Sign inversion of the third-order harmonic component

Fig. 1(A) shows the dynamic magnetization curves calculated for the anisotropic nanoparticles with randomly-oriented easy axes for various particle diameters, d, when an AC magnetic field H_ac·sin(2nπf·t) with an amplitude H_ac = 20 kA/m is irradiated at a frequency f = 50 kHz. The physical quantities regarding the anisotropy of these nanoparticles are summarized in Table 1. There is no hysteresis found for d = 10 and 13 nm, which is consistent with the fact that the Néel relaxation time, τ_N (shown in Table 1), is much shorter than the characteristic time of the field variation (2πf)⁻¹ ∼ 3 μs. For d = 13 nm, the system is more easily magnetized compared with d = 10 nm. This is reasonable because the Zeeman energy given by Eq. (1) is greater for larger volumes. In addition, it is noteworthy that the magnetization curve for d = 13 nm is apparently convex at H > 0, while it is concave at H < 0, as indicated by the arrows. In other words, the nonlinear susceptibility χ² of d = 13 nm is negative and its magnitude is significantly larger than that of d = 10 nm, where the definition of χ² is given as M = χ⁰H + χ²H³ + χ⁴H⁵ +… Therefore, while also regarding the Fourier series of M(t), , this difference in the χ² between the two particle diameters indicates that the in-phase component of the third-order harmonics M_3f^′ at d = 13 nm is fairly large in comparison with M_3f^′ at d = 10 nm, because M_3f^′ is expressed as (2) using the substitution H = H_ac·sin(2nπf·t). The upper cross-section view in Fig. 2 shows the size-dependence of M_3f^′, where the value of M_3f^′ seems to increase threefold as d increases from10 to 13 nm. The degree of this increase is weaker than is expected for the χ² = –(1/45)(μ₀M_sV/k_BT)³ ∞ d⁹ term of the Langevin function. One of the reasons is that the increase in V affects the higher-order terms in Eq. (2) more remarkably.

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Fig 1. Dynamic magnetization curves.

Dynamic magnetization curves calculated for the anisotropic nanoparticles with randomly-oriented easy axes when an AC magnetic field H_ac·sin(2nπf·t) is irradiated at a frequency f = 50 kHz. (a) Size d-dependence and (b) amplitude H_ac-dependence. The arrows indicate the representative curvatures seen at d = 13 and 20 nm.

https://doi.org/10.1371/journal.pone.0118156.g001

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Fig 2. Contour plot of M_3f^′ for mono-disperse nanoparticles.

Contour plot of the third-order harmonics of magnetization, M_3f^′, for mono-disperse nanoparticles with different sizes, d, in AC magnetic fields at various amplitudes, H_ac, and f = 50 kHz. The upper and right-side panels show the cross-sections taken at the broken and solid lines, respectively.

https://doi.org/10.1371/journal.pone.0118156.g002

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Table 1. Physical quantities of magnetite nanoparticles of different sizes.

https://doi.org/10.1371/journal.pone.0118156.t001

Before turning to the results for larger anisotropic nanoparticles, we note that H_ac is much smaller than the anisotropy field H_K of 72 kA/m for the assumed effective uniaxial anisotropy constant K_eff of 20 kJ/m³ (details in the Models section). In other words, these results correspond to minor loops in the conventional category. In Fig. 1(A), it can be seen that the magnetization curve for d = 16.5 nm exhibits an apparent hysteresis in the vicinity of zero magnetic field. The reason for this hysteresis is that the equilibration by the thermally-activated reversals cannot keep up with the variation of the AC magnetic field because τ_N of 8.5 μs at H = 0 is longer than (2πf)^-1 ∼ 3 μs (see Table 1.) On the other hand, the curve is anhysteretic in the range from 15–20 kA/m, although these fields are much lower than H_K. A notable point is that τ_N gets shorter as the barrier height is reduced by the Zeeman energy; consequently, it becomes shorter than (2πf)^{- 1} at 15 kA/m (250 ns in the case of n // H, where n is the unit vector along the easy axis). Thus, the anhysteretic properties is reasonable in the field range. On the other hand, at d = 20 nm, τ_N is still 7.4 μs at 20 kA/m. Certainly, no anhysteretic region can be seen in the results simulated for nanoparticles at this size. It is noteworthy that, as indicated by the arrows in Fig. 1(A), the shape of the magnetization curve for d = 20 nm seems concave at H > 0, while it is convex at H < 0, indicating that χ² for d = 20 nm has the opposite sign to χ² for d = 13 nm. This sign inversion exhibited by M_3f (∞ χ²) can be clearly seen in the upper cross-section view in Fig. 2. To illustrate the size-dependence of M_3f at other amplitudes of H_ac, a contour plot of M_3f is shown in the main panel of Fig. 2, where we find the same kind of the sign inversions occurring with smaller d particles when H_ac is lowered. This variation of the critical size with lowering H_ac is reasonable because the lower energy barriers of smaller particles are easily surmountable even with the lesser assist by the weak Zeeman energy at lower H_ac. The H_ac dependence of M_3f exhibits an interesting property at the critical size, wherein the M_3f from nanoparticles with d = 16.5 nm is clearly negative at H_ac = 10 kA/m, is zero at H_ac = 15 kA/m, and is positive at H_ac = 20 kA/m, as shown in the right cross-section in Fig. 2. This can be attributed to the change of the predominant curvature from a hysteresis loop at lower fields to an anhysteretic curve at higher fields, as shown in Fig. 1(B).

At this stage, it is helpful to recall the inhomogeneity of H_ac using actual diagnoses. For example, H_ac is 30 kA/m at the center of the coil when an AC current of 1 kA is supplied to a 0.20 m-diameter coil with 6 turns. The value of H_ac weakens with distance from the coil, so that it becomes roughly 20, 15 and 10 kA/m at a distance of 0.07, 0.10, and 0.14 m on the central axis from the coil, respectively. This fact clearly indicates that, if the above-discussed nanoparticles (at d = 16.5 nm) are selected on the grounds that the signal is relatively large at the center of the coil, the sign of M_3f measured in the field-free points near the body surface will be different from those measured at points deep inside the body, while there will be no signal at all in the intermediate region, although the states of the accumulated superparamagnetic nanoparticles are identical with each other. Therefore, the image reconstruction process will cause artifacts unless we take into account this kind of sign inversion.

A major reason why little attention has been given to this phenomenon of sign inversion with variation in H_ac is that the nanoparticles used in previous studies are polydispersive. Fig. 3 shows a contour plot of M_3f for poly-disperse nanoparticles with a standard deviation σ = 0.2. In this case, the positive contributions from smaller particles offset the negative ones from larger particles, and the phenomena observed with mono-disperse nanoparticles is wiped out. Consequently, we can find no sign inversion in the cross-section from C to D in Fig. 3, and we did not need to consider such sign inversion. Mono-disperse nanoparticles are strong candidates for use in MPI in the near future, however, because the M_3f for poly-disperse nanoparticles is much weaker compared with that of mono-disperse nanoparticles (see again Figs. 2 and 3.) Therefore, it is important at this stage of mono-disperse nanoparticle selection to recall the sign inversion of M_3f with variation in H_ac and prepare nanoparticles with diameters slightly smaller than the size for which the sign of M_3f changes from positive for anhysteretic magnetization curves to negative for hysteretic ones. The desirable size for the case exemplified here is roughly 13–15 nm (< 16.5 nm) and the value of M_3f is highly positive at any H_ac, as shown in Fig. 2.

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Fig 3. Contour plot of M_3f^′ for poly-disperse nanoparticles.

Contour plot of the third-order harmonics of magnetization, M_3f^′, for poly-disperse nanoparticles with different mean sizes, d_m, in AC magnetic fields at various amplitudes, H_ac, and f = 50 kHz. The upper and right-side panels show the cross-sections taken at the broken and solid lines, respectively.

https://doi.org/10.1371/journal.pone.0118156.g003

History dependence of the third-order harmonic component

The above section emphasizes the benefit of using superparamagnetic nanoparticles whose diameters are slightly smaller than the critical size. This raises the question whether the use of such particles could avoid any influence from the anisotropy of actual nanoparticles. Fig. 4(A) shows static magnetization curves calculated for the above-exemplified nanoparticles whose d = 13 nm, where the results for the cases when the easy axes are aligned at a certain direction are shown in addition to that of the case of randomly-oriented easy axes. We can see that, in the case where n // H, the system is easily magnetized in comparison with the prediction expressed by the Langevin functions, while the magnetizing process is relatively hard for n ⊥ H. To consider the reason for this difference, we note the Boltzmann factor of 260 and τ_N of 26 ns for nanoparticles with d = 13 nm (Table 1). These values indicate that e is fairly bound around the two directions parallel/antiparallel to the easy axis n, while the probabilities are instantly redistributed between them with changing Zeeman energy in the time scale of (2πf)^{- 1} ∼ 3 μs. This signifies that, for n // H, the redistribution causes a significant increase in M(t) with increasing H because they occur from the direction antiparallel to H to the parallel direction. On the other hand, the reversals for n ⊥ H contributes little to the increase in M(t) because they occur from the direction perpendicular to H to the other perpendicular direction. For these reasons, the static magnetization curves of anisotropic superparamagnetic nanoparticles highly depend upon the orientations of the easy axes, in contrast to the prediction obtained using the Langevin functions [13–15]. Consequently, the nonlinearity for n // H is much greater than that for n ⊥ H, as is apparently found in Fig. 4(A).

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Fig 4. Magnetization curves calculated for anisotropic nanoparticles with a size of d = 13 nm.

(A) Static magnetization curves calculated for nanoparticles whose easy axes are aligned at certain directions as well as for those with randomly-oriented easy axes. The dot-dashed line indicates the curve for the isotropic nanoparticles calculated using Langevin function. (B) Evolution of the dynamic magnetization curves with randomization of the directions of the easy axes for the field histories I and II (see also Fig. 5), where H_ac = 10 kA/m and f = 50 kHz.

https://doi.org/10.1371/journal.pone.0118156.g004

As sketched in Fig. 5(A), the history of the magnetic field while scanning the field-free point can be typified by the longitudinal scan (history I) and the transversal scan (history II). In the former case the AC magnetic field H_ac = (0, 0, H_ac·sin(2πf·t)) is parallel to the DC bias magnetic field H₀ = (0, 0, H₀) that is applied just before the field-free point comes, while the H_ac is perpendicular to H₀ = (H₀, 0, 0) in the latter case. In this study, the magnetic responses of the above-discussed nanoparticles are simulated for these two model histories (details in the Models section). Fig. 5(B) shows the calculated variation of the average for the absolute value of the Z-axis component of n, |n_z|_ave for the nanoparticles with d = 13 nm after H₀ disappears at t = 0. The initial value of |n_z|_ave is found to be larger than 0.5 for history I, which indicates that the easy axes, on average, are slightly oriented toward the z direction parallel to H₀ at t = 0. This is quite reasonable because there is competition between the magnetic torque that turns the easy axis toward the field direction and the Brownian torque that randomizes the directions of the easy axes. Also, we are certain that in history II, the easy axes are initially oriented toward the direction perpendicular to the z direction, because |n_z|_ave is smaller than 0.5 at t = 0. When considering their relaxations, we see that all of the |n_z|_ave values approach 0.5, wherein the time required for this process increases with the viscosity. In other words, the directions of the easy axes are gradually randomized at the field-free point. This behavior is also reasonable because the steady magnetic torque vanishes while the Brownian torque continues the randomization in the time corresponding to the Brownian relaxation time.

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Fig 5. Maxwell coil pair scan schematics and calculated values for two field histories.

(A) Typical scans of a DC bias field-free point in a Maxwell coil pair. In the longitudinal scan, the direction of the AC magnetic field, H_ac, is parallel to the DC bias magnetic field, H₀, applied just before the field-free point comes (history I), while H_ac is perpendicular to H₀ in the transversal scan (history II). (B) Calculated variation of the average of the absolute value of the z-axis component of the unit vector along the easy axis n, |n_z|_ave, for nanoparticles with d = 13 nm for the histories I and II after H₀ disappears at t = 0.

https://doi.org/10.1371/journal.pone.0118156.g005

At this stage, it is useful to recall the fact discussed earlier, that the magnetization curves of anisotropic superparamagnetic nanoparticles depend upon the orientations of the easy axes. In this context, we can expect that the relaxations of the orientations at the field-free point are accompanied by an evolution of the magnetization curve. Fig. 4(B) shows the results of the simulations for the field histories I and II. We find that the magnetization curve for the field history I gradually approaches from above the curve calculated earlier for the case of randomly-oriented easy axes, while that for the field history II does so from below. With these evolutions, the nonlinearity decreases and increases, respectively. Consequently, M_3f^′ also varies and reaches a steady magnitude in a time comparable to the Brownian relaxation time, t ∼ τ_B, as shown in Fig. 6(A). To sum up, in the case of d = 13 nm, the third-order harmonic response of the superparamagnetic nanoparticles at the field-free point apparently depends upon the history of the magnetic field in the time-scale of τ_B, despite the fact that τ_N = 26 ns is much less than τ_B. This history dependence may lead to another artifact in MPI image if the scanning of the field-free point is faster than the Brownian relaxation. Presently it is unclear how long we should wait for the randomization of the easy axes practically, because there is little knowledge concerning the local environments of magnetic nanoparticles accumulated in tumor tissue. A simple method to avoid this problem is the use of smaller particles, because the history-dependence of M_3f^′ becomes weak with decreasing the particle size, as shown in Fig. 6(B). The reason for this weak history-dependence is that the initial alignment of the easy axes at t = 0 is more randomized in smaller particles because the Brownian torque predominates the magnetic torque in smaller particles. It should be noted, however, that the magnitude of M_3f^′ steeply declines with decreasing particle size (Fig. 6(B)). In other words, this easy method has a trade-off relation between the high-speed scanning of the field-free point for a rapid diagnosis and the use of larger nanoparticles for an intensified signal. For this reason, new scanning protocols need to be constructed with consideration of the dependence upon the history of the magnetic field.

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Fig 6. Evolution of M_3f^′ with easy axes randomization for two field histories.

Evolutions of the third-order harmonics of magnetization, M_3f^′, with randomization of the directions of the easy axes for the field histories I and II (see also Fig. 5). (a) Viscosity-dependence at d = 13 nm (b) Size-dependence at η = 1.4 Pa s.

https://doi.org/10.1371/journal.pone.0118156.g006

In the present MPI technique, the detected third-order harmonics of the electromotive force, indicating the existence of hidden tumor fragments, are being analyzed on the assumption that the magnetization curve is expressed by the Langevin function. In other words, the nanoparticles are considered to be completely isotropic. Actual nanoparticles, however, do not satisfy this condition. Therefore, we simulate the magnetic responses of anisotropic magnetic nanoparticles in high-frequency AC magnetic fields with sufficiently large amplitudes, and attempt to clarify the features of the third-order harmonic component, wherein the magnetic/thermal torque-driven rotation of the easy axes as well as the thermally-activated magnetization reversals over magnetic hard planes are incorporated. Consequently, we find a slight delay of the magnetic response to alternations of the magnetic fields when larger nanoparticles with strong signals are employed. In this condition, the third-order harmonic response for the minor loop is negative while that for the major loop is positive. This fact clearly indicates the possibility that the sign of the signal is reversed when the field-free point scans from the body surface to points deep inside the body, although the states of the magnetic nanoparticles are identical. This may cause artifacts in the reconstructed image. We also find that, even in a superparamagnetic state where no hysteresis loops are registered under the AC field, the amplitude of the third harmonic component strongly depends upon the history of the DC bias magnetic field that aligns the easy axes of the nanoparticles. This is due to the fact that the equilibrium magnetization curve for the field applied at the magnetically hard plane is significantly different from the Langevin function and its non-linear component is extremely small. This simulation study finds, therefore, that MPI signals are seriously affected by the inevitable deviations from the Langevin function that exist with actual nanoparticles.

In this study, we paid attention to the fact that each actual nanoparticle is more or less anisotropic, and pointed the possibilities of artifacts in magnetic particle imaging using numerical simulations for individual anisotropic nanoparticles. At this stage, we must mention that aggregated superparamagnetic nanoparticles may be selected for MPI [5]. If the nanoparticles are not magnetically isolated, dipolar interactions between the nanoparticles would play an important role in magnetic responses to AC magnetic fields in rotatable cases [16] as well as in non-rotatable cases [17]. For this reason, in the next stage, numerical simulations considering such interactions are highly desired for further progress in the development of new tracers using aggregated nanoparticles.

Actual nanoparticles and local environment

The shapes of actual nanoparticles are not completely spherical, which causes a variation in the amount of surface magnetic charge with the direction of e. This is the shape anisotropy energy, U_d. On the other hand, faceted nanoparticles have surface magnetic anisotropy energy, U_s, because of the breaking of the local symmetry at the surface. In addition, atomic lattices without spherical symmetry induce a crystalline anisotropy energy U_c. Consequently, the total potential energy U is given by-μ₀M_sV e· H + U_d + U_c + U_s instead of that given in Eq. (1). Because the details of these anisotropies are still unknown [4], this formula has conventionally been simplified to (3) using a lower-order term as a first approximation, where K_eff is the effective uniaxial anisotropy constant and n is the unit vector along the easy axis. Considering iron oxide nanoparticles as a typical example, the reported value of K_eff varies from paper to paper in a range from 10–30 kJ/m³ [4,18,19]. This diversity may arise from the differences of the shape or the crystallinity, but the true reason is still unclear. If we choose the intermediate value of 20 kJ/m³, the magnitude of K_effV and the related quantities for nanoparticles with effective diameters of d = (6V/π)^1/3 are calculated and shown in Table 1. Compared with the thermal energy k_BT ∼ 4×10^–21 J at room temperature, the magnetic anisotropy expressed by the second term is not negligible for nanoparticles in the size range d ≥ 10 nm. On the other hand, the calculated Néel relaxation time, τ_N, indicates that a hysteresis loop should be observed for nanoparticles with d = 23 nm when the measurement using a vibrating sample magnetometer takes one hour. Because they have not been customarily regarded as superparamagnetic, we studied the properties of anisotropic nanoparticles in a size range of 10–20 nm diameters. When calculating M_s, the value of 450 kA/m obtained for bulk magnetite was used as is [18], because a difference of M_s does not seriously affect the results of the simulations. The poly-disperse nature of real nanoparticles was considered as a logarithmic normal distribution of d with a standard deviation σ = 0.2, where the cutoff is ±2σ.

Little is known about the local environments of magnetic nanoparticles accumulated in tumor tissue. If a passive targeted drug delivery system is used, the nanoparticles may be dispersed in intracellular media whose viscosity η is not constant, but whose η is 1–3 ×10^–3 Pa s in aqueous domains while the η = 8×10^–2 Pa s in other areas [20]. Surprisingly, there is a report giving the value of η = 1.4 Pa∙s [21]. Nanoparticles used in conjunction with tumor-homing peptides or antibodies for active targeting, however, should be anchored to a cell membrane. Recently, in-situ X-ray diffractometry on a single nanoparticle showed that a nanoparticle anchored to a substrate by an antigen-antibody combination or by a protein can slowly rotate on the time scale of milliseconds to seconds [22,23]. Assuming a kind of Brownian mechanism, η can be nominally estimated to be roughly 100 Pa s. To sum up, the nanoparticles in tumor tissue are, in many cases, rotatable, although the nominal viscosity is highly sensitive to the accumulated state. Therefore, we incorporated rotations into the simulation in the latter part, where values of η of 2×10⁻³, 8×10⁻², 1.4 and 100 Pa∙s were employed as typical examples.

Simulation of Néel relaxation

The orientation probability ρ(e) of the direction of e is given by a Boltzmann factor exp(−U(n, e, H)/k_BT) using Eq. (3). As indicated by their values for d ≥ 10 nm in Table 1, ρ(e) at e // n is higher by more than one digit compared with that at e ⊥ n at H = 0. Therefore, the direction of e can be considered to be localized at e₁ parallel to the easy axis (or the antiparallel direction e₂). In other words, we can simply assume that ρ(e₂) = 1 – ρ(e₁). In this case, we can use the two-level approximation that considers thermally-activated reversals between the minima via the midway saddle point, e₃, at the hard plane. The reversal probability from e₁ to e₂, υ₁₂, is given by 0.5f₀ exp[–(U(n, e₃, H) – U(n, e₁, H))/k_BT], while the backward reversal probability υ₂₁ is 0.5f₀·exp[–(U(n, e₃, H) – U(n, e₂, H))/k_BT], where f₀ is the attempt frequency of 10¹⁰ s^–1. In this framework, τ_N is given by (υ₁₂ + υ₂₁) ^–1. In this work, the time evolution of ρ(e₁) was computed using Δρ(e₁) = [υ₂₁ρ(e₂) – υ₁₂ρ(e₁)]Δt for an AC magnetic field (0, 0, H_{ac ·}sin(2πfm·t)) with various H_ac and an f of 50 kHz. The time step Δt was typically 10⁻⁴/f s, but was shorter when υ₁₂Δt (or υ₂₁Δt) became large compared with one. At each step, the magnetization M(t) was obtained as NM_sV [ρ(e₁)e₁·(H/H) + ρ(e₂)e₂·(H/H)]. The higher-order harmonics of M(t) were analyzed using the Fourier series .

Simulation of Brownian relaxation

In this study, we simulated rotations of magnetite nanoparticles smaller than d = 13 nm, where f was again set at 50 kHz. For these nanoparticles, τ_N was shorter than 26 ns, while the switching of the field occurred at every 10 μs at f = 50 kHz. In other words, ρ(e) can be regarded to be always equilibrated in U, although the Zeeman energy varies slowly. In this case [24], the magnetic torque on the easy axis T_r is given by the average, (4) where Ω_e is the solid angle of e. On the other hand, the frictional torque is expressed as πηd_H³∙ω using the assumption of Newtonian fluid for simplicity, where d_H is the hydrodynamic diameter of the nanoparticles including surface modification layers and ω is the angular velocity of rotation. In this simulation, d_H was assumed to be 30 nm. If the inertia of the nanoparticles is negligible, the torques balance each other as follows: (5) (6) (7) where λ(t) is the Brownian random torque, i and j indicate the Cartesian vector components, δ_ij is the Kronecker delta and δ(t₁ – t₂) is the Dirac delta function. Using these equations, the time evolution of n(t) was computed by the following steps: (i) ρ(e(t)) was calculated by the Boltzmann factor for U(n, e, H) with the latest n(t) and H(t), (ii) T_r was computed by substituting ρ(e(t)) into Equation (4), (iii) ω was calculated by substituting T_r into Equation (5), and (iv) n(t) was finally replaced by n(t +Δt) = n(t) + [ω(t) × n(t)]Δt, where the time step Δt was 2×10^–4/f = 4 ns. At each step, the magnetization M(t) was obtained as . The higher-order harmonics of M(t) were analyzed as stated above.

In this study, the scanning of the field-free point was roughly classified into the longitudinal scan and the transversal scan, where the history of the magnetic field in the former case (history I) was modeled as (0, 0, H₀) at—∞ < t < 0 and (0, 0, H_ac·sin(2πf·t)) at t > 0, while the history in the latter case (history II) was (H₀, 0, 0) at -∞ < t < 0 and (0, 0, H_ac·sin(2πf·t)) at t > 0. The magnitude of a DC bias magnetic field H₀ was set at 100 kA/m, considering the magnetic field required for saturation in Fig. 4 (A), while H_ac was at 10 kA/m. Because of the sufficiently long period in H₀, the system could be considered to be equilibrated well at t = 0, and the distribution of n(t = 0) and e(t = 0) should be independent of η. For this reason, the rotations of 18000 nanoparticles in an aqueous phase with η of 2×10⁻³ Pa s were simulated in H₀ beforehand, and their steady state was employed as the common initial state of the main simulations for the behavior in the field-free point in various viscous media with η values of 2×10⁻³, 8×10⁻², 1.4 and 100 Pa s at t > 0.