Construction of a three-dimensional finite element

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Dental Science - Review Article Construction of a three‑dimensional finite element model of maxillary first molar and it’s supporting structures M. Sameena Begum, M. R. Dinesh1, Kenneth F. H. Tan, Vani Jairaj, K. Md Khalid, Varun Pratap Singh2

Department of Orthodontics and Dentofacial Orthopaedics, The Oxford Dental College and Hospital, Bommanahalli, Bangalore, 1D.A.P.M.R.V. Dental College and Hospital, Bangalore, India, 2Noble Medical College Teaching Hospital and Research Centre, Kanchanbari, Biratnagar, Nepal

ABSTRACT The finite element method (FEM) is a powerful computational tool for solving stress‑strain problems; its ability to handle material inhomogeneity and complex shapes makes the FEM, the most suitable method for the analysis of internal stress levels in the tooth, periodontium, and alveolar bone. This article intends to explain the steps involved in the generation of a three‑dimensional finite element model of tooth, periodontal ligament (PDL) and alveolar bone, as the procedure of modeling is most important because the result is based on the nature of the modeling systems. Finite element analysis offers a means of determining strain‑stress levels in the tooth, ligament, and bone structures for a broad range of orthodontic loading scenarios without producing tissue damage.

Address for correspondence: Dr. M. Sameena Begum, E‑mail: drsameena@ gmail.com Received : 28-04-15 Review completed : 28-04-15 Accepted : 22-05-15

KEY WORDS: Finite element model, orthodontic forces, stress and strain

F

inite element models have their current origin and real use in mechanical engineering analysis and design.[1] It is as indispensable to an engineer who designs an airplane frame, as is a pair of arch wires to an orthodontist. The finite element method (FEM) has been successfully applied to the mechanical study of stress and strain in the field of engineering. It is a method for numerical analysis based on material properties.[2]

The FEM was introduced into dental biomechanical research in 1973[3] and since then has been applied to analyze the stress and strain fields in the alveolar support structures.[4]

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Finite element modeling is the representation of geometry in terms of a finite number of elements and their connection points known as nodes. These are the building blocks of numerical representation of the model. The “elements” present are of the finite number as opposed to a theoretical model with complete continuity. The object of interest has to be broken up into a “meshwork” that consists of a number of nodes on and in the object. These nodes or points are then connected to form a system of elements. For a two‑dimensional example, if the brick wall is the network, the bricks are the elements and the four corners where the bricks meet each other are the “nodes”.

DOI: 10.4103/0975-7406.163496

How to cite this article: Begum MS, Dinesh MR, Tan KF, Jairaj V, Md Khalid K, Singh VP. Construction of a three-dimensional finite element model of maxillary first molar and it's supporting structures. J Pharm Bioall Sci 2015;7:S443-50.

© 2015 Journal of Pharmacy And Bioallied Sciences | Published by Wolters Kluwer - Medknow

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By knowing the mechanical properties of the object, such as modulus of elasticity and Poisson’s ratio, one can determine how much distortion each part of the cube undergoes when other part is moved by a force.[2]

Review of Literature on the Finite Element Method In the general field of medicine, FEM has been applied mainly to orthopedic research[5‑10] in which the mechanical responses of bony structures relative to external forces were studied. Furthermore, some research[11‑13] has been carried out in order to investigate the soft‑tissue and skeletal responses to mechanical forces. The applications of the FEM in dentistry have been found in studies by Thresher and Saito,[14] Knoell,[15] Tanne and Sakuda,[16] Atmaram and Mohammed,[17] Cook et al.[18] Tanne,[19] Rubin et al.,[20] Moss et al.,[21] and Miyasaka et al.[22] However, the application of this theory is relatively new in orthodontic research. It has been shown in previous studies [16‑20] that the FEM can be applicable to the problem of the strain‑stress levels induced in internal structures. This method also has the potential for equivalent mathematic modeling of a real object of complicated shape and different materials. Thus, FEM offers an ideal method for accurate modeling of the tooth‑periodontium system with its complicated three‑dimensional geometry. Experimental techniques are limited in measuring the internal stress levels of the PDL. Strain gauge techniques[23,24] may be useful in measuring tooth displacement; however, they cannot be directly placed in the PDL without producing tissue damage. The photoelastic techniques[25] are also limited in determining the internal stress levels because of the crudeness of modeling and interpretation. The force systems that are used on an orthodontic patient can be complicated. The FEM makes it possible to apply analytically various force systems at any point and in any direction. Experimental techniques on patients or animals are usually limited in applying known complex force systems. It is very important to keep in mind that the FEM will give the results based on the nature of the modeling systems and, for that reason, the procedure for modeling is most important.

Steps Involved in the Generation of Finite Element Model • Construction of a geometric model • Conversion of the geometric model to a finite element model • Material property data representation • Defining the boundary condition • Loading configuration • Solving the system of linear algebraic equation • Interpretation of the results. 

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Construction of a geometric model The purpose of the geometric modeling phase is to represent a geometry in terms of points (grids), lines, surfaces (patches) and volume (hyper patches). In this study, the analytical model of maxillary first molar was developed according to dimensions and morphology found in a standard textbook of dental anatomy, physiology, and occlusion by Wheeler’s. The buccal aspect of maxillary first molar is constructed using key points, which are identified from Wheeler’s textbook. The key points are represented at different co‑ordinate positions. The following key Table 1: The keypoints of the buccal aspect of maxillary first molar K, 1 K, 2, −.5, −2 K, 3, −1, −4 K, 4, −.5, −6 K, 5, 0.25, −7.5 K, 6, 2, −8 K, 7, 3, −7.75 K, 8, 3.5, −7 K, 9, 4.25, −6.5 K, 10, 5, −7 K, 11, 5.5, −7.5 K, 12, 6.5, −8 K, 13, 8, −7 K, 14, 8.5, −6.75 K, 15, 9.75, −5.5 K, 16, 9.75, −3.5 K, 17, 9.5, −3 K, 18, 9.25, −1.5 K, 19, 9 K, 20, 8.75, 1 K, 21, 8.25, 2 K, 22, 8.15, 3 K, 23, 8.15, 5 K, 24, 8.1, 7 K, 25, 8.075, 9 K, 26, 7.5, 11 K, 27, 7, 12 K, 28, 6, 13

K, 29, 5.1, 13.25 K, 30, 4.6, 13.15 K, 31, 4.1, 14 K, 32, 3.1, 14.25 K, 33, 2.1, 13.75 K, 34, 1.1, 13 K, 35, 0.5, 11 K, 36, 0.1, 10 K, 37.9 K, 38, 0.5, 7 K, 39, 0.75, 5 K, 40, 1, 4 K, 41, 0.75, 2 K, 42, 0.5, 1 Y1=KY (33) X1=KX (33) K, 100, X1+0.5, Y1−0.75 K, 101, X1+0.25, Y1−1.75 K, 102, X1+0.35, Y1−3 K, 103, X1+0.5, Y1−3.75 K, 104, X1+1, Y1−5.75 K, 105, X1+1.5, Y1−7 K, 106, X1+2.5, Y1−7 K, 107, X1+3.5, Y1−6 K, 108, X1+4, Y1−5 K, 109, X1+3.5, Y1−3 K, 110, X1+3, Y1−2 K, 111, X1+2.75, Y1−1

Table 2: BSPLINE used to join keypoints BSPLINE, 1, 2, 3, 4 BSPLINE, 4, 5, 6, 7 BSPLINE, 7, 8, 9, 10 BSPLINE, 10, 11, 12, 13 BSPLINE, 13, 14, 15, 16 BSPLINE, 16, 17, 18, 19 BSPLINE, 19, 20, 21, 22 BSPLINE, 22, 23, 24, 25 BSPLINE, 25, 26, 27, 28 BSPLINE, 28, 29, 30, 31 BSPLINE, 31, 32, 33, 34 BSPLINE, 34, 35, 36, 37 BSPLINE, 37, 38, 39, 40 BSPLINE, 40, 41, 42, 1 BSPLINE, 33, 100, 101, 102, 103 BSPLINE, 103, 104, 105, 106 BSPLINE, 106, 107, 108, 109 BSPLINE, 109, 110, 111 BSPLINE, 111, 30 BSPLINE: Boundary smooth plane line Journal of Pharmacy and Bioallied Sciences August 2015 Vol 7 Supplement 2


Figure 1: Key point representation of the buccal aspect of maxillary first molar

Figure 3: Key point representation of the distal aspect of maxillary first molar

Figure 5: Key point representation of the occlusal aspect of maxillary first molar

points mentioned below were plotted on the grids on the work plane to build the buccal surface of the maxillary first molar [Table 1]. The scale of the grid in the work plane measured 1 mm in X‑axis and 1 mm in Y‑axis.(K is the command to build Journal of Pharmacy and Bioallied Sciences August 2015 Vol 7 Supplement 2

Figure 2: Key point representation of the mesial aspect of maxillary first molar

Figure 4: Key point representation of the palatal aspect of maxillary first molar

Figure 6: Line point representation of the buccal aspect of maxillary first molar

key point, syntax of k command k, n, x, y, z where n is key point number, x is the x‑coordinate, y is the y‑coordinate, Z is the Z‑coordinate) [Figures 1-5]. PREP7 (Activation of preprocessor) S445 


The key points positions of the buccal aspect are joint to form lines. [Figures 6-10] The key points numbering used for the built up is shown below; (BSPLINE ‑ boundary smooth plane line) [Table 2].

The key points positions of the buccal aspect are joint to form lines and from lines to areas. The area plot of buccal aspect is built using ANSYS preprocessor [Figures 11-15].

Figure 7: Line point representation of the mesial aspect of maxillary first molar

Figure 8: Line point representation of the distal aspect of maxillary first molar

Figure 9: Line point representation of the palatal aspect of maxillary first molar

Figure 10: Line point representation of the occlusal aspect of maxillary first molar.

Figure 11: Area representation of the buccal aspect of maxillary first molar

Figure 12: Area representation of the mesial aspect of maxillary first molar

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Figure 13: Area representation of the distal aspect of maxillary first molar

Figure 15: Area representation of the occlusal aspect of maxillary first molar

Figure 14: Area representation of the palatal aspect of maxillary first molar

Figure 16: Oblique view of the geometric model of the maxillary first molar

Figure 17: The geometric model of dentin

All the aspects of the maxillary first molar are built up in the similar manner, extruded and Boolean operations are carried out to form three‑dimensional geometric model [Figure 16]. Individual models of enamel, dentin, periodontal ligament (PDL) and bone structure are built up [Figures 17-20].


Figure 18: The geometric model of dentin

The coordinates defining the shape of the PDL was simulated as a 0.20 mm thick ring around the model of the tooth and bone. The software used for the geometric modelling was [Figure 21].

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Figure 19: The geometric model of periodontal ligament

Figure 20: The geometric model of alveolar bone

Figure 22: The geometric model of enamel was converted into the three-dimension finite element Figure 21: The geometric model comprises of tooth, periodontal ligament and alveolar bone

Figure 24: The geometric model of periodontal ligament was converted into the three-dimension finite element Figure 23: The geometric model of dentin was converted into the three-dimension finite element

Conversion of geometric model to finite element model Individual models of enamel, dentin, PDL and bone structure are converted to finite element models using [Figures 22-25] hypermesh 7.0. 

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The geometric models were converted into the finite element models using hypermesh 7.0 [Figure 26]. The finite element model generation was achieved with the help of ANSYS 10 software. The element shape which was described in the model was a solid with a 4 noded tetra hedra (solid45) with 3 degrees of freedom (translations in the nodal x, y, and Journal of Pharmacy and Bioallied Sciences August 2015 Vol 7 Supplement 2


Figure 25: The geometric model of alveolar bone was converted the three-dimension finite element

z directions). These elements were connected to adjacent elements with the help of nodes. The more the number of nodes and elements, greater will be the accuracy of the result. Hence, finite element model was constructed, which approximately consisted of 1, 69,036 elements and 29,518 nodes.

Figure 26: The geometric model was converted into the threedimensional finite element model

Table 3: Material parameters used in the finite element model Material Enamel Dentin Periodontal ligament Alveolar bone

Young’s modulus (Gpa)

Poisson’s ratio

65.00 15.00 0.05 10

0.32 0.28 0.30 0.33

Material property data representation The different structures in the finite element models are enamel, dentin, PDL, alveolar bone. Each structure is then assigned a specific material property.[26] These material properties were the average values reported in the literature. Each material is defined to be homogenous and isotropic[5] [Table 3].

Defining the boundary condition The boundary condition, in the finite element models were defined at all the peripheral nodes of the bone with 0° of movement in all directions (the nodes of the base of the models were fixed to prevent free body displacement of the model). The final model was confirmatory from an engineering point of view for this study. It is very important to keep in mind that the finite element model will give the results based on the nature of the modeling system and for that reason, the procedure for modeling is most important.

periodontium and the FEM has frequently been the method of choice. The FEM provides the orthodontist with quantitative data that can extend the understanding of the physiologic reactions that take place. In particular, such numerical techniques may yield an improved understanding of the reactions and interactions of individual tissues. Such detailed information on stresses and strains in tissues is difficult to obtain accurately by any other experimental techniques because of the interaction of the surrounding tissues, which may then distort the data obtained for any individual material response. By applying new techniques such as FEM can theoretically predict the stress and strain fields in the tooth, PDL and bone structures.

Financial support and sponsorship Nil.

Application of forces

Conflicts of interest

Once the models were constructed, a force is applied.

There are no conflicts of interest.

Solving the system of linear algebraic equations

References

The sequential application of the above steps leads to a system of algebraic equations where the nodal displacements are unknown. These equations are solved by the frontal solver technique present in the ANSYS software (version 11.0, ANSYS, Canonsburg, Pennsylvania, US, 1970).

Conclusion Over the last few decades, numerical methods have been extensively used to calculate the stress and strain fields in the Journal of Pharmacy and Bioallied Sciences August 2015 Vol 7 Supplement 2

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