A mathematical model of performance describing aerobic and anaerobic energy production during exercise was applied to middle-distance running data from world records (WR) and from a group of elite runners (NL). The model is based on the assumption that, above a critical power (Pc), a continuous rate of anaerobic energy production occurs, until the entire anaerobic stores (W') are depleted. The fraction of metabolic power above Pc provided by anaerobic metabolism is denoted alpha. A second power threshold (Pt) sets the limit above which any further increase in power is met exclusively by anaerobic sources. The oxygen uptake kinetics was described by a monoexponential equation with time constant tau. The results show that the model successfully fits the WR over 1,500-5,000 m. However, in the range of distances from 800 to 5,000 m the performance over 800 and 1,000 m were overestimated. Contrary to Pc and the anaerobic contribution at steady state oxygen uptake, the estimate of W' was sensitive to the value assigned to tau in the range from 0 to 30 s. Using best performances from 1,500 to 5,000 m in NL resulted in Pc estimates not significantly different from the metabolic power at the lactate threshold. The anaerobic contribution at steady state oxygen uptake increased from zero at Pc to 8.3% (WR) and 7.8+/-3.1% (NL) at Pt. This suggests that a substantial contribution of anaerobic processes occurs in the range between Pc and Pt, even though the exercise does not elicit maximal aerobic power.

The purpose of this study was to analyze the relevance of introducing the maximal power (P(m)) into a critical-power model. The aims were to compare the P(m) with the instantaneous maximal power (P(max)) and to determine how the P(m) affected other model parameters: the critical power ( P(c)) and a constant amount of work performed over P(c)(W'). Twelve subjects [22.9 (1.6) years, 179 (7) cm, 74.1 (8.9) kg, 49.4 (3.6) ml/min/kg] completed one 15 W/min ramp test to assess their ventilatory threshold (VT), five or six constant-power to exhaustion tests with one to measure the maximal accumulated oxygen deficit (MAOD), and six 5-s all-out friction-loaded tests to measure P(max) at 75 rpm, which was the pedaling frequency during tests. The power and time to exhaustion values were fitted to a 2-parameter hyperbolic model (NLin-2), a 3-parameter hyperbolic model (NLin-3) and a 3-parameter exponential model (EXP). The P(m) values from NLin-3 [760 (702) W] and EXP [431 (106) W] were not significantly correlated with the P(max) at 75 rpm [876 (82) W]. The P(c) value estimated from NLin-3 [186 (47) W] was not significantly correlated with the power at VT [225 (32) W], contrary to other models ( P <0.001). The W' from NLin-2 [25.7 (5.7) kJ] was greater than the MAOD [14.3 (2.7) kJ, P < 0.001] with a significant correlation between them (R = 0.76, P <0.01). For NLin-3, computation of W (P > P c), the amount of work done over P(C), yielded results similar to the W' value from NLin-2: 27.8 (7.4) kJ, which correlated significantly with the MAOD (R = 0.72, P <0.01). In conclusion, the P(m) was not related to the maximal instantaneous power and did not improve the correlations between other model parameters and physiological variables.

The aim of this study was to compare the lactate indices provided by single- and double-breakpoint models with lactate thresholds obtained with conventional methods. Arterial samples for the determination of lactate concentrations were drawn from eight participants at rest and every minute during a ramp test (15 W x min(-1)) on a cycle ergometer. Lactate thresholds were determined from a blood lactate concentration equal to 4 mM (LT(4)), from an increase of 1 mM above the resting level (Delta1 mM), and from indirect methods using ventilatory parameters. Other indices were computed from the modelling of the lactate curve using an exponential function (LSI), a polynomial function (Dmax), a semi-log model (SLog), a parabola plus delay model (Mod P), and a two-breakpoint model (Mod M). Mod P and Mod M showed poor agreement with the other methods. LT(4), Dmax, LSI, and respiratory exchange ratio equal to 1 were correlated with each other (0.81

An extension of the original hyperbolic model (Model-2) was proposed by using power output required to elicit maximal oxygen uptake (Pt). This study aimed to test this new model (Model-alpha) using mechanical work produced during cycle ergometry. Model alpha assumed that power exceeding a critical power (Pc) was met partly by the anaerobic metabolism. The parameter alpha was the proportion of the power exceeding Pc provided by anaerobic metabolism, while power exceeding Pt was exclusively met by anaerobic metabolism. Aerobic power was assumed to rise monoexponentially with a time constant tau. The exhaustion was assumed to be reached when the anaerobic work capacity W' was entirely utilised. Twelve subjects performed one progressive ramp test to assess the power at ventilatory threshold (P(VT)) and Pt and five constant-load exercise to exhaustion within 2-30 min, with one to estimate the maximal accumulated oxygen deficit (MAOD). Parameters from Model alpha were fitted with tau equal to 0, 10, 20 and 30 s. Results in goodness-of-fit was better than Model-2 whatever the value assumed for tau (P < 0.05). The value of tau did not affect much the estimates for P (c) and alpha. P (c) estimates were significantly correlated with Pc from Model-2 and with P(VT). W' estimates, which were dependent on the value ascribed to tau, were not statistically different than MAOD. These two variables were, however, not significantly correlated. In conclusion, Model alpha could provide useful information on the critical power and the anaerobic contribution according to exercise intensity, whereas W' estimates should be used with care because of the sensitivity to the assumption on aerobic power kinetics tau.

This study aimed to test the consistency of using the power required to elicit maximal oxygen uptake during incremental test (P (t)) to demarcate the range of power intensity in the modelling of the power-exhaustion time relationship. Different mathematical procedures were tested using data from ten subjects exercising on a cycle ergometer. After the determination of P (t) and the power at the ventilatory threshold, the subjects did six tests at constant power to exhaustion within 2-15 min. Estimates were obtained from a segmented model using two distinct equations of the anaerobic contribution to power below and above P (t), respectively. This model fit the overall data with a better adequacy than the simple hyperbolic model (standard error of 29.2 +/- 25.2 vs. 42.3 +/- 25.2 s). The power asymptotes were 225.7 +/- 27.3 W from the segmented model, 226.2 +/- 27.3 and 283.3 +/- 20.5 W from the simple model applied to data below and above P (t), respectively. The estimates from the segmented model were strongly correlated with their analogues from the simple model applied only to data below P (t) (R = 1.00 for power asymptote and curvature coefficient). They were not correlated with their analogues from the simple model applied only to data above P (t). These discrepancies between modelling procedures could arise from the method used to determine P (t) and the oversimplification of the oxygen uptake kinetics. These limitations could lead the segmented model to an overestimation of the anaerobic contribution which was around 15% of total energy expended at P (t).