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Analytical Modelling of the Tooth Translational Motions: Comparative Analysis for Different Shapes of Root

  • Kirill Yurkevich
  • Sergei Bosiakov
  • Holm AltenbachEmail author
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 80)

Abstract

The orthodontic treatment planning may be carried out based on the finite element and analytical models of periodontal ligament (PDL). For analytical modelling of the PDL behavior the shape of the tooth root mainly was approximated by circular or elliptical paraboloid. Another shape of the tooth root is the elliptical two-sheeted hyperboloid. Semi-axes of the tooth root cross-section in the shape of the elliptical paraboloid and two-sheeted hyperboloid on the alveolar crest level are the same, but the shape of a two-sheeted hyperboloid allows employing the additional parameter for describing the root apex rounding. The aim of this study is the comparative analysis of the hydrostatic stresses patterns during the tooth root translational displacements in the almost incompressible PDL for the root in the shape of the elliptical paraboloid and two-sheeted hyperboloid. As a result, patterns of the hydrostatic stresses in the PDL during translational displacement are nearly identical for the tooth root in the shape of a paraboloid and the tooth root in the shape of a two-sheeted hyperboloid with the rounded apex of the tooth root. The translational movement of the tooth root with a pointed apex leads to the higher hydrostatic stresses in the PDL compared with the tooth root with a rounded apex. The obtained results indicated that the rounding of the tooth root should be considered during planning of orthodontic treatment.

Keywords

Orthodontics Periodontal ligament Initial translational tooth motion Idealized shape Stress patterns Analytical model 

Notes

Acknowledgements

The authors acknowledge the support of FP7 IRSES Marie Curie grant No 610547 TAMER and DAAD grant No A/12/87820.

References

  1. 1.
    Bosiakov, S., Mselati, A., Krupoderov, A.: Mathematical modelling of initial displacements of the tooth root in the hyperboloid of two sheets form. Russ. J. Biomech. 19(2), 161–176 (2015)Google Scholar
  2. 2.
    Bourauel, C., Vollmer, D., Jager, A.: Application of bone remodeling theories in the simulation of orthodontic tooth movements. J. Orofac. Orthop. 61(4), 266–279 (2000)CrossRefGoogle Scholar
  3. 3.
    Cattaneo, P., Dalstra, M., Melsen, B.: The finite element method: a tool to study orthodontic tooth movement. J. Dent. Res. 84(5), 428–433 (2005)CrossRefGoogle Scholar
  4. 4.
    Chang, Y., Shin, S., Baek, S.: Three-dimensional finite element analysis in distal en masse movement of the maxillary dentition with the multiloop edgewise archwire. Eur. J. Orthod. 26, 339–345 (2004)CrossRefGoogle Scholar
  5. 5.
    Clement, R., Schneider, J., Brambs, H., Wunderlich, A., Geiger, M., Sander, F.: Quasi-automatic 3D finite element model generation for individual single-rooted teeth and periodontal ligament. Comp. Meth. Prog. Biomed. 73(2), 135–144 (2004)CrossRefGoogle Scholar
  6. 6.
    De Pauw, G., Dermaut, L., De Bruyn, H.: The value of the centre of rotation in initial and longitudinal tooth and bone displacement. Eur. J. Orthod. 25, 285–291 (2003)CrossRefGoogle Scholar
  7. 7.
    Hayashia, K., Arakib, Y., Uechia, J., Ohnoc, H., Mizoguchi, I.: A novel method for the three-dimensional (3-d) analysis of orthodontic tooth movement calculation of rotation about and translation along the finite helical axis. J. Biomech. 2002, 45–51 (2002)CrossRefGoogle Scholar
  8. 8.
    Hohmann, A., Kober, C., Young, P., Dorow, C., Geiger, M., Boryor, A., Sander, F.M., Sander, C., Sander, F.: Influence of different modeling strategies for the periodontal ligament on finite element simulation results. Am. J. Orthod. Dentofac. Orthop. 139(6), 775–783 (2011)CrossRefGoogle Scholar
  9. 9.
    Jeon, P., Turley, P., Moon, H., Ting, K.: Analysis of stress in the periodontium of the maxillary first molar with a three-dimensional finite element model. Am. J. Orthod. Dentofac. Orthop. 115(3), 267–274 (1999)CrossRefGoogle Scholar
  10. 10.
    Kojima, Y., Fukui, H.: A numerical simulation of tooth movement by wire bending. Am. J. Orthod. Dentofac. Orthop. 130, 452–459 (2006)CrossRefGoogle Scholar
  11. 11.
    Kusy, R., Tulloch, C.: Analysis of moment/force ratios tooth movement. Am. J. Orthod. Dentofac. Orthop. 90, 127–131 (1986)CrossRefGoogle Scholar
  12. 12.
    Maceri, F., Marino, G., Vairo, G.: Mechanical behaviour of endodontic restorations with multiple prefabricated posts: a finite-element approach. J. Biomech. 40, 2386–2398 (2007)CrossRefGoogle Scholar
  13. 13.
    Middleton, J., Jones, M., Wilson, A.: The role of the periodontal ligament in bone modeling: the initial development of a time-dependent finite element model. Am. J. Orthod. Dentofac. Orthop. 109, 155–162 (1996)CrossRefGoogle Scholar
  14. 14.
    Mohandesan, H., Ravanmehr, H., Valaei, N.: A radiographic analysis of external apical root resorption of maxillary incisors during active orthodontic treatment. Eur. J. Orthod. 29(2), 134–139 (2007)Google Scholar
  15. 15.
    Natali, A.N., Pavan, P.G., Scarpa, C.: Numerical analysis of tooth mobility: formulation of a nonlinear constitutive law for the periodontal ligament. Dent. Mater. 20, 623–629 (2004)CrossRefGoogle Scholar
  16. 16.
    Nikolai, R.: Rigid-body kinematics and single-tooth displacements. Am. J. Orthod. Dentofac. Orthop. 110, 88–92 (1996)CrossRefGoogle Scholar
  17. 17.
    Provatidis, C.G.: A comparative fem-study of tooth mobility using isotropic and anisotropic models of the periodontal ligament. finite element method. Med. Eng. Phys. 22, 359–370 (2000)CrossRefGoogle Scholar
  18. 18.
    Provatidis, C.G.: An analytical model for stress analysis of a tooth in translation. Int. J. Eng. Sci. 39(12), 1361–1381 (2001)CrossRefGoogle Scholar
  19. 19.
    Qian, H., Chen, J., Katona, T.R.: The influence of pdl principal fibers in a 3-dimensional analysis of orthodontic tooth movement. Am. J. Orthod. Dentofac. Orthop. 120, 272–279 (2001)CrossRefGoogle Scholar
  20. 20.
    Qian, Y., Fan, Y., Liu, Z., Zhang, M.: Numerical simulation of tooth movement in a therapy period. Clin. Biomech. 23, S48–S52 (2008)CrossRefGoogle Scholar
  21. 21.
    Rees, J., Jacobsen, P.: Elastic modulus of the periodontal ligament. Biomaterials 18, 995–999 (1997)CrossRefGoogle Scholar
  22. 22.
    Smith, R.J., Burstone, C.J.: Mechanics of tooth movement. Am. J. Orthod. 85(4), 294–307 (1984)CrossRefGoogle Scholar
  23. 23.
    Tanne, K., Nagataki, T., Inoue, Y., Sakuda, M., Burstone, C.J.: Patterns of initial tooth displacement associated with various root lengths and alveolar bone heights. Am. J. Orthod. Dentofac. Orthop. 100, 66–71 (1991)CrossRefGoogle Scholar
  24. 24.
    Toms, S.R., Eberhardt, A.W.: A nonlinear finite element analysis of the periodontal ligament under orthodontic tooth loading. Am. J. Orthod. Dentofac. Orthop. 123, 657–665 (2003)CrossRefGoogle Scholar
  25. 25.
    Van Schepdael, A., Geris, L., Vander Sloten, J.: Analytical determination of stress patterns in the periodontal ligament during orthodontic tooth movement. Med. Eng. Phys. 35, 403–410 (2013)Google Scholar
  26. 26.
    Van Schepdael, A., Vander Sloten, J., Geris, L.: Mechanobiological modeling can explain orthodontic tooth movement: Three case studies. J. Biomech. 46, 470–477 (2013)Google Scholar
  27. 27.
    Vollmer, D., Bourauel, C., Maier, K., Jager, A.: Determination of the center of resistance in an upper human canine and idealized tooth model. Eur. J. Orthod. 21(6), 633–648 (1999)CrossRefGoogle Scholar
  28. 28.
    Yashiro, K., Montero, E.D.C., Takada, K.: Simulation of initial tooth displacement by inverse kinematic modeling: Prediction of anchorage-loss for orthodontic tooth movement. Int. Cong. Ser. 1284, 49–54 (2005)CrossRefGoogle Scholar
  29. 29.
    Ziegler, A., Keilig, L., Kawarizadeh, A., Jager, A., Bourauel, C.: Numerical simulation of the biomechanical behaviour of multi-rooted teeth. Eur. J. Orthod. 27(4), 333–339 (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Kirill Yurkevich
    • 1
  • Sergei Bosiakov
    • 1
  • Holm Altenbach
    • 2
    Email author
  1. 1.Belarusian State UniversityMinskBelarus
  2. 2.Institute of Mechanics, Faculty of Mechanical EngineeringOtto-von-Guericke-University MagdeburgMagdeburgGermany

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