A Reduced-Order Reaction-Diffusion Model of Localised Nanoparticle-Mediated Payload Delivery and Tumor Response in Osteosarcoma

Osteosarcoma treatment is influenced not only by tumor proliferation but also by the spatial delivery of therapeutic agents and the extent to which delivered material becomes therapeutically active inside tumor cells. We present a deliberately reduced one-dimensional reaction-diffusion model that separates extracellular nanoparticle transport from an effective intracellular active-payload exposure state. The model couples normalised tumor density, extracellular nanoparticle concentration, and intracellular exposure through logistic tumor growth, Fickian diffusion of tumor cells and nanoparticles, first-order nanoparticle clearance, cell-associated uptake, intracellular payload loss, and payload-dependent tumor-cell death. A localised Gaussian source is normalised by its spatial integral, making total input independent of mesh resolution. Symmetric outward Robin leakage is imposed at the domain boundaries. The method-of-lines system is solved with a stiff BDF solver, and numerical reliability is assessed by mesh refinement, tolerance checks, nonnegativity monitoring, and a matched untreated control. At the illustrative baseline, centre tumor density falls from 0.8000 initially to 0.718891 at day 5, whereas the untreated control reaches 0.9359. The corresponding endpoint centre reduction is 23.18%; the spatially averaged burden is 12.70% lower, and the five-day integrated burden is 5.56% lower. Refinement from 161 to 321 nodes changes the centre value by 0.023%, with an observed second-order refinement rate. A structural comparison with the predecessor two-state formulation shows why the interpretation of the uptake coefficient β depends on model structure: β has a negative normalised sensitivity (-0.177) in the three-state model but a positive sensitivity (+0.128) when uptake is treated only as extracellular nanoparticle removal. The model is therefore positioned as a transparent mechanistic and numerical framework for hypothesis generation, not as a clinically validated predictive tool.