Yield strain behavior of trabecular bone

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Abstract

If bone adapts to maintain constant strains and if on-axis yield strains in trabecular bone are independent of apparent density, adaptive remodeling in trabecular bone should maintain a constant safety factor (yield strain/functional strain) during habitual loading. To test the hypothesis that yield strains are indeed independent of density, compressive (n=22) and tensile (n=22) yield strains were measured without end-artifacts for low density (0.18±0.04 g cm-3) human vertebral trabecular bone specimens. Loads were applied in the superior–inferior direction along the principal trabecular orientation. These ‘on-axis’ yield strains were compared to those measured previously for high-density (0.51±0.06 g cm-3) bovine tibial trabecular bone (n=44). Mean (±S.D.) yield strains for the human bone were 0.78±0.04% in tension and 0.84±0.06% in compression; corresponding values for the bovine bone were 0.78±0.04 and 1.09±0.12%, respectively. Tensile yield strains were independent of the apparent density across the entire density range (human p=0.40, bovine p=0.64, pooled p=0.97). By contrast, compressive yield strains were linearly correlated with apparent density for the human bone (p<0.001) and the pooled data (p<0.001), and a suggestive trend existed for the bovine data (p=0.06). These results refute the hypothesis that on-axis yield strains for trabecular bone are independent of density for compressive loading, although values may appear constant over a narrow density range. On-axis tensile yield strains appear to be independent of both apparent density and anatomic site.

Introduction

It has been hypothesized that bone adapts to produce uniform functional apparent strains in both cortical and trabecular bone in response to habitual loads (Turner et al., 1997). If yield strains were also uniform, this would imply that the set point for remodeling is some ratio of the yield strain to the functional strain or a ‘safety factor’. Wolff’s law implies that trabecular orientation aligns itself to the direction of the functional principal stresses (Cowin, 1986). As a consequence, the relevant yield strains in consideration of bone adaptation are those for ‘on-axis’ loading of the bone (i.e. along the principal trabecular orientation). It has also been hypothesized that aging and disease may decrease the yield strain of bone and thus its safety factor (Biewener et al., 1993). One prerequisite to establishing the above hypotheses is to determine the dependency of the on-axis yield strains of trabecular bone on apparent density and to do this for a range of anatomic sites. A more complete understanding of the yield strains in trabecular bone is also fundamental to continued progress in computer modeling of whole bones (Keyak et al., 1993; Lotz et al., 1991; Silva et al., 1996), which in turn may improve diagnosis and treatment of pathologies that weaken trabecular bone such as osteoporosis.

While there is mounting evidence that apparent failure strains in trabecular bone are independent of apparent density, the data are not conclusive. A number of studies have shown no dependence of compressive failure strains on apparent density (Ford and Keaveny, 1996; Hansson et al., 1987; Keaveny et al., 1994; Lindahl, 1976; Rohl et al., 1991), but a number of others have shown failure strains to increase (Hvid et al., 1989; Keaveny et al., 1994; Turner, 1989) or decrease (Hvid et al., 1985; Mosekilde et al., 1987) with increasing apparent density. It is not clear whether the former studies lacked statistical power to show a real dependence or if differences in anatomic site, trabecular orientation, definition of the failure strain, or experimental testing techniques accounted for these different findings. For example, accurate data on apparent failure strains for trabecular bone are difficult if not impossible to measure if end-artifacts are present since strain measures are highly sensitive to this artifact (Keaveny et al., 1993; Keaveny et al., 1997; Odgaard and Linde, 1991). In the context of understanding trabecular bone failure as it pertains to bone adaptation, very few data exist for on-axis apparent failure strains since specimens are usually machined along anatomic directions which rarely align with the principal trabecular orientation. Thus, there is a need for a study on trabecular failure strains that minimizes end-artifact errors, uses bone from different anatomic sites, and tests the bone in the on-axis orientation.

Based on predictions of an axial strut cellular solid model for trabecular bone (Fig. 1) (Christensen, 1986; Gent and Thomas, 1959; Gibson, 1985; Gibson and Ashby, 1988; Rajan, 1985), the hypothesis was developed in this study that, for on-axis loading, compressive apparent yield strains should be positively correlated with apparent density due to underlying buckling mechanisms. Since the slenderness (length/thickness) ratio of individual trabeculae decreases as apparent density increases (Snyder et al., 1993), the significance of this relationship should diminish as density increases due to a lower propensity for trabeculae to buckle. Since buckling cannot occur in tension, it was also hypothesized that the tensile apparent yield strains should remain constant regardless of anatomic site or species due to axial yielding of trabeculae. The following questions were addressed specifically: (1) What are the relationships between the compressive and tensile yield strains vs apparent density for human vertebral trabecular bone? (2) Are these yield strains and strain–density relationships different from those measured previously for bovine tibial trabecular bone (Keaveny et al., 1994)? and (3) For a single anatomic site, is the variance in yield strains small enough to reasonably assume constant yield strains in bone adaptation simulations? The results were then discussed in the context of bone adaptation, aging, and disease.

Section snippets

Methods

Seventeen fresh frozen vertebrae, T10-L4, without radiographic evidence of bone pathologies, were obtained from 11 cadavers (Table 1). Forty-eight cylindrical specimens (8 mm diameter, 25 mm length nominally) were cored in water with marrow in situ along the superior–inferior direction. With the central portion (∼15 mm) wrapped in damp gauze, marrow was removed from the ends (∼5 mm), which were then glued into pre-aligned brass end-caps. The specimens were then randomly assigned to a tension or

Results

Tensile yield strains for the human vertebral bone were independent of apparent density (p=0.31) while compressive yield strains showed a weak but highly significant positive correlation with density (r2=0.52,p=0.0002) (Table 2). Ultimate strains were independent of apparent density in both compression (p=0.64) and tension (p=0.19), although these data showed considerably more scatter than the yield strains. As expected, elastic modulus (Fig. 3), yield stress (Fig. 4), and ultimate stress for

Discussion

The goal of this work was to provide comprehensive data on the failure strains of trabecular bone, with specific reference to their role in bone adaptation, disease, and aging. Our results provide strong evidence that (1) the on-axis compressive apparent yield strains correlate positively but weakly with apparent density, the correlation being stronger in less dense bone; and (2) the on-axis tensile apparent yield strains represent a uniform failure property independent of anatomic site and

Acknowledgements

Funding was provided by NIH AR41481 and AR43784. Bone tissue was provided by the Harvard Anatomical Gifts Program, the UCSF School of Medicine, and the Southwestern Medical Center at the University of Texas, Dallas. Thanks to Peter Kim and Albert Lou for technical assistance.

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