Vol. 87, Issue 2, 853-861, August 1999
MODELING IN PHYSIOLOGY
A metabolic limit on the ability to make up for lost time in
endurance events
Yoshiyuki
Fukuba and
Brian J.
Whipp
Department of Physiology, St. George's Hospital Medical School,
University of London, London SW17 0RE, United Kingdom
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ABSTRACT |
It has been repeatedly demonstrated that the
tolerable duration (t) of
high-intensity 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 hyperbolic P-t
relationship has also been confirmed in running and swimming, when
speed (V) is used instead of
P; that is,
(V 
VF) · t = D', where
VF is the
V at the fatigue threshold, and
D' is the curvature constant.
Therefore, we theoretically analyzed herein the consequences of an
athlete performing the initial part of an endurance event at a
V different from the constant rate
that would allow the performance time to be determined by the
hyperbolic
V-t
relationship. We considered not only the
V-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. Our analysis demonstrates that both the
VF and
D' parameters of the athlete's
V-t
curve 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 below
VF, the athlete
can never attain the goal of achieving the time equivalent to that of
running the entire race at constant maximal
V (i.e., that determined by one's own
best
V-t
curve); and 2) the "endurance
parameter ratio"
D'/VF is
especially important in determining the flexibility of the race pace
that the athlete was able to choose intentionally.
power-duration hyperbolic curve; speed-duration hyperbolic curve; endurance athlete performance; race-pace strategy; theoretical analysis
| |
INTRODUCTION |
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 race-pace 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 distance X. Athlete
A runs the race at a constant speed;
athlete B runs the first one-half of
the race at one-half the speed of athlete A. At what speed must athlete
B run the second one-half 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 catch
athlete A. The reason is simply that,
in running one-half the distance at one-half 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 constant-speed 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
(
F) and a curvature constant
W' 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.
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Fig. 1.
Typical example of power-duration relationship for high-intensity
cycling exercise in a single human subject studied in our laboratory
(A) and estimated hyperbolic curve
from its linear transformation (B).
W' and
F, parameters of hyperbolic
curve that are represented by model equations
(P F) · t = W' [Eq. 1a (A)] and
P = W' · (1/t) + F [Eq. 1b (B)],
respectively. Dotted line was extrapolated from the estimation. See
text for further details.
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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 and
2) 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 power-duration 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 power-duration 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 power-duration curve are likely to be
important in establishing the limits of the pace allocation strategy of
the athlete.
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THE HYPERBOLIC POWER-DURATION RELATIONSHIP FOR CYCLING |
It is well known that the relationship between power
(P; W or J/s) and its endurance time
(t; s) for high-intensity cycling may
be characterized as a hyperbolic function (e.g., see Ref. 2 for
review). That is
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(1a)
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or
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(1b)
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where
F (W) is the power asymptote
for the
P-t
hyperbola and consequently represents a unique power above which
the time to fatigue is predictably determined (11, 14).
W' represents a constant amount
of work that can be performed above
F. It has been considered to
represent the energy store components:
O2 stores, phosphagen, and
anaerobic glycolysis (22).
The total amount of work done
(Wtot),
however, is
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(2)
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Therefore,
Eq. 1b is rewritten as
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(3)
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where
Wtot > W'. In the case of the subject
in Fig. 1, for example, F and
W' are 168 W and 15.6 kJ, respectively.
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THE HYPERBOLIC VELOCITY-DURATION RELATIONSHIP FOR RUNNING |
Whereas the hyperbolic
P-t
relationship 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 of
P (3-5, 12, 18). In the following
discussion, we therefore use the term
V instead of
P. That is
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(4a)
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or
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(4b)
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where
VF is now the
speed asymptote (m/s), and distance
(D') is expressed in meters. As
the total running distance [D
(m)] is
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(5)
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Eq.
4 may be rewritten as
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(6)
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The
relationships among Eqs. 4a,
4b, and
6 are graphically represented in Fig.
2, A,
B, and
C, respectively.
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Fig. 2.
Schematic example of speed-duration
(V-t)
curve (A), speed vs. reciprocal of
duration (B), and distance vs.
duration (C). Distance curves to
X m
(X = 1,000, 3,000, and 5,000 m) are
also shown. Crossing point between these curves corresponds to maximal
average speed
[Vmax (X)]
and/or its running time
[tmax (X)].
VF, speed at
fatigue threshold; D', distance
curvature. See text for further details.
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MAXIMAL AVERAGE SPEED |
Each individual is assumed to have a specific hyperbolic relationship
between V and
t. Each is therefore able to run some specific distance X at the maximal
average speed
[Vmax (X)], which is determined by the crossing point of the particular
V-t curve defined by Eqs. 4a,
4b, or
6 (i.e., the "best
V-t
curve to D'") and the
V-t
relationship curve to some specific distance X defined by Eq. 5 (i.e., the "distance curve to
X"). That is, Eq. 5 with X substituted
for D puts in Eq. 4b
![<IT>V</IT><SUB>max (<IT>X</IT>)</SUB> = <IT>V</IT><SUB>F</SUB>/[1 − (<IT>D</IT>′/<IT>X</IT>)]](/content/vol87/issue2/fulltext/853/img009.gif)
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(7)
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where
X > D'.
The running time t
[tmax (X)]
to distance X run with a constant
speed
Vmax (X)
is
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(8)
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As a simple example, consider the
Vmax (X)
and
tmax (X)
of some specific subject whose
VF and
D' are 4.4 (m/s) and 300 (m),
respectively, i.e., when the subject runs at the constant
Vmax (X)
throughout the 5,000-m distance. According to Eqs.
7 and 8,
Vmax (X)
and
tmax (X) of this subject for a 5,000-m run are 4.68 (m/s) and 1,068 (s: 17 min
48 s), respectively. Figure 2, A-C,
provides the graphic representations of this example, i.e., the
crossing point between both curves derived from Eqs.
4a, 4b, or
6, and the line from
Eq. 5, respectively.
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ALLOCATION OF RUNNING PACE |
In general, runners change the running speed during a race according to
their own race-pace strategy, although other runners' tactics can
naturally alter this. We shall consider the simple situation in which
the runner runs at different speeds
(V1 and V2) during the
parts of divided distances
(X1 and
X2) of total distance X.
During the initial part
(X1), the
runner is only able to choose a speed
(V1) below
Vmax (X1),
which is determined as the crossing point of the
V-t
curve to D' and the distance curve to X1. If
V1 is above the
VF of the runner,
the maximum speed
(V2), which the
runner can choose during the second part of the race
(X2), is
automatically determined by the amount of D' remaining [i.e.,
D'(X2)],
because a specific amount of D'
[D'(X1)]
is already expended during
X1. However, if
V1 is below
VF, the runner
can subsequently run at a speed (V2,) which
utilizes the entire D' to the
limit of the best
V-t curve. Note that, in this case, the runner cannot run above the speed
[Vmax (X2)],
which is determined by the best
V-t curve to
D'(X2)
(= D') and distance curve to
X2. In the following discussion, therefore, we consider
V1 in greater detail.
VF 
V1 < Vmax (X1)
In the initial part
X1, the runner
uses the
D'(X1)
and the time
(t1) necessary
to run distance
X1 at a speed
V1, which is
![<IT>t</IT><SUB>1</SUB> = <IT>X</IT><SUB>1</SUB>/<IT>V</IT><SUB>1</SUB> = [<IT>X</IT><SUB>1</SUB> − <IT>D</IT>′<SUB>(<IT>X</IT>1)</SUB>]/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img011.gif)
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(9)
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where
0 D'(X1) D' and
D'(X1) < X1. In the
second part X2,
V2 is
consequently determined by the remaining D'
[D'(X2)].
Therefore, the time
(t2) that is
taken to run X2
at a speed V2
[VF < V2 < Vmax (X2)]
is
![= [(<IT>X</IT> − <IT>X</IT><SUB>1</SUB>) − (<IT>D</IT>′ − <IT>D</IT>′<SUB>(<IT>X</IT>1)</SUB>)]/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img013.gif)
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(10)
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where
D'(X2) < X2. If the
runner runs the total distance X with
this pace allocation, then the total time for the race
(ttot)
is
![= (1/<IT>V</IT><SUB>F</SUB>) ⋅ {(<IT>X</IT><SUB>1</SUB>+<IT>X</IT><SUB>2</SUB>)−[<IT>D</IT>′<SUB>(<IT>X</IT>1)</SUB>+<IT>D</IT>′<SUB>(<IT>X</IT>2)</SUB>]}=(<IT>X</IT>−<IT>D</IT>′)/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img015.gif)
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(11)
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That is,
ttot is exactly
same as the time
[tmax (X)]
when the runner runs at the constant speed
Vmax (X), i.e., the speed determined by the best
V-t
curve to X and distance curve to
X.
V1 < VF
In the initial part
X1, if the runner
chooses a speed
(V1) that is
below VF, the
time (t1) that
is taken to run distance
X1 with
V1
(V1 < VF) is
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(12)
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In the second part
X2, the runner
can now utilize the entire D' as
D'(X2),
at the speed V2
determined by the best V-t
curve to D' and the distance
curve to X2. The
t2 [=
tmax (X2)], which is taken for
X2 with
V2 [=
Vmax (X2)],
is
![<IT>t</IT><SUB>2</SUB> = <IT>X</IT><SUB>2</SUB>/<IT>V</IT><SUB>2</SUB> = (<IT>X</IT><SUB>2</SUB> − <IT>D</IT>′)/<IT>V</IT><SUB>F</SUB> = [(<IT>X</IT> − <IT>X</IT><SUB>1</SUB>) − <IT>D</IT>′]/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img017.gif)
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(13)
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where
D' < X2. Therefore,
total running time
(ttot*)
![= <IT>X</IT><SUB>1</SUB>[(1/<IT>V</IT><SUB>1</SUB>) − (1 − <IT>V</IT><SUB>F</SUB>)] + [(<IT>X</IT> − <IT>D</IT>′)/<IT>V</IT><SUB>F</SUB>]](/content/vol87/issue2/fulltext/853/img019.gif)
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(14)
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In a comparison of
ttot*
to the time
[tmax (X)],
when the runner runs at the constant
Vmax (X)
throughout the whole distance X, the
ttot*
is always longer than
tmax (X),
where X1 > 0, because the second term of Eq. 14 is
equal to
tmax (X).
As a consequence, when the runner runs at
V1 below
VF, it is not
possible to make up for the lost time during the second part of the
race (X2). That
is, even if the runner runs the
X2 distance with
V2 at
Vmax (X2),
the athlete can never attain the goal of achieving the time equivalent
to that of running the entire race at the average speed
Vmax (X),
i.e., that determined by his or her own best
V-t
curve to D' and distance curve
to X. A more general theory of the
pace allocation problem is presented in the
APPENDIX.
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NUMERICAL AND GRAPHIC EXAMPLES |
Here we consider the 5,000 m by using the same subject from
MAXIMAL AVERAGE SPEED from a numerical
and graphic standpoint. We choose the subject's
VF and
D' to be 4.4 (m/s) and 300 (m), respectively, and the
Vmax and
tmax for this
race are 4.68 (m/s) and 1,068.2 (s), respectively (see Fig.
2A).
We used one of a number of plausible allocations of running pace in
VF V1 < Vmax (X1). The
initial part X1
is 3,500 m with a speed
(V1), which is
relatively slow but still above VF, e.g., 4.50 (m/s). This speed corresponds to 3.6% below
Vmax at 5,000 m
[Vmax (5,000)].
For the initial 3,500 m,
t1 and
D'(3,500) are 777.8 (s) and 77.8 (m), respectively. These values are determined by the crossing point of the runner's
V-t
curve to
D'(3,500)
(= 77.8 m) and the distance curve for 3,500 m (Fig.
3A). For
the second 1,500 m, because the runner has access only to the remaining D'(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 best
V-t
curve to
D'(1,500)
(= 222.2 m) and the distance curve to 1,500 m (Fig.
3B). This requires the second 1,500 m to be run in a
t2 of 290.4 (s)
(i.e., "required time") and at a speed
V2 of 5.165 (m/s). This is +10.3% greater than
Vmax (5,000).
As a result,
ttot, i.e., the
race record, is 1,068.2 (s), and is exactly the same to
tmax, which is
derived from maximal average speed,
Vmax (5,000).
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Fig. 3.
Example of race-pace allocation
(X1 = 3,500, X2 = 1,500 m) in
5,000-m run for runner whose
VF and
D' are 4.4 (m/s) and 300 (m),
respectively. If runner runs at
Vmax (5,000m)
by using entire D' (hatched bar)
throughout race,
tmax (5,000m)
(= 1,068.2 s) and
Vmax (5,000m)
(= 4.68 m/s) are determined, respectively, by crossing point of
runner's best
V-t
curve and distance curve for 5,000 m ( ).
A: if chosen speed is 4.50 m/s
(V1  VF) during
initial 3,500 m, runner uses 77.8 m as
D'(3,500)
(the white area of bar).
t1 (= 777.8 s) is
determined by crossing point (i.e., large  ) of runner's
V-t
curve to
D'(3,500)
(= 77.8 m) and distance curve for 3,500 m.
B: consequently, for second 1,500 m,
runner has access only to remaining
D'(1,500)
(= 222.2 m, hatched rectangle) for last spurt.
V2 (= 5.165 m/s)
and t2 (= 290.4 s) are also determined by crossing point (i.e., small of runner's
V-t
curve and distance curve to 1,500 m, respectively). Total running time
(ttot) is,
naturally, the sum of
t1 and
t2 (= 1,068.2 s):
this is achievable as described in ALLOCATION OF
RUNNING PACE. C: if
runner chooses initial speed of 4.21 m/s
(V1 < VF) for
X1,
t1 is 831.4 s
(large ). Note, however, that runner has reserved the whole
D' (hatched area) for last
spurt. D: in second 1,500 m, runner
can run at speed,
Vmax (1,500m)
(= 5.501 m/s), which is determined by crossing point (small )
between runner's best
V-t
curve to D' (= 300 m,
hatched area) and distance curve to 1,500 m
(t2 = 272.7 s).
To attain goal of achieving time equivalent to that of running entire
distance at constant
Vmax (1,500m),
runner has to choose a speed of ~6.33 m/s ( ). However, it is
impossible due to "metabolic" limitation. As a result,
ttot (= 1,104.1 s) is necessarily longer than
tmax. See text
for further details.
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Consider, however, if the runner chooses the speed during the initial
part that is below
VF, e.g., 4.21 (m/s) as in V1 < VF. This
corresponds to 10.0% below
Vmax (5,000)
for the initial 3,500 m of the 5,000-m distance. Throughout
X1, the runner
can reserve the whole D' for the
last spurt (i.e., remaining energy). Note, however, that 831.4 (s) is
already taken as
t1 (Fig.
3C). This requires the second 1,500 m to be run in a
t2 of 236.8 (s)
(i.e., required time). In the second 1,500 m, the athlete
can run at the maximal speed
[Vmax (1,500)],
which is determined by the crossing point between the runner's best
V-t
curve to D' (= 300 m) and the
distance curve to 1,500 m. Throughout the second part,
t2 is 272.7 (s)
and V2 is 5.501 (m/s), which is about +25.0% greater than
Vmax (5,000)
(Fig. 3D). As a result,
ttot is 1,104.1 (s); this is some 36 s longer than
tmax. This is the
case for all durations of suboptimal early race pace that are below
VF.
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GENERAL CONSIDERATIONS |
Here we simulate these relationships systematically for the
two-division pace-allocation strategy in the 5,000 m. Furthermore, we
also examine the effect of
VF and
D' 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.,
VF and
D' are ~5.0 (m/s) and ~150
(m), respectively.
Consider the expected race performance when the athlete runs a 5,000-m
distance at a maximal average speed, i.e., as determined by the
V-t
hyperbola: in this case
Vmax is 5.155 (m/s) and ttot is
970 (s). We examined the systematic effect of pace allocation by
utilizing two different speeds
(V1,
V2) in
different combinations of
X1-X2
to the total 5,000 m.
X1 and
X2 (m) were
chosen to be 1,000-4,000, 2,000-3,000, 3,000-2,000, and 4,000-1,000. In
each X1-X2
combination, we calculated the parameters according to
Eqs. 9-14.
V1 was varied
systematically, from 20% of
Vmax (5,000)
to the maximum within the limit set by
D', as described in
ALLOCATION OF RUNNING PACE.
We confirmed by simulation that, if
V1 is below
VF, the final
race time (i.e.,
ttot) is always
longer than
tmax (5,000)
(see Fig. 4). In the range of
V1 below
VF,
ttot was
gradually increased as a function of decreasing
V1. This effect
was, naturally, more pronounced the longer was
X1. The upper
limits of V1 are
automatically determined by the limit of the best
V-t
curve of the subject and each distance curve to
X1. That is, by
definition no one can run at the speed above the intersection of both
curves.
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Fig. 4.
ttot for 5,000-m
race and speed V1
in initial part
X1 for simulated
2-division-pace allocation (for several combinations of
X1-X2).
See text for further details.
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If t1 was
relatively longer for
X1, the runner
would have to run considerably faster in
X2 as shown in
the relationship between t1 and
V2 in Fig.
5 (top
left). The dotted line shows the
V2 required to
keep the same
ttot to
tmax (X),
and the bold line shows the calculated
V2. For example,
in the specific case of the combination of
X1-X2,
at the shortest
t1
(point 1 in Fig. 5,
top left),
V2 is exactly the
same as VF as a
result. This point means that, the runner having run at
Vmax (X1)
for the initial
X1, the required
speed for the remaining
X2 distance is VF, i.e., the
upper (theoretical) limit of "fatigue-free" running speeds (i.e.,
strategy 1, Fig. 5). If
t1 took longer,
the athlete would have to run faster than
VF for
X2, i.e., at
V1 given by the
dotted line (Fig. 5, top left),
which is the required
V2 for the same
running time to
tmax (X).
However, in the range of
t1 above the
deflection point (point 2 in Fig. 5,
top left) of the bold line, the
actual calculated
V2 is dissociated
from the required
V2 (dotted line),
due to the "metabolic" limitation. This means that
V2 cannot exceed
the maximal speed set by the best
V-t
curve, as determined by the whole
D' and distance
X2 (i.e.,
strategy 3, Fig. 5, e.g.,
point 3 in Fig. 5,
top left). As a result,
this deflection point ( point 2 in Fig. 5, top left) means
that the athlete must run at
VF for
X1 and at
Vmax(X2) for X2 (i.e.,
strategy 2, Fig. 5). Therefore, the
range between these points [i.e., the range over
which ttot can
equal to
tmax (X)] represents the maximal possible range for the strategic modification of
the race pace.
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Fig. 5.
Schematic representation of relationship of
t1 and
V2 to
"t range," i.e.,
"permitted" pace allocation for single change of pace.
t-Range is determined by 2 extreme
strategy conditions: 1)
V1 = Vmax (X1)
and V2 = VF and
2)
V1 = VF and
V2 = Vmax (X2).
If runner chooses strategy 3 {bottom right,
left and
right;
[V1 < VF and
V2 = Vmax (X2)]}
intentionally, V2
is metabolically limited from achieving required speed (dotted line on
top left). Each "strategy,"
shown as point nos. (1-3;
top left), is represented in
graphical detail on right (top to
bottom, respectively). Symbols are
defined as in Fig. 3.
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These conditions are described by the following equations. In
strategy 1 ( point 1 in Fig. 5,
top left),
V1 is
Vmax(X1) and V2 is
VF,
t1 and
t2 are
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(15)
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In the other extreme condition, strategy
2 ( point 2 in
Fig. 5, top left),
V1 is
VF and
V2 is
Vmax(X2),
t1 and
t2 are
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(16)
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The range of
t1 between these
two extreme strategic conditions is
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(17)
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and
also
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(18)
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This demonstrates an important property of
D' and
VF on the
possible time range
(
t1 = t2) for a
single strategic change of pace. Interestingly, the range is not
affected by the total distance, X, or
by any combination of
X1-X2
or
V1-V2;
it is purely determined by the runner's "endurance parameter
ratio":
D'/VF.
Consider also the difference of speeds between two extreme strategic
conditions (i.e., conditions 1 and
2)
![&Dgr;<IT>V</IT><SUB>1</SUB> = <IT>V</IT><SUB>1(<IT>1</IT>)</SUB> − <IT>V</IT><SUB>1(<IT>2</IT>)</SUB> = <IT>V</IT><SUB>F</SUB> ⋅ [<IT>D</IT>′/(<IT>X</IT><SUB>1</SUB> − <IT>D</IT>′)]](/content/vol87/issue2/fulltext/853/img024.gif)
|
(19)
|
![&Dgr;<IT>V</IT><SUB>2</SUB> = <IT>V</IT><SUB>2(<IT>2</IT>)</SUB> − <IT>V</IT><SUB>2(<IT>1</IT>)</SUB> = <IT>V</IT><SUB>F</SUB> ⋅ [<IT>D</IT>′/(<IT>X</IT><SUB>2</SUB> − <IT>D</IT>′)]](/content/vol87/issue2/fulltext/853/img025.gif)
|
(20)
|
where
X1 > D' and
X2 > D'. This possible speed
V2 range for the
distance X2
(V2;
V2 range) that is
permitted for the strategic pace allocation (including the last spurt)
is dependent on
X2.
To examine the effects of
VF and
D' on the time range and
V2 range, we
plotted the
t1-V2
relationship in which
VF or
D' was changed systematically.
The effect of D' on the
t1-V2
relationship under the same
VF condition,
i.e., 5.0 (m/s) is shown in Fig. 6A. As a
result, the larger D'
dramatically extended the time range and
V2 range for the
5,000-m run. However, compared with the effect of
D', the effect of
VF was relatively
small (Fig. 6B). This indicates that
D' is relatively important for
the runner's possible range of pace change: the larger the
D', the larger the potential
pace change in the race.
VF is, of course,
an essential determinant of
tmax (X),
i.e., the actual level of the race time (Fig.
6B). Therefore, although
VF seems to be
the major determinant of the level of race performance,
D' is an important determinant
of the flexibility of race-pace strategy.
View larger version (24K):
[in this window]
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|
Fig. 6.
V2-t1
relationship for establishing effect of
D'
(A) and
VF
(B) on
t range and
V2 range in
2-division-pace simulation. See text for further details.
|
|
| |
ADDITIONAL REMARKS ON THE ANALYSIS |
P-V
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 (6-8) 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 the
V-t
relationship for treadmill running (by using cross-country 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 middle-distance
running (i.e., from 800 to 5,000 m). It was shown that the
V-t
curve 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, the
V-t
hyperbolic relationships for treadmill running in subelite middle-distance 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 middle-distance running is likely to be within about ±1.0 m/s. For
example, by using the world-class record, the last lap of 52 s in a
3,000-m 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 the
P-t
curve 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
The
P-t
relationship for high-intensity 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 F
and W' remain conjectural, the
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 acid-base status, and blood
lactate concentration, given sufficient time (14). Therefore,
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 (13-15), although 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 O2 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 F,
regardless of its rate of performance.
W' was not increased by
endurance training, whereas F
increased systematically; the
W-t
curves 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 glycogen-depleted conditions.
Application Limits of the Hyperbolic Curve
The mechanical limitations of the muscle for high-intensity (i.e.,
short-duration) exercise is relevant to the condition of X < D' in our analysis. For such
short distances, there is no intersection between the
V-t
curve 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 whose
VF and
D' are 5.0 (m/s) and 150 (m),
respectively, as discussed in GENERAL
CONSIDERATIONS, runs 200 m at constant
Vmax (200),
the Vmax (200)
and
tmax (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 influence
D' or
VF. 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 entire
W' (or
D') to continue to perform by
decreasing the power output to
F (or
VF),
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 available
W' (or
D'), then the sudden reduction
of the power output to F (or
VF) with W' (or
D') being depleted would, of
course, leave the mechanism(s) of fatigue fully expressed.
Repleting W' (or
D') would require a
recovery power that is, presumably, below
F (or
VF), with the repletion rate likely to be greater the lower the recovery power below F (or
VF). Further
studies are required, however, to determine the intensity-time features
of this repletion process.
Characteristics of
D'/VF
As stated in GENERAL CONSIDERATIONS,
the index
D'/VF
(or
W'/F,
which we term the endurance parameter ratio) appears to be an important
determinant of the race-pace strategy. Here, we further consider the
meaning of this index. One might rewrite Eq.
1 as
![<IT>W</IT>′/&thgr;<SUB>F</SUB> = <IT>t</IT> ⋅ [(<IT>P</IT>/&thgr;<SUB>F</SUB>) − 1]](/content/vol87/issue2/fulltext/853/img026.gif)
|
(21)
|
Interestingly, the endurance parameter ratio
(W'/F)
corresponds to the time for which
P/F
is equal to two: that is, the limiting time for an athlete to perform a
power output that is twice F
(i.e., t at which
W'/F = 2). This metabolic relationship is similar to that of chronaxie to
rheobase in neurophysiology. If
F of two competitors are the
same, W'
(D' for the running performance)
will determine the flexibility of the intentional race-pace change.
In summary, our results demonstrate that the speed or
power at the fatigue threshold
(VF or
F) and the curvature
constant (D' or
W') parameters of the
athlete's
V-t
or
P-t
hyperbolic 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 below
VF, 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; and
2)
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'/VF
(i.e., the endurance parameter ratio) may therefore be considered to be
an important determinant of race-pace strategy.
| |
APPENDIX |
Theoretical Aspects of General Race-Pace Allocation
In ALLOCATION OF RUNNING PACE, we
considered only the simple situation in which the runner runs at two
different speeds
(V1 and
V2) during two
distances (X1 and
X2) of the
total distance X. Here we expand that
into more general situations in which the runner runs at
n different speeds
(Vi) during
n distances
(Xi) of total
distance X. The total distance
(X) is divided by
n parts (Xi,
i = 1,..,j,..,n).
The runner can run at the different speeds Vi in each part
Xi, the only
assumption being that the total amount of
D' 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 of
D', during the one part
Xj, where Vj is
1) above
VF or
2) below
VF, whereas the
runner chooses speed Vi that equals
or exceeds VF
[i.e.,
V(i 
j) VF]
during the remaining
Xi
Situation 1:
V(i j) VF,
Vj > VF, i = 1,..,j,..,n
This situation means that
Vi VF excluding
Vj, and
Vj > VF during at
least one arbitrary part,
Xj. In such a
situation, ti during any part
Xi is
![<IT>t</IT><SUB><IT>i</IT></SUB> = [<IT>X</IT><SUB><IT>i</IT></SUB> − <IT>D</IT>′<SUB>(<IT>Xi</IT>)</SUB>]/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img027.gif)
|
(22)
|
where
D'(Xi) < Xi, and if
Vi = VF then
D'(Xi) = 0. As an assumption
|
(23)
|
Therefore, the total time for the race
(ttot)
is
|
(24)
|
Compared with Eq. 8
|
(25)
|
That
is, as long as the runner keeps the speed
Vi that exceeds
VF during all
Xi, including
Xj, by using the
whole D' according to his or her
own strategy in the race, the runner can reach the goal at the same
time to
tmax(X).
As a very specific example from the theoretical consideration, if the
runner runs at
Vj above
VF by using the
entire D' during one part,
Xj, the runner
can reach the goal of achieving the same time as for the constant speed
Vmax(X)
race, even if the runner chooses every remaining
Vi to be the
same as VF
Situation 2:
V(i j) VF,
Vj < VF,
i = 1,..,j,..,n
This situation means that
Vi VF excluding
Vj, but
Vj < VF during one part,
Xj. As
Vj is below
VF
|
(26)
|
and
tj is
|
(27)
|
In each remaining part i (i.e.,
i = 1,..,n,
i 
j), as
V(i j)
is above VF
![<IT>t</IT><SUB>(<IT>i</IT>≠<IT>j</IT>)</SUB> = {<IT>X</IT><SUB>(<IT>i ≠ j</IT>)</SUB> − <IT>D</IT>′<SUB>[<IT>X</IT>(<IT>i</IT>≠<IT>j</IT>)]</SUB>}/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img034.gif)
|
(28)
|
where
D'[X(i j)] < X(i j).
Therefore, total running time
(ttot)
is
|
(29)
|
where k = 1,...,n,
k j, and D' = 
Time difference between ttot and
tmax(X)
![− (1/<IT>V</IT><SUB>F</SUB>) ⋅ (<IT>X</IT> − <IT>D</IT>′) = <IT>X<SUB>j</SUB></IT> ⋅ {[1/<IT>V</IT><SUB>F</SUB> − &Dgr;<IT>V</IT>)] − (1/<IT>V</IT><SUB>F</SUB>)}](/content/vol87/issue2/fulltext/853/img039.gif)
|
(30a)
|
From Eq. 26,
V > 0, so
1/VF < 1/(VF V); therefore
|
(30b)
|
As a
consequence, if a runner runs at a speed
Vi which is less
than VF during
any one part of the race,
Xi (i.e.,
Vj in
Xj), the
"lost" time cannot be made up during the remaining parts, even if
the runner chooses all remaining
Vi that are
above VF (but
within the limit of his or own
D'). That is, the runner can never attain the goal of achieving the time equivalent to that of
running the entire race at the constant
Vmax(X).
| |
ACKNOWLEDGEMENTS |
Y. Fukuba was supported by an Overseas Research Fellowship of the
Uehara Memorial Foundation, Japan.
| |
FOOTNOTES |
Present address of Y. Fukuba: Dept. of Exercise Science and Physiology,
School of Health Sciences, Hiroshima Women's University, Hiroshima
734-8558, Japan.
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, Ujina-higashi, Minami-ku,
Hiroshima 734-8558, Japan.
Received 13 March 1995; accepted in final form 26 April 1999.
| |
REFERENCES |
1.
Hill, A. V.
Muscular Movement in Man. New York: McGraw-Hill, 1927.
2.
Hill, D. W.
The critical power concept.
Sports Med.
16:
237-254,
1993[Medline].
3.
Housh, T. J.,
G. O. Johnson,
S. L. McDowell,
D. J. Housh,
and
M. Pepper.
Physiological responses at the fatigue threshold.
Int. J. Sports Med.
12:
305-308,
1991[Medline].
4.
Hughson, R. L.,
C. J. Orok,
and
L. E. Staudt.
A high velocity treadmill running test to assess endurance running potential.
Int. J. Sports Med.
5:
23-25,
1984[Medline].
5.
Kan, A.,
T. Inamizu,
N. Yasuda,
and
Y. Fukuba.
The velocity-duration time hyperbolic curve as an index of middle distance performance (Abstract).
Physiologist
39:
A78,
1996.
6.
Komi, P. V.,
A. Ito,
B. Sjödin,
R. Wallenstein,
and
J. Karlsson.
Muscle metabolism, lactate breaking point, and biomechanical features of endurance running.
Int. J. Sports Med.
2:
148-153,
1981[Medline].
7.
Luhtanen, P.,
P. Rahkila,
H. Rusko,
and
J. T. Viitasalo.
Mechanical work and efficiency in treadmill running and aerobic and anaerobic thresholds.
Acta Physiol. Scand.
139:
153-159,
1990[Medline].
8.
McMahon, T. A.
Mechanics of locomotion.
In: Muscles, Reflexes, and Locomotion. Princeton, NJ: Princeton Univ. Press, 1984, p. p.189-233.
9.
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.
Ergonomics. In press.
10.
Monod, H.,
and
J. Scherrer.
The work capacity of a synergic muscular group.
Ergonomics
8:
329-338,
1965.
11.
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[Medline].
12.
Pepper, M. L.,
T. J. Housh,
and
G. O. Johnson.
The accuracy of the critical velocity test for predicting time to exhaustion during treadmill running.
Int. J. Sports Med.
13:
121-124,
1992[Medline].
13.
Peronnet, F.,
and
G. Thibault.
Mathematical analysis of running performance and world running records.
J. Appl. Physiol.
67:
453-465,
1989[Abstract/Free Full Text].
14.
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[Medline].
15.
Poole, D. C.,
S. A. Ward,
and
B. J. Whipp.
The effects of training on the metabolic and respiratory profile of high-intensity cycle ergometer exercise.
Eur. J. Appl. Physiol.
59:
421-429,
1990.
16.
Roston, W. L.,
B. J. Whipp,
J. A. Davis,
D. A. Cunningham,
R. M. Effros,
and
K. Wasserman.
Oxygen uptake kinetics and lactate concentration during exercise in humans.
Am. Rev. Respir. Dis.
135:
1080-1084,
1987[Medline].
17.
Sjödin, B.,
and
I. Jacobs.
Onset of blood lactate accumulation and marathon running performance.
Int. J. Sports Med.
2:
23-26,
1981[Medline].
18.
Wakayoshi, K.,
K. Ikuta,
T. Yoshida,
M. Udo,
T. Moritani,
Y. Mutoh,
and
M. Miyashita.
The determination and validity of critical velocity as swimming performance index in the competitive swimmer.
Eur. J. Appl. Physiol.
64:
153-157,
1992.
19.
Ward-Smith, A. J.
A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world-class athletes.
J. Biomech.
18:
337-349,
1985[Medline].
20.
Whipp, B. J.,
D. J. Huntsman,
N. Stoner,
N. Lamarra,
and
K. Wasserman.
A constant which determines the duration of tolerance to high-intensity work.
Federation Proc.
41:
1591,
1982.
21.
Whipp, B. J.,
and
S. A. Ward.
Pulmonary gas exchange dynamics and the tolerance to muscular exercise: Effects of fitness and training.
Ann. Physiol. Anthrop.
11:
207-214,
1992[Medline].
22.
Whipp, B. J.,
and
S. A. Ward.
Respiratory response of athletes to exercise.
In: Oxford Textbook of Sports Medicine, edited by M. Harries,
L. J. Micheli,
W. D. Stanish,
and C. Williams. New York: Oxford Univ. Press, 1994, p. 13-25.
23.
Wilkie, D. R.
Equations describing power input by humans as a function of duration of exercise.
In: Exercise Bioenergetics and Gas Exchange, edited by P. Cerretelli,
and B. J. Whipp. New York: Elsevier, 1980, p. 75-80.
J APPL PHYSIOL 87(2):853-861
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Copyright © 1999 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
![<IT>t</IT><SUB>2</SUB> = <IT>X</IT><SUB>2</SUB>/<IT>V</IT><SUB>2</SUB> = [<IT>X</IT><SUB>2</SUB> − <IT>D</IT>′<SUB>(<IT>X</IT>2)</SUB>]/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img012.gif)
![<IT>t</IT><SUB>tot</SUB> = <IT>t</IT><SUB>1</SUB> + <IT>t</IT><SUB>2</SUB> = [<IT>X</IT><SUB>1</SUB> − <IT>D</IT>′<SUB>(<IT>X</IT>1)</SUB>]/<IT>V</IT><SUB>F</SUB> + [<IT>X</IT><SUB>2</SUB> − <IT>D</IT>′<SUB>(<IT>X</IT>2)</SUB>]/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img014.gif)
![<IT>t</IT><SUB>tot</SUB><SUP>*</SUP> = <IT>t</IT><SUB>1</SUB> + <IT>t</IT><SUB>2</SUB> = (<IT>X</IT><SUB>1</SUB>/<IT>V</IT><SUB>1</SUB>) + [(<IT>X</IT> − <IT>X</IT><SUB>1</SUB>) − <IT>D</IT>′]/<IT>V</IT><SUB>F</SUB>](/content/vol87/issue2/fulltext/853/img018.gif)
![<IT>t</IT><SUB>tot</SUB> = <LIM><OP>∑</OP><LL><IT>i</IT>=1</LL><UL><IT>n</IT></UL></LIM> <IT>t</IT><SUB><IT>i</IT></SUB> = <LIM><OP>∑</OP><LL><IT>i</IT>=1</LL><UL><IT>n</IT></UL></LIM> {[<IT>X</IT><SUB><IT>i</IT></SUB> − <IT>D</IT>′<SUB>(<IT>Xi</IT>)</SUB>]/<IT>V</IT><SUB>F</SUB>}](/content/vol87/issue2/fulltext/853/img029.gif)
