Modelling of running performances: comparison of power laws and Endurance Index Henry Vandewalle1, Badrane Zinoubi2, Tarak Driss2 1
Laboratoire de Physiologie, Faculté de Médecine, Université Paris XIII, Bobigny, France
2
CeRSM, EA-2931, UFR STAPS, Université Paris Ouest Nanterre-La Défense, France Oral communication to the14th Annual Conference of The Society of Chinese Scholars on Exercise Physiology and Fitness (SCSEPF) in Macau on 22-23 July 2015
Abstract Objective Power laws have been proposed between running speed (S), time (tlim) and distance (Dlim) for world running records (Kennely 1906): Dlim = k tlim and S = k/t 1- = k tlim - 1 For Péronnet and Thibault (1989), the slope of the relationship between the logarithm of tlim and the fractional utilization of Maximal Aerobic Speed (MAS) is an index of endurance capability: 100 S/MAS = C – E ln(tlim/420)] Where C is a constant close to 100 and E an Endurance Index. E was assumed to be independent of the values of tlim used in its computation. In the present paper, we applied the concepts of power law and Endurance Index to the individual running performance of two exceptionnal runners (Nurmi and Gebreselassie) and 11 physical education students (PES) and we verified that E is independent of the range of tlim. Methodology Péronnet and Thibault assumed that MAS corresponds to the maximal velocity that can be sustained over 7 min (420 s). Therefore MAS was computed from the Dlim-tlim power laws: MAS = k 420 - = S420. The power laws between tlim and S or Dlim normalized to S420 by were computed from the regression between the logarithms of Dtlim/S420, S/S420 and tlim. The relation between E and tlim was calculated for different values of [E = 100(1 – Stlim/S420)/ln(tlim/420) where Stlim = k tlim - 1]. Intercept C of the model of Péronnet and Thibault was computed from two values of running speed corresponding to tlim equal to 14 and 60 min. Results & Discussion The running performances of Nurmi, Gebrselassie and PES could be described by power-law curves. For tlim lower than 15 min, the power-law relationships were: Dlim = 10.24 tlim0.914
(Nurmi); Dlim = 9.03 tlim0.952 (Gebrselassie); Dlim = 15.82 tlim0.781 (PES)
For Gebrselassie, E was equal to 4.90. Intercept C of the model of Péronnet and Thibault was close to 100 for Nurmi and Gebrselassie but not for sujects whose value of is much lower (PES). E was almost independent of the range of tlim for Gebrselassie and Nurmi but largely depended on the range of tlim for PES. Conclusion The present study suggests that power laws can describe the indivudal performances and that E can be used as an estimator of endurance capability provided that it is computed for the same ranges of tlim in all the subjects.
Diapositives of the oral communication in the next pages
Modelling of running performances: comparison of Power Laws and Endurance Index Henry Vandewalle1, Badrane Zinoubi2, Tarak Driss2
1
Laboratoire de Physiologie, Faculté de Médecine, Université Paris XIII, Bobigny 2
CeRSM, E.A. 2931, UFR STAPS, Université Paris Ouest, Nanterre
Corresponding author: H Vandewalle,
[email protected]
Model of Péronnet-Thibault (1989)
Péronnet and Thibault, proposed a model of running performances (from 100 m to marathon) based on bioenergetics, including anaerobic and aerobic metabolisms. In this model, the power of aerobic metabolism decreases according to a logarithmic relation for exhausting running exercises longer than 7 min.
Endurance Index In this model, the slope E of the relationship between the logarithm of tlim and the fractional utilization of Maximal Aerobic Speed (MAS) was proposed as an index of endurance capability:
VO2 average / VO2max = 1 – E ln(tlim)
(for tlim > 7 min)
S/MAS = C – E ln(tlim/ tMAS) Where C is a constant close to 1, and E is an Endurance Index and tMAS the exhaustion time corresponding to MAS.
S/MAS 1.0
E
0.9
0.046
0.8 0.086
0.7 0.6
0.15
0.5
0.20
0.4 0.3
Péronnet-Thibault
0.2
S 0$6= 1 - E ln(tlim / tMAS)
0.23
0.1 0.0 0
2
4
6
8
10
12
tlim / tMAS
14
16
18
20
S/MAS 1.0
Elite runner
0.8 0.6
Untrained 0.4
S/MAS = 1 - E ln(t lim/t MAS) Péronnet-Thibault
0.2 0.0 0.0
0.5
1.0
1.5
2.0
ln(t lim/t MAS)
2.5
3.0
Power Law model In 1906, Kennelly proposed Power laws for the modelling of world record in running:
Dlim = k tlim S = k/t
1-
=k
-1 tlim
where S is running speed, tlim time of the world record and Dlim the running distance.
S/MAS 1.0
g
0.9
0.95
0.8 0.90
0.7 0.6
0.80
0.5
0.70
0.4 0.3
Kennelly
0.2
0.60
S / SMAS = kN(tlim / tMAS)
0.1
g -1
0.0 0
2
4
6
8
10
12
tlim / tMAS
14
16
18
20
S/MAS
almost equal
1.0
g
0.9
E
0.95 0.046
0.8 0.90 0.086
0.7 0.6
0.80 0.15
0.5
0.70 0.20
0.4 0.3
Kennelly Péronnet-Thibault
0.2
0.60 0.23
0.1 0.0 0
2
4
6
8
10
12
tlim / tMAS
14
16
18
20
Either the best fit of running data corresponds to Kennelly’s model (power laws);
or the best fit of running data corresponds to the model
proposed
by
Peronnet
and
Thibault
(Endurance index);
or the best fit of running data corresponds to another model.
If the best fit of running data corresponds to Kennelly’s model
S/MAS Slope E is almost independent of tlim
1.0
g 0.95
0.8
Slope E depends on tlim
0.6 0.4
0.60
g -1
S/MAS = kN(t lim/t MAS)
0.2
Kennelly's model 0.0 0.0
0.5
1.0
1.5
ln(t lim/t MAS)
2.0
2.5
3.0
Objectives of the study “Is it possible to describe the individual running performances with both models for a large range of running distance?”
We applied the concepts of Power Law and Endurance Index to individual performances instead of world running records.
First, we applied the concepts of Power Law and Endurance Index to the performances of two exceptionnal runners (Nurmi and Gebreselassie)
We applied the concepts of Power Law and Endurance Index to individual performances instead of world running records.
First, we applied the concepts of Power Law and Endurance Index to the performances of two exceptionnal runners (Nurmi and Gebreselassie)
Secondly, we applied these concepts to the performances of 11 physical education students (PES). These subjects performed runs upto exhaustion at 3 running speeds on an indoor track.
Results
S
(m.s -1 )
Gebrselassie
7.5 S = 8.978 - 0.367 Ln(t lim)
Péronnet-Thibault
7.0 - 0.0580
S = 9.5867 t lim
Kennelly
6.5
6.0
5.5 0
1000 2000 3000 4000 5000 6000 7000 8000
t lim (s)
Dlim (m) 5000
Gebrsellasie Dlim = 9.03 tlim
Dlim = 10.24 tlim
4000
Nurmi
0.952
Dlim = 15.82 tlim
0.914
0.781
PES
3000 2000 1000 0 0
200
400
tlim (s)
600
800
Maximal Aerobic Speed (MAS) Péronnet and Thibault assumed that MAS corresponds to the maximal velocity that can be sustained over 7 min (420 s). Therefore MAS was computed from the Dlim-tlim power laws: MAS = k 420 - 1 = S420.
MAS Nurmi = 6.135 m.s-1 MAS Gebrselassie = 6.75 m.s-1 MAS PES = 4.20 m.s-1
S / S420 S 1.3
S/S420 = 1.34 tlim 1.2
S/S420 = 1.67 tlim S/S420 = 3.77 tlim
1.1
- 0.0479 - 0.086 - 0.0219
1.0
Gebrsellasie Nurmi
0.9
PES 0.8 140
280
420
560
tlim (s)
700
840
Effects of tlim on slope E (Péronnet-Thibault’s model) The relation between E and tlim was calculated for different values of : Stlim = k tlim
- 1
E = (1 – Stlim/S420)/ln(tlim/420) &ZDVDVVXPHGWREHHTXDOWR
E 0.40
g 0.60
0.35 0.30 0.25 0.20
0.65
For high values of g and tlim slope E is almost constant
0.70 0.75 0.80
0.15 0.10
0.85 0.90
0.05
0.95
0.00 2
5
8
11
tlim / tMAS
14
17
20
E 0.40
g 0.60
0.35 0.30 0.25 0.20
For low values of g , slope E largely depends on tlim
0.65 0.70 0.75 0.80
0.15 0.10
0.85 0.90
0.05
0.95
0.00 2
5
8
11
tlim / tMAS
14
17
20
Conclusions 1) Both models of Kennelly and Péronnet-Thibault can accurately describe running performance in elite endurance runners. 2) If the best fit corresponds to Kennelly’s model: - exponent is an index of endurance; - slope E of Péronnet-Thibault’s model can also be used as an Endurance index in well trained runners provided that the ranges values of tlim are similar in all the runners.
谢谢
Appendix
Modelling of world records The world records can be accurately described by both models although they correspond to different equations: S/MAS = C – E ln(tlim/tMAS) S =MAS[C – E ln(tlim/tMAS)] = MAS[C + E ln(tMAS) – E ln(tlim)] S = A1 – B1 ln(tlim)
Péronnet-Thibault model
S = k tlim - 1 ln(S) = ln(k) – (1 - ) ln(tlim ln(S) = A2 – B2 ln(tlim) where A1,A2,B1 and B2 are constants.
Kennelly model
E (calculated from 2 values of tlim) 0.35
0.35
1-2
0.30 0.25
E
2-4
E = 0.87 - 0.86 g 2 r = 0.999
0.30 0.25
3-6 0.20 0.15 0.10
0.20
5-10 7-14
0.15
10-20
0.10
tlim1 / tMAS - tlim2 / tMAS
0.05
0.05 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
g Hypothesis: the best model corresponds to Kennelly's model
g
g
1.0
= 1.01 - 1.11 E 2 r = 0.999
0.9
tlim1 / tMAS - tlim2 / tMAS
0.8
1-2 2-4 3-6 5-10
0.7 0.6 0.5
7-14
0.4
10-20
0.05
0.10
0.15
0.20
0.25
E (calculated from 2 values of t ) lim
Hypothesis: the best model corresponds to the model of Péronnet-Thibault
