Modelling human endurance: Power laws vs critical power

Abstract

Thepower–duration relationshipdescribes the time to exhaustion for exercise at different intensities. It is generally believed to be a “fundamental bioenergetic property of living systems” that this relationship is hyperbolic. Indeed, thehyperbolic(a.k.a.critical-power) model which formalises this belief is the dominant tool for describing and predicting high-intensity exercise performance, e.g. in cycling, running, rowing, or swimming. However, the hyperbolic model is now the focus of two heated debates in the literature because: (a) it unrealistically represents efforts that are short (< 2 minutes) or long (> 15 minutes); (b) it contradicts widely-used performance predictors such as the so-calledfunctional threshold power(FTP) in cycling. We contribute to both debates by demonstrating that the power–duration relationship is more adequately represented by an alternative,power-lawmodel. In particular, we show that the often observed good fit of the hyperbolic model between 2 and 15 minutes should not be taken as proof that the power–duration relationship is hyperbolic. Rather, in this range, a hyperbolic function just happens to approximate a power law fairly well. We also prove mathematical results which suggest that the power-law model is a safer tool for pace selection than the hyperbolic model and that the former better models fatigue than the latter. Finally, we use the power-law model to shed light on popular performance predictors in cycling, running and rowing such as FTP and Jack Daniels’“VDOT” calculator.