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Modeling: optimal marathon performance
on the basis of physiological factors
MICHAEL J. JOYNER
Department of Anesthesiology, Mayo Clinic and Mayo Foundation, Rochester, Minnesota 55905
JOYNER, MICHAEL J.
Modeling: optimal marathon perfor-
mance on the basis
of
physiological factors.
J. Appl. Physiol.
70(Z): 683-687, 1991.-This paper examines current concepts
concerning “limiting” factors in human endurance perfor-
mance by modeling marathon running times on the basis of
various combinations of previously reported values of maximal
0, uptake (VO,,, ) lactate threshold, and running economy in ,
elite distance runners. The current concept is that vo2,, sets
the upper limit for aerobic metabolism while the blood lactate
threshold is related to the fraction of vo2 max that can be sus-
tained in competitive events greater than -3,000 m. Running
economy then appears to interact with \io, max and blood lac-
tate threshold to determine the actual running speed at lactate
threshold, which is generally a speed similar to (or slightly
slower than) that sustained by individual runners in the mar-
athon. A variety of combinations of these variables from elite
runners results in estimated running times that are signifi-
cantly faster than the current world record (2:06:50). The fast-
est time for the marathon predicted by this model is 15758 in a
hypothetical subject with a VO, 1118X of 84 ml
l
kg-‘. min-‘, a lac-
tate threshold of 85% of VO 2 max, and exceptional running econ-
omy. This analysis suggests that substantial improvements in
marathon performance are “physiologically” possible or that
current concepts regarding limiting factors in endurance run-
ning need additional refinement and empirical testing.
maximal oxygen uptake; lactate threshold; running economy;
human performance
PHYSIOLOGISTS
have long been interested in modeling
optimal human performance in various running events
on the basis of world records (23,26). Although a variety
of approaches to this problem has been used (23, 26),
recent mathematical models of running performance
have been improved by the recognition that maximal 0,
uptake
(90,
,,)
cannot be sustained in competition for
S-10 min (23,26). This approach is supported by experi-
mental data demonstrating that submaximal variables
including the blood lactate threshold and running econ-
omy (0, cost to run a given speed) are powerful predic-
tors of endurance running performance (3-5, 7,
11, 14,
16, 22, 25). Along these lines, this paper attempts to ex-
tend the current models of human distance running per-
formance by considering how
VO,
m8X, blood lactate
threshold, and running economy interact as determi-
nants of performance in the marathon.
The emerging concepts concerning the limits of mar-
athon performance are that
vo2
max sets the upper limit
for aerobic metabolism and that the blood lactate thresh-
old is related to the fraction of
Vo2max
that can be sus-
tained in competitive events of 2-3 h. Running economy
then appears to interact with
Vo2mar
and blood lactate
threshold to determine the actual running speed at lac-
tate threshold, which is generally a speed similar to (or
slightly slower than) that sustained by individual run-
ners in the marathon (14).
With this information as a background, the purpose of
this paper is to estimate an “optimal” human perfor-
mance in the marathon on the basis of the following sim-
ple physiological model
marathon running speed
.
=
vo
2 max (ml
l
kg-l
l
min-‘)
X %802 max
at LT
X
RE [km
l
h-l
l VOW’
(ml
l
kg-l . min-‘)I
where LT is lactate threshold, RE is running economy,
and
VO,
is 0, uptake. This model is then used in conjunc-
tion with a range of well-accepted values for these vari-
ables in elite male distance runners to estimate the physi-
ologically optimal marathon performance.
METHODS
Previously reported values for
VO,
max, lactate thresh-
old, and running economy in highly trained and elite en-
durance athletes were used to establish three estimates
(low, average, and high) for each of the three variables.
These values were used in different combinations to esti-
mate the running speed at lactate threshold.
Maximal O2 Uptake
The
VO
2 max values from Pollock’s study (24) of 19 elite
runners were used. These ranged from 71.3 to 84.4
ml
l
kg-l
l
min-l and averaged 76.9 ml
l
kg-’
l
min-‘. On
the basis of these data, the low, average, and high
vo2 max
values used for predictions of running speed at lactate
threshold were 70,77, and 84 ml. kg-l. min-‘. It is recog-
nized that there have been occasional reports of
Vo2max
values >84 ml
l
kg-‘. min-’ in humans; however, this
value is at or near the upper limit of values usually re-
ported for elite runners in textbooks (1). The 70
ml
l
kg-‘. min-l value used for the lower limit was se-
lected
1)
so that symmetrical changes in
VO,
mru among
the low, average, and high values would occur and 2)
because there have been reports of a marathon world
record holder with a
VO
2 m8X value of ~70 ml
l
kg-’
l
min-l
6, 24).
0161-7567191 $1.50
Copyright 0
1991
the American Physiological Society
683
684
LIMITING FACTORS IN THE MARATHON
TABLE
1.
Predicted
vo2
at lactate threshold
vo2 -, ml
l
kg-’
l
min-’
%VO,, at LT 70 77 84
75 52.5 57.8 63.0
80 56.0 61.6 67.2
85 59.5 65.5 71.4
Values are expressed in ml
l
kg-’
l
min-‘. LT, lactate threshold.
Blood Lactate Threshold
Lactate threshold data from several sources were used
(4, 7,
14, 20, 22, 25).
Numerous criteria, techniques, and
nomenclature systems for the lactate threshold and re-
lated physiological events have been used
(12).
However,
it appears that, regardless of definition, most runners
sustain a pace in the marathon that elicits blood lactate
levels between
2
and 3 mmol/dl(4,14). This may explain
why actual running speed for the marathon is generally
above the onset of plasma lactate accumulation value
(first increase in blood lactate above baseline) used by
Farrell et al. (14) and generally below the onset of blood
lactate accumulation (i.e., 4 mmol/dl) value used by Sjo-
din and Jacobs (25). Additionally, regardless of the no-
menclature system used, it does appear that elite runners
are able to run the marathon at speeds that require
435% of Vo2mm (7, 14, 23-25). On the basis of these
considerations, it seemed reasonable to set the low, aver-
age, and high values for the lactate threshold at 75, 80,
and 85% of
Vo2
max, respectively.
It should be noted that there are anecdotal reports of
elite runners who appear to be able to sustain roughly
90% of their
VO
2 max values during the marathon (5). Fur-
thermore, slower runners may not be able to sustain run-
ning speeds associated with increases in blood lactate
levels much above those observed at rest. This factor
could act to lower the percent
V02mar
utilized to run the
race and further prolong the time required to complete
the distance by such individuals (4,14).
The 0, uptake
(90~)
ml
l
kg-’
l
min-‘) at lactate thresh-
old was then calculated by multiplying the percent
vo 2 max at lactate threshold and VO, max (Table
1) (7, 14,
25). These values then served as estimates of the relative
60,
values that could be sustained for a marathon.
Running Economy
Three running economy curves relating running speed
to
V,Z
were established. Raw data for each of the
12
subjects studied by Conley and Krahenbuhl (G. S. Kra-
henbuhl, personal communication; see Ref. 3, Fig.
1)
were examined. Running economy values from the two
most economical subjects (lowest 0, cost for a given run-
ning speed) were averaged, and a linear regression equa-
tion between running speed and VO, was calculated. Like-
wise, running economy values from the two least econom-
ical subjects (highest 0, cost for a given running speed)
were averaged, and a regression equation was calculated.
Finally, an average linear regression equation between
running speed and
60,
was established using the mean
values of all 12 subjects (Fig.
1).
This allowed estimates
of running economv at faster sneeds. Use of a linear
model appears justified on the basis of the work of Hagen
et al.
(15).
Because the highest speed used in the measurement of
running economy by Conley was 17.7 km/h (which is be-
low the current world record speed of -20 km/h for the
marathon) and because reports of running economy data
at higher speeds are generally anecdotal, individual ex-
amples of running economy in world class athletes were
obtained to support the extrapolation of the Conley data
to higher speeds. These individual values are plotted in
Fig.
1
and represent previously unpublished observations
for best running economy observations made in several
individuals between -19 and -24 km/h (J. T. Daniels,
personal communication). These values and those of
Conley were obtained during brief periods (5-10 min) of
treadmill running, so the effects of wind resistance and
the upward drift in
VO,
that occurs during prolonged ex-
ercise are not considered in the regression equations used
to construct Fig. 1 (10, 17).
Running speed at lactate threshold for each of the
three values of running economy was then estimated us-
ing the
i702
values at lactate threshold (Table 1) and
three regression equations relating running speed and
vo,
(see
RESULTS).
These estimates of running speed
and time were then slowed by - 10% to account for added
effects of the
1)
7-8% reduction in running economy that
would probably occur as a result of wind resistance dur-
ing overground (compared with treadmill) running (10)
and
2)
the 2-3% increase in
VO,
that would occur from
10
min to 2 h of running (2, 6, 8, 17). This resulted in 27
estimates of running speed at lactate threshold (Table 2)
ranging from a combination of the three “lowest” values
to a combination of the three “highest” values for each
variable. Marathon time was estimated by dividing the
marathon distance (42.195 km) by the calculated values
for running speed at lactate threshold and converting to
- 28
';, 26
1
9 24
ti
c3
z
z
Z
3
CY
12- -*'
a . - .
Low
10 I I 1 I I I I I I
1
35 40 45 50 55 60 65 70 75 80 85
i/o, (ml
l
kg-l
. min-
‘1
FIG. 1.
High, average, and low running economy curves plotted
from raw data of Conley and Krahenbuhl (Ref. 3, Fig. 1). Actual data
were collected on treadmill at speeds of 14.48,16.09, and 17.70 km/h. A,
Unpublished observations of best individual running economy values
for running speeds >17.7 km/h observed by Dr. J. T. Daniels in more
than 20 years of testing elite runners. These data suggest that linear
extrapolation of treadmill running economy curves to speeds >17.70
km/h is generally appropriate and that regression equation for high
running economy values used in calculations is accurate for high-speed
running. A discussion of how treadmill and overground running may
1.m ’ l 1 1 1. 1 1
tuner is inciuaea in text.
LIMITING FACTORS IN THE MARATHON
685
TABLE
2.
Estipated marathon running speeds and times
on the basis
of
VO,
lactate threshold and running economy
fro, at LT,
ml
l
kg-’
l
min-’ Low RE
Running Speed, km/h
Avg RE High RE
52.5 14.40 15.18 16.42
(2:55:49) (2:46:47) (2:34:11)
56.0
15.28 16.10 17.35
(2:45:41) (2:37:15) (2:25:55)
57.8
15.74 16.56 17.84
(2:40:51) (2:32:53) (2:21:55)
59.5
61.6
63.0
65.5
67.2
16.16 17.00
18.29
2:36:40) (2:28:55)
2:18:25)
16.70
17.56 18.85
2:31:36)
(2:24:10) 2:14:18)
17.05
17.93 19.23
2:28:29)
(2:21:12) 2:11:39)
17.68 18.58
19.89
2:23:12)
(2:16:16) 2:07:17)
18.11
19.03 20.35
(2:19:48)
(2:13:02) (2:04:24)*
71.4 19.17 20.13 21.46
(2:12:45) (2:05:46)* (1:57:48)*
Values in parentheses (hours: minutes: seconds) represent time to
complete a marathon at estimate of marathon running speed directly
above. All values were obtained using data from Table 1 along with Eqs.
1-3. They were corrected (slowed) by - 10% to account for effects of
wind resistance and upward drift in
Vo2
that occurs with time. For
details see text. * Performances that exceeded current world record.
hours, minutes, and seconds. It is assumed for discussion
purposes that the weather and race course would also be
“ideal.”
RESULTS
The calculations of
vo2
at the lactate threshold are
presented in Table 1. Values ranged from 52.5
ml
l
kg-’
l
min-l, when the lowest estimates of Vop max and
lactate threshold were used, to 61.6 ml
l
kg-‘. min-‘,
when the “average” estimates for
Vo2max
and lactate
threshold were used, and 71.4 ml. kg-l
l
min-‘, when the
highest estimates were used. Three running economy re-
gression equations relating treadmill running speed (RS)
to
vo2
were generated on the basis of the data of Conley
and Krahenbuhl (3) “high” running economy
RS (km/h) =
VO,
(ml
l
kg-‘. min-‘) X 0.2936 + 2.6481
(1
average running economy
RS (km/h) =
vo2
(ml. kg-’
l
min-‘) X 0.2878 + 1.5867
(2
“low” running economy
RS (km/h) =
vo2
(ml. kg-’
l
min-‘) X 0.2779 + 1.2499
(3)
Data from the treadmill running economy regression
curves and predicted values above 17.70 km/h are plotted
in Fig. 1. Also plotted are the previously unpublished indi-
vidual data points collected at faster running speeds in
elite runners. The correlation coefficient for
Eqs.
1-3
was 0.99.
Running speed at lactate threshold during actual mar-
athon running was then estimated for each of the nine
VO, values at lactate threshold in Table 1 by using the
three equations above and correcting the estimates by
10% to account for the effects of wind resistance and
VO,
drift (2,6,10,17). The 27 combinations of running speed
at lactate threshold (3 X 3 X 3) and estimated marathon
time on the basis of these values are presented in Table 2.
Values ranged from 14.40 km/h (estimated marathon
time = 2:55:49) for the three low values to 17.56 km/h
(2:24:10) for the three average values and 21.46 km/h
(1:57:58) when the high values were used.
DISCUSSION
In this paper, a set of reasonable but upper-limit as-
sumptions was used to predict the fastest possible mar-
athon time on the basis of.currently available informa-
tion about the interplay of VO, mar, lactate threshold, and
running economy as “limiting” factors in endurance ex-
ercise performance (4, 5, 11, 14, 15, 20, 22-25). This ap-
proach yielded a predicted “best” marathon time nearly
9 min faster than the present world record, with 3 of the
27 estimates of marathon time being faster than the pres-
ent world record (2:06:50).
On the basis of these estimates, two basic interpreta-
tions of the model presented in this paper seem reason-
able: First, one could argue that substantial improve-
ments in the marathon world record are “physiologi-
cally” possible at this time. Second, one could argue that
the disparity between the present world record and the
predictions in Table 2 suggests that either factors in ad-
dition to the three explored in this paper limit elite run-
ning performance or the linear extrapolation of data col-
lected at slower running speeds in less-gifted athletes is
open to question. In either case, the possible shortcom-
ings of the proposed model must be addressed.
Potential Limits
of
the Model
Genetics.
Little information is available concerning the
genetic factors required to attain the extremely high
vo
2 max?
lactate threshold, and running economy values
needed to run the faster times in Table 2. Although a
large part of championship athletic performance has
been attributed to genetic endowment, there is no infor-
mation about the population frequency of the character-
istics that, in combination with prolonged intense train-
ing, predispose individuals for success in endurance run-
ning. If, for example, the genetic likelihood of a very high
.
vo 2 max, lactate threshold, or running economy value is
1 X 10D3, then the probability of the same individual hav-
ing all three values is 1 X lo-‘! (For discussion see Ref. 1,
p. 291-292, and Ref. 5.)
Are exceptional values
for
more than one variable mutu-
ally exclusive?
It may be that exceptional values for one
variable are mutually exclusive of exceptional values for
another of the factors. Two possibilities come to mind.
First, the ideal marathoner may fail to achieve the ex-
tremely fast predicted times (<2 h) because of an inabil-
ity to increase fat utilization enough to avoid glycogen
depletion or intracellular acidosis during an effort re-
quiring a
VO,
of >70 ml. kg-l
l
min-’ for several hours
(18, 19). Second, it may be that high
VO,
mLII values are
686
LIMITING FACTORS IN THE MARATHON
incompatible with excellent running economy or lactate
threshold values (5, 22). In the study of Pollock (24), a
group of elite runners successful at 1,500-lQ,OOO m had
higher (-79 ml
l
kg-’
l
min-l) values for
Vozmax
than
elite marathon runners (~74 ml. kg-’
l
min-l), with the
marathon runners averaging ~2 ml
l
kg-l
l
min-’ of
VO,
less than the distance runners at 19.3 km/h. In summary,
on the basis of currently available information, it is not
possible to determine whether exceptional values for one
variable exclude exceptional values for another variable.
.
Running economy.
Although the assumptions about
vo 2 max and lactate threshold are well documented in the
literature, less is known about running economy. It must
be emphasized that the
VO,
vs. running speed regression
lines in Fig. 1 are based on treadmill running data col-
lected during lo-min trials at speeds of 14.48, 16.09, and
17.70 km/h (241,268,295 m/min, and 9,10, and 11 mph,
respectively) (3). Figure 1 and the equations used to cal-
culate running speed at lactate threshold are based on
the assumption that running economy continues to in-
crease in a linear manner at speeds ~17.70 km/h (15).
This assumption, although supported by limited data
(J. T. Daniels, unpublished observations; Refs. 21,24), is
tenuous because of the lack of systematically collected
running economy data on large numbers of elite subjects
at running speeds between 18.0 and 22.5 km/h (300-375
m/min, 12-14 mph).
In this context, it was assumed that two factors would
distort the running economy curves during faster run-
ning for *periods >lO min. First, there is a slow upward
drift in
VO,
during prolonged exercise (2, l7), while the
running economy data used in this model were collected
during 5- to lo-min exercise bouts. The experimental ob-
servations suggest that
Vo2
during cycling or running at
70% of maximum increases ~0.1 l/min (6,8) from 10 min
to 2 h of exercise. Such an increase might slow the time
required to complete the marathon by 2-3%. Second, the
best available evidence suggests that the 0, cost of sus-
tained (5-7 min) high-speed overground running in elite
runners is 7-8% higher than for treadmill running, proba-
bly because of the addition of wind resistance (10,ll). It
would therefore seem likely that these two factors might
operate together to slow the ideal runner on the order of
10% during overground running for 2 h. Additionally,
when the estimates of performance are not corrected for
the effects of
VO,
drift and wind resistance, the fastest
predicted time falls to an unrealistically low l:47:13 and 9
of the 27 estimates of performance surpass the current
world record. Finally, this discussion demonstrates the
need for the systematic collection of running economy
data at speeds >300 m/min in elite performers. The ef-
fects of wind resistance and the duration of the exercise
bout on running economy at these speeds should also be
considered.
AdditionaL sites of fatigue.
The traditional concept has
been that metabolic processes are the key determinants
of fatigue during marathon running (18, 19). Although
the large training-induced increases in muscle fiber oxi-
dative enzymes (18, 19) seen across fiber types in the
trained muscle of endurance athletes suggest a high de-
gree of fatigue resistance, there are other potentially im-
portant sites of fatigue causing failure upstream from the
muscle fiber in the neuromuscular apparatus (13). Along
these lines,
1)
Are the “highest threshold” motor units
recruited during fast running? and 2) If recruited, can
they be trained to fire and contract repeatedly for several
hours without fatigue?
Summary
A variety of approaches has been used in previous at-
tempts to model human performance. These approaches
have recently been evaluated in detail, and the newer
models have been improved by the recognition that
.
vo
2 max
cannot be sustained indefinitely (23, 26). How-
ever, in these models, further improvement in world
record marathon running is predicated on increases in
.
vo
.
2 max, because the values for the fraction of sustainable
vo
2 max
and running economy presented are held fairly
constant (23, 26). In the model presented in this paper,
the effects of altering various submaximal variables
(running economy and lactate threshold) known to affect
performance have been evaluated along with changes in
.
vo 2max (3-5, 7, 9,
12,
15,
17,
21, 23). The principal new
concept advanced using this approach is that, on the ba-
sis of a set of well-documented and reasonable assump-
tions, an ideal marathon runner may be able to run sub-
stantially faster than the present world record. The fact
that the current world record (2:06:50) is nearly 9 min
slower than the predicted best time indicates that either
the genetic probabilities against such a performance are
immense or our level of knowledge about the determi-
nants of human performance is inadequate. In either
case, studies of how
VO,
maX, lactate threshold, and (par-
ticularly) running economy interact as possible determi-
nants of performance in elite athletes are needed to pro-
vide new insight into the physiological determinants and
limitations of human performance.
The author thanks Dr. Gary Krahenbuhl, Arizona State University,
for generous use of the raw running economy data (Ref. 3, Fig. 1) and
Dr. Jack Daniels, SUNY Cortland, for the use of previously unpub-
lished and unique running economy data (Fig. 1). Actual reference
times were obtained from
Truck and Field
News. The author also
thanks Janet Beckman for secretarial assistance.
The author was supported by The Upjohn Medical Education Fund,
Medical Student Research Training Grant HL-07479 and Postdoctoral
Training Grant GM-08288 from the National Institutes of Health, and
the Mayo Foundation.
Received 20 February 1990; accepted in final form 11 September 1990.
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A good analogy for VO$_{2}$ max and LT is to think about VO$_{2}$ as the size of the car's engine and LT as the redline or the maximum engine speed.
Note that now only one of these values is below the World Record (2:02:57)
**Blood Lactate Threshold is the exercise intensity at which the blood concentration of lactate and/or lactic acid begins to exponentially increase** (often expressed as a percentage of VO$_{2}$ max)
During power exercises such as sprinting, when the rate of demand for energy is high, glucose is broken down and oxidized to pyruvate, and lactate is then produced from the pyruvate faster than the body can process it, causing lactate concentrations to rise. High lactate concentrations can be toxic to the body and results in an excessively low pH in the bloodstream.
A lot of experts believe one of the biggest factors that will allow runners to improve their times in the future is a coordinated pacing effort (cooperative draft).
This would be a result of teams unifying under the same country or sponsor and then running in a formation. Even if they do it just for a portion of the race this would reduce substantially the overall wind drag and save a lot of time (similarly to what happens in cycling).
Currently the World Record is 2:02:57 and was set by [Dennis Kimetto](https://en.wikipedia.org/wiki/Dennis_Kipruto_Kimetto) in 2014.
On May 5th 2017, Nike organized a marathon to [attempt to beat the 2 hour mark](https://www.wired.com/story/nike-breaking2-marathon-eliud-kipchoge/) (Breaking2 project). The Nike Sports Research Laboratory worked for 3 years with 3 of the best marathon runners on the planet and planned a race under ideal conditions: special shoes to improve running efficiency, flat track without many sharp turns so that athletes wouldn't lose time on curves, ideal temperature and even finely tuned the configuration of the group of athletes to minimize wind drag.
[Eliud Kipchoge](https://en.wikipedia.org/wiki/Eliud_Kipchoge) ended up running the marathon in 2:00:25, only 25 seconds away from breaking the 2 hour mark (due to the controlled conditions of the race the record is unofficial).
If you want to learn more about this experiment watch this video
[](https://www.youtube.com/watch?v=c6FS3D4a_kA)
"The current world record time for men over the distance is 2 hours 1 minute and 39 seconds, set in the Berlin Marathon by Eliud Kipchoge of Kenya on 16 September 2018, an improvement of 1 minute 18 seconds over the previous record also set in the Berlin Marathon by Dennis Kipruto Kimetto" - wikipedia
Marathon records have been dropping consistently over the past years and there's a lot of discussion on whether we are approaching some sort of biological limit (a lot of people believe a sub-2 hour marathon is impossible) or if it's still humanly possible to keep beating records.
In this paper Michael J. Joyner models the marathon running time according to several physiological factors and is able to predict a time of 1:57:58 under limit conditions.
Note that although this might look like an achievable improvement, getting to 1:57:48 means decreasing 4.1% the current world record time! When Usain Bolt smashed the record for the men’s 100-meter, bringing it down from 9.74 sec to 9.58 sec it represented "just" a 1.7% improvement!
In fact there has never been a single 4.1% drop or higher in the history of Marathon records.
We make ATP in a three-step process: Glycolysis, Krebs Cycle, and Electron Transport Chain (ETC). The products of Glycolysis feed into Krebs, which subsequently feeds its products into ETC. When one glucose molecule is broken down completely, a small amount of ATP is made during both Glycolysis and Krebs, but most of the ATP is generated at the end of ETC. The drawback is that ETC is much slower than Glycolysis. ETC is very effective at making enough ATP to sustain low to moderate intensity exercise. When intensity is high we need more ATP that ETC can produce as its maximal output. When the concentration of lactate in the blood starts to climb, our brain senses this and we start to feel nauseous. Within a few minutes we are forced to drop the intensity, ATP demand reduces, Glycolysis is slowed, lactate is cleared from the blood, and all is back to normal.
A plot of lactate concentration vs. percentage of V02 max is produced and the lactate threshold is identified as the point of inflection.
VO$_{2}$ max is the **maximum rate of oxygen consumption measured during incremental exercise**. VO$_{2}$ max is reached when oxygen consumption remains at a steady state despite an increase in workload. VO$_{2}$ max is usually expressed as a relative rate in millilitres of oxygen per kilogram of body mass per minute (e.g., mL/(kg·min)).
***Why is VO$_{2}$ important for running performance?***
Oxygen is used in a cellular process called aerobic respiration to convert biochemical energy from nutrients into adenosine triphosphate (ATP). The chemical energy stored in ATP can then be used to drive processes requiring energy including locomotion. Thus VO$_{2}$ max basically sets maximum value for the rate of energy available for running.
***Measurement***
Accurately measuring VO$_{2}$ max involves a physical effort sufficient in duration and intensity to fully tax the aerobic energy system. In general clinical and athletic testing, this usually involves a graded exercise test (either on a treadmill or on a cycle ergometer) in which exercise intensity is progressively increased while measuring:
1) Ventilation
2) Oxygen and carbon dioxide concentration of the inhaled and exhaled air.
Since pushing the body to VO$_{2}$ max can be dangerous there have been developed several ways to estimate the VO$_{2}$ without reaching the maximum of the respiratory and cardiovascular systems.
The Cooper test is an example:
$$
\text{VO$_{2}$ max}=\frac{d_{12}-504.9}{44.73}
$$
where $d_{12}$ is the distance (in metres) covered in 12 minutes.
There are also several consumer fitness devices (Garmin Forerunner 620, PulseOn) that use similar methods to estimate the VO$_{2}$ max.