# Mathematical Prediction for Geometry‐Mediated Cell 3D In‐Growth on Bone Tissue Engineering Scaffolds

## TL;DR

This paper presents a mathematical model to predict how cell infiltration into bone tissue scaffolds is influenced by pore size and geometry.

## Contribution

The study introduces a novel Porous-Fisher model that quantitatively predicts cell coverage rates and the impact of scaffold geometry on tissue growth.

## Key findings

- Small pores promote horizontal bridging while large pores favor vertical migration of cells into scaffolds.
- Convex geometries accelerate infiltration, while concave geometries allow spatiotemporal control of tissue growth.
- Smaller pores in low diffusion environments may benefit elderly patients by delaying coverage.

## Abstract

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.

This study identifies a fundamental pore size dependent pattern of three dimensional bone marrow derived mesenchymal stem cell (BMSC) infiltration within porous scaffolds, where small pores promote horizontal cellular bridging and large pores facilitate vertical migration. An analytical Porous Fisher mathematical framework is developed, employing a superposition approach tailored to boundary specific solutions. This model quantitatively predicts cellular coverage rates by examining grid dimensions and diffusion coefficients, while mathematically elucidating the role of pore curvature and geometric design. It reveals that convex topological configurations accelerate cellular infiltration, whereas concave geometries allow spatiotemporal modulation of tissue growth. The model further indicates that reduced diffusion environments delay coverage, suggesting scaffold designs with smaller pores may benefit elderly patients. Finally, the predictions are successfully validated in a rat cranial critical size defect model.