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The influence of an elastic tendon on the force producing capabilities of a muscle during dynamic movements
Zachary J. Domire a; John H. Challis a a Biomechanics Laboratory, The Pennsylvania State University, PA 16802, USA First Published on: 18 June 2007 To cite this Article: Domire, Zachary J. and Challis, John H. (2007) 'The influence of an elastic tendon on the force producing capabilities of a muscle during dynamic movements', Computer Methods in Biomechanics and Biomedical Engineering, 10:5, 337 - 341 To link to this article: DOI: 10.1080/10255840701379562 URL: http://dx.doi.org/10.1080/10255840701379562
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Computer Methods in Biomechanics and Biomedical Engineering, Vol. 10, No. 5, October 2007, 337–341
The influence of an elastic tendon on the force producing capabilities of a muscle during dynamic movements ZACHARY J. DOMIRE and JOHN H. CHALLIS* Biomechanics Laboratory, The Pennsylvania State University, PA 16802, USA (Received 29 August 2006; in final form 22 March 2007)
With increasing computer power, computer simulation of human movement has become a popular research tool. However, time to complete simulations can still be long even on powerful computers. One possibility for reducing simulation time, with models of musculo-skeletal system, is to simulate the muscle using a rigid tendon rather than the more realistic compliant tendon. This study examines the effect of tendon elasticity on muscle force output under different dynamic conditions. A single muscle, point mass model was used and simulations were performed varying the mass, the tendon length, the initial position, and the task. For simulations for relatively slow motion, as experienced for example in upper limb reaching motions or rising from a chair, tendon properties had little influence on muscle force, in contrast simulations of an explosive task similar to jumping or throwing tendon had a much larger effect. Keywords: Computer simulation; Biomechanics; Tendon; Musculo-skeletal system
1. Introduction Computer simulation has become a very powerful research tool in biomechanics. Simulation allows for the examination of factors, which cannot be examined experimentally, for example changing the origin of the gastrocnemius so that it becomes an uniarticular muscle (Van Soest et al. 1993). Simulation time for complex models can be extremely long even on powerful parallel processing computers (Anderson and Pandy 2001). The time for a simulation depends on the complexity of the model, and the size of the integration time step for solving the ordinary differential equations in the model. There are two coupled sets of ordinary differential equations for musculo-skeletal models, one set relating to the muscle model, the other to the mechanical system. Which set of equations requires the smallest time step depends on the task, for activities not involving an impact typically the muscle model requires the smallest time step. Therefore for certain tasks one possibility for reducing simulation time for models of the musculo-skeletal system is to perform the simulations with a rigid tendon. By using a rigid tendon one ordinary differential equation per modeled muscle can be eliminated from the model. The numerical solution of ordinary differential equations is sensitive to the time step used (Press et al. 1992); so there remains the
potential to save additional time, when using a rigid tendon, by increasing the time step used in the simulation. The effect tendon has on a muscle’s force producing capabilities depends on the activity and muscle architecture. Van Soest et al. (1995) examined isometric muscle action and showed that the tendon resting length relative to the optimal fiber length influenced how large the effect of tendon was on force output. Tendon had very little effect on the force a muscle produces unless the neural input varied greatly. Bobbert (2001) showed that squat jump height is enhanced when simulated with more compliant tendons, suggesting that tendon influences muscle force output during dynamic movements. As having a rigid tendon in a musculo-skeletal model is advantageous in terms of computer simulation time it is profitable to know under what circumstances, if any, this assumption is justifiable. Therefore, the purpose of this study was to use a muscle model to examine the effect of an elastic tendon on the force producing capabilities of the model under different dynamic conditions.
2. Methods The model consists of a single muscle with non-linear force – length and force – velocity properties as well as
*Corresponding author. Email:
[email protected] Computer Methods in Biomechanics and Biomedical Engineering ISSN 1025-5842 print/ISSN 1476-8259 online q 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/10255840701379562
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Z. J. Domire and J. H. Challis Table 1. Muscle model parameters.
Fmax (N) 9000
Lfopt (m)
Ltr (m)
W
Vmax (Lfopt/s)
K
0.093
0.16 or 0.36
0.56
5.2
2.44
Fmax, maximum isometric force (values are for both legs); Lfopt, optimal fiber length; Ltr, resting tendon length; W, spread of the force – length curve; Vmax, maximum unloaded shortening velocity; K, force – velocity curvature constant.
activation dynamics (Gallucci and Challis 2002). The muscle model was given parameters similar to the vasti group (Van Soest et al. 1993). The muscle model parameters are given in table 1. Two tasks were simulated. In both tasks the muscle moved a point mass a distance of 6 cm against gravity (figure 1). This corresponds to approximately 908 of motion at the knee. The system can be represented by the following equations: x€ ¼
FM 2g m
ð1Þ
where F_ M , rate change of muscle force from the previous time step; k, tendon stiffness; VT, tendon velocity. In this study, this ordinary differential equation was solved using a 4th order Runge – Kutta integrator. The minimum time step used for all simulations was 0.0005 s, smaller time steps than this did not influence simulation accuracy. For the simulations with a rigid tendon, the length and velocity of the muscle fibers is equal to the muscle– tendon complex velocity, which can be determined directly from the kinematics; this obviates the requirement of using numerical integration for this part of the muscle model. In the first task, neural excitation was maximum throughout the movement. This would be a similar activation that may be seen during jumping or throwing. In the second task, the performance criterion was to minimize the rate change of muscle force squared. The objective function can be represented by the following equation, ð tF min
F M ¼ q·F max ·F L ðLF Þ·F V ðV F Þ
ð2Þ
where x€ , acceleration of the point mass; FM, muscle force; m, mass; g, acceleration due to gravity; q, current active state of muscle model; Fmax, maximum isometric force possible by muscle model; FL(LF), fraction of the normalized force – length curve that the model can produce at its current fiber length (LF); and FV(VF) is the fraction of normalized force – velocity curve that the model can produce at its current fiber velocity (VF). For simulations with a compliant tendon, the length and velocity of the muscle fibers is determined by subtracting tendon length and velocity from muscle – tendon complex length and velocity. Muscle force is then estimated by numerically integrating the following equation: F_ M ¼ k·V T
Figure 1. Schematic of the single muscle point mass system.
ð3Þ
0
2 F_ M dt
ð4Þ
where tF is the time at which the desired position is reached. This criterion has been used to simulate sit to stand (Pandy et al. 1995). Here, the optimal sequence of neural excitation was selected by a genetic search algorithm (Goldberg 1989). In this simple model, this objective function is effectively the same as minimizing jerk, which has been proposed as an objective function for reaching (Flash and Hogan 1985). The optimization was constrained to find neural excitation patterns that resulted in the mass being stationary at the end of the movement. Initial muscle activation for both cases was selected to put the system in static equilibrium. In addition to the task, several other factors were examined to determine their effect on the contribution of tendon to muscle force output. Three different masses (100, 200 and 300 kg) were used. Two different initial muscle lengths were used (optimal length þ2 cm, optimal length þ 4 cm). As a result of tendon compliance, the starting muscle fiber lengths were slightly different between rigid and compliant tendon simulations. No attempt was made to correct for this, as this was considered part of the effect of using the rigid tendon. The effect of tendon length was examined by performing simulations with two different tendon resting lengths (16, 36 cm). All simulations were performed first with a rigid tendon. After determining muscle neural excitations for the movement with a rigid tendon, the same neural excitations were used to simulate the movement with an elastic tendon that would stretch four percent beyond its resting length under maximum isometric force (Van Soest et al. 1993); such an assumption is common in models of the musculo-skeletal system (Van Soest et al. 1993, Pandy et al. 1995, Anderson and Pandy 2001, Gallucci and Challis 2002). The changes in the force production throughout the range of motion were investigated. Simulations were compared by examining differences between force-position curves.
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Influence of an elastic tendon
3.2 Effect of task
Table 2. Change in muscle force when varying time step.
Rigid tendon (N) Elastic tendon (N)
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0.0005
0.001
0.002
0 0
1.5 3.1
5.3 11.3
Two comparisons were made to examine the possible time savings as a result of using a rigid tendon. First, the time to run a single iteration of each model was calculated using a 0.0005 s time step. This reflects the time savings as a result of eliminating the ordinary differential equation. Second, the effect of using larger time steps on muscle force was examined in both models for a representative condition.
During the slow movement tendon elasticity had very small effects on force output. The maximum difference in muscle force between elastic and inelastic models was 1.8% in the 300 kg condition starting from optimal length 2 2 cm (figure 2). The effect in this condition was more than twice as large as in any other slow condition. During the fast movement tendon had a much larger effect on muscle force output. The largest difference between elastic and inelastic models was a 6.4% increase in force in the 100 kg condition starting from optimal length 2 2 cm (figure 3). All conditions had an increase of at least 5%. 3.3 Effect of mass
The results are presented in the following sub-sections: time savings when using a rigid tendon, effect of task, effect of mass, effect of initial position and effect of tendon length.
Tendon’s effect on force production varied slightly when changing the mass moved. There was no clear pattern that could be determined during the slow condition. However, during the fast condition tendon’s influence on force production decreased as the mass increased. The largest difference seen as a result of mass was 3.6%.
3.1 Time savings
3.4 Effect of initial position
Simulations run with a rigid tendon were 35% faster than those run with elastic tendon. Table 2 shows the effect of varying the time step on representative condition for both models. Simulations with an elastic tendon were more sensitive to use of a larger time step.
When starting from the longer length, the early part of the movement happened on a steeper part of the force – length curve. This means an increase in the rate of change of force and consequently an increase in tendon velocity. Therefore, the effect of tendon elasticity on force
3. Results
Figure 2. Force plotted against position for the 300 kg condition starting from optimal length 22 cm when objective function which minimizes the rate change of muscle force, for models with a rigid or an elastic tendon.
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Z. J. Domire and J. H. Challis
Figure 3. Force plotted against position for the 100 kg condition starting from optimal length 22 cm when the neural excitation was maximum, for models with a rigid or an elastic tendon.
production in the early part of the movement was larger when starting from the longer length when compared with starting from the shorter length. The largest effect of this was seen in the fast condition when using the 300 kg mass. When starting from a longer length with these conditions, tendon elasticity produced a 4.2% larger increase in muscle force than when staring from the shorter length. The rate of decrease of force late in the movement is larger when starting from the shorter position; therefore the decrease in force as a result of tendon elasticity is larger when compared to starting from the longer length. The largest effect of this was seen in the fast condition when using the 100 kg mass. When starting from a shorter length with these conditions, tendon elasticity produced a 6.7% larger decrease in muscle force than when the staring from the longer length. 3.5 Effect of tendon length The effect of tendon length on the ability of tendon elasticity to alter force production was dependant on the starting position. The effect on the early part of the movement was greater when starting from the shorter position, and the effect on the later part of the movement was greater when starting from the longer position. When starting from the longer position, tendon elasticity produced an additional increase of muscle force in the force early in the movement, between 0 and 0.4%, when tendon length was increased. There was an additional decrease in muscle force, between 1.2 and 6.6%, late in the movement. When starting from the shorter position,
tendon elasticity produced an additional increase of muscle force in the force early in the movement, between 0.1 and 4.9%, when tendon length was increased. There was an additional decrease in muscle force, between 0 and 2.6%, late in the movement.
4. Discussion The purpose of this study was to use a muscle model to examine the effect of an elastic tendon on the force producing capabilities of the muscle under different dynamic conditions. This was done to determine when, if ever, the assumption of a rigid tendon is justifiable. The results indicate the importance of including tendon compliance in musculo-skeletal system simulations depends on the task, with such an assumption being appropriate for slow movements but not for the fast movements simulated in this study. The time savings that can be accomplished by using a rigid tendon were found to be significant. A 35% time reduction was accomplished by eliminating one ordinary differential equation. In more complicated models where many different muscles are modeled, the time savings could be much larger. The results also indicate the potential exists to perform simulations with a larger time step when using a rigid tendon. The reason for this is that when simulating with a compliant tendon, it is not possible to know the contributions of the muscle fibers and the tendon to the muscle – tendon complex length velocity. One state variable for each modeled muscle must be
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Influence of an elastic tendon
estimated from the previous time step. This introduces a small error into the model. The size of this error is reduced by decreasing the time step. When simulating with a rigid tendon, the length and velocity of the muscle fibers is equal to the muscle – tendon complex length and velocity, which can be determined directly by the kinematics. An elastic tendon can change force output by its influence on the length and velocity of the muscle fibers. For a given muscle–tendon complex length as an elastic tendon stretches the fibers must be shorter compared with the inelastic tendon case. This length change in the muscle fiber moves it to a different position on its force–length curve; this effect was very small in all conditions analysed. The reason for this is that the largest shifts in the force– length curve occur when the muscle is on the plateau region of the force–length curve where changes in length produce very small changes in muscle force. For the model used in this study tendon velocity is proportional to the rate change of muscle force divided by the stiffness of the tendon, therefore whenever muscle force is decreasing, tendon is shortening. This allows the muscle fibers to shorten at a reduced velocity, therefore muscle force is enhanced. Whenever muscle force is increasing, tendon is lengthening, this forces the muscle fibers to shorten at an increased velocity, therefore muscle force is decreased. Muscle activation as well as the interaction of force–length and force–velocity parameters determine the rate change of muscle force throughout a movement. In this study the effect of tendon velocity on muscle force output varied throughout the movement and depended highly on the task. Tendon compliance is given in this study as a constant strain that the tendon experiences under maximum isometric muscle force. The use of a longer tendon increased the tendon compliance in absolute terms. This means an objective function can exploit the muscle properties so that for any given muscle –tendon complex velocity, tendon velocity can be relatively high and fiber velocity low, and thus the fiber operates on a portion of their force –length curve permitting the production of greater force. These results are in agreement with the finding of Bobbert (2001), who found increased tendon compliance increased jumping performance.
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While all factors examined had an effect on outcome, the largest influence came from the task. Reaching and sit to stand can be very time consuming movements to simulate. A primary reason for this is that these movements take much longer for subjects to execute than an explosive movement such as jumping, therefore these movements take a longtime to perform, muscle velocities are low. The results from this study show that given these conditions the effect of tendon elasticity on muscle force output is minimal. When simulating these movements, the use of a rigid tendon should be considered as a means of reducing simulation time. Investigators can run similar simulations for different muscles and tasks to examine the feasibility of ignoring tendon elasticity, and therefore reducing simulation time. In explosive movements such as running or jumping an elastic tendon has larger effects on force output and simulations of these movements should include tendon elasticity. References F.C. Anderson and M.G. Pandy, “Dynamic optimization of human walking”, J. Biomech. Eng., 123, pp. 381–390, 2001. M.F. Bobbert, “Dependence of human squat jump performance on the series elastic compliance of the triceps surae: a simulation study”, J. Exp. Biol., 204, pp. 533–542, 2001. T. Flash and N. Hogan, “The coordination of arm movements: an experimentally confirmed mathematical model”, J. Neurosci., 5, pp. 1688–1703, 1985. J.G. Gallucci and J.H. Challis, “Examining the role of the gastrocnemius during the leg curl exercise”, J. Appl. Biomech., 18, pp. 15 –27, 2002. D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Reading, MA: Addison-Wesley Pub. Co., 1989. M.G. Pandy, B.A. Garner and F.C. Anderson, “Optimal control of nonballistic muscular movements: a constraint-based performance criterion for rising from a chair”, J. Biomech. Eng., 117, pp. 15–26, 1995. W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., New York: Cambridge University Press, 1992. A.J. Van Soest, A.L. Schwab, M.F. Bobbert and G.J. van Ingen Schenau, “The influence of the biarticularity of the gastrocnemius muscle on vertical-jumping achievement”, J. Biomech., 26, pp. 1–8, 1993. A.J. Van Soest, P.A. Huijing and M. Solomonow, “The effect of tendon on muscle force in dynamic isometric contractions: a simulation study”, J. Biomech., 28, pp. 801– 807, 1995.
