Tension-Compression Viscoelastic Behaviors of the Periodontal Ligament PDF

Journal of the Formosan Medical Association (2012) 111, 471e481 Available online at www.sciencedirect.com journal homepage: www.jfma-online.com ORIGINAL ARTICLE Tension-compression viscoelastic behaviors of t...

Journal of the Formosan Medical Association (2012) 111, 471e481 Available online at www.sciencedirect.com journal homepage: www.jfma-online.com ORIGINAL ARTICLE Tension-compression viscoelastic behaviors of the periodontal ligament Chen-Ying Wang a,b, Ming-Zen Su a,b, Hao-Hueng Chang a,b, Yu-Chih Chiang a,b, Shao-Huan Tao c, Jung-Ho Cheng c, Lih-Jyh Fuh d,*, Chun-Pin Lin a,b,** a Graduate Institute of Clinical Dentistry, School of Dentistry, National Taiwan University, Taipei, Taiwan, ROC b Department of Dentistry, National Taiwan University Hospital, Taipei, Taiwan, ROC c Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan, ROC d School of Dentistry & Graduate Institute of Dental Sciences, China Medical University, Taichung, Taiwan, ROC Received 1 April 2011; received in revised form 20 June 2011; accepted 27 June 2011 KEYWORDS Background/Purpose: Although exhaustively studied, the mechanism responsible for tooth creep source; support and the mechanical properties of the periodontal ligament (PDL) remain a subject finite element of considerable controversy. In the past, various experimental techniques and theoretical analysis; analyses have been employed to tackle this intricate problem. The aim of this study was to periodontal ligament; investigate the viscoelastic behaviors of the PDL using three-dimensional finite element viscoelastic behavior analysis. Methods: Three dentoalveolar complex models were established to simulate the tissue behaviors of the PDL: (1) deviatoric viscoelastic model; (2) volumetric viscoelastic model; and (3) tension-compression volumetric viscoelastic model. These modified models took into consideration the presence of tension and compression along the PDL during both loading and unloading. The inverse parameter identification process was developed to determine the mechanical properties of the PDL from the results of previously reported in vitro and in vivo experiments. Results: The results suggest that the tension-compression volumetric viscoelastic model is a good approximation of normal PDL behavior during the loading-unloading process, and the deviatoric viscoelastic model is a good representation of how a damaged PDL behaves under loading conditions. Moreover, fluid appears to be the main creep source in the PDL. * Corresponding author. DDS, School of Dentistry & Graduate Institute of Dental Sciences, China Medical University, Number 91, Hsueh- Shih Road, Taichung, Taiwan 40402, R.O.C. ** Corresponding author. School of Dentistry and Graduate Institute of Clinical Dentistry, National Taiwan University and National Taiwan University Hospital, Number 1, Chang Te Street, Taipei 10016, Taiwan, ROC. E-mail addresses: [email protected] (L.-J. Fuh), [email protected] (C.-P. Lin). 0929-6646/$ - see front matter Copyright ª 2012, Elsevier Taiwan LLC & Formosan Medical Association. All rights reserved. doi:10.1016/j.jfma.2011.06.009 472 C.-Y. Wang et al. Conclusion: We believe that the biomechanical properties of the PDL established via retro- grade calculation in this study can lead to the construction of more accurate extra-oral models and a comprehensive understanding of the biomechanical behavior of the PDL. Copyright ª 2012, Elsevier Taiwan LLC & Formosan Medical Association. All rights reserved. Introduction theoretical biomechanical applications. It has been successfully applied to numerous biomechanical problems The periodontal ligament (PDL) is a dense, fibrous, connec- since its debut in the orthopedic biomechanics literature in tive tissue that surrounds the root of the tooth and attaches 1972.10 Although several finite element models of teeth the tooth to the alveolus in order to provide tooth support.1 (both with and without dental restorations) have been Like soft fibrous connective tissue elsewhere in the body, the published, some models are two-dimensional11,12 or PDL also consists of a fibrous stroma in a gel of ground axisymmetric approximations13,14 that do not completely substance that contains cells, blood vessels, and nerves. The represent the three-dimensional geometry or actual loading fibrous stroma is composed primarily of collagen, while the conditions associated with dental structures. In studies that cells are mainly fibroblasts. incorporated three-dimensional representations of teeth, Because the PDL is generally considered to be a suspen- either the geometry or the material’s properties were sory ligament, several ideas such as tensile, compression, simplified. In some cases, parts of the dental structure such fluid-filled, and viscoelastic theories, have evolved to as the pulp, cementum, PDL, or the alveolar bone were explain the intricacies of tooth support.2e4 However, all of disregarded altogether.15 In addition, the interfacial inter- the current theories consider this to be a multiphase system actions between the root and its surrounding periodontium that consists of fibers, ground substance, blood vessels, and have rarely been addressed. Because the PDL is believed to fluid that interacts to resist mechanical forces. The results act as a viscoelastic, shock-absorbing, load-transfer of past studies suggest that the viscoelastic theory best medium between the tooth and the alveolar bone,16 the depicts the mechanisms within the PDL. After comparing interfacial load-transfer mechanism of the PDL and its PDL behavior with that of various types of mechanical surrounding tooth and alveolar bone should be taken into springs and dampers, Bien arrived at the conclusion that account when modeling the PDL. We have taken these the PDL behaves as a viscoelastic gel.5 Moreover, Picton and factors into consideration when constructing our mathe- Will listed five characteristics of a stressed periodontal matical models. ligament, all of which imply that the PDL possesses visco- Ross7 and Walker9 conducted experiments to examine elastic properties.6 creep behavior and determined its dependency on time, Although many experimental techniques and theoretical displacement, and intraperiodontal pressure (IPP). Although analyses have been employed in the past to aid in the they were unable to fully explain the complex mechanisms understanding the PDL, experiments concerning the PDL of the PDL with their data, they suggested that an are especially difficult to perform due to its complex explanation could be derived via mathematical modeling. structure. With the hope of creating a more accurate Thus, the primary goal of this study was to investigate the representation of the PDL, we incorporated mathematical PDL creep source using previously reported human and models into our data analysis. Mathematical modeling animal tooth behaviors and finite element analysis to has been widely used and accepted as a supplementary validate the data gathered from Walker’s and Ross’s research tool. Although not an exact representation of the experiments and to ascertain the mechanical properties of human body, mathematical modeling has successfully the PDL. A series of models relating the functions and answered questions regarding orthopedics, sports medi- properties of the PDL were proposed to examine the creep cine, and the specifics of human movement. Numerous behaviors of the PDL, and, ultimately, the most suitable researchers have also become proponents of mathematical finite element model for simulating PDL behavior was modeling because of its precision, credibility, and accurate identified. results. Ross et al asserted that man can never know all the parts and transformations of a real system, but he may hope to establish a model that mimics it; e.g., when Materials and methods simulating a dynamical system, the model should satisfac- torily mimic observed real-time behavior.7 Yoshida et al Construction of the finite element models took in vivo measurements of the elastic modulus of the human PDL and used the finite element method to simulate The construction of the finite element models began by orthodontic tooth movement.8 In addition, Walker et al designing the geometric mesh of a tooth based on the gathered a great deal of perplexing experimental data and dimensions of a sample tooth. In this study, a maxillary suggested that mathematical modeling might be able to central incisor was embedded and sectioned into 19 slices clarify many of the unresolved questions regarding the in order to construct a three-dimensional finite element PDL.9 Clearly, mathematical modeling has steadily gained model. The three-dimensional finite element model, which its share of advocates in dental research. was comprised of the maxillary central incisor, pulp Finite element analysis is a specific type of mathemat- chamber, cementum, PDL, cortical bone, and cancellous ical modeling that has become the favored choice in bone, consisted of 3772 nodes and 3721 eight-node Viscoelastic behaviors of the PDL 473 isoparametric solid elements. The construction of the finite Table 1 Material constants of the finite element model element model begins with the design of a geometric model (Goel, 1992). of the tooth, which was based on the dimensions of a sample tooth and the general morphology of the maxillary Tissue Modulus of elasticity Poisson’s central incisors.6 Since the width of the PDL ranges from (Mpa) ratio 0.15e0.38 mm at different locations along the tooth,17 Enamel 84,000 0.31 a 0.25-mm-thick layer was created to represent the PDL. Dentin 18,600 0.31 The alveolar bone was modeled according to the reported Cementum 18,600 0.31 size and shape determined in a related study (Fig. 1).18 Two Cortical Bone 13,700 0.3 analytic programs, ABAQUS 6.2.1 (ABAQUS Inc.) and I-DEAS Trabecular Bone 1370 0.3 8.0 (Siemens PLM Software), were used to construct the Pulp 2 0.45 finite element models.17 The mechanical properties of the enamel, dentin, pulp chamber, cementum, cortical bone, and cancellous bone, which were based on the results of where eij and 3 jj are the mechanical deviatoric and volu- previous studies, are listed in Table 1. The boundary metric strains, respectively, and G(t) and K(t) are the conditions at the bottom of the alveolar bone are fixed and appropriate relaxation functions of the reduced time, t, asymmetrical, as shown in Fig. 2. that denote differentiation with respect to t.19 The relax- ation functions, G(t) and K(t), can be individually defined in The constitutive PDL equation terms of a series of exponents known as the Prony series: ! The basic hereditary integral formulation of linear isotropic X N t=ti viscoelasticity, in terms of its deviatoric and dilatational GðtÞZG0 gN þ gi e ; ð2Þ components, is defined as follows: iZ1 Zt ! veij ðtÞ X N t=ti sij ðtÞZ2 Gðt  tÞ dt KðtÞZK0 kN þ ki e ; ð3Þ vt iZ1 N ð1Þ Zt v3 jj ðtÞ where G0 and K0 are the instantaneous shear and bulk sii ðtÞZ3 Kðt  tÞ dt vt moduli, respectively, and gi and ki are the relative moduli N of i. Note the following: Figure 1 Finite element mesh of the maxillary central incisor. 474 C.-Y. Wang et al. Figure 2 Boundary and loading conditions. (A) Boundary condition. (B) Lateral loading condition. (C) Intrusive loading condition.   X N X N GðtÞZG0 gN þ g1 et=t1 þ g2 et=t2 gN þ gi et=ti ZkN þ ki et=ti Z1 ð4Þ ð6Þ KðtÞZK0 Zconstant; iZ1 iZ1 In the tension-compression volumetric viscoelastic The volumetric viscoelastic, deviatoric viscoelastic, and model, the temporal behavior is derived from the volu- tension-compression volumetric viscoelastic models were metric portion of the strain tensor, and separate rates of proposed in order to represent the mechanical properties volumetric change were set to differentiate between areas of the PDL, as shown in Table 2. of tension and compression. The relaxation functions, G(t) In the volumetric viscoelastic model, temporal behavior and K(t), were chosen to describe the temporal behavior: is derived from the volumetric element of the strain tensor, and the following relaxation functions, G(t) and K(t), were  0 Zconstant GðtÞZG   chosen to describe the temporal behavior: K k þ k1 et=t1 þ k2 et=t2 ; for3 ii  0 ð7Þ kðtÞZ C0  N t=t1 t=t2 KT0 kN þ k1 e þ k2 e ; for3 ii > 0 GðtÞZG0 Zconstant where KC0 and KT0 are the instantaneous bulk moduli in the   ð5Þ KðtÞZK0 kN þ k1 et=t1 þ k2 et=t2 areas of compression and tension, respectively. Hence, G(t) is constant and K(t) is set to be different in areas of tension In the deviatoric viscoelastic model, the temporal and compression, respectively. behavior is derived from the deviatoric portion of the strain In the deviatoric viscoelastic model, we assumed that tensor, and the following relaxation functions, G(t) and the PDL was incompressible and that the shear-time effect K(t), were chosen to describe the temporal behavior: would dominate creep behavior. Meanwhile, the elastic Table 2 PDL mechanical models and their corresponding creep sources. Model type Creep source Source of material constants Model assumptions Volumetric viscoelastic Free fluid flow Loading part of the creep Volumetric temporal behavior only; experiment by Ross linear elastic behavior Deviatoric viscoelastic Distortion Loading part of the creep Almost incompressible; deviatoric experiment by Ross temporal behavior only; linear elastic behavior Tension-compression Free fluid flow Entire creep experiment by Ross Volumetric temporal behavior only; volumetric viscoelastic linear elastic behavior Viscoelastic behaviors of the PDL 475 behavior was assumed to be nonlinear because the magni- displacement error. In general, the objective value equa- tude of the shear stress may be influenced by the surface of tion is written as a function of least squares: the collagen fibrils and the degree of cross-linking. X N  2 MinQ Z bi DFEA i  DRoss i ; ð8Þ Inverse parameter identification process iZ1 where bi represent the weighted factors necessary to scale Because the environmental conditions inside the mouth are the objective value. too dissimilar from the conditions outside the mouth, To minimize this nonlinear objective function, the ascertaining the true mechanical properties of the PDL via complex method, which only uses function values, can be standard means is exceedingly difficult. The inverse iden- implemented. The complex method20 is the constrained tification process, diagrammed in Fig. 3, is a feasible simplex method that incorporates a direct search algorithm alternative. Based on the viscoelastic constitutive equa- to more efficiently solve the optimization problem associ- tions, we used the constant loading displacement-time ated with the results generated by the ABAQUS software. curve from the lateral tooth movement experiment7 to In order to study the pressure stress within the PDL obtain the mechanical properties of the PDL and, in during loading, data from Walker’s study7 were used as accordance with that experiment, a 0.05 N lingually a reference source for the actual PDL pressure stress directed load was applied, maintained for 2.5 seconds, and responses during loading. In Walker’s animal study, pres- then released. The relaxation parameters were defined sure recordings in the PDL of canine teeth were divided to correspond to the parameters of the Prony series in according to pattern type, and the most common patterns ABAQUS, and the optimal parameters were obtained from were the positive-peaking (P-response) and sustained retrograde calculations using Ross’s experimental data. The pressure (S-response) types. The P-response represents the retrograde calculation method was based on an optimiza- pressure changes in physiologically normal PDL tissue, tion method that minimizes the square of the tooth whereas the S-response is the product of the traumas sus- tained by the ligament. In this study, the pressure stresses in the PDL were investigated by modeling a 397 g vertical load to the crown of the tooth in both the deviatoric viscoelastic and tension-compression volumetric visco- elastic models. The pressure stresses from the apex of tooth root to the cementoenamel junction were recorded. Finite element stress analysis Because neither in vivo nor in vitro studies are capable of determining the internal mechanics of the PDL under loading conditions, finite element models were used in this study to depict the internal stress distribution of the PDL. Using the tension-compression volumetric and deviatoric viscoelastic models, the application of two loading condi- tions to the crown of the tooth was modeled. First, a 0.05 N lingually directed load was applied, maintained for 2.5 seconds, and then released. Next, a vertical load of the same magnitude was applied and sustained for the same duration. The stress distributions of the dentoalveolar complex during the application of loading and sustained loading were investigated using both models and compared with each other. Results Fig. 4 illustrates the creep test data for the volumetric viscoelastic, deviatoric viscoelastic, and tension-compression volumetric viscoelastic models, compared with the data of Ross’ Experiment. Most significantly, the volumetric visco- elastic, deviatoric viscoelastic, and tension-compression volumetric viscoelastic models that were implemented in this experiment closely simulated the results from Ross’s experiment when loading was applied to the tooth and corresponding PDL. This result indicates that the hereditary Figure 3 Flow chart of inverse parameters identification integral formulation model for linear isotropic viscoelas- procedure. ticity is capable of approximating the behavior of the PDL 476 C.-Y. Wang et al. Figure 4 Typical curves of creep test data for the volumetric viscoelastic, deviatoric viscoelastic, and tension-compression volumetric viscoelastic models, compared with the data of Ross’ Experiment. when subjected to loading forces. The viscoelastic prop- unloading phase of Ross’s experiment. More precisely, the erties of the PDL, represented as an elastic modulus and PDL’s rate of recovery during unloading seems to be slower Prony constants in Table 3, were obtained via an inverse than its rate of compression during loading. Thus, the parameter identification process. The relaxation functions volumetric viscoelastic model was modified to incorporate can be expressed as follows: tensile and compressive relaxation functions to describe PDL behavior during loading and unloading. The elastic 1. Volumetric viscoelasticity modulus and Prony constants obtained via inverse param- GðtÞZG0 Z0:0634 eter identification from Ross’s data are listed in Table 3. ð9Þ KðtÞ Z0:076 þ 0:454et=0:0873 þ 0:0833et=0:694 When these constants were used, the tension-compression volumetric viscoelastic model was able to emulate the 2. Deviatoric viscoelasticity behavior of the PDL during both loading and unloading, as shown in Fig. 4. GðtÞZ0:00376 þ 0:0391et=0:0209 þ 0:0117et=0:457 ð10Þ Data on IPP were also collected in this study. Fig. 5(A) KðtÞZK0 Z0:89062 shows the IPP observed during loading and unloading for the 3. Tension-compression volumetric viscoelasticity volumetric viscoelastic model. Before loading, IPP was measured at the initial pressure level. When a loading force Tensile area : was applied to the tooth, the volumetric viscoelastic model GðtÞZG0 Z0:0133 indicated that the IPP of the corresponding PDL increased KðtÞ Z0:1032 þ 0:4368et=0:11 þ 0:06et=1:48 sharply before rapidly subsiding to the initial pressure ð11Þ level. Upon unloading, the PDL experienced a similar but Compressive area : GðtÞZG0 Z0:0133 inverse effect to its IPP. The resulting IPP pattern during KðtÞZ0:00167 þ 0:00708et=0:11 þ 0:000973et=1:48 the loading-unloading process bears a striking resemblance to the waveform of the P-response observed in Walker’s The volumetric and deviatoric viscoelastic mathematical study. models were not able to use these same sets of constants to Fig. 5(B) shows the IPP pattern observed during loading accurately reproduce the empirical data from the and unloading of the deviatoric viscoelastic model. In Table 3(a) Mechanical parameters of the volumetric viscoelastic model and deviatoric viscoelastic models for the loading period. Volumetric viscoelastic model Deviatoric viscoelastic model Elastic behavior K0 0.613 0.891 G0 0.0634 0.0545 Temporal behavior Prony constants N 1 2 N 1 2 ti 0.0873 0.694 ti 0.0209 0.0566 ki 0.740 0.136 ki 0 0 gi 0 0 gi 0.717 0.214 Viscoelastic behaviors of the PDL 477 Upon vertical loading in the tension-compression volu- Table 3(b) Mechanical parameters of the tension- metric viscoelastic model, as shown in Fig. 8, the pressure compression volumetric viscoelastic model. immediately increased in both the root apex and the apical Tension-compression volumetric third of the PDL. During sustained vertical loading, the viscoelastic model pressure at the apical third of the PDL decreased substan- tially, and the remaining stress appeared to be uniformly Tensile area Compressive area distributed over the entire PDL, as shown in Fig. 8. Elastic K0 0.6 0.00973 The deviatoric viscoelastic model, however, suggests behavior G0 0.0133 0.0133 a different scenario for PDL behavior when subjected to Temporal Prony N 1 2 vertical loading. Not only did pressure immediately behavior constants ti 0.11 1.48 increase at the crown of the tooth, but the entire PDL also ki 0.728 0.10 experienced an increase in pressure rather than just at the apical third as indicated by the volumetric viscoelastic model. The stress concentration along the PDL, as shown in contrast to the IPP levels obtained by the volumetric model, Fig. 9, increased and reached its maximum intensity along the IPP pattern observed in the deviatoric viscoelastic the apical third of the PDL, near the root apex. During model resembles the waveform of the S-response observed sustained loading, the pressure did not dissipate but rather in Walker’s experiment. Although IPP rapidly increased increased in magnitude. Although stress at the coronal third when loading forces were applied, the pressure did not of the PDL decreased slightly, the pressure at the apical decay as quickly as in the volumetric viscoelastic model. third of the PDL increased in both intensity and coverage Instead, the IPP remained steady until unloading, during area, as shown in Fig. 9. which it immediately decreased to the preloading level. When lingual loading was applied to the crown of the tooth, the tension-compression volumetric viscoelastic Discussion model (Fig. 6) indicated that the pressure stress immediately increased near the cervical area of the PDL. Fig. 6 illustrates The time dependency of the mechanical behavior of the that over time the stress dissipated to a slight degree, and PDL is well known.21e23 Based on histological findings in the the remaining pressure was distributed in a relatively PDL and theories on continuity mechanics, we attributed uniform fashion over the entire PDL. tooth creep behavior to two possible sources. One source is According to the deviatoric viscoelastic model, applying the volumetric effect, which represents the free fluid in the lingual loading to the crown of the tooth also created PDL that flows in the space between the PDL and the immediate pressure at the coronal third of the PDL as well alveolar bone; the other source is the deviatoric effect, at the cervical area, as shown in Fig. 7. During sustained which corresponds to changes in the shape of the PDL loading, however, the pressure did not dissipate, as sug- material that occur due to loading effects over time. To gested by the volumetric viscoelastic model, but rather study these two sources of tooth creep behavior, we increased in magnitude as well as coverage area, as illus- initially constructed two models: the volumetric visco- trated in Fig. 7D. In addition, the pressure due to loading elastic model and the deviatoric viscoelastic model. By was not distributed along the entire PDL. Instead, the implementing Prony constants and using the results from increased stress remained localized near the coronal third Ross’s lateral tooth movement experiment as a reference, of the PDL, as shown in Fig. 7B. the creep behavior of the PDL was modeled. Both the Figure 5 Typical PDL pressure stress in the middle of the root for (A) the tension-compression volumetric viscoelastic model and (B) the deviatoric viscoelastic model. 478 C.-Y. Wang et al. Figure 6 Pressure stress distribution pattern under lateral loading of the tension-compression volumetric viscoelastic model. (A) Pressure stress at t Z 0.01 seconds in the PDL. (B) Pressure stress at t Z 2.5 seconds in PDL. (C) Pressure stress at t Z 0.01 seconds in the labial-lingual direction. (D) The pressure stress at t Z 2.5 seconds in the labial-lingual direction. volumetric viscoelastic and deviatoric viscoelastic models viscoelastic behavior of the PDL during loading with a good were used to simulate tooth creep behavior, and the cor- degree of accuracy, neither model was able to accurately responding mechanical constants were obtained via retro- predict the displacement-time relationship during unload- grade calculation, as shown in Table 3. ing. We surmise that this discrepancy can be attributed to A slight discrepancy was found between the displacement- our assumption that PDL behavior is consistent throughout time relationship observed in Ross’s experiment and the the entire PDL during both loading and unloading. In fact, same relationship that was predicted by both the volu- the PDL experiences both tension and compression during metric viscoelastic and deviatoric viscoelastic models.7 loading as well as unloading, and in areas of tension fluid Although both models were able to approximate the enters the PDL while fluid is leaving the PDL in areas of Figure 7 Pressure stress distribution pattern under lateral loading of the deviatoric viscoelastic model. (A) Pressure stress at t Z 0.01 seconds in the PDL. (B) Pressure stress at t Z 2.5 seconds in the PDL. (C) The pressure stress at t Z 0.01 seconds in labial- lingual direction. (D) Pressure stress at t Z 2.5 seconds in the labial-lingual direction. Viscoelastic behaviors of the PDL 479 Figure 8 Pressure stress distribution pattern under intrusive loading of the tension-compression volumetric viscoelastic model. (A) Pressure stress at t Z 0.01 seconds in the PDL. (B) Pressure stress at t Z 2.5 seconds in the PDL. (C) Pressure stress at t Z 0.01 seconds in the labial-lingual direction. (D) Pressure stress at t Z 2.5 seconds in the labial-lingual direction. compression. In other words, fluid may simultaneously tension. Another implication is that during loading, the enter and leave the PDL through different areas during both majority of the areas along the PDL are in compression, loading and unloading. The tension-compression volumetric while more areas are tense during unloading. viscoelastic model, which was constructed to take the While the volumetric viscoelastic model was modifiable, aforementioned point into account, was able to portray the the deviatoric viscoelastic model could not be adapted displacement-time relationship during the entire loading- to account for the presence of tension and compression in unloading process. Moreover, the data indicated that the the PDL during both unloading and unloading. The devia- fluid flow rate of fluid leaving the PDL is greater than the toric viscoelastic model assumes that the PDL is incom- flow rate of the fluid entering the PDL. The successful pressible and that the shear-time effect dominates PDL approximation achieved by the tension-compression volu- creep behavior. Thus, no applicable parameters could metric viscoelastic model, along with this observed flow be added or changed in the deviatoric viscoelastic model rate difference, implies that the inflow rate in areas of by linear means. Although not included in this study, compression is greater than the outflow rate in areas of perhaps a nonlinear method could be utilized to Figure 9 Pressure stress distribution pattern under intrusive loading of the deviatoric viscoelastic model. (A) Pressure stress at t Z Z 0.01 seconds in the PDL. (B) Pressure stress at t Z 2.5 seconds in the PDL. (C) Pressure stress at t Z 0.01 seconds in the labial-lingual direction. (D) Pressure stress at t Z 2.5 seconds in the labial-lingual direction. 480 C.-Y. Wang et al. construct an appropriate deviatoric viscoelastic model. To describe the properties of the PDL, different While the tension-compression volumetric viscoelastic investigators have suggested various roles for each model incorporates such a parameter, the deviatoric component of the PDL. Parfitt hypothesized that the viscoelastic model does not. Consequently, the volumetric displacement of the tooth is largely controlled by fluids in effect is most likely the dominant mechanism that affects both the vascular and tissue fluid systems.25 Bien also PDL behavior during loading and unloading. suggested that the tooth is supported by vascular Based on the pressure stress patterns observed during elements. Bien proposed that the fibers are controlled by loading and unloading, we found that the pressure stress the vascular system and that the fibers play only nones- patterns in the volumetric-based models (Fig. 4) more sential roles.5 Packman recorded pastille changes at rest closely resemble the fluid pressure changes in the PDL that and found that the pastille changes are synchronous with were observed in Walker’s experiment.9 The pressure stress the heartbeat. Packman also reported that loading on changes in the PDL of the volumetric-based models match a tooth produces a decrease in blood volume in areas the P-response, and the pressure stress changes in the under compression, while in areas under tension an initial deviatoric model are similar to the typical S-response. increase in blood volume was followed by a decrease as According to Walker’s study, the P-response is representa- the magnitude of force increased. The ground substance tive of the behavior of a normal, undamaged PDL, while of the PDL consists of 70% water, much of which is the S-response represents the behavior of a damaged PDL. bound.24 In Wills’s study, the volume of the vascular Based on the consistency between this study and Walker’s components of the PDL was found to be approximately 1% experiments with regard to both responses and Walker’s of the total volume. However, these systems significantly reasoning of their significance, several key implications contributed to tooth support only when forces were < 1.0 can be inferred. Because the volumetric-based models N. They also showed that the fluid systems take the emphasize the fluidic aspect of the PDL’s pressure response responsibility of 30% of the tooth in the tension- system, the observed P-response suggests that an undam- compression volumetric viscoelastic model, which is the aged PDL responds to a loading force by releasing fluid to most appropriate interpretation of normal PDL behavior relieve the added pressure. This rationalization explains under loading conditions, and that the deviatoric visco- why the P-response elicited from the volumetric model elastic model is a good representation of how a damaged demonstrates a rapid increase in pressure followed by PDL behaves under loading conditions.26 a rapid decrease. The fluid release commences after the Essentially, this study demonstrates that because it is loading force has been applied. In contrast, the deviatoric difficult to ascertain the in vivo stress distribution through model emphasizes the shape-changing aspect of the PDL’s laboratory experimentation, the method used in this study pressure response system. The observed S-response provides an opportunity to observe stress distributions suggests that a loading force affects the damaged PDL by throughout the tooth and its surrounding tissues. The compressing its nonfluid parts. This explanation justifies effects of the distribution of tooth stress on the assump- why the pressure level observed in the deviatoric model did tions of various properties of the periodontium will be not subside like the pressure level in the volumetric-based addressed in a subsequent study. Further experimental models. Because a damaged PDL has less fluid to release, its work is required to verify the validity of the assumptions only feasible response to a loading force is to compress its used for model derivation as well as the numerical values nonfluid components, such as tissue and collagen fibers. of the PDL’s material constants. We believe that the To distinguish between volumetric time effects and biomechanical properties of the PDL that were established deviatoric time effects, we needed to look at the stress via retrograde calculation in this study can lead to the patterns and pressure stress changes in a PDL under construction of more accurate extra-oral models and the a constant load. For the tension-compression volumetric comprehensive understanding of the biomechanical viscoelastic model (Figs. 8 and 9), the pressure stress in the behavior of the PDL. PDL under both horizontal and intrusive loading decreased over time. These results concur with the results of both Acknowledgments Packman24 and Walker’s9 studies. In Packman’s experi- ment, circulatory activities were monitored. In areas of the PDL that were compressed, horizontal and axial forces The authors gratefully acknowledge the financial support were found to generate a decrease in blood volume. provided by National Taiwan University Hospital and China Meanwhile, in areas under tension, the initial increase in Medical University (97F008-107&98F008-211). blood volume was followed by a decrease as the magnitude In addition, Shao-Huan Tsao, who recently died of of the force rose above the critical level of 90e180 g. The leukemia, was one of the authors of this paper. Many of the pulse volume, however, increased during both phases.24 creative ideas in this paper were inspired and completed by Walker’s study recorded fluid pressure changes in the PDL him. This paper would not have been possible without his of the canine teeth of dogs during and following the contributions. Our deepest gratitude goes to Shao-Huan application of loads up to 5.0 N. Moreover, the application Tsao. of a load caused a sudden pressure increase, which quickly decayed (half-time < 1 second). On the other hand, pres- References sure stresses in the deviatoric viscoelastic model increased over time, which suggests that the major source of visco- 1. Nanci A, Ten Cate AR. Development of tooth and its supporting elasticity in the PDL comes from the fluid/vascular system tissue in Ten Cate’s oral histology: development, structure, of the PDL.9 and function. 7th ed. New York: Mosby Inc; 2008. p.79e108. Viscoelastic behaviors of the PDL 481 2. Bergomi M, Cugnoni J, Botsis J. The role of the fluid phase in 14. Rieger MR, Adams WK, Kinzel GL, Brose MO. Finite element the viscous response of bovine periodontal ligament. 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