A 2‐D continuum model of a mixture of bone tissue and bio‐resorbable material for simulating mass density redistribution under load slowly variable in time

The bio-mechanical phenomena occurring in bones grafted with the inclusion of artificial materials demand the formulation of mathematical models which are refined enough to describe their not trivial behavior. A 3D theoretical model, previously developed and used in 1D space, is employed to investigate and explain possible effects resulting from 2D interactions, which may not be present in 1D case so more realistic situations are approached and discussed. The enhanced model was used to numerically analyze the physiological balance between the processes of bone apposition and resorption and material resorption in a bone sample under plain stress state. The specimen was constituted by a portion of bone living tissue and one of bio-resorbable material and was acted by an in-plane loading condition. The signal intensity between sensor cells and actor cells was assumed to decrease exponentially with their distance; the effects of adopting two different laws, namely an absolute and a quadratic functions, were compared. Ranges of load magnitudes were identified within which physiological states are established. A parametric analysis was carried out to evaluate the sensitivity of the model to changes of some critical quantities within physiological ranges , namely resorption rate of bio-material, load level and homeostatic strain. In particular the spatial distribution of mass densities of bone tissue and of resorbable bio-material and their time evolution were considered in order to analyze the biological effects due to the parameter's changes. Synthetically, these biological effects can be associated to different ratios between bone and bio-material densities at the end of the process and to different delays in the bone growth and material resorption. These numerical analyses allowed for finding the most desirable situations in which a gradual resorption of the artificial graft occurs together with the simultaneous formation of new bone, finally leading to an almost complete substitution of the bio-resorbable material with living tissue.

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