Swimming is widely recognized as a popular sport and it has been part of the Olympic program since the first modern Olympic Games in 1896. Most of the swimming events at the Olympic program are performed between 50 and 200 m (or about 21–150 s), demanding a high rate of ATP resynthesis by aerobic and anaerobic energy systems (Capelli et al., 1998; Pyne and Sharp, 2014). During swimming training, sets of interval exercises at low and high intensities are interspersed with relief periods lasting generally less than 60 s (Nugent et al., 2017). However, the relief time is prescribed by coaches with scarce scientific support. Given environmental and technological constraints in swimming, feasible technologies of training prescription and assessment are helpful to athletes, coaches, and sports scientists.

At the beginning of the last century, Hill (1925) observed a hyperbolic relationship between work rate or speed and performance time. This power/speed-time relationship is characterized by two parameters: critical power (CP) or critical speed (CS) demarcating the boundary between heavy and severe exercise domains, and the maximum amount of work/distance that can be performed above CP/CS represented by the mathematical expression W′ or D′, respectively (Poole et al., 2016). Both CP and CS as well as W′ and D′ are analogous but expressed in different units of measurement. Although the precise mechanisms of W'/D′ have remained elusive (Broxterman et al., 2016; Hureau et al., 2016; Poole et al., 2016), the exercise tolerance provides similar amounts of work/distance performed above CP/CS and similar attainment of a critical level of intramuscular phosphocreatine, inorganic phosphate and/or pH (Fukuba et al., 2003; Vanhatalo et al., 2010; Jones and Vanhatalo, 2017). Therefore, to any severe intensity exercise the task failure coincides with the complete depletion of W'/D′ during constant and intermittent exercises, while the replenishing of W'/D′ necessitates a sub-CP/CS intensity (Coats et al., 2003; Chidnok et al., 2012).

Skiba et al. (2012) proposed a mathematical model to determine the balance of W′ remaining at any given time during an intermittent exercise session (W'_BAL) where some amount of W′ is expended and reconstituted during periods performed above and below CP, respectively. This mathematical model was initially developed for cycling exercise and provides a novel approach for coaches to determine the optimal training intervals and intensity (Skiba et al., 2014a) or for athletes to perform the best pace during a competitive race (Patton et al., 2013). Such a model assumes a linear expenditure and a curvilinear reconstitution of W′ comprising two equations: Eq. 1 determines W'_BAL considering the work intensity and duration, the relief intensity and duration, and the time constant of the exponential reconstitution of the W' (τW′), whereas Eq. 2 estimates τW′ to be inserted into Eq. 1. In the second equation, the difference between power output at recovery and CP (D_CP) is fitted to each relief interval and participant, while the mathematical constants are arbitrary parameters from cycling exercise determined by plotting D_CP with actual τW' (found by an iterative process until modeled W'_BAL equaled zero at the time to exhaustion) (Skiba et al., 2012). Therefore, in theory, whether CP is unchanged and relief intensity is the same for different interval training sessions, the τW′ should be the same for these exercise sessions regardless of work interval intensity and duration as well as relief interval duration. However, it is unclear whether τW′ remains constant during different interval training sessions with the same work interval intensity and duration as well as relief intensity but different relief interval durations. W ′ b a l = W ′ − ∫ 0 t ( W ′ ⁡ exp ) ( e − ( t − u ) / τ W ′ ) (1) τ W ′ = 546 e ( − 0.01 D c p ) + 316 (2) where the W'_BAL at any point during a training session or race is the difference between the known W′ and the total W′ expended, W′ equals the subject’s known W′ as calculated from CP model, W’exp is equal to the expended W′, (t - u) is equal to the time in seconds between segments of the exercise session that resulted in a depletion of W′, τW′ is the time constant of the reconstitution of the W′, and D_CP is the difference between the recovery power output and the CP.

Although the W'_BAL model has been proposed to characterize the expenditure and reconstitution of W′ during intermittent cycling exercises, there are few studies investigating this model for other exercise modalities (Galbraith et al., 2015; Broxterman et al., 2016). Galbraith et al. (2015) applied the W'_BAL model during intermittent running (i.e. D'_BAL) and showed a D'_BAL negative on average (−21.2 m) at interval training session termination. The authors also reported a time constant of the exponential reconstitution of the D' (τD′), determined interactively until modeled D'_BAL equaled zero at the time to exhaustion. Apparently the τD′ was lower compared with the previously reported τW′ in intermittent cycling (∼376 vs. 578 s) (Skiba et al., 2012). During severe intensity handgrip exercises, τW′ was affected by different contraction–relaxation cycles, ranging between 580 and 2,450 s (Broxterman et al., 2016). In addition, the authors reported an exponential decay relationship between τW′ and CP (Broxterman et al., 2016). Taken together, these data indicated that τW'/τD′ could be modality-specific and should be directly determined.

Based on the aforementioned statements, the application of the D'_BAL model for swimming exercise would be useful as a feasible training prescription tool not requiring any sophisticated apparatus. Therefore, the purposes of this investigation were to 1) test the applicability of the D'_BAL model for swimming; 2) determine an equation to estimate τD′ for swimming; and 3) verify if τD′ is constant during two swimming interval training sessions with the same work interval intensity and duration as well as recovery intensity but with different recovery interval durations. We hypothesized that 1) D'_BAL could be suitable for swimming exercises; 2) τD′ would be related to CS; and 3) τD′ would be similar between different interval trainings sessions. The implications of confirming these hypotheses for coaches and sports scientists would be a cost-free practical tool to consistently determine optimal intervals and intensities for swimming improving exercise prescription and experimental designs.

The performances for 200, 400, 600, and 800 m races lasted 136 ± 8, 297 ± 21, 461 ± 35, and 624 ± 45 s, respectively. The distance-time relationship provided average values of 1.23 ± 0.09 m s⁻¹ (91.2 ± 2.7% of the 400 m pace) and 33.69 ± 8.65 m for CS and D′, respectively. The goodness of fit of the distance–time relationship was 0.999 ± 0.001 (range: 0.998—0.999). The mean standard error of the estimate were 0.01 ± 0.01 m s⁻¹ (1.1 ± 0.7%) for CS and 5.95 ± 3.87 m (18.7 ± 12.4%) for D'. Using Eq. 3, the constant swimming speed that would result in exhaustion in 180 s during continuous exercise was estimated to be 1.42 ± 0.08 m s⁻¹ (105.2 ± 2.3% of the 400 m pace). The work interval duration was 35 ± 2 s, while the recovery durations were 18 ± 1 s for T2:1 and 9 ± 1 s for T4:1. The TTE (work plus recovery intervals) were 856 ± 355 s for T2:1 and 301 ± 72 s for T4:1.

The mean and individual values of actual τD′ found by an iterative process from T2:1 and T4:1 are shown in Table 1. The τD′ was similar between T2:1 and T4:1 [t (12) = -1.13, p > 0.05; 95% CI = -994 to 312 s] but it showed a within-subject coefficient of variation of 306%. Figure 2A shows the bias ±95% limits of agreement of actual τD′ found interactively in T2:1 and T4:1. The D'_LOW was lower in T2:1 (−1.86 ± 1.73 m) compared with T4:1 (−0.06 ± 0.23 m) [t (12) = −4.13, p < 0.05; 95% CI = −2.74 to—0.85 m]. The D'_LOW was lower than zero for twelve swimmers in T2:1 and for two swimmers in T4:1, consequently it was equal to zero for one swimmer in T2:1 and eleven swimmers in T4:1. The Total D′ reconstituted during the passive rests was higher in T2:1 (105 ± 63 m) compared with T4:1 (20 ± 17 m) [t (12) = 5.76, p < 0.05; 95% CI = 52–117 m], as well as total D′ expended during exercise was higher in T2:1 (138 ± 64 m) compared with T4:1 (53 ± 18 m) [t (12) = 5.77, p < 0.05; 95% CI = 52–117 m]. An example of individual D'_BAL model data for a single swimmer in both training sessions is shown in Figures 3A,B (as these τD′ were calculated by iterative processes D'_END has to be equal to zero).

When the τD′ determined for T2:1 was applied in T4:1 and vice versa, the D'_END were similar between T2:1 (-10.8 ± 35.8 m; 95% CI = −32.5 to 10.8 m) and T4:1 (−2.6 ± 7.4 m; 95% CI = −7.0 to 1.9 m) [t (12) = −0.71, p > 0.05; 95% CI = −33.5 to 16.9 m]. The bias and 95% limits of agreement between actual D'_END (i.e. interactively determined and equal to zero) and estimated D'_END (i.e. τD′ inverted) was 10.8 m and -59.3–81.0 m for T2:1 and 2.6 m and −11.8–17.0 m for T4:1, respectively.

An example of individual D'_BAL model data when the τD′ was inverted for a single swimmer in both training sessions is shown in Figures 3C,D. It was not possible to predict the TTE with τD′ inverted in T2:1 for seven swimmers because the D'_END did not approach zero. In the other six swimmers, the actual TTE was 653 ± 267 s, while the predicted TTE was 261 ± 71 s. The bias and 95% limits of agreement between actual and predicted TTE for T2:1 are shown in Figure 4A and the coefficient of variation was 29%. For T4:1 it was possible to predict the TTE (261 ± 75 s) for all swimmers when applied the τD′ found by an iterative process from T2:1. The bias and limits of agreement between actual and predicted TTE for T4:1 are shown in Figure 4B and the coefficient of variation was 28%. No linear or nonlinear relationships were found between τD′ and CS (all R ² < 0.04 or not converged; Figures 5A,B).

The swimmer 11 showed a very different τD′ value in T4:1 (4,000 s) compared with T2:1 and other swimmers (Table 1). This swimmer exhibited no difference during the data collect. Thus, the source for this discrepancy is unclear (e.g. physiological response or random error), but results remain similar when reanalyzed excluding this swimmer. As a result of such reanalyzing, the within-subject coefficient of variation of τD′ was 80.4% with no agreement between the two τD′ values (Figure 2B). The bias and 95% limits of agreement between actual and predicted TTE was 339 s and −186–863 s for T2:1 (n = 5) as well as 49 s and −182–281 s for T4:1. Coefficient of variation between actual and predicted TTE was 43% for T2:1 (n = 5) and 29% for T4:1. No linear or nonlinear relationships between τD′ and CS were found without the swimmer 11 (all R ² < 0.03 or not converged; Figures 5C,D).

This was the first study to model the D′ expenditure and reconstitution during swimming exercise. The main finding of this study was that τD′ is not constant during two similar high-intensity interval trainings, showing high variability between sessions. Thus, when the τD′ determined for T2:1 was applied in T4:1 and vice versa, the D'_BAL model was inconsistent to predict the exhaustion of swimmers. In addition, τD′ was not related to CS regardless of the linear or nonlinear equations used. The initial hypothesis has been refuted, suggesting that the current form of D'_BAL model is inconsistent to track the dynamic response of D′ during intermittent swimming exercises.

Skiba et al. (2012) were the first to develop the CS/CP model for intermittent exercise using linear expenditure and curvilinear reconstitution of W′ during cycling. According to model theory, the curvilinear reconstitution of the D'/W′ occurs below CS/CP and it is dependent on the difference between recovery intensity and CS/CP (Chidnok et al., 2012; Skiba et al., 2012). Therefore, different training sessions with the same recovery intensity should produce the same τD'/τW'. However, Skiba et al. (2014b) reported that decreasing the recovery duration from 30 to 20 s resulted in an additional reduction of τW′ during cycling exercise with the same recovery intensity. Recently, Caen et al. (2019) and Lievens et al. (2020) confirmed that recovery characteristics can affect W′ reconstitution during cycling exercise. In particular, the model seems to underestimate the reconstitution of W′ after shorter recovery intervals (Caen et al., 2019). In addition, Chorley et al. (2019) and Chorley et al. (2020) reported that the reconstitution of W′ is subject to fatigue following successive bouts of maximal exercise and related to aerobic fitness. Taken together, these results demonstrate that the reconstitution of D'/W′ is more complex than the current model considers (Equations (1) and (4)). As τD′ represents the rate of D′ reconstitution, any physiological changes related to D′ reconstitution should affect τD'. However, the physiological parameters related to D'/W′ have yet to be fully elucidated (Broxterman et al., 2015; Vanhatalo et al., 2016; Raimundo et al., 2019) to improve understanding of D'/W′ reconstitution. In the present study, although not statistically different, τD′ had high variability between training sessions, which resulted in low predictability of D'_END and TTE when the τD′ determined for T2:1 was applied in T4:1 and vice versa. Notably, most studies reporting τD′ and the predictive ability of the D'_BAL model have only reported systematic changes (Skiba et al., 2014b; Broxterman et al., 2016; Shearman et al., 2016). Despite being an important component for analyzing the robustness of the model, systematic changes do not indicate the consistency of τD′ and D'_BAL model as a coefficient of variation and limits of agreement (Hopkins, 2000). Therefore, the dynamics of D′ reconstitution need to be better understood and mathematically described for τD′ to be widely applicable during different swimming interval trainings.

The present study observed that τD′ was not related to CS regardless of linear or nonlinear equations used. On the other hand, τW′ was previously related to D_CP or CP in cycling, running, and handgrip exercises (Skiba et al., 2012; Broxterman et al., 2016; Vassallo et al., 2020). As previously mentioned, it is possible that no relationship was found because of the high within-subject variability of τD'. Also, the discrepancies between results might, to a large extent, be due to these studies employing recovery intensity based on different exercise domains (Skiba et al., 2012; Vassallo et al., 2020) or different contraction–relaxation cycles (Broxterman et al., 2016). In accordance, when performing a visual inspection in figures presented by Skiba et al. (2012), Vassallo et al. (2020), and Broxterman et al. (2016), the relationships would likely have a worse or no fit if only one exercise domain were used for recovery intensity (Skiba et al., 2012; Vassallo et al., 2020) or contraction–relaxation cycle (Broxterman et al., 2016). Therefore, although passive rests are usually employed during swimming interval trainings, future studies should relate τD′ and D_CS with recovery intensities of different exercise domains.

Considering the aspects mentioned above, the current form of D'_BAL model was inconsistent for the swimming interval training sessions tested herein. Before incorporating the D'_BAL model into common practices of swimming teams, this model should include other physiological variables as τD′ not being constant during a training session (Chorley et al., 2019). For instance, phosphocreatine seems to be one of the determinants of D'/W' (Miura et al., 1999) as intramuscular phosphocreatine is depleted during high-intensity exercise (Vanhatalo et al., 2010). However, a model for phosphocreatine resynthesis showed a higher concentration of phosphocreatine after exercise, with the phosphocreatine concentration rising ∼5% above that recorded at rest (Nevill et al., 1997). Furthermore, priming exercise can increase CS/CP and/or D'/W' (Miura et al., 2009; Burnley et al., 2011), overestimating the amount of D'/W′ depleted during exercise above CS/CP and underestimating the replenishment during exercise below CS/CP. Collectively, all these physiological factors should be considered to provide a greater practical application of D'_BAL model in swimming.

In summary, this study confirmed that τD′ is not constant during two swimming interval training sessions, although the same recovery intensity has been used. Consequently, when the τD′ determined for T2:1 was applied in T4:1 and vice versa, the D'_BAL model was not able to predict the exhaustion of swimmers. Therefore, the current form of D'_BAL model was inconsistent to track the dynamic response of D′ during swimming, at least for the interval workouts tested herein.