Quantification of trabecular bone microdamage using the virtual internal bond model and the individual trabeculae segmentation technique

Abstract

Trabecular bone microdamage significantly influences the skeletal integrity and bone remodelling process. In this paper a novel constitutive model, called the virtual internal bond model (VIB), was adopted for simulating the damage behaviour of bone tissue. A unique 3D image analysis technique, named individual trabeculae segmentation, was used to analyse the effects of microarchitectures on the damage behaviours of trabecular bone. We demonstrated that the process of initiation and accumulation of microdamage in trabecular bone samples can be captured by the VIB-embedded finite-element method simulation without a separate fracture criterion. Our simulation results showed that the microdamage can occur at as early as about 0.2-0.4% apparent strain, and a large volume of microdamage was accumulated around the apparent yield strain. In addition we found that the plate-like trabeculae, especially the longitudinal ones, take crucial roles in the microdamage behaviours of trabecular bone.

Figure 1

The microstructure of trabecular bone and the virtual internal bond model. The force-extension relation of the virtual bond is used for the description of the interaction between material microstructures of bone tissue.

Figure 2

Illustrations of the evolution of the damage process in a transverse trabecular rod. (a) A 4 × 4 × 4 mm sample where the position of the transverse rod is shown in the ellipse; (b) the apparent stress and strain curve on which the typical loading points are marked corresponding to the panels (c)–(g); (c)–(g) snapshots of stress and deformation of elements in the rod at 0, 0.5, 0.77, 0.80 and 0.90% apparent strains, respectively, and green-to-red colours on the trabecula represents the magnitude of stress (S₃₃) from low to high. The elements contained in the white box in (f) and (g) are damaged.

Figure 3

Damage evolution in a local region of a trabecular bone sample at different loading strains. (a) A little damage is nucleated at 0.5% apparent strain; more damage has accumulated at (b) 1.0% apparent strain and (c) 1.5% apparent strain. (d) Catastrophic failure occurs near 2.5% apparent strain.

Figure 4

(a) The apparent strain of damage initiation of trabecular bone samples. (b) The apparent yield strain of trabecular bone samples.

Figure 5

(a) The linear correlation of the apparent yield strength and the apparent elastic modulus of the trabecular bone samples; (b) the linear correlation of yield strength and the bone volume fraction (BV/TV) of the bone samples; (c) A local magnification of the panel (b) at small BV/TV with the sample ID marked and (d) the linear correlation of yield strength and the axial bone volume fraction (aBV/TV).

Figure 6

(a) A comparison of the damaged number fraction (DNF) between trabecular plates and rods; (b) A comparison of the damage tissue fraction (DTF) between trabecular plates and rods (Mean ± SE). Asterisk indicates significantly higher damaged plate fraction than rod fraction (p < 0.05).

Figure 7

(a) The DNF of total trabeculae in longitudinal, oblique and transverse directions. (b) The DTF of total trabeculae in longitudinal, oblique and transverse directions (mean ± SE). Asterisk indicates significantly higher damaged plate fraction than rod fraction (p < 0.05).

Figure 8

(a) and (b) Comparison of DNF and DTF of trabecular plates along longitudinal, oblique and transverse directions; (c) and (d) comparison of DNF and DTF of trabecular rods along longitudinal, oblique and transverse directions (Mean ± SE). Asterisk indicates significantly higher damaged longitudinal plate fraction than either oblique or transverse plate fraction (p < 0.05). Plus indicates significantly higher damaged transverse rod than either oblique or longitudinal rod fraction (p < 0.05).