ABSTRACT

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.

INTRODUCTION

The load system delivered by an orthodontic appliance to a specific tooth is the most important determinant factor in the quality of the resulting orthodontic movement. The resulting tooth movement in three dimensions (3D) will depend on specific 3D moment-to-force ratio (M:F) combinations. The current method to describe the type of tooth movement consists of measuring the distance from the tooth's projected axis of rotation (center of rotation; CROT) to the virtual intersection of the axes of resistance (center of resistance; CRES). Some recent studies14  and many previous studies59  investigated the role of M:F associated with different tooth movements. One classic study focused on applying a force perpendicular to a canine long axis with a parabolic shaped root, obtaining the Burstone formula (M:F = 0.068 × h2/y), where h is the distance from the alveolar crest to the apex and y is the distance between the CRES and the CROT.5 

A previous study demonstrated a nonlinear relationship between tooth movement and force system directions and that the Burstone formula must be modified depending on them. In addition, different types of nonlinear behavior depend on the applied force and moment directions. This behavior was demonstrated for an upper first premolar with average dimension.10 

The hypothesis of the current article was that this phenomenon is due to root asymmetries and specific differences in tooth morphology. The purpose was to describe which tooth morphological characteristics determine this nonlinear behavior and statistically test if tooth movement in any direction can be predicted if data on root morphology and the original force system are provided to the orthodontist.

The specific objective was to determine the relationships between all meaningful permutations of M:F and the CROT, for different force directions, in each spatial plane (XY, YZ, ZX), for each tooth. Through finite element analysis (FEA), comparative maps of the effects of relevant M:F combinations on each tooth were built. Then, the statistical evaluation of the resulting maps was used to determine if tooth morphological features could be used to predict how the tooth will move with a specific force system. The significance of this result, under the limitations of the method, is that a clinician could derive the biomechanical behavior of any tooth if the force system and tooth dimensions are given, without having to resort to FEA.

To complete the tooth movement laws, the aim was to retrieve a statistically based 3D mathematical relationship between CRES and tooth morphology. The CRES coordinates were analyzed for the previous data set to also introduce a set of covariates to calculate the 3D CRES coordinates based on the tooth dimensions.

MATERIALS AND METHODS

The 3D models for each tooth were obtained from two patients according to the method described by Savignano et al.10  The teeth dimensions are reported in Table 1. The root dimensions were measured considering the root part attached to the bone through the periodontal ligament (PDL). The Loma Linda University Institutional Review Board (IRB) committee determined that this research did not require IRB approval because data or specimens were not collected specifically for this study and private individually identifiable information were not received.

Table 1.

Tooth Dimensions for the Two Data Sets

Tooth Dimensions for the Two Data Sets
Tooth Dimensions for the Two Data Sets

Each maxillary and mandibular model was manually sliced to obtain seven tooth-PDL-bone multibody models from each arch for the FEA. A total of 28 submodels were extracted. Bone and teeth were modeled as simplified homogenous bodies without discerning between cortical and cancellous bone and enamel, pulp and dentin, as was done also in previous studies because the data differences are insignificant.912  A linear elastic model was used for each structure to test the movements under the assumption of low PDL strains (<7.5%),13  as summarized in Table 2.

Table 2.

Material Properties Used for the Finite Element Analysis

Material Properties Used for the Finite Element Analysis
Material Properties Used for the Finite Element Analysis

The geometries were imported in the Finite Element Software ANSYS Workbench 16 (ANSYS, Canonsburg, Penn), where all the bodies were meshed with solid elements.

The coordinate system was defined for each tooth according to the occlusal plane. The x-axis was congruent with the buccolingual tooth dimension, whereas the y-axis was congruent with the mesiodistal and the z-axis was perpendicular to the occlusal plane (Figure 1).

Figure 1.

The cementoenamel junction (CEJ) intersection (yellow) used to refer the CRES coordinates was defined on the plane XY as the intersection between the mesiodistal and the linguobuccal tooth dimensions. On the z-axis, the coordinate system was located at the average between CEJ on the mesiodistal views and CEJ on the linguobuccal views.

Figure 1.

The cementoenamel junction (CEJ) intersection (yellow) used to refer the CRES coordinates was defined on the plane XY as the intersection between the mesiodistal and the linguobuccal tooth dimensions. On the z-axis, the coordinate system was located at the average between CEJ on the mesiodistal views and CEJ on the linguobuccal views.

To find the CRES, three simulations were run for each tooth, one for each coordinate plane, applying a moment with the specific values shown in Table 3. The moment amount was different for each tooth because each tooth requires a different load14  to keep the PDL strain <7.5%, allowing for a linear PDL model within this range.15 

Table 3.

Moment and Forces Applied to Each Tooth to Locate the Approximate Center of Resistance, Keeping the PDL Principal Strain Value Below 7.5%

Moment and Forces Applied to Each Tooth to Locate the Approximate Center of Resistance, Keeping the PDL Principal Strain Value Below 7.5%
Moment and Forces Applied to Each Tooth to Locate the Approximate Center of Resistance, Keeping the PDL Principal Strain Value Below 7.5%

The CRES coordinates were measured with the coordinate system located at the center of the cementoenamel junction (CEJ), as shown in Figure 1. For each tooth, the different M:F tested were applied at the respective CRES to provide a generalized map of tooth movements that could be transferred to any appliance (eg, brackets, aligners).

A tooth-specific constant force was applied at the CRES for each tooth as reported in Table 3, while the M:F varied from −12 mm to 12 mm. As for the moment, the forces used during the simulations were also proportionally different for each tooth, so as to keep the same strain <7.5% in the PDL for ascertaining the validity of the linear model. All of the bones' nodes were assigned zero displacement to simulate a rigid body due to the transient nature of tooth displacement solely attributed to bone deformation, as reported previously.9  The simulations were performed on the three spatial planes (plane XY, plane YZ, plane ZX). For each plane, 17 equivalent force systems were applied at the CRES of each tooth, as shown in the example of Figure 2. Figure 3 shows a schematic of the experimental design used for each tooth.

Figure 2.

Example, plane XY, representation of the incremental directional force system changes (left) and correspondent force systems tipped 10° and 40° with respect to the x-axis for a maxillary first molar (right). Each of the different force and moment combinations at the CRES was equivalent to one of the 17 single forces spaced 2 mm or 1 mm from each other, where the central force is applied at the CRES. The higher resolution force increment (1 mm) was applied in the region M:F = [−4:4]. To vary directions, the force was constant, but its direction for each position was changed in the spatial plane of interest in 10° increments, resulting in 10 different simulations for each M:F value at the CRES.

Figure 2.

Example, plane XY, representation of the incremental directional force system changes (left) and correspondent force systems tipped 10° and 40° with respect to the x-axis for a maxillary first molar (right). Each of the different force and moment combinations at the CRES was equivalent to one of the 17 single forces spaced 2 mm or 1 mm from each other, where the central force is applied at the CRES. The higher resolution force increment (1 mm) was applied in the region M:F = [−4:4]. To vary directions, the force was constant, but its direction for each position was changed in the spatial plane of interest in 10° increments, resulting in 10 different simulations for each M:F value at the CRES.

Figure 3.

Experiment workflow.

Figure 3.

Experiment workflow.

The projected axis of rotation (CROT) was evaluated for each scenario through the displacement vectors of two nodes of the tooth.8  The resulting CROT coordinates and the distances from the CRES were evaluated and analyzed to obtain a mathematical relationship between M:F and CROT. The D (distance CRES − CROT) vs M:F was analyzed using CurveExpert Basic software (CurveExpert, Madison, Ala), dividing the analysis by tooth and spatial plane. The starting model was set as simple hyperbolic, as previously validated for different teeth:  
formula

Statistical analyses were performed with the aim of obtaining generalized formulae to locate the tooth CRES and calculate an approximate k for every tooth and force system.

Multiple linear regression analysis was conducted to assess the influence of the teeth morphological data reported in Table 1 for the first data set. All hypotheses were tested at an α = .05 level for the second k data set. A sample size of two patients was considered appropriate for a pilot study. For statistical analysis, normality was assumed for the data even if it could not be proven because of the sample size, and the bootstrap method was applied for coefficient convergence. Multicollinearity between the independent variables was tested, discarding variables with variance inflation factor >1.5.

To evaluate the performance of the regression models, the covariates were tested on a randomly chosen tooth: a mandibular central incisor was reconstructed through a combination of cone-beam computed tomography (CBCT) and optical scanner for this purpose. The process shown in Figure 3 was applied to the tooth to calculate its k values. Then, the CRES coordinates and k values were derived using the covariates obtained by the regression models, and the two FEA data sets were then compared.

RESULTS

The statistical analysis demonstrated that the CRES coordinates can be calculated using the set of covariates shown in Table 4 at a significance level of .05. Different equations were found for each spatial coordinate stratified by maxillary and mandibular teeth. The statistical analysis showed that k can be predicted by using the root dimensions, the force direction, and the spatial plane, as hypothesized.

Table 4.

CRES Coefficients for Each Tootha

CRES Coefficients for Each Tootha
CRES Coefficients for Each Tootha

The results also might imply that a different set of covariates should be considered for each plane, with additional distinction placed upon the tooth morphology.

The width of the 95% confidence intervals for the k covariates varied considerably and lack the power to be considered as reliable for several of the coefficients.

Each hyperbolic equation was characterized by the constant of proportionality (k), which depended on the plane, the force's direction, and the tooth (Table 5), as classically defined by Burstone5  and generalized in 3D by Savignano and Viecilli.10 

Table 5.

k Coefficients for Each Tootha

k Coefficients for Each Tootha
k Coefficients for Each Tootha

Figure 4 shows the k shapes for the first patient and indicated that k increased greatly when the force direction became parallel to the tooth long axis (z-axis). In addition, the k values were larger on the YZ and ZX planes than on the XY plane, which contained the mesiodistal and buccolingual dimensions of the teeth. These findings generalized for every tooth confirmed the results obtained by a previous study on a first maxillary premolar.10 

Figure 4.

Graphical representation of k vs force direction for each tooth on the different planes for patient 1. The force direction is reported on the horizontal axis and the k value on the vertical axis. The direction is referred as the angle between the force and the first coordinate axis for each plane.

Figure 4.

Graphical representation of k vs force direction for each tooth on the different planes for patient 1. The force direction is reported on the horizontal axis and the k value on the vertical axis. The direction is referred as the angle between the force and the first coordinate axis for each plane.

After calculating the CROT location for all of the different load systems, it was confirmed that the M:F and the had an hyperbolic relationship in every tooth, characterized by the parameter k, which changed for each tooth and each force system.

The k variability was different among different teeth and among different planes for the same tooth. The analysis of the k values on the different planes showed that they depended not only on the force direction but also on the moment direction.

K could be predicted with the set of covariates shown in Table 5 at a statistically significant level (P < .05). It was possible to retrieve different equations for each tooth and planes, and hence, it was possible to estimate k for every tooth in any direction in this manner.

A random mandibular central incisor was chosen to show the covariates' efficacy.

Table 6 and Figure 5 show the k data sets obtained on a random mandibular central incisor through FEA and morphological covariates, respectively.

Table 6.

Comparison of Predicted and Calculated k Values for a Random Mandibular Central Incisor With MD = 3.3 mma

Comparison of Predicted and Calculated k Values for a Random Mandibular Central Incisor With MD = 3.3 mma
Comparison of Predicted and Calculated k Values for a Random Mandibular Central Incisor With MD = 3.3 mma
Figure 5.

Graphical representation of predicted k vs calculated k for a random mandibular central incisor on each plane.

Figure 5.

Graphical representation of predicted k vs calculated k for a random mandibular central incisor on each plane.

DISCUSSION

The relationship between M:F and initial tooth movement was previously investigated by other researchers. The limitation of the previous studies2,57,9,16  was that they did not relate the effect of M:F variations with tooth morphology, thus making it a necessary requirement to run additional and advanced computations to obtain tooth response curves. Only one study showed in two different teeth that they had a different orthodontic response to M:F variations.7  This study, although providing more comprehensive 3D data, also had a limitation, which was the simplified linear PDL model. This choice was justified because the loads modeled were below the strain threshold where the PDL starts to stiffen. In a PDL nonlinear model, the responses and k values would likely be different for different loads.15 

The statistical analysis provided a set of covariates to calculate the CRES coordinates and an approximate k for a random tooth, which described the asymmetric mechanical response of every tooth to an applied load, because of the tooth morphology and force system (Figure 6). The root morphology defined the tooth-specific response to an applied force system. This was likely because the PDL is the most deformable structure and approximately follows the root shape, thus being mathematically related to tooth movement. In addition, the root dimensions weigh differently on the response curves depending on the plane on which the movement occurs and on the number of roots. Both factors define the size of the active contact region between the tooth and PDL (the region where the PDL is in tension or compression).

Figure 6.

The different incremental changes in CROT with two M:F values (4 mm and 6 mm) and with two different force directions on the plane YZ for a maxillary central incisor and a maxillary second molar.

Figure 6.

The different incremental changes in CROT with two M:F values (4 mm and 6 mm) and with two different force directions on the plane YZ for a maxillary central incisor and a maxillary second molar.

Orthodontic appliances are usually tested with an electromechanical transducer, capable of measuring in 3D the load system delivered by orthodontic appliances.1719  However, the results obtained by transducers cannot be reliably used to predict or approximate the tooth behavior since limited information is available on the different relationship between M:F values and 3D tooth movements for each tooth.

During the evaluation of the appliance's effectiveness, it is recommended to calculate the k for each tooth. Afterward, the errors in the force system delivered to each tooth could be weighted to understand how each error really affected the final expected results, and then a decision can be made as to whether it is an acceptable error or changes are needed in the appliance design.

It is not practical to calculate k for each tooth using FEA before every orthodontic treatment. Hence, the set of equations provided by the statistical analysis can be used to calculate an approximate k knowing the tooth morphology. In particular, it would be necessary to provide a patient's CBCT to obtain the appropriate variables.

Comparing the results obtained on a random mandibular central incisor through FEA and covariates, respectively, it was noticed that the results were more accurate on the XY (16% average error) than on the ZX plane (26% average error) and on the YZ plane (24% average error). Despite that, the method introduced in the present article showed an overall acceptable accuracy (22% average error), using only the linguobuccal size at the CEJ, mesiodistal size at the CEJ, and root length.

Previous authors have investigated the CROT location features. Yang and Tang12  compared the effect of different M:F on the CROT location for a canine tooth, with and without a bracket. Cattaneo et al.7  tested different M:F values on a premolar and canine and showed how the CROT moved closer to the apex with an increase in M:F value.

Provatidis20  demonstrated that the position of the CROT was affected by root dimensions, PDL thickness, and material properties. On the other hand, the present study aimed at finding a generalized set of predictors for every tooth to estimate CRES and CROT locations, which will allow comparison of the effect of different orthodontic appliances in 3D. The estimation of the expected CRES and CROT coordinates for each scenario could be helpful to easily understand which appliance design provides the best expected movement.

It could be desirable to develop a set of equations that require only tooth crown features and features retrievable by a panoramic radiograph, but it would evidently result in more error. This would allow calculation of the k data set for a random tooth using a less invasive method based on an optical scan and less radiation.

CONCLUSIONS

  • The current study showed that it is possible to estimate the approximate CRES coordinates and the entire tooth movement curve for the initial phase of orthodontic movement, knowing the root dimensions, using separate equations for maxillary and mandibular teeth.

  • The results showed that variations of the M:F have a different influence depending on the tooth morphology and the force system features, and the k values are comprised between 1 and 119.4. Therefore, those differences cannot be neglected when estimating tooth movement response curves.

  • The geometrical features necessary to calculate k are linguobuccal size at the CEJ, mesiodistal size at the CEJ, root length on the ZX plane, root length on the YZ plane, and average root length. All of these dimensions can be measured using CBCT.

  • Using k, it is possible to estimate the distance between the expected CROT and the CROT obtained by any force system applied to a tooth. This method allows quantification of the potential effectiveness of any orthodontic appliance and selection of the most appropriate one. It is necessary to determine the desired orthodontic movement (expected CROT), the force system delivered by the appliance under examination (actual CROT), and the tooth morphology.

  • Further studies should analyze a larger data set and attempt to apply the same method for the whole orthodontic movement, while also accounting for nonlinear PDL behavior.

ACKNOWLEDGMENTS

The study was supported by the American Association of Orthodontists Foundation. The authors declare no potential conflicts of interest with respect to the authorship and/or publication of this article.

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Author notes

a

Assistant Professor, Center for Dental Research, School of Dentistry, Loma Linda University, Loma Linda, Calif.

b

Adjunct Associate Professor, Center for Dental Research, School of Dentistry, Loma Linda University, Loma Linda, Calif.

c

Assistant Professor, Dental Education Services, School of Dentistry, Loma Linda University, Loma Linda, Calif.