Physics of heat generation using magnetic nanoparticles for hyperthermia

Abstract

Magnetic nanoparticle hyperthermia and thermal ablation have been actively studied experimentally and theoretically. In this review, we provide a summary of the literature describing the properties of nanometer-scale magnetic materials suspended in biocompatible fluids and their interactions with external magnetic fields. Summarised are the properties and mechanisms understood to be responsible for magnetic heating, and the models developed to understand the behaviour of single-domain magnets exposed to alternating magnetic fields. Linear response theory and its assumptions have provided a useful beginning point; however, its limitations are apparent when nanoparticle heating is measured over a wide range of magnetic fields. Well-developed models (e.g. for magnetisation reversal mechanisms and pseudo-single domain formation) available from other fields of research are explored. Some of the methods described include effects of moment relaxation, anisotropy, nanoparticle and moment rotation mechanisms, interactions and collective behaviour, which have been experimentally identified to be important. Here, we will discuss the implicit assumptions underlying these analytical models and their relevance to experiments. Numerical simulations will be discussed as an alternative to these simple analytical models, including their applicability to experimental data. Finally, guidelines for the design of optimal magnetic nanoparticles will be presented.

• Hyperthermia
• hysteresis modelling
• linear response theory
• magnetic nanoparticle

Introduction

Exposing diseased tissue to elevated temperatures for a period of time, or hyperthermia, has long been regarded as a promising treatment for cancer and other diseases. Cells are susceptible to heat damage because heat is energy that indiscriminately disrupts cellular pathways, especially through its effects on protein structure and function [1,2]. Depending upon the thermal dose, defined as ‘time-at-temperature’, hyperthermia induces a heat-shock response in cells, leading to cell death via a series of biochemical changes within the cell [1,3]. Heat also enhances the therapeutic effects of anti-cancer agents such as ionising radiation, often producing a combined cytotoxicity that is significantly greater than that arising from either agent alone [4–6]. It has been observed that heat inhibits DNA damage repair, rendering cells unable to recover from the effects of ionising radiation, culminating in cell death or senescence [1,2,4,7]. Heat, however, is neither intrinsically cancer-specific, nor are cancer cells intrinsically more sensitive to heat. On the other hand, aberrant tumour physiology has been implicated to provide an advantage for hyperthermia as an effective cancer therapy [1,4–6,8]. Tumours typically possess a more heterogeneous tissue structure than their normal counterparts, due to less regulated or unregulated growth. Tumours often display chaotic vascularity, high interstitial fluid pressure, significant deposits of dense and fibrous stroma, and regions chronically deprived of oxygen (hypoxia) – all contributing to resistance to standard therapies. Hyperthermia ameliorates these effects by increasing blood perfusion and modifying tissue structure, thereby enhancing therapy and enabling penetration of therapeutic agents.

Clinical application of hyperthermia requires selective delivery to the tumour(s) with an effective dose to cause a measurable response. Certainly, a corollary of selective delivery is that the deposited dose to nearby healthy cells and tissues must be minimised to avoid deleterious side effects [6,8]. Many strategies to deliver heat for cancer hyperthermia have emerged, with varying success [1,5,6,9]. Direct energy devices expose tissues to mechanical (ultrasound) or electromagnetic energy (laser, radio-frequency, or microwave) that is directly absorbed by tissue creating heat via several potential mechanisms depending upon the frequency of the energy [10–13]. Because the energy must be directed or focused to the appropriate region, issues arise from insufficient tissue-specific control producing over-treatment of some areas and under-treatment in others.

Another general approach combines specific properties of materials to preferentially absorb an externally applied energy, converting it to heat locally. Such susceptive materials couple more efficiently with the applied energy than the surrounding tissue, and provide local heating via a physical process. Examples include metallic implants or ‘seeds’ and nanometer-scale magnetic materials [13–15]. Specifically, magnetic nanoparticle hyperthermia (MNH) is a cancer therapy whereby magnetic nanoparticles are delivered either systemically through targeting or by direct injection to a tumour and then are subjected to an alternating magnetic field (AMF). This method, also known as magnetic fluid hyperthermia (MFH) or magnetic particle hyperthermia (MPH), was first proposed by Gilchrist [15] and holds promise for tissue-specific hyperthermia because it offers the potential for precise control of dose and tissue-specificity [16–19]. In general, magnetic materials heat via loss power through hysteresis when exposed to AMFs [20]. Because tissue is weakly diamagnetic, it does not attenuate or scatter static or low frequency magnetic fields, enabling imaging and therapy strategies that use magnetic energy. AMF at radio frequencies does, however, deposit energy indiscriminately to tissues via Joule heating through induced eddy currents [21–23]. The extent of this power absorption by tissues depends upon both the frequency and magnitude of the AMF. Therefore, if the promise offered by the combination of magnetic materials with AMF for heat-based therapies is to be a clinical reality, the challenges to provide sufficient magnetic material that is localised to the treatment area enabling beneficial heating with non-injurious AMF must be overcome.

The success of MNH depends upon development of magnetic nanoparticles having precisely controlled physical and magnetic properties [16,24]. To effectively optimise the properties of magnetic nanoparticles for hyperthermia, it is necessary to develop an understanding of the primary parameters that produce heat. Therefore, a theoretical evaluation of the mechanisms responsible for heating by magnetic nanoparticles is necessary to find optimal magnetic and structural properties to tailor synthesis methods for specific AMF frequency and amplitude conditions. This will provide reliable performance and predictive capability for thermal dosimetry to enable treatment planning and clinical translation. Calculating the heat dissipated by magnetic nanoparticles that are subjected to an AMF has, therefore, been under intense investigation recently [25].

In this review, we provide a brief summary of the literature describing the magnetic properties of nanometer-scale magnetic materials suspended in biocompatible fluids, the properties and mechanisms understood to be responsible for heating, and more recent directions in research that hold promise for clinical and scientific applications. Specifically, we will evaluate models developed for MNH, including their assumptions and limitations. We will also review results obtained from other scientific disciplines, such as magnetic thin films, transformer cores, and geology. Well-developed models (e.g. for magnetisation reversal mechanisms and pseudo-single domain formation) exist from these other fields of research that inform magnetic nanoparticle heat generation, yet these are often overlooked. Finally, we highlight some recent results that suggest primary parameters controlling heat generation, such as collective magnetic behaviour. We explore how these properties can be included in the theoretical models and exploited for therapy.

General properties of magnetic materials and magnetic nanoparticles

Magnetism [20] arises from interactions among the orbital and spin motions of electrons in atoms, producing an atomic (dipole) magnetic moment. Thus, all matter is ‘magnetic’. However, in some materials, there exists no interaction between atomic magnetic moments (called the exchange interaction), whereas for others there is a very strong exchange interaction (either positive or negative). In particular, diamagnets and paramagnets have no exchange interaction between atomic magnetic moments. Ferromagnets and antiferromagnets show strong negative and positive exchange interaction, resulting in parallel and antiparallel alignment, respectively. The way to identify the different types of magnetic materials is by their response to an external magnetic field as a function of temperature, as described below.

There are three classes of magnetic materials which are classified as ‘non-magnetic’ because they have no net magnetic moment in zero field. They are diamagnets, paramagnets, and antiferromagnets. All materials are diamagnetic, since every material has paired electrons. At zero applied magnetic field, all the electrons are paired, so their spin and angular momentums cancel out, resulting in zero net moment. In non-zero applied magnetic field (based on a classical explanation), each electron is assumed to act independently and expands or contracts its orbit (thereby changing its angular momentum) to oppose the applied magnetic field. Therefore, diamagnets display a negative (positive) magnetic moment to a positive (negative) applied magnetic field. This magnetic moment response, which is linear with a negative slope with applied magnetic field, does not display a strong temperature dependence. In contrast, paramagnetic materials have unpaired electrons, so they display a positive magnetic moment response to a positive applied magnetic field in addition to the diamagnetic negative response from their paired electrons. This paramagnetic response originates from the alignment of the unpaired electron with the applied magnetic field, and dominates the diamagnetic response from the change in angular momentum. The individual atomic magnetic moments, however, do not have any exchange interaction so magnetic moments point in all directions, resulting in zero net moment in zero applied magnetic field. Paramagnets also show a strong temperature dependence of the magnetic moment. Finally, in antiferromagnetic materials which have a strong positive exchange interaction, there is a (spontaneous) alignment of the atomic magnetic moments antiparallel with their neighbours below a critical temperature, called the Néel temperature (T_N). This exchange interaction dominates the diamagnetic change in angular momentum also present. Above T_N, the thermal energy is sufficient to overcome this exchange interaction, and the material acts as a paramagnet. Below T_N, since the neighbouring atomic magnetic moments are equal in magnitude, an antiferromagnet has zero net magnetic moment in zero applied magnetic field. Under the influence of a (very large) applied magnetic field, the exchange interaction can again be overcome, and the atomic magnetic moments will align parallel to the applied magnetic field. For most laboratory achievable fields, only a partial parallel alignment is possible, which is typically obtained via spin canting or spin flop. For completeness, it should be noted that there exists a special class of antiferromagnets, called geometrical frustrated magnets (or just frustrated magnets for short). These occur when the crystal structure inhibits all the antiferromagnetic alignments desired. A triangular lattice is an example of a frustrated system; the first two moments at any two vertices can be arranged with antiferromagnetic alignment. The third moment, however, cannot have an antiferromagnetic alignment with one of the other vertices, regardless of orientation (up or down). For MNH, however, this special class of magnetic system has not been studied, and is unlikely to be relevant, as low temperatures are required to observe the frustration.

There are two classes of magnetic materials which are classified as ‘magnetic’ because they have a net magnetic moment in zero field. These are ferromagnets and ferrimagnets. In both cases, the exchange interaction present dominates over the diamagnetic change in angular momentum which is also present. In ferromagnetic materials which have a strong negative exchange interaction, there is a (spontaneous) alignment of the atomic magnetic moments parallel with their neighbours below a critical temperature, called the Curie temperature (T_C). Above T_C, the thermal energy is sufficient to overcome this exchange interaction, and the material acts as a paramagnet. Below T_C, since the neighbouring atomic magnetic moments are parallel, a ferromagnet can have a net magnetic moment in zero applied magnetic field. Under the influence of an applied magnetic field, the atomic magnetic moments will align parallel to the applied magnetic field. Ferrimagnets are a special class of antiferromagnets, where the antiparallel atomic moments have different magnitudes. This means that ferrimagnets can exhibit a non-zero net magnetic moment at zero applied magnetic field. (The two ‘magnetic’ iron oxides, magnetite, Fe₃O₄, and maghemite, γ-Fe₂O₃ are both actually ferrimagnets, not ferromagnets.) Again, in ferrimagnetic materials which have a strong positive exchange interaction, there is a (spontaneous) alignment of the atomic magnetic moments antiparallel with their neighbours below the antiferromagnetic critical temperature T_N. Above T_N, the thermal energy is sufficient to overcome this exchange interaction, and the material acts as a paramagnet. Below T_N, since the neighbouring atomic magnetic moments are unequal in magnitude, a antiferrimagnet can have a net magnetic moment in zero applied magnetic field. Under the influence of an applied magnetic field, the exchange interaction can again be overcome, and the atomic magnetic moments will align parallel to the applied magnetic field. For most applications, only ferromagnets or ferrimagnets are of interest. Furthermore, ferrimagnets are often treated as just weaker ferromagnets. There is another special class of ferromagnet or anti-ferromagnet, called a spin glass [26]. Simply described, a spin glass results from a collection of moments (either from atoms or nanoparticles) whose low temperature state is a frozen disordered one, rather than the periodic pattern typically found in crystalline materials. For MNH applications, this special class is also not important, as low temperatures are required to observe spin-glass behaviour.

In addition to possessing an atomic magnetic moment (m), all magnetic materials exhibit additional characteristic features when exposed to an external field (see Figure 1A). These include (but are not limited to): 1) saturation magnetisation (M_S), 2) remanent magnetisation (M_R), 3) coercivity or coercive field (H_C), 4) anisotropy energy density (K), and 5) a characteristic response time of the moments of the material to a change in the applied magnetic field. This last property, as will be discussed later, is critically important for MNH. The saturation magnetisation (M_s), or moment per unit volume (mass), is the maximum value of the magnetisation observed only when all magnetic moments in the material are aligned parallel to the direction of an external applied field. The remanent magnetisation (M_R) is the value of the magnetisation remaining when the applied magnetic field returns to zero, whereas the coercivity (H_C) is the magnetic field that must be applied to ‘coerce’ the magnetisation back to a zero value. (When m = 0, the coercivity is also known in the magnetics literature as the ‘intrinsic coercivity’ H_ci.) H_C is actually a dynamic property that depends upon the frequency of the time-varying applied field. The ‘dynamic coercivity’ is the value for the coercivity at a specified frequency versus the ‘coercivity’ which is measured with time-invariant or static field conditions. The switching field (H_SW) is the appropriate term to describe the magnetic field required to reverse the moment orientation of a magnetic material, and it is an intrinsic property of the material. The coercivity (dynamic or static) of a magnetic material is easier to measure, but it is an extrinsic property of the material; therefore, the terms switching and coercive fields are commonly used interchangeably although they are quite different by definition.

Figure 1. (A) Schematic of a hysteresis loop showing the important parameters. (B) Hysteresis loops as a function of frequency on a magnetic nanoparticle system. (Reprinted with permission from Eggeman et al. [48].).

The anisotropy energy density, K, is the energy required to change the magnetisation of the material from a particular direction with respect to its physical structure. In other words, a material that exhibits a preferred direction of the magnetic moment with respect to other physical parameters (e.g. crystal orientation) is said to exhibit anisotropy, and the energy required to change this orientation is the anisotropy energy. In nanoparticles it typically describes the energy required to reverse the direction of magnetisation, or change it by 180°. The anisotropy energy density is a net energy per unit volume (mass) arising from effects of the crystalline lattice (e.g. structure, shape, volume, defects) and the collective magnetic interactions (e.g. exchange, interparticle, self-field generation). Thus, contributions to K are influenced by the unique features of the material under study and its energy is proportional to the anisotropy field (H_K) or the field required to completely change the magnetisation direction (see Figure 1A) given by Equation 1. where μ₀ is the permeability of free space and M_S is the saturation magnetisation with respect to volume. Here we have implicitly assumed the simplest form of magnetic anisotropy: uniaxial anisotropy, where there is a preferred axis of orientation to which the moment will orient parallel or antiparallel with equal probability. The relative influence of sources of anisotropy to the overall magnetic behaviour of materials depends upon many factors. Four sources of the anisotropy energy (most influential for nanoparticles) will be highlighted in this review – magnetocrystalline anisotropy, shape anisotropy, surface/interface anisotropy, and colloidal anisotropy. Magnetocrystalline anisotropy occurs when the crystal structure dictates a preferred orientation for the atomic magnetic moments relative to the crystallographic axes. Shape anisotropy occurs when the shape of the magnetic material (e.g. an ellipsoid, rod, or cube) dictates a preferred orientation of the magnetic moment with respect to the major/minor axes of the sample, due to energy minimisation of magnetostatic fields produced by the material. Surface/interface anisotropy occurs due to simple structural asymmetries such as those between the crystal and the vacuum at the surface or due to compositional changes at crystal boundaries/interfaces. Finally, colloidal anisotropy arises from interactions among magnetic nanoparticles suspended in a liquid (i.e. magnetic colloid) that enables translational and rotational degrees of freedom permitting formation of higher order structures such as chains. Interactions of the moments comprising nanoparticles in a chain create a preferred orientation of the magnetic moments of the particles along the chain [27]. Additional contributions may arise from stress or strain in the crystal structure, and other factors.

As mentioned when describing the different types of magnetic materials, another important property of a material is the exchange interaction that originates from overlap of the electronic (quantum mechanical) wavefunctions, and determines the magnitude and sign of the interaction between adjacent spins. Competition between exchange energy and magnetocrystalline anisotropy in combination with the energy from the applied magnetic field (called the Zeeman energy) and the energy from the net magnetic field produced by the material (magnetostatic field) determines the length scale of the collective parallel orientation of atomic magnetic moments. When the energy required to maintain this parallel orientation exceeds the available energy from the applied magnetic field, a domain wall [28] or boundary (e.g. Bloch walls or Néel walls) will form producing separate magnetic domains. Each domain has a collective parallel orientation of atomic magnetic moments within the domain, but different domains have different orientations of their collective parallel atomic magnetic moments. The domain wall or boundary represents the transition of atomic magnetic moments between the two domains. The formation of domain boundaries requires energy, but the formation of domains can reduce the total energy of the material.

Whether the material will form multiple domains or one single domain depends upon the specific nature of the material – its chemical and structural composition, size, and temperature, for example, and on the magnitude of the applied field. Domain walls can spontaneously form in materials having a size that exceeds the characteristic length scale for domain walls for that material. However, when the size of a magnetic material is reduced below a critical value, a unique property emerges as a consequence of this energy competition. Below this critical size regime, the exchange energy for domain formation becomes prohibitively expensive, leading to single domain magnetic materials. Therefore, even in zero applied magnetic field, such a magnetic material is a single domain magnet, wherein all the atomic magnetic moments in the material are aligned. For many magnetic materials, the dimensional limit for domain walls is in the range of about 2 nm to about 800 nm; for magnetite (Fe₃O₄, one of the two ‘magnetic’ forms of iron oxide), it is in the order of 50 nm [29]. Thus, by virtue of their size (i.e. 2–100 nm), magnetic nanoparticles are often single domain magnets. As the size increases beyond this lower limit, a pseudo-single domain structure is observed in which two large domains may be present. These domains are able to easily reverse in small applied magnetic fields to form a single domain structure. These domains can reverse by a wide variety of methods [29–31], such as coherent rotation in which all the moments remain parallel while rotating, curling, buckling, or domain nucleation and growth. (As will be discussed later, these reversal mechanisms play a major role in determining hysteresis behaviour in all magnetic systems.)

These reversal mechanisms are influenced by many physical properties of the material such as crystallinity, presence of crystal defects (number and type), strain, and surface/interface effects. The surface/interface effects in particular can have a significant influence on magnetic nanoparticles, because decreasing grain size is accompanied by a corresponding increase in the fraction of atoms comprising the grains that lie on the surface and interface regions. The latter is closely related to the detailed chemical nature of surface and grain boundaries influencing the intrinsic properties (i.e. K, H_SW) as well as the extrinsic properties (i.e. H_C, reversal mechanisms). In particular, the total anisotropy energy may increase with decreasing grain size, within a defined size range, because of the increased contribution of surface anisotropy. This, and other enhanced magnetic properties, make nanoscale magnetic materials interesting for study and particularly well-suited for a variety of applications.

Interaction of magnets with time-varying fields: Hysteresis and heating

Magnetic materials possess atomic magnetic moments (m) or dipoles that have both magnitude and direction. The magnitude and direction of the atomic magnetic moments is affected, in turn, by other properties that depend upon the chemical and crystalline composition of the material, such as atomic composition, shape, and volume. When exposed to an external magnetic field, the atomic magnetic moments of the material are forced to align with the external field. The degree of alignment depends upon the magnitude of the external field and the material properties. As the external field polarity changes, the direction of m will also change, thereby changing the measured magnitude of the net magnetic moment along a fixed measurement direction. As the frequency of the changing external field increases, the material's magnetic moment may, at some frequency, cease to change ‘instantaneously’ with a change in magnetic field, resulting in a lag in the magnetic response. If the external alternating magnetic field has sufficient amplitude to force m to oscillate, open hysteresis loops are the result of this lag. By definition, magnetic hysteresis is the property of a material in which the magnetisation vector of the sample fails to retrace a single trajectory when exposed to an oscillating magnetic field. Instead, the magnetisation traces a ‘loop’. The mechanism leading to hysteresis differs for magnets possessing multiple or single domains. When a material possessing multiple domains is exposed to an oscillating or time-varying magnetic field, hysteresis of the magnetisation of the sample occurs, which arises from domain wall motion, or reorganisation, For a single-domain magnet, however, magnetisation reversal can also occur via coherent rotation of all the atomic magnetic moments within the sample. When the anisotropy energy of the sample is high relative to the frequency and amplitude of the external applied field, this reversal is inhibited, resulting in hysteresis. Hysteresis for both multi- and single-domain magnets is accompanied by losses, which manifest themselves as heat. The enhanced anisotropy present in magnetic nanoparticles contributes to increased hysteretic losses of these materials when they are subjected to AMFs, leading to much higher specific loss power (SLP), or heat. Indeed, nanoscale magnetic materials can generate significantly higher SLP than their bulk counterparts because the surface anisotropy contributes significantly to the total anisotropy energy [32–37].

Heat generated by hysteresis losses arises as a consequence of the first law of thermodynamics for magnetic systems, because the total change of internal energy results from a combination of both heat (Q) and work (W), i.e. ΔU = Q + W. The work necessary to magnetise a unit volume of a magnetic material is understood [20] to be as in Equation 2 where H is the applied magnetic field and M is the magnetisation. The magnetic flux density B is related to M and H by: B = µ₀(H + M). This work is related to the energy that must be absorbed by the system to align the atomic magnetic moments with the applied field, and to overcome any interactions (e.g. exchange, anisotropy, dipolar) of the atomic magnetic moment with other atomic magnetic moments within the material. Because energy is conserved, one traversal of the hysteresis (M versus H) loop must return to the same total energy, i.e. ΔU = 0 and thus Q = W. In other words, when the magnetic field changes from any field H1 to any field H2 causing a change in the magnetisation along a specific path, does the magnetisation change via that same path if the field is reversed (i.e. from H2 back to H1)? If so, there is no net work. If not, then there is net work performed. For example (see Figure 1A), in going from +H_K to –H_K and then back to +H_K, does the magnetisation take the same path? The answer is no, so net work is performed. Any net work performed is converted into heat (W_heat). This can be written as Equation 3.

Therefore, W_heat is equal to the area enclosed by the hysteresis loop (see Figure 1A). There are two types of hysteresis loop:

1. A major loop that begins at magnetic saturation (where all the dipoles are aligned parallel to the applied magnetic field). It goes to negative saturation and back.

2. A minor loop, in which the maximum field applied is less than that required to achieve magnetic saturation.

The maximum heat generated by any magnetic material, including nanoparticle systems, occurs by tracing the major loop (height = 2(M_S) and width = 2(μ₀H_k)), and is:

Using the equation for the anisotropy field (Equation 1), this can be simplified to = 8 K. For the case of hyperthermia, the heat originates from the area enclosed by the minor loop because the accessible magnetic field amplitude and frequency are limited by practical considerations of device design for safety and economy. Biological constraints limit the available choice of nanoparticle material and size for biocompatibility. Field conditions are limited by considerations of (patient) safety to avoid over-exposure and non-specific heating by eddy currents and to avoid nerve stimulation [22,38,39]. Power generation requirements, cost of equipment, and efficiency limit choices for accessible frequency and amplitude conditions.

The specific loss power (SLP) is the (sample) mass normalised measured power loss reported as W/mass sample. Unfortunately, an ab initio calculation of the work performed, without a measurement of the hysteresis loops, is non-trivial and often impractical [40]. Except in a few simple cases no analytical model exists. Numerical simulations are possible, and commonly performed by researchers of micromagnetic systems, with software packages such as the Object Oriented MicroMagnetic Framework (OOMMF) [41], providing an alternative method to calculate the highly non-linear M–H (or B–H) loops. Therefore, modelling efforts have been focused on methods to approximate the work performed over a full hysteresis loop. These will be detailed below, including the assumptions underlying the theories and their efficacy.

For the purposes of this review, Joule heating of magnetic nanoparticles by eddy currents is ignored. Biocompatible magnetic materials are typically oxides, having high electrical resistivity. This, and their size (diameter << 1 µm) limit eddy currents and thus Joule heating effects to negligible values [42].

Superparamagnetism

Single-domain magnets exhibit many properties observed in their bulk (i.e. macroscopic or multi-domain) counterparts. However, superparamagnetism is a unique feature of single-domain magnets, but not all single-domain magnets are necessarily superparamagnetic. Indeed, it was the observation of hysteresis in single-domain magnets that confounded interpretation for some time, leading to the development of time-dependent relaxation theory by Néel [30]. Despite this, there exists a prevailing notion in the magnetic nanoparticle hyperthermia literature that single-domain magnets are superparamagnetic [43]. This is inconsistent with the body of literature describing single-domain magnets [29,44].

Considering that superparamagnetism has dominated the nanoparticle hyperthermia literature, it is useful to begin with this special case in calculating the heat generated. In addition, analytical solutions exist to describe superparamagnetism contributing, perhaps, to the prevalence of this special case in the magnetic hyperthermia literature. By definition, superparamagnetic particles have zero coercivity and zero remanent magnetisation. Thus, there exists no net magnetostatic field from the nanoparticle and they do not interact with each other. They do, however, possess larger magnetic moments than paramagnets, hence the ‘super’, so their saturation magnetisation can be quite substantial. Agglomeration due to magnetic interactions does not occur, enabling ready stabilisation in solution. For this reason, superparamagnetic particles are often considered to be desirable for biomedical applications, making this system an excellent test case [43]. The analytical form [45] for the total moment (m_T) as a function of applied magnetic field in the classical case is also known as the Langevin function, and is given by Equation 5: where m is the moment per particle, N is the number of particles having magnetic moment m, T is the temperature, k_B is the Boltzmann constant. The analytical form [46] for the quantum mechanical description of superparamagnetism in Equation 6 is similar. where g is the spectroscopic splitting factor, and J is the total angular moment. For the special case where there is no orbital contribution to the moment (L = 0), so J = S where S is the spin angular moment, and S = 1/2, this reduces to Equation 7: since g = 2 when J = 1/2 and L = 0. In both cases, when the magnetic field is zero, the magnetisation is zero; hence there is no remanent magnetisation and the coercive field is zero. A plot of M versus H reveals that the magnetisation traverses the same path as the magnetic field is oscillated between ±H_max, the maximum magnetic field magnitude. Therefore and no heat is generated from the work performed by the magnetic field on a superparamagnetic nanoparticle system. Thus for MNH, in the thermal sense, superparamagnetism is undesirable.

A critical feature of superparamagnetic single-domain magnets is the characteristic timescale [47] for spontaneous, or thermally driven moment reversal. If this characteristic reversal time is shorter than the measurement time, then the magnetic moment will spontaneously reverse during the measurement, resulting in an average value for the magnetic moment of zero. If, on the other hand this characteristic reversal time is longer than the measurement time, then the magnetic moment appears stable. Typical observations of ‘zero remanent moment’, linear, or an ‘S’ shaped hysteresis loop lead to a common designation of the sample as a ‘superparamagnetic’ system which is inferred from measurements of magnetisation reversal under experimental conditions that access long times. Thus, this inference is only approximately true for measurements conducted using static or slowly oscillating external fields (∼10 Hz). At frequencies relevant for hyperthermia (>100 kHz), the measurement time is now 1,000 to 10,000 times shorter, and may thus be short compared to the characteristic timescale of the system. For such systems, the characteristic relaxation time is of the right order of magnitude to produce (apparently) stable magnetisation when measured at higher frequencies where the nanoparticle system is no longer superparamagnetic. Thus, both coercivity and remanent magnetisation are observed (Figure 1B). This has been demonstrated [48] experimentally, and can explain observed ‘heating by superparamagnetic nanoparticles’.

An additional confounding feature of single-crystal magnetic systems arises from the long-range nature of magnetic interactions. When single crystal (nanometer scale) magnets aggregate into clusters, magnetic moment reversal may be hindered. The reversal of an individual crystal or grain is no longer free to occur independently of the other moments present in the aggregate. Instead, individual moments comprising the aggregate move cooperatively, leading to increased relaxation time and thus hysteresis. Such ‘cooperative’ behaviour of magnetic nanoparticle systems has been well-described [29]; however, satisfactory models describing this behaviour have not been developed and its implications for hyperthermia have not been fully explored, particularly with regard to observations of ‘superparamagnetic’ behaviour. This is discussed further below.

Linear response theory

The first attempt to approximate the heat produced by an alternating magnetic field acting on a magnetic nanoparticle system was introduced by Rosensweig [49]. In this analysis, he considered the dissipation relationships based solely on the rotational relaxation of non-interacting single domain particles. A linear response of the magnetisation to the applied magnetic field is implicitly assumed, leading to Equation 9: where χ is the susceptibility (χ = M/H) that is assumed to be constant through the considered magnetic field amplitude. Hence, this approach is commonly referred to as linear response theory. This response profile implies very low applied magnetic fields relative to the magnetocrystalline anisotropy of the magnetic nanoparticle and their relationship to the frequency (f) of oscillation of the applied magnetic field. With these assumptions, the power dissipation (P) is where H_max is the amplitude of the oscillating magnetic field. A significant amount of research [50] has analysed magnetic nanoparticle systems using this equation as the beginning point. It is from this analysis that a scale factor of H²f has been proposed [24] to provide a basis for comparing nanoparticle heating data measured using a variety of applied magnetic field and frequency combinations [25]. When examining heating data obtained from magnetic nanoparticle systems over a wide range of amplitude at fixed frequency, it becomes apparent that this scaling relationship may be inadequate [51] (Figure 2). This is particularly evident for iron oxide nanoparticles dispersed in a dextran matrix (e.g. Ferridex or Nanomag-D-spio). The magnetic nanoparticles comprising these systems are very weakly interacting, thus they are often labelled superparamagnetic. In magnetic particle imaging, for which Ferridex is an example [52], it is apparent that all nanoparticle systems have highly non-linear hysteresis loops when characterised over amplitudes that achieve or exceed M_S. This non-linearity [53] violates the basic assumption of linear response theory. While the Rosensweig analysis provides an excellent basis to begin modelling heat dissipation from magnetic nanoparticles arising from multiple potential mechanisms (Néel or hysteresis losses and rotational or friction losses), it is critical to understand its limitations, which are detailed in Rosenweig’s report [49]. Another detailed discussion of these limitations can be found in Carrey et al. [54].

Figure 2. (A) SLP generated by four different magnetic nanoparticle systems dispersed in water as a function of maximum magnetic field at a fixed frequency of 150 kHz. Highlighted area in (A) is shown in (B) as a close-up of the SLP in low field amplitudes. (Figure courtesy of Anilchandra Attaluri.).

Thermal relaxation

Magnetic colloids possess many degrees of freedom (translational and rotational); therefore, thermal energy randomises the orientation of individual moments (i.e. via Brownian motion [55,56]). Brownian motion of colloids introduces additional complexity to understanding the time-dependent behaviour of single-domain magnets and their response to external time-varying fields. Attempts to develop models from theoretical analysis that incorporate thermal relaxation mechanisms in single-domain magnetic nanoparticles, given the relative magnetic and thermal energies available, are described in Coffey and Kalmykov [44]. In this comprehensive review, models are described that include explicit analytical calculations of the anisotropy constants for specific cases. Rosensweig’s analysis includes thermal relaxation in linear response theory by calculating a relaxation time constant (τ) determined by two different thermal relaxation mechanisms: Brownian relaxation (τ_B) in which the nanoparticle and the magnetic moment rotate together, and Néel relaxation (τ_N) in which the magnetic moment rotates with respect to the crystal. The functional form for Brownian relaxation is shown in Equation 11: where η is the viscosity of the fluid and V_H is the hydrodynamic volume [49]. For a 100-nm diameter particle suspended in water, Brownian relaxation occurs with a time of ∼10⁻³ s, about 100 times longer than the time of field oscillation used for hyperthermia (∼10⁻⁵ s). The functional form for Néel relaxation is shown in Equation 12: where τ_A is the attempt frequency (generally taken to be 10⁻⁹ s, from ferromagnetic resonance data), and V_M is the magnetic volume. Note that the anisotropy constant K includes all contributions to the anisotropy. This functional form for Néel relaxation time only applies for the special case of null magnetic field. In the presence of a non-zero magnetic field, the magnetic energy term (which is only the anisotropy energy KV_M in zero field) must account for the Zeeman energy arising from the applied magnetic field. The expression for the Néel relaxation time then becomes Equation 13: where θ is the angle between the magnetisation and the applied magnetic field. It is important to note that Brownian relaxation is not affected by the magnetic properties or the applied field (amplitude or frequency). The effective time constant τ is therefore with the implicit assumption that only coherent rotation of the magnetisation is possible. It is also explicitly assumed that the system comprises identical and non-interacting nanoparticles. Magnetic interactions among particles will change the anisotropy (see above) and require an additional dipolar interaction [57] term to the magnetic energy. These interactions can modify the time constant [58] by multiple orders of magnitude (i.e. from 10⁻⁹ to 10⁵s). The expression for power dissipation in the modified form of the (restricted case) linear response theory that includes thermal relaxation then becomes Equation 15.

As mentioned, thermal relaxation serves primarily to disorder an ordered system. This can reduce the hysteresis losses by spontaneously changing the direction of the magnetisation independent of field, or increase the hysteresis losses by spontaneously rotating the nanoparticle, thereby changing the direction of the magnetisation so that it exhibits hysteresis. The latter is typically negligible for most operating conditions.

It is possible, however, to observe small oscillations on the same frequency as the field, with a slow drift towards equilibrium. A stable colloid in equilibrium will display [59] the strongest increase in the hysteresis losses that arise from these spontaneous particle rotations. This mechanism also provides a lower bound to the usable frequency because disorder occurring on the same timescale as the measurement will result in an effective magnetisation of zero reducing the work done by the system to zero. This lower bound on the frequency (ω = 2πf) is given by the higher of ωτ_N = 1 and ωτ_B = 1, where ωτ_N = 1 marks the transition between magnetically fluctuating and magnetically stable nanoparticle systems. Similarly, ωτ_B = 1 marks the transition point between physically fluctuating and physically stable nanoparticle systems. Therefore, the physically fluctuating nanoparticles require a ‘torque’ to perturb significantly the randomising Brownian motion. The torque that results from dynamic effects must be explicitly taken into account to properly evaluate the fluid dynamics of a magnetic colloid that is subjected to an AMF.

Anisotropy

Another method proposed [53,54,60] to calculate the dissipated power begins with the analytical form for the energy of a magnetic nanoparticle where K_U is the uniaxial anisotropy energy density, θ is the angle between the magnetisation and the crystalline easy axis, and φ is the angle between the easy axis and the magnetic field. If μ₀H << 2 K/M_S = μ₀H_k for randomly oriented easy axes, then an analytical expression for the energy barrier ΔE is shown in Equation 17.

The switching probability is now determined by the relationship between the measurement time and the relaxation time, as given by the Néel-Brown model for magnetisation reversal [61]: where τ_meas is the measurement time. This is a generic form for Equation 12. Assuming that ΔE > k_BT, the power loss is given by Equation 19.

This differs significantly from the power loss equation obtained from linear response theory, even after incorporating the Néel relaxation time. Calculated values plotted as a function of field show [60] the same curvature and transition to saturation that is observed experimentally in many nanoparticle systems [51] (see Figure 2). Other single domain models that include anisotropy barriers include the Stoner-Wohlfarth model [54] for single domain particles, which predicts a sudden onset of heating not observed experimentally, and the two-state model. Other models have been developed and are described in detail in other reviews [54].

As stated previously, magnetic anisotropy is the energy barrier that must be overcome to reverse the magnetisation. This means that if the applied field does not exceed the anisotropy field, then the magnetisation will not reverse, resulting in a straight line for the hysteresis loop, rather than an open loop, so no heat is generated. So, for a given nanoparticle system, there is a minimum applied field for applications (H_appl > H_K). However, there is also a maximum applied frequency associated with this energy barrier. If the frequency is too high, then there is insufficient time to allow for enough attempts to cross the energy barrier to actually result in a reversal. If there is no reversal, again, no open hysteresis loop results, and no heat is generated. (This was shown numerically in Figure 3.) As described previously, this transition at ωτ_N = 1 is between magnetically fluctuating and magnetically stable nanoparticle systems. Therefore, the ratio of the magnetic energy (just KV in zero field or with the inclusion of either a magnetostatic or interaction term) to the thermal energy (k_BT) will define [62] the low barrier versus the high barrier regime. The optimal applied magnetic field will then be a very low field in the low barrier regime and a very high field in the high barrier regime, allowing for tuning of the magnetic nanoparticle properties for the specific application (e.g. whole body versus local exposure). As a result, the anisotropy becomes a critical parameter to control, and there are multiple ways to control it, as will be discussed further below.

Figure 3. Minor loops for randomly oriented nanoparticles with fixed magnetic field amplitude (10 Oe) and fixed frequency as a function of magnetic anisotropy energy density in ergs/cm³, where K = (A) 1 × 10⁵, (B) 2 × 10⁵, (C) 3 × 10⁵, (D) 4 × 10⁵, (E) 5 × 10⁵, (F) 6 × 10⁵, (G) 7 × 10⁵, (H) 8 × 10⁵, and (I) 9 × 10⁵. (Reprinted with permission from Sohn and Victora [60].).

Nanoparticles that possess a magnetic core comprising aggregated crystals will exhibit magnetic properties, namely magnetic anisotropy (K) that may be enhanced considerably [63–66]. A colloidal suspension of these nanoparticles will thus display complex colloidal features that depend upon the available thermal energy (k_BT), magnitude of an external field, and properties of the solvent (e.g. viscosity, ionic strength, surfactants, polymer) [63,64,65]. Recent attempts to synthesise particles using this knowledge in order to deliberately modify the heating properties of magnetic colloids for MNH present a promising direction for clinical applications [67–69,70]. Often, the synthesis of such multi-crystalline core particles produces high magnetic anisotropies that, while they provide significant heating (Figure 2, compare BNF particles), will also result in high coercivity [65]. High coercivity in turn manifests as heat production that occurs only with high amplitude at a given frequency, potentially limiting clinical application.

The realisation that additional control, or modulation, of the magnetic anisotropy of multi-crystalline core nanoparticles is needed to reduce magnetic anisotropy to enable significant heating at low amplitude fields has led to further innovation in synthesis. Hydrothermal ageing of nanoparticles post-synthesis has yielded nanoparticles that offer promise for clinical application (Figure 2, compare. JHU particles) [71]. Such techniques are emerging as a new direction for magnetic nanoparticle synthesis for MNH applications.

Collective magnetic behaviour

An understanding of the complex interaction between anisotropy, intra- and inter-particle dipole–dipole interactions, and AMF parameters is needed to realise the potential offered by MNH. Magnetic forces are long-range in nature, thus exerting their effects over multiple length scales. Magnetic colloids can exhibit rich and diverse behaviour, arising from interactions of these forces that, if poorly understood, can lead to confounding interpretations of data. Since magnetic forces exert their influence over multiple length scales, they modulate the time-dependent relaxations of individual magnetic moments or domains when the separation between individual single-domain magnets is reduced to distances allowing interactions of these moments [72]. Modulating the time-dependent relaxation will therefore modulate the heating properties of magnetic colloids, so interparticle interactions are an important parameter. For example, long stabilising ligands on small magnetic nanoparticles could cause the nanoparticles to be far enough apart that their interactions are negligible. This results in the properties of a single particle being the controlling parameter, like magnetocrystalline anisotropy. For example, iron oxide nanocrystals dispersed in a diamagnetic matrix (e.g. Ferridex or Nanomag-D-spio) do not usually display cooperative behaviour, because the individual magnetic nanocrystals within each nanoparticle are stabilised by the polymer matrix and remain far enough apart that their interactions are weak. In contrast, for shorter stabilising ligands, or for larger cores, the magnetic nanoparticles as they move around ‘see’ their neighbours. This results in the formation of more complex structures, like rings and chains. This effect can be enhanced when the applied field is large enough to drive in the close formation of magnetic nanoparticles with some neighbours (again into rings or chains). Now, the properties of the collective (like colloidal anisotropy) are the controlling parameter, not the properties of the individual.

However, biology also plays a role, and leads to difficulties for clinical translation. For example, the SLP of magnetic nanoparticles is typically measured in the nanoparticles’ native aqueous (biocompatible) suspension. The experimental conditions, i.e. solvent viscosity and temperature, generally enable nanoparticle rearrangements that minimise the total free energy of the colloid when it is subjected to an external field. As described above, the magnetic field enhances the formation of higher-order structures such as chains that can significantly alter the magnetic anisotropy through introduction of a colloidal anisotropy and therefore the heating properties [27,57,63–65,70,73]. When introduced into tissues or cells, however, the available degrees of freedom become severely restricted because the nanoparticles interact with proteins, are internalised by cells, or they become immobilised in the tissue microenvironment [65,68,74]. The effects of confinement and immobilisation reduce the available degrees of freedom, thereby modifying the colloid properties (and interparticle magnetic interactions), affecting the heating rate. For example, the stripping of stabilisation ligands upon endocytosis significantly changes the Brownian relaxation. If this is followed by packing of the nanoparticles into endosomes, then the lack of a stabilising ligand means aggregation [65]. The system then becomes more like a loosely packed transformer core of iron oxide, rather than a colloid. Heating rates that change when particles are immobilised in tissues as compared against the measured value in suspension present a challenge to clinical translation. Further, such behaviour has the potential to modulate the heating properties variably in different biological settings. Whether this occurs and its potential impact on heating properties has been inadequately addressed in research efforts, presenting a direction of study that has significant clinical relevance.

Incoherent rotation and multi-domain states

The models described in the preceding sections assume coherent rotation of the magnetisation, which, unfortunately, is often not the case. As stated earlier, the potential of a single magnetic domain grain (crystal) to generate heat via hysteresis losses when exposed to an alternating magnetic field is determined by the total anisotropy energy, KV. This anisotropy energy presents an energy barrier, E_B, to changes in orientation of the magnetic moment. Thus, the stability of the magnetic moment with respect to time increases with increasing values of E_B. The amount of heat realised through hysteresis losses of a single domain grain when exposed to an alternating magnetic field is the result of a combination of both the intrinsic properties of the grain and experimental conditions, namely frequency of field and temperature. Experimental temperature determines the relative difference between E_B and the thermal energy available to the system, thus setting an experimental relaxation time, or τ. Thus, the relationship of the period of oscillation of the AMF, 1/f, with τ becomes a critical experimentally observable quantity that leads directly to the amount of heat generated through hysteresis losses. For 1/f ≫τ, the available thermal energy (k_BT) is sufficient to overcome E_B, enabling the moment to spontaneously rotate, via non-coherent rotation, without exhibiting hysteresis losses, i.e. no heat will be generated. Conversely, if 1/f ≪ τ, the moment is stable with time and will rotate coherently with an applied field.

Anisotropy energy, or potential hysteretic loss, in a single domain grain is proportional, in first approximation, to the volume of the grain. Therefore, for large single magnetic domain grains, the anisotropy energy may be so high that the energy barrier for magnetisation reversal cannot be overcome by thermal energies for any reasonable experimental temperature. If the magnetic field is removed, the magnetic moment will retain the orientation imprinted by the magnetic field for a characteristic time. The time required for such an orientation change of the magnetic moment to occur after the field is removed is a relaxation time that is characteristic of the grain and is a consequence of both the anisotropy energy of the grain and k_BT. In the extreme case of intrinsically stable magnetic single domain grains, this time is greater than 10⁸s [75].

Conversely, as the volume of a grain decreases within the single domain regime, so does the anisotropy energy. Below a certain characteristic grain size, the anisotropy energy may become so low as to be comparable to or lower than k_BT for any reasonable experimental temperature. This implies that the energy barrier for magnetisation reversal may be overcome, and then the total magnetic moment of the grain can thermally fluctuate.

Between these two extremes lies a range of grain volumes for which the anisotropy energy is intermediate, and thus the timescale of magnetisation reversal depends explicitly upon the temperature and timescale of measurements. This seems to be generally the case for nanoparticle systems used in MNH, given that truly superparamagnetic particles exhibit no hysteresis at the clinically relevant experimental conditions (Figure 2, compare Ferridex particles). Theoretical models must incorporate incoherent rotation, given that this comprises some portion of magnetisation reversal in nanoparticle systems used for MNH.

The area enclosed by the hysteresis loop (especially minor loops) is very sensitive to the reversal processes. This is, in part, the basis underlying the first order reversal curves method [76] for analysing magnetic property distributions. Incoherent magnetisation reversal can be modelled [59,77] through the chain of spheres [78] model, which incorporates interactions between three identical spherical particles in a chain, and allows bending of the chain and non-symmetric tilt of the magnetisation off the chain axes. The energy of the nanoparticle system can be written as a sum of magnetostatic energy (first term in brackets), dipole interaction energy (second term in brackets), and (uniaxial) anisotropy energy (third term in brackets), as in Equation 20: where β is the angle between the moments and θ is the angle between the easy axis and magnetisation. To determine the full equations of motion, the magnetodynamic torque which originates from : and the viscous torque which originates from where F is the hydrodynamic drag acting on the particle from the solution it is dispersed in: where α is a shape parameter and is equal to 1 for a sphere, must be included. Integrating the torque with respect to the angle results in an energy loss term. Therefore, the energy dissipation from just the magnetodynamics (solid sample) and the energy dissipation from both the magnetodynamics and the hydrodynamics (liquid sample) can be calculated. The result yields curves that are very similar to those observed when the heating is measured as a function of applied magnetic field (Figure 4).

Figure 4. Heat (W, in dimensionless units) produced during one cycle as a function of field amplitude (p0, in dimensionless units where p = mH/U) in 1) liquid, and 2) solid dispersion. (Reprinted with permission from Kashevsky et al. [77].).

In addition, the dynamic coercivity (H_C,dynamic) of the oriented system in a longitudinal AMF can be estimated [79] to be where the applied frequency f = ω/2π. Thus, the dynamic coercivity increases as a function of volume, at fixed frequency. However, the dynamic coercivity also decays as a function of increasing particle volume at fixed frequency. This can be approximated [80] for oriented spherical particles with radius R and uniaxial anisotropy energy density K_U as Equation 24. where R_f is the radius of the spherical particle when ωτ_N = 1. The combination of the two results in narrow size range (if any) over which competition between domain formation/incoherent magnetisation reversal and thermal relaxation do not significantly reduce heating. Therefore, both size and anisotropy play an important role in determining heating.

Finally, pseudo-single domain [81] and multi-domain states may also be present. The pseudo-single domain state has been extensively discussed in the geology literature [82–89]. Multi-domain states add additional complexity through exchange interactions. Also, defects and impurities acting as pinning sites during the reversal process can contribute significantly to observed behaviour. Analytic solutions do not exist for these systems, and calculations require numerical methods. Additional work is needed to describe domain formation and pinning as well as incoherent reversal mechanisms and their effect on heating.

Polydispersity

Typical magnetic colloids used for hyperthermia exhibit heterogeneity of physical and magnetic properties, for example variations of particle size, shape, anisotropy. This heterogeneity introduces additional complexity, necessitating modifications to the treatment that incorporate ensembles of nanoparticles displaying a statistical distribution of each property. Typical values for small batch-volume synthesis methods (e.g. thermodecomposition, polyol) are ∼10%; commercial systems are typically higher at ∼30% with their larger batch quantities. (Herein, polydispersity is defined as the standard deviation of the diameter divided by the diameter multiplied by 100 to get a percentage.)

The effect of polydispersity on the heat generation, as well as on the magnetic properties in general, is an open question. Rosensweig [49], and Hergt and Dutz [50] claim that polydispersity in the size distribution of the magnetic nanoparticles will degrade the heat production of a magnetic nanoparticle system. Following the methodology of Chantrell [90], the effect of polydispersity can be determined by replacing the volume in any equation above with a lognormal (or Gaussian) distribution. This was done by Landi and Bakuzis [62] for linear response theory, who found that specific parameters dominate the heat generation for given field amplitude and frequency. (Other example modifications to models and analytical treatments to account for polydispersity have been described elsewhere [29].) If we consider individual magnetic nanoparticles, these results make sense. For given field and frequency, the most heat will be generated for a specific combination of parameters (e.g. size and anisotropy). Particles away from this optimal combination will generate less (or no) heat, thereby reducing the average heat production. However, strong interactions may mitigate this effect. In the latter case, once one nanoparticle reverses, the local field it produces may be large enough to switch its neighbours, resulting in a cascade effect. Therefore, while the majority of the magnetic nanoparticles are not optimised for the field amplitude and frequency, the particles that are optimised may drive the remaining particles to produce more heat than they would by themselves. In either case, the end result is that polydispersity will, of course, produce a distribution of the heat generated per particle corresponding to the distribution(s) in the relevant magnetic properties.

Numerical simulations

At present, the most reliable numerical solutions, and also the most time-consuming, are micromagnetic simulations [60,62,70,91]. They avoid the inherent limitations of the described analytical models, but require precise knowledge of the material parameters such as exchange energy and magnetocrystalline anisotropy. Micromagnetic calculations are based on the Landau-Lifshitz-Gilbert (LLG) equation [41] using a discretised mesh, the size of which is critical for accurate results. The stochastic LLG equation is shown in Equation 25 [60]. where γ is the gyromagnetic ratio, α is the damping constant, H_eff is the effective field (including the applied field and the anisotropy field), and H_th is the thermal fluctuation field. This method naturally incorporates interactions between nanoparticles, which have been demonstrated [65,92] to be critical to efficacy. Unfortunately, current micromagnetic algorithms do not explicitly include temperature, but efforts are underway to address this. An example [60] of the utility of micromagnetic calculations used to predict hysteresis losses contradicts the results obtained from linear response theory. In particular, micromagnetic simulations demonstrate that for fixed frequency and field amplitude hysteresis losses increase from zero to a maximum, and then decrease to zero with increasing magnetic anisotropy. For low anisotropy, a small value of applied field is sufficient to increase the magnetisation, which follows the magnetic field, resulting in a closed loop with zero coercivity and zero remanent magnetisation. As the anisotropy increases, however, the magnetisation lags, opening the loop and yielding hysteresis losses. As the anisotropy increases further, the field is insufficient to overcome the anisotropy energy barrier, and the magnetisation remains fixed, resulting in a line with a slope of zero (Figure 3). This behaviour, with heat production beginning at zero (low anisotropy), rising to a maximum (intermediate anisotropy), and then decreasing back to zero (high anisotropy), can be translated to either variable field or variable frequency. The predicted results have been experimentally observed and demonstrate that an optimal combination of anisotropy and interactions, field amplitude, and frequency are preferred [48,93] (Figure 1B). This is in direct contradiction to linear response theory; considering the case of variable frequency, the heat generation should increase as f² at low frequencies and saturating to a constant value at high frequency.

Conclusion

From a review of the available models, it becomes clear that the fundamental parameter [93] controlling nanoparticle heating is the magnetic anisotropy, regardless of origin (e.g. magnetocrystalline, shape, interactions). A rule of thumb to determine a useful value for the anisotropy is its ratio to the magnetic field amplitude. As the ratio decreases below 1, the area enclosed by the M–H loop decreases rapidly, resulting in negligible heat output. Given that magnetic nanoparticles often display unusually large magnetic anisotropy (K ∼ 13−36 × 10⁴J/m³ or H_k ∼ 0.14 T) [94–96], likely resulting from magnetic interactions [73], it is not a surprise that low heating is typically observed, due to the use of low magnetic field amplitudes, which are typically of the order of 0.01 T. Therefore, control of the magnetic anisotropy through interactions and crystallinity is a critical parameter for the development of magnetic nanoparticle systems for hyperthermia optimised for specific magnetic field amplitudes. For example, low field applications require [70] a reduced anisotropy.

In addition, the interplay between the magnetic nanoparticle size and anisotropy determines the optimal frequency at which a particular magnetic nanoparticle system can be useful. For a given frequency, increasing the anisotropy while fixing the size increases the heat generated, at least until the anisotropy field exceeds the maximum field applied. Interactions between nanoparticles, especially chain formation, create a transient ‘shape’ anisotropy or colloidal anisotropy that provides an increased anisotropy within specific experimental conditions. This however, can be potentially problematic because increasing anisotropy demands increased field amplitude to realise measurable heating. The effect of polydispersity remains an unresolved question, especially because all synthetic particle systems possess polydispersity. A general statement of the effect of polydispersity is that the larger particles dominate the power generation, so long as their anisotropy remains smaller than the applied magnetic field amplitude.

Finally, other suggestions have been proposed to increase the heat generation. Fundamentally, all of these serve to impact the total energy. Examples include 1) texturing [50] which will reduce the magnetostatic contribution since the particles' easy axes are already aligned with the field direction, 2) hydrothermal ageing [71], 3) applying a static magnetic field perpendicular [60] to the alternating magnetic field direction, which will affect the anisotropy energy directly, or 4) immobilising the magnetic nanoparticles [97] which removes viscous torques that only serve to misalign the easy axis of the nanoparticles from the magnetic field direction.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.