Role of Fem in Orthodontics / orthodontic courses by Indian dental academy

ROLE OF FEM IN ORTHODONTICS

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INTRODUCTION

In the last decade the application of a well proven predictive technique the Finite Element Method, originally used in structural analysis has revolutionized dental biomedical research. Finite Element analysis was introduced originally as a method for solving structural mechanical problems, which was later recognized as a general procedure for numerical approximation to all physical problems that can be modeled by a differential equation description.


Finite Element analysis has also been applied to the description of physical form changes in biologic structures particularly in the area of growth and development and various branches of dentistry. Finite element method which is an engineering method of calculating stresses and strains in all materials including living tissues has made it possible to adequately model the tooth and periodontal structure for scientific checking and validating the clinical assumptions.


Finite Element Method – General Review


Finite analysis solves a complex problem by redefining it as the summation of the solution by a series of interrelated simpler problems. The first step is to subdivide (i.e. discretize) the complex geometry into a suitable set of smaller "elements" of "finite" dimensions when combined from the "mesh" model of the investigated structures. Each element can adapt a specific geometric shape (i.e. triangle, square, tetrahedron etc) with a specific internal strain function. Using these functions and the actual geometry of the element, the equilibrium equations between the external forces acting on the element and the displacements occurring on its nodes can be determined.


Information required for the software used in the computer is as follows.

1) Coordinates the nodal points. 2) Number of nodes for each element. 3) Young's modulus and Poissons ratio of the

material modeled by different elements. 4) The initial and boundary conditions. 5) External forces applied on the structure.


The boundary condition of these models is defined so that

all the movements at the base of the model are restrained. This manner of restraining prevents the model from any rigid body motion while the load is acting.

The two-dimensional axisymmetric finite element

modeling has been used in most of the previous research. Although numerical results can be easily obtained in two-dimensional modeling, it has some significant shortcomings.


The human is highly irregular in shape, such that it cannot be represented in a two-dimensional space and the actual loading cannot be simulated without taking the third dimension into consideration. The distribution of various materials of the tooth structure does not show any symmetry. Therefore a three dimensional modeling with the actual dimension must be preferred for a reliable analysis.


Element Attributes


One can take finite elements of any kind one at a time. Their local properties can be developed by considering

them in isolation, as individual entities. This is the key to the programming of element libraries. In the Direct Stiffness Method, elements are isolated by

disconnection and localization steps. This procedure involves the separation of elements from their neighbors by disconnecting the nodes, followed by the referral of the element to a convenient local coordinate system.


Summary of the data associated with an individual finite element.

This data is used in finite element programs to carry out element level calculations.


Dimensionality: Elements can have one, two or three

space dimensions. (There are also special elements with zero dimensionality)


Nodal points: Each element possesses a set of

distinguishing points called nodal points or nodes for short. Nodes serve two purposes: definition of element geometry, and home for degrees of freedom. They are located at the corners or end points of elements; in the so-called refined or higher-order elements nodes are also placed on sides or faces.


Geometry: The geometry of the element is defined by

the placement of the nodal points. Most elements used in practice have fairly simple geometries. In one-dimension, elements are usually straight lines or curved segments. In two dimensions they are of triangular or quadrilateral shape. In three dimensions the three common shapes are tetrahedral, pentahedral (also called wedges or prisms), and hexahedra (also called cuboids or “bricks”).


Degrees of freedom: The degrees of freedom (DOF) specify the

state of the element. They also function as “handles” through which adjacent elements are connected. DOFs are defined as the values (and possibly derivatives) of a primary field variable at nodal points. For mechanical elements, the primary variable is the displacement field and the DOF for many (but not all) elements are the displacement components at the nodes.


Nodal forces: There is always a set of nodal

forces in a one-to-one correspondence with degrees of freedom. In mechanical elements the correspondence is established through energy arguments.


Constitutive properties: For a mechanical element these is

the relation that specifies the material properties. For example, in a linear elastic bar element it is sufficient to specify the elastic modulus E and the thermal coefficient of expansion


Fabrication properties: For a mechanical element these are

fabrication properties which have been integrated out from the element dimensionality. Examples are cross sectional properties of elements such as bars, beams and shafts, as well as the thickness of a plate or shell element.

This data is used by the element generation subroutines to compute element stiffness relations in the local system.


BOUNDARY CONDITIONS: A key strength of the FEM is the

ease and elegance with which it handles arbitrary boundary and interface conditions. This power, however, has a down side. One of the biggest hurdles a FEM newcomer faces is the understanding and proper handling of boundary conditions


Essential and Natural B.C. The important thing to remember is

that boundary conditions (BCs) come in two basic flavors:


Essential BCs are those that directly affect the degrees of freedom

Natural BCs are those that do not directly affect the degrees of freedom


Showing the boundary conditions of the model


Linear/non-linear FE-modeling: The stiffness matrix for a linear

problem remains constant .This means that throughout the analysis the relation between force and displacement is linear. If the relation between force and displacement is not constant at the different steps of the analysis, the problem to be solved is called non-linear.


There are different sources of non-linearity:


• Material non-linearity generated by non-linear relations between stresses and

strains.• Geometric non-linearity generated by

non-linear behavior of the strain to deformation and stress to force relations.


Non-linear boundary conditions that are generated when the boundary conditions are changing during analysis. A typical case is the contact problem where two separate objects are getting in contact with each other during the analysis.


To solve problems with one or more of the above types of non-linearity, the solution has to be calculated in steps where the load is gradually incremented. At every step the equations of equilibrium should be fulfilled within a prescribed error.


Finite element models are created by breaking the design in to numerous discrete parts or

elements.


Applications of finite element method


Finite element analysis has been applied to the description of form changes in biological structures (morphometrics), particularly in the area of growth and development.

Finite element analysis as well as other related morphometric techniques such as the macro-element and the boundary integral equation method (BIE) is useful for the assessment of complex shape changes.


the knowledge of physiological values of alveolar stresses is important for the understanding of stress related bone remodeling and also provides a guideline reference for the design of dental implants.

Finite element method is also useful for structures with inherent material Homogeneity and potentially complicated shapes such as dental implants.


Analysis of stresses produced in the periodontal ligament when subjected to orthodontic forces.

To study stress distribution in tooth in relation to different designs.

To optimize the design of dental restorations To investigate stress distribution in tooth with cavity preparation.


The type of predictive computer model described may be used to study the biomechanics of tooth movement, whilst accurately assessing the effect of new appliance systems and materials without the need to go to animal or other less representative models.


Advantages of FEM


It does not require extensive instrumentation

Any problems can be split into smaller number of problems

It is an non-invasive technique


3-D model of the object can be easily generated with FEM

The actual physical properties of the materials involved can be simulated

Reproducibility does not affect the physical properties involved


The study can be repeated as many times as the operator wants

There is close resemblance to natural conditions

Static and dynamic analysis can be done


Disadvantages of FEM


The tooth is treated as pinned to the supporting bone, which is considered to be rigid, and the nodes connecting the tooth to the bone are considered fixed. This assumption will introduce some error however maximum stresses are generally located in the cusp area of the tooth. The progress in the finite element analysis will be limited until better defined physical properties for enamel, dentin and periodontal ligament and cancellous and cortical bone are available.


REVIEW OF LITERATURE ON THE FINITE ELEMENT METHOD


In the general field of medicine, FEM has been applied mainly to orthopedic research in which the mechanical responses of bony structures relative to external forces were studied. Furthermore, some research has been carried out in order to investigate the soft-tissue and skeletal responses to mechanical forces.


First FEM study in dentistry appeared in 1974,where j.w.farah and r.g.craig did finite element stress analysis in a restored axisymmetric first molar


The applications of the FEM in dentistry have been found in studies by Thresher and Saito, Knoell, Tanne and Sakuda, Atmaram and Mohammed, Cook, Weinstein, and Klawitter, Tanne, Rubin and associates, Moss and associates, and Miyasaka and associates.


The application of this theory is relatively new in orthodontic research. It is the development of powerful mainframe computers with extensive memory and number of improved soft wares that has now placed finite element analysis in the hands of orthodontic researchers


It has been shown in previous studies that the finite element method can be applicable to the problem of the strain-stress levels induced in internal structures. This method also has the potential for equivalent mathematic modeling of a real object of complicated shape and different materials. Thus, FEM offers an ideal method for accurate modeling of the tooth-periodontium system with its complicated three-dimensional geometry.


Experimental techniques are limited in measuring the internal stress levels of the PDL. Strain gauge techniques may be useful in measuring tooth displacement; however, they can not be directly placed in the PDL without producing tissue damage. The photo elastic techniques are also limited in determining the internal stress levels because of the crudeness of modeling and interpretation.


The force systems that are used on an orthodontic patient can be complicated. The FEM makes it possible to analytically apply various force systems at any point and in any direction. Experimental techniques on patients or animals are usually limited in applying known complex force systems.


It is very important to keep in mind that the FEM will give the results based upon the nature of the modeling systems and, for that reason, the procedure for modeling is most important.


FEM studies in orthodontics


FEM has already been broadly applied in orthodontic research. Yettram et al. (1972) were amongst the first to employ a two-dimensional finite element model of a maxillary central incisor to determine the instantaneous centre of rotation of this tooth During translation. Halazonetis (1996) used a similar two dimensional model to determine periodontal ligament (PDL) stress distribution following force application at varying distances from the centre of resistance of a maxillary incisor.


Using more complex three dimensional models Wilson et al. (1992, 1994), Tanne et al. (1987, 1988) and McGuinness et al. (1990, 1991) have studied moment to force ratios and stress distributions during orthodontic tooth movement. Cobo et al. studied periodontal Stresses during tooth movement


In the field of dentofacial orthopedics, Finite Element models have been employed to evaluate the stress distribution induced within the craniofacial complex during the application of protraction headgear (Tanne et aI., 1988, 1989; Miyasaka- Hiraga et al., 1994), orthopedic chin cup forces (Tanne, 1993), and conventional headgear forces (Tanne and Matsubara, 1996).


Mechanical properties of bone and FEM


Knowing the mechanical properties of bone is' of the utmost importance when an FE-analysis of a bony structure has to be performed. Bone is a living tissue that models and remodels throughout life, and thus continuously changing its mechanical behavior .Moreover a clear discrimination between cortical and trabecular bone is not a straightforward procedure especially in the transition areas.


Having this in mind, it is anyway important to have a mathematical description (i.e. Young's moduli and Poisson's ratio) for both cortical and trabecular bone properties. In cortical bone the osteons are aligned to the bone's long axis or in case of short bones along the direction of forces, therefore cortical bone exhibits a higher Young's modulus along the direction following the osteon arrangement than in the other two transversal directions.


The differences in between the two transversal directions are much smaller so that cortical bone is often assumed to be transversely isotropic in case of trabecular bone a precise mechanical connotation is more problematic as mechanical properties are strongly dependent on the orientation of the trabeculae.


Cortical bone as well as trabecular bone has viscoelastic properties. This means that it has different values for ultimate strength and stiffness depending on the strain rate during loading. In addition to this the mechanical properties of the bone are also depend on age and thus the level of mineralization .These factors, together with the uncertainty in the determination of the mechanical properties, make it impossible to give an ultimate value for both trabecular and cortical bone.


Moss et al (1985 ajo) Finite element methods are able to provide

absolute quantitative descriptions of cranial skeletal shape and shape change with local growth significance, independent of any external frame of reference, and, by so doing, eliminate the principal source of methodological error in customary roentgenographic cephalometry.


Finite element methods uniquely describe growth locally. Given the coordinate information defining the location of the nodes of a series of individual finite elements at successive times, the FEM provides an invariant (unique) description of the time-related shape changes of each finite element of a given structure independently of the coordinate system used and referred to its own initial boundaries. While it is possible to integrate over the descriptions of all the individual finite elements so as to provide a summary (global) description of that same structure, as a whole, the utility of the FEM increases when the structures analyzed are subdivided into increasingly smaller and more numerous constituent elements.


Kazuo Tanne et al (angle 1991) Investigated stress distributions in the craniofacial

complex by means of the finite element analysis. An orthopedic 1.0 Kg force was applied on the first molars of the model in the anterior direction parallel to the occlusal plane The model was restrained at the region around the foramen magnum where no linear and angular displacements were allowed, The analysis was executed using a computer program, FEM 3 (Fujitu Corp., Tokyo, Japan). Three principal stresses were determined in the craniofacial bones and around the sutures.


Stress distributions produced only by an anteriorly directed force applied to the maxillary first molars were investigated. Large compressive stresses were found in the bones around the maxillofacial sutures in addition to tensile stresses in the maxillary bone. These biomechanical changes in the sutures were caused by counterclockwise rotation and upward displacement of the complex.


they concluded that orthopedic maxillary protraction forces applied in more downward directions and/or at more anteriorly located teeth may eliminate concomitant rotation of the skeleton and produce more efficient sutural modifications for subsequent maxillary growth and repositioning. These considerations will be effective in terms of normal maxillary growth direction


G.D. Singh, J.A. McNamara. (Angle 1998)... Traced cephalographs of 73 pre pubertal

children of European American descent with untreated Class III malocclusions and eight mandibular landmarks were digitized. The resulting eight-noded geometries were normalized, and the mean Class III geometry was compared with the equivalent Class I average. A color-coded finite-element (FEM) analysis was used to localize differences in morphology.


they Compared Class III and normal mandibular configuration for changes in size and FEM revealed positive allometry of the mandibular corpus and around supramentale (15% increase in size), with reductions (30%) between the incisor alveolus and menton. For changes in shape, mandibular configurations were predominantly isotropic, with the exception of the anisotropic anterior region in the Class III subjects. Incremental growth differences are consistent and concluded that the absence of physical restraint is associated with mandibular prognathism.


Iseri et al (1998ejo) Evaluated the biomechanical effect of rapid

maxillary expansion (RME) on the craniofacial complex by using a three-dimensional finite element model (FEM) of the craniofacial skeleton. The construction of the three-dimensional FEM was based on computer tomography (CT) scans of the skull of a 12-year-old male subject. The CT pictures were digitized and converted to the finite element model


The final mesh consisted of 2270 thick shell elements with 2120 nodes. The mechanical response in terms of displacement and stresses was determined by expanding the maxilla up to 5 mm on both sides. Viewed occlusally, the two halves of the maxilla were separated almost in a parallel manner during 1-, 3- and 5-mm expansions. The greatest widening was observed in the dento-alveolar areas, and gradually decreased through the superior structures. The width of the nasal cavity at the floor of the nose increased markedly.


However, the postero-superior part of the nasal cavity was moved slightly medially. No displacement was observed in the parietal, frontal and occipital bones. High stress levels were observed in the canine and molar regions of the maxilla, lateral wall of the inferior nasal cavity, zygomatic and nasal bones, with the highest stress concentration at the pterygoid plates of the sphenoid bone in the region close to the cranial base


Dermaut et al (ejo2001)

From 55 frontal tomograms (CT-scans) using the 'Patron' finite element processor, a three-dimensional finite element model (FEM) of a dog skull was constructed. The model was used to calculate bone displacements under orthopedic loads. This required good representation of the complex anatomy of the skull. Five different entities were distinguished: cortical and cancellous bone, teeth, acrylic and sutures. The first model consisted of 3007 elements and 5323 nodes, including three sutures, and the second model 3579 elements and 6859 nodes, including 18 sutures.


Prior to construction of the FEM, an in vivo study was undertaken using the same dog. The initial orthopedic displacements of the maxilla were measured using laser speckle interferometers. Under the same loading conditions, using the second FEM, bone displacements of the maxilla were calculated and the results were compared with the in vivo measurements. Compared with the initial displacement measured in vivo, the value of the constructed FEM to simulate the orthopedic effect of extra-oral force application was high for cervical traction and acceptable for anterior traction.


Jafari et al (angle 2005) Analyzed the stress distribution patterns

within the craniofacial complex during rapid maxillary expansion., a finite element model of a young human skull was generated using data from computerized tomographic scans of a dried skull. The model was then strained to a state of maxillary expansion simulating the clinical situation. The three-dimensional pattern of displacement and stress distribution was then analyzed.


Maximum lateral displacement was 5.313 mm at the region of upper central incisors. The inferior parts of the pterygoid plates were also markedly displaced laterally. But there was minimum displacement of the pterygoid plates approximating the cranial base. Maximum forward displacement was 1.077 mm and was seen at the region of the anteroinferior border of the nasal septum. In the vertical plane, the midline structures experienced a downward displacement. Even the ANS and point A moved downward.


The findings of this study provide some additional explanation of the concept of correlation between the areas of increased cellular activity and the areas of dissipation of heavy orthopedic forces. Therefore, the reason for the occurrence of sensation of pressure at various craniofacial regions, reported by the patients undergoing maxillary expansion could be correlated to areas of high concentration of stresses as seen in this study. Additionally, the expansive forces are not restricted to the intermaxillary suture alone but are also distributed to the sphenoid and zygomatic bones and other associated structures.


Tooth and periodontium


More recent work has attempted to quantify periodontal proper1ies during instantaneous tooth movement (Tanne, 1995; Volp et al., 1996). These studies have allowed the development of more clinically valid three-dimensional finite models of the tooth (Middleton et al.,, 1997; Jones e t al , 1998).


Bobak et al (1997ajo) The finite element method of analysis (FEM) was

used to analyze theoretically the effects of a transpalatal arch (TPA) on periodontal stresses of molars that were subjected to typical retraction forces. The purposes of this investigation were (1) to construct an appropriate finite element model, (2) to subject the model to orthodontic forces and determine resultant stress patterns and displacements with and without the presence of a TPA, and (3) to note any differences in stress patterns and displacements between models with and without a TPA.


A finite element model, consisting of two maxillary first molars, their associated periodontal ligaments and alveolar bone segments, and a TPA, was constructed. The model was subjected to simulated orthodontic forces (2 N per molar) with and without the presence of the TPA. Resultant stress patterns at the root surface, periodontal ligament, and alveolar bone, as well as displacements with and without a TPA, were calculated.


Analysis of the results revealed minute differences of less than 1% of the stress range in stress values with respect to the presence of a TPA. Modification of bone properties to allow for increased displacement levels confirmed the ability of the TPA to control molar rotations; however, no effect on tipping was noted. Results suggested that the presence of a TPA has no effect on molar tipping, decreases molar rotations, and affects periodontal stress magnitudes by less than 1%. The final results suggest an inability of the TPA to modify orthodontic anchorage through modification of periodontal stresses.


Tanne et al (1998bjo) This study was designed to quantify the magnitude

of tooth mobility in adolescents and adults, and to investigate the differences in the biomechanical response of tooth and periodontium to orthodontic forces. The initial displacement of the maxillary central incisor was measured in 50 adolescent and fifty adult patients and the biomechanical properties of the periodontium were examined using the finite element method (FEM) and supporting experimental data. The magnitude of tooth mobility was significantly greater in the adolescent group than in the adult group.


By integrating the differences in tooth mobility in both subject groups with analytical tooth displacements, the Young's modulus of the periodontal ligament (PDL) was demonstrated to be greater in the adults than in the adolescent subjects. The differing biomechanical properties of the PDL in adults were demonstrated to result in almost equivalent or somewhat increased stress levels in the PDL in adult subjects.


It is suggested that this might produce a reduction in the biological response of the PDL and thus lead to a delay in tooth movement in adults.


Provatidis et al (2000) reports on studies upon five different

hypothetical mechanical representations of the periodontal ligament (PDL) which plays the most significant role in tooth mobility. The first model considers the PDL as an isotropic and linear-elastic continuum without fibers; it also discusses some preliminary visco-elastic aspects.


The next three models assume a nonlinear and anisotropic material composed of fibres only that are arranged in three different orientations, two hypothetical that have appeared previously in the literature and one more consistent with actual morphological data. The fifth model considers the PDL as an orthotropic material consisting of both a continuum and of fibres. Results were obtained by applying the Finite Element Method (FEM) on a maxillary central incisor


It was found that the isotropic linear-elastic PDL leads to occlusal positions of both centres in comparison with those obtained through the well-known Burrstone’s theoretical formula, while histological anisotropic fibres locate them apically and eccentrically.


Jones et al (jo2001) In this study they developed a 3 D computer model of the movement of the maxillary incisor tooth when subjected to orthodontic load .this was to be used to validate the finite element based computer model.

The design took the form of a prospective experiment at a laboratory at the University of Wales


A laser apparatus, was used to sample tooth movement every. 0.01 seconds .over a one minute cycle for ten healthy volunteers when a constant load of 0.39N load was applied "This process was repeated on eight occasions and five consistent readings were made this data was used to calculate the physical properties of the PDL. This was formed by 15000 four noded tetra hedral elements. Tooth displacements ranged from 0.012 to 0.133 mm the maximum strain located at the alveolar bone was thirty five times less than that of the PDL


Computer-generated 3-dimensional finite element ‘‘meshwork’’ of a maxillary central incisor, periodontal ligament, and alveolar

bone.


Geramy et al (ejo2002) Investigated the stress components that appear

in the periodontal membrane (PDM), when subjected to transverse and vertical loads equal to 1 N. Six three-dimensional (3D) finite element models (FEM) of a human maxillary central incisor were designed. The models were of the same configuration except for the alveolar bone height. Special attention was paid to changes of the stress components produced at the cervical, apical, and sub-apical levels


results showed that alveolar bone loss caused increased stress production under the same load compared with healthy bone support (without alveolar bone resorption). Tipping movements resulted in an increased level of stress at the cervical margin of the PDM in all sampling points and at all stages of alveolar bone loss. These increased stress components were found to be at the sub-apical and apical levels for intrusive movement.


Materials


The finite element method has only recently been applied to the evaluation of orthodontic attachment. Ghosh et al(1995) have used three-dimensional FEM models of ceramic orthodontic bracket designs to determine the stress distribution and likely mode of cohesive failure within the bracket when a full dimension stainless steel arch wire is engaged within the bracket slot. Katona. (1994, 1997), and Katona and Moore (1994) have used a two dimensional finite element model of the bracket tooth interface to assess the stress distribution in the system when bracket removing forces are applied.


Similarly, Rossouw and Tereblanche (1995) have used a simplified three dimensional finite element model to evaluate the stress distribution around orthodontic attachments during debonding. Katona (1997b) compared different methods of bracket removal and suggested that different loading methods resulted in significantly different stress patterns.

In addition, peak stress concentrations were suggested to be responsible for attachment failure indicating that mean stress values were of little value in quantifying the quality of attachment.


Ghosh and nanda (1995ajo) This investigation was designed to generate finite

element models for selected ceramic brackets and graphically display the stress distribution in the brackets when subjected to arch wire torsion and tipping forces. Six commercially available ceramic brackets, one monocrystalline and five polycrystalline alumina, of twin bracket design for the permanent maxillary left central incisor were studied. Three-dimensional computer models of the brackets were constructed and loading forces, similar to those applied by a full-size (0.0215 ´ 0.028 inch) stainless steel arch wire in torsion and tipping necessary to fracture ceramic brackets, were applied to the models.


Stress levels were recorded at relevant points common among the various brackets. High stress levels were observed at areas of abrupt change in geometry and shape.

The Starfire bracket ("A" Company, San Diego, Calif.) showed high stresses and irregular stress distribution, because it had sharp angles, no rounded corners, and no isthmus.


The finite element method proved to be a useful tool in the stress analysis of ceramic orthodontic brackets subjected to various forces.


Finite element model of unit cell


Finite element mesh of cement part of unit cell


katona et al (1994ajo) A finite element model (FEM) of an

orthodontic bracket bonded to enamel with glass ionomer cement was developed. The loading on the model simulated tensile loading conditions associated with the testing of bonding system strength.


The results indicate that peak stress values increase as the load deflection angulation increases. If the tensile load is inadvertently applied entirely on one wing of the bracket, the stress components nearly double in magnitude.


Conclusion: In future with this proviso, computer models of

various types can be used increasingly for fundamental biomechanics research in dentistry. They also provide an ideal "test-bed“ for research and development of new materials for use in mouth.


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Role of Fem in Orthodontics / orthodontic courses by Indian dental academy