Rebound Resilience of Tendons in the Feet of Sheep (Ovis Aries)

A running mammal decelerates and accelerates in each step and its centre of mass falls and rises (Alexander, 1977). Both the kinetic energy and the potential energy of a biped are highest between steps, when neither foot is on the ground. They are lowest in the middle of a step, as the centre of mass passes over the supporting foot. Energy which would otherwise be lost as heat can be stored at this stage as elastic strain energy, and re-converted to kinetic and potential energy in an elastic recoil. The same principle applies to a bouncing ball, which stores elastic strain energy briefly when it hits the ground. It also applies to a quadruped, which resembles two bipeds running in tandem.

There is strong evidence that in running men, about half of the power which would otherwise be needed is saved by elastic storage (Cavagna, Saibene & Margaria, 1964). There is similar evidence for kangaroos (Alexander & Vernon, 1975) and it seems reasonable to assume the importance of elastic storage in mammals in general. Rough calculations indicate that the strain energy must be stored predominantly in tendons : very little of it can be stored in the contractile tissue of the muscles (Alexander & Bennet-Clark, 1977).

Suppose an elastic material is deformed and allowed to recoil. The rebound resilience of the material is the energy recoverable in the recoil, expressed as a fraction of the work done in the deformation. The efficacy of a material for elastic storage depends on its having a reasonably high rebound resilience. We have devised a simple experiment to measure the rebound resilience of a group of tendons which are believed to be important in locomotion. They are the tendons of the interosseous muscle of the fore-limb of the sheep, Ovis aries. The functions of the corresponding tendons in horses were discussed by Camp & Smith (1942).

The interosseous muscle runs along the posterior face of the fore cannon bone (the fused metacarpals) and is a flexor of the metacarpo-phalangeal joints (Fig. 1). It originates on the proximal part of the cannon bone. It has four tendons of insertion each of which bifurcates; one branch of each attaches through a sesamoid to one of the proximal phalanges while the other crosses to the anterior face of the foot and merges with the digital extensor tendon.

The muscle is enclosed in a sheath of tough connective tissue which also encloses the tendons of the digital flexor muscles. This sheath is attached proximally to the carpals and to the cannon bone. Thickenings in its medial and lateral faces merge distally with the superficial flexor tendon.

The interosseous muscle is multipennate, with muscle fibres only about 1 mm long. The volume of the contractile tissue is so small compared to that of the tendons that it is easily overlooked, and the tendons are often referred to as the suspensory ligament. Since the muscle fibres are so short their contraction can cause little movement : the muscle and its tendons must function essentially as a ligament. For instance, 30% shortening of the muscle fibres would bend the metacarpo-phalangeal joint through only 2°.

When a fore foot is on the ground in locomotion the metacarpo-phalangeal joints are hyperextended. The force exerted by the ground on the foot exerts a moment about these joints which is presumably balanced partly by the interosseous muscle, and partly by the digital flexor muscles.

The fore foot of a freshly killed sheep was amputated by sawing through the radius and ulna near their distal ends. The foot was then skinned from its proximal end to about 4 cm distal to the carpals. The digital flexor muscles and their tendons were removed from the skinned portion. A hole of diameter 3 mm was bored transversely through the exposed part of the cannon bone. After skinning and throughout the experiment the skinned part of the foot was kept moist by applying pads of cotton wool soaked in 0·94% aqueous sodium chloride.

The soft tissues between the digits were cut and the foot was clamped in the holder shown in Fig. 2. The posterior surface of the foot is uppermost and the bolt passes between the digits. The holder was fixed firmly by large G-cramps (not illustrated) to a concrete bench. A nail was passed through the hole previously bored in the cannon bone and a mass, which was changed as required, was suspended from it by a chain. A jubilee clip (as used for fastening hoses in automobile engines) was tightened round the bone and interosseous tendon immediately proximal to the nail. A cloth pad moistened with saline was placed under it, to protect the tendon. The clip was used to hold a light pointer which wrote on a smoked-drum kymograph.

Any vertical displacement of the mass was necessarily accompanied by a change in length of the interosseous muscle and/or its tendons. The elastic stiffness of the muscle and tendons interacted with the mass so that the system had a natural frequency of vibration which was 5−7 Hz in most of our experiments. With the kymograph drum revolving, the mass was given a small vertical displacement and released. The train of damped oscillations which followed was recorded on the drum. Particular care was taken to ensure that the pointer pressed only very lightly against the kymograph drum, so we are confident that energy losses through friction on the drum were very small compared to the energy content of the oscillations. Several records were made with each mass for each specimen. They were subsequently measured to determine rebound resilience.

Let the amplitudes of successive half cycles of the oscillation be x₂, x₃,…, x_n. The rebound resilience is then (x_n+1/x_n)² (see, for instance, Alexander, 1968). If it is independent of amplitude, a graph of log x against n will be a straight line : we found that this was approximately the case. The gradient of such a graph is the logarithm of the square root of the rebound resilience.

The natural frequency of a spring-mass system which is not too heavily damped is approximately where m is the mass and S is the stiffness (i.e. the force required to effect unit displacement of the mass; see, for instance, Alexander, 1968). Our records gave frequencies and the applied masses were known, so stiffness could be calculated. The mass used in the calculation was the suspended mass plus the mass of the chain, and amounted to 1·94, 5·02 or 10·56 kg. The mass of the preparation was about 0·16 kg, and was ignored.

The stiffness so calculated was converted to torsional stiffness (i.e. to the moment required to bend the joint through unit angle) by multiplying by the square of the distance of the nail from the axis of the metacarpo-phalangeal joint. This distance was measured from X-radiographs.

The stiffness calculated from the natural frequency was a dynamic stiffness. The static torsional stiffness was also determined for a few specimens, from observations of the displacement of the pointer caused by hanging weights from the nail.

The static stiffness of the apparatus was investigated by setting it up with a pair of stout brass bars in place of the specimen. Weights were suspended from the bars at the same distance from the holders as in the experiments with feet. It was found that the static torsional stiffness was 300 N m rad⁻¹. The static and dynamic torsional stiffnesses of our preparations were always less than (and usually much less than) 50 N m rad⁻¹. The elastic compliance of the equipment cannot have affected our results appreciably.

The masses were suspended on a chain 0·35 m long so the natural frequency of mass and chain, swinging as a pendulum, was about 0·8 Hz. This is much lower than the frequencies of 5−7 Hz at which the feet vibrated so it is unlikely that our results were affected appreciably by transfer of energy between the modes of vibration. Nevertheless we took care to displace the mass vertically at the start of each train of oscillations, trying to avoid setting it swinging as a pendulum.

Our principal experiments, of which the results are shown in Table 1, were performed precisely as described above. Four additional experiments were performed in which the same foot was tested both at 37 and 20 °C. For these experiments we used feet amputated at the carpo-metacarpal joint, obtained from a slaughterhouse and stored at −20 °C for up to 15 days before use. These experiments were performed on a wooden bench which was a little less rigid than the concrete bench used for our principal experiments.

Temperatures were changed as required, by heating the laboratory.

Table 1 shows the results of our principal experiments. The mean of all the values of rebound resilience recorded in it is 0·62.

The experiments in which the same feet were tested both at 37 °C and at 20 °C failed to demonstrate any effect of temperature on rebound resilience. We used a 5 kg mass and found that the rebound resilience was 0·52−0·71 (mean 0·64) at 37 °C, and 0·65−0·69 (mean 0·67) at 20 °C.

Table 1 and Fig. 3 show that both the dynamic and the static stiffness increase as the suspended mass is increased. Our values of static stiffness are uniformly lower than the corresponding values of dynamic stiffness for the same specimen. The experiments at 37 °C and 20 °C showed no significant effect of temperature on stiffness.

We have to consider to what extent the properties which have been observed are properties of the tendons, and to what extent they may be properties of the muscle fibres. The loads which were applied in the experiments extended the metacarpophalangeal joints through 0·2− 0·5 rad (Fig. 3 a). We estimate from X-radiographs and by manipulating dissections that the moment arms about these joints of the tendons of the interosseous muscle are about 10 mm, so 0·5 rad extension of the joint requires 5 mm extension of the muscle. It seems inconceivable that the muscle fibres, which are only about 1 mm long, could stretch by more than a small fraction of this : most of the observed extension must be due to extension of the tendons.

Another rough calculation supports this view. The area of tendon in any crosssection of the muscle is about 20 mm² and the length of the muscle is about 10 cm. Hence if only the tendons stretch and if Young’s modulus for tendon is 10⁹ N m⁻² (Matthews & Ellis, 1968), the tensile stiffness of the muscle is about 2 × 10⁶ N m⁻¹. If the moment arm about the joint is 10 mm, the torsional stiffness of the joint should be about 20 N m rad⁻¹, or a little more if the sheath contributes appreciably to the stiffness. This is in the range of observed values (Fig. 3b).

Consider next whether the moments and frequencies involved in the experiments are appropriate to experiments designed to throw light on locomotion. Jayes & Alexander (1978) recorded forces exerted by a 45 kg sheep crossing a force platform. It can be shown from their records and films that in a typical example of trotting at 1·4 m s⁻¹, the force on a fore foot exerted moments up to 12 N m about the metacarpophalangeal joints. In an example of cantering at 2·4 m s⁻¹, moments up to 23 N m occurred. Most of the experiments reported in Table 1 were on legs from young sheep of mass about 20 kg so the moments which would have acted about their metacarpophalangeal joints in similar gaits would have been much smaller, probably 4−5 N m and 8−10 N m. These moments would presumably have been counteracted partly by the digital flexor muscles and only partly by the interosseous muscle and its sheath. The moments applied in the experiments ranged from 1·3 to 7 N m, and had to be counteracted by the interosseous muscle and sheath alone. They were probably similar in magnitude to the moments which the muscle and sheath have to counteract in locomotion.

Jayes & Alexander (1978) found that the duration of contact with the ground of the forefoot of their sheep ranged from 0·6 s in a slow walk (at 0·7 m s⁻¹) to 0·2 s in a slow gallop (at 3·4 m s⁻¹). It would presumably have been slightly less for smaller sheep in similar gaits (see Pennycuick, 1975). Jayes & Alexander made no observations of sheep galloping fast. Alexander, Langman & Jayes (1977) observed contact durations down to 0·05 s for 20 kg gazelles galloping fast but we do not suppose sheep could achieve this. The interosseous muscle is stretched and recoils once while the foot is on the ground so the duration of contact corresponds to half a cycle of oscillation in our experiments. The frequencies of oscillation ranged from 4·6 to 7·2 Hz so the duration of a half cycle was 0·07−0·11 s. This is possibly a little too short to be entirely realistic as simulation of locomotion.

The experiments at elevated temperatures were carried out because the temperature of the muscle in living sheep is presumably well above ordinary room temperatures. A sheep kept indoors on wire mesh at an ambient temperature of 9 °C was found to have an interdigital temperature of 37 °C. However, differences of temperature in the range 20−37 °C had no significant effect on the results of our experiments.

The rebound resiliences of about 0·62 which have been measured in experiments are far lower than the values for resilin, abductin and feather shafts (over 0·9 in each case, Jensen & Weis-Fogh, 1962; Alexander, 1966; Pennycuick & Lock, 1976). They are nevertheless high enough, if they apply to tendons in general, to make substantial savings of energy possible in locomotion. We are not aware of any previous determinations of the rebound resilience of tendons.

A weakness of this investigation is that the effect of the muscle fibres on our results is unknown, though it is thought likely to be small since the fibres are so short. Further experiments should be performed with isolated tendons and might yield higher values of rebound resilience. A strength of the investigation is that it has been possible to make the measurements with the metacarpo-phalangeal joints and the skin covering them intact: any energy losses caused in life by joint friction, skin viscosity etc. will have occurred also in our experiments.

We are grateful to Mr K. G. Towers for supplying legs and Dr C. G. Scanes for help in measuring interdigital temperature. This work is part of a programme of research supported by a grant to R.McN.A. from the Science Research Council.