J. Zool., Lond. (1988) 216, 309-324

Why are mammalian tendons so thick?

R. F. KER, R. McN. ALEXANDER AND M. B. BENNETT Department of Pure and Applied Biology, University of Leeds, Leeds, LS2 9JT, UK

(Accepted 25 January 1988)

(With 4 figures in the text)

The maximum stresses to which a wide range of mammalian limb tendons could be subjected in life were estimated by considering the relative cross-sectional areas of each tendon and of the fibres of its muscle. These cross-sectional areas were derived from mass and length measurements on tendons and muscles assuming published values for the respective densities. The majority of the stresses are low. The distribution has a broad peak with maximum frequency at a stress of about 13 MPa, whereas the fracture stress for tendon in tension is about I00 MPa. Thus, the majority of tendons are far thicker than is necessary for adequate strength. Much higher stresses are found among those tendons which act as springs to store energy during locomotion. The acceptability of low safety factors in these tendons has been explained previously (Alexander, 1981). A new theory explains the thickness of the majority of tendons. The muscle with its tendon is considered as a combined system which delivers mechanical energy: the thickness of the tendon is optimized by minimizing the combined mass. A thinner tendon would stretch more. To take up this stretch, the muscle would require longer muscle fibres, which would increase the combined mass. The predicted maximum stress in a tendon of optimum thickness is about 10 MPa, which is within the main peak of the observed stress distribution. Individual variations from this value are to be expected and can be understood in terms of the functions of the various muscles.

Contents Page

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 12 Results and comments . . . . . . . . . . Theory . . . . . . . . . . . . . . . . Discussion of the theory . . . . . . . . . .

Factors other than the minimization of mass Muscles with very short fibres . . . . . . Other muscles . . . . . . . . . . . .

Conclusion . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . 313

. . . . . . . . . . . . . . . . . . 317

. . . . . . . . . . . . . . . . . . 320

. . . . . . . . . . . . . . . . . . 321

. . . . . . . . . . . . . . . . . . 321

. . . . . . . . . . . . . . . . . . 322

. . . . . . . . . . . . . . . . . . 323

. . . . . . . . . . . . . . . . . . 323

Symbols

The following symbols are used: a cross-sectional area of tendon e extension of tendon f tension in tendon I length of tendon r area ratio (muscle to tendon: A / a )

309 0952-8369/88/010309+ 16 $0340 0 1988 The Zoological Society of London

310

A D E F L r: M R &

e P 0

R. F . KER, R. McN. ALEXANDER A N D M . B . BENNETT

total cross-sectional area of the fibres of a muscle overall length of a muscle and its tendon tangent modulus of tendon force developed by a muscle length of the fibres of a muscle fibre length factor: the ratio of the length of a muscle's fibres to the extension of its tendon (p. 322). mass of a muscle range of lengths over which a muscle fibre operates strain angle of pennation (Fig. 1) density stress

subscripts: m refers to muscle t refers to tendon s 1, & in, are the length and mass, respectively, of a se-xted portion of tendon i Li, Mi and Ri are values which would apply to a muscle if its tendon were inextensible opt refers to optimum values (Theory): i.e. those giving minimum mass 0 c0 is the intercept on a strain axis (Fig. 3).

Introduction

Some of the tendons of mammals have low safety factors (Alexander, 1981): i.e. the stresses in strenuous activities may be only a little less than the fracture stress. Table I lists values of stresses or strains in life for a selection of leg tendons. The fracture stress of tendon is about 100 MPa (Bennett, Ker, Dimery & Alexander, 1986) and the corresponding strain is about 8%. Certain tendons may well be somewhat stronger; none the less, some of the values in Table I are clearly high. The tendons of Table I are a biased sample for they all function as springs to save energy in locomotion (Alexander, 1984). A pre-requisite for this function is that the tendon is under tension when the foot is on the ground: this is also a pre-requisite for the method of estimating stresses used by the authors referred to in Table I. For functionally less specialized tendons, the tensions achieved in life have been assessed in very few cases. Our purpose is to enquire as to whether low safety factors are typical of mammalian limb tendons in general.

One of these few cases for which data is available from direct measurement of load is the human flexor pollicis longus. Rack & Ross (1984) give a tendon stress of 15 MPa for a strenuous activity for the flexor pollicis longus muscle. The loading conditions of Rack & Ross' experiment can be imitated by hanging a weight of about 70 N from near the tip of the thumb (palmer surface of the thumb upwards) with the interphalangeal joint at about 150". Although Rack & Ross were not aiming at an absolutely maximal load, imitating the load in this way makes it seem most unlikely that the stress in the tendon could be increased by a factor of 3 or 4 times, as would be implied if the safety factor were low. The tendon is too thick and the muscle too weak for a high tendon stress to be reached.

Tendons are in series with muscles. The peak tension in a tendon is limited by the maximum load its muscle can develop. Figure 1 is a diagrammatic view of a muscle-tendon combination. The arrangement shown is unipennate, but the same principles apply to other muscle geometries. The

THICKNESS O F TENDONS

TABLE I Strains or stresses in lower limb tendons measured in vivo

31 1

Activity Stress

Tendon Strain (MPa) Reference

Horse galloping, moderate speed Horse galloping Wallaby hopping, 2.4 m s-'

Dog jumping, 1.3 m scale jump Camel pacing, 6.2 m s-' Deer galloping, fast

Donkey trotting

Human running, 4.5 m s-'

Forefoot toe flexor Toe flexors Ankle extensors

Ankle extensors Ankle extensors Toe and wrist flexors and ankle extensors Forefoot toe flexors Hind foot toe flexors Achilles

10% 4-9 Yo

15-41

84 18

28-74

33-44 28-37

53

Herrick et al. (1978) Dimery, Alexander & Ker (1986) Alexander & Vernon (1975)* Ker, Dimery &Alexander (1 986) Alexander (1974)* Alexander et al. (1982) Dimery, Ker &Alexander (1986)

Alexander & Dimery (1985) Dimery & Alexander (1985) Ker et al. (1987)

* The values from these papers have been adjusted to allow for a different assumed value of tendon density (see Ker, Dimery & Alexander, 1986).

- Muscle showing direction of fibres

- Aponeurosis over muscle belly

- I

tendon - External

FIG. I . A muscle-tendon combination. The effective length of the tendon is (D-L) , where D is the overall length from origin to insertion and L is the length of the muscle fibres. This assumes cos 0 to be near unity, where 0 is the angle of pennation.

- i

angle of pennation, 0, is the angle between the muscle fibres and the aponeurosis. If F is the total tension developed by the muscle fibres andfis the tension in tendon,f=Fcos 8. For mammalian muscle, 0 < 30" and often 0 < 20" (see Alexander, 1974, for dogs and Alexander & Vernon, 1975,

312 R. F. KER, R. McN. ALEXANDER AND M. B. BENNETT

for wallabies): cos 30" = 0.87. For the precision required here, the factor c o d may be omitted. Then the maximum tension in the tendon is o,A, where om is the maximum stress developed by the muscle fibres and A is their cross-sectional area. The maximum stress in the tendon, ot, is therefore:

where a is the cross-sectional area of the tendon. All mammalian muscles seem to be able to develop about the same maximum stress, in which

case the maximum stress in the tendon is proportional to the area ratio, r=A/a . The maximum isometric stress in muscle fibres (Wells, 1965; Jayes &Alexander, 1982) is about 0.3 MPa. This is the value we will use. However, when an active muscle is being rapidly stretched, higher stresses can be developed, up to about 0.6 MPa (Flitney & Hirst, 1978; Cavagna, Citterio & Jacini, 1981), which should be borne in mind when considering some of the results to be given below.

The values of cross-sectional areas required for equation (1) were measured during dissections. The same method has previously been used by Alexander, Dimery & Ker (1985) to estimate the maximum likely stresses on an aponeurosis of the back of a deer and a dog. (The estimates were 25 and 17 MPa, respectively.) A similar method was used by Elliott & Crawford (1965) with seven muscles from the hind legs of rabbits. They express cross-sectional areas in terms of indices based on dry mass, so water contents and densities of wet materials have to be estimated to convert their values to tendon stresses in MPa. On this basis, their results indicate tendon stresses ranging from 15 to 26 MPa when the muscles are exerting maximum isometric tetanic tension.

Materials and methods

We examined the muscles and tendons of the lower leg of the following mammals using fore- and/or hind limbs. Opossum (Didelphis virginiana, body mass 3.5 kg) Possum (Trichosurus vulpecula, 5 kg) Macaque monkey (Macaca nemestrina 2 specimens, (a) 15 kg and (b) 11 kg) Vervet monkey (Cercopithecus aethiops, 7 kg) Human (Homo sapiens, mass unknown) Dalmatian dog (Canis familiaris, 21 kg) Badger (Meles meles, 10 kg) Cow (Bos taurus, mass unknown) Horse (Equus caballus, 570 kg) The human leg had been amputated because of irreparable vascular disease. For the other specimens, carcasses were made available to us after the mammal had been killed for reasons unconnected with this research. The badger and the possum were victims of road accidents. All specimens were stored at -20 "C until required. This selection of mammals provides a wide range regarding the degree of specialization of the limbs for running.

The cross-sectional areas, A and a, required for equation (I), were found gravimetrically. Using fresh (i.e. unfixed) material, the foliowing measurements were made during dissections: the overall

length, D, from origin to insertion; the mass, m,, of a uniform sample of tendon of length ls; the mass, M , of the muscle belly; the length, L, of the muscle fibres. To measure L, a cut was required through the muscle in the plane of the muscle fibres. The muscle fibres may become distorted and therefore the muscle fibre lengths were also measured on the other limb, when this was available, which had been injected with fixative prior to dissection. Despite this precaution, the value of L has an uncertainty of up to perhaps 25% due to the range of lengths over which muscle fibres operate. This uncertainty could have been overcome by measuring

THICKNESS OF TENDONS 313

sarcomere lengths to establish where each muscle fibre was within its range of operation (Elliott & Crawford, 1965). However, as will be seen below, an uncertainty of 25% is not critical to our conclusions.

The effective length of the tendon (see Fig. 1) is given by:

I = ( D - L) (2)

The cross-sectional area of muscle fibres is given by:

M A = - 1060L (3)

using S.I. units. This assumes the density of muscle to be 1060 kg m-3 (Mendez & Keys, 1960). For most muscles, the fibre length was nearly constant throughout any one muscle head. The humeral head of the deep digital flexor of the forelimb of the cow and of the horse has 2 parts, the fibre length in one part being 10 times those in the other. In these cases, we cut between the 2 parts and weighed them separately to obtain a proper value for the total cross-sectional area.

The cross-sectional area of tendon is given by:

using S.I. units. This assumes the density of tendon to be 1120 kg m-3 (Ker, 1981).

Results and comments

Table I1 gives maximum tendon stresses calculated from the measured area ratios using equation (1) with om = 0.3 MPa. The variability of the stresses is obvious and is emphasized by including two macaque hind limbs which differ markedly. However, the majority of the stresses are very much less than the few higher values. The distribution of values is shown by the histogram (Fig. 2) which includes all the measurements on which Table I1 is based, except for the macaque hind limbs where average values for the two specimens have been used rather than individual values for both. (This avoids bias towards macaque hind limbs in the distribution.) The most common tendon stress in the sample is about 13 MPa, whereas the more extreme stresses are four to eight times greater.

For any tendon, the peak stress in life would be lower if the relevant muscle was never fully used. Such partial use of muscles could result in a more even distribution of stresses in life than appears in Fig. 2. However, Table I, which gives measured stresses, indicates that this is unlikely; high tendon stresses (> 50 MPa) occur in the horse, the dog, the deer and the human-and, no doubt, also in other mammals.

The majority of tendons have a safety factor of about 8. (Even allowing for om to reach 0.6 MPa during the rapid stretching of an active muscle, the safety factor remains about 4.) Such a safety factor is not necessary because, as the muscle is in series with the tendon, an untoward tension cannot be applied: the muscle will give way before the tendon breaks. A tendon can have a lower safety factor than most structural components (Alexander, 1981), as is demonstrated by the existence of such tendons.

Perhaps tendons which are never subject to high stresses are weaker than the others. To investigate this possibility, we carried out a tensile test on a ‘low stress’ tendon, the common digital extensor tendon from an adult cow (ot= 12 MPa), using the methods of Bennett, Ker, Dimery & Alexander (1986). The tendon broke at 80 MPa. This value is typical of the results obtained in strength tests on tendon: higher values are consistently obtained only when special precautions are

314 R. F . KER, R. McN. ALEXANDER A N D M. B . BENNETT

TABLE I1 Maximum stresses in mammalian limb tendons, assuming the maximum stress from a muscle to be 0.3 MPa. The muscles investigated act on the wrist or ankle and include the extrinsic digital muscles. Most of the data are gathered into ‘muscle assemblies’. For example, the assembly of ‘Wrist flexors’ covers the palmaris longus. the flexor carpi radialis, the flexor carpi ulnaris and, for some mammals, ulnaris lateralis. (In others, the latter is a wrist extensor.) Additionafstress vaiues are available when a muscle and its tendon can be divided into separate heads: for example, the flexor carpi ulnaris of the dog. However, stress values are not available for those muscles which lack a uniform portion of tendon external to the muscle belly: for example, theflexor carpi ulnaris and the ulnaris lateralis of several mammals and the soleus of all except the human. Furthermore. in certain mammals, some of the muscles are lacking. In this table. whenever more than two values are available, only the range of the values is given. For example, the stresses among the wrist flexors of the macaque are 12,21

and 22 MPa, respectively; this is given as the range 12-22 MPa

Maximum stresses, ut, in forelimb tendons in MPa

Superficial Deep Other digital digital Digital Wrist wrist

Mammal flexor flexors extensors flexors muscles

Opossum Possum Dog Badger Sheep Horse cow Macaque, A

9 20 24 33

4 1-74 15 69

14-20

22 1 1 21 9

12-35 37,39 15-46 19,20

8-14 3-8 16

8-1 1 15-22 15, 36 8-1 1 6-18

5-14 9-51

27-49 17

11,22 21 8

21,22

7-18 8-15 6, 9

10, 17 12, 14 17,49 6, I I 12-22

Maximum stress, utr in hind limb tendons in MPa

Deep Other digital Digital Gastroc- ankle

Mammal Plantaris flexors extensors nemius* muscles

Horse 105 36 47 25 Sheep 41 29 12-16 49 16,20 Macaque, A 1 1 14, 18 12, 14 30 13-26 Macaque, B 12 6 , s 8, 8 20 11-17 Vervet 8 8 , 9 2 22 4-22 Human* 36 16,23 11, 13 61 14-25

For the human, the value under ‘Gastrocnemius’ refers to the Achilles tendon, which is the common tendon of the gastrocnemius and soleus muscles.

taken to avoid the effects of stress concentrations in the clamps. The result is to be considered as a lower limit for the strength, but even this is many times greater than the maximum stress applied in life. A healthy cow digital extensor tendon cannot break in tension in life, because its muscle cannot exert anywhere near sufficient stress. This conclusion is likely to apply to all the tendons with low stress values in Table 11, say crt < 50 MPa. The more highly stressed exceptions include the digital flexors of ungulates and dogs and the human Achilles tendon. Allowing for the possibility of values of om > 0.3 MPa, a few tendons appear to be at risk. However, these high muscle stresses may not always be achievable because of the need to accommodate the extension of the tendon.

THICKNESS OF TENDONS 315

1

50 1 00 Stress (Mpa)

FIG. 2. Distribution of maximum tendon stresses among limb tendons of mammals. aapt is the theoretical optimum stress. The results from tendons whose muscles have relatively very short fibres (i < 2) are shown diagnonally hatched (a); those with relatively slightly longer fibres (2 C i <4) are shown stippled (El). (i is included in the list of symbols.)

The tendon cannot break if the available muscle and joint displacements are insufficient to stretch the tendon to its limit.

These expectations are in accord with clinical experience. McMaster (1933), considering human tendons in general, states: ‘When a normal muscle-tendon system is subjected to severe strain, the tendon does not rupture’. Any failure is in the bone or muscle or in a junction to the tendon. However, many authors (cited by Barfred, 1973) consider rupture in tension of the healthy Achilles tendon to be possible. The difficulty is in assessing whether the tendon had degenerated in some way prior to rupture. Clinical records are also available for horses and the situation appears to be similar. Rupture of healthy toe flexor tendons is a possibility, but prior degeneration cannot easily be ruled out (Webbon, 1977). Barfred (1973), using anaesthetized rats, carried out experiments in which the Achilles tendon and muscles were pulled to breaking point via the attached bones. The muscles were maximally stimulated by an electrode on the sciatic nerve. Rupture sometimes occurred in the tendon.

Figure 3, obtained using the extensometer described by Ker (1981), is the lower stress part of the stress-strain curve for the cow tendon referred to above. This plot is typical of adult mammalian tendons (Bennett et al., 1986). The lower curved portion is called the toe region. At higher stresses,

316 R. F. KER, R. McN. ALEXANDER A N D M. B. BENNETT

I

3t

i? z w 2 5

l!

%Dt-

I

t 1 2 EO

Strain (YO)

FIG. 3. The stress-strain curve for an adult cow forelimb common digital extensor tendon obtained by the methods of Ker (1981). The tendon was subjected to sinusoidal straining at a frequency of 2.2 Hz. This is the ‘loading curve’: i.e. for increasing stress. The ‘unloading curve’ would not be grossly different, since hysteresis in tendons is small (Ker, 1981; Bennett et al., 1986). At higher stresses than shown here, the plot is linear up to fracture. The tangent at the theoretical optimum stress, uopl, is shown: its intercept on the strain axis is at 60.

above 30 MPa, the plot becomes effectively linear. The maximum stress values in Fig. 2 are mostly well below 30 MPa, so the majority of tendons operate entirely within the toe region where the tangent modulus (i.e. the slope of the stress-strain curve) is less than at higher stresses. In contrast, for the most highly stressed tendons during vigorous activities, the toe region represents only a small departure from linear properties.

Figure 3 will be used in the following Theory section to provide illustrative numerical values relating strain to stress. We chose this particular plot because the tangent modulus in the linear region (1 500 MPa) is the same as the average found by Bennett et al. (1986) for a range of adult tendons. In this sense, Fig. 3 is an average stress-strain plot for tendon. However, the tangent moduli vary considerably: the values given by Bennett et al. (1986) show a standard deviation of 250 MPa. The consequences of this spread for the theory are discussed on p. 320. Compared to these variations between tendons, other possible factors affecting the stress-strain curve are likely to be of lesser importance. For example, any uncertainty in the position of the origin of strain is unimportant to the theory for which the critical quantity is the tangent modulus around the value of maximum stress (see paragraph following equation 11). The stress-strain curve may vary with

THICKNESS OF TENDONS 317

the rate of straining: however, Herrick, Kingsbury & Lou (1978), Ker (1981), Riemersma & Schamhardt (1985) and Bennett et al. (1986) show this effect to be small over a range of rates from 1 to 100% s- '. Tendons from immature mammals have markedly different mechanical properties, (Torp, Arridge, Armeniades & Baer, 1975): we are concerned here with tendons from skeletally mature mammals.

Despite low stress, a long tendon may, none the less, extend significantly in relation to the displacement at its joint. For example, Rack & Ross (1984) give data for a human flexor pollicis longus subject to a stress of 15 MPa. If Fig. 3 applies, the strain is 1.7%. The tendon has an average length of about 155 mm and therefore stretches about 2.6 mm. This corresponds to rotation of the interphalangeal joint of the thumb through about 20". Rack & Ross (1984) and Rack (1985) discuss the significance of this extension for the functioning of the muscle-tendon system.

Theory

For many structural components, optimum design calls for the minimum thickness consistent with adequate strength, due allowance being made for the variability of loads and materials. This appears a promising approach for those tendons subject to large stresses (say ot > 50 MPa, in Table TI), but it cannot explain the much lower stresses found for the majority of tendons. Why are tendons so thick?

Muscles can contract while under load, thus acting as a source of energy. A tendon extends under load and therefore some of the contraction of the muscle fibres serves only to take up the extension of its tendon without contributing to the displacement at the joint. A thinner tendon extends more and so impairs the ability of the muscle-tendon system to displace the joint. The required mechanical capability can be restored by using a muscle with longer fibres. Increasing the fibre length increases the mass of the muscle, since the cross-sectional area of the fibres is fixed, being determined by the force the muscle is required to develop. Minimum mass for the combination of the tendon and its muscle is achieved at an intermediate tendon thickness: too thin and the muscle must be unnecessarily heavy; too thick and the tendon itself is unnecessarily heavy.

The theory outlined above applies to muscle acting as a source of energy. An active muscle being stretched absorbs energy. Then the extension of the tendon is in the same direction as the lengthening of the muscle fibres: the tendon assists the muscle in absorbing energy. The optimal design for this function has a tendon which is as thin as possible. This limits the applicability of the theory (see Discussion). However, the characteristic function of the majority of muscles is to supply mechanical energy: the need to fulfil the demand for energy is likely to be of greatest importance in determining the dimensions of the muscle and its tendon.

It is convenient to think in terms of the area ratio (muscle fibres to tendon), r = A/a. r is inversely proportional to a, because A is a constant. The maximum stress in the tendon is ro, (equation 1). The corresponding strain, Et, can be found for a typical tendon from Fig. 3. The extension of the tendon, e= h,. The subscript 'i' will be used to denote quantities which would apply to themuscle if its tendon were inextensible: thus the mass would be Mi, the fibre length would be Li and the range through which the fibre length can alter would be Ri. Ri is the displacement required at the distal end of the tendon. With a real (i.e. extensible) tendon the range, R, over which the muscle fibres operate is (e+ Ri) instead of Ri. To achieve this the muscle fibres must be proportionateIy longer. Since A is constant, the muscle mass also increases proportionally. Thus, the actual muscle mass, M , is given by:

318 R . F. KER, R . McN. ALEXANDER A N D M. B. BENNETT

The extra mass of muscle, ( M - Mi), is Mi (e/Ri). Mi = pmALi, where pm is the density of muscle. The mass of the tendon is pt(A/r)l where pt is the density of tendon. Add the mass of the tendon to the extra mass of muscle, with substitutions for e and Mi, to give:

Total extra mass = p,A ZE, + - PtAl (2) r

Only the ratio of Li to Ri appears here and not the absolute values. Thus, (Li/Ri) can be replaced by (LIR). Finally divide by ptAZ to give dimensionless terms:

Total extra mass 1 PtAl (7)

This step is only useful if Zcan be considered as a constant, independent of r . Strictly, 1 is a variable for whenever L is altered, 1 must also be altered to keep the same overall length. However, these changes in 1 have very little effect and can be neglected. This statement will be justified at the end of this section.

Values have already been given for the densities (pm = 1060 & pt = 1 120 kg mW3). We will again take om to be 0.3 MPa. The possibility of achieving a greater stress by stretching an active muscle does not arise here for the theory applies to muscles which work by contracting, rather than stretching, their fibres. RIL is equal to the fractional range over which the sarcomeres of muscles operate in life. Dimery (1 985) has examined R / L and the corresponding range of sarcomere length changes in the leg muscles of galloping rabbits and she cites other work on the jaws of rabbits and rats; see also Cutts (1986), on the flight muscles of birds. On the basis of this evidence, we will take the working range of sarcomeres to be about 0-6 pm. The relevant sarcomere length is that which gives peak stress, which is 2.3 pm for mammalian muscle (Dimery, 1985). Therefore, we will assume R / L ~ 0 . 2 5 . However, the range of operation must be expected to be different in different muscles; in particular, some muscles operate through a smaller range, but many of these may be muscles to which this theory does not apply. With these values, and using Fig. 3 to relate tendon strain to stress, the plots shown in Fig. 4 are obtained for the two terms on the right-hand side of equation (7) and for their sum. The minimum total extra mass occurs when r = 34 and the corresponding stress is 10.2 MPa. This stress value is marked on Figs 2 and 3 as Copt, the theoretical optimum stress.

From Fig. 4, the minimum value of (Extra mass)/(p,AC) is 0-078. pt = 1 120 kg m-3 and, therefore, for any particular tendon:

Minimum extra mass = 87A1

in S.I. units. As is to be expected, the extra mass is proportional to the cross-sectional area of the muscle fibres (and hence to the force available) and to the length of the tendon. From Fig. 4,62% of the extra mass is associated with muscle and 38% is the mass of the tendon.

With hindsight, an algebraic expression can be obtained for the optimum area ratio by using a straight-line approximation to the stress-strain curve in the neighbourhood of the stress of 10.2 MPa. (This is acceptable, even though the stress is in the toe region, because it is only the immediate neighbourhood of this stress which is relevant in determining the position of minimum mass.) The equation of the straight-line is:

(8)

THICKNESS OF TENDONS 319

, \ \ , 0 ' ' "Muscle

, , \ / ' .' ,\

0.05

/ I I

20 40 60 Area ratio (muscle to tendon), r

FIG. 4. Calculation of the extra mass required because of the extensibility of tendon, for a muscle-tendon combination doing positive work. The extra mass has a minimum when the area ratio (muscle to tendon) is 34. The extra mass is shown on the left-hand axis as a dimensionless ratio: the right-hand axis gives the extra mass of the common digital flexor of the forelimb of a cow.

61 El = Eo + -

E (9)

where E is the tangent modulus at a stress of 10.2 MPa and EO is the intercept of the tangent on the strain axis. The tangent is shown in Fig. 3. Substitute equation (9) into (7):

Total extra mass =(?) (i) P o + ( % ) + ; 1

PtAl

Differentiate with respect to r and equate to zero to obtain the optimum area ratio, rOpl, at which the mass is a minimum. This gives:

J(2) (;) (:) In this approximation, the relevant feature of the stress-strain curve for tendon is the tangent

modulus, E, at the optimum stress. This can be approximated or found by iteration. However,

320 R. F. KER, R. McN. ALEXANDER A N D M. B. BENNETT

because of the square root, ropt is not very sensitive to changes in the parameters: in particular, using E= 1500 MPa (the tangent modulus in the linear region) rather than E= 1300 MPa (the tangent modulus at a stress of 10.2 MPa) has only a limited effect. Even in the full theory, the area ratio giving minimum mass is determined by the form of the stress-strain curve in the neighbourhood of the optimum stress. In particular, the position of the origin of strain, which is often ill-defined in tensile tests on tendon, is irrelevant to the theory.

Equation (1 1) gives an (approximate) indication of the variations in ropt to be expected for different tendons due to individual variations in the parameters. Bennett et al. (1986) found the standard deviation of tangent modulus of tendons (in the linear region) to be 250 MPa. The mean is 1500 MPa so 95% of the values are expected to be between 1000 and 2000 MPa. (Some of this variability may well reflect differences in collagen content; see Riemersma & Schamhardt, 1985.) The two-fold spread in stiffness will spread the values of ropt through a factor of only 1.4. The fractional range over which muscles function, R/L, can be much less than 0.25. Some of the muscles of Dimery (1985) have R/L nearer 0.1, in which case equation (1 1) indicates ropt = 22 rather than 34. The corresponding optimum stress is then 6.6 rather than 10.2 MPa. R / L is more likely to be much less than 0.25 than much greater than 0.25. The effect is to favour lower area ratios and stresses.

The decision to treat las a constant remains to be justified. In considering different designs of the muscle-tendon system, any change in L requires an equal and opposite change in 1 to keep the overall length constant. But the tendon is of much smaller cross-sectional area than the muscle fibres and so equal lengths have very different masses. With r = 34, the effect on mass of a change in 1 is only about 3% of that due to the accompanying change in L. Therefore, to the accuracy relevant here, changes in 1 can be neglected.

Discussion of the theory

How general is this theory and its result? ropt is independent of the length of the tendon, so a single, general result (rapt = 34) is obtained. However, individual variations are to be expected for three main reasons: (i) the parameters of equation (7), including the stress-strain curve Fig. 3, differ for different muscle-tendon systems (see Theory); (ii) the optimization is based only on minimization of mass (see below: sub-section Factors other than the minimization of mass); (iii) muscles are not always sources of energy.

The theory applies to a muscle doing positive work. The fibres are contracting under load, extending the tendon and displacing the joint. The energy flow is from the muscle to the tendon (as strain energy) and to the external system. This direction of energy flow is reversed, or partially reversed, in two circumstances. One of these is considered below in the sub-section on Muscles with very shortfibres. The other is when the muscle is absorbing mechanical energy from the external system (doing ‘negative work’). Then the external system supplies the strain energy to the tendon. The muscle fibres are being lengthened, so the theory can be modified to cover negative work simply by giving a negative value to R. This alters the properties of equation (7), which is now minimized by making r as large as possible. If a muscle only undertook negative work, strength would become the limiting factor in the optimization of tendon thinness.

This prediction for negative work was stated early in the Theory section, but the requirements of positive work were thereafter assumed to be the more important in determining the size of the muscle and its tendon. The agreement achieved between the prediction for ropt and the measured values gives weight to this assumption, which anyway seems reasonable. A muscle can absorb

THICKNESS OF TENDONS 32 I

more energy than it can supply, which mirrors the greater maximum stress achievable during negative work. If the muscle is large enough to fulfil the requirements as an energy source, it will also be able to do substantial negative work. Its tendon, with r x 34, will be amply thick enough to avoid risk of fracture even though the muscle stress can be greater than 0-3 MPa during negative work. Since all muscles can be used both to supply and absorb energy, the proper design of tendons will be a compromise between the two predictions. The arguments here point towards the optimum value being nearer the prediction for positive work: for most muscles, r will be shifted to only slightly higher values.

Factors other than the minimization of mass

Natural selection can be viewed as a process of minimizing costs (Alexander, 1981). The theory of this paper has treated mass as a cost and has implicitly assumed equal cost for equal mass of tendon and of extra muscle. Extra mass implies costs in terms of energy for growth, for maintenance and for use: the use cost includes the energy required to carry around the extra mass. The growth cost is likely to be comparable for tendon and muscle. In use, muscle requires energy for activation and contraction. The ‘extra’ muscle (mass M-Mi) uses energy each time it is activated; most of this energy is dissipated as heat, some becomes the strain energy in the tendon. This factor favours a lower area ratio. A factor favouring a higher area ratio, for leg tendons, is the increased disadvantage of mass and volume in the more distal parts of the leg, where most of the tendon is usually situated. (This was pointed out to us by Dr Peter Purslow.) A similar factor is likely to apply to most muscles with long tendons for the tendons often go through regions where space is at a premium. We have not estimated maximum stress in the aponeuroses of muscles lacking an external tendon, because measurement of the cross-sectional area would be inconvenient; but the theory should apply rather better to such aponeuroses than to tendons whose thickness may be constrained by the narrow spaces through which they must fit. Finally, a larger area ratio will be encouraged by any advantage gained from the extension of the tendon. Rack (1985) has pointed out the benefit for the accurate control of forces which accrues from the extension of the tendon.

Strength, which is an important factor in the design of many structural components, has no effect in determining the thickness of those tendons to which the theory applies. The probability of failure is effectively zero. About a four-fold reduction in safety factor (to a value of about 2) would be required to change this situation. Thus, using equation (ll), strength would only become relevant if tendon was made from a material of modulus 16 times greater and of unchanged strength.

Muscles with very short Jibres

The fibres of some muscles are too short to allow the muscle to stretch its tendon fully. The muscle cannot provide the tendon’s strain energy, at high loads, let alone supply energy to the external system. These muscles cannot control the displacement at their joints in the way which was assumed when developing the theory in the previous section.

An example is of the human Achilles tendon and its muscles. The following dimensions are for the specimen included in Table I1 under ‘Gastrocnemius’. The maximum stress is 67 MPa. From Fig. 3 (and the linear region at higher stresses), the corresponding strain is about 5.2%. The muscles are the two heads of the gastrocnemius, each with a fibre length of about 35 mm, and the

322 R. F. KER, R. McN. ALEXANDER A N D M. B . BENNETT

soleus, with a fibre length of 45 mm. The length of the tendon, found as in equation (2), is 41 5 mm for the gastrocnemius and, on average, about 300 mm for the soleus muscle. (The origin of the soleus is on bone, so different parts of the tendon are of different lengths.) Assuming the tendon strain is the same throughout, the maximum extension of the tendon is about 22 mm for the gastrocnemius and 16 mm for the soleus. With R/L=0.25, the gastrocnemius muscle would contract through about 9 mm and the soleus through about 11 mm. Neither could fully stretch its tendon; some, at least, of the tendon’s strain energy must come from the external system.

A dimensionlessfibre length factor, i, will be introduced to describe the relative importance of the muscle as against its tendon in determining the displacement at the joint at full load. Define i as the ratio of the length of a muscle’s fibres to the extension of its tendon, when the muscle stress is 0.3 MPa. For the gastrocnemius muscle the fibre length factor is 1.6 and for the soleus muscle it is 2.8. A muscle is less important than its tendon when l <4, if R is assumed be 0-25L. ( R is under voluntary control and so can take smaller values at will. But our purpose here is to consider the potential of a muscle to control the position of its joint and, for this, R = 0.25L seems reasonable.) In Fig. 2, the diagonally hatched blocks represent tendons with t < 2 and the stippled blocks have 2 <l <4. For all the other blocks, i 2 4 . (The muscles involved often have two or more heads with very different fibre lengths. The blocks have been marked according to the smallest length factor among the heads. Thus the block for the human Achilles tendon has been hatched, not stippled.) The hatched and stippled blocks include all those at the highest stresses. None of the strikingly high stresses is for a tendon to which the theory is expected to apply. With three exceptions, to be referred to in the next paragraph, the tendons with < 4 seem likely to function as springs in running (Alexander, 1984). For this function, the optimum tendon thickness is expected to be determined by strength, with a safety factor which can be quite small (Alexander, 1981). This expectation accords well with the high stresses of the marked blocks in Fig. 3, especially when allowance is made for values of g,,, > 0.3 MPa.

When its tendon acts as a spring, the only necessary role for the muscle is to maintain tension. Short fibres are mechanically adequate and bring the benefits of low activation energy to maintain tension and of small mass. The ungulates take this design to extremes. The lowest value of in our sample is for the horse forelimb superficial toe flexor muscle with t = 0-09. Even this is surpassed by the equivalent muscle from the hind limb (the plantaris), for in adult horses this muscle is virtually absent ( i = O ) although the tendon is massive. It has had to be omitted from Table I1 because our method cannot be applied when the muscle is absent! The three exceptions referred to above are all from the horse: two are toe extensors and the third is a wrist extensor. These cannot act as springs in the way described by Alexander (1984), because they are not under load when the foot is on the ground. Their thinness may simply reflect the great emphasis in the design of the horse on minimizing mass in the distal parts of the limbs. Or do these tendons act as springs, in a mechanism yet to be discovered?

Other muscles

A sharp cut-off in the applicability of the theory is not to be expected. Values of cover a full range. Even among those tendons which act as springs in running, most are also used to help in positioning and loading the joint during other activities. The applicability of the theory is likely to increase with the importance of the ‘other activities’. In Table 11, note the virtual absence of high stresses among the tendons of monkeys, whose feet are particularly manoeuvrable. The badger forelimb is interesting for the contrast between the stress in the deep and the superficial digital

THICKNESS OF TENDONS 323

flexors. It is tempting to suggest the superficial flexor is used for running and the deep flexor for digging. Except for the horse tendons mentioned above, all toe extensors have low stresses.

The evolved design of each tendon is a compromise. Compare the human Achilles tendon with the gastrocnemius and plantaris tendons of the monkeys (Table 11). The human Achilles tendon is used in a range of activities, but the fairly high stress indicates that the role in running is given considerable emphasis. The lower stresses in the monkey tendons (and higher values o f t ) , indicate a greater emphasis on other activities. None the less, monkeys may use their ankle extensor tendons as springs in running. A tendon subject only to low stresses can function effectively as a spring as has been explained by Alexander (198 1). The drawback is the unnecessarily high mass (for this function) of the tendon.

Finally, the theory of thick tendons may also apply to other groups of animals. Some birds have calcified leg tendons with E much greater than for collagenous tendon (Bennett & Stafford, 1988) and ropt would then also be expected to be greater. Insects have tendons which are up to 10 times stiffer than mammalian tendons (Bennet-Clark, 1975). Equation (1 1) then gives ropt as about 100. This would bring no risk of fracture as insect tendons have a much higher ultimate tensile stress than mammalian tendons. Only those tendons to which the theory does not apply might fail in tension. Such a tendon is the tibia1 extensor tendon of the locust, which is a spring for powering the jump. Bennet-Clark notes that, if the tibia of a locust is pulled away from the femur, the extensor tendon usually snaps somewhat distal to the muscle insertions, but the flexor tendon comes out intact surrounded by torn muscle. The flexor tendon is stronger than its muscle.

Conclusion

In a muscle-tendon system, the extension of the tendon may be disadvantageous, as when the muscle is supplying energy to the external system, or it may be advantageous, as when the tendon acts as a spring. These circumstances lead to very different predictions for the optimum tendon thickness. The tendon should be as thin as possible for negative work or for acting as a spring. For positive work, a much thicker tendon is optimal in minimizing the combined mass of the muscle and the tendon. Both these extremes are observed among the leg tendons of mammals and also everything in between, which reflects the range of muscle functions. However, for the majority of muscles the ability to perform positive work is the more significant in determining tendon thickness. For these, the ‘theory of thick tendons’ introduced in this paper quantitatively explains the observed ratio of areas of muscles and tendons. This ratio determines the maximum stress to which a tendon can be subjected. The majority of tendons, when healthy, are not at risk from fracture in tension. The few tendons for which strength is the determining factor and for which the probability of failure is therefore finite, include the human Achilles and plantaris tendons.

This work was supported by a grant from the Science and Engineering Research Council. We thank Mr R. C. Kester and Mr S. R. Bibby for arranging for the human leg to be available for this investigation. Permission was granted by the patient. We thank Dr Caroline Pond for supplying most of the other mammals.

REFERENCES

Alexander, R. McN. (1974). The mechanics of jumping by a dog (Canis familiaris). J . Zoo/.. Lond. 173 549-573. Alexander, R. McN. (1981). Factors of safety in the structure of animals. Sci. Prog., Ox$ 67: 109-130.

324 R . F. KER, R . McN. ALEXANDER AND M. B. BENNETT

Alexander, R. McN. (1984). Elastic energy stores in running vertebrates. Am. Zoo/. 24: 85-94. Alexander, R. McN. & Dimery, N. J. (1985). Elastic properties of the forefoot of the Donkey, Equus asinus. J. Zool., Lond.

Alexander, R. McN., Dimery, N. J. & Ker, R. F. (1985). Elastic structures in the back and their r6le in galloping in some

Alexander, R. McN., Maloiy, G. M. O., Ker, R. F., Jayes, A. S. & Warui, C. N. (1982). The role of tendon elasticity in the

Alexander, R. McN. &Vernon, A. (1975). The mechanics of hopping by kangaroos (Macropodidae). J. Zool., Lond. 177:

Barfred, T. (1973). Achilles tendon rupture: aetiology and pathogenesis of subcutaneous rupture assessed on the basis of

Bennet-Clark, H. C. (1975). The energetics of the jump of the locust Schistocerca gregaria. J. exp. Biol. 63: 53-83. Bennett, M. B., Ker, R. F., Dimery, N. J. & Alexander, R. McN. (1986). Mechanical properties of various mammalian

Bennett, M. B. & Stafford, J. A. (1988). Tensile properties of calcified and uncalcified avian tendons. J. Zool., Lond. 214

Cavagna, G . A., Citterio, G. & Jacini, P. (1981). Effects of speed and extent of stretching on the elastic properties of active

Cults, A. (1986). Sarcomere length changes in the wing muscles during the wing beat cycle of two bird species. J. Zool.,

Dimery, N. J. (1985). Muscle and sarcomere lengths in the hind limb of the rabbit (Oryctolugus cuniculus) during a

Dimery, N. J. &Alexander, R. McN. (1985). Elastic properties of the hind foot of the Donkey, Equus asinus. J. Zool., Lond.

Dimery, N. J., Alexander, R. McN. & Ker, R. F. (1986). Elastic extension of leg tendons in the locomotion ofhorses (Equus

Dimery, N. J., Ker, R. F. &Alexander, R. McN. (1986). Elastic properties ofthe feet ofdeer (Cervidae). J. Zool., Lond. ( A )

Elliott, D. H. & Crawford, G. N. C. (1965). The thickness and collagen content of tendon relative to the strength and cross-

Flitney, F. W. & Hirst, D. G. (1978). Cross-bridge detachment and sarcomere ‘give’ during stretch of active frog’s muscle.

Herrick, W. C., Kingsbury, H. B. & Lou, D. Y. S. (1978). A study of the normal range of strain, strain rate and stiffness of

Jayes, A. S. &Alexander, R. McN. (1982). Estimates of mechanical stresses in leg muscles of galloping Greyhounds (Canis

Ker, R. F. (1981). Dynamic tensile properties of the plantaris tendon of sheep (Ouis aries). J. exp. Biol. 93: 283-302. Ker, R. F., Bennett, M. B., Bibby, S. R., Kester, R. C. & Alexander, R. McN. (1987). The spring in the arch of the human

Ker, R. F., Dimery, N. J. & Alexander, R. McN. (1986). The role of tendon elasticity in hopping in a wallaby (Macropus

McMaster, P. E. (1933). Tendon and muscle ruptures: clinical and experimental studies on the causes and location of

Mtndez, J. & Keys, A. (1960). Density and composition of mammalian muscle. Metabolism 9 184-188. Rack, P. M. H. (1985). Stretch reflexes in man: the significance of tendon compliance. In Feedback and motor control in

invertebrates and vertebrates: 217-229. Barnes, W. J. P. & Gladden, M. H. (Eds). London: Croom Helm. Rack, P. M. H. & Ross, H. F. (1984). The tendon of flexor pollicis longus; its effects on the muscular control of force and

position at the human thumb. J. Physiol., Lond. 351: 99-110. Riemersma, D. J. & Schamhardt, H. C. (1985). In oitro mechanical properties of equine tendons in relation to cross-

sectional area and collagen content. Res. uet. Sci. 3 9 263-270. Torp, S., Arridge, R. G. C., Armeniades, C. D. & Baer, E. (1975). Structure-property relationship in tendon as a function of

age. In Structure ofJibrous biopolymers: 197-221. Atkins, E. D. T. & Keller, A. (Eds). London: Butterworths. Webbon, P. M. (1977). A post mortem study of equine digital flexor tendons. Equine vet J. 9 61-67. Wells, J. B. (1965). Comparison ofmechanical properties between slow and fast mammalian muscle. J. Physiol., Lond. 178:

( A ) 205 51 1-524.

mammals. J. Zool., Lond. ( A ) 207: 467482.

locomotion of the camel (Camelus dromedarius). J. Zool.. Lond. 198: 293-313.

265-303.

the literature and rupture experiments in rats. Acta orthop. scand. (Suppl.) 152 1-124.

tendons. J. Zool., Lond. ( A ) 209: 537-548.

343-351.

frog muscle. J. exp. Biol. 91: 131-144.

Land. ( A ) 209: 183-185.

galloping stride. J. Zool., Lond. ( A ) 205: 373-383.

( A ) 207: 9-20.

caballus). J. Zool., Lond. ( A ) 210 415425.

208: 161-169.

sectional area of muscle. Proc. R. SOC. ( B ) 162 137-146.

J. Physiol.. Lond. 276 449465.

tendon. J. biomed. Materials Res. 12: 877-894.

familiaris). J. Zool., Lond. 198: 315-328.

foot. Nature, Lond. 325 147-149.

rufogriseus). J. Zool., Lond. ( A ) 208: 417428.

subcutaneous ruptures. J. Bone Jt Surg. 1 5 705-722.

252-269.