On the motion of superparamagnetic particles in magnetic drug targeting

Abstract

A stochastic ODE model is developed for the motion of a superparamagnetic cluster suspended in a Hagen-Poiseuille flow and guided by an external magnet to travel to a target. The specific application is magnetic drug targeting, with clusters in the range of 10-200 nm radii. As a first approximation, we use a magnetic dipole model for the external magnet and focus on a venule of 10 −4 m radius close to the surface of the skin as the pathway for the clusters. The time of arrival at the target is calculated numerically. Variations in release position, background flow, magnetic field strength, number of clusters, and stochastic effects are assessed. The capture rate is found to depend weakly on variations in the velocity profile, and strongly on the cluster size, the magnetic moment, and the distance between the magnet and the blood vessel wall. A useful condition is derived for the optimal capture rate. The case of simultaneous release of many clusters is investigated. Their accumulation in a neighborhood of the target at the venule wall follows a normal distribution with the standard deviation roughly half of the distance between the magnet and the target. Ideally, this deviation should equal the tumor radius, and the magnet should be beneath the center of the tumor. The optimal injection site for a cluster is found to be just prior to arrival at the target. Two separate mechanisms for capturing a cluster are the magnetic force and, for radii smaller than 20 nm, Brownian motion. For the latter case, the capture rate is enhanced by Brownian motion when the cluster is released near the wall.

Modeling Superparamagnetic Particles in Blood Flow for Applications in Magnetic Drug Targeting

“…The advantage of this is to both verify the consistency of the simulations with measured values from previous experiments as well as to produce estimates for quantities that are, practically, extremely difficult to measure. Some universal constants that are used include 10 −2 m for the tumor radius, the release location as 0.1 m upstream from the tumor from the center of the vessel (except when a distribution of starting positions is applied), 10 −4 m as the radius of the vessel consistent with venules and arterioles, as desired, and particle radius of 10 −7 m consistent with commercially available products and previously conducted theoretical studies [10,22].…”

“…with M signifying the particle's magnetization and V the volume of the particle [10]. The Clausius-Mossotti formula for a spherical particle is then used to produce a uniform magnetization throughout the particle…”

“…Previous research has shown that it is insufficient to only consider the force balance between cross-stream magnetic attraction and drag given a uniform flow across the cross-section of the pipe in light of the significance of diffusion and no-slip near the vessel walls [15][16][17]. As such, studies have evolved from accounting only for magnetic and drag forces [9] to tracking particle trajectories when subjected to an external magnet under power-law flow [10]. Recent models have even considered diffusion and sources of stochasticity with Brownian motion and interactions with red blood cells [11,12].…”

“…There have been numerous promising trials [1,2] and models of the behavior of ferrofluids intended for applications in retinal detachment restoration [3][4][5], magnetic separation for fluid filtering [6,7], drug therapy for in-stent restenosis [8], and localized cancer treatment [9][10][11][12][13][14] validating the plausibility of this method.…”

Ferrofluids and magnetically guided superparamagnetic particles in flows: a review of simulations and modeling

“…In this section, we summarize the set-up and results of [42]. One of the simplest model problems for predicting the accuracy of drug targeting concerns superparamagnetic nanoparticles of at most several nanometers diameter, and flowing through a venule of order 10 −4 m radius; see figures 4 (a -b).…”