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A method to predict muscle control in the kinematically and mechanically indeterminate human masticatory system
Journal of Biomechanics 34 (2001) 1179–1188
A method to predict muscle control in the kinematically and mechanically indeterminate human masticatory system J.H. Koolstra*, T.M.G.J. van Eijden Department of Functional Anatomy, Academic Centre for Dentistry Amsterdam (ACTA), Meibergdreef 15, 1105 AZ Amsterdam, Netherlands Accepted 18 March 2001
Abstract A method is proposed to generate muscle activation patterns for goal-directed movements of the human masticatory system. This system is special because apart from a larger amount of muscles than degrees of freedom its joints do not restrict its movements a priori. Therefore, each muscle is able to influence all six degrees of freedom which makes the system kinematically and mechanically indeterminate. Furthermore, its working space is principally determined by the dynamical properties of its muscles and not by passive constraints. The presented method determines the contribution of each degree of freedom to a movement of a reference point on the mandible. It avails of straightforward mathematical techniques like Linear Programming. It does not require a separate trajectory planning step. It was applied in a six degrees of freedom dynamical mathematical model of the human masticatory system. This model which was based upon rigid-body dynamics incorporating skull morphology and muscle architecture including dynamical properties. Movements were exclusively defined by a goal position of the mandibular reference point. The method proved to be robust in generating muscle activation patterns for both feasible and infeasible movement tasks. Generally, they were accomplished faster than habitually observed. If the task was infeasible the movement stopped at the outer boundary of the working space at the side of the unreachable goal. The method, therefore, enables to explore the working space of the mandible and the factors that are relevant for its boundaries. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: Control; Masticatory muscles; Jaw movement; Biomechanics
1. Introduction The human masticatory system is a typical example of a kinematically and mechanically indeterminate system. Two segments, the mandible and the skull, are able to move with respect to each other. These movements are guided by two mutually linked temporomandibular joints. In each joint a mandibular condyle articulates incongruently with the articular surface of the temporal bone. The articular capsule is slack. Due to this construction both joints allow for movements with six degrees of freedom (Koolstra and van Eijden, 1999). Consequently, jaw movements are not limited to rotations about one or more axes defined by the joint (Andrews and Hay, 1983). If the joint surfaces are assumed to be undeformable and maintain contact all *Corresponding author. Tel.: +31-20-5665370; fax: +31-206911856. E-mail address: [email protected] (J.H. Koolstra).
the time, the mandible still is able to move with four degrees of freedom (Schumacher, 1961). Jaw movements can be defined by the three-dimensional path travelled by the lower central incisor (Lewin, 1985). This can be accomplished in various ways with the system that is able to move with at least four degrees of freedom. Consequently, the masticatory system must be considered as kinematically redundant. Jaw movements are accomplished by a large number of masticatory muscles. The majority is relatively short with large attachment areas. While these muscles can be activated heterogeneously (Blanksma et al., 1997) each muscle is able to influence more than one degree of freedom (van der Helm and Veenbaas, 1991). All muscle portions together generate a resultant force and torque (six degrees of freedom) with respect to the centre of gravity of the lower jaw (Koolstra and van Eijden, 1995). The distribution of the forces and torques necessary to perform any movement over the different muscle portions is not a priori
0021-9290/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 1 ) 0 0 0 5 3 - 7
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established. Consequently, the system is mechanically redundant. Various theories have been proposed to predict suitable muscle recruitment patterns in redundant biomechanical systems. Mostly they were based upon minimization of some cost function such as fatigue, energy consumption, joint load or jerk (Zajac and Winters, 1990) to determine a distribution of joint torques over the relevant muscles. In the human masticatory system, however, this is not applicable while movements cannot be described by rotations about a priori defined joint axes. In an alternative theory, muscles are activated proportional to the amount of required muscle shortening (Feldman et al., 1990). It requires a complete kinematic image at the final position and has been applied for jaw movements restricted to two degrees of freedom (Laboissi"ere et al., 1996). For unrestricted three-dimensional jaw movements, however, the kinematical redundancy prevents to obtain the required unambiguous kinematic image at the final position. Jaw opening, protrusion, retrusion and laterodeviation are limited to a few centimeters maximally as described by the so-called envelope of jaw motion (Posselt, 1957). While the dynamical properties of the masticatory muscles contribute to a large extent to excursion limitations of the mandible (Koolstra and van Eijden, 1996), it is to be expected that the reachability of a putative goal cannot be determined from the geometry of the joint only. Generally, available methods for muscle recruitment prediction, are not able to handle such uncertainties in goal reachability. The aim of the present study was to develop a suitable method for generating appropriate muscle activation patterns for putative jaw movements. Prerequisitions were: robust towards kinematic redundancy and not dependent on a kinematic image of putative situations. It was to be implemented in a six degree of freedom dynamical biomechanical model of the human masticatory system (Koolstra and van Eijden, 1995, 1996, 1997, 1999), and applied to simulate goal-directed jaw movements and to explore their spatial limits.
2. Materials and methods 2.1. The model Jaw movement simulations were performed using a biomechanical model of the human masticatory system (Fig. 1). This model has been described extensively in Koolstra and van Eijden (1995, 1996). Briefly, it consists of a lower jaw which is accelerated by forces and accompanying torques with respect to its centre of
Fig. 1. Overview of the model. Ventro-lateral view. Continuous lines: muscle lines of action. Cross-bar: muscle origin. Circle: muscle insertion. MAS S: superficial masseter, MAS P: deep masseter, MPT: medial pterygoid, TEM A: anterior temporalis, TEM P: posterior temporalis, LPT S: superior lateral pterygoid, LPT I: inferior lateral pterygoid, DIG: digastric, GEH: geniohyoid, MYH: mylohyoid. Dots: position of centre of right and left condyle and incisor point.
gravity applied by muscles (represented by their lines of action running from fixed origins at the skull and the hyoid bone), joint surfaces, bite points and ligaments. The forces generated by the muscle portions which were modelled as a lumped sarcomere (Fig. 2) were dependent on the amount of activation, sarcomere dynamics and the activation dynamics. The included muscle portions (Fig. 1) were assumed to be homogeneous, physiologically similar and subjected to mutually independent activation. Their fibre lengths, sarcomere lengths and physiological cross-sections were obtained from van Eijden et al. (1995, 1996, 1997), the dynamical muscle properties from van Ruijven and Weijs (1990) and the activation dynamics from Winters and Stark (1987). 2.2. Muscle activation The muscles were activated in such a manner that a reference point on the mandible would move towards a goal. This movement was halted if this point reached its destination or if passive constraints and/or muscle insufficiency failed to move it any further. During the movement the instantaneous forces and torques generated by muscles ðFm Þ; joints ðFj Þ and bite points ðFb Þ are not in equilibrium. A rest force and torque ðFr Þ remains present according to: X X X Fm þ Fj þ Fb þ Fr ¼ 0; ð1Þ
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Fig. 2. Muscle model. Total force is the sum of the forces produced by the sarcomeres ðFsarcomeres Þ: The active force ðFactive Þ is dependent on the activation through the activation dynamics, the instantaneous sarcomere length and contraction velocity. The parallel elastic force ðFpassive Þ is dependent on the instantaneous sarcomere length.
through Fm by applying an adequate instantaneous activation pattern Am ð0pAm p1Þ while X X Fm ¼ ðFm;min þ Am Fm;max Þ; ð2Þ
Fig. 3. Forces and torques controlling the linear and angular accelerations of the lower jaw. Dashed lines: axes of the Cartesian system through the centre of gravity of the lower jaw, v: linear velocity, o: angular velocity, m: mass, I: moment of inertia. Dot: incisor point.
in which F expresses a six degrees of freedom force tensor (Fig. 3). Fb and Fj are dependent on the instantaneous location of the mandible. Fr controls the future direction of movement and can be adapted
in which Fm;min is the minimum force of a muscle equal to its instantaneous passive elastic force and Fm;max is the maximum active force dependent on its physiological cross-sectional area and the instantaneous dynamical properties. The activation pattern is selected from the collection that provides a rest force that would move the lower incisor (Fig. 1) from the current position ðx; y; zÞc to a desired position ðx; y; zÞd : It is not known a priori whether the desired position can be reached or not. The difference between the desired and current position can be treated as a measure for the effort which has to be exerted. The three components ðxd xc ; yd yc ; zd zc Þ of this difference were related to the linear angular displacements of the lower jaw. From Fig. 3 it is clear that the azimuth primarily influences a y-displacement of the incisor point and the elevation a z-displacement as long as these rotations are less than 451. The roll does not contribute unambiguously to the incisor point movement direction. An adequate Fr will diminish the difference between desired and present position and the required effort will be reduced. From the numerous ways to create a rest force which answers the require-
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Fig. 4. Simulation of a feasible jaw movement to ðx; y; zÞ=(0.4, 0.7, 1.6) relative to the starting position. Top panels: lateral view. Middle panels: frontal view. Bottom panels: superior view. P: posterior. A: anterior. M: medial, R: right. L: left. Dark curved lines: path of the centre of both condyles and the incisor point during the movement. Dots: starting point of the condylar and incisor paths. Cross-hairs: position of the incisor after 0.1 s. Inset: lateral, frontal and superior views of the mandible with the position of incisor and condylar points. Orientation: parallel to upper occlusal plane.
ments, the most effective rest force can be found by optimization. This can be accomplished using Linear Programming methods by maximizing the Objective Function: aðxd xc ÞFx þ bðyd yc ÞFy þ gðzd zc ÞFz þdðyd yc ÞFazimuth eðzd zc ÞFelevation
ð3Þ
subject to Eqs. (1) and (2). Although this way the shortest route from ðx; y; zÞc to ðx; y; zÞd is attempted it can be disturbed by constraints in the system. The method optimizes instantaneously and therefore adapts continuously to new situations. The activation pattern, however, cannot be changed instantaneously. If it produces a submaximal active force and is switched off, a part of this active force is still present after a short period of time. In contrast, if the activation should have been switched to 100% the active force will be less than the maximum possible force after this short time interval (Winters and Stark, 1987). This property was implemented in the boundary conditions of the optimization problem by adapting Fm;min and Fm;max in such a way that they would equal the muscle force at 0.1 s after
complete inactivation or complete activation, respectively. In the present study the constants a; b; g; d and e were set to 1. The model was implemented on an IBM RS/6000 SP computer system. It was written in AIX XL FORTRAN and parallelized using Message Passing Interface (MPI) library routines. The Linear Programming was performed using a revised SIMPLEX algorithm (DDLPRS from the IMSL library). 2.3. Simulations Simulations of both feasible and border position movements were performed. For each of them 0.5 s was reserved to approach the desired position as close as possible. For the first a jaw-open movement starting from the closed position in combination with a laterodeviation was chosen. The desired position of the lower incisor was 0.4 cm backwards, 0.7 cm to the left and 1.6 cm downwards relative to the start. Border position movements of the lower central incisor were simulated by defining desired positions far beyond the
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lower incisor working space. They were located inferiorly, anteriorly, posteriorly and laterally with respect to the resting position. Thus, a maximum open, protrusion, retrusion and laterodeviation movement were forced, respectively.
3. Results The desired position for the feasible movement was reached with an accuracy of about 0.03 cm. After 0.1 s the incisor was less than 1 mm separated from its destination (Fig. 4), and the muscle recruitment pattern was still subject to considerable changes in order to reach the destination as close as possible (Fig. 5). After about 0.4 s the muscle recruitment pattern became stationary, although some minor adaptations occurred. The mandibular incisor travelled along a slightly curved line without unexpected major deviations from the shortest route. The four simulated border position movements are presented in Fig. 6. The model reached a maximum opening of 28 mm. The obtained maximum protrusion, retrusion and laterodeviation were 10, 0.5 and 14 mm, respectively. In Fig. 7, the relevant muscle recruitment patterns underlying these movements are shown. The muscle activation patterns and their resultant forces necessary to maintain the extreme positions are depicted in Fig. 8. Except for the retrusion movement the border position movements were relatively straightforward. After 0.1 s they reached their end positions closely and the changes in muscle activation after 0.2 s could almost all be attributed to the limits caused by the muscle activation dynamics (Fig. 7). The retrusion movement approached its limit within about 0.1 s but the path was more capricious (Fig. 6C). Before the actual retrusion occurred the mandible first opened about 13 mm. The condyles moved forward and then backward again. The relevant muscle activation strategy needed about 0.1 s more than the others to become stationary. It is striking that the superior lateral pterygoid muscle, which pulls the mandibular condyles forward, was activated maximally, while through the force–length relationship it could generate less force than 0.5 N (Fig. 8C). Also the medial pterygoid muscles, which pull upward and forward, were recruited. The muscles contributing primarily to the border movements were: lateral pterygoid, mylohyoid and digastric muscles at maximal open, inferior lateral pterygoid muscles at maximal protrusion, mylohyoid muscle portions at maximal retrusion and ipsilateral temporalis portions and contralateral medial pterygoid and mylohyoid muscles at a maximal laterodeviation. In the open, retrusion and laterodeviation position the mylohyoid muscle portions were fully activated and
Fig. 5. Muscle activation pattern generated in order to accomplish the feasible jaw movement. Top panel: right sided muscles. Bottom panel: left-sided muscles. Muscles as in Fig. 1. a: anterior muscle portion. p: posterior muscle portion. Muscles not shown were not activated. Vertical axis: activation relative to the maximum. Horizontal axis: time in s.
produced almost maximum forces (Fig. 8). This also counts for the superior lateral pterygoid muscle in the open position. All other muscles, although sometimes activated maximally, produced only a part of their maximum force due to the force–length relationship. The maximum protrusion excursion (Fig. 8B) required the least muscle force (about 36 N). This can be attributed to the presence of the inferior lateral pterygoid muscles which hardly have unfavourable force and torque components regarding this movement. In contrast, the maximum laterodeviation excursion (Fig. 8D) required a total of about 650 N muscle force. The passive forces generated by the muscles at the border positions are relatively small. The largest amount (approximately 65 N in total) was generated in the maximum open position by the jawclosing muscles.
4. Discussion Generally, the working space of musculo-skeletal systems is defined by constraints applied through joint construction (Blankevoort et al., 1988; Wang et al., 1998). The action of the muscles is generally described
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Fig. 6. Simulated border position movements. A: maximum open excursion. B: maximum protrusion excursion. C: maximum retrusion excursion. D: maximum laterodeviation excursion.
by a torque about one or more joints (Gribble and Ostry, 1998; Lindbeck et al., 1997). In contrast, in the human masticatory system the working space is not primarily constrained by passive structures, but by limitations for force production of its muscles and the action of the muscles cannot be described by a torque about its joints. The proposed method for muscle activation has been proven able to
generate muscle recruitment patterns without a priori knowledge about on the one side the feasibility of the desired movement and on the other a kinematic image of the system at any putative position. Therefore, this method can be regarded to extend the possibilities for development of control strategies from constrained musculo-skeletal systems to unconstrained ones.
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Fig. 7. Muscle activation patterns generated in order to accomplish border position movements. A: maximum open excursion. B: maximum protrusion excursion. C: maximum retrusion excursion. D: maximum laterodeviation excursion. In A, B and C a single side is shown, the activation patterns at the other side were equal.
4.1. Performance considerations Although the model applied in this study is the most complete biomechanical model of the human masticatory system that has been published as far as we know, it still includes many simplifications (for an extensive overview see Koolstra and van Eijden, 1995, 1996, 1997) which may have affected the results. Its joint shapes were simplified, especially in the mediolateral direction. Articular discs were not included and the capsules were modelled by the temporomandibular ligaments only. This may enable a somewhat larger joint movability than generally observed. The muscle portions were modelled by straight (or curved) line elements with sarcomeres differing by resting length only. The recruitment of motor units which may be dependent on their size and dynamical properties (van Eijden and Turkawski, 2001), therefore, could not be studied and
internal muscle torsions were neglected. The muscle portions were sized according to elderly subjects which may have caused lesser strength than on average. This could have been partly responsible for a reduced amount of maximum jaw opening. The model was based upon azimuth and elevation rotations of less than 451 in order to influence displacements of the lower incisor primarily in the yand z-directions, respectively. The largest azimuth (7.41) was obtained at the maximum laterodeviation position. The largest elevation (18.91) was found in the maximum open position. Both were amply within the required limits. Optimization of a linear combination of the rest force components in the objective function enables to use standard Linear Programming algorithms, but it makes a precise conduction of the system more difficult. The constants a; b; g; d and e can be adapted to influence the
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move beyond its possible workspace it ended up on the border of this workspace but not necessary as close to the (unreachable) destination as possible. The optimization of muscle activation patterns was performed by a (revised) SIMPLEX algorithm. The Objective Function and the constraints used in the optimization procedure were convex functions. The optimization problem, therefore, is convex also and the SIMPLEX method will converge to a global optimum (Dantzig, 1963). Functionally, such a global optimum only has a significance at the border of the workspace. It predicts that no other pattern will be able to move the mandible beyond this limit. Consequently, the predicted boundary coincides with the boundary as defined by the governing morphology. 4.2. Activation patterns
Fig. 8. Muscle activation patterns and resulting muscle forces at border positions at 0.5 s after onset. A: maximum open excursion. B: maximum protrusion excursion. C: maximum retrusion excursion. D: maximum laterodeviation excursion. Black bars: muscle activation relative to maximum. White bars: active force relative to maximum isometric force. Grey bars: passive force. Numbers: total muscle force in N. In A, B and C a single side is shown. In D both sides are shown, right and left side muscles alternating.
algorithm to a certain extent but never completely. While we were not aiming to mimic habitual movements this drawback was not dominant. A consequence of this feature is that when the lower incisor was forced to
In order to reach a goal muscle activation patterns were generated without a priori knowledge about their normal use. This can be compared with a situation where the brain has to learn new movements. Only the instantaneous contributions for force production regarding the six degrees of freedom of the lower jaw were known. The brain is assumed to know in what direction the jaw will move if certain muscles are activated. This can be compared with the so-called motor primitives, which transform putative directions of limb movements into (primitive) motor commands (Thoroughman and Shadmehr, 2000). During motor learning they are combined to a sensorimotor map in the brain which eventually contains complete recruitment patterns. The present method optimized muscle use to apply for every possibility to move closer to a destination and to maintain the final position. It created stiff systems requiring a relatively large amount of co-contraction. This approach led to adequate but not always intuitive solutions. For instance, the muscles expected to produce a maximum retrusion are digastric, mylohyoid and posterior temporalis. They are mainly jaw openers and produce a posteriorly directed force vector. Apart from a retrusion this combination may cause jaw opening which was counteracted by activation of the (vertical) posterior deep masseter and the (anteriorly directed) medial pterygoid. In a sagittal analysis the latter is unexpected. However, this muscle has also a strong medially directed component which may help in cocontraction with its contralateral equivalent to stabilize the jaw in the horizontal direction. The superior lateral pterygoids acted likewise. The model did not recognize that they were almost insufficient and applied them as muscles with some (albeit small) maximum forces. In a maximum laterodeviation for this movement relatively inefficient temporalis parts were activated after the lateral pterygoids had become insufficient for force production due to the force–length relationship. In
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contrast to the other border positions in this position the majority of the active muscles were not stretched or shortened to a length where their maximum force is strongly reduced. While they were activated as much as possible the total amount of force was relatively large. The maintainance of a final position by a stiff system with much co-contraction is not always necessary in habitual movements. Presumably, learned efficient movements have a more ballistic nature, applying not more co-contraction than necessary. Habitual jaw movements are also slower than in our simulations (Blanksma and van Eijden, 1995) requiring less muscle force. 4.3. Movement trajectories The path travelled by the mandibular incisor point to reach a certain destination was almost never straight. This may be caused on the one hand by perturbations by passive structures and on the other, by the dependence on position information only. No mechanism has been implemented to decelerate the jaw at the final approach. Deceleration beyond this approach will be delayed even more due to the activation dynamics of the muscles. This may cause an overshoot at the destination before the jaw is halted. In the present simulations this effect was only minor. In order to predict bell-shaped velocity profiles as observed usually (Feldman et al., 1990), some active damping has to be added by implementation of a negative feedback on velocity in the Objective Function (3). Applying such feedback will slow down the movements to more habitually speeds. 4.4. Validation Border position movements of the lower jaw have been measured and published abundantly in the dental literature. The extreme positions reached during the protrusion, retrusion and laterodeviation simulations are similar to concomitant data from Brown (1975) and Nishigawa et al. (1991). They reported 10–12, 1–1.3 and 13–17 mm for maximum excursions in these directions, respectively. The extreme amount of jaw opening of these measured border position movements is about 1.7 times larger than reached in our simulations. The difference is in agreement with the previous findings (Koolstra and van Eijden, 1997) that the model was unable to open the mouth as much as measured experimentally. This could be contributed to an absence of depression of the hyoid bone during full mouth opening excursions in the model (Langenbach and Hannam, 1999; Slager et al., 1997). While different individuals chew differently (Yamashita et al., 1999), each one exploits a different muscle contraction strategy for a similar masticatory task. This makes it impossible to predict universal muscle activa-
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tion strategies, although it remains possible that each individual optimizes its strategy on the basis of muscle morphology, skull geometry, food properties and the task to be performed. A comparison of the patterns from Fig. 7 with patterns obtained during similar movements (Vitti and Basmajian, 1977) can be summarized as follows. The unexpected contribution of the medial pterygoid to maximum open excursions (Fig. 7A) has also been observed in vivo. In contrast, the use of only the inferior lateral pterygoid in a protrusive excursion (Fig. 7B) was not found, although its contribution to such jaw movements was demonstrated recently (Murray et al., 1999). The large number of muscles contributing to jaw retrusion was also confirmed although the observed dominant action of the anterior temporalis was not predicted by our method (Fig. 7C). The activation pattern for a lateral excursion was, except for the contralateral masseter, roughly according to our predictions (Fig. 7D).
Acknowledgements We gratefully thank Academic Computing Services Amsterdam (SARA) for technical support and Dr G.E.J. Langenbach and L.J. van Ruijven for reading the manuscript. This research was institutionally supported by the Interuniversity Research School of Dentistry, through the Academic Centre for Dentistry Amsterdam (ACTA).
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A method to predict muscle control in the kinematically and mechanically indeterminate human masticatory system J.H. Koolstra*, T.M.G.J. van Eijden Department of Functional Anatomy, Academic Centre for Dentistry Amsterdam (ACTA), Meibergdreef 15, 1105 AZ Amsterdam, Netherlands Accepted 18 March 2001
Abstract A method is proposed to generate muscle activation patterns for goal-directed movements of the human masticatory system. This system is special because apart from a larger amount of muscles than degrees of freedom its joints do not restrict its movements a priori. Therefore, each muscle is able to influence all six degrees of freedom which makes the system kinematically and mechanically indeterminate. Furthermore, its working space is principally determined by the dynamical properties of its muscles and not by passive constraints. The presented method determines the contribution of each degree of freedom to a movement of a reference point on the mandible. It avails of straightforward mathematical techniques like Linear Programming. It does not require a separate trajectory planning step. It was applied in a six degrees of freedom dynamical mathematical model of the human masticatory system. This model which was based upon rigid-body dynamics incorporating skull morphology and muscle architecture including dynamical properties. Movements were exclusively defined by a goal position of the mandibular reference point. The method proved to be robust in generating muscle activation patterns for both feasible and infeasible movement tasks. Generally, they were accomplished faster than habitually observed. If the task was infeasible the movement stopped at the outer boundary of the working space at the side of the unreachable goal. The method, therefore, enables to explore the working space of the mandible and the factors that are relevant for its boundaries. r 2001 Elsevier Science Ltd. All rights reserved. Keywords: Control; Masticatory muscles; Jaw movement; Biomechanics
1. Introduction The human masticatory system is a typical example of a kinematically and mechanically indeterminate system. Two segments, the mandible and the skull, are able to move with respect to each other. These movements are guided by two mutually linked temporomandibular joints. In each joint a mandibular condyle articulates incongruently with the articular surface of the temporal bone. The articular capsule is slack. Due to this construction both joints allow for movements with six degrees of freedom (Koolstra and van Eijden, 1999). Consequently, jaw movements are not limited to rotations about one or more axes defined by the joint (Andrews and Hay, 1983). If the joint surfaces are assumed to be undeformable and maintain contact all *Corresponding author. Tel.: +31-20-5665370; fax: +31-206911856. E-mail address: [email protected] (J.H. Koolstra).
the time, the mandible still is able to move with four degrees of freedom (Schumacher, 1961). Jaw movements can be defined by the three-dimensional path travelled by the lower central incisor (Lewin, 1985). This can be accomplished in various ways with the system that is able to move with at least four degrees of freedom. Consequently, the masticatory system must be considered as kinematically redundant. Jaw movements are accomplished by a large number of masticatory muscles. The majority is relatively short with large attachment areas. While these muscles can be activated heterogeneously (Blanksma et al., 1997) each muscle is able to influence more than one degree of freedom (van der Helm and Veenbaas, 1991). All muscle portions together generate a resultant force and torque (six degrees of freedom) with respect to the centre of gravity of the lower jaw (Koolstra and van Eijden, 1995). The distribution of the forces and torques necessary to perform any movement over the different muscle portions is not a priori
0021-9290/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 1 ) 0 0 0 5 3 - 7
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established. Consequently, the system is mechanically redundant. Various theories have been proposed to predict suitable muscle recruitment patterns in redundant biomechanical systems. Mostly they were based upon minimization of some cost function such as fatigue, energy consumption, joint load or jerk (Zajac and Winters, 1990) to determine a distribution of joint torques over the relevant muscles. In the human masticatory system, however, this is not applicable while movements cannot be described by rotations about a priori defined joint axes. In an alternative theory, muscles are activated proportional to the amount of required muscle shortening (Feldman et al., 1990). It requires a complete kinematic image at the final position and has been applied for jaw movements restricted to two degrees of freedom (Laboissi"ere et al., 1996). For unrestricted three-dimensional jaw movements, however, the kinematical redundancy prevents to obtain the required unambiguous kinematic image at the final position. Jaw opening, protrusion, retrusion and laterodeviation are limited to a few centimeters maximally as described by the so-called envelope of jaw motion (Posselt, 1957). While the dynamical properties of the masticatory muscles contribute to a large extent to excursion limitations of the mandible (Koolstra and van Eijden, 1996), it is to be expected that the reachability of a putative goal cannot be determined from the geometry of the joint only. Generally, available methods for muscle recruitment prediction, are not able to handle such uncertainties in goal reachability. The aim of the present study was to develop a suitable method for generating appropriate muscle activation patterns for putative jaw movements. Prerequisitions were: robust towards kinematic redundancy and not dependent on a kinematic image of putative situations. It was to be implemented in a six degree of freedom dynamical biomechanical model of the human masticatory system (Koolstra and van Eijden, 1995, 1996, 1997, 1999), and applied to simulate goal-directed jaw movements and to explore their spatial limits.
2. Materials and methods 2.1. The model Jaw movement simulations were performed using a biomechanical model of the human masticatory system (Fig. 1). This model has been described extensively in Koolstra and van Eijden (1995, 1996). Briefly, it consists of a lower jaw which is accelerated by forces and accompanying torques with respect to its centre of
Fig. 1. Overview of the model. Ventro-lateral view. Continuous lines: muscle lines of action. Cross-bar: muscle origin. Circle: muscle insertion. MAS S: superficial masseter, MAS P: deep masseter, MPT: medial pterygoid, TEM A: anterior temporalis, TEM P: posterior temporalis, LPT S: superior lateral pterygoid, LPT I: inferior lateral pterygoid, DIG: digastric, GEH: geniohyoid, MYH: mylohyoid. Dots: position of centre of right and left condyle and incisor point.
gravity applied by muscles (represented by their lines of action running from fixed origins at the skull and the hyoid bone), joint surfaces, bite points and ligaments. The forces generated by the muscle portions which were modelled as a lumped sarcomere (Fig. 2) were dependent on the amount of activation, sarcomere dynamics and the activation dynamics. The included muscle portions (Fig. 1) were assumed to be homogeneous, physiologically similar and subjected to mutually independent activation. Their fibre lengths, sarcomere lengths and physiological cross-sections were obtained from van Eijden et al. (1995, 1996, 1997), the dynamical muscle properties from van Ruijven and Weijs (1990) and the activation dynamics from Winters and Stark (1987). 2.2. Muscle activation The muscles were activated in such a manner that a reference point on the mandible would move towards a goal. This movement was halted if this point reached its destination or if passive constraints and/or muscle insufficiency failed to move it any further. During the movement the instantaneous forces and torques generated by muscles ðFm Þ; joints ðFj Þ and bite points ðFb Þ are not in equilibrium. A rest force and torque ðFr Þ remains present according to: X X X Fm þ Fj þ Fb þ Fr ¼ 0; ð1Þ
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Fig. 2. Muscle model. Total force is the sum of the forces produced by the sarcomeres ðFsarcomeres Þ: The active force ðFactive Þ is dependent on the activation through the activation dynamics, the instantaneous sarcomere length and contraction velocity. The parallel elastic force ðFpassive Þ is dependent on the instantaneous sarcomere length.
through Fm by applying an adequate instantaneous activation pattern Am ð0pAm p1Þ while X X Fm ¼ ðFm;min þ Am Fm;max Þ; ð2Þ
Fig. 3. Forces and torques controlling the linear and angular accelerations of the lower jaw. Dashed lines: axes of the Cartesian system through the centre of gravity of the lower jaw, v: linear velocity, o: angular velocity, m: mass, I: moment of inertia. Dot: incisor point.
in which F expresses a six degrees of freedom force tensor (Fig. 3). Fb and Fj are dependent on the instantaneous location of the mandible. Fr controls the future direction of movement and can be adapted
in which Fm;min is the minimum force of a muscle equal to its instantaneous passive elastic force and Fm;max is the maximum active force dependent on its physiological cross-sectional area and the instantaneous dynamical properties. The activation pattern is selected from the collection that provides a rest force that would move the lower incisor (Fig. 1) from the current position ðx; y; zÞc to a desired position ðx; y; zÞd : It is not known a priori whether the desired position can be reached or not. The difference between the desired and current position can be treated as a measure for the effort which has to be exerted. The three components ðxd xc ; yd yc ; zd zc Þ of this difference were related to the linear angular displacements of the lower jaw. From Fig. 3 it is clear that the azimuth primarily influences a y-displacement of the incisor point and the elevation a z-displacement as long as these rotations are less than 451. The roll does not contribute unambiguously to the incisor point movement direction. An adequate Fr will diminish the difference between desired and present position and the required effort will be reduced. From the numerous ways to create a rest force which answers the require-
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Fig. 4. Simulation of a feasible jaw movement to ðx; y; zÞ=(0.4, 0.7, 1.6) relative to the starting position. Top panels: lateral view. Middle panels: frontal view. Bottom panels: superior view. P: posterior. A: anterior. M: medial, R: right. L: left. Dark curved lines: path of the centre of both condyles and the incisor point during the movement. Dots: starting point of the condylar and incisor paths. Cross-hairs: position of the incisor after 0.1 s. Inset: lateral, frontal and superior views of the mandible with the position of incisor and condylar points. Orientation: parallel to upper occlusal plane.
ments, the most effective rest force can be found by optimization. This can be accomplished using Linear Programming methods by maximizing the Objective Function: aðxd xc ÞFx þ bðyd yc ÞFy þ gðzd zc ÞFz þdðyd yc ÞFazimuth eðzd zc ÞFelevation
ð3Þ
subject to Eqs. (1) and (2). Although this way the shortest route from ðx; y; zÞc to ðx; y; zÞd is attempted it can be disturbed by constraints in the system. The method optimizes instantaneously and therefore adapts continuously to new situations. The activation pattern, however, cannot be changed instantaneously. If it produces a submaximal active force and is switched off, a part of this active force is still present after a short period of time. In contrast, if the activation should have been switched to 100% the active force will be less than the maximum possible force after this short time interval (Winters and Stark, 1987). This property was implemented in the boundary conditions of the optimization problem by adapting Fm;min and Fm;max in such a way that they would equal the muscle force at 0.1 s after
complete inactivation or complete activation, respectively. In the present study the constants a; b; g; d and e were set to 1. The model was implemented on an IBM RS/6000 SP computer system. It was written in AIX XL FORTRAN and parallelized using Message Passing Interface (MPI) library routines. The Linear Programming was performed using a revised SIMPLEX algorithm (DDLPRS from the IMSL library). 2.3. Simulations Simulations of both feasible and border position movements were performed. For each of them 0.5 s was reserved to approach the desired position as close as possible. For the first a jaw-open movement starting from the closed position in combination with a laterodeviation was chosen. The desired position of the lower incisor was 0.4 cm backwards, 0.7 cm to the left and 1.6 cm downwards relative to the start. Border position movements of the lower central incisor were simulated by defining desired positions far beyond the
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lower incisor working space. They were located inferiorly, anteriorly, posteriorly and laterally with respect to the resting position. Thus, a maximum open, protrusion, retrusion and laterodeviation movement were forced, respectively.
3. Results The desired position for the feasible movement was reached with an accuracy of about 0.03 cm. After 0.1 s the incisor was less than 1 mm separated from its destination (Fig. 4), and the muscle recruitment pattern was still subject to considerable changes in order to reach the destination as close as possible (Fig. 5). After about 0.4 s the muscle recruitment pattern became stationary, although some minor adaptations occurred. The mandibular incisor travelled along a slightly curved line without unexpected major deviations from the shortest route. The four simulated border position movements are presented in Fig. 6. The model reached a maximum opening of 28 mm. The obtained maximum protrusion, retrusion and laterodeviation were 10, 0.5 and 14 mm, respectively. In Fig. 7, the relevant muscle recruitment patterns underlying these movements are shown. The muscle activation patterns and their resultant forces necessary to maintain the extreme positions are depicted in Fig. 8. Except for the retrusion movement the border position movements were relatively straightforward. After 0.1 s they reached their end positions closely and the changes in muscle activation after 0.2 s could almost all be attributed to the limits caused by the muscle activation dynamics (Fig. 7). The retrusion movement approached its limit within about 0.1 s but the path was more capricious (Fig. 6C). Before the actual retrusion occurred the mandible first opened about 13 mm. The condyles moved forward and then backward again. The relevant muscle activation strategy needed about 0.1 s more than the others to become stationary. It is striking that the superior lateral pterygoid muscle, which pulls the mandibular condyles forward, was activated maximally, while through the force–length relationship it could generate less force than 0.5 N (Fig. 8C). Also the medial pterygoid muscles, which pull upward and forward, were recruited. The muscles contributing primarily to the border movements were: lateral pterygoid, mylohyoid and digastric muscles at maximal open, inferior lateral pterygoid muscles at maximal protrusion, mylohyoid muscle portions at maximal retrusion and ipsilateral temporalis portions and contralateral medial pterygoid and mylohyoid muscles at a maximal laterodeviation. In the open, retrusion and laterodeviation position the mylohyoid muscle portions were fully activated and
Fig. 5. Muscle activation pattern generated in order to accomplish the feasible jaw movement. Top panel: right sided muscles. Bottom panel: left-sided muscles. Muscles as in Fig. 1. a: anterior muscle portion. p: posterior muscle portion. Muscles not shown were not activated. Vertical axis: activation relative to the maximum. Horizontal axis: time in s.
produced almost maximum forces (Fig. 8). This also counts for the superior lateral pterygoid muscle in the open position. All other muscles, although sometimes activated maximally, produced only a part of their maximum force due to the force–length relationship. The maximum protrusion excursion (Fig. 8B) required the least muscle force (about 36 N). This can be attributed to the presence of the inferior lateral pterygoid muscles which hardly have unfavourable force and torque components regarding this movement. In contrast, the maximum laterodeviation excursion (Fig. 8D) required a total of about 650 N muscle force. The passive forces generated by the muscles at the border positions are relatively small. The largest amount (approximately 65 N in total) was generated in the maximum open position by the jawclosing muscles.
4. Discussion Generally, the working space of musculo-skeletal systems is defined by constraints applied through joint construction (Blankevoort et al., 1988; Wang et al., 1998). The action of the muscles is generally described
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Fig. 6. Simulated border position movements. A: maximum open excursion. B: maximum protrusion excursion. C: maximum retrusion excursion. D: maximum laterodeviation excursion.
by a torque about one or more joints (Gribble and Ostry, 1998; Lindbeck et al., 1997). In contrast, in the human masticatory system the working space is not primarily constrained by passive structures, but by limitations for force production of its muscles and the action of the muscles cannot be described by a torque about its joints. The proposed method for muscle activation has been proven able to
generate muscle recruitment patterns without a priori knowledge about on the one side the feasibility of the desired movement and on the other a kinematic image of the system at any putative position. Therefore, this method can be regarded to extend the possibilities for development of control strategies from constrained musculo-skeletal systems to unconstrained ones.
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Fig. 7. Muscle activation patterns generated in order to accomplish border position movements. A: maximum open excursion. B: maximum protrusion excursion. C: maximum retrusion excursion. D: maximum laterodeviation excursion. In A, B and C a single side is shown, the activation patterns at the other side were equal.
4.1. Performance considerations Although the model applied in this study is the most complete biomechanical model of the human masticatory system that has been published as far as we know, it still includes many simplifications (for an extensive overview see Koolstra and van Eijden, 1995, 1996, 1997) which may have affected the results. Its joint shapes were simplified, especially in the mediolateral direction. Articular discs were not included and the capsules were modelled by the temporomandibular ligaments only. This may enable a somewhat larger joint movability than generally observed. The muscle portions were modelled by straight (or curved) line elements with sarcomeres differing by resting length only. The recruitment of motor units which may be dependent on their size and dynamical properties (van Eijden and Turkawski, 2001), therefore, could not be studied and
internal muscle torsions were neglected. The muscle portions were sized according to elderly subjects which may have caused lesser strength than on average. This could have been partly responsible for a reduced amount of maximum jaw opening. The model was based upon azimuth and elevation rotations of less than 451 in order to influence displacements of the lower incisor primarily in the yand z-directions, respectively. The largest azimuth (7.41) was obtained at the maximum laterodeviation position. The largest elevation (18.91) was found in the maximum open position. Both were amply within the required limits. Optimization of a linear combination of the rest force components in the objective function enables to use standard Linear Programming algorithms, but it makes a precise conduction of the system more difficult. The constants a; b; g; d and e can be adapted to influence the
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move beyond its possible workspace it ended up on the border of this workspace but not necessary as close to the (unreachable) destination as possible. The optimization of muscle activation patterns was performed by a (revised) SIMPLEX algorithm. The Objective Function and the constraints used in the optimization procedure were convex functions. The optimization problem, therefore, is convex also and the SIMPLEX method will converge to a global optimum (Dantzig, 1963). Functionally, such a global optimum only has a significance at the border of the workspace. It predicts that no other pattern will be able to move the mandible beyond this limit. Consequently, the predicted boundary coincides with the boundary as defined by the governing morphology. 4.2. Activation patterns
Fig. 8. Muscle activation patterns and resulting muscle forces at border positions at 0.5 s after onset. A: maximum open excursion. B: maximum protrusion excursion. C: maximum retrusion excursion. D: maximum laterodeviation excursion. Black bars: muscle activation relative to maximum. White bars: active force relative to maximum isometric force. Grey bars: passive force. Numbers: total muscle force in N. In A, B and C a single side is shown. In D both sides are shown, right and left side muscles alternating.
algorithm to a certain extent but never completely. While we were not aiming to mimic habitual movements this drawback was not dominant. A consequence of this feature is that when the lower incisor was forced to
In order to reach a goal muscle activation patterns were generated without a priori knowledge about their normal use. This can be compared with a situation where the brain has to learn new movements. Only the instantaneous contributions for force production regarding the six degrees of freedom of the lower jaw were known. The brain is assumed to know in what direction the jaw will move if certain muscles are activated. This can be compared with the so-called motor primitives, which transform putative directions of limb movements into (primitive) motor commands (Thoroughman and Shadmehr, 2000). During motor learning they are combined to a sensorimotor map in the brain which eventually contains complete recruitment patterns. The present method optimized muscle use to apply for every possibility to move closer to a destination and to maintain the final position. It created stiff systems requiring a relatively large amount of co-contraction. This approach led to adequate but not always intuitive solutions. For instance, the muscles expected to produce a maximum retrusion are digastric, mylohyoid and posterior temporalis. They are mainly jaw openers and produce a posteriorly directed force vector. Apart from a retrusion this combination may cause jaw opening which was counteracted by activation of the (vertical) posterior deep masseter and the (anteriorly directed) medial pterygoid. In a sagittal analysis the latter is unexpected. However, this muscle has also a strong medially directed component which may help in cocontraction with its contralateral equivalent to stabilize the jaw in the horizontal direction. The superior lateral pterygoids acted likewise. The model did not recognize that they were almost insufficient and applied them as muscles with some (albeit small) maximum forces. In a maximum laterodeviation for this movement relatively inefficient temporalis parts were activated after the lateral pterygoids had become insufficient for force production due to the force–length relationship. In
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contrast to the other border positions in this position the majority of the active muscles were not stretched or shortened to a length where their maximum force is strongly reduced. While they were activated as much as possible the total amount of force was relatively large. The maintainance of a final position by a stiff system with much co-contraction is not always necessary in habitual movements. Presumably, learned efficient movements have a more ballistic nature, applying not more co-contraction than necessary. Habitual jaw movements are also slower than in our simulations (Blanksma and van Eijden, 1995) requiring less muscle force. 4.3. Movement trajectories The path travelled by the mandibular incisor point to reach a certain destination was almost never straight. This may be caused on the one hand by perturbations by passive structures and on the other, by the dependence on position information only. No mechanism has been implemented to decelerate the jaw at the final approach. Deceleration beyond this approach will be delayed even more due to the activation dynamics of the muscles. This may cause an overshoot at the destination before the jaw is halted. In the present simulations this effect was only minor. In order to predict bell-shaped velocity profiles as observed usually (Feldman et al., 1990), some active damping has to be added by implementation of a negative feedback on velocity in the Objective Function (3). Applying such feedback will slow down the movements to more habitually speeds. 4.4. Validation Border position movements of the lower jaw have been measured and published abundantly in the dental literature. The extreme positions reached during the protrusion, retrusion and laterodeviation simulations are similar to concomitant data from Brown (1975) and Nishigawa et al. (1991). They reported 10–12, 1–1.3 and 13–17 mm for maximum excursions in these directions, respectively. The extreme amount of jaw opening of these measured border position movements is about 1.7 times larger than reached in our simulations. The difference is in agreement with the previous findings (Koolstra and van Eijden, 1997) that the model was unable to open the mouth as much as measured experimentally. This could be contributed to an absence of depression of the hyoid bone during full mouth opening excursions in the model (Langenbach and Hannam, 1999; Slager et al., 1997). While different individuals chew differently (Yamashita et al., 1999), each one exploits a different muscle contraction strategy for a similar masticatory task. This makes it impossible to predict universal muscle activa-
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tion strategies, although it remains possible that each individual optimizes its strategy on the basis of muscle morphology, skull geometry, food properties and the task to be performed. A comparison of the patterns from Fig. 7 with patterns obtained during similar movements (Vitti and Basmajian, 1977) can be summarized as follows. The unexpected contribution of the medial pterygoid to maximum open excursions (Fig. 7A) has also been observed in vivo. In contrast, the use of only the inferior lateral pterygoid in a protrusive excursion (Fig. 7B) was not found, although its contribution to such jaw movements was demonstrated recently (Murray et al., 1999). The large number of muscles contributing to jaw retrusion was also confirmed although the observed dominant action of the anterior temporalis was not predicted by our method (Fig. 7C). The activation pattern for a lateral excursion was, except for the contralateral masseter, roughly according to our predictions (Fig. 7D).
Acknowledgements We gratefully thank Academic Computing Services Amsterdam (SARA) for technical support and Dr G.E.J. Langenbach and L.J. van Ruijven for reading the manuscript. This research was institutionally supported by the Interuniversity Research School of Dentistry, through the Academic Centre for Dentistry Amsterdam (ACTA).
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