The role of periodontal ligament elasticity in periodontal changes - numerical simulation

The clinical applications of oral mucosal biomechanics are particularly important in determining how stimulation can be performed for tissue remodeling, in understanding the level of pressure that can be applied to the painful threshold, in osseous resorption, and the availability of tissue movement. This study shows, through the finite element, the way in which the periodontal ligament behaves at the level of all dental surfaces in nonlinear model and 3 levels of alveolar bone retraction (0 %, 50%, 75%). From the point of view of geometric construction, each model consists of 3 independent components: the alveolar bone, the periodontal ligament (PL) and the tooth. The deformities on the lingual surface are very small compared to the other surfaces. The deformity increases progressively with the loss of the supporting bone. From a physiological point of view, the nonlinear model is much closer to reality because it simulates the reaction of the tissue to its immediate loading by a very low rigidity after which, due to the hydrostatic pressure created in the interstitial fluid, the value of tissue rigidity increases expo-nentially.


InTRoduCTIon
The periodontal ligament (PL) is a fibrous tissue, with thicknesses between 0.15 and 0.38 mm. The purpose of PL is to attach the dental root, specifically the root cementum, to the surrounding alveolar bone [1,2]. Its main role is to transmit the forces acting on the teeth to the alveolar bone [3,4].
The clinical applications of oral mucosal biomechanics are particularly important in determining how stimulation can be performed for tissue remodelling, in understanding the level of pressure that can be applied to the painful threshold, in bone resorption, and the availability of tissue movement. In the mastication process, the soft tissues adjacent to the teeth play a decisive role in the distribution of occlusal forces to the maxillary bone area. The distribution is in the form of a functional hydrostatic pressure that occurs in the soft tissues and deter-mines the pressure centers according to the loading [5][6][7].
The compression of the mucosa leads to a reduction in blood flow within that area, leading to local pain and discomfort. Also, higher or prolonged compression for a longer period of time can lead to intracellular edema or cellular inflammation. If the hydrostatic pressure reaches a level that exceeds the capillary pressure, then the circulation is reduced, and in its absence bone resorption occurs, a phenomenon that is progressive and harmful to oral health [8,9]. This is due to a lack of essential nutrients.
The removal of mechanical stress generates an elastic recovery of the mucosa, especially in young patients. Reducing the surface pressure will allow the interstitial fluids to irrigate the area again, sometimes generating a higher flow pressure than the initial one. In the elderly, however, the phenom-enon does not occur in the same way due to the loss of elasticity of the mucosa as the person is aging. If the mechanical stress is significant and for a long time, even tissue ischemia may occur [10][11][12][13].
Studies in the biomechanics of the oral mucosa have shown that the oral mucosa is much more tolerant of cyclic stress than of continuous stress. Thus, the resorption threshold of the alveolar ridge was identified at 19.6 kPa for cyclic stresses and only 6.8 kPa for continuous stress [14,15,6].
These values are of reference, they are strongly influenced by the pathophysiological condition of the individual, to the same extent as the level of continuous pressure applied.
Elasticity is one of the fundamental properties that define the behavior of a material. This is quantified by the modulus of elasticity (Young's modulus) which shows the tendency of a body to deform proportionally with the applied stress. In the case of the mucosa, it is very deformable to compression, presenting modulus of elasticity in a very wide range [16] Due to the inhomogeneity of the mucosa, its initial rigidity derives from two components: the epithelial layer, blood vessels and collagen fibers on the one hand and the fluids (blood, interstitial fluid) on the other.

MATeRIAl And MeThod
For geometric models, linear elastic materials were defined by their two properties (longitudinal modulus of elasticity and Poisson's ratio), based on the bibliographic study. These are presented in Table I, in this way the property being considered as homogeneously distributed in the model. Because the periodontal ligament (PL) is the key element of this study, the nonlinear material models were considered for it (figure 1). The nonlinear model materializes a nonlinear change of the modulus of elasticity from small to large, a change that takes place from a physiological point of view as the structure is loaded. Thus, at low stress values, the tissue reacts with a strong and less elastic deformation, after which the modulus of elasticity increases due to the hydrostatic pressure that is generated inside the tissue. Such a material model can be simulated as a hyper elastic one, in this case being opted for a Mooney-Rivlin model defined by 5 parameters. In PL compression it is described by a very low modulus of elasticity up to almost 100% deformation, after which the modulus increases exponentially about 1000 times. This increased value simulates the precontact between the tooth and the alveolar bone. In extension, the modulus of elasticity has a different variation, increasing from 0.05 MPa corresponding to a zero deformation, gradually, up to 50% deformation where it reaches a value 10 times higher. This deformation value in stretching is a breaking point of PL tissue fibers [22].
Poisson's coefficients indicating the PDL deformation ratio for each material model were extracted from the literature [19,[23][24][25].  The analysis was performed on 3 geometric models that materialize 3 levels of alveolar bone retraction. From the point of view of geometric construction, each model consists of 3 independent components: the alveolar bone, the (PL) and the tooth. The tooth is a geometric pattern reconstructed based on radiographic images to have a geometry as accurate as possible with the actual tissue. The PL ligament was constructed geometrically with the help of Boolean operations of addition and subtraction of reconstructed dental volume. This allowed a coincidental assembly between the tooth and the PL. The alveolar bone was constructed in the shape of a cylinder from which the volume corresponding to the ligament was extracted. Thus, 3 independent volumes 1, 2 and 3 (figure 2.) were obtained, which could be assembled by coincidence. In the same way we worked for all the components of the 3 analysis models: model A, B and C, the result being shown in figure 3. The differences in height of the alveolar bone and implicitly of the ligament were measured as a percentage from the top of it (gingival level). Also in figure 3 are shown the directions corresponding to the anatomy of the tooth, directions that will be referred to in the simulation.
The discretization of the structure was performed with automatic tetrahedral elements for the tooth and alveolar bone components. For PL, due to the special role that this component plays, the discretization was achieved with elements of constant size and value of 0.2 mm. This allows on the one hand a finer observation of the spectrum of stresses and strains by defining several control elements and nodes, and on the other hand facilitates the success of nonlinear analysis (figure 4).

ReSulTS
PL ligament deformation is an indicator of how stress is transmitted from the tooth to the alveolar bone. The deformation is closely correlated with the tension in the ligament, respectively with the hydrostatic pressure that appears with the stress.
The results in Figure 5 reveal the behavior of PL in case of lack of bone retraction for nonlinear material model. To see how the PL loading takes place, the results were presented according to the anatomical directions: distal, mesial, vestibular, and lingual.  Due to the way in which the loading was applied, the results show a symmetry of the vestibular and oral surfaces, respectively an asymmetry of movement on the distal and mesial surfaces. Naturally, the most important displacement of the structure is in the area closest to the occlusal surface, which will dictate the position of the center of rotation of the tooth. Figures 6 and 7 show the situations in which there is a retraction of the alveolar bone by 50 and 75% respectively from the complete anatomical position. This entails a corresponding retraction of the PL, i.e. a change in the distribution of stresses and displacements of the structure. Thus, considering the integral structure (0% loss) as a reference of the PL deformation, it can be seen that it increases greatly in the case of bone loss of 75%. The values obtained are significantly high and indicate a displacement of the tooth in relation to the alveolar bone, considered fixed. This displacement of the tooth is usually rotational. To determine the center of rotation of this movement it is necessary to identify the maximum and minimum values of stresses and deformations.
As a result of the change in the physical size of both the alveolar bone and the PL ligament, it is observed that the displacements of the structure are increasing with the loss of bone (at the same load). This can be explained by a lever effect at which the fulcrum has changed. With this change, the value of the load together with the distance from the support point to the right support of the load produce a momentum of greater value, precisely due to the increase of this distance. Therefore, in order to have a mechanical balance at the level of the assembly, in the remaining contact points between the tooth/PL and the alveolar bone, reactions will be generated which in turn will produce a strong momentum of high value. Figures 8,9 and 10 show the results of sampling the voltage values in the nodes. For this, a number of 30 nodes were selected in the form of a curved line on the distal, mesial, vestibular and lingual surfaces of the PL ligament. The sampling line on each surface was drawn in such a way that it was always the median line of that surface. This was systematically repeated for the simulations of the 3 geometric models. Thus, the maximum and minimum aspects of the deformations can be observed, as well as the way in which the deformation transition is performed on the longitudinal direction of the tooth. It is found for the intact model (0% loss) how the minimum deformation is approximately in the middle of the median distance on any of the 4 surfaces stud-

Distal Mezial Vestibular Lingual
Mooney-Rivlin nonlinear The displacement values from the PL for the 50% loss model Figures 11 (a,b,c) show the stress results at the upper extremity of the PL. Also, the shape and dimensions of these sections resulting from the conjugation of the alveolar bone with the tooth on the 3 levels explained above can be identified here. Considering the significantly higher dimensions of the upper surface of the PL in the case of the integral model compared to the dimensions of the models with bone loss, it can be understood why the value of stress and deformation increases with the bone loss, the stress being defined as the ratio between the value of the external stress and the area of the surface on which it is distributed.
For the model without bone loss, the stresses appear almost evenly distributed on the upper surface of the PL. This is observed by the presence of the transition colors between the high voltage values (represented by warm colors: red-orange) and those of low value (represented here by cold colors: greenblue). The appearance of a sudden change from blue to orange is mainly found in the case of the 75% loss model and almost independent of the material model used. This sudden change in value and sign of tension occurs in the distal-mesial direction due to the fact that the stress has important components introduced in this direction (shear direction). Overall, the stresses in the PL surface adjacent to the dental crown are higher than those at the apex, due to the higher rotational displacement of this extremity.
According to the PL deformation analysis, the minimum and maximum values as well as their physical positions on the simulated tissue were extracted. The physical positioning of the maximum and minimum deformation is shown in Tables IV a,b,c. An important factor in the manifestation of these positions is the direction of the occlusal force. The decisive role in the occurrence of maximum and minimum in the simulation is played by the shape of the structure, which derives from both tooth anatomy and from simulating alveolar bone loss.
In the complete periodontium it is found that have maximum values in the mesial crown area, and the minimum values are located differently, at distal in ½ crown area.
In the case of a 50% gingival retraction, the changes are visible, the nonlinear model has the maximum distal crown and the minimum lingual crown.
Gingival recession of 75% shows a value of the maximal crown mesial and the minimum at the apex for the low elasticity model, the maximum distal-crown and the minimum mesial-crown for the linear model with high elasticity and the maximum crown-distal and the minimum at the apex for the nonlinear model.
The deformations inside the longitudinal sections of the PL are shown in Figures 12 a,b,c. Their appearance confirms the deformation distribution in the PL and also indicates the imbalances that occur with the stress and the change of geometry. Lower values of apex stresses and strains are recorded because the stress occurs in the form of rotation of the tooth in the support structure. Otherwise, at a pure compression occlusal stress, the apex of the structure will be most intensely stressed, due to its low volume compared to the rest of the model. In the present situation, the stress distribution to the area considered fixed is made towards the mesial surface and in the upper third of the model.
The center of rotation is in the area of minimum displacement (blue), being located in the lower half of the section.

Distal
Mesial Labial Lingual a) Mooney-Rivlin nonlinear dISCuSSIonS PL has an important role in biological functions through fibrous connections with cementum and alveolar bone and a mechanical role in absorbing occlusal forces and maintaining alveolar bone height. PL plays a key role in alveolar bone remodeling and resorption during tooth movements, which is why it is absolutely necessary for the practitioner to fully understand the predictability of tooth mobility under functional dental loads, i.e. the mechanical behavior of PL, which continues to raise questions from a biomimetic point of view [26,4].
The morphological diversity of PL and its complex structure is a challenge when it comes to investigating its mechanical properties, as PL cannot be separated from the bone-PL-tooth complex without destroying the periodontal complex. That is why studies investigating the relationship between the mechanical properties of human PL and tooth position are limited [4].
These results have implications not only for understanding the stresses and strains that occur in PL under the action of forces generated by the dentomaxillary apparatus during the exercise of its functions, but also for providing complete information for practitioners and researchers on the role of material models used to address the mechanical behavior of PL and other components of the tooth.
The occlusal forces acting on the teeth depending on the somatic and psychic changes of each patient. Furthermore, the adaptive capacity of the periodontal complex, under the action of these forces, also varies significantly from individual to individual and even in the same individual in different periods of time.
One of the most used methods of theoretical analysis of biological structures is finite element analysis. Simulation of PL behavior by finite element analysis can be applied to calculate the stresses and quantitative deformations of PL under the action of static and dynamic forces but also to analyze the mechanical properties of the components implemented in the model [27].
The nonlinear model is much closer to reality because it simulates the reaction of the tissue to its immediate loading through a very low rigidity. After that, due to the hydrostatic pressure created in the interstitial fluid, the value of tissue stiffness increases exponentially.
The present study presents the results of the simulation with MEF of the behavior of the periodontal ligament in 3 clinical situations: for the integral biomechanical structure, in which the alveolar bone and the ligament have normal physiological dimensions and for the situations in which the bone undergoes a retraction by 50% and 75% of the initial value. These three variables of a geometric nature are added the type of material that simulates the PL, such as the non-linear model Mooney-Rivlin.
In this study we can see the deformity increases progressively with the loss of the supporting bone. Hemanth M et al. stated that the stresses found in the nonlinear analysis are higher than the stresses found in the linear analysis for the same magnitude of the force. Conversely, nonlinear analysis requires less force compared to linear analysis. [28].
PDL-induced apical stress increases as PDL thickness decreases. Alveolar bone stress decreases with increasing PDL thickness. Clinically, this distribution of stress may be useful in adult patients with increased PDL thickness and the use of optimal forces to reduce the susceptibility to root resorption at the root tip [29]. The viscoelastic biomechanical behavior of PDL can better address the various tasks of PDL, i.e. anchoring the tooth to the bone, amortization while chewing, swallowing or squeezing, transmitting force to the bone, and transforming long-term orthodontic force into biomechanical signals for bone remodeling and orthodontic movement of the teeth (Keilig et al., 2016) [30].
Reducing alveolar bone height had little effect on the tooth and supporting tissue when 25% of the bone support was lost, however the stress increased dramatically when 50 and 75% of the bone support was lost. Also, the stress moved apically on the tooth, with increasing bone resorption, which is explained by the increase in the amount of force distributed in relation to the surface area, as tissue loss leads to a decrease in the surface, according to this study [31].
In our study, at the level of the whole periodontium it is found have maximum values in the mesial -crown area, and the minimum values are located and distal in ½ crown.
In the case of a 50% gingival retraction, the changes are visible, so that in the nonlinear model shows the maximum distal-crown and the minimum crown-lingual.
Gingival retraction of 75% shows a value of the distal-crown maximum and the apex minimum for the nonlinear model.
Hemanth et al showed in their study that with the application of extrusive force, tensile stresses were observed at the apex, while the compression stress was distributed to the cervical margin. When the rotational force was applied, the maximum compression stress was distributed to the apex and the cervical third, while the tensile stress was distributed to the cervical third of the PL on the lingual surface [28].
Stress values were higher at the apex, regardless of the degree of bone loss. This may explain the appearance of the incipient periapical abscess, which appears as an apical radiolucency, in those teeth with primary occlusal trauma without periodontitis. Another result of the same study refers to the fact that the maximum stress values were recorded at the distal-vestibular level, near the enamel-cement junction, which may explain the occurrence of abrasion, due to excessive loads [31].
In most of the situations simulated in this study, minimal deformation occurs at or near the apex regardless of the material pattern used or the geometric pattern, which means that the effect of tooth rotation is lowest in that area. The maximum displacement appears physiologically on the mesial surface then the geometric model simulates the in-tegral structure. As bone loss occurs and develops, the maximal point changes position toward the distal surface, but also in the coronary third. This indicates that the strain deformations become larger than the compression deformations, the distal face being the one that opposes the direction of stress, and which is consequently the surface subject to stretching. Also, in our study, the deformities on the lingual surface of the PL are very small compared to the other faces, and the average 1/3 of each surface analyzed has smaller deformations.
The study by Reimann S. et al. who analyzed the role and mode of action of the occlusal forces on the PL found that when applying a normal force at the level of a healthy periodontal structure the minimum tensions were highlighted on the middle part of the PL, and the maximum tensions on the palatal side. Applying the same normal load on the PL resulted in the production of minimal stresses on the distal side in the present situation of 50% bone loss and maximum stresses are seen at the junction of the apical and middle third when the bone loss is 75% on the palatal side [32].
At a pysiological height of the alveolar bone, at a load of 150N (normal function), in PL the minimum stresses were observed on the mesial surface at the level of the enamel-cementum junction, and the maximum ones at the level of the palatal surfaces. As the load increased to 290N, the voltages also increased by 90%. In cases with bone resorption, at a load of 150N (normal function), the minimum stresses were observed at the superficial surface in the middle third of the root with 50% bone loss and the maximum tensions at the palatal face at the junction of the apical third with the middle third of the root with 75% bone loss. As the load increased to 290N, the voltages increased even by 90% [18].
The data from previous research studies are useful both for understanding the periodontal complex and how it works and responds to different stimuli, and for the evolution of treatment methods in terms of its pathology. However, it is difficult to make clear comparisons between the results of studies conducted so far due to variations in methodology, but also the proposed objectives.

ConCluSIonS
The nonlinear model is much closer to reality because it simulates the reaction of the tissue to its immediate loading by a very low rigidity after which, due to the hydrostatic pressure created in the interstitial fluid, the value of tissue stiffness increases exponentially.
The deformities on the lingual surface of the PL are very small compared to the other surfaces.
For the model without bone loss is found that maximum values are in the mesial crown area, and the minimum values are located at distal in ½ crown area.
Bone loss is associated with a change in the position of the center of rotation due to the lack of supporting bone volume. Its migration occurs to the apical area.
In the case of a 50% gingival recession, the changes are visible, the nonlinear model has the maximum distal crown and the minimum lingual crown.
Gingival recession of 75% shows a value of the maximum crown-distal and the minimum at the apex for the nonlinear model.
The increase in the value of external stress will lead to an increase in the hydrostatic pressure in the tissue and thus to exceeding on the one hand the painful threshold and on the other hand to the acceleration of the tissue retraction phenomenon.