Introduction
This study develops a finite element analysis method to define the 3-dimensional (3D) zone of center of resistance (ZCR) position for maxillary central incisors and first molars and validates its applicability under different alveolar bone levels.
Methods
Cone-beam computed tomography scans from 40 patients were grouped: group 1 (maxillary central incisors, no bone loss), group 2 (maxillary central incisors, bone loss), group 3 (maxillary first molars, no bone loss), and group 4 (maxillary first molars, bone loss). The 3D models of teeth, a simulated 0.2-mm-thick PDL, and alveolar bone were reconstructed using Mimics software (Materialise, Leuven, Belgium) and imported into ANSYS Workbench (ANSYS Inc, Canonsburg, Pa) to calculate the tooth axis of rotation (resistance axis). A 5-step method defined the ZCR: (1) set the crown center as origin, (2) apply pure 3 N·mm couples at the crown center along the x-, y-, and z-axes, (3) find the resistance axes using displacement data, (4) use an algorithm to find best point filtering (1%-3%) where axes meet; fit a sphere to these points to get the ZCR center and radius, and (5) verify the accuracy by measuring rotation angles after applying forces at the center of the ZCR, and the 1 and 2 times radius in the 3 directions.
Results
Optimal filtering percentages averaged 2.23% (group 1), 2.10% (group 2), 1.33% (group 3), and 1.63% (group 4). Alveolar resorption reduced the ZCR height. The central incisors decreased from 59.17% (standard deviation [SD]: 1.01) to 45.19% (SD: 1.61), whereas first molars decreased from 53.76% (SD: 3.03) to 46.76% (SD: 2.02) of root length. The ZCR radius decreased with alveolar bone loss—central incisors (from 0.55 to 0.49 mm) and first molars (from 0.58 to 0.48 mm). Forces applied at the center of the ZCR minimized rotation angles (x/y-axis: 0.12°; z-axis: 0.09°). Rotation increased significantly when forces were applied beyond the sphere, reaching 1.92° at twice the radius.
Conclusions
The finite element analysis method accurately and efficiently defined the 3D ZCR position and extent in central incisors and first molars. Alveolar bone loss induced apical displacement and a reduction in the ZCR extent.
Highlights
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Novel 3-dimensional method locates the tooth resistance center area via pure couples.
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Bone loss shifts the resistance center toward the root tips and shrinks its size.
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Forces applied at resistance centers minimize unwanted tooth rotations.
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This study provides a clinical reference for optimizing orthodontic forces in patients with alveolar bone loss.
In orthodontic biomechanics, the center of resistance (CR) serves as the key reference point for controlling tooth movement. A force vector passing through the CR produces pure translation, enhancing treatment precision. The CR position is influenced by root morphology, periodontal ligament (PDL) properties, and alveolar bone height. ,,, Furthermore, it varies with the direction of the applied force. Traditional theory simplifies the CR as a single point, but studies show that asymmetrical morphology and periodontal support cause CRs from different force directions to disperse in 3D space. Oh et al thus proposed the concept of a zone of center of resistance (ZCR), a small spatial region the CR occupies. Defining the ZCR is crucial for modern multidirectional force systems, especially with clear aligners applying forces from varied angles. ,
Finite element analysis (FEA) has become a reliable method for locating the CR because of its capacity to handle complex anatomic structures and provide reproducible mechanical simulations. ,,, Prior studies focused mainly on CR height along the long axis, neglecting its precise 3D position. ,, Traditional iterative methods for finding the CR were inefficient and labor-intensive. Yang et al pioneered the 3D ZCR quantification in mandibular teeth using FEA, applying forces in 4 horizontal directions and fitting a confidence sphere to the calculated CR points. This revealed the ZCR’s spherical nature in single-rooted teeth, but the method had limitations: a small sample limited to anterior teeth, no assessment of alveolar bone loss effects, and computational intensity from requiring CR calculation in multiple directions per tooth.
Classical biomechanics states that a pure couple rotates a tooth around its CR. This study proposes a novel method based on this principle: applying pure couples at the clinical crown center along the mesiodistal (MD), buccolingual (BL), and occlusogingival axes to derive the corresponding resistance axes. A custom Python algorithm then identifies their optimal 3D intersection zone, which is fitted to a sphere to define the ZCR. This approach is more efficient than previous methods as it avoids iterative force trials and multidirectional CR calculations.
Maxillary central incisors (esthetic, near-symmetrical) and first molars (functional, multirooted) have distinct biomechanics, making their ZCR comparison clinically valuable. This study aims to employ this couple-based method to define the ZCRs of these teeth and validate the method’s applicability under different alveolar bone levels (nonresorption vs resorption).
We hypothesized that (1) the intersection zone of the 3 resistance axes derived from pure couples applied along the principal clinical axes would constitute a well-defined, spherical 3D ZCR, and (2) alveolar bone loss would induce significant apical displacement and a reduction in the spatial extent (radius) of this ZCR in both tooth types.
Material and methods
This study obtained approval from the ethics committee of the affiliated Stomatological Hospital of Chongqing Medical University (Approval No. 2023199). Cone-beam computed tomography (CBCT) scans (120 kVp, 5 mA, and 0.3-mm voxel size; KaVo Dental GmbH, Biberach, Germany) were acquired from 40 patients at the same institution. Alveolar bone height was quantified as the mean vertical distance from root apex to mid, distal, buccal, and lingual alveolar crest. Subjects were divided into 4 groups: group 1 (maxillary central incisors, nonresorption, mean alveolar height: 10.78 mm) ( Fig 1 , A ), group 2 (maxillary central incisors, resorption, mean alveolar height: 8.12 mm) ( Fig 1 , B ), group 3 (maxillary first molars, nonresorption, mean alveolar height: 11.02 mm) ( Fig 1 , C ), and group 4 (maxillary first molars, resorption, mean alveolar height: 8.33 mm) ( Fig 1 , D ).
Schematic illustration of the experimental groups demonstrating 3D models of maxillary teeth and their surrounding alveolar bone: A, Group 1: maxillary central incisor with alveolar bone nonresorption (mean alveolar bone height: 10.78 mm); B, Group 2: maxillary central incisor with alveolar bone resorption (mean alveolar bone height: 8.12 mm); C, Group 3: maxillary first molar with alveolar bone nonresorption (mean alveolar bone height: 11.02 mm); D, Group 4: maxillary first molar with alveolar bone resorption (mean alveolar bone height: 8.33 mm).
Digital imaging and communications in medicine format images were imported into Mimics software (Materialise, Leuven, Belgium). Threshold-based segmentation reconstructed 3D models of the maxillary alveolar bone, central incisors, and first molars. Because CBCT cannot visualize the PDL, the tooth volumes were uniformly offset outward by 0.2 mm. Boolean intersection operations between the offset teeth and alveolar bone defined the simulated 0.2-mm-thick PDL space. , The assembled components (teeth, PDL, and alveolar bone) were exported to ANSYS Workbench (ANSYS Inc, Canonsburg, Pa). In this study, the teeth, alveolar bone, and PDL were assumed to be homogeneous, isotropic, and linear elastic materials. The material properties assigned to the tissues, based on values widely adopted in the orthodontic biomechanics literature, are summarized in Table I . ,,, Considering that orthodontic forces cannot cause a large amount of tooth deformation, the teeth were modeled with a Young modulus that was 100 times larger than that reported in a previous study. All structures were discretized using 4-node tetrahedral elements. The average number of elements and nodes for each group model is shown in Table II . Bonded contacts were established at the tooth-PDL and PDL-bone interfaces.
Table I
Material properties of the finite element model
| Structure | Elastic modulus (MPa) | Poisson ratio |
|---|---|---|
| Teeth | 1,860,000 | 0.31 |
| PDL | 0.69 | 0.45 |
| Bone | 2000 | 0.30 |
Table II
Average number of nodes and elements of the finite element model
| Model | Group 1 | Group 2 | Group 3 | Group 4 |
|---|---|---|---|---|
| Elements | 27,691 | 26,893 | 38,485 | 37,856 |
| Nodes | 42,830 | 41,967 | 64,756 | 63,986 |
To ensure that the simulation results were independent of mesh density, a convergence study was performed on a representative sample. Three mesh densities were generated: coarse, medium (which corresponds to the mesh setting used for all simulations in this study), and fine. The global average element sizes for the coarse, medium, and fine meshes were approximately 1.3, 1, and 0.75 mm, respectively, resulting in total element counts ranging from approximately 11,000 (coarse) to over 57,000 (fine) for the integrated tooth-PDL-bone models. The ZCR coordinates and radius were chosen as the key outcome measures for convergence. As summarized in Table III , for the representative sample, the changes in both the ZCR position and radius between the medium and fine meshes were below 2.0%, confirming that the analysis was considered converged. Therefore, the medium mesh setting was adopted for all simulations in this study to balance computational accuracy and efficiency.
Table III
Mesh convergence analysis for a representative maxillary central incisor (group 1)
| Mesh density | Number of elements (tooth, PDL, bone) | ZCR centroid (x, y, z) (mm) | ZCR radius (mm) | ΔCentroid (%) | ΔRadius (%) |
|---|---|---|---|---|---|
| Coarse | 11,723 | (10.50, 55.26, 14.77) | 0.60 | 27.44 | 5.26 |
| Medium | 28,765 | (10.55, 55.38, 14.86) | 0.57 | Reference | Reference |
| Fine | 57,689 | (10.55, 55.37, 14.85) | 0.56 | 1.12 | 1.75 |
The steps for determining the 3D ZCR are as follows:
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Step 1: To simulate clinical force application, the clinical crown center was designated as the origin of a local coordinate system: x-axis (positive: distal), y-axis (positive: lingual), and z-axis (positive: gingival).
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Step 2: Pure 3 N∙mm couples were applied at the origin along positive x-, y-, and z-axes ( Figs 2 , A and 3 , A ). The tooth rotates around the CR, generating 3 resistance axes ( Figs 2 , B and 3 , B ). The linearity of the model and the load-independence of the ZCR were verified by repeating the analysis on 4 representative samples with applied moments of 1 and 2 N·mm. The resulting ZCR centroid and radius exhibited negligible changes (variation in centroid coordinates <0.002 mm, radius constant within each group), confirming that the ZCR location is independent of the moment magnitude within the linear elastic range.
Fig 2 Biomechanical analysis of moments applied to maxillary central incisors and their derived axes of resistance: A, Pure 3 N∙mm couples were applied at the origin along positive x-, y-, and z-axes. x-axis ( red , positive: distal), y-axis ( blue , positive: lingual), and z-axis ( green , positive: gingival); B, Corresponding axes of resistance.
Fig 3 Biomechanical analysis of moments applied to maxillary first molars and their derived axes of resistance: A, Pure 3 N∙mm couples were applied at the origin along positive x-, y-, and z-axes. x-axis ( red , positive: distal), y-axis ( blue , positive: lingual), z-axis ( green , positive: gingival); B, Corresponding axes of resistance.
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Step 3: nodal displacements and coordinates along each axis were recorded and exported.
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Step 4: a custom Python algorithm was employed to isolate nodes with minimal displacement. Experimental observations revealed that when the filtering threshold was set below 1%, the number of identified nodes became insufficient to reliably fit a stable ZCR. Conversely, threshold values exceeding 3% resulted in an excessively broad spatial distribution of nodes, consistently yielding the ZCR radius greater than 1 mm—a finding inconsistent with established literature. Consequently, the algorithmic filtered nodal data at thresholds from 1.0% to 3.0% in 0.25% increments. This identified minimally displaced nodes forming cylindrical distributions ( Fig 4 , A ). Least-squares fitting determined axes through these node clusters. In physiological tooth morphology, the 3 fitted resistance axes do not intersect at 1 point but maintain spatial separation. Rank each filtering percentage by the maximum interaxis distance at that threshold. Find the intersecting nodes from the 3 filtered cylindrical distributions. Using the filtering percentage yielding the smallest maximum interaxis distance (optimal percentage), filter node data across all directions. Extract the intersection of these 3 node sets ( Fig 4 , B ). Least-squares spherical fitting defined the ZCR centroid and radius from these intersected points ( Fig 4 , C ).
Fig 4 Computational workflow for the ZCR determination using Python: A, Raw scatter plot of 3 resistance axes: initial visualization of nodal displacements showing cylindrical distributions of minimally displaced nodes ( yellow-red and green , cylindrical distributions of minimally displaced nodes); B, Intersected nodes at optimal threshold: points obtained from the intersection of 3 directionally filtered cylindrical distributions using the optimal screening percentage (determined by minimizing maximum interaxis distance across 1.0%-3.0% thresholds) ( black , converged resistance nodes after filtration); C, Final ZCR sphere fitting: least-squares spherical fit applied to intersected nodes from ( B) .
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Step 5: Validation forces (sufficient for 0.1 mm movement) were applied at the center of the ZCR, and the 1 and 2 times radius along 3 axes ( Fig 5 ). Resultant rotation angles assessed the ZCR accuracy. Specifically, lingual (+y) and distal (+x) movements were validated. For lingual movement (+y), verification forces were applied at the following points: the center of the sphere, 1 radius apical (–radius), 2 radii apical (–twice radius), 1 radius coronal (+radius), and 2 radii coronal (+twice radius). The labiolingual inclination angle of the tooth’s long axis was measured, as shown in Figure 5 , A . In addition, forces were applied at the center of the sphere, 1 radius mesially (–radius), 2 radii mesially (–twice radius), 1 radius distally (+radius), and 2 radii distally (+twice radius). The MD rotation angle of the tooth was measured, as shown in Figure 5 , C . For distal movement (+x), verification forces were applied at the center of the sphere, 1 radius apical (–radius), 2 radii apical (–twice radius), 1 radius coronal (+radius), and 2 radii coronal (+twice radius). The MD inclination angle of the tooth’s long axis was measured, as shown in Figure 5 , B . Forces were also applied at the center of the sphere, 1 radius buccally (–radius), 2 radii buccally (–twice radius), 1 radius lingually (+radius), and 2 radii lingually (+twice radius). The BL rotation angle of the tooth was measured, as shown in Figure 5 , D . The rotation angles resulting from forces applied at offset locations along the x and y axes were recorded for statistical comparison. Likewise, the inclination angles of the tooth’s long axis resulting from z-axis offsets were measured and recorded.
Fig 5 Validation forces (sufficient for 0.1 mm movement) were applied at the center of ZCR, and the 1 and 2 times radius along 3 axes: A, Apply a validation force in the +y direction with the point of force application offset along the z-axis, and measure the resulting rotation angle of the tooth about the x-axis; B, Apply a validation force in the +x direction with the point of force application offset along the z-axis and measure the resulting rotation angle of the tooth about the y-axis; C, Apply a validation force in the +y direction with the point of force application offset along the x-axis and measure the resulting rotation angle of the tooth about the z-axis; D, Apply a validation force in the +x direction with the point of force application offset along the y-axis and measure the resulting rotation angle of the tooth about the z-axis.
Statistical analysis
The ZCR parameters (height from apex and sphere radius) for all samples in each group were expressed as mean ± standard deviation. The normality of data distribution was confirmed using the Shapiro-Wilk test. To compare the ZCR height and radius between the bone resorption and nonresorption groups for each tooth type, an independent samples t test was employed. A P value of less than 0.01 was considered statistically significant. All statistical analyses were performed using SPSS (version 26.0; IBM, Armonk, NY).
Results
The optimal filtering percentages for the 3 resistance axes’ intersection zone are shown in Table IV . The data from 4 groups indicate that all optimal percentages fall between 1%-2.75% and that the optimal filtering percentage for maxillary first molars is lower than that for maxillary central incisors.
Table IV
Optimal threshold for axes of resistance intersection zone (1%-3%)
| Sample number | Optimal threshold | |||
|---|---|---|---|---|
| Group 1 | Group 2 | Group 3 | Group 4 | |
| 1 | 2.75 | 2.50 | 1.25 | 2.00 |
| 2 | 1.50 | 2.75 | 1.25 | 1.50 |
| 3 | 2.75 | 2.25 | 1.75 | 2.00 |
| 4 | 2.75 | 1.75 | 1.00 | 2.75 |
| 5 | 1.25 | 2.25 | 1.00 | 2.00 |
| 6 | 2.50 | 2.00 | 1.00 | 1.00 |
| 7 | 1.75 | 1.25 | 1.00 | 1.00 |
| 8 | 2.50 | 2.25 | 1.25 | 1.00 |
| 9 | 2.00 | 2.00 | 1.75 | 1.50 |
| 10 | 2.50 | 2.00 | 2.00 | 1.50 |
| Average | 2.23 | 2.10 | 1.33 | 1.63 |
| SD | 0.56 | 0.41 | 0.37 | 0.57 |
Note. All values are in percentages (%).
SD, standard deviation.
Table V summarizes the ZCR parameters: the center of the ZCR height and sphere radius for all groups. Figure 6 compares the mean of the center of ZCR heights and sphere radius for the 4 groups (the center of ZCR height measured as vertical distance from apex to sphere center as a percentage of root length). Data indicate that for both maxillary central incisors and maxillary first molars, the following phenomena were observed: (1) ZCR sphere center height: the ZCR sphere center height was significantly lower in the alveolar bone resorption groups compared with the nonresorption groups and (2) ZCR sphere radius: the ZCR radius was significantly smaller in the alveolar bone resorption groups than in the nonresorption groups. Statistical analysis revealed a significant difference in the ZCR height and radius between the bone resorption and nonresorption groups ( P <0.01). This indicates that alveolar bone resorption exerts a significant influence on the height and radius of the ZCR.
Table V
The heights and radius of the fitted ZCR
| Sample number | Height | Radius (mm) | ||||||
|---|---|---|---|---|---|---|---|---|
| Group 1 | Group 2 | Group 3 | Group 4 | Group 1 | Group 2 | Group 3 | Group 4 | |
| 1 | 59.50% | 45.12% | 53.70% | 48.70% | 0.60 | 0.54 | 0.51 | 0.49 |
| 2 | 59.07% | 43.93% | 51.01% | 46.49% | 0.51 | 0.47 | 0.67 | 0.51 |
| 3 | 58.36% | 48.31% | 58.54% | 47.51% | 0.48 | 0.54 | 0.66 | 0.60 |
| 4 | 58.18% | 46.39% | 50.38% | 48.25% | 0.49 | 0.50 | 0.57 | 0.52 |
| 5 | 58.46% | 46.44% | 57.02% | 48.66% | 0.57 | 0.46 | 0.53 | 0.46 |
| 6 | 61.58% | 43.24% | 51.59% | 47.99% | 0.55 | 0.41 | 0.57 | 0.43 |
| 7 | 59.88% | 45.08% | 51.71% | 46.60% | 0.56 | 0.46 | 0.52 | 0.41 |
| 8 | 58.42% | 43.16% | 57.14% | 46.17% | 0.61 | 0.54 | 0.58 | 0.45 |
| 9 | 59.09% | 45.82% | 55.44% | 42.05% | 0.58 | 0.49 | 0.56 | 0.51 |
| 10 | 59.20% | 44.41% | 51.08% | 45.21% | 0.57 | 0.49 | 0.58 | 0.44 |
| Average | 59.17% | 45.19% | 53.76% | 46.76% | 0.55 | 0.49 | 0.58 | 0.48 |
| SD | 1.01% | 1.61% | 3.03% | 2.02% | 0.05 | 0.04 | 0.05 | 0.06 |
| P value | <0.0001 | <0.01 | <0.0001 | <0.01 | ||||
SD, standard deviation.
The mean center of ZCR heights and sphere radius for the 4 groups (the center of ZCR height measured as vertical distance from apex to sphere center as a percentage of root length).
Validation experiment rotation angles are presented in Tables VI and VII . z-axis force application at the center of the ZCR produced minimal rotation about the tooth long axis (mean: 0.09°). Forces applied at the 1-time radius position increased rotation to 0.24°. Force application outside the ZCR sphere significantly increased rotation, reaching 0.6°. Forces applied along the x- and y-axes at the center of ZCR induced minimal rotation (mean: 0.12°). The 1-time radius position raised the rotation to 0.62°. Rotation exceeded 1° at the 2 times radius position, reaching 1.92°.
Table VI
Angles of rotation around the z-axis during movement of loading points along the x- and y-axes (°)
| Group number | Sample number | Loading direction | —Twice radius | —Radius | Center of ZCR | Radius | Twice radius |
|---|---|---|---|---|---|---|---|
| Group 1 | 1 | Lingual (y) | –1.75 | –0.92 | 0.12 | 0.87 | 1.68 |
| Distal (x) | –1.62 | –0.78 | –0.15 | 1.05 | 1.82 | ||
| 2 | Lingual (y) | –1.53 | –1.02 | 0.25 | 0.95 | 1.59 | |
| Distal (x) | –1.88 | –0.65 | 0.18 | 0.68 | 1.73 | ||
| 3 | Lingual (y) | –1.21 | –0.83 | –0.08 | 1.12 | 1.86 | |
| Distal (x) | –1.45 | –0.97 | 0.29 | 0.79 | 1.67 | ||
| 4 | Lingual (y) | –1.67 | –0.71 | 0.03 | 0.92 | 1.77 | |
| Distal (x) | –1.32 | –1.08 | –0.19 | 1.14 | 1.91 | ||
| 5 | Lingual (y) | –1.93 | –0.61 | 0.22 | 0.73 | 1.62 | |
| Distal (x) | –1.56 | –0.89 | 0.17 | 1.03 | 1.79 | ||
| 6 | Lingual (y) | –1.29 | –1.05 | –0.03 | 0.81 | 1.54 | |
| Distal (x) | –1.81 | –0.74 | 0.27 | 0.98 | 1.85 | ||
| 7 | Lingual (y) | –1.42 | –0.96 | 0.17 | 1.09 | 1.72 | |
| Distal (x) | –1.66 | –0.67 | –0.12 | 0.86 | 1.63 | ||
| 8 | Lingual (y) | –1.14 | –0.79 | 0.08 | 1.01 | 1.88 | |
| Distal (x) | –1.77 | –0.53 | 0.31 | 0.76 | 1.69 | ||
| 9 | Lingual (y) | –1.35 | –1.01 | –0.18 | 0.94 | 1.57 | |
| Distal (x) | –1.51 | –0.85 | 0.14 | 1.11 | 1.84 | ||
| 10 | Lingual (y) | –1.84 | –0.72 | 0.28 | 0.83 | 1.76 | |
| Distal (x) | –1.23 | –0.93 | –0.07 | 1.07 | 1.81 | ||
| Group 2 | 1 | Lingual (y) | –1.58 | –0.86 | –0.14 | 0.97 | 1.65 |
| Distal (x) | –1.72 | –1.07 | 0.26 | 1.13 | 1.89 | ||
| 2 | Lingual (y) | –1.39 | –0.77 | 0.09 | 0.71 | 1.58 | |
| Distal (x) | –1.96 | –0.63 | –0.21 | 0.88 | 1.74 | ||
| 3 | Lingual (y) | –1.25 | –1.03 | 0.19 | 1.04 | 1.83 | |
| Distal (x) | –1.47 | –0.88 | 0.02 | 0.75 | 1.61 | ||
| 4 | Lingual (y) | –1.82 | –0.69 | 0.11 | 0.91 | 1.71 | |
| Distal (x) | –1.31 | –0.95 | –0.16 | 1.08 | 1.87 | ||
| 5 | Lingual (y) | –1.16 | –0.81 | 0.23 | 0.84 | 1.55 | |
| Distal (x) | –1.64 | –1.09 | 0.06 | 1.15 | 1.92 | ||
| 6 | Lingual (y) | –1.78 | –0.59 | –0.09 | 0.78 | 1.64 | |
| Distal (x) | –1.43 | –0.99 | 0.30 | 1.02 | 1.78 | ||
| 7 | Lingual (y) | –1.27 | –0.90 | 0.15 | 0.89 | 1.66 | |
| Distal (x) | –1.55 | –0.75 | –0.24 | 0.72 | 1.59 | ||
| 8 | Lingual (y) | –1.91 | –0.64 | 0.21 | 1.10 | 1.90 | |
| Distal (x) | –1.36 | –1.04 | 0.05 | 0.96 | 1.75 | ||
| 9 | Lingual (y) | –1.49 | –0.80 | –0.20 | 0.80 | 1.70 | |
| Distal (x) | –1.68 | –0.94 | 0.13 | 1.06 | 1.80 | ||
| 10 | Lingual (y) | –1.33 | –0.87 | 0.07 | 0.93 | 1.68 | |
| Distal (x) | –1.79 | –0.70 | –0.11 | 0.85 | 1.72 | ||
| Group 3 | 1 | Lingual (y) | –0.81 | –0.30 | –0.03 | 0.16 | 0.79 |
| Distal (x) | –0.48 | –0.24 | 0.17 | 0.24 | 0.62 | ||
| 2 | Lingual (y) | –0.99 | –0.51 | 0.04 | 0.15 | 0.58 | |
| Distal (x) | –0.57 | –0.49 | 0.03 | 0.36 | 0.45 | ||
| 3 | Lingual (y) | –0.79 | –0.42 | 0.04 | 0.46 | 0.69 | |
| Distal (x) | –0.61 | –0.24 | –0.02 | 0.46 | 0.60 | ||
| 4 | Lingual (y) | –0.85 | –0.37 | –0.10 | 0.22 | 0.75 | |
| Distal (x) | –0.37 | –0.14 | 0.04 | 0.44 | 0.56 | ||
| 5 | Lingual (y) | –0.79 | –0.35 | 0.15 | 0.22 | 0.74 | |
| Distal (x) | –0.62 | –0.36 | –0.09 | 0.43 | 0.78 | ||
| 6 | Lingual (y) | –0.73 | –0.39 | 0.05 | 0.26 | 0.63 | |
| Distal (x) | –0.59 | –0.26 | 0.21 | 0.39 | 0.81 | ||
| 7 | Lingual (y) | –0.69 | –0.24 | –0.08 | 0.43 | 0.83 | |
| Distal (x) | –0.47 | –0.17 | 0.05 | 0.35 | 0.49 | ||
| 8 | Lingual (y) | –0.96 | –0.25 | 0.04 | 0.27 | 0.64 | |
| Distal (x) | –0.71 | –0.55 | 0.19 | 0.26 | 0.56 | ||
| 9 | Lingual (y) | –0.71 | –0.23 | 0.10 | 0.34 | 0.95 | |
| Distal (x) | –0.56 | –0.24 | 0.20 | 0.18 | 0.61 | ||
| 10 | Lingual (y) | –0.68 | –0.40 | 0.12 | 0.16 | 0.71 | |
| Distal (x) | –0.40 | –0.35 | 0.15 | 0.33 | 0.53 | ||
| Group 4 | 1 | Lingual (y) | –0.78 | –0.37 | –0.10 | 0.18 | 0.73 |
| Distal (x) | –0.57 | –0.37 | –0.14 | 0.35 | 0.59 | ||
| 2 | Lingual (y) | –0.78 | –0.39 | –0.14 | 0.49 | 0.84 | |
| Distal (x) | –0.59 | –0.38 | –0.16 | 0.29 | 0.57 | ||
| 3 | Lingual (y) | –0.95 | –0.47 | 0.06 | 0.45 | 0.54 | |
| Distal (x) | –0.47 | –0.14 | –0.03 | 0.15 | 0.84 | ||
| 4 | Lingual (y) | –0.58 | –0.28 | –0.07 | 0.14 | 0.96 | |
| Distal (x) | –0.54 | –0.36 | –0.13 | 0.30 | 0.60 | ||
| 5 | Lingual (y) | –0.73 | –0.52 | 0.08 | 0.37 | 0.71 | |
| Distal (x) | –0.61 | –0.17 | –0.03 | 0.30 | 0.83 | ||
| 6 | Lingual (y) | –0.80 | –0.18 | 0.03 | 0.40 | 0.65 | |
| Distal (x) | –0.56 | –0.47 | 0.00 | 0.23 | 0.68 | ||
| 7 | Lingual (y) | –0.85 | –0.52 | –0.03 | 0.44 | 0.75 | |
| Distal (x) | –0.63 | –0.42 | –0.04 | 0.18 | 0.74 | ||
| 8 | Lingual (y) | –0.71 | –0.43 | –0.19 | 0.32 | 0.66 | |
| Distal (x) | –0.63 | –0.18 | –0.13 | 0.16 | 0.64 | ||
| 9 | Lingual (y) | –0.79 | –0.48 | –0.09 | 0.29 | 0.60 | |
| Distal (x) | –0.76 | –0.59 | 0.01 | 0.23 | 0.69 | ||
| 10 | Lingual (y) | –0.82 | –0.30 | –0.11 | 0.47 | 1.02 | |
| Distal (x) | –0.60 | –0.23 | –0.08 | 0.17 | 0.75 | ||
| Average | 1.11 | 0.59 | 0.12 | 0.62 | 1.21 | ||
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