Assessment of trabecular bone yield and post-yield behavior from high-resolution MRI-based nonlinear finite element analysis at the distal radius of premenopausal and postmenopausal women susceptible to osteoporosis

Abstract

Rationale and objectives: To assess the performance of a nonlinear microfinite element model on predicting trabecular bone yield and post-yield behavior based on high-resolution in vivo magnetic resonance images via the serial reproducibility.

Materials and methods: The nonlinear model captures material nonlinearity by iteratively adjusting tissue-level modulus based on tissue-level effective strain. It enables simulations of trabecular bone yield and post-yield behavior from micro magnetic resonance images at in vivo resolution by solving a series of nonlinear systems via an iterative algorithm on a desktop computer. Measures of mechanical competence (yield strain/strength, ultimate strain/strength, modulus of resilience, and toughness) were estimated at the distal radius of premenopausal and postmenopausal women (N = 20, age range 50-75) in whom osteoporotic fractures typically occur. Each subject underwent three scans (20.2 ± 14.5 days). Serial reproducibility was evaluated via coefficient of variation (CV) and intraclass correlation coefficient (ICC).

Results: Nonlinear simulations were completed in an average of 14 minutes per three-dimensional image data set involving analysis of 61 strain levels. The predicted yield strain/strength, ultimate strain/strength, modulus of resilience, and toughness had a mean value of 0.78%, 3.09 MPa, 1.35%, 3.48 MPa, 14.30 kPa, and 32.66 kPa, respectively, covering a substantial range by a factor of up to 4. Intraclass correlation coefficient ranged from 0.986 to 0.994 (average 0.991); CV ranged from 1.01% to 5.62% (average 3.6%), with yield strain and toughness having the lowest and highest CV values, respectively.

Conclusions: The data suggest that the yield and post-yield parameters have adequate reproducibility to evaluate treatment effects in interventional studies within short follow-up periods.

Keywords: Finite element analysis; MRI; reproducibility and reliability; trabecular bone mechanics.

Figure 1

Hypothetical load deformation curve with definition of the μFE-derived mechanical parameters: yield point is the point on the stress-strain curve at which plastic deformation begins to occur, which is calculated here using the 0.2% offset rule; yield strain/strength is the corresponding strain/stress value at the yield point; ultimate strength is set as the peak stress value on the stress-strain curve and the corresponding strain value is referred to as the ultimate strain; modulus of resilience is calculated as the integral of the stress-strain curve from zero to the yield strain point and toughness as the integral from zero to the ultimate strain point.

Figure 2

MR images of the distal radius in a study subject at three time-points (top to bottom: baseline, follow-ups 1 and 2, visually illustrating similarities across.): (a) Cross-sectional view of acquired unprocessed images; (b) Bone volume fraction maps; (c) magnified 3D volume renderings of a small subregion (2.6 × 2.6 × 10.3 mm³).

Figure 3

Simulated stress-strain curves from two subjects (a, b) scanned at three time-points highlighting within-group similarities and between-subject differences.

Figure 4

Longitudinal maximum intensity projections of the simulated strain maps for a thin slab of 1.1 mm thickness at 0.8% applied strain for two subjects (a, b in Figure 3) evaluated at three time-points: baseline, follow-ups 1 and 2, highlighting within-group similarities and between-subject differences. Also note that subject 2 has relatively fewer failed trabeculae (shown in white) and overall higher strain values than subject 1, suggesting TB of subject 2 to exhibit greater ultimate strain, which is consistent with that shown in the simulated stress-strain curves (Figure 3).

Figure 5

Example MR images and simulated stress-strain curves from two subjects showing distinctly different mechanical features reflected by the different stress-strain behaviors. TB in subject 1 has considerably lower toughness and ultimate strength than that of subject 2 (also see Table 2). Distinctly different structural features are apparent with thicker but sparser trabeculation in the bone of subject 2.

Figure 6

Test-retest plots for nonlinear μFEA-estimated mechanical parameters from all twenty subjects; blue: follow-up 1 versus baseline; red: follow-up 2 versus baseline; light grey: line of identity (p < 0.0001 for all correlations).