## 6 Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Part.** & **Pmean (W)** & **Pmean(PRopt) (W)** & **A** & \(\tau\) & **c** & **R\({}^{2}\)** \\ \hline
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 \\ \hline \end{tabular}
\end{table} 
Table 2: Model parameters of the mono-exponential dynamic optimal cadence PRopt(t) with corresponding model quality R\({}^{2}\) as well as mean power output Pmean and theoretical mean power output Pmean(PRopt) in the 60-s sprint for all participants. 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). 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 of all nine athletes. With a bias of \(<\)1% at a standard deviation of 4%, the overall agreement was found to be satisfactory. It can be seen that larger deviations between modelled and measured data occurred mainly in the high-cadence range, which represents the first few seconds of the sprint. Mean power output P\({}_{\text{mean}}\) was statistically significantly smaller than theoretical mean power output P\({}_{\text{mean}}\)(PR\({}_{\text{opt}}\)) 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. There was a strong positive correlation between the average power P\({}_{\text{mean}}\) and the values for P\({}_{\text{max}}\) (r = 0.704, \(p<\) 0.05), F\({}_{\text{max}}\) (r = 0.512, \(p<\) 0.05), and \(\tau\) (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\({}^{2}\)\({}_{\text{adj}}\) = 0.900):  
\[P_{mean}=0.339\cdot P_{max}+0.054\cdot F_{max}+6.064\cdot\tau-68.833\]  
The statistical power for the model quality was calculated to be \(>\)0.99. The reproduction of the quality of prediction stated was thus guaranteed, even with very small samples. The power of the other statistics ranged from 0.26 to 0.77. In order to be able to reproduce all results presented with a probability of at least 80%, we recommend choosing a sample size of \(>\)32. 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\({}_{\text{opt,i}}\)) calculated using Equation (4). Grey curve: time-dependent optimal cadence (PR\({}_{\text{opt}}\)(t)) calculated using Equation (5). The amplitude A was 85 rpm, the time constant \(\tau\) 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.