Mathematical model of fluid flow in an osteon influence of cardiac system

Numerical simulations of the behavior of the osteonal structure are more and more acute and an important parameter is the pressure of the bony fluid. Haversian and Volkmann canals contain blood vessels that transport oxygen and nutrients necessary for the cellular activity. The pressure in these vessels must be taken into account. While it is possible to estimate the value of this pressure, there is no information on the effect of the vessel wall that may have in the transmission of pressure

Elaboration d'une modélisation mathématique du transfert multi-échelle des signaux mécaniques dans l'os cortical humain. Aspects théoriques et simulations numériques

Understand bone remodelling needs the knowledge of mechanical information transfer: what information receives a cortical bony cell when the bone is solicited? The purpose of this study is the elaboration of a modeling and of mathematical tools allowing to estimate, from a mechanical loading applied on a human bone, various fields existing in collagen, hydroxyapatite and bony fluid.Mathematical theories of homogenization and flow in porous media are used. Model is made by taking successively into account spatial organization of hydroxyapatite crystals, different mineralizations, a new behaviour's law, motion of fluid containing ions in each of architectural levels and homogenization of complex composite structures (lamellae, osteons, cortical bone).On a mathematically point of view, asymptotic developments method in a new piezoelectric framework (with threshold) is used. One establishes all necessary relationships, a property of convergence and a local analysis is made. The return to microscopic level is made directly via a technique of localization or indirectly when the effect of threshold occurs. Developped computational methods have been packed in two softwares.On a biomechanical point of view, it has been established that human cortical bone is a non piezoelectric orthotropic medium for which anisotropy is essentially involved by the nanoscopic architecture, that they are two types of flow in osteon and that flows in the osteons differ according to their architecture. A process being able to explain how cells know what architecture to give to the collagen tissue is thus pointed out.Comprendre le remodelage osseux nécessite la maîtrise du transfert des informations mécaniques: quelles informations reçoit une cellule osseuse de la partie corticale lorsque l'os est sollicité ? Ce mémoire est l'élaboration d'une modélisation et d'outils mathématiques permettant d'estimer, à partir d'un chargement mécanique appliqué à un os humain, divers champs existant dans le collagène, l'hydroxyapatite et le fluide environnant.On utilise les théories mathématiques de l'homogénéisation et des écoulements en milieux poreux. La modélisation est mise en place, étape par étape: organisation spatiale des cristaux d'hydroxyapatite, prise en compte de minéralisations différentes, d'une nouvelle loi de comportement, d'un fluide contenant des ions à chacun des niveaux architecturaux et homogénéisation de structures composites complexes (lamelles, ostéon, os cortical). Sur le plan mathématique, on reprend la méthode des développements asymptotiques dans un cadre piézoélectrique (avec seuil), on établit toutes les relations nécessaires, une propriété de convergence et une estimation de propriétés locales. Le retour au microscopique est fait directement via une technique de localisation ou indirectement lorsque l'effet de seuil se produit. Les méthodes numériques ont été implantées dans deux logiciels. Sur le plan biomécanique, on établit que l'os cortical humain est un milieu orthotrope non piézo électrique pour lequel l'anisotropie est due à l'architecture nanoscopique, que les ostéons sont le siège de deux types d'écoulement, que les écoulements y différent selon l'architecture : on voit comment les cellules savent quelle architecture donner au tissu collagènique

Collagen fibres effect on the mechanical properties of cortical bone. A numerical approach

International audienc

Nano and Macro Structure of Cortical Bone: Numerical Investigations

International audienceWe propose a multi-scale approach: cortical bone is structured in a hierarchy of five levels. The cortical part of a given bone is made of various areas having different physical properties adapted to locally existing solicitations. A Bony Elementary Volume (BEV) 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, fiber and fibril. The latter is composed of collagen and hydroxyapatite (Hap) occurring in a viscous liquid containing mineral ions. Mathematical homogenization theory is used to determine equivalent macroscopic properties of a BEV, knowing physical properties of collagen and Hap and architectural description of this bony structure. A new behavior's 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. The computational methods have been packed into software called SiNuPrOs. For a given organization of EVMC, mechanical properties are found at each level of the bony architecture and of course at the macroscopic level. The influence of EVMC's properties on the macroscopic properties of cortical bone has been studied and the influence of the different parameters variations is also presented

Elaboration of assumptions for the fluid problem at microscopic scale in Sinupros, mathematical model of cortical bone

International audienceCells live in a fluid environment located in a biological tissue, and the knowledge of the behavior of this fluid needs knowledge of the tissue itself. In the present study, the biological tissue is the cortical bone, which has a very complex architecture. In previous studies, a modelization of this bony structure has been made: the SiNuPrOs model [M. Racila, J.M. Crolet, Nano and macro structure of cortical bone: Numerical investigations, Mechanics of Advanced Materials and Structures, 14 (8) (2007) 655–663] whose description is based on several architectural levels. This model can be used in two ways. Firstly, both solid and fluid parts being coupled to compute the physical properties at each level of this bony structure, it is possible to determine realistic properties of cortical bone at the macroscopic scale by developments based on the homogenization theory. Then, it is easy, with classic finite element methods, to compute mechanical fields (displacement, strain or stress) in a human bone. In a second step, from these data obtained at a macroscopic scale, the SiNuPrOs model can be used to get information on sublevels: osteonal, lamellar or nanoscopic. It is the localization method which can be seen as an “inverse” way of the previous homogenization method. Bone can be considered as a porous medium and SiNuPrOs points out this aspect with three different levels: at the macroscopic levels, the pores are the Havers and Volkmann channels, at the osteonal scale, these previous channels are “large” channels (flow is modeled by Navier–Stokes equation) and the pores are the canaliculae and at the nanoscopic scale, the canaliculae are the “large” channels and the pores are made by the holes between the Hap crystals. Several applications can be simulated in the framework of healthy or pathological bones. In the present study, we investigate the case of a new osteon during its mineralization phase

SINUPROS: un modèle et un logiciel nano-maco pour les propriétés mécaniques de l'os cortical humain

International audienceSinupros is a model of the human cortical bone that takes into account the architectural, multiphysics and multi scale aspects of this medium. All the introduced parameters in each of the five levels are described, a behavior law allowing the modelisation of the mineralization effect is suggested and the homogenization theory is used in order to determine the physical properties at each level of this structure. The numerical algorithms are implemented in a software based on Matlab which allows easily simulating the effect of each parameter. The first analysis of the results which is pursued deals with the mineral composition of this medium and with the longitudinal and transverse elastic properties obtained at the macroscopic scale.Sinupros est un modèle de l’os cortical humain qui prend en compte les aspects architecturaux, multi physiques et multi-échelles de ce milieu. Tous les paramètres introduits dans chacun des cinq niveaux sont décrits, une loi de comportement permettant de prendre en compte l’effet de la minéralisation est suggérée et la théorie de l’homogénéisation est mise en œuvre pour déterminer les propriétés physiques de cette structure à chacun des niveaux concernés. Les schémas numériques évoqués sont implémentés dans un code écrit en Matlab, lequel permet aisément de simuler l’effet de chacun des paramètres. La première analyse des résultats de ce modèle qui est menée porte sur la composition minérale de ce milieu et les composantes élastiques longitudinales et transverses obtenues au niveau macroscopique

SINUPROS: human cortical bone multiscale model with a fluide-structure interaction

International audienceCells live in a fluid environment located in a biological tissue and the knowledge of the behaviour of this fluid needs the knowledge of the tissue itself. In the present study, the biological tissue is the cortical bone which has a very complex architecture

Mathematical modellisation of fluid flow in osteonal structures

International audienc

SiNuPrOs : mathematical model of human cortical bone

We present here a mathematical model of the human cortical bone which allows studying the mechanical behavior of this heterogeneous and complex structure knowing the properties of its basics components (collagen, hydroxyapatite (Hap) and bony fluid) and its architectural configuration. At long term It could be helpful to understand the bone remodeling which obviously needs the knowledge of the mechanical information transfer. One uses the asymptotic homogenization method (AHM) [1] in a piezoelectric framework (since collagen is piezoelectric) and a finite element method is developed for solving the cells problems obtained by homogenization. The developed computational methods have been packed into a software (called SiNuPrOs) made in Matlab; it is in free access on the website http://isifc.univ-fcomte.fr/SINUPROS/accueil.ht

Human cortical bone: the SiNuPrOs model Part I—description and elastic macroscopic results

International audienceMany 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