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.     

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.     

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.     

The effects of modeling simplifications on craniofacial finite element models: the alveoli (tooth sockets) and periodontal ligaments

Several finite element models of a primate cranium were used to investigate the biomechanical effects of the tooth sockets and the material behavior of the periodontal ligament (PDL) on stress and strain patterns associated with feeding. For examining the effect of tooth sockets, the unloaded sockets were modeled as devoid of teeth and PDL, filled with teeth and PDLs, or simply filled with cortical bone. The third premolar on the left side of the cranium was loaded and the PDL was treated as an isotropic, linear elastic material using published values for Young's modulus and Poisson's ratio. The remaining models, along with one of the socket models, were used to determine the effect of the PDL's material behavior on stress and strain distributions under static premolar biting and dynamic tooth loading conditions. Two models (one static and the other dynamic) treated the PDL as cortical bone. The other two models treated it as a ligament with isotropic, linear elastic material properties. Two models treated the PDL as a ligament with hyperelastic properties, and the other two as a ligament with viscoelastic properties. Both behaviors were defined using published stress-strain data obtained from in vitro experiments on porcine ligament specimens. Von Mises stress and strain contour plots indicate that the effects of the sockets and PDL material behavior are local. Results from this study suggest that modeling the sockets and the PDL in finite element analyses of skulls is project dependent and can be ignored if values of stress and strain within the alveolar region are not required.     

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.     

Effect of variable periodontal ligament thickness and its non-linear material properties on the location of a tooth’s centre of resistance

In orthodontics, the 3D translational and rotational movement of a tooth is determined by the force–moment system applied and the location of the tooth’s centre of resistance (CR). Because of the practical constraints of in-vivo experiments, the finite element (FE) method is commonly used to determine the CR. The objective of this study was to investigate the geometric model details required for accurate CR determination, and the effect of material non-linearity of the periodontal ligament (PDL). A FE model of a human lower canine derived from a high-resolution µCT scan (voxel size: 50 µm) was investigated by applying four different modelling approaches to the PDL. These comprised linear and non-linear material models, each with uniform and realistic PDL thickness. The CR locations determined for the four model configurations were in the range 37.2–45.3% (alveolar margin: 0%; root apex: 100%). We observed that a non-linear material model introduces load-dependent results that are dominated by the PDL regions under tension. Load variation within the range used in clinical orthodontic practice resulted in CR variations below 0.3%. Furthermore, the individualized realistic PDL geometry shifted the CR towards the alveolar margin by 2.3% and 2.8% on average for the linear and non-linear material models, respectively. We concluded that for conventional clinical therapy and the generation of representative reference data, the least sophisticated modelling approach with linear material behaviour and uniform PDL thickness appears sufficiently accurate. Research applications that require more precise treatment monitoring and planning may, however, benefit from the more accurate results obtained from the non-linear constitutive law and individualized realistic PDL geometry.     

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.     

Response of the tooth-periodontal ligament-bone complex to load: A microCT study of the minipig molar

One strategy evolved by teeth to avoid irreversible damage is to move and deform under the loads incurred during mastication. A key component in this regard is the periodontal ligament (PDL). The role of the bone underlying the PDL is less well defined. We study the interplay between the PDL and the underlying alveolar bone when loaded in the minipig. Using an Instron loading device we confirmed that the force-displacement curves of the molars and premolars of relatively fresh minipig intact mandibles are similar to those obtained for humans and other animals. We then used this information to obtain 3D images of the teeth before and after loading the tooth in a microCT such that the load applied is in the third linear part of the force displacement curve. We observed that at many locations there is a complimentary topography of the cementum and alveolar bone surface, strongly suggesting an active interplay between the tooth and the bone during mastication. We also observed that the loaded tooth does not come into direct contact with the underlying bone surface. A highly compressed layer of PDL is present between the tooth and the bone. The structure of the bone in the upper furcation region has a unique appearance with little obvious microstructure, abundant pores that have a large size range and at many locations the bone at the PDL interface has a needle-like shape. We conclude that there is a close interaction between the tooth, the PDL and the underlying alveolar bone during mastication. The highly compressed PDL layer that separates the tooth from the bone may fulfill a key shock absorbing function.     

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.     

Mass acquisition of human periodontal ligament stem cells

The periodontal ligament (PDL) is an essential fibrous tissue for tooth retention in the alveolar bone socket. PDL tissue further functions to cushion occlusal force, maintain alveolar bone height, allow orthodontic tooth movement, and connect tooth roots with bone. Severe periodontitis, deep caries, and trauma cause irreversible damage to this tissue, eventually leading to tooth loss through the destruction of tooth retention. Many patients suffer from these diseases worldwide, and its prevalence increases with age. To address this issue, regenerative medicine for damaged PDL tissue as well as the surrounding tissues has been extensively investigated regarding the potential and effectiveness of stem cells, scaffolds, and cytokines as well as their combined applications. In particular, PDL stem cells (PDLSCs) have been well studied. In this review, I discuss comprehensive studies on PDLSCs performed in vivo and contemporary reports focusing on the acquisition of large numbers of PDLSCs for therapeutic applications because of the very small number of PDLSCs available in vivo.     

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.     

A Bone-remodelling Scheme Based on Principal Strains Applied to a Tooth During Translation

This paper investigates the role of principal strains within the periodontal ligament (PDL) during bone remodelling in orthodontics and particularly in the case of bodily motion (pure translation). Using analytical formulas of stress and strains within the PDL for the particular case of a paraboloidal central incisor during translation, the strains are directly related to the motion of the interface between the alveolar bone and the PDL, called bone surface. It is shown that both normal and shear strains within the PDL are of the same importance for bone surface motion. Moreover, both “mean average” and “geometrical average” of principal strains within the PDL play a significant role in the bone remodelling process, as they contribute with the same proportionality. In summary, the proposed formulas differ than previous ones that had been successfully applied to describe remodelling within long bones. The proposed theory is also sustained by a linear finite element analysis.     

The divergent expression of periostin mRNA in the periodontal ligament during experimental tooth movement

A novel 90-kDa protein named periostin, which is preferentially expressed in the periosteum and the periodontal ligament (PDL), may play a role in bone metabolism and remodeling. However, the precise role of periostin in the PDL remains unclear. Therefore, we examined the expression of periostin mRNA during experimental tooth movement. Experimental tooth movement was achieved in 7-week-old male Sprague-Dawley rats. In control specimens without tooth movement, the expression of periostin mRNA was uniformly observed in the PDL surrounding the mesial and distal roots of the upper molars and was weak in the PDL of the root furcation area. The periostin mRNA-expressing cells were mainly fibroblastic cells in the PDL and osteoblastic cells on the alveolar bone surfaces. The divergent expression of periostin mRNA in the PDL began to be observed at 3 h and continued up to 96 h after tooth movement. The maximum changes, which showed stronger staining in the pressure sites than in the tension sites, were observed at 24 h. The expression of periostin mRNA in the PDL 168 h after tooth movement exhibited a similar distribution to that of the control specimens. These results suggest that periostin is one of the local contributing factors in bone and periodontal tissue remodeling following mechanical stress during experimental tooth movement.     

A bone-remodelling scheme based on principal strains applied to a tooth during translation

This paper investigates the role of principal strains within the periodontal ligament (PDL) during bone remodelling in orthodontics and particularly in the case of bodily motion (pure translation). Using analytical formulas of stress and strains within the PDL for the particular case of a paraboloidal central incisor during translation, the strains are directly related to the motion of the interface between the alveolar bone and the PDL, called bone surface. It is shown that both normal and shear strains within the PDL are of the same importance for bone surface motion. Moreover, both "mean average" and "geometrical average" of principal strains within the PDL play a significant role in the bone remodelling process, as they contribute with the same proportionality. In summary, the proposed formulas differ than previous ones that had been successfully applied to describe remodelling within long bones. The proposed theory is also sustained by a linear finite element analysis.     

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.     

Oscillatory shear loading of bovine periodontal ligament--a methodological study

This study examined the stress response of bovine periodontal ligament (PDL) under sinusoidal straining. The principle of the test consisted in subjecting transverse tooth, PDL and bone sections of known geometries to controlled oscillatory force application. The samples were secured to the actuator by support plates fabricated using a laser sintering technique to fit their contours to the tooth and the alveolar bone. The actuator was attached to the root slices located in the specimen's center. Hence the machine was able to push or pull the root relative to its surrounding alveolar bone. After determining an optimal distraction amplitude, the samples were cyclically loaded first in ramps and then in sinusoidal oscillations at frequencies ranging from 0.2 to 5 Hz. In the present study the following observations were made: (1) Imaging and the laser sintering technique can be used successfully to fabricate custom-made support plates for cross-sectional root-PDL-bone sections using a laser sintering technique, (2) the load-response curves were symmetric in the apical and the coronal directions, (3) both the stress response versus phase angle and the stress response versus. strain curves tended to "straighten" with increasing frequency, and (4) the phase lag between applied strain and resulting stress was small and did not differ in the intrusive and the extrusive directions. As no mechanical or time-dependent anisotropy was demonstrable in the intrusive and extrusive directions, such results may considerably simplify the development of constitutive laws for the PDL.     

Functional cues in the development of osseous tooth support in the pig, Sus scrofa

Alveolar bone supports teeth during chewing through a ligamentous interface with tooth roots. Although tooth loads are presumed to direct the development and adaptation of these tissues, strain distribution in the alveolar bone at different stages of tooth eruption and periodontal development is unknown. This study investigates the biomechanical effects of tooth loading on developing alveolar bone as a tooth erupts into occlusion. Mandibular segments from miniature pigs, Sus scrofa, containing M₁ either erupting or in functional occlusion, were loaded in compression. Simultaneous recordings were made from rosette strain gages affixed to the lingual alveolar bone and the M₂ crypt. Overall, specimens with erupting M₁s were more deformable than specimens with occluding M₁s (mean stiffness of 246 vs. 944 MPa, respectively, p=0.004). The major difference in alveolar strain between the two stages was in orientation. The vertically applied compressive loads were more directly reflected in the alveolar bone strains of erupting M₁s, than those of occluding M₁s, presumably because of the mediation of a more mature periodontal ligament (PDL) in the latter. The PDL interface between occluding teeth and alveolar bone is likely to stiffen the system, allowing transmission of occlusal loads. Alveolar strains may provide a stimulus for bone growth in the alveolar process and crest.     

A micromechanically-based, three-dimensional interface finite element for the modelling of the periodontal ligament

Some ideas are presented for the implementation of an interface finite element capable to model in 3-dimensions several mechanical features of the periodontal ligament. Such an element is based on a simple 2-cable micromechanical model, able to reproduce the periodontal ligament stiffness and strength under any loading condition, including the pure torsion of a tooth. A single cable represents a sufficiently populated sample of collagen fibres, each with an initially crimped geometry; a single collagen fibre can provide a mechanical response, in tension, only when it is completely uncoiled. The macroscopic interface behaviour is obtained by statistical integrations over the uncoiled length of each collagen fibre, up to the fibre failure. Such a model can reproduce the periodontal ligament anisotropy due to the variable fibre orientation along the tooth root, its different behaviour in tension/compression/shear, its different behaviour for extrusive/intrusive loading, and so forth. Some numerical examples illustrate the potentialities of this interface element, quite simple in essence but rather complete from an engineering viewpoint.     

Influence of the stiffness of bone defect implants on the mechanical conditions at the interface--a finite element analysis with contact

The study focused on the influence of the implant material stiffness on stress distribution and micromotion at the interface of bone defect implants. We hypothesized that a low-stiffness implant with a modulus closer to that of the surrounding trabecular bone would yield a more homogeneous stress distribution and less micromotion at the interface with the bony bed. To prove this hypothesis we generated a three-dimensional, non-linear, anisotropic finite element (FE) model. The FE model corresponded to a previously developed animal model in sheep. A prismatic implant filled a standardized defect in the load-bearing area of the trabecular bone beneath the tibial plateau. The interface was described by face-to-face contact elements, which allow press fits, friction, sliding, and gapping. We assumed a physiological load condition and calculated contact pressures, shear stresses, and shear movements at the interface for two implants of different stiffness (titanium: E=110GPa; composite: E=2.2GPa). The FE model showed that the stress distribution was more homogeneous for the low-stiffness implant. The maximum pressure for the composite implant (2.1 MPa) was lower than for the titanium implant (5.6 MPa). Contrary to our hypothesis, we found more micromotion for the composite (up to 6 microm) than for the titanium implant (up to 4.5 microm). However, for both implants peak stresses and micromotion were in a range that predicts adequate conditions for the osseointegration. This was confirmed by the histological results from the animal studies.     

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.