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First published online January 19, 2006
Journal of Experimental Biology 209, 455-465 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02029
Dynamically similar locomotion in horses
Department of Anatomy, University of Bristol, Southwell Street, Bristol, BS2 8EJ, UK
* Author for correspondence at present address: Human Performance Laboratory, Faculty of Kinesiology, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada (e-mail: sbullimore{at}kin.ucalgary.ca)
Accepted 6 December 2005
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Summary |
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Key words: ontogeny, allometry, gait, intraspecific, horse, Equus caballus
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Introduction |
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)
showed that mammalian species of very different sizes use the same values of
these dimensionless parameters when moving at speeds corresponding to equal
values of another dimensionless parameter, the Froude number
(u2/gh, where u is forward speed,
g is the acceleration due to gravity and h is hip
height).
All of the dimensionless parameters considered above are `mechanical'
dimensionless parameters, because the parameters from which they are
constructed can be defined in terms of forces, lengths and times. Systems that
have equal values of mechanical dimensionless parameters are said to be
`dynamically similar' (Isaacson and
Isaacson, 1975), so Alexander and Jayes
(1983) described animals that
have equal values of the above dimensionless parameters as moving in a
`dynamically similar fashion'.
Dynamically similar locomotion is not, however, inevitable in animals
moving at equal Froude number. For example, Alexander and Jayes
(1983) found that cursorial and
non-cursorial mammals do not move in a dynamically similar fashion at equal
Froude number and Donelan and Kram
(2000) found that humans do
not move in a dynamically similar fashion when running at equal Froude number
in different levels of simulated reduced gravity. Engineering theory tells us
that, for dynamic similarity to occur in some aspect of a system, all relevant
system parameters must scale appropriately with size
(Isaacson and Isaacson, 1975).
Previously, we have applied this to understanding dynamic similarity in animal
locomotion. We argued that, because tendon elastic modulus does not increase
with size as required for dynamic similarity, compensatory changes in other
parameters with size are required for dynamically similar locomotion to be
possible (Bullimore and Burn,
2004). We referred to these changes as `compensatory distortions',
after a term that has been used in engineering to describe the changes that
must be made in physical models in order to make them dynamically similar to
the systems that they represent (Baker et
al., 1973). We showed that the changes in limb posture that occur
with size in mammals (Biewener,
1989) should compensate for the size-independence of tendon
elastic modulus sufficiently for dynamically similar locomotion to be possible
in species of different sizes.
This is an example of a general principle that is frequently encountered in
studies of size effects in biology: where organisms of different sizes are
made of the same materials, they cannot remain functionally similar unless
systematic changes in form compensate for size effects. For example, for
animal bones and tree trunks to maintain the same resistance to elastic
buckling, their thickness must increase disproportionately with length
(McMahon, 1973). Conversely,
where form remains the same, differently sized organisms differ functionally.
For example, the functional characteristics of the hairy appendages of
crustaceans are size-dependent (Koehl,
2004). In the case of mammalian locomotion, changes in limb
posture with size allow functional similarity in locomotor mechanics to be
maintained despite the size-independence of tendon material properties.
However, if no factor compensates for size effects, changes in locomotor
patterns with size can be expected, so locomotion will not be dynamically
similar.
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Here we address both of these questions for the first circumstance under which intraspecific size differences occur: variation between adult individuals. We chose to study the domestic horse (Equus caballus) because it occurs in a wide range of sizes, allowing small deviations from dynamic similarity to be detected, and because there is a large volume of published anatomical and biomechanical literature on this species. To address the question of whether detectable deviations from dynamic similarity could occur (question 1 above), we modelled `idealised' horses that were identical except for size differences, so that there were no compensatory distortions, and predicted by how much locomotion would deviate from dynamic similarity in these animals. To address the question of whether real horses of different sizes move in a dynamically similar fashion (question 2 above), we measure RSL and DF in trotting horses of body mass 86-714 kg. The theoretical predictions are presented in Part 1 of this paper and the experimental results in Part 2.
A factor that could potentially decrease the effects of size on locomotion
is the nonlinear stress-strain relationship of tendon. Because tendons are
stiffer at higher stresses, a given increase in stress causes a smaller
increase in strain when it occurs at higher stresses. A consequence of this is
that tendon strain will scale less markedly with size than tendon stress
(Fig. 1). If tendon elastic
modulus does not increase with size as required for dynamic similarity, and
there are no compensatory distortions, then it would be expected that larger
animals would experience greater tendon strains, causing deviations from
dynamically similar locomotion (Bullimore
and Burn, 2004). The capacity of the nonlinear tendon
stress-strain relationship to decrease the scaling exponent for strain would
reduce this effect. This is not a compensatory distortion, because it does not
depend upon something changing with animal size. It also could not compensate
completely for a size-independent tendon elastic modulus, because it would
never give the same tendon strain for different stresses. However, it could
reduce deviations from dynamically similar locomotion to some extent. In order
to estimate the magnitude of this effect, we made the predictions in Part 1
both with and without taking these nonlinear tendon properties into
account.
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(Part 1) Predicted scaling exponents |
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and all
areas scaled in proportion to 
,
where Mb is body mass). This represents situation (i)
above, except that variation due to factors other than size is ignored. The
relationships of RSL and DF to body mass predicted for these `idealised'
horses were then compared to the relationships measured in real horses in Part
2. A difference between the predicted and measured relationships was taken to
indicate that systematic changes with size do occur, i.e. that possibility
(ii) above is correct. The predictions were made in four stages. We first predicted the scaling of tendon strain with size, and used this to estimate the scaling of joint angular excursion. From this, we predicted the scaling of limb stiffness, which we defined as the ratio of peak GRF to limb shortening during the stance phase. Lastly, we used the planar spring-mass model of locomotion to predict how this scaling of limb stiffness would affect RSL, DF and RPF. We made the predictions for both a linear tendon stress-strain relationship and a realistic nonlinear relationship, in order to determine how much this would affect deviations from dynamic similarity.
In order to predict limb stiffness, i.e. in stages (i) to (iii) of the
predictions below, we took peak GRF to be proportional to
Mb in horses moving at equal Froude number. In horses that
are not moving in a dynamically similar fashion, peak GRF will not be exactly
proportional to Mb. However, because the force-length
relationships of the limbs of animals tend to be approximately linear
(Farley et al., 1991;
McGuigan and Wilson, 2003),
limb stiffness is not sensitive to the GRF for which it is calculated. The
scaling exponents predicted below for RPF are small
(Table 1), justifying this
approximation.
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We also assumed that peak tendon force would be proportional to peak GRF, and therefore also to Mb. Again, this assumption will only be exact for dynamically similar horses. If larger horses experience greater joint angular excursions, as expected if tendon elastic modulus is size-independent and there are no compensatory distortions, the moment arms of the GRF about the limb joints will be relatively greater. This will increase the tendon forces required to counteract a given GRF and so could potentially increase deviations from dynamic similarity. From this point of view, the predictions made here are conservative.
(i) Predicted scaling of tendon strain
For a peak tendon force proportional to Mb, and a
tendon cross-sectional area proportional to
, tendon stress will increase with
size in proportion to . For
tendons with a linear stress-strain relationship, strain will scale in
proportion to stress and therefore also to
. In order to predict the scaling
exponent for strain in tendons with a nonlinear stress-strain relationship, we
used the data of Riemersma and Schamhardt
(1985), digitised from their
fig. 4. This gives stress-strain relationships for a superficial digital
flexor tendon (SDFT), a deep digital flexor tendon (DDFT) and a suspensory
ligament (SL) from an equine hindlimb. These structures cross the
metacarpophalangeal joint in the forelimb and the metatarsophalangeal joint in
the hindlimb. Hyperextension of these joints is responsible for a large
proportion of the shortening of the limb during the stance phase in trotting
horses (McGuigan and Wilson,
2003) and, because the muscle fibres associated with these
structures are relatively short (Ker et
al., 1988), it is likely that most of this hyperextension occurs
through tendon elongation. Therefore, the properties of these tendons have a
substantial influence on overall limb compliance. To determine whether the
data of Riemersma and Schamhardt
(1985) were typical we fitted
straight lines to the stress-strain relationships between strains of 3.5% and
6.5% (R2>0.99). This gave elastic moduli of 1.15 GPa,
1.44 GPa and 0.56 GPa and intercepts with the strain axis of 1.6%, 2.1% and
2.1%, for the SDFT, DDFT and SL, respectively. Comparison with published
values indicated that these are typical properties for mammalian tendons: mean
elastic modulus for mammalian limb tendons is 1.24±0.23 GPa
(Pollock and Shadwick, 1994)
and mean intercept strain for the equine SDFT is 1.5%
(Wilson, 1991). In order to
make predictions for horses with identical tendon material properties, we used
the stress-strain relationships measured by Riemersma and Schamhardt
(1985) for horses of all
sizes.
The relationship between the scaling exponents for stress and strain
depends upon which part of the tendon stress-strain relationship is used.
Biewener (1998) calculated peak
tendon stresses of approximately 16, 20 and 13 MPa in the SDFT, DDFT and SL,
respectively, in trotting horses of approximately 275 kg. We used these values
as a starting point for calculating the scaling exponent for strain, which
corresponds to a stress proportional to
. Tendon stresses for horses
between 80 and 800 kg were predicted using the equation:
 | (1) |
where the scaling constant, k, was calculated from the above
stresses for a 275 kg horse, giving values of 2.46, 3.08 and 2.00 for the
SDFT, DDFT and SL, respectively. The tendon stress-strain data of Riemersma
and Schamhardt (1985) were
fitted with third order polynomials (R2>0.99) and these
were used to predict tendon strains corresponding to the stresses predicted by
Eqn 1. Allometric equations relating these predicted strains to body mass were
obtained by log-transforming the data and fitting linear least-squares
regression equations (R2>0.99), as described by
Schmidt-Nielsen (1984). This
gave scaling exponents of 0.19, 0.16 and 0.18 for strain in the SDFT, DDFT and
SL, respectively. To determine how robust these predictions were, we varied
the stresses used for calculating k. Large variations (±5 MPa)
altered the predicted exponents by less than 0.01. The predicted exponents are
considerably lower than the exponent of 0.33 for a tendon with linear
properties, indicating that nonlinear tendon properties could substantially
reduce the effects of size on tendon strain.
(ii) Predicted scaling of joint angular excursion
If the joint is modelled as having a circular profile, the joint angular
excursion arising from a given tendon strain can be calculated as:
 | (2) |
For geometrically similar horses, joint radii and tendon lengths will be
proportional to , so that joint
angular excursions will be directly proportional to tendon strain. Therefore,
we take joint angular excursion to be proportional to
for a tendon with linear
properties and to 
(the mean of
the scaling exponents calculated above) for a tendon with nonlinear
properties. In both cases this represents an increase in joint angular
excursion with size, but the predicted increase is substantially smaller for a
tendon with nonlinear properties.
The above calculation ignores the contribution of muscle strain to joint
angular excursion. This seems reasonable in trotting horses because most of
the length change in the limb occurs distally
(McGuigan and Wilson, 2003)
where the muscle-tendon units have relatively short fibres and long tendons
(Ker et al., 1988). However,
muscle strains would also be expected to increase with size, by the following
argument. Muscle stress would increase with size in the same way as tendon
stress, i.e. as . The horses we
are modelling have identical muscle activation patterns. Therefore, due to the
nature of the force-velocity relationship of muscle, these higher stresses
would tend to decrease the ability of the muscle to shorten and to increase
the possibility of it lengthening, resulting in greater muscle strains in
larger animals.
(iii) Predicted scaling of limb stiffness
In order to predict the scaling of limb stiffness, we needed to be able to
predict limb shortening from joint angular excursion, where limb shortening is
defined as the change in the distance between the proximal and distal ends of
the limb between ground contact and midstance. This cannot be done without
knowledge of limb morphology because the relationship depends upon limb
segment lengths and initial joint angles. For this reason, we used
experimental data from three horses (horses 7 and 8 from
Table 2 and a third horse of
630 kg) as a basis for predicting the scaling exponent for limb shortening.
Reflective markers were placed over the joint centres of the right fore- and
hindlimbs and their positions were recorded by optical motion capture (240
frames s-1; Proreflex, Qualisys, Sweden) during trot. The marker
positions were used to calculate the limb segment lengths, joint angles at
ground contact and joint angular excursions between ground contact and
midstance. Each leg of each horse was then used separately as the basis for
calculating scaling exponents for limb shortening. Limb segment lengths for
horses of 80, 200, 400, 600 and 800 kg were predicted by scaling the measured
lengths in proportion to (so that
they were geometrically similar), and joint angular excursions were predicted
by scaling measured values in proportion to
to model a tendon with linear
properties, or to to model a
tendon with nonlinear properties, as predicted above. Limb shortening was then
calculated trigonometrically, assuming initial joint angles were equal to
measured values. This produced a total of twelve sets of values for predicted
limb shortening against Mb: six corresponding to the fore-
and hindlimbs of each of the three horses for linear tendon properties, and
another six for nonlinear tendon properties. For each of these twelve
datasets, an allometric equation relating predicted limb shortening to body
mass was calculated by log-transforming the data and fitting a linear
least-squares regression equation (R2>0.99). Because
the calculated scaling exponents were very similar for the data based on the
three different horses, and on the fore- and hindlimbs, the means of these
scaling exponents were used. The mean exponent for limb shortening with linear
tendon properties was 0.77±0.01 (± s.e.m.) which, in combination
with a GRF proportional to Mb, gives a limb stiffness
proportional to 
. The mean
exponent for limb shortening with nonlinear tendon properties was
0.58±0.01 (± s.e.m.), which gives a limb stiffness proportional
to 
. Therefore, the predicted
scaling exponent for limb stiffness is higher with nonlinear tendon
properties, but is still substantially lower than the exponent of 0.67 that
would be required for dynamic similarity
(Bullimore and Burn, 2004).
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(iv) Predicted scaling of dimensionless locomotor parameters
The effect of the above scaling of limb stiffness on the dynamics of
locomotion was predicted using the planar spring-mass model
(Blickhan, 1989;
McMahon and Cheng, 1990). This
is a simple model of running gaits, such as trot, in which the animal is
represented by a point mass bouncing on a spring. It describes the mechanical
relationships between basic locomotor parameters such as limb stiffness,
stride length, stance time and GRF, and has been shown to be able to model the
mechanics of trotting in mammalian species of a wide range of different sizes
remarkably well (Farley et al.,
1993).
GRF data (1000 Hz; model 9287 force plate, Kistler Instruments,
Switzerland) and kinematic data (240 frames s-1; Proreflex) from
one horse of intermediate size (horse 8;
Table 2) were used to determine
model parameter values representative of a horse trotting at Froude numbers of
0.5, 0.75 and 1.0. With the calculated parameter values, the model predicted
RSL, DF and RPF to within 5% of the values measured in this horse. Parameter
values representing horses of 80, 200, 400, 600 and 800 kg were generated by
scaling limb stiffness as calculated above (i.e. in proportion to
for linear tendon properties and
to for nonlinear properties) and
keeping the other model parameters dynamically similar to the values
calculated for horse 8. The parameter values that were used are listed in the
Appendix. Predictions of RSL, DF and RPF for the horses of different sizes
were then obtained by numerical integration of the equations of motion for the
model using a function written in Matlab (version 6.5, The MathWorks, Inc.,
MA, USA). The method used has been described in detail previously
(Bullimore and Burn, 2006).
Allometric equations relating the predicted values of RSL, DF and RPF to body
mass were obtained by log-transforming the data and fitting linear
least-squares regression equations (R2>0.98). The
scaling exponents predicted for both linear and nonlinear tendon properties
are shown in Table 1. The
exponents for RSL and DF are positive, indicating that these dimensionless
parameters are predicted to increase with size, while the exponents for RPF
are negative, indicating that it is predicted to decrease with size.
Incorporating the nonlinear characteristics of tendon into the predictions
substantially reduced the exponents predicted for RSL, while the exponents
predicted for DF and RPF were small for both linear and nonlinear tendon
properties.
In summary, the results obtained in Part 1 predict that: (a) in horses that
do not exhibit any systematic changes with size, RSL at equal Froude number
will increase with size in proportion to approximately
, while DF and RPF will be close
to independent of size, and (b) relative to a tendon with a linear
stress-strain relationship, a tendon with a realistic nonlinear relationship
could reduce the effects of size on tendon strain, joint angular excursion and
RSL.
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(Part 2) Measured scaling exponents |
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Anatomical measurements
Girth was defined as the circumference of the trunk just behind the
forelegs. Wither height was defined as the height above the ground of the
highest point at the base of the neck. Body mass was measured using a
weighbridge, girth was measured using a tape measure and the height of the
withers and the greater trochanter above the ground were determined using a
measuring stick.
To calculate RSL and Froude number, a measure of leg length is required.
Previous studies (e.g. Alexander and Jayes,
1983; Donelan and Kram,
2000) used greater trochanter height in the standing animal.
However, the greater trochanter is difficult to palpate in horses and its
height above the ground varies with standing posture. Preliminary calculations
based on a pilot study indicated that error in this measurement was likely to
be a significant source of interindividual variability. Therefore, instead we
measured radius and metacarpus length between the most distal prominence on
the lateral side of the elbow joint and the prominence on the lateral,
proximopalmar aspect of the proximal phalanx. This measurement proved to be
substantially more repeatable than greater trochanter height. For convenience,
we multiplied the measured radius and metacarpus lengths by a scale factor of
2.05 to obtain a `calculated leg length', which was approximately equal to
greater trochanter height. This was done to make our results comparable with
previous studies. The scale factor of 2.05 was the mean value of the ratios of
greater trochanter height to radius and metacarpus length for all the horses.
This ratio was independent of body mass
(Table 3). Because the same
scale factor was used for all horses it did not contribute to interindividual
variability and so did not affect the capacity to detect deviations from
dynamic similarity. The calculated leg lengths are shown in
Table 2.
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Experimental procedure
All experimental procedures were approved by the University ethics
committee. Reflective markers were attached to the medial side of the left
fore hoof, to the lateral side of the right fore hoof and overlying the dorsal
spinous process of the fifth thoracic vertebra ('t5 marker'). The horses were
trotted along a track by an experienced handler. The first section of the
track was used for acceleration, the central section for data collection and
the final section to slow the horses to a stop. The positions of the
reflective markers were recorded at 240 frames s-1 using a 3D
optical motion capture system (Proreflex) as the horses passed through the
data collection region. The field of view was sufficient for at least one
stride to be recorded from all horses at all speeds. For all except horses 7
and 8, one stance phase of the left or right fore leg was recorded at 250
frames s-1 using a high speed digital video camera (MotionCorder
SR-1000, Kodak, Herts, UK). Trials were obtained over as wide a range of
trotting speeds as possible while maintaining an approximately constant speed
through the data collection region.
Data analysis
Allometric equations relating girth, wither height, greater trochanter
height and radius and metacarpus length to body mass were calculated by
log-transforming the data and fitting linear regression lines. Reduced major
axis (RMA) regression was used because this method is appropriate for data in
which error occurs in both the dependent and independent variables
(Rayner, 1985). The fit of the
allometric equations was assessed by calculating the mean absolute percent
deviation (MAPD) of the data from the values predicted by the equations
(Prothero, 1986). 95%
confidence intervals for the scaling exponent were calculated using the
t-distribution. If these confidence intervals included 0.33, the horses were
considered to be geometrically similar in the measurement in question
(Schmidt-Nielsen, 1984). The
ratio of greater trochanter height to radius and metacarpus length was
calculated and an allometric equation relating this parameter to body mass was
calculated as above, except that least-squares regression was used because the
relationship obtained using RMA regression did not fit the data well, probably
because the range of the dependent variable was small.
Forward speed was calculated as the mean velocity of the t5 marker over one complete stride. Stride length was calculated as the distance travelled by the left or right fore-hoof marker between consecutive stance phases of that hoof. Stance time was determined from the high speed video. Stride time was calculated as stride length divided by forward speed. RSL was calculated as stride length divided by calculated leg length, DF was calculated as stance time divided by stride time and Froude number was calculated as u2/gh, where u is forward speed, g is the acceleration due to gravity (9.81 m s-2) and h is calculated leg length.
For each horse, the RSL and DF that would be used at Froude numbers of 0.5, 0.75 and 1.0 were predicted by fitting quadratic equations to the data for RSL and DF against Froude number. Quadratics were chosen on the basis of a pilot study in six horses in which linear, quadratic, power and logarithmic functions were tested and quadratics gave the highest R2 values. Higher order polynomials were not used, despite sometimes giving higher R2 values, because the resulting curves often exhibited multiple extrema, which were thought unlikely to be representative of the true relationship.
For each of the three Froude numbers considered, allometric equations relating RSL and DF to body mass were calculated as described above. Least-squares regression was used because RMA regression gave a poor fit to the data. MAPD and 95% confidence intervals were calculated as described above. If the confidence intervals included zero, the parameter in question was considered to be independent of body mass and therefore to be dynamically similar.
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Results |
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At all three of the Froude numbers considered, the scaling exponents for RSL and DF against body mass were not significantly different from zero (Fig. 2, Table 4). This conclusion was not altered by including horses 18-21 in the analysis. Including data from these horses increased the body mass range without greatly increasing the scatter of the data, so resulted in narrower confidence intervals (Table 4). The results support the hypothesis that RSL and DF are independent of size in horses and therefore that these parameters are dynamically similar in horses trotting at equal Froude number.
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Discussion |
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(Table 1). This illustrates the
argument that we have made previously that dynamically similar locomotion
would not be expected in geometrically similar animals with identical tendon
properties (Bullimore and Burn,
2004). The scaling exponent of 0.10 is well outside the 95%
confidence intervals that we obtained experimentally when all horses were
included in the analysis (Table
4), so would have been detectable in our experimental measurements
(Fig. 3). This suggests that
detectable deviations from dynamically similar locomotion are possible over
the size range present in adult horses.
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In this study, we have only considered the consequences of size differences
between adult individuals. However, the size changes that occur during growth
are often much greater. Therefore, it seems likely that detectable alterations
in locomotion could arise within an individual during growth. It would be of
interest to determine whether this occurs and to identify the biomechanical
consequences. In addition to the effects of size, locomotion will also be
influenced by changes in anatomical proportions and tendon properties during
growth. For example, the increase in tendon elastic modulus that occurs with
age in some species (Gillis et al.,
1995; Yamamoto et al.,
2004) may allow dynamically similar locomotion to be maintained
despite large size differences.
The question of the physiological significance of the predicted deviations from dynamic similarity is more difficult to answer. A useful first approach is to compare the magnitude of the predicted size effects with the amount of interindividual variability that arises from other sources. The values of RSL that we measured at a Froude number of 0.75 ranged between 1.50 and 1.69, although some of this variability probably arose from experimental error. With a scaling exponent of 0.10, mean RSL would increase from 1.50 to 1.85 over the size range of horses used in this study. This effect is larger than the measured interindividual variability, so could potentially be physiologically significant.
The scaling exponents predicted for DF and RPF, both with and without
taking nonlinear tendon properties into account, were very low. This indicates
that these parameters are unlikely to vary much with size, even without
compensatory distortions, and that they are insensitive to the scaling of leg
stiffness. This may explain why Alexander and Jayes
(1983) found greater
differences in RSL than in DF when comparing mammalian species of different
sizes. Because the exponents predicted for these parameters were within, or
close to, our measured confidence intervals, we could not distinguish them
statistically from zero.
The second question that we addressed was whether systematic deviations
from dynamically similar locomotion occur with size in adult horses. We found
that the scaling exponents for RSL and DF in horses trotting at equal Froude
number were not significantly different from zero. The 95% confidence
intervals for the scaling exponents were narrow
(Table 4), indicating that any
deviations from dynamic similarity that occur in RSL and DF are likely to be
very small. We did not measure the phase relationships of the limbs or the
RPF. However, all of the horses were trotting, so their diagonal limb pairs
must have had a phase difference of half a stride. Griffin et al.
(2004) have also shown that
the abrupt change in limb phase at the walk-trot transition occurs at similar
Froude numbers in horses of different sizes, as would occur in dynamically
similar locomotion. It can be deduced that RPF must have been close to
independent of size because, if vertical GRF rises and falls as a half
sinusoid during the stance phase, animals moving with equal DF must also have
equal RPF (Alexander et al.,
1979). Vertical GRF is approximately sinusoidal in horses
(Witte et al., 2004), and we
found that DF was independent of size, so RPF must have been independent, or
close to independent, of size.
Our results are therefore consistent with the hypothesis that horses of
different sizes move in a dynamically similar manner when trotting at equal
Froude number. This does not necessarily imply that the same would be found in
an undomesticated species that is subject to natural selection. However, it
does show that dynamic similarity in RSL and DF is possible within a species
over a more than eightfold range of body mass. Because we only considered
trotting, further work would be needed to establish whether horses of
different sizes move in a dynamically similar manner at other gaits. It is
also important to note that dynamic similarity in some mechanical parameters
cannot be taken to imply that other mechanical parameters are dynamically
similar (Bullimore and Burn,
2004).
We predicted theoretically that, in `idealised' horses which were identical
except for size differences, RSL at equal Froude number would increase with
size. However, we did not find this experimentally; the measured scaling
exponents were significantly different from the theoretically predicted values
and were not significantly different from zero. This discrepancy suggests that
some systematic change occurs with size in horses that compensates for the
effects of size differences on locomotion. It would be of interest to identify
this factor. One possibility is that, while tendon elastic modulus does not
increase with size across species (Pollock
and Shadwick, 1994), it does increase with size in horses.
Alternatively, a compensatory distortion may occur. For example, larger horses
may have relatively thicker tendons. We have discussed possible sites of
compensatory distortions previously
(Bullimore and Burn, 2004) and
concluded that an important compensatory distortion between species is the
scaling of the effective mechanical advantage (EMA) of the limb
(Biewener, 1989). Large animals
tend to have a more upright limb posture than smaller species, and Biewener
(1989) demonstrated that this
results in a systematic increase the ratio of muscle moment arms to GRF moment
arms (the EMA) with animal size. This compensates for the size-independence of
tendon elastic modulus in three ways: (i) by decreasing the muscle and tendon
forces required to counteract a given GRF, (ii) by decreasing the joint
angular excursion that arises from a given tendon elongation, and (iii) by
decreasing the limb shortening that arises from a given joint angular
excursion. Griffin et al.
(2004) have shown that no
gross changes in limb posture occur with size in horses. However the changes
in EMA that would be required to maintain dynamically similar locomotion in
horses would be small and therefore difficult to detect. To illustrate this,
the metacarpophalangeal joint can be taken as an example. In a 450 kg horse,
the digital flexors have a moment arm about this joint of approximately 30 mm
(Brown et al., 2003) and the
GRF moment arm at midstance is approximately 100 mm (S.R.B. and J.F.B.,
unpublished data). If EMA is independent of size, and muscle moment arms scale
in proportion to , then the
corresponding GRF moment arms would be 57 mm in an 80 kg horse and 116 mm in a
700 kg horse, while the muscle moment arms would be 17 mm and 35 mm,
respectively. However, a small change in muscle moment arms, to 11 mm and 39
mm respectively, would give an EMA that was proportional to
, as has been found between
species (Biewener, 1989). This
is a difference of only a few millimetres, suggesting that careful
measurements from a large number of individuals would be required to detect
it.
An important question is how such a compensation for the effects of size could occur in a species where the size range has developed through artificial selection by humans. Horses are often used for physically demanding activities and are selectively bred for a wide range of characteristics including speed, endurance, agility and resistance to injury. The fact that this has resulted in similar patterns of locomotion being maintained across a wide range of sizes suggests that this way of moving confers desirable qualities. The lack of even a small amount of locomotor scaling with size suggests that relatively slight alterations in movement would be detrimental. Similarly, the (different) selection pressures acting on wild species could potentially also result in dynamically similar locomotion by selecting for optimal locomotor patterns.
Our third aim was to estimate the extent to which nonlinear tendon
properties could compensate for the effects of size. We found that taking
nonlinear tendon properties into account substantially reduced the scaling
exponents predicted for tendon strain, joint angular excursion and RSL, so
that predicted deviations from dynamic similarity were smaller than for linear
tendon properties. These predictions were made for horses, but tendon
properties are similar among species
(Pollock and Shadwick, 1994),
and the tendons of many species are loaded predominantly in the nonlinear
region of their stress-strain relationships
(Ker et al., 1988). Therefore
these conclusions are likely to be widely applicable. The effect of tendon
nonlinearity will be greatest when tendons are used in the low stress region,
where the stress-strain relationship is at its most nonlinear. Therefore we
expect the effect of tendon nonlinearity to decrease as speed increases during
locomotion. Because the digital flexor tendons of horses are loaded at higher
stresses than many other tendons (Ker et
al., 1988), we also expect the effect of nonlinear tendon
properties to be greater in many other species and tendons than was found
here. Where tendons are used at low stresses, their nonlinear properties will
effectively help to `buffer' the effects of size differences on locomotion. In
addition to reducing locomotor differences between individuals, this could
also reduce the effects of body mass fluctuations within an individual due to,
for example, growth, pregnancy or seasonal variation.
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Appendix |
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TOPSummaryIntroduction(Part 1) Predicted scaling...(Part 2) Measured scaling...ResultsDiscussionAppendixReferences |
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0), where
K=kl0/Mbg,
(the square root of Froude number),
and k is spring stiffness, l0 is initial spring
length, Mb is mass, g is the magnitude of the
acceleration due to gravity, u0 is horizontal landing
velocity and v0 is vertical landing velocity
(McMahon and Cheng, 1990). The
values of K, U0 and V0 that were used
to simulate trotting horses of different sizes in Part 1(iv) are listed in
Table A1. 0
was chosen so that the model bounced symmetrically, by the method described
previously (Bullimore and Burn,
2006).
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Acknowledgments |
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References |
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.
Therefore, a nonlinear tendon stress-strain relationship can reduce the
effects of size differences on tendon strain.
,
broken line is relationship predicted with linear tendon properties
, circles are
relationship predicted with realistic nonlinear tendon properties
. The
predicted effect of size is substantially reduced by taking into account the
nonlinear properties of tendon, but is still greater than measured
experimentally. This suggests that additional factors compensate for size
effects in horses.