

INVITED SUBMISSION 

Year : 2018  Volume
: 52
 Issue : 6  Page : 142150 

Infinite applications of finite element method
Rohan Mascarenhas^{1}, Shahistha Parveen^{2}, B Satish Shenoy^{3}, G S. Stanly Selva Kumar^{4}, Vardhadharajula Venkata Ramaiah^{5}
^{1} Prof., Department of Orthodontics and Dentofacial Orthopaedics, Yenepoya University (Deemed to be University), Mangalore, Karnataka, India ^{2} Reader, Department of Orthodontics and Dentofacial Orthopaedics, Yenepoya University (Deemed to be University), Mangalore, Karnataka, India ^{3} Prof. and Head, Department of Aeronautical and Automobile Engineering, Manipal Institute of Technology, Manipal University (MAHE), Manipal, Karnataka, India ^{4} Resident, Department of Orthodontics and Dentofacial Orthopaedics, Yenepoya University (Deemed to be University), Mangalore, Karnataka, India ^{5} Asst. Prof., Department of Dental Hygiene, College of Dentistry and Applied Health Sciences, Qassim University, AlRass, Kingdom of Saudi Arabia
Date of Submission  12Nov2018 
Date of Acceptance  22Nov2018 
Date of Web Publication  7Dec2018 
Correspondence Address: Prof. Rohan Mascarenhas Department of Orthodontics and Dentofacial Orthopaedics, Yenepoya University (Deemed to be University), Mangalore, Karnataka India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/jios.jios_242_18
Different types of orthodontic tooth movement may produce different mechanical stresses in varying locations within the root. The quantification of stress in the periodontal ligament is an important concept as stress in this tissue is transmitted to alveolus with subsequent bone remodeling and tooth movement. Constructing complicated threedimensional shapes of tissues with different biomechanical properties and characteristics make the finite element method (FEM), the most suitable means of analysis. Finite element analysis is a computerbased numerical technique for calculating the strength and behavior of materials. This method uses elements, nodes, and contour plots to provide a good visual impression. The basic concept of FEM is the subdivision of a body into many smaller elements. The FEM is an engineering resource used to calculate stress and deformations in complex structures, and it has been widely applied in biomedical research. FEM helps to visualize stress, strain, and displacement and to study the materials of complex geometric shapes made of different materials and complex loading. FEM gives results with a reasonable degree of accuracy and has proved to be a less timeconsuming process in experimental research. Even though small differences may remain between reality and FEM, the numerical system can reveal otherwise inaccessible stress distribution. The FEM has reformed biomechanical research in Orthodontics, as it represents a noninvasive, accurate method that provides quantitative and detailed data regarding orthodontic biomechanics and physiological responses occurring in tissues, tooth. However, FEM has “inherent” errors where an object with smooth surfaces will not be completely included, as the FE model uses approximated domain which tends to leave out some areas at the boundaries.
Keywords: Deformation, finite element analysis, finite element method, finite element model, strain, stress
How to cite this article: Mascarenhas R, Parveen S, Shenoy B S, Kumar G S, Ramaiah VV. Infinite applications of finite element method. J Indian Orthod Soc 2018;52, Suppl S2:14250 
How to cite this URL: Mascarenhas R, Parveen S, Shenoy B S, Kumar G S, Ramaiah VV. Infinite applications of finite element method. J Indian Orthod Soc [serial online] 2018 [cited 2019 Jan 20];52, Suppl S2:14250. Available from: http://www.jios.in/text.asp?2018/52/6/142/247060 
Introduction   
Force dosage during orthodontic treatment is one of the most difficult problems in orthodontics.^{[1]} Different types of orthodontic tooth movement may produce different mechanical stresses in varying locations within the root.^{[2]} Stress in the periodontal ligament (PDL) is believed to be the initiating factor in tooth movement.^{[3]} Proffit stated that “tooth movement is primarily a PDL phenomenon.”^{[4]} The quantification of stress in the PDL is an important concept as stress in this tissue is transmitted to alveolus with subsequent bone remodeling and tooth movement.
Constructing complicated threedimensional shapes of tissues with different biomechanical properties is the major challenge. These characteristics of the periodontium make the finite element method (FEM) the most suitable means of analysis because of its ability to handle various shapes and material homogeneity.^{[5]}
Finite element analysis (FEA) is a computerbased numerical technique for calculating the strength and behavior of materials. This method uses elements, nodes, and contour plots to provide a good visual impression. The basic concept of FEM is the subdivision of a body into many smaller elements. The FEM is an engineering resource used to calculate stress and deformations in complex structures, and it has been widely applied in biomedical research. FEM helps to visualize stress, strain, and displacement and to study the materials of complex geometric shapes made of different materials and complex loading. FEM gives results with a reasonable degree of accuracy and has proved to be a less timeconsuming process in experimental research. Even though small differences may remain between reality and FEM, the numerical system can reveal otherwise inaccessible stress distribution.
FEA was introduced into orthodontics by Yettram et al., in 1972.^{[6]} The orthodontic displacement of a tooth is the result of a mechanical stimulus, generated by a force applied to the crown of a tooth, being turned into biological reactions. From a mechanical point of view, an alteration in the stressstrain distribution within the PDL, and the surrounding alveolar bone is the first reaction to the application of an orthodontic load. This leads to an intraalveolar displacement of the tooth and bending of the surrounding alveolar bone, provided the forces used are large enough. FEM has been used to study the effect of forces on teeth, PDL, bone, and orthodontic appliances. By applying the FEM, orthodontics can shape and analyze any material or dentomaxillofacial structures.^{[7]}
In the last decade, the application of a wellproven predictive technique, the FEM, FEA has also been applied to the description of physical form changes in biologic structures particularly in the area of growth and development and various branches of dentistry. FEM, which is an engineering method of calculating stresses and strains in all materials including living tissues, has made it possible to adequately model the tooth and periodontal structure for scientific checking and validating the clinical assumptions. FEA techniques can replace stereolithographic models for presurgical planning, thus provides an economical solution for the same.
However, every finite element is based on an assumedshape function expressing internal displacements as functions of nodal displacements. A certain element may give accurate answers for a particular type and location of support and loading, but inaccurate answers for another type and location.
The FEM has reformed biomechanical research in Orthodontics, as it can represent a noninvasive, accurate method that provides quantitative and detailed data regarding the physiological responses occurring in tissues, such as the PDL and the alveolar bone.^{[8]}
Limitations of Finite Element Method   
FEM has “inherent” errors where an object with smooth surfaces will not be completely included, as the FE model uses approximated domain which tends to leave out some areas at the boundaries [Figure 1].
Steps in finite element method
 Construction of the geometric model
 Conversion of the geometric model to a FEM model
 Material property data representation
 Defining the boundary condition
 Loading configuration
 Solving the system of linear algebraic equation
 Interpretation of the results.
Finite element method in orthodontics
 The FEM makes it possible to analytically apply various force systems at any point and in any direction
 The effect of altering the geometry of the bracket, wires and applied force on the quality of orthodontic tooth movement, employing a threedimensional finite element computer model is another application of FEM in orthodontics
 In the field of dentofacial orthopedics, finite element models have been employed to evaluate the stress distributioninduced within the craniofacial complex during the application of protraction, restriction forces on maxilla and mandible, with and without rapid and slow expansion
 Expansion (unilateral or bilateral) can also be studied individually.
Basic requirements
Material properties
Material properties greatly influence the stress and strain distribution in a structure. These properties can be modeled in FEA as isotropic, transversely isotropic, orthotropic, and anisotropic.
Bone
Is a living tissue that models and remodels throughout life, thus continuously changing its mechanical behavior. Moreover, a clear discrimination between cortical and trabecular bone is not a straightforward procedure, especially in the transition areas. Having this in mind, it is anyway important to have a mathematical description (i.e., Young's modulus and Poisson's ratio) for both cortical and trabecular bone properties. In cortical bone, the osteons are aligned to the bone's longaxis or in case of shortbones along the direction of forces; therefore, cortical bone exhibits a higher Young's modulus.
Cortical bone as well as trabecular bone has viscoelastic properties. This means that it has different values for ultimate strength and stiffness depending on the strain rate during loading. In addition to this, the mechanical properties of the bone also depend on age and thus the level of mineralization.
Tooth
The magnitude and the stress pattern in the PDL for different types of tooth movement can be studied using FEM and the optimal forces for tipping, and the bodily movement can also be determined.
Intrusive, extrusive displacement and resulting stress fields, canine retraction, en masse retraction in sliding mechanics and frictionless mechanics can also be studied.
Periodontal ligament
Development of more appropriate, but not necessarily complex, mathematical models may offer clinicians better insight into the biological processes of orthodontic tooth movement and lead to more accurate means of predicting reactions. A better understanding of the PDL's biomechanical behavior under physiologic and traumatic loading conditions might enhance the understanding of the PDL's biologic reaction in health and disease. Providing greater insight into the response of the PDL would be instrumental to orthodontists and engineers for designing more predictable and therefore more efficacious orthodontic appliances.
Nonlinear analysis – The simulation of the PDL with its complex geometry, varied biomechanical response to loading and internal component interaction require the determination of elastic, plastic, and viscoelastic material properties. The exact idea of stress and strain in the PDL for different types of tooth movements can be determined with the nonlinear simulation. The nonlinear models provide a more realistic description of behavior since they consider different phases as interacting continuous media.
Brackets
Can be designed by FEM and is useful in the evaluation of stress distribution when subjected to archwire torsion and tipping forces. Threedimensional computer models of the brackets have proved to be a useful tool in the stress analysis when subjected to various forces and can provide key information to the development of an optimum bracket design.
Wires
The mechanical property of a wire can be influenced or changed depending on the dimension, shape, and material. These archwires can be of different crosssections such as edgewise, ribbon arch, round, and square.
The materials that are used for these archwires may be stainless steel, nitinol, and TMA. Rectangular archwires can be analyzed in edgewise or ribbon mode. Different shapes of the archwires may be used such as ovoid, narrow, and square for labial orthodontics and mushroom, straight for lingual orthodontics. Archwires may be flat or reverse curve.
The Finite Element Method   
Radiographs and scans (cone beam computed tomography [CT]/CT) can be used to generate the finite element model [Figure 2] and [Figure 3].
Single tooth: Anteriors
A single tooth can consist of different material properties for pulp, dentin, enamel, PDL, and bone [Figure 4], etc. However, when more teeth are modeled, the tooth can be given a single material property. Multiple teeth can be generated with bone which may be created uniformly from a scan. The bone can be defined to represent different densities of compact bone, cancelous bone, and inner cortical bone. The PDL and surrounding bone may be modeled seperately so as to study and accurately represents clinical situations.  Figure 4: Different components of a tooth (a) pulp (b) enamel (c) periodontal ligament (d) tooth (e) bone (f) tooth, periodontal ligament and bone (g) meshed model of periodontal ligament, (h) meshed model of tooth, periodontal ligament, and bone
Click here to view 
Different types of simple tooth movement such intrusion, extrusion, tipping, rotation can be studied using FEM [Figure 5]. The PDL and surrounding may be modeled separately.  Figure 5: Finite element analysis of periodontal ligament during (a) extrusion (b) intrusion (c) retraction
Click here to view 
FEM can be used to analyze the stress, strain, deformation with different materials, dimensions, and forces [Figure 6].  Figure 6: Principal elastic strain on lower incisor periodontal ligament (a) maximum (b) minimum
Click here to view 
Multiple teeth: Anteriors
Movement, such as intrusion, extrusion, retraction, and rotation, can be studied on multiple teeth using FEM [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11], [Figure 12].  Figure 10: Maximum principal stress on the periodontal ligament of incisors
Click here to view 
 Figure 11: Maximum principal stress on periodontal ligament of anteriors, labial view
Click here to view 
 Figure 12: Maximum principal stress on periodontal ligament of anteriors, lingual view
Click here to view 
Single tooth: Posteriors
Single or multiple posterior teeth can be modeled for stress, strain, and deformation analysis using FEM [Figure 13], [Figure 14], [Figure 15].  Figure 13: Meshed model (a) maxillary first molar tooth (b) periodontal ligament of maxillary first molar (c) teeth, periodontal ligament and bone
Click here to view 
Stress and displacement generated by different intrusive arches can be studied using FEM [Figure 16].  Figure 16: (a) Utility arch (b) burstone intrusion arch (c) Ksir (d) deformation of archwire
Click here to view 
FEM can also be used to study fixed orthodontics [Figure 17] and effect of power arm during retraction [Figure 18].
Different loop designs with different dimensions and material properties can be analyzed using FEM [Figure 19] and [Figure 20].
Effect of corticotomy on retraction can be analyzed [Figure 21].
Temporary anchorage devices can be analyzed [Figure 22].  Figure 22: Absolute anchorage biomechanics can be analyzed. (a) Mini screw (b) mini screw in bone (c) different locations of temporary anchorage devices (d) mini plate – buccal (e) mini plate – palatal, (f) dental implants – tooth
Click here to view 
Finite Element Method in Adult Orthodontics   
The percentage of adult patients who seek orthodontic treatment has increased significantly in recent decades. Excessive orthodontic force with advanced periodontal bone loss may traumatize the periodontium. Therefore, the intensity and point of application of forces must be modified. FEM can modify geometry for the simulation at different levels of bone loss [Figure 23].  Figure 23: Alteration of moment to force ratio can be simulated with bone loss in the anterior and posterior region
Click here to view 
Although theoretical approaches have explained the differences between labial and lingual orthodontics, FEM is better suited to analyze these differences [Figure 24]. FEM can analyze the effect of both intrusive and retractive forces on the tooth using FEM [Figure 24] and [Figure 29].^{[9]}  Figure 24: Combination of intrusion and retraction forces can bring about (a) Intrusion (b) tipping
Click here to view 
When intrusion and retraction forces are applied in labial orthodontics, the net force vector passes directly through the CR which brings about the intrusion whereas the net force vector in lingual orthodontics produces lingual tipping.^{[10]}
Lingual biomechanics requires lesser forces as compared to labial biomechanics [Figure 25] and [Figure 26]. This difference can be studied using FEM.^{[11],[12]} Application of 33.6 g tipping force lingually brought similar kind of stress as compared to 50 g tipping force labially [Figure 25].^{[11]}  Figure 25: Tipping force (a) 50 g labially (b) 50g lingually (c) 33.6 g lingually
Click here to view 
 Figure 26: Multiple teeth biomechanics (a) labial forces (b) lingual forces
Click here to view 
The characteristics of brackets with different material properties can be studied using FEM [Figure 27].
Virtual digital setup can be used to arrange the teeth and apply different forces to analyze and to compare stress and deformation in PDL, tooth, and bone in orthodontics [Figure 28] and [Figure 29].  Figure 28: (a) Archwire and bracket system (b) Archwire, brackets and teeth with periodontal ligament
Click here to view 
 Figure 29: Effect of force with lingual appliance (a) Intrusion (b) Retraction
Click here to view 
FEM can also be studied for retraction using TAD such as implants and miniplate with lingual orthodontics [Figure 22].
Orthopedic forces
FEM can be used to analyze stress and displacement in the craniofacial structures on application of restrictive and protraction forces on maxilla [Figure 30]. The effect of expansion forces alone or with protraction can be analyzed using FEM [Figure 31]. Effect of advancement forces or restrictive forces in mandible can also be studied. Nodes can be selected in areas of interest such as sutures [Figure 32].  Figure 30: Effect of orthopedics (protraction and restriction) with and without expansion
Click here to view 
 Figure 31: Finite element analysis of cleft (a) protraction and slow maxillary expansion (b) protraction and rapid maxillary expansion
Click here to view 
 Figure 32: Finite element method has been used to analyze changes in mandible on application of protraction and restrictive forces (a) displacement (b) stress
Click here to view 
The effects of forces on cleft models can be analyzed and iterated for unilateral cleft.
Fixed functional devices can be modeled and the forces generated and their effects on the teeth and jaws can be studied [Figure 33].  Figure 33: Effect of fixed functional appliance can be analyzed (a) Forsus (b) Effect on maxilla (c) Effect on mandible
Click here to view 
Conclusion   
Although the FEM is a very effective tool to study orthodontics, it should be used appropriately. The analysis carried out should be interpreted keeping in mind the limitations and shortcomings.
Without doubt, it is a tool for future appliance designs where biomechanics have to be evaluated. It is best when used for comparisons of stress distribution using different appliances and force applications.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
References   
1.  Tanne K, Koenig HA, Burstone CJ. Moment to force ratios and the center of rotation. Am J Orthod Dentofacial Orthop 1988;94:42631. 
2.  Reitan K. Effects of force magnitude and direction of tooth movement on different alveolar bone types. Angle Orthod 1964;34:24455. 
3.  Storey E. Force in orthodontics and its relation to tooth movement. Aust J Dent 1952;56:118. 
4.  Proffit WR. Contemporary Orthodontics. 3 ^{rd} ed. St. Louis: Mosby Co.; 2000. p. 296. 
5.  Rudolph DJ, Willes PM, Sameshima GT. A finite element model of apical force distribution from orthodontic tooth movement. Angle Orthod 2001;71:12731. 
6.  Yettram AL, Wright KW, Houston WJ. Centre of rotation of a maxillary central incisor under orthodontic loading. Br J Orthod 1977;4:237. 
7.  Lotti RS, Machado AW, Mazzieiro ÊT, Landre J Jr. Scientific applicability of the method of finite elements. Rev Dent Press Ortodon Ortop Facial 2006;11:3543. 
8.  Kamble RH, Lohkare S, Hararey PV, Mundada RD. Stress distribution pattern in a root of maxillary central incisor having various root morphologies: A finite element study. Angle Orthod 2012;82:799805. 
9.  Mascarenhas R, Revankar AV, Mathew JM, Chatra L, Husain A, Shenoy S. Effect of intrusive and retraction forces in labial and lingual orthodontics: A finite element study. APOS Trends Orthod 2014;4:36. [Full text] 
10.  Geron S, Romano R, Brosh T. Vertical forces in labial and lingual orthodontics applied on maxillary incisors – A theoretical approach. Angle Orthod 2004;74:195201. 
11.  Mascarenhas R, Chatra L, Shenoy S, Husain A, Mathew JM, Parveen S. A comparative study of forces in labial and lingual orthodontics using finite element method. J Indian Orthod Soc 2015;49:15. [Full text] 
12.  Mascarenhas R, Shenoy S, Parveen S, Chatra L, Husain A. Evaluation of lingual orthodontic appliances. J Comput Methods Sci Eng 2017;17:25360. 
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10], [Figure 11], [Figure 12], [Figure 13], [Figure 14], [Figure 15], [Figure 16], [Figure 17], [Figure 18], [Figure 19], [Figure 20], [Figure 21], [Figure 22], [Figure 23], [Figure 24], [Figure 25], [Figure 26], [Figure 27], [Figure 28], [Figure 29], [Figure 30], [Figure 31], [Figure 32], [Figure 33]
