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Research Papers

Modeling Stress-Relaxation Behavior of the Periodontal Ligament During the Initial Phase of Orthodontic Treatment

[+] Author and Article Information
Jason P. Carey

e-mail: jason.carey@ualberta.ca
Department of Mechanical Engineering,
University of Alberta,
4-9 Mechanical Engineering Building,
Edmonton, AB T6G 2G8, Canada

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received December 10, 2012; final manuscript received May 14, 2013; accepted manuscript posted May 23, 2013; published online July 10, 2013. Assoc. Editor: James C. Iatridis.

J Biomech Eng 135(9), 091007 (Jul 10, 2013) (8 pages) Paper No: BIO-12-1607; doi: 10.1115/1.4024631 History: Received December 10, 2012; Revised May 14, 2013; Accepted May 23, 2013

The periodontal ligament is the tissue that provides early tooth motion as a result of applied forces during orthodontic treatment: a force-displacement behavior characterized by an instantaneous displacement followed by a creep phase and a stress relaxation phase. Stress relaxation behavior is that which provides the long-term loading to and causes remodelling of the alveolar bone, which is responsible for the long-term permanent displacement of the tooth. In this study, the objective was to assess six viscoelastic models to predict stress relaxation behavior of rabbit periodontal ligament (PDL). Using rabbit stress relaxation data found in the literature, it was found that the modified superposition theory (MST) model best predicts the rabbit PDL behavior as compared to nonstrain-dependent and strain-dependent versions of the Burgers four-parameter and the five-parameter viscoelastic models, as well as predictions by Schapery's viscoelastic model. Furthermore, it is established that using a quadratic form for MST strain dependency provides more stable solutions than the cubic form seen in previous studies.

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References

Fill, T. S., Carey, J. P., Toogood, R. W., and Major, P. W., 2011, “Experimentally Determined Mechanical Properties of, and Models for, the Periodontal Ligament: Critical Review of Current Literature,” J. Dent. Biomech., 10, p. 312980. [CrossRef]
Fill, T. S., Toogood, R. W., Major, P. W., and Carey, J. P., 2012, “Analytically Determined Mechanical Properties of, and Models for the Periodontal Ligament: Critical Review of Literature,” J. Biomech., 45, pp. 9–16. [CrossRef] [PubMed]
Proffit, W. R., Fields, H. W., and Sarver, D. M., 2007, Contemporary Orthodontics, Mosby Elsevier, St. Louis.
Jónsdóttir, S. H., Giesen, E. B. W., and Maltha, J. C., 2006, “Biomechanical Behaviour of the Periodontal Ligament of the Beagle Dog During the First 5 Hours of Orthodontic Force Application,” Eur. J. Orthod., 28, pp. 547–552. [CrossRef] [PubMed]
Owman-Moll, P., Kurol, J., and Lundgren, D., 1995, “Continuous Versus Interrupted Continuous Orthodontic Force Related to Early Tooth Movement and Root Resorption,” Angle Orthod., 65, pp. 395–401. [PubMed]
Bergomi, M., Cugnoni, J., Botsis, J., Belser, U. C., and Anselm Wiskott, H. W., 2010, “The Role of the Fluid Phase in the Viscous Response of Bovine Periodontal Ligament,” J. Biomech., 43, pp. 1146–1152. [CrossRef] [PubMed]
Pietrzak, G., Curnier, A., Botsis, J., Scherrer, S., Wiskott, A., and Belser, U., 2002, “A Nonlinear Elastic Model of the Periodontal Ligament and Its Numerical Calibration for the Study of Tooth Mobility,” Comput. Methods Biomech. Biomed. Eng., 5, pp. 91–100. [CrossRef]
Embery, G., 1990, “An Update on the Biochemistry of the Periodontal Ligament,” Eur. J. Orthod., 12, pp. 77–80. [CrossRef] [PubMed]
Carvalho, L., Moreira, R. A. S., and Simões, J. A., 2006, “Application of a Vibration Measuring Technique to Evaluate the Dynamic Stiffness of Porcine Periodontal Ligament,” Technol. Health Care, 14, pp. 457–465. [PubMed]
Picton, D. C. A., and Wills, D. J., 1978, “Viscoelastic Properties of the Periodontal Ligament and Mucous Membrane,” J. Prosthet. Dent., 40, pp. 263–272. [CrossRef] [PubMed]
Moxham, B. J., and Berkovitz, B. K. B., 1979, “The Effects of Axially-Directed Extrusive Loads on Movements of the Mandibular Incisor of the Rabbit,” Arch. Oral Biol., 24, pp. 759–763. [CrossRef] [PubMed]
Pini, M., Zysset, P., Botsis, J., and Contro, R., 2004, “Tensile and Compressive Behaviour of the Bovine Periodontal Ligament,” J. Biomech., 37, pp. 111–119. [CrossRef] [PubMed]
Dorow, C., Krstin, N., and Sander, F. G., 2003, “Determination of the Mechanical Properties of the Periodontal Ligament in a Uniaxial Tensional Experiment,” J. Orofac. Orthop., 64, pp. 100–107. [CrossRef] [PubMed]
Slomka, N., Vardimon, A. D., Gefen, A., Pilo, R., Bourauel, C., and Brosh, T., 2008, “Time-Related PDL: Viscoelastic Response During Initial Orthodontic Tooth Movement of a Tooth With Functioning Interproximal Contact-A Mathematical Model,” J. Biomech., 41, pp. 1871–1877. [CrossRef] [PubMed]
Sopakayang, R., and De Vita, R., 2011, “A Mathematical Model for Creep, Relaxation and Strain Stiffening in Parallel-Fibered Collagenous Tissues,” Med. Eng. Phys., 33, pp. 1056–1063. [CrossRef] [PubMed]
Nishihira, M., Yamamoto, K., Sato, Y., Ishikawa, H., and Natali, A. N., 2003, “Mechanics of Periodontal Ligament,” Dent. Biomech., pp. 20–34.
Nanda, R., and Kuhlberg, A., 1997, “Principles of Biomechanics,” Biomechanics in Clinical Orthodontics, R.Nanda, ed., Saunders, Philadelphia, pp. 1–20.
van Driel, W. D., van Lecuwen, E. J., von den Hoff, J. W., Maltha, J. C., and Kuijpers-Jagtman, A. M., 2000, “Time-Dependent Mechanical Behaviour of the Periodontal Ligament,” Proc. Inst. Mech. Eng. H, 214, pp. 497–504. [PubMed]
Toms, S. R., Lemons, J. E., Bartolucci, A. A., and Eberhardt, A. W., 2002, “Nonlinear Stress-Strain Behavior of Periodontal Ligament Under Orthodontic Loading,” Am. J. Orthod. Dentofacial Orthop., 122, pp. 174–179. [CrossRef] [PubMed]
Natali, A. N., Pavan, P. G., Carniel, E. L., and Dorow, C., 2004, “Viscoelastic Response of the Periodontal Ligament: An Experimental-Numerical Analysis,” Connect. Tissue Res., 45, pp. 222–230. [CrossRef] [PubMed]
Komatsu, K., Sanctuary, C., Shibata, T., Shimada, A., and Botsis, J., 2007, “Stress-Relaxation and Microscopic Dynamics of Rabbit Periodontal Ligament,” J. Biomech., 40, pp. 634–644. [CrossRef] [PubMed]
Wise, G. E., and King, G. J., 2008, “Mechanisms of Tooth Eruption and Orthodontic Tooth Movement,” J. Dent. Res., 87, pp. 414–434. [CrossRef] [PubMed]
Pilon, J. J., Kuijpers-Jagtman, A. M., and Maltha, J. C., 1996, “Magnitude of Orthodontic Forces and Rate of Bodily Tooth Movement. An Experimental Study,” Am. J. Orthod. Dentofacial Orthop., 110, pp. 16–23. [CrossRef] [PubMed]
Fung, Y. C., 1993, Biomechanics: Mechanical Properties of Living Tissues, Springer-Verlag, New York.
Wang, C. Y., Su, M. Z., Chang, H. H., Chiang, Y. C., Tao, S. H., Cheng, J. H., Fuh, L. J., and Lin, C. P., 2012, “Tension-Compression Viscoelastic Behaviors of the Periodontal Ligament,” J. Formosan Med. Assoc., 111, pp. 471–481. [CrossRef]
Komatsu, K., Shibata, T., Shimada, A., Viidik, A., and Chiba, M., 2004, “Age-Related and Regional Differences in the Stress-Strain and Stress-Relaxation Behaviours of the Rat Incisor Periodontal Ligament,” J. Biomech., 37, pp. 1097–1106. [CrossRef] [PubMed]
Findley, W. N., Lai, J. S., and Onaran, K., 1976, “Linear Viscoelastic Constitutive Equations,” Creep and Relaxation of Nonlinear Viscoelastic Materials, North-Holland, New York, pp. 57–64.
Shepherd, T. N., Zhang, J., Ovaert, T. C., Roeder, R. K., and Niebur, G. L., 2011, “Direct Comparison of Nanoindentation and Macroscopic Measurements of Bone Viscoelasticity,” J. Mech. Behav. Biomed. Mater., 4, pp. 2055–2062. [CrossRef] [PubMed]
Findley, W. N., Lai, J. S., and Onaran, K., 1976, “Multiple Integral Representation,” Creep and Relaxation of Nonlinear Viscoelastic Materials, North-Holland, New York, pp. 172–175.
Schapery, R. A., 1966, “An Engineering Theory of Nonlinear Viscoelasticity With Applications,” Int. J. Solids Struct., 2, pp. 407–425. [CrossRef]
Provenzano, P. P., Lakes, R. S., Corr, D. T., and Vanderby, R., Jr., 2002, “Application of Nonlinear Viscoelastic Models to Describe Ligament Behavior,” Biomech. Model. Mechanobiol., 1, pp. 45–57. [CrossRef] [PubMed]
Findley, W. N., Lai, J. S., and Onaran, K., 1976, “Nonlinear Creep (or Relaxation) Under Variable Stress (or Strain),” Creep and Relaxation of Nonlinear Viscoelastic Materials, North-Holland, New York, pp. 229–233.
Marangalou, J. H., Ghalichi, F., and Mirzakouchaki, B., 2009, “Numerical Simulation of Orthodontic Bone Remodeling,” Orthodont. Waves, 68, pp. 64–71. [CrossRef]
Romanyk, D. L., Liu, S. S., Lipsett, M. G., Toogood, R. W., Lagravère, M. O., Major, P. W., and Carey, J. P., “Towards a Viscoelastic Model for the Unfused Midpalatal Suture: Development and Validation Using the Midsagittal Suture in New Zealand White Rabbits,” J. Biomech., (in press).
Favino, M., Krause, R., and Steiner, J., 2012, “An Efficient Preconditioning Strategy for Schur Complements Arising From Biphasic Models,” Proceedings of the 9th IASTED International Conference on Biomedical Engineering, BioMed 2012, pp. 700–705.
Blake, M., Woodside, D. G., and Pharoah, M. J., 1995, “A Radiographic Comparison of Apical Root Resorption After Orthodontic Treatment With the Edgewise and Speed Appliances,” Am. J. Orthodont. Dentofac. Orthoped., 108, pp. 76–84. [CrossRef]
Purslow, P. P., Wess, T. J., and Hukins, D. W. L., 1998, “Collagen Orientation and Molecular Spacing During Creep and Stress-Relaxation in Soft Connective Tissues,” J. Exp. Biol., 201, pp. 135–142. [PubMed]

Figures

Grahic Jump Location
Fig. 1

Three-stage schematic of the PDL response due to applied orthodontic loading. Stage 1 (time 0–0+), is the initial load drop caused by the PDL compliance. Stage 2 (time 0+–T1) is characterized by an almost linear constant force with a nonlinear creep PDL displacement. This can be considered the transient stage. Stage 3 (time T1–T2) is characterized by the maximum PDL displacement and a decaying stress relaxation to the zero force level. This can be considered a stable or steady-state stage.

Grahic Jump Location
Fig. 2

Representation of test specimens utilized by Komatsu et al. [21]

Grahic Jump Location
Fig. 3

Spring/damper representation of the Burgers model

Grahic Jump Location
Fig. 4

Five-parameter spring/damper configuration

Grahic Jump Location
Fig. 5

Comparison of quadratic and cubic polynomial fit: (a) Gr(ε, t) = {0.4, 0.5, 0.58, 0.6) for time t = 200 s; (b) Gr(ε, t) = {0.38, 0.49, 0.57, 0.59) for time t = 200 s; (c) Gr(ε, t) = {0.41, 0.51, 0.585, 0.6) for time t = 200 s; (d) Gr(ε, t) = {0.39, 0.5, 0.57, 0.59) for time t = 300 s; (e) Gr(ε, t) = {0.38, 0.49, 0.54, 0.56) for time t = 300 s; (f) Gr(ε, t) = {0.37, 0.48, 0.53, 0.56) for time t = 300 s. The R2 for all curves range between 0.88 and 0.89 for both cubic and quadratic approaches.

Grahic Jump Location
Fig. 6

Evaluation of viscoelastic models that describe the behavior of the PDL with initial deformation of (a) 36 μm, (b) 57 μm, (c) 78 μm, and (d) 99 μm. Data of (a) were used to determine model constants. (b)–(d) show the predictive capabilities of the models for different deformation levels. Left figure is the full data from 0 to 300 seconds. The right is a close-up view of the first 20 seconds.

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