A novel mechanical parameter to quantify the microarchitecture effect on apparent modulus of trabecular bone: A computational analysis of ineffective bone mass

Abstract

Background: One of the characteristics of osteoporotic bone is the deterioration of trabecular microarchitecture. Previous studies have shown microarchitecture alone can vary the apparent modulus of trabecular bone significantly independent of bone volume fraction (BV/TV) from morphological and topological perspectives. However, modulus is a mechanical quantity and there is a lack of mechanical explanatory parameters. This study aims to propose a novel mechanical parameter to quantify the microarchitecture effect on the apparent modulus of trabecular bone.

Materials and methods: Fourteen human female cadaveric vertebrae were scanned with a dual-energy X-ray (DXA) equipment followed by a micro-CT (μCT) system at 18 μm isotropic resolution. Four trabecular bone specimens (3.46 × 3.46 × 3.46 mm) were obtained from each vertebral body and converted to voxel-based micro finite element (μFE) models. The apparent modulus (E) of the μFE model was computed using a linear micro finite element analysis (μFEA). The normalized apparent modulus (E*) was computed as E divided by BV/TV. The relationship between E and BV/TV was analyzed by linear, power-law and exponential regressions. Linear regression was performed between E* and BV/TV. Ineffective bone mass (InBM) was defined as the bone mass with a negligible contribution to the load-resistance and represented by elements with von Mises stress less than a certain stress threshold. InBM was quantified as the low von Mises stress ratio (LSVMR), which is the ratio of the number of InBM elements to the total number of elements in the μFE model. An incremental search technique with coarse and fine search intervals of 10 and 1 MPa, respectively, was adopted to determine the stress threshold for calculating LSVMR of the μFE model. Correlation between E* and LSVMR was analyzed using linear and power-law models for each stress threshold. The threshold producing the highest coefficient of determination (R2) in the correlation between E* and LSVMR was taken as the optimal stress threshold for calculating LSVMR. Linear regression was performed between E and LSVMR. Multiple linear regression of E against both BV/TV and LSVMR was further analyzed.

Results: E significantly (p < .001) correlates to BV/TV whereas E* has no significant (p = .75) correlation with BV/TV. Incremental search suggests 59 MPa to be the optimal stress threshold for calculating LSVMR. BV/TV alone can explain 59% of the variation in E using power-law regression model (E = 2254.64BV/TV1.04, R2 = 0.59, p < .001). LSVMR alone can explain 48% of the variation in E using linear regression model (E = 1696.4–1647.1LSVMR, R2 = 0.48, p < .001). With these two predictors taken into consideration, 95% of the variation in E can be explained in a multiple linear regression model (E = 1364.89 + 2184.37BV/TV – 1605.38LSVMR, adjusted R2 = 0.95, p < .001).

Conclusion: LSVMR can be adopted as the mechanical parameter to quantify the microarchitecture effect on the apparent modulus of trabecular bone.