The critical power (CP) concept was originally introduced by Monod and Scherrer (1965), and describes the relationship between sustainable power output and duration for severe-intensity exercise. A simple hyperbolic, two parameter model was proposed:

where W′ = total work accumulated above CP until task failure, t_lim = duration until task failure, P = power output, and CP = critical power, defined as a rate limited sustainable power output below which no net expenditure of W′ occurs. Equation (1) can be conceptualized according to a hydraulic model (Margaria, 1976; Morton, 2006), whereby the value W′ progressively depletes during exercise whenever P > CP and reconstitutes when P<CP. Additionally, it can also be seen that if W′ depletes to zero, then t_lim also reaches zero and hence task failure is attained. A continuous integral function has been proposed, termed the “work-balance” (WBAL′) model, which mathematically describes the depletion and reconstitution of W′ (Skiba et al., 2012, 2014a, 2015). A differential equation form of the WBAL′ has also been proposed (Skiba et al., 2015). As per the abovementioned hydraulic model, the WBAL′ models predict that task failure during an intermittent task is attained when W' depletes to 0 kJ. Both versions of the WBAL′ model require CP and W′ as input variables, hence accurate estimation of CP and W′ is a prerequisite for computing WBAL′. However, studies have reported that moderate hypoxia decreases CP (Dekerle et al., 2012; Simpson et al., 2015; Shearman et al., 2016) and severe hypoxia decreases both CP and W′ (Valli et al., 2011). Since it is impractical to measure CP and W′ at all possible altitudes, a prediction model which corrects CP and W′ for the effect of hypoxia would permit WBAL′ computation at any given altitude.

The mechanistic basis of the two parameter CP model has been extensively studied (Jones et al., 2010). Numerous investigations have demonstrated that CP corresponds to the highest exercise intensity at which pulmonary V˙O₂ (Poole et al., 1988; Jones et al., 2008; Vanhatalo et al., 2016), and intramuscular high energy phosphates (Jones et al., 2008; Chidnok et al., 2013), can achieve a steady state. The achievement of steady-state V˙O₂ kinetics indicates the net energetic demand of the task can be met by oxidative metabolism (Poole et al., 2016). When the task requirement increases beyond CP the additional energetic demand is supplemented by substrate level phosphorylation (Vanhatalo et al., 2016) which induces non-steady state V˙O₂ kinetics, accelerated degradation of high energy phosphates, and accumulation of metabolites involved in peripheral muscle fatigue (Jones et al., 2008; Poole et al., 2016). Since the total amount of work accumulated above CP is terminated at the moment of task failure, the mechanisms leading to task failure itself also determine the value of W′. Recent work examining the etiology of neuromuscular fatigue during high intensity exercise reveals that some combination of both central and peripheral fatigue mechanisms is always present at the moment of task failure, whilst the relative contribution of each depends on the task (Hureau et al., 2016). Observed values for [PCr], [P_i], and [H⁺] at task failure have been shown to remain similar despite manipulations in pacing strategy during continuous exercise (Burnley et al., 2010), or recovery duration (Chidnok et al., 2012) and recovery power (Chidnok et al., 2013) during intermittent exercise. Also, following exhaustive high-intensity exercise the magnitude of peripheral fatigue, assessed via twitch interpolation, remains consistent in normoxia and moderate hypoxia (Amann et al., 2006b; Romer et al., 2007). Recently it was demonstrated that changes in peripheral fatigue (assessed via twitch potentiation) was significantly correlated to changes in both [P_i] and [H⁺] during exercise when lower limb muscle afferent feedback was impaired using lumbar intrathecal fentanyl (Blain et al., 2016). Collectively, these findings have led to the theory that within a given task, peripheral muscle fatigue may be regulated via group III/IV afferent feedback, which limits central motor drive to the locomotor muscle (Hureau et al., 2016). The existence of such a feedback loop might explain why W′ appears to resemble a fixed capacity within a given task (Broxterman et al., 2015). From a mathematical perspective, a fixed value of W′ allows performance during high-intensity tasks to be predicted using the 2-parameter CP model.

During exercise in hypoxia, convective O₂ transport to the working muscle is reduced (Amann and Calbet, 2008), and multiple studies have reported a significant decrease in CP without a corresponding change in W′ (Dekerle et al., 2012; Simpson et al., 2015; Shearman et al., 2016). If CP is lower in hypoxia, then according to the CP model a given absolute exercise intensity in the severe domain will result in a faster rate of W′ depletion. Previously we reported a large error in modelled WBAL′ during intermittent exercise performed in hypoxia when the normoxic CP estimate is used (Shearman et al., 2016). Therefore, CP must either be tested in hypoxia, or estimated from measurements in normoxia. Various studies have examined the effect of increasing altitude on V˙O_2max reporting either a linear decrease (Wehrlin and Hallén, 2006; Clark et al., 2007), a curvilinear decrease (Péronnet et al., 1991; Bassett et al., 1999), or a curvilinear interaction between altitude and sea level V˙O_2max (MacInnis et al., 2015). To our knowledge, no studies have examined the dose-response effect of increasing altitude on CP and W′. Moreover, a large reduction in W′ was reported at high altitude (5,050 m) (Valli et al., 2011), whereas another study found no change at simulated altitude equivalent to 3,800 m (Simpson et al., 2015). Hence, the approximate threshold altitude where W′ begins to decline remains unclear. The purpose of this study was to examine the dose-response effect of increasing altitude on both CP and W′, and thereafter to develop a prediction equation enabling WBAL′ computation in hypoxia. A secondary aim was to compare the integral vs. the differential equation form of the WBAL′ model. We hypothesized that CP would decline in a curvilinear fashion commencing from the lowest altitude above sea level (1,250 m) tested, whereas W′ would only begin to decline at altitudes above ≈3,800 m.

Previous studies have quantified the decrease in V˙O_2max that occurs at altitude (Bassett et al., 1999; Wehrlin and Hallén, 2006; Clark et al., 2007; MacInnis et al., 2015), however this is the first to examine the effect of altitude on CP and W′. Consistent with data on the dose-response effect of hypoxia on V˙O_2max (Bassett et al., 1999; MacInnis et al., 2015), we observed a curvilinear decrease in CP. Secondly, the effect of hypoxia on W′ appears to display threshold type characteristics since there were no significant changes at lower altitudes, whereas a decrease occurred at the highest altitude only (4,250 m). Lastly, we have demonstrated for the first time that a prediction equation can be used in place of actual measurements of CP + W′ at simulated altitude of 2,250 m, to characterize intermittent high-intensity exercise using the WBAL′ model.

With increasing altitude, V˙O_2max has been shown to decrease linearly (Wehrlin and Hallén, 2006; Clark et al., 2007), or curvilinearly (Bassett et al., 1999; MacInnis et al., 2015). The curvilinear decrease in CP observed in this study (see Figures 1A,C) was similar to that for V˙O_2max reported by Bassett et al. (1999). Whilst the effect of hypoxia on the factors that determine V˙O_2max are well understood (Wagner, 1996), less is known about the determinants of CP in hypoxia. Traditionally, CP has been considered to reflect a rate limited aerobic energetic supply (Jones et al., 2010). However, it is important to note that V˙O₂ at CP is below V˙O_2max (Poole et al., 1988; Vanhatalo et al., 2016), and therefore oxidative metabolism at CP cannot be “rate limited.” Rather, CP is associated with the highest exercise intensity where a V˙O₂ steady state, and muscle “metabolic stability” can be achieved (Poole et al., 1988; Jones et al., 2008; Vanhatalo et al., 2016). Metabolic stability is characterized by minimal disturbance to intramuscular [PCr], [P_i], [H⁺], [ADP_free], [AMP_free] and Gibbs free energy of ATP hydrolysis (Grassi et al., 2011). In hypoxia, a decrease in convective O₂ transport to working muscle occurs (Amann and Calbet, 2008), and the V˙O₂ primary component decelerates (Hughson and Kowalchuk, 1995). Since the V˙O₂ primary component is considered an “epiphenomenon” of metabolic stability (Grassi et al., 2011), and has been shown to correlate with CP (Murgatroyd et al., 2011), then an O₂ supply limitation on V˙O₂ kinetics may impair metabolic stability, and thus explain why CP is reduced in hypoxia (Dekerle et al., 2012; Simpson et al., 2015; Shearman et al., 2016).

In the present study we found no significant differences in W′ at moderate altitudes up to 3,250 m, however a marked reduction (≈27%) occurred at 4,250 m (see Figure 1B). These results broadly align with several other investigations examining the effect of differing magnitudes of hypoxia on W′ (Valli et al., 2011; Dekerle et al., 2012; Simpson et al., 2015; Shearman et al., 2016). In recent years improved understanding of the mechanistic basis of W′ has developed. When the exercise intensity increases beyond CP, there is progressive recruitment of type IIx muscle fibers (Copp et al., 2010) and a slowing of V˙O₂ uptake kinetics (Brittain et al., 2001). Slower V˙O₂ uptake kinetics allows progressive deterioration of muscle metabolic stability to occur which has been demonstrated for both constant load (Jones et al., 2008), and intermittent exercise (Chidnok et al., 2013). The cellular changes associated with failure of metabolic stability are believed to be linked to the emergence of the V˙O₂ slow component, and to underlie mechanisms of peripheral muscle fatigue (Grassi et al., 2015). Murgatroyd et al. (Murgatroyd et al., 2011) reported a significant correlation between the V˙O₂ slow component magnitude and W′, which suggests the capacity to complete work above CP until the point of exhaustion is ultimately determined by the mechanisms of fatigue contributing to task failure. In moderate hypoxia (FiO₂: ≈0.15), the absolute magnitude of peripheral muscle fatigue, assessed via twitch interpolation, remains similar to normoxia following either constant load work to task failure (Amann et al., 2007), or self-paced TT exercise (Amann et al., 2006a). Additionally, Romer et al. (Romer et al., 2007) found the rate of peripheral fatigue development to increase in hypoxia (FiO₂: ≈0.13) compared with normoxia, but the absolute magnitude remained similar. These findings support the notion that the apparent fixed nature of W′ may be linked to the existence of a peripheral muscle fatigue limit which cannot be surpassed despite varying experimental conditions, including hypoxia (Amann et al., 2006b; Romer et al., 2007; Poole et al., 2016). However, since hypoxia reduces CP, a given absolute workrate in the severe domain corresponds to a higher intensity relative to CP, compared with normoxic conditions. Therefore, according to the CP model, W′ should deplete faster and time to exhaustion would decline. This would be associated with an exacerbated rate of fatigue development in hypoxia, as shown by Romer et al. (Romer et al., 2007). Hence, rather than conceptualizing hypoxia per se as the mechanism which exacerbates fatigue, it is the effect of hypoxia on decreasing CP that indirectly leads to a more rapid onset of fatigue, coincident with depletion of W′, at a given absolute workrate.

This study is the first to examine the effect of increasing altitude on W′. Our finding that W′ was markedly reduced only in severe hypoxia (≈27% at 4,250 m) led to the construction of a two-segment linear model (see Figure 1D). Using an average confidence interval across all trials, we estimated that a significant decrease in W′ would occur at altitudes beyond ≈3,500 m. Simpson et al. (2015) reported a small decrease in W′ at 3,800 m, but this did not reach statistical significance, whilst Valli et al. (2011) found a large decrease (≈55%) in W′ at 5,050 m. Thus, it appears likely that severe hypoxia reduces W′, yet some uncertainty remains regarding the lowest altitude at which this occurs. Measurement of W′ shows high within-subject variability (Karsten et al., 2014) though, which may confound attempts to accurately determine such a threshold altitude. Valli et al. (2011) suggested the decrease in W′ was consistent with reduced muscle-venous O₂ storage. More recent evidence reveals a decrease in central motor drive in severe hypoxia (≈5,250 m), but no change in moderate hypoxia (≈2,500 m), compared with sea level (Amann et al., 2007). Group III/IV afferent feedback from the locomotor muscles has been suggested to regulate central motor drive (Amann et al., 2006a, 2007), although evidence also suggests that a direct effect of cerebral hypoxia, independent of afferent feedback, may contribute to reduced performance and altered central motor drive in severe hypoxia (Millet et al., 2012). A direct inhibitory effect of cerebral hypoxia on central motor drive might explain the reduction in W′ found in this study and that of Valli et al. (2011), and also the finding that peripheral fatigue is significantly reduced at task failure only in severe hypoxia, but not moderate hypoxia (Amann et al., 2007).

In order to extend the applicability of the constant load CP model to intermittent high-intensity exercise, Skiba et al. (2012) introduced the WBALint′ model. This model includes the following assumptions, (1) expenditure of W′ occurs when the power output exceeds CP, (2) reconstitution of the W′ occurs when the power output falls below CP, and (3) the reconstitution of W′ follows a predictable monoexponential time course. The WBALint′ model has been validated empirically in normoxia (Skiba et al., 2012, 2014b), whilst a receiver-operator characteristic analysis found subjective rating of exhaustion to occur when the modelled WBAL′ fell below 1.5 kJ (Skiba et al., 2014a). Previously, we demonstrated the WBALint′ model to be applicable during intermittent high intensity exercise at ≈2,450 m (Shearman et al., 2016). However, this model was only valid when CP and W′ were also measured at the same FiO₂ (Shearman et al., 2016). In the present study, we have shown that a predictable decline in CP occurs with increasing altitude up to 4,250 m (Figures 1A,C), and therefore, WBAL′ can be calculated in hypoxia using measurements of CP and W′ in normoxia. We found no difference in computed WBALint′ or WBALdiff′ during intermittent high-intensity cycling at 2,250 m, when either actual measurements of CP and W′ at 2,250 m were used, or predicted values based on measures at 250 m (see Table 1).

A modified version of the WBALint′ model (Skiba et al., 2012) was recently published (Skiba et al., 2015). This newer model adopted principles of chemical reaction kinetics and takes the form of a differential equation, hence it was referred to as the WBAL′ “differential” model. The advantage of a differential equation is that the time constant of W′ recovery does not require prior fitting to empirical data (Skiba et al., 2015), or estimation from Equation 4. The present study though, is the first to directly compare the WBALint′ model (Skiba et al., 2012) vs. the WBALdiff′ model (Skiba et al., 2015), within a single subject sample (see Table 1). We found significantly higher values for computed WBALdiff′ from the fourth interval onwards during the HIIT, at which point WBALint′ had declined by ≈50% from initial W′. This difference can largely be explained by the smaller time constant (hence faster recovery kinetics) observed for WBALdiff′ vs. the WBALint′ model (Skiba et al., 2015). Upon completion of the 3TT at both 250 or 2250 m, WBALdiff′ values displayed only small effect sizes and non-significant differences compared with a theoretical criterion of 0 kJ. However, the difference in WBALint′ values was moderate at 250 m and large at 2,250 m. This finding contrasts our previous work (Shearman et al., 2016), in which the WBALint′ model showed good agreement with a criterion range of WBAL′ = 0 ± 1.5 kJ at task failure. In the present study however, we included all-out sprint efforts in addition to self-paced TT exercise during the HIIT. Whilst the WBALdiff′ model appeared better suited to the HIIT in this study, the short recovery time constant led to faster W′ reconstitution than reported by Ferguson et al. (2010) for longer recovery durations. Further research is required to understand the limitations of the current mathematical approaches and to develop a more robust model of intermittent exercise.

Since interval training and road cycling competition is highly stochastic in nature, there are limitless permutations of intensity and work to rest ratios. The application of WBAL′ approach though, enables analysis of all such permutations within a single unifying mathematical framework. A key justification for the present study was to extend the practical application of WBAL′ during dynamic environmental conditions such as a mountain climb in cycling. Figure 4 presents competition field data from the 2015 Giro d'Italia, during a stage which ascends beyond 1,400 m. The effect of increasing altitude can be seen by comparing the uncorrected CP vs. corrected (for altitude) parameter input into WBAL′ model. Interestingly, despite a maximum reported effort on the final hill climb, it appears as though WBAL′ is reconstituting since the power is below CP. The failure to deplete WBAL′ in this instance likely reflects prolonged accumulation of fatigue mechanisms such as glycogen depletion and/or increasing central fatigue (Thomas et al., 2015), which are not taken into account within the framework of the current models. Glycogen depletion has been shown to decrease the value of W′ (Miura et al., 2000), and fatigue induced inefficiency might decrease CP (Grassi et al., 2015). Accordingly, further research is warranted to develop a robust WBAL′ model for a variety of different task requirements. Prolonged endurance exercise would be one such example.

It is well known that endurance exercise performance is reduced upon ascent to altitude (Amann and Calbet, 2008). Secondly, recent progress has been made in the field of mathematical modelling intermittent high-intensity performance, which has practical applications in training load prescription and monitoring (Skiba et al., 2014a, 2015; Shearman et al., 2016). In the present investigation, we report a curvilinear decrease in CP with increasing altitude as well as a significant reduction in W′ occurring only at 4,250 m. The predictable decline in CP, combined with lack of change in W′ up to 3,250 m, enables modelling of WBAL′ in hypoxic environments without the requirement for testing at all altitudes. This enables the prescription of equivalent relative intensity interval training workouts in hypoxic conditions compared with normoxia. Whilst we validated use of the altitude correction factor within the WBAL′ at 2,250 m, since it is known that W′ contains relatively high typical error (Karsten et al., 2014, 2016), and there may be changes in W′ at higher altitudes, caution is required when interpreting modelled intermittent performance in severe hypoxia above ≈3,500 m.

NT, DN, PS, and JP contributed to experimental concept and design. NT, DN, SR, and JP contributed to data collection and analysis. NT, PS, SR, and JP contributed to manuscript preparation. The authors declare that the results of the study are presented clearly, honestly, and without fabrication, falsification, or inappropriate data manipulation.