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.
Osteoporosis is a degenerative disorder of the skeleton, leading to an increased risk of fracture. The most common sites of osteoporotic fractures are the hip, spine, and wrist; all sites with a significant fraction of trabecular bone (TB). Fractures of the distal radius (e.g., Colles fracture) account for approximately 15% of all fractures in adults . They are more prevalent at earlier age than hip fractures and have been shown to be a risk factor of future spine and hip fractures . The incidence of Colles fracture is greater in women than in men, especially after menopause . For these reasons, the distal radius is an anatomic site of great interest for early detection of osteoporosis.
The ability of bone to resist stresses without failure (often referred to as “bone mechanical competence”) is impaired in subjects with osteoporosis. Improvements in mechanical competence, for example via medication or exercise, can help prevent osteoporosis or alleviate its symptoms. In many earlier studies, bone structural and topological parameters have been used as surrogates for fracture susceptibility based on the associations found between microarchitecture and the presence or extent of osteoporotic fractures . However, structural parameters are only surrogates for the bone’s mechanical failure behavior. The most direct approach to evaluate bone mechanical competence is via image-based micro-finite element (μFE) analysis . Constructed from high-resolution images of bone microstructure comprising both trabecular and cortical compartments, μFE models are capable of directly estimating bone elastic moduli, predicting bone strength, and assessing osteoporotic fracture risk .
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Materials and methods
Image Acquisition
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Image Processing
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Simulation of Yield and Post-yield Behavior
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E(εtissue)=((sech((50×εtissue+0.53)1.4))0.6+0.05)×15GPa. E
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E with E = 15 GPa and U being the strain-energy density of that element. All other parameters in were empirically calibrated so that the resultant stress-strain curves were consistent with those observed by mechanical testing in work performed by others . Equation 2 is used to relate the local strain to the modulus of elasticity as illustrated in Zhang et al ( ; Fig 1 ). A nonlinear system, A(u)u = b , was thereby established in a way analogous to that in the linear case . This nonlinear system was solved for the resultant stress via an iterative algorithm . The maximum number of iterations was empirically set to 30. A total of 61 strain levels were applied in all experiments with the smallest strain and the step size both being 0.05%.
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Statistical Analysis
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Results
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Table 1
Parameter Means ± SD and Ranges for the 20 Subjects, with Average Coefficient of Variance (CV) and Intraclass Correlation Coefficient (ICC) of Estimated Mechanical Parameters from Nonlinear Simulations for Three Repeat Studies
Mean ± SD Range Average CV (%) ICC Yield strain (%) 0.78 ± 0.05 [0.70, 0.87] 1.01 0.988 Yield strength (MPa) 3.09 ± 1.01 [1.62, 5.11] 3.65 0.995 Ultimate strain (%) 1.35 ± 0.28 [0.90, 1.95] 3.68 0.987 Ultimate strength (MPa) 3.48 ± 1.05 [1.75, 5.51] 3.42 0.994 Modulus of resilience (kPa) 14.30 ± 5.39 [6.83, 25.85] 4.19 0.996 Toughness (kPa) 32.77 ± 12.22 [16.09, 60.22] 5.62 0.986
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Table 2
Yield and Post-yield Parameters for the Two Subjects in Figure 5
Subject 1 Subject 2 Yield strain (%) 0.74 0.87 Yield strength (MPa) 1.97 5.11 Ultimate strain (%) 0.90 1.50 Ultimate strength (MPa) 2.03 5.50 Modulus of resilience (kPa) 8.60 25.85 Toughness (kPa) 16.09 60.22
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Discussion
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Acknowledgments
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