Angiogenesis in Bone Fracture Healing: a Bioregulatory Model

The process of fracture healing involves the action and interaction of many cells, regulated by biochemical and mechanical signals. Vital to a successful healing process is the restoration of a good vascular network. In this paper, a continuous mathematical model is presented that describes the different fracture healing stages and their response to biochemical stimuli only (a bioregulatory model); mechanoregulatory effects are excluded here. The model consists of a system of nonlinear partial differential equations describing the spatiotemporal evolution of concentrations and densities of the cell types, extracellular matrix types and growth factors indispensable to the healing process. The model starts after the inflammation phase, when the fracture callus has already been formed. Cell migration is described using not only haptokinetic, but also chemotactic and haptotactic influences. Cell differentiation is controlled by the presence of growth factors and sufficient vascularisation. Matrix synthesis and growth factor production are controlled by the local cell and matrix densities and by the local growth factor concentrations. Numerical simulations of the system, using parameter values based on experimental data obtained from literature, are presented. The simulation results are corroborated by comparison with experimental data from a standardised rodent fracture model. The results of sensitivity analyses on the parameter values as well as on the boundary and initial conditions are discussed. Numerical simulations of compromised healing situations showed that the establishment of a vascular network in response to angiogenic growth factors is a key factor in the healing process. Furthermore, a correct description of cell migration is also shown to be essential to the prediction of realistic spatiotemporal tissue distribution patterns in the fracture callus. The mathematical framework presented in this paper can be an important tool in furthering the understanding of the mechanisms causing compromised healing and can be applied in the design of future fracture healing experiments.