The critical power model for intermittent exercise

Abstract

This paper develops and illustrates the critical power model for intermittent work. Model theoretic development reveals that total endurance time is always a step function of one or more of the four independent variables: work interval power output (P _w), rest interval power output (P _r), work interval duration (t _w), and rest interval duration (t _r). Six endurance-trained male athletes recorded their best performances during the season in 3-, 5-, and 10-km races, and performed three different intermittent running tests to exhaustion in random order, recording their total endurance times. These data were used to illustrate the model and compare anaerobic distance capacities (α) and critical velocities (β) estimated from each type of exercise. Good fits of the model to data were obtained in all cases: 0.954<R ²<0.999. Critical velocity was found to be significantly less when estimated using an intermittent versus continuous running protocol.

Appendix

Consider the situation of a subject characterized by anaerobic work capacity (α) and critical power (β), exercising intermittently at work and rest powers [(P _w) and (P _r), respectively]. The rest interval duration (t _r) is fixed, but the work interval duration (t _w) is allowed to vary.

For any one of a number of sub-ranges of t _w within the overall range permitted of t _w by the restrictions of the model, suppose the subject is able to perform n complete (work + rest) cycles before becoming exhausted some time into the next work interval.

t from Equation 5 is then given by:

$$ \begin{aligned} t = n{\left( {t_{{\text{w}}} + t_{{\text{r}}} } \right)} + \frac{{\alpha - n{\left[ {{\left( {P_{{\text{w}}} - \beta } \right)}t_{{\text{w}}} - {\left( {\beta - P_{{\text{r}}} } \right)}t_{{\text{r}}} } \right]}}} {{P_{{\text{w}}} - \beta }} \\ = \frac{{n{\left( {P_{{\text{w}}} - \beta } \right)}t_{{\text{w}}} + n{\left( {P_{{\text{w}}} - \beta } \right)}t_{{\text{r}}} + \alpha - n{\left( {P_{{\text{w}}} - \beta } \right)}t_{{\text{w}}} + n{\left( {\beta - P_{{\text{r}}} } \right)}t_{{\text{r}}} }} {{P_{{\text{w}}} {\text{ - }}\beta }} \\ = \frac{{\alpha + n{\left( {P_{{\text{w}}} - P_{{\text{r}}} } \right)}t_{{\text{r}}} }} {{P_{{\text{w}}} - \beta }} \\ \end{aligned} $$

(7)

which does not contain t _w.

We note that if P _w=P _r (or t _r=0 equivalently), this reduces to the hyperbolic form of the CP model for continuous constant power:

$$ t:\frac{\alpha } {{P_{{\text{w}}} - \beta }} $$

(8)

In the case of Fig. 2c, where α=20,000 J, β=200 W, P _w=400 W, P _r=100 W and t _r=20 s, the equation above reduces to:

$$ \begin{aligned} t = 100 + 30n & \\ & \\ \end{aligned} $$

(9)

where n = 1, 2 ... 9.