Effects of recovery interval duration on the parameters of the critical power model for incremental exercise

Abstract

Introduction

We tested the linear critical power ($\dot{w}_{\text{cr}}$) model for discrete incremental ramp exercise implying recovery intervals at the end of each step.

Methods

Seven subjects performed incremental (power increment 25 W) stepwise ramps to subject’s exhaustion, with recovery intervals at the end of each step. Ramps’ slopes (S) were 0.83, 0.42, 0.28, 0.21, and 0.08 W s⁻¹; recovery durations (t _r) were 0 (continuous stepwise ramps), 60, and 180 s (discontinuous stepwise ramps). We determined the energy store component (W′), the peak power ($\dot{w}_{\text{peak}}$), and $\dot{w}_{\text{cr}}$.

Results

When t _r = 0 s, $\dot{w}_{\text{cr}}$ and W′ were 187 ± 26 W and 14.5 ± 5.8 kJ, respectively. When t _r = 60 or 180 s, the model for ramp exercise provided inconsistent $\dot{w}_{\text{cr}}$ values. A more general model, implying a quadratic $\dot{w}_{\text{peak}}$ versus $\sqrt S$ relationship, was developed. This model yielded, for t _r = 60 s, $\dot{w}_{\text{cr}}$ = 189 ± 48 W and W′ = 18.6 ± 17.8 kJ, and for t _r = 180 s, $\dot{w}_{\text{cr}}$ = 190 ± 34 W, and W′ = 16.4 ± 16.7 kJ. These $\dot{w}_{\text{cr}}$ and W′ did not differ from the corresponding values for t _r = 0 s. Nevertheless, the overall amount of energy sustaining work above $\dot{w}_{\text{cr}}$, due to energy store reconstitution during recovery intervals, was higher the longer t _r, whence higher $\dot{w}_{\text{peak}}$ values.

Conclusions

The linear $\dot{w}_{\text{cr}}$ model for ramp exercise represents a particular case (for t _r = 0 s) of a more general model, accounting for energy resynthesis following oxygen deficit payment during recovery.

Appendix

In intermittent exercise, the amount of energy store component (W′) that is used during the ith exercise step above $\dot{w}_{\text{cr}}$ (W′ _out(i)) is, by axiom, the difference between step’s power output and $\dot{w}_{\text{cr}}$ multiplied for exercise duration (t _s):

$$W_{{{\text{out}}(i)}}^{'} = t_{\text{s}} (i{\text{ }}\Delta \dot{w} - \dot{w}_{\text{cr}} ) = t_{\text{s}}{\text{ }}\Delta \dot{w}(i - i_{\text{cr}} )$$

(A1)

where i is the step ordinal number, $\Delta \dot{w}$ is the difference in power between two consecutive steps and i _cr is the step ordinal number at $\dot{w}_{\text{cr}}$ (rounded). Equation (A1) tells that W′ deficit contraction has a linear kinetics, whereas in recovery, W′ is reconstituted following an exponential kinetics, in line with experimental data from literature (Ferguson et al. 2010), according to the following equation:

$$W_{{{\text{def}}(t_{\text{r}} ,i)}}^{'} = W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'}{\text{ }}{\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }}$$

(A2)

where $W_{{{\text{def}}(t_{\text{r}} ,i)}}^{'}$ and $W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'}$ define, respectively, the total amount of W′ deficit attained at the end of the ith recovery and at the end of the ith exercise step. With these premises, it is possible to calculate the time course of the W′ deficit during the ith exercise step and the ensuing recovery (Fig. 5; Box 1). Incidentally, it is of note that a similar conceptual analysis was proposed by Margaria et al. (1969) for intermittent exercise at constant power of supramaximal intensity. From the algebraic developments in Box 1, it derives that $W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'}$, for the properties of geometric series is equal to

Fig. 5

Time course of the deficit of the energy store component (W′_def(t)) during intermittent exercise above the critical power with $\Delta \dot{w}$ = 25 W, t _r = 60 s, t _s = 90 s, and W′ = 13 kJ. W′ _out(i) and W′ _in(i) represent the absolute amount of W′ erosion and reconstitution during the ith exercise-recovery cycle, while $W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'}$ represents the total deficit attained at the end of an exercise step. Except for the first step, $W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'} > W_{{{\text{out}}(i)}}^{'}$

Box 1 Development of a general rule for the W′ deficit by induction

$$W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'} = \Delta \dot{w} t_{\text{s}} \frac{{{\text{e}}^{{ - (i - i_{\text{cr}} + 1) t_{\text{r}}{\text{ }}\tau^{ - 1} }} - (i - i_{\text{cr}} + 1){\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} + i - i_{\text{cr}} }}{{(1 - {\text{e}}^{{ -t_{\text{r}}{\text{ }}\tau^{ - 1} }} )^{2} }}.$$

(A3)

If this is so, then it must exist a theoretical step number (i _peak) at which $W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'}$ is equal to W′ (theoretical because it may be not an integer):

$$\Delta \dot{w}{\text{ }}t_{\text{s}} \frac{{{\text{e}}^{{ - (i_{\text{peak}} - i_{\text{cr}} + 1) t_{\text{r}} \tau^{ - 1} }} - (i_{\text{peak}} - i_{\text{cr}} + 1){\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} + i - i_{\text{cr}} }}{{(1 - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} )^{2} }} = W^{'}$$

(A4)

$$\Delta \dot{w}{\text{ }}t_{\text{s}}\left[ {\frac{{i_{\text{peak}} - i_{\text{cr}} }}{{1 - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} }} - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1}}}\frac{{1 - {\text{e}}^{{ - (i_{\text{peak}} - i_{\text{cr}} ) t_{\text{r}}{\text{ }}\tau^{ - 1} }} }}{{(1 - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} )^{2} }}} \right] = W^{'}$$

(A5a)

$$i_{\text{peak}} - i_{\text{cr}} = (1 - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} )\frac{{W^{'} }}{{\Delta \dot{w}{\text{ }}t_{\text{s}} }} + {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} \frac{{1 - {\text{e}}^{{ - (i_{\text{peak}} - i_{\text{cr}} ) t_{\text{r}}{\text{ }}\tau^{ - 1} }} }}{{1 - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} }}$$

(A5b)

$$\dot{w}_{\text{peak}} = \dot{w}_{\text{cr}} + (1 - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} )\frac{{W^{'} }}{{t_{\text{s}} }} + \Delta \dot{w}\frac{{1 - {\text{e}}^{{ - (\dot{w}_{\text{peak}} - \dot{w}_{\text{cr}} ) \Delta \dot{w}^{ - 1} t_{\text{r}}{\text{ }}\tau^{ - 1} }} }}{{{\text{e}}^{{t_{\text{r}}{\text{ }}\tau^{ - 1} }} - 1}}.$$

(A5c)

Defining the ramp slope (S) as

$$S = \frac{{\Delta \dot{w}}}{{t_{\text{s}} }},$$

(A6)

we obtain

$$\dot{w}_{\text{peak}} = \dot{w}_{\text{cr}} + (1 - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} ){\text{ }}\frac{{W^{'} }}{{\Delta \dot{w}}}S + \Delta \dot{w}\frac{{1 - {\text{e}}^{{ - (\dot{w}_{\text{peak}} - \dot{w}_{\text{cr}} ) \Delta \dot{w}^{ - 1} t_{\text{r}}{\text{ }}\tau^{ - 1} }} }}{{{\text{e}}^{{t_{\text{r}}{\text{ }}\tau^{ - 1} }} - 1}}.$$

(A7)

Equation (A7) is not a function, since any given X value is associated with more than one Y value. However, we can perform the analysis of limit of $W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'}$ (Equation A3) and re-calculate the corresponding $\dot{w}_{\text{peak}}$–S relationship.

When t _r → +∞, the $\dot{w}_{\text{peak}}$–S relationship becomes equal to the critical power model for continuous constant-power exercise (Monod and Scherrer 1965):

$$\mathop { \lim }\limits_{{t_{\text{r}} \to + \infty }} W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'} = \frac{{i{\text{ }}\Delta \dot{w} - \dot{w}_{\text{cr}} }}{{t_{\text{s}} }}$$

(A8a)

$$\dot{w}_{\text{peak}} = \dot{w}_{\text{cr}} + \frac{{W^{'} }}{{\Delta \dot{w}}}S = \dot{w}_{\text{cr}} + \frac{{W^{'} }}{{t_{\text{s}} }}.$$

(A8b)

Equation (A9b) corresponds in fact to the second term of Eq. (A7) for t _r → +∞, and equivalent to the power output of a single time trial of duration t _s.

When t _r → 0, the $\dot{w}_{\text{peak}}$–S relationship becomes similar to the critical power model for ramp exercise (Morton 1994), with a distortion higher the higher $\Delta \dot{w}$:

$$\mathop {\lim }\limits_{{t_{\text{r}} \to 0}} W_{{{\text{def}}(t_{\text{s}} ,i)}}^{'} = \mathop { \lim }\limits_{{t_{\text{r}} \to 0}} \Delta \dot{w}{\text{ }}t_{\text{s}}{\text{ }}\frac{{{\text{e}}^{{ - (i - i_{\text{cr}} + 1){\text{ }}t_{\text{r}}{\text{ }}\tau^{ - 1} }} - (i - i_{\text{cr}} + 1) {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} + i - i_{\text{cr}} }}{{(1 - {\text{e}}^{{ - t_{\text{r}}{\text{ }}\tau^{ - 1} }} )^{2} }} = \Delta \dot{w}{\text{ }}t_{\text{s}}{\text{ }}\frac{1}{2}(i - i_{\text{cr}} )(i - i_{\text{cr}} + 1)$$

(A9a)

$$\dot{w}_{\text{peak}} = \dot{w}_{\text{cr}} + \sqrt {2W^{'}S + \frac{1}{4}\Delta \dot{w}^{2} } - \frac{1}{2}\Delta \dot{w}.$$

(A9b)

Equation (A9b) corresponds to the critical power model for continuous incremental stepwise exercise. In fact, when $\Delta \dot{w} \to 0$ (true ramp exercise), it turns out equal to Eq. (5). It can also be derived with the induction process described in Box 1 for t _r = 0.

Since the third term of Eq. (A7) increases with increasing $(\dot{w}_{\text{peak}} - \dot{w}_{\text{cr}} )$, and since, when t _r → 0, (a) the third term of Eq. (A7) becomes predominant, and (b) $\dot{w}_{\text{peak}}$ solution is provided by Eq. (A9a), we can oversimplify the third term of Eq. (A7) by assuming that $\dot{w}_{\text{peak}}$ is always equal to that of Eq. (A9b). This will transform Eq. (A7) into a function:

$$\dot{w}_{\text{peak}} = \dot{w}_{\text{cr}} + (1 - {\text{e}}^{{ - t_{\text{r}} \tau^{ - 1} }} )\frac{{W^{'} }}{{\Delta \dot{w}}}S + \Delta \dot{w} \frac{{1 - {\text{e}}^{{ - \left( {\sqrt {2 W^{'}S{\text{ }}\Delta \dot{w}^{ - 2} + \frac{1}{4}} - \frac{1}{2}} \right) t_{\text{r}}{\text{ }}\tau^{ - 1} }} }}{{{\text{e}}^{{t_{\text{r}}{\text{ }}\tau^{ - 1} }} - 1}}.$$

(A10)

The error included in Eq. (A10) is negligible (approximately 1 W when 0 < S < 1 W s⁻¹), because $(\dot{w}_{\text{peak}} - \dot{w}_{\text{cr}} )$ becomes significantly greater than $\sqrt {2 W^{'} S + \frac{1}{4}\Delta \dot{w}^{2} } - \frac{1}{2}\Delta \dot{w}$ only when is t _r large, but when t _r is large, the third term of the right branch of Eq. (A10) is small. For Eq. (A10), we can affirm that the relationship between $\dot{w}_{\text{peak}}$ and $\sqrt S$ for intermittent exercise is the sum of a quadratic and an exponential function, the former being predominant in our investigated t _r range.

It is, therefore, demonstrated that in discontinuous incremental stepwise exercise, peak power is a quadratic function of $\sqrt S$.