# Mathematical Comparison of Time‐Based and Stroke‐Based Fatigue Models in Elite Sprint Cycling: Convergence Analysis and Practical Applications

## TL;DR

This study compares two models for predicting fatigue in elite sprint cycling, finding that one model is more accurate while the other is more practical for real-world use.

## Contribution

The paper provides a novel mathematical comparison of time-based and stroke-based fatigue models in sprint cycling.

## Key findings

- Both models fit sprint power profiles well (R² > 0.98).
- PASA had lower prediction errors than standard PESA variants.
- Models converged at moderate cadences but diverged at high cadences and during late sprint phases.

## Abstract

Power decline during maximal cycling sprints is commonly modeled using either the Parallel Shift Approach (PASA), representing fatigue as a time‐dependent downward shift of the linear force‐velocity (F‐v) relationship, or the Pedal Stroke‐Based Approach (PESA), which assumes a constant relative power loss per pedal stroke (Δ). This study compared their predictive accuracy, convergence behavior, and practical applicability across sprint duration and cadences. Twelve elite track sprint cyclists (6 female and 6 male) performed 45 s maximal sprints at a fixed cadence of 135 rpm. Both models were calibrated using individual F‐v profiles and evaluated using RMSE and R2
. Model convergence was examined across sprint phases and cadence ranges using empirically derived and optimized parameters. Both models demonstrated excellent fit to individual sprint power profiles (R2
 > 0.98). PASA yielded lower prediction errors (RMSE: 32 ± 11 W) than standard PESA variants (RMSE: 57–60 W, p < 0.001). Optimizing Δ substantially improved PESA performance (35 ± 12 W). Models converged during early sprint phases (0–15 s: 4 ± 3 W difference, p = 0.065) and at moderate cadences (90–130 rpm) but diverged during late phases (45 s: 22 ± 11 W, p = 0.013) and at high cadences (> 150 rpm). PASA and PESA provide comparable predictions under short‐duration, moderate‐cadence conditions but diverge during prolonged sprints and at extreme cadences. PASA offers higher precision for detailed fatigue analysis, whereas PESA represents a computationally efficient alternative for practice. Model selection should be guided by analytical objectives and practical constraints. Further research should prioritize validation under variable‐cadence conditions and refinement of physiological assumptions underlying sprint fatigue modeling.