization saturation \(\sigma_{\rm s}\) = 0.3 emu.g\({}^{-1}\)). According to [PERSON] [33], the domain magnetization (\(M_{\rm d}\) in Equations (10) and (11)) is given by \(M_{\rm d}=M_{\rm s}/\varphi\), where substitution in the present case gives \(M_{\rm d}\approx\) 253 kA/m. If we select the value of 35 nm for the ferrous powder average size in the range 20-50 nm reported by [PERSON] et al. [91], and use the above-mentioned values, we find (from [PERSON]'s theory) 1.7 \(\times\) 10\({}^{6}\) W/m\({}^{3}\) of dissipated heat. The comparison of the present model tumor surface temperature, using this value, with the measurements by [PERSON] et al. [91] is shown in Figure 13c for Case A and Figure 13d for Case B. The blood perfusion value used for the [PERSON] et al. [86] comparison is also used here. Note that in [91] the temperature is recorded using thermal imaging. The exact location of the tumor surface at which temperature was measured is not reported. The thermal images presented in [PERSON] et al.