Editorial Type:
Article Category: Research Article
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Online Publication Date: 19 Jul 2016

Effect of apical portion of T-, sloped L-, and reversed L-closing loops on their force systems

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Page Range: 104 – 110
DOI: 10.2319/020316-95.1
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ABSTRACT

Objective: To investigate the effect of the position of the apical portion of closing loops on the force system at both loop ends.

Materials and Methods: T-loops were compared with backward-sloped L-loops (SL) and reversed L-loops (RL). SL-loops were directed toward the anterior side; RL-loops were directed toward the posterior side. Loop response to loop pulling was determined with finite element analysis at six positions of the apical loop portion for 12-mm interbracket distance and 8-mm loop length and height. Three-dimensional models of the closing loops were created using beam elements with the properties of stainless steel. Loop responses (horizontal load/deflection, vertical force, and moment-to-force ratio) at both loop ends were calculated as well as at 100 g and 200 g activation forces.

Results: T-, SL-, and RL-loops with the same position of the apical portion showed approximately the same force system at both loop ends. This behavior was found across the investigated range through which the loops were moved (interbracket center to posterior bracket).

Conclusions: The center of the apical portion determined the force system of the closing loops regardless of the position of the loop legs. The centers of the apical portion of the T-, SL-, and RL-loops acted like V-bend positions.

Keywords: T-loop; Reversed L-loop; Force system; V-bend; Finite element analysis

INTRODUCTION

All symmetric shape loops without gable bends, such as T-loops and vertical loops, have V-bend characteristics, through which off-center positioning produces differential moments, with the greater moment and extrusive vertical force acting on the closest tooth.1–7 In the continuous closing loop archwire, these planar loops are usually placed close to anterior teeth, and when activated they generate extrusive forces and high moment-to-force (M/F) ratios at the anterior end. At the posterior end they will generate intrusive forces and low M/F ratios in the same or opposite directions. When such a symmetric loop is placed in the center of the interbracket space, there are only moments in opposite directions without vertical forces on either wire end.1–3 However, for asymmetric loops, such as the L-loop, the response is different. When an upright Opus or L-loop is centered, the M/F ratio at the canine bracket (CB) end shows counterclockwise M/F ratios of 8.5 to 9.3, while the M/F ratio is almost zero at the premolar bracket (PB) end.6,7

It has been shown that for the same loop position between the brackets or at the same ratios between anterior and posterior ends, the force system of upright L-loops is different compared to that of sloped L-loops.6 Instead, a backward-sloped L-loop like the Opus 70 showed similar force system responses as the T-loop (load/deflection, vertical force, M/F ratio).6 Closer examination of the Opus 70 with the vertical legs sloped 70° backward and T-loop shows that, surprisingly, the apical portions of both loops are in approximately the same position.6 This has not been noted before in the academic literature.

In this study, we hypothesized that the apical portion of the loops plays an important role in the response of closing loop force systems. To test this hypothesis, force systems of T-, reversed L-, and sloped L-loops with apical portions at the same position were investigated using finite element analysis.

MATERIALS AND METHODS

Eighteen T-, backward-sloped L- (SL), and reversed L- (RL) closing loop models were evaluated in this analysis. Their dimensions are shown in Figure 1A. The T-loop sizes were 8 × 8 mm (length × height), and the interbracket distance (b) was 12 mm. The vertical legs of the SL-loops were sloped 69° toward the PB. This angle allowed a good match with the apical parts of the other two loops (Figure 1A) and is close to the angle used for Opus 70 loops. On the PB end, the distances from the center apical portion of the T-, SL-, and RL-loops to the wire end (a) were 2, 3, 4, 5, 6, and 7 mm. Therefore, the a:b ratios were 0.17, 0.25, 0.33, 0.42, 0.50, and 0.58, respectively. The CB and PB ends were placed in the same plane. The distance from the center of the vertical legs of the T-, SL-, and RL-loops to the PB end were x, y, and z, respectively. For a of 2, 3, 4, 5, 6, and 7 mm, the x dimensions were 2, 3, 4, 5, 6, and 7 mm; y dimensions were 1, 2, 3, 4, 5, and 6 mm; and z dimensions were 5.5, 6.5, 7.5, 8.5, 9.5, and 10.5 mm, respectively.

Figure 1. (A) T-, SL-, and RL-loop dimensions, where a is distance from center of the apical portion of the closing loop to PB end, and b is the interbracket distance (12 mm). (B) Definitions for forces and moments: Positive vertical forces move a tooth in the direction opposite of the loop (extrusion), negative forces move a tooth in the loop direction (intrusion); positive moments rotate a tooth clockwise, negative moments rotate a tooth counterclockwise.Figure 1. (A) T-, SL-, and RL-loop dimensions, where a is distance from center of the apical portion of the closing loop to PB end, and b is the interbracket distance (12 mm). (B) Definitions for forces and moments: Positive vertical forces move a tooth in the direction opposite of the loop (extrusion), negative forces move a tooth in the loop direction (intrusion); positive moments rotate a tooth clockwise, negative moments rotate a tooth counterclockwise.Figure 1. (A) T-, SL-, and RL-loop dimensions, where a is distance from center of the apical portion of the closing loop to PB end, and b is the interbracket distance (12 mm). (B) Definitions for forces and moments: Positive vertical forces move a tooth in the direction opposite of the loop (extrusion), negative forces move a tooth in the loop direction (intrusion); positive moments rotate a tooth clockwise, negative moments rotate a tooth counterclockwise.
Figure 1. (A) T-, SL-, and RL-loop dimensions, where a is distance from center of the apical portion of the closing loop to PB end, and b is the interbracket distance (12 mm). (B) Definitions for forces and moments: Positive vertical forces move a tooth in the direction opposite of the loop (extrusion), negative forces move a tooth in the loop direction (intrusion); positive moments rotate a tooth clockwise, negative moments rotate a tooth counterclockwise.

Citation: The Angle Orthodontist 87, 1; 10.2319/020316-95.1

The loop geometry was modeled in finite element analysis software (Marc Mentat 2007r1, MSC Software, Santa Ana, Calif) using three-dimensional beam elements with a 0.016 × 0.022–inch rectangular cross section. These beam elements used linear interpolation for displacements and rotations and included transverse shear provisions. 520, 440, and 440 elements were used to create the shapes of the T-, SL-, and RL-loops, respectively. The beam elements were given the material properties of stainless-steel wire by applying an elastic modulus of 157.6 GPa and a Poisson's ratio of 0.3.8

For modeling the loading and fixation, the ends of the closing loops were fixed in 6 degrees of freedom (three displacements, three rotations). During the analysis, activation was simulated by moving both ends up to 2 mm horizontally in opposite directions, thereby increasing the total distance between the wire ends up to 4 mm. We verified that all strains remained well below the stainless-steel yield strain based on a yield strength of 1400 MPa.8 The horizontal reaction forces, vertical forces, and moments at both ends were calculated during the simulated activation and recorded when the horizontal forces reached 100 g or 200 g. The load/deflection values were calculated by dividing the (horizontal) force by the corresponding horizontal deflection. Directions of vertical force and moment are defined in Figure 1B. Positive values of the vertical force are directed away from the loop (extrusion), and negative values are in the loop direction (intrusion). Positive moments turn the loop clockwise (distal crown tipping or mesial root tipping), negative moments counterclockwise (mesial crown tipping or distal root tipping).

RESULTS

Figure 2 shows the load/deflection at 100-g horizontal force for the three types of loops with increasing a:b ratios (ie, the center of the apical portion of the loop moves from close to the PB end toward and slightly past the center). In all positions, the differences in load/deflection ratios were small among the three loop types. At 100-g force, the load/deflection ranged from 124.3 g to 172.9 g per millimeter of deflection (Table 1). The load/deflection ratios were highest with the apical loop center closest to the PB end, lowest when centered, and increased again when moved from the center toward the CB end. Load/deflection values at the 200-g force were nearly the same as at 100 g (Table 1).

Figure 2. Load/deflection of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.Figure 2. Load/deflection of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.Figure 2. Load/deflection of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.
Figure 2. Load/deflection of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.

Citation: The Angle Orthodontist 87, 1; 10.2319/020316-95.1

Table 1. Force Systems of Three Loops for Various Loop Positions at 100- and 200-g Horizontal Activation Forcea
Table 1.
a SL indicates backward-sloped L-loop; RL, reversed L-loop; PB, premolar bracket; CB, canine bracket; a, distance of center of apical portion of the loop from PB end; b, interbracket distance; L/d, load deflection (g/mm); Vf, vertical force (g) (+ = extrusion, − = intrusion); and M/F, moment-to-force ratio (mm) (+ = clockwise, − = counterclockwise).

Vertical forces created by the horizontal activation forces were nearly zero when the center of the apical portion was centered (a:b ratio = 0.5) and highest when placed close to the PB end (Table 1; Figure 3). When the center of the apical portion was close to the PB end, an extrusion force was found at the PB end and an intrusion force at the CB end. Vertical forces were reversed at a:b ratios of approximately 0.5. Vertical forces increased almost proportional with the applied horizontal activation forces (100 or 200 g), but the general variation with the apical portion center position remained the same.

Figure 3. Vertical forces at the canine bracket of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.Figure 3. Vertical forces at the canine bracket of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.Figure 3. Vertical forces at the canine bracket of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.
Figure 3. Vertical forces at the canine bracket of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.

Citation: The Angle Orthodontist 87, 1; 10.2319/020316-95.1

Figure 4 shows the variation in M/F ratios at both loop ends when the apical centers of the loops moved. The differences among loop types were relatively small, and all loops showed the same pattern of changes. For T-, SL-, and RL-loops the maximum M/F ratios at the PB end were found at a:b ratios of 0.17–0.25, 0.25, and 0.17, respectively. At the CB end the maximum (negative) M/F ratios were found for the 0.58 a:b ratio (Table 1; Figure 4). Among the three loop models, the RL showed the highest M/F ratio of 6.3 at 100-g activation force, compared to 5.6 for the T-loop and 5.0 for the SL-loop. At this position, the M/F ratio on the other end was close to zero and in the opposite direction (SL). As loop position moved toward the CB (a:b ratio increased), the M/F ratio became less positive at the PB side and more negative at the CB side. Both ends had the same M/F ratio directions in all loop positions. Directional change of the M/F ratio was found at the CB end of the T- and RL-loops at a:b ratio 0.17.

Figure 4. Moment-to-force (M/F) ratios at premolar bracket (PB) and canine bracket (CB) ends of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.Figure 4. Moment-to-force (M/F) ratios at premolar bracket (PB) and canine bracket (CB) ends of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.Figure 4. Moment-to-force (M/F) ratios at premolar bracket (PB) and canine bracket (CB) ends of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.
Figure 4. Moment-to-force (M/F) ratios at premolar bracket (PB) and canine bracket (CB) ends of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.

Citation: The Angle Orthodontist 87, 1; 10.2319/020316-95.1

DISCUSSION

Before discussing the results, it is important to note that the transfer of forces and moments to attached teeth involves more than only the wire loop because the transfer is also affected by bracket attachment, in particular bracket play and curvature. However, the objective of this study was to evaluate the characteristic force system of the closing loop itself. Therefore, we fixed the loops in the planar bracket positions. Bracket play of a 0.016 × 0.022–inch wire in 0.018 × 0.025–inch slot has been shown to reduce the forces and moments generated by closing loops, but does not change their general characteristics.9 Furthermore, the reason for activating the closing loops in the anteroposterior plane was to eliminate out-of-plane effects that could interfere with the characterization of basic loop characteristics.10 While the general loop characteristics discussed in this study should thus remain valid, it is important to acknowledge that clinical application must always consider the combination of the unique factors present in each individual case.

The analysis of the three types of closing loops showed how the force systems evolved when placed between the center and PB end. When a T-loop is placed off center, close to the CB, for closing an extraction space, it affects the force system on both loop ends. For T-loops, off-center positioning had a significant effect on the moments produced, with the higher moment occurring at the bracket closest to the loop position.2 Loop placement was suggested to resemble a V-bend, for which off-center positioning produced differential moments where the greater moment acted on the tooth closest to the V-bend.11 A vertical extrusive force would occur at the short end, while the same amount of intrusive force occurred at the long end. Increasing horizontal activation increased the vertical force. For a centered V-bend, the M/F ratio was equal in value but different in direction. Other studies1,3,6,7 reported similar reactions. Comparing the force systems of the SL- and RL-loops with the T-loop showed that they had a similar behavior as the T-loop if the apical portions were placed in the same position, despite differences in shape and loop leg position. This not previously reported observation was confirmed across the investigated a:b ratios (0.17–0.58). Consequently, at 0.17 a:b ratio, when SL- and T-loop legs are positioned adjacent to the PB, the RL-loop legs are close to the center while still generating the same force system. And when the T- and SL-loop legs are close to the center at 0.58 a:b ratio (Figure 5), the RL-loop generated the same force system with its legs close to the CB.

Figure 5. Illustration of the similar force systems for T-, SL-, and RL-loops when the apical portions are at the same position (here the center) while loop legs are at different positions. The horizontal activation force in this example is 200 g. The solid line shows the activated stage; the gray line shows the inactivated stage.Figure 5. Illustration of the similar force systems for T-, SL-, and RL-loops when the apical portions are at the same position (here the center) while loop legs are at different positions. The horizontal activation force in this example is 200 g. The solid line shows the activated stage; the gray line shows the inactivated stage.Figure 5. Illustration of the similar force systems for T-, SL-, and RL-loops when the apical portions are at the same position (here the center) while loop legs are at different positions. The horizontal activation force in this example is 200 g. The solid line shows the activated stage; the gray line shows the inactivated stage.
Figure 5. Illustration of the similar force systems for T-, SL-, and RL-loops when the apical portions are at the same position (here the center) while loop legs are at different positions. The horizontal activation force in this example is 200 g. The solid line shows the activated stage; the gray line shows the inactivated stage.

Citation: The Angle Orthodontist 87, 1; 10.2319/020316-95.1

To our knowledge, the similarity between the investigated loop systems based on the position of the center of the apical portion has not been demonstrated in the literature before. L-loops with reversed direction are usually assumed to have a force system based on loop leg position rather than the location of the apical portion. But this study shows that closing loop force systems should not be taken for granted; the mechanics of each shape need to be analyzed to fully understand their behavior. Advances in understanding the force system of each shape can further improve closing loop design for effective clinical treatments (Figure 6). The finding that the center point of the apical portion of a loop acts like a V-bend offers clinicians a new option, who might use its position to obtain the desired properties for moving teeth to their planned positions. For example, the strategy to overcome the extrusive force from an off-centered closing T-loop archwire, such as an anterior step-up T-loop, can provide an effective intrusive force to the anterior. However, loop properties may change significantly depending on the combination of the position of a T-loop and the amount of step bend, which may render it less effective for moving a tooth or group of teeth in the desired direction.9 Based on the results of this study, we can use reversed L-loops that have the center of the apical portion close to the PB from the start. The noted V-bend–like effect will provide an intrusion force to the anterior teeth, while no extrusive force will occur throughout the space-closure procedure (Figure 6). These potential procedures will need further scientific verification and should be further investigated theoretically and experimentally.

Figure 6. (A) Anterior space closure performed by the planar 8 × 8–mm reversed-L closing loop archwire. The center of the apical portion (indicated by white dot) acted like a V-bend. At the beginning of space closure the V-bend was located a little off-centered to the premolar bracket. When the space was gradually closed, the V-bend (center of the apical portion) moved toward the posterior side. (B) Space was closed gradually, at which point the center of the apical portion was located above the premolar bracket. This force system provided intrusive forces thorough the space-closure period. As a result, there was no extrusion of the anterior teeth compared to the posterior teeth.Figure 6. (A) Anterior space closure performed by the planar 8 × 8–mm reversed-L closing loop archwire. The center of the apical portion (indicated by white dot) acted like a V-bend. At the beginning of space closure the V-bend was located a little off-centered to the premolar bracket. When the space was gradually closed, the V-bend (center of the apical portion) moved toward the posterior side. (B) Space was closed gradually, at which point the center of the apical portion was located above the premolar bracket. This force system provided intrusive forces thorough the space-closure period. As a result, there was no extrusion of the anterior teeth compared to the posterior teeth.Figure 6. (A) Anterior space closure performed by the planar 8 × 8–mm reversed-L closing loop archwire. The center of the apical portion (indicated by white dot) acted like a V-bend. At the beginning of space closure the V-bend was located a little off-centered to the premolar bracket. When the space was gradually closed, the V-bend (center of the apical portion) moved toward the posterior side. (B) Space was closed gradually, at which point the center of the apical portion was located above the premolar bracket. This force system provided intrusive forces thorough the space-closure period. As a result, there was no extrusion of the anterior teeth compared to the posterior teeth.
Figure 6. (A) Anterior space closure performed by the planar 8 × 8–mm reversed-L closing loop archwire. The center of the apical portion (indicated by white dot) acted like a V-bend. At the beginning of space closure the V-bend was located a little off-centered to the premolar bracket. When the space was gradually closed, the V-bend (center of the apical portion) moved toward the posterior side. (B) Space was closed gradually, at which point the center of the apical portion was located above the premolar bracket. This force system provided intrusive forces thorough the space-closure period. As a result, there was no extrusion of the anterior teeth compared to the posterior teeth.

Citation: The Angle Orthodontist 87, 1; 10.2319/020316-95.1

CONCLUSIONS

  • The apical portion of closing loops, rather than the position of their vertical legs, is one of the principal determinants that affects the force system of closing loops.

  • The center point of the apical portion of a loop acts like a V-bend.

ACKNOWLEDGMENT

This research was supported by the Ratchadaphiseksomphot Endowment Fund of Chulalongkorn University (RES560530256-AS).

REFERENCES

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    Techalertpaisarn P,
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    Mechanical properties of Opus closing loops, L-loops, and T-loops investigated with finite element analysis. Am J Orthod Dentofacial Orthop. 2013;143:675–683.

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    Techalertpaisarn P,
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    How do mechanical responses at closing loop ends vary when loop position changes? A systematic analysis of vertical, T-, and L-loops. Oral Sci Int. 2013;10:58–64.

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    Hayashi K,
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    Techalertpaisarn P,
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Copyright: © 2017 by The EH Angle Education and Research Foundation, Inc.
Figure 1.
Figure 1.

(A) T-, SL-, and RL-loop dimensions, where a is distance from center of the apical portion of the closing loop to PB end, and b is the interbracket distance (12 mm). (B) Definitions for forces and moments: Positive vertical forces move a tooth in the direction opposite of the loop (extrusion), negative forces move a tooth in the loop direction (intrusion); positive moments rotate a tooth clockwise, negative moments rotate a tooth counterclockwise.

Figure 2.
Figure 2.

Load/deflection of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.

Figure 3.
Figure 3.

Vertical forces at the canine bracket of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.

Figure 4.
Figure 4.

Moment-to-force (M/F) ratios at premolar bracket (PB) and canine bracket (CB) ends of the T-, SL-, and RL-loops for various loop positions at 100-g horizontal activation force.

Figure 5.
Figure 5.

Illustration of the similar force systems for T-, SL-, and RL-loops when the apical portions are at the same position (here the center) while loop legs are at different positions. The horizontal activation force in this example is 200 g. The solid line shows the activated stage; the gray line shows the inactivated stage.

Figure 6.
Figure 6.

(A) Anterior space closure performed by the planar 8 × 8–mm reversed-L closing loop archwire. The center of the apical portion (indicated by white dot) acted like a V-bend. At the beginning of space closure the V-bend was located a little off-centered to the premolar bracket. When the space was gradually closed, the V-bend (center of the apical portion) moved toward the posterior side. (B) Space was closed gradually, at which point the center of the apical portion was located above the premolar bracket. This force system provided intrusive forces thorough the space-closure period. As a result, there was no extrusion of the anterior teeth compared to the posterior teeth.

Contributor Notes

Corresponding author: Dr Antheunis Versluis, Department of Bioscience Research, College of Dentistry, University of Tennessee Health Science Center, 875 Union Avenue, Memphis, TN 38163 (e-mail: antheun@uthsc.edu)
Received: 01 Feb 2016
Accepted: 01 Jun 2016
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