# A Computational Study on Magnetic Nanoparticles Hyperthermia of Ellipsoidal Tumors
## 2 Materials and Methods
### Heat Generation by the Magnetic Nanoparticles  
As explained in the work by [PERSON] and [PERSON] [49], the superparamagnetic heating phenomenon is the base of magnetic nanoparticles hyperthermia. [PERSON] [33] formulated a theoretical model for the heating of colloidal magnetic fluid (ferrofluid) due to application of an AFM. In his work, the power dissipation density (in W/m\({}^{3}\)) from a monodispersed magnetic fluid filled with superparamagnetic particles subjected to an alternating magnetic field [15] is given by:  
\[Q_{\text{s}}=\pi\mu_{0}\chi_{0}{H_{0}}^{2}f\frac{2\pi f\tau}{1+\left(2\pi f \tau\right)^{2}} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Tissue** & \(\rho\) (**kg/m\({}^{3}\)**)** & \(c\) (J/kg\(\cdot\)K) & \(k\) (W/m\(\cdot\)K) & \(w_{\text{b}}\) (s\({}^{-1}\)) & \(Q_{\text{met}}\) (W/m\({}^{3}\)) \\ \hline Tumor & 1045 & 3760 & 0.5 & \(1.3\times 10^{-3}\) & 540 \\ Healthy tissue & 1045 & 3760 & 0.5 & \(1.3\times 10^{-3}\) & 540 \\ Blood & 1060 & 3770 & – & – & – \\ \hline \hline \end{tabular}
\end{table}
Table 2: Physical properties of the tissues [36; 63; 67; 71; 94].  
where \(\mu_{0}=4\pi\times 10^{-7}\) H\(\cdot\)m\({}^{-1}\) is the vacuum magnetic permeability, \(f\) is the frequency of the magnetic field with magnetic field intensity \(H_{0}\) (A\(\cdot\)m\({}^{-1}\)), \(\chi_{0}\) corresponds to the magnetic susceptibility of the magnetic fluid defined by  
\[\chi_{0}=\chi_{\rm i}\bigg{(}\coth\xi-\frac{1}{\xi}\bigg{)} \tag{9}\]  
where \(\chi_{\rm i}\) the initial susceptibility given by  
\[\chi_{\rm i}=\frac{\mu_{0}\ \varphi\ M_{\rm d}{}^{2}\ V_{m}}{3{\rm k}_{\rm B}T} \tag{10}\]  
and  
\[\xi=\frac{\mu_{0}\ M_{\rm d}\ H_{0}\ V_{\rm m}}{3{\rm k}_{\rm B}T} \tag{11}\]  
with \(M_{\rm d}\) the domain magnetization of a suspended particle, \(V_{\rm m}=4/3R^{3}\) is the magnetic volume for a nanoparticle of radius \(R\), \({\rm k}_{\rm B}=1.381\times 10^{-23}\) J\(\cdot\)K\({}^{-1}\) is the Boltzmann constant and \(T\) is the absolute temperature. In Equation (10), \(\varphi\) is the solid volume fraction of the nanoparticles. The power dissipation is accomplished via two physical processes, the Neel and Brownian relaxation. Because these processes take place in parallel, the effective relaxation time in Equation (5) is given by  
\[\frac{1}{\tau}=\frac{1}{\tau_{\rm B}}+\frac{1}{\tau_{\rm N}} \tag{12}\]  
where \(\tau_{\rm B}\) the Brownian characteristic relaxation time  
\[\tau_{\rm B}=\frac{3\eta V_{\rm H}}{{\rm k}_{\rm B}T} \tag{13}\]  
with \(\eta\) the viscosity of the matrix fluid and \(V_{\rm H}\) is taken as the hydrodynamic volume of the nanoparticle related to \(V_{\rm m}\) as \(V_{\rm H}=(1+\delta/R)^{3}V_{\rm m}\) where \(\delta\) is the thickness of a sorbed surfactant layer (\(\delta=2\) nm according to [PERSON] [33]). In Equation (13) \(\tau_{\rm N}\) is the Neel characteristic relaxation time given by [15]:  
\[\tau_{\rm N}=\frac{\sqrt{\pi}}{2}\tau_{0}\frac{\exp(\Gamma)}{\Gamma^{1/2}}, \ \ \ \Gamma=\frac{\mathcal{K}V_{\rm m}}{{\rm k}_{\rm B}T} \tag{14}\]  
where \(\tau_{0}\approx 10^{-9}\) s is an attempt time [15; 49] and \(\mathcal{K}\) is the anisotropy constant (J/m).