The algorithm used for the calculation of gas exchange affects the estimation of O2 uptake kinetics at the onset of moderate‐intensity exercise

Abstract At the start of a moderate‐intensity square‐wave exercise, after a short delay, breath‐by‐breath O2 uptake at the mouth is approximated to a mono‐exponential function, whose time constant is considered matched to that of the O2 uptake of the working muscles. We compared the kinetic parameters obtained from the breath‐by‐breath gas exchange data yielded by the ‘Independent‐breath’ algorithm (IND), which accounts for the changes in lung gas stores, with those obtained with the classical ‘Expiration‐only’ algorithm (EXP). The two algorithms were applied on the same flow and gas fraction traces acquired on 10 healthy volunteers, performing 10 times the same moderate‐intensity exercise transition. Repeated O2 uptake responses were stacked together and the kinetic parameters of a mono‐exponential function were estimated by non‐linear regression, removing the data pertaining to 1‐s progressively longer initial periods (ΔT r). Independently of ΔT r, the mean response time (time constant + time delay) obtained for the IND data was faster compared to the EXP data (∼43 s vs. ∼47 s, P < 0.001), essentially because of shorter time delays. Between ΔT r = 16 s and ΔT r = 29s, the time constants of the IND data decreased (30.7 s vs. 28.0 s, P < 0.05; drop = 10%), but less than those of the EXP data (32.2 s vs. 26.2 s, P < 0.001; drop = 23%); with the same ΔT r, the time constants of the two algorithms’ data were not different (P > 0.07). The different decrease in the time constant, together with the different mean response time, suggests that the data yielded by the two algorithms provide a different picture of the phenomena occurring at the beginning of the exercise.


INTRODUCTION
At the start of a moderate-intensity square-wave exercise, breathby-breath O 2 uptake ( VO 2 ) as assessed at the mouth lags behind the requested mechanical power for a few minutes (Ferretti et al., 2022).Two main phases are deemed to occur throughout this transient period.The first phase (Φ 1 ) is believed to represent the cardiodynamic adjustment to the exercise, while the second one (Φ 2 , called primary phase) is assumed to be the image of the O 2 uptake at the muscle level, and to be related to endurance performance (Bowen et al., 2019;Rossiter, 2011).Phase 2 is investigated by approximating the oxygen uptake at the mouth to a mono-exponential function and the time constant (τ) of its kinetics, obtained by the non-linear regression fitting procedure, is mostly the only evaluated parameter.However, the estimated τ might be affected by the amount of information for Φ 1 included or excluded in or from the fitting window.Theoretically, if the mono-exponential function adequately fits the time course of the original data, stable τ values should be reached after an initial decrease, as soon as the information pertaining to Φ 1 is completely excluded (Rossiter et al., 1999).However, no stable τ values were found either in clean simulated VO 2 data (Benson et al., 2017) or in experimental data showing a low signal-to-noise ratio (Francescato & Cettolo, 2021).
The time course of oxygen consumption at the muscle level is believed to be better approximated by the gas exchange assessed at the mouth if the changes in lung gas stores are accounted for, as occurs with the VO 2 values yielded by the 'Independent-breath' algorithm (Cettolo & Francescato, 2018a).This algorithm is based on the alternative view of the respiratory cycle of Grønlund (1984), and identifies the start and end points of the cycle on the basis of equal ratios between the end-tidal fractions of exchanged and not exchanged respiratory gases (Cettolo & Francescato, 2015).Conversely, the most commonly used algorithm is likely the 'Expiration-only' one (Golja et al., 2018), which calculates gas exchange from information collected only during expiration, estimating the inspiratory volume through the Haldane transformation (Roecker et al., 2005;Ward, 2018) and identifying the start and end points of the respiratory cycle from the changes in the flow direction.
Changes in lung gas stores are expected mainly at the start of a square-wave exercise (Wüst et al., 2008), and thus the use of different calculation algorithms might affect the temporal behaviour of gas exchange during this transient period.As a matter of fact, by comparing the kinetic parameters obtained for the VO 2 yielded by the 'Independent-breath' algorithm with those provided by more commonly available algorithms, a faster kinetics was found when all the data pertaining to the transient period were included in the fitting window (Francescato & Cettolo, 2019).In that experimentation, however, the signal-to-noise ratio was poor because only one repetition of the square-wave exercise was performed by each volunteer.It can be hypothesized that, by improving the signal-tonoise ratio, a faster kinetics will be found for the data yielded by the 'Independent-breath' algorithm compared to those of the 'Expirationonly' algorithm, even excluding from the fitting window the data

Highlights
• What is the central question of this study?
Is the kinetics of breath-by-breath gas exchange at the start of a square-wave moderate intensity exercise faster when accounting for the changes in lung gas stores?
• What is the main finding and its importance?
Accounting for the changes in lung gas stores, the time constant of the VO 2 kinetics changed less when the data pertaining to the initial period were progressively removed; time delays and mean response times were faster.Consequently, using different gas exchange calculation algorithms, the physiological phenomena occurring at the beginning of moderate-intensity exercise are characterized differently.
pertaining to the initial time period (e.g. the data included in the first 20-25 s).
The aim of the present work was to test the above hypothesis by pooling together the breath-by-breath gas exchange data obtained during 10 repeated transitions performed by each volunteer, in order to increase the signal-to-noise ratio.For both algorithms, the parameters describing the timing of the kinetics were estimated several times by changing the length of the period excluded from the fitting window, in order to evaluate the effects of the inclusion/exclusion of interfering information, likely pertaining to Φ 1 .

Experimental protocol
Ten healthy and moderately active adults (5 males and 5 females) The metabolic unit automatically controlled the timings of the protocol, in particular the transitions between the 10W-P and the MI-P conditions.Before each experimental session, the analysers were calibrated following the procedures indicated by the manufacturer.
Mean actual mechanical power during the experimental sessions was 71.5 ± 13.1 W.

Gas exchange calculations
Breath-by-breath oxygen consumption ( VO 2 ) was calculated by means of the 'Independent-breath' algorithm (IND) according to the following equation: where F N 2 is the instantaneous nitrogen fraction (calculated as 1 -F O 2 -F CO 2 ); t 1 and t 2 correspond to the start and the end time points of the respiratory cycle, respectively, both identified on the trace of the (Cettolo & Francescato, 2018a).
Using the same originally acquired flow and gas fraction traces, VO 2 was calculated also by means of the 'Expiration-only' algorithm (EXP) according to the following equation: where times t i and t e correspond to the time points where flow changes direction and the inspiration and expiration start, respectively; and ) introducing the appropriate changes into the above equations, as well as the corresponding respiratory exchange ratios (i.e., RER IND and RER EXP ).
Both algorithms and details of the computations were described in Francescato and Cettolo (2019).As a result, three time series (namely, VO 2 , VCO 2 and RER) were obtained for the IND and EXP algorithms, and thus two triplets were obtained for each experimental session for each volunteer, expressed in STPD conditions.No data were discarded (or filtered) on any of the time series before the subsequent analyses.

Statistics on the steady states of the individual time series
All the data were analysed with the R environment (R Core Team, 2020).
For all the time series (namely, and RER EXP ), mean values, together with the corresponding standard deviations (SD), were calculated over the following four time periods: (a) baselines BL1 and BL2, namely the last 2 min just preceding each of the two transitions between 10W-P and MI-P, and (b) steady-states SS1 and SS2, that is, the last 2 min of each of the two MI-P bouts (Figure 1).
Significant differences for the means and the SDs were assessed applying an analysis of variance for repeated measures (2 × 2 × 5 ANOVA), with the following effects: between the values obtained during BL and SS (Load effect), between the two repetitions within the same experimental session (Repetition effect), and among the five experimental sessions (Session effect), as inter-subject effects.
In addition, the possible significant differences between the data yielded by the IND and the EXP algorithms were assessed (Algorithm effect).Post hoc pairwise comparison was used to detect significant differences among the various conditions.

Data treatment and statistics on the assembled time series
For each subject and each experimental session, all the obtained VO 2 , VCO 2 and RER time series were split at t = 16 min, and the times were shifted setting the start of the two MI-P bouts at t = 0. Subsequently, for each volunteer, the corresponding 10 repeated time series were assembled together using the stacking procedure (Francescato et al., 2014b).
Mean values and corresponding standard deviations were calculated over the same time periods described above, namely, for baseline (the last 2 min of 10W-P just preceding t = 0 s; A b ) and for steady-state (the last 2 min of MI-P just preceding t = 360 s; A s ).
These data allowed calculation of the 'functional gain' as the ratio between the net O 2 uptake (i.e., A s − A b ) and the net mechanical power above baseline.In addition, over the same time periods, slopes of VO 2 against time were calculated by linear regression and their statistical significance was evaluated to assess the attainment of a steady condition.
The kinetic parameters of VO 2 and VCO 2 corresponding to the square-wave exercise transition were obtained running the non-linear regression procedure, using the R routine nls.lm (included in the minpack.lmlibrary and minimizing the residual sum of squares by the Levenberg-Marquardt algorithm) without cleaning any outlier.The following mono-exponential model was used: The initial values to run the iterative non-linear regression procedure were set to 25 s and 0 s for the time constant (τ) and the time delay (T d ), respectively.Baseline signal was set to A b (as calculated above), while the initial value for the signal change (ΔA) was set to A s − A b , that is, the difference between the two signal intensities calculated above.The fitting procedure yielded the estimated values for τ, T d and ΔA, and their asymptotic standard errors (ASE τ , ASE Τd and ASE ΔA , respectively), which allow evaluation of the specific confidence intervals of each estimated parameter (Francescato et al., 2014a,b).
In addition, the mean response time (MRT) of the overall kinetics was calculated for each data set as follows: The fitting procedure was run 51 times (always applying Equation 3), removing each time 1 s progressively longer time period (ΔT r ) from the fitting window, starting from t = 0 s (i.e., ΔT r ∈ [0 s, 50 s]).At the end of this procedure, 51 sets of estimated parameters (one for each ΔT r ) were obtained for each volunteer and for the VO 2 and VCO 2 time series obtained for the two algorithms.
Student's two-tailed t-test for paired data was applied for each of the different ΔT r , to detect the statistically significant differences between the kinetic parameters obtained for the gas exchange time series provided by the two algorithms for each volunteer (n = 10).When a statistically significant difference was detected, its magnitude was evaluated by calculating the absolute value of the Cohen's d Z effect size, where the Z alludes to the fact that the unit of analysis is the difference between the paired data, and the following absolute effect size classification was considered:

The experimental sessions
For all the investigated time series (calculated either by the IND and the EXP algorithms), no significant difference was observed among the mean values, or the corresponding standard deviations, when comparing the five different experimental sessions of each volunteer (repeated measures ANOVA, Session effect, F < 1.7, P > 0.17).
Similarly, no significant difference was detected when comparing the two repetitions of the same experimental session (repeated measures ANOVA, Repetition effect, F < 3.6, P > 0.09).The mean values obtained for the moderate intensity pedalling condition, as well as their corresponding standard deviations, were significantly higher than those obtained for the 10W-P condition (during SS vs. BL; repeated measures ANOVA, Load effect, F > 16.7, P < 0.003), with the exception of the standard deviations for RER (F = 1.2, P = 0.31).
The mean values obtained for the IND algorithm were not significantly different from those obtained for the EXP algorithm (repeated measures ANOVA, Algorithm effect, F < 2.2, P > 0.17).Attainment of a steady state condition was confirmed for all the stacked VO 2 time series, their slopes being never significantly different from zero (t < 2.0, P > 0.05 in all cases).

3.2
The time period removed from the fitting window: effects on the VO 2 kinetic parameters

3.3
The time period removed from the fitting window: effects on the VCO 2 kinetic parameters The time constants of the stacked VIND

DISCUSSION
The main result of the present investigation was that the time parameters that characterize the oxygen uptake (or carbon dioxide exhalation) kinetics during moderate intensity exercise differed depending on the algorithm used to calculate the gas exchange data.
In particular, our results suggest that the temporal behaviour of the VIND

Comparison of the two algorithms under steady conditions
In the present experimentation the IND and EXP algorithms provided similar mean gas exchange values despite the EXP algorithm (Equation 2) was neglecting the dead space of the breathing apparatus.It should be noted, however, that the latter adjustment as proposed by Beaver et al. (1973) is under debate, considering that it should account even for breathing frequency and/or tidal volume (Ward, 2018).The mean oxygen uptake values are in agreement with the literature.Indeed, applying Equation 4of Formenti et al. (2015), the estimated VO 2 will amount to 1.15 litres min −1 , which is practically equal to the values reported in Table 1 (∼1.16litres min −1 ).Moreover, the 'functional gain' of the responses (∼9.5 ml min −1 W −1 ) is not far from 9.47 ± 0.85 and ∼10.0 ml min −1 W −1 , as reported by Benson et al. (2017) and Rossiter (2011), respectively.Although we are aware that the RER values are affected by the duration of fasting before the exercise (Liu & Chen, 2022), no attention was paid to performing all the experimental sessions at the same day time, since RER was not a main outcome.The obtained RER values, however, were in line with the values reported in the literature for similar exercise conditions (Baldassarre et al., 2022;Liu & Chen, 2022;Miyamoto et al., 2022).
As already observed under several different exercise conditions (Cettolo & Francescato, 2018a;Francescato & Cettolo, 2019;Koschate et al., 2019) .Indeed, the IND algorithm accounts for the changes of the pulmonary O 2 stores, which typically occur for the anomalous breaths (Francescato et al., 2019), thus reducing the fluctuations around the mean of the oxygen consumption as measured at the mouth.Similar standard deviations were observed between the VIND CO 2 and VEXP CO 2 values, in particular at rest.It is to be noted, however, that the alveolar CO 2 stores are dynamically balanced with the amount of HCO 3 − dissolved in the blood (Cettolo & Francescato, 2018b); accordingly, when an anomalous breath occurs, the CO 2 lung stores might remain quite constant, making negligible the correction for their changes.

The VO 2 ON transient phase
Theoretically, as soon as all the information pertaining to Φ 1 is excluded, the behaviour of the time constant of VO 2 as a function of ΔT r should reach a stable value (Rossiter et al., 1999).This behaviour was not followed by the mean τ value of VEXP was observed for ΔT r > 16 s suggesting that, for these fitting windows, information pertaining to phase 1 was likely excluded.Since the flow and gas traces, as collected at the mouth, were the same for the calculations with both algorithms, it could be expected that the cardiodynamic adjustment to the exercise (e.g. the increase in cardiac output) would be excluded for the same ΔT r , which was not the case.Consequently, the observed discrepancy can be explained only by the fact that the two algorithms account (or not) for the changes in lung gas stores in their calculations.
The model commonly used to estimate the kinetic parameters by non-linear regression includes also the time delay, which mathematically is the back projection of the fitted data to the baseline.
This parameter is correlated to the Φ 1 to Φ 2 transition time, although it does not correspond to the latter (Rossiter, 2011).However, it should be noted that the sum of T d and τ reflects the MRT of the overall O 2 uptake kinetics during the transient and allows calculation of the O 2 deficit (O 2def ) as follows (Whipp et al., 1982): Accordingly, there is a direct proportion between MRT and O 2 deficit with equal ΔA (i.e. the net increase in O 2 uptake at the asymptote).The O 2def includes the changes in the pulmonary O 2 stores, which are theoretically compensated for by the IND algorithm.
A faster MRT can thus be expected for the data of the IND algorithm compared to the EXP one, as was observed in the present work (43.3 ± 7.1 s vs. 47.5 ± 6.2 s).
The O 2 deficit is a physiological variable that should be independent of the amount of information used by the non-linear fitting procedure.
Consequently, a quite constant MRT should be predicted irrespective of ΔT r , as was the case for both algorithms up to ΔT r ≈ 40 s (Figure 2, lower panel), mainly because of complementary time constants and time delays.
For all the estimated parameters, the asymptotic standard error, namely the statistical descriptor that allows evaluation of the confidence intervals of the corresponding estimated value (Francescato et al., 2014a,b), increased on average with increasing ΔT r , likely because fewer data points were included in the non-linear regression.Smaller ASE values were obtained for VIND , in particular for the smaller ΔT r (Figure 2, right panels).
These results suggest that the IND algorithm provides less noisy gas exchange data also during the transient phases and/or the VIND O 2 data fit better the mono-exponential model.

The VCO 2 ON transient phase
To the best of our knowledge, the kinetics of carbon dioxide exhalation at the start of a moderate intensity exercise has received much less attention in the literature compared to the kinetics of oxygen uptake, despite its providing insights into the inherent control features of ventilation and gas exchange (Ward et al., 2023;Whipp et al., 1982).It can be hypothesized that the cardiodynamic adjustment

4.4
The commonly used procedure of analysis , respectively), which is suggestive of a faster overall kinetics.(open blue circles; dashed line) time series of the same subjects as in Figure 1.For graphical purposes, data of the 10 repetitions pertaining to the same algorithm were first stacked together, sorted in ascending time and then averaged both in time and in signal intensity over 10 consecutive data points (i.e. the number of repetitions).For both algorithms, the parameters estimated by non-linear regression with ΔT r = 20 s were used to draw the two lines showing the corresponding kinetics.
In the present experimentation, volunteers repeated 10 times the same exercise transient.Nevertheless, when the number of repetitions was limited to four, as suggested by Benson et al. (2017), results were comparable to those obtained on 10 repeated bouts, as concerned all the investigated areas, that is, the mean values and corresponding standard deviations, the statistically significant differences between the two algorithms, and, finally, the behaviours according to the excluded time period (ΔT r ).As expected, however, the average ASE values obtained for the data yielded by both algorithms were somewhat greater than those obtained for 10 repetitions, while the VIND

Fraction of pulmonary O 2 stores accounted for by the IND algorithm
As already stated above, the MRT of VO 2 at the start of a squarewave exercise of moderate intensity allows calculation of the O 2 deficit (Whipp et al., 1982).A smaller O 2 deficit could be estimated for the VIND Indeed, the overall changes in pulmonary O 2 stores (∼90 mL) were estimated assuming that: (a) the splitting of the high-energy phosphates is the mirror image of oxygen consumption at the muscle level (Ferretti et al., 2022;Rossiter et al., 1999), with a time constant of 25 s without any time delay (Barker et al., 2008;Francescato et al., 2013;Jones et al., 2008), and (b) the changes in the venous blood O 2 stores can be estimated according to the arterio-to mixed venous blood oxygen difference at rest and at steady state (Francescato et al., 2003), using the values reported for an exercise similar to that of the present investigation (Fontolliet et al., 2021).

Strengths and limitations
Volunteers started abruptly the moderate intensity exercise bout after a 10-W pedalling period, ensuring that the increase in O 2 uptake was essentially due to the increase in mechanical power and not to the inertial effects of the ergometer flywheel at exercise onset (Whipp et al., 1982) or to the work needed to move the limbs (Francescato et al., 1995).Even possible interferences of anticipation were avoided (Miyamoto et al., 2022), since volunteers were not informed about the timings of the protocol, in particular of the changes in workload, that were automatically set by the software of the metabolic unit.
This allowed us to circumvent the need to apply manual re-alignments specific for each volunteer.
Exercise intensity was chosen according to volunteer's body mass, not relative to an individual physiological threshold during exercise (e.g.gas exchange threshold or lactate threshold).During a constant intensity exercise below the above thresholds, however, following the cardiodynamic phase and the primary phase, gas exchange remains quite constant and does not show a temporal drift (the so-called slow component).The non-statistically significant slopes found in the gas exchange data during steady-state are suggestive of the absence of a slow component in all the volunteers.Moreover, since the same original flow and gas fraction traces were used to calculate gas exchange by means of the two algorithms, the slow component would have had a similar effect on whatever kinetic parameter (e.g.lengthening the time constant) obtained for both algorithms.
Each volunteer repeated 10 times the same moderate intensity pedalling bout and repeatable data were collected making us confident that our experimental data show an adequate signal-to-noise ratio for kinetic analyses.Moreover, the different repetitions were stacked, without discarding or filtering the originally acquired data, maintaining the validity and meaning of every single value in determining the kinetic parameters and the corresponding uncertainty (Francescato et al., 2014b(Francescato et al., , 2015(Francescato et al., , 2017)).
The group of volunteers was rather uniform in age, allowing us to obtain more homogeneous results and to make inferences on the basis of literature data for volunteers of similar characteristics (i.e., the phosphocreatine breakdown time constant and the arterio-to mixed venous blood oxygen difference at rest and during exercise).
Conversely, the narrow range of ages might be considered a limitation, since older subjects might lead to different results.
Finally, the comparison of gas exchange kinetic parameters for VO 2 data obtained with different calculation methods/algorithms has led to contrasting results (e.g., Aliverti et al., 2009;Beaver et al., 1981;Cautero et al., 2002), likely because of different experimental protocols (e.g.different baselines) and/or methods of analysis (e.g.fitting with a bi-exponential model).Consequently, a systematic comparison with our results could only introduce further confusion.
It remains to be evaluated whether the two algorithms at stake might show a different sensitivity in detecting intervention-induced changes in the gas exchange kinetic parameters (e.g.exercise training).
It can be expected that the IND algorithm might detect differently the interventions that change the Φ 1 Φ 2 transition time and/or that change the involvement of the lung gas stores.

Conclusions
Results of the present investigation suggest that, using a monoexponential model, the kinetic parameters of breath-by-breath oxygen uptake and carbon dioxide exhalation might differ depending on whether gas exchange is calculated accounting for the changes in lung gas stores or not.The IND and EXP algorithms resulted in comparable oxygen uptake time constants only for a portion of the investigated fitting windows (i.e., for ΔT r between 16 s and 29 s), while the time constants of carbon dioxide exhalation were comparable at all the investigated ΔT r .Nevertheless, the IND algorithm yielded significantly faster mean response times for VO 2 over the whole range of investigated ΔT r , to which corresponded a smaller O 2 deficit.
Consequently, the physiological phenomena occurring at the start of a moderate intensity exercise might be interpreted differently according to the gas exchange calculation algorithm used.
volunteered to be subjects.Their mean (± SD) age, stature and body mass were 24.6 ± 3.6 years, 1.73 ± 0.09 m, and 73.5 ± 15.1 kg, respectively.The experimental protocol, design and methods conformed to the standards set by the Declaration of Helsinki, except for registration in a database, and were approved by the Institutional Review Board of the Department of Medicine of the University of Udine (Italy) (no.07/2020_IRB issued on 5 March 2020).After having been thoroughly informed about the nature, purpose and possible risks of the investigation, written informed consent was obtained by all the volunteers prior to their participation.All the volunteers were required to have completed the Covid-19 vaccination protocol at least 15 days before the first experimental session and no one reported long-term Covid-19 symptoms.In addition, the last experimental session of one volunteer and the first one of the next volunteer were scheduled at least 1 week apart.Each volunteer was submitted to respiratory gas collection at the mouth while pedalling as close as possible to 60 rpm on an ergometer (Corival; Lode B.V., Groningen, the Netherlands) during: (a) 5 min pedalling at 10 W (10W-P), (b) the first bout of 6 min pedalling at ∼1 W kg −1 body mass (moderate intensity pedalling; MI-P), (c) 10 min pedalling at 10 W, (d) the second bout of 6 min pedalling at ∼1 W kg −1 body mass, and (e) 8 min pedalling at 10 W. Overall duration of the experimental session was thus of 35 min, during which the volunteer repeated twice the same square-wave exercise transition, each preceded by at least 5 min pedalling at 10 W. The experimental session was repeated 5 times by each volunteer, at least 1 day apart, thus allowing a total of 10 repetitions for each of them to be obtained.At the start of the first experimental session, each participant chose the cycling position that met his/her comfort requirements and the same position was subsequently used in all the following sessions.Mechanical power, pedalling frequency, heart rate (HR), and flow ( V), as well as O 2 and CO 2 fractions (F O 2 and F CO 2 , respectively) in inspired and expired air at the mouth were continuously acquired throughout the trial (Metalyzer 3B, Cortex GmbH, Liepzig, Germany).

F
I G U R E 1 Example of gas exchange data obtained from one volunteer during an experimental session.The experimental protocol included: (a) an initial 5-min pedalling period at 10 W (10W-P), (b) two bouts of moderate intensity exercise, lasting 6 min each (1 W/kg, i.e., 55 W in this volunteer; MI-P), separated by 10 min of pedalling at 10 W, and (c) a final recovery period lasting 8 min while pedalling at 10 W. Pedalling frequency was kept as close as possible to 60 rpm throughout.VO 2 (upper panel) and VCO 2 (lower panel) data obtained applying the IND (continuous red line) or the EXP (dashed blue line) algorithms are illustrated.Vertical lines correspond to the start and end times of the two bouts of moderate-intensity pedalling periods.Shaded areas correspond to the time periods over which the data were averaged, namely, two periods during 10W-P (BL1 and BL2) and two periods during MI-P (SS1 and SS2).
resulted in a faster kinetics compared to the VEXP O 2 time series (Figure 2, left panels).As a matter of fact, up to ΔT r < 16 s significantly shorter τ values were obtained for the stacked VIND t > 2.32, P < 0.046, d Z > 0.73; the difference being >4.0%).It is interesting to be noted that, between ΔT r = 16 s and ΔT r = 29 s, the time constants obtained for VIND t < 2.05, P > 0.07).Finally, for ΔT r ≥ 30 s, the mean τ obtained for the stacked VEXP 2.50, P < 0.034, d Z > 0.79, the difference being >9.1%).Conversely, the mean T d (Figure 2, middle left panel) and the MRT (Figure 2, lower panel) obtained for the VIND O 2 data were significantly shorter for all the ΔT r values up to 50 s (t > 2.60, P < 0.03, d Z > 0.82 for both parameters).The difference was greater than 17% and 8% for T d and MRT, respectively.The VIND O 2 data provided more stable kinetic parameters as a function of ΔT r .In particular, by comparing ΔT r = 0 s with ΔT r = 40 s, the mean time constants (Figure 2, upper left panel) decreased with increasing ΔT r , with a maximum drop of 35% for the stacked VIND 6.86, P < 0.001, d Z = 2.17) which reached 61% for the VEXP = 15.4,P < 0.001, d Z = 4.86).Focusing on the ΔT r interval between 16 s and 29 s, the fall was still less pronounced for the VIND O 2 data (t = 2.82, P < 0.05, d Z = 0.89; drop of 10%) compared to the VEXP O 2 data (t = 11.18,P < 0.001, d Z = 3.54; drop of 23%).For both algorithms, the time delays increased by more than 100% from ΔT r = 0 s to ΔT r = 40 s (t > 5.70, P < 0.001, d Z > 1.80).The resulting MRT increased significantly for the VEXP t = 2.83, P = 0.02, d Z = 2.66), whereas it remained constant for the VIND 1.55, P = 0.15).The mean ASE τ values (Figure 2, upper right panel) obtained for the stacked VIND to ΔT r = 27 s (t > 2.53, P < 0.05, d Z > 0.79, the difference being >10%).The corresponding mean ASE Td values (Figure 2, middle right panel) were statistically lower up to ΔT r = 16 s (t > 2.36, P < 0.05, d Z > 0.62, the difference being >13%).All the mean ASE values increased with increasing ΔT r .F I G U R E 2 Mean time constants (τ; upper left panel) and mean time delays (T d ; middle left panel) of oxygen uptake kinetics are illustrated as a function of the time period removed from the fitting window (ΔT r ), as well as the corresponding mean asymptotic standard errors, ASE τ (upper right panel) and ASE Td (middle right panel).The corresponding mean response times (MRT) are illustrated in the lower panel.Results obtained for the data yielded by the IND (filled red circles) and the EXP (open blue circles) algorithms are illustrated in all panels.Vertical bars are the standard deviations of the individual parameters obtained running the non-linear regression procedure with the mono-exponential model.The differences between the two algorithms with the same ΔT r are illustrated as Cohen's d Z effect sizes (right vertical axis, on a logarithmic scale only for graphical purposes), using different symbols according to the statistical significance (n = 10; two-tailed paired t-test; * P < 0.05; ** P < 0.01).The 5 areas of decreasing colour intensity from bottom to top highlight the magnitude of Cohen's d Z according to the commonly used thresholds (namely, 'trivial' < 0.2 < 'small' < 0.5 < 'medium' < 0.8 < 'large' < 1.3 < 'very large' with white background).

F
Mean time constants (τ; upper left panel) and mean time delays (T d ; middle left panel) of carbon dioxide exhalation kinetics are illustrated as a function of the time period removed from the fitting window (ΔT r ), as well as the corresponding mean asymptotic standard errors ASE τ (upper right panel) and ASE Td (middle right panel).The corresponding mean response times (MRT) are illustrated in the lower panel.Results obtained for the data yielded by the IND (filled red circles) and the EXP (open blue circles) algorithms are illustrated in all panels.Vertical bars are the standard deviations of the individual parameters obtained running the non-linear regression procedure with the mono-exponential model.The differences between the two algorithms with the same ΔT r are illustrated as Cohen's d Z effect sizes (right vertical axis, on a logarithmic scale only for graphical purposes), using different symbols according to the statistical significance (n = 10; two-tailed paired t-test; * P < 0.05; ** P < 0.01).The 5 areas of decreasing colour intensity from bottom to top highlight the magnitude of the Cohen's d Z according to the commonly used thresholds (namely, 'trivial' < 0.2 < 'small' < 0.5 < 'medium' < 0.8 < 'large' < 1.3 < 'very large' with white background).The mean ASE values (Figure 3, upper and middle right panels) obtained for the stacked VIND occasionally, mainly for the time constants and for ΔT r up to 21 s.For both parameters, the mean ASE values increased with increasing ΔT r .
faster overall kinetics, as suggested by a shorter mean response time.
up to ΔT r ≈ 40 s.Conversely, a rather stable mean τ value for VIND O 2 at the beginning of the exercise affects also the breath-by-breath VCO 2 values, involving changes in lung gas stores.Indeed, similar to the behaviour of the kinetic parameters of VO 2 , in the present investigation, the time constant of VCO 2 decreased with increasing ΔT r , while the T d increased (Figure3, left panels).Nevertheless, no quite steady VCO 2 kinetic parameters were found by removing from the fitting window 1-s longer initial periods, suggesting that the behaviour of VCO 2 at the start of the exercise departs from the mono-exponential function much more than the kinetics of VO 2 .In addition, results of the present work suggest that the changes in lung CO 2 stores are less pronounced compared to the changes in lung O 2 stores (both accounted for by the IND algorithm).Indeed, these changes were not sufficient to affect the values of the VCO 2 time constant, and/or the corresponding ASE values, which were not significantly different between the two algorithms at stake for the majority of the explored ΔT r .The lung CO 2 stores, however, changed to an extent detectable by the IND algorithm, and influenced slightly the start of the kinetics, resulting in a lower time delay and/or mean response time.
Figure 4 illustrates the two stacked VO 2 time series of the same subject as for Figure 1, together with the regression lines corresponding to the two kinetics obtained for ΔT r = 20 s.It clearly appears that the kinetics for the VIND for graphical reasons, in Figure 4 the data of the 10 repetitions were first stacked together, sorted in ascending time and then averaged both in time and in signal intensity over 10 consecutive data points (i.e. the number of repetitions).Finally, it should be noted that, for ΔT r = 20 s, the VIND O 2 data resulted in smaller ASE values, suggesting that the corresponding estimated parameters are more precise (Figure 2 right panels).

F
in lower ASE values compared to the VEXP O 2 data (data not shown).
.e., ∼420 mL vs. ∼470 mL), because of the ∼10% difference between the MRT values obtained for the two algorithms.The difference of ∼50 mL in the O 2 deficit might be associated to the changes in lung O 2 stores, likely accounted for by the IND algorithm, but not by the EXP one.This volume might seem small and thus negligible, but this is not the case, since it represents about 56% of the overall changes in the pulmonary O 2 stores (50 mL/90 mL).
IO 2 and F IN 2 correspond to the inspired ambient oxygen and nitrogen fractions and were set to 20.93% and 79.02%, respectively.
Mean values and means of the corresponding standard deviations (SD) of the stacked oxygen uptake ( VO 2 ), carbon dioxide exhalation ( VCO 2 ) and respiratory exchange ratio (RER) data yielded by the EXP and IND algorithms, for baseline and steady state time periods.The table summarizes also the mean values of functional gain, calculated on the basis of the individual actual mechanical power.n = 10 (number of volunteers).Data in brackets are the standard deviations of the corresponding mean.Statistically significant different values by comparing the IND and EXP algorithms (two-tailed paired t-test, *P < 0.05; **P < 0.01; ***P < 0.005).The corresponding d Z was >0.94 for the standard deviations, and >2.86 for the RER values.P = 0.02, d Z = 0.94).Significantly smaller standard deviations were obtained for RER EXP compared to RER IND (t > 9.0, P < 0.001, d Z > 2.87).
TA B L E 1(repeated measures ANOVA, Algorithm effect, F = 20.5, P = 0.001).No significant difference was detected for the VCO 2 values obtained with the two algorithms (F = 4.8, P = 0.06), whereas the RER EXP showed significantly lower standard deviations compared to RER IND (F = 476.3,P<0.001).Since no Repetition or Session effects were detected, the 10 time series of VO 2 , VCO 2 and RER as obtained by each of the two algorithms were assembled together, thus yielding six stacked time series for each volunteer.Overall mean values and standard deviations were then calculated on the stacked time series and are summarized in Table1, together with the corresponding 'functional gains' .No significant differences were detected between the corresponding average values (t < 1.6, P > 0.14).The standard deviations were significantly smaller for the VIND , the standard deviations of the VIND