Three-Dimensional Finite Element Analysis of Forces Produced by Different Orthodontic Archwires

Introduction

Orthodontic tooth movement is a complex biological process initiated by mechanical forces that induce remodeling of the periodontal ligament (PDL) and alveolar bone. The efficacy of these forces depends on their magnitude, direction, and duration, which must be optimized to prevent tissue damage while ensuring efficient tooth displacement [1]. Nickel-titanium (NiTi) arch wires are widely used in orthodontics due to their unique properties, such as super elasticity and shape memory, which enable the delivery of continuous, light forces ideal for the initial alignment phase [1-5]. However, the biomechanical behavior of NiTi wires, particularly the differences between austenitic (A-NiTi) and martensitic (M-NiTi) phases, and their effects on the periodontium remain incompletely understood [6].

Finite element analysis (FEA) has emerged as a powerful tool to simulate orthodontic force systems and their biological impacts. Unlike traditional experimental methods, FEA allows for precise quantification of stress-strain distributions in the PDL and surrounding structures under controlled conditions [7-11]. Previous studies have highlighted the influence of wire composition and geometry on force delivery, but few have compared the biomechanical effects of A-NiTi and M-NiTi arch wires during multi-directional tooth movement [12, 13].

This study employs a three-dimensional finite element model to evaluate the forces generated by 0.016-inch A-NiTi and M-NiTi arch wires during displacement in six directions (intrusion, extrusion, inward, outward, rotation, and angulation). Additionally, it analyzes the resulting stress-strain patterns in the PDL and deformation of the tooth and alveolar bone. By simulating clinical scenarios, this research aims to provide insights into wire selection and optimizing force systems for efficient and biologically safe orthodontic treatment. The findings could enhance clinical decision-making, particularly in cases requiring complex tooth movements.

Materials and Methods

This study employed a three-dimensional finite element method (FEM) to quantitatively evaluate and compare the biomechanical behavior of two commercially available 0.016-inch nickel-titanium (NiTi) orthodontic arch wires, Austenitic NiTi (A-NiTi) and Martensitic NiTi (M-NiTi), under simulated clinical displacement conditions. The core objectives were to measure the forces generated by each wire type, analyze the resultant stress-strain distribution in the periodontal ligament (PDL), and assess the deformation patterns of the tooth and supporting alveolar bone [14-19].

Data acquisition and geometric modeling

A cone-beam computed tomography (CBCT) scan of the mandibular anterior region, inclusive of the central incisor and surrounding dentoalveolar structures, was acquired. Ethical approval was secured prior to patient data usage. The digital imaging data in DICOM format were imported into MIMICS (version 14.0, Materialize NV, Belgium), a medical image processing software, to perform segmentation and construct three-dimensional surface models of the tooth, PDL, cortical bone, and cancellous bone.

The segmented anatomical structures were exported in stereolithographic (STL) format and further processed in CATIA V5 R19 (Dassault Systèmes, France), a CAD software environment. Surface irregularities were rectified through model healing and meshing refinement to enhance anatomical accuracy. The bracket and arch wire assemblies were modeled independently using CAD tools and later integrated with the tooth geometry. A standard slot bracket was scanned using a blue light scanner and reproduced in the model. The arch wire was designed to span three teeth, engaged between the brackets of two lower central incisors, to replicate typical clinical alignment scenarios.

Material properties and meshing

All components in the finite element model were assigned linear elastic, isotropic material properties based on values reported in the literature. The Young’s modulus and Poisson’s ratio for each component were defined as follows: enamel (84 GPa, 0.3), dentin (18.6 GPa, 0.31), cortical bone (13.7 GPa, 0.3), cancellous bone (1.37 GPa, 0.3), PDL (0.07 MPa, 0.45), and stainless steel bracket (200 GPa, 0.3) (Table 1).

Table 1. Material Properties Assigned to Finite Element Components

The NiTi arch wires were modeled with distinct mechanical characteristics: A-NiTi was considered super elastic, while M-NiTi was treated as thermally activated with shape memory properties. These were incorporated through user-defined material models in the ANSYS Workbench 15.0 environment.

The entire assembly, consisting of the tooth, PDL, alveolar bone, bracket, and wire, was discretized using tetrahedral elements. A convergence test was conducted to ensure mesh density adequacy. The final model comprised approximately 300,000 nodes and 220,000 elements. Special emphasis was placed on mesh refinement at the PDL and root interface to ensure accurate stress resolution in this biologically critical zone (Figure 1).

Boundary conditions and loading protocol

Two sequential analyses were conducted: the wire deformation study and the biomechanical tooth response analysis.

In the first phase, the arch wire was analyzed in isolation. Boundary conditions were applied to simulate engagement within two bracket slots. The length of the wire spanned three teeth, with the central segment undergoing displacement in six clinical directions: extrusive, intrusive, inward (lingual), outward (labial), rotational, and angular. Displacements of 0.5 mm to 2 mm were applied incrementally. A coefficient of friction (μ = 0.28) was defined between the arch wire and bracket to enable realistic sliding behavior. No constraints were placed on the wire ends, allowing free sliding within the bracket slots.

In the second phase, the quantified displacement forces from the first analysis were applied to the tooth model in corresponding directions. Boundary conditions were set by fixing the base of the alveolar bone to simulate physiological constraints. The contact between the archwire and the bracket/tooth was modeled as bonded to simplify computational requirements in the deformation analysis [20-29].

Finite element analysis procedure

The finite element simulations were performed using ANSYS Workbench 15.0. A two-step solver approach was utilized, comprising static structural analysis for both the wire deformation and the tooth-PDL-bone system. For the first analysis, reaction forces generated due to specific arch wire displacements were recorded. These were subsequently applied as input loads in the second analysis phase on the tooth model.

Stress and strain distributions within the PDL were assessed for each movement direction and for varying displacement magnitudes. Von Mises stress, principal stress values, and maximum shear stress were calculated. Similarly, displacement vectors were recorded to assess tooth and alveolar bone deformation. Additional simulations were performed to evaluate combined displacements (e.g., simultaneous extrusive and rotational forces) to reflect clinical multi-vectorial movement scenarios.

Validation and reproducibility

The modeled material properties and boundary conditions were cross-validated with published orthodontic and biomechanical literature. The geometric fidelity of the tooth model was corroborated through visual comparison with clinical CBCT slices and morphological anatomical references. To ensure consistency, all simulations were repeated thrice and analyzed for variation in peak stress values; results varied within a 5% margin, confirming computational reliability.

Results and Discussion

The results revealed significant differences in the force magnitudes produced by each wire under varied displacements, as well as in the resultant stress-strain responses within the periodontal ligament (PDL) and associated deformation patterns of the tooth and alveolar bone (Table 2).

Table 2. Force Output of A-NiTi and M-NiTi Archwires by Direction and Displacement

In the first phase of the study, involving the isolated wire model, the A-NiTi arch wire consistently exhibited greater force output across all directions and magnitudes of displacement when compared to the M-NiTi wire. Force levels increased linearly with each incremental displacement, and this pattern was evident in both wires. Figure 2 shows the deformation behavior of the NiTi arch wires under varied directional loads. However, the magnitude of forces recorded in the A-NiTi model was markedly higher, reflecting the higher elastic modulus and superelastic properties inherent to the Austenitic phase.

Among the six directions of displacement analyzed, extrusive, intrusive, inward (lingual), outward (labial), rotational, and angular, the inward and outward movements elicited the highest force levels from both arch wire types. For instance, at a 2 mm displacement, the A-NiTi arch wire generated a peak force of 138.2 cN in the outward direction and 134.6 cN in the inward direction. By contrast, the M-NiTi wire demonstrated significantly lower force values for the same displacements, recording 107.8 cN and 102.3 cN, respectively (Table 3). These findings suggest that labiolingual deflections result in nearly double the force output compared to vertical (intrusive/extrusive) or rotational displacements, irrespective of the wire type.

Table 3. Maximum PDL Stress and Strain (2 mm Displacement)

Extrusive and intrusive displacements were associated with moderate force magnitudes. In the extrusive direction, A-NiTi delivered forces ranging from 22.1 cN to 88.3 cN across the 0.5 mm to 2 mm displacement spectrum. In contrast, M-NiTi produced slightly lower values, ranging from 17.6 cN to 65.8 cN. Intrusive forces followed a similar trend, with A-NiTi generating between 20.3 cN and 83.4 cN, and M-NiTi producing 15.2 cN to 62.0 cN. These results reflect the stiffness differential between the two materials and underscore the relatively greater force capability of the superelastic A-NiTi variant.

Rotational and angular displacements yielded the lowest force outputs in both arch wire types. A-NiTi recorded forces ranging from 8.6 cN to 26.7 cN, while M-NiTi generated values between 6.9 cN and 20.4 cN across the displacement range. This reduction in force magnitude for rotational and angular movements is attributable to the geometric configuration of displacement and the limited structural resistance offered by the arch wire in those planes.

The second part of the analysis, involving the application of these displacement-derived forces to a fully modeled dentoalveolar unit, elucidated the differential stress-strain behavior in the PDL and revealed distinct deformation patterns in the alveolar bone and tooth structure. Stress analysis indicated that the highest von Mises stress values were observed in the inward and outward directions, particularly when force vectors were aligned with the center of resistance of the tooth. A-NiTi produced maximum PDL stress values nearing 0.038 MPa during inward displacement, while M-NiTi demonstrated a peak of approximately 0.028 MPa under the same conditions. This trend was consistent across all displacements, with A-NiTi invariably resulting in greater stress deposition in the PDL compared to M-NiTi.

In the extrusive and intrusive movements, stress levels were comparatively lower. For A-NiTi, von Mises stress ranged from 0.015 MPa to 0.027 MPa, while M-NiTi recorded slightly attenuated values between 0.012 MPa and 0.022 MPa. Notably, the PDL displayed a more uniform stress distribution in vertical displacements than in labiolingual or combined vector movements, suggesting a more predictable biomechanical response in such clinical scenarios.

Rotational and angular movements induced the lowest stress responses in the PDL for both wire types. The values recorded were substantially reduced, with peak stresses not exceeding 0.015 MPa in A-NiTi and 0.010 MPa in M-NiTi. Correspondingly, these displacements also generated the least strain within the PDL, reaffirming their biomechanical mildness in terms of tissue loading.

The highest strain values were observed during extrusive and inward displacements, particularly with the A-NiTi wire. Inward-directed forces from A-NiTi induced strain levels approaching 10.2%, whereas M-NiTi resulted in peak strain values around 7.6% for the same direction and magnitude. The extrusive forces yielded strain magnitudes between 6% and 9% in A-NiTi and between 4.5% and 7.2% in M-NiTi. These strain values are considered biologically relevant, as they remain within the optimal loading range for inducing orthodontic tooth movement without pathological compression or ischemia in the ligament.

Tooth and bone deformation analyses revealed a consistent trend: A-NiTi generated higher displacement magnitudes in both tooth and alveolar bone compared to M-NiTi across all directions. Maximum deformation occurred in labiolingual movements, where tooth displacement exceeded 0.18 mm in A-NiTi models at 2 mm wire deflection, whereas M-NiTi models recorded displacements under 0.15 mm. Similar proportional differences were observed in bone deformation (Table 4). The lowest deformation values occurred during rotational and angular movements, further confirming their minimal biomechanical impact.

Table 4. Tooth and Bone Deformation at 2 mm Displacement

In multi-vectorial displacement simulations, involving combined directional forces (e.g., extrusive with rotational), stress accumulation in the PDL and deformation values increased multiplicatively. A-NiTi continued to produce more pronounced effects, with multi-directional displacements generating composite stress peaks that were approximately 1.5 to 2 times higher than those recorded during uniaxial loading. Comparative strain and stress effects of A-NiTi and M-NiTi under multi-directional forces are illustrated in Figure 3.

M-NiTi, although less potent in magnitude, exhibited more diffuse and evenly distributed stress patterns, indicating a potentially more favorable biological response under complex force systems.

In summary, A-NiTi wires were shown to deliver consistently higher forces and induce greater mechanical stimuli within the PDL, tooth, and surrounding bone. M-NiTi, while producing lighter forces, offered a more uniform stress distribution and lower strain values, suggesting suitability for initial alignment phases or in patients with compromised periodontal support. The directionality of displacement significantly influenced the mechanical response, with inward and outward movements generating the highest stress, strain, and deformation, followed by extrusive and intrusive, and lastly by angular and rotational movements. These findings offer a comprehensive biomechanical profile of two commonly employed NiTi arch wires, providing valuable insights for optimizing wire selection based on the desired therapeutic outcome.

The present study employed a three-dimensional finite element model (FEM) to evaluate and compare the mechanical behavior of austenitic (A-NiTi) and martensitic (M-NiTi) nickel-titanium arch wires under controlled displacements, with a focus on the resultant force magnitudes, periodontal ligament (PDL) stress-strain distribution, and dentoalveolar deformation. The findings reinforce the foundational role of wire composition and mechanical phase behavior in orthodontic biomechanics and underscore the necessity of tailoring wire selection to the intended clinical application. These results are best interpreted in light of the current literature on NiTi biomechanics and finite element modeling.

One of the primary insights emerging from this investigation is the clear difference in force magnitudes delivered by A-NiTi and M-NiTi wires under equivalent deflection conditions. A-NiTi wires consistently generated higher forces across all displacement directions, reflecting their superelastic properties and elevated plateau stress range. This finding is consistent with prior experimental observations by Kapila et al. who reported that superelastic NiTi alloys exhibit high spring back and sustained activation forces over extended ranges of deflection compared to heat-activated variants, which engage primarily within temperature-dependent stress thresholds [30].

The implications of this force differential extend into the clinical context. The elevated forces associated with A-NiTi may prove advantageous during early alignment phases involving significant deflections, where efficiency in overcoming severe crowding is desired. However, this advantage must be balanced against the biological tolerance of the PDL and surrounding alveolar structures. As McGuinness et al. demonstrated through three-dimensional finite element modeling, stress concentrations exceeding 0.025 N/mm² in the cervical and apical PDL regions could predispose to adverse responses, including root resorption or hyalinization [31]. In the present study, A-NiTi wires reached or slightly exceeded this stress threshold during inward and outward displacement scenarios, warranting caution during their prolonged use in highly displaced dentitions [32-34].

M-NiTi wires, in contrast, yielded lower stress values and strain amplitudes, suggesting a more biologically favorable force profile. This aligns with the findings of Yang Y et al. who showed that lower force systems, when combined with appropriate moment-to-force (M/F) ratios, can achieve efficient tooth translation while minimizing localized PDL compression [35]. Furthermore, M-NiTi wires are thermally activated and exhibit a broader transformation temperature range (TTR), which renders them less sensitive to mechanical fluctuations during insertion. Their gradual activation within the oral cavity ensures a gentler force application, a quality particularly beneficial for patients with compromised periodontal support or low pain thresholds.

The directional dependency of stress distribution observed in this study reinforces the complexity of orthodontic force systems and their interaction with anatomical geometry. Inward and outward displacements produced the highest stress and strain values in the PDL. This finding corresponds with earlier simulations by Tanne et al. and Raghuveer H et al. who observed that labiolingual forces induced significantly higher stress concentrations in the cervical third of the PDL compared to vertical or rotational forces [36, 37]. Such directional sensitivity suggests that orthodontic tooth movement cannot be generalized across all axes and that force vector orientation plays a crucial role in determining biological response.

Rotational and angular displacements, although producing the least force, also induced minimal deformation and stress, consistent with observations by Yettram et al. who reported that off-axis movements, due to their limited contact area and indirect force propagation, typically result in diffuse and low-magnitude stress fields [38]. These findings imply that while A-NiTi may be appropriate for rapid initial alignment, M-NiTi could be the wire of choice when aiming to achieve delicate movements such as rotation or torque control, minimizing the risk of overloading.

It is notable that finite element analysis, as a modeling approach, allowed for high-resolution examination of stress fields within highly complex geometries such as the root-PDL-bone interface. Several prior authors have highlighted FEM’s capacity to provide clinically relevant approximations of force transmission and tissue response. Katta M et al. noted that FEM simulations, when appropriately parameterized, reliably mirror in vivo stress-strain responses, especially when derived from patient-specific computed tomography data [39]. In this study, the use of high-fidelity anatomical models derived from CBCT scans added validity to the simulated responses and reinforced the applicability of these findings in clinical orthodontics.

While the absolute stress values differ between studies due to variation in material property assignment and boundary conditions, the relative patterns observed in this investigation mirror those in the broader literature. For example, Toms et al. demonstrated through cadaveric testing and FEM modeling that nonlinear PDL behavior significantly impacts stress accumulation, particularly at the apical and cervical margins [40]. The current study also observed stress peaks at these sites, particularly under inward and outward displacement forces with A-NiTi wires.

From a biomechanical standpoint, the differential stress response is not merely a product of the material properties of the wires but also of the complex interaction between the bracket slot geometry, friction coefficients, and the elastic response of adjacent tissues. Cattaneo et al. emphasized that finite element simulations must incorporate realistic PDL modeling and anisotropic bone behavior to predict clinical outcomes faithfully. Although the current model employed isotropic material definitions, the trends observed are nonetheless consistent with more advanced simulations that incorporate heterogeneity and nonlinearity.

Clinically, the strain thresholds identified in this study are of particular interest. The maximum PDL strains recorded, approaching 10% in some A-NiTi inward displacement simulations, reside near the upper boundary of physiological strain tolerance. Prolonged exposure to such magnitudes could lead to ischemia and sterile necrosis, as previously suggested by Attik e et al., who showed that intrusive and labial forces correlated strongly with increased risk of root resorption in maxillary incisors [41]. This underscores the necessity of time-modulated force application, and perhaps the intermittent use of superelastic wires in high-stress displacement cases.

Furthermore, the finding that combined (multi-vectorial) displacements amplified stress synergistically is in agreement with findings from Zhong et al. who demonstrated that compounded force systems generate non-linear increases in PDL load, often exceeding the sum of individual vector components [42]. In practice, this means that clinicians should anticipate and manage load interactions in crowded arches where simultaneous labiolingual and vertical correction is required.

Finally, this study affirms the broader clinical observation that M-NiTi wires may serve as a more biologically adaptive choice during early stages of orthodontic treatment. Their low continuous force delivery, combined with enhanced patient comfort and favorable stress distribution, makes them particularly suitable for initiating alignment in cases involving fragile periodontal conditions or low anchorage tolerance. On the other hand, A-NiTi wires, though capable of producing more rapid displacement due to higher force levels, must be employed judiciously, with regular monitoring to mitigate potential risks such as root resorption, as well as patient-reported discomfort or pulp sensitivity.

In summary, the present FEM-based analysis contributes valuable biomechanical insight to the comparative evaluation of A-NiTi and M-NiTi arch wires. The evidence supports a phase-specific approach to wire selection that considers not only the magnitude and direction of displacement but also the physiological tolerance of the supporting structures .

As Cattaneo et al. aptly observed, patient-specific modeling remains the future of precision orthodontics, wherein FEM tools may increasingly guide individualized treatment strategies. Future studies incorporating time-dependent bone remodeling algorithms and nonlinear viscoelastic modeling of the PDL will further refine our understanding of force-biologic response relationships in orthodontic therapy.

Conclusion

This three-dimensional finite element analysis underscores the critical influence of arch wire composition on the biomechanics of orthodontic tooth movement. Austenitic NiTi (A-NiTi) wires exhibited higher force output and induced greater stress and strain within the periodontal ligament and dentoalveolar complex, particularly during labiolingual displacements. In contrast, Martensitic NiTi (M-NiTi) wires generated lighter, more biologically favorable forces with more uniform stress distribution, suggesting their appropriateness in early or sensitive treatment phases.

The direction of displacement emerged as a decisive factor in biomechanical response, with inward and outward movements producing the most pronounced mechanical effects, irrespective of wire type. These findings support a strategic, patient-specific approach to wire selection, emphasizing the need to balance mechanical efficiency with biological tolerance. By integrating realistic anatomical modeling with advanced simulation, this study contributes to the optimization of force systems in orthodontics. Future research incorporating time-dependent remodeling and viscoelastic PDL properties will be essential to enhance predictive precision and treatment individualization.

Acknowledgments: None

Conflict of interest: None

Financial support: None

Ethics statement: Ethical approval for this study was obtained from the Institutional Ethics Committee of Yenepoya (Deemed to be University), Mangalore, Karnataka, India (YUEC 294/31/12/2014)