A stochastic model of cell transformations at bone tissue regeneration

Recently, there has been increasing interest in simulating of basic phenomena appearing during regeneration of bone. These phenomena include diffusion, proliferation, differentiation and apoptosis processes for the several cell types. The traditional set of cells includes four cell types (mesenchymal stem cells, fibroblasts, chondrocytes and osteoblasts). The damage space is represented as a “cell pool” where these elements can coexist. Consequently, the evolution of above-mentioned system can be described by a set of four differential equations with the corresponding boundary and initial conditions. However, the process parameters, comprising of diffusion, proliferation, differentiation and apoptosis of various cells cannot be described precisely and simplifications are necessary. As a result, various mathematical instruments are used for numerical solution of the differential equations. In the present paper, a stochastic approach based on a Markov chains is first developed for the cell-phenotype specific modelling. The Markov chain approach assumes the division of the area, which is under observation, into finite discrete intervals. The process evalutions is observed in a descrete time moments. In the context of the present work, one-dimensional model of a damaged bone area is considered as a constant cross-section cylinder. Although this model is only based on the start-up cells distributions, it provides a full description of spacetime changes of the “cell pool”. The parametric identification of the model is carried out using the experimental data borrowed from the literature.