Research Article
Modelling of Running Performances:
Comparisons of Power-Law, Hyperbolic, Logarithmic,
and Exponential Models in Elite Endurance Runners
H. Vandewalle
UFR de Sant´e, M´edecine et Biologie Humaine, Universit´e Paris XIII, Bobigny, France
Correspondence should be addressed to H. Vandewalle; henry.vandewalle@club-internet.fr
Received 1 April 2018; Revised 2 August 2018; Accepted 2 September 2018; Published 3 October 2018
Academic Editor: Ronald E. Baynes
Copyright © 2018 H. Vandewalle. Thisis an open accessarticle distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Many empirical and descriptive models have been proposed since the beginning of the 20th century. In the present study, the power-
law(Kennelly)andlogarithmic (P´eronnet-Thibault)modelswere comparedwithasymptotic modelssuchas2-parameter hyperbolic
models (Hill and Scherrer), 3-parameter hyperbolic model (Morton), and exponential model (Hopkins). These empirical models
were compared from the performance of 6 elite endurance runners (P. Nurmi, E. Zatopek, J. V¨a¨at¨ainen, L. Vir´en, S. Aouita, and H.
Gebrselassie) who were world-record holders and/or Olympic winners and/or world or European champions. These elite runners
were chosen because they participated several times in international competitions over a large range of distances (1500, 3000, 5000,
and 10000 m) and three also participated in a marathon. The parameters of these models were compared and correlated. The less
accurate models were the asymptotic 2-parameter hyperbolic models but the most accurate model was the asymptotic 3-parameter
hyperbolic model proposed by Morton. The predictions of long-distance performances (maximal running speeds for 30 and 60 min
and marathon) by extrapolation of the logarithmic and power-law models were more accuratethan the predictions by extrapolation
in all the asymptotic models. The overestimations of these long-distance performances by Morton’s model were less important than
the overestimations by the other asymptotic models.
1. Introduction
Many models [1–11] of running performances based on
biomechanics and physiology have been proposed. These
models are generally complex. For example, the physiological
model proposed by P´eronnet and Thibault [7] included the
inertia, power, and capacity of the anaerobic and aerobic
metabolisms.
Empirical and descriptive models have also been pro-
posed since the beginning of the 20th century and presented
in many reviews [12–21]. Empirical models are derived by
observation and experimentation rather than by theoretical
considerations [14]. The empirical models are less complex
than the biomechanical and physiological models but are also
less explicative. The most famous empirical models corre-
sponded to a power-law model (Kennelly, 1906), asymptotic
hyperbolic models (Hill, 1927; Scherrer, 1954), and, more
recently, a logarithmic model (P´eronnet and Thibault, 1987)
and 3-parameter asymptotic models (Hopkins, 1989; Morton,
1996). The asymptotic models correspond to horizontal
asymptote equations: the functions approach a horizontal line
when tlim tends to infinity. In these models, it is assumed that
the speeds lower than these asymptotes can be maintained
infinitely.
The empirical models of running exercises are often used
to estimate
(i) the improvement in performance [22]
(ii) the effects of age [23, 24] and sex [25, 26] on running
performance
(iii) the future performances and running speeds over
given distances
(iv) the endurance capability [7, 8], that is, “the ability
to sustain a high fractional utilization of maximal
oxygen uptake for a prolonged period of time”
Hindawi
BioMed Research International
Volume 2018, Article ID 8203062, 23 pages
https://doi.org/10.1155/2018/8203062
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Table 1: Individual performances (in seconds) of elite endurance runners.
1500
3000
5000
10000
Marathon
Nurmi
233
500
868
1806
Zatopek
233
488
837
1734
8583
V¨a¨at¨ainen
224
473
808
1672
Vir´en
222
463
796
1658
7991
Aouita
209
449
778
1646
Gebrselassie
214
445
759
1583
7439
(v) the speed of training sessions [27]
(vi) the maximal aerobic speed [7, 8]
The maximal aerobic speed, otherwise known as MAS, is
the lowest running speed at which maximum oxygen uptake
(V02 max) occurs, and is also referred to as the velocity at
V02 max (vV02 max). MAS is useful for training prescrip-
tion and monitoring training loads. P´eronnet and Thibault
suggested estimating MAS by computing the maximal speed
corresponding to 7 min [8]. The maximal lactate steady state,
defined as the highest constant power output that can be
maintained without a progressive increase in blood lactate
concentration, is usually sustainable for 30 to 60 min. [28–
30].
The first studies on the modelling of running perfor-
mances were based on the world records because these
records measured under standard external conditions repre-
sent the most reliable index of human performance [31, 32].
The running times of the slower runners are more variable
than those of the faster runners [33]. The best performances
of world elite runners are probably very close to their
maximal performances because they generally correspond to
the results of many competitions against other elite runners
and the motivation is probably optimal during these races.
Now, the best performances of elite endurance runners who
ran on different distances and were the best of their times
can be found on the Internet (Wikipedia, etc.). Therefore, it
is possible to study the characteristics of the different models
which have been proposed for endurance exercises with the
best performances of elite endurance runners.
The performances of different runners were used in each
study on the modelling of world and Olympic records [7,
22, 31, 32, 34, 35]. In contrast, in the present investigation,
each model was computed only from the performances
of a single runner. The computations of each model were
repeated for different world elite endurance runners (P.
Nurmi, E. Zatopek, J. V¨a¨at¨ainen, L. Vir´en, S. Aouita, and
H. Gebrselassie) who were world-record holders and/or
Olympic winners and/or world or European champions.
They participated several times in international competitions
over the same distances (1500, 3000, 5000, and 10000 m)
that corresponded to a large range of distances. Their best
individual performances are presented in Table 1.
Moreover, if a model is not perfect for a large range of
performances, the values of its parameters computed from
different ranges of distances will be significantly different. In
the present study, the parameters of the different models were
computed with 3 ranges of distances:
(i) 1500-3000-5000-10000 m for the largest range
(ii) 1500-3000-5000 m, which is equivalent to the range
of tlim generally used in the studies on critical speed
or critical power (from 3 to 15 min)
(iii) 3000-5000-10000 m, which corresponds to exercises
slower than maximal aerobic speed
Several previous investigations studied the evolution of
the parameters in the models of running performances at
different times [22, 34]. Similarly, the six elite endurance
athletes of the present study ran at different times and
their performances were performed in different conditions
(cinder tracks versus synthetic tracks, nutrition, etc.) and
were the results of different running exercises (for example,
an equivalent of fartleck for Nurmi, an equivalent of interval-
training for Zatopek, and altitude training for Gebrselassi´e),
which could partly explain the evolution of the performances
in these world elite runners and could also change the best
model of individual running performances.
The present study (1) applied the power-law and logarith-
mic models and four asymptotic models (two 2-parameter
hyperbolic models, a 3-parameter hyperbolic model, and a
3-parameter exponential model) to the individual perfor-
mances of the elite runners, (2) compared the accuracy of
these models and the effects of the range of performances on
their parameters to assess which is the best model, and (3)
compared the predictions of MAS by interpolation and the
prediction of maximal running speeds for long distances (30,
60 min and also marathon in 3 runners) by extrapolation.
2. History of the Power-Law, Hyperbolic,
Logarithmic, and Exponential Models
2.1. Power-Law Model (Kennelly). In 1906, Kennelly [12]
studied the relationship between running speed (S) and the
time of the world records (tlim) and proposed a power law:
Dlim = ktlim
g
(1)
where k is a constant and g an exponent. This power law
between distance and time corresponds to a power law
between time and speed (S):
S = Dlim
tlim
= ktlim
g
tlim
= ktlim
g - 1
(2)
Exponent g is probably an expression of endurance capability.
Indeed, the tlim-Dlim relationship would be perfectly linear if
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g is equal to 1. It is likely that the curvatures of the tlim-S and
tlim-Dlim relationships depend on the decrease in the fraction
of maximal aerobic metabolism that can be sustained during
long lasting exercises. The value of exponent g is independent
of scaling as it is independent of the expression of tlim, S, and
Dlim.
In theory, parameter k should be correlated to maximal
running speed because k is equal to the maximal running
speed corresponding to one second. Indeed, when tlim is equal
to 1s
S = ktlim
g – 1 = k ∗ 1g - 1 = k ∗ 1 = k
(3)
In 1981, a similar power-law model was proposed by Riegel
[36]:
tlim = aDlim
b
(4)
S = Dlim
tlim
=
Dlim
aDlim
b =
(Dlim
1 - b)
a
(5)
As Dlim = ktlim
g
Dlim
1/g = (ktlim
g)1/g = k1/gtlim
tlim = Dlim
1/g
k1/g
=
(Dlim
1 - b)
a
a = k1/g
(6)
and
Dlim
1/g = (Dlim
1 - b)
1
g = 1 - b
b = 1 - 1
g = (g – 1)
g
(7)
These equations of Riegel have recently been applied to a large
study on 2303 recreational endurance runners [37].
2.2. Hyperbolic Model (Hill, Scherrer). In 1927, Hill [1] pro-
posed a hyperbolic model to describe the world-record curve
in running and swimming. Hill observed that the “running
curve,” or the relationship between a runner’s power output
(P) and the total duration of a race (T), can be described by a
hyperbolic function:
P = (A
T ) + R
(8)
where A and R represent the capacity of anaerobic
metabolism and the rate of energy release from aerobic
metabolism, respectively. In 1954, Scherrer et al. proposed a
linear relationship [38] between the exhaustion time (tlim)
of a local exercise (flexions or extensions of the elbow or the
knee) performed at different constant power outputs (P) and
the total amount of work performed at exhaustion (Wlim) for
tlim ranging between 3 and 30 minutes:
Wlim = a + btlim
(9)
Consequently, the relationship between P and tlim is hyper-
bolic:
Wlim = Ptlim = a + btlim
tlim =
a
(P – b)
(10)
After the publication of an article in English (1965) by Monod
and Scherrer [39], Ettema (1966) applied the critical-power
concept to world records in running, swimming, cycling,
and skating exercises [40] and proposed a linear relationship
between Dlim and tlim for world records from 1500 to 10000
m:
Dlim = a + btlim
(11)
where tlim corresponded to the world record for a given
distance (Dlim). It was assumed that the energy cost of
running, i.e., the energy expenditure per unit of distance, was
almost independent of speed under 20 km.h−1. Consequently,
Dlim and parameter a were equivalent to amounts of energy.
Therefore, parameter a has been interpreted as equivalent
to an energy store and an estimation of maximal Anaerobic
Distance Capacity (ADC expressed in metres) for running
exercises whereas slope b was considered as a critical velocity
(SCrit).
Dlim = ADC + SCrit1tlim
(12)
tlim =
ADC
(S – SCrit1)
(13)
However, the linear Wlim-tlim was an approximation as
indicated by Scherrer and Monod (1960): “The relationship
W = f(t) is not perfectly linear as shown on Figure 2(a), where
the curves tend towards abscissa beyond 30 minutes” [41]. In
the study by Ettema in 1966, SCrit and ADC depended on the
range of tlim, which was confirmed by more recent studies
[42, 43].
In 1981, the linear Wlim-tlim relationship was adapted to
exercises on a stationary cycle ergometer and it was demon-
strated that slope b of the Wlim-tlim relationship was highly
correlated with the ventilatory threshold [44]. Therefore,
slope b was proposed as an indicator of general endurance
and the concept of critical power or critical velocity was again
studied. Different equations were proposed for the estimation
of SCrit (or CP). For example, SCrit on a treadmill [45] was
computed from the linear relationship between Dlim and the
inverse of tlim (1/tlim):
S = a ( 1
tlim
) + b = ADC2 ( 1
tlim
) + SCrit2
(14)
More recently, Morton [15] proposed a fourth model for
the critical power, a nonlinear model including a third
parameter corresponding to maximal instantaneous power
(Pmax). This model has been adapted to running exercises
with an instantaneous maximal running speed (SMax):
tlim =
ADC3
(S – SCrit3) –
ADC3
(SMax - SCrit3)
(15)
Actually, the different asymptotic hyperbolic models are the
most used and studied [46].
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2.3. Logarithmic Model (P´eronnet-Thibault). The metabolic
model proposed by P´eronnet and Thibault [7, 8] included
factors that took into account the contributions of aerobic and
anaerobic metabolism to total energy output according to the
duration of the race. The inertia of the aerobic metabolism at
the beginning of the exercise was also included in the model.
In addition, the use of anaerobic store SA was assumed to
decrease beyond TMAP (exhaustion time corresponding to
maximal aerobic power):
SA = A
for T ≤ TMAP
SA = A – 0.233A ln (
T
TMAP
)
for T > TMAP
(16)
A runner is only capable of sustaining his maximal aerobic
power for a finite period of time. The performances in
long distance events depend on the ability to utilize a large
percentage of VO2max over a prolonged period of time
(endurance capability). P´eronnet and Thibault [7, 8] assumed
that tlim corresponding to maximal aerobic speed (tMAS) is
equal to 7 min. They proposed the slope (E) of the relationship
between the fractional utilization of MAS and the logarithm
of tlim/7min (420 s) as an index of endurance capability:
S = MAS – E7min ln ( tlim
420)
100 S
MAS = 100 – E ln ( tlim
420)
(17)
where MAS is the maximal running speed corresponding
to 7 min and E is the endurance index corresponding to
MAS (E =100 E7min/MAS). There was a significant correla-
tion between the ventilatory threshold and E in marathon
runners [47], which suggested that E was an index of aerobic
endurance. The values of E and MAS7min can be estimated
from two running performances with a nomogram [48].
2.4. Exponential Model. Hopkins et al. [13] have presented
an asymptotic exponential model for short-duration (10 s - 3
min) running exercises on a treadmill with 5 different slopes
(9 to 31%). This model was
It = I∞ + (I0 – I∞) exp (–tlim
𝜏 )
(18)
where I∞ is the slope corresponding to infinite time, I0 the
slope corresponding to a time equal to zero, It the slope
corresponding to tlim, and 𝜏 is a time constant. This model
can be adapted to running exercises on a track:
S = S∞ + (S0 – S∞) exp (–tlim
𝜏 )
(19)
This asymptotic exponential model derived from Hopkins’
model has been used and compared to the different asymp-
totic hyperbolic models in several studies [49–52].
3. Methods
The logarithmic, power-law, and hyperbolic models which
are 2-parameter models were computed by linear least-square
regressions between time data and speed data (or distance
data). Time data correspond to tlim or the logarithm of tlim.
Speed data correspond to speed or the logarithm of speed.
The models by Morton and Hopkins are 3-parameter models
whose individual regressions were computed by an iterative
least square method.
3.1. Computation of the Empirical Models
3.1.1. Computation of the Power-Law Model. If Y = A∗X, the
logarithm of Y is equal to
ln (Y) = ln (A) + ln (X)
(20)
If Y = X-B, the logarithm of Y is equal to
ln (Y) = -B ln (X)
(21)
If Y = A∗X- B, the logarithm of Y is equal to
ln (Y) = ln (A) - B ln (X) = C - B ln (X)
(22)
where C = ln(A) and exp(C) = exp[ln(A)] = A.
Therefore, the power laws between tlim and Dlim or S
can be determined by computing the regression between the
natural logarithms of Dlim and tlim:
ln (Dlim) = 𝛼 + 𝛾 ln (tlim) = ln (k) + g ln (tlim)
k = eln(k) = e𝛼
(23)
3.1.2. Computation of the Hyperbolic Models. In the present
study, three estimations of critical velocity (SCrit1, SCrit2, and
SCrit3) were computed:
Dlim = ADC1 + SCrit1tlim
Y = 𝛼1 + 𝛽1X
(12 bis)
where Y = Dlim; X = tlim; 𝛼1 = ADC1; 𝛽1 = SCrit1
S = a + b ( 1
tlim
) = SCrit2 + ADC2 ( 1
tlim
)
Y = 𝛼2 + 𝛽2X
(14 bis)
where Y = S; X = 1/tlim; 𝛼2 = SCrit2; 𝛽2 = ADC2
In the 3-parameter model by Morton
tlim =
ADC3
(S – SCrit3) –
ADC3
(SMax - SCrit3)
(15)
Let C = ADC3/(SMax - SCrit3)
tlim =
ADC3
(S – SCrit3) – C
SMax = SCrit3 + ADC3
C
(24)
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First, this equation was computed by an iterative least square
method for a hyperbolic decay formula with 3 parameters
(Y0, a, and b):
Y = Y0 +
ab
(x + b)
(25)
where Y0 = - C, b = - SCrit3, and ab = ADC3
Unfortunately, there was no convergence of the iteration.
Therefore, an iteration was tested for another equation:
tlim + C =
ADC3
(S – SCrit3)
S – SCrit3 =
ADC3
(tlim + C)
S = SCrit3 +
ADC3
(tlim + C)
(24 bis)
This equation was computed with an iterative least square
method for a similar hyperbolic decay formula with 3
parameters (Y0, a, and b):
Y = Y0 +
ab
(x + b)
(26)
where Y = S, Y0 = SCrit3, ab = ADC3, and b = C.
As the value of Smax = SCrit3 + ADC/C
Smax = Y0 + ab
b = Y0 + a
(27)
Fortunately, there was a convergence in the iteration for this
equation.
3.1.3. Computation of the Logarithmic Model. The value of E
was estimated by computing the regression between S and the
logarithm of tlim/420 for the different distances:
S = 𝛼 – 𝛽 ln ( tlim
420)
(28)
When tlim = 420, S is equal to MAS and ln(tlim/420) is equal
to 0. Therefore
S = MAS = 𝛼 + 0
E = 100𝛽
MAS = 100𝛽
𝛼
(29)
3.1.4. Computation of the Exponential Model. At least three
distances are necessary to compute Hopkins’ model (see
(19)) which is a three-parameter model (S∞, a1, and b1) like
Morton’s model.
S = S∞ + (S0 – S∞) exp (−tlim
𝜏 )
= S∞ + a exp (–btlim)
(30)
The regressions were computed by an iterative least square
method for a single exponential decay formula with 3 param-
eters (Y0, a, and b):
Y = Y0 + 𝛼 exp (−𝛽X)
(31)
where X = tlim, Y0 = S∞, 𝛼 = a, and 𝛽 = b
3.2. Estimations of Maximal Running Speeds corresponding
to 7, 30, and 60 Minutes. The estimations of the individual
maximal running speeds corresponding to 7 minutes (esti-
mation of maximal aerobic speed, MAS) were performed by
interpolation from the 1500-3000-5000m performances.
The estimations of the maximal running speed during 30
min were done by extrapolation from the 1500-3000-5000m
performances. The 30-min running times were compared
with the 10000 m performances (S10000).
The estimations of the maximal running speed during 60
min were done by extrapolation from the 1500-3000-5000-
10000 m performances.
3.3. Accuracy of the Estimations of Running Speed. The
individual running speeds corresponding to the different
distances (1500, 3000, 5000, and 10000 m) were estimated
from the individual regressions of the different models and
compared with the actual speeds for the same distances.
First, for each model, the individual running speeds corre-
sponding to tlim between 1 and 1900 s were computed from
the individual regressions with an increment equal to 1 s.
Secondly, the individual relationships between distance and
the estimated value of tlim were computed by multiplying tlim
and the corresponding estimated speed (distance = speed
x time). Then, the individual estimated values of running
speed corresponding to 1500, 3000, 5000, and 10000 m were
registered and compared with the actual values of running
speeds.
Thereafter, the ratios of estimated speed to actual speed
were computed for each distance and each runner.
3.4. Statistics. All the computations of the model and the
statistics were performed with the SigmaPlot software (Systat,
Chicago, USA).
3.4.1. Comparisons of the Parameters. The comparisons of
the parameters, computed from different ranges of distances
or from different running models (SCrit1, SCrit2, SCrit3, S∞,
SMax, and S0), were studied with a nonparametric paired test
(Wilcoxon signed rank test) since the sample sizes were low
(6 runners). Significance was accepted at critical P<0.05. The
probability was equal to 0.031 in Wilcoxon signed rank test
when all the individual values of a parameter are either lower
or higher than all the corresponding individual values of a
parameter in another model (or another performance range).
3.4.2. Comparison of the Accuracy in the Different Models.
In statistics, the sum of the squares of residuals (deviations
predicted from actual empirical values of data) is a measure
of the discrepancy between the data and an estimation model.
A small sum of the squares of residuals indicates a tight fit of
the model to the data.
However, in the present study, the comparisons of the
accuracy in the different models cannot be based on the
differences in the sums of the squares of residuals because
the residuals in the power-law model corresponded to the
logarithm of the residuals and because the individual regres-
sion of the first hyperbolic model (SCrit1) did not correspond
to regressions between tlim and running speeds (S) but
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regressions between tlim and distances (Dlim). Moreover, it
would be assumed that there was a homoscedasticity in
the residuals of the running speeds, which could not be
tested with only 4 datasets in an individual regression. In
addition, the residuals of computed running speeds could
be more important in the faster runners. In the present
study, the residuals were computed as equal to the differences
between 1 and the ratios of estimated speed to actual speed
for each distance and each runner. For a given running
model, the squares of these residuals were computed for each
distance and each runner, which corresponded to 24 squares
(4 distances x 6 runners). The values of the squares of a
model were compared with the values of squares for the same
distances and same runners in another model. The statistical
significance values of the 24 paired differences between two
running models were tested with paired Student’s t-tests
after normality tests (Kolmogorov-Smirnov tests). When
the normality tests failed, the paired Student’s t-tests were
replaced with the Wilcoxon signed rank tests.
In addition, for each runner, the sum of squared errors
for the four distances was computed for each model. The
square root of the mean of this sum (root mean square error,
RMSE) was computed for each runner and each model. A
large error has a disproportionately large effect on RMSE
which is, consequently, sensitive to outliers.
4. Results
4.1. Power-Law Model Applied to Elite Runners. The effects of
the distance range were not significant for exponent g (0.063
< P < 0.125) as well as parameter k (0.063 < P < 0.094).
The estimations of the logarithm of running speeds (S)
were close to the logarithm of actual speeds (Figure 1(a)). The
correlation coefficients of the individual linear relationships
(see (5)) between ln(S) and ln(tlim) or ln(Dlim) and ln(tlim)
were higher than 0.999 in all the runners for 1500-10000m.
Similarly, the ratios of estimated to actual speeds (Table 3)
for the four distances were accurate: the errors were lower
than 1%, except the 10000 m performance by Nurmi (error
equal to 1.1%).
Marathon performances were under the extrapolation of
the lines of regression computed from the 1500-10000m track
performances (Figure 1(b)).
4.2. Hyperbolic Model Applied to Elite Endurance Runners
4.2.1. SCrit1 Model. The linear relationships between time
(tlim) and distance (Dlim) are presented in Figure 2. For all the
runners, the correlation coefficients of the linear regression
between tlim and Dlim were higher than 0.999 for the different
ranges of Dlim. Parameters SCrit1 and ADC1 are presented in
Table 4. As in previous studies on critical power [42, 43],
the values of SCrit1 depended of the range of tlim. All the
differences in SCrit1 and ADC1 were significant (P = 0.031 in
the Wilcoxon signed rank test): the values SCrit1 computed
from 1500 to 5000m were significantly higher than SCrit1
computed from 3000 to 10000m. The ratios of the estimated
running speeds to the actual speed estimated from SCrit1
model are presented in Table 5. The errors are moderate (<
2%) except for 1500 m.
The values of ADC1 largely depended on the range of
performances as shown in Figure 3. When the individual
critical speeds decreased because of a change in the range
of performances, the corresponding ADC1 increased. These
increases in ADC1 were much more important than the
decrease in SCrit1. For example, SCrit1 computed from 3000-
10000 m was 3.8% lower than SCrit1 computed from 1500-5000
m (Table 3) whereas the corresponding increase in ADC1 was
equal to 79% (319 ± 53 m versus 178 ± 39 m, Figure 3).
4.2.2. SCrit2 Model. The individual S-1/tlim relationships were
not linear (Figure 4(a)) when long distances (10 km) were
included. The correlation coefficients of the linear regressions
between 1/tlim and Dlim were equal to 0.976 ± 0.0126.
Parameters SCrit2 and ADC2 depended on the range of
distances (Table 6). All the differences in SCrit2 and ADC2 in
function of the distance ranges were significant (P = 0.031).
When SCrit2 decreased because of a change in the range
of performances, the corresponding ADC2 increased. These
variations in ADC2 were much more important than the
variation in SCrit2 (Table 6).
4.2.3. Comparison of the SCrit1 and SCrit2 Models. As in previ-
ous studies [49–52], the estimates of SCrit differed according
to the mathematical model used to describe the speed-tlim
relationships. The values of SCrit2 (Table 6) were significantly
higher (P = 0.031) than SCrit1 (Table 4). Indeed, the values of
SCrit1 were slightly lower in all the elite endurance runners
than the value of SCrit2 when they were computed with three
(3-5-10km) or four (1.5-3-5-10km) distances (Figure 5(a)).
When short distances (1500 m) were included, the differences
between SCrit1 and SCrit2 increased as demonstrated in Fig-
ure 5(a). However, SCrit1 and SCrit2 computed from the same
range of performance were highly correlated (P ≥ 0.996). The
values of ADC2 (Table 6) were significantly lower (P = 0.031)
than ADC1 (Table 4) but were significantly correlated (0.940
< r < 0.992; P <0.001).
Interestingly, as shown in Figure 5(b), the values of SCrit1
were equal to SCrit2 when both were computed from the same
two distances, only (for example, 1.5 and 10 or 3 and 10 km).
Similarly, ADC1 and ADC2 were equal when both were only
computed from the same two distances.
For all the runners, the correlation coefficients for the
linear regressions between 1/tlim and Dlim in SCrit2 model were
lower than for the tlim-Dlim regressions in SCrit1 model. In
contrast, the ratios of estimated to actual speeds (Table 7)
were more accurate in the SCrit2 model: the errors on 1500 m
and RMSE were lower (P = 0.031) than in the SCrit1 model. On
the other hand, the errors on 10000 m were higher (P = 0.031)
in the SCrit2 model.
4.2.4. Morton’s Model Applied to Elite Runners. In all the
runners, the performances estimated from Morton’s model
were very close to their actual performances (Figure 6).
When the 3-parameter model by Morton was computed
with 4 distances (from 1500 m to 10000 m), the correlation
coefficient was very high (0.999 ± 0.000752) in all the
BioMed Research International
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7.5
7.0
6.5
6.0
5.5
S (m.s-1)
200
300
400 500
700
1000
2000
tlim (s)
Gebrselassie
Aouita
V ̈a ̈at ̈ainen
Zatopek
Nurmi
Vir ́en
(a)
7.5
7.0
6.5
6.0
5.5
5.0
4.5
S (m.s-1)
200
500
1000
2000
5000
10000
tlim (s)
(b)
Figure 1: (a) Individual linear relationships (power-law model) with logarithmic scales for running speed and tlim. The performances
by Nurmi and Zatopek were the same for the 1500 m distance. (b) Extrapolation of the linear relationships (dashed lines) to marathon
performances.
10000
8000
6000
4000
2000
0
10000
8000
6000
4000
2000
0
Dlim (m)
200
400
600
800
1000 1200 1400 1600 1800
200
400
600
800
1000 1200 1400 1600 1800
tlim (s)
Aouita
V ̈a ̈at ̈ainen
Nurmi
Gebrselassie
Zatopek
Vir ́en
Figure 2: Linear relationships between exhaustion time (tlim) and
distance (Dlim).
runners. When this model was computed with 3 distances
(1500-3000-5000 m or 3000-5000-10000 m), the correlation
coefficients were equal to 1 in all the runners.
The differences in SCrit, SMax, and ADC between the
ranges of distances (Table 8) were all significant (P = 0.031).
450
400
350
300
250
200
150
100
ADC1(m)
5.25
5.50
5.75
6.00
6.25
6.50
SCrit1 (m.s-1)
N
Z
Va
V
A
G
Figure 3: Relation between critical speed and Anaerobic Distance
Capacity (ADC1) for different ranges of distances: 1500 to 5000 m
(black dots), 3000 to 10000 m (empty circles), and 1500 to 10000 m
(grey dots).
The ratios of estimated to actual speeds are presented in
Table 9. In all the runners, the errors were very low (< 0.5%)
for all the distances, from 1500 to 10000 m. However, the val-
ues of S corresponding to a marathon were overestimated in
the three runners who participated in this road competition
(Figure 6(b)).
4.3. Logarithmic Model Applied to Elite Runners. The values of
parameters E and MAS in the logarithmic model depended
on the range of running distance (Table 10) but these
differences were not significant for MAS between 1500-10000
and 1500-5000 ranges and for E between 1500-5000 range and
the two other distance ranges (P = 0.063).
The correlation coefficients were high, 0.995 ± 0.005, for
the logarithmic model including the four distances from 1500
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7.4
7.2
7.0
6.8
6.6
6.4
6.2
6.0
5.8
5.6
5.4
0.000
0.001
0.002
0.003
0.004
0.005
1/tlim
20 km
10 km
5 km
3 km
1.5 km
S (m.s-1)
Gebrselassie
Aouita
V ̈a ̈at ̈ainen
Zatopek
Nurmi
Vir ́en
(a)
0
500
1000
1500
2000
7.5
7.0
6.5
6.0
5.5
S (m.s - 1)
tlim (s)
(b)
0
500
1000
1500
2000
7.5
7.0
6.5
6.0
5.5
tlim (s)
S (m.s - 1)
(c)
Figure 4: (a) Individual S-1/tlim relationships in elite endurance runners. ((b) and (c)) Individual hyperbolic curves corresponding to SCrit1
model (dashed curves) and SCrit2 model (solid curves).
6.4
6.2
6.0
5.8
5.6
5.4
5.2
1.5, 3, 5, 10 km (All)
3, 5, 10 km
6.4
6.2
6.0
5.8
5.6
5.4
5.2
SCrit1 (m.s-1)
SCrit2 (m.s-1)
SCrit computed from
(a)
1.5 & 10 km
3 & 10 km
6.4
6.2
6.0
5.8
5.6
5.4
5.2
6.4
6.2
6.0
5.8
5.6
5.4
5.2
SCrit1 (m.s-1)
SCrit2 (m.s-1)
SCrit computed from
(b)
Figure 5: (a) Relationships between the individual values of SCrit1 and SCrit2 computed from 3 distances (black dots) or 4 distances (empty
circles). (b) Relationships between SCrit1 and SCrit2 computed from 2 distances, only.
BioMed Research International
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Aouita
V ̈a ̈at ̈ainen
Nurmi
8.0
7.5
7.0
6.5
6.0
5.5
0
500
1000
1500
2000
tlim (s)
S (m.s- 1)
(a)
Gebrselassie
Zatopek
Vir ́en
7.0
6.5
6.0
5.5
5.0
0
2000
4000
6000
8000
Marathon
tlim (s)
S (m.s- 1)
(b)
Figure 6: Relationship between running speed (S) and time (tlim) in Morton’s model computed from 1500 to 10000 m. (b) The same model
in the three subjects who ran the marathon.
7.2
7.0
6.8
6.6
6.4
6.2
6.0
5.8
5.6
5.4
200
400
600
800 1000
2000
S (m.s-1)
tlim (s)
Gebrselassie
Aouita
V ̈a ̈at ̈ainen
Zatopek
Nurmi
Vir ́en
(a)
7.0
6.6
6.2
5.8
5.4
5.0
4.6
500
1000
5000
10000
S (m.s-1)
tlim (s)
(b)
Figure 7: (a) Individual linear regressions between the logarithms of tlim and running speeds. The data corresponding to 1.5 km were
not included in the computation of the regressions. The performances by Nurmi and Zatopek were the same for the 1500 m distance. (b)
Extrapolation of the speed-ln(tlim) relationships of the 3000-10000 m performances to tlim corresponding to a marathon (dashed lines). The
scale of tlim is a logarithmic scale.
to 10000 m. The ratios of estimated to actual speeds for the
four distances were accurate (Table 11): all the errors were
lower than 1%.
When the 1500 m distance was not included as suggested
by P´eronnet and Thibault [7, 8], the correlation coefficient
was higher (0.999 ± 0.002). The individual running perfor-
mances between 3000 and 10000 m were well described by
the logarithmic model as shown by the linear regressions
between speed and the logarithm of tlim (Figure 6(a)).
All the individual 1500m performances were above the
individual regression lines computed from 3000 to 10000 m
(Figure 6(a)) as in the logarithmic model including the 1500
m performances (Table 10).
On the other hand, marathon performances were under
the extrapolation of the lines of regression computed from the
3000-10000 m track performances (Figures 7(a) and 7(b)).
4.4. Exponential Models Applied to Elite Runners. The rela-
tionships between tlim and S in the exponential model are
presented in Figure 8.
As for the other models, the values of parameters S∞, S0,
and 1/𝜏 depended on the range of tlim-Dlim (Table 12).
When computed from 4 distances (Figure 8), the individ-
ual regressions were accurate (r = 0.998 ± 0.0014). Similarly,
the ratios of estimated to actual speeds for the four distances
were highly accurate (Table 13): all the errors were lower
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Table 2: Parameters k and g according to the ranges of distances used in the computation of the power-law model.
1500-10000 m
1500-5000 m
3000-10000 m
k
g
k
g
k
g
Nurmi
9.55
0.926
10.2
0.915
8.80
0.938
Zatopek
8.65
0.945
8.86
0.941
8.39
0.950
V¨a¨at¨ainen
8.99
0.944
9.36
0.938
8.45
0.953
Vir´en
9.17
0.943
9.20
0.943
9.14
0.944
Aouita
11.0
0.920
11.3
0.915
10.5
0.927
Gebrselassie
9.24
0.949
9.12
0.951
9.25
0.948
Mean
9.43
0.938
9.67
0.934
9.08
0.943
SD
0.81
0.012
0.90
0.015
0.76
0.010
8.2
7.8
7.4
7.0
6.6
6.2
5.8
5.4
8.2
7.8
7.4
7.0
6.6
6.2
5.8
5.4
0
300
600
900
1200
1500
1800
0
300
600
900
1200
1500
1800
1.5 km
1.5 km
3 km
3 km
5 km
5 km
10 km
10 km
Aouita
V ̈a ̈at ̈ainen
Nurmi
Gebrselassie
Zatopek
Vir ́en
S (m.s-1)
tlim (s)
Figure 8: Individual relationships between running speed and tlim
in the Hopkins model computed with 4 distances (1500-10000 m).
than 0.75%. As expected, the 3-parameter model was more
accurate (r = 1) for the description of the elite runner
performances when it was computed from 3 distances (1.5-
3-5 km or 3-5-10 km), only.
4.5. Prediction of Running Speeds
4.5.1. Prediction of Maximal Aerobic Speed. Maximal aerobic
speed (MAS) can be estimated by computing the maximal
speed corresponding to 7 min [7, 8] from the different
models. These estimations (Table 14) were performed by
interpolation from the 1500-5000m performances.
The effect sizes were small for all the differences (0.037
< Cohen’s d < 0.218). The estimations of MAS were almost
equal for SCrit1 and SCrit2 models that were significantly lower
than the estimations of all the other models. The differences
between all the other models were not significant (P ≥ 0.063).
The correlations between the different estimations were
highly significant (r > 0.998; P < 0.001).
4.5.2. Prediction of Maximal Speed during 30 Min. The
estimations of the maximal running speed during 30 min
done by extrapolation from the 1500-5000m performances
are compared with the 10000 m performances (S10000) in
Table 15. The correlations between the different estimations
were highly significant (r ≥ 0.860; P < 0.0025). All the
different estimations were significantly correlated with S10000
(r ≥ 0.989; P < 0.001). The effect sizes were small for the
power-law and logarithmic models (Cohen’s d = 0.131) or
for the hyperbolic and exponential models (Cohen’s d =
0.033) but large for the difference between power-law and
exponential models (Cohen’s d = 0.742). The 30-minute
running speed estimated from asymptotic models was sig-
nificantly higher than those estimated from power-law and
logarithmic models (P = 0.031). The 30-min running speed
was overestimated by the hyperbolic and exponential models
because these estimations were approximately 2.5% higher
than S10000 (P = 0.031) although the individual values of tlim
corresponding to 10000 m (Table 2) were lower than 1800
s (from 1583 to 1734 s) except for Nurmi (1806 s). On the
contrary, the 30-minute estimated speeds computed with the
logarithmic and power-law models were probably close to the
actual 30-minute performances since they were slightly lower
(0.7 and 1.4%) than S10000.
4.5.3. Prediction of Maximal Speed during 60 Min. The esti-
mations of maximal running speed during 60 min (Table 16)
were done by extrapolation from the 1500-10000 m perfor-
mances. The effect size between power-law and logarithmic
models was small (Cohen’s d = 0.073). All the predictions
of the 60-min speeds from the different models were signif-
icantly correlated (r ≥ 0.964; P < 0.002). However, the 60-
minute running speed predicted from the asymptotic models
was significantly higher (P = 0.031) than those estimated from
BioMed Research International
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Table 3: Ratios of estimated to actual speeds for the different distances in the power-law model. RMSE = root mean square of the errors
between estimated running speed and actual speed.
1500
3000
5000
10000
RMSE
Nurmi
0.9909
1.0058
1.0057
0.9899
0.00794
Zatopek
0.9958
1.0016
1.0005
0.9954
0.00323
V¨a¨at¨ainen
0.9919
1.0051
0.9994
0.9924
0.00612
Viren
0.9974
0.9974
0.9975
0.9963
0.00290
Aouita
0.9965
1.0077
1.0022
0.9981
0.00448
Gebrselassi´e
1.0022
1.0039
0.9995
1.0042
0.00309
Mean
0.996
1.004
1.001
0.996
0.00463
SD
0.0041
0.0037
0.0029
0.00495
0.00203
Table 4: Values of SCrit1 and ADC of the SCrit1 model according to the range of distances. ∗: P = 0.031 for all the differences between the
different ranges.
1500-10000 m
1500-5000 m
3000-10000 m
SCrit1
ADC1
SCrit1
ADC1
SCrit1
ADC1
Nurmi
5.39
284
5.51
228
5.35
339
Zatopek
5.65
226
5.79
160
5.61
282
V¨a¨at¨ainen
5.86
220
5.99
161
5.83
262
Vir´en
5.90
245
6.09
160
5.85
314
Aouita
5.89
332
6.14
225
5.83
413
Gebrselassie
6.19
230
6.42
133
6.13
301
Mean
5.81∗
256∗
5.99∗
178∗
5.77∗
319∗
SD
0.27
44
0.31
39
0.26
53
Table 5: Ratios of estimated to actual speeds for the different distances in the SCrit1 model. RMSE = root mean square of the errors between
estimated running speeds and actual speeds.
1500
3000
5000
10000
RMSE
Nurmi
1.032
0.992
0.992
1.002
0.0173
Zatopek
1.033
0.994
0.990
1.002
0.0175
V¨a¨at¨ainen
1.026
0.997
0.991
1.002
0.0138
Vir10n
1.044
0.992
0.988
1.003
0.0231
Aouita
1.056
0.993
0.983
1.004
0.0296
Gebrselassi´e
1.043
0.995
0.985
1.003
0.0231
Mean
1.039
0.994
0.988
1.003
0.021
SD
0.011
0.002
0.004
0.001
0.006
Table 6: Values of SCrit2 and ADC2 according to the range of distances. ∗: P = 0.031 for all the differences between the different ranges.
1500-10000 m
1500-5000 m
3000-10000 m
SCrit2
ADC2
SCrit2
ADC2
SCrit2
ADC2
Nurmi
5.47
233
5.54
211
5.37
318
Zatopek
5.74
171
5.82
146
5.64
257
V¨a¨at¨ainen
5.93
176
6.00
156
5.86
235
Vir´en
6.02
174
6.14
141
5.88
283
Aouita
6.04
248
6.18
210
5.89
368
Gebrselassie
6.32
157
6.44
123
6.19
257
Mean
5.92∗
193∗
6.02∗
165∗
5.81∗
286∗
SD
0.29
38
0.31
37
0.28
49
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Table 7: Ratios of estimated to actual speeds for the different distances in the SCrit2 model. RMSE = root mean square of the errors between
estimated running speed and actual speed.
1500
3000
5000
10000
RMSE
Nurmi
1.005
0.989
0.996
1.011
0.00834
Zatopek
1.005
0.990
0.994
1.012
0.00855
V¨a¨at¨ainen
1.003
0.994
0.994
1.009
0.00659
Viren
1.006
0.987
0.993
1.015
0.0112
Aouita
1.007
0.986
0.989
1.019
0.0132
Gebrselassi´e
1.006
0.989
0.990
1.016
0.0110
Mean
1.005
0.989
0.993
1.014
0.00981
SD
0.0013
0.0027
0.0025
0.0035
0.00241
Table 8: Values of SCrit3, SMax and ADC of Morton’s model according to the range of distances. ∗: P = 0.031 for all the differences between the
different ranges.
1500-10000 m
1500-5000 m
3000-10000 m
SCrit3
SMax
ADC
SCrit3
SMax
ADC
SCrit3
SMax
ADC
Nurmi
5.29
7.74
504
5.31
7.85
470
5.27
7.45
549
Zatopek
5.51
7.05
539
5.61
7.25
388
5.44
6.74
760
Vaatainen
5.78
7.69
393
5.94
9.70
211
5.59
6.77
982
Viren
5.67
7.23
793
5.77
7.31
605
5.60
7.08
995
Aouita
5.69
8.16
772
5.98
9.07
410
5.41
7.34
1666
Gebrselassi´e
5.92
7.40
868
6.28
7.85
292
5.46
7.05
2961
Mean
5.64∗
7.55∗
645∗
5.82∗
8.17∗
396∗
5.46∗
7.07∗
1319∗
SD
0.22
0.40
193
0.33
0.99
137
0.12
0.29
888
Table 9: Ratios of estimated to actual speeds for the different distances in Morton’s model. RMSE = root mean square of the errors between
estimated running speed and actual speed.
1500
3000
5000
10000
RMSE
Nurmi
1.0000
1.0004
0.9995
1.0003
0.00036
Zatopek
0.9997
1.0012
0.9985
1.0005
0.00100
V¨a¨at¨ainen
0,9996
1.0027
0.9965
1.0014
0.00236
Viren
0.9997
1.0008
0.9990
1.0003
0.00069
Aouita
0.9993
1.0038
0.9953
1.0017
0.00315
Gebrselassi´e
0.9992
1.0033
0.9968
1.0010
0.00241
Mean
0.9996
1.0020
0.9976
1.0009
0.0017
SD
0.0003
0.0014
0.0016
0.0006
0.0011
Table 10: Values of MAS and E in the logarithmic model according to the range of distances. a: P = 0.031 between 1500-10000 and 3000-10000
m; b: P = 0.031 between 1500-5000 and 3000-10000 m.
1500-10000 m
1500-5000 m
3000-10000 m
MAS
E
MAS
E
MAS
E
Nurmi
6.13
7.18
6.12
8.48
6.05
5.90
Zatopek
6.22
5.35
6.22
5.87
6.19
4.83
V¨a¨at¨ainen
6.44
5.47
6.43
6.24
6.38
4.49
Vir´en
6.52
5.54
6.52
5.72
6.51
5.39
Aouita
6.77
7.82
6.76
8.52
6.71
6.96
Gebrselassie
6.78
5.05
6.78
4.94
6.77
4,98
Mean
6.48
6.07
6.47
6.63
6.43a,b
5.42a
SD
0.27
1.14
0.27
1.51
0.29
0.90
BioMed Research International
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Table 11: Ratios of estimated to actual speeds for the different distances in the logarithmic model. RMSE = root mean square of the errors
between estimated running speeds and actual speeds.
1500
3000
5000
10000
RMSE
Nurmi
0.992
1.009
1.009
0.990
0.00916
Zatopek
0.997
1.004
1.003
0.996
0.00352
V¨a¨at¨ainen
0.993
1.008
1.002
0.994
0.00622
Viren
0.999
1.000
1.001
0.998
0.00135
Aouita
0.994
1.008
1.003
0.995
0.00594
Gebrselassi´e
0.999
1.002
0.998
1.001
0.00167
Mean
0.996
1.005
1.003
0.996
0.00460
SD
0.0031
0.0037
0.0038
0.0037
0.00302
Table 12: Values of S∞, S0 and 1/𝜏 of the exponential model according to the range of distances. ∗: P = 0.031 for all the differences between
the different ranges.
1500-10000 m
1500-5000 m
3000-10000 m
S∞
S0
1/𝜏
S∞
S0
1/𝜏
S∞
S0
1/𝜏
Nurmi
5.52
7.06
0.00224
5.64
7.24
0.00298
5.48
6.68
0.00167
Zatopek
5.73
6.81
0.00187
5.87
6.96
0.00280
5.68
6.57
0.00132
Vaatainen
5.97
7.17
0.00228
6.13
7.50
0.00397
5.86
6.69
0.00115
Vir´en
5.96
7.10
0.00163
6.11
7.20
0.00234
5.91
6.94
0.00127
Aouita
6.03
7.76
0.00202
6.31
8.13
0.00354
5.87
7.23
0.00114
Gebreselassie
6.23
7.29
0.00151
6.50
7.52
0.00323
5.99
7.03
0.00073
Means
5.91∗
7.20∗
0.00193∗
6.09∗
7.43∗
0.00314∗
5.80∗
6.86∗
0.00121∗
SD
0.25
0.32
0.00032
0.31
0.40
0.00057
0.19
0.25
0.00031
Table 13: Ratios of the estimated to actual speeds in the different distances for the exponential model. RMSE = root mean square of the errors
between estimated running speeds and actual speeds.
1500
3000
5000
10000
RMSE
Nurmi
0.999
1.004
0.996
1.002
0.00293
Zatopek
0.999
1.003
0.997
1.001
0.00227
V¨a¨at¨ainen
0.998
1.005
0.994
1.002
0.00405
Viren
0.999
1.002
0.998
1.001
0.00162
Aouita
0.998
1.007
0.993
1.002
0.00530
Gebrselassi´e
0.998
1.005
0.996
1.001
0.00309
Mean
0.999
1.004
0.996
1.001
0.00321
SD
0.0005
0.0018
0.0018
0.0006
0.00131
Table 14: Estimation of maximal running speed (m.s−1) corresponding to 420 s computed from the different models. ∗: P = 0.031 for the
differences with Morton’s model, exponential, logarithmic and power-law models.
SCrit1
SCrit2
Morton
Exponential
Log
Power
Nurmi
6.0491
6.0454
6.09
6.10
6.12
6.11
Zatopek
6.1714
6.1669
6.20
6.21
6.22
6.20
V¨a¨at¨ainen
6.3716
6.3737
6.39
6.39
6.43
6.41
Vir´en
6.4693
6.4690
6.52
6.52
6.52
6.53
Aouita
6.6776
6.6812
6.72
6.72
6.76
6.75
Gebrselassie
6.7348
6.7349
6.76
6.76
6.78
6.79
Means
6.412∗
6.412∗
6.45
6.45
6.47
6.47
SD
0.272
0.274
0.27
0.27
0.27
0.28
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Table 15: Maximal running speed (m.s−1) during 30 min, computed from the different models. S10000: running speed over 10000 m; ∗: P =
0.031 for the differences with logarithmic and power-law models. 1: P = 0.031 for the differences with SCrit1 model. 3: P = 0.031 for the differences
with Morton’s model. S: P = 0.031 for the differences with S10000.
Log
Power
Morton
SCrit1
Exp
SCrit2
S10000
Nurmi
5.36
5.40
5.55
5.63
5.65
5.66
5.54
Zatopek
5.69
5.69
5.80
5.88
5.88
5.90
5.77
V¨a¨at¨ainen
5.84
5.85
6.06
6.08
6.13
6.09
5.98
Vir´en
5.98
6.01
6.05
6.18
6.13
6.21
6.03
Aouita
5.92
5.96
6.19
6.27
6.31
6.30
6.07
Gebrselassie
6.29
6.32
6.43
6.49
6.50
6.52
6.32
Mean
5.85S
5.87
6.01∗,S
6.09∗, 3, S
6.10∗,3, S
6.11∗,1,3,S
5.95
SD
0.31
0.31
0.30
0.30
0.31
0.30
0.27
Table 16: Maximal runningspeed (m.s−1) during 60 min computed from the different models. ∗: P = 0.031 for the differences with logarithmic
model. P: P = 0.031 for the differences with power-law model. 1: P = 0.031 for the differences with SCrit1 model. 3: P = 0.031 for the differences
with Morton’s model. E: P = 0.031 for the differences with exponential model.
Log
Power
Morton
SCrit1
Exp
SCrit2
Nurmi
5.18
5.21
5.42
5.47
5.52
5.53
Zatopek
5.50
5.52
5.65
5.71
5.73
5.78
V¨a¨at¨ainen
5.68
5.69
5.88
5.92
5.96
5.98
Vir´en
5.74
5.75
5.86
5.97
5.95
6.07
Aouita
5.63
5.69
5.89
5.99
6.02
6.11
Gebrselassie
6.04
6.08
6.13
6.25
6.22
6.36
Mean
5.63
5.66∗
5.81∗,P
5.88∗,P,3
5.91∗,P,3
5.97∗,P,1,3,E
SD
0.28
0.29
0.24
0.27
0.25
0.29
power-law and logarithmic models. Moreover, the prediction
of the 60-minute running speed from the power-law model
was higher than that from the logarithmic model (P =
0.031). It is possible that the 60-minute running speeds
estimated from power-law and logarithmic models were
slightly overestimated because the world record on one hour
by Gebrselassie was about 2.5% slower (5.913 m.s−1 instead of
6.04 m.s−1 for the logarithmic model and 6.08 m.s−1 for the
power-law model). On the other hand, the record by Zatopek
on 20 km (3591 s; 5.57 m.s−1) was slightly faster than the 60-
minute running speeds S estimated from the power-law (5.52
m.s−1) and logarithmic (5.50 m.s−1) models.
4.5.4. Prediction of Marathon Performances. The overestima-
tions of the marathon running speed (Figure 9) by the dif-
ferent models were similar in the 3 runners. The predictions
of marathon running speeds from the logarithmic model
(red curves in Figure 9) were 5.216 m.s−1 for Zatopek, 5.457
m.s−1 for Viren, and 5.792 m.s−1 for Gebrselassi´e, which
corresponded to overestimations equal to 6.1%, 3.4%, and
2.1%, respectively. The overestimations by the power-law
model (blue curves in Figure 9) were slightly higher than
those of the logarithmic model in the 3 runners.
On the other hand, the overestimations were more impor-
tant with the four asymptotic models (hyperbolic models and
exponential model). These overestimations by the asymptotic
models were similar for the 3 runners who ran the marathon
distance. The large overestimations were similar for the
Table 17: Average values of the 6 runners Roots Mean Square Errors
(RMSE) for the different models.
RMSE
Morton’s model
0.00166 ± 0.00113
Exponential model
0.00321 ± 0.00131
Power-law model
0.00463 ± 0.00203
Logarithmic model
0.00464 ± 0.00302
SCrit2 model
0.00981 ± 0.00241
SCrit1 model
0.0207 ± 0.00566
SCrit1 and SCrit2 models (orange curves) and exponential
model (black curve). In the 3 marathon runners, the lowest
overestimations by an asymptotic model corresponded to
Morton’s model (green curves).
4.6. Comparison of the Accuracies of the Different Models.
For the modelling of the four distances (from 1500 to 10000
m), the lowest mean values of the RMSE of the six runners
corresponded to Morton’s model (Table 17).
The statistical significance values of the differences of
the squared errors between the different models for the four
distances and six runners (n = 24) are presented in Table 18.
The accuracy of Morton’s model was significantly better than
those of all the other models. The accuracies of the power-
law and logarithmic models were not statistically different.
The accuracies of SCrit1 and SCrit2 models were not statistically
BioMed Research International
15
Table 18: Values of paired Student’t test (underlined) or Wilcoxon signed rank test for the difference in squared errors between the running
models.
Power law
SCrit1
SCrit2
Morton
Logarithmic
Exponential
Power law
X
SCrit1
0.003
X
SCrit2
< 0.001
0.484
X
Morton
0.001
0.001
< 0.001
X
Logarithmic
0.830
< 0.001
< 0.001
0.005
X
Exponential
0.061
< 0.001
< 0.001
< 0.001
0.017
X
Table 19: Correlation coefficients of the linear regressions between the different endurance indices. ∗: P = 0.05; ∗∗∗: P <0.001.
SCrit1
SCrit3
g
E
SCrit1/S420
SCrit3/S420
SCrit1
X
SCrit3
0.965∗∗
X
g
0.551
0.513
X
E
0.538
0.499
0.999∗∗∗
X
S∞
0.985∗∗∗
0.984∗∗∗
0.435
0.422
SCrit1/S420
0.976∗∗∗
0.973∗∗∗
X
SCrit3/S420
0.683
0.676
0.775
X
S∞/S420
0.720
0.711
0.824∗
0.991∗∗∗
different but were significantly lower than those of all the
other models.
4.7. Correlations between the Parameters of the
Different Models
4.7.1. Correlations of the Endurance Indices. In Table 19, the
comparisons of the endurance indices concern the indices
computed with the running performances from 1500 to
5000 m that corresponded to the usual range of tlim (3.5 to
15 min) in the studies on the modelling of the individual
performances in nonelite runners. The correlations between
the dimensionless indices (E and g) and either SCrit1 or SCrit3
or S∞ were not significant. In contrast, SCrit1, SCrit3, and S∞
were significantly correlated.
When SCrit1 was normalised to an estimate of maxi-
mal aerobic speed (S420) computed from the same model
(Table 14), its correlations with the dimensionless indices g
and E became significant (Table 19). After normalisation to
S420 computed from the same model (Table 14), the correla-
tion coefficients between SCrit3 or S∞ and the dimensionless
indices (E and g) increased but were not significant.
4.7.2. Correlations between S𝑀𝑎𝑥, S0, and k. When k, SMax, and
S0 were computed from the performances in the 4 distances
(from 1500 to 10000 m, Tables 2, 8, and 12), these parameters
were significantly correlated (P ≤ 0.044):
Smax = 0.0617 + 1.040S0
r = 0.824
k = −3.923 + 1.770Smax
r = 0.862
k = −7.55 + 2.357S0
r = 0.910
(32)
Parameter SMax was significantly higher than S0 (P = 0.031).
Parameter k was significantly higher than SMax and S0 (P =
0.031).
When SMax, S0, and k were computed from 3 distance per-
formances (1500-3000-5000) their values were significantly
higher (P = 0.031) for SMax and S0 but there was no significant
correlation between SMax, S0, and k (r ≤ 0.788; P ≥ 0.063).
5. Discussion
Interestingly, for a given distance and a given model, the
ratios of estimated to actual speeds were similar for the six
runners (Tables 3, 5, 7, 9, 11, and 13). Indeed, for a given
distance and a given model, the ratios of estimated to actual
speed were not spread around 1 but either all the ratios were
higher than 1 or all were lower (except several runners in
the power-law model and one in the logarithmic model).
Therefore, the modelling of the running performances was
probably similar for the six elite runners although they
ran in different conditions and they were probably trained
according to different programmes. However, it cannot be
excluded that there were submaximal performances in some
runners. Indeed, the models would be similar if the ratios of
submaximal speeds to maximal speeds are the same for each
distance in a runner.
5.1. Effects of the Range of t𝑙𝑖𝑚. In the present study, there were
significant differences in the parameters computed from the 3
different ranges of distances for the 3 hyperbolic models and
the exponential model.
The effect of the range of tlim on a parameter is the
most important for parameter ADC computed from the 3
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7.0
6.5
6.0
5.5
5.0
2000
4000
6000
8000
SCrit2
SCrit1
Exponential
Morton
Power-law
Logarithmic
S (m.s-1)
tlim (s)
7.0
6.5
6.0
5.5
5.0
2000
4000
6000
8000
S (m.s-1)
tlim (s)
7.0
6.5
6.0
5.5
5.0
2000
0
0
0
4000
6000
8000
S (m.s-1)
tlim (s)
Gebrselassie
Zatopek
Viren
Figure 9: Comparisons of the relationship between (tlim) and
running speed (S) of the logarithmic model, power-law model,
SCrit1 and SCrit2 models, Morton’s model, and exponential model
computed from 4 distance performances (1500, 3000, 5000, and
10000 m; empty circles) in the three runners who participated in
marathon (black dots).
different hyperbolic models (Figure 3 and Tables 4, 6, and 8).
When the individual critical speeds decreased because of a
change in the range of performances, the corresponding ADC
increased. These increases in ADC1 (79%) were much larger
than the decreases in Scrit1 (3.8%) in the present study. The
dependence of ADC on the range of performances can be
verified (Figure 10) with the data of 19 elite endurance runners
530
490
450
410
370
330
290
250
210
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
N
C
H
Ku
I
K
Z
R
A
Va
W
V
J
Wa
B
M
G
O
E
ADC1 (m)
SCrit1 (m.s-1)
Figure 10: Relation between SCrit1 and ADC1 in the 19 elite runners
whose ranges of performances were different: 1500-10000 m (black
dots), 5000-10000 m (empty circles), and 3000-5000 m (grey dots).
who were world-record holders and/or Olympic winners
and/or world champions: Aouita (A), Bekele (B), Coe (C), El
Gerrouj (E), Gebreselassie (G), Halberg (H), Ifter (I), Jazy (J),
Keino (K), Kuts (Ku), Mo Farah (M), Nurmi (N), Ovett (O),
Ryun (R), V¨a¨at¨ainen (Va), Viren (V), Wadoux (W), Walker
(WA), and Zatopek (Z). The values of ADC1 were high (448
± 67 m) in elite runners whose data included 5000 and 10000
m, only (empty circles). The values of ADC1 were lower (254
± 38 m) in elite runners whose data included all the distances
from 1500 to 10000 m (black dots). In elite runners whose
data did not include the 10000 m performances, ADC1 were
intermediate (263 ± 43 m). Moreover, the values of ADC
are much higher in Morton’s model (Table 8) than in SCrit1
and SCrit2 models (Tables 4 and 6). Therefore, the anaerobic
capacity cannot be estimated from the hyperbolic models.
5.2. Endurance Indices. Parameter E of the logarithmic model
by P´eronnet and Thibault is an estimation of endurance
capability [7, 8]. However, the validity of parameter E as an
endurance index is questionable because MAS is computed
assuming that the value of tlim corresponding to MAS (tMAS)
is equal to 7 min (420s) [7], which is contested. Indeed, in
a review on the exhaustion time at VO2max [53], the value
of tMAS was 6 min. In another study on the energetics of the
best performances in middle distance running [9] the value of
tMAS was estimated as equal to 14 min. Therefore, the interest
of parameter E as an endurance index can be questioned
because it depends on tMAS.
The effect of tMAS on the endurance index by P´eronnet-
Thibault can be calculated [54]:
S = 𝛼1 - 𝛽1 ln ( tlim
420)
MAS420 = 𝛼1
E420 = 100𝛽1
𝛼1
(33)
BioMed Research International
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If T = tMAS
S = 𝛼2 - 𝛽2 ln (tlim
T )
MAST = 𝛼2
ET = 100𝛽2
𝛼2
S = (𝛼1 + 𝛽1 ln (420)) - 𝛽1 ln (tlim)
S = (𝛼2 + 𝛽2 ln (T)) - 𝛽2 ln (tlim)
(34)
The slopes between S and tlim are the same. Therefore
𝛽1 = 𝛽2
S = (𝛼1 + 𝛽1 ln (420)) - 𝛽1 ln (tlim)
= (𝛼2 + 𝛽1 ln (T)) - 𝛽1 ln (tlim)
𝛼1 + 𝛽1 ln (420) = 𝛼2 + 𝛽1 ln (T)
𝛼2 = 𝛼1 + 𝛽1 ln (420) - 𝛽1 ln (T)
= 𝛼1 - 𝛽1 ln ( T
420)
ET = 100𝛽2
𝛼2
=
100𝛽1
(𝛼1 - 𝛽1 ln (T/420)
ET
E420
= [100𝛽1/ (𝛼1 -𝛽1 ln (T/420)]
[100𝛽1/𝛼1]
=
𝛼1
(𝛼1 - 𝛽1 ln (T/420))
ET
E420
=
1
(1 - (𝛽1/𝛼1) ln (T/420))
=
1
(1 - E420 ln (T/420) /100)
(35)
In Figure 11, this relationship between ratio ET/E420 and T (see
(35)) is computed for 3 theoretical runners: an elite endurance
runner (E420 = 4), a medium level endurance runner (E420 =
8), and a low level endurance runner (E420 = 16). The effect
of tMAS is much more important in the low level endurance
runner than in the elite endurance runner (Figure 10).
Large variations in tMAS have small effects on the classi-
fication of runners because the differences in E420 between
elite and medium or low level runners are very large (from 4
to 16). For example, if tMAS is equal to 14 min instead of 7 min,
the medium level endurance runner would still be considered
as a medium level endurance runner in spite of the increase
of E (8.47 instead of 8). Similarly, the elite endurance runner
would still be considered as an elite runner in spite of the
increase in E (4.11 instead of 4) if tMAS is also equal to 14
min instead of 7 min. On the other hand, if tMAS is equal to
4 min instead of 7 min, the medium level endurance runner
would still be considered as a medium level endurance runner
in spite of the decrease in E (7.66 instead of 8.00). Similarly,
1.14
1.10
1.06
1.02
0.98
0.94
0.90
240
360 420 480 540 600 660 720 780 840
300
ET/E420
E420 = 16
E420 = 8
E420 = 4
T = tMAS (s)
Figure 11: Effect of tMAS (T) on the ratio ET/E7min for an elite
endurance runner (E7min = 4), a medium level endurance runner
(E7min = 8), and a low-level endurance runner (E7min = 16).
the low level endurance runner would still be considered as a
low level endurance runner in spite of the decrease in E (14.7
instead of 16) if tMAS is also equal to 4 min instead of 7 min.
The endurance capability can also be estimated by the
asymptotic models if parameters SCrit1, SCrit2, SCrit3, and S∞
are normalised to maximal aerobic speed (MAS). However,
the values of MAS computed from the asymptotic models also
depend on tMAS. Therefore, the validity of these endurance
indices is questionable.
Parameter g of the power-law model by Kennelly has a
high interest because it can be demonstrated that exponent g
is a dimensionless index of endurance that does not depend
on tMAS unlike parameter E in the logarithmic model. The
curvature of the Dlim-tlim equation depends on exponent
g. In the elite endurance runners the Dlim-tlim equation is
almost perfectly linear (Figure 2) whereas this equation is
more curved in runners who are not endurance athletes. For
example, exponent g was close to 1 in elite endurance runners
and lower than 0.9 in physical education students [55]. It can
be demonstrated that exponent g is equal to the ratio of the
slope of the Dlim-tlim equation to MAS when tlim is equal
to tMAS. Indeed, the slope of Dlim-tlim is equal to the first
derivative of the power-law equation. Therefore, the slope of
the Dlim–tlim equation is equal to
dDlim
dtlim
= d (ktlim
g)
dtlim
= kgtlim
g – 1
(36)
For tlim equal to tMAS, the running speed corresponds to MAS:
S = MAS = ktMAS
g - 1
k =
MAS
(tMAS
g – 1) = MAStMAS
1 – g
(37)
Therefore
dDlim
dtlim
= (MAStMAS
1 – g) (gtlim
g – 1)
(38)
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2.0
0.85
1.0
0.0
0.0
1.0
2.0
tlim/tMAS
Slope = g = 0.85
Dlim = 15.8 tlim
Dlim/Dlim at MAS
(a)
5000
4000
3000
2000
1000
0
Distance (m)
0
200
400
600
800
time (s)
tlim 1 = 0.40 tMAS
tlim2 = 4.23 tlim1
tlim 2
tMAS
tlim 1
Dlim = 15.8 tlim
0.85
tangent tMAS
SCrit(tlim 1 tlim 2)
(b)
Figure 12: (a) Slope of the tangent at tMAS of the curve corresponding to the power-law model with tlim normalised to tMAS and Dlim normalised
to Dlim at maximal aerobic speed (MAS). (b) Comparison of a critical speed computed from two values of tlim with the tangent at tMAS (420s).
When tlim = tMAS,
dDlim
dtMAS
= (MAStMAS
1 – g) (gtMAS
g – 1)
= gMAS
(dDlim/dtMAS)
MAS
= g
(39)
Consequently, the ratio of the Dlim-tlim slope to MAS corre-
sponding to tMAS is equal to exponent g and is independent of
tMAS unlike the endurance indices computed from the other
models. In Figure 12(a), Dlim and tlim are normalised to DMAS
(Dlim at MAS) and tMAS, respectively.
Dlim
DMAS
=
Dlim
(tMASMAS) =
ktlim
g
(tMASMAS)
=
(MAS/ (tMAS
g – 1)) tlim
g
(tMASMAS)
= ( tlim
tMAS
)
g
(40)
The slope of the line joining two points corresponding to
tlim1 and tlim2 of the Dlim-tlim curve in Figure 12(b) is equal
to exponent g when it is parallel to the tangent of the curve
at tMAS. In Figure 12(b), ratio tlim1/tmas is equal to 0.4 and
ratio tlim2/tlim1 is equal to 4.23. In many studies on SCrit (or
PCrit) the range of tlim is from 3 to 15 min, which corresponds
to tlim1 equal to about 0.4-0.5 tMAS (if tMAS corresponds to
7 or 6 min) and ratio tlim2/tlim1 about 4-5. This range of tlim
also corresponds to the performances on 1500 and 5000 m
in endurance runners. In the present study, when SCrit1 is
computed from 1500-3000-5000m and is normalised to S420
(Table 14), the value of SCrit1/S420 is equal to 0.934 ± 0.016
and is significantly correlated (r = 0.976; P < 0.001) to g
(0.934 ± 0.16). The product of exponent g and MAS is the
equivalent of a critical speed computed from a 3-15-minute
tlim range. For example, the product of exponent g and S420
estimated from power-law model (Table 14) is equal to 6.04
± 0.30 m.s−1 and is significantly correlated (r = 0.998; P <
0.001) with SCrit1 that is slightly but significantly (P = 0.031)
lower (5.99 ± 0.31 m.s−1). The similar values of SCrit/S420 and
g and the close values of SCrit1 and product g∗S420 and their
significant correlation confirm the hypothesis that exponent
g is an endurance index.
5.3. Correlations between the Parameters of the Different Mod-
els. The correlation between g and E was highly significant
(r = 0.999, Table 19), which confirms the hypothesis that
exponent g is an endurance index. Parameters SCrit1, SCrit2,
SCrit3, and S∞ were highly correlated (P ≥ 0.965). These
parameters that depend not only on endurance capability
but also on maximal aerobic speed were not correlated
with dimensionless parameters g and E (r ≤ 0.551). When
SCrit1, SCrit3, and S∞ were normalised to an estimate of
maximal aerobic speed (S420) computed from their model
(Table 14), these parameters became dimensionless. The value
of SCrit1/S420 was significantly correlated with the dimension-
less indices g, and E (Table 19). After normalisation to S420,
the correlation coefficients between SCrit3/S420 or S∞/S420 and
E or g increased (r ≥ 0.676) but were not significant perhaps
because of the small number of runners. Indeed, a correlation
coefficient equal to 0.6664 would have been significant if
there were 9 runners.
A study [56] compared the critical speeds from different
mathematical models in 12 middle- or long-distance male
runners on a track in order to determine which model
provides the most accurate prediction of performance in 1
hour. In this latter study, the parameters SCrit1, SCrit2, SCrit3,
and S∞ were also significantly correlated (0.85 < r < 0.99,
p < 0.01) and the differences between these different critical
speeds were the same as in the present study for the 1500-5000
m range: SCrit3 < SCrit1 < SCrit2 < S∞.
The meaning of parameters SMax (Morton’s Model) and
S0 (exponential model) is identical and corresponds, in
theory, to maximum running speed. When SMax and S0
BioMed Research International
19
were computed from the 4 distance performances (from
1500 to10000 m, Tables 8 and 12), these parameters were
significantly correlated (r = 0.824; P = 0.044). However, SMax
was significantly higher than S0 (P = 0.31). When SMax and
S0 were computed from the 3 distance performances (from
1500 to 5000 m) their values were higher. A previous study
[57] compared which parameter (SMax or S0) is closest to
maximum speed by measuring maximal velocity during a
sprint. The values of SMax and S0 were well correlated (r =
0.93, P<0.001) but they were significantly different. As in the
present study, SMax (7.80 ± 0.93 m.s−1) was higher than S0
(7.49 0.90 m.s−1) but lower than the actual maximum speed
(8.43 ± 0.33 m.s−1) on a track. However, SMax and S0 were
computed from the performances on a treadmill whereas the
actual maximum running speed was measured on a track
during short sprints with photocells placed at 30 and 40 m.
It is likely that it would be better to measure actual maximum
speed during a 60 m sprint on a track with a laser apparatus
and to compare it with SMax and S0 from Morton’s model and
exponential models computed from performances on a track
instead of a treadmill.
In the present study, parameter k of the power-law model
was 25% higher than SMax and 31% higher than S0. However,
k was significantly correlated with SMax and S0. These results
confirm the hypothesis that parameter k should be correlated
with the maximal running speed because it is equal to
the running speed corresponding to one second. However,
the value of k depends on the time unit. If the running
performances are evaluated in minutes, parameter k would
be equal to the maximal speed corresponding to 1 minute
whereas SMax and S0 would still correspond to maximal
running speed but expressed in m.min−1.
5.4. Prediction of Long Distances. The asymptotes of hyper-
bolic and exponential model correspond to SCrit1, SCrit2,
SCrit3, and S∞, respectively. In these models, the speeds
lower than these asymptotes can be maintained infinitely.
Therefore, the extrapolations of the asymptotic hyperbolic
and exponential models overestimate the running speeds
on very long distances (Figure 9). In fact, power-law and
logarithmic models are also asymptotic models but these
asymptotes are equal to zero.
The overestimations of marathon performances from
the extrapolations of power-law and logarithmic models
(Figures 1(b), 6(b), and 9) are much smaller. Similarly, the
computations of 30-minute and 60-minute running speeds
by extrapolation of the asymptotic models (Table 7) were
probably overestimations whereas the extrapolations of the
power-law and logarithmic models were probably close to the
actual running speeds.
The overestimations of marathon performances by the
logarithmic and power-law models (Figures 1, 6, and 9)
are probably due not only to the causes of fatigue in long
distances [58] but, perhaps, also to the effects of ground (track
versus road, slopes, etc.), wind, shoes, and age.
5.5. Which Is the Optimal Empirical Model? The optimal
running model is an accurate, useful, and practical model.
5.5.1. Which Is the Most Accurate Model? When computed
from 4 distances, the individual correlation coefficients of all
the models were high in all the elite runners. The correlation
coefficients were the highest for the 3-parameter models
by Morton and Hopkins and they were equal to 1 when
they were computed from 3 distances only. These correlation
coefficients equal to 1 were expected. Similarly, the regression
coefficients of all the 2-parameter running models would
have been equal to 1, if they were computed with only two
distances.
The values of RMSE were the lowest for the 3-parameter
models (Table 17). Morton’s model was the most accurate as
demonstrated by the ratios of estimated to actual running
speeds which were very close to 1 for each distance (Table 9).
Indeed, the differences between the estimated to actual
running speeds were lower than 0.5% in each distance for all
the runners. This model was significantly more accurate than
all the other models as shown in Table 18.
However, if a running model is perfect, there should be no
significant difference between its parameters computed from
different ranges of distances. Morton’s model was probably
not perfect because its parameters were significantly different
(P = 0.031) when they were computed from different ranges
of distances. In the present study, the empirical models
consist of single equations and are less complex than the
physiological and biomechanical models, which probably
explained that the parameters of all these empirical models
depended on the range of tlim. Indeed, the causes of fatigue
differ for short, medium, and long distances [58].
The SCrit1 and SCrit2 models and the concepts of critical
speed (or critical power) are by far the most used and
taught [21, 46]. Nonetheless, SCrit1 and SCrit2 models were the
less accurate models for the relationship between running
speed and tlim. The curves derived from (12) and (14) did
not describe accurately the relationships between speed and
tlim (Figures 4(b) and 4(c)). The only points corresponding
to 10000 m performances were close to the curves derived
from (12) whereas the only points corresponding to 1500 m
performances were close to the curves derived from (14).
Consequently, the speed-tlim relationship would be better
described by the mean values of ADC and SCrit:
ADC = (ADC1 + ADC2)
2
= (𝛼1 + 𝛽2)
2
SCrit = (SCrit1 + SCrit2)
2
= (𝛼2 + 𝛽1)
2
(41)
Even if the description of the individual speed-tlim relation-
ships was better with the curves computed from the mean
values of ADC and SCrit in (12) and (14) (Figure 13), this new
hyperbolic model is not optimal when it is compared with the
figures of the other models.
5.5.2. Which Is the Most Useful Model? The empirical models
of running exercises are often used to estimate the running
speeds over given distances, the endurance capability, and
MAS. The race performance calculation requires 2 or 3
parameters depending on the model used. On the other hand,
for each running model in the present study, there is only one
20
BioMed Research International
0
500
1000
1500
2000
7.5
7.0
6.5
6.0
5.5
tlim (s)
S (m.s - 1)
Figure 13: Individual relationships between speed and tlim com-
puted from the mean values of ADC and SCrit in (12) and (14).
parameter that is an expression of the long-distance running
capability. Indeed, parameter ADC in the hyperbolic models
is not reliable and parameters k, SMax, and S0 that are maximal
speed indices are probably not useful for endurance runners.
Similarly the parameter corresponding to the time constant
(𝜏) in Hopkins’ model is not useful.
The useful parameters of the asymptotic model cor-
respond to SCrit1, SCrit2, SCrit3, and S∞. In theory, these
parameters represent the fastest speed that can be maintained
for a very long time. However, when SCrit1 was computed from
exercises shorter than 20 min, the subjects were generally
only able to maintain SCrit1 for less than 30 min and the
running velocities that could be maintained for 60 minutes
on a treadmill were largely overestimated by SCrit1 [59]. In
another study on the relationship between critical velocity
and marathon performance [60], SCrit1 (4.43 m.s−1) was
44% faster than the marathon running speed (3.07 m.s−1).
Nonetheless, the correlation between marathon performance
and SCrit1 was more significant than the correlations with
the other physiological parameters. In this latter study, it
was possible to calculate an approximation of the marathon
performance from SCrit1 (r = 0.87 and SEE = 14 min).
Approximations of long-distance performances (> 10000 m)
are probably also possible with SCrit2, SCrit3, and S∞ since they
are highly correlated with SCrit1 (P ≥ 0.965). For example,
in the study on 12 trained middle- and long-distance male
runners [56], the correlation coefficients of SCrit1, SCrit2, and
S∞ with the maximal running speed during 60 min were
equal to 0.90, 0.91, and 0.93, respectively. Amazingly, the
correlation coefficient with the 60-min running speed was
the lowest (0.80) for SCrit3 in these middle- and long-distance
runners but the overestimation was the smallest (0.13 ± 0.21
m.s−1) as in the present study.
It is likely that the logarithmic and power-law models
that are not asymptotic are the best empirical models for
the predictions of very long distances by extrapolation as
suggested in Table 15 and Figure 9. The predictions of
the running speeds corresponding to 30 min, 60 min, and
marathon by extrapolation of Morton’s model were higher
than the same predictions from the logarithmic and power-
law models. But the overestimations of the running speeds
corresponding to 30 min, 60 min, and marathon by Morton’s
model were lower than the overestimations by the other
asymptotic models (Tables 15 and 16 and Figure 9). On the
other hand, the predictions of competition performances
between 1500 and 10000 m (for example, one or two miles
or 2000 m) by interpolation should be better with the 3-
parameter models by Morton or Hopkins whose accuracies
were the best. Similarly, the running speed corresponding to
6 or 7 min (an estimation of MAS) should be more accurate
when computed with these 3-parameter models.
The endurance index of the power-law model (exponent
g) should be the most useful since it is the only endurance
index that does not depend on tMAS (Section 5.2).
5.5.3. Which Is the Most Practical? The most practical model
should be the less sensitive to a slightly submaximal perfor-
mance and the easiest to compute.
Unfortunately, no study compares the sensitivity of the
different models to submaximal performances. However, in
a previous study [61], some results were assumed to be the
effect of submaximal performances on SCrit1 model whose
sensitivity was discussed in a review on the critical power
concept [16]. Similarly, the values of parameter k that is
an index of maximal running speed were overestimated
in several physical education students in a previous study
[55], which was probably the effect of submaximal running
performances. Indeed, in 4 physical education students,
parameters k were largely overestimated since they were
higher than 20 m.s−1, whereas the maximal running speed is
about 12.2 m.s−1 for the best world sprinter U. Bolt [62]. The
comparison of parameters k of Ovett and Coe [63] is also a
demonstration of the effects of submaximal performances on
the modelling of running performances with the power-law
model. Indeed, the differences between Ovett and Coe for the
performances over 800, 1500, and 2000 m are around 1 second
but the inclusion of longer distances (3000 m and 5000 m)
causes large differences in the values of k and g. The value of k
was largely higher than 12 m.s−1 for Coe but not for Ovett. The
best performance for a given distance is probably maximal if
the elite runner has run this distance many times, which was
not the case for Coe in the 3000 m and 5000 m distances.
In the present study, the sensitivity of Morton’s model to
submaximal performances could be not negligible. Indeed,
the parameters of this model were significantly different when
they were computed from different distance ranges although
the differences between the estimated and the actual speeds
were very low (< 0.5%). The sensitivity of Morton’s model
to submaximal performances could also explain why the
correlation coefficient of SCrit3 with the 60 min speed was the
lowest in the study on the twelve middle- and long-distance
runners [56].
Many runners compete over two distances, only (either
800 and 1500 m or 5000 and 10000 m or half-marathon and
marathon). Their performances on the other distances could
be slightly submaximal and, consequently, the 3-parameter
BioMed Research International
21
models by Morton or Hopkins could be not optimal for these
runners.
The 3-parameter models need a software that can com-
pute the parameters by iteration. The 2-parameter models
are easier to compute either by a nomogram [48] or by
the current database software (Microsoft Excel, LibreOffice
Calc, etc.). The calculation of SCrit1 is much easier than the
parameters of the other models. Particularly, it is very easy to
calculate SCrit1 from two running performances:
SCrit1=(Dlim2 – Dlim1)
(tlim2 – tlim1)
(42)
In addition, the SCrit1 model is the only model that can
directly predict the performance corresponding to a distance
from its parameters (ADC1 and SCrit1):
Dlim = ADC1 + SCrit1 ∗ tlim
tlim = (Dlim - ADC1)
SCrit1
(43)
In the present study, the other models can only predict
performances corresponding to a value of tlim. In these
models, the protocol presented in Section 3.3 is necessary for
the prediction of a performance corresponding to a distance.
6. Conclusion
The comparison of the accuracies of the different models
in the six elite endurance runners suggests that the most
accurate model is the asymptotic 3-parameter hyperbolic
model proposed by Morton and that the less accurate models
are SCrit1 and SCrit2 models which are the most often used.
However, it is likely that logarithmic and power-law models
are the most accurate models for the predictions of long-
distance performances (maximal running speeds for 30 and
60 min or marathon) by extrapolation. In addition, exponent
g of the power-law model is an interesting endurance index
that does not depend on tMAS. The comparison of the sen-
sitivity of the different models to submaximal performances
should be studied to select the most practical model.
Data Availability
All the “experimental” data are presented in Table 1. All the
results of the computations according to the different models
are presented in the next 15 tables (from Table 2 to Table 16).
Conflicts of Interest
The author declares that they have no conflicts of interest.
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Modelling of Running Performances: Comparisons of Power-Law, Hyperbolic, Logarithmic, and Exponential Models in Elite Endurance Runners.
Vandewalle, H