Bone remodeling process as an optimal design

The functional adaptation of bone to mechanical usage implies the existence of a physiological control process: once the structure has sufficiently adapted, the feedback signal is diminished and further changes to shape and properties are stopped. In bone tissue it has been widely accepted that mineral component 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 daily 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 goals. Maximizing stiffness is equivalent to minimize the compliance or minimize the strain energy in the bone. From this point of view, the bone remodeling process is analogous to the topology optimization in structural design including the cellular automaton (CA) technique and the solid isotropic material with penalization (SIMP) approach. The process of bone remodeling can be analytically described, integrated with the finite element method and numerically simulated. With a proper control strategy, an iterative process drives the overall modelled structure to an optimal configuration. The optimized controllers in this investigation regard proportional, integral and derivative strategies. A local remodeling rule iteratively updates the value of the modulus (or relative mass) of the cellular automata, in which the structure is discretized, individually based on the difference between a current stimulus value and a target value, relative to the external load. The purpose is to obtain a constant, optimal value for the strain energy per unit bone mass, by adapting the mass density. In medical applications, 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. The aim of this thesis is to perform a two-step procedure of control and optimization in order to solve an optimization problem of lightweight stiffened structures; various objective functions and constraints are considered so that different design requirements are compared. The selection of the optimized parameters in the evolution rules, not yet faced in an in-depth study, is successfully studied and the convergence is improved. The set of optimal parameters includes the control gains, the target of the error signal and the weight of the cost index J1, defined as the sum of the total energy and mass of the domain of interest. Two-dimensional bone samples, subjected to an in-plane constant and linear loading, are analyzed. The bone samples are discretized in 25, 625 and 1250 cellular automata, but the proposed model is suitable to be used not only in bone mechanics but in many other fields of artificial materials discretized in whatever number of elements. The contents of the Ph.D. thesis appear as follows: in Chapter 1 the fundamental cellular mechanisms responsible for bone remodeling are briefly described; Chapter 2 contains an overview on several important bone remodeling models (optimization, phenomenological and mechanistic models) of the last forty years; Chapter 3 is made up of theoretical and numerical tools (FEM analysis, topology optimization of structures, cellular automaton model, adaptive elasticity theory and PID control) that are transferred into a unified numerical code and used for modeling bone adaptation effects. Chapter 4 includes the complete control and optimization procedure that predicts an optimal distribution of mass and energy of a bone structure under specific constraints and loading conditions. To conclude, final remarks of the proposed study are reported.