Mechanical work and muscular efficiency in walking children
Unité de Physiologie et Biomécanique de la Locomotion, Université Catholique de Louvain, 1 Place Pierre de Coubertin, B-1348 Louvain-la-Neuve, Belgium
* Author for correspondence (e-mail: patrick.willems{at}loco.ucl.ac.be)
Accepted 17 November 2003
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Summary |
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Key words: walking children, mechanical work, energy cost, muscular efficiency
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Introduction |
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In order to take into account the difference in size between children and
adults, the speed of progression can be normalised using the dimensionless
Froude number,
(gl),
where
f is mean walking
speed, g is acceleration of gravity and l is leg
length (Alexander, 1989
). In
this case, the difference in the cost of transport between children and adults
for the most part disappears. This indicates that, after the age of 34
years, the difference in the cost of transport may be explained mostly on the
basis of body size (DeJaeger et al.,
2001
).
As previously observed in running
(Schepens et al., 2001), body
size can also affect the positive muscletendon work
(Wtot) performed during walking. Wtot
naturally falls into two categories: the external work
(Wext), which is the work necessary to sustain the
displacement of the centre of mass of the body (COM) relative to the
surroundings, and the internal work (Wint), which is the
work that does not directly lead to a displacement of the COM. Only
some of Wint can be measured: (1) the internal work done
to accelerate the body segments relative to the COM
(Wint,k) and (2) the internal work done during the double
contact phase of walking by the back leg, which generates energy that will be
absorbed by the front leg (Wint,dc). On the contrary, the
internal mechanical work done for stretching the series elastic components of
the muscles during isometric contractions, to overcome antagonistic
co-contractions, to overcome viscosity and friction cannot be directly
measured (although this unmeasured internal work will affect the efficiency of
positive work production; Willems et al.,
1995
).
Walking is characterised by a pendulum-like exchange between the kinetic
and potential energy of the COM. In children, the `optimal speed' at
which these pendulum-like transfers are maximal increases progressively with
age from 0.8 m s1 in 2-year-olds up to 1.4 m
s1 in 12-year-olds and adults
(Cavagna et al., 1983). At all
ages, the optimal speed is close to the speed at which the mass-specific work
to move the COM a given distance, Wext, is at a
minimum. Above the optimal speed, the energy transfers decrease. This decrease
is greater the younger the subject. The decreased transfers result in a
greater power required to move the COM: at 1.25 m
s1, the mass-specific external power
(
ext) is twice as great in
a 2-year-old child than in an adult. When normalising the speed with the
Froude number,
ext is
similar in children and in adults.
The work done by one leg against the other (Wint,dc)
was not counted in the `classic' measurements of the positive muscular work
done during walking, which was calculated as
Wtot=Wext+Wint,k
(Cavagna and Kaneko, 1977;
Willems et al., 1995
);
consequently, Wint,k has previously been referred to
simply as Wint. Using force platforms, Bastien et al.
(2003
) studied the effect of
speed and age (size) on Wint,dc in 312-year-old
children and in adults. Wint,dc as a function of speed
shows an inverted U-shape curve, attaining a maximum value of
approximately 0.150.20 J kg1 m1,
which is independent of size but occurs at higher speeds in larger subjects.
The differences due to size disappear for the most part when
Wint,dc is normalised with the Froude number.
These observations indicate that, as for the energy expenditure, the speed-dependent changes in Wext and Wint,dc are primarily a result of body size changes. To our knowledge, the internal work due to the movement of the limbs relative to the COM (Wint,k) has never been measured in walking children. In the present study, we measure simultaneously Wext, Wint,k and Wint,dc in children and in adults walking at different speeds and calculate Wtot and efficiency.
Wtot is calculated from Wext,
Wint,k and Wint,dc over a complete
stride, taking into account any possible energy transfers that would reduce
the muscular work done. Transfers between Wext and
Wint,k were analysed by Willems et al.
(1995). Energy transfers
between the back and the front legs in the computation of
Wint,dc were discussed by Bastien et al.
(2003
). In the present study,
we analyse the possible transfers between Wint,k and
Wint,dc; we show that, during the double contact phase,
some positive work done by the back leg in pushing the body forwards can
result in an increase of the kinetic energy of the front leg moving backwards
relative to the COM.
The total mechanical work is compared with the energy expenditure to evaluate the efficiency of positive work production. It is shown that children younger than 6 years are less efficient than adults in producing positive work during walking.
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Materials and methods |
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Subjects and experimental procedure
Twenty-four healthy children of 312 years of age and six healthy
adults participated in the experiments. They were divided into six age groups:
the 34-year-old group included subjects 3 years to <5 years old; the
56-year-old group included subjects 5 years to <7 years old, etc.
Each group comprised 46 children; the averaged physical characteristics
of these groups are given in table
1 of Schepens et al.
(2001). Written informed
consent of the subjects and/or their parents was obtained. The experiments
involved no discomfort, were performed according to the Declaration of
Helsinki and were approved by the local ethics committee.
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Subjects were asked to walk across a 6 m-long force platform at different
speeds. The mean speed (f)
was measured by two photocells placed at the level of the neck and set
0.75.5 m apart depending upon the speed. In each age group, the data
were gathered into speed classes of 0.130.14 m s1
(0.5 km h1).
Measurement of positive mechanical work done per stride
The kinetic internal work (Wint,k), the external work
(Wext) and the work done during the double contact phase
(Wint,dc) were measured simultaneously on 531 complete
strides according to the procedures described below. A stride was selected for
analysis only when the subject was walking at a relatively constant average
height and speed. Specifically, the sum of the increments in both vertical and
forward velocity of the COM could not differ by more than 25% from
the sum of the decrements (Cavagna,
1975). According to these criteria, the average vertical force was
within 4% of the body weight, and the difference in the forward velocity of
the COM, from the beginning to the end of the selected stride, was
less than 5% of
f in 95%
of the trials (at very low speeds, it was less than 10% of
f).
Measurement of positive internal work due to the segment movements per stride
Wint,k was computed from the segment movements and
anthropometric parameters. The body was divided into 11 rigid segments
(Willems et al., 1995): one
head/neck/trunk segment and two thigh, two shank, two foot, two upper arm and
two lower-arm/hand segments. The head/neck/trunk segment and the right limb
segments were delimited by infrared emitters placed at their extremities (see
table 2 in Schepens et al.,
2001
). The coordinates of the infrared emitters in the forward and
vertical directions were measured every 5 ms by means of a SELSPOT
II® system (SELCOM®, Göteborg, Sweden). The
coordinates were smoothed with a cubic spline function
(Dohrmann et al., 1988
).
A `stick man' of the position of each segment relative to the
head/neck/trunk segment was constructed every frame
(Fig. 1). The movements of the
head/neck/trunk segment relative to the COM were ignored because
their contribution to Wint,k is negligible
(Willems et al., 1995). The
left side of the subject, opposite to the camera, was reconstructed from the
right-side data on the assumption that the movements of the segments of one
side were equal and 180° out of phase with the other side. The angular
velocity of each segment and the translational velocity of its centre of mass
relative to the head/neck/trunk segment were calculated from the derivative of
their position versus time relationship. The position of the centre
of mass and the moment of inertia of the body segments were calculated using
the anthropometric parameters of table 2 in Schepens et al.
(2001
).
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The kinetic energy of each segment due to its displacement relative to the
head/neck/trunk segment and due to its rotation was then calculated as the sum
of its translational and rotational energy. The kinetic energy versus
time curves of the segments in each limb were summed. The internal work due to
the movements of the upper limbs,
, was then calculated by
adding the increments in their kinetic energytime curves
(Fig. 1). In order to minimise
errors due to noise, the increments in kinetic energy were considered to
represent positive work actually done only if the time between two successive
maxima was greater than 20110 ms, according to the speed of
progression. The same procedure was used with the kinetic energytime
curves of the lower limbs to compute the internal work due to their movements,
(Fig. 1).
Wint,k was then computed as the sum of
and
. This procedure allowed
energy transfers between segments of the same limb but disallowed any energy
transfers between different limbs (Willems
et al., 1995
).
Measurement of positive external work per stride
Wext was calculated from the vertical and forward
components of the force exerted on a 6 mx0.4 m force platform mounted 25
m from the beginning of a 40 m walkway. The platform was made of 10 different
plates, similar to those described by Heglund
(1981). The plates measured
the foreaft and vertical components of the forces exerted by the feet
on the ground. The responses were linear within 1% of the measured value for
forces up to 3000 N. The natural frequency of the plates was 180 Hz.
The signals from the platform were digitised synchronously with the camera
system. The integration of the vertical and forward components of the ratio
force/mass yielded the velocity changes of the COM, from which the
kinetic energy (Ek) was calculated after evaluation of the
integration constants (Cavagna,
1975; Willems et al.,
1995
). The kinetic energy of the COM is equal to
Ek=
m(Vf2+Vv2),
where m is body mass and Vf and
Vv are the forward and vertical components, respectively, of
the velocity of the COM. A second integration of the vertical
velocity yielded the vertical displacement of the COM, from which the
gravitational potential energy (Ep) was calculated.
Potential energy of the COM is equal to
Ep=mgSv, where
g is the gravitational constant and Sv is
the vertical displacement of the COM.
The mechanical energy of the COM (Eext) was the sum of the Ek and Ep curves over a complete stride. Wext was the sum of the increments in the Eext curve (Fig. 1). Similarly, Wk, the positive work done to sustain the velocity changes of the COM, was the sum of the increments of the Ek curve, and Wp, the positive work done against gravity, was calculated from the increments in the Ep curve (Fig. 1). The increments in mechanical energy were considered to represent positive work actually done only if the time between two successive maxima was greater than 20 ms.
Walking can be compared to a pendular mechanism where potential energy is
transformed into kinetic energy and vice versa
(Cavagna et al., 1976). The
recovery (R) of energy due to this pendular mechanism was estimated
by:
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Measurement of positive internal work made by one leg against the other during double contact
In walking, during the double contact phase, positive work is done by the
back leg pushing forwards while negative work is done by the front leg pushing
backwards. The forces exerted by each lower limb on the ground were measured
separately. The powers generated against the external forces by the front and
back legs were calculated from the dot product of the vertical and horizontal
components of the ground reaction forces acting under each leg multiplied,
respectively, by the vertical and horizontal velocity of the COM
(Donelan et al., 2002;
Bastien et al., 2003
). The
positive work done by the ground reaction forces was calculated independently
for the back (Wback) and the front
(Wfront) limb from the time-integral of the power curves,
taking into account any energy transfers
(Bastien et al., 2003
). Part of
the positive work done by the limbs results in an acceleration and/or an
elevation of the COM. In order not to count the same work twice, the
positive muscular work realised by one leg against the other during double
contact Wint,dc was evaluated by:
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Since Wint,dc was computed from the individual limb ground reaction forces, it was necessary that the two feet were on different plates during the double contact phase. This requirement could not often be fulfilled over consecutive double contact phases. For this reason, Wint,dc was measured on a single double contact phase of the stride and the result was doubled to obtain the Wint,dc for the whole stride. In 10% of the trials, two successive measurements of Wint,dc were possible within a stride; the two measurements were not statistically different (t=0.995, P<0.32, N=53).
Evaluation of total positive muscular work done each stride
In order to compute the total positive muscular work done
(Wtot), it is necessary to account for the possible energy
transfers between Wext, Wint,k and
Wint,dc. Willems et al.
(1995) showed that
Wtot was best evaluated when no transfers of energy were
allowed between Wext and Wint,k.
Bastien et al. (2003
) carefully
analysed which part of the positive mechanical work done by the legs can be
attributed to Wext and to Wint,dc. In
the following paragraphs, we analyse the possible energy transfers between
Wint,k and Wint,dc.
During the double contact phase, the push of the back leg that increases
the forward speed of the COM relative to the surroundings also
increases the backward speed of the front limb relative to the COM.
This is shown, for example, in the first period of double contact in
Fig. 1. The back leg (in this
case, the left leg) does positive muscular work to lift and to accelerate the
COM (increment a in Eext;
Fig. 1). The back leg also does
positive muscular work on the front leg (increment b in
Wint,dc; Fig.
1). The work done on the front leg can appear as an increase in
the rotational and translational kinetic energy of the front leg relative to
the COM (increment c in
;
Fig. 1) rather than just being
absorbed and dissipated as negative work in the muscles of the front leg
(decrement d in Wint,dc;
Fig. 1). In order to allow this
transfer, the
and
Wint,dc curves of each leg are added instant-by-instant.
Due to this transfer during the double contact phase, the increment e
in
(Fig. 1) is smaller than
increment c in
. The sum of the
increments of the resulting curve,
, is the internal work
done on a lower limb
(
).
The total positive work done by the muscles during walking, after allowing
reasonable energy-saving transfers, is:
![]() | (4) |
![]() | (5) |
Normalisation of the mechanical work done during a stride
In order to compare subjects of different body size, the work done per
stride was divided by the subject's body mass. This mass-specific work can
then be either divided by the stride length to obtain the work done per unit
distance or divided by the stride period to obtain the mean mechanical power
expended during walking. In the sections that follow, the work symbols
(W) usually refer to the mass-specific work done per unit distance (J
kg1 m1), and the symbols with a dot
() refer to the mass-specific power (W
kg-1).
Efficiency of positive work production
Efficiency of positive work production is calculated as the ratio of the
total positive muscular mechanical power to the net steady-state energy
consumption rate (net). The
net is the energetic
equivalent of the total oxygen consumption rate minus the standing oxygen
consumption rate. The total oxygen consumption rate for children was taken
from the data of DeJaeger et al.
(2001
).
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Results |
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Since the stride frequency (f) at a given speed is higher in
children than in adults (middle row, Fig.
2), the mass-specific internal power
(int,k = work per stride
multiplied by stride frequency) is greater in the children (bottom row,
Fig. 2). The difference is
greater at high speeds and in the young subjects and becomes negligible at
speeds less than 1 m s1 and in subjects older than 10
years.
External, internal and total mechanical work
At all ages, the recovery (R;
equation 1) of mechanical energy
via the pendulum-like transfer between Ep and
Ek attains a maximum of 65% at intermediate walking
speeds (circles in upper row of Fig.
3), although the speed of the maximum R increases with
age during growth. In the same panels of
Fig. 3, the crosses show the
recovery Rc (equation
2). R and Rc are very similar because
the variations of Ekv are small compared with the
variations of Ep and Ekf. Since the
vertical velocity of the COM is nil when it reaches its highest and
lowest point, the maximum and minimum of the Ep curve are
likely to be the same as those of the
Ep+Ekv curve.
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The mass-specific external work per unit distance
(Wext; second row, Fig.
3) reaches a minimum at a speed slightly lower than where
R is maximal. Above this `optimal' speed, Wext
increases more in children than in adults since the decrease of R
with speed is greater in children than in adults (the effect of speed and
growth on the link between R and Wext was
discussed in detail by Cavagna et al.,
1983).
Both in children and in adults, the mass-specific
Wint,dc per unit distance shows an inverted U-shape
curve as a function of speed (third row of panels in
Fig. 3).
Wint,dc attains a maximum value of 0.15 J
kg1 m1, independent of age, although the
speed at which this maximum occurs increases from
1.1 m
s1 at the age of 3 years to
1.6 m s1
above the age of 10 years. As a consequence, the maximum mass-specific power
developed by one leg against the other (which is the product of mass-specific
Wint,dc per unit distance multiplied by speed) increases
with age, from 0.15 W kg1 at the age of three to 0.25 W
kg1 above the age of 10.
The mass-specific Wint,k per unit distance (fourth row,
Fig. 3) represents the work
done to move the limbs relative to the centre of mass; it is the sum of
.
The difference between Wint,k in children and in adults is
greater the younger the subject and the higher the speed and becomes
negligible after the age of 10.
The mass-specific internal work per unit distance, Wint
(fifth row, Fig. 3), is not
equal to the sum of Wint,dc and
Wint,k. Indeed, due to the energy transfer between the
Wint,dc and
curves
(Fig. 1),
Wint is very similar to Wint,k.
Compared with adults, Wint is noticeably greater in
subjects younger than 11 years and at speeds higher than 1.5 m
s1.
The total mass-specific muscular work per unit distance,
Wtot (bottom row, Fig.
3), is calculated as the sum of the mass-specific external work
per unit distance, Wext, plus the mass-specific internal
work per unit distance, Wint. The arrows in the bottom
panels indicate the speed at which the net energy cost is minimal (see
fig. 3 of
DeJaeger et al., 2001).
Contrary to the net energy cost of walking, the curve of
Wtot as a function of speed does not have a well-defined
minimum, although this could occur at speeds lower than the ones explored.
Above 0.5 m s1, Wtot increases with
walking speed more steeply in children than in adults. A two-way
repeated-measures analysis of variance with contrasts (SuperANOVA, 1.11) was
made to determine the speed at which Wtot differs between
children and adults. Specifically, the effect of speed was analysed within
each age group, and the speed at which Wtot in children
became significantly different from that in adults was determined
(Table 1). In children younger
than 11 years, Wtot is always greater at high walking
speeds. The contrast analysis also shows a statistical difference in
Wtot between the 56 years group and the adults,
even at very low speeds. This difference is of the order of 0.04 J
kg1 m1, which represents
5% of the
adult values. Even if this difference is statistically significant it is
unlikely to be biologically significant.
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Discussion |
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Efficiency of positive work production during walking
The efficiency of positive work production by the muscles and tendons
during walking is calculated as the ratio of the total mechanical power
(tot; i.e. the work per
stride multiplied by the stride frequency) to the net energy consumption rate
(
net; i.e. the gross energy
consumption rate minus the standing energy expenditure rate):
![]() | (6) |
The total mechanical power is shown as a function of walking speed in the
top row of Fig. 4.
net, the cost of operating
the locomotory machinery, is presented in the middle row of
Fig. 4 (data from
DeJaeger et al., 2001
). At all
ages, both
tot and
net increase with walking
speed, although the increase is greater the younger the subject. The
differences in
tot and in
net between adults and
children disappear after the age of 10.
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The efficiency of positive work production is presented in the bottom row
of Fig. 4. Assuming the
precision of the mechanical power measurement is as good as 0.1 W
kg1, then at low speeds, where the total mechanical power is
small, a change in tot of
0.1 W kg1 would result in a change in the efficiency of
>0.05. For this reason, the values of efficiency are considered to be
robust only at speeds above
0.75 m s1.
In adults, the efficiency reaches a maximum of 0.300.35 at 1.25
m s1; at lower and higher speeds the efficiency decreases.
These values are in good agreement with those of Cavagna and Kaneko
(1977
) and Willems et al.
(1995
). At speeds greater than
1 m s1, the efficiency of positive work production is
greater than the maximal efficiency of the conversion of chemical energy into
positive work by muscles (
0.25;
Dickinson, 1929
), suggesting
that elastic energy is stored during the phase of negative work to be
recovered during the following phase of positive work
(Willems et al., 1995
).
Before the age of seven, the increase in
net cannot be explained
only by an increase in
tot.
Part of the extra cost of walking in young children appears to be due to a
reduction in the efficiency of positive work production. For example, in
34-year-old children, the efficiency is 0.150.25, while after
the age of six the efficiency is similar in children and adults. The lower
efficiency in young children could be explained, at least in part, by the
immature muscular pattern observed during walking before the age of five
(Sutherland et al., 1988
),
which may require more isometric and/or antagonistic contractions to stabilise
the body segments. These contractions would result in an increased energy
expenditure without any increase in the mechanical work
(Griffin et al., 2003
).
Contribution of external and internal power to the total mechanical power during walking
At speeds below 1 m s1, the internal power is smaller
than the external power in all age groups
(Fig. 5). This is due to the
fact that the two components of the internal power tend towards zero as speed
approaches zero. On the contrary, the external power, specifically the power
necessary to sustain the vertical movements of the COM, does not tend
towards zero as speed approaches zero
(Cavagna et al., 1983). As
speed is increased above 1 m s1, Wint
increases faster than Wext, and at high walking speed it
is 2040% greater than Wext (except in the
34-year olds).
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In the present study, the two components of the internal power, and the
energy transfers between them, are taken into account for the first time in
the computation of the total muscular power of walking.
tot, based upon
equation 4 (i.e. allowing all
reasonable energy transfers, as explained in the Materials and methods), is
shown by the solid line in Fig.
5. If, on the other hand, the total power is based upon
equation 5 (i.e. assuming no
energy transfers), then the resulting total power is shown by the upper broken
line in Fig. 5. At the other
limit, if the total power is calculated simply as the sum of
ext+
int,k,
ignoring
int,dc as has been
done in the past, the result is the lower broken line in
Fig. 5. It can be seen that at
all ages,
int,dc represents
a small fraction of the total muscular power spent during walking:
int,dc represents
10%
of the total power at intermediate speeds, decreasing to zero at low speeds
and <5% at high speeds.
Normalisation for body size
At a given speed, Wint,k per unit body mass and per
stride is the same in all age groups (top row,
Fig. 2) in spite of large
differences in stride frequency, movement amplitude and limb dimensions. In
other words, this normalisation of Wint,k makes it
independent of the amplitude/duration of the oscillation and takes into
account the different dimensions of the children and adults.
Different size subjects can be compared at equivalent, size-independent
speeds if the mean velocity is normalised using the Froude number
(Alexander, 1989). This assumes
that children and adults move in a dynamically similar manner, i.e. all
lengths, times and forces scale by the same factors. In a situation such as
walking, where inertia and gravity are of primary importance, size-dependent
speed differences should disappear if the assumption of dynamic similarity is
justified.
The upper panel of Fig. 6
shows int as a function of
the Froude speed. For the most part, the differences between children and
adults disappear, although at the same Froude speed the data of the smaller
subjects tend to be lower than those of the larger subjects, indicating that
not all differences can be explained simply on the basis of size. The same can
be seen for
tot
(Fig. 6, lower panel).
Likewise, when
ext,
net and
int,dc are expressed as a
function of the Froude number, the differences between children and adults
also tend to disappear (Cavagna et al.,
1983
; DeJaeger et al.,
2001
; Bastien et al.,
2003
). These observations indicate that, after the age of three,
the differences observed in the mechanics and energetics of walking during
growth may be explained, for the most part, on the basis of dynamic
similarity. The fact that efficiency is lower in very young children compared
with in adults suggests that factors other than size scaling, such as
developmental changes in the neuromuscular system, may play a role before the
age of six.
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List of symbols |
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Acknowledgments |
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