Nomogram to predict performance equivalence for distance runners. Mercier D, Léger L, Desjardins M. Track Technique 1986; 94: 3004-3009, ...
Modelling of running performances in elite endurance runners
Nurmi
Zatopek
Väätäinen
Virén
Eastbourne 10/07/2017
Aouita
Gebrselassie
Biomechanical and biochemical models have been proposed: - Hill (1925) - Henry (1954), - Wilkie (1960 and 1980), - Ettema (1966) - Morton (1986), - Ward-Smith (1985 and 1999), - Péronnet and Thibault (1989), - di Prampero et al. (1993).
The biophysical and biochemical model are complex. For example, the model proposed by Péronnet and Thibault is:
Mathematical and empirical models are generally simpler
Mathematical models of the running performances can be easily computed and applied to individual performances: - power laws (Kennelly 1906), - hyperbolic (Hill 1925, Scherrer 1958, Ettema 1966) - logarithmic (Péronnet-Thibault 1989), - exponential decay (Hopkins et al. 1989).
What are the advantages of modelling the performances of endurance races? - Estimation of future performances and running speeds over given distances. - Estimation of Endurance Capability. - Estimation of the speed at maximal lactate steady state. - Estimation of Maximal Aerobic Speed.
Endurance Capability The performances in long distance events depends not only on a runner’s VO2max and running efficiency but also on the ability to utilize a large percentage of VO2max over a prolonged period of time. This ability, which is subject to wide variation amongst runners of similar performance levels, can be described as “endurance capability”. Péronnet and Thibault 1989
Why is it interesting to study eliterunner performances ? The interpretation of the different models assumes that the running data correspond to the maximal performance for each distance. The study of the best performances of world elite runners is interestingEHFDXVH: - the best performances of world elite runners generally correspond to the results of many competitions against other elite runners; - the motivation is probably optimal during these races.
Nomogram of running performances A nomogram (from Greek νόμος nomos, "law" and γραμμή grammē, "line"), also called a nomograph, alignment chart or abaque, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. Wikipedia
Nomogram to predict performance equivalence for distance runners. Mercier D, Léger L, Desjardins M. Track Technique 1986; 94: 3004-3009,
Mercier D, Léger L, Desjardins M. Nomogram to predict performance equivalence for distance runners. Track Technique 1986; 94: 3004-3009,
The line that describes the athlete’s performance at two distances allows one to predict the performance at a third distance Mercier D, Léger L, Desjardins M. Nomogram to predict performance equivalence for distance runners. Track Technique 1986; 94: 3004-3009,
The line that describes the athlete’s performance at two distances is assumed to allows one to predict the performance at a third distance Mercier D, Léger L, Desjardins M. Nomogram to predict performance equivalence for distance runners. Track Technique 1986; 94: 3004-3009,
The line that describes the athlete’s performance at two distances (3000-10000m) cannot predict the performances at 5000 m and marathon of an elite endurance runner (Nurmi). Mercier D, Léger L, Desjardins M. Nomogram to predict performance equivalence for distance runners. Track Technique 1986; 94: 3004-3009,
The line that describes the athlete’s performance at two distances (3000-10000m) cannot predict the performances at marathon of elite endurance runners (Gebrselassie, Virèn, Zatopek, Radcliffe). Mercier D, Léger L, Desjardins M. Nomogram to predict performance equivalence for distance runners. Track Technique 1986; 94: 3004-3009,
The line that describes the athlete’s performance at two distances allows one to predict an index of endurance, obtained by subtracting the value in column B from the value in column A. Mercier D, Léger L, Desjardins M. Nomogram to predict performance equivalence for distance runners. Track Technique 1986; 94: 3004-3009,
The maximal oxygen uptake (VO2max) is predicted from a horizontal line passing by the 3km performance Mercier D, Léger L, Desjardins M. Nomogram to predict performance equivalence for distance runners. Track Technique 1986; 94: 3004-3009,
Hyperbolic model Hill (1925, 1927) Scherrer (1958) Monod & Scherrer (1965) Ettema (1966)
Velocity (yards.s-1) 11 10 9 8 7 6 5 4
Best time (running)
3 2 1 0
0
2
4
6
8
10
time (x 100 s)
12
14
Adapted from ”Muscular movement in man” A.V. Hill, 1927, quoted by Scherrer and Monod 1960
16
Archibald Vivian Hill
Bristol 1886 - Cambridge 1977 Prix Nobel 1922
Hill proposed the following relationships between Velocity, Distance and time: Velocity = S/(B*time) + (R – A)/B Distance = Velocity* time = S/B + time*(R – A)/B S = the energetic capacity or supply R = the energy debit rate during exercise. A = VO2 at rest about 3.5 ml/min/kg (1 MET) B = running economy (VO2 equivalent for a given velocity )
From critical power to critical velocity
Jean Scherrer
Hugues Monod
Wlim (kJ) 24
Quadriceps femoris
18 Biceps brachii
12 6
Triceps brachii 0 0
6
12 18 tlim (min)
24
30
Relationships between exhaustion time (tlim) and the amount of work performed at the end of exercise (Wlim) for local exercises Adapted from Monod and Scherrer, 1965, Ergonomics 8:329-338
In the study by Scherer and Monod, the total work done at the end of an exhausting exercise (Wlim) was linearly related to exhaustion time (tlim): Wlim = a + b*tlim Therefore the relationship between the average power during the exhausting exercise (P) and tlim was hyperbolic: P = Wlim/tlim = (a + b*tlim)/tlim = a/tlim + b tlim = a/(P – b)
Wlim (kJ) 24
C
18
B
A
b
12 6
a
0 0
6
12 18 tlim (min)
24
30
Points A, B and C are on the same ligne: Wlim = a + b . t lim
When P = b tlim = a/(P – b) = a/(b – b) = a/(0) = infinite Therefore b was considered as a power that can be maintained during a very long time and was named critical power.
Wlim (kJ) 24
C
18
B
A
12 6
b = critical power
0 0
6
12 18 tlim (min)
24
30
Points A, B and C are on the same ligne: Wlim = a + b . t lim
Distance (m) 1500 Arne Borg
1000
World records
Taris (1929)
500 0
French records
Swimming 0
500 1000 time (seconds)
1500
Adapted from J. Scherrer 1958 Méd Ed Phys Sport 32: 7-12.
Significance of
parametre a (Y intercept)
Wlim (kJ) 24
C
18
B
A
12
A’ B’
C’
with tourniquet
6 0 0
6
12 18 tlim (min)
24
30
After the article published in English (1965) by Monod and Scherrer, Ettema (1966) applied the critical-power concept to world records in running, swimming, cycling and skating exercises: Dlim = a + b tlim where tlim corresponded to the world record for a given distance (Dlim).
Ettema proposed a linear relationship between Dlim and tlim for world records from 1500 to 10000 m:
Dlim = a + b tlim Slope b was equivalent to critical power and was considered as a critical velocity (SCrit). Parameter “a” was considered as an estimation of maximal anaerobic distance capacity (ADC in meter) for running exercises:
Dlim = ADC + SCrit tlim
D lim (m) 10000 8000 6000 4000
Gebrselassie Virén Zatopek
2000 0 200
400
600
800 1000 1200 1400 1600 1800
t lim (s)
Dlim (m) 10000
2.9%
1.7%
Gebrselassie
8000
Nurmi 6000 0.1%
0.2%
4000 - 0.6%
- 0.4%
2000 0.4%
0.9% 0.7%
0 0
300
600
900
t lim (s)
1200
1500
1800
D lim (m) 5000
4000
3000 Ovett Coe
2000
1000 200
400
600
t lim (s)
800
D lim (m) 20000
15000
10000 Gebrselassie
Zatopek
5000
0 0
500 1000 1500 2000 2500 3000 3500 4000
t lim (s)
Different equations have been proposed for the computation of SCrit: Dlim = SCrit tlim + ADC S = Dlim/tlim = SCrit + ADC / tlim= SCrit + ADC*(1/ tlim) tlim = ADC/(S – SCrit) or (S – SCrit) = ADC/ tlim
-1
S (m.s ) 7.4 1.5 km
7.2 7.0 6.8
Slope = ADC
3 km
6.6 6.4
5 km
6.2 6.0 5.8
SCrit
S = Dlim / tlim= ADC / tlim + SCrit
5.6 5.4 0.000
0.001
0.002
0.003
1/t lim
0.004
0.005
-1
S (m.s ) 7.4
1.5 km
7.2 7.0 3 km
6.8
5 km
6.6 10 km
6.4 6.2
Gebrselassie Aouita Virén Väätäinen Zatopek Nurmi
20 km
6.0 5.8 5.6 5.4 0.000
0.001
0.002
0.003
1/t lim
0.004
0.005
Significance of
Critical speed
[lactate] mmol/L (post exercise) 16 12 8 4 0 95
100
105
110
115
120 125
Running velocity (% Vcrit) Blood lactate at the end of exhaustive running exercises with running velocities expressed as percent of critical velocity Sid-Ali B, Vandewalle H, Chaîr K, Moreaux A, Monod H Arch Int Physiol Biochem Biophys1991, 99:297-301
Heart rate drift -1 (bpm.min ) 16
12 8 4 0
95
100
105
110
Running velocity (% Vcrit)
115
Heart rate drift during constant-velocity running exercises. Running velocities expressed as percent of critical velocity Sid-Ali B, Vandewalle H, Chaîr K, Moreaux A, Monod H Arch Int Physiol Biochem Biophys1991, 99:297-301
Critical velocity (km.h-1)
20
19 18 17 16 15 15
16
17
18
19
20
Maximal lactate steady-state velocity -1 9&KDVVDLQ(km.h )
Relationship between maximal lactate steady-state velocity (adapted from Chassain) and Critical velocity in 8 elite algerian runners Sid-Ali B, Vandewalle H, Chaîr K, Moreaux A, Monod H Arch Int Physiol Biochem Biophys1991, 99:297-301
Power-law model Kennelly (1906)
Scherrer and Monod, 1960, J Physiol (Paris) 52:419-501
“ La relation T = f(t) n’est pas strictement linéaire comme le montre d’ailleurs la figure 2A où les courbes tendent à s’infléchir vers l’abscisse au delà d’une trentaine de minutes.” “The relationship W = f(t) is not perfectly linear as shown on figure 2A, where the curves tend towards abscissa beyond 30 minutes” Scherrer and Monod, 1960, J Physiol (Paris) 52:419-501
In 1906, Kennelly studied the relationship between the world records (tlim) and running distance (Dlim). He proposed power laws of fatigue for the different types of exercise in humans and horses: Dlim = k tlim
g
S = Dlim / tlim = k tlim
g-1
This model was adapted in cycling exercises by GrosseLordemann & Müller (1937) and Tornvall (1963): Work = tlim g Power = Work/ tlim = tlim g - 1
The value of exponent g in power laws is a dimensionless parameter (independent of scaling). Exponent g is an expression of the endurance capability. Indeed, it is likely that the curvature of the tlim-Dlim relationship depends on the decrease in the fraction of maximal aerobic metabolism that can be sustained during long lasting exercises.
Dlim (m) 0.4%
10000
- 0.7%
Gebrselassie
8000
Nurmi
6000 0.04%
0.5%
4000 0.4%
2000
0.2%
0.3%
0.8% - 1.3%
0 0
300
600
900
t lim (s)
1200
1500
1800
The values of parameter k and exponent g in power laws can determined by computing the regressions between the natural logarithms of S, Dlim and tlim: ln(Dlim) = ln(k) + g ln(tlim) ln(S) = ln(k) + (g -1) ln(tlim) with k = exp(ln(k))
For people who are not mathematicians, the use of a nomogram will be easier and more useful.
It is possible to design a nomogram based on the power law model proposed by Kennelly. From two values of distances and times it is possible to estimate the performances over other distances. By estimating the slope (B - A), it is also possible to estimate the exponent g and, consequently, an index of endurance capability
Gebrselassie B-A = 8 - 3,5 = 4.5 g = 0.948 g computed = 9.49
Nurmi B-A = 11.75 - 5.25 = 6.5 g = 0.927 g computed = 9.26
Radcliffe B-A = 11.0 - 6.75 = 4.25 g = 0.954 g computed = 9.50
Virén B-A = 9.2 - 4.3 = 4.9 g = 0.940 g computed = 9.43
Väätäinen B-A = 9.5 - 4.5 = 5 g = 0.943 g computed = 9.44
Aouita B-A = 9.3 - 2,5 = 6.8 g = 0.925 g computed = 9.21
Zatopek B-A = 11.3 - 5.6 = 5.7 g = 0.935 g computed = 9.45
El Guerrouj B-A = 9.7 - 1.8 = 7.9 g = 0.913 g computed = 9.12
Gebrselassie B-A 3000-5000 = 7.5 - 4.0 = 3.5 g 3000-5000 = 0.960 B-A 5000-10000 = 8.1 - 3.0 = 5.1 g 1500-3000 = 9.45 B-A 5000-10000 = 8.0 - 3.6 = 4.4 g 1500-3000 = 9.49
Nurmi B-A 1500-10000 = 11.6 - 5.1 = 6.5 g 1500-10000 = 0.917 B-A 1500-3000 = 12.3 - 5.0 = 7.3 g 1500-3000 = 9.20
Radcliffe B-A = 11.0 - 6.75 = 4.25 g = 0.954 g computed = 9.50 B-A 3000-10000 = 11.2 - 6.2 = 5.0 g 3000-10000 = 0.943
Logarithmic model Péronnet & Thibault(1989)
The fractional utilisation of VO2max generally decreases linearly when time is expressed as natural logarithm: %VO2max = a - b ln(tlim) Therefore, Péronnet and Thibault (1989) proposed the following model: Running Speed / MAS = 100 – E * ln(tlim / 7 min) where - MAS was Maximal Aerobic Speed, - 7 min was the exhaustion time at MAS, - slope E was an index of aerobic endurance.
The Péronnet-Thibault model was similar to the model previously proposed by Gleser & Vogel (1973) for exercises on a cyle-ergometre : ln(tlim) = A *P/MAP + B where P was power MAP was Maximal Aerobic Power A and B were parametres of individual endurance capacity
-1
S (m.s )
Gebrselassie Aouita Virén Väätäinen Zatopek Nurmi
7.2 7.0 6.8 6.6 6.4 6.2 6.0 5.8 5.6 5.4 200
400
600
800 1000
tlim (s)
2000
-1
S (m.s )
Gebrselassie Aouita Virén Väätäinen Zatopek Nurmi
Logarithmic model
7.2
Péronnet-Thibault 1989
7.0 3 km
6.8
5 km
6.6 10 km
6.4 6.2 6.0 5.8 5.6 5.4 0
250
500
750 1000 1250 1500 1750 2000
tlim (s)
S/S420
Logarithmic model
3 km
1.00
Gebrselassie Aouita Virén Väätäinen Zatopek Nurmi
Péronnet-Thibault 1989 5 km
0.95
10 km
0.90 400
800
1200
tlim (s)
1600
2000
Exponential decay Hopkins, Edmund, Hamilton, MacFarlane, Ross Relation between power and endurance for treadmill running of short duration. Ergonomics 1989;32:1565-1571.
Hopkins et al. (1989) have proposed a model for short-duration (10 s-3 min) running exercises on a treadmill with 5 different inclinations (9 to 31 %).
15 km.h
9% (5.14°)
-1
31% (17.2°)
Inclinaison (%) 40
I0
35 30 25 20 15 10
I∞
5 0 0
20
40
60
time (s)
80
100
The model proposed by Hopkins et al. (1989) Ls:
It = I∞ + (I0 - I∞) exp( - tlim/ ) where I∞ Ls the inclination corresponding to infinite time, I0 the inclination corresponding to an exhaustion time equal to zero, It the inclination corresponding to tlim and is a time constant.
The model proposed by Hopkins et al. (1989) is:
It = I∞ + (I0 - I∞) exp( - tlim/ )
This model can be modified for running exercises on a track:
S = S∞ + (S0 -S∞) exp( - tlim/ ) S = S∞ + a exp( - b tlim)
At least, performances over 3 distances are necessary to compute the regressions between speed and time according to the model of Hopkins. When computed from performances over only 3 distances (1.5-3-5 km or 3-5-10 km), the model is very accurate in elite endurance runners.
-1
S (m.s ) 8.2 7.8 7.4
1.5 km
1.5-3-5 km
7.0
3 km 5 km
6.6
10 km
6.2
Aouita Väätäinen
5.8
Nurmi 5.4 0
300
600
900
t lim (s)
1200
1500
1800
-1
S (m.s ) 8.2 7.8 7.4
1.5 km
3-5-10 km
7.0
3 km 5 km
6.6
10 km
6.2
Aouita Väätäinen
5.8
Nurmi
5.4 0
300
600
900
t lim (s)
1200
1500
1800
When computed from performances over 4 distances (1.5-3-5-10 km km), the individual regressions are less accurate: - all the points corresponding to 3000 m are slightly below the individual regression curves; - all the points corresponding to 5000 m are slightly above the individual regression curves.
-1
S (m.s )
Exponential decay (Hopkins)
8.2 7.8 7.4
1.5 km
7.0
3 km 5 km
6.6
10 km
6.2 5.8
Aouita Väätäinen Nurmi
5.4 0
300
600
900
t lim (s)
1200
1500
1800
-1
S (m.s )
Exponential decay (Hopkins)
8.2 7.8 7.4 1.5 km
7.0
1.5-3-5-10 km
3 km 5 km
6.6
10 km
Gebrselassie Virén Zatopek
6.2 5.8 5.4 0
300
600
900
t lim (s)
1200
1500
1800
-1
Average Running Speed during the race (m.s ) 7.0
1.5 km
Mathematical models computed from 1.5-3-5-10 km (empty circles)
3 km 5 km 6.5 10 km
Hill-Scherrer Hopkins 6.0
Kennelly Péronnet-Thibault
21.09 km
5.5
42.19 km
Running performances of H. Gebrselassie 5.0
1000 2000 3000 4000 5000 6000 7000
Duration of the race(s)
