Published in Corpus, Psyche et Societas 5, 22-48 1998

Modelling Skeletal Muscle-Tendon Units as Actuators in Motor Behaviour: Morphological Aspects

Gertjan Ettema Department of Sport Sciences Norwegian University of Science and Technology 7034 Trondheim Norway phone: + 47 73590616 fax:

+ 47 73591770

E-mail: [email protected] Note: figures at end of paper

Summary: This paper discusses a number of morphological aspects of skeletal muscle and muscle-joint lever system that should be considered when modelling skeletal muscles as actuators of the bodies’ motor system. It is recognised that, to implement muscle models in whole body models, these should be relatively simple. Thus, whenever possible, relatively simple algorithms are proposed that describe the major impacts of the morphological features of muscle on their mechanical behaviour. This way, it should be possible to model and describe the mechanical properties and behaviour of skeletal muscle in a more realistic way, without compromising the simplicity of models. INTRODUCTION Modelling intrinsic properties of skeletal muscle-tendon units in the scope of coordinated movements has gained momentum in recent years (see Winters and Woo, 1990; Beek, 1991; Bobbert, 1991; Otten, 1991; Winters, 1995; Huijing, 1995). With regard to motor behaviour, the role and behaviour of the nervous and musculoskeletal systems rely heavily on each other’s properties and each other’s organisation. Thus, a proper understanding of the physiological and structural properties of skeletal muscle is essential to understand motor control and organisation in a general way (e.g. van Ingen Schenau et al., 1995; Winters, 1995). Although many structural and biophysical models have been developed (see Zahalak, 1990), in motor control studies, the phenomenological modelling approach (e.g. Hill, 1938) has prevailed, be it mainly for high intensity tasks (see Winters, 1995). This is not only because of the relative simplicity, but also because of the phenomenological approach itself: the Hill-type models seem to best fit the purpose of describing the mechanics of one component of the integrated (sensor–)controller–actuator system in a functional way. Beek (1991), arguing Otten (1991), posed that the phenomenological approach does not exclude explanation of the phenomena studied. An example is the clarification of the effects of tendinous structures on muscle mechanical behaviour (e.g. Hof et al., 1983; Avis et al., 1986; Alexander, 1988; Ettema et al., 1990; Griffiths, 1991; Ettema and Huijing, 1994a). In principle, a phenomenological Hill-type model does not bare any relationship with the physical muscle structure (Zahalak, 1990). Therefore, phenomenological models often have limited generalisation power as their design relies on a restricted number of experimental observations. Irrespective of the modelling approach used, a strong need for model improvements on many aspects is apparent in the field of motor behaviour (see Winters, 1995 & Huijing, 1995 for reviews). This paper deals with a number of mainly morphological aspects of muscle –2–

that may improve the accuracy of these models of muscles as the mechanical actuators of the motor system. If such factors are included in phenomenological muscle models, the relationship between phenomenon and physical entity causing the phenomenon is enhanced. A further aim is to not compromise the simplicity of the model significantly (see Bobbert, 1991; Winters, 1995). This paper does not discuss the neural organisation and the proprioceptive function of muscle. CONTRACTILE PROPERTIES The sarcomere within a muscle fibre is the building block of contraction. Consisting of actin and myosin filaments, a sarcomere contracts under electrical stimulation of the fibre by forming cross-bridges between actin and myosin. Each cross-bridge generates force, depending on the actual biochemical state of the actin-myosin binding. Length-tension and force-velocity The length-tension curve of muscle describes the relationship between the length of a muscle and the isometric tension it can generate under full activation. The typical inverted Ushaped curve is explained by the sliding-filament theory (Huxley, 1957). The theory assumes independent cross-bridges that perform cross-bridge cycles in a pseudo-random, stochastic manner. The number of cross-bridges that are able to generate force than depends on the overlap of actin and myosin. The number of cross-bridges that are in an attached state, in which they are able to generate force, depends in the sliding velocity (Huxley, 1957). Thus, the force generated by a sarcomere and an entire fibre is velocity dependent. For isokinetic and isotonic contractions, a typical hyperbolic curve describes the force-velocity relationship. Hill-type modelling Hill-type models, or lumped phenomenological models (after A.V. Hill, 1938) are used to describe a number of features of mechanical behaviour of muscle. The basic model consists of three elements, contractile element (CE), series elastic element (SEE), and parallel elastic element (PEE). The terms ‘series’ and ‘parallel’ refer to the placement of the respective elastic elements with reference to the CE as the central component. In the original Hill model, the CE behaves according the described length-tension and force-velocity properties, i.e. representing the cross-bridge behaviour in a lumped manner. CE does not represent a morphological structure but behaviour. In models of entire muscle-tendon units, –3–

SEE usually represents tendinous tissue connected in series with the fibres, and PEE represents connective tissue. It should be noted however, that in the phenomenological model these elements represent muscle behaviour, not a morphological structure of a muscle. One of the major problems of the Hill model is the accuracy and general applicability (see also Winters, 1995 & Huijing, 1995). The model is quite successfully used in explosive movements, with near maximal activation levels of short duration. At submaximal activity levels however, the model fails, primarily because the CE properties are dramatically altered as compared to maximal activation levels (Winters, 1995; Huijing, 1995). The reason may be that the affinity of different cross-bridge cycle states is non-linearly related to [Ca2+] levels in the sarcoplasm, altering the cross-bridge dynamics. But also, at high activity levels, a number of discrepancies occur between models and experimental data. Especially, the length tension curve as derived from single sarcomeres does not comply with the length tension curves of whole muscles. Some of the reasons for this discrepancy are discussed in the next section within the scope of the aim of modelling. The classic muscle contraction models (Huxley’s sliding filament model and Hillmodel) do not predict force responses in full detail. In the last decade or so, the force response of skeletal muscle on length changes and particularly its deviation from classic model predictions has gained momentum (e.g. Morgan, 1990; Herzog and Leonard, 1997; Ettema, 1998; Meijer et al., 1998; van der Linden et al., 1998). Although this field of study is beyond the scope of this paper, a mention is required because of it functional implications. With regard to modelling muscle as whole-body-movement actuators the challenge is to improve classic models or provide alternatives that are based on structure and contraction mechanisms, yet that remain relatively simple. ARCHITECTURE: FROM FIBRE TO MUSCLE Pennation Many models describing the functional implications of muscle architecture consider the alignment of muscle fibres relative to the line of pull of the entire muscle. Often, the pennation of muscle fibres is presented as the cause of reduced force that is generated by the entire muscle compared to the force capacity of the muscle fibres. In other words, muscle force is calculated as Fm = Ff∗cos(α), α being the pennation angle. Yet, this model is incorrect and incomplete. Pennation does not only affect the force but also the change of muscle length during contraction. It is well known that the isometric length-tension curve of muscle generally has a wider active length-range than is to be expected on the basis of the –4–

corresponding sarcomere length-tension curve. First, the length-tension curve of single mammalian muscle fibres does not fully correspond to the sarcomere length-tension curve as described by Gordon et al. (1966) (Zuurbier et al., 1995). But even the length-tension curve of isolated fibres does not predict the length-tension curve of an intact muscle. Huijing and Woittiez (1984) presented a planimetric model describing the effect of pennation on the functional length-force characteristics. Otten (1988) revised the mathematical representation of the model. He applied the principle of equilibrium of forces and assumed that muscle volume remains constant during contraction. This resulted in the appropriate relationship between muscle and fibre force and length changes, fulfilling the requirement of balance of work between fibres and muscle (all work done by the fibres during shortening is also done by the muscle as a lumped entity): dl m = dl f ∗

cos( β ) cos(α + β )

(1a)

Fm = F f ∗

cos(α + β ) cos( β )

(1b)

where Fm and Ff are muscle and fibre force, α and β are the angles of fibre and aponeurosis with respect to the line of pull, and dlm and dlf are length changes of muscle and fibre. As mentioned, the basic assumptions of this model are constant muscle volume and balance of work between fibres and muscle Otten (1988). In other words, no work is lost from fibres to entire muscle (we assume a frictionless condition), yet a trade-off between force and displacement occurs. Note that the pennation angles are in fact a function of muscle length. Experimental geometrical changes are well predicted by this model. When describing the length-tension diagram of a muscle, not equation 1a, but the primitive should be applied, which is given by Otten (1988): 2 2 2 2 l m = ( d + (l a + l f − d )

(1c)

d = l fo sin(α o + β o ) d is the distance between the aponeuroses; the subscript o refers to optimum length. Figure 2 shows the main effects of pennation on the transition from fibre to muscle as well as the geometrical changes during shortening. At the transition of aponeurosis to tendon, an angle in the tendinous structures occurs (Ettema and Huijing, 1990; Huijing et al., 1994), which normally cannot be maintained in a –5–

flexible element. The model explains how the angle is maintained. Figure 1B shows the free body diagram of the aponeurosis. Fp is the force representing the overall pressure in the muscle that is resisting the fibre force pulling the aponeuroses towards each other, and thus keeping the aponeuroses at constant distance (this the 2-D analogue of constant volume). If we assume that the fluid volume in the muscle belly acts without friction on the tendon plates (i.e. no shear stresses occur), Fp is perpendicularly directed to the aponeurosis (Otten, 1988). However, an analysis of equilibrium of momentum at point I, i.e. the insertion of aponeurosis into the tendon, shows that the point of application of Fp is not in the middle of the tendon plate. The summed fibre force does have its point of application in the middle of the aponeurosis because for a homogeneous muscle it is assumed that all fibres exert the same force in the same direction. The deviation of Fp from the middle point of the aponeurosis indicates a pressure gradient underneath the aponeurosis surface. The point of application of Fp, and thus the pressure gradient depends on the angles α and β according to d=

la cos( β ) ∗ sin(α + β ) ∗ 2 sin(α )

(2)

Since d is larger than ½la, the gradient under the aponeurosis runs from low pressure at tendon insertion to high pressure at the other end of the aponeurosis (Fig 1C). Similar but opposite gradients were modelled and experimentally found (Otten, 1988). The reason for this opposite pressure gradient is unclear and needs further attention. The current model explains the angle between tendon and aponeurosis. This angle occurs at the transition point of the tendonaponeurosis continuum where fibre forces and pressure forces are starting to be applied to this continuum (i.e. the crucial difference between free tendon and aponeurosis). Of course the model is a simplification of reality: the fibres and aponeurosis are modelled as elements with absolute stiffness or elasticity in exclusively longitudinal direction. Furthermore, a frictionless condition is assumed among the elements, including the fluid volume of the muscle. In the simplest version of the model the aponeurosis is a stiff element. Otten (1988) and Ettema and Huijing (1990) implemented elastic aponeuroses (see below), studying its functional consequences. In the present version of the model the fibres and aponeuroses do not curve. The mechanical and morphological validity of models is of importance for heuristic and general applicability reasons. For example, Otten (1988) argued that a three-dimensional model presented by Woittiez et al. (1984) was unrealistic because of the different distal and proximal aponeurosis areas available for muscle fibre attachment. Van Leeuwen and Spoor –6–

(1992) argued that model presented above (Otten, 1988; Huijing and Woittiez, 1984) cannot exist because of the tendon-aponeurosis angle. However as argued above, the model explains how this experimentally demonstrated angle is maintained. I would argue that the Otten’s model is in fact compatible with the pressure based alternative model by van Leeuwen and Spoor (1992). The main differences between the two models as presented in the literature are the complexity and the detail of description, which becomes important when the purpose of the model is considered (see Winters 1990; Bobbert, 1991). The advantage of Otten’s model is its mathematical simplicity, which is obtained by a global analysis, i.e. considering only a handful of elements. Zuurbier and Huijing (1992) showed that the model predicts shortening velocity of the muscle on basis of fibre and aponeurosis velocities very well (note that muscle velocity exceeds the summed fibre and aponeurosis velocity). Recently, Ettema and Huijing (1994b) successfully applied the model in combination with distribution of fibre length (see below) to predict the length-force characteristics of rat gastrocnemius. The advantage of the simplification in the model is of heuristic nature: in the pennate muscle, fibres do not curve for reasons of constant volume (and thus lack of space in shortened position), but probably because of internal pressure forces in perpendicular direction, which the fibres cannot resist. The increase in pennation angle during shortening creates more perpendicular space in the muscle belly, allowing thickening of the fibres. Muscle fibres always curve outwards but, except at extremely short muscle lengths (unpublished results), curving appears to be reasonably constant and independent of muscle length. This indicates that lack of space is not the principle cause for fibre curving. In a parallel fibred muscle, fibres must curve when the muscle shortens, because there is no increment of pennation angle creating more crosssectional space for the thickening fibres. The difference between aponeurosis and fibre curving can be explained with aid of the model described by Otten (1988). Pressure forces, perpendicular on the aponeuroses, are counteracted (and originally created) by fibre forces; such counteraction does not occur for pressure forces on the muscle fibres, and thus in reality they must curve outwards. (In the model an infinite bending stiffness is implicitly assumed, preventing any curving). Experimental data indicate that the aponeurosis often remains reasonably straight during activation. In some cases curving is even inwards and opposite than predicted from the pressure model by van Leeuwen and Spoor (1992). Heterogeneity of fibre lengths and forces (Zuurbier and Huijing, 1993; Ettema and Huijing, 1994b) may be a crucial factor in curving of the aponeurosis. –7–

The model also indicates that the specific tension (stress) that is usually calculated as isometric force over physiological cross-sectional area (PCSA) is affected by pennation. To obtain a valid estimate of fibre specific tension the following equation should be used. Tf =

Ff cos( β ) Fm = ∗ PCSA PCSA cos(α + β )

(3)

Although the planimetric model (Woittiez and Huijing, 1984) and Otten’s improvement is an extreme simplification of reality, they are very useful because of their predictive power for muscle length-tension curves and mathematical simplicity. Another advantage of the planimetric model for unipennate muscles described in equation (1) is that it applies to bipennate and parallel muscles as well. In bipennate muscles angle β, and in parallel muscle both α and β equal zero. Fibre Distribution Although geometrical changes during muscle shortening are very well described by Otten’s model, it appears that pennation effects can only partly explain the discrepancy between the width of the length-force curve of entire muscle and a single fibre (Huijing and Woittiez, 1984; Huijing et al., 1989). Another factor influencing the length-tension curve is the distribution of lengths of fibres and motor units within the muscle (Lewis et al., 1972; Bagust et al., 1973; Stephens et al., 1975; Huijing, et al., 1989; Bobbert et al., 1990; van Eijden and Raadsheer, 1992). Ettema and Huijing (1994b) estimated the effect of fibre distribution with reasonable success using a sarcomere-number-distribution model. The distribution of the number of sarcomeres connected in series within a single fibre, i.e. the distribution of optimum fibre lengths (lfo) within the muscle, was described by a Gaussian distribution using a mean (µlfo) and a standard deviation (σlfo) as parameters: y = ( 2πσ

1 2 −2 )

( −(( x − µ ) σ ) 2 ) 2 ∗e

{

}

(4)

where y is the probability of occurrence of fibre length x. The integration of y with respect to x results in unity. In the model, y was multiplied by maximal isometric force for the entire population of muscle fibres (Fo), so that the summation of all forces was equal to Fo. Three distribution models were tested shown in Fig 3A. In model I all fibres in the muscle were assumed to work at the same absolute length, giving the same distribution of sarcomere lengths for all muscle lengths. In model II all fibres were assumed to act at their own optimum

–8–

length at muscle optimum length, resulting in a changing distribution of sarcomere length as a function of muscle length; at muscle optimum length there is no distribution of sarcomere length. In model III no distribution of fibre optimum length was incorporated (σlfo=0), but a Gaussian distribution was applied to their absolute length (lf) at muscle optimum length. Fibre optimum length was taken as the average fibre length at muscle optimum (µlf), and the standard deviation (σlf) was taken the same as σlfo. Thus, in model III there is a changing sarcomere length distribution at all muscle lengths. The fibre distribution model was applied in combination with Otten’s pennation model and lumped effects of elasticity of tendinous structures. The main findings are shown in Fig. 3B. Considering these findings and data on fibre and sarcomere distribution (Stephens et al., 1975; Holewijn et al., 1984; Otten, 1988; Bobbert et al., 1990; Heslinga and Huijing, 1990; Zuurbier and Huijing, 1993), the authors concluded that a combination of models I and III seemed most likely. The main consequence of fibre length distributions is that a muscle has an enhanced active length range but a reduced force generation capacity. The reduced force generation capacity has an effect on specific tension of muscle, which is calculated as the force over physiological cross-sectional area. ELASTIC BEHAVIOUR In this section, only the elastic behaviour that is of importance in active muscle will be considered. In a Hill-type model, this behaviour is represented by the SEE. The connective tissue modelled as PEE will be disregarded. The series elastic component is of functional importance as it affects the dynamics of the contractile machinery and allows storage and release of elastic energy (e.g. Griffiths, 1991; Ettema et al., 1990; Cavagna, 1977). Intra- and extracellular series elasticity A series elastic component is by definition any structure in the muscle that is aligned in series with the contractile machinery and that behaves elastically. Thus, one distinguishes extra- and intracellular series elastic structures. The extracellular structures comprise all tendinous tissue, i.e. isolated tendon and aponeuroses (tendon plates), while the intracellular structures constitute myofilaments and the cross-bridge connections. Quite some debate has been going on about the relative contribution of these components to series elasticity. Most of the debate is based on the confusion about the object studied: whole muscle, isolated fibres or single sarcomeres.

–9–

A general model for series elasticity is shown in Fig. 4. The most important issue regarding modelling whole muscle-tendon units, is the different behaviour of the components and thus their contribution to the total elasticity. In Fig. 4, three components are identified, tendinous (extracellular) compliance, myofilament compliance, and cross-bridge compliance. The tension on the tendon is independent of the force-generation site, while the cross-bridge and myofilament compliance is affected by the number of cross-bridges in attached state. Morgan (1977) developed a quantitative model that was adapted by Ettema and Huijing, (1993). For a full isometric contraction the cross-bridge kinetics can be assumed to be linear, independent and stochastic (Blangé et al., 1972; Ford et al., 1977; 1981; 1985). Compliance of cross-bridge elastic component, as measured in a step-response, is C cb =

dl cb E cb = dF Fi

(5)

Fi is the initial isometric force level, and Ecb is the total elastic elongation of the cross-bridge. Total compliance of the series elastic component can be described as Cm =

dl m = C t + C cb dFm

(6)

Ct represents compliance of the tendinous structures (i.e. tendon plus aponeurosis). From equations (5) and (6) it follows that C = Ct +

E cb ⇔ C ∗ Fm = C ts ∗ Fm + E cb Fi

(7)

Morgan (1977) assumed Ct to be constant, and used equation (7) for direct distinction between tendon and cross-bridge compliance. Ettema and Huijing (1993; 1994c) used equation (8) to distinguish a force dependent compliance from a constant compliance: C ∗ Fi = C c ∗ Fi + E d

(8)

Ettema and Huijing (1993) calculated that in rat muscle-tendon units (EDL and gastrocnemius) most of the series-elastic compliance was located in the tendinous structures, contributing to approximately 85-90% of series elastic elongation and about 70% of series elastic compliance at isometric force. Furthermore, it appeared that the tendinous structures did not show a linear stress-strain curve at high forces, up to maximal isometric force, which was in agreement with data from isolated tendon (Bennett et al., 1986). In conclusion, in muscles with relatively long tendons, the series elastic element is reasonably well modelled – 10 –

by is single non-linear elastic element, which behaves independently of the mode of force generation. In muscles with short tendons, the cross-bridge compliance plays a far more significant role and SEE should be modelled in a more complex manner (Ettema and Huijing, 1994c). Pennation In pennate muscle, not only the contractile machinery but also the series elastic components in aponeurosis and cross-bridges are aligned under an angle with the line of pull. Thus, equation (1) directly applies to the compliance estimates of the series elastic component. In the simplest version, ignoring possible interactions between aponeurosis and fibres, muscle compliance is Cm =

dl m 1 cos 2 (α + β ) C = C tendon + C apon ∗ + ∗ cb dFm cos 2 β cos 2 β

(9)

For many muscles, the effect of pennation on compliance will be minimal and can be ignored. For a typical rat gastrocnemius muscle (lf =12mm, la =20mm, lt =10mm, α =20 degrees, β =10 degrees) the pennation effect is about 8%, i.e. Cm is about 0.92(Ctendon + Capon + Ccb). LEVERS Most skeletal muscles generate a moment at a joint and thus may generate rotations of body segments around a centre of rotation in the joint. This centre of rotation is usually not fixed but its location changes with movement (e.g. Lieber and Boakes, 1988; Smidt, 1973). Thus, often one refers to the instantaneous centre of rotation (ICR). Furthermore, the line of pull of the muscle in relation to the ICR may change dramatically with movement. As a consequence, the moment arm, or leverage, of a muscle is usually not constant but changes with joint angle (Grieve et al., 1978; Bobbert et al., 1986; Spoor et al., 1990; Visser et al., 1990; Mai and Lieber, 1990; Smidt, 1973; Ettema, 1997). The moment arm (κ) is the derivative of the relationship between muscle length and joint angle (Spoor et al., 1990): κ=

dl m dα

(10),

and thus can be determined experimentally without information about the location of the ICR. Accurate modelling of the moment arm-joint angle relationship is of crucial importance for the quantitative analysis of muscle moments, especially when using inverse dynamics (Spoor

– 11 –

et al., 1990). Often a second or third order polynomial is used to fit muscle-length to joint angle data (e.g. Grieve et al., 1978; Bobbert et al., 1986; Visser et al., 1990). However, it has been shown that these polynomial fits lead to erroneous and sometimes unrealistic values for moment arms. Where possible one should use models that are based on the morphometry of the muscle-joint system, particularly where biarticular muscle are involved. In biarticular systems an interaction may occur between the two joints (Ettema, 1997; Ettema et al., 1998). Because a morphometric model is highly dependent on joint configuration, specifics of such models are not presented here. The reader is referred, however, to Spoor et al. (1990) and Ettema (1997) on a further elaboration of the topic. MOTOR CONTROL – STIFFNESS In studies of motor control and co-ordination, stiffness is often considered the most relevant property of the movement actuators. This can be easily imagined with the following illustration in Fig. 5, which is based on the equilibrium point hypothesis (see e.g. Feldman, 1986; Latash, 1993). Two active antagonistic muscles around the elbow are modelled as springs behaving according to their length-tension properties. Fig. 5C illustrates what happens when a perturbation is applied (Fig.5 B&C, fat arrow) to two systems with different stiffness (i.e. slope of the length-tension curves). The system with a low stiffness more susceptible to the external perturbation than the stiff system, indicated by the size of the resulting movement (Fig. 5C). The default stiffness of a joint is determined by muscle stiffness of the actuators under maximal activity, but the system is able to adjust this stiffness by the level of cocontraction of antagonists. If a person intentionally wishes to make a movement, the situation is different. One can move the elbow by altering the stiffness of one of the springs (or both) through change of the level of activation. In case of high default stiffness (e.g. because of the muscles morphology: large PCSA and motor units, and short fibres), one is required to change the activation level to large extent for a small movement (Fig. 5D). In other words, a stiff system has a low ∆ position/∆ activation ratio or ‘activation to movement’ gain. A very compliant system has a high gain. An ‘activation to movement’ gain that is too high will lead to over-sensitivity for internal errors (i.e. a small error in activation level will cause a large error in movement outcome). A low gain causes low movement resolution: the smallest change in activity level (i.e. activity of a single motor unit) will lead to a relatively large movement. Muscle and SEE do not show fully elastic behaviour, but are rather visco-elastic in nature. In other words, muscle and SEE force do not only depend on length change of the – 12 –

respective structures, but also on the rate of length change [F=f(l,dl/dt)]. The viscous properties allow joint-muscle systems to act as damped systems, and thus, are of high importance for normal movement and control. The above remarks on stiffness also apply, in principle, to the rate dependent behaviour. Functional vs SEE stiffness Many misunderstandings and misconceptions evolve from the lack of notion that two essentially different types of stiffness exist. Functional stiffness is the slope of the actual length-tension curve during a particular motor action (usually not the isometric length-tension curve). Series elastic stiffness is the stiffness of the series elastic component, which, regarding motor control, is only relevant during fast perturbation-like actions of short amplitude. During such actions, the stiffness of the contractile element is extremely high (e.g. Morgan, 1977; Ettema and Huijing, 1993). Thus the stiffness, of the whole muscle is equal to SEE stiffness. The lowest possible stiffness of muscle is set by the isometric-length tension curve and activation level. Muscle vs Joint stiffness In most movements, joint stiffness rather than muscle stiffness is the essential parameter in motor control (see Fig. 5C and D, where muscle properties are converted to joint properties). As mentioned earlier, the moment arm of a muscle is the key for the conversion from muscle to joint behaviour. Particularly for stiffness, the influence of the moment arm κ cannot be ignored. Below the deduction for equation 11, the relationship between muscle and joint stiffness (Sm and Sj) is given. l m = g (α ); Fm = f (l m ) = f ( g (α )); κ = Sm = Sj =

dl m = g ' (α ); M = κ Fm dα

dFm = f ' (l m ) = f ' ( g (α )) dl m

dM d { f ( g (α )) ∗ g ' (α )} = = f ' ( g (α )) ∗ g ' (α ) ∗ g ' (α ) + f ( g (α )) ∗ g ' ' (α ) dα dα

= f ' ( g (α )) ∗ g ' (α ) 2 + f ( g (α )) ∗ g ' ' (α ) S J = S m ∗ κ 2 + Fm ∗

dκ dα

(11a)

– 13 –

S J = Sm ∗κ 2 ;

κ = constant;

dκ =0 dα

(11b)

Equation 11b indicates the conversion for constant moment arm. The relationship shows that the moment arm κ is to be taken squared. Furthermore, the assumption that moment arm is constant may lead to disastrous interpretations of muscle activity patterns. An actually negative joint stiffness (i.e. an unstable joint system) may be computed as a small positive stiffness (i.e. a relatively stable system) on the basis of the assumption of a constant moment arm (Fig. 6). The interpretation of muscle activation patterns during a particular task execution may be completely opposite as a result of such computation error. In an unstable system, the muscle actions do not oppose an error in the execution (as would happen in a stable system), but rather amplify the error. Thus, accurate information about the system stability and therefore muscle moment arms is essential in the study of motor behaviour and control.

Concluding remarks Detailed morphology and related biomechanics of the musculoskeletal system can often be modelled effectively for the purpose of implementation in whole body models. Moreover, the modelling exercise enhances the insight of the impact that details of the morphological design of the body and its components may have on human motor actions and their control.

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REFERENCES Alexander R.McN. (1988) Elastic Mechanisms in Animal Movement. Cambridge University Press, Cambridge. Avis F.J., Toussaint H.M., Huijing P.A., van Ingen Schenau G.J. (1986) Positive work as a function of eccentric load in maximal leg extension movements. European Journal of applied Physiology 55, 562-568. Bagust, J., Knott, S., Lewis, D.M., Luck, J.C. and Westerman, R.A. (1973) Isometrics contractions of motor units in a fast twitch muscle of the cat. Journal of Physiology 231, 87-104. Beek P.J. (1991) Modelling complexity: a complicated business. In: eds Jacobs, R. and Rikkert, W.E.I. Movement Control. An Interdisciplinary Forum. VU University Press, Amsterdam. pp. 91-96. Bennett, M.B., Ker, R.F., Dimery, N.J. and Alexander, R. McN. (1986) Mechanical properties of various mammalian tendons. Journal of Zoology 209, 537-548. Blangé, T., Karemaker, J.M. and Kramer, A.E.J.L. (1972) Elasticity as an expression of cross-bridge activity in rat muscle. Pflügers Archiv 336, 277-288. Bobbert M.F. (1991) Required performance. In: eds Jacobs, R. and Rikkert, W.E.I. Movement Control. An Interdisciplinary Forum. VU University Press, Amsterdam. pp. 85-89. Bobbert, M.F., P.A. Huijing and G.J. van Ingen Schenau (1986) A model of the human triceps surae muscle-tendon complex applied to jumping. Journal of Biomechanics 19, 887-898. Bobbert, M.F., Ettema, G.J.C. & Huijing, P.A. (1990) The force-length relationship of a muscle-tendon complex: experimental results and model calculations. European Journal of applied Physiology 61, 323-329. Cavagna, G.A. (1977) Storage and utilization of elastic energy in skeletal muscle. Exercise and Sport Sciences Reviews 5, 89-129. Ettema, G.J.C. (1997) Gastrocnemius muscle length in relation to knee and ankle joint angles: verification of a geometric model and some applications. Anatomical Record 247, 1-8. Ettema, G.J.C. (1998) Modelling muscle contraction history: an alternative for Hill? in: eds Matsuzaki, Y., Nakamura, T. and Tanaka, E. Third World Congress of Biomechanics – August 2-8, 1998, Sapporo. Abstracts p. 88b.

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Ettema, G.J.C. and Huijing, P.A. (1990) Architecture and elastic properties of the series elastic element of muscle-tendon complex. in: eds Winters, J.M. and Woo, S.L.-Y. Multiple Muscle Systems. Biomechanics and Movement Organization. Springer-Verlag, New York. pp. 57-68. Ettema, G.J.C. and P.A. Huijing (1993) Series elastic properties of rat skeletal muscle: distinction of series elastic components and some implications. Netherlands Journal of Zoology 43, 306-325. Ettema G.J.C. and Huijing P.A. (1994a). Frequency response of rat gastrocnemius medialis in small amplitude vibrations. Journal of Biomechanics 27, 1015-1022. Ettema, G.J.C. and Huijing, P.A. (1994b) Effects of distribution of muscle fiber length on active length-force characteristics of rat gastrocnemius medialis. Anatomical Record 239, 414-420. Ettema, G.J.C. and Huijing, P.A. (1994c) Skeletal muscle stiffness in static and dynamic contractions. Journal of Biomechanics 27, 1361-1368. Ettema G.J.C., van Soest A.J. and Huijing P.A. (1990) The role of series elastic structures in prestretch induced work enhancement during isotonic and isokinetic contractions. Journal of experimental Biology 154, 121-136. Ettema, G.J.C., Styles, G. and Kippers, V. (1998) The moment arms of 23 muscles of the upper limb in different elbow and forearm positions. Human Movement Science 17, 201220. Feldman, A.G. (1986) Once more on the equilibrium point hypothesis (lambda model) for motor control. Journal of Motor Behaviour 18, 17-54. Ford, L.E., Huxley, A.F. and Simmons, R.M. (1977) Tension responses to sudden length change in stimulated frog muscle fibres near slack length. Journal of Physiology 269, 441-515. Ford, L.E., Huxley, A.F. and Simmons, R.M. (1981) The relation between stiffness and filament overlap in stimulated frog muscle fibres. Journal of Physiology 311, 219-249. Ford, L.E., Huxley, A.F. and Simmons, R.M. (1985) Tension transients during steady shortening of frog muscle fibres. Journal of Physiology 361, 131-150.

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Gordon, A.M., Huxley, A.F. and Julian, F. (1966) The variation in isometric tension with sarcomere length in vertebrate muscle fibres. Journal of Physiology 184, 170-192. Grieve, D.W., Pheasant, S. and Cavanagh, P.R. (1978) Prediction of gastrocnemius length from knee and ankle joint posture. In: eds Asmussen, E. and Jorgensen, K. Biomechanics VI-A, International Series on Biomechanics, Vol. 2A. University Park Press, Baltimore. pp. 405-412. Griffiths, R.I. (1991) Shortening of muscle fibres during stretch of the active cat medial gastrocnemius muscle: the role of tendon compliance. Journal of Physiology 436, 219-236. Herzog, W. and Leonard, T.R. (1997) Depression of cat soleus forces following isokinetic shortening. Journal of Biomechanics 30, 865-872. Heslinga, J.W. and Huijing, P.A. (1990) Effects of growth on architecture and functional characteristics of adult rat gastrocnemius muscle. Journal of Morphology 206, 119-132. Hill, A.V. (1938) The heat of shortening and the dynamic constants of muscle. Proceedings of the Royal Society, Series B 126, 136-195. Hof, A., Geelen, B.A. and van den Berg, J.W. (1983) Calf muscle moment, work and efficiency in level walking; role of series elasticity. Journal of Biomechanics 16, 523-537. Holewijn, M., Plantinga, P., Woittiez, R.D. and Huijing, P.A. (1984) The number of sarcomeres and architecture of the m. gastrocnemius of the rat. Acta Morphologica Neerlando-Scandinavica 22, 257-263. Huijing, P.A. (1995) parameter interdependence and success of skeletal muscle modelling. Human Movement Science 14, 443-486. Huijing, P.A. and Woittiez, R.D. (1984) The effect of architecture on skeletal muscle performance: A simple planimetric model. Netherlands Journal of Zoology 34, 21-32. Huijing, P.A., van Lookeren Campagne, A.A.H. and Koper, J.F. (1989) Muscle architecture and fibre characteristics of rat gastrocnemius and semimembranosus muscles during isometric contractions. Acta Anatomica 135, 46-52. Huijing, P.A., Nieberg, S.M., van de Veen, E.A., and Ettema, G.J.C. (1994) A comparison of rat extensor digitorum longus and gastrocnemius medialis muscle architecture and lengthforce characteristics. Acta Anatomica 149, 111-120.

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Huxley, A.F. (1957) Muscle structure and theories of contraction. Progresss in Biophysics and Biophysical Chemistry 7, 255-318. Latash, M.L. (1993) Control of Human Movement. Human Kinetics Publishers, Champaign, Illinois. Lewis, D.M., Luck, J.C. and Skott, S. (1972) A comparison of isometric contractions of the whole muscle with those of motor units in a fast-twitch muscle of the cat. Experimental Neurology 37, 68-85. Lieber, R.L. and J.L. Boakes (1988) Sarcomere length and joint kinematics during torque production in frog hindlimb. American Journal of Physiology 254, C759-C768. Mai, M.T. and R.L. Lieber (1990) A model of semitendinosus muscle sarcomere length, knee and hip joint interaction in the frog hindlimb. Journal of Biomechanics 23, 271-279. Meijer, K., Grootenboer, H.J., Koopman, H.J.F.M., van der Linden, B.J.J.J. and Huijing, P.A. (1998) A Hill type model of rat medial gastrocnemius muscle that accounts for shortening history effects. Journal of Biomechanics (in press). Morgan, D.L. (1977) Separation of active and passive components of short-range stiffness of muscle. American Journal of Physiology 232, C45-C49. Morgan, D.L. (1990) New insights in the behavior of muscle during active lengthening. Biophysics Journal 57, 209-221. Otten, E. (1988) Concepts and models of functional architecture in skeletal muscle. Exercise and Sport Sciences Reviews 16, 89-137. Otten, E. (1991) Modelling movement control. In: eds Jacobs, R. and Rikkert, W.E.I. Movement Control. An Interdisciplinary Forum. VU University Press, Amsterdam. pp. 6984. Smidt, G.L. (1973) Biomechanical analysis of knee flexion and extension. Journal of Biomechanics 6, 79-92. Spoor, C.W., J.L. van Leeuwen, C.G.M. Meskers, A.F. Titulaer, and A. Huson (1990) Estimation of instantaneous moment arms of lower-leg muscles. Journal of Biomechanics 23, 1247-1259.

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Stephens, J.A., Reinking, R.M. and Stuart, D.G. (1975) The motor units of cat medial gastrocnemius: Electrical and mechanical properties as a function of muscle length. Journal of Morphology 146, 495-512. van Eijden, T.M.G.J. and Raadsheer, M.C. (1992) Heterogeneity of fiber and sarcomere length in the human masseter muscle. Anatomical Record 232, 78-84. van der Linden, B.J.J.J., Meijer, K., Huijing, P.A., Koopman, H.J.F.M. and Grootenboer, H.J. (1998) A reducing cross-bridge model explains experimentally observed contraction history effects. Biological Cybernetics, submitted. van Ingen Schenau, G.J., van Soest, A.J., Gabreëls, F.J.M. and Horstink, M.W.I.M. (1995) The control of multi-joint movements relies on detailed internal representations. Human Movement Science 14, 511-538. van Leeuwen, J.L. and Spoor, C.W. (1992) Modelling mechanically stable muscle architectures. Philosophical Transactions of the Royal Society, London B 336, 275-292. Visser, J.J., Hoogkamer, J.E., Bobbert, M.F. and Huijing, P.A. (1990) Length and moment arm of human leg muscles as a function of knee and hip-joint angles. European Journal of applied Physiology 61, 453-460. Winters, J.M. (1990) Hill-based muscle models: a systems engineering perspective. in: eds Winters, J.M. and Woo, S.L.-Y. Multiple Muscle Systems: Biomechanics and Movement Organization. Springer-Verlag, New York. pp. 69-93. Winters, J.M. (1995) How detailed should muscle models be to understand multi-joint movement coordination? Human Movement Science 14, 401-442. Winters, J.M. and Woo, S.L-Y. (1990) Multiple muscle systems: biomechanics and movement organization. New York, NY, Springer-Verlag. Woittiez, R.D., Huijing, P.A., and Boom, H.B.K. (1984) A three-dimensional muscle model: A quantified relation between form and function of skeletal muscles. Journal of Morphology 182, 95-113. Zahalak, G.I. (1990) Modeling muscle mechanics (and energetics). In: eds Winters, J.M. and Woo, S.L.-Y. Multiple Muscle Systems: Biomechanics and Movement Organization. New York, Springer-Verlag. pp 1 - 23.

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Zuurbier, C.J. and Huijing, P.A. (1992) Influence of muscle geometry on shortening speed of fibre, aponeurosis and muscle. J. Biomechanics 25, 1017-1026. Zuurbier, C.J. and Huijing, P.A. (1993) Changes in geometry of actively shortening unipennate rat gastrocnemius muscle. J. Morphol. 218, 167-180. Zuurbier, C.J., Heslinga, J.W., Lee-de Groot, M.B.E. and van der Laarse, W.J. (1995) Mean sarcomere length-force relationship of rat muscle fibre bundles. J. Biomechanics 28, 83-87.

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A la β

lf α

lm

Fp

B Fm

I

d Ff

½l a

C

Figure 1

A. Planimetric model of a unipennate muscle. α and β are angles between fibre (lf) and line of pull (free tendon) and aponeurosis (la) and line of pull, respectively. B. Free body diagram of the aponeurosis. Fp is the summation of pressure forces along the aponeurosis, Ff the summed fibre forces, and Fm muscle force along the free tendon. Fp does not apply in the middle of the aponeurosis, as Ff does (distance d is not equal 0.5la), indicating a pressure gradient. C. Qualitative presentation of pressure gradient along the aponeuroses.

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A Force (Ffo-1)

1.0 0.8 0.6 0.4 Ff

0.2

Fm

0

Angle (deg.)

100 80

0

B

60

α+β β

40 20 0

0

∆lf Figure 2

A. length-tension diagram of fibre and muscle in a unipennate muscle. The area under the curve (i.e. tension-length integral or virtual work) for muscle and fibre are equal for any shortening distance (shaded areas). A shift in optimum length from fibre to muscle is indicated, i.e. at muscle optimum length, the fibres do act slightly beyond their own optimum length. B. Pennation angles as a function of fibre length. Diamonds indicate the shape of the muscle.

– 22 –

A

Model I

lo Model II

Force

lo Model III

tendon sarcomere

lo

B

force

Muscle Length

Experiment III sd=16% Uniform I sd=24% I sd=16%

Figure 3

length

A. Left hand side diagrams: Fibre length-force curves of five fibre groups for models I, II, and III. Fibre forces are plotted against muscle length. Right hand side diagrams: schematic representation of three fibres at three different muscle lengths relative to optimum length (lo). B. Normalised length-tension curves of uniform and fibre-distributed pennate muscles, compared with experimental data. (Adapted from Ettema and Huijing, 1994b.)

– 23 –

1

i

2 n

ii

1 2 n

N

1 2 n

SEEt Figure 4

SEEmyo

CE

SEEcb

Model of the series elastic element (SEE) of skeletal muscle, and its relationship with the contractile element (CE). Each component of CE acts as an independent force generator, acting on a single SEEcb component, a single SEEmyo component and on SEEt.

– 24 –

as ce

nd

ing

li m

b

Force

B

A

External Torque

C Flexor

Torque

Torque

Length

Extensor

Extension

Flexion

Elbow Angle Figure 5

D Flexor

Extensor

Extension

Flexion

Elbow Angle

Spring model of antagonists at the elbow. A. Length-tension curve of the muscle, indicating active, passive, and total force. B. Biceps Brachii (flexor) and Triceps Brachii (extensor) represented as springs, acting on the ascending limb of the length-tension curve. C&D. Torque-joint angle diagrams (simplified conversion from length-tension curves) of the two muscles. The equilibrium point is where both torques are equal but opposite in direction (not indicated), determining the joint position. C: Fat, vertical arrow indicates an external torque perturbation (see diagram B), resulting in a joint movement indicated by the thin horizontal arrows. The system with low stiffness (grey lines) encounters a larger movement than the stiff system (black lines). D. Fat arrow indicates an intended movement, which requires a change in stiffness (by muscle activation) of at least one muscle. The stiff system (grey lines) requires a larger activity change than the more compliant system. See text for further explanation. – 25 –

Mj κ Fm +

0

Sj Sj κ = c 50

75

100

125

150

175

Joint angle (degrees) Figure 6

Mechanical properties of a muscle lever system, shown in the inset. With increasing joint angle, muscle force increases, i.e. muscle stiffness is positive. The joint moment shows an inverted U-shape, The bottom diagram shows the correctly calculated joint stiffness (eq. 11a) and joint stiffness, assuming constant moment arm (eq. 11b). Vertical dotted line indicates the transition from positive to negative joint stiffness.

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