This research presents a new modelling procedure which allows the computation of the physical properties of the human cortical bone, considered as a strongly heterogeneous medium consisting of bony architecture and the physical properties of the two basic components: the collagen and the hydroxyapatite (Hap). The numerical simulations are based on the homogenisation theory, however, since the size of the Hap crystals are small compared to the size of a collagen stick, a new entity (the elementary volume of mineral content (EVMC)) is defined at the nanoscopic scale. This model permits the testing of all the possible structural configurations that may be present and suggests that the anisotropy of the bone is not only induced by the haversian structure but by the properties of the Hap crystals and their special organisation.

Many models that have been developed for cortical bone oversimplify much of the architectural and physical complexity. With SiNuPrOs model, a more complete approach is investigated: it is multiscale because it contains five structural levels and multi physic because it takes into account simultaneously structure (with various properties: elasticity, piezoelectricity, porous medium), fluid and mineralization process modelization. The multiscale aspect is modeled by using 18 structural parameters in a specific application of the mathematical theory of homogenization and 10 other physical parameters are necessary for the multi physic aspect. The modelization of collagen as a piezoelectric medium has needed the development of a new behaviour law allowing a better simulation of the effect of a medium considered as evolving during a mineralization process. Then the main interest of SiNuPrOs deals with the possibility to study, at each level of the cortical architecture, either the elastic properties or the fluid motion or the piezoelectric effects or both of them. All these possibilities constitute a very large work and all this mass of information (fluid aspects, even at the nanoscopic scale, piezoelectric phenomena and simulations) will be presented in several papers. This first one is only devoted to the presentation of this model with an application to the computation of elastic properties at the macroscopic scale. The computational methods have been packed into software also called SiNuPrOs and allowing a large number of predictive simulations corresponding to various different configurations.

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This paper is a contribution to a plausible explanation of the mechanotransduction phenomenon in cortical bone during its remodelling. Our contribution deals only with the mechanical processes and the biological aspects have not been taken into account. It is well known that osteoblasts are able to generate bone in a suitable bony substitute only under fluid action. But the bone created in this manner is not organised to resist specific mechanical stress. Our aim was to suggest the nature of the physical information that can be transmitted - directly or via a biological or biochemical process - to the cell to initiate a cellular activity inducing the reconstruction of the osteon that is best adapted to local mechanical stresses. For this, the cell must have, from our point of view, a good knowledge of its structural environment. But this knowledge exists at the cellular scale while the bone is loaded at the macroscopic scale. This study is based on the SiNuPrOs model that allows exchange of information between the different structural scales of cortical bone. It shows that more than the fluid, the collagen - via its piezoelectric properties - plays an essential role in the transmission of information between the macroscopic and nanoscopic scales. Moreover, this process allows us to explain various dysfunctions and even some diseases.