Christophe Desceliers

Influence of a gradient of material properties on ultrasonic wave propagation in cortical bone: Application to axial transmission

The aim of this work is to evaluate the effect of a spatial gradient of material properties (mass density and stiffness coefficients) of cortical bone on its ultrasonic response obtained with an axial transmission device. Therefore, a two-dimensional finite element time-domain method is derived to model transient wave propagation in a three-layer medium composed of an inhomogeneous transverse isotropic solid layer sandwiched between two acoustic fluid layers and excited by an acoustic linear source located in one fluid layer, delivering broadband ultrasonic pulses. The model couples the acoustic propagation in both fluid media with the elastodynamic response of the solid layer. A constant spatial gradient of material properties is considered for two values of bone thicknesses corresponding to relatively thick and thin bone widths. For a thin bone (0.6 mm) compared to wavelength (around 4 mm at 1 MHz), the results are in good agreement with a S(0) Lamb wave assuming a homogeneous material with spatially averaged material properties. For a thick bone (4 mm), the results are in agreement with the propagation of a lateral wave and allow the derivation of an equivalent contributing depth in the case of a transverse isotropic inhomogeneous solid layer.

Influence of viscoelastic and viscous absorption on ultrasonic wave propagation in cortical bone: Application to axial transmission

Cortical bone and the surrounding soft tissues are attenuating and heterogeneous media, which might affect the signals measured with axial transmission devices. This work aims at evaluating the effect of the heterogeneous acoustic absorption in bone and in soft tissues on the bone ultrasonic response. Therefore, a two-dimensional finite element time-domain method is derived to model transient wave propagation in a three-layer medium composed of an inhomogeneous transverse isotropic viscoelastic solid layer, sandwiched between two viscous fluid layers. The model couples viscous acoustic propagation in both fluid media with the anisotropic viscoelastic response of the solid. A constant spatial gradient of material properties is considered for two values of bone thicknesses (0.6 and 4 mm). In the studied configuration, absorption in the surrounding fluid tissues does not affect the results, whereas bone viscoelastic properties have a significant effect on the first arriving signal (FAS) velocity. For a thin bone, the FAS velocity is governed by the spatially averaged bone properties. For a thick bone, the FAS velocity may be predicted using a one-dimensional model.

Determination of the random anisotropic elasticity layer using transient wave propagation in a fluid-solid multilayer: Model and experiments

The aim of this paper is to introduce a simplified model for an uncertain solid layer sandwiched between two acoustic fluid layers and using the ultrasonic characterization with an acoustic source placed in one fluid layer. Uncertainties are taken into account with a probabilistic model of the elasticity tensor. Its parameters are the mean value of the random tensor and a dispersion parameter that controls the statistical fluctuation level. The characterization of the solid layer given a database of actual measurements consists in the determination of the (i) elastic parameters of the mean elasticity model, (ii) the dispersion parameter, and (iii) mass density of the solid. This is performed with a numerical solver of wave propagation and for in vivo data collected previously. The model is representative of measurements of human bone properties with the so-called axial transmission technique. The capability of the model to predict the velocity of the first experimental arriving signal in the statistical sense is proved. The identified anisotropic elasticity tensor of cortical bone from actual data based on the simplified model is given.

Computational Aspects for Constructing Realizations of Polynomial Chaos in High Dimension

This paper deals with computational aspects related to the construction of realizations of polynomial chaos expansion in high dimension. The method proposed consists of (1) constructing the realizations of the multivariate monomials using a generator of independent realizations of the germs whose probability distribution is the given arbitrary measure and (2) performing an orthogonalization of the realizations of the multivariate monomials with an algorithm different from the Gram–Schmidt orthogonalization algorithm which is not stable in high dimension. A brief review of polynomial chaos expansion with arbitrary measure is given. The statistically independent realizations of multivariate monomials are introduced. The centered statistically independent realizations of orthonormal multivariate polynomials are developed. Finally, a quantification of the errors induced by the usual methods is given.

Equivalent contributing depth investigated by a lateral wave with axial transmission in viscoelastic cortical bone

Cortical bone is a viscoelastic heterogeneous medium which may be assessed with axial transmission. This work aims at evaluating the average depth investigated by the lateral wave for radial variations of material properties in relatively thick cortical bone. The equivalent contributing depth (ECD) is derived from the finite element simulation results for spatial variations of a viscoelastic coefficient (η(11)) and of porosity. A value of ECD equal to around 1.6 mm is obtained for a spatial variation of η(11). The method fails to predict accurate values of the ECD for a spatial variation of porosity, because all parameters vary simultaneously.

Experimental multiscale measurements for the mechanical identification of a cortical bone by digital image correlation

The implementation of the experimental methodology by optical measurements of mechanical fields, the development of a test bench, the specimen preparation, the experimental measurements, and the digital image correlation (DIC) method, have already been the object of research in the context of biological materials. Nevertheless, in the framework of the experimental identification of a mesoscopic stochastic model of the random apparent elasticity field, measurements of one specimen is required at both the macroscopic scale and the mesoscopic scale under one single loading. The nature of the cortical bone induces some difficulties, as no single speckled pattern technique is available for simultaneously obtaining the displacement at the macroscopic scale and at the mesoscopic scale. In this paper, we present a multiscale experimental methodology based on (i) an experimental protocol for one specimen of a cortical bone, (ii) its measuring bench, (iii) optical field measurements by DIC method, (iv) the experimental results, and (v) the multiscale experimental identification by solving a statistical inverse problem.

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

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.

Modeling of transient wave propagation in a heterogeneous solid layer coupled with fluid: Application to long bones

The aim of this work is to evaluate the effects of the heterogeneity and anisotropy of material properties of cortical bone on its ultrasonic response obtained by using axial transmission method. The heterogeneity and anisotropy of material properties are introduced by using a parametric probabilistic model. The geometrical configuration of the tested sample is described by a tri-layer medium composed of a heterogeneous and anisotropic solid layer sandwiched between two acoustic fluid layers of which one of these layers is excited by an acoustic linear source. The numerical results focus on studying of an interest quantity, called velocity of the first arriving signal, showing that it strongly depends on the dispersion induced by statistical fluctuations of stochastic elasticity field.

On the uncertainty propagation in multiscale modeling of cortical bone elasticity

Uncertainty in experimental data is a major concern in biomechanical modeling – even more in a multiscale framework, where details of material microstructure are often hardly accessible. Since input data are uncertain, deterministic modeling is limited and new approaches are required. Probability theory can be used to account for uncertain model parameters and propagate this uncertainty through the scales (Soize 2008). We have recently proposed a stochastic multiscale model of cortical bone based on continuum micromechanics and the maximum entropy (MaxEnt) principle (Sansalone et al. 2014). This model was used to predict bone elasticity at the organ scale while accounting for the uncertainties in bone elasticity. In this contribution, we extend this approach to account for the uncertainties in the composition of a human bone sample. It is shown that, unlike a deterministic model which only provides nominal results, the stochastic model can provide statistics (mean, confidence intervals ...) of the elastic coefficients of the cortical tissue which allow assessing the reliability of these results.

Robustness analysis of an uncertain computational model to predict well integrity for geologic CO2 sequestration

Geologic storage of CO2 must respond to demonstrations of safety, control, and acceptability with authorities and public. The wells are essential elements of the storage system and constitute the only man-made intrusive element in the geologic systems. The role of containment of components of wells must then be ensured for hundreds of years, despite degradation mechanisms that affect their properties. Probabilistic approaches are used to take into account the uncertainties on the quantities of CO2 which migrate from the reservoir of CO2 towards the surface and towards the aquifer. Uncertainties are taken into account by using the generalized probabilistic approach which allows both the system-parameter uncertainties and the model uncertainties induced by modeling errors to be performed in the stochastic computational model. These probabilistic tools, applied to industrial projects, allow owners and operators to set up decisions and provide a strong support to long-term safety demonstration with a high level of confidence, even in presence of uncertainties in the computational models.