Analytical model of magnetic nanoparticle transport and capture in the microvasculature

An analytical model is presented for predicting the transport and capture of therapeutic magnetic nanoparticles in the human microvasculature. The nanoparticles, with surface bound drug molecules, are injected into the vascular system upstream from malignant tissue, and are captured at the tumor site using a local applied magnetic field. The applied field is produced by a rare-earth cylindrical magnet positioned outside the body. An analytical expression is derived for predicting the trajectory of a particle as it flows through the microvasculature in proximity to the magnet. In addition, a scaling relation is developed that enables the prediction of the minimum particle radius required for particle capture. The theory takes into account the dominant magnetic and fluidic forces, which depend on the position and properties of the magnet, the size and magnetic properties of the nanoparticles, the dimensions of the microvessel, the hematocrit level of the blood, and the flow velocity. The model is used to study noninvasive drug targeting, and the analysis indicates that submicron particles can be directed to tumors that are several centimeters from the field source.

Figure 1

Noninvasive magnetophoretic drug targeting in a microvessel.

Figure 2

Geometry and reference frame for analysis: (a) coordinate systems and reference frames, (b) magnetic flux lines for a cylindrical magnet, cross section of microvessel with reference frame.

Figure 3

Relative apparent viscosity vs blood vessel diameter and hematocrit.

Figure 4

Magnetic field components along the axis of a microvessel (analytical vs FEA analysis).

Figure 5

Magnetic force components on a ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle (analytical vs FEA analysis).

Figure 6

Trajectories of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles in a microvessel with $\alpha =0.25$ (cross section of bias magnet shown for reference).

Figure 7

Trajectories of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles in a microvessel: (a) $\alpha =0.5$, (b) $\alpha =0.75$.

Figure 8

Trajectories of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticles in a microvessel: (a) $\alpha =1.0$, (b) $\alpha =1.25$.

Figure 9

Minimum particle capture radius for $x_0=Rv$ and $x_0=0$.