Characterization of Magnetic Hyperthermia in Magnetic Nanoparticles

Abstract

Certain magnetic nanoparticles are able to generate heat through magnetic moment reversal processes under the action of an adequate alternating magnetic field. This ability, together with biocompatibility and nanosize of the particles, makes them promising materials for biomedical applications. Among the potential applications is magnetic hyperthermia, an oncological therapy expected to battle malignant tumors with minimal side effects by using localized heating. The success of the therapy requires, among others, accurate quantification of the released heat leading to the prediction of the temperature increase in and around the treatment area. This chapter is devoted to the recent advances in the determination of this heating ability.

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1 Definition of the Topic

Certain magnetic nanoparticles are able to generate heat through magnetic moment reversal processes under the action of an adequate alternating magnetic field. This ability, together with biocompatibility and nanosize of the particles, makes them promising materials for biomedical applications. Among the potential applications is magnetic hyperthermia, an oncological therapy expected to battle malignant tumors with minimal side effects by using localized heating. The success of the therapy requires, among others, accurate quantification of the released heat leading to the prediction of the temperature increase in and around the treatment area. This chapter is devoted to the recent advances in the determination of this heating ability.

2 Overview

The heating ability of magnetic nanoparticles is quantified by means of the Specific Absorption Rate (SAR) , also referred to as Specific Loss Power (SLP) , which accounts for the heat power released per unit mass of magnetic material. While magnetic hyperthermia is nowadays an approved anticancer therapy, there exist neither standardized experimental setups nor protocols to determine the heating ability of magnetic nanoparticles under the action of alternating magnetic fields. A wide variety of special-purpose mostly noncommercial setups are currently used to determine SAR either by magnetic or calorimetric approaches. These methods and setups are revised in this chapter, together with their inaccuracies and their possible minimization.

On the other hand, the SAR value of a magnetic nanoparticle increases with the amplitude, H ₀ , and frequency, f, of the applied alternating magnetic field, since more electromagnetic energy is invested in magnetic reversal processes. But these parameters are limited to human-tolerated values in magnetic hyperthermia. The prediction of SAR as a function of H ₀ and f in this range is challenging and strongly dependent on the nature, geometry, and size of the nanoparticles, which govern their magnetic state. In this chapter, both experimental and theoretical results on this subject are displayed and discussed. Also, particular examples highlighting the interest of measuring SAR as a function of the temperature are provided. Eventually, the unavoidable interparticle magnetic interactions, arising from the concentration and acquired arrangement of nanoparticles and responsible for the nonnegligible modification of SAR values, are tackled.

3 Introduction

Cancer is, by statistics from 2010, the second most common cause of death across the European Union and the United States, as reported by Eurostat [1] and the National Center for Health Statistics [2]. Classical anticancer therapies, i.e., surgery, chemo- and radiotherapy, face harmful side effects due to the lack of selectivity of the cytotoxic agents or cannot be applied upon some tumor locations and conditions. Therefore, major efforts are devoted to boosting the effectiveness of these therapies or to developing new ones in order to increase the survival rate and improve the quality of life of cancer patients under treatment.

Clinical hyperthermia is currently the fourth most important therapeutic approach for cancer treatment. It is based on the mild heating (between 42 °C and 50 °C) of the area under treatment for a desired amount of time, inducing a heat-shock response in the metabolism of the cells that depends on the thermal dose, the target tissue, and other physiological parameters [3]. A localized heating is desired in the case of small and/or well defined tumors. In particular, magnetic hyperthermia , also known as magnetic fluid hyperthermia , is a promising local hyperthermia therapy, in which magnetic nanoparticles (MNPs) are responsible for supplying heat to the tissue using electromagnetic to thermal energy conversion. If the MNPs are dispersed only within the tumor region, the heat generated will produce a localized temperature increase during the time of application of an externally generated alternating magnetic field, AMF .

Experimental investigations on magnetic hyperthermia started in 1957 with the work of Gilchrist et al. [4]. Although there has been ongoing research since then, the last 20 years have been the most productive for this research field, nourished by the optimization of MNP synthetic procedures, a better understanding of cell-level biology and development of more accurate characterization methods, high-quality AMF applicators, and realistic thermal simulations. In 2011, the European Union approved magnetic hyperthermia for treatment of glioblastoma multiforme after the promising results obtained in the phase II clinical trial carried out by the company MagForce (Berlin, Germany) [5], and up-to-date, magnetic hyperthermia is offered to the public as a complement to radiotherapy.

The NanoTherm® therapy is currently performed in several steps. First, the ferrofluid containing the MNPs is injected into the tumor. Afterwards, the acquired MNP distribution is studied by computed tomography. Using this information, the therapy is planned by means of special-purpose software that uses the so-called bioheat transfer equation to simulate the spatial and temporal temperature distribution in and around the tumor area. Once the adequate H ₀ value and duration are established, the therapy is finally applied. The achieved temperatures are monitored by thermometers previously installed using catheters.

From this process, it is easy to infer that the success of the therapy is partially based on the capability of simulating as precisely as possible the acquired maximum temperatures , in order to achieve a therapeutic effect where required, while preserving the healthy tissue. Simulations require quantification of several parameters: geometrical, physiological or thermal, both of the tissues and of the MNPs. In particular, accurate SAR values are crucial for feeding the simulations, as well as for tailoring the MNP characteristics for optimal heating ability. This heating ability depends on the AMF parameters [6] and on the individual characteristics of the MNPs [7–10], but also on the concentration and geometrical arrangement acquired by the MNPs in the tissue [11]. All these factors must be taken into account when evaluating SAR.

4 Experimental and Instrumental Methodology

At present, many research groups are involved worldwide in the characterization of magnetic nanoparticle hyperthermia. Two main approaches are used to measure SAR: magnetic detection or calorimetric determination , the latter being much more widespread. Although some specific-purpose setups are commercially available [12], most groups have chosen to build their own setups or to use other experimental facilities leading to the determination of SAR data such as commercial magnetometers. In this section, the currently used methods and setups for SAR determination are described.

4.1 Alternating Magnetic Field Generation

Regardless of the technique used for SAR measurement, an AMF generation means is required, since the magnetic hyperthermia phenomenon occurs when magnetic nanoparticles turn the AMF electromagnetic energy into heat.

The more used AMF source is the copper coil [12–15], with a variable number of turns (2 to 70). Also, the AMF can be created within the gap of ferrite-core electromagnets [16–19], as well as inside superconducting coils immersed in a proper cryogenic liquid [20, 21].

4.2 Magnetic Detection

One way of determining SAR is measuring the magnetic response of a sample subjected to an applied AMF. The SAR can be then calculated using theoretical expressions that relate such response with the released heat power. In particular, a MNP subjected to an external AMF takes electromagnetic energy from the field to reverse its magnetization , which in turn results in a release of heat [22]. This magnetic response can be determined by different types of magnetic measurements, based on the quantification, through different sensors, of the current induced in a gradiometric inductive coil due to the changes over time in the magnetic flux density produced by the sample [23].

4.2.1 Hysteresis Loops

When an AMF is applied to a ferro/ferrimagnetic material, the magnetization , M, vs. applied field, H, describes a hysteresis loop due to the nonlinearity and delay of M with respect to H. The area entrapped within the cycle accounts for the heat dissipated per cycle, and SAR can be calculated from quantification of this area as [24],

$$ \mathrm{S}\mathrm{A}\mathrm{R}=\frac{f{\mu}_0}{\rho_{MNP}}{\displaystyle \oint } M(H) d H $$

(8.1)

where μ ₀  = 4π × 10⁻⁷ T · m/A is the permeability of free space, ρ _MNP is the density of the magnetic material, and M and H are expressed in SI units (A/m). According to Eq. 8.1, SAR can be computed by calculating the value of the closed integral of the experimental M(H) curve and taking into account the AMF frequency.

Hysteresis loops can be obtained using commercial magnetometers such as, for example, vibrating sample magnetometers (VSM) [25] or superconducting quantum interference device (SQUID) magnetometers [26], as well as homemade gradiometric coils [27].

4.2.2 Ac Susceptibility

When a MNP is subjected to an AMF whose amplitude, H ₀ , is small enough as to fulfill the requirements of the linear response theory (LRT), then the value of its magnetization is linearly proportional to H ₀ . This means that its magnetic susceptibility , χ, depends on the AMF frequency, but not on H ₀ . Considering χ in its complex form, χ = χ′−iχ″, where χ′ and χ″ are the in-phase and out-of-phase components of the magnetic susceptibility, respectively, and assuming a sinusoidal AMF [24], Eq. 8.1 changes into,

$$ \mathrm{S}\mathrm{A}\mathrm{R}=\frac{\pi {\mu}_0}{\rho_{MNP}} f{H}_0^2\chi {\prime \prime} $$

(8.2)

which relates χ″ and the AMF parameters with the released heat power.

According to Ref. [28] and considering an isolated MNP, the LRT is only valid when,

$$ {H}_0<\frac{k_B T}{\mu_0{M}_S{V}_M} $$

(8.3)

where M _s and V _M are, respectively, the saturation magnetization and the magnetic volume of the MNP in SI units, k _B is the Boltzmann constant and T is the temperature. Equation 8.3 indicates that the LRT is only valid for small H ₀ values and that the maximum H ₀ value for LRT validity decreases with increasing V _M .

Some of the most widespread commercial devices for magnetic measurements may work as ac-susceptometers, which allow performing χ″ high-sensitivity measurements as a function of temperature, T, f, and H ₀ .

4.3 Calorimetric Detection

The other approach to determine SAR is measuring the temperature increase of a sample subjected to an AMF and calculating SAR afterwards, this time using thermal models that describe the thermal conditions in which the experiment takes place. In a general thermal model for SAR measurement, a sample of volume V, specific heat capacity c, thermal conductivity k, and density ρ contains a mass of magnetic material m _MNP . This sample is initially at the same temperature than its environment, T ₀ . At t = t ₀ the AMF is switched on and the MNPs generate heat, so that the sample can be considered a material with heat sources inside. The total released power P is SAR·m _MNP . If we assume that the heat sources (MNPs) are homogeneously distributed across the sample, then heat power per volume unit is P/V. The heat generated by the MNPs is diffused across the sample, and the heat flux arriving to the sample limits is continuously transferred to the sample environment (container, insulator, air…) by conduction, convection and radiation. These phenomena originate a spatial and temporal temperature distribution within the sample, governed by,

$$ \rho \cdot c\cdot \frac{\partial T\left(\overrightarrow{r}, t\right)}{\partial t}= k\cdot {\nabla}^2 T\left(\overrightarrow{r}, t\right)+\frac{P}{V} $$

(8.4)

The analytical solution of this differential equation in partial derivatives is complex, making difficult the establishment of a theoretical expression relating T and P.

This thermal model is appreciably simplified if, in an idealization of the system, the temperature gradients across the sample are neglected, so that the sample temperature is considered always homogeneous. This assumption gives reasonable estimations when the internal (inside the sample) thermal relaxation time is about ten times lower than the external (sample to its environment) one [29]. Within this approximation, the temporal evolution of the sample temperature can be inferred from the power balance between the sample and its environment. This balance obviously depends on the thermal conditions of the experimental setup.

4.3.1 Adiabatic Conditions

In ideal adiabatic conditions , no heat transfer occurs between the sample and its environment. To achieve such measuring conditions, the temperature of the sample environment must be continuously kept equal to that of the sample, which results in minimization of thermal losses and, as a consequence, in the investment of all the released heat in the sample heating. Given that thermal relaxation to the environment does not occur, it is reasonable to use power balances to describe the system is these conditions. This balance can be written as,

$$ P= C\cdotp \frac{dT(t)}{dt} $$

(8.5)

where C is the heat capacity of the sample. Assuming that P and C do not vary with T, which can be achieved using small heating intervals, the integration of Eq. 8.5 derives a simple analytical expression,

$$ \Delta T=\frac{P}{C}\Delta t\to \mathrm{S}\mathrm{A}\mathrm{R}=\frac{\Delta T}{\Delta t}\frac{C}{m_{MNP}} $$

(8.6)

relating linearly SAR and T, where ΔT is the temperature increment undergone by the sample upon application of a heating pulse of duration Δt. At present and to our knowledge, there is only one experimental setup capable of measuring SAR in adiabatic conditions [13]. A scheme of this setup is shown in Fig. 8.1a. The AMF is provided by a 30-turn coil placed outside the glass vessels (vacuum and liquid nitrogen), so that the eventual heating at high field amplitudes does not interfere in the adiabatic control. Adiabatic conditions are achieved by holding the sample (and its container) using poorly conductive means (e.g., thin threads) in the vacuum environment, to avoid conduction and convection. Sample and container are surrounded by a nonmetallic radiation shield with heating means (resistive-alloy thin film heater) whose temperature must be continuously controlled to the same temperature of the sample, so that thermal radiation losses to the shield cannot occur due to absence of temperature differences between them. Holding threads and temperature sensor wires (thermocouples) are thermalized in the radiation shield, to reduce thermal gradients.

Fig. 8.1

SAR determination in adiabatic conditions: (a) scheme of the setup described in Ref. [13]; (b) calculation of ΔT by the pulsed heating method

In real systems, achieving strict adiabatic conditions is impossible, and a very weak heat transfer between the sample and its environment is always present. To account for this interchange, the magnitude ΔT in Eq. 8.6 is calculated by the pulse heating method , used in traditional adiabatic calorimetry. Once in adiabatic conditions, the sample temperature is registered before, during, and after application of a heating pulse of duration Δt. An example of a T-vs.-t characteristic is shown in Fig. 8.1b, in which the linear drifts due to the presence of very small heat losses (or gains) are indicative of acceptable adiabatic conditions. Corrections of these small thermal losses are performed by linear fitting the temperature drifts in equilibrium and subtracting the extrapolations of both drift rates toward the midpoint of Δt, as indicated in Fig. 8.1b. This method then calculates SAR using stationary states of the sample.

4.3.2 Isoperibol Conditions

In isoperibol conditions , the temperature of the sample varies, but the temperature of its environment is always constant. Assuming that the internal (inside the sample) thermal relaxation time is about ten times lower than the external (sample to its environment), the power balance that can be used in this case is

$$ P= C\cdot \frac{dT(t)}{dt}+ L\cdot \left[ T(t)-{T}_0\right] $$

(8.7)

where L(W/K) is a coefficient accounting for losses. This balance indicates that the released heat is partly invested in the sample heating and partly transferred to the environment. Assuming that the losses between the sample and its environment are linear with T, (i.e., that L is constant) and that P and C do not vary with T, the integration of Eq. 8.7 derives an analytical expression that relates SAR and T by means of the thermal losses to the environment,

$$ T(t)={T}_0+\Delta {T}_{\max}\left[1-{e}^{- \frac{t}{\tau_i}}\right] $$

(8.8)

where ΔT _max = P/L, and τ _i  = C/L. In the stationary state, the heat generated and lost become equal, and the sample temperature remains constant at a value T _max = T ₀  + ΔT _max.

To simplify the calculations and avoid the use of L, SAR determination in isoperibol conditions is often performed using the initial-slope method , which is based on the additional assumption of that the heat losses are negligible during a certain time interval at the beginning of the heating process. The derivative of Eq. 8.8 at the onset of heating then is

$$ {\left.\frac{dT}{dt}\right|}_{t\to 0}=\frac{\Delta {T}_{\max }}{\tau}{\left.\left({e}^{-\frac{t}{\tau}}\right)\right|}_{t\to 0}=\frac{P}{C}\to {\left.\mathrm{SAR}=\frac{C}{m_{MNP}}\cdot \frac{dT}{dt}\right|}_{t\to 0}=\frac{C}{m_{MNP}}\cdot \beta $$

(8.9)

where the initial slope, β, is calculated within the time interval in which heat losses are negligible (see Fig. 8.2a). This method then calculates SAR in transient states of the sample temperature.

Fig. 8.2

SAR determination in isoperibol conditions: (a) T-vs-t characteristic and calculation of β; (b) scheme of a typical setup

Currently, the most part of the research groups working on MFH have adopted this initial-slope method [14, 16, 18–21, 30–61]. A scheme of one of the used setups is shown in Fig. 8.2b. In it, the sample is placed into a circular two-turn coil inside a test tube, and temperature variations are measured by means of a fiber optic. Similar setups have been built in several laboratories and some are also commercially available. Samples are placed in different containers: glass or plastic test tubes, plastic or glass vials, glass capillaries, centrifuge or microcentrifuge tubes, plastic capsules and also especial-purpose holders in polypropylene or Teflon, glass flasks with vacuum shield, etc. The volume of the container is either partially or fully occupied by the sample. The sample container is often partially or fully thermally linked with room air by different poorly heat-conducting means, such as vacuum vessels, 2-walled glass vessels with air flow, polymeric foams, asbestos, polypropylene, or Teflon. The temperature of the sample is registered by different contact or noncontact sensors, usually fiber optic probes but also thermistors, thermocouples, organic liquid thermometers, pyrometers, and IR cameras. T vs. t data within a certain initial time interval, which varies from several tens of seconds to several minutes, are used to calculate the initial slope. This can be done by differentiation of a fit (linear [37, 41, 62], polynomial [30, 46]) or by direct calculation of ΔT/Δt [18, 35, 40, 51]. Some authors use numerical differentiation, and given that the initial T-vs.-t trend is not strictly linear and the slope is nonconstant, they use maximum slopes [60, 63] or constant slopes [45] to determine SAR. In summary, there is a large variety of setups and measuring conditions.

4.3.3 In Vivo Conditions

When MNP heating takes place in vivo, it is not possible to neglect temperature gradients across the sample (the body tissue). Accordingly, temporal power balances cannot be used to describe the temperature evolution of the treated area.

The reached temperature does not only depend on the heat generated by the MNPs, but also on the geometry of the target region, the MNPs spatial distribution, the thermal properties of the tissues, and other factors such as blood convection. Equation 8.4 must be then adapted to biological systems, giving rise to the so-called bioheat transfer equation [64, 65].

The simplest useful equation deriving physically sound results for magnetic hyperthermia is known as the Pennes heat transfer equation [64]. It describes the spatial and temporal evolution of the temperature in biological systems neglecting blood convection and assuming that the heat transfer between blood and surrounding tissues mainly occurs in the capillary bed. It then considers heat losses just by conduction and blood perfusion. The Pennes bioheat equation reads,

$$ \uprho \mathrm{c}\frac{\partial \mathrm{T}}{\partial \mathrm{t}}=\nabla \cdotp \mathrm{k}\nabla \mathrm{T}-{W}_b{\rho}_b{c}_b\left(\mathrm{T}-{T}_b\right)+\frac{P_m}{V}+\frac{\mathrm{P}}{V} $$

(8.10)

where the magnitudes without subscripts are relative to the tissue, W _b , ρ _b , c _b , and T _b are the perfusion constant, mass density, specific heat, and temperature of the blood, respectively, and P _m is the metabolic heat power generation.

Pennes equation can be solved analytically for simple cases. However, numerical methods such as finite element, finite difference, or Monte Carlo methods are currently used [66] to solve this equation for realistic tumor geometries in practical applications.

At this point, it seems rather evident that in vivo conditions are not adequate to provide accurate SAR values. On the contrary, SAR values obtained with other accurate techniques should be used to feed the bioheat transfer equation (P = SAR · m _MNP ) and calculate the temperature distribution in and around the target region as function of the AMF characteristics and the exposure time. This would help in providing the right therapeutic temperature where required, while avoiding overheating and damaging of the surrounding healthy tissue [67].

5 Key Research Findings

5.1 Inaccuracies of Available Experimental Methods

One of the concerning consequences of the use of a large variety of setups and measuring conditions for SAR determination is the always presumed and sometimes demonstrated existence of important inaccuracies of the obtained data. In this section, the sources of uncertainties, as well as their possible minimization, will be discussed.

5.1.1 Alternating Magnetic Field Generation

The first source of data inaccuracy is the characteristics of the generated AMF . Three main requirements should be expected from a suitable AMF source:

• It must have a sample space with a constant and well-known H ₀ .

• It must minimize the thermal interchange with the sample.

• It must generate H ₀ and f values in the biological range of field application.

Given that the heating generated by MNPs under an AMF is strongly dependent on H ₀ , it is thus very important that the whole sample volume is subjected to the same AMF, and that the heating output is assigned to the correct H ₀ value. Then the first requirement should be ensured by special-purpose careful designs of the AMF source and corroborated by simulations [13, 34, 68, 69]. For example, for assuring a constant H ₀ in coils, it is important that their height is much larger than the sample space, since H ₀ decreases towards the coil limits [70].

The second requirement is aimed at avoiding any temperature increase of the sample due to an eventual heating of the AMF source. This can be achieved either by thermal isolation of the sample or by refrigeration of the coil. In the first case, vacuum or an effective insulator layer can be used. In the second, the coil may be cooled by means of a circulating liquid (water, air, nonane) or directly be immersed in a cryogenic liquid.

Eventually, the values of H ₀ and f that can be safely used at in vivo applications are restricted to a certain range due to unwanted responses of the organism. Measuring SAR out of this range necessarily implies the use of extrapolations but, as explained later in the chapter, the SAR values obtained with a given H ₀ and f values cannot be easily extrapolated to other H ₀ and f values. For this reason, measurements in the correct H ₀ and f range are highly preferable.

5.1.2 Magnetic Methods

Let us first discuss the calculation of SAR from hysteresis loops as explained in Sect. 4.2.1. VSM or SQUID magnetometers are commercial equipment present in many laboratories involved with magnetism. These setups allow recording hysteresis loops up to sufficiently high H ₀ values, and therefore, they should be suitable to calculate SAR. However, these commercial setups measure magnetization in presence of a magnetic field varying at very low frequency, much smaller than those used for MFH. Going back to Eq. 8.1, we find at first glance that SAR depends linearly with frequency, so that extrapolations to the right f do not seem a source of error in SAR determination. But as it will be explained in next sections, the area of the loop, represented by the closed integral in Eq. 8.1, may also depend on f. In these cases, measuring hysteresis cycles with the wrong f value may drive to a misleading SAR determination.

Ideally, the area of the hysteresis loops should be determined applying AMFs with the typical parameters for MFH, to avoid uncertainties derived from frequency effects. Up to the authors’ knowledge, there is no commercial equipment committed to the measurement of M(H) cycles in the biological range of AMF application, due to the difficulty in measuring magnetization at frequencies in the range of the kilohertz [25]. Nevertheless, such homemade devices have been developed by some research groups [17, 27, 37, 69, 71–73]. It is worth highlighting the devices in Refs. [27, 69] and [74] which measure temperature while recording M(t) and H(t) during AMF application at typical f and H ₀ for MFH.

A similar problem is encountered when SAR is calculated from χ″ measurements. The typical AMF parameters involved in commercial setups (e.g., for MPMS and PPMS devices from Quantum Design Inc., respectively, (f = 10 Hz–10 kHz, H ₀  ≤ 1.2 kA/m or f = 0.1 Hz–1 kHz, H ₀  ≤ 0.2 kA/m) are outside the MFH typical range. However, high-frequency ac-susceptometers have been built [27, 38, 39, 75–78], with frequencies up to 1 MHz, more suitable for MFH applications. One of them is currently commercially available [75]. They use H₀ ≤ 0.5 kA/m, with an exception [77], where H ₀ reaches 3.2 kA/m. Even in these setups, extrapolations are generally needed to obtain SAR values at the H ₀ for MFH applications, with the inherent inaccuracies that this implies. It must be pointed that the system described in Ref. [27] determines simultaneously χ″ and the hysteresis loop area from the temporal evolution of H and M. The SAR obtained with both magnetic methods displays a good agreement.

In addition to typical experimental considerations for general magnetic measurements (e.g., diamagnetic corrections for sample holders and dispersive media), the T at which the measurement is performed must be monitored. Since the studied MNPs are designed for heating applications, some temperature increase is expected during measurements, due to heat generated by the MNPs. As SAR depends on T, it is important to control T during measurements, or at least assign each SAR value with the T during each measuring process.

5.1.3 Calorimetric Methods

The pulse heating method in adiabatic conditions is considered the unique “absolute” and the most accurate and direct method for the determination of heat capacities [79, 80] and, by analogy, of SAR. Indeed, for heat capacity determination, a heat pulse of duration Δt is provided to the sample by a calibrated heater, and the heat capacity of the sample is obtained and, in SAR determination, the heat capacity of the sample is known, and the heat provided by MNPs upon application of an AMF pulse is calculated. However, the main drawback for the applicability of this method is the availability of an adequate setup providing adiabatic conditions .

The accuracy of the pulse heating method in adiabatic conditions for SAR determination was experimentally evaluated with the setup described in Sect. 4.3.1 by measuring the induction heating power of a copper cylinder [13]. The experimental value was compared to the theoretical one and calculated using the analytical expression for the heating power dissipated due to Foucault currents by a metallic semi-infinite cylinder in a uniform axial AMF. Both values were found to differ only in 3 %.

The initial-slope method in isoperibol conditions should also provide accurate results if the requirements assumed in Sect. 4.3.2 were fulfilled. These are:

• The internal (inside the sample) thermal relaxation time is about ten times lower than the external (sample to its environment).

• The temperature of the sample environment is always constant.

• The losses between the sample and its environment are linear with T.

These requirements highly depend on thermal and geometrical parameters of the sample and its environment and vary for each experimental setup . For the first requirement, the most suitable case of application would involve a highly conductive sample with a weak thermal link to its environment. This would allow assuming a homogeneous temperature distribution inside the sample, a very important fact for the initial-slope determination, which takes place in transient conditions. The second would imply temperature control means, and the third, study and adequacy of the sample-environment thermal link.

The main problem with this technique is that the widespread utilization of the initial-slope method has driven to an incorrect systematic use of Eq. 8.9 as a quasi-exact definition of SAR, and the requirements exposed above are seldom checked. As a consequence, obtained results may present from small to large unknown uncertainties, and for this reason, data so-obtained must be handled with care, and considered in many cases an estimate.

The inaccuracies arising from this method have been studied in some cases. For example in Ref. [81], the same setup , sample, and temperature sensor used in the determination of the accuracy of the pulse heating method in adiabatic conditions was utilized to carry out measurements without temperature control . It was confirmed that the use of the initial-slope method could underestimate the theoretical SAR by 21 %.

Also in Ref. [81], the effect of the SAR calculation method, as well as insulating conditions, temperature sensors, and sample-sensor contacts, was studied. Taking the accurate SAR value obtained by the pulse heating method as a reference, the uncertainties of the initial-slope method were determined. Using as temperature sensor a glued thermocouple, SAR underestimations between 2 and 34 % were obtained under different isolation conditions. Higher underestimations of 48–50 % were obtained using a pyrometer or a fiber-optics thermometer through the container wall, with polystyrene foam as insulator. Finally, with polystyrene foam as insulator, and the fiber-optics thermometer inside the ferrofluid, the SAR was either overestimated (5 %, using a Δt = 40 s for linear regression) or underestimated (12 %, using a Δt = 120 s for linear regression).

More recently, the sources of errors in measuring SAR using the initial-slope method were studied by numerical simulations [70, 82]. The temperature distribution across a ferrofluid volume was studied using parameters typical of experimental setups working under isoperibol conditions . It was found that the sample volume required to obtain a reduced error must be relatively high, e.g., 2.5 ml or more for a power density of 3.8 W/cm³, and that the inaccuracies depend on the generated heat power, the specific heat of the sample, and the thermal losses of the system. Among the studied cases, the best results were obtained when the temperature sensor was placed at the position where the sample temperature was maximal. This position varied depending on the experimental conditions. Eventually, short interval times (of the order of several seconds) provide in general less error.

In sum, the initial-slope method in isoperibol conditions would certainly improve accuracy through (i) the development of a more accurate thermal model describing more precisely a reference experimental setup and (ii) the modification of the currently used setups to mimic that reference one.

5.2 Effects of the Alternating Magnetic Field Amplitude and Frequency

One of the present challenges of magnetic hyperthermia is reaching therapeutic temperatures with minimal MNP doses [83]. In other words, MNPs able to generate high heat power are continuously pursued. However, the SAR values one can encounter across the scientific literature vary several orders of magnitude. Then, some questions arise. Are those differences reliable? Or are MNPs displaying appealing SAR values of up to 10⁴ W/g always more suitable for magnetic hyperthermia than those showing overlookable SAR values of 1 W/g? To answer these questions properly, the relationship between the MNP properties and the parameters (H ₀ , f) of the applied AMF must be taken into consideration.

5.2.1 Magnetic States of a MNP

Let us consider a single, immobilized MNP. The main parameters that influence its heating ability are: the applied magnetic field , the temperature, the magnetization , and the magnetic anisotropy constant, K, the last two being MNP properties that depend on temperature, composition , shape, and size . For a given temperature (e.g., room temperature), a given MNP composition (e.g., magnetite, Fe₃O₄) and a given shape (e.g., spherical), then M _S and K are governed by the MNP size. Large MNPs (>83 nm for Fe₃O₄ [84]) show a ferro/ferrimagnetic multidomain (FM-MD) magnetic structure. In each domain, all the magnetic moments are coupled together, but depending on the applied magnetic field, the magnetization of different domains may or may not be parallel. As MNP size is reduced, an intermediate vortex state [85] leads to a single-domain (FM-SD) structure, in which the magnetic moments of the whole MNP are coupled together. And as the MNP size decreases below a certain value (≅25 nm for Fe₃O₄ [84]), the magnetization reversal processes are not only governed by H, but also by thermal fluctuations. The MNP is still single-domain, but enters the superparamagnetic (SPM-SD) state in which, due to size reduction, the magnetostatic energy of the particle becomes low enough that it is comparable to its thermal energy, allowing thermally induced magnetization relaxations, which are characterized by the Néel relaxation time, τ _N . Other materials different from Fe₃O₄ require other MNP sizes to enter the FM-SD or the SPM-SD state at a fixed temperature (see Table 8.1). The size for the onset of the SPM-SD state mainly depends on K, while that for the FM-SD state depends also on M and on the stiffness of the exchange interaction between magnetic moments [85, 86]. But in general, the sizes of FM-MD nanoparticles are too high to be interesting for magnetic hyperthermia.

Table 8.1 Critical sizes for the SPM/FM-SD and FM-SD/MD transitions in common magnetic materials at room temperature [84]

5.2.2 Effects of f and H ₀ in the SAR of SD Nanoparticles Near the SPM/FM Transition

Due to the thermally induced magnetization reversal processes in superparamagnets, the M(H) loops recorded for one of these MNPs under slowly varying AMFs have a sigmoidal shape in which no hysteresis is observed. This does not imply that such MNPs are unable to undergo hysteresis losses when subjected to an AMF, since dynamic M(H) loops at adequate frequencies may show hysteresis. According to the Rosensweig model of superparamagnetism [24], the out-of-phase ac-susceptibility , indicative of dissipative processes, can be expressed as,

$$ {\chi}^{{\prime\prime} }={\chi}_0\cdot \frac{2\pi f\tau}{1+{\left(2\pi f\tau \right)}^2} $$

(8.11)

where χ ₀ is the static susceptibility and τ is the magnetic moment relaxation time, which stands for the Néel relaxation time in case of an immobilized MNP. The simplified expression for this time is,

$$ \tau \equiv {\tau}_N={\tau}_0 \exp \left(\frac{K{ V}_M}{k_B T}\right)={\tau}_0 \exp \left(\frac{E_{b0}}{k_B T}\right) $$

(8.12)

where τ ₀ is the attempt time, τ ₀  = 10⁸–10¹⁰ s and E _b0 is the barrier energy for magnetization reversal , created by the anisotropy energy of the MNPs.

It is easily deduced from Eq. 8.11 that χ″ has a maximum at 2πfτ = 1, whose value is χ ₀ /2. This maximum occurs at the SPM/FM transition . In particular, the T at which this transition takes place is called the blocking temperature (T _B ) , which delimits the FM (T < T _B ) and the SPM (T > T _B ) behavior. In order to locate T _B at therapeutic temperatures and optimize heat dissipation, the frequency of the AMF must be tuned with respect to the MNP properties, or vice versa. For a given f, MNP composition, and shape, the MNP size can be tuned as

$$ {V}_M^{Opt}=\frac{k_B{T}_B}{K} \ln \frac{1}{2\pi f{\tau}_0} $$

(8.13)

where V _M ^Opt corresponds to the MNP volume that has its SPM/FM transition at T _B . The optimal conditions are much more sensitive to the size of the MNPs than to the f of the AMF. For example, using T _B  = 300 K, K = 32 kJ/m³ and τ ₀  = 10⁻⁹ s and assuming spherical MNPs, Eq. 8.13 is fulfilled at 100 kHz with a particle diameter of 12.2 nm, while for 400 kHz the optimal diameter is 11.4 nm: a fourfold increase of the frequency leads to only a 6 % decrease of the optimal particle diameter.

As seen in Eq. 8.2, χ″ and SAR are linearly related in the LRT approach. In this framework, χ ₀ can be approximated to the initial susceptibility of the Langevin equation [24],

$$ {\chi}_0\cong \frac{\mu_0{M}_S^2{V}_M}{3{k}_B T} $$

(8.14)

and SAR can be expressed as,

$$ \mathrm{S}\mathrm{A}\mathrm{R}=\frac{\pi {\mu}_0}{\rho_{M NP}}\cdot \frac{\mu_0{M}_S^2{V}_M}{3{k}_B T}\cdot {H}_0^2\cdot f\frac{2\pi f\tau}{1+{\left(2\pi f\tau \right)}^2} $$

(8.15)

This theoretical development highlights the dependency of SAR on f and H ₀ within the LRT limits. Although χ″ presents a maximum at 2πfτ = 1, SAR presents a nonlinear growing trend with f , due to the increasing number of the AMF cycles described per time unit (see Fig. 8.3a). Also, SAR increases with H ₀ ². The difference in using H ₀ values of, for example, 4 kA/m instead of 1 kA/m is obtaining a SAR value 16 times higher. So, the effect of both magnitudes, but especially H ₀ , is notorious, and it is then imperative to provide the H ₀ and f value with each SAR datum. Eventually, within the LRT limits, the SAR value obtained with a given H ₀ value can be easily extrapolated to other H ₀ values, but the same statement is not true for f, where a more complicated dependence is found.

Fig. 8.3

SAR of SD MNPs near the SPM/FM transition: (a) theoretical dependency on f of χ″ and SAR (LRT limit); (b) effect of MNP size at constant H ₀ , f; (c) effect of f at constant H ₀ , size; (d, e, and f) effect of H ₀ at constant f and size for iron oxide (d, e) and non-iron-oxide (f) MNPs. Curves aiding to identify H ₀ , H ₀ ² and H ₀ ³ dependencies are also included in figures d, e and f

The limits of LRT (see Eq. 8.3) are not, however, wide enough to enclose the whole biological range of AMF application. For example, the size of a magnetite nanoparticle (M _S  = 446 kA/m) subjected to an AMF of H ₀  = 5 kA/m and f = 100 kHz at 300 K should be under 14 nm to fulfill the LRT. The use of higher H ₀ values would require smaller MNPs for the LRT to be valid, and vice versa. Out of these limits, Eqs. 8.11, 8.12, and 8.13 are still valid, but not Eqs. 8.14 and 8.15, and the relationship between SAR and the AMF parameters is not straightforward.

To our knowledge, no analytical expressions of the dependence of SAR on H ₀ or f are available out of the LRT. However, numerical simulations aiming to calculate dynamic hysteresis loop areas can be used to infer some trends. With respect to the influence of H ₀ , for example, numerical calculations based on a two-level approximation [28] predict that the LRT model overestimates the hysteresis area of SPM nanoparticles right above the LRT limits, indicating that the SAR would be proportional to H ₀ to a power lower than 2. Further results arising from this model will be tackled in Sect. 5.2.3. Another interesting result of numerical simulations [87] is that, at large enough H ₀ values, increasing H ₀ implies shifting the dissipation maximum to higher frequencies. This is due to the fact that in that strong nonlinear regime, the barrier energy for magnetization reversal gets appreciably reduced by H ₀ , so that τ depends not only on the materials properties, but also on H ₀ . This would make SAR dependence on H ₀ stronger or weaker than predicted by the LRT, at higher or lower frequencies, respectively, than that corresponding to the condition of the maximum of Eq. 8.11. Also, it has been found [88, 89] that an increase in H ₀ also entails a widening of the SAR maxima as a function of the MNP size, this implying that in this situation, even MNP assemblies with a certain size distribution will present a good performance, due to the less restricted tuning condition between f and V _M .

Let us now qualitative contrast these theoretical findings with recent experimental data . For this purpose, SAR/f (and not SAR) is studied in order to highlight just the influence of H ₀ and f on the hysteresis area , and not the linear dependence on f resulting of increasing the number of the AMF cycles. Figure 8.3b and c illustrate the effects of the fτ product on SAR. In Fig. 8.3b, experimental SAR/f values are depicted as a function of H ₀ . These data include iron oxide SPM-SD nanoparticles and some FM-SD nanoparticles next to the SMP/FM limit. For each data series, H ₀ and f are constant, but the MNP size varies (see Table 8.2). This size variation induces a great change in τ, thus making SAR/f vary several orders of magnitude, according to Eq. 8.15. Figure 8.3c displays the variation of SAR/f on f for SPM-SD nanoparticles with various compositions (see Table 8.2). For each data series, H ₀ and V _M are constant. It is observed that some of these series show the maximum that delimits the FM (above the maximum) and the SPM (below the maximum) regime for each particular MNP size. All these results stand out the relevant and nonlinear dependence of SAR on the AMF frequency in the SPM-SD regime, indicating that extrapolating SAR data according to a linear dependence on f is not always correct.

Table 8.2 Experimental details of the data collected in Fig, 8.3

Figure 8.3 also collects recent experimental evidence of the effect of H ₀ on SAR/f data in the SPM-SD regime or in the FM-SD regime next to the SMP/FM limit. As a comparison reference between values, the same series of SAR/f data [90] is included in some graphics. It corresponds to FM-SD nanoparticles and will be discussed in next section. In addition, in order to estimate the exponent of the H ₀ ⁿ dependence of SAR, trends proportional to H ₀ , H ₀ ², and H ₀ ³ are also included.

Figure 8.3d and e shows data corresponding to iron-oxide materials . It is first observed that when H ₀ is sufficiently high so that major loops are described, SAR becomes independent of H ₀ due to saturation of the magnetization. For minor loops, different dependencies are found, but most data can be fitted to H ₀ ⁿ curves with n ranging between 1.8 and 2.5. They thus present deviations from the LRT according to the numerical simulations referred above. Figure 8.3f, which displays data of materials other than iron oxides (see Table 8.2), derives similar results. In addition, Fig. 8.3e, which shows the most recent data, highlights the evolution in the optimization of SAR in iron-oxide materials, since SAR values are obtained at relatively low H ₀ values. Comparatively, non-iron-oxide MNPs (Fig. 8.3f) require greater AMF amplitudes to produce similar heating.

In sum, most of the available experimental data of SD nanoparticles near the SPM/FM transition do not follow the LRT , due to the mostly high H ₀ values used. This reinforces the abovementioned requirement of the adequacy of measuring SAR with H ₀ and f values in the biological range of field application, since no analytical expressions are available out of the LRT. For example, measuring SAR at high H ₀ values (e.g., 20 kA/m) and extrapolating it to lower value (e.g., 5 kA/m) using a H ₀ ² law would either underestimate or overestimate this value, if the real dependency is, respectively, weaker or stronger on H ₀ . Eventually, Fig. 8.3 points out that the same MNP can display a wide range of SAR values and that outstanding SAR values are often indicative of high H ₀ values.

5.2.3 Effects of f and H ₀ in the SAR of FM Nanoparticles

A MNP in the FM state does not undergo thermally induced magnetization relaxations, but its magnetization describes a hysteresis loop upon application of a slowly varying AMF cycle, i.e., a quasi-static hysteresis loop, whose coercive field (H _C ) value accounts for the FM hardness or softness of the MNP.

The Stoner–Wohlfarth model [112, 113] is a simple analytical model that describes the main features of the quasi-static hysteresis of FM-SD nanoparticles considering each particle as a macrospin, assuming uniaxial anisotropy and neglecting thermal fluctuations. According to this model, the maximum energy per volume unit that a MNP can dissipate during a hysteresis loop is

$$ {E}_{\mathrm{hyst}, \max } = 4\cdotp {\mu}_0\cdotp {M}_S\cdotp {H}_K = 8\cdotp K $$

(8.16)

where H _K  = 2 · K/μ ₀  · M _S is the anisotropy field (in SI units). This is the area of an ideal major square loop, in which the magnetic moment of the MNP is parallel to H and then H _C  = H _K . For other orientations between M and K, H _C  < H _K and E _hyst decreases. In particular, for a randomly oriented nanoparticle assembly, this area is [114],

$$ {E}_{\mathrm{hyst}, \max}\cong 2\cdotp {\mu}_0\cdotp {M}_S\cdotp {H}_C $$

(8.17)

which in terms of SAR becomes

$$ {\left(\mathrm{SAR}/ f\right)}_{\max}\cong 2\cdotp {\mu}_0\cdotp {M}_S\cdotp {H}_C/{\rho}_{M NP} $$

(8.18)

According to these results, it seems straightforward to consider that for an improved heating ability, FM MNPs must present a large anisotropy constant leading to a high H _C value. This can be achieved, for example, controlling the size, since it is well known that H _C increases as the size of a MNP grows from the SPM/FM-SD to the FM-SD/MD transition size, showing a maximum at the latter and decreasing for larger sizes [85]. However, the Stoner–Wohlfarth model also predicts that a minimum critical field is necessary for the magnetization to get oriented with the applied field. The existence of this critical field, related to the anisotropy constant, implies that lower H ₀ values will not produce appreciable losses.

Some early data [90, 115] will be here used as a starting point to illustrate the above consideration. Figure 8.4 collects SAR/f values of four samples determined as a function of H ₀ through quasi-static minor loops. These samples have H _C and M _S values ranging between 2.6 and 32.8 kA/m and 333 and 450 kA/m, respectively. Data [90], the series already included in Fig. 8.3, correspond to the heating ability of bacterial magnetosomes , MNPs fabricated by magnetotactic bacteria and widely considered as very good performing material for magnetic hyperthermia. Figure 8.4 reveals the following phenomena:

Fig. 8.4

Dependence on H ₀ of the SAR of iron oxide FM-SD nanoparticles (Adapted from Ref. [90, 115]). Both figures show the same data, but Figure: (a) highlights the low-H ₀ dependency; (b) displays the significant SAR/f increase at H ₀  ≅ H _C . Each H _C is marked with a vertical line of the same color than the symbols

• SAR/f may vary several orders of magnitude when H ₀ ranges from 1 to 100 kA/m, being the SAR/f values obtained at a certain H ₀ value hardly extrapolable to other H ₀ values if the particular SAR/f (H ₀ ) trend is unknown.

• For H ₀ >> H _C and when major hysteresis loops are considered, the higher H _C is, the larger SAR/f is. It is observed that SAR/f first increases with H ₀ and then reaches a plateau that corresponds to the description of a major loop, due to the saturation of M at this H ₀ value. The SAR/f value in this plateau is larger/smaller for the sample with higher/lower H _C .

• For H ₀  < H _C , the higher H _C is, the smaller SAR/f is. It is easily inferred from Fig. 8.4b that SAR/f increases significantly only when H ₀  > H _C .

These phenomena are essentially in accordance with the Stoner–Wohlfarth model, although some aspects are not. For example, nonzero SAR/f data are obtained for H ₀ values below the critical field of the Stoner–Wohlfarth model, showing H ₀ ⁿ laws. This highlights the limitations of this model to describe precisely the SAR/f dependency on H ₀ . Another limitation in this model is the absence of dynamic parameters, thus neglecting frequency effects in SAR/f, i.e., in the area of dynamic hysteresis loops.

Some more recent works have looked deeper into the influence of H ₀ and f in FM-SD nanoparticles through analytical models and numerical calculations [28, 89, 114, 116–118]. For example, Usov et al. [118] obtained, through a modification of the Stoner–Wohlfarth model , an analytical expression to account for the variation of H _C with V _M , f, and T. For an oriented MNP assembly,

$$ {H}_C={H}_K\cdot \left(1-{\kappa}^{\raisebox{1ex}{$1$}\left/ \raisebox{-1ex}{$2$}\right.}\right)\ \mathrm{with}\ \kappa =\frac{k_B T}{K{ V}_M}\cdot \ln \left(\frac{k_B T}{K{ V}_M}\cdot \frac{H_k{t}_m}{\tau_0\Delta H}\right) $$

(8.19)

where t _m is the time elapsed between the fields H and H + ΔH during the hysteresis loop. Equation 8.19 is valid for κ < 0.7 and when major loops are described [28]. According to it, H _C increases when t _m decreases, i.e., when f grows.

The influence of H ₀ in SAR/f has been inferred from numerical calculations of dynamic hysteresis loops , obtained using a two-level approximation [28]. This model allows calculating major and minor hysteresis loops for both SPM and FM-SD particles. Figure 8.5a illustrates the H ₀ dependence of the ratio SAR/SAR(H _K /2) obtained within this model for K = 13 kJ/m³, M _S  = 10³ kA/m, f = 100 kHz, T = 300 K, and τ ₀  = 5 × 10⁻¹¹ s. With these data, H _K  = 20.69 kA/m, and the volume that maximize χ″ at 100 kHz according to the LRT (i.e., that defines the SPM/FM transition) is V _M  = 3.30 × 10⁻²⁴ m³, which corresponds to a particle size of 18.5 nm. For this particle, the LRT is just fulfilled for H ₀  < 1 kA/m.

Fig. 8.5

(a) H ₀ dependence of the ratio SAR/SAR(H _K /2) for SD nanoparticles of different sizes and K = 13 kJ/m³, M _S  = 10³ kA/m, f = 100 kHz, T = 300 K and τ ₀  = 5 × 10⁻¹¹ s (Adapted from Ref. [28]). (b) n values obtained from fitting to a H ₀ ⁿ law the data of the medium-H ₀ range of Fig. 8.5a

It is observed that the curves present three main ranges: (i) low-H ₀ , in which SAR increases slowly, (ii) medium-H ₀ , in which SAR experiments a more or less sharp increase, (iii) high-H ₀ , in which SAR saturates. Figure 8.5b displays the n values obtained from fitting the data of the medium-H ₀ range to an H ₀ ⁿ law, showing that this exponent trend undergoes an inflexion point at the SPM/FM transition . Also, it is found that 1.3 < n < 2.2 in the SPM regime, which is in accordance with the experimental results of Sect. 5.2.2. It becomes obvious that H ₀ is a highly influent parameter and that SAR data obtained at a particular H ₀ value are not easily extrapolable to other H ₀ values, considering in addition that all trends in Fig. 8.5 will be quantitatively different for other material properties. The above results also highlight the suitability of nanoparticles with sizes next to the SPM/FM limit (18–20 nm) for low-H ₀ applications and that of larger particles for higher fields.

Eventually, Fig. 8.6 collects recent experimental SAR/f data of FM-SD nanoparticles as a function of f and of H ₀ . Sample details are collected in Table 8.3. Figure 8.6a displays constant or decreasing SAR/f curves with increasing f, according to the H _C shift predicted by Eq. 8.19, except for data [119] −3.1 kA/m and [119] −4.6 kA/m, recorded with H ₀  < H _C . Figure 8.6 also shows the variation of SAR/f with H ₀ for iron oxide (b) and non-iron-oxide (c, d) FM-SD particles. Similar trends than those of Figs. 8.4 and 8.5 are found, and especially for non-iron-oxide materials, we observe curves revealing high H _C values and outstanding SAR/f values for H ₀  > H _C .

Fig. 8.6

SAR of FM-SD MNPs: (a) effect of f at constant H ₀ , size; (b, c, and d) effect of H ₀ at constant f and size for iron oxide (b) and non-iron-oxide (c, d) MNPs. Curves aiding to identify H ₀ , H ₀ ², and H ₀ ³ dependencies are also included

Table 8.3 Experimental details of the data collected in Fig. 8.6

5.2.4 Biological Limits of f and H ₀

The above results and discussion help answering the two questions posed at the beginning of Sect. 5.2. On the one hand, most of the SAR value range found in the literature could be reliable even for the same material, given that correct f and H ₀ values are supplied with these data. But, on the other hand, appealing SAR values of up to 10⁴ W/g may not be more suitable for magnetic hyperthermia than those showing overlookable SAR values of 1 W/g, if they are achieved with H ₀ values well above the safety limits. This is especially true for FM MNPs with H _C values that overcome these limits. The question is: where are those limits?

The deleterious effects of AMF in the f, H ₀ range suitable for hyperthermia have not been systematically studied. Hence, the exposure limits of patients during hyperthermia therapies are not well established. The dominant physiological adverse response under AMF between 100 and 1000 kHz is the nonspecific temperature increase due to the induction of eddy currents in the tissue, which can lead to burns and blisters in extreme cases. The power dissipation due to Joule losses depends on f and H ₀ , the duration of the exposure, the electrical conductivity of the tissue, and the external radius, i.e., larger and more conductive parts of the body will generate more heat due to eddy currents. For this reason, the limbs usually tolerate higher f, H ₀ values than other parts of the body. Other complications are muscle stimulations, including cardiac stimulation or arrhythmia.

In general, the tolerated H ₀ decreases with increasing f, more or less steeply depending on the f range. The guidelines from the International Commission on Non-Ionizing Radiation Protection establish that for frequencies between 65 and 1000 kHz the f · H ₀ product should not exceed 1.6 kHz · kA/m. However, this reference level is a very conservative limit, focused on protect people from everyday exposure to radiation. A more widespread limit within the magnetic hyperthermia community is the product f · H ₀  = 485 kHz · kA/m, derived from the work of Atkinson et al. [135]. In this study, the patient comfort was evaluated increasing H ₀ under a frequency of 13.56 MHz on the thorax. However, this product has to be taken as a rule-of-thumb for the order of magnitude of the fields involved in magnetic hyperthermia.

During the recent clinical studies on magnetic hyperthermia, a trial-and-error evaluation under medical supervision was carried out. Using a fixed AMF frequency of 100 kHz, the magnetic field strength was increased until the patients reported discomfort, to ensure the maximum possible heating ability for the injected MNPs. In clinical trials on prostate cancer [136], 10 patients experienced discomfort at H ₀  > 4 kA/m, corroborating the Atkinson limit. However, the H ₀ tolerated in the treatment of glioblastoma multiforme (brain tumor) went from 3.8 to 13.5 kA/m in a group of 14 patients [137]. Obviously, in these studies the sensitivity of the patients is also another variable.

In spite of the lack of exhaustive and reliable guidelines for the application of AMF in humans, the above experimental evidences lead to be conservative and expect that H ₀ values higher than 15 kA/m with f in the 100–500 kHz range are probably not suitable for human application or, at least, for all parts of the human body. Coming back to Figs. 8.3 and 8.6, it can be concluded that some of the experimental results, especially those with the most outstanding SAR values, are obtained at probably too high H ₀ values for human application, and that given the difficulties found for their extrapolation to lower H ₀ values, they may not be that interesting for magnetic hyperthermia.

5.3 Interest of Measuring SAR as a Function of Temperature

Given that magnetic hyperthermia therapy involves temperatures from 36 up to about 50 °C, it is essential to evaluate SAR in this particular temperature range. As seen in the previous sections, the heating ability at given H ₀ , f values is determined by the magnetic state of the MNP, which in turn depend on temperature. Some materials may show a weak SAR variation in this range, making reasonable the use of average values for therapy planning. But in other cases, SAR values may change appreciably as temperature increases, and applications may require the use of SAR(T) functions. Moreover, the determination of SAR does not need to be restricted to this narrow temperature range, since measurements over wider T ranges may provide valuable information about the magnetic state of the MNPs, which is helpful to the optimization of MNP heating ability.

Regarding magnetic detection, some SQUID magnetometers allow measuring hysteresis loops and ac-susceptibility in a wide T range (1.8 K to 1000 K). Also, some homemade setups [27, 39, 138] have succeeded in performing dynamic magnetic measurements up to 50 K above room temperature. Among calorimetric methods, the pulse heating method in adiabatic conditions is able to determine SAR(T) in the 120–370 K range [139, 140].

5.3.1 Self-regulating MNPs

The control of the maximum temperatures acquired by the tissues during the hyperthermia therapy is an open problem nowadays. In pursue of giving a solution to this problem, the so-called self-regulating MNPs have been considered. Such self-regulation is based on the transition undergone by FM materials at the Curie temperature, T _C . When T > T _C the nanoparticles become paramagnetic and stop releasing heat, as no hysteresis can take place. By modifying the chemical composition of some MNPs, their T _C can be tuned so that it is located right above the maximum temperature desired for therapeutic hyperthermia. Such particles would then self-regulate the temperature during the therapy.

This is the case, for example, of lanthanum manganites such as La_1−xAg_yMnO_3+δ [39] or La_1−xSr_xMnO_3+δ [140]. The SAR(T) of those MNPs was measured, between 20 and 50 °C using magnetic methods in the former and between −20 and 100 °C by the pulse heating method in adiabatic conditions in the later. In both cases, SAR(T) characterization allowed finding a dissipation peak right below T _C , assigned to a Hopkinson peak , giving experimental evidence to the theoretical predictions of this phenomenon in MNPs [141]. Also, a sharp reduction in SAR was observed at T _C . Figure 8.7a shows this behavior for La_1−xSr_xMnO_3+δ MNPs with different T _C values. Field-cooled M(T) curves are also depicted in the figure to show the magnetic transition at T _C . This accurate SAR(T) determination also allowed relating the unexpected maximum temperatures achieved in nonadiabatic heating experiments with the thermal losses of the experimental system and the MNP heating ability [140].

Fig. 8.7

(a) Lines: field-cooled M(T) of La_1−xSr_xMnO_3+δ MNPs with different T _C values, recorded with H = 4 kA/m. Lines + symbols: SAR(T) of the same samples measured by the pulse heating method in adiabatic conditions, with H ₀  = 2 kA/m and f = 108 kHz (Adapted from Ref. [140]). (b) Temperature dependence of the out-of-phase ac magnetic susceptibility obtained from magnetic measurements (MPMS, 4.64–476 Hz, and PPMS, 4642 Hz) and from calorimetric measurements (SAR, 47 and 410 kHz). χ″(T) was calculated from the SAR(T) measurements using Eq. 8.2 (Adapted from Ref. [139])

5.3.2 Location of the FM/SMP Transition

As concluded in Sect. 5.2, MNPs with sizes next to the SPM/FM limit are the most suitable for low-H ₀ applications. This optimum MNP volume (Eq. 8.13) depends on f and K and takes place at a given T _B . Then, for a fixed frequency, χ″ (T) measurements may be used to determine T _B , since at this temperature χ″ presents a peak (Eq. 8.11). The location of T _B helps to conclude whether smaller (if T _B  > 36 °C) or larger (if T _B  < 36 °C) MNPs will optimize SAR for hyperthermia applications, thus serving as feedback for the synthesis of optimized MNPs. In the low-field range where the LRT applies (Eq. 8.3), SAR is linearly proportional to χ″ (Eq. 8.2) and χ″(T) provides equivalent information than SAR(T).

Figure 8.7b shows the good agreement between χ″(T) and SAR(T) measurements on a system of iron oxide nanoparticles. χ″(T) was determined using the MPMS and PPMS devices from Quantum Design Inc., and SAR(T) was obtained by the pulse heating method in adiabatic conditions [139]. The clear advantage of calorimetric SAR(T) measurements is that the AMF parameters used are typical of magnetic hyperthermia.

Eventually, when the H ₀ values overcome the limits of the LRT, and the relationship between χ″ and SAR is unclear, it becomes evident that the direct determination of SAR(T) will provide more reliable information than χ″ about the temperature at which the dissipation peak takes place.

5.3.3 Dynamic Effects in Ferrofluids

The quantification of SAR at laboratory level is mainly performed on ferrofluids. However, in magnetic hyperthermia, MNPs are not dispersed in liquid, but trapped inside solid matrices (tissues). This may drive to discrepancies between the measured SAR values and the real MNP heating performance during therapies.

One possible source of discrepancy is the Brown relaxation mechanism. This is, like Néel relaxation, a thermally induced magnetization relaxation mechanism. The difference between them is that in Néel relaxation the magnetic moment reverses but the particle is fixed, while in Brown relaxation the whole MNP reverses. Both mechanisms take place in parallel and it is the faster one which dominates dissipation. In general, Brown relaxation is only relevant for large MNPs.

However, other effects may occur. MNPs dispersed in a liquid media can get reoriented or form structures, particularly when subjected to a magnetic field [127, 142]. In general, the formed structures depend on the properties of the MNPs, dispersive media, H ₀ and f, and will affect the heating ability of the assembly, as will be later explained in Sect. 5.4. SAR(T) measurements can unveil these processes [11, 143]. For example, Fig. 8.8a shows the SAR(T) of an assembly of Fe₃O₄ nanoparticles dispersed in n-dodecane, measured by the pulse heating method in adiabatic conditions. The sample presents a sharp heating peak below the melting temperature of the solvent and a trend change above the melting. This peak is a consequence of the premelting of the solvent at the interface between the particles and the dispersive media. The formed viscous layer allows the MNP to rotate and get oriented along the applied magnetic field, producing an increased magnetization (and SAR) with respect to that of the randomly oriented case. The increase of the magnetization at the premelting stage also appears in zero-field-cooled/field-cooled (ZFC/FC) M(T) measurements (Fig. 8.8b) under a static magnetic field. However, both figures are not fully comparable, since SAR is consequence of a dynamic magnetic field.

Fig. 8.8

(a) SAR(T) of Fe₃O₄ MNPs dispersed in n-dodecane, measured by the pulse heating method in adiabatic conditions under two different AMFs. (b) ZFC(green)/FC(blue) M(T) of the same sample measured under a static magnetic field of 3 kA/m

5.4 Influence of Magnetic Nanoparticle Concentration and Arrangement

All the theoretical considerations made in previous sections about the effect of f, H ₀ , and T on SAR can be strictly applied to a single MNP or, in other words, to a system of noninteracting MNPs. However, due to the magnetic nature of these particles, magnetic interparticle interactions are unavoidable. As MNPs are usually covered with a surfactant, they have no physical contact and thus the most relevant interparticle interaction is the dipolar one. Since dipolar interactions are long range in nature, a minimum interparticle distance of about 20 times the particle diameter is needed (volume concentration of 0.01 %) to achieve a noninteracting MNP system [144]. Such low concentrations are nonrealistic in magnetic hyperthermia. It has been observed that, when biocompatible MNPs are internalized by the cells, they are usually enclosed in vesicles containing hundreds of closely packed particles [145]. So even if the average volume concentration in the cell is low, the local concentration at the vesicles may be very high. The study of the effect of interparticle interactions on SAR is thus relevant for magnetic hyperthermia, since the SAR of MNPs well dispersed on ferrofluids can differ from that of densely packed arrangements.

The effect of magnetic interactions is modifying the energy barrier established by the magnetic anisotropy of the individual MNPs (Eq. 8.12). This explains that MNPs with lower anisotropy energy (i.e., MnFe₂O₄) are more affected by interparticle interactions than those with high anisotropy energy (i.e., CoFe₂O₄). Most analytical models including interparticle interactions deal with the calculation of a modified energy barrier, E _b , and the direct computation of τ [146–152]. Except for Ref. [148], all analytical models predict an increase of E _b and thus τ, with magnetic interactions, leading to an increase of the blocking temperature, T _B . But although the calculation of τ(T) gives useful information about T _B , the relaxation time alone does not give direct information on SAR, unless analytical or empirical expressions are obtained relating both quantities in the case of magnetic interactions.

When considering realistic assemblies with millions of individual particles, developing an analytical theoretical framework is an extremely challenging task. Dipolar interactions are long-ranged and depend on the relative orientation of the MNP magnetic moments and on the interparticle distances. For this reason, the theoretical consideration of magnetic interactions has been recently carried out by means of numerical simulations of the hysteresis cycles . Most simulations deal with assemblies of randomly distributed MNPs also with random easy axis distribution, dispersed in a solid matrix (i.e., the position and orientation of the MNPs are fixed). In these conditions, the effect of magnetic interactions on the heating ability of the MNPs will depend on the state of the particle (i.e., FM-SD or SPM-SD), the amplitude of the applied field, the average distance between MNPs, and the intrinsic properties of the particles: size, shape, polydispersity, M _S , and K.

Neglecting the effect of f, the static hysteresis loops of systems of randomly distributed nanoparticles in the FM-SD regime have been simulated at different MNP volume concentrations [128]. Figure 8.9a depicts the obtained results. It is observed that the behavior of SAR/f with interaction strength was found to depend on H ₀ : for low fields, increasing the interaction strength increased SAR/f, while for fields above H _C  ≅ 0.48 · H _K , SAR/f decreased with increasing interactions. Moreover, the interactions increased the field at which saturation of SAR/f occurred.

Fig. 8.9

(a) Numerical simulation of the variation of SAR/f on H ₀ for the same FM-SD MNPs with different volume concentrations (Adapted from Ref. [128]). Dashed lines indicate the position of H _C (≅0.48 H _K ) and H _K . (b) Numerical simulation of SAR of SD MNPs as a function of KV _M /k _B T at fixed f, V _M and M _S at different values of the interaction parameter γ and low H ₀ (Adapted from Ref. [153]). The arrow indicates the direction of increasing concentration

Dynamic hysteresis loops have been also numerically simulated [153] to calculate the SAR of SD particles as a function of KV _M /k _B T at a fixed f and high and low H ₀ , using a mean field approximation. In this work, the strength of the interactions is quantified by the parameter γ, being γ = 0 the noninteracting case. Figure 8.9b displays the obtained results for low H ₀ values. A displacement of T _B towards higher temperatures takes place with increasing magnetic interactions. To illustrate the effect of this displacement on SAR, let us consider a MNP system tested at 36 °C. If this system is in the SPM regime (T _B  < 36 °C) in the noninteracting case, decreasing interparticle distances leads to an initial increase of SAR (T _B  ≅ 36 °C), steeper in the case of low H ₀ , followed by a decrease of SAR (T _B  > 36 °C), since the MNP system enters the FM state. Consequently, if the size of a MNP system has been designed to show optimum heating ability at 36 °C in the noninteracting case, the presence of magnetic interactions can drive the energy barrier of the system out of the optimum value, thus diminishing the SAR. Figure 8.9b also shows that the value of the maximum SAR (height of the curves) decreases with increasing γ.

Variations of SAR with concentration have also been found experimentally . For example, the SAR at 3 kA/m (H ₀  < H _K ) and 109 kHz of Fe₂O₃ ferrofluids in the SPM regime decreased a 57 % with increasing volume concentration from 0.04 % to 0.18 % [154]. Also, the SAR of polydisperse Fe₃O₄ MNPs suspensions in the SPM regime was found to decrease with increasing volume concentrations up to 6.6 % under AMFs with f = 1500 kHz and H ₀  < H _K [104]. The SAR of Fe₃O₄ suspensions of magnetite MNPs of different sizes measured under H ₀  > H _C at 765 kHz [93] was reported to behave differently at increasing concentration (0.15–1.2 mg/ml) depending on their size, in qualitative accordance with simulations in Ref. [153]. In contrast to previous results, a concentration-independent SAR was found for Fe₂O₃ suspensions in water with different mean sizes measured under H ₀  < H _C and 522.7 kHz with increasing concentration from 5 to 260 mg_Fe/ml [96].

This variety of theoretical and experimental results reflects the intrinsic complexity of the problem. But there is a further factor increasing difficulty: MNP arrangement . The above-referred works make the assumption that an increase in concentration leads to a homogeneous decrease of the interparticle distance, i.e., that the MNPs are homogeneously distributed in the dispersive media. However, in most systems, the interparticle distance is not controlled, and concentration is just a statistical value. In dispersion, MNPs can form aggregates, columns, chains, rings, etc. In sum, a wide variety of 1D, 2D, or 3D structures. This can occur in absence or presence of either a static or an alternating magnetic field [132]. And it has been found that such arrangements influence the heating ability of a MNP system in a different way than homogeneous concentration.

Recent numerical simulations on chains of SPM nanoparticles [155] point to a reduction of the maximum SAR and a shifting of T _B towards higher temperatures as the chain length increases, while simulations of MNP chains well within the FM regime [156] reveal that the maximum SAR value at high H ₀ values grows as the chain length increases, as well as the required H ₀ value to achieve this high SAR. Eventually, numerical simulations of anisotropic columns of 6x6 particles and increasing length under saturating fields H ₀  > H _K [132] predict an increase of SAR with interactions for low K particles, while a reduction for high K particles. These three works [132, 155, 156] show also experimental results with good qualitative agreement with their simulations.

These studies, together with others on clusters and rings [156–158], highlight the nonnegligible influence of the MNP arrangement on the heating ability of MNP systems, which should not be overlooked. Given that at present it is not possible to predict the geometrical arrangement that MNPs will adopt in tissues, maybe the fabrication of nanoobjects preserving a favorable MNP superstructure would be a promising path to achieve MNP assemblies whose SAR will scarcely vary once they are internalized by cells [159].

6 Conclusions and Future Perspective

The heat generated by certain magnetic nanoparticles under the action of adequate alternating magnetic fields is used as active principle in the magnetic hyperthermia cancer therapy. A successful localized therapy must be able to provide therapeutic temperatures in tumors, while keeping reasonable low temperatures in the healthy tissue. In addition, the parameters of the applied AMF should be limited to human-tolerated values, although these values are only roughly defined at present.

Obtaining precise SAR values of the used MNPs is necessary to determine the spatial and temporal temperature evolution of the maximum temperatures acquired during the therapies. This implies using experimental setups and methods deriving the most accurate results. It has been concluded that the methods using magnetic detection often derive highly accurate data, but the main drawback of commercial setups is that they operate with H ₀ , f values not suitable for magnetic hyperthermia. However, recent homemade magnetometers and susceptometers are starting to overcome this problem. With respect to calorimetric methods, the widely used initial-slope method in isoperibol conditions often derives unquantified inaccuracies due to the disagreement between the theoretical assumptions of the thermal model and the experimental setups and measuring conditions. Eventually, the only realization of a setup capable of determining SAR by the pulse heating method in adiabatic conditions has proven the good accuracy of the method, considered the only “absolute” method in adiabatic calorimetry.

Furthermore, several factors affect the heating ability of MNPs and must be therefore taken into account when evaluating SAR. Among them, the parameters of the applied AMF can make the SAR of the same MNP vary several orders of magnitude. SAR increases with H ₀ and f, since more electromagnetic energy is absorbed by the MNP and afterwards released as heat. But the dependencies of SAR on these parameters are not straightforward and vary with the magnetic state of the MNP. Such dependencies have been studied through the area of the dynamic hysteresis loop described in each AMF cycle, SAR/f. It has been shown that for SD MNPs whose blocking temperature is near the temperature of interest (≅36 °C) for magnetic hyperthermia, i.e., MNPs that are near the FM/SPM transition at 36 °C, SAR/f depends on H ₀ ² if H ₀ is low enough as to fulfill the LRT condition. For higher H ₀ values, SAR/f varies as H ₀ ⁿ, with n ranging between 1.8 and 2.5. When H ₀ is sufficiently high (H ₀  > H _k ) so that major loops are described, SAR becomes independent of H ₀ . For FM-SD MNPs (T _B >> 36 °C), SAR/f is found to be negligible for H ₀  < H _C . For H _C  < H ₀  < H _k , SAR experiments a sharp increase, following a H ₀ ⁿ law with n values up to even 60. For H ₀  > H _k SAR/f saturates (major loops). These dependencies drive to two main conclusions: (i) SAR data obtained at a particular H ₀ value are not easily extrapolable to other H ₀ values, so that it is imperative to determine SAR with the H ₀ values suitable for magnetic hyperthermia; (ii) MNPs with sizes next to the SPM/FM limit at 36 °C are suitable for low-H ₀ applications (e.g., prostate cancer), while larger particles should be used for higher fields, provided that their H _C value do not overcome the H ₀ limit for biological application. The variation of SAR/f with f, while being less pronounced than that of H ₀ , is nonnegligible, nonextrapolable, and also dependent on the MNP magnetic state.

SAR must be evaluated between 36 up to about 50 °C, and in the general case, the use of SAR values averaged over this T range may be reasonable for planning the therapies. However, in some cases, SAR values may change appreciably as temperature increases as, for example, in self-regulating MNPs, so that SAR(T) data must be used. In addition, the determination of SAR over wider T ranges may provide valuable information, such as the location of the FM/SPM transition, helpful to the optimization of MNP heating ability at low H ₀ values, or the presence of dynamic effects of MNPs in fluids, leading to an overestimation of their heating ability compared to its in vivo performance.

Also, many analytical and numerical models neglect the unavoidable magnetic interparticle interactions, given the MNP concentrations involved in magnetic hyperthermia. But recently, many studies have highlighted the great influence of this phenomenon in SAR. Among them, dipolar interaction is the most relevant in MNPs systems. This is a long-range interaction which depends on the relative orientation of the MNP magnetic moments and on the interparticle distances, so that computing its effects in realistic systems with millions of individual particles is a hard task. Different analytical models, numerical simulations, and experimental measurements have revealed the complexity of this problem. Dipolar interactions modify the energy barrier established by the magnetic anisotropy of the individual MNPs and, in general, generate a decrease in SAR, although some magnetic states of the noninteracting MNPs may result in a SAR increase under moderate interaction strength. In addition, the formation of different 1D, 2D and 3D arrangements of MNPs increases difficulty, since it has been found that such arrangements influence SAR differently than MNP arrangements with homogeneous interparticle distances.

The study of the influence of the concentration and arrangement of MNPs in SAR is expected to remain a very active research line in the next future, since it aims at reducing the often encountered discrepancies between the SAR values, usually measured in ferrofluids, and the temperature distributions acquired in hyperthermia applications, in which MNPs are internalized by cells and often confined in densely packed vesicles. In this sense, a more accurate determination of SAR on more adequate arrangements (e.g., solid matrices, phantoms, biopsies) would improve the accordance between simulations and real temperatures during therapies. Also, the development of further numerical simulations deriving general trends and conclusions would help increasing this accordance. And, eventually, the use of nanoobjects with an initial favorable MNP arrangement could also aid reducing these discrepancies, as well as the deleterious effects of the uncontrolled arrangement of MNPs on SAR.

7 Tables of Acronyms and Symbols

See Tables 8.4 and 8.5

Table 8.4 Table of acronyms

Table 8.5 Table of physical quantity symbols