J. Biomechanics Vol. 25, No. 2, pp. 207-209, Printed in Great Britain

co21-9290/92 15.00+.lxl Pergamon Press plc

1992.

LETTERS TO THE EDITOR

ON THE ROLE OF BIARTICULAR

MUSCLES

IN HUMAN JUMPING

J. L. VAN LEEUWEN* and C. W. SPOORS *Neuroregulation Group, Department of Physiology, Leiden University, Wassenaarseweg 62, P.O. Box 9604, NL-2300 RC, The Netherlands and tDcpartment of Biomaterials, School of Medicine, Building 55, Leiden University, Rijnsburgerweg 10, NL-2333 AA Leiden, The Netherlands

Muscle function in the human squat jump has been studied in various ways. van Ingen Schenau and colleagues (reviewed by van Ingen Schenau, 1989) have used the inverse dynamics approach in combination with measurements of ground reaction forces and EMGs from some of the major leg muscles. Calculated peak net joint power (Bobbert and van Ingen Schenau, 1988) was highest in the knee (x 2500 W), it was only slightly lower in the ankle and much lower in the hip (x 1500 W). This is in contrast to the volumes of the muscles crossing the joints. A consideration of muscle volume showed that the muscle fibrcs of the ankle plantar flexors would not be able to generate the high peak net power output. The discrepancy was explained by (i) the actions of the biarticular muscles, which evoke a mechanical coupling ofjoints and (ii) a rapid release of stored elastic energy in the Achilles tendon and tendinous sh&s of the m. gastrocnemius (GAS) and m. soleus. The authors concluded that jumping performance is largely influenced by the action of biarticular muscles. Power generated by the voluminous hip and knee extensors would be transported to more distal joints by a timely activation of biarticular muscles like rectus femoris (RF) and GAS. In our opinion, the coupling action of a biarticular muscle is complicated owing to its complex dynamic properties (e.g. Zajac, 1989; Van Lecuwen, 1992). The coupling is influenced by the activation of muscle fibres, muscle fibre properties, instantaneous muscle-tendon complex length and velocity, strain energy storage and other factors. Thus, coupling is by no means as strict as it would be with an inextensible string. A disadvantage of the usage of biarticular muscles as force transmitters is an energy loss owing to viscous properties of the tendons and the muscle belly. Most of the absorbed energy by tendons and tendinous sheets can be recovered in elastic recoil (Ker, 1981).This is different for muscle fibres. To avoid a substantial energy loss, it is of crucial importance that forcible stretching of muscle fibres is avoided as much as possible. A better understanding of the co-ordination of muscle activity during jumping could be obtained by a direct dynamics analysis with appropriate models for all leg muscles and including joint-angular dependence of muscle moment arms. This would show the complete flow of mechanical power in the jumping apparatus. Search techniques could be employed to find optimum muscle activation patterns. A valuable step towards this goal was recently made in this journal by Pandy et al. (1990) and Pandy and Zajac (1991). They formulated a four-segment model, jointed together with frictionless rcvolutes, and driven by 8 musculo-tendon actuators. The authors challenged the idea that jumping pcrformance could be increased by the unique biarticular action of the GAS by comparing simulated jumps (i) with all muscles, (ii) with all biarticular muscles removed and (iii) with all biarticular muscles removed, but with the addition of a monoarticular GAS (Pandy and Zajac, 1991). Jumping performance was found highest in case (iii) and lowest in case (ii).

For the following reasons, we doubt whether these findings allow Pandy and Zajac’s conclusion about the GAS: (a) The moment arms of the simulated muscles were not always very accurate, resulting in particular in a vanishing flexor knee moment of GAS towards knee extension (Pandy et al., 1990; Fig. 4b). Hence, close to full knee extension, the simulated biarticular GAS works effectively as a monoarticular muscle. Since GAS is mainly active just before toeoff, we would indeed expect only a small calculated difference between the simulated biarticular and monoarticular GAS. In simulation (i), RF was calculated to have a significant negative work contribution. Therefore, the absence of RF in case (iii) gives the highest simulated jumping performance.. In contrast to Pandy and Zajac’s model assumptions, Spoor et al. (1990) have measured increasing moment arms of GAS with knee extension. Muscle moment arms about the knee vary such that knee angulation is decelerated towards full extension, and leg segmental kinetic energy (associated with rotation) is used to further lift the centre of mass through enhanced net ankle power output. Via the force transmitting limb bones, this power is transported to proximal segments. As explained by Pandy and Zajac (1991), the massive trunk segment gains most energy during the jump, so that one can speak of an energy flow from the leg muscles in a proximd direction. (b) The linear stress-strain relationship of the tendons used by Pandy and Zajac (1991) might have led to a more than realistic initial muscle fibre stretching and, therefore, to an increased energy loss in the biarticular muscles. Furthermore, maximum contraction speed and activation dynamics were chosen the same for all muscle-tendon actuators, whereas the well-known fast properties of the GAS may be essential to avoid, initially, forcible muscle fibre stretching. We conclude that: (i) A leg with a combination of mono and biarticular muscles is still likely to have a higher effective energy output than a leg with monoarticular muscles only (and equal total muscle mass) would have, if forcible muscle fibre stretching in the biarticular muscles is avoided by a fine tuning of activation. Nevertheless, it remains to be proven quantitatively that the energetic advantages more than compensate for the energy losses in biarticular muscles. (ii) The direct dynamics approach is very promising, but it requires implementation of more accurate model descriptions of muscles and joints before the issue of the role of

207

biarticular

muscles can be settled.

REFERENCES

Bobbert, M. F. and Ingen Schenau, G. J. van (1988) Coordination in vertical jumping. J. Biomechanics 21, 249-262.

208

Letters to the Editor

Ingen Schenau, G. J. van (1989) From rotation to translation: constraints on multiple joint movements and the unique action of biarticular muscles. Hum. Mvmt. Sci. 8,301-337. Leeuwen, J. L. van (1992) Muscle function in locomotion. In mechanics #“A&tat Locomotion (Edited by Alexander, R. McN.), Advances in Comparative and Environmental Physiology, Vol. 11, pp. 191-250. Springer, Heidelberg. Ker, R. F. (1981) Dynamic tensile properties of the plantaris tendon of sheep (Ovis aries). J. expl Biol. 93, 2831302. Pandy, M. G. and Zajac, F. E. (1991) Optimal muscular coordination strategies for jumping. .I. Biomec~u~jcs 24,

Pandy M. G., Zajac, F. E., Sim, E. and Levine, W. S. (1990) An- optimal control model for maximum-height human iumdna. J. Biomechanics 23. 1185-l 198. Spoor;C.W., Leeuwen, J. L. van, Meskers, C. G. M., Titulaer; A. F., Huson, A. (1990) Estimation of instan~neous moment arms of lower-leg muscles. J. Biomechanics 23, 1247-1259.

Zajac, F. E. (1989) Muscle and tendon: properties, models, scaling and application to biomechanics and motor control. In CRC Crit. Rev. Biomed. Engng (Edited by Bourne, J. R,), vol. 17, pp. 359-411. CRC press, Boca Raton.

l-10.

AUTHORS’

RESPONSE

FELIX E. ZAJAC* and MARCUS G. PANDY~ *M~hanical Engin~~ng Department, Biom~h~ical Ensnaring Program, Stanford University, Stanford, CA 943054021, U.S.A., and the Rehabilitation Research and Development Center (153), Veterans Affairs Medical Center, Palo Alto, CA 94305-1200, U.S.A.;f Department of Kinesiology and Health Education, University of Texas at Austin, Austin, TX 78712, U.S.A.

We chose to study ‘vertical’ jumping because experimental and computer-m~eling techniques can be used together to understand how leg muscles work in synergy to coordinate the body segments of this, a reasonably complex, motor task (Zajac and Levine, 1979). A four-segment, eight-muscle model of the human body was developed, and used together with optimal control theory, to quantify muscle and segmental coordination during jumping (Pandy et af., 1990). With this forward-dynamics approach, we showed that our model is able to reproduce the major features of a maximumheight squat jump; specifically, the time histories of all bodysegmental displacements, vertical and fore-aft ground reaction forces, timing and sequence of muscular activity, jump height, and total ground contact time (Pandy and Zajac, 1991). Thus, we felt justified in conducting a detailed analysis of the optimal control solution to better understand how intermu~ular control, inertial interactions among body segments, and mu~~otendon dynamics coordinate a maximum-height jump. Obviously, we are pleased that van Leeuwen and Spoor also support a forward-dynamics, optimal-control approach to studying intermuscular coordination. van Leeuwen and Spoor argue that we should reexamine our suggestion that the biarticular muscle gastrocnemius (GAS) has no unique role in jumping. The basis for their argument seems to be: (1) Had we modeled GAS tendon elasticity and knee moment arm more accurately, we would have found GAS to contribute more to jumping performance because more force and power would have been developed by GAS. (2) Jumping performance was found by us to be enhanced when GAS was modeled as a monoarticular muscle and rectus femoris (RF) and hamstrings were removed from the model, not because GAS was now monoarticular, but because RF, having been removed from the model, was no longer generating negative work. We believe their second reason to be invalid. While it is true that RF was found to generate negative work (Fig. 7, Pandy and Zajac, 1991), it is not true that we would have found jumping performance to be enhanced had we removed only RF from the model. In fact, Zajac (1985) used the same flawed reasoning as van Leeuwen and Spoor to explain the cessation in excitation found in posterior biarticular thigh muscles in cat prior to lift-off during maximum-height jumps. But this logic, although intuitively appealing, is not necessarily correct. The optimal-control solution found by Pandy et at. (1990) was to excite RF, rather than to keep it unexcited, although this option was allowed. Thus, jumping per-

formance will decrease when RF alone is removed from the model, even though RF generates negative work. What this means is that the negative work generated by RF is more than offset by the additional positive work of the other muscles resulting from the participation of RF to the overall coordination pattern. We can also argue against their first reason (inaccurate modeling of GAS). First, although all the co-states (sensitivities of states to performance) were not published, all of them were found (e.g., Figs 10 and 11, Pandy et al., 1990). Specifically, the sensitivity of jumping performance to a newton increase in force at any time during propulsion is about the same among all the muscles. As a force generator, therefore, GAS does not have any distinct advantage in how an increase in its force would affect performance. However, GAS is at a disadvantage, compared to the uniarticular plantarflexors, because it generates a knee flexion moment, which hinders leg extension and performance (e.g. VAS is fully excited during propulsion; thus, a knee flexion moment generated by any muscle hinders propulsion). We believe that the retarding effect on performance, caused by the knee flexion moment developed by GAS, is the reason why the excitation onset of GAS is delayed compared to the uniarticular plantartlexors (Fig. 3, Pandy and Zajac, 1991). Thus, any power au~entation that might arise by GAS operating on a more favorable position of its force-length and force-velocity curves (e.g. by modeling its knee moment arm so that it increases with knee extension, Spoor et al., 1990) would probably still be short-lived (about 80% to 100% of ground contact time) and the contribution of GAS to total muscle power output would be small (see Fig. 7, Pandy and Zajac, 1991). Second, we have found that jumping height is primarily sensitive to muscle strength, secondarily to intrinsic speed of shortening of muscle fibers, and hardly sensitive to tendon elasticity (Pandy and Zajac, 1989).Thus, increases in tendon elasticity of GAS (or its speed of shortening) are expected to hardly affect overd jumping performance, although such increases may indeed enhance the work produced by GAS. Nevertheless, we agree with van Leeuwen and Spoor that ‘. . _more accurate model descriptions of muscles and joints (are needed) before the issue of the role of biarticular muscles can be settled’. Our arguments above, as well as those of van Leeuwen and Spoor, are based on our own intuitions and logic, and, as stated above, we both can go awry. Certainly, the fidelity of our (or anyone’s) optimal-control model for jumping depends on the structure and the parameters assumed (e.g. body segmental and mu~ulotendon structure and parameters). Thus, exactly how coordination would be