Finite element solution of coupled multiphysics reaction-diffusion equations for fracture healing in hard biological tissues

Abstract

This study proposes a computational framework to investigate the multi-stage process of fracture healing in hard tissues, e.g., long bone, based on the mathematical Bailon-Plaza and Van der Meulen formulation. The goal is to explore the influence of critical biological factors by employing the finite element method for more realistic configurations. The model integrates a set of variables, including cell densities, growth factors, and extracellular matrix contents, managed by a coupled system of partial differential equations. A weak finite element formulation is introduced to enhance the numerical robustness for coarser mesh grids, complex geometries, and more accurate boundary conditions. This formulation is less sensitive to mesh quality and converges smoothly with mesh refinement, exhibiting superior numerical stability compared to previously available strong-form solutions. The model accurately reproduces various stages of healing, including soft cartilage callus formation, endochondral and intramembranous ossification, and hard bony callus development for various sizes of fracture gap. Model predictions align with the existing research and are logically coherent with the available experimental data. The developed multiphysics simulation clarifies the coordination of cellular dynamics, extracellular matrix alterations, and signaling growth factors during fracture healing.