IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006
Dynamic Characteristics of Superparamagnetic Iron Oxide Nanoparticles in a Viscous Fluid Under an External Magnetic Field Min-Cheol Kim1 , Do-Kyung Kim2 , Se-Hee Lee3 , M. Shahrooz Amin3 , Il-Han Park4 , Charn-Jung Kim1 , and Markus Zahn3 School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-744, Korea Institute for Science and Technology in Medicine, Keele University, Hartshill, Stoke-on-Trent, ST4 7QB, U.K. Department of Electrical Engineering and Computer Science (EECS), Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 USA School of Information and Communication Engineering, Sungkyunkwan University, Suwon 440-746, Korea Predicting the motion of superparamagnetic iron oxide nanoparticles (SPION) in biological systems is one of the most relevant parameters for the biomedical application of SPION. Recently, magnetic nanoparticles such as SPION have been utilized as the MR image contrast agent, targeted drug delivery, magnetic separation, biosensor, etc. Because the size of SPION is in the nanometer scale, it is difficult to test their dynamics under an external magnetic field experimentally. Moreover, when the SPION moves in a viscous fluid such as blood, the situation can be more complicated. To overcome these difficulties, we proposed a fast-solving technique combined with the finite-element method for analyzing the dynamic characteristics of SPION in a viscous fluid under an external magnetic field. Using this numerical modeling, the effective system dimension, external magnetic field, particle size, and trajectory of SPION can be estimated. To verify the proposed method, three magnetic systems with permanent magnet and micro capillary were tested. Index Terms—Superparamagnetic iron oxide nanoparticles (SPION), targeted drug delivery, trajectory, viscous fluid.
AGNETIC microparticles and nanoparticles offer some attractive possibilities for biomedical applications –. Particularly speaking, the superparamagnetic iron oxide nanoparticles (SPION) have been applied in the active drug delivery, hyperthermia, magnetic cell separation, and in enhancing the MR image, etc. . SPION can be easily isolated from cerebrospinal fluid (CSF) by applying an external magnetic field after intravenous or intracellular administration. Therefore, successful transport of immunomagnetic particles through biological membranes [blood–brain–barrier (BBB)] will be a frontier for developing therapies for psychological and neurodegenerative disorders. Schizophrenia, depression, and Alzheimer’s diseases are strongly dependent upon molecular mechanisms, at a network level (intra- and intercellular communications) as well as at a unicellular level (genes and proteins) and can be analyzed with biomolecules coupled to SPION. Interaction between the organic ligands and the surface of inorganic nanoparticles paves the way for the coupling of biomolecular recognition systems to generate novel materials. This represents a new concept for noninvasive monitoring of cancer/tumor by both in vivo SPION injection and in vitro microdialysis . An essential aspect of this paper is considering the development of a theoretical model that can simulate the in vivo motion of SPION under an applied external magnetic field. Numerical models illustrated in this paper are intended to provide rough
estimation and general guidelines for transporting SPION under a nonuniform external magnetic field through micro-sized biological capillary. General guidelines are determined by considering the interaction of competing forces in the system. Specifically, magnetic, drag, gravitational, and buoyant forces are taken into account to analyze the trajectories of SPION under the external magnetic field in biological membrane. This numerical modeling of the SPION trajectory is inevitable because the dimensions of the system, such as the diameter of particles, are in the nanometer regime; thus, they cannot be traced by using the existing instrumentation techniques. Using the given force and dynamic equations, the simultaneous ordinary differential equation (ODE) was formulated and solved by employing a four-stage Rosenbrock method . The migration velocity and elapsed time of single SPION were calculated under different magnetic-field strength, gaps between two magnets, widths of permanent magnet, and particle sizes. We can also note the trapping position and time spanned by SPION in a capillary along with the fluid flow. II. MATHEMATICAL MODELING FOR SPION MIGRATION IN A VISCOUS FLUID The fundamental forces acting on SPION in a viscous solu, gravtion under an external magnetic field are magnetic , buoyant , and drag resistance forces. itational When magnetic particles are immersed in a fluid such as blood, the magnetic force on magnetic particles is expressed as , 
Digital Object Identifier 10.1109/TMAG.2006.872032 0018-9464/$20.00 © 2006 IEEE
IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006
where denotes the volume of the SPION, the volume the volume sussusceptibility of the SPION in the medium, ceptibility of the fluid, the magnetic permeability of vacuum, the external magnetic field, and the del operator. The net buoyant force considering the gravitational force on the SPION in a viscous fluid is expressed as (2) is the density of the SPION in the liquid medium, where the density of the liquid medium, and the acceleration due to gravity. When the SPION moves adequately slow, i.e., the , the particle motion is governed Reynolds number by Stokes’ law, and of the SPION in the aqueous medium is expressed as (3)
Fig. 1. Ordered counterclockwise definition of polygon with n vertices.
III. NUMERICAL INTEGRATION SCHEME AND PARTICLE SEARCHING ALGORITHM
Usually, integration of a stiff ODE has been encountered by the stability. For more stringent parameters, the Rosenbrock method remains a more reliable and offers simpler algorithms than the Runge–Kutta method . Both Rosenbrock and Runge–Kutta methods have been applied to solve the SPION’s movement in a capillary for a simple case with analytic magnetic-field solution . In that case, the Runge–Kutta method is more complicated to control the time steps for solving the stiff ODE. The numerical data obtained from the Runge–Kutta method was deviated in most of the cases and required more steps of iteration (more than 50 000 in this case) to avoid any further deviation. On the other hand, the Rosenbrock method exhibited a more reliable data trend and consumed less time in computing and exhibited no deviation. For this reason, a four-stage Rosenbrock method has been introduced to calculate the motion of SPION in a viscous fluid. Hence, the particle-search algorithm is required to evaluate the velocity of the neighboring fluid at the location of the particle. Once information about location of the particle is obtained, spatial and temporal interpolations of velocity fields and other fluid properties at the location of the particle are performed; using neighboring cell connectivity, the forces acting on the particle can then be calculated. In case of a structured Cartesian grid, the location of the particle along with obtained carrier properties can be readily solved because faces of control volumes are parallel to the axes of the Cartesian coordinates. This simple problem, however, cannot be applied in complicated geometries, such as nonorthogonal unstructured grids. Therefore, the particle-search algorithm plays a crucial role in enhancing the computation. In this paper, we used the Zhou–Leschziner (ZL) method for searching SPION in an analysis region . The ZL method defines an arbitrary two-dimensional (2-D) polygon by the Cartesian coordinates vertices ordered of its in an counterclockwise manner as shown in Fig. 1. A scalar test variable , which determines whether a particle is located within the specified polygon, is defined as
, and where the relative velocity are the and components of magnetic field density, respectively; , . and
denotes the fluid viscosity, the SPION radius, and the velocity of the SPION relative to the . Therefore, the force balance equation for fluid velocity the SPION in the aqueous solution in the presence of external magnetic field can be expressed as where
(4) Equation (4) can be a stiff ODE due to the extremely small dimension of SPION size. The velocity of a single SPION under an external magnetic field can be obtained from (4) assuming as (5) Finally, the simultaneous state equations in the – plane can be obtained to solve the dynamic characteristics of the SPION migration in a capillary as
KIM et al.: DYNAMIC CHARACTERISTICS OF SPION IN A VISCOUS FLUID UNDER AN EXTERNAL MAGNETIC FIELD
Fig. 3. Velocity profile of fully developed flow in the microcapillary. The inlet and outlet boundary condition are applied at x = 0 and x = 0.02 m, respectively. No-slip condition is used in the wall of the capillary so that the blood cannot pass through the wall and cannot move along the wall. The maximum magnitude of the velocity is 0.002 m/s at the center line of the capillary, y = 0. Fig. 2. Schematic representation of the microcapillary model tested with different radius of the SPION, magnet sizes (D1), distances between two magnets (D2), and remnant magnetic flux densities (B ) in a blood flow. The mean inlet velocity is 1 mm/s, the viscosity of blood is chosen to be 0.00345 Pa s, and the mass densities of the SPION and blood are chosen to be 5200 k and 1045 kg/m . The volumetric susceptibilities of the SPION and 10 and 9:05, respectively. blood are chosen to be 2
where means is located on the left-hand side means is located of the line segment , and on the right-hand side of the line segment means is located on the line segment . When , the search is continued to the next line segment of the , the tracking is moved same polygon, and then in case to the neighboring polygon. Finally, the tracking finishes until the entire test variable in the polygon yields positive values. IV. NUMERICAL RESULTS Fig. 2 shows the diagram of analysis model with microchannel and permanent magnets as an external applied magnetic-field source. In this model, the assumption that the movement of SPION has no relation with fluid flow can be introduced because the particle Reynolds number when 10 nm is much less than 1.0. We, therefore, can decouple the magnetic field and fluid flow field. The velocity profile in the microcapillary was analyzed by using the computational fluid dynamics (CFD) solver as shown in Fig. 3. We put the single SPION at (0, 0) position so that the trajectory of the single SPION is a function of alone. To verify the proposed method, we compared the velocity profiles obtained from the analytic velocity expression (5) and the proposed algorithm tested with different remnant magnetic 1, 0.5, and 0.1 T, as shown in Fig. 4. They flux densities generally agree well with each other. The results of the analytic calculation are slightly higher than those of the proposed 1 T in the proposed method, the SPION method. In case of 0.0117 m by the exare captured and oscillated at around ternal magnetic field because the magnetic force is balanced 0.0117 m. Because with other mechanical forces at around (5) was obtained from when we neglect the inertia term, the analytic expression cannot consider the effects of motion.
Fig. 4. Comparisons between the analytic solutions by using (5) and the numerical solutions by using the proposed method in case of B = 1, 0.5, and 0.1 T along with y = 0 line.
To show the effects of the important variables, the parametric analyses were conducted for calculating the velocity profile and elapsed time as a function of the position as shown in Fig. 5. When the magnetic force acts in the negative direction, the velocity reaches at 0 m/s because the forces acting on the SPION are balanced. At this point, the SPION is trapped. After being trapped, when the SPION moves in the positive or in the negative direction following the flow of fluid, the negative or positive values of drag and magnetic forces act in the negative or in the positive direction respectively. For this reason, this trapped single SPION oscillates horizontally. Fig. 5(a)–(f) shows that the maximum velocities are strongly dependent on the external magnetic field. The axis of the peak positions also shifted toward the direction of the peak positions of magnetic forces as shown in Fig. 5(c) and (d). When the size of the SPION increased, an increase of the velocity is observed because of increased moment of inertia. Figs. 6 and 7 show the effects of shape factor of permanent magnet. The cubic squared permanent magnet produces a higher
IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006
Fig. 7. Comparisons between cubic square and circle shapes of permanent magnets: (a) velocity profile and (b) elapsed time of the SPION as a function of x position with y = 0.
Rosenbrock method and electromagnetic and CFD finite-element solvers. The SPION movement was successfully calculated by considering the magnetic field and fluid flow in a microcapillary. The most important factors, such as the distributions of an external magnetic field and particle size, have been considered for the movement of single SPION in a microcapillary. The optimum conditions for the migration of SPION in biological systems, such as central nervous system capillaries and blood vessels, can be effectively predicted by utilizing this method. The numerical modeling in this paper has considered only single-particle motion, but it can be further extended by considering more complicated systems, including the interaction between the particles, surface tension, three-dimensional geometry, etc. ACKNOWLEDGMENT Fig. 5. Velocity profile and elapsed time of the SPION in a blood flow as a function of x position with y = 0. (a) and (b) resulted from different B = 0.1, 0.5, and 1 T when 2R = 10 nm and D1 = D2 = 0.01 m; (c) and (d) resulted from different D1 = 0.001, 0.005, 0.01, 0.015, and 0.02 m when 2R = 10 nm, D2 = 0.01 m, and B = 1 T; (e) and (f) resulted from different D2 = 0.001, 0.005, 0.01, 0.015, and 0.02 m when 2R = 10 nm, D1 = 0.01 m, and B = 1 T; (g) and (h) resulted from different 2R = 8, 10, 20, and 30 nm when D1 = D2 = 0.01 m, and B = 1 T. The maximum velocity, approximately 0.0025 m/s, was obtained in case of D2 = 0.001, as shown in Fig. (e).
Fig. 6. Analysis models tested with different permanent magnet shapes: (a) cubic square, and (b) circle shape in 2-D geometry.
magnitude of magnetic force field than the circle shape at around 0.0125 m so that the SPION can be trapped. V. CONCLUSION To analyze the dynamic characteristics of SPION, we proposed a new fast-solving technique by employing the four-stage
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Manuscript received June 20, 2005 (e-mail: [email protected]