## 3 Results and Discussion
### Characteristic Observables of CoFe\({}_{2}\)O\({}_{4}\) Nanoparticles and Study Method
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(\mathbf{M_{\rm s}}\) & \(\mathbf{K}\) & \(\mathbf{\rho}\) & \(\mathbf{c}\) & \(\mathbf{\varepsilon}\) & \(\mathbf{H}\) & \(\mathbf{f}\) & \(\mathbf{D}\) \\ (\(\times\)10\({}^{3}\) A m\({}^{-1}\)) & (\(\times\)10\({}^{3}\) J m\({}^{-3}\)) & (\(\times\)10\({}^{3}\) kg m\({}^{-3}\)) & (Jkg\({}^{-1}\) K\({}^{-1}\)) & & (\(\times\)10\({}^{3}\) A m\({}^{-1}\)) & (\(\times\)10\({}^{3}\) Hz) & (\(\times\)10\({}^{-9}\) m) \\ \hline
425 & 200 & 5.29 & 700 & 0.01–0.15 & 5–180 & 50–1000 & 1–20 \\ \hline \end{tabular}
\end{table} 
Table 1: Characteristic observables of the CoFe\({}_{2}\)O\({}_{4}\) ferrite nanoparticles and parameters of magnetic field. anisotropy is given by the magnetocrystalline component (K\({}_{\rm eff}\)-K) [83], as we will consider in this paper (Table 1) [55]. Additionally, the surface effects and spins disorder in surface layer of nanoparticles can significantly reduce the spontaneous/saturation magnetization at room temperature. However, the concrete value of the decrease of spontaneous/saturation magnetization depends on many other factors, besides the size of the nanoparticles, such as the type of magnetic material, the preparation conditions of the sample, the surfactant or not of the nanoparticles, different heat treatments, previous history of sample, etc. Therefore, it is difficult to predict an exact value of the spontaneous magnetization of nanoparticles. Thus, in order not to introduce ambiguities in this theoretical study, we consider in the calculations the standard value of the spontaneous magnetization (M\({}_{\rm s}\)) at room temperature (20 \({}^{\circ}\)C) of the CoFe\({}_{2}\)O\({}_{4}\) ferrite (Table 1) [54]. Of course, in the case of concrete applications, it is beneficial to determine experimentally a priori the values of spontaneous/saturation magnetization and effective magnetic anisotropy of nanoparticles for better accuracy. In our study, we considered two fundamental observables used in superrparamagnetic hyperthermia: (i) the specific loss power (P\({}_{\rm s}\)) (Equation (2) with Equations (3), (5)-(7)) and (ii) the heating temperature (T\({}_{\rm h}\)) (Equation (8)) for CoFe\({}_{2}\)O\({}_{4}\) nanoparticles, which are key parameters in the analysis of the efficacy and effectiveness of superparamagnetic hyperthermia for cancer therapy. The Maximum Specific Loss Power in the Case of Superparamagnetic Hyperthermia with CoFe\({}_{2}\)O\({}_{4}\) Nanoparticles  
The most important observable that allows the anticipation of the characteristics that CoFe\({}_{2}\)O\({}_{4}\) ferrimagnetic nanoparticles must have in order to be successfully used in superparamagnetic hyperthermia for the destruction of tumor cells is the specific loss power P\({}_{\rm s}\) (or SLP). Using Equation (2) and Equations (3), (5) and (6), with the values of the characteristic parameters in Table 1, the specific loss power P\({}_{\rm s}\) was calculated as a function of the diameter of nanoparticles (D) and the characteristic observables of magnetic field: frequency (f) and amplitude (H). For this it was taken into account that the size of magnetic nanoparticles is a key parameter in magnetic hyperthermia, therefore it was considered primarily the dependence of the specific loss power P\({}_{\rm s}\) on the diameter D of nanoparticles. Thus, the 3D dependence of the specific loss power (P\({}_{\rm s}\)) in the CoFe\({}_{2}\)O\({}_{4}\) ferrimagnetic nanoparticles as a function of the nanoparticle diameter (D) and the frequency of the magnetic field (f) for the values included in the domains D = 1-20 nm and f = 100-1000 kHz (Table 1), and for volume packing fraction \(\varepsilon\) = 0.15, having as parameter the amplitude of magnetic field H (in the range specified in Table 1), is shown in Figure 1. Here two values for H were considered: 15 kA/m (Figure 1a) and 30 kA/m (Figure 1b).