## 2 Basic Theoretical Aspects  
The standard physical observables that indicate if different magnetic nanoparticles can be used in superparamagnetic hyperthermia to destroy tumor cells are: specific loss power \(P_{s}\) (or denoted symbolic SLP) and heating rate of nanoparticles \(\Delta T_{i}/\Delta t\), where \(T_{i}\) is the temperature and \(t\) is the time, having in view that the specific absorption rate (\(SAR\)) [67] for an adiabatic system is  
\[SAR=c\Delta T_{i}/\Delta t \tag{1}\]  
where \(c\) is the specific heat capacity of the nanoparticles.  
The specific loss power can be expressed by the formula [1,53]  
\[P_{s}=\frac{3\pi\mu_{0}\chi_{i}}{\rho\xi}\bigg{(}\coth\xi-\frac{1}{\xi}\bigg{)} \frac{2\pi f\tau}{1+\left(2\pi f\tau\right)^{2}}fH^{2} \tag{2}\]  
where \(\mu_{0}\) is the magnetic permeability of vacuum (4\(\pi\)\(\times\)\(10^{-7}\) H/m), \(f\) and \(H\) are the frequency and the amplitude of magnetic field, \(\rho\) is the density of nanoparticle material. The observable \(\xi\) in the above formula is the parameter of the [PERSON] function [68], expressed as a function of the diameter of the nanoparticles (\(D\)) considered spherical.  
\[\xi=\frac{\pi\mu_{0}M_{s}D^{3}}{6k_{B}T}H \tag{3}\]  
that describing the magnetization of magnetic nanoparticles (\(M\)) in the external magnetic field \(H\)[69],  
\[M=M_{sat}\bigg{(}\coth\xi-\frac{1}{\xi}\bigg{)} \tag{4}\]  
where \(M_{sat}\) is the saturation magnetization of the nanoparticle system. The observable \(\xi\) is the initial magnetic susceptibility of magnetic nanoparticles [4,53], which is given by the formula:  
\[\chi_{i}=\frac{\varepsilon\pi\mu_{0}M_{s}^{2}D^{3}}{18k_{B}T} \tag{5}\]  
and strongly depends on the diameter of the magnetic nanoparticles (the diameter at the third power). The observable \(\tau\) in Equation (2) is the Neel magnetic relaxation time [2]  
\[\tau=\tau_{0}exp\bigg{(}\frac{\pi KD^{3}}{6k_{B}T}\bigg{)} \tag{6}\]  
where \(\tau_{0}\) is a time constant with a value of \(10^{-9}\)[70].  
Formula given by Equation (2) was determined by taking into account the dependence of static magnetic susceptibility on the external magnetic field applied when reaching high values and the magnetic susceptibility can no longer be considered constant and equal to the initial magnetic susceptibility (\(\chi_{i}\)) [53].  
In small amplitude fields the formula is simplified, being reduced to [53]  
\[P_{s}=\frac{\pi\mu_{0}\chi_{i}}{\rho}\frac{2\pi f\tau}{1+\left(2\pi f\tau \right)^{2}}fH^{2} \tag{7}\]  
Taking into account the specific absorption rate given by Equation (1) and the specific loss power in an adiabatic system (\(SAR=P_{s}\)) given by Equation (2) (or Equation (7)) it is possible to obtain the heating temperature \(\Delta T_{h}\) of the nanoparticles in a finite time interval \(\Delta t\).  
\[\Delta T_{h}=\frac{1}{c}P_{s}\Delta t \tag{8}\]  
In magnetic hyperthermia of tumors, the heating temperature must reach approx. 43 \({}^{\circ}\)C in a short period of time in order to be effective on tumor cells, leading to their destruction at least by cellular apoptosis [7,10], and not to affect healthy cells. However,heating temperatures during magnetic hyperthermia can increase to values higher than 43 \({}^{\circ}\)C. In such cases, automatic control systems with an electronic reaction of the magnetic field generator (reverse reaction) can be used which allows limiting and maintaining the heating temperature constant at the required value of 43 \({}^{\circ}\)C.  
In conclusion, if during the process of heating the magnetic nanoparticles in superparamagnetic hyperthermia a temperature of least 42.5-43 \({}^{\circ}\)C is obtained in a short period of time so that the healthy cells are not affected, then those nanoparticles could be successfully used in cancer therapy via superparamagnetic hyperthermia.