Passive elements in series

Morphologically myofilaments, Z-disks, tendons and aponeurosis (i.e. intramuscular tendinous sheets) have been identified as passive elements connected in series with contractile apparatus. In addition, the contractile apparatus itself, the cross bridges, have elastic properties and their compliance makes up part of the total series compliance of the muscle-tendon complex. Because of this, although the cross bridges are ‘active’ elements, their elastic

properties are briefly discussed in this section. Originally, observations that muscle shortening or elongation is the result of sliding of the myofilaments with respect to each other and not due to changes in their lengths (Huxley and Hanson, 1954; Huxley and Niedergerke, 1954), led to the conclusion that myofilaments were inextensible. More recent evidence however suggests that this is not true and that myofilament compliance does make up a large part of the total sarcomere compliance (e.g. Linari et al, 1998; Herzog, 2000a). However, this compliance is usually assumed to be very small compared to the compliance of the tendon and its aponeurosis. For example, using the values from Linari et al (1988) obtained from by applying step length changes in isolated frog fibres contracting at the plateau of their force-length relation, the approximate compliance of the thin and thick filament and of the cross bridge was estimated to be ~2, ~1 and ~1 nm pm'^ Tq’^ (units are normalised to half sarcomere length and to the maximal isometric force (To)). Adding all these values together makes up a total half sarcomere compliance of ~4 nm pm'^ To'\ The optimal half sarcomere length for tension generation is approximately 1 pm in frog fibres, and therefore at that length under isometric conditions cross bridge and total sarcomere compliance would be ~1 and ~4 nm To'\ respectively. Extrapolating this result to human sarcomeres, which have longer actin filaments than frog ones, predicts that the equivalent values would be approximately 30% larger (see Rassier et al, 1999) than those for frog half sarcomeres i.e. -1.3 and -5.2 nmTo’^ for the cross bridge and total half sarcomere compliance. The compliance of a tendinous structure would depend on its Young’s modulus and dimensions. Taking the human tibialis anterior as an example (e.g. Maganaris and Paul, 1999), tendon strained at -2.5% by its

muscle’s maximal isometric force. For such a tendon, the compliance of 1.3 pm (i.e. length approximately equal to one thin filament) length of tendon would be 0.025 1.3 pm = 32.5 nm at its maximal isometric force. This compliance is more than twenty five times greater than that in the cross bridge and approximately six times greater than the total sarcomeric compliance. Of course the results of these calculations are very rough approximations and the ratios of sarcomeric to tendinous compliance may vary within the same and between different muscle-tendon complexes as a result of different contraction conditions (such as for example differences in filament overlap and activation levels) and also due to dimensional and structural differences. Interestingly, it has recently been shown using ultrasonography that when the human ankle joint is moved passively the tibialis anterior and gastrocnemious tendons take up more than 50% of the total MTC length change, even though the forces experienced by the MTC are low (Herbert et al, 2002).

The force (load)-extension tendon properties -or stress-strain properties, when force and extension are normalised for the tendon’s cross sectional area and unloaded length respectively - have been investigated both in vitro (e.g. Rigby

et al, 1959; Diamant et al, 1972; Ker, 1981; Ker et al, 1988) and in vivo (e.g. Maganaris and Paul, 1999; 2000a,b; Kubo et al, 2000). The load-extension behaviour of a tendon changes irreversibly at strains exceeding a certain limit (Rigby et al, 1959) which is approximately 4-5 % (see Herzog and Gall, 1999) or when a tendon is subjected to prolonged static or repetitive cyclic loading (Wang and Kerr, 1995; Wang et al, 1995). At loads and strains lower than those corresponding to the tendons’ elastic limits, tendons behave in a

reproducible manner. The tendon stiffness (or Young’s modulus when the stiffness is normalised for the tendon cross sectional area and initial length) shows a gradual increase with tendon elongation at strains up to 2-4% (Rigby

et al, 1959; Ker, 1981; Proske and Morgan, 1987). This region of the load- elongation curve is called the ‘toe’ region and has been associated with straightening of the waviness of the collagen that is observed when the tendon is slack (Rigby et al, 1959; Diamant et al, 1972). At even greater strains ranging from 2-3% up to 5-6% however, tendon stiffiiess remains almost constant (Rigby et al, 1959; Ker, 1981; Proske and Morgan, 1987). This region of the tendon load-elongation relationship is referred to as the linear region and it reflects the elastic properties of straightened collagen fibres.

The area under the load-elongation curve for a tendon represents the energy expended to stretch the tendon. When a tendon is allowed to recoil after elongation back to its original resting length, the force (stress) for a given value of elongation from its original length (strain) is less compared to when the tendon is being stretched. The energy recovered from the tendon, i.e. the area under the load-elongation curve during recoiling, is therefore less than the energy expended to stretch it (e.g. Ker, 1981; Lieber et al, 1991). However, the difference is very small and it has been estimated that 89-94% of the energy expended to stretch a tendon is recovered during recoiling (see Herzog and Gall, 1999).

In the past, intramuscular tendon (or aponeurosis) has been incorporated in muscle models as being inextensible (e.g. Woittiez et al, 1984). However it is

now known that this is not the case and that aponeurosis have elastic properties (e.g. Lieber et al^ 1991; Zuurbier et al, 1994; Kawakami and Lieber, 2000; Maganaris and Paul, 2000a,b). Interestingly, some studies have shown that aponeurosis undergoes a greater elongation compared to the tendon for the same level of force and therefore being more compliant than the tendon (Lieber et al, 1991;Maganaris and Paul, 2000a,b). Extension non-uniformities along the length of the aponeurosis under loading conditions have also been reported, indicating regional differences in aponeurosis compliance and/or non-uniform force distribution along its length (Zuurbier et al, 1994; Maganaris and Paul, 2000a,b).

Tendon structure appears to be highly adaptable depending on functional demands (Ker, 1999). Although mechanical tests on mammalian tendon properties have not revealed systematic differences between species or anatomical sites their dimensions can vary considerably depending on the function of the muscle-tendon complex (see Alexander and Ker, 1990). Ker et al (1988) measured the mass and length of mammalian muscles and tendons obtained from a wide range of species and from these measurements they estimated their cross sectional areas and the stress in the tendon while subjected to the maximal isometric force of the muscle. They found a wide continuum of stresses ranging between 2-105 MPa, having a skewed frequency distribution with the most commonly occurring value being approximately 13 MPa. This stress is very low compared to the maximal stress a tendon can withstand before it is ruptured (>100 MPa; see Ker, 1999). They suggested a theory according to which tendons are adapted so that the total mass of muscle

and tendon is minimised; they calculated the minimum of this sum to occur when the ratio of muscle-to-tendon cross sectional area is equal to 34. This ratio is found to vary widely but the value of 34 appears to be the most commonly encountered value (see Ker, 1999). This theory applies for muscle- tendon complexes doing positive work and tendons that do not act as energy- saving springs during locomotion. Tendon adaptability has been examined in relation to the function and location of the muscle-tendon complex (Alexander and Ker, 1990). Alexander and Ker (1990) classified muscle-tendon complexes in the limbs of a wide range of mammalian species into three main categories. Type I have long fascicles and relatively short tendons. The muscles have a large volume and are therefore capable of generating and absorbing/degrading large amounts of work. Such muscles are located in the proximal segments of limbs such that the moment of inertia of the limb does not increase as much as if they were located more distally. In addition this location is advantageous during explosive movements such as, for example, jumping where large moments around more proximal limb joints such as the hip have to be generated. Type II muscle-tendon complexes have tendons that are long relatively to the muscle fascicles and are also relatively thick. The long tendons allow muscles located more proximaly (thus keeping limb inertia relatively low) to act on remote joints located distally, such as for example the finger joints moved by muscles located in the forearm. Such muscle-tendon complexes are commonly involved in performance of precise movements. In order for muscle fascicle shortening to be precisely transmitted to the joint, the compliance of the tendon has to be low. As these tendons are long, they are also relatively thick to minimise compliance. Type III muscle-tendon

complexes have tendons that are very long relative to the muscle fascicles and also relatively thin so they become highly stressed. Type III muscle-tendon complexes are involved in locomotion and are designed so that metabolic cost of locomotion is kept relatively low. As locomotion involves stretch- shortening cycles, in the absence of series compliance the length changes would have to be taken up by muscles. In the absence of tendons, joint movements would have to be completely taken up by muscle fascicles length changes. If a compliant tendon is introduced between the muscle and the limb, joint movements are taken up by combined length changes of muscle and tendon and as a result smaller length changes would be required by the muscle fascicles. Alexander and Ker (1990) argue that if the increased metabolic energy expended by the muscle while doing work was assumed to be exactly balanced by the reduction in energy expended while absorbing work, so the energy spent during a stride would be proportional to the time integral of the force, a long and/or fast muscle would spend more metabolic energy than a short and slow one to maintain the same force output. Thus, the length changes of the tendon during locomtion reduces the metabolic cost by allowing for shorter and slower muscle fibres. Again, in order not to add unnecessarily to the inertia of the limb, the muscle is located more proximally than the tendon. The tendons are long, but since a relatively large compliance is required, the tendons are also slender. This results in these tendons becoming highly stressed and hence strained during locomotion. An example of such a muscle- tendon complex is the triceps surae acting as an ankle plantarflexor during running.