The characteristic invisibility of lingual appliances,

Transcription

1 ORIGINAL ARTICLE A comparative evaluation of different compensating curves in the lingual and labial techniques using 3D FEM Sang Jin Sung, DDS, PhD, a Hyoung Seon Baik, DDS, PhD, b Yoon Shik Moon, DDS, PhD, c Hyung Seog Yu, DDS, PhD, d and Young Soo Cho, PhD e Seoul, Korea Because adults dislike the visibility of orthodontic appliances, the use of the lingual orthodontic technique has increased over time. But few studies compare tooth movement of the lingual technique with that of the labial technique. In this study, human mandibular left teeth were aligned, and a 3-dimensional finite element model was made (consisting of nodes and 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. 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. (Am J Orthod Dentofacial Orthop 2003;123:441-50) The characteristic invisibility of lingual appliances, because the bracket is bonded to the lingual surface of the teeth, has fascinated the public and many orthodontists. 1,2 However, the lingual fever of the early 1980s began to diminish because cases finished by the lingual technique did not meet the same standards of excellence as those finshed by the labial technique. 3 Recently, with the improvement in the indirect bonding technique of lingual brackets, the development of new archwire material, and the application of computer systems, the lingual technique has become more precise and simpler than before. 4-7 Therefore, the lingual technique has once again gained popularity with a Assistant professor, Department of Dentistry, University of Ulsan, Seoul. b Professor and Chairman, Department of Orthodontics, College of Dentistry, University of Yonsei, Seoul. c Professor and chairman, Department of Dentistry, University of Ulsan, Seoul. d Assistant professor, Department of Orthodontics, College of Dentistry, University of Yonsei, Seoul. e Research professor, Department of Mechanical Engineering, University of Hanyanng. Reprint requests to: Hyoung Seon Baik, Department of Orthodontics, College of Dentistry, Yonsei University, 134 Shinchon-Dong, Seodaemun-ku, Seoul, Korea, ; , baik@yumc.yonsei.ac.kr. Submitted, January 2002; revised and accepted, April Copyright 2003 by the American Association of Orthodontists /2003/$ doi: /mod patients who want to receive less visible orthodontic care. Compared with the labial technique, the lingual technique has some restrictions in applying orthodontic biomechanics or using various orthodontic appliances. Additionally, it is known that tooth movement along the archwire is different. For example, for mandibular incisor intrusion, because the bracket is placed on the lingual surface of the teeth, and the application of the intrusion force is directed near or lingual to the center of resistance, labial tipping of the mandibular incisor might be reduced or even tipped lingually. 3,8 During canine retraction with the lingual technique, vertical bowing can result from lingual tipping of the incisors and mesial tipping of the molars. Transverse bowing can also occur from rotation of the canine and buccal displacement of the premolars. To reduce the vertical and transverse bowing effects, a compensating curve (CC) or reverse curve of Spee can be placed in the archwire. 9 But there have been little data concerning the effects and amount of CC required to reduce the vertical and horizontal bowing effects. If we look at tooth movement caused by applying orthodontic forces as mechanical engineers analyze an object and its supporting structure, we can take advantage of laboratory work 10 or theoretical computation, such as the finite element method (FEM), to 441

2 442 Sung et al April 2003 Fig 1. A, Alignment of teeth with straight-wire appliance; B, cross-sectional tooth image on CT image. analyze the reaction of the structure (teeth, periodontal ligament [PDL], and alveolar bone) to the loading (force). In this study, a 3-dimensional (3D) FEM model of the mandibular left dentition was made to analyze the effect of the CC with the lingual technique compared with the labial technique. The direction and the amount of CC to reduce the vertical and horizontal bowing effects during canine retraction were also estimated. MATERIAL AND METHODS Human teeth were selected according to Wheeler s dental anatomy. 14 Brackets with Roth prescription (.018-in slot) were bonded to the midpoint of the facial axis of the clinical crown of each tooth. A preformed archwire with a Tru-arch (Ormco, Orange, Calif) form of in stainless steel (SS) was ligated to the brackets, and the tooth axis was aligned (Fig 1, A). After completing the alignment, the teeth were fixed in elastic rubber impression material, and computerized tomography (CT) images were taken with 1 mm intervals (Fig 1, B). To confirm the position of the bracket and the height of the alveolar bone, beaded restorative resin was bonded to the FA point, and a 1-mm wide resin band was bonded 1.5 to 2 mm below the cementoenamel junction. 15 After taking the CT images, Fujita lingual brackets (Citizen, Iwate, Japan) were bonded to the teeth. The lingual brackets served as a reference for the lingual arch form. For the 3D FEM modeling, the outline of a crosssectional tooth image was constructed with 40 coordinates. All measurements were converted to real size that was based on the vertical and horizontal scale in the CT image (Fig 1, B). The X-axis is the midsagittal line of the dental arch on the occlusal view. The Y-axis is perpendicular to the X-axis, and its origin is settled at a point 3.6 mm distal from the second molar (Fig 2, A). The origin of the Z-axis is the cementoenamel junction. A positive value indicates crown direction, and a negative value indicates root direction (Fig 2, B). A computer program was used to refine the geometric morphology and the 3D alignment of the teeth by manipulating key points. The corrected morphology and alignment of the tooth model can be visualized and inspected in the finite element program. The node coupling technique is introduced to represent the contact point effect between the second premolar and the first molar in the direction of anchor loss (X-axis, Fig 2, A). Teeth, PDL, and alveolar bone are assumed to be isoparametric and homogeneous materials. With reference to previous research, the PDL was considered to have a uniform thickness (0.25 mm). 12 The type of finite element used in the analysis was an 8-noded hexahedron. The element division was performed with the mapped mesh scheme. The properties of different materials in this study were based on a review of the literature (Table I). 12,13,16 Because the interbracket distance is shorter in the lingual technique than in the labial technique, an archwire with less stiffness was used. 17 To compare the different tooth reactions between the labial and lingual techniques under the same conditions, we assumed the main archwire to be a.016-in SS round wire and defined it as the 3D beam element. To permit sliding of the bracket along the archwire and the reciprocal 3D displacement between the bracket and the archwire, the link element is defined between the node on the mesial and distal ends of the bracket and the corresponding node of the labial or lingual archwire parallel to the direction of normal to the FA point on the X-Z plane. The definition of the CC put in the archwire is as follows: The wire between the distal of the canine bracket and the distal of the second molar tube has a

3 Volume 123, Number 4 Sung et al 443 Fig 2. 3D FEM model and boundary conditions in X-Y and X-Z planes. A, Occlusal view; B, lingual view. T, Teeth; B, bone; Lab, labial bracket; Lib, lingual bracket; W, labial wire; Sc, supporting conditions; Nc, node coupling. Table I. Material properties Young s modulus (g/mm 2 ) Poisson s ratio Teeth 2.6E6 0.3 Periodontal ligament 5.0E Alveolar bone 1.4E6 0.3 Stainless steel 19.2E6 0.3 constant compensation curvature. As shown in Figure 3, the curvatures in the X-Y and X-Z planes can be determined by equation. 18 The calculated radii of arc as a function of position of the archwire, amount of CC, and archwire length are summarized in Table II. The end of the CC was a direct line of constant curvature to the lingual side of the dental arch and to the apex of the second molar in the labial archwire, to the buccal side of the dental arch and to the apex of the second molar in the lingual arch wire (Fig 4). The 150 g of distal force was applied from the distal wing of the canine to the mesial wing of the first molar bracket. The deactivation forces from the archwire with CC were applied to the mesial and distal wings of each bracket according to the FEM test conditions. The entire model is fixed at the distal surface of the model. Fig 3. Definition of CC put in archwire. L, Length of archwire; H, amount of compensating curve: 8Hr 3 - (20H 2 L 2 )r 2 16H 3 r-4 H 4 0. In Figure 2, the supporting conditions are indicated by many small triangles distal to the second molar area. The amount of deformation of teeth and alveolar bone was far less than the displacement of teeth that resulted from the deformation of the PDL. Therefore, the amount of tooth displacement after orthodontic force application was analyzed. To express the displacement analyzed by FEM

4 444 Sung et al April 2003 Table II. Radii of arc as function of position of archwire, amount of compensating curve, and arch length (unit, mm) Position of archwire Amount of compensating curve (H) Archwire length (L) Radius of arc (r) Labial Lingual schematically, coordinates of all nodes that construct the outline of the crown at the level of 5.4 mm on the Z-axis and each node in the center of the root apex were graphed on the occlusal view (X-Y plane) and the sagittal view (X-Z plane) (Fig 5). The X, Y, and Z coordinates of each node after the displacement of teeth were assumed and calculated by adding the initial displacement of the node that was magnified 10,000 times to the X, Y, and Z coordinates of each node before displacement (Fig 6). The direction of tooth rotation follows the righthanded rule in physics. If the direction of the thumb coincides in a positive direction of the Z axis, then the direction of the clenching 4 fingers was defined as positive ( ) rotation. The counterclockwise tooth rotation on the occlusal plane is described as the Z direction (Fig 2). The distribution of compressive and tensile stresses was observed by a contour plot for principal stress. The area displaying the maximum positive principal stress was analyzed as the maximum tensile stress area, and the minimum negative principal stress was analyzed as the maximum compressive stress area (described as the negative pressure). The number of nodes (elements) comprising the model was as follows: teeth 5460 (3744), PDL 4224 (1920), alveolar bone 9019 (6264), bracket 648 (192), and archwire 31 (30). For the analysis of FEM, Ansys version 5.3 (ANSYS, Canonsburg, Pa) software and a Pentium II personal computer were used. RESULTS The deactivation force from the wire that corresponded to the mesial and distal node of each bracket is summarized according to wire diameter and amount of CC in Table III. The result that the total summation of force developed in the Y axis approaches zero proves that the deactivation force from the wire satisfies an equilibrium condition. A similar result is also obtained in the case of force developed in the Z axis. The wire has zero torsional reactions because its cross section is circular, and the friction between the wire and the bracket is ignored. The deactivation force acting on the brackets of the second premolar and the first molar was less than that of the canine and the second molar (Table III). Thus, it could be deduced that the deactivation force from an archwire with a CC initially acts on the brackets engaged on each end of the CC. When the amount of deactivation force on a node of each bracket was compared in the same direction, the.016-in SS with a 4-mm CC was about 1.6 times greater than the 2-mm CC (Table III). This almost agreed with the theory that the deactivation force of the archwire is proportional to its curvature (1:radius). 19 By increasing the diameter of the labial archwire (to.018 in), the deactivation force also increased. The deactivation force from the lingual archwire was less than from the labial archwire. In the labial technique, the canine was rotated mesially out ( Z), and the first molar was rotated mesially in ( Z) on the occlusal view (X-Y plane). The canine was tipped distally ( X), and the first molar was tipped mesially ( X) on the sagittal view (Fig 6). Compressive stress of g/mm 2 was observed on the distolingual surface of the canine cervix, and tensile stress ( 5.36 g/mm 2 ) was seen on the apex (Fig 7, A). On the mesial surface of the first molar cervix, compressive stress of g/mm 2 was observed, and the stress was distributed to the level of the apical half (Fig 7, B). In the lingual technique, the canine was rotated mesially in ( Z), and the first molar was rotated mesially out ( Z) on the occlusal view (X-Y plane). The mandibular central and lateral incisors were displaced lingually ( X). On the sagittal view, the canine and the first molar were tipped similarly as in the labial technique but more severely (Fig 6). Greater compressive stress was distributed more narrowly on the root cervix of the canine ( g/mm 2 ) and the first molar ( g/mm 2 ) than in the labial technique (Fig 7, C and D). In the labial technique, the incisors were displaced lingually ( Y), and the canine was rotated mesially ( Z) on the occlusal view. On the sagittal view, the canine was tipped mesially ( X), and the second molar and the lateral incisor were tipped distally ( X). In the lingual technique, the incisors were displaced labially ( Y) and buccally ( X), and the canine was rotated in the counterclockwise ( Z) direction on the occlusal view. But, in the sagittal view, the antitip effect of the CC was slight.

5 Volume 123, Number 4 Sung et al 445 Fig 4. 3D FEM model of archwire with 4-mm CC. A, Occlusal view; B, posteroanterior view. LaW, Labial archwire; LiW, lingual archwire. Fig 5. Schematic displacement graph. Coordinates of all nodes that construct outline of crown and all nodes in center of root apex graphed on occlusal (X-Y plane) and sagittal (X-Z plane) views. The second premolar and the first molar were rarely displaced in the labial and lingual techniques (Fig 8). In comparison to the canine retraction on a.016-in SS archwire with a 0-mm CC, as the amount of CC increased (2 and 4 mm) in the labial archwire, counterclockwise rotation of the canine was reduced, and the lateral incisor and the canine were displaced lingually ( Y) on the occlusal view. On the sagittal view, the distal tipping ( X) of the canine was slightly reduced (Fig 9). During canine retraction with a 4-mm CC, on the labial archwire, the compressive stress on the distolingual surface of the canine cervix was reduced to g/mm 2 and distributed evenly to the level of the apical half (Fig 7, E). As the amount of CC increased in the lingual archwire (0, 2, 4 mm), the lingual displacement ( X) Fig 6. Schematic displacement graph during canine retraction on.016-in SS labial and lingual archwire with 0-mm CC (unit, mm). of the incisors and the counterclockwise rotation of the canine were reduced on the occlusal view. The tipping of the canine and the second molar was slightly reduced, but there was little effect on the second premolar and the first molar in the sagittal view (Fig 10). During canine retraction with a 4-mm CC, more compressive stress was observed on the canine cervix in the lingual technique ( g/mm 2 ) than in the labial technique ( g/mm 2 ) (Fig 7, F). The amount of compressive stress on the canine cervix was not reduced, but the tensile stress on the distal of the canine apex disappeared. DISCUSSION Moran 20 reported that the lingual to labial interbracket distance ratio was reduced to 1:1.47; therefore,

6 446 Sung et al April 2003 Table III. Deactivation force from.016-in and.018-in stainless steel archwire with 2- and 4-mm compensating curve (cc) corresponding to mesial and distal node of each bracket (unit, g)..016-in.018-in Archwire Node of bracket CC2mm CC4mm CC2mm CC4mm Fy Fz Fy Fz Fy Fz Fy Fz Labial Central incisor mesial distal Lateral incisor mesial distal Canine mesial distal Second premolar mesial distal First molar mesial distal Second molar mesial distal Lingual Central incisor mesial distal Lateral incisor mesial distal Canine mesial distal Second premolar mesial distal First molar mesial distal Second molar mesial distal Fy, deactivation force acting in direction of Y axis. Fz, deactivation force acting in direction of Z axis. the lingual archwire size during the initial phase of treatment should be reduced. However, archwires used for space closure and final leveling must be selected on the basis of the traditional stiffness requirements necessary in the posterior segments. In this study, Fujita lingual brackets were used for FEM modeling, but various slot directions (occlusal, horizontal, vertical) and size were not considered. Only the actual bracket width was applied to observe the effects of the reduced interbracket distance. The lingual archwire form is mushroom shaped with a prominent premolar offset and a weak canine and molar offset bend, along with a vertical step bend between the canine and the molar. 21,22 In this study, the lingual archwire was simply formed with only a premolar offset and no vertical step. In space closure with the lingual technique, en masse retraction of the 6 anterior teeth is preferred because it is difficult to completely move the canine when there is a premolar offset, and the space distal to the lateral incisors after canine retraction seems to be unesthetic. However, with anterior crowding, regardless of the orthodontic technique, partial canine retraction should be done to relieve the crowding. In these cases, various sizes of archwires, such as.016 in,.018 in, in, and in SS, are preferred for the lingual technique. 9,21,22 In this study, to establish a condition similar to that of the labial technique and to compare the differences during canine retraction, we used a.016-in SS archwire. The frictional resistance of sliding archwires against brackets and ligature material should also be considered. The contact, or gap, element has been used to simulate the friction condition These elements restrict compression displacement between its master (bracket) and slave (wire) nodes, while tensile displacement is free. But because it is impossible to measure the real friction between wires and brackets and to simulate in vivo conditions, 26 so the friction element for wire and bracket has not been developed. The material

7 Volume 123, Number 4 Sung et al 447 Fig 7. Distribution of minimum principal stress on root during canine retraction with.016-in SS archwire. A, Compressive stress of g/mm 2 on distolingual surface of canine cervix; B, compressive stress of g/mm 2 on mesial surface of first molar cervix; C, greater compressive stress of g/mm 2 on distolingual surface of canine cervix (more narrow than in Fig 7, A); D, greater compressive stress of g/mm 2 on mesial surface of first molar cervix than in Fig 7, C; E, compressive stress on distolingual surface of canine cervix reduced to g/mm and distributed evenly to level of apical half; F, amount of compressive stress on canine cervix ( g/mm 2 ) was not reduced compared to Fig 7, C, but tensile stress on apex disappeared (Labial, labial archwire; Lingual, lingual archwire; CC, compensating curve). properties and the geometries of the model should be evaluated correctly to obtain appropriate results because these elements have nonlinear behavior. Therefore, to exclude the contradiction of a nonlinear analysis of materials defined as linear property elements and to reduce calculation time, it was assumed

8 448 Sung et al April 2003 Fig 8. Schematic displacement graph after application of deactivated.016-in SS labial or lingual archwire with 4-mm CC (unit, mm). Fig 10. Schematic displacement graph during canine retraction on deactivated.016-in SS lingual archwire with 2- and 4-mm CC (unit, mm). Fig 9. Schematic displacement graph during canine retraction on deactivated.016-in SS labial archwire with 2- and 4-mm CC (unit, mm). that there was no frictional resistance to sliding archwires against the brackets and no clearance between them. For the FEM analysis of the teeth and the surrounding complex, the correct definitions of Young s modulus and Poisson s ratio are critical, especially the PDL, because the deformation of the PDL was greater than the hard tissues (teeth and bone) under the 100 to 200 g of orthodontic force. Because it was reported that Young s modulus of the PDL ranges variously from 7 to 175,000 g/mm 2, it could be 1000 times less than the other hard tissues. 27,28 In these circumstances, it would have an effect on the accuracy of the data. Rees and Jacobsen 16 modeled 2 different in vivo tooth-loading systems described independently by Tanne 11 (horizontal loading) and Picton 29 (vertical loading) using the FEM. They reported that an elastic modulus of 50 Mpa (5000 g/mm 2 ) gave a good correlation between the FEM model and the experimental systems. In this study, those data were used also. In this study, during canine retraction on.016-in SS labial archwire with a 0-mm CC, the central and lateral incisors showed a tendency to displace buccally along with the archwire in the occlusal view. This might be caused by the buccal ( Y) deflection of archwire from the mesial of the canine bracket followed by the counterclockwise rotation of canine. In the lingual technique, this effect was the exact reverse of that in the labial technique (Fig 6). The canine tipped more (about 39 ) in the lingual archwire than in the labial archwire (Fig 6). Even though the interbracket distance between the canine and

9 Volume 123, Number 4 Sung et al 449 Fig 11. Arch rolling. A, Lingual arch form is shaped with prominent premolar offset (buccolingual adjustment); B, activated CC has little power to provide intended deactivation force by arch rolling. the second premolar was shorter in the lingual archwire than in the labial archwire, the premolar offset bend increased the length of the lingual archwire to 1.27 times more than that of the labial archwire (Figs 2 and 4). Therefore, the reduced wire stiffness might have led to the tipping of the canine. For these reasons, CC should be used to control the side effects of canine retraction, especially in the lingual technique. Berman 30 commented that he applies a CC of about 6 or 7 mm (at the premolar part) in the maxillary labial arch and 3 or 4 mm in the mandibular arch. But the amount of CC for the lingual archwire has not yet been reported. In this study, the effect of an archwire with a 4-mm CC on the teeth and arch form was observed (Fig 8). In the occlusal view of the labial archwire with a 4-mm CC, there was lingual displacement of the canine and the anterior teeth rather than buccal expansion of the premolar area. But in the lingual archwire, there was buccal displacement of the canine and the anterior teeth. This might be caused by a greater anchorage value on the posterior teeth than on the anterior teeth. In the sagittal view of the labial archwire with a 4-mm CC, the canine tipped mesially, and the lateral incisor tipped distally. This might be caused by the deflection of archwire from the mesial of the canine bracket that generated a rebounding force to the lateral incisor bracket. But the canine showed almost no mesial tipping by a lingual archwire with a 4-mm CC. This effect might be caused by arch rolling. Subtle effects occur when a mesiodistal tipping action is combined with buccolingual adjustment, as in Figure 11. The long span that has been twisted distal to the offset has little power to return the archwire to its original form to provide the intended compensation. 31 Table III also shows that the deactivation force (Fz: 8.23 g) from a lingual archwire with a 4-mm CC corresponds to the distal of the canine bracket that was weak. Therefore, there is a need to develop a new lingual appliance to reduce the size of the premolar offset or an archwire design that can resist the twisting force on the X-axis. The effect of CC on the molars that extends to the second molar does not efficiently control the rotation and tipping movements of the first molar against the canine retraction force. Therefore, a rigid SS lingual arch connecting the first molars or a sectional arch splinting the first molar with the second molar on the buccal side will be helpful. The principal (compressive or tensile) stress distributed on the canine root surface during canine retraction with 150 g of force was displayed below 60 g/mm 2 on the contour plot. The direction of the principal stress was not normal to the root surface, and the color only expresses the amount of force (Fig 7). Thus, the compressive or tensile stress on the cervix might imply rotation of a tooth, and the total amount of force distributed on the compressive area of the canine root was not equal to the retraction force of 150 g. Principal stress is fairly clear in engineering for certain applications such as static loading or fatigue failure, but the most appropriate type of stress to represent the biological conditions of teeth, PDL, and alveolar bone is unknown. To simplify the results in this FEM study, we simultaneously applied CC in the apical and the labiolingual directions. Because the results were similar to that of an archwire compensated separately in either direction, there seems to be no significant difference in the data collected. We also used the half model of dentition and set up the boundary condition to shorten the calculation time based on the mechanical engineering concept for the analysis of a half model for a symmetric structure. The schematic displacement graph magnified 10,000 times is useful for understanding the relative displacements and direction of the teeth because the

10 450 Sung et al April 2003 initial displacement calculated by FEM was restricted within narrow limits ( mm). If the empirical tooth movement on a continuous archwire disagrees with the schematic displacement graph, an important consideration is that the magnified displacement is not actual tooth movement, but an initial reaction that does not simulate a time-dependent reaction. The tooth reaction from lingual mechanics can be analyzed by tooth displacement and stress distribution on the root using the FEM. Therefore, harmonizing the results of the FEM with the clinical experience of labial orthodontics will be very useful in predicting the clinical effects of certain lingual mechanics that are derived from conventional labial mechanics. CONCLUSIONS 1. As the amount of CC increased (0, 2, 4 mm) in the archwire, the rotation and the distal tipping of the canine were reduced. 2. When the canine was retracted on a.016-in SS archwire with a 4-mm CC, the compressive stress on the distal root surface of the canine was reduced and distributed evenly to the level of the apical half with the labial technique, and the tensile stress affecting the apical area disappeared. 3. The antitip and antirotation action of the CC on the canine retraction was greater in the labial archwire than in the lingual archwire. REFERENCES 1. Fujita K. New orthodontic treatment with lingual bracket and mushroom archwire appliance. Am J Orthod 1979;76: Kurz C, Swartz ML, Andreiko C. Lingual orthodontics: a status report. Part 2. Research and development. J Clin Orthod 1982; 16: Kurz C, Romano R. Lingual orthodontics: historical perspective. In: Romano R, editor. Lingual orthodontics. Lewiston (NY): B. C. Decker; p Hong RK, Soh BC. Customized indirect bonding method for lingual orthodontics. J Clin Orthod 1996;30: Kim TW, Bae GS, Cho J. New indirect bonding method for lingual orthodontics. 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