_{1}

Our purpose in this study was to present a method for estimating the specific loss power (SLP) in magnetic hyperthermia in the presence of an external static magnetic field (SMF) and to investigate the SLP values estimated by this method under various diameters (
*D*) of magnetic nanoparticles (MNPs) and amplitudes (
*H _{0}*) and frequencies (

*f*) of an alternating magnetic field (AMF). In our method, the SLP was calculated by solving the magnetization relaxation equation of Shliomis numerically, in which the magnetic field strength at time

*t*(

*H*(

*t*)) was assumed to be given by , with

*H*being the strength of the SMF. We also investigated the SLP values in the case when the SMF with a field-free point (FFP) generated by two solenoid coils was used. The SLP value in the quasi steady state (

_{s}*SLP*) decreased with increasing

_{qss}*H*. The plot of the

_{s}*SLP*values against the position from the FFP became narrow as the gradient strength of the SMF (

_{qss}*G*) increased. Conversely, it became broad as

_{s}*G*

_{s}decreased. These results suggest that the temperature rise and the area of local heating in magnetic hyperthermia can be controlled by varying the

*H*and

_{s}*G*values, respectively. In conclusion, our method will be useful for estimating the SLP in the presence of both the AMF and SMF and for designing an effective local heating system for magnetic hyperthermia in order to reduce the risk of overheating surrounding healthy tissues.

_{s}Hyperthermia is one of the promising approaches to cancer therapy [

MNPs generate heat in an alternating magnetic field (AMF) as a result of hysteresis and relaxational losses, which results in heating of the tissue in which MNPs accumulate [

The estimation of SLP is important for evaluating the heating efficiency of MNPs, for optimizing the parameters of AMF, and for the optimal design of MNPs in an attempt to establish the effectiveness of magnetic hyperthermia. Rosensweig’s model [

It is also important to heat the targeted tumor to the desired temperature without damaging the surrounding healthy tissues in order to enhance the effectiveness of magnetic hyperthermia. In 2009, Tasci et al. [

Our purpose in this study was to present a method for estimating the SLP in magnetic hyperthermia in the presence of an external SMF and to investigate the SLP values estimated by this method under various conditions of MNPs, AMF, and SMF.

The magnetization relaxation equation of Shliomis [

where M is the magnetization of MNPs under the magnetic field H, Ω is the flow velocity, f is the volume fraction, and η is the viscosity of the suspending fluid. When there is no bulk flow and M and H are collinear, Equation (1) is reduced to the following equation [

In Equation (2), τ is the effective relaxation time given by

where τ_{N} and τ_{B} are the Néel relaxation and Brownian relaxation time, respectively [_{N} and τ_{B} are given by the following relationships [

where τ_{0} is the average relaxation time in response to a thermal fluctuation, k_{B} is the Boltzmann constant, T is the temperature, and_{H} is taken as the hydrodynamic volume of MNP that is larger than the magnetic volume _{H}, it is assumed that

where χ_{0} and

where H_{0} and f denote the amplitude and frequency of AMF, respectively, and H_{s} denotes the strength of an external SMF. Because the actual equilibrium susceptibility (χ_{0}) is dependent on the magnetic field, χ_{0} was assumed to be the chord susceptibility corresponding to the Langevin equation, given by [

where χ_{i} is the initial susceptibility given by_{d} is the domain magnetization of a suspended particle, and μ_{0} is the permeability of free space. It should be noted that ξ is magnetic field (H) dependent and thus time dependent.

Solving Equation (2) and using Equation (5) and Equation (6) yield

where

The average rate of energy dissipation per cycle of the period, i.e.,

Substituting Equation (6) into Equation (9) yields

The rate of energy dissipation per unit mass of MNPs, i.e., SLP can be obtained from

where ρ is the density of suspending fluid.

In this study, we considered a method for estimating the SLP, in which Equation (8) was used for _{i}) can be given by

It should be noted that when i is sufficiently large, the second term of the right-hand side of Equation (12) can be neglected and SLP_{i} approaches the steady state. We denote the SLP_{i} value in the quasi steady state by SLP_{qss}. Actually, SLP_{qss} was taken as the SLP_{i} value in the case when the relative error (RE) was less than 10^{−10}. The RE was defined by

where

In this study, we assumed that MNPs consisted of two kinds of iron oxide nanoparticles, i.e., maghemite (γ-Fe_{2}O_{3}) and magnetite (Fe_{3}O_{4}). We fixed τ_{0}, δ, M_{d}, K, η, ρ, f, and T to be 10^{−}^{9} s, 2 nm, 414 kA/m, 4.7 kJ/m^{3}, 0.00235 kg/m/s, 4600 kg/m^{3}, 0.003, and 37˚C, respectively, for maghemite [_{0}, δ, M_{d}, K, η, ρ, f, and T to be 10^{−}^{9} s, 2 nm, 446 kA/m, 9.0 kJ/m^{3}, 0.00235 kg/m/s, 5180 kg/m^{3}, 0.003, and 37˚C, respectively [_{0}, f, and D were fixed, they were taken as 20 mT, 300 kHz, and 20 nm, respectively. It should be noted that the unit of mT can be converted to kA/m by use of the relationship 1 mT = 0.796 kA/m.

When considering the control of the temperature rise using the SMF with a gradient strength of G_{s}, the strength of the SMF at a distance of x from the FFP (H_{s}(x)) was given by

As shown in Equation (12), the SLP_{i} value depends on the cycle number of the M-H curve. Thus, we calculated the SLP_{i} value in the quasi steady state, i.e., the SLP_{qss} value under the condition in which the RE given by Equation (13) was less than 10^{−10}, as previously described. ^{−8} s and 1.71 × 10^{−6} s for maghemite and magnetite, respectively. As shown in _{i} value reached the quasi steady state depended on the τ value, i.e., the larger the τ value, the more slowly the SLP_{i} value reached the quasi steady state. As shown in _{s}. Note that the plateau regions in

_{s} was varied from 0 to 30 mT with steps of 10 mT. For comparison, _{s}) given by_{0} was fixed at 20 mT and D was assumed to be 20 nm. _{s} and a large difference in the M-H curve was observed between maghemite and magnetite.

_{qss} values calculated from Equation (12) as a function of H_{s} with D being varied from 10 nm to 30 nm with steps of 5 nm for maghemite, whereas _{0} and f were fixed at 20 mT and 300 kHz, respectively. As shown in _{qss} value decreased with increasing H_{s} in all cases.

_{qss} values calculated from Equation (12) as a function of H_{s} with H_{0} being varied from 5 mT to 25 mT with steps of 5 mT for maghemite, whereas _{qss} value decreased with increasing H_{s} in all cases.

_{qss} values calculated from Equation (12) as a function of H_{s} with f being varied from 200 kHz to 1000 kHz with steps of 200 kHz for maghemite, whereas _{0} were

fixed at 20 nm and 20 mT, respectively. As in _{qss} value decreased with increasing H_{s} in all cases.

_{qss} values calculated from Equation (12) as a function of the distance from the FFP (x) at various values of G_{s} (1, 2, 5, and 10 T/m) for maghemite, whereas _{s} at x, i.e., H_{s}(x) was calculated from G_{s} and x using Equation (14), and D, H_{0}, and f were fixed at 20 nm, 20 mT, and 300 kHz, respectively. As shown in _{qss} values against x became narrow as G_{s} increased in both cases.

In this study, we presented a method for estimating the SLP in magnetic hyperthermia in the presence of both the AMF and SMF, which was derived by solving the magnetization relaxation equation of Shliomis [

As shown in ^{−}^{10} in this study) faster than that for magnetite (^{−}^{8} s and 1.71 × 10^{−}^{6} s for maghemite and magnetite, respectively, under the condition described in the “Simulation Studies” section). Thus, the above difference appears to be due to a difference in the τ value. Similarly, a large difference in the M-H curve was observed between maghemite (

As shown in _{s}. The area of the M-H curve directly represents the power loss during one cycle of the hysteresis loop. Thus, the above finding corresponds to the fact that SLP_{qss} decreases with increasing H_{s} (

We previously made an experimental device that allows for magnetic hyperthermia in the presence of the SMF that was generated using a Maxwell coil pair [_{s} [_{s}. Conversely, it became broad with decreasing G_{s} [_{s} increased, whereas it became broad as G_{s} decreased. These results also appear to support our experimental results described above [

In our previous study [_{s}, whereas it became narrow with increasing G_{s} (_{s}. The G_{s} value can also be calculated from the Biot-Savart law [

In this study, we derived Equation (12) by solving the magnetization relaxation equation of Shliomis [

Recently, Dhavalikar et al. [

We presented a method for estimating the SLP in magnetic hyperthermia in the presence of an external SMF, which is based on the numerical solution of the magnetization relaxation equation of Shliomis. We also presented the SLP values estimated under various conditions of MNPs, AMF, and SMF. Our method will be useful for estimating the SLP in the presence of both the AMF and SMF and for designing an effective local heating system using the SMF for magnetic hyperthermia in order to reduce the risk of overheating surrounding healthy tissues.

This work was supported by a Grant-in-Aid for Scientific Research (Grant Number: 25282131 and 15K12508) from the Japan Society for the Promotion of Science (JSPS).

Murase, K. (2016) A Simulation Study on the Specific Loss Power in Magnetic Hyperthermia in the Presence of a Static Magnetic Field. Open Journal of Applied Sciences, 6, 839-851. http://dx.doi.org/10.4236/ojapps.2016.612073