A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials

The multiform bio-mechanical phenomena occurring in bones grafted with the addition of artificial materials urge for the formulation of models which are sophisticated enough to describe their complexity. In the present paper we present a continuum poro-elastic mixture model in which two apparent mass densities are introduced to describe, at a macroscopic length scale, situations in which bone tissues and artificial materials coexist and interact. We focus on the final healing stage process when the bone remodelling becomes the dominant phenomenon. Artificial materials used are obviously to be bio-compatible and must resist to externally applied mechanical loads. More recently in order to favour bone tissue re-growth in grafts, which improves the long term performances of grafted bones, it has been conceived to use substitute materials which may be, similarly to bone tissue, bio-resorbed by osteoclasts and eventually replaced by newly synthesised living tissue. To account for resorption and synthesis phenomena suitable evolution equations are introduced for Lagrangian mass densities of the mixture constituents in which an integrodifferential operator defined on deformation fields appears. This operator is chosen to model some features of the coupling between mechanical compliance and biological bone tissue activity. The obtained system of integrodifferential equations is not trivial also when one considers one dimensional cases. Treating this simplified situations will allow us to individuate more easily some important remodelling scenarios. The numerical simulations which we present here show that the introduced model is promising and deserves to be developed to give previsions in more realistic applications.