Original PaperNoninvasive prediction of vertebral body compressive strength using nonlinear finite element method and an image based technique
Introduction
Clinical reports suggest that Vertebral body Compressive Fracture (VCF) represents distinctive characteristics of osteoporosis with an annual incidence of approximately 700 000 patients in the US alone [1]. The important role of vertebral fractures in the reduction of life expectancy has motivated researchers to look for noninvasive methods for predicting the strength of vertebrae [2], [3], [4].
A number of common radiographic methods, such as the plain radiography and computed tomography or densitometric methods like the Dual X-ray Absorbtiometry (DXA) and quantitative computed tomography (QCT), have been used for predicting the VCF [2], [3], [4], [5]. Studies based on these methods have generally shown modest correlations (R2 = 0.40–0.69) from in vitro measurements of vertebral strength [2], [3], [4], [5].
Other researchers [6], [7], [8] have claimed that using apparent morphological properties, such as trabecular thickness (Tb.Th) and trabecular separation (Tb.Sp) and BMD will be useful for evaluating bone apparent mechanical properties by means of patient-specific biomechanical models. In this regard, Diament et al. [6] developed computational orthogonal lattice models of trabecular micro architecture and analyzed the lattices with different trabecular characteristics using the FE method. They determined apparent elastic modulus of any trabecular lattice depending on three trabecular characteristics (Tb.Th, Tb.Sp and BMD) under uniaxial compression loading using the parametric FE method and validated their model in vitro against experimental tests. Although this kind of modeling can improve noninvasive prediction of vertebral strength and stiffness, it is not possible to measure the morphological parameters of cancellous bone at the tissue level due to the low spatial resolution available with available clinical imaging systems such as current QCT scanners. To overcome this limitation, other researchers have recommended the use of specimen-specific models with macroscopic characteristics of bone available with current QCT scanners [9], [10], [11], [12], [13], [14], [15].
Recent studies, based on a combination of the QCT and FE method, have led to strong correlation (R2 = 0.78–0.87) between predicted strengths and experimental results [9], [10], [11], [12]. This QCT voxel-based FE method is an intrinsically specimen-specific method in which subtle geometric and densitometric differences among patients are considered.
Crawford et al. [10] showed a strong correlation (R2 = 0.86) between the experimental uniaxial compression test and predicted vertebral strength using a QCT voxel-based linear FE method. However, because of using a linear elastic model for defining the material properties of trabecular bone and ignoring the BMDQCT dependence on trabecular yield strain, this method could not reflect the nonlinear behavior of vertebral trabecular bone. Hence, the predicted values of vertebral strength resulted from this study showed an underestimation about 25% compared to that of the experimental values.
Imai et al. [13] tried to estimate vertebral body compressive strength using a nonlinear FE method based on computed tomography. However, they considered a constant thickness of cortical shell and used a tetrahedral meshing and reported an excellent correlation between predicted strengths and experimental results (R2 = 0.93). Although using tetrahedral meshing could improve complex geometry modeling, contrary to the QCT voxel-based models, this kind of mapping is not perfect since the QCT voxels have brick shape geometry. Moreover, even though considering the cortical shell in the vertebral body model leads to better results for strength measurements, it would cast a shadow on the specimen-specificity of the method since the material property and geometrical data of the cortical shell are not obtained from QCT scans. The low spatial resolution of QCT systems do not permit to model the cortical shell explicitly. On the other hand, modeling the cortical shell with a constant thickness, as assumed by Imai et al. [13], does not match with the specifications of specimen-specific models and cannot reflect the real mechanical behavior of specimens. Finally, they predicted the compressive strength by an overestimation of 10% being contrary to other reported results [9], [10], [11], [12], [14], [15].
In the method used by Buckley et al. [12], [14] for the prediction of vertebral body compressive load, no effort was made to model the cortical shell because the clinical resolution of their QCT scans was not able to produce a discrete image of the cortical shell. Furthermore, by using a linearly elastic–perfectly plastic material model for trabecular bone, the ultimate properties of trabecular bone were ignored. However, this type of modeling showed a correlation (R2 = 0.80) less than that of Imai et al. [13]. In addition, their results showed an underestimation about 12% for the predicted values of vertebral compressive load. They also demonstrated that using a nonlinear QCT-based FE method can improve the prediction of vertebral strength under bending loads more than other methods such as the BMD or MOS [14]. However, their results indicated a weak correlation between the predicted FE strength and experimental strength under bending loads (R2 ∼ 0.40) [14]. Finally, they concluded that the poor predictive capacities of QCT-based FE measures may be due to its inability to account for inelastic failure mechanisms that occur in the vertebral body for anterior bending. Although using both of the FE methods (linear and nonlinear) have led to strong correlations (R2 = 0.78–0.93) between predicted strengths and experimental results [9], [10], [11], [12], [13], [14], [15], due to the ignorance of BMDQCT dependence of trabecular yield strain in linear models and equating trabecular yield stress and ultimate stress in nonlinear models, a more comprehensive material model seems to be needed for the prediction of vertebral compressive strength with less errors in estimations.
In this study, we have tried to introduce a new material model, linearly elastic–linearly plastic model, in which inelastic failure behaviour of trabecular bone can be considered better than other material models. We believe that applying a linearly elastic–linearly plastic material model in QCT voxel-based nonlinear FE analyses could improve the prediction of human vertebral body compressive strength because it seems to be more compatible with the real inelastic material behavior of vertebral trabecular bones. It could also decrease the level of error occurred in the estimation of the predicted vertebral body compressive load without casting a shadow on the specimen-specificity of the method.
The purpose of this study was to apply the proposed linearly elastic–linearly plastic material model along with an appropriate mechanical test to a number of human vertebral specimens to ensure the efficiency of this model for the prediction of vertebral compressive strength.
As time saving and computational simplicity are also important factors in clinical situations, we have also introduced a simple image based parameter for the prediction of vertebral compressive strength.
Section snippets
Materials and methods
This section describes the specifications of the specimens used and the experimental and numerical methods implemented.
Results
Fig. 12 shows a sample load–displacement diagram derived from the nonlinear finite element analyses carried out for 0.25 × 0.25 × 1, 0.5 × 0.5 × 1 and 1 × 1 × 1 mm3 voxel based element size meshes. From this diagram and similar diagrams obtained for other samples in our study two main issues were inferred:
- 1)
Although regrouping the original elements (0.25 × 0.25 × 0.25 mm3) to larger sizes for reducing the computational cost leads to coarser models and surface serration, it does not seem to have any significant
Discussion
A recent study [10] showed that QCT voxel-based linear FE models predict in vitro vertebral body compressive strength better than QCT-derived parameters (BMDQCT and BMDQCTAmin) due to subtle geometric and densitometric inhomogenities embedded in such models. However, due to intrinsic characteristics of the linear FE method, the vertebral body compressive strength cannot be calculated explicitly and is calculated implicitly based on the FE model stiffness [9], [10]. Hence, because of this
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