Modelling human power and endurance

Individualized physiology-based digital twin model for sports performance prediction: a reinterpretation of the Margaria-Morton model

Performance in many racing sports depends on the ability of the athletes to produce and maintain the highest possible work i.e., the highest power for the duration of the race. To model this energy production in an individualized way, an adaptation and a reinterpretation (including a physiological meaning of parameters) of the three-component Margaria-Morton model were performed. The model is applied to the muscles involved in a given task. The introduction of physiological meanings was possible thanks to the measurement of physiological characteristics for a given athlete. A method for creating a digital twin was therefore proposed and applied for national-level cyclists. The twins thus created were validated by comparison with field performance, experimental observations, and literature data. Simulations of record times and 3-minute all-out tests were consistent with experimental data. Considering the literature, the model provided good estimates of the time course of muscle metabolite concentrations (e.g., lactate and phosphocreatine). It also simulated the behavior of oxygen kinetics at exercise onset and during recovery. This methodology has a wide range of applications, including prediction and optimization of the performance of individually modeled athletes.

Mathematical modeling of exercise fatigability in the severe domain: A unifying integrative framework in isokinetic condition

Muscle fatigue is the decay in the ability of muscles to generate force, and results from neural and metabolic perturbations. This article presents an integrative mathematical model that describes the decrease in maximal force capacity (i.e. fatigue) over exercises performed at intensities above the critical force Fc (i.e. severe domain). The model unifies the previous Critical Power Model and All-Out Model and can be applied to any exercise described by a changing force F over time. The assumptions of the model are (i) isokinetic conditions, an intensity domain of Fc Collapse

Key Words

• All-out
• Anaerobic capacities
• Critical power
• Endurance
• Exercise fatigability
• Fatigue
• Maximal force capacity
• Muscular capacities
• Power-time relationship
• Time to exhaustion

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MESH Headings

• Exercise/physiology
• Muscle Fatigue
• Muscles/physiology
• Models, Theoretical
• Muscle, Skeletal/physiology

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Grants Collapse

Affiliation(s)

M Bowen Laboratoire Interuniversitaire de Biologie de la Motricité LIBM, EA 7424, Savoie Mont Blanc University, F-7300, Chambéry, France.
P Samozino Laboratoire Interuniversitaire de Biologie de la Motricité LIBM, EA 7424, Savoie Mont Blanc University, F-7300, Chambéry, France
M Vonderscher Laboratoire Interuniversitaire de Biologie de la Motricité LIBM, EA 7424, Savoie Mont Blanc University, F-7300, Chambéry, France
D Dutykh Mathematics Department, Khalifa University of Science and Technology, PO Box 127788, Abu Dhabi, United Arab Emirates; Causal Dynamics Pty Ltd, WA 6009, Perth, Australia
B Morel Laboratoire Interuniversitaire de Biologie de la Motricité LIBM, EA 7424, Savoie Mont Blanc University, F-7300, Chambéry, France

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Development and validation of dynamic bioenergetic model for intermittent ergometer cycling

PURPOSE

The aim of this study was to develop and validate a bioenergetic model describing the dynamic behavior of the alactic, lactic, and aerobic metabolic energy supply systems as well as different sources of the total metabolic energy demand.

METHODS

The bioenergetic supply model consisted of terms for the alactic, lactic, and aerobic system metabolic rates while the demand model consisted of terms for the corresponding metabolic rates of principal cycling work, pulmonary ventilation, and accumulated metabolites. The bioenergetic model was formulated as a system of differential equations and model parameters were estimated by a non-linear grey-box approach, utilizing power output and aerobic metabolic rate (MRae) data from fourteen cyclists performing an experimental trial (P2) on a cycle ergometer. Validity was assessed by comparing model simulation and measurements on a similar follow-up experimental trial (P3).

RESULTS

The root mean square error between modelled and measured MRae was 61.9 ± 7.9 W and 79.2 ± 30.5 W for P2 and P3, respectively. The corresponding mean absolute percentage error was 8.6 ± 1.5% and 10.6 ± 3.3% for P2 and P3, respectively.

CONCLUSION

The validation of the model showed excellent overall agreement between measured and modeled MRae during intermittent cycling by well-trained male cyclist. However, the standard deviation was 38.5% of the average root mean square error for P3, indicating not as good reliability.

Validity and Reliability of Hydraulic-Analogy Bioenergetic Models in Sprint Roller Skiing

Purpose: To develop a method for individual parameter estimation of four hydraulic-analogy bioenergetic models and to assess the validity and reliability of these models' prediction of aerobic and anaerobic metabolic utilization during sprint roller-skiing. Methods: Eleven elite cross-country skiers performed two treadmill roller-skiing time trials on a course consisting of three flat sections interspersed by two uphill sections. Aerobic and anaerobic metabolic rate contributions, external power output, and gross efficiency were determined. Two versions each (fixed or free maximal aerobic metabolic rate) of a two-tank hydraulic-analogy bioenergetic model (2TM-fixed and 2TM-free) and a more complex three-tank model (3TM-fixed and 3TM-free) were programmed into MATLAB. The aerobic metabolic rate (MR _ae ) and the accumulated anaerobic energy expenditure (E _an,acc ) from the first time trial (STT1) together with a gray-box model in MATLAB, were used to estimate the bioenergetic model parameters. Validity was assessed by simulation of each bioenergetic model using the estimated parameters from STT1 and the total metabolic rate (MR _tot ) in the second time trial (STT2). Results: The validity and reliability of the parameter estimation method based on STT1 revealed valid and reliable overall results for all the four models vs. measurement data with the 2TM-free model being the most valid. Mean differences in model-vs.-measured MR _ae ranged between -0.005 and 0.016 kW with typical errors between 0.002 and 0.009 kW. Mean differences in E _an,acc at STT termination ranged between -4.3 and 0.5 kJ and typical errors were between 0.6 and 2.1 kJ. The root mean square error (RMSE) for 2TM-free on the instantaneous STT1 data was 0.05 kW for MR _ae and 0.61 kJ for E _an,acc , which was lower than the other three models (all P < 0.05). Compared to the results in STT1, the validity and reliability of each individually adapted bioenergetic model was worse during STT2 with models underpredicting MR _ae and overpredicting E _an,acc vs. measurement data (all P < 0.05). Moreover, the 2TM-free had the lowest RMSEs during STT2. Conclusion: The 2TM-free provided the highest validity and reliability in MR _ae and E _an,acc for both the parameter estimation in STT1 and the model validity and reliability evaluation in the succeeding STT2.

Effect of Mathematical Modeling and Fitting Procedures on the Assessment of Critical Speed and Its Relationship With Aerobic Fitness Parameters

An accurate estimation of critical speed (CS) is important to accurately define the boundary between heavy and severe intensity domains when prescribing exercise. Hence, our aim was to compare CS estimates obtained by statistically appropriate fitting procedures, i.e., regression analyses that correctly consider the dependent variables of the underlying models. A second aim was to determine the correlations between estimated CS and aerobic fitness parameters, i.e., ventilatory threshold, respiratory compensation point, and maximal rate of oxygen uptake. Sixteen male runners performed a maximal incremental aerobic test and four exhaustive runs at 90, 100, 110, and 120% of the peak speed of the incremental test on a treadmill. Then, two mathematically equivalent formulations (time as function of running speed and distance as function of running speed) of three different mathematical models (two-parameter, three-parameter, and three-parameter exponential) were employed to estimate CS, the distance that can be run above CS (d'), and if applicable, the maximal instantaneous running speed (s max ). A significant effect of the mathematical model was observed when estimating CS, d', and s max (P < 0.001), but there was no effect of the fitting procedure (P > 0.77). The three-parameter model had the best fit quality (smallest Akaike information criterion) of the CS estimates but the highest 90% confidence intervals and combined standard error of estimates (%SEE). The 90% CI and %SEE were similar when comparing the two fitting procedures for a given model. High and very high correlations were obtained between CS and aerobic fitness parameters for the three different models (r ≥ 0.77) as well as reasonably small SEE (SEE ≤ 6.8%). However, our results showed no further support for selecting the best mathematical model to estimate critical speed. Nonetheless, we suggest coaches choosing a mathematical model beforehand to define intensity domains and maintaining it over the running seasons.

The Application of Critical Power, the Work Capacity above Critical Power (W'), and its Reconstitution: A Narrative Review of Current Evidence and Implications for Cycling Training Prescription

The two-parameter critical power (CP) model is a robust mathematical interpretation of the power–duration relationship, with CP being the rate associated with the maximal aerobic steady state, and W′ the fixed amount of tolerable work above CP available without any recovery. The aim of this narrative review is to describe the CP concept and the methodologies used to assess it, and to summarize the research applying it to intermittent cycle training techniques. CP and W′ are traditionally assessed using a number of constant work rate cycling tests spread over several days. Alternatively, both the 3-min all-out and ramp all-out protocols provide valid measurements of CP and W′ from a single test, thereby enhancing their suitability to athletes and likely reducing errors associated with the assumptions of the CP model. As CP represents the physiological landmark that is the boundary between heavy and severe intensity domains, it presents several advantages over the de facto arbitrarily defined functional threshold power as the basis for cycle training prescription at intensities up to CP. For intensities above CP, precise prescription is not possible based solely on aerobic measures; however, the addition of the W′ parameter does facilitate the prescription of individualized training intensities and durations within the severe intensity domain. Modelling of W′ reconstitution extends this application, although more research is needed to identify the individual parameters that govern W′ reconstitution rates and their kinetics.

A survey of mathematical models of human performance using power and energy

The ability to predict the systematic decrease of power during physical exertion gives valuable insights into health, performance, and injury. This review surveys the research of power-based models of fatigue and recovery within the area of human performance. Upon a thorough review of available literature, it is observed that the two-parameter critical power model is most popular due to its simplicity. This two-parameter model is a hyperbolic relationship between power and time with critical power as the power-asymptote and the curvature constant denoted by W′. Critical power (CP) is a theoretical power output that can be sustained indefinitely by an individual, and the curvature constant (W′) represents the amount of work that can be done above CP. Different methods and models have been validated to determine CP and W′, most of which are algebraic manipulations of the two-parameter model. The models yield different CP and W′ estimates for the same data depending on the regression fit and rounding off approximations. These estimates, at the subject level, have an inherent day-to-day variability called intra-individual variability (IIV) associated with them, which is not captured by any of the existing methods. This calls for a need for new methods to arrive at the IIV associated with CP and W′. Furthermore, existing models focus on the expenditure of W′ for efforts above CP and do not model its recovery in the sub-CP domain. Thus, there is a need for methods and models that account for (i) the IIV to measure the effectiveness of individual training prescriptions and (ii) the recovery of W′ to aid human performance optimization.

Circadian rhythm of peripheral perfusion during 10-day hypoxic confinement and bed rest

INTRODUCTION

Future planetary habitats will be hypobaric and hypoxic to reduce the risk of decompression sickness during preparation for extra-vehicular activities. This study was part of a research programme investigating the combined effects of hypoxia and microgravity on physiological systems.

PURPOSE

We tested the hypothesis that hypoxia-induced peripheral vasoconstriction persists at night and is aggravated by bed rest. Since sleep onset has been causally linked to nocturnal vasodilatation, we reasoned that hypoxia-induced vasoconstriction at night may explain sleep disturbances at altitude. Peripheral perfusion alterations as a consequence of bed rest may explain poor sleep quality reported during sojourns on the International Space Station.

METHODS

Eleven males underwent three 10-day interventions in a randomised order: (1) hypoxic ambulatory confinement; (2) hypoxic bed rest; (3) normoxic bed rest. During each intervention we conducted 22-h monitoring of peripheral perfusion, as reflected by the skin temperature gradient. Measurements were conducted on the first (D 1) and last day (D 10) of each intervention.

RESULTS

All interventions resulted in a decrease in daytime toe perfusion from D 1 to D 10. There was no difference in the magnitude of the daytime reduction in toe perfusion between the three interventions. There was a significant vasodilatation of the toes in all interventions by 11 pm. The fingertips remained well perfused throughout.

CONCLUSIONS

Daytime vasoconstriction induced by hypoxia and/or bed rest is abolished at night, lending further support to the theory that changes in peripheral skin temperature may be functionally linked to sleep onset.

Integrating the Internal and External Training Loads in Soccer

Purpose:

This study aimed to assess the relationships of fitness in soccer players with a novel integration of internal and external training load (TL).

Design:

Ten amateur soccer players performed a lactate threshold (LT) test followed by a soccer simulation (Ball-Sport Endurance and Sprint Test [BEAST90mod]).

Methods:

The results from the LT test were used to determine velocity at lactate threshold (vLT), velocity at onset of blood lactate accumulation (vOBLA), maximal oxygen uptake (VO2max), and the heart rate–blood lactate profile for calculation of internal TL (individualized training impulse, or iTRIMP). The total distance (TD) and high intensity distance (HID) covered during the BEAST90mod were measured using GPS technology that allowed measurement of performance and external TL. The internal TL was divided by the external TL to form TD:iTRIMP and HID:iTRIMP ratios. Correlation analyses assessed the relationships between fitness measures and the ratios to performance in the BEAST90mod.

Results:

vLT, vOBLA, and VO2max showed no significant relationship to TD or HID. HID:iTRIMP significantly correlated with vOBLA (r = .65, P = .04; large), and TD:iTRIMP showed a significant correlation with vLT (r = .69, P = .03; large).

Conclusions:

The results suggest that the integrated use of ratios may help in the assessment of fitness, as performance alone showed no significant relationships with fitness.

Differential modeling of anaerobic and aerobic metabolism in the 800-m and 1,500-m run

This study examined the hypothesis that running speed over 800- and 1,500-m races is regulated by the prevailing anaerobic (oxygen independent) store (ANS) at each instant of the race up until the all-out phase of the race over the last several meters. Therefore, we hypothesized that the anaerobic power that allows running above the speed at maximal oxygen uptake (V̇o2max) is regulated by ANS, and as a consequence the time limit at the anaerobic power (tlim PAN = ANS/PAN) is constant until the final sprint. Eight 800-m and seven 1,500-m male runners performed an incremental test to measure V̇o2max and the minimal velocity associated with the attainment of V̇o2max ( vV̇o2max), referred to as maximal aerobic power, and ran the 800-m or 1,500-m race with the intent of achieving the lowest time possible. Anaerobic power (PAN) was measured as the difference between total power and aerobic power, and instantaneous ANS as the difference between end-race and instantaneous accumulated oxygen deficits. In 800 m and 1,500 m, tlim PAN was constant during the first 70% of race time in both races. Furthermore, the 1,500-m performance was significantly correlated with tlim PAN during this period ( r = −0.92, P < 0.01), but the 800-m performance was not ( r = −0.05, P = 0.89), although it was correlated with the end-race oxygen deficit ( r = −0.70, P = 0.05). In conclusion, this study shows that in middle-distance races over both 800 m and 1,500 m, the speed variations during the first 70% of the race time serve to maintain constant the time to exhaustion at the instantaneous anaerobic power. This observation is consistent with the hypothesis that at any instant running speed is controlled by the ANS remaining.

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.

New racing equation for championship performance

A separate investigation of swimming endurance has led to a family of expressions relating maximal human locomotor activity to time and/or distance having such wide application to medicine, physiology, and sports as to merit independent publication. Specifically, variations of a single racing equation with one variable parameter suffice to predict record times, speeds, energy, power, and endurance accurately for cycling, running, and swimming, provided only that performance be categorized as steady-state, aerobic effort.

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.

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

This article traces the study of interrelationships between power output, work done, velocity maintained or distance covered and the endurance time taken to achieve that objective. During the first half of the twentieth century, scientists examined world running records for distances from < 100 m to > 1000 km. Such examinations were empirical in nature, involving mainly graphical and crude curve-fitting techniques. These and later studies developed the use of distance/time or power/time models and attempted to use the parameters of these models to characterise the endurance capabilities of athletes. More recently, physiologists have proposed theoretical models based on the bioenergetic characteristics of humans (i.e. maximal power, maximal aerobic and anaerobic capacity and the control dynamics of the system). These models have become increasingly complex but they do not provide sound physiological and mathematical descriptions of the human bioenergetic system and its observed performance ability. Finally, we are able to propose new parameters that can be integrated into the modelling of the power/time relationship to explain the variability in endurance time limit at the same relative exercise power (e.g. 100% maximal oxygen uptake).

The relationship between power output and endurance: a brief review

It is well established that for work requiring high power output, endurance time is short, and that low power outputs can be maintained for long periods. Parameters describing this relationship are important in characterising work performance and the capacity of humans as a source of mechanical power. The purpose of this paper is to provide a brief review of the available literature investigating this relationship and its parameters. Most experimental data reflect measurements of endurance times over a range of constant power outputs on the cycle ergometer. Early graphical analyses of these data have been superseded by curve fitting, which in turn has led to establishment of the two component hyperbolic model now embodied in the critical power test. This model has been modified and extended in various ways to account for its shortcomings. In addition, a number of different exercise forms have been studied, and the effects of a variety of secondary factors (training status, age, sex, for example) on the parameters have also been investigated.

A 3-parameter critical power model

The critical power test is a well-established procedure that provides estimates of two important parameters characterizing work performance; anaerobic work capacity (AWC) and critical power (CP). The concept proscribes a hyperbolic relationship between power output (P) and time to exhaustion (t), given by (P - CP)t = AWC. Since evidence now exists that the procedure overestimates CP and underestimates AWC, this study was undertaken to investigate the effect of relaxing the requirement that the time asymptote necessarily be at zero. Using data from a previous study, it is shown that in so doing, (1) a time asymptote significantly less than zero is obtained, (2) significantly smaller estimates of CP and larger estimates of AWC are obtained, (3) a third parameter is introduced that theoretically represents maximal instantaneous power, (4) it implies that the maximal power that could be developed at any instant is proportional to the amount of AWC remaining at that instant, which in turn implies that (5) at exhaustion not necessary all of AWC is consumed.

A mathematical model for the force and energetics in competitive running

A simple mathematical model for competitive running is developed. This model contains the force and energy reserves as key variables and it described their relationship and dynamics. It is made up of three submodels for the biomechanics of running the energetics and the optimization. The model for the energetics is an extension of the hydraulic model of Margaria and Morton. The key geometric parameters of this piecewise linear, three compartment model are determined on the basis of well known physiological facts and data.