This study presents a novel numerical approach for deriving the optimal body mass exponent in the power-to-mass ratio for performance prediction, determined by the primary resistive forces acting on a cyclist. The approach utilized the power-duration relationship to estimate time-trial performance across five typical elite-level road cyclist profiles, constructed using previously published normative data, over both simplified hypothetical and complex real-world time-trial courses. A schematic overview of the method is presented in Figure 1, and the following subsections describe each step in detail.

As the first step in deriving the power-to-mass ratio resulting in optimized performance prediction, five elite-level cyclist profiles were created according to cyclist typology. These were constructed using previously published data from 144 male World Tour and Pro Continental cyclists, collected during training and competition, over several years (10). These cyclist profiles were designed to represent “typical” riders in the professional peloton based on both morphology and power profile characteristics. The cyclist profiles represented the following rider categories: sprinter, climber, all-rounder, general-classification (GC)-contender, and time trialist. These profiles were intended to reflect the key performance characteristics commonly observed among elite-level road cyclists (Table 1).

The subsequent step in the process of deriving the power-to-mass ratio reflecting cycling performance was to estimate the performance of the five defined cyclist profiles over the analysed race courses. This step was essential for estimating the power required to counteract aerodynamic drag and for calculating the time the cyclist would spend on the total course as well as on each course section. Firstly, simplified performance estimations were carried out over hypothetical flat and uphill courses representing constant inclines over a 10 km distance. For the flat conditions (α=0∘), both windstill and headwind scenarios were modeled. As uphill conditions, moderate (α=2∘) and steep (α=7∘) inclines were examined. Subsequently, complex time-trial courses of two recent Grand Tours were analysed. In addition, comprehensive analyses were performed, aiming to reveal the underlying relationships between the optimal mass exponent of the power-to-mass ratio and such factors as incline, wind velocity, power output and equipment mass.

To obtain the power-duration relationship parameters of the five created typical cyclist profiles, the following critical power model was fitted to the respective power data for each cyclist profile:(1)P(t)=CP+Pan1+t/τ,where P is power output, CP is critical power, Pan is power capacity above CP, t is duration, τ represents the time constant corresponding to the depletion of half of Pan (i.e., CP+Pan/2), and the product of Pan and τ equals the work capacity above CP (i.e., W′).

Neglecting marginal losses such as frictional loss in the drive train and wheel bearings, cycling power output when traveling in a straight line at a constant speed can be expressed as:(2)P=Pgrav+Proll+Pair,where Pgrav, Proll and Pair denote power against gravity, rolling resistance and aerodynamic drag, respectively and P is the propulsive power output generated by the cyclist (16). Assuming constant system mass, road conditions, no inertia and a constant drag area, expressing each component in Equation 2 results in:(3)P=(msysg(sin(α)+Crrcos(α))+0.5CdAρ(v+vwindcos(ϕ))2)v,where msys is the system mass, amounting to the sum of body mass (m) and equipment mass (mequip), g is the gravitational acceleration (i.e., 9.81 m/s2), α denotes the incline of the road, Crr is the rolling resistance coefficient, Cd is the drag coefficient, A is the projected frontal area of the cyclist-bicycle system, ρ is the air density, the v+vwind⋅cos(ϕ) expression is the headwind velocity relative to the cyclist (i.e., relative velocity), where vwind is the velocity of wind, ϕ denotes the angle of wind with the direction of travel and v is the cyclist’s speed of travel (for a more refined formulation refer to Equation A1 in the Appendix). To determine the drag area, the drag coefficient and the projected frontal area of the cyclist-bicycle system were allometrically scaled (CdA=0.0725⋅m0.312), assuming a standard bicycle geometry, as described by Heil (17).

The mathematical model for estimating the speed (v) of the typical cyclist profiles over the hypothetical time-trial courses was derived by substituting Equation 1, into the left-hand side of Equation 3, resulting in:(4)CP+Pan1+svτ=(msysg(sin(α)+Crrcos(α))+0.5CdAρ(v+vwindcos(ϕ))2)v,where s denotes course distance, while CP and Pan values corresponding to each typical cyclist profile are presented in Table 1. Thus, time-trial performance was defined as the speed corresponding to the estimated average power output over the given time-trial course, assuming an even pacing strategy. This equation was solved numerically for speed using the secant method implemented in Python v3.12.5 (Python Software Foundation, Wilmington, DE, USA). Iterations continued until the difference between successive approximations of v met the predetermined convergence criterion |vk+1−vk|<10−8 m/s. Equipment mass (mequip) was set to 6.8 kg, while Crr and ρ were set to 0.005 and 1.225 kg/m³, respectively, for time-trial performance estimations on Grand Tour stage courses.

To demonstrate the proposed method for deriving optimally normalized power output for complex courses, the course profiles of recent Grand Tour individual time trial stages (ITTs) were analyzed, namely stage 7 of the 2024 Giro d’Italia (Foligno—Perugia, 10th May) and stage 21 of the 2024 Tour de France (Monaco—Nice, 21st July). Both stages featured significant climbs, they measured 40.9 and 37.8 km in length, with total elevation gains of 341 and 663 m, and total elevation losses of 107 and 660 m, respectively.

To simplify time-trial performance estimations on these real-world courses, we assumed two-dimensional time-trial courses in a vertical plane, consistent with the methods used in previous studies (18–21). Course profiles were parsed using a custom-made algorithm implemented in Python v3.12.5. This process involved the identification of surface points, i.e., course coordinates, extracted from two-dimensional course profile schematics obtained online (22). The extracted coordinates were interpolated using cubic spline interpolation and filtered using a Savitzky-Golay filter, to generate the virtual course profile models used in the further analyses (23, 24). Each course profile was then divided into distinct sections concerning terrain profile (i.e., flat, uphill, and downhill sections). This process resulted in 11 and 9 sections for the Giro d’Italia and the Tour de France stages, respectively (Figure 2). Section boundaries were defined at points where substantial changes in the course inclination occurred, and attributes for each section, such as distance and elevation gain, were extracted. The duration (t) required to complete each section of the course was determined using the estimated mean speed of each typical cyclist profile over the specific section, based on Equation 4, and assuming an estimated constant mean power output across the entire course. The total finishing time was calculated as the sum of section durations.

When aiming to mass normalize power output for optimized performance prediction (i.e., to define the optimal mass exponent in the power-to-mass ratio), a key challenge is to establish a power metric that reflects performance independently of body mass. The underlying assumption is that, under consistent external conditions and for cycling at a given speed, the value of an optimally normalized power metric should indicate equivalent performance across cyclists of different body masses. In other words, a higher optimally mass-normalized power output should indicate superior road-cycling performance.

To achieve this, the mass-exponent x in the power-to-mass ratio P/mx, was determined iteratively, aiming to eliminate the effect of body size (m), such as that P/mx remains constant across riders of different body masses. This was accomplished by minimizing the absolute slope |a| of the regression line:(5)P/mx=am+b,where b is the y-intercept of the regression line. The mass exponent in Equation 5 fulfilling this criterion was designated as the optimal mass exponent (xopt):(6)xopt=arg minx∈[0,1]|a|,and the slope of the regression line in Equation 6 was expressed as:(7)a=∑i=1n(mi−m¯)(Pnorm,i−P¯norm)∑i=1n(mi−m¯)2,where m¯ and P¯norm denotes the mean body mass and mean optimally normalized power output, respectively, and mi∈Z∩[45,100] (i.e., body mass takes up all integers between 45 and 100 kg). The optimization algorithm iterated Equation 7 over each xj∈[0,1] using increments of 10−5. The algorithm tracked the slope of the regression line and its corresponding body mass exponent across all iterations. If the current |aj| that corresponded to xj was smaller than the current minimal slope corresponding to the stored exponent, this stored exponent was overwritten to xj. After completing all iterations for x, the algorithm returned the value of xj=xopt, which corresponded to the minimal absolute slope of the regression line (Figure 3).

For complex courses that include varying terrain types or fluctuating wind conditions, we introduced the concept of the duration-weighted optimal mass exponent (x¯opt). In practice, any time-trial course can be segmented into a discrete number of arbitrary sections (n), and the estimated performance of the typical cyclist profiles over these sections can be characterized by mean section durations {t¯i}i=1,2,…,n. By determining the optimal mass exponent {xopti}i=1,2,…,n for each section, the overall course-specific exponent was calculated as the duration-weighted arithmetic mean of these section-specific exponents as:(8)x¯opt=∑i=1nxopti⋅t¯i∑i=1nt¯i,where the denominator represents the estimated mean total finishing time of the group of typical cyclists.

By definition, the magnitude of a power metric that accurately reflects performance independently of body mass should remain constant across different body masses at a given speed of travel (see Section 2.5 for details concerning methodology). However, it is crucial to note that the complete mass independence of propulsive power output is technically unattainable due to the inherent non-linearity of the power balance function. As a result, the minimal absolute slope of the regression line is always nonzero. Although this violates the theoretical assumption of perfect mass independence, it does not significantly compromise the method’s accuracy, since the slope of the regression line remains close to zero, resulting in a minimal variation in P/mxopt (see Figure 3). To quantify this variation, the typical error for consecutive calculations of P/mxopt was assessed according to the method described by Hopkins (33). Across the constant arbitrary velocities, within the studied body mass range, the typical error was in the order of 10−3, corresponding to a coefficient of variation of approximately 0.3% for P/mxopt across all investigated courses. It should also be noted that performance estimation based on power output profiles is inherently limited by fluctuations in athletes’ record power outputs, which can vary both throughout a competitive season and within a single race (5, 6, 34).

In this study, we applied a simplified approach for scaling the CdA, similar to the work of Sundström et al. (35), for instance. It is important to emphasize, however, that CdA is influenced by several factors, including wind direction, bicycle geometry and configuration, equipment characteristics (e.g., helmet and shoe aerodynamics), and the cyclist’s posture and positioning on the bike. Therefore, for practical implementation of the presented methods and performance metrics, accurate measurements (36, 37), or refined calculations of CdA, as well as modeling changes in CdA as a function of wind direction and drafting are necessary [for a detailed explanation see Martin et al. (16) and Blocken et al. (30)]. It is believed that this potential discrepancy between the scaled and actual CdA values was the main reason why the estimated average speeds over the modeled ITTs appeared to be considerably slower than the speed of today’s elite-level cyclists over such efforts. For example, the later winner of the race, Tadej Pogačar (UAE Team Emirates) won the analysed ITT of the 2024 Tour de France (i.e., stage 21) with an average speed of 44.5 km/h, demonstrating an almost 4 km/h positive difference compared to the fastest estimated average speed of 40.7 km/h for this course, which would have resulted in a 25th place. Additionally, the normative power profiles used to create the typical cyclist profiles in the current study may no longer reflect the performance capacity of the sport’s top performers, given the rapid advancements in cycling. Since these profiles were derived from averaged historical data, they likely underestimate the capabilities of podium-level athletes.

Another limitation lies in the assumption of a constant power output distribution during the modeled time trials. In this case, power output was not optimally distributed according to terrain profile (18, 20, 27). Instead, a constant mean power output corresponding to the estimated total finishing times was applied across all points of the course for each typical cyclist profile. Additionally, the model did not consider inertia in the direction of travel which may introduce further inaccuracies when analysing courses with numerous turns, or frequent accelerations. As a result of these simplifications, the intermittent expenditure and reconstitution of W′ were also omitted from the model, likely contributing to the relatively low estimated average speeds across the time-trial courses (38).

Further in-field research is needed to validate whether the magnitude of optimally normalized power output more accurately reflects road cycling performance than previously used metrics (e.g., W/kg0.32 and W/kg). Additionally, as the present study focused exclusively on determining x¯opt for male cycling races, it is important to derive and evaluate optimal mass exponents during ITTs for female cyclist profiles to assess potential sex-specific differences.

Despite the various limitations that influenced the results of this study, it remains highly relevant to assess whether the race course profile suits the attributes and abilities of specific cyclists, particularly in the preparation for an upcoming race. Identifying duration-weighted optimal mass exponents of the power-to-mass ratio for complex race courses or key course segments provides valuable insights into the specific demands of a given course in terms of power output relative to body mass. This approach can enhance the interpretation of power output profiles in relation to performance capacity, support the development of targeted training strategies, course profile categorization, and refine cyclist assignment models.