This referenced article serves as a broad exploration into the power duration relationship and the parameters that result from the hyperbolic powerTlim characteristic.
I. INTRODUCTION
It is well established that work requiring high speed and power output is short lived but that at low speed and power can be prolonged. This relationship has been shown in a number of living species, including humans, horses, mouse and salamanders. In human activities, 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.
Within this powerduration curve, there is a maximum level of speed or power that can be tolerated beyond which exercise tolerance until termination can be predicted.
That particular threshold value of speed or power is called "critical velocity" or "critical power". The literature provides an "expanded" definition to be the highest steady state metabolic rate (i.e intensity) that can be sustained solely by oxidative energy provision beyond which homeostasis is lost and exercise tolerance is limited.
Although in layspeak, we tend to associate "thresholds" with points beyond which things "blow up " in the body, the transitions between intensity regions as far as fatigue related variables go is much more "gradual", as one recent study showed [33].
Regardless, any exercise crossing critical power happens on borrowed time as the organism shifts away from sustaining muscle activity exclusively through aerobic pathways and starts concurrently relying on finite anaerobic stores. A slow rise in VO2 kicks in accelerating the drive towards VO2max and eventual exercise termination.
For sports, critical power is an index of aerobic endurance. It was found to have strong positive correlations with skeletal muscle capillarity, particularly around type I fibers, and type I fiber composition [14].

The association of Critical Power and capillarity in two athletes with different CPs. Source [14]. 
In a recent review, Jones et.al called CP the "gold standard" when the goal was to determine the maximal metabolic steady state [11]. This appears to be one of the few resolutions to some long standing criticisms of the CP paradigm and lack of validity in relation to metabolic steady state.
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 bioenergetic 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 endexercise 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.
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 fasttwitch (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 1015 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 interindividual 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, nonsteadystate intensity that can be maintained for a period in excess of 20 minutes, but generally no longer than 40 minutes." [2]
CP has been found to be influenced by the carbohydrate availability. Researchers found that 2 hours of high intensity activity can decrease CP over time and that carbohydrate feeding negated some of the decrease [12]. The time rate of fall in muscle glycogen also exhibits interindividual differences, so the time course of decrease of CP in turn also varies from person to person depending on their physiology.
III. CP MODELS
Traditionally, CP is determined from multiduration tests conducted over several days in the laboratory. The resulting work rate vs duration (Powertime, or pt) relation can be mathematically modeled various fits :
1) The
exponential CP model (Hopkins et.al)  Nonlinear
2) The
3parameter CP model (Morton et.al)  Nonlinear
3) The
2parameter CP model (Hill et.al)  Nonlinear
4) The
linear model (Moritani et.al)  Linear
5) The
inverse time CP model (Whipp et.al)  Linear
These mathematics behind the model 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 2parameter 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].
2PARAMETER MODEL
The 2parameter hyperbolic form of the pt 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, CP : This is the horizontal asymptote of the hyperbola, which when read off the yaxis, 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 prime, 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.

Range of attainable power output in a young male along with the oxygen consumption attained. Shown in the intensity range are lactate threshold (LT) and critical Power (CP) along with VO2max, the point which results in termination of exercise. Source [8] 
The hyperbola may also be linearized, in which case the linear relationship becomes one between work done and time duration. The yintercept 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.
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 supramaximal 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) Estimated CP and real CP maybe different : Critical power is a parameter estimated from the hyperbolic relationship between power and time or the linear relationship from work and time. There is no guarantee that the estimation from model fits consisting of limited test points actually point to the "real CP", i.e the real physiological boundary demarcating heavy and severe intensities. Unless ofcourse, the parameter is experimentally validated in the lab against the real procedure to determine CP (multiple lab visits at different test durations).
2) Model and protocol dependency : In a very practical research study, scientists compared several models for estimating CP using different combinations of timetoexhaustion 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 overestimated and W' underestimated [9].

Absolute difference (watt) between CP modeling techniques and the criterion model (3P CP) are presented along with 90% confidence interval around each difference and effect size calculation. Source [9]. 
In particular to our interest, the 2parameter 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 subclasses of athletes such as elite athletes [6].
Subdiscussions that arise from model time duration dependencies are as follows :
2.1) Effect of exclusively very short durations in the model : When critical power is calculated from slope of the workduration relationship using short supramaximal 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].
2.2) Effect of exclusively long durations in the model : When critical power is calculated from very long submaximal 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].
2.3) Weakness of some nonlinear models : Due to the hyperbolic form of the CP 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. The nonlinear 2 parameter CP model suffers from a distinct weakness : As time approaches 0 seconds, power becomes infinite and at exhaustion, all of the muscular energy reserves associated with W' are exhausted. This is ofcourse, not necessarily true and the 3 parameter model was formulated to address this weakness, by bringing in an additional Pmax term.
2.4) Weakness of linear models : Linear models are often linearized from nonlinear observations and as such introduce statistical errors simply from the linearization process. For example, it is possible that the fit parameters computed from linearized models yield higher values compared to their original nonlinear forms.
VI. MANAGING TESTING AND USE OF CP MODELS
To get around some of the weaknesses of CP models, careful application is necessary. I can suggest a few things :
1) Getting the right "intensity" : Critical power is a "heavy" work output above which a slow rise in VO2 can speed the approach to VO2max and eventual exhaustion. As such, it has been suggested that critical power should only be calculated from exhaustion times corresponding to "heavy submaximal 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.
This idea that CP is a heavy intensity also brings a concern of long tests like 20 minute tests. The submaximal nature of a 20 minute test brings into question its reliability.
The data that is fed into the model matters. Garbage in, garbage out. While conducting tests, effort must be feel "strong" and motivation needs to be very high. Deflated and/or inflated values of power or speed will skew the results one way or the other when modeling.
2) Which CP model to use : Since the nature of the powerduration curve is a nonlinear hyperbola, statistically speaking the best model fit would be a nonlinear fit without transformation of any of the variables for linearization.
The nonlinear 2 and 3 parameter CP models should be preferred over a linear model. However, the 3 parameter CP model was proposed by Morton as a way to get around flaws of the 2 parameter CP model (see section IV). Therefore, of all the 5 models, the 3 parameter CP model would be the sound choice. But this also means 4 or more trials need to be conducted.
Linear models systematically inflate CP than do nonlinear models and there is ample evidence from literature that time to fatigue is drastically shortened when testing at work intensities estimated from linear models. This alone would support the move away from linear models.
3) Choice of test durations : Owing to research done in [9], it is best to include a mix of test durations in order to balance the short supramaximal with the long submaximal. 2 and 3 duration tests can be analyzed by linear CP and 2 parameter model. 4 durations or greater can be analyzed with the 3 parameter CP model and linear CP models.
 For 2 durations : Pick from a range of 1020 minutes. Avoid very short and very long durations like 3 and 20 minutes.
 For 3 durations : 7, 12 and 15 minutes. If glyoclytic capacity needs to be tested, 3, 10 and 15 minutes is a good spread.
 Pacing : All short duration trials should be done in time trial mode to exhaustion but not "ALLOUT". Example, a 3 minute test is an aerobic time trial to exhaustion, not a maximal sprint mixed with an aerobic effort.
 The 3parameter hyperbolic CP model (Morton model) is deemed protocol independant and works with 4 or more test durations. But I've also been told that when you have a trial that is too close to 20 min, you might get odd values for the Pmax parameter (too high values), which is not realistic.
 Ideally, testing within specific durations would be conducted on different days. For a single day test, maximum 2 or 3 tests are recommended spaced by ample break but done this way, the impact of prior testing on a subsequent test performance has to be assessed.
5) Validate the model : Experimental validation is the only way to check if a model derived estimates of CP is representative of physiological CP. Until that happens, a model calculated value holds a presumption that it is accurate when it may not be. For example, a model might yield an inflated estimate of CP which would be above true physiological CP as measured in a lab leading to loss of maintenance of homeostasis.
VII. INTERVENTION STUDIES
CP and W' are not just parameters of a fitting model. That there are some underpinning physiological relations to them are shown by intervention studies designed to manipulate either one independently of each other. For example, studies show that training adaptations are specific to either CP or W'. Nutritional and external gaseous interventions also affect the parameters.
Broadly, some of the studies and their references are listed below for further exploration :
1. Endurance training in normal subjects results in an increase in critical power with little or no change in W' [16, 17, 18].
2. Endurance training enhances critical power and end test power in a 3min all out test [19, 20].
3. Sprint cycle training with long rest intervals improves W′ [21].
4. W' is sensitive to, and modified by resistance training with no change in CP [22, 23].
5. W' is sensitive to creatine supplementation [24, 25, 26, 27].
6. Hypoxia systematically reduces CP with no significant impact W' [28]. Conversely, hyperoxia improves CP [31].
7. SupraCP fatigue inducing work with different recovery durations affects the reconstitution dynamics of W' in different ways, without having an effect on CP [29].
8. Glycogen depletion has been shown to result in a decrease in W' [30].
VIII. FIELD MEASUREMENT OF CP
1) Multiduration 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 120 minutes. The time to exhaustion in these exercises is plotted power output. The hyperbolic 2parameter 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 allout 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]. 
3) Software CP modeling ; Traditionally, CP is determined by acquiring powertime series data over several visits and fitting a chosen model to the data. However, lab visits are expensive and time consuming. With the proliferation of GPS and power meters, these can be reproduced by acquiring mean maximal power and duration data in a given sport over a time span such as recent weeks or months. Once that data is acquired, software can be used to plot the data as a pt chart. Model fitting is done to solve for the parameters.
IX. TRAINING APPLICATIONS
1) Predicting Time to Exhaustion : The most fundamental application for the critical power (or velocity) model is to help determine the time to exhaustion during work performed above CP. The very purpose of modeling is to find out parameters that can be used with the power output to determine time to exhaustion.
With the simple 2 parameter hyperbolic form using power and work done, 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.
Similarly, in the distance and speed domain :
Tlim = (D  D')/CS
Setting Critical Speed (CS) = 6 m/s, D = 1600m, D' = 200m :
Tlim = (1600m  200m) / 6m/s = 233.3s = 3.88 minutes.
This way, the time duration using a given estimated critical power or speed can be predicted.
2) Training Zone Descriptions : Once the critical power (or critical speed) has been determined, training descriptions can be communicated to an athlete.
The following training levels described by Dr. Skiba can be a decent start. These levels may have to be modified depending on the athlete and race performances and/or tests.
Recovery (Light): Less than 56% (or go by feel)
Level 2 (Moderate), Endurance : 5675% of CP
Level 3 (Heavy), Tempo : 7690% CP
Level 4 (Very Heavy), Critical Power : 91105% of CP
Level 5 (Severe), VO2max : 106120% of CP
Anaerobic Capacity (Extreme) : Greater than 120% of CP
Bettina Karsten in a wellwritten thesis summed up CP training zones or intensity domains as defined in the literature. She gave extensive scientific references for these "zones". Background reading can be done beginning at section 2.3.1 in reference [32].
As she also highlighted in her work, exercise is a continuum and therefore the absolute "strictness" of these demarcation markers have not been fully demonstrated within research literature to date [32].

Broad CP based training intensity domains. Source [32]. 

Training zones and exercise intensity domains. Source [32]. 
3) Interval Training Prescription : One of the promising areas for using the critical power concept is to explore promoting targeted anaerobic and aerobic effects in an athletes. HIIT training can be prescribed for individuals proportionate to their D' or W'. By setting intervals to deplete a fixed percentage of W' and controlling the rest, individuals can complete a fixed distance at different speeds relative to their criticals. Examples of approaches are provided in [15].
4) Race Pacing Strategy : 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 95100% CP. A 5K race performance maybe targeted within a range of 100105% 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 daytoday variations.
For prolonged duration high intensity events, CP is purported to decrease over time. If that is true, it is not clear how effectively one could employ CP to set pace prescription [12]. However, the studies reveal that carbohydrate feeding of around 60g/hour should be an important strategy to negate considerable decreases in CP over long durations [12]. I also suggest the use of a multipronged approach for marathons and ultramarathons, involving the use of pace, heart rate and perceived exertion.
5) Potential in antidoping : A paper was published arguing that the CP model could be useful for doping detection mainly based on the predictable sensitivities of its parameters to ergogenic aids and other performanceenhancing interventions [13]. I understand this proposal is still in its early stages and needs to be vetted.
6) Educative value : Critical power models have educative value behind them. They can teach concepts underpinning human endurance and record performances.
Curve fitting is easily done in Microsoft Excel.
Filipe Maturana, a PhD candidate, built
an app developed on R 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 whatif analyses.
Using that app, I ran a simple example of how two different combinations of time and power data yield two different values of CP and W' estimate when using a simple linear CP model. The power and time to exhaustion data was taken from the reference in [9] and the duration combinations used as inputs to the two scenarios were 3+20 minutes and 12+20 minutes.

Model output showing how different combinations of power duration data can yield different values of CP. 
X. 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, with several lab studies validating the model for athletes. The number of parameters are small (CP and W') and they have physiological meanings.
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 that 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.
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