Modeling the Power-Duration Relationship in Professional Cyclists During the Giro d’Italia

Abstract Vinetti, G, Pollastri, L, Lanfranconi, F, Bruseghini, P, Taboni, A, and Ferretti, G. Modeling the power-duration relationship in professional cyclists during the Giro d’Italia. J Strength Cond Res 37(4): 866–871, 2023—Multistage road bicycle races allow the assessment of maximal mean power output (MMP) over a wide spectrum of durations. By modeling the resulting power-duration relationship, the critical power (CP) and the curvature constant (W′) can be calculated and, in the 3-parameter (3-p) model, also the maximal instantaneous power (P0). Our aim is to test the 3-p model for the first time in this context and to compare it with the 2-parameter (2-p) model. A team of 9 male professional cyclists participated in the 2014 Giro d’Italia with a crank-based power meter. The maximal mean power output between 10 seconds and 10 minutes were fitted with 3-p, whereas those between 1 and 10 minutes with the 2- model. The level of significance was set at p < 0.05. 3-p yielded CP 357 ± 29 W, W′ 13.3 ± 4.2 kJ, and P0 1,330 ± 251 W with a SEE of 10 ± 5 W, 3.0 ± 1.7 kJ, and 507 ± 528 W, respectively. 2-p yielded a CP and W′ slightly higher (+4 ± 2 W) and lower (−2.3 ± 1.1 kJ), respectively (p < 0.001 for both). Model predictions were within ±10 W of the 20-minute MMP of time-trial stages. In conclusion, during a single multistage racing event, the 3-p model accurately described the power-duration relationship over a wider MMP range without physiologically relevant differences in CP with respect to 2-p, potentially offering a noninvasive tool to evaluate competitive cyclists at the peak of training.


Introduction
The hyperbolic relationship between power (P) and duration (T lim ) is a well-established framework for human performance modeling (4,22,23). Commercially available power meters greatly expanded the knowledge on the P-T lim relationship in competitive cycling by recording the maximal mean power output (MMP) over different durations and allowing the collection of a large amount of data from training, testing, and competition (14,25). Racing MMPs have been often collected without further modeling attempts, both in single-day events (6,18) and in multistage "Grand Tours" (7,26,30,37,38), with the main aim of quantifying the physical strain imposed by the race. When such modeling occurred, the 2-parameter (2-p) hyperbolic model was generally used, where the power asymptote (critical power [CP]) is the upper limit of the metabolic steady state (8,12), whereas the curvature constant (W9) is the amount of energy available above CP in the severe exercise-intensity domain (i.e., where maximal values of oxygen consumption, intramuscular metabolites, and blood lactate concentration are reached at exhaustion) (4,34).
In experienced cyclists, CP and W9 estimates were comparable between laboratory tests and racing MMPs gathered from multiple national and international competitions (28). Moreover, higher MMP profiles-and a higher CP-were recorded during racing than training (15). Grand Tours are particularly suitable for this approach because they are characterized by numerous and various stages during which prolonged periods of submaximal cycling are interspersed with supramaximal bursts for intermediate and short durations, resulting in a wide-ranging P spectrum (30). However, the simplicity of the 2-p model (the time asymptote is constrained to 0, meaning that it predicts infinite P when T lim 5 0) makes it overestimate the performance below a T lim of 1-2 minutes (14,23,35), which indeed may represent a great portion of the MMP profile expressed in a Grand Tour (30). A solution for this issue is offered by the 3-parameter (3-p) model, where the time asymptote is the third, unconstrained, parameter (k), so that the predicted P for T lim 5 0 takes a finite value (P 0 ), theoretically corresponding to the maximal instantaneous power (20). In so doing, performance with a T lim below 1-2 minutes is no more overestimated, whereas estimations of CP are comparable with that of 2-p (35). This allows 3-p to accurately describe performance not only in the severe but also in the extreme exercise-intensity domain, i.e., where T lim is so low that exhaustion occurs before of oxygen consumption and blood lactate concentration can reach their maximal values (4,35). Although only a few studies tested the 3-p in the extreme intensity domain, in particular with T lim , 60 seconds (27,35), they promisingly showed a well-preserved goodness of fit to experimental data up to a T lim of 20 (35) to 1 second (27). Therefore, 3-p has the potential of encompassing a greater part of the P-T lim spectrum expressed in a Grand Tour with respect to 2-p.
Extracting CP, W9, and P 0 from a single multiday event is attractive, as it can provide insights into the physiological characteristics of professional cyclists at their peak of physical fitness without subjecting them to impractical exhaustive tests near competitions. Therefore, the main aim of this study was to test the hypothesis that the power profile generated by professional cyclists during a multistage road bicycle race are compatible with a hyperbolic P-T lim relationship, in particular the 3-p model. With this aim, we tested both models in the prediction of racing MMPs with longer T lim than those used for curve fitting. Secondary aims were to further test the 3-p model with data points including T lim , 60 seconds and to compare the obtained parameter estimates with the 2-p model and previously published data (7,15,28,31,35).

Experimental Approach to the Problem
Power output data from a professional cycling team competing in the 2014 Giro d'Italia (21 stages in 24 days: 3 time-trial, 6 flat, 7 medium mountain, and 5 mountain stages) partially published in a previous study (26) were retrospectively analyzed. The assumption was that the wide-ranging environmental conditions and team strategies of the Giro d'Italia allowed cyclists to perform a maximal effort particularly for middle and short durations (and thus selected time windows are assumed to reflect T lim ).

Subjects
The study involved a team of 9 professional road cyclists (age: 28 6 5 years, range: 22-34, height: 176 6 6 cm, average body mass throughout the race: 64.5 6 3.3 kg). After being informed of the risks and the benefits, athletes gave signed written informed consent to participate. The study conformed to the Declaration of Helsinki and was approved by the Ethics Committee the University of Milano-Bicocca.

Procedures
All bicycles were equipped with a crank-based power meter (Power2Max, Chemnitz, Germany) with a precision within 62% (16). Before each stage, power meters were calibrated according to manufacturer's recommendations, including the reset of their zero offset. Because of device malfunctions and the drop-out of 1 athlete after stage 17, 162 athlete 3 stage events were available (86%), with an average of 18 stages per athlete (range 14-21). Eight MMPs calculated over 8 predefined durations (10,15,30,60, 300, 600, 1,200, and 1800 seconds) were available for each stage, whereas raw power output time courses were no longer available because of privacy restrictions. For the purposes of this study, we selected the athlete's highest MMPs for every duration (MMP T , where T is the duration in seconds). Six P-T lim points for each athlete (MMP 10 , MMP 15, MMP 30 , MMP 60 , MMP 300 , and MMP 600 ) were retained for the analysis of the P-T lim relationship, as they were within the recommended T lim range (12,23). MMP 1200 and MMP 1800 developed during time-trial stages, where the effort can be assumed steadily maximal, were used to test the models' predictions by means of the Bland-Altman plot.

Statistical Analyses
All 6 P-T lim points were fitted with a 3-p model (20) by means of the nonlinear regression analysis. For comparison, the lowest 3 MMPs (MMP 60 , MMP 300 , and MMP 600 ) were fitted also with a 2-p model, as performed by Quod et al. (28). The general form of the fitted model was the 3-p (20): where k is the time asymptote (i.e., the extent of the shift of the hyperbola along the T lim axis), which allows the curve to cross the P-axis at P 0 ( Figure 1). Therefore, P 0 was calculated by setting T lim 5 0 s and solving for P. The 2-p was derived as a particular case of 3-p, by constraining k 5 0 s, where P 0 cannot be determined because it becomes infinite. Contrary to the laboratory conditions, a random measurement error must be acknowledged not only for T lim (biological variability of endurance time (22), plus the use of predefined time windows which may misestimate real T lim ) but also for P because outdoor conditions and the development of high P both affect the precision of power meter technology (16). Therefore, the geometric mean regression method (36)

Results
The P-T lim relationship and the parameter estimates with their respective SEE of the 2 models are displayed in Figure 1 and Table 1, respectively. CP estimates were significantly lower in 3-p with respect to 2-p by a trivial amount (24 6 2 W or 21.0 6 0.5% p , 0.001, d 5 0.13), because of very small but systematically negative individual differences (range 20.2% to 21.8%), resulting in an almost perfect correlation (r 5 1.00). W9 estimates in the 3-p model were significantly higher than 2-p (2.3 6 1.1 kJ or 21 6 8%, p , 0.001, d 5 0.62), with excellent correlation (r 5 0.98). CP estimates of 2-p and 3-p were significantly related to MMP 1200 (r 5 0.89 and 0.90, respectively) and MMP 1800 (r 5 0.71 for both), being not significantly different between them. Bias 6 95% limits of agreement between 2-p and 3-p predictions and the longer MMPs developed during time-trial stages were, respectively, 2 6 8 W and 0 6 9 W for MMP 1200 , whereas 10 6 38 W and 12 6 38 W for MMP 1800 (Figure 2).

Discussion
The P-T lim relationship obtained in the field from data collected during an extensive and multifaceted road bicycle race such as the Giro d'Italia resulted compatible with the hyperbolic model, both 2-p and 3-p. Short-term performance (10-60 seconds), which is usually analyzed separately (31,32), can be included in the same theoretical framework if the 3-p model is used instead of the 2-p. A strength of this study is having selected MMPs from a large amount of data generated in a relatively short period of time (21 stages in 24 days), which represents a trade-off between homogenous testing conditions and a high probability of catching the "true" best performances. Model predictions highly agree with measured MMP 1200 but overestimate MMP 1800 , a reasonable finding given the environmental and tactical characteristics of the race that limit's the possibility to perform long-lasting maximal cycling bouts without planned, forced, or unexpected slowdowns. The present findings are restricted to male athletes; however, there is high likelihood that the model can be applied also to women's professional multistage races, where relative exercise intensities were found to be greater than for men (29).
A limitation of this study was the lack of access to raw power meter data, which restricted the number of available time windows, in particularly those of 2-3 and 12-15 minutes (23). Nonetheless, concerning the 2-p model, almost the same time windows (1, 4, and 10 minutes) were able to yield similar CP and W9 estimates between racing and laboratory conditions (28) and were recently recommended when assessing CP from the power profile of a cyclist (24). Moreover, the 6 available P-T lim data points were adequate for the 3-p model, in line with previous studies (average 6 data points, range 4-9) (2,3,5,9,10,17,20,35), as well as with the recent recommendation of one sprint effort 10-15 seconds long plus at least 3 maximal efforts between 2 and 15 minutes (14).
The fact that 3-p provides a slightly lower CP with respect to 2p and a higher W9 is a universal finding (2,3,5,9,10,17,20,35), and it has been ascribed to mathematical reasons as demonstrated elsewhere (35). In brief, because the 3-p shifts a portion of the curve in the negative T lim -axis quadrant (Figure 1), the portion of W9 with negative T lim coordinates becomes nonavailable, thus, to fit the same data points, W9 must take a higher value, and this occurs partly at the expense of CP (35). However, the observed CP decrease is more of statistical than practical relevance, given the trivial effect size and the fact that its relative size (1%) is lower than the precision of the power meter (2%). When compared with previous studies, the CP estimated from our data is among the highest values ever reported (15), much higher than that usually reported for noncyclists (34,35) and slightly higher than that estimated from racing data in non-professional-experienced cyclists (28), whereas the opposite trend appears for W9 (Table 2). Under the assumption that the difference between the CP and the maximal aerobic power (MAP) is a constant (1), one might expect that athletes with elevated MAP also have elevated CP. Moreover, assuming a difference between MAP and CP similar to the one reported in a previous study (35), the MAP estimates are comparable with those measured in a similar athletic cohort of professional cyclists (7,31) ( Table 2).
The use of 3-p instead of 2-p model also provides the P 0 estimate, which is a challenging factor to accurately establish. In fact, the local slope of the P-T lim curve at extreme P (Figure 1) implies that a small uncertainty greatly influences the P-axis intercept; as a consequence, P 0 has a higher relative SEE compared with CP and W9 (Table 1). Moreover, we lack external validators such as maximal instantaneous muscular power measurements, using either a force platform (35) or a cycle ergometer (33). Nonetheless, a comparison with previously published data has been  attempted (Table 2). Although it is likely that P 0 is inherently lower than maximal instantaneous muscular power, as demonstrated for untrained subjects (35), the predicted P for T lim 5 1 second (P 1 ) does not underestimate actual measurements in similarélite cyclists cohorts (7,31). Thus, as previously hypothesized (35), the validity of the 3-p model can be extended up to T lim of few seconds, but not near instantaneous. Interestingly, although the absolute values of CP and P 0 are clearly higher than those observed in untrained subjects, the difference (P 0 2 CP) is not and tends to be even lower (Table 2). This could be explained by the fact that the higher relative proportion of type I muscle fiber iń elite road cyclists contributes to the enhancement of CP (19) but decreases the peak power per unit of muscle mass. Of course, whole-body P 0 is still enhanced, thanks to higher quadriceps muscle volume and pennation angle (13), but by a lower extent with respect to CP. This increase of CP "at the expense" of other parameters could be responsible also for the lower W9 with respect to untrained subjects ( Table 2), but the physiological explanation of this difference is less clear, and the role of muscle fiber composition as a determinant of W9 is unproven to date (19).  Since W9 has both anaerobic and aerobic components (34), we interpret these results as a relative reduction of the former with respect to the latter. From a physiological perspective, k remains the most enigmatic parameter of the 3-p model. Starting from the observation that the utilization of anaerobic energy reserves has finite kinetics, Morton introduced this parameter to account for a linear feedback control system on the availability of W9 (21). Therefore, when P is high (and T lim is short), W9 cannot be entirely exploited. In this context, k represents the amount of decrease in the available W9 per every increment of P above CP, as formally demonstrated by Equation 10 of Vinetti et al. (35). It seems to be less negative iń elite cyclists with respect to normal subjects (35), probably as an effect of the correlation that exists between k and W9 ( Figure 3). Strikingly, this correlation holds also with the W9 calculated from the 2-p model, where k is absent. Therefore, it seems that k is dependent on the intrinsic curvature of experimental data, so that its correlation with W9 is more than a mere statistical artifact, possibly reflecting different athletic phenotypes. We speculate that the higher the maximal anaerobic lactic capacity (i.e., the higher W9), the lower is its "exploitability" at short T lim (i.e., the more negative is k), because the maximal rate of anaerobic lactic energy release (maximal lactic power) is finite. Vice versa, subjects with a low maximal lactic capacity (lower W9) have "less to lose" at short T lim (k is less negative). Moreover, k seems to have a narrow acceptable "physiological" range: the Gaussian distribution of current and previous data shows that the 95 th and fifth percentiles of k estimates are 27 and 221 seconds, respectively ( Figure 3). However, when the quality or quantity of data points with T lim , 60 seconds is scarce, the 3-p model can generate unreliable k and P 0 estimates, with negative repercussions also on the other parameters (35). This is well exemplified in Table 1, subject D, where the only value of k beyond the proposed upper limit of 27 was associated with an exceptionally high SEE of P 0 . In this setting, constraining k into this "physiological" range during the fitting procedure may avoid excessive distortions, although, at least with data of Figure 3, this will not abolish, but only attenuate, the correlation with W9. Future studies should better define the "physiological" range of k, possibly including highly specialized athletes of both sexes.
In conclusion, the 3-p model can efficiently describe MMPs in the 10 seconds-10 minutes duration range collected in a single multistage road bicycle race, with acceptable predictions up to 1 second and 20 minutes. Although the uncertainty regarding P 0 remains high, the time asymptote of the power-duration relationship (k) shows a Gaussian distribution; therefore, it can be potentially constrained to a "normal" range to reduce the variability of P 0 . The resulting parameter estimates offer an objective and noninvasive assessment of cyclists' physical fitness and performance.

Practical Applications
With the methods and limitations described above, it is possible to reasonably assess the power-duration relationship during a multistage racing event. Using such an approach, the resulting parameter estimates are representative of the peak in physical fitness ofélite cyclists (in particular, CP for the maximal sustainable aerobic power, whereas W9 and P 0 for the short-term maximal power output), a condition where multiple time-to-exhaustion tests are intolerable or impractical. These findings may be useful for coaches and sports scientists to obtain a "snapshot" of a cycling team's condition.  (35) to increase statistical power and generalizability.