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The inflection point: a torque reference for lingual bracket positioning on the palatal surface curvature of the maxillary central incisor
Progress in Orthodontics volume 19, Article number: 39 (2018)
Abstract
Background
Contrary to buccal orthodontics, lingual orthodontics has no reference for vertical bracket positioning on the maxillary central incisor. The aim of this study was to provide a reference point in relation to torque for lingual bracket positioning on the palatal surface curvature (PSC) of the maxillary central incisor.
Methods
Cone beam computed tomography (CBCT) radiographs of 50 right maxillary central incisors from archives of a dental radiographic center were transferred to Photoshop, where their PSC was traced using pentool. The PSC torque angle values of the incisors were calculated in Excel using cubic polyBezier curves at 0.5mm increments and at the inflection point of PSC. Descriptive statistics for the torque angle values of the increments and for the inflection point for the 50 incisors were then calculated. Oneway ANOVA test was used to detect systematic differences between the increments, and Tukey test was used posthoc.
Results
For all incisors, increments incisal to inflection point exhibited progressive decrease in torque angle values from the firstcalculated increment to inflection point while increments cervical to inflection point exhibited progressive increase from inflection point to lastcalculated increment. Mean torque angle values of all the increments and inflection point showed high standard deviations and vast range of values. Oneway ANOVA test was highly statistically significant (p < 0.0001) and most pairwise comparisons of the increments using Tukey test were significant.
Conclusions
Inflection point can be used as a reference for bracket positioning on PSC. Cervically oriented shifts in vertical bracket position cause crown labial/root palatal movement cervical to inflection point and crown palatal/root labial movement incisal to it. A scientific mathematical justification for customized bracket torque prescriptions on PSC of maxillary central incisor was also provided.
Background
The torque of a certain crown site can be assessed by viewing the proximal tooth aspect and then determining the inclination of the tangent at that site [1,2,3,4,5,6,7,8,9,10,11,12,13]. Consequently, each bracket site on a tooth will have an associated torque angle value (\( {Torque}_{Value}^{Angle} \)) determined by the tangent’s inclination at that site. The difference in torque between two bracket positions can be found by subtracting their associated \( {Torque}_{Values}^{Angle} \) [6,7,8,9,10,11,12,13]. The effect of vertical bracket position on torque in labial/buccal orthodontics has been discussed thoroughly in the literature [6,7,8,9,10,11,12,13] compared to a single study in lingual orthodontics [2]. Yet that lingual study used only four different vertical bracket positions to investigate that effect instead of using 0.5 mm or 1 mm tooth increments, akin to the studies of the labial/buccal orthodontics [6,7,8, 10,11,12,13]. In their turn, Kurz et al. [1] calculated the \( {Torque}_{Value}^{Angle} \) of only one bracket site on the lingual surfaces of a set of maxillary and mandibular dentition, while Bryant et al. [4] with a mathematical equation of a parametric survival model could only calculate the maximum slope found at the inflection point of the palatal surface curvature (PSC) of the maxillary central incisor. The inflection point of a mathematical function is the point where the curvature of that function changes from convex to concave or vice versa [14].
\( {Torque}_{Value}^{Angle} \) calculation is generally accomplished by drawing tangents directly on the crown [8, 9]. Miethke pointed out that this method “is more or less subjective depending on the crown curvature” [8]. The error in angle measurement which can occur upon the use of drawn tangents to assess lingual surface curvature can exceed 4° [2], which calls for more accurate mathematical methods of angle measurement. The pentool in Adobe Photoshop Creative Cloud 2013 (Adobe Systems Inc., San Francisco, CA) can create cubic Bezier curves, which are parametric mathematical equations where the tangent at any point on these curves could be calculated by using the curve’s first derivative [15]. A cubic Bezier curve is formed by four control points and mathematically it is represented by two equations [16]. The two equations could be found by substituting the coordinates of the four control points in the mathematical formula of the cubic Bezier curve [16]. The initial and terminal control points of the cubic Bezier curve lie on the curve and are always its endpoints while the other two intermediate control points which determine its curvature do not generally lie on the curve [16].
Contrary to lingual orthodontics, conventional labial orthodontics has a reference point for bracket positioning reflected in the long axis point [17]. As there is no reference in lingual orthodontics for bracket positioning on PSC, the aim of this study was to find if the inflection point of PSC can be used as a torque reference for lingual bracket positioning (the inflection point of PSC is the anatomical landmark where the intersection between the convex and concave portions of PSC occurs). The use of cone beam computed tomography (CBCT) to calculate the \( {Torque}_{Values}^{Angle} \) on the labial and buccal surface curvatures were made previously in two studies [12, 13]. To our knowledge, no study has been done using CBCT to assess the PSC of the maxillary central incisor through cubic polyBezier curves in Photoshop.
Methods
Ethical committee approval was obtained from the university’s ethical board before beginning the study (preapproval code: 2016H0040DM0155). To fulfill the aim of the study, 50 right maxillary central incisors (RMCI) were selected from CBCT archives of a dental radiographic center and then the \( {Torque}_{Values}^{Angle} \) of their Photoshoptraced PSC were calculated at 0.5mm increments and at the inflection point using the first derivative of their cubic polyBezier curve. A total of 50 CBCT radiographs containing both jaws were selected randomly from the archives of a radiographic center in a private office. Those radiographs were made for nonorthodontic reasons and were taken by a Kodak 9500C Cone Beam 3D machine (Kodak Dental Systems, Carestream Health Inc., Rochester, NY) at 10 mA, 80 KV and an exposure time of 10.8 s with a voxel dimension of 300 μm. The inclusion/exclusion criteria for the selection of each CBCT radiograph is detailed below.
Inclusion criteria

Radiographs should belong to individuals aged between 15 and 30.

The absence of interincisal contact on the palatal surface of the RMCI on radiograph. Contact of the lower incisors on the PSC of the RMCI would not allow proper tracing of PSC.

RMCI presenting palatal surface and incisal edge integrity on radiograph.
Exclusion criteria

Intraoral presence of metal or amalgam restorations shown in radiograph.

Intraoral presence of labial or lingual brackets in radiograph.

RMCI with attrition or caries or a dilacerated root in radiograph.
The manipulation of each RMCI followed the procedures detailed below (all procedures were made by one orthodontist):

1.
Using CS 3D Imaging Software 3.1.9 (Carestream Health, Rochester, NY), the axial slice in “Oblique Slicing” tab was selected and the indicator which represented the sagittal plane was oriented with the labiopalatal axis of the RMCI (Fig. 1a). In the sagittal slice, a line with a known measurement (calibration line) was drawn that was used later for calibration (Fig. 1b). A screenshot image of the workspace at double magnification was made and then the TIFF image was opened with Photoshop.

2.
In Photoshop, the scale from pixels to millimeter was calibrated using the calibration line. The long axis of the crown was then drawn in Photoshop (Fig. 1c). The long axis of the crown was defined similarly to Bryant et al. [4] and van Loenen et al. [9] as a line drawn from the incisal edge of the incisor to the midpoint of the line joining the palatal and labial CEJ. The image was then rotated until the long axis of the crown became horizontal (parallel to Photoshop’s xaxis) (Fig. 1d). This rotation allows superimposition of all incisors on their crown’s long axis, enabling direct comparison of the torque angles.

3.
As a single cubic Bezier curve failed in accurately describing the PSC, 2 cubic polyBezier curves were used. On the crown’s long axis, at a distance of 2 mm from the incisal edge, a line perpendicular to the crown’s long axis was drawn that intersected PSC at point P_{1} (Fig. 2a). The initial anchor point (first control point) of the first cubic polyBezier curve of PSC was P_{1}, while its terminal anchor point (fourth control point) was point P_{4}, a point located 1.5 to 3 mm cervical to the visually estimated position of the inflection point of PSC (Fig. 2a–c). Choosing the terminal anchor point of the first cubic polyBezier curve as previously described ensures that the inflection point of PSC is contained in the first cubic polyBezier and gives the ability to objectively determine the true position of inflection point on PSC as well as its \( {Torque}_{Value}^{Angle} \) through accurate mathematical procedures.

4.
To allow the first and second cubic polyBezier curves to be continuous and differentiable at P_{4}, the second control point (P_{5}) of the second cubic polyBezier was positioned so that line P_{3}P_{4} and line P_{4}P_{5} have equal lengths and slopes (Fig. 2c) [16]. The terminal anchor point of the second cubic polyBezier (P_{7}) was placed in a position cervical and labial to the palatal CEJ (Fig. 2c), in order to allow the second cubic polyBezier to more accurately trace the part of PSC cervical to P_{4}.

5.
The origin of the 2 cubic polyBezier curves was set at P_{1}, with the xaxis parallel to the long axis of the crown and the yaxis perpendicular to the xaxis (Fig. 2d). The xaxis and yaxis were positive in the right and upwards directions, respectively. The coordinates of the four control points that are needed to obtain the equations of each of the 2 cubic polyBezier curves were found using the ruler tool in Photoshop. Figure 3 gives the cubic Bezier curve formulas and the formulas of their first and second derivatives [15] that were used to find the slope at inflection point and at the 0.5mm increments from P_{1} to palatal CEJ (Fig. 2e, f). Figures 4 and 5 list the procedures done in Microsoft Excel 2013 (Microsoft, Redmond, Washington) to calculate the \( {Torque}_{Values}^{Angle} \) at the 0.5mm increments and inflection point.

6.
After calculating the \( {Torque}_{Values}^{Angle} \) of all the 0.5mm increments and inflection point, the incisor was divided into two incisor parts, a part incisal to the inflection point and a second part cervical to it. Incremental subtractions in each incisor part were done in Excel, where an incremental subtraction was defined as follows: the difference in the \( {Torque}_{Value}^{Angle} \) between two successive 0.5 mm increments, where the more incisal increment was always subtracted from the more cervical increment (Table 1).
Statistical analysis
The statistical analysis was performed using the Statistical Package for Social Sciences SPSS (IBM SPSS Statistics version 23, Armonk, NY). Intraobserver reliability in tracing the RMCI and in calculating the \( {Torque}_{Values}^{Angle} \) of their increments was determined using the Dahlberg formula, by randomly selecting 10 incisors and repeating the tracing and measuring procedures after 1 month. Descriptive statistics for the \( {Torque}_{Values}^{Angle} \) of the 50 RMCI at the inflection point and at the 0.5mm increments between [P_{1}] and palatal CEJ were calculated. The frequency of positive and negative incremental subtractions in each incisor part of the 50 incisors was found. Since the data did not violate assumption of normality as detected by Shapirowilk test, oneway ANOVA was done to detect systematic differences between the mean \( {Torque}_{Values}^{Angle} \) of the increments and when significant differences exist Tukey test was used posthoc. The level of significance was set at p < 0.05 for all statistical tests.
Results
The Dahlberg error for repetitive tracing and measuring procedures was 1.18°.
The mean \( {Torque}_{Values}^{Angle} \) at all 0.5mm increments and at inflection point showed highstandard deviations and a wide range of values for the 50 RMCI (Table 2).
All 50 incisors studied showed that their 0.5mm increments had a progressive decrease in \( {Torque}_{Values}^{Angle} \) from P_{1} to inflection point and an opposite progressive increase in \( {Torque}_{Values}^{Angle} \) from inflection point to the most cervical calculated increment, with the inflection point exhibiting the most negative \( {Torque}_{Value}^{Angle} \). All incremental subtractions in the incisor part cervical to inflection point were positive (478 incremental subtractions), and all incremental subtractions in the incisor part incisal to inflection point were negative (350 incremental subtractions) (Additional file 1).
The oneway ANOVA test showed a highly statistically significant difference between the increments, F(22,885) = 137.60, p < 0.0001 (Table 3). The results of the posthoc Tukey test were mostly significant and are presented in Table 4.
Discussion
The inflection point of PSC of maxillary central incisor has utmost importance in understanding the directional change in torque which will occur upon a 0.5mm shift in a vertical bracket position. As incremental subtractions were always positive cervical to inflection point and negative incisal to it, the following can be derived: Cervically oriented shifts or errors in vertical bracket position in an RMCI cause crown labial/root palatal torque changes cervical to inflection point (Fig. 6) and crown palatal/root labial torque changes incisal to it, while incisally oriented shifts cause movements opposite to the mentioned ones cervical and incisal to inflection point, respectively. Furthermore, the more cervical a bracket is placed on the incisor part cervical to inflection point, the more is the potential of crown labial/root palatal torque expression. Conversely, the more cervical a bracket is placed on the incisor part incisal to inflection point, the more is the potential of crown palatal/root labial torque expression. As the inflection point exhibits the most negative \( {Torque}_{Value}^{Angle} \) on PSC, it is the site with most crown palatal/root labial torque expression potential. The characteristics of the inflection point mentioned in the four previous sentences justify its use as a torque reference for lingual bracket positioning on the PSC of the maxillary central incisor. Bracket position on PSC specified as either incisal or cervical to inflection point allows the orthodontist to recognize the inherent characteristics of the bracket site rather than it being specified arbitrary and thus devoid of this recognition.
The vast extent of \( {Torque}_{Values}^{Angle} \) at all the 0.5mm increments of the 50 RMCI studied (Table 2) and the statistically significant differences between the \( {Torque}_{Values}^{Angle} \) of the increments (Tables 3 and 4) are a scientific justification through a mathematical model (cubic Bezier) for the use of customized bracket torque prescription on PSC. The adoption of a preestablished bracket torque prescription is inappropriate for embracing the extremely varying PSC morphology of the maxillary central incisor. The most common maxillary central incisor lingual bracket torque prescriptions of 40°, 55°, and 68° are not sufficient to cover the wide spectrum of \( {Torque}_{Values}^{Angle} \) at each of the 0.5mm increments.
This study is in agreement with other studies that reported on the wide variability in PSC morphology [1, 2, 4]. The PSC form of the incisors in this study varied from slight to moderate to complex Sshaped curvatures (Fig. 7). The aforementioned difference in PSC form justifies the approach in lingual orthodontics to individualize the base of each maxillary central incisor lingual bracket [18]. The wide range in the forms of PSC is responsible for the broad variation in the \( {Torque}_{Values}^{Angle} \) of the 0.5mm increments. Also contributing to this broad variation of \( {Torque}_{Values}^{Angle} \) is the anatomical location of the 0.5mm increment being measured. For example, the increments [P_{1} + 3] and [P_{1} + 3.5] of the RMCI in Fig. 7b are cervical to the inflection point while in the incisor of Fig. 7d those two increments are incisal to it. The anatomical variation in increment location with respect to the inflection point between incisors leads to distinctly different \( {Torque}_{Values}^{Angle} \).
Conclusions

1.
The inflection point is the anatomical landmark on PSC where directional change in torque occurs in a maxillary central incisor, as of this it can be used as a torque reference for lingual bracket positioning on PSC.

2.
Cervically oriented shifts in vertical bracket position in an RMCI cause crown palatal/root labial torque changes incisal to inflection point and crown labial/root palatal torque changes cervical to inflection point, while incisally oriented shifts cause opposite movements incisal and cervical to inflection point, respectively.

3.
The highstandard deviation of the mean \( {Torque}_{Values}^{Angle} \) of all the 0.5mm incremental PSC sites of the 50 studied incisors calls for the fabrication of customized brackets that incorporates individualized torque prescriptions appropriate to vertical bracket position.
Abbreviations
 CBCT:

Cone beam computed tomography
 PSC:

Palatal surface curvature
 RMCI:

Right maxillary central incisor
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Acknowledgements
The authors of this study did not receive an aid or advice from other persons.
Funding
The authors funded the study.
Availability of data and materials
The dataset supporting the conclusions of this article is included within the article and its additional file.
Author information
Affiliations
Contributions
AHK designed the study, performed all the procedures on the right maxillary central incisors, executed the statistical analysis, and wrote the manuscript. JB, EO, and AAMES designed the study, evaluated the statistical results, and proofread the manuscript. All authors read and approved the final manuscript.
Corresponding author
Correspondence to Abdel Hadi Kanj.
Ethics declarations
Ethics approval and consent to participate
Ethical committee approval was obtained from the university’s ethical board before beginning the study (Preapproval code: 2016H0040DM0155).
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Not applicable
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The authors declare that they have no competing interests.
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Additional file
Additional file 1:
The torque angle value of each 0.5 mm increment and of each inflection point for each of the 50 RMCI used in this study are found in this additional file. Furthermore, the location of each inflection point of each of the 50 RMCI is disclosed here. Each value for each incremental subtraction in the incisor part incisal to inflection point or in the incisor part cervical to the inflection point for each RMCI is also shown in this additional file. (XLSX 76 kb)
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Kanj, A.H., Bouserhal, J., Osman, E. et al. The inflection point: a torque reference for lingual bracket positioning on the palatal surface curvature of the maxillary central incisor. Prog Orthod. 19, 39 (2018). https://doi.org/10.1186/s4051001802340
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Keywords
 Bracket positioning
 Lingual orthodontics
 Maxillary central incisor
 Torque