A Model of Bone Adaptation Using a Global Optimisation Criterion Based on the Trajectorial Theory of Wolff

Abstract

Julius Wolff originally proposed that trabecular bone was influenced by mechanical stresses during the formative processes of growth and repair such that trabeculae were required to intersect at right angles. In this work, we have developed an analytical parametric microstructural model, which captures this restriction. Using homogenisation theory, a global material model was obtained. An optimal structure constructed of the homogenised material could then be found by optimising a cost function accounting for both the structural stiffness and the biological cost associated with metabolic maintenance of the bone tissue. The formulation was applied to an example problem of the proximal femur. Optimal densities and orientations were obtained for single load cases. The situation of multiple loads was also considered. In this case, we observe that the alignment of principal strains with the material orthotropy direction is, in general, not possible for all load cases. Thus less restrictive microstructures (nonorthotropic) will yield higher structural stiffnesses than strictly orthotropic microstructures.

A quasi-brittle continuum damage finite element model of the human proximal femur based on element deletion

“…Several recent approaches have been developed to assign the anisotropic orientation of bone as a function of its cortical and trabecular structural morphology and mechanical behavior. These approaches include orientation methods using anatomical directions corresponding to the bone shape [7,26,38,90], variation in the CT Hounsfield unit values based on micromechanical considerations [30,66,73,76], bone remodeling simulation prior to fracture prediction to obtain the bone orthotropic orientation and elastic assignment [11,17,20,40,51], and a procedure to orientate orthotropic properties in a proximal femur FE model using the directions of the principal stresses produced by a physiological load scheme [64]. Empirical relations between the orthotropic constants and bone density have been suggested by several authors [29][30][31][32][33][34].…”

Bone Implant Design Using Optimization Methods

“…The remodeling process used was developed by Fernandes et al [45] and assumes that bone adapts to the mechanical loads in order to maximize the stiffness and minimize the metabolic cost to maintain bone tissue. An algorithm for bone ingrowth is proposed, taking into account the five interfacial conditions that must be satisfied to obtain osseointegration into the porous coating.…”

“…The remodeling model presented by Fernandes et al was used [12,45]. In this model bone tissue is considered a porous material with a periodic microstructure that is obtained by the repetition of cubic cells with prismatic holes with dimensions a 1 , a 2 , and a 3 , as shown in Figure 10.17.…”

Topology Optimization for Human Proximal Femur Considering Bi-modulus Behavior of Cortical Bones

“…They claimed that topology optimization (with minimal structural compliance) and the bone remodeling (SED distributing uniformly) are equivalent. Fernandes et al (1999) an analytical parametric micro-structural model for trabecular bone in proximal femur was proposed according to the homogenization theory, and optimal densities and orientations were obtained by topology optimization. Similar work was given by Kowalczyk (2010).…”