Measuring Maxillary Posterior Tooth Movement: A Model Assessment using Palatal and Dental Superimposition

Summary

This manuscript presents a comprehensive protocol to evaluate the three-dimensional (3D) movement of maxillary posterior teeth with clear aligners using digital model superimposition, an invaluable tool in orthodontics and dentofacial orthopedics.

Abstract

Since the introduction of Invisalign by Align Technology, Inc. in 1999, questions and debates have persisted regarding the precision of Invisalign (clear aligner) therapy, particularly when compared to the use of traditional fixed appliances. This becomes particularly significant in cases involving anteroposterior, vertical, and transverse corrections, where precise comparisons are of paramount importance. To address these inquiries, this study introduces a meticulously devised protocol, placing a primary emphasis on digitally superimposing the movement of maxillary posterior teeth to facilitate accurate analysis. The sample included 25 patients who had completed their first series of Invisalign (clear) aligners. Four maxillary digital models (pre-treatment, post-treatment, ClinCheck-initial, and final models) were digitally superimposed using the palate rugae and dentitions as stable references. A software combination was used for model superimposition and tooth segmentation. Transformation matrices then expressed the differences between the achieved and predicted tooth positions. Thresholds for clinically relevant differences were at ±0.25 mm for linear displacement and ±2° for rotation. Differences were assessed using Hotelling’s T-squared tests with Bonferroni correction. The mean differences in rotation (2.036° ± 4.217°) and torque (-2.913° ± 3.263°) were significant statistically and clinically, with p-values of 0.023 and 0.0003 respectively. De-rotation of premolars and torque control for all posterior teeth were less predictable. All mean differences for the linear measurements were statistically and clinically insignificant, except that the first molars seemed slightly (0.256 mm) more intruded than their predicted position. The clear aligner system appears to meet its prediction for most translational tooth movements and mesial-distal tipping in maxillary posterior teeth for non-extraction cases with mild to moderate malocclusions.

Introduction

In 1999, digitally fabricated removable orthodontic appliances were made commercially available by Align (Align Technology Inc., Tempe, AZ). Originally, this system was designed to solve non-growing cases with mild to moderate crowding or close small spaces as an aesthetic alternative to traditional fixed edgewise appliances. With decades of improvements in computer-aided design and manufacturing (CAD/CAM), dental materials, and treatment planning, clear aligner therapy (CAT) has since been used to treat over 10 million patients with various malocclusions worldwide¹. A recent retrospective study suggested that CAT is as effective as fixed appliance therapy for the teenage population with mild malocclusions, with significantly improved results in tooth alignment, occlusal relations, and overjet². The number of appointments, emergency visits, and overall treatment time also had better outcomes for clear aligner therapy patients. Although CAT can be used to treat non-extraction, mild-to-moderate malocclusions in non-growing patients^3,4, and shorten treatment duration and chair time⁵, it remains unclear whether the treatment is as effective as the gold standard of conventional labial braces^4,6,7,8,9, especially for anteroposterior and vertical correction¹⁰.

ClinCheck is a software platform developed by Align to provide clinicians with virtual three-dimensional (3D) simulations of prospective tooth movements. Primarily concerned with the patient's initial status and the clinician's prescribed treatment plan, it can also be a visual communication tool for the patient. Any mismatch between the predicted and achieved results may require a mid-course correction, refinement, or conversion to fixed appliance therapy. Consequently, the reliability of software predictions has drawn increasing attention from investigators. Since Lagravere and Flores-Mir's systematic review published in 2005¹¹, investigations into the concordance between predicted models and post-treatment models have been measured in different ways, measurement methods including arch-length, inter-canine distance, overbite, overjet, midline deviation¹², the American Board of Orthodontics objective grading system (ABO-OGS) reduction score¹³, upper and lower interdental width¹⁴, and measures derived from cone-beam computed tomography¹⁵.

Comparisons have also been made by superimposing 3D models^{16,17,18,19,20,21}. For example, many current software platforms, such as ToothMeasure (internal software developed by Align Technology), can reproducibly superimpose two digital models using user-selected reference points on untreated teeth, palatal rugae, or dental implants. Since the predicted and achieved models usually do not include the palatal surfaces, many previous studies^15,16,17,18 have used the untreated posterior teeth as references for superimposition, including the possibility of adding errors due to the relative movements of these teeth. These studies have been confined to anterior regions of the arch in relatively simple cases with spacing or mild to moderate crowding.

Grünheid et al. used mathematical superimposition to quantify the discrepancies between virtual treatment plans and actual treatment outcomes to evaluate the accuracy of full-dentition CAT without stable anatomic structures in digital models²⁰. Haouili et al. used the same method in a best-fit algorithm within the Compare software to conduct a prospective follow-up study on the efficacy of tooth movement with CAT²¹. The aim was to provide an update on the accuracy associated with emerging technology, i.e., SmartForce, SmartTrack aligner materials, and digital scans. Their findings of an improved overall accuracy from 41%¹⁷ to 50%²¹ were encouraging but do not negate the possibility that some tooth movements are still not satisfactorily achievable with the clear aligner system.

When predicted and achieved, digital models include a common 3D reference independent of the dentition, such as palatal rugae, dental implants, or tori; they can be co-registered within the coordinate system of many suitable software platforms. If a tooth of interest is then segmented from one and transformed mathematically to match its displaced version in the other, the transformation matrix contains the complete information needed to describe the entire 3D transposition. Its content can be expressed as three translations and three rotations described by a formal convention. An example is found on the Invisalign ClinCheck Pro 3D control software, where the numerical parameters indicating the 3D tooth movements needed to move teeth to their predicted positions are shown in a tooth movement table.

While the initial and final (predicted) models from the planning software share a common coordinate system provided by the same software platform, their absence of palates restricts the possibility of co-registering with any other digital dentition model unless they possess identical dentition. In this context, it was hypothesized that the superimposition of software-predicted and post-treatment (achieved) models would be feasible. This feasibility arises from the availability of two pairs: initial and final (automatically superimposed during export from planning software) and another pair of pre-treatment and achieved models (superimposed using palatal rugae). These pairs could be registered using the pre-treatment dentition as a reference to align them with the Invisalign-initial model. Subsequently, the segmentation of individual teeth could be performed to assess differences in their positions and orientations. This assessment involves transposing teeth between the models, and the transformation matrices would enable a numerical quantification of the translations and re-orientations.

In this protocol, an approach to evaluate the effectiveness of CAT in addressing mild to moderate malocclusions in both adolescents and adults was introduced, specifically focusing on the maxillary posterior teeth. The null hypothesis was that there was no difference between the achieved and the planning software-predicted tooth position in the maxillary posterior teeth after the first series of clear aligners.

Protocol

This study received ethical approval from the Institutional Review Board at the University of British Columbia (No. H19-00787). To uphold confidentiality, all samples utilized in the study underwent de-identification procedures. Furthermore, prior to their inclusion in the research, informed consent was appropriately obtained from all participating patients.

NOTE: Each participant contributed four maxillary digital models, which encompassed the following:

1. Pre-treatment digital model, with the palate scanned using iTero
2. Post-treatment digital model, with the palate scanned using iTero
3. Pre-treatment model, exported from the planning software.
4. Predicted model, exported from the planning software.

This protocol leveraged a combination of several software tools, which included CloudCompare, Meshmixer, and Rhinoceros. These software platforms played a pivotal role in facilitating the registration process and enabling the segmentation of individual teeth for the purpose of analyzing their movements and orientations. It is worth noting that these software tools may be replicable with other open-source software options, provided that they can achieve similar objectives. A workflow illustrating the software sequence is presented in Figure 1.

1. Preparation

1. Obtain the initial and final (predicted) models as stereolithographic (STL) files from the planning software by clicking on Tools > Export > STL.
   NOTE: Models exported from the planning software present only clinical crowns and virtual gingiva without the palate.
2. Obtain the pre-treatment and post-treatment digital models as STL files from the scanned model software (OrthoCAD) by selecting the scan, clicking and choosing Export > Export Type (open shell), Data Format (File per Arch [arches oriented in occlusion]).
   ​NOTE: Models exported from the model scanning software include not only dentition but also gingiva and the whole palate.

2. Palatal superimposition of pre- and post-treatment digital models in CloudCompare

1. Open the software and drag and drop the STL files of the pre-treatment and post-treatment digital models.
2. Select each model and click Edit > Colors > Set Unique to change the colors of the selected models.
3. Select the post-treatment digital model and click the Translate/Rotate icon. Right-click to drag the model so they are side by side. Click the Green Checkmark.
4. Select the pre-treatment digital model and click the Segment icon.
5. Click four points on the palatal rugae and right-click to deselect. Click Segment In, then click the Green Checkmark. Repeat steps 2.4 to 2.5 for the post-treatment digital model.
6. Hide the PostTreatModel.remaining and PreTreatModel.remaining models and select both the PostTreatModel.part and PreTreatModel.part models.
7. Click the Rough Registration alignment icon (point pair picking) and place at least three corresponding landmarks on the palate on each side of the midline for both the pre- and post-treatment palates. Click Align, then click the Green Checkmark.
8. Unhide the meshes for both models and move the untransformed PostTreatModel.remaining model by copying the transformation matrix, clicking Edit > Apply Transformation, and pasting the transformation matrix.
   NOTE: The transformation matrix outputs are shown in the console.
9. Hide the PostTreatModel.remaining and PreTreatModel.remaining models and select the PostTreatModel.part and PreTreatModel.part models.
10. Click the Fine Registration alignment icon and make sure the PreTreatModel.part model is selected as the reference. Click Ok.
    NOTE: Confirm the resulting root mean square (RMS) in the registration information window. A deviation of ≤ 0.05 RMS is acceptable.
11. Unhide the meshes for both models and move the untransformed PostTreatModel.remaining model by copying the transformation matrix, clicking Edit > Apply Transformation, and pasting the transformation matrix.
12. Save the superimposed PostTreatModel.remaining and PreTreatModel.remaining models as STL files.

3. Software-model preparation for superimposition with Rhinoceros software

1. Import the STL files of the planning software pre-treatment and predicted models separately.
   NOTE: When importing software models into measurement software such as Rhinoceros or CloudCompare, the orientation and models' registration are preserved
2. Select the simulated gingiva and press Delete to remove it.
3. Click MeshTools, select Meshplane. Draw a plane around the teeth and move the plane to the occlusal 1/3 of the tooth crowns. This wil improve the superimposition precision.
4. Double click the button Right to expand the right view.
5. Enter the command MeshBooleanSplit and select the plane and all the teeth then press Enter.
6. Delete the plane and cervical portions of the teeth leaving the 1/3 occlusal tooth crowns.
7. Save the split model as an STL file.
8. Repeat all steps for the other model.

4. Superimposition of software-predicted and post-treatment digital models with CloudCompare

1. Drag and drop the STL files of the previously palatally superimposed pre-treatment and post-treatment digital models, and the split pre-treatment and split predicted models.
2. Select each model and click Edit > Colors > Set Unique to change the colors of the selected models.
3. Select both the pre-treatment and post-treatment digital models and click the Translate/Rotate icon. Right-click to drag the models so they are side by side.
4. Request the software hide the split predicted model and post-treatment digital model by uncheckmarking the corresponding boxes. Select the split pre-treatment model and pre-treatment digital model.
5. Click the Rough Registration alignment icon and place corresponding landmarks on the crowns' cusps on both the split pre-treatment model and the pre-treatment digital model. Click Align, then click the Green Checkmark.
6. Unhide the split predicted model and post-treatment model and move the untransformed post-treatment model by copying the transformation matrix, clicking Edit > Apply Transformation, and pasting the transformation matrix.
7. Hide the post-treatment and split predicted models. Select the pre-treatment and split pre-treatment models. Click the Fine Registration alignment icon for the best fit between the split pre-treatment model and the pre-treatment digital model.
8. Unhide the meshes and move the untransformed model by copying the transformation matrix, clicking Edit > Apply Transformation, and pasting the transformation matrix.
9. Unhide the split predicted and post-treatment digital models, then Hide the split pre-treatment model and pre-treatment digital model to display the superimposition (Figure 2).
10. Save the models as STL files.

5. Crown segmentation using Meshmixer

1. Import the split predicted model and post-treatment digital model to Meshmixer.
2. Click Edit > Duplicate to duplicate the models for the number of teeth to be segmented. Label each model with the corresponding tooth number to be segmented.
3. Hide the split predicted model by clicking the Eye icon, keeping the post-treatment digital model visible.
4. On the post-treatment model, click Select and adjust the size of the Brush. To segment the selected crown, drag the Brush tool on the occlusal surface of the selected tooth, paying close attention to the cusp tips.
5. Click Modify > Invert, then Edit > Discard to delete the rest of the model, leaving the segmented crown.
6. Unhide the split predicted model and hide the post-treatment model by clicking the corresponding Eye icons.
7. Repeat steps 5.4-5.5 for the split predicted model.
8. Export each selected crown as STL files.
9. Repeat all steps for each tooth segmentation.

6. Dental superimposition with CloudCompare

1. Import the segmented post-treatment digital crowns and split software-predicted crowns into the software. Ensure that the orientation and cloud registration remain consistent. Establish the World Coordinate Grid to standardize the orientation of both right and left teeth, enhancing the reliability of the methodology. The grid's center should represent the (0,0,0,0,0,0) coordinate of the CloudCompare software cloud.
2. Select both crowns and click Edit > Normals > Compute > Per-Vertex.
3. Select each tooth and click Edit > Colors > Set Unique to change the colors of the selected models.
4. Hide the post-treatment tooth by unchecking the box and select both the hidden post-treatment tooth and the visible predicted tooth.
5. Select the bottom view, click the Translate/Rotate icon, and use the planes X, Y, and Z to rotate the tooth such that the buccal cusp lines up with the vertical line.
6. Select the left side view, click the Translate/Rotate icon, and line the buccal and lingual cusps with the horizontal line.
7. Select the back view, click the Translate/Rotate icon, and line the buccal and lingual cusps with the horizontal line.
   NOTE: Aim to align its occlusal and facial surfaces with the world axes and planes. Ensure the bounding box center of the tooth is positioned at the world origin. By adhering to the World Coordinate Grid, the positions of all teeth will be standardized. This step ensures a consistent and accurate conversion of X, Y, and Z translations across all axes, irrespective of an individual tooth's specific position.
8. Once all the cusps line up, click the Translate/Rotate icon to center the tooth on the grid in all views.
9. Unhide the post-treatment tooth and select the predicted tooth and post-treatment tooth.
10. Click the Fine Registration alignment icon to register the post-treatment tooth over the predicted tooth. Click OK.
    NOTE: Upon completion, CloudCompare will show the registration information, including superimposition RMS (Figure 3).
11. To determine the positional and rotational differences between the two teeth, select the post-treatment tooth, copy the transformation matrix, click Edit > Apply Transformation, and paste the transformation matrix.
12. Select the Euler Angles icon to display the rotational and linear movements between the predicted tooth and the post-treatment tooth.
13. Document all translation and rotation measurements in a spreadsheet. Repeat this process for all remaining posterior teeth.
    NOTE: Use the American Board of Orthodontics (ABO) model grading system¹² to identify clinically significant measurement differences. Differences greater than 0.5 mm linearly and 2 degrees angularly are considered clinically relevant.
14. Adjust measurement values for the anterior-posterior direction of right-side teeth in a spreadsheet. This adjustment accounts for the standardized orientation of right-side teeth to left-side teeth.

7. Measurement specifications

1. Understand the sequence of rotation and measurement conventions: CloudCompare employs the Tait-Bryan ZYX extrinsic (world origin) convention for its measurements.
   NOTE: For translation, the axes represent X (buccolingual direction), Y (mesiodistal direction), and Z (vertical direction: intrusion/extrusion). Angular movements are represented by the X-axis (Psi – mesiodistal tipping), the Y-axis (Theta – buccolingual torque), and the Z-axis (Phi – mesiodistal rotation)²². Tooth movements are expressed in terms of the tooth's anatomy, regardless of its position in the arch. The sign of measures (+, -) indicates the direction from the world origin and rotation around its axes.
2. Importance of contextual relevance: Note that the directional terms describing tooth movements (e.g., mesial, distal, buccolingual) reference the specific tooth and do not account for alterations relative to the dental arch.

8. Statistical analysis

1. Use the R statistical package (v 3.2.3, RStudio Inc.) via RStudio (version 1.4.1103) for all the analyses.
2. Select 32 teeth at random and perform duplicate measurements at a 1 month interval.
3. Test intra-examiner reliability with Intra-class correlation coefficients (ICC) and Bland Altman analyses for both sets of measurements.
4. Apply Hotelling's T-squared tests to test mean prediction differences between the predicted and achieved tooth positions for both angular and linear parameters.
5. Adjust for multiple tooth comparisons using a Bonferroni correction on p-values, aiming for a family-wise error rate of 0.05.
6. Conduct a post-hoc Hotelling's T-squared test if any significant differences are detected to determine if prediction differences for each tooth type and movement parameter are significant. Consider discrepancies of 0.25 mm or more in linear measurements and 2° or above for angular measurements as clinically relevant.

Representative Results

A minimum sample size of 24 cases was required to detect an effect change of 0.6° for the average tip and torque angles, with an 80% power and an alpha of 0.05²³. The inclusion criteria were as follows: (1) full permanent dentition through the first molars, (2) Class I malocclusions, or less than 2 mm Class II /III malocclusions with spacing, or mild to moderate crowding that had undergone non-extraction Invisalign treatment, (3) completion of the first series of Invisalign aligners at least, and (4) palatal rugae presented on both the initial and refinement intra-oral scans. The exclusion criteria were: (1) previous exposure to auxiliary expansion and distalization appliances, (2) visible wear facets in the dentition during treatment, (3) history of trauma, craniofacial syndrome, or missing teeth, and (4) poor compliance with aligner wearing documented on the chart. The second molars of several cases were absent or erupting and therefore excluded from the analysis. Accordingly, this study comprised 150 teeth (50 first premolars, 50 second premolars, and 50 first molars) selected from 25 participants (17 females and 8 males), aged from 12 to 44 years old with a mean age of 24.8 ± 8.8 years. Of the 25 patients, 4 were Class I, 15 were Class II, and 6 were Class III malocclusions, all less than 2 mm. The average number of trays was 24.8 ± 11.2, and the average treatment duration was 214 ± 131 days. Among the 150 maxillary posterior teeth, there were 63 teeth without any attachment, 7 with conventional, and 80 with optimized attachments.

Mean ICCs for intra-examiner reliability were greater than 0.990, suggesting the intra-examiner agreement was excellent (Table 1). The results of Bland-Altman analyses are reported in Table 2, which also showed high intra-examiner agreement.

Table 3 shows the angular and linear differences between the predicted and achieved tooth position in maxillary posterior teeth. In general, the angular measures for rotation, torque, and tip had notably greater variation than the distance measures for buccal-lingual, mesial-distal, and occlusal-gingival translations. The mean rotation differences for first premolars and second premolars were greater than 2° and the 95% confidence interval did not include zero. This suggests that clinically, the maxillary first and second premolars were significantly rotated mesially. Torque for all tooth types substantially deviated from zero, while the mean difference for second premolars and first molars was less than -2°, suggesting all maxillary posterior teeth, especially the second premolars and first molars, had a more clinically relevant buccal crown torque relative to the predicted position.

Hotelling's T-squared test results and the 95% overall confidence intervals with Bonferroni correction for each parameter are presented in Table 4. The results indicate the mean differences in rotation (2.036° ± 4.217°) and torque (-2.913° ± 3.263°) were statistically significantly different from zero, with p-values of 0.023 and 0.0003 respectively.

To further explore the possible effects of attachment use on the accuracy of the prediction, a primary investigation can be visualized in Figure 4, which showed minor differences across different attachments (No, conventional, or optimized attachment). However, this is likely due to low frequencies of conventional attachment.

Figure 1: A workflow of the software usage sequence. Please click here to view a larger version of this figure.

Figure 2: Superimposition of the software-final (predicted) and the post-treatment (achieved) models. (A) Four models from one subject registered in the same coordinate system. Color-coding indicates the pre-treatment and post-treatment models with palates but different dentitions, the software-initial model without a palate and the same dentition as the pre-treatment model, and the softwarefinal model without a palate and the predicted dentition. The method for superimposition is described in the text. (B) The software-predicted final and post-treatment models shown alone. Differences in their tooth positions and orientations were measured in this study. Please click here to view a larger version of this figure.

Figure 3: Dental superimposition with measurements. (A) A segmented first molar from the achieved (post-treatment) model registered to the software-predicted version. The transformation matrix for registration and the root mean square (RMS) of the fit is from CloudCompare's pop-up window. (B) Euler angles and displacements derived from the transformation matrix. Please click here to view a larger version of this figure.

Figure 4: Comparison of prediction differences without attachment with conventional and optimized attachments. Please click here to view a larger version of this figure.

Parameter	Mean	95%	CI	Significance
Rotation (°)	1	1	1	0
Torque (°)	0.991	0.982	0.996	0
Tip (°)	0.992	0.983	0.996	0
Buccal-Lingual (mm)	0.999	0.997	0.999	0
Mesial-Distal (mm)	0.99	0.979	0.995	0
Occlusal-Gingival (mm)	0.998	0.996	0.999	0

Table 1: Intra-class correlation coefficients (ICC) for intra-examiner reliability (n = 32 teeth). CI: confidence interval.

Parameter	Mean difference	95%	CI
Rotation (°)	0.032	-0.045	0.137
Torque (°)	0.182	-0.099	0.503
Tip (°)	0.061	-0.08	0.218
Buccal-Lingual (mm)	-0.011	-0.043	0.012
Mesial-Distal (mm)	0.008	-0.033	0.048
Occlusal-Gingival (mm)	0.011	-0.002	0.026

Table 2: Results of Bland-Altman analyses for intra-examiner agreement (n = 32 teeth). CI: confidence interval.

Para-meter	First Premolar (n=50)				Second Premolar (n=50)				First Molar (n=50)
	Mean	SD	95%	CI	Mean	SD	95%	CI	Mean	SD	95%	CI
Rotation (°)	2.801	3.881	1.767	4.023	2.472	5.265	1.195	4.148	0.835	3.004	0.098	1.74
Torque (°)	-1.261	1.912	-1.765	-0.722	-3.597	3.586	-4.588	-2.512	-3.881	3.413	-4.895	-2.934
Tip (°)	0.746	2.851	-0.079	1.632	0.409	3.015	-0.434	1.238	-0.326	1.917	-0.582	0.506
Buccal-Lingual (mm)	-0.18	0.455	-0.311	-0.046	-0.156	0.516	-0.307	-0.018	-0.048	0.619	-0.203	0.132
Mesial-Distal (mm)	0.143	0.535	-0.006	0.309	0.155	0.56	-0.01	0.299	0.213	0.618	0.041	0.392
Occlusal-Gingival (mm)	-0.141	0.407	-0.256	-0.031	-0.206	0.408	-0.323	-0.09	-0.256	0.398	-0.363	-0.147

Table 3: Descriptive statistics for angular and linear differences between the predicted and the achieved tooth position for maxillary first premolars, second premolars, and first molars. Positive values indicated an achieved tooth position more buccal, distal, or occlusal, or with more mesial rotation, more distal crown tip, or more lingual crown torque than the predicted tooth position. SD: standard deviation; CI: confidence interval.

Parameter	Mean	SD	95%	CI	P
Rotation (°)	2.036	4.217	1.408	2.756	0.023*
Torque (°)	-2.913	3.263	-3.411	-2.388	0.0003*
Tip (°)	0.374	2.641	-0.049	0.8	1
Buccal-Lingual (mm)	-0.128	0.534	-0.216	-0.041	0.186
Mesial-Distal (mm)	0.17	0.569	-0.076	0.258	1
Occlusal-Gingival (mm)	-0.201	0.405	-0.266	-0.136	0.123

Table 4: Comparisons of angular and linear mean prediction differences in all the maxillary posterior teeth measured with Hotelling's T-squared tests with Bonferonni correction. Positive values indicated an achieved tooth position more buccal, distal, or occlusal, or with more mesial rotation, more distal crown tip, or more lingual crown torque than the predicted tooth position. *P < 0.05.

Discussion

The palatal rugae have a unique configuration at adolescence; they remain constant during growth, are authentic markers for personal identification, and are considered stable anatomic references for maxillary model superimposition^24,25,26,27. Dai et al. used this method to compare the achieved and predicted tooth movement of maxillary first molars and central incisors with clear aligners after the first premolar extraction²⁸. The achieved post-treatment model was registered via Rapidform software to the pre-treatment and the planned post-treatment model. They reported statistically significant differences between the predicted and achieved tooth movements. The pre- and post-treatment models were acquired from different sources (alginate impressions and intraoral scans). The measurements of tooth movement were expressed within a coordinate system using the posterior occlusal plane as a transverse plane and the palatal suture as a guide to constructing a midsagittal plane. Since angular and translational parameters for the respective maxillary first molars and upper central incisors were projected onto these planes, it is difficult to compare their results with investigations using different coordinate systems (e.g., a coordinate origin based on a tooth's approximate center of resistance)^{17,18,19,20,21}.

The study was limited to the maxillary arch so the palate and its rugae could be used to register the predicted and achieved models. Previously, model registration has been done using untreated or assumedly stable posterior teeth^16,17,18,19, best-fit algorithms^20,21, miniscrews^26,27, tori²⁹, implants³⁰, the cranial-base^31, or other bony structures³². In one of the few previous studies using palatal superimposition to assess clear aligner efficacy, Dai et al. compared the achieved and predicted tooth movements of maxillary first molars and central incisors in extraction cases²⁸, although they did not report individual differences in tooth position and orientation with the six degrees of freedom used in this and other studies^20,21.

Anterior teeth were not included in this protocol because the shapes of their clinical crowns were too difficult to orient within the world coordinate system used by CloudCompare to express its transformations. However, the premolars and molars had sufficient occlusal landmarks and facial surface regularity to allow each treated tooth's occlusal surface, long axis, and bounding box to be oriented to the software's origin and world coordinates.

Using a transformation matrix to provide the translational and rotational information describing tooth movement during registration requires adherence to specific conventions. CloudCompare employs the Tait-Bryan convention, in which a ZYX axis rotation sequence is adopted first, followed by the 3D translations from world zero needed to complete the match. As the angles reported in this study reflect the Tait-Bryan convention³³, studies using different conventions would produce different results. Aligning the post-treatment tooth to the world origin and coordinates ensured the measurements indicated translation from the tooth's original position and directions determined by each tooth's specific surface anatomy.

Overall, our results showed that the rotation and torque of the achieved tooth position were statistically and clinically significantly different from the predictions, with more mesial rotation and buccal torque after treatment with clear aligners. While rotational movement in maxillary first molars was relatively successful, de-rotation in the first and second premolars was more problematic, suggesting the morphology of premolar crowns might contribute to this difference. These findings are similar to a recent study conducted by Al-Nadawia et al., who found that the accuracy of posterior teeth movements with a 7-day protocol is not as accurate as the 14 days protocol for intrusion, distal intrusion, distal-crown tip, and buccal-crown torque³⁴. In order to determine whether the accuracy of Invisalign had improved with the newer technology, Haouili et al. updated Kravitz et al.'s pioneering study¹⁷ and they also found the least overall accuracy with rotation, particularly challenging for canines, premolars, and molars ²¹.

Translation data showed no statistically significant differences for all three directions, in agreement with previous studies. Simon et al. found that upper molar distalization was the most effectively predicted movement¹⁸. A non-statistical but clinically significant prediction difference was noticed for occlusal-gingival movement in the first molars in the current study, which tended to be slightly intruded relative to their predicted positions. Haouili et al. also indicated that although the extrusion of the maxillary incisors improved with the use of optimized extrusion attachments, extrusion of the maxillary and mandibular molars had the lowest accuracy²¹.

In the present study, tooth movements with attachments were not different from teeth with no attachments in achieving the desired tooth movements. Tooth movements with optimized attachment appeared to be slightly inaccurate with rotational movement. Although teeth with optimized attachment showed more accurate torque tooth movement compared with conventional or no attachments, the overall torque tooth movements were challenging. Kravitz et al. stated that attachments may provide minute clinical improvements with rotational movements compared to no attachments; however, not with a statistical significance¹⁶. On the other hand, Simon et al. found that attachments are significantly beneficial when derotating premolars¹⁸. Cortona et al. also addressed with a finite study that the most efficient way to derotate mandibular round-shaped teeth was to add a single attachment with a 1.2° of aligner activation³⁵. Nucera et al. systematically reviewed the effects of composite attachments on clear aligner therapy and outlined the contradictory results in the current literature³⁶. The lack of evidence warrants further clinical trials to clarify the influence of attachments and their number, size, shape, and position on each orthodontic movement.

Overall, Invisalign achieved most of its predicted posterior tooth movements in adolescents and adults with mild to moderate malocclusions. Specifically, the predicted de-rotation of the maxillary premolars, and especially the first premolar, was more challenging. All maxillary posterior teeth tended to torque buccally without adequate torque control. The more distally positioned the tooth, the more unpredictable the result. With or without attachments or different types of attachments seemed to make no difference in the prediction. Generally, additional refinements or overcorrection would be necessary to achieve all predictions. Comparisons of Invisalign-predicted with clinically achieved digital models in the maxillary arch can benefit from model registration using both palatal and dental features, individual tooth segmentation, and the mathematical transformations used to match them.

Materials

CloudCompare	GPL software	Version 2.11	open-source software (https://www.cloudcompare.net/)
Meshmixer software	Autodesk, Inc.
Rhinoceros 5.0	Robert McNeel & Associates	Version 5.0