Mechanical work and muscular efficiency in walking children

Children consume more energy per unit body mass to walk at a given speed than do adults (DeJaeger et al.,2001). The difference in the net mass-specific metabolic energy cost per unit distance (i.e. the cost of transport, the energy required to operate the locomotory machinery) between adults and children is greater the higher the speed and the younger the subject. For example, at a speed of 1 m s^–1, a 3–4-year-old has a net oxygen consumption 33%greater than adults. This difference disappears by the age of 11–12 years.

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,

As previously observed in running(Schepens et al., 2001), body size can also affect the positive muscle–tendon work(W_tot) performed during walking. W_totnaturally falls into two categories: the external work(W_ext), 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 (W_int), which is the work that does not directly lead to a displacement of the COM. Only some of W_int can be measured: (1) the internal work done to accelerate the body segments relative to the COM(W_int,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 (W_int,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 s^–1 in 2-year-olds up to 1.4 m s^–1 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, W_ext, 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 s^–1, the mass-specific external power(Ẇ_ext) is twice as great in a 2-year-old child than in an adult. When normalising the speed with the Froude number, Ẇ_ext is similar in children and in adults.

The work done by one leg against the other (W_int,dc)was not counted in the `classic' measurements of the positive muscular work done during walking, which was calculated as W_tot=W_ext+W_int,k(Cavagna and Kaneko, 1977; Willems et al., 1995);consequently, W_int,k has previously been referred to simply as W_int. Using force platforms, Bastien et al.(2003) studied the effect of speed and age (size) on W_int,dc in 3–12-year-old children and in adults. W_int,dc as a function of speed shows an inverted U-shape curve, attaining a maximum value of approximately 0.15–0.20 J kg^–1 m^–1,which is independent of size but occurs at higher speeds in larger subjects. The differences due to size disappear for the most part when W_int,dc is normalised with the Froude number.

These observations indicate that, as for the energy expenditure, the speed-dependent changes in W_ext and W_int,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 (W_int,k) has never been measured in walking children. In the present study, we measure simultaneously W_ext, W_int,k and W_int,dc in children and in adults walking at different speeds and calculate W_tot and efficiency.

W_tot is calculated from W_ext, W_int,k and W_int,dc over a complete stride, taking into account any possible energy transfers that would reduce the muscular work done. Transfers between W_ext and W_int,k were analysed by Willems et al.(1995). Energy transfers between the back and the front legs in the computation of W_int,dc were discussed by Bastien et al.(2003). In the present study,we analyse the possible transfers between W_int,k and W_int,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.

Measurements of the mechanical work during walking were performed on the same subjects and during the same sessions as the previously published measurements of the mechanical work during running(Schepens et al., 2001). The details of the methods used to compute W_ext are given in Cavagna (1975) and Willems et al. (1995), those used to compute W_int,k are given in Cavagna and Kaneko(1977) and Willems et al.(1995), and those used to compute W_int,dc are given in Bastien et al.(2003); these methods are only summarised here.

Twenty-four healthy children of 3–12 years of age and six healthy adults participated in the experiments. They were divided into six age groups:the 3–4-year-old group included subjects 3 years to <5 years old; the 5–6-year-old group included subjects 5 years to <7 years old, etc. Each group comprised 4–6 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.

Subjects were asked to walk across a 6 m-long force platform at different speeds. The mean speed (V̄_f)was measured by two photocells placed at the level of the neck and set 0.7–5.5 m apart depending upon the speed. In each age group, the data were gathered into speed classes of 0.13–0.14 m s^–1(0.5 km h^–1).

The kinetic internal work (W_int,k), the external work(W_ext) and the work done during the double contact phase(W_int,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 V̄_f in 95%of the trials (at very low speeds, it was less than 10% of V̄_f).

W_int,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 SelspotII® 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 W_int,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).

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 versustime curves of the segments in each limb were summed. The internal work due to the movements of the upper limbs,

W_ext was calculated from the vertical and forward components of the force exerted on a 6 m×0.4 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 fore–aft 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 (E_k) was calculated after evaluation of the integration constants (Cavagna,1975; Willems et al.,1995). The kinetic energy of the COM is equal to E_k=½m(V_f²+V_v²),where m is body mass and V_f and V_v 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 (E_p) was calculated. Potential energy of the COM is equal to E_p=mgS_v, where g is the gravitational constant and S_v is the vertical displacement of the COM.

The mechanical energy of the COM (E_ext) was the sum of the E_k and E_p curves over a complete stride. W_ext was the sum of the increments in the E_ext curve (Fig. 1). Similarly, W_k, the positive work done to sustain the velocity changes of the COM, was the sum of the increments of the E_k curve, and W_p,the positive work done against gravity, was calculated from the increments in the E_p 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.

Since W_int,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, W_int,dc was measured on a single double contact phase of the stride and the result was doubled to obtain the W_int,dc for the whole stride. In 10% of the trials, two successive measurements of W_int,dc were possible within a stride; the two measurements were not statistically different(t=–0.995, P<0.32, N=53).

In order to compute the total positive muscular work done(W_tot), it is necessary to account for the possible energy transfers between W_ext, W_int,k and W_int,dc. Willems et al.(1995) showed that W_tot was best evaluated when no transfers of energy were allowed between W_ext and W_int,k. Bastien et al. (2003) carefully analysed which part of the positive mechanical work done by the legs can be attributed to W_ext and to W_int,dc. In the following paragraphs, we analyse the possible energy transfers between W_int,k and W_int,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 E_ext; Fig. 1). The back leg also does positive muscular work on the front leg (increment b in W_int,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

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 kg^–1 m^–1), and the symbols with a dot(Ẇ) refer to the mass-specific power (W kg^-1).

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 (Ṁ_net). The Ṁ_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).

As walking speed goes up, the velocity of the head/neck/trunk segment relative to the surroundings increases. As a consequence, the backward velocity of the supporting limb relative to the COM increases. Furthermore, the stride length becomes longer and the time to reset the limbs becomes shorter. As a result, the forward velocity of the swing leg increases relative to the COM. For these reasons, the mass-specific W_int,k done per stride increases with speed, both in children and in adults (top row, Fig. 2). Note that, at a given speed, children and adults do the same amount of W_int,k per unit body mass each stride.

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(Ẇ_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 s^–1 and in subjects older than 10 years.

At all ages, the recovery (R; equation 1) of mechanical energy via the pendulum-like transfer between E_p and E_k 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 R_c (equation 2). R and R_c are very similar because the variations of E_kv are small compared with the variations of E_p and E_kf. Since the vertical velocity of the COM is nil when it reaches its highest and lowest point, the maximum and minimum of the E_p curve are likely to be the same as those of the E_p+E_kv curve.

The mass-specific external work per unit distance(W_ext; second row, Fig. 3) reaches a minimum at a speed slightly lower than where R is maximal. Above this `optimal' speed, W_extincreases more in children than in adults since the decrease of Rwith speed is greater in children than in adults (the effect of speed and growth on the link between R and W_ext was discussed in detail by Cavagna et al.,1983).

Both in children and in adults, the mass-specific W_int,dc per unit distance shows an inverted U-shape curve as a function of speed (third row of panels in Fig. 3). W_int,dc attains a maximum value of ∼0.15 J kg^–1 m^–1, independent of age, although the speed at which this maximum occurs increases from ∼1.1 m s^–1 at the age of 3 years to ∼1.6 m s^–1above 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 W_int,dc per unit distance multiplied by speed) increases with age, from 0.15 W kg^–1 at the age of three to 0.25 W kg^–1 above the age of 10.

The mass-specific W_int,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 mass-specific internal work per unit distance, W_int(fifth row, Fig. 3), is not equal to the sum of W_int,dc and W_int,k. Indeed, due to the energy transfer between the W_int,dc and

The total mass-specific muscular work per unit distance, W_tot (bottom row, Fig. 3), is calculated as the sum of the mass-specific external work per unit distance, W_ext, plus the mass-specific internal work per unit distance, W_int. 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 W_tot 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 s^–1, W_tot 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 W_tot differs between children and adults. Specifically, the effect of speed was analysed within each age group, and the speed at which W_tot in children became significantly different from that in adults was determined(Table 1). In children younger than 11 years, W_tot is always greater at high walking speeds. The contrast analysis also shows a statistical difference in W_tot between the 5–6 years group and the adults,even at very low speeds. This difference is of the order of 0.04 J kg^–1 m^–1, which represents ∼5% of the adult values. Even if this difference is statistically significant it is unlikely to be biologically significant.

In the present study, the internal work done to accelerate the limb segment relative to the COM (W_int,k) is measured for the first time in children walking at different speeds. W_int,kis measured simultaneously with the external work (W_ext)and the work done by one leg against the other (W_int,dc)per stride. The W_int,k, W_ext and W_int,dc are combined, taking into account possible energy transfers, to get the total mechanical work (W_tot) per stride. The results obtained for the mass-specific W_extand W_int,dc per unit distance are in agreement with those of Cavagna et al. (1983) and Bastien et al. (2003),respectively.

The total mechanical power is shown as a function of walking speed in the top row of Fig. 4. Ṁ_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 Ẇ_tot and Ṁ_net increase with walking speed, although the increase is greater the younger the subject. The differences in Ẇ_tot and in Ṁ_net between adults and children disappear after the age of 10.

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 kg^–1, then at low speeds, where the total mechanical power is small, a change in Ẇ_tot of 0.1 W kg^–1 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 s^–1.

In adults, the efficiency reaches a maximum of 0.30–0.35 at ∼1.25 m s^–1; 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 s^–1, 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 Ṁ_net cannot be explained only by an increase in Ẇ_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 3–4-year-old children, the efficiency is 0.15–0.25, 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).

At speeds below 1 m s^–1, 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 s^–1, W_intincreases faster than W_ext, and at high walking speed it is 20–40% greater than W_ext (except in the 3–4-year olds).

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. Ẇ_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 Ẇ_ext+Ẇ_int,k,ignoring Ẇ_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, Ẇ_int,dc represents a small fraction of the total muscular power spent during walking: Ẇ_int,dc represents ∼10%of the total power at intermediate speeds, decreasing to zero at low speeds and <5% at high speeds.

At a given speed, W_int,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 W_int,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. 6shows Ẇ_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 Ẇ_tot(Fig. 6, lower panel). Likewise, when Ẇ_ext, Ṁ_net and Ẇ_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.

 
• COM

  centre of mass of the body

 
• E_ext

  mechanical energy of the COM

 
• \(E_{\mathrm{int}}^{\mathrm{l}}\)

  internal energy of the lower limb

 
• \(E_{\mathrm{int,k}}^{\mathrm{l}}\)

  kinetic energy change of the lower limbs due to their velocity relative to the COM

 
• \(E_{\mathrm{int,k}}^{\mathrm{u}}\)

  kinetic energy change of the upper limbs due to their velocity relative to the COM

 
• E_k

  kinetic energy of the COM

 
• E_kf

  kinetic energy due to the forward motion of the COM

 
• E_kv

  kinetic energy due to the vertical motion of the COM

 
• E_p

  gravitational potential energy of the COM

 
• f

  stride frequency

 
• g

  gravitational acceleration

 
• l

  leg length

 
• m

  body mass

 
• Ṁ _net

  net steady-state energy consumption rate

 
• R

  recovery of mechanical energy through a pendular mechanism

 
• S_v

  vertical displacement of the COM

 
• V̄ _f

  mean walking speed

 
• V_f

  forward component of the velocity of the COM

 
• V_v

  vertical component of the velocity of the COM

 
• W_back

  positive muscular work done by the back leg during double contact

 
• W_ext

  external work

 
• Ẇ _ext

  mass-specific external power

 
• W_f

  work done to sustain the forward motion of the COM

 
• W_front

  positive muscular work done by the front leg during double contact

 
• W_int

  internal work

 
• Ẇ _int

  mass-specific internal power

 
• W_int,dc

  positive muscular work realised by one leg against the other during double contact

 
• Ẇ _int,dc

  mass-specific power expended by one leg against the other during the double contact phase

 
• W_int,k

  work required to move the limbs relative to the COM

 
• Ẇ _int,k

  mass-specific internal power expended to move the limbs relative to the COM

 
• \(W_{\mathrm{int}}^{\mathrm{l}}\)

  internal work done on a lower limb

 
• \(W_{\mathrm{int,k}}^{\mathrm{l}}\)

  internal work due to the movements of the lower limbs

 
• \(W_{\mathrm{int,k}}^{\mathrm{u}}\)

  internal work due to the movements of the upper limbs

 
• W_k

  work done to accelerate the COM

 
• W_p

  potential gravitational energy changes of the COM

 
• W_tot

  total mechanical work

 
• Ẇ _tot

  total mass-specific mechanical power

 
• W_v

  work done to sustain the vertical motion of the COM

This study was supported by the Fonds National de la Recherche Scientifique of Belgium.