Introduction
Although many attempts have been made to study the mechanical behavior of closing loops, most have been limited to analyses of the magnitude of forces and moments acting on the end of the closing loop. The objectives of this study were to simulate orthodontic tooth movement during the activation of a newly designed closing loop combined with a gable bend and to investigate the optimal loop activation condition to achieve the desired tooth movement.
Methods
We constructed a 3-dimensional model of maxillary dentition reproducing the state wherein a looped archwire combined with a gable bend was engaged in brackets and tubes. Orthodontic tooth movements were simulated for both anterior and posterior teeth while varying the degree of gable bend using the finite element method.
Results
The incorporation of a 5° gable bend into the newly designed closing loop produced lingual crown tipping for the central incisor and bodily movement for the first molar. The incorporation of 10° and 15° gable bends produced bodily movement and root movement, respectively, for the central incisor and distal tipping for the first molar.
Conclusions
Torque control of the anterior teeth and anchorage control of the posterior teeth can be carried out effectively and simply by reducing by half the thickness of a teardrop loop with a height of 10 mm and a 0.019 × 0.025-in cross-section, to a distance of 3 mm from its apex, and by incorporating various degrees of gable bend into the loop corresponding to the treatment plan.
Highlights
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We simulated orthodontic tooth movement for correction of Class II Division 2 malocclusion.
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A newly designed closing loop combined with a gable bend was studied.
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A 5° gable bend produced lingual crown tipping of the central incisor.
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A 10° bend produced bodily movement of central incisor and distal tipping of the first molar.
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A 15° gable bend produced root movement of the central incisor.
Since the establishment of the Tweed technique, closing loops have been widely used for space closure in the treatment of premolar extraction patients. , Contrary to sliding mechanics, wherein friction generated during space closure can reduce the rate of tooth movement, , loop mechanics is a frictionless treatment system. Therefore, this technique has the potential to produce predetermined moment-to-force (M/F) ratios to achieve controlled movement of the anterior teeth accurately. , In contrast, the loop mechanics system has several disadvantages. That is, the M/F ratio produced by conventionally designed loops is too low to achieve bodily or root movement of the anterior teeth. Another shortcoming is that closing loops made of stainless steel wire could produce an excessively heavy force, especially with the incorporation of gable bends. , , , Furthermore, the use of an archwire with a larger cross-section could generate a greater retraction force in the case of the 0.022-in bracket slot system.
To generate a low force magnitude and increase the M/F ratio for achieving better control of the anterior teeth during space closure, many loop designs with complicated shapes have been developed by extending both the horizontal length and the vertical height. Sumi et al developed a simple design of a 10-mm-high teardrop loop with a wire cross-section reduced in thickness by 50% for a distance of 3 mm from the apex. This newly designed closing loop produces a gentle force and a higher M/F ratio of 9.3 without adding a gable bend. However, the M/F ratios obtained from these numerical studies do not apply directly to the incisor but rather to the canine bracket adjacent to the loop. However, because of the difficulty of conducting numerical analysis for the mechanical behavior of a looped archwire, the tooth movement generated by activating this new closing loop design has not yet been investigated. The purposes of the present study were to simulate orthodontic tooth movement of not only the anterior but also the posterior teeth after the application of the newly designed loop with a partially reduced wire cross-section in the 0.022-in bracket slot system and to investigate the effect of the degree of gable bend on the tooth movement pattern using the finite element method (FEM).
Material and methods
Three-dimensional (3D) images of a maxillary dentition, taken with a multi-image cone-beam computed tomography scanner (3DX; J. Morita, Kyoto, Japan), were exported as digital imaging and communication in medicine file to 3D image processing and editing software (Mimics, version 10.02; Materialise, Leuven, Belgium). After reconstructing the digital imaging and communication in medicine data to 3D surface data, they were imported to FEM preprocessing and postprocessing software (Patran 2017; MSC Software, Los Angeles, Calif). After remeshing, the teeth were aligned to the ideal positions of the normal dentition, assuming that the case model was diagnosed as maxillary protrusion, the first premolar extractions were indicated ( Fig 1 , A ).
Each tooth model was constructed using thin shell elements with a thickness of 3 mm to develop a more simplified model for reducing the analysis time. A Young modulus of 18,600 MPa and a Poisson ratio of 0.3 were assigned in accordance with previous studies. , The periodontal ligament (PDL) was constructed using wedge elements with a uniform thickness of 0.2 mm, Young’s modulus of 0.05 MPa, and a Poisson ratio of 0.3.
The fixed orthodontic appliance was composed of 0.022 × 0.028-in slot brackets and a 0.019 × 0.025-inch stainless steel archwire. These were constructed using solid modeler software (Solidworks 2016; Dassault Systèmes, Vélizy-Villacoublay, France) and imported to the preprocessing software. An archwire and brackets were modeled using 8-node hexahedral elements with Young’s modulus of 200 GPa and a Poisson ratio of 0.3. , All materials were set to be isotropic. Each bracket was attached to the corresponding tooth with rigid body elements. A symmetric model of the maxillary dentition was developed to reduce the number of elements and minimize the analysis time. The front end of the archwire, on the midsagittal plane, was constrained in the transverse (x) direction for symmetric analysis.
Long-term orthodontic tooth movements were simulated using a bone remodeling algorithm that has been described in detail elsewhere. Simply, in the first step, an orthodontic force is applied to the tooth while displacements are constrained at the outer surface of the PDL. After an initial tooth displacement is produced, each node forming the outer surface of the PDL is displaced so that the PDL is restored to its original configuration and a thickness of 0.2 mm in the next step. In the following step, a second orthodontic force is applied to the tooth while displacements are retrained at the outer nodes of the PDL, and again, the geometry of the PDL is updated. Then, 2 steps—the initial displacement and bone remodeling—are iterated to simulate orthodontic tooth movement on the assumption that initial tooth displacement could be a predictor of long-term tooth movement. All analyses were performed using a FEM package (Marc 2016; MSC Software).
The basic design of the closing loop examined in this study was a teardrop that was 10 mm in height and 2 mm in width, bent from a 0.019 × 0.025-in stainless steel archwire. The thickness of the cross-section of the loop was reduced by 50% for a distance of 3 mm from the apex ( Fig 1 , B ).
The tooth movement during space closure was simulated by activating the loop by 1 mm using an actuator element connecting the distal end of the second molar tube and a point distal to the tube on the archwire ( Fig 2 ). The actuator element is a truss element with variable length. Lengthening of the actuator element causes distal movement of the posterior portion of the archwire, which enables activation of the loop in the same manner as a tie-back in clinical practice.
We tested 5 different mechanical conditions of loop activation. First, we investigated the effect of a reduction in the cross-section at the loop apex by comparing the tooth movements when the thickness of the cross-section of the loop apex was reduced by 0% and 50%. Next, we compared the effect of different degrees of gable bend (5°, 10°, and 15°) incorporated into the loop on the tooth movement pattern.
The loading condition, wherein the loop with gable bend is activated, was simulated in the following steps. First, we constructed the model representing the state of neutral activation, namely, the state before activation of the loop, instead of engaging the bent archwire with the closing loop combined with a gable bend into the brackets and tubes. To construct the model in which the closing archwire combined with a gable bend is passively inserted into the brackets and tubes on the posterior teeth ( Fig 3 , A ), 2 coordinate systems were established on the basis of the loop configuration ( Fig 3 , B ). One is the coordinate system whose origin is located at the corner of the loop base of the anterior portion of the archwire, and the other is the coordinate system whose origin is located at the corner of the loop base of the posterior portion of the archwire (CS2). The original model was divided into 3 segments. The anterior segment consisted of the anterior portion of the archwire mesial to the loop, including the anterior teeth models and corresponding PDLs, the middle segment was the closing loop, and the posterior segment included the posterior portion of the archwire distal to the loop, including the posterior teeth models and corresponding PDLs.
The loop and posterior segments of the model were rotated in 2 steps ( Fig 3 , C-E ). First, the posterior segment was rotated around the z-axis at the origin of CS2 by one-half of the angulation of the gable bend ( Fig 3 , C ). That is, each node forming the archwire distal to CS2 was rotated. At the same time, the posterior teeth and corresponding brackets were also rotated while maintaining their relative position to the archwire. Next, the loop and posterior segments were rotated around the z-axis at the origin of the coordinate system whose origin is located at the corner of the loop base of the anterior portion of the archwire by one-half of the angulation of the gable bend ( Fig 3 , D ). The model before activating the loop with a gable bend was constructed in this manner ( Fig 3 , A and E ).
Next, the mechanical conditions of loop activation were reproduced as follows in the case that a gable bend is incorporated into the loop. Every node of the loop and posterior segments of the model was displaced to its original position ( Fig 4 , A and B ). Because the archwire was engaged into the brackets, it also caused the displacement of the archwire, including the loop combined with a gable bend. Then, the loop was activated by 1 mm using the actuator element ( Fig 4 , C ), and space was closed after the movements of the anterior and posterior teeth ( Fig 4 , D ). At this time, displacements were constrained at the outer surface of the PDL, and consequently, long-term tooth movement was simulated on the basis of a bone remodeling algorithm. Tooth movements of the central incisor and first molar were represented as a combination of translation and rotational displacement at the center of resistance (CR) in the sagittal plane.
