Affiliations: [a] Department of Mathematics, University of Franche-Comté, Besançon, France | [b] Institut Supérieur d'Ingénieurs de Franche-Comté, France | [c] Department of Mathematics, University of Craiova, Romania | [x] Mechanical Engineering Faculty, Laboratory for Materials Technology, University of Applied Science, Regensburg, Germany | [y] University Clinic, Department of Traumatology, Regensburg, Germany
Abstract: It is well known that long term behavior of implants depends on bone remodeling. In the absence of a model of this phenomenon, few numerical simulations take into account bone remodeling. Some laws have been proposed but they cannot be used in the essential area surrounding the implant. We propose a multi-scale approach: cortical bone is structured in a hierarchical way consisting of five levels. The cortical part of a given bone is made up of various areas having different physical properties adapted to locally existing conditions. A Bony Elementary Volume denotes the elementary part of such a zone which constitutes our first level. The other levels are in conformity with our previous studies: osteon, lamella, fibre and fibril. This latter is composed by collagen and hydroxyapatite (Hap) occurring in a viscous liquid containing mineral ions. Mathematical homogenisation theory is used to determine equivalent macroscopic properties of a BEV, knowing the physical properties of collagen and Hap and the architectural description of this bony structure. For improving the performance of our simulation software, a new behavior law has been introduced with no continuity between the various levels. The effect of the fluid at the nanoscopic scale is modeled by a constant pressure. Recent developments allow us to determine the magnitude of various entities at nanoscopic scale from information at the macroscopic level. Realized simulations show that the assumption of constant pressure is not sufficient to characterize the nanoscopic mechanical behaviour. This point needs a more complex model with the introduction of a coupling between structure and fluid. This aspect is in development.
