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Singularity-free finite element model of bone through automated voxel-based reconstruction a
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L. Esposito , P. Bifulco , P. Gargiulo & M. Fraldi
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Department of Structures for Engineering and Architecture (DiSt), School of Engineering, University of Naples Federico II, via Claudio 21, 80125 Naples, Italy b
Department of Electric Engineering and Information Technologies (DIETI), School of Engineering, University of Naples Federico II, via Claudio 21, 80125 Naples, Italy c
Department of Science – Vísindadeild Lndspitali (LSH), Biomedical Engineering Centre, Reykjavik University and Landspitali, Menntavegi 1, 101 – Reykjavik, Iceland
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Interdisciplinary Research Centre for Biomaterials (CRIB), University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy Published online: 27 Feb 2015.
To cite this article: L. Esposito, P. Bifulco, P. Gargiulo & M. Fraldi (2015): Singularity-free finite element model of bone through automated voxel-based reconstruction, Computer Methods in Biomechanics and Biomedical Engineering, DOI: 10.1080/10255842.2015.1014347 To link to this article: http://dx.doi.org/10.1080/10255842.2015.1014347
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Computer Methods in Biomechanics and Biomedical Engineering, 2015 http://dx.doi.org/10.1080/10255842.2015.1014347
SHORT COMMUNICATION Singularity-free finite element model of bone through automated voxel-based reconstruction
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L. Espositoa*, P. Bifulcob1, P. Gargiuloc2 and M. Fraldia,d3 a Department of Structures for Engineering and Architecture (DiSt), School of Engineering, University of Naples Federico II, via Claudio 21, 80125 Naples, Italy; bDepartment of Electric Engineering and Information Technologies (DIETI), School of Engineering, University of Naples Federico II, via Claudio 21, 80125 Naples, Italy; cDepartment of Science – Vı´sindadeild Lndspitali (LSH), Biomedical Engineering Centre, Reykjavik University and Landspitali, Menntavegi 1, 101 – Reykjavik, Iceland; dInterdisciplinary Research Centre for Biomaterials (CRIB), University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy
(Received 8 September 2014; accepted 29 January 2015) Computed tomography (CT) provides both anatomical and density information about tissues. Bone is segmented by raw images and Finite Element Method (FEM) voxel-based meshing technique is achieved by matching each CT voxel to a single finite element (FE). As a consequence of the automated model reconstruction, unstable elements – i.e. elements insufficiently anchored to the whole model and thus potentially involved in partial rigid body motion – can be generated, a crucial problem in obtaining consistent FE models, hindering mechanical analyses. Through the classification of instabilities on topological connections between elements, a numerical procedure is proposed in order to avoid unconstrained models. Keywords: voxel-based; FEM; DICOM; automated model reconstruction; singularity; instability
1.
Introduction
The most common way of constructing patient-specific finite element (FE) bone models is to derive information from computed tomography (CT) images. Some research groups have focused their activities on the development of a rapid and automated FE preprocessing (Keyak et al. 1990). In principle, there are two basic modalities to generate FE models from CT scans: geometry-based and voxel-based, which are still controversially discussed (Marks and Gardner 1993). In particular, geometry based meshing is widely adopted, but the inherent difficulty of extracting inner and outer bone contours inhibits its use in fully automated reconstructions. On the other hand, voxelbased meshing, introduced by Keyak et al. (1990), is achieved by arranging nodes in the form of a cubic lattice, directly converting each voxel into an eight-node isoparametric standard brick element: this method does not require any interpolation of data and takes full advantage of all available CT information. Lengsfeld et al. (1996) compared geometry-based with voxel-based FE models by testing five femurs subjected to nine load cases with reference to strain gauge based model. Very good agreement of the first principal stress was observed between strain gauges and FE results, suggesting that the validity of the voxel-based FE model is comparable with geometry-based one (Keyak et al. 1993).
*Corresponding author. Email:
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However, automated procedure of voxel-based meshing might generate unstable elements, a crucial problem in obtaining consistent FE models. A novel procedure, based on the analysis of topological connections between elements, is presented with the aim of avoiding singularities. As an example, FE models of two patients’ femurs, extracted from the Vı´sindadeild Landspitali (Reykjavik) CT database, are presented in order to show the effectiveness of the proposed strategies.
2.
Methods
Modern CT scan provides images compliant to Digital Imaging and Communications in Medicine (DICOM) standard; each image contains densitometric data related to the absorption of X-ray into patient’s body expressed as Hounsfield Units (HU) and geometric information about patient’s 3D positioning. The DICOM header of each image (i.e. upper left corner, tilt orientation, pixel spacing and slice thickness) provides all the information needed to correctly reconstruct the geometric model, while HU values provides densitometric values for each voxel useful to mechanically characterize the material. HU values were transformed into actual bone density values by using the calibration function obtained by CT images of a phantom enclosing different known materials (Steinarsdottir et al. 2012).
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Figure 1.
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An example of unstable elements arising from bone segmentation (transverse section of femur diaphysis).
Finally, each bone density value was associated with a material, whose mechanical properties (i.e. Young’s modulus), were deducted by relationships reported in literature (Galante et al. 1970; Keller 1994; Keyak et al. 1994; Helgason et al. 2008). For each point P the following correlation between Young’s modulus and ash density has been chosen (Keller 1994): Eðrash ðPÞÞ ¼ 10:5r2:29 ash ðPÞ: The import of raw CT data was realized by integrating Mathematica 8.0 (Wolfram, Champaign, IL, USA) functions in Ansys 10 (Ansys Inc., Canonsburg, PA, USA) environment by means of an external call to Mathematica and by creating CSV text files containing data to be exchanged (Esposito 2013). Bone tissue was segmented out of the raw CT images by selecting HU units ranging from 226 to 4000 HU (Hsieh 2003). Once imported, each voxel was modeled as a standard eight-nodes hexahedral isoparametric element associated with three degrees of freedom for each node, whose geometrical and material properties resemble CT information. Completely isolated elements were eliminated. Nevertheless, the automated model reconstruction may generates unstable elements, that is finite elements insufficiently anchored to the whole model and, thus, potentially involved in partial rigid body motion; these elements, attached to the surrounding elements only through one or two nodes as shown in Figure 1, will determine mechanical instabilities during the FE analyses and any numerical solution would diverge due to insufficient constraints. To avoid these instabilities, new elements were ad hoc added to the model. A vanishing stiffness (in the present case chosen to be corresponding to the lowest cancellous bone density value of about 200 HU, associated with a
Figure 2. node.
Basic unit of eight elements surrounding a generic
Table 1. Total and unstable combinations for groups of 2, 3 or 4 elements.
2 elements 3 elements 4 elements
Total combinations
Unstable combinations
Nodes
28 56 70
16 24 6
. 12 18 21
Young modulus of the order of 1 GPa) was assigned to these additional elements preserving, as much as possible, the bone structure obtained by CT segmentation and not appreciably modifying the mechanical behaviour.
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Table 2. List of all possible unstable and corresponding stabilizing combinations where the bold number means an element to be added with vanishing stiffness.
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Unstable combination
Figure 3. Example of an unstable combination of two elements (i.e. 1– 3).
Given that the maximum number of elements attached to a generic node is eight, a basic unit of eight surrounding elements is associated to each node; these elements were numbered from 1 to 8 (following a clockwise path, from front to back), as shown in Figure 2. A node is thus said a candidate to be ‘unstable’ only if it is surrounded by 2, 3 or 4 elements. The instability depends on the mutual geometric arrangement of elements and nodes. The number of the total possible combination of 2, 3 or 4 elements in eight places can be readily
Figure 4. The stable combination associated with that in Figure 3 obtained by adding the fictitious element in position 4.
2 elements 13 16 17 18 24 25 27 28 35 36 38 45 46 47 57 68 3 elements 127 128 135 137 146 147 157 167 178 235 238 246 248 258 268 278 345 346 356 357 358 456 467 468 4 elements 1278 1357 1467 2358 2468 3456
Binary unstable combination
Element to be added
Binary stabilized combination
10100000 10000100 10000010 10000001 01010000 01001000 01000010 01000001 00101000 00100100 00100001 00011000 00010100 00010010 00001010 00000101
4 5 48 4 1 1 3 15 48 2 4 1 15 8 8 5
10110000 10001100 10010011 10010001 11010000 11001000 01100010 11001001 00111001 01100100 00110001 10011000 10011100 00010011 00001011 00001101
11000010 11000001 10101000 10100010 10010100 10010010 10001010 10000110 10000011 01101000 01100001 01010100 01010001 01001001 01000101 01000011 00111000 00110100 00101100 00101010 00101001 00011100 00010110 00010101
4 4 4 4 5 8 8 5 4 1 1 1 1 1 5 4 1 2 2 8 4 1 8 7
11010010 11010001 10111000 10110010 10011100 10010011 10001011 10001110 10010011 11101000 11100001 11010100 11010001 11001001 01001101 01010011 10111000 01110100 01101100 00101011 00111001 10011100 00010111 00010111
11000011 10101010 10010110 01101001 01010101 00111100
4 4 2 1 1 1
11010011 10111010 11010110 11101001 11010101 10111100
computed. The unstable combinations are in fact only those in which nodes are more than 12 for two-element combination, and equal to 18 and 21 for three- and fourelement combinations respectively, as shown in Table 1.
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In order to easily represent the existing elements around a node, a binary array (8-bit length) was used: each bit is set to 1 if the corresponding element does exist, 0 otherwise. As an example, with reference to Figure 3 where only the elements numbered 1 and 3 are present around the node, the binary array results equal to 10100000. For each unstable combination, fictitious elements are added in order to eliminate the instability. The mechanical property of any fictitious element was referred to vanishing stiffness, while the geometrical properties resemble those of the voxels. Figure 4 shows the stable combination corresponding to that unstable shown in Figure 3. A look-up table, matching each unstable combination to corresponding stabilized one, obtained including specific fictitious elements, was generated (see Table 2). This control is iterated for all nodes of the model with the aim of obtaining a stable model suitable for FE analyses. The flow-chart (Figure 5) summarizes the steps of the algorithm. Finally, models of two real patients were numerically analyzed positioning the femurs 138 in adduction and 78 in flexion (Simo˜es et al. 2000), constraining distal diaphysis and applying loads related to the weight of each patient to the femur heads (see Figure 6 for magnitude and direction of forces).
3. Results Two human femurs were reconstructed with 267,184 and 258,109 elements, respectively. Final stable models were obtained by adding 2103 and 1936 (about 0.8% of the total number of elements) stabilizing elements without any operator’s intervention. Almost all unstable elements were found at the periphery of the model (i.e. in proximity of the boundaries), where stresses along the normal direction have to be zero in absence of applied tractions. The resulting stress distributions inside bones are shown in Figure 7. As expected, no significant influence has resulted by the additional elements: due to their lower stiffness, they very weakly participate to the mechanical response of the model, in terms of both stress magnitudes and distributions. The use of the proposed look-up table requires low computational load and ensures high running speed.
4.
Discussion and conclusions
FE model from CT scan data can be generated by geometry-based and voxel-based meshing. The geometrybased approach, typically realized by means of tetrahedral meshes, necessarily requires interpolation of densitometric CT data, with a consequent homogenization of the
Figure 5. Flow-chart describing the proposed procedure.
mechanical properties. This results in a smoothing effect on data, and implies a lower resolution with respect to the original CT data. On the contrary, the voxel-based
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Figure 6.
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Magnitude and directions of loads applied to the femurs.
Figure 7. Results of FE analysis showing Von Mises 3D stress maps of the two femurs (upper row: isoparametric view; lower row: coronal section).
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approach fully exploits the resolution and the accuracy of the CT densitometric information. In order to avoid unconstrained models due to voxelbased FE automated model reconstruction, an ad hoc procedure was developed, by focusing the attention on the classification of each geometrical instability, potentially hindering FE analyses. The presented procedure offers a simple way to produce voxel-based bone model ready for FE analyses, without eliminating any of the original data and without any concern about possible local instabilities. The fictitious elements, because of their vanishing stiffness, do not affect at all the results of the mechanical analyses.
Conflict of interest disclosure statement No potential conflict of interest was reported by the authors.
Notes 1. 2. 3.
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