A minimal power model for human running performance

Abstract

Models for human running performances of various complexities and underlying principles have been proposed, often combining data from world record performances and bio-energetic facts of human physiology. The purpose of this work is to develop a novel, minimal and universal model for human running performance that employs a relative metabolic power scale. The main component is a self-consistency relation for the time dependent maximal power output. The analytic approach presented here is the first to derive the observed logarithmic scaling between world (and other) record running speeds and times from basic principles of metabolic power supply. Our hypothesis is that various female and male record performances (world, national) and also personal best performances of individual runners for distances from 800m to the marathon are excellently described by this model. Indeed, we confirm this hypothesis with mean errors of (often much) less than 1%. The model defines endurance in a way that demonstrates symmetry between long and short racing events that are separated by a characteristic time scale comparable to the time over which a runner can sustain maximal oxygen uptake. As an application of our model, we derive personalized characteristic race speeds for different durations and distances.

Fig 1. Definition of endurance for long and short duration, E_l and E_s, respectively, from the duration T(p) over which a relative power p can be sustained.

Shown is a typical range of endurances for long and short duration (gray regions, with lower and upper limits for γ_l and γ_s) and an example curve that visualizes the definition of E_l and E_s.

Fig 2. Mean race velocity v¯(d) as function of race distance.

Velocity is re-scaled by v_m, and distance d is re-scaled by d_c = v_mt_c. Shown are the male and female world records (WR, dots), model prediction from Eq (12) (solid lines), and a typically expected maximal range of velocities (gray regions). Indicated are the lower and upper limits of γ_s and γ_l for these regions. Due to the re-scaling of $\overline{v}\left(d\right)$ and d, this graph highlights endurance for short and long duration, independently of the velocity v_m at maximal aerobic power.

Fig 3. Plot of the supplemental factor of Eq (15) for as predicted by our model for male and female world records (WR), US records (US), and European records (EU).

The cusp in the curves occurs at the time t_c.

Fig 4. Same visualization of endurance as in Fig 2 but for individual male (top) and female (bottom) runners, see Tables 3 and 4.

Colors label different runners.