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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
J Magn Magn Mater. Author manuscript; available in PMC Sep 1, 2011.
Published in final edited form as:
J Magn Magn Mater. Sep 2010; 322(17): 2607–2617.
doi:  10.1016/j.jmmm.2010.03.029
PMCID: PMC2901184
NIHMSID: NIHMS210166

Simulating Magnetic Nanoparticle Behavior in Low-field MRI under Transverse Rotating Fields and Imposed Fluid Flow

Abstract

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, B0. Theoretical analysis and simulation is used to predict the effect of both alternating-sinusoidal and rotating magnetic fields transverse to B0. 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 B0 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 B0 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 Ωτ 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.

Keywords: Magnetic nanoparticles, MRI, rotating magnetic field, interactive magnetization, magnetic particle imaging

1 Introduction

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 [14]. Following the analysis of Shliomis, [1], much work has sought to validate his theory through experiments [57]. 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 [1720], 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, B0; a transverse RF field, B1; and spatially-varying encoding fields or gradients, G, also z-directed. The B0-field (~1.5 T) induces polarization of nuclear spins, the much weaker (~0.01 mT) and transient (~ms) B1 field is used to drive the induced magnetization into a transverse component for imaging, and the gradients (~10 mT/m) are used to spatially encode the transverse magnetization during relaxation to a resting state (~10 ms). Under the large B0 of conventional clinical MRI systems, it is expected that the equilibrium magnetization of a water-based ferrofluid would be saturated and virtually all the magnetic particles in the suspension would rigidly align with B0. However, low-field MRI platforms exist, typically employed for better patient access or intervention by physicians during scanning. Commercial low-field MRI scanners operate typically between 0.1 T and 0.35 T, and this is the field-range examined in the simulations which follow. With this lower B0, ferrofluid saturation is not complete, as determined by the Langevin relation for magnetic suspensions [2]. Physical nanoparticle rotation in an additional rotating magnetic field is therefore possible. In our analysis, we derive the expression for the ferrofluid spin velocity, ω, induced by imposing a fourth field component, Be, to a conventional MRI. Although not easily measureable, ω is used in this work as a quantifiable analytical indication of the non-alignment between the field exciting the ferrofluid, Be, which is a rotating field applied in the transverse xy plane, and the associated transverse magnetization vector. While both Be and B1 are transverse, rotating fields, f0, the rotation frequency of B1, known as the Larmor frequency, is proportional to B0 as given by (1), where γ is the gyromagnetic ratio of the nucleus under study, which for conventional MRI is the 1H proton with γ = 2π · 42.58 × 106 rad · (T · s)−1.

equation M1
(1)

This work examines a wide range of rotation frequencies for Be including those typical of f0 in open MRI systems (where the Larmor frequency is 8.5 MHz for 0.2 T and 14.9 MHz for 0.35 T). However, the amplitude of Be is set between 1% and 10% of B0, which is much larger than typical B1 amplitudes although it is of the same order as fields proposed in MPH [11]. Interactions with the transverse magnetization is achieved by selecting the transverse Be excitation frequency, termed Ω, to be of the same order as the reciprocal of the ferrofluid relaxation time τ. For typical magnetic nanoparticles in biomedical applications, τ is on the order of 1 μs.

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 B0 field. Having established this linearization, the scenario of a concurrent B0 and transverse rotating Be 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.

Fig. 1
A planar ferrofluid layer between rigid and stationary walls (at y = 0 and y = d) is magnetically stressed by a uniform DC z-directed magnetic field H0, an x-directed small-signal sinusoidal magnetic field with complex amplitude ĥx and a y-directed ...

2 Theory

2.1 Linearization of the Langevin Relation

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, α, is a function of the magnetic field magnitude, |H|, within the ferrofluid, as given by (3) where Md is the single domain magnetization of the particle (typically with a value of 446 kA/m for magnetite [2]), Vp is the particle magnetic volume, μ0 is the magnetic permeability of free space, k is Boltzmann’s constant and T is the absolute temperature. The ferrofluid saturation magnetization Ms is related to Md by the fraction of solid magnetic volume in the suspension, denoted [var phi] where Ms = [var phi]Md.

equation M2
(2)
equation M3
(3)

Linearization of the Langevin function can be performed about a DC operating point defined by the magnetic field intensity, H0. In this work, H0 represents the large DC magnetic field which characterizes MRI. It is assumed that there are no large-signal DC components directed along ix or iy which are unit vectors along the x and y axes respectively so that H0 is directed along iz, the unit vector along z. Additional small-signal perturbations about that operating point along ix, iy and iz are denoted by hx, hy and hz respectively leading to the expression in (4).

equation M4
(4)

The associated total equilibrium ferrofluid magnetization vector is Meq where the components of magnetization due to the perturbations along ix, iy and iz are denoted mx, my and mz respectively.

equation M5
(5)

Since H and Meq are necessarily collinear, Meq is written in terms of (4).

equation M6
(6)

The following analysis considers first-order, linearized small-signal perturbations to the magnetic field components (denoted hx, hy, hz) and the associated small-signal contributions to the magnetization components (mx, my, mz), added to the large, z-directed equilibrium DC field component, H0 with associated DC magnetization component along z, denoted M0. The equilibrium magnetization, given by (2), will be linearized for the imposed small-signal magnetic field perturbations to yield linearized, first-order expressions for mx, my and mz. The total resultant magnetization associated with the nanoparticle suspension consists of the DC component, M0, along z, added to the small-signal contributions, mx, my and mz along x, y and z respectively.

Considering a first-order expansion of the denominator of (6), one can rewrite the expression for Meq as simplified in (7) when all second-order terms (e.g., hxhz and hyhz) are ignored. Note that there is no first order perturbation to the z-directed component of the Langevin magnetization.

equation M7
(7)

The Langevin parameter, α, is written as the linear sum of contributions due to the Langevin function evaluated at the operating point (denoted α0) and contributions due to the small-signal perturbations (denoted α′). The terms α0 and α′ are evaluated by a first-order expansion of (2) and (3) where |H| is found from (4) to yield (10) and (11).

equation M8
(8)
equation M9
(9)
equation M10
(10)
equation M11
(11)

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

equation M12
(12)
equation M13
(13)
equation M14
(14)
equation M15
(15)
equation M16
(16)
equation M17
(17)

Substituting for L(α) in (7) (as shown in (15) through (17)) allows the constituent components of Meq to be expressed as follows by comparison with (5) where the component due to the zeroth order DC H0 field, which is only z-directed, is given by (21). The first-order, small-signal magnetization components, mx, my and mz, follow from substituting for α′ and ignoring second and higher order terms.

equation M18
(18)
equation M19
(19)
equation M20
(20)
equation M21
(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 H0 = B0/μ0M0. When the perturbations are either x or y-directed, the small signal magnetic field (hx and hy) and the resultant magnetizations (mx and my) are related by the ratio M0/H0. Comparing these two relations in Figure 2, it is easily shown that the two relations are approximately equal in the low-field limit where α0 [double less-than sign] 1 and M0 is proportional to H0.

Fig. 2
The values of the chord susceptibility, M0/H0, and the tangent susceptibility, equation M84, are plotted for the parameters of Table 1 including a particle core radius of 4 nm. As expected, the values converge in the low-field limit where α0 [double less-than sign] 1 ...

This work examines the case of small-signal magnetic field perturbations along ix and iy which are temporally displaced by equation M22 radians, to create a rotating field in the transverse {xy} plane. No small-signal perturbations are considered along iz (hz = 0). The DC field is limited to variations between 0.1 T and 0.35 T in the results which follow since nanoparticles with a magnetic core radius of 4 nm (typical of MRI contrast agents) are already 90% saturated (L(α0) ≈ 0.9) at B0 = 0.35 T.

2.2 Governing Equations

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.

2.2.1 Magnetization Constitutive Law

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 ω and the ferrofluid relaxation time [2] is τ.

equation M23
(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, ω. The equation states that deviations away from the equilibrium magnetization in a magnetic fluid have a characteristic time constant, denoted τ. The time constant, τ, is given by (23) and is due to the contributions from Brownian (τB) and Néel (τN) relaxation times [2,25].

equation M24
(23)

The relative expressions for τB and τN are given by (24) and (25) respectively where Vh is the nanoparticle hydrodynamic volume in m3 (including the surfactant contribution), ηc is the dynamic viscosity of the carrier liquid (assumed that of water in this analysis) in N·s·m−2, k is Boltzmann’s constant (1.38×10−23 m2·kg·s−2·K−1), T is the absolute temperature in K (assumed 295 K unless stated), Vp is the nanoparticle volume in m3 (excluding surfactant contribution), Ka is the anisotropy constant in J·m−3 and τ0 is the characteristic Néel time given by Rosensweig [2] as 1 ns.

equation M25
(24)
equation M26
(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.

equation M27
(26)
equation M28
(27)

2.2.2 Magnetic Fields

Considering the geometric arrangement of Fig. 1, the imposed magnetic field intensities along ix and iz are spatially uniform with corresponding zero spatial derivatives. This follows from Ampère’s Law, given by (28), which states that in the absence of any conduction (J = 0) or displacement current densities ( equation M29), the curl of the magnetic field intensity in the ferrofluid channel is necessarily zero, as given by (28). For the geometry of Fig. 1, this means that both the x and z components of magnetic field intensity are uniform in the channel since H0 is invariant in space. The convenience of the planar channel should now be apparent where field derivatives with respect to x and z are taken to be zero while the pressure gradient, equation M30 can remain non-zero, allowing for the imposition of linear flow.

equation M31
(28)
equation M32
(29)

The imposed magnetic flux density along iy is governed by Gauss’ Law.

equation M33
(30)
equation M34
(31)

In light of these observations, it can be seen that by and hx can only be spatially constant, independent of y while hy may depend on y if there is a y-dependence in my. The total instantaneous magnetic flux density, B, the total instantaneous magnetic field, H, and the total instantaneous magnetization, M are given by (32), (33) and (34) respectively where hx and hy are now sinusoidally time-varying quantities and hz = 0. Complex amplitude notation is used for convenience. The ∧ symbol denotes a complex amplitude, Re signifies the real component of a complex quantity, Ω is the frequency of excitation in rad·s−1 and equation M35. The relationship between B, H and M is given by the familiar expression of (35).

equation M36
(32)
equation M37
(33)
equation M38
(34)
equation M39
(35)

The relationship between M0 and H0, 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 iz. Therefore, the z-component of magnetization is DC and given by the equilibrium value of M0. To solve for the instantaneous magnetization, the imposed fields, by and ĥx, are assumed known small-signal sources. Substituting for Meq from (18) through (21) and for M from (34), the x and y (transverse) components of (22) are given by (36) and (37) respectively. Making use of (35), (36) and (37) are used to write solutions for mx and my which are functions of spin-velocity, as given by (38) and (39).

equation M40
(36)
equation M41
(37)
equation M42
(38)
equation M43
(39)

The complex amplitudes of transverse magnetization, mx and my, are a function of the yet undetermined spin-velocity ωz. As will be shown in the subsequent analysis, in the absence of imposed flow, the spin-velocity (and hence mx and my) is spatially invariant. With the imposition of flow, the spin-velocity, as well as mx and my, then vary with channel width, y. The second term of (22) does not contribute any terms since the flow velocity is along ix and equation M44 of the magnetization is zero. The magnetization in the ferrofluid gives rise to a torque which, in turn, is responsible for fluid motion causing a non-zero value for spin-velocity ωz which changes the magnetization. Therefore, as well as the ferrofluid relaxation, one should consider the mechanical equations of interest to arrive at a consistent solution for mx, my and ωz.

2.2.3 Fluid Mechanics

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 iy in m·s−2, ρ is the ferrofluid mass density in kg·m−3, ζ is the ferrofluid vortex viscosity in N·s·m−2 which is approximately given by equation M45 for [var phi] [double less-than sign] 1 [2], η is the dynamic shear viscosity of the ferrofluid in N·s·m−2 and Tm is the magnetic torque density given by (43) in N·m−2. The small-signal expressions consider the situation of sinusoidal steady state with viscous dominated flow conditions (so that inertia is negligible) and the ferrofluid only responds to force and torque densities which have a time-averaged, non-zero component. Also ignored are the coefficients of shear and bulk viscosity (η′ and λ′) [2] discussed by Elborai [27] and He [28] in reference to (42).

equation M46
(40)
equation M47
(41)
equation M48
(42)
equation M49
(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 [nabla] × v term) and (ii) due to a magnetic torque, Tm, arising when the magnetic field and the associated magnetization in the suspension, are no longer parallel. In the absence of a magnetic torque density (e.g., in a non-magnetic fluid), (42) reduces to the rotational momentum equation, neglecting inertial terms, where spin-velocity is due only to vorticity in the flow. For a magnetic fluid, in the absence of flow, the spin-velocity is proportional to the magnetic torque density and, thereby a measure of misalignment between the magnetic field and fluid magnetization. After time averaging, denoted by left angle bracket right angle bracket, the only remaining non-zero component of magnetic torque density for the system, now expressed in terms of the applied fields from (43), is given by (44) where * denotes the complex conjugate.

equation M50
(44)

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

equation M51
(45)

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

equation M52
(46)

Considering (40), one should note that the second non-zero term, μ0(M · [nabla])H, is the magnetic force density Fm associated with the ferrofluid. However for the geometric arrangement of Fig. 1, the field components can only vary with y so the only non-zero component of magnetic force density is y-directed and considering (31) and (35) given by (47). Again, only the non-zero, time-averaged terms are retained. The ∧ symbol indicates the complex, small-signal amplitude.

equation M53
(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 ix and iy, given by (48) and (49) respectively. There is no component along iz. Assuming that pressure may vary as a function of channel width and length (i.e., p = p(x, y)), then integration of (49) leads to (50) where f(x) is due to the x-directed flow velocity while the pressure’s y-dependence is due (a) to gravity, as noted in (41) and (b) the y-dependence in my.

equation M54
(48)
equation M55
(49)
equation M56
(50)

Equations (38), (39), (46) and (48) now constitute a closed system of equations with 4 unknown quantities: spin-velocity, ωz, flow velocity, vx, and the two complex amplitudes of magnetization along ix and iy, denoted mx and my, respectively. It is again noted that by and ĥx are imposed uniform small-signal fields and therefore known. While (50) is an additional equation of interest which accounts for changes in fluid pressure, it does not govern fluid flow. However, (50) does provide a way to measure |my| from pressure measurements.

2.3 General Solutions

2.3.1 Boundary Conditions

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 in is the unit vector normal to the boundary in the second medium. Ks is the boundary surface current density in A·m−1 at y = 0 and y = d for Figure 1 which is the source for the x-directed and z-directed magnetic fields. The magnetic flux density and magnetic field intensity, given by Bin and Hin respectively, correspond to the fields on the inside of the ferrofluid region (i.e., 0 < y < d) while Bout and Hout correspond to the fields outside the ferrofluid (i.e., y < 0 and y > d). The surface currents at y = 0 and y = d must be opposite in direction (x-directed DC surface current to generate a uniform DC z-directed field and z-directed time-varying surface current to generate a time-varying x-directed field) so that the magnetic fields outside the ferrofluid region are zero. For the y-directed B field, the field is imposed upon the channel when magnetic pole faces are located at y = 0 and y = d.

equation M57
(51)
equation M58
(52)

There are no boundary conditions on the spin velocity, ωz, as the spin viscosity is not considered in (42), while the velocity, vx, at the stationary boundaries (y = 0 and y = d) is also zero.

2.3.2 Poiseuille Flow Solution

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 equation M59 allows for simplification of (48) by substitution of the equation M60 term from (53).

equation M61
(53)

This leads to the expression of (57) after twice integrating (48) with respect to y. K1 and K2 are constants of integration while left angle bracketTm,zright angle bracket is given by (45).

equation M62
(54)
equation M63
(55)
equation M64
(56)
equation M65
(57)

Applying the non-slip boundary conditions on vx at y = 0 and d allows for the solution of K1 and K2.

equation M66
(58)
equation M67
(59)

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

equation M68
(60)

Rewriting (46) as (61) and substituting for the derivative of vx leads to the following expression for the spin-velocity, ωz, which is independent of vx and where left angle bracketTm,zright angle bracket is given by (45).

equation M69
(61)
equation M70
(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 left angle bracketTm,zright angle bracket is negligibly small, in which case, spin-velocity is given by (63) and the mid-channel spin-velocity is zero (i.e., equation M71). This is the case of “vorticity-driven” spin-velocity.

equation M72
(63)

The resulting Poiseuille flow has flow velocity profile given by (64) with a maximum value, Up, at y = d/2.

equation M73
(64)
equation M74
(65)

Another interesting limiting case is conditions of negligible flow in the channel (i.e., vx(y) = 0). Then ωz, mx, my and left angle bracketTm,zright angle bracket are independent of y so that ωz reduces to (67).

equation M75
(66)
equation M76
(67)

Substituting for mx and my from (38) and (39) respectively yields a cubic solution for the spin-velocity, ωz. The no-flow solution (vx = 0) is examined in the Results of Section 3 since it represents the contribution to spin-velocity and transverse fluid magnetization due only to the applied transverse magnetic field in the absence of any imposed fluid flow, a situation typical in low-perfusion MR imaging (e.g., breast tissue imaging).

3 Results

3.1 Physical Parameters

The physical parameters for the simulated magnetic nanoparticles are shown in Table 1 as well as the nominal field conditions. Up = vx(y = d/2), from (65), denotes the maximum imposed inflow velocity under conditions of Poiseuille flow.

Table 1
Table of Nominal Physical Parameters

The main DC B0 field is directed along iz. An approximately rotating field of constant amplitude is generated in the transverse xy plane by setting ĥx = He and by = jBe where Be and He are real-valued quantities. A purely rotating field is difficult to achieve since the transverse magnetization components, mx and my, will clearly influence the fields within the fluid. However, when He [dbl greater-than sign] mx, my and Be is set to be μ0He, one can assume an approximate rotating magnetic field. The resultant transverse field rotates in a clockwise sense about the z axis for He, Be > 0. The magnetic particle properties are typical of those used as superparamagnetic contrast agents in MRI [8,10,30] and similar to those proposed in magnetic nanoparticle hyperthermia [11] and targeted drug delivery within MRI [12]. The magnetic core is assumed to be magnetite (Fe3O4). The radian frequency of excitation, denoted Ω, of the transverse rotating field varies but is normalized with respect to the ferrofluid time constant τ in the resulting plots. The non-dimensional spin-velocity is defined as ωzτ. The ferrofluid dynamic viscosity, η, is that recorded by He [28] using 3% solid volume MSG-W11 water-based ferrofluid from the Ferrotec Corporation, Bedford, New Hampshire. The nominal concentration of 0.03 solid volume fraction is approximately three orders of magnitude greater than the recommended in vivo concentration of Feridex, the most widely used commercial SPIO MRI contrast agent, used for liver imaging [29] although such high concentrations may well prove necessary in applications such as nanoparticle hyperthermia. The reasons for this adjustment are outlined in the discussion of Section 4.

The effect of varying the physical system parameters is examined for the normalized spin-velocity, |ωz|τ, and the magnitude of the time-average transverse magnetization, left angle bracket|Mtrans(y)|right angle bracket. Referring to (34), one might write left angle bracket|Mtrans(y)|right angle bracket as given by (71). For non-dimensionalized plotting, the transverse magnetization is normalized with respect to Be by dividing by Be/μ0.

equation M77
(68)
equation M78
(69)
equation M79
(70)
equation M80
(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 mx and my can differ in both magnitude and phase.

3.2 Results with No Imposed Flow

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 B0 field on both the normalized spin velocity, ωzτ, and the time-average transverse magnetization, normalized with respect to Be were investigated. The results are shown in Fig. 3(a) and (b). Also investigated was the magnetization’s dependence on ferrofluid concentration, [var phi], shown in Fig. 4(a) versus normalized time, Ωt (shown for Ωτ = 1 over one full cycle), and in Fig. 4(b) versus normalized frequency, Ωτ. The plots were achieved using Mathematica 5.2 (Wolfram Research, Champaign, IL) with reference to (38), (39) and (67) where Mathematica was used to express and plot the solutions in closed form. In each case, the x-axis is the non-dimensional transverse rotating field frequency Ωτ, the rotating field amplitude, Be, is 5% of B0 and [var phi] = 3%.

Fig. 3
Frequency dependence of |ωz|τ and the normalized, time-average transverse magnetization on B0 is shown for [var phi] = 3% and Be = 0.05B0. Note that for τ = 1 μs, Ω/(2π) ≈ 160kHz when Ωτ ...
Fig. 4
Time dependence of the transverse magnetization magnitude (a) for various values of Ωτ plotted versus normalized time, Ωt for [var phi] = 0.05 and (b) frequency dependence of the time-average normalized transverse magnetization ...

3.3 Imposing Poiseuille Flow

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 Up for Poiseuille flow conditions. Finite element simulation and solution of the variables in the channel is achieved using Comsol Multiphysics where (38) through (44) are the equations of interest, in the case of Poiseuille flow. Boundary conditions of zero flow are imposed upon the upper and lower plates as is the case for no imposed flow. However, inlet flow (defined from the left side) is x-directed and defined by the parabolic distribution of (72) where d is the channel width while the outlet boundary condition defines the pressure, p = 0. The channel length was set to 100d in the numerical channel simulations using Comsol Multiphysics.

equation M81
(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, left angle bracketTm,zright angle bracket. Since the flow-induced spin-velocity changes sign at y = d/2, the two contributions to the spin-velocity (from (i) the imposed fields and (ii) the imposed flow) will sum positively for d/2 < y < d and sum negatively for 0 < y < d/2. For the geometry of Fig. 1 where flow is directed along +ix, the contributions to ωz from the imposed rotating field, Be, and that due to Poiseuille flow, add at the lower plate and subtract at the top so that the maximum value of |ωz|τ occurs at y = 0 and y = d. In the results of Fig. 5, the channel width (1 μm) and imposed maximum flow velocity (0.1 < Up <1 m/s) are chosen so that the spin-velocity, and hence changes in the transverse magnetization as a function of channel width, y, are almost entirely due to the imposed flow rather than the applied rotating field. Fig. 5 shows the normalized spin-velocity and normalized, time-average transverse magnetization across the channel width for various maximum imposed flow velocities: Up =0.1 m/s, 0.5 m/s and 1 m/s and when Ωτ = 1 to maximize the effect of spin-velocity and B0 = 0.2 T. These flow velocities represent the maximum range of blood flow velocities in vivo where the magnetic nanoparticle suspension might potentially flow.

Fig. 5
The normalized spin-velocity and the normalized, instantaneous, transverse magnetization are shown as a function of y, the channel width in μm for various maximum imposed Poiseuille flow velocities, Up. B0 = 0.2 T, Be = 5% of B0 and [var phi] ...

4 Discussion

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 B0) 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 Ωτ → 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 Ωτ. The ferrofluid spin-velocity, ωz, was introduced as a measure of nanoparticle spin or rotation [1]. In the absence of fluid flow (v = 0) in a magnetic fluid, the spin-velocity is proportional to the magnetic torque density of the fluid suspension, as given in (42) (or in physical terms, the spin-velocity is proportional to the instantaneous misaligment which occurs between the fluid’s magnetization and the applied magnetic field). However, with the introduction of linear flow, the particle rotation (and hence the spin-velocity in the fluid) will be altered due to any vorticity that may be present in the flow as well as due to the magnetic torque density, as governed by (42).

4.1 Spin-velocity dependence on rotating field frequency Ω

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, ωz, is closely related to the ferrofluid time constant, τ, as should be expected from the expressions of (38) and (39) since ωzτ is a term that appears in both the numerator and denominator of each expression. For low-frequencies where Ωτ [double less-than sign] 1 and ωzτ [double less-than sign] 1, the effect of spin-velocity on the fluid’s transverse magnetization is negligible. Physically this low-frequency limit corresponds to a fluid with a small-signal transverse susceptibility given by M0/H0 and the instantaneous transverse M and H vectors are almost collinear.

As Ωτ 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 H for a particular rotating field frequency Ω. Mathematically, this is shown by the on-axis magnetization (both mx and my) given by (38) and (39) becoming increasingly dependent on the orthogonal transverse field component, ĥx, for the case of my, and by for the case of mx. (i.e., the effect of by in (38) and the effect of ĥx in (39) increases). This cross-coupling results in a non-symmetric, small-signal transverse susceptibility which causes Mtrans to lag transverse magnetic field excitation as previously noted.

In the high-frequency limit, where Ωτ [dbl greater-than sign] 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 Ωτ [dbl greater-than sign] 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/(2π) on the order of 1.6 MHz before Ωτ ≈ 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 Ωτ [dbl greater-than sign] 1 so that the plots should be treated with some caution beyond Ωτ ≈ 10. The z-directed magnetization, M0, is DC as previously noted (hz = 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 Ωτ = 1 within the MRI, as indicated by the peak in spin-velocity, as shown in Figure 3(a).

4.2 No Imposed Flow

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 Ωτ = 1 and increases with the main z-directed B0 field, as shown in Fig. 3(a). However, the time-average transverse magnetization seen in Figure 3(b) decreases with B0. This is because when B0 is increased, the small-signal susceptibility for the transverse components, M0/H0 shown in Figure 1, decreases. So while the value of spin-velocity increases, the underlying coupling between the small-signal transverse field perturbation (hx or hy) and the associated instantaneous magnetization components (with complex amplitudes given by (38) and (39)), is weakened with larger B0. Physically, this effect with increasing B0 can be interpreted as the increasing probability (as determined by the Langevin relation) of each individual magnetic nanoparticle remaining parallel to B0 rather than rotating in the transverse plane in response to the transverse Be field. The result is a decrease in the fluid’s transverse magnetization component with increasing B0.

The time-dependence of the transverse magnetization is shown in Figure 4(a) for various values of Ωτ and [var phi] = 0.05. Clearly, as predicted by (71), the phase lag behind the sinusoidal Be field is more pronounced as Ωτ increases where the instantaneous transverse magnetization has an inflection point that lags that of the sinusoidal Be field. It is this phase-lag between the transverse M and H fields which gives rise to a non-zero magnetic torque density in the magnetic fluid. The effect of [var phi], shown in Figure 4(b) on the time-average transverse magnetization is pronounced and arises from the dependence of the saturation magnetization, Ms = [var phi]Md, causing the normalized time-average transverse magnetization left angle bracket|Mtrans|right angle bracket increases with [var phi].

In no case does the value of normalized transverse magnetization become comparable to Be/μ0. Even for the conditions of Figure 3(b) where B0 = 0.1 T and Be = 5% of B0, the normalized magnetization is only 11% of He for the low-frequency limit discussed previously. This indicates that it may be difficult to observe the frequency-dependence of the transverse magnetization in low-field MRI since the change in the transverse magnetization is in addition to (i) the main B0 and (ii) the applied rotating field used to generate it. However, non-MRI imaging modalities, such as magnetic particle imaging (MPI) [14] may be suitable. In MPI, it is the magnetization associated with the magnetic nanoparticles themselves which give rise to signal rather than the imaging spin of the hydrogen proton, as is the case in MRI.

The transverse magnetization decreases as Ωτ 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 Ωτ 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 B0 (which is not controllable in an interactive sense). However, imaging modalities do exist where the B0 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 B0 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 Ωτ 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 Ωτ ~ 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 Ωτ approaches unity.

4.3 Imposing Poiseuille Flow

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 Up evaluated (i.e., the maximum normalized spin velocity, ωzτ, has values of 0.2, 1 and 2 when Up = 0.1, 0.5 and 1 m/s respectively where d = 1μm and τ = 1μs).

equation M82
(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 μm 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 B0 is increased (e.g., 1.5 T or 3 T) due to magnetic saturation. The dependence on spin-velocity (and hence on Up for a vorticity-driven system) is involved as seen from (38) and (39) in conjunction with (62). For small Up, typical of physiological flow (~0.1 m/s), the normalized spin velocity term (ωzτ) in the numerator dominates and the magnetization is approximately a linear function of the channel width, y. With increasing Up ~1 m/s, it is found the equation M83 term of the denominator becomes significant, resulting in the non-linear functions of y shown in Fig. 5(b). The point here is that what was previously a uniform fluid magnetization in the transverse plane is now non-uniform due to the nanoparticle flow in narrow channels. The effect provides a second mechanism of interaction with the fluid’s magnetization. While effects due to flow vorticity many not be readily applicable in the physiological environment (due to (i) the high particle concentrations of nanoparticles required to see effects in the vasculature and (ii) the combination of high-flow velocities and narrow channels necessary to create the necessary vorticity), the effect is noted and may be more relevant to non-clinical scenarios such as pipe-flow monitoring with magnetic or magnetically-doped flow.

5 Conclusion

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 Ωτ 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.

Acknowledgments

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.

Footnotes

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