Neural network prediction of cortical bone damage using a stochastic computational mechanical model

This work aims at predicting the damage of a human cortical bone specimen from in vivo actual data measurements and using a neural network trained and validated from a stochastic computational mechanical model. Within the context of fracture risk prediction and/or osteoporosis diagnostics, the ultrasonic characterization of the human cortical bone properties can be performed using the so-called experimental axial transmission (AT) technique [1], that consists in measuring the velocity of the fastest acoustic signal delivered by a transmitter and recorded at several receivers. In this work, we consider a simplified random elasto-acoustic multi-layer model to simulate the propagation of ultrasonic waves axially transmitted along the bone axis through a biological system (made of cortical bone, soft tissues and marrow bone), while taking into account the variabilities on the ultrasound velocity measurements (due to uncertainties induced by modeling errors on the experimental configuration) by considering an ad hoc probabilistic model of the (tensor-valued) elasticity field [2] in the cortical bone. This simplified model is representative of the experimental measurements obtained with the ultrasonic AT device. It corresponds to an elasto-acoustic three-layer system that is composed of a random heterogeneous damaged (porous) elastic solid layer (referred to as cortical bone layer) sandwiched between two deterministic homogeneous acoustic fluid layers (referred to as soft tissues and marrow bone layers) and excited by an acoustic line source located in the soft tissues layer [3]. A simple spatial gradient model (fluid-solid mixture) of the material properties (mass density and mean elasticity field) along the thickness direction is introduced in the damaged cortical bone layer in order to take into account the alteration of the cortical bone properties caused by the degradation of the microstructure in the neighborhood of the marrow bone layer [3]. The underlying stochastic computational model is then parameterized by two geometrical parameters (corresponding to the thicknesses of the “pure” and “porous” elastic solid parts), a dispersion parameter (controlling the level of statistical fluctuations of the random elasticity field) and a correlation length along the thickness direction (characterizing the spatial correlation structure of the random elasticity field). A very large database is constructed by solving the transient elasto-acoustic wave propagation problem for different selected values of hyperparameters. The data set is used to train, validate and test a neural network [4]. This latter can then be employed for the direct characterization of the geometrical and mechanical properties and the real-time damage prediction of human cortical bone specimens from in vivo experimental measurements, allowing for a straightforward and automatic discrimination between healthy and osteoporotic subjects. References [1] E. Bossy, M. Talmant, M. Defontaine, F. Patat, and P. Laugier. Bidirectional axial transmission can improve accuracy and precision of ultrasonic velocity measurement in cortical bone: a validation on test materials. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 51(1):71–79, Jan 2004. [2] C. Soize. Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Computer Methods in Applied Mechanics and Engineering, 195(1–3):26–64, 2006. [3] C. Desceliers, C. Soize, S. Naili, and G. Haiat. Probabilistic model of the human cortical bone with mechanical alterations in ultrasonic range. Mechanical Systems and Signal Processing, 32:170–177, 2012. Uncertainties in Structural Dynamics. [4] Martin T. Hagan, Howard B. Demuth, and Mark H. Beale. Neural Network Design. PWS Publishing Co., Boston, MA, USA, 1996.