The heating performance of magnetic nanoparticles (MNPs) under an alternating magnetic field (AMF) is dependent on several factors. Optimizing these factors improves the heating efficiency for cancer therapy and meanwhile lowers the MNP treatment dosage. AMF is one of the most easily controllable variables to enhance the efficiency of heat generation. This paper investigated the optimal magnetic field strength and frequency for an assembly of magnetite nanoparticles. For hyperthermia treatment in clinical applications, monodispersed NPs are forming nanoclusters in target regions where a strong magnetically interactive environment is anticipated, which leads to a completely different situation than MNPs in ferrofluids. Herein, the energy barrier model is revisited and Néel relaxation time is tailored for high MNP packing densities. AMF strength and frequency are customized for different magnetite NPs to achieve the highest power generation and the best hyperthermia performance.

## I. INTRODUCTION

Magnetic hyperthermia treatment using magnetic nanoparticles (MNPs) is often considered as an alternative therapy for cancer treatment, currently in clinical trials. Magnetite (*FeO* ⋅ *Fe*_{2}*O*_{3}) nanoparticles with low toxicity and high self-heating capacity are among the most promising candidates for hyperthermia treatment. To date, lots of work has been done on investigating the heat generation efficiency of magnetic fluids, which may not be applicable to clinical hyperthermia treatments. In biomedical applications, particles are administered directly to tumor tissues and the interaction between MNPs and target cells induces aggregation. Hence, there are single nanoparticles coexisting with nanoclusters in tumor tissues. For those particles in clusters, they are in close contact and are highly interactive.^{1} The random and competing interparticle interactions among MNP clusters alter the magnetic dynamic properties by affecting the energy barrier, thence changing the relaxation time and giving rise to a collective magnetic behavior.^{2} For this reason, an understanding of the MNP aggregation effect on hyperthermia performance is crucial.

For hyperthermia treatments in clinical applications, the magnetic field strength is limited by physiological considerations because the induced eddy current in non-magnetic tissues will cause necrosis or carbonization of healthy tissues.^{3} Brezovich^{4} has experimentally determined a safety threshold for AMF with field strength *H* and frequency f as follows: *H* ⋅ *f* < 5 × 10^{9} Am^{−1}s^{−1}. Technical consideration limits the field frequencies f to a narrow range,^{3} namely, from 100 kHz to 300 kHz, which, as a result, sets the upper limits of field strength *H* under different scenarios.

## II. MODEL AND METHOD

Specific absorption rate (SAR) is used as a figure of merit to characterize the hyperthermia performances of magnetic nanoparticles under an AMF. SAR values are defined as heat generation per mass of nanoparticles:^{5}

where *m*_{s}, *c*_{p} are the mass and heat capacity of solution, *m*_{np} is the mass of nanoparticles, and $\Delta T/\Delta t$ is the slope of temperature rise *vs.* time. Nanoparticles with high SAR are favored in order to lower treatment dosage and meanwhile achieving essential temperature rise to induce tumor cell apoptosis.

The power generation *P* of nanoparticles under an AMF is defined by the Rosensweig equation:^{6}

where $\mu 0$ is the vacuum permeability, $\chi 0$ is the magnetic susceptibility of nanoparticles, $\tau eff$ is the effective relaxation time, *H* and *f* are the magnetic field strength and frequency, respectively. The fraction term leads to a global maximum of *P* and SAR at $2\pi f\tau eff=1$, which defines the critical frequency for the system.

Heat is generated from each particle during the process of rotating magnetic moment against energy barrier^{3} upon an AMF. For free nanoparticles in ferrofluids, there are three independent mechanisms that can result in hyperthermia upon the stimulation of AMF: Néel relaxation, Brownian relaxation, and hysteresis loss mechanism. Néel relaxation is the rotation of magnetic moment inside a stationary particle and Brownian relaxation is the rotation of the entire particle along with the magnetic moment. Hysteresis loss comes from the movement of domain walls, which is responsible for heating in larger sized ferromagnetic particles.^{5} Herein, we are investigating on single domain magnetite NPs where the hysteresis loss is nonexistent. Magnetite NPs with diameters below 50 nm are single domain, and thus we can work within the macrospin approximation.^{7} Mathematical modeling results^{5} limited the particle diameter within 20 nm to maintain its superparamagnetism. So magnetite NPs with an average diameter range from 7 nm to 20 nm are studied in this paper.

We consider *N* identical magnetite NPs with perfect spherical shape and magnetic core diameter *D*. Upon the injection of magnetite NPs, NPs are taken up by tumor cells and fixed within cells, forming nanoclusters. The contribution of Brownian relaxation can be neglected^{8} and the heating power comes from Néel relaxation exclusively. An analytical expression for Néel relaxation time is:^{9}

where *h* is the time-varying parameter modulated by AMF: $H(t)=H0\u22c5cos(2\pi ft).\gamma $ is the electron gyromagnetic ratio $(1.76\xd71011radT\u22121s\u22121)$, and the damping constant $\alpha \u2032$ is assumed to be 0.1 for magnetite NPs.^{10} $\u27e8\Delta E\u27e9$ is the energy barrier height a nanoparticle needs to overcome before flipping from one potential well to another. *k*_{B} is Boltzmann constant, *T* is temperature in Kelvin, *σ* is energy barrier and Magnetic core volume *V*_{c} equals to $\pi D3/6$. Equation (3) is valid for $\sigma \u22650$ and $0\u2264h\u22641$.

Due to surface spin disorder,^{11} the saturation magnetization *M*_{s} of NPs is significantly smaller than the bulk material. This reduced *M*_{s} can be well explained by taking the magnetically dead layer into consideration. The equation below is used to estimate *M*_{s} of NPs:

where $Msb=4.6\xd7105A/m$ is the saturation magnetization of the bulk magnetite,^{12} d is the thickness of magnetically anomalous shell. P. Dutta *et al*.^{13} have reported a linear fit of M_{s} from experimental results which yield a magnetically anomalous shell thickness of d = 0.68 nm.

The anisotropy constant is an effective value including surface contribution:

where *K*_{v} and *K*_{s} are the bulk and surface anisotropy constants, respectively. $Kv=1.04\xd7104J/m3$ for bulk magnetite, and $Ks=3.9\xd710\u22125J/m2$ from A. Demortiere *et al.*^{14}

For an isolated NP in free space with magnetic anisotropy energy given by:

where *θ* is the angle between its magnetic moment and easy axis. The energy minimums are separated by an energy barrier of *K*_{eff}*V*_{c} and $\sigma \u2032=KeffVckBT$. For MNP nanoclusters with high packing densities, interparticle interactions can have a strong influence on the magnetic dynamics of each MNP.^{15–20} Since the MNPs for hyperthermia applications are coated with surfactants, the distance between MNP surfaces exceeds the range of exchange coupling. Herein, exchange coupling is neglected and long-range dipolar interaction is taken into consideration.^{21,22} For an ensemble of randomly distributed NPs, this energy barrier is thus revised by the mean-field approximation:^{23,24}

where *N*_{i} is the number of the *i*th nearest neighbor to a reference particle, *p*_{2} is the average of a Legendre polynomial $(p2\u223c0.9)$, and $\u27e8Ri\u27e9$ is the statistical average of interparticle distance between the *i*th nearest neighbor and the reference particle. Each particle with a rotating magnetic moment $\mu \u2192(\u27e8\mu \u27e9=MsVc)$ possesses uniaxial anisotropy with easy axis oriented in a random direction. J.-O. Andersson *et al.*^{2} reported that this dipolar interaction field reaches a plateau as inter-particle distance meets $\u27e8Ri\u27e9/D\u22654$. For an ensemble of closely packed congruent spherical NPs with surfactant coating thickness of *d*′ (varies from 5 nm up to 25 nm), the highest packing density is achieved by forming a face-centered cubic (FCC) structure. Interparticle distance of the nearest neighbor (nn) is $\u27e8R1\u27e9=D+2\xd7d\u2032$, and the maximum number of NPs that can occupy this first layer is $N1=12$. The interparticle distance and maximum occupancy of higher-order layers are listed in Table I.

. | Interparticle . | Maximum . | . | Interparticle . | Maximum . | . | Interparticle . | Maximum . |
---|---|---|---|---|---|---|---|---|

Layer . | Distance . | Occupancy . | Layer . | Distance . | Occupancy . | Layer . | Distance . | Occupancy . |

1 | $\u27e8R1\u27e9$ | 12 | 7 | $7\xd7\u27e8R1\u27e9$ | 48 | 13 | $13\xd7\u27e8R1\u27e9$ | 72 |

2 | $2\xd7\u27e8R1\u27e9$ | 6 | 8 | $8\xd7\u27e8R1\u27e9$ | 6 | 14 | $14\xd7\u27e8R1\u27e9$ | 0 |

3 | $3\xd7\u27e8R1\u27e9$ | 24 | 9 | $3\xd7\u27e8R1\u27e9$ | 36 | 15 | $15\xd7\u27e8R1\u27e9$ | 48 |

4 | $2\xd7\u27e8R1\u27e9$ | 12 | 10 | $10\xd7\u27e8R1\u27e9$ | 24 | 16 | $4\xd7\u27e8R1\u27e9$ | 12 |

5 | $5\xd7\u27e8R1\u27e9$ | 24 | 11 | $11\xd7\u27e8R1\u27e9$ | 24 | 17 | $17\xd7\u27e8R1\u27e9$ | 48 |

6 | $6\xd7\u27e8R1\u27e9$ | 8 | 12 | $12\xd7\u27e8R1\u27e9$ | 24 |

. | Interparticle . | Maximum . | . | Interparticle . | Maximum . | . | Interparticle . | Maximum . |
---|---|---|---|---|---|---|---|---|

Layer . | Distance . | Occupancy . | Layer . | Distance . | Occupancy . | Layer . | Distance . | Occupancy . |

1 | $\u27e8R1\u27e9$ | 12 | 7 | $7\xd7\u27e8R1\u27e9$ | 48 | 13 | $13\xd7\u27e8R1\u27e9$ | 72 |

2 | $2\xd7\u27e8R1\u27e9$ | 6 | 8 | $8\xd7\u27e8R1\u27e9$ | 6 | 14 | $14\xd7\u27e8R1\u27e9$ | 0 |

3 | $3\xd7\u27e8R1\u27e9$ | 24 | 9 | $3\xd7\u27e8R1\u27e9$ | 36 | 15 | $15\xd7\u27e8R1\u27e9$ | 48 |

4 | $2\xd7\u27e8R1\u27e9$ | 12 | 10 | $10\xd7\u27e8R1\u27e9$ | 24 | 16 | $4\xd7\u27e8R1\u27e9$ | 12 |

5 | $5\xd7\u27e8R1\u27e9$ | 24 | 11 | $11\xd7\u27e8R1\u27e9$ | 24 | 17 | $17\xd7\u27e8R1\u27e9$ | 48 |

6 | $6\xd7\u27e8R1\u27e9$ | 8 | 12 | $12\xd7\u27e8R1\u27e9$ | 24 |

## III. RESULTS AND DISCUSSION

In Fig. 1 we have demonstrated that the energy barrier and relaxation time of one particle increases as the number of its neighboring NPs increases. For a single nanoparticle immobilized in tumor tissues, the energy barrier is its anisotropy energy competing with thermal fluctuation *k*_{B}*T*. If this particle sits in a nanocluster which lots of neighbors are in close contact with, then dipolar interaction energy plays an important role in altering energy barrier and relaxation time. For a specific type of magnetite NP with $D=d\u2032=15nm$ under a step field of $30000A/m$, its relaxation time increases from 1.01 μs to 1.12 μs from single nanoparticle to nanocluster.

Interparticle distance is largely dependent on the surfactant coating thickness of NPs, so we varied the surfactant coating thickness *d*′ from 5 nm to 25 nm in the simulation. In Fig. 2a, we found that the relaxation time reaches a plateau in all of these different situations. When $d\u2032=5nm$, dipolar interactions from neighboring NPs no longer affect the magnetic behavior of reference particle as it reaches to the 13^{th} layer $(\u27e8Ri\u27e9/D\u226513)$. While for *d*′ = 10 *nm*, 15 *nm*, 20 *nm* and 25 *nm*, the critical layers are the 6th, 4th, 3rd, and the 2nd, respectively.

The particle aggregation effect has varying degrees of impact on the relaxation time of MNPs with different core diameters. For those magnetite NPs with core diameters smaller than 12 nm, their relaxation times are nearly invariant (Fig. 2b, their percentage changes are less than 1%). As the core diameter increases, relaxation time becomes more susceptible to the aggregation effect. For instance, the relaxation time of a single nanoparticle with *D* = 20 *nm* is 73.8 μs, while it is 468.9 μs in a nanocluster. There is a 535% difference in relaxation time.

As is shown in equations (3)–(8), Néel relaxation time is modulated by the strength of the applied field. An AMF is applied and the real-time relaxation times of different NPs and nanoclusters are recorded in Fig. 3. The global maximums of relaxation time appear when the AMF sweeps across 0, and the global minimums appear when AMF reaches troughs or crests. In Fig. 3(b), the relaxation time of single nanoparticle and nanocluster are almost identical when magnetic core diameter is 7 nm. However, it is a totally different scenario for NPs with *D* = 17 *nm*, where there is at least a 40% difference at any field strength.

We collected $|ln(2\pi f\tau eff)|$ as a function of field strength and frequency and plotted it in Fig. 4 to search for the critical AMF for an ensemble of NPs according to the golden rule of $2\pi f\tau eff=1$. A smaller value of $|ln(2\pi f\tau eff)|$ indicates a higher SAR and a higher efficiency of heat generation. For magnetite NPs with core diameter *D* = 10 *nm*, the critical point is found at *H*_{0} = 2500 *A*/*m* and *f* = 300 *kHz* for both single nanoparticle and nanocluster. When *D* = 15 *nm*, the critical point moves from *H*_{0} = 30000 *A*/*m*, *f* = 160 *kHz* for single nanoparticle to *H*_{0} = 30000 *A*/*m*, *f* = 140 *kHz* for nanocluster. When *D* = 20 *nm*, the critical point moves from *H*_{0} = 45000 *A*/*m*, *f* = 120 *kHz* for single nanoparticle to *H*_{0} = 50000 *A*/*m*, *f* = 100 *kHz* for nanocluster.

## IV. CONCLUSIONS

In this paper, we have studied the magnetic response of magnetite NPs with average diameter varying from 7 nm to 20 nm. A mean-field approximation method is used to calculate energy barrier and relaxation time. For a reference particle in nanocluster, dipolar fields exerted from neighboring NPs alter the energy barrier. And the energy barrier increases as its number of neighbors increases until it reaches a plateau (Fig. 1). Dipolar interaction strength is inversely proportional to the cube of interparticle distance. For an identical NP with different surfactant coating thicknesses, this dipolar interaction vanishes after the 2^{nd} layer of neighbors for *d*′ = 25 *nm*, while it doesn’t disappear until the 16^{th} layer for *d*′ = 5 *nm* (Fig. 2a). We fixed the surfactant coating thickness *d*′ to 15 *nm* and changed core diameters from 7 nm to 20 nm. For those NPs with core diameters smaller than 12 nm, the relaxation time is almost invariant. However, the relaxation time of NPs with *D* = 15 *nm* increased by 11% due to this aggregation effect. The percentage increase is 93% for *D* = 18 *nm* and 535% for *D* = 20 *nm* (Fig. 2b). This result is reasonable because dipolar interaction forces are dependent on magnetic moment, and thus the neighboring particles are exerting larger stray fields upon the reference particle, which contributes a noticeable difference in relaxation time of single particles and particles in a nanocluster. Finally, we plotted a figure of merit to judge whether an AMF can drive one specific type of MNP into the highest SAR, thus giving rise to the best hyperthermia performance, and $|ln(2\pi f\tau eff)|$ is introduced as the characterization factor, where a smaller value indicates that $2\pi f\tau eff$ is closer to 1. We conclude that for smaller MNPs, the critical point of the AMF doesn’t change regardless of whether it’s a single particle or a nanocluster. However, for MNPs with *D* = 20 *nm*, the critical point moved from *H*_{0} = 45000 *A*/*m*, *f* = 120 *kHz* to *H*_{0} = 50000 *A*/*m*, *f* = 100 *kHz* due to the aggregation effect.

## ACKNOWLEDGMENTS

The authors thank the support from the Institute of Engineering in Medicine, National Science Foundation MRSEC facility program, the Distinguished McKnight University Professorship, UROP program, MNDrive STEM program, MNDrive program, and the Interdisciplinary Doctoral Fellowship from the University of Minnesota. Authors thank the fruitful discussion with Yipeng Jiao from Department of Electrical Engineering, University of Minnesota.