Stochastic multi-scale prediction on the apparent elastic moduli of trabecular bone considering uncertainties of biological apatite (BAp) crystallite orientation and image-based modelling

Abstract

An assessment of the mechanical properties of trabecular bone is important in determining the fracture risk of human bones. Many uncertainty factors contribute to the dispersion of the estimated mechanical properties of trabecular bone. This study was undertaken in order to propose a computational scheme that will be able to predict the effective apparent elastic moduli of trabecular bone considering the uncertainties that are primarily caused by image-based modelling and trabecular stiffness orientation. The effect of image-based modelling which focused on the connectivity was also investigated. A stochastic multi-scale method using a first-order perturbation-based and asymptotic homogenisation theory was applied to formulate the stochastically apparent elastic properties of trabecular bone. The effective apparent elastic modulus was predicted with the introduction of a coefficient factor to represent the variation of bone characteristics due to inter-individual differences. The mean value of the predicted effective apparent Young's modulus in principal axis was found at approximately 460 MPa for respective 15.24% of bone volume fraction, and this is in good agreement with other experimental results. The proposed method may provide a reference for the reliable evaluation of the prediction of the apparent elastic properties of trabecular bone.

Keywords::

• trabecular bone
• uncertainties
• apparent elastic moduli
• image-based modelling
• stochastic multi-scale analysis

Acknowledgements

The authors would like to thank Prof. Yuji Nakajima and Prof. Hiroshi Kiyama (Osaka City University) for providing the bone specimen. We also wish to thank the donor's family for their generosity in the face of their bereavement. Next, the dedicated help from Dr Takuya Ishimoto and Dr Sayaka Miyabe (both from Osaka University) in preparing the micro-CT images for this research is acknowledged with gratitude. This study was also supported in part by the Ministry of Higher Education (MOHE), Malaysia under a Scholarship for Academicians Training Scheme.