Determination of stress distribution on periodontal ligament and alveolar bone by various tooth movements – A 3D FEM study

Abstract

Aim

The purpose of this study was to evaluate the stress distribution on the maxillary central incisors by various tooth movements using three-dimensional finite element modeling with varying periodontal ligament (PDL) thickness and different alveolar bone height (at the apex and alveolar crest).

Material and methods

A Finite Element Modeling model was created using surface data of the tooth using SolidWorks Software. Different types of force (intrusion, extrusion, tipping, and bodily movement) were applied on the maxillary central incisor, with two different periodontal ligament thickness (0.15 mm and 0.24 mm) and alveolar bone height (at the apex and alveolar crest). Stress generated due to force applied due to different types of tooth movement was calculated and compared.

Results

Maximum stresses generated under intrusion, extrusion, tipping, bodily movement were 9.0421 E⁻⁰⁰³ N/mm² for 0.15 mm pdl at alveolar bone, 7.2833 E⁻⁵ N/mm²for 0.24 mm pdl labio-lingually, 9.1792 E⁻⁰⁰² N/mm² at 0.15 mm pdl at alveolar bone height and 6.2208 E⁻⁶ N/mm² for 0.24 mm pdl at alveolar crest respectively.

Conclusion

The stress pattern seen was nearly the same in all the cases in both PDL thickness. The maximum stress pattern was found to be at the apex of the central incisor, reducing from apex to the cervical region. Intrusion, extrusion, and tipping movement showed the greatest amount of relative stress at the apex of the maxillary central incisor. The bodily movement produced forces at root apex and distributed it all over.

Highlights

• •The stress pattern for almost all types of tooth movements was nearly the same in all cases in both PDL thickness.
• •The apical stress-induced in PDL increases as the thickness of PDL decreases. The stress in alveolar bone decreases with an increase in PDL thickness.
• •Intrusion, extrusion, and tipping showed the greatest amount of relative stress at the apex of the maxillary central incisor.
• •Clinically, this stress distribution can be used as a means, that in adult patients with increased PDL thickness at the root apex.

1. Introduction

Orthodontic tooth movement occurs primarily due to the periodontal ligament phenomenon. When prolonged pressure is applied, bone remodeling takes place and tooth movement occurs in the bone. The quantity of stress in the periodontal ligament is significant and this stress is transferred to the alveolus which results in bone remodeling and subsequent tooth movement.¹ As there is altered crown-to- root ratios in many adult orthodontic patients due to alveolar bone loss induced by periodontal disease, the effect of these variables on the biomechanical behaviour of a tooth may be more important in adults than adolescents.²The force applied on the tooth should be a controllable variable, and careful study of the mechanics underlying our clinical applications can help in reducing undesirable side effects.³ The available literature is very less, which can help us to accurately determine the level of stress in various areas of the periodontal ligament (PDL). Accurate estimation of this force might be the best means of relating the application of a force and the effects of it on the tooth, PDL and bone.^4,5

Various experimental techniques like animal experiments,⁶ histological methods,^7,8 strain gauge techniques,^9,10 photoelastic methods¹¹ have been used to determine the effects of orthodontic forces on teeth and the periodontal tissues. Many studies have analyzed periodontal stresses caused by orthodontic loading and these stresses are very much often determined by finite element analysis (FEA).¹² Different studies have included multi-rooted teeth along with alveolar bone for stress analysis and have concluded FEM to be a treasured, non-invasive tool for analyzing the mechanical stress distribution within the periodontium during orthodontic force application.¹³

The basic principle of “Finite element analysis” is based on dividing or breaking a complex structure into very small and simple sections, called “Elements”. These elements have specified features (such as Young's modulus) known for their physical response to an external load (as orthodontic forces) or displacement (such as bending).¹⁴ Finite element analysis (FEA) is used in dentistry to inspect a wide range of topics, such as the structure of teeth, biomaterials, restorations, dental implants, and root canals.¹⁵

A better understanding of how stress distribution leads to alveolar bone loss and apical root resorption is needed. During alveolar bone loss, the center of resistance of the tooth is lowered which also alters the distribution of stress patterns on the root.¹⁶ Alteration in root length also shows similar changes. The maxillary central incisors were chosen for this study because they have the highest incidence of root resorption in the apical region, suggesting that the periodontal ligament stresses could be highest in the apical region. From a clinical perspective the study helps the clinician in planning the mechanics (minimal force) needed for intrusion of the anterior teeth as well as the helps in deciding which tooth movement should be done to achieve the ideal correction. Hence the purpose of this study was to determine the level of stress on the maxillary central incisor with various tooth movements using three-dimensional finite element modeling with varying periodontal ligament thickness in the alveolar bone.

2. Material and methods

This study was approved by the Institutes Ethical Committee Saraswati Dental College and Hospital (SDC/RES/2018/xxxx). In the present study, the maxillary central incisor analytical model was developed from the Dental Anatomy textbook by Wheeler's.¹⁷ The height of the central incisor (distance from the apex of the root to the incisor edge) was taken as 23.5 mm and the mesiodistal and labio-palatal width of the crown was taken to be 8.75 mm and 7 mm respectively. The root length was 13 mm. The periodontal ligament was simulated as 0.15 mm and 0.24 mm. These widths of the PDL were taken as the average width of the PDL is 0.2 mm and the width ranges from 0.15 to 0.38 mm with the thinnest portion around the middle third of the root. Since there is a variation in the PDL widths with regards to the age of the patient, we decided to assess the stress with 2 different PDL widths.

Surface data of the tooth was generated using SolidWorks Software. The analysis was carried out using ANSYS WORKBENCH 14.0 software. The geometric models were then imported to ANSYS software for meshing (Fig. 1). Meshing is the phenomenon of converting the geometric model into a finite element model. This finite element model consists of 30452 elements and 61900 nodes. To reciprocate the natural anatomy, the tooth was connected to the surrounding alveolar bone through the periodontal ligament. Therefore, the convergences of all these nodes were recognized to make the connectivity of the model. The structures such as periodontal ligament and the alveolar bone in the finite element model of the human maxillary central incisor were allocated and the model was completed (Fig. 2). The properties of the material (Young's modulus and Poisson's ratio) of the tooth (cortical bone, cancellous bone, cementum, and enamel) were entered in the pre-processing stage. The material characteristics were taken from the data available in the literature which were assumed to be linear, elastic, homogeneous, and isotropic (Table 1).¹⁸ The PDL was set as a hyper-elastic material to simulate its real mechanical characteristics.¹⁹

Fig. 1.

Meshed maxillary central incisor.

Fig. 2.

–Complete maxillary central incisor FEM model.

Table 1.

Material properties of Central incisor and surrounding structures.

Material	Young's modulus	Poisson's ratio
Enamel	8.41 × 104	0.33
Dentin	1.83 × 104	0.31
Cementum	1.5 × 104	0.31
PDL	6.9 × 10−1	0.45
Alveolar bone	1.37 × 104	0.30

Four different types of forces were applied on the tooth with two different periodontal ligament thickness (0.15 mm and 0.24 mm) and at the alveolar bone (at the apex and alveolar crest). The force applied was to achieve four different types of tooth movements-intrusion, extrusion, tipping, and bodily movement. A force of 15 gm, 30 gm, 30 gm, 100 gm was applied at different areas to achieve the intended tooth movement (intrusion, extrusion, tipping, and bodily movement) (Fig. 3a-d). This force was applied to the simulated maxillary central incisor with two different periodontal ligament thickness i.e. 0.15 mm and 0.24 mm and two alveolar bone heights (at the apex and alveolar crest).

Fig. 3.

–Force levels applied. a-intrusion - 15 gm, b-extrusion – 30 gm, c-tipping – 30 gm, d-bodily movement −100 gm

3. Results

The result of finite element structural analysis is represented in colourful contours. The spectrum of different colour bands in the figures depicts the stress pattern and level of stress distribution in the alveolar bone and the periodontal ligament under different types of tooth movements. The analysis of stresses produced under intrusion, extrusion, tipping, and bodily displacing forces at two different periodontal ligament thickness (0.15 mm, 0.24 mm) and alveolar bone height was performed and can be seen in Table 2, Table 3, Table 4, Table 5, Fig. 4a–d respectively.

Table 2.

Stresses generated under Intrusion forces at 0.15 mm and 0.24 mm PDL thickness.

Stress Levels	Thickness of PDL 0.15 mm	Thickness of PDL 0.24 mm
Maximum stress seen on PDL labio-lingually	3.5843 E−5 N/mm2	2.185E−5 N/mm2
Minimum stress on PDL at alveolar crest	1.057 E−7 N/mm2	8.4367E−8 N/mm2
Maximum stress on alveolar bone at alveolar crest	9.0421E−003 N/mm2	7.3535E−003 N/mm2
Minimum stress on alveolar bone at apical region	1.7948E−004 N/mm2	1.7029E−004 N/mm2

Table 3.

–Stresses generated under Extrusion forces at 0.15 mm and 0.24 mm PDL thickness.

Stress Levels	Thickness of PDL 0.15 mm	Thickness of PDL 0.24 mm
Maximum stress seen on PDL labio-lingually	1.1948E−4 N/mm2	7.2833E−5 N/mm2
Minimum stress on PDL at alveolar crest	3.5246E−7 N/mm2	2.8122E−7 N/mm2
Maximum stress on alveolar bone at alveolar crest	2.8535E−002 N/mm2	2.2906E−002 N/mm2
Minimum stress on alveolar bone at apical region	6.0508E−004 N/mm2	5.7212E−004 N/mm2

Table 4.

Stresses generated under Tipping forces at 0.15 mm and 0.24 mm PDL thickness.

Stress Levels	Thickness of PDL 0.15 mm	Thickness of PDL 0.24 mm
Maximum stress seen on PDL labio-lingually	2.6532E−4 N/mm2	2.7807E−4 N/mm2
Minimum stress on PDL at alveolar crest	3.1109E−7 N/mm2	5.1122E−7 N/mm2
Maximum stress on alveolar bone at alveolar crest	9.1792E−002 N/mm2	6.5855E−002 N/mm2
Minimum stress on alveolar bone at apical region	2.0396E−003 N/mm2	2.313E−003 N/mm2

Table 5.

Stresses generated under Bodily movement forces at 0.15 mm and 0.24 mm PDL thickness.

Stress Levels	Thickness of PDL 0.15 mm	Thickness of PDL 0.24 mm
Maximum stress seen on PDL labio-lingually	2.7213E−4 N/mm2	2.8737E−3 N/mm2
Minimum stress on PDL at alveolar crest	4.6591E−6 N/mm2	6.2208E−6 N/mm2
Maximum stress on alveolar bone at alveolar crest	3.3532E−1 N/mm2	2.5144E−1 N/mm2
Minimum stress on alveolar bone at apical region	3.6845E−003 N/mm2	3.196E−003 N/mm2

Fig. 4.

–Von mises stress. a - 0.15 mm pdl at alveolar bone for intrusion, b - 0.24 mm pdl labio-lingually for extrusion, c - 0.15 mm pdl at alveolar bone height for tipping, d - 0.24 mm pdl at the alveolar crest for bodily movement.

4. Discussion

Over the years, it has been realized that many types of orthodontic tooth movements can produce different mechanical stresses (loading) at varying locations along the root.6, 7, 8^,13 Burstone,⁴ Reitan⁶ and many other authors had studied the phenomena of tissue reaction to various levels of force application.^7,10,18,20 Hack et al.¹⁶ had studied the distribution of force in the periodontal ligament on a 2-D model of an incisor with a parabolic shaped root. Same study was done by a different author.²¹ Toms²² made a 2-D plane finite element model of a mandibular premolar. The logical results of the FEM are highly dependable on the models developed; hence, they have to be precisely constructed to be equivalent to real objects in different aspects. From a structural point of view, geometry idealization, materials data selection, and definition of boundary conditions of the present analysis were sound. The pattern of initial displacement of a tooth may be influenced by various anatomic variables such as the dimension of the tooth, periodontal ligament space, and the alveolar bone. The biomechanical behaviour of teeth when subjected to orthodontic forces² can be modified by variations in periodontal ligament thickness. The present study showed that there was a greater displacement of the apex which can probably lead to root resorption and excessive injury to the periodontal ligament which leads to tooth mobility.

In this study 15 gm of an intrusive force was applied and the stress distribution observed was maximum near the apex and was minimum around the incisal edge, in both 0.15 mm and 0.24 mm PDL thickness. The results of this study were similar to the ones done by many other authors.^15,23, 24, 25 The present study results were not similar to those reported by Vikram et al.¹³ The distribution of stress in the cementum with thickness from 200 μm to 600 μm was concentrated maximum around the mid root region and minimum at the root tip. Geramy²⁶ also found that for the normal model no significant stress concentration was found by the application of intrusive force at the root apex. The present study also showed that the highest concentration of stress was found to be at the alveolar crest and minimum at the apex in a healthy alveolar bone when intrusive force was applied on both 0.15 mm and 0.24 mm PDL thickness. During intrusive tooth movement, the maximum force is concentrated over a small area at an apex, therefore, extremely light forces are required to produce an appropriate pressure with the PDL during the intrusion. For this movement, most of the previous studies^2,27,28 have evaluated the distribution of stress in periodontium by using the same force values which were greater than the optimal force values.

In the present study, when an extrusive force of 30 gm was applied as parallel to the long axis and in the direction of the incisal edge of the tooth from the center of the crown, the distribution of stress in the periodontal ligament with both 0.15 mm and 0.24 mm, was concentrated highest towards the root tip (apex) and lowest at the incisal edge.

In the present study, on applying a tipping force of 30 gm in a mesial direction and perpendicular to the long axis of the 3-DFEM of a tooth, the distribution of stress in the periodontal ligament with both 0.15 mm and 0.24 mm thickness, was concentrated maximum towards the root tip and minimum at the incisal edge. The highest periodontal ligament stress (7.9767 E⁻⁰⁷) is seen in 0.15 mm of PDL thickness as compared to 0.24 mm periodontal ligament thickness (3.2767 E⁻⁰⁶). The previous study was done by Andersen et al. ⁹also observed a noticeable change or variation in the stress distribution pattern from incisal edge to apex, on the application of tipping force. Tanne et al.²⁹ reported that while tipping movement, stresses are non-uniformly varied with a large difference from the incisal edge to the root apex. The present study shows that stresses generated were within the optimal range as suggested by Lee²⁰(0.015–0.26 n/mm²) with the loading configuration as given by Proffit⁵ (35–60 gm). The distribution of stress patterns and the values coincide exactly with the previous study done by David Rudolph et al.¹⁵ (0.013 N/mm²) at the root tip. The tipping type of tooth movement introduces several variables that mask the study of the relationship between the force and the rate of tooth movement. In addition to the initial compression of PDL, the most obvious variable is the unequal distribution of the force along the root. The highest force is at the alveolar crest, which decreases to zero at the axis of rotation, and it is also capable of deforming the wall socket.³⁰ The distribution of stress patterns and the values coincide with the previous study done by David Rudolph et al.¹⁵ (0.013 N/mm²) and Hemanth et al.¹(0.17152 N/mm²) at the tip of the root.

In the present study for bodily movement, when a force of 100 gm in lingual direction with a couple in the buccal crown direction was applied on a central incisor, the stress distribution was observed in the middle of the periodontal ligament and at the apex. The distribution of stress in the PDL with both 0.15 mm and 0.24 mm thickness was concentrated maximum towards the root tip and minimum at the incisal edge. The highest periodontal ligament stress (7.5886 E⁻⁰⁴) is observed in 0.15 mm periodontal ligament thickness as compared to 0.24 mm periodontal ligament thickness (6.3312 E⁻⁰⁴). The present study also shows that various patterns of stress distribution at two levels of alveolar bone (alveolar crest and apex). The highest stress value was found in the apical area as compared to alveolar crest in both 0.15 mm and 0.24 mm periodontal ligament thickness. Tanne et al.²⁹reported that patterns of the distributions of stress were different in different alveolar bone heights in both the qualitative and the quantitative aspects; i.e., apico-gingival level of stress distribution shifted more apically, and the stress levels also increased with the reduction of alveolar bone. On applying two simultaneous forces to the crown of a tooth, the tooth can be moved bodily. Hence, to achieve the same stress response, greater force is required for bodily tooth movement as compared to intrusion, extrusion, and tipping movement.¹

The present study also shows that various patterns of stress distribution at two levels of alveolar bone (alveolar crest and apex). The highest stress value was found at the apical area as compare to alveolar crest in both 0.15 mm (3.6845 E⁻⁰³) and 0.24 mm (3.1960E⁻⁰³) periodontal ligament thickness. The observations of the present study exactly coincide with the previous study done by Hemanth et al.¹ and Tanne et al.²⁸ These findings tell that the bodily movement can be more physiologic, as was suggested in the previous histological studies because of the lower and more uniform stress distribution in the periodontium.⁷ The present study was a linear study and therefore the accuracy in detecting a non-uniform stress distribution on the periodontium and alveolar bone was decreased which was explained by Toms et al.²² in their study. A non-linear study could have been very cumbersome to find the accurate force ratio for each situation. A similar study can be performed by considering all four incisors for extrusion, intrusion, tipping, and the bodily movement. Further studies should compare increasing force levels from lighter (eg, intrusion arch) to heavier (headgear and face masks). FEM studies are done in an in vitro environment. Everything is simulated using specifically assigned parameters. Actual readings seen will vary with the ones seen in the oral cavity. This is the biggest disadvantage of the method.

5. Conclusion

• •For almost all types of tooth movements, the stress pattern seen was nearly the same in all cases in both PDL thickness which is maximum at the apex of the central incisor reducing from apex to cervical region.
• •The apical stress-induced in PDL increases as the thickness of PDL decreases. The stress in alveolar bone decreases with an increase in PDL thickness. This will be helpful in planning intrusive movements in adult patients who may have compromised periodontium.
• •Intrusion, extrusion, and tipping movements showed the greatest amount of relative stress at the apex of the maxillary central incisor. The bodily movement produced forces at root apex and distributed it all over.
• •Clinically, this stress distribution can be used as a means, that in adult patients with increased PDL thickness at the root apex, the susceptibility of root resorption is more and care should be taken to use optimum orthodontic force levels (25 g/cm2) for tooth movement.

Ethical approval and consent to participate

Ethical approval for the study was taken from the Institutes Ethical Committee. - Consent for publication.

All authors give consent for publication of the article.

Availability of supporting data

Not applicable.

Conflicts of interest

• All authors state there is no competing interest.

Funding

There was no funding for this study.

Authors' contributions

MG and KM carried out the research work in gathering the articles for the study, thought of the concept and drafted the manuscript. RK, AY and SC participated in the structuring the research, defined the intellectual content and performed the quality check. RK, HK and MG conceived of the study and participated in its design, coordination and helped to draft the final manuscript. RK, HK, AY and KM were part of the manuscript preparation, editing and reviewing. All the authors read and approved the final manuscript.

Acknowledgements

Not applicable.