The study of the oxygen uptake (V˙O2) kinetics deals with the ability of data modeling to describe, in mathematical terms, the temporal characteristics of the aerobic response to the challenge of a step increase in work rate (Ẇ) (Hughson, 2009). Time-domain kinetic analysis has some limitations due to the white Gaussian noise associated with breath-by-breath fluctuation (Lamarra et al., 1987) that adds uncertainty to time-domain index predictions estimated from a single test dataset. To increase signal-to-noise ratio, studies commonly repeat similar tests multiple times within the same session (Ozyener et al., 2001; Beltrame et al., 2016; Christensen et al., 2016) or on different days (Whipp et al., 1982; Keir et al., 2014a) and average repetition-like transitions before time-domain data modeling. However, it is time consuming to repeat multiple similar exercise protocols until reproducible data are obtained. In addition to multiple repetition-like transitions, studies have applied frequency-domain data filtering before time-domain kinetic analysis (Harper et al., 2006; Schlup et al., 2015). Despite reducing the confidence interval of estimated time-domain parameters, the exponential data fitting procedure after filtering still deals with explicit modeling where model parameters are assumed a priori (Eßfeld et al., 1987). Therefore, new methods for the extraction of indexes related to aerobic system temporal dynamics without the need of model assumption should be investigated.

An attractive alternative to multiple repetitions of step transitions for evaluating the kinetic behavior of the aerobic energy supply system is the pseudo-random binary sequence (PRBS) in which Ẇ varies between two levels which are normally constrained to the light to moderate intensity exercise domains (Eßfeld et al., 1987; Hughson et al., 1990; Beltrame and Hughson, 2017a). The V˙O2 response to PRBS protocols is evaluated in the frequency-domain filtering out non-periodic signals associated with white noise, improving the extraction of parameters related to the aerobic system dynamics. The influence of inter-breath noise in the V˙O2 dynamics commonly occurs at higher frequencies that can be neglected during the frequency-domain analysis, theoretically increasing the biological significance of the estimated indexes. The attractiveness of the PRBS approach results from the potential to gain a quantitative index of kinetics from fewer exercise testing sessions in comparison to time-domain approaches (Hughson et al., 1991; Yoshida et al., 2008).

To date, there are few studies of the variability of the methodology and the requirements for precision in estimation from PRBS testing (Edwards et al., 2001, 2003; Koschate et al., 2016). In support of recent findings (Beltrame and Hughson, 2017a,b; Beltrame et al., 2017a), the purpose of this study was to describe in detail the computation of the mean normalized gain (MNG) and to test its consistency to characterize the V˙O2 kinetics during random exercise in humans. The MNG will be validated against the time-domain approach, and checked for intra-subject consistency by applying multiple PRBS protocols on different days and times of the day. In addition, the MNG was also evaluated considering different number of repeated tests averaged together before data modeling and different filtering techniques.

The hypothesis of this study was that the V˙O2 dynamics characterized by MNG during random exercise would be similar to the dynamic indices obtained by time-domain analysis, even with fewer exercise repetitions. In addition, we hypothesized that MNG was independent of the testing day and the time of the day, demonstrating therefore that it can be used to evaluate the individual aerobic response during random exercise in humans. These results could set the stage for advancing frequency domain analyses outside the confines of the research laboratory to assess kinetics, and therefore an index of physical fitness, in activities common to daily living or athletic training.

In agreement with our initial hypothesis, the calculation of MNG was able to characterize the temporal dynamics of V˙O2 to random exercise input being strongly correlated with the time-domain indicator, τ, obtained in the same persons. The MNG eliminated the expected differences in static gain between individuals by expressing the dynamic response as a percentage of the fundamental harmonic value. The comparison of MNG against the time-domain V˙O2 kinetics analysis was shown to be independent of the period of the day, the day of the test, the filtering technique used and the number of exercise repetitions ensemble-averaged before data analysis. The detection of differences in MNG was independent of the number of exercise repetitions for differences higher than ≈8% which correspond to a τ variation of ≈15 s. Further, these data are important for the experimental design of further studies by informing the number of repetitions necessary according to an expected effect size (Figure 7). In addition, as reported in Figure 8, MNG seemed to be, in comparison with τ, ≈50% less susceptible to noise than time-domain analysis thus isolating better the temporal dynamics of the aerobic response to changes in energy demand.

The breath-by-breath fluctuation (Lamarra et al., 1987) during exercise transitions adds uncertainty to time-domain parameter prediction, mainly from a single test dataset. The confidence interval of the estimated τ, and therefore the “sensitivity” to identify aerobic fitness differences, depends on the V˙O2 signal-to-noise ratio (Lamarra et al., 1987; Keir et al., 2014b), the model complexity (i.e., the degree of freedom) (Motulsky and Ransnas, 1987) and the selected data window (Bell et al., 2001; Murias et al., 2011). To increase signal-to-noise ratio, studies commonly repeat similar tests multiple times within the same session (Ozyener et al., 2001; Christensen et al., 2016) or on different days (Whipp et al., 1982; Keir et al., 2014a) and average repetition-like transitions before time-domain data modeling.

The frequency-domain analysis has some advantages over the time-domain approach. Firstly, no explicit data modeling with a degree of arbitrariness is necessary (Eßfeld et al., 1987) since the V˙O2 time series can be decomposed, and therefore rebuilt from the infinite sum of its harmonic components (Hughson et al., 1990). Second, the random noise associated with V˙O2 measured at the mouth (Lamarra et al., 1987) is filtered when transferred into frequency space, diminishing the impact of the inter-breath oscillations over the V˙O2 dynamics characterization.

The early studies from Eßfeld et al. (1991) and Hoffmann et al. (1992) were the first to normalize the system gain amplitudes by the amplitude at the fundamental harmonic (i.e., gAmp₁ in Equation 4). They successfully showed that a faster V˙O2 kinetics maintained a higher normalized gain across the frequency spectrum. However, since the focus of their experiments was to investigate possible aerobic system controllers, no further comparisons were carried out to explore the applicability of this normalization for the V˙O2 temporal dynamics assessment. Other studies used the absolute system gain to infer about V˙O2 dynamics (Eßfeld et al., 1987; Hughson et al., 1990, 1991). In fact, the use of absolute gains may be sufficient for intra-subject comparisons since the system static gain seems to remain constant as the aerobic system “speeds up” after training (Christensen et al., 2016). However, for the comparison between subjects by an absolute index such as τ, the gain must be normalized.

We demonstrated by computer simulations (Table 2, Figure 2) that MNG was able to characterize the temporal characteristics of the aerobic system by comparing MNG with τ_s. The MNG refined the ability of the Fourier transformation to separate the system dynamic gain from the static gain, isolating therefore the rate at which the aerobic system supplies the energy demand (i.e., power) from the capacity of the aerobic system to supply the demand at steady state. The latter is susceptible to inter-individual variability which confounds the interpretation of the temporal dynamics based on the system absolute gains (as demonstrated in Figure 2B).

In the experimental data (Figure 4), we demonstrated that the MNG was significantly correlated to τ (used as reference). The τ calculated from eight repetitions and low-pass filter still has an intrinsic non-physiological variability that could be associated with the low signal-to-noise ratio as a consequence of noise, short data window and/or elevated modeling degrees of freedom. The CI₉₅ of the τ estimated from eight repetitions and low-pass filtered (0.075 Hz) data between all participants (n = 8) was 19.8 s which represented 57% of the average τ-value. In contrast, the MNG presented a lower variability in comparison to τ possibly due to the inherent noise reduction and the lower degrees of freedom of the proposed method. Consequently, only 75% of the MNG variation could explain the variations in τ, both calculated based on eight exercise repetitions (Figure 4). However, we demonstrated (Figure 8) that ≈30% of the τ variability (from total of ≈60%) seems to be a consequence of data modeling by comparing MNG and CI₉₅. This might be occurring because τ, as an explicit parameter, has an intrinsic degree of freedom originating from V˙O2 data modeling (Motulsky and Ransnas, 1987). Therefore, based on Figure 8, the MNG seemed to “isolate” aerobic system temporal dynamics from noise better than τ obtained from V˙O2 time-domain analysis, even after eight repetitions and filtering. In addition, in agreement with previous research (Keir et al., 2014b), the CI₉₅ of τ was more dependent on the number of exercise repetitions in comparison to MNG where the CI₉₅ was independent of exercise repetitions.

The biological significance of τ and MNG was also demonstrated in Figures 5, 6 by estimating these indexes at different conditions described in Table 1. The temporal dynamics of the V˙O2 response, and presumably the aerobic fitness, should not suffer major changes between periods of day or the day of testing. However, MNG (Figure 5) seems to have lower variability associated to the method of choice in comparison to τ (Figure 6) which is in accordance to Figures 7, 8. In addition, the participant identified as an outlier (arrow in Figure 6A) was not apart from the group response in Figure 5A (MNG). Therefore, the unexpected behavior of the V˙O2 dynamics during the exercise transition used for τ estimation was not “transferred” to MNG because the source of this distortion was not periodic.

As demonstrated in Figure 4, the relationship between τ and MNG was systematically below the simulated data (MNG vs. τ_s, Figures 4, 2C). There are two possible explanations for this. Firstly, following Hoffmann et al. (2013) and as expected to occur in the muscle, the simulations were generated based on a non-delayed exponential response (single time constant τ_s, no TD). However, the V˙O2 response at the mouth is classically described as a delayed exponential response (single time constant τ with a TD). The addition of the TD term to the fitting model is a mathematical way to account for the “latency” period when the muscle responses have not started to be expressed at the mouth level due to circulatory transit time. Like the phase shift obtained from frequency-domain analysis (Eßfeld et al., 1987), the parameter TD has an elevated variability between subjects without main physiological relevance. Therefore, comparing the exponential characteristics (i.e., τ) of the V˙O2 response at the mouth to the simulated data τ_s appears to show an incorrect gain amplitude generated at the muscle in higher frequencies effectively “slowing down” the response in frequency domain (i.e., lower MNG-values). A possible way to account for this issue is to consider the sum of τ and TD, or the mean response time (MRT), as the “effective” muscular V˙O2 time constant measured at the mouth level (Linnarsson, 1974; Whipp and Ward, 1990). In comparison to Figure 4 and as depicted in Figure 9, the addition of TD term brings the relationship between MRT and MNG in line with the simulated data. Despite the apparent differences in r and p-values for the MNG vs. MRT compared to the MNG vs. τ (r = −0.802, p = 0.016 and r = −0.855, p = 0.006 respectively for 8 repetitions), the CI₉₅ and the squared error were not statistically different (p > 0.05 by paired t-test) considering the individual responses. Therefore, the inclusion of the TD did not alter the relationship between MNG and the time-domain dynamics indicators (τ or MRT). It is known that TD does not carry biological information (Eßfeld et al., 1987; Hoffmann et al., 2013) and its inclusion is commonly related to data modeling strategy (Whipp and Ward, 1990).

An alternative explanation of why the MNG based on the experimental data is below the simulated data (Figure 4) is based on the V˙O2 system linearity. The V˙O2 measured at the mouth presented a certain degree of energy dispersion across the spectrum due to circulatory distortions and/or all the assumptions that are necessary to obtain an estimate of V˙O2 from ventilatory and gas concentration signals. In contrast, the data simulation was based on a purely linear system that did not present any source of distortion beyond the one related to the exercise stimulus. In the simulations, all energy applied to the system was perfectly converted into the same-order output response by the superposition law, maintaining a higher gain across the frequencies. The possibility exists that even in the range of input stimulation frequencies assumed to result in linear output (V˙O2) (Hoffmann et al., 1992) that non-linearities exist effectively lowering the system response at the higher frequencies. It was previously speculated (Hughson et al., 1990; Hoffmann et al., 1992) that distortions of the circulatory system which includes O₂ stores oscillations, variable muscle-to-lungs transit time and blood venous volume (Hoffmann et al., 2013) might influence the expression of the V˙O2 dynamics at the mouth level at high frequencies.

Another advantage of MNG over time-domain-derived temporal indexes is that it does not require an arbitrary decision for data modeling regarding the length of the cardio-dynamic phase that is variable between participants (Murias et al., 2011). The MNG is estimated from periods longer than 112 s thus this index does not reflect the cardio-dynamic components during exercise transitions.

The intrinsic degree of uncertainty associated with τ estimated from the exponential modeling precludes the use of τ as a “gold standard” method to validate the use of MNG to assess the system temporal dynamics. The CI₉₅ of the relationship between τ (and MRT) with MNG might be influenced by the elevated CI₉₅ of τ estimation (and TD for MRT). Therefore, there is an expected source of error also in the reference method (time-domain) which complicates the validation method. Our data showed that a faster V˙O2 response will be translated to a higher MNG or a lower τ and MRT; however, the ability of the MNG to extract this information from V˙O2 data seemed optimized and more sensitive to detect differences in the system temporal dynamics due to its inherent filtering characteristics and the lower degrees of freedom.

As a Fourier transformation criterion, the proposed method assumes a symmetrical V˙O2 dynamic between the exercise onset and recovery transitions. However, the V˙O2 signal may be composed of asymmetries between these two phases during exercise intensities higher than moderate (Ozyener et al., 2001; Markovitz et al., 2004). The highest intensity used in the current study (100 watts) was restricted to moderate intensity (Bennett et al., 1981; Eßfeld et al., 1987; Hughson et al., 1988); therefore, the MNG can be compared to the τ obtained from the V˙O2 response during the onset exercise transition.

Although the frequency range selected in the current study limits the V˙O2 response to a range where the system linearity is reportedly preserved (Hoffmann et al., 1992) the MNG might still be susceptible to system non-linearities originating from circulatory distortions or some sort of periodic noise can be present at higher frequencies. Further studies might explore the application of specific filtering techniques (Eßfeld et al., 1987; Hoffmann et al., 1992) to remove noises/responses uncorrelated to exercise in order to increase even more the precision of the proposed index to characterize the temporal dynamics of the V˙O2 response. However, consistent with the purpose of this index, we successfully showed that a faster aerobic response can be characterized by a higher MNG since that the majority of the evaluated harmonics were probably linear. These results should also be verified across a wider range of participants with differing levels of physical fitness and health status.

Characterization of physical fitness has classically been conducted by measurement of maximal V˙O2 (Astrand and Saltin, 1961; Drake et al., 1968). Varying levels of physical fitness and the effects of training programs are also associated with differing kinetics of adaptation of V˙O2, expressed by τ, to the challenge of a step increase in Ẇ (Phillips et al., 1995). The temporal characteristics of the oxygen uptake (V˙O2) dynamic during moderate exercise have previously been related to maximal aerobic power (Beltrame and Hughson, 2017b) and a faster V˙O2 response was associated with a better aerobic fitness (Powers et al., 1985; Norris and Petersen, 1998), functional mobility (Alexander et al., 2003), and disease prognosis (Borghi-Silva et al., 2012). This study, beyond demonstrating how to compute, validated a new method to assess V˙O2 dynamics in random exercises more typical of daily life.

The MNG can be used to identify changes in the temporal aerobic system dynamics. The applicability of our findings may extend beyond controlled exercise protocols as shown with simulated activities of daily living (Beltrame et al., 2017a) and in freely moving daily life (Beltrame et al., 2017b). Indeed, MNG has the potential importance to rehabilitation programs, exercise prescription and fitness evaluation where the temporal dynamics of the aerobic response might be related to aerobic power (Beltrame and Hughson, 2017b). The inherent filtering characteristics, the need for no model assumption and the low variability between days and time of the day seems to make MNG attractive for the evaluation of the aerobic system temporal dynamics. Additionally, because MNG is expressed as a percent of the fundamental harmonic, it can be applied to comparisons of system dynamics across the variables contributing to the delivery and utilization of oxygen independent of their formal units. In conclusion, the present study successfully validated the use of the MNG as a tool for aerobic system analysis based on random exercise stimulus.

TB and RH conceived and designed research, performed experiments, analyzed data, interpreted results of experiments, prepared figures, drafted manuscript, edited and revised manuscript, and approved final version of manuscript.