Bi-exponential modelling of \(W^{^{\prime}}\) reconstitution kinetics in trained cyclists

The critical power model introduced by Monod and Scherrer (1965) describes the hyperbolic relationship between constant power output and tolerable duration within the confines of the ‘severe’ intensity domain (Eq. 1). The model consists of two parameters: critical power (CP), which is the asymptote of the hyperbola, and the curvature constant (W′). Furthermore, the model can also be rearranged mathematically (Morton 1994) to predict the tolerable duration of ramp exercise (Eq. 2) where the S is the ramp rate.where T_lim is the time to limit of tolerance (s); W′ is the work capacity above CP (J); P is the power output (W); CP is the critical power (W); S is the ramp rate (W s⁻¹).

CP represents the highest power output that can be sustained by the provision of adenosine triphosphate from wholly aerobic means (Coats et al. 2003; Poole et al. 1988), and the maximum work rate at which metabolic homeostasis can be maintained. As such, it denotes the physiological boundary between the ‘heavy’ and ‘severe’ intensity domains (Jones et al. 2019). W′ is the finite capacity of work that can be performed above CP (Jones and Vanhatalo 2017), However, the underlying biochemistry that comprises W′ remains only partially understood. Initially thought of as ‘anaerobic work capacity’ and believed to be dependent upon levels of phosphocreatine (PCr), stored glycogen and oxygen within the muscle (Moritani et al. 1981), W′ is now considered to be at least partly dependent upon the accumulation of fatiguing metabolites such as adenosine diphosphate, inorganic phosphates and hydrogen ions (Fukuba et al. 2003; Johnson et al. 2014; Jones et al. 2008). Most recently, exercise-based investigations have associated the magnitude of W′ with the development of the oxygen uptake (\(\dot{V}{\text{O}}_{2}\)) slow component (Burnley and Jones 2018; Murgatroyd et al. 2011), muscle glycogen availability (Clark et al. 2019; Miura et al. 2000), and leg morphology (Byrd et al. 2017). The kinetics of W′ are of particular interest within competitive cycle sport as the outcomes of many races are decided by the efficacy of riders’ intermittent efforts above CP interspersed with short recovery periods below CP (Craig and Norton 2001; Vogt et al. 2007) that allow for the partial reconstitution of W′ (Chidnok et al. 2013a).

Like W′, intramuscular PCr stores deplete when exercising above CP and reconstitute when power output is reduced below CP (Chidnok et al. 2013b). Indeed, there is a significant relationship between the two (Chidnok et al. 2013a), albeit that W′ recovers at a slower rate than PCr (Ferguson et al. 2010). Furthermore, both PCr (Chidnok et al. 2013a) and W′ reconstitution kinetics (Chorley et al. 2019) are slowed following repeated severe intensity efforts that culminate at the limit of tolerance, suggesting that W′ reconstitution processes are partially dependent upon PCr regeneration. The time-course of W′ reconstitution has been described as curvilinear by Ferguson et al. (2010) following observations of its recovery to 37%, 65% and 86% of baseline levels resulting from respective recovery durations of 2, 6 and 15 min. More extensive modelling of W′ reconstitution was subsequently carried out by Skiba et al. (2012) to produce a mono-exponential model of W′ reconstitution (Eq. 3) derived from a short intermittent exercise protocol (60 s work, 30 s recovery) to the limit of tolerance using untrained cyclists:where W′_bal is the balance of W′ at time t (J); W′ is the work capacity above CP (J); W′_exp is the W′ expended (J); t − u is the recovery duration (s); τ_W′ is the W′ reconstitution time constant (s).

The time constant (τ_W′) was found to be inversely correlated to the difference between CP and recovery power output (D_CP) and fitted to the model via non-linear regression to produce Eq. 4 (Skiba et al. 2012).where τ_W′ is the W′ reconstitution time constant (s); DCP is the difference between CP and recovery power output.

It has, however, been suggested that the model underestimates W′ reconstitution in elite cyclists (Bartram et al. 2017) and does not account for the slowing of W′ reconstitution with repeated maximal incremental exercise (Chorley et al. 2019). Furthermore, large individual variations in τ_W′ were observed in both the modelling of τ_W′ in untrained cyclists (Skiba et al. 2012), and the modified τ_W′ model for elite cyclists (Bartram et al. 2017). As other research into W′ reconstitution kinetics has similarly found high inter-individual variability of W′ reconstitution, it has been argued that τ_W′ should be determined on an individual basis (Caen et al. 2019; Chorley et al. 2019; Skiba et al. 2015) rather than the use of Eq. 4. This reliance solely upon D_CP for the determination of τ_W′ has been questioned (Chorley and Lamb 2020) following significant differences being found between predicted W′ reconstitution and experimental measurements (Chorley et al. 2019; Lievens et al. 2021) and several markers of aerobic fitness together with age and body composition have been correlated with W′ reconstitution (Chorley et al. 2020). Hence, it is posited that such individual factors might contribute to the accuracy of W′ reconstitution modelling.

The time course of W′ reconstitution has yet to be completely elucidated, such that it remains unknown whether a mono-exponential or a multi-exponential model best describes W′ reconstitution kinetics, and accounts for its slowing due to repeated efforts. Therefore, the main aim of this study was to investigate the individual W′ reconstitution kinetics of trained cyclists, specifically over several short duration (< 6 min) time points following repeated maximal incremental exercise, and to determine an optimal non-linear model to describe W′ reconstitution. It was hypothesised that W′ reconstitution will be best explained by a multi-exponential mathematical model incorporating variables that account for high inter-individual variations of the W′ reconstitution time-course. The secondary aim of the study was to determine if the W′ reconstitution model parameters could be adequately determined via fewer (two) exercise testing sessions.

Individual measurements of CP, W′, \(\dot{V}{\text{O}}_{2\max }\), body mass, and fat mass are shown in Table 1.

This study has demonstrated the time-course of W′ reconstitution tracked a curvilinear path for all participants following both the first and second recovery periods, extending to approximately 75% of W′ reconstitution within the first 4 min of recovery. Our data mirrors previous findings (Ferguson et al. 2010); however, the additional data over the short (< 2 min) recovery times from the present study revealed a bi-phasic pattern of W′ reconstitution kinetics comprising a distinct initial fast phase of W′ reconstitution before noticeably slowing from 60 s onwards. The new bi-exponential model proved to be an excellent fit and superior to existing mono-exponential models of W′ reconstitution. Furthermore, this study demonstrated the novel finding that the fatiguing effect of repeated bouts (Chorley et al. 2019) is confined to the slower phase of W′ reconstitution, evident beyond 180 s (see Fig. 2). That the extent of W′ reconstitution in the current study was notably greater than previously shown by Ferguson et al. (2010) after 120 s of recovery (~ 63% versus ~ 37%) is likely explained by the different training status and the resultant critical power (~ 300 W versus ~ 213 W) of the participants (Chorley et al. 2020; Skiba et al. 2012); however, the effect of the differing ramp and constant load exercise during the W′ depletion phase remains to be determined. Despite the homogeneity of the participants in the present study in terms of CP (CV: 12.4%), W′ (CV: 19.6%) and \(\dot{V}{\text{O}}_{2\max }\) (CV: 11.5%), the reconstitution of W′ (CV: 28.8% at 30 s) varied markedly between individuals, particularly over the shorter durations, with W′ reconstitution ranging between 24 and 60% (absolute values: 2.5–4.8 kJ) after the first 30 s. Large differences in W′ reconstitution rates have previously been reported via the τ of mono-exponential models (Caen et al. 2019; Skiba et al. 2014b).

We hypothesised that a multi-exponential model would best represent the curvilinear reconstitution of W′ following exhaustive exercise, and indeed when fitted against the measurements of W′ reconstitution the mono-exponential function proved a poor fit, even when individually fitted for each participant, whilst the bi-exponential model proved to fit well when individually parameterised (yielding an adjusted R² > 0.94 in all cases). That the bi-exponential model was not preferred in all individual cases based on AICc analysis was likely due to the relatively high number of model parameters of the bi-exponential model compared and relatively low number of W′ reconstitution data points. Indeed, the bi-exponential model was preferred for all those participants completing the additional four test sessions. The W′_bal models (Skiba et al. 2012, 2015) and modifications (Bartram et al. 2017) previously explored are based on mono-exponential reconstitution of W′, originally devised following an intermittent 60-s work, 30-s recovery protocol to exhaustion, with no intermediate measurements of W′ possible. The mono-exponential W′_bal model has been validated using similar intermittent protocols in hypoxia (Shearman et al. 2016; Townsend et al. 2017), and by retrofitting to the point of exhaustion during training and race data (Skiba et al. 2014a), where the mono-exponential model proved a successful fit against the measurements of W′ reconstitution over the short intermittent recoveries. Validations of the mono-exponential W′_bal model via different protocols have, however, found significant differences against longer recovery durations (Chorley et al. 2019) and partial prior depletion of W′ (Lievens et al. 2021; Sreedhara et al. 2020), albeit without τ being individually fitted. Where τ has been individualised, it has only been done so against W′ reconstitution at specific measured time points (Caen et al. 2019; Chorley et al. 2020) rather than against a time-course of W′ reconstitution.

Whilst the mono-exponential model can be resolved such that τ is adjusted to fit any single time point, it cannot follow the reconstitution of W′ over time as well as the bi-exponential model does since the latter accommodates the compartmental fast and slow phases observed in the W′ reconstitution time course. The fast and slow components for both the group fit and the mean of the individual fits are of similar magnitudes (each being ~ 50% of the overall recovery magnitude); however, as with their respective τ there is high variability between individuals. The underlying determinants of W′ were originally thought to comprise phosphate and intramuscular stores of glycogen and oxygen (Moritani et al. 1981), yet later findings have suggested an accumulation of fatiguing metabolites and muscle metabolic perturbations (Jones et al. 2008; Vanhatalo et al. 2016). It would seem plausible that the complex mechanisms that underpin W′ reconstitution are at least partially dependent upon both the replenishment of energy stores and the removal of muscle metabolites. Indeed, the exponential recovery of PCr has been previously evaluated as τ = 29.4 s in the vastus lateralis (van den Broek et al. 2007) and τ = 25 s in the calf (Haseler et al. 1999) both comparing closely to τ_FC in the present study, whilst τ_SC is comparable to blood lactate clearance following a repeated sprint protocol (Menzies et al. 2010). Attributing the fast and slow components of the bi-exponential model to these two factors may be oversimplistic, given that PCr recovery alone may follow a more complex bi-phasic time course (Harris et al. 1976; Iotti et al. 2004), and that blood lactate is at best a delayed proxy for muscle metabolism (Rusko et al. 1986), hence the need for a greater understanding of the interactions that comprise W′. As PCr recovery is an oxygen dependent process (Haseler et al. 1999), it is likely that \(\dot{V}{\text{O}}_{2}\) kinetics will influence the restoration rate of PCr during recovery and consequently the reconstitution of W′. Prior exercise has been shown to alter \(\dot{V}{\text{O}}_{2}\) responses for up to 45 min (Burnley et al. 2006) and repeated sprint performance which is almost certainly dependent upon W′ reconstitution during recovery, is better maintained by those with faster \(\dot{V}{\text{O}}_{2}\) kinetics during the recovery phase (Dupont et al. 2010). Whilst detailed \(\dot{V}{\text{O}}_{2}\) kinetics were beyond the scope of the current study, the relationship between \(\dot{V}{\text{O}}_{2}\) recovery and FCamp suggests further investigation is warranted.

The bi-exponential model presented demonstrates that fatigue due to repeated efforts is confined to the slowing of W′ reconstitution kinetics represented by an increase in τ. The relative amplitudes of the fast and slow components remain unchanged between the recovery periods following the modelling optimisation process, suggesting that small variations in amplitude do not warrant a more complex model. Given that W′ reconstitution has been shown to slow following repeated efforts (Chorley et al. 2019, 2020), that τ_FC in the present study did not increase was somewhat unexpected. Colloquial cycling terminology refers to cyclists “burning matches” when they perform high-intensity surges and having a limited number of “matches” available; it is feasible that τ_FC may increase after a greater number of repetitions than undertaken in the present study. Contrastingly, τ_SC did increase significantly during the recovery period following a single repeated bout. Neither body composition nor physiological characteristics in this homogenous group were found to be related to τ_SC itself. However, the difference in τ_SC between the first and the second recoveries (\(\Delta \tau_{{{\text{SC}}}}\)), which describes the extent of the slowing of W′ reconstitution with repeated bouts of exercise, was related to the measures of aerobic fitness (CP and \(\dot{V}{\text{O}}_{2\max }\)), heart rate recovery and EPOC, as found previously (Chorley et al. 2020). Additionally, the delta τ_SC was related to thigh muscle girth, which has previously been shown to correlate with W′ (Kordi et al. 2018; Miura et al. 2002) independently of muscle fibre type distribution (Vanhatalo et al. 2016). Interestingly, one participant had a greater W′ reconstitution following the second recovery across all time points (15–360 s) which stood out from the correlates of aerobic fitness. Whilst the individual demonstrated high \(\dot{V}{\text{O}}_{2\max }\) and CP it was notable that he was alone in having previously competed as an elite road cyclist, indicating that fatigue resistance may have a hitherto unexplained component that influences race performance and selection.

The present study demonstrated a bi-exponential, rather than a mono-exponential, model provides a superior fit to W′ reconstitution kinetics during active recovery at a nominal 50 W. Exponential models have been used to describe physiological processes such as PCr recovery (Iotti et al. 2004; van den Broek et al. 2007) and the goodness of fit of the bi-exponential model supports its selection in the present study. Other mathematical models could also be generated to describe W′ reconstitution kinetics; however, it is likely a larger number of model parameters would be required to do so. A secondary finding was that when the bi-exponential model parameters were calculated using measured W′ and its fractional reconstitution from only the 30-s and 240-s time points, this provided an excellent fit against the W′ reconstitution kinetics for everyone that was no different to that of using multiple recovery time points. That the prediction model produces comparable results from just the 30-s and 240-s time points allows the test burden to be reduced considerably (to a baseline and two experimental tests), although it should be noted that the prediction equations have yet to be tested against a different data set. As the effect of changing recovery power output is known to affect W′ reconstitution below CP (Caen et al. 2019; Skiba et al. 2012), future studies should seek to establish a three-dimensional model that explains W′ reconstitution kinetics at varying recovery power outputs as would be encountered under race conditions, enabling its application to competitive cycle sport.

Understanding the reconstitution kinetics of W′ of individual athletes can describe a performance characteristic which can be used to influence race outcomes tactically by manipulating severe intensity attacks and recovery durations. The present data has shown that the reconstitution kinetics of W′ among trained cyclists were best described by a new bi-exponential model based on a fast component and a slow component, the parameters of which varied markedly for individual cyclists despite similar training status. A further novel finding was that the slow component alone exhibited impaired W′ reconstitution kinetics following a repeated bout of exercise, the magnitude of which was related to measures of aerobic fitness. The awareness of such individual characteristics can be used to inform training programmes and race tactics. Additionally, for assessment and monitoring purposes, using only 30-s and 240-s recoveries were found to be effective in determining W′ reconstitution kinetics when compared to modelling using a wider range of recovery durations.