Trabecular bone modulus–density relationships depend on anatomic site

https://doi.org/10.1016/S0021-9290(03)00071-XGet rights and content

Abstract

One outstanding issue regarding the relationship between elastic modulus and density for trabecular bone is whether the relationship depends on anatomic site. To address this, on-axis elastic moduli and apparent densities were measured for 142 specimens of human trabecular bone from the vertebra (n=61), proximal tibia (n=31), femoral greater trochanter (n=23), and femoral neck (n=27). Specimens were obtained from 61 cadavers (mean±SD age=67±15 years). Experimental protocols were used that minimized end-artifact errors and controlled for specimen orientation. Tissue moduli were computed for a subset of 18 specimens using high-resolution linear finite element analyses and also using two previously developed theoretical relationships (Bone 25 (1999) 481; J. Elasticity 53 (1999) 125). Resultant power law regressions between modulus and density did depend on anatomic site, as determined via an analysis of covariance. The inter-site differences were among the leading coefficients (p<0.02), but not the exponents (p>0.08), which ranged 1.49–2.18. At a given density, specimens from the tibia had higher moduli than those from the vertebra (p=0.01) and femoral neck (p=0.002); those from the trochanter had higher moduli than the vertebra (p=0.02). These differences could be as large as almost 50%, and errors in predicted values of modulus increased by up to 65% when site-dependence was ignored. These results indicate that there is no universal modulus–density relationship for on-axis loading. Tissue moduli computed using methods that account for inter-site architectural variations did not differ across site (p>0.15), suggesting that the site-specificity in apparent modulus–density relationships may be attributed to differences in architecture.

Introduction

A key concept in the study of skeletal structure–function relationships is that bones adapt to their mechanical environment, perhaps optimizing stiffness (Currey, 1984; Fyhrie and Vashishth, 2000) or strain energy density (Carter et al., 1987; Huiskes et al., 1987). By describing changes in stiffness that occur due to the addition or removal of bone, relationships between trabecular bone elastic modulus and apparent density reflect the mechanics of this adaptive strategy. It is not clear, however, if the modulus–density relationship for human trabecular bone along its main loading axis is universal or if it depends on anatomic site. For example, inter-site differences in trabecular architecture (Hildebrand et al., 1999) and tissue modulus (Hoffler et al., 2000) may serve to alter the modulus–density relationship across site. Characterizing to what extent this quantitative relationship can be generalized throughout the skeleton would provide a broader understanding of factors that influence the functional adaptation process. Use of site-specific relations, if determined necessary, would improve the fidelity of finite element models of whole bones (Keyak and Rossi, 2000; Lotz et al., 1991; Silva et al., 1998) and bone-implant systems (Skinner et al., 1994) that address clinical issues such as hip fracture, kyphosis, and total joint replacement.

Although many studies have compared modulus–density across multiple human anatomic sites, the potential site-dependence of this relationship requires further study for two reasons. First, few specific differences among sites have been reported, nor have estimates of the error incurred by ignoring the site dependence been given. Studies by Ciarelli et al. (1991) and Goulet et al. (1994) indicated that regression coefficients differed across anatomic site but provided no post-hoc statistical comparisons. Keller (1994) reported that modulus–density regressions were different for the vertebra than for the proximal femur, but the femoral group included cortical bone specimens. Differences between these two sites were also reported by Majumdar et al. (1998); however, the regression model in this study assumed a linear relationship between modulus and bone mineral density as opposed to a power law model recommended by Currey (1986) for both statistical and mechanistic reasons. Second, traditional compression testing and specimen machining protocols used in many of these studies introduce end artifact (Keaveny et al., 1997; Odgaard and Linde, 1991) and specimen misalignment (Turner and Cowin, 1988) errors in the data that can confound inter-site comparisons.

Both types of errors have been eliminated in recent studies by computing elastic constants through high-resolution finite element analyses (Kabel et al., 1999; Ulrich et al., 1999; Yang et al., 1999). Using specimens pooled from multiple anatomic sites, the performance of theoretical relationships has been assessed between the orthotropic elastic constants, density, and tissue modulus (Yang et al., 1999) and between these nine constants, density, fabric eigenvalues, and tissue modulus (Kabel et al., 1999). Both theories, which were developed by Cowin and Yang (1997) and Cowin (1985), respectively, treat the tissue modulus as isotropic and uniform throughout each specimen, and both will predict site-specific apparent modulus–density relationships if the tissue moduli are site-specific. In addition, the earlier theory allows for site-specificity through the relative magnitudes of the fabric eigenvalues. Two outstanding issues regarding these theoretical relationships are whether, when combined with experimental data for apparent moduli, they predict tissue moduli that are consistent with direct experimental measurements, and whether these predicted tissue moduli are site-specific.

The overall goal of this study was to characterize the site-dependence of the relationship between elastic modulus and apparent density for human trabecular bone. To allow for inter-site comparisons, specimen machining and testing protocols were used that loaded the specimens along the principal trabecular orientation (“on-axis”) and that minimized end artifacts. Specifically, our objectives were to: (1) compare the on-axis modulus–density regressions for human trabecular bone from the vertebra, proximal tibia, femoral greater trochanter, and femoral neck; (2) determine the effects of using a pooled rather than site-specific regression on the precision of the modulus predictions; and (3) use the data together with existing theoretical relationships and high-resolution finite element analyses to investigate the roles of tissue modulus and architecture in the site-specificity of apparent modulus–density relationships.

Section snippets

Methods

All human tissue used in this experiment was obtained from 61 donors with no medical history of metabolic bone disease or cancer and showed no radiographic evidence of damage or bone pathologies (Table 1). Tissue was frozen within 24 h post-mortem. Previously published protocols were used to obtain 8 mm diameter on-axis specimens from the human vertebra, proximal tibia, and proximal femur (Kopperdahl and Keaveny, 1998; Morgan and Keaveny, 2001). For each anatomic site, specimens were randomly

Results

Relationships between elastic modulus and apparent density did depend on anatomic site (Table 2, Fig. 1). No differences in power law exponents were found among sites (p>0.08, β=0.73), but significant differences were found among the leading coefficients. As a result, at a given apparent density, specimens from the proximal tibia had higher moduli than those from the vertebra (p=0.01) and femoral neck (p=0.002), and specimens from the greater trochanter had higher moduli than those from the

Discussion

The goal of this study was to investigate the site-dependence of on-axis modulus–density relationships for human trabecular bone. Although regressions of modulus against density measures have been a subject of research efforts for over two decades, whether these relationships depend on site has remained unresolved. This is despite the extensive use of such regressions for analyses of bone adaptation (Carter et al., 1987), fracture risk (Keyak and Rossi, 2000; Silva et al., 1998), and

Acknowledgements

Funding was provided by NIH Grant No. AR43784 and the National Science Foundation Graduate Fellowship Program. Tissue was obtained from the Anatomic Gift Foundation and the National Disease Research Interchange. The authors would like to thank Andrew Burghardt Jacob Pollock, David Kopperdahl, Yves Arramon, and David Barone for their technical assistance, and Maureen Lahiff for her advice on the statistical analyses. Computing time was made available through NPACI UCB266.

References (45)

  • T.M. Keaveny et al.

    Trabecular bone exhibits fully linear elastic behavior and yields at low strains

    Journal of Biomechanics

    (1994)
  • T.S. Keller

    Predicting the compressive mechanical behavior of bone

    Journal of Biomechanics

    (1994)
  • J.H. Keyak et al.

    Prediction of femoral fracture load using finite element modelsAn examination of stress- and strain-based failure theories

    Journal of Biomechanics

    (2000)
  • D.L. Kopperdahl et al.

    Yield strain behavior of trabecular bone

    Journal of Biomechanics

    (1998)
  • S. Majumdar et al.

    High-resolution magnetic resonance imagingthree-dimensional trabecular bone architecture and biomechanical properties

    Bone

    (1998)
  • E.F. Morgan et al.

    Dependence of yield strain of human trabecular bone on anatomic site

    Journal of Biomechanics

    (2001)
  • A. Odgaard

    Three-dimensional methods for quantification of cancellous bone architecture

    Bone

    (1997)
  • A. Odgaard et al.

    The underestimation of Young's modulus in compressive testing of cancellous bone specimens

    Journal of Biomechanics

    (1991)
  • J.Y. Rho et al.

    Young's modulus of trabecular and cortical bone materialUltrasonic and microtensile measurements

    Journal of Biomechanics

    (1993)
  • J.C. Rice et al.

    On the dependence of the elasticity and strength of cancellous bone on apparent density

    Journal of Biomechanics

    (1988)
  • C.H. Turner et al.

    The elastic properties of trabecular and cortical bone tissues are similarresults from two microscopic measurement techniques

    Journal of Biomechanics

    (1999)
  • D. Ulrich et al.

    Finite element analysis of trabecular bone structurea comparison of image-based meshing techniques

    Journal of Biomechanics

    (1998)
  • Cited by (925)

    • Quantification of long-term nonlinear stress relaxation of bovine trabecular bone

      2024, Journal of the Mechanical Behavior of Biomedical Materials
    View all citing articles on Scopus
    View full text