The original data are those of a previous experimentation (Francescato and Cettolo 2024); however, according to the purpose of the present investigation, a different data analysis was performed.

Five females and 5 males (n = 10), all healthy and moderately active, with mean (± SD) age, stature and body mass of 24.6 ± 3.6 years, 1.73 ± 0.09 m and 73.5 ± 15.1 kg, volunteered to be subjects. Experimental protocol, design, and methods were approved by the Institutional Review Board of the Department of Medicine of the University of Udine (Italy) (#07/2020_IRB issued on March 5th 2020) and conformed to the standards set by the Declaration of Helsinki, except for registration in a database. Volunteers were thoroughly informed about the nature, purpose, and possible risks of the investigation and, thereafter, gave written informed consent to their participation.

In brief, each volunteer repeated the experimental session 5 times at least one day apart. On each experimental session, volunteers performed twice the same 6-min square-wave moderate-intensity exercise bout while continuously pedaling as close as possible to 60 rpm on the ergometer (Corival; Lode B.V., the Netherlands); mechanical power was set to 1.0 W∙kg−1 of body mass when Body Mass Index (BMI) was less than 25.0 kg∙m−2, whereas for a greater BMI, mechanical power was reduced to 0.95 W∙kg−1. The two exercise bouts were preceded by at least 5 min pedaling at 10 W and were separated by no less than 10 min. In agreement with previous experimentations from our workgroup (Francescato and Cettolo 2019; Francescato et al. 2004), the power for the moderate-intensity exercise was chosen according to body mass since we believed that the exercise intensity relative to a precise physiological threshold was not strictly necessary; moreover, this choice allowed avoiding a further visit of each participant to the physiology laboratory for the assessment of lactate threshold or gas exchange threshold, facilitating the recruitment of volunteers for an already demanding experimental protocol.

Respiratory gas collection at the mouth was performed throughout all experimental sessions. The metabolic unit (Metalyzer 3B, Cortex GmbH, Liepzig, Germany) automatically controlled the timings of the protocol and acquired continuously mechanical power, pedaling frequency, heart rate, flow, as well as O2 and CO2 fractions in inspired and expired air. The analyzers were calibrated according to the procedures indicated by the manufacturer. Breath-by-breath gas exchange was calculated by means of the “Expiration-only” algorithm using the acquired flow and gas fraction traces; details of the computations were described previously (Francescato and Cettolo 2024, 2019).

All the original data as well as the used gas exchange calculation software are available from the corresponding author upon request.

All data were analyzed using the R environment (R Core Team 2020).

For each subject and each experimental session, all the obtained oxygen uptake time series were split at t = 16 min, and the times were shifted setting the start of the two moderate-intensity exercise bouts at t = 0. As a result, a total of 10 distinct \(\dot}_\) time series were obtained for each volunteer.

The “1-s-bins” procedure was applied on all the time series to obtain evenly spaced values at 1-s time intervals (starting from t = 0 s), where the new time points were paired with a clone (i.e., a copy) of the \(\dot}_\) value of the closest native time point, allowing obtaining 10 distinct uniformly spaced time series for each volunteer (Fig. 1, upper panels).

Oxygen uptake data of one volunteer during the first bout of the first experimental session (upper panels) and for all the 10 repetitions assembled together (lower panels). Left panels illustrate the results obtained from the assembling by means of the “stacking” procedure, whereas the right panels illustrate the results of the “1-s-bins” assembling procedure. One repetition included an initial baseline period lasting 3 min while pedaling at 10 W, and a bout of moderate-intensity exercise (55 W for this volunteer) lasting 6 min. Vertical dashed lines correspond to the start of the exercise bout (t = 0 s)

Following the performance order, an increasingly greater number (Nr) of \(\dot}_\) time series were assembled together for each volunteer, assuming the start of the square-wave exercise as t = 0 s. Two assembling procedures were used: (a) the “stacking” procedure, where the native data pertaining to the distinct time series were simply stacked up to the already included data, and (b) the “1-s-bins” procedure, where the evenly spaced distinct time series were averaged over an increasing number of time series (Bringard et al. 2014; Francescato et al. 2014b). For each volunteer and both assembling procedures, 10 assembled time series were obtained, i.e., taking the first repetition alone (Nr = 1), assembling the first two (Nr = 2), assembling the first three (Nr = 3), the first four (Nr = 4), and so on until Nr = 10 (Fig. 1, lower panels).

The kinetic parameters of \(\dot}_\) during the square-wave exercise transition were estimated for all the distinct and all the assembled time series by non-linear regression, using the nls.lm routine, without cleaning any outlier, and evaluating the fit by means of the chi-square (χ2) value and its statistical significance.

The following mono-exponential model was used in all cases:

$$\dot}_\left( } \right) = }_}} + \Delta }\left( }^} - }}}}}} \right)}} \ge }}$$

(2)

The starting values for the time constant (τ) and the time delay (Td) were set to 25 s and 0 s, respectively; baseline signal (Ab) was set to the mean of all the data pertaining to the 3 minutes just before t = 0 s; signal change (ΔA) was set to the difference between the corresponding steady-state signal (mean of all the data pertaining to the 3 min just before t = 360 s) and the baseline one.

The non-linear regression procedure was run 41 times (always applying Eq. 2) on all the \(\dot}_\) time series, excluding each time a 1-s progressively longer time period (ΔTr) from the fitting window, starting from t = 0 s (i.e., ΔTr ∈ [0 s, 40 s]), thus yielding for each ΔTr the estimated values for τ, Td, and ΔA, and their Asymptotic Standard Errors (ASEτ, ASETd, and ASEΔA, respectively). The subsequent analyses were mainly focused on the results obtained with ΔTr = 0 s or ΔTr = 20 s.

The behavior of the ASEτ (or of the ASETd) values, with the same volunteer and same algorithm, was evaluated by non-linear regression as a function of the number of assembled repetitions (Nr, ranging from 1 to 10) according to the following equation:

where k represents the ASE value extrapolated for Nr = 1.

The Analysis of Variance for repeated measures (2 × 10 ANOVA) was used to detect the significant differences for the kinetic parameters and their corresponding ASE values, with the following Within-Subjects effects: between the two data treatments (Treatment effect) and among the number of repetitions, both distinct (Repetition effect) and assembled (Repeats effect). Helmert post hoc contrast was used to assess the specific differences within the Repetition or Repeats effect.

Finally, the “Coverage of the Confidence Interval” was calculated for τ, and Td, as the percentage responses where the range of values calculated by Expression 1 (i.e., estimated value ± tdf(α) ∙ ASE, for two-tails α probability = 95%) included the “surrogate of the true” value. The latter was assumed to be the corresponding kinetic value estimated after the assembling, with the same procedure, of all the 10 repetitions. Of note, theoretically, for the two-tails α probability = 95%, the ASE values allow getting the width of the Confidence Interval, satisfying its statistical meaning, only if the “coverage” results ≅95%.

Significance level was set at p < 0.05. Summarized values are reported as means ± SD.