Abstract
Bones of the murine 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 Turing's reaction-diffusion equations, 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 hypothesize 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 was 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 provide precise three dimensional (3D) measures of murine 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 individual vault bones, the pattern of cranial vault bone growth over time, and the position of cranial vault sutures.
Highlights
The part of the cranium that surrounds the brain is known as the neurocranium and can be divided into two parts
Considering the relation between strain and bone formation we observed in Sec. 2.1, we model the process of bone growth as an expansion of regions of high concentration of differentiated osteoblasts according to the diffusion of the morphogen (H3) and the distribution of strain (H1) (Eqs. (4) and (5)):
Comparing these results from two models, we conclude that a reaction–diffusion model is sensitive to the location of the initial perturbations of molecules
Summary
The part of the cranium that surrounds the brain is known as the neurocranium and can be divided into two parts. The cranial vault covers and protects the superior and lateral aspects of the brain and consists of a series of flat bones that form through intramembranous ossification, where undifferentiated mesenchymal cells of either neural crest or mesoderm origin form condensations and differentiate directly into osteoblasts and begin to form bone.[1] Intramembranous ossification is based on epithelial-mesenchymal interactions that are controlled by a cascade of reactions of locally produced and circulating factors (e.g., signaling molecules and their receptors, extracellular matrix proteins and receptors)[2,3,4] that play significant roles in the differentiation of cells into osteoblasts that make bone.[5,6] Though much is known about the details of signaling required for mesenchymal cells to differentiate and begin to produce collagen and mineralize the collagen matrix, much less is known about how single bones acquire their characteristic shapes and how they cooperate to form an integrated skull. Research in tissue regeneration is revealing that biomechanical input, though not completely understood, plays an important role in these processes.[7,8,9,10,11]
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