# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors
## 5 Concluding Remarks  
A computational study for magnetic hyperthermia using nanoparticles of ellipsoidal tumors has been presented. The tumors were approximated as equal volume prolate and oblate spheroids of various aspect ratios, surrounded by a large spherical healthy tissue region. The nanoparticles are assumed to be uniformly distributed in the whole tumor. The bio-heat transfer analysis is carried out using the Pennes bio-heat equation. The results indicate that the highest temperature is achieved in the ellipsoidal tumor center, the value of which decreases by increasing the aspect ratio of the tumor. This value appears to be insensitive to whether the ellipsoidal tumor is a prolate or oblate spheroid. Probing the temperature at the tumor surface at two locations, one along the major and one along the minor axis, reveals that oblate tumors have generally higher surface temperatures than oblate ones, the values of which strongly depend on the aspect ratio. Using the Arrhenius kinetic model for thermal damage, we find that the thermal damage in the tumor center is unaffected by whether the tumor is oblate or prolate and decreases for increasing aspect ratio. Also, the computational model produces results for the extent of the tumor necrotic area, which is affected by the aspect ratio as well as the prolateness and oblateness of the ellipsoid tumors. The numerical model was compared with three different sets of experimental measurements involving nanoparticle hyperthermia in animal tumors which are available in the literature. In all comparisons, we have approximated each tumor shape with two prolate spheroid geometries of slightly different aspect ratios to describe as best as possible the tumor shape. Both case geometries produced results reasonably close to the measured ones. Model predictions were generally in satisfactory or perhaps good agreement with the experiments when uncertainties in the measured properties of the nanoparticles are taken into account. Also, although the parameters of the tissue used in the model are derived from different tissues (muscle [86], liver [91], prostate [92]), the comparisons show good agreement with the experimental measurements presented by other authors with the proposed numerical method. It should be pointed out that according to [PERSON] et al. [113], available technologies that convey heat to tumors, such as RF, microwave, ultrasound and conductive, have  
Figure 13: Two cases approximating the tumor shape from a histological cross-section by [PERSON] et al. [91], 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 \(\approx\) 2.5, on top of the redrawn tumor and (**b**) prolate spheroid shape, case B with AR \(\approx\) 2.82, on top of the redrawn tumor. Comparison of the present model assuming two nanoparticle size values, with experimental temperature measurements at the tumor surface for (**c**) Case A and (**d**) Case B.  
not been able to target heat specifically to tumors in an effective manner, especially to metastatic ones. Hyperthermia using magnetic nanoparticles is a minimally invasive treatment that appears to offer more localized heating of the tumors as reported by many authors cited in the present work. Nonetheless, the treatment needs to be precisely assessed by taking into account several factors, such as temperature dependence over time, treatment time, tumor shape, tissue damage and nanoparticle dosage. Computational modeling is absolutely necessary for an in-depth understanding on the effect of the above parameters on the treatment outcome. The present computational model has been focused on the determination of such parameters, specifically for ellipsoidal (prolate and oblate) tumors with several aspect ratios. The model can be easily implemented and reproduced for various treatment scenarios and may be useful for further treatment planning. Conceptualization, N.D.P. and A.A.G. ; methodology, N.D.P., A.A.G., I.E.S. and L.A.S. ; software, N.D.P.