3-d solid and finite element modeling of biomechanical ... - CiteSeerX

7th EUROMECH Solid Mechanics Conference J. Ambrósio et.al. (eds.) Lisbon, Portugal, September 7-11, 2009

3-D SOLID AND FINITE ELEMENT MODELING OF BIOMECHANICAL STRUCTURES - A SOFTWARE PIPELINE N. S. Ribeiro, P.C. Fernandes, D.S. Lopes, J.O. Folgado and P.R. Fernandes IDMEC – Instituto Superior Técnico, Technical University of Lisbon, Portugal Av. Rovisco Pais, 1049-001 Lisboa, Portugal {nribeiro, pcfernandes, danlopes, jfolgado, prfernan}@dem.ist.utl.pt

Keywords: Software Pipeline, Computed Tomography, Image Segmentation, Surface Mesh Adjustment, Solid Model Generation, Finite Element Mesh. Abstract. One of the major difficulties of the finite element method applied to computational biomechanics is the complexity associated with the development of patient-specific anatomical models. In this paper, a geometric modeling pipeline developed to create 3-D models from medical image data is presented. Several image and geometric processing blocks make part of the pipeline: image acquisition, image segmentation, surface mesh adjustment, solid model generation and finite element modeling. The input of the pipeline is an ordered stack of medical images with high spatial resolution and tissue contrast. The pipeline outputs geometric models, such as triangular surface meshes and solid models, and finite element meshes suitable for stress analysis, among other applications like visualization and rapid prototyping. The examples here presented are modeled based on computed tomography data, them being, the solid models and the finite element meshes of vertebrae C5 and C6, scapula, humerus, clavicle, mandible and teeth.

Nelson S. Ribeiro, Paula C. Fernandes, Daniel S. Lopes, João O. Folgado and Paulo R. Fernandes

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INTRODUCTION

Nowadays modern technology is in permanent expansion in all areas of knowledge. In medical sciences, new developed concepts have an important contribute to comprehend biological systems. One of the greatest achievements in medical systems has occurred with medical imaging. Modalities such as computed tomography (CT) and magnetic resonance imaging (MRI) provide 3-D anatomophysiological data that allow clinicians and surgeons to carry out important medical decisions. For a great number of pathologies, medical images are the starting point for the clinical evaluation of the patients and the establishment of their diagnoses. Images are then an intermediate containing relevant structural or functional information of the human body, dictating in many cases future treatment, surgical planning or palliative care. Medical images provide abundant geometrical information of anatomical tissues, revealing the inner parts of the human body. From this type of information it is possible to accurately reconstruct 3-D anatomical models. These models are well suited for finite element (FE) solvers in order to study bioengineering problems that involve the determination of stresses, strains, deformations, thermal or electromagnetic fields among many other biophysical parameters. Quantitative data obtained from FE analysis may be an important resource for increasing the probability of successful therapeutic treatments, and improving surgical planning and injury prevention. Anatomical geometric models can also be used for 3-D visualization, rapid prototyping and contain resourceful amount of information for medical device design. 3-D anatomical models are used in a wide spectrum of applications, but one of the most exciting domains of its application is the biomechanical area, principally in FE analysis. The study of the mechanical behaviour of human tissues (hard and soft tissues like bone, muscle, cartilage and ligaments) in normal, pathologic and implant conditions are just some of the applications of this vast biomedical field [1-3]. The process of developing geometric models based on medical images is typically associated to a pipeline formed by digital image processing and mesh processing blocks connected in a sequential manner. The input of the pipeline is an ordered stack of medical images that is submitted to a cascade of computational operations. The outputs are 3-D geometric models, digital equivalents to the human tissue being modeled. According to the applications, various geometric modeling systems have been proposed, however they all share in common some indispensable blocks such as: image acquisition, digital image processing and mesh processing [4-7]. In this work, a software pipeline that generates patient-specific anatomical 3-D models from medical images is proposed. One of the motivations for this work comes from the difficulties in applying the FE method in biomechanics due to the complexity associated in creating subject-specific anatomical models. The suggested pipeline relies on open source freeware, easily accessible on the internet, and commercial software, commonly available in most academic communities. The following section describes the several blocks that constitute the developed software pipeline. In the results section, the models generated by the pipeline are shown, namely, shoulder joint, cervical spine C5 and C6 vertebrae, mandible and respective teeth. At last, geometric modeling and computational aspects are discussed.

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2

METHODOLOGY

In this section, the blocks that integrate the mounted software pipeline are described. Figure 1 shows a schematic diagram of the mounted pipeline that is used to model anatomical structures. The process of developing solid and FE models begins with the acquisition of medical images that contain the geometric data of the biological tissue of interest. The examples presented in this paper are based in CT data. Geometric modeling is initiated with the identification of the tissues and their boundaries by segmentation of a 3-D image dataset. To convert the segmented data into a surface, a mesh-based technique, named marching cubes, is used to extract a polygonal isosurface from the voxelized image dataset. The isosurface needs then to be smoothed and decimated so that unwanted geometric features are attenuated or eliminated. In order to generate an accurate solid model from the filtered isosurfaces, these are interpolated with freeform surface patches. The last geometric modeling step consists on generating a volume mesh from the previously created solid models. It is important to emphasize that despite the usage of various computational methods, modeling human structures requires keen visual capabilities and a profound anatomical knowledge. At all stages of the pipeline the cervical spine C5 vertebra is presented as an example to demonstrate the various transformations that the pipeline inflicts upon the model being developed.

Figure 1: Diagram of the software pipeline. The boxes in the left column contain the software tools and the right column boxes indicate the data file extension

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2.1

Image Acquisition

Each medical imaging modality is based on a specific physical principle that provides characteristic in vivo image data for different tissue types within a body section. The selection of a proper imaging modality is a cornerstone stage for geometric modeling of patient-specific anatomy. Contrast-to-noise ratio, signal-to-noise ratio and image artefact types are some of the features to care in mind whenever opting for the imaging modality. For orthopaedic applications, CT images are more suitable for bone modeling since hard tissue has a high contrast relatively to soft tissue. On the other hand, MRI images are appropriate for soft tissue modeling due to the greater tissue resolution, in which it is possible to distinguish tendons from muscles and cartilage from bone. The small differences in signal intensity between these types of tissue do not allow a clear separation in a CT image, and therefore reasonable segmentation cannot be conducted. The differences between CT and MRI images derive from the physical principle behind these medical imaging modalities. CT is based on X-ray radiation and the pixel values quantify tissue density, also known as attenuation coefficient or Hounsfield unit; hard tissues greatly attenuate high energy radiation and therefore appear in the image with a high signal. However soft tissues, such as muscle and fat, are more transparent to X-rays and appear with a lower signal intensity and less image contrast, so CT can not discriminate different types of soft tissue that are contiguous. Relatively to MRI, this method uses a powerful magnetic field to align the nuclear magnetization of the hydrogen atoms of the water molecules within the body. MRI images present a high soft tissue contrast relatively to CT images since there is enough variation in tissue-specific parameters (T1, T2*, proton density) that affect the MR signal and permit acquisition of images with greater contrast between different soft tissues. Furthermore, the effect on the contrast in an MRI image of the tissue-specific parameters can be suppressed or enhanced by another set of operator-selectable parameters (such as repetition time, echo time and flip angle). To generate a model of a specific anatomical structure it is necessary, first of all, to identify the tissues that compose the organ(s). Then, according to the desired image features, an appropriate medical imaging modality is chosen for data acquisition. For all cases here reported CT imaging is the proper selection. CT images of the cervical spine were acquired from a 34 year old male subject without any local degeneration. For the shoulder joint, images from a 24 year old healthy male subject were acquired. Mandible CT data was obtained from a 60 year old male patient. Some of the image parameters for the three cases are grouped in Table 1. Image Parameters Number of Slices X,Y Voxel Spacing Z Voxel Spacing Width x Heigth Color Type Bit Depth

Cervical Spine 157 0.375 1.25 512 x 512 grayscale 16

Shoulder Joint 363 0.905 0.8 512 x 512 grayscale 16

Mandible 166 0.361 0.5 512 x 512 grayscale 16

Table 1: Image parameters of each volume data

In presence of signal noise and image artifacts it is necessary to improve the resolution and quality of the CT images by appropriate acquisition parameters tuning as slice thickness and spatial resolution. Also, several digital image processing techniques can be applied in order to enhance the visual information contained in an image. 4


2.2

Image Segmentation

The next modelling step of the pipeline is called image segmentation, which can be defined as a partition of an image into non-overlapping regions, in which each region is the locus of an object. It plays a crucial role in many medical imaging applications by facilitating the delineation of anatomical structures and other regions of interest. Usually, this delimitation is based on a specific image feature, such as intensity or texture, that is homogeneous within a region [8]. Segmentation plays a fundamental role in the designed pipeline for anatomical modeling as it establishes the transition between image data and 3-D mesh data. However, when dealing with medical images, segmentation is one of the most difficult tasks to be performed within the entire pipeline. One should take into account that a medical image comprises the information relative to the inner parts of the human body, but also of undesired signals. Artifacts and blurred edges are some of the characteristics common to any medical image. Various different segmentation methods exist and can be selected according to the input image. Taking into account that CT images are the source of 3-D medical data for the construction of shoulder joint, cervical spine and mandible models, three techniques are used in conjunction to solve the bone segmentation problem: global thresholding, active contour method based on region competition and manual segmentation. Considering the volume histogram, the intensities of the tissue of interest lie between two values, or thresholds, defined by the user. Hence, thresholding consists of selecting the intensities that identifies a tissue (image foreground) and cancel all the other values (image background), estimating in this way the locus occupied by the object. The values of a thresholded image range from -1 to 1, where the background and foreground pixels correspond, respectively, to the extremities. Any pixel with a value near zero indicates an edge between foreground and background. This type of global threshold can be performed as a point processing operation changing each voxel value according to a function that maps the original Hounsfield values to new values. This mapping is also called intensity region filter. So, by partitioning an image into regions according to the voxel intensity value, usually, a coarse but overall segmented object will result (Figure 2). Due to the partial volume effect, the established intensity interval encloses not only bone tissue but also muscle, fasciae and bone marrow. Despite this limitation, global threshold constitutes a necessary pre-processing stage for the application of active contour method.

A

B

Figure 2: Transversal section of the C5 vertebra. A – Original CT image; B – Segmented CT image with global thresholding

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The active contour method is a semi-automatic approach which combines the efficiency and repeatability of automatic segmentation with the user’s judgement and experience [9]. It is based on deformable models, i.e., closed parametric curves or surfaces with physical properties that, under the influence of external and internal mechanical forces, deform adapting to image features. The models can also be referred to as 2-D or a 3-D snakes and, as a result, the deformation process is referred to as a snake evolution. The modeling pipeline exploits the 3-D active segmentation method known as region competition [9]. A tissue can be outlined by placing an initial set of closed surfaces, such as spherical surfaces, in the proximity of the region of interest. The contour initialization is a rough estimate of the anatomical structure of interest. In this deformable model procedure, each snake is parameterized by spatial variables u, v and by a time variable t. When submitted to forces, every (u,v) point moves in an iterative manner according to the following partial differential equation: ∂ (1) c ( u , v; t ) = Fn ∂t where n is the unit normal to contour c at the point (u,v) and F is the sum of the forces that act on the contour in the normal direction. For each temporal iteration t, internal forces derived from surface geometry guarantee smooth surface variations, while external forces originate the snake deformation leading the model to adjust to the object boundaries [9].The region competition method calculates, for every point of the image, the internal force as the mean curvature of c(u,v;t) and the external force based on the difference between the probability of a voxel belonging to the foreground or to the background. The voxel probability maps derive from the previously described intensity region filter. The process stops when the snakes enclosure all the voxels with greater probability value or until the user finds a suitable solution. To perform the segmentation step an open source freeware program called ITK-SNAP is used [10]. Figure 3 represents the iterative process of snake evolution, where t = 0 shows a very rough estimate of the anatomical structure of interest and t = 120 represents a very close approximation of the structure.

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t

t

t

t

t

t

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t

Figure 3: A) Iterative process of 2-D snake evolution in different time instants (transversal section of the segmented image data); (continued.)

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t

t

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t

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Figure 3: B) Iterative process of 3-D snake evolution in different time instants.

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The results obtained by global thresholding and the active contour method are not totally accurate due to segmentation errors. In order to correct such errors, it was necessary to proceed to manual segmentation. To improve the accuracy of this process, the user must have a priori anatomical knowledge of the structure to be modeled. The output of the segmentation stage is a binary volume (i.e., each voxel is either 0 or 1) containing the anatomical structure of interest in a voxelized format, which can also be interpreted as a volume mesh formed by parallelepiped elements. ITK-SNAP automatically generates a 3-D surface mesh of the segmented data using the marching cubes algorithm, a technique that creates a triangular surface mesh from a volume scalar field. The voxels of the 3-D image are considered as three dimensional points with an associated intensity value. If the voxels of a cube have intensity values lower and other intensities above a user-specified value, also known as isovalue, then the voxels that form that cube contribute to the isosurface. The algorithm marches through each and every parallelepiped and establishes a surface boundary composed by triangular elements. By connecting all the triangular facets an isosurface is created. For a detailed description of the algorithm consult Lorensen and Cline [11]. The surface model generated by the segmentation stage is then exported to the next block of the pipeline. Figure 4 presents the 3-D surface mesh of the segmented data created by the marching cube algorithm.

Figure 4: 3-D surface mesh of the segmentation output of the C5 vertebra.

2.3

Surface Mesh Adjustments

After the segmentation stage, processing turns from image to mesh data. The modeling pipeline is now designed to improve surface meshes that are suitable to generate solid models. The model created by the marching cube algorithm (Figure 4) presents two major features that require a careful approach: (i) a characteristic stair-step shape surface, which obviously does not correspond to the natural surface curvature, and (ii) an excess of nodes and facets that expresses irrelevant information and hampers further computational processes or simulations. In order to suppress these undesired features, smoothing and decimation are

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some of the commonly used techniques for mesh adjustment. Usually, mesh adjustment techniques are not applied individually; instead they are sequentially combined for mesh quality improvement, e.g., the surface is first smoothed then decimated and smoothed again to remove decimation artefacts. As shown in Figure 4, the surface model is strongly corrupted by a jagged artifact. Therefore, the surface requires a low-pass filtering, i.e., smoothing the node’s positions relatively to each other without modifying the mesh topology. By adjusting the node coordinates, the overall mesh appearance and the shape of the triangular elements will be modified. Consequently, the mesh geometry will not be preserved but the number of nodes and triangular elements remains equal. Smoothing is a local processing operation which means that the new coordinates of a given node will depend on the position of the adjoined nodes. Laplacian smoothing is a very common and effective filter used to rectify the step-like artifacts produced by the reconstruction algorithm, improving the surface mesh appearance. For a single node pi at position xi the Laplacian smoothing filter will allocate pi to a new position xi+1 according to the following equation: n

xi +1 = xi + λ ∑ ( x j − xi )

(2)

j =1

where xj are the positions of the n neighboring nodes pj connected to pi, and λ a user-specified parameter that controls the amount of “smoothness” to be performed upon the surface mesh. The greater the value of λ, the smoother the mesh will be. The smoothed reconstructed surface contains a significant amount of triangle elements that assure a realistic appearance to the model when rendered. For 3-D visualization purposes a large number of nodes and surface elements are required to achieve high resolutions, especially for complicated curvatures. Conversely, large data sets easily surpass computational capabilities and as a result this excess of data is superfluous for computer simulations. The decimation operation [12] is a mesh simplification technique that reduces the total number of nodes and surface triangles. The reduction process, parameterized by the percentage of nodes to be eliminated, is userspecified. Usually, the reduction percentage is very high, in-between 50-90%; the resulting mesh is a good approximation to the original geometry, although mesh topology is not preserved. The decimation algorithm is an iterative process that performs node removal at each pass. Every node is submitted to three steps in order to classify a node as a candidate or non candidate for removal: at first, the node’s local vertex geometry and topology are characterized; depending on the type of node characterization a certain decimation criterion is evaluated, if the node satisfies the criterion, the node and all triangles that use the node will be removed creating a hole in the surface mesh; finally, it is necessary to recoat the hole with triangles. Decimation iterates again over all the nodes until the reduction percentage is reached. This process implies a resultant model with a lower surface resolution. This stage of the pipeline is assured by the open source freeware ParaView [13], which is responsible for the surface mesh adjustments by the smooth and decimate operations. Figure 5 shows the effect of these operations applied to the C5 vertebra. For visualization and rapid prototyping purposes model A is appropriated. On the other hand, model B, which is a decimated surface of model A, is the more adequate for volume mesh generation.

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A

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Figure 5: (A) Smoothed and (B) smoothed-decimated meshes of the surface model that appears in Figure 4.

2.4

Solid Model Generation

This stage of the pipeline deals with the generation of accurate solid models from smoothed and decimated isosurfaces. The reason for freeform surface fitting relies on the need for a manageable geometric representation for volume mesh generation in a FE environment. Although there are adequate software for volume mesh generation from triangular surface models, some mesh features affect negatively the application of such method: (i) in some cases, defining the boundary and loading conditions in volume FE models imported on FE solvers is a cumbersome task to carry out due to the graphical interface difficulties associated on selecting nodes (manually, one by one); (ii) constructing hexahedral meshes from triangular surface meshes gives, in most cases, undesired results. One way to overcome these difficulties consists on generating, by interpolation of the adjusted isosurface mesh, a discrete set of surface patches that form a solid model. A solid model built this way can be considered as mesh of freeform patches arranged, preferably, in an adaptive manner. These solid models are easily imported to a FE software and readily prompt for FE tetrahedral or hexahedral mesh generation. In freeform surfaces modeling, non uniform rational basis-splines (NURBS) surfaces are widely used due to their ability to handle large surface patches and to accurately represent free form geometries, so omnipresent in human anatomy. A NURBS surface is defined by the following equation [14]: nu

nv

∑∑ B

ui

S (u , v) =

i =1 j =1 nu nv

(u ) Bvj (v)ωij cij (3)

∑∑ Bui (u) Bvj (v)ωij i =1 j =1

where cij are control points, Bui(u) and Bvj(v) are the normalized B-spline functions in the u and v direction, respectively, and ωij are the weights of the control points. The commercial software SolidWorks® contains the ScanTo3D® toolbox which automatically creates solid models based on the user-defined amount of patch detail. Surfaces created by ScanTo3D® are composed by NURBS patches. Groups of patches form regions on the surfaces, which are delimitated by feature lines, entities that define the boundaries between regions. It is possible to choose the feature lines location and, therefore, control, in an direct way, the distribution of patches in the solid model. Ideally, feature lines should be placed on mesh curvature

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transitions to obtain a more regular patch distribution. In this step of the pipeline the input is a decimated surface mesh and the output is a solid model suitable for volume meshing on FE software. Figure 6 presents the solid model generated for the surface model B of Figure 5.

Figure 6: Solid model generated from the surface mesh in Figure 5 B.

2.5

Finite Element Modeling

The last step of the pipeline consists on generating a volume mesh from the solid models. The examples showed in this paper were obtained using the commercial software ABAQUS®. Taking into account the mesh tools supplied by this software, generating a tetrahedral mesh is a simple assignment to fulfill. In this process, mesh construction can ignore or consider the NURBS patches boundaries for mesh seeding, which can be useful to obtain a regular and unstructured tetrahedral mesh on the outer surface of the model. In Figure 7 it is presented a FE mesh with origin on the previously created solid model. On the other hand, when one wants to make a hexahedral mesh, the process gets a lot more complicated due to the complex anatomical curvatures of the structures being modeled.

Figure 7: FE tetrahedral mesh of the solid model from Figure 6.

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3

GEOMETRIC MODELING STRUCTURES

RESULTS

OF

PATIENT-SPECIFIC

BONE

In this section, three examples of solid and FE geometric models, obtained by the application of the suggested software pipeline (Figure 8, Figure 9 and Figure 11), are presented. C5 and C6 vertebrae, shoulder joint (scapula, humerus and clavicle) and mandible with teeth are just some of the biomechanical structures that can be modeled using the mounted pipeline.

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B

Figure 8: Geometric models of the C5 vertebra, C6 vertebra and intervertebral discs: A – exploded view (left) and assembled view (right) of the solid models; B - exploded view (left) and assembled view (right) of the tetrahedral meshes.

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A

B

Figure 9: Geometric models of the mandible and teeth: A - exploded view (left) and assembled view (right) of the solid models (view without NURBS patches); B - exploded view (left) and assembled view (right) of tetrahedral meshes.

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A

B

Figure 10: A – Solid models of humerus and scapula; B - FE tetrahedral meshes of humerus and scapula;

A

B

C

Figure 11: Cut view of (A) C6 vertebra, (B) humerus and (C) mandible FE volumetric meshes.

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Table 2 shows some statistic measures obtained from the generated volumetric meshes. These values were returned by ABAQUS® using the standard analysis check supplied by this software tool. Analysis warnings represent the number of elements that seem inappropriately distorted, and analysis errors indicate the number of elements severely distorted that impede ABAQUS® to go on with a FE analysis. FE Mesh C5 C6 Mandible Scapula Humerus

Nº Elements Analysis Warnings Analysis Errors 26226 25465 48473 43776 27655

2 (0.008%) 5 (0.020%) 6 (0.012%) 35 (0.080%) 1 (0.004%)

0 (0%) 0 (0%) 0 (0%) 0 (0%) 0 (0%)

Table 2: A few statistic measures of the various FE meshes

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DISCUSSION

Errors can occur at any stage of the pipeline which will degrade the fidelity of surface and volume meshes, and limiting the applicability at which they are aimed for. Image acquisition plays an important role in the accuracy of the obtained models. If anatomical data is corrupted in some way with artifacts and noise the accuracy of subsequent reconstructed models will be in jeopardize. Another determinant factor for the previous argument is low image resolution. Image filtering or resampling techniques are plausible solutions to attenuate signal noise and image artefacts. In the presence of spatial aliasing, e.g, slice thickness with greater values compared to inplane resolution values, the resampling technique is a good option to improve spatial resolution. In this technique a group of discrete images are interpolated and transformed in a continuous signal, and an up sampling technique is applied [15]. The output is an increased number of slices in the region of interest and, therefore, an increase of spatial resolution. Relatively to the CT images of the cervical spine, a low Z resolution was observed. Therefore, a resampling method was applied between the segments C5 and C6 (45 slices constituted the region of interest). MeVisLab [16] is used for this task. Applying this upsampling technique, the discrete images number increased approximately three times (147 slices). However, there is no such thing as the ideal medical image. Even with digital image processing, the most prominent noise and artifacts, such as partial volume effect and streaks at the girdle regions (e.g., shoulder girdle), still persist. Some tissue boundaries remain ambiguous, which will induce the semi-automatic segmentation procedure to select voxels of non interest. With respect to manual segmentation, this method alone guarantees a highly accurate result but the time, effort and training involved are impractical for large-population studies. Combining semi-automatic methods followed by manual segmentation provides a powerful and reliable instrument for accurate 3-D image segmentation. There is a fact that must be remarked. In the cervical spine model, the intervertebral discs were obtained only by manual segmentation due to the insufficient discriminative power of correspondent CT images to segmentation by active contour method. The stage of mesh adjustment may instigate errors by unbalanced mesh smoothing and decimation. Although the smoothing operation conserves the structure’s volume and topology, the geometry is altered, and with an inappropriate λ the produced surface model can have an abrasive representation comparatively to the real structure being modeled. On the other hand, mesh simplification consists on eliminating nodes and elements. So, if data is deleted, then 16


the resolution of the surface curvature will be at risk. Therefore, when dealing with this type of mesh operators, a careful and balanced approach is recommended. In surface mesh interpolation with NURBS, ScanTo3D® only permits the variation of a single parameter, surface patches detail, that affects the accuracy of the solid models. There is a minimum limit to ensure that if is not achieved, consequent solid models will not present anatomical landmarks and other fine details. In this CT based pipeline, besides accurate FE meshes generation, there is another factor that contributes for the subject-specific FE modeling: the assignment of bone mechanical properties. With the finite element meshes obtained from the pipeline, it is possible to extract bone density from CT images and make a correspondence between CT intensity data and material properties relative to each element. Nowadays, it is of a great importance to effectively derive the distribution of mechanical properties in the bone tissue from CT data, and to properly map it into subject-specific FE models [17]. Finally, for result validation of the designed pipeline one can recur to visual comparison between virtual 3-D models with ex vivo models, like dried bones. However this analysis is entirely qualitative and prone to subjective interpretation. The lack of well established methods to evaluate qualitative and quantitative error imposes this type of visual validation.

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CONCLUSIONS

The proposed pipeline is one among many different possible approaches to solve the problem of anatomical structures modeling based on medical images. It is a relatively efficient approach and the results are truthful and quite accurate. For medical visualization purposes, the anatomical particularities of the modeled structures are well defined and the resemblances are quite evident. Relatively to the created FE meshes, these maintain a good level of anatomical detail, a factor which is extremely important to produce accurate and real results when carrying out FE analysis to study the mechanical behaviour and other type of studies. Moreover, the generated meshes exhibit a very low element distortion, which is recommended for the FE method.

ACKNOWLEDGEMENTS The first author would like to acknowledge the support given by the Fundação para a Ciência e a Tecnologia (FCT) through project “Orthodontic Mechanics to Control Force and Tooth Movement - Design of a Novel Orthodontic Appliance” (PTDC/EMEPME/65749/2006. The second author would like to acknowledge the support given by the FCT through project “Computational Modeling of Bone Structure - Application to Bone Tissue Engineering” (PTDC/EME-PME/71436/2006). The third author would like to acknowledge the support given by FCT through projects ProPaFe (PTDC/EMEPME/67687/2006) and DACHOR (MIT-Pt/BS-HHMS/0042/2008), and for the PhD grant SFRH/BD/47750/2008.

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