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

Published in final edited form as:

J Magn Magn Mater. 2010 September; 322(17): 2607–2617.

doi: 10.1016/j.jmmm.2010.03.029PMCID: PMC2901184

NIHMSID: NIHMS210166

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

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 The goal of this work is to examine, by means of analysis and simulation, the concept of interactive fluid magnetization using the dynamic behavior of superparamagnetic iron oxide nanoparticle suspensions in the MRI environment. In addition to the usual magnetic fields associated with MRI, a rotating magnetic field is applied transverse to the main *B*_{0} field of the MRI. Additional or modified magnetic fields have been previously proposed for hyperthermia and targeted drug delivery within MRI. Analytical predictions and numerical simulations of the transverse rotating magnetic field in the presence of *B*_{0} are investigated to demonstrate the effect of Ω, the rotating field frequency, and the magnetic field amplitude on the fluid suspension magnetization. The transverse magnetization due to the rotating transverse field shows strong dependence on the characteristic time constant of the fluid suspension, *τ*. The analysis shows that as the rotating field frequency increases so that Ω*τ* approaches unity, the transverse fluid magnetization vector is significantly non-aligned with the applied rotating field and the magnetization’s magnitude is a strong function of the field frequency. In this frequency range, the fluid’s transverse magnetization is controlled by the applied field which is determined by the operator. The phenomenon, which is due to the physical rotation of the magnetic nanoparticles in the suspension, is demonstrated analytically when the nanoparticles are present in high concentrations (1 to 3% solid volume fractions) more typical of hyperthermia rather than in clinical imaging applications, and in low MRI field strengths (such as open MRI systems), where the magnetic nanoparticles are not magnetically saturated. The effect of imposed Poiseuille flow in a planar channel geometry and changing nanoparticle concentration is examined. The work represents the first known attempt to analyze the dynamic behavior of magnetic nanoparticles in the MRI environment including the effects of the magnetic nanoparticle spin-velocity. It is shown that the magnitude of the transverse magnetization is a strong function of the rotating transverse field frequency. Interactive fluid magnetization effects are predicted due to non-uniform fluid magnetization in planar Poiseuille flow with high nanoparticle concentrations.

The ferrohydrodynamics of suspensions of superparamagnetic iron-oxide (most usually magnetite-dominated) in carrier liquids such as oil or water, commonly termed ferrofluids, is well-understood [1–4]. Following the analysis of Shliomis, [1], much work has sought to validate his theory through experiments [5–7]. With the work of Weissleder [8] among others [9], [10], water-based ferrofluids have found application as imaging contrast agents in magnetic resonance imaging (MRI), where they are known as superparamagnetic iron oxide (SPIO) contrast agents. More recently, magnetic nanoparticles have received much attention as the heat source in magnetic particle hyperthermia (MPH) [11] and as a mechanism for targeted drug delivery *in vivo* [12]. Also, modifications and additions to the existing MRI gradient fields have been proposed as a method for targeted particle delivery [12,13]. Most recently, an alternate imaging modality to MRI known as magnetic particle imaging (MPI) [14] has been proposed which uses the non-linear magnetic response of magnetic nanoparticles (*i.e.*, the Langevin relation) for direct imaging of their distribution. While much important work has been undertaken to characterize the biomedical, physical and non-dynamic magnetic properties of such SPIO-type agents [15], [16], including the relaxivity critical to contrast in MRI [17–20], there has yet been no attempt to analyze the potential effect of nanoparticle dynamics within MRI. This work represents a preliminary investigation of the dynamic behavior by means of analysis and numerical simulations. Potential application of the described phenomena for interactive fluid magnetization is outlined, including a proposed experimental investigation of the effect. Similarities between the physical characteristics of the nanoparticles investigated in this work with other biomedical applications of magnetic nanoparticles are also noted, including those applicable to MPH and targeted drug delivery systems.

Fundamental to conventional MRI are three applied magnetic flux densities: a strong, homogenous *z*-directed DC field, *B*_{0}; a transverse RF field, ** B_{1}**; and spatially-varying encoding fields or gradients,

$${f}_{0}=\frac{\gamma}{2\pi}{B}_{0}$$

(1)

This work examines a wide range of rotation frequencies for *B** _{e}* including those typical of

A simple but important channel geometry introduced by Zahn and Greer [21] shown in Fig. 1 is analyzed, which allows imposition of boundary conditions on both magnetic fields and fluid flow. A novel linearization of the Langevin relation for magnetic fluid suspensions is presented for small signal field variations around the operating point determined by the *B*_{0} field. Having established this linearization, the scenario of a concurrent *B*_{0} and transverse rotating *B** _{e}* field is examined for the case of Poiseuille flow [22] where fluid flow is due to a pressure differential along the channel length. Two critical parameters of interest are examined. These are the ferrofluid spin-velocity in the channel and the resultant change in the transverse ferrofluid magnetization that arises due to the simultaneous relaxation and realignment of the magnetic nanoparticles with the applied transverse rotating field. Results for both quantities are examined for magnetic nanoparticles with physical characteristics typical of magnetic nanoparticle contrast agents used in MRI as well as those proposed for magnetic nanoparticle hyperthermia and targeted drug delivery.

The Langevin relation relates the ratio of the magnetic and thermal energy densities in a ferrofluid [2]. It takes the form of (2) and *L*(*α*) describes the degree of alignment of the ferrofluid magnetic nanoparticles with the applied magnetic field, of magnitude |** H**|. The Langevin parameter,

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

(2)

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

(3)

Linearization of the Langevin function can be performed about a DC operating point defined by the magnetic field intensity, *H*_{0}. In this work, *H*_{0} represents the large DC magnetic field which characterizes MRI. It is assumed that there are no large-signal DC components directed along *i** _{x}* or

$$\mathit{H}={h}_{x}{\mathit{i}}_{x}+{h}_{y}{\mathit{i}}_{y}+({H}_{0}+{h}_{z}){\mathit{i}}_{z}$$

(4)

The associated total equilibrium ferrofluid magnetization vector is *M** _{eq}* where the components of magnetization due to the perturbations along

$${\mathit{M}}_{eq}={m}_{x}{\mathit{i}}_{x}+{m}_{y}{\mathit{i}}_{y}+({M}_{0}+{m}_{z}){\mathit{i}}_{z}$$

(5)

Since ** H** and

$${\mathit{M}}_{eq}={M}_{s}L(\alpha )\frac{{h}_{x}{\mathit{i}}_{x}+{h}_{y}{\mathit{i}}_{y}+({H}_{0}+{h}_{z}){\mathit{i}}_{z}}{\sqrt{{({h}_{x})}^{2}+{({h}_{y})}^{2}+{({H}_{0}+{h}_{z})}^{2}}}$$

(6)

The following analysis considers first-order, linearized small-signal perturbations to the magnetic field components (denoted *h _{x}*,

Considering a first-order expansion of the denominator of (6), one can rewrite the expression for *M** _{eq}* as simplified in (7) when all second-order terms (

$$\begin{array}{l}{\mathit{M}}_{eq}\approx {M}_{s}L(\alpha )\frac{{h}_{x}{\mathit{i}}_{x}+{h}_{y}{\mathit{i}}_{y}+({H}_{0}+{h}_{z}){\mathit{i}}_{z}}{{H}_{0}}(1-\frac{{h}_{z}}{{H}_{0}})\\ \approx {M}_{s}L(\alpha )\frac{{h}_{x}{\mathit{i}}_{x}+{h}_{y}{\mathit{i}}_{y}+({H}_{0}){\mathit{i}}_{z}}{{H}_{0}}\end{array}$$

(7)

The Langevin parameter, *α*, is written as the linear sum of contributions due to the Langevin function evaluated at the operating point (denoted *α*_{0}) and contributions due to the small-signal perturbations (denoted *α*′). The terms *α*_{0} and *α*′ are evaluated by a first-order expansion of (2) and (3) where |** H**| is found from (4) to yield (10) and (11).

$$\alpha ={\alpha}_{0}+{\alpha}^{\prime}$$

(8)

$$\begin{array}{l}\alpha =\frac{{M}_{d}{V}_{p}{\mu}_{0}}{kT}\sqrt{{({h}_{x})}^{2}+{({h}_{y})}^{2}+{({H}_{0}+{h}_{z})}^{2}}\\ \approx \frac{{M}_{d}{V}_{p}{\mu}_{0}}{kT}{H}_{0}(1+\frac{{h}_{z}}{{H}_{0}})\end{array}$$

(9)

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

(10)

$$\Rightarrow {\alpha}^{\prime}={\alpha}_{0}\frac{{h}_{z}}{{H}_{0}}$$

(11)

Simplifying the expression for the Langevin Relation in (12) [2] using a first-order Taylor Series expansion, leads to (14).

$$L(\alpha )=coth({\alpha}_{0}+{\alpha}^{\prime})-\frac{1}{{\alpha}_{0}+{\alpha}^{\prime}}$$

(12)

$$L(\alpha )\approx L({\alpha}_{0})+\frac{dL(\alpha )}{d\alpha}{\mid}_{\alpha ={\alpha}_{0}}{\alpha}^{\prime}$$

(13)

$$L(\alpha )\approx coth({\alpha}_{0})-\frac{1}{{\alpha}_{0}}-\frac{{\alpha}^{\prime}}{{sinh}^{2}({\alpha}_{0})}+\frac{{\alpha}^{\prime}}{{\alpha}_{0}^{2}}$$

(14)

$${m}_{x}\approx {M}_{s}(coth({\alpha}_{0})-\frac{1}{{\alpha}_{0}}-\frac{{\alpha}^{\prime}}{{sinh}^{2}({\alpha}_{0})}+\frac{{\alpha}^{\prime}}{{\alpha}_{0}^{2}})\frac{{h}_{x}}{{H}_{0}}$$

(15)

$${m}_{y}\approx {M}_{s}(coth({\alpha}_{0})-\frac{1}{{\alpha}_{0}}-\frac{{\alpha}^{\prime}}{{sinh}^{2}({\alpha}_{0})}+\frac{{\alpha}^{\prime}}{{\alpha}_{0}^{2}})\frac{{h}_{y}}{{H}_{0}}$$

(16)

$${M}_{eq,z}\approx {m}_{z}+{M}_{0}\approx {M}_{s}(coth({\alpha}_{0})-\frac{1}{{\alpha}_{0}}-\frac{{\alpha}^{\prime}}{{sinh}^{2}({\alpha}_{0})}+\frac{{\alpha}^{\prime}}{{\alpha}_{0}^{2}})$$

(17)

Substituting for *L*(*α*) in (7) (as shown in (15) through (17)) allows the constituent components of *M** _{eq}* to be expressed as follows by comparison with (5) where the component due to the zeroth order DC

$${m}_{x}\approx {M}_{s}L({\alpha}_{0})\frac{{h}_{x}}{{H}_{0}}\approx \frac{{M}_{0}}{{H}_{0}}{h}_{x}$$

(18)

$${m}_{y}\approx {M}_{s}L({\alpha}_{0})\frac{{h}_{y}}{{H}_{0}}\approx \frac{{M}_{0}}{{H}_{0}}{h}_{y}$$

(19)

$${m}_{z}\approx {M}_{s}(\frac{-{\alpha}_{0}}{{sinh}^{2}{\alpha}_{0}}+\frac{1}{{\alpha}_{0}})\frac{{h}_{z}}{{H}_{0}}\approx \frac{d{M}_{0}}{d{H}_{0}}{h}_{z}$$

(20)

$${M}_{0}={M}_{s}L({\alpha}_{0})$$

(21)

The expressions of (18) through (21) show that for *z*-directed perturbations, the relationship between the small signal magnetic field and the resultant magnetization is due to the slope of the Langevin relation evaluated at the DC operating point defined by *H*_{0} = *B*_{0}/*μ*_{0} − *M*_{0}. When the perturbations are either *x* or *y*-directed, the small signal magnetic field (*h _{x}* and

This work examines the case of small-signal magnetic field perturbations along *i** _{x}* and

The geometry shown in Fig. 1 follows the analysis of Zahn and Greer [21] and is carefully selected to allow the imposition of fields directed along all three axes.

The magnetization relaxation equation for a ferrofluid under the action of a rotating magnetic field, thereby undergoing simultaneous magnetization and reorientation with the applied field, is given by (22). The fluid linear flow velocity vector is ** v**, the ferrofluid spin-velocity vector is

$$\frac{\partial \mathit{M}}{\partial t}+\mathit{v}\xb7\nabla \mathit{M}-\mathit{\omega}\times \mathit{M}+(1/\tau )(\mathit{M}-{\mathit{M}}_{eq})=0$$

(22)

The left-side of the relaxation equation is simply a generalized convective derivative of magnetization for linear motion (at linear velocity ** v**) and angular or spin velocity,

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

(23)

The relative expressions for *τ _{B}* and

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

(24)

$${\tau}_{N}={\tau}_{0}{e}^{{\scriptstyle \frac{{K}_{a}{V}_{p}}{kT}}}$$

(25)

In (23), the smaller time constant dominates in determining *τ*. Thus, while both Brownian and Néel relaxation times increase with particle radius [2], Néel relaxation, which describes the rotation of the magnetization vector within the particle, generally dominates for small particles with core radius less than 4 nm while Brownian relaxation, due to particle rotation in the carrier liquid, dominates for particles larger than 4 nm [25]. If the nanoparticle is constrained, for example by attachment to a surface, Néel relaxation is still operative while Brownian relaxation is not.

For the planar geometry shown in Fig. 1, the flow velocity can only be *x*-directed and the time-averaged spin-velocity can only be z-directed. Both quantities may vary spatially with y.

$$\mathit{v}={v}_{x}(y){\mathit{i}}_{x}$$

(26)

$$\mathit{\omega}={\omega}_{z}(y){\mathit{i}}_{z}$$

(27)

Considering the geometric arrangement of Fig. 1, the imposed magnetic field intensities along *i** _{x}* and

$$\nabla \times \mathit{H}=\mathit{J}+\frac{\partial \mathit{D}}{\partial t}=\mathbf{0}$$

(28)

$$\begin{array}{c}\Rightarrow \frac{\partial ({H}_{0})}{\partial y}-\frac{\partial {h}_{y}}{\partial z}=0\phantom{\rule{0.38889em}{0ex}}\Rightarrow \frac{\partial ({H}_{0})}{\partial y}=0,\phantom{\rule{0.38889em}{0ex}}\frac{\partial {h}_{y}}{\partial z}=0\\ \Rightarrow \frac{\partial {h}_{x}}{\partial z}-\frac{\partial ({H}_{0})}{\partial x}=0\phantom{\rule{0.38889em}{0ex}}\Rightarrow \frac{\partial ({H}_{0})}{\partial x}=0,\phantom{\rule{0.38889em}{0ex}}\frac{\partial {h}_{x}}{\partial z}=0,\\ \Rightarrow \frac{\partial {h}_{y}}{\partial x}-\frac{\partial {h}_{x}}{\partial y}=0\Rightarrow \frac{\partial {h}_{y}}{\partial x}=0,\frac{\partial {h}_{x}}{\partial y}=0\end{array}$$

(29)

The imposed magnetic flux density along *i** _{y}* is governed by Gauss’ Law.

$$\nabla \xb7\mathit{B}=\frac{\partial {b}_{x}}{\partial x}+\frac{\partial {b}_{y}}{\partial y}+\frac{\partial ({B}_{0})}{\partial z}=0$$

(30)

$$\Rightarrow \frac{\partial {b}_{y}}{\partial y}=0\phantom{\rule{0.38889em}{0ex}}\text{since}\phantom{\rule{0.38889em}{0ex}}\frac{\partial {b}_{x}}{\partial x}=\frac{\partial ({B}_{0})}{\partial z}=0$$

(31)

In light of these observations, it can be seen that *b _{y}* and

$$\mathit{B}=\mathfrak{R}e\{({\widehat{b}}_{x}(y){\mathit{i}}_{x}+{\widehat{b}}_{y}{\mathit{i}}_{y}){\text{e}}^{j\mathrm{\Omega}t}\}+{B}_{0}{\mathit{i}}_{z}$$

(32)

$$\mathit{H}=\mathfrak{R}e\{({\widehat{h}}_{x}{\mathit{i}}_{x}+{\widehat{h}}_{y}(y){\mathit{i}}_{y}){\text{e}}^{j\mathrm{\Omega}t}\}+{H}_{0}{\mathit{i}}_{z}$$

(33)

$$\mathit{M}=\mathfrak{R}e\{({\widehat{m}}_{x}(y){\mathit{i}}_{x}+{\widehat{m}}_{y}(y){\mathit{i}}_{y}){\text{e}}^{j\mathrm{\Omega}t}\}+{M}_{0}{\mathit{i}}_{z}$$

(34)

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

(35)

The relationship between *M*_{0} and *H*_{0}, the *z*-directed components of magnetization and magnetic field intensity respectively, has been established in (21). There is no *z*-directed, small-signal magnetic field which, in turn means that there is no associated small-signal magnetization component along *i** _{z}*. Therefore, the

$$j\mathrm{\Omega}{\widehat{m}}_{x}+{\omega}_{z}{\widehat{m}}_{y}+\frac{1}{\tau}({\widehat{m}}_{x}-\frac{{M}_{0}}{{H}_{0}}{h}_{x})=0$$

(36)

$$j\mathrm{\Omega}{\widehat{m}}_{y}-{\omega}_{z}{\widehat{m}}_{x}+\frac{1}{\tau}({\widehat{m}}_{y}-\frac{{M}_{0}}{{H}_{0}}{h}_{y})=0$$

(37)

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

(38)

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

(39)

The complex amplitudes of transverse magnetization, * _{x}* and

Applying the principles of conservation of linear and angular momentum to an incompressible ferrofluid leads to the simplified expressions of (40) and (42) respectively [1], [2] where *p*′ is the modified pressure along the channel, given by (41), *p* is the absolute pressure in Pa, *g* is the gravitational acceleration acting along *i** _{y}* in m·s

$$-\nabla {p}^{\prime}+{\mu}_{0}(\mathit{M}\xb7\nabla )\mathit{H}+2\zeta \nabla \times \mathit{\omega}+(\zeta +\eta ){\nabla}^{2}\mathit{v}=0$$

(40)

$${p}^{\prime}=p+\rho gy$$

(41)

$${\mathit{T}}_{m}+2\zeta (\nabla \times \mathit{v}-2\mathit{\omega})=0$$

(42)

$${\mathit{T}}_{m}={\mu}_{0}(\mathit{M}\times \mathit{H})$$

(43)

The conservation of linear momentum (neglecting inertial terms), shown in (40), contains terms due to (i) pressure and gravity, (ii) magnetic force density (given by the Kelvin force density), (iii) coupling to spin-velocity and (iv) viscosity. The spin-velocity, as indicated by the conservation of angular momentum (42), can be caused by two factors; (i) vorticity in the flow (due to the × ** v** term) and (ii) due to a magnetic torque,

$$\langle {T}_{m,z}\rangle =\frac{{\mu}_{0}}{2}\mathfrak{R}e({\widehat{m}}_{x}{\widehat{h}}_{y}^{\ast}-{\widehat{h}}_{x}{\widehat{m}}_{y}^{\ast})$$

(44)

Considering (35), one can rewrite the time-averaged torque density in terms of the imposed fields and the complex transverse magnetization amplitudes.

$$\langle {T}_{m,z}\rangle =\frac{1}{2}\mathfrak{R}e({\widehat{m}}_{x}{\widehat{b}}_{y}^{\ast}-{\mu}_{0}({\widehat{h}}_{x}+{\widehat{m}}_{x}){\widehat{m}}_{y}^{\ast})$$

(45)

Using (42), one can relate vorticity, spin-velocity and torque density as given by (46).

$$-2\zeta \frac{\partial {v}_{x}}{\partial y}-4\zeta {\omega}_{z}+\langle {T}_{m,z}\rangle =0$$

(46)

Considering (40), one should note that the second non-zero term, *μ*_{0}(** M** · )

$$\begin{array}{c}{F}_{m,y}={\mu}_{0}({m}_{x}\frac{\partial {h}_{y}}{\partial x}+{m}_{y}\frac{\partial {h}_{y}}{\partial y}+({M}_{0})\frac{\partial {h}_{y}}{\partial z})\\ \Rightarrow {F}_{m,y}={\mu}_{0}({m}_{y}\frac{\partial {h}_{y}}{\partial y})=-{\mu}_{0}({m}_{y}\frac{\partial {m}_{y}}{\partial y})\\ {F}_{m,y}=-\frac{{\mu}_{0}}{2}\frac{\partial ({m}_{y}^{2})}{\partial y}\\ \langle {F}_{m,y}\rangle =-\frac{{\mu}_{0}}{4}\frac{\partial {\mid {\widehat{m}}_{y}\mid}^{2}}{\partial y}\end{array}$$

(47)

In light of the simplification of (47), one might write the non-zero components for (40) as given by (48) and (49). Conservation of linear momentum yields components along both *i** _{x}* and

$$-\frac{\partial {p}^{\prime}}{\partial x}+2\zeta \frac{\partial {\omega}_{z}}{\partial y}+(\zeta +\eta )\frac{{\partial}^{2}{v}_{x}}{\partial {y}^{2}}=0$$

(48)

$$-\frac{\partial (p+\rho gy)}{\partial y}-\frac{{\mu}_{0}}{4}\frac{\partial {\mid {\widehat{m}}_{y}(y)\mid}^{2}}{\partial y}=0$$

(49)

$$\Rightarrow p(x,y)+\rho gy+\frac{{\mu}_{0}}{4}{\mid {\widehat{m}}_{y}(y)\mid}^{2}=f(x)$$

(50)

Equations (38), (39), (46) and (48) now constitute a closed system of equations with 4 unknown quantities: spin-velocity, *ω _{z}*, flow velocity,

Generalized analytical solutions of (38), (39), (46) and (48) can be obtained subject to boundary conditions. From Maxwell’s Equations, the relevant boundary conditions on the magnetic field components are given by (51) and (52) where *i** _{n}* is the unit vector normal to the boundary in the second medium.

$${\mathit{i}}_{n}\xb7({\mathit{B}}_{in}-{\mathit{B}}_{\mathit{out}})=0$$

(51)

$${\mathit{i}}_{n}\times ({\mathit{H}}_{in}-{\mathit{H}}_{\mathit{out}})={\mathit{K}}_{s}$$

(52)

There are no boundary conditions on the spin velocity, *ω _{z}*, as the spin viscosity is not considered in (42), while the velocity,

Poiseuille flow is achieved in the channel by means of an *x*-directed pressure differential which, in the absence of a magnetic torque density in the fluid, results in a parabolic flow profile with *y* which is approximated by blood flow in the medium to large human vessels. The *x*-component of (40) has pressure gradient, spin velocity, and viscous flow contributions.

Differentiating (46) and writing in terms of ${\scriptstyle \frac{\partial {\omega}_{z}}{\partial y}}$ allows for simplification of (48) by substitution of the ${\scriptstyle \frac{\partial {\omega}_{z}}{\partial y}}$ term from (53).

$$\frac{\partial {\omega}_{z}}{\partial y}=\frac{1}{4\zeta}\frac{\partial \langle {T}_{m,z}\rangle}{\partial y}-\frac{1}{2}\frac{{\partial}^{2}{v}_{x}}{\partial {y}^{2}}$$

(53)

This leads to the expression of (57) after twice integrating (48) with respect to *y. K*_{1} and *K*_{2} are constants of integration while *T _{m}*

$$-\frac{\partial {p}^{\prime}}{\partial x}+2\zeta (\frac{1}{4\zeta}\frac{\partial \langle {T}_{m,z}\rangle}{\partial y}-\frac{1}{2}\frac{{\partial}^{2}{v}_{x}}{\partial {y}^{2}})+(\zeta +\eta )\frac{{\partial}^{2}{v}_{x}}{\partial {y}^{2}}=0$$

(54)

$$\Rightarrow -\frac{\partial {p}^{\prime}}{\partial x}+\frac{1}{2}\frac{\partial \langle {T}_{m,z}\rangle}{\partial y}+\eta \frac{{\partial}^{2}{v}_{x}}{\partial {y}^{2}}=0$$

(55)

$$\Rightarrow -\frac{\partial {p}^{\prime}}{\partial x}y+\frac{1}{2}\langle {T}_{m,z}\rangle +\eta \frac{\partial {v}_{x}}{\partial y}+{K}_{1}=0$$

(56)

$$\Rightarrow -\frac{\partial {p}^{\prime}}{\partial x}\frac{{y}^{2}}{2}+\frac{1}{2}{\int}_{0}^{y}\langle {T}_{m,z}\rangle d{y}^{\prime}+\eta {v}_{x}+{K}_{1}y+{K}_{2}=0$$

(57)

Applying the non-slip boundary conditions on *v _{x}* at

$${v}_{x}(0)=0\Rightarrow {K}_{2}=0$$

(58)

$${v}_{x}(d)=0\Rightarrow {K}_{1}=\frac{d}{2}\frac{\partial {p}^{\prime}}{\partial x}-\frac{1}{2d}{\int}_{0}^{d}\langle {T}_{m,z}\rangle d{y}^{\prime}$$

(59)

Substituting the results of (58) and (59) into (57) and rearranging one arrives at (60).

$$\Rightarrow {v}_{x}=\frac{1}{2\eta}\frac{\partial {p}^{\prime}}{\partial x}({y}^{2}-yd)-\frac{1}{2\eta}\left({\int}_{0}^{y}\langle {T}_{m,z}\rangle d{y}^{\prime}-\frac{y}{d}{\int}_{0}^{d}\langle {T}_{m,z}\rangle d{y}^{\prime}\right)$$

(60)

Rewriting (46) as (61) and substituting for the derivative of *v _{x}* leads to the following expression for the spin-velocity,

$$\frac{\partial {v}_{x}}{\partial y}=\frac{1}{2\zeta}\langle {T}_{m,z}\rangle -2{\omega}_{z}$$

(61)

$${\omega}_{z}=-\frac{1}{4\eta}\frac{\partial {p}^{\prime}}{\partial x}(2y-d)+\langle {T}_{m,z}\rangle \frac{\eta +\zeta}{4\zeta \eta}-\frac{1}{4d\eta}{\int}_{0}^{d}\langle {T}_{m,z}\rangle d{y}^{\prime}$$

(62)

While (62) is not solvable analytically, it can be solved numerically and the results which follow use *Comsol Multiphysics* (COMSOL AB, Stockholm, Sweden) to do that.

The simplest case is when *T _{m}*

$$\begin{array}{c}{U}_{p}=-\frac{{d}^{2}}{8\eta}\frac{\partial {p}^{\prime}}{\partial x}\\ \Rightarrow {\omega}_{z}(y=0)=-{\omega}_{z}(y=d)=\frac{d}{4\eta}\frac{\partial {p}^{\prime}}{\partial x}=-\frac{2{U}_{p}}{d}\end{array}$$

(63)

The resulting Poiseuille flow has flow velocity profile given by (64) with a maximum value, *U _{p}*, at

$$\Rightarrow {v}_{x}(y)=\frac{1}{2\eta}\frac{\partial {p}^{\prime}}{\partial x}y(y-d)=\frac{4}{{d}^{2}}{U}_{p}y(y-d)$$

(64)

$$\Rightarrow {v}_{x}(d/2)=-\frac{{d}^{2}}{8\eta}\frac{\partial {p}^{\prime}}{\partial x}={U}_{p}$$

(65)

Another interesting limiting case is conditions of negligible flow in the channel (*i.e.*, *v _{x}*(

$${\omega}_{z}=\langle {T}_{m,z}\rangle \frac{\eta +\zeta}{4\zeta \eta}-\frac{1}{4d\eta}{\int}_{0}^{d}\langle {T}_{m,z}\rangle dy=\frac{1}{4\zeta}\langle {T}_{m,z}\rangle $$

(66)

$${\omega}_{z}=\frac{1}{8\zeta}\mathfrak{R}e({\widehat{m}}_{x}{\widehat{b}}_{y}^{\ast}-{\mu}_{0}({\widehat{h}}_{x}+{\widehat{m}}_{x}){\widehat{m}}_{y}^{\ast})$$

(67)

Substituting for * _{x}* and

The physical parameters for the simulated magnetic nanoparticles are shown in Table 1 as well as the nominal field conditions. *U _{p}* =

The main DC *B*_{0} field is directed along *i** _{z}*. An approximately rotating field of constant amplitude is generated in the transverse

The effect of varying the physical system parameters is examined for the normalized spin-velocity, |*ω _{z}*|

$${\mathit{M}}_{\mathit{trans}}(y,t)=\mathfrak{R}e\{({\widehat{m}}_{x}(y){\mathit{i}}_{x}+{\widehat{m}}_{y}(y){\mathit{i}}_{y}){\text{e}}^{j\mathrm{\Omega}t}\}$$

(68)

$$\mid {\mathit{M}}_{\mathit{trans}}(y,t)\mid =\sqrt{{(\mathfrak{R}e\{{\widehat{m}}_{x}(y){\text{e}}^{j\mathrm{\Omega}t}\})}^{2}+{(\mathfrak{R}e\{{\widehat{m}}_{y}(y){\text{e}}^{j\mathrm{\Omega}t}\})}^{2}}$$

(69)

$$\langle \mid {\mathit{M}}_{\mathit{trans}}(y,t)\mid \rangle =\frac{1}{\sqrt{2}}\sqrt{{|{\widehat{m}}_{x}(y)|}^{2}+{|{\widehat{m}}_{y}(y)|}^{2}}$$

(70)

$$\langle \mid {\mathit{M}}_{\mathit{trans}}(y,t)\mid \rangle =\frac{1}{\sqrt{2}}\sqrt{{(\mathfrak{R}e\{{\widehat{m}}_{x}(y)\})}^{2}+{(\mathfrak{J}m\{{\widehat{m}}_{x}(y)\})}^{2}+{(\mathfrak{R}e\{{\widehat{m}}_{y}(y)\})}^{2}+{(\mathfrak{J}m\{{\widehat{m}}_{y}(y)\})}^{2}}$$

(71)

The results which follow examine the magnitude of the time-average transverse magnetization. The transverse magnetization will, in general, be elliptically polarized in the *xy* plane since * _{x}* and

The case of no imposed flow was examined for a 5 mm wide channel. This represents an approximate resolution limit for low-field MRI [23]. The effect of changing the MRI’s DC *B*_{0} field on both the normalized spin velocity, *ω _{z}τ*, and the time-average transverse magnetization, normalized with respect to

Frequency dependence of |*ω*_{z}|*τ* and the normalized, time-average transverse magnetization on *B*_{0} is shown for = 3% and *B*_{e} = 0.05*B*_{0}. Note that for *τ* = 1 *μ*s, Ω/(2*π*) ≈ 160kHz when Ω*τ* **...**

Poiseuille flow conditions were implemented by means of an imposed flow profile on the left-most entry to the channel of Fig. 1 rather than an imposed pressure differential across the channel. This approach allowed the maximum flow velocity in the channel to be defined as *U _{p}* for Poiseuille flow conditions. Finite element simulation and solution of the variables in the channel is achieved using

$${v}_{in,x}(y)=\frac{4{U}_{p}y(d-y)}{{d}^{2}}$$

(72)

The result of the imposed parabolic flow profile in the channel is that the spin-velocity is no longer spatially invariant as in the previous case. The total spin-velocity in the channel is now the sum of contributions from the Poiseuille imposed flow condition and the time-averaged magnetic torque density, *T _{m}*

The unusual behavior of magnetic fluid in the presence of rotating magnetic fields is well understood [1,2]. This work now adds the strong DC field associated with MRI and examines the simulated behavior of these fluids in a planar channel. The effect of nanoparticle rotation in the presence of a transverse rotating field (amplitudes between 1% to 10% of *B*_{0}) was considered. While this represents a significant modification of the low-field MRI system, one should bear in mind the recent additions and modifications proposed to MRI magnetic fields for a variety of biomedical applications including hyperthermia and targeted drug delivery [11,12]. These include the addition of an RF hyperthermia coil to 1.5 T MRI [32] which operated a time-sharing arrangement between image acquisition and therapy.

As the transverse rotating magnetic field frequency increases such that Ω*τ* → 1, the fluid’s transverse magnetization component and the applied rotating field become increasingly misaligned due to the magnetic torque density on the fluid and the transverse magnetization’s magnitude is a strong function of Ω*τ*. The ferrofluid spin-velocity, *ω _{z}*, was introduced as a measure of nanoparticle spin or rotation [1]. In the absence of fluid flow (

Spin-velocity was introduced as a measure of magnetic nanoparticle rotation within the fluid suspension due to the rotating transverse magnetic field in the MRI. The dependence of the fluid spin-velocity, *ω _{z}*, is closely related to the ferrofluid time constant,

As Ω*τ* approaches unity, the spin-velocity becomes increasingly significant. This is because the time constant of the suspension, *τ*, is no longer fast enough to allow the particles to reestablish transverse equilibrium before the excitation changes direction, due to the rotating transverse field. Instead, synchronism is maintained at a constant lag angle between the transverse components of ** M** and

In the high-frequency limit, where Ω*τ* 1, the denominators of (38) and (39) become the dominant terms. The spin-velocity is less significant in the transverse magnetization and in the limit of Ω*τ* 1, the transverse magnetization becomes negligible so that the ferrofluid no longer appears magnetic in the transverse *xy* plane. It is noted that this scenario is very difficult to achieve in reality since the typical time constant of 1 *μs* requires Ω/(2*π*) on the order of 1.6 MHz before Ω*τ* ≈ 10. However, it can be intuitively expected that finite fluid viscosity limits nanoparticle rotation at high frequency. Shliomis [1] notes that his formulation is no longer valid in the frequency range of Ω*τ* 1 so that the plots should be treated with some caution beyond Ω*τ* ≈ 10. The *z*-directed magnetization, *M*_{0}, is DC as previously noted (*h _{z}* = 0) and is given by (21) throughout. The point of this portion of the analysis is that the interaction between the rotating magnetic field and nanoparticle’s transverse magnetization is optimized at Ω

In the absence of externally imposed flow, the ferrofluid spin-velocity is spatially constant in the channel of Fig. 1. As noted, the maximum spin-velocity occurs when Ω*τ* = 1 and increases with the main *z*-directed *B*_{0} field, as shown in Fig. 3(a). However, the time-average transverse magnetization seen in Figure 3(b) decreases with *B*_{0}. This is because when *B*_{0} is increased, the small-signal susceptibility for the transverse components, *M*_{0}/*H*_{0} shown in Figure 1, decreases. So while the value of spin-velocity increases, the underlying coupling between the small-signal transverse field perturbation (*h _{x}* or

The time-dependence of the transverse magnetization is shown in Figure 4(a) for various values of Ω*τ* and = 0.05. Clearly, as predicted by (71), the phase lag behind the sinusoidal *B _{e}* field is more pronounced as Ω

In no case does the value of normalized transverse magnetization become comparable to *B _{e}*/

The transverse magnetization decreases as Ω*τ* approaches unity since the increasing misalignment of the *M* and *H* fields is accompanied by a decrease in the absolute magnitude of the transverse magnetization. This decrease results from the dominance of the denominator in (38) and (39) as Ω*τ* increases. The point here is that the applied transverse magnetic field controls the transverse magnetization of the fluid, where the frequency, Ω, effectively acts as the amplitude dial on the transverse magnetization. This presents the possibility of an interactive fluid magnetization mechanism, controlled by the applied magnetic field frequency and amplitude. It would be evident that the role of the transverse magnetization is in addition to that of the DC magnetization due to *B*_{0} (which is not controllable in an interactive sense). However, imaging modalities do exist where the *B*_{0} field is pulsed (*e.g.*, prepolarized MRI (pMRI) [31] where pulsed electromagnets produce diagnostic quality 0.5 to 1.0 T images with significantly reduced cost and susceptibility artifacts) or eliminated entirely (*e.g.*, magnetic particle imaging (MPI) [14] uses strong magnetic field gradients to generate a moving point of zero-field where the magnetic nanoparticles are unsaturated for direct imaging of their distribution in the absence of a *B*_{0} field). In MPI, the signal is not the imaging protons of the hydrogen proton (as in MRI) but rather the magnetization of the magnetic particles, as investigated in this work. The addition of an oscillating field (10 mT drive field arbitrarily chosen at a frequency of 25 kHz) to the gradient fields move the zero-field point over the field of view, inducing a signal where the nanoparticles are present. Since the oscillating field frequency was determined without regard to *τ*, it may be possible to reconfigure the system to allow Ω*τ* approach unity and examine the modified effects on the fluid magnetization. A proposed investigative system might consist of the commercial MRI contrast agent, *Feridex* (Advanced Magnetics, Cambridge, MA), a rotating field amplitude of 10 mT and a frequency range of 50–300 kHz, such that operation in the Ω*τ* ~ 1 range might be achieved. Under such conditions, one should expect significant changes in the fluid magnetization (the signal source in MPI) as a function of rotating field frequency and amplitude where Ω*τ* approaches unity.

Imposing conditions of Poiseuille flow on the channel of Fig. 1 shows significant change from the preceding case of no imposed flow since, now, the spin-velocity and transverse magnetization are functions of the channel width along *y*. As already noted, the channel width and flow velocities are chosen such that the flow vorticity, and not the magnetic torque density, is the dominant source for the spin-velocity and subsequent changes in transverse magnetization. Comparing (63) with (64) yields the limit of the maximum spin-velocity in a vorticity-driven planar flow system, given by (73). Inspection of Figure 5 shows that this is the case for the three values of *U _{p}* evaluated (

$$\mid {\omega}_{z,\mathit{max}}\mid \tau \to \frac{2{U}_{p}}{d}\tau $$

(73)

The transverse magnetization’s dependence on the spin-velocity now gives rise to a non-uniform magnetization across the channel width for vorticity-dominated spin-velocity, as shown in Figure 5(b). Although channel widths on the order of *μ*m simulated here are not currently resolved in low-field MRI due to decreased SNR, this non-uniformity in the magnetization is a function of the flow velocity. The effect does not persist as *B*_{0} is increased (e.g., 1.5 T or 3 T) due to magnetic saturation. The dependence on spin-velocity (and hence on *U _{p}* for a vorticity-driven system) is involved as seen from (38) and (39) in conjunction with (62). For small

This work examines the dynamic behavior of magnetic nanoparticle suspensions in the low-field MRI environment with the addition of a strong rotating magnetic field transverse to the MRI’s main DC field. It is shown that under circumstances of high local nanoparticle concentrations (~0.05 solid volume fraction) and low field MRI (~0.1 to 0.35 T), the fluid’s transverse magnetization shows strong dependence on the rotating field frequency, as shown in Figures 3(b) and and4.4. At high MRI field strengths (1.5 T and 3 T) assuming the rotating field amplitude remains in the mT range considered in this work, the effects decrease significantly (as evident from Figure 3(b)) due to the ferrofluid’s magnetic saturation. However, at low field MRI where rotating field amplitudes in the mT range are more easily realized, the transverse ferrofluid magnetization magnitude can be significantly decreased by decreasing field frequency. It is proposed that the phenomenon might be investigated by means of the recently proposed imaging modality of magnetic particle imaging [14], where the system frequency is modified to allow Ω*τ* approach unity, thus enabling nanoparticle rotation dynamics to alter the fluid magnetization, and potentially, image contrast in MPI where fluid magnetization is the signal source.

It was further shown that the fluid’s transverse magnetization can be significantly altered by the presence of the ferrofluid in the channel of Fig. 1 as a function of increasing flow velocity since vorticity in the flow induces spin-velocity. Application of this phenomenon may prove difficult in the physiological environment due to the significant vorticity, not typical of clinical flow, and the high concentrations of magnetic nanoparticles required to see the effect.

The effects outlined in this work have not been previously examined in the MRI environment and so, this work should be of interest, not only to those using magnetic nanoparticles as MRI contrast agents, but also those exploiting magnetic nanoparticles for other *in vivo* applications. These include targeted drug delivery and magnetic nanoparticle hyperthermia but, perhaps most significantly, for MPI, where the effects outlined here may give rise to interactive ferrofluid magnetization due to the dynamic behavior of the magnetic nanoparticles in response to rotation.

The authors would like to thank the R.J. Shillman Career Development Award, Thomas and Gerd Perkins Professorship Award, the MIT Dean’s Fellowship, the Bushbaum Foundation at MIT and National Institutes of Health Award R01 EB007942. The authors would also like to thank the reviewers for their thorough review and thoughtful suggestions.

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