One goal of orthodontic treatment is to create an

Transcription

1 ONLINE ONLY Subjective classification and objective analysis of the mandibular dental-arch form of orthodontic patients Kazuhito Arai a and Leslie A. Will b Tokyo, Japan, and Boston, Mass Introduction: Our objective was to evaluate the relationship between subjective classification of dental-arch shape, objective analyses via arch-width measurements, and the fitting with the fourth-order polynomial equation. Methods: Twenty-seven pretreatment mandibular dental casts (from 13 males and 14 females; ages, years) were selected. Standardized photographs of the arches were serially organized from tapered to square by 10 examiners. The mean position in the ranking of each cast was calculated as a rank of each arch form. The dental casts were analyzed with a 3-dimensional laser scanning system. Dental-arch widths at the canines and molars were measured, and then a fourth-order polynomial equation was fit to each arch. Correlations between the rank of arch shape and the objective measurements were statistically tested. Results: The arch forms having the greatest variations among the examiners were those with an intermediate (ovoid) ranking. Statistically significant correlations were found between the ranks of arch shape, arch dimensions, and the polynomial equation analyses. Conclusions: Subjective clinical assessments were generally in agreement at the extremes of tapered and square arch forms; the exceptions were arches with an ovoid shape. There were statistically significant correlations between subjective dental-arch classifications and dental-arch dimensions, as well as the ratio determined from these variables and polynomial equation analyses. Therefore, fourth-order polynomial equations might be an important factor in the quantitative analysis of dental-arch form in orthodontic patients. (Am J Orthod Dentofacial Orthop 2011;139:e315-e321) One goal of orthodontic treatment is to create an individualized dental arch that is ideal for the patient. 1-3 The original dental-arch form of the patient is mimicked to achieve stable treatment results because an arch form that has been orthodontically modified has a tendency to return to its original width. 4,5 Therefore, reliable evaluation and accurate analysis of the patient s pretreatment dental-arch form are essential steps for orthodontic diagnosis. In general, subjective classification methods with 3 or 5 simple shape categories have been commonly applied to evaluate the initial dental-arch form during a Professor and chair, Department of Orthodontics, Nippon Dental University, Tokyo, Japan. b Chair and Anthony A. Gianelly Professor of Orthodontics, Boston University Goldman School of Dental Medicine, Boston, Mass. The authors report no commercial, proprietary, or financial interest in the products or companies described in this article. Reprint requests to: Kazuhito Arai, Department of Orthodontics, Nippon Dental University, Fujimi, Chiyoda-ku, Tokyo , Japan; , drarai@ tky.ndu.ac.jp. Submitted, April 2009; revised and accepted, December /$36.00 Copyright Ó 2011 by the American Association of Orthodontists. doi: /j.ajodo the orthodontic diagnostic process. 6-8 For example, the subjective classification method, which uses 3 recommended shapes of tapered, ovoid, and square forms, has been widely used in the clinic to select prefabricated orthodontic archwires for a specific patient. 8 However, general human error can be expected in subjective analysis; therefore, the intraoperator and interoperator reproducibility of these evaluations might be inaccurate. Additionally, although a mathematical definition of these square, ovoid, and tapered dental-arch forms has been proposed, clinical application of this evaluation method for computer analysis is still rather limited. 9 Another common analysis of the dental-arch form is measuring canine and molar widths for clinical and research purposes. 10 These transverse dimensions are usually measured at the cusp tips or other anatomic structures of the tooth crown. One considerable advantage of this method is the numeric analysis of the dental-arch form. Additionally, determination of the canine-molar width ratio has been also used as a simple quantification method and has been widely used in a clinical setting. 11 e315

2 e316 Arai and Will In comparison, objective numeric analysis methods, which use particular geometric and mathematical models, have been developed to quantitatively describe the dental-arch forms of orthodontic patients For instance, a parabola or second-order polynomial, 19 beta function, 18 cubic spline function, 20 and fourthorder or larger polynomial equations have all been applied. 13,21-25 Recently, AlHarbi et al 26 compared these functions and concluded that the fourth-order polynomial function (y 5 ax 4 1 bx 3 1 cx 2 1 dx 1 e) was the most reasonable equation for analysis when the objective was to describe the general smooth curvature of the dental arch. Application of fourth-order polynomials to represent the dental-arch form has several advantages for dentalarch form analysis. 13,21-25 According to Lu, 13 1 advantage of the fourth-order polynomial curve-fitting method is that the coefficients of each term can be simply associated with specific aspects of the arch form. The coefficients of the fourth (x 4 or quartic) and second (x 2 or quadratic) terms describe the square and tapered shapes of dental-arch forms, respectively. Although a large fourth-order coefficient a indicates a square arch form, a large second-order coefficient c describes a tapered arch form. Consequently, the fourth-order polynomial equation has been applied extensively on the basis of this hypothesis. 13,21-25 For example, Hayama et al 23 evaluated the relationship between maxillary and mandibular dental-arch forms and found statistically significant positive correlations for all coefficients. Richards et al 16 applied the fourth-order polynomial equation to analyze the correlation between twin subjects and concluded that this equation can accurately represent dental-arch shape. Additionally, Ferrario et al 17 investigated dental-arch size differences between the sexes using this equation. However, little research has been conducted to compare the mathematical description of the dental-arch form with the subjective evaluations and the objective width measurements made by clinicians as orthodontic diagnostic tools. The purpose of this study was to evaluate the relationship between the results of both a subjective classification by human judgment and 2 objective analytic methods by using dental-arch dimension measurements and the fourth-order polynomial equation for analysis of the dental-arch form of orthodontic patients. MATERIAL AND METHODS A total of 27 pretreatment casts from 13 male and 14 female subjects, ranging in age from 12 to 31 years (mean age, years), were selected from 720 records at the Harvard School of Dental Medicine Teaching Clinic in Boston, Mass. The following inclusion Fig 1. A, Digital photograph of a dental cast (cast 11); B, reference points determined on the cloud data in shaded display (cast 11). criteria were required: (1) complete dentition, excluding third molars; (2) no prosthetic crowns and minimal restorations; (3) minimal signs of occlusal attrition; and (4) minimal spacing. Subjects with severe crowding (.10 mm) andtransposedteethwereexcludedfromtheanalysis. This sample size was first analyzed and determined to provide an adequate statistical power (80%, b 5 0.2) for detection of a correlation coefficient of 0.5 for a and 0.6 for a All selected subjects signed informed consent forms granting permission for their records to be used in research. A detailed protocol of this project was approved by the Committee on Human Studies of Harvard Medical School and Harvard School of Dental Medicine (X ). Standardized occlusal photographs of the mandibular dental casts of the 27 mandibular arches were taken with a digital camera (DSC-F55V, Sony, Japan) and printed in actual size (Fig 1, A). Photographs of the mandibular dental arches were organized in a series from tapered to square shapes by 10 members of the April 2011 Vol 139 Issue 4 American Journal of Orthodontics and Dentofacial Orthopedics

3 Arai and Will e317 Department of Orthodontics (5 teaching staff members and 5 fourth-year and third-year residents) from the Department of Growth and Development, Harvard School of Dental Medicine, Boston, Mass, and Department of Orthodontics, Nippon Dental University, Tokyo, Japan. Each examiner evaluated the 27 dental arches and ranked them from most tapered (1) to most square (27). The average rank given to a particular cast by all 10 examiners was determined to generate a mean rank for the dental arch. The standard deviation of the mean rank was also calculated for each dental arch. For example, a dental cast photograph ranked as most tapered (1) by 7 examiners and the second most tapered (2) by the other 3 examiners would receive an overall mean rank of This dental arch was then placed into the shape continuum near the most tapered side. The examiners were asked to judge the dental-arch form using their usual clinical evaluation method, and no discussion or calibration sessions occurred before the evaluations. The dental casts were also scanned and analyzed with a 3-dimensional dental cast-measuring system. This system consisted of a laser-scanning unit (Surflacer VMS- 100F/ UNISN, Osaka, Japan) and a computeraided design (CAD) software program (Surfacer version 9.0. Imageware Inc., Ann Arbor, Mich). 27,28 The reference points digitized on the image of each cast were the midpoints of the incisal edges, the canine cusp tips, and the buccal cusps of the premolars and the first and second molars (Fig 1, B). The midpoint between the mesiobuccal and distobuccal cusp tips of the first and second molars was then computed and used to define the molar position. Only 1 point was used for each molar to eliminate the effects of rotation, resulting in a single point for the position of the tooth. X and y coordinate data were extracted, and the distances between bilateral reference points for the canines and the first and second molars were calculated as canine width, first molar width, and second molar width, respectively. The canine-first molar and canine-second molar ratios (percentages) were also calculated. Then the means and standard deviations of canine width, first molar width, second molar width, canine-first molar ratio, and canine-second molar ratio were calculated. Additionally, these coordinate sets were used to fit a fourth-order polynomial equation (y 5 ax 4 1 bx 3 1 cx 2 1 dx 1 e) to the 14 reference points for a dental arch by using the least squares method (Fig 2). Then, the coefficients of fourth-order and second-order terms ( a and c, respectively) could be determined from this equation for each dental arch. The means and standard deviations of the a and c terms were also calculated. The nonparametric Spearman rank correlations (r s ) between the mean rankings of the dental-arch forms, the canine and molar widths and ratios, and the a and c coefficients derived from the equations were statistically analyzed. The Pearson correlation coefficients (r) between the dental-arch width and ratios, as well as the a and c terms, were also calculated and statistically analyzed with the Fisher z-transformation. 29 Two statistical analysis methods for correlations were used in this research: (1) the nonparametric Spearman rank correlation to rank data and (2) the Pearson coefficient of correlation for continuous data. To evaluate the reliability of the dental-arch form ranking, photographs of 10 casts were randomly selected from the sample. These standardized photographs were ranked twice from tapered to square shapes by the same examiner (K.A.), with an interval of 2 weeks between evaluations, to determine intraexaminer reliability. The same determination was made once by another examiner (L.A.W.) to evaluate interexaminer reliability. Nonparametric Spearman rank correlations of intraexaminer and interexaminer reliability were 0.71 (P ) and 0.93 (P \0.01), respectively. To evaluate the reliability of landmark location, 10 dental casts were randomly selected from the sample. The following 10 reference points were determined: incisal edges of the central incisors, canine cusp tips, buccal cusp tips of the first premolars, and mesial and distal buccal cusp tips of the first molar on the left side of the dental arch. The same examiner (K.A.) determined 100 reference points in duplicate to evaluate intraexaminer reliability, with 2 weeks between evaluations. The same determination was made once by another examiner (L.A.W.) to evaluate interexaminer reliability. The mean differences between the 2 determinations by the same examiner were 0.21 mm (SD, 0.21 mm) in the sagittal plane and 0.17 mm (SD, 0.19 mm) in the transverse plane. The mean differences between the 2 examiners were 0.19 mm (SD, 0.22 mm) in the sagittal plane and 0.21 mm (SD, 0.19 mm) in the transverse plane. RESULTS The correlations between the means and standard deviations of the subjective evaluation rankings for all dental-arch forms are shown in Figure 3. The dentalarch forms ranked as most tapered or most square had small standard deviations, and interexaminer agreement among examiners was obtained. In contrast, the arch forms exhibiting relatively greater standard deviations, indicating interexaminer variation in the evaluation, were those with an intermediate ranking (ovoid arch form). Statistically significant positive correlations were found between the mean rankings of the dental-arch forms, the widths and ratios of the canine and the first and second molars, and the fourth-order term a of American Journal of Orthodontics and Dentofacial Orthopedics April 2011 Vol 139 Issue 4

4 e318 Arai and Will second-order term c, and between the canine-second molar ratio and the fourth-order term a. Statistically significant negative correlations were found between canine width, first molar width, second molar width, and canine-second molar ratio and c (Table II). Fig 2. Example of dental-arch form analyses. Reference points for each tooth and the polynomial equation with the curve on the dental-arch form are shown (cast 11). The results of the analyses were (1) mean rank 6 SD, ; (2) canine width, mm; (3) first molar width, mm; (4) canine-first molar ratio, 59.56%; (5) second molar width, mm; (6) canine-second molar width ratio, 54.41%; (7) a, ; and (8) c, Fig 3. Correlation between rank of dental-arch form and standard deviation of the ranking. the equation. Statistically significant negative correlations were found between the mean rankings, the canine-molar ratios, and the second-order term c of the equation (Table I). The highest statistically significant positive correlation was observed between the mean rank and canine width. In contrast, the highest negative correlation was obtained between the mean rank and the second-order term c (Table I). Statistically significant positive correlations were found between the canine-first molar ratio and the DISCUSSION Several subjective methods of classification have been used to evaluate the characteristics of a patient s pretreatment dental-arch form for orthodontic diagnosis and treatment planning. 6-8 In this study, arch forms were subjectively ranked from tapered to square by 10 examiners, according to their own clinical evaluation methods without any calibration sessions among examiners. The arches ranked closest to either end of the continuum varied the least among examiners, suggesting that dental-arch forms that were distinctively tapered or square were easier to classify. This result indicates that the visual cognitive ability of the trained clinician can almost instantaneously recognize the differences between characteristics of complex dentalarch shapes that are subtly distinct without a measuring device, resulting in consistent classifications of arches with the simple description of tapered or square. These results also indicate that this type of subjective classification with ranking becomes relatively more difficult for dental-arch forms that are intermediate, such as the ovoid shape. Therefore, this difficulty in classifying intermediate arch forms might result in unreliable classification of the ovoid shape and suggests that calibration should be performed among examiners before classification and that quantitative analysis can be important, especially for the boundaries between tapered and ovoid and between ovoid and square shapes. 7 Additionally, relatively high interevaluator and intraevaluator reliabilities of a subjective classification of dental-arch forms by using a set of standard arch form templates was recently reported. 30 Therefore, additional study might be necessary to compare these classification methods. There were statistically significant positive correlations between rank and dental-arch widths (Table I). This result suggests that wider dental arches had a tendency to be ranked as square shapes. Also, there were statistically significant negative correlations between the ranks and ratios of the canines and the first and second molars. Therefore, dental-arch forms with canines that are wide relative to both the first and second molars had a tendency to be square. In comparison, the highest positive correlation was obtained between rank and canine width. These results indicate that the examiners might predominantly focus on individual variations in April 2011 Vol 139 Issue 4 American Journal of Orthodontics and Dentofacial Orthopedics

5 Arai and Will e319 Table I. Means and standard deviations of the canines, first and second molar widths, canine-first and canine-second molar ratios, and fourth-order and second-order terms of coefficients of the equation ( a and c ) of the dental arch forms Canine width (mm) First molar width (mm) Canine-firstmolar ratio (%) Second molar width (mm) Canine-second molar ratio (%) Fourth-order term a Second-order term c Mean E SD E r s Z-transformation P value \ y y y * y y \ y Results of the statistical analyses by nonparametric Spearman rank correlations between the mean rank and canine width, first and second molar widths, canine-first molar and canine-second molar ratios, and a and c. *P \0.05; y P \0.01. Table II. Pearson coefficient of correlation (r) between the dental arch width and ratios and a and c were also calculated and statistically analyzed with the Fisher z-transformation a c r P value r P value Canine width \ y First molar width y Canine-first molar ratio y Second molar width * Canine-second molar ratio y y *P \0.05; y P \0.01. canine width during classification and also support a clinical system of preformed archwires that varies the canine widths only. 8 Additionally, canine width measurements made with calipers might serve as a quantitative analysis of dental-arch form for preformed orthodontic archwire selection in the clinic. For example, based on this study, the mean 6 standard deviation of canine width was mm. Therefore, the ovoid arch shape can be mathematically defined as a canine width between and mm. To apply this range of canine width for ovoid dental-arch shapes as a standard in preformed archwire selection, 8 further studies in Class I normal occlusion and other Angle classifications 7 might be required. Statistically significant positive and negative correlations between the subjective rankings of dental-arch forms and objective quantitative analyses by using the coefficients a and c of the fourth-order polynomial equation were found, respectively (Table I). These findings indicate that a tapered dental-arch form has a large c, and a square dental-arch form has a large a, supporting the hypothesis of Lu. 13 Therefore, dental-arch forms can be quantitatively analyzed, including the reference points on every tooth, by the result of the fitted fourth-order term a and the second-order term c of the dental arch, even in intermediate ovoid arches. Additionally, the curve created by the equation results in a smooth flexible curve, which represents all potential tooth positions for the dental arch and, therefore, can be used as an archwire template for each patient. 13,26 There were statistically significant correlations between all arch-width measurements and canine-molar ratios and the second-order term c (Table II). For example, wider canines had smaller c values in the polynomial equation fitted to the dental arch. On the other hand, only the canine-second molar ratio was significantly correlated with the fourth-order term a (Table II). These results suggest that the more posterior teeth exert a greater influence on the polynomial curve by the fourth-order term a. Therefore, the second-order term c is the main element for consideration in the mathematical analysis with the polynomial equation curve and of greater significance than the fourth-order term a for the anterior region. Additionally, the fourth-order term a can serve as a modifier for the curve fitting in the posterior area of the dental-arch form. Although we analyzed dental-arch forms without much crowding, mathematical curve fitting for the American Journal of Orthodontics and Dentofacial Orthopedics April 2011 Vol 139 Issue 4

6 e320 Arai and Will dental arch with crowding has also been reported. 31 Therefore, how crowding and asymmetries affect the arch form and whether this observation has an implication on the ranking method can be studied by using the fourth-order polynomial equation in a future study. The disadvantage of the fourth-order polynomial equation is the current lack of simplicity for daily use in a clinical setting to fit the curve onto the dental arch. However, a software program fitting polynomial equations to the dental arch is available, 32 and some computer systems with 3-dimensional technology for dental cast analysis, diagnosis, and treatment planning have recently been introduced and are rapidly spreading in orthodontics Soon, it is likely that this technology will be readily available in orthodontic offices. Based on the results of this study, the fourthorder polynomial equation can be applied to estimate the dental-arch forms of orthodontic patients with an accurate and a flexible mathematical expression. CONCLUSIONS 1. Subjective clinical assessments were generally in agreement at the extremes of tapered and square dental-arch forms, but the exceptions were arches with an ovoid shape. 2. There were statistically significant correlations between subjective dental-arch classifications and dental-arch dimensions, as well as the ratio determined from these variables and polynomial equation analyses. 3. Coefficients of fourth-order polynomial equations were significantly correlated with individual variations in the size and shape of dental-arch forms and might be a reliable tool for quantitative analysis of dental-arch form in orthodontic patients. REFERENCES 1. Hawley CA. Determination of the normal arch, and its application to orthodontia. Dent Cosmos 1905;47: Angle EH. Treatment of malocclusion of the teeth and fractures of the maxillae. Angle s system. 7th ed. Philadelphia: S.S. White; p Sarver DM, Proffit WR, Ackerman JL. Diagnostic and treatment planning in orthodontics. In: Graber TM, Vanarsdall RL Jr, editors. Orthodontics: current principles and techniques. 3rd ed. St Louis: Mosby; p Case CS. Principles of retention in orthodontia. Int J Orthod Oral Surg 1920;6: Riedel RA. A review of the retention problem. Angle Orthod 1960; 30: Ricketts RM. Design of arch form and details for bracket placement (catalog number P-365). Denver, Colo: Rocky Mountain Orthodontics; Nojima K, McLaughlin RP, Isshiki Y, Sinclair PM. A comparative study of Caucasian and Japanese mandibular clinical arch forms. Angle Orthod 2001;71: McLaughlin RP, Bennett JC. Arch form considerations for stability and esthetics. Rev Esp Ortod 1999;29(Suppl 2): Noroozi H, Nik TH, Saeeda R. The dental arch form revisited. Angle Orthod 2001;71:386-9; erratum, Moorrees CFA. The dentition of the growing child. A longitudinal study of dental development between three and eighteen years of age. Cambridge, Mass: Harvard University Press; Williams PN. Dental engineering and the normal arch. Dent Cosmos 1918;60: de la Cruz A, Sampson P, Little RM,Artun J, Shapiro PA. Long-term changes in arch form after orthodontic treatment and retention. Am J Orthod Dentofacial Orthop 1995;107: Lu KH. An orthogonal analysis of the form, symmetry and asymmetry of the dental arch. Arch Oral Biol 1966;11: Pepe SH. Polynomial and catenary curve fits to human dental arches. J Dent Res 1975;54: Sampson PD. Dental arch shape: a statistical analysis using conic sections. Am J Orthod 1981;79: Richards LC, Townsend GC, Brown T, Burgess VB. Dental arch morphology in south Australian twins. Arch Oral Biol 1990;35: 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: Braun S, Hnat WP, Fender DE, Legan HL. The form of the human dental arch. Angle Orthod 1998;68: Battagel JM. Individualized catenary curves: their relationship to arch form and perimeter. Br J Orthod 1996;23: BeGole EA. Application of the cubic spline function in the description of dental arch form. J Dent Res 1980;59: Ferrario VF, Sforza C, Schmitz JH, Colombo A. Quantitative description of the morphology of the human palate by a mathematical equation. Cleft Palate Craniofac J 1998;35: Ferrario VF, Sforza C, Colombo A, Carvajal R, Duncan V, Palomino H. Dental arch size in healthy human permanent dentitions: ethnic differences as assessed by discriminate analysis. Int J Adult Orthod Orthognath Surg 1999;14: Hayama K, Arai K, Ishikawa H. Correlation between upper and lower dental arch forms by fitting of fourth-order polynomials. Orthod Waves 2000;59: Uzuka S, Arai K, Ishikawa H. Polynomial curve superimpositions on dental arch forms with normal occlusions. Orthod Waves 2000;59: Shikano C, Arai K, Ishikawa H. Evaluation of dental arch form in normal occlusion fitting of fourth-order polynomials on FA points. Orthod Waves 2001;61: AlHarbi S, Alkofide EA, AlMadi A. Mathematical analyses of dental arch curvature in normal occlusion. Angle Orthod 2008;78: Arai K, Ishikawa H. Application of non-ontact three-dimensional shape measuring system to dental cast reduction of blind region. Orthod Waves 1999;58: Ronay V, Miner RM, Will LA, Arai K. Mandibular arch form: the relationship between dental and basal anatomy. Am J Orthod Dentofacial Orthop 2008;134: Norman GR, Steiner DL. Biostatistics. The bare essentials. 2nd ed. Hamilton, Ontario, Canada: B. C. Decker; Trivi~no T, Siqueira DF, Scanavini MA. A new concept of mandibular dental arch forms with normal occlusion. Am J Orthod Dentofacial Orthop 2008;133:10.e April 2011 Vol 139 Issue 4 American Journal of Orthodontics and Dentofacial Orthopedics

7 Arai and Will e Wellens H. Applicability of mathematical curve-fitting procedures to late mixed dentition patients with crowding: a clinical-experimental evaluation. Am J Orthod Dentofacial Orthop 2007; 131:160.e Noroozi H, Djavid GE, Moeinzad H, Teimouri AP. Prediction of arch perimeter changes due to orthodontic treatment. Am J Orthod Dentofacial Orthop 2002;122: Boyd RL, Miller RJ, Vlaskalic V. The Invisalign system in adult orthodontics: mild crowding and space closure cases. J Clin Orthod 2000;34: Marcel TJ. Three-dimensional on-screen virtual models. Am J Orthod Dentofacial Orthop 2001;119: Mah J, Sachdeva R. Computer-assisted orthodontic treatment: the SureSmile process. Am J Orthod Dentofacial Orthop 2001;120:85-7. American Journal of Orthodontics and Dentofacial Orthopedics April 2011 Vol 139 Issue 4