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

The crucial role of periodontal ligament's Poisson's ratio and tension-compression asymmetric moduli on the evaluation of tooth displacement and stress state of periodontal ligament

The hydrostatic stress in the periodontal ligament (PDL) evaluated by finite element analysis is considered an important indicator for determining an appropriate orthodontic force. The computed result of the hydrostatic stress strongly depends on the PDL material model used in the orthodontic simulation. This study aims to investigate the effects of PDL Poisson's ratio and tension-compression asymmetric moduli on both the simulated tooth displacement and the PDL hydrostatic stress. Three tension-compression symmetric and two asymmetric PDL constitutive models were selected to simulate the tensile and compressive behavior of a PDL specimen under uniaxial loading, and the resulting numerical results were compared with the in-vitro PDL experimental results reported in the literature. Subsequently, a tooth model was established, and the selected constitutive models and parameters were employed to assess the hydrostatic stress state in the PDL under two distinct loading conditions. The simulated results indicate that PDL Poisson's ratio and tension-compression asymmetry exert substantial influences on the simulated PDL hydrostatic stress. Conversely, the elastic modulus exhibits minimal impact on the PDL stress state under the identical loading conditions. Furthermore, the PDL models with tension-compression asymmetric moduli and appropriate Poisson's ratio yield more realistic hydrostatic stress. Hence, it is imperative to employ suitable Poisson's ratio and tension-compression asymmetric moduli for the purpose of characterizing the biomechanical response of the PDL in orthodontic simulations.

Functional non-uniformity of periodontal ligaments tunes mechanobiological stimuli across soft- and hard-tissue interfaces

The bone-periodontal ligament-tooth (BPT) complex is a unique mechanosensing soft-/hard-tissue interface, which governs the most rapid bony homeostasis in the body responding to external loadings. While the correlation between such loading and alveolar bone remodelling has been widely recognised, it has remained challenging to investigate the transmitted mechanobiological stimuli across such embedded soft-/hard-tissue interfaces of the BPT complex. Here, we propose a framework combining three distinct bioengineering techniques (i, ii, and iii below) to elucidate the innate functional non-uniformity of the PDL in tuning mechanical stimuli to the surrounding alveolar bone. The biphasic PDL mechanical properties measured via nanoindentation, namely the elastic moduli of fibres and ground substance at the sub-tissue level (i), were used as the input parameters in an image-based constitutive modelling framework for finite element simulation (ii). In tandem with U-net deep learning, the Gaussian mixture method enabled the comparison of 5195 possible pseudo-microstructures versus the innate non-uniformity of the PDL (iii). We found that the balance between hydrostatic pressure in PDL and the strain energy in the alveolar bone was maintained within a specific physiological range. The innate PDL microstructure ensures the transduction of favourable mechanobiological stimuli, thereby governing alveolar bone homeostasis. Our outcomes expand current knowledge of the PDL's mechanobiological roles and the proposed framework can be adopted to a broad range of similar soft-/hard- tissue interfaces, which may impact future tissue engineering, regenerative medicine, and evaluating therapeutic strategies. STATEMENT OF SIGNIFICANCE: A combination of cutting-edge technologies, including dynamic nanomechanical testing, high-resolution image-based modelling and machine learning facilitated computing, was used to elucidate the association between the microstructural non-uniformity and biomechanical competence of periodontal ligaments (PDLs). The innate PDL fibre network regulates mechanobiological stimuli, which govern alveolar bone remodelling, in different tissues across the bone-PDL-tooth (BPT) interfaces. These mechanobiological stimuli within the BPT are tuned within a physiological range by the non-uniform microstructure of PDLs, ensuring functional tissue homeostasis. The proposed framework in this study is also applicable for investigating the structure-function relationship in broader types of fibrous soft-/hard- tissue interfaces.

Identification of the periodontal ligament material parameters using response surface method

The orthodontic treatment can be guided by the finite element (FE) simulation of periodontal ligament (PDL) mechanical properties, and the biomimetic degree of FE simulation can be primarily affected by the material properties of the PDL. According to the principle of parameter inverse, a method: response surface (RS) method and FE inverse method were proposed to identify the material parameters of PDL. The Prony series viscoelastic FE model was established based on the relaxation experiment. With root mean square error of simulation results and experimental results as the objective function, the optimal parameter combination was obtained by RS method, and the FE simulation result were compared with the experimental result. The result showed that the optimal parameters of the PDL were elastic modulus: 3.791 MPa, Poisson's ratio: 0.42, temperature: 29.294°C separately, and the simulation result of optimal combination maintained consistency with experiment with the correlation coefficient of 0.97258, indicating that the method proposed in this paper could well identify of PDL material parameters. The parameter identification method used in this paper can significantly improve the calculation efficiency, and reduce the parameter identification error compared with the simple FE inverse method, which has scientific significance and theoretical value.

Biomechanical Modelling for Tooth Survival Studies: Mechanical Properties, Loads and Boundary Conditions-A Narrative Review

Recent biomechanical studies have focused on studying the response of teeth before and after different treatments under functional and parafunctional loads. These studies often involve experimental and/or finite element analysis (FEA). Current loading and boundary conditions may not entirely represent the real condition of the tooth in clinical situations. The importance of homogenizing both sample characterization and boundary conditions definition for future dental biomechanical studies is highlighted. The mechanical properties of dental structural tissues are presented, along with the effect of functional and parafunctional loads and other environmental and biological parameters that may influence tooth survival. A range of values for Young's modulus, Poisson ratio, compressive strength, threshold stress intensity factor and fracture toughness are provided for enamel and dentin; as well as Young's modulus and Poisson ratio for the PDL, trabecular and cortical bone. Angles, loading magnitude and frequency are provided for functional and parafunctional loads. The environmental and physiological conditions (age, gender, tooth, humidity, etc.), that may influence tooth survival are also discussed. Oversimplifications of biomechanical models could end up in results that divert from the natural behavior of teeth. Experimental validation models with close-to-reality boundary conditions should be developed to compare the validity of simplified models.

Finite Element Study of Periodontal Ligament Properties for a Maxillary Central Incisor and a Mandibular Second Molar Under Percussion Conditions

Purpose

The quantitative percussion diagnostics (QPD) response of a mandibular second molar and a maxillary central incisor including their supporting ligament/bone structure was simulated using dynamic 3D finite element analysis (FEA). The focus of the work was on the role of the periodontal ligament (PDL) which acts as a damper in the dental structure and dissipates occlusal forces transmitted from the tooth surface to the surrounding bone.

Methods

Several FEA models were developed to examine the effects of mechanical characteristics that have been reported for the PDL. Specifically, the effects of changing the PDL’s quasi-static elastic modulus and Rayleigh damping properties were predicted.

Results

The present FEA simulations indicate that the PDL can significantly reduce forces for both the incisor and the molar compared to when there is no PDL (i.e. ankylosed tooth) as long as the quasi-static elastic modulus of the PDL is among the lowest reported (~ 0.1 MPa). In addition, the FEA simulations for both the incisor and molar with this lower value of the PDL quasi-static elastic modulus are also in reasonably good agreement with experimental percussion data. A simple approximation for partitioning Rayleigh damping properties between the hard and soft tissues was also found to provide reasonable values of overall damping that are consistent with experimental data.

Conclusion

The overall findings indicate that using a quasi-static elastic modulus of approximately 0.1 MPa for the PDL in combination with Rayleigh damping gives realistic predictions of the mechanical response of a tooth under QPD loading conditions.

Development and verification of a constitutive model for human periodontal ligament based on finite element analysis

This study aimed to develop a constitutive model for human periodontal ligament (PDL) by combining the hyperelastic and viscosity models. We performed the finite element analysis (FEA) to simulate the experimental processes of the PDL in vitro and in vivo tests to verify the developed model. The FEA results indicated that the simulative curves were consistent with the experimental curves in the PDL in vitro tests. Moreover, for the in vivo measurements, the simulative result of 0.6258 N was similar to the experimental value of 0.65 N. The study results can help orthodontists better understand the biomechanical characteristics of PDL.

An in-silico approach to modeling orthodontic tooth movement using stimulus-induced external bone adaptation

Bone remodeling process has been used in orthodontics to treat malposition of teeth in patients by applying stimuli outside of usual everyday loads to promote tooth movement by affecting equilibrium state of the surrounding bone tissue. Accurate modeling of long term orthodontic tooth movement (OTM) is crucial in the field of dental biomechanical research since it allows to predict the behavior and interaction of bone-tooth environment in a non-destructive way, and helps to gain more insight on how exactly tooth motion progresses over time. Existence of such predictive tools might help to avoid the adverse effects of OTM on teeth and the surrounding tissues during this clinical procedure. In this study a new numerical approach to simulating long-term OTM is proposed, that involves external bone adaptation with strain energy density of the bone taken as the stimulus parameter and bone adaptation modeled by nodal movements at the bone-tooth interface using Abaqus UMESHMOTION subroutine. Contrary to conventional re-meshing algorithms, where the mesh of resorbed-apposed bone region is constantly updated and element deletion/creation is performed for each increment, the proposed method only moves nodes without changing the initial mesh topology. For this study, a 3D model of right central maxillary incisor tooth and its surrounding maxillary bone was used for the modeling of OTM for a duration of 1 week. Two test cases were performed and the results from induced tooth motion were investigated. Results indicate tooth movement values that were quite close to clinical values provided in the literature and this method is easily applicable to validate various postulates of OTM via adapting the stimulus-adaption rate relation and patient-specific planning of orthodontic patients as well.

Tensile creep mechanical behavior of periodontal ligament: A hyper-viscoelastic constitutive model

OBJECTIVE

In orthodontic treatment, the biomechanical response of periodontal ligament (PDL) induces tooth movement. Coupling modeling of PDL can effectively reflect its biomechanical response. The nonlinear creep mechanical behavior of PDL was studied by uniaxial tensile creep test and a new hyper-viscoelastic constitutive model. Two coupling modeling methods with limitations were excluded.

METHODS

PDL specimens were prepared from the central incisors of pig mandible. The theoretical step function was replaced by static loading with a total loading time of 1 s. The creep loading with the constant stresses of 0.05, 0.1, and 0.15 MPa was selected and kept unchanged for 1000 s. The instantaneous hyperelastic mechanical behavior and time-dependent nonlinear viscoelastic mechanical behavior of PDL were characterized by coupled instantaneous third-order Ogden hyperelastic and time-dependent nonlinear creep models.

RESULTS

The results showed that the instantaneous elastic curve of PDL increases in the form of hyperelastic index. The creep strain and creep compliance curves increase rapidly before 200s, and then increase slowly in steady state. The creep strain increased with an increase in the constant stress; conversely, the creep compliance decreased with an increase in the constant stress. The results showed that the experimental data were highly consistent with the hyper-viscoelastic constitutive model (R²>0.97).

SIGNIFICANCE

We normalize the framework of hyper-viscoelastic coupling modeling (Instantaneous hyperelastic model + time-dependent nonlinear viscoelastic model). Which can be extended to other nonlinear viscoelastic biomaterials.

Where is the center of resistance of a maxillary first molar? A 3-dimensional finite element analysis

INTRODUCTION

The center of resistance (C_Res) is regarded as the fundamental reference point for predictable tooth movement. Accurate estimation can greatly enhance the efficiency of orthodontic tooth movement. Only a handful of studies have evaluated the C_Res of a maxillary first molar; however, most had a low sample size (in single digits), used idealized models, or involved 2-dimensional analysis. The objectives of this study were to: (1) determine the 3-dimensional (3D) location of the C_Res of maxillary first molars, (2) evaluate its variability in a large sample, and (3) investigate the effects of applying orthodontic load from 2 directions on the location of the C_Res.

METHODS

Cone-beam computed tomography scans of 50 maxillary molars from 25 patients (mean age, 20.8 ± 8.7 years) were used. The cone-beam computed tomography volume images were manipulated to extract 3D biological structures via segmentation. The segmented structures were cleaned and converted into virtual mesh models made of tetrahedral triangles having a maximum edge length of 1 mm. The block, which included the molars and periodontal ligament, consisted of a mean of 7753 ± 2748 nodes and 38,355 ± 14,910 tetrahedral elements. Specialized software was used to preprocess the models to create an assembly and assign material properties, interaction conditions, boundary conditions, and load applications. Specific loads were applied, and custom-designed algorithms were used to analyze the stress and strain to locate the C_Res. The C_Res was measured in relation to the geometric center of the buccal surface of the molar and the trifurcation of the molar roots.

RESULTS

The average location of the C_Res for the maxillary first molar was 4.94 ± 1.39 mm lingual, 2.54 ± 2.7 mm distal, and 7.86 ± 1.66 mm gingival relative to the geometric center of the buccal surface of the molar and 0.136 ± 1.51 mm lingual (P <0.01), 1.48 ± 2.26 mm distal (P <0.01), and 0.188 ± 1.75 mm gingival (P >0.01) relative to the trifurcation of the molar roots. In the anteroposterior (y-axis) and the vertical (z-axis) planes, the C_Res showed significant association with root divergence (P <0.01).

CONCLUSIONS

The C_Res of the maxillary first molar was located apical and distal to the trifurcation area. It showed significant variation in its location. The 3D location of and also varied with the force direction. In some samples, this deviation was large. For accurate and predictable movement, tooth-specific C_Res need to be calculated.

Orthodontic Tooth Movement Studied by Finite Element Analysis: an Update. What Can We Learn from These Simulations?

PURPOSE OF REVIEW

To produce an updated overview of the use of finite element (FE) analysis for analyzing orthodontic tooth movement (OTM). Different levels of simulation complexity, including material properties and level of morphological representation of the alveolar complex, will be presented and evaluated, and the limitations will be discussed.

RECENT FINDINGS

Complex formulations of the PDL have been proposed, which might be able to correctly predict the behavior of the PDL both when chewing forces and orthodontic forces are simulated in FE models. The recent findings do not corroborate the simplified view of the classical OTM theories. The use of complex and biologically coherent FE models can help understanding the mechanisms leading to OTM as well as predicting the risk of root resorption related to specific force systems and magnitudes.

Tissue Engineering for Periodontal Ligament Regeneration: Biomechanical Specifications

The periodontal biomechanical environment is very difficult to investigate. By the complex geometry and composition of the periodontal ligament (PDL), its mechanical behavior is very dependent on the type of loading (compressive versus tensile loading; static versus cyclic loading; uniaxial versus multiaxial) and the location around the root (cervical, middle, or apical). These different aspects of the PDL make it difficult to develop a functional biomaterial to treat periodontal attachment due to periodontal diseases. This review aims to describe the structural and biomechanical properties of the PDL. Particular importance is placed in the close interrelationship that exists between structure and biomechanics: the PDL structural organization is specific to its biomechanical environment, and its biomechanical properties are specific to its structural arrangement. This balance between structure and biomechanics can be explained by a mechanosensitive periodontal cellular activity. These specifications have to be considered in the further tissue engineering strategies for the development of an efficient biomaterial for periodontal tissues regeneration.

Numerical simulation of optimal range of rotational moment for the mandibular lateral incisor, canine and first premolar based on biomechanical responses of periodontal ligaments: a case study

OBJECTIVES

The objective of this study was to investigate the optimal range of rotational moment for the mandibular lateral incisor, canine and first premolar to determine tooth movements during orthodontic treatment using hydrostatic stress and logarithmic strain on the periodontal ligament (PDL) as indicators by numerical simulations.

MATERIAL AND METHODS

Teeth, PDL and alveolar bone numerical models were constructed as analytical objects based on computed tomography (CT) images. Teeth were assumed to be rigid bodies, and rotational moments ranging from 1.0 to 4.0 Nmm were exerted on the crowns. PDL was defined as a hyperelastic-viscoelastic material with a uniform thickness of 0.25 mm. The alveolar bone model was constructed using a non-uniform material with varied mechanical properties determined based on Hounsfield unit (HU) values calculated using CT images, and its bottom was fixed completely. The optimal range values of PDL compressive and tensile stress were set as 0.47-12.8 and 18.8-51.2 kPa, respectively, whereas that of PDL logarithmic strain was set as 0.15-0.3%.

RESULTS

The rotational tendency of PDL was around the long axis of teeth when loaded. The optimal range values of rotational moment for the mandibular lateral incisor, canine and first premolar were 2.2-2.3, 3.0-3.1 and 2.8-2.9 Nmm, respectively, referring to the biomechanical responses of loaded PDL. Primarily, the optimal range of rotational moment was quadratically dependent on the area of PDL internal surface (i.e. area of PDL internal surface was used to indicate PDL size), as described by the fitting formula.

CONCLUSIONS

Biomechanical responses of PDL can be used to estimate the optimal range of rotational moment for teeth. These rotational moments were not consistent for all teeth, as demonstrated by numerical simulations.

CLINICAL RELEVANCE

The quantitative relationship between the area of PDL internal surface and the optimal orthodontic moment can help orthodontists to determine a more reasonable moment and further optimise clinical treatment.

In vivo effects of different orthodontic loading on root resorption and correlation with mechanobiological stimulus in periodontal ligament

Orthodontic root resorption is a common side effect of orthodontic therapy. It has been shown that high hydrostatic pressure in the periodontal ligament (PDL) generated by orthodontic forces will trigger recruitment of odontoclasts, leaving resorption craters on root surfaces. The patterns of resorption craters are the traces of odontoclast activity. This study aimed to investigate resorptive patterns by: (i) quantifying spatial root resorption under two different levels of in vivo orthodontic loadings using microCT imaging techniques and (ii) correlating the spatial distribution pattern of resorption craters with the induced mechanobiological stimulus field in PDL through nonlinear finite-element analysis (FEA) in silico. Results indicated that the heavy force led to a larger total resorption volume than the light force, mainly by presenting greater individual crater volumes ( p < 0.001) than increasing crater numbers, suggesting that increased mechano-stimulus predominantly boosted cellular resorption activity rather than recruiting more odontoclasts. Furthermore, buccal-cervical and lingual-apical regions in both groups were found to have significantly larger resorption volumes than other regions ( p < 0.005). These clinical observations are complemented by the FEA results, suggesting that root resorption was more likely to occur when the volume average compressive hydrostatic pressure exceeded the capillary blood pressure (4.7 kPa).

Molecular Basis for Periodontal Ligament Adaptation to In Vivo Loading

A soft food diet leads to changes in the periodontal ligament (PDL). These changes, which have been recognized for more than a century, are ascribed to alterations in mechanical loading. While these adaptive responses have been well characterized, the molecular, cellular, and mechanical mechanisms underlying the changes have not. Here, we implicate Wnt signaling in the pathoetiology of PDL responses to underloading. We show that Wnt-responsive cells and their progeny in the PDL space exhibit a burst in proliferation in response to mastication. If an animal is fed a soft diet from the time of weaning, then this burst in Wnt-responsive cell proliferation is quelled; as a consequence, both the PDL and the surrounding alveolar bone undergo atrophy. Returning these animals to a hard food diet restores the Wnt signaling in PDL. These data provide, for the first time, a molecular mechanism underlying the adaptive response of the PDL to loading.

A biomechanical case study on the optimal orthodontic force on the maxillary canine tooth based on finite element analysis

Excessive forces may cause root resorption and insufficient forces would introduce no effect in orthodontics. The objective of this study was to investigate the optimal orthodontic forces on a maxillary canine, using hydrostatic stress and logarithmic strain of the periodontal ligament (PDL) as indicators. Finite element models of a maxillary canine and surrounding tissues were developed. Distal translation/tipping forces, labial translation/tipping forces, and extrusion forces ranging from 0 to 300 g (100 g=0.98 N) were applied to the canine, as well as the force moment around the canine long axis ranging from 0 to 300 g·mm. The stress/strain of the PDL was quantified by nonlinear finite element analysis, and an absolute stress range between 0.47 kPa (capillary pressure) and 12.8 kPa (80% of human systolic blood pressure) was considered to be optimal, whereas an absolute strain exceeding 0.24% (80% of peak strain during canine maximal moving velocity) was considered optimal strain. The stress/strain distributions within the PDL were acquired for various canine movements, and the optimal orthodontic forces were calculated. As a result the optimal tipping forces (40-44 g for distal-direction and 28-32 g for labial-direction) were smaller than the translation forces (130-137 g for distal-direction and 110-124 g for labial-direction). In addition, the optimal forces for labial-direction motion (110-124 g for translation and 28-32 g for tipping) were smaller than those for distal-direction motion (130-137 g for translation and 40-44 g for tipping). Compared with previous results, the force interval was smaller than before and was therefore more conducive to the guidance of clinical treatment. The finite element analysis results provide new insights into orthodontic biomechanics and could help to optimize orthodontic treatment plans.

Nonlinear Biomechanical Characteristics of the Schneiderian Membrane: Experimental Study and Numerical Modeling

Objective

The aim of this study is to quantify the nonlinear mechanical behavior of the Schneiderian membrane.

Methods

Thirty cadaveric maxillary sinus membrane specimens were divided into the elongation testing group and the perforation testing group. Mechanical experimental measurements were taken via ex vivo experiments. Theoretical curves were compared with experimental findings to assess the effectiveness of the nonlinear mechanical properties. The FE model with nonlinear mechanical properties was used to simulate the detachment of the Schneiderian membrane under loading.

Results

The mean thickness of the membrane samples was 1.005 mm. The mean tensile strength obtained by testing was 6.81 N/mm². In membrane perforation testing, the mean tensile strength and the linear elastic modulus were significantly higher than those in membrane elongation testing (P < 0.05). The mean adhesion force between the Schneiderian membrane and the bone was 0.052 N/mm. By FE modeling, the squared correlation coefficients of theoretical stress-strain curves for the nonlinear and linear models were 0.99065 and 0.94656 compared with the experimental data.

Conclusions

The biomechanical properties of the Schneiderian membrane were implemented into the FE model, which was applied to simulate the mechanical responses of the Schneiderian membrane in sinus floor elevation.

3-D finite element analysis of the effects of post location and loading location on stress distribution in root canals of the mandibular 1st molar

Objective

The purpose of this study was to evaluate, by using finite element analysis, the influence of post location and occlusal loading location on the stress distribution pattern inside the root canals of the mandibular 1^st molar.

Material and Methods

Three different 3-D models of the mandibular 1^st molar were established: no post (NP) – a model of endodontic and prosthodontic treatments; mesiobuccal post (MP) – a model of endodontic and prosthodontic treatments with a post in the mesiobuccal canal; and distal post (DP) – a model of endodontic and prosthodontic treatments with a post in the distal canal. A vertical force of 300 N, perpendicular to the occlusal plane, was applied to one of five 1 mm² areas on the occlusal surface; mesial marginal ridge, distal marginal ridge, mesiobuccal cusp, distobuccal cusp, and central fossa. Finite element analysis was used to calculate the equivalent von Mises stresses on each root canal.

Results

The DP model showed similar maximum stress values to the NP model, while the MP model showed markedly greater maximum stress values. The post procedure increased stress concentration inside the canals, although this was significantly affected by the site of the force.

Conclusions

In the mandibular 1^st molar, the distal canal is the better place to insert the post than the mesiobuccal canal. However, if insertion into the mesiobuccal canal is unavoidable, there should be consideration on the occlusal contact, making central fossa and distal marginal ridge the main functioning areas.