Modelling human endurance: Power laws vs critical power

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 viewed as the "gold standard" for assessing exercise capacity, e.g. in cycling, running, rowing, and 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 a mathematical result which suggests that the power-law model is a safer tool for pace selection than the hyperbolic model. 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.

Can We Predict Long-Duration Running Power Output? Validity of the Critical Power, Power Law, and Logarithmic Models

“…Gaesser et al (6) observed that the CP 3hyp reported the lowest CP value which did not differ with the work rate of the so-called ventilatory threshold for longterm exercise (i.e., the highest power output that could be maintained for 40 minutes without a rise in ventilation from the minute 20). This position the CP at work rates close to the 30-minute (CP 2hyp : 98 [4]%; CP 3hyp : 99 [4]%) and 60-minute (CP 2hyp : 103 [5]%; CP 3hyp : 104 [5]%) power output, which consequently enhance the accuracy of the estimates.…”

“…Predicting long-distance running performance has been always of great interest to athletes and practitioners. It is well known that the capacity to sustain a work rate decreases with the increase of time, and different functions have been proposed to model it (5,14,20). On the one hand, the critical power (CP) models define the power-duration curve (PDC) along the severe-intensity domain.…”