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Effects of transpalatal arch on molar movement produced by mesial force: A finite element simulation
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Effects of transpalatal arch on molar movement produced by mesial force: A finite element simulation Yukio Kojimaa and Hisao Fukuib Nagoya, Japan Introduction: The transpalatal arch (TPA), which splints together 2 maxillary molars, has been believed to preserve anchorage. The purpose of this study was to clarify this effect from a mechanical point of view. Methods: The finite element method was used to simulate the movement of anchor teeth subjected to mesial forces with and without a TPA. Results: In the initial movement produced by elastic deformation of the periodontal ligament, stress magnitude in the periodontal ligament was not changed by the TPA. In the orthodontic movement produced by bone remodeling, the mesial force tipped the anchor teeth irrespective of the TPA. The tipping angles of anchor teeth with and without the TPA were almost the same. The anchor teeth without the TPA were rotated in the occlusal plane and moved transversely. Conclusions: The TPA had no effect on the initial movement. In the orthodontic movement, the TPA had almost no effect, preserving anchorage for mesial movement. However, the TPA prevented rotational and transverse movements of the anchor teeth. These results are valid when the assumptions used in this calculation are satisfied. (Am J Orthod Dentofacial Orthop 2008;134:335.e1-335.e7)
T
he transpalatal arch (TPA), by which 2 maxillary molars are splinted together, has many functions in orthodontic treatment.1-7 When an activated TPA is placed between molars, forces and moments produced by the TPA correct arch width and inclination of the molars.3-7 When a passive TPA is placed, it prevents both rotation and buccolingual tipping of the molars, and also maintains the transverse distance of the molars. These functions are expected because of the mechanical rigidity of the TPA. On the other hand, a function that the TPA preserves anchorage for mesial movement is not obvious, because molars can move or tip mesially together with the TPA.1,2 In this study, we focused on the “preserve anchorage” function of the TPA, which has not been verified from a mechanical point of view. The first quantitative analysis on the “preserve anchorage” function was carried out by Bobak et al.8 They calculated periodontal stresses in molars with and without a TPA during initial tooth movement, and a
Associate professor, Department of Mechanical Engineering, Nagoya Institute of Technology, Nagoya, Japan. b Professor, Department of Dental Materials Science, School of Dentistry, Aichi-Gakuin University, Nagoya, Japan. Reprint requests to: Yukio Kojima, Department of Mechanical Engineering, Shikumi College, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan; e-mail, [email protected]. Submitted, December 2007; revised and accepted, March 2008. 0889-5406/$34.00 Copyright © 2008 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2008.03.011
showed that a TPA affected periodontal stress magnitude by less than 1%. This result suggested that the TPA had no effect on the “preserve anchorage” function. However, estimates based on initial movement might be inappropriate, because the initial force system changes as the teeth move. This was shown in our numeric simulations with the finite element method.9-11 The TPA should be estimated on the basis of orthodontic tooth movement. In this study, to develop the results of Bobak et al,8 we calculated orthodontic movements of anchor teeth with and without a TPA subjected to mesial forces using a method from our previous articles.9-11 The effects of the TPA on the “preserve anchorage” function were examined from a mechanical point of view. MATERIAL AND METHODS Tooth element for calculating stresses in the periodontal ligament
All biologic reactions produced in the periodontal tissue during orthodontic tooth movement have not been clarified. Therefore, we proposed a phenomenologic method to simulate tooth movement induced by a mechanical stimulus in the periodontal ligament (PDL). In this method, a tooth moves as a result of absorption and apposition of the alveolar bone, which is produced by the stress distribution in the PDL. By accepting some assumptions described in Figure 1, the stress distribution in the PDL can be calculated by a 335.e1
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Fig 1. Assumptions and a surface model of a tooth used to calculate stress distribution in the PDL. By using the surface model, the stiffness matrix of the tooth element is calculated.
surface model of the tooth. For this purpose, the relationship between movements u and forces f at the bracket position is calculated. This is written as Ku ⫽ f, where K is the stiffness matrix in the finite element method. In other words, the mechanical response of a tooth supported by the PDL can be replaced by a single element having the K. This is a substructure element in the finite element method and called the “tooth element” in this article. The K is obtained by the procedure below. 1. Displacement (translation or rotation) imposes on the tooth at the bracket position. 2. Strains (tension or compression) for each triangular region are calculated from the rigid kinematic motion of the root surface, and the stress distribution in the PDL is calculated by using Young’s modulus and Poisson’s ratio of the PDL. 3. Summing up the stress in each triangular region over the root, forces and moments acting on the tooth are obtained. Detailed mathematical formulation for this procedure was explained our previous article.11 Analysis model
The maxillary first molars were splinted together by a TPA with a 5-mm omega loop (Fig 2). Width and depth of the TPA were 38 and 14 mm, respectively. The TPA was made of 0.036-in round wire of stainless steel (Young’s modulus: 200 GPa). The TPA was fixed
Fig 2. Finite element model of maxillary molars splinted with a TPA. The tooth element is connected to the TPA with a rigid beam element. Mesial forces (1 N) are applied to the molar brackets.
firmly to the molars. Then the molars were connected to the second premolars with a 0.016-in square stainless steel wire. When the TPA was placed, no forces or moments acted on the molars; the TPA was passive. Mesial forces of 1 N were applied to the molars at the bracket positions. Forces and moments from adjacent
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teeth are not considered. Also, forces from the tongue and the palate did not act on the TPA. The left half of the arch was modeled because of symmetry at the centerline. Surface models of the tooth were made based on a dental study model (AM-10, Nissin Dental Products, Kyoto, Japan). Root surface areas of the molar and the premolar were 430 and 223 mm2, respectively. The TPA was divided into 3-dimensional elastic beam elements. A tooth element for the first molar was connected to the TPA with a rigid beam element. The degrees of freedom of the tooth element were 6 and equal to that of the 3-dimensional beam element. Therefore, the tooth element could be connected directly to the beam element. Then a tooth element for the molar was connected to that for the second premolar with a wire divided into elastic beam elements. By using these finite element models, stress distributions in the PDL were calculated when the mesial forces act on the molars. Calculation of orthodontic tooth movement
Bone remodeling was assumed to be produced in proportion to mean stress m ⫽ (1 ⫹ 2 ⫹ 3)/3 in the PDL and to occur in the normal direction to the outer surface of the PDL. The amount of absorption (apposition) per unit stress and unit time was denoted by a coefficient C (m/kPa per day). This coefficient was assumed to be the same in all teeth. During a small time increment, ⌬T, tooth movement produced by bone remodeling was achieved by the procedure below. 1. The amount of absorption or apposition of the alveolar bone at each triangular region, ⌬t, was calculated in proportion to the mean stress m in the PDL. By this bone remodeling, the outer surface of the PDL was moved by ⌬t, and the PDL was stretched or compressed. This deformation produced stresses in the PDL. 2. Summing up the stresses induced by the bone remodeling, a force to move the tooth, ⌬R, was calculated. 3. A movement of the tooth, ⌬u, is calculated by K⌬u ⫽ ⌬R, and then the tooth was moved by ⌬u. The detailed calculation for this procedure was explained in our previous article.11 By repeating these 2 steps— calculating stress in the PDL and then tooth movement—the tooth position can be calculated to any specified amount of time, T. We developed a computer program of the finite element method including these calculation steps. Tooth movement is in proportion to the amount of bone remodeling, and the amount of bone remodeling is
Fig 3. Mean stress distributions in the PDL of molars with and without the TPA, immediately after force was applied (initial tooth movement). Stress distributions were almost the same irrespective of the TPA. The TPA had no effect on the stress in the PDL. A, Without the TPA; B, with the TPA.
in proportion to parameter CT (m/kPa), which is the product of the bone remodeling rate C (m/kPa per day) and time T (day). Therefore, it is obvious that tooth movement is controlled by parameter CT. In this calculation, CT is used to indicate the progress of tooth movement, because a value of C is uncertain. RESULTS
Immediately after a unit force was applied to the molar, initial movement was produced by elastic deformation of the PDL. At this time, stress distributions in the PDL with and without the TPA were almost the same (Fig 3, A and B). Maximum and minimum values of mean stress, m, in the PDL, and mesial and transverse movements of the molar with the TPA were almost the same as those without the TPA (Table I). In orthodontic movements, the parameter CT was used to indicate the progress of movement. The move-
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Table I.
In initial movement, stress in the PDL and molar movement with and without the TPA (these values were not affected by the TPA)
Maximum mean stress in PDL Minimum mean stress in PDL Mesial movement of molar Transverse movement of molar
With TPA
Without TPA
23.5 kPa ⫺27.4 kPa 7.6 m 1.3 m
23.8 kPa ⫺27.6 kPa 7.6 m 1.3 m
ments were calculated until CT ⫽ 200 m/kPa, when the movement patterns were clearly visible. If a value of C (m/kPa per day) were estimated, a real time T (day) would be obtained by substituting C into CT ⫽ 200 m/kPa. At CT ⫽ 200 m/kPa, crown mesial tipping was produced irrespective of the TPA (Fig 4, A and B). The mesial force rotated the molar without the TPA in the occlusal plane (Fig 5). The TPA decreased the magnitude of mean stress m in the PDL and prevented molar rotation (Table II). When a premolar was connected to the molar, these teeth were tipped at CT ⫽ 200 m/kPa irrespective of the TPA (Fig 6, A and B). The tipping angles (13° and 14°) were about half of those without the premolars (28° and 31° in Fig 4, A and B). Without the TPA, the anchor teeth were rotated centering on the molar, so that the premolar was moved transversely in the occlusal plane (Fig 7). The TPA decreased the magnitude of mean stress m in the PDL and prevented rotation of the anchor teeth (Table III). DISCUSSION Mechanical effect of the TPA
In the initial movement, the difference in stress magnitude in the PDL between the molars with and without the TPA was less than 1%. The presence of the TPA had no effect on the stress in the PDL. If we adopt a hypothesis that stress in the PDL controls orthodontic tooth movement, the TPA has not effect. This result is the same as that of Bobak et al.8 Their result was also confirmed by our calculation. In orthodontic movement at CT ⫽ 200 m/kPa, the mesial force tipped the molar irrespective of the TPA (Fig 4, A and B). The tipping angle with the TPA (28°) was about the same as that without the TPA (31°). The TPA could not prevent crown mesial tipping. Therefore, the TPA has no effect on the anchorage potential for mesial movement. The mesial force rotated the molar without the TPA in the occlusal plane (Fig 5). This is because the direction of the mesial force did not pass through the molar’s center of
Fig 4. Movements and mean stress distributions in the PDL of molars with and without the TPA, in orthodontic movement at CT ⫽ 200 m/kPa. Both molars were tipped by mesial forces, and the tipping angles were almost the same irrespective of the TPA. The TPA could not prevent crown mesial tipping and had almost no effect on the “preserve anchorage” function for mesial movement. A, Without the TPA; B, with the TPA.
resistance. In addition, due to the transverse component of the mesial force, the molar was moved transversely, and so the distance between the left and right molars decreased (Fig 5). On the other hand, the mesial force hardly rotated the molar with the TPA in the occlusal plane, and the molar was not moved transversely. The magnitude of mean stress in the PDL decreased to about 75% of that without the TPA (Table II). It was verified that the TPA was effective for preventing these movements. Although the TPA had almost no effect on initial tooth movement, it prevented molar rotation after a long time. A moment preventing molar rotation was produced by elastic deflection of the TPA. In the initial
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Fig 5. Movements of the molar in the occlusal plane at CT ⫽ 200 m/kPa. The molar without the TPA was rotated and moved transversely. These movements were almost prevented by the TPA. Table II. In orthodontic movement at CT ⫽ 200 m/ kPa, stress in the PDL and movement of the molar with and without the TPA (tipping angles in both cases were almost the same, and the molar with the TPA hardly rotated)
Maximum mean stress in PDL Minimum mean stress in PDL Tipping angle of molar Rotational angle of molar Mesial movement of molar Transverse movement of molar
With TPA
Without TPA
19.7 kPa ⫺16.2 kPa 28° 1.4° 5.0 mm 0.4 mm
23.4 kPa ⫺22.5 kPa 31° 18.5° 6.7 mm 2.4 mm
movement, movement of the molar with the TPA was slight (several millimeters), so that the moment produced in the TPA was small. However, it increased with the orthodontic movement of the molar and prevented molar rotation. This effect of the TPA could not be estimated from initial tooth movement. When a premolar was connected to the molar, the mesial force tipped the anchor teeth irrespective of the TPA (Fig 6, A and B). Therefore, the TPA has no effect on anchorage potential for mesial movement. However, the TPA was effective for preventing rotation of anchor teeth in the occlusal plane (Fig 7). Tipping angles of the anchor teeth (13° and 14° in Table III) were about half of those without the premolar (28° and 31° in Table II). Also, maximum stresses in the PDL (11.7 and 17.7 kPa in Table III) were smaller than those without the premolar (19.7 kPa and 23.4 kPa in Table II). Although the root surface area of the premolar (223 mm2) was
Fig 6. Movements of the anchor teeth at CT ⫽ 200 m/kPa when a premolar was connected to the molar. Addition of the premolar decreased the tipping angle of the anchor teeth, in comparison with Fig 4, A and B. Tipping angles were almost the same irrespective of the TPA. The TPA had almost no effect on the “preserve anchorage” function for mesial movement. A, Without the TPA; B, with the TPA.
about half that of the molar (430 mm2), the addition of the premolar was useful for increasing anchorage potential. Recently, Zablocki et al12 carried out a reliable cephalometric study to quantify the anchorage capabilities of the TPA. Their results indicated that the TPA had no significant effect on the mesial movement of the maxillary first molars during extraction treatment. This finding is consistent with the results from our calculations. If an activated TPA were placed on the molars, the orthodontic tooth movement would be calculated with these methods. Such tooth movements produced by bent wires were reported in our previous article.10 Also,
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Fig 7. Movements of the anchor teeth in the occlusal plane at CT ⫽ 200 m/kPa when a premolar was connected to the molar. The anchor teeth without the TPA were rotated, and the premolar was moved transversely. These movements were almost prevented by the TPA. Table III.
When a premolar was connected to the molar, stress in the PDL and movement of the anchor teeth with and without the TPA at CT ⫽ 200 m/kPa (tipping angles in both cases were almost the same, and the anchor teeth with the TPA hardly rotated; addition of the premolars decreased the tipping angle in comparison with Table II)
Maximum mean stress in PDL Minimum mean stress in PDL Tipping angle of anchor teeth Rotational angle of anchor teeth Mesial movement of molar Transverse movement of premolar
With TPA
Without TPA
11.7 kPa ⫺16.1 kPa 13° 1.8° 2.7 mm 0.5 mm
17.7 kPa ⫺17.6 kPa 14° 13.6° 4.0 mm 2.0 mm
molar movement produced by tongue pressure acting on the TPA would be estimated, if magnitude, frequency, and direction of tongue pressure were clarified. Calculation method
Since the mechanism of orthodontic tooth movement is not fully understood, we adopted the simplest assumptions in the calculation method.13,14 These assumptions might be insufficient to simulate tooth movements in clinical situations. For example, cortical bone in the maxilla might impede orthodontic tooth movement. Also, the nonlinear property or the fiber structure of the PDL could affect tooth movement.
These phenomena have not been fully clarified, and experimental data required for this calculation are not available. Including uncertain factors in the calculation would complicate the method and the results. Hence, such effects were ignored in this calculation. The assumptions used in this calculation were discussed in previous articles.9-11 The validity of the assumptions should be confirmed by comparison of the calculated tooth movements with clinical ones. When the calculated results are different from clinical experiences, the assumptions in the calculations should be reconsidered and modified. By repetition of modifications, tooth movements in clinical situations would be simulated by this method. This is the final purpose of our studies. Since we cannot measure clinical tooth movements in our facilities, only calculated results are shown in this article. We hope these results will be criticized from a clinical point of view. CONCLUSIONS
Orthodontic movements of anchor teeth splinted with and without a TPA subjected to mesial forces were calculated by the finite element method. In the initial movement, the stress magnitude in the PDL was not changed by the TPA. Therefore, the TPA had no effect on initial tooth movement. In the orthodontic movement, the mesial force tipped the anchor teeth irrespective of the TPA. The tipping angles of the anchor teeth with and without the TPA were almost the same. Therefore the TPA had almost no effect on the “preserve anchorage” function for mesial movement. However, the mesial force produced rotation and transverse movement of the anchor teeth without the TPA. These movements could be prevented by the TPA. Many assumptions were used to calculate the orthodontic tooth movement. The calculated results are valid when these assumptions are satisfied.
REFERENCES 1. Proffit WR. Contemporary orthodontics. St Louis: Mosby; 1986. 2. Williams JK, Cook PA, Isaacson KF, Thom AR. Fixed orthodontic appliances–principles and practice. Oxford, United Kingdom: Butterworth-Heinemann; 1995. 3. Burstone CJ, Koenig HA. Precision adjustment of the transpalatal lingual arch: computer arch form predetermination. Am J Orthod 1981;79:115-33. 4. Baldini G, Luder HU. Influence of arch shape on the transverse effects of transpalatal arches of the Goshgarian type during application of buccal root torque. Am J Orthod 1982;81:202-8. 5. Gunduz E, Zachrisson BU, Honigl KD, Crismani AG, Bantleon HP. An improved transpalatal bar design. Part I. Comparison of moments and forces delivered by two bar designs
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6.
7.
8.
9.
for symmetrical molar derotation. Angle Orthod 2003;73: 239-43. Gunduz E, Crismani AG, Bantleon HP, Honigl KD, Zachrisson BU. An improved transpalatal bar design. Part II. Clinical upper molar derotation— case report. Angle Orthod 2003;73:244-8. Dahlquist A, Gebauer U, Ingervall B. The effect of a transpalatal arch for the correction of first molar rotation. Eur J Orthod 1996;18:257-67. Bobak V, Christiansen RL, Hollister SJ, Kohn DH. Stress-related molar responses to the transpalatal arch: a finite element analysis. Am J Orthod Dentofacial Orthop 1997;112:512-8. Kojima Y, Fukui H. Numerical simulation of canine retraction by sliding mechanics. Am J Orthod Dentofacial Orthop 2005;127: 542-51.
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10. Kojima Y, Fukui H. Numerical simulation of tooth movement by wire bending. Am J Orthod Dentofacial Orthop 2006;130: 452-9. 11. Kojima Y, Mizuno T, Fukui H. Numerical simulation of tooth movement produced by molar uprighting spring. Am J Orthod Dentofacial Orthop 2007;132:630-8. 12. Zablocki HL, McNamara JA Jr, Franchi L, Baccetti T. Effect of the transpalatal arch during extraction treatment. Am J Orthod Dentofacial Orthop 2008 (in press). 13. Quinn RS, Yoshikawa DK. A reassessment of force magnitude in orthodontics. Am J Orthod 1985;88:252-60. 14. Ren Y, Maltha JC, Kuijpers-Jagtman AM. Optimum force magnitude for orthodontic tooth movement: a systematic literature review. Angle Orthod 2003;73:86-92.
Effects of transpalatal arch on molar movement produced by mesial force: A finite element simulation Yukio Kojimaa and Hisao Fukuib Nagoya, Japan Introduction: The transpalatal arch (TPA), which splints together 2 maxillary molars, has been believed to preserve anchorage. The purpose of this study was to clarify this effect from a mechanical point of view. Methods: The finite element method was used to simulate the movement of anchor teeth subjected to mesial forces with and without a TPA. Results: In the initial movement produced by elastic deformation of the periodontal ligament, stress magnitude in the periodontal ligament was not changed by the TPA. In the orthodontic movement produced by bone remodeling, the mesial force tipped the anchor teeth irrespective of the TPA. The tipping angles of anchor teeth with and without the TPA were almost the same. The anchor teeth without the TPA were rotated in the occlusal plane and moved transversely. Conclusions: The TPA had no effect on the initial movement. In the orthodontic movement, the TPA had almost no effect, preserving anchorage for mesial movement. However, the TPA prevented rotational and transverse movements of the anchor teeth. These results are valid when the assumptions used in this calculation are satisfied. (Am J Orthod Dentofacial Orthop 2008;134:335.e1-335.e7)
T
he transpalatal arch (TPA), by which 2 maxillary molars are splinted together, has many functions in orthodontic treatment.1-7 When an activated TPA is placed between molars, forces and moments produced by the TPA correct arch width and inclination of the molars.3-7 When a passive TPA is placed, it prevents both rotation and buccolingual tipping of the molars, and also maintains the transverse distance of the molars. These functions are expected because of the mechanical rigidity of the TPA. On the other hand, a function that the TPA preserves anchorage for mesial movement is not obvious, because molars can move or tip mesially together with the TPA.1,2 In this study, we focused on the “preserve anchorage” function of the TPA, which has not been verified from a mechanical point of view. The first quantitative analysis on the “preserve anchorage” function was carried out by Bobak et al.8 They calculated periodontal stresses in molars with and without a TPA during initial tooth movement, and a
Associate professor, Department of Mechanical Engineering, Nagoya Institute of Technology, Nagoya, Japan. b Professor, Department of Dental Materials Science, School of Dentistry, Aichi-Gakuin University, Nagoya, Japan. Reprint requests to: Yukio Kojima, Department of Mechanical Engineering, Shikumi College, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan; e-mail, [email protected]. Submitted, December 2007; revised and accepted, March 2008. 0889-5406/$34.00 Copyright © 2008 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2008.03.011
showed that a TPA affected periodontal stress magnitude by less than 1%. This result suggested that the TPA had no effect on the “preserve anchorage” function. However, estimates based on initial movement might be inappropriate, because the initial force system changes as the teeth move. This was shown in our numeric simulations with the finite element method.9-11 The TPA should be estimated on the basis of orthodontic tooth movement. In this study, to develop the results of Bobak et al,8 we calculated orthodontic movements of anchor teeth with and without a TPA subjected to mesial forces using a method from our previous articles.9-11 The effects of the TPA on the “preserve anchorage” function were examined from a mechanical point of view. MATERIAL AND METHODS Tooth element for calculating stresses in the periodontal ligament
All biologic reactions produced in the periodontal tissue during orthodontic tooth movement have not been clarified. Therefore, we proposed a phenomenologic method to simulate tooth movement induced by a mechanical stimulus in the periodontal ligament (PDL). In this method, a tooth moves as a result of absorption and apposition of the alveolar bone, which is produced by the stress distribution in the PDL. By accepting some assumptions described in Figure 1, the stress distribution in the PDL can be calculated by a 335.e1
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Fig 1. Assumptions and a surface model of a tooth used to calculate stress distribution in the PDL. By using the surface model, the stiffness matrix of the tooth element is calculated.
surface model of the tooth. For this purpose, the relationship between movements u and forces f at the bracket position is calculated. This is written as Ku ⫽ f, where K is the stiffness matrix in the finite element method. In other words, the mechanical response of a tooth supported by the PDL can be replaced by a single element having the K. This is a substructure element in the finite element method and called the “tooth element” in this article. The K is obtained by the procedure below. 1. Displacement (translation or rotation) imposes on the tooth at the bracket position. 2. Strains (tension or compression) for each triangular region are calculated from the rigid kinematic motion of the root surface, and the stress distribution in the PDL is calculated by using Young’s modulus and Poisson’s ratio of the PDL. 3. Summing up the stress in each triangular region over the root, forces and moments acting on the tooth are obtained. Detailed mathematical formulation for this procedure was explained our previous article.11 Analysis model
The maxillary first molars were splinted together by a TPA with a 5-mm omega loop (Fig 2). Width and depth of the TPA were 38 and 14 mm, respectively. The TPA was made of 0.036-in round wire of stainless steel (Young’s modulus: 200 GPa). The TPA was fixed
Fig 2. Finite element model of maxillary molars splinted with a TPA. The tooth element is connected to the TPA with a rigid beam element. Mesial forces (1 N) are applied to the molar brackets.
firmly to the molars. Then the molars were connected to the second premolars with a 0.016-in square stainless steel wire. When the TPA was placed, no forces or moments acted on the molars; the TPA was passive. Mesial forces of 1 N were applied to the molars at the bracket positions. Forces and moments from adjacent
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teeth are not considered. Also, forces from the tongue and the palate did not act on the TPA. The left half of the arch was modeled because of symmetry at the centerline. Surface models of the tooth were made based on a dental study model (AM-10, Nissin Dental Products, Kyoto, Japan). Root surface areas of the molar and the premolar were 430 and 223 mm2, respectively. The TPA was divided into 3-dimensional elastic beam elements. A tooth element for the first molar was connected to the TPA with a rigid beam element. The degrees of freedom of the tooth element were 6 and equal to that of the 3-dimensional beam element. Therefore, the tooth element could be connected directly to the beam element. Then a tooth element for the molar was connected to that for the second premolar with a wire divided into elastic beam elements. By using these finite element models, stress distributions in the PDL were calculated when the mesial forces act on the molars. Calculation of orthodontic tooth movement
Bone remodeling was assumed to be produced in proportion to mean stress m ⫽ (1 ⫹ 2 ⫹ 3)/3 in the PDL and to occur in the normal direction to the outer surface of the PDL. The amount of absorption (apposition) per unit stress and unit time was denoted by a coefficient C (m/kPa per day). This coefficient was assumed to be the same in all teeth. During a small time increment, ⌬T, tooth movement produced by bone remodeling was achieved by the procedure below. 1. The amount of absorption or apposition of the alveolar bone at each triangular region, ⌬t, was calculated in proportion to the mean stress m in the PDL. By this bone remodeling, the outer surface of the PDL was moved by ⌬t, and the PDL was stretched or compressed. This deformation produced stresses in the PDL. 2. Summing up the stresses induced by the bone remodeling, a force to move the tooth, ⌬R, was calculated. 3. A movement of the tooth, ⌬u, is calculated by K⌬u ⫽ ⌬R, and then the tooth was moved by ⌬u. The detailed calculation for this procedure was explained in our previous article.11 By repeating these 2 steps— calculating stress in the PDL and then tooth movement—the tooth position can be calculated to any specified amount of time, T. We developed a computer program of the finite element method including these calculation steps. Tooth movement is in proportion to the amount of bone remodeling, and the amount of bone remodeling is
Fig 3. Mean stress distributions in the PDL of molars with and without the TPA, immediately after force was applied (initial tooth movement). Stress distributions were almost the same irrespective of the TPA. The TPA had no effect on the stress in the PDL. A, Without the TPA; B, with the TPA.
in proportion to parameter CT (m/kPa), which is the product of the bone remodeling rate C (m/kPa per day) and time T (day). Therefore, it is obvious that tooth movement is controlled by parameter CT. In this calculation, CT is used to indicate the progress of tooth movement, because a value of C is uncertain. RESULTS
Immediately after a unit force was applied to the molar, initial movement was produced by elastic deformation of the PDL. At this time, stress distributions in the PDL with and without the TPA were almost the same (Fig 3, A and B). Maximum and minimum values of mean stress, m, in the PDL, and mesial and transverse movements of the molar with the TPA were almost the same as those without the TPA (Table I). In orthodontic movements, the parameter CT was used to indicate the progress of movement. The move-
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Table I.
In initial movement, stress in the PDL and molar movement with and without the TPA (these values were not affected by the TPA)
Maximum mean stress in PDL Minimum mean stress in PDL Mesial movement of molar Transverse movement of molar
With TPA
Without TPA
23.5 kPa ⫺27.4 kPa 7.6 m 1.3 m
23.8 kPa ⫺27.6 kPa 7.6 m 1.3 m
ments were calculated until CT ⫽ 200 m/kPa, when the movement patterns were clearly visible. If a value of C (m/kPa per day) were estimated, a real time T (day) would be obtained by substituting C into CT ⫽ 200 m/kPa. At CT ⫽ 200 m/kPa, crown mesial tipping was produced irrespective of the TPA (Fig 4, A and B). The mesial force rotated the molar without the TPA in the occlusal plane (Fig 5). The TPA decreased the magnitude of mean stress m in the PDL and prevented molar rotation (Table II). When a premolar was connected to the molar, these teeth were tipped at CT ⫽ 200 m/kPa irrespective of the TPA (Fig 6, A and B). The tipping angles (13° and 14°) were about half of those without the premolars (28° and 31° in Fig 4, A and B). Without the TPA, the anchor teeth were rotated centering on the molar, so that the premolar was moved transversely in the occlusal plane (Fig 7). The TPA decreased the magnitude of mean stress m in the PDL and prevented rotation of the anchor teeth (Table III). DISCUSSION Mechanical effect of the TPA
In the initial movement, the difference in stress magnitude in the PDL between the molars with and without the TPA was less than 1%. The presence of the TPA had no effect on the stress in the PDL. If we adopt a hypothesis that stress in the PDL controls orthodontic tooth movement, the TPA has not effect. This result is the same as that of Bobak et al.8 Their result was also confirmed by our calculation. In orthodontic movement at CT ⫽ 200 m/kPa, the mesial force tipped the molar irrespective of the TPA (Fig 4, A and B). The tipping angle with the TPA (28°) was about the same as that without the TPA (31°). The TPA could not prevent crown mesial tipping. Therefore, the TPA has no effect on the anchorage potential for mesial movement. The mesial force rotated the molar without the TPA in the occlusal plane (Fig 5). This is because the direction of the mesial force did not pass through the molar’s center of
Fig 4. Movements and mean stress distributions in the PDL of molars with and without the TPA, in orthodontic movement at CT ⫽ 200 m/kPa. Both molars were tipped by mesial forces, and the tipping angles were almost the same irrespective of the TPA. The TPA could not prevent crown mesial tipping and had almost no effect on the “preserve anchorage” function for mesial movement. A, Without the TPA; B, with the TPA.
resistance. In addition, due to the transverse component of the mesial force, the molar was moved transversely, and so the distance between the left and right molars decreased (Fig 5). On the other hand, the mesial force hardly rotated the molar with the TPA in the occlusal plane, and the molar was not moved transversely. The magnitude of mean stress in the PDL decreased to about 75% of that without the TPA (Table II). It was verified that the TPA was effective for preventing these movements. Although the TPA had almost no effect on initial tooth movement, it prevented molar rotation after a long time. A moment preventing molar rotation was produced by elastic deflection of the TPA. In the initial
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Fig 5. Movements of the molar in the occlusal plane at CT ⫽ 200 m/kPa. The molar without the TPA was rotated and moved transversely. These movements were almost prevented by the TPA. Table II. In orthodontic movement at CT ⫽ 200 m/ kPa, stress in the PDL and movement of the molar with and without the TPA (tipping angles in both cases were almost the same, and the molar with the TPA hardly rotated)
Maximum mean stress in PDL Minimum mean stress in PDL Tipping angle of molar Rotational angle of molar Mesial movement of molar Transverse movement of molar
With TPA
Without TPA
19.7 kPa ⫺16.2 kPa 28° 1.4° 5.0 mm 0.4 mm
23.4 kPa ⫺22.5 kPa 31° 18.5° 6.7 mm 2.4 mm
movement, movement of the molar with the TPA was slight (several millimeters), so that the moment produced in the TPA was small. However, it increased with the orthodontic movement of the molar and prevented molar rotation. This effect of the TPA could not be estimated from initial tooth movement. When a premolar was connected to the molar, the mesial force tipped the anchor teeth irrespective of the TPA (Fig 6, A and B). Therefore, the TPA has no effect on anchorage potential for mesial movement. However, the TPA was effective for preventing rotation of anchor teeth in the occlusal plane (Fig 7). Tipping angles of the anchor teeth (13° and 14° in Table III) were about half of those without the premolar (28° and 31° in Table II). Also, maximum stresses in the PDL (11.7 and 17.7 kPa in Table III) were smaller than those without the premolar (19.7 kPa and 23.4 kPa in Table II). Although the root surface area of the premolar (223 mm2) was
Fig 6. Movements of the anchor teeth at CT ⫽ 200 m/kPa when a premolar was connected to the molar. Addition of the premolar decreased the tipping angle of the anchor teeth, in comparison with Fig 4, A and B. Tipping angles were almost the same irrespective of the TPA. The TPA had almost no effect on the “preserve anchorage” function for mesial movement. A, Without the TPA; B, with the TPA.
about half that of the molar (430 mm2), the addition of the premolar was useful for increasing anchorage potential. Recently, Zablocki et al12 carried out a reliable cephalometric study to quantify the anchorage capabilities of the TPA. Their results indicated that the TPA had no significant effect on the mesial movement of the maxillary first molars during extraction treatment. This finding is consistent with the results from our calculations. If an activated TPA were placed on the molars, the orthodontic tooth movement would be calculated with these methods. Such tooth movements produced by bent wires were reported in our previous article.10 Also,
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Fig 7. Movements of the anchor teeth in the occlusal plane at CT ⫽ 200 m/kPa when a premolar was connected to the molar. The anchor teeth without the TPA were rotated, and the premolar was moved transversely. These movements were almost prevented by the TPA. Table III.
When a premolar was connected to the molar, stress in the PDL and movement of the anchor teeth with and without the TPA at CT ⫽ 200 m/kPa (tipping angles in both cases were almost the same, and the anchor teeth with the TPA hardly rotated; addition of the premolars decreased the tipping angle in comparison with Table II)
Maximum mean stress in PDL Minimum mean stress in PDL Tipping angle of anchor teeth Rotational angle of anchor teeth Mesial movement of molar Transverse movement of premolar
With TPA
Without TPA
11.7 kPa ⫺16.1 kPa 13° 1.8° 2.7 mm 0.5 mm
17.7 kPa ⫺17.6 kPa 14° 13.6° 4.0 mm 2.0 mm
molar movement produced by tongue pressure acting on the TPA would be estimated, if magnitude, frequency, and direction of tongue pressure were clarified. Calculation method
Since the mechanism of orthodontic tooth movement is not fully understood, we adopted the simplest assumptions in the calculation method.13,14 These assumptions might be insufficient to simulate tooth movements in clinical situations. For example, cortical bone in the maxilla might impede orthodontic tooth movement. Also, the nonlinear property or the fiber structure of the PDL could affect tooth movement.
These phenomena have not been fully clarified, and experimental data required for this calculation are not available. Including uncertain factors in the calculation would complicate the method and the results. Hence, such effects were ignored in this calculation. The assumptions used in this calculation were discussed in previous articles.9-11 The validity of the assumptions should be confirmed by comparison of the calculated tooth movements with clinical ones. When the calculated results are different from clinical experiences, the assumptions in the calculations should be reconsidered and modified. By repetition of modifications, tooth movements in clinical situations would be simulated by this method. This is the final purpose of our studies. Since we cannot measure clinical tooth movements in our facilities, only calculated results are shown in this article. We hope these results will be criticized from a clinical point of view. CONCLUSIONS
Orthodontic movements of anchor teeth splinted with and without a TPA subjected to mesial forces were calculated by the finite element method. In the initial movement, the stress magnitude in the PDL was not changed by the TPA. Therefore, the TPA had no effect on initial tooth movement. In the orthodontic movement, the mesial force tipped the anchor teeth irrespective of the TPA. The tipping angles of the anchor teeth with and without the TPA were almost the same. Therefore the TPA had almost no effect on the “preserve anchorage” function for mesial movement. However, the mesial force produced rotation and transverse movement of the anchor teeth without the TPA. These movements could be prevented by the TPA. Many assumptions were used to calculate the orthodontic tooth movement. The calculated results are valid when these assumptions are satisfied.
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