From primary retention to saturation: an Avrami-type framework for magnetic nanoparticle capture in flow

Magnetic targeting requires quantitative models linking magnetic-field geometry, flow conditions, and nanoparticle retention. Here, dose-dependent capture of mixed Fe₃O₄/γ-Fe₂O₃ nanoparticles was investigated in a flow-through in vitro system at magnet-to-tube distances ranging from 0 to 5 cm. The particles had an X-ray-derived crystallite size of approximately 14 nm and a saturation magnetisation of 48.5 emu g^-1 after normalisation to the magnetically active fraction. Mean-flow force-balance analysis showed that individual nanoparticles cannot be stably retained under the applied conditions; at zero magnet-to-tube separation, stable retention requires effective magnetic objects with an equivalent hydrodynamic diameter of approximately 7-8 µm. The dose-response curves exhibited primary retention, cooperative accumulation, and saturation. In the working range of 0-2 cm, the Avrami-type exponent remained between 1.35 and 1.44. The limiting captured mass decreased from 0.110 g at magnet contact to 0.0116 g at 5 cm and followed a sublinear power-law dependence on the local magnetic force-related parameter, with an exponent of 0.445. At a fixed magnet-to-tube distance of 2 cm, the tissue-to-control capture ratios were 0.94 for adipose tissue and 0.91 for muscle tissue, with no clear evidence of additional tissue-specific attenuation beyond experimental variability. The model therefore provides a compact physical description of finite-capacity magnetic trap filling and identifies field geometry as a key optimization parameter.