High-Precision Temperature Estimation Based on Magnetic Nanoparticles Dominated by Brownian Relaxation under Combined AC and DC Magnetic Fields

Brownian relaxation is one of the primary mechanisms that allows magnetic nanoparticles (MNPs) to convert magnetic energy into thermal energy under an excitation magnetic field. Accurately characterizing the MNPs' magnetization dynamics dominated by Brownian relaxation is crucial for achieving high-precision temperature estimation. However, the lack of a readily applicable analytical expression remains a major obstacle to the advancement of magnetic nanoparticle hyperthermia (MNPH). In this paper, the perturbation method was applied to derive analytical expressions from the Fokker-Planck equation, which characterized MNPs' magnetization behaviors under the AC and DC magnetic fields. Numerical simulations were conducted to validate the accuracy of the analytical expressions and to explore the correlation between temperature and the magnetization response. Then, a temperature analysis model based on magnetization harmonics was constructed. The first and second harmonic ratios and first harmonic phase were used to calculate MNPs' temperature, respectively. The experimental results demonstrate that within 310 K to 320 K, the estimation error of the temperature using the amplitude ratio of the first to second harmonics is below 0.0151 K, while the error using the first harmonic phase is below 0.0218 K. The derived analytical expressions are expected to enhance the accuracy of MNP-based temperature measurements and facilitate their broader applications in MNPH and MNP imaging.