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The Curvature Constant Parameter of the Power-Duration Curve for Varied-Power Exercise


Medicine & Science in Sports & Exercise: August 2003 - Volume 35 - Issue 8 - pp 1413-1418

FUKUBA, Y., A. MIURA, M. ENDO, A. KAN, K. YANAGAWA, and B. J. WHIPP. The Curvature Constant Parameter of the Power-Duration Curve for Varied-Power Exercise. Med. Sci. Sports Exerc., Vol. 35, No. 8, pp. 1413–1418, 2003.

Introduction: The tolerable duration (t) for high-intensity cycle ergometry bears a hyperbolic relationship to the power output (P) with an asymptote termed the critical power (CP), and a curvature constant (W′) that is numerically equivalent to an amount of work that can be performed above CP. The physiological nature of W′ has received little consideration compared with CP, e.g., whether the total amount of work above CP remains constant when the power actually changes during the high-intensity task.

Purpose: The purpose of this study was to compare W′ derived from the standard estimation method, consisting of several different constant-P tests, and the total amount of work above CP during an exhausting exercise bout using a variable-P protocol.

Methods: Eleven healthy male subjects (age: 21–40 yr) volunteered to participate in this study. Each initially performed four-to-six high-intensity square-wave exercise bouts for estimation of CP [mean (SD); 213.3 (22.4) W] and W′ [12.68 (3.08) kJ]. The subjects subsequently performed two variable-P tests to the limit of tolerance. During the first part, P was 117% or 134% of CP for a duration that expended approximately half of W′. The work rate was then abruptly increased to 134% (UP protocol) or decreased to 117% (DOWN protocol) of CP for the second part.

Results: There were no significant differences between W′ [12.68 (3.08) kJ] and the total amount of work above CP during the UP [12.14 (4.18) kJ] and DOWN [12.72 (4.05) kJ] protocols (P > 0.05).

Conclusion: We conclude that the work equivalent of W′ is not affected by power variations during exhausting cycle ergometry, at least in the P range of 100–134% of CP.

It has been repeatedly demonstrated that the tolerable duration (t) of high-intensity constant-load cycling decreases hyperbolically as a function of power (P), with an asymptote that is appreciably greater than that equivalent to the lactate threshold (θL) and that has been termed the critical power (CP) (7,10,12,26,33), MATH

In this formulation, W′ represents the curvature constant of the relationship and, having the units of work, is equivalent to a constant amount of work that can be performed above CP, at least for constant rates of performance (see the upper panel in Fig. 1) (7,10,12,33). This hyperbolic relationship between power and its tolerable duration has also been demonstrated for exhausting dynamic contractions in small muscle groups (24), and also treadmill running (16,18) and swimming (30). In the running and swimming tests, speed is used as an analog of P to circumvent the difficulty of assessing the actual power being generated.

The CP has been shown to be the highest constant work rate that can be sustained without V̇O2, blood lactate and [H+] increasing inexorably to, or toward, maximal attainable levels (28). It presumably represents an inherent characteristic of the aerobic energy supply system, the physiological significance of which has already received considerable attention (e.g., 7,10,28,33). The curvature constant W′, being equivalent to a constant amount of energy above CP, has been postulated to reflect a finite available energy store comprised of a phosphagen pool, an anaerobic glycolytic component, and an O2 store. Consequently, W′ has been considered to be equivalent to the O2 deficit (10) or the subject’s “anaerobic work capacity” (AWC) (3,13,23,26).

Notionally W′ may be utilized rapidly by exercising at high power outputs, or may be sustained for longer durations by exercising at lower work rates. However, the parameters of the P-t hyperbolic relationship (i.e., CP and W′) are currently based on studies using constant work rate exercise. It seems appropriate to ask, therefore, whether W′ would be depleted in the same manner or even to the same extent if the work rate was varied during the bout.

It was, therefore, the purpose of this study to compare W′ derived from the standard estimation method, consisting of several different constant-P exercise bouts, with the total amount of work above CP during two variable-P exercise protocols. We hypothesized that the total amount of work above CP would not be significantly different among the three protocols, such that its verification or falsification might benefit subsequent experimental designs.

1Department of Exercise Science and Physiology, School of Health Sciences, Hiroshima Prefectural Women’s University, Hiroshima, JAPAN; and

2Division of Respiratory and Critical Care Physiology and Medicine, Harbor-UCLA Medical Center, Torrance, CA

Address for correspondence: Yoshiyuki Fukuba, Ph.D., FACSM, Professor of Human Physiology, Department of Exercise Science and Physiology, School of Health Science, Hiroshima Prefectural Women’s University, 1-1-71, Ujina-higashi, Minami-ku, Hiroshima 734-8558 Japan; E-mail: address:

Submitted for publication October 2002.

Accepted for publication April 2003.

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Eleven healthy male subjects volunteered to participate in this study. Their mean (± SD) age, height, and body weight were 27 (±7) yr, 176 (±6) cm, and 68.7 (±4.8) kg, respectively. After the protocols and possible risks associated with participation in the study were explained, subjects signed an informed consent form, which had been approved by the ethics committees of the institution (in accordance with the Declaration of Helsinki). All subjects (Table 1) were familiar with the equipment and procedures for the exhaustive exercise testing.

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All exercise bouts were performed in the seated position on an electrically braked iso-power cycle ergometer (232c-XL, Combi) at a constant pedal frequency of 60 rpm to avoid the cadency effect (15) and at an individually standardized seat height and handle position. Each subject initially performed a ramp-incremental exercise test (25 W·min−1) to the limit of tolerance for the estimation of the lactic acidosis threshold (θL) and maximum V̇O2 (V̇O2max). The lactic acidosis threshold (θL) was estimated using the V-slope method (2) and gas exchange criteria to detect the break points at which there was a systematic increase in the ventilatory equivalent for V̇O2 (V̇E/V̇O2) and end-tidal PO2, with no concomitant increase in the ventilatory equivalent for CO2 output (V̇E/V̇CO2) or decrease in end-tidal PCO2 (31). All exercise tests were performed in an air-conditioned laboratory (24 ± 1°C) situated at sea level.

The subjects initially performed a series of four to six different high-intensity square-wave exercise bouts allowing us to estimate CP and W′ (Fig. 1). Each bout was preceded by a control phase of 20 W exercise for 4 min. The work rate (P) for each square-wave exercise bout was selected to induce exhaustion within a range of approximately 2–10 min. The pedal cadence was 60 rpm as cued by the audio signal from an electric metronome. Exhaustion was considered to occur when the subject could no longer maintain a pedal revolution of even 50 rpm despite investigator encouragement. Each test was followed by a 5-min “cooling down” period at 20 W. Only one exercise bout was performed in a given week to avoid or minimize any training effect: the order of the tests was randomized. The hyperbolic P-t relationship for each subject was linearized by plotting (1/t) versus P: (1/t) = (1/W′)·P − (CP/W′) by the least-squares linear regression technique (see the lower panel in Fig. 1). The slope of this relationship provides the curvature constant (W′) and its P intercept corresponds to CP (12,26).

The subjects subsequently performed two variable-power exercise bouts to the limit of tolerance at work rates well within the accurately calibrated work rate range of the cycle. Consequently, the initial part of the tests was at a power output of either 117% or 134% of CP for a duration which expended approximately half of W′ (the time integral of the power above CP utilizing half of W′, as schematized in Fig. 2). The work rate was then abruptly increased to 134% (“UP” protocol) or decreased to 117% (“DOWN” protocol) of CP for the second part, without any cue, and the subject (who was informed that the work rate would change once slightly during the exhausting exercise protocol) continued to the limit of tolerance. Care was taken that the subjects maintained the pedal cadence across the transition: again exhaustion was considered to occur when the subject could no longer maintain the pedaling cadence of 50 rpm. The order of UP and DOWN protocols was randomized (six subjects were assigned to perform UP protocol as a first one) and both varied-P tests were separated by a week. The power-time integral above CP (units of work) was then calculated and termed W′u for the “UP” protocol, W′d for the “DOWN” protocol; these were compared to W′ for the work equivalent of the constant-load P-t relationship as shown in Figure 2.

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Ventilatory and gas exchange responses were determined breath-by-breath by a computerized metabolic measuring system (Aero-Monitor, Minato Medical Co.). Before each exercise test, the flow-sensor and gas analyzers were calibrated by inputting a known volume of air at several mean flow rates and gas mixtures of known concentration, respectively. The heart rate was monitored by cardiotachogram (BP-306, Colin). The second-by-second time course was calculated for each variable by interpolation of the breath-by-breath data. Data were stored on disk for further analysis.

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Statistical analysis.

The comparison of the variables derived from three conditions (the standard procedure, the UP protocol, and the DOWN protocol) was tested using one-way ANOVA with repeated measures. If a significant difference was detected, these were further evaluated by post hoc Duncan’s multiple-range test. The statistical significance was declared when P < 0.05. The data are expressed as mean ± SD unless otherwise specified. All statistical procedure was performed using SPSS for Windows.

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The time to fatigue during the constant-load tests was hyperbolically related to P in all subjects (for example, Fig. 1), such that all subjects evidenced a high degree of linearity in the relationship between the inverse of time and P; the correlation coefficients were extremely high (r = 0.982–0.999;Table 1). The average values for CP and W′ were 213.3 ± 22.4 W and 12.68 ± 3.08 kJ, respectively (Table 1).

In the UP protocol, subjects first exercised at 117% of CP for individual durations of 120–257 s (the predetermined duration which expended approximately half of W′, as described in methods). Then P was abruptly increased to 134% of CP for the second part until exhaustion (for example, see in Fig. 2). The total amount of work performed above CP during the UP protocol (W′u) averaged 12.14 ± 4.18 kJ. In the DOWN protocol, W′d averaged 12.72 ± 4.05 kJ. There were no significant differences among W′, W′u, and W′d for the different protocols (Fig. 3).

The peak V̇O2 values attained at exhaustion during the UP and DOWN protocols were not statistically different from each other (UP: 3.00 ± 0.49; DOWN: 3.08 ± 0.58 L·min−1) or the average of the constant power tests (2.98 ± 0.44 L·min−1) as shown in Fig. 4; the same was true of heart rate (UP: 173 ± 10.0; DOWN: 176.9 ± 10.7; constant-load: 176 ± 10.2 beat·min−1), suggesting a similarly maximum effort had been produced for each protocol.

As peak V̇E and V̇CO2 have been shown to be inversely proportional to the time to exhaustion for constant-load tests (27), we only compared the maximum values for V̇E and V̇CO2 between the UP and DOWN protocols: there was no significant difference between either the peak V̇CO2 (UP: 3.85 ± 0.42 vs DOWN: 3.77 ± 0.48 L·min−1) or peak V̇E (UP: 121.45 ± 12.99 vs DOWN: 121.82 ± 10.62 L·min−1) values.

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Our results confirm our hypothesis that the total amount of work performed to the limit of tolerance above CP during a varied-power protocol was not significantly different from that predicted from establishing the curvature constant of a standard power-duration curve.

However, it is, perhaps, easier to conceptualize the physiological mechanism(s) responsible for the critical power parameter of the power-duration curve than for its curvature constant. In the former case, CP has been demonstrated to characterize the upper limit for stable aerobic energy exchange (i.e., requiring no continuously developing slow phase of the V̇O2 kinetics with its trajectory to, or toward, the maximum O2 uptake) and also the upper limit for the muscle lactate production being matched by its predominantly oxidative clearance (28) resulting in a stable acidic pH. In contrast, the curvature constant (that some have equated to an anaerobic work capacity or maximum O2 deficit (3,13,26)) with its units of work or energy is suggestive of a limited supplemental, presumably anaerobic, resource that may be depleted at a rate proportional to the magnitude of the power demands above the CP. This notion is supported by the demonstration that hypoxic or hyperoxic inspiration altered CP without discernibly influencing W′ (25,32). By the same token, increased muscle phosphocreatine (20) and decreased muscle glycogen stores (21) each influences W′ without a significant change of CP. However, it seems less likely, in this intensity domain, that the limit of tolerance results from an available energy resource being depleted than to a build-up of fatigue metabolites such as hydrogen ions and di-protonated inorganic phosphate (or of factors proportionally coupled to them) impairing muscle contractility and producing limiting symptoms (29). If this is the case, however, it would require that the metabolite build-up be proportionally related to the rate of W′ utilization. Although our study does not directly provide information on the mechanism(s) responsible for W′, it does address, we believe, an important related issue: whether the amount of limiting “work” above CP is the same when the work rate changes within the supra-CP domain as when it is constant, as schematized in Figure 5.

We are unaware of any studies that have experimentally addressed the influence of tactical pace change during an endurance event on the optimum performance limit. In a previous theoretical study (8), however, we analyzed the consequences of an athlete performing the initial part of an endurance event at a work rate different from the constant rate that would allow the performance time to be determined by the hyperbolic P-t relationship. We considered not only the P-t constraints that limit the athlete’s ability to make up the time lost by too slow an early pace but also the consequences of a more rapid early pace. This analysis demonstrated that both CP and W′ have an important role in the pace allocation strategy. For example, when P for any part of the endurance race is below CP, the athlete can never attain the goal of achieving the time equivalent to that of performing the entire race at the constant optimal P. Furthermore, W′ was shown to be especially important in determining the flexibility of the race pace that the athlete is able to choose intentionally.

These conclusions, however, were based on the assumption that the total amount of work (i.e., that equivalent to W′) that can be performed above CP remains constant when P changes during the exhausting exercise. The results of our present study are consistent with this assumption, i.e., the total amount of work expended above CP during the varied-P exercise was essentially the same as that predicted from the standard procedure using several high-intensity square-wave tests (i.e., W′), at least for work rates between CP to 134% of CP. This seems to rule out a significant beneficial effect of the first component of the varied power protocol on the second one. Such an effect has been previously demonstrated as a result of a “priming” bout of heavy-intensity exercise to the V̇O2 response during the following identical heavy bout (i.e., the repeated bouts protocol;1,4,9,11); we have not used the term “warm-up” for the preliminary exercise as it has been consistently demonstrated that the altered kinetics are not influenced by experimental alterations of muscle temperature (6,19). In this respect, Miura et al. (22) recently reported that the tolerable duration of supra-CP work rates was increased by such a priming bout (as a result of CP being increased rather than W′). However, there was a recovery period to a low metabolic demand before the second exercise bout in their study. This differs from the present study for which both components were continuously above CP. It is relevant in this regard that Coats et al. (5) recently reported that after exhausting exercise the work rate needs to be reduced to a level below CP for the subject to be able to continue to exercise. Hill and Smith (14) have demonstrated a 27% improvement in exercise time on a second bout at CP, performed on a subsequent day, compared with the first. It is unlikely that our results are influenced by such a “learning” or “sequence” effect, however, as our subjects were all experienced in performing component tests of the power-duration curve on different days, with no distortions evident in the hyperbolic profile occasioned by the tests subsequent to the first. Furthermore, had the initial phase of the varied power protocol created such a beneficial effect on the subsequent component, then the total work calculated as the power-time integral above CP would have been systematically increased; as shown in Figure 3, this was not the case.

But, had there been any consequent beneficial effect of the initial component of our varied-power test on the subsequent component, then exercise tolerance at this intensity would presumably be increased as a result of changes in one or both parameters of the P-t hyperbolic relationship; i.e., we would expect that the total “work” expended above the original CP to be increased due to a prolonged second phase of the test. The result of this study, however, did not support this hypothesis. In addition, our results suggest that there was no functional replenishing of W′ as the work rate was decreased in the DOWN protocol. This seems a much more plausible explanation for the findings than both CP and W′ being changed in exactly the appropriate inverse proportion, although this cannot be conclusively ruled out from the available evidence.

Further work is clearly needed to resolve the nature of the W′ determinant(s), the basis of its apparent constancy, as the pattern of the supra-CP power output is varied, and its precise role in pace allocation strategies for competitive athletic performance (17).

This study was supported in part by Grant in-Aid for Scientific Research from the Ministry of Educations, Science, Sports, and Culture in Japan (no. 12680048) and the Uehara Memorial Life Science Foundation to YF.

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1. Bearden, S. E., and R. J. Moffatt. VO2 and heart rate kinetics in cycling: transitions from an elevated baseline. J. Appl. Physiol. 90: 2081–2087, 2001.
2. Beaver, W. L., K. Wasserman, and B. J. Whipp. A new method for detecting anaerobic threshold by gas exchange. J. Appl. Physiol. 60: 2020–2027, 1986.
3. Bulbulian, R., J. W. Jeong, and M. Murphy. Comparison of anaerobic components of the Wingate and critical power tests in males and females. Med. Sci. Sports Exerc. 28: 1336–1341, 1996.
4. Burnley, M., A. M. Jones, H. Carter, and J. H. Doust. Effects of prior heavy exercise on phase II pulmonary oxygen uptake kinetics during heavy exercise. J. Appl. Physiol. 89: 1387–1396, 2000.
5. Coats, E. M., H. B. Rossiter, J. R. Day, A. Miura, Y. Fukuba, and B. J. Whipp. Intensity dependent tolerance to exercise after attaining VO2max in humans. J. Appl. Physiol. 95: 483–490, 2003.
6. Fukuba, Y., N. Hayashi, H. Sato, and T. Yoshida. Effects of diathermic-warming of working muscles on V̇O2-kinetics during heavy exercise (Abstract). Med. Sci. Sports Exerc. 30: S189, 1998.
7. Fukuba, Y., and B. J. Whipp. The “fatigue threshold”: its physiological significance for assigning the transition from heavy to severe exercise. In: Recent Advances in Physiological Anthropology, M. Sato, H. Tokura, and S. Watanuki (Eds.). Fukuoka: Kyushu University Press, 1999, pp. 341–346.
8. Fukuba, Y., and B. J. Whipp. A metabolic limit on the ability to make up for lost time in endurance events. J. Appl. Physiol. 87: 853–861, 1999.
9. Fukuba, Y., N. Hayashi, S. Koga, and T. Yoshida. VO2 kinetics in heavy exercise is not altered by prior exercise with a different muscle group. J. Appl. Physiol. 92: 2467–2474, 2002.
10. Gaesser, G. A., and D. C. Poole. The slow component of oxygen uptake kinetics in humans. Exerc. Sport Sci. Rev. 24: 35–71, 1996.
11. Gerbino, A., S. A. Ward, and B. J. Whipp. Effects of prior exercise on pulmonary gas-exchange kinetics during high-intensity exercise in humans. J. Appl. Physiol. 80: 99–107, 1996.
12. Hill, D. W. The critical power concept. Sports Med. 16: 237–254, 1993.
13. Hill, D. W., and J. C. Smith. A comparison of methods of estimating anaerobic work capacity. Ergonomics 36: 1495–1500, 1993.
14. Hill, D. W., and J. C. Smith. Determination of critical power by pulmonary gas exchange. Can. J. Appl. Physiol. 24: 74–86, 1999.
15. Hill, D. W., J. C. Smith, J. L. Leuschel, S. D. Chasteen, and S. A. Miller. Effect of pedal cadence on parameters of the hyperbolic power-time relationship. Int. J. Sports Med. 16: 82–87, 1995.
16. Hughson, R. L., C. J. Orok, and L. E. Staudt. A high velocity treadmill running test to assess endurance running potential. Int. J. Sport Med. 5: 23–25, 1984.
17. Jones, A. M., and B. J. Whipp. Bioenergetic constraints on tactical decision making in middle distance running. Br. J. Sports Med. 36: 102–104, 2002.
18. Kan, A., T. Inamizu, N. Yasuda, and Y. Fukuba. The velocity-duration time hyperbolic curve as an index of middle distance running performance (Abstract) Physiologist 39: 78A, 1996.
19. Koppo, K., A. M. Jones, L. Vanden Bossche, and J. Bouckaert. Effect of prior exercise on V̇O2 slow component is not related to muscle temperature. Med. Sci. Sports Exerc. 34: 1600–1604, 2002.
20. Miura, A., F. Kino, S. Kajitani, H. Sato, H. Sato, and Y. Fukuba. The effect of oral creatine supplementation on the curvature constant parameter of the power-duration curve for cycle ergometry in humans. Jpn. J. Physiol. 49: 169–174, 1999.
21. Miura, A., H. Sato, H. Sato, B. J. Whipp, and Y. Fukuba. The effect of glycogen depletion on the curvature constant parameter of the power-duration curve for cycle ergometry. Ergonimics 43: 133–141, 2000.
22. Miura, A., T. Shiragiku, Y. Hirotoshi, T. J. Barstow, B. J. Whipp, and Y. Fukuba. The effect of prior heavy exercise on the parameters of the power-duration curve for cycle ergometry (Abstract). Physiologist 43: 324, 2000.
23. Miura, A., M. Endo, H. Sato, H. Sato, T. J. Barstow, and Y. Fukuba. Relationship between the curvature constant parameter of the power-duration curve and muscle cross-sectional area of the thigh for cycle ergometry in humans. Eur. J. Appl. Physiol. 87: 238–244, 2002.
24. Monod, H., and J. Scherrer. The work capacity of a synergic muscular group. Ergonomics 8: 329–338, 1965.
25. Moritani, T., A. Nagata, H. A. deVries, and M. Muro. Critical power as a measure of physical work capacity and anaerobic threshold. Ergonomics 24: 339–350, 1981.
26. Nebelsick, G. L. J., T. J. Housh, G. O. Johnson, and S. M. Bauge. Comparison between methods of measuring anaerobic work capacity. Ergonomics 31: 1413–1419, 1988.
27. Neder, J. A., P. W. Jones, L. E. Nery, and B. J. Whipp. Determinants of the exercise endurance capacity in patients with chronic obstructive pulmonary disease: the power-duration relationship. Am. J. Respir. Crit. Care Med. 162: 497–504, 2000.
28. Poole, D. C., S. A. Ward, G. W. Gardner, and B. J. Whipp. Metabolic and respiratory profile of the upper limit for prolonged exercise in man. Ergonomics 31: 1265–1279, 1988.
29. Rossiter, H. B., S. A. Ward, J. M. Kowalchuk, F. A. Howe, J. R. Griffiths, and B. J. Whipp. Dynamic asymmetry of phosphocreatine concentration and O2 uptake between on- and off-transients of moderate- and high-intensity exercise in humans. J. Physiol. (Lond.) 541: 991–1002, 2002.
30. Wakayoshi, K., K. Ikuta, T. Yoshida, et al. Determination and validity of critical velocity as an index of swimming performance in the competitive swimmer. Eur. J. Appl. Physiol. 64: 153–157, 1992.
31. Whipp, B. J. The bioenergetic and gas exchange basis of exercise testing. Clin. Chest Med. 15: 173–192, 1994.
32. Whipp, B. J., D. J. Huntsman, N. Storer, N. Lamarra, and K. Wasserman. A constant which determines the duration of tolerance to high-intensity work (Abstract). Fed. Proc. 41: 1591, 1982.
33. Whipp, B. J., and S. A. Ward. Respiratory responses of athletes to exercise. In: Oxford Textbook of Sports Medicine, M. Harries, L. J. Michel, W. D. Stanish, and C. Williams (Eds.). Oxford: Oxford University Press, 1994, pp.13–25.

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