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Exercise intensity domains and phase transitions: the power-duration relationship

August 31, 2020

The exercise intensity spectrum has been a fascination of mine for as long as I can remember, and I’ve been lucky enough to work on it experimentally for the last 16 years or so.  In doing so, I made contributions to the development of the 3 minute all-out cycling test, the examination of the relationship between neuromuscular fatigue and critical torque, and the study of physiological complexity during fatiguing exercise above the critical torque.  Most of these investigations are available for free in the public domain (references at the bottom).  Most recently, we investigated a thing that has been bugging me for ages.  That thing being whether or not critical power (or speed, torque, force etc.) can be described as a sudden threshold or a more gradual phase transition.  In other words, do the physiological thresholds we all discuss so much represent a single point in the intensity spectrum (e.g., 300 W), or is there a band of intensity, a grey area if you will, that the landmark is characterised by (e.g., 290-310 W)?  Before we get to that, I’d like to outline what the exercise intensity spectrum and its physiological significance looks like to me.


Physiological landmarks


When athletes perform tests of aerobic function, such as incremental exercise (often called a “VO2max” test), several fitness parameters can be determined.  The point at which lactate begins to increase in the blood is one such parameter, known as the lactate threshold.  This threshold also separates two exercise intensity domains: exercise performed below the lactate threshold is known as ‘moderate-intensity exercise’, and exercise above the lactate threshold is known as ‘heavy-intensity exercise’.  The upper limit of heavy-intensity exercise is generally agreed to be the highest running speed or power output at which a steady state can be attained.  There are two landmarks that can estimate this boundary, namely the maximal lactate steady state, and the so-called critical power.  Exercise performed above this boundary is known as ‘severe-intensity exercise’.  Below we will see how the physiological responses behave in each exercise intensity domain, focusing mostly on how the body matches the energy requirements with aerobic metabolism (if it can).


Moderate exercise:

This domain is perhaps the simplest, because exercise at these intensities the body rapidly reaches a steady state, in which the energy demand of the task is relatively rapidly met by aerobic metabolism.  For this to happen, the body adjusts the rate of oxygen uptake after exercise starts until oxidative metabolism provides essentially all of the energy required.  This is the “steady state”.  If lactate increases in the blood, it does so only transiently.  The respiratory exchange ratio stays below 1.0, meaning that less carbon dioxide is exhaled per unit time than oxygen taken up.  Because the amount of air you breathe in and out every minute is closely coupled, one way or another, to the rate of CO2 output, the demand on the lungs and respiratory muscles is relatively small.  The typical oxygen uptake (VO2) response to moderate exercise looks like this:


Figure 1: the oxygen uptake response to moderate-intensity exercise. Notice the attainment of a steady state in VO2 in less than ~3 minutes.  Also notice the relatively low VO2 response (~50% of VO2max. In this participant, VO2max was 4.25 L.min-1).


The tolerable duration of moderate-intensity exercise has never been precisely established, but we do know that you should be able to sustain it for several hours provided you don’t succumb to injury or hyperthermia.  The fatigue mechanisms are also unclear, although the weight of evidence suggests that, without the involvement of heat stress or high-altitude, central mechanisms of fatigue are dominant, including disruptions to brain neurotransmission resulting in a loss of ‘drive’ to exercise.


Although the moderate-intensity domain is regularly frequented by team sports athletes in recovery from sprints or high-intensity phases, or in repositioning manoeuvers, endurance sports that occupy this domain are almost exclusively ultra-endurance events lasting more than 2 hours.  For example, in a 4-5 hour flat or ‘sprint’ stage of a cycling race, the riders will spend the vast majority of their time in the moderate domain, with the main sprinters only exiting this domain in the final sprint itself.  Prior to that, they will spend the entire stage in the peloton, protected by teammates from exerting any significant effort.  In short, events longer than the marathon are characterized by being performed in the moderate-intensity domain, and the predominant fatigue mechanism is likely to be central in nature.  At these intensities, feeding and hydration are usually easy to ensure with correct planning.  In fact, some of these events are so long that the problem could be access to too much food and fluid, rather than too little!


Heavy exercise:

When athletes exceed the so-called lactate threshold, they enter the ‘heavy-intensity domain’. In this domain, blood lactate concentration is elevated above resting levels but stabilizes, and oxygen uptake reaches a steady state too.  Importantly, it takes 10-20 minutes for oxygen uptake to reach a steady state, and when it reaches this steady state, the oxygen uptake is higher than would be expected if the exercise was moderate.  This slowly increasing O2 uptake is known as the “VO2 slow component”.  This higher oxygen cost of exercise has a number of consequences for fatigue and exercise tolerance, as we will see below.  A typical heavy-intensity oxygen uptake response looks like this:

Figure 2

Figure 2: the oxygen uptake response to heavy-intensity exercise. Notice the greater VO2 response than moderate exercise (~80% VO2max), and the delayed attainment of a steady state, in which VO2 is higher than would be predicted from the responses to moderate exercise.



The fatigue processes that occur during heavy-intensity exercise include both central and peripheral mechanisms.  Peripheral fatigue is not likely to be caused by high-energy phosphate depletion or accumulation, since a steady state in O2 uptake, blood lactate concentration and, when measured, phosphorylcreatine (PCr), inorganic phosphate, muscle lactate and pH.  Classically, the depletion of muscle glycogen has been identified as the likely cause of fatigue during prolonged heavy exercise.  Multiple lines of evidence support this assertion.  Most recently, it has been demonstrated that there are specific pools of glycogen in a muscle fibre, and the pool depleted most rapidly plays a critical role in excitation-contraction coupling.  This may explain the gradual, rather than catastrophic, loss of muscle force output as exercise progresses and individual fibres become unresponsive to excitatory input.  In the heavy-intensity domain, the higher O2 cost that occurs as a result of the development of the VO2 slow component will result in a faster utilization of the glycogen stores.  It is not surprising, therefore, that marathon runners select a pace that is just above the lactate threshold when racing.  If they ran any faster, they would increase the energy demand of running and risk ‘hitting the wall’.


Central fatigue also occurs in the heavy-intensity domain, but its cause is far from certain.  Some of the mechanisms described above for moderate intensity exercise could also occur during heavy exercise, but because alterations in brain neurochemistry, for example, take several hours to develop, it seems unlikely to explain the central fatigue that develops within minutes of beginning heavy exercise.  It is more likely that the repetitive activity of the muscles makes them, or their motoneurones, harder to drive (that is, they are less responsive to excitatory input from the motor cortex and spinal cord).  Whether these structures are also affected in this way, putting the cause of central fatigue further ‘upstream’ in the CNS, is currently unclear.  An additional component of central fatigue may be related to muscle glycogen depletion: in the latter stages of heavy exercise, blood glucose concentration begins to fall.  This may rob the brain itself of fuel, leading to the confusion, lethargy, and irritability characteristic of somebody ‘hitting the wall’.


Endurance events that take place in the heavy-intensity domain include half-marathon and marathon running, as well as many cycling time trials and 10,000 m swimming.  The first (and so far only) human-powered flight across the English Channel took 2 hours and 49 minutes in 1979.  This supreme feat of both engineering and endurance is described in Wilkie’s biographical sketch, and it is clear that this would only have been possible in an individual sufficiently fit to produce more than 250 W and maintain a steady state. Bryan Allen was that individual.


The upper limit of the heavy intensity domain is, by definition, the highest power output or speed at which a steady state can be maintain.  This boundary point is surprisingly difficult to measure, but two methods have been used extensively.  The first is to perform a series of constant speed trials of up to 30 minutes and identify the point at which a steady state is no longer possible.  The other is to perform several exhaustive exercise bouts in the severe-intensity domain and construct the hyperbolic power-duration relationship.  The point at which this curve flattens out (i.e., when it reaches an asymptote) is known as the ‘critical power’ (analogously, the ‘critical speed’ in running).  Exercise above this point will undoubtedly be non-steady state, and therefore severe.  An analysis of elite marathon runners illustrates the importance of this upper boundary: a selection of the fastest marathon runners of all time shows that all of them performed the marathon at or below the critical speed.  Their ability to run so close to this boundary is likely to be a consequence of them also possessing a very high lactate threshold running speed as a fraction of their VO2max.


The most common and accepted means of countering fatigue during heavy-intensity exercise is to regularly ingest carbohydrates during performance, since it has been shown that late in exercise almost all of the carbohydrates used in the muscle come from blood glucose.  As dehydration can also occur in prolonged heavy exercise, these carbohydrates are often consumed in drinks, or as gels washed down with water.  However, ingesting food and fluids during heavy exercise is not easy, and feeding strategies must be trialled in training to avoid underfeeding or the discomfort of overfeeding.


Severe exercise:

Above the critical power (or critical speed), it is not possible to achieve a metabolic steady state in.  Oxygen uptake, blood lactate, muscle PCr, inorganic phosphate and pH fail to stabilise, and exercise duration is limited to 40 minutes or less.  In this intensity domain, the VO2 slow component drives VO2 upward until VO2max is attained.  Task failure occurs soon after VO2max is attained.  The higher the power output above critical power, the more rapidly the slow component increases, and the more rapidly muscle metabolites deplete and accumulate.  All else being equal, exercise terminates when the aforementioned metabolites reach surprisingly similar levels, despite wide variations in exercise duration.  An example of the oxygen uptake response to severe-intensity exercise is shown below.

Figure 3

Figure 3: the oxygen uptake response to severe-intensity exercise.  Notice the absence of a steady state and VO2 rising until VO2max is attained.  The power output in this test was 295 W, 45 W higher than the critical power, and resulted in exhaustion in approximately 12 minutes.


The peripheral mechanisms of fatigue during severe-intensity exercise appear to be directly related to the depletion of high-energy phosphates (PCr) and the accumulation of metabolites associated with them (inorganic phosphate and protons).  Both reduced muscle pH and elevated inorganic phosphate concentrations, alone or in combination, have been shown to reduce muscle force output and/or shortening velocity.  This appears to be the result of direct effects on muscle crossbridge function as well as diminished calcium release and uptake (for a review see Allen et al., 2008).  These findings are often derived from experiments performed in vitro, but importantly it has been shown that similar metabolite changes can be observed during exercise in vivo using biopsy samples or magnetic resonance spectroscopy.


Central fatigue has also been shown to occur during severe-intensity exercise, although again the mechanism underpinning this is unclear.  In addition to reduced motoneurone excitability mentioned above, a further potential source of reduced voluntary activation in this domain is afferent feedback from the fatiguing muscle.  It has been shown that populations of thinly myelinated or unmyelinated afferent fibres are sensitive to substances produced during muscular contraction.  These ‘metaboreceptors’ are thought to produce inhibitory input to various parts of the CNS, reducing voluntary activation and thus the drive to the muscle.  The combination of direct and indirect effects of metabolic changes during severe-intensity exercise therefore appears to be pivotal in the fatigue processes above the critical power.


The tolerable duration of severe-intensity exercise ranges from ~2-40 minutes, meaning that the majority of athletic events labelled “endurance” occur in this intensity domain.  This is why the fatigue processes identified for 5,000 m and 10,000 m running in a previous sub-section were so similar.  The common fatigue mechanisms in this intensity domain also makes the tolerable duration of severe exercise highly predictable using as few as two parameters: the critical power (the asymptote in power output) and the degree of curvature in the relationship between power output and time.  This hyperbolic relationship theoretically provides a measure of the highest power output that can be sustained without fatigue (the critical power), and a measure of the amount of work that can be performed above critical power before task failure occurs (labelled W’).  In reality, of course, fatigue is not absent when exercising at critical power.  Rather, it would be more accurate to say that the critical power represents the highest power output that can, in principle, be maintained without experiencing a progressive metabolically-mediated fatigue. The power-duration relationship is shown below.

Figure 4

Figure 4: the power-duration relationship.  In this example, a participant performed four exhaustive constant-load cycling bouts on separate days.  Plotting the time to exhaustion against power produces a hyperbolic relationship of the form: Tlim = W’/(Power – CP), where Tlim is time to exhaustion, W’ is the parameter that defines the curvature of the relationship and CP is the critical power (the asymptote of the power-duration relationship, dashed line).


The power-duration relationship seen above is a very important concept in exercise physiology, because the critical power represents the ‘red line’ in performance.  Exceeding the critical power means that an athlete has only minutes of sustainable exercise left before ‘exhaustion’ occurs.  Consequently, this concept is often used by athletes to develop pacing strategies for races in order to optimise performance, or to exploit perceived weaknesses of opponents.  If athletes get their pacing strategies wrong, and use up the work capacity reflected in the W’ before the finish line, they will slow down dramatically.  This is frequently seen in the final lap of races lasting 1,500 m or longer on the track.  It is also commonly seen in events like the 4000 m team pursuit, in which one of the riders may fail to make the finish line due to depleting their W¢ too early.  On the other hand, this can sometimes be a deliberate strategy to use up that rider and allow the other riders to draft them before the final effort.


Thresholds or phase transitions?


As useful as exercise intensity domains and their landmarks are, there has been a tendency in the literature, and particularly in practice, to assume or demand unrealistic levels of precision in their measurement.  It has also been assumed that maximal lactate steady state (MLSS) is THE gold standard measure of the heavy-severe domain boundary, and that CP is inferior.  We have disputed this in a previous viewpoint, but the bigger issue, for me, is the assumption that either MLSS or CP can be recorded to the nearest watt or km/h.  In reality, the above hyperbolic curve is always subject to measurement error, and these errors can be surprisingly large.  For example, if you perform 3 predicting trials, and use a 2-parameter equation to fit the data, you are left with 1 degree of freedom.  This means that the standard error (say, 5 W) must be multiplied by ~12.71. This means the 95% confidence limits in this case would be ~64 W in each direction! Adding predicting trials would reduce both the standard error and the confidence limits, but there would always be some error in the prediction of CP.  One thing we were interested in exploring was how big this error is, functionally speaking.  In other words, how far above CP can you go before you observe consistent severe-intensity behaviour?


To do this, we used a well-established intermittent isometric contraction model to study the “critical torque”.  The whole study is free to view here, but the key observation was that we failed to observe consistent severe-intensity behaviour in a range of fatigue-related variables when contractions were performed at least two standard errors above the point estimate of the critical torque.  This may not seem too surprising given that we were still inside the 95% confidence limits in at this point, but previous literature has suggested that critical power itself should be non-steady state.  We also saw some evidence of severe behaviour on very rare occasions below the critical torque, and plenty of evidence of apparently heavy intensity behaviour above critical torque.  This led us to conclude that the critical torque is best described as a phase transition rather than a sudden threshold.  The grey area is not just a statistical quirk; it is reflected in the underlying physiological responses.


Where does this leave the exercise intensity spectrum, its landmarks and especially the critical power concept? First, if you perform a “coarse grained” analysis of exercise intensity, the domains remain distinct and the transitions between them abrupt.  They are still the foundation of many other concepts in exercise physiology, and the distinct physiological and fatigue responses within each domain are still crucial to appreciate, understand and apply.  But a fine-grained analysis of the physiological landmarks show that the abrupt transitions are anything but.  Critical power, lactate threshold, or indeed any other threshold marker, will always be associated with a band of uncertainty around them, of the order of about ±10-15 W in the case of cycling.  This uncertainty probably has a physiological basis that you will never avoid: variations in metabolism, blood flow, muscle recruitment will tend to smear these thresholds across a band of power or speed.  This means that if you want to exercise strictly “below critical power” it should be at least two standard errors below critical power, and vice versa above it.


Where does it leave my view on the power-duration relationship and its physiological basis? I personally think the concept is strengthened by the phase transition idea, since this agrees with, rather than is at odds with, the underlying heterogeneities of the physiological response to exercise.  The grey area around the point estimate of these landmarks makes them feel more real to me. And I like that.



Burnley, M., Doust, J.H. & Vanhatalo, A.  (2006).  A 3 min all-out test to determine peak oxygen uptake and the maximal steady state.  Medicine and Science in Sports and Exercise 38, 1995-2003.


Burnley, M.  (2009).  Estimation of critical torque using intermittent isometric maximal voluntary contractions of the quadriceps in humans.  Journal of Applied Physiology, 106, 975-983.


Burnley, M., Vanhatalo, A. & Jones, A.M.  (2012).  Distinct profiles of neuromuscular fatigue during muscle contractions below and above the critical torque in humans.  Journal of Applied Physiology, 115, 215-223.


Jones AM, Burnley M, Black MI, Poole DC & Vanhatalo A. (2019). The maximal metabolic steady state: redefining the ‘gold standard’. Physiological Reports, 7, e14098.


Pethick, J., Winter, S.L. & Burnley, M. (2015). Fatigue reduces the complexity of knee-extensor torque fluctuations during maximal and submaximal intermittent isometric contractions in humans. Journal of Physiology, 593, 2085-2096.


Pethick, J., Winter, S.L. & Burnley, M. (2016).  Loss of knee extensor torque complexity during fatiguing contractions occurs exclusively above the critical torque.  American Journal of Physiology Regulatory Integrative and Comparative Physiology, 310, R1144-R1153.


Pethick J, Winter SL & Burnley M. (2020). Physiological evidence that the critical torque is a phase transition not a threshold. Medicine and Science in Sports and Exercise, in press.


Vanhatalo, A., Doust, J.H. & Burnley, M.  (2007).  Determination of critical power using a 3-min all-out cycling test.  Medicine and Science in Sports and Exercise 39, 548-555.


Vanhatalo, A., Doust, J.H. & Burnley M.  (2008).  Robustness of a 3 min all-out cycling test to manipulations of power profile and cadence in humans.  Experimental Physiology 93, 383-390.


One Comment leave one →
  1. April 10, 2022 10:09 am

    thanks alot of information

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