journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
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Research Paper
Finite element analysis of the human mastication cycle Maria S. Commisson, Javier Martı´nez-Reina, Joaquı´n Ojeda, Juana Mayo Department of Mechanical Engineering, University of Seville, Camino de los Descubrimientos s/n, E-41092 Seville, Spain
art i cle i nfo
ab st rac t
Article history:
The aim of this paper is to propose a biomechanical model that could serve as a tool to
Received 27 June 2014
overcome some difficulties encountered in experimental studies of the mandible. One of these
Received in revised form
difficulties is the inaccessibility of the temporomandibular joint (TMJ) and the lateral pterygoid
18 September 2014
muscle. The focus of this model is to study the stresses in the joint and the influence of the
Accepted 23 September 2014
lateral pterygoid muscle on the mandible movement. A finite element model of the mandible,
Available online 7 October 2014
including the TMJ, was built to simulate the process of unilateral mastication. Different
Keywords:
activation patterns of the left and right pterygoid muscles were tried. The maximum stresses
Mastication
in the articular disc and in the whole mandible during a complete mastication cycle were
Temporomandibular joint
reached during the instant of centric occlusion. The simulations show a great influence of the
Lateral pterygoid
coordination of the right and left lateral pterygoid muscles on the movement of the jaw during
FE simulation
mastication. An asynchronous activation of the lateral pterygoid muscles is needed to achieve a
Articular disc
normal movement of the jaw during mastication.
Quasi-linear viscoelastic model
1.
Introduction
Biomechanical models of the human mandible have been extensively used to explore the functioning of dental implants (Huang et al., 2014; Holberg et al., 2013; Lan et al., 2012; Ojeda et al., 2011), other fixation devices (Huang et al., 2012; Bohluli et al., 2010), mandible reconstruction (Narra et al., 2014; Li et al., 2014), and temporomandibular disorders (Cheng et al., 2013; Commisso et al., 2014) among other problems. This paper uses a model of the mandible, including the temporomandibular joint (TMJ), to simulate the process of unilateral mastication. Two basic aspects of this process constitute the main focus of this study: the articular disc of n
Corresponding author. Tel.: þ34 954481365; fax: þ34 954460475. E-mail address: [email protected] (M.S. Commisso).
http://dx.doi.org/10.1016/j.jmbbm.2014.09.022 1751-6161/& 2014 Elsevier Ltd. All rights reserved.
& 2014 Elsevier Ltd. All rights reserved.
the TMJ and the muscular activity, in particular that of the lateral pterygoid muscle. The main component of the TMJ is the articular disc, a plate of fibrocartilage that facilitates the relative movement between the mandible and the temporal bone. The articular disc acts as a load absorber and distributes the loads over larger contact areas to prevent the damage of the articulating surfaces. In turn, damage of the articular disc can be one of the causes of temporomandibular disorders (TMD). Some researchers have shown that acute mechanical overloads in vivo can cause severe cartilage damage (Radin et al., 1984; Thompson et al., 1991). Particularly, shear stresses are believed to alter the cells in the tissue and might lay behind the damage in the cartilage (Smith et al., 2004). So, the
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journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
analysis of the shear stresses in the disc during chewing can be of interest for a study of TMD. The stresses and strains in the articular disc during loading are very difficult to measure experimentally. Therefore, several studies have investigated the stress and strain distribution in the TMJ during mastication by means of finite element (FE) models (see Beek et al., 2001; Koolstra and van Eijden, 2007, among others). To the authors’ knowledge, only Koolstra and van Eijden (2007) studied a complete mastication cycle and considered the viscoelastic behaviour of the articular disc. However, the movement of the mandible was defined with a multibody model and the jaw was assumed as a rigid body, so that the stress distribution within the bone was not accessible. The present study tries to overcome that limitation by using a FE model of the mandible with a detailed treatment of the TMJ, including the ligaments and a viscoelastic model of the articular disc. In regard to the second pillar of this study, the muscular activity during mastication, it must be noted that the human masticatory system has a very complex performance and requires the balanced and coordinated activities of the masticatory muscles on the left and right sides (Yamaguchi et al., 2011). In particular, the human lateral pterygoid muscle plays an important role in the control of the jaw movements (Murray, 2012) and stabilization of the TMJ (Murray et al., 1999). Nevertheless, there is very limited understanding on how it performs these functions. Furthermore, studies do not even agree on the specific functions of the two major bellies of the muscle (Murray et al., 1999): the superior and inferior heads. Only one thing seems to be clear: the inferior head is active in jaw-opening, jaw protrusion and contralateral jaw movements (Phanachet et al., 2001; Murray, 2012). One of the main reasons for the lack of understanding of the function of this muscle is the difficulty of studying it with electromyography (EMG) recordings, arising from the deep location of the muscle, which can result in a misplacement of the electromyograph. Another reason is that the condylar movement should be simultaneously recorded with the lateral pterygoid activity. This is essential in this muscle, in light of the observations of several authors (Sessle and Gurza, 1982; Murray et al., 1999), who noticed that the position of the jaw is an important determinant of its EMG activity. As an alternative to these EMG studies, some previous studies have simulated the mastication cycle to predict the activity of the lateral pterygoid from an inverse analysis. Hannam et al. (2008), Koolstra and van Eijden (2005) and de Zee et al. (2007) proposed multibody system models of the mandible and the muscles attached to it to simulate masticatory movements. They proposed an inverse analysis to fit the activity of the lateral pterygoid, such that it produced the normal movement of the jaw during mastication. This normal movement is characterized by a trajectory of the incisal edge with a “tear drop shape” when projected on a frontal plane (Buschang et al., 2000; Bhatka et al., 2004). However, those models contain simplifications with a major influence on the masticatory movement. Apart from the use of a multibody system model (rigid) for the mandible, the model of the TMJ was very simplified and the activities of the left and right lateral pterygoid muscles were synchronous (Koolstra and van Eijden, 2005) or nearly synchronous
(Hannam et al., 2008). It will be shown here that synchronous activation patterns result in abnormal movements when the constraints imposed by the TMJ are more accurately modelled. The lateral pterygoid muscle and its relationship with TMD have also been studied and reviewed extensively (Juniper, 1984; Widmalm et al., 1987; Hiraba et al., 2000; Fujita et al., 2001; Okeson, 2003; Desmons et al., 2007), though no consensus has been achieved to establish the origin of that relationship. In order to shed light on this matter it is essential to understand well its normal functioning. In this work, a FE model of the mandible and TMJs is proposed to simulate a mastication cycle by applying the loads exerted by the jaw opening and jaw closing muscles during the cycle. A special treatment is given to the forces applied by the lateral pterygoid muscle. Since there is no agreement in defining the activity of that muscle, different temporal activation patterns were tried to analyse its influence on the jaw movement. The main objective of this work is to propose a comprehensive biomechanical model of the masticatory system that allows the study of a mastication cycle and the stresses generated in the mandible and in the TMJ during it. Another objective is to check if the activity of the lateral pterygoid muscle has a significant influence on the movement of the jaw during chewing.
2.
Materials and methods
2.1.
Finite element model
A FE model of the mandible and TMJ was built using the commercial software Abaqus FEAs v6.10. The model was previously described in more detail in Commisso et al. (2014). The modelled parts of the joint were the articular disc, the collateral and temporomandibular (TML) ligaments and the posterior part of the articular capsule (see Fig. 1).
2.2.
Muscle model
The estimation of the muscle forces was made using a Hilllike model (Hill, 1938; Thelen, 2003), with the force being applied by the muscle the sum of two terms: the active force, due to the contraction of the muscles fibres, and the passive force, due to the stiffness of the connective tissue: F ¼ FA þ FP
ð1Þ
The active forces are generally expressed as a function of the maximum force in isometric contraction, FM 0 , the length of the muscular-tendon unit, l, the contraction velocity, v, and the activation level, a. For example, in the Thelen (2003) model, FA ¼ a FM 0 f L ðlÞ f V ðvÞ
ð2Þ
where fL(l) and fV(v) describe the dependency on the length and the contraction velocity respectively and are dimensionless functions of the so-called optimal fibre length, lM 0 , among other variables. The function fL(l) has a maximum f L ðlM 0 Þ ¼ 1. Besides, lM 0 was assumed to occur when the interincisal gap is 2 mm, like in Langenbach and Hannam (1999). Mouth opening during
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journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
Fig. 1 – (a) Complete FE model. (b) Detail of the TMJ.
Table 1 – Data of the forces exerted by the masticatory muscles. Only the orientation of the forces of the right side is given (symmetry with respect to the sagittal plane is assumed). The magnitude of the muscle force peak in centric occlusion for the ipsilateral (I) and contralateral (C) sides is also provided. The þx-axis is directed anteriorly, the þy-axis rightward and the þz-axis downward. CP: closing phase, OP: opening phase. Muscle
Portion
Length (mm)a
cos-xb,c
cos-yb,c
cos-zb,c
Peak of activity amax d,e I
C
b,c FM 0 ðNÞ
Masseter
Superficial Deep
48.0 25.7
0.419 0.358
0.207 0.546
0.885 0.758
0.56 0.56
0.20 0.20
190.4 81.6
Temporalis
Anterior Middle Posterior
51.9 52.7 52.9
0.044 0.5 0.855
0.149 0.221 0.208
0.988 0.837 0.474
0.65 0.60 0.54
0.51 0.53 0.54
158.0 95.6 75.6
42.7
0.372
0.486
0.791
0.97
0.47
174.8
32.6
0.757
0.63
0.645
0.761
0.36 0.10 0.026
66.9
31.3
0.50 0.10 0.039
Medial pterygoid Lateral pterygoid (CP) Lateral pterygoid (OP)
Inferior Inferior Superior
Anterior belly digastric
43.2
Geniohyoid Mylohyoid
40.0 Anterior Posterior
23.0 42.6
0.053 0.995 0.176 0.223
0.174 0.074 0.576
0.815 0.103
0.018
0.831 0.616
0.528 0.756
5 10
4
2.6 10
4 4
1.6 10 4.7 10 4
5 10
28.8 4
2.6 10
40.0
4
38.8
4
63.6 21.2
1.6 10 4.7 10 4
a
Length with the mouth closed (van Eijden et al., 1997). Korioth et al. (1992) for the jaw closing muscles. c van Eijden et al. (1997) for the jaw opening muscles. d Nelson (1986) for the jaw closing muscles. e Proportional to the values adopted by Hannam et al. (2008) for the jaw opening muscles. b
mastication is a movement of small amplitude and, consequently, the length of the muscles varies little around its value with that gap, lM 0 . Therefore, it can be assumed that f L ðlÞ 1 during the masticatory movement. On the other hand, this movement is relatively slow and it can be also assumed that f V ðvÞ f V ð0Þ ¼ 1. Then, the active muscle force is simplified as FA ðtÞ ¼ aðtÞ FM 0
ð3Þ
and was calculated throughout the cycle with the values of FM 0 given in Table 1 and the evolution of a(t) was given in Figs. 4 and 5.
The passive forces throughout the cycle were calculated as in Thelen (2003): P
M
FP ðtÞ ¼ FM 0 F ðl ðtÞÞ M
ð4Þ
where l ðtÞ ¼ lM ðtÞ=lM 0 is the normalized fibre length. The P normalized passive force, F , was defined as 8 M kP M > > > Þ ; l 41 þ εM 1 þ M l ðtÞ ð1 þ εM > 0 0 < ε0 P M ð5Þ F l ðtÞ ¼ P M M > > M ek ðl ðtÞ 1Þ=ε0 > M > ; l r1 þ ε : P 0 ek
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journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
Fig. 2 – Insertions of the different portions of the jaw closing muscles. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
where εM 0 ¼ 0:6 is the passive muscle strain corresponding to the maximum isometric force and kP ¼ 4 is a shape factor for the passive force–length relationship.
2.3.
Force boundary conditions
A complete unilateral mastication cycle with the right molars was simulated by applying the forces of the masticatory muscles. These forces were imposed as external loads, distributed over the insertion area of each muscle. Figs. 2 and 3 highlight in different colours the groups of nodes where the different muscles were inserted. The orientation of the muscle forces was taken from Korioth et al. (1992) and van Eijden et al. (1997) (see Table 1). The closing (CP) and opening (OP) phases of the cycle were simulated by applying the muscle activity patterns given by Hylander (1992) and shown in Fig. 4, plus the activity of the lateral pterygoid muscle in the opening phase, not shown in that figure but discussed later on. These activity patterns provide a(t), needed in Eq. (3), normalized by the peak of activity, amax, given in Table 1. Note that centric occlusion (CO) corresponds to the instant of maximum mastication force (around t ¼ 0.19 s) indicated with a vertical arrow in Fig. 4. Note also that the closing and opening phases are named after which muscles are active, either the jaw closing or jaw opening muscles, but not after the movement of the mandible. The values of amax corresponding to the jaw closing muscles (see Table 1) were taken from Nelson (1986) and used in a previous model (Reina et al., 2007). For the jaw opening muscles (anterior digastric, geniohyoid, anterior and posterior mylohyoid) they were defined based on the values adopted by Hannam et al. (2008). Those authors simulated an opening of the mandible of 20 mm, which is beyond 6–14 mm observed in chewing (Bhatka et al., 2004; Hedjazi et al., 2013). So, the values of amax for these muscles were chosen proportional to the values adopted by Hannam et al. (2008), with a unique constant of proportionality adjusted to get an opening of 8 mm. During a mastication cycle the direction of the forces exerted by the jaw closing muscles can be assumed: (1) uniform, because their fibres run approximately parallel
Fig. 3 – Insertions of the different portions of jaw opening muscles. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
close to the insertion to the mandible and (2) constant throughout the cycle, because the motion of their insertion points is of small amplitude (the jaw is approximately rotating around the TMJ, close to those insertion points). However, this is not true for the jaw opening muscles. First, these are fan-shaped muscles and each fibre has a different orientation and, second, they are inserted far from the TMJ. Thus, the direction of their forces and/or their moment arms with respect to the joint center varies with the mouth opening. This made necessary to consider the movement of the mandible to estimate the orientation of the forces exerted by those muscles. Such orientations were approximated by the direction of a segment joining the origin of the whole muscle and the insertion point of each individual fibre, thus, neglecting the pennation angle.1
1 The muscle force dependency on this pennation angle is proportional to its cosine and the maximum pennation angle for the mastication muscles is 161 (van Eijden et al., 1997). This implies a maximum error of 4% in the estimation of the force, which justifies the adopted simplification.
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
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Fig. 4 – Activation pattern of the jaw closing and opening muscles during unilateral mastication. The ipsilateral (I) muscles are above, in the upper y-axis, while the contralateral (C) ones are represented in the inferior y-axis. The amplitudes were normalized to the corresponding peak of muscle force, given in Table 1 for each muscle. The activity of the lateral pterygoid muscle during the OP has been omitted for clarity (see Fig. 5 for this activity). Adapted from Hylander (1992).
The position of the unique point representing the origin of each jaw opening muscle was estimated as follows. A representative insertion point was approximately located in the FE model (see Fig. 3) by choosing a single point placed in the center of the corresponding insertion area. Then, the origin was estimated with the orientation and the length with the mouth closed, both given in Table 1. This origin was assumed to be fixed for moderate mouth openings. For instance, the hyoid bone is the origin of several jaw opening muscles and moves during jaw opening. However, the displacement of the hyoid bone can be neglected for openings of small amplitude (Hannam et al., 2008). Once the origin was located, the position of the insertion points of each fibre, calculated throughout the simulation of the movement, allowed assessing the instantaneous directions of the forces and the instantaneous length of each fibre, lM ðtÞ, needed to calculate the passive forces (see Eq. (5)). In addition, the length of the fibres estimated for an interincisal gap of 2 mm was assumed to correspond to lM 0 (Langenbach and Hannam, 1999), as stated before.
2.3.1.
Activity of the lateral pterygoid during the OP
In addition to the muscle activity described in Fig. 4, different activation patterns of the lateral pterygoid muscle during the opening phase were tried. Different activities of the right and left lateral pterygoid muscles were proposed in cases 1, 2 and 3 to analyse its influence on the movement of the mandible. These cases were compared with the activation patterns proposed by Hannam et al. (2008) (case 4) and Koolstra and van Eijden (2005) (case 5). Case 1 was synchronous, like case 4, but with the peak of activity at t ¼ 0.55 s, as proposed by Hylander (1992), and the activation level reduced to 0.7, so as to get an opening gap similar to that obtained in the other cases. Cases 2 and 3 were asynchronous and symmetric to each other. The activation and deactivation profiles of these cases were defined arbitrarily with a change
of slope, like other muscles had in the activation patterns given by Hylander (1992) (see Fig. 4). Additional asynchronous cases (6 and 7) were studied to analyse the influence of the activation and deactivation profiles. Both cases have the peaks of activation in the same instants as that of case 2 (see Fig. 5), but different activation and deactivation rates.
2.4.
Displacement boundary conditions
To simulate the teeth–food contact during the CP, the vertical displacements were constrained in the occlusal face of the corresponding molars. With this boundary condition the grinding forces were simulated as reactions, like in Reina et al. (2007). This boundary condition was active during the CP and relaxed during the OP. The temporal bone was fixed during the whole cycle. And a friction coefficient of μ ¼ 0.015 was assumed in the contact between the disc and the bony parts. The simulations started with the CP, afterwards the OP was performed by activation of the jaw opening muscles. Given that the articular disc was modelled as a viscoelastic material, a second mastication cycle was simulated to evaluate the influence of that viscoelasticity on repeated cycles.
2.5.
Material models
The TMJ disc was modelled with a quasi-linear viscoelastic (QLV) model with the material constants obtained from experimental test of porcine TMJ discs (Commisso, 2012). The ligaments of the TMJ were modelled following the approach of Gardiner and Weiss (2003), proposed to describe the behaviour of the medial collateral ligament of the knee. The TMJ model used in this paper (including the boundary conditions) is the same one used in a previous work (Commisso et al., 2014), which can be consulted for more
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journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
3.
Results
3.1. Analysis of the activation patterns of the lateral pterygoid muscle during the OP
Fig. 5 – Activation patterns tried for the lateral pterygoid muscles during the OP of a unilateral mastication. The ipsilateral (I) muscles are above, in the upper y-axis, while the contralateral (C) ones are represented in the inferior y-axis. The amplitudes are normalized to the corresponding peak of muscle force, given in Table 1.
details. The mechanical properties of the mandible bone depend on the local density and local anisotropy and this dependency was modelled with the equations given by Doblaré and García (2002). The distribution of both variables, and, thus, of the mechanical properties was estimated in Reina et al. (2007).
The activation patterns of the right and left lateral pterygoid muscles shown in Fig. 5 produced different opening movements. Fig. 6 shows the resultant displacement of the incisal edge projected onto the frontal plane. The synchronous activation of the right and left muscles of case 1 produced an almost identical movement in the ongoing and return displacement of the jaw. On the contrary, asynchronous activations (cases 2 and 3) led to the typical “tear drop shape” movement. A later activation of the left (contralateral) muscle (case 2) generated a movement of the jaw from the left to the right. Instead, the prior activation of that muscle (case 3) produced a movement in the opposite direction. The activation patterns proposed in the literature for the lateral pterygoid (cases 4 and 5) were analysed separately. The frontal movement at the incisal edge obtained in those cases was similar to that of the case 1, given that muscles on both sides were activated almost simultaneously, like in case 1. Finally, the influence of the activation and deactivation rates is analysed by comparing cases 2, 6 and 7. It can seen that the trajectory of the incisal edge is very sensitive to the variations of the activation and deactivation profiles of the lateral pterygoid. In addition, the trajectories of the ipsilateral and contralateral condyles are presented in Fig. 7 for cases 1–3. Analogous to the incisal movement, the displacement of the condyles was almost identical in the ongoing and the return movements in case 1. Instead, cases 2 and 3 presented a lateral deviation from the ongoing to the return movement. Moreover, these trajectories are followed in opposite directions in cases 2 and 3, just like the trajectory of the incisal edge. Another important observation is that the contralateral condyle experimented an ampler movement than the ipsilateral one in all the simulated cases. Fig. 8 shows the temporal evolution of the vertical displacement of the incisal edge obtained in cases 1–3. In case 1, the ongoing and return movements occurred fast and the instant of maximum opening coincided with the peak of activity of the lateral pterygoid of both sides (t ¼ 0.55 s, recall Fig. 5). Instead, in cases 2 and 3, the mandible dropped rapidly until t ¼ 0.55 s, but then this fall slowed down. The mouth shifted mainly in the lateral direction (see Fig. 6) for approximately 0.1 s, with only a slight increase in the opening gap. Fig. 9 shows the distribution of the strain energy density (SED) obtained in the mandible at the instant of maximum opening in cases 1–3. SED has been chosen for its importance on the behaviour of bone and particularly on bone remodelling and thus on density distribution (Huiskes et al., 1987). Despite the notable differences in the movement of the jaw, there were only slight differences in the distribution of SED in the mandible, except for the condyles. An animation of the movement of the jaw with the variation of SED during the OP for case 2 is available in the Online Resource 1. The maximum shear stress (computed from the difference between the maximum and minimum principal stresses) was
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
29
Fig. 6 – Trajectory of the incisal edge projected onto the frontal plane for different activation patterns of the lateral pterygoid.
Fig. 7 – Trajectory of the condyles projected onto the frontal plane for different activation patterns of the lateral pterygoid.
estimated for representative nodes at different locations of the disc. The evolution of that shear stress during the OP is plotted in Fig. 10 for cases 1–3. The maximum shear stress was achieved in each disc, at the instant of maximum activation of the lateral pterygoid of the same side, both in
cases 2 and 3. In case 1, it was achieved at the instant of maximum opening, because it coincided, in this case, with the instant of maximum activity of both lateral pterygoid muscles. In all cases, the maximum shear stresses were reached in the intermediate zone, followed by the medial
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journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
region. The lateral region of the ipsilateral disc was subjected to the highest shear stresses during the ongoing and return movement, while those stresses remained almost constant at the anterior and posterior regions.
3.2.
Analysis of the complete mastication cycle
The activation of the lateral pterygoid muscle of case 2 was chosen to simulate the complete mastication cycle. The instant of maximum mastication force coincides with the centric occlusion (CO) and this was the instant when the highest stress level was reached in most of the mandible. The SED distribution in the mandible at that instant is shown in Fig. 11. Fig. 12 shows the maximum and minimum principal stresses at representative nodes of the articular discs throughout the mastication cycle. As mentioned before, a second cycle was simulated to evaluate the influence of the viscoelasticity of the articular disc. The distribution of SED in the mandible at the instant of CO of the first and second cycles is compared in Fig. 13. The minimum principal stresses in the articular disc (the most relevant) are also compared in Fig. 14.
Fig. 8 – Temporal evolution of the vertical displacement of the incisal edge for cases 1–3. The instant of maximum opening is identified as top .
4.
Discussion
The maximum bite force obtained at the instant of CO, 458 N, was within the range measured in experimental studies: 430– 650 N (Van Der Bilt et al., 2008; Sathyanarayana et al., 2012). Regarding the activity of the lateral pterygoid muscle during the OP, it was seen that the asynchronous activation of the right and left lateral pterygoid was responsible for the occurrence of the ”tear drop shape” movement. The synchronous or almost synchronous cases (1, 4 and 5) all produced a movement pattern that was similar in the ongoing and return movements and that occurred despite case 1 had a lower peak of activity of both pterygoid muscles (0.7 against 1.0 of the other two cases). The muscle activation of case 4, proposed by Hannam et al. (2008), was synchronous but with an uneven activation of the muscles of both sides (25% less in the contralateral muscle), which was not enough to get an appreciable lateral deviation. The trajectory of the incisal edge obtained in case 2 was similar to the normal movement during unilateral mastication with the right molars: with a “tear drop shape” of amplitude similar to that measured experimentally and followed in the right direction (Buschang et al., 2000; Bhatka et al., 2004). Case 3 produced a similar movement though in the opposite direction. Cases 6 and 7 produced a movement in the correct direction, thus, indicating that the right lateral pterygoid should be activated prior to the left one in a mastication with the right molars. However, the shape of the movement was far from the typical “tear drop shape”, showing the sensitivity of the movement to the activation and deactivation profiles. A later but faster activation (case 7) produced a high lateral deviation (0.6 mm) while decreased the maximum opening gap (around 5 mm). On the contrary, a sustained but slower activation (case 6) reduced the lateral deviation (0.25 mm) and increased the maximum opening gap (9.5 mm). Consequently, case 2, with an intermediate activation profile, was considered as the most adequate one and taken as the reference for the subsequent simulations. The condyles of the TMJ followed a movement pattern analogous to that observed in the incisal edge. The movement of the mandible is asymmetric during unilateral chewing and this resulted in uneven displacements of the left and
Fig. 9 – Distribution of the SED in the mandible bone at maximum opening for cases 1–3.
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
31
Fig. 10 – Evolution of the maximum shear stress during the OP in representative nodes of the articular disc for cases 1–3. The instant of maximum opening, top, is indicated with a vertical dashed line.
right condyles, exhibiting the contralateral condyle (left in this case) an ampler movement, as confirmed by Hylander (1992). The shear stresses in the disc during the OP were closely related to the activation of the lateral pterygoid muscle as well. The maximum shear stress in each disc was obtained at the instant of maximum activation of the lateral pterygoid of the same side. This result provides arguments for the hypothesis of several authors that related temporomandibular disorders with the force exerted by the lateral pterygoid muscle (Juniper, 1984; Hiraba et al., 2000; Fujita et al., 2001; Okeson, 2003). The SED is the mechanical variable usually employed to measure the intensity of loads in bones. The SED was much lower during the OP than during the CP, when the food is ground (compare the scales of Figs. 9 and 11). Regarding the principal stresses in the articular disc during the complete chewing cycle (Fig. 12), it can be noticed that the discs were subjected to the highest stresses during the CP. This is in accordance with some experimental
Fig. 11 – Distribution of the SED in the mandible bone at maximum mastication force: t ¼ 0.22 s.
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journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
Fig. 12 – Evolution of the principal stresses during a mastication cycle at representative nodes of different zones of the disc: posterior (P), anterior (A) and intermediate (I). Top: ipsilateral disc. Bottom: contralateral disc.
Fig. 13 – Distribution of SED in the mandible at the instant of CO of (a) the first cycle and (b) the second cycle.
measurements (Fushima et al., 2003), where an analysis of the joint space was performed for different mouth opening gaps. The reduction of the joint space can be related to the compressive loading of the TMJ (Fushima et al., 2003). In that study, it was concluded that the condyle–fossa distance was reduced around 5–10% more during the CP than during the OP. During the CP, the contralateral disc underwent higher stresses than the ipsilateral one. Therefore, the reaction force was lower in the ipsilateral joint as confirmed by Mongini et al. (1981) and Kuboki et al. (1996). The different joint reaction forces could be due to an asymmetrical displacement of the condyles and the consequent uneven compression of the
discs. The ipsilateral condyle experienced a displacement of 0.21 mm in magnitude in antero-superior direction (at an angle of 421 with respect to the anteroposterior axis), while the contralateral condyle was subjected to a displacement of 0.43 mm (at 381 with respect to the anteroposterior axis). This result agrees with the measurements of Kuboki et al. (1996) who reported a shift of the condyle against the disc around 50% higher in the contralateral side than in the ipsilateral one. The ipsilateral disc exhibited higher stresses at the intermediate and lateral regions, while the contralateral disc was subjected to higher stresses at its medial portion, which is in accordance with the results presented by Koolstra and van Eijden (2005). These authors obtained the highest stresses
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
33
Fig. 14 – Minimum principal stresses in the contralateral disc at the instant of CO of (a) the first cycle and (b) the second cycle.
around the intermediate zone extending in mediolateral direction. Another important remark was that in the intermediate zone only compressive stresses were seen during the CP (the maximum principal stresses were negative). Acute mechanical overload can cause severe damage of the articular cartilage (Radin et al., 1984; Thompson et al., 1991). More precisely, the shear stresses may induce molecular changes associated with apoptosis of the chondrocytes (Smith et al., 2004), that could affect the disc. During the mastication cycle, the maximum shear stress obtained in the simulations was around 1 MPa, which is much lower than that obtained during abnormal activities like bruxism. For example, a shear stress of 4 MPa was obtained in a simulation of sustained clenching that used this model (Commisso et al., 2014). The presence of a viscoelastic material (the articular disc) suggested the necessity of analysing a second mastication cycle. The stresses (or equivalently the SED) obtained in the mandible in the first and second cycles were very similar (see Fig. 13) due to the fast stress relaxation occurring in the disc (see Fig. 14). The third and further cycles produced identical results to those obtained in the second cycles. This result has important implications, since it allows simplifying the study of the stresses in the mandible during mastication by analysing only one cycle. The limitations of the model are commented next. First, the anisotropy of the articular disc arising from the oriented network of collagen fibres was not considered. The material properties were fitted from experimental tests performed in pieces of disc under uniaxial compression. The collagen fibres are arranged mainly in anteroposterior direction. Thus, in a compression of the disc in vertical direction the fibres are stretched, contributing with their stiffness to resist the external load. However, the influence of the fibres is different under shear or traction in vertical direction. In this last case, the fibres are shortened adding no stiffness and its influence can be neglected. If the disc was subjected to different types of loading, its anisotropy should be considered, but in the cases simulated here it was subjected mainly to a compression in the vertical direction, like in the experimental tests
from which their properties were fitted. Therefore, the simplified material model adopted here can be valid in this particular case. Another limitation is not considering the articular cartilage that covers the surfaces of the articulating bones. The articular cartilage helps to reduce the stresses within the condyle, having little influence on reducing the stresses within the disc (Hu et al., 2003). The mechanical properties of this layer have not been well established, but it is generally accepted that it is more flexible than the articular disc (Beek et al., 2001). For this reason, it is thought to have an effect on the redistribution of stresses over the condyle that needs to be addressed in future works. Finally, Eq. (3) may be a strong simplification in case of masticatory movements of wide amplitude or rapid. Additionally, lM 0 was assumed to occur for an interincisal gap of 2 mm, following the work by Langenbach and Hannam (1999). These authors proposed that gap based on the passive forces needed to keep the mouth open in a resting position, without activation of the masticatory muscles. However, the resulting passive forces were too high to allow their model performing wider movements. So, those authors proposed a second assumption: to estimate lM 0 for an interincisal gap of 12 mm. The effect of this change is limited in a movement of small amplitude but should be investigated in any case. Other limitations are the simplified definition of the teeth-food contact, omission of the tongue and no consideration of the pennation angle of the muscles.
5.
Conclusions
A FE model of the mandible including the TMJ has been used to simulate chewing. Although biomechanical models of the human jaw are difficult to validate, mainly due to the inaccessibility of the joint (Peck and Hannam, 2007), the model presented here tries to overcome the limitations of previous studies and to present a powerful methodology to analyse the functioning of the human jaw. More precisely, the effect of some muscles, such as lateral pterygoid, whose
34
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
activity is difficult to measure in experimental tests and the stresses induced in the mandible bone and in the disc during the masticatory cycle. The most relevant features of the model are (1) a visco-hyperelastic model fitted from experimental tests was used for the articular disc and (2) the mandible bone was considered a deformable body, in contrast to similar studies, where it was modelled as a rigid body (Hannam et al., 2008; Koolstra and van Eijden, 2005). The mandible and the articular discs were subjected to the highest stresses at CO. In the OP, the maximum stresses were observed at the instant of maximum opening and were located at the rami and chin. However, the mechanical stimulus is much lower than that obtained during the CP. The intermediate zone of the articular disc was the most loaded region, specially during the CP and more markedly in the contralateral disc. During the OP the discs were subjected to the highest stresses when the lateral pterygoid muscle of the same side reached the maximum activation level. The activation of the lateral pterygoid muscle was seen to affect significantly the opening movement of the jaw. A prior activation of the ipsilateral muscle is needed in order to obtain the typical “tear drop shape” movement of the incisal edge. In any case, the results obtained here support the notion that the lateral pterygoid muscle provides the principal driving force for moving the jaw laterally, which is in agreement with the observations made by Murray et al. (1999).
Acknowledgements Funding was provided by the Junta de Andalucía for the research project P07-TEP-03115 entitled Biomecánica de la mandíbula humana for which this paper has been prepared.
Appendix A.
Supplementary material
Supplementary data associated with this paper can be found in the online version at http://dx.doi.org/10.1016/j.jmbbm. 2014.09.022
references
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Available online at www.sciencedirect.com
www.elsevier.com/locate/jmbbm
Research Paper
Finite element analysis of the human mastication cycle Maria S. Commisson, Javier Martı´nez-Reina, Joaquı´n Ojeda, Juana Mayo Department of Mechanical Engineering, University of Seville, Camino de los Descubrimientos s/n, E-41092 Seville, Spain
art i cle i nfo
ab st rac t
Article history:
The aim of this paper is to propose a biomechanical model that could serve as a tool to
Received 27 June 2014
overcome some difficulties encountered in experimental studies of the mandible. One of these
Received in revised form
difficulties is the inaccessibility of the temporomandibular joint (TMJ) and the lateral pterygoid
18 September 2014
muscle. The focus of this model is to study the stresses in the joint and the influence of the
Accepted 23 September 2014
lateral pterygoid muscle on the mandible movement. A finite element model of the mandible,
Available online 7 October 2014
including the TMJ, was built to simulate the process of unilateral mastication. Different
Keywords:
activation patterns of the left and right pterygoid muscles were tried. The maximum stresses
Mastication
in the articular disc and in the whole mandible during a complete mastication cycle were
Temporomandibular joint
reached during the instant of centric occlusion. The simulations show a great influence of the
Lateral pterygoid
coordination of the right and left lateral pterygoid muscles on the movement of the jaw during
FE simulation
mastication. An asynchronous activation of the lateral pterygoid muscles is needed to achieve a
Articular disc
normal movement of the jaw during mastication.
Quasi-linear viscoelastic model
1.
Introduction
Biomechanical models of the human mandible have been extensively used to explore the functioning of dental implants (Huang et al., 2014; Holberg et al., 2013; Lan et al., 2012; Ojeda et al., 2011), other fixation devices (Huang et al., 2012; Bohluli et al., 2010), mandible reconstruction (Narra et al., 2014; Li et al., 2014), and temporomandibular disorders (Cheng et al., 2013; Commisso et al., 2014) among other problems. This paper uses a model of the mandible, including the temporomandibular joint (TMJ), to simulate the process of unilateral mastication. Two basic aspects of this process constitute the main focus of this study: the articular disc of n
Corresponding author. Tel.: þ34 954481365; fax: þ34 954460475. E-mail address: [email protected] (M.S. Commisso).
http://dx.doi.org/10.1016/j.jmbbm.2014.09.022 1751-6161/& 2014 Elsevier Ltd. All rights reserved.
& 2014 Elsevier Ltd. All rights reserved.
the TMJ and the muscular activity, in particular that of the lateral pterygoid muscle. The main component of the TMJ is the articular disc, a plate of fibrocartilage that facilitates the relative movement between the mandible and the temporal bone. The articular disc acts as a load absorber and distributes the loads over larger contact areas to prevent the damage of the articulating surfaces. In turn, damage of the articular disc can be one of the causes of temporomandibular disorders (TMD). Some researchers have shown that acute mechanical overloads in vivo can cause severe cartilage damage (Radin et al., 1984; Thompson et al., 1991). Particularly, shear stresses are believed to alter the cells in the tissue and might lay behind the damage in the cartilage (Smith et al., 2004). So, the
24
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
analysis of the shear stresses in the disc during chewing can be of interest for a study of TMD. The stresses and strains in the articular disc during loading are very difficult to measure experimentally. Therefore, several studies have investigated the stress and strain distribution in the TMJ during mastication by means of finite element (FE) models (see Beek et al., 2001; Koolstra and van Eijden, 2007, among others). To the authors’ knowledge, only Koolstra and van Eijden (2007) studied a complete mastication cycle and considered the viscoelastic behaviour of the articular disc. However, the movement of the mandible was defined with a multibody model and the jaw was assumed as a rigid body, so that the stress distribution within the bone was not accessible. The present study tries to overcome that limitation by using a FE model of the mandible with a detailed treatment of the TMJ, including the ligaments and a viscoelastic model of the articular disc. In regard to the second pillar of this study, the muscular activity during mastication, it must be noted that the human masticatory system has a very complex performance and requires the balanced and coordinated activities of the masticatory muscles on the left and right sides (Yamaguchi et al., 2011). In particular, the human lateral pterygoid muscle plays an important role in the control of the jaw movements (Murray, 2012) and stabilization of the TMJ (Murray et al., 1999). Nevertheless, there is very limited understanding on how it performs these functions. Furthermore, studies do not even agree on the specific functions of the two major bellies of the muscle (Murray et al., 1999): the superior and inferior heads. Only one thing seems to be clear: the inferior head is active in jaw-opening, jaw protrusion and contralateral jaw movements (Phanachet et al., 2001; Murray, 2012). One of the main reasons for the lack of understanding of the function of this muscle is the difficulty of studying it with electromyography (EMG) recordings, arising from the deep location of the muscle, which can result in a misplacement of the electromyograph. Another reason is that the condylar movement should be simultaneously recorded with the lateral pterygoid activity. This is essential in this muscle, in light of the observations of several authors (Sessle and Gurza, 1982; Murray et al., 1999), who noticed that the position of the jaw is an important determinant of its EMG activity. As an alternative to these EMG studies, some previous studies have simulated the mastication cycle to predict the activity of the lateral pterygoid from an inverse analysis. Hannam et al. (2008), Koolstra and van Eijden (2005) and de Zee et al. (2007) proposed multibody system models of the mandible and the muscles attached to it to simulate masticatory movements. They proposed an inverse analysis to fit the activity of the lateral pterygoid, such that it produced the normal movement of the jaw during mastication. This normal movement is characterized by a trajectory of the incisal edge with a “tear drop shape” when projected on a frontal plane (Buschang et al., 2000; Bhatka et al., 2004). However, those models contain simplifications with a major influence on the masticatory movement. Apart from the use of a multibody system model (rigid) for the mandible, the model of the TMJ was very simplified and the activities of the left and right lateral pterygoid muscles were synchronous (Koolstra and van Eijden, 2005) or nearly synchronous
(Hannam et al., 2008). It will be shown here that synchronous activation patterns result in abnormal movements when the constraints imposed by the TMJ are more accurately modelled. The lateral pterygoid muscle and its relationship with TMD have also been studied and reviewed extensively (Juniper, 1984; Widmalm et al., 1987; Hiraba et al., 2000; Fujita et al., 2001; Okeson, 2003; Desmons et al., 2007), though no consensus has been achieved to establish the origin of that relationship. In order to shed light on this matter it is essential to understand well its normal functioning. In this work, a FE model of the mandible and TMJs is proposed to simulate a mastication cycle by applying the loads exerted by the jaw opening and jaw closing muscles during the cycle. A special treatment is given to the forces applied by the lateral pterygoid muscle. Since there is no agreement in defining the activity of that muscle, different temporal activation patterns were tried to analyse its influence on the jaw movement. The main objective of this work is to propose a comprehensive biomechanical model of the masticatory system that allows the study of a mastication cycle and the stresses generated in the mandible and in the TMJ during it. Another objective is to check if the activity of the lateral pterygoid muscle has a significant influence on the movement of the jaw during chewing.
2.
Materials and methods
2.1.
Finite element model
A FE model of the mandible and TMJ was built using the commercial software Abaqus FEAs v6.10. The model was previously described in more detail in Commisso et al. (2014). The modelled parts of the joint were the articular disc, the collateral and temporomandibular (TML) ligaments and the posterior part of the articular capsule (see Fig. 1).
2.2.
Muscle model
The estimation of the muscle forces was made using a Hilllike model (Hill, 1938; Thelen, 2003), with the force being applied by the muscle the sum of two terms: the active force, due to the contraction of the muscles fibres, and the passive force, due to the stiffness of the connective tissue: F ¼ FA þ FP
ð1Þ
The active forces are generally expressed as a function of the maximum force in isometric contraction, FM 0 , the length of the muscular-tendon unit, l, the contraction velocity, v, and the activation level, a. For example, in the Thelen (2003) model, FA ¼ a FM 0 f L ðlÞ f V ðvÞ
ð2Þ
where fL(l) and fV(v) describe the dependency on the length and the contraction velocity respectively and are dimensionless functions of the so-called optimal fibre length, lM 0 , among other variables. The function fL(l) has a maximum f L ðlM 0 Þ ¼ 1. Besides, lM 0 was assumed to occur when the interincisal gap is 2 mm, like in Langenbach and Hannam (1999). Mouth opening during
25
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
Fig. 1 – (a) Complete FE model. (b) Detail of the TMJ.
Table 1 – Data of the forces exerted by the masticatory muscles. Only the orientation of the forces of the right side is given (symmetry with respect to the sagittal plane is assumed). The magnitude of the muscle force peak in centric occlusion for the ipsilateral (I) and contralateral (C) sides is also provided. The þx-axis is directed anteriorly, the þy-axis rightward and the þz-axis downward. CP: closing phase, OP: opening phase. Muscle
Portion
Length (mm)a
cos-xb,c
cos-yb,c
cos-zb,c
Peak of activity amax d,e I
C
b,c FM 0 ðNÞ
Masseter
Superficial Deep
48.0 25.7
0.419 0.358
0.207 0.546
0.885 0.758
0.56 0.56
0.20 0.20
190.4 81.6
Temporalis
Anterior Middle Posterior
51.9 52.7 52.9
0.044 0.5 0.855
0.149 0.221 0.208
0.988 0.837 0.474
0.65 0.60 0.54
0.51 0.53 0.54
158.0 95.6 75.6
42.7
0.372
0.486
0.791
0.97
0.47
174.8
32.6
0.757
0.63
0.645
0.761
0.36 0.10 0.026
66.9
31.3
0.50 0.10 0.039
Medial pterygoid Lateral pterygoid (CP) Lateral pterygoid (OP)
Inferior Inferior Superior
Anterior belly digastric
43.2
Geniohyoid Mylohyoid
40.0 Anterior Posterior
23.0 42.6
0.053 0.995 0.176 0.223
0.174 0.074 0.576
0.815 0.103
0.018
0.831 0.616
0.528 0.756
5 10
4
2.6 10
4 4
1.6 10 4.7 10 4
5 10
28.8 4
2.6 10
40.0
4
38.8
4
63.6 21.2
1.6 10 4.7 10 4
a
Length with the mouth closed (van Eijden et al., 1997). Korioth et al. (1992) for the jaw closing muscles. c van Eijden et al. (1997) for the jaw opening muscles. d Nelson (1986) for the jaw closing muscles. e Proportional to the values adopted by Hannam et al. (2008) for the jaw opening muscles. b
mastication is a movement of small amplitude and, consequently, the length of the muscles varies little around its value with that gap, lM 0 . Therefore, it can be assumed that f L ðlÞ 1 during the masticatory movement. On the other hand, this movement is relatively slow and it can be also assumed that f V ðvÞ f V ð0Þ ¼ 1. Then, the active muscle force is simplified as FA ðtÞ ¼ aðtÞ FM 0
ð3Þ
and was calculated throughout the cycle with the values of FM 0 given in Table 1 and the evolution of a(t) was given in Figs. 4 and 5.
The passive forces throughout the cycle were calculated as in Thelen (2003): P
M
FP ðtÞ ¼ FM 0 F ðl ðtÞÞ M
ð4Þ
where l ðtÞ ¼ lM ðtÞ=lM 0 is the normalized fibre length. The P normalized passive force, F , was defined as 8 M kP M > > > Þ ; l 41 þ εM 1 þ M l ðtÞ ð1 þ εM > 0 0 < ε0 P M ð5Þ F l ðtÞ ¼ P M M > > M ek ðl ðtÞ 1Þ=ε0 > M > ; l r1 þ ε : P 0 ek
26
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
Fig. 2 – Insertions of the different portions of the jaw closing muscles. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
where εM 0 ¼ 0:6 is the passive muscle strain corresponding to the maximum isometric force and kP ¼ 4 is a shape factor for the passive force–length relationship.
2.3.
Force boundary conditions
A complete unilateral mastication cycle with the right molars was simulated by applying the forces of the masticatory muscles. These forces were imposed as external loads, distributed over the insertion area of each muscle. Figs. 2 and 3 highlight in different colours the groups of nodes where the different muscles were inserted. The orientation of the muscle forces was taken from Korioth et al. (1992) and van Eijden et al. (1997) (see Table 1). The closing (CP) and opening (OP) phases of the cycle were simulated by applying the muscle activity patterns given by Hylander (1992) and shown in Fig. 4, plus the activity of the lateral pterygoid muscle in the opening phase, not shown in that figure but discussed later on. These activity patterns provide a(t), needed in Eq. (3), normalized by the peak of activity, amax, given in Table 1. Note that centric occlusion (CO) corresponds to the instant of maximum mastication force (around t ¼ 0.19 s) indicated with a vertical arrow in Fig. 4. Note also that the closing and opening phases are named after which muscles are active, either the jaw closing or jaw opening muscles, but not after the movement of the mandible. The values of amax corresponding to the jaw closing muscles (see Table 1) were taken from Nelson (1986) and used in a previous model (Reina et al., 2007). For the jaw opening muscles (anterior digastric, geniohyoid, anterior and posterior mylohyoid) they were defined based on the values adopted by Hannam et al. (2008). Those authors simulated an opening of the mandible of 20 mm, which is beyond 6–14 mm observed in chewing (Bhatka et al., 2004; Hedjazi et al., 2013). So, the values of amax for these muscles were chosen proportional to the values adopted by Hannam et al. (2008), with a unique constant of proportionality adjusted to get an opening of 8 mm. During a mastication cycle the direction of the forces exerted by the jaw closing muscles can be assumed: (1) uniform, because their fibres run approximately parallel
Fig. 3 – Insertions of the different portions of jaw opening muscles. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)
close to the insertion to the mandible and (2) constant throughout the cycle, because the motion of their insertion points is of small amplitude (the jaw is approximately rotating around the TMJ, close to those insertion points). However, this is not true for the jaw opening muscles. First, these are fan-shaped muscles and each fibre has a different orientation and, second, they are inserted far from the TMJ. Thus, the direction of their forces and/or their moment arms with respect to the joint center varies with the mouth opening. This made necessary to consider the movement of the mandible to estimate the orientation of the forces exerted by those muscles. Such orientations were approximated by the direction of a segment joining the origin of the whole muscle and the insertion point of each individual fibre, thus, neglecting the pennation angle.1
1 The muscle force dependency on this pennation angle is proportional to its cosine and the maximum pennation angle for the mastication muscles is 161 (van Eijden et al., 1997). This implies a maximum error of 4% in the estimation of the force, which justifies the adopted simplification.
journal of the mechanical behavior of biomedical materials 41 (2015) 23 –35
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Fig. 4 – Activation pattern of the jaw closing and opening muscles during unilateral mastication. The ipsilateral (I) muscles are above, in the upper y-axis, while the contralateral (C) ones are represented in the inferior y-axis. The amplitudes were normalized to the corresponding peak of muscle force, given in Table 1 for each muscle. The activity of the lateral pterygoid muscle during the OP has been omitted for clarity (see Fig. 5 for this activity). Adapted from Hylander (1992).
The position of the unique point representing the origin of each jaw opening muscle was estimated as follows. A representative insertion point was approximately located in the FE model (see Fig. 3) by choosing a single point placed in the center of the corresponding insertion area. Then, the origin was estimated with the orientation and the length with the mouth closed, both given in Table 1. This origin was assumed to be fixed for moderate mouth openings. For instance, the hyoid bone is the origin of several jaw opening muscles and moves during jaw opening. However, the displacement of the hyoid bone can be neglected for openings of small amplitude (Hannam et al., 2008). Once the origin was located, the position of the insertion points of each fibre, calculated throughout the simulation of the movement, allowed assessing the instantaneous directions of the forces and the instantaneous length of each fibre, lM ðtÞ, needed to calculate the passive forces (see Eq. (5)). In addition, the length of the fibres estimated for an interincisal gap of 2 mm was assumed to correspond to lM 0 (Langenbach and Hannam, 1999), as stated before.
2.3.1.
Activity of the lateral pterygoid during the OP
In addition to the muscle activity described in Fig. 4, different activation patterns of the lateral pterygoid muscle during the opening phase were tried. Different activities of the right and left lateral pterygoid muscles were proposed in cases 1, 2 and 3 to analyse its influence on the movement of the mandible. These cases were compared with the activation patterns proposed by Hannam et al. (2008) (case 4) and Koolstra and van Eijden (2005) (case 5). Case 1 was synchronous, like case 4, but with the peak of activity at t ¼ 0.55 s, as proposed by Hylander (1992), and the activation level reduced to 0.7, so as to get an opening gap similar to that obtained in the other cases. Cases 2 and 3 were asynchronous and symmetric to each other. The activation and deactivation profiles of these cases were defined arbitrarily with a change
of slope, like other muscles had in the activation patterns given by Hylander (1992) (see Fig. 4). Additional asynchronous cases (6 and 7) were studied to analyse the influence of the activation and deactivation profiles. Both cases have the peaks of activation in the same instants as that of case 2 (see Fig. 5), but different activation and deactivation rates.
2.4.
Displacement boundary conditions
To simulate the teeth–food contact during the CP, the vertical displacements were constrained in the occlusal face of the corresponding molars. With this boundary condition the grinding forces were simulated as reactions, like in Reina et al. (2007). This boundary condition was active during the CP and relaxed during the OP. The temporal bone was fixed during the whole cycle. And a friction coefficient of μ ¼ 0.015 was assumed in the contact between the disc and the bony parts. The simulations started with the CP, afterwards the OP was performed by activation of the jaw opening muscles. Given that the articular disc was modelled as a viscoelastic material, a second mastication cycle was simulated to evaluate the influence of that viscoelasticity on repeated cycles.
2.5.
Material models
The TMJ disc was modelled with a quasi-linear viscoelastic (QLV) model with the material constants obtained from experimental test of porcine TMJ discs (Commisso, 2012). The ligaments of the TMJ were modelled following the approach of Gardiner and Weiss (2003), proposed to describe the behaviour of the medial collateral ligament of the knee. The TMJ model used in this paper (including the boundary conditions) is the same one used in a previous work (Commisso et al., 2014), which can be consulted for more
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3.
Results
3.1. Analysis of the activation patterns of the lateral pterygoid muscle during the OP
Fig. 5 – Activation patterns tried for the lateral pterygoid muscles during the OP of a unilateral mastication. The ipsilateral (I) muscles are above, in the upper y-axis, while the contralateral (C) ones are represented in the inferior y-axis. The amplitudes are normalized to the corresponding peak of muscle force, given in Table 1.
details. The mechanical properties of the mandible bone depend on the local density and local anisotropy and this dependency was modelled with the equations given by Doblaré and García (2002). The distribution of both variables, and, thus, of the mechanical properties was estimated in Reina et al. (2007).
The activation patterns of the right and left lateral pterygoid muscles shown in Fig. 5 produced different opening movements. Fig. 6 shows the resultant displacement of the incisal edge projected onto the frontal plane. The synchronous activation of the right and left muscles of case 1 produced an almost identical movement in the ongoing and return displacement of the jaw. On the contrary, asynchronous activations (cases 2 and 3) led to the typical “tear drop shape” movement. A later activation of the left (contralateral) muscle (case 2) generated a movement of the jaw from the left to the right. Instead, the prior activation of that muscle (case 3) produced a movement in the opposite direction. The activation patterns proposed in the literature for the lateral pterygoid (cases 4 and 5) were analysed separately. The frontal movement at the incisal edge obtained in those cases was similar to that of the case 1, given that muscles on both sides were activated almost simultaneously, like in case 1. Finally, the influence of the activation and deactivation rates is analysed by comparing cases 2, 6 and 7. It can seen that the trajectory of the incisal edge is very sensitive to the variations of the activation and deactivation profiles of the lateral pterygoid. In addition, the trajectories of the ipsilateral and contralateral condyles are presented in Fig. 7 for cases 1–3. Analogous to the incisal movement, the displacement of the condyles was almost identical in the ongoing and the return movements in case 1. Instead, cases 2 and 3 presented a lateral deviation from the ongoing to the return movement. Moreover, these trajectories are followed in opposite directions in cases 2 and 3, just like the trajectory of the incisal edge. Another important observation is that the contralateral condyle experimented an ampler movement than the ipsilateral one in all the simulated cases. Fig. 8 shows the temporal evolution of the vertical displacement of the incisal edge obtained in cases 1–3. In case 1, the ongoing and return movements occurred fast and the instant of maximum opening coincided with the peak of activity of the lateral pterygoid of both sides (t ¼ 0.55 s, recall Fig. 5). Instead, in cases 2 and 3, the mandible dropped rapidly until t ¼ 0.55 s, but then this fall slowed down. The mouth shifted mainly in the lateral direction (see Fig. 6) for approximately 0.1 s, with only a slight increase in the opening gap. Fig. 9 shows the distribution of the strain energy density (SED) obtained in the mandible at the instant of maximum opening in cases 1–3. SED has been chosen for its importance on the behaviour of bone and particularly on bone remodelling and thus on density distribution (Huiskes et al., 1987). Despite the notable differences in the movement of the jaw, there were only slight differences in the distribution of SED in the mandible, except for the condyles. An animation of the movement of the jaw with the variation of SED during the OP for case 2 is available in the Online Resource 1. The maximum shear stress (computed from the difference between the maximum and minimum principal stresses) was
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Fig. 6 – Trajectory of the incisal edge projected onto the frontal plane for different activation patterns of the lateral pterygoid.
Fig. 7 – Trajectory of the condyles projected onto the frontal plane for different activation patterns of the lateral pterygoid.
estimated for representative nodes at different locations of the disc. The evolution of that shear stress during the OP is plotted in Fig. 10 for cases 1–3. The maximum shear stress was achieved in each disc, at the instant of maximum activation of the lateral pterygoid of the same side, both in
cases 2 and 3. In case 1, it was achieved at the instant of maximum opening, because it coincided, in this case, with the instant of maximum activity of both lateral pterygoid muscles. In all cases, the maximum shear stresses were reached in the intermediate zone, followed by the medial
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region. The lateral region of the ipsilateral disc was subjected to the highest shear stresses during the ongoing and return movement, while those stresses remained almost constant at the anterior and posterior regions.
3.2.
Analysis of the complete mastication cycle
The activation of the lateral pterygoid muscle of case 2 was chosen to simulate the complete mastication cycle. The instant of maximum mastication force coincides with the centric occlusion (CO) and this was the instant when the highest stress level was reached in most of the mandible. The SED distribution in the mandible at that instant is shown in Fig. 11. Fig. 12 shows the maximum and minimum principal stresses at representative nodes of the articular discs throughout the mastication cycle. As mentioned before, a second cycle was simulated to evaluate the influence of the viscoelasticity of the articular disc. The distribution of SED in the mandible at the instant of CO of the first and second cycles is compared in Fig. 13. The minimum principal stresses in the articular disc (the most relevant) are also compared in Fig. 14.
Fig. 8 – Temporal evolution of the vertical displacement of the incisal edge for cases 1–3. The instant of maximum opening is identified as top .
4.
Discussion
The maximum bite force obtained at the instant of CO, 458 N, was within the range measured in experimental studies: 430– 650 N (Van Der Bilt et al., 2008; Sathyanarayana et al., 2012). Regarding the activity of the lateral pterygoid muscle during the OP, it was seen that the asynchronous activation of the right and left lateral pterygoid was responsible for the occurrence of the ”tear drop shape” movement. The synchronous or almost synchronous cases (1, 4 and 5) all produced a movement pattern that was similar in the ongoing and return movements and that occurred despite case 1 had a lower peak of activity of both pterygoid muscles (0.7 against 1.0 of the other two cases). The muscle activation of case 4, proposed by Hannam et al. (2008), was synchronous but with an uneven activation of the muscles of both sides (25% less in the contralateral muscle), which was not enough to get an appreciable lateral deviation. The trajectory of the incisal edge obtained in case 2 was similar to the normal movement during unilateral mastication with the right molars: with a “tear drop shape” of amplitude similar to that measured experimentally and followed in the right direction (Buschang et al., 2000; Bhatka et al., 2004). Case 3 produced a similar movement though in the opposite direction. Cases 6 and 7 produced a movement in the correct direction, thus, indicating that the right lateral pterygoid should be activated prior to the left one in a mastication with the right molars. However, the shape of the movement was far from the typical “tear drop shape”, showing the sensitivity of the movement to the activation and deactivation profiles. A later but faster activation (case 7) produced a high lateral deviation (0.6 mm) while decreased the maximum opening gap (around 5 mm). On the contrary, a sustained but slower activation (case 6) reduced the lateral deviation (0.25 mm) and increased the maximum opening gap (9.5 mm). Consequently, case 2, with an intermediate activation profile, was considered as the most adequate one and taken as the reference for the subsequent simulations. The condyles of the TMJ followed a movement pattern analogous to that observed in the incisal edge. The movement of the mandible is asymmetric during unilateral chewing and this resulted in uneven displacements of the left and
Fig. 9 – Distribution of the SED in the mandible bone at maximum opening for cases 1–3.
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Fig. 10 – Evolution of the maximum shear stress during the OP in representative nodes of the articular disc for cases 1–3. The instant of maximum opening, top, is indicated with a vertical dashed line.
right condyles, exhibiting the contralateral condyle (left in this case) an ampler movement, as confirmed by Hylander (1992). The shear stresses in the disc during the OP were closely related to the activation of the lateral pterygoid muscle as well. The maximum shear stress in each disc was obtained at the instant of maximum activation of the lateral pterygoid of the same side. This result provides arguments for the hypothesis of several authors that related temporomandibular disorders with the force exerted by the lateral pterygoid muscle (Juniper, 1984; Hiraba et al., 2000; Fujita et al., 2001; Okeson, 2003). The SED is the mechanical variable usually employed to measure the intensity of loads in bones. The SED was much lower during the OP than during the CP, when the food is ground (compare the scales of Figs. 9 and 11). Regarding the principal stresses in the articular disc during the complete chewing cycle (Fig. 12), it can be noticed that the discs were subjected to the highest stresses during the CP. This is in accordance with some experimental
Fig. 11 – Distribution of the SED in the mandible bone at maximum mastication force: t ¼ 0.22 s.
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Fig. 12 – Evolution of the principal stresses during a mastication cycle at representative nodes of different zones of the disc: posterior (P), anterior (A) and intermediate (I). Top: ipsilateral disc. Bottom: contralateral disc.
Fig. 13 – Distribution of SED in the mandible at the instant of CO of (a) the first cycle and (b) the second cycle.
measurements (Fushima et al., 2003), where an analysis of the joint space was performed for different mouth opening gaps. The reduction of the joint space can be related to the compressive loading of the TMJ (Fushima et al., 2003). In that study, it was concluded that the condyle–fossa distance was reduced around 5–10% more during the CP than during the OP. During the CP, the contralateral disc underwent higher stresses than the ipsilateral one. Therefore, the reaction force was lower in the ipsilateral joint as confirmed by Mongini et al. (1981) and Kuboki et al. (1996). The different joint reaction forces could be due to an asymmetrical displacement of the condyles and the consequent uneven compression of the
discs. The ipsilateral condyle experienced a displacement of 0.21 mm in magnitude in antero-superior direction (at an angle of 421 with respect to the anteroposterior axis), while the contralateral condyle was subjected to a displacement of 0.43 mm (at 381 with respect to the anteroposterior axis). This result agrees with the measurements of Kuboki et al. (1996) who reported a shift of the condyle against the disc around 50% higher in the contralateral side than in the ipsilateral one. The ipsilateral disc exhibited higher stresses at the intermediate and lateral regions, while the contralateral disc was subjected to higher stresses at its medial portion, which is in accordance with the results presented by Koolstra and van Eijden (2005). These authors obtained the highest stresses
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Fig. 14 – Minimum principal stresses in the contralateral disc at the instant of CO of (a) the first cycle and (b) the second cycle.
around the intermediate zone extending in mediolateral direction. Another important remark was that in the intermediate zone only compressive stresses were seen during the CP (the maximum principal stresses were negative). Acute mechanical overload can cause severe damage of the articular cartilage (Radin et al., 1984; Thompson et al., 1991). More precisely, the shear stresses may induce molecular changes associated with apoptosis of the chondrocytes (Smith et al., 2004), that could affect the disc. During the mastication cycle, the maximum shear stress obtained in the simulations was around 1 MPa, which is much lower than that obtained during abnormal activities like bruxism. For example, a shear stress of 4 MPa was obtained in a simulation of sustained clenching that used this model (Commisso et al., 2014). The presence of a viscoelastic material (the articular disc) suggested the necessity of analysing a second mastication cycle. The stresses (or equivalently the SED) obtained in the mandible in the first and second cycles were very similar (see Fig. 13) due to the fast stress relaxation occurring in the disc (see Fig. 14). The third and further cycles produced identical results to those obtained in the second cycles. This result has important implications, since it allows simplifying the study of the stresses in the mandible during mastication by analysing only one cycle. The limitations of the model are commented next. First, the anisotropy of the articular disc arising from the oriented network of collagen fibres was not considered. The material properties were fitted from experimental tests performed in pieces of disc under uniaxial compression. The collagen fibres are arranged mainly in anteroposterior direction. Thus, in a compression of the disc in vertical direction the fibres are stretched, contributing with their stiffness to resist the external load. However, the influence of the fibres is different under shear or traction in vertical direction. In this last case, the fibres are shortened adding no stiffness and its influence can be neglected. If the disc was subjected to different types of loading, its anisotropy should be considered, but in the cases simulated here it was subjected mainly to a compression in the vertical direction, like in the experimental tests
from which their properties were fitted. Therefore, the simplified material model adopted here can be valid in this particular case. Another limitation is not considering the articular cartilage that covers the surfaces of the articulating bones. The articular cartilage helps to reduce the stresses within the condyle, having little influence on reducing the stresses within the disc (Hu et al., 2003). The mechanical properties of this layer have not been well established, but it is generally accepted that it is more flexible than the articular disc (Beek et al., 2001). For this reason, it is thought to have an effect on the redistribution of stresses over the condyle that needs to be addressed in future works. Finally, Eq. (3) may be a strong simplification in case of masticatory movements of wide amplitude or rapid. Additionally, lM 0 was assumed to occur for an interincisal gap of 2 mm, following the work by Langenbach and Hannam (1999). These authors proposed that gap based on the passive forces needed to keep the mouth open in a resting position, without activation of the masticatory muscles. However, the resulting passive forces were too high to allow their model performing wider movements. So, those authors proposed a second assumption: to estimate lM 0 for an interincisal gap of 12 mm. The effect of this change is limited in a movement of small amplitude but should be investigated in any case. Other limitations are the simplified definition of the teeth-food contact, omission of the tongue and no consideration of the pennation angle of the muscles.
5.
Conclusions
A FE model of the mandible including the TMJ has been used to simulate chewing. Although biomechanical models of the human jaw are difficult to validate, mainly due to the inaccessibility of the joint (Peck and Hannam, 2007), the model presented here tries to overcome the limitations of previous studies and to present a powerful methodology to analyse the functioning of the human jaw. More precisely, the effect of some muscles, such as lateral pterygoid, whose
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activity is difficult to measure in experimental tests and the stresses induced in the mandible bone and in the disc during the masticatory cycle. The most relevant features of the model are (1) a visco-hyperelastic model fitted from experimental tests was used for the articular disc and (2) the mandible bone was considered a deformable body, in contrast to similar studies, where it was modelled as a rigid body (Hannam et al., 2008; Koolstra and van Eijden, 2005). The mandible and the articular discs were subjected to the highest stresses at CO. In the OP, the maximum stresses were observed at the instant of maximum opening and were located at the rami and chin. However, the mechanical stimulus is much lower than that obtained during the CP. The intermediate zone of the articular disc was the most loaded region, specially during the CP and more markedly in the contralateral disc. During the OP the discs were subjected to the highest stresses when the lateral pterygoid muscle of the same side reached the maximum activation level. The activation of the lateral pterygoid muscle was seen to affect significantly the opening movement of the jaw. A prior activation of the ipsilateral muscle is needed in order to obtain the typical “tear drop shape” movement of the incisal edge. In any case, the results obtained here support the notion that the lateral pterygoid muscle provides the principal driving force for moving the jaw laterally, which is in agreement with the observations made by Murray et al. (1999).
Acknowledgements Funding was provided by the Junta de Andalucía for the research project P07-TEP-03115 entitled Biomecánica de la mandíbula humana for which this paper has been prepared.
Appendix A.
Supplementary material
Supplementary data associated with this paper can be found in the online version at http://dx.doi.org/10.1016/j.jmbbm. 2014.09.022
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