Mechanisms of hyperthermia in magnetic nanoparticles

Magnetic hyperthermia occurring in colloids of magnetic nanoparticles has been demonstrated to reduce tumour size in human beings [1]. However, at the present time the mechanism by which the heat is generated is not fully understood [2]. In this Fast Track Communication we report a detailed study of the heating properties of magnetic nanoparticles of different sizes for hyperthermia applications. In particular we have modified the colloidal properties for three samples with different particle sizes so as to control the mechanisms leading to heat dissipation.

Several heating mechanisms are possible, associated with susceptibility loss, hysteresis loss and viscous heating, i.e. stirring. Susceptibility loss occurs in superparamagnetic particles and has two relaxation times associated with Néel relaxation and Brownian rotation of the particles as they are in a liquid environment [3]. The Néel (τ_N) and Brownian (τ_B) relaxation times are given by

$$\begin{equation} \tau_{{\rm N}} =\frac{1}{f_{0}}\exp \left[ {\frac{KV(1-H/H_{K} )^{2}}{k_{{\rm B}} T}} \right] \end{equation} \tag{ 1 }$$

$$\begin{equation} \tau_{{\rm B}} =\frac{3V_{{\rm h}} \eta}{k_{{\rm B}} T}, \end{equation} \tag{ 2 }$$

where f₀ = 10⁹ s⁻¹, K is the anisotropy constant of the particles (3 × 10⁵ ergs cm⁻³ in the case of magnetite particles) [4], V their volume, H is the applied field, H_K is the anisotropy field, k_B is Boltzmann's constant and T is the temperature of measurement. Equation (2) for the Brownian relaxation is controlled by hydrodynamic parameters. Hence V_h is the hydrodynamic volume and η is the viscosity of the medium. The hydrodynamic volume V_h is not a well defined parameter because in colloidal dispersions the particles are coated with one or more dispersants which may form multiple layers on the surface. Also as the particles move, entrained molecules of the carrier liquid will also contribute to the overall value of V_h. Generally V_h is measured using techniques such as photon correlation spectroscopy (PCS). In practice a 10 nm diameter particle coated with oleic acid, one of the most commonly used dispersants, has a typical hydrodynamic size of about 25–75 nm.

The existence of two relaxation times leads to a combined relaxation time τ given by

$$\begin{equation} \tau =\frac{\tau_{{\rm N}} \tau_{{\rm B}}}{\tau_{{\rm N}} +\tau_{{\rm B}}}. \end{equation} \tag{ 3 }$$

Above a critical diameter in zero or a small field (D_p(0)) the particles will be blocked and hence heating due to susceptibility losses will not occur. This critical diameter is given by [5],

$$\begin{equation} D_{{\rm p}} (0)=\left( {\frac{6k_{{\rm B}} T\ln (tf_{0} )}{\pi K}} \right)^{1/3}, \end{equation} \tag{ 4 }$$

where t is the time of measurement. Above D_p(0) the particles will switch and heating due to hysteresis loss will occur. D_p(0) denotes the critical size in zero field. For hyperthermia applications an ac field of up to 200 Oe at a frequency of around 100 kHz is generally used [6]. The low amplitude of the field will limit the fraction of blocked particles that undergo reversal and, hence, hysteresis heating. At 111.5 kHz, the value used in our experiments, the coercivity (H_c) variation with size (D) is given by [7],

$$\begin{equation} H_{{\rm c}} =H_{K} \left[ {1-\left( {\frac{54.6k_{{\rm B}} T}{\pi KD^{3}}} \right)^{1/2}} \right], \end{equation} \tag{ 5 }$$

where the factor 54.6 is the value of 6 ln(tf₀) evaluated at 111.5 kHz. This means that particles above a given diameter (D_p(H)) will not contribute to either susceptibility or hysteresis loss heating. The value of this diameter depends upon the applied field H and can be written as

$$\begin{equation} D_{{\rm p}} (H)=\left[ {1-\frac{HM_{{\rm s}}}{0.96K}} \right]^{-2/3}D_{{\rm p}} (0), \end{equation} \tag{ 6 }$$

where M_s is the saturation magnetisation of the particles and the factor 0.96 arises due to the random distribution of easy axes i.e. H_K = 0.96K/M_s [8]. We have used a value of K of 3 × 10⁵ erg cm⁻³ which corresponds to an average particle elongation of r = 1.8. From transmission electron microscopy (TEM) images (e.g. figure 2) this choice seems reasonable and is the value reported for particles prepared by a similar route [4]. Only those particles with a diameter D_p(0) < D < D_p(H) will contribute to hysteresis loss heating. This is shown in figure 1.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For particles with D > D_p(H) the moments cannot switch in the 250 Oe field. The majority of the particles are not aligned with the solenoidal field and viscous heating due to stirring of the particles in the colloid will occur. At a frequency of 100 kHz (6 × 10⁶ rpm) this effect, which will be more pronounced for aggregated particles, probably accounts for some of the anomalously large heating effects that have been reported (e.g. [9]). The presence of a solenoidal field with a finite RMS value will also lead to the gradual alignment of bigger particles and aggregates thus accounting for non-linear heating rates that have also been reported (e.g. [10]).

This leads to three contributions to the heat generated in the system: susceptibility losses (P_ac), hysteresis loss (P_hys) and magnetic stirring (P_stir). The power generated through susceptibility losses is given by [3],

$$\begin{equation} P_{{\rm ac}} =\pi f\chi''H^{2}, \end{equation} \tag{ 7 }$$

where f is the frequency of measurement (= 1/t) and χ'' is the complex part of the ac susceptibility. χ'' can be written as

$$\begin{equation} \frac{\chi''}{\chi_{0}}=\frac{2\pi f\tau}{1+(2\pi f\tau )^{2}}, \end{equation} \tag{ 8 }$$

where χ₀ is the dc initial susceptibility and τ is the combined relaxation time from superparamagnetic Néel and Brownian processes. In real systems a distribution of particle volumes and hydrodynamic sizes and, hence, relaxation times occurs given by

$$\begin{equation} \tau_{{\rm N}}^{{\rm eff}} =\int_0^{V_{{\rm p}} (0)} {\tau_{{\rm N}} (V)f(V)\,{\rm d}V} \end{equation} \tag{ 9 }$$

$$\begin{equation} \tau_{{\rm B}}^{{\rm eff}} =\int_0^{V_{\rm h}(0)} {\tau_{{\rm B}} (V_{{\rm h}} )f(V_{{\rm h}} )\,{\rm d}V_{{\rm h}}} , \end{equation} \tag{ 10 }$$

where $\tau_{{\rm N}}^{{\rm eff}}$ and $\tau_{{\rm B}}^{{\rm eff}}$ are the effective relaxation times, f(V) is the distribution of physical volumes and f(V_h) is the distribution of hydrodynamic volumes, V_p(0) is the volume of a particle having a diameter D_p(0) and V_h(0) is the hydrodynamic volume of a particle having a physical diameter D_p(0).

In the case of hysteresis heating the amount of heat generated is proportional to the frequency multiplied by the area of the loop. However, it is only the irreversible part of the magnetisation that contributes to this effect. Since the particles are randomly oriented and, hence, their remanence is equal to 0.5M_s, and only particles with a diameter D[D_p(0) < D < D_p(H)] are blocked and can be switched in the field H, P_hys can be written as

$$\begin{equation} P_{{\rm hys}} =2M_{{\rm s}} \int_{V_{{\rm p}} (0)}^{V_{{\rm P}} (H)} {H_{{\rm c}} (V)f(V)\,{\rm d}V} , \end{equation} \tag{ 11 }$$

where V_p(H) is the volume of a particle having a diameter D_p(H).

A distribution of anisotropy constants is likely to occur although a uniform value of K has been assumed in our calculations. The effect of this assumption is to imply that the actual switching field distribution (SFD) will be broader than the particle size distribution. A simple qualitative consideration of the effect of a distribution of K shows that it would broaden the overall energy barrier distribution. Hence calculations based on a constant value of K will give an overestimate of the expected heating. Assuming a Gaussian distribution of K with a standard deviation of 0.2 times the mean, a reduction in the calculated value of the specific absorption rate (SAR) of about 50% would result depending on the region of the SFD that generated the hysteresis heating. For each grain size the limitation of hysteresis heating would be at a value of V associated with the median value of the anisotropy constant. There is no known technique for the measurement of the distribution of anisotropy constants in fine particle systems. Hence at the present time our theory is as accurate as possible.

The calculation of the heat generated by the frictional drag on the particle (P_stir) is extremely complex. We have no value for the drag coefficient between a coated nanoparticle and the liquid. However, the flow of the liquid around such particle will almost certainly be turbulent and hence none of the standard equations of hydrodynamics apply. Nonetheless magnetic stirring of the liquid will lead to a frictional heating effect. This effect is not only impossible to calculate but also impossible to control. Stirring may also lead to a lag between the alignment of the moment and the field which will create another form of magnetic heating similar to the susceptibility loss.

The amount of heat generated by magnetic nanoparticles is normally quantified in terms of the SAR which represents the power generated per unit mass of magnetic material in the solution. Depending on the exact procedure followed to measure the SAR, widely differing results may be obtained. Furthermore factors such as container shape and sample volume can lead to different SAR values for the same particles [11]. SAR is given by

$$\begin{equation} {\rm SAR}=\frac{C\rho}{\phi}\frac{\Delta T}{\Delta t}, \end{equation} \tag{ 12 }$$

where C is the specific heat of the colloid, φ is the concentration of Fe per ml of solution, ρ is the density of the colloid and ΔT/Δt represents the heating rate. Equation (12) does not take into account the heat capacity of the sample holder. The heating of the sample holder will use a significant proportion of the power generated. Hence, it is crucial that all measurements are made using identical conditions in the same volume of sample. However, this also suggests that it might not be possible to compare measurements made in different measurement systems.

All samples studied in this work were supplied by Liquids Research Ltd [12]. The particles were prepared by the co-precipitation method [13], with controlled growth conditions giving a narrow size distribution. The particles are nominally Fe₃O₄ but the exact composition will lie between Fe₃O₄ and Fe₂O₃. The samples were dispersed in two solvents with different viscosities to study the effect of the viscosity of the suspension on the heating properties of the samples and did not have identical surfactant coatings. The solvents used were the isoparaffin oils Isopar M and Isopar V [14], which have viscosities of 3.0 cP and 10.8 cP, respectively. The samples were also dispersed in a wax to eliminate heating effects arising from Brownian relaxation of the particles and viscous heating. The critical diameter above which hysteresis heating will occur is independent of the colloid environment. The matrix used will only affect Brownian and viscous heating effects. For in vivo applications the viscosity is likely to be significantly greater than in a colloid, for example the viscosity of blood is 10 cP, and hence dynamic effects will be suppressed if not eliminated.

The physical median particle size (D_m) for each sample was measured by TEM using a 200 keV JEOL 2011 TEM. Samples were prepared by placing one drop of a dilute suspension of the particles in hexane on a carbon coated grid and allowing the solvent to evaporate. Over 500 particles were measured using a Zeiss particle size analyser to ensure good statistics. This technique uses an equivalent area process to obtain the particle diameter. The data were fitted to a lognormal distribution.

$$\begin{equation} f(D)\,{\rm d}D=\frac{1}{\sqrt {2\pi} \sigma D}\exp \left[ {-\frac{(\ln D-\overline {\ln D} )^{2}}{2\sigma^{2}}} \right]{\rm d}D \end{equation} \tag{ 13 }$$

Measuring a large number of particles is essential to determine the standard deviation of the distribution (σ) accurately. Note that σ is the standard deviation of ln(D). Figures 2(a), (b) and (c) show typical bright field TEM images for the samples studied in this work. Figure 2(d) shows the particle size distribution for each sample. The median particle size and standard deviation for each distribution are summarized in table 1. We note that the TEM images show a proportion of elongated particles. This is a facet of the preparation process. High resolution TEM imaging (not shown) indicates that the elongated particles are not polycrystalline.

A PCS (Malvern Instruments Zetasizer) was used to measure the hydrodynamic size distribution for each sample. The viscosity of the colloids was measured at 27 °C using a Wells-Brookfield cone and plate viscometer. Details of the median hydrodynamic size (D_h) and the standard deviation of the distribution (σ_h) as well as the viscosity (η) are summarized in table 1.

Figure 3 shows the susceptibility loss peak given by the complex part of the ac susceptibility (χ'') for the samples dispersed in Isopar V. The position of the ac loss peak varies over an order of magnitude depending on the physical/hydrodynamic properties of the samples. Interestingly, there is no monotonic variation of the position of the peak with the median physical particle size. This highlights the importance of the distribution of hydrodynamic sizes when calculating susceptibility losses. The solid lines in figure 3 are calculated fits from equations (8), (9) and (10). The calculation of χ''(f) from equation (8) used a value of χ₀ measured for each sample using an alternating gradient force magnetometer. This data shows that for typical frequencies of ∼100 kHz used in hyperthermia applications the susceptibility losses are due to Brownian relaxation. It is well established that in these colloids two susceptibility mechanisms occur as given by equations (1) and (2). The volume dependence in these formulae indicates that a transition from Néel to Brownian reversal occurs at ∼8 nm depending on the viscosity of the colloid. At this size the Néel relaxation frequency is of the order of MHz and will make little contribution at ∼100 kHz. This susceptibility loss is beyond the range of our susceptometer but its contribution at 111.5 kHz is negligible as shown in figure 3. For samples dispersed in wax Néel relaxation will be the only susceptibility loss mechanism as the particles will not be able to physically rotate.

Zoom In Zoom Out Reset image size

The heating properties of the colloids were measured using a Nanotherics Magnetherm system [15]. The system consists of a sample coil enclosure, a function generator and a power amplifier forming a resonant circuit. The system is monitored by an output from the coil to an oscilloscope. The samples were placed in a 15 ml tube (10 mm × 100 mm) which was then placed inside an expanded polystyrene jacket which acted as an insulator. The ac frequency is varied by changing the capacitor and coil configuration. This allowed us to study the heating characteristics of the samples at different fields and constant frequency. All measurements were made under identical conditions and concentrations and are therefore comparable. Figure 4 shows the SAR for all samples as a function of the applied field. For Isopar M and Isopar V, C = 2206 J kg⁻¹ K⁻¹ and for the wax C = 2140 J kg⁻¹ K⁻¹. The mass ratio of particles to liquid φ = 20 mg Fe ml⁻¹ for all samples. This concentration of iron corresponds to approximately 28 mg of Fe₃O₄ per ml or a ferrofluid with a concentration of 30 G. In colloids of these concentrations and with the dispersion quality that can be seen in the TEM images, the effect of dipole–dipole interactions will be small. However, we cannot exclude the possibility that aggregates or flocculates of particles would affect the hysteresis. These results together with those for hysteresis loss are shown in table 2.

Zoom In Zoom Out Reset image size

Table 2 shows the contribution to the SAR from hysteresis and susceptibility loss effects for all samples which were calculated from equations (11) and (7), respectively. From the calculated data it is clear that the contribution to the SAR from susceptibility loss is very small except for sample A where it is 17% of the total. For samples B and C it is insignificant. Hence for in vivo use with tissue having a minimum viscosity of 10 cP there will be little or no contribution to the SAR from susceptibility loss.

The other immediate result from table 2 is the significant contribution of viscous heating (stirring) to the SAR which reduces with viscosity. The values in wax indicate that 30–40% of the SAR is generated by viscous heating. The dynamic effects are complex because they are determined by hydrodynamic parameters. Reference to table 1 shows that sample B has the largest hydrodynamic size which accounts for the larger contribution of stirring to the SAR for this sample. One clear conclusion that can be drawn form these results is that measurements made in water based colloids will not be representative of heating effects in vivo.

Table 2 also shows the measured values of SAR together with calculated values of hysteresis heating from equation (11). The results for the samples dispersed in wax can be compared with the calculated values for hysteresis loss because in the solid matrix and at a frequency of 111.5 kHz only hysteresis heating can occur. On first inspection the agreement between the measured and calculated values is quite poor. However the integral across the active region of the particle size distribution is very sensitive to the value of K used (K = 3.0 × 10⁵ ergs cm⁻³) which affects the values of V_p(0) and V_p(H) in equation (11). Depending where these parameters lie on the particle size distribution large variations are possible. This problem is due to the extreme sensitivity of the value of the shape anisotropy constant (K_s) to particle elongation at low aspect ratios (r). For example a change in r from 1.8 to 3.0 varies K_s by 70% leading to a reduction in the SAR value for sample C by a factor 7. Inspection of the TEM image for this sample shown in figure 2(c) shows that there is a significant fraction of the particles having aspect ratios of 3.0 or greater with the majority having aspect ratios of greater than the value of 1.8 used to obtain the data in table 2. Hence numerical agreement between calculated and measured values would not be expected. This is particularly true for sample C with D_m = 15.2 nm because at this size the colloid is unstable and sedimentation was seen to occur. This accounts for the very low value of SAR observed. For a complete agreement to be achieved it will be necessary for the distribution of particle elongation and hence the distribution of K_s to be included in the model. An attempt at this challenging measurement is being undertaken.

In conclusion, heating arising in magnetic nanoparticles dispersed in solvents of different viscosity can be understood when the distributions of physical and hydrodynamic sizes are known to high accuracy. For the materials described here the contribution of susceptibility loss to the SAR is small at 111.5 kHz and hysteresis heating is dominant. Measurements of samples dispersed in wax show that the contribution of stirring is about 43% of the heating effect. These effects are uncontrollable and will not occur in vivo. Meaningful SAR values should be obtained from measurements of samples in a solid form.

OW acknowledges financial support from the UK Institute of Physics under the Top 40 work placements scheme. AGR acknowledges financial support from the Spanish Ministerio de Educación, through Programa Nacional de Movilidad de Recursos Humanos of Plan Nacional of I-D+i 2008–2011. This work was funded in part by the MULTIFUN FP7 project under contract number 262943.