Hyperthermia is one of the promising approaches to cancer therapy. The most commonly used heating method in the clinical setting is capacitive heating that uses a radiofrequency (RF) electric field [1] . However, a major technical problem with hyperthermia is the difficulty of heating the targeted tumor to the desired temperature without damaging the surrounding tissues, as the electromagnetic energy must be directed from an external source and penetrate normal tissue. Other hyperthermia modalities, including ultrasound hyperthermia, have been reported [2] , but the efficacy of these modalities depends on the size and depth of the tumor, and disadvantages include the ability to target the tumor and control the exposure.

Hyperthermia with use of magnetic nanoparticles (MNPs) (magnetic hyperthermia) was developed in the 1950s [3] and is still under development in the effort to overcome the above disadvantages [4] [5] . 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 [6] . For small MNPs, the relaxational losses caused by a delay in magnetization relaxation are dominant for heat dissipation [6] . With the development of precise methods for synthesizing functionalized MNPs [7] , MNPs with functionalized surfaces, which have high specificity for tumor tissue, have been developed as heating elements for magnetic hyperthermia [8] . Recently, MNPs with a higher heating efficiency, i.e., specific loss power (SLP), have also been actively developed [9] . Furthermore, there is renewed interest in magnetic hyperthermia as a treatment modality for cancer, especially when it is combined with other, more traditional therapeutic approaches such as the co-delivery of anticancer drugs [10] or radiation therapy [11] . From these aspects, magnetic hyperthermia has received much recent attention.

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 [6] has often been used for the estimation of SLP. His model, however, is based on the so-called linear magnetization assumption [12] , and thus it is said that his model is strictly valid only in the limit of small amplitude and frequency of AMF. In this study, we presented three methods for estimating SLP and compared the SLP values estimated by the three methods under various conditions of MNPs and AMF. Especially, we investigated the validity of Rosensweig’s model in comparison with the numerical solution of the magnetization relaxation equation of Shliomis [13] .

As shown in Equation (14), the value depends on the cycle number of the M-H curve. Thus, we calculated the value in the quasi steady state, i.e., the SLP₁ value under the condition of with ε being taken as 10⁻⁶, as previously described. When we neglected the second term in the right-hand side of Equation (14), the value reached the steady state after a few cycles in all the cases studied. Figure 1(a) shows the M-H curves in the quasi steady state calculated from Equation (8) for various frequencies of AMF. For comparison, Figure 1(b) shows the M-H curves calculated from Equation (9). It should be noted that was normalized by the saturation magnetization (M_s) given by. In these simulations, H₀ was fixed at 20 mT and D was assumed to be 20 nm. Figure 2 shows the case when D was varied from 10 nm to 30 nm with steps of 5 nm. As shown in Figure 1 and Figure 2, a large difference between the M-H curves obtained by Equation (8) and Equation (9) was observed and it increased with increasing f and D.

Figure 3(a) shows the SLP₃ values calculated from Equation (16) as a function of H with f being varied from 200 kHz to 1000 kHz with steps of 200 kHz, whereas Figure 3(b) shows the case when D was varied from 10 nm to 30 nm with steps of 5 nm. In

these simulations, H₀ was fixed at 20 mT. As shown in Figure 3, the SLP₃ value became maximum, i.e., when H was zero. When |H| was the maximum, i.e., H₀, the SLP₃ value became minimum, i.e.,.

Figure 4(a) shows the comparison of SLP₁, SLP₂, , , and as a function of D, whereas Figure 4(b) shows the RD₂, , , and values as a function of D. As shown in Figure 4(b), was the largest and

increased with increasing D, whereas was the smallest and was almost constant regardless of D.

Figure 5(a) shows the comparison of SLP₁, SLP₂, , , and as a function of H₀, whereas Figure 5(b) shows the RD₂, , , and values as a function of H₀. As shown in Figure 5(b), was the largest and increased with increasing H₀, whereas was the smallest and was almost constant regardless of H₀.

Figure 6(a) shows the comparison of SLP₁, SLP₂, , , and as a function of f, whereas Figure 6(b) shows the RD₂, , , and values as a function of f. In this case, the RD₂, , , and values were almost constant regardless of f (Figure 6(b)).

In this study, we presented three methods for the estimation of SLP in magnetic hyperthermia and compared the SLP values estimated by the three methods (SLP₁, SLP₂, and SLP₃). SLP₁ was derived by solving the magnetization relaxation equation of Shliomis [13] numerically. SLP₂ was derived by solving Shliomis’ relaxation equation [13] using the complex susceptibility. SLP₃ was derived based on Rosensweig’s model, in which the complex susceptibility with χ' and χ'' (basically χ₀) being assumed to be constant has been used.

As previously described, Rosensweig’s model [6] has often been used for the estimation of SLP. To the best of our knowledge, however, few studies have been performed to validate the SLP estimation based on Rosensweig’s method [6] in comparison with that based on the numerical solution of the magnetization relaxation equation of Shliomis [13] .

As shown in Figure 1 and Figure 2, a large difference was observed between the M-H curves calculated from Equation (8) and Equation (9), especially when H is zero, and the difference increased with increasing f and D. When using Equation (9), becomes equal to at, i.e., and it becomes equal to at, i.e.,. On the other hand, becomes equal to at, i.e., and it becomes equal to at, i.e.,. Thus, the above difference in the M-H curves shown in Figure 1 and Figure 2 may suggest that given by Equation (10) is overestimated compared to the case when using Equation (8). Furthermore, the area of the M-H curve calculated from Equation (9) (Figure 1(b) and Figure 2(b)) was larger than that calculated from Equation (8) (Figure 1(a) and Figure 2(a)). 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 SLP₂ is larger than SLP₁ (Figure 4(a), Figure 5(a), and Figure 6(a)).

The SLP₃ given by Equation (16) has often been used for characterizing the heating property of MNPs [16] . As previously described, SLP₃ has been derived with an assumption that χ₀ is constant. However, χ₀ is actually magnetic field (H) dependent, because χ₀ is the function of the Langevin parameter (ξ) as shown in Equation (7) and ξ is the function of H. To investigate to what extent SLP₃ depends on H, we showed the SLP₃ values as a function of H in Figure 3. As shown in Figure 3, the SLP₃ value changed largely depending on H. χ₀ in Equation (16) is the monotonically decreasing function of |ξ| or |H| (data not shown). Thus, the SLP₃ value becomes maximum when H is zero (Figure 3). In this case, χ₀ becomes equal to χ_i, because in Equation (7) approaches ξ/3 when ξ approaches zero. On the other hand, when |H| is maximum, i.e., H is equal to ±H₀, the SLP₃ value becomes minimum.

To compare the SLP values estimated by Equation (15) and Equation (16) with that estimated using the numerical solution of the magnetization relaxation equation of Shliomis [13] , we calculated the relative differences given by Equation (18) to Equation (21). As shown in Figure 4 and Figure 5, the value was the largest and increased with increasing D and H₀, whereas the value was the smallest and was almost constant regardless of D and H₀. These results suggest that when estimating SLP using Rosensweig’s model [6] , H in ξ should be taken as H₀.

In this study, we solved the magnetization relaxation equation of Shliomis [13] (Equation (1)) with an assumption that there is no bulk flow and the magnetization of MNPs and magnetic field are collinear. In this case, Equation (1) is reduced to Equation (2), which can be easily solved using convolution integral as shown in Equation (8). Although Equation (2) appears to be valid in considering the magnetic hyperthermia with use of small MNPs in the superparamagnetic state and we believe that this study will provide the basis for establishing the effectiveness of such magnetic hyperthermia, it will be necessary to solve Equation (1) without any assumptions or another magnetization equation derived microscopically from the Fokker-Planck equation [12] [17] for more detailed analysis. These studies are currently in progress. As previously described, we targeted the MNPs consisting of maghemite with the magnetic and physical properties described in the “Simulation Studies” section, because maghemite is the core iron oxide of Resovist®, which is a commercially-available organ-specific contrast agent for magnetic resonance imaging and has been approved for clinical use in Japan [15] . We will also perform further studies for other MNPs.

We presented three methods for estimating SLP in magnetic hyperthermia and compared the SLP values estimated by the three methods under various conditions of MNPs and AMF. This study will be useful for optimizing the parameters of AMF in magnetic hyperthermia and for developing the MNPs suitable for magnetic hyperthermia. We also investigated the validity of Rosensweig’s model in comparison with the numerical solution of the magnetization relaxation equation of Shliomis, suggesting that when estimating SLP using Rosensweig’s model, the magnetic field strength in the Langevin parameter should be taken as the amplitude of AMF.

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) Methods for Estimating Specific Loss Power in Magnetic Hyperthermia Revisited. Open Journal of Applied Sciences, 6, 815-825. http://dx.doi.org/10.4236/ojapps.2016.612071