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The Finite Element Method: a Tool to Study Orthodontic Tooth MovementDept. of Orthodontics, Royal Dental College, University of Aarhus, Vennelyst Boulevard 9, DK-8000, Aarhus C, Denmark; Correspondence: * corresponding author, pcattaneo{at}odont.au
Orthodontic tooth movement is achieved by (re)modeling processes of the alveolar bone, which are triggered by changes in the stress/strain distribution in the periodontium. In the past, the finite element (FE) method has been used to describe the stressed situation within the periodontal ligament (PDL) and surrounding alveolar bone. The present study sought to determine the impact of the modeling process on the outcome from FE analyses and to relate these findings to the current theories on orthodontic tooth movement. In a series of FE analyses simulating teeth subjected to orthodontic loading, the influence of geometry/morphology, material properties, and boundary conditions was evaluated. The accurate description of alveolar bone morphology and the assignment of non-linear mechanical properties for the PDF elements demonstrate that loading of the periodontium cannot be explained in simple terms of compression and tension along the loading direction. Tension in the alveolar bone was far more predominant than compression.
Key Words: finite element analysis • periodontal ligament • tooth movement • orthodontics • biomechanics
The orthodontic displacement of a tooth is the result of a mechanical stimulus, generated by a force applied to the crown of a tooth, being turned into biological reactions. This transformation involves all the processes of mechanotransduction typical of bone modeling and remodeling: mechanocoupling, biomechanical coupling, cell-to-cell signaling, and effector response (Turner and Pavalko, 1998). From a mechanical point of view, the first reaction to the application of an orthodontic load is an alteration in the strain-stress distribution within the periodontal ligament (PDL) and the surrounding alveolar bone. This leads to an intra-alveolar displacement of the tooth and a bending of the surrounding alveolar bone, provided the forces used are large enough. In the past, two theories explaining alveolar tissue reactions related to tooth movement have been proposed: the ‘pressure-tension’ theory (Schwartz, 1932; Reitan, 1951), and the distortion or bending of the alveolar bone (Baumrind, 1969; Heller and Nanda, 1979). Recently, a third theory has been proposed, suggesting that bone apposition could be induced by (1) the load exerted by stretched fibers of the PDL, which may also induce a slight bending of the alveolar wall; (2) direct resorption by unloading of the alveolar wall in the case of low forces; and (3) indirect resorption as repair due to ischemia following the application of high forces (Melsen, 2001). A way to verify the validity of the abovementioned theories would be to determine the stress and strain levels in the periodontium. Finite element (FE) analyses offer a means for the calculation of these quantities. The FE method was introduced into dental biomechanical research in 1973 (Farah et al., 1973) and since then has been extensively applied to analyze the stress and strain fields in the alveolar support structures (Tanne et al., 1987, 1998; Middleton et al., 1990, 1996; Cobo et al., 1993; Bourauel et al., 1999; van Driel et al., 2000; Provatidis, 2000; Qian et al., 2001; Toms and Eberhardt, 2003). Though the FEM is a powerful tool for the analysis of complex structures, the outcome of the FE analysis is dependent on the formulation of the problem. Thus, FE analyses of the load transfer from the tooth through the PDL to the alveolar bone must account for the physical properties and morphology of the periodontium. Despite the fact that the PDL is known to be a non-linear visco-elastic material, most of the previous FE models incorporate homogeneous, linear elastic, isotropic, and continuous PDL properties. At the same time, the morphology of the alveolar structures has been considered a ‘solid’ and has not been adjusted for differences in micromorphology. The aim of the present study was therefore to evaluate the influence of morphology, material properties, and boundary conditions on the outcome of FE analyses and to interpret the results in the light of existing theories on orthodontic tooth movement.
FE Model Generation A segment of a lower left human mandible, including a canine and a first premolar, obtained from autopsy (approval obtained from the University Ethical Committee) was scanned with a micro-CT (µCT) scanner with a voxel dimension of 37 µm (µCT40, Scanco Medical, Bassersdorf, Switzerland). Image-processing software (Mimics 7.10, Materialise, Leuven, Belgium) was used to generate three-dimensional outer shapes of the alveolar bone and teeth. The PDL tissue was modeled as the space between the alveolar socket and root of the tooth. With the use of the automatic mesh generator of COSMOS/M (Structural Research & Analysis Corp., Los Angeles, CA, USA), the different anatomical parts were meshed with ten-node tetrahedral elements. This produced a FE model of the mandibular segment consisting of 197,186 elements, 253,309 nodes, and 754,347 degrees of freedom (Fig. 1D  ).
Material Properties The material properties of bone were assigned based on two different procedures: (1) All bone elements were either assigned a Young’s modulus of 12,200 MPa and a Poisson’s ratio of 0.3, thereby representing a homogeneous distribution of bone properties (referred to as the homogeneous model) (Fig. 1B  ); or (2) different Young’s moduli were assigned to each element individually, based on the true morphology of the bone as obtained from the µCT-scans using a slightly modified version of the procedure implemented by Cattaneo et al.(2001) (the density-based model). Three different Young’s moduli were considered to represent full cortical bone (17,500 MPa, Poisson’s ratio of 0.3), partly cortical bone (i.e., the elements which have their centroid lying on the border between bone and bone marrow; 5000 MPa, Poisson’s ratio of 0.3), and bone marrow (200 MPa, Poisson’s ratio of 0.3), respectively (Fig. 1C). The value of 12,200 MPa, chosen for the homogeneous model, was calculated as the mean of the values given to the bone elements in the density-based model.
For the PDL, 3 material behaviors were assumed. In the first and second cases, the PDL was considered to be linear with a Young’s modulus of 0.17 MPa (linear-high model) and 0.044 MPa (linear-low model), respectively. In the third case, the PDL was assumed to be non-linear (non-linear model): In compression, the PDL was described with a Young’s modulus of 0.005 MPa up to the 93% strain level, after which a Young’s modulus of 8.5 MPa was used to simulate pre-contact between the roots and the surrounding bone. In tension, the Young’s moduli gradually increased from 0.044 MPa at zero strain level to 0.44 MPa at about 50% strain, after which a smaller Young’s modulus of 0.032 MPa was used to simulate fiber disruption (Fig. 2
Loading and Boundary Conditions Two types of orthodontic tooth movement (uncontrolled tipping and translation) were simulated. The loading regimes were defined by the moment-to-force (M/F) ratio applied to the bracket. Uncontrolled tipping was simulated by the application of a force of 0.5 N acting in the buccal-lingual direction for the premolar (M/F = 0), and the same force but in the opposite direction for the canine. For translation, a force of 100 cN in the buccal-lingual direction with a M/F ratio of 10 was applied to the premolar, while the same force but in the opposite direction with a M/F force ratio of 12 was used for the canine (Christiansen and Burstone, 1969; Burstone and Pryputniewicz, 1980). Movement in all directions was suppressed for the nodes situated on the bottom edge of the bone segment.
The material properties of the PDL and the alveolar bone morphology significantly influenced the transfer of orthodontic forces, as demonstrated in the following sets of analyses. (1) Non-linear PDL vs. linear PDL behaviors with 2 different stiffnesses, while assuming the density-based alveolar bone model:
During uncontrolled tipping, the compression and tension sites in the PDL could be clearly identified in the models with the linear material behavior. Though regions of compressive and tensile normal stresses in the PDL were also present in the non-linear model, the stress levels in the ‘compression’ areas were significantly lower than for the linear PDL, while the tensile stresses were substantially higher (Fig. 3A
During translation, the overall displacements of both teeth, expressed at the crown and at the apex levels, were smaller (0.01 mm for the canine in the case of the non-linear model) than in the case of uncontrolled tipping. In the non-linear model, a uniform distribution of tensile stresses on the lingual surface of the PDL (about 0.02 MPa) and a very mild compression on the buccal surface of the PDL (about 0.001 MPa) were seen on the canine. Therefore, the non-linear model did not transfer load along the buccal surface. For both linear models, the magnitude of the compressive stresses was higher than in the case of the non-linear model (Fig. 3B ).
(2) Density-based vs. homogeneous alveolar bone, while assuming non-linear PDL behavior: The homogeneous model showed basically the same load transfer at the bone/PDL interface as the density-based model, for both tipping and translation (Fig. 3
In evaluations of the biological reaction of the periodontal tissues to tooth displacement, it is generally agreed that the restructuring of the periodontium may actually consist of two or more processes occurring simultaneously, and that the cascade of events leading to the remodeling of the osseous alveolus is not necessarily the same as the one leading to the changes within the PDL. The remodeling of the alveolus in the direction of the applied force may be only an indirect consequence of the cellular action, leading to a breakdown and a remodeling of the fibrous tissue of the PDL. In the past, it has been remarked that, in some cases, the "FEM was regarded as a magic tool to solve all problems", and that the distribution of stresses in a model obtained by a FE analysis is not only dependent on the loading configuration but also on the geometry of the structure and the properties of its material (Huiskes and Chao, 1983). Despite this, in some FE analyses, the alveolar bone has been modeled as a homogenous tissue, approximating a regular geometric shape (cube, cylinder, etc.), and the roots shaped as cones or paraboloids (Vollmer et al., 1999; Provatidis and Kanarachos, 2000; Qian et al., 2001). Moreover, the material properties of the PDL have been modeled as linear elastic, with elastic moduli ranging from 0.07 to 100 MPa (Tanne et al., 1987; Andersen et al., 1991; Jones et al., 2001). In recent studies, the PDL was modeled as either a non-linear fiber-reinforced material or with visco-elastic properties (Bourauel et al., 1999; Vollmer et al., 1999; Provatidis, 2000; van Driel et al., 2000; Qian et al., 2001; Pietrzak et al., 2002; Poppe et al., 2002). In the present study, the FE models are based on a µCT dataset providing detailed descriptions of both the external geometry and the internal morphology of the alveolar bone. A non-linear stress-strain relationship based on experimental results was chosen for modeling the PDL (Vollmer et al., 1999; Poppe et al., 2002). Its basic shape, with a low-stiffness toe region and a high-stiffness slope, closely resembles both experimentally (Toms et al., 2002) and mathematically determined relationships (Pietrzak et al., 2002). Visco-elasticity was not considered, because our goal was to determine the stress/strain distribution in the alveolar bone when a visco-elastic steady-state in the PDL is reached. This steady-state is usually attained about 5 hrs after the teeth are loaded, when the flow of the PDL’s interstitial fluid through the surrounding alveolar bone will cause the pressure within the PDL to decrease and the solid phase to carry the load alone (van Driel et al., 2000). Since the concern of this study was to describe the overall behavior of the PDL, rather than its inner composition, no attempt was made to model the PDL as a fiber-reinforced material. Our analyses have shown that the transfer mechanism of orthodontic loads through the alveolar supporting structures cannot be explained in simple terms of compression and tension. The ‘traditional’ compression and tension zones were observed during tipping and translation only when the PDL was modeled as a linear material. With non-linear properties of the PDL, tension was by far more predominant than compression, and the alveolar bone situated in the ‘traditional’ compressive areas was loaded significantly less than the bone in the tension side. This is consistent with the hypothesis of Melsen (2001) and may indicate that direct bone resorption in front of the roots along the movement direction starts as a consequence of a hypo-physiological loading characterized by strain values below the minimum effective strain as defined by Frost (1992). The anatomical shape of the jaw sets the boundaries for the three-dimensional load transfer, and it is therefore of the utmost importance that the real geometry be modeled precisely. In particular, the presence of a thin cortical shell at the alveolar edges gives rise to secondary load transfer mechanisms, resulting in stress distributions that are exactly opposite those that might be expected based on the ‘pressure-tension’ theory (Schwartz, 1932; Reitan, 1951). These phenomena have never been described before, probably because it would be impossible to encounter them in non-anatomical FE models. The FE model of the present study was based on the anatomy of a single donor, and the results should be interpreted accordingly. Therefore, some characteristics of the load transfer might not apply in general, yet the forte of this study was to examine the various aspects of orthodontic load transfer for an actual morphology rather than a generic, and therefore non-specific, model. The reliability of the FE model cannot be checked directly, due to the fact that the jaw segment was obtained at autopsy. However, when the calculated deflections of the crowns are compared with previously reported experimental studies (Christiansen and Burstone, 1969; Jones et al., 2001), the amounts of deflections are of similar magnitude. Whether the steering signals for alveolar bone remodeling are the stresses and strains in the PDL rather than in the bone itself (Bourauel et al., 1999; Jones et al., 2001), or whether alveolar bone remodeling can be explained by mechano-biological principles derived from Frost’s mechanostat theory (Melsen, 2001) remains an open question. Nevertheless, the present FE analyses indicate that alveolar bone remodeling cannot be based on the simplified, yet still generally accepted, concept of resorption due to compression and formation due to tension.
This research was financed by the Aarhus University Research Fund. This paper is based on a thesis submitted to the Faculty of Health Science, University of Aarhus, Denmark, in partial fulfillment of the requirements for the PhD degree. Received for publication February 10, 2004. Revision received January 18, 2005. Accepted for publication January 21, 2005.
Journal of Dental Research, Vol. 84, No. 5,
428-433 (2005) This article has been cited by other articles:
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