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 (ẇ₀) thanks to a negative time asymptote (k). We aimed to validate 3-p for short T_lim and to test the hypothesis that ẇ₀ 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 ẇ₀ 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 ẇ₀ and measured maximal instantaneous power, a modification of the model has been proposed.

Appendix

The classic form of the 3-p is

$${T_{{\text{lim}}}}=\frac{{W^{\prime}}}{{\dot {w} - {{\dot {w}}_{{\text{cr}}}}}}+k.$$

(7)

Inserting T_lim = 0 and ẇ = ẇ₀ into (7) as already pointed out by Morton (1996), we can define k as

$$k= - \frac{{W^{\prime}}}{{{{\dot {w}}_0} - {{\dot {w}}_{{\text{cr}}}}}}.$$

(8)

Then, we define W⁺ as the effective work capacity above ẇ_cr:

$${W^+}=\left( {\dot {w} - {{\dot {w}}_{{\text{cr}}}}} \right)~{T_{{\text{lim}}}}.$$

(9)

Substituting T_lim with its definition in (7) yields

$${W^+}=\left( {\dot {w} - {{\dot {w}}_{{\text{cr}}}}} \right)~\left( {\frac{{W^{\prime}}}{{\dot {w} - {{\dot {w}}_{{\text{cr}}}}}}+k} \right)=W^{\prime}+k\left( {\dot {w} - {{\dot {w}}_{{\text{cr}}}}} \right).$$

(10)

Since k is negative, it is evident that in 3-p W⁺ decreases linearly with increasing ẇ. Substituting k with its definition in (8) yields

$${W^+}=W^{\prime} - W^{\prime}\frac{{\dot {w} - {{\dot {w}}_{{\text{cr}}}}}}{{{{\dot {w}}_0} - {{\dot {w}}_{{\text{cr}}}}}}=W^{\prime}\left( {1 - \frac{{\dot {w} - {{\dot {w}}_{{\text{cr}}}}}}{{{{\dot {w}}_0} - {{\dot {w}}_{{\text{cr}}}}}}} \right)=W^{\prime}\frac{{{{\dot {w}}_0} - \dot {w}}}{{{{\dot {w}}_0} - {{\dot {w}}_{{\text{cr}}}}}}$$

(11)

that corresponds to Eq. (4a) of the main text. So, when ẇ → ẇ₀, W⁺ → 0, while when ẇ → ẇ_cr, W⁺ → W’. It is of note that in 2-p, W⁺ = W’.

To analyse the relationship between W⁺ and T_lim, we can re-arrange (7) as

$$\dot {w} - {\dot {w}_{{\text{cr}}}}=\frac{{W^{\prime}}}{{{T_{{\text{lim}}}} - k}}.$$

(12)

Substituting (12) into (9) yields W⁺ as a function T_lim:

$${W^+}=W^{\prime}\frac{{{T_{{\text{lim}}}}}}{{{T_{{\text{lim}}}} - k}}$$

(13)

that corresponds to Eq. (4b) of the main text. Since k is negative, it is evident that W⁺ increases hyperbolically with T_lim, starting from 0 when T_lim = 0 and tending towards W′ when T_lim → +∞.

It is of note that also the net (above resting) V̇O_2peak attained at the exhaustion of a constant-power trial above ẇ_cr depends on T_lim:

$$\dot {V}{{\text{O}}_{2{\text{peak}}\left( {{\text{net}}} \right)}}=\dot {V}{{\text{O}}_{2{\text{max}}\left( {{\text{net}}} \right)}}\left( {1 - {{\text{e}}^{ - \frac{{{T_{{\text{lim}}}}}}{\tau }}}} \right),$$

(14)

where τ is the mean time constant of the V̇O₂-on kinetics. Expressing (14) as a function of T_lim yields

$${T_{{\text{lim}}}}=\tau ~\ln \left( {1 - \frac{{\dot {V}{{\text{O}}_{2{\text{peak}}\left( {{\text{net}}} \right)}}}}{{\dot {V}{{\text{O}}_{2{\text{max}}\left( {{\text{net}}} \right)}}}}} \right).$$

(15)

Then, it is possible to substitute (15) into (13) to obtain the relationship between W⁺ and V̇O_2peak:

$${W^+}=W^{\prime}\frac{{\tau ~\ln \left( {1 - \frac{{\dot {V}{{\text{O}}_{2{\text{peak}}\left( {{\text{net}}} \right)}}}}{{\dot {V}{{\text{O}}_{2{\text{max}}\left( {{\text{net}}} \right)}}}}} \right)}}{{\tau ~\ln \left( {1 - \frac{{\dot {V}{{\text{O}}_{2{\text{peak}}\left( {{\text{net}}} \right)}}}}{{\dot {V}{{\text{O}}_{2{\text{max}}\left( {{\text{net}}} \right)}}}}} \right) - k}}.$$

(16)

Dividing both arms by W′ and simplifying for τ, we obtain the Equation used to describe Fig. 3b:

$$\frac{{{W^+}}}{{W^{\prime}}}=\frac{{\ln \left( {1 - \frac{{\dot {V}{{\text{O}}_{2{\text{peak}}\left( {{\text{net}}} \right)}}}}{{\dot {V}{{\text{O}}_{2{\text{max}}\left( {{\text{net}}} \right)}}}}} \right)}}{{\ln \left( {1 - \frac{{\dot {V}{{\text{O}}_{2{\text{peak}}\left( {{\text{net}}} \right)}}}}{{\dot {V}{{\text{O}}_{2{\text{max}}\left( {{\text{net}}} \right)}}}}} \right) - \frac{k}{\tau }}}.$$

(17)

Fitting Eq. (17) with our data (Fig. 3b) yields τ = 19 s, probably as a result of the speeding of V̇O₂ kinetics during explosive supramaximal exercise and the fact that starting V̇O₂ was higher than resting metabolic rate due to the warm-up.