# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors
## 1 Introduction
These can be predicted usually by solving the Pennes bio-heat transfer equation [50], but other models have also been proposed to predict heating effects such as, for instance, porous media based models [51; 52]. Treatment temperature and duration may also affect the mortality of the malignant or healthy cells, which can be evaluated by the Arrhenius formulation description; the treatment thermal dose usually evaluated by the Cumulative Equivalent Minutes is at the 43 \({}^{\circ}\)C (CEM43) parameter [21; 53; 54; 55; 56; 57] or the temperature threshold method [58; 59]. Other cell death models have been also developed [60; 61]. The most frequently assumed tumor shape, in numerous earlier modeling studies, is a perfect sphere. Such an ideal geometry allows for derivation of analytical expressions for the therapeutic temperature profile [62; 63; 64; 65; 66; 67; 68; 69]. Also, for spherical tumors, numerical models have been formulated using the finite element, finite volume and Lattice Boltzmann methods [70; 71; 72; 73; 74; 75; 76; 77; 78; 79]. However, tumor shapes usually depart from the perfect sphericity assumption. Elongated shapes, reminiscent of geometric ellipsoids, are frequently encountered. According to [PERSON] et al. [80] and [PERSON] et al. [81], in vitro experiments have shown that the studied tumors have grown into highly elongated ellipsoidal-like shapes. Similar elongated shapes have also been reported for primary breast carcinoma [82], primary gastric gastrointestinal stromal tumors [83] and glioblastoma related tumors [84]. In fact, in the last three studies relating to overall survival in patients, the tumor volume was calculated by the shape of an ellipsoid. Relevant to magnetic hyperthermia using nanoparticles, several in vivo experimental investigations in rodents show histological cross-sections of subcutaneous and deep-seated tumors that have grown into ellipsoidal-shaped geometries with mild to high aspect ratios [85; 86; 87; 88; 89; 90; 91]. In these studies, the tumor volume was estimated based on the geometric shape of an ellipsoid. A few researchers have developed some computational models assuming ellipsoid tumor geometry. [PERSON] et al. [92] report in vivo and 3D computational results of temperature distribution and cell damage of adenocarcinoma subcutaneous tumors in the fore-shoulders of female mice. The shape of the tumors was approximated by an ellipsoid. [PERSON] et al. [93] studied the magnetic nanoparticle hyperthermia of a subcutaneous sarcoma tumor both experimentally and computationally. The shape of the tumor was also approximated by an ellipsoidal shape. [PERSON] et al. [94] proposed a computational model for the utilization of power modulation for magnetic nanoparticle hyperthermia of elliptic (2D) and ellipsoidal (3D) tumors. In the above-mentioned studies, the aspect ratio of the ellipsoid tumors was fixed. [PERSON] et al. [95] developed an analytical model for the transient temperature evolution in three tumor shapes of equal volume: a perfect spherical, a prolate spheroid with an aspect ratio of approximately three and an oblate spheroid with an aspect ratio of eight. Spatial temperature distributions in the tumor and the surrounding healthy tissue were neglected. Their results show that the uniform temperature in the spherical tumor was higher than in the prolate spheroid tumor and much higher than the oblate spheroid tumor. [PERSON] et al. [96] studied numerically oblate and prolate spheroid tumors of equal volumes for the treatment of microwave ablation using a coaxial antenna. In their work the aspect ratio of the ellipsoids varied from one to five. Their results show that the aspect ratio has a significant effect on the extent of the ablation zone in the tumor. The objective of the present investigation is to provide a systematic study for magnetic nanoparticles hyperthermia of ellipsoidal tumors of various aspect ratios and to compare the results of the numerical predictions to experimental data.