A model of bone adaptation as a topology optimization process with contact

Topology optimization is presently used in most diverse scientific, technologic and industrial areas, including biomechanics. Bone remodelling models and structural optimization has mutually provided inspiration for new developments in biomechanics and biomedicine. Considering that bone has the ability to adapt its internal structure to mechanical loading (Wolff’s law and Roux’s paradigm), it is possible to model the behaviour of the bone structure by the use of a topology optimization methodology whose optimization variables can be the relative densities and the orthotropic directions. In this work, the internal bone adaptation of a proximal femur is considered. The bone-remodelling scheme is numerically described by a time-dependent evolutionary procedure with anisotropic material parameters. The remodelling rate equation is obtained from the structural optimization task of maximizing the stiffness subject to a biological cost associated with metabolic maintenance of bone tissue in time. The situation of multiple load conditions is considered for a three-dimensional finite element model of the proximal femur. The bone density distribution of a real femur is used as the initial design for the onset of the remodelling mechanism. Examples of bone adaptation resulting from load changes are presented. The three-dimensional finite element model of the proximal femur with initial bone density distribution was adapted to implant a cementless stem. A remeshing technique is used to assign the bone relative density distribution to the new geometry and mesh. The time adaptation of the bone is assessed considering contact with friction at the bone-stem interface. Results of bone density evolution and osteointegration distribution are obtained.