Abstract
It has been repeatedly demonstrated that the tolerable duration (t) of highintensity cycling is well characterized as a hyperbolic function of power (P) with an asymptote that has been termed the “fatigue threshold” and with a curvature constant. This hyperbolicPtrelationship has also been confirmed in running and swimming, when speed (V) is used instead ofP; that is, (V −V _{F}) ⋅ t= D′, whereV _{F} is theV at the fatigue threshold, andD′ is the curvature constant. Therefore, we theoretically analyzed herein the consequences of an athlete performing the initial part of an endurance event at aV different from the constant rate that would allow the performance time to be determined by the hyperbolicVtrelationship. We considered not only theVtconstraints 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. Our analysis demonstrates that both theV _{F} andD′ parameters of the athlete’sVtcurve play an important role in the pace allocation strategy of the athlete. That is, 1) when the running V during any part of the whole running distance is belowV _{F}, the athlete can never attain the goal of achieving the time equivalent to that of running the entire race at constant maximalV (i.e., that determined by one’s own bestVtcurve); and 2) the “endurance parameter ratio” D′/V _{F} is especially important in determining the flexibility of the race pace that the athlete was able to choose intentionally.
 powerduration hyperbolic curve
 speedduration hyperbolic curve
 endurance athlete performance
 racepace strategy
 theoretical analysis
the choice of speed, and the variations of speed, that will maximize an endurance athlete’s ability to succeed in winning a race involves a complex interplay of physiological and psychological factors. An inappropriate allocation of pace can reduce the athlete’s likelihood of success in the event. One aspect of the physiological basis of racepace strategy has not, we believe, received appropriate consideration.
This is, perhaps, best exemplified by the physiological inferences that may be drawn from the answer to the following question. Imagine that two athletes engage in a race of some specified distanceX. Athlete A runs the race at a constant speed;athlete B runs the first onehalf of the race at onehalf the speed of athlete A. At what speed must athlete B run the second onehalf of this race to finish at the same time as athlete A?
The answer, of course, is that it is impossible:athlete B cannot possibly catchathlete A. The reason is simply that, in running onehalf the distance at onehalf the speed,athlete B has used all of the winning time of athlete A. By extension, athletes may use the strategy of relatively slow running early in the race, either in an attempt to husband their reserves for a subsequent sprint or as a tactic to induce suboptimal performance in other athletes. That is, the strategy requires disproportionately higher subsequent velocities to achieve the same effective time that would have resulted from constantspeed running. This increases the likelihood of draining the limited and, it has been proposed, constant metabolic energy reserve that is available to the athlete.
The tolerable duration of a particular work rate has been shown to be an inverse function of the work rate, as shown in Fig.1. It has further been demonstrated (10,11, 14, 15, 20) that this relationship is hyperbolic, with an asymptote on the power axis that has been termed “the critical power” (CP) or “fatigue” threshold (θ_{F}) and a curvature constantW′ that, being the product of power and time, has the units of work, i.e., equivalent to a constant amount of energy above CP. This constant amount of energy may notionally be utilized rapidly by exercising at high power outputs, or may be eked out for longer durations by exercising at lower work rates.
It is important to emphasize, first, that the hyperbolic relationship is unlikely to provide a precise representation of the actual physiological behavior at the very extremes of performance, because of distorting factors such as 1) limitations of mechanical force generation in muscle for the very highest power or running speed and2) constraints resulting from substrate provision and thermoregulatory or body fluid requirements for markedly prolonged exercise (21). The duration limit for the application of the hyperbolic relationships has not been precisely established, but it does appear to be applicable up to ∼20 or 30 min.
We consider here the consequences of an athlete performing the initial part of an endurance event at a speed different from that of the constant rate that would allow the performance time to be determined by the powerduration relationship. In this context, by endurance we mean those events for which deliberate pace changes might be a plausible competitive strategy; i.e., sprints are not considered. We consider not only the powerduration constraints that limit the athlete’s ability to make up the time lost by keeping too slow an early pace but also the consequences of keeping a more rapid early pace. In other words, we consider the pace allocation to achieve the best possible time for a runner. Our results demonstrate that the threshold and curvature parameters of the subject’s powerduration curve are likely to be important in establishing the limits of the pace allocation strategy of the athlete.
THE HYPERBOLIC POWERDURATION RELATIONSHIP FOR CYCLING
It is well known that the relationship between power (P; W or J/s) and its endurance time (t; s) for highintensity cycling may be characterized as a hyperbolic function (e.g., see Ref. 2 for review). That is
The total amount of work done (W
_{tot}), however, is
THE HYPERBOLIC VELOCITYDURATION RELATIONSHIP FOR RUNNING
Whereas the hyperbolicPtrelationship has been confirmed by several previous studies for cycling performance (11, 14, 15, 20), it has also been confirmed to be hyperbolic in running and swimming, when speed or veolicty (V) is used instead ofP (35, 12, 18). In the following discussion, we therefore use the termV instead ofP. That is
MAXIMAL AVERAGE SPEED
Each individual is assumed to have a specific hyperbolic relationship between V andt. Each is therefore able to run some specific distance X at the maximal average speed [V
_{max (}
_{X}
_{)}], which is determined by the crossing point of the particularVtcurve defined by Eqs. 4a
,
4b
, or
6
(i.e., the “bestVtcurve to D′”) and theVtrelationship curve to some specific distanceX defined by Eq.5
(i.e., the “distance curve toX”). That is, Eq.5
with X substituted for D puts in Eq.4b
The running time t[t
_{max (}
_{X}
_{)}] to distance X run with a constant speedV
_{max (}
_{X}
_{)}is
ALLOCATION OF RUNNING PACE
In general, runners change the running speed during a race according to their own racepace strategy, although other runners’ tactics can naturally alter this. We shall consider the simple situation in which the runner runs at different speeds (V _{1} andV _{2}) during the parts of divided distances (X _{1} andX _{2}) of total distance X.
During the initial part (X _{1}), the runner is only able to choose a speed (V _{1}) belowV _{max (} _{X} _{1)}, which is determined as the crossing point of theVtcurve to D′ and the distance curve to X _{1}. IfV _{1} is above theV _{F} of the runner, the maximum speed (V _{2}), which the runner can choose during the second part of the race (X _{2}), is automatically determined by the amount ofD′ remaining [i.e.,D′_{(} _{X} _{2)}], because a specific amount of D′ [D′_{(} _{X} _{1)}] is already expended duringX _{1}. However, ifV _{1} is belowV _{F}, the runner can subsequently run at a speed (V _{2},) which utilizes the entire D′ to the limit of the bestVtcurve. Note that, in this case, the runner cannot run above the speed [V _{max (} _{X} _{2)}], which is determined by the bestVtcurve toD′_{(} _{X} _{2)}(= D′) and distance curve toX _{2}. In the following discussion, therefore, we considerV _{1} in greater detail.
V_{F} ≤ V_{1} < V_{max (}_{X}_{1)}
In the initial partX
_{1}, the runner uses theD′_{(}
_{X}
_{1)}and the time (t
_{1}) necessary to run distanceX
_{1} at a speedV
_{1}, which is
V_{1} < V_{F}
In the initial partX
_{1}, if the runner chooses a speed (V
_{1}) that is below V
_{F}, the time (t
_{1}) that is taken to run distanceX
_{1} withV
_{1}(V
_{1} <V
_{F}) is
As a consequence, when the runner runs atV _{1} belowV _{F}, it is not possible to make up for the lost time during the second part of the race (X _{2}). That is, even if the runner runs theX _{2} distance withV _{2} atV _{max (} _{X} _{2)}, the athlete can never attain the goal of achieving the time equivalent to that of running the entire race at the average speedV _{max (} _{X} _{)}, i.e., that determined by his or her own bestVtcurve to D′ and distance curve to X. A more general theory of the pace allocation problem is presented in the .
NUMERICAL AND GRAPHIC EXAMPLES
Here we consider the 5,000 m by using the same subject frommaximal average speed from a numerical and graphic standpoint. We choose the subject’sV _{F} andD′ to be 4.4 (m/s) and 300 (m), respectively, and theV _{max} andt _{max} for this race are 4.68 (m/s) and 1,068.2 (s), respectively (see Fig.2 A).
We used one of a number of plausible allocations of running pace inV_{F} ≤ V_{1} < V _{max (X1)}. The initial part X _{1}is 3,500 m with a speed (V _{1}), which is relatively slow but still aboveV _{F}, e.g., 4.50 (m/s). This speed corresponds to 3.6% belowVmax at 5,000 m [V _{max (5,000)}]. For the initial 3,500 m,t _{1} andD′_{(3,500)}are 777.8 (s) and 77.8 (m), respectively. These values are determined by the crossing point of the runner’sVtcurve toD′_{(3,500)}(= 77.8 m) and the distance curve for 3,500 m (Fig.3 A). For the second 1,500 m, because the runner has access only to the remainingD′_{(1,500)}(= 300–77.8 = 222.2 m) for the last “spurt” (i.e., “remaining energy”), the athlete can run only at the speed that is determined by the crossing point of the bestVtcurve toD′_{(1,500)}(= 222.2 m) and the distance curve to 1,500 m (Fig.3 B). This requires the second 1,500 m to be run in at _{2} of 290.4 (s) (i.e., “required time”) and at a speedV _{2} of 5.165 (m/s). This is +10.3% greater thanV _{max (5,000)}. As a result,t _{tot}, i.e., the race record, is 1,068.2 (s), and is exactly the same tot _{max}, which is derived from maximal average speed,V _{max (5,000)}.
Consider, however, if the runner chooses the speed during the initial part that is belowV _{F}, e.g., 4.21 (m/s) as in V_{1}< V_{F} . This corresponds to 10.0% belowV _{max (5,000)}for the initial 3,500 m of the 5,000m distance. ThroughoutX _{1}, the runner can reserve the whole D′ for the last spurt (i.e., remaining energy). Note, however, that 831.4 (s) is already taken ast _{1} (Fig.3 C). This requires the second 1,500 m to be run in at _{2} of 236.8 (s) (i.e., required time). In the second 1,500 m, the athlete can run at the maximal speed [V _{max (1,500)}], which is determined by the crossing point between the runner’s bestVtcurve to D′ (= 300 m) and the distance curve to 1,500 m. Throughout the second part,t _{2} is 272.7 (s) and V _{2} is 5.501 (m/s), which is about +25.0% greater thanV _{max (5,000)}(Fig. 3 D). As a result,t _{tot} is 1,104.1 (s); this is some 36 s longer thant _{max}. This is the case for all durations of suboptimal early race pace that are belowV _{F}.
GENERAL CONSIDERATIONS
Here we simulate these relationships systematically for the twodivision paceallocation strategy in the 5,000 m. Furthermore, we also examine the effect ofV _{F} andD′ on this strategy. We used the actual values for the runner determined from the study of Hughson et al. (Fig. 3 in Ref. 4); i.e.,V _{F} andD′ are ∼5.0 (m/s) and ∼150 (m), respectively.
Consider the expected race performance when the athlete runs a 5,000m distance at a maximal average speed, i.e., as determined by theVthyperbola: in this caseV _{max} is 5.155 (m/s) and t _{tot} is 970 (s). We examined the systematic effect of pace allocation by utilizing two different speeds (V _{1},V _{2}) in different combinations ofX _{1}X _{2}to the total 5,000 m.X _{1} andX _{2} (m) were chosen to be 1,0004,000, 2,0003,000, 3,0002,000, and 4,0001,000. In eachX _{1}X _{2}combination, we calculated the parameters according toEqs. 914 .V _{1} was varied systematically, from −20% ofV _{max (5,000)}to the maximum within the limit set byD′, as described inallocation of running pace.
We confirmed by simulation that, ifV _{1} is belowV _{F}, the final race time (i.e.,t _{tot}) is always longer thant _{max (5,000)}(see Fig. 4). In the range ofV _{1} belowV _{F},t _{tot} was gradually increased as a function of decreasingV _{1}. This effect was, naturally, more pronounced the longer wasX _{1}. The upper limits of V _{1} are automatically determined by the limit of the bestVtcurve of the subject and each distance curve toX _{1}. That is, by definition no one can run at the speed above the intersection of both curves.
If t _{1} was relatively longer forX _{1}, the runner would have to run considerably faster inX _{2} as shown in the relationship betweent _{1} andV _{2} in Fig.5 (top left). The dotted line shows theV _{2} required to keep the samet _{tot} tot _{max (} _{X} _{)}, and the bold line shows the calculatedV _{2}. For example, in the specific case of the combination ofX _{1}X _{2}, at the shortestt _{1}(point 1 in Fig. 5,top left),V _{2} is exactly the same as V _{F} as a result. This point means that, the runner having run atV _{max (} _{X} _{1)}for the initialX _{1}, the required speed for the remainingX _{2} distance isV _{F}, i.e., the upper (theoretical) limit of “fatiguefree” running speeds (i.e.,strategy 1, Fig. 5). Ift _{1} took longer, the athlete would have to run faster thanV _{F} forX _{2}, i.e., atV _{1} given by the dotted line (Fig. 5, top left), which is the requiredV _{2} for the same running time tot _{max (} _{X} _{)}. However, in the range oft _{1} above the deflection point (point 2 in Fig. 5,top left) of the bold line, the actual calculatedV _{2} is dissociated from the requiredV _{2} (dotted line), due to the “metabolic” limitation. This means thatV _{2} cannot exceed the maximal speed set by the bestVtcurve, as determined by the wholeD′ and distanceX _{2} (i.e.,strategy 3, Fig. 5, e.g.,point 3 in Fig. 5,top left). As a result, this deflection point ( point 2in Fig. 5, top left) means that the athlete must run atV _{F} forX _{1} and atV _{max(} _{X} _{2)}for X _{2} (i.e.,strategy 2, Fig. 5). Therefore, the range between these points [i.e., the range over which t _{tot} can equal tot _{max (} _{X} _{)}] represents the maximal possible range for the strategic modification of the race pace.
These conditions are described by the following equations. Instrategy 1( point 1 in Fig. 5,top left),V
_{1} isV
_{max(}
_{X}
_{1)}and V
_{2} isV
_{F},t
_{1} andt
_{2} are
Consider also the difference of speeds between two extreme strategic conditions (i.e., conditions 1 and2)
To examine the effects ofV _{F} andD′ on the time range andV _{2} range, we plotted thet _{1}V _{2}relationship in whichV _{F} orD′ was changed systematically. The effect of D′ on thet _{1}V _{2}relationship under the sameV _{F} condition, i.e., 5.0 (m/s) is shown in Fig.6 A. As a result, the larger D′ dramatically extended the time range andV _{2} range for the 5,000m run. However, compared with the effect ofD′, the effect ofV _{F} was relatively small (Fig. 6 B). This indicates thatD′ is relatively important for the runner’s possible range of pace change: the larger theD′, the larger the potential pace change in the race.V _{F} is, of course, an essential determinant oft _{max (} _{X} _{)}, i.e., the actual level of the race time (Fig.6 B). Therefore, althoughV _{F} seems to be the major determinant of the level of race performance,D′ is an important determinant of the flexibility of racepace strategy.
ADDITIONAL REMARKS ON THE ANALYSIS
PV Relationship During Running
Running, either on a treadmill or on a track, provides a major difficulty in terms of assessing the actual power output. However, from the few studies (68) that have attempted to determine the actual mechanical work during running, it has been proposed that the efficiency was similar over the range of speeds utilized in this study (i.e., 5–7 m/s), that is, justifying the use of speed as a functional analog of power output. Hughson et al. (4) showed that theVtrelationship for treadmill running (by using crosscountry runners) could be well described by a hyperbolic function at least within the speed range of 5.2–6.3 m/s. The endurance time was 2–12 min, which approximates the range of performance times for middledistance running (i.e., from 800 to 5,000 m). It was shown that theVtcurve for the treadmill running, by using the subjects who are not as regularly trained as those of Hughson et al., could also be well fitted the hyperbolic function within the exhaustion time range of 1.9–9.4 min (approximately, within the speed range of 4.2 to 5.5 m/s) (3). Furthermore, theVthyperbolic relationships for treadmill running in subelite middledistance runners of a university track team were well recognized, with a speed range of ∼4.7–6.5 m/s (5). Although there is no experimental evidence that demonstrates that the relationship remains hyperbolic beyond these speed and/or endurance times, the change of speed for the last spurt in actual middledistance running is likely to be within about ±1.0 m/s. For example, by using the worldclass record, the last lap of 52 s in a 3,000m race means the speed of ∼7.6 m/s for the last spurt, compared with the average running speed of ∼6.6 m/s (∼3,000 m/7 min 30 s). We have, therefore, assumed that within this running speed range, the speed will be proportional to the power. It should be recognized that, although we have treated thePtcurve as if it is hyperbolic for running, this needs further verification. If it proves not to be the case, however, an additional term(s) will need to be added to the characterization (e.g., 13, 19, 23), but this will not alter the conceptual issue being addressed in this study.
Physiological Interpretation of the Hyperbolic Curve
ThePtrelationship for highintensity exercise was described by Hill (1) as early as 1927, and more recently it has been characterized as a hyperbolic function (see Ref. 2 for review). Although the precise physiological determinants of θ_{F}and W′ remain conjectural, the θ_{F} (or CP) has been shown to represent the highest work rate for which a steady state can be attained in pulmonary gas exchange, blood acidbase status, and blood lactate concentration, given sufficient time (14). Therefore, θ_{F} can be regarded as reflecting a rate of energy pool reconstitution that dictates the maximum power that can be sustained without a continued and progressive anaerobic contribution. In many subjects, this highest sustainable lactate level occurs at ∼4 meq/l (1315), although θ_{F} can occur at a wide range of lactate levels among individual subjects up to 8 meq/l or more, for example (21).
W′ can be regarded as an energy store composed of O_{2} stores, a phosphagen pool, and a source related to anaerobic glycolysis. It has the units of work and hence represents a constant amount of work that can be performed above θ_{F}, regardless of its rate of performance.W′ was not increased by endurance training, whereas θ_{F}increased systematically; theWtcurves were well fit by the hyperbola in both conditions (15). Furthermore, Miura et al. (9), recently presented evidence consistent with this view: W′ was significantly reduced under glycogendepleted conditions.
Application Limits of the Hyperbolic Curve
The mechanical limitations of the muscle for highintensity (i.e., shortduration) exercise is relevant to the condition ofX <D′ in our analysis. For such short distances, there is no intersection between theVtcurve and distance curve to X as seen in Eqs. 7 and 8 . This is consequently beyond the application of the hyperbolic theory. For example, if the runner whoseV _{F} andD′ are 5.0 (m/s) and 150 (m), respectively, as discussed in general considerations, runs 200 m at constantV _{max (200)}, theV _{max (200)}andt _{max (200)}are 20 (m/s) and 10 (s), respectively. No athlete can run at such high speed. Therefore, the duration range over which the hyperbolic theory can be applied, might be from 40–50 s to 20–30 min. In other words, these correspond approximately to distances between 400 and 10,000 m of a race for an athlete. Accordingly, we have further assumed in this analysis that the manner in which the race is run (i.e., different pace strategies) does not influenceD′ orV _{F}. Although we know of no experimental evidence to justify a different assumption, other studies that address this issue could well educe such an influence; in which case, the theory could be modified accordingly.
One further concern is that this theory would appear to allow the athlete who has “consumed” the entireW′ (orD′) to continue to perform by decreasing the power output to θ_{F} (orV _{F}), which represents the upper limit for wholly aerobic energy transfer. However, as the magnitude or intensity of the consequent fatigue is presumably some inverse function of the availableW′ (orD′), then the sudden reduction of the power output to θ_{F} (orV _{F}) withW′ (orD′) being depleted would, of course, leave the mechanism(s) of fatigue fully expressed. Repleting W′ (orD′) would require a recovery power that is, presumably, below θ_{F} (orV _{F}), with the repletion rate likely to be greater the lower the recovery power below θ_{F} (orV _{F}). Further studies are required, however, to determine the intensitytime features of this repletion process.
Characteristics of D′/V_{F}
As stated in general considerations, the indexD′/V
_{F}(orW′/θ_{F}, which we term the endurance parameter ratio) appears to be an important determinant of the racepace strategy. Here, we further consider the meaning of this index. One might rewrite Eq. 1 as
In summary, our results demonstrate that the speed or power at the fatigue threshold (V _{F} or θ_{F}) and the curvature constant (D′ orW′) parameters of the athlete’sVtorPthyperbolic curve each play an important role in the pace allocation strategy. That is 1) when the running speed during any part of the whole running distance is belowV _{F}, the runner can never attain the goal of achieving the time equivalent to that of running the entire race at constant optimal speed even if the runner attempts to make up for the time lost with a final spurt; and2)D′ is especially important in determining the flexibility of the race pace that the runner is able to choose intentionally. The ratio of these parameters:D′/V _{F}(i.e., the endurance parameter ratio) may therefore be considered to be an important determinant of racepace strategy.
Acknowledgments
Y. Fukuba was supported by an Overseas Research Fellowship of the Uehara Memorial Foundation, Japan.
Footnotes

Address for reprint requests and other correspondence: Y. Fukuba, Dept. of Exercise Science and Physiology, School of Health Sciences, Hiroshima Women’s Univ., 1–1–71, Ujinahigashi, Minamiku, Hiroshima 734–8558, Japan.

Present address of Y. Fukuba: Dept. of Exercise Science and Physiology, School of Health Sciences, Hiroshima Women’s University, Hiroshima 7348558, Japan.
 Copyright © 1999 the American Physiological Society
Appendix
Theoretical Aspects of General RacePace Allocation
In allocation of running pace, we considered only the simple situation in which the runner runs at two different speeds (V _{1} andV _{2}) during two distances (X _{1} andX _{2}) of the total distance X. Here we expand that into more general situations in which the runner runs atn different speeds (V_{i} ) duringn distances (X_{i} ) of total distance X. The total distance (X) is divided byn parts (X_{i} ,i = 1,..,j,..,n). The runner can run at the different speedsV_{i} in each partX_{i} , the only assumption being that the total amount ofD′ has to be used in the entire race. From a theoretical viewpoint, however, it is sufficient to consider two specific situations, that is, within the limit ofD′, during the one partX_{j} , whereV_{j} is1) aboveV _{F} or2) belowV _{F}, whereas the runner chooses speedV_{i} that equals or exceeds V _{F}[i.e.,V _{(} _{i} _{ ≠ } _{j} _{)}≥ V _{F}] during the remainingX_{i}
Situation 1: V_{(}_{i}_{ ≠ }_{j}_{)}≥ V_{F},V_{j} > V_{F}, i = 1,..,j,..,n
This situation means thatV_{i}
≥V
_{F} excludingV_{j}
, andV_{j}
>V
_{F} during at least one arbitrary part,X_{j}
. In such a situation, t_{i}
during any partX_{i}
is
Situation 2: V_{(}_{i}_{ ≠ }_{j}_{)}≥ V_{F}, V_{j} < V_{F}, i = 1,..,j,..,n
This situation means thatV_{i}
≥V
_{F} excludingV_{j}
, butV_{j}
<V
_{F} during one part,X_{j}
. AsV_{j}
is belowV
_{F}