Effect of including damage at the tissue level in the nonlinear homogenisation of trabecular bone

Relation between mechanical and densimetric properties to fractal dimension in human rib cortical bone

BACKGROUND

Numerous prior studies hypothesized a power-law relationship (E∝ρ^α) between cortical bone Young's modulus (E) and density (ρ) with an exponent 2.3≤α≤3.0, that has not been previously justified in the literature on a theoretical level. Moreover, despite the fact microstructure have been extensively studied, the material correlate of Fractal Dimension (FD) as a descriptor of bone microstructure was not clear in previous studies.

METHODS

This study examined the effect of mineral content and density on the mechanical properties of a large number of human rib cortical bone samples. The mechanical properties were calculated using Digital Image Correlation and uniaxial tensile tests. CT scans were used to calculate the Fractal Dimension (FD) of each specimen. For each specimen, the mineral (f_min), organic (f_org) and water (f_wat) weight fractions were determined. In addition, density was measured after a drying-and-ashing process. Then, Regression Analysis was employed to investigate the relationship between anthropometric variables, weight fractions, density and FD, as well as its impact on the mechanical properties.

FINDINGS

Young's modulus exhibited a power-law relationship with an exponent of α>2.3 when using the conventional density (wet density), but α=2 when using dry density (desecated specimens). In addition, FD increases with decreasing cortical bone density. A significant relationship has been found between FD and density, whereby FD is correlated with the embedding of low density regions in cortical bone.

INTERPRETATION

This study provides a new insight in the exponent value of the power-law relation between Young's Modulus and density, and relates bone behavior with the fragile fracture theory in ceramic materials. Moreover, the results suggest that Fractal Dimension is related to presence of low-density regions.

Mesh-independent damage model for trabecular bone fracture simulation and experimental validation

We propose in this study a two-dimensional constitutive model for trabecular bone combining continuum damage with embedded strong discontinuity. The model is capable of describing the three failure phases of trabecular bone tissue which is considered herein as a quasi-brittle material with strains and rotations assumed to be small and without viscous, thermal or other non-mechanical effects. The finite element implementation of the present model uses constant strain triangle (CST) elements. The displacement jump vector is implicitly solved through a return mapping algorithm at the local (finite element) level, while the global equilibrium equations are dealt with by Newton-Raphson method. The performance, accuracy and applicability of the proposed model for trabecular bone fracture are evaluated and validated against experimental measurements. These comparisons include both global and local aspects through numerical simulations of three-point bending tests performed on 10 single bovine trabeculae in the quasi-static regime.

Prediction of mechanical properties of trabecular bone in patients with type 2 diabetes using damage based finite element method

Type-2 diabetic (T2D) and osteoporosis (OP) suffered patients are more prone to fragile fracture though the nature of alteration in areal bone mineral density (aBMD) in these two cases are completely different. Therefore, it becomes crucial to compare the effect of T2D and OP on alteration in mechanical and structural properties of femoral trabecular bone. This study investigated the effect of T2D, OP, and osteopenia on bone structural and mechanical properties using micro-CT, nanoindentation and compression test. Further, a nanoscale finite element model (FEM) was developed to predict the cause of alteration in mechanical properties. Finally, a damage-based FEM was proposed to predict the pathological related alteration of bone's mechanical response. The obtained results demonstrated that the T2D group had lower volume fraction (-18.25%, p = 0.023), young's modulus (-23.47%, p = 0.124), apparent modulus (-37.15%, p = 0.02), and toughness (-40%, p = 0.001) than the osteoporosis group. The damage-based FE results were found in good agreement with the compression experiment results for all three pathological conditions. Also, nanoscale FEM results demonstrated that the elastic and failure properties of mineralised collagen fibril decreases with increase in crystal size. This study reveals that T2D patients are more prone to fragile fracture in comparison to OP and osteopenia patients. Also, the proposed damage-based FEM can help to predict the risk of fragility fracture for different pathological conditions.

Using Non-linear Homogenization to Improve the Performance of Macroscopic Damage Models of Trabecular Bone

Realistic macro-level finite element simulations of the mechanical behavior of trabecular bone, a cellular anisotropic material, require a suitable constitutive model; a model that incorporates the mechanical response of bone for complex loading scenarios and includes post-elastic phenomena, such as plasticity (permanent deformations) and damage (permanent stiffness reduction), which bone is likely to experience. Some such models have been developed by conducting homogenization-based multiscale finite element simulations on bone micro-structure. While homogenization has been fairly successful in the elastic regime and, to some extent, in modeling the macroscopic plastic response, it has remained a challenge with respect to modeling damage. This study uses a homogenization scheme to upscale the damage behavior from the tissue level (microscale) to the organ level (macroscale) and assesses the suitability of different damage constitutive laws. Ten cubic specimens were each subjected to 21 strain-controlled load cases for a small range of macroscopic post-elastic strains. Isotropic and anisotropic criteria were considered, density and fabric relationships were used in the formulation of the damage law, and a combined isotropic/anisotropic law with tension/compression asymmetry was formulated, based on the homogenized results, as a possible alternative to the currently used single scalar damage criterion. This computational study enhances the current knowledge on the macroscopic damage behavior of trabecular bone. By developing relationships of damage progression with bone's micro-architectural indices (density and fabric) the study also provides an aid for the creation of more precise macroscale continuum models, which are likely to improve clinical predictions.

A study of the progression of damage in an axially loaded Branta leucopsis femur using X-ray computed tomography and digital image correlation

This paper uses X-ray computed tomography to track the mechanical response of a vertebrate (Barnacle goose) long bone subjected to an axial compressive load, which is increased gradually until failure. A loading rig was mounted in an X-ray computed tomography system so that a time-lapse sequence of three-dimensional (3D) images of the bone’s internal (cancellous or trabecular) structure could be recorded during loading. Five distinct types of deformation mechanism were observed in the cancellous part of the bone. These were (i) cracking, (ii) thinning (iii) tearing of cell walls and struts, (iv) notch formation, (v) necking and (vi) buckling. The results highlight that bone experiences brittle (notch formation and cracking), ductile (thinning, tearing and necking) and elastic (buckling) modes of deformation. Progressive deformation, leading to cracking was studied in detail using digital image correlation. The resulting strain maps were consistent with mechanisms occurring at a finer-length scale. This paper is the first to capture time-lapse 3D images of a whole long bone subject to loading until failure. The results serve as a unique reference for researchers interested in how bone responds to loading. For those using computer modelling, the study not only provides qualitative information for verification and validation of their simulations but also highlights that constitutive models for bone need to take into account a number of different deformation mechanisms.