Extrapolating Metabolic Savings in Running: Implications for Performance Predictions

Abstract

Training, footwear, nutrition, and racing strategies (i.e., drafting) have all been shown to reduce the metabolic cost of distance running (i.e., improve running economy). However, how these improvements in running economy (RE) quantitatively translate into faster running performance is less established. Here, we quantify how metabolic savings translate into faster running performance, considering both the inherent rate of oxygen uptake-velocity relation and the additional cost of overcoming air resistance when running overground. We collate and compare five existing equations for oxygen uptake-velocity relations across wide velocity ranges. Because the oxygen uptake vs. velocity relation is non-linear, for velocities slower than ∼3 m/s, the predicted percent improvement in velocity is slightly greater than the percent improvement in RE. For velocities faster than ∼3 m/s, the predicted percent improvement in velocity is less than the percent improvements in RE. At 5.5 m/s, i.e., world-class marathon pace, the predicted percent improvement in velocity is ∼2/3rds of the percent improvement in RE. For example, at 2:04 marathon pace, a 3% improvement in RE translates to a 1.97% faster velocity or 2:01:36, almost exactly equal to the recently set world record.

Full text

fphys-10-00079 February 8, 2019 Time: 17:57 # 1
MINI REVIEW
published: 11 February 2019
doi: 10.3389/fphys.2019.00079
Edited by:
Davide Malatesta,
Université de Lausanne, Switzerland
Reviewed by:
Leonardo Alexandre
Peyré-Tartaruga,
Universidade Federal do Rio Grande
do Sul (UFRGS), Brazil
Fabrice Vercruyssen,
Université de Toulon, France
Andrew Mark Jones,
University of Exeter, United Kingdom
*Correspondence:
Rodger Kram
rodger.kram@colorado.edu
Specialty section:
This article was submitted to
Exercise Physiology,
a section of the journal
Frontiers in Physiology
Received: 24 October 2018
Accepted: 22 January 2019
Published: 11 February 2019
Citation:
Kipp S, Kram R and
Hoogkamer W (2019) Extrapolating
Metabolic Savings in Running:
Implications for Performance
Predictions. Front. Physiol. 10:79.
doi: 10.3389/fphys.2019.00079
Extrapolating Metabolic Savings in
Running: Implications for
Performance Predictions
Shalaya Kipp1,2 , Rodger Kram1*and Wouter Hoogkamer1
1Department of Integrative Physiology, University of Colorado, Boulder, CO, United States, 2School of Kinesiology, University
of British Columbia, Vancouver, BC, Canada
Training, footwear, nutrition, and racing strategies (i.e., drafting) have all been shown
to reduce the metabolic cost of distance running (i.e., improve running economy).
However, how these improvements in running economy (RE) quantitatively translate
into faster running performance is less established. Here, we quantify how metabolic
savings translate into faster running performance, considering both the inherent rate
of oxygen uptake-velocity relation and the additional cost of overcoming air resistance
when running overground. We collate and compare five existing equations for oxygen
uptake-velocity relations across wide velocity ranges. Because the oxygen uptake vs.
velocity relation is non-linear, for velocities slower than ∼3 m/s, the predicted percent
improvement in velocity is slightly greater than the percent improvement in RE. For
velocities faster than ∼3 m/s, the predicted percent improvement in velocity is less
than the percent improvements in RE. At 5.5 m/s, i.e., world-class marathon pace, the
predicted percent improvement in velocity is ∼2/3rds of the percent improvement in RE.
For example, at 2:04 marathon pace, a 3% improvement in RE translates to a 1.97%
faster velocity or 2:01:36, almost exactly equal to the recently set world record.
Keywords: energetic cost, locomotion, marathon, oxygen uptake, running economy
INTRODUCTION
The remarkable 2:00:25 exhibition marathon in Monza, Italy in 2017 and the current world record
time of 2:01:39 set in Berlin in 2018 by Eliud Kipchoge raise an intriguing question: can we predict
improvements in endurance running performance based on improvements in running economy
(RE)? Together with lactate threshold and maximal oxygen uptake ( ˙
VO2max), RE is one of the
three primary physiological determinants of performance (Daniels, 1985;Joyner, 1991;Foster and
Lucia, 2007). RE is traditionally defined as the rate of oxygen uptake ( ˙
VO2, in mlO2/kg/min)
for running at a specified submaximal velocity1. Improvements in RE allow athletes to run at a
faster velocity for the same oxygen uptake and thus achieve superior performances (Joyner, 1991;
Hoogkamer et al., 2016, 2017). RE can also be expressed in oxygen uptake per unit distance (in
mlO2/kg/km), by dividing ˙
VO2by the running velocity at which it was assessed. From ∼2.2 to
5.6 m/s (8–20 km/h), net ˙
VO2(gross minus rest or standing) per distance remains fairly constant
1We and others prefer to express RE in units of energy utilization (W/kg or kcals/min/kg) (Fletcher et al., 2009;Shaw et al.,
2014;Beck et al., 2018;Kipp et al., 2018) to account for differences in substrate utilization and therefore, in the amount of
energy liberated per liter of oxygen uptake. Here, we needed to incorporate several classic studies that only reported oxygen
uptake rates and thus we express RE in units of oxygen uptake rate.
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Kipp et al. Extrapolating Metabolic Savings in Running
(Margaria et al., 1963;di Prampero et al., 1986;Saibene and
Minetti, 2003;Lacour and Bourdin, 2015). Accordingly, 1%
improvements in RE (lower rates) should directly translate
to 1% faster running performances (Daniels and Daniels,
1992; McLaughlin et al., 2010). Indeed, we demonstrated that
laboratory-measured percent changes in RE translate to similar
percent changes in distance running performance (assessed by
3 km time trials) (Hoogkamer et al., 2016).
Recently, we have used these insights and models to
translate metabolic savings reported in the literature
(Hoogkamer et al., 2017) and measured in our laboratory
(Hoogkamer et al., 2018a) into predicted improvements
in elite marathon running performances. Unfortunately,
the heterogeneity of racecourses, meteorological and
competitive conditions, combined with fluctuations in
the training status of elite marathon runners preclude
controlled experiments on racing performance. Recent
marathon race results suggest that finishing times may
not match the theoretically predicted improvements from
drafting (Hoogkamer et al., 2017) or advances in shoe
technology (Hoogkamer et al., 2018a). Here, we examine
the assumptions underlying our extrapolations and derive a
revised model for extrapolating metabolic savings into running
performance improvements.
RUNNING VELOCITY AND ˙
VO2
To extrapolate how changes in RE will impact performance,
we focus on the gross ˙
VO2-velocity relation. If the relation is
directly proportional with a zero ˙
VO2-intercept, running 1%
faster exacts a 1% higher metabolic rate and we can expect
that a 1% improvement in RE would allow for a 1% faster
race performance (Hoogkamer et al., 2016). There are many
reports of linear gross ˙
VO2-velocity relations for treadmill
running. The relations have either positive (Pugh, 1970;Léger
and Mercier, 1984;Helgerud et al., 2010) or negative ˙
VO2-
intercepts (Joyner, 1991;Daniels and Daniels, 1992;Jones and
Doust, 1996), depending on the velocity ranges considered. In
the case of a linear gross ˙
VO2-velocity relation with a positive
˙
VO2-intercept, running 1% faster requires less than 1% more
oxygen. In the case of a linear gross ˙
VO2-velocity relation with
a negative ˙
VO2-intercept, running 1% faster requires more than
1% more oxygen. It is therefore critical to base any extrapolation
of metabolic savings to running performance on the best available
˙
VO2-velocity relation data.
More recent treadmill running studies have indicated
that both the gross ˙
VO2-velocity relation and the metabolic
rate (Watts or kcal/min)-velocity relations are actually better
described as inherently curvilinear, especially over wide
ranges in velocity (Steudel-Numbers and Wall-Scheffler, 2009;
Batliner et al., 2018;Black et al., 2018;Kipp et al., 2018).
Figure 1A illustrates both linear and curvilinear regressions
to treadmill running data from 10 high-level male runners
(<30-min 10 km) for velocities spanning 1.78–5.14 m/s
and measured at ∼1600 m altitude (Batliner et al., 2018).
The upward curvilinear relation explains why a positive
gross ˙
VO2-intercept is observed when a linear regression
line is fitted to slow velocity gross ˙
VO2data (Bransford and
Howley, 1977;Maughan and Leiper, 1983) and a negative
intercept when fitted to fast velocity gross ˙
VO2data (Conley
and Krahenbuhl, 1980;Joyner, 1991;Daniels and Daniels,
1992;Jones and Doust, 1996). A critical implication of a
curvilinear gross ˙
VO2-velocity relation is that at fast running
velocities, a 1% improvement in RE translates to smaller
(<1%) improvements in running velocity and thus a less than
directly proportional performance benefit. This inherent upward
curvilinearity, has not previously been accounted for in models
to predict running performance (di Prampero et al., 1986;
McLaughlin et al., 2010).
Air resistance is a second important consideration when
translating metabolic savings quantified in treadmill studies to
overground running performance. Most studies that show a
curvilinear ˙
VO2-velocity relation have actually been conducted
on treadmills, with negligible air resistance (Steudel-Numbers
and Wall-Scheffler, 2009;Black et al., 2018;Batliner et al.,
2018;Kipp et al., 2018). However, as described by Pugh
(1970, 1971), the oxygen cost of overcoming air resistance
can be expected to increase more than proportionally at
faster running velocities, since air drag force is proportional
to air (running) velocity squared (du Bois-Reymond, 1925;
Hill, 1928) and hence mechanical power (force ×velocity)
is proportional to velocity cubed. Specifically, Pugh (1971)
related the metabolic cost of overcoming air resistance to the
mechanical power needed to overcome the air drag forces
during running: ˙
VO2(L/min) = 0.00354·Ap·v3for an athlete
of projected frontal area Ap(m2), running at velocity v(m/s),
through still air. Throughout this paper, we will use an Ap
of 0.45 m2, for an elite male marathoner (58 kg and 1.71 m)
(DuBois and DuBois, 1916;Hoogkamer et al., 2017). Léger
and Mercier (1984) added Pugh’s cubic air resistance term to
the linear equation they had derived from a regression on
data from 10 separate treadmill studies over various moderate
velocity ranges. Velocity (v) is expressed in m/s for all
equations below.
Léger and Mercier, 1984 (including Pugh’s cubic term):
˙
VO2ml/kg/min=0.02724v3+11.39v+2.209 (1)
In Eq. 2 and Figure 1B (sold line), we added Pugh’s cubic air
resistance term to the inherent curvilinear equation from Batliner
et al., 2018.
Batliner et al., 2018 +Pugh’s cubic term:
.
VO2(ml/kg/min)=0.02724v3+1.5355v2+1.5374v+15.661
(2)
In Figure 1C we have depicted how this curvilinear ˙
VO2-
velocity relation affects the predicted improvements in running
velocity with a consistent hypothetical 10% improvement
in RE. The percent velocity enhancement resulting from
an improvement in RE depends on the baseline running
velocity itself.
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FIGURE 1 | Oxygen uptake ( ˙
VO2) increases curvilinearly with running velocity. (A) Linear (solid line) and curvilinear (dashed line) regressions to treadmill running data
from 10 high-level male runners (<30-min 10 km) over a wide range of velocities (1.78–5.14 m/s) (Batliner et al., 2018). (B) Batliner et al. (2018) quadratic equation
(dashed line) and the quadratic equation combined with Pugh’s cubic term for overcoming air resistance (solid line), as per Eq. [2]. (C) Based on this cubic Eq. [2]
(black line), a 10% improvement in running economy (RE; gray line) allows for percent improvements in running velocity which depend on running velocity itself. At
slower running velocities (∼<3.0 m/s), ˙
VO2increases gradually with increases in running velocity, and, as a result at 2.5 m/s a 10% improvement in RE should
facilitate running 12.6% faster. At faster running velocities, ˙
VO2increases steeply with running velocity and as a result at 5.5 m/s, a 10% improvement in RE should
allow for running only 6.7% faster.
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RUNNING VELOCITY AND POTENTIAL
IMPROVEMENTS IN VELOCITY
Multiple long-term interventions, such as endurance, interval,
resistance, and plyometric training, have been shown to
improve RE (for review: Saunders et al., 2004;Barnes
and Kilding, 2015). Other factors such as the racecourse
elevation profile (e.g., downhill) (Minetti et al., 2002), favorable
meteorological conditions and innovations in footwear can
also improve RE (Hoogkamer et al., 2017). Recently, we
showed that a prototype of the Nike Vaporfly 4%, a shoe
with exceptionally compliant and resilient midsole in which
a stiff carbon-fiber plate is embedded improved RE by an
average of 4%, compared to two well-established racing
shoes (Hoogkamer et al., 2018a). The mechanisms behind
the energy savings have been detailed in Hoogkamer et al.
(2019). How much faster could an athlete wearing these
shoes run, assuming their response is equal the average
response of our group; i.e., a consistent improvement in
RE of 4%?
We quantified the possible improvements in running velocity
using Eqs 1 and 2, by Léger and Mercier (1984) and Batliner et al.
(2018), treadmill data from Black et al. (2018) and Kipp et al.
(2018), and overground running data from Tam et al. (2012).
Figure 2A shows how the improvements in running velocity that
are possible with a 4% improvement in RE depend on running
velocity for each of these studies.
We fit a quadratic equation through the ˙
VO2data of Black
et al. (2018) who studied 14 male and 10 female athletes running
at 10 different velocities from 2.22 to 4.72 m/s at sea level. Then,
we added Pugh’s cubic air resistance term (Eq. 3).
Black et al., 2018 +Pugh’s cubic term:
.
VO2ml/kg/min=0.02724v3+1.9128v2−3.2483v
+ 25.806 (3)
Similarly, we fit a quadratic equation through Kipp et al.’s (2018)
data measured at ∼1600 m altitude for 10 high-level male athletes
at six running velocities ranging from 2.22 to 5.00 m/s and added
Pugh’s cubic air resistance term (Eq. 4).
Kipp et al., 2018 +Pugh’s cubic term:
.
VO2ml/kg/min=0.02724v3+1.7321v2−0.4538v
+ 18.91 (4)
Uniquely, Tam et al. (2012) measured ˙
VO2in 10 elite male
Kenyan athletes (<2:09-h marathon) running overground on a
clay track at ∼2,000 m altitude at four running velocities ranging
from 3.33 to 5.00 m/s. They constrained their regression to
have a linear and a cubic term, without a square term, similar
to Léger and Mercier (1984).Tam et al. (2012) expressed their
metabolic data in net energy cost of transport (J/kg/km) (gross –
upright resting) and then fit a line through the data plotted
against the square of velocity. We repeated this analysis for the
data expressed in ml O2/kg/km, converted this to rate of oxygen
uptake in ml O2/kg/min and then added the reported upright
resting rate of oxygen uptake, to get gross ˙
VO2values at each
velocity (Eq. 5).
Tam et al., 2012:
.
VO2ml/kg/min=0.0537v3+9.8158v+5.7 (5)
The equations with a square term (Eqs. 2–4) all follow a
similar trend and concur closely for running velocities faster
than 4 m/s (Figure 2A). While the cubic term in Eq. 1 is
identical to that in Eqs. 2–4, Eq. 1 predicts fairly consistent
velocity improvements over the presented velocity range (2.5–
6.0 m/s), as opposed to the increasingly smaller percent velocity
improvements predicted with Eqs. 2–4. This indicates that the
square term (which represents the inherent curvilinearity of the
˙
VO2-velocity relation) substantially alters the relation between
baseline running velocity and the possible improvements in
running velocity. This is also demonstrated by the dashed line,
which is based on a linear fit through Batliner et al.’s (2018) data
with Pugh’s cubic air resistance term added.
Linear fit of Batliner et al., 2018 +Pugh’s cubic term:
.
VO2ml/kg/min=0.02724v3+12.2v−1.11 (6)
Interestingly, this line closely resembles the running velocity
improvements predicted using Leger and Mercier’s (1984) and
Tam et al.’s (2012) equations, which do not have a square term.
In short, ignoring the inherent curvilinearity of the ˙
VO2-velocity
relation results in over-prediction of the percent improvements
in velocity at the faster velocities.
It is important to realize that Eqs. 1–4 are used to predict
changes in performance at sea level. If one wants to apply these
equations to predict changes in performance at other altitudes,
Pugh’s cubic air resistance term should be adjusted for the
difference in air density. While second order polynomials are fit
through treadmill ˙
VO2data collected at altitude (∼1600 m for
Batliner et al., 2018 and Kipp et al., 2018), we believe that the
effect of air density on the relation between ˙
VO2and treadmill
running speed is small, since externally it would only affect
the cost of moving the extremities through the air relative to
the torso.
Eqs. 2–4 take into account the inherent curvilinearity of the
˙
VO2-velocity relation, but their coefficients differ slightly. This
is likely due to differences in the subject populations tested
and the experimental setups. One of the major determinants
of the equation coefficients is the velocity range over which
the data were collected. Narrower velocity ranges result in
less pronounced curvilinearity of the ˙
VO2-velocity relation.
A narrower velocity range is what has led many previous studies
to describe the ˙
VO2-velocity relation as linear (Menier and Pugh,
1968;Daniels and Daniels, 1992;Helgerud et al., 2010). Here, we
have utilized the Batliner et al., 2018 equation (Eq. 2) because it
is derived from the widest running velocity range. Interestingly,
even though it was collected over the widest range of velocity, it
has the most conservative inherent curvilinearity term (as seen in
the square term of the equation).
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FIGURE 2 | Predicted percent improvements in running velocity depend on the baseline running velocity. (A) Predicted percent improvements in running velocity vs.
running velocity, based on a 4% improvement in RE, using several equations from the recent scientific literature. The solid green is based on a quadratic fit through
Batliner et al. (2018) data with Pugh’s cubic air resistance term. The green dashed line is based on a linear fit through Batliner et al. (2018) data combined with
Pugh’s cubic air resistance term. The difference between the two green lines highlights the importance of the inherent curvilinearity of the ˙
VO2-velocity relation which
substantially alters the magnitude of percent improvement in velocity. (B) Predicted percent improvements in running velocity vs. running velocity, based on 1 to 4%
improvements in RE, using Eq. [2], which combines the quadratic equation from Batliner et al. (2018) with Pugh’s cubic air resistance term. Beyond the velocity
range of Batliner et al. (2018) (>5.14 m/s) prediction lines are dashed.
IMPLICATIONS FOR RUNNING
PERFORMANCE
Figure 2B depicts the relation between the baseline running
velocity and the percent increases in running velocity possible for
different percent improvements in RE, based on Eq. 2. With an
improvement in RE of 1% (due to training, footwear, nutrition,
tailwind, etc.) a recreational athlete who could typically run at
2.60 m/s (4:30:00 marathon) would be predicted to run their race
1.17% faster, finishing in ∼4:26:53, a 3 min and 7 s improvement.
Alternatively, with the same 1% improvement in RE, an elite
marathoner running at 5.72 m/s (2:03:00 marathon), would be
able to run only 0.65% faster, finishing in 2:02:13, only a 47 s
improvement. A similar trend is apparent for all improvements
in RE (Figure 2B). Generally, for velocities slower than ∼3 m/s,
the percent improvement in velocity are expected to be slightly
greater than the percent improvement in RE. For velocities faster
than ∼3 m/s, percent improvements in velocity are expected to
be less than the percent improvements in RE. At velocities faster
than ∼5.5 m/s (∼2:08 marathon pace), percent improvements
in velocity are expected to be less than 2/3rds of the percent
improvements in RE.
We used this same approach to go back to our 2016
study (Hoogkamer et al., 2016), where we demonstrated that
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lab-measured changes in RE translate to similar changes in
distance running performance, assessed by 3 km time trials. The
metabolic data indicated that adding 100 g mass to each shoe
worsened RE on average by 1.11%, while it slowed 3 km time
trial performance by 0.78%. The discrepancy in those percent
changes can now be explained by the inherent curvilinearity of
the ˙
VO2-velocity relation and the additional curvilinear cost of
overcoming air resistance. Eq. 2 predicts that a 1.11% worsening
in RE at a running velocity of 4.79 m/s (i.e., the average
running velocity during the 3 km time trials) would result in a
0.78% slower time, exactly matching the experimentally observed
average slowing of the time trial performances.
Calculating predicted improvements in running velocity based
on baseline running velocity and percent improvements in RE
based on Eq. 2, requires the non-trivial solving of a third-order
polynomial for running velocity (v). To allow readers to calculate
their own comparisons/predictions, we provide a spreadsheet
that solves the cubic equation (see Supplementary Material).
The spreadsheet predicts marathon, half-marathon, and 10 km
performances based on only four inputs: height, weight, percent
improvement in RE and baseline performance. When using this
calculator, it is important to realize that it provides a general
prediction that does not take into account individual variability in
the ˙
VO2-velocity relation. Furthermore, percent improvements
in RE due to footwear innovations (Hoogkamer et al., 2018a) or
long-term training interventions (Saunders et al., 2004;Barnes
and Kilding, 2015) also differ between individuals. Finally, Pugh’s
cubic air resistance term is dependent on a runner’s projected
frontal area, which can be estimated based on the runner’s height
and body mass. In this paper, we have assumed those to be 1.71 m
and 58 kg, respectively. In the Supplementary Spreadsheet, these
numbers can be adjusted at will.
At the previous world record marathon pace of 5.72 m/s, a 4%
improvement in RE translates to a 2.64% faster running velocity,
allowing a marathon time of 1:59:47. Yet, with the introduction
of a 4% more economical running shoe, the marathon world
record has only been broken by 1.03%. It is important to note
that Dennis Kimetto, the previous holder of the world record has
not competed in the newly developed shoe. The fastest marathon
by Eliud Kipchoge (current marathon world record holder) prior
to adopting the shoes with an average of 4% RE enhancement
was 2:04:00 at Berlin in 2015. According to our calculations,
starting with a 2:04 baseline, a 3% improvement in RE translates
to a 1.97% faster velocity or 2:01:36, almost exactly equal to the
recently set world record. It is unknown how much of a RE
enhancement Kipchoge experiences in the new shoes.
POSSIBLE CONFOUNDING FACTORS
The major assumption in our approach to predict improvements
in running performance based on improvements in RE is that all
other performance related factors remain the same. This might
not always be the case. For example, when RE is improved
through drafting behind other competitors or pacemakers, the
reduced air flow over the skin might negatively affect the
runner’s thermoregulation (less heat convection/evaporation),
which could impair running performance and, at least partly,
counter the gains in RE (Hoogkamer et al., 2018b). Although
small body size provides thermoregulatory advantages (via a
greater surface area to volume ratio) (Joyner et al., 2011),
the aerodynamic drag force per kg body mass is greater for
smaller individuals.
Similarly, when RE is improved by running an overall
downhill course, it can be expected that the repeated eccentric
loading will result in additional muscle damage (Hikida et al.,
1983), which will negatively affect running performance. Muscle
damage is likely to occur in elite marathon runners due to
the distance and fast speeds, but it is not well understood
how RE changes with muscle damage or fatigue. Indeed, there
are several reports of worsening RE during the marathon and
ultra-marathon distance (Petersen et al., 2007;Vernillo et al.,
2017), which might be related to muscle damage, fatigue (Millet
et al., 2011) or substrate utilization shifts (Vernillo et al., 2017).
However, as long as those RE changes during the marathon
are consistent and do not change the curvilinearity of the
˙
VO2-velocity relation, deterioration in RE during a race should
not affect our predictions. Theoretically, running faster per se,
independent of the source of the improvement in RE, might
result in more muscle damage during a race, which would impair
running performance. However, more cushioned running shoes
can be expected to reduce muscle damage. It may also be that
the extensive training of elite marathoners mitigates the muscle
damage common in slower marathoners. Further, some data
suggest that RE differences between shoes might be affected by
fatigue (Vercruyssen et al., 2016).
A potential limitation of our approach is that we do not have
direct measurements of the relation between running velocity and
metabolic energy cost (i.e., W/kg or kcal/min) at elite marathon
pace. If this relation is steeper beyond the tested velocity range,
percent improvements in performance will be even smaller.
Distance runners in shorter races (e.g., half-marathons and
10 km) compete at velocities above their lactate threshold, where
it is not possible to measure RE due to contributions from non-
oxidative sources. It is not completely understood how the total
metabolic demands (oxidative and non-oxidative) change at these
high intensities.
Unlike elite runners, slower runners should have a greater
percent improvement from technological advancements in
footwear. As shown in Figure 1C, at slower speeds, there is a
greater improvement in velocity for a given improvement in RE.
Thus, it is likely that improvements in RE from footwear will
produce a wave of recreational runners setting personal records
(Quealy and Katz, 2018).
FUTURE PERSPECTIVES
Our analysis here focused solely on the oxygen cost of running.
Expressing RE in units of rates of energy utilization (W/kg or
kcals/min/kg) accounts for differences in substrate utilization
and, therefore, in the amount of energy liberated per liter
oxygen. To be most relevant to elite marathoners, future
investigations should quantify how the energy cost of running
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changes during overground running at world-class marathon
velocities on pavement surfaces at sea level.
AUTHOR CONTRIBUTIONS
SK and WH were responsible for conception of the review. SK
drafted the manuscript. RK and WH revised it. RK conceived of
the calculator in the Supplementary Material while SK and WH
developed it. SK, RK, and WH approved the final version of the
manuscript. All authors agreed to be accountable for all aspects
of the work.
ACKNOWLEDGMENTS
We thank Erik K. Johnson for pointing out the existence of
analytical solutions for third order polynomials and Dr. Matthew
I. Black for providing us with the data from his article.
SUPPLEMENTARY MATERIAL
The Supplementary Material for this article can be found
online at: https://www.frontiersin.org/articles/10.3389/fphys.
2019.00079/full#supplementary-material
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Conflict of Interest Statement: RK is a paid consultant to Nike, Inc.
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conflict of interest.
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