The effects of the periodontal ligament on mandibular stiffness: a study combining finite element analysis and geometric morphometrics

It is generally accepted that the periodontal ligament (PDL) plays a crucial role in transferring occlusal forces from the teeth to the alveolar bone. Studies using finite element analysis (FEA) have helped to better understand this role and show that the stresses and strains in the alveolar bone are influenced by whether and how PDL is included in FE models. However, when the overall distribution of stresses and strains in crania and mandibles are of interest, PDL is often not included in FE models, although little is known about how this affects the results. Here we study the effect of representing PDL as a layer of solid material with isotropic homogeneous properties in an FE model of a human mandible using a novel application of geometric morphometrics. The results show that the modelling of the PDL affects the deformation and thus strain magnitudes not only of the alveolar bone around the biting tooth, but that the whole mandible deforms differently under load. As a result, the strain in the mandibular corpus is significantly increased when PDL is included, while the strain in the bone beneath the biting tooth is reduced. These results indicate the importance of the PDL in FE studies. Thus we recommend that the PDL should be included in FE models of the masticatory apparatus, with tests to assess the sensitivity of the results to changes in the Young's modulus of the PDL material.     

Investigation of effective intrusion and extrusion force for maxillary canine using finite element analysis

Abstract

Orthodontic tooth movement is mainly regulated by the biomechanical responses of loaded periodontal ligament (PDL). We investigated the effective intervals of orthodontic force in pure maxillary canine intrusion and extrusion referring to PDL hydrostatic stress and logarithmic strain. Finite element analysis (FEA) models, including a maxillary canine, PDL and alveolar bone, were constructed based on computed tomography (CT) images of a patient. The material properties of alveolar bone were non-uniformly defined using HU values of CT images; PDL was assumed to be a hyperelastic–viscoelastic material. The compressive stress and tensile stress ranging from 0.47 to 12.8?kPa and 18.8 to 51.2?kPa, respectively, were identified as effective for tooth movement; a strain 0.24% was identified as the lower limit of effective strain. The stress/strain distributions within PDL were acquired in canine intrusion and extrusion using FEA; root apex was the main force-bearing area in intrusion–extrusion movements and was more prone to resorption. Owing to the distinction of PDL biomechanical responses to compression and tension, the effective interval of orthodontic force was substantially lower in canine intrusion (80–90?g) than in canine extrusion (230–260?g). A larger magnitude of force remained applicable in canine extrusion. This study revised and complemented orthodontic biomechanical behaviours of tooth movement with intrusive–extrusive force and could further help optimize orthodontic treatment.     

A visco-elastic model for the prediction of orthodontic tooth movement

This study presents a biomechanical model of orthodontic tooth movement. Although such models have already been presented in the literature, most of them incorporate computationally expensive finite elements (FE) methods to determine the strain distribution in the periodontal ligament (PDL). In contrast, the biomechanical model presented in this work avoids the use of FE methods. The elastic deformation of the PDL is modelled using an analytical approach, which does not require setting up a 3D model of the tooth. The duration of the lag phase is estimated using the calculated hydrostatic stresses, and bone remodelling is predicted by modelling the alveolar bone as a viscous material. To evaluate the model, some typically used motion patterns were simulated and a sensitivity analysis was carried out on the parameters. Results show that despite some shortcomings, the model is able to describe commonly used motion patterns in orthodontic tooth movement, in both single- and multi-rooted teeth.     

Mechanical responses of the periodontal ligament based on an exponential hyperelastic model: a combined experimental and finite element method

The V–W exponential hyperelastic model is adopted to describe the instantaneous elastic response of the periodontal ligament (PDL). The general theoretical framework of constitutive modeling is described based on nonlinear continuum mechanics, and the elasticity tensor used to develop UMAT subroutine is formulated. Nanoindentation experiment is performed to characterize mechanical properties of an adult pig PDL specimen. Then the experiment is simulated by using the finite element (FE) analysis. Meanwhile, the optimized material parameters are identified by the inverse FE method. The good agreement between the simulated results and experimental data demonstrates that the V–W model is capable of describing the mechanical behavior of the PDL. Therefore, the model and its implementation into FE code are validated. By using the model, we simulate the tooth movement under orthodontic loading to predict the mechanical responses of the PDL. The results show that local concentrations of stress and strain in the PDL are found.     

Tensile testing of the mechanical behavior of the human periodontal ligament

Background

The periodontal ligament (PDL) plays a key role in alveolar bone remodeling and resorption during tooth movements. The prediction of tooth mobility under functional dental loads requires a deep understanding of the mechanical behavior of the PDL, which is a critical issue in dental biomechanics. This study was aimed to examine the mechanical behavior of the PDL of the maxillary central and lateral incisors from human. The experimental results can contribute to developing an accurate constitutive model of the human PDL in orthodontics.

Methods

The samples of human incisors were cut into three slices. Uniaxial tensile tests were conducted under different loading rates. The transverse sections (cervical, middle and apex) normal to the longitudinal axis of the root of the tooth were used in the uniaxial tensile tests. Based on a bilinear simplification of the stress–strain relations, the elastic modulus of the PDL was calculated. The values of the elastic modulus in different regions were compared to explore the factors that influence the mechanical behavior of the periodontal ligament.

Results

The obtained stress–strain curves of the human PDL were characterized by a bilinear model with two moduli (E₁ and E₂) for quantifying the elastic behavior of the PDL from the central and lateral incisors. Statistically significant differences of the elastic modulus were observed in the cases of 1, 3, and 5 N loading levels for the different teeth (central and lateral incisors). The results showed that the mechanical property of the human incisors’ PDLs is dependent on the location of PDL (ANOVA, P?=?0.022, P?<?0.05). The elastic moduli at the middle planes were greater than at the cervical and apical planes. However, at the cervical, middle, and apical planes, the elastic moduli of the mesial and distal site were not significantly different (ANOVA, P?=?0.804, P?>?0.05).

Conclusions

The values of elastic modulus were determined in the range between 0.607 and 4.274 MPa under loads ranging from 1 to 5 N. The elastic behavior of the PDL is influenced by the loading rate, tooth type, root level, and individual variation.

     

The Biomechanical Function of Periodontal Ligament Fibres in Orthodontic Tooth Movement

Orthodontic tooth movement occurs as a result of resorption and formation of the alveolar bone due to an applied load, but the stimulus responsible for triggering orthodontic tooth movement remains the subject of debate. It has been suggested that the periodontal ligament (PDL) plays a key role. However, the mechanical function of the PDL in orthodontic tooth movement is not well understood as most mechanical models of the PDL to date have ignored the fibrous structure of the PDL. In this study we use finite element (FE) analysis to investigate the strains in the alveolar bone due to occlusal and orthodontic loads when PDL is modelled as a fibrous structure as compared to modelling PDL as a layer of solid material. The results show that the tension-only nature of the fibres essentially suspends the tooth in the tooth socket and their inclusion in FE models makes a significant difference to both the magnitude and distribution of strains produced in the surrounding bone. The results indicate that the PDL fibres have a very important role in load transfer between the teeth and alveolar bone and should be considered in FE studies investigating the biomechanics of orthodontic tooth movement.     

Mechanical behaviour of endodontic restorations with multiple prefabricated posts: a finite-element approach

This paper investigates some mechanical aspects of a new endodontic restoration technique, based on the idea that the root cavity can be more efficiently filled if multiple prefabricated composite posts (PCP) are employed. Multi-post technique increases bearing capacity and durability of endodontically treated teeth, as shown by numerical simulations performed through three-dimensional elastic finite-element static analyses of a lower premolar, constrained by a non-linearly elastic spring system representing the periodontal ligament, under several parafunctional loads. The influence of PCPs' number, material and dimensions is investigated by comparison of the resulting stress fields with those obtained in cases of traditional restorations (cast metal post and cemented single-PCP) and natural tooth, highlighting the advantages of the proposed technique when standard restorative materials are considered. A risk-analysis of root-fracture and interface-failure shows that cast gold-alloy post produces high stress concentrations at post-dentin interface, whereas multi-post solution leads to a behaviour closer to the natural tooth's, exhibiting some advantages with respect to single-PCP restorations. As a matter of fact, whenever PCPs' overall cross-section area increases, multi-post solution induces a significant reduction of stress levels into the residual dentin (and therefore the root-fracture-risk decreases) as well as of the expected polymerization shrinkage effects. Moreover, interfacial stress values in multi-post restorations can be higher than the single-PCP ones when carbon-fibre posts are considered. Nevertheless, the interfacial adhesive/cohesive failure-risk is certainly acceptable if glass-fibre posts are employed.     

Time-related PDL: viscoelastic response during initial orthodontic tooth movement of a tooth with functioning interproximal contact-a mathematical model

Most anteroposterior orthodontic movements of posterior teeth have to overcome the "resistance" of adjacent teeth with functioning interproximal contacts. The aim of this study was to develop a mathematical model describing initial posterior tooth movement associated with functioning interproximal contacts in relation to the viscoelastic mechanical behavior of the human periodontal ligament (PDL). A linear viscoelastic 2D mathematical model was modified to depict tipping movement around the center of rotation (C(rot)) of a premolar where tipping is restrained by adjacent teeth. Equilibrium equations were applied taking into account the sagittal moment developed around the C(rot). The constants of the model were analyzed and applied to a numerical model that can simulate short-term tooth creep movement caused by a tipping force. Changes in force magnitude (0.5-3N) and crown length (6-10mm) were analyzed until no movement was observed (steady state). Premolar displacement in contact with adjacent teeth showed a non-linear progression over time with an initial sharp tipping movement followed by a transient period of 2.6-7.1min. As tipping force increased the transient period increased. A similar but smaller effect was observed with an increase in crown length. The premolar initial displacement within the arch (3.2-19.5microm) is about seven-fold smaller than retraction/protraction movement of an incisor. These suggest reduction in tooth displacement when functioning interproximal contact is present and clinically recommend establishing a space in the direction of tooth displacement before tooth movement.     

Role of regulator of G-protein signaling 2 (RGS2) in periodontal ligament cells under mechanical stress

Mechanical stress is thought to regulate the expression of genes in the periodontal ligament (PDL) cells. Using a microarray approach, we recently identified a regulator of G-protein signaling 2 (RGS2) as an up-regulated gene in the PDL cells under compressive force. The RGS protein family is known to turn off G-protein signaling. G-protein signaling involves the production of cAMP, which is thought to be one of the biological mediators in response to mechanical stress. Here, we investigated the role of RGS2 in the PDL cells under mechanical stress. PDL cells derived from the ligament tissues of human premolar teeth were cultured in collagen gels and subjected to static compressive force. Compressive force application time-dependently enhanced RGS2 expression and intracellular cAMP levels. To examine the interrelationship between RGS2 and cAMP, the PDL cells were treated with 2',5'-dideoxyadenosine (DDA), an inhibitor of adenyl cyclase, or antisense S-oligonucleotide (S-ODN) to RGS2 under compressive force. DDA dose-dependently inhibited RGS2 stimulated by compressive force. Blockage of RGS2 by antisense S-ODN elevated the cAMP levels compared with controls. These results indicate that cAMP stimulates RGS2 expression, which in turn leads to a decrease in the cAMP production by inactivating the G-protein signaling in the mechanically stressed PDL cells.     

Mechanics of the maxillary central incisor. Influence of the periodontal ligament represented by beam elements

This study aimed to evaluate the influence of loading on a maxillary central incisor with the periodontal ligament (PDL) represented by 2D elastic beam elements using a 2D finite element analysis. Two models (M) were built varying the PDL representation: Mh (homogeneous PDL) and Mht (heterogeneous PDL with beam3 elements). Stress and displacements were determined for three loading conditions (L): Ll, lingual face loading at 45° with the tooth long axis; Li, perpendicular to the incisal edge; and Lip, on the incisal edge, parallel to the tooth long axis. Evaluation was performed on ANSYS software. Lip provided lower stress variation on the tooth and support structures when compared to Ll and Li. PDL's influence on stress values was lower for Lip. Oblique loading showed stress and displacement not observed in parallel loading condition through PDL's heterogeneous representation and it is probably incompatible with the in vivo condition.     

Numerical study of tension and strain distribution around rat molars]

A knowledge of the mechanical processes triggered in the bone and periodontal ligament (PDL) by orthodontic forces applied to a tooth is of decisive importance for an understanding of the subsequent remodelling around the tooth. To investigate these mechanical relationships, three-dimensional finite element (FE) models of the first lower molar in the rat were established. On the basis of digitized serial histological sections, these FE models were generated semi-automatically. Using various simplified geometrical variations, an appropriate FE model for the analysis of the stress and strain distributions was established. The numerical analyses were carried out under a mesially directed force of 0.1 N. Stress distributions in the bone and PDL showed a similar pattern, while strains in the bone were lower than in the PDL by a factor of 10-5. The data confirm the assumption that strain patterns in the PDL may be the key stimulus of bone remodelling.     

Finite element simulation of the behavior of the periodontal ligament: A validated nonlinear contact model

Due to its significance in tooth movement, the stress/deformation field of periodontium and the alveolar bone remodeling process, periodontal ligament (PDL) cannot be excluded from the studies investigating dental biomechanics regarding its excessive deformability. Therefore, many analytical and numerical researches are carried out to simulate its response and to create a constitutive model via experiments intending to discover the material properties of PDL. The aim of this study is to formulate a user specified contact model that can be used in conjunction with finite element (FE) software and reflects PDL’s influence on neighboring structures based on the currently available information, without requiring an actual volumetric finite element mesh of ligament. The results show good agreement with available experimental tooth mobility data. Smooth stress fields are obtained on the tooth root and alveolar bone, which is a significant aspect in bone-remodeling studies. The advantage of simulating PDL as a contact model at the interface of tooth root and the alveolar process instead of a solid-meshed FE model with poor geometric morphology and/or very dense mesh is expected to save pre/post-processing workforce, to increase the accuracy and to contribute to the smoothness of interface stress distributions.     

Development of a novel intraoral measurement device to determine the biomechanical characteristics of the human periodontal ligament

Periodontal diseases like gingivitis and periodontitis have damaging effects on the periodontium and commonly affect the mechanical properties of the periodontal ligament (PDL), which in the end might lead to loss of teeth. Monitoring tooth mobility and changes of the material properties of the PDL might help in early diagnosis of periodontal diseases and improve their prognosis. It was the aim of this study to develop a novel intraoral device to determine the biomechanical characteristics of the periodontal ligament. This includes the measurement of applied forces and resulting tooth displacement in order to investigate the biomechanical behaviour of the periodontium with varying loading protocols with respect to velocity and tooth displacement. The developed device uses a piezoelectric actuator to apply a displacement to a tooth's crown, and the resulting force is measured by an integrated force sensor. To measure the tooth displacement independently and non-invasively, two magnets are fixed on the teeth. The change in the magnetic field caused by the movement of the magnets is measured by a total of 16 Hall sensors. The displacement of the tooth is calculated from the movement of the magnets. The device was tested in vitro on premolars of four porcine mandibular segments and in vivo on two volunteers. The teeth were loaded with varying activation curves. Comparing the force progression of different activation velocities, the forces decreased with decreasing velocity. Intensive testing demonstrated that the device fulfils all requirements. After acceptance of the ethical committee, further testing in clinical measurements is planned.     

Semi-automatic generation of finite element meshes fo dental preparations]

The mechanical properties and elastic behaviour of periodontal tissue are a decisive factor in understanding initial tooth mobility and bone remodelling processes in orthodontics. An experimental set-up was designed to precisely determine a tooth's elastic response to different loading conditions. Segments of pig's maxilla bearing separated molars were used, and their mechanical response to loading was recorded. Subsequently, finite element analysis (FEA) was performed on the basis of the experimental data. The combination of experimental and numerical methods was used to determine the material properties of the periodontal ligament (PDL). The geometries of the preparations were reconstructed and FE meshes generated semi-automatically with the aid of the special computer program, CAGOG (Computer Aided Generator for Orthodontic Geometries) to optimally match the experimental geometry. Nonlinear material parameters were determined for the PDL and verified by comparing experimental and numerical results obtained in other specimens with an error of about 10%. This good correlation indicates that the selected method of mesh generation is appropriate for creating realistic FE models that can be compared with experimental results.     

Dynamic shear properties of the porcine molar periodontal ligament

The role of the periodontal ligament (PDL) is to support the tooth during function and resist external forces applied to it. The dominant vertical component of these forces is associated with shear in the PDL. Little information, however, is available on the dynamic behavior of the PDL in shear. Therefore, the present study was designed to determine the dynamic shear properties of the PDL in the porcine molar (n=10). From dissected mandibles transverse sections of the mesial root of the first molar were obtained at the apical and coronal levels and used for dynamic shear tests. Shear strain (0.5%, 1.0%, and 1.5%) was applied in superoinferior direction parallel to the root axis with a wide range of frequencies (0.01-100 Hz). The dynamic complex and storage moduli increased significantly with the loading frequency, the dynamic loss modulus showed only a small increase. The dynamic elasticity was significantly larger in the coronal region than in the apical region although the dynamic viscosity was similar in both regions. The present results suggest that non-linearities, compression/shear coupling, and intrinsic viscoelasticity affect the shear material behavior of the PDL, which might have important implications for load transmission from tooth to bone and vice versa.     

A transversely isotropic hyperelastic constitutive model of the PDL. Analytical and computational aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

Continuum description of the Poisson׳s ratio of ligament and tendon under finite deformation

Ligaments and tendons undergo volume loss when stretched along the primary fiber axis, which is evident by the large, strain-dependent Poisson?s ratios measured during quasi-static tensile tests. Continuum constitutive models that have been used to describe ligament material behavior generally assume incompressibility, which does not reflect the volumetric material behavior seen experimentally. We developed a strain energy equation that describes large, strain dependent Poisson?s ratios and nonlinear, transversely isotropic behavior using a novel method to numerically enforce the desired volumetric behavior. The Cauchy stress and spatial elasticity tensors for this strain energy equation were derived and implemented in the FEBio finite element software (www.febio.org). As part of this objective, we derived the Cauchy stress and spatial elasticity tensors for a compressible transversely isotropic material, which to our knowledge have not appeared previously in the literature. Elastic simulations demonstrated that the model predicted the nonlinear, upwardly concave uniaxial stress–strain behavior while also predicting a strain-dependent Poisson?s ratio. Biphasic simulations of stress relaxation predicted a large outward fluid flux and substantial relaxation of the peak stress. Thus, the results of this study demonstrate that the viscoelastic behavior of ligaments and tendons can be predicted by modeling fluid movement when combined with a large Poisson?s ratio. Further, the constitutive framework provides the means for accurate simulations of ligament volumetric material behavior without the need to resort to micromechanical or homogenization methods, thus facilitating its use in large scale, whole joint models.     

A Transversely Isotropic Hyperelastic Constitutive Model of the PDL. Analytical and Computational Aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.