1. Introduction
Molar distalization is a commonly used procedure to obtain aligning space in orthodontic treatment, where doctors utilize the obtained space to correct crowding, Angle’s class II molar relationship, or incisor proclination. Various types of appliances have been proposed to distalize molars efficiently, such as the headgear, the pendulum appliance, the Greenfield molar distalizer, the Carriere distalizer, etc. Today, distalization using orthodontic anchor screws is becoming more common because it is less dependent on patients’ adherence, more predictable than the conventional appliances [1] by providing solid skeletal anchorage, and its force system can be modified easily by changing the position of the anchor screws.
For efficient distalization, it is preferable to apply well-distributed distalization force over premolars and molars, and to ensure that stress concentration does not occur at a specific tooth. If such stress concentration occurs on a premolar, the molars cannot receive sufficient distalization force, while if it occurs on a molar, the premolars will be left behind. Moreover, high compressive stress caused by the stress concentration in the periodontal ligament (PDL) slows down the tooth movement due to hyalinization of the PDL. To avoid such a phenomenon, some techniques employ two-step distalization [2,3,4,5], where they first distalize the molars using the screw anchorage directly or indirectly, and then distalize the rest of teeth. This method can apply force to molars more securely, but the two-step procedure will increase the total time required for whole-arch distalization, since the premolars are suspended while the molars are being distalized. On the other hand, some previous studies have reported that it is possible to distalize the whole arch as one group by using anchor screws [6,7,8]. This method is called grouped distalization and can shorten the required treatment period if the force distribution is appropriate, i.e., all premolars and molars receive the ideal level of distalization force, and no stress concentration occurs on a specific tooth. However, Ammoury et al. [9] reported that higher stress was observed in the periodontal ligaments of canines and premolars while lower stress was observed in molars when the grouped distalization method was employed, suggesting that grouped distalization does not generate optimal force distribution.
To improve the force distribution over posterior teeth and achieve efficient distalization, we propose a novel technique for grouped distalization. In the conventional technique for grouped distalization, distalization force is applied to a single tooth, which is typically a canine or a premolar, and the rest of teeth receive force through tooth–tooth contact and friction between the bracket and archwire. However, the contact force between teeth gradually decreases as the force is transmitted to the distal tooth because part of the contact force is supported by the PDL. This is why canines and premolars receive more force and molars receive less force in the grouped distalization method, and the molars will not move until the premolars are distalized. To overcome this issue, we propose a novel method where we insert open-coil springs with different spring rates between the posterior teeth. A retraction force of 4 N is applied to the first premolar, an open-coil spring of 3 N is inserted between the premolars, another spring of 2 N is inserted between the second premolar and the first molar, and a 1 N spring is inserted between the molars (Figure 1). Since adjacent springs cancel one another’s forces, each of the premolars and molars should receive 1 N, meaning that the distalization force will be applied more directly and securely to the molars.
Although the proposed technique generates better force distribution theoretically, its force system might not be as ideal as expected. Since the brackets of the posterior teeth are not in a straight line, the actual force distribution might be different from what is expected. Moreover, the posterior teeth will rotate and tip distally when buccal distalization forces are applied. This rotation and tipping causes frictional resistance between the archwire and the bracket, which might affect the balance of the force system. Furthermore, although Ammoury et al. [9] reported that the tooth movement pattern caused by the conventional grouped distalization was employed, they only investigated the initial displacement of teeth in the PDL. A previous study [10] suggested that the force system generated by a multibracket appliance drastically changed over time due to tooth movement caused by bone remodeling. Thus, their results might be insufficient to understand the actual behavior of the conventional grouped distalization.
The objective of the present study was to clarify whether our proposed technique improves force distribution over posterior teeth and supports efficient distalization, using the finite element method (FEM). The FEM is broadly used to evaluate force systems generated by orthodontic appliances because of its reproducibility [10,11,12] and capability of visualizing force and stress distribution easily. We also employed an analysis method that considers the bone remodeling process to reveal more realistic behavior of both our proposed technique and the conventional grouped distalization method.
2. Materials and Methods
We obtained a three-dimensional(3D) model of upper left dentition (Figure 2) for the FEM from a dry skull by taking computed tomography (CT) scans using a multi-image micro-CT scanner (3DX, J. Morita, Kyoto, Japan) with a voxel size of 80 µm. Image segmentation of the upper left dentition was performed on 3D image processing and editing software (Mimics 10.02, Materialize Software, Leuven, Belgium) to obtain 3D geometric data of the dentition, and then 3D surface data were exported in a Standard Triangulated Language (STL) format. The 3D surface data were imported to pre- and post-processing software for FEM (Patran 2017, MSC Software Corp., Los Angeles, CA, USA) as shell elements. Each tooth was aligned in the pre-processing software so that it had normal inclination, and each interproximal distance became less than 0.01 mm to perform appropriate contact analysis between teeth. Subsequently, we constructed PDLs of 0.2 mm as solid elements on the root surface of each tooth. On top of that, 0.022-inch brackets and a 0.019 × 0.025-inch archwire were added as orthodontic appliances.
The material properties were configured according to previous articles. We assumed that the teeth were rigid bodies, which was the same assumption as in [10], and set the Young’s moduli and the Poisson’s ratio to 204 GPa and 0.3, respectively, to make the teeth rigid enough. The PDL was configured as a bilinear elastic material whose elastic moduli ranged from 0.03 to 0.3 [13]. The Young’s moduli and the Poisson’s ratio of the brackets and the archwire were set to 204 GPa and 0.3, respectively, which are the material properties of stainless steel. The alveolar bone was also assumed to be a rigid body since it deforms little under small loads such as orthodontic force, as described in a previous study [10]. Thus, we constrained the displacement of outer surface of the PDL to simulate rigid bone and reduce the number of elements for efficient analysis.
We compared two models, namely, a model without open-coil springs (NoSp model), which represents the conventional grouped distalization technique, and one with open-coil springs (Sp model), which represents our proposed technique. Elastic chains and open-coil springs were represented by external forces (Figure 3). A retraction force of 4 N to simulate an elastic chain was applied to the first premolar. The force was directed towards an anchor screw between the roots of the upper molar and the second molar, and the height of the anchor screw was 5.0 mm from the bracket level. In the Sp model, external forces of 3 N were applied reciprocally to the distal surface of the first premolar’s bracket and the mesial surface of the second premolar’s brackets, simulating an open-coil spring between them. External forces of 2 N were applied between the second premolar’s bracket and the first molar’s bracket, and forces of 1 N were applied between the first molar’s bracket and the second molar’s bracket in the same manner.
Contact boundary conditions were applied between the teeth and between the archwire and the brackets. The friction coefficient between the archwire and the brackets was determined to be 0.08 according to previous studies [14,15,16], based on the assumption that friction coefficient becomes smaller under the vibration caused by mastication than that under static conditions.
A symmetric boundary condition was applied to perform analysis unilaterally. A contact wall was placed at the midsagittal plane of the model, and the mesial end of the archwire was constrained in the midsagittal plane while the mesial end was able to slide on the midsagittal plane. Using a symmetric boundary condition, the number of elements was reduced by half compared to the analysis using a full dentition, enabling us to reduce the analysis time.
The finite element analysis was performed with a commercial finite element analysis solver (Marc 2017.1, MSC Software Corp., Los Angeles, CA, USA), and simulation of long-term tooth movement considering the bone remodeling process was performed using a previously proposed method [10], where the constraint of the outer surface of the PDL was iteratively modified so that the teeth would move according to the initial displacement in the PDL. The bone remodeling simulation on the software was iterated 30 times after initial tooth displacement occurred in the PDL, to evaluate the change in the force system over time. The bone remodeling simulation was implemented using the Fortran language as a custom subroutine on the finite element analysis solver.
To compare the proposed technique and the conventional technique, displacements of the center of resistance (CR), forces acting on teeth, frictional resistance between the brackets and the archwire, and stress on the PDLs were evaluated. The positions of the center of resistance were computed using a previously proposed method [10]. Forces acting on teeth were calculated as the sum of contact forces between teeth, contact forces between the bracket and the archwire, frictional forces between the bracket and the archwire, and the external force of the retraction force and open-coil springs. The displacement and the force were evaluated along the mesiodistal axis defined by the bracket center of the upper first premolar and that of the upper second molar.
3. Results
Figure 4 shows the achieved distal displacement of the CR of each tooth. The Sp model achieved more distalization than the NoSp model on the second premolar, the first molar, and the second molar. On the other hand, the NoSp model showed slightly more distalization on the first premolar. The amounts of distalization in the Sp model were 0.57, 0.54, 0.36, and 0.33 mm for the first premolar, the second premolar, the first molar, and the second molar, respectively, at the end of analysis, while they were 0.59, 0.52, 0.31, and 0.22 in the NoSp model, respectively. In addition, although the applied distalization forces were 4 N for both the Sp model and the NoSp model, the Sp model showed greater movement, meaning that more frictional loss and force reduction occurred on the NoSp model. In fact, a higher level of contact normal force was generated in the NoSp model, as shown in Figure 5, generating more frictional force.
The molars in the NoSp model received less force than those in the Sp model. Figure 6 indicates the change in the transmitted distalization force on each tooth throughout the simulation. As for the NoSp model, the force transmitted to the second molar was 0.2 N, while the first premolar received 2.5 N at the stage of initial displacement within the PDL. The forces transmitted to the molars began to increase at a certain point after the bone remodeling operations, reaching 0.6 N. On the other hand, all premolars and molars in the Sp model received forces of 1 N at the initial displacement. The forces began to decrease at a certain point after the bone remodeling operations, dropping to 0.7 N.
Slight frictional force between the bracket and the archwire was observed in both the NoSp and Sp models at the initial displacement step, and it increased after a certain number of iterations of the bone remodeling operation in the simulation. Figure 7 indicates the changes in the frictional force on each bracket throughout the simulation. Positive values indicate that the frictional force was directed toward the distal direction, while negative values indicate that it was directed toward the mesial direction. For the NoSp model, frictional force that resisted distalization force was observed at the premolars, while frictional force that supported distalization was observed at the second molar. On the other hand, in the Sp model, frictional force that resisted distalization force was observed at both premolars and molars, and higher friction was observed at the molars than at the premolars.
Figure 5 shows the distribution of the contact normal force between the teeth and between the brackets and the archwire. The color and size of the arrows indicate their force magnitude. The arrows at the interface between the archwire and the brackets show how heavily the archwire pushed the brackets, while between teeth show how heavily each tooth pushed its adjacent teeth. More contact forces between teeth were generated in the NoSp model than in the Sp model. As for the contact force between the brackets and the archwire, the distal edge of first premolar’s bracket slot in the NoSp model received buccal force, which resisted distal rotation of the premolar, while the other brackets received an anti-distal-tipping moment from the archwire. On the other hand, all brackets in the Sp model received an anti-distal-tipping moment and an anti-distal-rotation moment from the archwire.
Figure 8 shows the stress distribution over the PDL. The red-colored regions indicate highly compressed regions where the minimum principal stress (i.e., compression stress) was lower than −40 kPa. As for the NoSp model, a large highly compressed region was observed in the PDL of the first premolar, while a smaller region was observed in that of the molars at the initial displacement. The minimum principal stress of the most highly compressed region in the first premolar reached −159.1 kPa. As for the Sp model, a highly compressed region was observed on every tooth—especially on the distobuccal side at the initial displacement. The most highly compressed region was observed in the second premolar, and its minimum principal stress was −82.1 kPa. After a number of iterations of the bone remodeling operation in the simulation, the stress distribution became similar in all teeth. However, the compression level was still slightly higher at the first premolar in the NoSp model than in the Sp model when the simulation was finished.
4. Discussion
The Sp model achieved more distalization on the molars and the second premolar, while the NoSp model showed slightly more distalization the on first premolar, and the distalization was achieved more efficiently with the open-coil springs; 1.5 times more distalization was observed in the Sp model than the NoSp model, and the first and second molars were more evenly distalized in the Sp model, while the second molar was distalized significantly less than the first molar in the NoSp model.
The distal displacements of the molars were smaller than those of the premolars in both the Sp and NoSp models. Since the force levels acting on the molars and premolars were not significantly different in the Sp model, this difference in the amount of displacement was probably caused by the difference in the surface area of the PDL. The PDL area of molars is larger than that of premolars, and moving molars requires more force than moving premolars at the same rate. As a result, the molars acted as a speed controller of the grouped distalization.
Most of the distalization force was absorbed by the PDL of the first premolar in the NoSp model. The first premolar received 60% of the distalization force in the NoSp model, and the minimum principal stress in the PDL of the first premolar was also significantly higher in the NoSp model—nearly twice as high as that in the Sp model. This high load on the first premolar may cause hyalinization of the PDL and root resorption [17]. At the same time, less than 5% of the distalization force was transmitted to the second molar. This low load on the second molar may cause no movement of the second molar. Although we constructed the model so that the teeth had tight interproximal contact, the contact surface could not transmit the moment, because it only transmitted pushing force and could not pull the adjacent surface. This was because the premolar was rotated by the retraction force and the posterior teeth did not move as one unit. Furthermore, since the simulation using FEM did not consider the threshold of stress that causes bone remodeling in real PDLs, the simulated bone remodeling would occur even if the force was smaller than the threshold. Therefore, the displacement of the second molar in a real patient might be smaller than that in the simulation.
The force distribution in the NoSp model was gradually improved over time; however, it was still worse than that in the Sp model at the end of the simulation. After some steps of bone remodeling in the simulation, the force transmitted to the molars gradually increased due to contact between the teeth and frictional forces between the archwire and the brackets.
Compared to the NoSp model, the force was more evenly distributed over the premolar and the molars in the Sp model. The force transmitted to each tooth was 1.0 N at the initial displacement, and little frictional force existed, as we had expected. This was probably because that the archwire and the brackets were not completely engaged until the bone remodeling operation was executed a certain number of times. The stress was also distributed more evenly than in the NoSp model. The peak stress level at the premolar in the Sp model was significantly lower than that in the NoSp model, while higher stress was observed at the molars, suggesting that the Sp model could transmit forces more efficiently.
More compressive stress was observed on the distobuccal side of the PDL in the Sp model during the initial phase of the distalization. This compressive stress indicates that the distalization force was transmitted by open-coil springs in the Sp model, while it was transmitted by the contact between the teeth and the friction in the NoSp model, and more moment was generated, which caused distal rotation. This rotational force in the Sp model might be helpful for the treatment of class II cases because molars in such cases usually show mesial rotation [18] and have to be rotated distally to achieve a class I molar relationship.
The proposed technique has great advantages when one or more teeth do not respond to the treatment. Since the results of the present study suggested that a tooth would not be distalized until its mesial teeth were distalized, if the mesial tooth does not move for some reason—such as ankylosis—the tooth movement sequence will stop at the tooth. This makes it difficult to find which tooth is causing the problem, because all that a doctor can observe is the fact that the grouped distalization failed. On the other hand, it is easy to find which tooth causes the problem with the proposed method, because even if a tooth in the arch absorbs all force transmitted from its mesial teeth, the distal coil spring still applies distalization force to its distal tooth, resulting in a gap between the problematic tooth and the distal tooth.
To apply this technique in real clinical practice, it must be considered that the actual spring rate is not always at the same level. In the present study, we simulated the force generated by springs and elastics using external forces, so the force level was always the same regardless of the distance between brackets. However, the force generated by a spring will differs based on the amount of activation in the real world. A previous study showed that even if a NiTi coil spring was advertised that could generate continuous force of a certain level, its force increased significantly as the amount of activation increased [19]. Thus, the force-level springs should be measured accurately outside of the mouth before the application of the present mechanics.
The simulation of a long-term tooth movement was necessary to clarify the force system of the proposed method. Since the results suggested that the distalization force was not transmitted to the second molar until bone remodeling of the mesial teeth occurred, and that the stress distribution dramatically changed throughout the analysis, we would have overestimated the efficiency of the proposed method if we had evaluated the initial displacement only. The simulation of a long-term tooth movement made it possible to compare the force systems of the conventional grouped distalization method and our proposed method more impartially and accurately.
A limitation of the present study was that it did not take bone density into account. It takes more time to resorb alveolar bone of high density. For example, there is a wall of cortical bone in the distal region of the second molar, meaning that more force is required to move the second molar distally or completely block the movement of the second molar. However, the bone remodeling algorithm that we employed in the present study assumed that bone density was uniform everywhere. Thus, the molars could be distalized infinitely for as long as the distalization force was applied and the number of bone remodeling operations was iterated. This behavior was obviously different from that in the real alveolar bone. This is was why we finished the bone remodeling operation after 30 iterations and avoided excessive distalization, which would probably be seen as unrealistic anyway. However, even with this limitation and the use of the simplest way of simulating bone remodeling, we believe that simulated bone remodeling provided more information than an analysis of the initial displacement in the PDL, which most of the other FEM studies employed. If the relationship between tooth movement rate, bone density, and the stress in the periodontal ligament is investigated more deeply in the future, we would like to implement it in our bone remodeling simulation for more accurate prediction.
We configured the rate of open-coil springs so that each tooth would receive a uniform force of 1 N in the present analysis; however, this force might not be ideal. Since the PDL area of a molar is larger than that of a premolar, higher force is required to move a molar. In fact, there were significant differences in the amount of the displacement between the molars and the premolars in the present analysis. The molars acted as a bottleneck of the distalization, and the tooth movement rate was restricted by the molars. With our proposed technique, the force system can be modified simply by changing the spring rate between the brackets, so that it generates such ideal forces. For example, we can use the set of 3.3 N, 2.6 N, and 1.4 N springs with distalization force of 4 N to apply 0.7 N, 0.7 N, 1.4 N, and 1.4 N to the first premolar, the second premolar, the first molar, and the second molar, respectively. It is our future intent to find best combinations of the spring rates and the distalization force.
5. Conclusions
In the present study, we proposed a novel technique to distalize the molars efficiently. Inserting open-coil springs between the posterior teeth significantly improved the distribution of the distalization force, and was able to achieve greater amounts of distalization than the conventional grouped distalization in the finite element analysis. The results suggest that the conventional grouped distalization method causes stress concentration at the first premolar and carries the risk of hyalinization of PDL and root resorption. Further improvement for more efficient distalization can be achieved by modifying the spring rates between teeth, which will be the topic of our future research perspectives.
R.H., J.-y.T. and N.Y., designed the study; R.Y., D.K. and H.K. executed the finite element analysis; S.J. visualized of the results. All authors have read and agreed to the published version of the manuscript.
The data used in the current study are available from the corresponding author upon request.
The authors declare that they have no competing interest.
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Enlarge this image.Figure 1. An overview of the mechanics that we propose for distalizing molars efficiently. Each tooth will theoretically receive a distalization force of 1 N.
Enlarge this image.Figure 2. FE model of the upper left dentition with 0.022-inch brackets and a 0.019 × 0.025-inch archwire.
Enlarge this image.Figure 3. The two simulated force systems: the model without open-coil springs (NoSp model), and the model with three open-coil springs (Sp model), which we proposed in the present study.
Enlarge this image.Figure 4. Changes in the distal displacement of the center of resistance (CR) of each tooth over time. More distalization was observed in the Sp model. Line colors indicate tooth type.
Enlarge this image.Figure 5. Contact forces between the teeth and between the brackets and the archwire. Higher levels of contact force were observed in the NoSp model.
Enlarge this image.Figure 6. Changes in the distalization force transmitted to each tooth over time. Forces were more evenly distributed in the Sp model.
Enlarge this image.Figure 7. Changes in the frictional force at the bracket slots on each tooth over time.
Enlarge this image.Figure 8. Distribution of compressive stress in the PDL at the initial and final steps. Higher compressive stress was observed in the initial displacement (left). A large, highly compressed region existed at the first premolar in the NoSp model.
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