Multiple Verification in Complex Biological Systems: The Bone Remodelling Case Study

We present a set of formal techniques and a methodology for a composite formal analysis at the tissue and organ level, focusing on the verification of quantitative properties in the process of bone remodelling. Starting from a differential equation model, we derive a stochastic model and a piecewise multi-affine approximation in order to perform model checking of stabilisation properties for the biological tissue, and to assess the differences between a regular remodelling activity and a defective activity typical of pathologies like osteoporosis. The complex nonlinear dynamics of bone remodelling is analysed with a variety of techniques: sensitivity analysis for the differential equation model; quantitative probabilistic model checking for the stochastic model; and classical model checking and parameter synthesis on the piecewise multi-affine model. Such analyses allow us to extract a wealth of information that is not only useful for a deeper understanding of the biological process but also towards medical diagnoses.