Journal of
Anatomy
J. Anat. (2011) 218, pp40–46
doi: 10.1111/j.1469-7580.2010.01289.x
An experimentally validated micromechanical model
of a rat vertebra under compressive loading
Naomi Tsafnat and Stephen Wroe
Evolution and Ecology Research Centre, School of Biological, Earth and Environmental Sciences, University of
New South Wales, NSW, Sydney, Australia
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.
Key words 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
Correspondence
Naomi Tsafnat, Evolution and Ecology Research Centre, School of
Biological, Earth and Environmental Sciences, University of New
South Wales, NSW 2052, Sydney, Australia. E: n.tsafnat@unsw.edu.au
or n.tsafnat@unswalumni.com
Accepted for publication 29 July 2010
Article published online 31 August 2010
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 lm 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
ª 2010 The Authors
Journal of Anatomy ª 2010 Anatomical Society of Great Britain and Ireland
Validated Model of Rat Vertebra, N. Tsafnat and S. Wroe 41
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 lA, at a voxel resolution of 15 lm. 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
Fig. 1 Schematic diagram of the MTS (reproduced from the SKYSCAN
MTS manual version 1.1). The specimen is placed inside the thinwalled 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.
materials were located on top of each other rather than sideby-side; nevertheless, some distortion at the interface is
unavoidable.
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
ª 2010 The Authors
Journal of Anatomy ª 2010 Anatomical Society of Great Britain and Ireland
42 Validated Model of Rat Vertebra, N. Tsafnat and S. Wroe
100
90
80
70
Load (N)
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.
60
50
Experimental data
40
Linear
y = 81.6x
R 2 = 0.9759
30
Yield point 1
Yield point 2
20
Yield point 3
10
Linear (liner)
0
0
0.2
0.4
0.6
0.8
Displacement (mm)
1
1.2
Fig. 3 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.
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.
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.
By using the values of Eeff coupled with the measured
strain and force, an effective cross-sectional area Aeff of the
3 mm
Fig. 2 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.
ª 2010 The Authors
Journal of Anatomy ª 2010 Anatomical Society of Great Britain and Ireland
Validated Model of Rat Vertebra, N. Tsafnat and S. Wroe 43
Fig. 4 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 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.
vertebra can be calculated (Eq. 1), and was found to be
6.8 mm)2.
Aeff ¼
F
Eeff e
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 r = 62 N ⁄ 6.8 mm)2, or 9.1 MPa.
ð1Þ
Discussion and conclusions
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
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
ª 2010 The Authors
Journal of Anatomy ª 2010 Anatomical Society of Great Britain and Ireland
44 Validated Model of Rat Vertebra, N. Tsafnat and S. Wroe
Fig. 6 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.
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.
Acknowledgements
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
ª 2010 The Authors
Journal of Anatomy ª 2010 Anatomical Society of Great Britain and Ireland
Validated Model of Rat Vertebra, N. Tsafnat and S. Wroe 45
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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Journal of Anatomy ª 2010 Anatomical Society of Great Britain and Ireland