Optimality conditions in modeling of bone adaptation phenomenon

Continuous bone remodeling consists in simultaneous resorption of tissues and synthesis of a new matrix. If, due to variable external or internal conditions, the equilibrium is disrupted, significant rearrangements of the micro-structure and bone shape are possible. Many mathematical and computational models of this adaptation phenomenon can be assingned one of the two categories; namely, theoretical models originating from the theory of adaptive elasticity and computational models making use of the optimization theory.

In the present paper the approach based on the hypothesis of optimal response of a bone is proposed. It enables derivation of various adaptation laws associated with extremum of the objective functional under a set of appropriate constraints and makes a bridge between the aforementioned categories. In order to illustrate possible application of the proposed general approach the specific formulation is presented and mathematical relations governing the adaptation process are derived. Four numerical examples illustrating some of possible applications of the presented relations are included.