Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity

Abstract

<jats:p>With ageing and various diseases, the vascular pore volume fraction (porosity) in cortical bone increases, and the morphology of the pore network is altered. Cortical bone elasticity is known to decrease with increasing porosity, but the effect of the microstructure is largely unknown, while it has been thoroughly studied for trabecular bone. Also, popular micromechanical models have disregarded several micro-architectural features, idealizing pores as cylinders aligned with the axis of the diaphysis. The aim of this paper is to quantify the relative effects on cortical bone anisotropic elasticity of porosity and other descriptors of the pore network micro-architecture associated with pore number, size and shape. The five stiffness constants of bone assumed to be a transversely isotropic material were measured with resonant ultrasound spectroscopy in 55 specimens from the femoral diaphysis of 29 donors. The pore network, imaged with synchrotron radiation X-ray micro-computed tomography, was used to derive the pore descriptors and to build a homogenization model using the fast Fourier transform (FFT) method. The model was calibrated using experimental elasticity. A detailed analysis of the computed effective elasticity revealed in particular that porosity explains most of the variations of the five stiffness constants and that the effects of other micro-architectural features are small compared to usual experimental errors. We also have evidence that modelling the pore network as an ensemble of cylinders yields biased elasticity values compared to predictions based on the real micro-architecture. The FFT homogenization method is shown to be particularly efficient to model cortical bone.</jats:p>

A review of nonlinear FFT-based computational homogenization methods

“…Closely related are applications to microstructures of bound sand used for casting processes [109,281]. FFT-based computational methods may be used to calculate the elastic properties of bones [282,283], for estimating stiffness and strength of medium-density fiberboards [284][285][286] and gaining insight into the orientations of fiber bundles in natural fiber based microstructures [285,287].…”

Maximum effect of the heterogeneity of tissue mineralization on the effective cortical bone elastic properties

“…We have used a collection of synchrotron radiation micro-computed tomography (SR-µCT) 3D images of volumes of interest (VOI) from a previous study (Cai et al 2019a) (Fig. 2).…”

“…Thereafter, axis x 3 was approximately along the direction of osteons and axes x 1 and x 2 were perpendicular to osteons. The images were then binarized by simple thresholding to obtain two phases : pores and mineralized matrix (Cai et al 2019a). In each specimen, a VOI of approximately 2.8 × 3.9 × 4.8 mm 3 corresponding to one representative volume element (RVE) for the homogenization was selected manually.…”

“…The VOI size was much larger than the minimum RVE size for cortical bone (Grimal et al 2011a). After a convergence study (Cai et al 2019a), the voxel size was then increased to 35 µm to reduce the computational cost, which lead to about 1.2 million voxels per specimen.…”

“…The heterogeneity of the elastic properties of the matrix, i.e., the differences between osteonal and interstitial tissues, has been considered in some studies (Crolet et al 1993;Dong and Guo 2006;Grimal et al 2008;Hamed et al 2010). However, many other authors have considered an homogeneous matrix (Hellmich and Ulm 2004;Sevostianov and Kachanov 2000;Parnell and Grimal 2009;Baumann et al 2012;Gagliardi et al 2017;Cai et al 2019a). The errors on predicted mesoscale properties when the differences in elasticity between osteonal and interstitial tissues are disregarded have not been quantified.…”