Mathematical Prediction for Geometry-Mediated Cell 3D In-Growth on Bone Tissue Engineering Scaffolds.

3D cell infiltration into porous scaffolds constitutes a fundamental prerequisite for bone tissue engineering. Though pore size and curvature are known to dictate this process, their mathematical coupling remains elusive. Herein, we identified a size-dependent bone marrow-derived mesenchymal stem cells 3D in-growth pattern in which small pores promoted horizontal bridging, while large pores favored vertical cellular migration into the scaffold core. An analytical framework of Porous-Fisher model was developed using a superposition approach tailored to boundary-specific solutions. This approach not only enabled quantitative prediction of coverage rates through examination of grid dimensions and diffusion coefficients but also mathematically elucidated curvature and strategic geometric design. Furthermore, the prediction of cellular diffusion patterns on porous scaffolds was achieved through the alteration of boundary conditions and diffusion coefficients. Convex topological configurations were shown to accelerate cellular infiltration, whereas concave geometries permitted spatiotemporal modulation of tissue growth. Additionally, lower diffusion environments delayed coverage, suggesting scaffold designs with reduced pore sizes might benefit elderly patients. Consequently, the accuracy of model was in vivo validated by a rat cranial defect model. Overall, the mathematical model provided an effective way for ideal pore structure prediction in advance and propel the application of porous scaffolds in tissue engineering.