Feedback Control for Transportation of Magnetic Fluids With Minimal Dispersion: A First Step Toward Targeted Magnetic Drug Delivery

In targeted magnetic drug delivery, drugs attached to magnetic nanoparticles are delivered to targeted regions of the body (e.g., tumors) at high concentrations by the application of external magnetic fields. Beyond many technical difficulties posed by the complexities of the human body, the performance of this scheme is limited in concept by the inherent tendency of the magnetic fields to disperse any constellation of magnetic particles moving under their applied forces. This tendency causes a gradual loss of concentration when a spot of ferrofluid (suspension of magnetic nanoparticles in water) moves inside a magnetic field. To minimize this undesirable effect to an acceptable level, a feedback control policy for the dynamic control of electromagnets is presented to drive a ferrofluid spot from the edge of a domain to a central target with minimal dispersion. Representing the spot of ferrofluid by its center of mass and the covariance matrix of its distribution, these control goals are formulated as an optimal control problem that constrains the motion of the former, while minimizing the trace of the latter (as a measure of dispersion). To simplify the solution of this problem, the ferrofluid dynamics, which is precisely governed by a partial differential equation, is approximated by a finite-dimensional set of state-space equations, and a suboptimal control is developed for this simpler model by further approximation of the Hamilton-Jacobi-Bellman equation. Simulation results are presented for the closed-loop system, which demonstrate that the proposed control is able to move the ferrofluid spot to a central target with reasonably small dispersion.