Experiments to determine the time dependent material properties of the periodontal ligament]

The periodontal ligament is a tissue that attaches the tooth (root) to its alveolar socket, and thus plays an important role in the regulation of tooth movements. Detailed knowledge of the material properties of the periodontal ligament is therefore essential to an understanding of tooth reaction to forces applied during orthodontic treatment. A knowledge of material parameters can also be used in simulations of long-term tooth movements with the aim of improving orthodontic treatment. To this end, this study investigated time-dependent material properties, namely the hysteresis behaviour of the periodontal ligament under constant-velocity loading, the influence of loading velocity on the hysteresis, and its failure under constant loading. Specimens obtained from pigs were used for testing purposes, and the experiments were conducted in a special test setup using a material testing device. The material behaviour of the periodontal ligament was shown to be viscoelastic, and the elastic parameters of material behaviour were also determined. Under constant-velocity loading, material behaviour showed a nonlinear course of the stress-strain curve, also known as hysteresis. When loading was repeated several times, the maximum stress of the hysteresis decreased with each cycle. Determination of the deflection of the specimen at different velocities showed maximum stress to be dependent on the loading rate. The measured stress-strain curves were approximated by bilinear behaviour, permitting the use of finite element calculations. Also investigated was the failure behaviour of the periodontal ligament, which revealed tissue rupture to be inconstant.     

Mechanical Response of Bone under Short-term Loading of a Dental Implant with an Internal Layer Simulating the Nonlinear Behaviour of the Periodontal Ligament

We consider a non-standard design for a fixed dental implant, incorporating a soft layer which simulates the presence of the periodontal ligament (PDL). Instead of being aimed at causing an a priori defined stress/strain field within the surrounding bone, upon loading, such a design simply tries to better reproduce the natural tooth–PDL configuration. To do this, the mechanical properties of the internal layer match those of the PDL, determined experimentally to be strongly nonlinear. Three-dimensional finite element analyses show that the presence of such a layer produces (i) a prosthesis mobility very similar to that of a healthy tooth, for several loading conditions, and (ii) a stress/strain distribution substantially different from that arising, upon loading, around a conventional implant. The lack of knowledge of the real mechanical fields existing, under loading, in the bone around a healthy tooth makes it very difficult to state that the stress distribution produced by the modified implant is “better” than that produced by the standard one. Nevertheless, the comparison of the results obtained here, with those of previous refined analyses of the tooth–PDL–bone system, indicates that the modified implant tends to produce a stress distribution in the bone, upon loading, closer to “natural” than that given by the standard one, within the limits imposed by the presence of threads coupling the implant with the bone.     

Mechanical response of bone under short-term loading of a dental implant with an internal layer simulating the nonlinear behaviour of the periodontal ligament

We consider a non-standard design for a fixed dental implant, incorporating a soft layer which simulates the presence of the periodontal ligament (PDL). Instead of being aimed at causing an a priori defined stress/strain field within the surrounding bone, upon loading, such a design simply tries to better reproduce the natural tooth-PDL configuration. To do this, the mechanical properties of the internal layer match those of the PDL, determined experimentally to be strongly nonlinear. Three-dimensional finite element analyses show that the presence of such a layer produces (i) a prosthesis mobility very similar to that of a healthy tooth, for several loading conditions, and (ii) a stress/strain distribution substantially different from that arising, upon loading, around a conventional implant. The lack of knowledge of the real mechanical fields existing, under loading, in the bone around a healthy tooth makes it very difficult to state that the stress distribution produced by the modified implant is "better" than that produced by the standard one. Nevertheless, the comparison of the results obtained here, with those of previous refined analyses of the tooth-PDL-bone system, indicates that the modified implant tends to produce a stress distribution in the bone, upon loading, closer to "natural" than that given by the standard one, within the limits imposed by the presence of threads coupling the implant with the bone.     

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.     

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.     

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.     

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.     

Tensile and compressive behaviour of the bovine periodontal ligament

The mechanical response of the bovine periodontal ligament (PDL) subjected to uniaxial tension and compression is reported. Several sections normal to the longitudinal axis of bovine incisors and molars were extracted from different depths. Specimens with dimensions 10 x 5 x 2 mm including dentine, PDL and alveolar bone were obtained from these sections. Scanning electron microscopy suggested a strong similarity between the bovine PDL and the human PDL microstructure described in the literature. The prepared specimens were tested in a custom made uniaxial testing machine. They were clamped on their bone and dentine extremities and immersed in a saline solution at 37 degrees C. Stress-strain curves indicated that the PDL is characterized by a non-linear and time-dependent mechanical behaviour with the typical features of collagenous soft tissues. The curves exhibited hysteresis and preconditioning effects. The mechanical parameters evaluated in tension were maximum tangent modulus, strength, maximizer strain and strain energy density. For the molars, all these parameters increased with depth except for the apical region. For the incisors, all parameters increased with depth except ultimate strain which decreased. It was assumed that collagen fibre density and orientation were responsible for these findings.     

Tooth periodontal ligament: Direct 3D microCT visualization of the collagen network and how the network changes when the tooth is loaded

The periodontal ligament (PDL), a soft tissue connecting the tooth and the bone, is essential for tooth movement, bone remodeling and force dissipation. A collagenous network that connects the tooth root surface to the alveolar jaw bone is one of the major components of the PDL. The organization of the collagenous component and how it changes under load is still poorly understood. Here using a state-of-the-art custom-made loading apparatus and a humidified environment inside a microCT, we visualize the PDL collagenous network of a fresh rat molar in 3D at 1 μm voxel size without any fixation or contrasting agents. We demonstrate that the PDL collagen network is organized in sheets. The spaces between sheets vary thus creating dense and sparse networks. Upon vertical loading, the sheets in both networks are stretched into well aligned arrays. The sparse network is located mainly in areas which undergo compressive loading as the tooth moves towards the bone, whereas the dense network functions mostly in tension as the tooth moves further from the bone. This new visualization method can be used to study other non-mineralized or partially mineralized tissues, and in particular those that are subjected to mechanical loads. The method will also be valuable for characterizing diseased tissues, as well as better understanding the phenotypic expressions of genetic mutants.     

A mechanistic damage model for ligaments

PurposeThe accuracy of biomechanical models is predicated on the realism by which they represent their biomechanical tissues. Unfortunately, most models use phenomenological ligament models that neglect the behaviour in the failure region. Therefore, the purpose of this investigation was to test whether a mechanistic model of ligamentous tissue portrays behaviour representative of actual ligament failure tests.ModelThe model tracks the time-evolution of a population of collagen fibres in a theoretical ligament. Each collagen fibre is treated as an independent linear cables with constant stiffness. Model equations were derived by assuming these fibres act as a continuum and applying a conservation law akin to Huxley’s muscle model. A breaking function models the rate of collagen fibre breakage at a given displacement, and was chosen to be a linear function for this preliminary analysis.MethodsThe model was fitted to experimental average curves for the cervical anterior longitudinal ligament. In addition, the model was cyclically loaded to test whether the tissue model behaves similarly.ResultsThe model agreed very well with experiment with an RMS error of 14.23 N and an R² of 0.995. Cyclic loading exhibited a reduction in force similar to experimental data.Discussion and conclusionThe proposed model showcases behaviour reminiscent of actual ligaments being strained to failure and undergoing cyclic load. Future work could incorporate viscous effects, or validate the model further by testing it in various loading conditions. Characterizing the breaking function more accurately would also lead to better results.     

An interface model for the periodontal ligament

A nonlinear interface constitutive law is formulated for modeling the mechanical behavior of the periodontal ligament. This gives an accurate interpolation of the few available experimental results and provides a reasonably simple model for mechanical applications. The model is analyzed from the viewpoints of both mathematical consistency and effectiveness in numerical calculations. In order to demonstrate the latter, suitable two- and three-dimensional nonlinear interface finite elements have been implemented.     

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.     

Structure of the periodontal ligament during tooth eruption in the dog: descriptive aspects

Serially stained uncalcified sections of young dog mandibles were examined to study the structure of the periodontal ligament of the erupting first right molar. The periodontal ligament around tooth crown presents three zones. The first, near the dental follicle, is a tick layer of parallel collagen bundles with numerous flattened fibroblasts. The second, intermediate, contains a blood vessels network, particularly veins and capillaries. The third, outer, is occupied by a continuous layer of osteoclasts and osteoblasts. Also the periodontal ligament around the tooth presents three layers, the outer and the intermediate rich of cells more than the inner. Particularly, the outer layer shows numerous osteoblasts surrounding the developing trabeculae of the alveolar bone and the collagen fiber bundles of the periodontal ligament. These penetrate into the trabeculae and appear similar to the osteoid layer. These results indicate that the alveolar bone increases by ossification of the connective tissue of the periodontal ligament.     

Mechanical identification of layer-specific properties of mouse carotid arteries using 3D-DIC and a hyperelastic anisotropic constitutive model

The role of mechanics is known to be of primary order in many arterial diseases; however, determining mechanical properties of arteries remains a challenge. This paper discusses the identifiability of the passive mechanical properties of a mouse carotid artery, taking into account the orientation of collagen fibres in the medial and adventitial layers. On the basis of 3D digital image correlation measurements of the surface strain during an inflation/extension test, an inverse identification method is set up. It involves a 3D finite element mechanical model of the mechanical test and an optimisation algorithm. A two-layer constitutive model derived from the Holzapfel model is used, with five and then seven parameters. The five-parameter model is successfully identified providing layer-specific fibre angles. The seven-parameter model is over parameterised, yet it is shown that additional data from a simple tension test make the identification of refined layer-specific data reliable.     

Preparation of Unfixed and Undecalcified Frozen Sections of Adult Rat Periodontal Ligament During Experimental Tooth Movement

The upper first molars of adult male rats were moved for 7 days and unfixed, undecalcified frozen sections of the molar periodontal ligament were prepared and observed. The upper jaws of the rats were immersed rapidly in liquid nitrogen and sectioned with a cryostat using a super hard knife. Five micrometer serial sections were cut, collected, freeze-dried and observed with both light and scanning electron microscopy. Electron probe microanalysis (EPMA) was also performed on the sections. On the tension side of the periodontal ligament, periodontal fibers were stretched and the osteoblasts were aligned on the osteoid, which showed metachro-masia with the toluidine blue stain. On the pressure side where the periodontal ligament was extremely compressed, tissue degeneration was caused by tooth movement and the osteoclasts were observed on the bone surface adjacent to the degenerating tissues. Scanning electron microscopy revealed a network arrangement of the collagen fiber bundles on the tension side, but not on the pressure side of the periodontal ligament. The spectrum obtained from EPMA of the osteoid demonstrated X-ray (Ka) peaks of Na, P, S, K and Ca.