Abstract
The power–duration relationship describes 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, the hyperbolic (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-called functional 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-law model. 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.
Competing Interest Statement
The authors have declared no competing interest.
Footnotes
Modelling fatigue [Section 5]. In the revised version, we have added an entire new section which shows that, unlike the hyperbolic model, the power-law model is automatically consistent with the empirical observation that the power-duration curve moves downwards as the athlete becomes fatigued from prolonged exercise. We also explain why the often-made assumption that critical power decreases with fatigue is contradictory. Range of validity [Section 2.2.2]. In the revised version, we now explain why the assumption that the critical-power model is valid for exercise in the "severe-intensity domain" is impractical. Applying the critical-power model only to 2-15 minutes is not unfair [Appendix B.2]. In the revised version, we have added additional figures to illustrate that restricting the critical-power model to data in the 2-15 minute range is not unfair to the model. That is, the new figures illustrate that the prediction error inside the 2-15 minute range (where the model is thought to be valid) become much worse if the model was applied to all available data instead. Power-law models and world-record data [Section 2.3.1]. In the revised version, we now explain why findings from Garcia-Manso et al. (2012) that power-law models do not perfectly fit world-record performances across different athletes in running are not applicable to our setting. Acknowledgements. We have added an acknowledgements section.
