JOURNALOF APPLIED PHYSIOLOGY Vql. 39, No. 4, October 1975. Prinki
Using
body
in U.S.A.
size to understand
design of animals: THOMAS
the structural
quadrupedal
locomotion
A. McMAHON
Division of Engineeringand Applied Physics,Harvard University, Cambridge,Massachusetts 02138
celerating the limbs with respect to the body. If a limb motion is accomplished in time At by a muscle whose change in length is Al, the force produced in that muscle must be proportional to mass of the limb X (Al/At’)) by Newton’s second law. For strict geometric similarity, Al is proportional to some characteristic length I of the animal, muscle crosssectional area A is proportional to 12, and limb mass is proportional to 13. Then muscle force
MCMAHON, THOMAS A. Using boa) size to understand the structural design of animals: quadrupedal locomotion. J. Appl. Physiol. 39(4) : 619627. 1975.Many parameters of gait and performance, including stride frequency, stride length, maximum speed, and rate of 02 uptake are experimentally found to be powerlaw functions of body weight in running quadrupeds. All of these parameters are reasonably easy to measure except maximum speed, where the question arises whether one means top sprinting speed or top speed for sustained running. Moreover, differences in training and motivation make comparisons of top speed difficult. The problem is circumvented by comparing animals running at the transition between trotting and galloping, a physiologically similar speed. Theoretical models are proposed which preserve either geometric similarity, elastic similarity, or static stress similarity between animals of large and small body weights. The model postulating elastic similarity provides the best correlation with published data on body and bone proportions, body surface area, resting metabolic rate, and basal heart and lung frequencies. It also makes the most successful prediction of stride frequency, stride length, limb excursion angles, and the metabolic power required for running at the trotgallop transition in quadrupeds ranging in size from mice to horses.
scale effects; running; galloping; oxygen similarity; muscle stress; muscle velocity
consumption;
The power deli vered by the muscle is proportional muscle force t imes shorte ning vel ocity, or power
a F(l/At)
a 12(l/At)3
to
(2)
Three important conclusions arguments. The first is that
may be derived from these since the maximum stress (F/A) a given muscle may bear is independent of body size, so is the overall muscle shortening velocity, (Z/At) from (I). Stride length is also proportional to I, so the maximum running speed an animal may attain is independent of body size. At this maximum running speed, the stride period At is proportional to I (e.g., to limb length). Lastly, from (2), the metabolic power expended in running at top speed should be proportional to Z2 (e.g., limb crosssectional area, body surface area, or W213, where W is body weight)‘, and furthermore, the power a particular animal must supply through his muscles when running at any speed should increase as the cube of speed. In the years since Hill published these predictions, experimental evidence testing each of the above points has become available (13, 23, 26). This work has shown that it is not true that all quadrupeds have the same top speed, nor that stride frequency is inversely proportional to limb length, nor that rate of oxygen consumption increases with the cube of running speed. Evidence has also shown that animals of different size are not geometrically similar, but regularly become thicker and more stout as one considers adult animals of larger and larger body size (20, 21).
elastic
MATHEMATICAL MODELS of animals are onlv imitations of what is found in nature and therefore cannot hope to be as subtle and fascinating as animals themselves. Yet models have their usefulness. A. V. Hill (16) employed a mathematical model when he asked the important question, “How can the easily measured parameters of gait and performance (e.g., stride frequency, stride length, maximum speed) be expected to change as a function of body size, all other matters being equal?” Starting with the fundamental assumptions that 1) muscle stress, e.g., tension per unit crosssectional area, is the same in animals of different size, and 2) that the work per stroke a muscle is capable of performing on its surroundings is proportional to the muscle weight, and 3) that geometric similarity is maintained between the animals to be compared, Hill developed a powerful model which he used to predict the body size dependence of a variety of animal gait and performance parameters. One of these parameters was the metabolic power required for running at constant speed. Hill reasoned that this power was mainly required for accelerating and de
l In this paper, the theoretical predictions of the models will be expressed as the whole fractional powers of W, e.g., IV3 or W3j8. Experimental results will be expressed as decimal fractional powers, e.g., M/O.33 or
W0.38
.
619
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620
T.
A.
McMAHON
In what follows, three alternative schemes are proposed for comparing animals of dissimilar size but grossly similar appearance. Alternative predictions based on these models are tested against experimental data for limb and body proportions, basal metabolic rate, and several parameters of quadrupedal running in an effort to choose between the models. The eventual goal is to understand the principle which has dictated the observed regular change in body separate those biological shape with size, and thereby parameters which necessarily change as a function of body size from those which do not. ANALYTICAL
METHODS
A familiar result from the engineering theory of models is that an exact scale model, made of the same materials as the prototype, may not work the same way the prototype works. It is frequently necessary to introduce regular distortions in mechanical geometry or material properties in order to ensure that some important parameter or condition is the same between model and prototype. Thus the gears of a small pendulum clock cannot have the same ratio as a large one if they are both to keep the same tirne, because the periods of their pendulums are different. Flow in a model river basin may employ distortions in either channel depth or density of working fluid to ensure similarity of wave patterns. Before construction of an experimental model can be attempted, it is always necessary to state which physical parameters or conditions will be kept the same between model and prototype, and which will necessarily change as a function of size. Geometric similarity. Not only Hill ( 16)) but Thompson (27) and many others have discussed geometrically similar
Ap]
@Geometric
@Static
or
FIG. 1. A: schematic illustration similar beams,
for
which
&/12 = stress,
(&/dl)
l 12.
Similarity
5 = /x&d
of large and small geometrically = d2/& . B: elastically similar 61//l , but Z& = (d2/&)2/3. C: beams of so that ij2 = z’l , which requires 12/11 =
quadrupeds,showing12//l
equalaverage static
Stress t a dth
Similarity
60
I I
56 FIG.
Least Figures height
JIllI
8 10
I
I
I
20 30 Diameter, d (mm)
IllIll
50
I 70
loo
Midshaft 2. Length vs. midshaft diameter of artiodactyl humerus bones. mean squares fit shows I = 24.09 d.655with I and d in mm. represent elastically similar schematic quadrupeds different in by a factor of ‘2 and weight by a factor of 16.
models (scale models) of animals of dissimilar size, where there is a single length scale. If I is the length of a particular bone or muscle and d is its diameter, geometric similarity requires I a d, so that both length and diameter are multiplied by the same factor as we go from a small animal to a large one (Fig. 1, A). If large and small geometrically similar animals are drawn to the same scale, it is not possible to distinguish between them. Elastic similarity. As I have discussed in a previous paper (21) elastically similar anirnals are those whose structures are similarly threatened by elastic failure under their own weight. When a structure buckles, the elastic forces produced by deformation are not suficient to overcome the applied forces (in this case due to the structure’s weight). Visualize two tall columns of different size made of the same material just at the critical buckling condition. If their lengths are in the ratio Z&l = 2, the critical buckling condition is just preserved when their diameters are in the ratio dz/dl = 2 ~~ When these two columns are turned on their sides and held by their ends (Fig. 1, B) so that they become simply supported beams, the deflection at the center 6 divided by the length 1 is the same in the two beams. Whether failure occurs by buckling or bending, elastic similarity is guaranteed by keeping Z a d2j3, whatever values I is allowed to take. Static stress similarity. Let us calculate the average COIWpressive stress 5 in the top half of a beam of circular crosssection, diameter d, shown in Fig. 2, C. For this calculation, we assume the deflection of the beam is small. Cutting the beam at the center and summing the applied moments on the segment to the right of the cut to zero, the weight pgrd21/8 acting through the moment arm 1/2 is balanced by twice the moment contributed by 8 acting through d/4 (distance from the center line to the center of pressure). Thus a = pg12/d, where pg is the weight density. If 5 is to be maintained the same between selfloaded beams of the same material, I a d112. In the case shown, where l&l = 2, stress instead &/dl = 4. The rule is true when maximum of average stress is to be kept the same between beams of
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SCALE
EFFECTS
IN
QUADRUPEDAL
621
RUNNING
different size, or when any arbitrary crosssectional shape is substituted for the circular one considered above. Note that this result is true only for bending: if the beams are stood on end to become columns, the stress at the base of each can be equal only when both are made the same absolute height, since in this case @base = pgl. RESULTS
repreBody proportions. In Fig. 2, there is a schematic sentation of two quadruped specimens different in weight by a factor of 16. The larger animal is a distorted model of the smaller, since the length of each quasicylindrical element is different by a factor of 2, while diameters are different by a factor 2dr The larger animal is thus more robustly built, in such a way as to maintain elastic similarity between the two. If the animals were to be made similar with respect to breaking stress, the larger would become would have to be even more robust, i.e., the diameters made different by a factor of 4. Let us derive how the somatic exponent b in the allometric relation y = aWb depends on the lengthdiameter exponent p for such regularly distorted models, where I a dp. Here y is a somatic parameter, such as body height, W is total body weight, and a is a constant. The weight of any limb or other structural element is always a specified fraction of W for any body size, and since W a ld2 if I a dp, then /
a
WPiPf2;
da
W
llPv2
(3)
As detailed in Table 1, for geometric similarity there is only one length scale because both I and d are proportional to the fS power of body weight. For elastic similarity, where p = 36, all body lengths should scale up as W1i4, while all diameter or girth dimensions are proportional to W3j8. For static stress similarity, p = $4, so that I a W115, d o= PV5. To summarize geometric similarity, I a 1/1/’i3; d a W1/3 elastic similarity, I o= W1j4; d a W318 static stress similarity, I a PV5; d a W2i5 The study of comparative morphometry use of the fact that the somatic dimensions TABLE
1. Three alternative similarity
(4)
has long made of many species
models Geometric
Slenderness
exponent,
0:
I a dp
Similarity
Crosssection
A a
Period
for first
Muscle
work
Basal
metabolic
s
= 0.125;
geometry, per
stroke, power,
s
= 0.200;
At
At
0: W/A
e 0: umuscle A AZ
s
M/l
I3
I
M/l/3
M/2/3 W//3
a: Wl13
s
= 0.333;
B
Static
= 0.375;
Wli4 M/‘3/8
Aa
M/3/4
0:
Z
W518
W’14
= 0.400;
g
a
Similarity
1/2
M/T215
A a:
w4/5
M/“/b
ea:W p#74/5
= 0.600;
s
= 0.625;
(Ref)
= 0.66, artiodactyls (McMahon, (20)) fj a W”z4, cattle (Brody, (2)) G a WO.36, cattle (Brody, (2)) S a WO.65, mammals (Stahl (24) > Aortic area a: WO.72, mammals (Clark, (5) > Heart period o= W”a25, mammals (Stahl, (24)) Skeletal muscle e 0: W (Hill (16)) W”J5, mammals (Kleiber, (18))
w3/5
a
Expt
p
M/“/s
do: Sa
At
w/44
s
Stress
p =
eaW
W2/3
= 0.250;
a
da So=
At
ea:W
0: A
Similarity
P = 2/3
I a do: S a
A a d2
Elastic
p=l
Somatic length exponent, la M/Tp’(p ’ 2, Somatic diameter exponent, d a Wl’(* + 2, Total body surface, S 0: dl area,
bear a power law relation to body weight. Does such information allow us to choose which of the three models provides the most realistic prediction of scale effects? Brody (2) measured chest girth and height at withers in more than 3,000 Holstein cattle. Comparison with the elastically similar model is good: his regression shows chest girth proportional to W” e36 ( W3’* predicted, Table 1) Stahl and while height goes as Wo24 ( W1j4 predicted). Gummerson (25) studied 35 primate specimens representing five species over a weight range from 0.28 to 22 kg. Most of the Stahl and Gummerson data on gross body dimensions agree well with the elastically similar model: the somatic exponent b is 0.28 for trunk height (x predicted) and 0.38 for maximum thigh girth (34 predicted), with correlation coefficients for the leastmeansquares fit generally above 0.95 (0.99 in the case of chest circumference). Agreement between their measurements of individual bone proportions and the rule of elastic similarity is less satisfactory (femur length o= W”*34, fibula length o= 1/1/O.24). It may be that the bones of the lower limbs in these primate specimens make up an increasing fraction of the total skeletal weight as body weight increases. The mean skeleton b value was between 1.O and 1.1. In other animal families, a skeletal weight/body weight ratio independent of body size has been found. Davis (7) reports skeletal weight proportional to wo*gg in 14 cats over a weight factor of 60, (making skeletal weight a fixed fraction of body weight, as required in each of the presently proposed models). In a recent study of the overall length and midshaft diameters of 5 limb bones in 118 skeletal specimens representing 95 species of adult ungulates, I found very reasonable agreement with the predictions of the elastically similar model (20). In Fig. 2, data are shown for the humerus of 103 artiodactyl specimens in which p = 0.66 (95 predicted) with a correlation coefficient of r = 0.93. Since the bones differ in length by a factor of seven, they should represent animals different in weight by 74 = 2,500. Table 2 summarizes the leastmeansquares powerlaw fits to the artiodactyl data including the body weight exponents for length and diameter calculated according to Eq. 3 assuming that any individual bone is the same fraction of body weight in each animal. Body surface area. The surface area S of each quasicylindrical element increases as the product of d and I,
s
= 0.667;
x
= 0.750;
g
= 0.800.
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622
T. A. McMAHON
Somatic
Humerus Ulna Femur Tibia Metatarsus Forelimb Hindlimb
I = I = 1 = I = I= 1 = I =
24.09d 35.86d 23.34d 36.64d 25.73d 51.84d 91.83d
One hundred and three weight range factor = 2,500.
Exponent,
b
r
ld Relation
0.655 0.633 0.747 0.647 0.685 0.670 0.647 specimens
0.926 0.865 0.954 0.889 0.819 0.937 0.911
la, Wb
d 00 Wb
0.247 0.241 0.272 0.245 0.255 0.251 0.245
0.377 0.380 0.364 0.378 0.372 0.375 0.378
representing
83 species
;
neglecting the ends. Thus S a IV /3W1/3 = IV213 for geometric similarity, S a IV1 j41V3’* = Mr5/* for elastic similarity, and S a IV1’5MJ2/5 = W415 for static stress similarity. In data spanning the range from rats to humans, Stahl (24) found that S increases as PV”.65. Hemmingsen, (14) reported surface area measurements over a weight range of 106, and his data can be fitted by a somatic exponent between 0.63 and 0.67 (21). Thus the available body surface area data are in agreement with the elastically similar model, but do not support this model to the exclusion of the others. Muscle mechanics and energetics. Observers since Galileo (9) have hypothesized that metabolic power must somehow be limited by total body surface area. The data of Stahl (24) and Hemmingsen (14) cited above show that total body surface area increases less rapidly with size than the surface area of a sphere of the same weight (e.g., the somatic exponent is less than 0.67), but Kleiber (18) has pointed out that basal metabolic rate is proportional to M/o*75, and Hemmingsen (14) has extended this result to maximal energy metabolism. The difliculty in relating area and metabolic power is resolved if we replace total body surface area S by body or limb crosssectional area A in Galileo’s hypothesis, since in the elastically similar model, crosssectional area is proportional to d2 and thus to Mj3j4 (A o= Mrzj3 for geometric similarity, A a W4j5 for static stress similarity) . power be proporIsotonic power. Why should metabolic tional to A? Two possible reasons may be considered. First, following the Hill argument introduced earlier, suppose a area is proportiona 1 to A and muscle whose crosssectional tendons is proportional to I, whose overa .ll length including shortens a length AI against a cons ta nt external load a,,, c iJ in time At. The power the muscle provides to its environment is u musclJ(Al/At). If both gmuscie, timeaveraged stress developed in the muscle, and AZ/At, the timeaveraged shortening velocity, are invariant of body size, then the metabolic power developed by homologous muscles in animals of different size depends only on A, as we set out to show. In a recent review article, Close (6) concludes “the peak twitch tension per unit crosssectional area of muscle is about the same for fast and slow limb muscles of the same and different animal species despite marked differences in the isometric twitch contraction time. ” Thus there is experimental evidence for taking to be independent of body size, as we needed to cmuscle
assume above. The size dependence of shortening velocity is a point we shall return to later. Each of the three models is consistent with the widely observed fact that work per stroke under conditions of maximal energy metabolism is proportional to a muscle’s weight, and thus to body weight Mr. This is because work a urn u sc leAA/, and AAl is proportional to muscle volume while om u sc ie is independent of size. For example, data reported by Stahl (24) is presented in Table 3 showing that respiratory work/breath, a parameter fairly easy to observe, is proportional to body weight. Isometric power. A second, but perhaps more important consideration is that muscle requires a metabolic power input to maintain tension, independently of whether or not it does work. Studies by Hartree and Hill, (12) among others quoted by Ruegg (ZZ), show that isometric metabolic power increases directly as muscle force, hence directly as crosssectional area for muscles bearing the same stress. The equivalent statement for cardiac muscle is that the tensiontime index is found to be a more important determinant of 02 consumption than is cardiac work. Since comparative evidence exists (19) that ventricular wall stress is the same in mammals of widely different size, the power required to run the heart increases directly as cardiac muscle crosssectional area, just as the power required to maintain a given level of skeletal muscle tone (a given fraction of the total number of muscle fibers active) increases directly as limb crosssectional area. Application to lung and heart frequencies. Many physiological periods are found to be powerlaw functions of body weight with exponents close to 0.25 (see Table 3). Kleiber (17) has proposed a metabolic explanation for the dependence of heart rate and respiratory frequency upon body weight: since ventricular stroke volume and respiratory tidal volume are found to be proportional to PP”, and cardiac output and minute ventilation have body weight exponents near TABLE
3. Mammalian Variable
Respiration
Heart
allometric rules ~.~._~_____~~ Mass Exponent,
rate
0.26, 0.28
rate
0.26,
0.27, 0.25, 0.23* 0.29, 0.19*, 0.23*
Lifespan Time for 50% growth Time for 98% growth Halflife of indicator injected into plasma Tidal volume
0.25 0.26 0.25
1.04,
1.01,
Heart
weight
0.98,
0.98*
Power Volume
of breathing 02 uptake
0.78 0.76,
0.73
Cardiac output Inulin clearance Work/breath Y
= aWb
b
0.81 0.77 1.08 * Data
1.08
Stahl (24), Gunther & Guerra (1 l), Adolph (1) Adolph, (l), Stahl, (24), Calder (3) Lindstedt, cited by Calder (3) Stahl (25a) Stahl (25a) Dedrick et al. (8)
Stahl (24), Adoph Gunther & Guerra Stahl, (24), Gunther Guerra (11) Stahl (24) Stahl, (24), Gunther Guerra (11) Stahl (24) Stahl (25a) Stahl (24)
(l), (11) &
&
for birds.
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SCALE
EFFECTS
IN
QUADRUPEDAL
RUNNING
623
E is the slope of the stressstrain curve for the stimulated muscle, A its crosssectional area, and I its length. Setting m proportional to pld2, where p is the mass density, we write the equation of motion for vibrational displacement x of the mass
F A
spring
const.
k =g
=z
mass X acceleration = force pld2j;. =  (AE/l)x
= AE/I
(5)
We solve the equation by assuming the form x = x0 cos ZY~ It, where ~0 is the initial displacement from the equilibrium position. Thus j; = x0(27& i)2 cos 271~It. Substituting in (I), we obtain, by setting A o= d2
f nl
FIGURE 3. A: first geometry : a simple massspring combinauon. Contracted muscle is a spring, k = AE/Z. With m proportional to d2Z, natural frequency fIl is proportional to 1/Z. B: second geometry: a muscle acting to extend a joint. Natural frequencyf,, is now multiplied by the mechanical advantage of the joint, i.e., fn a d/Z2. C: third geometry: a system of joint, links, and muscles represented by a beam in its lowest mode of vibration. Natural frequency is still proportional to d/i2.
s/4, then heart rate and respiratory frequency must go as M/r’ j4. Clark (5) measured the aortic crosssectional area in mammals over a weight range of lo6 and found it proportional to PV” .72, so that mean blood velocity and thus overall (apex to base) shortening speed Al/At of cardiac tissue are calculated to be approximately body weight independent. Only the elastically similar model is compatible with all these observations, as the reader may review for himself in Tables 1 and 3. Three basic mechunicul geometries. In Fig. 3, three mechanical geometries representative of various special cases of animal motion are shown. None of these geometries are representative of walking, because the influence of gravity is neglected Cavagna and Margaria (4) and others in each problem. have shown that walking is characterized by small exchanges between the kinetic and gravitational potential ene rgy of the whole body. Most walking is thus well modeled the limbs plu s the body as an inverted bY considering pendulum. The limb acts as polevaulting pole during the interval when the foot is on the ground, lifting the body as it moves forward. Galloping is quite different. In galloping, the accelerations provided to the limbs by the muscles are much larger than those due to gravity. If energy is to be exchanged between a kinetic and potential form in galloping, the potential form must involve stretching of elastic elements within the bodv. The first geometry (Fig. 3, A) shows a simple massless spring, representing a muscle attached to an inertial load, m. This situation is thus roughly representative of the heart, where the inertial mass is the intraventricular blood volume plus the effective mass of the myocardium. The spring is the tensionlength relation of the myocardium (whose fibers may be thought for the moment to run directly from apex to base). As illustrated beside the figure, the spring constant k is proportional to EA/Z, where
1 O= l
E J
i
(6)
Thusf, is proportional to PV1/3, M/T_l 14, and PP1j5 in the geometric, elastic, and static stress models. Recall from an earlier paragraph that the basal frequency of the heart and lungs may be expected to decrease as W1/4 for elastically similar animals on metabolic grounds; it is interesting to note that the natural frequency decreases the same way. The second mechanical geometry (Fig. 3, B) shows the muscle acting to extend a single joint. The center of mass of the swinging limb is located a distance proportional to I from the pivot, making the moment of inertia of the limb proportional to Z2(ld2). Th e moment acting to extend the joint is provided by the muscle force F acting through the joint radius, which is proportional to d. Balancing the reaction moment with the extending moment EAd2 p13d28 a 18 The natural
frequency
(7)
of this system is
d 1 E (8) I I J ; Notice that the joint has introduced a scaledependent mechanical advantage for the muscle. This effect causes the natural frequency to increase by a factor d/l over the first mechanical geometry, giving the result that fn is proportional to I/Vl 13, M/l / 8, and W” in the geometric, elastic, and breaking stress models. Now consider the case when there are many joints instead of just one. The third mechanical geometry (Fig. 3, C) shows a continuous beam oscillating in its lowest mode of vibration. The beam represents the limiting condition when there are a large number of muscles and joints, as shown schematically. Appendix 1 shows that the natural frequency fn for the lowest mode of vibration of a beam of anv crosssectional shape is
f n2
Oc
Ed4 pZ4d2
=
(9) just as was found above for a single joint. Thus, fn3 is proportional to Wl 13, Wl Is, and W” for the geometric, elastic and static stress models, respectively. Quadrupedal galloping. Hill (16) concluded, on the basis of the geometrically similar model, that if overall muscleplustendon shortening speed was body size independent in homologous muscles, all quadrupeds should be able to
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624
T.
reach the same speed. The agreement with animal data is satisfactory over a narrow range of body weights (whippet to race horse), but poor over the range from mouse to horse. It is a great disadvantage to have to compare the top speeds of animals, since the question arises whether one means top sprinting speed or top speed for sustained running. There are also the questions of training and motivation: how is similarity ensured here? Heglund et al. (13) found that running animals make a transition in gait under conditions which are physiologically similar. Our hypothesis at present is that muscle stress is the same at the trotgallop transition in homologous muscles of animals of different size. In what follows, animals of dissimilar size running exactly at the speed where their gait changes from trot to gallop will be compared according to the three similarity models. Stride frequency. The findings by Heglund et al. (13) that stride frequency increases only slightly as an individual quadruped increases its speed in a gallop is evidence in favor of the idea that animals gallop at frequencv determined by their lowest mode of vibration (third mechanical geometry). If this is true, stride frequency at the trotgallop transition should be proportional to lV1 13, VP1 j8, and M/O in the three models. The experiments (Fig. 4) showed frequency proportional to PP”*14, which is substantially different from the weight independence predicted by static stress similarity or the VVV1’3 predicted by geometric similarity, by elastic similarity. but quite close to the 1V1’8 predicted Speed. Running speed at the trotgallop transition in30g MOUSE
A. McMAHON
creases as the product of stride frequency and stride length, As Hill noted, geometric similarity reor VTG = fTGDw quires speed to be independent of body weight. Heglund et al. (13) found uTG a ?V”*z4, in sharp disagreement with Hill’s prediction (Fig. 4). Stride length. In geometric similarity, where there is only one length scale, stride length is constrained to be proportional to VV lj3. What happens when there are two length scales, as there are in elastic similarity and static stress similarity? Stride length measures the distance an animal travels over the ground in one stride cycle. This distance depends not only on the length which the toe of a single foot moves during the period the foot is on the ground, but also on the degreeto which motions of the two hindlimbs (or two forelimbs) are asynchronous and thus add their steps in series. It also depends on the distance traveled by the body during freeflight phases of the motion. Thus we are not encouraged to look for a simple kinematic answer to the question of which length scale to associate with DTG, the stride length at the trotgallop transition, but fortunately there is an energetic principle which can be a guide. Gold (10) and Herschman (15) have discussed the energy per kilogram of body weight which an animal requires for each step of running. This would be the wholeanimal analog of the specific work per muscle operation, which we noted earlier is found to be body size independent in isolated muscle experiments and in measurements of the work performed in breathing. Taylor et al. (26) present an empirical relation giving the specific metabolic power cost of running at any arbitrary soeed v
!$ =
(8.46 W“*40)v+ 6.0 Wo*25
w9
360 g RAT
where to2/W has the units ml 02. gl hl and v is in km. hl. The first term of this equation may be called an incremental cost, since it represents the increase over the second term (postural cost) required to run at the speed v. We may calculate from the first term the specific energy required to run per step at the trotgallop transition l
I 15 SPEED
W, BODY
I 20 KM/FIR
WEIGHT
I
I
I
I
25
30
35
40
Kg
FIG. 4. A: stride frequency vs. speed for mouse, rat, dog, and horse. There is a pronounced discontinuity in the slope of the curves at the transition between trot and gallop. B: speed at trotgallop transition as a function of body size. Data fit a powerlaw regression in agreement with elastic similarity, UTG = 5.5 WO.24, r = (0.93), which txedicts UTC, a w1j4.
work =.gstep
work g* h
0.40 h a ~VW step VIDTG
=
DTG Wo*4o
(II)
The expression on the right can only be independent of body size if we associate D TG with the diameter dimension d in our similarity models; this would require DTG to be proportional to W’ i3, VV318, and W215 in the geometric, elastic, and static stress models. In the recent experiments reported by Heglund et al. (13) DTG was computed from VTG and
[email protected] AS shown in Table 4, DTG was proportional to Wo38. Thus the principle of the constancy of muscle work per stroke has led us to predict that DTG is associated with the diameter length scale, which increases with W318 in elastic similarity, and this prediction agrees with our observations. Peak muscle stress. As noted earlier, there is extensive evidence that the peak tension per unit crosssectional area, with all motor units active, is about constant between limb muscles within a given animal, or even between limb mus
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SCALE TABLE
EFFECTS
IN
4. Kinematics
QUADRUPEDAL
of trotgallop Geometric Similarity
Natural Stride
frequency, frequency,
fII fTo
625
RUNNING
transition Static Stress Similarity ~._
Elastic Similarity
w113
W11
8
w1/3
W11
8
W0 W0
Expt
(Ref)
wo.14
weg
lund
et
al.
(13)) Stride Running
length,
DT~
speed,
UTG
w1/3
W3’
W0 W0 W0
8
M/215
Wo38 (Heglund
J+71/4
M/‘2I 6
W” .24 (Heglund
MI0
W0
Wl/8
w1’6
et al et al.
Peak muscle stress Limb excursion angle, Al Muscle stroke length, Stride efficiency, DTG/ ZAO Intrinsic muscle velocity,
/TG(A~)/~
w0
(13)) (13))
.lO
(this
study)
w1/3
w1/4
M/“/6
W0
M/TV4
M/2/6
WO.29 (this
Wl/”
W118
W0
wo.13
study) from
(data Close
(6))
cles of animals of different species (6). Thus the peak muscle stress at the trotgallop transition is taken to be a constant in each of the models, implying that the same fraction of the total number of muscle fibers are active under these conditions. No direct measurements during galloping of this parameter are yet available. Limb excursion angle. In Fig. 3, where the action of a muscle at a joint is shown in the second and third mechanical geometries, one may see that muscle stress, F/A, depends directly on the slope of the stimulated stressstrain curve, E, and the product AM/Z. Thus for constant muscle stress, the excursion angle A0 for any joint must be proportional to I/d, and therefore W”, Wl 18, and Wl I5 in the three models. ‘In Fig. 5, this prediction is tested against experiments. I traced individual frames of motion pictures taken by N. Heglund of animals running at the trotgallop transition and measured the maximum angular excursion of a line drawn from the head of the femur to the toe of the foot, as shown in the figure for mouse and horse. I also measured the length of the foot in the extreme forward and back positions, II and 12. Both the maximum excursion angle of the hindlimb, 02  81, and the cumulative angle of the hindlimb joints, (/p  11)/11, were greater in the mouse than in the rat, dog, or horse. The former was proportional to w*1o, and the latter proportional to Wsog, in reasonable agreement with the predictions of elastic similarity. A4uscle stroke length. Since the overall distance AZ a muscle shortens is proportional to da& AZ is proportional to I or WI3 WI4 and lW5 in the three models . StAde ej&ency. It is apparent from all the above that the distance an animal’s body moves in one stride, DTG, is not the same as the arc length AB through which the toe of the foot moves with respect to the head of the femur in one stride. The ratio of these two lengths, D,,/lAe, may be called the stride efficiency, which is therefore predicted to be proportional to W”, lW4, VW5 in the three models. Since all the parameters which enter into the stride efhciency have been measured in the paragraphs above, we may compute the experimental stride efficiency as propor
tional to W (.38  .25 + .lO> = W”*23. The precise kinematics by which animals achieve this bodyweightdependent stride efhciency must be a subject for further research. Part of the effect may be due to the sequence of hindlimb foot falls,c which are closer to being synchronous in smaller animals than in larger ones. Another part may be due to the fact that the chord length AB, which is a measure of the horizontal distance traveled during the time a single foot is on the ground, is not the same as the arc length AB, and in fact is a smaller part of the arc length in smaller animals. This is a consequence of the fact that the angular excursion of all joints is larger in smaller animals at the trotgallop transition, as noted earlier. Intrinsic muscle velocig. We may calculate that if all the muscles involved in locomotion execute a length change AZ in time lIfTG, then the intrinsic muscle velocity, which is proportional to the rate of change of length of individual sarcomeres, should be proportional to J,,A///, or 11” /3, M/l I8 > and M/O in the three models. This is because sarcomeres basically occur in one length, and are arranged in series. The sarcomere velocity is the overall muscle velocity divided by the number of sarcomeres, and this number is proportional to the resting length of the muscle. Close (6) has collected data on the intrinsic speed of shortening of the soleus and extensor digitorum longus muscles in isolated muscle experiments in mice, rats, and cats. His data are plotted in Fig. 5. On the basis of this Angular dJ
8
excursicn
of rear
limb
= 37.2
W‘lo
/ 0 0
W,
BODY
WEIGHT,
kg
FIG. 5. A: maximum angular excursion of the rear limb at trotgallop transition as shown for tracings of mouse and horse. Elastic similarity predicts A0 a Wq118; here angular excursion = 37.2 WoJO. B: intrinsic speed of sarcomere shortening; data from Close (6). EDL = extensor digitorum longus. SOL = soleus. Elastic similarity predicts sarcomere speed a W1/8.
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626 limited data, it appears that intrinsic muscle velocity is proportional to WO ml3 in the soleus muscle, in agreement with the model postulating elastic similarity. How does a sarcomere decide whether it is in a large or a small animal? Perhaps the Golgi tendon organs provide a feedback mechanism, resulting in alteration of the local myosin ATPase activity, which prevents muscles from contracting against inertial loads at rates which would induce unbearably great stresses. Recall that in all of the presently proposed models, muscle stroke length and thus shortening speed has been set to insure that peak muscle stress is scale independent. Metabolic power for galloping. According to each of the present models, animals gallop under conditions of frequency, speed, etc. determined by the elastic properties of muscles acting as springs against the individual masses of all of the various links of the body. The stimulated muscles are not ordinary springs, but are springs for which hire must be paid, since the metabolic cost increases directly as the muscle force. Thus the metabolic power required for running at the trotgallop transition or any other physiologically similar speed (for which peak muscle stress is body size independent) should increase as limb crosssectional area or W 13, W3 j4, and W4j5 in the three models. In his studies on the cost of locomotion, Taylor et al. (26) found that the absolute metabolic cost of locomotion in ml 02. hl is made up of two terms (Eq. 10). Multiplying through by W, we find that the second term, which inof the speed and therefore creases as M/0.75, is independent may represent the cost of maintaining all the muscles with sufficient tone to be ready to take part in the running activity. The first term depends directly on the running speed, which illustrates how increasing speed leads to increasing muscle forces and thus increasing metabolic costs. At the where the running speed is proportrotgallop transition, tional to W” .24, (13) the first term contributes a factor proportional to WC0 e6+ O.24) = W”ws4. Thus the total cost of locomotion at the trotgallop transition is empirically given by the sum of two terms of approximately equal magnitude, with a combined body weight dependence in the neighborhood of W”.75.
T.
elastic and static stress models is the running speed, which is greater in the larger species. This greater speed for greater size (not found in geometric similarity) may have provided evolutionary pressures for the development of larger forms, since it comes a,bout as a natural consequence of the sizeinduced distortions of an animal’s shape. APPENDIX Calculating the Natural Frequency third Mechanical Geometry2
of the
In the third mechanical geometry, Fig. 3, C, several joints and their interconnecting links are considered. The situation represents an animal galloping, where elastic energy may be stored in the stretched muscles of the trunkflexing vertebral column as well as those of the limbs. Note that here, as in the first and second geometries, the contracted muscles are the springs. When the number of links and joints becomes suitably large, the mechanical situation is that of a beam in its lowest mode of vibration. As shown schematically in Fig. 3, C, the strain c of a fiber a distance y from the neutral axis is c =
[(I

y>dO

rde]/rde
= y/r
(A1)
where r is the local radius of curvature. The local stress c is c E, where E is the slope of the stimulated muscle stressstrain curve, as explained in the text. Summing over the crosssectional area of the beam gives the elastic restoring moment A4
s
A4=
E = 
CEybdy
EI
r s A y26dy
A
(A2)
= r
where I is the local moment of inertia of the cross section, whose breadth is b(y). If we assume a form 6(s, t) = S(s) cos wt for the deflection 6 of the beam from its equilibrium position, where s is the distance from the end, the instantaneous acceleration of any small segment during vibration is a(s,
t> =
 026(s)
cos wt
(A3’
and the inertial reaction force per unit length is w’ = apbd, where p is the mass density and d is the (assumed uniform) thickness of the beam. Therefore, the vibrational bending moment is A4’ Summing zero
= (s 
the restoring EI r
(A4)
cos 6’ d.z
A4 and the bending
moment
M’
to
s 
&‘(z)
(s 
s0
Setting d26
.dw’(z)
moment
CONCLUSION
Three separate models are presented for comparing the morphometric and physiologic consequences of body size, all other factors being equal. In geometric similarity, only one length scale appears, since all dimensions are multiplied by the same factor as body size changes. Elastic similarity and similarity with respect to static stress both require distorted models where length dimensions are multiplied by one factor and diameters by another as body size changes. Elastic similarity provides the best comparison with data on gross rnorphometry, body surface area, metabolic power, and the kinematics of locomotion. In all of the models, peak contractile stress and specific work/stroke are the same for homologous muscles in animals of different size running at the trotgallop transition, which we take to be a physiologically similar speed. Among the several physiologic parameters which are a function of body size at the trotgallop transition in the
A. McMAHON
1  rv
EI
d26 z,
r
2 cos wt pbd
ds2=
(A5)
edz
cos
(A 6)
’ s o
b 