

ORIGINAL RESEARCH 

Year : 2017  Volume
: 8
 Issue : 3  Page : 7073 

Mathematical ratio in defining arch form
Johan A Budiman
Orthodontic Department, Dental Technician Study, Health Polytechnic Jakarta II, Ministry of Health, Republic of Indonesia
Date of Web Publication  8Aug2017 
Correspondence Address: Johan A Budiman Orthodontic Department, Dental Technician Study, Health Polytechnic Jakarta II, Ministry of Health Republic of Indonesia
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/denthyp.denthyp_26_17
Introduction: The treatment of Class I malocclusion aims to arrange teeth position in a good arch form. The arch form consists of tooth size and arch dimension (intercanine width, canine depth, intermolar width, molar depth). Numerous methods have been used to describe the arch form quantitatively. The aim of this study was to develop a mathematical ratio for identifying arch form (square, oval, tapered) using arch dimension variables (intercanine width, canine depth, intermolar width, molar depth). Materials and Methods: Dental cast pre and postorthodontic treatments from 190 Indonesian patients were scanned to obtain digital data. All data were measured using “Image Tool.” The measured data (tooth size, intercanine width, intercanine depth, intermolar width, intermolar depth, and arch perimeter) were analyzed statistically using ordered logistic to find out determining variables to the arch form. Results: The validity, reliability, and normality of all the data were analyzed using Stata. From analyzing the data using ordered logistic, intercanine width and intermolar depth showed a reverse relation to the arch form. The shape of the arch form (square, oval, and tapered) can be described quantitatively by using ratio (CD/CW)/(MD/MW); a ratio less than 45.30% indicates square, 45.30–53.37% indicates oval, and more than 53.37% indicates tapered. Conclusions: (CD/CW)/(MD/MW) ratio can be used to describe arch form quantitatively. Keywords: Arch dimensions, arch form, mathematical ratio
How to cite this article: Budiman JA. Mathematical ratio in defining arch form. Dent Hypotheses 2017;8:703 
Introduction   
Despite wide acceptance of the idea that arch forms vary among individuals, there is a long orthodontic tradition of seeking a single ideal arch form. Dental arches are correlated with the dimensions and shape of the face. This variation was caused by variation in tooth size so that it is not the goal of orthodontic treatment to produce dental arches of a single ideal size and shape for everyone.^{[1]} The treatment of Class I malocclusion is to arrange the teeth position in a good arch form. The arch form consists of tooth size and arch dimension (intercanine width, canine depth, intermolar width, and molar depth).
Three basic qualitative arch forms have been repeatedly described in the literature, namely, tapered, oval, and square. Many geometric forms and mathematical functions have been proposed as quantitative models of the human dental arch, such as parabolic equation. Several factors influence arch form such as malocclusion, Bolton’s ratio,^{[2],[3]} shape of a face, tooth size, habits, and muscular and patient’s profile.^{[4]} The objective of this research is to develop diagnostic reference using mathematical ratio for identifying arch form (square, oval, tapered).
Materials and Methods   
This is a diagnostic research using a scanneddental cast, upper and lower, before and after orthodontic treatment of Class I malocclusion. Dental casts were gathered from orthodontically treated patients at Orthodontic clinic of Faculty of Dentistry, University of Indonesia and three other private orthodontic practices in Jakarta.
The dental cast before and after orthodontic treatment was scanned using a scanner (ScanMaker Microtec 5800, Hsinchu, Taiwan) and Image tool (the Department of Dental Diagnostic Science at The University of Texas Health Science Center, San Antonio, Texas) used to measure the scanning result. Measurement variables can be seen in [Figure 1]. The measured variables were tooth size, arch dimensions, kinds of treatment, and gender. All data were analyzed using Stata (Lakeway Drive, College Station, Texas USA).
Results   
One hundred and ninety pairs of the dental casts before and after orthodontic treatment were scanned using ScanMaker Microtec 5800, and Image tool was used to measure the scanning result. The dental casts originated from 42 males (22.1%) and 148 females (77.9%). Types of orthodontic treatment are as follows: nonextraction (32.63%), premolars extraction (48.42%), upper premolar extraction (11.05%), and atypical extraction (7.90%).
The validity of measurement was tested with variance analysis (Barlett’s test and Schaffer’s table) showed that there is no difference in means within and among these measurements (P > 0.05). Reliability of measurements before and after orthodontic treatment was tested using Bland Altman’s method. The result showed that there is no significant difference in all upper and lower, before and after measurement. Normality test showed that all the data distribute normally.
ttest was used for proving that tooth size and arch dimensions before and after orthodontic treatment of Class I malocclusion can be used as a diagnostic reference in deciding arch form from different genders. Analysis of Variance (ANOVA) was used to compare arch forms (square, oval, tapered) and gender (male and female), with each component of arch dimension upper and lower jaw, before and after treatment. It indicated that there is no significant difference from all component of arch dimensions upper and lower, before and after treatment with gender (P > 0.05).
The ordered logistic analysis to compare arch dimensions and gender to arch forms (square, oval, tapered) was carried out. This test was repeated until all variables showed a significant difference to arch forms, as seen in [Table 1].  Table 1: Ordered logistic result on arch form with Arch dimensions variable using Stata
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Ways on describing arch forms use fundamental mathematic equations, such as parabolic shape equation. The reference parameter used for this equation is the same as reference point in this research; therefore, ttest was carried out to find the significant difference between the result on this research and the result calculated by parabolic equation. All variables were statistically different between tour calculations and results of hyperbolic equation, except for anterior perimeter variable after treatment. Therefore, the parabolic equation can only be used for anterior perimeter variables after treatment.
Describing arch forms qualitatively with oval, tapered and square, the data can be grouped into oval 227 casts (59.78%), square 125 casts (32.89%), and tapered 28 casts (7.37%). Ordered logistic analysis was performed to determine influence variables on arch forms. The results are shown in [Table 2].
Negative value showed reverse correlation among variables. From this analysis, probability model of ordered logistic can be found. Upper and lowerjaw data was mixed based on the fact of occlusion, so that upper and lower jaw should occlude with the same arch form pattern. The model of ordered logistic is illustrated in [Figure 2].
The probability model of ordered logistic is as follows:
Using the probability model of ordered logistic, the probability of the arch form (oval, tapered, square) can be calculated and the results compared with the result of this research are illustrated in [Table 3].
These findings can be concluded as follows: Canine depth and intermolar width show high correlation
 Intercanine width and molar depth show reverse correlation.
Those findings can be defined in a ratio as shown in [Figure 3]. This rational ratio can be applied in daily practice. However, the cutoff point for each arch form should be denoted. Using dummy variables, this ratio was calculated using the research data. The result can be seen in [Table 4].
The application of this ratio for predicting arch form is as follows: If the ratio less than 45.30% square arch form is expected
 If the ratio of 45.30% to 53.37% oval arch form is expected
 If the ratio more than 53.37% tapered arch form is expected
Discussion   
Despite the use of threedimensional scanners nowadays, this study gave an insight of using a twodimensional scanner for defining arch form. Dental casts of upper and lower jaw, before and after treatment were scanned using the same scanner, Microtek 5800. The best scanning result can be obtained by using the scanner with resolution more than 1200 dpi.^{[5],[6]} Measurements were performed using Image Tool.^{[7]} This program can be used for measuring using a computer. For the calibration system, the dental casts are digitized while sitting on a sheet of millimetre graph paper in transforming by the scanner.
Fundamental mathematic equations, such as parabolic shape equation and the beta function were used to describe arch form mathematically. For some researchers, the beta function is an accurate mathematical model of the dental arch, but using the tip of the canine as a reference point was found difficult to digitize using a software compared to proximal contact point used by the parabolic shape equation, especially for the abrasive canine. The reference parameter used for parabolic equation is the same as the reference point in this research. The use of parabolic equation was also compared to the data of this study, using anterior perimeter, intercanine width, and canine depth.^{[8]} This analysis concludes that anterior perimeter of the upper jaw after the treatment is in accordance with the result calculated by the parabolic equation. It implies that anterior perimeter of the upper jaw after the treatment can be predicted using parabolic equation.
The majority of researchers recognize that there is variability in the size and shape of the human arch form.^{[9]} The form of human dental arch describes in oval, tapered and square has been used traditionally.^{[10],[11]} These variations in arch form are reflected in arch wire. There is still a long debate on subjectivity in deciding arch form. The arch form can also be patterned using a template of oval, tapered and square.^{[10],[11]} The data was analysed using the ologit command in Stata. In this study, there is a nonequal distribution of arch form, 227 oval forms (59.78%), 125 square forms (32.89%), and 28 tapered forms (7.37%).
The optimal likelihood is 92.46 with intercanine width, canine depth, intermolar width, and molar depth variables giving significant difference from ordered logistic analysis with negative cutoffs (−5.2582 and −1.3239). This concludes intercanine width, canine depth, intermolar width and molar depth give influence for judging arch forms.
By computing probability models from ordered logistic and compared with the data gathered in this study, it is found that there are still deviations in judging arch forms, especially for tapered and square. The small amount of data, tapered (7.37%) and square (32.89%), might cause this deviation. From ordered logistic study, this study can also develop (CD/CW)/(MD/MW) ratio.The square, oval, and tapered arch forms have not yet been mathematically defined. One solution may be to define them based on relative ratios of the canine and molar width along with their relative arch depth.^{[12]} When the CW/MW ratio increases the CD/MD ratio decreases, the arch becomes squarer. On the contrary, when CW/MW ratio decreases and CD/MD ratio increases, the arch gets a more tapered form.^{[13]}
The application of this ratio, found in this study, is as follows when the ratio is less than 45.30% the arch form is square. When the ratio of 45.30% to 53.37% the arch form is oval, and when the ratio is more than 53.37% the arch form is tapered.
Conclusions   
It can be concluded that a mathematical ratio of (CD/CW)/(MD/MW) can be used to describe dental arch form quantitatively. A ratio of less than 45.30% square is expected; if the ratio is 45.30% to 53.37% oval is expected; and if the ratio is more that 53.37% tapered shape is expected.
Acknowledgement
The author thank Prof. Dr. Bambang Sutrisna, dr, MHSc for his statistical review, Prof. Dr. M. Suharsini, drg, Sp.KGA and Prof. Dr. Retno Hayati, drg, SKM, SpKGA for their review of manuscript.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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9.  Othman SA, Xinwei ES, Lim SY, Jamaludin M, Mohamed NH, Yusof ZY et al. Comparison of arch form between ethnic Malays and Malaysian Aborigines in Peninsular Malaysia. Korean J Orthod 2012;42:4754. 
10.  McLaughlin RP, Bennet JC. Arch Form Considerations for Stability and Esthetics. Rev Esp Orthod 1999;29:4663. 
11.  McLaughlin RP, Bennet JC, Trevisi HJ. Systemized Orthodontic Treatment Mechanics. 1st ed, Edinburgh: Mosby International Ltd; 2001. pp 7491. 
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[Figure 1], [Figure 2], [Figure 3]
[Table 1], [Table 2], [Table 3], [Table 4]
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