**(First posted to the internet in 2002.)**

**by Andrew R. Coggan, Ph.D. -**A lengthy, but hopefully informative and useful, post...

There have been a number of equations presented in the scientific literature describing human power output as a function of time, some derived from first principles modeling based on the underlying physiology, and some simply derived empirically. One of the simplest and most robust, though, is the original "critical power" concept of Monod, first proposed around 1960. Various formulations of this have been presented, but the original equation is a hyperbolic of the form:

t = W'/(W(dot) - W(dot)cp)

where t = time to exhaustion, W(dot)cp is the work rate (i.e., power)asymptote, and W' represents the degree of curvature of the relationship.In this form, time is the dependent variable, being determined by the rate of doing work, i.e., power output (W(dot)) in relation to the individual's critical power (W(dot)cp).

While this is logical (i.e., how long you can go is determined by how hard you are going relative to your own ability), actually fitting data to such a curvilinear relationship isn't esp. convenient. Hence, it is common to rearrange the equation to yield a linear equivalent, i.e.,:

Wlim = W' + W(dot)cp * t

In this expression, Wlim is the total amount of work accomplished during a given exercise bout and is the product of W(dot) and t (i.e.,since power = work/time, work = power * time). The above equation is of the form y = mx + b (or y = a +bx, if you prefer), with the slope of the line (i.e., W(dot)cp) being a person's"critical power".

Conceptually, critical power is a power that can be sustained "for a very long time without fatigue", and is "an inherent characteristic of the aerobic energy supply system". On a practical basis, critical power has been shown to correlate very closely with power at lactate threshold, although it may in fact be significantly (both statistically speaking, and from an athlete's perspective) higher depending on exactly how critical power and lactate threshold are measured/determined. On the other hand, the y-intercept of this relationship, W', represents a fixed amount of

*work*that can be accomplished during an exercise task to fatigue, but is non-renewable. Conceptually, this parameter reflects anaerobic capacity (not power), i.e., the total amount of energy that can be liberated from non-aerobic energy sources, i.e., from breakdown of high energy phosphate stores (ATP and PCr) and via production and accumulation of lactate. Support for this interpretation comes from experiments showing a close correlation between W' and total work performed during an all-out 30 second exercise test (i.e., a Wingate test), and between W' and maximal accumulated oxygen deficit. Moreover, critical power has been shown to be influenced by interventions that would be expected to affect aerobic energy production, e.g., hypoxia, whereas W' is not. Conversely, interventions expected to influence anaerobic capacity, such as creatine loading or resistance training (in untrained persons), have been shown to alter W' without changing critical power. Finally, whereas most of the research involving critical power has been performed using cycle ergometry (due to the convenience with which power can be measured), this conceptual formulation has been shown to closely describe performance in other sports, as well as the performance or function of isolated muscles/muscle groups.

The critical power concept is not without its limitation...in particular, it tends to greatly overestimate the maximal power that can be generated for only a few seconds, and it predicts that there should be a power output below which fatigue will never occur. In addition, the exact values obtained for W' and critical power depend in part on the testing protocol, e.g., the exact combination of powers and durations used to define the curve, how fatigue is defined, etc. Nonetheless, despite its simplicity this equation describes the power vs duration curve quite well over a wide range of exercise intensities/durations,i.e., from perhaps 20 seconds out to several hours.

Relevant to the purposes and interests of this group, I think the critical power concept is useful for two reasons. One, it provides a very good conceptual framework for understanding the most basic factors determining exercise performance/power output (i.e., anaerobic and aerobic energy production), and how the contribution of each varies as a function of time. Two, actually measuring critical power is well within the capacity of anyone who owns a power meter (and understands a little bit about math), and thus provides a means of quantifying changes in fitness beyond just even "I was able to sustain X watts for Y seconds!". In other words, determination of critical power allows one to determine whether it is chances in anaerobic capacity or aerobic function (lactate threshold)accounting for any changes in performance, w/o requiring a trip to a laboratory or invasive measurements (blood sampling). This would be especially true for people using the SRM software, since it allows you to "extract"the necessary power and duration data from regular training and racing. (Although it should be noted that W' declines in direct proportion to W(dot) during the period of time immediately before the effort...as demonstrated by how hard it is to sprint very fast at the end of a TT!).

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