Comptes Rendus
A stochastic homogenization approach to estimate bone elastic properties
Comptes Rendus. Mécanique, Volume 342 (2014) no. 5, pp. 326-333.

The mechanical properties of bone tissue depend on its hierarchical structure spanning many length scales, from the organ down to the nanoscale. Multiscale models allow estimating bone mechanical properties at the macroscale based on information on bone organization and composition at the lower scales. However, the reliability of these estimates can be questioned in view of the many uncertainties affecting the information which they are based on. In this paper, a new methodology is proposed, coupling probabilistic modeling and micromechanical homogenization to estimate the material properties of bone while taking into account the uncertainties on the bone micro- and nanostructure. Elastic coefficients of bone solid matrix are computed using a three-scale micromechanical homogenization method. A probabilistic model of the uncertain parameters allows propagating the uncertainties affecting their actual values into the estimated material properties of bone. The probability density functions of the random variables are constructed using the Maximum Entropy principle. Numerical simulations are used to show the relevance of this approach.

Les propriétés mécaniques du tissu osseux dépendent de sa structure hiérarchisée, de l'échelle de l'organe à celle de ses constituants élémentaires (nano-échelle). En se basant sur la connaissance de la morphologie, de l'organisation et des propriétés mécaniques de ces derniers, des modèles multi-échelles permettent d'estimer les propriétés mécaniques d'ensemble du tissu osseux. Cependant, ces informations sont souvent partielles ou incertaines, rendant peu fiables lesdites estimations. Dans cet article, nous proposons une stratégie originale permettant de prendre en compte ces difficultés de façon efficace. Plus précisément, un modèle multi-échelles du tissu osseux basé sur la théorie de la micromécanique des milieux continus est associé à un traitement probabiliste de certaines des variables du modèle (notamment, les propriétés mécaniques des constituants élémentaires du tissu osseux). Le modèle multi-échelle permet de prendre en compte la microarchitecture et l'organisation du tissu osseux aux petites échelles pour estimer les coefficients élastiques de l'ultrastructure osseuse (la matrice solide du tissu osseux). Les incertitudes sur les variables d'entrée sont prises en compte en construisant des lois de probabilités pertinentes basées sur le principe du maximum d'entropie. Quelques résultats numériques sont montrés pour étayer l'intérêt de cette approche.

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Published online:
DOI: 10.1016/j.crme.2013.12.007
Keywords: Biomechanics, Continuum micromechanics, Stochastic modeling, Maximum entropy principle, Elastic properties
Mot clés : Biomécanique, Micromécanique des milieux continus, Modélisation stochastique, Principe du maximum d'entropie, Propriétés élastiques

Vittorio Sansalone 1, 2; Salah Naili 1, 2; Christophe Desceliers 2

1 Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Échelle, MSME UMR 8208 CNRS, 61, Av. du Général de Gaulle, 94010 Créteil, France
2 Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Échelle, MSME UMR 8208 CNRS, 5, bd Descartes, 77454 Marne-la-Vallée, France
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Vittorio Sansalone; Salah Naili; Christophe Desceliers. A stochastic homogenization approach to estimate bone elastic properties. Comptes Rendus. Mécanique, Volume 342 (2014) no. 5, pp. 326-333. doi : 10.1016/j.crme.2013.12.007. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.12.007/

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