Mechanotransduction in cortical bone and the role of piezoelectricity: a numerical approach.

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.

Collagen's role in the cortical bone's behaviour: a numerical approach.

Literature devoted to experimental measurements of the elastic properties of the human cortical bone gives us a relatively wide spectrum of values. This proves that the result depends on the bone itself and on the area from where the sample has been taken. Ultrasonic measurement, which is a fine technology, points out complex maps. The reason of this very strong heterogeneity is not completely explained. The present study is based on a numerical model of the human cortical bone, the SiNuPrOs model and aims to suggest an explanation. If one admits that mineral apposition occurs around collagen fibres, the spatial orientation of these fibres would have an important consequence on the elastic properties of the medium. On the basis of homogenisation theory allowing to compute all the components of the elasticity tensor, this study quantifies the main influence of this architectural orientation and its effect on the anisotropy of the cortical bone.

Human cortical bone: the SiNuPrOs model. Part II--a multi-scale study of permeability.

Cortical bone is more and more considered as a porous medium and this induces the necessity of the determination of the physical properties associated with such a concept: the porosity and the permeability. If porosity does not present a major problem, at least for the order of magnitude, there is a difficulty for the permeability. According to experimental sources, values vary between 10(- 13) and 10(- 23) m(2): it seems obvious that the same entities have not been measured. This article proposes a new vision of the permeability based on a concept of multi-scale medium corresponding to the scales already introduced in the SiNuPrOs model which has been developed for cortical bone. According to this model, several architectural levels are proposed and a mathematical development based on the homogenisation theory, which can be applied to each of these levels, allows a numerical computation of the permeability tensor coefficients. A comparative analysis of our simulations and some experimental results (already published) shows a good accordance with the literature.

Human cortical bone: the SINUPROS model.

Several modelizations have been investigated on human cortical bone in our team and we have often observed that the introduction of a new geometrical parameter induces significant perturbations on the numerical values obtained with previous models. We have therefore decided to take into account the totality of all possible parameters in a modelization which is physically and physiologically plausible. In order to do this, we have analyzed the architecture of cortical bone and exhibited all parameters that occur. To determine physical properties at each architectural level, the best adapted tool is without any doubt the mathematical theory of homogenization. All the necessary algorithms have been implemented into SINUPROS software (websites of the Universities). Its main interest is the evaluation of macroscopic physical properties for a given configuration. It can also be used to seek, by successive tests, configurations corresponding to properties experimentally measured. Computation time being too high (10 to 45 minutes according to tested configurations), a fast version based on approximation theory has been developed and thus the obtaining of the results is immediate. The researched configuration being thus obtained, it has then to be validated by the original version.

Human cortical bone: the SiNuPrOs model.

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.

Human cortical bone: Computer method for physical behavior at nano scale constant pressure assumption.

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.

A new numerical concept for modeling hydroxyapatite in human cortical bone.

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.

On the mechanical characterization of compact bone structure using the homogenization theory.

In a previous paper (Crolet et al., 1993, J. Biomechanics 26, 677-687), a modelling of the mechanical behavior of compact bone was presented, in which the homogenization theory was the basic tool of computation. In this simulation, approximations were used for the modelling of the lamellae and the osteons: the lamella and the osteon were divided into cylindrical sectors, each sector being approximated as a parallelepiped having a periodic structure (fibrous composite for the lamella, superimposition of plates for the osteon). The present study deals with a new model without these approximations. First, it can be proved that the homogenized elasticity tensor for a lamella, which has non-periodic structure, is obtained at each geometrical point as a homogenized tensor of a periodic problem. Similarly, for the osteonal structure, the components of the homogenized tensor are determined at each point as the result of a periodic homogenization. The software OSTEON, which is the computational method associated with this model, allows one to obtain a better understanding of the effects of many bony parameters. The obtained results are in accordance with experimental data.

Compact bone: numerical simulation of mechanical characteristics.

One of the main difficulties encountered in the numerical simulation of the anisotropic elastic characteristics of compact bone is to account for the Haversian microstructure when determining the overall macroscopic behavior. Engineering analyses of such problems are usually based on 'homogenized approximations'. Compact bone is not exactly a composite material, but rather a heterogeneous medium which exhibits a multiscale composite structure. If the homogenized approximation is precise enough (and this is true for the mathematical theory of homogenization), it is then possible to simulate the macroscopic behavior from the microscopic mechanical characteristics. The present paper is devoted to such mathematical developments. Moreover, the 'inverse simulation' allows the computation of the microscopic stress fields in the haversian structure from the macroscopic stress fields, taking into account bone microstructure.