Elsevier

Journal of Biomechanics

Volume 44, Issue 8, 17 May 2011, Pages 1566-1572
Journal of Biomechanics

Fast and accurate specimen-specific simulation of trabecular bone elastic modulus using novel beam–shell finite element models

https://doi.org/10.1016/j.jbiomech.2011.02.082Get rights and content

Abstract

Elastic modulus and strength of trabecular bone are negatively affected by osteoporosis and other metabolic bone diseases. Micro-computed tomography-based beam models have been presented as a fast and accurate way to determine bone competence. However, these models are not accurate for trabecular bone specimens with a high number of plate-like trabeculae. Therefore, the aim of this study was to improve this promising methodology by representing plate-like trabeculae in a way that better reflects their mechanical behavior. Using an optimized skeletonization and meshing algorithm, voxel-based models of trabecular bone samples were simplified into a complex structure of rods and plates. Rod-like and plate-like trabeculae were modeled as beam and shell elements, respectively, using local histomorphometric characteristics. To validate our model, apparent elastic modulus was determined from simulated uniaxial elastic compression of 257 cubic samples of trabecular bone (4 mm×4 mm×4 mm; 30 μm voxel size; BIOMED I project) in three orthogonal directions using the beam–shell models and using large-scale voxel models that served as the gold standard. Excellent agreement (R2=0.97) was found between the two, with an average CPU-time reduction factor of 49 for the beam–shell models. In contrast to earlier skeleton-based beam models, the novel beam–shell models predicted elastic modulus values equally well for structures from different skeletal sites. It allows performing detailed parametric analyses that cover the entire spectrum of trabecular bone microstructures.

Introduction

Elastic modulus and strength of bones are negatively affected by a number of metabolic bone diseases, in particular osteoporosis. Evidence from prospective studies, using markers of bone formation and bone resorption, indicates that an excessive rate of bone remodeling is one of the major determinants of age-related bone loss and osteoporosis (Bauer et al., 1999, Cummings et al., 1993, Garnero et al., 1996). Excessive bone remodeling leads to changes in microarchitecture with accumulation of microdamage and some degree of hypomineralization (Mori et al., 1997), resulting in bone fragility.

Experimental mechanical tests are considered the gold standard to determine bone competence and have been performed extensively to quantify the effects of osteoporosis and of potential treatments. But these tests have practical limitations and a high sensitivity to measurement errors (Keaveny et al., 1997).

Elastic moduli can be derived accurately from micro-computed tomography (μCT)-based voxel-FE (μFE) models that mimic the microstructure of the bone in detail by representing every voxel as a hexahedral element (Van Rietbergen et al., 1995). These models are necessarily large to capture the complex architecture and thus are computationally demanding, especially in nonlinear analysis. Furthermore, even though these models incorporate the intricate trabecular architecture, the structure–mechanics relationships remain unclear. One reason is that these models are not readily manipulated to independently test the influence of different local structural properties.

Beam FE models have been proposed as an alternative to μFE analyses (Pothuaud et al., 2004, Stauber et al., 2004, van Lenthe et al., 2006). Typically, these models represent trabecular bone as a three-dimensional (3D) network of beams. Beam properties are derived from local analyses of the individual trabeculae. These models have the advantage that they require far less CPU-time than μFE models, because the number of elements is greatly reduced. Furthermore, beam properties are easily manipulated to parametrically asses the influence of specific trabecular features.

Beam FE models have been shown to accurately predict apparent elastic modulus of human trabecular bone samples (van Lenthe et al., 2006) as well as failure for an aluminum foam (Stauber et al., 2004). We hypothesized that for plate-like structures the use of beam-models is insufficient, and that a better representation of the plate-like trabeculae was needed in order to capture their specific nature. Specifically, in this paper a new beam–shell model is presented and its superiority over beam-only models is demonstrated.

Section snippets

Materials and methods

Through skeletonization, classification and meshing, voxel-based models of cubic trabecular bone samples were simplified to a construction of beam and shell elements (Fig. 1). Based on these structural skeletons FE meshes were constructed. All procedures were implemented in MATLAB (The Mathworks Inc., Natick, MA, USA).

Results

The trabecular bone samples covered a wide range of bone volume fractions (BV/TV), ranging from 4% to 55% (Table 1); the samples contained 2% to 98% plate-like trabecular volume. All the femoral samples (FRA) had more than 50% plate-like trabecular volume, while all lumbar spine samples (L4A) had less than 50% plate-like trabecular volume. Calcaneus (CAB) and iliac crest (ICF) samples covered a wider range of plate-like trabecular volume (Table 1). We found a strong linear correlation between

Discussion

In this study we proposed a skeleton-based FE-model for parametric and fast analysis of trabecular bone structures. Rod- and plate-like trabeculae were modeled as beam and shell elements. The results show that at the level of global elastic characteristics this model is equivalent to μFE models.

In comparison to previously developed beam models, the strength of the beam–shell model is the accurate representation of plates. Earlier beam-models did not implement meshing of plate-like trabeculae or

Conflict of interest statement

The authors have no conflict of interest concerning this work.

References (24)

  • J.M. Bland et al.

    Statistical-methods for assessing agreement between 2 methods of clinical measurement

    Lancet

    (1986)
  • J. Dequeker

    Assessment of quality of bone in osteoporosis—BIOMED I: fundamental study of relevant bone

    Clin. Rheumatol.

    (1994)
  • Cited by (18)

    • Computational analysis of primary implant stability in trabecular bone

      2015, Journal of Biomechanics
      Citation Excerpt :

      Beam and shell elements provide an elegant way to represent the trabecular struts and plates. It has been shown that such models can accurately represent the apparent mechanical behavior of bone (Van Lenthe et al., 2006; Vanderoost et al., 2011). Such models would have the potential to analyze the mechanical behavior of bone–implant systems as well.

    • Trabecular plates and rods determine elastic modulus and yield strength of human trabecular bone

      2015, Bone
      Citation Excerpt :

      In addition, these PR μFE models provided 83-fold reduction in model size and 324-fold reduction in nonlinear FEA computational time to determine yield strength of trabecular bone. In Vanderoost et al., they achieved a 7-fold reduction in the number of elements, and a 33-fold reduction in the central processing unit (CPU) time in linear analyses for estimating elastic modulus [35]. One of their pilot nonlinear analyses indicated a 45-fold reduction in CPU time.

    • A combined experimental and computational study of mechanical properties after balloon kyphoplasty

      2021, Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine
    View all citing articles on Scopus
    View full text