RESEARCH ARTICLE
A minimal power model for human running
performance
Matthew Mulligan1, Guillaume Adam2, Thorsten EmigID2,3*
1 Claremont McKenna College, W.M. Keck Science Department, Claremont, California, United States of
America, 2 Massachusetts Institute of Technology, MultiScale Materials Science for Energy and Environment,
Joint MIT-CNRS Laboratory (UMI 3466), Cambridge, Massachusetts, United States of America, 3 Laboratoire
de Physique The´orique et Modèles Statistiques, CNRS UMR 8626, Baˆt. 100, Universite´ Paris-Saclay, Orsay
cedex, France
* emig@mit.edu
Abstract
Models for human running performances of various complexities and underlying principles
have been proposed, often combining data from world record performances and bio-ener-
getic facts of human physiology. The purpose of this work is to develop a novel, minimal and
universal model for human running performance that employs a relative metabolic power
scale. The main component is a self-consistency relation for the time dependent maximal
power output. The analytic approach presented here is the first to derive the observed loga-
rithmic scaling between world (and other) record running speeds and times from basic prin-
ciples of metabolic power supply. Our hypothesis is that various female and male record
performances (world, national) and also personal best performances of individual runners
for distances from 800m to the marathon are excellently described by this model. Indeed,
we confirm this hypothesis with mean errors of (often much) less than 1%. The model
defines endurance in a way that demonstrates symmetry between long and short racing
events that are separated by a characteristic time scale comparable to the time over which a
runner can sustain maximal oxygen uptake. As an application of our model, we derive per-
sonalized characteristic race speeds for different durations and distances.
Introduction
Scientists have been fascinated by trying to explain running performance and to predict its
limitations for more than 100 years. A purely descriptive approach was employed by Kennelly
as early as 1906 for speeds in racing events of animals and humans. For men running events
from 20 yards up to a few hundred miles he found a power law relation between distance d
and duration T with T * d9/8 with a relative large error of up to 9% for distances from 100m
to 50 miles (and larger errors for shorter and longer distances) [1].
Almost a century ago, in 1925 noted mathematician and physiologist A.V. Hill proposed
a power model based on metabolic energy considerations to describe the maximal power
output Pmax(T) over a given duration T by a hyperbolic function Pmax(T) = P0 + P1/T with
constants P0 and P1 (known as the “running curve”) [2]. Ward-Smith introduced a model,
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November 16, 2018
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OPEN ACCESS
Citation: Mulligan M, Adam G, Emig T (2018) A
minimal power model for human running
performance. PLoS ONE 13(11): e0206645.
https://doi.org/10.1371/journal.pone.0206645
Editor: Barbora Piknova, National Institutes of
Health, National institute of Diabetes and Digestive
and Kidney Diseases, UNITED STATES
Received: July 18, 2018
Accepted: October 16, 2018
Published: November 16, 2018
Copyright: © 2018 Mulligan et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: All relevant data are
within the paper and its Supporting Information
files.
Funding: This work was supported by Centre
national de la recherche scientifique, grant
EMERGENCE2017 of INP, www.cnrs.fr (T.E.) and
Agence nationale de la recherche, grant ANR-11-
IDEX-0001-02 (T.E.). The funders had no role in
study design, data collection and analysis, decision
to publish, or preparation of the manuscript.
Competing interests: The authors have declared
that no competing interests exist.
based on the first law of thermodynamics, to describe performances at Olympic Games from
1960 to 1976 with an average absolute error for the predicted times of 0.86% for distances
from 100m to 10,000m [3]. In 1973 the mathematician Keller formulated a purely mechani-
cal model that is based on the runner’s equation of motion with a damping term [4]. The
propulsive force is connected to the mechanical power utilized for running which is different
from the overall metabolic power requirement. In analogy to purely mechanical problems,
Keller assumed that the damping is linear in velocity and that the damping coefficient is con-
stant over time. The justification for these assumptions is not validated given that a compari-
son of his model to world track records from 50yards to 10,000m yields a relative large
errors of about 3% for distances larger than 5000m. Furthermore, both Hill’s and Keller’s
models predict the existence of a maximal speed that can be sustained for an infinite dura-
tion, which is not possible from a physiological point of view and incompatible with data on
running records. Similarly, a threshold power has been proposed by Jones et al. in the critical
power model [5].
In fact, existing models appear to be unable to explain an important observation that has
been made already by Hill in the context of his above mentioned model: The average frac-
tional utilization of maximal power (or the average running speed) of world record perfor-
mances scales linearly with the logarithm of the duration of the performance [2]. An
interesting model that interpolates between fundamental knowledge of human bioenergetics
during exercise and actual world record running performance was proposed by Peronnet
and Thibault [6, 7]. Their model combines characteristics of energy metabolism, based on
Hill’s hyperbolic “running curve” and the dynamics of oxygen uptake. However, the frac-
tional utilization of maximal power over a given duration is described in their model by a
phenomenological logarithmic term that is based on observations in running records. The
latter term accounts for endurance limited sustainability of maximal aerobic power. Cur-
rently, this model is most effective in reproducing world record running performances.
However, it uses a number of fixed parameters that are assumed to be equal for all world
record performances although they have been achieved by different athletes. In fact, many
parameters can be different among individuals. For example, running economy, i.e., the
energy cost of running at a given velocity, shows substantial inter-individual variation [8].
These variations are observed even among well trained elite runners. Another quantity that
is modeled as a constant in Peronnet’s and Thibault’s model is the duration over which max-
imal aerobic power (or VO2max) can be maintained during running which they assumed to
be 7 minutes. However, direct measurements of oxygen uptake have demonstrated varia-
tions of the order of one to two minutes among individuals [9, 10]. From a fundamental per-
spective it is desirable to derive a model from basic principles of metabolic power generation
and utilization that predicts human performances without additional phenomenological
input. This is the objective of the present work.
For the development of our model it is instructive to review some facts and experimental
observations from exercise physiology. When developing a model that can describe run-
ning performances as obtained in world records up to the marathon distance one should
realize at what relative intensities these races are performed. All Olympic endurance events
require intensities above 85% of VO2max which corresponds to the effort reached approxi-
mately in the marathon [11]. When looking at record performances, we can also assume
that runner has followed an optimal carbohydrate loading strategy so that the stored
amount of glycogen is permitting best possible performance. This is of importance for the
half marathon and in particular the marathon distance which is raced predominantly on
carbohydrate fuel with an average respiratory gas exchange ratio of close to one for faster
runners [12].
A minimal power model for human running performance
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An important physiological observation is that the total energy cost of running increases lin-
early with the covered distance with no or a very small dependence on the running velocity [13,
14]. Hence the power output changes linearly with speed, with the slope quantifying running
economy. It is known that this running economy can vary about 30–40% among individuals
[11]. An important observation that is essential for the construction of our model is that run-
ning economy usually becomes worse with the duration of a running event. The magnitude of
the change in economy increases with duration and intensity. The actual change is probably
subject dependent and also influenced by external conditions. We shall see below that this is an
important factor in determining race velocities and endurance. This drift in running economy
has been quantified in treadmill studies with a change of 4.4% for 40min at 80% VO2max, a
change of 6.6% for 60min at 70% VO2max, and a change of 9.5% for 60min at 80% VO2max [15].
An other study found for 60min treadmill running near 80% of VO2max a shift of about 3% in
oxygen uptake [16]. Changes in running economy have been also observed during a 5km run
at a constant pace eliciting about 80–85% of VO2max with an average increase in oxygen uptake
of 3.3% for men and 2.0% for women [17]. The reason for the increase in oxygen uptake and
reduction in running economy is unknown. A number of mechanisms have been postulated in
the literature but most of them are speculative [12, 18–20], including an increase in oxygen
uptake due to neuromuscular fatigue [21]. Without discussing here the various attempts that
have been made for explaining this observation, we just conclude that every activated physio-
logical system increases its own particular energy consumption with the duration of exercise.
Methods
A minimal model for running performance
In view of the current status of theoretical descriptions of human running performances, it
appears useful to construct a minimal and universal model for human running performance
that fulfills the following two requirements:
1. Based on basic concepts and observations on metabolic power generation and utilization
during running
2. Minimal number of physiological parameters that are not fixed a priori
In order to eliminate irrelevant normalization parameters from the model (that would
depend on the choice of units for energy, power, etc.), we express our model in terms of rela-
tive quantities. We shall base the model on expedited power measured as oxygen uptake per
time since this quantity can be measured directly under real conditions by mobile spirometry.
This implies a slight time dependence of oxygen uptake during prolonged exercise, even when
the power output is constant, due to a change of the respiratory quotient with substrate utiliza-
tion [22]. Also, since body weight usually changes during prolonged exercise, we measure
power or oxygen uptake always per body weight.
While the basal metabolic rate Pb is close to 1.2W/kg [6], its actual value is not required in
the following. In fact, in the parameterization of running economy to be employed below, we
chose to associate Pb with the power that is obtained by linearly extrapolating the running
economy to zero velocity. Hence we neglect the non-linear dependence of the energy cost on
sub-running (walking) velocities which causes no problem since our model uses the energy
cost of motion only in the linear running regime. In our model there exists a crossover power
Pm that we expect to be close to the maximal aerobic power associated with maximal oxygen
uptake VO2max which is typically in the range of 75 to 85ml/(kg min) for elite runners [6]. The
power Pm should not be confused with the critical or the maximal power that occurs in the
3-parameter critical power model of Morton [23].
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We measure power relative to the base value Pb, in units of the aerobic power reserve Pm −
Pb that is available to the runner, hence defining the relative running power (or intensity) as
p ¼ P To construct our model, we start from the following self-consistency relation
PmaxðTÞ þ PsupðTÞ ¼ 1
T
Z T
0
PTðtÞdt ;
ð4Þ
which states that the sum of the nominal average power and the additional supplemental
power Psup equals the time average of the instantaneously utilized power. We make the impor-
tant conjecture that the instantaneous power utilized at time t equals the maximal power that
can be sustained for the remaining time T − t of the event [26], i.e.,
PTðtÞ ¼ PmaxðT leading to
TðvÞ ¼
tc exp
vmbetween mean race velocity v and race distance d is not a power law as assumed in some stud-
ies [28, 29]. For example, Riegel’s formula corresponds in above notation to τ(δ) = αδ − L, υ(δ)
= −(α − 1)δ + L with a constant L and an exponent α close to 1.06. Our model predicts that α =
1 exactly and that the very small deviation from α = 1, observed by Riegel and others, is due to
a hierarchy of logarithmic corrections, giving rise to a non-constant L. It is interesting to
observe from Eq (13) that the endurance measuring parameter γl or γs is the only quantity
which determines the time to distance and velocity to distance relations when time is mea-
sured in units of tc and velocity in units of vm. We note that for the comparison of our model
to record performances and personal best performances of individual runners, we always use
the exact expressions involving the Lambert W-function.
Interpretation of supplemental power Psup, and of γl, γs
The supplemental power defined in Eq (3) can be expressed relative to the aerobic power
reserve Pm − Pb as
PsupðTÞ
Pm qualitative difference in interpretation. We shall come back to these endurance measures
when we discuss personalized characteristic race paces.
Estimation of physiological model parameters
Our model depends on the four independent parameters vm, tc, γs and γl that characterize a
group of runners (for example world record holders) or individual runners. Otherwise our
model is universal in the sense that it contains no additional fixed parameters or constants.
The four parameters can be estimated from a given set of results (distance and time) from
exercise performed at maximal intensity, i.e., races. These sets can be either records, like world
records, involving a group of different runners or personal records (best performances) from
individual runners. To check the accuracy of our model and to compute the model parameters,
we minimize numerically the sum of the squared differences between the actual race time and
the one predicted by Eq (11) for all results in a given set. This method will be used to recon-
struct individual physiological profiles (running economy and endurance) from race perfor-
mances in Application 1 below.
Prediction of race times and characteristic paces for given times and
distances
Once the model parameters for a given set of performance results have been determined, the
model can be applied to compute a number of interesting quantities that could guide racing
and training of a runner. For example, by comparing the time difference between the actual
Fig 1. Definition of endurance for long and short duration, El and Es, respectively, from the duration T(p) over which a
relative power p can be sustained. Shown is a typical range of endurances for long and short duration (gray regions, with lower
and upper limits for γl and γs) and an example curve that visualizes the definition of El and Es.
https://doi.org/10.1371/journal.pone.0206645.g001
A minimal power model for human running performance
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November 16, 2018
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race time and the model’s prediction for all raced distances, preferred or optimal distances for
a runner can be identified. For distances that have not been raced before, or only prior to a
newly focused training program, the formula of Eq (11), or its approximative version in Eq
(13), can be used to predict racing times.
Another application of our model is the estimation of characteristic velocities that corre-
spond to a prescribed relative power output ^p, measured in percent of aerobic power reserve
that is available over a given duration. Generally, running velocities v in training units depend
on the purpose of the training session and hence on duration T or distance d of the workout
intervals. Suppose that a runner trains at a relative power ^p. This relative power relates the tar-
get power output P(v) to the maximal power above basal power, Pmax(T) − Pb, that can be
maintained for the duration T by the relation
PðvÞ ¼ ^pðPmaxðTÞ this requirement, and in this section we shall validate its accuracy by comparing it to various
record performances.
World and other records have been analyzed before and found to follow an approximate
power law. However, the exponent of this power law shows variations with gender and dis-
tance which renders its universality and general applicability questionable. Also, there is no
physiological foundation for a simple power law. In fact, the existence of a crossover velocity
vm implies different scaling of performances below and above this velocity due to distinct phys-
iological and bio-energetic processes involved.
We have analyzed record performances for eight distances, from 1000m to the marathon,
for world records (current as of Oct. 2018, 2000, 1990, and 1980), current European records,
and current national records (USA, Germany) see Table 1 for male records, and Table 2 for
female records. Following the method described in the previous section, we have estimated the
parameters of our model for each group of records. The resulting parameters tc, vm, γs, and γl
together with the endurances Es and El are summarized in Tables 1 and 2. The mean relative
error between our model prediction and the VDOT prediction for the race times for 13 dis-
tances between 1000m and the marathon are 0.15%, 0.11%, and 0.18% for VDOT = 40, 60, and
80, respectively. These small errors suggest that the race times predicted by the VDOT model
are mutually consistent. This presumably reflects that the times were obtain from a mathemati-
cal model that is based on physiological observations made by Daniels among well trained and
elite runners.
A number of interesting observations can be made from the results: There is a high level of
agreement between actual and predicted times with the relative error being larger than 1%
only for a single event (Half-marathon, WR 1980) for male records, and four events for female
records. The mean of the absolute value of the relative error is always smaller than 1% with the
exception of the female WR from 1990 where it is 1.05%. For the male WR a decrease of the
absolute value of the relative error from 1980 to today can be observed, indicating an increas-
ing optimization towards the maximally possible performance (within current level of technol-
ogy and training methods) that is described by our model. Hence, the record times have
become more consistent with our model over time which might be also due to an increasing
number of attempts to achieve best possible performances. A similar observation is made for
the female WR from 1990 to 2000. However, from the 2000 WR some results (Chinese run-
ner’s results for 1500m, 3.000m, 5.000m, and 10.000m) have been excluded due to the use of
performance-enhancing drugs [30], and the current WR for 1500m and 10.000m are also con-
troversial [31]. For the latter two distances our model predicts more than 0.5% slower times
than actually raced. It is interesting to observe that our predictions are very sensitive to excep-
tional performances for a particular distance compared to the other distances, and hence is
able to identify suspicious race results. Due to the women’s shorter history of endurance run-
ning, the female world records for 1980 are less consistent than more recent records and hence
have been excluded them from our analysis.
It also instructive to compare the physiological model parameters obtained from the record
performances. For the male records, the obtained values for tc vary between five and six min-
utes, which is in very good agreement with laboratory testing [32]. However, for female rec-
ords, we observe a larger variation in tc with values around 10min being not unusual.
However, in cases with such long tc the crossover velocity vm is reduced proportionally. The
endurance parameter El for long distances varies between 5 and 6 for male records, implying
that 90% of maximal aerobic power can be maintained for a duration between approximately
25min and 36min, for the values of tc observed here. For female records, the endurance param-
eter El is significantly larger with variations in an interval of approximately 6 to 8.5, implying
that 90% of maximal aerobic power can be maintained for durations up to 85min.
A minimal power model for human running performance
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Table 1. Race times and model parameters for various male running records, as of Oct. 2018.
Record
WR men
WR 2000 men
WR 1990 men
tc[min]
6.26
5.50
5.90
vm[m/min]
411.72
417.07
405.00
100 γs
9.99
9.87
11.76
100 γl
5.36
6.19
5.93
Es
0.37
0.36
0.43
El
6.46
5.04
5.41
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
1000
02:11.96
02:11.94
-0.02
02:11.96
02:11.94
-0.02
02:12.80
02:12.82
+0.01
1500
03:26.00
03:26.24
+0.12
03:26.00
03:26.24
+0.12
03:29.46
03:29.26
-0.10
1609.34
03:43.13
03:42.91
-0.10
03:43.13
03:42.91
-0.10
03:46.32
03:46.50
+0.08
3000
07:20.67
07:20.99
+0.07
07:20.67
07:19.38
-0.29
07:29.45
07:30.88
+0.32
5000
12:37.35
12:37.10
-0.03
12:39.36
12:38.37
-0.13
12:58.39
12:56.88
-0.19
10000
26:17.53
26:18.84
+0.08
26:22.75
26:33.98
+0.71
27:08.23
27:08.68
+0.03
21097.5
58:23.00
58:11.94
-0.32
59:22.00
59:18.82
-0.09
1:00:46.00
1:00:25.03
-0.58
42195
2:01:39.00
2:01:52.99
+0.19
2:05:42.00
2:05:26.57
-0.20
2:06:50.00
2:07:21.71
+0.42
mean
0.12
0.21
0.22
Record
WR 1980 men
US men
EU men
tc[min]
5.26
6.08
4.97
vm[m/min]
405.27
406.06
412.81
100 γs
12.74
10.35
12.24
100 γl
6.21
5.67
5.76
Es
0.46
0.38
0.44
El
5.00
5.83
5.67
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
1000
02:13.40
02:13.41
+0.01
02:13.90
02:13.87
-0.02
02:12.18
02:12.17
-0.01
1500
03:31.36
03:31.26
-0.05
03:29.30
03:29.61
+0.15
03:28.81
03:28.90
+0.04
1609.34
03:48.80
03:48.89
+0.04
03:46.91
03:46.62
-0.13
03:46.32
03:46.24
-0.04
3000
07:32.10
07:34.44
+0.52
07:29.00
07:28.52
-0.11
07:26.62
07:26.39
-0.05
5000
13:08.40
13:04.65
-0.48
12:53.60
12:51.55
-0.26
12:49.71
12:48.61
-0.14
10000
27:22.47
27:30.09
+0.46
26:44.36
26:53.60
+0.58
26:46.57
26:49.72
+0.20
21097.5
1:02:16.00
1:01:26.52
-1.32
59:43.00
59:41.08
-0.05
59:32.00
59:38.51
+0.18
42195
2:09:01.00
2:10:02.03
+0.79
2:05:38.00
2:05:26.28
-0.16
2:05:48.00
2:05:34.12
-0.18
mean
0.46
0.18
0.11
Record
GER men
tc[min]
4.79
vm[m/min]
411.05
100 γs
11.22
100 γl
6.11
Es
0.41
El
5.14
distance
T
Tmodel
%
1000
02:14.53
02:14.52
-0.01
1500
03:31.58
03:31.71
+0.06
1609.34
03:49.22
03:49.10
-0.05
3000
07:30.50
07:30.28
-0.05
5000
12:54.70
12:57.09
+0.31
10000
27:21.53
27:13.07
-0.52
(Continued)
A minimal power model for human running performance
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November 16, 2018
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The impact of endurance alone on running performances can be highlighted by measuring
the mean race velocity vðdÞ in units of the crossover velocity vm and the race distance d in
units of the crossover distance dc = vmtc. The resulting relation between vðdÞ=vm and d/dc is
shown in Fig 2 for the current world records. Our model predicts that this relation depends
only on the endurance parameters γl and γs, see Eq (12). The corresponding model curves are
also plotted in Fig 2, showing good agreement with the data from world records. The better
Table 1. (Continued)
21097.5
1:00:34.00
1:00:45.41
+0.31
42195
2:08:33.00
2:08:28.19
-0.06
mean
0.17
https://doi.org/10.1371/journal.pone.0206645.t001
Table 2. Race times and model parameters for various female records, as of Oct. 2018. † For the women WR of 2000 the result of Chinese runners for the distances
1500m, 3000m, 5000m and 10000m have been excluded due to use of performance-enhancing drugs [30].
Record
WR women
WR 2000 women†
WR 1990 women
tc[min]
8.30
10.01
5.50
vm[m/min]
361.37
352.14
364.74
100 γs
9.60
10.27
12.13
100 γl
4.85
5.53
5.74
Es
0.35
0.38
0.44
El
7.88
6.10
5.70
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
1000
02:28.98
02:28.78
-0.13
02:28.98
02:29.07
+0.06
02:30.67
02:30.17
-0.33
1500
03:50.07
03:52.05
+0.86
03:52.47
03:52.94
+0.20
03:52.47
03:57.27
+2.06
1609.34
04:12.56
04:10.70
-0.74
04:12.56
04:11.75
-0.32
04:21.68
04:16.95
-1.81
3000
08:20.68
08:18.13
-0.51
08:21.64
08:21.94
+0.06
08:22.62
08:25.93
+0.66
5000
14:11.15
14:12.40
+0.15
14:31.48
14:29.77
-0.20
14:37.33
14:31.08
-0.71
10000
29:17.45
29:29.07
+0.66
30:13.74
30:14.88
+0.06
30:13.74
30:24.18
+0.58
21097.5
1:04:51.00
1:04:50.60
-0.01
1:06:40.00
1:06:56.85
+0.42
1:08:32.00
1:07:34.87
-1.39
42195
2:15:25.00
2:15:00.95
-0.30
2:20:43.00
2:20:18.48
-0.29
2:21:06.00
2:22:16.17
+0.83
mean
0.42
0.20
1.05
Record
US women
EU women
GER women
tc[min]
10.80
10.19
5.87
vm[m/min]
347.42
351.63
356.56
100 γs
9.39
10.25
13.91
100 γl
5.17
4.63
5.01
Es
0.34
0.38
0.49
El
6.92
8.66
7.35
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
1000
02:31.80
02:32.01
+0.14
02:28.98
02:29.08
+0.07
02:30.67
02:30.48
-0.13
1500
03:56.29
03:56.68
+0.17
03:52.47
03:52.92
+0.20
03:57.71
03:59.58
+0.79
1609.34
04:16.71
04:15.62
-0.42
04:12.56
04:11.73
-0.33
04:21.59
04:19.83
-0.67
3000
08:25.83
08:26.40
+0.11
08:21.42
08:21.75
+0.07
08:29.89
08:34.62
+0.93
5000
14:38.92
14:37.27
-0.19
14:23.75
14:27.22
+0.40
14:42.03
14:41.99
-0.00
10000
30:13.17
30:24.74
+0.64
29:56.34
29:56.03
-0.02
30:57.00
30:34.60
-1.21
21097.5
1:07:34.00
1:07:03.57
-0.75
1:06:25.00
1:05:40.14
-1.13
1:07:58.00
1:07:25.42
-0.80
42195
2:19:36.00
2:20:00.22
+0.29
2:15:25.00
2:16:23.60
+0.72
2:19:19.00
2:20:46.06
+1.04
mean
0.34
0.37
0.70
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endurance of women for distances longer than the crossover distance dc is clearly visible. The
gray cones in the figure indicate the range of endurance parameters that could potentially be
realized in practice by runners from the recreational to the elite level with suitable event spe-
cialization. This type of visualization of race performances allows one to evaluate runner’s
endurance independently of their maximal aerobic power and running economy which are
described by the parameters vm and tc.
Estimate of supplemental power
We have seen that supplemental power is responsible for a slow logarithmic decline of racing
velocities with distance. In Fig 3 the supplemental factor of Eq (15) (square brackets in this
equation) is plotted for various record performances as function of the race duration T. The
variation range of the factor implies a supplemental power between  6% and 10% above the
nominal power, with the European male records (EU men) being an outlier. The curves have
their maximum at the crossover time T = tc. During supra-maximal exercise (for times shorter
than tc), the oxygen uptake cannot stabilize and continues to increase until the end of the race
[33]. Hence we observe an increasing deviation from the nominal power with increasing
Fig 2. Mean race velocity vðdÞ as function of race distance. Velocity is re-scaled by vm, and distance d is re-scaled by
dc = vmtc. Shown are the male and female world records (WR, dots), model prediction from Eq (12) (solid lines), and a
typically expected maximal range of velocities (gray regions). Indicated are the lower and upper limits of γs and γl for
these regions. Due to the re-scaling of vðdÞ and d, this graph highlights endurance for short and long duration,
independently of the velocity vm at maximal aerobic power.
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duration. However, at very short times below about 1 minute, oxygen uptake kinetics limit oxy-
gen supply, and the energy deficit is compensated by the anaerobic system. After 30 to 60 sec-
onds, the oxygen uptake can reach 90% of VO2max [33]. This short term kinetic effect is not
included in our model. Above tc, i.e., for sub-maximal velocities, oxygen uptake stabilizes and
the supplemental factor decreases. However, it does not decrease to one and this is likely related
to the fact that the energy cost of running starts to increase above a nominal linear curve when
the lactate threshold is approached [34]. For even longer race durations, we observe a slight
increase in the supplemental factor that is presumably linked to the increase of the energy cost
of running with increasing distance, as discussed in the Introduction. For a marathon or a 2
hour run at about 80% VO2max the supplemental power was measured to be between 5% and
7% in terms of oxygen uptake [35, 36] which is consistent with our model prediction for
T * 120min. We note that for male records, the supplemental factor shows a shallow minimum
around one hour. For female records this minimum is displaced to times above two hours.
Application 1: Reconstruction individual physiological profiles
After we have validated the accuracy of our model against record performances, we would like
to find out if it can be also applied to individual runners. If that is the case then one could
Fig 3. Plot of the supplemental factor of Eq (15) for as predicted by our model for male and female world records
(WR), US records (US), and European records (EU). The cusp in the curves occurs at the time tc.
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compute from their personal best performances their individual physiological parameters that
characterize their training state and future performance potential. The assessment of the train-
ing state of an individual is important not only for performance optimization but also beyond
competitive athletics for the monitoring of the health status of recreational runners.
There have been performance models developed for individual runners. A popular model is
the so-called VDOT model by Daniels [37]. This model and other approaches employ maximal
oxygen uptake as single factor determining performance [38, 39] and this parameter is then
used to determine the training state and to predict running performances. A notable exception
is the model Peronnet and Thibault which has been also applied to individual runners [7]. It
turns out that their model yields comparable but somewhat larger errors than the present
model. Partially, this might be due their model’s assumption that the energy cost of running
and the crossover time tc would be identical for all runners. Other physiological factors that
determine an individual’s performance include blood lactate concentration, and the anaerobic
threshold. However, these parameters require laboratory measurements that are not always
available, particularly on sufficiently short time intervals and for recreational athletes.
With the advent of large online databases for personal best performances, it becomes possi-
ble to probe the accuracy of performance models for a large set of individual athletes. Similar to
our analysis of running records, our model predictions for individual runners can be validated
through comparison with their personal best performances. First we reconstruct running econ-
omy and endurance profiles of an individual runner from personal best performances for a few
race distances and then estimate projected race times for other distances and also some charac-
teristic paces. This eliminates physiological uncertainties that result from the use of universal,
typical physiological parameters in previous models. In fact, the present model provides a gen-
eral scheme that can be applied to any endurance runner over a range of distances and it is not
based on observations made for only a small sample of trained athletes. Our approach also
yields individual relative intensities, in percent of the aerobic power reserve Pm − Pb, at which a
runner performs races. This is important for the relative use of fat and carbohydrate as fuels,
and hence the total carbohydrate consumption for a given race distance.
In the following, we apply our model to personal best performances of British runners that
are available online in the database www.thepowerof10.info [40]. As a first test of our model
for individual runners, we have considered the personal bests of the top nine male and female
marathon runners from this database, according to the 2015 ranking. Their personal best
times for seven distances from 800m to the marathon are summarized in Tables 3 and 4. With
the same methodology that we used for running records above, we obtain the four model
parameters for each runner that are also listed in the tables. From these parameters we com-
pute the predicted race times. We find that the agreement between the predicted and actual
race times are the most accurate to date, with an average mean error of less than 1% for each
individual runner for all seven distances, see Tables 3 and 4. This suggests that our model can
describe the running performance of individual runners with reliable accuracy. The slightly
larger mean error for individuals than for groups of runners (record holders) appears natural
since an individual runner can hardly reach optimized performance for all distances. When
analyzing personal bests of an individual runner one should also realize that the best times on
various distances have been probably obtained over a large time span of many years. Especially
at the beginning of the career of a runner, when he races predominantly shorter distances, per-
formance might not be optimal. Alternatively, one could consider only best performances
obtained within a short time interval like a year which however limits presumably the available
distances.
Hence the individual variations of the parameters tc and vm can be large but they are
strongly correlated. This suggests that tc gives a rather precise estimate of the time over which
A minimal power model for human running performance
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November 16, 2018
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Table 3. Personal best times and model parameters for individuals (Leading male marathon runners from UK, ranking 2015, http://www.thepowerof10.info/
rankings).
Runner
01
02
03
tc[min]
23.84
11.28
4.57
vm[m/min]
353.28
360.35
373.49
100 γs
8.16
11.65
10.26
100 γl
4.67
5.07
4.85
Es
0.29
0.42
0.38
El
8.52
7.20
7.86
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
800
01:52.08
01:52.52
+0.39
01:49.98
01:49.94
-0.04
01:58.32
01:58.32
+0.00
1500
03:41.88
03:41.06
-0.37
03:40.80
03:40.95
+0.07
03:57.48
03:57.48
-0.00
3000
07:48.90
07:46.84
-0.44
08:00.48
08:00.34
-0.03
08:16.62
08:16.24
-0.08
5000
13:28.32
13:31.65
+0.41
13:57.66
14:01.83
+0.50
14:13.32
14:09.91
-0.40
10000
28:49.02
28:32.80
-0.94
29:23.04
29:09.19
-0.79
29:18.48
29:26.13
+0.43
21097.5
1:01:25.02
1:02:32.11
+1.82
1:04:07.02
1:04:12.25
+0.14
1:04:30.00
1:04:49.91
+0.51
42195
2:10:55.02
2:09:41.73
-0.93
2:13:40.98
2:13:52.45
+0.14
2:15:51.00
2:15:11.79
-0.48
mean
0.76
0.24
0.27
Runner
04
05
06
tc[min]
19.88
8.57
6.88
vm[m/min]
357.87
349.94
382.82
100 γs
8.27
4.84
6.93
100 γl
5.70
4.19
5.70
Es
0.30
0.13
0.24
El
5.78
10.86
5.79
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
800
01:51.78
01:52.20
+0.38
02:09.48
02:08.54
-0.73
01:55.20
01:55.20
+0.00
1500
03:41.94
03:40.71
-0.56
04:05.22
04:08.44
+1.31
03:45.66
03:45.66
-0.00
3000
07:46.74
07:46.78
+0.01
08:40.50
08:34.37
-1.18
08:00.12
07:53.93
-1.29
5000
13:31.20
13:32.52
+0.16
14:38.58
14:36.90
-0.19
13:33.00
13:35.29
+0.28
10000
28:42.18
28:31.86
-0.60
30:04.02
30:10.06
+0.33
27:57.24
28:25.13
+1.66
21097.5
1:02:22.98
1:03:06.46
+1.16
1:04:46.98
1:05:55.65
+1.77
1:03:00.00
1:03:04.39
+0.12
42195
2:12:57.00
2:12:10.52
-0.58
2:18:21.00
2:16:24.13
-1.41
2:13:40.02
2:12:33.95
-0.82
mean
0.49
0.99
0.60
Runner
07
08
09
tc[min]
8.44
8.17
5.28
vm[m/min]
355.63
367.03
347.39
100 γs
5.72
8.15
15.25
100 γl
5.62
5.81
4.82
Es
0.17
0.29
0.52
El
5.93
5.59
7.95
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
800
02:05.10
02:04.97
-0.11
01:57.42
01:57.12
-0.26
02:00.42
02:00.42
+0.00
1500
04:02.40
04:02.87
+0.19
03:49.98
03:51.04
+0.46
04:10.08
04:10.08
-0.00
3000
08:28.62
08:26.15
-0.49
08:14.04
08:10.42
-0.73
08:47.70
08:51.43
+0.71
5000
14:35.94
14:30.06
-0.67
14:01.02
14:04.01
+0.36
15:18.30
15:09.93
-0.91
10000
30:04.02
30:17.72
+0.76
29:32.70
29:26.30
-0.36
31:30.90
31:30.08
-0.04
21097.5
1:06:04.02
1:07:09.06
+1.64
1:04:28.02
1:05:23.05
+1.42
1:09:12.00
1:09:20.92
+0.21
42195
2:22:55.98
2:20:56.89
-1.39
2:18:49.02
2:17:31.77
-0.93
2:24:31.02
2:24:32.68
+0.02
mean
0.75
0.65
0.27
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Table 4. Personal best times and model parameters for individuals (Leading female marathon runners from UK, ranking 2015, http://www.thepowerof10.info/
rankings).
Runner
01
02
03
tc[min]
13.01
9.48
3.85
vm[m/min]
319.66
316.60
368.12
100 γs
5.45
6.80
5.82
100 γl
4.70
4.28
4.41
Es
0.16
0.23
0.18
El
8.39
10.34
9.64
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
800
02:17.28
02:17.17
-0.08
02:18.60
02:18.31
-0.21
02:05.94
02:05.94
-0.00
1500
04:25.56
04:25.95
+0.15
04:29.58
04:30.60
+0.38
04:05.40
04:05.12
-0.11
3000
09:13.08
09:12.71
-0.07
09:32.82
09:28.54
-0.75
08:22.20
08:26.50
+0.86
5000
15:44.22
15:47.11
+0.31
16:13.02
16:09.74
-0.34
14:29.10
14:25.36
-0.43
10000
32:39.36
32:42.02
+0.14
33:01.98
33:23.16
+1.07
30:01.08
29:51.82
-0.51
21097.5
1:12:36.00
1:11:45.67
-1.16
1:12:28.02
1:13:01.34
+0.77
1:05:40.02
1:05:30.03
-0.25
42195
2:28:04.02
2:29:05.72
+0.69
2:32:40.02
2:31:12.50
-0.96
2:15:25.02
2:16:00.78
+0.44
mean
0.37
0.64
0.37
Runner
04
05
06
tc[min]
9.45
5.92
12.36
vm[m/min]
317.48
297.47
303.80
100 γs
6.93
9.70
9.00
100 γl
5.06
4.62
6.19
Es
0.24
0.36
0.33
El
7.23
8.71
5.02
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
800
02:18.72
02:17.68
-0.75
02:28.80
02:28.80
-0.00
02:17.40
02:17.16
-0.18
1500
04:26.04
04:29.59
+1.33
04:57.42
04:57.42
-0.00
04:30.84
04:31.69
+0.31
3000
09:36.72
09:26.96
-1.69
10:22.86
10:21.12
-0.28
09:40.44
09:39.64
-0.14
5000
16:08.10
16:11.38
+0.34
17:43.02
17:42.23
-0.07
16:47.82
16:46.53
-0.13
10000
33:24.72
33:39.59
+0.74
36:40.02
36:42.61
+0.12
35:18.00
35:11.88
-0.29
21097.5
1:13:21.00
1:14:10.86
+1.13
1:19:55.02
1:20:39.07
+0.92
1:17:43.02
1:18:25.22
+0.90
42195
2:36:39.00
2:34:47.30
-1.19
2:48:55.98
2:47:45.34
-0.70
2:46:19.02
2:45:29.11
-0.50
mean
1.03
0.30
0.35
Runner
07
08
09
tc[min]
16.50
5.40
14.64
vm[m/min]
281.73
300.16
272.55
100 γs
7.43
16.76
7.25
100 γl
4.28
5.16
4.45
Es
0.26
0.55
0.25
El
10.33
6.93
9.47
distance
T
Tmodel
%
T
Tmodel
%
T
Tmodel
%
800
02:29.82
02:29.38
-0.29
02:20.22
02:20.22
-0.00
02:37.26
02:36.54
-0.46
1500
04:51.42
04:52.95
+0.52
04:55.20
04:55.20
-0.00
05:04.32
05:06.82
+0.82
3000
10:18.72
10:17.25
-0.24
10:08.70
10:20.47
+1.93
10:48.48
10:46.06
-0.37
5000
17:58.98
17:48.35
-0.98
18:13.98
17:44.85
-2.66
18:26.70
18:32.44
+0.52
10000
36:31.98
36:45.40
+0.61
37:07.98
36:59.37
-0.39
38:34.98
38:19.98
-0.65
21097.5
1:19:07.02
1:20:19.99
+1.54
1:20:39.00
1:21:45.39
+1.37
1:24:06.00
1:23:55.81
-0.20
42195
2:48:16.02
2:46:13.19
-1.22
2:51:46.02
2:51:06.21
-0.39
2:53:25.02
2:53:58.66
+0.32
mean
0.77
0.96
0.48
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a runner can sustain the velocity vm which, however, can deviate slightly from the actual veloc-
ity at VO2max, depending on the available personal best performances in the vicinity of this
crossover point. In order to measure individual endurances independently of aerobic capacity,
we have computed and plotted the relation between the re-scaled race velocity vðdÞ=vm and
distance d/dc in analogy to our analysis of running records, see Fig 4. Two important observa-
tions can be made from this graph: (1) For each individual runner, there are two distinct rela-
tions between velocity and distance above and below the crossover velocity vm and distance dc.
(2) Even within the group of top UK marathon runners, there is a large variation in endur-
ances as quantified by the different slopes of the re-scaled velocity-distance curves and the
parameters γs and γl. They gray cones of expected maximal variations shown in Fig 4 are
almost completely covered by the performances of the studied runners.
For one of the female runners included in Table 4, runner 03 which is Paula Radcliffe, phys-
iological data are available for a long time span of about 12 years [41]. While her personal rec-
ords have been obtained over a similar period of time (800m in 1993 and marathon in 2003),
and her physiological data have progressed during this time, in particular running economy,
we can compare our model prediction for the speed vm to Radcliffe’s speed at VO2max, aver-
aged over the time period from 1993 to 2003 which is about 22.5 km/h or 375.0 m/min [41].
This value compares very well with our finding of vm = 373.5 m/min, see Table 4.
Our findings show that individual performances do not follow a unique power law as sug-
gested, for example, by Riegel’s formula. There are more complex variations of physiological
metrics among runners and those have to be taken into account for describing and predicting
accurately performances and presumably optimal training. Our computational approach
reveals the physiological parameters that determine individual performance and explains how
they can be used in praxis to guide training and racing.
Application 2: Personalized characteristic paces
We expect that our four parameter model can measure an individual runner’s performance
status for distances from 800m to the marathon more accurately than previous performance
models that often assume for all runners the same (average) values for certain characteristics
like running economy or endurance. An example for the latter type of models is the popular
VDOT model of J. Daniels which assumes a fixed running economy and endurance curves for
all runners [37, 42]. Although the VDOT model represents a good first approximation of char-
acteristic paces based on a single race performance, the ability to monitor individual perfor-
mances with more than just one parameter allows the runner to ascertain a better
understanding of their training status and potential performance. It then becomes beneficial to
have a model that makes use of larger available data sets. In the same way that one may better
understand current fitness by examining relative oxygen consumption at different paces rather
than absolute oxygen consumption, [43] developing an approach that makes use of perfor-
mance over several races describes an individual runner better than a single race.
Characteristic paces are often defined by the pace that a runner can race (at current training
status) for a prescribed duration or distance. When the physiological model parameters of a
runner are known from sufficiently many recent race performances, the running velocities for
a prescribed intensity and duration, or intensity and distance can be computed from Eqs (17)
and (18), respectively. In the following we consider race paces for a given duration or distance,
corresponding to ^p ¼ 1 in these equations. In order to compare our model predictions to the
characteristic paces of the VDOT model, we consider three hypothetical runners that are
assumed to have achieved race performances as predicted by the VDOT model with model
parameter values VDOT = 40, 60, and 80. (VDOT can be regarded as an effective value for
A minimal power model for human running performance
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Fig 4. Same visualization of endurance as in Fig 2 but for individual male (top) and female (bottom) runners, see
Tables 3 and 4. Colors label different runners.
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VO2max, see [37] for details.) From these race performances we obtain the four parameters of
our model. These parameters are given in the captions of Tables 5, 6 and 7. These tables pro-
vide race paces (time per km) for various distances and durations specified in the first column.
Some of the paces correspond the specific paces named in the VDOT model, and they are
labeled correspondingly as R-, I-, T- and M-pace. The paces proposed by the VDOT model are
given in the second column. The remaining columns provide the predictions of our model.
The third column lists the paces as obtained from the values of the four model parameters that
result from the hypothetical race performances of the runner with the given VDOT score.
There is agreement within a few seconds per kilometer. It should be kept in mind that our
model, unlike the VDOT model, does not implement any fixed parameters or constants a pri-
ori. We observe that the fixed parameters of the VDOT model correspond to rather superior
endurance with γl  0.05 for long distances and average endurance with γs  0.09 for short
distances. As we have seen above, there is substantial variation in these parameters among
individuals. Hence, characteristic paces should also determined individually. We have modi-
fied the endurance parameters γl and γs independently within their typical minimal and maxi-
mal values while keeping vm and tc unchanged. The resulting paces are shown in the last four
columns of the tables. The fast paces for short distances (1mile and 5min paces) can change up
to ±10sec/km compared to the original VDOT model which is substantial. For the slower
paces (for time tc and longer) the variation can be even larger with a maximum change for the
Table 5. Paces per km for a runner with VDOT = 40 score for different endurances. The original physiological parameters are tc = 12.35min, vm = 214.88m/min, γl =
0.051 and γs = 0.096. In last 4 columns the endurances El and Es are given only when they are different from the original values.
pace at
max. power for
Ref. [37]
original
γl = 0.04
γl = 0.08
γs = 0.15
γs = 0.05
El = 7.1
El = 12.2
El = 3.5
Es = 0.35
Es = 0.51
Es = 0.14
1 mile (R-pace)
04:20
04:25.21
orig.
orig.
04:16.72
04:32.06
5min
—
04:16.96
orig.
orig.
04:05.87
04:27.14
time tc (I-pace)
04:42
04:39.22
orig.
orig.
orig.
orig.
5.000m
04:49
04:49.06
04:46.79
04:55.54
orig.
orig.
10.000m
05:00
05:00.67
04:55.58
05:15.85
orig.
orig.
60min (T-pace)
05:06
05:03.67
04:58.07
05:19.64
orig.
orig.
Half marathon
05:15
05:14.30
05:05.68
05:41.31
orig.
orig.
marathon (M-pace)
05:29
05:28.16
05:15.70
06:09.16
orig.
orig.
https://doi.org/10.1371/journal.pone.0206645.t005
Table 6. Paces per km for a runner with VDOT = 60 score for different endurances. The original physiological parameters are tc = 12.67min, vm = 298.51m/min, γl =
0.052 and γs = 0.092. The meaning of the columns is the same as in Table 5.
pace at
max. power for
Ref. [37]
original
γl = 0.04
γl = 0.08
γs = 0.15
γs = 0.05
El = 6.8
El = 12.2
El = 3.5
Es = 0.34
Es = 0.51
Es = 0.14
1 mile (R-pace)
03:05
03:05.04
orig.
orig.
02:54.93
03:12.36
5min
—
03:05.15
orig.
orig.
02:56.39
03:12.07
time tc (I-pace)
03:23
03:21.00
orig.
orig.
orig.
orig.
5.000m
03:25
03:24.14
03:23.36
03:26.00
orig.
orig.
10.000m
03:32
03:32.41
03:29.49
03:39.64
orig.
orig.
60min (T-pace)
03:40
03:38.78
03:34.33
03:49.55
orig.
orig.
Half marathon
03:42
03:42.12
03:36.53
03:56.62
orig.
orig.
marathon (M-pace)
03:52
03:51.99
03:43.51
04:15.08
orig.
orig.
https://doi.org/10.1371/journal.pone.0206645.t006
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November 16, 2018
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marathon pace (M-pace). For a VDOT = 40 runner, the M-pace window between slowest and
fastest pace is about 55sec/km, for a VDOT = 60 runner it is about 30sec/km and even for a
high level runner with VDOT = 80 it is still about 20sec/km. These variations result from dif-
ferent endurances, with the crossover speed vm unchanged. We have also studied the effect of a
modification of the time tc from the original VDOT model value which appears rather long
with 12 to 13min. The results are shown in Tables 8, 9 and 10. The first three columns have the
same meaning as in the three tables before. The last four columns list the paces that correspond
to a reduction or an increase of tc by 10% or 20%, respectively. Here we observe a smaller varia-
tion by a few seconds around the original paces, relatively independent of the duration or dis-
tance that defines the pace. This shows that racing paces are more dependent on endurance
than on the time over which runners can sustain their crossover speed at VO2max. The reason
for that is the exponential dependence on γs, γl of the duration T(p) over which a relative
power p can be maintained, independently of tc and vm, see Fig 1.
It is interesting to relate this observation to physiological parameters that can be measured
in the laboratory and have been linked to endurance capacity, like blood lactate concentration.
It is known that the running speed at the lactate threshold can improve independently of
VO2max and so can the runner’s endurance. Often the lactate threshold pace is identified with
the running velocity that a runner can race for about 60min. The corresponding paces are
shown in Tables 5–10 as “T-pace”. The relative intensity or power output in percent of the aer-
obic power reserve [see Eq (1)] at the lactate threshold is given by pLT = 100[1 − γl log(60/tc)].
For example, for a recreational runner (with VDOT = 40), described by the parameters of
Table 7. Paces per km for a runner with VDOT = 80 score for different endurances. The original physiological parameters are tc = 12.92min, vm = 376.85m/min, γl =
0.053 and γs = 0.088. The meaning of the columns is the same as in Table 5.
pace at
max. power for
Ref. [37]
original
γl = 0.04
γl = 0.08
γs = 0.15
γs = 0.05
El = 6.6
El = 12.2
El = 3.5
Es = 0.32
Es = 0.51
Es = 0.14
1 mile (R-pace)
02:25
02:23.98
orig.
orig.
02:13.52
02:30.46
5min
—
02:26.99
orig.
orig.
02:19.37
02:32.00
time tc (I-pace)
02:41
02:39.22
orig.
orig.
orig.
orig.
5.000m
02:40
02:39.45
02:39.39
02:39.59
orig.
orig.
10.000m
02:46
02:45.91
02:44.14
02:49.88
orig.
orig.
60min (T-pace)
02:54
02:53.33
02:49.64
03:01.52
orig.
orig.
Half marathon
02:53
02:53.50
02:49.59
03:02.65
orig.
orig.
marathon (M-pace)
03:01
03:01.22
02:54.99
03:16.46
orig.
orig.
https://doi.org/10.1371/journal.pone.0206645.t007
Table 8. Paces per km for a runner with VDOT = 40 score for different variations of the time tc. The original physiological parameters are tc = 12.35min, vm =
214.88m/min, γl = 0.051 and γs = 0.096.
pace at
max. power for
Ref. [37]
original
tc = 12.35min
0.8tc
0.9tc
1.1tc
1.2tc
1 mile (R-pace)
04:20
04:25.21
04:31.27
04:28.03
04:22.70
04:20.46
5min
—
04:16.96
04:22.12
04:19.37
04:14.82
04:12.90
time tc (I-pace)
04:42
04:39.22
04:42.43
04:40.73
04:36.70
04:34.43
5.000m
04:49
04:49.06
04:52.69
04:50.76
04:47.53
04:46.15
10.000m
05:00
05:00.67
05:04.62
05:02.52
04:59.02
04:57.52
60min (T-pace)
05:06
05:03.67
05:07.47
05:05.45
05:02.07
05:00.63
Half marathon
05:15
05:14.30
05:18.63
05:16.33
05:12.49
05:10.86
marathon (M-pace)
05:29
05:28.16
05:32.89
05:30.37
05:26.18
05:24.39
https://doi.org/10.1371/journal.pone.0206645.t008
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November 16, 2018
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Table 5, one has pLT = 91.94% for the original value γl = 0.051, while pLT = 93.68% for γl = 0.04,
and pLT = 87.35% for γl = 0.08. These values appear rather large when compared to the lactate
threshold estimates from current world records: pLT = 87.08% for male and pLT = 90.41% for
female records. This implies again that the VDOT model assumes a rather optimized
endurance.
Conclusion
Modern performance testing is often based on laboratory testing of athletes with the goal of
identifying physiological metrics that correlate with performance and can be linked to funda-
mental physiological processes. However, measuring physiological metrics requires time con-
suming and expensive testing, often under rather idealized laboratory conditions. Hence, it
appears to be very useful to extract information on power characteristics for individual run-
ners or certain groups of runners from performance results in racing events or time trails. This
is of particularly great interest for analyzing the effect of aging on human performance, consid-
ering the enormous improvement of performance in older age groups. As stated already by A.
V. Hill, world and other records constitute very interesting data sets since their accuracy by far
exceeds that of laboratory measurements and they correspond to best human performances at
a given time in history under realistic conditions.
The model presented here provides a quantitative method for extracting characteristic
parameters from race performances of a group of runners or of an individual runner. The key
equations and computational steps of our model are as follows:
Table 9. Paces per km for a runner with VDOT = 60 score for different variations of the time tc. The original physiological parameters are tc = 12.67min, vm =
298.51m/min, γl = 0.052 and γs = 0.092.
pace at
max. power for
Ref. [37]
original
tc = 12.67min
0.8tc
0.9tc
1.1tc
1.2tc
1 mile (R-pace)
03:05
03:05.04
03:08.94
03:06.86
03:03.42
03:01.97
5min
—
03:05.15
03:08.72
03:06.82
03:03.67
03:02.34
time tc (I-pace)
03:23
03:21.00
03:23.37
03:22.11
03:19.25
03:17.68
5.000m
03:25
03:24.14
03:26.73
03:25.36
03:23.06
03:22.08
10.000m
03:32
03:32.41
03:35.22
03:33.72
03:31.23
03:30.17
60min (T-pace)
03:40
03:38.78
03:41.60
03:40.10
03:37.60
03:36.54
Half marathon
03:42
03:42.12
03:45.20
03:43.56
03:40.83
03:39.66
marathon (M-pace)
03:52
03:51.99
03:55.36
03:53.57
03:50.58
03:49.31
https://doi.org/10.1371/journal.pone.0206645.t009
Table 10. Paces per km for a runner with VDOT = 80 score for different variations of the time tc. The original physiological parameters are tc = 12.92min, γl = 0.053
and γs0.088.
pace at
max. power for
Ref. [37]
original
tc = 12.92min
0.8tc
0.9tc
1.1tc
1.2tc
1 mile (R-pace)
02:25
02:23.98
02:26.80
02:25.30
02:22.81
02:21.76
5min
—
02:26.99
02:29.69
02:28.25
02:25.87
02:24.85
time tc (I-pace)
02:41
02:39.22
02:41.12
02:40.11
02:37.90
02:36.71
5.000m
02:40
02:39.45
02:41.48
02:40.40
02:38.17
02:36.87
10.000m
02:46
02:45.91
02:48.11
02:46.94
02:44.99
02:44.16
60min (T-pace)
02:54
02:53.33
02:55.60
02:54.39
02:52.38
02:51.53
Half marathon
02:53
02:53.50
02:55.91
02:54.63
02:52.49
02:51.58
marathon (M-pace)
03:01
03:01.22
03:03.85
03:02.45
03:00.11
02:59.12
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• The key equation for the comparison of our model to race results is the expression for the
race time T(d) as function of the race distance d, given in Eq 11.
• We minimize the sum of the squared relative deviations in percent between actual race times
and the function T(d) by varying the four model parameters vm, tc, γs, and γl in T(d) for all
distances raced by a runner or a group of runners (records). The final model parameters are
those that result from this minimization.
• From the four model parameters the values of which were obtained from race performances
or by other input like physiological data, the function T(d) predicts the race times and Eq 12
the mean race velocities for arbitrary distances.
The model parameters quantify the runner’s performance status and can be used to predict
personalized fastest possible but realistic and safe racing paces for a wide range of race distances
and durations. Our model provides an unified description of running events at sub- and supra-
maximal velocities that are separated by a time scale tc whose value is in good agreement with
independent measurements. On a fundamental level, for the first time our approach provides a
derivation of the previously observed but unexplained linear relation between the mean velocity
and the logarithm of the duration for running records. The mechanism underlying this loga-
rithmic relation could be identified as the necessity of a supplemental power, beyond the nomi-
nal power cost of running, for maintaining the mean velocity. Our findings are different from
the previously postulated power law relation between the mean race speed v and distance d,
v  dbreak 2 hours in the marathon. For example, the latest update of the world record in the marathon
by Eliud Kipchoge in Berlin on September 16, 2018 which is included in our results of Table 1,
has increased the endurance for long duration from El = 5.98 to El = 6.46, i.e., by 8%, while the
speed that can be raced for 6min (413.5 m/min) and the short term endurance Es remained basi-
cally unchanged. Our model predicts that the endurance for long duration had to be increased to
El = 7.49 with all other parameters unchanged to obtain a marathon time of 1:59:56. This corre-
sponds to another increase of 16% compared to the just updated value which appears unrealistic
in near future. Another possibility, however, would be to assume the endurance of the current
world record, El = 6.46, and an increased speed at VO2max. For example, increasing the speed
that a runner can sustain for 6min by 1.3% to vm = 418.7 m/min would yield a marathon time of
1:59:58. This could be achieved by an increase in running economy by only *1% which seems
feasible, at least by material improvements and/or suitable racing conditions (course, climate).
Future studies based on our model could include the dependence of the performance state
on distance specialization, altitude, air temperature, age, and other factors. With the availabil-
ity of big data set on running performances, these studies could be performed with much bet-
ter statistics than studies with much smaller groups of runners participating in laboratory and
clinical studies. Our model could be applied to other endurance sports after a modification of
the running specific dependence of power on velocity.
Supporting information
S1 Appendix. Solution of the integral equation for Pmax(T).
(PDF)
S2 Appendix. Comparison to oxygen uptake measurement.
(PDF)
Acknowledgments
Valuable discussions with Veronique Billat and Francois Pe´ronnet on various physiological
aspects of the model and with Jack Daniels on the methodology of the VDOT model are
acknowledged.
Author Contributions
Conceptualization: Thorsten Emig.
Data curation: Guillaume Adam.
Formal analysis: Matthew Mulligan, Thorsten Emig.
Funding acquisition: Thorsten Emig.
Investigation: Matthew Mulligan, Guillaume Adam, Thorsten Emig.
Methodology: Thorsten Emig.
Software: Matthew Mulligan, Guillaume Adam.
Supervision: Thorsten Emig.
Validation: Guillaume Adam, Thorsten Emig.
Writing – original draft: Matthew Mulligan, Thorsten Emig.
Writing – review & editing: Matthew Mulligan, Guillaume Adam, Thorsten Emig.
A minimal power model for human running performance
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November 16, 2018
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A minimal power model for human running performance
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A minimal power model for human running performance.
Mulligan, Matthew,Adam, Guillaume,Emig, Thorsten