Evaluation of the Force System Acting Within an Orthodontic Appliance

1. Introduction

Implant-prosthetic rehabilitation of single-tooth edentulism is a functionally and aesthetically predictable solution, but its success depends essentially on a well-designed interdisciplinary (prosthetic and orthodontic) treatment plan. One of the determining factors for correct implant placement is the availability of adequate dimensions of the potential implant-prosthetic space—mesiodistally, vertically, and buccal-orally—so that the final restoration complies with the biomechanical, aesthetic, and biological principles of implant-supported prosthetic restoration [1,2].

In many clinical situations, early tooth loss is followed by migration of adjacent teeth, tooth tilting, changes in occlusal contacts, and a reduction in the space required for implant placement. In these cases, orthodontic treatment plays a fundamental role by realigning dental axes, converging or parallelizing the roots, and restoring the size of the edentulous space to a position favorable for prosthetic restoration [2,4]. Thus, orthodontics serves not only to “create space” but also to optimize the three-dimensional conditions for the implant, contributing to the correct positioning of the future prosthetic abutment and the final crown.

The importance of this stage is particularly heightened in the anterior regions, where aesthetic demands are high, and any discrepancy in position or volume can compromise the final result. The literature indicates that implants placed without adequate orthodontic preparation may be associated with difficulties in prosthetic emergence, gingival margin asymmetries, or compromises in the aesthetic contour of the restoration [1,2]. Conversely, through proper treatment sequencing, orthodontics facilitates implant placement in an area with sufficient space and favorable root relationships, reducing the risk of prosthetic and surgical complications.

Furthermore, in growing patients, the timing of implant placement must be carefully considered, as implants behave like ankylosed structures and do not follow the physiological growth of the jawbones. For this reason, in certain situations, orthodontic treatment also serves to maintain or reopen the space until skeletal maturity is reached, at which point implantation can be performed under conditions of long-term stability [3]. This stage is essential for preventing vertical discrepancies and for maintaining dentoalveolar harmony over time.

Consequently, the role of orthodontics in creating space for implants goes beyond simple dental alignment, integrating biological, functional, and aesthetic principles into an interdisciplinary treatment plan [4,5]. However, to successfully achieve these objectives, it is necessary to understand and precisely adhere to the system of orthodontic forces capable of moving the teeth to align and reshape the implant-prosthetic space in accordance with bone biology, without causing excessive bone resorption. Only in this way can the primary and secondary stability of the dental implant be guaranteed [6,7].

In this context, the aim of the present study is to evaluate the entire system of forces acting on metal bracket-type components in an orthodontic setup designed to reshape the potential implant-prosthetic space.

2. Materials and Methods

2.1. CBCT Imaging Data

This study was based on a set of CBCT images acquired at a specialized dental clinic on January 25, 2024, for a 25-year-old female patient presenting with maxillary and mandibular lateral edentulism. This study was approved by Decision No. 156/August 30, 2023, of the Ethics Committee and was conducted with the patient’s consent.

As part of the study, a geometric model of the tissues was created based on the set of CBCT (Cone Beam Computed Tomography) images. The data was processed using software that converts CBCT images into primary geometries for the constituent tissues using specific filters based on different shades of gray. The initial geometry, which consisted of a “point cloud,” was processed using techniques and methods specific to reverse engineering. Initially, this “point cloud” was transformed into a multitude of spatial triangular surfaces, which subsequently formed perfectly closed surfaces for the bone components (maxilla and mandible) and for the specific dentition. Using techniques specific to both Reverse Engineering and Direct Engineering, based on “offset” surfaces, models of the periodontal ligaments for each tooth were obtained.

The models were loaded, one by one, into a CAD (Computer-Aided Design) program, where they were aligned based on a common coordinate system and converted into virtual solids. Within the same software, the metal components for the orthodontic wires and bracket-type elements were modeled using techniques specific to Direct Engineering. These models were added to the models of the bone, tooth, and periodontal ligament components. Elastic deformations were measured on the orthodontic archwire models in the virtual environment. Based on specific equations from the field of Strength of Materials, the elastic forces were determined, followed by the reactions; finally, the entire system of forces acting on the bracket-type components was evaluated.

Figure 1 shows a screenshot from the CS 3D Imaging program for the set of tomographic images containing 658 CBCT images for the patient under study.

2.2. Orthodontic Materials

For the orthodontic assembly, we analyzed, from a dimensional perspective, Orthofocus-type orthodontic wires for the maxilla and mandible with a circular cross-section and a diameter of 0.012 inches (approx. 0.3 mm), as shown in Figure 2.

Similarly, the bracket-type elements shown in Figure 3 were studied.

2.3. Software Used

2.3.1. InVesalius

InVesalius is open-source research software used for the three-dimensional reconstruction of medical images obtained through computed tomography (CT) or magnetic resonance imaging (MRI). The software converts tomographic images into 3D models of the body’s anatomical tissues to better understand internal biological structures. The resulting 3D models are available in standard formats (such as STL, OBJ, or PLY) and can be used for virtual prototyping and 3D printing. The software is used for medical diagnosis, surgical planning, dentistry, research, education, veterinary medicine, and even in engineering for the virtual prototyping of medical devices. It was developed by the Renato Archer Information Technology Center (CTI) in Brazil (CTI, Campinas, Brazil) [i4].

2.3.2. Geomagic

Geomagic is a professional software program for reverse engineering, 3D scanning, and dimensional inspection, used in the medical, automotive, and aerospace industries. Geomagic enables the conversion of 3D scans—which initially consist of “point clouds”—into CAD models. The software enables the conversion of 3D scans into CAD models consisting of surfaces or solids, the reconstruction of a digital model from a physical part, the comparison of manufactured parts with the original CAD model, and the “cleaning and repair” of meshes such as STL, OBJ, etc. (3D Systems, Rock Hill, SC, USA) [i4], [t1].

2.3.3. SolidWorks

SolidWorks is a professional computer-aided design (CAD) program used to generate 3D models of components, assemblies, and technical documentation specific to mechanical engineering and industrial design. It also allows for virtual testing and simulations for strength analysis (Simulation module), kinematic analysis (Motion module), thermal analysis, vibration analysis, and fluid flow analysis (Flow module), as well as fatigue testing (Fatigue module) [i4], [t1].

This software uses methods specific to Direct Engineering and may contain specialized modules (Dassault Systèmes, Velizy-Villacoublay, France).

2.3.4. Microsoft Office Software Components

Microsoft Office is a suite of software modules developed by Microsoft, used for working with data, documents, presentations, and diagrams and charts.

The Microsoft Office suite was used to draft documents, organize, manage, and analyze data, as well as to implement formulas and perform repetitive calculations (Microsoft Corporation, Redmond, One Microsoft Way, 1, 98052, USA).

2.3.5. Artificial Intelligence (AI) Modules

These were used to explain terms, techniques, concepts, and methods, or to detail the advantages and disadvantages of certain software modules.

2.4. Hardware Systems

A Hewlett Packard Z2 workstation with the following technical specifications was used for 3D modeling:

- 13th-generation Intel® Core™ i9 processor;

- 64 GB DDR5-4800 RAM (4 x 16 GB);

- Graphics: NVIDIA RTX™ A4500 (20 GB dedicated GDDR6 memory), memory slots: 4 DIMM, internal storage: 1 TB HP Z Turbo Drive PCIe® NVMe™ TLC SSD;

- Graphics: Intel® UHD Graphics, memory and storage: 64 GB memory / 1 TB SSD storage;

- Processor: Intel® Core™ i9-13900K (up to 5.8 GHz, with Intel® Turbo Boost technology);

- 36 MB L3 cache, 24 cores, processor L3 cache: 36 MB, processor family:

- Operating system: Windows 11 Pro.

To generate charts and graphs and for computational data analysis, several desktop computers with the following technical specifications were used:

- Intel Core i3 processor with a clock speed of 3.7 GHz;

- 8 GB of RAM;

- 466 GB hard drive;

- 64-bit Windows 10 operating system.

For similar purposes, a Lenovo laptop was also used with the following technical specifications:

- Intel Core i5 processor with a clock speed of 2.9 GHz;

- 476 GB SSD;

- 930 GB hard drive;

- 16 GB RAM;

- 64-bit Windows 10 operating system.

2.5. Methods

2.5.1. Methods Specific to Medical Imaging

Medical imaging methods are investigative techniques that allow for the visualization of the body’s structures, tissues, and functions without (or with minimal) surgical intervention. They are very important for medical diagnosis, treatment guidance, and patient monitoring [8] [i4].

The main medical imaging methods are based on radiography (X-ray), computed tomography (CT or CBCT), magnetic resonance imaging (MRI), ultrasound, nuclear medicine, fluoroscopy, or angiography.

2.5.2. Methods of Elasticity Theory

The methods of Elasticity Theory constitute a set of mechanical and mathematical procedures used to analyze the behavior of bodies considered elastic (deformable solids) when subjected to a system of forces or other external actions [9,10,11].

The use of these methods leads to the determination of strains, stresses, and displacements in a deformable solid [12,13,14].

The main methods of the Theory of Elasticity are: the analytical (exact) method, the energy method (energetic methods), the approximate method, the numerical method, and the experimental method [15,16,17].

2.5.3. Methods of Strength of Materials

The methods of strength of materials consist of a set of computational procedures, principles, and assumptions used to analyze the behavior of deformable solid bodies (plates, beams, bars, shafts) when they are subjected to a system of forces and moments [18,19].

By applying these methods, it is possible to determine the deformations, displacements, and stresses in the deformable solid under analysis. Furthermore, the conditions for strength, stability, and stiffness can be established [20,21,22]

These methods include: the section method, the method of static equilibrium, the allowable stress method, the deformation method, the deformation energy method, the superposition of effects method, and (modern) numerical methods [23,24,25].

2.5.4. Reverse Engineering Methods

Reverse engineering methods consist of a set of processes and techniques used to study an existing product, system, or software in order to understand its structure, operation, and design principles without access to the original documentation.

This study will utilize methods specific to mechanical engineering, which may be based on 3D scanning, manual measurements, CAD reconstruction, materials analysis, and physical disassembly. Analytical and functional methods may also be used, such as functional analysis, mathematical modeling, numerical simulation, benchmarking, or AI-based reverse engineering.

2.5.5. Direct Engineering Methods

Direct Engineering Methods (also known as Forward Engineering) refer to the set of techniques and methods used to design and build a system, starting from requirements or design briefs and gradually moving toward actual implementation. In practice, the process begins with an idea, leads to a model, and then to a final product.

Forward engineering is a system development process that begins with requirements and specifications, proceeds to the creation of conceptual and design models, and culminates in implementation, testing, and operation. It is the opposite of reverse engineering, which starts with an existing system.

In this study, these methods were used to model orthodontic wires and bracket-type elements.

2.6. Three-Dimensional Model of an Orthodontic System

2.6.1. Three-Dimensional Models of the Bony Components (Maxilla and Mandible)

The patient’s CBCT scans were imported into the InVesalius program. Figure 4 shows the program interface after importing the set of scans.

To obtain a primary “point cloud” geometry, the dental enamel (adult) filter was used, as shown in Figure 5.

Because the shades of gray in enamel-type dental tissues are similar to those of bone components, the model was imported into the Geomagic software for editing and processing. Figure 6 shows the Geomagic interface after importing the model from InVesalius as a “point cloud.” In the first phase, this software automatically transforms the “point cloud” into spatial triangular surfaces. Initially, this model contained 5,093,918 primary triangular surfaces.

This model also contains elements that are part of the dentition due to the similar densities of the tooth enamel. These components were manually removed using specific techniques. Figure 7 shows the steps involved in these removal operations for the anterior region.

Next, the left lateral dentition was removed; some of these steps are shown in Figure 8.

Next, various reverse engineering techniques were used to process the model; certain gaps were filled; non-conforming surfaces were identified and removed; finishing techniques were applied; and, finally, the number of elementary triangular surfaces was reduced to 24,198. Figure 9 illustrates some of these operations.

Finally, the models were imported into SolidWorks, where they were converted into virtual solids, as shown in Figure 10.

2.6.2. Three-Dimensional Models of the Dentition

To create the tooth models, the dental enamel filter in the InVesalius program was used, and the initial model shown in Figure 11 was generated.

Initially, the model contained 2,795,770 triangular element surfaces in Geomagic, as shown in Figure 12.

In the first step, the areas corresponding to the bone components were removed, as shown in Figure 13.

The steps for removing bone components continued, and Figure 14 shows some of them.

After applying several techniques for finishing, filling gaps, reducing the number of triangular faces, and eliminating non-conforming faces, the model shown in Figure 15 was obtained, which had 100,110 element faces.

Finally, the dental model was imported into SolidWorks, where it was automatically converted into virtual solids, as shown in Figure 16.

2.6.3. Three-Dimensional Models of Dental Ligaments

To generate the models of the dental ligaments, we started with the dentition model and used Offset surfaces applied to the entire model in Geomagic. Subsequently, the surfaces corresponding to the dental crown areas were removed, then filled using Fill techniques to obtain perfectly closed surfaces; finally, the models were imported into SolidWorks, where they were converted into virtual solids, as shown in Figure 17. The cavities corresponding to the contact surfaces with the dentition were subsequently created using CAD techniques.

2.6.4. Three-Dimensional Model of the Dento-Maxillary Apparatus of the Patient Under Study

Next, the models of the dentition and the dental ligaments were loaded into the Assembly module of SolidWorks, as shown in Figure 18.

The models are derived from the same set of CBCT images and, for this reason, have identical coordinate systems. Figure 19 shows the models and the six reference planes, three for each model.

To ensure that the models are in the correct position, the corresponding reference planes were aligned. Figure 20 shows the three alignment operations based on the coincidence of the planes.

The result of the plane alignment operation is shown in Figure 21.

In the context of the assembly, a volume subtraction operation was performed between the ligament and the teeth, resulting in the dental ligament cavities, as shown in Figure 22.

Next, in the SolidWorks Assembly module, the model of the two bone components was loaded. Similar operations were performed to align the reference planes for the models, as shown in Figure 23.

Next, two volume subtractions were performed between the bone components and the dentition and dental ligaments, resulting in the dental alveoli within the bone components, as shown in Figure 24.

2.6.5. Three-Dimensional Models of the Bracket-Type Components and the Orthodontic Adhesive

Next, the bracket-type components were modeled using CAD and direct engineering techniques. The following section details the steps involved in generating a model of a bracket for the mandibular incisors. The sketch shown in Figure 25 was drawn on one of the initial reference planes.

Since this is a symmetrical component, we chose to model an extruded solid with a midplane, as shown in Figure 26.

The sketch shown in Figure 27 was drawn in a reference plane perpendicular to the original one.

Using this sketch, the basic shape was cut out from the outside (Flip side to cut – enabled), as shown in Figure 28.

In the initial planes sketch from Figure 29 was drawn.

This sketch was used to add an extrusive plane using Mid Plane, as shown in Figure 30.

In the same plane sketch from Figure 31 was drawn.

Using the sketch, a cut was created using the Mid Plane option, as shown in Figure 32.

Next, a reference plane parallel to the initial one was defined, as shown in Figure 33.

In this plane sketches from Figure 34 were drawn.

With this sketches cutouts from Figure 35 were obtained.

Next, additional fillet shape were generated, as shown in Figure 36.

Next, a new reference plane was defined at a distance from the base of the bracket – type component, as shown in Figure 37.

Next, the sketch shown in Figure 38 was defined in this plane.

This sketch was used to create a new extruded shape, as shown in Figure 39.

This extruded shape was duplicated using a Linear Pattern command, and the resulting model is shown in Figure 40.

This shape was cut out using a surface located at a distance from the base of the bracket-type component, and the result is shown in Figure 41.

Using an outline taken from the sole of the pattern, the pattern was cut out from the outside, as shown in Figure 42.

To generate the orthodontic wire model, three points were drawn on the bracket component model (Figure 43).

Models of all bracket-type components were created using similar techniques (Figure 44). Mirroring techniques were also used for the symmetrical components.

To model the orthodontic adhesive, the contour of the bracket-type component’s base was copied and then extruded, as shown in Figure 45.

The bracket-type components were positioned on the corresponding teeth; Figure 46 shows, for example, their placement on a canine.

All bracket-type components were positioned in a similar manner with the adhesive, resulting in the model shown in Figure 47.

By applying a volume reduction between the adhesive models and each individual tooth, the final adhesive models were obtained.

2.6.6. Three-Dimensional Models of Orthodontic Wires

Next, using the points corresponding to the bracket-type components, a spline curve was drawn for the maxilla, as shown in Figure 48.

A circle with a diameter of 0.012 inches (approx. 0.3 mm) was drawn in a plane perpendicular to this curve, as shown in Figure 49.

Next, using a Sweep feature, the virtual solid modeling the orthodontic wire for the maxilla was defined (Figure 50).

Figure 51 shows the model of the orthodontic wire for the maxilla.

Similarly, the orthodontic wire for the mandible was also modelled (Figure 52).

2.7. Application of Techniques and Methods Specific to the Theory of Elasticity and Strength of Materials in Evaluating the Force System in the Orthodontic System Under Study

Introductory Theoretical Concepts

From the Strength of Materials and the Theory of Elasticity, it is known that a bar with a constant cross-section, supported at both ends and subjected to a force F, undergoes an elastic deformation s. This deformation is called deflection (Figure 53-a). Figure 53-b shows the equivalent model based on reactions.

An orthodontic archwire can be considered a bar with a constant cross-section. Furthermore, points A, B, and C can correspond to bracket-type components. We can thus map the model in Figure 2.52 onto the bracket-type elements on the orthodontic archwire. Now, the problem is to determine the elastic forces F and the reactions V_A and V_B.

It is also known that:

there is a functional relationship between the force F and the deflection s. This is:

(1)

where

F – elastic force;

k – elastic constant;

s – elastic deformation (deflection).

From the Principles of Strength of Materials and the Theory of Elasticity, the elastic force F can be calculated as follows:

$$F={\displaystyle \frac{9\sqrt{3}·L·E·I·s}{a·{(L^2-a^2)}^{3/2}}}$$

(2)

where: s is the deflection of the bar;

F is the elastic force;

L is the length of the bar;

E is Young’s modulus (the modulus of elasticity of the orthodontic archwire);

I is the axial moment of inertia, and, in the case of a bar with a circular cross-section, it is:

$$I={\displaystyle \frac{\pi ·d^4}{64}}$$

(3)

where d is the diameter of the bar (in this case, the diameter of the orthodontic archwire).

From equations (1) and (2), the elastic constant k can be determined as follows:

$$k={\displaystyle \frac{9\sqrt{3}·L·E·I}{a·{(L^2-a^2)}^{3/2}}}$$

(4)

If the elastic force F is known, the reactions V_A and V_B can be estimated as follows:

$$V_{A=}{\displaystyle \frac{F\bullet b}{L}}$$

(5)

$$V_{B=}{\displaystyle \frac{F\bullet a}{L}}$$

(6)

Table 1 shows how mathematical models are applied to the jaw.

Similarly, the mathematical models were adapted for the mandibular dentition, as shown in Table 2.

where: M_1dr – right primary molar; M_2dr – right secondary molar; P_1dr – right primary premolar; P_2dr – right secondary premolar; C_dr – right canine; I_2dr – right secondary incisor; I_1dr – right primary incisor; M1st – left primary molar; M2st – left second molar; P1st – left first premolar; P2st – left second premolar; C_st – left canine; I2st – left second incisor; I1st – left first incisor.

To determine the lengths L, a, b, and s, points of type A and B were connected by line segments; then, from points of type C, perpendiculars were drawn to the lines of type AB to highlight the arrows of type s, as illustrated for the orthodontic wire on the maxilla in Figure 54. Point A corresponds to M_1dr, B corresponds to C_dr, and point C corresponds to I_2dr.

For the maxillary orthodontic wire the lines are shown in Figure 55.

For the mandibular orthodontic wire these lines are shown in Figure 56.

Figure 57 illustrates how to measure the L-type length, that is, segment AB (M_1dr - I_2dr).

Figure 58 illustrates how to measure a length of type a, that is, the distance from point A (M_1dr) to the foot of the perpendicular drawn from point C (C_dr).

To determine distances of type s, the length of the perpendicular drawn from point C (C_dr) to line AB (M_1dr - I_2dr) was measured, as shown in Figure 59.

b-type distances are calculated using the formula:

b=La

3. Results

3.1. Determination of the L, a, and S Distances by Measurement in the Virtual Environment on the Orthodontic Wire Curve

For the maxilla, the distances measured on the orthodontic wire curve are shown in Table 3. and for the mandible, they are shown in Table 4.

3.2. Determination of the Elastic and Reaction Forces Acting on Bracket-Type Components

To apply the mathematical formulas (2) through (6), it was necessary to also consider the values of d and E.

The diameter of the spring is 0.012 inches, or 0.3048 mm, so:

d = 0.0003048 m

The spring is made of a nickel-titanium alloy, known as nitinol. For this material, the modulus of elasticity is:

E = 34,500,000,000 Pa

By applying the mathematical formulas that were incorporated into Microsoft Excel, the specific forces of the maxilla presented in Table 5 and those of the mandible in Table 6 were determined.

These values are summarized in the graphs in Figure 60 and Figure 61.

4. Discussion

This study aimed to evaluate the system of forces acting within an orthodontic appliance through an integrated engineering and medical approach, based on three-dimensional modeling, virtual measurements, and the application of classical methods from the Theory of Elasticity and Strength of Materials. The results obtained are consistent with the general direction described in the specialized literature, according to which orthodontic biomechanics must be analyzed by correlating the geometry of the appliance, the material properties, and the biological response of the dento-periodontal structures [26,27].

An essential element of the study is the construction of the three-dimensional geometric model, derived from CBCT images of a real patient.

In this regard, the results are consistent with the studies by Kapila et al., as well as those by Scarfe and Farman, which show that CBCT offers major advantages in orthodontics by providing precise three-dimensional information about dento-skeletal relationships [28,29]. However, unlike these studies, which focus primarily on diagnosis and planning, the present study extends the use of CBCT to a personalized mechanical analysis, in which imaging data are transformed into a numerical model capable of estimating force distribution.

The integration of bone, dental, and periodontal ligament models into a common coordinate system ensured the spatial consistency of the assembly and enabled the precise positioning of orthodontic components modeled using direct engineering. This step is critical from an engineering standpoint, as any misalignment would lead to erroneous values for the deformations and, consequently, for the calculated forces. Compared to the idealized models used in classical analysis of orthodontic systems, the present approach has the advantage of preserving the anatomical specificity of the case, which increases the clinical relevance of the results [30,31].

Modeling orthodontic wires using spline-type trajectories, defined based on the actual position of the brackets, represents a significant advantage of this study. This choice is consistent with the ideas formulated by Burstone, who demonstrated the importance of geometric control of the wire and the positioning of force application points in achieving predictable biomechanical systems [32]. The main difference from classical approaches lies in the fact that here the wire curves are not merely theoretically assumed but reconstructed from the actual position of the brackets, which reduces reliance on idealized assumptions and makes the analysis more closely aligned with the clinical situation.

From a mechanical standpoint, orthodontic wires have been considered elastic elements with a constant cross-section, and the material’s behavior has been approximated as linearly elastic. This assumption is useful for comparisons and for basic mechanical interpretation; however, the literature shows that modern Ni-Ti wires exhibit superelastic behavior and a significant dependence on activation and stress history [33]. Miura et al.[34] demonstrated as early as in classical studies that the Ni-Ti alloy possesses clinically relevant superelastic properties, and Kusy showed that the properties of orthodontic wires vary significantly depending on composition, diameter, and conditions of use [35].

Therefore, the current model should be interpreted as a valid engineering approximation for comparative analysis, not as a complete description of the material’s actual behavior. Analysis of the results reveals a non-uniform distribution of forces along the orthodontic wires, in both the maxilla and the mandible. This result is consistent with the principles described by Burstone [32], according to which even small geometric changes can significantly alter the distribution of moments and forces. Furthermore, studies in orthodontic biomechanics emphasize that the efficiency of tooth movement depends not only on the magnitude of the force but also on its direction, duration, and point of application [36,37]. From this perspective, the non-uniform distribution observed in the present model confirms the sensitivity of the orthodontic system to the geometric configuration of the assembly.

It is important to note that the developed model does not include effects such as spring–bracket friction, nonlinear contact, the influence of the viscoelastic properties of periodontal ligaments, or biological adaptation over time. In the literature, these elements are recognized as important factors that can significantly alter the system’s actual response [38,39]. However, their simplification in the present study is justified by the need to obtain a clear, controllable, and interpretable model capable of highlighting the direct influence of geometry on the forces developed.

From a clinical perspective, the results are consistent with the classic observations of Reitan and Storey, who demonstrated that effective tooth movement is achieved through moderate, well-controlled forces, and that overloading can have undesirable biological effects [40,41]. In this regard, the present study provides a useful framework for understanding how appliance geometry influences mechanical loading and, consequently, for optimizing therapeutic design.

Through the complexity of the geometric model and the integration of classical engineering methods into a three-dimensional CAD environment, the study demonstrates the feasibility of a rigorous interdisciplinary approach capable of providing a solid foundation for further numerical or experimental developments. Compared to existing studies, the main contribution lies in the shift from a general description of orthodontic biomechanics to a customized analysis based on the patient’s actual geometry and direct virtual measurements.

5. Conclusions

Based on the research conducted in this study, the following main conclusions can be drawn.

A detailed three-dimensional model of the dento-maxillary system and the orthodontic appliance was developed, based on real CBCT imaging data and advanced methods of inverse and direct engineering.

The proposed methodology allows for a direct correlation between the actual geometry of orthodontic wires and the distribution of elastic forces, eliminating some of the geometric simplifications used in traditional approaches.

The application of methods from the Theory of Elasticity and Strength of Materials led to the quantitative determination of the forces and reactions acting on bracket-type components, highlighting the non-uniform nature of the mechanical stresses.

The results confirm that the distribution of forces in an orthodontic system is strongly dependent on the geometry of the archwire and the positioning of the components, an aspect of major engineering significance.

The study demonstrates the feasibility of using a rigorously engineered framework for the mechanical analysis of orthodontic systems, providing a solid theoretical foundation for extending the analysis to advanced numerical models.

Future research directions may include incorporating the nonlinear behavior of materials, analysis using other virtual methods, consideration of friction and archwire–bracket contact, as well as experimental validation of the obtained results.

In conclusion, this study contributes to strengthening engineering approaches in the field of orthodontics, demonstrating that the integration of three-dimensional modeling with classical mechanics methods is an effective tool for the analysis and optimization of orthodontic systems.

The study demonstrates that the integration of CBCT data with three-dimensional modeling and classical methods from the Theory of Elasticity and Strength of Materials allows for a rigorous evaluation of the force system developed within an orthodontic setup. The geometric model obtained through reverse engineering provided an accurate representation of the dento-maxillary structures and orthodontic components, ensuring the necessary conditions for a relevant mechanical analysis.

The results revealed an uneven distribution of forces along the orthodontic wires, determined by the geometric characteristics of the assembly and the positioning of the support points. Despite the simplifications made, the proposed methodology provides a solid foundation for understanding the relationship between the appliance’s geometry and the resulting mechanical stresses.

Therefore, the interdisciplinary approach used confirms the usefulness of three-dimensional CAD modeling in orthodontic biomechanical analysis and supports the development of further methods for numerical and experimental evaluation, with potential applications in optimizing individualized treatment.