Bone metastatic growth as a phase transition

In the first chapter we describe the main characteristics of bone tissue, from skeletogenesis to bone remodelling. In the second chapter we describe the metastatic process, a multistep process involving the colonisation of bone by tumour cells. Tumour cells in order to detach themselves from the primary tumour and invade foreign tissue will have to migrate through very different microenvironments and undergo changes in their mechanical and physical properties. In the third chapter, through the analysis of the results obtained from experiments conducted at the Rizzoli, we have developed the idea that the tumour propagation and spread can be described as a non-linear dynamic system with two attractors, where the transition from the 'normal state' to the 'cancerous state' can be described by a first-order phase transition. Then a fourth chapter deals with the basic theory of the calculus of variations, giving us the necessary tools for the model that we will propose. In the last chapter, we propose a simple 1D model describing the energy of the system in which healthy and tumour cells are present. The aim of the chapter is to show that the tumour can be described as a phase transition. In this setting, we show that if the free energy of the system is non-convex with respect to a suitably chosen order parameter, then homogeneous configurations (representing mixtures of healthy and cancerous cells) become unstable, thus favoring instead the onset of inhomogeneous configurations, where the tumor localizes. In this first stage, the analysis of the theory of 1D phase transitions is abstract and does not account for important elements such as the tissue elasticity, which is well known to play a major role. At the same time, however, the results of this thesis provide an interesting direction for future studies, where considerations of energetic type could bring a deeper physical and biological understanding of the complex process of tumor invasion in bones.