Optimal body mass normalization of power output for accurate prediction of estimated cycling performance over complex time-trial courses

Abstract

Introduction: Power profiling is widely used in cycling performance analysis, but both absolute and mass-normalized power outputs have limitations as performance indicators, as they neglect external factors such as terrain, wind, aerodynamic drag, and pacing strategy. To address these limitations, this study introduced a numerical method to quantify how external forces acting on the cyclist influence the conversion of power output into race velocity. Thus, the study aimed to enable accurate prediction of cycling performance based on estimated mean power output over complex time-trial courses.

Methods: Time-trial performances of five elite-level road cyclist profiles-a sprinter, climber, all-rounder, general classification (GC) contender, and a time trialist-were estimated using the power-duration relationship and previously published normative data. These performance estimates were applied to both simplified hypothetical courses and complex real-world time-trial courses. Optimal mass exponents for the power-to-mass ratio were determined based on the estimated average speeds over the respective course sections, cyclist morphology, and external factors such as gradient and wind velocity.

Results: Across two recent Grand Tour individual time-trial courses, stage 21 of the 2024 Tour de France and stage 7 of the 2024 Giro d'Italia, the duration-weighted optimally mass-normalized power output metrics were $\text{W}/{\text{kg}}^{0.6068}$ and $\text{W}/{\text{kg}}^{0.4891}$ , respectively. These metrics accurately predicted the estimated performances of the five defined cyclist profiles ( $R^2=0.99$ for both).

Discussion: The results indicate that the duration-weighted optimal mass exponents for the power-to-mass ratio are course-specific. By deriving optimal mass exponents across various modeled courses and wind conditions, the study was able to precisely quantify the influence of road gradient, headwind speed, and bicycle mass on the conversion of power output relative to body mass into speed. Further research is needed to validate the presented method for determining optimal mass exponents in real-world performance settings.

Keywords: allometric scaling; critical power; numerical methods; performance prediction; power-duration relationship; sports engineering; time-trial performance.

Figure 1

Schematic overview of the developed process for deriving optimally normalized power output metrics, enabling accurate performance prediction on complex individual time trial (ITT) courses.

Figure 2

Schematic representation of the analysed Grand Tour individual time trial (ITT) courses, with course sections identified based on terrain profile. $s_i$ marks the $i$th section of the course.

Figure 3

Illustration of the derived absolute power output ($P$) as a function of body mass ($m$) (orange solid curve), and its transformation into the derived power-to-mass ratio ($P/m^{x_{\mathrm{opt}}}$) (blue dashed curve). Values are shown across a body mass range of 45–100 kg, assuming constant speed ($v=10$ m/s) and incline ($\alpha =3^{\circ }$). The derived metric ($P_{\mathrm{norm}}=4.7\times {10}^{-7}m+25.72$) demonstrates approximate body mass independence across the studied range.

Figure 4

Relationships between the optimal mass exponent of the power-to-mass ratio ($x_{\mathrm{opt}}$) and relative velocity at different inclines (A), and between $x_{\mathrm{opt}}$, power output, and incline (B), for a cyclist with body mass $m=70$ kg.

Figure 5

Derived power-to-mass ratios ($P/m^{{\overline{x}}_{\mathrm{opt}}}$) and total finishing times for the defined typical cyclist profiles over stage 21 of the 2024 Tour de France (A) and stage 7 of the 2024 Giro d’Italia (B).