REVIEW ARTICLE

R. Hugh Morton

The critical power and related whole-body bioenergetic models

Accepted: 13 October 2005 / Published online: 12 November 2005� Springer-Verlag 2005

Abstract This paper takes a performance-based ap-proach to review the broad expanse of literature relatingto whole-body models of human bioenergetics. It beginswith an examination of the critical power model and itsassumptions. Although remarkably robust, this modelhas a number of shortcomings. Attention to these hasled to the development of more realistic and more de-tailed derivatives of the critical power model. Themathematical solutions to and associated behaviour ofthese models when subjected to imposed ‘‘exercise’’ canbe applied as a means of gaining a deeper understandingof the bioenergetics of human exercise performance.

Keywords Anaerobic work capacity Æ Energy ÆEndurance Æ Exercise Æ Metabolism Æ Performance

Introduction

Over recent decades there appears to have been a drifttowards a more ‘‘micro’’ view of human bioenergetics,focussing more towards the cellular and even molecularend of the spectrum. Although the ‘‘macro’’ view has notbeen neglected, it is nevertheless timely to examinemodels of human bioenergetics from the whole-bodyperspective. The human body is a highly complexorganism and has been the subject of intense scientificscrutiny for several centuries. Despite considering abroad picture of only one aspect of its functioning, I donot make any special attempt to be comprehensive inthis review. Rather, I attempt to blend the essentialtheoretical elements of these models and their mathe-matical solutions with the practical and performance-related interpretations of their properties. In so doing, I

hope their value as an investigative research vehicle or asa framework for practical aspects of the study of humanperformance, or even as a teaching tool for undergrad-uate or postgraduate students, can be evaluated.

The review begins with an examination of the simplecritical power model and its assumptions. When sub-jected to scrutiny, this model is remarkably robust, but itnevertheless has a number of shortcomings. Remedialattention to these has led to the development of morerealistic and more detailed derivatives of the criticalpower model. The mathematical solutions to thesemodels under a variety of exercise scenarios providefertile ground for gaining a better understanding of thebioenergetics of human exercise performance.

This review is not intended to update the literature oncritical power (Hill 1993; Vandewalle et al. 1997) nor thaton the relationship between power output and endurancetime (Billat et al. 1999a; Morton and Hodgson 1996) norits relationship to whole-body fatigue (Walsh 2000), butrather to set the critical power model and its derivatives inthe wider bioenergetic modelling context.

The critical power model

Origins, assumptions and development

It was Monod and Scherrer (1965) who first observed ahyperbolic relationship between the level of constantpower output (P) and corresponding time to exhaustion(t) in a single muscle group. This relationship, whentransformed knowing that total work performed (W) isgiven by the productPt, can be expressed in a linear form.The intercept of this line was termed the ‘‘anaerobic workcapacity’’ (AWC) and its slope termed the ‘‘criticalpower’’ (CP). Mathematically the two equations:

W ¼ AWCþ CP� t ð1aÞ

and

t ¼ AWC=(P � CP) ð1bÞ

R. H. MortonInstitute of Food, Nutrition and Human Health,Massey University, Private Bag 11-222,Palmerston North, New ZealandE-mail: h.morton@massey.ac.nzTel.: +64-6-3504265Fax: +64-6-3505781

Eur J Appl Physiol (2006) 96: 339–354DOI 10.1007/s00421-005-0088-2

are equivalent. There are in fact four other equivalentequations:

P ¼ AWC=t þ CP, ð1cÞ

a transformation first proposed by Whipp et al. (1982)

t ¼ (W �AWC)=CP ð1dÞ

and

P ¼ W � CP=(W �AWC), ð1eÞW ¼ AWC� P=(P� CP) ð1f Þ

although only the first three are in common use. Giventhat t is the most natural dependent variable, and P themost natural explanatory variable, Eq. 1b is the pre-ferred version from an exercise performance viewpoint,though a similar case could be made for Eq. 1f.

Moritani et al. (1981) extended those ideas for the firsttime to whole-body exercise on the cycle ergometer. Thefour essential assumptions of the whole-body CP conceptas it became known (Hill 1993) are the following:

1. There are only two components to the energy supplysystem for human exercise, termed aerobic andanaerobic.

2. The aerobic supply is unlimited in capacity, but israte limited. This limiting parameter is the criticalpower, CP.

3. The anaerobic supply conversely is not rate limited,but is capacity limited by the parameter AWC.

4. Exhaustion, and by implication termination of exer-cise, occurs when all of AWC has been utilised.

From these assumptions it can be deduced that forconstant power P £ CP, the notion of exhaustion isvacuous and t is undefined or infinite, whereas forP>CP, Eq. 1b and its equivalents follow directly. Overthe last 20 or more years, a large number of studies havefound good fits of data to Eqs. 1a, 1b, 1c, at least withincertain bounds.

It is important nevertheless to realise that implicitlyembedded in these assumptions are a number of others,specifically:

5. Aerobic power is available at its limiting rate CP themoment the exercise begins and remains so right upuntil the end of the exercise.

6. The power domain over which the model applies is allof CP<P<¥.

7. The time domain over which the model applies is allof 0<t<¥, and endurance at CP is infinitely long.

8. The efficiency of transformation of metabolic energyto mechanical energy is constant across the wholepower (and time) domain(s).

9. CP and AWC are constants, independent of P (and/or of t).

All models of real processes are necessarily simplifi-cations and the above assumptions are indicative ofthem. There are of course unrealities to varying degreesin all of them, and these will be discussed in Extending

the critical power model when extensions to the CPmodel and the reasons behind them will be considered.

Parameter interpretation and estimation

The assumptions above suggest clearly enough that CPis aerobic in nature and that AWC is anaerobic, butempirical verifications are vital. It was Moritani et al.(1981) who first provided evidence for these interpreta-tions. They found CP to be highly correlated with theventilatory anaerobic threshold and AWC to be unaf-fected by hypoxia. Several other early studies (Greenand Dawson 1993; Green et al. 1994; Green 1994, 1995;Housh et al. 1991; Jenkins and Quigley 1990, 1991;Talbert et al. 1991) have provided similar evidences,though the evidence is not all conclusive (Housh et al.1992). Perhaps the most convincing evidence of theaerobic and anaerobic natures of these parameters canbe found amongst the collection of studies reportinginterventions and their impact on the CP model, whichwill be discussed in Interventions and their impact onthe model.

In order to estimate the parameters CP and AWC forany subject, it is necessary to complete at least twoseparate trials, cycling to exhaustion at two different butconstant power outputs. In practice four or more arecommon, spread across a spectrum, though six or moreare uncommon. On each occasion P and t (and henceW = Pt also) are recorded. Any one of the six equationforms can be fitted using any standard statistical curve-fitting software. It is important to choose these experi-mental design aspects carefully, for the CP and AWCestimates may differ according to how many trials areperformed (Housh et al. 1990; Taylor and Batterham2002), what durations (and therefore also what powerselections) are planned (Bishop et al. 1998; Jenkins et al.1998; Poole et al. 1986) and which of the mathematicalforms of the model may be used (Bull et al. 2000;Gaesser et al. 1990, 1995; Hill et al. 1993; Hill 2004;Housh et al. 2001). Furthermore, the rest interval be-tween trials may affect the estimates also (Bishop andJenkins 1995; Hill 1993) varying between 30 min andover 24 h in different studies.

It has been observed too that variations in repeatedperformances are larger when durations are longer(McLellan et al. 1995; Poole et al. 1988). For this reasonthe exact nature of the curve-fitting routine employedmay also affect the estimates. It may be statisticallyadvisable to use weighted regression, giving less weightto long-duration observations and vice versa (Mortonand Hodgson 1996).

Performance implications

Application of the CP model using data from any sub-ject clearly provides an assessment of two dimensions ofthe bioenergetic characteristics of that individual. Then,

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employing Eqs. 1b, 1f for example and substituting anygiven value of P provides predictions of that individual’sperformance (i.e. endurance) time and amount of workthat could be accomplished, respectively. Such perfor-mance aspects are obvious and have been employed indetermining the ability of humans generally as sources ofmechanical power (Wilkie 1960) or specifically in activ-ities such as human-powered flight (Nadel 1988, 1996).

Of far more relevance in today’s world of competitivesports performance is the question of optimal strategy.For example, can performance time (and the amount ofwork accomplished) be improved by working at otherthan the constant power throughout determined by thesolution to the CP model equations for that individual?Fukuba and Whipp (1996, 1999) have solved thisproblem (albeit for the running analogue of the CPmodel). The answer, or at least part of it, is ‘‘No’’. Theyalso demonstrate that any one of an infinite number ofstrategies involving stagewise steps of periods of con-stant power produces an identical endurance time, pro-vided P never drops below CP. This can also be shownto be true when P changes (always above CP) in anysmooth and continuous fashion (Morton 2004).

Conceptualisation as a hydraulic model

This conceptualisation is originally due to Margaria(1976) and can be described as follows. If we envisagefluid contained in a vessel or vessels to represent meta-bolic energy and the flow of fluid therefore to representpower, the CP model can be diagrammatically repre-sented as in Fig. 1.

The system is comprised of two vessels representingthe aerobic (Ae) and anaerobic (An) energy supplycomponents (assumption 1). Ae is of infinite capacity(assumption 2). R1 is a connecting tube of fixed diame-ter, which determines the limit to the fluid flow fromAe to An. This limit is denoted by CP (assumption 2).Vessel An is of limited capacity, denoted by AWC(assumption 3). T is a tap which can be opened to allowan unlimited positive flow P (assumption 6). This in turn

(since CP is finite) implies that the rate of decrease of thefluid level in An is also unlimited (assumption 3). Pro-vided P>CP (assumption 6), the fluid level in An mustdrop to make up the difference P–CP, and P can nolonger be maintained when An becomes empty(assumption 4). This event occurs at time t>0(assumption 7). The flow through R1 is always equal toexactly CP, determined only by the unchanging depth ofvessel Ae (assumption 5). When P=CP or less, fluidfrom An is not required and An will never be emptied,i.e. t is infinite (assumption 7). Since the whole hydraulicmodel is a physical construct with fixed dimensions andcontaining an incompressible fluid, the volumetric andrate (and therefore also efficiency) parameters are fixedand independent of P (assumptions 9 and 8).

All the above lead directly to the derivation of Eq. 1bfrom first principles. Knowing also thatW = Pt enablesall other mathematical equivalences to Eq. 1b to bederived algebraically.

Interventions and their impact on the model

The CP model has been the object of a large number ofintervention studies designed to discover what happenswhen it is exposed to deliberate changes in conditions.These studies have assisted significantly in our under-standing of the model. In general, their results conformto what physiological logic might lead us to expect andaffirms the overall nature of the concept and its com-ponent structure.

An early study of Jenkins and Quigley (1990) showedthat a group of trained subjects had a significantlyhigher average CP and AWC than another group ofuntrained subjects. Since that time, more specificallydesigned training studies have examined the direct ef-fects of various training strategies on changes in CP and/or AWC in the same subjects. Given the aerobic natureof CP, it would be expected that endurance or pre-dominantly aerobic training would increase the CP ofthose undertaking such training. A number of studieshave confirmed that this indeed occurs (Gaesser andWilson 1988; Jenkins and Quigley 1992; Jones andCarter 2000; Poole et al. 1986). Likewise it would beexpected that strength or predominantly anaerobictraining would increase AWC and, from the outset,studies do confirm this (Jenkins and Quigley 1992; Pooleet al. 1990; Stokes et al. 1993). Interval training, in whichperiods of low-intensity training alternate with periodsof high intensity, is generally regarded as a particularlyeffective overall training strategy (Billat 2001; Laursenand Jenkins 2002) because it taxes energy supplymechanisms across the spectrum. Inferring from theabove, it would be expected and has been shown thatinterval training improves both CP and AWC (Gaesserand Wilson 1988; Poole et al. 1990).

From the hydraulic representation of Fig. 1, aerobictraining appears to increase the diameter of pipe R1. Itwould not likely increase the depth of vessel Ae, as thisFig. 1 The CP concept as a hydraulic model

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depth could be viewed as the analogue of the O2 partialpressure in the inspired air. Strength training on thecontrary appears to increase the volume of vessel An,probably via a cross-sectional increase rather than adepth change.

Again, given the aerobic nature of CP, it would beexpected that subjects tested under hypoxic conditionswould evidence reduced CP, while comparative testsunder hyperoxic conditions would evidence an elevatedCP. This appears to be the case (Moritani et al. 1981;Whipp et al. 1982), whereas AWC is unchanged. Fromthe hydraulic model perspective similarly therefore, hy-poxia probably represents reduced depth of vessel Ae

rather than a constriction of R1, whereas hyperoxiarepresents the converse.

Since AWC is regarded as an anaerobic store repre-senting energy derived from high-energy phosphates andglycolysis leading to net lactate formation, it may beexpected that creatine and/or carbohydrate supplemen-tation would increase AWC. Studies show that this is thecase (Miura et al. 1999; Smith et al. 1998; Stout et al.1999). Creatine supplementation has no effect on CP,but its effects on AWC are augmented by carbohydratesupplementation. No studies have reported the effects ofcarbohydrate supplementation alone. Like strengthtraining, these types of supplementation appear to in-crease the volume of vessel An.

Likewise, if prior exercise is undertaken with little orno recovery or if the timing is only started once VO2

max is reached, a diminution of AWC would be ex-pected. Once again, empirical evidence confirms theseexpectations (Billat et al. 1999a, b; Heubert et al. 2003,2004; Swanson et al. 1992).

Differing pedal cadences have been considered as aconfounding factor with the potential to cause changesin the CP and/or AWC estimates. The available evidenceis to some extent contradictory (Carnevale and Gaesser1991; Green et al. 1995) but it is believed that it is best toallow subjects to select a cadence at which they arecomfortable, at least between the limits of 60–100 rpm(Hill et al. 1994; Hill 2004).

Another confounding factor which does appear toaffect CP and/or AWC estimates is a learning effect,manifesting under repeat performance or test/retestconditions (Scarborough et al. 1991; Smith et al. 1991).As would be expected, performances are better andestimates higher on subsequent testing occasions.

In summary, this array of interventions serves tostrongly affirm the robust nature of the concept inassigning an aerobic character to CP and an anaerobiccharacter to AWC. The interpretation of these param-eters is now firmly established.

Applications to other constant power exercise modalities

Although Lloyd (1966) first suggested application ofthese ideas to running, it was Hughson et al. (1984) whofirst published the details in which critical velocity (CV,

measured in metre/second) and anaerobic distancecapacity (ADC, measured in metres) were consideredand evaluated as analogues of CP and AWC, respec-tively. While not subject to quite the same degree ofscrutiny as the original form of the CP concept, studiesdo in general confirm those results reported previously.For example, CV has been found no different from themaximal lactate steady-state running velocity or fromthe lactate turn point velocity (Smith and Jones 2001),and to be a threshold velocity above which exercise ofsufficient duration will lead to attainment of VO2max

(Hill and Ferguson 1999). The success with which therunning version of the concept is able to predict times atrunning various distances does, however, seem in somedoubt (Kolbe et al. 1995; McDermott et al. 1993) andthe choice of which equation of the model to use alsoseems unclear (Housh et al. 2001).

In the same way as for running, CV and ADC can beevaluated for swimming (Biggerstaff et al. 1992; Stewardet al. 1994; Wakayoshi et al. 1992a, b). Although subjectto even less scrutiny than running, the critical swimmingvelocity corresponds to the maximal lactate steady-stateswimming velocity (Wakayoshi et al. 1993) and corre-lates significantly with swimming VO2 at the anaerobicthreshold and the onset of blood lactate accumulation(Wakayoshi et al. 1992a, b) and with the lactatethreshold (Martin and Whyte 2000). Critical swimvelocity has been found to correspond to a maximalstroke rate which can theoretically be maintained con-tinuously without exhaustion (Dekerle et al. 2002). Thechoice of equation for swimming, however, seems to beof little consequence (Lane et al. 1994), but there isconflicting evidence on whether the anaerobic swimmingdistance capacity provides a reliable estimate of anaer-obic capacity or not (Biggerstaff et al. 1992; Dekerleet al. 2002; Lane et al. 1994).

In theory of course, any exercise modality in whichphysical work can be accomplished, or which involvessome form of locomotion, offers scope for application ofthe CP concept. Several of these have been briefly re-ported, such as kayaking (Clingeleffer et al. 1994),wheelchair propulsion (Arabi et al. 1999), weightlifting(Morton et al. 2005) and rowing (Hill et al. 2003; Ken-nedy and Bell 2000). Cycling when regarded as a meansof locomotion appears to have not been considered.

In all these cases, whatever the form of locomotion,the relationship between energy cost of transport andspeed over the applicable range must be known andexplicitly introduced into the calculations (di Prampero1999).

Applications to variable power output exercise

There is nothing particularly special about continuousexercise at constant power, except perhaps that themathematical solutions (and the associated curve fittingto data) to the CP model under such conditions areextremely simple. One exercise protocol common in

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laboratory testing settings is ramp exercise, where poweroutput starts at zero (though in general this need notnecessarily be the case) and increases linearly at somefixed rate, referred to as the ramp slope, s. The CP modelhas been adapted for ramp exercise (Morton 1994), ex-cept that endurance time becomes a function of s, albeitmathematically slightly more complex than linear orhyperbolic. Like the original CP model, it too is avail-able in several different mathematically equivalent forms(Morton 1997). Its validity has been questioned (Van-dewalle 1995) despite the fact that its assumptions areprecisely the same as for the constant power CP model(Morton 1995). Indeed, investigation has shown thatestimates of both CP and AWC from ramp tests are notsignificantly different from those derived from constantpower tests in the same subjects (Morton et al. 1997).

Similarly, there is nothing special about continuousexercise per se. Many sports (particularly team games)involve exercise of an intermittent nature, where periodsof relative rest and intense exercise are interspersed. In-deed intermittent exercise (Christensen 1960) has for longbeen regarded as a particularly beneficial form of exercise(Billat 2001; Laursen and Jenkins 2002). Can the CPconcept therefore be applied to intermittent exercise?

Examination of the hydraulic representation of theCP model depicted in Fig. 1 suggests that if tap T wereto be closed, corresponding to the cessation of work (ora period of relative rest), then vessel An would refill byvirtue of the flow through R1. Only when An had beencompletely refilled would the flow through R1 cease.Thus for intermittent exercise one can regard the intenseexercise periods as drains on An and the relative restperiods as refilling An. This is exactly the approach ta-ken by Morton and Billat (2004), who find good fits ofthe derived model equation to data, but the CV esti-mated from continuous running was significantly higherthan the CV estimated from intermittent running in thesame subjects. This discrepancy may be due to increasedsensitivity of the model to the implicit assumption of theinfinitely fast step increases and decreases in VO2

occurring many times in this mode or exercise. Inaddition, it should be noted that this particular appli-cation of the CV concept is quite different to that ofKachouri et al. (1996) and others, who simply apply thestandard model approach to aggregated intermittentrunning times. Rather than being an example of thelimits of the CP concept as claimed, Kachouri’s appli-cation is perhaps more an example of an inappropriateor inaccurate procedure, not based on proper consider-ation of the CP concept assumptions detailed above.

Extending the critical power model

Questioning the assumptions

All models purporting to represent aspects of the humansystem are necessarily simplifications, and the CP modelis no exception. All nine of the assumptions of Origins,

assumptions and development can be questioned. Inbrief, the following critiques can be made of them.

1. There are only two components to the energy supplysystem for human exercise, termed aerobic andanaerobic. While it may be true that there are onlytwo biochemical pathways, aerobic and anaerobic, tohuman energy metabolism, there are certainly morethan two components to the energy source or supply.The aerobic pathway taps into both glycogen and fatas fuel sources. The anaerobic pathway comprises alactic component in which glycogen metabolisesresulting in net lactic acid production and an alacticcomponent (essentially the high energy phosphatecompounds or phosphagens) whose metabolism doesnot result in lactic acid production. Incorporating allthese aspects realistically will of necessity complicatethe modelling. It has been attempted in a simplefashion, described in Applications and developmentsof the 3-component hydraulic model.

2. The aerobic supply is unlimited in capacity, but is ratelimited. This limiting parameter is the CP. Since theaerobic pathway requires an energy substrate to beoxidised, the amount of which is of course limited inany human, it is clearly untrue to assert that theaerobic capacity is unlimited. It may well be relativelylarge (in energy units) in comparison to other avail-able energy substrates by an order of magnitude ormore, and this may well explain the robust characterof the CP model at least for exercise of moderateduration. However, it is well established that theaerobic supply is rate limited. Whether this rate iscalled the critical power, maximum aerobic power(MAP) or VO2max, or indeed some other convenientterm, is not the issue. Some upper limit clearly exists.

3. The anaerobic supply conversely is not rate limited, butis capacity limited by the parameter AWC. Theanaerobic supply component likewise requires energysubstrates, and so it is quite reasonable to assume alimited capacity. While explosive muscular powermay well be quite high, it is clearly not unlimited.Humans cannot run with instantaneous velocity ofmore than about 13 m/s, and if the resistance on thepedal cranks of a cycle ergometer is too high, thepedals simply cannot be turned. No physical work ispossible and no power is generated. The notion of anendurance time at such intensities is clearly vacuous.

4. Exhaustion, and by implication termination of exercise,occurs when all of AWC have been utilised. While itmay be reasonable for a mechanical engine to ceasewhen its usable fuel is fully depleted (provided there isno other malfunction), the human engine appears tobe different. It has for long been known that atexhaustion, significant amounts of unused glycogenremain, and that the higher is the power requirementat exhaustion, the more it is that remains (Saltin andKarlsson 1971). Furthermore, it is well establishedthat if the power requirement at the point ofexhaustion were to be significantly reduced, exercise

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could continue. It is clear therefore that some linkageor dependency exists between the power demanded atthe point of exhaustion and the residual unused fuelsupply available at that time. Thus while it may bereasonable to assert that the anaerobic capacity is ofa fixed and limited value, the usable amount clearly isnot. Consequently any estimates of this value basedon the CP model assumptions will obviously beunderestimates.

5. Aerobic power is available at its limiting rate CP themoment exercise begins and remains so right up untilthe end of exercise. The kinetics of oxygen uptake,whether regarded as mono- or bi-exponential andwith or without any delay(s) (Barstow and Mole1991), clearly dictate that the establishment of theaerobic supply at any given rate (not just at CP) is notinstantaneous. Three minutes or so may be required.What this means is that the capacity-limited anaero-bic supply as reasoned in the CP model must be inreality larger than the model prescribes. Again, asabove, AWC must be underestimated by the CPmodel.

6. The power domain over which the model applies is all ofCP<P<¥. If P £ CP, the anaerobic supply is neverrequired, since under assumption 5, aerobic power atCP (or less) is immediately available. The left-handinequality therefore is obviously reasonable. At theother end of the scale, human muscle is not all-powerful and, as indicated above, a finite upper limitto the power that can be developed must exist and tothe velocity of running, etc. This upper limit could bedenoted Pmax and regarded as a maximum ‘‘instan-taneous’’ power.

7. The time domain over which the model applies is all of0<t<¥, and endurance at CP is infinitely long. Thenotion of ‘‘infinite’’ endurance is a somewhat nebu-lous one. It is not uncommon that long-durationcontinuous exercises of moderate power output areterminated for a variety of non-energy-substrate re-lated reasons, such as boredom, psychological issues,the necessity to eat, drink or toilet. This thereforeplaces an imprecise upper limit to the time domainover which the CP model applies. In similar vein veryshort time intervals at very high power (even if lessthan Pmax) are not realistic either. Factors like over-coming inertia and acceleration come into consider-ation. Strictly speaking of course, if exercise at Pmax

or more was to be attempted, endurance time shouldbe zero.

8. The efficiency of transformation of metabolic energy tomechanical energy is constant across the whole power(and time) domain(s). This assumption, implyingthat the energy cost of locomotion per unit distance isconstant, seems to depend critically on the form oflocomotion (di Prampero 1999). It seems a reason-able assumption for running and cycle ergometry (atleast within the sorts of bounds discussed at variousplaces above), but it changes substantially with speedin swimming and cycling on the road or track. If the

form of such dependency was known, it would needto be explicitly introduced into the model. While thiscould be done, the result would be cumbersome.

9. CP and AWC are constants, independent of P (and/orof t). The constancy (or otherwise) of these twoparameters of the model introduces issues of inter-pretation. For example both CP and AWC are notconstant over time with respect to changes in exer-cising habits (training in particular is discussed inInterventions and their impact on the model above)and, in any event, are nevertheless probably subjectto the usual intra-subject variations (other factorsbeing equal) typically inherent in any biologicalorganism. In another sense, Bishop et al. (1998) haveshown that the estimates of CP and AWC differ whendiffering sets of P are used in the testing sessions.While in some sense this non-constancy might seemdisturbing, relative stability of CP and AWC doesseem reasonable over a short term and indeed with-out such an assumption not much progress can bemade.

Towards a more realistic critical power model

Bearing in mind the many deficiencies in the assump-tions as described above, the CP model can neverthelessbe improved without a great deal of added complication.In several instances this has been done.

For example, Wilkie (1980) focussed on assumption5, introducing a correction factor for oxygen uptakekinetics based on a single exponential with time constants and without delay. Wilkie’s version of Eq. 1a nowbecomes:

W ¼ aþ bt � bs(1� e�t=s), ð2aÞ

where a and b correspond to AWC and CP, respectively,and the third term represents the amount of energy re-leased from anaerobic sources before the attainment ofan aerobic steady state at CP. This factor is the degreeby which AWC underestimates the ‘‘true’’ anaerobiccapacity and makes up for the difference in energy termsbetween the assumed infinitely fast (square wave) VO2

kinetics of the CP model and real VO2 kinetics with atime constant equal to s. Wilkie also presents a versionof Eq. 1c with a corresponding correction term, butderivations of corresponding equations 2a to 1b, 1d, 1eand 1f are mathematically not straightforward.

The bioenergetics of Wilkie’s correction can bedemonstrated by Fig. 2. Here the ‘‘real’’ anaerobiccapacity is not a as it would have been under the CPmodel, but a plus the approximately triangular areabelow the dotted line and above the curve (the correctionfactor in Eq. 2a).

Wilkie showed his correction to be appropriate fortimes to exhaustion between 50 s and 10 min, giving s avalue of about 10 s. Although 10 s is rather too fast(30 s might be more reasonable), this term does adjust

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appropriately for oxygen kinetics in exercise of fairlyshort duration. For exercise of longer duration, its effectis relatively lessened.

Wilkie’s correction can be modelled by a simplechange to the hydraulic model presented in Fig. 1. Allthat is required is to lower the vessel Ae and its con-necting tube R1 such that when An is full, the fluid levelsin An and Ae are in alignment (Fig. 3)

Following Conceptualisation as a hydraulic model,this model operates as follows. When T is opened tosome value P>0 the level in An immediately begins todrop, creating a differential in levels between An and Ae.This differential induces a flow through R1 directlyproportional to h, corresponding to a rise in VO2 aboverest. This flow in turn slows the rate at which the level inAn drops, slowing the rate at which VO2 began to rise. IfP £ CP, then h and VO2 will reach a steady state withthe level in An no lower than the connecting tube R1.That is, the height h is exactly sufficient to induce a flowthrough R1 exactly equalling the flow P through T.Mathematically, the flow through R1, and hence VO2,can be shown to be mono-exponential, whether or not asteady state is reached.

Of course if P>CP, h will increase beyond the depthof vessel Ae, the flow through R1 will have reached its

limit at CP, and An will continue to empty in order tosustain the flow P>CP. In the model context, exhaus-tion occurs when An is fully depleted as P can no longerbe sustained. Mathematically, this can be shown to beprecisely the Wilkie formulation of the CP model.

While other corrections may be feasible focussing onany one or more of the problems associated with theoriginal CP model, probably the most important from aphysiological point of view is the way in which AWCappears to not necessarily all be usable.

A feedback control system and the 3-parameter model

It is well recognised that the last 100 m of a longer race(say 5,000 m) cannot be run at the same speed as thatwhich could be achieved had the runner been sprinting100 m afresh. Taking this in conjunction with the workof Saltin and Karlsson (1971), there appears to be someform of control or feedback system at work, whichoperates to limit the maximum power output achievableinversely according to how fresh the runner is or moreprecisely according to how much energy store remains.

Therefore we can retain assumption 2 and modifyassumption 6 such that CP<P<Pmax and introduce asimple linear feedback control system in accordancewith the amount of the An store remaining.

This means that the maximum achievable power atany time becomes a function of the amount of An, spe-cifically:

CP + An(Pmax � CP)=AWC. ð3Þ

If the individual is fully rested and replete,An = AWC and Pmax can be achieved. On the contraryif An is empty then only CP can be achieved, and asbefore the notion of exhaustion in vacuous unlessP>CP. This assumption of a linear control feedbackforms the basis of the 3-parameter CP model and is fullydescribed by Morton (1996).

It is immediately apparent that this model avoids, orat least mitigates, the problems associated withassumptions 3, 4 and 6. Research at the time (Gaesseret al. 1995; Morton et al. 1997) and subsequently forexample Bull et al. (2000) and Hill et al. (2003) suggestthat AWC estimates are higher and CP estimates lowerthan those obtained from the 2-parameter model, andthat endurance at CP (while not infinite) is significantlylonger than previously (assumption 7).

Diagrammatically, the 3-parameter model can berepresented by the hydraulic model of Fig. 1, but withthe addition of an appropriate linkage between the fluidlevel in An and the tap T in accordance with Eq. 3above.

Further developments

The most obvious development of the above discussionis to combine Wilkie’s correction with the feedbackFig. 3 Wilkie’s correction to the CP hydraulic model

Fig. 2 Wilkie’s correction to the CP model

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assumption of the 3-parameter CP model. The details ofthis combination have not previously been published, soit can be achieved quite simply as follows.

di Prampero (1999) gives a version of Wilkie’s cor-rection as his Eq. 3 on p. 163, which since W = Pt canbe adapted to

An ¼ ðP � CPÞt þ CPsð1� e�t=sÞ

and when combined with Eq. 3 above produces

CPþ ðPmax � CPÞðAWC� ðP � CPÞt � CPsð1� e�t=sÞÞ=AWC ¼ P

in the manner described by Morton (1996), where t is thetime to exhaustion.

This equation is transcendental (t occurs both linearlyand in an exponent), so cannot be solved explicitly for t.Numerical methods need to be employed. However,since Wilkie asserts s is about 10 s, and even if it is amore realistic 30 s, then provided t is much greater thans (by a factor of say 4, which it usually is), the exponentcan be neglected to a good order of agreement.

Neglecting the exponential term and solving for tproduces the following equation for time to exhaustion:

t¼AWC=(P �CP)þAWC=(CP�Pmax)�CPs(P �CP).

Here we note that if VO2 kinetics is assumed infinitelyfast (a square wave), then s=0 and Eq. 4 reverts exactlyto the 3-parameter model equation. In addition, we notethat Wilkie’s correction depends, as indeed it should, onthe feedback property of the extent by which P exceedsCP, i.e. more so if P greatly exceeds CP, precisely when tis short.

A different development of the original CP model,adapted for running, is due to Keller (1973), who posesthe question of an optimal running velocity in order tocover any given distance (from 50 to 10,000 m andcorresponding time 6 s to 30 min). This treatise is quitemathematical, requiring the calculus of variations, butits results are straightforward and logical. Keller makesno adjustments to the original CP model, but does allowfor the work required to accelerate the body in the earlystages of a race (a point mentioned earlier). Interest-ingly, the optimal solution is in two parts. For races lessthan about 291 m, an all-out sprint is optimal. Forlonger races an all-out sprint is required only up to acertain velocity (lower for longer distances), followed byconstant maintenance of this velocity almost to the veryend of the race (effectively to when An is emptied), fin-ishing with the briefest ‘‘collapse’’ over the finish linewhere CP and residual momentum are the only drivingforces.

The general notion that constant pace throughout arace represents the duration-minimising strategy (asdistinct from a winning one) is widely accepted. Whilethis appears to be but one among countless such optimalstrategies for the 2-parameter CP model (Performance

implications), it appears to be sub-optimal for the 3-parameter model. It has been shown (Morton 2004) thatunder the linear feedback system assumed for the 3-parameter model, an all-out effort is the duration-minimising strategy. Consequently it can be conjecturedthat for any linear feedback of the type in which themaximum power attainable is linearly dependant on theamount of fuel source remaining, an all-out strategywould be optimal. Observers of human performance andathletes themselves will surely be sceptical. Thus al-though the 3-parameter CP model has certain advanta-ges over the original CP model, it still has propertiesand/or assumptions open to further questioning.

More detailed hydraulic models

Assumption 1, as indicated in Towards a more realisticcritical power model, is one which though appropriatefrom a broad biochemical pathway point of view, ismetabolically inadequate. There should be a method ofincorporating the aerobic, alactic anaerobic and lacticanaerobic pathways into a model scheme. This section isdevoted to showing how this has been done, elucidatingsome of the properties of these models and consideringtheir further development.

Margaria’s seminal ideas

It was Margaria (1976) who first proposed a 3-compo-nent hydraulic model of human bioenergetics. It tookthe following form (Fig. 4).

The fluid in vessel P (representing the alactic anaer-obic phosphagens) is directly connected with the outsidethrough the tap T, which regulates the flow (net energyexpenditure above rest). At rest, with T closed, the upperlevel of fluid in P is the same as in the communicatingvessel O (representing the oxidative or aerobic source).The vessel O is of infinite capacity and is connectedthrough tube R1. The second communicating vessel L(representing the lactic or glycolytic anaerobic source) isof finite capacity, with upper level the same as the bot-tom level of vessel O, apart from a very narrow exten-sion tube, B. The fluid in B, corresponding to the restingblood lactic acid, is of very small volume relative to Land does not contribute to the flows to any significantdegree. It is, however, measurable and corresponds tothe so-called ‘early lactate’ which is produced at theonset of even mild aerobic exercise. L is connected to Pthrough a wider, but one-way tube R2, and P is con-nected back to L by another, but very much smaller one-way tube, R3.

If T is partly opened, corresponding to a workloadW, the level in P falls, inducing a flow through R1

(oxygen consumption, VO2) in accordance with thedifference in levels, h, between the two vessels. Thisinduced flow slows the rate at which the level in P falls

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and, provided W is not too large, equilibrium will bereached at a level above the outlet R1. The level in P isbelow the resting level, and fluid flows continuouslyfrom O to P and out through T. If the equilibriumlevel is exactly at the level of R1, then the oxidativemechanism is at its maximum, denoted VO2max. Oncethe equilibrium is established, the only energy mecha-nism contributing is the oxidative; the exercise ispurely aerobic and, in theory, could continue indefi-nitely. Prior to equilibrium of course, P has contrib-uted some of its supply, and the empty volume in Pabove the equilibrium level is known as the alacticoxygen debt.

If T is now closed, i.e. exercise ceases, P will begin torefill through R1, but at a slower and slower rate as thelevel in P returns to normal. When it does so, the flow inR1 ceases and the subject is said to have repaid hisoxygen debt during this recovery period.

If T had been widely opened (severe exercise), theinitial situation would be as described above, but thelevel in P would fall below R1. This happens after about50% of the fluid in P has been utilised, and the subject issaid to have crossed his anaerobic threshold. As soon asthis happens, two things occur: the flow in R1 hasreached and continues at its maximum determined onlyby the height of the vessel O and a flow through R2 isinduced. This flow is also in accordance with the dif-ference in levels between vessels L and P (the level in Llagging behind the level in P). The flow through R2 willslow the fall of level in P, but since the flow through R1

is insufficient and the capacity in L is limited, the levelsin both L and P will continue to fall. If exercise is pro-longed, L and P will be emptied and the subject will beexhausted.

If T is closed at or before exhaustion, P will again berefilled. Initially it will be filled both through R1 at themaximal rate and through R2 until the lag in levels

between L and P has been eliminated. This latter flowthrough R2 is a delayed lactic acid formation, which hasbeen observed to occur for a short period after cessationof exercise (di Prampero et al. 1973). Once the levelshave been equated, P will fill through R1, initially at themaximal rate and thereafter at a progressively slowerrate as described previously. L will be refilled from Pthrough R3 at a rate in accordance with the difference inlevels between the two. As R3 is so small, the level in Lwill lag behind the level in P and the repayment of this,the lactic oxygen debt, is very slow. Finally both P and Lare refilled and the subject has fully recovered.

Although Margaria’s model was not quantified, suf-ficient information was available to do so. No mathe-matical solution was offered therefore, though apurported graphical solution (Margaria’s Figs. 1.27 and1.28) was presented. A detailed examination of, andmathematical solution to, Margaria’s model has beenundertaken from several perspectives (Morton 1984,1985a, b, 1986a, b). This reveals that the actual solutiondiffers from both Margaria’s ‘‘sketch solution’’ and fromwhat actually happens under experimental conditions.

Thus although Margaria’s model clearly represents asignificant step forward, the observed discrepancies aresuch that it must be rejected in its original form andreformulated in an attempt to alleviate the discrepancies.This has been done by Morton (1986a, b) and has led todevelopment of the 3-component M–M model.

The 3-component hydraulic model

It is clear from Fig. 4 that many different configurationsof the hydraulic model are possible depending on thevarious heights of the tops and/or bottoms of the vessels,relative locations of the connecting tubes, etc. Morton(1986a, b) argues that only 4 of 16 are consistent with

Fig. 4 Margaria’s hydraulicbioenergetic model

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known physiological facts and later (Morton 1990) re-duces the number to one unique configuration shownbelow in Fig. 5.

The operation of this configuration can be deducedfrom Fig. 4 and the description in Margaria’s seminalideas, with appropriate amendments, and is describedfully by Morton (1986a, b, 1990). Nevertheless it isworth noting some of its salient features.

The top of vessel AnL, apart from the very narrowtube B, is below the level of fluid in vessel O, but abovethe inlet tube R1 (as is the bottom of vessel AnL). Thismeans that the anaerobic threshold (as signalled by thecommencement of lactic acid production represented bya flow through R2) does not commence until VO2

reaches some value above rest and clearly below VO2max,say at 40% of VO2max.

This commencement of flow through R2 changes thedynamics of the flow through R1, i.e. changes VO2

dynamics at this point, as described in some detail byCerretelli and di Prampero (1987). This ‘‘elbow’’ can bededuced from a breath-by-breath VO2 plot and is thecommencement of the VO2 ‘‘slow component’’ (Gaesserand Poole 1996). In other words, the crossing of theanaerobic threshold can be modelled using the hydraulicmodel (Morton and Gass 1987). What happens is that atthis delay point VO2 dynamics change from mono-exponential to bi-exponential as has since been empiri-cally verified by Barstow and Mole (1991). Thehydraulic model formulation differs from the Barstowand Mole formulation only in that the former proposesa smooth transition across the threshold, whereas thelatter does not.

A steady state may or may not be reached thereafterdepending on whether W is so large as to drive the levelin AnA below R2. If it does then AnL will ultimatelyempty and exhaustion is sure to occur. The maximalaerobic steady state is clearly the point where h is exactlyopposite to R2 and with AnL empty, the flow through R1

exactly equals P. This is the true maximal sustainable

aerobic power and is clearly less than VO2 max. Con-sidering Fig. 5, this may be at 85% of VO2max, say.

Of course if T is closed, the system refills, initiallythrough R1 and R2, then through R1 and R3. Thereversal of the flow from R2 to R3 suggests that recoveryVO2 kinetics has two bi-exponential phases, a phenom-enon which does not seem to have been examined indetail. Since R3 is relatively so narrow, AnL takes aconsiderable time to refill, during the early stages ofwhich AnA is almost completely filled. Thus the modelpredicts that for some long time after cessation ofexercise, VO2 would not in fact return fully to the restinglevel, but remain just above it. Indeed this phenomenonwas described by Hill in the 1920s and is further re-viewed in di Prampero (1981). This phenomenon is akinto the well-known post-prandial minor elevation in VO2.

It is a quite straightforward exercise to programme acomputer with the equations of this hydraulic model.For any specified time course of W, running the pro-gramme would simulate VO2, phosphagen depletion,lactic acid production (or glycogen depletion) and thereplenishment thereof if W reduces significantly or takesthe value zero. Given the right scaling constants andoffsets for resting values, quite realistic smooth ‘‘data’’can be generated.

What the model of course does not incorporate is anycriterion for exhaustion, other than AnL emptying whenVO2 exceeds MAP. There is no feedback system of thetype described in A feedback control system and the3-parameter model.

Feedback control systems, maximal and optimalperformance

Since the 3-component hydraulic model has two anaer-obic vessels which partially empty during exercise, thereare two fuel starvation feedback systems possible.Morton (1990) has investigated both. A phosphagen-based limitation does not lead to the right sort of time

Fig. 5 The generalised M–Mhydraulic model

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course for all-out power decline such as in Wingate testor prolonged sprint, despite the observation that chan-ges in these muscle metabolites do tend to correlate withpower decline. Much better correspondence occurs if themaximum attainable power is assumed directly propor-tional to the remaining glycogen store. The propertiesand predictions of the M–M model under this assump-tion are examined in detail by Morton (1990).

All-out power is constant at its maximum (governedin the model by the diameter of tap T when open) foronly a few seconds. This is the time taken for the level inAnA to drop to the top level of vessel AnL, during whichof course there is no flow through R2. Then only doesthe level in AnL begin to drop and the feedback beginsto operate. Using published data for fit males, all-outpower declines after about 6 s at 972 W to very lowlevels after about 2 min. Plenty of data are available tocompare this prediction for efforts of 30 s or so, but datalasting 2 min or more appear to be sparse. Very wellmotivated subjects would be needed to provide it.

Endurance at constant power declines from ad infi-nitum at 208 W (about 89% VO2 max) to 6 s at 972 Win a ‘‘hyperbolic’’ shaped curve (actually it is the solu-tion to a ratio between two bi-exponentials). At VO2max

an endurance of around 9 min is predicted. The modelalso allows investigation of endurance at ramp tests ofvarious incremental rates. For rates between 30 and60 W/min, endurance reduces from 14 to 9 min, and theanaerobic threshold is reached in times between 6 and3 min. Interestingly, in these ramp tests the model pre-dicts that power output at exhaustion (peak power) in-creases with increasing ramp slope, but terminal oxygenconsumption (VO2 peak) is unaffected.

Contrarily, the M–M model has not yet been inves-tigated to determine any optimal strategy for completinga given amount of work in minimum time (or running afixed distance in minimum time). It is conjectured (as aninference from the work by Morton (2004) reported inFurther developments) that this feedback system couldlead to implicating an all-out strategy. However, itshould be noted that the form of the original glycogen-based limitation [Eq. 7 in Morton (1990)] is slightlydifferent from the form of feedback Eq. 3 above, and sothis implication is by no means assured.

Applications and developments of the 3-componenthydraulic model

One obvious application of the M–M model is its use asa teaching and learning tool to give a macroscopicoverview of the operation of the human bioenergeticsystem during exercise and recovery. At the simplestlevel the concepts only can be used, since understandingthe flows of a fluid between interconnecting vessels isquite intuitive. Indeed I believe this to be the motivationbehind Margaria’s original 3-component model. Theanalogies between the hydraulic system’s attributes(capacities, flow rates, height differentials, etc.) and the

corresponding human bioenergetic attributes (energystores, power, etc.) are also most intuitive. While notabsolutely realistic in its properties, the M–M modelpresents a significant advance on the earlier describedCP model.

At an intermediate level, graphical illustrations ofchanges in various components of the system (forexample of the flow through R1, analogous to VO2) canprovide more insight. These can be enhanced by theselection of corresponding real data sets to verify theaccuracy (at least to a reasonable degree) of the work-ings of the model.

At the highest level, those with appropriate knowl-edge of ordinary differential equations can gain an evendeeper comprehension by working through the mathe-matics to obtain solutions to the system under a varietyof conditions.

Taking this application one step further and into theresearch context, the model provides a means ofanswering a wide variety of ‘‘what if’’ questions. Theresulting model predictions can then be checked againstappropriate existing data or data from suitably designedexperiments. Just how many of the M–M model pre-dictions conform to reality or not remains to be seen.For example, as already observed, the bi-exponentialnature of the model prediction for VO2 for constantpower exercise above the anaerobic threshold is nowclearly established (Barstow and Mole 1991).

For ramp exercise, the M–M model predicts a fastexponential trend towards linearity of VO2, but only upuntil the AT, followed by a second exponential trendtowards lesser sloped linearity until exhaustion.Unequivocal verification of this has not yet been estab-lished, for available data seems to be contradictory(Morton and Bocquet 2003).

As a further example, glycogen loading obviouslyenlarges the capacity of vessel AnL, but whether this isaccomplished by increasing the cross-sectional area and/or the depth (either by raising the top or lowering thebottom or both) is unclear. Changes in VO2 dynamicsforecast by the model as a result of such structuralchange may provide clues.

Behncke (1993) has incorporated the M–M model asone submodel in a larger mathematical model for theforce and energetics of competitive running. The othersubmodels include the biomechanics of running and anoptimisation determination. Behncke modifies the M–Mmodel first by regarding vessel O as finite, though verylarge, because glycogen depletion and fatigue are seriousproblems for long-distance events like 10 km or mara-thon runs. As a second modification, vessel O is con-sidered as the union of two vessels, the smaller anduppermost having lesser cross-sectional area, althoughboth are large. This is done because even though it maybe reasonable to consider separate compartments forglycogen and lipids, it cannot be done becausethese processes are not additive as far as their power isconcerned. They run in parallel using the same substrate(oxygen). As a third modification Behncke regards

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maximal available power at any time to be a function ofthe level of fluid in vessel AnA, but only after theanaerobic threshold is reached. Specifically, maximumavailable power declines linearly from Pmax to zero asAnA empties beyond the anaerobic threshold.

However the time course for changes in maximumavailable power starting from 0.85Pmax (Behncke’sFig. 2, p. 865) does not entirely conform to the experi-mental data described in Morton (1990).

The optimisation submodel is mathematically com-plex, but in essence produces a strategy very similar tothat of Keller (1973) as described in Performanceimplications. Constant power throughout most of therace appears to negate the conjecture of all-out effortsuggested above. Such matters are clearly in need of amore careful general solution.

Behncke (1997) then applies this model to worldrunning records of the time in order to determine keyparameters. The values so obtained were in goodagreement with those published in the literature andthose obtained by other means. The model is also ap-plied to determine the velocity profile for a 100 m raceand used to compare the effects of adverse wind andaltitude. In as much as there are data available, Behnckeevidences good agreement.

Other bioenergetic models

A model based on the first law of thermodynamics

Ward-Smith (1985a) proposed a theory of runningbased on the first law of thermodynamics. In confor-mity with this law, the energy released by chemicalconversion, C, must ultimately be accounted for eitherin the form of physical work, W, or degraded intoheat, H. The overall energy balance equation istherefore

C ¼ H þ W

and by differentiation with respect to time, a powerequation can be derived. Ward-Smith traces the originsof his modelling back to Lloyd (1967) who proposed Cto consist of two terms: one expressing the energyavailable from a store and the other expressing the en-ergy available at a steady state. The similarity toassumptions 1–3 of the CP model are immediatelyapparent.

Of interest here are Ward-Smith’s assumptions aboutthe derivative dC/dt, regarding the two terms as anaer-obic and aerobic, respectively, based on Margaria’s(1976) earlier work. Specifically

dC=dt ¼ Pmaxe�kt þ Rð1� e�ktÞ;

where Pmax is the maximal anaerobic power, k a rateparameter governing energy release and R the steady-state maximum aerobic power (the critical power). Thereaerobic power, Paer, given by the expression

R(1� e�kt)

is the same as that previously referred to as Wilkie’scorrection, and the robust nature of this expression hasalready been noted. Ward-Smith does argue (p. 341) thatthe anaerobic energy store has two components, alacticand glycolytic, but nevertheless combines them as one. Aseparation is investigated in more recent papers (Ward-Smith and Radford 2000a, b).

Where a departure from most earlier work is evidentis in the expression for anaerobic power, Pan, given by

Pmaxe�kt

rather than by the difference between dC/dt (whichcommonly is constant, but is by no means necessarilyso), as determined by the individual exerting the effort,and Paer. What this means is that under the Ward-Smithassumption dC/dt always begins at Pmax, decliningexponentially thereafter to an asymptotic value of R andquite independent of the power requirement of theexercise, constant or otherwise. This of course is notnecessarily so, for the athlete may choose to start atsome value higher than R, but less than Pmax. Further-more if the expression for Paer is adopted, then Pan mustbe a function of both the power requirement of theexercise and of t, not just of t alone. Closer investigation,however, reveals that the departure is only an illusion,for Ward-Smith’s equations are mathematically equiv-alent to those of Lloyd.

The feedback nature in the Ward-Smith model is thehypothesis of Margaria that prior to the activation ofthe glycolytic mechanism, the rate of energy release dueto oxidation is dependent on the amount of splitphosphagen. Overall, the bioenergetic components ofthe Ward-Smith model are therefore a compilation ofprior model themes.

Where there is originality is the manner in which thebiomechanical equations of running are linked to thebioenergetics, which is the incorporation of runningresistance and the thermodynamic equation for increaseof thermal energy. When all is said and done, the Ward-Smith model predictions (given appropriate selection ofparameter values) conform very well with actual runningdata from real athletes. Indeed the robust nature ofWard-Smith’s model has been further demonstrated in anumber of successful applications (Ward-Smith 1985a,b, 1999a, b; Ward-Smith and Radford 2000a, b).

Other bioenergetic models

Another detailed and often quoted model is due toPeronnet and Thibault (1989) which describes the aver-age power output sustained over time t. It is based on a2-component (aerobic and anaerobic) system, closelyrelated to the models described above, and based on thework of Lloyd (1966) and Ward-Smith (1985a, b). Theaerobic component is again a single exponential analo-gous to Wilkie’s correction as described previously.

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However, Peronnet and Thibault argue that themaximal aerobic power can only be sustained for a finitelength of time (about 7 min) and that the fraction ofMAP that can be sustained for longer times than thisdeclines linearly with the natural logarithm of t, ln(t).The amplitude of the aerobic component is thereforereduced accordingly for times in excess of 7 min.

In the case of the anaerobic component, Peronnetand Thibault adopt essentially the same assumption asWard-Smith that the available anaerobic componentcontribution declines exponentially with t (again withthe same time constant as the aerobic component) forany power output, provided it is sufficient to lead toexhaustion. Peronnet and Thibault recognise that not allthe anaerobic store is necessarily available for races ofany duration. They argue that all is available for races ofup to 7 min duration, but that for longer races theavailability decreases with ln(t).

This last assumption represents a significant contrastto one important property of the 3-parameter CP model,where availability of AWC increases more and morewith exercise lasting longer and longer. Nevertheless,Peronnet and Thibault evidence very good fits of theirmodel to men’s world records of the time from 60 m tothe marathon.

di Prampero et al. (1993) published a model to predictrunning performances rather similar to that of Peronnetand Thibault, showing good agreement between actualand calculated best times over distances from 800 to5,000 m, not only for their own data, but also forindependent data of Lacour et al. (1990). A summary ofthis and related studies can be found in di Prampero(2003).

Olds et al. (1993) incorporate a bioenergetic compo-nent into their modelling of road cycling performance.This component consists of a fixed anaerobic contribu-tion, referred to as the maximal accumulated oxygendeficit (their equivalent to AWC), together with a vari-able aerobic contribution. In the manner of Peronnetand Thibault, this aerobic contribution increases mono-exponentially at exercise onset and reaches a steady stateat some fraction of VO2max depending on the durationof exercise. The only difference is that whereas Peronnetand Thibault regard this fraction as decreasing linearlywith ln(t), Olds et al. regard it as unity for times£ 10 min, equal to the individual anaerobic thresholdfor times ‡30 min, and as declining linearly with t be-tween the two. Empirically derived field and laboratorydata were used to assess the accuracy of the model,yielding mean (observed�predicted) differences of 1.3%,very good, but marginally worse that the 0.73% ofPeronnet and Thibault.

Further application of this model investigated theimpact of a number of biophysical, environmental andphysiological variables on cycling performance (Oldset al. 1995). The most important physiological factorswere found to beVO2max, fractional utilisation ofVO2max,mechanical efficiency and projected frontal area. Anumber of practical issues, such as the effect of drafting

and the normalisation of performances under differentconditions, were also considered. Maronski (1994) alsoconsiders an optimising problem in cycling, and a feed-back mechanism limits force development according tothe cyclist’s velocity. In amore recent review of modellinghuman locomotion, Olds (2001) makes a further modifi-cation to the modelling of the anaerobic contribution, byasserting (in the manner of Ward-Smith) that the oxygendeficit may not all be utilised by the end of exercise. Inother words, it is regarded as declining exponentially witha fixed time constant, once again independent of thepower requirement of the exercise.

In the more complex (at least for physiologists) ap-plied mathematical literature can also be found somemodels appropriate to mention. These can be viewedalongside the Keller (1973) model, as most seek to ob-tain optimal strategies in some way. Like Keller, Cooper(1990) adopts no feedback system to a model for opti-mising wheelchair propulsion. On the contrary, Mathis(1989) adapts Keller’s model by introducing a feedbackdue to a fatigue effect in such a way that the propulsiveforce available to the runner declines linearly with theoxygen deficit of the runner. The similarity with thefeedback mechanism of the 3-parameter critical powermodel is immediately apparent. Of interest is that thisdependency leads to an optimal solution in which theconstant velocity segment is run at a slightly lower paceand which includes a period of perceptible accelerationover the last 10–40 s of the race. The slower pace is toensure that the oxygen deficit does not become too high,and the stronger the dependency the lower the velocityand the shorter the duration of the final sprint. As far asI can determine, this is the only application where themathematically optimal strategy includes a final sprint.Readers will no doubt take note that this is an extremelycommon strategy in real races. Woodside (1991) adoptsa similar approach, using a feedback fatigue constantwhich relates the rate of energy loss due to fatigue to theenergy already spent in running. This analysis does notresult in a finishing sprint.

Both Tibshirani (1997) and Mureika (1997) followMathis’ example in that the propulsive force duringsprinting declines linearly, but with t rather than withthe oxygen deficit. As it turns out, this difference is oflittle consequence, evidencing just as good fits to data asdoes Mathis.

Harman (2002) modifies the Ward-Smith and Rad-ford formulation to include three anaerobic compo-nents: ATP, phosphagen and glycolysis. Eachcomponent is regarded as bi-exponential in nature withits own rise and fall rate parameters. In this model thelimited ATP supply is largely depleted within the firstfew seconds, albeit that in reality ATP concentration isessentially constant (though maybe low) throughout theexercise intensity range because of the equilibrium con-stant of the creatine kinase reaction. In this model allthree anaerobic mechanisms have peaked within 10 s.Nevertheless, as with many of the models describedpreviously, very good fits to real data are evidenced.

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Conclusions and further considerations

I have traced the development of whole-body bioener-getic models over the last 50 years, from the early simple2-component ideas of Monod and Scherrer (1965) andLloyd (1966) and others; through Margaria (1976) andMorton’s (1986a, b) 3-component extensions; to themore complex recent models of Behncke (1993, 1997),Ward-Smith (1985a, b) and Harman (2002). It is usefulnow to view the current situation in summary with ref-erence back to the ten assumptions of the original CPmodel (Origins, assumptions and development andQuestioning the assumptions).

It would seem that significant progress has been madeon all counts, simply by an evolutionary development ofexisting ideas. Indeed such is the nature of science and theadvancement of knowledge. Nevertheless it appears thatthere is at present no single published model whichincorporates all the developments. Herein lies a challenge.

One issue of note where it appears there has been nodevelopment is with respect to assumption 6 (and to acertain extent 7 also). Why should the bioenergeticmodels available so far and surveyed herein apply onlyto/or have been tested only against) exercises of amaximal nature or which ultimately lead to exhaustion?Why should they not equally be applied to and testedagainst data from submaximal work which does notnecessarily lead to exhaustion. Herein lies a secondchallenge.

A second issue of note is that a fundamental char-acteristic of the bioenergetic process is that it is demanddriven by a single element, the power output required. Itis supply limited obviously, though this may not neces-sarily be the only feedback mechanism. However, it isclear that all amplitude parameters must depend on thedemand (if power is high, VO2 will be high, etc.) and allrate parameters depend on the structure (i.e. capacities,maximal rates, etc.) of the system. No single model sofar presented is of this character, though the work ofBehncke comes closest. Herein lies a third challenge.

Finally, the single most remarkable conclusion is onthe one hand such a plethora of models fit the real worldso well, yet there is so much more to discover.

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