Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids

In this paper, a continuum mixture model with evolving mass densities and porosity is proposed to describe the process of bone remodeling in the presence of bio-resorbable materials as driven by externally applied loads. From a mechanical point of view, both bone tissue and biomaterial are modeled as linear elastic media with voids in the sense of Cowin and Nunziato (J Elast 13:125-147, 1983). In the proposed continuum model, the change of volume fraction related to the void volume is directly accounted for by considering porosity as an independent kinematical field. The bio-mechanical coupling is ensured by the introduction of a suitable stimulus which allows for discriminating between resorption (of both bone and biomaterial) and synthesis (of the sole natural bone) depending on the level of externally applied loads. The presence of a 'lazy zone' associated with intermediate deformation levels is also considered in which neither resorption nor synthesis occur. Some numerical solutions of the integro-differential equations associated with the proposed model are provided for the two-dimensional case. Ranges of values of the parameters for which different percentages of biomaterial substitution occur are proposed, namely parameters characterizing initial and maximum values of mass densities of bone tissue and of the bio-resorbable material.