# Recent Advances in Understanding Magnetic Nanoparticles in AC Magnetic Fields and Optimal Design for Targeted Hyperthermia

Authors: Hiroaki Mamiya
Journal: Journal of Nanomaterials
 (2013)

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## Abstract

Targeted hyperthermia treatment using magnetic nanoparticles is a promising cancer therapy that enables selective heating of hidden microcancer tissues. In this review, I outline the present status of chemical synthesis of such magnetic nanoparticles. Then, the latest progress in understanding their heat dissipation mechanisms under large magnetic fields is overviewed. This review covers the recently predicted novel phenomena: magnetic hysteresis loops of superparamagnetic states and steady orientations of easy axes at the directions parallel, perpendicular, or oblique to the AC magnetic field. Finally, this review ends with future prospects from the viewpoint of optimal design for efficacy with a low side-effect profile.


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## Body

## 1. Introduction

Hippocrates said, “those diseases which drugs cannot cure, the knife cures; those which the knife cannot cure, fire cures; those which fire does not cure must be considered incurable.” In one respect, medicine has not changed over time; even today, several treatments are used in combination to treat illnesses that have no established effective treatment protocol, the most prominent example being cancer. The current standard treatments for cancer include surgery, chemotherapy, and radiotherapy. Beyond these treatments, much research is being undertaken to create several new treatment options such as immunotherapy, and the modern equivalent of Hippocrates’ “fire”: thermotherapy. Thermotherapy is a treatment method that exploits the lowered heat resistance of cancerous tissues compared with that of normal tissues. Cancerous tissues undergo cell death even at temperatures within the range of 42 to 43°C, thus rendering thermotherapy as a promising option to reduce the disease burden in a patient [1].To reduce damage to normal tissues using standard treatments, endo-/laparoscopic surgical techniques have been developed as a modern equivalent of Hippocrates’ “knife.” For chemotherapy, much effort has focused on drug delivery to selectively transport anticancer agents to tumor tissues using antigen-antibody reactions. Such drug delivery systems are also used to concentrate boron compounds in tumor tissues. In the treatment known as boron neutron capture therapy, the patient is radiated with epithermal neutrons, which selectively induceα-decay of boron nuclei concentrated inside hidden tumors, thus specifically destroying cancer cells. Along these lines, could thermotherapy also utilize drug delivery technology to specifically deliver thermal seeds to cancer cells at unknown locations? For example, if it were possible to develop miniature induction-heating cooking pans and selectively send these to hidden tumors, then would this result in selective heating of the tumor tissues in a human body on an induction-heating cooker? In contrast to a microwave oven, we know that placing one’s hand over the induction-heating cooker will not immediately result in a burn. (Usually, we cannot confirm this feature using the commercial IH cooker, because it automatically stops working when we take off a metal pan from it.) This experience shows that a radio-waveband AC magnetic field generated in the cooker can easily penetrate deep into tissues where a tumor may be embedded. Therefore, we believe that the simple idea mentioned here, targeted hyperthermia using magnetic nanoparticles, has the potential to selectively destroy cancer cells hidden deep in the body [2–8].Of course, there are many questions about this concept that need to be addressed. Is it safe to put magnetic nanoparticles in the body? Can magnetic nanoparticles really be concentrated in hidden tumor tissues? Can magnetic nanoparticles be heated within the body? To answer these questions, DeNardo et al. injected iron oxide nanoparticles conjugated with monoclonal antibodies into mouse tails and found that they accumulated at a concentrationc of approximately 0.3 kg/m3 (0.3 mg/cm3) in tumors [9]. (The side effects of iron oxide nanoparticles as an MRI contrast agent had been previously studied and approved for intravenous injection.) Wust et al. showed that injecting high concentrations (c~10 kg/m3) of magnetic nanoparticles directly into tumors at known locations and irradiating them with an AC magnetic field caused the temperature of the tumors to increase to 43°C [10]. Therefore, it is known that iron oxide nanoparticles are safe, can be selectively accumulated in hidden tumors to some degree, and can be adequately heated when existing at high concentrations. Nevertheless, problems with targeted hyperthermia using magnetic nanoparticles are still evident. Does the density of magnetic nanoparticles delivered to tumors increase as it does when they are directly injected into tumors? If not, is it possible to compensate for a lower density of magnetic nanoparticles by maximizing their heat dissipation? The first problem is primarily a biochemical one, so materials researchers have primarily focused on improving the heating performance of magnetic nanoparticles [11–30]. Consequent advances in chemical synthesis technology have resulted in the fabrication of magnetic nanoparticles of engineered size, shape, and structure. With respect to physical heating mechanisms, the nature of the nonlinear response and nonequilibrium dissipation in AC magnetic fields of magnetic nanoparticles, which are in contrast to the properties of cooking pans, have been uncovered. This review addresses this progress as follows. In Section 2, conventional models that are the basis of the traditional design of hyperthermia treatments are introduced. In Section 3, advances in the synthesis of magnetic nanoparticles are described and limitations in the conventional models when the monodisperse nanoparticles are used in actual thermotherapy are considered. In Section 4, recent advances in the knowledge of heating mechanisms provided by numerical simulations are explained. Finally, we summarize the optimal design of magnetic nanoparticles for hyperthermia treatment and discuss their potential as an effective and safe version of Hippocrates’ “fire” in Section 5.


## 2. Conventional Models for Magnetic Response to AC Magnetic Fields [31–34]

The main advantage of hyperthermia treatment using magnetic nanoparticles is that the nanoparticles can reach the cancer tissue directly by travelling through the sub-micrometer spaces between blood cell walls. Therefore, for practical use, the nanoparticles should not form long chains or large clusters. Even though the many-body effects caused by dipole-dipole couplingJdd are not fully understood [35–38], it is known that a dispersion becomes unstable if Jdd between the closest nanoparticles is more than five times the thermal energy [35, 38]. Under these conditions, the minimum allowable distance between iron nanoparticles with diameter d of 12 nm is roughly 27 nm, and that between ferrite nanoparticles with d of 25 nm is almost 40 nm. In contrast, small-angle neutron scattering experiments have indicated that the thickness of an absorbed layer is normally several nanometers [39]. Therefore, the upper limit of d is estimated to be roughly 12 nm for iron and 25 nm for ferrite. These values would be references for considering the criteria for easy delivery of the nanoparticles, although agglomeration, aggregation, or flocculation may occur depending on the surface charge of biofunctionalized nanoparticle or on the interaction between tumor-targeting ligands. Note that these values are smaller than the critical diameters for the transition into a single-domain configuration and for the coherent reversal of all spins [40]. Therefore, it has been considered that a magnetic nanoparticle used in hyperthermia treatment has only one magnetic moment, μ=MsV, where Ms is the spontaneous magnetization and V=π/6·d3 is the volume of the magnetic core of the nanoparticle. Such magnetic nanoparticles have been conventionally classified as “ferromagnetic” or “superparamagnetic,” depending on whether the direction of μ thermally fluctuates or not.Firstly, a ferromagnetic nanoparticle with uniaxial magnetic anisotropy, anisotropy constantK, is considered, where V is large enough that its magnetic anisotropy barrier with a height of KV blocks the thermal fluctuations; accordingly, the remanent state appears to be permanent [41]. If a magnetic field H is applied in the direction antiparallel to μ, the state becomes metastable, as depicted in Figure 1(a). Then, μ reverses when the barrier disappears at the anisotropy field HK=2K/(μ0Ms); consequently, the Zeeman energy falls from μ0μHK to -μ0μHK and the energy corresponding to the difference dissipates, where μ0 is permeability of vacuum. In this case, the work done in one cycle of the AC magnetic field Hacsin(2πf·t) is 0 for Hac<HK and 4μ0μHK for Hac>HK. This kind of heat dissipation has been termed “hysteresis loss.” Briefly, the heat dissipation from nanoparticles with unit weight during unit time, also called specific loss power PH, abruptly increases from zero to 4μ0μHK·f·w-1(=4μ0MsHK·f·ρ-1) when Hac becomes higher than HK, where w and ρ-1 are the weight and density of the magnetic core of the nanoparticles, respectively. Then, PH flattens out even if Hac is strengthened further. According to this argument, the guiding principle for maximizing PH of ferromagnetic nanoparticles is that Hac is adjusted to HK and the number of cycles f is maximized.Schematic diagrams of conventional models for magnetic loss. (a) Hysteresis loss equivalent to the area ofM-H loop, and the potential energy in magnetic fields. (b) Relaxation loss given by the out-of-phase component of AC susceptibility. As illustrated in the sketches, in Néel relaxation, the magnetic moment shown by the yellow arrow reverses (the particle does not rotate), while in Brownian relaxation, the magnetic core (the red sphere) rotates with absorbed molecules (the green chains) as a whole (the magnetic moment does not reverse).
(a)
Hysteresis loss
(b)
Relaxation lossNext we move to smaller superparamagnetic nanoparticles with thermally fluctuating reversal ofμ [42]. The reversal probability in a zero magnetic field is expressed as
(1)τN-1=f0·exp(-KVkBT),
where τN is the Néel relaxation time, f0 is the attempt frequency of 109s-1, kB is the Boltzmann constant, and T is the temperature. We must also consider Brownian rotation of the nanoparticles if they are dispersed in a liquid phase. In this case, the characteristic time of the rotation, Brownian relaxation time, in a zero magnetic field is given by
(2)τB=3ηVH(kBT),
where η is the viscosity of the liquid phase and VH is the hydrodynamic volume of the nanoparticles including surface modification layers. If reversal and rotation occur in parallel, the characteristic time of relaxation τ could be expressed as the following equation:
(3)τ-1=τN-1+τB-1.
For very small superparamagnetic nanoparticles, τ is determined only by τN because τN-1 increases exponentially with decreasing V, while the increase of τB-1 is inversely proportional to VH.If a linear response of the thermodynamic equilibrium state of such nanoparticles is assumed for smallHac, the average out-of-phase component of AC susceptibility, χ′′, contributed from each nanoparticle is given by
(4)χ′′=μ0μ2(3kBT)·2πf·τ[1+(2πf·τ)2].
Consequently, “relaxation loss” occurs and its heat dissipation PH is expressed as
(5)PH=πμ0χ′′·Hac2·f·w-1=12[μ02Ms2V/(3kBTτρ)]·Hac2·(2πf·τ)2[1+(2πf·τ)2].
Equation (5) indicates that PH increases in proportion to f2 in the low frequency range, 2πf≪τ, whereas it flattens out at 1/2[μ02Ms2V/(3kBTτρ)]·Hac2 even if f is increased further in the high frequency range, 2πf≫τ. According to this argument, the guiding principle for maximizing PH of superparamagnetic nanoparticles is that f is adjusted to τ-1 and Hac is maximized.


## 3. Progress in Synthesis of Magnetic Nanoparticle and Their Use in Thermotherapy

### 3.1. Size-, Shape- and Composite-Controlled Synthesis of Magnetic Nanoparticles

As discussed above, to improve hysteresis loss, ferromagnetic nanoparticles with an anisotropy field(HK=2K/(μ0Ms)) matching the amplitude Hac of the AC magnetic field generated in the oscillator of realistic medical equipment need to be synthesized. In contrast, increasing relaxation loss involves the synthesis of superparamagnetic nanoparticles that have τN matching f of the AC magnetic field. For these reasons, a large number of studies have focused on controlling the size, shape, or composite structure of nanoparticles to optimize HK and τN.The history of colloids (magnetic fluids) stably dispersing magnetic nanoparticles in solution goes back to the 1960s, when magnetic suspensions were prepared by pulverizing bulk iron oxide and used for fuel delivery in a weightless environment [43], such as those involving NASA expeditions. Elsewhere, Sato of Tohoku University created magnetic fluids from minute iron oxide particles using chemical methods [44]. There have also been several major subsequent advances in magnetic fluid development, such as the monodisperse iron nitride-based magnetic fluids developed by Nakatani et al. [45]; however, because the industrial applications of magnetic fluids at that time did not require precise control of size, shape, or structure, more extensive research was not conducted in this field. However, in 2000, Sun et al. from IBM described an ordered self-assembled film of monodisperse iron-platinum nanoparticles that could serve as an ultrahigh-tech magnetic recording medium [46]. Since then, researchers have focused on developing methods to synthesize well-controlled nanoparticles, which have been reviewed extensively [47, 48]. Next, we briefly summarize these methods.Generally, formation of nanoparticles starts with nucleation in a supersaturated melt, solution, or vapor. Particle growth continues until the concentration of solute atoms falls below the saturation solubility. If nucleation and growth proceed in parallel, nanoparticles formed initially have already grown when the last nanoparticle is formed, thus resulting in nanoparticles of variable size. Furthermore, processes such as coarsening and aggregation simultaneously occur in many cases. One way to obtain monodisperse nanoparticles is the two-stage growth method: in the first stage, rapid heating causes fast supersaturated-burst nucleation, and in the second stage, the gradual precipitation of solute atoms at a temperature below the critical point of supersaturation allows only the existing nanoparticles to grow slowly. In this process, surfactants are often introduced to the solution to prevent coarsening and aggregation. Because all nanoparticles follow the same growth process in this method, their size after growth should in principle be uniform. In practice, different groups have developed particular methods to produce nanoparticles of specific composition and size.With respect to controlling the shape of a nanoparticle, growth kinetics is essential in addition to thermodynamic stability to minimize surface free energy. For example, if the growth rate for cubic{111} surfaces is slower than for {100} surfaces, the surface area of {100} facets will decrease with growth, and the particles finally become octahedrons of {111} facets only. Similarly, if the growth of {001} surfaces in a hexagonal crystal system is fast, rods or, conversely, plates can be formed. For this reason, the adsorption of surfactants on particular surfaces has been intensively studied to fabricate a desired shape by controlling the growth rate of each surface. Figure 2 shows examples of regular octahedral and cubic nanoparticles [49, 50]. With regard to compositing, nanoparticles dispersed in solution are regularly conjugated by substances such as surfactants to lower their surface energy or prevent aggregation, forming a kind of composite material. Advanced compositing techniques have been developed to protect easily oxidized metal cores or to enable the simultaneous expression of multiple functions. For example, dumbbell-shaped junctions in different kinds of nanoparticles [51] and core-shell structures [27] have been produced recently (Figure 3).Figure 2
Transmission electron micrographs of shape-controlled magnetic nanoparticles with different projection shapes: (a) hexagonal outlines of octahedron-shaped Fe3O4 nanoparticles (zone axis: 〈111〉), and (b) parallelogram outlines of the same Fe3O4 nanoparticles as in (a) (zone axis: 〈110〉). *Reproduced from Li et al. [49] with permission (Copyright 2010 American Chemical Society). (c) Hexagonal outlines of cube-shaped Ni-Pt nanoparticles (zone axis: 〈111〉). Private communication (Copyright 2011 B. Jeyadevan).Figure 3
Electron energy-loss spectroscopy (EELS) mapping analysis of CoFe2O4@MnFe2O4 nanoparticles, where Co, Fe, and Mn atoms are indicated as green, red, and blue, respectively. *Reproduced from Lee et al. [27] with permission (Copyright 2011 Nature).


### 3.2. Magnetic Nanoparticles to Maximize Heat Dissipation

Using these advanced synthesis techniques, researchers have fabricated magnetic nanoparticles to maximize heat dissipation based on the guiding principles described above. As an example, I highlight the recent report by Lee et al. [27], who fabricated novel superparamagnetic nanoparticles with a uniform diameter d of 15 nm (see Figure 3 again). One of the reasons why they chose such a size may be to avoid aggregation. In addition, the oscillator of their equipment can generate an AC magnetic field of frequency f=500 kHz. As discussed above, superparamagnetic nanoparticles that have a Néel relaxation time (τN) that matches f are required to maximize relaxation loss. Briefly, τN should be (2πf)-1=318 ns. (Overall, τ needs to be set to 400 ns when also considering the Brownian relaxation time τB=1.6 μs.) Substituting τN=318 ns in (1), the required energy barrier height (KV) is calculated to be 2.4×10-20 J. This value corresponds to a uniaxial anisotropic particle with d=15 nm and K=1.4×104 J/m3. However, examination of parameters such as bulk crystalline magnetic anisotropy [40] revealed that no suitable candidate substances had been reported. For substances with cubic symmetry, the magnitude of K1 and the barrier height, -(1/12)K1V for negative K1 or (1/4)K1V for positive K1, calculated using d=15 nm are as follows: −1.2 × 104 J/m3 and 0.18 × 10−20 J (Fe3O4); −0.46 × 104 J/m3 and 0.08 × 10−20 J (γ-Fe2O3); −0.25 × 104 J/m3 and 0.04 × 10−20 J (MnFe2O4); and 18 × 104 J/m3 and 8.0 × 10−20 J (CoFe2O4). As a result, shape control, which affects shape and surface magnetic anisotropy, or composition control or composite structure control, which influences the crystalline magnetic anisotropy, are therefore required. From among the possibilities mentioned, Lee et al. selected core-shell structures of cobalt and manganese ferrites and used a core-and-shell exchange coupling to control the magnitude of effective magnetic anisotropy. As a result, they obtained the core-shell structure shown in Figure 3, with a measured magnetic anisotropy constant K of 1.7 × 104 J/m3 (Table 1). When these core-shell nanoparticles were irradiated with an AC magnetic field of frequency f=500 kHz and amplitude Hac=37.3 kA/m, the heat dissipation (PH) per unit weight reached 3 MW/kg (3 kW/g), which was significantly higher than that using nanoparticles of cobalt ferrite (0.4 MW/kg) or manganese ferrite (0.2 MW/kg). The heat generation of these core-shell nanoparticles is unprecedented so they have received widespread attention.Table 1
Size, saturation magnetization (Ms), anisotropy constant (K), and heat dissipation rate per unit weight PH (at Hac=37.3 kA/m, f=500 kHz) of ferrite nanoparticles experimentally determined in [27].
Sample
Size (nm)
M
s (kA/m)
K (kJ/m3)
P
H (MW/kg)
CoFe2O4
12
510
200
0.4
MnFe2O4
18
700
3
0.2
MnFe2O4@CoFe2O4
15
570
17
3.0This example suggests optimized design of nanoparticle synthesis has succeeded in producing nanoparticles that generate large amounts of heat. However, further consideration revealed two notable points. First, the actual amplitude ofHac reached 37.3 kA/m or 80% that of the anisotropic magnetic field, HK=2K/Ms=47.3 kA/m. This is large enough for the energy barrier to magnetization reversal to disappear because of the Zeeman energy in cases where the direction of the AC magnetic field is not completely parallel to the easy axis of nanoparticles. Thus, these conditions do not permit the application of the guiding principles given in (2)–(5) because these assume a linear response for superparamagnetic nanoparticles in zero magnetic field limit. This raises the question of whether irradiation with an AC magnetic field with f of 500 kHz and Hac of 37.3 kA/m for core-shell structured nanoparticles with d of 15 nm and K of 1.7 × 104 J/m3 are really the optimum conditions. However, it is difficult to apply the other guiding principle to maximize hysteresis loss of ferromagnetic nanoparticles because thermally assisted reversals of μ occur stochastically before the barrier disappears at HK. Recalling that the characteristic time of thermal fluctuation was estimated to be a few hundred nanoseconds even in a zero magnetic field, the conditions used by Lee et al. are outside the scope of applicability of conventional models for ferromagnetic nanoparticles at a temperature of absolute zero and for superparamagnetic nanoparticles in a zero magnetic field. Consequently, new guiding principles to maximize heat dissipation PH are required. The second point is that Hac=37.3 kA/m is much larger than the exposure restriction for this waveband [52]. This point is examined further in Section 5. The next section will present results of recent numerical studies on the behavior of nanoparticles under conditions outside the scope of applicability of conventional models. This knowledge will be useful to establish sophisticated guiding principles that are adapted to advanced technologies that control the size, shape, and composite structure of nanoparticles.


## 3.1. Size-, Shape- and Composite-Controlled Synthesis of Magnetic Nanoparticles

As discussed above, to improve hysteresis loss, ferromagnetic nanoparticles with an anisotropy field(HK=2K/(μ0Ms)) matching the amplitude Hac of the AC magnetic field generated in the oscillator of realistic medical equipment need to be synthesized. In contrast, increasing relaxation loss involves the synthesis of superparamagnetic nanoparticles that have τN matching f of the AC magnetic field. For these reasons, a large number of studies have focused on controlling the size, shape, or composite structure of nanoparticles to optimize HK and τN.The history of colloids (magnetic fluids) stably dispersing magnetic nanoparticles in solution goes back to the 1960s, when magnetic suspensions were prepared by pulverizing bulk iron oxide and used for fuel delivery in a weightless environment [43], such as those involving NASA expeditions. Elsewhere, Sato of Tohoku University created magnetic fluids from minute iron oxide particles using chemical methods [44]. There have also been several major subsequent advances in magnetic fluid development, such as the monodisperse iron nitride-based magnetic fluids developed by Nakatani et al. [45]; however, because the industrial applications of magnetic fluids at that time did not require precise control of size, shape, or structure, more extensive research was not conducted in this field. However, in 2000, Sun et al. from IBM described an ordered self-assembled film of monodisperse iron-platinum nanoparticles that could serve as an ultrahigh-tech magnetic recording medium [46]. Since then, researchers have focused on developing methods to synthesize well-controlled nanoparticles, which have been reviewed extensively [47, 48]. Next, we briefly summarize these methods.Generally, formation of nanoparticles starts with nucleation in a supersaturated melt, solution, or vapor. Particle growth continues until the concentration of solute atoms falls below the saturation solubility. If nucleation and growth proceed in parallel, nanoparticles formed initially have already grown when the last nanoparticle is formed, thus resulting in nanoparticles of variable size. Furthermore, processes such as coarsening and aggregation simultaneously occur in many cases. One way to obtain monodisperse nanoparticles is the two-stage growth method: in the first stage, rapid heating causes fast supersaturated-burst nucleation, and in the second stage, the gradual precipitation of solute atoms at a temperature below the critical point of supersaturation allows only the existing nanoparticles to grow slowly. In this process, surfactants are often introduced to the solution to prevent coarsening and aggregation. Because all nanoparticles follow the same growth process in this method, their size after growth should in principle be uniform. In practice, different groups have developed particular methods to produce nanoparticles of specific composition and size.With respect to controlling the shape of a nanoparticle, growth kinetics is essential in addition to thermodynamic stability to minimize surface free energy. For example, if the growth rate for cubic{111} surfaces is slower than for {100} surfaces, the surface area of {100} facets will decrease with growth, and the particles finally become octahedrons of {111} facets only. Similarly, if the growth of {001} surfaces in a hexagonal crystal system is fast, rods or, conversely, plates can be formed. For this reason, the adsorption of surfactants on particular surfaces has been intensively studied to fabricate a desired shape by controlling the growth rate of each surface. Figure 2 shows examples of regular octahedral and cubic nanoparticles [49, 50]. With regard to compositing, nanoparticles dispersed in solution are regularly conjugated by substances such as surfactants to lower their surface energy or prevent aggregation, forming a kind of composite material. Advanced compositing techniques have been developed to protect easily oxidized metal cores or to enable the simultaneous expression of multiple functions. For example, dumbbell-shaped junctions in different kinds of nanoparticles [51] and core-shell structures [27] have been produced recently (Figure 3).Figure 2
Transmission electron micrographs of shape-controlled magnetic nanoparticles with different projection shapes: (a) hexagonal outlines of octahedron-shaped Fe3O4 nanoparticles (zone axis: 〈111〉), and (b) parallelogram outlines of the same Fe3O4 nanoparticles as in (a) (zone axis: 〈110〉). *Reproduced from Li et al. [49] with permission (Copyright 2010 American Chemical Society). (c) Hexagonal outlines of cube-shaped Ni-Pt nanoparticles (zone axis: 〈111〉). Private communication (Copyright 2011 B. Jeyadevan).Figure 3
Electron energy-loss spectroscopy (EELS) mapping analysis of CoFe2O4@MnFe2O4 nanoparticles, where Co, Fe, and Mn atoms are indicated as green, red, and blue, respectively. *Reproduced from Lee et al. [27] with permission (Copyright 2011 Nature).


## 3.2. Magnetic Nanoparticles to Maximize Heat Dissipation

Using these advanced synthesis techniques, researchers have fabricated magnetic nanoparticles to maximize heat dissipation based on the guiding principles described above. As an example, I highlight the recent report by Lee et al. [27], who fabricated novel superparamagnetic nanoparticles with a uniform diameter d of 15 nm (see Figure 3 again). One of the reasons why they chose such a size may be to avoid aggregation. In addition, the oscillator of their equipment can generate an AC magnetic field of frequency f=500 kHz. As discussed above, superparamagnetic nanoparticles that have a Néel relaxation time (τN) that matches f are required to maximize relaxation loss. Briefly, τN should be (2πf)-1=318 ns. (Overall, τ needs to be set to 400 ns when also considering the Brownian relaxation time τB=1.6 μs.) Substituting τN=318 ns in (1), the required energy barrier height (KV) is calculated to be 2.4×10-20 J. This value corresponds to a uniaxial anisotropic particle with d=15 nm and K=1.4×104 J/m3. However, examination of parameters such as bulk crystalline magnetic anisotropy [40] revealed that no suitable candidate substances had been reported. For substances with cubic symmetry, the magnitude of K1 and the barrier height, -(1/12)K1V for negative K1 or (1/4)K1V for positive K1, calculated using d=15 nm are as follows: −1.2 × 104 J/m3 and 0.18 × 10−20 J (Fe3O4); −0.46 × 104 J/m3 and 0.08 × 10−20 J (γ-Fe2O3); −0.25 × 104 J/m3 and 0.04 × 10−20 J (MnFe2O4); and 18 × 104 J/m3 and 8.0 × 10−20 J (CoFe2O4). As a result, shape control, which affects shape and surface magnetic anisotropy, or composition control or composite structure control, which influences the crystalline magnetic anisotropy, are therefore required. From among the possibilities mentioned, Lee et al. selected core-shell structures of cobalt and manganese ferrites and used a core-and-shell exchange coupling to control the magnitude of effective magnetic anisotropy. As a result, they obtained the core-shell structure shown in Figure 3, with a measured magnetic anisotropy constant K of 1.7 × 104 J/m3 (Table 1). When these core-shell nanoparticles were irradiated with an AC magnetic field of frequency f=500 kHz and amplitude Hac=37.3 kA/m, the heat dissipation (PH) per unit weight reached 3 MW/kg (3 kW/g), which was significantly higher than that using nanoparticles of cobalt ferrite (0.4 MW/kg) or manganese ferrite (0.2 MW/kg). The heat generation of these core-shell nanoparticles is unprecedented so they have received widespread attention.Table 1
Size, saturation magnetization (Ms), anisotropy constant (K), and heat dissipation rate per unit weight PH (at Hac=37.3 kA/m, f=500 kHz) of ferrite nanoparticles experimentally determined in [27].
Sample
Size (nm)
M
s (kA/m)
K (kJ/m3)
P
H (MW/kg)
CoFe2O4
12
510
200
0.4
MnFe2O4
18
700
3
0.2
MnFe2O4@CoFe2O4
15
570
17
3.0This example suggests optimized design of nanoparticle synthesis has succeeded in producing nanoparticles that generate large amounts of heat. However, further consideration revealed two notable points. First, the actual amplitude ofHac reached 37.3 kA/m or 80% that of the anisotropic magnetic field, HK=2K/Ms=47.3 kA/m. This is large enough for the energy barrier to magnetization reversal to disappear because of the Zeeman energy in cases where the direction of the AC magnetic field is not completely parallel to the easy axis of nanoparticles. Thus, these conditions do not permit the application of the guiding principles given in (2)–(5) because these assume a linear response for superparamagnetic nanoparticles in zero magnetic field limit. This raises the question of whether irradiation with an AC magnetic field with f of 500 kHz and Hac of 37.3 kA/m for core-shell structured nanoparticles with d of 15 nm and K of 1.7 × 104 J/m3 are really the optimum conditions. However, it is difficult to apply the other guiding principle to maximize hysteresis loss of ferromagnetic nanoparticles because thermally assisted reversals of μ occur stochastically before the barrier disappears at HK. Recalling that the characteristic time of thermal fluctuation was estimated to be a few hundred nanoseconds even in a zero magnetic field, the conditions used by Lee et al. are outside the scope of applicability of conventional models for ferromagnetic nanoparticles at a temperature of absolute zero and for superparamagnetic nanoparticles in a zero magnetic field. Consequently, new guiding principles to maximize heat dissipation PH are required. The second point is that Hac=37.3 kA/m is much larger than the exposure restriction for this waveband [52]. This point is examined further in Section 5. The next section will present results of recent numerical studies on the behavior of nanoparticles under conditions outside the scope of applicability of conventional models. This knowledge will be useful to establish sophisticated guiding principles that are adapted to advanced technologies that control the size, shape, and composite structure of nanoparticles.


## 4. Recent Numerical Simulations for Novel Responses to AC Magnetic Fields

To further improve the guiding principles for the design of magnetic nanoparticles, we must clarify the behavior of nanoparticles under conditions outside the scope of applicability of conventional models. However, it is difficult to discuss nonlinear nonequilibrium responses algebraically; as an alternative, numerical simulation has been performed extensively because of recent advances in computing speed. Noteworthy results obtained from these studies will be introduced in this section. To fully discuss their features from the viewpoint of efficiency, the results are shown as the ratio of the simulated value ofPH to the theoretical upper limit of PH : PHMax, where PHMax is expressed as 4μ0MsHac·f·ρ-1 for irradiating AC magnetic field Hacsin(2πf·t), because the loss dissipated in one cycle is the area of the hysteresis loop.In most of the simulations, it was assumed that magnetic nanoparticles were individually delivered to tumor tissues and accumulate randomly inside them, apart from the present status of this treatment [53]. Because the actual concentration of nanoparticles in tumors c does not exceed 10 kg/m3 (10 mg/cm3) as stated above, effects caused by dipole-dipole interactions Jdd between the accumulated nanoparticles were considered insignificant at room temperature. For example, at the mean distance 〈r〉≈d·ρ1/3·c-1/3,Jdd/kB≈μ0μ2/(〈r〉3kB) is estimated to be 25 K for magnetite nanoparticles with d=15 nm, Ms=450 kA/m, and c=10 kg/m3. Thus, the nanoparticles in this hyperthermia treatment simulation were considered magnetically isolated from each other.


### 4.1. Néel Relaxation in Magnetic Fields

In a magnetically isolated nanoparticle, the potential energy,U, with respect to the direction of μ is simply given by the sum of magnetic anisotropy energy and Zeeman energy. As a first approximation, uniaxial magnetic anisotropy has usually been assumed for the former term, although it contains  contributions from various kinds of magnetic anisotropy such as shape, crystalline, and surface anisotropy. In this case, U can be expressed as
(6)U(ϕ,ψ)=KVsin2ϕ-μ0μHacsin(2πf·t)cosψ,
where ϕ is the angle between the easy axis and μ, and ψ is the angle between μ and H. The detailed trajectories of μ in this potential can be precisely simulated by solving the stochastic Landau-Lifshitz-Gilbert equations [53–57]. However, we are only interested in the reversal of μ once every microsecond because the frequency used for hyperthermia treatment is limited. Carrey et al. calculated the behavior of μ using a well-known coarse-grained approach or “two-level approximation” [58, 59], which considers thermally activated reversals between the metastable directions via the midway saddle point in the energy barrier. In this calculation, easy axes of the accumulated nanoparticles were assumed to be fixed. This assumption seems valid when the nanoparticles are strongly anchored to structures resembling organelles.Figures4(a), 5(a), and 6(a) show contour plots of PH/PHMax calculated for cobalt ferrite, manganese ferrite and their core-shell nanoparticles introduced above, respectively, where the time evolution of the occupation probabilities of the directions parallel to the randomly oriented easy axes are simulated in the same way as Carrey et al. using the parameters given in Table 1. At low Hac of 1 kA/m, PH/PHMax of the core-shell nanoparticles increases with f, and a single maximum is found at a peak frequency, fp, of 110 kHz (Figure 6(a)). This behavior is consistent with the above prediction that PH is maximized when f-1 is adjusted to the Néel relaxation time. It is notable that fp shifts to higher frequency as Hac increases. This acceleration of Néel relaxation can be attributed to lowering of the energy barrier by the Zeeman energy. As indicated by the dashed line in Figure 6(a), the shift of fp can be explained by τN(Hac) calculated using the conventional Brown’s equation as follows [60]:
(7)[τN(Hac)]-1=f0·(1-h2)×{(1+h)exp[(-KVkBT)(1+h)2]×+(1-h)exp[(-KVkBT)(1-h)2]},
where h is H/HK. In Figure 6(a), fp at Hac=20 kA/m, a typical Hac for hyperthermia treatment, is about 40 times faster than that in a zero magnetic field. This fact clearly indicates that maximum heat dissipation cannot be obtained if we prepare nanoparticles according to the conventional guiding principles expressed in (1)–(5). This problem becomes serious when monodisperse nanoparticles are synthesized, although we barely noticed the problem because we used polydisperse nanoparticles with a broad distribution of τN.Calculated efficiency of heat dissipation by CoFe2O4 nanoparticles that are (a) nonrotatable and (b) rotatable. Dashed lines represent the Néel relaxation time (2πτN)-1 and the solid line indicates fp, which was calculated using (11). Diamonds denote the conditions used in the experiment.
(a)
(b)Calculated efficiency of heat dissipation by MnFe2O4 nanoparticles that are (a) non-rotatable and (b) rotatable. Dashed lines represent the Néel relaxation time (2πτN)-1 and the solid line indicates fp, which was calculated using (11). Diamonds denote the conditions used in the experiment.
(a)
(b)Calculated efficiency of heat dissipation by core-shell nanoparticles that are (a) non-rotatable and (b) rotatable. Dashed lines represent the Néel relaxation time(2πτN)-1, the solid line indicates fp, which was calculated using (11), and the dashed-dotted line shows the value calculated using (12). Diamonds denote the conditions used in the experiment.
(a)
(b)It is very important that these calculated results are compared with experimental data, even under only one set of conditions withf=500 kHz and Hac=37.3 kA/m. In Figure 6(a), 40% of PHMax, that is 1.4 MW/kg, is expected for the core-shell nanoparticles at f=500 kHz and Hac=37.3 kA/m (diamonds), whereas a larger value of 3.0 MW/kg was actually observed. In Figure 4(a), almost zero dissipation was calculated for the cobalt ferrite nanoparticles under the same conditions, because these nanoparticles are ferromagnetic, so no hysteresis loss is dissipated when Hac=37.3 kA/m because it is sufficiently lower than HK=630 kA/m. In contrast, considerable dissipation of 0.4 MW/kg was experimentally reported for the cobalt ferrite nanoparticles. In Figure 5(a), a small amount of dissipation is expected for the manganese ferrite nanoparticles under the same conditions, because these nanoparticles are typically superparamagnetic and little relaxation loss dissipates at f=500 kHz that is sufficiently lower than [2πτN(Hac)]-1 of several tens of megahertz. However, a considerable dissipation of 0.2 MW/kg was experimentally reported for the manganese ferrite nanoparticles. Some of these inconsistencies may be attributed to the fact that the magnetic nanoparticles were easily rotatable in a low viscous liquid of toluene. Hence, Brownian rotations would be described next.


### 4.2. Brownian Relaxation in Magnetic Fields

In this subsection, ferromagnetic nanoparticles in Newtonian fluids are considered because toluene is a typical Newtonian fluid (η=0.55 mPa·s), although the actual microviscoelasticity of the local environment in cancer cells is still unknown. In this case, the inertia of nanoparticles with a typical size of 10 nm can be neglected in considering their rotation by Brownian dynamics simulation [61, 62]. In the inertia-less limit, frictional torque for the rotation of a sphere balances with magnetic torque μ(t) × H(t) and Brownian torque λ(t) as follows:
(8)6ηVH·ω(t)=μ0μ(t)×H(t)+λ(t)(9)〈λi(t)〉=0,(10)〈λi(t1)λi(t2)〉=2kBT·(6ηVH)·δ(t1-t2),
where ω(t) is the angular velocity of rotation for the unit vector e(t) along the easy axis given by de/dt = ω(t) × e(t), and δ(t1-t2) is the Dirac delta function. Yoshida and Enpuku [63] simulated the rotation of ferromagnetic nanoparticles using the Fokker-Planck equation equivalent to the above relationships; they assumed that μ(t) was permanently fixed at the direction parallel to e(t) as long as Hac<HK. As a result, they confirmed that, at zero magnetic field limit, the frequency-dependence of heat dissipation exhibits a single maximum at fp=(2πτB)-1, as predicted by (2)–(5). They also found that fp increases with Hac according to the equation:
(11)2πfp≈τB-1[1+0.07(μ0μHackBT)2]0.5≈{τB-1atμ0μHac≪kBT0.5(μ0μHac6ηVH)atμ0μHac≫kBT.
This equation indicates that the driving force of the rotation changes from Brownian random torque to magnetic torque as Hac increases.As an example, this equation is applied to the cobalt ferrite nanoparticles discussed above. The solid curve in Figure4(b) shows the values of 2πfp calculated using (11) with the parameters in Table 1. The obtained line is close to the position of the diamond located at f=500 kHz and Hac=37.3 kA/m. In other words, the magnetic torque from the magnetic field at 37.3 kA/m happened to satisfy the conditions of rotating the cobalt ferrite nanoparticles with an appropriate delay to the alternation at 500 kHz; consequently, a considerable amount of heat, 3.7 MW/kg, dissipates. Apart from the magnitude, this is the reason why PH=0.4 MW/kg was experimentally observed for the cobalt ferrite nanoparticles, despite the conventional prediction of no hysteresis loss under the experimental conditions. As exemplified here, delayed rotations are caused by magnetic torque (not Brownian torque) even at Hac much lower than HK, resulting in significant heat dissipation.Researchers are also interested in the magnetic response whenHac becomes comparable to HK. In this case, the above-mentioned assumption that μ(t) is permanently fixed at the direction parallel to e(t) is invalid, because μ(t) is canted from the easy axis by the Zeeman energy. Furthermore, μ(t) stochastically reverses by thermal fluctuations even in ferromagnetic nanoparticles, because the Zeeman energy lowers the barrier height sufficiently. Therefore, I simultaneously computed the rotations of the nanoparticles using (8)–(10) with the thermally activated reversals of μ(t) on the potential given by (6) [64]. Note that (8) is valid within the two-level approximation [65]. The results calculated for these cobalt ferrite nanoparticles are shown as the contour lines (and color difference) in Figure 4(b). Firstly, we are certain that, at Hac≪HK≈630 kA/m, the location of the ridge in the contour plot of PH/PHMax is consistent with the solid line given by (11). This result indicates that ferromagnetic nanoparticles are rotated by the magnetic torque before the reversal of μ(t) occurs within it. However, the ridge seems turn to the position extrapolated from the dashed curve given by (7) when Hac becomes comparable to HK. In other words, μ(t) is promptly reversed before the rotation because the Néel relaxation is accelerated enough in this Hac range. These relationships can be written as
(12)2πfp≈[τN(Hac)]-1+τB-1[1+0.07(μ0μHackBT)2]0.5.
This equation is an extended relationship of τ-1=τN-1+τB-1 ((3)) for a large AC magnetic field. It is noteworthy that the first term τN(Hac) usually becomes extremely small for ferromagnetic nanoparticles at Hac≈HK in an aligned case (e//H) or at Hac≈HK/2 in tilted cases, while the second term is approximately expressed as 0.5(μ0μHac/6ηVH) when μ0μHac≫kBT. Therefore, the changeover from rotation to reversal occurs at 2πf≈0.5(μ0μHK/6ηVH)=KV/(6ηVH) or KV/(12ηVH) for aligned and tilted cases, respectively. For example, this changeover frequency corresponds to 4 MHz for the aligned cobalt ferrite nanoparticles with d=12 nm, V/VH=0.63,K=200 kJ/m3, and η=0.55 mPa·s. Importantly, the changeover frequency is independent of the size of nanoparticles as long as the ratio V/VH is constant. In other words, rotations predominate over the magnetic response at 1 MHz even for much larger cobalt ferrite nanoparticles (d=120 nm, (2πτB)-1=200 Hz). We must keep in mind that, even when ferromagnetic nanoparticles are large enough for their Brownian relaxation to be negligible, magnetic torque can easily rotate such nanoparticles at a time scale of microseconds if they are in a liquid phase. This knowledge is helpful when considering the optimal frequency for hyperthermia treatment, even if it is for a simplified system.


### 4.3. Easy Axes Oriented to the Directions Parallel, Perpendicular, or Oblique to the AC Field

As described above, the fast reversals ofμ(t) are predominant in the magnetic response of ferromagnetic nanoparticles at frequencies higher than the changeover frequency. The simulations, however, revealed that, at the frequencies, the rotation induces various kinds of stationary orientations of the easy axes e(t), which critically affect the reversals [64, 66]. In this section, we also examine the results determined for cobalt ferrite nanoparticles with d=12 nm, V/VH=0.63,K=200 kJ/m3, and η=0.55 mPa·s. In the initial state before irradiation with the AC magnetic field, the easy axes are set to be randomly oriented in the fluid, as shown in Figure 7(a). Therefore, in the first cycle, the major hysteresis loop obtained at Hac=640 kA/m >HK is consistent with the magnetization curve predicted by the Stoner-Wohlfarth model (see the inset). If the irradiation of the AC magnetic field at Hac=640 kA/m is continued in the simulation, the easy axes gradually turn toward the direction parallel to H. Note that, in the case where the easy axis is not parallel to H, the direction of μ is not completely parallel to H even though μ is already reversed at H≥HK. Therefore, a large magnetic torque proportional to sin ψ can turn the easy axis even if the magnetization seems almost saturated at H≈HK. For example, sin ψ is 0.43 when cos ψ is 0.9. Consequently, a longitudinally oriented structure of the easy axes is formed in the fluid (see Figure 7(d)). The formation of this nonequilibrium structure makes the dynamic hysteresis loop squarer than the initial curve, as shown in the inset of upper panel of Figure 7(d).Figure 7
Calculated orientation distribution of the easy axesρ(θ) of CoFe2O4 nanoparticles in (a) thermal equilibrium at H=0, and (b)–(d) nonequilibrium steady states under AC magnetic field at various Hac and f=30 MHz.The inset shows the dynamic hysteresis loops. Diagrams of the magnetic torques in the AC field are depicted in (e) and (f), where the ellipsoid in each figure shows a nanoparticle, and the broken line, open and closed arrows indicate the directions of the easy axis, magnetic moment of the particle, and the AC magnetic field, respectively. The nonequilibrium structures under the high-frequency AC magnetic field are illustrated in the sketches in the lower column.In contrast, the magnetization curve atHac=300 kA/m <HK/2 is a minor hysteresis loop, as shown in Figure 7(b). In this case, the easy axis turns toward the direction perpendicular to H and they maintain planar orientations if the ferromagnetic nanoparticles are continuously irradiated by an AC magnetic field at Hac=300 kA/m. A question now arises because we know that the longitudinal orientation is preferred when the Zeeman energy is considered. To clarify the reason for this, we consider an initial state in which a nanoparticle with an easy axis at angle θ has a magnetic moment μ at a parallel direction ψ=θ. When a small magnetic field H<HK/2 is applied to the nanoparticle, μ immediately tilts to ψ=θ-ϕ without reversals (see Figure 7(e)), because the position of the local minimum on U(ϕ,ψ) is changed. Then, the magnetic torque, -μ0μHsin(θ-ϕ), rotates μ toward the longitudinal direction: ψ→0. Because μ drags the easy axis, θ also decreases. In other words, the easy axis turns toward the direction parallel to H. If H is reversed subsequently, the direction of μ at this moment is almost antiparallel to H at ψ=θ+π-ϕ. Then ψ instantly changes to θ+π+ϕ because of the effect of variation of the minimum on U(ϕ,ψ) (see Figure 7(e)). The magnetic torque at this stage, -μ0μHsin(θ+π+ϕ)=μ0μHsin(θ+ϕ), forces μ to rotate toward the direction ψ=2πviaψ=(3/2)π. Because μ is bound on the easy axis, θ also increases. In other words, the easy axis starts to turn toward the plane perpendicular to H. If the direction of H alternates at a high frequency, a planar orientation of the easy axis is formed on average, because μ0μHsin(θ+ϕ) is larger than μ0μHsin(θ-ϕ). This reduces the remanence of the hysteresis loop. In contrast, a longitudinal orientation is formed in a large AC magnetic field H≥HK as discussed above, because μ is always reversed to the direction parallel to H immediately after H is reversed. Overall, θ decreases toward 0 when the reversal of μ occurs with alternation of the direction of H, whereas θ increases toward π/2 without reversal of μ.This feature leads to formation of novel nonequilibrium structures such as the obliquely oriented state found at an intermediate amplitude ofHac=340 kA/m. Without considering thermal fluctuations, the reversals should occur in the range of θ from 0.15π to 0.35π for Stoner-Wohlfarth nanoparticles with HK=630 kA/m, while μ never reverses in the other ranges. If this feature simply applies, θ should decrease with time in the range between 0.15π and 0.35π, whereas it should increase both between 0 and 0.15π and between 0.35π and π/2. These variations certainly lead to formation of a bimodal ρ(θ) with double maxima at θ=0.15π and π/2, as found in Figure 7(c). Consequently, the easy axes are oriented in both the planes perpendicular and oblique to the magnetic field.Concisely, in ferromagnetic nanoparticles in toluene or an aqueous phase, longitudinal, conical, or planar orientations are formed, irrespective of the free energy, as nonequilibrium structures under a high-frequency AC magnetic field. As a result, the major hysteresis loop becomes squarer and the minor loop becomes narrower compared with the magnetization curve calculated for randomly oriented nanoparticles. These variations of the area of the loops cause the maximum ofPH/PHMax to shift towards higher Hac from the optimal conditions predicted by the conventional models in Section 2. This kind of averaging of the oscillating rotations, discussed using the cobalt ferrite nanoparticles as an example, should always occur as long as the alternation of the magnetic field is much more frequent than the characteristic time of rotation, 0.5(μ0μHac/6ηVH). For this reason, these nonequilibrium structures would form in the radio-waveband used for hyperthermia treatment if the amplitude is somewhat smaller (~10 kA/m) or the viscosity is considerably higher (~10 mPa·s). Therefore, we must keep in mind the important effects of nonequilibrium structures on heat dissipation when establishing the optimal design of ferromagnetic nanoparticles for hyperthermia treatment.


### 4.4. Magnetic Hysteresis of Superparamagnetic States

Let us leave ferromagnetic nanoparticles and move on to superparamagnetic manganese ferrite nanoparticles, from which a considerable amount of heat dissipation, 0.2 MW/kg, was experimentally reported atf=500 kHz. The orientation of μ on these nanoparticles is easily equilibrated in the magnetic potential expressed in (6) within the scale of the Néel relaxation time τN(Hac=0) of 1 × 10−8 s. Therefore, little relaxation loss is expected using the conventional model. For this reason, I wish to examine this inconsistency from the viewpoint of the effects of slow rotations on the fast reversals in superparamagnetic nanoparticles.The contour lines (and color difference) in Figure5(b) show the results obtained from the simultaneous simulation of rotations and reversals for the manganese ferrite nanoparticles. Firstly, we find a secondary maximum of PH/PHMax around f=100 kHz in addition to the primary ridge of PH/PHMax indicated by the dashed curve at frequencies of several tens of megahertz, which is explained by (7) for τN(Hac) above. To clarify the origin of the new kind of heat dissipation, the magnetization curve calculated under the conditions of Hac=4 kA/m and f=100 kHz is presented in Figure 8(a). An S-shaped hysteresis loop without remanence is observed. In this cycle, the directions of the easy axes have butterfly-shaped hysteresis, as shown in Figure 8(b). This behavior is explained by the following atypical magnetic response in the period f-1 (10 μs). Initially (at t=0), no magnetization exists because the occupation probabilities of μ in the two stable directions parallel to the easy axis are equalized in a zero magnetic field. As H increases, the occupation probability in the more stabilized direction immediately increases because of reversals on a time scale of τN (≤10 ns). The reversed μ in the stabilized direction is not completely parallel to H,  ψ≠0, and the magnetic torque μ0μHsinψ turns the easy axis towards the direction of the field. The time constant of this process is approximately expressed as [0.5(μ0μHac/6ηVH)]-1 using the second term in (11). For the manganese ferrite nanoparticles, it is 3 μs when H is 4 kA/m. Therefore, rotation is not negligible in the peak period of the oscillations of H. Subsequently, H decreases to zero at t=0.5f=5 μs, and the occupation probabilities are again equalized because reversal is rapid, so the magnetic torque disappears. Alternatively, the Brownian torque randomizes the orientation of the easy axis on a time scale of τB (= 2 μs). Therefore, competition between the magnetic and Brownian torques can cause the butterfly-shaped hysteresis of 〈cosθ〉. Because the equilibrium magnetization of the superparamagnetic nanoparticles with easy axes parallel to H is higher than that of randomly oriented ones [58, 67], the magnetization curve shows hysteresis without remanence. Consequently, a secondary maximum appears even though τN≪τB if the nanoparticles are rotatable. As discussed here, we should remove the stereotype of a single peak at a 2πfp value of τ-1(=τN-1+τB-1).Calculated magnetic response of MnFe2O4 nanoparticles with an applied AC field with Hac=4 kA/m and f=100 kHz. (a) Steady magnetization curves, and (b) mean orientation of the easy axis of the nanoparticles, 〈cosθ〉. In the inset in (a), the ellipsoid shows a nanoparticle, and the broken line, open and closed arrows indicate the directions of the easy axis, magnetic moment of the particle, and the AC magnetic field, respectively. The variation of easy axis orientations is illustrated in the sketches in (b).
(a)
M
-
H curves
(b)
Mean orientationNeedless to say, there is still room for further study. For example,PH simulated at f=500 kHz and Hac = 37.3 kA/m is 0.13 MW/kg, which is inconsistent with the observed PH of 0.2 MW/kg. At present, it is unclear whether the difference can be attributed to the nontrivial polydisperse nature of the prepared sample or the accuracy of the simulations, because the experiment was performed under only one set of conditions with f=500 kHz and Hac=37.3 kA/m. Thus, measurement of PH under various conditions will be helpful to establish a model of heat dissipation in superparamagnetic nanoparticles. In addition, it is certain that the protocols of these simulations are also improvable, because it has been assumed that the direction of μ is fixed at one of the local minima in the energy potential given by (6), although we know μ stochastically explores all over the potential well [65]. Briefly, the magnetic torque is overestimated. Recently, more strict calculations were carried out and they also show the same kind of butterfly-shaped hysteresis [67]. As described here, much still remains to be done.


### 4.5. Intermediate State between Ferromagnetic and Superparamagnetic Nanoparticles

Core-shell nanoparticles, which can generate the largest amount of heat out of various nanoparticle structures, fit neither ferromagnetic(τN(Hac=0)≫f-1) nor superparamagnetic (τN(Hac=0)≪f-1) conditions. This is because the value of the Néel relaxation time τN(Hac=0) calculated using the parameters in Table 1 is 1 μs, which is comparable with the alternation time of the AC magnetic field used in hyperthermia treatment. Furthermore, the Brownian relaxation time τB is also estimated to be 1 μs. Therefore, it is worth discussing this intermediate case before concluding this section. Figure 6(b) shows the results obtained by simultaneous simulation of rotation and reversal as contour lines (and color difference). In this figure, we are certain that location of the ridge in the contour plot of PH/PHMax is consistent with neither the dashed curve (7) nor the solid curve (11), but instead with the dashed-dotted curve given by (12). Furthermore, the iso-height contour lines, for example, the boundary between yellow and light green, shift toward lower frequency compared with the randomly fixed case in Figure 6(a). Figure 9 shows the magnetization curve and variation of the directions of the easy axes calculated for the core-shell nanoparticles under the conditions of Hac=37.3 kA/m and f=500 kHz. We observe eyeglass-shaped hysteresis in the variation of the directions of the easy axes. This behavior is attributed to complicated competition between normal rotations when μ is parallel to H, counter-rotations when μ is antiparallel to H, and randomization at H≈0. The major point is that the first term seems to dominate the other terms, because the baseline of the eyeglass-shaped oscillations of the easy axes is considerably higher than the 0.5 expected for randomly oriented nanoparticles. This longitudinal orientation makes the dynamic hysteresis loop squarer and leads to an increase in PH (see Figure 9(a)). In addition to this effect, on average, oscillation of the directions of the easy axes induced by the alternation of the counter-rotations and randomization further increases PH. Indeed, we can observe that the hysteresis loop of the rotatable nanoparticles in Figure 9(a) opens even in the higher magnetic field where the loop of the non-rotatable nanoparticles in Figure 9(a) is closed after all μ are reversed. Overall, both the phenomena discussed for ferromagnetic and superparamagnetic nanoparticles contribute to amplification of the hysteresis loop area in this intermediate state; as a result, PH increases from 1.4 MW/kg for the non-rotatable case to 2.4 MW/kg for the rotatable one. We can say that this value is fairly consistent with the observed PH of 3 MW/kg in consideration that the simulation was carried out for completely isolated monodisperse nanoparticles with uniform uniaxial anisotropy.Figure 9
Calculated magnetic response of core-shell nanoparticles with an applied AC field withHac=37.3 kA/m and f=500 kHz. (a) Steady magnetization curves, and (b) mean orientation of the easy axis of the nanoparticles, 〈cosθ〉.


## 4.1. Néel Relaxation in Magnetic Fields

In a magnetically isolated nanoparticle, the potential energy,U, with respect to the direction of μ is simply given by the sum of magnetic anisotropy energy and Zeeman energy. As a first approximation, uniaxial magnetic anisotropy has usually been assumed for the former term, although it contains  contributions from various kinds of magnetic anisotropy such as shape, crystalline, and surface anisotropy. In this case, U can be expressed as
(6)U(ϕ,ψ)=KVsin2ϕ-μ0μHacsin(2πf·t)cosψ,
where ϕ is the angle between the easy axis and μ, and ψ is the angle between μ and H. The detailed trajectories of μ in this potential can be precisely simulated by solving the stochastic Landau-Lifshitz-Gilbert equations [53–57]. However, we are only interested in the reversal of μ once every microsecond because the frequency used for hyperthermia treatment is limited. Carrey et al. calculated the behavior of μ using a well-known coarse-grained approach or “two-level approximation” [58, 59], which considers thermally activated reversals between the metastable directions via the midway saddle point in the energy barrier. In this calculation, easy axes of the accumulated nanoparticles were assumed to be fixed. This assumption seems valid when the nanoparticles are strongly anchored to structures resembling organelles.Figures4(a), 5(a), and 6(a) show contour plots of PH/PHMax calculated for cobalt ferrite, manganese ferrite and their core-shell nanoparticles introduced above, respectively, where the time evolution of the occupation probabilities of the directions parallel to the randomly oriented easy axes are simulated in the same way as Carrey et al. using the parameters given in Table 1. At low Hac of 1 kA/m, PH/PHMax of the core-shell nanoparticles increases with f, and a single maximum is found at a peak frequency, fp, of 110 kHz (Figure 6(a)). This behavior is consistent with the above prediction that PH is maximized when f-1 is adjusted to the Néel relaxation time. It is notable that fp shifts to higher frequency as Hac increases. This acceleration of Néel relaxation can be attributed to lowering of the energy barrier by the Zeeman energy. As indicated by the dashed line in Figure 6(a), the shift of fp can be explained by τN(Hac) calculated using the conventional Brown’s equation as follows [60]:
(7)[τN(Hac)]-1=f0·(1-h2)×{(1+h)exp[(-KVkBT)(1+h)2]×+(1-h)exp[(-KVkBT)(1-h)2]},
where h is H/HK. In Figure 6(a), fp at Hac=20 kA/m, a typical Hac for hyperthermia treatment, is about 40 times faster than that in a zero magnetic field. This fact clearly indicates that maximum heat dissipation cannot be obtained if we prepare nanoparticles according to the conventional guiding principles expressed in (1)–(5). This problem becomes serious when monodisperse nanoparticles are synthesized, although we barely noticed the problem because we used polydisperse nanoparticles with a broad distribution of τN.Calculated efficiency of heat dissipation by CoFe2O4 nanoparticles that are (a) nonrotatable and (b) rotatable. Dashed lines represent the Néel relaxation time (2πτN)-1 and the solid line indicates fp, which was calculated using (11). Diamonds denote the conditions used in the experiment.
(a)
(b)Calculated efficiency of heat dissipation by MnFe2O4 nanoparticles that are (a) non-rotatable and (b) rotatable. Dashed lines represent the Néel relaxation time (2πτN)-1 and the solid line indicates fp, which was calculated using (11). Diamonds denote the conditions used in the experiment.
(a)
(b)Calculated efficiency of heat dissipation by core-shell nanoparticles that are (a) non-rotatable and (b) rotatable. Dashed lines represent the Néel relaxation time(2πτN)-1, the solid line indicates fp, which was calculated using (11), and the dashed-dotted line shows the value calculated using (12). Diamonds denote the conditions used in the experiment.
(a)
(b)It is very important that these calculated results are compared with experimental data, even under only one set of conditions withf=500 kHz and Hac=37.3 kA/m. In Figure 6(a), 40% of PHMax, that is 1.4 MW/kg, is expected for the core-shell nanoparticles at f=500 kHz and Hac=37.3 kA/m (diamonds), whereas a larger value of 3.0 MW/kg was actually observed. In Figure 4(a), almost zero dissipation was calculated for the cobalt ferrite nanoparticles under the same conditions, because these nanoparticles are ferromagnetic, so no hysteresis loss is dissipated when Hac=37.3 kA/m because it is sufficiently lower than HK=630 kA/m. In contrast, considerable dissipation of 0.4 MW/kg was experimentally reported for the cobalt ferrite nanoparticles. In Figure 5(a), a small amount of dissipation is expected for the manganese ferrite nanoparticles under the same conditions, because these nanoparticles are typically superparamagnetic and little relaxation loss dissipates at f=500 kHz that is sufficiently lower than [2πτN(Hac)]-1 of several tens of megahertz. However, a considerable dissipation of 0.2 MW/kg was experimentally reported for the manganese ferrite nanoparticles. Some of these inconsistencies may be attributed to the fact that the magnetic nanoparticles were easily rotatable in a low viscous liquid of toluene. Hence, Brownian rotations would be described next.


## 4.2. Brownian Relaxation in Magnetic Fields

In this subsection, ferromagnetic nanoparticles in Newtonian fluids are considered because toluene is a typical Newtonian fluid (η=0.55 mPa·s), although the actual microviscoelasticity of the local environment in cancer cells is still unknown. In this case, the inertia of nanoparticles with a typical size of 10 nm can be neglected in considering their rotation by Brownian dynamics simulation [61, 62]. In the inertia-less limit, frictional torque for the rotation of a sphere balances with magnetic torque μ(t) × H(t) and Brownian torque λ(t) as follows:
(8)6ηVH·ω(t)=μ0μ(t)×H(t)+λ(t)(9)〈λi(t)〉=0,(10)〈λi(t1)λi(t2)〉=2kBT·(6ηVH)·δ(t1-t2),
where ω(t) is the angular velocity of rotation for the unit vector e(t) along the easy axis given by de/dt = ω(t) × e(t), and δ(t1-t2) is the Dirac delta function. Yoshida and Enpuku [63] simulated the rotation of ferromagnetic nanoparticles using the Fokker-Planck equation equivalent to the above relationships; they assumed that μ(t) was permanently fixed at the direction parallel to e(t) as long as Hac<HK. As a result, they confirmed that, at zero magnetic field limit, the frequency-dependence of heat dissipation exhibits a single maximum at fp=(2πτB)-1, as predicted by (2)–(5). They also found that fp increases with Hac according to the equation:
(11)2πfp≈τB-1[1+0.07(μ0μHackBT)2]0.5≈{τB-1atμ0μHac≪kBT0.5(μ0μHac6ηVH)atμ0μHac≫kBT.
This equation indicates that the driving force of the rotation changes from Brownian random torque to magnetic torque as Hac increases.As an example, this equation is applied to the cobalt ferrite nanoparticles discussed above. The solid curve in Figure4(b) shows the values of 2πfp calculated using (11) with the parameters in Table 1. The obtained line is close to the position of the diamond located at f=500 kHz and Hac=37.3 kA/m. In other words, the magnetic torque from the magnetic field at 37.3 kA/m happened to satisfy the conditions of rotating the cobalt ferrite nanoparticles with an appropriate delay to the alternation at 500 kHz; consequently, a considerable amount of heat, 3.7 MW/kg, dissipates. Apart from the magnitude, this is the reason why PH=0.4 MW/kg was experimentally observed for the cobalt ferrite nanoparticles, despite the conventional prediction of no hysteresis loss under the experimental conditions. As exemplified here, delayed rotations are caused by magnetic torque (not Brownian torque) even at Hac much lower than HK, resulting in significant heat dissipation.Researchers are also interested in the magnetic response whenHac becomes comparable to HK. In this case, the above-mentioned assumption that μ(t) is permanently fixed at the direction parallel to e(t) is invalid, because μ(t) is canted from the easy axis by the Zeeman energy. Furthermore, μ(t) stochastically reverses by thermal fluctuations even in ferromagnetic nanoparticles, because the Zeeman energy lowers the barrier height sufficiently. Therefore, I simultaneously computed the rotations of the nanoparticles using (8)–(10) with the thermally activated reversals of μ(t) on the potential given by (6) [64]. Note that (8) is valid within the two-level approximation [65]. The results calculated for these cobalt ferrite nanoparticles are shown as the contour lines (and color difference) in Figure 4(b). Firstly, we are certain that, at Hac≪HK≈630 kA/m, the location of the ridge in the contour plot of PH/PHMax is consistent with the solid line given by (11). This result indicates that ferromagnetic nanoparticles are rotated by the magnetic torque before the reversal of μ(t) occurs within it. However, the ridge seems turn to the position extrapolated from the dashed curve given by (7) when Hac becomes comparable to HK. In other words, μ(t) is promptly reversed before the rotation because the Néel relaxation is accelerated enough in this Hac range. These relationships can be written as
(12)2πfp≈[τN(Hac)]-1+τB-1[1+0.07(μ0μHackBT)2]0.5.
This equation is an extended relationship of τ-1=τN-1+τB-1 ((3)) for a large AC magnetic field. It is noteworthy that the first term τN(Hac) usually becomes extremely small for ferromagnetic nanoparticles at Hac≈HK in an aligned case (e//H) or at Hac≈HK/2 in tilted cases, while the second term is approximately expressed as 0.5(μ0μHac/6ηVH) when μ0μHac≫kBT. Therefore, the changeover from rotation to reversal occurs at 2πf≈0.5(μ0μHK/6ηVH)=KV/(6ηVH) or KV/(12ηVH) for aligned and tilted cases, respectively. For example, this changeover frequency corresponds to 4 MHz for the aligned cobalt ferrite nanoparticles with d=12 nm, V/VH=0.63,K=200 kJ/m3, and η=0.55 mPa·s. Importantly, the changeover frequency is independent of the size of nanoparticles as long as the ratio V/VH is constant. In other words, rotations predominate over the magnetic response at 1 MHz even for much larger cobalt ferrite nanoparticles (d=120 nm, (2πτB)-1=200 Hz). We must keep in mind that, even when ferromagnetic nanoparticles are large enough for their Brownian relaxation to be negligible, magnetic torque can easily rotate such nanoparticles at a time scale of microseconds if they are in a liquid phase. This knowledge is helpful when considering the optimal frequency for hyperthermia treatment, even if it is for a simplified system.


## 4.3. Easy Axes Oriented to the Directions Parallel, Perpendicular, or Oblique to the AC Field

As described above, the fast reversals ofμ(t) are predominant in the magnetic response of ferromagnetic nanoparticles at frequencies higher than the changeover frequency. The simulations, however, revealed that, at the frequencies, the rotation induces various kinds of stationary orientations of the easy axes e(t), which critically affect the reversals [64, 66]. In this section, we also examine the results determined for cobalt ferrite nanoparticles with d=12 nm, V/VH=0.63,K=200 kJ/m3, and η=0.55 mPa·s. In the initial state before irradiation with the AC magnetic field, the easy axes are set to be randomly oriented in the fluid, as shown in Figure 7(a). Therefore, in the first cycle, the major hysteresis loop obtained at Hac=640 kA/m >HK is consistent with the magnetization curve predicted by the Stoner-Wohlfarth model (see the inset). If the irradiation of the AC magnetic field at Hac=640 kA/m is continued in the simulation, the easy axes gradually turn toward the direction parallel to H. Note that, in the case where the easy axis is not parallel to H, the direction of μ is not completely parallel to H even though μ is already reversed at H≥HK. Therefore, a large magnetic torque proportional to sin ψ can turn the easy axis even if the magnetization seems almost saturated at H≈HK. For example, sin ψ is 0.43 when cos ψ is 0.9. Consequently, a longitudinally oriented structure of the easy axes is formed in the fluid (see Figure 7(d)). The formation of this nonequilibrium structure makes the dynamic hysteresis loop squarer than the initial curve, as shown in the inset of upper panel of Figure 7(d).Figure 7
Calculated orientation distribution of the easy axesρ(θ) of CoFe2O4 nanoparticles in (a) thermal equilibrium at H=0, and (b)–(d) nonequilibrium steady states under AC magnetic field at various Hac and f=30 MHz.The inset shows the dynamic hysteresis loops. Diagrams of the magnetic torques in the AC field are depicted in (e) and (f), where the ellipsoid in each figure shows a nanoparticle, and the broken line, open and closed arrows indicate the directions of the easy axis, magnetic moment of the particle, and the AC magnetic field, respectively. The nonequilibrium structures under the high-frequency AC magnetic field are illustrated in the sketches in the lower column.In contrast, the magnetization curve atHac=300 kA/m <HK/2 is a minor hysteresis loop, as shown in Figure 7(b). In this case, the easy axis turns toward the direction perpendicular to H and they maintain planar orientations if the ferromagnetic nanoparticles are continuously irradiated by an AC magnetic field at Hac=300 kA/m. A question now arises because we know that the longitudinal orientation is preferred when the Zeeman energy is considered. To clarify the reason for this, we consider an initial state in which a nanoparticle with an easy axis at angle θ has a magnetic moment μ at a parallel direction ψ=θ. When a small magnetic field H<HK/2 is applied to the nanoparticle, μ immediately tilts to ψ=θ-ϕ without reversals (see Figure 7(e)), because the position of the local minimum on U(ϕ,ψ) is changed. Then, the magnetic torque, -μ0μHsin(θ-ϕ), rotates μ toward the longitudinal direction: ψ→0. Because μ drags the easy axis, θ also decreases. In other words, the easy axis turns toward the direction parallel to H. If H is reversed subsequently, the direction of μ at this moment is almost antiparallel to H at ψ=θ+π-ϕ. Then ψ instantly changes to θ+π+ϕ because of the effect of variation of the minimum on U(ϕ,ψ) (see Figure 7(e)). The magnetic torque at this stage, -μ0μHsin(θ+π+ϕ)=μ0μHsin(θ+ϕ), forces μ to rotate toward the direction ψ=2πviaψ=(3/2)π. Because μ is bound on the easy axis, θ also increases. In other words, the easy axis starts to turn toward the plane perpendicular to H. If the direction of H alternates at a high frequency, a planar orientation of the easy axis is formed on average, because μ0μHsin(θ+ϕ) is larger than μ0μHsin(θ-ϕ). This reduces the remanence of the hysteresis loop. In contrast, a longitudinal orientation is formed in a large AC magnetic field H≥HK as discussed above, because μ is always reversed to the direction parallel to H immediately after H is reversed. Overall, θ decreases toward 0 when the reversal of μ occurs with alternation of the direction of H, whereas θ increases toward π/2 without reversal of μ.This feature leads to formation of novel nonequilibrium structures such as the obliquely oriented state found at an intermediate amplitude ofHac=340 kA/m. Without considering thermal fluctuations, the reversals should occur in the range of θ from 0.15π to 0.35π for Stoner-Wohlfarth nanoparticles with HK=630 kA/m, while μ never reverses in the other ranges. If this feature simply applies, θ should decrease with time in the range between 0.15π and 0.35π, whereas it should increase both between 0 and 0.15π and between 0.35π and π/2. These variations certainly lead to formation of a bimodal ρ(θ) with double maxima at θ=0.15π and π/2, as found in Figure 7(c). Consequently, the easy axes are oriented in both the planes perpendicular and oblique to the magnetic field.Concisely, in ferromagnetic nanoparticles in toluene or an aqueous phase, longitudinal, conical, or planar orientations are formed, irrespective of the free energy, as nonequilibrium structures under a high-frequency AC magnetic field. As a result, the major hysteresis loop becomes squarer and the minor loop becomes narrower compared with the magnetization curve calculated for randomly oriented nanoparticles. These variations of the area of the loops cause the maximum ofPH/PHMax to shift towards higher Hac from the optimal conditions predicted by the conventional models in Section 2. This kind of averaging of the oscillating rotations, discussed using the cobalt ferrite nanoparticles as an example, should always occur as long as the alternation of the magnetic field is much more frequent than the characteristic time of rotation, 0.5(μ0μHac/6ηVH). For this reason, these nonequilibrium structures would form in the radio-waveband used for hyperthermia treatment if the amplitude is somewhat smaller (~10 kA/m) or the viscosity is considerably higher (~10 mPa·s). Therefore, we must keep in mind the important effects of nonequilibrium structures on heat dissipation when establishing the optimal design of ferromagnetic nanoparticles for hyperthermia treatment.


## 4.4. Magnetic Hysteresis of Superparamagnetic States

Let us leave ferromagnetic nanoparticles and move on to superparamagnetic manganese ferrite nanoparticles, from which a considerable amount of heat dissipation, 0.2 MW/kg, was experimentally reported atf=500 kHz. The orientation of μ on these nanoparticles is easily equilibrated in the magnetic potential expressed in (6) within the scale of the Néel relaxation time τN(Hac=0) of 1 × 10−8 s. Therefore, little relaxation loss is expected using the conventional model. For this reason, I wish to examine this inconsistency from the viewpoint of the effects of slow rotations on the fast reversals in superparamagnetic nanoparticles.The contour lines (and color difference) in Figure5(b) show the results obtained from the simultaneous simulation of rotations and reversals for the manganese ferrite nanoparticles. Firstly, we find a secondary maximum of PH/PHMax around f=100 kHz in addition to the primary ridge of PH/PHMax indicated by the dashed curve at frequencies of several tens of megahertz, which is explained by (7) for τN(Hac) above. To clarify the origin of the new kind of heat dissipation, the magnetization curve calculated under the conditions of Hac=4 kA/m and f=100 kHz is presented in Figure 8(a). An S-shaped hysteresis loop without remanence is observed. In this cycle, the directions of the easy axes have butterfly-shaped hysteresis, as shown in Figure 8(b). This behavior is explained by the following atypical magnetic response in the period f-1 (10 μs). Initially (at t=0), no magnetization exists because the occupation probabilities of μ in the two stable directions parallel to the easy axis are equalized in a zero magnetic field. As H increases, the occupation probability in the more stabilized direction immediately increases because of reversals on a time scale of τN (≤10 ns). The reversed μ in the stabilized direction is not completely parallel to H,  ψ≠0, and the magnetic torque μ0μHsinψ turns the easy axis towards the direction of the field. The time constant of this process is approximately expressed as [0.5(μ0μHac/6ηVH)]-1 using the second term in (11). For the manganese ferrite nanoparticles, it is 3 μs when H is 4 kA/m. Therefore, rotation is not negligible in the peak period of the oscillations of H. Subsequently, H decreases to zero at t=0.5f=5 μs, and the occupation probabilities are again equalized because reversal is rapid, so the magnetic torque disappears. Alternatively, the Brownian torque randomizes the orientation of the easy axis on a time scale of τB (= 2 μs). Therefore, competition between the magnetic and Brownian torques can cause the butterfly-shaped hysteresis of 〈cosθ〉. Because the equilibrium magnetization of the superparamagnetic nanoparticles with easy axes parallel to H is higher than that of randomly oriented ones [58, 67], the magnetization curve shows hysteresis without remanence. Consequently, a secondary maximum appears even though τN≪τB if the nanoparticles are rotatable. As discussed here, we should remove the stereotype of a single peak at a 2πfp value of τ-1(=τN-1+τB-1).Calculated magnetic response of MnFe2O4 nanoparticles with an applied AC field with Hac=4 kA/m and f=100 kHz. (a) Steady magnetization curves, and (b) mean orientation of the easy axis of the nanoparticles, 〈cosθ〉. In the inset in (a), the ellipsoid shows a nanoparticle, and the broken line, open and closed arrows indicate the directions of the easy axis, magnetic moment of the particle, and the AC magnetic field, respectively. The variation of easy axis orientations is illustrated in the sketches in (b).
(a)
M
-
H curves
(b)
Mean orientationNeedless to say, there is still room for further study. For example,PH simulated at f=500 kHz and Hac = 37.3 kA/m is 0.13 MW/kg, which is inconsistent with the observed PH of 0.2 MW/kg. At present, it is unclear whether the difference can be attributed to the nontrivial polydisperse nature of the prepared sample or the accuracy of the simulations, because the experiment was performed under only one set of conditions with f=500 kHz and Hac=37.3 kA/m. Thus, measurement of PH under various conditions will be helpful to establish a model of heat dissipation in superparamagnetic nanoparticles. In addition, it is certain that the protocols of these simulations are also improvable, because it has been assumed that the direction of μ is fixed at one of the local minima in the energy potential given by (6), although we know μ stochastically explores all over the potential well [65]. Briefly, the magnetic torque is overestimated. Recently, more strict calculations were carried out and they also show the same kind of butterfly-shaped hysteresis [67]. As described here, much still remains to be done.


## 4.5. Intermediate State between Ferromagnetic and Superparamagnetic Nanoparticles

Core-shell nanoparticles, which can generate the largest amount of heat out of various nanoparticle structures, fit neither ferromagnetic(τN(Hac=0)≫f-1) nor superparamagnetic (τN(Hac=0)≪f-1) conditions. This is because the value of the Néel relaxation time τN(Hac=0) calculated using the parameters in Table 1 is 1 μs, which is comparable with the alternation time of the AC magnetic field used in hyperthermia treatment. Furthermore, the Brownian relaxation time τB is also estimated to be 1 μs. Therefore, it is worth discussing this intermediate case before concluding this section. Figure 6(b) shows the results obtained by simultaneous simulation of rotation and reversal as contour lines (and color difference). In this figure, we are certain that location of the ridge in the contour plot of PH/PHMax is consistent with neither the dashed curve (7) nor the solid curve (11), but instead with the dashed-dotted curve given by (12). Furthermore, the iso-height contour lines, for example, the boundary between yellow and light green, shift toward lower frequency compared with the randomly fixed case in Figure 6(a). Figure 9 shows the magnetization curve and variation of the directions of the easy axes calculated for the core-shell nanoparticles under the conditions of Hac=37.3 kA/m and f=500 kHz. We observe eyeglass-shaped hysteresis in the variation of the directions of the easy axes. This behavior is attributed to complicated competition between normal rotations when μ is parallel to H, counter-rotations when μ is antiparallel to H, and randomization at H≈0. The major point is that the first term seems to dominate the other terms, because the baseline of the eyeglass-shaped oscillations of the easy axes is considerably higher than the 0.5 expected for randomly oriented nanoparticles. This longitudinal orientation makes the dynamic hysteresis loop squarer and leads to an increase in PH (see Figure 9(a)). In addition to this effect, on average, oscillation of the directions of the easy axes induced by the alternation of the counter-rotations and randomization further increases PH. Indeed, we can observe that the hysteresis loop of the rotatable nanoparticles in Figure 9(a) opens even in the higher magnetic field where the loop of the non-rotatable nanoparticles in Figure 9(a) is closed after all μ are reversed. Overall, both the phenomena discussed for ferromagnetic and superparamagnetic nanoparticles contribute to amplification of the hysteresis loop area in this intermediate state; as a result, PH increases from 1.4 MW/kg for the non-rotatable case to 2.4 MW/kg for the rotatable one. We can say that this value is fairly consistent with the observed PH of 3 MW/kg in consideration that the simulation was carried out for completely isolated monodisperse nanoparticles with uniform uniaxial anisotropy.Figure 9
Calculated magnetic response of core-shell nanoparticles with an applied AC field withHac=37.3 kA/m and f=500 kHz. (a) Steady magnetization curves, and (b) mean orientation of the easy axis of the nanoparticles, 〈cosθ〉.


## 5. Optimized Design and Future Outlook

Magnetic nanoparticles for thermotherapy, particularly rotatable nanoparticles, have been predicted to exhibit various novel responses to AC magnetic fields, as described above. Examples include magnetic hysteresis observed for superparamagnetic states and nonequilibrium structures with easy axes oriented to the directions parallel, perpendicular, or oblique to the magnetic field. These nonlinear and nonequilibrium phenomena cannot be explained using conventional models. Further systematic simulations and their experimental verification are required to establish sophisticated guiding principles for such magnetic nanoparticles. However, some feel that the heat generation of 3 MW/kg achieved by Lee et al. is sufficient for practical use in hyperthermia treatment, so more sophisticated guidelines may not be necessary. In this final section, we discuss this issue.Tumors less than 0.01 m (= 1 cm) in size are considered difficult to find with existing diagnostic methods, so here we examine whether or not the heat dissipation from current magnetic nanoparticles is enough to treat hidden tumors of such size. According to Andrä et al. [68], raising the temperature of a tumor by ΔT requires heat generation of approximately 3λΔTR-2 without considering blood flow, where λ is thermal conductivity and 2R is the diameter of a tumor. If we assume λ=0.6 WK−1m−1, ΔT=5K, and 2R=0.005 or 0.01 m, the required heat generation would be 1.5 or 0.4 MW/m3, respectively. The rate of blood flow in tumor tissues is typically 1% per second by volume (60 mL/min/100 g) [69]; thus, when ΔT=5K, the heat transport caused by blood flow is estimated to be 0.2 MW/m3 using a value of ~4 MJ-m−3 K−1 for the specific heat of blood. Therefore, the total cooling power of hidden tumors is between 0.6 and 2 MW/m3 for ΔT=5K. This assessment indicates that the amount of heat dissipation PH required to kill metastatic cancer cells is estimated to be within 0.3 and 1 MW/kg if we can expect a nanoparticle concentration within tumors of approximately 2 kg/m3. The developed core-shell magnetic nanoparticles thus clearly enable adequate heat dissipation. However, are they actually suitable for use in hyperthermia treatment?Note that Section4 described how nanoparticles with PH of 3 MW/kg was obtained from irradiation using an AC magnetic field of Hac=37.3 kA/m and f=500 kHz. When this AC magnetic field is irradiated on a simple model body composed of a homogenous column with electrical conductivity σ=0.2 Sm−1 and radius r=0.1 m, the maximum voltage generated on the outer circumference is V=πr22πf(μ0Hac) = 4,600 V per revolution, at which point the eddy current loss Pe=1/2π2μ02σr2f2Hac2 is 5 MW/m3 (5 W/cm3). This heat generation is sufficient to raise the temperature of thermally insulated tissues by 10 K or more in 10 seconds. For this reason, we cannot ignore the side effects of Pe on normal tissues, although the model assuming a constant σ is oversimplified. According to guidelines published by the International Commission on Non-Ionizing Radiation Protection [52], the upper limit for work-related exposure of the torso is 10 W/kg (corresponding to 0.01 MW/m3). However, because this value is the upper limit for routine exposure, it may not be indicative of the maximum exposure in medical treatment. A slightly more specific value can be calculated as follows. Heat generation only occurs in the outer edge of a human body if a magnetic field is irradiated over the whole body; therefore, the heated region can be considered as a cylinder that is a few centimeters thick. This region can be cooled from the body surface area in medical treatment. Its cooling power, λ∂2T/∂r2~λΔT(Δr)-2, is roughly estimated to be 0.03 MW/m3 under the conditions of ΔT=20 K and Δr=0.02 m. Because blood vessels expand when temperature rises, blood flow increases even in tissues with little blood flow normally. In subcutaneous tissues, for example, a blood flow rate of approximately 0.2% per second by volume (12 mL/min/100 g) has been reported at 42°C [69, 70]. Under these conditions, calculating the heat transport caused by blood flow using the same method yields a value of 0.03 MW/m3 when the temperature difference from the outside of the irradiated region is set at ΔT=4 K. These values are the result of rough calculations that do not consider the detailed structure of a human body [71], but their sum, which is about 0.06 MW/m3, can be used as an approximation of cooling ability. When a magnetic field is applied to the model body mentioned above, this value corresponds to Pe for the condition Hacf = 2 × 109 Am−1 s−1. Calculating the behavior of the above-mentioned core-shell nanoparticles (d=15 nm) within this restriction (see Figure 10) shows that PH does not reach the requirement of 0.3 MW/kg. However, Figure 10 indicates that if the size of the particle is increased slightly, sufficient PH can be obtained from the rotatable nanoparticles at higher Hac (equivalent to lower f) even under this restriction, and adequate heating is expected inside hidden tumors with a diameter of 0.01 m without serious side effects on normal tissues from Pe.Calculated heat dissipation by core-shell nanoparticles that are (a) non-rotatable and (b) rotatable, whereHacf is always 2 × 109 Am−1 s−1 (corresponding to the restriction that the eddy current loss Pe is 0.06 MW/m3 in normal tissue). The size d is changed in the simulation, but the other parameters were fixed at the values shown in Table 1.
(a)
Nonrotatable
(b)
RotatableOur discussion up to this point applies to treatment using continuous irradiation where heat balance holds. Irradiation time and interval can be controlled in medical treatment. For example, when tumors with a specific heat of 4 MJm−3K−1 containing the above-mentioned core-shell nanoparticles with a concentration c of 2 kg/m3 were irradiated with an AC magnetic field of Hac = 37.3 kA/m and f=500 kHz, heat of approximately cPH = 6 MW/m3 was generated. Relative to this value, the quantity of heat diffused to the surrounding areas from 10 mm tumors is negligible when ΔT<5 K; thus, the temperature will rise by 5 K after approximately 3 seconds. Because the eddy current loss Pe in this case is 5 MW/m3, it will take approximately 4 seconds for the temperature of normal tissue to rise by 5 K. Stopping irradiation after 3 seconds will thus enable selective heating of tumors by 5 K or more. This is an extreme example; however, it does indicate that there is another option apart from continuous irradiation. The ratio of cPH to Pe is important. Although obtaining robust values requires detailed protocol, a factor of 4-5 or so might be a criterion for cPH/Pe. As an example, we calculated cPH/Pe for the core-shell nanoparticles and found that this condition is satisfied for lower frequencies/smaller amplitudes than those indicated by the solid line in Figure 11 [72]. This finding reflects the fact that PH is the area of the M-H curve × frequency, which is proportional to Hacf at most, whereas Pe increases in proportion to (Hacf)2, as previously described. Because it is impossible to attain a rise in temperature of 5 K if cPH is at least 0.6 (or 2) MW/m3, irradiation must therefore be conducted using a higher frequency and larger amplitude to ensure that this condition is met (see dashed lines in Figure 12 [72]). Ultimately, stronger, faster conditions are needed to destroy cancer cells, and weaker, slower conditions are needed to limit damage to normal tissue. Using the core-shell nanoparticles of d=15 nm, a frequency of f=500 kHz is thus acceptable, but Hac needs to be maintained at 9  kA/m to resolve the conflicting requirements.Figure 11
Calculated selection ratiocPH/Pe for core-shell nanoparticles in AC magnetic fields with various Hac and f. Rotatable nanoparticles are compared with randomly oriented ones. The dashed lines show the isoplethic curves at PH=0.3 and 1 MW/kg (see Figure 12), while the solid lines show the isoplethic curves at cPH/Pe=4. *Reproduced from Mamiya [72] with permission (Copyright 2012 TIC).Figure 12
Calculated magnetic lossPH for core-shell nanoparticles in AC magnetic fields for various Hac and f. Rotatable nanoparticles are compared with randomly oriented ones. The dashed lines show the isoplethic curves at PH=0.3 and 1 MW/kg, while the solid lines show the isoplethic curves at the selection ratio cPH/Pe=4 (see Figure 11). *Reproduced from Mamiya [72] with permission (Copyright 2012 TIC).As discussed above, the combination of the core-shell nanoparticles ofd=15 nm and K=1.7×104 J/m3 with an AC magnetic field of f=500 kHz and Hac=37.3 kA/m may not be optimal. A narrow range of combinations of these parameters will facilitate efficient hyperthermia treatment and prevent side effects. We have not yet optimized the conditions for hyperthermia treatment; however, establishing the optimal combinations may be difficult, particularly if a trial and error approach is used. The routes used to synthesize magnetic nanoparticles of controlled size, shape, and composite structure have become increasingly advanced, as described in this paper. Dramatic advances in computing speed have also promoted numerical simulation of nonlinear nonequilibrium responses to AC magnetic fields. If we continue to improve material design on the bases of such advanced nanotechnology and computer simulations, optimal conditions will eventually be clarified. Remarkable advances are still continually being reported in clinical trials are being conducted, even though the combination of nanoparticles and oscillation of the equipment has not been optimized yet [73, 74]. Once optimization improves local heating ability, then thermotherapy should be established as a fourth or fifth standard cancer treatment method to reduce the disease burden of a patient.



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*Source: 752973-2013-07-28.xml*