Une approche multi-échelle pour la modélisation des fibrilles de l’os

Les propriétés mécaniques du tissu osseux sont dues à sa structure hiérarchique. À l’échelle de quelques micromètres, la structure que l’on peut observer est la fibrille de collagène minéralisée. Dans ce travail, nous présentons un modèle mécanique multi-échelle de la fibrille basé sur une technique d’homogénéisation variationnelle. Des simulations numériques montrent la dépendance des propriétés élastiques de la fibrille vis-à-vis de sa structure à l'échelle nanométrique

Influence de l’anisotropie sur le comportement des ondes ultrasonores dans l’os trabéculaire

L’objectif de cette étude est de contribuer à l’analyse des résultats expérimentaux obtenus in vitro en utilisant la technique de transmission transverse (utilisée en clinique) sur des échantillons d’os trabéculaire. Le système est décrit par une couche de milieu poreux élastique anisotrope entre deux fluides. La réponse transitoire du système en hautes fréquences est étudiée, montrant l’influence de l’anisotropie sur la propagation des ondes lorsque le système est sollicité par une onde plane

Multiphysics modelling of transport phenomena within cortical bone treated as a biporous material

Interstitial fluid and ionic transport taking place in the fluid compartments of bone are thought to play a major role in bone mechanotransduction. In this study, we present a three-scale model of the multiphysical transport phenomena taking place within the vasculature porosity and the lacuno-canalicular network of cortical bone. These two fluid porosity levels exchange mass and ions through the permeable outer wall of the Haversian-Volkmann channels. Thus, coupled equations of electro-chemo-hydraulic transport are derived from the nanoscale of the canaliculi toward the cortical tissue, considering the intermediate scale of the intra-osteonal tissue. In particular, the Onsager reciprocity laws that govern the coupled transport are checked

A thermodynamics framework to describe bone remodeling: a 2D study

Introduction Bone is a living material which is continuously reorganized by bone cells in response to their mechanical and biochemical environment. This process, known as bone remodeling, is of major importance in everyday life, in case of fractures and to allow osseointegration phenomena around implants. Bone remodeling can be described as a stress- and chemistry-driven evolution of the mechanical properties of bone tissue. The interplay between mechanics and biochemistry, as well as the multiple scales involved in this process, represent serious challenges for the development of realistic bone adaptation models. This study describes a novel, thermodynamically sound model of bone remodeling which, while being mechanistic in nature, is able to account for the above issues.   Methods The theory of material remodeling [1] offers a firm basis to build a model of bone remodeling. Bone is described as an orthotropic elastic medium whose elastic properties evolve in time according to the prevailing mechanical and chemical stimuli. In this study, we focus on a special class of evolution, namely the stress-driven rotation of the elastic principal axes of the bony material. Our model is based on balance laws derived through a suitable statement of the virtual power principle, as well as the description of a constitutive theory. The latter is based on the definition of a strain-energy density depending only on the elastic strain, and on a formulation of the dissipation principle incorporating the dissipation due to remodeling. This modeling framework leads to a remodeling evolution law giving an explicit relationship coupling the dissipation related to the remodeling, the stress and strain tensors, the rotation of the material axes and its evolution. Remodeling equilibrium is achieved when material properties no longer evolve, corresponding to a stationary state of the rotation. It is worth noting that this model predicts the principal axes of the strain and stress tensors to be collinear at the remodeling equilibrium [2]. Thus, a physically sound condition for remodeling equilibrium [3] is recovered without any ad-hoc assumption.   Results and Discussion The model was studied in 2D where the rotation of the elastic principal axes is parameterized by an angle. In case of uniform stress/strain conditions, stable remodeling equilibrium states were found to correspond to strain energy minima (imposed displacements). Finite element simulations were also performed to study the evolution of the elastic principal axes resulting from different boundary conditions. The prediction of the rotation of the principal axes of the material in simple loading configurations is consistent with the superimposed boundary conditions confirming the ability of the model to simulate the material response to non-uniform stress configurations.   Conclusion A novel, thermodynamically sound model of bone remodeling was proposed. Preliminary results obtained in 2D are promising and show the potential of this approach. Model predictions for in vivo biomechanical loading configurations need to be further tested. Suitable experimental data will be identified. The particular case of tissue surrounding an implant will be also studied. Our model can also integrate the mechanobiological phenomena regulating bone remodeling. However, this would require a reliable description of the biochemical stimuli of bone remodeling. This matter is out of the scope of this paper and will be addressed in future works.   Acknowledgement This project has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation program (grant agreement No 682001, project ERC Consolidator Grant 2015 BoneImplant).   References [1] DiCarlo A, Naili S, and Quiligotti S (2006) C. R. Mecanique, 334:651-661 [2] Sansalone V, Naili S, and Di Carlo A (2011) Comput Methods Biomech Biomed Engin, 14(s1):203-204 [3] Cowin S (1986) J Biomech Eng, 108:83-8

Prise en compte des incertitudes dans la modélisation multiéchelle de l'élasticité de l'os à partir de l'imagerie

Un modèle stochastique multiéchelle de l'os cortical est présenté. Les fractions volumiques des constituants de l'os sont représentées par des variables aléatoires. Les fonctions de densité de probabilité sont obtenues en utilisant le principe du Maximum d'Entropie. Nous utilisons la théorie de la micromécanique pour propager les incertitudes de l'échelle des constituants jusqu'à l'échelle de l'organe. Ce modèle micro mécanique est utilisé sur des échantillons d'os provenant de la partie inférieure du col du fémur à partir desquels des images ont été acquises. Ces informations statistiques ont été utilisées pour obtenir les valeurs moyennes et les intervalles de confiance des coefficients d'élasticité de l'os cortical

Influence des flux de calcium sur la contrainte de cisaillement agissant sur les ostéocytes dans l'os cortical

Nous avons modélisé l'écoulement du fluide encerclant les cellules osseuses mécano-sensibles (ostéocytes). Le but de cette étude est d'améliorer nos modèles précédents en incluant des flux de calcium apparaissant lors de la dissolution ou de la précipitation de la matrice osseuse. Même si ces flux ne semblent pas altérer de manière significative la vitesse du fluide, ils peuvent changer le cisaillement ressenti par les ostéocytes. Par conséquent, nous avons examiné la façon dont de tels échanges chimiques affectent l'écoulement du fluide et donc la mécano-sensibilité des ostéocytes

A multiscale approach for composite materials as multifield continua

A continuum model for composite materials made of short, stiff and tough fibres embedded in a more deformable matrix with distributed microflaws is proposed. Based on the kinematics of a lattice system made of fibres, perceived as rigid inclusions, and of microflaws, represented by slit microcracks, the stress-strain relations of an equivalent multifield continuum is obtained. These relations account for the shape and the orientation of the internal phases and include internal scale parameters, which allow taking into account size effects. Some numerical analyses effected on a sample fibre-reinforced composite pointed out the influence of the size and orientation of the fibres on the gross behaviour of the material