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Finite Element Analysis in Orthodontics

INDIAN DENTAL ACADEMY Leader in continuing dental education

was developed in early 1960’s to solve the structural problems of aerospace the term ‘ Finite Element’ was coined by Argyris and Clough in1960. It was introduced in implant dentistry in 1976 by Weinstein.

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FEM is a technique for obtaining a solution to a complex mechanical problem by dividing the problem domain into a collection of much smaller and simpler domains (elements) in which the field variables can be interpolated with the use of shape function. These elements are connected at specific (eg corner) locations called nodal points Every element is assigned one or more parameters (eg modulus of elasticity) that defines its material (stiffness) behavior.

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The computer program calculates the stiffness characteristics of each element and assembles the element mesh through mutual forces and displacements in each node.

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Components of Finite Element Method Broadly divided into two parts: A) Finite element modeling B) Finite element analysis

Finite element modeling Geometric modeling is of three types: 1.Wire frame modeling 2.Surface modeling 3.Solid modeling

Once geometry is created it is transferred into a finite element model by the processor. generation is used to describe this procedure

Mesh generation It forms the back bone of the finite element analysis. It refers to the generation of nodal coordinates and elements. It also includes the automatic numbering of nodes and elements based on the minimal amount of user supplied data.  The ‘Element’ forms the basic constituent of the finite element modeling. 

The Element type defines the following: Element Dimensionality Element Shape Element Order Element Degree of Freedom Physical Properties

Finite Element Analysis There are 3 main components of Finite Element Analysis package: •Preprocessor Generates nodes and elements Generates element connectivity Applies material properties Applies boundary conditions Applies load •Processor Generates element matrices Computes nodal values and derivatives Solves governing matrix equations Compute parameters for memory / file management. •Post processor Analysis of results

Plots displacement contours Plots stress contours Plot contours of failure index Plots deformed mesh over undeformed mesh

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Interpreting the Results and Analysis Display

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The output from the Finite Element Analysis is primarily in the numerical form. It usually consists of nodal values of the field variables and its derivatives. For example in solid mechanical problems, the output is nodal displacement and element stresses. In heat transfer problems, the output is nodal temperatures and element heat fluxes. Graphic outputs and displays are usually more informative. The curves and contours of the field variable can be plotted and displayed. Also deformed shapes can be displayed and superimposed on unreformed shapes. The output is primarily in the form of color-coded maps. The quantitative analysis is determined by interpreting these maps. The result involves calculation of stresses by Von Misses criteria for each node.

Physical Model- Constitutive Properties of Tissues The tissues to be considered in the model are enamel, dentin, periodontal ligament, pulp and bone. This last tissue can be classified in two categories: cortical and cancellous. According to Anusavice (1998) most material properties of human teeth have been measured, but their values vary from author to author. Enamel and Dentin ď Ž Enamel is formed by parallel prisms of hydroxyapatite, oriented with its axis normal to the surface of dentin. As a consequence, it exhibits an orthotropic fragile mechanical behavior. ď Ž

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Dentin microstructure consists of tubules connected by organic material, provoking anisotropy, as the organic material connecting the fibers is less resistant than the hydroxyapatite. Both tissues exhibit fragile behavior under tensile stress, especially enamel.

Craig and Peyton, presented new values for dentin using a compressive test. They found the limit of proportionality, ultimate strength and Young’s modulus to be 167 MPa, 297 MPa and 16 GPa. In a later work, Craig and co-workers verified, using similar compression tests of enamel specimens, that no relevant variation occurred when testing enamel samples from different tooth locations, and obtained average values of 353MPa, 384MPa, 84.1 GPa for the limit of proportionality, ultimate strength and Young’s modulus of the enamel. In 1959 Tyldesley reported 66.2 MPa (limit of proportionality) and 267 MPa (ultimate strength) for the dentin and E to be 131 GPa for enamel, and 12.3 GPa for dentin.

Periodontal ligament and pulp 

Rees and Jacobsen(1997) found values of E as 50 MPa, using a finite element simulation and adjusting the ligament modulus to get measured values of displacements in two experimental systems. They found in the literature a large dispersion in the values of modulus of elasticity used in finite element models, ranging from 0.07 to 1750 MPa. Ko et al(1992) and Ho et al (1994) used a value of E=68.9MPa. and 0.45 for the Poisson ratio ν. For pulp, Rubin and Capilouto (1983) used in their work of an elasticity modulus for the pulp of 2.07 MPa and Poisson’s ratio of 0.45.

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Due to its very small stiffness, the pulp can be ignored when examining the mechanical behavior of dental structures. In some works, the periodontal ligament is not included in the models, and the geometry of the pulp cavity is not considered, with the section of the tooth solid. Authors such as Darendeliler et al (1992) Thresher and Saito(1973) point to the importance of including the pulp cavity in computational models for the tooth.

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Bone In the region of interest, the bone is heterogeneous and anisotropic, and is constituted of two different parts: the cortical layer and the spongeous internal basis. The former is a compact and stiff tissue with modulus of elasticity of 13.7 GPa, and the Poisson’s ratio of 0.3. The latter, named cancellous bone, is much more flexible, with a Young modulus of only 1.37 GPa, and 0.3 for Poisson coefficient.. Many authors choose not to consider the bone in their models, applying fixed boundary conditions directly to the upper surface of the dentin.

Occlusion  the subject occlusion envelopes all factors that cause, affect, influence or result from the mandibular position, its function, parafunction or disfunction, and not only the dental contact relationships 

Due to occlusal contact nature, horizontal and axial loads are produced during mastication. This load combination leads to tooth movement in all directions Contact angle is affected by occlusal morphology. In humans, bite load was measured to be 500 N in the molar region and 100 to 200 N in the incisor region. Maximum axial loads are of 70 to 150 N. Occlusal load during bruxism can reach values up to 1000 N according to Peters et al.

Modeling ď Ž

Stress analysis is performed using the Finite Element Method, which is particularly suitable to biological structures since it allows easier modeling of geometrically complex and multi-material domains. The commercial code ANSYS is used as the FEM analysis platform on this work.

Properties E v Enamel 4.1x104 0.30 Dentin 1.86x 104 0.30 PDL 68.9 0.45 Cortical bone 1.37x 104 0.30 Cancellous bone 1.37x104 0.30 TABLE Tooth tissue material mechanical properties

Discretization ď Ž

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Figure shows model element discretization used for the tooth model, resulting in a total of 4,996 elements of 8 nodes (15,117 nodes) and a system with 30,234 degrees of freedom. Numerical Analysis

Application of FEM in orthodontics Stress related responses of molar to TPA Bobak et al (1997) FEM analysis was used to analyze the effects of the transpalatal arch on periodontal stresses and displacements when subjected to orthodontic forces. The forces simulated were analogous to those used clinically when a transpalatal arch is used to "increase anchorage" (e.g., mesially directed forces produced by an elastomeric chain during canine retraction). To accomplish this analysis, an appropriate finite element model first was constructed. Resultant stresses and displacements in a model without a transpalatal arch were compared qualitatively with stresses produced in a model with a transpalatal arch to address the hypothesis that the TPA can indeed modify periodontal stresses. ď Ž

Results showed that 1. Fringe plots of intermediate principal stresses possessing a high level of sensitivity failed to resolve any effects due to the presence of a transpalatal arch on root surface, periodontal ligament, or alveolar bone stress patterns; 2. Scatter graphs of stresses calculated at element centroids distinguished differences in stress values due to the presence of a transpalatal arch. When normalized, however, the values differed by less than 1%; and 3. The results of analyses with altered physical bone properties that allowed simulation of increased molar displacements suggest that the transpalatal arch is effective in controlling molar rotations. The results of the finite element analysis therefore suggest that the presence of a TPA induces only minor changes in the dental and periodontal stress distributions.

Stresses induced in the periodontal ligament ď Ž

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Mc Guiness et al (1992) examined initial stresses within the periodontal ligament of a tooth when subjected to an orthodontic force when using an edgewise appliance. To achieve this, a finite element model of a human maxillary canine tooth was constructed and subjected to load. The Finite element analysis revealed that stress in the periodontal ligament due to most orthodontic forces, including those produced by edgewise appliances, is largely concentrated at the cervical margin and at the apex.

Stress in PDL of molar due to simulated bone loss

Minimum principal stress in PDL of maxillary first molar without bone loss under mesializing force (300 g)

Minimum principal stress in PDL of maxillary first molar with 3.5-mm bone loss under mesializing force (300 g)

Finite Element based Cephalometric Analysis ď Ž ď Ž

Sameshima, Melnick (1994) The finite element method has proven to be a useful tool for morphometric analysis in craniofacial biology. However, few attempts have been made to adapt this method for routine use by clinicians. The CEFEA program incorporates the advanced features of the finite element method but bypasses the detailed understanding of the engineering and mathematics previously required to interpret results. The program uses the color graphics display of common personal computers to show size change, shape change, and angle of maximum change. These are pictured as colored triangles of clinically relevant regions between pre- and mid- or posttreatment lateral headfilms. The program is designed to have features of interest in both clinical practice and research.

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The user first selects SIZE to visualize where major size change occurred and to determine if the change was nearly the same for groups of related elements, e.g., body of the mandible. Next, the user assesses SHAPE to determine the pattern of craniofacial development (shape change) in the same way. Finally, the user accesses ANGLE to observe common direction with respect to known gradients, e.g., growth parallel to the body of the mandible. A control panel allows the user to switch back and forth among the three screens, or to select a different case at will to make a swift and meaningful assessment of the effects of treatment, either during treatment or at its completion

Mandibular changes in Class II Div 1 patients treated with twin block appliance ď Ž Singh, Clark (2001) ) ď Ž

Comparing male pubertal configurations, finite element scaling analysis revealed marked positive allometry(=27%) in the condylar neck and negative allometry (=16%) at the apex of the coronoid process. For the female pubertal configurations, local increases in size were noticeable at the condylar neck (=15%), with negative allometry (=9%) in the coronoid process. For shape change, all configurations were highly isotropic over the entire mandibular nodal mesh. Therefore, in growing patients treated for Class II Division 1malocclusions with Twin-block appliances, condylar growth, coronoid process remodeling, and osteogenesin corpus and dentoalveolar regions may reflect the correction of the underlying skeletal dysmorphology.

A)Pubertal male configuration. Note white/purple area of condylar neck region, indicating increase in size of up to ≈28%. Green coronoid process shows decrease in size of ≈16%. Reddish coloration of angle, ramus, corpus, and symphyseal regions indicates ≈1%-5% increase in size in those regions. B) Pubertal female configuration. Note white/purple coloration in condylar neck region, indicating ≈15% increase in local size. Green coloration of coronoid process, angle, and symphyseal regions indicates decrease in local size of ≈9%. Red regions of ramus show little change in size, whereas purple coloration of corpus indicates ≈2% increase in size.

Bond strength offered by various bracket base design ď Ž

Knox et al (2001) determined the effect of altering the geometry of the bracket base mesh on the quality of orthodontic attachment employing a threedimensional finite element computer model

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They showed that in the double-mesh design, the relatively coarse outer mesh is shielded from the applied load by the increased stiffness of the deeper mesh layer. In addition, there is more of a gradient in stiffness from the bracket base to the fine mesh and ultimately the coarse mesh resulting in a less abrupt change in physical properties, and this reduces stress concentration at the adhesive interface.

FEM Analysis of distal en masse movement of maxillary dentition with multiloop edgewise archwire Chang et al(2004) ď Ž The authors compared the effects of multiloop edgewise archwire on distal en masse movement with a continuous plain ideal archwire. The stress a distribution and displacement of maxillary dentition was analysed when Class II intermaxillary elastics(300 gm/ side) and 5 degree tip back bends were applied to the ideal archwire and multiloop edgewise archwire

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The results showed that compared with ideal archwire the multiloop edgewise a archwire showed less discrepancy in the amount of tooth movement and individual tooth movements were more uniform and balanced. There was minimal vertical displacement or rotation of teeth with multiloop edgewise archwire.

Stress differences between sliding and sectional mechanics ď Ž

Vasquez et al(2001

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A 3D mathematical model was constructed which simulated an endosseous implant upper canine with its PDL and cortical and cancellous bone. Levels of initial stress were measured 2 types of canine retraction mechanics

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The results showed that the area with the highest stress was the cervical margin of the osseointegrated implant and its cortical bone but they were of such low magnitude that were unable to produce any permanent failure of the implant. Stress distribution was similar in the PDL and in the cortical bone around the canine, with a larger magnitude and more irregular stress distribution in the cortical bone. The zero stress area is associated with the centre of rotation of tooth.

Stress distribution and displacement of various craniofacial structures following application of transverse orthopedic forces ď Ž

Jafari, Sadashiv Shetty, Kumar M(2003)

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The study was to evaluate the pattern of stress accumulation, dissipation, and displacement of various craniofacial structures after RME, using a three-dimensional FEM study.

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The results of the study using the threedimensional FEM of a human skull indicted that the transverse orthopedic forces not only produced an expansive force at the intermaxillary suture but also high forces on various structures on the craniofacial complex, particularly the sphenoid and zygomatic bones. The confining effect of the pterygoid plates of the sphenoid minimizes dramatically the ability of the palatine bones to separate at the midsagittal plane.

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Further posteriorly, the pterygoid plates can bend only to a limited extent because pressure is applied to them, and their resistance to bending increases significantly in the parts closer to the cranial base where the plates are much more rigid. Therefore, the clinician should realize that with activation of the RME appliance he/she is producing not only an expansion force at the intermaxillary suture but also forces on other structures within the craniofacial complex that may or may not be beneficial for the patient.

Optimization of unilateral overjet management by FEM ď Ž Geramy (2002) ď Ž The main goal of this research was to introduce, evaluate, and mathematically optimize the treatment procedure of unilateral overjet cases. Patients with Class II subdivision malocclusions usually reach a point with canines in a Class I position, and a unilateral overjet remains to be treated at the next stage of treatment.

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This study tried to prepare an archwire design that combines the midline-shift correction and the unilateral overjet reduction simultaneously. The analyses of displacements were carried out by the finite element method. The upper dental arch was designed three-dimensionally. Three archwire designs that were thought to be useful in these cases were modeled and engaged the dental-arch model separately

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The use of an archwire containing a closed vertical loop with a helix distal to the lateral incisor on the affected (excess overjet) side and an open vertical loop without a helix distal to the lateral incisor on the normal side (normal overjet) while lacing the four incisors can be suggested as an optimum procedure to treat a unilateral overjet that is combined with a midline shift. The archwire cross-section depends on the initial position of the incisors. This mechanotherapy can be prescribed for both dental arches

Transfer of occlusal forces through maxillary molars ď Ž Cattaneo, Dallastra, Melsen B (2003) ď Ž The morphology of the skeleton is known to reflect functional demand. A change in the intramaxillary position of molars can be expected to influence the transfer of occlusal forces to the facial skeleton. A finite element analysis was used to simulate the displacement of a molar in relation to the welldefined morphology of the maxilla.

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Three 3-dimensional unilateral models of a maxilla from a skull with skeletal Class I and neutral molar relationships were produced based on CT-scan data. The maxillary first molar was localized so that the contour of the mesial root continued into the infrazygomatic crest.

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When the molar was loaded with occlusal forces, the stresses were transferred predominantly through the infrazygomatic crest. This changed when mesial and distal displacements of the molars were simulated. In the model with mesial molar displacement, a larger part of the bite forces were transferred through the anterior part of the maxilla, resulting in the buccal bone being loaded in compression.

Fig 3. Lateral view of deformed plots (actual deformations are magnified by factor of 60) of 3 maxilla models (mesial, left; neutral, middle; distal, right) with distribution of superior-inferior displacements (positive values denote superior movement). Units given in millimeters.

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In the model with distal molar displacement, the posterior part of the maxilla was deformed through compression; this resulted in higher compensatory tensile stresses in the anterior part of the maxilla and at the zygomatic arch. This distribution of the occlusal forces might contribute to the posterior rotation often described as the orthopedic effect of extraoral traction.

Comparative evaluation of different compensating curves in labial and lingual techniques (Sung et al 2003) ď Ž

In this study, human mandibular left teeth were aligned, and a 3-dimensional finite element model was made (consisting of 19382 nodes and 12150 elements). To compare the effect of compensating curves on canine retraction between the lingual and the labial orthodontic techniques, the compensating curve was increased on the .016-in stainless steel labial or lingual archwire, and a 150-g force was applied distally on the canine. The relative direction and the amount of tooth displacement of the finite element model were compared on a schematic displacement graph (magnified 10,000 times), and the compressive stress distributed on the root surface was observed.

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The pattern of tooth movement (with or without a compensating curve) was different between the labial and the lingual techniques. As the amount of compensating curve increased (0, 2, and 4 mm) in the archwire, the rotation and the distal tipping of the canine was reduced. The antitip and antirotation action of compensating curve on the canine retraction was greater in the labial archwire than in the lingual archwire.

Anchorage effects of various shape palatal osseointegrated implants ď Ž

Chen et al (2005)

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Anchorage effects of three types of cylinder osseointegrated implants (simple implant, step implant, screw implant) were investigated. Three finite element models were constructed. Each consisted of two maxillary second premolars, their associated periodontal ligament (PDL) and alveolar bones, palatal bone, palatal implant, and a transpalatal arch. Another model without an implant was used for comparison.

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The results showed that the palatal implant could significantly reduce von Mises stress in the PDL (maximum von Mises stress was reduced 24.3–27.7%). The von Mises stress magnitude in the PDL was almost same in the three models with implants. The stress in the implant surrounding bone was very low. These results suggested that the implant is a useful tool for increasing anchorage. Adding a step is useful to lower the stress in the implant and surrounding bone, but adding a screw to a cylinder implant had little advantage in increasing the anchorage effect.

References Tanne et al. Three dimensional FEM analysis for stress in the periodontal tissue by orthodontics forces AJODO 1987; 92; 499 Cobo et al. Initial stress induced in periodontal tissuewith diverge degree of bone loss by an orthodontic force: 3D FEM analysis AJODO 1993; 104; 448 McGuiness NJP, Wilson A, Jones ML, Middleton J, et al. Stress induced by edgewise appliances in the periodontal ligament—a finite element study. Angle Orthod. 1991;62:15–21.

Vasquez et al.Initial Stress Differences Between Sliding and Sectional Mechanics with an Endosseous Implant as Anchorage: A 3Dimensional Finite Element Analysis(Angle Orthod 2001;71:247–256.) Geramy A. Initial stress produced in the periodontal membrane by orthodontic loads in the presence of varying loss of alveolar bone: a three-dimensional finite element analysis. Eur J Orthod. 2002;24(1):21–33.

Jeon et al Analysis of stress in the periodontium of maxillary first molar with a 3D FEM model AJODO 1999;115;267

Knox et alAn Evaluation of the Quality of Orthodontic Attachment Offered by Single- and Double-Mesh Bracket Bases Using the Finite Element Method of Stress Analysis. Angle orthod 2001 ;71; 149 Sameshima,Melnik. Finite element based cephalometric analysis Angle orthod 1994; 5 ; 343 Fotos et al Orthodontic forces generated by simulated archwire applince evaluated by FEM Angle Orthod 1990, 4 ; 277 Transverse Jafari, Sadashiv Shetty, Kumar : Study of Stress Distribution and Displacement of VariousCraniofacial Structures Following Application of Orthopedic Forces—A Three-dimensional FEM Study(Angle Orthod 2003;73:12–20

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Sung et al.A comparative evaluation of different compensating curves in the lingual and labial techniques using 3D FEM (Am J Orthod Dentofacial Orthop 2003;123:441-50)

Chang et al. 3D FEM Analysis in distal en masse movement of the maxillary dentition with multiloop edgewise archwire. Europ Journ of orthod 2004;26; 339

Bobak et al.Stress related molar responses to TPA: an FEM analysis AJODO 1997; 112;512

Chen et al. Anchorge effects of various shape palatal osseointegrated implant. Angle Orthod 2005; 75; 378

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