A Computational Analysis Of Bone Formation In The Cranial Vault Using A Reaction Diffusion Strain Model

Bones of the mouse cranial vault are formed by differentiation of mesenchymal cells into osteoblasts, a process that is primarily understood to be controlled by a cascade of reactions between extracellular molecules and cells. We assume that the process can be modeled using Turings reaction-diffusion model, a mathematical model describing the pattern formation controlled by two interacting molecules (activator and inhibitor). In addition to the processes modeled by reaction-diffusion equations, we propose that mechanical stimuli of the cells due to growth of the underlying brain contribute significantly to the process of cell differentiation in cranial vault development. Structural analysis of the surface of the brain is conducted to explore the effects of the mechanical strain on bone formation. We propose a mechanobiological model for the formation of cranial vault bones by coupling the reaction-diffusion model with structural mechanics. The mathematical formulation was solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data that provides precise three-dimensional (3D) measures of mouse cranial geometry and cranial vault bone formation for specific embryonic time points. The results of this study suggest that mechanical strain contributes information to specific aspects of bone formation. Our mechanobiological model predicts some key features of cranial vault bone formation that were verified by experimental observations including the relative location of ossification centers of the individual vault bones, the pattern of cranial vault bone growth over time, and the position of cranial vault sutures.