Orthotropic remodelling of cancellous bone based on a parametric constitutive model - discussion of local microstructure constraints

Formation, maintenance and repair processes of bone tissues are only possible due to the internal ability of bones to remodel their microstructure. The mechanobiological factors of tissue formation, differentiation and regeneration in bone have been investigated by many authors. Here, the simplified yet well-justified Wolff's hypothesis will be recalled, postulating that processes of remodelling are mainly driven by mechanical factors. Its consequence is e.g. a very efficient distribution of stresses within the bone microstructure. Attempts to numerically simulate the process of adaptive bone remodelling were reported in the literature since eighties of the last century. Local relations between bone density and stress corresponding to different load cases were formulated in [1,2]. Remodelling was understood as local density changes occuring in response to changed level of local stresses. Another approach was proposed in [3,4] in which remodelling problem was formulated as optimization of density distribution in bone to ensure minimum value of a certain global quality functional postulated as total strain energy. This approach appears in fact to be a generalized problem of structural topology optimization [5]. The approach was reformulated in [6] in terms of time rates, to include bone remodelling in response to loads variable in time. Initially, publications in the subject were based on the assumption of bone isotropy, with only one parameter (apparent density) describing the bone microstructure. The models did not thus allow to include directionality in the bone microstructure and, in consequence, in its remodelling. Attempts to introduce anisotropy were made since late nineties by e.g. [7,8]. The main problem is definition of appropriate anisotropic microstructural parameters and relating them to elastic material constants. In this study, a numerical model that allows to simulate the process of anisotropic remodelling of cancellous bone is presented. The bone is treated as continuum with linear elastic anisootropic mechanical properties that result from predefined microstructure topology and parameterized geometric proportions of trabeculae. Elastic constants and relative density are explicitly known functions of the geometric parameters [9]. The microstructure parameters (including orientation directions) are non-uniformly distributed in the bone volume and subject to changes in time according to a remodelling rule that takes the form of minimization of the elastic strain energy rate. Special attention is focused on constraints in the optimization procedure that allow to eliminate unrealistic microstructure instances.