The curvilinear relationship between work rate and performance time for a sporting event involving whole-body exercise was first noted by Hill (^{18} ) in 1925. However, a formal mathematical framework for the fatigue of synergistic muscle groups was not developed until Monod and Scherrer (^{30} ) presented the critical power (CP) model in 1965. The equation is hyperbolic and consists of two parameters: CP and the W ′ (^{22} ):

In this model, P is equal to power output, and t is equal to time to exhaustion at that power output. For sports such as swimming or running, P and CP can be substituted with speed and critical speed (CS), respectively, and the W ′ can be substituted with distance.

The CP represents an asymptote, a power output that could theoretically be maintained indefinitely on the basis of principally “aerobic” metabolism. The CP is unlimited in capacity but limited in rate (^{22} ). It is an important predictor of endurance exercise performance (^{34} ). In contrast, the W ′ represents a finite work capacity (J) available to the athlete once he or she attempts a power output above CP, i.e., it is theoretically unlimited in rate but limited in capacity (^{22} ). The CP equation also implies that the W ′ (power–time integral > CP) remains constant regardless of the rate of its discharge.

The W ′ is of substantial importance to athletic performance because complete depletion of the W ′ results in the inability to perform supra-CP exercise (for review, see Jones et al. [^{22} ] and Vanhatalo et al. [^{37} ]). Knowledge of both the CP and W ′ can assist in performance optimization (^{14,22} ), particularly in situations where the athlete is required to make surges in power output above CP. However, it has been difficult to apply the power–duration relationship in real time because there has traditionally been no way of tracking power output and thus W ′ expenditure during competition. This has changed with the advent of the on-bicycle power meter (e.g., SRM, Colorado Springs, CO; Saris PowerTap, Saris, Madison, WI [^{3,17} ]), which could theoretically permit dynamic modeling of W ′ utilization during an exercise task given an appropriate mathematical framework.

Despite the ease with which the CP and W ′ can be calculated, the precise physiological determinants of the W ′ remain uncertain (^{12,22,38} ). The CP defines the boundary between the heavy and severe exercise intensity domains (^{7,19,20,22,32,36} ). Interestingly, the slow component of pulmonary O_{2} uptake (V˙O_{2} ) in the severe domain has been demonstrated to be related to the W ′ during both constant work rate and “all-out” exercise (^{7,13,23,38} ). Although any intrinsic relationship between the W ′ and V˙O_{2} kinetics is at odds with the traditional interpretation of the W ′ as a fixed “anaerobic” work capacity (^{29,30} ), Vanhatalo et al. (^{36} ) have recently reported that hyperoxia both reduces the W ′ and increases the CP during knee extensor exercise. This suggests that the CP and W ′ are interrelated and that the nature of the W ′ might need to be reconsidered (^{7} ).

The aforementioned studies have only examined “all-out” exercise or constant work rate exercise in the severe domain. However, many endurance athletic competitions require frequent changes in power, with surges above CP and periods of recovery below it. Many studies indicate a correlation between maximal V˙O_{2} (V˙O_{2max} ) and the ability to repeat sprint exercise (^{1,4,11} ) and between the primary (phase II) time constant of V˙O_{2} kinetics and the ability to repeat sprint exercise (^{10} ). However, despite the probability that the ability to perform and sustain intermittent severe exercise is related to the charge/discharge state of the W ′, these studies have not addressed a possible dynamic temporal relationship between V˙O_{2} and the W ′.

The primary purpose of the present investigation was to develop a dynamic model that tracks the discharge and recharge of the W ′ during severe intermittent exercise. We also investigated the possibility of a link between V˙O_{2} kinetics and the discharge of the W ′. Finally, we wished to consider whether such a model might be useful in the analysis of real-world bicycle race power data.

MATHEMATICAL FRAMEWORK
Morton and Billat (^{31} ) presented a novel model that permitted the application of the CP model to intermittent exercise:

where t is equal to total endurance time, P_{w} and P_{r} are equal to the work and rest interval power, and T_{w} and T_{r} are equal to the work and rest interval time. This model was successfully applied to intermittent cycling exercise by Chidnok et al. (^{8} ). However, although this model makes assumptions that are mathematically plausible (i.e., linear kinetics of W ′ discharge and recharge), it has been recently reported that the W ′ may actually be reconstituted in a curvilinear manner, with a calculated t _{1/2} of approximately 234 s (assuming exponential recovery, time constant = 336 s) for cycle ergometer exercise (^{12} ). This suggests the possibility of developing a simplified continuous function that would account for the depletion and reconstitution kinetics of the W ′ during intermittent exercise.

Assuming that 1) the expenditure of the W ′ begins the moment a subject exceeds CP, 2) the reconstitution of the W ′ begins the moment the subject falls below CP, and 3) the reconstitution of the W ′ follows a predictable exponential time course, it is possible to formulate an equation describing the balance of W ′ remaining at any given time during an exercise session (

) where some amount of W ′ was expended.

where W ′ equals the subject’s known W ′ as calculated from the two-parameter CP model,

is equal to the expended W ′, (t − u ) is equal to the time in seconds between segments of the exercise session that resulted in a depletion of W ′, and τ_{W} _{′} is the time constant of the reconstitution of the W ′. In other words, the amount of W ′ remaining at any time t is equal to the difference between the known W ′ and the total sum of the joules of the W ′ expended before time t in the exercise session, each joule of which is being recharged exponentially during recovery < CP (Fig. 1 ).

FIGURE 1: Model ofW′bal (dashed line) and power output (continuous line) for a representative subject in the S20 trial. The dotted and horizontal solid lines indicate the subject’s GET and CP, respectively. Hashmarks on the right axis indicate power outputs for the SS, SH, andSM recovery trials. In this example, the subject exercises in the severe domain for 60 s and recovers at 20 W, repeating the process until exhaustion.

METHODS
Protocol.
Full details of the experimental procedures are given in the companion article (^{8} ). Briefly, seven healthy males (mean ± SD: age = 26 ± 5 yr, height = 1.79 ± 0.06 m, body mass = 81 ± 6 kg) volunteered to participate in this study. The subjects were recreational athletes but were not highly trained. They were familiar with laboratory exercise testing procedures, having previously participated in studies using similar procedures in our laboratory. The study was approved by the University of Exeter Research Ethics Committee. After the experimental procedures, associated risks, and potential benefits of the study protocol had been explained to the subjects, they were required to give their written informed consent to participate. Subjects were instructed to arrive at the laboratory in a rested and fully hydrated state and at least 3 h postprandial. They were also asked to avoid strenuous exercise in the 24 h preceding each testing session and were asked to refrain from caffeine and alcohol for 3 h before each test. All tests were performed at the same time of day (±2 h) at sea level in an air-conditioned laboratory at 20°C. At least 48 h separated each test.

The gas exchange threshold (GET) and V˙O_{2max} were estimated for each subject from data collected on a Lode cycle ergometer (Lode Excalibur Sport, Groningen, The Netherlands) during a standard ramp incremental protocol (30 W·min^{−1} ). After an advance familiarization trial, the subject’s CP and W ′ were determined using a 3-min all-out test (^{35} ). The CP was determined as the mean power output during the final 30 s of the test, and the W ′ was estimated as the power–time integral above the CP. In subsequent visits, the subjects performed a constant work rate trial to exhaustion in the severe domain and four intermittent exercise trials to exhaustion. In each case, the intermittent exercise consisted of 60-s work intervals at the power output predicted to result in exhaustion in 6 min (P _{6} ; equation 1) + 50% of the difference between P _{6} and the athlete’s CP and 30-s recovery intervals at a predetermined intensity (Fig. 1 ). The recovery intervals were devised as follows:

20 W (S _{20} ).
Moderate-intensity recovery (S_{M} ) at a power output of 90% of the GET.
Heavy-intensity recovery (S_{H} ) at a power output of GET + 50% of the difference between the GET and CP.
Severe-intensity recovery (S_{S} ) at a power output equal to P _{6} − 50% of the difference between the CP and P _{6} .
During all tests, pulmonary gas exchange was measured breath by breath (Jaeger Oxycon Pro; Hoechberg, Germany) with subjects wearing a nose clip and breathing through a low dead space (90 mL), low-resistance (0.75 mm Hg·L^{−1} ·s^{−1} at 15 L·s^{−1} ) mouthpiece, and impeller turbine assembly (Jaeger Triple V). The analyzer was calibrated before each test with gases of known concentration, and the turbine volume transducer was calibrated using a 3-L syringe (Hans Rudolph, Shawnee, KS). V˙O_{2} , carbon dioxide output, and minute ventilation were calculated using standard formulae (^{2} ).

Analyses.
The work/time data from intermittent bouts 1–4 were fit to equation 3 by inputting the number of joules expended above CP each second. The time constant was varied by an iterative process until modeled

= 0 at the time of exhaustion. Derived time constants were then plotted against the difference between recovery power and CP (D _{CP} ).

The breath-by-breath V˙O_{2} data collected during each of the work bouts were processed to exclude errant breaths, and values lying >4 SDs from the local mean V˙O_{2} were removed. These data were then linearly interpolated to provide second-by-second data. V˙O_{2baseline} was defined as the mean V˙O_{2} measured during the final 90 s of unloaded cycling before the onset of the protocol, whereas the work interval V˙O_{2} was defined as the mean V˙O_{2} measured during the entire 60-s work interval. This was plotted against the modeled end

for each corresponding interval. Regression analysis was performed using computer software (GraphPad Prism; GraphPad Software, San Diego, CA). The relationship between τ_{W} _{′} and CP was assessed by linear regression. The relationship between τ_{W} _{′} and D _{CP} was assessed by both linear and nonlinear regression. Significance was accepted at the P < 0.05 level, and data are reported as mean ± SD.

RESULTS
The W ′, CP, and V˙O_{2max} for each of the subjects are reported in Table 1 .

TABLE 1: Physiological data for each subject and group mean ± SD.

τ_{W} _{′} was inversely correlated with CP in the S_{M} condition (r ^{2} = 0.64, P = 0.03). There was a strong trend toward a significant inverse correlation in the S _{20} and S_{H} conditions (r ^{2} = 0.53, P = 0.06 and r ^{2} = 0.48, P = 0.08), suggesting a relationship between τ_{W} _{′} and CP irrespective of sub-CP intensity domain. There was no correlation in the S_{S} condition (r ^{2} = 0.07, P = 0.56).

τ_{W} _{′} was inversely correlated with D _{CP} in the S _{20} , S_{M} , and S_{H} trials (r ^{2} = 0.67, P < 0.0001). There was no linear correlation when recovery power exceeded CP (S_{S} trial) (r ^{2} = 0.05, P = 0.55). Above CP, τ_{W} _{′} increased to nonphysiological values, indicating no net recharge of the W ′ and merely a slightly lower rate of depletion during the recovery interval. The τ_{W} _{′} -versus-D _{CP} data were best fit by an exponential regression of the form y = ae ^{(−} ^{kx} ^{)} + b , yielding a close correlation for the S _{20} , S_{M} , and S_{H} trials (r ^{2} = 0.77; SE: a = 86.11, k = 0.004, b = 61.8) (Fig. 2 ). The precise equation as determined by nonlinear regression (GraphPad Prism; GraphPad Software) would be written as follows:

FIGURE 2: Graphical depiction of time constant ofW′ reconstitution (τW′) as a function of DCP. Individual recovery levels are represented bya common symbol, where S20 = triangles, SM = circles, and SH = diamonds. Note that the overall relationship could also be well described by a bilinear fit.

The mean τ_{W} _{′} for the S _{20} trial was 377 ± 29 s. The majority of the W ′ repletion time constants clustered near 380 s during the S _{20} trial. There was greater variation in the S_{M} (452 ± 81 s) and S_{H} conditions (580 ± 105 s) (Table 2 ). The S_{S} condition yielded an average time constant of 7056 ± 11,969 s.

TABLE 2: Calculated time constants ofW′ repletion (τW′) for each recovery condition for each subject and group mean ± SD.

Modeled W ′ depletion was strongly related to the rise in V˙O_{2} above baseline during each successive interval in the severe domain (r ^{2} = 0.82–0.96, P < 0.0002–0.0049) (Figs. 3 A–C).

FIGURE 3: ModeledW′ expended versus increase in V˙O2 above CP during intermittent exercise for a representative subject (subject 2). Top panel: 20-W recovery (r2 = 0.91). Middle panel: moderate recovery (r2 = 0.87). Bottom panel: heavy recovery (r2 = 0.88).

Equation 3 was also used to analyze preexisting data collected from a well-trained cyclist participating in a mass start race to determine whether model-predicted depletion of the W ′ was coincident with athlete exhaustion. τ_{W} _{′} was estimated using equation 4, using the mean of all power values less than CP (75 W) as the recovery power for calculating D _{CP} . D _{CP} was held constant for the purposes of the simulation. The model demonstrated that the cyclist was forced to reduce power output below CP (227 W) as the calculated W ′ balance fell below 1.5 kJ (Fig. 4 ). After executing two major attacks during the race, the athlete was forced to retire after 55.4 min.

FIGURE 4: ModeledW′ expended (heavy solid line) and athlete power output (thin solid line). The athlete’s CP (227 W) is denoted by the dashed line. Peak power output was 409 W. Numbers indicate important points as race unfolds. 1: athlete establishes position in pack. 2: athlete has attacked but has severely depleted the W′, forcing recovery. 3: athlete attacks again. 4: athlete again depletes W′ and is forced to withdraw from race as lead pack escapes.

DISCUSSION
This is the first study to mathematically characterize the discharge and reconstitution kinetics of the W ′ during intermittent exercise over a range of recovery power outputs. The τ_{W} _{′} was negatively correlated with D _{CP} and was well fit by an exponential function for all power outputs below CP (Fig. 3 ). A likely explanation for the inverse correlation between τ_{W} _{′} and D _{CP} is the presence of a smaller “oxidative reserve” with increasing recovery power output. In other words, the smaller the difference between the V˙O_{2} required to maintain the recovery power and the V˙O_{2} at CP, the smaller the capacity to reconstitute the W ′. As expected, the modeled time constants became unreasonably large when the recovery interval power output exceeded CP, indicating that no recharge of the W ′ occurred within the 30 s of “recovery” time permitted.

The mean τ_{W} _{′} for the S _{20} trial is compatible with results presented previously (^{12} ). The τ_{W} _{′} seemed to cluster between 370 and 380 s during the S _{20} trial for most of the subjects, with one τ_{W} _{′} of 320 s for a subject (subject 4), who had the highest CP of the group. The greater variation in the time constants calculated in the S_{M} and S_{H} conditions and the observation that there was only a trend toward correlation between CP and τ_{W} _{′} in the S_{H} condition and no correlation in the S_{S} suggest that the process of W ′ repletion may become more complex with increasing recovery power. With this in mind, it is interesting to note that the relationship between τ_{W} _{′} and D _{CP} is better fit by an exponential than a linear regression, and it is also possible that the relationship is bilinear. This suggests that the reconstitution of the W ′ may be related to different physiological factors with increasing recovery power output. This would help explain the variability of the relationship between CP and τ_{W} _{′} in the different recovery conditions. Further investigations with respect to the mechanisms underpinning the W ′ should consider this possibility.

The correlation between the rise in V˙O_{2} during intermittent exercise and the calculated net discharge of the W ′ is most interesting, particularly in light of the fact that it is difficult to perform conventional V˙O_{2} modeling because of the short work and rest durations (Figs. 3 A–C). The progressive loss of efficiency noted with increasing repetitions is most likely representative of the V˙O_{2} slow component (V˙O_{2sc} ) that has been described for constant work rate exercise above GET (^{21,25,26} ). In this context, these data lend support to previous findings that link the W ′ to V˙O_{2} kinetics. Not only does it seem that the W ′ is related to the “size” of the severe domain (^{7,22,36,38} ), but also, its expenditure seems to correlate well with the temporal course of the rise in V˙O_{2} in the severe domain during intermittent exercise. Because theV˙O_{2sc} has been linked to the recruitment of Type II fibers and the development of fatigue (^{21} ), this may suggest the possibility that the W ′ is related to such recruitment.

It has been suggested that CP may differentiate exercise intensities that are principally limited by the availability of glycogen (below CP) from other mediators of fatigue (^{34} ). Indeed, time to exhaustion above CP is correlated with time to attain V˙O_{2max} (^{19} ). This may be explained by the observation that discharge of the W ′ is associated with a depletion of muscle phosphocreatine (PCr) stores (^{22,24} ). In turn, this may explain the progressive rise in V˙O_{2} as the fall in PCr predicts increased stimuli to mitochondrial respiration (^{33} ). It has been proposed (^{24,32,36} ) that the depletion of the W ′ may reflect the predictable rate of PCr degradation and/or increase in other metabolites toward some limiting value, which would coincide with exhaustion. However, data indicating a definitive causative relationship are lacking.

There is other evidence suggesting a relationship between [PCr] and the W ′ from biopsy studies (^{5,6} ). For example, restoration of [PCr] after a 30-s sprint was highly correlated with recovery of peak power output, 6- and 10-s maximal power output, and maximal pedal speed after 1.5 and 3 min of recovery. This suggests a much shorter time constant than that reported for W ′ reconstitution. However, a “plateau” in the recovery pattern has also been noted (^{6} ). Assuming exponential reconstitution, 30-s sprint power output recovered with a time constant of approximately 333 s (^{6} ), close to the 377-s average time constant calculated in this study for the S _{20} condition and closer still to the 336 s extrapolated from a previous investigation (^{12} ). It has been suggested that this “plateau” in 30-s sprint power recovery reflects fatigue of the fast-twitch fiber pool because of inherently slower PCr recovery kinetics (^{6} ). We may also come to this supposition independently in light of recent work linking the V˙O_{2sc} (and thus the slow component of PCr on-kinetics) with the recruitment of Type II muscle fibers (^{25–27} ). In addition, disproportionate perfusion of predominantly glycolytic (fast twitch) muscle regions has been reported when rats were exercised above CS (^{9} ). Because the CP construct seems highly conserved among vertebrates (^{9,15,28} ) and even arthropods (^{16} ), there is reason to believe that similar phenomena and mechanisms might be observed in humans.

Taken together, the above suggests that the W ′ may be primarily representative of the recruitment of (and relative fatigue state of) a separate “compartment” of the exercising muscle mass, i.e., the Type II fiber pool. The present model can therefore be rewritten with two components (i.e., in a bottom-up approach) to satisfy the following conditions:

Depletion of the W ′ begins the instant the subject exceeds CP, and some portion of the W ′ comes from two separate compartments that are representative of the Type I and Type II fiber pools, both of which are activated simultaneously.
The portions describing compartment I and compartment II have different time constants to reflect different rates of W ′ repletion during recovery.
The absolute contribution to the W ′ of the two compartments is different, either because of purely energetic considerations or because of the absolute size of the compartment providing the work. In either (or both) case(s), there must be a gain term in addition to the exponential term.
The sum of W ′ expended by I and II at exhaustion must equal the known W ′.
In the general case, we would write the new equation as follows:

where k _{1} and k _{2} are gain terms and τ _{1} and τ _{2} are the time constants for the two different compartments.

The benefit of using the simpler model tested in this work is that it can be calculated through noninvasive, easily performed tests such as the 3-min all-out test to determine CP and W ′ and the time constants derived using regression equation 3. If more detailed information is required, the athlete needs only to perform intermittent severe work to exhaustion at several different recovery power outputs such as in the present study to enable the calculation of a personalized regression model. Such a process may be more important in highly trained athletes: we noted an outlier in our data set (subject 4) with a high V˙O_{2max} (>5 L·min^{−1} ), whose time constant of W ′ repletion did not change from the S _{20} to the S_{M} condition. It is possible that highly aerobically fit individuals differ from recreationally active individuals in terms of W ′ recovery.

Practical applications.
Because this model accounts for both the expenditure and repletion of the W ′, it permits the possibility of intracompetition performance management. We retrospectively analyzed power meter records from a competitive amateur cyclist who collected the data during a road race (Fig. 4 ). Using a time constant calculated by the use of equation 4 (440 s), we found that the athlete was forced to reduce power output because of the perception of impending exhaustion any time the calculated W ′ balance approached 1.5 kJ. This simulation is limited in that the τ_{W} _{′} will, in fact, vary according to the instantaneous D _{CP} . Despite this limitation, the model performed adequately for the purposes of this simulation. However, future studies should investigate the use of a variable τ_{W} _{′} to take into account variable power output_{.} The present data do suggest a novel technological application: the equations could be programmed into an on-bike power monitoring device for cycling (e.g., SRM or PowerTap). This would permit the athlete knowledge of the real-time state of the W ′ and thus help inform important decisions, e.g., the optimal amount of drafting and recovery in advance of a sprint. Similarly, adapting the equation to CS and distance would allow it to be programmed into a wrist-worn GPS or accelerometer devices for use in running races.

CONCLUSIONS
The principal novel finding of this work is the development of a simplified, continuous equation that describes the dynamic state of the W ′ during intermittent exercise. This model may be of significant practical value to competitive athletes who are interested in managing and optimizing training and racing performances.

This research was not supported by external funding. W.C. was supported by a Ph.D. scholarship from the National Science and Technology Development Agency of the Royal Thai Government. P.F.S. has no conflicts of interest to report.

P.F.S. thanks Dr. David Clarke and Kevin Joubert for discussion of this work.

W.C. has no conflicts of interest to report. A.V. has no conflicts of interest to report. A.M.J. has no conflicts of interest to report.

The results of the present study do not constitute endorsement by the American College of Sports Medicine.

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