High-resolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone
Introduction
Computational modeling of trabecular bone has a number of potentially important applications. While the multiaxial and off-axis failure properties of trabecular bone play an important role in fracture (Lotz et al (1991a), Lotz et al (1991b)) and prosthesis design (Cheal et al., 1992; Keaveny and Bartel (1993), Keaveny and Bartel (1994)), experimental data on multiaxial strength has been limited (Fenech and Keaveny, 1999; Keaveny et al., 1999; Stone et al., 1983) due to technical difficulties, and the problem remains somewhat intractable. The problem is further compounded when applied to human tissue since this tissue is scarce and many specimens are required to statistically account for interspecimen heterogeneity. Use of high-resolution finite element models, which have been used extensively to gain insight into the elastic properties of trabecular bone (Hollister et al., 1994; Van Rietbergen et al., 1995), could circumvent these problems if they were sufficiently accurate to qualify them as a surrogate for the real specimen. Such a computational tool would also provide predictions of tissue strains and regions of failure, information that could provide insight into mechanisms of multiaxial failure, as well as bone adaptation and repair.
Previous generic numerical (Pugh et al., 1973; Silva and Gibson, 1997) and analytical (Gibson, 1985) studies of apparent level trabecular bone failure have assumed tissue yield properties based on cortical bone. When these properties were used in high-resolution finite element models, the predicted apparent yield stresses were highly sensitive to the assumed tissue elastic properties (Fyhrie and Hou, 1995). Van Rietbergen et al. (1998) used specimen-specific calibration of the tissue modulus and tissue yield properties based on fatigue properties of trabecular tissue. This protocol resulted in ultimate stresses within 15% of experimental measurements for five human tibial specimens, but ultimate strain was underpredicted by more than 35%. One limitation of these previous studies is the use of the von Mises criterion to model tissue yielding. This criterion assumes equal tensile and compressive yield properties. As Pugh (1973) had long ago noted, this assumption is probably inappropriate for trabecular tissue because cortical tissue is stronger in compression than tension (Reilly and Burstein, 1975), and similar behavior is likely for trabecular tissue (Keaveny, 1997; Morgan et al., 1999; Niebur et al., 1999a). The use of a yield criterion that exhibits strength asymmetry at the tissue level has not been investigated for high-resolution finite element models, and it is not clear whether such a model would result in improved predictions of apparent yield behavior.
In the context of developing a computational tool that might qualify as a surrogate for destructive mechanical testing of trabecular bone, the goal of this study was to develop and validate a non-linear high-resolution finite element analysis technique for failure analysis of trabecular bone. The novelty of the approach was that we treated the trabecular tissue as though it had yield strains similar to those of cortical tissue. In particular, we hypothesized that incorporation of a bilinear, asymmetric yield criterion for the trabecular tissue into a high-resolution finite element model could successfully describe the apparent yield behavior of trabecular bone for a variety of apparent loading conditions. Our specific objectives were to: (1) calibrate the effective tissue elastic moduli for each of seven specimens of bovine tibial trabecular bone; (2) calibrate the tensile and compressive yield strains of the trabecular tissue by performing a parametric finite element analysis on a single specimen; and (3) test the model validity by comparing against literature data the predictions of the apparent yield properties of seven specimens for three different loading cases (tension, compression, and shear).
Section snippets
Specimen preparation
Seven cylindrical bovine tibial trabecular bone specimens from seven steers (approximately 2 years old) were studied. The specimens were harvested with the principal trabecular orientation approximately aligned with the axis of the cylinder as determined by contact radiographs (Keaveny et al., 1994). The apparent elastic modulus of each specimen was measured by six repeated nondestructive mechanical tests to 0.2% strain in compression using endcaps to minimize testing artifacts (Keaveny et al.,
Results
The calibrated effective tissue moduli were typical of modulus values for cortical bone. The mean (±S.D.) effective tissue modulus calculated for the seven specimens was 18.7±3.4 GPa, ranging from 14.4 to 23.8 GPa. Effective tissue modulus did not depend on the measured tissue density, but was negatively correlated with both apparent modulus (Etissue=29.73–5.63Eapparent GPa, R2=0.63, p<0.05 where Eapparent is the apparent modulus in GPa) and volume fraction (Etissue=36.57–78.84νf GPa, R2=0.72,p
Discussion
To establish the basis for computational study of trabecular bone failure, we developed and validated a technique for simulating trabecular bone yielding using high-resolution finite element models. The main novelty was that our approach incorporated a bilinear asymmetric failure criterion for the trabecular tissue, under the assumption that trabecular tissue has yield strains similar to those of cortical bone. This tissue level criterion was uniform for all specimens, although the effective
Acknowledgements
This study was supported by Grants from the National Science Foundation (BES-9625030), National Institutes of Health (AR43784), National Partnership for Advanced Computing Infrastructure (UCB 254); and Lawrence Livermore National Laboratory (ISCR B291837, 97-06 and 98-04). We would like to thank Sean Haddock for specimen imaging.
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