Superparamagnetic iron-oxide nanoparticles (SPIONs) carrying a drug payload are injected into a vessel and swept along by blood flow. Their axial speed follows the Poiseuille profile for laminar flow in a tube of radius R:
v(r) = v_max · (1 − (r/R)²)
An external magnet near the tumor creates a field gradient. The force it exerts on a magnetizable particle scales with particle volume (∝ diameter³) and falls off with distance d from the magnet — modeled here as:
F_mag ∝ B_strength · d_particle² / d²
Bigger particles feel a stronger net pull (volume grows faster than the drag they add), while smaller particles undergo more Brownian jitter, per the Stokes–Einstein relation:
D = k_B·T / (6π·η·r_particle) → jitter amplitude ∝ 1/r_particle
The tumor's vasculature is abnormally leaky (the Enhanced Permeability and Retention, or EPR, effect), so a particle that strays close enough to the vessel wall inside the tumor zone extravasates into the tissue and is captured. Magnetic targeting raises the local capture rate by physically dragging particles toward that wall. Once captured, the payload decays by first-order release kinetics:
dM/dt = −k · M(t) → M(t) = M₀ · e^(−k·t)
Particles that reach the end of the vessel without being captured recirculate with a fresh payload, and their original drug counts as lost to systemic circulation — the quantity magnetic targeting is designed to minimize.