Modèles mathématiques et physiologiques de la performance humaine

Modelling of Running Performances: Comparisons of Power-Law, Hyperbolic, Logarithmic, and Exponential Models in Elite Endurance Runners

Many empirical and descriptive models have been proposed since the beginning of the 20th century. In the present study, the power-law (Kennelly) and logarithmic (Péronnet-Thibault) models were compared with asymptotic models such as 2-parameter hyperbolic models (Hill and Scherrer), 3-parameter hyperbolic model (Morton), and exponential model (Hopkins). These empirical models were compared from the performance of 6 elite endurance runners (P. Nurmi, E. Zatopek, J. Väätäinen, L. Virén, S. Aouita, and H. Gebrselassie) who were world-record holders and/or Olympic winners and/or world or European champions. These elite runners were chosen because they participated several times in international competitions over a large range of distances (1500, 3000, 5000, and 10000 m) and three also participated in a marathon. The parameters of these models were compared and correlated. The less accurate models were the asymptotic 2-parameter hyperbolic models but the most accurate model was the asymptotic 3-parameter hyperbolic model proposed by Morton. The predictions of long-distance performances (maximal running speeds for 30 and 60 min and marathon) by extrapolation of the logarithmic and power-law models were more accurate than the predictions by extrapolation in all the asymptotic models. The overestimations of these long-distance performances by Morton's model were less important than the overestimations by the other asymptotic models.

Metabolic factors limiting performance in marathon runners

Each year in the past three decades has seen hundreds of thousands of runners register to run a major marathon. Of those who attempt to race over the marathon distance of 26 miles and 385 yards (42.195 kilometers), more than two-fifths experience severe and performance-limiting depletion of physiologic carbohydrate reserves (a phenomenon known as ‘hitting the wall’), and thousands drop out before reaching the finish lines (approximately 1–2% of those who start). Analyses of endurance physiology have often either used coarse approximations to suggest that human glycogen reserves are insufficient to fuel a marathon (making ‘hitting the wall’ seem inevitable), or implied that maximal glycogen loading is required in order to complete a marathon without ‘hitting the wall.’ The present computational study demonstrates that the energetic constraints on endurance runners are more subtle, and depend on several physiologic variables including the muscle mass distribution, liver and muscle glycogen densities, and running speed (exercise intensity as a fraction of aerobic capacity) of individual runners, in personalized but nevertheless quantifiable and predictable ways. The analytic approach presented here is used to estimate the distance at which runners will exhaust their glycogen stores as a function of running intensity. In so doing it also provides a basis for guidelines ensuring the safety and optimizing the performance of endurance runners, both by setting personally appropriate paces and by prescribing midrace fueling requirements for avoiding ‘the wall.’ The present analysis also sheds physiologically principled light on important standards in marathon running that until now have remained empirically defined: The qualifying times for the Boston Marathon.

Marathon running, historically perceived as testing the physiologic limits of human endurance, has become increasingly popular even among recreational runners. Of those runners who test their endurance by racing the marathon distance, however, more than two in five report ‘hitting the wall,’ the rapid onset of severe fatigue and inability to maintain a high-intensity pace, resulting from the near-complete depletion of carbohydrate stores in the leg muscles and liver. An apparent paradox of long-distance running is that even the leanest athletes store enough fat to power back-to-back marathons, yet small carbohydrate reservoirs can nevertheless catastrophically limit performance in endurance exercise. In this study I develop and validate a mathematical model that facilitates computation of personalized estimates of the distances at which runners will exhaust their carbohydrate stores while running at selected paces. In addition, I provide a systematic approach to estimating personalized maximum speeds at which runners can safely complete a marathon, based on accessible physiologic parameters such as heart rate and running speed. This analysis provides a quantitative basis for improving the safety and optimizing the performance of endurance runners, evaluating midrace fueling requirements, and estimating limits of performance in human endurance running, for elite and recreational runners alike.

A Prediction Model for Peak Power Output From Different Incremental Exercise Tests

Context:

There is currently no model that predicts peak power output (PPO) thereby allowing comparison between different incremental exercise test (EXT) protocols. In this study we have used the critical power profile to develop a mathematical model for predicting PPO from the results of different EXTs.

Purpose:

The purpose of this study was to examine the level of agreement between actual PPO values and those predicted from the new model.

Methods:

Eleven male athletes (age 25 ± 5 years, VO2max 62 ± 8 mL · kg–1 · min–1) completed 3 laboratory tests on a cycle ergometer. Each test comprised an EXT consisting of 1-minute workload increments of 30 W (EXT30/1) and 3-minute (EXT25/3) and 5-minute workload increments (EXT25/5) of 25 W. The PPO determined from each test was used to predict the PPO from the remaining 2 EXTs.

Results:

The differences between actual and predicted PPO values were statistically insignificant (P > .05). The random error components of the limits of agreement of ≤30 W also indicated acceptable levels of agreement between actual and predicted PPO values.

Conclusions:

Further data collection is necessary to confirm whether the model is able to predict PPO over a wide range of EXT protocols in athletes of different aerobic and anaerobic capacities.