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A new concept of anatomic lingual arch form
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A new concept of anatomic lingual arch form Luca Lombardo,a Luca Saba,b Giuseppe Scuzzo,c Kyoto Takemoto,d Lola Oteo,e Juan Carlos Palma,f and Giuseppe Sicilianig Ferrera, Italy, Tokyo, Japan, and Madrid, Spain Introduction: The aim of this study was to describe a natural and anatomic lingual arch form obtained from subjects with normal occlusion that could be used, with other criteria, in the construction of personalized setups for the lingual straight-wire technique. Methods: The study sample comprised 58 pairs of dental casts of the arches of 58 southern Europeans (37 women, 21 men) with ideal natural occlusions. After the reference points of the dental arches were identified and marked, the dental casts were scanned. The exact position of the models on the scanner was established by using an acetate sheet with a Cartesian reference system. For each image, 14 reference points (x, y) were measured and recorded. The measurements were processed with software to select the polynomial function that best described the shape of the dental arches. The ninthdegree polynomial function was selected to represent the lingual arch form of both arches. Distribution analysis of the x and y values of each tooth in each arch resulted in the creation of 3 groups (small, medium, and large) to verify the most appropriate measures of the central tendencies of our data. Results: Statistical analysis showed no significant sex difference in the medians of the 6 parameters used to measure depth and width in both arches. A representation of the variability of the lingual curve of our sample was created to document at least 3 sizes of the representative curve of the central tendency for our data. No statistically significant differences in shape were found between men and women, considering the medians as a measure of the central tendencies. Conclusions: Three lingual curves (small, medium, and large) for the maxillary and mandibular arches, representing the mean values of our sample, were developed and can be used as guides for the setup in the lingual straight-wire technique. (Am J Orthod Dentofacial Orthop 2010;138:260.e1-260.e13)
T
he form of the dental arch is considered a fundamental element in orthodontic diagnosis and treatment planning because of its ability to influence not only available space and dental and smile esthetics, but also long-term occlusal stability.1-7 Furthermore, respecting this important parameter reduces the possibility of crowding relapse and periodontal damage.5,8 Moreover, definition of the shape of the arch aids the clinician in achieving results that agree with the natural laws of biologic a
Research assistant, Department of Orthodontics, University of Ferrara, Ferrara, Italy. b Resident, Department of Orthodontics, University of Ferrara, Ferrara, Italy. c Adjunct Professor, Department of Orthodontics, University of Ferrara, Ferrara, Italy. d Adjunct Professor, Department of Orthodontics, University of Ferrara, Ferrara, Italy. e Professor, Department of Orthodontics, University Complutense, Madrid, Spain. f Adjunct Professor, Department of Orthodontics, University Complutense, Madrid, Spain. g Chairman, Department of Orthodontics, University of Ferrara, Ferrara, Italy. The authors report no commercial, financial, or proprietary interest in the products or companies described in this article. Reprint requests to: Luca Lombardo, Contrada Nicolizia, 92100 Licata, AG, Italy; e-mail, [email protected]. Submitted, October 2009; revised and accepted, April 2010. 0889-5406/$36.00 Copyright Ó 2010 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2010.04.022
variability.1,5 Andrews,9 father of the labial straightwire technique, cited the arch form as the seventh of his keys to achieving Class I occlusion. Hence, in recent decades, numerous studies have analyzed the arch form from anatomic and anthropologic perspectives10-16 to evaluate its implications in orthodontic therapy17-21 or its modifications after orthodontic treatment.18,22-25 Even though the necessity of individualizing each patient’s arch form during treatment is widely recognized, after the evolution of labial straight-wire appliances and because of the superelastic properties of the latest generation of archwires, it seems to be clinically reasonable to stereotype the constructed arch shapes from subjects with normal occlusion into a few sets of preformed arches.1,5 The most similar to that of the patient before treatment could therefore be selected on the basis of geometric similarity, ethnicity, and type of malocclusion.1,3-5 Nevertheless, no objective study has thus far elucidated the choice of a particular arch form,3 although various studies have sought, by using the incisal and occlusal margins of the teeth as anatomic reference points, to represent it graphically from the labial side.1-6,24,25 Numerous authors used geometric curves to describe the dental arch.24-35 Moreover, various authors have observed that polynomial functions can simply and symmetrically describe the shape of the dental arch, and have exploited these 260.e1
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mathematic equations as an accurate method for its description.2,12,24-35 Many articles in the literature have dealt with the shape of the dental arch from the labial perspective, although nothing has yet been published regarding the lingual side. Many systems for representing the normal arch form have been proposed, including qualitative descriptions, geometric constructions, and mathematic curves; although there is consensus on the importance of this factor in orthodontics, a definitive geometric shape is elusive, as demonstrated recently by Trivin˜o et al.2 However, the introduction of the straight-wire to the lingual technique36 and the possibility of replacing Fujita’s ‘‘mushroom form’’37 with an arch form without insets has led to the question of which arch form to choose and which criteria to use in the lingual straight-wire setup and treatment planning. Because a ‘‘straight’’ arch form has not previously been described from a lingual perspective and this is fundamentally important in orthodontic treatment planning, in this study it is described with an objective, standardized, and reproducible methodology: a natural and anatomic arch form obtained from subjects with normal occlusion. This can be used, with other criteria, in the construction of personalized setups for the lingual straight-wire technique. MATERIAL AND METHODS
The sample was carefully selected from the records of the University of Ferrara Postgraduate School of Orthodontics in Italy and the Department of Orthodontics at the Complutense University of Madrid, Spain. It included southern European white adults with the following characteristics2,26: (1) age not less than 18 years (range, 19.08-70.25 years); (2) no previous orthodontic treatment; (3) regular arch form with little or no crowding; (4) complete dentition up to at least the second molars; (5) no extensive restoration and implants; (6) at least 4 of the 6 Andrews’ keys9 for optimal occlusion and bilateral Class I molar and canine relationship; (7) normal (within 2 6 1 mm) overbite and overjet; (8) no deviations of the interincisal lines; (9) no gingival recession; (10) no muscular or joint pathologies; (11) no ectopic teeth, tooth aplasia, or anomalies in tooth shape; (12) no supernumerary or congenitally missing teeth; (13) no anterior or posterior crossbite; (14) no visible intraoral or extraoral asymmetry; and (15) minimal diastemas, premolar rotations, and incisor irregularities (present in several subjects). Each subject participating in the study gave informed written consent.
American Journal of Orthodontics and Dentofacial Orthopedics September 2010
Mandibular and maxillary dental casts of each subject, according to Tweed prescriptions, were obtained, for a total of 75 pairs. The regularity of these models was first determined clinically and then quantified by using Little’s irregularity index38; 9 pairs of models had a value greater than 3 and were therefore excluded, as were a further 8 pairs, which, although having a value less than 3, had spaces greater than 1 mm or a slightly asymmetrical contraction of the arch in the premolar area. Thus, the definitive sample comprised 58 pairs of dental casts of the arches of 58 white subjects (37 women, 21 men) of southern European ancestry (mainly Italian and Spanish). The mean age of the sample was 29.21 years (SD, 8.73 years); the youngest was 19.08, and the oldest was 70.25 years of age. Reference points for calculation of the shape of the arch were identified and marked on the lingual surface of each tooth (from the left second molar to the right second molar) of each dental cast by 1 operator (L.S.) using an indelible marker (.08-mm Pilot drawing pen; Pilot Corp., Tokyo, Japan). In the mandibular arch, these points, which trace the lingual straight plane, were situated at the center of the clinical crown (vertical position) along the central lingual axis and at the most prominent point on the lingual surface of each tooth (horizontal position) on the molars and premolars and corresponding to the middle third of the anterior teeth. In the maxillary arch, the reference points were marked at the most prominent point on the lingual surface of each tooth (horizontal position) at the intersection between the middle third and the gingival third of the anterior teeth, along their central axes, and at the center of the clinical crown of the posterior teeth. Before permanently marking these points on the lingual surface, the median axis of the clinical crown of each tooth was traced in pencil to determine its horizontal position, and a gauge was used to measure the height of the clinical crown to determine its vertical position. Although the same operator carried out this procedure for each tooth, it was an important source of error, because the accuracy of identification of the reference points necessarily influences the tracing of the curves on which the calculations of arch shape are based.33 After the reference points had been marked, the dental casts were scanned with an Expression 1680 Pro scanner (Epson, Cinisello, Balsamo, Italy), and images in TIF format at a resolution of 300 dpi were obtained. To avoid distortion of the image, and hence the reference points, during the scanning procedure, the models were positioned so that both the base and the occlusal plane were parallel to the scanner’s surface.39 The exact position of the models on the scanner was established with a specially created acetate sheet onto which a sheet of
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Fig 1. Cartesian system and reference points (black) used to establish lingual dental arch form; each point is described by a copy of x and y coordinates: x measurements, distance between the reference point and the y-axis; y measurements, distance between the reference point and the x-axis.
millimeter paper had been photocopied. After the photocopies were made, the prepared acetate and millimeter papers were superimposed to verify the lack of distortion in the acetate copy. On this acetate sheet, reference axes—a horizontal line and a perpendicular vertical line—were drawn with an indelible marker (.05-mm Pilot drawing pen). The sheet was subsequently placed on the glass surface of the scanner under the plaster models to align the posterior margin of the second molars with the abscissa and the median line with the ordinate, thus creating a Cartesian reference system (Fig 1).2 For each image, 14 reference points, corresponding to the distance between each reference point and the abscissa and ordinate were measured and recorded to provide their Cartesian coordinates (x, y). Four weeks later, to calculate the method error, 15 models were selected at random, and the coordinates of each reference point on both arches were replotted by the same operator as above. Analysis of the reliability of measurements was performed by using DahlP berg’s coefficient,40 Se 5 O d2/2n, where S is the Dahlberg coefficient, d is the difference between the first and second measurements, and n is the number of repeated measurements. The values corresponding to the coordinates of the central and lateral incisors and the second premolars were compared singly with their control measurements. The statistical significance was evaluated by using the Student t test for paired data. The systematic error of measurement of each value considered was shown to be not significant at P 5 0.05, thereby confirming the reliability of the results. First, the dimensions of both dental arches were evaluated via 3 transversal and 3 sagittal measurements: the transverse diameters and arch depths at the canines, first molars, and second molars, as described by Raberin et al.6 The lingual reference points considered were (1) the interincisor point; (2) the most prominent point on the central axis of the lingual surface of the canine
crown; (3) the most prominent point on the lingual surface of the first molar at the center of its clinical crown; and (4) the most prominent point on the lingual surface of the second molar at the center of its clinical crown. Thus, 6 measurements of distance were taken for each arch to analyze their width and depth. To measure arch width, the following were considered: intercanine diameter (D3), intermolar diameter at the first molars (D6), and intermolar diameter at the second molars (D7). To measure the arch length, the following were considered: (1) canine depth, the distance between the interincisor point and the line connecting the canine reference points (L3); (2) molar depth at the first molars, the distance between the interincisor point and the line connecting the reference points on the first molars (L6); and (3) molar depth at the second molars, the distance between the interincisor point and the line connecting the reference points on the second molars (L7). Subsequently, the shape of the dental arch was evaluated from the lingual side. The measurements (x and y coordinates) of each reference point were all obtained and processed by using Curve Expert software (version 1.3, http://curveexpert.webhop.biz) by the same operator. This software was used to select the polynomial function that best described the shape of the dental arches. After the dental arches were viewed and the presence of slight asymmetry was established, even though patients with normal occlusion had been used as models, the reference point coordinates of the 58 dental casts were divided into the left and right sides. Each arch half, described by 7 reference points (7 x and y coordinates), was then ‘‘mirrored’’ to obtain 116 whole maxillary and 116 whole mandibular arches with symmetrical curves.2 Based on the number of reference points (7 for each semiarch mirrored) and after the graphs obtained of the various mathematical functions were viewed, the ninthdegree polynomial function was selected to represent the lingual surfaces of both arches.
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Table I.
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Descriptive statistics of the 6 parameters used to measure dental arch depth and width by arch and sex
Mandibles in men D3-3 D6-6 D7-7 L3 L6 L7 Mandibles in women D3-3 D6-6 D7-7 L3 L6 L7 Maxillae in men D3-3 D6-6 D7-7 L3 L6 L7 Maxillae in women D3-3 D6-6 D7-7 L3 L6 L7
Maximum
Percentile 25
18 28 30 2 22 31
26 42 48 6 30 42
20 30 36 4 24 34
22 34 40 5 26 37
1.928 3.034 3.783 0.899 1.914 2.472
20 34 40 4 25 35
18 30 34 2.5 21 30
26 40 48 8 31 43
20 32 38 4 24 33
22 36 42 5 26 36
1.826 2.399 3.153 0.871 1.960 2.647
27.1 36.7 42.2 6.9 29.3 38.5
26 36 42 7 30 39
24 30 34 4 22.5 32
34 50 52 9 41 48
26 34 40 6 28 37
28 38 44 8 30.5 40
2.510 3.502 3.769 1.173 2.870 2.987
26.6 36.9 42.4 6.6 28.9 42.1
26 36 42 7 29 39
22 29 34 5 25 33
34 46 50 9 34 41
24 34 40 6 27 36
28 38 46 7 30 40
2.447 3.367 4.028 1.035 2.193 3.328
Mean
Median
21.2 32.7 38.5 4.1 25.2 35.3
21 32 38 4 25 355
21.0 33.4 39.6 4.1 25.2 35.1
Minimum
Percentile 75
SD
D3-3, Intercanine diameter; D6-6, intermolar diameter at the first molars; D7-7, intermolar diameter at the second molars; L3, canine depth; L6, depth at the first molars; L7, depth at the second molars.
This polynomial function yielded the curve most representative of the shape of the lingual dental arch. It was first selected according to visual inspection criteria, and then, to confirm this choice objectively, we determined which marker could be considered to establish the suitability of the model.2,6 With the residual analysis, we ascertained whether the observed values fell outside the trend of the expected curve. The residual plot graphically depicts the difference between the data points and the model evaluated at the data points. The residual at point i is defined by residual 5 yi – f(xi), where yi is the measured value at xi, and f(xi) is the predicted value at xi. These distances are shown as bars or points on the residual plot; the magnitudes of the data points are simply replaced by the residual defined above. If the residual is positive, then the data point is above the model’s prediction; if the residual is negative, then the data point is below the model’s prediction. The residuals can provide an indication of a particular model’s performance. If there are runs of like-signed residuals, then a better model for the data is likely to exist.
Optimally, the residuals should show a random scatter around zero, which indicates that the data points are randomly distributed around the curve. A ‘‘run’’ is a sequence of like-signed residuals, which stand out on the residual plot. A large number of runs indicates that the data systematically deviate from the curve. The critical reference measurement for a residual value is usually 1.96. If this value is exceeded at 1 point or more, the fit is unacceptable, and adaptations must be made. The measurement error was compared with the standard error values of various models to ascertain which model better interpolated the data—ie, that characterized by the lowest standard error was considered the best; in our case, it was the ninth-degree polynomial. The standard error was calculated with minimization of the so-called merit function (http://mathworld. wolfram.com/MeritFunction.html). Statistical analysis
The sample was initially subjected to descriptive statistical analysis including mean, standard deviation, variance, minimum, and maximum values.
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Table II. Results of the Mann-Whitney nonparametric U test used to evaluate the sex differences of the 6 parameters used to measure dental arch depth and width Variable Mandible D3-3 D6-6 D7-7 L3 L6 L7 Maxilla D3-3 D6-6 D7-7 L3 L6 L7
Rank sum for women
Rank sum for men
U
P value
N valid women
N valid men
4273.5 4584 4598 4156 4272 4163
2512.5 2202 2188 2630 2514 2623
1498.5 1299 1285 1381 1497 1388
0.752040 0.143744 0.122974 0.321717 0.745508 0.341744
74 74 74 74 74 74
42 42 42 42 42 42
4158.5 4387 4401.5 4083 4132.5 4146
2627.5 2399 2384.5 2703 2653.5 2640
1383.5 1496 1481.5 1308 1357.5 1371
0.328780 0.741164 0.679161 0.158455 0.260194 0.294463
74 74 74 74 74 74
42 42 42 42 42 42
D3-3, Intercanine diameter; D6-6, intermolar diameter at the first molars; D7-7, intermolar diameter at the second molars; L3, canine depth; L6, depth at the first molars; L7, depth at the second molars. Significance level, P \0.05.
The analysis of sexual dimorphism was carried out with a comparative analysis of the lingual distances D3-3, D6-6, D7-7, L3, L6, and L7 to verify whether there were statistically significant differences between the measures of the central trend in the men and women. Due to the abnormal distribution of data, the nonparametric Mann-Whitney U test was applied to compare the 2 independent groups.41 To provide a visible representation of a hypothetical ideal arch form curve, first the means and standard deviations of the 14 coordinates of the abscissa and ordinate were calculated for all curves relative to the lingual sides of both arches. Three groups—small, medium, and large—were created by a distribution analysis of the x and y values of each tooth in each arch to verify the most appropriate measure of the central tendencies for our data. It was verified that all x and y values were characterized by strongly nonnormal, asymmetrical, and unimodal distributions. This led us to choose the median value as the reference measure of the central tendency, and then the values of the 25th (first quartile) and 75th (third quartile) percentiles as measures of variability. Assuming that the median was an adequate representation of the measure of the central tendencies for the medium group, the measures of the central tendency for the small and large groups were identified. The objective was to identify 3 curves, corresponding to the 3 groups, that interpolated the measures of central tendency in a parallel fashion. Because of the many anomalous values and the variability that characterized each tooth, it was not possible to
consider the first and third quartiles as measures of the central tendency for each tooth in the small and large groups. Hence, the coordinates characterized by the minimum variability (x7, y1) were considered, and a maximum deviation around the median equal to 1 mm in both arches was identified. Thus the following measures were obtained. Central tendency for x7 small 5 median (x7) – 1. Central tendency for x7 large 5 median (x7) – 1. Central tendency for y1 small 5 median (y1) – 1. Central tendency for y1 large 5 median (y1) – 1. Thus, the central tendency of the small and large groups for the other coordinates were obtained, applying the 1-mm displacement to the median of all x and y values. In this way, constant ‘‘distances’’ along all points of the mouth between the central tendency measures of the medium group and those of the small and large groups were created. RESULTS
The descriptive statistics of each of the 6 parameters used to measure depth and width, subdivided by arch and sex, are reported in Table I. The values for mean, median, minimum, maximum, 25th percentile, 75th percentile, and standard deviation are also reported. The results of the Mann-Whitney nonparametric U test are reported in Table II. The P values show no significant statistical sex difference for the medians measured in both arches.
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Table III.
Descriptive statistics of the values of the coordinates of the reference points for the 2 arches Minimum
Mandible 1X 2X 3X 4X 5X 6X 7X* 1Y* 2Y 3Y 4Y 5Y 6Y 7Y Maxilla 1X 2X 3X 4X 5X 6X 7X* 1Y* 2Y 3Y 4Y 5Y 6Y 7Y
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Percentile 25
Median
Percentile 75
Maximum
Mean
24 21 18 16 13 8 4 7 17 28 36 44 46 47
20.5 17 15 13 11 7 3 5 15 24 30 37 40 41
19 16 14.25 13 10 7 2 4.5 14 23 29 35 38 39
19 16 14 12 10 6 2 4 13.5 22 28 34 36 37.5
15 14 12 11 9 5.5 2 3 11 19 24.5 30 33 34
19.67 16.58 14.62 12.78 10.53 6.84 2.45 4.40 14.31 23.01 29.19 35.42 38.15 39.56
26 25 22 20 17 12 5 3 10 18 24 27 35 37
22.75 19 17.25 15 14 10 4 4 13 21 27 34 38 40
21 18 16 14 13 10 4 4 13 22 29 35.5 40 42
20 17 15.75 13.5 13 9 3.5 3.5 14 23 30 37 42 44
17 14.5 13.5 12 11 7 2 2 17 29 35 44 50 52
21.12 18.46 16.55 14.33 13.4 9.65 3.85 4.09 13.21 22.25 28.79 34.89 40.25 42.32
*These coordinates varied the least.
Initially, a simple graphic representation of the lingual side of the arch form was obtained by using the means and standard deviations of the coordinates for the maxillary and mandibular arches, respectively. Subsequently, a representation of the variability of the lingual curve of our sample was created to document at least 3 sizes of the representative curve. Table III documents the descriptive statistics of the 14 coordinates corresponding to the reference points used to describe the curves that best represent the shapes of the arches, respectively. Figure 2 depicts the data in Table III; it is possible to observe the difference in variation of the coordinates. Thus, the x and y coordinates that varied the least were considered, as shown in Table III. After identifying 1 mm as the constant measure to be subtracted and added to the median measures, the central tendencies relative to the small and large groups were obtained, as described in Table IV. Figure 3 shows the 3 curves, large, medium, and small, superimposed onto the scatter plot. Figure 4 displays the values of all coordinates relative to both arches for each group (large, medium, and small) and the corresponding curves. The values reported in
Table IV were determined by the Curve Expert software to visualize the ninth-degree polynomial curve that interpolates the points relative to the large, medium, and small groups, respectively. Figure 5 illustrates the 3 representative curves describing the arch forms from the lingual side in the mandibles and the maxillae of our sample. At this point, the presence or absence of differences in shape between the arches and between the sexes was verified, considering the medians as a measure of central tendencies. The Mann-Whitney U test was used to compare the central tendencies (medians) of the 2 arches (Table V) and the 2 sexes (Table VI). Table V shows the significant differences between the medians of the 2 arches. For instance, the median of x1 in the maxillary arch (21) was significantly greater (P \0.000) than the corresponding measurement in the mandibular arch (19). Likewise, the median of y1 in the maxillary arch (4) was significantly lower (P 5 0.002) than that measured in the mandibular arch (4.5), and so on. No statistically significant differences were found between the central tendencies of the sexes as shown in Table VI.
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Line Plot of multiple variable Descriptive.sta 6v*14c 0
-10
-20
-30 Minimum Percentile 25% Median Percentile 75% Maximum Mean
-40
-50 X1
X2
X3
X4
X5
X6
X7
Y1
Y2
Y3
Y4
Y5
Y6
Y7
4X
5X
6X
7X
1Y
2Y
3Y
4Y
5Y
6Y
7Y
60
50
Minimum Percentile 25% Median Percentile 75% Maximum Mean
40
30
20
10
0
-10
-20
-30 1X
2X
3X
Fig 2. Difference in variation of the measures of central tendency of the coordinates of the reference points: A, mandibular arch; B, maxillary arch. In the x-axis are reported the coordinates (7X and 7Y), and in the y-axis is reported the range of variation in values of the coordinates for the maxillary and mandibular arches (Table I).
DISCUSSION
Many studies in the literature document analyses of the shape of the dental arches, with different methodologies, of similar samples of healthy subjects with normal occlusion to obtain clinical data pertinent to the labial edgewise technique.1-6,10-16 All of these
authors concluded that it was impossible to represent 1 ideal arch form. However, in the literature, no study has reported reference points to describe the dental arch from the lingual perspective. The introduction of straight-wire concepts to the lingual technique has led clinicians to
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Table IV.
Median central tendencies of the 3 groups Large
Mandible 1X 2X 3X 4X 5X 6X 7X 1Y 2Y 3Y 4Y 5Y 6Y 7Y 1X 2X 3X 4X 5X 6X 7X 1Y 2Y 3Y 4Y 5Y 6Y 7Y Maxilla 1X 2X 3X 4X 5X 6X 7X 1Y 2Y 3Y 4Y 5Y 6Y 7Y 1X 2X 3X 4X 5X 6X 7X 1Y 2Y 3Y 4Y 5Y 6Y 7Y L, Left; R, right.
American Journal of Orthodontics and Dentofacial Orthopedics September 2010
Medium
Small
Side
20 17 15.25 14 11 8 3 5.5 15 24 30 36 39 40 20 17 15.25 14 11 8 3 5.5 15 24 30 36 39 40
19 16 14.25 13 10 7 2 4.5 14 23 29 35 38 39 19 16 14.25 13 10 7 2 4.5 14 23 29 35 38 39
18 15 13.25 12 9 6 1 3.5 13 22 28 34 37 38 18 15 13.25 12 9 6 1 3.5 13 22 28 34 37 38
L L L L L L L L L L L L L L R R R R R R R R R R R R R R
22 19 17 15 14 11 5 5 14 23 30 36.5 41 43 22 19 17 15 14 11 5 5 14 23 30 36.5 41 43
21 18 16 14 13 10 4 4 13 22 29 35.5 40 42 21 18 16 14 13 10 4 4 13 22 29 35.5 40 42
20 17 15 13 12 9 3 3 12 21 28 34.5 39 41 20 17 15 13 12 9 3 3 12 21 28 34.5 39 41
L L L L L L L L L L L L L L R R R R R R R R R R R R R R
pose the important questions of which form should be used in setting up indirect bonding and according to which criteria.36 Thus, from our sample of 58 subjects with normal occlusion, we described, with an objective, standardized, and reproducible method, a natural arch form for the lingual straight wire that can be used as a guide, along with other criteria, for the personalized setup; this is fundamentally important in the lingual technique.42-44 Adults were therefore selected so that the influence of growth on the arch form was minimized, although not entirely excluded, since the literature suggests that alterations can also occur in adulthood.15 In the literature, there is great diversity in the reference points that authors used to evaluate and describe the size and shape of the arches; most previous studies used conventional anatomic reference points such as the incisal margin for the anterior teeth and the cusps of the canines, molars, and premolars,6,12,34,45,46 whereas others chose the contact points,47 the edge of the alveolar bone,1 the mesiodistal diameters of the teeth,1,2 and cranial structures.2 Despite their anatomic significance, however, these points cannot provide a clinical representation that could feasibly be used as an arch form guide. In contrast, reference points on the surface of the teeth (FA points) are a direct representation of the clinical arch form that can serve as a template for the manufacture of archwires for orthodontic treatment.1-3,9,26,30,31 As in studies analyzing arch forms from a labial perspective, the reference points in this study were selected according to Takemoto and Scuzzo,36 so that the FA points could provide a direct clinical representation of the shape of the lingual side of the arch.3,9,26 After the reference points were identified, we evaluated the forms and dimensions of the lingual side of the arch according to a method previously described by Trivin˜o et al2; likewise, an article by Raberin et al6 was used as a reference for the dimensional evaluation. Not surprisingly, the results of descriptive analysis of the mean values relative to the width and depth of the arches were lower that those regarding the labial side reported by Raberin et al6 and other authors13,19 because the reference points considered were on the lingual surfaces of the arches, which necessarily have a smaller circumference. Thus, a direct comparison with other studies could not be made. The results of this study of possible sexual dimorphism in the diameters between the intercanine, interfirst molar, and inter-second molar diameters, and the depth of the arch on the lingual side showed no statistical differences between the medians in men and women in either arch. These results contrast with previous reports that document evident sex dimorphism in
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Fig 3. Three curves (not mirrored): A, mandibular arch; B, maxillary arch. Large, medium, and small are superimposed on the scatter plot. In the x-axis is reported the range of variation of values of the x coordinates, and in the y-axis is reported the range of variation of values of the y coordinates.
dimension (men have wider arches than women, with more marked differences generally seen in the interfirst molar than the intercanine and inter-second molar diameters).6,12,26,31 This difference is probably due to the choice of reference points on the lingual sides of the teeth, which excluded the dimensional differences in the buccolingual diameter of the molars that presumably exist between the sexes, and have been
documented in anthropologic studies on the shape of the arch by Ferrario et al.11,12 To obtain an accurate graphic representation of the shape of the lingual arch, in this study, the curves that interpolated the measure of the central tendency as precisely as possible were used. By using Curve Expert software, the median values of the x and y coordinates of our sample were interpolated with various
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0 -5
LARGE MEDIUM SMALL
-10 -15
Y
-20 -25 -30 -35 -40 -45 -25
-20
-15
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-5
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20
25
X
50
40
Y
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20
LARGE MEDIUM SMALL
10
0 -20
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0
10
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X
Fig 4. Central tendencies relative to the values (mirrored): A, mandibular arch; B, maxillary arch. In the x-axis is reported the range of variations of values of the x coordinates, and in the y-axis is reported the range of variations of values of the y coordinates.
polynomial curves, from the fourth to sixth degrees, as described in the literature.2,12,30-35 The most appropriate curve was first determined by visual means, taking into account the regularity of the shape of the curve and its proximity to the values of the median of the coordinates.2 Subsequently, to confirm this visual assessment objectively, it was determined which markers to consider to establish the goodnessof-fit of our curve. Residual analysis permitted us to verify whether the observed values fell outside the expected curve. The measurement error was calculated
by using the standard error method in the Curve Expert software manual. The polynomial function characterized by the lowest standard error was that of the ninth degree, and, even though it had not previously been described in the literature, it was adopted to represent the shape of the arches from the lingual side. The anterior portion of this curve is considerably flattened in correspondence to the frontal group and fairly straight toward the posterior; it is reminiscent of the ‘‘flat form’’ of Raberin et al6 and the ‘‘Form B’’ of Trivin˜o et al2 to describe the labial
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Fig 5. Mandibular and maxillary lingual dental arch forms for natural, normal occlusion in small, medium, and large sizes.
Table V.
1X 1Y 2X 2Y 3X 3Y 4X 4Y 5X 5Y 6X 6Y 7X 7Y
Mann-Whitney nonparametric U test used to evaluate differences in shape between the arches Median for the mandible
Median for the maxilla
U
P value
N valid lower
N valid upper
19* 4.5* 16* 14* 14.25* 23* 13* 29 10* 35 7* 38* 2* 39*
21* 4* 18* 13* 16* 22* 14* 29 13* 36 10* 40* 4* 42*
3817* 5183* 2463.5* 3552.5* 2136.5* 4968* 2358* 6061 420* 6553 108.5* 3853.5* 750.5* 3165.5*
0 0.0025 0 0 0 0.0006 0 0.1923 0 0.7328 0 0 0 0
116* 116* 116* 116* 116* 116* 116* 116 116* 116 116* 116* 116* 116*
116* 116* 116* 116* 116* 116* 116* 116 116* 116 116* 116* 116* 116*
*Significant at P \0.05.
side of the arch. This shape is probably attributable to the choice of reference points on the lingual surfaces, which annul the difference in buccolingual diameter between the teeth in the anterior and posterior sections. Three groups, small, medium, and large, were created; the 3 sizes allowed more accurate individualization of forms and dimensions, since our objective was to describe a natural and anatomic arch form from subjects with normal occlusion that, we hope, can be used,
with other criteria, in the construction of personalized setups for the lingual straight-wire technique, thus reducing errors in the selection of the best initial dental arch form for a patient. To this end, a distributive analysis of the x and y values for each tooth was performed to determine the measure of the central tendency most appropriate for our data. The aim of our research was to identify 3 curves, corresponding to the 3 groups (small, medium, and large) that could interpolate the
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Table VI.
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American Journal of Orthodontics and Dentofacial Orthopedics September 2010
Mann-Whitney nonparametric U test used to evaluate differences in shape between the arches of the sexes Median for the mandible
Median for the maxilla
U
P value
N valid women
N valid men
20 4 17 14 15.75 22 13 29 12 35 8 39 3 40.5
20 4 17 14 15 23 13 29 12 36 8 39.5 3 42
5840 5893.5 5753 5661.5 5770 5617.5 5913.5 5419.5 5907.5 5561.5 5885 5584 6026.5 5584.5
0.6220 0.7019 0.5016 0.3896 0.5241 0.3417 0.7326 0.1744 0.7233 0.2864 0.6889 0.3078 0.9131 0.3083
152 152 152 152 152 152 152 152 152 152 152 152 152 152
80 80 80 80 80 80 80 80 80 80 80 80 80 80
1X 1Y 2X 2Y 3X 3Y 4X 4Y 5X 5Y 6X 6Y 7X 7Y
measures of the central tendencies to represent our sample. Because of many anomalous values and the variability that characterized each tooth, it was decided arbitrarily to consider the coordinates characterized by minimum variability (x7, y1); we identified a maximum deviation around the median equal to 1 mm in both arches. When we analyzed the differences in shape, we considered the values of the medians of our coordinates, and no significant differences between the arches in either sex were found. This finding agrees with those of Ferrario et al,11 who found no sexual dimorphism between the arches when evaluated via Euclidean distance matrix analysis, and Camporesi et al,26 who reached the same conclusions using thin-plate spline analysis, a morphometric system. The samples examined in these 2 studies, like ours, included subjects with natural ideal occlusion from southern Europe, and thus the studies are comparable. Other investigators found significant morphologic differences in the shape of the labial arch among different ethnic groups; this is a possible topic for future research into the shape of the dental arch from the lingual side.4,48 Our results should be considered specific only for European populations in the Mediterranean area. CONCLUSIONS
We analyzed the shape of the lingual sides of the maxillary and mandibular dental arches in a sample of adults from southern Europe with natural ideal occlusions. Analysis of the shape of the arches calculated from the lingual side indicated the following. 1.
There is no sexual dimorphism for the dimensions of the lingual diameter in the intercanine, interfirst molar, and inter-second molar measurements.
2. 3.
4.
There are no significant differences in shape between the arches or between the sexes. Three lingual curves, small, medium, and large, for the 2 arches, representing the mean values of our sample, were developed. The maxillary curves coordinate with the mandibular ones and can be used as a guide for the setup in the lingual straight-wire technique.
REFERENCES 1. Ronay V, Miner MR, Will LA, Arai K. Mandibular arch form: the relationship between dental and basal anatomy. Am J Orthod Dentofacial Orthop 2008;134:430-8. 2. Trivin˜o T, Siqueira DF, Scanavini MA. A new concept of mandibular dental arch forms with normal occlusion. Am J Orthod Dentofacial Orthop 2008;133:10.e15-22. 3. Fujita K, Takada K, QianRong G, Shibata T. Patterning of human dental arch wire blanks using a vector quantization algorithm. Angle Orthod 2002;72:285-94. 4. Nojima K, McLaughlin RP, Isshiki Y, Sinclair PM. A comparative study of Caucasian and Japanese mandibular clinical arch forms. Angle Orthod 2001;71:195-200. 5. Lee RT. Arch width and form: a review. Am J Orthod Dentofacial Orthop 1999;115:305-13. 6. Raberin M, Laumon B, Martin JL, Brunner F. Dimensions and form of dental arches in subjects with normal occlusions. Am J Orthod Dentofacial Orthop 1993;104:67-72. 7. de Rysky S, Collesano V, Avril C, Slavicek R. Clinical functional analysis. Riv Ital Stomatol 1983;52(2):111-24. ˚ rtun J, Shapiro PA. 8. De La Cruz AR, Sampson P, Little RM, A Long-term change in arch form after orthodontic treatment and retention. Am J Orthod Dentofacial Orthop 1995;107: 518-30. 9. Andrews LF. Composites. I: straight-wire, the concept and appliance. San Diego, Calif: L.A. Wells; 1989. 10. Brader AC. Dental arch form related with intraoral forces: PR 5 C. Am J Orthod 1972;61:541-61. 11. Ferrario VF, Sforza C, Miani A Jr, Tartaglia G. Human dental arch shape evaluated by Euclidean-distance matrix analysis. Am J Phys Anthropol 1993;90:445-53.
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12. Ferrario VF, Sforza C, Miani A Jr, Tartaglia G. Mathematical definition of the shape of dental arches in human permanent healthy dentitions. Eur J Orthod 1994;16:287-94. 13. Ferrario VF, Sforza C, Poggio CE, Serrao G, Colombo A. Threedimensional dental arch curvature in human adolescents and adults. Am J Orthod Dentofacial Orthop 1999;115:401-5. 14. Ferrario VF, Sforza C, Colombo A, Ciusa V, Serrao G. Threedimensional inclination of the dental axes in healthy permanent dentitions—a cross-sectional study in a normal population. Angle Orthod 2000;71:257-64. 15. Harris EF. A longitudinal study of arch size and form in untreated adults. Am J Orthod Dentofacial Orthop 1997;111:419-27. 16. Carter GA, McNamara JA Jr. Longitudinal dental arch changes in adults. Am J Orthod Dentofacial Orthop 1998;114:88-99. 17. Ricketts RM. Research in factors of appliance design and arch form. Foundation for Orthodontic Research 1979;9:45-57. 18. Felton MJ, Sinclair PM, Jones DL, Alexander RG. A computerized analysis of the shape and stability of mandibular arch form. Am J Orthod Dentofacial Orthop 1987;92:478-83. 19. Braun S, Hnat WP, Fender DE, Legan HL. The form of the dental arch. Angle Orthod 1998;68:29-36. 20. Noroozi H, Nik TH, Saeeda R. The dental arch form revisited. Angle Orthod 2001;71:386-9. 21. McLaughlin RP, Bennett JC. Considerazioni sulla forma d’arcata per ottenere stabilita` ed estetica. Ortognatodonzia Italiana 2001; 10:217-35. 22. BeGole EA, Fox DL, Sadowsky C. Analysis of change in arch form with premolar expansion. J Dent Res 1998;113:307-15. 23. Poggio CE, Mancini E, Salvato A. Valutazione degli effetti sulla forma d’arcata della terapia fissa e della recidiva mediante thin-plate spline analysis. Ortognatodonzia Italiana 2000;9:345-50. 24. Mutinelli S, Manfredi M, Cozzani M. A mathematic-geometric model to calculate variation in mandibular arch form. Eur J Orthod 2000;22:113-25. 25. Mutinelli S, Cozzani M, Manfredi M, Siciliani G. Dental arch analysis system. Prog Orthod 2004;5:200-8. 26. Camporesi M, Franchi L, Baccetti T, Antonini A. Thin-plate spline analysis of arch form in a southern European population with an ideal natural occlusion. Eur J Orthod 2006;28:135-40. 27. Lele S, Richtsmeier JT. Euclidean distance matrix analysis: a coordinate-free approach for comparing biological shapes using landmark data. Am J Phys Anthropol 1991;86:415-27. 28. McAlarnay ME, Chiu WH. Comparison of numeric techniques in the analysis of cleft palatal dental arch form change. Cleft Palate Craniofac J 1997;34:281-91. 29. McIntyre GT, Mossey PA. Size and shape measurement in contemporary cephalometrics. Eur J Orthod 2003;25:231-42. 30. Trivin˜o T, Vilella OV. Form and dimensions of the lower dental arch. Rev Soc Bras Orthod 2005;5:19-28.
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31. Kageyama T, Domı`nguez-Rodrı`guez GC, Vigorito JW, Deguchi T. A morphological study of the relationship between arch dimensions and craniofacial structures in adolescents with Class II Division 1 malocclusions and various facial types. Am J Orthod Dentofacial Orthop 2006;129:368-75. 32. Lu KH. An orthogonal analysis of the form, symmetry and asymmetry of the dental arch. Arch Oral Biol 1966;11:1057-69. 33. Pepe SH. Polynomial and catenary curve fits to human dental arches. J Dent Res 1975;54:1124-33. 34. Wakabayashi K, Sohmura T, Takahashi J, Kojima T, Akao T, Nakamura T. Development of the computerized dental cast form analyzing system—three dimensional diagnosis of dental arch form and the investigation of measuring condition. Dent Mater J 1997;16:180-90. 35. Ferrario VF, Sforza C, Dellavia C, Colombo A, Ferrari RP. Threedimensional hard tissue palatal size and shape: a 10-year longitudinal evaluation in healthy adults. Int J Adult Orthod Orthognath Surg 2002;17:51-8. 36. Takemoto K, Scuzzo G. The straight-wire concept in lingual orthodontics. J Clin Orthod 2001;35:46-52. 37. Fujita K. New orthodontic treatment with lingual bracket and mushroom archwire appliance. Am J Orthod 1979;76:657-75. 38. Little RM. The irregularity index: a quantitative score of mandibular anterior alignment. Am J Orthod 1975;68:554-63. 39. Robinson DL, Blackwell PG, Stillman EC, Brook AH. Impact of landmark reliability on the planar Procustes analysis of tooth shape. Arch Oral Biol 2002;47:545-54. 40. Dahlberg G. Statistical methods for medical and biological students. London: George Allen and Unwin; 1940.p.12-22. 41. Flowler J, Jarvis P, Chevannes M. Statistica per le professioni sanitarie. 2006, EdiSES S.r.l.-Napoli. 42. Siciliani G, Terranova S. Chapter 8. In: Masson, editor. Ortodonzia linguale. Milan, Italy: Vivendi Universal Publishing; 2001. 43. Scuzzo G, Takemoto K. Invisible orthodontics: current concepts and solutions in lingual orthodontics. Berlin, Germany: Quintessenz Verlags; 2003. 44. Fillion D, Buso-Frost L. An overall view of the different laboratory procedures used in conjunction with lingual orthodontics. Semin Orthod 2006;12:203-10. 45. Taner T, Ciger S, El H, Germec¸ D, Es A. Evaluation of dental arch width and form changes after orthodontic treatment and retention with a new computerized method. Am J Orthod Dentofacial Orthop 2004;126:464-76. 46. Nie Q, Lin J. A comparison of dental arch forms between Class II Division 1 and normal occlusion assessed by Euclidean distance matrix analysis. Am J Orthod Dentofacial Orthop 2006;129:528-35. 47. Savostin-Asling I. The geometric analysis of mandibular dental arch form. Ann Dent 1980;39:3-11. 48. Burris BG, Harris FH. Maxillary arch size and shape in American blacks and whites. Angle Orthod 2000;70:297-302.
A new concept of anatomic lingual arch form Luca Lombardo,a Luca Saba,b Giuseppe Scuzzo,c Kyoto Takemoto,d Lola Oteo,e Juan Carlos Palma,f and Giuseppe Sicilianig Ferrera, Italy, Tokyo, Japan, and Madrid, Spain Introduction: The aim of this study was to describe a natural and anatomic lingual arch form obtained from subjects with normal occlusion that could be used, with other criteria, in the construction of personalized setups for the lingual straight-wire technique. Methods: The study sample comprised 58 pairs of dental casts of the arches of 58 southern Europeans (37 women, 21 men) with ideal natural occlusions. After the reference points of the dental arches were identified and marked, the dental casts were scanned. The exact position of the models on the scanner was established by using an acetate sheet with a Cartesian reference system. For each image, 14 reference points (x, y) were measured and recorded. The measurements were processed with software to select the polynomial function that best described the shape of the dental arches. The ninthdegree polynomial function was selected to represent the lingual arch form of both arches. Distribution analysis of the x and y values of each tooth in each arch resulted in the creation of 3 groups (small, medium, and large) to verify the most appropriate measures of the central tendencies of our data. Results: Statistical analysis showed no significant sex difference in the medians of the 6 parameters used to measure depth and width in both arches. A representation of the variability of the lingual curve of our sample was created to document at least 3 sizes of the representative curve of the central tendency for our data. No statistically significant differences in shape were found between men and women, considering the medians as a measure of the central tendencies. Conclusions: Three lingual curves (small, medium, and large) for the maxillary and mandibular arches, representing the mean values of our sample, were developed and can be used as guides for the setup in the lingual straight-wire technique. (Am J Orthod Dentofacial Orthop 2010;138:260.e1-260.e13)
T
he form of the dental arch is considered a fundamental element in orthodontic diagnosis and treatment planning because of its ability to influence not only available space and dental and smile esthetics, but also long-term occlusal stability.1-7 Furthermore, respecting this important parameter reduces the possibility of crowding relapse and periodontal damage.5,8 Moreover, definition of the shape of the arch aids the clinician in achieving results that agree with the natural laws of biologic a
Research assistant, Department of Orthodontics, University of Ferrara, Ferrara, Italy. b Resident, Department of Orthodontics, University of Ferrara, Ferrara, Italy. c Adjunct Professor, Department of Orthodontics, University of Ferrara, Ferrara, Italy. d Adjunct Professor, Department of Orthodontics, University of Ferrara, Ferrara, Italy. e Professor, Department of Orthodontics, University Complutense, Madrid, Spain. f Adjunct Professor, Department of Orthodontics, University Complutense, Madrid, Spain. g Chairman, Department of Orthodontics, University of Ferrara, Ferrara, Italy. The authors report no commercial, financial, or proprietary interest in the products or companies described in this article. Reprint requests to: Luca Lombardo, Contrada Nicolizia, 92100 Licata, AG, Italy; e-mail, [email protected]. Submitted, October 2009; revised and accepted, April 2010. 0889-5406/$36.00 Copyright Ó 2010 by the American Association of Orthodontists. doi:10.1016/j.ajodo.2010.04.022
variability.1,5 Andrews,9 father of the labial straightwire technique, cited the arch form as the seventh of his keys to achieving Class I occlusion. Hence, in recent decades, numerous studies have analyzed the arch form from anatomic and anthropologic perspectives10-16 to evaluate its implications in orthodontic therapy17-21 or its modifications after orthodontic treatment.18,22-25 Even though the necessity of individualizing each patient’s arch form during treatment is widely recognized, after the evolution of labial straight-wire appliances and because of the superelastic properties of the latest generation of archwires, it seems to be clinically reasonable to stereotype the constructed arch shapes from subjects with normal occlusion into a few sets of preformed arches.1,5 The most similar to that of the patient before treatment could therefore be selected on the basis of geometric similarity, ethnicity, and type of malocclusion.1,3-5 Nevertheless, no objective study has thus far elucidated the choice of a particular arch form,3 although various studies have sought, by using the incisal and occlusal margins of the teeth as anatomic reference points, to represent it graphically from the labial side.1-6,24,25 Numerous authors used geometric curves to describe the dental arch.24-35 Moreover, various authors have observed that polynomial functions can simply and symmetrically describe the shape of the dental arch, and have exploited these 260.e1
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mathematic equations as an accurate method for its description.2,12,24-35 Many articles in the literature have dealt with the shape of the dental arch from the labial perspective, although nothing has yet been published regarding the lingual side. Many systems for representing the normal arch form have been proposed, including qualitative descriptions, geometric constructions, and mathematic curves; although there is consensus on the importance of this factor in orthodontics, a definitive geometric shape is elusive, as demonstrated recently by Trivin˜o et al.2 However, the introduction of the straight-wire to the lingual technique36 and the possibility of replacing Fujita’s ‘‘mushroom form’’37 with an arch form without insets has led to the question of which arch form to choose and which criteria to use in the lingual straight-wire setup and treatment planning. Because a ‘‘straight’’ arch form has not previously been described from a lingual perspective and this is fundamentally important in orthodontic treatment planning, in this study it is described with an objective, standardized, and reproducible methodology: a natural and anatomic arch form obtained from subjects with normal occlusion. This can be used, with other criteria, in the construction of personalized setups for the lingual straight-wire technique. MATERIAL AND METHODS
The sample was carefully selected from the records of the University of Ferrara Postgraduate School of Orthodontics in Italy and the Department of Orthodontics at the Complutense University of Madrid, Spain. It included southern European white adults with the following characteristics2,26: (1) age not less than 18 years (range, 19.08-70.25 years); (2) no previous orthodontic treatment; (3) regular arch form with little or no crowding; (4) complete dentition up to at least the second molars; (5) no extensive restoration and implants; (6) at least 4 of the 6 Andrews’ keys9 for optimal occlusion and bilateral Class I molar and canine relationship; (7) normal (within 2 6 1 mm) overbite and overjet; (8) no deviations of the interincisal lines; (9) no gingival recession; (10) no muscular or joint pathologies; (11) no ectopic teeth, tooth aplasia, or anomalies in tooth shape; (12) no supernumerary or congenitally missing teeth; (13) no anterior or posterior crossbite; (14) no visible intraoral or extraoral asymmetry; and (15) minimal diastemas, premolar rotations, and incisor irregularities (present in several subjects). Each subject participating in the study gave informed written consent.
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Mandibular and maxillary dental casts of each subject, according to Tweed prescriptions, were obtained, for a total of 75 pairs. The regularity of these models was first determined clinically and then quantified by using Little’s irregularity index38; 9 pairs of models had a value greater than 3 and were therefore excluded, as were a further 8 pairs, which, although having a value less than 3, had spaces greater than 1 mm or a slightly asymmetrical contraction of the arch in the premolar area. Thus, the definitive sample comprised 58 pairs of dental casts of the arches of 58 white subjects (37 women, 21 men) of southern European ancestry (mainly Italian and Spanish). The mean age of the sample was 29.21 years (SD, 8.73 years); the youngest was 19.08, and the oldest was 70.25 years of age. Reference points for calculation of the shape of the arch were identified and marked on the lingual surface of each tooth (from the left second molar to the right second molar) of each dental cast by 1 operator (L.S.) using an indelible marker (.08-mm Pilot drawing pen; Pilot Corp., Tokyo, Japan). In the mandibular arch, these points, which trace the lingual straight plane, were situated at the center of the clinical crown (vertical position) along the central lingual axis and at the most prominent point on the lingual surface of each tooth (horizontal position) on the molars and premolars and corresponding to the middle third of the anterior teeth. In the maxillary arch, the reference points were marked at the most prominent point on the lingual surface of each tooth (horizontal position) at the intersection between the middle third and the gingival third of the anterior teeth, along their central axes, and at the center of the clinical crown of the posterior teeth. Before permanently marking these points on the lingual surface, the median axis of the clinical crown of each tooth was traced in pencil to determine its horizontal position, and a gauge was used to measure the height of the clinical crown to determine its vertical position. Although the same operator carried out this procedure for each tooth, it was an important source of error, because the accuracy of identification of the reference points necessarily influences the tracing of the curves on which the calculations of arch shape are based.33 After the reference points had been marked, the dental casts were scanned with an Expression 1680 Pro scanner (Epson, Cinisello, Balsamo, Italy), and images in TIF format at a resolution of 300 dpi were obtained. To avoid distortion of the image, and hence the reference points, during the scanning procedure, the models were positioned so that both the base and the occlusal plane were parallel to the scanner’s surface.39 The exact position of the models on the scanner was established with a specially created acetate sheet onto which a sheet of
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Fig 1. Cartesian system and reference points (black) used to establish lingual dental arch form; each point is described by a copy of x and y coordinates: x measurements, distance between the reference point and the y-axis; y measurements, distance between the reference point and the x-axis.
millimeter paper had been photocopied. After the photocopies were made, the prepared acetate and millimeter papers were superimposed to verify the lack of distortion in the acetate copy. On this acetate sheet, reference axes—a horizontal line and a perpendicular vertical line—were drawn with an indelible marker (.05-mm Pilot drawing pen). The sheet was subsequently placed on the glass surface of the scanner under the plaster models to align the posterior margin of the second molars with the abscissa and the median line with the ordinate, thus creating a Cartesian reference system (Fig 1).2 For each image, 14 reference points, corresponding to the distance between each reference point and the abscissa and ordinate were measured and recorded to provide their Cartesian coordinates (x, y). Four weeks later, to calculate the method error, 15 models were selected at random, and the coordinates of each reference point on both arches were replotted by the same operator as above. Analysis of the reliability of measurements was performed by using DahlP berg’s coefficient,40 Se 5 O d2/2n, where S is the Dahlberg coefficient, d is the difference between the first and second measurements, and n is the number of repeated measurements. The values corresponding to the coordinates of the central and lateral incisors and the second premolars were compared singly with their control measurements. The statistical significance was evaluated by using the Student t test for paired data. The systematic error of measurement of each value considered was shown to be not significant at P 5 0.05, thereby confirming the reliability of the results. First, the dimensions of both dental arches were evaluated via 3 transversal and 3 sagittal measurements: the transverse diameters and arch depths at the canines, first molars, and second molars, as described by Raberin et al.6 The lingual reference points considered were (1) the interincisor point; (2) the most prominent point on the central axis of the lingual surface of the canine
crown; (3) the most prominent point on the lingual surface of the first molar at the center of its clinical crown; and (4) the most prominent point on the lingual surface of the second molar at the center of its clinical crown. Thus, 6 measurements of distance were taken for each arch to analyze their width and depth. To measure arch width, the following were considered: intercanine diameter (D3), intermolar diameter at the first molars (D6), and intermolar diameter at the second molars (D7). To measure the arch length, the following were considered: (1) canine depth, the distance between the interincisor point and the line connecting the canine reference points (L3); (2) molar depth at the first molars, the distance between the interincisor point and the line connecting the reference points on the first molars (L6); and (3) molar depth at the second molars, the distance between the interincisor point and the line connecting the reference points on the second molars (L7). Subsequently, the shape of the dental arch was evaluated from the lingual side. The measurements (x and y coordinates) of each reference point were all obtained and processed by using Curve Expert software (version 1.3, http://curveexpert.webhop.biz) by the same operator. This software was used to select the polynomial function that best described the shape of the dental arches. After the dental arches were viewed and the presence of slight asymmetry was established, even though patients with normal occlusion had been used as models, the reference point coordinates of the 58 dental casts were divided into the left and right sides. Each arch half, described by 7 reference points (7 x and y coordinates), was then ‘‘mirrored’’ to obtain 116 whole maxillary and 116 whole mandibular arches with symmetrical curves.2 Based on the number of reference points (7 for each semiarch mirrored) and after the graphs obtained of the various mathematical functions were viewed, the ninthdegree polynomial function was selected to represent the lingual surfaces of both arches.
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Table I.
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Descriptive statistics of the 6 parameters used to measure dental arch depth and width by arch and sex
Mandibles in men D3-3 D6-6 D7-7 L3 L6 L7 Mandibles in women D3-3 D6-6 D7-7 L3 L6 L7 Maxillae in men D3-3 D6-6 D7-7 L3 L6 L7 Maxillae in women D3-3 D6-6 D7-7 L3 L6 L7
Maximum
Percentile 25
18 28 30 2 22 31
26 42 48 6 30 42
20 30 36 4 24 34
22 34 40 5 26 37
1.928 3.034 3.783 0.899 1.914 2.472
20 34 40 4 25 35
18 30 34 2.5 21 30
26 40 48 8 31 43
20 32 38 4 24 33
22 36 42 5 26 36
1.826 2.399 3.153 0.871 1.960 2.647
27.1 36.7 42.2 6.9 29.3 38.5
26 36 42 7 30 39
24 30 34 4 22.5 32
34 50 52 9 41 48
26 34 40 6 28 37
28 38 44 8 30.5 40
2.510 3.502 3.769 1.173 2.870 2.987
26.6 36.9 42.4 6.6 28.9 42.1
26 36 42 7 29 39
22 29 34 5 25 33
34 46 50 9 34 41
24 34 40 6 27 36
28 38 46 7 30 40
2.447 3.367 4.028 1.035 2.193 3.328
Mean
Median
21.2 32.7 38.5 4.1 25.2 35.3
21 32 38 4 25 355
21.0 33.4 39.6 4.1 25.2 35.1
Minimum
Percentile 75
SD
D3-3, Intercanine diameter; D6-6, intermolar diameter at the first molars; D7-7, intermolar diameter at the second molars; L3, canine depth; L6, depth at the first molars; L7, depth at the second molars.
This polynomial function yielded the curve most representative of the shape of the lingual dental arch. It was first selected according to visual inspection criteria, and then, to confirm this choice objectively, we determined which marker could be considered to establish the suitability of the model.2,6 With the residual analysis, we ascertained whether the observed values fell outside the trend of the expected curve. The residual plot graphically depicts the difference between the data points and the model evaluated at the data points. The residual at point i is defined by residual 5 yi – f(xi), where yi is the measured value at xi, and f(xi) is the predicted value at xi. These distances are shown as bars or points on the residual plot; the magnitudes of the data points are simply replaced by the residual defined above. If the residual is positive, then the data point is above the model’s prediction; if the residual is negative, then the data point is below the model’s prediction. The residuals can provide an indication of a particular model’s performance. If there are runs of like-signed residuals, then a better model for the data is likely to exist.
Optimally, the residuals should show a random scatter around zero, which indicates that the data points are randomly distributed around the curve. A ‘‘run’’ is a sequence of like-signed residuals, which stand out on the residual plot. A large number of runs indicates that the data systematically deviate from the curve. The critical reference measurement for a residual value is usually 1.96. If this value is exceeded at 1 point or more, the fit is unacceptable, and adaptations must be made. The measurement error was compared with the standard error values of various models to ascertain which model better interpolated the data—ie, that characterized by the lowest standard error was considered the best; in our case, it was the ninth-degree polynomial. The standard error was calculated with minimization of the so-called merit function (http://mathworld. wolfram.com/MeritFunction.html). Statistical analysis
The sample was initially subjected to descriptive statistical analysis including mean, standard deviation, variance, minimum, and maximum values.
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Table II. Results of the Mann-Whitney nonparametric U test used to evaluate the sex differences of the 6 parameters used to measure dental arch depth and width Variable Mandible D3-3 D6-6 D7-7 L3 L6 L7 Maxilla D3-3 D6-6 D7-7 L3 L6 L7
Rank sum for women
Rank sum for men
U
P value
N valid women
N valid men
4273.5 4584 4598 4156 4272 4163
2512.5 2202 2188 2630 2514 2623
1498.5 1299 1285 1381 1497 1388
0.752040 0.143744 0.122974 0.321717 0.745508 0.341744
74 74 74 74 74 74
42 42 42 42 42 42
4158.5 4387 4401.5 4083 4132.5 4146
2627.5 2399 2384.5 2703 2653.5 2640
1383.5 1496 1481.5 1308 1357.5 1371
0.328780 0.741164 0.679161 0.158455 0.260194 0.294463
74 74 74 74 74 74
42 42 42 42 42 42
D3-3, Intercanine diameter; D6-6, intermolar diameter at the first molars; D7-7, intermolar diameter at the second molars; L3, canine depth; L6, depth at the first molars; L7, depth at the second molars. Significance level, P \0.05.
The analysis of sexual dimorphism was carried out with a comparative analysis of the lingual distances D3-3, D6-6, D7-7, L3, L6, and L7 to verify whether there were statistically significant differences between the measures of the central trend in the men and women. Due to the abnormal distribution of data, the nonparametric Mann-Whitney U test was applied to compare the 2 independent groups.41 To provide a visible representation of a hypothetical ideal arch form curve, first the means and standard deviations of the 14 coordinates of the abscissa and ordinate were calculated for all curves relative to the lingual sides of both arches. Three groups—small, medium, and large—were created by a distribution analysis of the x and y values of each tooth in each arch to verify the most appropriate measure of the central tendencies for our data. It was verified that all x and y values were characterized by strongly nonnormal, asymmetrical, and unimodal distributions. This led us to choose the median value as the reference measure of the central tendency, and then the values of the 25th (first quartile) and 75th (third quartile) percentiles as measures of variability. Assuming that the median was an adequate representation of the measure of the central tendencies for the medium group, the measures of the central tendency for the small and large groups were identified. The objective was to identify 3 curves, corresponding to the 3 groups, that interpolated the measures of central tendency in a parallel fashion. Because of the many anomalous values and the variability that characterized each tooth, it was not possible to
consider the first and third quartiles as measures of the central tendency for each tooth in the small and large groups. Hence, the coordinates characterized by the minimum variability (x7, y1) were considered, and a maximum deviation around the median equal to 1 mm in both arches was identified. Thus the following measures were obtained. Central tendency for x7 small 5 median (x7) – 1. Central tendency for x7 large 5 median (x7) – 1. Central tendency for y1 small 5 median (y1) – 1. Central tendency for y1 large 5 median (y1) – 1. Thus, the central tendency of the small and large groups for the other coordinates were obtained, applying the 1-mm displacement to the median of all x and y values. In this way, constant ‘‘distances’’ along all points of the mouth between the central tendency measures of the medium group and those of the small and large groups were created. RESULTS
The descriptive statistics of each of the 6 parameters used to measure depth and width, subdivided by arch and sex, are reported in Table I. The values for mean, median, minimum, maximum, 25th percentile, 75th percentile, and standard deviation are also reported. The results of the Mann-Whitney nonparametric U test are reported in Table II. The P values show no significant statistical sex difference for the medians measured in both arches.
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Table III.
Descriptive statistics of the values of the coordinates of the reference points for the 2 arches Minimum
Mandible 1X 2X 3X 4X 5X 6X 7X* 1Y* 2Y 3Y 4Y 5Y 6Y 7Y Maxilla 1X 2X 3X 4X 5X 6X 7X* 1Y* 2Y 3Y 4Y 5Y 6Y 7Y
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Percentile 25
Median
Percentile 75
Maximum
Mean
24 21 18 16 13 8 4 7 17 28 36 44 46 47
20.5 17 15 13 11 7 3 5 15 24 30 37 40 41
19 16 14.25 13 10 7 2 4.5 14 23 29 35 38 39
19 16 14 12 10 6 2 4 13.5 22 28 34 36 37.5
15 14 12 11 9 5.5 2 3 11 19 24.5 30 33 34
19.67 16.58 14.62 12.78 10.53 6.84 2.45 4.40 14.31 23.01 29.19 35.42 38.15 39.56
26 25 22 20 17 12 5 3 10 18 24 27 35 37
22.75 19 17.25 15 14 10 4 4 13 21 27 34 38 40
21 18 16 14 13 10 4 4 13 22 29 35.5 40 42
20 17 15.75 13.5 13 9 3.5 3.5 14 23 30 37 42 44
17 14.5 13.5 12 11 7 2 2 17 29 35 44 50 52
21.12 18.46 16.55 14.33 13.4 9.65 3.85 4.09 13.21 22.25 28.79 34.89 40.25 42.32
*These coordinates varied the least.
Initially, a simple graphic representation of the lingual side of the arch form was obtained by using the means and standard deviations of the coordinates for the maxillary and mandibular arches, respectively. Subsequently, a representation of the variability of the lingual curve of our sample was created to document at least 3 sizes of the representative curve. Table III documents the descriptive statistics of the 14 coordinates corresponding to the reference points used to describe the curves that best represent the shapes of the arches, respectively. Figure 2 depicts the data in Table III; it is possible to observe the difference in variation of the coordinates. Thus, the x and y coordinates that varied the least were considered, as shown in Table III. After identifying 1 mm as the constant measure to be subtracted and added to the median measures, the central tendencies relative to the small and large groups were obtained, as described in Table IV. Figure 3 shows the 3 curves, large, medium, and small, superimposed onto the scatter plot. Figure 4 displays the values of all coordinates relative to both arches for each group (large, medium, and small) and the corresponding curves. The values reported in
Table IV were determined by the Curve Expert software to visualize the ninth-degree polynomial curve that interpolates the points relative to the large, medium, and small groups, respectively. Figure 5 illustrates the 3 representative curves describing the arch forms from the lingual side in the mandibles and the maxillae of our sample. At this point, the presence or absence of differences in shape between the arches and between the sexes was verified, considering the medians as a measure of central tendencies. The Mann-Whitney U test was used to compare the central tendencies (medians) of the 2 arches (Table V) and the 2 sexes (Table VI). Table V shows the significant differences between the medians of the 2 arches. For instance, the median of x1 in the maxillary arch (21) was significantly greater (P \0.000) than the corresponding measurement in the mandibular arch (19). Likewise, the median of y1 in the maxillary arch (4) was significantly lower (P 5 0.002) than that measured in the mandibular arch (4.5), and so on. No statistically significant differences were found between the central tendencies of the sexes as shown in Table VI.
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Line Plot of multiple variable Descriptive.sta 6v*14c 0
-10
-20
-30 Minimum Percentile 25% Median Percentile 75% Maximum Mean
-40
-50 X1
X2
X3
X4
X5
X6
X7
Y1
Y2
Y3
Y4
Y5
Y6
Y7
4X
5X
6X
7X
1Y
2Y
3Y
4Y
5Y
6Y
7Y
60
50
Minimum Percentile 25% Median Percentile 75% Maximum Mean
40
30
20
10
0
-10
-20
-30 1X
2X
3X
Fig 2. Difference in variation of the measures of central tendency of the coordinates of the reference points: A, mandibular arch; B, maxillary arch. In the x-axis are reported the coordinates (7X and 7Y), and in the y-axis is reported the range of variation in values of the coordinates for the maxillary and mandibular arches (Table I).
DISCUSSION
Many studies in the literature document analyses of the shape of the dental arches, with different methodologies, of similar samples of healthy subjects with normal occlusion to obtain clinical data pertinent to the labial edgewise technique.1-6,10-16 All of these
authors concluded that it was impossible to represent 1 ideal arch form. However, in the literature, no study has reported reference points to describe the dental arch from the lingual perspective. The introduction of straight-wire concepts to the lingual technique has led clinicians to
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Table IV.
Median central tendencies of the 3 groups Large
Mandible 1X 2X 3X 4X 5X 6X 7X 1Y 2Y 3Y 4Y 5Y 6Y 7Y 1X 2X 3X 4X 5X 6X 7X 1Y 2Y 3Y 4Y 5Y 6Y 7Y Maxilla 1X 2X 3X 4X 5X 6X 7X 1Y 2Y 3Y 4Y 5Y 6Y 7Y 1X 2X 3X 4X 5X 6X 7X 1Y 2Y 3Y 4Y 5Y 6Y 7Y L, Left; R, right.
American Journal of Orthodontics and Dentofacial Orthopedics September 2010
Medium
Small
Side
20 17 15.25 14 11 8 3 5.5 15 24 30 36 39 40 20 17 15.25 14 11 8 3 5.5 15 24 30 36 39 40
19 16 14.25 13 10 7 2 4.5 14 23 29 35 38 39 19 16 14.25 13 10 7 2 4.5 14 23 29 35 38 39
18 15 13.25 12 9 6 1 3.5 13 22 28 34 37 38 18 15 13.25 12 9 6 1 3.5 13 22 28 34 37 38
L L L L L L L L L L L L L L R R R R R R R R R R R R R R
22 19 17 15 14 11 5 5 14 23 30 36.5 41 43 22 19 17 15 14 11 5 5 14 23 30 36.5 41 43
21 18 16 14 13 10 4 4 13 22 29 35.5 40 42 21 18 16 14 13 10 4 4 13 22 29 35.5 40 42
20 17 15 13 12 9 3 3 12 21 28 34.5 39 41 20 17 15 13 12 9 3 3 12 21 28 34.5 39 41
L L L L L L L L L L L L L L R R R R R R R R R R R R R R
pose the important questions of which form should be used in setting up indirect bonding and according to which criteria.36 Thus, from our sample of 58 subjects with normal occlusion, we described, with an objective, standardized, and reproducible method, a natural arch form for the lingual straight wire that can be used as a guide, along with other criteria, for the personalized setup; this is fundamentally important in the lingual technique.42-44 Adults were therefore selected so that the influence of growth on the arch form was minimized, although not entirely excluded, since the literature suggests that alterations can also occur in adulthood.15 In the literature, there is great diversity in the reference points that authors used to evaluate and describe the size and shape of the arches; most previous studies used conventional anatomic reference points such as the incisal margin for the anterior teeth and the cusps of the canines, molars, and premolars,6,12,34,45,46 whereas others chose the contact points,47 the edge of the alveolar bone,1 the mesiodistal diameters of the teeth,1,2 and cranial structures.2 Despite their anatomic significance, however, these points cannot provide a clinical representation that could feasibly be used as an arch form guide. In contrast, reference points on the surface of the teeth (FA points) are a direct representation of the clinical arch form that can serve as a template for the manufacture of archwires for orthodontic treatment.1-3,9,26,30,31 As in studies analyzing arch forms from a labial perspective, the reference points in this study were selected according to Takemoto and Scuzzo,36 so that the FA points could provide a direct clinical representation of the shape of the lingual side of the arch.3,9,26 After the reference points were identified, we evaluated the forms and dimensions of the lingual side of the arch according to a method previously described by Trivin˜o et al2; likewise, an article by Raberin et al6 was used as a reference for the dimensional evaluation. Not surprisingly, the results of descriptive analysis of the mean values relative to the width and depth of the arches were lower that those regarding the labial side reported by Raberin et al6 and other authors13,19 because the reference points considered were on the lingual surfaces of the arches, which necessarily have a smaller circumference. Thus, a direct comparison with other studies could not be made. The results of this study of possible sexual dimorphism in the diameters between the intercanine, interfirst molar, and inter-second molar diameters, and the depth of the arch on the lingual side showed no statistical differences between the medians in men and women in either arch. These results contrast with previous reports that document evident sex dimorphism in
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Fig 3. Three curves (not mirrored): A, mandibular arch; B, maxillary arch. Large, medium, and small are superimposed on the scatter plot. In the x-axis is reported the range of variation of values of the x coordinates, and in the y-axis is reported the range of variation of values of the y coordinates.
dimension (men have wider arches than women, with more marked differences generally seen in the interfirst molar than the intercanine and inter-second molar diameters).6,12,26,31 This difference is probably due to the choice of reference points on the lingual sides of the teeth, which excluded the dimensional differences in the buccolingual diameter of the molars that presumably exist between the sexes, and have been
documented in anthropologic studies on the shape of the arch by Ferrario et al.11,12 To obtain an accurate graphic representation of the shape of the lingual arch, in this study, the curves that interpolated the measure of the central tendency as precisely as possible were used. By using Curve Expert software, the median values of the x and y coordinates of our sample were interpolated with various
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0 -5
LARGE MEDIUM SMALL
-10 -15
Y
-20 -25 -30 -35 -40 -45 -25
-20
-15
-10
-5
0
5
10
15
20
25
X
50
40
Y
30
20
LARGE MEDIUM SMALL
10
0 -20
-10
0
10
20
X
Fig 4. Central tendencies relative to the values (mirrored): A, mandibular arch; B, maxillary arch. In the x-axis is reported the range of variations of values of the x coordinates, and in the y-axis is reported the range of variations of values of the y coordinates.
polynomial curves, from the fourth to sixth degrees, as described in the literature.2,12,30-35 The most appropriate curve was first determined by visual means, taking into account the regularity of the shape of the curve and its proximity to the values of the median of the coordinates.2 Subsequently, to confirm this visual assessment objectively, it was determined which markers to consider to establish the goodnessof-fit of our curve. Residual analysis permitted us to verify whether the observed values fell outside the expected curve. The measurement error was calculated
by using the standard error method in the Curve Expert software manual. The polynomial function characterized by the lowest standard error was that of the ninth degree, and, even though it had not previously been described in the literature, it was adopted to represent the shape of the arches from the lingual side. The anterior portion of this curve is considerably flattened in correspondence to the frontal group and fairly straight toward the posterior; it is reminiscent of the ‘‘flat form’’ of Raberin et al6 and the ‘‘Form B’’ of Trivin˜o et al2 to describe the labial
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Fig 5. Mandibular and maxillary lingual dental arch forms for natural, normal occlusion in small, medium, and large sizes.
Table V.
1X 1Y 2X 2Y 3X 3Y 4X 4Y 5X 5Y 6X 6Y 7X 7Y
Mann-Whitney nonparametric U test used to evaluate differences in shape between the arches Median for the mandible
Median for the maxilla
U
P value
N valid lower
N valid upper
19* 4.5* 16* 14* 14.25* 23* 13* 29 10* 35 7* 38* 2* 39*
21* 4* 18* 13* 16* 22* 14* 29 13* 36 10* 40* 4* 42*
3817* 5183* 2463.5* 3552.5* 2136.5* 4968* 2358* 6061 420* 6553 108.5* 3853.5* 750.5* 3165.5*
0 0.0025 0 0 0 0.0006 0 0.1923 0 0.7328 0 0 0 0
116* 116* 116* 116* 116* 116* 116* 116 116* 116 116* 116* 116* 116*
116* 116* 116* 116* 116* 116* 116* 116 116* 116 116* 116* 116* 116*
*Significant at P \0.05.
side of the arch. This shape is probably attributable to the choice of reference points on the lingual surfaces, which annul the difference in buccolingual diameter between the teeth in the anterior and posterior sections. Three groups, small, medium, and large, were created; the 3 sizes allowed more accurate individualization of forms and dimensions, since our objective was to describe a natural and anatomic arch form from subjects with normal occlusion that, we hope, can be used,
with other criteria, in the construction of personalized setups for the lingual straight-wire technique, thus reducing errors in the selection of the best initial dental arch form for a patient. To this end, a distributive analysis of the x and y values for each tooth was performed to determine the measure of the central tendency most appropriate for our data. The aim of our research was to identify 3 curves, corresponding to the 3 groups (small, medium, and large) that could interpolate the
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Table VI.
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Mann-Whitney nonparametric U test used to evaluate differences in shape between the arches of the sexes Median for the mandible
Median for the maxilla
U
P value
N valid women
N valid men
20 4 17 14 15.75 22 13 29 12 35 8 39 3 40.5
20 4 17 14 15 23 13 29 12 36 8 39.5 3 42
5840 5893.5 5753 5661.5 5770 5617.5 5913.5 5419.5 5907.5 5561.5 5885 5584 6026.5 5584.5
0.6220 0.7019 0.5016 0.3896 0.5241 0.3417 0.7326 0.1744 0.7233 0.2864 0.6889 0.3078 0.9131 0.3083
152 152 152 152 152 152 152 152 152 152 152 152 152 152
80 80 80 80 80 80 80 80 80 80 80 80 80 80
1X 1Y 2X 2Y 3X 3Y 4X 4Y 5X 5Y 6X 6Y 7X 7Y
measures of the central tendencies to represent our sample. Because of many anomalous values and the variability that characterized each tooth, it was decided arbitrarily to consider the coordinates characterized by minimum variability (x7, y1); we identified a maximum deviation around the median equal to 1 mm in both arches. When we analyzed the differences in shape, we considered the values of the medians of our coordinates, and no significant differences between the arches in either sex were found. This finding agrees with those of Ferrario et al,11 who found no sexual dimorphism between the arches when evaluated via Euclidean distance matrix analysis, and Camporesi et al,26 who reached the same conclusions using thin-plate spline analysis, a morphometric system. The samples examined in these 2 studies, like ours, included subjects with natural ideal occlusion from southern Europe, and thus the studies are comparable. Other investigators found significant morphologic differences in the shape of the labial arch among different ethnic groups; this is a possible topic for future research into the shape of the dental arch from the lingual side.4,48 Our results should be considered specific only for European populations in the Mediterranean area. CONCLUSIONS
We analyzed the shape of the lingual sides of the maxillary and mandibular dental arches in a sample of adults from southern Europe with natural ideal occlusions. Analysis of the shape of the arches calculated from the lingual side indicated the following. 1.
There is no sexual dimorphism for the dimensions of the lingual diameter in the intercanine, interfirst molar, and inter-second molar measurements.
2. 3.
4.
There are no significant differences in shape between the arches or between the sexes. Three lingual curves, small, medium, and large, for the 2 arches, representing the mean values of our sample, were developed. The maxillary curves coordinate with the mandibular ones and can be used as a guide for the setup in the lingual straight-wire technique.
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