Comparison of the Linear Finite Element Prediction of Deformation and Strain of Human Cancellous Bone to 3D Digital Volume Correlation Measurements

Abstract

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.

High Performance Digital Volume Correlation

“…Various mini-mization algorithms have been used for DVC, including Levenberg-Marquardt [5] and BFGS [73], both of which use first derivative information to approximate the Hessian in Newton's method, thereby reducing the cost per iteration. Steepest descent has also been used [28,79], but it converges slowly unless the problem is very well conditioned, so is not recommended in general.…”

“…Choices include C 0 trilinear interpolation [12,28,42], C 1 tricubic Hermite interpolation [5,73,79], and C 2 tricubic spline interpolation [8,21]. In 2D DIC, C 4 biquintic spline interpolation has also been used [71].…”

“…Previously, representative volume elements have been determined by using 2D DIC to extract a representative surface element, and then assuming some type of extension into the third dimension [17,45,53]; DVC now offers the opportunity to attain a true 3D representative volume element. DVC is also useful to analyze complex materials, such as bone [5,46,73,79], rock [42], synthetic foams [67], wood [21], sugar [20] and sand grains [35]; as well as complex behaviors, such as material fatigue after repeated loading cycles [10,62], which are difficult to simulate. Thus, although only a relatively small number of researchers have studied DVC, it is a powerful tool with significant future potential.…”

“…A wide variety of materials such as bone [5,46,73,79], rock [42], synthetic foams [67], and metals [62,34] have been studied using these techniques. Faithfully duplicating realistic imaging conditions, such as image size, resolution, and particularly noise, is an inherent part of the robustness of the DVC technique.…”

“…ied triaxial compression of argillaceous rock, computing 60,000 points, but using only three DOF and trilinear interpolation. Zauel et al [79] used three DOF and C 1 tricubic interpolation, with steepest descent and a quadratic line search. Their code was optimized by using SIMD instructions to achieve 0.47 seconds per point on a 1.8 GHz Pentium 4.…”