COMSOL USER’S CONFERENCE ABSTRACT

NUMERICAL SIMULATIONS OF MAGNETIC NANOPARTICLE SUSPENSIONS AS INTERACTIVE MRI CONTRAST AGENTS PÁDRAIG CANTILLON-MURPHY1, PROFESSOR ELFAR ADALSTEINSSON1,2 AND PROFESSOR MARKUS ZAHN1 1

MASSACHUSSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MA 02139, 2MIT DIVISION OF HEALTH SCIENCES AND TECHNOLOGY, CAMBRIDGE, MA 02139

Introduction: Magnetic nanoparticle suspensions, or ferrofluids, are synthesized, superparamagnetic, colloidal mixtures of non-magnetic carrier liquid, typically water or oil, containing single domain permanently magnetized particles, typically magnetite, with diameters of order 5-15 nm and volume concentration up to 10%. Brownian motion keeps the nanoparticles from settling under gravity and a surfactant or polymeric layer, such as oleic acid, surrounds each particle to prevent agglomeration. Industrial applications of ferrofluids in loudspeakers and computer disk drives are extensive. However, with modern advances in fabrication and biotechnology, ferrofluids (referred to as superparamagnetic iron oxides or SPIOs) have found application as contrast agents [1] in magnetic resonance imaging (MRI). However, there is, as yet, no means of controlling the signal intensity associated with the nanoparticles by means of an external magnetic field. This current work seeks to utilize the ferrofluid’s complex magnetic susceptibility to impact the signal intensity in MRI. Applications of the proposed technique may include more accurate agent detectability in vivo and as well as the prospect of interaction with hyperthermia or other therapeutic treatments. Theory: When a DC magnetic field H is applied to a ferrofluid, each magnetic nanoparticle with magnetic moment m experiences a torque, μ 0 m × H , which tends to align m and H with a characteristic time constant, τ , which, depends on the particle size. Ferrofluid equilibrium magnetization, Meq, is accurately described by the Langevin equation for paramagnetism (1), where M eq and H are collinear and MS = Nm = Mdφ is the saturation magnetization when all magnetic dipoles with moment m = MdVp are aligned with the magnetic field, H. N is the number of magnetic dipoles per unit volume and φ is the volume fraction of magnetic nanoparticle material in the ferrofluid.

Meq = MS [coth α −1/ α], α = μ0mH / kT

(1)

In the presence of uniform rotating field, the ferrofluid time constant ensures that the instantaneous magnetic field leads the ferrofluid magnetization by a constant phase angle and the susceptibility of the ferrofluid is a complex vector quantity, as can be shown from Shliomis’ Relaxation Equation (2), where v is the ferrofluid flow velocity vector and ω is the ferrofluid spin velocity vector [2],[3].

∂M ∂t + (v.∇) M − ω × M + ( M − Meq ) / τ = 0

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(2)

The relaxation equation in conjunction with conservation of linear (3) and angular (4) momentum defines a closed system of equations where ζ is the ferrofluid vortex viscosity and η is the kinematic viscosity of the carrier liquid.

−∇p + μ0 ( M .∇) H + 2ζ (∇× ω) + (ζ + η )∇2v = 0

(3)

2ζ (∇× v − 2ω) + μ0 M × H = 0

(4)

Problem Definition and Results: In order to demonstrate an effect on MRI signal intensity, the arrangement of concentric cylinders, shown in Figure 1, has been simulated using Comsol Multiphysics. A large DC field, B0, along the z axis, is orthogonal to a second rotating field, Brot in the xy plane such that the two excitations are uncoupled by Shliomis’ Relaxation Equation and the problem is reduced to a 2D simulation of two subdomains consisting of ferrofluid and water. Due to the differing susceptibilities between the ferrofluid and the water, the applied rotating field results in a changing dipolar field in the water, dependent on the frequency of excitation of Brot. This change in the local B field in the water can be used to impact the MRI image signal intensity under careful selection of MRI excitation techniques. B

Figure 1. A typical problem of concentric cylinders of water and ferrofluid with a large DC field, B0, and a rotating field, Brot, in the transverse plane.

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Figure 2. Screenshot of the instantaneous solution for transverse magnetic flux density with Ωτ = 0.001

References: [1] D.D. Stark, R. Weissleder and G. Elizondo, Radiology 1988; 168, p.297. [2] M. Shliomis, JETP, 1972 ; 34, p.1291, [3] R.E. Rosensweig, Ferrohydrodynamics. New York: Dover Publications. [4] M. Zahn, Electromagnetic Field Theory: a problem solving approach. Florida: Krieger Publishing Company, 2003. [5] S.M. Elborai, “Ferrofluid Surface and Volume Flows in Uniform Rotating Magnetic Fields,” PhD Thesis. Massachusetts Institute of Technology, 2006. [6] X. He, “Ferrohydrodynamic Flows in Uniform and Non-uniform Rotating Magnetic Fields,” PhD Thesis. Massachusetts Institute of Technology, 2006.

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