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J Magn Magn Mater. Author manuscript; available in PMC 2011 March 1.

Published in final edited form as:

J Magn Magn Mater. 2010 March 1; 322(6): 727–733.

doi: 10.1016/j.jmmm.2009.10.050PMCID: PMC2811342

NIHMSID: NIHMS158269

P. Cantillon-Murphy: ude.tim@giardap; L.L. Wald: ude.dravrah.hgm.rmn@dlaw; E. Adalsteinsson: ude.tim@rafle; M. Zahn: ude.tim@nhaz

See other articles in PMC that cite the published article.

In the presence of alternating-sinusoidal or rotating magnetic fields, magnetic nanoparticles will act to realign their magnetic moment with the applied magnetic field. The realignment is characterized by the nanoparticle’s time constant, τ. As the magnetic field frequency is increased, the nanoparticle’s magnetic moment lags the applied magnetic field at a constant angle for a given frequency, Ω, in rad/s. Associated with this misalignment is a power dissipation that increases the bulk magnetic fluid’s temperature which has been utilized as a method of magnetic nanoparticle hyperthermia, particularly suited for cancer in low-perfusion tissue (e.g., breast) where temperature increases of between 4°C and 7°C above the ambient *in vivo* temperature cause tumor hyperthermia. This work examines the rise in the magnetic fluid’s temperature in the MRI environment which is characterized by a large DC field, *B*_{0}. Theoretical analysis and simulation is used to predict the effect of both alternating-sinusoidal and rotating magnetic fields transverse to *B*_{0}. Results are presented for the expected temperature increase in small tumors (~1 cm radius) over an appropriate range of magnetic fluid concentrations (0.002 to 0.01 solid volume fraction) and nanoparticle radii (1 to 10 nm). The results indicate that significant heating can take place, even in low-field MRI systems where magnetic fluid saturation is not significant, with careful selection of the rotating or sinusoidal field parameters (field frequency and amplitude). The work indicates that it may be feasible to combine low-field MRI with a magnetic hyperthermia system using superparamagnetic iron oxide nanoparticles.

Recent years have seen the development of an entire field of research dedicated to the treatment of cancer patients, based on the localized heating effects of magnetic nanoparticles *in vivo* [1,2,4–6]. Treatment falls under one of two categories, hyperthermia and thermoablation.

In hyperthermia, the target area is subjected to a local temperature in the range of 42°C to 45°C for periods of up to a few hours [4]. This temperature range is destructive to cancerous cells without harming healthy cells. The end result is usually not sufficient to eradicate all the cancerous cells and is coupled with other techniques such as irradiation or chemotherapy [6]. Thermoablation aims at creating *in vivo* temperatures in excess of 50° in the tumor region and exposure time is limited to just minutes. Although this approach would appear preferred, there are concerns regarding the physiological effects of such a rapid, localized heating effect [7]. To date, particles that have been used are either injected *in situ* or are designed to bind selectively to cancerous cells. The majority of investigations use superparamagnetic magnetite (Fe_{3}O_{4}) or maghemite (γ-Fe_{2}O_{3}) in water based suspensions since these are well metabolized. Creating *in vivo* concentrations to allow for sufficient heating is a major challenge due to heat conduction away from the target area as well as blood perfusion around the tumor [4]. A second significant challenge is the undesired heating associated with eddy currents in surrounding healthy tissue [8]. Brezovich [8] found that “subjects had a sensation of warmth but were able to withstand the treatment for more than an hour” when the product of the field amplitude *H _{e}* and the frequency

Rosensweig [2] examined the heating effect of alternating-sinusoidal RF fields on superparamagnetic fluid suspensions. His work developed dissipation relationships based on the rotational relaxation of single-domain magnetic nanoparticles dispersed in a tetradecane solvent due to sinusoidal magnetic fields. Eddy current heating due to ohmic dissipation is assumed to be negligible due to the small size of the particles (< 20 nm core diameter) compared to the skin depth. Rosensweig expressed the volumetric power dissipation using a simplified version of Shliomis’ Relaxation Equation [17] in the absence of so-called “spin-velocity” effects for a magnetic fluid suspension in an alternating-sinusoidal RF field. This present work follows Rosensweig’s approach and introduces two new significant developments: (i) the analysis is revised for the presence of rotating as well as alternating-sinusoidal magnetic fields and (ii) the analysis is examined in the MRI environment, where, it is expected that the heating effect will be limited due to the magnetic fluid approaching saturation in the presence of the large DC magnetic field, denoted *B*_{0}, characteristic of MRI. The notion of a combined MRI/hyperthermia unit has existed for some time [10,11] using RF ablation techniques. This work represents the first known analytical study accounting for both magnetic saturation effects due to the MRI and the heating effects of superparamagnetic nanoparticles in a rotating field, for purposes of hyperthermia treatment.

The analysis is performed in two distinct phases. Firstly, the fluid magnetization is solved as a function of all the variables of interest following and extending Rosensweig’s approach [2]. Next, the resultant temperature increase in low perfusion tissue is solved using established results for heat conduction from a spherical volume (representing a small tumor) with uniform heat generation (due to the heating effect of the magnetic nanoparticles in the rotating magnetic field) to the surrounding medium (see Carslaw and Jaeger,[3] pp.232), in a similar manner to the results obtained by Andra et al. [18].

The governing equation on a magnetic fluid’s equilibrium magnetization, **M**_{eq}, in a magnetic field is the Langevin Relation [1] given by (1) and (2).

In the presence of either alternating-sinusoidal or rotating magnetic fields, the fluid’s magnetization is governed by:

In general, the Langevin relation determines the extent of magnetic saturation of the nanoparticles in the suspension. This depends on the strength of the applied field. Two fields are considered in this work, one either alternating-sinusoidal or rotating in the transverse *xy* plane and the second, the large *z*-directed DC *B*_{0} field associated with the MRI. In the absence of *B*_{0}, the Langevin function will be determined by the “chord susceptibility”, following Rosensweig’s approach [2]. In the presence of both transverse and *z*-directed fields, only *B*_{0} will be considered in determining the Langevin relation since, for the analysis which follows, the alternating-sinusoidal or rotating fields do not exceed 10% of *B*_{0}, when it is present. Therefore, the *z*-directed component of the equilibrium magnetization, **M**_{eq}, is denoted *M*_{0}, and in the presence of a *z*-directed, DC magnetic field with magnitude *H*_{0}, is given by (1) where $\alpha =\frac{{\mu}_{0}{V}_{p}{M}_{d}{H}_{0}}{kT}$ and *M*_{0} *H*_{0} for large field strengths (μ_{0}*H*_{0} > 0.1 T). The nanoparticle magnetic volume is *V _{p}*, the single-domain magnetization,

$${\mathbf{M}}_{\mathit{\text{eq}}}={\mathbf{M}}_{s}L(\alpha )={\mathbf{M}}_{s}(\text{coth}(\alpha )-1/\alpha )$$

(1)

$$\alpha =\frac{{\mu}_{0}{V}_{p}{M}_{d}|\mathbf{H}|}{kT}$$

(2)

$${\mu}_{0}(\mathit{M}\xb7\nabla )\mathit{H}+2\zeta \nabla \phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}\mathit{\omega}+(\zeta +\eta ){\nabla}^{2}\mathit{\upsilon}=0$$

(3)

$${\mu}_{0}(\mathit{M}\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}\mathit{H})+2\zeta (\nabla \phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}\mathit{\upsilon}-2\mathit{\omega})=0$$

(4)

$$\frac{\partial \mathbf{M}}{\partial t}+\mathit{\upsilon}\xb7\nabla \mathbf{M}-\mathit{\omega}\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}\mathbf{M}+(\mathbf{M}-{\mathbf{M}}_{\mathit{\text{eq}}})/\tau =0$$

(5)

For the simplified expressions of (3) and (4), the fluid linear flow velocity vector is **υ**, the magnetic fluid spin-velocity vector is **ω**, η is the dynamic shear viscosity of the ferrofluid in N·s·m^{−2}, measured as 0.002 by Elborai when ϕ = 0.035 [13], ζ is the ferrofluid vortex viscosity in N·s·m^{−2}, given by $\frac{3\eta \varphi}{2}$ for ϕ 1, and the μ_{0}(** M** ×

Magnetic relaxation of the superparamagnetic nanoparticles is governed by Shliomis’ Relaxation Equation, given by (5) where **M** is the instantaneous fluid magnetization and the magnetic fluid relaxation time [1] is τ, given by (6). The time constant’s dependence on nanoparticle radius [1,15] is plotted in Figure 1.

$$\frac{1}{\tau}=\frac{1}{{\tau}_{B}}+\frac{1}{{\tau}_{N}}$$

(6)

The magnetic fluid time constant, τ, is usually dominated by either the Néel (τ_{N}) or Brownian (τ_{B}) relaxation times, depending on the particle’s radius.

As seen from (6), τ, is due to two distinct contributions. These are the Brownian (τ_{B}) and Néel (τ_{N}) relaxation times. For Brownian relaxation, which dominates at larger particle sizes, relaxation is due to particle rotation in the carrier liquid. For Néel relaxation, which is the dominant mechanism for smaller particle sizes (≤ 5 nm), the mechanism is due to rotation of the magnetic vector within the particle. If the nanoparticle is constrained, for example by attaching it to a tumor surface, Néel relaxation is still operative although Brownian is not. Therefore, in the context of hyperthermia treatment, it is the Néel relaxation which accounts for the majority of the therapeutic effect. The relative expressions for τ_{B} and τ_{N} are given by (7) and (8) respectively where *V _{h}* is the nanoparticle hydrodynamic volume in m

$${\tau}_{B}=\frac{3{V}_{h}\eta}{kT}$$

(7)

$${\tau}_{N}={\tau}_{0}\phantom{\rule{thinmathspace}{0ex}}{e}^{\frac{{K}_{a}{V}_{p}}{kT}}$$

(8)

Complex notation is employed for convenience where the complete field solutions to (3) through (5) have sinusoidal time variation as given by (9) through (11) and *i*_{x}, *i*_{y} and *i*_{z} are the unit vectors in a rectangular coordinate system. To facilitate analytical solutions, consider the *x* and *z*-directed magnetic fields to be applied by means of *z* and *x*-directed planar surface current sources at *y* = 0, *d* respectively in a planar channel of width *d* along *y*, and of infinite extent in *x* and *z*. In this case, the magnetic fields are not functions of *x* and *z*, but, at most, only a function of *y*. However, since the magnetic field in a current-free media has a curl × ** H** =

In a realistic MRI/hyperthermia system, the transverse field might be applied by means of birdcage or planar current-driven coil and the *z*-directed DC field is the *B*_{0} field of the MRI. However, the resulting fluid magnetization and associated spin-velocity are not analytically solvable. Therefore, the notion of a planar channel is introduced for convenience to facilitate analytical solutions.

$$\mathbf{M}=\Re e\{({\hat{m}}_{x}(y){\mathit{i}}_{x}+{\hat{m}}_{y}(y){\mathit{i}}_{y}){\mathrm{e}}^{j\mathrm{\Omega}t}\}+{M}_{0}{\mathit{i}}_{z}$$

(9)

$$\mathbf{H}=\Re e\{({\hat{h}}_{x}{\mathit{i}}_{x}+{\hat{h}}_{y}(y){\mathit{i}}_{y}){\mathrm{e}}^{j\mathrm{\Omega}t}\}+{H}_{0}{\mathit{i}}_{z}$$

(10)

$$\mathbf{B}=\Re e\{({\hat{b}}_{x}(y){\mathit{i}}_{x}+{\hat{b}}_{y}{\mathit{i}}_{y}){\mathrm{e}}^{j\mathrm{\Omega}t}\}+{B}_{0}{\mathit{i}}_{z}$$

(11)

In the absence of any applied pressure differential or imposed linear flow (**υ** = 0), the solutions of (5) for the *x* and *y* directed transverse magnetization components, associated with the applied transverse rotating field are given by (12) and (13) [15].

$${\hat{m}}_{x}=\frac{{M}_{0}}{{H}_{0}}\frac{(j\mathrm{\Omega}\tau +1+{M}_{0}/{H}_{0}){\hat{h}}_{x}-({\omega}_{z}\tau ){\hat{b}}_{y}/{\mu}_{0}}{(j\mathrm{\Omega}\tau +1)(j\mathrm{\Omega}\tau +1+{M}_{0}/{H}_{0})+{({\omega}_{z}\tau )}^{2}}$$

(12)

$${\hat{m}}_{y}=\frac{{M}_{0}}{{H}_{0}}\frac{({\omega}_{z}\tau ){\hat{h}}_{x}+(j\mathrm{\Omega}\tau +1){\hat{b}}_{y}/{\mu}_{0}}{(j\mathrm{\Omega}\tau +1)(j\mathrm{\Omega}\tau +1+{M}_{0}/{H}_{0})+{({\omega}_{z}\tau )}^{2}}$$

(13)

In the limit of very low frequency excitation (Ωτ 1) and zero spin-velocity (ω_{z}τ ~ 0), the transverse *xy* magnetization components, _{x} and _{y}, are related to the fields causing them (*i.e.*, *h _{x}* and

$$\begin{array}{c}{\hat{m}}_{x}\approx \frac{{M}_{0}}{{H}_{0}}{h}_{x}\hfill \\ {\hat{m}}_{y}\approx \frac{{M}_{0}}{{H}_{0}}\frac{{b}_{y}}{{\mu}_{0}}\hfill \end{array}$$

(14)

The spin-velocity, **ω**, shown in the expressions of (12) and (13) was solved in the absence of imposed flow (**υ** = 0) by analysis of the conservation of angular (4) and linear momentum (3) with the solution being *z*-directed (**ω** = ω_{z}*i*_{z}) and given by (15) [1,15]. In the thermodynamic analysis which follows there is no imposed flow.

$${\omega}_{z}=\frac{1}{8\zeta}\Re e({\hat{m}}_{x}{\hat{b}}_{y}^{*}-{\mu}_{0}({\hat{h}}_{x}+{\hat{m}}_{x}){\hat{m}}_{y}^{*})$$

(15)

Since _{y} and *ĥ*_{x} are known imposed source terms, (12), (13) and (15) can be solved for the transverse magnetization complex amplitudes, _{x} and _{y} considering the relationship of (16) where μ_{0} = 4π × 10^{−7}, the magnetic permeability of free space.

$$\mathbf{B}={\mu}_{0}(\mathbf{H}+\mathbf{M})$$

(16)

From the first law of thermodynamics, the differential internal heat density, *dU* (which is independent of process path) for a closed system of constant density is given by (17) where δ*Q* is the differential energy input and δ*W* is the differential work done on the system. Both δ*Q* and δ*W* are process path-dependent, leading to the partial derivative notation.

$$\mathit{\text{dU}}=\delta Q+\delta W$$

(17)

For an adiabatic process, it can be assumed that δ*Q* is zero so that all generated heat remains within the system. This ignores, for example, blood perfusion.

$$\mathit{\text{dU}}=\delta W$$

(18)

The internal energy is then given by δ*W*, which, for a magnetic fluid with negligible conductivity is given by (19) [1].

$$\delta W=\mathbf{H}\xb7d\mathbf{B}$$

(19)

$$\delta W=\mathbf{H}\xb7d(\mathbf{H}+\mathbf{M})$$

(20)

$$\delta W={\mu}_{0}(\mathbf{H}\xb7d\mathbf{H}+\mathbf{H}\xb7d\mathbf{M})$$

(21)

For cyclic variations in ** H** and

$$\begin{array}{c}\hfill \Rightarrow \mathrm{\Delta}U=\mathrm{\Delta}W={\displaystyle \oint \delta W}\hfill \\ \hfill \Rightarrow \mathrm{\Delta}U={\displaystyle \oint (\mathbf{H}\xb7d\mathbf{H}+\mathbf{H}\xb7d\mathbf{M})}\hfill \end{array}$$

(22)

The first term on the right side of (22) is zero for cyclic variations in ** H** while the second term on the right side can be rewritten using the Chain Rule of Differentiation.

$$\mathrm{\Delta}U={\mu}_{0}{\displaystyle \oint d(\mathbf{M}\xb7\mathbf{H})-{\mu}_{0}{\displaystyle \oint \mathbf{M}\xb7d\mathbf{H}}}$$

(23)

Again, the first term on the right side is zero for cyclic variations in ** M** and

$$\mathrm{\Delta}U=-{\mu}_{0}{\displaystyle \oint (\mathbf{M}\xb7d\mathbf{H}})$$

(24)

As noted by Rosensweig [2], Δ*U* is positive in a magnetic fluid solution when **M** lags behind **H** as occurs when the alternating-sinusoidal or rotating field frequency is such that Ωτ ~ 1. The solution is expressed in terms of one complete time period *P* = 2π/Ω, in (25) for time-averaged heating. It is noted that the partial time derivative of (25) is correct for a stationary medium.

$$\langle \mathrm{\Delta}U\rangle =-{\mu}_{0}{\displaystyle {\int}_{0}^{P}(\mathbf{M}\xb7\frac{\partial \mathbf{H}}{\partial t})dt}$$

(25)

The expression of (25) is evaluated where **M** is the vector whose components are the complex amplitudes, _{x} and _{y}, given by (12) and (13). The time derivative of the **H** field is evaluated in (27) where **H** is given by (26).

$$\begin{array}{c}\mathit{H}=\Re e({\mathrm{e}}^{j\mathrm{\Omega}t}({\hat{h}}_{x}{\mathit{i}}_{x}+{\hat{h}}_{y}{\mathit{i}}_{y}))+{H}_{0}{\mathit{i}}_{z}\hfill \\ \frac{\partial \mathbf{H}}{\partial t}=\Re e(j\mathrm{\Omega}{\mathrm{e}}^{j\mathrm{\Omega}t}({\hat{h}}_{x}{\mathit{i}}_{x}+{\hat{h}}_{y}{\mathit{i}}_{y}))\hfill \end{array}$$

(26)

$$=\Re e(j\mathrm{\Omega}{\mathrm{e}}^{j\mathrm{\Omega}t}({\hat{h}}_{x}{\mathit{i}}_{x}+({\hat{b}}_{y}/{\mu}_{0}-{\hat{m}}_{y}){\mathit{i}}_{y}))$$

(27)

Substituting the expression of (27) and the transverse components of ** M** into (25) eventually yields (31). There can be no contribution from the

$$\langle \mathrm{\Delta}U\rangle =-\frac{{\mu}_{0}}{2}{\displaystyle {\int}_{0}^{P}\Re e(({\hat{m}}_{x}{\mathit{i}}_{x}+}{\hat{m}}_{y}{\mathit{i}}_{y}){\mathrm{e}}^{j\mathrm{\Omega}t}\xb7{\{j\mathrm{\Omega}{\mathrm{e}}^{j\mathrm{\Omega}t}({\hat{h}}_{x}{\mathit{i}}_{x}+({\hat{b}}_{y}/{\mu}_{0}-{\hat{m}}_{y}){\mathit{i}}_{y})\}}^{*})dt$$

(28)

$$=-\frac{{\mu}_{0}}{2}\Re e({\displaystyle {\int}_{0}^{P}({\hat{m}}_{x}{(j\mathrm{\Omega}{\hat{h}}_{x})}^{*}+{\hat{m}}_{y}{(j\mathrm{\Omega}({\hat{b}}_{y}/{\mu}_{0}-{\hat{m}}_{y}))}^{*})dt)}$$

(29)

$$=-\frac{{\mu}_{0}}{2}\Re e({\displaystyle {\int}_{0}^{P}({\hat{m}}_{x}(-j\mathrm{\Omega}{\hat{h}}_{x}^{*})+{\hat{m}}_{y}(-j\mathrm{\Omega}({\hat{b}}_{y}^{*}/{\mu}_{0}-{\hat{m}}_{y}^{*})))dt)}$$

(30)

$$=\frac{{\mu}_{0}}{2}\Re e(j\mathrm{\Omega}{\hat{m}}_{x}{\hat{h}}_{x}^{*}+j\mathrm{\Omega}{\hat{m}}_{y}({\hat{b}}_{y}^{*}/{\mu}_{0}))P$$

(31)

The last term of (30) contains _{y} multiplied by its complex conjugate (purely real) which is then multiplied by *j* so this term is purely imaginary and thus has no real part. It is eliminated from the expression in (31). The period *P*, is given by 2π/Ω so further simplification is possible.

$$\langle \mathrm{\Delta}U\rangle ={\mu}_{0}\Re e(j\pi ({\hat{m}}_{x}{\hat{h}}_{x}^{*}+{\hat{m}}_{y}({\hat{b}}_{y}^{*}/{\mu}_{0})))$$

(32)

Substituting for *ĥ*_{x} = *H _{e}* and

$$\langle \mathrm{\Delta}U\rangle ={\mu}_{0}\Re e(j\pi ({\hat{m}}_{x}{H}_{e}+{\hat{m}}_{y}(-j{B}_{e}/{\mu}_{0})))$$

(33)

The volumetric, time-averaged, power dissipation, *P*_{υ}, in W/m^{3} is then given by *P*_{υ} = ΩΔ*U*/(2π) as shown in (34). This expression equals equation (6) of Rosensweig’s work [2] when *B _{e}* is zero and the spin velocity in (12) and (13) is ignored.

$${P}_{\upsilon}={\mu}_{0}\Re e(\frac{j\mathrm{\Omega}}{2}({\hat{m}}_{x}{H}_{e}+{\hat{m}}_{y}(-j{B}_{e}/{\mu}_{0})))$$

(34)

While Rosensweig related this expression to the rate of temperature rise, the specific absorption rate (SAR) is the conventional measure of heating effects in human tissue. SAR is given by the volumetric power dissipation divided by the mass density, ρ_{m}, in kg/m^{3}, as in (35). As noted by Andra et al. [18], the value of ρ_{m} should be given by the mass of magnetic nanoparticles per unit mass of tissue rather than the value of the tissue mass density.

$$\mathit{\text{SAR}}={P}_{\upsilon}/{\rho}_{m}={\mu}_{0}\Re e(\frac{j\mathrm{\Omega}}{2{\rho}_{m}}({\hat{m}}_{x}{H}_{e}+{\hat{m}}_{y}(-j{B}_{e}/{\mu}_{0})))$$

(35)

The relation between SAR and temperature rise *in vivo* has been evaluated by Andra [18] using the result of Carslaw [3] for the particular case of a small, spherical tumor in relatively homogenous tissue with low blood perfusion (*e.g.*, muscle tissue or fat). Carslaw derives and states the radial and temporal dependencies inside (*r* < *R*) and outside (*r* > *R*) a sphere of radius *R* containing a uniform power dissipation source, *P*_{υ} for *r* < *R*. In this work, *P*_{υ} is due to the time-averaged power dissipation of the magnetic nanoparticles in the transverse magnetic field. The situation is approximately realized in breast tumors. While the full expressions ([3], pp.232) detail spatial and temporal variations in temperature, the steady-state temperature rise above ambient temperature, Δ*T*, on the surface of a spherical tumor of radius, *R*, is given by (36) where λ is the heat conductivity of tissue with an approximate value of 0.64 W·K^{−1}·m^{−1} [18].

$$\mathrm{\Delta}T=\frac{\mathit{\text{SAR}}\xb7{R}^{2}{\rho}_{m}}{3\lambda}=\frac{{P}_{\upsilon}{R}^{2}}{3\lambda}$$

(36)

The variation in the steady-state temperature rise is investigated for a number of cases of interest. The work represents two significant developments over that of Rosensweig [2]. Firstly, the case of rotating rather than purely alternating-sinusoidal magnetic field excitation is investigated. Secondly, the effect of excitation, with both sinusoidal and rotating fields, is investigated in the presence of a low-field 0.2 T MRI system. The rotating and sinusoidal fields are applied orthogonal to the DC *B*_{0} ≈ μ_{0}*H*_{0} field. In each case, the effect of variation in (i) particle radius, *r*, (ii) field amplitude, *H _{e}* =

This section examines heating in the absence of MRI with an alternating-sinusoidal magnetic field excitation. Following the analysis of Rosensweig, spin-velocity is zero (*i.e.*, ω_{z} = 0) in the case of an alternating-sinusoidal driving magnetic field. This leads to a simplification in the expressions for _{x} and _{y} in (12) and 13) as given by (37) and (38) where the sinusoidal excitation has amplitude *H _{e}* and is applied along the

$${\hat{m}}_{x}=\chi \frac{{\hat{h}}_{x}}{(j\mathrm{\Omega}\tau +1)}$$

(37)

$${\hat{m}}_{y}=0$$

(38)

Following Rosensweig’s formulation, the actual susceptibility is field-dependent and therefore, also time-dependent when the field is time-varying. In the absence of the *B*_{0} field characteristic of MRI, the only field is the alternating-sinusoidal field with amplitude, *H _{e}*. The field-dependent susceptibility, (denoted χ

$$\chi =\frac{\varphi {M}_{d}}{{H}_{e}}(\text{coth}(\alpha )-1/\alpha )$$

(39)

The field-dependent susceptibility, χ is plotted in Figure 2 as a function of particle radius for various values of field excitation, denoted *H _{e}*, in units of A/m. Clearly, as particle radius increases, the field-dependent susceptibility, increases. As already noted, in the absence of the MRI’s

The change in steady state temperature rise, Δ*T*, on the surface of a tumor of radius *R* = 1 cm is shown versus nanoparticle radius for changing values of particle concentration, ϕ, and μ_{0}*H _{e}* = 10 mT in Figure 4(a). It is assumed that all nanoparticles are confined within the tumor (

The effect on Δ*T* is shown versus particle radius with varying magnetic nanoparticle concentration (a) without MRI due to an alternating-sinusoidal magnetic field of 10 mT, (b) with a 0.2 T MRI field and an alternating-sinusoidal magnetic field **...**

This section examines the effect of heating due to an alternating-sinusoidal magnetic field in the presence of the *B*_{0} field of MRI. In the MRI, the large, *z*-directed *B*_{0} field is applied in addition to the transverse *x*-directed sinusoidal magnetic field. The primary effect of *B*_{0} on the magnetic fluid’s magnetization is that the saturation of the nanoparticles is determined primarily by *B*_{0} when *B*_{0} μ_{0}*H _{e}*. Since physically achievable alternating-sinusoidal or rotating fields are unlikely to exceed tens of mT [4] due to power requirements, the assumption that saturation is determined by

The effect of changing particle radius on the ratio of *M*_{0} to *H*_{0} is shown for increasing values of *B*_{0}, 0.1 T, 0.2 T and 0.35 T. The ratio saturates at large values of *r* as might be expected since the onset of magnetic saturation is earlier with increasing **...**

The change in steady state temperature rise, Δ*T*, on the surface of a tumor of 1 cm radius is shown versus nanoparticle radius for changing values of particle concentration, ϕ, and μ_{0}*H _{e}* = 10 mT in Figure 4(b) for an MRI field strength of 0.2 T. The steady state temperature rise is shown in Figure 5(b) for changing values of field amplitude,

The alternating-sinusoidal field of the previous section is now replaced by a rotating field of amplitude *H _{e}* and frequency of 1.88 × 10

Figure 4(a) and Figure 5(a) complement those of Rosensweig although the solution is now a water-based rather than an hydrocarbon-based (tetradecane) solution [2] as was the case in Rosensweig’s analysis. The time constant, τ, as plotted in Figure 1, varies with particle radius as shown in Figure 1. In the case of a constrained nanoparticle (such as the case of a particle which is bound to a tumor site) Néel relaxation is expected to dominate. Therefore, for the results of Figure 4 and Figure 5, the Brownian relaxation time in infinite (τ_{B} → ∞) and τ_{N} = τ.

As is evident from Figure 4 and Figure 5 the steady-state temperature rise is a maximum at a particle radius of approximately 5nm for a rotating field frequency of 300 kHz. As one might suspect from the expressions of (12) and (13), this is where the product of Ωτ approaches unity when Ω = 1.88 × 10^{6} rad/s or 300 kHz. This is true for both alternating-sinusoidal and rotating fields and is true regardless of the presence of the MRI’s *B*_{0} field.

After the addition of the *z*-directed *B*_{0} field which characterizes MRI, the most striking result is the large decrease in Δ*T*. This is because, for reasons of practical realizability, *B*_{0} is chosen to dominate the sinusoidal or rotating field amplitude, μ_{0}*H _{e}*, throughout this work. This decrease is greater as

Δ*T* will increase linearly in the presence of *B*_{0} although the dependence on *H _{e}* is non-linear as seen in Figure 5.

Adding a second orthogonal field component to the previous case of an alternating-sinusoidal field has the effect of creating a rotating field if the two components are temporally displaced by a quarter cycle. The results take account of the spin-velocity in the ferrofluid although the effects of spin-velocity only become significant in narrow channels (μm) or with large, imposed channel flow velocities (≥ 10 m/s) [15]. The term is nonetheless included for completeness. Therefore, the results for a rotating field in addition to the MRI’s *B*_{0} field show an approximate factor of two increase in Δ*T*, as might be expected, compared to the case of a purely alternating-sinusoidal field albeit at the cost of a second RF power amplifier for the *y* component of the rotating field.

This paper examines the potential for magnetic nanoparticle hyperthermia in low-field MRI (0.2 T) where the nanoparticles are still free to rotate due to incomplete magnetic saturation, as determined by the Langevin function. The conclusion of the analysis is that, perhaps contrary to intuition, significant nanoparticle heating is still conceivable, due to Néel relaxation effects, even at 0.2 T MRI field strength. Nanoparticles with a magnetic core radius corresponding to Ωτ = 1 show the most significant effect. This is clear from the results in Figure 4 and Figure 5 where, although the steady-state temperature increase in a 1 cm radius spherical tumor (assuming no perfusion) is decreased significantly by the addition of the MRI field, temperature increases of up to 10°C are still possible with reasonable values of rotating field amplitude (~ 10% of *B*_{0}) and particle concentration (ϕ ~ 0.01). This is well within the range of 4°C to 7°C typical in hyperthermia treatment for breast cancer.

To the author’s best knowledge, this work represents the first time that an analysis of this kind has been undertaken which bridges the rotational dynamics of magnetic nanoparticle suspensions (ferrofluids) [1,2] with two of the primary applications of these nanoparticles in modern medicine: magnetic nanoparticle hyperthermia [4–6] and MRI contrast agents. This work raises the possibility of combining these two applications in a single therapeutic environment, where, for example, magnetic nanoparticle hyperthermia for breast cancer treatment could be combined with low-field (*e.g.*, 0.2 T) MR imaging for real-time, non-invasive temperature measurement.

The authors would like to acknowledge the reviewer’s insightful and informed comments which greatly improved the form and consistency of the publication.

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