Potential of in vivo MRI‐based nonlinear finite‐element analysis for the assessment of trabecular bone post‐yield properties

Abstract

Purpose: Bone strength is the key factor impacting fracture risk. Assessment of bone strength from high-resolution (HR) images have largely relied on linear micro-finite element analysis (μFEA) even though failure always occurs beyond the yield point, which is outside the linear regime. Nonlinear μFEA may therefore be more informative in predicting failure behavior. However, existing nonlinear models applied to trabecular bone (TB) have largely been confined to micro-computed tomography (μCT) and, more recently, HR peripheral quantitative computed tomography (HR-pQCT) images, and typically have ignored evaluation of the post-yield behavior. The primary purpose of this work was threefold: (1) to provide an improved algorithm and program to assess TB yield as well as post-yield properties; (2) to explore the potential benefits of nonlinear μFEA beyond its linear counterpart; and (3) to assess the feasibility and practicality of performing nonlinear analysis on desktop computers on the basis of micro-magnetic resonance (μMR) images obtained in vivo in patients. Methods: A method for nonlinear μFE modeling of TB yield as well as post-yield behavior has been designed where material nonlinearity is captured by adjusting the tissue modulus iteratively according to the tissue-level effective strain obtained from linear analysis using a computationally optimized algorithm. The software allows for images at in vivo μMRI resolution as input with retention of grayscale information. Associations between axial stiffness estimated from linear analysis and yield as well as post-yield parameters from nonlinear analysis were investigated from in vivo μMR images of the distal tibia (N = 20; ages: 58-84) and radius (N = 20; ages: 50-75). Results: All simulations were completed in 1 h or less for 61 strain levels using a desktop computer (dual quad-core Xeon 3.16 GHz CPUs equipped with 40 GB of RAM). Although yield stress and ultimate stress correlated strongly (R 2 > 0.95, p < 0.001) with axial stiffness, toughness correlated moderately at the distal tibia (R 2 = 0.81, p < 0.001) and only weakly at the distal radius (R 2 = 0.34, p = 0.007). Further, toughness was found to vary by up to 16% for bone of very similar axial stiffness (<2%). Conclusions:The work demonstrates the practicality of nonlinear μFE simulations at in vivo μMRI resolution, as well as its potential for providing additional information beyond that obtainable from linear analysis. The data suggest that a direct assessment of toughness may provide information not captured by stiffness.

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

“…Trabecular bone post-yield behavior was assumed to follow that of an elastic-plastic material. Thus, the dependence of C on ε was expressed in our model via a nonlinear relationship between E and ε d erived on an element-by-element basis from the stress-strain relation for an elastic-plastic body (36) via the following function (24): $$Efalse({\epsilon }_{\mathit{tissue}}false)=false({false(sechfalse({false(50\times {\epsilon }_{\mathit{tissue}}+0.53false)}^{1.4}false)false)}^{0.6}+0.05false)\times 15\mathrm{italicGPa}.$$…”

“…All other parameters in (2) were empirically calibrated so that the resultant stress-strain curves were consistent with those observed by mechanical testing in work performed by others (14, 20, 21, 38). Equation [2] used to relate the local strain to the modulus of elasticity is illustrated in Figure 1 of (24). A nonlinear system, A(u)u = b , was thereby established in a way analogous to that in the linear case (25).…”

“…In this work, the performance of a recently developed nonlinear μFE model (24) is examined in terms of serial reproducibility and reliability of TB yield and post-yield estimates at the distal radius of twenty women using high-resolution in-vivo MR images. The nonlinear algorithm and program takes advantage of a computationally optimized algorithm for linear μFE analysis (25) complemented by a strain-based criteria allowing dynamic adjustment of tissue-level modulus.…”

Advances in imaging approaches to fracture risk evaluation

“…Recently, μMRI has been used with machine learning to improve the accuracy of discriminating between fracture and non-fracture cases (106). μMRI can also be used with μFEA to predict mechanical properties from structure (107, 108). For example, in renal transplant patients, μFEA showed significantly lower mechanical properties after transplantation (109), even when structural properties did not change significantly.…”

3D Imaging Bone Quality: Bench to Bedside

“…Since the voxel sizes were large (0.410 to 1.0 mm slice thickness) such a step would be necessary. To date only one in vivo MRI based study has attempted to corroborate the computer modelled mechanical properties, comparing values measured using hrMRI and micro-CT as the gold standard 32 . Cadaveric distal tibia were MRI scanned at 0.160 3 mm voxel size but the resolution of the Micro-CT scans are excellent for visualising, measuring and describing trabecular architecture e.g.…”