Sprint running: a new energetic approach

Since the second half of the 19th century, the energetics and biomechanics of running at constant speed have been the object of many studies, directed towards elucidating the basic mechanisms of this most natural form of locomotion; but the results of these studies have also had direct practical applications, e.g. for the assessment of the overall metabolic energy expenditure, or for the prediction of best performances (e.g. see Alvarez-Ramirez, 2002; Lacour et al., 1990; Margaria, 1938; Margaria et al., 1963; Péronnet and Thibault,1989; di Prampero et al.,1993; Ward-Smith,1985; Ward-Smith and Mobey,1995; Williams and Cavanagh,1987).

In contrast to constant speed running, the number of studies devoted to sprint running is rather scant. This is not surprising, since the very object at stake precludes reaching a steady state, thus rendering any type of energetic analysis rather problematic. Indeed, the only published works on this matter deal with either some mechanical aspects of sprint running(Cavagna et al., 1971; Fenn, 1930a,b; Kersting, 1998; Mero et al., 1992; Murase et al., 1976; Plamondon and Roy, 1984), or with some indirect approaches to its energetics(Arsac, 2002; Arsac and Locatelli, 2002; van Ingen Schenau et al., 1991, 1994; di Prampero et al., 1993; Summers, 1997; Ward-Smith and Radford, 2000). The indirect estimates of the metabolic cost of acceleration reported in the above-mentioned papers are based on several assumptions that are not always convincing. In the present study we therefore propose a novel approach to estimate the energy cost of sprint running, based on the equivalence of an accelerating frame of reference (centred on the runner) with the Earth's gravitational field. Specifically, in the present study, sprint running on flat terrain will be viewed as the analogue of uphill running at constant speed, the uphill slope being dictated by the forward acceleration(di Prampero et al., 2002). Thus, if the forward acceleration is measured, and since the energy cost of uphill running is fairly well known (e.g. see Margaria, 1938; Margaria et al., 1963; Minetti et al., 1994, 2002), it is a rather straightforward matter to translate the forward acceleration of sprint running into the corresponding up-slope, and thence into the corresponding energy cost. Knowledge of this last and of the instantaneous forward speed will then allow us to calculate the corresponding metabolic power, which is presumably among the highest values attainable for any given subject.

Summarising, sprint running can be considered equivalent to constant speed running on the Earth, up an equivalent slope ES, while carrying an additional massΔ M=M_b(g′/g–1), so that the overall equivalent mass EM becomes EM=ΔM+M_b.

Both ES and EM are dictated by the forward acceleration (Eq. 4, 8);therefore they can be easily calculated once a_f is known. The values of ES and EM so obtained can then be used to infer the corresponding energy cost of sprint running, provided that the energy cost of uphill running at constant speed per unit body mass is also known.

It should be pointed out that the above analogy is based on the following three simplifying assumptions, which will be discussed in the appropriate sections. (i) Fig. 1 is an idealised scheme wherein the overall mass of the runner is assumed to be located at the centre of mass. In addition, (ii) Fig. 1 refers to the whole period during which one foot is on the ground, as such it denotes the integrated average applying to the whole step (half stride). (iii) The calculated ES and EM values are those in excess of the values applying during constant speed running, in which case the subject's body is not vertical, but leans slightly forward (Margaria,1975).

The aim of the present study was that to estimate the energy cost and metabolic power of the first 30 m of an all-out run from a stationary start,from the measured forward speed and acceleration.

The experiments were performed on an outdoor tartan track of 100 m length,at an average barometric pressure and temperature of about 740 mmHg and 21°C, using 12 medium-level male sprinters whose physical characteristics are reported in Table 1. The subjects were informed on the aims of the study and gave their written consent to participate.

The individual values of speed and acceleration were calculated for each subject over one run. The values so obtained were then pooled and the means calculated. Values are reported as means ± 1 standard deviation(s.d.), where N=12.

It is also immediately apparent that, when ES=0 and EM=1, C_sr reduces to that applying at constant speed running on flat terrain, which amounted to about 3.6 J kg^–1min^–1 (Minetti et al.,2002), a value close to that reported by others (e.g. see Margaria et al., 1963; di Prampero et al., 1986, 1993).

The speed increased to attain a peak of 9.46±0.19 m s^–1 about 5 s from the start. The highest forward acceleration was observed immediately after the start (0.2 s): it amounted to 6.42±0.61 m s^–2. The corresponding peak ES and EM values amounted to 0.64±0.06 and to 1.20±0.03(Table 2). The behavior of ES and EM, throughout the entire acceleration phase for a typical subject, as calculated from a_f (see Fig. 4) on the bases of Eq. 4 and 8, is reported in Fig. 5,which shows that, after about 30 m, ES tended to zero and EM to one, which correspond to constant speed running.

The energy cost of sprint running (C_sr), as obtained from Eq. 13 on the basis of the above calculated ES and EM, is reported in Fig. 6 for a typical subject. This figure shows that the instantaneous C_sr attains a peak of about 50 J kg^–1 m^–1 immediately after the start; thereafter it declines progressively to attain, after about 30 m, the value for constant speed running on flat terrain (i.e. about 3.8 J kg^–1 m^–1). This figure shows also that ES is responsible for the greater increase of C_sr whereas EM plays only a marginal role. Finally, Fig. 6 also shows that the average C_sr over the first 30 m of sprint running in this subject is about 11.4 J kg^–1 m^–1, i.e. about three times larger than that of constant speed running on flat terrain.

The product of C_sr and the speed yields the instantaneous metabolic power output above resting; it is reported as a function of time for the same subject in Fig. 7, which shows that the peak power output, of about 100 W kg^–1, is attained after about 0.5 s and that the average power over the first 4 s is on the order of 65 W kg^–1.

The instantaneous values of forward acceleration were obtained from the first derivative of exponential equations describing the time course of the speed. Linear regressions between measured and modelled speed values(Fig. 3) were close to the identity line for all 12 subjects (r²>0.98; P<0.01), showing the high accuracy of this kind of speed modelling during sprint running (Chelly and Denis,2001; Henry, 1954; Volkov and Lapin, 1979). Even so, it should be noted that: (i) at the start of the run the centre of mass is behind the start line and (ii) whereas the centre of mass rises at the very onset of the run, the radar device does not; as a consequence, (iii) the initial speed data are slightly biased. However, after a couple of steps this effect becomes negligible, as such it will not be considered further. Finally,it should also be pointed out that filtering the raw speed data, while retaining the general characteristics of the speed vs time curve(Fig. 3), leads to substantial smoothing of the speed swings that occur at each step and are a fundamental characteristic of locomotion on legs.

The number of subjects of this study (12) may appear small. However the coefficients of variation of peak speeds and peak accelerations for this population (0.02 and 0.095) were rather limited, and the subjects were homogeneous in terms of performance (Tables 1, 2). Finally, the present approach is directed at obtaining a general description of sprint running,rather than at providing accurate statistical descriptions of specific groups of athletes.

The main assumptions on which the calculations reported in the preceding sections were based are reported and discussed below.

1. The overall mass of the runner is assumed to be located at the centre of mass of the body. As such, any possible effects of the motion of the limbs,with respect to the centre of mass, on the energetics of running were neglected. This is tantamount to assuming that the energy expenditure associated with internal work is the same during uphill running as during sprint running at an equal ES. This is probably not entirely correct, since the frequency of motion is larger during sprint than during uphill running. If this is so, the values obtained in this study can be taken to represent a minimal value of the energy cost, or metabolic power, of sprint running.

2. The average force applied by the active muscles during the period in which one foot is on the ground is assumed to be described as in Fig. 1B, thus neglecting any components acting in the frontal plane. In addition, the assumption is also made that the landing phase (in terms of forces and joint angles) is the same during uphill as during sprint running at similar ES, a fact that may not be necessarily true, and that may warrant ad hoc biomechanical studies.

3. The calculated ES and EM values are assumed to be in excess of those applying during constant speed running, in which case the subject's body is not vertical, but leans slightly forward(Margaria, 1975) and the average force required to transport the runner's body mass is equal to that prevailing under the Earth's gravitational field. Indeed, the main aim of this study was to estimate the energy cost and metabolic power of sprint running,and since our reference was the energy cost of constant speed running per unit body mass, the above simplifying assumptions should not introduce any substantial error in our calculations.

4. The energy cost of running uphill at constant speed, as measured at steady state up to inclines of +0.45, was taken to represent also the energy cost of sprint running at an equal ES. Note that the energy cost of running per unit of distance, for any given slope, is independent of the speed (e.g. see Margaria et al., 1963; di Prampero et al., 1986; 1993). Thus the transfer from uphill to sprint running can be made regardless of the speed. Even so, the highest values of ES attained by our subjects (about 0.70) were greater than the highest slopes for which the energy cost of uphill running was actually measured (0.45). Thus the validity of our values for slopes greater than 0.45 is based on the additional assumption that, also above this incline, the relationship between C_sr and ES is described by Eq. 14. Graphical extrapolation of the Minetti et al.(2002) equation does seem to support our interpretation of their data; however, stretching their applicability as we did in the present study may seem somewhat risky. We would like to point out, however, that the above word of caution applies only for the peak C_sr and metabolic power values, i.e. to the initial 3 m (Fig. 5), which represent about 1/10 of the distance considered in this study. Thus, the majority of our analysis belongs to a more conservative range of values.

5. Minetti et al. (2002)determined the energy cost of uphill running from direct oxygen uptake measurements during aerobic steady state exercise. In contrast, the energy sources of sprint running are largely anaerobic. It follows that the values of C_sr and metabolic power (P_met), as calculated in this study, should be considered with caution. Indeed, they are an estimate of the amount of energy (e.g. ATP units) required during the run,expressed in O₂ equivalents. The overall amount of O₂consumed, including the so-called `O<sub>2</sub> debt payment' for replenishing the anaerobic stores after the run, may well be different, a fact that applies to any estimate of energy requirement during `supramaximal exercise'. Finally, the calculated values of C_sr and P_met represent indirect estimates rather than `true'measured values. However, the actual amount of energy spent during sprint running cannot be easily determined with present day technology, thus rendering any direct validation of our approach rather problematic. However,in theory at least, computerised image analysis of subjects running over series of force platforms could be coupled with the assessment of the overall heat output by means of thermographic methods. Were this indeed feasible, one could obtain a complete energetic description of sprint running to be compared with the present indirect approach.

The peak metabolic power values reported in Table 3 are about four times larger than the maximal oxygen consumption(V_O2max) of elite sprinters which can be expected to be on the order of 25 W kg^–1 (70 ml O₂kg^–1 min^–1 above resting). This is consistent with the value estimated by Arsac and Locatelli(2002) for sprint elite runners, which amounted to about 100 W kg^–1, and with previous findings showing that, on the average, the maximal anaerobic power developed while running at top speed up a normal flight of stairs is about four times larger than V_O2max(Margaria et al., 1966).

The same set of calculations was also performed on one athlete (C. Lewis,winner of the 100 m gold medal in the 1988 Olympic games in Seoul with the time of 9.92 s) from speed data reported by Brüggemann and Glad(1990). The corresponding peak values of ES and EM amounted to 0.80 and 1.3, whereas the peak C_sr and metabolic power attained 55 J kg^–1 m^–1 and 145 W kg^–1. The overall amount of metabolic energy spent over 100 m by C. Lewis was also calculated by this same approach. It amounted to 650 J kg^–1,very close to that estimated for world record performances by Arsac(2002) and Arsac and Locatelli(2002). However, these same authors, on the basis of a theoretical model originally developed by van Ingen Schenau (1991), calculated a peak metabolic power of 90 W kg^–1 for male world records, to be compared with the 145 W kg^–1 estimated in this study for C. Lewis. The model proposed by van Ingen Schenau is based on several assumptions, among which overall running efficiency plays a major role. Indeed, the power values obtained by Arsac and Locatelli(2002) were calculated on the bases of an efficiency (η) increasing with the speed, as described byη _t=0.25+0.25. v_t/v_max whereη _t and v_t are efficiency and speed at time t, respectively, and v_max is the maximal speed. However, Arsac and Locatelli point out that, if a constant efficiency of 0.228 is assumed, then the estimated peak metabolic power reaches 135 W kg^–1, not far from that obtained above for C. Lewis. Thus, in view of the widely different approaches, we think it is the similarity between the two sets of estimated data that should be emphasized, rather than their difference.

In the preceding paragraphs, the effects of the air resistance on the energy cost of sprint running were neglected; they will now be briefly discussed. The energy spent against the air resistance per unit of distance(C_aer) increases with the square of the air velocity(v): C_aer=k′v²,where the proportionality constant (k′) amounts to about 0.40 J s² m^–3 per m² of body surface area(Pugh, 1971; di Prampero et al., 1986, 1993). This allowed us to calculate that C_aer attained about 0.86 J kg^–1 m^–1 at the highest speeds. Thus,whereas in the initial phase of the sprint at slow speeds and high ES, C_aer is a negligible fraction of the overall energy cost,this is not so at high speeds with ES tending to zero. Indeed, at the highest average forward speed (v_f) attained in this study (9.46 m s^–1), C_aer amounted to about 20% of the total energy cost and required about 8 additional W kg^–1 in terms of metabolic power. This is substantially equal to the data estimated by Arsac (2002) for sea level conditions.

Finally, the analysis presented in Fig. 1 shows that, in the deceleration phase, sprint running can be viewed as the analogue of downhill running at constant speed. According to Minetti et al. (2002), Eq. 13 can also be utilised to describe the energetics of downhill running. So, the negative values of ES, obtained when a_f is also negative,can be inserted into Eq. 14 to estimate the corresponding C_sr values in the deceleration phase. Quantitatively,however, the effects of deceleration on C_sr are much smaller than those described above for the acceleration phase, because throughout the whole range of the downhill slopes (from 0 to –0.45), the energy cost of running changes by a factor of 2, to a minimum of 1.75 J kg^–1 m^–1 at a slope of –0.20, rising again for steeper slopes, to attain a value about equal to that for level running (3.8 J kg^–1 m^–1) at a slope of–0.45. This is to be compared with an increase of about fivefold from level running to +0.45 (see Eq. 13 and Minetti et al., 2002).

When, as is often the case, the sprint occurs in calm air and hence v=s, these two equations can be easily solved at any point in time, provided that the time course of the ground speed is known.

 
• A_ns

  amount of energy derived from anaerobic stores

 
• a_f

  forward acceleration

 
• g′

  overall acceleration acting on the runner's body

 
• C

  energy cost

 
• C_aer

  energy spent against the air resistance per unit of distance

 
• C_sr

  energy cost of sprint running

 
• d

  distance

 
• g

  acceleration of gravity

 
• E_tot

  overall energy expenditure

 
• EM

  equivalent normalised body mass

 
• ES

  equivalent slope

 
• F

  average force

 
• F′

  average force exerted by active muscles during the stride cycle

 
• k′

  proportionality constant

 
• M_b

  body mass

 
• P_met

  metabolic power

 
• Pcr

  phosphocreatine

 
• s

  ground speed

 
• s_max

  maximal velocity reached during the sprint

 
• t

  time

 
• t_e

  performance time

 
• v

  speed

 
• v_f

  forward speed

 
• v_max

  maximal speed

 
• V_O2

  oxygen consumption

 
• x

  incline angle between g′ and the terrain

 
• τ

  time constant

 
• η

  efficiency

S.F. is the recipient of a FIDAL Fellowship, Regional Committee of Friuli Venezia Giulia, Trieste, Italy. The financial support of FIDAL is gratefully acknowledged.