Background
In this study, the effect of open-arch mechanics on the displacement and von Mises stress distributions in mandibular teeth was investigated using a finite element analysis.
Methods
After mandibular bone, teeth, and periodontal ligament formation, 0.022-in brackets and 0.016 × 0.022-in tubes were placed on the buccal equatorial line. Four scenarios were modeled using 2 materials (nickel-titanium [NiTi] and stainless steel [SS]) and 2 arch wire sizes (0.016-in and 0.016 × 0.022-in). Displacement and von Mises stresses were analyzed via finite element analysis (Algor Fempro, ALGOR Inc, Pittsburgh, Pa) in models including teeth up to the second molar, with the mandibular left first premolar missing.
Results
The highest von Mises stress was found in the 0.016-in NiTi wire, and the lowest in the 0.016-in SS wire. Across scenarios, peak root surface stress was at the apical region of the mandibular left second premolar, whereas the highest periodontal ligament stress was in its gingival third. The buccal tubercle of this tooth showed the greatest displacement. Among archwires, the highest stress occurred in the 0.016-in SS wire.
Conclusions
Von Mises stresses decreased with distance from the missing tooth site. The highest stress occurred at the apical end of the mandibular left second premolar. Except for the 0.016-in round NiTi wire, this tooth showed the greatest tubercle apex displacement. In all scenarios, except the mandibular left first molar, mesial root stresses were greater in the mandibular roots than in the distal roots.
Highlights
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Archwire material and cross-sectional geometry influenced stress distribution.
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Peak von Mises stresses were observed using 0.016-in nickel-titanium wires.
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Rectangular stainless steel wires (0.016 × 0.022-in) minimized uncontrolled movement.
The dynamic and static harmony of intraoral and extraoral structures is important for the maintenance of health. Tooth loss disrupts the balance of intraoral and extraoral muscles, resulting in nonideal functional occlusion. Therefore, it is necessary to organize the gaps that occur in tooth deficiencies to provide long-term stability with muscle balance and ideal functional occlusion.
Tooth deficiencies can be appropriately organized with different mechanics by opening or closing gaps. Wire bends, loops, chain elastics, the sleeve push technique, and most commonly open coil spring mechanics are used to open the space. The opening of the missing tooth gap can be achieved by pushing the anterior teeth forward, pushing the posterior teeth backward, or both movements together. During this process, the teeth may exhibit controlled or uncontrolled tipping, parallel movement, as well as extrusion or intrusion. Orthodontic treatments using archwires of different materials and thicknesses can cause different levels of clinical effects on incisor proclines, tooth rotations, tipping, and vertical tooth movements. ,,,,, Depending on the amount of these movements, complications, such as root resorption, gingival recession, and alveolar bone loss, may occur. ,
For the orthodontist to make the most effective treatment plan, to choose the appropriate and correct mechanics, to provide only the planned tooth movements, and to control unwanted side effects, biomechanical principles should be analyzed and applied in the best way. However, orthodontic force systems used even in the simplest mechanics are considered as unmeasurable force systems. This means that the forces acting on each tooth are 3-dimensional (3D) and there is a wide range of unknowns to predict the expected tooth movement. Moreove, 3D finite element analysis (FEA), an effective computer simulation technique, simulates the orthodontic force systems used in the clinic and provides the opportunity to examine the biomechanical changes caused by various external forces in living structures in 3 dimensions. It enables the prediction of stress, deformation, and strain in different structures, such as alveolar bone, teeth, and periodontal ligament (PDL), with orthodontic force application. According to fundamental biomechanical principles, stress is directly proportional to the applied force and inversely proportional to the cross-sectional area over which it acts. In cases that cannot be simulated or standardized in clinical conditions, FEA is used to eliminate individual differences and compare different biomechanical properties under standard conditions. , In this study, although it is not possible to examine the effect of open spring mechanics on each tooth by compressing it to apply a 100-g force in clinical conditions, it was standardized in a computer environment.
In the literature, there are no FEA studies that analyze the displacement method of open arc mechanics. There are no studies analyzing the displacement and stress distribution of archwires of different thicknesses and materials on the teeth. Therefore, the aim of our study was to evaluate the stress distribution and displacements of mandibular teeth during the replacement of a missing mandibular left first premolar by open spring mechanics with archwires of different materials and thicknesses using FEA.
Material and methods
In this study, the stress distribution and displacements of mandibular teeth in the process of obtaining space with open-arch mechanics using archwires of different materials and thicknesses were evaluated by 3D FEA in patients in which the mandibular left first premolar was missing and the second molars were included. Mandibular bone tissue was modeled after obtaining a 3D model using the 3D complex render method. Spongiosis bone was created with the offset method to separate the bone tissue, and the continuity of the structure was ensured by making the necessary adjustments. During the FEA, creating geometric models, creating the network structure, determining the material properties, defining the boundary conditions were followed.
Teeth were individually modeled manually using Rhinoceros software (version 4.0; McNeel Inc, Seattle, Wash) according to the detailed morphology and dimensions found in the Wheeler atlas of dental anatomy and physiology. The PDL is modeled between the tooth roots and mandibular alveolar bone, with an average linear thickness of 0.25 mm and an isotropic and homogeneous structure. , 0.022 × 0.025-in slotted brackets (Ormco, Orange, Calif), molar tubes, and archwires were modeled using Rhinoceros software (McNeel Inc) and arranged individually manually. The brackets are positioned on the vestibular crown surfaces of the teeth so that the center of the geometric shape of the bracket in relation to the base of the bracket corresponds to the middle part of the crown of the tooth.
After the mandibular cortical and compact bone, wires, brackets, and spongiosis bone were transferred to the model to reflect the actual morphologic structure, the modeling process was completed by placing the models in the correct positions in 3D space in the Rhinoceros software (McNeel Inc), and the study model was obtained ( Fig 1 ).
Digital image of the study model.
Four different scenarios were created for patients in which the mandibular left first premolar was missing. All teeth, including the second molars on the mandibular dental arch, were included in the model. To ensure symmetry of the result, both mandibular condyles were modeled and analyzed with symmetry conditions by fixing the head and the inferior border of the corpus ( Fig 2 ).
Sites of mandibular stabilization.
In 4 different scenarios, archwires of varying materials and dimensions were inserted into bracket slots, and a force of 100 g was applied using open springs. In the first scenario, a 0.016-in full-round nickel-titanium (NiTi) archwire was used, whereas in the second scenario, a 0.016-in full-round stainless-steel (SS) archwire was used under the same force conditions. The third scenario involved a 0.016 × 0.022-in NiTi archwire, whereas the fourth scenario used a 0.016 × 0.022-in SS archwire. In all patients, the application of force remained consistent, allowing for a comparative evaluation of the effects of wire material and geometry on orthodontic force transmission. In 4 scenarios, the parameters were evaluated at the first application of forces.
The displacement values occurring in the teeth were evaluated in the vertical, transverse, and sagittal directions at specific nodal points, including the apices, tubercle apexes, and the midpoint of the incisal margins of mandibular teeth. In addition, stress values, specifically von Mises stress, were measured at these same selected nodal points, which include the apices, tubercle apexes, and the midpoint of the incisal edges of mandibular teeth. This analysis provides insight into both the displacement and stress distribution in response to the applied forces. Furthermore, von Mises stress values were also analyzed at selected nodes in the apical and gingival one-third of the PDL of mandibular teeth, providing a comprehensive understanding of stress distribution within the periodontal structures ( Fig 3 ).
In figures in which displacements are shown, negative displacements along the x-axis indicate movement toward the patient’s right side, whereas positive displacements indicate movement toward the patient’s left side. Negative displacements along the y-axis represent anterior movement, whereas positive displacements indicate posterior movement. Negative displacements along the z-axis indicate intrusion, whereas positive displacements represent extrusion.
The displacement findings of the tooth tissues, according to their initial positions, are expressed in millimeters. Stress values are given in N/mm 2. The amount of displacement and von Mises stresses in the teeth, determined as a result of FEA, are visualized with a color scale. Displacement values are interpreted according to the coordinate system, in which negative and positive values correspond to specific directions of tooth movement: negative x-axis values indicate movement toward the patient’s right, and positive values toward the left; negative y-axis values indicate anterior movement, and positive values posterior movement; negative z-axis values indicate intrusion, and positive values extrusion. Blue-colored areas indicate low von Mises stresses, whereas red-colored areas indicate high von Mises stresses. To create the network structure, the geometric models finalized with the VR Mesh Studio software (VirtualGrid Inc, Bellevue City, Wash) were converted into a mesh structure and transferred to the Algor Fempro (ALGOR Inc, Pittsburgh, Pa) FEA software. The mathematical model was created by using tetrahedral or brick elements with 8 nodes, as many as possible, in the construction of the network structure. The models and structures used in our study were assumed to be isotropic, homogeneous, and linearly elastic. The physical properties of these structures were defined by Poisson ratio and modulus of elasticity values determined based on previous studies. ,,,
Results
The space-opening mechanics using different archwires demonstrated variable effects depending on the wire type. Among the 4 scenarios designed, the following observations were made.
In scenario 1, the highest displacement value in all 3 directions—transversal, sagittal, and vertical—was observed in the buccal tubercle of the mandibular left second premolar (0.911 mm), whereas the lowest value was observed in the distolingual tubercle of the mandibular right second molar (0.013 mm). The highest von Mises stress was observed at the root apex of the second premolar (5387 N/mm 2). Stress areas in the PDL are concentrated in the PDL surrounding the mesial root of the canine, second premolar, and mandibular left second molars adjacent to the missing mandibular left first premolar.
In scenario 2, the highest displacement value in all 3 directions was observed at the buccal tubercle of the mandibular left second premolar (0.830 mm), whereas the smallest value was observed at the distolingual tubercle of the mandibular left second molar (0.189 mm). The highest von Mises stress was observed at the root apex of the mandibular left second premolar (4697 N/mm 2). The areas of stress in the PDL are the root of the second premolar adjacent to the cavity of the mandibular left first premolar and the PDL surrounding the mesial root of the mandibular left first molar.
In scenario 3, the highest displacement value in all 3 directions was observed at the buccal tubercle of the mandibular left second premolar (0.764 mm), whereas the lowest value was observed at the mesial root apex of the mandibular right first molar (0.011 mm). The highest von Mises stress was observed at the root apex of the mandibular second premolar (4284 N/mm 2). Stress areas in the PDL showed high von Mises stresses in the root of the mandibular left second premolar and the mesial root of the mandibular left first molar.
In scenario 4, the highest displacement value in all 3 directions was observed at the buccal tubercle of the mandibular left second premolar (0.693 mm), whereas the smallest value was observed at the buccal tubercle apex of the mandibular right first premolar (0.008 mm). The highest von Mises stress was observed at the root apex of the mandibular left second premolar (3774 N/mm 2). The areas of stress in the PDL are the root of the mandibular left second premolar and the PDL surrounding the mesial root of the mandibular left first molar ( Tables I-III ).
Table I
Displacements
| Mean displacement | Displacement along the x-axis | Displacement along the y-axis | Displacement along the z-axis | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Tubercle apex | Apical | Tubercle apex | Apical | Tubercle apex | Apical | Tubercle apex | Apical | |||||||||
| CS | ES | CS | ES | CS | ES | CS | ES | CS | ES | CS | ES | CS | ES | CS | ES | |
| 0.016-in NiTi | ||||||||||||||||
| Central incisor | 0.365 | 0.428 | 0.102 | 0.148 | –0.243 | –0.222 | 0.047 | 0.045 | –0.270 | –0.362 | 0.085 | 0.123 | –0.036 | –0.054 | 0.033 | 0.069 |
| Lateral incisor | 0.238 | 0.568 | 0.067 | 0.176 | –0.192 | –0.378 | 0.041 | 0.072 | –0.135 | –0.423 | 0.051 | 0.110 | –0.041 | –0.021 | 0.013 | 0.117 |
| Canine | 0.172 | 0.502 | 0.049 | 0.175 | –0.158 | –0.499 | 0.039 | 0.174 | –0.061 | –0.035 | 0.030 | –0.022 | –0.028 | 0.045 | 0.006 | 0.009 |
| First premolar | 0.018 | N/A | 0.014 | N/A | –0.006 | N/A | 0.010 | N/A | ||||||||
| Buccal | 0.122 | N/A | –0.122 | N/A | –0.010 | N/A | –0.007 | N/A | ||||||||
| Palatal | 0.093 | N/A | –0.087 | N/A | –0.032 | N/A | 0.013 | N/A | ||||||||
| Second premolar | 0.021 | 0.334 | 0.004 | 0.205 | –0.014 | –0.257 | –0.016 | –0.060 | ||||||||
| Buccal | 0.074 | 0.911 | –0.069 | –0.545 | 0.016 | 0.724 | –0.019 | 0.099 | ||||||||
| Palatal | 0.050 | 0.769 | –0.048 | –0.574 | –0.007 | 0.487 | –0.011 | –0.155 | ||||||||
| First molar | ||||||||||||||||
| Mesiobuccal | 0.049 | 0.518 | 0.017 | 0.171 | –0.054 | –0.240 | –0.007 | –0.011 | 0.022 | 0.485 | –0.016 | –0.146 | 0.009 | 0.182 | 0.008 | 0.088 |
| Distobuccal | 0.045 | 0.521 | 0.016 | 0.156 | –0.029 | –0.076 | 0.007 | –0.066 | 0.032 | 0.523 | –0.008 | –0.119 | –0.012 | 0.052 | –0.012 | 0.076 |
| Mesiolingual | 0.044 | 0.413 | –0.042 | –0.017 | 0.008 | 0.403 | 0.009 | 0.088 | ||||||||
| Distolingual | 0.028 | 0.452 | –0.024 | –0.095 | 0.014 | 0.433 | –0.006 | –0.084 | ||||||||
| Second molar | ||||||||||||||||
| Mesiobuccal | 0.043 | 0.404 | 0.013 | 0.089 | –0.031 | 0.058 | –0.004 | –0.029 | 0.028 | 0.392 | –0.011 | –0.065 | 0.006 | 0.078 | –0.005 | –0.054 |
| Distobuccal | 0.033 | 0.422 | 0.013 | 0.149 | –0.012 | –0.091 | 0.004 | –0.092 | 0.030 | 0.409 | –0.005 | –0.069 | –0.005 | –0.046 | –0.012 | –0.116 |
| Mesiolingual | 0.026 | 0.213 | –0.025 | 0.021 | 0.007 | 0.211 | 0.004 | 0.023 | ||||||||
| Distolingual | 0.013 | 0.298 | –0.006 | –0.134 | 0.010 | 0.244 | –0.007 | 0.106 | ||||||||
| 0.016-in SS | ||||||||||||||||
| Central incisor | 0.301 | 0.343 | 0.081 | 0.113 | –0.204 | –0.209 | 0.034 | 0.038 | –0.220 | –0.269 | 0.069 | 0.091 | –0.031 | –0.037 | 0.025 | 0.057 |
| Lateral incisor | 0.214 | 0.443 | 0.056 | 0.128 | –0.168 | –0.355 | 0.031 | 0.061 | –0.126 | –0.265 | 0.046 | 0.070 | –0.039 | –0.003 | 0.010 | 0.089 |
| Canine | 0.158 | 0.492 | 0.042 | 0.158 | –0.143 | –0.481 | 0.031 | 0.153 | –0.060 | 0.080 | 0.027 | 0.038 | –0.029 | 0.067 | 0.004 | –0.008 |
| First premolar | 0.016 | N/A | 0.011 | N/A | –0.006 | N/A | 0.010 | N/A | ||||||||
| Buccal | 0.109 | N/A | –0.108 | N/A | –0.013 | N/A | –0.009 | N/A | ||||||||
| Palatal | 0.088 | N/A | –0.082 | N/A | –0.030 | N/A | 0.010 | N/A | ||||||||
| Second premolar | 0.017 | 0.284 | 0.002 | 0.180 | –0.011 | –0.215 | –0.013 | –0.047 | ||||||||
| Buccal | 0.069 | 0.830 | –0.066 | –0.535 | 0.009 | 0.627 | –0.017 | 0.095 | ||||||||
| Palatal | 0.050 | 0.716 | –0.048 | –0.513 | –0.010 | 0.0482 | –0.009 | –0.132 | ||||||||
| First molar | ||||||||||||||||
| Mesiobuccal | 0.055 | 0.450 | 0.015 | 0.142 | –0.051 | –0.065 | –0.007 | –1027 | 0.019 | 0.417 | –0.013 | –0.120 | 0.006 | 0.154 | –0.002 | 0.077 |
| Distobuccal | 0.042 | 0.443 | 0.015 | 0.125 | –0.031 | –0.100 | 0.007 | –0.016 | 0.026 | 0.431 | –0.007 | –0.110 | –0.010 | –0.020 | –0.012 | –0.057 |
| Mesiolingual | 0.042 | 0.387 | –0.041 | –0.076 | 0.006 | 0.375 | 0.008 | 0.060 | ||||||||
| Distolingual | 0.028 | 0.406 | –0.025 | –0.103 | 0.012 | 0.387 | –0.005 | –0.069 | ||||||||
| Second molar | ||||||||||||||||
| Mesiobuccal | 0.039 | 0.318 | 0.013 | 0.072 | –0.029 | 0.021 | –0.005 | –0.021 | 0.025 | 0.312 | –0.010 | –0.043 | 0.004 | 0.057 | –0.007 | –0.053 |
| Distobuccal | 0.031 | 0.342 | 0.014 | 0.120 | –0.012 | –0.086 | 0.003 | –0.066 | 0.028 | 0.328 | –0.004 | –0.007 | –0.006 | –0.040 | –0.012 | –0.100 |
| Mesiolingual | 0.024 | 0.189 | –0.024 | –0.015 | 0.005 | 0.188 | 0.002 | –0.001 | ||||||||
| Distolingual | 0.014 | 0.262 | –0.006 | –0.122 | 0.010 | 0.209 | –0.008 | –0.098 | ||||||||
| 0.016 × 0.022-in NiTi | ||||||||||||||||
| Central incisor | 0.291 | 0.327 | 0.082 | 0.114 | –0.196 | –0.199 | 0.033 | 0.035 | –0.212 | –0.257 | 0.072 | 0.093 | –0.036 | –0.036 | 0.023 | 0.055 |
| Lateral incisor | 0.213 | 0.439 | 0.059 | 0.111 | –0.161 | –0.398 | 0.029 | 0.072 | –0.135 | –0.186 | 0.049 | 0.063 | –0.038 | –0.007 | 0.014 | 0.055 |
| Canine | 0.148 | 0.511 | 0.041 | 0.151 | –0.129 | –0.505 | 0.028 | 0.151 | –0.067 | 0.031 | 0.030 | –0.004 | –0.028 | 0.068 | 0.004 | 0.013 |
| First premolar | 0.012 | N/A | 0.007 | N/A | –0.007 | N/A | 0.007 | N/A | ||||||||
| Buccal | 0.099 | N/A | –0.098 | N/A | –0.011 | N/A | –0.008 | N/A | ||||||||
| Palatal | 0.080 | N/A | –0.075 | N/A | –0.024 | N/A | 0.008 | N/A | ||||||||
| Second premolar | 0.017 | 0.248 | –0.004 | 0.099 | –0.011 | –0.225 | –0.013 | –0.029 | ||||||||
| Buccal | 0.068 | 0.764 | –0.066 | –0.411 | 0.007 | 0.636 | –0.017 | 0.099 | ||||||||
| Palatal | 0.052 | 0.671 | –0.051 | –0.360 | –0.007 | 0.558 | –0.009 | 0.094 | ||||||||
| First molar | ||||||||||||||||
| Mesiobuccal | 0.054 | 0.422 | 0.011 | 0.141 | –0.052 | –0.070 | –0.006 | –0.006 | 0.012 | 0.386 | –0.011 | –0.111 | 0.006 | 0.157 | 0.001 | 0.086 |
| Distobuccal | 0.038 | 0.393 | 0.012 | 0.115 | –0.033 | –0.072 | 0.006 | –0.007 | 0.018 | 0.386 | –0.006 | –0.109 | –0.008 | –0.002 | –0.007 | –0.038 |
| Mesiolingual | 0.044 | 0.380 | –0.043 | –0.071 | 0.009 | 0.365 | 0.010 | 0.079 | ||||||||
| Distolingual | 0.031 | 0.374 | 0.030 | –0.074 | 0.006 | 0.364 | –0.002 | –0.049 | ||||||||
| Second molar | ||||||||||||||||
| Mesiobuccal | 0.033 | 0.290 | 0.013 | 0.074 | –0.025 | –0.031 | –0.005 | –0.008 | 0.021 | 0.287 | –0.006 | –0.030 | –0.002 | 0.023 | –0.010 | –0.067 |
| Distobuccal | 0.026 | 0.320 | 0.014 | 0.115 | –0.012 | –0.105 | 0.001 | –0.042 | 0.022 | 0.297 | –0.002 | –3087 | –0.009 | –0.055 | –0.014 | –0.107 |
| Mesiolingual | 0.022 | 0.194 | –0.021 | –0.046 | 0.004 | 0.186 | –0.009 | –0.030 | ||||||||
| Distolingual | 0.014 | 0.270 | –0.007 | –0.130 | 0.007 | 0.205 | –0.009 | –0.116 | ||||||||
| 0.016 × 0.022-in SS | ||||||||||||||||
| Central incisor | 0.256 | 0.281 | 0.072 | 0.089 | –0.176 | –0.196 | 0.022 | 0.029 | –0.182 | –0.200 | 0.068 | 0.073 | –0.035 | –0.030 | 0.010 | 0.043 |
| Lateral incisor | 0.194 | 0.387 | 0.052 | 0.086 | –0.144 | –0.372 | 0.022 | 0.063 | –0.125 | –0.107 | 0.045 | 0.042 | –0.037 | 0.002 | 0.011 | 0.041 |
| Canine | 0.137 | 0.517 | 0.037 | 0.145 | –0.116 | –0.497 | 0.022 | 0.143 | –0.067 | 0.112 | 0.029 | –0.024 | –0.027 | 0.087 | 0.003 | 0.011 |
| First premolar | 0.008 | N/A | 0.004 | N/A | –0.004 | N/A | 0.005 | N/A | ||||||||
| Buccal | 0.090 | N/A | –0.088 | N/A | –0.016 | N/A | –0.008 | N/A | ||||||||
| Palatal | 0.074 | N/A | –0.070 | N/A | –0.025 | N/A | 0.006 | N/A | ||||||||
| Second premolar | 0.014 | 0.213 | –0.003 | 0.100 | –0.008 | –0.188 | –0.011 | –0.015 | ||||||||
| Buccal | 0.063 | 0.693 | –0.061 | –0.351 | 0.001 | 0.549 | –0.015 | 0.101 | ||||||||
| Palatal | 0.050 | 0.615 | –0.048 | –0.410 | –0.011 | 0.500 | –0.007 | –0.070 | ||||||||
| First molar | ||||||||||||||||
| Mesiobuccal | 0.050 | 0.411 | 0.011 | 0.126 | –0.049 | –0.135 | –0.007 | 0.007 | 0.008 | 0.362 | –0.009 | –0.101 | 0.004 | 0.140 | 0.008 | 0.076 |
| Distobuccal | 0.037 | 0.367 | 0.010 | 0.117 | –0.034 | –0.107 | 0.004 | 0.032 | 0.014 | 0.351 | –0.004 | –0.109 | –0.007 | 0.001 | –0.007 | –0.028 |
| Mesiolingual | 0.042 | 0.381 | –0.042 | –0.118 | –0.002 | 0.359 | 0.008 | 0.049 | ||||||||
| Distolingual | 0.029 | 0.372 | –0.029 | –0.096 | 0.003 | 0.353 | –0.005 | –0.067 | ||||||||
| Second molar | ||||||||||||||||
| Mesiobuccal | 0.031 | 0.259 | 0.012 | 0.068 | –0.025 | –0.055 | –0.006 | –0.004 | 0.017 | 0.253 | –0.005 | –0.022 | –0.004 | 0.015 | –0.010 | –0.064 |
| Distobuccal | 0.025 | 0.286 | 0.014 | 0.097 | –0.013 | –0.108 | –0.003 | –0.024 | 0.019 | 0.260 | –0.002 | –0.005 | –0.010 | –0.054 | –0.014 | –0.094 |
| Mesiolingual | 0.021 | 0.195 | –0.021 | –0.064 | 0.002 | 0.181 | –0.004 | –0.036 | ||||||||
| Distolingual | 0.014 | 0.255 | –0.007 | –0.125 | 0.004 | 0.193 | –0.011 | –0.112 | ||||||||
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