- Open Access
Analysis of the influences of bracket and force system in different directions on the moment to force ratio by finite element method
© The Author(s). 2018
- Received: 10 May 2018
- Accepted: 19 June 2018
- Published: 4 July 2018
The human teeth face a variety of ways to move during orthodontic treatment. The types of orthodontic tooth movements have linked to the moment to force (M/F) ratio. This paper aims to determine the precise M/F ratios that generate different types of movements for the models with and without bracket, and the M/F ratios are compared between the force systems applied in buccal-lingual and distal-mesial direction for the mandible canine by using finite element method (FEM). A segment of mandible canine obtained from autopsy had been scanned with microcomputed tomography, and two finite element models with and without brackets were established. Each canine model was subjected to a force of 100 cN and varied M/F ratios from 0 ~ 12 in buccal-lingual and distal-mesial directions. We find that the model with a bracket required larger M/F ratio compared to the model without a bracket for the same tooth movement. For the different directions, the value of M/F ratio is larger in the buccal-lingual than in the distal-mesial direction. The geometry of the tooth and the PDL are gained from a regular patient. Therefore, the results are applicable to any canine, and the precise M/F values provide a theoretical basis for the orthodontist design optimal force system.
- M/F ratios
- Tooth movement
- Finite element analysis
Tooth movement occurs due to the changes in the strain-stress distribution within the periodontal ligament (PDL) after the application of a force system to the bracket. The types of tooth movements can be summarized as follows: tipping, translation, rotation, and root movement. In addition, tipping can be further classified into uncontrolled tipping and controlled tipping. As an important parameter, the moment to force (M/F) ratio plays a key role in determination of the type of tooth movement [1, 2].
According to the classic theory, Burstone et al.  studied the localization of the center of resistance (CRe) at an approximate of 40% from the apex to the measured length between the alveolar crest and the tooth root based on a mathematical model. Moreover, the center of rotation (CRot) can be calculated by the Burstone’s theoretical formula (y = 0.068 h2 (F/M)), where h is the distance between apex and the alveolar crest and y denotes the distance from the CRe to the CRot. Provatidis CG  validated the formula through the finite element method (FEM). Christiansen  confirmed that the CRot is related to the M/F ratio by application of different force levels on maxillar central incisors, but the CRot seemed to be positioned more inclined to apical direction by FEM compared to mathematical calculation. These results are confirmed by many clinical cases. The different locations of CRe between theoretically calculated and experimental results can be attributed mainly to the limitation of the model, 2D model established and the assumption of linear behavior for the PDL using FEM.
In recent years, numeric methods have been widely used to calculate the stress/strain in the PDL, and FEM has been the choice of orthodontic researchers frequently. Some previous literatures [6, 7] confirmed the limitations on the finite element (FE) results due to the complexity of shape and composition of the periodontal tissue. Many attempts on the relationship between the force system and tooth movement have been made as well as the reaction of the surrounding tissues have been made; however, it is still an open question. Cattaneo et al.  simulated the pure translation of the tooth with an M/F ratio of 10 and 12 respectively for premolar and canine. Athicha et al.  suggested that bodily movement of incisor happened when the ratio is 11.65 at zero inclination. Zhipeng Liao et al.  proposed that an M/F ratio of approximately 10.2, 9.7, and 8.8 can generate the bodily movements of maxilla canine, incisor, and first premolar. Cai et al.  attempted to find appropriate M/F ratio for translation and controlled tipping movement. However, they neglected the influence of a bracket and the direction of external force on the M/F ratio. In quantitative analysis, the influences can help the orthodontist design ideal orthodontic force system for different patients.
Application of FEM fitted to a dental biomechanic commenced in the mid 1970s. This technology has particularly attracted a number of orthodontic investigators since they can establish an elaborate 3D model, simulate tooth movement accurately when subjected to various orthodontic forces, and provide an excellent visualization profile of the displacement and stress fields that occur in the PDL and surrounding tissues. In finite element (FE) analysis, each material component is subdivided into a large number of elements. An accurate tooth model including the physical properties and geometry of the component is required for calculation. As an important connective soft tissue, the PDL transmits orthodontic force from the tooth to the alveolar bone, resulting in tooth movement. Even though the PDL is known to represent a nonlinear viscoelastic behavior, many literatures [12–15] assumed a linear elastic behavior for the PDL in FE analysis. The hyperelastic, viscoelastic, and biphasic poroelastic behaviors are also adopted to descript the mechanical properties of the PDL under different loading conditions [16–19]. The PDL’s nonlinear viscoelastic behavior can be considered as an instantaneous elastic and time-dependent nonlinear viscous behavior, while a hyperelastic model  can describe the short-term instantaneous behavior.
Orthodontic treatment is a long-term process with repeated occurrences and even side effects (if you do not expect to have tooth movement). The reason for this is that the orthodontists can not accurately control the orthodontic force. Studying orthodontics and the way the tooth moves can help orthodontists to develop a more rational orthopedic plan and design to ultimately improve orthodontic treatment efficiency. Previous studies have focused on teeth movements when a force or moment is applied to the tooth directly, the effect of bracket, and the direction of the force on the M/F ratios based on suitable material assumption for the PDL which are rarely performed in orthodontics. The aim of this study is to determine the precise M/F ratio that can generate uncontrolled tipping, controlled tipping, and translation movements for the mandible canine models with and without brackets, as well as the force systems applied in buccal-lingual and distal-mesial directions.
The results of mesh convergence
Mechanical property of four materials
Elastic modulus (MPa)
2.1 × 105
Parameters of the Ogden model
The types of canine movements correspond to the M/F ratios
0 ~ 5.6
0 ~ 6.8
0 ~ 4.2
0 ~ 4.9
Orthodontists usually apply orthodontic force and moment to the bracket. The orthodontic force system is mostly derived from arch wire, elastic rubber, bands, or various screws and so on. On the other hand, more and more people are inclined to use invisible appliance without bracket. The results of this study can lay a theoretical foundation for medical researchers to design more individualized orthodontic tools.
The target of orthodontic treatment is to shift maloccluded teeth a predetermined distance in a prescribed direction. The accuracy of M/F ratios and the location of the CRe are the two key factors to obtain satisfaction treatment result. Many researchers reported that CRe is located at 24 to 50% of the distance from the alveolar crest to the apex based on mathematical calculation and FEM [26–28]. Nevertheless, in reality, clinicians preferred to employ M/F ratios for a specific type of tooth movement because of clinical experience, without fully understanding the precise location of CRe for a tooth. It is difficult for the orthodontists to determine the exact location of CRe by mathematical deduction and FE calculation. However, the relationship between M/F ratios and the type of tooth movement remains inaccurate attributing to the difference form patient to patient, repeated visits to orthodontists for adjustment of orthodontic force system have to be necessitated.
The aim of this study is to determine the precise value of M/F ratios that generate different types of tooth movements and compare the M/F ratios between the models with bracket and without bracket, as well as different directions (buccal-lingual and distal-mesial directions). A patient-specific canine has chosen as the model to study. For the same type of tooth movement, the value of M/F ratio is larger for the model without bracket than for the model with bracket. In additional, the displacement amplitude is lower for the model with bracket. This can be explained that the bracket model has larger area to sustain orthodontic force. By looking at the force system applied in different directions, one can notice that the displacements of the crown and the apex for the models in distal-mesial direction are far less than for the models in buccal-lingual direction. More M/F ratios are needed for the models in buccal-lingual direction (Figs. 4 and 5). There are principal results of two causes, in which there is a smaller contact area of the canine at the lingual side compared to the mesial side (Fig. 3) and the force system did not pass though the CRe when the force system applied to the models in buccal-lingual direction.
A number of studies have been investigating the influence of different factors on the position of the CRot. An FEM conducted by Provatidis , concluded that the position of the CRot is linked not only to root length but also the root diameter, the material properties of the PDL, as well as its thickness. D. Vollmer et al.  compared the difference of position of CRot between an upper human canine and idealized tooth model. By applying a range of values of M/F ratio, Cattaneo, P. M  found that the position of the CRot of premolar and canine gradually approaches to the apex with an increase of M/F value (before translational movement occurred) applied to the model, which is corresponding with this study. Fig. 3 displays the relationship between the position of CRot and the M/F ratio for the models with and without bracket when the force system applied in buccal-lingual and distal-mesial direction. The position of CRot is highest for the model without bracket, followed by the model with bracket, and is higher in buccal-lingual than in distal-mesial direction when equal force system is applied to the models (uncontrolled tipping).
One limitation of this study is that the FE results are conducted based on one human sample. However, the difference of tooth movement linked to M/F ratio between the models with and without bracket, as well as in the buccal-lingual and distal-mesial direction, this method is applicable to any canine. In additional, the results have not compared with clinic experiments. We assume the PDL as a uniform thickness of 0.2 mm that represents the average thickness. In reality, the PDL’s thickness range from 0.15 to 0.38 mm , and it exhibits anisotropic and nonuniform character. Toms SR  suggests the stress distributions of the PDL are different for the uniform and nonuniform models. This paper has studied how the bracket and orthodontic force influence M/F ratio and the way of tooth movement. Further research is to investigate what will happen to M/F ratio and the way of tooth movement when we exert orthodontic force system on different position of tooth crown.
This study revealed the influences of bracket and force system from different directions on M/F ratios, which led to various tooth movements. The M/F ratios of 0~ 5.6, 6.2, and 10.6 which are generally accepted for uncontrolled tipping, controlled tipping, and translation movements of mandible canine, respectively. These values are smaller for the canine model without bracket, and are larger in buccal-lingual direction than in distal-mesial direction. Therefore, a universal M/F ratio is not recommended. The results of this study have guiding significances for the orthodontist formulate correct therapeutic plan to deal with different situations. Due to the orthodontic forces derived from bracket, wires, and elastic band, it is expected that future tools for orthodontic treatment may be based on the FE results.
The research presented in this paper was supported by School of mechanical Engineering, Southeast University, Nanjing, P.R. China.
The authors acknowledge the Fundamental Research Funds for the Central Universities, (Grant: 2242017K3DN02).
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
YY was the main writer of this paper. He proposed the main idea, and completed the simulation, and analyzed the result. WT gave some important suggestions for the performance of this article. All authors read and approved the final manuscript.
Yu Yang is pursuing his Ph.D. degree in School of Mechanical Engineering, Southeast University. His research interests include biomechanics, Orthodontic simulation.
Wencheng Tang was serving as a professor, and a Ph.D. supervisor. He is currently at work in the School of Mechanical Engineering, Southeast University, Nanjing, China. His research interests include advance manufacture technology, element finite analysis, and topology optimization.
The authors declare that they have no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
- R Savignano, RF Viecilli, A Paoli, AV Razionale, S Barone, Nonlinear dependency of tooth movement onforce system directions. Am. J. Orthod. Dentofac. Orthop 149(6), 838–846 (2016)View ArticleGoogle Scholar
- A Geramy, K Tanne, M Moradi, H Golshahi, YF Jalali, Finite element analysis of the convergence of the centres of resistance and rotation in ectreme moment-to-force ratios. Int. Orithod. 14(2), 161–170 (2016)Google Scholar
- CJ Burstone, in Vistas in Orthodontics, ed. by B S Kraus, A Ripamonti. The biomechanics of tooth movement (Lea and Febinger, Philadelphia, 1962), pp. 197–213Google Scholar
- CG Provatidis, Numerical estimation of the centres of rotation and resistance in orthodontic tooth movement. Comput. Methods Biomech. Biomed. Engin 2(2), 149–156 (1999)View ArticleGoogle Scholar
- RL Christiansen, CJ Burstone, Centers of rotation within the periodontal space. Am. J. Orthod 55(4), 353–369 (1969)View ArticleGoogle Scholar
- KR Williams, JT Edmundson, Orthodontic tooth movement analyzed by the finite element method. Biomaterials 5(6), 347–351 (1984)View ArticleGoogle Scholar
- K Tanne, HP Bantleon, Stress distribution in the periodontal ligament induced by orthodontic forces. Use of finite-element method. Inf. Orthod. Kieferorthop 21(2), 185–194 (1989)Google Scholar
- PM Cattaneo, M Dalstra, B Melsen, The finite element method: a tool to study orthodontic tooth movement. J. Dent. Res 84(5), 428–433 (2005)View ArticleGoogle Scholar
- A Kanjanaouthai, K Mahatumarat, P Techalertpaisarn, Effect of the inclination of a maxillary central incisor on periodontal stress finite element analysis. Angle Orthodonist 82(5), 812–819 (2012)View ArticleGoogle Scholar
- Z Liao, J Chen, W Li, Biomechanical investigation into the role of the periodontal ligament in optimising orthodontic force: a finite element case study. Arch. Oral Biol 66, 98–107 (2016)View ArticleGoogle Scholar
- Y Cai, B He, X Yang, J Yao, Optimization of configuration of attachment in tooth translation with transparent tooth correction by appropriate moment-to-force ratios: biomechanical analysis. Biomed. Mater Eng 26(s1), S507–S517 (2015)Google Scholar
- Y Kojima, H Fukui, A finite element simulation of initial movement, orthodontal movement, and the centre of resistance of the maxillary teeth connected with an archwire. Eur. J. Orthod 36(3), 255–261 (2014)View ArticleGoogle Scholar
- A Merdji, R Mootanah, BAB Bouiadjra, L Aminallah, Stress analysis in single molar tooth. Mater. Sci. Eng 33(2), 691–698 (2013)View ArticleGoogle Scholar
- MR Matson, HR Lewgoy, DA Barros Filho, Finite element analysis of stress distribution in intact and porcelain veneer restored teeth. Comput. Methods Biomech. Biomed. Engin 15, 795–800 (2012)View ArticleGoogle Scholar
- J Yan, H Xianglong, B Cheng, B Ding, Three-dimensional FEM analysis of stress distribution in dynamic maxillary cnaine movement. Chin. Sci. Bull. BiomedicaEng 58(20), 2454–2459 (2013)View ArticleGoogle Scholar
- R Aversa, D Apicella, L Perillo, R Sorrentino, F Zarone, M Ferrari, A Apicella, Non-linear elastic three-dimensional finite element analysis on the effect of endo-crown material rigidity on alveolar bone remodeling process. Dent. Mater 25, 678–690 (2009)View ArticleGoogle Scholar
- IZ Oskui, A Hashemi, Dynamic tensile properties of bovine periodontal ligament: a nonlinear viscoelastic model. J. Biomech 49(5), 756–764 (2016)View ArticleGoogle Scholar
- IZ Oskui, A Hashemi, H Jafarzadeh, Biomechanical behavior of bovine periodontal ligament: experimental tests and constitutive model. J. Mech. Behav. Biomed. Mater 62, 599–606 (2016)View ArticleGoogle Scholar
- M Bergomi, J Cugnoni, M Galli, J Botsis, UC Belser, HW Wiskott, Hydro-mechanical coupling in the periodontal ligament: a porohyperelastic finite element model. J. Biomechem 44(1), 34–38 (2011)View ArticleGoogle Scholar
- W Li, MV Swain, Q Li, GP Steven, Towards automated 3D finite element modeling of direct fiber reinforced composite dental bridge. J Biomed Mater Res Part B 74(1), 520–528 (2005)View ArticleGoogle Scholar
- HH Ammar, P Ngan, RJ Crout, VH Mucino, OM Mukdadi, Three-dimensional modeling and finite element analysis in treatment planning for orthodontic tooth movement. Am. J. Orthod. Dentofac. Orthop 139(1), 59–71 (2011)View ArticleGoogle Scholar
- YQ Cai, XX Yang, BW He, J Yao, Numerical analysis of tooth movement in different plans of transparent tooth correction therapies. Technol. Health. Care 23, 299–305 (2015)View ArticleGoogle Scholar
- YQ Cai, XX Yang, BW He, Influence of friction in transparent tooth correction treatment: finite element method. J. Mech. Med. Biol 15, 1550052 (2015)View ArticleGoogle Scholar
- Z Wei, X Yu, X Xu, X Chen, Experiment and hydro-mechanical coupling simulation study on the human periodontal ligament. J. Mech. Med. Biol 15(04), 749–759 (2015)Google Scholar
- WR Proffit, HW Fields, in Contemporary orthodontics, 3rd edn., ed. by P Rudolph. The biological basis of orthodontic therapy (Mosby, St Louis, 2000), pp. 296–325Google Scholar
- ME Geiger, BG Lapatki, Locating the center of resistance in individual teeth via two- and three-dimensional radiographic data. J. Orofac. Orthop 75(2), 96–106 (2014)View ArticleGoogle Scholar
- T Kondo, H Hotokezaka, R Hamanaka, Types of tooth movement, bodily or tipping, do not affect the displacement of the tooth’s center ofresistance but do affect the alveolar bone resorption. Angle Orthod 87(4), 563–569 (2017)View ArticleGoogle Scholar
- T Ouejiaraphant, B Samruajbenjakun, E Chaichanasiri, Determination of the centre of resistance during en masse retraction combined with corticotomy: finite element analysis. J. Orthod. 45(1), 11–15 (2018)View ArticleGoogle Scholar
- PM Cattaneo, M Dalstra, B Melsen, Strains in periodontal ligament and alveolar bone associated with orthodontic tooth movement analyzed by finite element. Orthod. Craniofac. Res 12(2), 120–128 (2009)View ArticleGoogle Scholar
- S Chandra, S Chandra, M Chandra, N Chandra, Textbook of dental and oral histology with embryology (Jaypee Brothers Medical Publishers (P) Ltd., New Delhi, 2007)MATHGoogle Scholar
- SR Toms, JE Lemons, AA Bartolucci, AW Eberhardt, Nonlinear stress-strain behavior of periodontal ligament under orthodontic loading. Am. J. Orthod. Dentofacial. Orthop 122, 174–179 (2002)View ArticleGoogle Scholar