On the dynamics of the growth plate in primary ossification

In this work a mathematical model for the interaction of two key signalling molecules in rat tibia ossification is presented and discussed. The molecules under consideration are Indian hedgehog (Ihh) and parathyroid hormone-related peptide (PTHrP). These are known to be major agents in the dynamics of the so-called growth plate, where transition from pristine cartilage to advancing bone takes place. Our model consists in a steady-state linear approximation to a reaction-diffusion system where only diffusion and absorption mechanisms are retained. Estimates on some system parameters are given, on the basis of the knowledge of a few measurable quantities. This allows for explicitly solving our model, whereupon a discussion on robustness and regulatory properties thereof is provided. In particular, we show that the size of the Proliferative Zone in the growth plate is rather insensitive to variations in the flux coefficients for Ihh and PTHrP at their boundaries. Besides, we also show that the model is also insensitive to large changes in the (comparatively small) critical value of the PTHrP concentration which marks the transition form Proliferative to Hyperthropic Regions within the Growth Plate. These results hold irrespective of the particular diffusivities selected for Ihh and PTHrP.