A Finite Element Simulation of the Temporomandibular Joint of a Pig A

Home , Articular disk, Human head

A Finite Element Simulation of the Temporomandibular Joint of a Pig

A thesis presented to

the faculty of

the Russ College of Engineering and Technology of Ohio University

In partial fulfillment

of the requirements for the degree

Master of Science

Sarang G. Dalne

November 2009

This thesis titled

A Finite Element Simulation of the Temporomandibular Joint of a Pig

SARANG G. DALNE

has been approved for

the Department of Mechanical Engineering

and the Russ College of Engineering and Technology by

John R. Cotton

Assistant Professor of Mechanical Engineering

Dennis Irwin

Dean, Russ College of Engineering and Technology 3

ABSTRACT

DALNE, SARANG G., M.S., November 2009, Mechanical Engineering

A Finite Element Simulation of the Temporomandibular Joint of a Pig (98 pp.)

Director of Thesis: John R Cotton

Temporomandibular joint (TMJ) disorders affect 20 to 25% of the population and

include pain dysfunction, arthritis, and internal derangements. The pig is a common

model due to its similar structure and mechanics to humans. This work represents the first

finite element analysis (FEA) of pig TMJ. The primary objective was to investigate the

stress distribution pattern in the TMJ articulating disc. The TMJ disc was defined as a

Mooney-Rivlin material. Geometry was obtained from CT scans. Retrodiscal and TMJ

ligaments were included as linear springs. The disc was considered glued to the mandible, but had frictionless contact with the temporal bone. Muscle loads were used to simulate clenching. It was observed that the TMJ disc followed the condylar motion.

Also, significant lateral movement of the mandible was observed. The intermediate- lateral part of the disc was most stressed. The observed stresses were comparable to the

human TMJ models in magnitudes and locations.

Approved: ______

John R. Cotton

Assistant Professor of Mechanical Engineering 4

ACKNOWLEDGMENTS

I am very thankful to number of people for their support and guidance to help me

accomplish this thesis work. I am grateful to Dr. John R Cotton; who is my mentor, for all his efforts and guidance. I would also like to thank Dr. Betty Sindelar for her valuable inputs to the present study.

I am also thankful to my other committee members, Dr. Betty Sindelar,

Dr.Hajrudin Pasic and Dr. Robert Williams II for being a constructive critic for my work.

I would also like to acknowledge the help of Dr. Larry Witmer and the Ohio University

µCT facility for providing us with their data.

At last, I would like to thank all the Mechanical Engineering faculties and my fellow students for their help and support.

TABLE OF CONTENTS

Page

Abstract ...... 3

Acknowledgments...... 4

List of Tables ...... 7

List of Figures ...... 9

Chapter 1: Introduction ...... 11

1.1 Purpose of study ...... 11

1.2 Introduction to the TMJ ...... 12

1.2.1 What is the TMJ? ...... 12

1.2.2 TMJ Geometry ...... 13

1.2.2.1 The Temporal Bone ...... 15

1.2.2.2 Mandible ...... 16

1.2.2.3 Articular Disc ...... 17

1.2.2.4 Articular Cartilage ...... 18

1.2.2.5 Muscles ...... 18

1.2.2.6 Ligaments ...... 20

1.2.3 TMJ Disorders ...... 21

1.3 Background ...... 21

Chapter 2: Methods ...... 29

2.1 Geometry ...... 29

2.2 Mesh ...... 38 6

2.3 Material properties ...... 43

2.3.1 Linear material model ...... 43

2.3.2 Non-Linear model ...... 44

2.5 Contact ...... 53

Chapter 3: Results and discussion...... 57

3.1 Results ...... 57

3.2 Discussion ...... 73

Chapter 4: Conclusion...... 83

References ...... 85

Appendix A : Calculation of couple forces ...... 93

Appendix B: Calculation of stiffness for the TMJ ligaments ...... 95

LIST OF TABLES

Page

Table 1. The information regarding the muscles of mastication. This information includes the name of muscle with meaning, the place of origin, insertion and the function performed by each [16]…………………………………………………………………………..20 Table 2. Summary of the individual solid mesh information for the different entities ………………………………………………………………………………………………….…...39 Table 3. The values of the material properties for the temporal bone, articular disc and the mandible. E is the elastic modulus specified in MPa, ʋ stands for the Poisson’s ratio and k for the spring stiffness. Source column indicated the original source from where these values have been taken...... 44 Table 4. The information regarding the muscle forces used. Position of the muscle forces are given as the x, y, z coordinates of the node referred as node id. The magnitude of force is given in Newtons. Inclinations are the angles made by the muscle force vectors with the occlusal plane. In the last column the x, y, z components of muscles forces is specified...... 47 Table 5. The magnitude of the couple forces. Calculations for these couple forces are given in Appendix A...... 52 Table 6. Type of contact used with respective material model...... 54 Table 7. TMJ models used in this study...... 57 Table 8. The von Mises stress details for the linear elastic TMJ disc. Averages and standard deviations for every band were calculated by considering the both the surface and the internal nodes of the TMJ disc...... 60 Table 9. The major principal stress details for the linear elastic TMJ disc. All stresses are in MPa. Averages and standard deviations for every band were calculated by considering the both the surface and the internal nodes of the TMJ disc...... 62 Table 10. The von Mises stress details for the Mooney-Rivlin TMJ disc. Note: Averages and standard deviations for every band were calculated by considering the both the surface and the internal nodes of the TMJ disc ...... 67 Table 11. The major principal stress details for the Mooney-Rivlin TMJ disc. All stresses are in MPa. Averages and standard deviations for every band were calculated by considering the both the surface and the internal nodes of the TMJ disc...... 69 Table 12. The compressive/tensile response of the TMJ disc in different regions for two material models used in this study. C- Compression, T- Tension. This table reports the C 8 or T for a region depending upon the number of nodes in compression or tension in that region...... 77 Table 13. The maximum von Mises stress and their location for different TMJ models...... 80 Table 14. The maximum and minimum principal stresses and their location for different TMJ models...... 82 9

LIST OF FIGURES

Page

Figure 1. The location of the temporal bone as highlighted by the yellow ring [4] ...... 12 Figure 2. The articular disc with the ligament attachments [1,7]...... 15 Figure 3. The skull with different bones forming the cranium along with the location of the temporal bone [10]...... 16 Figure 4.The mandible with the condylar process which fits into mandibular fossa [10] 17 Figure 5. The articular disc (sagittal view) shows the various zones of the articular disc. The posterior band refers to posterior region of disc with maximum thickness; intermediate zone is the central portion of the disc where it is thinnest. Posterior attachments are the posterior bilaminar retrodiscal tissues [12]...... 18 Figure 6. Muscles of mastication [10] ...... 19 Figure 7. The 2D model created by J Chen [21]. Model showing the mandibular condyle, different ligament attachments, articular disk and fossa eminence complex...... 23 Figure 8. (A) A TMJ model with the disc attachments in the form of the springs. (B) TMJ model with disk being attached posteriorly. (C) Model with no attachments to the disk. [24] ...... 24 Figure 9. The 3D model of the J H Koolstra study [13] showing the muscle loads (A). (B) Shows the cartilaginous material. (C) Highlights the TMJ in the sagittal view [13] ...... 25 Figure 10. A 3D model of Perez Del Palomar [16], with highlighted healthy and displaced TMJ...... 27 Figure 11. The 3D TMJ models used in the Miho Hirose and Tanaka E study [25] ...... 28 Figure 12. A single slice of the CT image of the pig head. Brighter pixels representing denser material. Bone is represented by the brightest pixels...... 30 Figure 13. The surface model of pig head developed from the CT images without any segmentation...... 32 Figure 14. The CT image after performing the threshold segmentation...... 33 Figure 15. The CT image after refining the threshold segmentation result (A) Mandible (B) Temporal bone ...... 33 Figure 16. The surface model generated with Amira after the segmentation (A) Mandible (B) Temporal bone...... 34 Figure 17. (A) The mandible surface extracted from the surface model of mandible. (B) The temporal surface extracted from the surface model of the temporal bone. (C) The mandible and the temporal bone surface together create the surfaces of the disc model. (D) The solid model of the articular disc...... 35 Figure 18. (A) Retrodiscal Ligaments superior and inferior (B) TMJ Ligaments ...... 36 Figure 19. Shows the line of action of the spring in the current model and the ideal line action of the inferior retrodiscal ligaments ...... 37 Figure 20 Shows the surface over imposition of the extracted temporal and mandible surface on the solid meshes (A) Overlap with mandible solid mesh (B) Overlap with 10

temporal solid mesh (C) Extracted mandible surface overlap on the TMJ disc solid mesh (D) Extracted temporal surface overlap on the TMJ disc solid mesh ...... 40 Figure 21. (A) The solid mesh of the mandible. (B) The solid mesh of the articular disc. (C) The solid mesh of the temporal bone...... 41 Figure 22. The combined solid mesh of the articular disc and the mandible (E) The combined solid mesh of the articular disc and the mandible...... 42 Figure 23. The approximate Force diagram of the muscle forces applied to the mandible. (A) Medial view (B) Lateral view...... 48 Figure 24. The contact triangle for the medial pterygoid with the centroid as the point of application of the muscle tension...... 49 Figure 25. The muscle forces applied as point loads with MSC. Marc. (A) Lateral pterygoid (B) Masseter (C) Medial Pterygoid...... 50 Figure 26. The muscle forces applied as point loads with MSC. Marc. (A) Temporalis in (B) Temporalis out...... 51 Figure 27. The food resistance force with couple. (A) Shows the resistance force in Y direction, the x and z forces contributing to the couple (B) x and z forces of the couple. Note: In (A) the z force is not visible, this z force forms couple with z force in (B)...... 52 Figure 28. (Clockwise from the top left) Front view, side view, top view and top view of the constrained temporal bone and the zygomatic arch...... 53 Figure 29. Contact bodies...... 55 Figure 30. The TMJ disc division into anterior, intermediate and posterior bands with the sub divisions. CP- central part, MP- medial part and LP- lateral part [9]...... 58 Figure 31. The Equivalent von Mises stress distribution in the linear elastic TMJ disc. The stresses are given in N/cm2. (100 N/cm2 = 1 MPa)...... 60 Figure 32. The major principal stress distribution in the linear elastic TMJ disc. The stresses are given in N/cm2...... 61 Figure 33. The compressive stress distribution across the medial to lateral direction of the anterior band of the linear elastic TMJ disc. All stresses are in N/cm2...... 63 Figure 34. The compressive stress distribution across the medial to lateral direction of the intermediate band of the linear elastic TMJ disc. All stresses are in N/cm2...... 64 Figure 35. The compressive stress distribution across the medial to lateral direction of the posterior band of the linear elastic TMJ disc. All stresses are in N/cm2. Grey is tension. 65 Figure 36. The Equivalent von Mises stress distribution in the linear elastic TMJ disc. Note: The stresses are given in N/cm2...... 67 Figure 37. The major principal stress distribution in the Mooney-Rivlin TMJ disc. The stresses are given in N/cm2...... 68 Figure 38. The compressive stress distribution across the medial to lateral direction of the anterior band of the Mooney-Rivlin TMJ disc. All stresses are in N/cm2...... 70 Figure 39. The compressive stress distribution across the anterior to posterior direction of the medial part of the Mooney-Rivlin TMJ disc. All stresses are in N/cm2...... 71 Figure 40. The compressive stress distribution across the medial to lateral direction of the posterior band of the linear elastic TMJ disc. All stresses are in N/cm2...... 72 Figure 41. Pig mandible with the distances used for the couple force calculations...... 93 11

CHAPTER 1: INTRODUCTION

1.1 Purpose of study

This study analyzes the stress distribution patterns in the temporomandibular joint

(TMJ) disc of the pig when the muscle loads are applied to simulate a static case of occlusion or biting. A finite element technique was used to achieve this purpose. The goals of this technique are, first the development of the first finite element model of a pig TMJ and the surrounding bone, and second is to analyze the stress/strain patterns developed in the porcine TMJ when the physiological loads are applied and compare to human models of TMJ. The first goal is comprised of two stages: the first stage develops a solid model from a computed tomography (CT) scan and the second stage creates a finite element mesh using this solid model and assigns suitable materials to this mesh.

The second goal involves selecting relevant loads and boundary conditions. The final step is to obtain the stress/strain patterns within the TMJ by using a suitable force scenario and then compare those patterns with those obtained from human models.

Thus, the steps involved in achieving the purpose of this study can be summarized as

1. To develop the first finite element model of the TMJ of a pig

2. To analyze the stress/strain patterns in the porcine TMJ

¾ Obtaining the stress/strain patterns in the pig’s TMJ

¾ Comparing these patterns with those of the human TMJ will be the

secondary objective 12

This finite element model of the pig TMJ will predict the stresses. This will help clinicians get a better understanding of how the TMJ behaves for a particular loading condition. So that in turn, they can develop a better clinical treatment for the TMJ disorders.

1.2 Introduction to the TMJ

1.2.1 What is the TMJ?

The TMJ is a synovial joint which allows the movement of the jaw with respect to

the skull [1,2]. There is one TMJ on each side of the face [1]. These joints are

symmetric in geometry [1]. They are located just in front of the ears as shown in Figure

1; one can feel the movement in these joints just by pressing their fingers on the skin

just in front of ears and opening and closing the mouth [3].

Figure 1. The location of the temporal bone as highlighted by the yellow ring [4] 13

1.2.2 TMJ Geometry

The geometry of the TMJ is discussed here as we go from the superior (top) to

inferior (bottom) direction. The upper part of the TMJ is the temporal bone [1,2]. There is a depression in the temporal bone which is called the articular fossa or mandibular fossa as shown in Figure 2 [1,2]. The temporal bone is covered with articular cartilage on the articular fossa [1,2]. In this depression the articular disc and the mandible fit to form the

TMJ [1,2]. The temporal cartilage through synovial fluid is in contact with the articular disc [1,2]. The articular disc on the inferior side is in contact with the condyle of the mandible [1,2]. This condyle is also covered with articular fibrocartilage [1,2]. The

articular disc facilitates and stabilizes the smooth motion within the joint and acts to distribute stress more uniformly [1,2]. The disc is held in its position with the help of the ligaments as shown in Figure 2[1,2]. Collateral ligaments are most significant in holding

the disc on to the condyle [1,2]. Muscles also play an important role in the operation of

the joint; they facilitate the opening, closing, lateral movement and the balancing action

of the mandible [1,2].

Even though pigs and humans are said to have similar TMJ anatomy [5] there are

few differences in them [6]. Pigs have a longer and narrower mandible as compared to

humans. Pigs also have a sharper coronoid process. Also, pigs have a more rounded

mandibular condyle as compared to humans. Herring [6] reported the difference between

articulation of the condyle with the temporal bone between humans and pigs. She has

reported that in the case of pigs the condyle fits with the articular eminence, whereas in

humans the condyle fits in the mandibular fossa [6]. Thus, in humans there is a well 14

rounded concave cavity called mandibular fossa in the temporal bone, whereas this

curved cavity is absent in the pig temporal bone [10,55]. Instead of the rounded

mandibular fossa pigs have a curved articular eminence. The absence of mandibular fossa

in pigs has an effect on the mandibular motions. For example, where opening movement

the mandible has to translate in the forward direction inside the fossa whereas no such

translation is required for the opening movement in pigs [6]. In humans, the articulation

of the mandible and the temporal bone is located just after the zygomatic arch ends,

whereas in pigs the zygomatic arch is present in the lateral direction of the articulation

[55]. Pigs also have a very large medial pterygoid muscle as compared to humans. Also, pigs have a muscle called zygomaticomandibularis which is absent in humans. [5-

6,10,55]

Higher primates like monkeys or apes would be the closest to model a human

TMJ. But, except great apes all others do not have the mandibular fossa. Also, using great apes as models is not economical. Thus, apart from the differences it is suitable to use pig models for the study of human as it has been reported that they have very similar TMJ anatomy, soft tissue structure and jaw movements [5-6] and are also very economical to use. [5-6]

Figure 2. The articular disc with the ligament attachments [1,7].

The main parts of the TMJ are described in greater detail below.

1.2.2.1 The Temporal Bone

The temporal bone is one of eight bones which form the cranium, which is the collective name given to the bones forming the skull except the jaw [8]. The temporal bone is located at the inferior side of the cranium, as shown in Figure 3[9]. The temporal bone has a depression referred to as the mandibular fossa located between the squamous

and petrous portions [10].

Figure 3. The skull with different bones forming the cranium along with the location of the temporal bone [10].

1.2.2.2 Mandible

The mandible is the lower half of the face as shown in Figure 4. The condylar process is the rounded portion at the top of the mandible [1]. The section of the mandible along the superior inferior axis is called the ramus [10]. The mandible is the

lone bone in human head capable of motion [10]. 17

Figure 4.The mandible with the condylar process which fits into mandibular fossa [10]

1.2.2.3 Articular Disc

The articular disc shown in Figure 5 is located in the region between the

temporal bone and the mandible of the TMJ [2,11]. The articular disc is tightly fixed

over the edge of the mandible and approximates the shape of the cavity or capsule

[2,11]. This disc is not uniform in its thickness as shown in Figure 5 [11]. The articular

disc is thinnest in its intermediate zone [2]. This is the zone where the disc is in contact

with the mandible [11]. The disc is thickest in its posterior and anterior region [2]. The

disc gets thick as we move towards anterior, medial and lateral directions [11]. 18

Figure 5. The articular disc (sagittal view) shows the various zones of the articular disc. The posterior band refers to posterior region of disc with maximum thickness; intermediate zone is the central portion of the disc where it is thinnest. Posterior attachments are the posterior bilaminar retrodiscal tissues [12].

1.2.2.4 Articular Cartilage

The articular cartilage covers the articulating surfaces of bones [1]. The

articulating cartilage acts as a buffer, distributing stresses that would otherwise

concentrate on the specific region of the bone [13]. It also facilitates the load distribution

within TMJ [14]. The fibrocartilage is found in TMJ [2,15].

1.2.2.5 Muscles

The muscles of mastication are the lateral pterygoid, medial pterygoid, masseter

and temporalis as shown in Figure 6 [15]. Masticatory muscles facilitate the various

movements of the TMJ by activating themselves in different patterns [10,16]. 19

Table 1 gives the information regarding the origin, insertion and function of each of the above listed muscle.

Figure 6. Muscles of mastication [10]

Table 1 The information regarding the muscles of mastication. This information includes the name of muscle with meaning, the place of origin, insertion and the function performed by each [10].

1.2.2.6 Ligaments

The ligaments in the TMJ help to keep together the capsule of the TMJ as the

capsule by itself cannot check the moving parts of the TMJ [11]. The TMJ ligaments

perform the function of limiting the motion either of the articular disc or of the mandible

[2,11]. There are three types of ligaments present in the TMJ [2]. They are

temporomandibular ligament, sphenomandibular and stylomandibular ligaments [2]. The

temporomandibular ligament functions to control the condylar motion [11]. The

sphenomandibular checks the angle of the mandible whereas the stylomandibular checks

protrusion on opening [11]. 21

1.2.3 TMJ Disorders

As the TMJ is one of the profusely operational joints in the body, its disorders are also quite common [1]. It is an estimate that 20 to 25% of the population suffers from a temporomandibular disorder (TMD), whereas only 3 to 4% of sufferers get suitable treatment [17]. Commonly TMDs are caused by malfunctioning of the following parts of the TMJ: articular disc, articular cartilage, muscles at the joint, ligaments, teeth [18].

These may lead to TMDs like the pain dysfunction, Bruxism, arthritis or internal irregularities [1]. The malfunctioning of the parts of the TMJ can be as follows:

• Internal disarrangement of the articular disc causing it to slip out of its

normal position in the joint capsule. [1]

• Wearing out of the articular disc due to continuous joint overloads, which

might be due to clenching of teeth during sleep or teeth grinding, improper

chewing pattern or excessive use of chewing gums [3].

• Wearing out of the joint cartilage may be caused due to arthritis, fatigue in

the muscles of mastication, ligament tear or irregularities in teeth

alignment. [3,20]

1.3 Background

Computational finite element studies of human TMJ biomechanics started in the early nineties with development of a 2D model by Chen [21,22] and 3D model by

Korioth [23]. Both Chen and Korioth used linear elastic material properties and used condylar displacements for loading the joint [21-23]. I have identified 43 published works on finite element analysis of TMJ with ISI search. Out of these works I identified 22

five critical works. These works are of Chen [21] which was the first 2D model of the

TMJ, Koolstra [13] which was the first to use rigid body analysis with finite element

analysis, DeVocht [24] which looked at the effect of the disc attachments towards the disc movement, Perez del Palomar [16] which addressed the asymmetry within the TMJ and the last one by Hirose and Tanaka [25] which addressed the clenching stresses in

TMJ. These works are described in detail below.

The first 2D finite element model of the human TMJ [21] was published in 1994

by Chen. This model was developed with the primary objective of analyzing the joint

stresses [21]. Until then studies were mainly concerned about the joint reaction forces

[26-32]. Chen [21] included the articular disc, mandibular condyle, temporal bone and

the ligaments as shown in Figure 7 in this model. For the material properties the articular

disc was assumed as a linear elastic solid with the bones as rigid bodies [21]. This is

justified because the bone material is much harder material than that of the articular disc

[21]. The ligaments were modeled as nonlinear spring elements [21].

Figure 7. The 2D model created by J Chen [21]. Model showing the mandibular condyle, different ligament attachments, articular disk and fossa eminence complex.

Chen [21] reported that with condylar displacement, the disc deformed while moving along the condyle. The superior and inferior surfaces of the disc were stressed with maximum compressive stress being 12 MPa, whereas maximum tensile stress was 4

MPa [21].

J DeVocht [24] examined the movement of the disc during the functioning of the

TMJ. His group created 2D models of the TMJ with and without the discal ligaments as shown in Figure 8 [24]. The objectives of their study was to monitor the disc movement, obtain the stress distribution in the joint while simulating jaw opening and quantify the modulus of elasticity of the articular disc [24]. For the material properties, they have considered both the bones and the disc as linear elastic materials and used springs to represent ligaments [24]. They have performed static analysis at small steps of condylar displacement to simulate the complete motion of the joint [24].

Figure 8. (A) A TMJ model with the disc attachments in the form of the springs. (B) TMJ model with disk being attached posteriorly. (C) Model with no attachments to the disk. [24]

The result of this study [24] showed that the disc behaves in the same way with and without any attachments. The maximum von Mises stress in the articular disc reported by DeVocht [24] is 0.4 MPa. The stresses in the bones varied with the variation in the value of elastic modulus of the disc [24].

Koolstra [13] used rigid body analysis and finite element analysis techniques together in one model shown in Figure 9. The primary objective of this study was to analyze the compatibility of this combined technique in a biomechanical system [13]. For this they have created a 3D finite element model of the TMJ with cartilage [13]. For material properties a Mooney-Rivlin model was used to describe properties of both the disc and the cartilage [13]. The bones were modeled as rigid bodies [13]. They have used 25 muscle forces as the loads on the joint instead of using the condylar displacements as used by Chen [21] and DeVocht [24] [13]. The reported results suggest that the disc experiences maximum von Mises stress of 2.5 MPa in the intermediate and the medial region during the opening and closing movement of the mandible [13]. There are mostly compressive stresses found in the cartilage [13].

Figure 9. The 3D model of the J H Koolstra study [13] showing the muscle loads (A). (B) Shows the cartilaginous material. (C) Highlights the TMJ in the sagittal view [13]

One of the most recent studies published by Perez del Palomar and Doblare [16] addresses the asymmetric nature of the TMJ. The primary aim of this study was to obtain and compare the disc behavior in the two situations [16]. They have created two finite element models one of a healthy and the other of a TMJ with an anteriorly displaced articular disc as shown in Figure 10 [16]. For the material properties they have considered bones as rigid bodies, the disc as a fiber reinforced poroelastic material and 26 ligaments as isotropic hyperelastic materials [16]. They have loaded the joint by doing percent activation of the muscles of mastication as given in 27

Table 1 [16]. The muscle forces are used at the steps of 25, 50 and 100% muscle

activation to simulate jaw opening and closing movement of the jaw [16]. The result reported by this study says that unhealthy joint experienced larger stresses than the

healthy one [16]. In a healthy disc the maximum compressive stresses were in intermediate region which was about 1.5 MPa [16].

Figure 10. A 3D model of Perez Del Palomar [16], with highlighted healthy and displaced TMJ.

One of the most recent studies by Miho Hirose and Tanaka E [25] investigates the

stress distribution in a healthy and anteriorly displaced human TMJ disc during prolonged clenching. For this study, they have created two TMJ models using the magnetic resonance images of two females as shown in Figure 11 [25]. For the material properties they have considered bones as linear elastic, the disc and the ligaments were modeled as linear viscoelastic material using Kelvin model [25]. They have loaded the joint using the

20% simultaneous activation of the muscles of mastication as given in 28

Table 1 [25]. The results reported that, for the healthy TMJ disc the maximum von Mises stress of 0.85 MPa was in the central and lateral part of the intermediate zone whereas the max von Mises stress of 0.41 MPa in the posterior region of the disc [25].

They have also reported stress relaxation in the retrodiscal tissue in case of the healthy disc [25].

Figure 11. The 3D TMJ models used in the Miho Hirose and Tanaka E study [25]

CHAPTER 2: METHODS

Any finite element study needs five basic inputs to accomplish the analysis [33].

These are the geometry, mesh, material properties, loads and boundary conditions [33].

The geometry refers to the solid model of the object one is interested in studying [33]. In

meshing, the solid model is discritized into a finite number of small elements for

numerical solution [33]. Then after the discretization, the solid model has to be assigned

material properties that approximate its real world material [33]. Then one can define the

loads and the boundary conditions to simulate the real world working condition of the

object [33]. After defining these five basic steps the FE analysis can be performed. [33]

2.1 Geometry

I developed the first 3D model of a pig’s TMJ. The geometry for the model was

taken from the temporomandibular joint of the Hanford pig skull [34]. A combination of

the CT images [16,35,36] shown in Figure 12 and a manual technique discussed later in

this chapter was used to obtain the desired geometry of the TMJ [24]. The CT images

were taken at the Ohio University Micro CT facility by Tickhill [34].

A CT scan uses X-rays to map an object [37]. The CT image displays a material

with higher X-ray attenuation as brighter pixels as shown in Figure 12 [37]. Depending upon the attenuation coefficient the brightness of the pixels is decided [37]. The brightness of the pixels is quantified in terms of the Hounsfield Units (HU) [37].

Standard Hounsfield units are defined for materials like water, air with zero for water and

-1000 for air [37,38]. The HU for bone is in the order of 1000 or more [39]. Bones are 30 represented by the brighter pixels in a CT image as they have higher attenuation coefficient i.e. scatter more x-rays than the other materials [38].

The CT images used for the current model were of 512 x 512 pixels per slice [34].

In total, 246 CT slices of the pig head was taken with slice thickness of 1mm [34].

Figure 12. A single slice of the CT image of the pig head. Brighter pixels representing denser material. Bone is represented by the brightest pixels.

As discussed earlier in section 1.2.2, the TMJ is comprised of the temporal bone, articular cartilage, articular disc, ligaments and the mandible arranged as shown in Figure

2 [1,2]. In this study, assuming symmetry between the right and left TMJs only the left

TMJ of a pig was modeled [40,41]. The current model includes the mandible, the articular disc, the surrounding part of the temporal bone and the ligaments 31

[13,16,22,25,40,42,43,36]. The articular cartilage was ignored in the current model

[21,24,40,41].

Chen [21] discussed how the effects of both the mandibular and the temporal cartilages are secondary to the TMJ mechanics. He considered cartilage to be parts of the mandible and the temporal bone in his model [21,22]. Thus, their effect can be ignored without much loss in accuracy. Koolstra [13] reports that the stresses in the disc are much

larger than those in the cartilage, cartilage functions just as cushion before the stresses get transferred to bones. In this study, we are more interested in the stresses experienced by the articular disc rather than the bones. Thus, the articular cartilage was ignored.

The bony part of the TMJ was determined from the CT scans [16,35,36]. The CT images were viewed using medical image viewing software Amira (Visage Imaging,

Carlsbad, CA, USA). Amira enables users to develop a 3D model from the stack of CT images, as shown in Figure 13 [44]. With a view to facilitate solid meshing, surface models of the mandible and part of the temporal bone surrounding the TMJ were created separately.

The process of selecting a structure in the CT image, in this case the bone in the region of interest, is called segmentation [44]. The mandible and the part of the temporal bone were segmented separately. The first step in the segmentation process was to perform a threshold segmentation using the threshold segmentation tool [44]. The threshold segmentation tool is a 3D tool which automatically selects the pixels within the specified brightness range for all slices or CT images in the stack [44]. In this case the 32 range was 2278 to 238. The threshold segmentation results were very coarse. The segmentation results after using the threshold segmentation tool are shown in Figure 14.

The second step was to select only the left mandible as shown in Figure 15 (A).

While selecting the mandible the threshold segmentation results were refined manually.

In the refinement process stray pixels were removed. These are the pixels which do not belong to the segmented object but are displayed as brighter pixels. The bone contour was smoothed and the holes within the mandible were considered as the part of the bone.

Figure 15 (A) shows the CT images after refined segmentation for the mandible.

Figure 13. The surface model of pig head developed from the CT images without any segmentation. 33

Figure 14. The CT image after performing the threshold segmentation.

(A) (B) Figure 15. The CT image after refining the threshold segmentation result (A) Mandible (B) Temporal bone

After the segmentation was refined, the 3D surface model of the mandible was

developed using the surfacegen tool as shown in Figure 16 (A) [44]. The same procedure

was followed for segmenting the temporal bone shown in Figure 15 (B) and developing

its surface model shown in Figure 16 (B). These surface models were used as the basis

for building a solid model using Marc (MSC Software, Santa Ana, CA, USA). The

surface models were saved in STL (stereo lithography) format to export them to Marc.

(A) (B) Figure 16. The surface model generated with Amira after the segmentation (A) Mandible (B) Temporal bone.

The geometry of the disc was created manually from the surface models of the mandible and part of the temporal bone. The articulating surface of the mandibular condyle was extracted from its surface model as shown in Figure 17 (A), using Marc.

This surface formed the lower surface of the articulating disc. The articulating surface of 35

the temporal bone shown in Figure 17 (B) was extracted similarly to the mandible. After extracting the surfaces for the lower and upper region, the next step was to combine the two surfaces in a single model as shown in Figure 17 (C) and connect them. Both the extracted surfaces were imported into one file. The medial, lateral, anterior and posterior geometry was approximated [16,40,43] by manually generating elements to connect the two surfaces and form closed surface defining the disc, as shown in Figure 17 (D).

(A) (B)

(C) (D) Figure 17. (A) The mandible surface extracted from the surface model of mandible. (B) The temporal surface extracted from the surface model of the temporal bone. (C) The mandible and the temporal bone surface together create the surfaces of the disc model. (D) The solid model of the articular disc.

As it was discussed in the introduction section, the articular disc is not of uniform

thickness, as was the case here. The disc thickness varied as we moved from anterior to

posterior and lateral to medial directions. The disc was thinnest in the intermediate zone

where its thickness was 0.25 cm. It was thickest in the posterior region where its thickness was 0.63 cm. The thickness in the anterior part was 0.42 cm.

To complete the TMJ geometry, the final addition was the ligaments. In this study, the posterior retrodiscal ligaments both superior and inferior [21,22] and the TMJ

ligaments [16,35,42] were included, as shown in Figure 18. The ligaments were added to

the model after the solid meshes of the bones and the disc were generated and combined.

The procedure of generating and combining the solid meshes will be described in section

2.2.

(A) (B) Figure 18. (A) Retrodiscal Ligaments superior and inferior (B) TMJ Ligaments

The ligaments were introduced as linear springs. The node to node spring connections were used, as shown in Figure 18. Nodes for the insertion and origin of 37

ligaments were picked according to Almora [2] and visual inspection of human TMJ

models [16,35,42]. These attachments were then reviewed by Dr. Sindelar, who has

experience of numerous TMJ dissections. In all 62 linear springs were used to model the

ligaments, out of which 39 represented the retrodiscal ligaments and 23 springs represented the TMJ ligaments. As ligaments were added, it was very important to avoid

passing the line of action for any of the spring through the solid model. Thus, the

attachment points were moved closer along the line of action to avoid passing of any

spring through the solid model. This adjustment was primarily done for the inferior

retrodiscal ligaments, as shown in Figure 19.

Figure 19. Shows the line of action of the spring in the current model and the ideal line action of the inferior retrodiscal ligaments

2.2 Mesh

As the surface models of the mandible, the temporal bone and articular disc were ready, the next step was to generate 3D solid meshes of them and combine them into one model. For this purpose surface models of each of the three entities were imported as

STL files into Marc one at a time.

Meshing divides an entity into a finite number of small elements [33]. In solid meshing the hollow solid is meshed internally to form a solid mesh [33]. Solid meshing was done using the Patran tetrahedral mesher provided in Marc. Patran tetrahedral mesher preserved the triangles of the surface mesh while the solid mesh was generated

[45]. The solid mesh developed contained 4 noded tetrahedral elements; details about the solid mesh are given in Table 2.

To generate the mandible solid mesh, the mandible surface model was imported in Marc. Then this surface model was checked for presence of any cross elements and inside out elements. The overlapping elements in Marc are referred as cross elements and the elements a having surface normal directing inwards to the solid are referred to as inside out elements [45]. Then, using the sweep command the nodes within a distance of

0.001 cm from each other were combined as one node, to form the closed surface so that solid meshing can be done. Sweeping these nodes also helps to remove cross elements.

After these checks on the surface model, the Patran tetrahedral mesher was invoked and a solid mesh of the mandible was generated with 4 noded tetrahedral elements as shown in

Figure 21 (A). The same procedure was followed to develop the solid meshes of the articular disc and temporal bone from their respective surface models as shown in Figure 39

21 (B, C). The details of the solid meshes of the individual and the complete model are

given in Table 2.

Table 2 Summary of the individual solid mesh information for the different entities. Entity Number of nodes Number of elements Degrees of freedom

Articulating disc 311 979 933 Temporal Bone 16,718 76,261 50,154 Mandible 8,481 39,388 25,443 Complete Model 25,510 116,628 76,530

The next step was to combine these three different solid meshes into one model. It

was critically important to make sure that these solid models will have node to node

contact on articulating surfaces when combined together. This will ensure proper contact

between the solid models and will also ensure that, no penetration of bodies into one another occurs during the simulation run. The articulating surfaces of the temporal bone

and the mandible were extracted from their respective surface models. These extracted

temporal and the mandible surfaces were then imported into the temporal and the

mandible solid meshes respectively, as shown in Figure 20A and B and it was checked

whether the nodes on each surface match up or not. This surface over imposition check was also performed on the articular disc solid mesh as shown in Figure 20 C and D. From these Figure 20 (A,B,C,D), it can be seen that there is perfect overlap between the surface of solid meshes and the extracted surfaces. Thus, it confirms that when combined these solid meshes will have perfect node to node contact on the articulating surfaces.

(A) (B)

(C) (D) Figure 20 Shows the surface over imposition of the extracted temporal and mandible surface on the solid meshes (A) Overlap with mandible solid mesh (B) Overlap with temporal solid mesh (C) Extracted mandible surface overlap on the TMJ disc solid mesh (D) Extracted temporal surface overlap on the TMJ disc solid mesh

To combine solid meshes into one model, first the solid mesh of the mandible was opened in Marc. The next step was to combine the solid mesh of the articular disc and the mandible. This was done using the merge command in Marc. The merge command is used for merging two solid meshes into one [45]. Thus, the solid mesh of the mandible and the articular disc was combined together as shown if Figure 22 (A).

(A) (B)

(C) Figure 21. (A) The solid mesh of the mandible. (B) The solid mesh of the articular disc. (C) The solid mesh of the temporal bone.

(A) (B) Figure 22. The combined solid mesh of the articular disc and the mandible (E) The combined solid mesh of the articular disc and the mandible.

The next step was to incorporate the temporal bone solid mesh into the combined mandible and the articular disc solid mesh. For that, the same procedure of combining the articular disc and mandible solid mesh was followed. This resulted in completed geometry of the right TMJ as shown in Figure 22 (B).

The mesh quality of the complete model was determined by checking the aspect ratio of the solid elements. An angle of 15 degrees between the adjacent edges of an element was used as threshold to check the aspect ratio [46]. Thus, any element having the cosine of the angle between their adjacent edges greater than the cosine of threshold angle was termed as a bad element, as its ratio of surface area to volume becomes more 43 that unity [45]. There were five bad elements detected in the complete model. These elements were of the temporal bone solid mesh. Thus, they were located away from the articular disc. Since, the objective of this study is to investigate the stress distribution only the articular disc, it was safe to ignore them.

2.3 Material properties

As done in early model, the material properties of the articular disc were considered to be linear elastic [21,24,40,47,48]. In the enhanced model, a non-linear material model defined the articular disc as Mooney-Rivlin solid [13,22,49]. The disc was considered to be homogenous and isotropic for both the early as well as the enhanced model [13,21,22,24,40,47,48,49]. The mandible and the temporal bone were considered to be linear elastic for both the early and enhanced models [21,22,24,40,47,48].

2.3.1 Linear material model

The linear material properties are given in Table 3. The material properties were assigned to the individual solid meshes of the articular disc, the mandible and the temporal bone using Marc material properties option. The stiffness values for the linear springs representing the ligaments are also listed in Table 3. The calculation for the stiffness of the TMJ ligaments is in Appendix B.

Table 3 The values of the material properties for the temporal bone, articular disc and the mandible. E is the elastic modulus specified in MPa, stands for the Poisson’s ratio and k for the spring stiffness. Source column indicated the original source from where these values have been taken.

Entity E (MPa) k (N/cm) Source Temporal 13700 0.3 M Beek (1999) Bone

Articular disc 100 0.4 J Chen (1994)

Mandible 13700 0.3 M Beek (1999)

Retrodiscal 2.72 J Chen(1998) Ligaments TMJ J T Blackburn 2.812 12.87 Ligaments (2009)

2.3.2 Non-Linear model

The TMJ disc material in vivo is anisotropic, non-homogenous and non-

linear in nature [2,21,50]. The disc material exhibits viscoelastic properties in vivo

[25,50]. As a step forward from the linear elastic model, this study considered the TMJ

disc to be non-linear, homogenous and isotropic [22,25]. To consider the nonlinearity of

the TMJ disc, it was represented by a two constant Mooney-Rivlin material model

[13,22,49]. This is a non-linear elastic, non time dependent model [23]. The Mooney

Rivlin constants were defined in Marc using the mooney option in the material properties

section. In general the Mooney-Rivlin constants C10 and C01 are derived from the stress-

strain curve of a uniaxial tension test [45]. The Mooney-Rivlin constants C10 and C01 45

were defined as 27.91 MPa and -20.81 MPa respectively, which were derived from a

canine mandibular disc [22].

The constitutive equation for the two constant Mooney-Rivlin solid in Marc is given as [45]:

ܹ= C10 (I1-3) + C01 (I2-3)

Where, W is the Strain energy C10 and C01 are the Mooney-Rivlin constants I1 is the first invariant of Left Cauchy Green deformation tensor I2 is the second invariant of Left Cauchy Green deformation tensor

2.4 Loads and Boundary Conditions

There are two loading conditions used in the literature for TMJ finite element

simulations. The first uses condylar displacement [22,24,40, 35,41,42] and second uses

muscle forces [13,16,25,47,48,49,51]. This study uses muscle forces as point loads for

loading the joint. The primary muscles of mastication are the masseter, the lateral and

medial pterygoid and the temporalis [10,15,52]. The function of each of these muscles is

given in brief in 46

Table 1. All four muscles are used in this study.

The magnitudes of the muscle forces are obtained from Herring [53] in which she reports the maximum possible muscle tension for each muscle. Herring first calculated the cross sectional area of each muscle [53]. The maximum muscle force was taken as being directly proportional to the cross sectional area of the muscle [53]. Herring used the constant value of 40 N/cm2 to calculate the maximum possible muscle tension for each one [53]. The inclination of these muscle forces were obtained from Herring [52] and

McDevitt [11]. The inclination is the angle between the line of action of the muscle forces and the occlusal (bite) plane [11,52]. The line of action for each muscle was obtained by tracing their respective points of insertion and origin [11,52]. The angles were measured from the right lateral direction, in an anticlockwise sense by a protractor

[52]. Herring reports that the accuracy of these measurements is within 5% [52-53]. The details about the muscle forces are mentioned in Table 4 .

Table 4 The information regarding the muscle forces used. Position of the muscle forces are given as the x, y, z coordinates of the node referred as node id. The magnitude of force is given in Newtons. Inclinations are the angles made by the muscle force vectors with the occlusal plane. In the last column the x, y, z components of muscles forces is specified. Position Force Inclination Force Force Muscle (x,y,z) magnitude (degrees) Component Component (cm) (N) Fy (N) Fz (N) ( 4.94, Masseter 304.4 60 -263.62 -152.2 5.06, 1.94)

Lateral (3.79, 2.97, 100.4 20 -34.33 -94.34 Pterygoid 2.60)

Medial (4-91, 226.8 60 -196.4 -113.4 Pterygoid 5.03, 3.39) Temporalis (4.02, - 105.6 115 -95.7 -44.62 (out) 0.38, 0.36)

Temporalis (3.93, - 105.6 115 -95.7 -44.62 (in) 0.46, 0.90)

(B)

Figure 23 shows all the muscle forces acting considered in this study from a right

lateral view [53]. The angles reported by Herring [52] are given in this plane. The solid

model surface for the mandible is in the y-z plane as shown in

(B)

Figure 23. Thus, only y and z components of forces exist, which were calculated

using simple vector calculations. There is no force component in the x-direction.

A (A) (B) Figure 23. The approximate Force diagram of the muscle forces applied to the mandible. (A) Medial view (B) Lateral view.

In this study, the muscle forces are applied as point loads. To decide the point of application, first the region of contact of each muscle with the mandible was determined.

This region of contact for every muscle was approximated as a triangle. These regions for every muscle were decided by observation using acrobat movies of the virtual pig head for the muscles of mastication [54], an artificial human skeleton [55] with the muscle attachments drawn on it, and a book on functional anatomy of the masticatory system by

W.E McDevitt[11].

The first muscle considered was the medial pterygoid. As discussed earlier, the first step in determining the point of application of muscle tension was to determine its region of contact with the mandible. The region of contact for the medial pterygoid was assumed as shown in Figure 24. This region of contact was then extrapolated to the solid 49

model of the mandible so as to get the coordinates of the vertices of the contact triangle.

Then, for the point of application the centroid of the contact triangle was calculated. Last, on the solid mesh the node having the coordinates closest to that of the centroid was finally decided as the point of application of the medial pterygoid force as shown in

Figure 25 (C). Similarly, the points of applications were calculated for lateral pterygoid, masseter, temporalis in and temporalis out. Temporalis in and out are the temporalis muscle forces acting on the inner and outer surface of the mandible. Figure 25 and

Figure 26, shows all the muscle forces applied as point loads on the solid model.

Figure 24. The contact triangle for the medial pterygoid with the centroid as the point of application of the muscle tension.

(A) (B)

(C) Figure 25. The muscle forces applied as point loads with MSC. Marc. (A) Lateral pterygoid (B) Masseter (C) Medial Pterygoid.

(A) (B) Figure 26. The muscle forces applied as point loads with MSC. Marc. (A) Temporalis in (B) Temporalis out.

In addition to the muscle forces a resistance force of 50 N was introduced [16].

Originally, Perez del Palomar [16] applied this resistance force between the central incisors. As discussed in the section 2.2, this study considers the cut mandible model as shown in Figure 21. Thus, this resistance force was applied on the cut plane with appropriate moments caused due to change in the point of application. Figure 27 (A,B) shows the resistance force with the associated moments. Since, only the left half of the mandible is considered the food resistance force was reduced to 25 N.

As the point of application of the food resistance force is moved from central incisors to the point on the cut plane, it moved in both x and z directions producing x- moment and z-moment. Both these moments are applied in the form of couple forces.

Table 5 presents all the magnitude of these couple forces.

(A) (B) Figure 27. The food resistance force with couple. (A) Shows the resistance force in Y direction, the x and z forces contributing to the couple (B) x and z forces of the couple. Note: In (A) the z force is not visible, this z force forms couple with z force in (B).

Table 5 The magnitude of the couple forces. Calculations for these couple forces are given in Appendix A. Magnitude Force (N) x-couple 13.14 z-couple 67.17

After applying the loading conditions the next step was to assign the boundary

conditions for the study. In this study I have decided to use zero displacement for the

medial cut surface of the temporal bone and the zygomaticus [36,40]. Figure 28

represents the boundary conditions used in this study. This boundary condition simulates

the real world situation of the temporal bone being held stationary while the mandible

moves [36,40]. 53

Figure 28. (Clockwise from the top left) Front view, side view, top view and top view of the constrained temporal bone and the zygomatic arch.

2.5 Contact

As discussed earlier, there are two contact regions in the TMJ. The first contact region is between the TMJ disc and the surface of the temporal bone and the second region is between the TMJ disc and the mandibular condyle surface [2]. The contact 54

definitions used to define the nature of contact in these regions were touching contact

model between the temporal bone and the articular disc [16,36] and glue contact between

mandible and articular disc. These contact definitions were used with each of the material

model. Table 6 lists all the TMJ models examined under this study.

Table 6 Type of contact used with respective material model. Material Simulation Model for Contact Definition No. TMJ disc Mandible - Temporal bone - Disc Disc 1 Linear Elastic Glued Touching 2 Mooney-Rivlin Glued Touching

In Marc, to define the contact between the articulating surfaces, individual solid

meshes must be first defined as contact bodies [45]. Thus, three contact bodies were

defined as shown in Figure 29. Defining solid meshes as contact bodies helps the Marc

solver identify a node to a particular contact body [45]. All the contact bodies were defined as deformable bodies. 55

Figure 29. Contact bodies.

As stated earlier this study uses glue contact and touching contact. The glue contact option was used to define contact between the mandible and the articular disc as

DeVotch (1996) reported that, in a healthy TMJ joint the articular disc fits firmly over the mandible and is able to follow the mandibular motion even if no there are no discal attachments [24]. Since, the friction coefficient between the TMJ articulating surface is very small friction contact model was not used [13,22]. These contact definitions are discussed below.

Glue contact model simulates the situation of two contact bodies glued to each other [45]. Thus, glued contact definition prohibits any relative tangential motion between contact bodies [45]. This Marc feature is specifically devised to assemble non- matching meshes in to one model [45]. With this contact definition, the articulating disc was considered glued to the condyle. To specify the properties of the nature of contact between the contact bodies Marc uses Contact Table Entry Properties chart. 56

Touching contact model simulates the condition of two bodies just touching each other [45]. In case of touching contact model, nodes in the contact region are restrained in the direction normal to the contact surface [45]. These nodes are however allowed to have relative motion in the tangential direction [45]. With this contact definition, the articulating disc was considered touching to the temporal bone. An important parameter that is required with this contact definition is the Hard to Soft Ratio. This is the ratio of the stiffness values of the contact bodies between the contact is defined [45]. Hard to soft ratio helps mark to identify the slave contact body. A slave contact body is the one with lower stiffness [45].

CHAPTER 3: RESULTS AND DISCUSSION

3.1 Results

The five basic inputs of the finite element study: geometry, mesh, material properties, loads and boundary conditions were discussed in chapter 2 [33]. Using these inputs, the TMJ models were subjected to analysis. As discussed in chapter 1, the main objective of this thesis is to analyze the stress distribution in TMJ disc. Table 7summaries the TMJ models analyzed in this thesis work.

Table 7 TMJ models used in this study. Contact Material Contact Material Definition Model Model Definition Properties Forces Ligaments with No. for TMJ With for Bones Temporal Disc Mandible Bone

Muscle Linear Linear Not 1 forces as Bonded Bonded Elastic Elastic present point loads

Muscle Linear Linear Linear 2 forces as Bonded Bonded Elastic Elastic Springs point loads

Muscle Linear Linear forces as Linear Glue Touching 3 Elastic Elastic point loads Springs Contact Contact

Muscle Mooney- Linear forces as Linear Glue Touching 4 Rivlin Elastic point loads Springs Contact Contact

To facilitate the presentation of stress distributions in the TMJ disc, it was first divided into three bands: anterior, intermediate and posterior as shown in Figure 30 [35].

Each of these bands was then divided in to three parts namely: medial, central and lateral, as shown in Figure 30. The von Mises and the major principal stress are reported. The major principal stress is the maximum magnitude principal stress at a node [45]. The von

Mises stress was reported; as it was reported in the earlier studies made on human TMJ.

The major principal stress was selected as it portrays the tensile and compressive response of the TMJ disc. Results for models listed in Table 7 are discussed in this section.

Figure 30. The TMJ disc division into anterior, intermediate and posterior bands with the sub divisions. CP- central part, MP- medial part and LP- lateral part [9].

The maximum von Mises stress in case of the model 1 was 1.88 MPa. This peak stress was located in the central part of the anterior band. The maximum tensile stress 59

was 0.94 MPa. This tensile stress was located in the posterior region of the articular disc.

The maximum compressive stress was 1.61 MPa. The peak compressive stress was

located in the central part of the anterior band.

The maximum von Mises stress in case of the model 2 was 1.41 MPa. This peak

stress was located in the central part of the anterior band. The maximum tensile stress

was 0.87 MPa. This tensile stress was located in the posterior region of the articular disc.

The maximum compressive stress was 1.44 MPa. The peak compressive stress was

located in the central part of the anterior band.

The results of simulation of model 3 having linear elastic material definition for the TMJ disc are shown in Figure 31 and Figure 32. In this case it was observed that, as loads were applied the mandible moved backward with some lateral movement. This lateral movement of the condyle was 0.0014 cm. It was observed that the TMJ disc follows the mandible during the closing movement of the mandible.

The von Mises stress distribution in the model 3TMJ disc is shown in Figure 31.

The maximum von Mises stress in the articular disc for this model was 1.33 MPa. This peak von Mises stress was found in the central part of the anterior band. The average von

Mises stress for each band was calculated. Table 8 reports these average von Mises

stresses along with the standard deviation for every band. From the Table 8, it can be

observed that the average von Mises stress was highest in the anterior band of the

articular disc. The posterior band of the disc was the least stressed band.

Figure 31. The Equivalent von Mises stress distribution in the linear elastic TMJ disc. 2 2 The stresses are given in N/cm . (100 N/cm = 1 MPa).

Table 8 The von Mises stress details for the linear elastic TMJ disc. Averages and standard deviations for every band were calculated by considering the both the surface and the internal nodes of the TMJ disc. Anterior Intermediate Posterior Band Band Band Average von Mises stress 0.42 0.37 0.23 (MPa) Standard deviation (MPa) 0.22 0.15 0.13 Peak von Mises stress 1.33 0.64 0.61 (MPa)

The major principal stress distribution in the model 3 TMJ disc is shown in

Figure 32. Mean major principal stresses and the corresponding standard deviations were

calculated for each subdivision of the disc, as listed in

Table 9. The variation in major principal stress, moving from medial towards the

lateral direction for every band is also reported using the cutting planes feature of Marc

[13,36,45].

Figure 32. The major principal stress distribution in the linear elastic TMJ disc. The stresses are given in N/cm2.

Table 9 The major principal stress details for the linear elastic TMJ disc. All stresses are in MPa. Averages and standard deviations for every band were calculated by considering the both the surface and the internal nodes of the TMJ disc. Anterior Band Intermediate Band Posterior Band

Mean Standard Mean Standard Mean Standard

stress deviation stress deviation stress deviation

Medial Part 0.06 0.42 0.06 0.32 0.11 0.19 Central Part -0.34 0.52 -0.24 0.56 0.20 0.25 Lateral Part -0.43 0.27 -0.21 0.37 0.20 0.22

It was observed that, the anterior band of the articular disc was predominantly

under compression with 64% of its nodes in compression. The peak compressive stress in

this band was 1.44 MPa. This peak compressive stress was located in the central part of

this band. The peak tensile stress in this band was 0.69 MPa. This peak tensile stress was

located towards the medial part.

The distribution of major principal stress across the anterior band starting from

medial direction moving towards lateral direction is shown in Figure 33. It can be

observed that, in the anterior band as we move from medial towards lateral direction, the articular disc goes from tension to compression. Thus, the medial part of the anterior band was under tension. Whereas, central and lateral parts were under compression.

Figure 33. The compressive stress distribution across the medial to lateral direction of the anterior band of the linear elastic TMJ disc. All stresses are in N/cm2.

The intermediate band was also predominantly under compression with 58% of its

node in compression. The peak compressive stress in the intermediate band was 1.44

MPa. This peak compressive stress was located in the central region of the band close to

the anterior band boundary. The peak tensile stress in this band was 0.69 MPa. This was

located in the medial region of the intermediate band.

The distribution of major principal stress across the intermediate band starting from medial direction moving towards lateral direction is shown in Figure 34. It can be

observed that, in the intermediate band as we move from medial towards lateral direction, 65 the articular disc goes from tension to compression. The central and the lateral part of the intermediate band were under compression. The medial part of the intermediate band experienced tension.

Figure 34. The compressive stress distribution across the medial to lateral direction of the intermediate band of the linear elastic TMJ disc. All stresses are in N/cm2.

The posterior band was predominantly in tension with 80% of its node in tension.

The peak tensile stress in this band was 0.76 MPa, located in the central part of the band.

The peak compressive stress was found to be 0.41 MPa, in the medial region of the band.

The distribution of major principal stress across the posterior band starting from medial direction moving towards lateral direction is shown in Figure 35. It was observed 66

that, in the posterior band as one move from medial direction towards lateral direction the

articular disc goes from compression to tension. Thus, the medial part of the posterior

band was mostly in compression. Both the central and the lateral part of this band were in

tension.

Figure 35. The compressive stress distribution across the medial to lateral direction of the posterior band of the linear elastic TMJ disc. All stresses are in N/cm2. Grey is tension.

Thus, the results of model 3 with a linear material model for the TMJ disc can be summarized as: The anterior and intermediate bands of the articular disc experienced more stresses as compared to the posterior band, which was least stressed. Both anterior 67

and intermediate bands were predominantly in compression whereas the posterior band

was mostly in tension.

The results of simulation of the model 4 having Mooney-Rivlin material definition for the TMJ disc are shown in Figure 36 and Figure 37. It was observed that, with closing movement of the mandible there was significant lateral movement. This lateral movement of the condyle was increased to 0.18 cm. It was also observed that, the

TMJ disc followed the motion of the mandible.

The von Mises stress distribution in the Mooney-Rivlin TMJ disc is shown in

Figure 36. The maximum von Mises stress in the articular disc for this model was 3.10

MPa. This peak von Mises stress was found in the lateral part of the intermediate band.

The average von Mises stress for each band was calculated. Table 10 reports these average von Mises stresses along with the standard deviation for every band.

Figure 36. The Equivalent von Mises stress distribution in the linear elastic TMJ disc. Note: The stresses are given in N/cm2.

Table 10 The von Mises stress details for the Mooney-Rivlin TMJ disc. Note: Averages and standard deviations for every band were calculated by considering the both the surface and the internal nodes of the TMJ disc. Intermediate Posterior Anterior Band Band Band Average von Mises stress 0.44 0.29 0.10 (MPa) Standard deviation (MPa) 0.64 0.54 0.06 Peak von Mises stress (MPa) 2.95 3.10 0.30

From the Figure 36 and Table 10, it can be observed that the anterior and

intermediate bands of the articular disc are having similar von Mises stress distribution.

The posterior band of the disc is least stressed.

The major principal stress distribution in the Mooney-Rivlin TMJ disc is shown in

Figure 37. Mean major principal stresses and the corresponding standard deviations were

calculated for each subdivision of the disc, as listed in

Table 11. The variation in major principal stress, moving from medial towards

the lateral direction for every band is also reported using the cutting planes feature of

Marc [13,36,45].

Figure 37. The major principal stress distribution in the Mooney-Rivlin TMJ disc. The stresses are given in N/cm2.

Table 11 The major principal stress details for the Mooney-Rivlin TMJ disc. All stresses are in MPa. Averages and standard deviations for every band were calculated by considering the both the surface and the internal nodes of the TMJ disc.

Anterior Band Intermediate Band Posterior Band

Mean Standard Mean Standard Mean Standard

stress deviation stress deviation stress deviation

Medial Part -0.01 0.07 0.01 0.05 0.06 0.10 Central Part -0.09 0.23 -0.02 0.15 0.06 0.15 Lateral Part -0.77 1.62 -0.61 1.48 0.04 0.30

It was observed that, the anterior band of the articular disc was predominantly under compression with 65% of its nodes in compression. The peak compressive stress in this band was 4.44 MPa. This peak compressive stress was located in the lateral part of this band. The peak tensile stress in this band was 1.73 MPa. This peak tensile stress was located in the lateral part near the boundary of the central part.

The distribution of major principal stress across the anterior band starting from medial direction moving towards lateral direction is shown in Figure 38. It can be observed that, in the anterior band the lateral most part of the articular disc experienced high compression. As one moves towards the central part, some tension was reported.

The medial part of the anterior band was mostly under compression.

Figure 38. The compressive stress distribution across the medial to lateral direction of the anterior band of the Mooney-Rivlin TMJ disc. All stresses are in N/cm2.

It was observed that, the intermediate band was mostly under tension in

the central part. The lateral part also experienced some tension towards the medial

direction. The peak compressive stress in this band was 5.70 MPa. This peak compressive stress was located in the lateral part of this band. The peak tensile stress in this band was

1.73 MPa. This peak tensile stress was located in the lateral part near the central part

boundary.

The distribution of major principal stress across the intermediate band starting from medial direction moving towards lateral direction is shown in Figure 39. It can be 73

observed that, in the intermediate band most of the lateral part experiences compression,

but as one move towards the central part, some tension was reported. The compression in

the central part was observed in the medial direction, whereas tension toward the lateral

direction. The medial part of the intermediate band was mostly under compression.

Tension in the medial part was reported near the posterior band boundary.

Figure 39. The compressive stress distribution across the anterior to posterior direction of the medial part of the Mooney-Rivlin TMJ disc. All stresses are in N/cm2.

The posterior band was predominantly in tension. The peak tensile stress in this

band was 1.10 MPa, located in the central part of the band. The peak compressive stress was found to be 0.67 MPa, in the lateral region of the band. 74

The distribution of major principal stress across the posterior band starting from

medial direction moving towards lateral direction is shown in Figure 40. It was observed

that, the posterior band was almost completely under tension. Some compression was

reported near the superior boundary in the central part of this band.

Figure 40. The compressive stress distribution across the medial to lateral direction of the posterior band of the linear elastic TMJ disc. All stresses are in N/cm2.

Thus, the results of the model 4 with Mooney-Rivlin material model for the TMJ disc can be summarized as: The anterior and intermediate bands of the articular disc experienced more stresses as compared to the posterior band, which was least stressed. 75

The anterior band was predominantly in compression. Intermediate band experienced both tension and compression, whereas the posterior band was mostly in tension.

An approximation error was quantified for the current models. An approximation

error is caused during the approximation of the displacements between two nodes [33].

To quantify this error the results of the current linear model was compared with the

results obtained for a model using higher order elements. These higher order elements

uses 2nd degree polynomials to estimate the displacements between nodes as compared to

linear polynomials as used in current model [56]. The average approximation error was

30.5%. The discretization error has the significant contribution to this error.

3.2 Discussion

In this study four Pig TMJ models were analyzed. First two models were analyzed

to study the effect of the ligaments on the stress distribution pattern in the TMJ disc.

Models 2-4 had the same geometry, mesh, loads and boundary conditions. Ligaments

were not included in model 1. No contact was defined between the articulating surface

for models 1 and 2. For models 3 and 4 contact between articular surfaces of the

mandible and disc was defined as a glued contact, whereas the contact between the

articular surfaces of the temporal bone and disc was defined as a touching contact. For

models 3 and 4, only thing that differed was the material model used to define the TMJ

disc. As stated earlier, for model 3 the TMJ disc material was defined as linear elastic and

for model 4 it was defined as a Mooney-Rivlin solid. All the models were loaded using

masticatory muscle forces. The temporal bone was held stationary in all the models. The

retrodiscal and the TMJ ligaments were modeled using linear springs. 76

Due to the assumptions like simultaneous activation of all muscle forces, defining the TMJ disc as isotropic, and no articular cartilage between the articulating surfaces, the models used in this study represents simplified a in vivo TMJ condition. Thus, to make current models closer to in vivo TMJ and to give future direction to this research some recommendations are made.

The first step in the future direction would be to have a finer mesh for the articular disc. This would help in reducing the present discretization error. Current models use maximum masticatory muscle forces to simulate clenching. Thus, the second step to improve these models in the future would be to use a percent of these maximum muscle forces [13,16]. This is because, all the masticatory muscle are not used to their full capacity to produce condylar motion [13,16]. Thus, instead of using the maximum possible force that can be generated by a muscle only some fraction of it should be used.

The third step for the future improvement should be using a viscoelastic material model to define the material properties for the TMJ disc. It was reported by some researchers that, the articular disc material is better represented by using a viscoelastic material [25,50]. The TMJ disc also exhibits anisotropy and inhomogeneity [2,21,50].

The last step in the future directions would be to add to the geometry of the

current model. The collateral ligaments and the muscle attachments to the TMJ disc

should be added. These collateral ligaments and the muscle attachments like lateral

pterygoid to the TMJ disc are reported to hold the TMJ disc in proper position during the

condylar motion [16]. 77

From models 1 and 2, it can be stated that by adding the ligaments the peak stresses in the articular disc was reduced up to 10%. The locations of all the peak stresses were same irrespective of the presence of ligaments. The lateral movement in case of the model with ligaments was increased to 0.057 cm from 0.0058 cm, which was in case of the model without ligaments.

The results from the model 3 and 4 used in this study are now discussed and compared with each other. Later, these results are compared with the results of other studies done on human TMJ models.

In the case of model 3 which used a linear elastic TMJ disc it was observed that, as loads were applied the mandible moved backward with some lateral movement pressing the articular disc against the glenoid fossa in the central and lateral parts. This mandibular motion was the primary reason that compression was seen in the central and lateral part of the anterior and the intermediate bands and in some part of the posterior band towards the medial direction. The tension in the medial part of the anterior and the intermediate bands was due to heavy compression in their central and lateral parts.

Tension in the posterior band of the disc was primarily due to the retrodiscal ligaments.

In the case of model 4 which used a Mooney-Rivlin TMJ disc it was observed that, as loads were applied the mandible moved backward with significant lateral movement. This lateral movement of the mandible in case of the model 4 was approximately 100 times the lateral movement obtained in case of model 3. The increase in lateral movement was due to the increased softness of the TMJ disc, glued contact definition between the mandible and the articular disc and due to the absence of the 78

collateral ligaments [36]. The increased lateral movement of the mandible shifts the point

of maximum compression in the TMJ disc in the lateral direction as compared to the

model 1, causing tension in the central part. Increased stresses in the lateral direction

were also reported by Palomar (2006) who observed that the absence of the collateral

ligaments causes the disc to be significantly more stressed in the lateral direction [36].

The posterior band was the articular disc was in tension primarily due to the retrodiscal ligaments.

From the results obtained, it can be stated that the material definition has a significant impact on the mandibular motion and the stress distribution pattern in the TMJ disc. The model with Mooney-Rivlin TMJ disc had increased lateral movement as compared to the model with a linear TMJ disc of model 3. This caused the stresses to concentrate in the lateral part of the TMJ disc. The comparison of the stress distribution in these two models is given in

Table 12.

Table 12 The compressive/tensile response of the TMJ disc in different regions for two material models used in this study. C- Compression, T- Tension. This table reports the C or T for a region depending upon the number of nodes in compression or tension in that region. Model Anterior Band Intermediate Band Posterior Band No. Med Centr Later Medi Centr Later Medi Centr Later ial al al al al al al al al

Model T C C T C C C T T 3

Model C C C C T C T T T 4

The peak stresses were increased for the Mooney-Rivlin TMJ disc. The maximum

compressive stress in Mooney-Rivlin TMJ disc was 5.75 MPa as compared to 1.44 MPa in the linear elastic TMJ disc. The maximum tensile stress in Mooney-Rivlin TMJ disc was 1.73 MPa as compared to 0.76 MPa in the linear elastic TMJ disc.

After comparing models 3 and 4 it can be concluded that Mooney-Rivlin material model causes the stress to be concentrated in the lateral part of the TMJ disc. The stresses are less uniformly distributed across the TMJ disc in case of the Mooney-Rivlin TMJ disc when compared to the linear elastic TMJ disc. Thus, it can be concluded that the using

Mooney-Rivlin material definition for TMJ disc reduces its role of distributing the stresses uniformly within the TMJ capsule [1].

The exact values for the stiffness of the TMJ ligaments are not presented in the literature. In this study the TMJ ligament elastic modulus is considered to be 2.81 MPa

[59]. Gupta [57] uses 0.49 MPa. To quantify the effect of the varying the stiffness of the

TMJ ligaments model 2 was run again using the elastic modulus value of 0.49 MPa for the TMJ ligaments. The results showed the stress distribution pattern in the articular disc

remains similar. The peak von Mises stress increased to 1.80 MPa from 1.71 MPa. Martin

[58] suggests that in general the elastic modulus values for the ligaments to be around 1

GPa. To quantify the effect of the varying the stiffness of the TMJ ligaments model 2 was

run again using the elastic modulus value of 1000 MPa for the TMJ ligaments. The

results showed that the peak von Mises stress moves towards the lateral part of the disc.

The peak von Mises stress was 0.91 MPa. 81

The secondary objective of this study was to compare the results to the results of the studies on human TMJ models. Available human studies simulate different aspects of the TMJ movement, they can be opening movement [13,16,36,40,41], lateral [42], closing movement [13,16,21,22] or clenching [25]. They use different material definitions for the TMJ disc, different loading conditions and constraints. The current models are loaded to simulate clenching. Thus, they will be more closely compared to clenching study of Hiorse M and Tanaka [25].

It should also be noted that, there is no specified guidelines to divide the disc into three bands. Thus, the regions included in the anterior, intermediate and the posterior band may vary from one researcher to another.

Hirose M [25], Beek [40], DeVotch [24], Koolstra [13] observed that the von

Mises stresses were mainly concentrated in the intermediate band of the TMJ disc whereas, Chen [21-22] obtained this stress concentration in the posterior region of the disc. In the case of the linear elastic pig TMJ disc, it was observed that the peak stress was in the central part of the TMJ disc. In the case of the Mooney-Rivlin pig TMJ disc the peak stress was in the lateral part of the intermediate band, which is in agreement with the maximum von Mises location reported by Hirose M [25], who also simulates the clenching. Thus, the von Mises stress distribution observed in the TMJ disc for models 3 and 4 were in good agreement with Hirose M [25], Beek [40], DeVotch [24], Koolstra

[13]. 82

Table 13 compares the maximum von Mises stresses and their location for the above reported studies.

Table 13 The maximum von Mises stress and their location for different TMJ models. Maximum Author TMJ Movement von Mises Location (Year) Model simulated stress (MPa) J Chen Posterior band towards the Human Closing 5.28 (1994) articular surfaces DeVotch Human Opening 0.38 Intermediate (1996) J Chen Posterior band towards the Human Closing 8 (1998) articular surfaces M Beek Lateral part of the intermediate Human Opening 2.79 (2000) zone

Central to lateral part of intermediate band for opening Koolstra Opening and Human 2.5 movement. Moves towards (2005) closing central part for closing movement.

Hirose Central and lateral part of the Human Clenching 0.85 (2006) intermediate band Present study Central part of the anterior Pig Clenching 1.61 model 3 band (2009)

Present study Lateral part of the intermediate Pig Clenching 4.04 model 4 zone (2009)

Table 14 lists the compressive and tensile stresses observed in the human and pig

TMJ discs. From the Table 14, it can be observed that, the region peak compressive stress

reported by both the pig TMJ discs and Palomar [16,35-36] was the intermediate region. 84

The tensile stress response of model 3 was in good agreement with Palomar [16,36]. The tensile stress response the model 4 was in good agreement with that reported by J Chen

[21-22] and Palomar [35-36].

Thus, from the above comparisons it can be said that the peak stress values observed in the current pig TMJ models are within the range observed from the human

TMJ models. Also, the stress distribution patterns are in good agreement with the studies investigating closing movements and clenching in human TMJ models.

Table 14 The maximum and minimum principal stresses and their location for different TMJ models. Tensile [Peak Author TMJ Movement Compressive [Peak value value MPa (Year) Model Simulated MPa (location)] (location) 3.97 (Superior 12.07 (Posterior band J Chen Human Closing surface of the towards the articular (1994) central region) surfaces) 3.70 (Superior 8.00 (Posterior band J Chen Human Closing surface of the towards the articular (1998) central region) surfaces) 6.00 (Lateral part of Palomar Human Opening the disc and in the 8.00 (Intermediate zone) (2006) posterior region) Palomar 8.50 (Anterior and Human Opening 11.00 (Intermediate zone) (2007) posterior region) Palomar 7.00 (intermediate and Human Closing - (2007) posterior) Palomar Human Opening 6.20 (Lateral part) 8.50 (intermediate zone) (2007) Present Study 0.86 (Posterior Pig Clenching 1.65 (Central part) model 3 region) (2009) Present Study Pig Clenching 1.10 (lateral part) 6.65 (intermediate zone) model 4 (2009)

CHAPTER 4: CONCLUSION

After studying and comparing the results for various models in this study some general and comparison specific conclusions were drawn. The general conclusions were those which were observed in every model included in the study. The comparison specific conclusions were regarding the effects of the ligaments, contact definition and disc material properties on the stress distribution in the articular disc.

From the obtained results it can be concluded that the articular disc experiences mostly compression. The anterior and the intermediate bands of the articular disc are always the most stressed, whereas the posterior band is always the least stressed. It can also be concluded that the region of maximum compression in the articular disc always remains in or near the intermediate band.

From the comparison of the results of models 1 and 2, it can be concluded that the location of peak stresses remains the same irrespective of the ligaments. But, the addition of ligaments had an impact on the peak stress magnitudes. On average the magnitudes of peak stresses were reduced by around 10%. This comparison also indicates that addition of the TMJ ligaments increases the lateral movement of the mandible. From the comparison of the results of models 2 and 3 it can be concluded that addition of contact definition reduces the tensile stress in the articular disc by almost 17%.

From the results of model 4 it can be concluded that the material property of the articular disc has the most significant effect on the disc stresses. The increased softness of the articular disc caused an increase in the lateral movement of the mandible. It can also 87 be said that the use of Mooney-Rivlin material model for the articular disc reduced the stress-distributor role of the articular disc.

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APPENDIX A : CALCULATION OF COUPLE FORCES

A food resistance force of 50 N was introduced [16]. Originally, Perez del

Palomar [16] applied this resistance force between the central incisors. In this study this

force was applied at a point on the cut plane as shown in Figure 41.

As the point of application of the food resistance force is moved from central incisors to the point on the cut plane, it moved in both x and z directions producing x- moment and z-moment. These moments were introduced as couple forces. This appendix describes all the calculations for these couple forces.

Figure 41. Pig mandible with the distances used for the couple force calculations.

Now, x = 2.25 cm; z = 11.5 cm; The distance‘d’ between the couple forces was 4.28 cm; Food Resistance force F = 25 N. [16] Moment caused by shifting the force F in X direction by distance ‘x’ = Mx Moment caused by shifting the force F in Z direction by distance ‘z’ = Mz Couple forces in the X direction = CFX Couple forces in the Z direction = CFZ

Thus, Mx = F x; = 25 * 2.25; Mx = 56.25 N-cm Now, CFX = Mx/d; = 56.25/4.28; CFX = 13.14 N Thus, Mz = Fz; = 25 * 11.5; Mz = 287.5 N-cm Now, CFZ = Mz/d; = 287.5/4.28; CFZ = 67.17 N

APPENDIX B: CALCULATION OF STIFFNESS FOR THE TMJ LIGAMENTS

This appendix discusses the calculation for the spring stiffness of linear springs

used to model the TMJ ligaments. The elastic modulus was obtained from the literature,

whereas the cross sectional area and the length of the TMJ ligament was obtained from

the current model.

Modulus of elasticity E = 2811.8 KPa [59] Stiffness k = (AE)/L Where, A = cross sectional area of the ligament L = Length Cross sectional area A = 2 cm2 The average length of the TMJ ligaments was 1.9 cm Thus, k = [2 cm2 x (281.18 N/cm)]/1.9 cm, k = 295.9789 N/cm Total number of springs used to represent the TMJ ligaments = 23 Thus, The stiffness per spring ki = k/23 ki = 12.87 N/cm