Correcting method for correcting ground covering weather by combining FR3 functional appliance with invisible tooth socket

Abstract

The invention relates to the field of stomatology, and discloses a correction method for correcting denture by combining an FR3 functional appliance with an invisible tooth socket, and the correction method for correcting denture by combining the FR3 functional appliance with the invisible tooth socket comprises the following steps: three-dimensional modeling of an oral cavity, extraction of a force application point, dynamic displacement modeling, energy functional construction, path optimization control and reverse determination of parameters. The combined correction of the FR3 appliance and the invisible tooth socket is realized; the correction system for correcting the ground covering by combining the FR3 functional appliance with the invisible tooth socket comprises a three-dimensional reconstruction module, a force vector field extraction module, a mechanical modeling module, a trajectory optimization module, a parameter output module and an implementation feedback module. According to the method, the dentition and jaw model is accurately constructed through a three-dimensional reconstruction technology, and precise control of FR3 and invisible tooth socket combined correction is realized by combining vector field extraction, mechanical modeling and system energy functional optimization. And through real-time feedback and path optimization, the correction effect, efficiency and precision are greatly improved.

Description

Technical Field

[0001] The present invention relates to the field of stomatology, specifically a correction method for correcting crossbite by combining FR3 functional appliance and invisible braces. Background Art

[0002] As a common malocclusion deformity, crossbite has become one of the main problems affecting people's appearance and function. Usually, the correction of crossbite requires a comprehensive treatment plan considering teeth, jaws, and muscles. Currently, FR3 functional appliance mainly treats such malocclusion deformities by improving jaw and muscle functions, but its control over tooth displacement is relatively limited. The design of FR3 appliance can solve problems at the muscle and jaw levels to a certain extent, but its precise regulation of teeth is relatively weak, resulting in an inability to fully achieve the ideal treatment effect when dealing with more complex crossbite cases.

[0003] On the other hand, invisible orthodontic appliances are in rapid development. It can effectively perform three-dimensional regulation on teeth and show high precision in tooth displacement. However, although invisible correction can better adjust the tooth position, its effect in improving jaw morphology and solving muscle function problems is relatively limited. The design of invisible braces only applies a tiny orthodontic force on teeth, and since its thickness is only about 1 mm, it cannot effectively act on tissues other than teeth, such as jaws and muscles, resulting in limited treatment effects.

[0004] Based on the deficiencies of the existing technology, in mild crossbite or functional crossbite cases, combining the advantages of FR3 appliance and invisible braces becomes a potential treatment option. By using these two orthodontic tools simultaneously, it is possible to correct tooth crossbite while improving jaw deformities and performing myofunctional training. Invisible braces can precisely control tooth displacement, while FR3 functional appliance can improve jaw and muscle functions, thus achieving a more comprehensive treatment effect. However, this treatment option that combines the advantages of both has not been fully developed and utilized in the existing technology, and there is also a lack of effective orthodontic mechanics and path optimization mechanisms. Therefore, how to reasonably combine these two orthodontic tools and simultaneously handle the orthodontic problems of teeth and jaws through precise path control. Therefore, the present invention proposes a correction method for correcting crossbite by combining FR3 functional appliance and invisible braces to solve the deficiencies of the existing technology. Summary of the Invention

[0005] Aiming at the deficiencies of the existing technology, the present invention provides a correction method for correcting crossbite by combining FR3 functional appliance and invisible braces, which solves the dual requirements of precise tooth displacement control and jaw function improvement, and avoids the limitation of the inability of a single orthodontic tool to comprehensively treat crossbite problems.

[0006] To achieve the above objectives, the present invention is implemented through the following technical solutions: A correction method for correcting crossbite by combining an FR3 functional appliance and an invisible aligner, comprising the following steps:

[0007] S1. Collect the three-dimensional data of the patient's oral cavity, construct a spatial model including the dentition and the jawbone, and obtain a three-dimensional model;

[0008] S2. Based on the three-dimensional model, obtain the force application point distribution of the FR3 functional appliance and the preset path of the invisible aligner, and map the two to the same spatial reference system;

[0009] S3. In the reference system, establish a dynamic displacement model under the action of tooth forces, which is used to describe the movement state of the teeth under the combined action of the FR3 and the invisible aligner;

[0010] S4. Construct a system energy functional for describing the overall movement of the teeth, and the energy functional includes energy components representing tooth deformation energy, periodontal tissue response energy, and force application deviation terms;

[0011] S5. On the basis of the force application vector and path feasibility, optimize and control the tooth movement path. The optimization control is solved with the system energy functional as the objective, and constraint terms that meet clinical biological conditions are set;

[0012] S6. Based on the optimized tooth movement path, reversely determine the structural design parameters of the FR3 appliance and the phased control parameters of the invisible aligner.

[0013] The present invention also provides a correction system for correcting crossbite by combining an FR3 functional appliance and an invisible aligner, comprising:

[0014] A three-dimensional reconstruction module for constructing the spatial structure of the dentition and the jawbone;

[0015] A force vector field extraction module for obtaining the force application information of the FR3 and the invisible aligner;

[0016] A mechanical modeling module for constructing tooth movement equations and a system energy functional;

[0017] A trajectory optimization module for solving the optimal path that meets the constraint conditions;

[0018] A parameter output module for inversely calculating the design parameters of each appliance;

[0019] An implementation feedback module for periodically collecting trajectories and performing error correction.

[0020] The present invention provides a correction method for correcting crossbite by combining an FR3 functional appliance and an invisible aligner. It has the following beneficial effects:

[0021] 1. The present invention adopts three-dimensional reconstruction technology to construct an accurate spatial structure of the dentition and jaws. Through high-precision three-dimensional modeling, the geometric features of teeth and alveolar bone can be accurately reproduced, providing reliable data support for subsequent orthodontic appliance design and path optimization. Compared with the simple two-dimensional images or rough three-dimensional modeling methods in the prior art, the present invention solves the common errors and inaccuracies in the process of tooth structure reconstruction by traditional methods, ensuring the accuracy of the entire orthodontic process.

[0022] 2. The present invention accurately obtains the force application points and directions of the FR3 functional orthodontic appliance and invisible braces through the force vector field extraction module. This technical solution can provide detailed data support for the reasonable distribution of orthodontic forces, ensuring the accuracy and effectiveness of force application during the orthodontic process. Different from the traditional fixed force application design, the present invention avoids the problems of excessive force concentration or uneven distribution, thus greatly improving the orthodontic effect and patient comfort.

[0023] 3. The present invention, through the mechanical modeling module, combines the tooth movement equation and the system energy functional to accurately simulate the dynamic behavior of teeth under the action of orthodontic forces. This solution enables the real-time optimization of displacement and mechanical balance during the orthodontic process. Compared with the models in the prior art that cannot fully consider the biomechanical characteristics of teeth, the present invention greatly improves the orthodontic effect and controllability, avoiding unnecessary damage to teeth and surrounding tissues.

[0024] 4. The implementation feedback module of the present invention can monitor and adjust the tooth movement path in real time to ensure dynamic optimization during the treatment process. By periodically collecting the actual movement trajectory and performing error correction, compared with the traditional method that only relies on the initial path planning, the present invention provides a flexible feedback mechanism that can correct the path and force application in a timely manner according to the actual reaction of the patient, thereby effectively shortening the treatment cycle, improving the treatment efficiency and reducing the uncertainty during the orthodontic process. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 is the flowchart of the method of the present invention;

[0026] Figure 2 is the system architecture diagram of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0027] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0028] Please refer to Figure 1, the embodiment of the present invention provides a correction method for correcting anterior crossbite by combining FR3 functional appliance and invisible braces, including the following steps:

[0029] S1. Collect the three-dimensional oral data of the patient, construct a spatial model including dentition and jaw bone, and obtain a three-dimensional model;

[0030] In the correction method for correcting anterior crossbite by combining FR3 functional appliance and invisible braces proposed by the present invention, step S1 plays a crucial fundamental role. This step not only serves as the input premise for subsequent tooth dynamics models and path optimization algorithms, but also undertakes the key guarantee tasks of model accuracy, registration consistency, and parameter identification quality. In order to ensure that the correction logic and mechanical response model of the overall system have personalized adaptability and high fitting ability, the constructed oral spatial model must achieve multi-dimensional accurate expression of tooth shape, root structure, periodontal connection interface, and jaw bone mass parameters.

[0031] Therefore, step S1 is not limited to conventional three-dimensional point cloud acquisition, but involves a complete set of spatial modeling processes, parameter mapping mechanisms, and fusion reconstruction processes of multi-source heterogeneous data, which are specifically disclosed as follows:

[0032] In this embodiment, the three-dimensional data of the oral region of the patient is collected by combining optical oral scanning and cone beam CT.

[0033] The optical scanning part provides high-density dentition surface point cloud data, and the CBCT scanning part provides root region structure, periodontal ligament boundary, and jaw bone density voxel information.

[0034] The two types of data are integrated through feature registration and voxel interpolation to form a unified individual oral model of the patient.

[0035] Generally, the collected dentition point cloud data is represented in the form of a dense discrete point set, denoted as where N represents the number of points, and each point p i =(x i , y i , z i ) corresponds to the sampling position on the tooth surface, and the unit of all coordinate components is millimeter, constructing the surface morphology of the crown area.

[0036] As an option, the point cloud data generates a continuous tooth surface mesh model through the Poisson reconstruction algorithm. This algorithm takes the position and normal vector of the points as input and outputs a closed mesh structure, which is used to calculate second-order geometric features such as the normal direction of the tooth occlusal surface, the curvature of the incisal edge morphology, and the tooth inclination angle.

[0037] Specifically, the periodontal ligament boundary is extracted by an active contour model based on CBCT gray-scale segmentation. The initial seed points are manually marked in the apical proximal area, and the closed boundary Γ is obtained according to the regional density gradient and morphological potential energy convergencePDL , for subsequent mechanical boundary condition setting. This boundary is used to simulate the elastic thin layer in the connection area between the tooth root and the alveolar bone.

[0038] Generally, its thickness is set to d PDL ∈[0.15, 0.30] mm, and this area is modeled as an anisotropic material with a non-linear stress-strain response.

[0039] In a possible implementation, the tooth root area adopts a voxel grid form Denote each voxel v j as a cuboid element with a volume of ΔV = Δx·Δy·Δz, where Δx, Δy, and Δz represent the spatial resolutions in three directions respectively, with the unit of millimeter.

[0040] This dataset is used to construct the volume grid model of the tooth root and provide the basis for impedance estimation during tooth movement.

[0041] The reconstruction of the jawbone structure is based on the voxel gray value ρ in the CBCT image k , defined in the three-dimensional voxel grid set where each b k corresponds to a bone voxel in space, and the gray value ρ k has the unit of Hounsfield Unit (HU).

[0042] The bone density parameter can be converted into the elastic modulus E k , and the mapping formula is as follows:

[0043] E k = a·ρ k + b;

[0044] where: E k represents the bone elastic modulus corresponding to the k-th voxel, with the unit of MPa; ρ k represents the gray value of the k-th voxel, with the unit of HU; a and b are empirical fitting parameters obtained according to clinical calibration, generally a ∈[0.3, 1.2], b ∈[50, 150].

[0045] The Poisson's ratio ν k can be set according to the bone density range classification. For example, when the density range ρ k > 800 HU, set ν k = 0.28, and when the low-density area ρ k < 400 HU, set ν k = 0.35, and the rest of the area is obtained by linear interpolation.

[0046] In some embodiments, to ensure the continuity of the tooth-bone connection in the model, a connection relationship is established between the root surface and the jawbone voxel boundary through node remapping. This relationship is determined by the minimum Euclidean distance between adjacent points, such that the boundary nodes of the root belong to the nearest bone voxel.

[0047] Meanwhile, a set of reference points for the occlusion relationship is collected for subsequent coordinate registration, force application direction constraint, and path error correction.

[0048] The reference points include but are not limited to:

[0049] the incisal edge point c1 of the maxillary central incisor,

[0050] the lingual point c2 of the mandibular central incisor,

[0051] the mesial buccal cusp points c3, c4 of the first molar

[0052] the occlusal plane corner points c5, c6, etc.

[0053] These points are represented in three-dimensional coordinate form, and the coordinate of each point is c i =(x i , y i , z i ), with the unit of millimeters.

[0054] In the path planning or the construction of the system orthodontic energy functional, the above point set is used to define the constraint energy term ε constr , and the specific definition is as follows:

[0055]

[0056] where: φ(c i ) is the coordinate of the i-th reference point after displacement transformation after orthodontics; is the coordinate of the expected target position; ∥·∥ is the Euclidean norm, used to measure the spatial distance between two points, with the unit of millimeters, and n is the number of tooth individuals.

[0057] To enhance the binding force of the occlusion relationship points in the entire model space, in some embodiments, a weight factor ω i is introduced, and the correction formula is as follows:

[0058]

[0059] where ω i ∈(0, 1], used to express the importance weight of this point for the overall registration. Clinically, a relatively high weight is often set for the midline point and the occlusion contact point to enhance the control accuracy, and n is the number of tooth individuals.

[0060] In a possible implementation manner, all the above model data are expressed in a unified Cartesian coordinate system, and multi-source data integration is performed by the rigid body registration method. This registration method solves the minimization of the rigid body transformation error based on the reference point set C:

[0061]

[0062] where: R ∈ SO(3) is the rotation matrix, satisfying R T R = I and det(R) = 1; is the translation vector; p i is the reference point coordinate in the initial model; q i is the target reference point coordinate, and n is the number of tooth individuals.

[0063] The above registration result is used as the initial input for the subsequent path optimization and energy functional minimization problems, so that all subsequent calculations are carried out in the decoupled unified physical coordinate framework, avoiding path drift and force application deviation caused by local modeling errors.

[0064] After this step S1 is completed, the system will obtain a high-fidelity three-dimensional space model including tooth shape, root topology, periodontal connection and jaw continuum structure; each region in the model has a clear mechanical parameter mapping path and coordinate reference basis, which can be directly used as the input basis for subsequent displacement function solution, force field construction and multi-stage planning of the orthodontic path, meeting the actual needs of industrial implementation and clinical deployment.

[0065] S2. Based on the three-dimensional model, obtain the force application point distribution of the FR3 functional appliance and the preset path of the invisible braces, and map the two to the same spatial reference system;

[0066] In step S2, based on the three-dimensional oral cavity model constructed in step S1, further determine the force application point distribution of the FR3 functional appliance and the invisible braces, and map the action paths of the two to the same spatial reference system. This process plays a crucial role in the correction process, ensuring the coordination and consistency between the FR3 functional appliance and the invisible braces, preventing the force application conflict caused by the spatial position deviation between the two, and ensuring the uniform application of the correction force and the smooth movement of the teeth.

[0067] In the foregoing step S1, a detailed three-dimensional model of the patient's teeth, alveolar bone, and occlusion relationship has been obtained through high-precision three-dimensional data acquisition and construction. At this time, the patient's oral data has been standardized in the spatial reference system. Next, the task of step S2 is to accurately determine the force application point distribution of the FR3 functional appliance and the invisible aligner, and through coordinate transformation, unify the force application paths of the two to the same spatial reference system. This mapping process is crucial because it will ensure that the appliance and the invisible aligner can work together during the treatment process, avoiding inconsistent force application and misalignment of the paths.

[0068] In this embodiment, first, according to the patient's three-dimensional oral model obtained in step S1, the force application point distribution of the FR3 functional appliance is determined. When determining the force application points, precise positioning needs to be carried out according to the position, shape of each tooth, and the contact relationship with the alveolar bone.

[0069] The force application points of the FR3 functional appliance are generally located on the tooth surfaces of the upper and lower jaws, especially in key areas such as the upper anterior tooth area and the lower first molar area. Larger orthodontic forces are usually required in these areas to correct underbite or other dental malocclusions.

[0070] For the FR3 functional appliance, its force application points are usually determined through the following steps:

[0071] According to the tooth surface morphology, calculate the curvature and surface normal of each tooth through point cloud data;

[0072] When selecting the force application points, consider the relative positions between the teeth and the direction of force application to ensure uniform distribution of the torque.

[0073] Generally, the selection of force application points needs to combine with a mechanical model to ensure that the acting force of the appliance can act evenly on the target teeth and can effectively adjust the position of the teeth. For the FR3 functional appliance, the number of force application points is usually between 10 and 15, and the specific number depends on the individual differences of the patient and the complexity of the treatment plan.

[0074] In some embodiments, the preset path of the invisible aligner is calculated. The invisible aligner plays a role in gradually applying force and finely adjusting the tooth position during the entire treatment process. The force application point distribution and path of the invisible aligner need to ensure uniform force field distribution during the treatment process, and each minor corrective movement can ensure smooth movement of the teeth without causing excessive deformation. The path planning of the invisible aligner first needs to consider the distance between the initial position and the target position of the teeth, and calculate the optimal movement trajectory. Specifically, the path planning of the invisible aligner involves the following steps:

[0075] Calculate the initial position and target position of each tooth, and determine the displacement amount of each correction stage through a mathematical optimization algorithm (such as the least squares method);

[0076] Stage - plan the movement of each tooth to ensure a smooth transition for each adjustment and avoid over - correction or reverse displacement.

[0077] In some embodiments, the path planning of the invisible braces is not only based on the position and movement trajectory of the teeth, but also takes into account the contact forces between the teeth and the surrounding hard and soft tissues, ensuring that the application of the orthodontic force will not cause discomfort or harm to the periodontal tissues. The path planning process combines the patient's oral anatomical characteristics, mechanical requirements, and correction goals to accurately determine the path for each stage.

[0078] Specifically, the force - application points of the FR3 functional appliance and the force - application path of the invisible braces need to be uniformly mapped in the same spatial reference system. To achieve this goal, first, the positional relationship between the two needs to be standardized into a unified three - dimensional coordinate system through coordinate transformation techniques.

[0079] In this coordinate transformation, a rigid - body transformation model is used to ensure the spatial consistency of the force - application points and the path. Assume that the set of force - application points of the FR3 functional appliance is where p i =(x i , y i , z i ) is the spatial coordinate of the i - th force - application point; the set of force - application path points of the invisible braces is where q j =(x′ j , y′ j , z′ j ) is the spatial coordinate of the invisible - braces path point.

[0080] To uniformly map the two to the same spatial reference system, we can use a rigid - body transformation to solve for a rotation matrix R and a translation vector t such that the set of force - application points of the FR3 functional appliance and the set of invisible - braces path points can be registered in the same reference system. The calculation formula for the rigid - body transformation is as follows:

[0081] P′ FR3 =R·P FR3 +t;

[0082] where: P′ FR3 is the set of force - application points of the transformed FR3 functional appliance; R is a 3×3 rotation matrix used to rotate the force - application points; t is a translation vector used to translate the force - application points; P FR3 is the original set of force - application points; P Invisalign is the set of invisible - braces path points.

[0083] During this coordinate transformation process, the rotation matrix R and the translation vector t are calculated by minimizing the Euclidean distance between two point sets. That is, R and t are solved through the following optimization problem:

[0084]

[0085] where ∥·∥ represents the Euclidean norm, indicating the distance between the force application points.

[0086] This transformation ensures that the force application points of the FR3 functional appliance and the path points of the invisible aligner are accurately aligned in the same three-dimensional space coordinate system.

[0087] As an option, to further improve the accuracy of spatial mapping, local constraint conditions can be added on this basis, such as the physical contact of specific teeth, the geometric limitations of the appliance, etc. These constraint conditions can be processed through the following formula:

[0088]

[0089] where: ω i is the weighting coefficient, used to adjust the weight of specific force application points to ensure the priority of key force application points; P′ FR3 and P Invisalign respectively represent the coordinates of the force application points of the FR3 functional appliance and the path points of the invisible aligner after coordinate transformation.

[0090] Through these local constraints, the model accuracy can be further improved to ensure the perfect coordination of the force application path and the displacement trajectory.

[0091] Through step S2, the force application points and paths of the FR3 functional appliance and the invisible aligner are unifiedly mapped, ensuring the spatial consistency of the contact force and movement trajectory between the appliance and the teeth. This step provides stable basic data for subsequent path optimization and mechanical analysis, greatly improving the accuracy and efficiency of treatment.

[0092] S3. In the reference system, establish a dynamic displacement model of the teeth under the action of force, which is used to describe the movement state of the teeth under the combined action of the FR3 and the invisible aligner;

[0093] In step S3, based on the work results of the aforementioned steps S1 and S2, further establish a dynamic displacement model of the teeth under the action of force. The core of this step is to describe the dynamic movement state of the teeth under the combined action of the FR3 functional appliance and the invisible aligner through a mechanical model, providing the necessary theoretical basis and calculation foundation for subsequent path optimization, the construction of energy functionals, and real-time feedback during the treatment process.

[0094] Specifically, the goal of step S3 is to establish a dynamic displacement model, which is used to describe the movement state of teeth under the combined action of FR3 functional appliance and invisible braces. This model is based on the principles of continuum mechanics and predicts the forces, deformations, and final displacements of teeth by solving partial differential equations related to mechanics.

[0095] In this embodiment, when establishing the dynamic displacement model, the displacement state u(x,t) of the teeth is first considered, which represents the displacement of the teeth at position x at time t. This dynamic displacement model is established based on the following kinetic equation:

[0096]

[0097] where: u(x,t) represents the displacement of the teeth at position x at time t, with the unit of millimeter (mm); ρ is the equivalent density of the teeth, with the unit of kilogram per cubic millimeter (kg / mm 3 ) is the acceleration of the teeth at time t, with the unit of millimeter per second squared (mm / s 2 ) represents the divergence of the stress tensor, reflecting the stress distributed inside the teeth; F FR3 (x,t) is the spatial orthodontic force exerted on the teeth by the FR3 functional appliance at time t, with the unit of Newton (N); F A (x,t) is the spatial orthodontic force exerted on the teeth by the invisible braces at time t, with the unit of Newton (N).

[0098] This equation describes the dynamic behavior of the teeth under the action of two main external force fields.

[0099] F FR3 (x,t) and F A (x,t) are the orthodontic forces exerted by the FR3 functional appliance and the invisible braces respectively, and σ(x,t) is the stress tensor generated by the force on the teeth, representing the internal force distributed inside the teeth and reflecting the deformation of the teeth under the external force.

[0100] As an option, in practical applications, the stress tensor σ(x,t) can be described by the elastic properties of the material, specifically given by the following formula:

[0101] σ(x,t) = C(x,t):∈(x,t);

[0102] where: C(x,t) is the stiffness matrix of the material, representing the rigidity of the tooth material at time t, with the unit of Newton per square millimeter (N / mm 2 );∈(x,t) is the strain tensor, describing the deformation of the teeth under the external force.

[0103] The strain tensor ∈(x,t) is calculated from the displacement gradient of the tooth and the specific formula is as follows:

[0104]

[0105] where is the gradient of the tooth displacement, representing the rate of change of the tooth displacement caused by the external force at the spatial position x; is the transpose of the displacement gradient, representing the deformation of the tooth in different directions.

[0106] In some embodiments, the stress tensor σ(x,t) in the dynamic displacement model is defined by the elastic properties of the material. The material stiffness matrix C(x,t) of the tooth needs to be adjusted individually according to the patient's tooth type, bone density, and other anatomical structures.

[0107] In addition, according to the force application points and paths mapped in the previous step S2, in the dynamic displacement model, F FR3 (x,t) is the set of force application points of the FR3 functional appliance, including multiple appliance force application points. The mechanical properties of each force application point, such as the magnitude, direction, and distribution of the applied force, are calculated based on the design parameters of the FR3 functional appliance. F A (x,t) is the force applied by the invisible aligner, usually a continuous force application model. This model controls the displacement amount of each orthodontic stage by gradually adjusting the small movement of the tooth. The magnitude of the force applied by the invisible aligner is related to the path planning, and usually the least squares method or other optimization algorithms are used to solve the force at each stage of the path.

[0108] In a possible implementation, the force application field and the stress field are combined to form a comprehensive mechanical field, comprehensively considering the force application and the deformation response of the tooth. To ensure the accuracy of the model, for each force application point, the following energy function can be used to describe it:

[0109]

[0110] where: F i (x,t) is the applied force at the i-th force application point, with the unit of Newton (N); Δu i (x,t) is the displacement amount at the force application point i, with the unit of millimeter (mm); T is the total time of the orthodontic process, with the unit of day (days), and n is the number of tooth individuals.

[0111] By optimizing this energy function, the force application paths of the appliance and the invisible aligner can be accurately adjusted to ensure the efficiency and accuracy of the orthodontic process.

[0112] In some embodiments, to further improve the accuracy of the model, an error feedback mechanism can be introduced. When there is an error between the actual tooth displacement and the preset trajectory, the dynamic displacement model will automatically adjust the orthodontic path for error correction. By calculating the error function:

[0113]

[0114] where: p i is the actual position of the tooth, q i is the target position, ∥·∥ represents the Euclidean distance in millimeters (mm), and n is the number of tooth individuals.

[0115] By real-time monitoring the deviation between the tooth and the orthodontic path, the model can dynamically adjust the force field distribution at each treatment stage to ensure the smooth progress of the orthodontic process.

[0116] Through the establishment of this dynamic displacement model, step S3 not only accurately describes the movement process of the tooth under the combined action of the FR3 functional appliance and the invisible braces, but also can dynamically adjust the treatment at different stages to ensure that the orthodontic force and path at each stage can achieve the expected effect. This provides a solid foundation for subsequent path optimization, mechanical simulation, and correction and feedback during the treatment process.

[0117] S4. Construct a system energy functional for describing the overall movement of the tooth, and the energy functional includes energy components representing tooth deformation energy, periodontal tissue response energy, and force application deviation terms;

[0118] In step S4, the energy evolution of the overall movement of the tooth is further described by constructing a system energy functional. This energy functional not only considers the energy of tooth deformation, but also introduces the influence of periodontal tissue response energy and force application deviation terms to ensure precise control of mechanical behavior during the orthodontic process. The goal of constructing this energy functional is to determine the optimal movement path of the tooth by minimizing the total energy, while ensuring that the efficiency and effect of tooth movement are maximized under the combined action of the FR3 functional appliance and the invisible braces.

[0119] In this embodiment, the energy functional is constructed as an integral with respect to the tooth displacement function u(x,t), where the displacement function u(x,t) represents the displacement of the tooth at position x at time t. The construction of this energy functional takes into account multiple factors, including tooth deformation energy, periodontal tissue response energy, and force application deviation terms. The specific formula is as follows:

[0120]

[0121] where, represents the tooth displacement vector function at position x at time t; represents the spatial gradient of tooth displacement; k is the equivalent elastic coefficient of the periodontal ligament; λ is the force application deviation penalty coefficient; represents the actual resultant force vector field under the combined action of the FR3 functional appliance and the invisible braces; is the target control orthodontic force vector field; Ω is the three-dimensional space region where the dentition is located; dx represents the volume integral over the region Ω.

[0122] This energy functional consists of the following main components:

[0123] Tooth deformation energy:

[0124] The first term reflects the deformation energy of the tooth under the action of external forces, specifically the sum of the squares of the displacement gradients of the tooth under the action of forces. This energy is closely related to the elastic properties of the tooth material and represents the degree of deformation of the tooth due to the applied forces.

[0125] represents the displacement gradient of the tooth, which describes the deformation of the tooth in space under the action of external forces.

[0126] This term of energy reflects how the material of the tooth responds to the externally applied forces during the force application process and provides an energy measure for the shape change of the tooth.

[0127] Periodontal tissue response energy:

[0128] The second term reflects the response energy of the periodontal ligament.

[0129] The elastic coefficient k of the periodontal ligament is an important factor affecting tooth movement, which represents the resistance of the periodontal ligament to tooth displacement. This term of energy is calculated through the displacement u(x,t), showing the deformation of the tooth supported by the periodontal ligament, and affecting the movement of the tooth under the applied force and the burden on the alveolar bone.

[0130] k is usually obtained from biomechanical experiments or clinical measurements, and its value depends on the individual's periodontal health status and the elastic properties of the teeth.

[0131] This term of energy mainly considers the influence of tooth displacement on the surrounding tissues, especially the response of the periodontal ligament.

[0132] Force application deviation term:

[0133] The third term λ|F cxt (x,t)-F target (x,t)| 2 reflects the deviation between the applied force and the target force.

[0134] During the orthodontic treatment process, the forces exerted by the FR3 functional appliance and the invisible braces are not always perfectly matched with the target force. Therefore, an energy term is needed to penalize this deviation. This energy term ensures the coordination of the forces exerted by the appliance and the invisible braces, and promotes the gradual approach of the forces during the treatment process to the target force in the treatment plan.

[0135] F cxt (x, t) represents the external force applied to the teeth at time t, including the forces exerted by the FR3 functional appliance and the invisible braces; F target (x, t) is the target applied force, representing the desired orthodontic force;

[0136] λ is the weight coefficient of this energy term, used to adjust the penalty degree of the force application deviation. A larger λ value will increase the penalty for the deviation, making the applied force closer to the predetermined target.

[0137] In some embodiments, through the optimization of the energy functional , the mechanical optimal solution in the tooth correction process can be achieved. In this optimization process, the goal is to minimize the total energy of the entire system to obtain the tooth displacement path that meets the treatment goals. Specifically, the optimization process obtains the optimal solution by solving the following variational problem:

[0138]

[0139] Through the variational method or other optimization algorithms (such as the gradient descent method), the tooth displacement function u(x, t) that minimizes can be obtained, that is, the optimal movement path that the teeth experience during the entire orthodontic treatment process. This process can ensure that in each orthodontic stage, the applied force and deformation of the teeth can meet the biomechanical constraints, and the deviation between the applied force and the target force is minimized.

[0140] In a possible implementation, the optimization process not only considers the minimization of the total energy, but may also incorporate a clinical feedback mechanism. In each treatment stage, the optimization process is corrected using the patient's feedback data (such as real-time measurement of the applied force or actual monitoring results of tooth displacement), thereby dynamically adjusting the orthodontic path.

[0141] Through the accurate modeling and optimization of the above energy functional, step S4 provides a solid theoretical basis for path optimization, mechanical simulation, and real-time adjustment in the entire treatment process. This energy functional ensures the coordinated action of the FR3 functional appliance and the invisible braces, maximizes the treatment effect, and also takes into account the biological limitations of the periodontal tissue.

[0142] S5. Based on the applied force vector and path feasibility, optimize the control of the tooth movement path. The optimization control is solved with the system energy functional as the goal, and a constraint term that meets the clinical biological conditions is set;

[0143] In step S5, the core objective of the optimization control is to precisely optimize the movement path of the teeth, such that this path not only satisfies the feasibility of the force application vector, but also minimizes the energy functional under biological and physical conditions. Through this optimization control, the force applied during the orthodontic treatment can ensure that the teeth move along the predetermined path, while taking into account the mechanical rationality and the effectiveness of the treatment effect. This step is an important link in specifically applying the system energy functional established in the aforementioned step S4 to the orthodontic treatment process.

[0144] In this embodiment, in the aforementioned step S4, we have constructed an energy functional that includes the tooth deformation energy, the periodontal tissue response energy, and the force application deviation term. In step S5, the objective of the optimization control is to obtain the optimal path of the tooth displacement function u(x,t) by solving the optimal solution of the system energy functional and the orthodontic force applied on this path satisfies the clinical biological conditions.

[0145] Specifically, the objective of the optimization control is to minimize the value of the energy functional to obtain the optimal movement path of the teeth. The mathematical form of this optimization problem can be expressed as:

[0146]

[0147] where: is the optimized minimum value of the energy functional; u(x,t) is the tooth displacement function, representing the movement path of the teeth during the entire treatment process.

[0148] To ensure that the optimization process meets the clinical requirements, this optimization control problem needs to satisfy the following several constraint conditions:

[0149] Initial condition: The initial state and position of the teeth at the start of the treatment, usually set as the tooth position before orthodontics.

[0150] Time boundary condition: Specifies the boundary restrictions within each time period during the orthodontic treatment, such as the maximum displacement in each stage, the maximum range of the applied force, etc.

[0151] Biological constraint condition: This part of the constraint ensures that the mechanics during the treatment process conforms to biological feasibility, such as the maximum range of tooth displacement, the intensity of the applied force, etc. Biological constraints are usually based on clinical experience and the individual differences of patients.

[0152] In some embodiments, the optimization problem can be realized by numerical solution methods, such as the variational method or the gradient descent method. The variational method can obtain the optimal solution by calculating the variational derivative of the energy functional, while the gradient descent method approaches the minimum energy point by gradually adjusting the parameters.

[0153] Furthermore, when solving the optimization problem, the contribution of each term included in the energy functional to the result will affect the process of path optimization. Specifically:

[0154] Tooth deformation energy: contributed by the deformation energy term It ensures that the displacement of the tooth under the orthodontic force is compatible with the elastic behavior of the tooth itself;

[0155] Periodontal tissue response energy: contributed by the square of the elastic coefficient k of the periodontal ligament and the displacement u(x, t). During the optimization process, the goal is to ensure the natural response of the periodontal ligament during tooth movement while preventing excessive stretching or compression;

[0156] Force application deviation term: by minimizing the deviation λ|F cxt (x, t) - F target (x, t)| 2 , it ensures that the force application distribution of the orthodontic appliance and the invisible aligner during treatment is as close as possible to the preset target.

[0157] By optimizing the combination of these energy terms, the rationality of tooth displacement and the mechanical balance can be ensured.

[0158] In a possible implementation, feedback on the actual force application during the treatment process needs to be considered during optimization. During each treatment stage, there may be a deviation between the force application of the orthodontic appliance and the invisible aligner and the target force application. Therefore, step S5 not only requires minimizing the energy functional but also needs to ensure the rationality of the force application and the dynamic adjustment of the path. In this case, the feedback mechanism plays a crucial role.

[0159] For this reason, step S5 also includes the following error correction links:

[0160] Real-time path feedback: When there is a significant deviation between the actual tooth movement trajectory and the target path, it is necessary to reconstruct the energy functional and return to step S5 to perform path optimization.

[0161] Error calculation and adjustment: The error calculation is based on the position difference of the key points of the tooth. For example, the incisal edge point of the central incisor, the cusp point of the canine, and the central fossa point of the first molar are usually selected as key points for error calculation. The error formula is as follows:

[0162]

[0163] where: p i and q i respectively represent the coordinates of the i-th key point on the actual path and the target path, and ∥·∥ represents the Euclidean distance norm, with the unit of millimeters (mm).

[0164] Error Threshold and Path Correction: If the error value exceeds the set threshold, it is necessary to reconstruct the energy functional to ensure that the applied force and the result after path correction can meet the new orthodontic goals.

[0165] In step S5, based on the optimal control of the energy functional, it can not only minimize the energy of the tooth movement path, but also ensure that the orthodontic force is within a reasonable range at each stage. By introducing biological constraints and feedback mechanisms, the optimal control process can precisely adjust the tooth orthodontic path to ensure the effectiveness and safety of the orthodontic process. This optimization process ultimately enables the teeth to achieve the expected displacement trajectory under the combined action of the FR3 functional appliance and the invisible aligner, achieving the optimal treatment effect.

[0166] S6. Based on the optimized tooth movement path, inversely determine the structural design parameters of the FR3 appliance and the phased control parameters of the invisible aligner;

[0167] In step S6, based on the optimized tooth movement path in the foregoing step S5, inversely deduce the structural design parameters of the FR3 appliance and the phased control parameters of the invisible aligner. The goal of this process is to convert the optimal control result into the specific design parameters of the actual appliance and aligner to ensure that the orthodontic force and path applied during the treatment can meet the biological and mechanical requirements. The inverse deduction process includes not only the physical design of the FR3 functional appliance but also the phased adjustment of the invisible aligner to precisely control the tooth movement.

[0168] In this embodiment, in the foregoing step S5, the optimal tooth movement path u(x, t) is obtained through optimal control. This path describes the expected movement of the teeth at each stage during the treatment. Next, based on this optimized path, it is necessary to inversely deduce the design parameters of the FR3 appliance and the invisible aligner so that they can precisely achieve this optimized path in actual treatment.

[0169] Specifically, the inverse deduction process of the optimized path includes the following main parts:

[0170] FR3 Functional Appliance Design Parameters:

[0171] Based on the tooth displacement data in the optimized path, it is necessary to deduce the specific structural parameters of the FR3 functional appliance. The deduction of these design parameters mainly depends on the following factors:

[0172] Vertical Height h FR3 : According to the tooth displacement and the magnitude of the orthodontic force, it is necessary to design the vertical height of the FR3 appliance to ensure that the applied force can effectively act on the target tooth group.

[0173] This parameter can be determined by calculating the relationship between the applied force and the target tooth displacement.

[0174] The specific formula is as follows:

[0175] F vertical = k vertical ·h FR3 ;

[0176] Wherein, F vertical is the vertical force applied to the teeth, with the unit of Newton (N); h FR3 is the vertical height of the FR3 appliance, with the unit of millimeter (mm); k vertical is the control coefficient of the vertical force, usually obtained through experiments or simulation calculations.

[0177] The thickness t of the resin retainer in the anterior tooth area rcsin : For the anterior tooth area that requires a large displacement amount, the thickness of the resin retainer of the FR3 appliance needs to be appropriately increased to provide sufficient orthodontic support.

[0178] The selection of the thickness is usually combined with mechanical analysis to ensure that the resin retainer can provide sufficient supporting force without generating excessive displacement. The thickness can be deduced through the following relationship:

[0179] t resin = k resin ·Δu fronttecth ;

[0180] Wherein, t resin is the thickness of the resin retainer, with the unit of millimeter; k rcsin is the adjustment coefficient of the resin retainer thickness, depending on the individual differences of the patient; Δu fronttecth is the displacement amount of the anterior tooth area in the optimization path.

[0181] The design parameter w of the mandibular positioning acid wax wax : The orthodontic design of the mandibular teeth involves the thickness and shape of the bite wax. The purpose of the bite wax design is to correctly position the mandibular teeth and effectively apply the predetermined orthodontic force. The bite wax design is usually derived based on the displacement requirements of the mandibular teeth. The bite wax thickness w wax can be expressed by the following formula:

[0182] w wax = k wax ·Δu lowerjaw ;

[0183] Wherein, w wax is the thickness of the mandibular bite wax, with the unit of millimeter (mm); k wax is the coefficient of the bite wax design, depending on the treatment design; Δu lowerjaw is the displacement amount of the mandibular teeth.

[0184] Phased control parameters of invisible braces:

[0185] The design of invisible braces is divided into multiple stages according to the tooth displacement data in the optimization path. The control parameters for each stage need to be designed based on the tooth displacement amount and the orthodontic requirements. Specifically, the design parameters of invisible braces include:

[0186] Number of stages N stagcs : The number of stages of invisible braces is determined based on the treatment goal and time requirements. Usually, the number of stages is determined by the progress of the treatment goal.

[0187] This number is determined by the displacement vector allocation in the optimization process, so that each stage can correct the teeth at an appropriate speed and accuracy. The calculation formula is:

[0188]

[0189] where, T total is the total time of the entire treatment cycle, in days; T pcrstage is the time length of each stage, in days.

[0190] Displacement vector u of each stage i (x,t): The displacement vector of each stage is divided by the total displacement function obtained from the aforementioned optimization. The displacement vector of each stage represents the movement weight of the teeth within that stage. Specifically, the stage displacement vector is calculated by the following formula:

[0191]

[0192] where, u i (x,t) is the displacement vector of the i-th stage; u(x,t) is the total tooth displacement function obtained after optimization; N stages is the number of stages of invisible braces.

[0193] Tooth replacement cycle T cycle : The tooth replacement cycle of each stage determines the duration of each stage of invisible braces. The tooth replacement cycle is usually set according to the movement speed of the patient's teeth and the expected treatment effect. The setting of the tooth replacement cycle ensures that the displacement within each stage can be carried out step by step to ensure the accuracy of treatment. This cycle can be calculated by the following formula:

[0194]

[0195] where, T cycle is the tooth replacement cycle of each stage, in days; T total is the total time of the entire treatment cycle, in days; N stagcs is the number of stages of invisible braces.

[0196] In some embodiments, these design parameters are matched with the patient's oral structure, treatment goals, and feedback from the treatment stage. Each parameter is derived through numerical simulation or mechanical models and dynamically adjusted in combination with clinical trials and the actual situation of the patient. The inverse derivation process can also be further optimized based on the patient's feedback during the treatment process to ensure the maximization of the orthodontic effect.

[0197] The inverse derivation process in step S6 converts the aforementioned optimization path and displacement function into the structural design parameters of the FR3 functional appliance and the phased control parameters of the invisible braces. Through precise design parameter calculation, it is ensured that each orthodontic stage can effectively apply orthodontic forces along the optimized path and achieve the treatment goals. At the same time, the biological and physical limitations considered in the optimization process ensure the mechanical rationality and biological feasibility during the treatment process, thereby providing a personalized and precise orthodontic solution.

[0198] Please refer to Figure 2 , the present invention also provides an orthodontic system for correcting crossbite by combining the FR3 functional appliance and invisible braces, including:

[0199] A three-dimensional reconstruction module, which is mainly used to construct the three-dimensional spatial structure of the dentition and jaws based on oral scan data (such as CT or intraoral scan). By obtaining the three-dimensional surface point cloud data of the upper and lower jaws, the periodontal ligament boundary information, the grid data of the apical region, and the jaw bone morphology data, an accurate three-dimensional model is constructed. These data provide a basis for subsequent mechanical modeling, path optimization, and appliance design. Through fine three-dimensional modeling, it is ensured that the geometric features of each tooth and the surrounding bone structure are accurately reproduced, laying the foundation for the accuracy of the entire treatment process;

[0200] A force vector field extraction module, which is used to extract the information of the force application points of the FR3 functional appliance and the invisible braces, including the magnitude, direction, and action point of the force field applied by each orthodontic tool. By analyzing the relationship between the displacement of the teeth and the force application points, the spatial position and the direction of the acting force of each force application point are extracted and recorded. This information is crucial for subsequent mechanical modeling and path optimization and can accurately simulate how the appliance affects the movement of the teeth during the correction process;

[0201] A mechanical modeling module, which is responsible for constructing the tooth movement equation and the system energy functional to describe the dynamic behavior of the teeth under the action of orthodontic forces. Through a model based on continuum mechanics, considering the elasticity, displacement, and stress state of the teeth, the tooth movement equation is generated. The energy functional will include the deformation energy, the response energy of the periodontal ligament, and the force application deviation term. The purpose of this module is to accurately simulate the movement trajectory of the teeth at different stages and provide the mechanical data support required during the optimization process;

[0202] Trajectory optimization module, which performs path optimization based on the system energy functional and satisfies specific correction constraints by solving the optimal path. The optimization goal is usually to minimize the total energy to ensure that the movement path of the tooth in the process of applying correction force is in line with the biomechanical principles and can effectively achieve the expected correction goals. This module also needs to ensure the feasibility of the path and consider various constraints, such as displacement restrictions and reasonable distribution of force, to ensure that the optimization results have practical clinical application value;

[0203] Parameter output module, which reversely derives the design parameters of the FR3 functional appliance and invisible braces based on the optimal path obtained by the trajectory optimization module. These parameters include the structural characteristics of the appliance, such as vertical height, thickness of the resin support in the anterior teeth area, wax bite design, etc., and the staged control parameters of the invisible braces, such as the number of stages, displacement vectors in each stage, and tooth replacement cycle. Through reverse calculation, these design parameters will ensure that the correction tool can accurately implement the optimized path and achieve the predetermined treatment goals;

[0204] Implement a feedback module, which is used to periodically collect the actual movement trajectory of the teeth during the treatment process and compare the error with the optimized path. At regular intervals (for example, two weeks), the deviation between the actual path and the target path is calculated, usually evaluated by the position error of key points. If the error exceeds the set threshold, the system will automatically trigger the path correction, recalculate the optimal path and adjust the design parameters of the orthodontic appliance. This feedback mechanism ensures that the path and force can be adjusted in real time during the treatment process to adapt to the individual changes of the patient and optimize the treatment effect.

[0205] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. The orthodontic method for correcting crossbite by combining FR3 functional appliance with invisible braces, characterized in that, It includes the following steps: S1. Collect the three-dimensional oral data of the patient, construct a spatial model including the dentition and the jawbone to obtain a three-dimensional model; S2. Based on the three-dimensional model, obtain the force application point distribution of the FR3 functional appliance and the preset path of the invisible braces, and map the two to the same spatial reference system; S3. In the reference system, establish a dynamic displacement model under the action of tooth force, which is used to describe the movement state of the teeth under the combined action of the FR3 and the invisible braces; S4. Construct a system energy functional for describing the overall movement of the teeth, and the energy functional includes energy components representing tooth deformation energy, periodontal tissue response energy, and force application deviation terms; S5. On the basis of the force application vector and path feasibility, optimize and control the tooth movement path. The optimization control is solved with the system energy functional as the objective, and limit terms that meet the clinical biological conditions are set; S6. Based on the optimized tooth movement path, inversely determine the structural design parameters of the FR3 appliance and the phased control parameters of the invisible braces.

2. The orthodontic method for correcting crossbite by combining the FR3 functional appliance and invisible braces according to claim 1, wherein, The three-dimensional oral data includes the following structural components: Three-dimensional surface point cloud data of the maxillary and mandibular dentitions, which is used to construct a tooth shape model; Periodontal ligament boundary information and apical region grid data, which are used to establish a tooth-alveolar bone connection model; Jawbone morphological structure data, including voxel representations of the maxilla and mandible and bone density parameters; Occlusal relationship reference point set, which is used for coordinate matching and constraint correction of the subsequent force application vector and displacement function.

3. The orthodontic method for correcting crossbite by combining the FR3 functional appliance and invisible braces according to claim 1, characterized in that The dynamic displacement model is constructed using the principles of continuum mechanics and includes the following motion balance equations: Among them, \(u(x,t)\) represents the tooth displacement vector at position \(x\) at time \(t\); \(\rho\) is the equivalent density; \(\sigma(x,t)\) is the stress tensor; \(F\) FR3 (x,t) is the space orthodontic force applied by FR3; \(F\) A (x,t) is the control force field generated by the invisible dental aligner.

4. The orthodontic method for correcting crossbite by combining the FR3 functional appliance and invisible braces according to claim 1, characterized in that, The energy functional is an integral functional with respect to the displacement function and is expressed as: Among them, represents the tooth displacement vector function at position x at time t; represents the spatial gradient of tooth displacement; k is the equivalent elastic coefficient of the periodontal ligament; λ is the force application deviation penalty coefficient; represents the actual resultant force vector field under the combined action of the FR3 functional appliance and the invisible braces; is the target control orthodontic force vector field; Ω is the three-dimensional space region where the dentition is located; dx represents the volume integral over the region Ω.

5. The orthodontic method for correcting crossbite by combining the FR3 functional appliance and invisible braces according to claim 1, characterized in that, The optimization control takes the system energy functional as the objective function to solve the optimal path of the tooth displacement function. The path meets the initial orthodontic conditions and time boundary conditions and is obtained by minimizing the energy functional value.

6. The orthodontic method for correcting crossbite by combining the FR3 functional appliance and invisible braces according to claim 1, wherein The limiting conditions set in the optimization process of the tooth movement path include the following forms: The tooth displacement amplitude is less than a preset threshold; The occlusal contact area maintains geometric consistency during time evolution; The distance between the tooth midline position and the reference midline is lower than the set threshold; The limiting conditions are added in the form of constraint equations in the path optimization.

7. The orthodontic method for correcting crossbite by combining the FR3 functional appliance and invisible braces according to claim 1, characterized in that, The phased control parameters include the vertical height of the FR3 functional appliance, the resin rest thickness in the anterior tooth area, and the mandibular positioning bite wax design parameters, and also include the number of stages of the invisible braces, the displacement vector of each stage, and the tooth replacement cycle.

8. The orthodontic method for correcting crossbite by combining the FR3 functional appliance and invisible braces according to claim 1, characterized in that After the structural parameter design of the FR3 functional appliance is completed and put into actual use, obtain the actual movement trajectory of the teeth, and calculate the difference between it and the target tooth movement path obtained in step 5. When the difference exceeds the preset error threshold, reconstruct the system energy functional and return to step 5 to perform path optimization.

9. The orthodontic method for correcting crossbite by combining the FR3 functional appliance and invisible braces according to claim 8, characterized in that, The error between the actual movement trajectory and the target tooth movement path is calculated every two weeks, and the error value is measured by the Euclidean space distance of the key tooth positions. The key points include the incisal edge points of the central incisors, the cusp points of the canines, and the central fossa points of the first molars.

10. An orthodontic system for correcting crossbite by combining an FR3 functional appliance with an invisible aligner, which is applied to the method for correcting crossbite by combining an FR3 functional appliance with an invisible aligner according to any one of claims 1-9, characterized in that, It includes: A three-dimensional reconstruction module for constructing the spatial structure of the dentition and the jawbone; A force vector field extraction module for obtaining the force application information of the FR3 and the invisible braces; A mechanical modeling module for constructing tooth motion equations and system energy functionals; A trajectory optimization module for solving the optimal path that satisfies the constraint conditions; A parameter output module for inversely deriving the design parameters of each orthodontic appliance; An implementation feedback module for periodically collecting trajectories and performing error correction.

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