Validity and reliability of a trigonometry-based method for the measurement of tooth movement on digital models

ABSTRACT Objective: The objectives of the present study were to develop a method for longitudinally measuring tooth rotation, inclination and angulation on digital models, and to test the method validity and reliability. Methods: The initial and final planned models of 14 patients treated with Invisalign® (386 teeth) were exported from ClinCheck®. The rotation, inclination and angulation values were assessed for the incisors, canines, premolars and molars, in both models, using trigonometry. An application was developed in Python 2.7 to automate the measurements. The ∆planned (variation in the position between the initial and final planned models) was obtained for each tooth and each type of movement. To test the validity, the degree of agreement between the ∆planned and the values available in the Invisalign® Table of Movements was assessed using the Intraclass Correlation Coefficient (ICC) and Bland-Altman analysis. For intra and inter-rater reliabilities, the ∆planned was obtained again. Results: Excellent ICCs (> 0.9) and limits of agreement with narrow and clinically acceptable discrepancies were obtained for the rotation of all teeth (except maxillary canines, which had broader limits: -3.47 - 5.43) and for the inclination of premolars and molars. The inclination of anterior teeth and angulation of all teeth had ICCs and limits that were not indicative of great agreement. The reliability was high for the three movements (discrepancy <2°). Conclusions: The method developed is reliable and suitable for longitudinally measuring inclination (posterior teeth) and rotation (except maxillary canines). It has limited value for the other movements measurements.


INTRODUCTION
Dental inclinations, angulations, and rotations are essential aspects to be evaluated during orthodontic treatment and are included in Andrews's six keys to normal occlusion. 1 Longitudinal evaluations of inclination and angulation between the beginning and conclusion of treatment are typically measured with the aid of lateral head films [2][3][4] or panoramic radiographs. 5,6 However, only incisor inclination and posterior teeth angulation can be measured using lateral radiographs, while only dental angulation can be assessed in panoramic radiographs. Moreover, superimposition remains a problem in both types of exams. 7 Although cone-beam computed tomography (CBCT) may overcome these limitations, it is not indicated as a routine exam. 8 The measurement of dental rotations is a significant factor for predicting posttreatment stability. 9 Little's irregularity index 10 expresses the degree of anterior segment alignment, but has limited value for the expression of rotation because the results are a combination of rotation and inclination. 11 The American Board of Orthodontics (ABO) introduced the objective grading system (OGS) to evaluate finished cases according to eight criteria. 12 Among these, the buccolingual inclination is used to indirectly assess the inclination of posterior teeth. Alignment is another criterion, but for the same reasons of Little's irregularity index, has limited value for expressing rotation reading.
Santos RF, Santos BFO, Fernandes VM, Caldas LD, Baldo TO, Dominguez GC -Validity and reliability of a trigonometry-based method for the measurement of tooth movement on digital models 5 Longitudinal evaluations of dental positioning are important both clinically and scientifically. Some studies have measured tooth inclination and angulation using plaster or digital models in a cross-sectional manner, 7,[13][14][15] with the longitudinal assessment being restricted to the use of radiographic exams.
Digital models can allow obtaining angular measurements, both cross-sectionally and longitudinally. Huanca et al. 13 and Lombardo et al. 16 described a method based on trigonometry that allowed the assessment of teeth inclination, 13 angulation, 13 and rotation 16 on digital models. The idea of obtaining dental positioning by means of trigonometry seems to be a viable option, but since these methodologies were not validated, the purposes of the present study were (1) to develop a method to longitudinally measure tooth rotation, inclination and angulation on digital models and (2) to test its validity and reliability. The initial and final planned models (predicted by the software as an estimate of treatment results) were exported from ClinCheck ® (Align Technology, Santa Clara, CA, USA). Therefore, for each patient, four models were exported in Standard Triangle Language (.stl): two initial (one upper; one lower) and two final planned (one upper; one lower). The initial models were obtained through polyvinyl siloxane impressions.
The methodology development was based on the description of Huanca et al. 13 for quantifying inclination and angulation, and Lombardo et al. 16 for rotation. The values were obtained through trigonometry for each tooth in the initial and final models, and angular variation was calculated (∆rotation, ∆inclination, and ∆angulation) for each tooth of the 14 patients (n=386 teeth). An application was developed in Python 2.7 to automate the measurements. The .stl models obtained from ClinCheck ® were imported into Geomagic Control ® (North Carolina, USA) software. Five points were marked per tooth on the initial model (Fig 1).
Subsequently, the "best fit" alignment was performed for each tooth from the initial model with the respective tooth from the final planned model. The points were thus copied from the initial to the final model (Fig 2A).

Establishing the Reference Plane
The establishment of a reference plane is necessary to make angular measurements. A "best fit" alignment of all teeth (except second molars, Fig 2B) was performed. Then, the reference plane was defined on the initial model as the best adjustment of the lingual gingival points of all teeth ( Fig 2C) and labeled Plane 1. A median reference plane was created and referred to as Plane 2 ( Fig 2D). Planes 1 and 2 were copied to the final         Since ∆rotation, ∆inclination, and ∆angulation variations were obtained between the initial and final planned models of patients undergoing treatment with Invisalign ® , the proposed method was expected to produce ∆s similar to those presented in the Invisalign ® 's Table of Movements.

STATISTICAL ANALYSIS
The validity was verified by the Intraclass Correlation Descriptive statistics of the difference (∆Invisalign Table -∆Method) were computed, and a 1-sample Student t-test was applied to verify the presence of systematic error. The random error was assessed by the Dahlberg formula.
For the intra-rater reliability analysis, 50% of the models were randomly selected and remarked after one month, and the ∆Method was obtained again. The same 50% models were remarked by a second rater for the assessment of inter-rater reliability. The ICC (2-way mixed, single measurement, absolute agreement) and Bland-Altman were used for both analyses. Descriptive statistics of the difference, a 1-sample Student t-test, and the Dahlberg formula were also applied.
Variables were expressed as the mean and standard deviation.
The normality assumption was investigated by the Kolmogorov-Smirnov test. Tables 1 and 2 describe the results of the ICC, whereas Tables   3 and 4  Considering the data provided by Invisalign ® as the gold standard, the method for calculating rotation was shown to be accurate for maxillary and mandibular teeth because it presented excellent ICC agreement values (greater than 0.9 for all groups) and because the Bland-Altman limits of agreement showed narrow and clinically acceptable (≤ 3º) discrepancies (except maxillary canines). The difference between the method and the For these teeth, the method tended to measure an average of 0.98° (0.10 - 1.86) less than the Invisalign ® Table. When evaluating the inclination, results have shown high overall ICCs (> 0.90) and narrow limits of agreement (≤ 3°) for posterior teeth. Systematic errors were not found. Anterior teeth, however, presented limits of agreement with discrepancies higher than the clinically relevant difference. Angulation presented ICCs that were not always indicative of a great relationship, and the limits of agreement were not as interesting; additionally, systematic errors were present for premolars and molars.

VALIDITY
Santos RF, Santos BFO, Fernandes VM, Caldas LD, Baldo TO, Dominguez GC -Validity and reliability of a trigonometry-based method for the measurement of tooth movement on digital models Table 4: Descriptive statistics, 1-sample t test, and 95% CI values for differences and limits of agreement (positive numbers represent underestimations, and negative represent overestimations of measurements obtained by the method with respect to Invisalign Table). Mandibular arch. were mainly less than 3°, and considering the clinical plausibility, only differences higher than 3° were considered relevant.

RELIABILITY
The intra-examiner reliability is described in Tables 5 and 6, while the inter-examiner is described in Tables 7 and 8. The method was shown to be highly reliable for measuring the three kinds of movements, since the discrepancies were less than 1° for intra-rater, and less than 2° for inter-rater reliabilities.  Table 6: Descriptive statistics, 1-sample t test, and 95% CI values for differences and limits of agreement (positive numbers represent underestimation, and negative represent overestimation of measurements obtained by the test with respect to retest).

DISCUSSION
In the present study, dental rotation, inclination and angulation were longitudinally obtained by calculating the ∆planned.
To the best of our knowledge, this is the first study to test the validity and reliability of a method for longitudinal evaluation of teeth positioning using digital models.
Bearing in mind that correlation coefficients lower than 0.5, between 0.5 and 0.75, between 0.75 and 0.9, and higher than 0.9 are indicative of low, moderate, good and excellent relationships, 18 respectively, it can be noticed that rotation    (Tables 1 and 2). Similarly, the evaluation of inclination for premolars and molars also presented excellent values. The same results were not found for the inclination of anterior teeth and angulation of all teeth, which mostly presented low or moderate indexes.
The correct statistical approach to measure the degree of agreement between two methods is not obvious. 19,20 It has been discussed that the ICC must not be considered alone for this analysis because high correlation does not necessarily mean that two methods agree. 19 The Bland-Altman limits of agreement were applied to overcome this problem.
We presume that limits with discrepancies >3° are clinically relevant, and therefore consider the method as having limited value when longitudinally measuring the maxillary canine's rotation, the inclination of anterior teeth and angulation of all teeth (Tables 3, 4). For the maxillary canines, for example, the method tends to underestimate the real rotation value up to 5.43°, or overestimate up to 3.47 (-3.47 -5.43). In cases in which there is no problem in accepting this discrepancy, the method to perform this measurement can be applied.
Santos RF, Santos BFO, Fernandes VM, Caldas LD, Baldo TO, Dominguez GC -Validity and reliability of a trigonometry-based method for the measurement of tooth movement on digital models 21 The fact that the Table of Movements and the method yielded different results for some measurements might be related to limitations in the stability of the reference plane between the initial and final planned models. When Invisalign ® receives the initial digital models, or when they create it using impressions, the model has its own reference plane. It happens that probably a setup is performed on this model to generate the final model, and the ∆measurements are obtained in relation to this plane, which remains stable. The stability of the plane is guaranteed because the first file generates the second. Nevertheless, a problem is found when the task is needed in the opposite way: create a stable reference plane for the two files.
According to Ferrario et al. 21  Ghislanzoni et al. 13 , who advocated that constructing the plane closer to the gingival region would make it less susceptible to changes caused by alterations in dental inclination, curve of Spee or Wilson. All these approaches, however, appeared to be insufficient for guaranteeing plane stability.
Although it cannot be affirmed, we believe that the reason Although it is not the purpose of the study, one of the possible applications of this method is to quantify how predictable the digital planning of clear aligners is. The ∆planned and ∆achieved movements of patients undergoing treatment with aligners can therefore be compared. We believe that, in this specific situation, even the methods that presented limited validity could be applied. The explanation is that if both ∆s are calculated by the same method, any presumed error will most likely be maintained for both variations, making the comparison valid. Besides, the method proved to be highly reliable for all the measurements.

CONCLUSIONS
The method developed is highly reliable for evaluating the three movements, and valid for the longitudinal evaluation of the rotation of all teeth (except maxillary canines) and the inclination of premolars and molars, even when a small anteroposterior change in the reference plane is present.
It has limited value to measure the rotation of maxillary canines, the inclination of incisors and canines or angulation of all teeth, at least in models in which an alteration in the overbite is present. Overall responsibility: RFS.