Stochastic multiscale modelling of cortical bone elasticity based on high-resolution imaging

Abstract

Accurate and reliable assessment of bone quality requires predictive methods which could probe bone microstructure and provide information on bone mechanical properties. Multiscale modelling and simulation represent a fast and powerful way to predict bone mechanical properties based on experimental information on bone microstructure as obtained through X-ray-based methods. However, technical limitations of experimental devices used to inspect bone microstructure may produce blurry data, especially in in vivo conditions. Uncertainties affecting the experimental data (input) may question the reliability of the results predicted by the model (output). Since input data are uncertain, deterministic approaches are limited and new modelling paradigms are required. In this paper, a novel stochastic multiscale model is developed to estimate the elastic properties of bone while taking into account uncertainties on bone composition. Effective elastic properties of cortical bone tissue were computed using a multiscale model based on continuum micromechanics. Volume fractions of bone components (collagen, mineral, and water) were considered as random variables whose probabilistic description was built using the maximum entropy principle. The relevance of this approach was proved by analysing a human bone sample taken from the inferior femoral neck. The sample was imaged using synchrotron radiation micro-computed tomography. 3-D distributions of Haversian porosity and tissue mineral density extracted from these images supplied the experimental information needed to build the stochastic models of the volume fractions. Thus, the stochastic multiscale model provided reliable statistical information (such as mean values and confidence intervals) on bone elastic properties at the tissue scale. Moreover, the existence of a simpler "nominal model", accounting for the main features of the stochastic model, was investigated. It was shown that such a model does exist, and its relevance was discussed.

Scale-invariant multilevel Monte Carlo method

“…As an example, we consider the linear-elastic material model of the human femoral bone, the constitutive law of which is assumed to be uncertain. In particular, the bone material symmetry is described as uncertain due to lack of conclusive verification of the class of elastic symmetry the human femur belongs to [23,17]. Hence, the entire elasticity tensor is constructed as random with predefined elastic symmetry in the mean (orthotropy [10] in this case) and purely arbitrary symmetries in each stochastic realization.…”

Foreword to the special issue on Bone and cartilage mechanobiology across the scales

“…Resonant ultrasound spectroscopy was used to measure the transverse isotropic stiffness tensor. The second paper on this topic presents a stochastic multiscale model of cortical bone elasticity based on high-resolution imaging (Sansalone et al 2015). This approach allows accurate and reliable assessment of bone quality and bone mechanical properties.…”

Stochastic Reorder Point-Lot Size (r,Q) Inventory Model under Maximum Entropy Principle

“…Taking into account some of the most recent works, we can cite the following contexts: urban planning [6]; queueing systems [7]; structural dynamics [8]; insurance [9]; computer vision and multimedia [10]; computer algebra [11]; geology [12]; biomechanics [13]; signal processing [14]. The literature gives some reviews concerning applications in ecology, finance, physics, chemistry, and biology [15][16][17][18].…”