British Milers’ Club News: An introduction to our model for endurance performance

Models to predict performances in running can be useful for both racing and training. It has been almost hundred years since the first systematic study of the connection between physiological principles and world record running performances was performed by Nobel prize winner Archibald Hill (1925). He proposed a mathematical model, based on metabolic energy considerations, for the maximal power output during a race. His so-called running curve predicts that the maximal power output first decreases rapidly with race time but then remains constant. This implies the existence of maximal speed that can be sustained for any duration. Variations of Hill’s model have been proposed to predict performances. However, the existence of a maximal speed that is sustainable for an arbitrarily long race contradicts running records. In fact, existing models appear to be unable to explain a fundamental observation that has been made already by Hill: The average running speed during a race keeps decreasing with the duration of the race but rather slowly, namely according to a logarithmic time scale, see Figure. For distances raced below VO2max speed, this means that the difference between racing speed and speed at VO2max increases two times when the race duration is squared. For example, if the speed at VO2max is 400m/min, and the average speed during a 10min race is 370m/min, then the speed difference is 30m/min, and the runner can sustain a speed of 340m/min for 102 min = 100min. This observation has been employed by Peronnet and Thibault in 1989 to deduce physiological characteristics from running records and to predict running performances with high accuracy. They introduced an endurance index for long distances that accounts for fatigue effects that are not related to VO2max, and not accounted for by an effective VO2max like the VDOT index in Daniels’ popular running formula. The endurance index measures the amount of the above mentioned speed difference to the speed at VO2max. Given these observations, one might ask what are the important physiological parameters that determine running performances over a wide range of distances. To answer this question, we recently developed a theoretical model from basic principles of metabolic power generation and utilisation. An important observation that is essential for the construction of our model is that running economy (the linear relation between power output and running speed) usually becomes worse with the duration of a running event.