# Experimental validation of the 3-parameter critical power model in cycling

## Abstract

### Purpose

The three-parameter model of critical power (3-p) implies that in the severe exercise intensity domain time to exhaustion (*T*_{lim}) decreases hyperbolically with power output starting from the power asymptote (critical power, *ẇ*_{cr}) and reaching 0 s at a finite power limit (*ẇ*_{0}) thanks to a negative time asymptote (*k*). We aimed to validate 3-p for short *T*_{lim} and to test the hypothesis that *ẇ*_{0} represents the maximal instantaneous muscular power.

### Methods

Ten subjects performed an incremental test and nine constant-power trials to exhaustion on an electronically braked cycle ergometer. All trials were fitted to 3-p by means of non-linear regression, and those with *T*_{lim} greater than 2 min also to the 2-parameter model (2-p), obtained constraining *k* to 0 s. Five vertical squat jumps on a force platform were also performed to determine the single-leg (i.e., halved) maximal instantaneous power.

### Results

*T*_{lim} ranged from 26 ± 4 s to 15.7 ± 4.9 min. In 3-p, with respect to 2-p, *ẇ*_{cr} was identical (177 ± 26 W), while curvature constant *W’* was higher (17.0 ± 4.3 vs 15.9 ± 4.2 kJ, *p* < 0.01). 3-p-derived *ẇ*_{0} was lower than single-leg maximal instantaneous power (1184 ± 265 vs 1554 ± 235 W, *p* < 0.01).

### Conclusions

3-p is a good descriptor of the work capacity above *ẇ*_{cr} up to *T*_{lim} as short as 20 s. However, since there is a discrepancy between estimated *ẇ*_{0} and measured maximal instantaneous power, a modification of the model has been proposed.

## Keywords

Physical work capacity Power–time relationship Non-linear model Hyperbolic model Endurance Anaerobic alactic metabolism## Abbreviations

- [La]
_{max} Capillary blood lactate concentration at the end of the incremental test

- [La]
_{peak} Capillary blood lactate concentration at the end of a constant-power trial

- 2-p
2-parameter critical power model

- 3-p
3-parameter critical power model

*k*Time asymptote of the critical power model

*k*^{ATP}Time asymptote of the power–time hyperbola provided by the immediately available ATP

*T*_{lim}Time to exhaustion

*V̇*O_{2}Oxygen consumption

*V̇*O_{2max}Maximal oxygen consumption of the incremental test

*V̇*O_{2peak}Peak oxygen consumption of a constant-power trial

*ẇ*Power output

- \(\widehat {{\dot {w}}}\)
Maximal instantaneous muscular power

*ẇ*_{cr}Critical power

*ẇ*_{0}Power limit of the critical power model as time to exhaustion approaches 0 s

*ẇ*_{max}Maximal aerobic mechanical power

*W’*Energy store component (mechanical)

*W*^{+}Mechanical work capacity above the critical power

*W*^{ATP}Mechanical work above the critical power provided by the immediately available ATP

## Notes

### Acknowledgements

The authors thank all the volunteers who participated in this study. This work was supported by Grant No. 2015 − 1080 from Cariplo Foundation, and by grants of University of Brescia to Guido Ferretti.

### Author contribution statement

GF conceived the hypothesis and the design of the study. GV, AT, SC, MD, NF, and CM performed the experiments. GV, AT, PB, and GF contributed to interpretation of data. GV conducted statistical analysis and wrote the first draft of the manuscript. All authors reviewed the draft and approved the final version of the manuscript.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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