Journal of Biomechanics 31 (1998) 1187—1192
Technical Note
Finite element analysis of trabecular bone structure: a comparison of
image-based meshing techniques
D. Ulrich , B. van Rietbergen , H. Weinans, P. Rüegsegger *
Institute for Biomedical Engineering, University of Zurich and Swiss Federal Institute of Technology of Zurich, Switzerland
Department of Orthopaedics, Erasmus University Rotterdam, The Netherlands
Received 4 December 1997; accepted 5 August 1998
Abstract
In this study, we investigate if finite element (FE) analyses of human trabecular bone architecture based on 168 lm images can
provide relevant information about the bone mechanical characteristics. Three human trabecular bone samples, one taken from the
femoral head, one from the iliac crest, and one from the lumbar spine, were imaged with micro-computed tomography (micro-CT)
using a 28 lm resolution. After reconstruction the resolution was coarsened to 168 lm. First, all reconstructions were thresholded and
directly converted to FE-models built of hexahedral elements. For the coarser resolutions of two samples, this resulted in a loss of
trabecular connections and a subsequent loss of stiffness. To reduce this effect, a tetrahedral element meshing based on the marching
cubes algorithm, as well as a modified hexahedron meshing, which thresholds the image such that load carrying bone mass is
preserved, were employed. For each sample elastic moduli and tissue Von Mises stresses of the three different 168 lm models were
compared to those from the hexahedron 28 lm model. For one sample the hexahedron meshing at 168 lm produced excellent results.
For the other two samples the results obtained from the hexahedral models at 168 lm resolution were poor. Considerably better
results were attained for these samples when using the mass-compensated or tetrahedron meshing techniques. We conclude that the
accuracy of the FE-models at 168 lm strongly depends on the bone morphology, in particular its trabecular thickness. A substantial
loss of trabecular connections during the hexahedron meshing process indicates that poor FE results will be obtained. In this case the
tetrahedron or mass-compensated hexahedron meshing techniques can reduce the loss of connections and produce better results than
the plain hexahedron meshing techniques. 1998 Elsevier Science Ltd. All rights reserved.
Keywords: Finite element analysis; Trabecular bone structure; Meshing; CT imaging; Mechanical properties
1. Introduction
Recently, micro-finite element (FE) techniques based
on high-resolution images (&50 lm resolution) have
been introduced that allow modeling of trabecular bone
structures in detail. Such images can be obtained from
micro-CT scanning (Feldkamp et al., 1989; Rüegsegger
et al., 1996) or serial sectioning (Odgaard and Linde,
1989). Meshing procedures have been developed to create
complex FE-models in an automated way; the most
commonly applied being the voxel conversion technique
providing meshes with brick elements (Fyhrie and
Hamid, 1993; Hollister et al., 1994; Van Rietbergen et al.,
1995) and the marching cube algorithm providing meshes
* Corresponding author. Tel.: 0041 1 632 45 98; fax: 0041 1 632 1214;
e-mail: [email protected]
with tetrahedral elements (Frey et al., 1994; Müller and
Rüegsegger, 1995). It has been demonstrated that such
models allow calculation of the tissue loading and the
anisotropic elastic properties of trabecular bone regions
(Hollister et al., 1994; Van Rietbergen et al., 1996; Ulrich
et al., 1997a). However, presently no imaging methods
exist to obtain images from human bones at that resolution in vivo.
Recently, new three-dimensional CT and MR imaging
techniques have been introduced that produce high-resolution images in vivo (Majumdar and Genant, 1997;
Rüegsegger et al., 1994). Although the resolution of these
images (170—300 lm) is not as good as that obtained
from micro-imaging techniques, it is sufficient to visualize
the trabecular network (Müller et al., 1994; Ulrich et al.,
1997b). Using the earlier developed automated meshing
techniques, such images could provide a basis for the
generation of FE-models that represent the trabecular
0021-9290/98/$ — see front matter 1998 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 1 1 8 - 3
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D. Ulrich et al. / Journal of Biomechanics 31 (1998) 1187—1192
morphology of entire bones in vivo. Since, however, the
voxel size of these images is substantially larger than that
used in earlier models, it is not clear if the same meshing
and FE-techniques can produce accurate results.
The purpose of this study is to investigate the effect of
image resolution and meshing techniques on the results
from FE-analyses of trabecular architectures. We will
investigate if FE-analyses based on images with a resolution of 168 lm, which is currently the highest available in
vivo resolution, can yield relevant results about the
mechanical characteristics of trabecular bone.
2. Methods
Three human trabecular bone cubes (side length 4 mm)
were used. Specimen 1 was taken from the femoral head,
specimen 2 from the iliac crest, and specimen 3 from the
lumbar spine. Sequential images of the bone samples
were obtained with a micro-tomographic system at a resolution of 28 lm (Rüegsegger et al., 1996). The images
were filtered and thresholded to obtain 3-D binary 28 lm
reconstructions of the micro-architecture. The bone volume fractions (BV/TV) were found to be 26.3%, 16.0%,
9.18%, respectively. Furthermore, sample 1 (trabecular
thickness 219 lm, trabecular spacing 712 lm) and
sample 2 (trabecular thickness 120 lm, trabecular spacing 671 lm) showed a plate-like architecture, whereas
sample 3 (trabecular thickness 127 lm, trabecular spacing 667 lm) showed a rod-like architecture.
Using the binary 28 lm images, coarser gray-scale
images with resolutions of 56, 84, 112, 140, and 168 lm
were generated. For the 56 lm images sets of 2;2;2
voxels in 28 lm resolution images were condensed to new
56 lm voxels. Similarly, for the 84, 112, 140, and 168 lm
images sets of 3;3;3, 4;4;4, 5;5;5, and 6;6;6
voxels, respectively, were condensed to new voxels. The
gray-scale level of each new voxel in the coarser images
was assigned corresponding to the number of 28 lm
bone voxels comprised in the new voxel.
FE-models were generated from the gray-scale images
using three different meshing approaches. The first, referred to as hexahedron method employs a straightforward
voxel conversion. With this method voxels with a grayscale value beyond a specified threshold level are converted to brick elements in a finite element model (Fig. 1,
top). The threshold is chosen such that the best possible
agreement between the BV/TV in the FE-model and that
of the binary 28 lm image is obtained. FE-models were
generated for all images from 28 to 168 lm (Table 1). For
the coarser FE-models, this thresholding procedure
caused a loss of trabecular connections which resulted in
unconnected bone parts which were removed because
they do not contribute to stiffness. This removal induced
a decrease of the BV/TV which was considerable for the
iliac crest and lumbar spine samples (Table 1).
The second approach, referred to as a compensated
hexahedron method, is a modification of the first meshing method. With this approach the threshold level is set
such that the BV/TV of the model after removal of
unconnected parts best agrees with that of the binary
28 lm image. Consequently, the threshold level with this
method was lower than that for the models created with
the first approach. This approach was applied only to the
168 lm images (Table 1).
The third, more advanced, meshing approach, referred
to as tetrahedral method, makes use of the marching
cubes algorithm (Frey et al., 1994; Müller and Rüegsegger, 1995) to generate finite element models built of
tetrahedron elements (Fig. 1, bottom). A smooth triangular bone surface is generated from the voxel gray-scale
image by interpolating the nodes of the triangles depending on a specified threshold. A tetrahedron mesh is then
generated which matches the interpolated triangle surface. The threshold is chosen such that the best agreement between the BV/TV of the resulting model and that
of the binary 28 lm image is reached. The tetrahedron
method was applied to all 168 lm images (Table 1). Since
this method can model details which are smaller than the
voxel size, few trabecular connections were lost after
thresholding and consequently, the decrease of the
BV/TV due to the removal of unconnected parts is small.
Additionally, to validate the procedures, one tetrahedron
model was created for the femoral head image with
a 84 lm voxel size. The results from the 84 micron tetrahedron model were compared with the results from the
84 lm hexahedron model for the femoral head image.
The FE-problems for the hexahedron models were
solved with a special-purpose solver (Van Rietbergen
et al., 1995). To solve the FE-problems for the tetrahedron models a commercial software package was used
(Ansys 5.2, SAS IP) which implements an iterative solver.
For all models, the element material properties were
considered to be isotropic, linear elastic, and uniform
with a tissue Young’s modulus of 10 GPa and a tissue
Poisson’s ratio of 0.3. The orthotropic elastic properties
(Young’s and shear moduli, Poison’s ratios) were derived
from six FE-analyses per model (Van Rietbergen et al.,
1996). Table 1 summarizes the BV/TV, number of elements, and the CPU time required to calculate the overall stiffness matrix (6 analyses) on a DECAlphaServer
8420 of all models.
The accuracy of the elastic constants calculated from
the coarser models was quantified by the average deviation from those calculated from the hexahedron 28 lm
models, which are considered as reference models. The
tissue Von Mises stresses were calculated at all element
midpoints of each model for a uniaxial 1%-strain case.
The tissue Von Mises stress distribution was characterized by its average and standard deviation. The accuracy
of these values was quantified by calculating the deviation from those calculated from the reference models.
D. Ulrich et al. / Journal of Biomechanics 31 (1998) 1187—1192
1189
Fig. 1. Finite element (FE) models of the femoral head specimen created at a voxel resolution of 84 lm (left) and 168 lm (right) using either the
hexahedron meshing (top) or the tetrahedron meshing approach (bottom).
Table 1
Summary of the models created with the hexahedron meshing, tetrahedron meshing and compensated hexahedron meshing. Numbers in brackets
indicate the relative bone volume (BV/TV) after removal of unconnected parts
Hexahedron method
Specimen 1
(femoral head)
BV/TV (%)
C Elements
CPU (min)
Specimen 2
(iliac crest)
BV/TV (%)
C Elements
CPU (min)
Specimen 3
(lumbar spine)
BV/TV (%)
C Elements
CPU (min)
Compensated Tetrahedron
hexahedron method
method
168 lm
168 lm
28 lm
56 lm
84 lm
112 lm
140 lm
168 lm
26.3
(26.3)
770 419
410
26.4
(26.4)
98 661
58
26.4
(26.3)
29 200
14
26.3
(26.2)
12 283
4
26.3
(26.2)
6 424
1.5
26.2
(26.1)
3 626
1
26.2
(26.2)
3 626
1
26.3
(26.2)
46 922
111
16.0
(16.0)
478 293
272
16.1
(16.0)
60 121
39
16.2
(16.0)
17 877
9
16.1
(14.8)
6 882
8
16.1
(10.4)
2 547
2
16.2
(4.1)
569
0.3
20.1
(16.0)
2 209
1
15.8
(15.5)
37 035
84
9.18
(9.18)
272 182
236
9.26
(9.19)
34 568
59
9.15
(8.60)
9 520
13
9.05
(7.56)
3 524
5
9.20
(2.96)
552
1
9.11
(1.89)
209
0.2
13.8
(10.5)
1 451
3
9.15
(8.86)
22 780
41
This was the best-possible agreement of the BV/TV with the reference model; the BV/TV could not be regulated in smaller steps due to the large
voxel size.
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D. Ulrich et al. / Journal of Biomechanics 31 (1998) 1187—1192
Table 2
Average deviations of stiffness moduli and tissue von Mises stress distribution characteristics from the reference values for the different finite element
(FE) models at a 168 micron resolution. By using either the tetrahedron or the compensated hexahedron meshing the deviations are clearly less
168 lm
Hexahedron meshing
Specimen 1
(femoral head)
Specimen 2
(iliac crest)
Specimen 3
(lumbar spine)
Moduli
Degree of anisotropy
Von Mises stress
Moduli
Degree of anisotropy
Von Mises stress
Moduli
Degree of anisotropy
Von Mises stress
Compensated
hexahedron meshing
Tetrahedron meshing
Average
St. deviation
!3%
!4%
!9%
!12%
!3%
!4%
!9%
!12%
!6%
#9%
!12%
#4%
Average
St. deviation
!84%
#69%
!83%
#537%
!14%
#1%
!27%
!1%
!40%
!3%
!38%
!4%
Average
St. deviation
!86%
#18%
!93%
#1513%
!8%
!19%
!26%
!11%
!41%
#22%
!46%
!19%
3. Results
The comparison of the results from the hexahedron
and tetrahedron models at 84 lm resolution of the bone
sample from the femoral head showed differences of 3%
in the elastic properties, 1% in the average Von Mises
tissue stress, and 22% in its standard deviation. The
degree of anisotropy differed only 2% between the two
models. From this comparison we conclude that the
two meshing techniques yield almost the same results at
higher resolutions.
The elastic moduli and the average Von Mises stress
calculated from the plain hexahedron models decreased
with increasing voxel size (Fig. 2). For the specimen with
the highest trabecular thickness and spacing (from the
femoral head), however, only a minor decrease was found
in the moduli (less than 3%) and in the average tissue
stress (less than 9%) and its standard deviation (less
than 12%). Also, the degree of anisotropy was underestimated by less than 3%. For the other two specimens
(from the iliac crest and lumbar spine), moduli and average tissue stresses were close to the reference values for
models with a voxel size of less than 56 lm for the one
sample, and less than 84 lm for the other. For larger
voxel sizes a continuous decrease was found. At a resolution of 168 lm, the moduli and average Von Mises stresses were severely underestimated by 86% and 93%, respectively. The standard deviations for the tissue stresses
were very different from those calculated from the reference models and deviations in the degree of anisotropy
were as large as 69%. It was found that the underestimation in the mechanical properties is correlated with
the loss of load carrying bone mass due to thresholding
(Fig. 2, Table 1).
The results of the two specimens with lower trabecular
thickness and spacing (from the iliac crest and lumbar
spine) at a 168 lm resolution were clearly improved when
using the compensated hexahedron or tetrahedron models (Table 2). Best results at 168 lm micron were obtained
when using the compensated hexahedron models, for
which the underestimation of the moduli and the average
Von Mises stress was limited to 14% and 27%, respectively. The standard deviations of the Von Mises stresses
differed less than 11% and deviations in the degree of
anisotropy were limited to 19%. When using the tetrahedron method at a 168 lm resolution the moduli and
average Von Mises stresses were underestimated by less
than 41% and 46%, respectively, and the standard deviation of the Von Mises stresses was underestimated by
less than 19%. Deviations in the degree of anisotropy
were limited to 22%.
4. Discussion
The results of this study indicate that the accuracy of
FE-models based on images with a resolution of 168 lm
is dependent on the trabecular bone structure. For one
sample (femoral head with a trabecular thickness of
219$52 SD lm) the straightforward hexahedron model
with a voxel size of 168 lm produces excellent results.
For two other samples (iliac crest and lumbar spine with
a trabecular thickness of &120$30 SD lm ), however,
the 168 lm hexahedron modeling is not accurate. At
a 168 lm resolution, considerably better results for these
two samples are obtained, when using a mass-compensated hexahedron or a tetrahedron meshing technique,
basically because both reduce the loss of trabecular
D. Ulrich et al. / Journal of Biomechanics 31 (1998) 1187—1192
Fig. 2. Top: elastic stiffness range defined by the primary and tertiary
Young’s moduli versus voxel size calculated from finite element (FE)
models generated with the plain voxel-conversion meshing technique.
Bottom: average von Mises tissue stress versus voxel size calculated
from FE-models generated with the plain voxel-conversion meshing
technique.
connections. The elastic properties and tissue stresses
calculated from these models are still lower than those
calculated from the high-resolution models, but differences are less than 41% in the moduli, less than 22% in
the degree of anisotropy, less than 46% in the average
tissue stresses, and less than 19% in its standard deviation. The standard deviations of the two ‘critical’
samples were extremely overestimated when using the
hexahedral meshing (Table 2) because of the scarce
remaining structure with high stress concentrations.
A clear disadvantage of the mass-compensated hexahedron method is that the loss of unconnected parts is
compensated by thickening of the remaining structure.
Although this approach worked well for the specimens
used in the present study, it might not always provide
1191
accurate results e.g. if the bone micro-architecture is
substantially changed due to the thickening of the remaining structure.
The tetrahedron meshing method can model trabecular connections that are smaller than the voxel size of the
images. For this reason, the loss of connectivity is less
dramatic and the original geometry is better represented
than with hexahedron elements. Another advantage of
this meshing techniques is the fact that a smooth surface
is generated (Fig. 1, bottom), thus providing a more
accurate calculation of the bone tissue loading at the
trabecular surface. The numerical requirements for the
generation and solving of FE-models with this method,
however, are much larger than those for the hexahedron
models (Table 1).
Some of the limitations of this study have to be discussed. First, only three bone samples were investigated
in this study. These specimens, however, were chosen
such that they represent trabecular bone with different
morphologies and BV/TV. Second, bone marrow was
not included in the models since, in an earlier study, bone
marrow was found to influence the mechanical properties
significantly only at strain rate 10 s\ (Carter and Hayes,
1977) and we assumed strain rate zero to study elastic
bone properties. Third, linear elasticity was assumed in
all models. This assumption is supported by earlier studies on the mechanical behavior of trabecular bone which
have not revealed signs of nonlinear elasticity (Odgaard
and Linde, 1991; Rohl et al., 1991). Fourth, constant
tissue properties were chosen for all models although
there can be local changes in the tissue properties. We
aimed to study the variation in the image-based FE
results due to the measurement of the trabecular bone
microarchitecture exclusively. Therefore, in this study, we
excluded the assessment of the variation in the tissue
properties. Fifth, the images with the coarser resolutions
were artificially generated from high-resolution images.
The quality of real in vivo images would be lower than
the one of the images used in this study because of
bluring effects due to the finite size of the X-ray beam as
well as the quantum noise due to the low X-ray dose
which is required for in vivo measurements. The results
found here thus represent an upper limit for the accuracy
that can be obtained at a 168 lm resolution. However, it
is expected that the resolution and consequently the
quality of in vivo imaging will increase in the near future.
In conclusion, our results indicate that the preferred
meshing method for creating micro-FE models of trabecular architectures at a resolution of 168 lm depends
on the trabecular morphology, in particular its trabecular thickness. The dilemma is that the trabecular thickness usually is not known since it cannot be accurately
measured from images at this resolution. However,
it is possible to evaluate the meshing technique before
the actual FE analysis based on the percentage of unconnected trabecular tissue that is created during the
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D. Ulrich et al. / Journal of Biomechanics 31 (1998) 1187—1192
thresholding and meshing procedure. In situations where
a plain hexahedron meshing technique results in a substantial loss of connected bone tissue, the tetrahedron or
mass-compensated hexahedron method will yield more
accurate results. The bone volume fraction, which is
needed to define the appropriate threshold level for creating the model, can be measured accurately, even in vivo,
from images created by calibrated pQCT scanners.
Although the calculated mechanical properties can be
inaccurate for certain bone morphologies, even when
using advanced meshing techniques, the methods presented here provide mechanically relevant information
which cannot be obtained from methods based on bone
density measurements alone. Furthermore, it is possible
that the results can be improved by using correcting
factors depending on the bone morphology. The determination of such factors, however, will require the analysis of a much larger set of bone specimens.
Acknowledgements
This study was supported by the Swiss National
Science Foundation grant 31-458111.95.
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