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Scientific Section |
Department of Dental Health and Development, University of Wales College Of Medicine, Dental School, Heath Park, Cardiff CF4 4XY, U.K.
Department of Basic Dental Science, University of Wales College Of Medicine, Dental School, Heath Park, Cardiff CF4 4XY, U.K.
Welsh Centre for Biomechanics, University College Swansea, U.K.
Jeremy Knox, Department of Dental Health and Development, University of Wales College of Medicine, Dental School, Heath Park, Cardiff CF4 4XY, U.K.
Abstract
The objective of the investigation was to develop a clinically valid three-dimensional computer model of the orthodontic bracket-cement-tooth continuum, and determine the magnitude and distribution of stresses generated by three different load cases.
A three-dimensional finite element model of the bracket-cement-tooth system was constructed consisting of 15324 nodes and 2971 finite elements. The stresses induced in the bracket-tooth interface by a masticatory load, a peel force and a twisting couple were recorded.
The maximum principal stresses resulting from occlusal and ‘twisting’ forces are distributed toward the lute periphery. Peel forces, applied to the bracket tie wing, are concentrated beneath the bracket stem. Twisting forces result in the highest enamel stresses.
The quality of orthodontic attachment can be explained by the magnitude and distribution of major principal stresses within the cement and impregnated bracket base. Shear and shear/peel forces are most likely to induce crack propagation within the adhesive layer. However, when a twisting action is used to remove orthodontic brackets, enamel failure is most likely. A clearer insight into the complexity of the bracket-cement-tooth system has been provided by numerical and finite element investigations. Further investigations, evaluating the influence of bracket base designs and orthodontic cement physical and geometric properties are indicated.
Refereed Scientific Paper
Key words: Orthodontic attachment, finite element method, stress analysis
Introduction
Engineering design begins with an attempt to analyse the stresses and strains acting on and within the structure under consideration before producing and testing a prototype. Unfortunately, the development of orthodontic cements and bracket bases has, largely, relied on relatively imprecise ex vivo experiments (Fox et al., 1991) which measure only the weakest component in the bracket-cement-tooth system. As the quality of attachment is primarily determined by the stresses generated in response to applied load, computer models are ideally suited to provide some insight into the structural behaviour of this system.
A number of methods of numerical analysis have been employed in bioengineering, in particular the Finite Element Method of stress analysis (FEM). FEM is a computer-aided mathematical technique for obtaining approximate numerical solutions to the abstract equations of calculus that predict the response of physical systems subjected to external influences (Burnett, 1987
). FEM has already been broadly applied in orthodontic research. Yettram et al. (1972) were amongst the first to employ a two-dimenstional 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, 1988a,b) and McGuinness et al. (1990, 1991) have studied moment to force ratios and stress distributions during orthodontic tooth movement. Cobo et al. (1993) studied periodontal stresses during tooth movement when attachment levels are reduced.
More recent work has attempted to quantify periodontal properties during instantaneous tooth movement (Tanne, 1995; Volp et al., 1996). These studies have allowed the development of more clinically valid three-dimensional FEM models of the tooth/PDL continuum (Middleton et al., 1996; Hickman, 1997; Jones et al., 1998).
In the field of dentofacial orthopaedics, finite element models have been employed to evaluate the stress distribution induced within the craniofacial complex during the application of protraction headgear (Tanne et al., 1988a,b, 1989a,Tanne et al., b; Miyasaka-Hiraga et al., 1994), orthopaedic chin cup forces (Tanne, 1993), and conventional headgear forces (Tanne and Matsubara, 1996).
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, 1997a), 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 de-bonding. 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.
Materials and Methods
The development of a valid three-dimensional computer model of the bracket tooth interface, requires the quantification of the physical and geometric properties of each of the system components. The geometric properties of a maxillary first premolar tooth were determined by preparing serial longitudinal sections of a representative tooth. Using computer imaging techniques, the three-dimensional co-ordinates of the tooth were recorded from serial longitudinal sections of a representative tooth, and a finite element mesh generated (15324 nodes each with 3 degrees of freedom, assembled into 2971 20-noded hexahedral elements) using a commercial mesh generating programme (PATRAN - PDA Engineering USA). To keep the size of the overall model reasonably small, only the area of enamel local to the orthodontic attachment was modelled. The remainder of the tooth was represented by the appropriate boundary conditions (Kralj et al., 1996).
A maxillary first premolar bracket was modelled. The bracket slot, tie wings, stem and thin foil base were considered separately from the bracket base mesh as cement impregnation of the base mesh produced a complex, non-homogeneous, area with physical properties which lay somewhere between those of the stainless steel bracket and the cement lute. The cement had an average thickness of approximately 271 µm, and the material was considered to be homogeneous and isotropic.
Although none of the materials considered in this model can be considered to be truly (microscopically) homogeneous, all but the impregnated wire mesh base and etched enamel surface, are homogeneous at the macroscopic level. The impregnated wire mesh (IWM) layer consists of a thin (94 µm diameter) stainless steel wire mesh (Figure 1) embedded in orthodontic cement. To determine the physical properties of this macroscopically non-homogeneous layer a theory of composite materials was employed to homogenize the layer and represent it with a mechanically equivalent, but homogeneous material (Hübsch et al., 1994, 1996).
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). The application of this theory relies on the microstructure of the material analysed being locally periodic or self-repeating. A small but representative area (unit cell) of the heterogeneous solid is selected and the behaviour of the unit cell studied under load. An extrapolation of the results of this process was made to represent the global material. Microscopic examination determined the smallest repeatable units of the IWM and finite element models (Hübsch et al., 1994, 1996) were constructed for the bracket base mesh (Figure 2) and orthodontic cement unit cells (Figure 3). The bracket base wires were considered to be prismatic bars of circular cross-section running in perpendicular sinusoidal courses. Where the wires crossed and were welded, rigid links of half the wire diameter were modelled. Both the wire and the orthodontic adhesive were considered to be linear elastic, homogeneous, and isotropic materials.
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. It was assumed that there were no air voids within the body of the cement, that cement penetration of the bracket base undercut was complete and that there was a perfect bond between the two materials. The impregnated wire mesh layer (IWM) was assumed to have transversely isotropic mechanical properties in planes parallel to and perpendicular to the mesh base. This demanded the introduction of elements with orthotropic properties in this region, i.e. material properties which differ in each of the three dimensions. In addition, the IWM is a double curved structure resulting in changes in the material property principal axes relative to the global co-ordinates (Figure 4).
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). The material properties of the IWM are presented in Table 2.
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) consisted of 15324 nodes and 2971 finite elements. To keep the size of this complex model within reasonable limits, only the relevant areas of the tooth were modelled, the remainder being substituted by the appropriate boundary conditions.
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) were selected to represent masticatory forces (shear), a ‘peel’ force (similar to that prescribed by a LOCI, lift off de-bonding instrument; North West Orthodontics, Seattle, Washington, U.S.A.) and the twisting force, similar to that used by some clinicians to remove brackets.
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All of the results presented are for unit loads. An insight is given into the stress distribution within the bracket-cement-tooth system and, assuming that all materials behave elastically, the location of initial failure can be predicted.
The maximum principal stresses are illustrated in all areas of the bracket tooth system. However, attention should be focused on the enamel, cement, and IWM areas as these primarily determine the quality of attachment. Stress concentrations are highest within the bracket wings and stem, but these are not considered critical due to the strength and ductility of stainless steel. In all analyses, maximum principle stress distributions are demonstrated as these are most likely to initiate crack propagation within the brittle cement, enamel and IWR layers. Figures 7–9 demonstrate the major principal stresses in the bracket-cement-tooth continuum resulting from the different load cases.
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Figure 7
presents the maximum principal stresses resulting from the application of a shear force to the occlusal surface of the bracket stem. It can be appreciated that the peak stress concentrations are distributed towards the periphery of the cement/IWM layer and underlying enamel surface. At first glance, it might appear that a masticatory load presents a case of shear loading. Unfortunately, pure shear almost never occurs and this is reflected in the stress distribution recorded here. The shear stress is largest under the middle of the bracket while the normal stress increases as the margins are approached. The normal stress is compressive under the gingival half of the bracket and tensile under occlusal section. Some relief of these normal stresses is achieved through flexing of the bracket base. However, maximum stresses are recorded at the bracket, cement and enamel margins.
Figure 8 shows the maximum principle stress distribution resulting from a peel force applied under the bracket tie-wing. Clearly, the peak stress is located under the bracket wing which is being loaded. This is due to the load-transfer mechanism through the relatively stiff bracket stem. Because the bracket base is relatively flexible, very little load is transferred from the bracket stem to the periphery of the foil. Instead, the load is channelled straight down from the body of the bracket and the adhesive into the tooth,
The force couple, presented in Figure 9, results in stress concentrations at the lute periphery, particularly in the enamel. As the ratio of maximum principle stress to ultimate tensile strength approaches 1 within a material, the likelihood of material failure increases. At values below this the material has some residual strength. Knowing the ultimate tensile strength values for each material and the level and distribution of major principle stress within the bracket-tooth interface allows the site of attachment initial failure to be predicted for each load case. Of primary concern during braket removal is the maintenance of enamel integrity. Table 3 shows the predicted site of failure for bracket removal by masticatory loads, ‘lift off device’ and twisting couple. In each case only the enamel and cement are considered. These results show that masticatory loads and ‘lift off devices’ are most likely to induce crack propagation within the adhesive layer. Care must be taken, however, when a twisting action is used to remove orthodontic brackets as enamel failure is likely.
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; Katona and Moore, 1994). In addition, suggestions were made that the use of tensile forces to remove orthodontic brackets would result in a reduced risk of enamel fracture when compared to shear/peel forces (Katona, 1997a). Rossouw and Tereblanche (1995) used a simplified three simensional finite element model to evaluate the stress distribution around orthodontic attachments during de-bonding. Their conclusions differed from those of Katona (1994, 1997a) in that enamel damage was suggested to be mroe likely when tensile forces were applied, rather than shear-torque. The current investigation demonstrated that shear and shear/peel forces are most likely to induce crack propagation within the adhesive layer. However, when a twisting action is used to remove orthodontic brackets, enamel failure is most likely. It is interesting to note that a ‘wrench’ was supplied for the removal of some of the early ceramic brackets by twisting. As can be appreciated, the use of a finite element model during the design of these brackets would have revealed that this method of bracket removal was inappropraite.
Conclusions
Acknowledgments
This project was funded by The Welsh Scheme for Development of Health and Social Research. The dimensions of the maxillary premolar bracket were provided by American Orthodontics Ltd, Sheboygan, U.S.A.
References
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