Effect of core size distribution on magnetic nanoparticle harmonics for thermometry

We investigated the effect of core size distribution on the performance of a magnetic nanoparticle thermometer (MNPT) in circumstances when Néel relaxation dominates the dynamic behavior of particles. Numerical simulations revealed the effects of excitation field strength and core size distribution on the temperature dependence of the amplitude and phase of harmonics. In MNPT, the field dependences of sensitivity deviated significantly from those calculated when the core size distribution was neglected. These simulation results were compared with those from experiments for which reasonable agreement was obtained. These findings must be carefully considered when designing an optimal MNPT system.

M agnetic nanoparticles (MNPs) have been widely studied for biomedical applications such as magnetic particle hyperthermia (MPH) [1][2][3] and magnetic particle imaging (MPI). [4][5][6] MPH involves locally heating up cancer/tumor tissue above 43°C to kill cancer/ tumor cells. 7) In MPH, temperatures around the MNPs must be controlled so that the normal surrounding tissue is not damaged. Therefore, accurate temperature monitoring is crucial in realizing this non-/low-invasive cancer treatment. 8) MPI is a new modality for imaging the spatial distribution of MNPs. Temperature monitoring is also important in realizing high quality MPI because the magnetization of the MNPs is affected by the physical state of the surroundings such as temperature and viscosity. [9][10][11] The magnetic nanoparticle thermometer (MNPT) is a novel tool using MNPs for non-invasive temperature measurements. [12][13][14][15][16][17][18][19] Weaver et al. proposed a method for measuring temperatures using the ratio of the intensities of the 5th (M 5 ) and 3rd (M 3 ) harmonic magnetizations of MNPs. 12) A MNPT using the phase lag of the fundamental (f 1 ) or 3rd (f 3 ) harmonic magnetization of MNPs was proposed in Refs. 17 and 15, respectively. Accurate temperature measurements were achieved in these previous studies. Nonetheless, little study has been done on how the amplitude of the excitation field and the core size distribution affect harmonic magnetizations and the sensitivity of temperature measurement. Pi et al. investigated the influence of the core size distribution on the magnetization signals for MNPT. 18) Because the Langevin model used in Ref. 18 ignores the relaxation time of MNPs, this approach is restricted to a low-frequency excitation field.
In this study, we investigated the effect of core size distribution on the performance of the MNPT when the frequency and amplitude of the excitation field are relatively high. Numerical simulations based on the Fokker-Planck equation were performed to reveal the effects of excitation field strength and core size distribution on the temperature dependence of the amplitude and phase of the harmonics. The field dependences of the sensitivity in temperature measurement diverged from those calculated when the core size distribution was neglected. The results were compared with those of experiments, and a reasonable agreement was obtained between the two.
In the present study, we used a sample called MS1 (Meito Sangyo, Japan). MS1 is magnetically fractionated MNPs from a Ferucarbotran sample, [20][21][22] and one of the promising candidates for both MNHT and MPI applications because it has a large number of MNPs, that become the source of heat generated and MPI signals. 20,21) Figure 1(a) represents the distribution of the core size d c of the MS1 sample, which was estimated from the static M-H curve. 23,24) The distribution is represented as a nV c versus d c curve, where n is the number density (in unit of m −1 kg Fe −1 ) of MNPs having the core size d c per unit particle length and iron mass, and V c = (π/6)d c 3 is the core volume. As shown, d c distributes from 5 to 40 nm, and nV c has a peak value at d c_typ = 22.3 nm. We calculated the field-dependent  27] was used to calculate τ N (H), and we used the anisotropy constant K = 4 kJ m −3 , which was obtained in a previous study. 20) In Fig. 1(b), τ N1 , τ N2 and τ N3 were calculated for d c = 22, 30 and 35 nm, respectively.
As shown in Fig. 1 This means that the dynamic behavior of MNPs is dominated by the Néel relaxation when m > H 6 mT. 0 The horizontal broken line in Fig. 1(b) represents the time T m = 1/(2πf ) when f = 20 kHz. As shown, τ B is much longer than T m , i.e., f = 20 kHz is well above the Brownian frequency 1/(2πτ B ). This result indicates that the effect of the Brownian relaxation can be almost neglected when MS1 is magnetized with high amplitude and high frequency. 28) Figure 1(c) represents the M-H curve measured at 20 kHz when the viscosity of the solution was changed. The measurements were performed using a measurement system developed at Kyushu University. 29) Symbols and lines represent the results obtained for η 1 = 0.917 mPa · s (water) and η 2 = 13.4 mPa · s (mixture of water and 60% glycerol), respectively. The results for μ 0 H ac = 6 and 20 mT are shown, where μ 0 is the permeability of the vacuum. As shown, the M-H curves are almost independent of viscosity, confirming Brownian-relaxationindependent properties. Therefore, we performed numerical simulations by taking account of only the Néel magnetization process. Furthermore, for simplicity, we assumed the easy axes of the MNPs are aligned in the direction of the AC excitation field. This is because the alignment of the easy axes is caused when the amplitude of the AC excitation field is large, as reported in Refs. 9,30,31.  Fig. 2(a), M 1 , M 3 , and M 5 increase with increasing H ac . We observed interesting behavior in the field dependences of the phase lags (f 1 , f 3 , f 5 ) as shown in Fig. 2(b). For example, f 3 increased with H ac for H ac < 10 mT μ 0 and then decreased with H ac for H ac > 10 mT μ 0 . A similar behavior for f 5 was found as shown in Fig. 2(b). We note that, for an ensemble of identical MNP, the phase lag monotonically decreases with increasing H ac due to the field-dependent relaxation time shown in Fig. 1(b). Therefore, Fig. 2(b) indicates that the field dependences of the phase lags are considerably affected by the core size distribution.
Figures 2(c) and 2(d) represent the experimental results of M i and f i (i = 1, 3, 5). In the experiment, the frequency was fixed at f = 20 kHz, and μ 0 H ac was changed from 2 to 20 mT. As shown, measured field dependences of M i and f i are similar to those obtained with the numerical simulation. Especially, the interesting behavior of f 3 and f 5 was confirmed in the experiment. We also note that similar values of M i and f i were obtained in both the experiment and simulation.
We next studied the temperature dependences of M 5 /M 3 and f 3 to explore the sensitivity of the temperature measurements when they are used in the MNPT. Figures 3(a) and 3(b) represent the simulation results of the M 5 /M 3 versus T and f 3 versus T curves, respectively. In the simulation, we assumed, for simplicity, that M s and K were independent of the temperature. We also set f = 25 kHz to compare the experimental results (to be shown later). As shown in Fig. 3(a), the values of M 5 /M 3 were almost independent of T for μ 0 H ac = 10 mT. This means that the sensitivity of the temperature measurement in MNPT is very low in this case. In contrast, M 5 /M 3 increased linearly with T for μ 0 H ac = 15 and 20 mT. This indicates that these excitation field amplitudes can be used for MNPT.  To investigate the sensitivity when using M 5 /M 3 , we determine the value S M53 using the slope of the M 5 /M 3 versus T curve [ Fig. 3(a)]. The solid line in Fig. 4(a) represents the dependence of S M53 on H ac . The sensitivity S M53 is evidently very small and becomes negative for μ 0 H ac < 10 mT. We obtain large S M53 values for μ 0 H ac > 10 mT, having a maximum value of 6.8 × 10 −4 K −1 at μ 0 H ac = 13.3 mT.
The solid line in Fig. 4(b) represents the simulation result on the H ac dependence of the sensitivity S f3 ; its value was obtained from the slope of f 3 versus T curve [ Fig. 3(b)]. As shown, | | f S 3 first increases with H ac for m  H 11.8 mT, ac 0 has a maximum value of 0.135 deg K −1 at μ 0 H ac = 11.8 mT, and then decreases with H .

ac
To clarify the effect of the core size distribution on the MNPT performance, we calculated S M53 and S f3 for monodisperse MNP. We considered a particle diameter d c_typ = 22.3 nm, which is a typical core diameter of MS1 because the nV c showed a peak at d c_typ [ Fig. 1(a)]. The dashed lines in Fig. 4 show the numerical simulation results for d c_typ = 22.3 nm. As can be seen, the field dependences of S M53 and S f3 are completely different from those calculated when the core size distribution was taken into account. The optimal value of H ac , at which | | f S M53 becomes maximum, differs between the two cases. The polarity of S M53 is also opposite between the two cases. For S f3 , there is no clear optimal value of H ac for the case of the monodisperse particle. These differences indicate that the properties of S M53 and S f3 are considerably affected by the core size distribution.
To validate the simulation results, we measured the M 5 /M 3 versus T and f 3 versus T curves of MS1, and obtained the values of S M53 and S f3 from the slope of the curves. The measurements were performed at f = 25 kHz from 296 to 326 K (23 to 53°C) using the magnetic particle spectroscopy equipment developed at TU Braunschweig. 32) The circles in Fig. 4 represent the experimental results. As shown, S M53 and | | f S 3 first increase with H ac for H ac < 12 mT/μ 0 , become maximum at around 12 mT/μ 0 , and finally decrease with H ac . These behaviors are similar to those of the simulation ones. In particular, the optimum value of H ac ≈ 12 mT/μ 0 agrees with that obtained from the numerical simulation. The polarity of the measured S M53 also agrees with those of the simulations. This agreement indicates that the core size distribution significantly affects the performance of the MNPT.
We note, however, that there are obvious deviations between experimental and simulated data. For example, the measured S f3 is about two times larger than the simulated one [ Fig. 4(b)]. We note that the temperature dependences of M s and K were neglected in the simulation. These dependences need to be included for a quantitative evaluation of S M53 and S f3 . We also note that S M53 smoothly changed with H ac in the experiment, while S M53 rapidly decreased when H ac < 10 mT/μ 0 in the simulation [ Fig. 4(a)]. In the present simulations, we assumed that all of the easy axes are aligned in the field direction for simplicity. This assumption will become inaccurate at low H ac values because the alignment of the easy axes due to the AC field considerably depends on H ac . 9) Therefore, it is also necessary to clarify the effect of H ac on the degree of alignment of the easy axes.
In conclusion, we investigated the effect of core size distribution on the performance of MNPT when MS1 was magnetized with relatively large amplitude (>6 mT) and high frequency (>20 kHz). In this case, dynamic behavior of particles is dominated by the Néel relaxation. It was shown   Fig. 1(a) is taken into account, while dashed lines represent the simulation results when only a typical core size of 22.3 nm was considered. Circles mark experimental results measured at f = 25 kHz. that the sensitivities S M53 and S f3 in temperature measurement depend on the amplitude of the excitation field, and the field dependences are affected by the core size distribution. Therefore, it is important to take into account the core size distribution for appropriately designing a MNPT system.