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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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.

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Fig. 2

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

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Fig. 3

Spring/damper representation of the Burgers model

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Fig. 4

Five-parameter spring/damper configuration

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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.

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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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