# Orthodontic Overcorrection Scheme Generation Based on Improved Multiparticle Swarm Optimization

Authors: Tian Ma; Jiahao Lyu; Qin Yang; Zhanli Li; Yuancheng Li; Yanping Chen; Xincheng Ren
Journal: Journal of Healthcare Engineering
 (2021)

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## Abstract

The orthodontic treatment often relies on the experience of doctors in traditional methods, and there are often fewer doctors with good experience, which is not conducive to improving the efficiency of patient consultation. Therefore, it has become the mainstream research direction in recent years to assist doctors in improving the efficiency of diagnosis by simulating the dental orthodontic process through computers. The orthodontic process is a multiobjective and high-dimensional path planning problem. To optimize the movement path of multiobjective orthodontics and compensate for the movement efficiency of invisible appliances, a preovercorrection orthodontic motion path scheme based on an improved multi-PSO algorithm was proposed to reduce the dimension disaster and the movement cost and improve the success rate of orthodontic surgery. Firstly, the solution set of the multiobjective particle swarm optimization (MOPSO) algorithm is introduced into the multi-PSO path planning algorithm to obtain the orthodontic movement path. Secondly, by analyzing the movement efficiency of the invisible appliance, tooth displacement compensation is designed and evaluated, and the final orthodontic scheme was generated for patients through the overcorrection method. Finally, the scheme is visualized by VTK visualization. The experimental results show that compared with the multi-PSO algorithm, the improved algorithm can reduce the length of the motion path by 10%, and the rotation angle is reduced by 4%. Meanwhile, the preovercorrection scheme designed can provide movement allowance for the orthodontic process, which guarantees that the optimal orthodontic path obtained by the scheme conforms to the clinical experiment results, ensuring that the tooth can move according to the expected path in the clinical experiment and assuring the success rate of orthodontic treatment.


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## Body

## 1. Introduction

Invisible orthodontics use computer-aided technology to obtain orthodontic movement paths and generate orthodontic solutions. The path of each tooth was recorded from the initial position to the ideal position movement, and the amount of movement and rotation was calculated. Finally, the doctor judges the rationality of each step of the orthodontic solutions and determines the final orthodontic solution, which is used for the creation of a virtual tooth orthodontic invisible brace model [1]. Invisible orthodontics can improve the accuracy of orthodontic solutions, which is an important step to help doctors develop orthodontic solutions.Invisible orthodontic technology includes tooth path planning and orthodontic solution customization. In 1998, invisalign, independently developed by the ALIGN company [2], achieved great success in orthodontic clinics. In 2003, Cadent developed the orthodontic software OrthoCAD [3], which can perform automatic tooth placement by analyzing internal parameters and predicting the entire orthodontic process. However, the data compatibility of the software is not good for widespread promotion. The invisible orthodontic system Invis-OrthoDS enhanced by Bai Yuxing of Capital Medical University [4] has the function of generating a preliminary orthodontic plan. This system divides the orthodontic process into several stages for treatment, which leads to increasing costs. Fan et al. [5] proposed a complete plan for orthodontics, especially in the segmentation and deformation of a single tooth, and achieved remarkable results, while the orthodontic effect of this method is not obvious in the presence of missing teeth. In 2016, Zhang [6] proposed a single tooth arrangement method and a collision detection method in the process of dental orthodontics, but there is still a gap between the collision method and real teeth. Chen et al. [7] proposed formulating a priority plan for orthodontics to find the breakthrough point of orthodontics in advance and then applied the advantage of full-probability coverage of the RRT algorithm to the orthodontic path planning. However, the random collision of the RRT algorithm increases the time complexity of the orthodontic process. With the in-depth study of heuristic algorithms such as genetic algorithm (GA) and simulated annealing algorithm (SAA), it is found that most of the above algorithms have problems such as heavy computation, redundant operation, and long algorithm running time. Therefore, more and more researchers propose using particle swarm optimization (PSO) algorithm to solve the related problems of path planning. PSO algorithm is a global optimization algorithm proposed by simulating bird behavior because its structure is simple and effective, thus widely used in various optimization problems. Therefore, in recent years, the PSO algorithm has been introduced in the research of invisible tooth correction to solve the multiobjective path planning problem. In 2020, Xu et al. [8] established the MOPSO algorithm based on the PSO algorithm which uses a particle to represent the whole teeth for orthodontics. However, this method does not consider the differences between different teeth, which easily leads to correction results that do not conform to conventional treatment methods. Our research group used the multi-PSO algorithm [9] in the early stage to achieve differential orthodontics for different types of teeth, but this method can only achieve local optimization. Considering the multiobjective problem of the orthodontic process, the total swarm communication optimization planning method was introduced to achieve orthodontic differentiation and improve optimization efficiency.In the process of clinical orthodontics, there is a lag between the actual tooth displacement and the expected tooth displacement, and different tooth movements are affected by periodontal tissues differently. By referring to a large number of medical studies and analyzing the motion efficiency of the appliance from different motion dimensions, Tuncay [10] showed that the average efficiency of Invisalign in incisor and canine depression was 79%. Kravitz et al. [11] showed that Invisalign has the lowest tooth elongation efficiency of all types of tooth movement, with an average of approximately 29.6%. The research group also found that the average correction efficiency of correcting the mesial and distal tilt of the tooth was 40.5%. Simon et al. [12] suggested that the amount of twisting has a significant effect on the efficiency of treatment. Invisalign is designed to twist less than 1.5° and has an orthodontic efficiency of 41.8%, and the efficiency of achieving proximal-medial and distal-medial lateral tooth movement is 88.4%. Djeu et al. [13] showed that the efficiency of controlling the movement of the buccal and lingual sides of the teeth is different; the former is 53.1%, and the latter is 37.6%. In response to this problem, the doctor usually applies 2 mm-3 mm of overcorrection to the orthodontic plan according to the condition of the patient’s tooth misalignment, which relies heavily on the orthodontic experience of the doctor. Therefore, an overcorrection method was proposed in this paper to overcome the problem of inefficiency of invisible orthodontics by adding an overcorrection allowance in the orthodontic process.In this paper, we combine a multiparticle swarm of individualized orthodontics with a single-particle swarm of total swarm particles as a guide, thus shortening the overall orthodontic path to reduce the orthodontic cost. Second, to solve the problem of insufficient performance of invisible aligners, we designed and evaluated orthodontic solutions according to different movement dimensions, which can generate more accurate orthodontic solutions.


## 2. Modeling of Orthodontic Problems

Orthodontics is a multiobjective movement in a three-dimensional obstructive environment, with the aim of moving the malocclusion to the ideal position through treatment. Due to the peculiarities of the physiological structure of teeth, there are various constraints in translational and rotational movements, and the motion of teeth is described by the motion of rigid bodies in this paper.Suppose the number of teeth of the patient isn, the current orthodontic stage is m, and the position of the tooth is indicated by the coordinates of the center of the tooth. The rotation center of tooth Ti at the mth stage is Cim=Cxim,Cyim,Czim, where the subscripts x, y, and z are the three-dimensional coordinates of a tooth. The posture angle is δim=αim,βim,γim, supposing that the local coordinates of a single tooth are X1, Y1, Z1 and the global coordinates of all teeth are X2, Y2, Z2. The projection of the aforementioned single tooth in the global coordinates is X1X2, Y1Y2, and Z1Z2, respectively, and the included angles are α, β, and γ. The position of tooth Ti in the mth orthodontic stage can be expressed as Pim=Cim,δim, and the displacement can be expressed by Euclidean distance as(1)Sim2=Cxim−Cxim−12+Cyim−Cyim−12+Czim−Czim−12.To simplify the rotation process of tooth, the rotation angle of toothTi in the mth orthodontic stage is expressed as(2)δim=αim−αim−1+βim−βim−1+γim−γim−1.The value of single-stage tooth translationSim shall not exceed 0.5 mm, and δim shall not exceed 3°.During the orthodontic process, if the orthodontic solutions are not appropriate, the teeth may collide with their adjacent teeth and interfere with the orthodontic effect. Therefore, considering the collision constraint can ensure that the teeth does not interfere with adjacent teeth when moving along the orthodontic path. LetKim be whether the tooth Ti in the mth stage collides or not:(3)Kim=0,Tidoes not collide with adjacent teeth,1,Ticollides with adjacent teeth.To have a significant change in the collision penalty function when adjacent teeth collide, then given a large constant termL, the penalty function when a collision occurs is expressed as L∗Kim.The orthodontic cost ofn teeth in m stages includes minimum moving path and minimum rotation angle, which are respectively expressed as F1 and F2. Under the premise of reasonable planning results, the smaller the value of the two is, the less pain the patient will suffer:(4)F1=min∑i=1n∑m=1mSim,(5)F2=min∑i=1n∑m=1mδim,where n and m represent the total number of teeth and the total orthodontic stage, respectively.


## 3. Orthodontic Path Design

### 3.1. Improved Multi-PSO Orthodontic Path Planning

#### 3.1.1. Multi-PSO Algorithm

To ensure the motion limitation problem of different orthodontic teeth, the research group adopted a multi-PSO approach for multistage orthodontic planning. By giving different types of teeth different inertial parameters to affect the particle flying speed, they can meet the requirements of orthodontic differentiation in the clinic. However, considering that orthodontics is a multiobjective path planning problem, the above algorithm only achieves the local optimum, and the algorithm is improved due to the above shortcomings.Suppose, in a D-dimensional search space, there is a multiparticle swarm community consisting ofZ∗n particles, the position of a current particle ij in the particle swarm of tooth Ti is Xij=xij1,xij2,…,xijD, the velocity of current particle ij is Vij=vij1,vij2,…,vijD, and the optimal position experienced by particle ij in the particle swarm of the tooth Ti is Pij=pij1,pij2,…,pijD. The optimal position experienced thus far by the particle swarm of the tooth Ti is Qig=qig1,qig2,…,qigD, and the different inertia parameters are ωi=hi/h, where h is the maximum depth of dental arch line, the maximum value of Beta curve on the Y-axis. hi is the value on the Y-axis of the center point of the ith tooth, as shown in Figure 1, and considering the different inertia parameters for different types of orthodontic velocities, the particle velocity and position update formulas for the tooth particle swarm are(6)vijDt+1=ωivijDt+c1r1pijDt−xijDt+c2r2qigDt−xijDt,(7)xijDt+1=xijDt+vijDt+1.Figure 1
Dental arch depth.Equations (6) and (7): t is the number of iterations, ω is the inertia weight, and the range of values is [0, 1]; c1 and c2 are learning factors, usually c1=c2=2; r1 and r2 are random numbers, and the range of values is [0, 1].According to orthodontic path planning, the teeth need to gradually move closer to the ideal position, the optimization purpose is the shortest path and the minimum rotation, and the constraint is no collision in the path, so the corresponding adaptation function can be constructed as follows:(8)Fij=LKim+λ1Cibest−Cijm+λ2δibest−δijm+λ1F1+λ2F2,where the first term is the penalty function of a collision, and the value of L in this paper is 10, the second term is the displacement of the tooth from the ideal position and Cibest is the rotation center for the ideal position, the third term is the rotation of the tooth from the ideal position and δibest is the posture angle for the ideal position, and the fourth and fifth terms are the sum of displacement and rotation, respectively, where λ1 and λ2 are the corresponding weights. According to orthodontic clinical indications, the path length accounts for a slightly larger proportion of patients in orthodontics than the rotation angle, so the weights in this paper are taken as empirical values 6 and 4, respectively.To solve the problem that the multi-PSO algorithm cannot achieve the global optimum, this paper combines the solution set of the MOPSO algorithm to realize the correct guidance of the orthodontic process, thereby reducing the planning cost and increasing the orthodontic efficiency.


#### 3.1.2. The Multi-PSO Algorithm with the MOPSO Solution Set

Orthodontic path planning is a complex multiconstrained combinatorial optimization problem in the target space, while meeting the clinical reality is an important prerequisite for orthodontic path planning. This paper improves the guidance strategy based on the previous multi-PSO algorithm of the research group and proposes an algorithm based on multiparticle swarm to introduce the total swarm interaction, which improves the convergence of the algorithm by continuously adding global optimal guidance individuals to the multiparticle swarm. This method takes into account the convergence and personalization of the algorithm to a certain extent in the evolution process.In the mechanism of the multiparticle swarm, toothTi can be coded as xiyiziαiβiγi, tooth Ti corresponds to a particle swarm i with an overall global best pigbest, and each particle has a historical best pijbest. According to the priority strategy to traverse n teeth and perform path optimization, the historical optimal solution set of each tooth in the mth stage is obtained as Rmp1gbest,p2gbest,…,pngbest, according to the termination conditions of the orthodontics to reach the ideal position for ending the orthodontic process. Otherwise, the orthodontic stage is added to continue the orthodontic process, and finally the orthodontic sequence RAR0,R1,…,Rl is obtained, where l is the orthodontic stage. Figure 2 shows the orthodontic flowchart of the mth stage of the multi-PSO.Figure 2
Path planning flowchart of themth stage of the multi-PSO.In the MOPSO,n teeth were encoded by x1y1z1α1β1γ1x2y2z2α2β2γ2…xnynznαnβnγn, input initial position and ideal position to obtain the solution set: RBR0,R1,…,Rl. Figure 3 shows the algorithm flow of MOPSO.Figure 3
Flowchart of MOPSO.The optimal solutionpigbest in each particle swarm in a multi-PSO algorithm is only the intraswarm optimal solution. To further improve the convergence of the algorithm, part of the learning from the overall optimum is added to the basic speed update mechanism so that the particles in the subpopulation can communicate with the overall optimum while they are searching individually, which helps the algorithm converge to the feasible domain quickly.By improving the flight properties of the particles to communicate with the global, the velocity update formula of the multi-PSO algorithm adds the position information of the MOPSO solution set as a guide to achieving the purpose of overall communication, and the improved velocity update formula is as follows:(9)vijDt+1=ωivijDt+c1r1pijDt−xijDt+c2r2qigDt−xijDt+c3r3qigD′t−xijDt,where the parameters c1=c2=c3=2, r1,r2 and r3 are random numbers between 0,1, and each particle is affected by the swarm optimum with the same weight as that of the MOPSO particle positional influence and it should be noted that the parameter qigD′ is the optimal position that the particle swarm of the tooth Ti has experienced thus far in the MOPSO algorithm, just like qigD in the multi-PSO algorithm. The mth stage orthodontic step of the improved multi-PSO algorithm is shown in Algorithm 1.Algorithm 1: Improved multi-PSO mth stage path planning algorithm flow.
Input: initial and ideal tooth position.Output: orthodontic path solution set.Step 1: administer orthodontic treatment of teeth by the MOPSO algorithm, and obtain the set of orthodontic solutionsRBR0,R1,…,RN.Step 2: calculate the set of dental positional solutionsRBmq1gD′,q2gD′,…,qngD′ for the mth stage of the MOPSO algorithm based on the path solution set RB obtained in Step 1.Step 3: orthodontics by the MOPSO algorithm in Section3.1.3.Step 3.1: initialize the swarm size to the number of teeth, and the number of particles in the particle swarm toZ.Step 3.2: calculate the value of the fitness function of the particle swarm according to (8) and determine the value of the current particle optimal position pijbest and the current particle swarm optimal position pigbest.Step 3.3: update the velocity and position of each particle by (9), and determine whether the maximum number of iterations is reached or whether the ideal position is reached; if not, return to Step 3.2, and if reached, terminate the planning.Step 4: output the solution setRmp1gbest,p2gbest,…,pngbest of each particle group in the mth phase of the particle swarm.


#### 3.1.3. Time Complexity Analysis of Algorithm

The time complexity analysis of a dental orthodontic problem in which the orthodontic stage isS, number of swarm P, number of particles in the swarm Z, and number of swarm iterations N (problem scale) is as follows: in each orthodontic stage, each swarm should be iterated to select the optimal swarm and the calculation of speed and position. The time complexity of the three algorithms is expressed by O(N) as follows.MOPSO:(10)OS,P,Z,N=S∗P+P∗3+ON2∗Z∗P+P+P,=S∗6P+ON2∗Z∗P,=S∗Z∗P∗ON2+6SP,≅S∗Z∗P∗ON2.Multi-PSO:(11)OS,P,Z,N=S∗P∗3+ON2∗Z+P,=S∗3P+ON2∗Z∗P+P,=S∗Z∗P∗ON2+4SP,≅S∗Z∗P∗ON2.The improved algorithm is the fusion of the above two algorithms, so the time complexity is approximately equal to the sum of the time complexity of the above two algorithms.Improved:(12)OS,P,Z,N=S∗Z∗P∗ON2+6SP+S∗Z∗P∗ON2+4SP,≅S∗Z∗P∗ON2.Although the time complexity of the three algorithms is approximately the same, it should be noted that the MOPSO algorithm takes a whole maxillary or mandibular tooth as a swarmP, P = 1. The coding format of a tooth is xyzαβλ, so each time a swarm is updated, the coding dimension of the algorithm reaches 6 times the number of teeth. The multi-PSO algorithm takes one tooth as a swarm, P = 14, and the tooth coding dimension is 6. In the MOPSO, the whole tooth needs to be calculated for every orthodontic stage, and the entire teeth also need to be calculated for swarm iteration, so the calculation dimension is relatively high. Therefore, in future clinical experiments, although the improved algorithm will achieve a better orthodontic result, it may consume relatively more time for tooth orthodontic.


### 3.2. Orthodontic Solution Design

#### 3.2.1. Analysis of Orthodontic Efficacy

Due to the “lag” in the orthodontic process, the concept of orthodontic efficacy (efficacy = (actual displacement/expected displacement) × 100%) is proposed. Figures4 and 5 show the orthodontic displacement and rotation efficacy analysis, respectively.Figure 4
Displacement effectiveness analysis chart.Figure 5
Rotation angle effectiveness analysis chart.In Figure4, assuming that the bracket box of the initial position of the tooth is A, the expected set orthodontic movement position is bracket box C. Due to the orthodontic effectiveness, the actual movement can only be to bracket B. The rotation angle effectiveness is the same as the movement effectiveness, the expected rotation angle is ∠1 in Figure 5, and the actual rotation angle is ∠2.


#### 3.2.2. Orthodontic Solution

When traditional orthodontic planning is completed, the doctor formulates exclusive invisible appliances for different patients based on the orthodontic results, and the final orthodontic effect will be different due to the different ages of the patients. In most cases, dental treatment often lasts as long as one year. During this process, different objective factors often cause dental orthodontics that does not meet the ideal expectations, which lead to the failure of the orthodontic treatment. Therefore, in response to this problem, this paper proposes the overcorrection method to increase the margin of movement of the teeth during orthodontic treatment by intervening in the orthodontic process in advance or during the process. This method reduces the risk of recurrent malocclusion due to improper use of the invisible orthodontic appliance and improves the success rate of orthodontic treatment.Orthodontic solutions can be divided into two types according to the order of the orthodontics. The first method considers the tooth movement efficiency before orthodontic movement planning, first to determine the ideal tooth position according to the arch line and second to compensate for the lack of tooth movement efficiency by adjusting the ideal tooth position and to compensate for the corresponding values for each of the 6 movement directions of the teeth. The second method does not require the adjustment of the ideal tooth position but adds a small amount of displacement and rotation at each stage to the path already planned in Section3.1 to ensure that the desired position is achieved at each stage of the movement.


### 3.3. Method 1: Preovercorrection

The core of the preovercorrection is the adjustment of the ideal orthodontic position of the teeth before the orthodontic path is planned. In this paper, based on medical statistics, the rotation of the following sample data was increased by 50% of the overcorrection, and the overcorrection displacement was increased by 70%, 10%, 60%, 40%, and 60% for the elongated, depressed, proximal and distal mesial, lingual and buccal sides of the teeth, respectively. Figure6 shows from left to right the current position of the tooth, the authentic ideal position of the tooth, and the corrected ideal position of the tooth. The preovercorrection process is outlined in Algorithm 2.Figure 6
Correction diagram of the ideal position of teeth.Algorithm 2: The process of preovercorrection.
Input: initial and ideal tooth position.Output: orthodontic solutions.Step 1: calculate the displacement valueSim and the rotation angle δim from the initial position of the tooth to the ideal position.Step 2: compensate in the direction of the displacement and the direction of the rotation, employing a reasonable efficiency compensation.Step 3: obtain the corrected ideal position by compensation.Step 4: using the initial position of the tooth and the modified ideal position as the input of the path planning, the tooth motion path was obtained using an improved multi-PSO combinatorial optimization algorithm.


### 3.4. Method 2: Process Overcorrection

Process overcorrection is the addition of excessive movement to the original orthodontic process. The initial and ideal positions of the teeth are known, and path planning is carried out using a combinatorial optimization algorithm to determine the amount of movement and rotation of the teeth at each stage, adding the corresponding motion compensation to the amount of movement at each stage. Figure7 shows a schematic chart of the orthodontics using the process overcorrection method to displacement motion compensation.Figure 7
Process overcorrection compensation chart.Assume that the initial position of the tooth isA and the ideal position is F. Using the path planning method to plan its movement path planning, the movement trajectory is A-B-C-D-E-F. Considering the movement effectiveness problem, the effectiveness compensation is carried out separately in the direction of the planned movement trajectory in order to enable the tooth to reach the established position at each stage after movement. The effectiveness compensation is carried out separately in the movement direction, the ratio of the compensation value. The same values are used as in method 1, i.e., the ideal position of the tooth at each stage of the correction using the process overcorrection method. For example, if A needs to move to the position of B, then the position of B–B′ needs to be overcompensated and the final actual trajectory of movement is A-B′-C′-D′-E′-F′. As there is excessive movement in each stage of the tooth, if the tooth movement in that stage exceeds the physiological tolerance, then the stage needs to be split. The process overcorrection is described in Algorithm 3.Algorithm 3:The process of process overcorrection.
Input: initial and ideal tooth position.Output: orthodontic solutions.Step 1: use the multi-PSO algorithm to obtain the set of path pointsTiX1,X2,…,Xl of each stage of the tooth Ti.Step 2: perform compensation calculations from the displacement and rotation of each stage of the different motion dimensions, and then obtain the motion point setTi′X1,X2,…,Xl after motion compensation.Step 3: determine whether the amount of tooth movement in the updated orthodontic phasem is more than the physiology can tolerate.Step 3.1: if the amount of movement in the phasem of the movement set Ti′ exceeds the maximum physiological tolerance, find the displacement and rotation values of all teeth in that phase.Step 3.2: find the midpoint of the displacement and half of the rotation angle of all of the teeth in this stage and use this as the motion position for the stagem+1. The motion position of the original stage m+1 is postponed by one stage.Step 3.3: update the exercise point setTi″X1,X2,…,Xm,Xm+1,…,Xl,Xl+1 when the amount of movement is exceeded.Step 4: generate orthodontic solutions based on the updated set of motion points.


## 3.1. Improved Multi-PSO Orthodontic Path Planning

### 3.1.1. Multi-PSO Algorithm

To ensure the motion limitation problem of different orthodontic teeth, the research group adopted a multi-PSO approach for multistage orthodontic planning. By giving different types of teeth different inertial parameters to affect the particle flying speed, they can meet the requirements of orthodontic differentiation in the clinic. However, considering that orthodontics is a multiobjective path planning problem, the above algorithm only achieves the local optimum, and the algorithm is improved due to the above shortcomings.Suppose, in a D-dimensional search space, there is a multiparticle swarm community consisting ofZ∗n particles, the position of a current particle ij in the particle swarm of tooth Ti is Xij=xij1,xij2,…,xijD, the velocity of current particle ij is Vij=vij1,vij2,…,vijD, and the optimal position experienced by particle ij in the particle swarm of the tooth Ti is Pij=pij1,pij2,…,pijD. The optimal position experienced thus far by the particle swarm of the tooth Ti is Qig=qig1,qig2,…,qigD, and the different inertia parameters are ωi=hi/h, where h is the maximum depth of dental arch line, the maximum value of Beta curve on the Y-axis. hi is the value on the Y-axis of the center point of the ith tooth, as shown in Figure 1, and considering the different inertia parameters for different types of orthodontic velocities, the particle velocity and position update formulas for the tooth particle swarm are(6)vijDt+1=ωivijDt+c1r1pijDt−xijDt+c2r2qigDt−xijDt,(7)xijDt+1=xijDt+vijDt+1.Figure 1
Dental arch depth.Equations (6) and (7): t is the number of iterations, ω is the inertia weight, and the range of values is [0, 1]; c1 and c2 are learning factors, usually c1=c2=2; r1 and r2 are random numbers, and the range of values is [0, 1].According to orthodontic path planning, the teeth need to gradually move closer to the ideal position, the optimization purpose is the shortest path and the minimum rotation, and the constraint is no collision in the path, so the corresponding adaptation function can be constructed as follows:(8)Fij=LKim+λ1Cibest−Cijm+λ2δibest−δijm+λ1F1+λ2F2,where the first term is the penalty function of a collision, and the value of L in this paper is 10, the second term is the displacement of the tooth from the ideal position and Cibest is the rotation center for the ideal position, the third term is the rotation of the tooth from the ideal position and δibest is the posture angle for the ideal position, and the fourth and fifth terms are the sum of displacement and rotation, respectively, where λ1 and λ2 are the corresponding weights. According to orthodontic clinical indications, the path length accounts for a slightly larger proportion of patients in orthodontics than the rotation angle, so the weights in this paper are taken as empirical values 6 and 4, respectively.To solve the problem that the multi-PSO algorithm cannot achieve the global optimum, this paper combines the solution set of the MOPSO algorithm to realize the correct guidance of the orthodontic process, thereby reducing the planning cost and increasing the orthodontic efficiency.


### 3.1.2. The Multi-PSO Algorithm with the MOPSO Solution Set

Orthodontic path planning is a complex multiconstrained combinatorial optimization problem in the target space, while meeting the clinical reality is an important prerequisite for orthodontic path planning. This paper improves the guidance strategy based on the previous multi-PSO algorithm of the research group and proposes an algorithm based on multiparticle swarm to introduce the total swarm interaction, which improves the convergence of the algorithm by continuously adding global optimal guidance individuals to the multiparticle swarm. This method takes into account the convergence and personalization of the algorithm to a certain extent in the evolution process.In the mechanism of the multiparticle swarm, toothTi can be coded as xiyiziαiβiγi, tooth Ti corresponds to a particle swarm i with an overall global best pigbest, and each particle has a historical best pijbest. According to the priority strategy to traverse n teeth and perform path optimization, the historical optimal solution set of each tooth in the mth stage is obtained as Rmp1gbest,p2gbest,…,pngbest, according to the termination conditions of the orthodontics to reach the ideal position for ending the orthodontic process. Otherwise, the orthodontic stage is added to continue the orthodontic process, and finally the orthodontic sequence RAR0,R1,…,Rl is obtained, where l is the orthodontic stage. Figure 2 shows the orthodontic flowchart of the mth stage of the multi-PSO.Figure 2
Path planning flowchart of themth stage of the multi-PSO.In the MOPSO,n teeth were encoded by x1y1z1α1β1γ1x2y2z2α2β2γ2…xnynznαnβnγn, input initial position and ideal position to obtain the solution set: RBR0,R1,…,Rl. Figure 3 shows the algorithm flow of MOPSO.Figure 3
Flowchart of MOPSO.The optimal solutionpigbest in each particle swarm in a multi-PSO algorithm is only the intraswarm optimal solution. To further improve the convergence of the algorithm, part of the learning from the overall optimum is added to the basic speed update mechanism so that the particles in the subpopulation can communicate with the overall optimum while they are searching individually, which helps the algorithm converge to the feasible domain quickly.By improving the flight properties of the particles to communicate with the global, the velocity update formula of the multi-PSO algorithm adds the position information of the MOPSO solution set as a guide to achieving the purpose of overall communication, and the improved velocity update formula is as follows:(9)vijDt+1=ωivijDt+c1r1pijDt−xijDt+c2r2qigDt−xijDt+c3r3qigD′t−xijDt,where the parameters c1=c2=c3=2, r1,r2 and r3 are random numbers between 0,1, and each particle is affected by the swarm optimum with the same weight as that of the MOPSO particle positional influence and it should be noted that the parameter qigD′ is the optimal position that the particle swarm of the tooth Ti has experienced thus far in the MOPSO algorithm, just like qigD in the multi-PSO algorithm. The mth stage orthodontic step of the improved multi-PSO algorithm is shown in Algorithm 1.Algorithm 1: Improved multi-PSO mth stage path planning algorithm flow.
Input: initial and ideal tooth position.Output: orthodontic path solution set.Step 1: administer orthodontic treatment of teeth by the MOPSO algorithm, and obtain the set of orthodontic solutionsRBR0,R1,…,RN.Step 2: calculate the set of dental positional solutionsRBmq1gD′,q2gD′,…,qngD′ for the mth stage of the MOPSO algorithm based on the path solution set RB obtained in Step 1.Step 3: orthodontics by the MOPSO algorithm in Section3.1.3.Step 3.1: initialize the swarm size to the number of teeth, and the number of particles in the particle swarm toZ.Step 3.2: calculate the value of the fitness function of the particle swarm according to (8) and determine the value of the current particle optimal position pijbest and the current particle swarm optimal position pigbest.Step 3.3: update the velocity and position of each particle by (9), and determine whether the maximum number of iterations is reached or whether the ideal position is reached; if not, return to Step 3.2, and if reached, terminate the planning.Step 4: output the solution setRmp1gbest,p2gbest,…,pngbest of each particle group in the mth phase of the particle swarm.


### 3.1.3. Time Complexity Analysis of Algorithm

The time complexity analysis of a dental orthodontic problem in which the orthodontic stage isS, number of swarm P, number of particles in the swarm Z, and number of swarm iterations N (problem scale) is as follows: in each orthodontic stage, each swarm should be iterated to select the optimal swarm and the calculation of speed and position. The time complexity of the three algorithms is expressed by O(N) as follows.MOPSO:(10)OS,P,Z,N=S∗P+P∗3+ON2∗Z∗P+P+P,=S∗6P+ON2∗Z∗P,=S∗Z∗P∗ON2+6SP,≅S∗Z∗P∗ON2.Multi-PSO:(11)OS,P,Z,N=S∗P∗3+ON2∗Z+P,=S∗3P+ON2∗Z∗P+P,=S∗Z∗P∗ON2+4SP,≅S∗Z∗P∗ON2.The improved algorithm is the fusion of the above two algorithms, so the time complexity is approximately equal to the sum of the time complexity of the above two algorithms.Improved:(12)OS,P,Z,N=S∗Z∗P∗ON2+6SP+S∗Z∗P∗ON2+4SP,≅S∗Z∗P∗ON2.Although the time complexity of the three algorithms is approximately the same, it should be noted that the MOPSO algorithm takes a whole maxillary or mandibular tooth as a swarmP, P = 1. The coding format of a tooth is xyzαβλ, so each time a swarm is updated, the coding dimension of the algorithm reaches 6 times the number of teeth. The multi-PSO algorithm takes one tooth as a swarm, P = 14, and the tooth coding dimension is 6. In the MOPSO, the whole tooth needs to be calculated for every orthodontic stage, and the entire teeth also need to be calculated for swarm iteration, so the calculation dimension is relatively high. Therefore, in future clinical experiments, although the improved algorithm will achieve a better orthodontic result, it may consume relatively more time for tooth orthodontic.


## 3.1.1. Multi-PSO Algorithm

To ensure the motion limitation problem of different orthodontic teeth, the research group adopted a multi-PSO approach for multistage orthodontic planning. By giving different types of teeth different inertial parameters to affect the particle flying speed, they can meet the requirements of orthodontic differentiation in the clinic. However, considering that orthodontics is a multiobjective path planning problem, the above algorithm only achieves the local optimum, and the algorithm is improved due to the above shortcomings.Suppose, in a D-dimensional search space, there is a multiparticle swarm community consisting ofZ∗n particles, the position of a current particle ij in the particle swarm of tooth Ti is Xij=xij1,xij2,…,xijD, the velocity of current particle ij is Vij=vij1,vij2,…,vijD, and the optimal position experienced by particle ij in the particle swarm of the tooth Ti is Pij=pij1,pij2,…,pijD. The optimal position experienced thus far by the particle swarm of the tooth Ti is Qig=qig1,qig2,…,qigD, and the different inertia parameters are ωi=hi/h, where h is the maximum depth of dental arch line, the maximum value of Beta curve on the Y-axis. hi is the value on the Y-axis of the center point of the ith tooth, as shown in Figure 1, and considering the different inertia parameters for different types of orthodontic velocities, the particle velocity and position update formulas for the tooth particle swarm are(6)vijDt+1=ωivijDt+c1r1pijDt−xijDt+c2r2qigDt−xijDt,(7)xijDt+1=xijDt+vijDt+1.Figure 1
Dental arch depth.Equations (6) and (7): t is the number of iterations, ω is the inertia weight, and the range of values is [0, 1]; c1 and c2 are learning factors, usually c1=c2=2; r1 and r2 are random numbers, and the range of values is [0, 1].According to orthodontic path planning, the teeth need to gradually move closer to the ideal position, the optimization purpose is the shortest path and the minimum rotation, and the constraint is no collision in the path, so the corresponding adaptation function can be constructed as follows:(8)Fij=LKim+λ1Cibest−Cijm+λ2δibest−δijm+λ1F1+λ2F2,where the first term is the penalty function of a collision, and the value of L in this paper is 10, the second term is the displacement of the tooth from the ideal position and Cibest is the rotation center for the ideal position, the third term is the rotation of the tooth from the ideal position and δibest is the posture angle for the ideal position, and the fourth and fifth terms are the sum of displacement and rotation, respectively, where λ1 and λ2 are the corresponding weights. According to orthodontic clinical indications, the path length accounts for a slightly larger proportion of patients in orthodontics than the rotation angle, so the weights in this paper are taken as empirical values 6 and 4, respectively.To solve the problem that the multi-PSO algorithm cannot achieve the global optimum, this paper combines the solution set of the MOPSO algorithm to realize the correct guidance of the orthodontic process, thereby reducing the planning cost and increasing the orthodontic efficiency.


## 3.1.2. The Multi-PSO Algorithm with the MOPSO Solution Set

Orthodontic path planning is a complex multiconstrained combinatorial optimization problem in the target space, while meeting the clinical reality is an important prerequisite for orthodontic path planning. This paper improves the guidance strategy based on the previous multi-PSO algorithm of the research group and proposes an algorithm based on multiparticle swarm to introduce the total swarm interaction, which improves the convergence of the algorithm by continuously adding global optimal guidance individuals to the multiparticle swarm. This method takes into account the convergence and personalization of the algorithm to a certain extent in the evolution process.In the mechanism of the multiparticle swarm, toothTi can be coded as xiyiziαiβiγi, tooth Ti corresponds to a particle swarm i with an overall global best pigbest, and each particle has a historical best pijbest. According to the priority strategy to traverse n teeth and perform path optimization, the historical optimal solution set of each tooth in the mth stage is obtained as Rmp1gbest,p2gbest,…,pngbest, according to the termination conditions of the orthodontics to reach the ideal position for ending the orthodontic process. Otherwise, the orthodontic stage is added to continue the orthodontic process, and finally the orthodontic sequence RAR0,R1,…,Rl is obtained, where l is the orthodontic stage. Figure 2 shows the orthodontic flowchart of the mth stage of the multi-PSO.Figure 2
Path planning flowchart of themth stage of the multi-PSO.In the MOPSO,n teeth were encoded by x1y1z1α1β1γ1x2y2z2α2β2γ2…xnynznαnβnγn, input initial position and ideal position to obtain the solution set: RBR0,R1,…,Rl. Figure 3 shows the algorithm flow of MOPSO.Figure 3
Flowchart of MOPSO.The optimal solutionpigbest in each particle swarm in a multi-PSO algorithm is only the intraswarm optimal solution. To further improve the convergence of the algorithm, part of the learning from the overall optimum is added to the basic speed update mechanism so that the particles in the subpopulation can communicate with the overall optimum while they are searching individually, which helps the algorithm converge to the feasible domain quickly.By improving the flight properties of the particles to communicate with the global, the velocity update formula of the multi-PSO algorithm adds the position information of the MOPSO solution set as a guide to achieving the purpose of overall communication, and the improved velocity update formula is as follows:(9)vijDt+1=ωivijDt+c1r1pijDt−xijDt+c2r2qigDt−xijDt+c3r3qigD′t−xijDt,where the parameters c1=c2=c3=2, r1,r2 and r3 are random numbers between 0,1, and each particle is affected by the swarm optimum with the same weight as that of the MOPSO particle positional influence and it should be noted that the parameter qigD′ is the optimal position that the particle swarm of the tooth Ti has experienced thus far in the MOPSO algorithm, just like qigD in the multi-PSO algorithm. The mth stage orthodontic step of the improved multi-PSO algorithm is shown in Algorithm 1.Algorithm 1: Improved multi-PSO mth stage path planning algorithm flow.
Input: initial and ideal tooth position.Output: orthodontic path solution set.Step 1: administer orthodontic treatment of teeth by the MOPSO algorithm, and obtain the set of orthodontic solutionsRBR0,R1,…,RN.Step 2: calculate the set of dental positional solutionsRBmq1gD′,q2gD′,…,qngD′ for the mth stage of the MOPSO algorithm based on the path solution set RB obtained in Step 1.Step 3: orthodontics by the MOPSO algorithm in Section3.1.3.Step 3.1: initialize the swarm size to the number of teeth, and the number of particles in the particle swarm toZ.Step 3.2: calculate the value of the fitness function of the particle swarm according to (8) and determine the value of the current particle optimal position pijbest and the current particle swarm optimal position pigbest.Step 3.3: update the velocity and position of each particle by (9), and determine whether the maximum number of iterations is reached or whether the ideal position is reached; if not, return to Step 3.2, and if reached, terminate the planning.Step 4: output the solution setRmp1gbest,p2gbest,…,pngbest of each particle group in the mth phase of the particle swarm.


## 3.1.3. Time Complexity Analysis of Algorithm

The time complexity analysis of a dental orthodontic problem in which the orthodontic stage isS, number of swarm P, number of particles in the swarm Z, and number of swarm iterations N (problem scale) is as follows: in each orthodontic stage, each swarm should be iterated to select the optimal swarm and the calculation of speed and position. The time complexity of the three algorithms is expressed by O(N) as follows.MOPSO:(10)OS,P,Z,N=S∗P+P∗3+ON2∗Z∗P+P+P,=S∗6P+ON2∗Z∗P,=S∗Z∗P∗ON2+6SP,≅S∗Z∗P∗ON2.Multi-PSO:(11)OS,P,Z,N=S∗P∗3+ON2∗Z+P,=S∗3P+ON2∗Z∗P+P,=S∗Z∗P∗ON2+4SP,≅S∗Z∗P∗ON2.The improved algorithm is the fusion of the above two algorithms, so the time complexity is approximately equal to the sum of the time complexity of the above two algorithms.Improved:(12)OS,P,Z,N=S∗Z∗P∗ON2+6SP+S∗Z∗P∗ON2+4SP,≅S∗Z∗P∗ON2.Although the time complexity of the three algorithms is approximately the same, it should be noted that the MOPSO algorithm takes a whole maxillary or mandibular tooth as a swarmP, P = 1. The coding format of a tooth is xyzαβλ, so each time a swarm is updated, the coding dimension of the algorithm reaches 6 times the number of teeth. The multi-PSO algorithm takes one tooth as a swarm, P = 14, and the tooth coding dimension is 6. In the MOPSO, the whole tooth needs to be calculated for every orthodontic stage, and the entire teeth also need to be calculated for swarm iteration, so the calculation dimension is relatively high. Therefore, in future clinical experiments, although the improved algorithm will achieve a better orthodontic result, it may consume relatively more time for tooth orthodontic.


## 3.2. Orthodontic Solution Design

### 3.2.1. Analysis of Orthodontic Efficacy

Due to the “lag” in the orthodontic process, the concept of orthodontic efficacy (efficacy = (actual displacement/expected displacement) × 100%) is proposed. Figures4 and 5 show the orthodontic displacement and rotation efficacy analysis, respectively.Figure 4
Displacement effectiveness analysis chart.Figure 5
Rotation angle effectiveness analysis chart.In Figure4, assuming that the bracket box of the initial position of the tooth is A, the expected set orthodontic movement position is bracket box C. Due to the orthodontic effectiveness, the actual movement can only be to bracket B. The rotation angle effectiveness is the same as the movement effectiveness, the expected rotation angle is ∠1 in Figure 5, and the actual rotation angle is ∠2.


### 3.2.2. Orthodontic Solution

When traditional orthodontic planning is completed, the doctor formulates exclusive invisible appliances for different patients based on the orthodontic results, and the final orthodontic effect will be different due to the different ages of the patients. In most cases, dental treatment often lasts as long as one year. During this process, different objective factors often cause dental orthodontics that does not meet the ideal expectations, which lead to the failure of the orthodontic treatment. Therefore, in response to this problem, this paper proposes the overcorrection method to increase the margin of movement of the teeth during orthodontic treatment by intervening in the orthodontic process in advance or during the process. This method reduces the risk of recurrent malocclusion due to improper use of the invisible orthodontic appliance and improves the success rate of orthodontic treatment.Orthodontic solutions can be divided into two types according to the order of the orthodontics. The first method considers the tooth movement efficiency before orthodontic movement planning, first to determine the ideal tooth position according to the arch line and second to compensate for the lack of tooth movement efficiency by adjusting the ideal tooth position and to compensate for the corresponding values for each of the 6 movement directions of the teeth. The second method does not require the adjustment of the ideal tooth position but adds a small amount of displacement and rotation at each stage to the path already planned in Section3.1 to ensure that the desired position is achieved at each stage of the movement.


## 3.2.1. Analysis of Orthodontic Efficacy

Due to the “lag” in the orthodontic process, the concept of orthodontic efficacy (efficacy = (actual displacement/expected displacement) × 100%) is proposed. Figures4 and 5 show the orthodontic displacement and rotation efficacy analysis, respectively.Figure 4
Displacement effectiveness analysis chart.Figure 5
Rotation angle effectiveness analysis chart.In Figure4, assuming that the bracket box of the initial position of the tooth is A, the expected set orthodontic movement position is bracket box C. Due to the orthodontic effectiveness, the actual movement can only be to bracket B. The rotation angle effectiveness is the same as the movement effectiveness, the expected rotation angle is ∠1 in Figure 5, and the actual rotation angle is ∠2.


## 3.2.2. Orthodontic Solution

When traditional orthodontic planning is completed, the doctor formulates exclusive invisible appliances for different patients based on the orthodontic results, and the final orthodontic effect will be different due to the different ages of the patients. In most cases, dental treatment often lasts as long as one year. During this process, different objective factors often cause dental orthodontics that does not meet the ideal expectations, which lead to the failure of the orthodontic treatment. Therefore, in response to this problem, this paper proposes the overcorrection method to increase the margin of movement of the teeth during orthodontic treatment by intervening in the orthodontic process in advance or during the process. This method reduces the risk of recurrent malocclusion due to improper use of the invisible orthodontic appliance and improves the success rate of orthodontic treatment.Orthodontic solutions can be divided into two types according to the order of the orthodontics. The first method considers the tooth movement efficiency before orthodontic movement planning, first to determine the ideal tooth position according to the arch line and second to compensate for the lack of tooth movement efficiency by adjusting the ideal tooth position and to compensate for the corresponding values for each of the 6 movement directions of the teeth. The second method does not require the adjustment of the ideal tooth position but adds a small amount of displacement and rotation at each stage to the path already planned in Section3.1 to ensure that the desired position is achieved at each stage of the movement.


## 3.3. Method 1: Preovercorrection

The core of the preovercorrection is the adjustment of the ideal orthodontic position of the teeth before the orthodontic path is planned. In this paper, based on medical statistics, the rotation of the following sample data was increased by 50% of the overcorrection, and the overcorrection displacement was increased by 70%, 10%, 60%, 40%, and 60% for the elongated, depressed, proximal and distal mesial, lingual and buccal sides of the teeth, respectively. Figure6 shows from left to right the current position of the tooth, the authentic ideal position of the tooth, and the corrected ideal position of the tooth. The preovercorrection process is outlined in Algorithm 2.Figure 6
Correction diagram of the ideal position of teeth.Algorithm 2: The process of preovercorrection.
Input: initial and ideal tooth position.Output: orthodontic solutions.Step 1: calculate the displacement valueSim and the rotation angle δim from the initial position of the tooth to the ideal position.Step 2: compensate in the direction of the displacement and the direction of the rotation, employing a reasonable efficiency compensation.Step 3: obtain the corrected ideal position by compensation.Step 4: using the initial position of the tooth and the modified ideal position as the input of the path planning, the tooth motion path was obtained using an improved multi-PSO combinatorial optimization algorithm.


## 3.4. Method 2: Process Overcorrection

Process overcorrection is the addition of excessive movement to the original orthodontic process. The initial and ideal positions of the teeth are known, and path planning is carried out using a combinatorial optimization algorithm to determine the amount of movement and rotation of the teeth at each stage, adding the corresponding motion compensation to the amount of movement at each stage. Figure7 shows a schematic chart of the orthodontics using the process overcorrection method to displacement motion compensation.Figure 7
Process overcorrection compensation chart.Assume that the initial position of the tooth isA and the ideal position is F. Using the path planning method to plan its movement path planning, the movement trajectory is A-B-C-D-E-F. Considering the movement effectiveness problem, the effectiveness compensation is carried out separately in the direction of the planned movement trajectory in order to enable the tooth to reach the established position at each stage after movement. The effectiveness compensation is carried out separately in the movement direction, the ratio of the compensation value. The same values are used as in method 1, i.e., the ideal position of the tooth at each stage of the correction using the process overcorrection method. For example, if A needs to move to the position of B, then the position of B–B′ needs to be overcompensated and the final actual trajectory of movement is A-B′-C′-D′-E′-F′. As there is excessive movement in each stage of the tooth, if the tooth movement in that stage exceeds the physiological tolerance, then the stage needs to be split. The process overcorrection is described in Algorithm 3.Algorithm 3:The process of process overcorrection.
Input: initial and ideal tooth position.Output: orthodontic solutions.Step 1: use the multi-PSO algorithm to obtain the set of path pointsTiX1,X2,…,Xl of each stage of the tooth Ti.Step 2: perform compensation calculations from the displacement and rotation of each stage of the different motion dimensions, and then obtain the motion point setTi′X1,X2,…,Xl after motion compensation.Step 3: determine whether the amount of tooth movement in the updated orthodontic phasem is more than the physiology can tolerate.Step 3.1: if the amount of movement in the phasem of the movement set Ti′ exceeds the maximum physiological tolerance, find the displacement and rotation values of all teeth in that phase.Step 3.2: find the midpoint of the displacement and half of the rotation angle of all of the teeth in this stage and use this as the motion position for the stagem+1. The motion position of the original stage m+1 is postponed by one stage.Step 3.3: update the exercise point setTi″X1,X2,…,Xm,Xm+1,…,Xl,Xl+1 when the amount of movement is exceeded.Step 4: generate orthodontic solutions based on the updated set of motion points.


## 4. Experimental Analysis

The experimental hardware environment was developed with an Intel i7 1.80 GHz CPU and 8 GB RAM, and the software environment was a Microsoft Windows 10 operating system, the invisible orthodontic system Orthodontic, MATLAB development environment, and VTK toolkit. In the experiments, the multi-PSO algorithm and MOPSO algorithm areZ = 50, the maximum number of iterations M = 50, c1=c2=2, r1 and r2 are random numbers between 0,1, in the improved multi-PSO c1=c2=c3=2, r1,r2,r3 is a random number between0,1 and the maximum number of iterations M = 50.


### 4.1. Comparison of the Orthodontic Path Planning Method

In orthodontics, the amount of movement and rotation of the teeth from the initial position to the target position determines the length of the orthodontic period, and the length of the orthodontic path affects the pain suffered by the patient during the orthodontic process; therefore, the smaller the movement and rotation, the better the orthodontic requirements.The test has been carried out on 30 real teeth data, and five typical groups are selected for experimental verification, which could be used to measure the performance of the algorithm from different angles. And the experimental data of the above three algorithms are the average after 10 independent runs. Table1 uses the MOPSO algorithm of Xu et al. [8], the previous method of this research group [9], and the improved multi-PSO of this paper to plan the path for 5 groups of mandibular teeth data, and twenty orthodontic stages are planned for the same example. Meanwhile, calculate the path cost of orthodontic stages and the error with the ideal position based on the same example data teeth. The orthodontic cost can be calculated according to (4) and (5) in Section 2 to obtain the total displacement F1, the total rotation F2, and the ideal position. The error includes the total error of displacement and the total error of the rotation angle, respectively, expressed as F3 and F4. l is the ideal orthodontic position, where the unit of displacement is millimeters (mm) and the unit of rotation angle is the degree (°).(13)F3=∑i=1nCxil−Cxim2+Cyil−Cyim2+Czil−Czim2,F4=∑i=1nαil−αim+βil−βim+γil−γim.Table 1
Statistical tables of 5 orthodontic results for the groups of arrays.
MethodPerformance indexE1E2E3E4E5MOPSO [8]F1 (mm)56.0834.247.4930.6856.33F2 (°)101.0790.9650.4343.49120.85F3 (mm)1.685.996.728.319.22F4 (°)4.9114.848.0414.233.31C1 (mm)57.7640.1954.2138.9965.55C2 (°)105.98105.858.4757.72124.16Multi-PSO [9]F162.1745.9349.813359.47F2106.0389.3457.2157.7125.04F30.344.372.215.664.32F42.7311.984.110.072.12C162.5150.352.0238.6663.79C2108.76101.3261.3167.77127.16ImprovedF156.1134.1247.7630.9156.43F2102.9989.0552.4653.71120.22F31.574.932.586.014.33F43.0612.264.7310.93.21C157.6839.0550.3436.9260.76C2106.05101.3157.1964.61123.43The bold values indicate that comparison is best in a single performance.At the same time, the total cost of the ideal position is adopted to evaluate the practicability of the algorithm from all aspects. The ideal total positional cost includes the total displacement costC1 and the total rotation angle cost C2.(14)C1=F1+F3,C2=F2+F4.Among the three path planning methods, the smaller the values ofF1 and F2 are, the less pain the patient suffers, and the smaller the values of F3 and F4 are, the better the orthodontic effect is. Smaller C1 and C2 mean that the total cost of orthodontics to the ideal posture is lower. It is shown in Table 1 that the MOPSO algorithm performs better than the other two algorithms in performance F1 and F2 in most samples since the algorithm realizes the globally optimal path and reduces unnecessary movement in the process. Similarly, the multi-PSO algorithm has overall advantages in performance F3 and F4, due to differentiated treatment among teeth, which optimizes the orthodontic effect of different types of teeth. The improved algorithm combines the advantages of the above two algorithms to realize both orthodontic global processes and differentiated tooth movement. The improved algorithm can achieve the majority of advantages in C1 and C2 performance, which indicates that the overall effect of the improved algorithm is obvious and contributes to the optimal clinical experiment results.Because the original data of the five mandibular teeth models had a great difference, the result of the same performance index had a great difference. To facilitate the analysis of the data in the tables, the results were tallied using statistical averaging, so that the characteristics and advantages of each algorithm can be seen intuitively. The average results are shown in Table2:Table 2
Analysis of the results of the three orthodontic algorithms.
MethodF¯1 (mm)F¯2 (°)F¯3 (mm)F¯4 (°)C¯1 (mm)C¯2 (°)MOPSO [8]44.9681.366.389.0751.3490.43Multi-PSO [9]50.0887.063.886.253.9693.26Improved45.0783.694.086.6349.1590.32The bold values indicate that comparison is best in a single performance.Analyzing the MOPSO algorithm shows that it does not provide differentiated movements for different types of tooth designs, and even though the values ofF1 and F2 indicate the lowest orthodontic cost, the values of F3 and F4 indicate large error values for the distance from the ideal posture; i.e., the algorithm is too idealistic and does not provide good orthodontic results. Both the multi-PSO algorithm and the improved multi-PSO algorithm can personalize orthodontics for patients, but the cost of the orthodontics before the improvement is too large and the orthodontics path is not globally optimal. Under the guidance of the MOPSO’s solution set, the improved multi-PSO algorithm reduces the orthodontic path length by approximately 10% and the rotation angle by approximately 4% on average, and also the path could reach the global optimal, which greatly reduces the cost of orthodontics. The difference between the orthodontic effect and the improvement is approximately 1%, which is negligible. The improved algorithm is in the second best in F¯1, F¯2, F¯3, and F¯4, and there is no big difference compared with the best, performing better than the other two algorithms for both C1 and C2, and the total cost of orthodontics is minimal, which is beneficial to doctors to achieve better treatment effect in clinical experiments and reduce the pain of patients.


### 4.2. Evaluation of the Overcorrection Method

In Section3.2, two orthodontic protocols are proposed, both of which used a 50% increase in the amount of rotation for overcorrection, a 70% increase in overcorrection displacement in the direction of tooth elongation, a 20% increase in overcorrection displacement in the direction of tooth depression, a 10% increase in overcorrection displacement in the mesial and distal mesial directions, a 60% increase in overcorrection displacement for lingual movement, and a 40% increase in overcorrection displacement for buccal movement. Since the values of the effective compensation of the displacement and rotation components are the same, the orthodontic results are the same for both groups of orthodontic solutions. The evaluation of the advantages and disadvantages of the orthodontic solutions focused on the cost of tooth movement, mainly in terms of the amount of rotation and displacement. Table 3 shows the evaluation of the two overcorrection methods and a nonovercorrection method based on improved multi-PSO algorithm path planning for five sample data sets, where the displacement is in millimeters (mm) and the rotation angle is in degrees (°). Note that the experimental results are also averages obtained after 10 independent runs.Table 3
Comparison table of overcorrection options.
MethodPerformance indexE1E2E3E4E5NonovercorrectionF1 (mm)56.1134.1247.7630.9156.43F2 (°)102.9989.0552.4653.71120.22PreovercorrectionF169.3242.2954.0441.3465.07F2122.7798.0363.0970.99150.40Process overcorrectionF177.9944.0160.1836.7868.28F2154.49133.5878.6980.57180.33According to Table3, the mean values of the displacement sum of the five groups of data for nonovercorrection, preovercorrection, and process overcorrection were 45.07, 54.41, and 57.45, respectively, and the mean values of the rotation angle sum were 83.69, 101.06, and 125.53, respectively. Compared with the two methods, the increments of the nonovercorrection displacement sum are 20.74% and 27.48%, respectively, and the increments of the rotating sum are 20.76% and 50.00%, respectively. Although the experimental results of the two overcorrection methods were much higher than those of the nonovercorrection method, there were many objective factors in the clinical experiments, such as teeth damage caused by improper eating and wearing of invisible orthodontic appliance, which led to the failure of the orthodontic process. Therefore, considering the existing objective factors, this paper is designed to use overcorrection methods for the above problems, which provides a movement margin for the orthodontic plan and ensures the effectiveness and variability of clinical experiments. The analysis showed that the displacement and rotation of the preovercorrection were significantly smaller than the values of the process overcorrection, and the preovercorrection was more in line with the actual clinical need after considering the pain suffered by the patients.


### 4.3. Visualization of the Orthodontic Solution

To visually see the results of the experiment, this section uses the preovercorrection method to visualize the orthodontic process of the teeth. Figure8 selects three sets of mandibular data to demonstrate the effect of the teeth before and after orthodontics. State A is the posture of the mandibular teeth before orthodontics, State B is the posture of the teeth after orthodontics, State C is the comparison of the posture of the teeth before and after orthodontics (in State C, the blue is the preorthodontic posture, and the white is the postorthodontic). The teeth are arranged neatly and beautifully after orthodontics.Figure 8
Before and after orthodontic treatment.Figure9 shows nine of the 24 stages of orthodontic treatment for a sample case. In each stage of orthodontic treatment, the teeth move toward the target position according to the tooth movement constraint until the algorithm reaches the termination condition. This method can plan a collision-free optimal path for tooth movement and generate an invisible correction plan that meets the requirements of clinical orthodontics.Figure 9
The orthodontic process.


## 4.1. Comparison of the Orthodontic Path Planning Method

In orthodontics, the amount of movement and rotation of the teeth from the initial position to the target position determines the length of the orthodontic period, and the length of the orthodontic path affects the pain suffered by the patient during the orthodontic process; therefore, the smaller the movement and rotation, the better the orthodontic requirements.The test has been carried out on 30 real teeth data, and five typical groups are selected for experimental verification, which could be used to measure the performance of the algorithm from different angles. And the experimental data of the above three algorithms are the average after 10 independent runs. Table1 uses the MOPSO algorithm of Xu et al. [8], the previous method of this research group [9], and the improved multi-PSO of this paper to plan the path for 5 groups of mandibular teeth data, and twenty orthodontic stages are planned for the same example. Meanwhile, calculate the path cost of orthodontic stages and the error with the ideal position based on the same example data teeth. The orthodontic cost can be calculated according to (4) and (5) in Section 2 to obtain the total displacement F1, the total rotation F2, and the ideal position. The error includes the total error of displacement and the total error of the rotation angle, respectively, expressed as F3 and F4. l is the ideal orthodontic position, where the unit of displacement is millimeters (mm) and the unit of rotation angle is the degree (°).(13)F3=∑i=1nCxil−Cxim2+Cyil−Cyim2+Czil−Czim2,F4=∑i=1nαil−αim+βil−βim+γil−γim.Table 1
Statistical tables of 5 orthodontic results for the groups of arrays.
MethodPerformance indexE1E2E3E4E5MOPSO [8]F1 (mm)56.0834.247.4930.6856.33F2 (°)101.0790.9650.4343.49120.85F3 (mm)1.685.996.728.319.22F4 (°)4.9114.848.0414.233.31C1 (mm)57.7640.1954.2138.9965.55C2 (°)105.98105.858.4757.72124.16Multi-PSO [9]F162.1745.9349.813359.47F2106.0389.3457.2157.7125.04F30.344.372.215.664.32F42.7311.984.110.072.12C162.5150.352.0238.6663.79C2108.76101.3261.3167.77127.16ImprovedF156.1134.1247.7630.9156.43F2102.9989.0552.4653.71120.22F31.574.932.586.014.33F43.0612.264.7310.93.21C157.6839.0550.3436.9260.76C2106.05101.3157.1964.61123.43The bold values indicate that comparison is best in a single performance.At the same time, the total cost of the ideal position is adopted to evaluate the practicability of the algorithm from all aspects. The ideal total positional cost includes the total displacement costC1 and the total rotation angle cost C2.(14)C1=F1+F3,C2=F2+F4.Among the three path planning methods, the smaller the values ofF1 and F2 are, the less pain the patient suffers, and the smaller the values of F3 and F4 are, the better the orthodontic effect is. Smaller C1 and C2 mean that the total cost of orthodontics to the ideal posture is lower. It is shown in Table 1 that the MOPSO algorithm performs better than the other two algorithms in performance F1 and F2 in most samples since the algorithm realizes the globally optimal path and reduces unnecessary movement in the process. Similarly, the multi-PSO algorithm has overall advantages in performance F3 and F4, due to differentiated treatment among teeth, which optimizes the orthodontic effect of different types of teeth. The improved algorithm combines the advantages of the above two algorithms to realize both orthodontic global processes and differentiated tooth movement. The improved algorithm can achieve the majority of advantages in C1 and C2 performance, which indicates that the overall effect of the improved algorithm is obvious and contributes to the optimal clinical experiment results.Because the original data of the five mandibular teeth models had a great difference, the result of the same performance index had a great difference. To facilitate the analysis of the data in the tables, the results were tallied using statistical averaging, so that the characteristics and advantages of each algorithm can be seen intuitively. The average results are shown in Table2:Table 2
Analysis of the results of the three orthodontic algorithms.
MethodF¯1 (mm)F¯2 (°)F¯3 (mm)F¯4 (°)C¯1 (mm)C¯2 (°)MOPSO [8]44.9681.366.389.0751.3490.43Multi-PSO [9]50.0887.063.886.253.9693.26Improved45.0783.694.086.6349.1590.32The bold values indicate that comparison is best in a single performance.Analyzing the MOPSO algorithm shows that it does not provide differentiated movements for different types of tooth designs, and even though the values ofF1 and F2 indicate the lowest orthodontic cost, the values of F3 and F4 indicate large error values for the distance from the ideal posture; i.e., the algorithm is too idealistic and does not provide good orthodontic results. Both the multi-PSO algorithm and the improved multi-PSO algorithm can personalize orthodontics for patients, but the cost of the orthodontics before the improvement is too large and the orthodontics path is not globally optimal. Under the guidance of the MOPSO’s solution set, the improved multi-PSO algorithm reduces the orthodontic path length by approximately 10% and the rotation angle by approximately 4% on average, and also the path could reach the global optimal, which greatly reduces the cost of orthodontics. The difference between the orthodontic effect and the improvement is approximately 1%, which is negligible. The improved algorithm is in the second best in F¯1, F¯2, F¯3, and F¯4, and there is no big difference compared with the best, performing better than the other two algorithms for both C1 and C2, and the total cost of orthodontics is minimal, which is beneficial to doctors to achieve better treatment effect in clinical experiments and reduce the pain of patients.


## 4.2. Evaluation of the Overcorrection Method

In Section3.2, two orthodontic protocols are proposed, both of which used a 50% increase in the amount of rotation for overcorrection, a 70% increase in overcorrection displacement in the direction of tooth elongation, a 20% increase in overcorrection displacement in the direction of tooth depression, a 10% increase in overcorrection displacement in the mesial and distal mesial directions, a 60% increase in overcorrection displacement for lingual movement, and a 40% increase in overcorrection displacement for buccal movement. Since the values of the effective compensation of the displacement and rotation components are the same, the orthodontic results are the same for both groups of orthodontic solutions. The evaluation of the advantages and disadvantages of the orthodontic solutions focused on the cost of tooth movement, mainly in terms of the amount of rotation and displacement. Table 3 shows the evaluation of the two overcorrection methods and a nonovercorrection method based on improved multi-PSO algorithm path planning for five sample data sets, where the displacement is in millimeters (mm) and the rotation angle is in degrees (°). Note that the experimental results are also averages obtained after 10 independent runs.Table 3
Comparison table of overcorrection options.
MethodPerformance indexE1E2E3E4E5NonovercorrectionF1 (mm)56.1134.1247.7630.9156.43F2 (°)102.9989.0552.4653.71120.22PreovercorrectionF169.3242.2954.0441.3465.07F2122.7798.0363.0970.99150.40Process overcorrectionF177.9944.0160.1836.7868.28F2154.49133.5878.6980.57180.33According to Table3, the mean values of the displacement sum of the five groups of data for nonovercorrection, preovercorrection, and process overcorrection were 45.07, 54.41, and 57.45, respectively, and the mean values of the rotation angle sum were 83.69, 101.06, and 125.53, respectively. Compared with the two methods, the increments of the nonovercorrection displacement sum are 20.74% and 27.48%, respectively, and the increments of the rotating sum are 20.76% and 50.00%, respectively. Although the experimental results of the two overcorrection methods were much higher than those of the nonovercorrection method, there were many objective factors in the clinical experiments, such as teeth damage caused by improper eating and wearing of invisible orthodontic appliance, which led to the failure of the orthodontic process. Therefore, considering the existing objective factors, this paper is designed to use overcorrection methods for the above problems, which provides a movement margin for the orthodontic plan and ensures the effectiveness and variability of clinical experiments. The analysis showed that the displacement and rotation of the preovercorrection were significantly smaller than the values of the process overcorrection, and the preovercorrection was more in line with the actual clinical need after considering the pain suffered by the patients.


## 4.3. Visualization of the Orthodontic Solution

To visually see the results of the experiment, this section uses the preovercorrection method to visualize the orthodontic process of the teeth. Figure8 selects three sets of mandibular data to demonstrate the effect of the teeth before and after orthodontics. State A is the posture of the mandibular teeth before orthodontics, State B is the posture of the teeth after orthodontics, State C is the comparison of the posture of the teeth before and after orthodontics (in State C, the blue is the preorthodontic posture, and the white is the postorthodontic). The teeth are arranged neatly and beautifully after orthodontics.Figure 8
Before and after orthodontic treatment.Figure9 shows nine of the 24 stages of orthodontic treatment for a sample case. In each stage of orthodontic treatment, the teeth move toward the target position according to the tooth movement constraint until the algorithm reaches the termination condition. This method can plan a collision-free optimal path for tooth movement and generate an invisible correction plan that meets the requirements of clinical orthodontics.Figure 9
The orthodontic process.


## 5. Conclusion

Orthodontic movement path generation is a system, which is an integral part of the invisible orthodontic technology and has received increasing attention from researchers in the fields of computer-aided design and graphics in recent years. This paper proposes a preovercorrection orthodontic motion path scheme based on an improved multi-PSO algorithm. Firstly, considering the personalized movement of different types of teeth, the solution set of the MOPSO algorithm is used as a guiding strategy on the basis of the multiparticle swarm algorithm, which reduces the cost of orthodontics while enhancing convergence and personalization. Secondly, we propose and evaluate two options for the lack of effectiveness of invisible aligners in moving teeth: preovercorrection and process overcorrection. Through experimental evaluation, the orthodontic cost of the preovercorrection option is significantly better than that of the process overcorrection. Then the optimization algorithm proposed in this paper adopts the preovercorrection method to visualize the orthodontic correction process by using the VTK toolkit. Finally, the experimental data verifies that the proposed optimization algorithm has better overall planning results than other algorithms and confirms the effect of clinical experiments to the greatest extent. Therefore, the dental orthodontic program proposed in this paper can help doctors assist in diagnosis, provide effective help in the future, and improve the dental orthodontic path planning in the relevant orthodontic treatment system. Because the improved algorithm is based on the fusion of the above two algorithms, the difference between the global and the local needs to be calculated in the orthodontic process, so the time complexity of the algorithm is high. Because of the physiological structure of the teeth, the occlusal conditions of upper and lower molars were not considered in the above scheme. Under this premise, the orthodontic process can only be divided into two parts: maxillary teeth and mandibular teeth, so the solution still has shortcomings and deserves further investigation.



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