Studying the rate-dependent specific absorption rate in magnetic hyperthermia through multiscale simulations

Abstract

In this work, the issue of whether the dynamic magnetic properties of monodispersed magnetic colloids, modeled using micromagnetic simulations, can be extrapolated to analyze magnetic particle hyperthermia data, i.e., specific absorption rate (SAR) values acquired at high frequencies of excitation fields, is addressed. Micromagnetic finite difference simulations were performed using the Object Oriented Micromagnetic Framework (OOMMF) software package in order to obtain the dynamic hysteresis loops under a 24 kA/m alternating magnetic field amplitude and for various frequencies (50–765 kHz). In OOMMF, the finite difference method was used to find the solution of the nonlinear Landau–Lifshitz Gilbert (LLG) equation, which describes the nanoparticles’ magnetization motion when applying an effective magnetic field. To create a system of randomly oriented magnetite nanoparticles having a certain volume fraction (0.02%) that coincides with the experimentally utilized concentration of 1 mg/ml, we start with a perfect simple cubic lattice with a large lattice spacing so that the particle–particle distance is large enough to neglect dipolar interactions (non-interacting nanoparticles). The system under study is a set of 40-nm magnetite nanoparticles with a lognormal size distribution. The simulations were performed assuming quasistatic conditions, an approach that is reasonable for ferromagnetic-like behavior. It is worth noting that the code considers not only the uniaxial anisotropy Ku but also the cubic magnetocrystalline one Kc as well. Kc is usually neglected in literature because the uniaxial contribution dominates, but this is not the case for magnetite since Ku = 9 kJ/m3 and Kc = −11 kJ/m3. Moreover, such an inclusion seems quite reasonable since the magnetocrystalline anisotropy is always present yet with a relative contribution. The SAR values at each frequency were determined after calculating hysteresis losses via the area of the simulated hysteresis loops. Interestingly, SAR values at low frequencies followed an exponential increase trend with a frequency indicating a deviation from the linear behavior usually reported in the literature. To validate our approach, we employed a coupled electromagnetic-thermal model based on COMSOL Multiphysics simulations that provides an accurate estimation of the magnetic field and temperature distribution within the ferrofluid. The time-dependent temperature curves are obtained after 30 min of magnetic particle hyperthermia treatment for the same alternating magnetic field amplitude used in OOMMF simulations (30 mT) and for two representative frequency values. One in the low (300 kHz) and one in the high (765 kHz) frequency regimes. The numerical curves were compared to the corresponding experimental ones and found to be in good agreement. Our findings provide new insight into the validity of dynamic micromagnetic simulation to analyze the frequency behavior of SAR within the framework of LLG and indicate that anisotropy selection plays a key role in the reliability of simulations.