Finite element analysis of the nonlinear response of oral mucosa in labial tissue induced by spherical versus square self-ligating brackets

Abstract

Objectives

This study employs finite element analysis (FEA) to compare the biomechanical effects of spherical versus square self-ligating brackets on the labial mucosa.

Methods

Two distinct tooth-bracket-lip models, featuring spherical and square brackets, were developed to simulate labial tissue mechanical responses under various incisor spatial configurations. Each model incorporated five clinical scenarios and employed a hyperelastic Ogden model, a two-stage simulation process (bracket placement followed by orthodontic sliding), and a hierarchical computational framework for localized stress analysis. Nonlinear interactions—including contact, material, and geometric couplings—were rigorously resolved.

Results

The stress concentration areas on the labial mucosa are primarily attributed to the clip groove of the spherical bracket and the edge angle of the square bracket, respectively. Under various loading conditions, the maximum stress exerted on the labial mucosa by the spherical bracket consistently remains lower than that induced by the square bracket. Furthermore, compared to the square bracket, the spherical bracket features a smoother surface profile, which significantly reduces mechanical stress on the labial mucosa and achieves a more uniform pressure distribution.

Conclusions

Spherical brackets significantly reduce biomechanical stress on the labial mucosa by optimizing surface topography. This advantage may result in a lower incidence of mucosal lesions and improved clinical comfort, supporting their preferential application in orthodontic practice.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12903-026-08826-1.

Introduction

Malocclusion negatively affects patients’ physical, social, and psychological well-being [1]. The demand for orthodontic treatment has surged due to changing aesthetic standards, increased awareness, technological advances, and marketing efforts. Fixed braces, effective for improving occlusal contacts, torque control, space closure, and correcting vertical and anteroposterior discrepancies, have been the standard orthodontic appliance for over a century [2]. Despite the rapid rise of clear aligners, fixed braces remain the most common clinical choice, with an average treatment duration of about 24.9 months [3]. Prolonged mechanical irritation from orthodontic appliances causes mucosal lesions in 60% to 81% of patients [4], disrupting oral tissue homeostasis. Damage to the mucous membrane exposes nerve endings, causing pain [5], which reduces treatment adherence, prolongs treatment, lowers patient motivation, and diminishes quality of life [6]. Therefore, strategies to reduce mechanical irritation are essential to improve patient comfort and treatment outcomes.

Previous research shows that injuries from fixed orthodontic appliances mainly affect the buccal and labial mucosa. The simplest solution is to identify and remove the source of irritation. Discomfort from extended wires can be reduced by carefully cutting and bending wire ends. Irritation from brackets is often relieved by applying dental wax to smooth the bracket surfaces [7]. However, wax must be removed before meals, may not adhere well to bracket protrusions, risks accidental ingestion, and can increase wire-slot friction and plaque buildup with prolonged use. To overcome these issues, customized arc-shaped bracket shields have been proposed to act as physical barriers between brackets and mucosa, improving surface texture perception. Pires et al. [8] found that these shields significantly enhanced patient comfort Yet, their routine use is limited by cost and failure rates, and bracket protectors may increase friction between the wire/ligatures and the bracket slot [9]. Despite potential stability concerns, improved comfort has led to interest in self-ligating brackets (SLBs) with spherical designs, which may reduce mucosal lesions by minimizing bracket profile and protrusions. Studies have examined the bond strength and morphological advantages of spherical SLBs compared to conventional brackets [10, 11]. Recent evidence also suggests that spherical SLBs provide better torque expression than passive SLBs (Damon Q) and conventional brackets (Discovery) [12]. However, there is a lack of theoretical evidence to confirm whether they are more comfortable than conventional square SLBs.

The finite element method (FEM) is a non-invasive, precise engineering tool widely used in orthodontic research to calculate deformation and stress distribution in complex structures [13]. It provides detailed quantitative data on tissue responses and has been used to evaluate forces from various orthodontic appliances, reducing the need for human clinical tests and associated ethical concerns. Finite element analysis (FEA) is also employed to estimate loads on masticatory mucosa under dentures, improving denture design [14, 15]. Similarly, analyzing the biological response of oral mucosa to different bracket types is vital to prevent soft tissue injuries. However, few studies have examined the interface stress between oral mucosa and orthodontic brackets. FEA in this area faces challenges such as characterizing labial tissue properties, modeling the complex functions of the lip, and capturing the contact mechanics between brackets and mucosa. This paper develops FEMs for spherical and square SLBs and compares their stress distributions on oral mucosa across various tooth alignments. The null hypothesis is that spherical self-ligating brackets (SLBs) will not reduce mechanical stress on the oral mucosa compared to conventional rectangular brackets.

Materials and methods

Two distinct self-ligating bracket systems were evaluated in this study: a spherical design (OO Dental, Guangzhou, China) and a square design (Damon, Ormco, Glendora, CA, USA). Their geometric parameters are detailed in Table 1.

Table 1.

Dimensional parameters of the brackets

Brackets	Mesiodistal width (mm)	Gingivo-occlusal width (mm)	Height (mm)
Spherical SLBs	4.0	3.4	2.5
Square SLBs	2.9	3.2	2.5

Finite element simulation scheme

Geometric modeling

This study selected a 20-year-old female as the research subject. Informed consent was obtained from the participant, and the study was approved by the relevant ethics committee (No. [2020]315-1). Pre-existing, anonymized CBCT data, originally acquired for the patient’s routine orthodontic diagnosis and treatment planning, were used for 3D reconstruction. No patient was exposed to additional radiation solely for the purpose of this study, strictly adhering to the ALARA (As Low As Reasonably Achievable) principle.

Cone beam computed tomography (CBCT) data were obtained using an i-CAT FLX scanner (Kavo Inc., USA) for 3D reconstruction of the maxillofacial region, while intraoral dentition scanning was performed with Dentrix60 (Fussen Inc., China). Medical modeling software Mimics (Materialise Inc., Belgium) and reverse engineering software Geomagic Studio (Geomagic Inc., USA) were used to extract geometric models of the upper labial tissue and right maxillary central incisor. These models were then imported into ANSYS Workbench (Version 19.2, ANSYS Inc., USA) to create the FEM for analysis.

Industrial modeling software SolidWorks (version: 2018; Dassault Systemes Inc., USA) was employed to accurately reconstruct the true morphology of square and spherical SLBs. The models were exported in STP format and imported into ANSYS Workbench to create two distinct appliance configurations: a square bracket variant, called the lip-incisor-square bracket case, and a spherical bracket variant, called the lip-incisor-spherical bracket case. Furthermore, the overall labial profile thickness (prominence from the tooth surface) of both brackets was strictly matched and controlled in the geometric modeling phase. This ensured that the simulated stress differences were primarily driven by the edge topology (curvature) rather than disparate bracket volumes.

Figure 1 shows the geometry models.

Fig. 1.

Geometric models. A The upper right central incisor. B The labial tissue. C The spherical bracket. D The square bracket

The appliance configurations and corresponding mesh are shown in Fig. 2. Second-order hexahedral elements were used for labial tissue to capture precise stress distribution. Linear tetrahedral elements were chosen for teeth and brackets due to their high stiffness and minimal deformation, fitting their geometric shapes.

Fig. 2.

Assembled appliance configurations. A The spherical bracket. B The square bracket

Materials property

Given the varying proportions of labial tissue across craniofacial regions and extensive research on adipose tissue properties [16], the labial tissue was uniformly modeled as adipose-like to simplify analysis [11]. Comley et al. [17, 18] described adipose tissue as an incompressible and inviscid fluid at 37 °C with an elastic modulus of about 1,000 Pa under low strain, also providing a stress-strain curve through experiments. Alkhouli et al. [16] proposed a double-asymptote model, with an initial elastic modulus of 1,000 Pa rising to 80,000 Pa at 0.3 strain, noting strains above 0.3 greatly increase tissue rupture risk, marking this as a damage threshold.

Hyperelastic material models were used in the FEA to capture the strain-dependent elastic modulus of adipose tissue. Common models include Neo-Hookean and Mooney-Rivlin, based on strain tensor invariants, and the Ogden model, formulated with principal stretches.

Sims et al. [19] showed the Neo-Hookean model suits adipose tissue at low-to-medium strains. Geerligs et al. [20] found the Mooney-Rivlin model applies under high strains where adipose tissue fully recovers after unloading. Comley et al. [18] confirmed the Ogden model captures the nonlinear, symmetric behavior and incompressibility of adipose tissue. Mihai et al. [21] compared these models, revealing that Neo-Hookean and Mooney-Rivlin poorly simulate adipose tissue under combined tension and shear, while the Ogden model aligns well with experimental data, making it ideal for 3D FEA.

Based on experimental results [19–21], this study uses the Ogden model to characterize labial tissue mechanics. Figure 3 shows the final fitted uniaxial stress-strain curve, where the initial elastic modulus is about 1,000 Pa and increases with strain, reaching 80,000 Pa at 0.32 strain, consistent with known material properties.

Fig. 3.

Stress-strain curve of the labial tissue [11]

The Ogden model offers various orders of formulation. Higher-order models improve data fitting accuracy but significantly increase computational complexity and convergence time, especially involved with nonlinear factors like contact and geometric nonlinearities. To balance accuracy and efficiency, this study uses a second-order Ogden model with six parameters to simulate labial tissue, detailed in Table 2.

Table 2.

Material parameters of the Ogden model for the labial tissue (Unit: Pa)

MU1	A1	MU2	A2	D1	D2
37.524	17.883	37.524	17.883	0	0

Given the much higher stiffness and minimal deformation of the tooth and brackets compared to adipose tissue, both the tooth and brackets were modeled as linear elastic materials. Their Young’s modulus and Poisson’s ratios are shown in Table 3.

Table 3.

Material parameters of the tooth and bracket

	Young’s modulus (Pa)	Poisson’s ratio
Tooth	20,000,000	0.3
Bracket	200,000,000	0.3

Boundary conditions and loading protocol

To simulate the nonlinear interaction between the brackets and the labial mucosa during functional oral activities (e.g., resting, smiling, speaking, and chewing), a continuous two-stage loading protocol was implemented (as illustrated in the cumulative displacement axis of Fig. 4). The tooth root was modeled as a fixed support. Given that the magnitude of labial tissue deformation (in millimeters) is significantly greater than the micro-mobility of the tooth within the periodontal ligament (PDL), the tooth was treated as a rigid boundary to ensure computational convergence in the high-strain contact analysis.

Fig. 4.

The maximum stress induced by spherical and square SLBs during the continuous loading protocol in the condition of an aligned tooth. The graph is demarcated into two distinct phases: the Compression Phase (0–3 mm vertical displacement) and the Sliding Phase (3–5 mm cumulative, representing the 2-mm mesiodistal sliding)

Stage 1: bracket penetration (compression phase)

This phase simulates the initial contact and resting pressure of the lip against the orthodontic appliance, and the simulation was initiated when the labial tissue first contacted the bracket’s top surface. Subsequently, the tissue was displaced 3 mm perpendicularly toward the tooth surface. Since the bracket profile height is approximately 2 mm, this 3-mm total displacement ensured that the bracket fully penetrated the mucosa, with an additional 1-mm of penetration into the surrounding tooth surface. This setup effectively captures the “clamping” effect and the static compressive state of the lip. The resulting compressive stresses (ranging from 1,000 to 10,000 Pa) in our model are consistent with in-vivo intraoral lip pressure measurements reported in clinical studies [22–24], providing a physiologically anchored baseline for the subsequent dynamic analysis.

Stage 2: bracket sliding (dynamic phase)

Following the initial 3-mm compression, a 2-mm orthogonal sliding displacement was applied to the labial tissue relative to the bracket-tooth complex (totaling 5 mm of cumulative displacement in the FEA process). This stage simulates the dynamic excursions of the oral mucosa during functional movements like speech or mastication. Accounting for the most unfavorable loading conditions, an orthogonal displacement configuration is adopted. The horizontal sliding is used to analyze mesial torsion condition and the vertical sliding is used to analyze labial torsion condition.

• ① Horizontal sliding: The labial tissue was displaced 2 mm mesiodistally across the bracket surface. This mimics the lateral excursion of the mucosa that occurs during actions such as smiling and speaking.

• ② Vertical sliding: The tissue was displaced 2 mm inciso-gingivally. This mimics the vertical excursion that occurs during actions such as chewing and speaking.

While actual clinical mucosal movements can exceed 15 mm, preliminary mesh-sensitivity simulations demonstrated that the maximum von Mises stress in the labial tissue reaches a stable steady-state within the first 2 mm of sliding (see Supplementary Fig. S2, which provides stress results of 4-mm sliding. It is noted that, a separate model and mesh were created for the 4-mm sliding condition. So, the calculation result differs slightly from that of 2-mm sliding condition shown in Fig. 4.). Therefore, a 2-mm sliding distance was adopted to balance computational efficiency with the need for accurate peak stress capture. To reflect diverse clinical scenarios, this loading protocol was applied across five distinct tooth spatial configurations, including mesial torsion and labial inclination (Fig. 5; Table 4).

Fig. 5.

Models of five distinct tooth spatial configurations. A & B The standard condition of aligned tooth. C The condition of mesial torsion 15° of tooth. D The condition of mesial torsion 30° of tooth. E The condition of labial torsion 15° of tooth. F The condition of labial torsion 30° of tooth

Table 4.

Simulation groups and corresponding clinical scenarios

Group	Tooth spatial configuration	Moving mode	Clinical scenario (functional movement)
A	Standard	3 mm perpendicular (penetration) then 2 mm horizontal (sliding)	Smiling or speaking (lateral excursion)
B		3 mm perpendicular (penetration) then 2 mm vertical (sliding)	Chewing or speaking (vertical excursion)
C	Mesial torsion 15°	3 mm perpendicular (penetration) then 2 mm horizontal (sliding)	Smiling or speaking (lateral excursion)
D	Mesial torsion 30°	3 mm perpendicular (penetration) then 2 mm horizontal (sliding)	Smiling or speaking (lateral excursion)
E	Labial torsion 15°	3 mm perpendicular (penetration) then 2 mm vertical (sliding)	Chewing or speaking (vertical excursion)
F	Labial torsion 30°	3 mm perpendicular (penetration) then 2 mm vertical (sliding)	Chewing or speaking (vertical excursion)

Contact factors

During oral activities, contact occurs between the labial tissues, brackets, and teeth. The bracket-tooth interface, a bonded connection, was modeled as bonded contact. The lip-bracket and lip-tooth interfaces were treated as frictional contacts. Due to the presence of oral mucosa and saliva, which results in minimal friction, a frictional coefficient of 0.05 was assigned to both interfaces [25].

The labial tissue has significantly lower stiffness than the brackets and tooth, leading to asymmetric contact conditions at the lip-bracket and lip-tooth interfaces. Brackets and tooth serve as target surfaces, while the labial soft tissue was the contact surface. During simulation, the contact surface was prevented from penetrating the targets. Since the applied displacement was similar to the element size, the “small sliding” option was disabled.

The penetration tolerance for contacts must meet two criteria: it should not exceed 1/100 of the labial tissue thickness and must be less than 1/10 of the FE mesh size. In this model, with an 8 mm soft tissue thickness and a mesh size of about 1 mm, the penetration tolerance was set to 0.05 mm.

Additionally, the labial soft tissue exhibits nonlinear behavior. To ensure accuracy and convergence, normal contact stiffness was updated after each iteration.

Optimized simulation method

In FEA with interpolated displacement fields, stress results converge more slowly than displacement results. Thus, a fine mesh is needed to accurately simulate labial tissue stress. Moreover, small edge chamfers on the bracket create significant stress concentrations at contact points with the tissue, requiring an even finer mesh to capture maximum stress accurately.

Traditionally, mesh refinement is applied uniformly across the overall model, increasing degrees of freedom but also computational demands, slower processing, and convergence issues. For instance, under a 30° mesiodistal torsion of the square bracket, we analyzed the model using a workstation with a 32-core AMD EPYC 7452 processor and 256 GB RAM. Labial tissues were meshed at eight refinement levels from 1.0 to 0.3 mm. Computational time increases sharply from 30 min (1.0 mm mesh) to over 1,000 min (0.3 mm mesh), exceeding 17 h. This approach is impractical for repeated analyses across multiple tooth configurations and requires optimization.

This study employed two mesh models for displacement and stress calculations. The coarse-mesh overall model yielded displacement results, which served as boundary conditions for the fine-mesh submodel, enhancing stress calculation accuracy and efficiency.

To validate the accuracy of the submodeling technique, a formal mesh convergence study was performed (detailed in Supplementary Fig. S1). The tested submodel mesh size ranges from 1.0 mm to 0.3 mm. The maximum von Mises stress stabilized when the local mesh size was refined below 0.4 mm. To balance simulation accuracy and efficiency, a mesh size of 0.3 mm was selected.

Figure 6 illustrates the global-submodel hierarchical analysis scheme. Based on Saint-Venant’s principle, stress concentrations near contact areas minimally affect distant displacements and stresses. The study focused on a 9 mm × 6 mm × 2.5 mm soft tissue region around the contact area, combined with partial bracket geometry to form the submodel. This submodel employed finer mesh elements for detailed stress analysis.

Fig. 6.

Schematic diagram of the overall-submodel hierarchical analysis. A coarse global model captures overall lip displacement, transferring boundary conditions to a refined submodel for precise localized stress calculation at the bracket-tissue interface

Results

The six simulation groups underwent continuous computational analysis under two conditions: bracket penetration and sliding. The FEA process was divided into 40 steps. The first 12 steps simulated bracket penetration, each with a 0.25 mm displacement, while the remaining 28 steps simulated bracket sliding, each with about 0.071 mm displacement. Figure 7 shows the maximum von Mises stress of labial tissues throughout the process for each group.

Fig. 7.

Maximum von Mises stress on labial tissue induced by two SLBs under varying biomechanical conditions. A Horizontal sliding movement (2 mm distal displacement) with aligned tooth. B Vertical sliding movement (2 mm incisal displacement) with aligned tooth. C Horizontal sliding with mesial torsion 15° of tooth. D Horizontal sliding with mesial torsion 30° of tooth. E Vertical sliding movement with 15° labial crown inclination. F Vertical sliding movement with 30° labial crown inclination

Our simulation showed that the maximum von Mises stress in the spherical bracket case was significantly lower than in the square bracket case across all conditions. This suggests the spherical bracket enhances labial tissue comfort during orthodontic treatment compared to the square bracket. Results also indicate that labial tissue strain can reach 0.3 when local stress nears 7,000 Pa, highlighting a high risk of mucosal lesions with a square bracket under 30° mesial torsion.

For an aligned tooth under horizontal displacement, initial compression (0.7 mm) caused negligible von Mises stress. Subsequent loading showed a progressive stress increase until stabilization, peaking at 1,882 Pa for spherical brackets and 3,686 Pa for square brackets during mesiodistal movement (Figs. 4 and 7). As demonstrated in the corresponding curves, the shear stress on the mucosa reaches a steady state prior to 2 mm of sliding (Supplementary Fig. S2), justifying the 2 mm displacement protocol to minimize computational cost without sacrificing accuracy. In the simulation of inciso-gingival sliding, maximum stresses reached 1,497 Pa and 4,250 Pa for spherical and square brackets, respectively. Figure 8 shows the maximum stress distribution on labial tissue caused by both brackets in mesiodistal and inciso-gingival sliding. The stress distribution patterns were similar across different labial tissue deformation modes for both brackets. For the spherical bracket, maximum stress occurred mainly at the clip groove, while for the square bracket, it concentrated at the edge angle. Beyond evaluating the peak von Mises stress, the volumetric distribution of high stress was analyzed to quantitatively assess clinical comfort. The high-stress area—defined as the tissue subjected to stress exceeding the physiological capillary occlusion threshold (~ 4,700 Pa) [26]—exists in some of the square bracket conditions. Conversely, all the spherical bracket conditions do not exceed the threshold. Comparing to the square bracket, the spherical bracket distributed the contact force more uniformly. Take the case of aligned tooth, where the relative difference between stress of square and spherical bracket is the smallest, as an example. Figure 9 shows the stress distribution on tissue for square and spherical bracket at the moment of maximum stress, while Fig. 10 shows a higher-resolution image of the detailed stress distribution on labial tissue. As can be seen from the figure, square bracket is significantly larger than spherical bracket in the tissue region above 1,000 Pa.

Fig. 8.

Stress distribution patterns on labial tissue under aligned tooth. A The distribution of maximum stress effected by spherical bracket during mesiodistal sliding. B The distribution of maximum stress effected by spherical bracket during inciso-gingival sliding. C The distribution of maximum stress effected by square bracket during mesiodistal sliding. D The distribution of maximum stress effected by square bracket during inciso-gingival sliding

Fig. 9.

Distribution of region with higher stress on labial tissue under aligned tooth. A The distribution of region with stress over 500 Pa and 1,000 Pa effected by square bracket during mesiodistal sliding. B The distribution of region with stress over 500 Pa and 1,000 Pa effected by spherical bracket during mesiodistal sliding

Fig. 10.

Detailed stress distribution on labial tissue under aligned tooth. A The distribution of square bracket during mesiodistal sliding. B The distribution of spherical bracket during mesiodistal sliding

Tooth rotation cases show greater differences between the two brackets in stress impact on labial tissue. In mesial torsion, the labial tissue undergoes horizontal displacement due to vertical compression against the bracket. At 15° mesial torsion, the spherical bracket induced nearly constant stress, while the square bracket’s labial tissue stress rose significantly from 3,686 Pa to 6,493 Pa. As mesial torsion increased from 15° to 30°, maximum stress from the spherical bracket decreased from 1,756 Pa to 413 Pa, whereas stress from the square bracket increased from 6,493 Pa to 9,510 Pa. Figure 11 shows von Mises stress on labial tissue contacting both brackets at 15° mesial torsion. The spherical bracket produced a more uniform stress distribution, while the square bracket caused higher stress concentrations, indicated by more yellow and red areas.

Fig. 11.

The sub-models in the condition of mesial torsion 15° of brackets. A The horizontal view of stress distribution on lining mucosa effected by rotated spherical bracket. B The horizontal view of stress distribution on lining mucosa effected by rotated square bracket. C The saggital view of stress distribution on lining mucosa effected by rotated spherical bracket. D The saggital view of stress distribution on lining mucosa effected by rotated square bracket

In labial torsion, vertically compressing the labial tissue against the bracket surface by 3 mm produced a maximum stress pattern during vertical sliding similar to that of horizontal sliding. As the tooth’s labial misalignment increased from 15° to 30°, the spherical bracket’s maximum stress on the labial tissue decreased from 1,061 Pa to 852 Pa, while the square bracket’s maximum stress rose from 2,299 Pa to 2,715 Pa.

Discussion

The oral mucosa serves as the body’s primary barrier within the oral cavity, with its structural integrity and regional variations in adaptive capacity playing a critical role in responding to mechanical stimuli [27, 28]. While stress distribution in masticatory mucosa under prosthetic devices has been extensively investigated, the biomechanical behavior of the lining mucosa in response to orthodontic brackets remains less understood [28, 29]. Clinically, mucosal irritation and pain induced by bracket contact are common yet poorly quantified in biomechanical terms. To address this, the present study employed finite element analysis (FEA) to evaluate and compare stress distributions in labial mucosa resulting from interactions with spherical and square self-ligating brackets, providing mechanistic insight into the origin of localized tissue discomfort.

Biomechanical trials on human ex vivo tissues often involve limited samples due to legal and ethical constraints, making it challenging to source tissue and define mechanical properties of living human tissue [30]. Thus, labial tissue properties must be calibrated against a representative soft tissue model. Labial tissue is highly distensible and easily deformed, similar to adipose tissue, which is commonly used in lip deformity treatments [31]. Adipose tissue’s material properties are better studied experimentally and theoretically than other soft tissues [32]. For modeling, labial tissue was treated as adipose-like, employing hyperelastic materials since adipose tissue’s elastic modulus increases with strain. The Ogden model, based on prior experimental results [11], was used to describe labial tissue behavior. A six-parameter second-order Ogden model was chosen for accuracy and computational stability. Fitting the uniaxial stress-strain curve yielded an initial elastic modulus of 1,000 Pa and a final modulus of 80,000 Pa at 30% strain, consistent with Alkhouli’s findings [16], which reported initial and terminal moduli of 1,000 Pa and 80,000 Pa at 32% strain, respectively. The use of this hyperelastic Ogden model, which represents the stiffening behavior of oral mucosa as it deforms, is essential. Unlike linear elastic materials, the mucosa exhibits deformation-dependent stiffness [25], meaning linear models would severely underestimate peak stresses during large lip deformations.

Lip pressure increased significantly after placing orthodontic brackets [22], with persistent bracket-labial tissue interactions during mastication and swallowing. To simulate these conditions, our computational model applied sequential tissue displacements: (1) an initial 3-mm vertical compression of the labial tissue toward the bracket surface, and (2) a subsequent 2-mm planar sliding (horizontally or longitudinally. The vertical compression represents immediate post-placement tissue adaptation, while planar displacements mimic dynamic functional movements. The model showed that von Mises stress in the oral mucosa did not significantly increase during the first 0.7 mm of compression, likely due to the mucosa’s vascularized, cushioning properties [25]. Stress then increased, consistent with Hua et al. [11]. During the sliding phase (2 mm horizontal displacement), the peak stress calculated for the square bracket reached 3,686 Pa, which corresponds well with the elevated lip pressures (approximately 4,000 Pa) observed during functional activities such as swallowing [22]. FEA revealed maximum stress concentrated at the clip groove of the spherical bracket, confirming our prior experimental findings [33].Stress concentrations were also noted at the square bracket’s edges and clip groove, explaining why these areas are often covered with orthodontic wax clinically [4, 6, 7]. The square bracket induced more high-stress (yellow and red) regions on the lining mucosa, indicating greater risk to soft tissue health. For a mesially rotated tooth, stress under the square bracket was higher than under the spherical bracket; at 30° mesial torsion, local maximum stress reached 9,510 Pa. Clinically, patients often report two distinct types of mucosal discomfort: a sensation of “bulk” (the lip being pushed forward during initial compression) and “rubbing” (frictional trauma during dynamic sliding) [7, 22]. Physiologically, the capillary blood pressure in the oral mucosa is approximately 35 mmHg (about 4700 Pa) [26]. When localized mechanical stress—such as the 9,510 Pa induced by the square bracket—exceeds this capillary threshold, blood flow will be decreased and may even temporarily cease, it induces tissue trauma and triggers nerve pain and discomfort, thus compromising patients’ life quality [25]. The square bracket’s edges embed more deeply in the mucosa, increasing local stress and injury risk, whereas the spherical bracket’s rounded shape distributes stress more evenly and consistently maintains peak stresses closer to or below this critical threshold, explaining its superior clinical comfort [34]. Additionally, dentition unevenness creates a lock groove structure that spatially separates high contact pressure areas from oral tissue, lowering localized stress concentrations.

It is also worth noting that our simulation utilized a low friction coefficient (µ = 0.05) to represent ideal salivary lubrication. In clinical scenarios such as xerostomia (dry mouth) or variations in salivary mucin, “dry spots” can drastically increase interfacial friction [25]. Under higher friction conditions, the tangential shear stress exerted by the square edges would escalate exponentially, further exacerbating the risk of mucosal tearing. Furthermore, while the spherical bracket demonstrated overwhelming biomechanical advantages overall, our stress contour maps (Figs. 8, 9 and 10) identified that its remaining mild stress concentration primarily localized at the clip groove. Future manufacturing optimizations focusing on smoothing or re-contouring this specific functional area could further enhance patient comfort.

Our methodology introduces new ways to study stress distribution beneath orthodontic brackets on oral mucosa. We developed 3D finite element (FE) models of the tooth-bracket-lip system, applied the Ogden model to characterize lip soft tissue properties, and used sub-modeling for efficient partial analysis. The results align with clinical observations, validating our computational approach. Within the limitations of this in silico study, the present findings provide preliminary insights into the biomechanical responses of the oral mucosa to orthodontic brackets. These insights may enhance clinical and biological understanding and offer preliminary guidance for numerical modeling based on clinical data. When applied appropriately, these models could support the optimization of bracket design and help identify potential sites of mucosal irritation caused by brackets. Moreover, the biomechanical advantages of spherical surface topography are not limited to the labial mucosa. In lingual orthodontics, tongue irritation and speech impairment remain major patient complaints. Applying the rounded, spherical surface concept to lingual self-ligating brackets could similarly help diffuse peak contact pressures on the highly sensitive lingual mucosa and lateral borders of the tongue, thereby representing a promising direction for future appliance evolution [35–37]. Nevertheless, further experimental and clinical validation is necessary to confirm these computational predictions.

Several limitations of this computational model should be acknowledged when interpreting the clinical implications. First, the complex layered structure of the labial tissue was beyond the scope of standard CT resolution [38], the labial tissue was modeled as a homogeneous, adipose-like hyperelastic block. In reality, the labial tissue is a complex multilayered structure comprising the intraorally the mucosa, submucosal, muscle, subcutaneous tissue, and epidermis [39]. Because the superficial mucosa is stiffer than the underlying adipose tissue, treating the entire lip as homogeneous may actually underestimate the localized “sharpness” effect of the square brackets. Thus, our findings likely 1,500–6,500 kPa provide a conservative estimate of the mechanical irritation caused by conventional brackets. Second, individual anatomical variations—such as lip thickness and elasticity, which differ with age, sex, and ethnicity [40]—were not accounted for; patients with thinner lips may possess less mechanical “buffering” capacity [24], potentially amplifying the stress disparities between bracket types. Third, the current model isolated the bracket geometry and did not incorporate the archwire-ligation complex. While the archwire sits within the slot and its direct frictional contact with the mucosa is generally secondary to the prominent bracket wings, future multi-component 3D models should include the archwire to fully replicate intraoral interactions. Advancing toward more clinically predictive simulations, subsequent studies will benefit from incorporating high-resolution imaging techniques, diverse anatomical variables (e.g., dental arch morphology, tissue biomechanics, periodontal ligament), and the complete archwire-bracket assembly to enhance personalization and translational relevance. In summary, future studies should address the current model’s limitations by integrating multi-layer tissue modeling, the full archwire-bracket complex, and population-level anatomical variability.

Conclusions

The following conclusions could be drawn:

1. Across all clinical models, spherical self-ligating brackets consistently exhibited lower maximum von Mises stress than square brackets, suggesting a corresponding reduction in mucosal irritation and a potentially beneficial effect on periodontal soft tissue health.

2. For malposed teeth—whether mesially rotated or buccolingually inclined—spherical self-ligating brackets exhibit a trend of decreasing von Mises stress with increasing degree of misalignment, which suggests a protective advantage for the oral mucosa in patients with dental crowding.

Authors’ contributions

L.G., Y.L. and J.H conceived and designed the study, performed the experiments, collected and analyzed the data, and drafted the manuscript. T.Z. and Z.C. participated in the experimental design, and provided essential technical support. W.S. and Y.H. conducted statistical analyses, provided expertise on analytical methods, and drafted the data analysis section. B.G. and T.C. contributed to the conception and design of the study, and revised the manuscript critically. L.J. as the principal investigator, secured funding, supervised the overall direction and progress of the study, and reviewed and approved the final version of the manuscript.

Funding

This study was supported by the University-Industry Collaborative Education Program (240701316220512, 220901316063638, 220801316245152), Guangzhou Leading Talent Project for Innovation and Entrepreneurship (2021–L004).

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Ethics approval and consent to participate

The present study was approved by Ethics Committee of The First Affiliated Hospital of Sun Yat-sen University (No. 2020[315]-1). Informed consent was obtained from all of the participants in this study for clinical information and images.

Consent for publication

Available (if requested).

Competing interests

The authors declare no competing interests.