arge, the second term of the right-hand side of Equation (12) can be neglected and SLPi approaches the steady state. We denote the SLPi value in the quasi steady state by SLPqss. Actually, SLPqss was taken as the SLPi value in the case when the relative error (RE) was less than 10−10. The RE was defined by

, (13)

where denotes the absolute value. The integration in Equation (12) was performed by use of the trapezoidal rule  (“trapz” in MATLAB®; The MathWorks, Inc., Natick, MA, USA) and the convolution integral was calculated using the MATLAB function (“conv”).

2.2. Simulation Studies

In this study, we assumed that MNPs consisted of two kinds of iron oxide nanoparticles, i.e., maghemite (γ-Fe2O3) and magnetite (Fe3O4). We fixed τ0, δ, Md, K, η, ρ, f, and T to be 109 s, 2 nm, 414 kA/m, 4.7 kJ/m3, 0.00235 kg/m/s, 4600 kg/m3, 0.003, and 37˚C, respectively, for maghemite   . For magnetite, we fixed τ0, δ, Md, K, η, ρ, f, and T to be 109 s, 2 nm, 446 kA/m, 9.0 kJ/m3, 0.00235 kg/m/s, 5180 kg/m3, 0.003, and 37˚C, respectively  . When H0, 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 Gs, the strength of the SMF at a distance of x from the FFP (Hs(x)) was given by

. (14)

3. Results

As shown in Equation (12), the SLPi value depends on the cycle number of the M-H curve. Thus, we calculated the SLPi value in the quasi steady state, i.e., the SLPqss value under the condition in which the RE given by Equation (13) was less than 10−10, as previously described. Figure 1(a) shows the common logarithm of the RE given by Equation (13) as a function of the number of cycles for maghemite, whereas Figure 1(b) shows the case for magnetite. In these simulations, we neglected the second term in the right-hand side of Equation (12), because it approaches zero in the steady state. The τ value given by Equation (3) was 4.10 × 10−8 s and 1.71 × 10−6 s for maghemite and magnetite, respectively. As shown in Figure 1, the speed with which the SLPi value reached the quasi steady state depended on the τ value, i.e., the larger the τ value, the more slowly the SLPi value reached the quasi steady state. As shown in Figure 1(b), there was a tendency for the RE to increase with increasing Hs. Note that the plateau regions in Figure 1 correspond to the so-called “machine epsilon”.

Figure 2(a) shows the M-H curves in the quasi steady state calculated from Equation (8) at an f of 300 kHz for maghemite. In this case, Hs was varied from 0 to 30 mT with steps of 10 mT. For comparison, Figure 2(b) shows those for magnetite. It should be noted that was normalized by the saturation magnetization (Ms) given by. In these simulations, H0 was fixed at 20 mT and D was assumed to be 20 nm. Figure 3 shows the case when f was taken as 600 kHz. As shown in Figure 2 and Figure 3, the area of the M-H curve decreased with increasing Hs and a large difference in the M-H curve was observed between maghemite and magnetite.

Figure 4(a) shows the SLPqss values calculated from Equation (12) as a function of Hs with D being varied from 10 nm to 30 nm with steps of 5 nm for maghemite, whereas Figure 4(b) shows the case for magnetite. In these simulations, H0 and f were fixed at 20 mT and 300 kHz, respectively. As shown in Figure 4, the SLPqss value decreased with increasing Hs in all cases.

(a) (b)

Figure 1. (a) Common logarithm of the relative error (RE) given by Equation (13) at various values of the static magnetic field (SMF) strength (Hs) (0, 25, and 50 mT) for magnetic nanoparticles (MNPs) consisting of maghemite; (b) Common logarithm of the RE given by Equation (13) at various Hs values (0, 25, and 50 mT) for MNPs consisting of magnetite. In these simulations, the amplitude (H0) and frequency (f) of an alternating magnetic field (AMF) and the diameter of MNPs (D) were assumed to be 20 mT, 300 kHz, and 20 nm, respectively. Note that the unit of mT can be converted to kA/m by use of the relationship 1 mT = 0.796 kA/m. The magnetic and physical properties of maghemite and magnetite used in this study are described in the “Simulation Studies” section. The specific loss power in the quasi steady state (SLPqss) was taken as the SLPi value calculated from Equation (12) in the case when the RE was less than the threshold (10−10 in this study).

(a) (b)

Figure 2. (a) M-H curves (hysteresis loops) in the quasi steady state calculated from Equation (8) at various Hs values (0, 10, 20, and 30 mT) for maghemite; (b) M-H curves calculated from Equation (8) at various Hs values (0, 10, 20, and 30 mT) for magnetite. In these simulations, H0, f, and D were assumed to be 20 mT, 300 kHz, and 20 nm, respectively.

(a) (b)

Figure 3. (a) M-H curves in the quasi steady state calculated from Equation (8) at various Hs values (0, 10, 20, and 30 mT) for maghemite; (b) M-H curves calculated from Equation (8) at various Hs values (0, 10, 20, and 30 mT) for magnetite. In these simulations, H0, f, and D were assumed to be 20 mT, 600 kHz, and 20 nm, respectively.

(a) (b)

Figure 4. (a) Specific loss power in the quasi steady state (SLPqss) values calculated from Equation (12) as a function of Hs at various D values (10, 15, 20, 25, and 30 nm) for maghemite; (b) SLPqss values calculated from Equation (12) as a function of Hs at various D values (10, 15, 20, 25, and 30 nm) for magnetite. In these simulations, H0 and f were assumed to be 20 mT and 300 kHz, respectively. Note that the SLPqss values for D of 10 nm are too small to be seen in the figure.

Figure 5(a) shows the SLPqss values calculated from Equation (12) as a function of Hs with H0 being varied from 5 mT to 25 mT with steps of 5 mT for maghemite, whereas Figure 5(b) shows the case for magnetite. In these cases, D and f were fixed at 20 nm and 300 kHz, respectively. As in Figure 4, the SLPqss value decreased with increasing Hs in all cases.

Figure 6(a) shows the SLPqss values calculated from Equation (12) as a function of Hs with f being varied from 200 kHz to 1000 kHz with steps of 200 kHz for maghemite, whereas Figure 6(b) shows the case for magnetite. In these simulations, D and H0 were

(a) (b)

Figure 5. (a) SLPqss values calculated from Equation (12) as a function of Hs at various H0 values (5, 10, 15, 20, and 25 mT) for maghemite; (b) SLPqss values calculated from Equation (12) as a function of Hs at various H0 values (5, 10, 15, 20, and 25 mT) for magnetite. In these simulations, f and D were assumed to be 300 kHz and 20 nm, respectively.

(a) (b)

Figure 6. (a) SLPqss values calculated from Equation (12) as a function of Hs at various f values (200, 400, 600, 800, and 1000 kHz) for maghemite; (b) SLPqss values calculated from Equation (12) as a function of Hs at various f values (200, 400, 600, 800, and 1000 kHz) for magnetite. In these simulations,H0 and D were assumed to be 20 mT and 20 nm, respectively.

fixed at 20 nm and 20 mT, respectively. As in Figure 4 and Figure 5, the SLPqss value decreased with increasing Hs in all cases.

Figure 7(a) shows the SLPqss values calculated from Equation (12) as a function of the distance from the FFP (x) at various values of Gs (1, 2, 5, and 10 T/m) for maghemite, whereas Figure 7(b) shows the case for magnetite. In these simulations, Hs at x, i.e., Hs(x) was calculated from Gs and x using Equation (14), and D, H0, and f were fixed at 20 nm, 20 mT, and 300 kHz, respectively. As shown in Figure 7, the plot of the SLPqss values against x became narrow as Gs increased in both cases.

(a) (b)

Figure 7. (a) SLPqss values as a function of the distance from a field-free point (x) at various gradient strength values of the SMF (Gs) (1, 2, 5, and 10 T/m) for maghemite; (b) SLPqss values as a function of x at various Gs values (1, 2, 5, and 10 T/m) for magnetite. In these simulations, D, H0, and f were assumed to be 20 nm, 20 mT, and 300 kHz, respectively.

4. Discussion

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  numerically. We also presented the results obtained by simulation studies under various conditions of MNPs, AMF, and SMF. Our results shown in Figure 4 to Figure 7 suggest that the SLP in magnetic hyperthermia can be controlled using the SMF and that our method will be useful for selecting the optimal parameters for controlling the temperature rise in magnetic hyperthermia in order to reduce the risk of overheating surrounding healthy tissues.

As shown in Figure 1, the RE given by Equation (13) for maghemite (Figure 1(a)) became less than the threshold (1010 in this study) faster than that for magnetite (Figure 1(b)). As previously described, there was a large difference in the τ value given by Equation (3) between maghemite and magnetite (4.10 × 108 s and 1.71 × 106 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 (Figure 2(a)) and magnetite (Figure 2(b)). This difference also appears to be due to a difference in the τ value.

As shown in Figure 2 and Figure 3, the area of the M-H curve calculated from Equation (8) decreased with increasing Hs. 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 SLPqss decreases with increasing Hs (Figure 4 to Figure 6).

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  . This device can also generate an FFP and move it by varying the DC electric currents applied to the individual Maxwell coils  . We investigated the effect of SMF on the temperature rise in magnetic hyperthermia using this device and then evaluated the feasibility of controlling the temperature rise using an SMF with an FFP under various conditions of the AMF and SMF. Our experimental results demonstrated that the SLP value (referred to as “SAR” in our previous paper  ) decreased with increasing Hs  . Thus, the present results shown in Figure 4 to Figure 6 appear to support our experimental results  . Furthermore, our experimental results showed that the area of local heating became narrow with increasing Gs. Conversely, it became broad with decreasing Gs  . As shown in Figure 7, the plot of the SLP values against the position from the FFP became narrow as Gs increased, whereas it became broad as Gs decreased. These results also appear to support our experimental results described above  .

In our previous study  , we treated the case in which the FFP is moved in one dimension by use of a single Maxwell coil pair. However, it would be possible to extend it to two or three dimensions by use of multiple Maxwell coil pairs. Theoretically, the magnetic field strength generated by a Maxwell coil pair can be calculated from the Biot-Savart law  . Thus, when the electric currents applied to the coils are known, the position of the FFP can be determined by solving the equations describing the magnetic field strength based on the Biot-Savart law  . Conversely, when the position of the FFP is given, the electric currents applied to the coils can be determined by the similar procedure. If this could be realized, it would become easy to control the position of the FFP, i.e., the position of local heating. As previously described, the plot of the SLP values against the position from the FFP became broad with decreasing Gs, whereas it became narrow with increasing Gs (Figure 7). These results suggest that the area of local heating can be controlled by varying the value of Gs. The Gs value can also be calculated from the Biot-Savart law  . Thus, it would become possible to control not only the position but also the area of local heating by controlling the electric currents applied to the coils based on the Biot-Savart law  . This will further enhance the feasibility and effectiveness of the magnetic hyperthermia with the temperature rise being controlled for reducing the risk of overheating surrounding healthy tissues.

In this study, we derived Equation (12) by solving the magnetization relaxation equation of Shliomis  (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 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  for more detailed analysis. These studies are currently in progress. As previously described, we targeted the MNPs consisting of maghemite and magnetite with the magnetic and physical properties described in the “Simulation Studies” section. We will also perform further studies for the MNPs with other magnetic and physical properties and/or other MNPs.

Recently, Dhavalikar et al.  have theoretically investigated the feasibility of spatially-focused heating of MNPs guided by the SMF with a field-free region such as the FFP. In their study, the phenomenological magnetization equation derived by Martsenyuk et al.  was used instead of the Shliomis’ equation  . A comparative study of the present results and those based on the equation of Martsenyuk et al.  is also currently in progress.

5. Conclusion

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.

Acknowledgements

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).

Cite this paper
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. doi: 10.4236/ojapps.2016.612073.
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