, Volume 105, Issue 2, pp 229–236 | Cite as

Mathematical beta function formulation for maxillary arch form prediction in normal occlusion population

  • Morteza Mina
  • Ali Borzabadi-Farahani
  • Azita Tehranchi
  • Mahtab Nouri
  • Farnaz Younessian
Original Article


The aim of this study was to assess the dental arch curvature in subjects with normal occlusion in an Iranian population and propose a beta function formula to predict maxillary arch form using the mandibular intermolar widths (IMW) and intermolar depths (IMD). The materials used were study casts of 54 adolescents with normal occlusion and mean age of 14.1 years (25 males, 29 females, age range 12–16 years). Curve-fitting analyses were carried out and the curves passing through the facial-axis point of the canines, premolars, first molars, and the incisal edges of the anterior teeth were studied using a 3D laser scanner. Using the measured IMW and IMD of the dental arches at the maxillary and mandibular first molar region, a beta function formula proposed for predicting maxillary arch form. The accuracy of the proposed formula was assessed on 10 randomly selected dental casts. The mean (SD) of the maxillary and mandibular IMW and IMD were 57.92 (4.75), 54.19 (5.31), and 31.59 (2.90) and 28.10 (2.59) mm, respectively. There was no gender dimorphism (P > 0.05) for both variables (IMW, IMD). There was a strong positive association (n = 10, Pearson r = 0.98, P < 0.05) between the measured (actual) maxillary arch length and proposed arch length derived from generated formula. The goodness of fit (whole arch) for the proposed beta function formula, using adjusted r square measure and root mean square in 10 patients averaged 0.97 and 1.49 mm, respectively. The corresponding figures for the maxillary anterior arch (canine to canine) were 0.90 and 0.92 mm, respectively. The proposed beta function formula used for predicting maxillary arch form based on two mandibular measures (IMW, IMD) was found to have a high accuracy for maxillary arch prediction in the Iranian population and may be used as a guide to fabricate customized arch wires or as an aid in maxillary reconstructive surgery.


Dental arch form Orthodontics Normal occlusion Beta function 



This study was extracted from a postgraduate thesis at the School of Dentistry, Shahid Beheshti University of Medical Sciences. The authors wish to thank the Dentofacial Deformities Research Center, Research Institute of Dental Sciences, Shahid Beheshti University of Medical Sciences for the support of this project.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Ronay V, Miner RM, Will LA, Arai K. Mandibular arch form: the relationship between dental and basal anatomy. Am J Orthod Dentofacial Orthop. 2008;134:430–8.CrossRefPubMedGoogle Scholar
  2. 2.
    Felton MJ, Sinclair PM, Jones DL, Alexander RG. A computerized analysis of the shape and stability of mandibular arch form. Am J Orthod. 1987;92:478–83.CrossRefGoogle Scholar
  3. 3.
    De La Cruz AR, Sampson P, Little RM, Artun J, Shapiro PA. Long term changes in arch form after orthodontic treatment and retention. Am J Orthod. 1995;107:518–30.CrossRefGoogle Scholar
  4. 4.
    AlHarbi S, Alkofide EA, AlMadi A. Mathematical analyses of dental arch curvature in normal occlusion. Angle Orthod. 2008;78:281–7.CrossRefPubMedGoogle Scholar
  5. 5.
    BeGole EA. Application of the cubic spline function in the description of dental arch form. J Dent Res. 1980;59:1549–56.CrossRefPubMedGoogle Scholar
  6. 6.
    Sampson PD. Dental arch shape: a statistical analysis using conic sections. Am J Orthod. 1981;79:535–48.CrossRefPubMedGoogle Scholar
  7. 7.
    Pepe SH. Polynomial and catenary curve fits to human dental arches. J Dent Res. 1975;54:1124–32.CrossRefPubMedGoogle Scholar
  8. 8.
    Ferrario VF, Sforza C, Miani A Jr, Tartaglia G. Mathematical definition of the shape of dental arches in human permanent healthy dentitions. Euro J Orthod. 1994;16:287–94.CrossRefGoogle Scholar
  9. 9.
    Braun S, Hnat WP, Fender DE, Legan HL. The form of the human dental arch. Angle Orthod. 1998;68:29–36.PubMedGoogle Scholar
  10. 10.
    Maeda T, Oyama A, Okamoto T, Funayama E, Furukawa H, Hayashi T, Murao N, Sato Y, Yamamoto Y. Combination of Tessier clefts 3 and 4: case report of a rare anomaly with 12 years' follow-up. J Craniomaxillofac Surg. 2014;42:1985–9.CrossRefPubMedGoogle Scholar
  11. 11.
    Kim J, Lagravére MO. Accuracy of Bolton analysis measured in laser scanned digital models compared with plaster models (gold standard) and cone-beam computer tomography images. Korean J Orthod. 2016;46:13–9.CrossRefPubMedPubMedCentralGoogle Scholar
  12. 12.
    Stewart J. Essential Calculus. In: Applications of Integration. Arc length, chap 7, sec 7.4. 2011. p. 392–98.Google Scholar
  13. 13.
    Quimby ML, Vig KW, Rashid RG, Firestone AR. The accuracy and reliability of measurements made on computer-based digital models. Angle Orthod. 2004;74:298–303.PubMedGoogle Scholar
  14. 14.
    Nouri M, Massudi R, Bagheban AA, Azimi S, Fereidooni F. The accuracy of a 3-D laser scanner for crown width measurements. Aust Orthod J. 2009;25:41–7.PubMedGoogle Scholar
  15. 15.
    Kuntz TR, Staley RN, Bigelow HF, Kremenak CR, Kohout FJ, Jakobsen JR. Arch widths in adults with Class I crowded and Class III malocclusions compared with normal occlusions. Angle Orthod. 2008;78:597–603.CrossRefPubMedGoogle Scholar
  16. 16.
    Al-Khateeb SN. Abu Alhaija ES. Tooth size discrepancies and arch parameters among different malocclusions in a Jordanian sample. Angle Orthod. 2006;76:459–65.PubMedGoogle Scholar
  17. 17.
    Uysal T, Usumez S, Memili B, Sari Z. Dental and alveolar arch widths in normal occlusion and Class III malocclusion. Angle Orthod. 2005;75:809–13.PubMedGoogle Scholar
  18. 18.
    Slaj M, Spalj S, Pavlin D, Illes D, Slaj M. Dental archforms in dentoalveolar Class I, II and III. Angle Orthod. 2010;80:919–24.CrossRefPubMedGoogle Scholar
  19. 19.
    Suk KE, Park JH, Bayome M, Nam Y-O, Sameshima GT, Kook Y-A. Comparison between dental and basal arch forms in normal occlusion and Class III malocclusions utilizing cone-beam computed tomography. Korean J Orthod. 2013;43(1):15–22.CrossRefPubMedPubMedCentralGoogle Scholar
  20. 20.
    Bishara SE, Jakobsen RJ, Treder J, Nowak A. Arch width changes from 6 weeks to 45 years of age. Am J Orthod Dentofacial Orthop. 1997;111:401–9.CrossRefPubMedGoogle Scholar
  21. 21.
    Henrikson J, Persson M, Thilander B. Long-term stability of dental arch form in normal occlusion from 13 to 31 years of age. Eur J Orthod. 2001;23:51–61.CrossRefPubMedGoogle Scholar
  22. 22.
    Burris BG, Harris EF. Maxillary arch size and shape in American blacks and whites. Angle Orthod. 2000;70:297–302.PubMedGoogle Scholar
  23. 23.
    Chang HF, Shiau YY, Chen KC. The relationship of dental crowding to tooth size, dental arch width, and arch depth. Proc Natl Sci Counc Repub China B. 1986;10:229–35.PubMedGoogle Scholar
  24. 24.
    Moorrees CFA, Gron AM, Lebret LML, Yen PKJ, Frohlich FJ. Growth studies of the dentition: a review. Am J Orthod. 1969;55:600–16.CrossRefPubMedGoogle Scholar
  25. 25.
    Borzabadi-Farahani A. Orthodontic considerations in restorative management of hypodontia patients with endosseous implants. J Oral Implantol. 2012;38:779–91.CrossRefPubMedGoogle Scholar
  26. 26.
    Sillman J. Dimensional changes of the dental arches: longitudinal study from birth to 25 years. Am J Orthod. 1964;50:824–42.CrossRefGoogle Scholar
  27. 27.
    Knott VB. Longitudinal study of dental arch widths at four stages of dentition. Angle Orthod. 1972;42:387–94.PubMedGoogle Scholar
  28. 28.
    DeKock WH. Dental arch depth and width studied longitudinally from 12 years of age to adulthood. Am J Orthod. 1972;62:56–66.CrossRefPubMedGoogle Scholar
  29. 29.
    Lyotard N, Hans M, Nelson S, Valiathan M. Short-term post orthodontic changes in the absence of retention. Angle Orthod. 2010;80:1045–50.CrossRefPubMedGoogle Scholar
  30. 30.
    Proffit WR, Fields H. Contemporary orthodontics Mosby. St Louis. 2000;78(124):315.Google Scholar
  31. 31.
    Braun S, Hnat WP, Leschinsky R, Legan HL. An evaluation of the shape of some popular nickel titanium alloy preformed arch wires. Am J Orthod Dentofacial Orthop. 1999;116:1–12.CrossRefPubMedGoogle Scholar

Copyright information

© The Society of The Nippon Dental University 2016

Authors and Affiliations

  1. 1.Department of Orthodontics, School of DentistryNorth Khorasan University of Medical SciencesBojnourdIran
  2. 2.Warwick Medical SchoolUniversity of WarwickCoventryUK
  3. 3.Orthodontics, Department of Clinical Sciences and Translational MedicineUniversity of Rome “Tor Vergata”RomeItaly
  4. 4.Preventive Dentistry Research CenterResearch Institute of Dental Sciences, School of Dentistry, Shahid Beheshti University of Medical SciencesTehranIran
  5. 5.Department of Orthodontics, School of DentistryShahid Beheshti University of Medical SciencesTehranIran
  6. 6.Dentofacial Deformities Research CenterResearch Institute of Dental Sciences, School of Dentistry, Shahid Beheshti University of Medical SciencesTehranIran
  7. 7.Dental Research CenterResearch Institute of Dental Sciences, School of Dentistry, Shahid Beheshti University of Medical SciencesTehranIran

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