A reaction–diffusion model for long bones growth

Bone development is characterized by differentiation and growth of chondrocytes from the proliferation zone to the hypertrophying one. These two cellular processes are controlled by a complex signalling regulatory loop between different biochemical signals, whose production depends on the current cell density, constituting a coupled cell-chemical system. In this work, a mathematical model of the process of early bone growth is presented, extending and generalizing other earlier approaches on the same topic. A reaction–diffusion regulatory loop between two chemical factors: parathyroid hormone-related peptide (PTHrP) and Indian hedgehog (Ihh) is hypothesized, where PTHrP is activated by Ihh and inhibits Ihh production. Chondrocytes proliferation and hypertrophy are described by means of population equations being both regulated by the PTHrP and Ihh concentrations. In the initial stage of bone growth, these two cellular proceses are considered to be directionally dependent, modelling the well known column cell formation, characteristic of endochondral ossification. This coupled set of equations is solved within a finite element framework, getting an estimation of the chondrocytes spatial distribution, growth of the diaphysis and formation of the epiphysis of a long bone. The results obtained are qualitatively similar to the actual physiological ones and quantitatively close to some available experimental data. Finally, this extended approach allows finding important relations between the model parameters to get stability of the physiological process and getting additional insight on the spatial and directional distribution of cells and paracrine factors.