# Power-Cadence Relationships in Cycling: Building Models from a Limited Number of Data Points
## 2 Materials and Methods  
Twenty-three males (age: 26 \(\pm\) 5 years; height: 178 \(\pm\) 5 cm; body mass: 73 \(\pm\) 11 kg), all of whom regularly engaged in physical activity through various recreational activities (e.g., endurance running, sprinting, basketball, road cycling, martial arts, resistance training, rowing, rugby union, soccer), volunteered to participate. Informed consent was obtained from each participant, and the study was conducted in accordance with the principles of the Declaration of Helsinki. The study was performed as part of the corresponding author's PhD research [40]. All participants were tested on a stationary cycle ergometer (Monark 818E, Stockholm, Sweden) equipped with a strain gauge (200 N, bandwidth 500 Hz) to measure frictional force, and an optical encoder (1969.2 points/m of displacement or 11,815 points/revolution) to measure flywheel displacement [22]. Friction force and displacement signals were sampled at 200 Hz and recorded via a 12-bit analog-to-digital interface card (DAS-8, 12-bit, Keithley Metrabyte, Taunton, MA, USA). Flywheel velocity and acceleration were calculated as the first and second derivatives of displacement, respectively. External torque was computed as the sum of frictional torque (measured via the strain gauge) and the torque required to accelerate the flywheel [41, 42]. The flywheel's moment of inertia (I) was calculated as 0.927 kg\(\cdot\)m\({}^{2}\) (m = 22.5 kg) using the free deceleration method previously described [41]. The strain gauge was calibrated, and the inertia of the flywheel was verified both before and after data collection. All participants completed a standardized warm-up protocol followed by a 5-min rest period. They then performed four 6-s maximal cycling sprints, each separated by 5 min of rest. All sprints were initiated from a stationary start. Three sprints were performed against the combined resistance of the flywheel's inertia and frictional belt forces set at 0.3, 0.7, or 1.1 N\(\cdot\)kg\({}^{-1}\) of body mass. One sprint was performed against the inertia of the flywheel only. The order of sprints was randomized for each participant. Prior to each sprint, participants were instructed to generate maximal acceleration during every pedal cycle while remaining seated and keeping their hands on the drops of the handlebars. Strong verbal encouragement was provided throughout each sprint. Average torque and cadence were calculated over the downstroke phase of each pedal cycle, and average power was derived from these values [42]. Power-cadence pairs from all maximal efforts were pooled for each participant to construct their individual power cadence relationships [5]. To model the individual power-cadence relationship, third-order polynomial regression was applied to the data points considered. The general form of the equation used is:  
\[y(x)=ax+bx^{2}+cx^{3} \tag{1}\]  
where \(y\) is power (W), \(x\) is cadence (RPM), and \(a\), \(b\), and \(c\) are coefficients that define the shape of the curve. As a first step, we examined the distribution of data points across cadences (Figure 1). Figure 1: Heatmaps showing the distribution of data points used to build the power-cadence models for each participant across 5-RPM cadence intervals. (**a**) High-Number (HN) model includes all available data points. (**b**) Moderate-Number (MN) model includes the maximal power value recorded within each 5-RPM interval. (**c**) Low-Number (LN) model includes five data points selected using a residual-based filtering method from key regions of the power–cadence curve. Each row represents one participant; colors indicate the presence or frequency of data points within each interval. In addition to the uneven distribution of experimental data points across cadences (Figure 1a), the cadence-specific power levels varied considerably across pedal cycles and maximal efforts. While some data points appeared to reflect true maximal power output at a given cadence, others were clearly submaximal (Figure 2).