The theoretical limits to the power output of a muscle tendon complex with inertial and gravitational loads

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1 Receive 1 November Accepte 17 March 3 Publishe online 11 June 3 The theoretical limits to the power output of a muscle tenon complex with inertial an gravitational loas Apostolos Galantis * an Roger C. Wolege University College Lonon Institute of Human Performance, Royal National Orthopaeic Hospital Trust, Brockley Hill, Stanmore, Milesex HA7 4LP, UK When a muscle elivers power to an inertial loa through a spring, the peak power can excee the maximum that the muscle alone coul prouce. Using normalize ifferential equations relating imensionless quantities we show, by solving the equations either analytically or numerically, that one imensionless constant ( ), representing the inertial loa, is sufficient to specify the behaviour uring shortening of a muscle tenon complex with linear force velocity an force extension properties. In the presence of gravity, an aitional constant ( ), representing the gravitational acceleration, is require. Nonlinear force velocity an force extension relationships each introuce an aitional constant, representing their curvature. In the absence of gravity the power output elivere to an inertial loa is limite to approximately 1.4 times the maximum power of the muscle alone, an when gravity is present the power elivere is limite to approximately twice the power of muscle alone. These limits are foun for the purely inertial loa at ca. 1 an with gravity acting at =.5 with arbitrarily small. The effects of nonlinear muscle an tenon properties ten to cancel each other out an o not prouce large eviations from these optima. A lever system of constant ratio between muscle an loa oes not alter these limits. Cams an catches are require to excee these limits an attain the high power outputs sometimes observe uring explosive animal movement. Keywors: muscle; inertial loa; power output; tenon 1. INTRODUCTION The power output of skeletal muscle has a clear limit, but animal movements are sometimes foun to use a power beyon this limit (Bennett-Clark 1975; Peplowski & Marsh 1997; Aerts 1998). This coul be because extra power is elivere by the recoil of elastic structures in series with the muscle. As pointe out for example by Marsh (1999) the presence of an inertial loa is a mechanism that can allow a movement uring which the power output of a muscle tenon complex (MTC) temporarily excees that of the muscle alone. Although some examples like this have been moelle (Wilkie 195; Calwell 1995; Marsh 1999), there has, to our knowlege, been no previous systematic escription of the possible interactions of a muscle, a series compliance an an inertial loa. We show in this paper that there are clear limits to the amount of extra power that can be ae by tenon recoil in such a system.. THE MODEL The analysis will consier three elements that are mechanically in series: a muscle, an elastic element (a spring) an an inertial loa. The movements of the components are summe an the force experience by each is ientical. This force etermines the spee of muscle shortening, the length of the spring an the acceleration of the loa. * Author for corresponence (a.galantis@rfc.ucl.ac.uk). When the functions escribing how these quantities epen on the common force have been state, the system is fully efine, an the movements of the components can be escribe by appropriate solutions of the ifferential equations. We will also consier the case where an aitional constant force acts on the loa only, representing gravity. The initial conition consiere is always that the loa is at rest. When gravity is absent the initial common force is zero an in the presence of gravity it is equal to the weight of the loa. Time zero will represent the start of the muscle activation that prouces shortening an movement of the loa. We consier first examples that can be solve analytically an then some more complicate cases for which a numerical solution is require. (a) Dimensional analysis The use of imensionless terms to specify a physical system often leas to a simpler solution involving one or more imensionless constants. Well-known examples are the Froue number an the Reynols number. Buckingham s theorem (McMahon & Bonner 1983) states that the number of imensionless constants require is equal to the number of parameters neee to escribe the system minus the number of funamental quantities (mass, length, time, etc.) involve. We will seek to escribe each of the systems we iscuss using this minimum number of imensionless constants. In the linear case to be escribe, muscle velocity eclines linearly with force: V m = V max 1 F (.1) ; Proc. R. Soc. Lon. B (3) 7, The Royal Society DOI 1.198/rspb.3.43

2 1494 A. Galantis an R. C. Wolege Power output of an MTC Table 1. Choice of normalizing factors an efinition of normalize quantities. factor quantity imensionless force F velocity V max V stiffness k k = = F V V max 1 = k k length k L = L k acceleration kv max t (V ) ( ) = t (V) kv max time mass kv max V max 1 k t M = t kv max = M V k length change of the spring is proportional to force: L s = F/k, (.) an acceleration of the loa is proportional to force: t (V l) = F M. (.3) Length is conserve: L m L s L l =. (.4) The sign convention is that an increase in the length of the muscle or spring is consiere positive an movement of the loa towars the spring an muscle is also positive. This system is efine by four parameters, V max,, k an M, which represent, respectively, the maximum shortening velocity, the maximum isometric force, the spring stiffness an the mass, an involves the three funamental quantities mass, length an time. A imensionless statement of the system shoul thus be possible using a single parameter. There is no unique metho of normalizing the quantities to reach imensionless expressions. The metho we have use is shown in table 1. Because there are three funamental quantities involve, only three inepenent choices can be mae, represente here by the first three lines in table 1. The other normalizing factors are erive from these three factors by appropriate substitutions. Equations (.1) (.4) can now be written in imensionless form as follows: muscle: m = 1; (.5) spring: s = ; (.6) loa: ( l) = ; (.7) length conservation: m s l =. (.8) The four parameters have been reuce to one,, representing the imensionless mass in the system. A sol- ution of these equations will therefore provie a escription for all possible combinations of V max,, k an M. 3. RESULTS (a) First case: linear motion; spring an muscle characteristics linear The orinary ifferential equation escribing this system can be obtaine by substituting into the time erivative of equation (.8) the corresponing erivatives of equations (.5) (.7). This gives ( l) ( l) 1 ( l ) 1 =. The solution of this equation is l = 1 e (Qcos(Q ) sin(q )), Q where Q = 1/ 1/4. The value of Q is real when the mass is less than 4 an imaginary when is greater than 4. When = 4 the velocity is given by l = 1 e 1. The force can be foun by ifferentiating these equations an multiplying by to give = e if = 4, = 1 Q e sin(q ) otherwise. Q represents the angular velocity of the system an when is less than 4 the force will return to zero when t = /Q. The ways in which force an velocity epen on are illustrate in figure 1. Two special cases are of interest: when = 4 (ashe line) the system is critically ampe ; this is the lowest value of for which the velocity oes not excee unity. In the case that tens to Proc. R. Soc. Lon. B (3)

3 Power output of an MTC A. Galantis an R. C. Wolege 1495 (a) power force ( ϕ ) velocity ( υ l) (b) (c) time ( τ ) Figure 1. Time-courses of (a) loa velocity, (b) force an (c) power elivere to the loa for seven values of the normalize inertial loa ( ). From left to right the lines are for values of 1/64, 1/16, 1/4, 1 (soli line), 4 (ashe line), 16 an 64. infinity (not shown in the figure) the force exponentially approaches unity with unit rate constant, an the velocity remains infinitesimal. When is less than 4 the peak loa velocity excees V max. Peak loa velocity occurs when the force rops to zero, an at that instant l = 1 e /Q. For infinitely small normalize inertial loas, Q becomes infinitely large an peak 1 becomes two, i.e. the infinitesimal inertial loa can be accelerate up to a maximum spee of twice the muscle s maximal shortening velocity. The power elivere to the loa is the prouct of force an loa velocity. Some representative power-output curves are shown in figure 1c. The maximum power that the muscle alone can prouce is.5 normalize units, which is the prouct of force an velocity at the mipoint of their relationship (equation (.5)). Figure shows, as a function of, the peak power generate as a ratio to this value. The maximum value of this power amplification by the inertia spring system is (to three ecimal places) an occurs at a value of very close to one log ( Ξ ) Figure. Ratio of the peak power output of the MTC to the value for the muscle alone. Soli line, without gravity; otte curves, with gravity at = 1 (left) an = 1 (right). The full lines in figure 1 show the time-courses of the mechanical events uring a contraction that prouces the greatest possible power amplification. Early in the contraction the muscle moves faster than the loa an, as a result, while force is rising, the muscle power is mostly use to increase the elastic energy in the spring. After the peak force the loa is moving faster than the muscle an the spring recoil as energy from the spring to the continuing power output of the muscle, thus proviing the optimum power elivery to the loa. Peak power thus occurs while force is falling an at this point the muscle has shortene by 1.89 units. (b) Secon case: rotary motion inertial loa; spring an muscle characteristics linear The equations given in an 3a will apply if F represents the torque, V the angular velocity, L the angular movement an M the moment of inertia of the loa. The system for normalization works as above with giving the normalize moment of inertia. An interesting case is when the muscle an spring are connecte to the loa via a lever so that the force in the muscle an spring is R times the force on the loa. In this case the same equations will apply except that is given by M(V max / ) kr. It follows that in these cases the maximum power amplification is also (c) Thir case: linear motion; inertial an gravitational loaing; linear muscle an spring In this case the loa experiences an aitional acceleration cause by gravity an equation (.7) becomes ( l ) =, where is the normalize acceleration ue to gravity, which is the usual gravitational acceleration g ivie by the appropriate normalizing factor. The subscript inicates the presence of a gravitational force. The orinary ifferential equation escribing this system is Proc. R. Soc. Lon. B (3)

4 1496 A. Galantis an R. C. Wolege Power output of an MTC. Γ = 1 Γ =1 Γ = ΞΓ. 5.5 work one by spring Figure 3. Peak power output as a function of the normalize weight of an inertial an gravitational loa. Soli line, = 1; ashe line, = 1; otte line, = 1. The two lower values of are as in figure. ( l ) ( l ) 1 l 1 =. The solution of this equation (from the time at which the force in the muscle equals the weight of the loa) is l = (1 4 ) 1 e 1 if = 4, = (1 ) 1 e Q (Qcos(Q ) sin(q )) otherwise. The force in the spring is foun by ifferentiating the velocity an multiplying by to obtain = (1 4 ) e 4 if = 4, = (1 ) e sin(q ) otherwise. Q Notice that the above expressions for the loa velocity an force in the spring can be expresse as functions of the velocity an force, respectively, in the absence of gravity (see 3a): l = (1 an ) l = (1 ). Figure shows peak power as a function of normalize inertial loa for three values of. The peak power is higher in the presence of gravity an occurs at lower values of. Figure 3 shows the peak power as a function of the normalize weight (that is, ). Peak power for high values of occurs when the weight is.5. The limiting value of the peak power is. an occurs when is very large. In the presence of gravity, higher peak power amplifications can be achieve with less energy release from the spring than when gravity is absent (figure 4). Increasing the gravitational acceleration reuces the work that is Figure 4. Peak power ratio as a function of the work one by the spring. Soli curve, purely inertial loa; broken curve, optima for ifferent gravitational loas. Higher optimal values correspon to higher gravitational loas. Dots, optima from the three curves in figure 3. elivere from the spring to the loa uner conitions for optimum power. () Fourth case: linear motion inertial loa; spring an muscle nonlinear We have not foun useful analytical equations for escribing the behaviour of these systems when the muscle force velocity an/or the spring force extension relations are nonlinear, as they are in real animals. We have therefore use numerical methos to solve the appropriate ifferential equations for some interesting cases. Equation (.1) can be replace by the force velocity relation expresse by Hill s equation: 1 (F / ) V m = V max 1 G(F / ). When G =, the relation reuces to equation (.1) an the force velocity relation is linear; as G increases the force velocity relation becomes more curve. The maximum power that the muscle can prouce falls as G rises. Using the other ifferential equations as given above (equations (.) an (.3)) we have calculate the force an velocity of the loa for a range of loas using a fourth orer Runge Kutta metho implemente in Mathca. The accuracy of the solutions was checke against the analytical solution for the linear case. G values from to 1 6 were investigate. The peak power expresse relative to the maximum muscle power increases as G increases. For very high values of G a limiting value is reache, which is approximately 1.473, ca. 3% greater than the value of for the linear force velocity relation. The case shown in figure 5 is for G = 4, which is similar to the values foun in actual muscle (Wolege et al. 1985). The force extension relation of the spring is also likely to be nonlinear in real animals, where the spring consists largely of tenon. To investigate the influence of this Proc. R. Soc. Lon. B (3)

5 Power output of an MTC A. Galantis an R. C. Wolege log ( Ξ ) Figure 5. The relationship between the logarithm of the normalize inertial loa an the. Soli curve, muscle an spring linear; triangles, nonlinear muscle force velocity relationship (G = 4); circles, nonlinear spring force extension relationship (H = 4). factor we have replace equation (.) with the hyperbolic function: kl s F = 1 H[1 (kl s / )]. When H = the function is linear an with H = 4 the function is a reasonable approximation to the actual shape of tenon stress strain curves. The maximum power output (with linear force velocity relation) is change by this nonlinearity of the force extension curve of the spring. The maximum power is reuce by ca. 1% in this example. We have investigate a range of H values up to 3. All these values give peak powers of less than the optimum with a linear force extension curve. In actual animals both the force velocity an the force extension relationships will be curve. The opposite changes in maximum power output relative to the muscle power elivere to an inertial loa will ten to cancel an the resulting power output coul be expecte to be within a few per cent of the values for the linear case. (e) Fifth case: catches If the movement of the loa is restricte by a catch, the force in the spring can increase to the maximum force the muscle can evelop. If the catch is now release, the force in the spring will accelerate the loa an the energy in the spring will be converte to kinetic energy of the loa. Although the muscle can a some power uring the recoil process, the spee of shortening of the muscle is not limiting an the rate at which this energy will be transferre from spring to loa is etermine primarily by the mass an the spring stiffness as escribe by the familiar equation of simple harmonic motion: sin(t K/M), or in our imensionless form: sin( / ). The peak power therefore rises without limit as the value of approaches zero (figure 6). The ashe curve shows how the peak power of a system containing a catch changes with the value of. For all values of up to at _ 1. _ log ( Ξ ) Figure 6. The relationship between the logarithm of the normalize inertial loa an the. Soli curve, without catch; ashe line, with a catch release from. least 3 ( 1.5 units) a catch mechanism can eliver a higher peak power than the inertial mechanism alone. A cam is a mechanism by which the lever ratio (R in 3b) varies as the rotation procees. If R is mae to ecrease, the action of the mechanism is analogous to that of a catch: at early times the force in the muscle an spring is large compare with that acting on the loa, with the full force of the tenon applie to the loa only later in the process. The power outputs of cam mechanisms will then fall between those of the inertial mechanisms an those of the catch mechanism shown in figure DISCUSSION Muscle an tenon behaviours have been escribe here using only force velocity an force extension relationships, respectively. Not inclue are the force length relationship, activation relaxation time-course, contractile epression resulting from shortening, an tenon force extension hysteresis, all of which may limit power output in animals. Muscle shortening an loa movement have not been constraine by any mechanical arrangement, as may be the case uring animal movement. The simplicity of our moel reveals the funamental principles unerlying the mechanical interactions between muscle, tenon an loas, but it remains to be iscusse whether actual muscles, subject to the above aitional constraints, coul make use of the mechanism for power amplification. How far oes a muscle have to shorten to achieve the maximum velocity of the loa with a contraction giving the greatest possible power output? Is this shortening within the range permitte by the force length relationship? Is the time of muscle action sufficient to allow for its full activation? These questions are best aresse by consiering a imensione example. The human triceps surae muscle group can generate a maximal static moment of ca. Nm aroun the ankle an shorten at a maximal velocity of ca. 8 ra s 1 (Bobbert & Van Ingen Schenau 199). These values are Proc. R. Soc. Lon. B (3)

6 1498 A. Galantis an R. C. Wolege Power output of an MTC equivalent to 4 N an.4 m s 1 in line with the muscle, using an Achilles tenon moment arm of ca..5 m (Fukunaga et al. 1996). An estimate of the Achilles tenon stiffness (1 Nm 1 ) can be obtaine from fig. 6 in Kurokawa et al. (1). Using these parameter values the optimal mass for power amplification ( = 1) is ca. 8 kg applie irectly at the muscle. The optimal mass at the foot assuming a lever ratio of four from the Achilles tenon is ca. 5 kg, i.e. not far from the average boy mass. The fascicle shortening require to achieve maximal power is ca..4 m. The resting fibre length in the human triceps surae muscle seems to range from.35 to.75 m (Maganaris et al. 1998). It woul therefore be possible for the triceps surae, when oppose by an inertial loa, to be use in a way in which power amplification woul be effective, an not far from the optimum, although perhaps peak power elivery may be constraine by the imensions of the fibres. For optimum loaing the peak power woul be reache two normalize time units after muscle activation, which in this example correspons to ca. 17 ms, which is a similar interval to that reporte for the human push off using the triceps surae muscles (Kawakami et al. ). It has been shown that the most important eterminant of the imensionless behaviour of an MTC acting against an inertial loa, an, as a result, of the maximum power that can be elivere to that loa, is the imensionless inertial loa,. Its value epens not only on the magnitue of the inertia, but also on the muscle an spring properties in the way shown in table 1 for linear-motion loa. For rotational-motion loa the lever ratio is also involve in the way shown in 3b. This result has implications for normalizing loas use uring experiments. It is common to express the gravitational loa relative to the muscle s maximum isometric force. However, expressing the loa in imensionless terms must also inclue normalization of its inertial component relative to the muscle, spring an lever characteristics of the system. A.G. thanks Our Father, an the MRC for his PhD stuentship. REFERENCES Aerts, P Vertical jumping in Galago senegalensis: the quest for an obligate mechanical power amplifier. Phil. Trans. R. Soc. Lon. B 353, (DOI 1.198/ rstb ) Bennett-Clark, H. C The energetics of the jump of the locust Schistocerca gregaria. J. Exp. Biol. 63, Bobbert, M. F. & Van Ingen Schenau, G. J. 199 Isokinetic plantar flexion: experimental results an moel calculations. J. Biomech. 3, Calwell, G. E Tenon elasticity an relative length: effects on the Hill two-component muscle moel. J. Appl. Biomech. 11, 1 4. Fukunaga, T., Roy, R. R., Shellock, F. G., Hogson, J. A. & Egerton, V. R Specific tension of human plantar flexors an orsiflexors. J. Appl. Physiol. 8, Kawakami, Y., Muraoka, T., Ito, S., Kanehisa, H. & Fukunaga, T. In vivo muscle fibre behaviour uring countermovement exercise in humans reveals a significant role for tenon elasticity. J. Physiol. 54, Kurokawa, S., Fukunaga, T. & Fukashiro, S. 1 Behaviour of fascicles an teninous structures of human gastrocnemius uring vertical jumping. J. Appl. Physiol. 9, McMahon, T. A. & Bonner, J. T On size an life. New York: Scientific American Books Inc. Maganaris, C. N., Baltzopoulos, V. & Sargeant, A. J In vivo measurements of the triceps surae complex architecture in man: implications for muscle function. J. Physiol. 51, Marsh, R. L How o muscles eal with real-worl loas: the influence of length trajectory on muscle performance. J. Exp. Biol., Peplowski, M. M. & Marsh, R. L Work an power output in the hinlimb muscles of the Cuban tree frogs Osteopilus septentrionalis uring jumping. J. Exp. Biol., Wilkie, D. R. 195 The relation between force an velocity in human muscle. J. Physiol. 11, Wolege, R. C., Curtin, N. A. & Homsher, E Energetic aspects of muscle contraction. Monographs of the Physiological Society, no. 41. Lonon: Acaemic Press. Proc. R. Soc. Lon. B (3)