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. 2010 Aug 31;218(1):40–46. doi: 10.1111/j.1469-7580.2010.01289.x

An experimentally validated micromechanical model of a rat vertebra under compressive loading

Naomi Tsafnat 1, Stephen Wroe 1
PMCID: PMC3039779  PMID: 20819113

Abstract

In recent years, finite element analysis (FEA) has been increasingly applied to examine and predict the mechanical behaviour of craniofacial and other bony structures. Traditional methods used to determine material properties and validate finite element models (FEMs) have met with variable success, and can be time-consuming. An implicit assumption underlying many FE studies is that relatively high localized stress/strain magnitudes identified in FEMs are likely to predict material failure. Here we present a new approach that may offer some advantages over previous approaches. Recently developed technology now allows us to both image and conduct mechanical tests on samples in situ using a materials testing stage (MTS) fitted inside the microCT scanner. Thus, micro-finite element models can be created and validated using both quantitative and qualitative means. In this study, a rat vertebra was tested under compressive loading until failure using an MTS. MicroCT imaging of the vertebra before mechanical testing was used to create a high resolution finite element model of the vertebra. Load-displacement data recorded during the test were used to calculate the effective Young's modulus of the bone (found to be 128 MPa). The microCT image of the compressed vertebra was used to assess the predictive qualities of the FE model. The model showed the highest stress concentrations in the areas that failed during the test. Clearly, our analyses do not directly address biomechanics of the craniofacial region; however, the methodology adopted here could easily be applied to examine the properties and behaviour of specific craniofacial structures, or whole craniofacial regions of small vertebrates. Experimentally validated micro-FE analyses are a powerful method in the study of materials with complex microstructures such as bone.

Keywords: compression test, finite element analysis, microstructure, X-ray micro-computed tomography

Introduction

Following advances in computer and imaging technology, FEA has been increasingly applied as a predictor of mechanical behaviour in the vertebrate cranium. Commonly used in a comparative context (Rayfield et al. 2001; Dumont et al. 2005; McHenry et al. 2007a; Rayfield, 2007; Wroe et al. 2007a,b, 2008; Slater & van Valkenburgh, 2009; Tseng, 2009), it is often implicitly or explicitly assumed that high localized stress/strain indicates susceptibility to failure. A number of FE-based studies of crania include validation (Strait et al. 2005; Kupczik et al. 2007); however, we know of none that has clearly identified material failure. In such studies, the validation and the determination of material properties have been conducted independently. Here we report the use of microCT fitted with a materials testing stage which allows both imaging and mechanical testing in situ, and the subsequent experimental validation of an FEM. In this instance the specimen of interest has been a rat vertebra subject to compressive loading, but the approach could be applied to whole crania of small vertebrates, or select regions of larger crania. Our broader aim has been to examine whether this approach can predict the material failure of bone.

Micro-computed tomography (microCT) is a non-destructive method of three-dimensional imaging. Voxel resolutions of up to about 2 μm are achievable. The voxel grey-scale values correspond to the local material density. MicroCT imaging has been used extensively in the biomedical field (Ritman, 2004; Ananda et al. 2006; Uzun et al. 2007; Schambach et al. 2009), predominantly in imaging and characterization of bone (Judex et al. 2003; Jones et al. 2004; Gabet et al. 2006; Muller, 2009).

The digital nature of microCT datasets makes them ideal for conversion into numerical models, most commonly using the finite element (FE) method. The detailed three-dimensional microstructure of the material is captured, and different materials can be modelled by segmentation based on grey-value. Micro-finite element analysis (microFEA) has been applied in the study of a variety of materials, and especially in the area of bone biomechanics (Keyak et al. 1990; van Rietbergen et al. 1995; Viceconti et al. 1998; Garboczi et al. 1999; Keaveny et al. 2001; van Rietbergen, 2001; Taddei et al. 2004; Yosibash et al. 2007; Chen et al. 2009; Rhee et al. 2009). As computers become more powerful, high resolution models of increasingly large samples can now be created.

MicroFEA allows us to elucidate the relationship between a microstructure of a material and the effective properties of its constituent materials, and its apparent, or bulk, properties, encompassing both the material and its geometrical structure (van Lenthe & Muller, 2006). Experimental results of mechanical testing coupled with microFEA allow us to calculate, for example, the structure's effective Young's modulus Eeff using the inverse method (Arns et al. 2002), which is a good predictor of bone strength (Yeni & Fyhrie, 2001). As is the case with any numerical modelling technique, experimental validation is a key factor in determining the robustness of the model. Finite element analyses validated by ex-vivo experiments include the study of teeth (Magne et al. 1999; Palamara et al. 2000, 2002; Magne, 2007; Barak et al. 2009) and bone (Shefelbine et al. 2005; Yosibash et al. 2007; Kluess et al. 2009; Trabelsi et al. 2009).

Materials and methods

The fourth lumbar vertebra of a young adult male laboratory rat (Rattus rattus) was removed by dissection. The vertebra (length approximately 10.7 mm) was frozen at −20 °C for 4 weeks and thawed before use. The specimen was scanned using a SkyScan 1172 high resolution desktop microCT (SkyScan, Aartselaar, Belgium) fitted with an in situ material testing stage. Scanning was performed with a source voltage of 100 kV, a current of 100 μA, at a voxel resolution of 15 μm. The vertebra was rotated around 180° at angular increments of 0.22°. Two scans were performed: before and after compression testing. Mechanical testing was performed using the MTS while the specimen remained in the microCT chamber between the two scans. The MTS consists of a thin plexiglass tube through which the specimen can be scanned, and a 440 N load cell (see Fig. 1). The sample was centred in the test stage, and no adhesive material was used to hold it in place (the sample was in effect restrained only in the axial direction). The vertebra was placed under continuous compressive load at a displacement rate of 0.001 mm s−1. The total displacement reached was 1.13 mm, at a loading of 92.2 N. Load and displacement data were continuously recorded. The load was stopped at this point as obvious failure occurred, which is why there is no load plateau at the maximal load. At the end of the mechanical test the compressed vertebra was scanned once again, while being kept under load. The scanned region was chosen to include only the bone and not the metal clamps. This was done to minimize the distorting effects caused by scanning soft material located near the high density metal. These effects were not pronounced as the materials were located on top of each other rather than side-by-side; nevertheless, some distortion at the interface is unavoidable.

Fig. 1.

Fig. 1

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Schematic diagram of the MTS (reproduced from the skyscan MTS manual version 1.1). The specimen is placed inside the thin-walled plexiglass chamber. During compression testing, the bottom clamp moves up, while the top clamp remains static. Experimental data are recorded and displayed on-screen during testing.

While some degree of stress relaxation is unavoidable during the second scanning, the fact that the MTS is fitted within the microCT means that movement due to repositioning of the sample before and after testing is avoided. Positioning of the sample in the microCT is therefore effectively static, allowing for accurate comparison of the ‘before’ and ‘after’ images.

Two sets of microCT radiograph projections were reconstructed using skyscan software (NRecon version 1.4.4) to obtain axial slice image datasets. Three-dimensional rendering was performed using vgstudiomax (version 1.2, Volume Graphics, Heidelberg, Germany), as well as conversion of the ‘before’ image dataset into a DICOM dataset. Prior to conversion into DICOM format, the image size was reduced by voxel binning to 75-micron resolution. The DICOM images were imported into mimics (version 11.02, Materialise, Leuven, Belgium) to create a three-dimensional STL surface mesh. The surface mesh was exported to strand7 finite element software (version 2.3, Sydney, Australia) and used to generate a solid mesh of 482,749 four-node tetrahedral elements using previously established protocols (McHenry et al. 2007; Wroe et al. 2007a,b;). Boundary conditions were applied to simulate the compression testing that was performed: the nodes at the top of the vertebra were held in place (zero degrees of freedom) while a quasi-static displacement of 1.13 mm (a total strain of 10.6%) was applied to the nodes at the bottom of the vertebra. The slow compression rate of the MTS ensures a quasi-static load, minimizing any dynamic effects; this was reproduced in the FE model.

The inverse method (Arns et al. 2002) was used to determine the effective modulus from the quantitative MTS measurements of load versus displacement. To determine the effective Young's modulus of the bone, several simulations were run in the linear elastic region using arbitrary values of E. The sum of the reaction forces in the vertical direction vary linearly with E (by definition, in the elastic region), and so were used to interpolate the value of E that corresponds to the force applied during mechanical testing for the given displacement.

The calculated Eeff of 128.0 MPa was then used to define the material property of the bone in a simulation of compression testing at the maximum displacement of 1.13 mm. Results of this simulation were compared with both the quantitative data measured during testing and with the qualitative assessment of bending and failure in the vertebra from the 3D rendering of the ‘after’ microCT data.

Results

During the compression testing the vertebra was monitored on screen using the microCT built-in fluoroscope. Three instances of failure were noted toward the end of the test, at loads of approximately 62 N, 72 N and 80 N. As seen in the ‘before’ and ‘after’ 3D reconstructions shown in Fig. 2, obvious catastrophic material failure occurred in the left transverse process shown in the front of the image; this region bent considerably, then cracked. We consider it probable that the three failures occurred incrementally in this one location, although it is possible that failure occurred elsewhere but was not clearly evident. Force and displacement data were continually recorded during the test as shown in Fig. 3. The apparent stiffness of the vertebra, calculated from the slope of the graph, is 81.6 N mm−1. Total work done (the area under the load-displacement curve) is 52 mJ, calculated by assuming that the curve is a straight line, up to the final load of 92 N.

Fig. 2.

Fig. 2

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Three-dimensional reconstructions of microCT data for rat vertebra before (left) and after (right) compression testing. Scale bar: 3 mm. Note significant elastic bending and twisting, and a visible fracture in the left transverse process (arrow). Orientation: bottom = anterior.

Fig. 3.

Fig. 3

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Load and displacement data recorded by the MTS. Experimental data are shown by the black tick marks and line; the slope of the linear line-fit in the lower range of the data (pink line) was used to estimate the apparent stiffness (y = 81.6×, R2 = 0.9759); green, blue and orange diamonds mark the first, second and third yield points, respectively.

Applying the inverse method gave an effective modulus of Eeff = 128.0 MPa. This was validated by running an FE simulation of the experimental test using the calculated Eeff. The sum of the reaction forces resulting from the simulation was equal to the force that was applied during mechanical testing, and the maximum displacement was the same as in the test. In addition to this quantitative validation, qualitative analysis of the FE results reveals that the model was able to predict the location of failure in the vertebra. The shear stress contour plots show bands of high stress at the same areas where failure occurred, as can be seen in Fig. 4. The YZ shear plot is shown, as this is the plane in which the left transverse process failed. The maximum values of von Mises stress at the area that failed are in the order of 60 MPa (Fig. 5). Areas of high stress values at the top and bottom areas are most likely artefacts of the imposed boundary conditions and should not be interpreted as actual stress values there. Figure 6 shows the strain distribution in the vertebra. Tensile strains occur at the saddle points of the vertebra, while compressive strains occur at the left transverse process which failed.

Fig. 4.

Fig. 4

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Contour plot of the shear stresses in the YZ plane. Note the high stress concentrations at the left transverse process where failure occurred due to bending in the YZ plane (see Fig. 2), shown here at left. Orientation: bottom = anterior.

Fig. 5.

Fig. 5

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Contour plot of von Mises stress distribution in the vertebra. Note the high stresses present at the left transverse process, where failure occurred. The high stresses seen at the top and bottom boundaries are most likely due to boundary effect artefacts and should not be assumed to be actual values. Orientation: bottom = anterior.

Fig. 6.

Fig. 6

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Contour plot of mean brick strain distribution. Tensile strains occur at the saddle points of the vertebra (shown in red), while compressive strains (shown in blue) are present at the left transverse process which failed. The strain values seen at the top and bottom boundaries are most likely due to boundary effect artefacts and should not be assumed to be actual values. Orientation: bottom = anterior.

By using the values of Eeff coupled with the measured strain and force, an effective cross-sectional area Aeff of the vertebra can be calculated (Eq. 1), and was found to be 6.8 mm−2.

graphic file with name joa0218-0040-m1.jpg (1)

While it is impossible to estimate a cross-sectional area for the actual geometry of the vertebra, its dimensions are on the order of 12–14 mm per side, roughly twice the value of the effective area. By using the effective cross-sectional area, we can then estimate the stress at the first incidence of failure, which was at a load of 62 N. The stress at that point was σ = 62 N/6.8 mm−2, or 9.1 MPa.

Discussion and conclusions

We present a microFEA model of a rat vertebra under compressive loading. The model has been validated both quantitatively and qualitatively using results of mechanical testing performed using a materials testing stage fitted inside a microCT scanner. An effective Young's modulus was found by calibrating the model with the experimental data. By simulating the mechanical test, the FE model results yield excellent agreement with the measured data. Comparison of stress distribution plots with the image of the vertebra after the load was applied show predictive qualities as bands of high stress in the same region where failure occurred. In many FE-based studies of bony structures conducted to date, including investigations into craniofacial biomechanics, it is assumed that high localized stress/strain indicates a propensity to mechanical failure. Our findings can be seen as a useful first step in the experimental confirmation of this assumption.

The Young's modulus of various types of bone from different animals has been measured using nano-indentation and other techniques (Rho et al. 1999; Turner et al. 1999; Jamsa et al. 2002; Busa et al. 2005; Hoffler et al. 2005; Akhtar et al. 2006; Donnelly et al. 2006; McNamara et al. 2006; Teo et al. 2006). Measured values vary considerably among different species, locations and types of bone, but most are in the range of 2–28 GPa; considerably higher than the effective modulus found in this study. One possible reason for this is the slow, quasi-static loading in the experiment, which leads to a low effective modulus. Another reason may be that the experimental setup was not based on the actual loading that an in vivo vertebra would ever be subjected to, and as the bone's structure is suited for specific load-bearing capabilities, our results are not indicative of the vertebra's actual response to physiological loads. While the example presented here did not simulate a realistic physiological load, the method can be applied to any type of bone and loading condition to simplify the measurement of skeletal effective elastic properties.

The apparent cross-sectional area of the vertebra is a method for modelling the mechanical behaviour of the bone, and can be used to compare its response under compressive loading with other bones of differing geometry. Assuming a linear elastic material model for the bone is a simplifying assumption (Ulrich et al. 1997); however, there is good reason to believe that the results are accurate, as the brittle nature of bone means that there is little plastic deformation prior to failure. While bone is essentially anisotropic, the material tends toward isotropic microhardness properties (Ziv et al. 1996). A homogeneous effective modulus also assumes no difference in the properties of cortical and cancellous bone. In this case, the response was generally linear until failure occurred, and the full linear model gives excellent results when fitted with both measured and qualitative assessment of the experimental test. This approach is useful where measurement of the difference in material properties between the cortical and trabecular bone is difficult, such as in fossils or in small animals.

High resolution FE models that are based on microCT imaging, validated using results of mechanical testing from an in situ testing stage, offer an excellent method of studying biological structures such as bone.

Acknowledgments

This work was funded by Australian Research Council (DP0666374 and DP0987985) and University of New South Wales Internal Strategic Initiatives Grants to S.W. The authors acknowledge the facilities as well as scientific and technical assistance of the Australian Microscopy & Microanalysis Research Facility (AMMRF) at the Australian Key Centre for Microscopy and Microanalysis at The University of Sydney. The authors would like to thank Ms Toni Ferrara for her help with specimen preparation.

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