Time in human endurance models. From empirical models to physiological models

An empirical model for world record running speeds with distance, age, and sex: anaerobic and aerobic contributions to performance

The objective of this study is to derive mathematical equations that closely describe published data on world record running speed as a function of distance, age, and sex. Running speed declines with increasing distance and age. Over long distances, where aerobic metabolism is dominant, speed declines in proportion to the logarithm of distance. Over short distances, anaerobic metabolism contributes significantly to performance, and speed is increased relative to the trend of the long-distance data. Equations are derived that explicitly represent these effects. The decline in speed with age is represented by an age-dependent multiplicative factor, which exhibits increasing sensitivity to age as age increases. Using these equations, data are analyzed separately for males and females, and close fits to published data are demonstrated, particularly for younger age groups. These equations provide insight into the contributions of aerobic and anaerobic components of metabolism to athletic performance and a framework for comparisons of performance across wide ranges of distance and age.NEW & NOTEWORTHY World record speeds at different distances for men and women in different age categories are used to develop a model to predict running performance as a function of race distance, age, and sex. This empirical model quantifies the decline in running speed with distance and age in a way that provides insight into the aerobic and anaerobic contributions to running speed and may help with developing training strategies for different age groups at various distances.

Prediction of Marathon Performance using Artificial Intelligence

Although studies used machine learning algorithms to predict performances in sports activities, none, to the best of our knowledge, have used and validated two artificial intelligence techniques: artificial neural network (ANN) and k-nearest neighbor (KNN) in the running discipline of marathon and compared the accuracy or precision of the predicted performances. Official French rankings for the 10-km road and marathon events in 2019 were scrutinized over a dataset of 820 athletes (aged 21, having run 10 km and a marathon in the same year that was run slower, etc.). For the KNN and ANN the same inputs (10-km race time, body mass index, age and sex) were used to solve a linear regression problem to estimate the marathon race time. No difference was found between the actual and predicted marathon performances for either method (p>0,05). All predicted performances were significantly correlated with the actual ones, with very high correlation coefficients (r>0,90; p<0,001). KNN outperformed ANN with a mean absolute error of 2,4 vs 5,6%. The study confirms the validity of both algorithms, with better accuracy for KNN in predicting marathon performance. Consequently, the predictions from these artificial intelligence methods may be used in training programs and competitions.

Performance prediction, pacing profile and running pattern of elite 1-h track running events

Purpose

This study aimed at comparing the predictive accuracy of the power law (PL), 2-parameter hyperbolic (HYP) and linear (LIN) models on elite 1-h track running performance, and evaluating pacing profile and running pattern of the men’s best two 1-h track running performances of all times.

Methods

The individual running speed–distance profile was obtained for nine male elite runners using the three models. Different combinations of personal bests times (3000 m-marathon) were used to predict performance. The level of absolute agreement between predicted and actual performance was evaluated using intraclass correlation coefficient (ICC), paired t test and Bland–Altman analysis. A video analysis was performed to assess pacing profile and running pattern.

Results

Regardless of the predictors used, no significant differences (p > 0.05) between predicted and actual performances were observed for the PL model. A good agreement was found for the HYP and LIN models only when the half-marathon was the longest event predictor used (ICC = 0.718–0.737, p < 0.05). Critical speed (CS) was highly dependent on the predictors used. Unlike CS, PLV20 (i.e., the running speed corresponding to a 20-min performance estimated using the PL model) was associated with 1-h track running performances (r = 0.722–0.807, p < 0.05). An even pacing profile with minimal changes of step length and frequency was observed.

Conclusions

The PL model may offer the more realistic 1-h track running performance prediction among the models investigated. An even pacing might be the best strategy for succeeding in such running events.

Road to Tokyo 2020 Olympic Games: Training Characteristics of a World Class Male Triathlete

There is a growing interest in the scientific literature for reporting top-class endurance athletes training programs. This case study reports on the training program of a world-class male triathlete preparing to compete in the Tokyo 2020 Olympic Games. A macrocycle of 43 weeks is presented. The triathlete performed 14.74 ± 3.01 h of weekly endurance training volume. Training intensity distribution (TID) was 81.93% ± 6.74%/7.16% ± 2.03%/10.91% ± 6.90% for zones 1 (low intensity, VT2) respectively. Pyramidal TID model is observed during the initial stages of the periodization and Polarized TID model is observed at the end of the macrocycle. The triathlete's peak ⩒O2 was increased by 20% on cycling and by 14% on running. Peak power was increased by 3.13% on cycling test and peak speed by 9.71% on running test. Finally, the triathlete placed 12th in Olympic distance and 10th in Mixed Relay in Tokyo 2020 Olympic games.

Exercise Dose Equalization in High-Intensity Interval Training: A Scoping Review

Based on comparisons to moderate continuous exercise (MICT), high-intensity interval training (HIIT) is becoming a worldwide trend in physical exercise. This raises methodological questions related to equalization of exercise dose when comparing protocols. The present scoping review aims to identify in the literature the evidence for protocol equalization and the soundness of methods used for it. PubMed and Scopus databases were searched for original investigations comparing the effects of HIIT to MICT. A total of 2041 articles were identified, and 169 were included. Of these, 98 articles equalized protocols by utilizing energy-based methods or exercise volume (58 and 31 articles, respectively). No clear consensus for protocol equalization appears to have evolved over recent years. Prominent equalization methods consider the exercise dose (i.e., energy expenditure/production or total volume) in absolute values without considering the nonlinear nature of its relationship with duration. Exercises resulting from these methods induced maximal exertion in HIIT but low exertion in MICT. A key question is, therefore, whether exercise doses are best considered in absolute terms or relative to individual exercise maximums. If protocol equalization is accepted as an essential methodological prerequisite, it is hypothesized that comparison of program effects would be more accurate if exercise was quantified relative to intensity-related maximums.

Predictions of the distance running performances of female runners using different tools

This study examined the validity and compared the precision and accuracy of a distance-time linear model (DTLM), a power law and a nomogram to predict the distance running performances of female runners. Official rankings of French women for the 3000-m, 5000-m, and 10,000-m track-running events from 2005 to 2019 were examined. Each performance was predicted from two other performances. Between the actual and predicted performances, only DTLM showed a difference (p < 0.05). The magnitude of the differences in these predicted performances was small if not trivial. All predicted performances were significantly correlated with the actual ones, with a very high correlation coefficient (p < 0.001; r > 0.90), except for DTLM in the 3000-m, which showed a high correlation coefficient (p < 0.001; r > 0.895). Bias and 95% limits of agreement were acceptable because, whatever the method, they were ≤ -3.7 ± 10.8% on the 3000-m, 1.4 ± 4.3% on the 5000-m, and -2.5 ± 7.4% on the 10,000-m. The study confirms the validity of the three methods to predict track-running performance and suggests that the most accurate and precise model was the nomogram followed by the power law, with the DTLM being the least accurate.

A Fitness-Fatigue Model of Performance in Peripheral Artery Disease: Predicted and Measured Effects of a Pain-Free Exercise Program

Banister impulse-response (IR) model estimates the performance in response to the training impulses (TRIMPs). In 100 patients with peripheral artery disease (PAD), we tested by an IR model the predictability of the effects of a 6-month structured home-based exercise program. The daily TRIMPs obtained from prescribed walking speed, relative intensity and time of exercise determined the fitness-fatigue components of performance. The estimated performance values, calculated from the baseline 6-min and pain-free walking distance (6MWD and PFWD, respectively) were compared with values measured at visits through regression models. Interval pain-free walking at controlled speed prescribed during circa-monthly hospital visits (5 ± 1) was safely performed at home with good adherence (92% of scheduled sessions, 144 ± 25 km walked in 50 ± 8 training hours). The mean TRIMP rose throughout the program from 276 to 601 a.u. The measured 6MWD and PFWD values increased (+33 m and +121 m, respectively) showing a good fit with those estimated by the IR model (6MWD: R² 0.81; PFWD: R² 0.68) and very good correspondence (correlation coefficients: 0.91 to 0.95), without sex differences. The decay of performance without training was estimated at 18 ± 3 weeks. In PAD, an IR model predicted the walking performance following a pain-free exercise program. IR models may contribute to design and verify personalized training programs.

Prediction of distance running performances of female runners using nomograms

This study examined the validity, precision and accuracy of the predictions of distance running performances in female runners from three nomograms. Official rankings of French women for the 3000-m, 5000-m, and 10,000-m track-running events from 2005 to 2019 were examined. Only female runners who performed in the three distance events within the same year were included (n=158). Each performance over any distance was predicted using the three nomograms from the two other performances. The 3000-m, 5000-m and 10,000-m performances were 11min17 s± 1min20 s, 19min29 s ± 2min20 s, 41min18 s ± 5min7 s, respectively. No difference was found between the actual and predicted running performances regardless of the nomogram (p>0.05). All predicted running performances were significantly correlated with the actual ones, with a very high correlation coefficient (p<0.001; r>0.90). Bias and 95% limits of agreement were acceptable because, whatever the nomogram, they were less than or equal to -0.0±6.2% on the 3000-m, 0.0±3.7% on the 5000-m, and 0.1±9.3% on the 10,000-m. The study confirms the validity of the three nomograms to predict track-running performance with a high level of accuracy. The predictions from these nomograms are similar and may be used in training programs and competitions.

Under the Hood: Skeletal Muscle Determinants of Endurance Performance

In the past decades, researchers have extensively studied (elite) athletes' physiological responses to understand how to maximize their endurance performance. In endurance sports, whole-body measurements such as the maximal oxygen consumption, lactate threshold, and efficiency/economy play a key role in performance. Although these determinants are known to interact, it has also been demonstrated that athletes rarely excel in all three. The leading question is how athletes reach exceptional values in one or all of these determinants to optimize their endurance performance, and how such performance can be explained by (combinations of) underlying physiological determinants. In this review, we advance on Joyner and Coyle's conceptual framework of endurance performance, by integrating a meta-analysis of the interrelationships, and corresponding effect sizes between endurance performance and its key physiological determinants at the macroscopic (whole-body) and the microscopic level (muscle tissue, i.e., muscle fiber oxidative capacity, oxygen supply, muscle fiber size, and fiber type). Moreover, we discuss how these physiological determinants can be improved by training and what potential physiological challenges endurance athletes may face when trying to maximize their performance. This review highlights that integrative assessment of skeletal muscle determinants points toward efficient type-I fibers with a high mitochondrial oxidative capacity and strongly encourages well-adjusted capillarization and myoglobin concentrations to accommodate the required oxygen flux during endurance performance, especially in large muscle fibers. Optimisation of endurance performance requires careful design of training interventions that fine tune modulation of exercise intensity, frequency and duration, and particularly periodisation with respect to the skeletal muscle determinants.

Development and field validation of an omni-domain power-duration model

Purpose: To validate and compare a novel model based on the critical power (CP) concept that describes the entire domain of maximal mean power (MMP) data from cyclists.Methods: An omni-domain power-duration (OmPD) model was derived whereby the rate of W' expenditure is bound by maximum sprint power and the power at prolonged durations declines from CP log-linearly. The three-parameter CP (3CP) and exponential (Exp) models were likewise extended with the log-linear decay function (Om3CP and OmExp). Each model bounds W' using a different nonconstant function, W'eff (effective W'). Models were fit to MMP data from nine cyclists who also completed four time-trials (TTs).Results: The OmPD and Om3CP residuals (4 ± 1%) were smaller than the OmExp residuals (6 ± 2%; P < 0.001). W'eff predicted by the OmPD model was stable between 120-1,800 s, whereas it varied for the Om3CP and OmExp models. TT prediction errors were not different between models (7 ± 5%, 8 ± 5%, 7 ± 6%; P = 0.914).Conclusion: The OmPD offers similar or superior goodness-of-fit and better theoretical properties compared to the other models, such that it best extends the CP concept to short-sprint and prolonged-endurance performance.

Modelling of Running Performances: Comparisons of Power-Law, Hyperbolic, Logarithmic, and Exponential Models in Elite Endurance Runners

Many empirical and descriptive models have been proposed since the beginning of the 20th century. In the present study, the power-law (Kennelly) and logarithmic (Péronnet-Thibault) models were compared with asymptotic models such as 2-parameter hyperbolic models (Hill and Scherrer), 3-parameter hyperbolic model (Morton), and exponential model (Hopkins). These empirical models were compared from the performance of 6 elite endurance runners (P. Nurmi, E. Zatopek, J. Väätäinen, L. Virén, S. Aouita, and H. Gebrselassie) who were world-record holders and/or Olympic winners and/or world or European champions. These elite runners were chosen because they participated several times in international competitions over a large range of distances (1500, 3000, 5000, and 10000 m) and three also participated in a marathon. The parameters of these models were compared and correlated. The less accurate models were the asymptotic 2-parameter hyperbolic models but the most accurate model was the asymptotic 3-parameter hyperbolic model proposed by Morton. The predictions of long-distance performances (maximal running speeds for 30 and 60 min and marathon) by extrapolation of the logarithmic and power-law models were more accurate than the predictions by extrapolation in all the asymptotic models. The overestimations of these long-distance performances by Morton's model were less important than the overestimations by the other asymptotic models.

The Critical Power Model as a Potential Tool for Anti-doping

Existing doping detection strategies rely on direct and indirect biochemical measurement methods focused on detecting banned substances, their metabolites, or biomarkers related to their use. However, the goal of doping is to improve performance, and yet evidence from performance data is not considered by these strategies. The emergence of portable sensors for measuring exercise intensities and of player tracking technologies may enable the widespread collection of performance data. How these data should be used for doping detection is an open question. Herein, we review the basis by which performance models could be used for doping detection, followed by critically reviewing the potential of the critical power (CP) model as a prototypical performance model that could be used in this regard. Performance models are mathematical representations of performance data specific to the athlete. Some models feature parameters with physiological interpretations, changes to which may provide clues regarding the specific doping method. The CP model is a simple model of the power-duration curve and features two physiologically interpretable parameters, CP and W′. We argue that the CP model could be useful for doping detection mainly based on the predictable sensitivities of its parameters to ergogenic aids and other performance-enhancing interventions. However, our argument is counterbalanced by the existence of important limitations and unresolved questions that need to be addressed before the model is used for doping detection. We conclude by providing a simple worked example showing how it could be used and propose recommendations for its implementation.

The relationship between movement speed and duration during soccer matches

The relationship between the time duration of movement (t(dur)) and related maximum possible power output has been studied and modeled under many conditions. Inspired by the so-called power profiles known for discontinuous endurance sports like cycling, and the critical power concept of Monod and Scherrer, the aim of this study was to evaluate the numerical characteristics of the function between maximum horizontal movement velocity (HSpeed) and t(dur) in soccer. To evaluate this relationship, GPS data from 38 healthy soccer players and 82 game participations (≥30 min active playtime) were used to select maximum HSpeed for 21 distinct t(dur) values (between 0.3 s and 2,700 s) based on moving medians with an incremental t(dur) window-size. As a result, the relationship between HSpeed and Log(t(dur)) appeared reproducibly as a sigmoidal decay function, and could be fitted to a five-parameter equation with upper and lower asymptotes, and an inflection point, power and decrease rate. Thus, the first three parameters described individual characteristics if evaluated using mixed-model analysis. This study shows for the first time the general numerical relationship between t(dur) and HSpeed in soccer games. In contrast to former descriptions that have evaluated speed against power, HSpeed against t(dur) always yields a sigmoidal shape with a new upper asymptote. The evaluated curve fit potentially describes the maximum moving speed of individual players during the game, and allows for concise interpretations of the functional state of team sports athletes.

Modeling of Running Performances in Humans: Comparison of Power Laws and Critical Speed

Zinoubi, B, Vandewalle, H, and Driss, T. Modeling of running performances in humans: comparison of power laws and critical speed. J Strength Cond Res 31(7): 1859-1867, 2017-The concepts of power law and critical speed (SCrit) have been applied to the analysis of individual running performances. We have analyzed the results of 2 exceptional runners (Nurmi and Gebrselassie) and 11 physical education students (PESs) who performed 3 exhausting running exercises. Power laws can accurately describe the relationships between exhaustion time (tlim) and distance (Dlim) or speed (S) (Equation is included in full-text article.)in elite runners and PES. However, the validity of the application of power laws must be verified for higher values of tlim in nonelite runners. Exponent γ is close to 1 in elite runners and lower in PESs (from 0.625 to 0.872). The value of SCrit was computed from 2 values of tlim (3-14 minutes; SCrit 3-14) and was expressed as a fraction of maximal aerobic speed (MAS) which was assumed to correspond to the maximal speed that can be sustained over 7 minutes (Equation is included in full-text article.)The individual values of (Equation is included in full-text article.)(0.945 for Gebrselassie, 0.919 for Nurmi, and 0.764 ± 0.078 in PESs) were linearly correlated with γ (r > 0.999) and almost equal to γ (0.952 for Gebrselassie, 0.918 for Nurmi, and 0.779 ± 0.076 for PESs). The same results were observed when SCrit was computed for tlim equal to 6-28 minutes (SCrit 6-28) and MAS was the maximal velocity sustainable during 14 minutes (S14min). The fact that γ is linearly correlated and almost equal to (Equation is included in full-text article.)or (Equation is included in full-text article.)suggests that exponent γ can be considered as an index of aerobic endurance.

Exercise Tolerance Can Be Enhanced through a Change in Work Rate within the Severe Intensity Domain: Work above Critical Power Is Not Constant

INTRODUCTION

The characterization of the hyperbolic power-time (P-tlim) relationship using a two-parameter model implies that exercise tolerance above the asymptote (Critical Power; CP), i.e. within the severe intensity domain, is determined by the curvature (W') of the relationship.

PURPOSES

The purposes of this study were (1) to test whether the amount of work above CP (W>CP) remains constant for varied work rate experiments of high volatility change and (2) to ascertain whether W' determines exercise tolerance within the severe intensity domain.

METHODS

Following estimation of CP (208 ± 19 W) and W' (21.4 ± 4.2 kJ), 14 male participants (age: 26 ± 3; peak VO2: 3708 ± 389 ml.min(-1)) performed two experimental trials where the work rate was initially set to exhaust 70% of W' in 3 ('THREE') or 10 minutes ('TEN') before being subsequently dropped to CP plus 10 W.

RESULTS

W>CP for TEN (104 ± 22% W') and W' were not significantly different (P>0.05) but lower than W>CP for THREE (119 ± 17% W', P<0.05). For both THREE (r = 0.71, P<0.01) and TEN (r = 0.64, P<0.01), a significant bivariate correlation was found between W' and tlim.

CONCLUSION

W>CP and tlim can be greater than predicted by the P-tlim relationship when a decrement in the work rate of high-volatility is applied. Exercise tolerance can be enhanced through a change in work rate within the severe intensity domain. W>CP is not constant.

Rationale and resources for teaching the mathematical modeling of athletic training and performance

A number of professions rely on exercise prescription to improve health or athletic performance, including coaching, fitness/personal training, rehabilitation, and exercise physiology. It is therefore advisable that the professionals involved learn the various tools available for designing effective training programs. Mathematical modeling of athletic training and performance, which we henceforth call "performance modeling," is one such tool. Two models, the critical power (CP) model and the Banister impulse-response (IR) model, offer complementary information. The CP model describes the relationship between work rates and the durations for which an individual can sustain them during constant-work-rate or intermittent exercise. The IR model describes the dynamics by which an individual's performance capacity changes over time as a function of training. Both models elegantly abstract the underlying physiology, and both can accurately fit performance data, such that educating exercise practitioners in the science of performance modeling offers both pedagogical and practical benefits. In addition, performance modeling offers an avenue for introducing mathematical modeling skills to exercise physiology researchers. A principal limitation to the adoption of performance modeling is a lack of education. The goal of this report is therefore to encourage educators of exercise physiology practitioners and researchers to incorporate the science of performance modeling in their curricula and to serve as a resource to support this effort. The resources include a comprehensive review of the concepts associated with the development and use of the models, software to enable hands-on computer exercises, and strategies for teaching the models to different audiences.

Modelling decremental ramps using 2- and 3-parameter "critical power" models

The "Critical Power" (CP) model of human bioenergetics provides a valuable way to identify both limits of tolerance to exercise and mechanisms that underpin that tolerance. It applies principally to cycling-based exercise, but with suitable adjustments for analogous units it can be applied to other exercise modalities; in particular to incremental ramp exercise. It has not yet been applied to decremental ramps which put heavy early demand on the anaerobic energy supply system. This paper details cycling-based bioenergetics of decremental ramps using 2- and 3-parameter CP models. It derives equations that, for an individual of known CP model parameters, define those combinations of starting intensity and decremental gradient which will or will not lead to exhaustion before ramping to zero; and equations that predict time to exhaustion on those decremental ramps that will. These are further detailed with suitably chosen numerical and graphical illustrations. These equations can be used for parameter estimation from collected data, or to make predictions when parameters are known.

The critical velocity in swimming

In supra-maximal exercise to exhaustion, the critical velocity (cv) is conventionally calculated from the slope of the distance (d) versus time (t) relationship: d = I + St. I is assumed to be the distance covered at the expense of the anaerobic capacity, S the speed maintained on the basis of the subject's maximal O(2) uptake (VO2max) This approach is based on two assumptions: (1) the energy cost of locomotion per unit distance (C) is constant and (2) VO2max is attained at the onset of exercise. Here we show that cv and the anaerobic distance (d (anaer)) can be calculated also in swimming, where C increases with the velocity, provided that VO2max its on-response, and the C versus v relationship are known. d (anaer) and cv were calculated from published data on maximal swims for the four strokes over 45.7, 91.4 and 182.9 m, on 20 elite male swimmers (18.9 +/- 0.9 years, 75.9 +/- 6.4 kg), whose VO2max and C versus speed relationship were determined, and compared to I and S obtained from the conventional approach. cv was lower than S (4, 16, 7 and 11% in butterfly, backstroke, breaststroke and front crawl) and I (=11.6 m on average in the four strokes) was lower than d (anaer). The latter increased with the distance: average, for all strokes: 38.1, 60.6 and 81.3 m over 45.7, 91.4 and 182.9 m. It is concluded that the d versus t relationship should be utilised with some caution when evaluating performance in swimmers.

The distance – time relationship over a century of running Olympic performances: A limit on the critical speed concept

We analyse the evolution of the slope (critical speed) and the y-intercept (anaerobic distance capacity) of the linear distance-time relationship over a century of Olympic running performances. The distance-time relationship of each Olympic Games (1920-2004) was plotted using the performances in the 800-, 1500- and 5000-m track events. Values for critical speed and anaerobic distance capacity were determined by linear modelling. Mean performances for the 800, 1500 and 5000 m were 104.9 +/- 1.5 s (1.4%), 217.2 +/- 2.8 s (1.3%) and 808.9 +/- 18.4 s (2.3%), respectively. Critical speed improved during the first three-quarters of the twentieth century to reach a plateau in 1984. This is in accordance with the literature (Peronnet & Thibault, 1989) and suggests that "human aerobic endurance" has improved within the century (+13.4%) and tends to stabilize. Anaerobic distance capacity was highly variable over the century (coefficient of variation = 9.4%) and did not show a linear improvement over the years as has previously been suggested (Peronnet & Thibault, 1989). This could be due to an artefact in the application of the two-parameter model to only three Olympic performances. A limitation to the use of this linear mathematical model to fit physiological data may have been demonstrated.

Vmax estimate from three-parameter critical velocity models: validity and impact on 800 m running performance prediction

The purpose of this study was to evaluate the validity of maximal velocity (Vmax) estimated from three-parameter systems models, and to compare the predictive value of two- and three-parameter models for the 800 m. Seventeen trained male subjects (VO2max=66.54+/-7.29 ml min(-1) kg(-1)) performed five randomly ordered constant velocity tests (CVT), a maximal velocity test (mean velocity over the last 10 m portion of a 40 m sprint) and a 800 m time trial (V 800 m). Five systems models (two three-parameter and three two-parameter) were used to compute V max (three-parameter models), critical velocity (CV), anaerobic running capacity (ARC) and V800m from times to exhaustion during CVT. Vmax estimates were significantly lower than (0.19<Bias<0.24 m s(-1)) and poorly associated (0.44<r<0.49) with actual Vmax (8.43+/-0.33 m s(-1)). Critical velocity (CV) alone explained 40-62% of the variance in V800m. Combining CV with other parameters of each model to produce a calculated V800m resulted in a clear improvement of this relationship (0.83<r<0.94). Three-parameter models had a better association (0.93<r<0.94) and a lower bias (0.00<Bias<0.04 m s(-1)) with actual V800 m (5.87+/-0.49 m s(-1)) than two-parameter models (0.83<r<0.91, 0.06<Bias<0.20). If three-parameter models appear to have a better predictive value for short duration events such as the 800 m, the fact the Vmax is not associated with the ability it is supposed to reflect suggests that they are more empirical than systems models.

The critical power and related whole-body bioenergetic models

This paper takes a performance-based approach to review the broad expanse of literature relating to whole-body models of human bioenergetics. It begins with an examination of the critical power model and its assumptions. Although remarkably robust, this model has a number of shortcomings. Attention to these has led to the development of more realistic and more detailed derivatives of the critical power model. The mathematical solutions to and associated behaviour of these models when subjected to imposed "exercise" can be applied as a means of gaining a deeper understanding of the bioenergetics of human exercise performance.

Sprint performance-duration relationships are set by the fractional duration of external force application

We hypothesized that the maximum mechanical power outputs that can be maintained during all-out sprint cycling efforts lasting from a few seconds to several minutes can be accurately estimated from a single exponential time constant (k(cycle)) and two measurements on individual cyclists: the peak 3-s power output (P(mech max)) and the maximum mechanical power output that can be supported aerobically (P(aer)). Tests were conducted on seven subjects, four males and three females, on a stationary cycle ergometer at a pedal frequency of 100 rpm. Peak mechanical power output (P(mech max)) was the highest mean power output attained during a 3-s burst; the maximum power output supported aerobically (P(aer)) was determined from rates of oxygen uptake measured during a progressive, discontinuous cycling test to failure. Individual power output-duration relationships were determined from 13 to 16 all-out constant load sprints lasting from 5 to 350 s. In accordance with the above hypothesis, the power outputs measured during all-out sprinting efforts were estimated to within an average of 34 W or 6.6% from P(mech max), P(aer), and a single exponential constant (k(cycle) = 0.026 s(-1)) across a sixfold range of power outputs and a 70-fold range of sprint trial durations (R2 = 0.96 vs. identity, n = 105; range: 180 to 1,136 W). Duration-dependent decrements in sprint cycling power outputs were two times greater than those previously identified for sprint running speed (k(run) = 0.013 s(-1)). When related to the respective times of pedal and ground force application rather than total sprint time, decrements in sprint cycling and running performance followed the same time course (k = 0.054 s(-1)). We conclude that the duration-dependent decrements in sprinting performance are set by the fractional duration of the relevant muscular contractions.

Reliability of time to exhaustion analyzed with critical-power and log-log modeling

PURPOSE

The large variability of time to exhaustion between repeated tests at constant power output or speed gives an impression that these tests are unsuitable for monitoring athletic performance. We have addressed this issue using critical-power and log-log models of the speed-duration relationship to analyze treadmill runs.

METHODS

Application of differential calculus to the models provided factors for converting variability in time to exhaustion into variability in equivalent time-trial time. We estimated values for the factors and variabilities from a reliability study. Eight male competitive runners performed a test consisting of three constant-speed runs to exhaustion lasting approximately 2, 4, and 8 min, with a 30-min rest between runs. A pair of such tests 5 d apart was repeated 7 and 14 wk later within a summer competitive season. We also used the models to predict times for fixed distances from each set of three runs. Repeated-measures analysis of log-transformed times provided estimates of variability expressed as coefficients of variation.

RESULTS

Variabilities of time to exhaustion were 9.2, 13, and 16% (shortest to longest runs). Converted to their equivalents in time-trial time, the variabilities were 2.6, 1.7, and 1.0% via critical-power modeling, and 1.3, 1.7, and 2.2% via log-log modeling (90% likely limits x//1.2). The conversion factors varied typically by 28% (x//1.5) from runner to runner. Variabilities in times predicted for fixed distances were similar, but more uniform, for the log-log model.

CONCLUSION

Runs to exhaustion are inherently reliable, but conversion of changes in time to exhaustion at a single fixed speed into changes in equivalent time-trial time is model and individual specific, and therefore only approximate. Combining runs at several speeds with log-log modeling provides accurate conversion.

Energetics of high-speed running: integrating classical theory and contemporary observations

We hypothesized that the anaerobic power and aerobic power outputs during all-out runs of any common duration between 10 and 150 s would be proportional to the maximum anaerobic (E(an-max)) and aerobic powers (E(aer-max)) available to the individual runner. Seventeen runners who differed in E(an-max) and E(aer-max) (5 sprinters, 5 middle-distance runners, and 7 long distance runners) were tested during treadmill running on a 4.6 degrees incline. E(an-max) was estimated from the fastest treadmill speed subjects could attain for eight steps. E(aer-max) was determined from a progressive, discontinuous, treadmill test to failure. Oxygen deficits and rates of uptake were measured to assess the respective anaerobic and aerobic power outputs during 11-16 all-out treadmill runs that elicited failure between 10 and 220 s. We found that, during all-out runs of any common duration, the relative anaerobic and aerobic powers utilized were largely the same for sprint, middle-distance, and long-distance subjects. The similar fractional utilization of the E(an-max) and E(aer-max) available during high-speed running 1) provides empirical values that modify and advance classic theory, 2) allows rates of anaerobic and aerobic energy release to be quantified from individual maxima and run durations, and 3) explains why the high-speed running performances of different event specialists can be accurately predicted (R(2) = 0.97; n = 254) from two direct measurements and the same exponential time constant.

Cardiac output and oxygen release during very high-intensity exercise performed until exhaustion

Our objectives were firstly, to study the patterns of the cardiac output (Q(.)) and the arteriovenous oxygen difference [(a-nu(-))O(2)] responses to oxygen uptake (V(.)O(2)) during constant workload exercise (CWE) performed above the respiratory compensation point (RCP), and secondly, to establish the relationships between their kinetics and the time to exhaustion. Nine subjects performed two tests: a maximal incremental exercise test (IET) to determine the maximal V(.)O(2) (V(.)O(2)peak), and a CWE test to exhaustion, performed at p Delta50 (intermediate power between RCP and V(.)O(2)peak). During CWE, V(.)O(2) was measured breath-by-breath, Q(.) was measured beat-by-beat with an impedance device, and blood lactate (LA) was sampled each minute. To calculate ( a-nu(-)O(2), the values of V(.)O(2) and Q(.) were synchronised over 10 s intervals. A fitting method was used to describe the V(.)O(2), Q(.) and ( a-nu(-))O(2) kinetics. The ( a-nu(-)O(2) difference followed a rapid monoexponential function, whereas both V(.)O(2) and Q(.) were best fitted by a single exponential plus linear increase: the time constant (tau) V(.)O(2) [57 (20 s)] was similar to tau ( a-nu(-)O(2), whereas tau for Q(.) was significantly higher [89 (34) s, P <0.05] (values expressed as the mean and standard error). LA started to increase after 2 min CWE then increased rapidly, reaching a similar maximal value as that seen during the IET. During CWE, the rapid component of V(.)O(2) uptake was determined by a rapid and maximal ( a-nu(-)O(2) extraction coupled with a two-fold longer Q(.) increase. It is likely that lactic acidosis markedly increased oxygen availability, which when associated with the slow linear increase of Q(.), may account for the V(.)O(2) slow component. Time to exhaustion was larger in individuals with shorter time delay for ( a-nu(-)O(2) and a greater tau for Q(.).

Maximal endurance time at VO2max

INTRODUCTION

There has been significant recent interest in the minimal running velocity which elicits VO2max. There also exists a maximal velocity, beyond which the subject becomes exhausted before VO2max is reached. Between these limits, there must be some velocity that permits maximum endurance at VO2max, and this parameter has also been of recent interest. This study was undertaken to model the system and investigate these parameters.

METHODS

We model the bioenergetic process based on a two-component (aerobic and anaerobic) energy system, a two-component (fast and slow) oxygen uptake system, and a linear control system for maximal attainable velocity resulting from declining anaerobic reserves as exercise proceeds. Ten male subjects each undertook four trials in random order, running until exhaustion at velocities corresponding to 90, 100, 120, and 140% of the minimum velocity estimated as being required to elicit their individual VO2max.

RESULTS

The model development produces a skewed curve for endurance time at VO2max, with a single maximum. This curve has been successfully fitted to endurance data collected from all 10 subjects (R2 = 0.821, P < 0.001). For this group of subjects, the maximal endurance time at VO2max can be achieved running at a pace corresponding to 88% of the minimal velocity, which elicits VO2max as measured in an incremental running test. Average maximal endurance at VO2max is predicted to be 603 s in a total endurance time of 1024 s at this velocity.

CONCLUSION

Endurance time at VO2max can be realistically modeled by a curve, which permits estimation of several parameters of interest; such as the minimal running velocity sufficient to elicit VO2max, and that velocity for which endurance at VO2max is the longest.