Fast and accurate specimen-specific simulation of trabecular bone elastic modulus using novel beam–shell finite element models
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)
- et al.
Bone density at various sites for prediction of hip fractures. The Study of Osteoporotic Fractures Research Group
Lancet
(1993) - et al.
Effect of trabecular curvature on the stiffness of trabecular bone
J. Biomech.
(2005) - et al.
Trabecular bone volume and microdamage accumulation in the femoral heads of women with and without femoral neck fractures
Bone
(1997) - et al.
High-resolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone
J. Biomech.
(2000) - et al.
A new shape preserving parallel thinning algorithm for 3D digital images
Pattern Recognition
(1997) - et al.
Volumetric spatial decomposition of trabecular bone into rods and plates—a new method for local bone morphometry
Bone
(2006) - et al.
The effects of side-artifacts on the elastic modulus of trabecular bone
J. Biomech.
(2006) - et al.
Specimen-specific beam models for fast and accurate prediction of human trabecular bone mechanical properties
Bone
(2006) - et al.
A new method to determine trabecular bone elastic properties and loading using micromechanical finite-element models
J. Biomech.
(1995) - et al.
Biochemical markers of bone turnover and prediction of hip bone loss in older women: the study of osteoporotic fractures
J. Bone Miner. Res.
(1999)
Statistical-methods for assessing agreement between 2 methods of clinical measurement
Lancet
Assessment of quality of bone in osteoporosis—BIOMED I: fundamental study of relevant bone
Clin. Rheumatol.
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