The Energy Cost of Sprint Running and the Energy Balance of Current World Records from 100 to 5000 m

Abstract

The time course of metabolic power during 100–400 m top running performances in world class athletes was estimated assuming that accelerated running on flat terrain is biomechanically equivalent to uphill running at constant speed, the slope being dictated by the forward acceleration. Hence, since the energy cost of running uphill is known, energy cost and metabolic power of accelerated running can be obtained, provided that the time course of the speed is determined. Peak metabolic power during the 100 and 200 m current world records (9.58 and 19.19 s) and during a 400 m top performance (44.06 s) amounted to 163, 99 and 75 W kg⁻¹, respectively. Average metabolic power and overall energy expenditure during 100–5000 m current world records in running were also estimated as follows. The energy spent in the acceleration phase, as calculated from mechanical kinetic energy (obtained from average speed) and assuming 25% efficiency for the transformation of metabolic into mechanical energy, was added to the energy spent for constant speed running (air resistance included). In turn, this was estimated as: (3.8 + k′ v²) · d, where 3.8 J kg⁻¹ m⁻¹ is the energy cost of treadmill running, k′ = 0.01 J s² kg⁻¹ m⁻³, v is the average speed (m s⁻¹) and d (m) the overall distance. Average metabolic power decreased from 73.8 to 28.1 W kg⁻¹ with increasing distance from 100 to 5000 m. For the three shorter distances (100, 200 and 400 m), this approach yielded results rather close to mean metabolic power values obtained from the more refined analysis described above. For distances between 1000 and 5000 m the overall energy expenditure increases linearly with the corresponding world record time. The slope and intercept of the regression are assumed to yield maximal aerobic power and maximal amount of energy derived from anaerobic stores in current world records holders; they amount to 27 W kg⁻¹ (corresponding to a maximal O₂ consumption of 77.5 ml O₂ kg⁻¹ min⁻¹ above resting) and 1.6 kJ kg⁻¹ (76.5 ml O₂ kg⁻¹). This last value is on the same order of the maximal amount of energy that can be derived from complete utilisation of phosphocreatine in the active muscle mass and from maximal tolerable blood lactate accumulation. The anaerobic energy yield has also been estimated, throughout the overall set of distances (100–5000 m), assuming that, at work onset, the rate of O₂ consumption increases with a time constant of 20 s tending to the appropriate metabolic power, but stops increasing once the maximal O₂ consumption is attained. Hence the overall energy expenditure can be partitioned into its aerobic and anaerobic components. This last increases from about 0.6 kJ kg⁻¹ for the shortest distance (100 m) to a maximum close to that estimated above (1.6 kJ kg⁻¹) for distances of 1500 m or longer.

Appendix

The equations appearing in this Appendix are numbered (12.21)–(12.33), even if some of them have previously been mentioned in the text.

In a preceding section of this chapter (12.6) it was assumed that the overall energy (E_tot) spent during a supramaximal effort to exhaustion can be described by:

$$E_{tot} = AnS + \dot{V}O_{2} max \cdot t_{e} {-}\dot{V}O_{2} max \cdot (1 - {\text{e}}^{{ - t_{e} /\tau }} ) \cdot \tau$$

(12.21)

where AnS is the amount of energy derived from anaerobic sources, t_e is the time to exhaustion and $\uptau( \approx 20\;\text{s})$ is the time constant of the ${\dot{\text{V}}}{\text{O}_2}$ kinetics at work onset. The third term of this equation takes into account the fact that ${\dot{\text{V}}}{\text{O}_2}$max is not attained at the very onset of the exercise, but it is reached following an exponential function with the time constant $\uptau$ (Wilkie 1980). Thus:

$$AnS = E_{tot} - \dot{V}O_{2} max \cdot t_{e} {+}\dot{V}O_{2} max \cdot (1 - {\text{e}}^{{ - t_{\text{e}} /\tau }} ) \cdot \tau$$

(12.22)

Furthermore, if t_e is sufficiently long (i.e. $\ge 4\uptau$), the quantity $\text{e}^{{ - \text{t}_{\text{e}} /\uptau}}$ becomes vanishingly small and the third term of the equation reduces to ${\dot{\text{V}}}{\text{O}_2}$max · τ. In this range of exhaustion times therefore, AnS can be easily estimates as:

$$AnS = E_{tot} - \dot{V}O_{2} max \cdot t_{e} + \dot{V}O_{2} max \cdot \tau$$

(12.22′)

Equations (12.21) and (12.22) are based on the implicit assumption that the ${\dot{\text{V}}}{\text{O}_2}$ kinetics is a continuous exponential function from the value prevailing at work onset to ${\dot{\text{V}}}{\text{O}_2}$max. However, as discussed in detail elsewhere (di Prampero et al. 2015), at the onset of supramaximal exercise [in which case the metabolic power requirement (Ė) is greater than the subject’s ${\dot{\text{V}}}{\text{O}_2}$max] ${\dot{\text{V}}}{\text{O}_2}$ increases exponentially towards Ė with the same time constant $(\uptau \approx 20\;\text{s})$, but stops abruptly at the very moment (t₁) when ${\dot{\text{V}}}{\text{O}_2}$max is attained (Margaria et al. 1965). This is shown graphically in Fig. 12.8, where Ė (red horizontal line) and ${\dot{\text{V}}}{\text{O}_2}$max (blue horizontal line) are indicated as a function of the exercise time, together with the time course of ${\dot{\text{V}}}{\text{O}_2}$ before the attainment of ${\dot{\text{V}}}{\text{O}_2}$max (blue continuous curve) and with the hypothetical time course of ${\dot{\text{V}}}{\text{O}_2}$ above ${\dot{\text{V}}}{\text{O}_2}$max, were Ė = ${\dot{\text{V}}}{\text{O}_2}$max (green broken curve). Inspection of this figure makes it immediately apparent that, whereas Eq. (12.22) is correct for Ė = ${\dot{\text{V}}}{\text{O}_2}$max, whenever Ė > ${\dot{\text{V}}}{\text{O}_2}$max it leads to an overestimate of AnS, the more so, the greater the difference between Ė and ${\dot{\text{V}}}{\text{O}_2}$max.

Fig. 12.8

Overall metabolic power requirement (red horizontal line, Ė) as a function of time (t) during square wave exercise of constant intensity and duration t_e. Subject’s maximal O₂ consumption (${\dot{\text{V}}}{\text{O}_2}$max) is indicated by blue horizontal line. At work onset, ${\dot{\text{V}}}{\text{O}_2}$ increases exponentially towards Ė, but stops abruptly at t₁, i.e. when ${\dot{\text{V}}}{\text{O}_2}$max is attained. Actual ${\dot{\text{V}}}{\text{O}_2}$ before t₁ is indicated by continuous blue curve, whereas after t₁ ${\dot{\text{V}}}{\text{O}_2}$ = ${\dot{\text{V}}}{\text{O}_2}$max. Green broken curve denotes hypothetical ${\dot{\text{V}}}{\text{O}_2}$ time course, were Ė ≤ ${\dot{\text{V}}}{\text{O}_2}$max. Anaerobic energy yield is given by the sum of areas 1 + 2 + 3 + 3′ (red); aerobic yield by the sum of areas 4 + 5 (blue). See text for details and calculations (Color figure online)

The aim of the paragraphs that follow is to describe an approach yielding a more accurate estimate of the anaerobic energy yield (AnS) in running at supramaximal constant metabolic power (Ė), provided the subject’s ${\dot{\text{V}}}{\text{O}_2}$max, the exercise duration (t_e) and $\uptau$ are known. Indeed, on the one side, this set of data allows one to estimate Ė, as described by Eq. (12.18):

$$\dot{E} = E_{tot} /t_{e} = \{ [C_{0} + k^{{\prime }} \cdot (d/t_{e} )^{2} ] \cdot d + (d/t_{e} )^{2} (2\eta)^{-1} \} /t_{e}$$

(12.23)

where all terms have been previously defined (see Sect. 12.6). On the other, if ${\dot{\text{V}}}{\text{O}_2}$max, $\uptau$ and t_e are also known, Fig. 12.8 allows one to appreciate graphically that the anaerobic contribution to the overall energy expenditure is represented by the area delimited by Ė and the ${\dot{\text{V}}}{\text{O}_2}$ − ${\dot{\text{V}}}{\text{O}_2}$max curve, i.e. by the sum of the areas 1, 2, 3 and 3′, whereas the sum of the two areas 4 and 5, below the ${\dot{\text{V}}}{\text{O}_2}$ − ${\dot{\text{V}}}{\text{O}_2}$max curve, represents the aerobic energy yield.

What follows is devoted to assess quantitatively the anaerobic energy yield as given by the sum of the areas defined above, indicated numerically as in the Fig. 12.8.

The ${\dot{\text{V}}}{\text{O}_2}$ kinetics at work onset is described by:

$$\dot{V}O_{2} (t) = \dot{E} \cdot (1 - {\text{e}}^{ - t/\tau } )$$

(12.24)

However, as shown in Fig. 12.8, ${\dot{\text{V}}}{\text{O}_2}$ stops increasing abruptly at time t₁, i.e. when ${\dot{\text{V}}}{\text{O}_2}$max is attained. Thus at t₁:

$$\dot{V}O_{2} (t) = \dot{E} \cdot (1 - {\text{e}}^{ - t1/\tau } ) = \dot{V}O_{2} max$$

(12.25)

Rearranging Eq. (12.25), one obtains:

$$1{-}\dot{V}O_{2} max/\dot{E} = {\text{e}}^{ - t1/\tau }$$

(12.26)

or:

$$\ln (1{-}\dot{V}O_{2} max/\dot{E}) = - \, t_{1} /\tau$$

(12.27)

where from t₁ can be finally obtained as:

$$t_{1} = - \ln (1{-}\dot{V}O_{2} max/\dot{E}) \cdot \tau$$

(12.28)

It can therefore be concluded that Eq. (12.28) allows one to estimate the time necessary to attain ${\dot{\text{V}}}{\text{O}_2}$max, provided that ${\dot{\text{V}}}{\text{O}_2}$max itself, together with Ė and $\uptau$ are known.

It is now possible to estimate the anaerobic energy yield proceeding as follows. The area of the rectangle $( { {{\boxed{ 2 }}} + {{\boxed{ 3 }}} + {{\boxed{ 3 }}}^{\prime }} )$ in Fig. 12.8 can be easily calculated as:

$${{\boxed{ 2 }}} + {{\boxed{ 3 }}} + {{\boxed{ 3 }}}^{\prime } = ( {\dot{E} - \dot{V}O_{2} max} ) \cdot t_{e}$$

(12.29)

The area corresponding to the O₂ deficit incurred once ${\dot{\text{V}}}{\text{O}_2}$max is attained (1 + 2) can be estimated as:

$${{\boxed{ 1 }}} + {{\boxed{ 2 }}} = \dot{V}O_{2} max \cdot \tau$$

(12.30)

Finally the area of the rectangle 2 is given by the product of the time t₁ (Eq. 12.28) and the vertical distance between Ė and ${\dot{\text{V}}}{\text{O}_2}$max:

$${{\boxed{ 2 }}} = t_{1} \cdot ( {\dot{E} - \dot{V}O_{2} max} ) = - \ln ( {1{-}\dot{V}O_{2} max/\dot{E}} ) \cdot \tau \cdot ( {\dot{E} - \dot{V}O_{2} max} )$$

(12.31)

The overall amount of energy derived from anaerobic stores (AnS) is finally expressed by the algebraic sum of the Eqs. (12.29), (12.30) and (12.31):

$$\begin{aligned} AnS & = {{\boxed{ 2 }}} + {{\boxed{ 3 }}} + {{\boxed{ 3 }}}^{\prime } + {{\boxed{ 1 }}} + {{\boxed{ 2 }}} - {{\boxed{ 2 }}} \\ & = ( {\dot{E} - \dot{V}O_{2} max} ) \cdot t_{e} + \dot{V}O_{2} max \cdot \tau -[ - \ln ( {1{-}\dot{V}O_{2} max/\dot{E}} ) \cdot \tau \cdot ( {\dot{E}{-}\dot{V}O_{2} max} )] \\ \end{aligned}$$

(12.32)

It should finally be pointed out that Eq. (12.32) defines the anaerobic energy yield whenever the exercise duration is greater that that necessary for ${\dot{\text{V}}}{\text{O}_2}$max to be attained (t_e > t₁). Whenever t_e ≤ t₁, things become much simpler, since in this specific case the only energy derived from anaerobic stores (AnS′) corresponds to the O₂ deficit incurred, as given by the product of the ${\dot{\text{V}}}{\text{O}_2}$ attained at the very end of the exercise, $( = {\dot{\text{E}}} \cdot (1 - {\text{e}}^{{ - {\text{t}}_{\text{e}} /\uptau}} ))$, Eq. (12.24) and the time constant of the ${\dot{\text{V}}}{\text{O}_2}$ kinetics ($\uptau$):

$$AnS^{{\prime }} = \dot{E} \cdot (1 - {\text{e}}^{{ - t_{e} /\tau }} ) \cdot \tau$$

(12.33)

It should finally be pointed out that, whenever Ė = ${\dot{\text{V}}}{\text{O}_2}$max, Eqs. (12.32) and (12.33) reduce to Eq. (12.22) or (12.22′) depending whether, or not, t_e is sufficiently long for ${\dot{\text{V}}}{\text{O}_2}$max to be attained.

It can be concluded that the anaerobic energy yield in running at supramaximal constant intensity can be easily estimated (Eqs. 12.32 and 12.33), provided that the metabolic power requirement (Eq. 12.23), together with the subject’s ${{\text{V}}}{\text{O}_2\text{max}}$, the exercise duration (t_e) and the time constant of the ${\dot{\text{V}}}{\text{O}_2}$ kinetics at work onset ($\uptau$) are known.