Apparent Young's modulus of human radius using inverse finite-element method

https://doi.org/10.1016/j.jbiomech.2006.09.018Get rights and content

Abstract

The ability to assess the elastic and failure properties of cortical bone at the radial diaphysis has a clinical importance. A new generation of quantitative ultrasound (QUS) devices and peripheral quantitative computed tomography (p-QCT) has been developed to assess non-invasively bone material and structural properties at the distal radius. This anatomical site is characterized by a thin cortical thickness that complicates traditional mechanical testing methods on specimens. Until now, mechanical properties of cortical bone at distal radius (e.g., elastic modulus, yield stress and strain) remain rarely studied probably due to experimental difficulties. The present study introduces an inverse finite-element method strategy to measure the elastic modulus and yield properties of human cortical specimens of the radial diaphysis. Twenty millimeter-thick portions of diaphysis were cut from 40 human radii (ages 45–90) for biomechanical test. Subsequently the same portion was modeled in order to obtain a specimen-specific three dimensional finite-element model (3D-FEM). Longitudinal elastic constants at the apparent level and stress characterizations were performed by coupling mechanical parameters with isotropic linear-elastic simulations. The results indicated that the mean apparent Young's modulus for radial cortical bone was 16 GPa (SD 1.8) and the yield stress was 153 MPa (SD 33). Breaking load was 12,946 N (SD 3644), cortical thickness 2.9 mm (SD 0.6), structural effective strain at the yield (εy=0.0097) and failure (εu=0.0154) load were also calculated. The 3D-FEM strategy described here may help to investigate bone mechanical properties when some difficulties arise from machining mechanical sample.

Introduction

Osteoporotic fractures of the radius represent an early predictive sign of future fractures at other sites such as the proximal femur or the spine and the assessment of bone radial properties could be a promising diagnostic evaluation to predict fracture risk and to initiate or evaluate therapeutic actions (Cuddihy et al., 1999, Cuddihy et al., 2002; Klotzbuecher et al., 2000; Haentjens et al., 2003, Haentjens et al., 2004). In the last two decades several studies on ultradistal radius and mid-shaft have been performed using biomechanical tests, but their efforts were limited to the assessment of the ultimate load. An extensive review has been reported (Eckstein et al., 2004).

With the advent of a new generation of quantitative ultrasound (QUS) devices measuring the speed of sound at the distal radius (Barkmann et al., 2000; Muller et al., 2003; Bossy et al., 2004a, Bossy et al., 2004b; Hudelmaier et al., 2004), a parameter sensitive to porosity and materials properties (Bossy et al., 2004b; Raum et al., 2005), it became necessary to evaluate not only the ultimate load at the radial diaphysis but also the elastic properties at the structural (apparent) level such as the Young's modulus in order to gain a better insight into the relation between QUS measurements and elastic and yield properties.

To the authors’ knowledge, no published data related to apparent Young's modulus of cortical bone at the radius were found in the literature. This is probably due to the difficulty of cutting test samples from radial diaphysis for mechanical testing.

Apparent Young's modulus assessed by biomechanical testing at the midshaft of long bone, mainly the femur, with a large number of specimens (regularly shaped reduced section or cubic samples) varies between 8 and 22.8 GPa (Table 1) in longitudinal direction (Reilly et al., 1974; Reilly and Burstein, 1975; Keller, 1994; Kaneko et al., 2003; Bayraktar et al., 2004; Dong and Guo, 2004). The apparent Young's modulus of a macroscopic sample is evaluated based on both the intrinsic anisotropic elastic properties of the bone extracellular matrix and the porosity due to lacunae, vascular canals and resorption cavities (Hengsberger et al., 2003).

Techniques relying on Scanning Acoustic Microscopy (SAM) and nanoindentation accurately characterize the intrinsic mechanical properties of bone tissue down to the lamellar level and provide values in the range of 13 GPa in the transverse direction, 20–30 GPa along the bone axis from acoustic techniques (Turner et al., 1999; Raum et al., 2006) and 16–23 GPa from nanoindentation (Turner et al., 1999; Zysset et al., 1999). These techniques could be helpful for an accurate characterization of bone tissue properties. This remain true when cortical thickness is less than that required for testing samples machining, but the techniques do not take into account porosity. Moreover, they are extremely sensitive to the region where the test is carried out (Zysset et al., 1999), due to their high resolution from a few μm for SAM (Raum et al., 2004) down to 1 μm for latest indenter (Turner et al., 1999). It may be possible to infer the apparent Young's modulus from such measurements performed at the microscopic level. However, the main limitation of this approach lies in the homogenization assumption required to evaluate the structural properties of the material. This assumption may not be valid for orthotropic and non-homogeneous material like bone. In a recent validation study (Hengsberger et al., 2003), it was shown how the indentation modulus could be related to apparent Young's modulus but the complexity of these techniques and the equipment cost limit their use.

The aim of this study was to estimate the apparent Young's modulus and tissue equivalent stresses using an inverse numerical technique relying on specimen-specific 3D finite-element models (FEMs) and biomechanical experiments.

Section snippets

Materials and methods

Forty excised human radii with soft tissue removed were used in this study. The sample population includes 15 female and 24 male donors (mean age 73±10 years old; range, 45–90 years old). Sex and age were not known for one donor. The samples were removed from fresh cadavers and were kept frozen at a temperature of −20 °C before use. Thawing was requested before computed tomography (CT) measurements but was kept to a minimum. Ethics approval for collection samples was granted at Human Ethics

Results

Mean height (±SD) of the 39 specimens was found to be 19.95±0.21 mm. Cortical thickness was 2.9±0.6 mm and the intra-specimen SD due to cortical thickness calculation was found to range between 0.1 and 0.3 mm.

Regarding the biomechanical experiments, the breaking load values ranged between 6640 and 20,215 N with a mean (±SD) of 12,797±3526 N. The mean (±SD) of the εu, the εy and ε(u−y) were, respectively: εu=0.0154±0.0017, ranging from 0.0117 to 0.0181; εy=0.0097±0.0018, ranging from 0.0058 to

Discussion

The originality of this study was two-fold. First, both elastic and yield properties were characterized at the apparent level for cortical bone at the radial diaphysis. Second, whole portions of cortical bone structure were characterized in terms of the apparent Young's modulus and stresses by combining data from biomechanical experiments with 3D specimen-specific inverse finite-element analysis. Specimen-specific geometry is needed because the specimen shape cannot be approximated by a

Acknowledgements

The authors wish to thank Dr. V. Bousson for the CT-scan acquisitions and Dr. C. Bergot, E. Jolivet, Y. Lafon, and S. Persohn for their respective contributions. The authors wish to acknowledge Pr. P.K. Zysset for critically reading the manuscript and making several useful remarks.

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