Mathematical analysis of running performance and world running records

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.