  To optimize the heating properties of magnetic nanoparticles (MNPs) in
magnetic hyperthermia applications, it is necessary to calculate the area of
their hysteresis loops in an alternating magnetic field. The three types of
theories suitable for describing the hysteresis loops of MNPs are presented and
compared to numerical simulations: equilibrium functions, Stoner-Wohlfarth
model based theories (SWMBTs) and linear response theory (LRT). Suitable
formulas to calculate the hysteresis area of major cycles are deduced from
SWMBTs and from numerical simulations; the domain of validity of the analytical
formula is explicitly studied. In the case of minor cycles, the hysteresis area
calculations are based on the LRT. A perfect agreement between LRT and
numerical simulations of hysteresis loops is obtained. The domain of validity
of the LRT is explicitly studied. Formulas to calculate the hysteresis area at
low field valid for any anisotropy of the MNP are proposed. Numerical
simulations of the magnetic field dependence of the area show it follows
power-laws with a large range of exponents. Then, analytical expressions
derived from LRT and SWMBTs are used for a theoretical study of magnetic
hyperthermia. It is shown that LRT is only pertinent for MNPs with strong
anisotropy and that SWMBTs should be used for weak anisotropy MNPs. The optimum
volume of MNPs for magnetic hyperthermia as function of material and
experimental parameters is derived. The maximum specific absorption rate (SAR)
achievable is calculated versus the MNP anisotropy. It is shown that an optimum
anisotropy increases the SAR and reduces the detrimental effects of size
distribution. The optimum anisotropy is simple to calculate and depends on the
magnetic field used in the hyperthermia experiments and on the MNP
magnetization only. The theoretical optimum parameters are compared to the one
of several magnetic materials.
