Does the center of resistance depend on the direction of tooth movement?

Three-dimensional zone of the centers of resistance of the mandibular incisors and canines: A novel approach by finite element analysis

OBJECTIVES

The distribution and size of the zone of the centres of resistance (ZCR) are critical for accurate orthodontic treatments and minimizing unexpected tooth movements. However, this information remains unclear for mandibular incisors and canines. This study aims to address these gaps in knowledge.

METHODS

Finite element models of four incisors and canines from four individuals were created. Four centres of resistance (CRs) under four orthodontic directions (0° ∼ 45° ∼ 90° ∼ 135° to the sagittal plane in the horizontal plane) were assessed by a novel method. The height of the CRs was normalized to a percentage of the long axis, and the offsets were expressed as a distance value after normalization. The ZCR was obtained by fitting a 90% confidence sphere of the CR distribution. Validation was conducted to find the perturbations when the positions out of the zone were applied.

RESULTS

The maximum variation of CR in the heights under four directions was 5.17% and 3.70% for the incisors and canines, respectively. The maximum offset between the CR and long axis was 0.14mm in incisors and 0.99mm in canines. The height of the zone in the incisor and canine was 57.75% and 59.72%, and the radius of the zone was 0.60mm and 0.65mm, respectively. The force-acting point outside the zone produced a large rotation, which was unexpected.

CONCLUSIONS

The ZCR of mandibular incisors located slightly lower than that of canines, but they were almost the same size. The ZCR was recommended as the "gold reference" for orthodontics to reduce unexpected movement.

The osteoclastic activity in apical distal region of molar mesial roots affects orthodontic tooth movement and root resorption in rats

The utilization of optimal orthodontic force is crucial to prevent undesirable side effects and ensure efficient tooth movement during orthodontic treatment. However, the sensitivity of existing detection techniques is not sufficient, and the criteria for evaluating optimal force have not been yet established. Here, by employing 3D finite element analysis methodology, we found that the apical distal region (A-D region) of mesial roots is particularly sensitive to orthodontic force in rats. Tartrate-resistant acidic phosphatase (TRAP)-positive osteoclasts began accumulating in the A-D region under the force of 40 grams (g), leading to alveolar bone resorption and tooth movement. When the force reached 80 g, TRAP-positive osteoclasts started appearing on the root surface in the A-D region. Additionally, micro-computed tomography revealed a significant root resorption at 80 g. Notably, the A-D region was identified as a major contributor to whole root resorption. It was determined that 40 g is the minimum effective force for tooth movement with minimal side effects according to the analysis of tooth movement, inclination, and hyalinization. These findings suggest that the A-D region with its changes on the root surface is an important consideration and sensitive indicator when evaluating orthodontic forces for a rat model. Collectively, our investigations into this region would aid in offering valuable implications for preventing and minimizing root resorption during patients' orthodontic treatment.

Three-dimensional positions of the center of resistance of the maxillary canine distal movement under orthodontic force loading

Using finite-element analysis, we aimed to determine the center of resistance (CRes) of the maxillary canine for setting orthodontic forces. The inclination of the canine was measured by first loading from the mesial to the distal side of the mesial root surface, then the position and direction of the load that minimized the inclination were investigated. The CRes was defined as the set of midpoints of the minimum distances between two inclination lines. Twenty-one CRes values were calculated from a set of seven lines. These CRes data were then aggregated as a 95% confidence ellipsoid of width 0.170×0.016×0.009 mm with center points 4.269, 0.224, and 4.315 mm in the apical, mesial, and lingual directions from the origin, respectively. Further studies are required to effectively apply the CRes identified in this study to clinical applications.

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.

Three-dimensional location and distribution of the center of resistance in the maxillary first molar applied to occlusal force

We aimed to investigate the center of resistance (CRes) of the maxillary first molar to set the occlusal force through finite element analysis. The inclination of the molar was measured, with loading to the root on the crown, and the position and direction of the load that minimized the inclination were investigated. The CRes was defined as the set of midpoints of the minimum distances between the two lines. Nine hundred and ninety CRes points were estimated from forty-five lines. The CRes was estimated as the point 1.22 mm (Z), -0.74 mm (X), and 0.23 mm (Y) from the origin in the apical, distal, and buccal side directions, respectively, with an ellipsoid area of 1.578 (Z) mm×0.097 (X) mm×0.100 (Y) mm. Further research is required to make effective use of the CRes identified in this study for clinical applications.

The effects of different types of periodontal ligament material models on stresses computed using finite element models

INTRODUCTION

Finite element (FE) method has been used to calculate stress in the periodontal ligament (PDL), which is crucial in orthodontic tooth movement. The stress depends on the PDL material property, which varies significantly in previous studies. This study aimed to determine the effects of different PDL properties on stress in PDL using FE analysis.

METHODS

A 3-dimensional FE model was created consisting of a maxillary canine, its surrounding PDL, and alveolar bone obtained from cone-beam computed tomography scans. One Newton of intrusion force was applied vertically to the crown. Then, the hydrostatic stress and the von Mises stress in the PDL were computed using different PDL material properties, including linear elastic, viscoelastic, hyperelastic, and fiber matrix. Young's modulus (E), used previously from 0.01 to 1000 MPa, and 3 Poisson's ratios, 0.28, 0.45, and 0.49, were simulated for the linear elastic model.

RESULTS

The FE analyses showed consistent patterns of stress distribution. The high stresses are mostly concentrated at the apical area, except for the linear elastic models with high E (E >15 MPa). However, the magnitude varied significantly from -14.77 to -127.58 kPa among the analyzed patients. The E-stress relationship was not linear. The Poisson's ratio did not affect the stress distribution but significantly influenced the stress value. The hydrostatic stress varied from -14.61 to -95.48 kPa.

CONCLUSIONS

Different PDL material properties in the FE modeling of dentition do not alter the stress distributions. However, the magnitudes of the stress significantly differ among the patients with the tested material properties.

A new protocol to accurately track long-term orthodontic tooth movement and support patient-specific numerical modeling

Numerical simulation of long-term orthodontic tooth movement based on Finite Element Analysis (FEA) could help clinicians to plan more efficient and mechanically sound treatments. However, most of FEA studies assume idealized loading conditions and lack experimental calibration or validation. The goal of this paper is to propose a novel clinical protocol to accurately track orthodontic tooth displacement in three-dimensions (3D) and provide 3D models that may support FEA. Our protocol uses an initial cone beam computed tomography (CBCT) scan and several intra-oral scans (IOS) to generate 3D models of the maxillary bone and teeth ready for use in FEA. The protocol was applied to monitor the canine retraction of a patient during seven months. A second CBCT scan was performed at the end of the study for validation purposes. In order to ease FEA, a frictionless and statically determinate lingual device for maxillary canine retraction was designed. Numerical simulations were set up using the 3D models provided by our protocol to show the relevance of our proposal. Comparison of numerical and clinical results highlights the suitability of this protocol to support patient-specific FEA.

A multi-patient analysis of the center of rotation trajectories using finite element models of the human mandible

Studying different types of tooth movements can help us to better understand the force systems used for tooth position correction in orthodontic treatments. This study considers a more realistic force system in tooth movement modeling across different patients and investigates the effect of the couple force direction on the position of the center of rotation (CRot). The finite-element (FE) models of human mandibles from three patients are used to investigate the position of the CRots for different patients’ teeth in 3D space. The CRot is considered a single point in a 3D coordinate system and is obtained by choosing the closest point on the axis of rotation to the center of resistance (CRes). A force system, consisting of a constant load and a couple (pair of forces), is applied to each tooth, and the corresponding CRot trajectories are examined across different patients. To perform a consistent inter-patient analysis, different patients’ teeth are registered to the corresponding reference teeth using an affine transformation. The selected directions and applied points of force on the reference teeth are then transformed into the registered teeth domains. The effect of the direction of the couple on the location of the CRot is also studied by rotating the couples about the three principal axes of a patient’s premolar. Our results indicate that similar patterns can be obtained for the CRot positions of different patients and teeth if the same load conditions are used. Moreover, equally rotating the direction of the couple about the three principal axes results in different patterns for the CRot positions, especially in labiolingual direction. The CRot trajectories follow similar patterns in the corresponding teeth, but any changes in the direction of the force and couple cause misalignment of the CRot trajectories, seen as rotations about the long axis of the tooth.

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.

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.

Approach towards the porous fibrous structure of the periodontal ligament using micro-computerized tomography and finite element analysis

The periodontal ligament (PDL) is a porous and fibrous soft tissue situated around the tooth, which plays a key role in the transmission of loads from the tooth to the alveolar bone of the mandible. Although several studies have tried to characterize its mechanical properties, the behaviour of this tissue is not clear yet. In this study, a new simulation methodology based on a material model which considers the contribution of porous and fibrous structure with different material model formulations depending on the effort direction is proposed. The defined material model was characterized by a non-linear approximation of the porous fibrous matrix to experimental results obtained from samples of similar species and was validated by rigorous test simulations under tensile and compressive loads. The global PDL response was also validated using the parameters of the characterization in a finite element model of full human canine tooth obtained by micro-tomography. The results suggest that the porous contribution has high influence during compression because the bulk modulus of the material depends on the ability of interstitial fluid to drain. On the other hand, the collagen fibres running along the load direction are the main responsible of the ligament stiffness during tensile efforts. Thus, a material model with distinct responses depending of the load direction is proposed. Furthermore, the results suggest the importance of considering 3D finite element models based of the real morphology of human PDL for representing the irregular stress distribution caused by the coupling of complex material models and irregular morphologies.

Inclusion of periodontal ligament fibres in mandibular finite element models leads to an increase in alveolar bone strains

Alveolar bone remodelling is vital for the success of dental implants and orthodontic treatments. However, the underlying biomechanical mechanisms, in particular the function of the periodontal ligament (PDL) in bone loading and remodelling, are not well understood. The PDL is a soft fibrous connective tissue that joins the tooth root to the alveolar bone and plays a critical role in the transmission of loads from the tooth to the surrounding bone. However, due to its complex structure, small size and location within the tooth socket it is difficult to study in vivo. Finite element analysis (FEA) is an ideal tool with which to investigate the role of the PDL, however inclusion of the PDL in FE models is complex and time consuming, therefore consideration must be given to how it is included. The aim of this study was to investigate the effects of including the PDL and its fibrous structure in mandibular finite element models. A high-resolution model of a human molar region was created from micro-computed tomography scans. This is the first time that the fibrous structure of the PDL has been included in a model with realistic tooth and bone geometry. The results show that omission of the PDL creates a more rigid model, reducing the strains observed in the mandibular corpus which are of interest when considering mandibular functional morphology. How the PDL is modelled also affects the strains. The inclusion of PDL fibres alters the strains in the mandibular bone, increasing the strains in the tooth socket compared to PDL modelled without fibres. As strains in the alveolar bone are thought to play a key role in bone remodelling during orthodontic tooth movement, future FE analyses aimed at improving our understanding and management of orthodontic treatment should include the fibrous structure of the PDL.

How accurate is Invisalign in nonextraction cases? Are predicted tooth positions achieved?

OBJECTIVE

To evaluate the accuracy of Invisalign technology in achieving predicted tooth positions with respect to tooth type and direction of tooth movement.

MATERIALS AND METHODS

The posttreatment models of 30 patients who had nonextraction Invisalign treatment were digitally superimposed on their corresponding virtual treatment plan models using best-fit surface-based registration. The differences between actual treatment outcome and predicted outcome were computed and tested for statistical significance for each tooth type in mesial-distal, facial-lingual, and occlusal-gingival directions, as well as for tip, torque, and rotation. Differences larger than 0.5 mm for linear measurements and 2° for angular measurements were considered clinically relevant.

RESULTS

Statistically significant differences (P < .05) between predicted and achieved tooth positions were found for all teeth except maxillary lateral incisors, canines, and first premolars. In general, anterior teeth were positioned more occlusally than predicted, rotation of rounded teeth was incomplete, and movement of posterior teeth in all dimensions was not fully achieved. However, except for excess posttreatment facial crown torque of maxillary second molars, these differences were not large enough to be clinically relevant.

CONCLUSIONS

Although Invisalign is generally able to achieve predicted tooth positions with high accuracy in nonextraction cases, some of the actual outcomes may differ from the predicted outcomes. Knowledge of dimensions in which the final tooth position is less consistent with the predicted position enables clinicians to build necessary compensations into the virtual treatment plan.

Mechanical environment change in root, periodontal ligament, and alveolar bone in response to two canine retraction treatment strategies

OBJECTIVE

To investigate the initial mechanical environment (ME) changes in root surface, periodontal ligament (PDL), and alveolar bone due to two treatment strategies, low or high moment-to-force ratio (M/F).

SETTING AND SAMPLE POPULATION

Indiana University-Purdue University Indianapolis. Eighteen patients who underwent maxillary bilateral canine retraction.

MATERIAL AND METHOD

Finite element models of the maxillary canines from the patients were built based on their cone beam computed tomography scans. For each patient, the canine on one side had a specially designed T-loop spring with the M/F higher than the other side. Four stress invariants (1st principal/dilatational/3rd principal/von Mises stress) in the tissues were calculated. The stresses were compared with the bone mineral density (BMD) changes reported previously for linking the ME change to bone modeling/remodeling activities. The correlation was tested by the mixed-model anova.

RESULTS

The alveolar bone in the direction of tooth movement is primarily in tension, while the PDL is in compression; the stresses in the opposite direction have a reversed pattern. The M/F primarily affects the stress in root. Three stress invariants (1st principal/3rd principal/dilatational stress) in the tooth movement direction have moderate correlations with BMD loss.

CONCLUSIONS

The stress invariants may be used to characterize what the osteocytes sense when ME changes. Their distributions in the tissues are significantly different, meaning the cells experience different stimuli. The higher bone activities along the direction of tooth movement may be related to the initial volumetric increase and decrease in the alveolar bone.

Estimating the location of the center of resistance of canines

OBJECTIVE

To develop a method to quickly estimate the location of center of resistance (CR) in mesial-distal (MD) and buccal-lingual (BL) directions from the tooth's image.

MATERIALS AND METHODS

The maxillary cone-beam computed tomography (CBCT) scans of 18 patients were used. Finite element (FE) models of the canines and their surrounding tissues were built based on their CBCT scans to calculate the locations of CR. Root length, centroid of the contact surface (CCS), and centroid of projection of the contact surface (CPCS) were also obtained from the images. The CCS and CPCS locations were projected on the tooth's long axis, which were represented as percentages of the root length measured from the root's apex.

RESULTS

Using the FE results as the standards, the errors of using CCS or CPCS to estimate CR were calculated. The average location of CR calculated using the FE method was 60.2% measured from the root's apex in the MD direction and 58.4% in the BL direction. The location of the CCS was 60.9%. The difference in CR was 0.7% in the MD direction and 2.5% in the BL direction. The location of CPCS was 60.2% in the MD direction and 59.1% in the BL direction, which resulted in a 0.1% and 0.8% difference with the reference CR, respectively. The average difference of CR in the MD and BL directions was small but statistically significant (P < .05).

CONCLUSION

The locations of the CR of a human canine in the MD and BL directions can be estimated by finding the CPCSs in those directions.

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.

An analytical approach to 3D orthodontic load systems

OBJECTIVE

To present and demonstrate a pseudo three-dimensional (3D) analytical approach for the characterization of orthodontic load (force and moment) systems.

MATERIALS AND METHODS

Previously measured 3D load systems were evaluated and compared using the traditional two-dimensional (2D) plane approach and the newly proposed vector method.

RESULTS

Although both methods demonstrated that the loop designs were not ideal for translatory space closure, they did so for entirely different and conflicting reasons.

CONCLUSIONS

The traditional 2D approach to the analysis of 3D load systems is flawed, but the established 2D orthodontic concepts can be substantially preserved and adapted to 3D with the use of a modified coordinate system that is aligned with the desired tooth translation.

Load system of segmental T-loops for canine retraction

INTRODUCTION

The orthodontic load system, especially ideal moment-to-force ratios, is the commonly used design parameter of segmental T-loops for canine retraction. However, the load system, including moment-to-force ratios, can be affected by the changes in canine angulations and interbracket distances. We hypothesized that clinical changes in canine position and angulation during canine retraction will significantly affect the load system delivered to the tooth.

METHODS

The load systems of 2 T-loop groups, one for translation and the other for controlled tipping, from 9 bilateral canine retraction patients were made to the targeted values obtained from finite element analyses and validated. Each loop was tested on the corresponding maxillary dental cast obtained in the clinic. The casts were made before and after each treatment interval so that both initial and residual load systems could be obtained. The pretreatment and posttreatment interbracket distances were recorded for calculating interbracket distance changes.

RESULTS

As the interbracket distances decreased, the average retraction-force drop per interbracket distance reduction was 36 cN/mm, a 30% drop per 1 mm of interbracket distance decrease. The average antitipping-moment drops per interbracket distance reductions were 0.02 N-mm per millimeter for controlled tipping and 1.4 N-mm per millimeter for translation, about 0.6% and 17% drops per 1 mm of interbracket decrease, respectively. Consequently, the average moment-to-force ratio increases per 1 mm of interbracket distance reduction were 1.24 mm per millimeter for controlled tipping and 6.34 mm per millimeter for translation. There was a significant residual load, which could continue to move the tooth if the patient missed the next-scheduled appointment.

CONCLUSIONS

Clinical changes in canine position and angulation during canine retraction significantly affect the load system. The initial planned moment-to-force ratio needs to be lower to reach the expected average ideal value. Patients should be required to follow the office visit schedule closely to prevent negative effects because of significant moment-to-force ratios increases with time.

Effects of first- and second-order gable bends on the orthodontic load systems produced by T-loop archwires

OBJECTIVE

To measure the effects of first- and second-order gable bends on the forces and moments produced by a commercially available closing T-loop archwire.

MATERIALS AND METHODS

A dentoform-simulated space closure case was mounted on an orthodontic force tester. Sixteen gable bend combinations were placed in the archwires, which were then activated using standard clinical procedures. At each activation, the three force components and three moment components on the maxillary left lateral incisor and canine were simultaneously measured.

RESULTS

The first- and second-order gable bends showed low load coupling effects when used independently, but the load systems became unpredictable when bends were combined. Gable bends affect the magnitudes and directions of the forces and moments that are applied to teeth. The resulting moment to force ratios are sensitive to the bends.

CONCLUSION

Gable bends alter the orthodontic load systems; however, the three-dimensional interactions produce complex and unpredictable tradeoffs.

Experimentally determined mechanical properties of, and models for, the periodontal ligament: critical review of current literature

Introduction. This review is intended to highlight and discuss discrepancies in the literature of the periodontal ligament's (PDL) mechanical properties and the various experimental approaches used to measure them. Methods. Searches were performed on biomechanical and orthodontic publications (in databases: Compendex, EMBASE, MEDLINE, PubMed, ScienceDirect, and Scopus). Results. The review revealed that significant variations exist, some on the order of six orders of magnitude, in the PDL's elastic constants and mechanical properties. Possible explanations may be attributable to different experimental approaches and assumptions. Conclusions. The discrepancies highlight the need for further research into PDL properties under various clinical and experimental loading conditions. Better understanding of the PDL's biomechanical behavior under physiologic and traumatic loading conditions might enhance the understanding of the PDL's biologic reaction in health and disease. Providing a greater insight into the response of the PDL would be instrumental to orthodontists and engineers for designing more predictable, and therefore more efficacious, orthodontic appliances.