Reviewers' Comments: 
 
Reviewer #1: 
Remarks to the Author: 
The authors are correct in identifying that an explosion of data collection from wearable exercise 
apps has the potential to enable new insight into exercise bioenergetics and fatigue. However, 
while I like the general approach taken by the authors and appreciate the labour involved in their 
study, I believe they have 'missed a trick' in limiting their analysis to the 'universal model of 
running performance' described by Mulligan et al in 2018 (and not yet validated by others). The 
'critical power' model is well established in the field and has both theoretical and practical validity 
but is unfortunately not used by the authors; indeed, the authors are somewhat (and unjustifiably, 
in my opinion) dismissive of the CP concept. In summary, as presented, the study is limited to a 
novel and essentially unvalidated model of running performance and this calls into question at 
least some of the conclusions. While the approach (data mining) surely has merit, the analyses 
need to be less blinkered and more comprehensive in this first step. 
 
 
 
Reviewer #2: 
Remarks to the Author: 
In this study, the researchers show that a model they previously developed to predict future race 
performance from past race performances performs well in a large dataset of race times and 
distances estimated from a running watch. They extract parameters from the model related to (1) 
endurance and (2) running speed and VO2 Max and show that there are associations between 
these two parameters and training metrics extracted from the dataset. 
 
The dataset is exciting, the associations with training are interesting, and as far as I can tell, the 
analysis as performed was sound overall. But there are some issues with the manuscript as written 
and the analysis as performed that limit impact. Of most significance is that the novelty is not 
overwhelmingly clear. For example, the model has been previously published and shown to relate 
well to real-world performance data, so these aspects of the current paper are not particularly 
novel. The physiological parameters are shown to vary among the population, but this is not 
particularly novel besides the means in which the parameters were extracted. The correlations 
with training performance are just correlations and, as the authors acknowledge, it is not possible 
to determine whether the training measures associated with higher performance cause those 
higher performances or are merely associated with being a high-performing athlete. 
 
Given the size and, I expect, richness of the dataset, I imagine that there is much more that the 
investigators could have learned. I list a few questions in the following paragraphs that the authors 
might have explored. 
 
Do aspects of training help to explain errors in the model predictions? This could have helped us 
move toward a more causal link between training and race performance. There also seemed to be 
a systematic error in their model related to the endurance parameter. Is this related to training? 
Or errors in their data? Or a gap in their model? This question should have been explored more 
fully. 
 
Are there means to predict an athlete’s race performance from sub-maximal training performance 
(i.e., not races), using heart rate or any other measures the watch might provide? The current 
model requires subjects to performance two or more races at maximal effort to extract these 
parameters. While this is an improvement on physiological testing it is still a burden and does not 
seem to take advantage of the dataset. I presume heart rate is available, for example. Would 
heart rate and heart-rate variability during training help to detect some of the physiological 
parameters on training runs? 
 
Do the measures extracted from their model and the real-world dataset match measures extracted 
from gold standard lab assessments in a small (but heterogeneous) subset of the subjects? While 
the researchers do compare to previously published data, these tests would have provided more 
convincing evidence that their model is valid in a population with varying age, gender, ethnicity, 
and training status. 
 
Does their phenomenological model perform better than past models on this large dataset? What 
about a simple linear regression model? Comparing the models against additional baselines would 
have provided further confidence. 
 
Another major contribution could be to share the dataset with other researchers, which would be 
highly novel and a means to accelerate research on human performance, injury, and real-world 
training. I would not expect the researchers to tackle all of these problems, but I would expect 
more novel insights or contributions in some form. 
 
Another issue with the current submission is that I found the manuscript more challenging to read 
and understand than needed. Work is needed to improve the readability and clarity of the writing. 
As one example, the abstract as written contains very few specific details about the study that was 
performed. What parameters were predicted? What is performance? What are training modes? 
Given space, the abstract should not be exhaustive, but the key details should be described with 
enough specificity to give the reader a more clear understanding of what the study entailed. The 
Introduction, Results, and Discussion need similar improvements to more clearly and succinctly 
state what analysis was performed. 
 
 
 
Reviewer #3: 
Remarks to the Author: 
MAJOR 
In the Introduction, the authors challenge an axiom that has been characterizing exercise 
physiology since longer than a century, the axiom that measurement conditions should be 
standardized. I kindly disagree with this view. Existing models, validated experimentally in the 
laboratory, and applicable on the field and on large-scale numbers, come from standard 
experimental laboratory conditions. The theoretical models do exist indeed. They have been 
developed theoretically and validated by measuring V’O2max during exercise testing and the 
fraction of V’O2max utilization and the energy cost of the locomotion mode at stake at steady 
state. The basic formula id as follows: 
V = f * V’O2max/C (1) 
Where v is velocity, f the sustainable fraction of V’O2max over a given distance, and C the energy 
cost of the locomotion mode at stake. C has the dimension of a force and represents the metabolic 
energy that is to be spent to generate the force that is necessary to overcome the forces that 
oppose to the body movement. These are two forces: 1) air resistance (or water resistance in 
swimming) and 2) frictional forces. These can be expressed as follows: 
C = k v2 + Cf (2) 
This equation means that, if you plot C as a function of the square of speed, you obtain linear 
relationships with slope equal to constant k and y-intercept equal to the energy cost that is 
necessary to overcome frictional forces (high in running, low in cycling). Constant k is directly 
proportional to the frontal surface area of the moving body, to the aerodynamic (or hydrodynamic 
factor) Cx, and to the air (or water) density. With these simple relationships, it is possible to 
simulate and interpret all what happens in field conditions and during actual competitions. I just 
give the authors a few references (clearly overlooked by the authors) to make them aware of what 
I am saying: di Prampero PE, Int J Sports Med 7: 55-72, 1986; di Prampero PE, Eur J Appl Physiol 
82: 345-360, 2000; di Prampero et al, J Appl Physiol 47: 201-206, 1979; di Prampero et al, Eur J 
Appl Physiol 55: 259-266, 1986; Ferretti et al, Eur J Appl Physiol 111: 391-401, 2011; Margaria et 
al, J Appl Physiol 18: 367-370, 1963; Minetti et al, J Appl Physiol 93: 1039-1046, 2002; 
Pendergast et al, J Appl Physiol 43: 475-479, 1977; Tam et al, Eur J Appl Physiol 112: 3797-3806, 
2012; Zamparo et al, Eur J Appl Physiol 111: 367-378, 2011. All the elements that are necessary 
to set a model to be applied to large data sets and in field conditions are present in those and 
many other studies. 
What is the use of a variable like running economy, which the authors define as the steady state 
oxygen consumption at a given constant speed instead of C? Physically speaking, C looks more 
appropriate. 
At the end of page 2, the authors create artificially a dichotomy between laboratory tests and 
actual competition conditions: nevertheless, the theoretical background derived from laboratory 
tests is fully applicable to laboratory conditions. 
A sentence like “Unfortunately, these approaches predict that speeds below a critical velocity can 
be maintained for infinite duration which contradicts observation” has been criticized (see e.g. 
Ferretti, Energetics of Muscular Exercise, Springer 2015, chapter on critical power). Laboratory 
physiologists are perfectly aware of the fact that energy sources in the body are finite. 
Under Results, I read: “Main determinants of aerobic fitness and endurance in long distance 
runners (LDR) are maximal oxygen uptake per body weight, VO2;max , velocity dependent, sub-
maximal oxygen demand, known as running economy (RE), and the lactate or ventilatory 
threshold (LT) that sets the limit below which a steady state blood lactate concentration is 
maintained”. Although a reference is given, this statement is not correct, see equation 1 in this 
report. 
Equation 1 of the article is constructed in such a way to encompass a large variety of field 
conditions, independent of the physiological model. Frankly, I do not see a role for such a study, 
unless we create an artificial opposition between classical physiological studies and a kind of 
“modern” approach that the authors claim. Big numbers are fascinating, but their utility depend of 
the context into which they are inserted and interpreted. The context exists, but the authors seem 
to be unaware of it. 
The discussion id biased by the chosen approach and does not deserve to be discussed analytically 
MINOR 
Endurance running dates back much longer than the ancient Olympic Games: it is alike that pre-
historical nomadic societies used endurance running while hunting or for migrating. 
Page 2, line 7 : ranging instead of raging 
3
Rebuttal letter
“Novel insights on human exercise performance from big data mining”
(NCOMMS-20-02292)
Please ﬁnd below our point-to-point answers to the reviewer comments (C: comment, A: answer).
Answer to Reviewer #1
We thank the reviewer for her/his time spent looking over our manuscript and their comments that we address
point-by-point in the following.
C The authors are correct in identifying that an explosion of data collection from wearable exercise apps has
the potential to enable new insight into exercise bioenergetics and fatigue. However, while I like the general
approach taken by the authors and appreciate the labour involved in their study, I believe they have ’missed a
trick’ in limiting their analysis to the ’universal model of running performance’ described by Mulligan et al in
2018 (and not yet validated by others). The ’critical power’ model is well established in the ﬁeld and has both
theoretical and practical validity but is unfortunately not used by the authors; indeed, the authors are somewhat
(and unjustiﬁably, in my opinion) dismissive of the CP concept.
A We acknowledge that the ’critical power’ model is well established. A. V. Hill described the idea behind this
concept already in 1925 (The Physiological Basis of Athletic Records, Lancet, 1925). He derived the idea from
running and other world records as he noted that maximum speed or power over time T follows a hyperbolic
curve that can be described by the equation Pmax(T) = Pc + A/T, where Pc corresponds to critical power
(although Hill did not use that term) and A represent anaerobic power reserve. Furthermore, Hill noted that
this relation is limited to durations up to 12 minutes and called it short-term fatigue. He thought that short-term
fatigue originates from muscles whereas other forms of fatigue that take longer to develop have more complex
origins, such as neural fatigue, and are therefore much harder to describe. Thus, his idea of ’critical power’ was
not meant to describe human performance over long-duration exercise. Currently, the ’critical power’ model is
mainly applied to running distances up to 10km. For longer distances, the concept of a duration dependent
fractional utilization of maximal aerobic power is required, as pointed out also by Reviewer #3 (see also work
by di Prampero, and Peronnet & Thibault). In fact, Hill indicated in Figure 4 of his 1925 paper that the
average running velocity tends to decrease logarithmically with race duration (we have attached this ﬁgure as
an appendix to this rebuttal). Our universal model for running performance, as described in our previous paper
in 2018 and used in the present work, builds on Hills observation. It describes running performance over a
much broader exercise duration band. However, we do acknowledge that critical power has been useful in some
applications. For example, it has been used in cycling, where loading of the muscles and fatigue is di↵erent
from running. Below, we have attached a table in which we summarize the ’critical power’ and other models.
In fact, the ’critical velocity’ vc (corresponding to critical power) indeed occurs in our model (as described in
Mulligan et al in 2018) as a combination of parameters, i.e., vc = vm − D0/tc where we followed the notation of
the ’critical power’ model used in A. M. Jones, A. Vanhatalo, Sports Med 47, S65 (2017). In order to highlight
the di↵erence between the ’critical power’ model and our model, we have performed a detailed comparison of
the models, using running world records from 1987 and 2020, and also personal records from six elite marathon
runners taken from the above mentioned publication of A. M. Jones & A. Vanhatalo. Corresponding results
are attached below. As the relation between the models is not directly relevant to our present data analysis of
race distances between 5km and the Marathon, we have removed the reference to ’critical power’ to avoid any
confusion and to not give a false impression of this concept.
C In summary, as presented, the study is limited to a novel and essentially unvalidated model of running perfor-
mance and this calls into question at least some of the conclusions. While the approach (data mining) surely
4
has merit, the analyses need to be less blinkered and more comprehensive in this ﬁrst step.
A While we agree that our model is novel, we disagree with the conclusion that it is ”essentially unvalidated”. As
pointed out by Reviewer #2, ”the model has been previously published and shown to relate well to real-world
performance data”. We note that other researchers have checked their models also by comparison to athletic
records for a certain range of distances, and so did we for our model. To provide a constructive basis for further
review, and to clear up misunderstandings, but also to defend our mathematical model used for analyzing the
data, we provide below a new, rather detailed comparison of the ’critical power’ model and our model, including
new graphs and tables. Our new results show that our model agrees with current athletic world and personal
records with an error of less than 1%. We are not aware of any mathematical model that explains current world
records from 800m to the Marathon at better accuracy.
5
Rebuttal letter
“Novel insights on human exercise performance from big data mining”
(NCOMMS-20-02292)
Please ﬁnd below our point-to-point answers to the reviewer comments (C: comment, A: answer).
Answer to Reviewer #2
We thank the reviewer for her/his time spent looking over our manuscript and their comments and interesting
questions that we address point-by-point in the following.
C The dataset is exciting, the associations with training are interesting, and as far as I can tell, the analysis as
performed was sound overall. But there are some issues with the manuscript as written and the analysis as
performed that limit impact. Of most signiﬁcance is that the novelty is not overwhelmingly clear. For example,
the model has been previously published and shown to relate well to real-world performance data, so these aspects
of the current paper are not particularly novel.
The physiological parameters are shown to vary among the
population, but this is not particularly novel besides the means in which the parameters were extracted. The
correlations with training performance are just correlations and, as the authors acknowledge, it is not possible
to determine whether the training measures associated with higher performance cause those higher performances
or are merely associated with being a high-performing athlete.
A Novelty of the current paper is that it can explain running performance from 5km up to the marathon in a large
group of runners, with a wide range of performance levels, using two e↵ective parameters: the crossover velocity
vm and the endurance parameter El. Usually, running performance is measured by VO2max alone, which is a
poor indicator of performance as it ignores running economy (the energy cost of running per distance) which
shows a considerable variation among athletes. In addition, running economy changes (slowly) over time, which
is believed to be associated with various forms of fatigue and change of physiological parameters like, e.g.,
body temperature. Given the complexity of these mechanisms and their current poor understanding, it appears
interesting that two parameters are rather e↵ective in describing running performances of a few hours duration.
It should be stressed that our model is the ﬁrst to model running over these long time scales by a logarithmic
decay in fractional utilization (FU) of maximal aerobic power. Previous models considered constant or linear
decrease in FU, leading to systematic errors for distances over 10km (Only the model by Peronnet & Thibault
considers a combination of logarithmic and power law decays). We have added in the appendix to this rebuttal
a detailed comparison of our model to other existing models that were suggested by the other reviewers.
In our work, we think for the ﬁrst time, one can see how training history is associated with key performance
parameters on a large population level. It should be noted that endurance is impossible to measure in the
laboratory as it would require multiple hours of running on a treadmill which presumably involves a number of
artiﬁcial e↵ects compared to ’real world’ running. Hence, there is currently no reliable evaluation of correlations
between real-world running endurance and training beyond some studies of individual athletes. It is, however,
correct that our analysis does not determine if training is the cause of observed performance, and just associated
with higher performance level. While this is an interesting question for future analysis, even the here detected
correlations can be of practical importance: They can be useful for estimating realistic expectations for a race
for less experienced runners from their training intensity and volume, and hence prevent ”hitting the wall”
early in the race. In addition, our observation that endurance peaks at a given training load (in TRIMP), see
Fig. 5(c), should help preventing over-training, i.e., unproductive increase in training that can cause injury and
other health problems.
6
C Given the size and, I expect, richness of the dataset, I imagine that there is much more that the investigators
could have learned. I list a few questions in the following paragraphs that the authors might have explored.
Do aspects of training help to explain errors in the model predictions? This could have helped us move toward
a more causal link between training and race performance. There also seemed to be a systematic error in their
model related to the endurance parameter. Is this related to training? Or errors in their data? Or a gap in their
model? This question should have been explored more fully.
A We feel that we should ﬁrst clarify the content of our data set since it is less rich as your comments suggest.
Our model contains only the date, total distance and average velocity for all runs of the subjects. (See below
for a comment on heart rate.) While more data are recorded by GPS watches, these time series with one second
resolution could not be provided by our industrial partner for millions of kilometers of running. However, we
agree that separate work on a smaller number of subjects with higher data detail would be very interesting and
should be performed in the future.
Regarding observed deviations between actual race times and model predictions, we ﬁrst note that our model
has been shown to outperform all existing models when applied to personal records from 800m to the marathon
of elite runners (please see appendix to this rebuttal). The systematic error in the predicted marathon times in
Fig.4(a) at rather small and large endurance appear to be a consequence of the diﬃculty to measure endurance
from a few races at shorter distances when these races are not performed under prefect conditions or optimal
motivation of the athlete.
As Fig.
4(b) shows, this problem is most pronounced for slower runners.
Fast
runners demonstrated the smallest error between prediction and actual race time. This is consistent with the
observation that fast runners display also highest consistency in performance over all race distances (due to
higher experience and more racing attempts on a given distance), and hence their endurance parameter shows
less uncertainty. This is particularly the case for elite runners (see analysis in the appendix). As there were
associations between performance indicators and training background (Fig. 5), we can draw a similar conclusion:
Low relative training intensity and high training volume, typical for more experienced and faster runners, is
associated with smaller model error.
C Are there means to predict an athletes race performance from sub-maximal training performance (i.e., not races),
using heart rate or any other measures the watch might provide? The current model requires subjects to per-
formance two or more races at maximal e↵ort to extract these parameters. While this is an improvement on
physiological testing it is still a burden and does not seem to take advantage of the dataset. I presume heart rate
is available, for example. Would heart rate and heart-rate variability during training help to detect some of the
physiological parameters on training runs?
A Estimating race performance from sub-maximal training performance directly is impossible without additional
assumptions being made. An important quantity for endurance running performance is the decay of fractional
utilization of maximal aerobic power with duration which measures for how long an runner can maintain a
certain fraction of maximal aerobic power output. This quantity can be estimated from ’time to exhaustion’
experiments in the laboratory, i.e., by maximal tests. Without a precise knowledge of this quantity (measured
by El in our model), a ’typical’ value can only be assumed (depending on training status). Our dataset contains
for most athletes a number of races over 5km to the halfmarathon as these distances are used during training
as ’test races’. Hence, for marathon runners, this information on maximal e↵ort events is usually available and
provides a clear improvement over physiological testing in the laboratory where maximal e↵ort is impossible to
motivate for a distance of 20km or longer.
As far as heart rate is concerned, out data set does not contain heart rate data for all runs and athletes as not all
runners who wore a GPS watch wear a heart rate monitor (chest strap). But even if this data would be available,
there remains an important unknown: the maximal heart rate of the athlete which varies substantially among
7
individuals and cannot be determined accurately and easily from age-based formulas. Without the maximal
heart rate the important relative e↵ort (the quantity p in our model) cannot be determined accurately. Because of
this, our running model was built on the requirement that a priori no information about the runner’s physiology
is needed. Additional challenges are that heart rate is a↵ected by external factors, such as temperature, that
are often unknown. The same goes for heart rate variability as even less is understood about how di↵erent levels
of heart rate variability during exercise relate to e↵ort or athletic performance.
C Do the measures extracted from their model and the real-world dataset match measures extracted from gold
standard lab assessments in a small (but heterogeneous) subset of the subjects? While the researchers do compare
to previously published data, these tests would have provided more convincing evidence that their model is valid
in a population with varying age, gender, ethnicity, and training status.
A When it comes to endurance running tests conducted in the laboratory, there are several reasons why they
should not be considered as benchmark for running performance: (1) Maximal tests in laboratory are diﬃcult
to repeat, possibly due to lack of motivation to go all-out without opponent or competition or even price money
to win. As a result, the coeﬃcient of variation may be as high as 25% (Billat et. al., Med Sci Sports Exerc,
1994; Wigley et al., Int J Sports Med, 2007); (2) Running mechanics varies considerably between treadmill and
over ground running (Nigg et al., Med Sci Sports Exerc, 1994; Sinclair et al., Sport Biomech, 2012). One reason
may diﬃculty to simulate wind resistance; (3) Maximal laboratory tests are short-lasting and therefore fail to
account for reduction in running economy and subsequent increase in oxygen consumption at given speed that
occurs over long-distance running. We note also that all existing models for running performance have been
validated by comparison to athletic records, and not by laboratory testing. Demographic and other measures
that rely on user input are not reliable in big data sets from tracking platforms as ours as many users never
update default settings. Only location, which is given by GPS, is considered reliable.
C Does their phenomenological model perform better than past models on this large dataset? What about a sim-
ple linear regression model? Comparing the models against additional baselines would have provided further
conﬁdence.
A The most realistic test of models is their agreement with running world records and personal records of elite
athletes since those data are most consistent and obviously obtained under maximal e↵ort and controlled settings.
A variety of models have been proposed in the past. Only one of them, proposed by Peronnet & Thibault
[F. Peronnet, G. Thibault, Mathematical analysis of running performance and world running records, J Appl
Physiol. 67, 453 (1989)] employs a logarithmically decaying fractional utilization of maximal aerobic power,
based on empirical observations in athletic performances. Their model predicts world-records with an error of
less than 1% but the model is complicated by the fact that it requires many physiological parameters (body
weight, running economy, etc) that are unrealistically assumed to be the same for every athlete. While our
model is similar to the one by Peronnet & Thibault it is di↵erent in two essential points: (1) The logarithmic
decay of fractional utilization of maximal aerobic power emerges in our model from an exact solution of a
self-consistency equation and (2) our model is universal in the sense that it depends only on relative (rescaled)
quantities and hence can be applied to all athletes without knowing details like, e.g., body weight and size, and
running economy. Our model predicts world records from 800m to the marathon with an error slightly less than
the one observed for the model of Peronnet & Thibault. To provide a constructive basis for further review,
and to clear up misunderstandings indicated by the other reviewers, but also to defend our mathematical model
used for analyzing the data, we provide below a new, rather detailed comparison of the Peronnet & Thibault
model and some other models (mentioned by the other reviewers) and our model, including new graphs and
tables. The attached tables and graphs also show that a linear regression would not work since the race velocities
change on a logarithmic time scale, with a marked crossover at about 2000m race distance. While we think
that our new comparison can help the evaluation of our present work, it would not improve the manuscript
8
but would only make it more exhaustive to read. We note that details of our model and its validation against
world records have been published earlier [M. Mulligan, G. Adam, T. Emig, A minimal power model for human
running performance, PLoS ONE 13(11): e0206645 (2018)].
C Another major contribution could be to share the dataset with other researchers, which would be highly novel
and a means to accelerate research on human performance, injury, and real-world training. I would not expect
the researchers to tackle all of these problems, but I would expect more novel insights or contributions in some
form.
A Following general policy, our data set shall be made available to other researchers upon request once our work
has been published.
C Another issue with the current submission is that I found the manuscript more challenging to read and understand
than needed. Work is needed to improve the readability and clarity of the writing. As one example, the abstract
as written contains very few speciﬁc details about the study that was performed. What parameters were predicted?
What is performance? What are training modes? Given space, the abstract should not be exhaustive, but the
key details should be described with enough speciﬁcity to give the reader a more clear understanding of what
the study entailed. The Introduction, Results, and Discussion need similar improvements to more clearly and
succinctly state what analysis was performed.
A We have rewritten some parts of the manuscript to improve clarity of the description of performed analysis.
Speciﬁcally, the abstract contains now more details about our study.
9
Rebuttal letter
“Novel insights on human exercise performance from big data mining”
(NCOMMS-20-02292)
Please ﬁnd below our point-to-point answers to the reviewer comments (C: comment, A: answer).
Answer to Reviewer #3
We thank the reviewer for her/his time spent looking over our manuscript and their comments that we address
point-by-point in the following.
C In the Introduction, the authors challenge an axiom that has been characterizing exercise physiology since longer
than a century, the axiom that measurement conditions should be standardized.
I kindly disagree with this
view. Existing models, validated experimentally in the laboratory, and applicable on the ﬁeld and on large-scale
numbers, come from standard experimental laboratory conditions. The theoretical models do exist indeed. They
have been developed theoretically and validated by measuring VO2max during exercise testing and the fraction
of VO2max utilization and the energy cost of the locomotion mode at stake at steady state. The basic formula
id as follows:
V = F ⇤ ˙V O2max/C
(.1)
where v is velocity, F the sustainable fraction of VO2max over a given distance, and C the energy cost of the
locomotion mode at stake. C has the dimension of a force and represents the metabolic energy that is to be spent
to generate the force that is necessary to overcome the forces that oppose to the body movement. These are two
forces: 1) air resistance (or water resistance in swimming) and 2) frictional forces. These can be expressed as
follows:
C = kv2 + Cf
(.2)
This equation means that, if you plot C as a function of the square of speed, you obtain linear relationships
with slope equal to constant k and y-intercept equal to the energy cost that is necessary to overcome frictional
forces (high in running, low in cycling). Constant k is directly proportional to the frontal surface area of the
moving body, to the aerodynamic (or hydrodynamic factor) Cx, and to the air (or water) density. With these
simple relationships, it is possible to simulate and interpret all what happens in ﬁeld conditions and during actual
competitions. I just give the authors a few references (clearly overlooked by the authors) to make them aware
of what I am saying: di Prampero PE, Int J Sports Med 7: 55-72, 1986; di Prampero PE, Eur J Appl Physiol
82: 345-360, 2000; di Prampero et al, J Appl Physiol 47: 201-206, 1979; di Prampero et al, Eur J Appl Physiol
55: 259-266, 1986; Ferretti et al, Eur J Appl Physiol 111: 391-401, 2011; Margaria et al, J Appl Physiol 18:
367-370, 1963; Minetti et al, J Appl Physiol 93: 1039-1046, 2002; Pendergast et al, J Appl Physiol 43: 475-479,
1977; Tam et al, Eur J Appl Physiol 112: 3797-3806, 2012; Zamparo et al, Eur J Appl Physiol 111: 367-378,
2011. All the elements that are necessary to set a model to be applied to large data sets and in ﬁeld conditions
are present in those and many other studies.
A We thank the Reviewer for discussing the details of the model developed by P. .E. di Prampero et al. These
remarks suggest that there has been a misunderstanding which could be due to our very brief discussion of our
model and in particular its relation to other models. Let us hence clarify this point, by using your notation
for the model. Our model is exactly equivalent to above equations (.1) and (.2) with a particular form for the
sustainable fraction F and a constant, velocity independent C which has been used previously by others [see,
e.g., S. Lazzer et al., Eur J Appl Physiol, 112, 1709 (2012)] and is justiﬁed for the running velocities in our data
10
set (negligible air resistance). While the reviewer does not provide an explicit expression for F, this model has
been applied to half and full marathon races, using for the sustainable fraction
F(T) = f0 − f1T ,
(.3)
i.e., a linearly decreasing function of the duration T of the race, with constants f0, f1[P. E. di Prampero et al.,
Eur J Appl Physiol 55, 259 (1986)]. A related model has been developed by Peronnet & Thibault [F. Peronnet,
G. Thibault, J Appl Physiol, 67, 453 (1989)], using a logarithmic function for the sustainable fraction,
F(T) = 1 +
E
MAP log(T/TMAP) ,
(.4)
with maximal aerobic power MAP, a negative constant E and TMAP = 7min. Peronnet & Thibault motivated this
choice by empirical arguments based on world record performances up to the Marathon distance. Interestingly,
we have shown in our paper in 2018 [M. Mulligan, G. Adam, T. Emig, PLoS ONE 13(11): e0206645 (2018)]
that the form of Eq. (.4) can be derived mathematically from a self-consistent integral equation. In the notation
of our present manuscript, the sustainable fraction is given by
F(T) = Pmax(T)
Pm
= 1 − Pl
Pm
log T
tc
(T > tc) ,
(.5)
with Pm =MAP and tc = 6min, see Eq. (3) of our manuscript. Substituting this equation in your Eq. (.1)
yields exactly our model. This and additional details of the relation between the model you described above,
the so-called ’critical power’ model proposed by another Reviewer, and the model by Peronnet & Thibault
are summarized in the attached table. We have also performed a new, extensive comparison of these models
to current running world records and personal records of some elite marathon runners, including the model
proposed in above Eqs. (.1) and (.2) with F given by Eq. (.3). All results are attached below. They show
that our model has overall the smallest average error for the considered athletics records. We note that the
’critical power’ model and above model with F given by Eq. (.3) show substantial discrepancies with running
records for distances longer than ⇠ 10km (see attached plots). Hence, we believe that (1) a logarithmic decay
of F is essential and (2) our model is a very reasonable approach to analyze the race distances of 5km, 10km,
Halfmarathon and Marathon in our data set.
C What is the use of a variable like running economy, which the authors deﬁne as the steady state oxygen con-
sumption at a given constant speed instead of C? Physically speaking, C looks more appropriate.
A We agree that the energy cost of running, C, is the appropriate quantity. In fact, as pointed out in the previous
item, our model does employ this concept. The exact relation between C in your equation and our model is
v ⇤ C = Pb + Pm − Pb
vm
v
(.6)
where Pm =MAP and Pb is the resting (basal) metabolic rate (power). This relation means that we measure
the energy cost of running in our model by a parameter vm which is a velocity that is close to the running
speed that can be maintained for about 6min, equivalent to the time scale TMAP in the model of Peronnet &
Thibault. Please note that this implies the relation Cf = (Pm − Pb)/vm and vm ⇤ C = MAP.
C At the end of page 2, the authors create artiﬁcially a dichotomy between laboratory tests and actual competition
conditions: nevertheless, the theoretical background derived from laboratory tests is fully applicable to laboratory
conditions.
A It is not our intention to suggest a general discrepancy between laboratory testing and actual race performance.
Our explanations on the items above show that we indeed use the theoretical background that you suggest. The
11
crucial di↵erence between previous approaches and ours is based on our result for the duration dependence of the
sustainable fraction F. And hence the point we want to rise here is the often relative short duration of laboratory
testing. Incremental running test is the most common laboratory test that is conducted to determine aerobic and
anaerobic thresholds as well as maximal aerobic speed and maximal heart rate. However, incremental running
test is short-lasting and cannot account for the e↵ect of exercise duration on thresholds or general e↵ects of
fatigue. The maximal fractional utilization F(T) can be investigated in time-to-exhaustion test such as running
at certain fraction of VO2max, but the obtained results may have low test-retest repeatability as indicated by
a 25% coeﬃcient of variation [Billat et. al., Med Sci Sports Exerc, 1994; Wigley et al., Int J Sports Med, 2007].
Furthermore, running mechanics between treadmill and over ground running are di↵erent [Nigg et al., Med Sci
Sports Exerc, 1994; Sinclair et al., Sport Biomech, 2012]. In conclusion, laboratory tests are most suitable
for observing changes in running performance over relative short durations, but test results may not always
accurately predict actual race performance due to a lack of knowledge of the function F(T) and also due to
di↵erences in running mechanics that occur between treadmill and outdoor ground. Also, an important aspect
when comparing laboratory testing and actual races in which world records are set is the degree of motivation
of the athlete. This latter point seems particularly relevant to long lasting time-to-exhaustion tests performed
to determine F.
C A sentence like Unfortunately, these approaches predict that speeds below a critical velocity can be maintained
for inﬁnite duration which contradicts observation has been criticized (see e.g. Ferretti, Energetics of Muscular
Exercise, Springer 2015, chapter on critical power). Laboratory physiologists are perfectly aware of the fact that
energy sources in the body are ﬁnite.
A With this statement on the ’critical power’ model we wanted to point out the importance of using a fractional
utilization F(T) < 1 of maximal aerobic power when describing long lasting events like the marathon.
C Under Results, I read: Main determinants of aerobic ﬁtness and endurance in long distance runners (LDR) are
maximal oxygen uptake per body weight, VO2max , velocity dependent, sub-maximal oxygen demand, known as
running economy (RE), and the lactate or ventilatory threshold (LT) that sets the limit below which a steady
state blood lactate concentration is maintained. Although a reference is given, this statement is not correct, see
equation 1 in this report.
A As explained before, the di↵erence between equation (1) in your report and our model consists in the function
used to describe the fractional utilization F(T). The ”lactate or ventilatory threshold (LT)” is deﬁned in our
article from the duration dependence of F(T) as the fractional utilization of MAP that the runner can maintain
for one hour. We have changed the name and description of this threshold in our article accordingly to avoid
confusion with other concepts such as LT. As also explained before, the energy cost of running is measured in
our model by the velocity vm which is directly related to C in equation (1) in this report.
C Equation 1 of the article is constructed in such a way to encompass a large variety of ﬁeld conditions, independent
of the physiological model. Frankly, I do not see a role for such a study, unless we create an artiﬁcial opposition
between classical physiological studies and a kind of modern approach that the authors claim. Big numbers are
fascinating, but their utility depend of the context into which they are inserted and interpreted. The context
exists, but the authors seem to be unaware of it. The discussion id biased by the chosen approach and does not
deserve to be discussed analytically.
A Equation 1 of our article is in fact an exact mathematical solution of the eq. (1) given in this report, with the
fractional utilization of MAP given by F(T) = 1 − log(T/tc) for a race of duration T > tc with the time scale
tc = 6min in this article. This form for F(T) was derived in our earlier work [M. Mulligan, G. Adam, T. Emig,
PLoS ONE 13(11): e0206645 (2018)]. Hence, our chosen approach ﬁts fully into the existing context after the
importance of F(T) is understood.
12
C Endurance running dates back much longer than the ancient Olympic Games: it is alike that pre-historical
nomadic societies used endurance running while hunting or for migrating.
A Thank you for this interesting remark. We have modiﬁed the beginning of the introduction to give a more
general presentation.
C Page 2, line 7 : ranging instead of raging
A Thank you. We corrected this spelling error.
13
Appendix to Rebuttal Letter: New results from a comparison of existing mathematical models
A. V. Hill: The physiological basis of athletic records (1925)
In his seminal work, Hill posed the question ”how long a given e↵ort can be maintained”. To answer this question
he analyzed running records. In Figure 4 of his original article (reproduced in Fig. 1 below) he plotted the average
running speed over the a logarithmic time scale. It can be seen that for running (an other sports) the velocity decays
linearly with the logarithm of time, following two branches with di↵erent slopes. The analysis of Peronnet & Thibault
[F. Peronnet, G. Thibault, Mathematical analysis of running performance and world running records, J Appl Physiol.
67, 453 (1989)] and our mathematical model, applied to current world records, conﬁrm Hill’s observation with high
accuracy, as shown in the next section. This logarithmic decay is not reproduced by the models proposed by Reviewers
#1 and #3.
FIG. 1 Original ﬁgure from A. V. Hill, The physiological basis of athletic records, The Lancet, September 5, 1925, showing
average speed for running and other sports over a logarithmic time scale.
14
Comparison of mathematical models
The mathematical models for running performance mentioned by the reviewers (reviewer #1: critical power model,
reviewer #3: di Prampero’s approach) and the model by Peronnet & Thibault are summarized and compared in
Table I. The last column of this table provides the relation of those models to our model.
In order to assess and compare the accuracy of these models and our model we have performed detailed anal-
yses of men running world records (1987 as in Peronnet & Thibault, and current as of April 2020) and personal
records of six elite marathon runners (Antonio Pinto, Eliud Kipchoge, Felix Limo, Haile Gebrselassie, Mo Farah,
Steve Jones; choice of athletes taken from A. M. Jones, A.Vanhatalo, Sports Med 47, S65 (2017); Data from
https://www.worldathletics.org/athletes). For all models the unknown parameters were determined by minimizing
the mean squared relative error between the theoretically predicted time and the actual race time, i.e., the expression
Err = 1
N
N
X
j=1
✓Ttheory(dj) − Trace(dj)
Trace(dj)
◆2
(.7)
was minimized where the sum extends over N race distances dj. A numerical algorithm based on di↵erential evolution
was used for this purpose. The following models were analyzed:
MIT: Our model, here called the ’MIT model’
[M. Mulligan, G. Adam, T. Emig, PLoS ONE 13(11): e0206645 (2018)]
CP: The ’critical power’ model
[see e.g. M. Jones, A.Vanhatalo, Sports Med 47, S65 (2017)]
PT: The model of Peronnet & Thibault
[F. Peronnet, G. Thibault, J Appl Physiol. 67, 453 (1989)]
diP: The model of di Prampero (with F(T) given by Eq. (.3) with f0 = 1)
[see e.g. P. E. di Prampero et al., Eur J Appl Phys 55, 259 1986]
Following the analysis in A. M. Jones, A.Vanhatalo, Sports Med 47, S65 (2017), for the CP model the race distances
were restricted to dj < 15.000m for the determination of the model parameters.
The analyzed race distances and times are listed along with the obtained model parameters in the attached tables,
see Figs. 2 and 4. Shown are also the errors of the model predictions for each race distance and the average error
(av.error) for each model. The attached plots in Figs. 3 and 5 show the race results (open circles) and the four
model predictions for the average race velocity ¯v(d) as function of the race distance d as solid curves. The velocity is
measured in units of vm and the distance in units of dc = vmtc which corresponds to a simple linear rescaling of time
and distance.
We decided to plot average velocity (in units of the velocity vm at maximal aerobic power, MAP) since this shows
clearly the relative slow decay of velocity with racing distance. For example, the world records show that a marathon
is raced just ⇠ 18% below the velocity at MAP. This means that a mathematical model needs to achieve a rather
high precision in predicting the mean velocity in order to properly distinguish between endurance running distances.
Summary of results from analyzing running records
Our ﬁndings are as follows:
1. For all analyzed data sets, the average error between the model prediction and the actual race times is smallest for
the MIT model, followed by the PT model. It should be noted that both models describe the fractional utilization
15
of maximal aerobic power by a logarithmic function. The typical error of both models for the marathon is well
below 1%, and with the average error of our MIT model being less than half of the error of the PT model for
world records.
2. The CP model shows a systematic discrepancy for distances over 10km and below 1500m. The predicted average
velocity tends to a constant (”critical velocity”) with increasing distance, indicated by a dashed line in the plots.
The typical error for the marathon varies around 8%, both for world and personal records.
3. The diP model also shows a systematic error in the range of long race distances. The curve for the mean velocity
shows a non-monotonous curvature that bends towards too small velocities for larger distances, leading to a
typical error of a few percent for the half and full marathon.
4. Interestingly, for most data sets (in particular the world records), three model predictions converge (intersect)
on one particular point that is deﬁned in the MIT model by the velocity vm and the distance dc = vmtc,
corresponding approximately to the time scale TMAP ⇠ tc in the PT model over which the velocity vm ⇠ vMAP
can be maintained. This observation has important consequences: It shows that all four models tend to agree
with increasing accuracy when the velocity vm vMAP is approached. This implies that ’critical power’ or ’critical
velocity’ can be obtained from the MIT model. This is indeed the case, and the relation is summarized in the
attached Tab. I.
5. The data from world records are described by all models in general better than personal records of individual
athletes since world records are a result of optimized preparation and talent of an athlete for a given distance.
However, even on the level of individual athletes, the MIT model outperforms the other models, as shown by
the modeling of elite marathon runners (see Tab. 4 and Fig. 5).
We conclude that the models based on a constant or polynomial function F(T) for the fractional utilization of MAP
give an approximate description of running records that is valid only for distances below the 5km or the 10km race
or durations below 15 to 30min. This time scale is consistent with the 15min duration already observed by A. V. Hill
in his 1925 paper for the ending of a rapid decrease of race velocity and the beginning of a slower, logarithmic fall.
Indeed, for larger distances, a logarithmic function F(T), as used in the PT model and the our model, is essential for
a consistent description of real world running records.
16
TABLE I Summary of performance models.
model, reference
main variables and equations
relation to our model (’MIT model’)
Critical Power (CP)
Monod & Scherrer (1965)
The model is expected to describe races from 800m up to 5km or perhaps 10km. Power P(v) and velocity
v(T) sustainable over time T: [A. M. Jones, A. Vanhatalo, Sports Med 47, S65 (2017)]
P(T) = Pc + W 0
T ,
or
v(T) = vc + D0
T
with critical power Pc and critical speed (CS) vc, anaerobic capacity W 0 (in W/kg) or distance D0 (in m).
Fractional utilization is ﬁxed at unity: Power P < Pc or velocity v < vc can be maintained for “inﬁnite”
time but in praxis limited by substrate.
CP model close to our model around duration tc
with relation
vc ⇡ vm − D0
tc
and
D0 ⇡
Ps
Pm + Ps vmtc
No description of fractional utilization of MAP,
corresponding to Pl = 0 in our model.
di Prampero (diP)
di Prampero (1986)
Maximal velocity v(T) sustainable for time T [di Prampero et al., J Appl Physiol 74,2318 (1993)]:
v(T) =
F(T)
C(v(T))
˙Emax,
˙Emax = A
T + MAP − MAP ⌧
T (1 − e−T/⌧)
with work A from anaerobic sources, maximal aerobic power MAP, ⌧ = 10s, and the energy cost of
running [per distance and body weight in J/(m kg)] given by
C(v) = Cf + kv2 + 2v3/d
(v in m/s, d in m)
with k = 0.0103, Cf = 3.79. Fractional utilization F(T) of MAP over duration T is approximated by
F(T) = f0 − f1T
where f0 ⇡ 1, f1 ⇡ 0 for T < 20min, and f0 ⇡ 0.94, f1 ⇡ 10−3 for durations from a half to a full marathon
with T in min. [di Prampero et al., Eur J Appl Phys 55, 259 1986].
MAP b= Pm
Fractional utilization of MAP:
F(T) = 1 − Pl
Pm log T
tc
(T > tc)
Power output required to run at velocity v:
C(v)v = Pb + Pm − Pb
vm
v,
so that Cf = (Pm − Pb)/vm and k = 0. This
means C(vm)vm = Pm implying that vm is speed
at MAP.
Peronnet & Thibault (PT)
Peronnet & Thibault (1989)
Power output P(T) sustainable over time T and power Pv(v) required to at velocity v:
P(T) =
8
>
>
>
<
>
>
>
:
c2(T) A
T + MAP − c1(T)(MAP − BMR)
(T < TMAP)
c2(T) A
T
⇣
1 + f log
T
TMAP
⌘
+ c1(T)BMR + (1 − c1(T))
✓
MAP + E log
T
TMAP
◆
(T > TMAP)
Pv(v) = BMR + 3.86v + C0v3
(v = d/T in m/s)
" E < 0: Fractional utilization of MAP
with
c2(T) = 1 − e−T/k2,
c1(T) = k1
T
⇣
1 − e−T/k1⌘
.
with maximal aerobic power MAP (in W/kg), anaerobic capacity A in J/kg, TMAP = 7min the maximal
race duration for which the peak aerobic power is MAP, rate of peak decline E in W/kg, k1 = 30s,
k2 = 20s, f = −0.233, C0 = 0.0103 + 2/d with distance d in m and basal metabolic rate BMR=1.2W/kg
MAP b= Pm,
BMR b= Pb,
TMAP = tc
A, f, C0 = 0
Our model does not include kinetics of aerobic
and anaerobic metabolism at the beginning of
exercise (< 30s) so that c1(T) = 0, c2(T) = 1.
Fractional utilization of MAP is the same as in
our model, i.e., logarithmic decrease with
E = −Pl
and for T < TMAP, A/T is replaced by
−Ps log T
tc with A ⇡ Pstc.
17
ID
distance
time
MIT
error[%]
CP
error[%]
PT
error[%]
diP
error[%]
MIT parameters
CP parameters
PT parameters
diP parameters
WR1987men
800
01:41.73
01:42.12
+0.39
01:32.50
-9.07
01:41.67
-0.06
01:40.19
-1.51
1000
02:12.18
02:11.57
-0.46
02:05.83
-4.80
02:12.60
+0.31
02:11.69
-0.37
1500
03:29.46
03:29.09
-0.18
03:29.16
-0.14
03:30.23
+0.37
03:31.63
+1.03
1609
03:46.32
03:46.62
+0.13
03:47.33
+0.45
03:47.18
+0.38
03:49.18
+1.26
2000
04:50.81
04:51.15
+0.12
04:52.50
+0.58
04:48.07
-0.94
04:52.34
+0.53
3000
07:32.10
07:32.41
+0.07
07:39.16
+1.56
07:25.84
-1.38
07:34.92
+0.62
5000
12:58.39
12:59.38
+0.13
13:12.49
+1.81
13:04.46
+0.78
13:03.24
+0.62
10000
27:13.81
27:13.51
-0.02
27:05.82
-0.49
27:32.87
+1.17
27:00.38
-0.82
21100
1:00:55.00
1:00:35.14
-0.54
57:55.81
-4.90
1:00:49.44
-0.15
59:29.22
-2.35
42195
2:07:12.00
2:07:39.60
+0.36
1:56:31.62
-8.39
2:06:33.66
-0.50
2:08:44.79
+1.22
tc
05:28.54
vm
6.76 m/s
Es=T110 %MAP/tc
0.480
El=T90 %MAP/tc
5.493
Anaerobic & aerobic metabolism
A=Pstc+(Pm-Pb)25s
1820.0 J/kg
D'=vmtcPs/(Ps+Pm)
255.7 m
CS=vm-D'/tc
5.98 m/s
VO2max
78.4 ml/(kg min)
av.error
0.24 %
D'
245.0 m
CS
6.00 m/s
av.error
3.22 %
vMAP
6.36 m/s
E/MAP
-5.00 %
A
1742.00 J/kg
VO2max
82.1 ml/(kg min)
av.error
0.60 %
vMAP
5.96 m/s
f1
1.85 ×10-5/s
A
1348.00 J/kg
VO2max
74.6 ml/(kg min)
av.error
1.03 %
WR2020men
800
01:40.91
01:41.71
+0.79
01:33.77
-7.07
01:41.32
+0.41
01:40.20
-0.70
1000
02:11.96
02:10.58
-1.04
02:05.89
-4.60
02:11.49
-0.36
02:10.77
-0.90
1500
03:26.00
03:26.09
+0.04
03:26.18
+0.09
03:27.05
+0.51
03:28.12
+1.03
1609
03:43.13
03:43.08
-0.02
03:43.68
+0.25
03:43.53
+0.18
03:45.08
+0.87
2000
04:44.79
04:45.40
+0.21
04:46.47
+0.59
04:42.74
-0.72
04:46.09
+0.46
3000
07:20.67
07:20.91
+0.05
07:27.05
+1.45
07:15.31
-1.22
07:22.99
+0.53
5000
12:37.35
12:36.66
-0.09
12:48.20
+1.43
12:41.29
+0.52
12:39.51
+0.29
10000
26:17.53
26:16.99
-0.03
26:11.10
-0.41
26:33.79
+1.03
26:05.02
-0.79
21100
58:01.00
58:05.83
+0.14
55:53.52
-3.66
58:17.54
+0.48
57:11.12
-1.43
42195
2:01:39.00
2:01:34.01
-0.07
1:52:20.93
-7.65
2:00:36.40
-0.86
2:02:35.30
+0.77
tc
05:14.60
vm
6.93 m/s
Es=T110 %MAP/tc
0.435
El=T90 %MAP/tc
6.732
Anaerobic & aerobic metabolism
A=Pstc+(Pm-Pb)25s
1678.0 J/kg
D'=vmtcPs/(Ps+Pm)
224.4 m
CS=vm-D'/tc
6.21 m/s
VO2max
80.3 ml/(kg min)
av.error
0.25 %
D'
216.0 m
CS
6.23 m/s
av.error
2.72 %
vMAP
6.54 m/s
E/MAP
-4.48 %
A
1686.00 J/kg
VO2max
84.8 ml/(kg min)
av.error
0.63 %
vMAP
6.19 m/s
f1
1.77 ×10-5/s
A
1247.00 J/kg
VO2max
77.8 ml/(kg min)
av.error
0.78 %
FIG. 2 Application of four mathematical models to men running world records from 1987 and 2020: Predicted race times and model parameters (see Tab.I for models).
18
CP (fit for d≤15km): av.error=3.22%
D'=245.0m, CS=6.00m/s
PT: av.error=0.60%
MAP=28.58W/kg, A=1742.00J/kg
E=-5.00%
diP: av.error=1.03%
MAP=25.99W/kg, A=1348.00J/kg
f=1.85×10-5/s
MIT: av.error=0.24%
Es=0.480, El=5.493
vm=6.76m/s, dc=2220.0m
tc= 05:28.54



















/
()/
  	


CP (fit for d≤15km): av.error=2.72%
D'=216.0m, CS=6.23m/s
PT: av.error=0.63%
MAP=29.54W/kg, A=1686.00J/kg
E=-4.48%
diP: av.error=0.78%
MAP=27.10W/kg, A=1247.00J/kg
f=1.77×10-5/s
MIT: av.error=0.25%
Es=0.435, El=6.732
vm=6.93m/s, dc=2179.0m
tc= 05:14.60



















/
()/
  

distance
time
CP model
error [%]
PT model
error [%]
diP model
error [%]
MIT model
error [%]
800
01:41.73
01:32.50
-9.07
01:41.67
-0.06
01:40.19
-1.51
01:42.12
+0.39
1000
02:12.18
02:05.83
-4.80
02:12.60
+0.31
02:11.69
-0.37
02:11.57
-0.46
1500
03:29.46
03:29.16
-0.14
03:30.23
+0.37
03:31.63
+1.03
03:29.09
-0.18
1609
03:46.32
03:47.33
+0.45
03:47.18
+0.38
03:49.18
+1.26
03:46.62
+0.13
2000
04:50.81
04:52.50
+0.58
04:48.07
-0.94
04:52.34
+0.53
04:51.15
+0.12
3000
07:32.10
07:39.16
+1.56
07:25.84
-1.38
07:34.92
+0.62
07:32.41
+0.07
5000
12:58.39
13:12.49
+1.81
13:04.46
+0.78
13:03.24
+0.62
12:59.38
+0.13
10000
27:13.81
27:05.82
-0.49
27:32.87
+1.17
27:00.38
-0.82
27:13.51
-0.02
21100
1:00:55.00
57:55.81
-4.90
1:00:49.44
-0.15
59:29.22
-2.35
1:00:35.14
-0.54
42195
2:07:12.00
1:56:31.62
-8.39
2:06:33.66
-0.50
2:08:44.79
+1.22
2:07:39.60
+0.36
distance
time
CP model
error [%]
PT model
error [%]
diP model
error [%]
MIT model
error [%]
800
01:40.91
01:33.77
-7.07
01:41.32
+0.41
01:40.20
-0.70
01:41.71
+0.79
1000
02:11.96
02:05.89
-4.60
02:11.49
-0.36
02:10.77
-0.90
02:10.58
-1.04
1500
03:26.00
03:26.18
+0.09
03:27.05
+0.51
03:28.12
+1.03
03:26.09
+0.04
1609
03:43.13
03:43.68
+0.25
03:43.53
+0.18
03:45.08
+0.87
03:43.08
-0.02
2000
04:44.79
04:46.47
+0.59
04:42.74
-0.72
04:46.09
+0.46
04:45.40
+0.21
3000
07:20.67
07:27.05
+1.45
07:15.31
-1.22
07:22.99
+0.53
07:20.91
+0.05
5000
12:37.35
12:48.20
+1.43
12:41.29
+0.52
12:39.51
+0.29
12:36.66
-0.09
10000
26:17.53
26:11.10
-0.41
26:33.79
+1.03
26:05.02
-0.79
26:16.99
-0.03
21100
58:01.00
55:53.52
-3.66
58:17.54
+0.48
57:11.12
-1.43
58:05.83
+0.14
42195
2:01:39.00
1:52:20.93
-7.65
2:00:36.40
-0.86
2:02:35.30
+0.77
2:01:34.01
-0.07
FIG. 3 Application of four mathematical models to men running world records from 1987 and 2020: Log-normal plot of the ’running curves’ predicted by the models (average velocity ¯v as
function of race distance d, in units of vm and dc = vmtc given in the plot legend) and actual race data (red dots). The tables summarize the actual and predicted race times, along with
the relative errors in percent.
19
ID
distance
time
MIT
error[%]
CP
error[%]
PT
error[%]
diP
error[%]
MIT parameters
CP parameters
PT parameters
diP parameters
Antonio
1500
03:39.25
03:39.25
+0.00
03:31.80
-3.40
03:40.26
+0.46
03:37.58
-0.76
3000
07:41.33
07:38.67
-0.58
07:41.66
+0.07
07:34.21
-1.54
07:43.99
+0.58
5000
13:02.86
13:08.15
+0.68
13:14.80
+1.53
13:09.54
+0.85
13:16.00
+1.68
10000
27:12.47
27:25.68
+0.81
27:07.65
-0.30
27:33.80
+1.31
27:20.58
+0.50
21097
1:01:45.00
1:00:45.21
-1.61
57:56.09
-6.18
1:00:51.38
-1.45
59:54.66
-2.98
42195
2:06:36.00
2:07:26.14
+0.66
1:56:30.40
-7.97
2:07:01.20
+0.33
2:08:12.86
+1.28
tc
05:46.65
vm
6.64 m/s
Es=T110 %MAP/tc
0.217
El=T90 %MAP/tc
6.224
Anaerobic & aerobic metabolism
A=Pstc+(Pm-Pb)25s
1223.0 J/kg
D'=vmtcPs/(Ps+Pm)
135.4 m
CS=vm-D'/tc
6.25 m/s
VO2max
77.1 ml/(kg min)
av.error
0.72 %
D'
228.5 m
CS
6.00 m/s
av.error
3.24 %
vMAP
6.43 m/s
E/MAP
-5.47 %
A
1318.00 J/kg
VO2max
83.1 ml/(kg min)
av.error
0.99 %
vMAP
5.88 m/s
f1
1.63 ×10-5/s
A
1226.00 J/kg
VO2max
73.5 ml/(kg min)
av.error
1.29 %
Eliud
1500
03:33.20
03:33.10
-0.05
03:23.46
-4.57
03:33.94
+0.34
03:31.20
-0.94
3000
07:27.66
07:29.73
+0.46
07:30.51
+0.64
07:26.12
-0.34
07:32.61
+1.11
3218.68
08:07.39
08:05.23
-0.44
08:06.52
-0.18
08:01.92
-1.12
08:07.99
+0.12
5000
12:46.53
12:51.56
+0.66
12:59.91
+1.75
12:55.80
+1.21
12:57.53
+1.44
10000
26:49.02
26:42.72
-0.39
26:43.41
-0.35
26:57.17
+0.51
26:41.56
-0.46
21097
59:25.00
58:48.35
-1.03
57:11.09
-3.76
58:58.62
-0.74
58:12.28
-2.04
42195
2:01:39.00
2:02:34.67
+0.76
1:55:05.94
-5.39
2:01:48.73
+0.13
2:02:48.65
+0.95
tc
08:05.23
vm
6.63 m/s
Es=T110 %MAP/tc
0.260
El=T90 %MAP/tc
7.481
Anaerobic & aerobic metabolism
A=Pstc+(Pm-Pb)25s
1563.0 J/kg
D'=vmtcPs/(Ps+Pm)
213.2 m
CS=vm-D'/tc
6.19 m/s
VO2max
77.0 ml/(kg min)
av.error
0.54 %
D'
264.7 m
CS
6.07 m/s
av.error
2.37 %
vMAP
6.45 m/s
E/MAP
-4.30 %
A
1519.00 J/kg
VO2max
83.5 ml/(kg min)
av.error
0.63 %
vMAP
6.00 m/s
f1
1.36 ×10-5/s
A
1324.00 J/kg
VO2max
75.1 ml/(kg min)
av.error
1.01 %
Felix
1500
03:40.14
03:40.14
+0.00
03:30.81
-4.24
03:40.93
+0.36
03:37.48
-1.21
3000
07:40.67
07:40.67
-0.00
07:42.85
+0.47
07:36.52
-0.90
07:47.68
+1.52
5000
13:16.42
13:12.92
-0.44
13:18.90
+0.31
13:13.50
-0.37
13:24.72
+1.04
10000
27:04.54
27:34.04
+1.82
27:19.03
+0.89
27:40.64
+2.22
27:40.89
+2.24
15000
41:29.00
42:25.21
+2.26
41:19.17
-0.40
42:31.69
+2.52
42:16.77
+1.92
16093.4
46:41.00
45:43.06
-2.07
44:22.89
-4.93
45:48.98
-1.86
45:31.07
-2.50
20000
58:20.00
57:37.14
-1.22
55:19.30
-5.16
57:39.85
-1.15
57:13.80
-1.89
21097
1:02:05.00
1:00:59.48
-1.76
58:23.62
-5.94
1:01:01.00
-1.72
1:00:33.61
-2.45
42195
2:06:14.00
2:07:46.93
+1.23
1:57:28.64
-6.94
2:07:07.95
+0.71
2:08:43.21
+1.97
tc
09:59.56
vm
6.40 m/s
Es=T110 %MAP/tc
0.208
El=T90 %MAP/tc
6.118
Anaerobic & aerobic metabolism
A=Pstc+(Pm-Pb)25s
1563.0 J/kg
D'=vmtcPs/(Ps+Pm)
220.0 m
CS=vm-D'/tc
6.04 m/s
VO2max
74.5 ml/(kg min)
av.error
1.20 %
D'
245.4 m
CS
5.95 m/s
av.error
3.25 %
vMAP
6.39 m/s
E/MAP
-5.28 %
A
1336.00 J/kg
VO2max
82.5 ml/(kg min)
av.error
1.31 %
vMAP
5.78 m/s
f1
1.45 ×10-5/s
A
1332.00 J/kg
VO2max
72.0 ml/(kg min)
av.error
1.86 %
Haile
800
01:49.35
01:49.20
-0.14
01:37.69
-10.66
01:49.37
+0.02
01:47.02
-2.13
1500
03:31.76
03:33.07
+0.62
03:30.09
-0.79
03:34.56
+1.32
03:35.83
+1.92
2000
04:52.86
04:49.49
-1.15
04:50.38
-0.85
04:49.56
-1.13
04:54.19
+0.45
3000
07:25.09
07:25.09
-0.00
07:30.95
+1.32
07:21.03
-0.91
07:31.83
+1.51
3218.68
08:01.08
07:59.55
-0.32
08:06.07
+1.04
07:56.24
-1.01
08:06.44
+1.12
5000
12:39.36
12:45.27
+0.78
12:52.10
+1.68
12:46.64
+0.96
12:50.23
+1.43
10000
26:22.75
26:39.32
+1.05
26:14.96
-0.49
26:46.51
+1.50
26:24.07
+0.08
16093.4
44:24.00
44:15.99
-0.30
42:33.39
-4.15
44:23.01
-0.04
43:33.24
-1.91
20000
55:48.00
55:49.62
+0.05
53:00.68
-5.00
55:54.06
+0.18
54:57.24
-1.52
21097
58:55.00
59:06.27
+0.32
55:56.83
-5.04
59:09.71
+0.42
58:13.03
-1.19
25000
1:11:37.00
1:10:51.77
-1.05
1:06:23.55
-7.29
1:10:50.90
-1.07
1:10:03.89
-2.17
42195
2:03:59.00
2:04:06.73
+0.10
1:52:24.59
-9.33
2:03:35.91
-0.31
2:07:49.12
+3.09
tc
05:18.59
vm
6.87 m/s
Es=T110 %MAP/tc
0.201
El=T90 %MAP/tc
6.058
Anaerobic & aerobic metabolism
A=Pstc+(Pm-Pb)25s
1189.0 J/kg
D'=vmtcPs/(Ps+Pm)
123.0 m
CS=vm-D'/tc
6.48 m/s
VO2max
79.6 ml/(kg min)
av.error
0.49 %
D'
191.6 m
CS
6.23 m/s
av.error
3.97 %
vMAP
6.62 m/s
E/MAP
-5.62 %
A
1338.00 J/kg
VO2max
86.0 ml/(kg min)
av.error
0.74 %
vMAP
6.19 m/s
f1
2.21 ×10-5/s
A
985.80 J/kg
VO2max
77.7 ml/(kg min)
av.error
1.54 %
Mo
800
01:48.24
01:47.52
-0.67
01:35.74
-11.55
01:47.43
-0.75
01:46.05
-2.02
1500
03:28.81
03:31.18
+1.14
03:30.22
+0.67
03:33.14
+2.08
03:35.60
+3.25
3218.68
08:03.40
08:01.04
-0.49
08:11.29
+1.63
07:56.54
-1.42
08:08.16
+0.98
5000
12:53.11
12:51.26
-0.24
13:02.60
+1.23
12:49.49
-0.47
12:53.77
+0.09
10000
26:46.57
26:51.68
+0.32
26:40.29
-0.39
26:57.13
+0.66
26:31.30
-0.95
21097
59:32.00
59:33.22
+0.03
56:55.07
-4.39
59:39.84
+0.22
58:18.11
-2.07
42195
2:05:11.00
2:05:02.35
-0.12
1:54:25.40
-8.60
2:04:48.67
-0.30
2:06:34.96
+1.12
tc
10:14.01
vm
6.57 m/s
Es=T110 %MAP/tc
0.269
El=T90 %MAP/tc
5.700
Anaerobic & aerobic metabolism
A=Pstc+(Pm-Pb)25s
1820.0 J/kg
D'=vmtcPs/(Ps+Pm)
273.2 m
CS=vm-D'/tc
6.12 m/s
VO2max
76.3 ml/(kg min)
av.error
0.43 %
D'
214.6 m
CS
6.11 m/s
av.error
4.07 %
vMAP
6.58 m/s
E/MAP
-5.71 %
A
1430.00 J/kg
VO2max
85.3 ml/(kg min)
av.error
0.84 %
vMAP
6.13 m/s
f1
1.99 ×10-5/s
A
1048.00 J/kg
VO2max
76.9 ml/(kg min)
av.error
1.50 %
Steve
800
01:47.43
01:47.43
+0.00
01:39.51
-7.37
01:47.79
+0.33
01:47.18
-0.24
3000
07:49.80
07:48.88
-0.20
07:51.18
+0.29
07:43.98
-1.24
07:50.84
+0.22
3218.68
08:26.71
08:24.88
-0.36
08:28.13
+0.28
08:21.43
-1.04
08:27.34
+0.12
5000
13:18.60
13:22.44
+0.48
13:29.07
+1.31
13:28.36
+1.22
13:26.01
+0.93
10000
27:39.14
27:45.63
+0.39
27:33.79
-0.32
28:04.09
+1.50
27:37.21
-0.12
21097
1:01:14.00
1:01:03.67
-0.28
58:48.55
-3.96
1:01:15.79
+0.05
1:00:18.34
-1.52
42195
2:07:13.00
2:07:10.00
-0.04
1:58:12.92
-7.08
2:06:07.20
-0.86
2:08:03.76
+0.67
tc
07:35.31
vm
6.41 m/s
Es=T110 %MAP/tc
0.411
El=T90 %MAP/tc
7.838
Anaerobic & aerobic metabolism
A=Pstc+(Pm-Pb)25s
1883.0 J/kg
D'=vmtcPs/(Ps+Pm)
282.1 m
CS=vm-D'/tc
5.79 m/s
VO2max
74.5 ml/(kg min)
av.error
0.25 %
D'
211.0 m
CS
5.92 m/s
av.error
2.95 %
vMAP
6.16 m/s
E/MAP
-3.82 %
A
1584.00 J/kg
VO2max
79.2 ml/(kg min)
av.error
0.89 %
vMAP
5.81 m/s
f1
1.46 ×10-5/s
A
1154.00 J/kg
VO2max
72.5 ml/(kg min)
av.error
0.54 %
FIG. 4 Application of four mathematical models to personal records of six elite marathon runners (Antonio Pinto, Eliud Kipchoge, Felix Limo, Haile Gebrselassie, Mo Farah, Steve Jones):
Predicted race times and model parameters (see Tab.I for models).
20
CP (fit for d≤15km): av.error=3.24%
D'=228.5m, CS=6.00m/s
PT: av.error=0.99%
MAP=28.95W/kg, A=1318.00J/kg
E=-5.47%
diP: av.error=1.29%
MAP=25.59W/kg, A=1226.00J/kg
f=1.63×10-5/s
MIT: av.error=0.72%
Es=0.217, El=6.224
vm=6.64m/s, dc=2303.0m
tc= 05:46.65



















/
()/
  	
	
CP (fit for d≤15km): av.error=2.37%
D'=264.7m, CS=6.07m/s
PT: av.error=0.63%
MAP=29.08W/kg, A=1519.00J/kg
E=-4.30%
diP: av.error=1.01%
MAP=26.14W/kg, A=1324.00J/kg
f=1.36×10-5/s
MIT: av.error=0.54%
Es=0.260, El=7.481
vm=6.63m/s, dc=3219.0m
tc= 08:05.23



















/
()/
  







()/
distance
time
CP model
error [%]
PT model
error [%]
diP model
error [%]
MIT model
error [%]
1500
03:39.25
03:31.80
-3.40
03:40.26
+0.46
03:37.58
-0.76
03:39.25
+0.00
3000
07:41.33
07:41.66
+0.07
07:34.21
-1.54
07:43.99
+0.58
07:38.67
-0.58
5000
13:02.86
13:14.80
+1.53
13:09.54
+0.85
13:16.00
+1.68
13:08.15
+0.68
10000
27:12.47
27:07.65
-0.30
27:33.80
+1.31
27:20.58
+0.50
27:25.68
+0.81
21097
1:01:45.00
57:56.09
-6.18
1:00:51.38
-1.45
59:54.66
-2.98
1:00:45.21
-1.61
42195
2:06:36.00
1:56:30.40
-7.97
2:07:01.20
+0.33
2:08:12.86
+1.28
2:07:26.14
+0.66
distance
time
CP model
error [%]
PT model
error [%]
diP model
error [%]
MIT model
error [%]
1500
03:33.20
03:23.46
-4.57
03:33.94
+0.34
03:31.20
-0.94
03:33.10
-0.05
3000
07:27.66
07:30.51
+0.64
07:26.12
-0.34
07:32.61
+1.11
07:29.73
+0.46
3218.68
08:07.39
08:06.52
-0.18
08:01.92
-1.12
08:07.99
+0.12
08:05.23
-0.44
5000
12:46.53
12:59.91
+1.75
12:55.80
+1.21
12:57.53
+1.44
12:51.56
+0.66
10000
26:49.02
26:43.41
-0.35
26:57.17
+0.51
26:41.56
-0.46
26:42.72
-0.39
21097
59:25.00
57:11.09
-3.76
58:58.62
-0.74
58:12.28
-2.04
58:48.35
-1.03
42195
2:01:39.00
1:55:05.94
-5.39
2:01:48.73
+0.13
2:02:48.65
+0.95
2:02:34.67
+0.76
distance
1500
3000
5000
10000
15000
16093.4
20000
21097
42195
FIG. 5 (a) Application of four mathematical models to personal records of elite marathon runners (Antonio Pinto, Eliud Kipchoge): Log-normal plot of the ’running curves’ predicted by
the models (average velocity ¯v as function of race distance d, in units of vm and dc = vmtc given in the plot legend) and actual race data (red dots). The tables summarize the actual and
predicted race times, along with the relative errors in percent.
21
%
CP (fit for d≤15km): av.error=3.25%
D'=245.4m, CS=5.95m/s
PT: av.error=1.31%
MAP=28.73W/kg, A=1336.00J/kg
E=-5.28%
diP: av.error=1.86%
MAP=25.08W/kg, A=1332.00J/kg
f=1.45×10-5/s
MIT: av.error=1.20%
Es=0.208, El=6.118
vm=6.40m/s, dc=3840.0m
tc= 09:59.56



















/
()/
  

CP (fit for d≤15km): av.error=3.97%
D'=191.6m, CS=6.23m/s
PT: av.error=0.74%
MAP=29.94W/kg, A=1338.00J/kg
E=-5.62%
diP: av.error=1.54%
MAP=27.08W/kg, A=985.80J/kg
f=2.21×10-5/s
MIT: av.error=0.49%
Es=0.201, El=6.058
vm=6.87m/s, dc=2188.0m
tc= 05:18.59



















/
()/
  






()/
[%]
0.05
0.46
0.44
0.66
0.39
1.03
0.76
distance
time
CP model
error [%]
PT model
error [%]
diP model
error [%]
MIT model
error [%]
1500
03:40.14
03:30.81
-4.24
03:40.93
+0.36
03:37.48
-1.21
03:40.14
+0.00
3000
07:40.67
07:42.85
+0.47
07:36.52
-0.90
07:47.68
+1.52
07:40.67
-0.00
5000
13:16.42
13:18.90
+0.31
13:13.50
-0.37
13:24.72
+1.04
13:12.92
-0.44
10000
27:04.54
27:19.03
+0.89
27:40.64
+2.22
27:40.89
+2.24
27:34.04
+1.82
15000
41:29.00
41:19.17
-0.40
42:31.69
+2.52
42:16.77
+1.92
42:25.21
+2.26
16093.4
46:41.00
44:22.89
-4.93
45:48.98
-1.86
45:31.07
-2.50
45:43.06
-2.07
20000
58:20.00
55:19.30
-5.16
57:39.85
-1.15
57:13.80
-1.89
57:37.14
-1.22
21097
1:02:05.00
58:23.62
-5.94
1:01:01.00
-1.72
1:00:33.61
-2.45
1:00:59.48
-1.76
42195
2:06:14.00
1:57:28.64
-6.94
2:07:07.95
+0.71
2:08:43.21
+1.97
2:07:46.93
+1.23
distance
time
CP model
error [%]
PT model
error [%]
diP model
error [%]
MIT model
error [%]
800
01:49.35
01:37.69
-10.66
01:49.37
+0.02
01:47.02
-2.13
01:49.20
-0.14
1500
03:31.76
03:30.09
-0.79
03:34.56
+1.32
03:35.83
+1.92
03:33.07
+0.62
2000
04:52.86
04:50.38
-0.85
04:49.56
-1.13
04:54.19
+0.45
04:49.49
-1.15
3000
07:25.09
07:30.95
+1.32
07:21.03
-0.91
07:31.83
+1.51
07:25.09
-0.00
3218.68
08:01.08
08:06.07
+1.04
07:56.24
-1.01
08:06.44
+1.12
07:59.55
-0.32
5000
12:39.36
12:52.10
+1.68
12:46.64
+0.96
12:50.23
+1.43
12:45.27
+0.78
10000
26:22.75
26:14.96
-0.49
26:46.51
+1.50
26:24.07
+0.08
26:39.32
+1.05
16093.4
44:24.00
42:33.39
-4.15
44:23.01
-0.04
43:33.24
-1.91
44:15.99
-0.30
20000
55:48.00
53:00.68
-5.00
55:54.06
+0.18
54:57.24
-1.52
55:49.62
+0.05
21097
58:55.00
55:56.83
-5.04
59:09.71
+0.42
58:13.03
-1.19
59:06.27
+0.32
25000
1:11:37.00
1:06:23.55
-7.29
1:10:50.90
-1.07
1:10:03.89
-2.17
1:10:51.77
-1.05
42195
2:03:59.00
1:52:24.59
-9.33
2:03:35.91
-0.31
2:07:49.12
+3.09
2:04:06.73
+0.10
distance
800
1500
3218.68
5000
10000
21097
42195
FIG. 5 (b) Application of four mathematical models to personal records of elite marathon runners (Felix Limo, Haile Gebrselassie): Log-normal plot of the ’running curves’ predicted by
the models (average velocity ¯v as function of race distance d, in units of vm and dc = vmtc given in the plot legend) and actual race data (red dots). The tables summarize the actual and
predicted race times, along with the relative errors in percent.
22
%
CP (fit for d≤15km): av.error=4.07%
D'=214.6m, CS=6.11m/s
PT: av.error=0.84%
MAP=29.72W/kg, A=1430.00J/kg
E=-5.71%
diP: av.error=1.50%
MAP=26.80W/kg, A=1048.00J/kg
f=1.99×10-5/s
MIT: av.error=0.43%
Es=0.269, El=5.700
vm=6.57m/s, dc=4033.0m
tc= 10:14.01



















/
()/
  	
CP (fit for d≤15km): av.error=2.95%
D'=211.0m, CS=5.92m/s
PT: av.error=0.89%
MAP=27.59W/kg, A=1584.00J/kg
E=-3.82%
diP: av.error=0.54%
MAP=25.26W/kg, A=1154.00J/kg
f=1.46×10-5/s
MIT: av.error=0.25%
Es=0.411, El=7.838
vm=6.41m/s, dc=2917.0m
tc= 07:35.31



















/
()/
  
[%]
.14
.62
.15
.00
.32
.78
.05
.30
.05
.32
.05
.10
distance
time
CP model
error [%]
PT model
error [%]
diP model
error [%]
MIT model
error [%]
800
01:48.24
01:35.74
-11.55
01:47.43
-0.75
01:46.05
-2.02
01:47.52
-0.67
1500
03:28.81
03:30.22
+0.67
03:33.14
+2.08
03:35.60
+3.25
03:31.18
+1.14
3218.68
08:03.40
08:11.29
+1.63
07:56.54
-1.42
08:08.16
+0.98
08:01.04
-0.49
5000
12:53.11
13:02.60
+1.23
12:49.49
-0.47
12:53.77
+0.09
12:51.26
-0.24
10000
26:46.57
26:40.29
-0.39
26:57.13
+0.66
26:31.30
-0.95
26:51.68
+0.32
21097
59:32.00
56:55.07
-4.39
59:39.84
+0.22
58:18.11
-2.07
59:33.22
+0.03
42195
2:05:11.00
1:54:25.40
-8.60
2:04:48.67
-0.30
2:06:34.96
+1.12
2:05:02.35
-0.12
distance
time
CP model
error [%]
PT model
error [%]
diP model
error [%]
MIT model
error [%]
800
01:47.43
01:39.51
-7.37
01:47.79
+0.33
01:47.18
-0.24
01:47.43
+0.00
3000
07:49.80
07:51.18
+0.29
07:43.98
-1.24
07:50.84
+0.22
07:48.88
-0.20
3218.68
08:26.71
08:28.13
+0.28
08:21.43
-1.04
08:27.34
+0.12
08:24.88
-0.36
5000
13:18.60
13:29.07
+1.31
13:28.36
+1.22
13:26.01
+0.93
13:22.44
+0.48
10000
27:39.14
27:33.79
-0.32
28:04.09
+1.50
27:37.21
-0.12
27:45.63
+0.39
21097
1:01:14.00
58:48.55
-3.96
1:01:15.79
+0.05
1:00:18.34
-1.52
1:01:03.67
-0.28
42195
2:07:13.00
1:58:12.92
-7.08
2:06:07.20
-0.86
2:08:03.76
+0.67
2:07:10.00
-0.04
FIG. 5 (c) Application of four mathematical models to personal records of elite marathon runners (Mo Farah, Steve Jones): Log-normal plot of the ’running curves’ predicted by the models
(average velocity ¯v as function of race distance d, in units of vm and dc = vmtc given in the plot legend) and actual race data (red dots). The tables summarize the actual and predicted
race times, along with the relative errors in percent.
Reviewers' Comments: 
 
Reviewer #1: 
Remarks to the Author: 
Given that my involvement to the review process started at a later stage, I will avoid providing 
detailed comments on each section as I would typically do. However, I have read the manuscript 
in detail. Although I appreciate the value of exploring big data sets, I have a major concern with 
this manuscript as I do not think that any link can be made to physiological responses to exercise, 
when no physiological measures have been extracted. Additionally, I would like to mention that 
this manuscript is quite difficult to read, and that the authors should make an effort to improve the 
flow and logical order of the presentation. Regardless, please find below some general comments 
that I would hope will help the authors reflecting further on this manuscript. 
 
I think that the authors are not fully aware of the type of testing that takes place in many 
laboratories. I understand that they need to highlight the relevance of “real-world” data, and that 
laboratory settings have limitations. However, there are many experimental studies that have 
produced very solid performance data that, even though they do not belong to the “real world” 
category, they offer information that the “real world” conditions will never provide. I fully agree 
that the best measure of performance is performance itself. From a performance perspective, I do 
not care about who has the greatest VO2max or critical intensity of exercise. I care about who 
runs faster. Then, from a mechanistic perspective, I bring people to the lab to try to understand 
why differences in performance exist, but not necessarily to make people faster. The authors 
stated “The undeniable fact that the best test of running performance is an actual race and not 
laboratory tests” is only partly true. It is the best test to measure performance. However, it is not 
the best test to evaluate physiological responses and to elucidate the mechanisms that control the 
final performance. I think that the point that I am trying to make is that, at least to a given 
extent, the authors seem to be misrepresenting what happens in a laboratory setting. 
 
From what I have read in this manuscript, there is nothing that connects its content to 
physiological responses to exercise (which are often mentioned in this document). I could accept 
the claim that this analysis can help establishing non-physiological outcomes that could potentially 
help improving performance. However, there is no physiological value that can be seriously 
considered in this data set. At least in my view, the model requires accepting assumptions that 
might make some sense, but that are not necessarily correct. The authors seem to have almost a 
dislike for physiological evaluations. I am fine with that. However, there is no point in discussing 
physiology when no physiological outcomes are presented. I do not feel comfortable with all the 
assumptions that need to be accepted to believe some of the key components of the analysis 
(e.g., MAP). 
 
Once again, the authors might have gotten it right in terms of some predictors of performance. 
The problem is that we will never know as no real physiological data were collected. Perhaps, 
performing some physiological testing in a sub-sample of participants would add validity to the 
project. However, the authors have already disregarded this possibility when responding to other 
reviewers. In relation to this, I was interested in some responses. I am presenting below just a 
few examples: 
 
- The authors indicated that “As far as heart rate is concerned, our data set does not contain heart 
rate data for all runs and athletes as not all runners who wore a GPS watch wear a heart rate 
monitor (chest strap). But even if this data would be available, there remains an important 
unknown: the maximal heart rate of the athlete which varies substantially among individuals and 
cannot be determined accurately and easily from age-based formulas. Without the maximal heart 
rate the important relative effort (the quantity p in our model) cannot be determined accurately.” I 
would accept that the age-based formulas are not ideal, but they can be a good approximation. 
Additionally, the authors have plenty of data from the participants and I am sure that there has to 
be some high intensity interval or sprint training, or high intensity constant speed session from 
which HRmax could be derived. I mean, I would be the first arguing that, even if you had the 
actual HRmax, there are clear limitations with this approach. However, what I find a bit surprising 
is that the authors are willing to accept a lot of assumptions for other parameters in their model, 
but then they are too concerned about not getting the HRmax 100% right. This is surprising to 
me. 
- The authors argued that “Maximal tests in laboratory are difficult to repeat, possibly due to lack 
of motivation to go all-out without opponent or competition or even price money to win. As a 
result, the coefficient of variation may be as high as 25%”. Let’s clarify that performance 
outcomes have large variability in both the lab and on the field, but that the variability is greatly 
reduced with longer durations of performance. Additionally, if the lack of motivation because of the 
price money is an issue, then the author should eliminate most of these data because the vast 
majority of the performances in the people that the authors evaluated are not worth any money. 
Most people are engaged for other reasons and most of them would perform as well in the lab as 
they do in the “real world”. I am not convinced by this line of argumentation. 
- Then the authors stated that “Running mechanics varies considerably between treadmill and 
over ground running...One reason may difficulty to simulate wind resistance”. In fact, there are 
portable devices to test people in the “real world”. I know, the conditions will be slightly different. 
However, nothing is perfect (and this includes the assumptions in the model that is presented by 
the authors). 
- Finally, the authors said, “Maximal laboratory tests are short-lasting and therefore fail to account 
for reduction in running economy and subsequent increase in oxygen consumption at given speed 
that occurs over long-distance running.” Why would this need to be the case? I just read a paper 
in which participants performed quite long incremental tests achieving the same VO2max as in the 
shorter tests (J Appl Physiol 2019; 127(6):1519-1527). Maximal tests do not need to be short. 
Testing protocols are adapted to what one wants to evaluate. This type of comments makes me 
feel that the authors might not be very familiar with laboratory testing. 
 
As a side comment, I would say that the speed and endurance relationship presented in this 
document are quite similar to what is typically measured in the lab. So why emphasizing so much 
the idea that field data are better than lab data? Also, the fact that from training data one can 
predict performance is pretty obvious. What one can do in a race reflects what one can do in 
training. I know it is nice to confirm this with data, but there is nothing novel in this finding. 
 
As a final comment, I would like to say that I do not think that the authors have a full appreciation 
of the relevance that exercise intensity domains and their corresponding boundaries (i.e., 
thresholds) have in performance. I understand that measurements of VO2 and exercise thresholds 
have been largely bastardized in the world of exercise testing (to which the authors contribute by 
arbitrarily assigning names to parameters such as MAP or LT without having any physiological way 
of justifying them in this study). However, when things are done properly, very precise 
quantification of the metabolic stress of the system can be made. Unlike what the authors 
insinuate, these evaluations consider economy, fatigue, substrate depletion, etc. to make 
predictions about performance. All I am trying to say is that the authors might have an interesting 
story in relation to non-physiological predictors of running performance. However, they should be 
very careful with not overreaching beyond of what their data can say. 
 
 
 
Reviewer #2: 
Remarks to the Author: 
I appreciate the authors’ detailed rebuttal and the appendix that they have included to compare 
their model to other similar models. While the paper is improved, it is still hard to follow and 
ascertain exactly what the novel insights are. I think that many exciting findings have resulted 
from the analysis, but as the paper is presently written, many of the key insights do not stand out 
to the reader. 
 
It is also not clear whether the focus of the paper is to provide additional evidence to validate their 
previously published model or to show some of the novel insights that applying their model to the 
dataset can generate. It might be possible to do both things, but this should be framed more 
explicitly at the beginning and then discussed more explicitly in the results. If the goal is to 
provide additional support for their previous model, then the comparisons that they include in the 
appendix of the rebuttal would at least be helpful to include as supplementary material. I am 
personally more interested in a focus on the insights gained from the application of their model to 
the real-world dataset. If this is the desired focus, this should be made more clear in the 
manuscript. Even in this case, the comparisons to other models would still provide confidence that 
the author’s model is reasonable and thus could still be helpful to include in supplementary 
material. 
 
These and other comments are discussed in more detail below. 
 
A couple general comments on the review process: 
Line numbers are very helpful in the manuscript review process; then the authors can note line 
numbers where changes have been made in the response to reviewers. As a reviewer, I can also 
provide specific locations relevant for my comments. An annotated version of the manuscript 
showing exactly where changes have been made (e.g., via highlighting) is also very helpful to me 
as a reviewer. 
 
Title 
The paper should include a more meaningful title that highlights the specific novelty of the present 
work. The terms “novel” and “insights” do not convey much information about the present work. 
The term “novel” should be removed at minimum, as I believe is policy at least for Nature. The 
authors were also not performing data mining by most definitions of the term, since they were 
using a pre-existing physiology-based model (as a side note, I think this approach is preferable, in 
general, to a naïve data mining one). Instead, the real-world or free-living nature of the data is 
relevant to highlight in the title. The size of the data is also worth noting, as the title already does. 
 
Abstract 
(1) “We derived two variables that explain race performance: maximal aerobic power and 
endurance capability. Inclusion of endurance, which describes the decline in sustainable power 
over duration, offers novel insights to performance analysis since a realistic estimate of this 
parameter is impossible in conventional laboratory testing.” 
 
The mathematical model that the authors use was presented in the authors’ previously published 
paper. The abstract gives the impression that the mathematical model is something newly-created 
for the present paper. Please revise to make the novelty of the current paper more clear (i.e., the 
application of the model to free-living data and interpretation of the extracted parameters). 
 
(2) The abstract is much more clear than in the previous version, but it still does not include 
specific results. Novel insights are mentioned. But what were these novel insights? 
 
Introduction 
(3) In general, the introduction (along with other parts of the paper) is unnecessarily negative 
about in-lab testing. Both in-lab and out-of-lab testing have strengths and weaknesses and these 
could be acknowledged in a more even-handed way. 
 
(4) “important insights for a variety of populations ranging from elite athletes over recreational 
exercisers to patients in rehabilitation” 
 
change over -> to 
 
(5) “These approach predict that the average racing velocity tends to an constant value with 
increasing race distance which contradicts observation” 
 
Approach ->approaches 
Tends to an -> tends to be a 
 
(6) “Several empirical and physiological models have been put forward for explaining running 
world records in terms of a few physiological parameters.” 
 
Start a new paragraph here. 
 
(7) “Our minimal and universal model characterizes a runner’s physiology by two parameters that 
measure endurance capability and the velocity requiring maximal aerobic power output” 
 
The authors should make more clear that the model has already been proposed and evaluated 
with some data from (real-world) races. The application of the model to the present dataset (and 
to training data?) is what makes the current paper new. The previous paper by the authors should 
be mentioned and cited in the introduction, for example. This should also be made more clear in 
the last paragraph of the introduction that lays out the goals for the paper. 
 
 
Results 
 
(8) “Universal Performance Model” section: The authors should more directly state that they are 
using the model that they present in a previous publication. Something like: 
 
(1) In previous work we developed a model that does X. To summarize, this model …. (describe 
the key features of the model). For more details, see XXXX. 
(2) Here we do XXXX with the model. 
 
If there are differences between the author’s model published previously and the one in the 
present model, please make these differences more clear. 
 
(9) The results section and paper in general would also benefit from a tighter focus on the key, 
novel findings of the paper. For example, below are some excerpts from the paper that are novel, 
but don’t stand out in the present draft. Focusing paragraphs in the results on each of these 
topics, would be helpful. Specific paragraphs could be focused around asking the associated 
questions and discussing the study results. The key findings could also be explicitly enumerated in 
the discussion. 
• For all RS with three and more races (N=12,309), the mean error between model prediction and 
actual race time was only 2.0% … As a function of physiological parameters, in the most likely 
parameter range the model predicted the marathon performance with an overall accuracy of better 
than 10%. 
• The ”one-hour utilization” ratio p1hU = v1hU /vm had been estimated previously from laboratory 
measurements and races for a smaller group of 18 male LDR to be approximately 0.82 ± 0.05 35. 
Strikingly, our findings from the running data for ∼ 14,000 subjects corroborate this range without 
any invasive measurements, as demonstrated in Fig. 2(c). 
• Our findings demonstrate the strong sensitivity of performance to endurance. For example, a 
runner with a velocity of vm = 5m/sec can improve their marathon time from 3h27min38sec to 
2h53min8sec by doubling endurance from El = 3 to El = 6 (corresponding to a change in the ”one-
hour utilization” from 79% to 87% of VO2max), without any change in VO2,max or RE. 
• We observed an initial linear increase of El with TRIMP, a plateau around El = 7.5 ± 2 for TRIMP 
∼ 25,000, and a statistically significant final drop which may be due to over-training. This result 
suggests that there is an optimal TRIMP per TS, and the corresponding maximal endurance 
enables a close to optimal marathon race time for a given velocity vm (see Fig. 3(a)). 
 
(10) Minimize the use of acronyms where possible in the text to make it easier for readers to 
understand the paper. I suggest you remove the following: 
• RS (racing season) 
• TS (training season) 
• RE (running economy) 
• LDR (long distance runners?) 
If the abbreviations are needed in a figure/table they are OK to use there, as long as they are 
defined in the caption. 
 
(11) “by matching them with an universal, i.e., subject independent model” 
An universal -> a universal 
A comma is needed after “model” 
 
(12) “Our minimal model introduces effective parameters by measuring” It is not clear what the 
authors mean by “effective”. 
 
(13) “observations made by Hill in running world records” 
Reword to make it clear that it wasn’t Hill who was running the world records :-). 
 
(14) “Fig. 3 first shows a color coded plot of Tmarathon as function of the physiological 
parameters.” 
This type of sentence is a better fit for a caption. In the Results it is preferable to describe specific 
findings. There are several instances of this in the Results. 
 
(15) “To investigate the predictive power of our model in more detail, we applied our model also 
the RS with the marathon performance excluded” 
A word is missing from this sentence. 
 
(16) “Consistent and inconsistent runners can be identified from the relative difference between 
our model estimates and actual race times.” A better topic sentence (that covers the main focus of 
the paragraph) is needed to improve the logical flow of this section of the results. In general, a 
careful review of the entire paper to ensure each paragraph has a clear topic sentence would 
improve the quality of the manuscript. 
 
Discussion 
(17) First paragraph: this should be broken into multiple paragraphs. The discussion of the 
limitations would be a natural split point. 
 
(18) “This is an important advance over physiological testing in the laboratory where the required 
maximal effort is impossible to motivate for a distance of 20km or longer.” 
 
I don’t think the authors intend to mean that there is no use for lab-based testing. This is another 
place where the authors could soften their language. (e.g., important advance -> important 
complement). 
 
In general, the primary point that stands out from the discussion is that the real-world data is a 
big improvement over lab testing. I don’t think this is the most important point (as lab-based 
testing in a controlled environment still has great value). I would instead focus more on reviewing 
the specific new insights about running, training, and performance that were gleaned from the 
analysis. 
 
Methods 
(19) “Only TS with 30 or more runs were considered.” 
 
What is the rationale for this choice? Was there any requirement from the minimum chronological 
length of the training season? Was there any sensitivity to these or other threshold choices 
discussed in the paragraph? 
 
(20) Check for redundancy between material included in the Methods and Results 
 
(21) The following passage is a better fit for the results or discussion than the Methods. 
 
For our two parameter model, the quality of the fitting could be probed for all RS with more than 
two races. For those RS we found a rather low average error of only 2:0% between the computed 
and actual race times. Another applicability test of our model is the estimation of the marathon 
finishing time from equation(1) when the parameters vm and l are obtained from the RS without 
the marathon. Given all the possible uncertainties in marathon racing that are beyond the control 
of this study (e.g. weather, course profile, motivation of the athlete), the predictive power 
reflected by the results for marathon finishing time estimate in Fig. 4 is rather satisfying 
 
 
 
Reviewer #3: 
Remarks to the Author: 
The authors have provided detailed and convincing responses to most of the questions and 
comments that I forwarded to them. In particular, I am convinced by their response on the 
relationship between their model and the critical power model. However, this did not translate into 
a modification of the article accounting entirely for their responses to the reviewer. This is a pity. 
The changes in the manuscript are minor and clearly inadequate. I would like to see the reasoning 
that the author developed in replying to the reviewer's comments more adequately integrated in 
the menuscript, especially in the discussion, and I hope the suthors will show more consideration 
for the suggested references and comments. To respond is good, but it is not enough. 
2
Rebuttal letter
“Human running performance from real-world big data”
(NCOMMS-20-02292)
Please ﬁnd below our point-to-point answers to the reviewer comments (C: comment, A: answer). All changes in the
manuscript are marked by colour highlighting (deleted text in red, newly added text in blue). Also, we have included
line numbers in the manuscript (colour coded version) in order to make reference to changes in the point-by-point
rebuttal letter.
Answer to Reviewer #1
We thank the reviewer for her/his time spent looking over our manuscript and their comments that we address
point-by-point in the following.
C I think that the authors are not fully aware of the type of testing that takes place in many laboratories. I under-
stand that they need to highlight the relevance of real-world data, and that laboratory settings have limitations.
However, there are many experimental studies that have produced very solid performance data that, even though
they do not belong to the real world category, they o↵er information that the real world conditions will never
provide. I fully agree that the best measure of performance is performance itself. From a performance perspec-
tive, I do not care about who has the greatest VO2max or critical intensity of exercise. I care about who runs
faster. Then, from a mechanistic perspective, I bring people to the lab to try to understand why di↵erences in
performance exist, but not necessarily to make people faster. The authors stated The undeniable fact that the
best test of running performance is an actual race and not laboratory tests is only partly true. It is the best test
to measure performance. However, it is not the best test to evaluate physiological responses and to elucidate the
mechanisms that control the ﬁnal performance. I think that the point that I am trying to make is that, at least
to a given extent, the authors seem to be misrepresenting what happens in a laboratory setting.
A We agree that our presentation was not balanced between laboratory testing and our approach. This is regret-
table because that is not how we think. Therefore, we now highlight how wearables can complement laboratory
testing by expanding the size of population that can be tested. We also compare strengths and weaknesses of
both approaches. Please see lines 46↵, 61↵, 102↵, 345↵.
C From what I have read in this manuscript, there is nothing that connects its content to physiological responses
to exercise (which are often mentioned in this document). I could accept the claim that this analysis can help
establishing non-physiological outcomes that could potentially help improving performance. However, there is
no physiological value that can be seriously considered in this data set. At least in my view, the model requires
accepting assumptions that might make some sense, but that are not necessarily correct. The authors seem to
have almost a dislike for physiological evaluations. I am ﬁne with that. However, there is no point in discussing
physiology when no physiological outcomes are presented. I do not feel comfortable with all the assumptions that
need to be accepted to believe some of the key components of the analysis (e.g., MAP).
A We have replaced throughout the manuscript ”physiological parameters” by indices of performance (aerobic
power index and endurance index), extracted from running exercise data using our model. We would like to
point out that our model makes assumptions that are also contained in other models proposed by exercise
physiologists (e.g. Monod & Scherrer, di Prampero, Peronnet & Thibault, see our detailed comparison in the
appendix to our last rebuttal letter). Hence, we believe that the key parameters of our model do have some
physiological meaning. However, in order to avoid any confusion and to not make unnecessary assumptions, we
now refer to our parameters as ”performance indices” and just state to which physiological variables they might
be related. Please see lines 96↵.
3
C Once again, the authors might have gotten it right in terms of some predictors of performance. The problem
is that we will never know as no real physiological data were collected. Perhaps, performing some physiological
testing in a sub-sample of participants would add validity to the project. However, the authors have already
disregarded this possibility when responding to other reviewers.
A We agree that physiological testing on a smaller sample of subjects would be very useful. We plan to carry
out such testing for a new group of subject in the future, to compare our model parameters to actual lab
measurements. It should be noted that also previous models (e.g. Peronnet & Thibault) have been applied only
to world running records to extract physiological parameters without a direct comparison to lab tests for these
athletes.
C The authors indicated that As far as heart rate is concerned, our data set does not contain heart rate data for all
runs and athletes as not all runners who wore a GPS watch wear a heart rate monitor (chest strap). But even if
this data would be available, there remains an important unknown: the maximal heart rate of the athlete which
varies substantially among individuals and cannot be determined accurately and easily from age-based formulas.
Without the maximal heart rate the important relative e↵ort (the quantity p in our model) cannot be determined
accurately. I would accept that the age-based formulas are not ideal, but they can be a good approximation.
Additionally, the authors have plenty of data from the participants and I am sure that there has to be some high
intensity interval or sprint training, or high intensity constant speed session from which HRmax could be derived.
I mean, I would be the ﬁrst arguing that, even if you had the actual HRmax, there are clear limitations with this
approach. However, what I ﬁnd a bit surprising is that the authors are willing to accept a lot of assumptions for
other parameters in their model, but then they are too concerned about not getting the HRmax 100% right. This
is surprising to me.
A Our point is that HR would not add any additional beneﬁt to the extraction of our model parameters. When
maximal and resting HR for each runner are known, the entire analysis could be based on HR instead of running
velocity, yielding an expression for the maximal duration over which a given HR could be sustained, Tmax(HR).
Hence the parameters vm and El could be determined from observed relations between velocity and HR, and the
average HR sustained during maximal e↵ort of a given duration (races). However, an unpublished study that
we performed previously on a much smaller number of subjects (20) showed that HR ﬂuctuates more strongly
than velocity, presumably due to weather conditions, non-running related stress, nutrition status, sleep status,
etc. In addition, there is always a time delay between a rise (or fall) in velocity and HR which requires the
exclusion of time windows with this hysteresis e↵ects. The accuracy of our model for race time predictions was
on average 2%. This is deﬁnitely better than the typical error for age-based formulas for maximal HR. All these
considerations led us to use velocity instead of HR in our data analysis.
C The authors argued that Maximal tests in laboratory are diﬃcult to repeat, possibly due to lack of motivation to
go all-out without opponent or competition or even price money to win. As a result, the coeﬃcient of variation
may be as high as 25%. Lets clarify that performance outcomes have large variability in both the lab and on the
ﬁeld, but that the variability is greatly reduced with longer durations of performance. Additionally, if the lack of
motivation because of the price money is an issue, then the author should eliminate most of these data because
the vast majority of the performances in the people that the authors evaluated are not worth any money. Most
people are engaged for other reasons and most of them would perform as well in the lab as they do in the real
world. I am not convinced by this line of argumentation.
A In the revised version of the manuscript we do not state that poor repeatability would compromise laboratory
test results. We would like to point out that not only price money is motivation but also competing against
friends, team members, or for something like age group win etc., i.e., real-world situations.
4
C Then the authors stated that Running mechanics varies considerably between treadmill and over ground run-
ning...One reason may diﬃculty to simulate wind resistance. In fact, there are portable devices to test people in
the real world. I know, the conditions will be slightly di↵erent. However, nothing is perfect (and this includes
the assumptions in the model that is presented by the authors).
A Our data comes from consumer-product based measurements (GPS watches) and more advanced portable devices
were not available to the huge group of runners monitored. We agree on the general possibility of more advanced
measurements in the ”real world”. However, these additional data are not relevant to our model as an input,
and they would impose limitations on the number of available subjects.
C Finally, the authors said, Maximal laboratory tests are short-lasting and therefore fail to account for reduction in
running economy and subsequent increase in oxygen consumption at given speed that occurs over long-distance
running. Why would this need to be the case? I just read a paper in which participants performed quite long
incremental tests achieving the same VO2max as in the shorter tests (J Appl Physiol 2019; 127(6):1519-1527).
Maximal tests do not need to be short. Testing protocols are adapted to what one wants to evaluate. This type
of comments makes me feel that the authors might not be very familiar with laboratory testing.
A We are not saying that VO2max is declining with duration. We are only saying that running economy (energy
cost of running) deteriorates with duration. This is based on results in Ref. 18, 19 and 20. However, in order to
provide in this context a better balance between lab testing and our approach, we have modiﬁed the paragraph
with this statement, see lines 46 – 68.
C As a side comment, I would say that the speed and endurance relationship presented in this document are quite
similar to what is typically measured in the lab. So why emphasizing so much the idea that ﬁeld data are better
than lab data? Also, the fact that from training data one can predict performance is pretty obvious. What one
can do in a race reﬂects what one can do in training. I know it is nice to conﬁrm this with data, but there is
nothing novel in this ﬁnding.
A We are glad to hear that what is measured typically in the lab is quite similar to our model ﬁndings. We believe
that the novel part of ﬁndings are quantitative relations between our model indexes and training volume and
intensity for a very large group of runners, yielding also good statistics for typical variations in these relations.
We do not think that ﬁeld data are better than lab testing. We have made clear in the revised manuscript that
our approach is complementary to lab testing, see lines 324 – 326.
C As a ﬁnal comment, I would like to say that I do not think that the authors have a full appreciation of the relevance
that exercise intensity domains and their corresponding boundaries (i.e., thresholds) have in performance. I
understand that measurements of VO2 and exercise thresholds have been largely bastardized in the world of
exercise testing (to which the authors contribute by arbitrarily assigning names to parameters such as MAP or LT
without having any physiological way of justifying them in this study). However, when things are done properly,
very precise quantiﬁcation of the metabolic stress of the system can be made. Unlike what the authors insinuate,
these evaluations consider economy, fatigue, substrate depletion, etc. to make predictions about performance. All
I am trying to say is that the authors might have an interesting story in relation to non-physiological predictors
of running performance. However, they should be very careful with not overreaching beyond of what their data
can say.
A We accept this criticism. We have now renamed model parameters to aerobic power index and endurance index
to di↵erentiate them from laboratory parameters. Overall, we have rewritten our manuscript in order to provide
a more balanced presentation of what our model predicts and the concepts and measurements in the world of
exercise testing in the lab.
5
Rebuttal letter
“Human running performance from real-world big data”
(NCOMMS-20-02292)
Please ﬁnd below our point-to-point answers to the reviewer comments (C: comment, A: answer). All changes in the
manuscript are marked by colour highlighting (deleted text in red, newly added text in blue). Also, we have included
line numbers in the manuscript (colour coded version) in order to make reference to changes in the point-by-point
rebuttal letter.
Answer to Reviewer #2
We thank the reviewer for her/his time spent again looking over our manuscript and their very detailed comments
and suggestions that we address point-by-point in the following.
C It is also not clear whether the focus of the paper is to provide additional evidence to validate their previously
published model or to show some of the novel insights that applying their model to the dataset can generate. It
might be possible to do both things, but this should be framed more explicitly at the beginning and then discussed
more explicitly in the results. If the goal is to provide additional support for their previous model, then the
comparisons that they include in the appendix of the rebuttal would at least be helpful to include as supplementary
material. I am personally more interested in a focus on the insights gained from the application of their model
to the real-world dataset. If this is the desired focus, this should be made more clear in the manuscript. Even
in this case, the comparisons to other models would still provide conﬁdence that the authors model is reasonable
and thus could still be helpful to include in supplementary material.
A The main aim of our work is ”to show some of the novel insights that applying their model to the dataset can
generate”. We have made this more clear in the revised manuscript, please see lines 96↵. The model comparison
of our previous rebuttal letter is not included in this work as it would go much beyond its scope. Instead, we
are in the process of completing a separate publication on a detailed comparison of mathematical models for
running performance which shall include the results shown in our rebuttal letter.
C Title: The paper should include a more meaningful title that highlights the speciﬁc novelty of the present work.
The terms novel and insights do not convey much information about the present work. The term novel should
be removed at minimum, as I believe is policy at least for Nature. The authors were also not performing data
mining by most deﬁnitions of the term, since they were using a pre-existing physiology-based model (as a side
note, I think this approach is preferable, in general, to a naive data mining one). Instead, the real-world or
free-living nature of the data is relevant to highlight in the title. The size of the data is also worth noting, as
the title already does.
A We have modiﬁed the title accordingly.
C (1) We derived two variables that explain race performance: maximal aerobic power and endurance capability.
Inclusion of endurance, which describes the decline in sustainable power over duration, o↵ers novel insights to
performance analysis since a realistic estimate of this parameter is impossible in conventional laboratory testing.
The mathematical model that the authors use was presented in the authors previously published paper. The
abstract gives the impression that the mathematical model is something newly-created for the present paper.
Please revise to make the novelty of the current paper more clear (i.e., the application of the model to free-living
data and interpretation of the extracted parameters).
A We have revised the abstract accordingly.
6
C (2) The abstract is much more clear than in the previous version, but it still does not include speciﬁc results.
Novel insights are mentioned. But what were these novel insights?
A We now described the novel insights more speciﬁcally.
C (3) In general, the introduction (along with other parts of the paper) is unnecessarily negative about in-lab
testing. Both in-lab and out-of-lab testing have strengths and weaknesses and these could be acknowledged in a
more even-handed way.
A We have given a more balanced presentation of in-lab and out-of-lab testing, please see lines 46↵, 61↵, 102↵,
345↵.
C (4) important insights for a variety of populations ranging from elite athletes over recreational exercisers to
patients in rehabilitation: change over ! to
A done.
C (5) These approach predict that the average racing velocity tends to an constant value with increasing race
distance which contradicts observation: Approach ! approaches, Tends to an ! tends to be a
A done.
C (6) Several empirical and physiological models have been put forward for explaining running world records in
terms of a few physiological parameters.: Start a new paragraph here.
A done.
C (7) Our minimal and universal model characterizes a runners physiology by two parameters that measure en-
durance capability and the velocity requiring maximal aerobic power output. The authors should make more clear
that the model has already been proposed and evaluated with some data from (real-world) races. The application
of the model to the present dataset (and to training data?) is what makes the current paper new. The previous
paper by the authors should be mentioned and cited in the introduction, for example. This should also be made
more clear in the last paragraph of the introduction that lays out the goals for the paper.
A We have modiﬁed the introduction accordingly, and referenced our previous paper. Please see lines 77 – 83.
C (8) Universal Performance Model section: The authors should more directly state that they are using the model
that they present in a previous publication. Something like: (1) In previous work we developed a model that does
X. To summarize, this model . (describe the key features of the model). For more details, see XXXX. (2) Here
we do XXXX with the model. If there are di↵erences between the authors model published previously and the
one in the present model, please make these di↵erences more clear.
A We modiﬁed this section accordingly (lines 108 – 124). There is no di↵erence with the model itself published
previously. One of the parameters of the model (tc) was ﬁxed at 6 minutes, as we had explained already in the
previous version of the manuscript.
C (9) The results section and paper in general would also beneﬁt from a tighter focus on the key, novel ﬁndings of
the paper. For example, below are some excerpts from the paper that are novel, but dont stand out in the present
draft. Focusing paragraphs in the results on each of these topics, would be helpful. Speciﬁc paragraphs could be
focused around asking the associated questions and discussing the study results. The key ﬁndings could also be
explicitly enumerated in the discussion.
7
– For all RS with three and more races (N=12,309), the mean error between model prediction and actual
race time was only 2.0% As a function of physiological parameters, in the most likely parameter range the
model predicted the marathon performance with an overall accuracy of better than 10%.
– The one-hour utilization ratio p1hU = v1hU/vm had been estimated previously from laboratory measurements
and races for a smaller group of 18 male LDR to be approximately 0.82 ± 0.05. Strikingly, our ﬁndings
from the running data for ⇠ 14,000 subjects corroborate this range without any invasive measurements, as
demonstrated in Fig. 2(c).
– Our ﬁndings demonstrate the strong sensitivity of performance to endurance. For example, a runner with a
velocity of vm = 5m/sec can improve their marathon time from 3h27min38sec to 2h53min8sec by doubling
endurance from El = 3 to El = 6 (corresponding to a change in the one-hour utilization from 79% to 87%
of VO2max), without any change in VO2max or RE.
– We observed an initial linear increase of El with TRIMP, a plateau around El = 7.5 ± 2 for TRIMP ⇠
25,000, and a statistically signiﬁcant ﬁnal drop which may be due to over-training. This result suggests that
there is an optimal TRIMP per TS, and the corresponding maximal endurance enables a close to optimal
marathon race time for a given velocity vm (see Fig. 3(a)).
A We have modiﬁed the results section to make our key ﬁndings stand out more clearly by adding subsections for
each key ﬁnding. We could not add a itemized list of the key ﬁndings in the discussion section due to length
restriction.
C (10) Minimize the use of acronyms where possible in the text to make it easier for readers to understand the
paper. I suggest you remove the following:
– RS (racing season)
– TS (training season)
– RE (running economy)
– LDR (long distance runners?)
If the abbreviations are needed in a ﬁgure/table they are OK to use there, as long as they are deﬁned in the
caption.
A We have removed these acronyms.
C (11) by matching them with an universal, i.e., subject independent model: an universal ! a universal, a comma
is needed after model
A done.
C (12) Our minimal model introduces e↵ective parameters by measuring It is not clear what the authors mean by
e↵ective.
A We have removed ”e↵ective”.
C (13) observations made by Hill in running world records: Reword to make it clear that it wasn‘t Hill who was
running the world records :-).
A Thank you ;-) We have made this clear now.
C (14) Fig. 3 ﬁrst shows a color coded plot of Tmarathon as function of the physiological parameters. This type
of sentence is a better ﬁt for a caption. In the Results it is preferable to describe speciﬁc ﬁndings. There are
several instances of this in the Results.
8
A We have moved this type of sentences to the ﬁgure captions.
C (15) To investigate the predictive power of our model in more detail, we applied our model also the RS with the
marathon performance excluded: A word is missing from this sentence.
A We have added the word ”to” so it reads ”... also to the race season ...”.
C (16) Consistent and inconsistent runners can be identiﬁed from the relative di↵erence between our model esti-
mates and actual race times. A better topic sentence (that covers the main focus of the paragraph) is needed to
improve the logical ﬂow of this section of the results. In general, a careful review of the entire paper to ensure
each paragraph has a clear topic sentence would improve the quality of the manuscript.
A We have reviewed and modiﬁed the manuscript to ensure that each paragraph has a clear topic and we added
new subsections to the result section.
C (17) Discussion, ﬁrst paragraph: this should be broken into multiple paragraphs. The discussion of the limitations
would be a natural split point.
A done.
C (18) This is an important advance over physiological testing in the laboratory where the required maximal e↵ort
is impossible to motivate for a distance of 20km or longer. I dont think the authors intend to mean that there is
no use for lab-based testing. This is another place where the authors could soften their language. (e.g., important
advance ! important complement).
In general, the primary point that stands out from the discussion is that the real-world data is a big improvement
over lab testing. I dont think this is the most important point (as lab-based testing in a controlled environment
still has great value). I would instead focus more on reviewing the speciﬁc new insights about running, training,
and performance that were gleaned from the analysis.
A We agree. We have modiﬁed the manuscript in general to give a more balanced view of ”real-world” data and
lab testing, and focused more on the new insights from our analysis.
C (19) Methods: Only TS with 30 or more runs were considered. What is the rationale for this choice? Was there
any requirement from the minimum chronological length of the training season? Was there any sensitivity to
these or other threshold choices discussed in the paragraph?
A This minimum run condition for training season was applied so that runner had at least trained once per week
on average during the 180 day long training season. Smaller number of runs could mean an interrupted training
(e.g. due to injury), and hence relation to performance would be less reliable.
C (20) Check for redundancy between material included in the Methods and Results.
A We believe that the Methods section should be self-contained to allow a complete account of the applied pro-
cedures. However, we do have reduced some redundancy by combining some part of the Methods section with
the appropriate paragraph of the Result section, please see next point.
C (21) The following passage is a better ﬁt for the results or discussion than the Methods. ”For our two parameter
model, the quality of the ﬁtting could be probed for all RS with more than two races. For those RS we found a
rather low average error of only 2.0% between the computed and actual race times. Another applicability test of
our model is the estimation of the marathon ﬁnishing time from equation (1) when the parameters vm and γl
are obtained from the RS without the marathon. Given all the possible uncertainties in marathon racing that
are beyond the control of this study (e.g. weather, course proﬁle, motivation of the athlete), the predictive power
reﬂected by the results for marathon ﬁnishing time estimate in Fig. 4 is rather satisfying.”
9
A We have moved part of this passage to the Results section, please see lines 198 – 209 and 439 – 443.
10
Rebuttal letter
“Human running performance from real-world big data”
(NCOMMS-20-02292)
Please ﬁnd below our point-to-point answers to the reviewer comments (C: comment, A: answer). All changes in the
manuscript are marked by colour highlighting (deleted text in red, newly added text in blue). Also, we have included
line numbers in the manuscript (colour coded version) in order to make reference to changes in the point-by-point
rebuttal letter.
Answer to Reviewer #3
We thank the reviewer for her/his time spent looking again over our manuscript and their comments that we address
point-by-point in the following.
C I would like to see the reasoning that the author developed in replying to the reviewer’s comments more adequately
integrated in the manuscript, especially in the discussion, and I hope the authors will show more consideration
for the suggested references and comments. The respond is good, but it is not enough.
A As explained more clearly in the revised version, the aim of this work is neither a validation of our previously
published model nor a comparison of our model to other existing models (which however are mentioned in
our work). Rather, the aim of our work is to apply our model to real-world data and to extract performance
parameters and relate them to racing performance and training. Due to this focus and due to length restrictions,
we can not include all our reasoning from the previous reply in our manuscript. However, we have revised the
manuscript overall to give a more balanced view of lab testing and our approach. To avoid confusion, we have
changed the term ”physiological parameters” to ”performance indices”. In addition, we have added relevant
references to previous work on theoretical concepts from exercise physiology in the Discussion section, please
see lines 345 – 349.
Reviewers' Comments: 
 
Reviewer #1: 
Remarks to the Author: 
I would like to thank the authors for the changes made to the this manuscript. From a conceptual 
perspective, I would say that I still disagree with the use of the MAP construct, as I think this is a 
flawed concept. However, I understand that it is commonly used and accepted by many, and that 
it serves the purpose of the present analysis. Aside from this comment (which is nothing but just a 
way of expressing my view), I am satisfied with the responses that the authors have provided and 
with the updated version of the manuscript. I think that focusing on performance rather than 
physiology makes this a much more solid and believable story. Thus, I have no further comments 
to make. 

Human running performance from real-world big data.
Emig, Thorsten,Peltonen, Jussi