**I. INTRODUCTION**Work and the duration that work can be performed for has a definite relationship in most living species, including humans, horses, mouse and salamanders. In humans, it's validity has been shown for running, cycling, swimming and rowing. It is valid for any activity where the limits of sustainable oxygen consumption is sufficiently challenged.

If you considered running, the most practical observation of such a relation is what runners know as the decrease of running speed with increase in distance and vice versa. In other words, maximal work output is higher the shorter the distance (or time duration) and lower the longer the distance.

There is some high intensity value of movement speed between these two extremes, which could be held for a long time (in most published studies - well under one hour) without "blowing up".

Conceptually, critical velocity or critical power is approximately equal to the highest steady state speed or power output with the internal body state in homeostasis. In a recent review, Jones et.al called CP the "gold standard" when the goal is to determine maximal metabolic steady state [11].

__II. PHYSIOLOGICAL BASIS FOR CP__
Exercise concepts must have good descriptions that link back to what actually takes place in the body. A good model would have a bio-energetic basis. In this respect, critical power (CP) has well established scientific underpinnings, unlike "other" training concepts in commercial circulation today. (There are of course models that are simply empirical, and do not help us understand how model parameters relate to something within our own bodies)

CP is thought to represent the highest rate of aerobic energy supply available for exercise. On an intensity spectrum, it forms the lower limit for the severe exercise intensity regime and an upper limit for the heavy exercise intensity regime.

The breakdown of metabolic control variables when exercising above CP. Black dots = baseline values. Gray = new values at work > CP. Source [2].

In this severe intensity regime, intramuscular metabolic control breaks down, and such exhaustive exercise results in the attainment of low end-exercise pH, [bicarbonate] and [PCr] values irrespective of the chosen work rate and a continuous increase in blood [lactate], pulmonary VO2 rate and ventilation relative to baseline values.

CP becomes the "threshold" beyond which metabolic control is lost by the individual.

CP becomes the "threshold" beyond which metabolic control is lost by the individual.

Beyond CP, a slow component of VO2 that was previously under control, rises so steeply so as to speed up the body's breathing path to VO2max attainment within the span of a few minutes. The slow component of VO2 is thought to arise from the incremental use of fast twitch muscle fiber. Considering this, exercise above CP always happens on 'borrowed time'.

Some 85% of the slow VO2 rise is linked to the recruitment of energetically costly fast-twitch (FT) muscle fibres as work intensity increases. The energy cost per unit force output is higher for FT fibers than for slow twitch (ST) fibers. The slow component of VO2 is not unique to humans; the same has been demonstrated in horses when they are exercised above their lactate threshold. [3]

The steep rise of slow component of VO2 at work > CP. Source [1]

In the hyperbolic critical power model, the term W' (vocally called

*W prime*) represents a constant amount of work that can be performed above CP and is notionally equivalent to an energy store consisting of O2 reserves, high energy phosphates and a source related to anaerobic glycolysis. The higher the sustained power output above the CP, the more rapidly the W' will be expended, and the greater will be the rate at which metabolites which have been associated with the fatigue process accumulate.

The average time to exhaustion in work done above CP maybe in the order of 10-15 minutes at most depending on the size of the athlete's anaerobic reserves and motivation. In some laboratory tests, the average time to exhaustion in test subjects at work above CP was 13 minutes [1].

Even at CP, physiological steady state is not necessarily achieved. The time to failure at CP ranged from 25 minutes 1 second to 40 minutes 3 seconds [2]. This inter-individual variability hints to the obvious possibility that better trained athletes can sustain exercise at CP longer than less aerobically trained individuals. Some of this variation may also be linked to unfamiliarity with exercising at the estimated CP ("learning effect").

One definition of CP is that it is the "highest, non-steady-state intensity that can be maintained for a period in excess of 20 minutes, but generally no longer than 40 minutes." [2]

__III. CP MODELS____The work rate vs duration (Power-time, or p-t) relation has been mathematically represented in various forms by scientists over the course of the 19th century.__

They are listed as follows :

1) The

*exponential CP model*(Hopkins et.al)

2) The

*3-parameter CP model*(Morton et.al)

3) The

*2-parameter CP model*(Hill et.al)

4) The

*linear model*(Moritani et.al)

5) The

*inverse time CP model*(Whipp et.al)

These models and their mathematical representations are shown below :

CP models and their mathematical representation. Source [9]. |

Although there doesn't seem to a consensus on what is the best model, there has been relatively more attention and research on the hyperbolic forms [7]. This focus of this writeup is primarily in the use of the 2-parameter hyperbolic model which may not be the best model but is the most simple to apply.

Note : This year, a new paper was published detailing an "omni duration" power duration model. Basically, the authors describe an adopted discontinuous mathematical function that helps some of the traditional CP models achieve a better fit at very long durations (more on protocol and duration dependancies below). Details of this model is within the paper in reference [10].

__2-PARAMETER MODEL__
The 2-parameter hyperbolic form of the p-t relation is shown below from a paper on the topic, clearly demarcating boundaries of moderate, heavy and severe intensity domains [1].

Two parameters are of interest in this model :

1) Critical power : This is the horizontal asymptote of the hyperbola, which when read off the y-axis, yields a value of power that could "theoretically" be sustained for ever but in reality, corresponds to a maximal duration of 60 minutes or less. Its units are in Watts.

2) W' : This is curvature constant of the model, signifying a constant "work" that can be done above critical power. Its units are in kilojoules.

Below CP, physiological balance is attained. This corresponds to the heavy and moderate areas in the plot. Above CP, VO2 is driven towards maximum and eventual exercise failure. That area is shown as the severe intensity region.

Two parameters are of interest in this model :

1) Critical power : This is the horizontal asymptote of the hyperbola, which when read off the y-axis, yields a value of power that could "theoretically" be sustained for ever but in reality, corresponds to a maximal duration of 60 minutes or less. Its units are in Watts.

2) W' : This is curvature constant of the model, signifying a constant "work" that can be done above critical power. Its units are in kilojoules.

Below CP, physiological balance is attained. This corresponds to the heavy and moderate areas in the plot. Above CP, VO2 is driven towards maximum and eventual exercise failure. That area is shown as the severe intensity region.

The geometrical descriptions of CP. Source [1]

In terms of power output and oxygen consumption, the second plot shows the values represented on the exercise intensity regime.

The hyperbola may also be linearized, in which case the linear relationship becomes one between work done and time duration. The y-intercept would then correspond to W' while the slope of the line would be critical power or velocity. The linear Moritani model is not discussed further here.

__IV. ASSUMPTIONS IN THE 2 PARAMETER CP MODEL__

Any model is a mathematical simplification of a real world phenomena and by nature, is never fully correct. As far as whole body CP concept is concerned, four major assumptions in the simple 2 parameter CP model has been documented :

1. There are only two components to the energy supply system, termed aerobic and anaerobic.

2. Aerobic supply is unlimited in capacity but rate limited, the limiting parameter being CP.

3. The anaerobic capacity is not rate limited but capacity limited.

4. Exhaustion, by implication, termination of exercise, occurs when all of the anaerobic work capacity is exhausted.

The treatment of these assumptions has been done beautifully by Morton, and the reader interested in understanding the details of each assumption need to read the reference [5] below. My conclusions from Morton's paper is as follows :

Assumption 1 : There are only two components to the energy supply system, termed aerobic and anaerobic.

Assumption 2 : Aerobic supply is unlimited in capacity but rate limited, the limiting parameter being CP.

Assumption 3 : The anaerobic capacity is not rate limited but capacity limited.

All models have assumptions and to be able to validate the model also means that the assumptions should be correct. If they deviate from reality, the model is wrong, sometimes dead wrong. Like CP, similar assumptions can be generated the concept of FTP and the astute athlete and coach can treat each assumption and try to understand at what point the usage of the model fails and is inapplicable to the athlete.

Note : Around 9 total assumptions about the 2 parameter CP model have been treated in the paper by Morton [5].

4) Due to the hyperbolic form of the model, small errors in CP translate to large errors in sustainable time duration. This reduces the predictive validity of CP when the model is misapplied by practitioners.

To get around some of these duration specific weaknesses, I can suggest a few things :

a) Critical power is supposed to be a "heavy" work output very close to maximal running velocity, maximal power output and maximal oxygen uptake as measured in the lab. As such, it has been suggested that critical power should only be calculated from exhaustion times corresponding to "heavy sub-maximal exercises". The recommended exhaustion time range is suggested as 6 - 30 minutes [6]. Below and above this range, the validity of the classic CP models are questionable.

b) Owing to research done in [9], it is best to include a mix of test durations in order to balance the short supra-maximal with the long sub-maximal. When using 3 durations, something like 7, 12 and 20 minutes is recommended. When using just two durations, using a range from 12-20 minutes may provide more accurate CP and W' estimations. The 2-parameter CP model yields valid parameters with durations greater than 10 minutes. The 3-parameter hyperbolic CP model (Morton) is deemed protocol independant.

1. There are only two components to the energy supply system, termed aerobic and anaerobic.

2. Aerobic supply is unlimited in capacity but rate limited, the limiting parameter being CP.

3. The anaerobic capacity is not rate limited but capacity limited.

4. Exhaustion, by implication, termination of exercise, occurs when all of the anaerobic work capacity is exhausted.

The treatment of these assumptions has been done beautifully by Morton, and the reader interested in understanding the details of each assumption need to read the reference [5] below. My conclusions from Morton's paper is as follows :

Assumption 1 : There are only two components to the energy supply system, termed aerobic and anaerobic.

*Yes, this is largely true but only to an extent. The body has more than two energy systems.*Assumption 2 : Aerobic supply is unlimited in capacity but rate limited, the limiting parameter being CP.

*This is not true, the aerobic capacity clearly has a limit in all humans. However, the statement that it is rate limited is correct. There is clearly a limit and you might define it by CP.*Assumption 3 : The anaerobic capacity is not rate limited but capacity limited.

*True, explosive power generated from anaerobic capacity is limited. It is not true that it is rate limited.**Assumption 4 : Exhaustion, by implication, termination of exercise, occurs when all of the anaerobic work capacity is exhausted.*

*The human engine does not necessarily terminate exercise when all the glycogen stores, consequently, anaerobic work capacity, is exhausted. Research proves that at the point of exercise termination, there is still glycogen left in the body. The fine proof is that when nearing exhaustion, if the power output is just slightly lowered, subjects exercising should be able to continue on despite still working at supra-maximal power outputs.*All models have assumptions and to be able to validate the model also means that the assumptions should be correct. If they deviate from reality, the model is wrong, sometimes dead wrong. Like CP, similar assumptions can be generated the concept of FTP and the astute athlete and coach can treat each assumption and try to understand at what point the usage of the model fails and is inapplicable to the athlete.

Note : Around 9 total assumptions about the 2 parameter CP model have been treated in the paper by Morton [5].

__V. WEAKNESSES OF CP MODELS__
Like any mathematical model, GIGO principle applies. All models are wrong, being a simplistic representation of reality. The CP models are not immune from this deficiency. Other concepts such as FTP also suffer from model related errors.

Some of the weaknesses in CP modeling are listed as follows :

1) CP is protocol and model dependent : Critical power and its calculation has both model and protocol dependency. In a fantastic research study, scientists compared several models for estimating CP using different combinations of time-to-exhaustion exercise sessions in 13 young recreational cyclists. They not only found that the 3 parameter CP model fit the data best, but when they compared model fits from time duration combinations having more of the short durations, CP was over-estimated and W' under-estimated [9].

In particular to our interest, the 2-parameter CP model was closest to the criterion measure only when mean duration combinations such as 7, 12 and 19 minutes were chosen, whereas when durations were consistently < 10 minutes, the model values were far from accurate [9].

There has been reports of large variations in the calculated value of W' arising from different models, particularly in sub-classes of athletes such as elite athletes [6].

Some of the weaknesses in CP modeling are listed as follows :

1) CP is protocol and model dependent : Critical power and its calculation has both model and protocol dependency. In a fantastic research study, scientists compared several models for estimating CP using different combinations of time-to-exhaustion exercise sessions in 13 young recreational cyclists. They not only found that the 3 parameter CP model fit the data best, but when they compared model fits from time duration combinations having more of the short durations, CP was over-estimated and W' under-estimated [9].

Different model fits and differences in parameters compared to criterion measure. Source [9]. |

In particular to our interest, the 2-parameter CP model was closest to the criterion measure only when mean duration combinations such as 7, 12 and 19 minutes were chosen, whereas when durations were consistently < 10 minutes, the model values were far from accurate [9].

There has been reports of large variations in the calculated value of W' arising from different models, particularly in sub-classes of athletes such as elite athletes [6].

2) Effect of very short only duration : When critical power is calculated from slope of the work-duration relationship using short supra-maximal exercises, the resulting power from models is higher than the power output which corresponds to a lab measured lactate "steady state" work intensity. The critical power also tends to be lower than maximal aerobic power [6].

3) Effect of long only duration : When critical power is calculated from very long sub-maximal exercise durations, the resulting power from the models tends to be lower than the power output which corresponds to a lab measured lactate steady state work intensity such as OBLA (onset of blood lactate) [6].

3) Effect of long only duration : When critical power is calculated from very long sub-maximal exercise durations, the resulting power from the models tends to be lower than the power output which corresponds to a lab measured lactate steady state work intensity such as OBLA (onset of blood lactate) [6].

4) Due to the hyperbolic form of the model, small errors in CP translate to large errors in sustainable time duration. This reduces the predictive validity of CP when the model is misapplied by practitioners.

To get around some of these duration specific weaknesses, I can suggest a few things :

a) Critical power is supposed to be a "heavy" work output very close to maximal running velocity, maximal power output and maximal oxygen uptake as measured in the lab. As such, it has been suggested that critical power should only be calculated from exhaustion times corresponding to "heavy sub-maximal exercises". The recommended exhaustion time range is suggested as 6 - 30 minutes [6]. Below and above this range, the validity of the classic CP models are questionable.

b) Owing to research done in [9], it is best to include a mix of test durations in order to balance the short supra-maximal with the long sub-maximal. When using 3 durations, something like 7, 12 and 20 minutes is recommended. When using just two durations, using a range from 12-20 minutes may provide more accurate CP and W' estimations. The 2-parameter CP model yields valid parameters with durations greater than 10 minutes. The 3-parameter hyperbolic CP model (Morton) is deemed protocol independant.

__VI. FIELD MEASUREMENT OF CP__

**1) Multi-duration testing :**The established lab practice to model CP is done using several bouts of constant load exercise done at varying durations to failure over several days. These bouts are administered in random order and the recommended exercise duration to exhaustion range from 1-20 minutes. The time to exhaustion in these exercises is plotted power output. The hyperbolic 2-parameter Whipp model when fit through this data yields CP and W', where CP is the horizontal asymptote of the curve and W' is the area between the curve and CP which represents a fixed quantity of work that can be done above CP before approaching complete exhaustion. However, the choice of durations would need to be scrutinized to yield a critical power that resembles a severe intensity workload.

**2) A 3 minute all out test (3MAOT)**has been scientifically established to point towards critical power. The idea with this test is that it is possible to deplete W’ in reasonably short time. Therefore, the idea of the test is to perform work all-out in a span of 3 minutes and deplete W'. The last 30 seconds of the 3 min all out test is supposedly close to the critical power.

There are indications from the scientific community that the 3MAOT field test overestimates CP and underestimates W' so therefore, it is not a reliable measure of capacity in "well trained athletes".

CP calculated from a 3MAOT test. Source [4]. |

__VII. TRAINING APPLICATIONS__

**1) Training Prescription :**Once the critical power (or critical speed) has been determined, training prescription can be designed for an athlete using a percentage of critical power.

As a start, the following training levels can be constructed. It is a good start, atleast in my experience.

Recovery : Best to go by perceived exertion

Endurance : 49-65% of CP

Tempo : 66-79% CP

Threshold : 80-92% of CP

Aerobic Power : 93-105% of CP

Anaerobic Capacity : > 105% of CP

These levels may have to be modified on a case by case basis depending on the critical power test and the athlete.

Pacing prescription may also be set for races where the use of running power is prevalent. A 10K race for a talented runner maybe targeted using 95-100% CP. A 5K race performance maybe targeted within a range of 100-105% CP. Again, experimentation is necessary with these ranges and no guidance can be offered set in stone, as courses are different and CP itself may exhibit small day-to-day variations.

For very long duration events, where it is known that CP actually decreases over time, it is not clear how effectively one could employ CP to set race prescription [12]. I suggest the use of a multi-pronged approach for marathons and ultra-marathons, involving the use of pace, heart rate and perceived exertion.

**2) Predicting Time to Exhaustion :**A fairly fundamental application for the critical power (or velocity) model is to help determine the time to exhaustion during work performed above CP.

With the simple 2 parameter hyperbolic form, time to exhaustion can be represented as :

Tlim = W′ /(P − CP)

As an example, setting W' = 20 KJ, CP = 250W, P = 300W :

Tlim = 20,000 J / (300W - 250W) = 400s = 6.66 minutes.

This way, the optimum time duration to cover a distance without "blowing up" can be predicted.

As one can see from the above example, any errors in the estimation of W' and CP translates to errors in the predicted time to exhaustion.

**3) Use in Software :**Nowadays, software can easily fit the 2 parameter model to mean maximal exercise data yielding all the parameters from the applied model.

Golden Cheetah is an open source software that does this. I will describe more on using Golden Cheetah and how it treats data in different models in another post, simply because the learning curve involved in using the software is high. However, some introductory tutorials on modeling CP using GC is

**shown here**.

As of today, Mark Liveredge tells me that the upcoming version of GC will feature the ability to overlay several more models on data, which would help the practitioner assess which of the models converge the best.

Filipe Maturana, a PhD candidate, showed

**me an app**developed on Shiny which allows you to model CP using a number of time to exhaustion trials. This would be a good model to play around with for the sheer educative value.

Apart from this, the 2 and 3 parameter model can be programmed in Microsoft Excel. I have been using such models for some time but do find using Excel cumbersome.

__VIII. CONCLUSION__**While there are several exercise concepts out there, the critical power model has been one of the most rigorously studied one in scientific literature.**

In this post, only one form of this model - the hyperbolic 2 parameter model - was described in a somewhat broad manner. There are several other models including 3 parameter and extended CP models. In future, this post will be expanded to include a treatment of those other models.

The concern over test protocol, quality of data and error propagation carries across to any CP model. The practitioner must be careful in the use of these models to advise exercise prescription, specially to talented elite athletes. Lab based physiological profiles will be better suited to making informed decisions in these athletes.

However, in a vast majority of recreational athletes, proper use of the field based testing protocol and the modeling based on the data will yield a useful approximation of the endurance capacity of an individual. That it is conceptually the highest power output or speed at physiological steady state is useful in training prescription. Practitioners will also be pleased in utilizing a very scientifically vetted training concept.

What remains to be seen is how the critical power concept marries with the central nervous system theory of fatigue. That the ultimate limiter of exercise performance is not the muscle but the brain was introduced more than a century ago by scientists.

Implicit in the effectiveness of applying the critical power concept is this idea that the performance tat is analyzed must be the maximal in nature, implying that the central drive must be maximum for that performance. The role of motivation and internal drive is significant enough to warrant further investigations as part of the critical power concept.

Readers are advised to expand on their knowledge and read the papers referenced below.

__REFERENCES__
1. Jones, A. M., Vanhatalo, A., Burnley, M., Morton, R. H., & Poole, D. C. (2010). Critical power: implications for determination of VO2max and exercise tolerance. Med Sci Sports Exerc, 42(10), 1876-90.

2. Brickley, G., Doust, J., & Williams, C. (2002). Physiological responses during exercise to exhaustion at critical power. European journal of applied physiology, 88(1-2), 146-151.

3. Langsetmo, I., Weigle, G. E., Fedde, M. R., Erickson, H. H., Barstow, T. J., & Poole, D. C. (1997). VO2 kinetics in the horse during moderate and heavy exercise. Journal of Applied Physiology, 83(4), 1235-1241

4. Miller, M. C., & Macdermid, P. W. (2015). Predictive validity of critical power, the onset of blood lactate and anaerobic capacity for cross-country mountain bike race performance. Sport Exerc Med Open J, 1(4), 105-110.

5. Morton, R.H. The critical power and related whole-body bioenergetic models. Eur J Appl Physiol 96, 339–354 (2006).

5. Morton, R.H. The critical power and related whole-body bioenergetic models. Eur J Appl Physiol 96, 339–354 (2006).

6. Vandewalle, Henry & Vautier, J-F & Kachouri, M & Lechevalier, J & Monod, H. (1997). Work-exhaustion time relationships and the critical power concept. A critical review. The Journal of sports medicine and physical fitness. 37. 89-102.

7. H. Monod & J. Scherrer (1965) The Work Capacity Of a Synergic Muscular Group, Ergonomics, 8:3, 329-338, DOI: 10.1080/00140136508930810

8. Mark Burnley & Andrew M. Jones (2018) Power–duration relationship: Physiology, fatigue, and the limits of human performance, European Journal of Sport Science, 18:1,

1-12, DOI: 10.1080/17461391.2016.1249524

9. Mattioni Maturana, Felipe & Fontana, Federico & Pogliaghi, Silvia & Passfield, Louis & Murias, Juan. (2017). Critical power: How different protocols and models affect its determination. Journal of Science and Medicine in Sport. 21. 10.1016/j.jsams.2017.11.015.

10. Puchowicz, Michael & Baker, Jonathan & Clarke, David. (2020). Development and field validation of an omni-domain power-duration model. Journal of Sports Sciences. 38. 1-13. 10.1080/02640414.2020.1735609.

11. Jones, Andrew & Burnley, Mark & Black, Matthew & Poole, David & Vanhatalo, Anni. (2019). The maximal metabolic steady state: redefining the ‘gold standard’. Physiological Reports. 7. 10.14814/phy2.14098.

12. Clark, Ida & Vanhatalo, Anni & Thompson, Christopher & Joseph, Charlotte & Black, Matthew & Blackwell, Jamie & Wylie, Lee & Tan, Rachel & Bailey, Stephen & Wilkins, Brad & Kirby, Brett & Jones, Andrew. (2019). Dynamics of the power-duration relationship during prolonged endurance exercise and influence of carbohydrate ingestion. Journal of Applied Physiology. 127. 10.1152/japplphysiol.00207.2019.

7. H. Monod & J. Scherrer (1965) The Work Capacity Of a Synergic Muscular Group, Ergonomics, 8:3, 329-338, DOI: 10.1080/00140136508930810

8. Mark Burnley & Andrew M. Jones (2018) Power–duration relationship: Physiology, fatigue, and the limits of human performance, European Journal of Sport Science, 18:1,

1-12, DOI: 10.1080/17461391.2016.1249524

9. Mattioni Maturana, Felipe & Fontana, Federico & Pogliaghi, Silvia & Passfield, Louis & Murias, Juan. (2017). Critical power: How different protocols and models affect its determination. Journal of Science and Medicine in Sport. 21. 10.1016/j.jsams.2017.11.015.

10. Puchowicz, Michael & Baker, Jonathan & Clarke, David. (2020). Development and field validation of an omni-domain power-duration model. Journal of Sports Sciences. 38. 1-13. 10.1080/02640414.2020.1735609.

11. Jones, Andrew & Burnley, Mark & Black, Matthew & Poole, David & Vanhatalo, Anni. (2019). The maximal metabolic steady state: redefining the ‘gold standard’. Physiological Reports. 7. 10.14814/phy2.14098.

12. Clark, Ida & Vanhatalo, Anni & Thompson, Christopher & Joseph, Charlotte & Black, Matthew & Blackwell, Jamie & Wylie, Lee & Tan, Rachel & Bailey, Stephen & Wilkins, Brad & Kirby, Brett & Jones, Andrew. (2019). Dynamics of the power-duration relationship during prolonged endurance exercise and influence of carbohydrate ingestion. Journal of Applied Physiology. 127. 10.1152/japplphysiol.00207.2019.