Multiscale approach including microfibril scale to assess elastic constants of cortical bone based on neural network computation and homogenization method

Abstract

SUMMARYThe complexity and heterogeneity of bone tissue require a multiscale modelling to understand its mechanical behaviour and its remodelling mechanisms. In this paper, a novel multiscale hierarchical approach including microfibril scale based on hybrid neural network computation and homogenisation equations was developed to link nanoscopic and macroscopic scales to estimate the elastic properties of human cortical bone. The multiscale model is divided into three main phases: (i) in step 0, the elastic constants of collagen-water and mineral-water composites are calculated by averaging the upper and lower Hill bounds; (ii) in step 1, the elastic properties of the collagen microfibril are computed using a trained neural network simulation. Finite element (FE) calculation is performed at nanoscopic levels to provide a database to train an in-house neural network program; (iii) in steps 2 to 10 from fibril to continuum cortical bone tissue, homogenisation equations are used to perform the computation at the higher scales. The neural network outputs (elastic properties of the microfibril) are used as inputs for the homogenisation computation to determine the properties of mineralised collagen fibril. The mechanical and geometrical properties of bone constituents (mineral, collagen and cross-links) as well as the porosity were taken in consideration. This paper aims to predict analytically the effective elastic constants of cortical bone by modelling its elastic response at these different scales, ranging from the nanostructural to mesostructural levels. Our findings of the lowest scale"s output were well integrated with the other higher levels and serve as inputs for the next higher scale modelling. Good agreement was obtained between our predicted results and literature data.

A multiscale modelling of bone ultrastructure elastic proprieties using finite elements simulation and neural network method

“…In fact, in order to accomplish their biological and mechanical functions, bone hierarchical structure is constituted of many scale levels with specific interactions and a very complex architecture [1]. These structural scales can be distinguished as follows: macro-scale (whole bone), meso-scale (cortical and trabecular bone), micro-scale (single osteon and single trabecula), sub-micro-scale (lamella), nano-scale (microfibrils, fibrils and fibers), and sub-nanoscale (HA crystals and TC molecules) [2,3].…”

“…Many researchers in the literature have addressed this problem by developing analytical and numerical multiscale modelling of the bone mechanical behaviour [3,[7][8][9][10][11][12].…”

“…Some of them have estimated bone elastic properties. We can cite, as an example and not a limited list, the works of Martinez et al [7] and Hamed et al [3].…”

“…There are few studies that focused on the MCMs scale level [3,13,15,16,28]. In the work of Barkaoui et al [13] has been proposed a finite element geometrical model to study elastic mechanical properties and investigate the effect of some mechanical and geometrical parameters on the mechanical behaviour of mineralized collagen microfibril scale.…”

Study of the mechanical behavior of cortical bone microstructure by the finite element method

“…E col E min φ min E f ib Barkaoui et al [2014] 2.5 GPa 120 GPa 0.24 0.9 GPa Bar-On and Wagner [2013b] 1 GPa 100 GPa 0.3 4 GPa Yoon and Cowin [2008] 1. …”