Finite Element Modeling for a Morphometric and Mechanical Characterization of Trabecular Bone from High Resolution Magnetic Resonance Imaging

Abstract

Although further evaluation is needed about the usefulness of these methods in osteoporosis, it seems clear that computational generated 3D models of the cancellous bone from high resolution 3 Tesla MRI can be used to characterize bone in vivo, analyzing different mechanical conditions of the cancellous microstructure. The FE analysis could be performed in a more complex philosophy, if bone anisotropy is considered for bulk material properties definition (Hellmich et al., 2008), and also if not only the part with linear behaviour of the stress–strain curve is considered, but also the curve region showing plasticity or non-recoverable deformation. These studies should add more information to the mechanisms involved in bone fracture, like buckling phenomena, just before the breaking point arrives. Patients are classified nowadays using the World Health Organization (WHO) criteria, depending on the amount of bone loss, in osteopenic or osteoporotic patient. These are the clinical references used for new biomarkers evaluation. However, multivariate studies may provide different groups or classification patterns for the patient populations. Results of the mechanical simulations among a large population should provide knowledge for the establishment of new biomarkers of disease. New clinical trials and studies should help to analyze the sensitivity of these parameters to the treatment.

Full text

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Finite element modeling for a morphometric and mechanical characterization
of trabecular bone from high resolution magnetic resonance imaging195
x

Finite element modeling for a morphometric
and mechanical characterization
of trabecular bone from high resolution
magnetic resonance imaging

Angel Alberich-Bayarri1, Luis Martí-Bonmatí1,
M. Ángeles Pérez2, Juan José Lerma3 and David Moratal4
1Department of Radiology, Hospital Quirón Valencia, Valencia, Spain.
2Group of Structural Mechanics and Materials Modeling, Instituto de Investigación en
Ingeniería de Aragón, Universidad de Zaragoza, Zaragoza, Spain.
3Department of Rheumatology, Hospital Quirón Valencia, Valencia, Spain.
4Center for Biomaterials and Tissue Engineering, Universitat Politècnica de València,
Valencia, Spain.

1. Introduction
Trabecular bone disorders such as osteoporosis are currently becoming a worldwide
concern, affecting nearly 200 million of people (Reginster & Burlet, 2006). This syndrome is
characterized by a diminution of the bone density and also deterioration in the trabecular
bone architecture, both factors inducing an increased bone fragility and higher risk to
suffering skeletal fractures. The current diagnosis of osteoporosis disease is based on bone
mineral density (BMD) measurements, which are obtained using Dual emission X-ray
Absorptiometry (DXA) techniques and do not provide any information about the trabecular
bone properties at structural level. Although it is accepted that low BMD increases fracture
risk, not all patients who are osteoporotic will sustain a fracture. Low BMD as a risk factor
for fracture can be compared to high cholesterol as a risk factor for heart disease, being BMD
a rather poor predictor of fracture risk (Wehrli et al., 2007). In fact, a large analysis showed
that, on the average, BMD explains about 60% of bone strength (Wherli et al., 2002).
The majority of bone fractures associated with osteoporosis take place in vertebrae, hip or
wrist. In the European Union, in the year 2000, the number of osteoporotic fractures was
estimated at 3.79 million, of which 24% were hip fractures (Kanis & Johnell, 2004). Of all
fractures due to osteoporosis, hip fractures are the ones that are most disabling. In 1990,
about 1.7 million new hip fractures occurred worldwide, and this figure is expected to rise
to 2.6 million by 2025 (Gullberg et al., 1997). Furthermore, women who have sustained a hip
fracture have a 10% to 20% higher mortality than would be expected for their age
(Cummings & Melton, 2002).
8

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Finite Element Analysis196
Fractures are the consequence of mechanical overloading. The evaluation of elasticity and
fluency limits of different materials and structures is commonly assessed by mathematics
and engineering techniques. It seems reasonable to consider that bone fracture risk should
also be analyzed by exact and precise engineering methods. Nevertheless, although bone
mechanical properties can be directly extracted from ex vivo experiments, the mechanical
analysis of bone in vivo has always been limited to indirect measurements since there is no
physical interaction with the sample. These indirect measurements have evolved through
time, being initially age and sex two good indicators of bone health. Nowadays, the most
extended measure of bone quality is the BMD measured using DXA techniques. In fact, the
DXA technique has been shown to be accurate, precise and reproducible in the BMD
quantification. However, other imaging-based techniques that work at microstructural level
are of high interest in the evaluation of bone microarchitecture quality. Therefore, the
investigation in new methods to characterize trabecular bone microarchitecture alterations
produced by the disease is crucial. Such advances may not only be useful for a better
diagnosis of the disease, but also to evaluate the efficacy of the existing therapies.
The advances of digital radiology and modern medical imaging techniques like computed
tomography (CT), high-resolution peripheral quantitative computed tomography (HR-
pQCT) and magnetic resonance imaging (MRI) have allowed the research of in vivo
characterization of bone microarchitecture (Ito et al., 2005; Boutroy et al., 2005). Although a
high sensitivity to bone tissue structure is appreciated in X-Ray based techniques, like CT or
HR-pQCT which can achieve very high spatial resolutions, the high inherent contrast
between bone and bone marrow and its non-ionizing radiations places MRI as an
appropriate technique for the in vivo characterization of cancellous bone.
Recent high field MRI scanners permit the acquisition of images with a high spatial
resolution, in combination with satisfactory signal-to-noise ratio (SNR) and contrast-to-noise
ratio (CNR). In fact, the higher spatial resolutions achievable with high field 3 Tesla MR
scanners allow the development of reliable three-dimensional (3D) reconstructions of tissues
with the possibility of applying engineering-based methodology to simulate structures and
behaviours under certain conditions. The finite element (FE) method is the most extended
mathematical tool for complex modelling and simulations and it has been widely applied to
civil, spatial and many different engineering disciplines. This method can also be used in
biomedical engineering for the simulation of organs and tissues under different scenarios
(Prendergast, 1997). In the case of bone tissue, the combination of high quality image
acquisitions with proper image post-processing and computational based mechanical
simulations using FE with reduced element size, also called micro-FE (µFE), will improve
osteoporosis characterization, reinforce the diagnosis process, and perform an accurate
evaluation of the response to different therapies (Ito et al., 2005; Boutroy et al., 2005;
Alberich-Bayarri et al., 2008).
In the present chapter, the process for trabecular bone mechanical simulation and analysis
from high spatial resolution MRI acquisitions is detailed. Results of the FE application to
healthy and osteoporotic populations are also shown.

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Finite element modeling for a morphometric and mechanical characterization
of trabecular bone from high resolution magnetic resonance imaging197
2. Image acquisition and processing
Image acquisition
Due to the high spatial resolutions required for the assessment of trabecular bone tissue,
SNR tends to be significantly reduced in MRI acquisitions. There exists a number of imaging
problems inherent to the use of high spatial resolution MRI where technical parameters
have to be considered. The best MRI configuration for proper acquisitions of the trabecular
bone region is based on 3D gradient echo (GRE) pulse sequences with reduced echo-time
(TE), repetition time (TR) and flip angle (α) in order to obtain a high contrast between bone
and surrounding bone marrow. A high SNR is achieved with the shortest TE and the use of
phased array surface coils that can cover the entire region under analysis. In terms of
acquisition quality, an appropriate area for the trabecular bone examination is the wrist
region, which can be easily surrounded by the receiving coil and the sample – receptor
distance is minimized. Metaphyses of the ulna and radius in the wrist region are rich in
trabecular bone tissue. Acquired images obtained at metaphyseal level of the radius have a
high detail of the bone trabeculae, as it can be appreciated in figure 1.


Fig. 1. Images acquired with a 3 Tesla MRI system (Achieva, Philips Healthcare, Best, The
Netherlands). In a), wrist slice corresponding to the acquisition performed in a left-handed
62 years old female patient with osteoporosis. In b), image corresponding to a series
acquired in a left-handed 64 years old female healthy volunteer.

In the examples shown in figure 1, acquired images have a very high spatial resolution, with
an isotropic voxel size of 180 x 180 x 180 µm3, which is even higher than the achievable with
modern multi-detector CT scanners. However, the use of surface coils in the acquisitions
produce slight modulations of the signal intensities across the images, also known as coil
shading phenomena. Furthermore, although a high spatial resolution of 180µm was used,
achieved voxel size was slightly larger than typical thickness of the trabeculae, which is
about 100-150 µm. As it can be concluded, although the images have a very good quality,
some pre-processing must be applied before 3D reconstruction, meshing and simulation of
the trabecular bone structures.

Segmentation
Segmentation of the trabecular bone from MR images is performed by placing a rectangular
region of interest (ROI) in the first slice corresponding to the most proximal position and

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Finite Element Analysis198
thereafter propagated to the rest of slices. Segmented areas must be verified to exclusively
contain marrow and trabecular bone (Figure 2-a).


Fig. 2. Representative images resulting from the application of the different pre-processing
algorithms. Example of the isolated region exclusively containing trabecular bone and
marrow (a). Image resulting from the application of the coil shading correction through the
local intensities determination (b). Interpolated image after the execution of the sub-voxel
processing algorithm for an apparently increased spatial resolution (c). Binarized image
obtained at the end of the pre-processing chain where only complete bone or marrow voxels
exist (d).

Coil heterogeneities correction
Slight modulations of the signal intensities across the acquisition volume, also known as coil
shading phenomena, are corrected with nearest-neighbour statistics by the application of an
implemented 3D local thresholding algorithm (LTA), as a generalization from its 2D version
(Vasilic & Wehrli, 2005). Marrow intensity values in the neighbourhood of each voxel are
determined and bone voxels intensities are scaled using calculated local intensities.
Concretely, the method is based on the calculation of the average Laplacian values
S r
center of the sphere, which is displaced through all voxels of the volume. When the
calculated Laplacian equals zero
 

t
r
can be determined locally. After the marrow intensity is obtained, it is used as a threshold
and voxels can be directly classified into pure marrow voxels or scaled by its local threshold
value and labeled as partially occupied by bone voxels (Figure 2-b).
Sub-voxel processing
The extreme conditions in terms of low SNR and partial volume effects due to larger voxel
size than typical thickness of the trabeculae, which is about 100-150µm, forced the
 
I
r
the
L



in a sphere region ( )
with a radius R=15 pixels, being I the voxel intensity and r

0L

I r



, the corresponding marrow intensity
 
tIr

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Finite element modeling for a morphometric and mechanical characterization
of trabecular bone from high resolution magnetic resonance imaging 199
implementation of a method to increase the reconstructed spatial resolution. A subvoxel-
processing algorithm is applied to minimize partial volume effects and therefore improving
the cancellous bone structural quantification from MR images (Hwang & Wehrli, 2002). The
method consists on a two-pass algorithm where each voxel is initially divided into eight
subvoxels which are assigned a level of intensity conditioned by the corresponding level of
their voxel and near subvoxels, and also under the assumption that the amount of bone
intensities must be conserved. In the first pass of the algorithm, each subvoxel is assigned an
intensity value depending on the intensities of the adjacent voxels and the local sum of
intensities. The second pass of the algorithm consists on the refinement of the previously
calculated subvoxel intensities considering the intensities of the neighbouring subvoxels and
the total amount of intensity conservation. Finally, an increased apparent isotropic spatial
resolution of 90 µm is achieved and partial volume effects are minimized (Figure 2-c).

Binarization
Resulting images are binarized into exclusively bone or marrow voxels (Figure 2-d).
Histogram shape-based thresholding is applied through the Otsu’s method implemented in
3D. The method consists in the minimization of the intra-class variance of volume intensities
(Otsu N., 1979), which has been shown to be equivalent to maximizing the between-class
variance. Thus, the optimum separation threshold is calculated for the entire volume using:


T
B
t
t
2
2
max.arg*



(1)

where
binarization threshold is *
on their intensity value.

3. Volumetric reconstruction and meshing
T
2

is the total variance and
t
B
2

is the between-class variance. The optimum
. Finally, voxels can be classified as bone or marrow depending
Once the region under analysis has been binarized, a logical 3D matrix is obtained, with
bone voxels represented by 1’s and marrow voxels by 0’s. For visualization, the volume is
smoothed by a three-dimensional routine and the marching cubes algorithm (Lorensen &
Cline, 1987) is then applied in order to obtain a volumetric reconstruction of the structure, as
can be seen in figure 3.
Depending on the model to be simulated, different element types can be used to construct
the mesh. Two-dimensional (2D) surface based meshes are usually compound of triangles or
quadrilaterals. Considering a trabecular bone structure, a 3D volume mesh needs to be
generated for a volumetric simulation and evaluation of trabeculae mechanical resistance.
Volume meshes are usually compound of solid elements such as tetrahedrons and
hexahedrons. Isotropic hexahedrons with 8 nodes are usually referred as brick elements.
To develop an efficient meshing of the trabecular bone structure, brick elements are the best
option in terms of computational burden. The 3D reconstruction is conserved and a more
efficient stress analysis can be implemented (Alberich-Bayarri et al., 2007).

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Finite Element Analysis200

Fig. 3. Smoothed 3D reconstruction of the trabecular bone obtained from the application of
volume rendering algorithms to the binary volume of data.


Fig. 4. Example showing the algorithm basis for conversion from the 3D binarized matrix
into a 3D matrix containing the node information of the structure. Node coordinates and
element connectivity extraction from this process is straightforward.

An algorithm is needed to convert 3D reconstruction geometry information into FE
structural data where node coordinates and element interconnections are detailed. Although
a very intuitive algorithm would consist in the voxel-by-voxel sequential analysis of
structural coordinates and corresponding connectivity, an optimized algorithm
implemented by the authors is used for the mesh generation process.

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Finite element modeling for a morphometric and mechanical characterization
of trabecular bone from high resolution magnetic resonance imaging201
The idea of this algorithm is a direct detection of nodes of the structure working completely
with matrices and without sequentiality. Considering the 3D binarized reconstruction
matrix of size m x n x p, a new matrix nodal equivalent matrix of size (m+1) x (n+1) x (p+1)
is created and filled with zeros. Then, for each bone voxel found in the 3D binarized
reconstruction, eight ones representing the nodes of the element are situated on the
corresponding position of the node equivalent reconstruction (see Fig. 4). The extraction of
the node coordinates from the node equivalent matrix is direct. The relationship between
the original bone voxel and the corresponding calculated nodes is also conserved. The
implemented meshing algorithm can build a mesh with 1048706 nodes and 285148 elements
in 9.78 seconds using an Intel® Core™ 2 Quad CPU at 2.83 GHz and 8 Gb of RAM memory.
At the end of the process, the lists containing the nodes with the corresponding coordinates
and the elements with the corresponding node connectivity are completed and the FE mesh
is fully defined.
Once the mesh is strictly defined, it must be assembled in a file to ease the importation from
specific FE commercial software where the simulation is performed. Different FE tools like
ANSYS (Ansys Inc., Southpointe, PA, USA) or ABAQUS (Simulia, Providence, RI, USA) are
used for the FE model definition and posterior simulation.

4. FE model definition
The generated mesh must be prepared for the simulation after it has been imported in the
FE software, that is, a model must be fully generated considering the bulk material
properties, boundary conditions and loads application.
Initially, the bulk material properties for each element must be defined. In this case,
elements composition is supposed to be formed by compact bone, with linear, elastic and
isotropic behaviour. Numerically, the elastic properties of the material forming the elements
consist of a Young’s modulus given by Eb=10GPa and a Poisson’s ratio of v=0.3 (Fung, 1993;
Newitt et al., 2002).
Experimentally, resistance and elasticity properties of materials and other structures are
evaluated by a compression essay using specific equipment in laboratory. Stress–strain
relationship is analyzed and Young’s modulus of the whole structure can be easily
calculated from the linear slope of the stress–strain curve. Analogously, in the FE
simulation, null displacement is imposed on nodes from one side while a total strain (ε) of
10% of the edge length is specified on nodes from the opposite side for the compression
simulation. An iconography of the scenario to be modelled can be appreciated in figure 5.
The modelling of a trabecular bone uniaxial compression essay requires deciding the
loading direction. In our case, in the radius bone, the principal orientation of the trabecular
bone corresponds to the longitudinal dimension (Gullberg et al., 1997). Since an axial
acquisition is performed for the images generation, the longitudinal direction corresponds
to the MR scanner slice z direction. In this sense, a crucial advantage of 3 Tesla systems
comparing with 1.5 Tesla is the capability of performing high resolution acquisitions not
only in plane but also in slice direction (isotropic voxel size). In figure 6, a FE mesh of a
trabecular bone structure loaded to the commercial package ANSYS 10.0 is shown. The
boundary conditions and the strain application can be also appreciated in the figure.

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Finite Element Analysis 202

Fig. 5. Representation of the simulation applied using the FE method. A compression test is
applied to the trabecular bone sample with a side fixed to a null displacement and the
opposite side suffering an imposed deformation.


Fig. 6. In a), a FE mesh of trabecular bone imported in the commercial software ANSYS 10.0
is shown. The coordinates system is also shown, with the z direction corresponding to the
longitudinal direction of the bone. In b), lateral view of the cancellous bone structure with
boundary conditions definition, that is, null displacement on the left side (blue) and 10%
strain in the right side (red).

Once the model is completely defined, including forces application, material properties and
boundary conditions, simulation process is ready to begin.

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Finite element modeling for a morphometric and mechanical characterization
of trabecular bone from high resolution magnetic resonance imaging203
5. Simulation and results calculation
The calculation of the mechanical results from the simulation is a very demanding task in
terms of computational cost. The number of equations of our system is considerably high,
since there are 3 degrees of freedom per node, and each element has 8 nodes. As an
example, a mesh with 800000 nodes defines a system of 2400000 equations.
In order to solve the large systems of equations, different strategies can be used, depending
on the kind of solver (sparse, conjugate gradients, minimal residual). In the present case, for
the characterization of the trabecular bone response, systems are solved using a standard
sparse solver.
Although the different techniques for handling the matrices and approaching to the final
solution, the mathematical problem to be solved is summarized as follows (Zienkiewicz et
al., 2006):
The stiffness matrix of each element
K can be calculated by equation (2),

eT
KB DBd vol
e

· ()
e
V

(2)
where D is the elasticity matrix, which exclusively depends on Young’s modulus and
Poisson’s ratio (see equation (3)) and B is a matrix containing spatial information.



0












2 / )
v
1 (0
01
01
v
1
2
v
v
E
D
(3)

The global stiffness matrix of the structure under analysis can be assembled by equation (4),
taking into consideration the stiffness matrix of each element.


ij
K

Once the stiffness matrix is assembled, the objective of the FE problem is to solve the
structural equation (5), which relates the global stiffness matrix with the nodal
displacements (u ) and forces (f):

uK·


e
ij
K
(4)

f

(5)

After nodal displacements and forces are calculated, the calculation of stresses and strains is
straightforward.

A very interesting parameter that can be extracted from the FE results is the apparent
Young’s modulus of the whole structure by the application of the homogenization theory
(Hollister et al., 1991; Hollister & Kikuchi, 1992) and equation (6).

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Finite Element Analysis 204


n

app
F
A
E

1
(6)

Where F corresponds to the nodal reaction forces measured in the fixed side, A is the area of
the fixed side, and ε is the imposed strain.
Apparent Young’s modulus (Eapp) is used as an estimation of cancellous bone resistance to
compression.

6. Validation with sheep models
The mechanical trabecular bone biomarker parameters extracted from high resolution 3T
MR images can be validated by a comparison with 64-MDCT and micro-CT derived
parameters in the cancellous bone of sheep tibiae.
Five fresh legs were extracted from adult sheep cadavers in order to perform the trabecular
bone analysis of the tibia. The samples were prepared after skin removal and cleaning for
the MR and 64-MDCT acquisitions. The µCT examinations required the preparation of small
trabecular bone samples that were extracted from the metaphysis of the sheep tibiae. Results
of the Von Mises stress maps can be appreciated in an example in figure 7.


Fig. 7. Parametric reconstructions of the Von Mises nodal stress results for the compression
simulation using three different modalities (3T-MRI, MDCT and µCT).

The values obtained for the analyzed parameters in the 3 acquisition modalities can be
appreciated in Table 1. Setting micro-CT results as the reference, MR calculated elastic
modulus were slightly smaller while CT derived elastic modulus were clearly much higher.
This bone overestimation in MDCT acquisitions is probably due to the bone hyper
estimation effect of the point spread function (PSF) at high spatial resolutions.

Ex (MPa)
3T-MRI 79.910
MicroCT 105.920
MDCT-64 1122.509
Table 1. Results for the apparent Young’s modulus obtained in a trabecular bone sample
from the sheep legs.
The results obtained for the elastic modulus after compression FE test of the MR-derived
mesh are more proximal to the calculated for micro-CT than the obtained from 64-MDCT.
Compared to micro-CT, a bone overestimation is observed in the 64-MDCT calculated
parameters, as previously mentioned. Similar stress distributions can be appreciated in
Ey (MPa)
320.160
485.911
1430.551
Ez (MPa)
827.390
995.390
2577.300

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Finite element modeling for a morphometric and mechanical characterization
of trabecular bone from high resolution magnetic resonance imaging205
Figure 1 between 3T-MR and micro-CT while 64-MDCT derived stress map shows a high
and non-uniform stress arrangement.

7. Application to healthy subjects and patients with osteoporosis
The Young’s modulus parameter was initially evaluated in a healthy population in order to
define a rank of normality for the parameter and evaluate differences between sex or age.
The Young’s modulus results on a healthy population of 40 subjects showed significant
differences between males and females (p=0.012) with lower values in females (143.02 ±
24.83 MPa vs. 241.58 ± 28.07 MPa, mean ± standard error of the mean (SEM), female and
male respectively). Age showed no relationship with the Young’s modulus parameter
(r2=0.033) (Alberich-Bayarri et al., 2008).
An interesting point is that the volume of analyzed bone had an influence on the elasticity
results. There was a statistically significant difference (p<0.001, ANOVA test) between Eapp
results obtained for the large volume of interest (VOI) and those obtained for a small VOI
(189.84 ± 20.02 vs. 81.56 ± 16.86, mean ± standard deviation). Furthermore, as it can be
appreciated in figure 8, differences between Eapp results for the two different volumes
analyzed become greater when the values of the elastic parameter grow. The reason for
small VOI analysis was the minimization of the computational burden in compression
simulation. However, it was shown that the goodness of the results diminished as the
restricted volume dimensions decreased (Alberich-Bayarri et al., 2008). Authors detected
that the mechanical analysis results applied to the large sample volumes of trabecular bone
better characterized its properties, with significant differences between male and female
subjects for Young’s modulus. As expected, Eapp results were greater in men (Keaveny &
Yeh, 2002).

Fig. 8. Bland-Altman plot comparing the results of the Eapp for two different VOI analyzed,
the large volume (L) and the small volume (S). (adapted from Alberich-Bayarri et al.).

Authors’ results for the mean elastic modulus were lower than the value obtained for the
uniaxial compression (Newitt D.C. et al., 2002) (189.84 ± 126.62 MPa vs. 2050 ± 590 MPa,
respectively). However, these differences are probably due to different acquisition

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Finite Element Analysis206
configurations. In this sense, author’s results are close to those obtained by (Dagan et al.,
2002) with an Eapp of 150 MPa. In conclusion, acquisition technique and processing tools
clearly influence the Eapp values.
Also, the same methodology was applied to a population of 20 female osteoporosis patients in
order to evaluate Young’s modulus sensitivity to the disease. Mean Young’s modulus computed
for the osteoporotic population and a paired reference subjects is shown in table 2.

Parameter Healthy (n=21)
Eapp (MPa) 142 ± 25
Table 2. Young’s modulus results in a population of healthy volunteers and a group of
patients with Osteoporosis.

Resultant Von Mises stress parametric maps were also qualitatively different between
healthy and osteoporotic subjects. While healthy volunteers present a uniform structural
response to compression, with a distributed stresses, in the case of patients with
osteoporosis the parametric reconstructions showed a high heterogeneity with a specific
zone of high stress and resulting in a less resistant structure.

8. Conclusions and future challenges
Patients (n=20)
43.1 ± 24.8
Although further evaluation is needed about the usefulness of these methods in
osteoporosis, it seems clear that computational generated 3D models of the cancellous bone
from high resolution 3 Tesla MRI can be used to characterize bone in vivo, analyzing
different mechanical conditions of the cancellous microstructure.
The FE analysis could be performed in a more complex philosophy, if bone anisotropy is
considered for bulk material properties definition (Hellmich et al., 2008), and also if not only
the part with linear behaviour of the stress–strain curve is considered, but also the curve
region showing plasticity or non-recoverable deformation. These studies should add more
information to the mechanisms involved in bone fracture, like buckling phenomena, just
before the breaking point arrives.
Patients are classified nowadays using the World Health Organization (WHO) criteria,
depending on the amount of bone loss, in osteopenic or osteoporotic patient. These are the
clinical references used for new biomarkers evaluation. However, multivariate studies may
provide different groups or classification patterns for the patient populations.
Results of the mechanical simulations among a large population should provide knowledge
for the establishment of new biomarkers of disease. New clinical trials and studies should
help to analyze the sensitivity of these parameters to the treatment.

Acknowledgements
The support of the Spanish Ministry of Science and Innovation through project TEC2009-
14128 and of the Generalitat Valenciana through projects GV/2009/126 (grups
d'investigació emergents) and ACOMP/2010/022
acknowledged. The funding from Instituto de la Mediana y Pequeña Industria Valenciana
(IMPIVA) of the Generalitat Valenciana (IMIDTP/2009/334) is also acknowledged.

(ajudes complementàries) is

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Finite element modeling for a morphometric and mechanical characterization
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