Bone remodeling and topology optimization by PID control

The functional adaptation of bone to mechanical usage implies the existence of a physiological control process: once the structure has adapted sufficiently, the feedback signal is diminished and further changes to shape and properties are stopped. In bones it has been widely accepted that mineral tissue is resorbed in regions exposed to low mechanical stimulus, whereas new bone is deposited where the stimulus is high. This process of functional adaptation is thought to enable bone to perform its mechanical functions with a minimum of mass and with the strength necessary to support mechanical loads associated with everyday activity and to protect internal organs. This premise can be expressed as a global, multi-objective optimization problem in which stiffness and mass are conflicting objectives. Maximizing stiffness is equivalent to minimizing the compliance or minimizing the strain energy in the bone. From this point of view, the bone remodelling process is analogous to topology optimization in optimal structural design The process of adaptative bone remodelling can be described analytically and simulated numerically, integrated with the finite element method. In the model discussed here bone tissue is described as continuous materials with variable density and with characteristics of optimal mechanical resistance and minimal mass. With a proper control strategy, this process drives the overall structure to an optimal configuration. The controllers optimized in this investigation include proportional, integral and derivative strategies. In this approach, the material distribution problem is parameterized by the elastic modulus of the discrete isotropic finite elements. A local remodelling rule iteratively updates the value of the modulus of each element individually based on the difference between a current stimulus value and a target value, relative to the external load. Its purpose is to obtain a constant, optimal value for the strain energy per unit bone mass, by adapting the mass density. For sample’s sake, a major problem threatening the long-term integrity of total hip replacement is the loss of proximal bone often found around non-cemented, press-fitted and bonded implants.