S2 Appendix. Comparison to oxygen uptake
measurement
While it is assuring to see below that our model can explain and predict record and
individual racing times, a more direct comparison to power output during running is
desirable to probe the logarithmic decline of the maximal power output with exercise
duration, as predicted by Eq. (6). This is of particular importance in the anaerobic
range where different functional forms, e.g., exponential decays, have been proposed [6].
However, running power, as measured by oxygen utilization, can be directly determined
only in the aerobic regime. For (supra-maximal) exercise with substantial contributions
from anaerobic systems where power output exceeds maximal oxygen uptake, Medbo et
al. showed that the oxygen demand can be estimated by extrapolating each runner’s
individual nominal linear relationship between running speed and submaximal oxygen
uptake [1]. The difference between the extrapolated oxygen utilization and the
measured oxygen uptake is the accumulated oxygen deficit. Using this method, Medbo
et al. determined from treadmill exercise at speeds that caused exhaustion within
different predetermined durations the oxygen demand relative to the maximal uptake.
Translated to percent of maximal aerobic power output, this oxygen demand is given by
100 × Pmax(T)/Pm in our model, with Pmax(T) given in Eq. (6).
While a logarithmic dependence for Pmax(T) has been deduced from purely
empirical data analyses for world records for times above tc before [6], to our knowledge
a logarithmic scaling has not been proposed for shorter exercise with large anaerobic
involvement. Hence, it is interesting that there exists experimental estimates of the
maximal oxygen utilization that can be maintained for a given duration. As explained
above, Medbo et al. [1] obtained for 11 runners data that correspond to
100 × Pmax(T)/Pm which is shown as function of T < 5min ∼ tc in Fig. A. We have
fitted the prediction of our model to the data, and the results for the runner with
smallest and largest oxygen demand are shown in the same figure. The agreement
between the data and our model prediction appears to be rather convincing. This
suggests that there exists indeed a logarithmic relation between maximally sustainable
power and duration in the range of supra-maximal intensities, resembling observation
that were made before in the sub-maximal zone.
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Fig A. Relative nominal oxygen demand as function of the maximum
duration over which it can be sustained. Original plot and data for 11
runners from Ref. [1]. The two curves are fits of Eq. (6) to the data for the
runners with smallest and largest relative oxygen demand.
References
1. Medbo JI, Mohn AC, Tabata I, Bahr R, Vaage O, Sejersted OM. Anaerobic
capacity determined by maximal accumulated O2 deficit. J Appl Physiol.
1988;64(1):50–60.
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A minimal power model for human running performance.
Mulligan, Matthew,Adam, Guillaume,Emig, Thorsten