# Modeling Optimal Cadence as a Function of Time during Maximal Sprint Exercises Can Improve Performance by Elite Track Cyclists

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{opt}is a dynamic aspect of fatigue. It is currently unclear what cadence is optimal for an athlete’s performance in sprint races and how it can be calculated. We examined fatigue-induced changes in optimal cadence during a maximal sprint using a mathematical approach. Nine elite track cyclists completed a 6-s high-frequency pedaling test and a 60-s isokinetic all-out sprint on a bicycle ergometer with continuous monitoring of crank force and cadence. Fatigue-free force-velocity (F/v) and power-velocity (P/v) profiles were derived from both tests. The development of fatigue during the 60-s sprint was assessed by fixing the slope of the fatigue-free F/v profile. Fatigue-induced alterations in PR

_{opt}were determined by non-linear regression analysis using a mono-exponential equation at constant slope. The study revealed that PR

_{opt}at any instant during a 60-s maximal sprint can be estimated accurately using a mono-exponential equation. In an isokinetic mode, a mean PR

_{opt}can be identified that enables the athlete to generate the highest mean power output over the course of the effort. Adding the time domain to the fatigue-free F/v and P/v profiles allows time-dependent cycling power to be modelled independent of cadence.

## 1. Introduction

_{max,}as the F/v intersection of the y-axis) and cadence (PR

_{max}, as the F/v intersection of the x-axis), maximum power output (P

_{max}, as the P/v apex), and optimal cadence (PR

_{opt}, as the pedaling rate corresponding to P

_{max}) [5].

## 2. Methods

^{2}> 0.95) in previous tests and who had experience with sprint time trials at national or international championships were tested.

## 3. Exercise Protocol

^{−1}bodyweight), followed by a 3-s maximal sprint. Each athlete rested passively for 10 min between warm-up and testing.

## 4. Data Processing

- theoretical maximal force ${\mathrm{F}}_{\mathrm{max}}=\mathrm{F}\left(0\right)=\mathrm{b}$,
- theoretical maximal velocity of movement ${\mathrm{PR}}_{\mathrm{max}}=-\mathrm{b}\xb7$a
^{−1}, - optimal cadence ${\mathrm{PR}}_{\mathrm{opt}}=-\mathrm{b}\xb7(2$a)
^{−1,}and - maximum power output ${\mathrm{P}}_{\mathrm{max}}=-$b
^{2}$\xb7(4$a)^{−1}.

^{2}→$\mathbb{R}$ in maximal sprints no longer than ~ 60 s in duration:

_{opt}during the sprint was approximated by an exponential function of time t:

_{mean}(PR

_{opt}) was calculated as the mean power output that could be generated at the calculated mean optimal cadence.

## 5. Statistical Analyses

^{2}. For all tests, a post hoc power analysis was performed to determine the retrospective power of the observed effect based on the sample size at the indicated level of significance. All mathematical analyses 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).

## 6. Results

^{2}are presented in Table 1 for all athletes.

^{2}for this profile was >0.98 for all athletes. The results of the procedure for generating the fatigue-free F/v and P/v profiles are depicted in Figure 1 for one athlete.

_{opt}(t) showed an exponential decrease with time. Table 2 presents the parameters of PR

_{opt}(t) with corresponding model quality R

^{2}, as well as mean power output in the 60-s sprint P

_{mean}and the theoretical mean power output P

_{mean}(PR

_{opt}) for all participants.

^{2}=$\text{}0.97\text{}\pm \text{}0.01$. Figure 3a depicts a representative example of the fitting of the function describing the optimal dynamic pedaling rate to the data collected. The model was tested for correct reproduction of the time-specific optimal cadence. Figure 3b shows the deviation between modelled and measured data for the data points for all nine athletes. With a bias of <1% at a standard deviation of 4%, the overall agreement was found to be satisfactory.

_{mean}was statistically significantly smaller than theoretical mean power output P

_{mean}(PR

_{opt}) calculated on the mean optimal cadence (d = −0.872, p < 0.001). The cadence that maximizes the mean power output as a solution to the non-linear optimization problem was equal to the calculated mean optimal cadence. Figure 4a,b presents an example of the influence of cadence (a) and its optimization (b) on time-dependent power output and the work performed (calculated as the corresponding integral) in a 60-s isokinetic sprint test.

_{mean}and the values for P

_{max}(r = 0.704, p < 0.05), F

_{max}(r = 0.512, p < 0.05), and τ (r = 0.475, p < 0.05). Applying multiple linear regression analysis, the following statistically significant linear model could be utilized to estimate an athlete’s mean power output (R

^{2}= 0.935, R

^{2}

_{adj}= 0.900):

## 7. Discussion

_{max}and F

_{max}characterize an athlete`s fatigue-free level of performance, whereas τ represents his/her resistance to fatigue. With respect to the calculated linear influence of these parameters on P

_{mean}, the fatigue-free level of power appears to exert a greater impact than the time constant of the loss of power induced by fatigue. Taking the resistances acting on an athlete during the different phases of a race into account further supports the proposal that, in races lasting 60 s or less, the fatigue-free level of performance is more important than resistance to fatigue [7]. Furthermore, P

_{max}(which is correlated to a cyclist’s maximal speed [4]) increases P

_{mean}to a greater extent than does F

_{max}(which is associated with the ability to accelerate (ibid.)).

## 8. Limitations

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

F/v | Force-velocity |

F | Force |

P | Power |

PR | Pedaling rate, cadence |

P/v | Power-velocity |

v | Velocity |

rpm | Crank revolutions per minute |

P_{max} | Maximum power output |

P_{max(t)} | Time-dependent maximal power output |

F_{max} | Maximum force |

PR_{max} | Maximum pedaling rate |

P_{mean} | Mean power output |

P_{mean}(PR_{opt}) | Theoretical mean power output at mean optimal cadence |

P_{opt}(t) | Time-dependent power output at mean optimal cadence |

PR_{opt} | Optimal cadence, optimal pedaling rate |

PR_{opt}(t) | Optimal dynamic cadence; time-dependent optimal pedaling rate |

PR_{optI&IIa} | Common optimal cadence of type I and type IIa muscle fibers |

A | Amplitude of the dynamic optimal pedaling rate |

τ | Time constant of the dynamic optimal pedaling rate |

c | Limiting value for the dynamic optimal pedaling rate |

Part. | Participant |

## References

- Ferguson, H.A.; Harnsih, C.; Chase, J.G. Using field based data to model sprint track cycling performance. Sports Med.
**2021**, 7, 1–12. [Google Scholar] [CrossRef] - De Koning, J.J.; Bobbert, M.F.; Foster, C. Determination of optimal pacing strategy in track cycling with an energy flow model. J. Sci. Med. Sport
**1999**, 2, 266–277. [Google Scholar] [CrossRef] - Craig, N.P.; Norton, K.I. Characteristics of track cycling. Sports Med.
**2001**, 31, 457–468. [Google Scholar] [CrossRef] [PubMed] - Douglas, J.; Ross, A.; Martin, J.C. Maximal muscular power: Lessons from sprint cycling. Sports Med. Open
**2021**, 7, 48. [Google Scholar] [CrossRef] - Dorel, S.; Hautier, C.A.; Rambaud, O.; Rouffet, D.; van Praagh, E.; Lacour, J.-R.; Bourdin, M. 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 cadance selection during cycling. Int. J. Sports Med.
**2009**, 10, 1–15. [Google Scholar] - Dunst, A.K. Trends und Perspektiven im Radsport—Der Trend großer Übersetzungen und seine Konsequenz für das physiologische Anforderungsprofil im Bahnradsprint. Leistungssport
**2021**, 5, 34–37. [Google Scholar] - Hill, A.V. The maximum work and mechanical efficiency of human muscles, and their most economical speed. J. Physiol.
**1922**, 56, 19–41. [Google Scholar] [CrossRef] [PubMed] - Jaric, S. Force-velocity relationship of muscles performing multi-joint maximum performance tasks. Int. J. Sports Med.
**2015**, 36, 699–704. [Google Scholar] [CrossRef] [PubMed] - Bertucci, W.; Taiar, R.; Grappe, F. Differences between sprint tests under laboratory and actual cycling conditions. J. Sports Med. Phys. Fit.
**2005**, 45, 277–283. [Google Scholar] - Gardner, A.S.; Martin, J.C.; Martin, D.T.; Barras, M.; Jenkins, D.G. Maximal torque- and power-pedaling rate relationships for elite sprint cyclists in laboratory and field tests. Eur. J. Appl. Physiol.
**2007**, 101, 287–292. [Google Scholar] [CrossRef] [PubMed] - Debraux, P.; Manolova, A.V.; Soudain-Pineau, M.; Hourdé, C.; Bertucci, W.M. Maximal torque and power pedaling rate relationships for high level BMX riders in field tests. J. Sci. Cycl.
**2013**, 2, 51–57. [Google Scholar] - 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. Res.
**2015**, 29, 2635–2640. [Google Scholar] [CrossRef] [PubMed] - Dunst, A.K. Anwendung von Kraft-Geschwindigkeits-Profilen im Bahnradsport. In Kräftiger, Schneller, Ausdauernder—Entwicklung der Muskulären Leistung im Hochleistungstraining; Lehmann, F., Wenzel, U., Sandau, I., Eds.; Meyer&Meyer Verlag: Aachen, Germany, 2020; pp. 113–120. [Google Scholar]
- Martin, J.C.; Davidson, C.J.; Pardvjak, E.R. Understanding sprint-cycling performance: The integration of muscle power, resistance, and modelling. Int. J. Sports Physiol. Perform.
**2007**, 2, 5–21. [Google Scholar] [CrossRef] [PubMed] - Dorel, S. Maximal force-velocity and power-velocity characteristics in cycling: Assessment and relevance. In Biomechanics of Training and Testing; Morin, J.B., Samozino, P., Eds.; Springer: Cham, Switzerland, 2018; pp. 7–31. [Google Scholar]
- Buttelli, O.; Seck, D.; Vandewalle, H.; Jouanin, J.C.; Monod, H. Effect of fatigue on maximal velocity and maximal torque during short exhausting cycling. Eur. J. Appl. Physiol. Occup. Physiol.
**1996**, 73, 175–179. [Google Scholar] [CrossRef] [PubMed] - Bogdanis, G.C.; Papaspyrou, A.; Theos, A.; Maridaki, M. Influence of resistive load on power output and fatigue during intermittent sprint cycling exercise in children. Eur. J. Appl. Physiol.
**2007**, 101, 313–320. [Google Scholar] [CrossRef] - Dunst, A.K.; Hesse, C. Trends und Perspektiven im Radsport—Geschwindigkeitsbasiertes Training in der Praxis; Leistungssport: Münster, Germany, 2022; p. 1. (in print) [Google Scholar]
- Burnley, M.; Jones, A.M. Power-duration relationship: Physiology, fatigue and the limits of human performance. Eur. J. Sport Sci.
**2018**, 18, 1–12. [Google Scholar] [CrossRef] - Sargeant, A.J. Structural and functional determinants of human muscle power. Exp. Physiol.
**2007**, 92, 323–331. [Google Scholar] [CrossRef] - MacIntosh, B.R.; Herzog, W.; Suter, E.; Wiley, J.P.; Sokolosky, J. Human skeletal muscle fibre types and force: Velocity properties. Eur. J. Appl. Physiol. Occup. Physiol.
**1993**, 67, 499–506. [Google Scholar] [CrossRef] [PubMed] - Pette, D.; Staron, R.S. Transitions of muscle fiber phenotypic profiles. Histochem. Cell Biol.
**2001**, 115, 359–372. [Google Scholar] [CrossRef] - Bottinelli, R.; Pellegrino, M.A.; Canepari, M.; Rossi, R.; Reggiani, C. Specific contributions of various muscle fibre types to human muscle performance: An in vitro study. J. Electromyogr. Kinesiol.
**1999**, 9, 87–95. [Google Scholar] [CrossRef] - Bogdanis, G.C.; Nevill, M.E.; Boobis, L.H.; Lakomy, H.K.; Nevill, A.M. Recovery of power output and muscle metabolites following 30 s of maximal sprint cycling in man. J. Physiol.
**1995**, 482, 467–480. [Google Scholar] [CrossRef] - Hultén, B.; Thorstensson, A.; Sjödin, B.; Karlsson, J. Relationship between isometric endurance and fibre types in human leg muscles. Acta Physiol. Scand.
**1975**, 93, 135–138. [Google Scholar] [CrossRef] [PubMed] - Saltin, B.; Henriksson, J.; Nygaard, E.; Andersen, P.; Jansson, E. Fiber types and metabolic potentials of skeletal muscles in sedentary man and endurance runners. Ann. N. Y. Acad. Sci.
**1977**, 301, 3–29. [Google Scholar] [CrossRef] [PubMed] - Monod, H.; Scherrer, J. The work capacity of a synergic muscular group. Ergonomics
**1965**, 8, 329–338. [Google Scholar] [CrossRef] - Sargeant, A.J. Human power output and muscle fatigue. Int. J. Sports Med.
**1994**, 15, 116–121. [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. Respir. Environ. Exerc. Physiol.
**1983**, 55, 218–224. [Google Scholar] [CrossRef] - Beelen, A.; Sargeant, A.J. Effect of fatigue on maximal power output at different contraction velocities in humans. J. Appl. Physiol.
**1991**, 71, 2332–2337. [Google Scholar] [CrossRef] - Laube, W.; Kibittel, W.; Pieper, K.S. Is it possible to estimate the muscle fiber composition in a noninvasive way? In Sport und Medizin Pro und Contra/32. Deutscher Sportärzte-Kongress, München 1990; Zuckschwerdt: München, Germany, 1991; pp. 688–690. [Google Scholar]
- Hautier, C.A.; Linossier, M.T.; Belli, A.; Lacour, J.R.; Arsac, L.M. Optimal velocity for maximal power production in non-isokinetic cycling is related to muscle fibre type composition. Eur. J. Appl. Physiol. Occup. Physiol.
**1996**, 74, 114–118. [Google Scholar] [CrossRef] - Hansen, E.A.; Andersen, J.L.; Nielsen, J.S.; Sjøgaard, G. Muscle fibre type, efficiency, and mechanical optima affect freely chosen pedal rate during cycling. Acta Physiol. Scand.
**2002**, 176, 185–194. [Google Scholar] [CrossRef] - Brisswalter, J.; Hausswirth, C.; Smith, D.; Vercruyssen, F.; Vallier, J.M. Energetically optimal cadence vs. freely-chosen cadence during cycling: Effect of exercise duration. Int. J. Sports Med.
**2000**, 21, 60–64. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kohler, G.; Boutellier, U. The generalized force–velocity relationship explains why the preferred pedaling rate of cyclists exceeds the most efficient one. Eur. J. Appl. Physiol.
**2005**, 94, 188–195. [Google Scholar] [CrossRef] - Annaheim, S.; Boutellier, U.; Kohler, G. The energetically optimal cadence decreases after prolonged cycling exercise. Eur. J. Appl. Physiol.
**2010**, 109, 1103–1110. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bessot, N.; Moussay, S.; Laborde, S.; Gauthier, A.; Sesboüé, B.; Davenne, D. The role of the slope of oxygen consumption and EMG activity on freely chosen pedal rate selection. Eur. J. Appl. Physiol.
**2008**, 103, 195–202. [Google Scholar] [CrossRef] [PubMed] - Henneman, E.; Mendell, L.M. Functional organization of motoneurone pool and its inputs. In Handbook of Physiology, Section 1, The Nervous System, vol. II, Motor Control; Brookhart, J.M., Mountcastle, V.B., Brooks, V.B., Geiger, S.R., Eds.; American Physiological Society: Rockville, MD, USA, 1981; pp. 423–507. [Google Scholar]

**Figure 1.**The fatigue-free force-velocity and power-velocity profiles for a professional track cycling sprinter calculated on the basis of the force and velocity data derived from the first 3 s of acceleration during the motoric and the sprint test. The solid line represents the F/v and the dotted line represents the P/v function. The maximal mean pedal force was 1223 N and the maximal pedaling rate was 277 rpm, resulting in a fatigue-free optimal pedaling rate of 139 rpm. For both profiles, R

^{2}was >0.99.

**Figure 2.**Power output and pedaling rate of an elite male track cyclist during an isokinetic (120 rpm) all-out sprint on an SRM ergometer (sampling rate, 10 Hz).

**Figure 3.**(

**a**) The optimal dynamic cadence as a function of time reflects the increase in fatigue with prolonged cycling. Black points: current optimal cadence calculated for data point i (PR

_{opt,i}) calculated using Equation (4). Grey curve: time-dependent optimal cadence (PR

_{opt}(t)) calculated using Equation (5). The amplitude A was 85 rpm, the time constant τ was 38.20 s, and the limiting value was c = 55.43 rpm with a time delay of 2.98 s. (

**b**) Comparison of measured and modelled data via linear regression analysis.

**Figure 4.**(

**a**) Time-dependent maximal power output (P

_{max}(t)) at dynamic optimal cadence (PR

_{opt}(t)) and power output (P(t)) during a 60-s all-out sprint at a cadence of 120 rpm (PR) on a cycle ergometer. (

**b**) Comparison of the time-dependent power output (P(t)) at a pedaling rate of 120 rpm and the corresponding work performed (W(t)) to optimized power output (P

_{opt}(t)) at a mean optimal pedaling rate of 102 rpm and the corresponding work performed (W

_{opt}(t)) during a 60-s all-out sprint on a cycle ergometer. In fact, free optimization of the mean power output by altering the cadence during the isokinetic test resulted in a calculated mean optimal pedaling rate of 102 rpm. Changing the cadence to 102 rpm would result in a 7% higher mean power output of the athlete in the isokinetic test mode.

**Figure 5.**A set of curves illustrating the time-course of the relationship between the force-velocity and power-velocity curves during a 60-s all-out sprint on a cycle ergometer. The profiles were derived from the data at every fifth second of the sprint test shown in Figure 2.

**Table 1.**Anthropometric data and model parameters of the linear fatigue-free F/v profile with corresponding model quality R

^{2}of all participants.

Part. | Age (yrs) | Height (cm) | Weight (kg) | PR_{opt} (rpm) | F_{max} (N) | a | R^{2} |
---|---|---|---|---|---|---|---|

1 | 20 | 192 | 96.2 | 142.74 | 1459.65 | −5.11 | 1.00 |

2 | 29 | 178 | 81.6 | 144.35 | 1293.20 | −4.48 | 1.00 |

3 | 20 | 190 | 91.4 | 171.68 | 1076.19 | −3.13 | 0.99 |

4 | 25 | 177.5 | 83.1 | 138.71 | 1222.92 | −4.41 | 1.00 |

5 | 21 | 184 | 87.8 | 141.76 | 1426.53 | −5.03 | 0.99 |

6 | 18 | 181 | 80.1 | 160.39 | 1262.81 | −3.94 | 1.00 |

7 | 19 | 186.5 | 92 | 152.63 | 1202.52 | −3.94 | 1.00 |

8 | 18 | 183.5 | 97.9 | 163.07 | 1144.68 | −3.51 | 1.00 |

9 | 29 | 186 | 92.4 | 148.99 | 1313.48 | −4.41 | 0.98 |

**Table 2.**Model parameters of the mono-exponential dynamic optimal cadence PR

_{opt}(t) with corresponding model quality R

^{2}as well as mean power output P

_{mean}and theoretical mean power output P

_{mean}(PR

_{opt}) in the 60-s sprint for all participants.

Part. | P_{mean} (W) | P_{mean}(PR_{opt}) (W) | A | τ | c | R^{2} |
---|---|---|---|---|---|---|

1 | 844.12 | 904.72 | 87.03 | 33.80 | 59.73 | 0.964 |

2 | 716.36 | 764.50 | 95.23 | 23.24 | 66.93 | 0.960 |

3 | 798.82 | 828.65 | 157.53 | 39.14 | 45.00 | 0.956 |

4 | 729.39 | 784.15 | 85.12 | 38.20 | 55.43 | 0.972 |

5 | 732.48 | 781.05 | 95.62 | 21.93 | 63.69 | 0.960 |

6 | 685.92 | 765.41 | 121.79 | 26.13 | 63.00 | 0.987 |

7 | 712.23 | 747.11 | 97.14 | 34.40 | 60.00 | 0.994 |

8 | 639.80 | 674.70 | 123.87 | 18.47 | 70.61 | 0.951 |

9 | 757.23 | 825.44 | 80.43 | 27.28 | 70.39 | 0.965 |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/app112412105

**AMA Style**

Dunst AK, Grüneberger R, Holmberg H-C.
Modeling Optimal Cadence as a Function of Time during Maximal Sprint Exercises Can Improve Performance by Elite Track Cyclists. *Applied Sciences*. 2021; 11(24):12105.
https://doi.org/10.3390/app112412105

**Chicago/Turabian Style**

Dunst, Anna Katharina, René Grüneberger, and Hans-Christer Holmberg.
2021. "Modeling Optimal Cadence as a Function of Time during Maximal Sprint Exercises Can Improve Performance by Elite Track Cyclists" *Applied Sciences* 11, no. 24: 12105.
https://doi.org/10.3390/app112412105