An optimal control procedure for bone adaptation under mechanical stimulus

Abstract

The process of adaptive bone remodeling can be described mathematically and simulated in a computer model, integrated with the finite element method. In the model discussed here, cortical and trabecular bone are described as a continuous material with variable mass density and hence elastic modulus. The remodeling rule applied to simulate the adaptation process in each cell individually is, in fact, an evolution law for an optimization process, relative to the external load. Its purpose is to obtain a uniform value for the strain energy per unit bone mass, by adapting the mass density. The feedback mechanism in the process is self-enhancing; denser bone attracts more strain energy, whereby the bone becomes even denser. In addition, the process ensures that the discontinuous end configuration is a structure with a relatively low mass and high stiffness, inasmuch as this is an explicit objective in the optimization process.Thus, an integrated procedure of control and optimization is proposed herein in order to solve a constrained optimization problem of lightweight stiffened structures; two alternative objective functions were considered. The selection of the parameters to be optimized in the evolution rule was not yet studied in an in-depth study. The set of such parameters included the control gains, the target of the error signal and the weight of one of the two proposed cost indices. A two-dimensional bone sample, subjected to an in-plane loading condition, was analyzed. The adopted values of material characteristics were typical of bone-like tissues.