# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors
## 2 Materials and Methods  
### Geometrical Description  
The general equation of an ellipsoid is given by [97]:  
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \tag{1}\]  
where \(a\), \(b\) and \(c\) are the lengths of the principal semi-axes. For the case of all lengths equal \(a=s=c=R\), Equation (1) describes a perfect sphere with radius \(R\). In the present work we are interested for ellipsoids with \(a=c\) (symmetric around the \(y\) axis), while perfect spherical tumors constitute only a limit-case scenario. Such shapes are usually referred to as ellipsoids by revolution. Here, the \(y\)-axis is set as the axis of revolution.  
Two basic cases may be distinguished:  
1. oblate spheroids with semi-axis \(a>b\)
2. prolate spheroids with semi-axis \(a<b\)  
as shown in Figure 1.  
In addition, we define the aspect ratio AR for the generated ellipsoids using the following notation [96]:  
\[\mathrm{AR}=\frac{\mathrm{major\ axis\ length}}{\mathrm{minor\ axis\ length}} \tag{2}\]  
Increasing AR leads to ellipsoidal tumors with more elongated shapes. The surface \(S\) of the ellipsoids is expressed via the following formulation [98]:  
\[S=2\pi a^{2}\left[1+\left\{\begin{array}{ccc}\frac{b}{a\epsilon}\mathrm{ arcsin}e&,&e^{2}=1-\left(\frac{a}{b}\right)^{2},&b>a&(\mathrm{prolate})\\ \\ \frac{b^{2}}{a^{2}\epsilon}\mathrm{arctanh}e&,&e^{2}=1-\left(\frac{b}{a} \right)^{2},&b<a&(\mathrm{oblate})\end{array}\right\}\right] \tag{3}\]  
where \(e\) is the eccentricity of the ellipsoid. The volume of the ellipsoids is given by [95]:  
\[V=\frac{4}{3}\pi a^{2}b \tag{4}\]  
All the generated ellipsoidal tumors are set to have equal volumes.  
The dimensions of the ellipsoid tumors used in this work are shown in Table 1. The tumor geometries are taken to have the same volume, as calculated from Equation (4). The range of the selected tumor dimensions are within the range of earlier works [80; 86; 95; 96]. It is also assumed that the ellipsoidal tumors are surrounded by healthy tissue of spherical geometry, as shown in Figure 2. The region of the healthy tissue is assumed to be significantly larger than the tumor. In particular, the radius of the healthy tissue \(R_{\text{h}}\) is taken approximately eight times larger than the semi-major axis of the tumor with the highest aspect ratio. Due to the rotational symmetry of the geometries, the present thermal problem can be solved as an axisymmetric problem instead of a 3D one, which substantially decreases the computational cost of the numerical simulations [99].  
For the discretization of the computational domains, we used a combination of regular and unstructured meshes consisting of triangular cells. All meshes were constructed using GMSH software [100]. The unstructured mesh is used to discretize the tumor region as well as a healthy tissue layer around the tumor. We followed this approach to better capture the surface geometry of the tumors with high aspect ratios (e.g., AR = 8). Two sample meshes for AR = 2 are shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Prolate Tumors**} \\ \hline Aspect ratio (AR) & \(a\) (mm) & \(b\) (mm) \\ \hline
2 & 7.93 & 15.87 \\
4 & 6.29 & 25.19 \\
8 & 5.0 & 40.0 \\ \hline \multicolumn{3}{c}{**Oblate Tumors**} \\ \hline Aspect ratio (AR) & \(a\) (mm) & \(b\) (mm) \\ \hline
1 & 10.0 & 10.0 \\
2 & 12.5 & 6.29 \\
4 & 15.87 & 3.96 \\
8 & 20.0 & 2.50 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dimensions of the ellipsoidal tumors studied.  
Figure 1: (**a**) Virtual representation of tumors by ellipsoid geometries. (**b**) Notation of the major and minor axis length of the spheroids. All shapes shown have the same volume and are fully symmetric around the \(y\)-axis.