## 3 Results and Discussion
### The Specific Loss Power in the Linear Approximation
#### 3.3.1 Maximum Specific Loss Power
In our calculus, we took into account the saturation magnetization of the nanoparticles expressed by  
\[M_{sat}=\varepsilon M_{s} \tag{9}\]  
where \(\varepsilon\) is the magnetic packing fraction (considered 0.15 as in all calculations made until now) and \(M_{s}\) is spontaneous magnetization [54] (Table 1). The magnetization was calculated up to a magnetic field close to saturation (500 kA/m) in order to register the Langevin type variation of the magnetization. Figure 7: The magnetization M as a function of magnetic field H for CoFe\({}_{2}\)O\({}_{4}\) nanoparticles in (a) high and (b) low magnetic field. Using the data fit technique obtained with professional software and using a linear function in the range of low fields, we find a very good linear dependence (red line) of magnetization with the field (Figure 7b): \(M=\alpha\ H\), where \(\alpha\) is the slope of the line. Moreover, this linear dependence is well fulfilled up to large magnetic fields of \(\sim\)60 kA/m. In Figure 7b the confidence interval is marked with the light blue color. Thus, the results obtained above show that the linear approximation in the case of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles can be used up to large magnetic fields (up to 60 kA/m), compared to magnetite, for example, where the field did not exceed 5 kA/m [53]. The obtained result is also in agreement with the one shown in Figure 6, where it is clearly observed that up to the magnetic field of \(\sim\)60 kA/m the maximum specific loss power \(P_{SM}\) can be calculated in the linear approximation, and the Equation (7) can be used in this case instead of the Equation (2). All these results shows that the calculations for determining the specific loss power in the case of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles are greatly simplified. Additionally, this is a great advantage in the case of the practical implementation of superparamagnetic hyperthermia because it is no longer necessary to know the magnetic susceptibility (\(\chi_{0}\)) as the nonlinear field function \(\chi_{0}=f(H)\),  
\[\chi_{0}=\chi_{i}\frac{3}{\xi}\left(\coth\xi-\frac{1}{\xi}\right) \tag{10}\]  
where \(\xi\) and \(\chi_{i}\) are given by the Equations (3) and (5). This very different magnetic behavior in the case of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles is also due to the small size (diameter) of the nanoparticles that are used in superparamagnetic hyperthermia to obtain the maximum specific loss power (Figure 1). The small size of CoFe\({}_{2}\)O\({}_{4}\) nanoparticles is due to their very large magnetic anisotropy. Under these conditions, the parentheses in Equation (4) can be approximated by  
Figure 3:0  
and, therefore, the magnetic susceptibility used can be approximated with the initial one \(\chi_{i}\),  
Figure 3:1  
which is constant (for a given diameter of the nanoparticle (Equation (3)). Maximum Specific Loss Power in Superparamagnetic Hyperthermia under Optimal Conditions and within Biologically Permissible Limits  
The efficiency of the method of superparamagnetic hyperthermia with CoFe\({}_{2}\)O\({}_{4}\) ferrimagnetic nanoparticles must be analyzed withing the biologically permissible limits of a magnetic field that does not affect healthy tissues. Thus, taking into account the values for the amplitude and frequency of the magnetic field corresponding to the admissible biological limit [84],  
Figure 3:2  
where \(f_{l}\) is the limit frequency, and using the 3D representations for the specific loss powers as in Figure 1, the maximum values of \(P_{s}\) and the corresponding optimal diameters (\(D_{op}\)) of the CoFe\({}_{2}\)O\({}_{4}\) nanoparticles were determined that lead to a maximum power (\(P_{SM}\))\({}_{l}\). The values of (\(P_{SM}\))\({}_{l}\) have been determined for a very wide range of values of the amplitude of the magnetic field (5-180 kA/m).