Journal of Biomechanics 36 (2003) 1031–1038
Muscle mechanics and neuromuscular control A.L. Hof* Laboratory of Human Movement Analysis, University Hospital, Institute of Human Movement Sciences, University of Groningen, P.O. Box 196 196, NL-9700 AD Groningen, The Netherlands Accepted 13 January 2003
Abstract The purpose of this paper is to demonstrate that the properties of the mechanical system, especially muscle elasticity and limb mass, to a large degree determine force output and movement. This makes the control demands of the central nervous system simpler and more robust. In human triceps surae, a muscle with short fibres and a long tendon, the time courses of the total muscle+tendon length and of the length of the contractile component (CC) alone in running are completely different. The muscle tendon complex shows first an eccentric phase with negative work, followed by a concentric phase. The CC, on the other hand, is concentric all the time. Moreover, the work that is done, is done at a speed that guarantees a high energetic efficiency. It is argued that this high efficiency is an in-built property of the muscle mechanics for muscles with a compliant tendon and a low vmax : When a muscle, or a set of muscles, moves a mass, and the duration of the action is short with respect to the isometric time constant of the muscle, we may call it an ‘elastic bounce contraction’. In such a case the mass–spring interaction largely determines the time course of the force, and the efficiency of muscle contraction is most of the time close to optimum. In a similar way, whole limbs can be modelled as springs, with a stiffness that can be modulated by flexing the joints more or less. The motor control task of the central nervous system is simple for such elastic bounce contractions: a block-like activation is sufficient, in which timing is critical, but activation level is not. It seems possible that a whole class of actions can be generated by an identical timing sequence, with only a modulation in activation amplitude. An example is walking or running at different speeds. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Series elasticity; Force–length relation; Force–velocity relation; Muscle contraction; Efficiency
1. Introduction It seems obvious to view neuromuscular control in the top-down direction. Commands are generated at the highest control centre, the motor cortex, programmed into a movement program and sent to the motorneuron pool. The motorneuron pool activates the muscle, the muscle generates force and finally this results in a movement. In terms of the Hill muscle model, the output of the motorneuron pool is the active state F0 ; to be discriminated from the final muscle force F. Muscle force is ‘somewhat modulated’ by effects of muscle length change, that is by the muscle mechanics. Often this is considered a minor effect. In this paper we will proceed not top-down, but bottom-up. We will start at the very bottom: with the muscle elastic properties, thus
*Tel.: +31-50-363-2645; fax: +31-363-3150. E-mail address:
[email protected] (A.L. Hof).
in fact not even with the muscle fibres but with the tendon. The aim of this paper is to show that the muscle mechanics plays an essential role in this scheme, to the extent that the muscle mechanics materially simplifies the problem of neuromuscular control as seen from the central nervous system.
2. Experiments 2.1. Experiments The argument to follow is based on a series of experiments, all carried out on human triceps surae muscle with the same group of four subjects, which has been reported more extensively elsewhere (Hof et al., 2002b). The first experiment consisted of determining the properties of the series-elastic component (SEC) in a controlled release ergometer. While the subject makes an
0021-9290/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0021-9290(03)00036-8
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isometric contraction, the foot is very suddenly released at a high constant speed. In this way, the moment–angle curve of triceps surae SEC in living humans can be measured (Fig. 1) (Hof, 1998). This curve can be recalculated into muscle force and length, as the moment arm is known (Spoor et al., 1990). In the second experiment with the same subjects, a recording of running was made with a forceplate and an optokinetic measurement system at 200 Hz. From these data the ankle moment and the angles of ankle and knee were determined. These too can be converted into force and length of the muscles, soleus and gastrocnemius. Here only data on soleus will be presented. In the third set of experiments the force–length and force–velocity relations of the CC were determined, by means of a isokinetic dynamometer and electrical stimulation in the same four subjects (Zandwijk et al., 1998). 2.2. The work loop for MTC and CC in running Force and length of the total muscle–tendon complex (MTC) can be plotted against each other, to give the socalled work loop, Fig. 2. First the muscle lengthens, with increasing force, eccentric contraction, then it shortens, with decreasing force, concentric contraction. Such a figure is called a ‘work loop’ because the area under the curve represents the mechanical work done (Biewener, 1998), negative when the curve goes clockwise (21 J in Fig. 2), positive (+54 J) when anti-clockwise. This work loop represents the total length of muscle plus tendon, contractile component (CC) plus series elastic component. The length of the CC can be found by subtracting the stretch of the SEC (Fig. 1) from the total MTC length, Fig. 2. It should be stressed that there is hardly any muscle model involved here, all is derived from the kinematic measurements, ankle moment and angle, and from the experimental SEC curve.
Fig. 1. Controlled release of the series-elastic component, SEC moment as a function of ankle angle. Five releases from different initial isometric moments. Measurements in human triceps surae (Hof, 1998).
Fig. 2. Work loops for human soleus in running at 3.7 m s1. (-.-.-) Muscle force as a function of MTC length, as derived from the ankle angle. Drawn line: muscle force as a function of CC length. CC length is obtained by subtracting SEC length (cf. Fig. 1) from MTC length. Time proceeds in the direction of the arrows. (—-) Force–length relation of soleus in the same subject. The force–length relation of the parallel elastic component (PEC) is also shown. From (Hof et al., 2002b).
It is seen that the two curves of Fig. 2 are markedly different: the CC runs completely from right to left, which means that no negative work is done anymore. All negative MTC work goes into elastic energy and out again, but the CC does only positive work. The amount is about equal to the difference between positive and negative work: 5421=33 J. These effects are because SEC length change is of a similar magnitude as to the length changes of the complete MTC: 4.5 cm at the peak force of 3000 N. This causes a major change in the work loop. It is seen that the work loop of the CC neatly follows the force–length relation, in contrast to the MTC loop. In the case of Fig. 2, mainly the descending part of the force–length relation is traversed. 2.3. Force–velocity plots for MTC and CC Next to the force–length relation, also the parameters of the Hill force–velocity relation were determined in our subjects by a series of isokinetic contractions (Zandwijk et al., 1998). To compare these force–velocity relations with the measured data, muscle force was plotted against MTC shortening speed (Fig. 3). In the case of running this results in a clockwise loop (-+-+-), as MTC speed is first eccentric, then concentric. As has been reported before (Bobbert and Ingen Schenau, 1990; Finni et al., 2001), this loop does not fit with the Hill force–velocity curve, the maximum concentric MTC speed is even greater than the maximum contractile velocity nmax : But when the shortening speed of the CC is plotted in this figure (-o-o-), it is seen that it neatly fits in with Hill’s relation. It is also seen that the CC shortening speeds are rather low, up to about 0.2 m/s. This is
A.L. Hof / Journal of Biomechanics 36 (2003) 1031–1038
Fig. 3. Force–velocity plots for the same experiment as Fig. 2. (-+-+-) Muscle force as a function of MTC speed. (-o-o-) Muscle force as a function of CC speed. Time between symbols 5 ms. (- - - ) Hill force velocity relation for this subject, with F0 ¼ 7000 N: Thin dashed line: muscular efficiency as a function of CC speed vc ; as calculated from Hill relation, right hand scale. Note that maximum concentric MTC speed exceeds vmax of the Hill relation. From (Hof et al., 2002b).
favourable, because just at these lower speeds the efficiency of muscular contraction is maximal, see the efficiency curve in Fig. 3. It may thus be said that soleus muscle works very efficiently in running. This was also verified for walking and jumping and for gastrocnemius. 2.4. Conclusions from the experiments First, there is no negative work done by the CC, because this negative work is stored as elastic energy, and the amount of positive contractile work (closely connected with energy consumption) is thus minimum. Second, the contractile work that is done, is done at an efficiency close to maximum.
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consists of a muscle, with contractile and elastic properties, connected in series to a mass. In fact this is a mass–spring–damper system: SE stiffness is the spring and the CCs force–velocity relation is the damper. Because all relevant muscle parameters from our four subjects were known, a model of this system could be made, including all non-linear properties of SE and CC. For the argument, however, it is easier to consider only the linear approximation, which gives a very satisfactory approximation, Fig. 4. This linear approximation is the familiar second-order linear system. The response of such a system is a damped sine wave function, see Appendix A. The important thing to observe is that the sinewave is only lightly damped. This is the outcome of inserting the experimental muscle parameters in the model. For a lightly damped second-order linear system it is known that the form of the response, i.e. the eigenfrequency, is mainly determined by mass and stiffness (A.10), and not by the damping, which is related to the muscle’s force–velocity properties. The amplitude of the force depends mainly on the initial condition, that is the initial speed of the mass. This means, see the detailed solution of the differential Equation (A.13), that muscle force is, to first order, not determined by the muscle contractile properties or by the active state F0 : It is only the mass and the elastic properties that matter. The time course can be predicted to beppart ffiffiffiffiffiffiffiffiffiffiffiof a sinewave with a known frequency f0 ¼ k1 =m=2p and the peak force can easiest be calculated by equating the kinetic energy before and the peak elastic energy during the contraction: 1=2mv2 ¼ 1=2F 2 =k; which gives peak force when the initial speed is known (cf. McMahon et al., 1987;
3. Muscular efficiency The question is how this high muscular efficiency is achieved. It might be a function for the central nervous system to send commands precisely tuned to muscle and tendon properties, and to the unpredictable load, to the effect that the muscle may work efficiently. The notion of a ‘concerted contraction’, suggests such a tight control (Hof 1990; Hof et al., 1983). On the basis of the experiments presented here, however, we have come to a different opinion. It will be shown that, for a wide category of contractions, the mechanical properties of muscle and tendon automatically ensure a high efficiency. The first step in the argument is to show that, in situations like running or jumping, strange as it may seem, muscle force F is not directly related to activation F0 : 3.1. Model What triceps surae muscle does in running, is to move the mass and carry the weight of the body. A model thus
Fig. 4. Thick line: soleus muscle force in running, same experiment as Figs 2 and 3. Dashed line: muscle force as predicted from a simple linear second-order model, Eq. (A.13). Initial value for vð0Þ has been fitted to give the correct maximum force, other parameters were predicted. This subject was a ‘heel-striker’, this may partly explain the difference between model and measured curve in the first half of the contraction.
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McMahon and Cheng, 1990). Active state F0 only comes into consideration when allowance is made for the nonlinear effects. It turns out then that active state F0 should be greater than muscle force F itself. But in that case, with F0 > F ; Hill’s relation immediately predicts that the CC should shorten. As long as this condition is fulfilled, the CC thus always contracts concentrically and it does no negative work. Hill himself has already shown (Hill, 1964) that within a wide range of F0 =F ; from about 1.5–5, the efficiency is close to optimum, see Fig. 5. (A further discussion on the issue of the activation time course is given in Appendix B.) 3.2. Muscle properties ensure efficiency It can thus be concluded that the intrinsic muscle properties ensure that contractions are efficient. That is, in the condition that we have studied: in muscles with short fibres and long tendons, moving a large mass. This considerably mitigates the demands on neuromuscular control in cases like these. Timing is still critical. The effective mass that is moved can change drastically during a movement, e.g. for triceps surae in transition from swing, when it only moves the foot, to stance, when it has to move full body weight. For such reasons muscles should not be switched on too early or too late. But the amplitude of the activation is not critical. As long as F0 > F an efficient action is guaranteed.
Fig. 6. Thick line: muscle force vs. time in running, as in Fig. 4. Thin line: isometric development of force of the same subject. Electrical stimulation, supramaximal pulses of 0.1 ms, 32 Hz at the n. tibialis (Zandwijk et al., 1998). It is seen that the total duration T of the ‘elastic bounce’ contraction in running (207 ms) is shorter than the time constant tm of the isometric contraction (290 ms).
between positive and negative work) is done, that the action is done more powerfully. For running this means: running faster. EMG studies support this view to some extent, EMG levels rise with speed, but the timing is not changed in most cases (Hof et al., 2002a; Nilsson and Thorstensson 1987). 3.4. Elastic bounce contractions
3.3. What is controlled by the activation? After the remarkable finding that muscle force F hardly depends on activation F0 ; the next question is: what then is controlled by the activation. When F0 is increased in a muscle action in which the force F is determined by mass and stiffness, CC speed and work are seen to increase with F0 =F : An increased activation thus would mean that more net work (i.e. the difference
Fig. 5. CC speed vc (thick line), CC power production Fvc (dashed), and muscular efficiency (thin line), all as a function of F0 =F : Calculated from Hill’s Eq. (A.3). The efficiency is above 90% of the optimum for 1:5oF0 =F o5:
Among the infinity of actions a muscle can perform, isometric, isotonic, isokinetic, it may be a good thing to give the actions as discussed here their own name. In agreement with earlier literature (Cavagna, 1970; McMahon and Greene, 1979; McMahon et al., 1987) it is proposed to call them ‘‘elastic bounce contractions’’. With some minor calculation (A.14), it can be shown that a contraction belongs to this category when its duration T is less than two times the time constant tm of an isometric contraction of the same muscle. Fig. 6 shows the moment in an isometric contraction, recorded on the same subject as in Figs. 2–5 in a dynamometer and with electrical stimulation, together with the moment in running. In our example we had T ¼ 200 ms, while tm was experimentally determined as 290 ms, so the condition was amply fulfilled. Running and jumping fall evidently within this category. In walking the duration of triceps surae action is longer, 0.4–0.6 s.
4. Discussion Almost simultaneously with the experiments reported here, obtained with the quick release method, the main
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results have been confirmed by direct imaging of the tendon and aponeurosis with ultrasound (Fukunaga et al., 2001). Drawbacks of the ultrasound method are that it can only monitor local movements and that the tendon stretch measurement is not very accurate, so that the derived velocities are rather noisy. An important advantage is that the elasticity effects can now be studied in a variety of muscles other than triceps surae, e.g. tibialis anterior (Ito et al., 1998) or vastus lateralis (Kubo et al., 1999). Moreover, to really see in a movie that muscle fibre length remains constant during walking (Fukunaga et al., 2001) is even more convincing than inspecting Figs. 2 and 3. Recently, the controlled release experiments have been duplicated (Zee and Voight, 2001). Our results could be confirmed, but the measured stiffnesses were on average 50–75% higher. In spite of a thorough on-site comparison this discrepancy has not yet been explained. When we remade Figs. 2 and 3 with 75% higher stiffness values the effects were essentially the same as described. It may be concluded that they are valid over a considerable range of the stiffness. The findings discussed here are in agreement with the predictions by Alexander (1997), that to drive an inertial load at the lowest metabolic cost, the optimum muscle should have a high compliance and a low shortening speed. They give also support to the views of the late McMahon and associates (McMahon et al., 1987; McMahon, 1984; McMahon and Cheng, 1990) that in movement models leg muscles can in many cases be approximated by simple springs. McMahon usually included the option that part of the stiffness might be due to the effect that muscle force is increased after a stretch due to reflex activity. In the above we have shown that in the human leg muscles, if the ‘elastic bounce’ conditions are fulfilled, there is no need for such an assumption. An important difference between the effects of tendon and reflex elasticity is that tendon elasticity has all the advantages of high muscular efficiency due to elastic energy storage, etc., but this is not the case for reflex elasticity, as this effect is due to muscles actively contracting and shortening or lengthening. In several models of complex movements the complete human leg is modelled as a compliant rod, a kind of pogo-stick (Alexander, 1995; Blickhan, 1989; Farley and Morgenroth, 1999; Farley and Ferris, 1998). As long as leg mass can be considered small in comparison with trunk mass, these are attractively simple models. The connection between such a total limb stiffness and the stiffness of the participating muscles has been given by (Hogan, 1985). His result show, among other things, that total limb stiffness can be modulated by more or less flexing the joints, a fact that well agrees with experience.
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The very simple model presented here has implications for our view on the role of motor control. It suggests that the time course of force and movement are to a larger degree than usually assumed determined by the properties of the musculoskeletal system. To be more concrete: by the elasticity of the muscles and the mass of the limbs. The motor control task of the central nervous system is thus made simpler: a block-like activation is sufficient, in which timing is critical, but the actual time course and level of the activation are much less. It seems possible that a whole class of actions can be generated by an identical timing sequence, with only a modulation in activation amplitude. An example is walking or running at different speeds (Hof et al, 2002a). In fact the whole idea reflects a simple observation from linear systems theory: when the input pulse is shorter than the time response of the (low-pass) system, the latter determines the time course of the response. The level of the input pulse is mainly reflected in the amplitude of the response. A related effect is that EMG profiles can be rather variable, even if the movement itself is very repeatable. The ideas presented here obviously need more experimental support, e.g. by forward dynamics studies. Our prediction is that models that include simple muscle models, e.g. elasticities that can be switched on and off, similar to McMahons ‘spring-and-jack’ model (McMahon, 1984), will perform better, i.e. be less critical with regard to activation, than pure linked-segment models driven by torque generators. Maybe this simplicity in motor control is more relevant to the human lower limb that to the upper limb, which seems better suited for precisely programmed movements. Between muscles there is evidently a trade-off between those built for a high and efficient work output and those more suited for precision movements (Alexander and Ker, 1990).
Appendix A A very simple linearized version of the Hill muscle model as presented in Hof et al. (2002b) will be presented here. It is based on the relation between CClength lc ; MTC-length l and SEC length ls : lc ¼ l ls :
ðA:1Þ
When this is differentiated, using the linearized SEC elasticity with stiffness k1 : vc ¼ v þ
1 dF : k1 dt
ðA:2Þ
The CC is represented by the Hill equation (for the maximum of the force–length relation): F ¼ F0
1 nvc =b 1 þ vc =b
ðA:3Þ
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which can be linearized around F ¼ F0 to give b F0 F vc ¼ : 1 þ n F0
ðA:4Þ
Parameters in (A.3) and (A.4) are the speed parameter b and the form parameter n; while the activation or ‘active state’ is represented by F0 : The maximum muscle shortening speed is then vmax ¼ b=n: A.1. Isometric First the isometric case will be presented, in which the MTC has fixed length, with v ¼ 0: Combining (A.2) with (A.4) then gives dF tm þ F ¼ F0 ðA:5Þ dt with ð1 þ nÞF0 tm ¼ : ðA:6Þ bk1 The solution for this linear first order differential equation is F ¼ F0 1 eðt=tm Þ ðA:7Þ CC force F thus eventually approaches active state F0 ; with a time constant tm that is determined by CC and SEC parameters. The non-linear model, with (A.3) and the non-linear expression for SEC elasticity (Hof et al., 2002b), gives a slightly different form of the curve, but essentially the same features. A.2. Muscle loaded with mass Now the muscle is loaded with an inertial load with mass m and weight Fw : According to Newton’s law then dv m ¼ F Fw : ðA:8Þ dt Inserting the derivatives of (A.2) and (A.3) in (A.8) gives the second-order differential equation 2 m d F bm dF þ F ¼ Fw : þ ðA:9Þ k1 dt2 F0 ð1 þ nÞ dt The resonant frequency o0 is introduced, with rffiffiffiffiffi k1 o0 ¼ : ðA:10Þ m Together with the definition of tm from (A.6) this gives for (A.9): 1 d 2F 1 dF þ F ¼ Fw : þ o20 dt2 o20 tm dt
ðA:11Þ
This is the familiar linear second-order differential equation. Its response is characterized by two parameters, the (angular) eigenfrequency o0 and the damping factor b; which in our case can be ex-
pressed as 1 : b¼ 2o0 tm
ðA:12Þ
When bo1 the system is underdamped and the response is a damped sinusoid. It is important now to observe that in the cases of interest b is much smaller than 1, and as a result the system is only lightly damped. For the subject from Figs. 1–4, running, at F0 ¼ 8000 N, e.g. we have k1 ¼ 188 kN m1 and tm ¼ 0:372 s. For assessing the effective mass, the lever arm with which the body is lifted is important. It was estimated from the moment in standing on one foot (94 Nm) divided by body weight, giving 0.126 m. The moment arm of the muscle is around 0.05 m. For the effective mass moved by the muscle, body mass (76 kg) should be multiplied with (0.126/0.05)2, giving m ¼ 483 kg. This results in o0 ¼ 19:7 rad s1 and b ¼ 0:07: When the initial values for F and v are indicated as F ð0Þ and vð0Þ; respectively, the solution of (A.11) is, neglecting some small terms: F ðtÞ ¼ Fw þ et=2tm ½ðF ð0Þ Fw Þ cos ðo0 tÞ pffiffiffiffiffiffiffiffiffi þ ð2bF0 þ vð0Þ k1 mÞ sin ðo0 tÞ:
ðA:13Þ
On inspection, pffiffiffiffiffiffiffiffiitffi is found that the solution depends strongly on k1 m and on the initial value for MTC speed, vð0Þ: Active state F0 has only a minor influence as long as b is small. The conclusion is that the behaviour of a muscle– tendon-mass system, with parameters relevant to human triceps surae in running or jumping, is mainly determined by SEC stiffness and mass. The CC manifests itself only by a slight damping, the exponential term in (A.13), and in the term 2bF0 : When this simple linear model is compared to the non-linear Hill model, two shortcomings of the linear model emerge. First, the linear model predicts negative muscle forces, which a non-linear model does not. Second, the non-linear model requires that F0 should be higher than about F =1:5: Below this limit the CC will ‘give way’, not able to hold the required muscle force and a high negative vc will result. A.3. Elastic bounce ‘Elastic bounce contractions’ can be defined as contractions where b is small. A simple criterion to assess when a muscle contraction falls into the category is by observing that in such a contraction the force follows a sinusoid with angular frequency o0 ; and that its duration T is about one period=2p=o0 : Inserting this in (A.12) and allowing some rounding off gives a very simple rule To2tm :
ðA:14Þ
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A dynamic contraction can be called ‘elastic bounce’ when its duration is smaller than twice the exponential time constant of an isometric contraction of the same muscle, cf. Fig. 6.
Appendix B The anonymous reviewer of this paper pointed to an interesting issue. In the Hill model CC and SEC are in series, the parallel elasticity playing a minor role in running. Admitting that the time course of the muscle force may mainly be determined by the SEC–mass interaction, it still holds that CC force equals MTC force. The question then arises how the CC force can decrease down to zero at the end of stance. This is in contrast with the fast rise of force, compared to isometric (Fig. 6), which is well understood (Hill, 1949). This is really an intriguing problem. The first possibility is that activation is still high. A decreasing force would then lead to a high CC shortening velocity. It might even reach vmax ; which indeed would result in zero force. A related option, possibility 1b, might be that due to this fast shortening CC length would reach lengths at which the force-length relation becomes zero. These two theoretical options seem not happen in practice, however. Such a fast shortening is not found, cf. Figs. 2 and 3, and the related figures in (Hof et al., 2002b). Moreover, a decreasing force results in a decrease of SEC length, and this counteracts CC shortening due to MTC shortening, Eq. (A.2). The alternative is that the force decrease is related to the decrease of activation. In the classical Hill model the activation, ‘active state’ (Hill, 1949) is assumed to be independent of CC length changes. It is well known that active state remains high for a considerable time after neural activation, observable from the EMG, has ended (Hof and Berg, 1981; Jewell and Wilkie, 1960; Ralston et al., 1976). For soleus, this delay amounts to some 200–300 ms. In fact the EMGs of all three triceps surae muscles end about 70 ms before the force becomes zero, Fig. 7. In a model this leads to a very critical timing. When the activation would stop too early, the muscle might give way and be unable to resist the load, resulting in a fast CC extension. If the activation would remain on too long in running, a new redevelopment of force after toe-off might occur, resulting in a forceful plantarflexion of the foot. Fortunately, neither of these effects are observed in practice. To explain the drop in CC force some amendment to the Hill model seems needed. There are indeed experimental findings which indicate a coupling between activation and CC length changes, e.g. (Herzog 2001; Slager et al., 1998). After neural activation has ended, active state remains high. It decreases both by a slow time-dependent process but also due to CC shortening.
Fig. 7. Ankle moment and EMGs of soleus, gastrocnemius medialis and lateralis and antagonist tibialis anterior (plotted negative) for the running step depicted in Figs. 2–3. The EMG ends about 70 ms before the moment reaches zero. Antagonist tibialis anterior is not active at this time. EMGs were rectified and smoothed with a time constant of 22 ms.
The CC has, as it were, some millimetres of shortening left after activation has ended. Explanations by calcium or crossbridge kinetics suggest themselves. A wellestablished model for these phenomena, which are of obvious importance for biomechanics, is still lacking, however.
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