On the motion of superparamagnetic particles in magnetic drug targeting

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.