INTRODUCTION

Orthodontic anchorage provided by teeth or extraoral structures is used to resist undesirable tooth movement or reaction forces. Adequate anchorage may become difficult, if not impossible, to obtain when teeth are missing. Such cases would benefit from an anchor unit, such as an implant, which could receive forces of enough magnitude to produce movement without becoming displaced by the applied forces.1

Since the introduction of osseointegration, titanium implants have been used successfully in the clinical treatment of edentulous patients. In addition to the prosthetic application of endosseous implants, considerable interest has developed in their use as an anchorage system in orthodontics and dentofacial orthopedics.2 Their application under orthodontic forces has been demonstrated both clinically2–6 and experimentally.^1,7–11

Orthodontic tooth movement during space closure is achieved through 2 types of mechanics. The first type, frictionless, involves closing loops fabricated either in full or sectional arch wires. The teeth move because of the activation of the wire loop, which can be designed to provide a low load-deflection curve and a controlled moment to force ratio (M/F). The second type—sliding, or frictional, mechanics—involves either moving teeth along an arch wire or sliding the arch wire through brackets and tubes.12,13

Elastomeric chains are extensively used in orthodontics to apply forces for canine retraction with frictional mechanics. One of its characteristics is the inability to deliver a continuous force level over an extended period of time. In 1970, Andreasen and Bishara14 demonstrated that after 24 hours of activation, Alastik modules (Unitek, Monrovia, Calif) lose up to 74% of their force. In 1975, Hershey and Reynolds,15 in contrast with Andreasen and Bishara,14 found a 50% force loss after the first day, with 40% of the original force remaining after 4 weeks. In order to compensate for the high and rapid decay rates of elastomers, initial forces must often be 4 times greater than that which is desirable.14

Traditional stress analytical experimental methods, such as photoelasticity, interferometric holography, and strain gauges, have been reported in dental stress analyses.16,17 The finite element method (FEM), a modern tool of numerical stress analytical technique, has the advantage of being applicable to solids of irregular geometry that contain heterogeneous material properties. It is, therefore, ideally suited to evaluate the structural behavior of teeth.18

FEM provides the orthodontist with quantitative data that can extend the understanding of the physiologic reactions that occur within the dentoalveolar complex.19 More specifically, such numerical techniques may yield an improved understanding of the reactions and interactions of individual tissues.20 Such detailed information of the stresses and strains is difficult to obtain and analyze by other experimental techniques because of the interaction with the surrounding tissues.21

The purpose of this study was, therefore, to determine and compare the initial stress profiles in an upper cuspid, an endosseous implant, and their supporting structures with different moments and forces (simulating canine retraction with elastic chains and T-loops) by using the FEM.

MATERIALS AND METHODS

A 3-dimensional, orthogonal, Cartesian model was generated by using the COSMOS/M 1.75 of 64,000 nodes software system. The model was composed of a Branemark endosseous titanium implant with standard dimensions; the implant simulated the position of an upper first permanent molar and an upper cuspid of average size, with periodontal ligament (PDL) and cortical and cancellous bone (Table 1; Figure 1).

TABLE 1. Model Dimensions^a

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The model was divided into 14,953 nodes and 34,109 elements: all of them were tetra 4-R type, a solid element with 4 faces and 4 nodes. Each node had 6 degrees of freedom—3 rotations and 3 translations. Homogeneous, isotropic, and linearly elastic behavior was assumed for all the materials (Table 2). Boundary conditions were located at the floor of the nasal cavity and maxillary sinus and at the lateral surface of the maxillary bone (Figure 1c and d).

TABLE 2. Mechanical Properties for the Titanium, Tooth, Periodontal Ligament (PDL), and Cortical and Alveolar Bone^a

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To apply the forces, 2 rigid structures simulating the bracket slot and the tube were modeled (Table 3; Figure 2). In the center of each rigid structure, 14 different forces were applied in a mesio-distal direction: the first 2 simulated the retraction of the upper cuspid with elastic chains.14,28 The other 12 simulated the retraction by a segmented 0.017 ;ts 0.025 inch T-loop in a TMA wire (Ormco Corporation, Glendora, Calif) designed to produce equal and opposite moments when the loop is in the centered position29,30 (Tables 4 and 5).

TABLE 3. Point of Force Application

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TABLE 4. Force Application (Force and Moments)^a

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The stresses and deflections were determined at different mesio-distal and occluso-gingival levels. In addition, specific stress values were evaluated in the different nodes of the root surface, PDL, cortical bone, and implant from the apical zone to the cervical margin.

Results were expressed as Von Misses stresses (combined effect of the different stresses) and compressive and tensile stress in the x- and y-axes. Principal stresses—maximum (P₁), intermediate (P₂), and minimum (P₃)—were evaluated, but they were similar in shape and magnitude to the stresses examined in the x- and y-axes (P₁ stresses were mostly tensile, P₃ were compressive, and P₂ were an intermediate value between P₁ and P₂).

RESULTS

The behavior of the evaluated structures under single forces or moments were linearly or directly proportional to the applied load. To simplify the analysis, loads 2 and 4, which corresponded to the maximum force and moment, and the combined loads 51, 56, 78, and 91 with different M/F ratios were evaluated (Tables 4 and 5).

Von Misses stresses in the evaluated loads showed the highest stresses on the implant and the cortical bone at the cervical third. The lowest stresses appeared at the apical third of the implant. By structure, the highest stress was observed in the implant, followed by the cuspid, the cortical bone, and finally, the periodontal ligament (Figure 3). Stress magnitudes were higher with a single force or moment than when the combined loads with different M/F ratio were considered.

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The root, the implant, and its cortical bone exhibit large bending stresses in the y-axis (Figures 4–6). The implant behaved as a rigid structure, with the highest stresses concentrated in the first cervical screw, declining steadily to the apex. On the other hand, at the canine, the highest stresses were found between the middle and the apical third, with a gradual reduction to approximately 0 at the cervical margin and at the apex. This could be caused by the differences in rigidity between the titanium and the dentin.

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In Figures 4, 5, and 6, load 2 (4.9 N) produced compression, whereas the single moment (load 4; 20.6 N/mm) produced tension. Low stress magnitudes were seen in the different moment-force ratios analyzed. In load 51 (moment-force ratio of 6.1:1), the single force profile predominated, whereas in loads 56, 78, and 91, moment effects prevailed (Tables 4 and 5). Stress distribution on the mesial and distal sides showed almost symmetrical, but opposite, behavior.

Evaluation of the x-axis on the cortical bone around the implant revealed that the cervical margin and the bone around the first screw of the implant were the major stress points (Figure 7).

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An analysis of the structures showed that the PDL presented the least stress magnitude, having a similar pattern of stress distribution at both in the PDL-tooth and PDL-bone interfaces and on the x- and y-axes. At the same time, stress distributions for the PDL and lamellar bone were similar, although lamellar bone stresses were of greater magnitude and had a more irregular distribution (Figures 8 and 9). To establish a relation between stresses and displacements (Figure 10), the analysis was done on the x-axis.

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Stress curves of the PDL cross the vertical axis, representing the center of rotation during tooth movement. This area showing 0 stresses varies with the different moment-force ratios and with the type of movement (Figures 8 and 10; Table 6). The stress profiles for the evaluated loads in the mesial side of the root surface, PDL, and cortical bone were similar but opposite in sign.

TABLE 6. Periodontal Ligament Stresses and Displacements (Distal Side) in the X-Axis for Different Loads

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DISCUSSION

One of the remodeling theories argues that local mechanical signals stimulate or induce regulating cells to trigger remodeling events.35 This implies the importance of studying stress levels from orthodontic forces in the root surface, PDL, and cortical bone and those that occur in the anchor unit like bones, or, as in this case, in an osseointegrated implant.

Overall, the highest stresses occurred in the implant, followed by the canine root surface and cortical bone and finally, the PDL (Figure 3). This could have been caused by the geometric differences and the different mechanical properties of the implant, tooth, cortical bone, and PDL. In 3-dimensional FEM studies, Tanne et al20 and Puente et al31 reported the same distribution, with the exception that the osseointegrated implant that was not modeled.

The highest stress concentration in the implant was localized in the cervical margin and at the first screw (1.2 mm apically from the cervical margin). Similar findings had been presented by Meijer et al,27 Barbier et al,32 and Clelland et al.33 However, it is important to point out that these stresses are of such low magnitude that they are unable to produce a failure in the implant. Therefore, the osseointegrated implants are able to withstand orthodontic forces and may function as adequate anchor units. Their stress magnitude and distribution are better when a force and a moment are applied together (Figure 4).

Thus, in osseointegrated implants designed to serve as orthodontic anchors, the functional and structural union of titanium to the cervical bone should be free of bacterial plaque accumulation because this preserves cervical osseointegration (Figures 5 and 7).

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Large bending stresses were observed in the canine root surface, but not in the PDL. Tanne et al20 found similar types of stresses at the surface of the root and at the alveolar bone. The differences found could be because they did not model the cortical bone (Figure 6).

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A simple horizontal force produces tipping of the tooth, with its stress concentrated at the cervical margin and at the apex of the tooth. This force produced an area of zero stresses in the PDL and its cortical bone near the middle of the root (center of rotation).20,21,31 This result agrees with the histologic findings that show that the cell-free area and hyalinization of the PDL are typically induced at the cervical and apical zones,34 which are high-stress concentration sites. A single moment produced a similar tipping of the tooth, but in an opposite direction (Figures 8 and 10; Table 6).

There was a lower magnitude and more uniform stress distribution when the moment and force were applied together. The stress produced in the periodontal ligament was related to the tooth movement (Table 6). The M/F ratio of 6.1:1 induced an uncontrolled tipping of the tooth (with distal crown and mesial root movement), whereas the M/F ratio of 10.3:1 produced a translation movement, and the bending stresses were significantly reduced in magnitude (Figure 6). The stresses in the PDL were compressive and more constant in the translation movements than in the tipping ones (Figures 8 and 10), making the first more physiologic.20

When the M/F ratio increased to 13.9:1 or to 26.4:1, uncontrolled root movement with a mesial displacement of the crown was observed (Figure 10). The M/F ratio determined the type of movement and its center of rotation. Burstone et al36 described similar findings.

During tooth movement, nonlinear elastic, plastic, and viscoelastic phenomena can occur.34 In this article, only linear elastic behaviors of the tooth-periodontal structures are considered.20,35 Therefore, in the future, additional modeling may be needed along with nonlinear elastic analysis.20,35 However, the model does provide quantitative results of the complex 3-dimensional stresses caused by mesio-distal forces during orthodontic treatment. The model allows areas of high stress to be accurately located, and these are considered to be relevant in understanding the long-term behavior of orthodontic tooth movement.35

CONCLUSIONS

Overall, the area with the highest stress was the cervical margin of the osseointegrated implant and its cortical bone. These stresses are of such low magnitude that they are unable to produce a permanent failure of the implant. Large bending stresses were found along the surface of the canine and at 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 center of rotation of the tooth, which varies with the different moment-force ratios and with the type of movement.

The combined loads with different M/F ratios produced the lower and more uniform stress distribution. The M/F ratio 6.1:1 had the best stress distribution in the implant and its cortical bone. The same was observed in the canine and its supporting structures with a M/F ratio of 10.3:1. Then, when the anchor unit is an endosseous implant, it is better to use a precalibrated retraction system without friction (T-loop) and with a low load-deflection curve in which the M/F ratio is known.

In the future, when the cell reaction to variable stress levels is known for each structure in the stomatognathic system, studies like this will have a big influence in clinical orthodontics.