Mathematical analysis of running performance and world running records

Abstract

The objective of this study was to develop an empirical model relating human running performance to some characteristics of metabolic energy-yielding processes using A, the capacity of anaerobic metabolism (J/kg); MAP, the maximal aerobic power (W/kg); and E, the reduction in peak aerobic power with the natural logarithm of race duration T, when T greater than TMAP = 420 s. Accordingly, the model developed describes the average power output PT (W/kg) sustained over any T as PT = [S/T(1 - e-T/k2)] + 1/T integral of T O [BMR + B(1 - e-t/k1)]dt where S = A and B = MAP - BMR (basal metabolic rate) when T less than TMAP; and S = A + [Af ln(T/TMAP)] and B = (MAP - BMR) + [E ln(T/TMAP)] when T greater than TMAP; k1 = 30 s and k2 = 20 s are time constants describing the kinetics of aerobic and anaerobic metabolism, respectively, at the beginning of exercise; f is a constant describing the reduction in the amount of energy provided from anaerobic metabolism with increasing T; and t is the time from the onset of the race. This model accurately estimates actual power outputs sustained over a wide range of events, e.g., average absolute error between actual and estimated T for men's 1987 world records from 60 m to the marathon = 0.73%. In addition, satisfactory estimations of the metabolic characteristics of world-class male runners were made as follows: A = 1,658 J/kg; MAP = 83.5 ml O2.kg-1.min-1; 83.5% MAP sustained over the marathon distance. Application of the model to analysis of the evolution of A, MAP, and E, and of the progression of men's and women's world records over the years, is presented.

Energy System Interaction and Relative Contribution During Maximal Exercise

“…Energy System Interaction and Contributionvery successful in predicting performance [75,85,91,92] and have been suggested to be generally more consistent than the body of experimental data. [42] Pér-onnet and Thibault, [85] covering 16 distances from 60m to the marathon, correlated 1987 male records with an average absolute prediction error of only 0.7% and a maximum error of 2.3%.…”

“…[75,78,85,89,90] This relatively high aerobic contribution is indirectly supported by the strong relationship between laboratory-determined maximal oxygen deficit and performance times over 100 and 400m, but not 800m, which suggests that the importance of the anaerobic energy supply diminishes, and consequently the aerobic energy supply increases, as running distance increases. [86] This is clearly illustrated in figure 1, which summarises the findings of Spencer and Gastin [89] who attempted to evaluate the energy system contribution, using the accumulated oxygen deficit method, to simulated sprint and middle distance running events.…”

“…[42] Pér-onnet and Thibault, [85] covering 16 distances from 60m to the marathon, correlated 1987 male records with an average absolute prediction error of only 0.7% and a maximum error of 2.3%. Ward-Smith [92] reported similar prediction errors (0.9 and 1.5%, respectively) for Olympic Games results over 7 distances and 4 Olympiads.…”

Maximal aerobic power and anaerobic capacity in cycling across the age spectrum in male master athletes

“…This sort of models has been successfully utilized in the past to predict best performance times in 391 running (Péronnet and Thibault 1989;di Prampero et al 1993) and in track cycling (Capelli et al 1998; Olds et al…”

“…In all the cases, this approach has proved to 394 estimate theoretical best performance times with remarkable precision both in élite and medium level, young athletes 395 (Capelli et al 1998;Olds et al 199;Péronnet and Thibault 1989;di Prampero et al 1993). Therefore, it seems to be 396 based on a sound knowledge of the physiological mechanisms dictating metabolic energy production in humans and of 397 the bioenergetical aspects of human locomotion.…”