# Fatigue-Free Force-Velocity and Power-Velocity Profiles for Elite Track Sprint Cyclists: The Influence of Duration, Gear Ratio and Pedalling Rates

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Participants

^{2}> 0.95) in previous tests in the laboratory and who had already competed in track cycling sprint events at international championships were included. All participants used their own bicycle, cycling shoes and pedals during the test.

#### 2.2. Exercise Protocol

#### 2.3. Warm-Up

^{−1}bodyweight), followed by two 6-s maximal sprints on the track (starting from rolling at approximately 40 rpm). This warm-up and each series of sprints were separated by 10 min of cycling at low intensity and 10 min of passive recovery.

#### 2.4. Motoric Test

#### 2.5. Track Sprints

#### 2.6. Data Processing

_{max}= −b·a

^{−1}, optimal cadence PR

_{opt}= −b·(2a)

^{−1}and maximal power output P

_{max}= −b

^{2}(4a)

^{−1}. The best efforts were defined as those that indicated the highest peak performance. To evaluate the validity of the different approaches, the consistency of the profiles with raw data results was checked by P

_{max}≥ P

_{peak}.

#### 2.7. Statistical Analyses

^{2}. Mathematical analysis and statistical tests were processed using IBM SPSS statistics version 24 Software for Windows (SPSS Inc., Chicago, IL, USA) and Office Excel 2016 (Microsoft Corporation, Redmond, WA, USA).

## 3. Results

^{2}calculated with model I and model II are presented in Table 1 for all athletes.

_{max}(1499.54 ± 373.17 W and 1623.84 ± 84; p<0.017, d = −1.711) were statistically significant higher and the slope of the function (a

_{I}= −6.78 ± 1.17 and a

_{II}= −5.24 ± 1.11; p < 0.003, d = −2.401) and PR

_{max}(223.73 ± 27.11 rpm and 264.59 ± 23.17 rpm; p < 0.004; d = −2.427) statistically significant steeper with model I than model II. Both linear regressions produced high coefficients of determination, with R

^{2}amounting to 0.93 0.06 for model I and to 1.00 for model II. Model II thus showed a higher, almost ideal explained variance (p < 0.003, d = −2.427).

## 4. Discussion

^{2}values reported by Gardner et al. [8] and Debraux et al. [9] (R

^{2}< 0.984) support the possibility of a fatigue bias present in the data used to generate F/v and P/v profiles.

## 5. Limitations

## 6. Practical Applications

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

a | Slope of the F/v profile |

F/v | Force-velocity |

F | Force |

${\mathrm{F}}_{\mathrm{max}}$ | Theoretical maximal pedal force |

P | Power output |

${\mathrm{P}}_{\mathrm{max}}$ | Maximal power output |

PR | Pedalling rate; cadence |

${\mathrm{PR}}_{\mathrm{max}}$ | Maximal pedalling rate; maximal cadence |

${\mathrm{PR}}_{\mathrm{opt}}$ | Optimal pedalling rate; optimal cadence |

P/v | Power-velocity |

v | Velocity |

rpm | Revolutions per minute |

## References

- Hill, A. The Heat of Shortening and the Dynamic Constants of Muscle. Proc. R. Soc. B. Biol. Sci.
**1938**, 126, 136–195. [Google Scholar] - Jaric, S. Force-velocity Relationship of Muscles Performing Multi-joint Maximal Performance. Int. J. Sports Med.
**2015**, 36, 699–704. [Google Scholar] [PubMed] - Schleichardt, A.; Badura, M.; Lehmann, F.; Ueberschär, O. Comparison of force-velocity profiles of the leg-extensors for elite athletes in the throwing events relating to gender, age and event. Sports Biomech.
**2019**, 20, 720–736. [Google Scholar] [CrossRef] [PubMed] - Seck, D.; Vandewalle, H.; Decrops, N.; Monod, H. Maximal power and torque-velocity relationship on a cycle ergometer during accerlation phase of a single all-out exercise. Eur. J. Appl. Physiol.
**1995**, 70, 161–168. [Google Scholar] [CrossRef] - Arsac, L.M.; Belli, A.; Lacour, J.R. Muscle function during brief maximal exercise: Accurate measurements on a friction-loaded cycle ergometer. Eur. J. Appl. Physiol.
**1996**, 74, 100–106. [Google Scholar] [CrossRef] - Dorel, S.; Hautier, C.A.; Rambaud, O.; Rouffet, D.; Van Praagh, E.L. Torque and power–velocity relationships in cycling: Relevance to track sprint performance in world-class cyclists. Int. J. Sports Med.
**2005**, 26, 739–746. [Google Scholar] [CrossRef] - Abbiss, C.R.; Peiffer, J.J.; Laursen, P. Optimal cadence selection during cycling. Int. J. Sports Med.
**2009**, 10, 1–15. [Google Scholar] - Gardner, A.S.; Martin, J.C.; Martin, D.T.; Barras, M.; Jenkins, D.G. Maximal torque- and power-pedalling rate relationships for elite sprint cyclists in laboratory and field tests. Eur. J. Appl. Physiol.
**2007**, 101, 287–292. [Google Scholar] [CrossRef] - Debraux, P.; Manolova, A.V.; Soudain-Pineau, M.; Hourde, C.; Bertucci, W. Maximal torque and power pedaling rate relationships for high level BMX riders in field tests. J. Sci. Cycl.
**2013**, 2, 51–57. [Google Scholar] - Sargeant, A.J.; Hoinville, E.; Young, A. Maximal leg force and power output during short-term dynamic exercise. J. Appl. Physiol.
**1981**, 51, 1175–1182. [Google Scholar] [CrossRef] - McCartney, N.; Heigenhauser, G.J.; Jones, N.L. Power output and fatigue of human muscle in maximal cycling exercise. J. Appl. Physiol.
**1983**, 55, 218–224. [Google Scholar] [CrossRef] - Baron, R.; Bachl, N.; Petschnig, R.; Tschan, H.; Smekal, G.; Pokan, R. Measurement of maximal power output in isokinetic and non-isokinetic cycling. A comparison of two methods. Int. J. Sports Med.
**1999**, 20, 532–537. [Google Scholar] [CrossRef] - Hirntzy, F.; Belli, A.; Grappe, F.; Rouillon, J.-D. Optimal pedalling velocity characteristics during maximal and submaximal cycling in humans. Eur. J. Appl. Physiol.
**1999**, 79, 426–432. [Google Scholar] [CrossRef] - Bertucci, W.T.; Grappe, F. Differences between sprint tests under laboratory and actual cycling conditions. J. Sports Med. Phys. Fitness.
**2005**, 45, 277–283. [Google Scholar] - McDaniel, J.; Behjani, N.S.; Elmer, S.J.; Brown, N.A.; Martin, J.C. Joint-specific power-pedaling rate relationships during maximal cycling. J. Appl. Biomech.
**2014**, 30, 423–430. [Google Scholar] [CrossRef] - Rylands, L.; Roberts, S.; Hurst, H. Variability in laboratory versus field testing of peak power, torque and time of peak power production amongst elite BMX cyclists. J. Strength Cond.
**2015**, 29, 2635–2640. [Google Scholar] [CrossRef] - Bozic, P.R.; Bacvarevic, B.B. Force-Velocity Profiles of Elite Athletes Tested on a Cycle Ergometer. Monten. Sports Sci. Med.
**2018**, 7, 59–66. [Google Scholar] [CrossRef] - Rudsits, B.L.; Hopkins, W.G.; Hautier, C.A.; Rouffet, D.M. Force-Velocity test on a stationary cycle ergometer: Methodological recommendations. J. Appl. Physiol.
**2018**, 124, 831–839. [Google Scholar] [CrossRef] - Martin, J.C.; Wagner, B.M.; Cyle, E.F. Inertial load method determines maximal cycling power in a single exercise bout. Med. Sci. Sports Exerc.
**1997**, 29, 1505–1512. [Google Scholar] [CrossRef] - Wackwitz, T.A.; Minahan, C.L.; King, T.; Du Plessis, C.; Andrews, M.H.; Bellinger, P.M. Quantification of maximal power output in well-trained cyclists. J. Sports Sci.
**2021**, 39, 84–90. [Google Scholar] [CrossRef] - Sanchez-Medina, L.; Perez, C.E.; Gonzalez-Badillo, J.J. Importance of the propulsive phase in strength assessment. Int. J. Sports Med.
**2010**, 31, 123–129. [Google Scholar] [CrossRef] - Jovanović, M.; Flanagan, E. Researched applications of velocity based strength training. J. Aust. Strength Cond.
**2014**, 22, 58–69. [Google Scholar] - Mann, J.B.; Ivey, P.A.; Sayers, S.P. Velocity-based training in football. Strength Cond. J.
**2015**, 37, 52–57. [Google Scholar] [CrossRef] - Jaric, S. Two-Load Method for Distinguishing Between Muscle Force, Velocity, and Power-Producing Capacities. Sports Med.
**2016**, 46, 1585–1589. [Google Scholar] [CrossRef] [PubMed] - Sašek, M.; Mirkov, D.M.; Hadžić, V.; Šarabon, N. The Validity of the 2-Point Method for Assessing the Force-Velocity Relationship of the Knee Flexors and Knee Extensors: The Relevance of Distant Force-Velocity Testing. Front. Physiol.
**2022**, 13, 849275. [Google Scholar] [CrossRef] [PubMed] - Dunst, A.K.; Hesse, C.; Feldmann, A.; Holmberg, H.C. A novel approach to determine alactic timespan in the assessment of the maximal lactate accumulation rate in elite track cyclists. 2022; Under Review. [Google Scholar]
- Kordi, M.; Folland, J.; Goodall, S.; Barratt, P.; Howatson, G. Isovelocity vs. Isoinertial Sprint Cycling Tests for Power- and Torque-cadence Relationships. Int. J. Sports Med.
**2019**, 40, 897–902. [Google Scholar] [CrossRef] [PubMed] - Dunst, A.K.; Grüneberger, R.; Holmberg, H.C. Modeling optimal cadence as a function of time during maximal sprint exercises can improve performance by elite track cyclists. Appl. Sci.
**2021**, 11, 12105. [Google Scholar] [CrossRef] - Dunst, A.K.; Grüneberger, R. A Novel Approach of Modelling and Predicting Track Cycling Sprint Performance. Appl. Sci.
**2021**, 11, 12098. [Google Scholar] [CrossRef] - Sargeant, A.J. Human Power Output and Muscle Fatigue. Int. J. Sports Med.
**1994**, 15, 116–121. [Google Scholar] [CrossRef]

**Figure 1.**Comparison of the F/v profiles calculated for athlete A with model I (grey line) or model II (black line) reveals that the former overestimates and underestimates force development at slow and fast pedalling rates, respectively. The values obtained with model I differs from the measured mean pedal force of 176 N at 232 rpm by more than 40%, whereas the corresponding values obtained with model II shows a deviation of <2%.

**Figure 2.**(

**A**–

**C**) The results of linear regression analysis for the main profile parameters of the profiles derived from the best sprint and calculated with model II for the different series driven with different gear ratios. (

**A**) Theoretical maximal mean pedal force F

_{max}; (

**B**) maximal power output P

_{max}and (

**C**) theoretical maximal pedalling rate PR

_{max}.

**Table 1.**Anthropometric data and parameters of the F/v profiles derived from the best effort of series 2 calculated with model I and model II for each participant. The best efforts were defined as those with highest calculated maximal power output.

Model I | Model II | ||||||||
---|---|---|---|---|---|---|---|---|---|

Part. | Age (yrs) | Height (cm) | Bodyweight (kg) | a (N rpm^{−1}) | b (N) | R^{2} | a (N rpm^{−1}) | b (N) | R^{2} |

1 | 29 | 186 | 92 | −8.52 | 1750 | 0.88 | −6.16 | 1578 | 1.00 |

2 | 29 | 178 | 81.6 | −5.75 | 1406 | 0.98 | −5.12 | 1353 | 1.00 |

3 | 25 | 187 | 95 | −8.14 | 1852 | 0.97 | −6.77 | 1736 | 1.00 |

4 | 33 | 182 | 91.8 | −6.10 | 1710 | 0.92 | −7.19 | 1788 | 1.00 |

5 | 22 | 189 | 89 | −6.43 | 1561 | 0.95 | −5.50 | 1491 | 1.00 |

6 | 25 | 179 | 84.7 | −6.98 | 1516 | 0.84 | −4.88 | 1355 | 1.00 |

7 | 29 | 184 | 95 | −8.95 | 1863 | 0.82 | −4.70 | 1539 | 1.00 |

8 | 22 | 182 | 78 | −6.74 | 1637 | 0.94 | −5.91 | 1577 | 1.00 |

9 | 21 | 177 | 74 | −5.37 | 1252 | 0.88 | −4.77 | 1184 | 1.00 |

10 | 23 | 167 | 68 | −5.00 | 1036 | 0.99 | −3.10 | 862 | 1.00 |

11 | 23 | 176 | 77 | −6.70 | 1302 | 0.98 | −4.95 | 1163 | 1.00 |

12 | 20 | 180 | 73.2 | −6.66 | 1206 | 1.00 | −3.84 | 985 | 1.00 |

^{2}: coefficient of determination.

**Table 2.**Parameters of the F/v profiles derived from the best efforts of series 1 and series 2 calculated with model II for each subject. Best efforts were defined as those with highest calculated maximal power output.

Series 1 | Series 2 | |||||||
---|---|---|---|---|---|---|---|---|

Part. | a (N rpm^{−1}) | b (N) | R^{2} | Developm. (m) | a (N rpm^{−1}) | b (N) | R^{2} | Developm. (m) |

1 | −6.16 | 1571.07 | 1.00 | 7.0 | −6.16 | 1578.39 | 1.00 | 8.4 |

2 | −5.17 | 1361.52 | 1.00 | 6.8 | −5.12 | 1352.67 | 1.00 | 8.4 |

3 | −6.77 | 1750.52 | 1.00 | 6.8 | −6.77 | 1736.32 | 1.00 | 8.7 |

4 | −7.16 | 1777.11 | 1.00 | 7.0 | −7.19 | 1787.65 | 1.00 | 8.4 |

5 | −5.56 | 1500.26 | 1.00 | 6.7 | −5.50 | 1490.74 | 1.00 | 7.1 |

6 | −4.89 | 1361.38 | 1.00 | 6.7 | −4.88 | 1355.12 | 1.00 | 8.4 |

7 | −4.74 | 1546.91 | 1.00 | 6.7 | −4.70 | 1539.41 | 1.00 | 7.6 |

8 | −6.01 | 1598.01 | 1.00 | 6.5 | −5.91 | 1577.43 | 1.00 | 8.0 |

9 | −4.77 | 1183.16 | 1.00 | 6.7 | −4.77 | 1183.78 | 1.00 | 7.5 |

11 | −4.94 | 1157.78 | 1.00 | 7.0 | −4.95 | 1163.14 | 1.00 | 7.7 |

12 | −3.83 | 983.24 | 1.00 | 6.5 | −3.84 | 985.18 | 1.00 | 7.8 |

^{2}: coefficient of determination. One subject had to be excluded because of failed measurements in the first series.

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**MDPI and ACS Style**

Dunst, A.K.; Hesse, C.; Ueberschär, O.; Holmberg, H.-C.
Fatigue-Free Force-Velocity and Power-Velocity Profiles for Elite Track Sprint Cyclists: The Influence of Duration, Gear Ratio and Pedalling Rates. *Sports* **2022**, *10*, 130.
https://doi.org/10.3390/sports10090130

**AMA Style**

Dunst AK, Hesse C, Ueberschär O, Holmberg H-C.
Fatigue-Free Force-Velocity and Power-Velocity Profiles for Elite Track Sprint Cyclists: The Influence of Duration, Gear Ratio and Pedalling Rates. *Sports*. 2022; 10(9):130.
https://doi.org/10.3390/sports10090130

**Chicago/Turabian Style**

Dunst, Anna Katharina, Clemens Hesse, Olaf Ueberschär, and Hans-Christer Holmberg.
2022. "Fatigue-Free Force-Velocity and Power-Velocity Profiles for Elite Track Sprint Cyclists: The Influence of Duration, Gear Ratio and Pedalling Rates" *Sports* 10, no. 9: 130.
https://doi.org/10.3390/sports10090130