Numerical Estimation of the Centres of Rotation and Resistance in Orthodontic Tooth Movement

The center of resistance of a tooth: a review of the literature

The center of resistance is considered the fundamental reference point for controlled tooth movement. Accurate determination of its location can greatly enhance the efficiency of orthodontic treatment. The purpose of this review was to analyse the scientific literature related to the location of center of resistance of tooth determined by various approaches. The literature describes three essential approaches to identify the center of resistance point, one being experimental in nature, one based on an analytical physical approach, and one using a numerical physical approach that uses a finite element simulation. A review on data referring to the location of the center of resistance, limited to single rooted tooth has been performed from electronic databases. It showed variation in its location related to the assumptions used in the model. The center of resistance of tooth therefore cannot be considered a static point, but rather as the composite point of all factors offering resistance to the applied force such as the tooth morphology and mass distribution within the tooth, the structure of the periodontium, the alveolar bone level, the adjacent teeth and direction of force applied.

Predictability of lower incisor tip using clear aligner therapy

Background

Uprighting incisors is particularly important with clear aligner therapy as incisor tip determines the mesio-distal space needed in the arch, and consequently the fit of the aligner. The objective of this study was to investigate the accuracy of ClinCheck® software to predict lower incisor tip by comparing digitally prescribed movements with actual clinical outcomes and to determine whether the presence of a vertically orientated rectangular composite attachment influences the efficacy of incisor tip.

Methodology

This retrospective study included 66 lower incisors from 42 non-extraction adult patients treated using the Invisalign® appliance. Twenty-one incisors had vertical attachments, while 45 incisors did not have any attachments. Lower incisor tip was measured at T0 (pre-treatment), T1 (predicted post-treatment) and T2 (achieved post-treatment) on digital models using metrology software. The change in position from T0 to T1 and T0 to T2 was measured from the estimated centre of resistance (CRes) of each tooth. The estimated centre of rotation was plotted relative to the CRes to describe the type of orthodontic tooth movement (OTM) predicted and achieved.

Results

Predicted incisor tip and achieved incisor tip were positively correlated (R2 = 0.55; p < 0.001). For every degree of tip planned 0.4 degrees of tip was achieved. The presence of an attachment resulted in 1.2 degrees greater tip (F = 3.7; p = 0.062) and 0.5 mm greater movement of the predicted apex of the tooth (F = 4.3; p = 0.042) compared with the no attachment group. The type of OTM achieved differed from the type predicted. Sixty-seven percent of incisors investigated were predicted to move by root movement, while 46% achieved this type of movement.

Conclusions

The amount of lower incisor tip achieved was on average substantially less than the ClinCheck® displayed. Vertically orientated rectangular attachments are recommended where large root movement is planned, and their presence slightly improves apex movement.

Three-dimensional nonlinear prediction of tooth movement from the force system and root morphology

OBJECTIVES

To determine the different impact of moment-to-force ratio (M:F) variation for each tooth and spatial plane and to develop a mathematical model to predict the orthodontic movement for every tooth.

MATERIALS AND METHODS

Two full sets of teeth were obtained combining cone-beam computed tomography (CBCT) and optical scans for two patients. Subsequently, a finite element analysis was performed for 510 different force systems for each tooth to evaluate the centers of rotation.

RESULTS

The center of CROT locations were analyzed, showing that the M:F effect was related to the spatial plane on which the moment was applied, to the force direction, and to the tooth morphology. The tooth dimensions on each plane were mathematically used to derive their influence on the tooth movement.

CONCLUSION

This study established the basis for an orthodontist to determine how the teeth move and their axes of resistance, depending on their morphology alone. The movement is controlled by a parameter (k), which depends on tooth dimensions and force system features. The k for a tooth can be calculated using a CBCT and a specific set of covariates.

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.

A Force on the Crown and Tug of War in the Periodontal Complex

The load-bearing dentoalveolar fibrous joint is composed of biomechanically active periodontal ligament (PDL), bone, cementum, and the synergistic entheses of PDL-bone and PDL-cementum. Physiologic and pathologic loads on the dentoalveolar fibrous joint prompt natural shifts in strain gradients within mineralized and fibrous tissues and trigger a cascade of biochemical events within the widened and narrowed sites of the periodontal complex. This review highlights data from in situ biomechanical simulations that provide tooth movements relative to the alveolar socket. The methods and subsequent results provide a reasonable approximation of strain-regulated biochemical events resulting in mesial mineral formation and distal resorption events within microanatomical regions at the ligament-tethered/enthesial ends. These biochemical events, including expressions of biglycan, decorin, chondroitin sulfated neuroglial 2, osteopontin, and bone sialoprotein and localization of various hypertrophic progenitors, are observed at the alkaline phosphatase-positive widened site, resulting in mineral formation and osteoid/cementoid layers. On the narrowed side, tartrate-resistant acid phosphatase regions can lead to a sequence of clastic activities resulting in resorption pits in bone and cementum. These strain-regulated biochemical and subsequently biomineralization events in the load-bearing periodontal complex are critical for maintenance of the periodontal space and overall macroscale joint biomechanics.

Experimental–numerical analysis of minipig's multi-rooted teeth

The paper pertains to the analysis of the biomechanical behaviour of the periodontal ligament (PDL) by using a combined experimental and numerical approach. Experimental analysis provides information about a two-rooted pig premolar tooth in its socket with regard to morphological configuration and deformational response. The numerical analysis developed for the present investigation adopts a specific anisotropic hyperelastic formulation, accounting for tissue structural arrangement. The parameters to be adopted for the PDL constitutive model are evaluated with reference to data deducted from experimental in vitro tests on different specimens taken from literature. According to morphometric data relieved, solid models are provided as basis for the development of numerical models that adopt the constitutive formulation proposed. A reciprocal validation of experimental and numerical data allows for the evaluation of reliability of results obtained. The work is intended as preliminary investigation to study the correlation between mechanical status of PDL and induction to cellular activity in orthodontic treatments.

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 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.

Parametric finite element analysis and closed-form solutions in orthodontics

The goal and clinical relevance of this work was the development of closed formulas that are correct and simple enough for a fast decision making by the orthodontist in the daily praxis. This paper performs a parametric three-dimensional finite element linear analysis on a maxillary central incisor with a root of paraboloidal shape, which is subjected to typical orthodontic force-systems. Parameters of most importance, such as the tooth mobility in translation and in pure moment rotation including orthodontic centers, as well as the stresses inside the periodontal ligament are calculated for a large variety of over four hundred different couples of root lengths and root diameters around a nominal value. Regression analysis is afterwards performed and establishes closed-form solutions, which are also explained in terms of analytical strain energy and hydrostatic stress considerations within the periodontal ligament characterised by a small compressibility. The obtained expressions include both the root length as well as the root diameter.

A comparative FEM-study of tooth mobility using isotropic and anisotropic models of the periodontal ligament. Finite Element Method

Orthodontic tooth movement is usually characterized by two centres: the centre of resistance and the centre of rotation. A literature survey shows that both centres vary to a significant extent in both clinical and computational experiments. This paper reports on studies upon five different hypothetical mechanical representations of the periodontal ligament (PDL) which plays the most significant role in tooth mobility. The first model considers the PDL as an isotropic and linear-elastic continuum without fibres; it also discusses some preliminary visco-elastic aspects. The next three models assume a nonlinear and anisotropic material composed of fibres only that are arranged in three different orientations, two hypothetical that have appeared previously in the literature and one more consistent with actual morphological data. The fifth model considers the PDL as an orthotropic material consisting of both a continuum and of fibres. Results were obtained by applying the Finite Element Method (FEM) on a maxillary central incisor. It was found that the isotropic linear-elastic PDL leads to occlusal positions of both centres in comparison with those obtained through the well-known Burstone's theoretical formula, while histological anisotropic fibres locate them apically and eccentrically.