Modelling human endurance: power laws vs critical power

The power-duration relationship describes the time to exhaustion for exercise at different intensities. It is believed to be a "fundamental bioenergetic property of living systems" that this relationship is hyperbolic. Indeed, the hyperbolic (a.k.a. critical-power) model which formalises this belief is the dominant tool for describing and predicting high-intensity exercise performance, e.g. in cycling, running, rowing or swimming. However, the hyperbolic model is now the focus of a heated debate in the literature because it unrealistically represents efforts that are short (< 2 min) or long (> 15 min). We contribute to this debate by demonstrating that the power-duration relationship is more adequately represented by an alternative, power-law model. In particular, we show that the often-observed good fit of the hyperbolic model between 2 and 15 min should not be taken as proof that the power-duration relationship is hyperbolic. Rather, in this range, a hyperbolic function just happens to approximate a power law fairly well. We also prove mathematical results which suggest that the power-law model is a safer tool for pace selection than the hyperbolic model and that the former more naturally models fatigue than the latter.

A critical review of critical power

The elegant concept of a hyperbolic relationship between power, velocity, or torque and time to exhaustion has rightfully captivated the imagination and inspired extensive research for over half a century. Theoretically, the relationship's asymptote along the time axis (critical power, velocity, or torque) indicates the exercise intensity that could be maintained for extended durations, or the "heavy-severe exercise boundary". Much more than a critical mass of the extensive accumulated evidence, however, has persistently shown the determined intensity of critical power and its variants as being too high to maintain for extended periods. The extensive scientific research devoted to the topic has almost exclusively centered around its relationships with various endurance parameters and performances, as well as the identification of procedural problems and how to mitigate them. The prevalent underlying premise has been that the observed discrepancies are mainly due to experimental 'noise' and procedural inconsistencies. Consequently, little or no effort has been directed at other perspectives such as trying to elucidate physiological reasons that possibly underly and account for those discrepancies. This review, therefore, will attempt to offer a new such perspective and point out the discrepancies' likely root causes.

Hemopexin dosing improves cardiopulmonary dysfunction in murine sickle cell disease

Hemopexin (Hpx) is a crucial defense protein against heme liberated from degraded hemoglobin during hemolysis. High heme stress creates an imbalance in Hpx bioavailability, favoring heme accumulation and downstream pathophysiological responses leading to cardiopulmonary disease progression in sickle cell disease (SCD) patients. Here, we evaluated a model of murine SCD, which was designed to accelerate red blood cell sickling, pulmonary hypertension, right ventricular dysfunction, and exercise intolerance by exposure of the mice to moderate hypobaric hypoxia. The sequence of pathophysiology in this model tracks with circulatory heme accumulation, lipid oxidation, extensive remodeling of the pulmonary vasculature, and fibrosis. We hypothesized that Hpx replacement for an extended period would improve exercise tolerance measured by critical speed as a clinically meaningful therapeutic endpoint. Further, we sought to define the effects of Hpx on upstream cardiopulmonary function, histopathology, and tissue oxidation. Our data shows that tri-weekly administrations of Hpx for three months dose-dependently reduced heme exposure and pulmonary hypertension while improving cardiac pressure-volume relationships and exercise tolerance. Furthermore, Hpx administration dose-dependently attenuated pulmonary fibrosis and oxidative modifications in the lung and myocardium of the right ventricle. Observations in our SCD murine model are consistent with pulmonary vascular and right ventricular pathology at autopsy in SCD patients having suffered from severe pulmonary hypertension, right ventricular dysfunction, and sudden cardiac death. This study provides a translational evaluation supported by a rigorous outcome analysis demonstrating therapeutic proof-of-concept for Hpx replacement in SCD.

Determination of the speed-time relationship during handcycling in spinal cord injured athletes

This study aimed to determine the critical speed (CS) and the work above CS (D') from three mathematical models of para-athletes during a treadmill handcycling exercise. Nine hand-cyclists with spinal cord injuries performed a maximal incremental handcycling test and three tests to exhaustion at a constant speed to determine the speed-time relationship. The three tests to exhaustion were performed at intensities between 90% and 105% of peak speed derived from the incremental test. Then, the determination of CS and D' was modelled by linear and hyperbolic models. CS and D' did not present any significant differences among the three mathematical models. Low values in the standard error of estimate for CS were found for the three models (Linear: Distance-time: 1.7 ± 0.5%; Linear: Speed-1/time: 3.0 ± 1.9% and Hyperbolic: 1.2 ± 0.6%). Based on the simplicity to calculate, the CS modelled by linear-distance-time can be a practical method for handcyclist coaches.

Determinants of last lap speed in paced and maximal 1500-m time trials

PURPOSE

The present study identified the physiological and performance characteristics that are deterministic during a maximal 1500-m time trial and in paced 1500-m time trials, with an all-out last lap.

METHODS

Thirty-two trained middle-distance runners (n = 21 male, VO_2peak: 72.1 ± 3.2; n = 11, female, VO_2peak: 61.2 ± 3.7 mL kg^-1 min^-1) completed a 1500-m time trial in the fastest time possible (1500_FAST) as well as a 1500_MOD and 1500_SLOW trial whereby mean speed was reduced during the 0-1100 m by 5% and 10%, respectively. Anaerobic speed reserve (ASR), running economy (RE), the velocity corresponding with VO_2peak (_VVO_2peak), maximal sprint speed (MSS) and maximal accumulated oxygen deficit (MAOD) were determined during additional testing. Carnosine content was quantified by proton magnetic resonance spectroscopy in the gastrocnemius and expressed as a Z-score to estimate muscle fibre typology.

RESULTS

1500_FAST time was best explained by RE and _VVO_2peak in female runners (adjusted r² = 0.80, P < 0.001), in addition to the 0-1100-m speed relative to _VVO_2peak in male runners (adjusted r² = 0.72, P < 0.001). Runners with a higher gastrocnemius carnosine Z-score (i.e., higher estimated percentage of type II fibres) and greater MAOD, reduced their last lap time to a greater extent in the paced 1500-m trials. Neither ASR nor MSS was associated with last lap time in the paced trials.

CONCLUSION

These findings suggest that _VVO_{2 peak} and RE are key determinants of 1500-m running performance with a sustained pace from the start, while a higher carnosine Z-score and MAOD are more important for last lap speed in tactical 1500-m races.

Prescribing 6-weeks of running training using parameters from a self-paced maximal oxygen uptake protocol

Purpose

The self-paced maximal oxygen uptake test (SPV) may offer effective training prescription metrics for athletes. This study aimed to examine whether SPV-derived data could be used for training prescription.

Methods

Twenty-four recreationally active male and female runners were randomly assigned between two training groups: (1) Standardised (STND) and (2) Self-Paced (S-P). Participants completed 4 running sessions a week using a global positioning system-enabled (GPS) watch: 2 × interval sessions; 1 × recovery run; and 1 × tempo run. STND had training prescribed via graded exercise test (GXT) data, whereas S-P had training prescribed via SPV data. In STND, intervals were prescribed as 6 × 60% of the time that velocity at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot {V}{{\text{O}}_{{\text{2max}}}}$$\end{document}V˙O2max (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\text{v}}}\dot {V}{{\text{O}}_{{\text{2max}}}}$$\end{document}vV˙O2max) could be maintained (T_max). In S-P, intervals were prescribed as 7 × 120 s at the mean velocity of rating of perceived exertion 20 (_vRPE20). Both groups used 1:2 work:recovery ratio. Maximal oxygen uptake (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot {V}{{\text{O}}_{{\text{2max}}}}$$\end{document}V˙O2max), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\text{v}}}\dot {V}{{\text{O}}_{{\text{2max}}}}$$\end{document}vV˙O2max, T_{max, v}RPE20, critical speed (CS), and lactate threshold (LT) were determined before and after the 6-week training.

Results

STND and S-P training significantly improved \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot {V}{{\text{O}}_{{\text{2max}}}}$$\end{document}V˙O2max by 4 ± 8 and 6 ± 6%, CS by 7 ± 7 and 3 ± 3%; LT by 5 ± 4% and 7 ± 8%, respectively (all P < .05), with no differences observed between groups.

Conclusions

Novel metrics obtained from the SPV can offer similar training prescription and improvement in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot {V}{{\text{O}}_{{\text{2max}}}}$$\end{document}V˙O2max, CS and LT compared to training derived from a traditional GXT.

Reliability and validity of a modified 3-minute all-out swimming test in elite swimmers

Critical speed (CS) testing is useful in monitoring training in swimmers, however, completing a series of time trials (TTs) regularly is time-consuming. The 3-minute test may be a solution with positive initial findings. This investigation examined whether a modified 3-minute test (12 × 25 m) could assess CS and supra-CS distance capacity (D') in swimmers. A series of 12 × 25 m intervals were completed unpaced at maximal effort, interspersed with 5 s rest periods. The model speed = a e^bt + c was fitted to the data and integrated to calculate D'. The slowest two of the last four efforts were averaged to calculate CS. To assess reliability, 15 highly trained swimmers (9 females, 6 males) completed the 12 × 25 m twice within 72 h. Four measures were deemed reliable: peak velocity (0.01 m s^-1; 0.5%, typical error and % coefficient of variation), CS (0.02 m s^-1; 1.2%), D' (1.22 m; 5.7%) and drop off % (0.70% points; 4.5%). To assess criterion validity, 21 swimmers (9 from reliability, 12 other) completed two competition races within 2 weeks of a 12 × 25 m in the same stroke. Traditional CS and D' measures were calculated from competition performances (TT method). TT CS and 12 × 25 m CS were highly correlated (adj. R² = 0.92, p < .001). D' values were moderately correlated (adj. R² = 0.60, p < .01). Two TTs may have been too few to estimate D' accurately. The 12 × 25 m all-out swimming test is a reliable method for assessing CS and D' in swimmers, however, the validity of D' requires further investigation.

Physiological differences between a noncontinuous and a continuous endurance training protocol in recreational runners and metabolic demand prediction

This study investigated the physiological difference in recreational runners between a noncontinuous and a continuous endurance training protocol. It also aimed to determine physiological surrogate that could monitor metabolic demand of prolonged running in real‐time. For data collection, a total of 18 active male recreational runners were recruited. Physiological (HR, RR, RER, ṼO₂, BLa), and overall perceptual (RPE_O) responses were recorded against three designed test sessions. Session 1 included ṼO_2submax test to determine critical speed (CS) at anaerobic threshold (AT). Session 2 was the noncontinuous CS test until exhaustion, having 4:1 min work‐to‐rest ratio at CS, whereas session 3 was the continuous CS test till exhaustion. As 1‐min recovery during session 2 may change fatigue behavior, it was hypothesized that it will significantly change the physiological stress and hence endurance outcomes. Results reported average time to exhaustion (TTE) was 37.33(9.8) mins for session 2 and 23.28(9.87) mins for session 3. Participants experienced relatively higher metabolic demand (BLa) 6.78(1.43) mmol.l⁻¹ in session 3 as compared to session 2 (5.52(0.93) mmol.l⁻¹). RER was observed to increase in session 3 and decrease in session 2. Student's paired t‐test only reported a significant difference in TTE, ṼO₂, RER, RPE_O, and BLa at “End” between session 2 and 3. Reported difference in RPE_O and %HR_max at “AT” were 5 (2.2) and 89.8 (2.60)% during session 2 and 6 (2.5) and 89.8 (2.59)% during session 3, respectively. Regression analysis reported strong correlation of %HR_max (adj. R‐square = 0.588) with BLa than RPE_O (adj. R‐square = 0.541). The summary of findings suggests that decreasing RER increased TTE and reduced BLa toward “End” during session 2 which might have helped to have better endurance. The %HR_max was identified to be used as a better noninvasive surrogate of endurance intensity estimator.

Relationship between simulated extravehicular activity tasks and measurements of physical performance

The purpose was to evaluate the relationships between tests of fitness and two activities that simulate components of Lunar- and Martian-based extravehicular activities (EVA). Seventy-one subjects completed two field tests: a physical abilities test and a 10km Walkback test. The relationships between test times and the following parameters were determined: running V˙O2max, gas exchange threshold (GET), speed at V˙O2max (s-V˙O2max), highest sustainable rate of aerobic metabolism [critical speed (CS)], and the finite distance that could be covered above CS (D'): arm cranking V˙O2peak, GET, critical power (CP), and the finite work that can be performed above CP (W'). CS, running V˙O2max, s-V˙O2max, and arm cranking V˙O2peak had the highest correlations with the physical abilities field test (r=0.66-0.82, P<0.001). For the 10km Walkback, CS, s-V˙O2max, and running V˙O2max were significant predictors (r=0.64-0.85, P<0.001). CS and to a lesser extent V˙O2max are most strongly associated with tasks that simulate aspects of EVA performance, highlighting CS as a method for evaluating astronaut physical capacity.