- [Defraeye et al., 2010]
- Defraeye, T., Blocken, B., Koninckx, E., Hespel, P., and Carmeliet, J. (2010). Aerodynamic study of different cyclist positions: Cfd analysis and full-scale wind-tunnel tests. JOURNAL OF BIOMECHANICS, 43(7):1262–1268. [ DOI | bib ]
- [Jones et al., 2010]
Jones, A. M., Vanhatalo, A., Burnley, M., Morton, R. H., and Poole, D. C.
Critical power: Implications for determination of (v) over doto(2max)
and exercise tolerance.
MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 42(10):1876–1890.
[ DOI |
JONES, A. M., A. VANHATALO, M. BURNLEY, R. H. MORTON, and D. C. POOLE. Critical Power: Implications for Determination of (V) over dotO(2max) and Exercise Tolerance. Med. Sci. Sports Exerc., Vol. 42, No. 10, pp. 1876-1890, 2010. For high-intensity muscular exercise, the time-to-exhaustion (t) increases as a predictable and hyperbolic function of decreasing power (P) or velocity (V). This relationship is highly conserved across diverse species and different modes of exercise and is well described by two parameters: the “critical power” (CP or CV), which is the asymptote for power or velocity, and the curvature constant (W') of the relationship such that t = W'/(P - CP). CP represents the highest rate of energy transduction (oxidative ATP production, (V) over dotO(2)) that can be sustained without continuously drawing on the energy store W' (composed in part of anaerobic energy sources and expressed in kilojoules). The limit of tolerance (time t) occurs when W' is depleted. The CP concept constitutes a practical framework in which to explore mechanisms of fatigue and help resolve crucial questions regarding the plasticity of exercise performance and muscular systems physiology. This brief review presents the practical and theoretical foundations for the CP concept, explores rigorous alternative mathematical approaches, and highlights exciting new evidence regarding its mechanistic bases and its broad applicability to human athletic performance.
Keywords: FATIGUE; EXERCISE INTENSITY DOMAINS; (V) over dotO(2) KINETICS; ANAEROBIC CAPACITY; ATHLETIC PERFORMANCE; MAGNETIC RESONANCE SPECTROSCOPY
- [Hendrix et al., 2009]
- Hendrix, C. R., Housh, T. J., Mielke, M., Zuniga, J. M., Camic, C. L., Johnson, G. O., and Schmidt, R. J. (2009). Critical torque, estimated time to exhaustion, and anaerobic work capacity from linear and nonlinear mathematical models. MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 41(12):2185–2190. [ DOI | bib ]
- [Morton, 2009]
Morton, R. H. (2009).
A new modelling approach demonstrating the inability to make up for
lost time in endurance running events.
IMA JOURNAL OF MANAGEMENT MATHEMATICS, 20(2):109–120.
[ DOI |
The tolerable duration of high-intensity exercise can be described by a simple hyperbolic function of power or velocity, with an asymptote referred to as the `critical power/velocity' and a curvature constant referred to as the `anaerobic work/distance capacity'. More recently, this hyperbola has been generalized by permitting a non-zero temporal asymptote. Using this three-parameter model, we consider the consequences of running the initial part of a race at a speed different from the constant rate proscribed by the hyperbola. We show that for any distance split, an improved time is achievable and that the least time occurs when both parts of the race are run at speeds determined by applying the hyperbola to each part. Further improvement is possible by an appropriate selection of initial distance, with the first part being run at a higher speed than the second. Still further improvement is possible if the athlete follows an all-out running strategy, and we prove that for this model an all-out strategy is uniquely optimal. Significant performance gains appear possible for events of less than 10-min duration. Thus, under this model, an athlete, who at any time during a race drops below their all-out pace, can never make up for lost time. This result is contrary to conventional wisdom. Accordingly, we examine some recent empirical evidence which confirms the predicted nature of all-out power development over short time periods and suggests that pace variation, at least to some degree, may not be as suboptimal as previously assumed.
Keywords: all-out effort; critical power; hyperbolic model; optimal strategy; power-duration curve
- [Nakamura et al., 2009]
Nakamura, F. Y., Okuno, N. M., Perandini, L. A. B., de Oliveira, F. R.,
Buchheit, M., and Simoes, H. G. (2009).
Perceived exertion threshold: Comparison with ventilatory thresholds
and critical power.
SCIENCE & SPORTS, 24(3-4):196–201.
[ DOI |
Objective. - The aim of this study was to provide concurrent validity evidences to perceived exertion threshold (PET) by comparing and establishing relationships with aerobic fitness parameters derived from square-wave and incremental tests. Methods. - Eleven male college students performed one incremental test to determine first and second ventilatory thresholds (VT1 and VT2, respectively), maximal oxygen uptake (VO2max), and maximal aerobic power (MAP); four predictive trials for the critical power (CP) and PET estimations. Results. - Oxygen consumption (VO2) at VT1 and VT2 were 22.9 +/- 4.2 and 35.8 +/- 4.7 ml/kg per minute, respectively. The MAP and VO2max. were 267 +/- 34 W and 40.3 +/- 6.3 ml/kg per minute, respectively. The PET (146 31 W) and CP (146 +/- 33 W) did not differ from each other, and both estimates were between VT1 (121 +/- 28 W) and VT2 (228 +/- 36 W). The correlations between PET and CP, expressed in relative terms to body mass, were significant (P < 0.01, r = 0.84). The correlations between PET and relative VO2 at VT1 (r = 0.76), VT2 (r = 0.72) and VO2max (r = 0.73) were significant (P < 0.05). Conclusion. - PET did not significantly differ from CP, and presented significant correlations with VT1, VT2 and VO2max, derived from incremental test. Thus, it can be considered as a valid measure of aerobic capacity. (C) 2008 Elsevier Masson SAS. All rights reserved.
Keywords: Aerobic capacity; Maximal oxygen uptake; Validity
- [Swart et al., 2009]
Swart, J., Lamberts, R. P., Lambert, M. I., Lambert, E. V., Woolrich, R. W.,
Johnston, S., and Noakes, T. D. (2009).
Exercising with reserve: exercise regulation by perceived exertion in
relation to duration of exercise and knowledge of endpoint.
BRITISH JOURNAL OF SPORTS MEDICINE, 43(10):775–781.
[ DOI |
Objective: The purpose of this study was to examine ratings of perceived exertion (RPE) and performance during repetitive maximal effort 40 km time trials as well as after an intervention that aimed to decrease certainty about the remaining distance of the exercise bout. In addition, we examined the RPE during exercise bouts of markedly different duration. Methods: Part 1: 12 well-trained, competitive-level cyclists completed five 40 km time trials. During the final time trial all feedback was withheld until the final kilometre. In addition, to cause confusion about the remaining distance, they were asked to report their RPE at random intervals from 18 km to 38 km. Part 2: 6 well-trained, recreation-level cyclists randomly completed a 5 km, 10 km, 40 km and 100 km time trial. Results: Part 1: Mean RPE increased during the first four trials and decreased during the final trial. The rate of RPE progression increased in linearity during the first four trials and became more conservative in the final trial. These changes were directly related to performance. Part 2: Mean RPE for longer duration trials (40 km, 100 km) were lower during the first half of trial duration but matched those of shorter trials in the final 20%. Conclusions: Increased familiarity of the exercise bout and certainty about its endpoint are associated with a more aggressive RPE strategy that produces a superior exercise performance. Certainty about the endpoint and the duration of exercise affect both the RPE strategy and performance.
- [Abbiss and Laursen, 2008]
- Abbiss, C. R. and Laursen, P. B. (2008). Describing and understanding pacing strategies during athletic competition. SPORTS MEDICINE, 38(3):239–252. [ bib ]
- [Garcia-Lopez et al., 2008]
Garcia-Lopez, J., Antonio Rodriguez-Marroyo, J., Juneau, C.-E., Peleteiro, J.,
Cordova Martinez, A., and Gerardo Villa, J. (2008).
Reference values and improvement of aerodynamic drag in professional
JOURNAL OF SPORTS SCIENCES, 26(3):277–286.
[ DOI |
The aims of this study were to measure the aerodynamic drag in professional cyclists, to obtain aerodynamic drag reference values in static and effort positions, to improve the cyclists' aerodynamic drag by modifying their position and cycle equipment, and to evaluate the advantages and disadvantages of these modifications. The study was performed in a wind tunnel with five professional cyclists. Four positions were assessed with a time-trial bike and one position with a standard racing bike. In all positions, aerodynamic drag and kinematic variables were recorded. The drag area for the time-trial bike was 31% higher in the effort than static position, and lower than for the standard racing bike. Changes in the cyclists' position decreased the aerodynamic drag by 14%. The aero-helmet was not favourable for all cyclists. The reliability of aerodynamic drag measures in the wind tunnel was high (r > 0.96, coefficient of variation < 2%). In conclusion, we measured and improved the aerodynamic drag in professional cyclists. Our results were better than those of other researchers who did not assess aerodynamic drag during effort at race pace and who employed different wheels. The efficiency of the aero-helmet, and the validity, reliability, and sensitivity of the wind tunnel and aerodynamic field testing were addressed.
Keywords: biomechanics; aerodynamics; cycling; wind tunnel; time-trial
- [Atkinson et al., 2007]
- Atkinson, G., Peacock, O., and Passfield, L. (2007). Variable versus constant power strategies during cycling time-trials: Prediction of time savings using an up-to-date mathematical model. JOURNAL OF SPORTS SCIENCES, 25(9):1001–1009. [ DOI | bib ]
- [Johnson et al., 2007]
- Johnson, M. A., Sharpe, G. R., and Brown, P. I. (2007). Inspiratory muscle training improves cycling time-trial performance and anaerobic work capacity but not critical power. EUROPEAN JOURNAL OF APPLIED PHYSIOLOGY, 101(6):761–770. [ DOI | bib ]
- [Dekerle et al., 2006]
- Dekerle, J., Brickley, G., Hammond, A., Pringle, J., and Carter, H. (2006). Validity of the two-parameter model in estimating the anaerobic work capacity. EUROPEAN JOURNAL OF APPLIED PHYSIOLOGY, 96(3):257–264. [ DOI | bib ]
- [Morton, 2006]
Morton, R. (2006).
The critical power and related whole-body bioenergetic models.
EUROPEAN JOURNAL OF APPLIED PHYSIOLOGY, 96(4):339–354.
[ DOI |
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.
Keywords: anaerobic work capacity; energy; endurance; exercise; metabolism; performance
- [Abbiss and Laursen, 2005]
- Abbiss, C. and Laursen, P. (2005). Models to explain fatigue during prolonged endurance cycling. SPORTS MEDICINE, 35(10):865–898. [ bib ]
- [Morton and Billat, 2004]
Morton, R. and Billat, L. (2004).
The critical power model for intermittent exercise.
EUROPEAN JOURNAL OF APPLIED PHYSIOLOGY, 91(2-3):303–307.
[ DOI |
This paper develops and illustrates the critical power model for intermittent work. Model theoretic development reveals that total endurance time is always a step function of one or more of the four independent variables: work interval power output (P-w), rest interval power output (P-r), work interval duration (t(w)), and rest interval duration (t(r)). Six endurance-trained male athletes recorded their best performances during the season in 3-, 5-, and 10-km races, and performed three different intermittent running tests to exhaustion in random order, recording their total endurance times. These data were used to illustrate the model and compare anaerobic distance capacities (alpha) and critical velocities (beta) estimated from each type of exercise. Good fits of the model to data were obtained in all cases: 0.954<R-2<0.999. Critical velocity was found to be significantly less when estimated using an intermittent versus continuous running protocol.
Keywords: endurance; exhaustion; fatigue; hyperbolic model; running
- [Foster et al., 2003]
Foster, C., De Koning, J., Hettinga, F., Lampen, J., La Clair, K., Dodge, C.,
Bobbert, M., and Porcari, J. (2003).
Pattern of energy expenditure during simulated competition.
MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 35(5):826–831.
[ DOI |
Purpose: To determine how athletes spontaneously use their energetic reserves when the only instruction was to finish in minimal time. and whether experience from repeated performance changes the strategy of recreational athletes. Methods: Recreational road cyclists/speed skaters (N = 9) completed three laboratory time trials of 1500 in on a windload braked cycle. The pattern of energy use was calculated from total work and from the work attributable to aerobic metabolism, which allowed computation of anaerobic energy use. Regional level speed skaters (N = 8) also performed a single 1500-m time trial with the same protocol and measurements. Results: The serial trials were completed in (mean +/- SD) 133.8 +/- 6.6, 133.9 +/- 5.8, 133.8 +/- 5.5 s (P > 0.05 among trials): and in 125.7 +/- 10.9 s in the skaters (P < 0.05 vs cyclists). The (V) over dot O-2peak during the terminal 200 m was similar within trials (3.23 +/- 0.44, 3.34 +/- 0.44. 3.30 +/- 0.51 (P > 0.05)) versus 3.91 +/- 0.68 L.min(-1) in the skaters (P < 0.05 vs cyclists). In all events, the initial power output and anaerobic energy use was high and decayed to a more or less constant value (similar to25% of peak) over the remainder of the event. Contrary to predictions based on an assumed “all out” starting strategy, the subjects reserved some of their ability to perform anaerobic work for a terminal acceleration. The total work accomplished was not different between trials (43.53, 43.78, and 47.48 kJ in the recreational athletes, or between the cyclists and skaters (47.79 kJ). The work attributable to anaerobic sources was not different between the ride., (20.67. 20.53, and 21.12 kJ in the recreational athletes). In the skaters, the work attributable to anaerobic sources was significantly larger versus the cyclists (24.67 kJ). Conclusion: Energy expenditure during high-intensity cycling seems: 1) to be expended in a manner that allows the athlete to preserve an anaerobic energetic contribution throughout an event, 2) does not appear to have a large learning effect in already well trained cyclists, and 3) anaerobic energy expenditure may be the performance discriminating factor among groups of athletes.
Keywords: anaerobic exercise; sports performance; cycling; anaerobic capacity
- [Fukuba et al., 2003]
- Fukuba, Y., Miura, A., Endo, M., Kan, A., Yanagawa, K., and Whipp, B. (2003). The curvature constant parameter of the power-duration curve for varied-power exercise. MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 35(8):1413–1418. [ DOI | bib ]
- [Hill et al., 2003]
- Hill, D., Alain, C., and Kennedy, M. (2003). Modeling the relationship between velocity and time to fatigue in rowing. MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 35(12):2098–2105. [ DOI | bib ]
- [Harman, 2002]
- Harman, C. (2002). A biomechanical power model for world-class 400 metre running. In Sixth Australian conference on mathematics and computing in sport. [ bib ]
- [Jones and Whipp, 2002]
- Jones, A. and Whipp, B. (2002). Bioenergetic constraints on tactical decision making in middle distance running. BRITISH JOURNAL OF SPORTS MEDICINE, 36(2):102–104. [ bib ]
- [Miura et al., 2002]
- Miura, A., Endo, M., Sato, H., Sato, H., Barstow, T., and Fukuba, Y. (2002). Relationship between the curvature constant parameter of the power-duration curve and muscle cross-sectional area of the thigh for cycle ergometry in humans. EUROPEAN JOURNAL OF APPLIED PHYSIOLOGY, 87(3):238–244. [ DOI | bib ]
- [Jones and Carter, 2000]
- Jones, A. and Carter, H. (2000). The effect of endurance training on parameters of aerobic fitness. SPORTS MEDICINE, 29(6):373–386. [ bib ]
- [Keller, 2000]
- Keller, J. (2000). Optimal running strategy to escape from pursuers. AMERICAN MATHEMATICAL MONTHLY, 107(5):416–421. [ bib ]
- [Miura et al., 2000]
- Miura, A., Sato, H., Sato, H., Whipp, B., and Fukuba, Y. (2000). The effect of glycogen depletion on the curvature constant parameter of the power-duration curve for cycle ergometry. ERGONOMICS, 43(1):133–141. [ bib ]
- [Noakes, 2000]
- Noakes, T. (2000). Physiological models to understand exercise fatigue and the adaptations that predict or enhance athletic performance. SCANDINAVIAN JOURNAL OF MEDICINE & SCIENCE IN SPORTS, 10(3):123–145. [ bib ]
- [Sharp et al., 2000]
Sharp, R., Casanova, D., and Symonds, P. (2000).
A mathematical model for driver steering control, with design, tuning
and performance results.
VEHICLE SYSTEM DYNAMICS, 33(5):289–326.
[ bib ]
A mathematical model for the steering control of an automobile is described. The structure of the model derives from linear optimal discrete time preview control theory but it is non-linear. Its parameter values are obtained by heuristic methods, using insight gained from the linear optimal control theory. The driver model is joined to a vehicle dynamics model and the path tracking performance is demonstrated, using moderate manoeuvring and racing speeds. The model is shown to be capable of excellent path following and to be robust against changes in the vehicle dynamics. Application to the simulation of manoeuvres specified by an ideal vehicle path and further development of the model to formalise the derivation of its parameter values and to put it to other uses are discussed.
- [Casanova et al., 2000]
Casanova, D., Sharp, R., and Symonds, P. (2000).
Minimum time manoeuvring: The significance of yaw inertia.
VEHICLE SYSTEM DYNAMICS, 34(2):77–115.
[ bib ]
A formal method for the evaluation of the minimum time vehicle manoeuvre is described. The problem is treated as one of Optimal Control and is solved using a direct transcription method. The resulting Non Linear Programming problem is solved using a Sequential Quadratic Programming (SQP) algorithm for constrained minimisation of a multivariable function. The optimisation program is used to investigate the effect of the yaw moment of inertia on vehicle performance in a double lane change manoeuvre. The method is shown to have excellent capabilities to predict the vehicle maximum performance in transient conditions and to perform sensitivity analysis. The influence of yaw inertia on the minimum manoeuvre time is found to be surprisingly small. The extension of the method to larger problems, e.g., lap time simulation, is also discussed.
- [Ward-Smith and Radford, 2000]
Ward-Smith, A. and Radford, P. (2000).
Investigation of the kinetics of anaerobic metabolism by analysis of
the performance of elite sprinters.
JOURNAL OF BIOMECHANICS, 33(8):997–1004.
[ bib ]
The principal motivation for the present work was the study of the kinetics of anaerobic metabolism. A new mathematical model of the bioenergetics of sprinting, incorporating a three-equation representation of anaerobic metabolism, is developed. Results computed using the model are compared with measured data from the mens' finals of the 100 m event at the 1987 World Championships. The computed results closely predict the overall average performance of the competitors over the course of the entire race. Further calculations show the three-equation model of anaerobic metabolism to be a significant improvement over the previous one-equation model. Representative values of time constants that govern the rare of anaerobic energy release have been determined for elite male athletes. For phosphocreatine utilisation, values for lambda(2) = 0.20 s(-1) and psi(2) = 3.0 s(-1) are consistent with data previously reported in the literature. New values of lambda(3) = 0.033 s(-1) and psi(3) = 0.34 s(-1) are proposed as offering an improved representation of the kinetics of oxygen-independent glycolysis. For the first time, tentative values for the time constants of ATP utilisation, lambda(1) = 0.9 s(-1) and psi(1) = 20 s(-1), are suggested. The maximum powers developed during sprinting by oxygen-independent glycolysis, PCr utilisation and endogenous ATP utilisation were calculated as 34.1, 30.1 and 16.6 W kg(-1), respectively, with an overall maximum anaerobic power of 51.6 W kg(-1). Sample calculations show the mathematical model can be used in principle to derive data on the kinetics of anaerobic metabolism of individual athletes. (C) 2000 Elsevier Science Ltd. All rights reserved.
Keywords: running; sprinting; biomechanics; bioenergetics; anaerobic; glycolysis; metabolism
- [Billat et al., 1999]
Billat, L., Koralsztein, J., and Morton, R. (1999).
Time in human endurance models - from empirical models to
SPORTS MEDICINE, 27(6):359–379.
[ bib ]
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 <100m to >1000km. 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).
- [Katz and Katz, 1999]
- Katz, J. and Katz, L. (1999). Power laws and athletic performance. JOURNAL OF SPORTS SCIENCES, 17(6):467–476. [ bib ]
- [de Koning et al., 1999]
- de Koning, J., Bobbert, M., and Foster, C. (1999). Determination of optimal pacing strategy in track cycling with an energy flow model. Journal of Science and Medicine in Sport, 2(3):266 – 277. [ DOI | bib | http ]
- [Ward-Smith, 1999]
Ward-Smith, A. (1999).
The kinetics of anaerobic metabolism following the initiation of
MATHEMATICAL BIOSCIENCES, 159(1):33–45.
[ bib ]
A mathematical study is made of the kinetics of anaerobic metabolism following the initiation of high-intensity exercise. Power and energy relationships are proposed for oxygen-independent glycolysis, phosphocreatine utilisation and the utilisation of endogenous ATP. The power relations consist of two components, ne describing the build-up phase, the other the controlled-utilisation phase. The controlled-utilisation phase of oxygen-independent glycolysis and the build-up of aerobic metabolism are shown to be closely inter-related. The theoretical relations display trends consistent with published experimental results. Some property values are derived, but because of the scatter of the experimental results, the values are, in general, to be regarded as tentative. (C) 1999 Elsevier Science Inc. All rights reserved.
Keywords: athletics; running; biomechanics; bioenergetics; phosphocreatine; glycolysis
- [Martin et al., 1998]
Martin, J., Milliken, D., Cobb, J., McFadden, K., and Coggan, A. (1998).
Validation of a mathematical model for road cycling power.
JOURNAL OF APPLIED BIOMECHANICS, 14(3):276–291.
[ bib ]
This investigation sought to determine if cycling power could be accurately modeled, A mathematical model of cycling power was derived, and values for each model parameter were determined. A bicycle-mounted power measurement system was validated by comparison with a laboratory ergometer. Power was measured during road cycling, and the measured values were compared with the values predicted by the model. The measured values Tor power were highly correlated (R-2 = .97) with, and were not different than, the modeled values. The standard error between the modeled and measured power (2.7 W) was very small. The model was also used to estimate the effects of changes in several model parameters on cycling velocity. Over the range of parameter values evaluated, velocity varied linearly (R-2 > .99). Thr results demonstrated that cycling power can be accurately predicted by a mathematical model.
Keywords: aerodynamic drag; rolling resistance; air velocity gradient
- [Behncke, 1997]
- Behncke, H. (1997). Optimization models for the force and energy in competitive running. JOURNAL OF MATHEMATICAL BIOLOGY, 35(4):375–390. [ bib ]
- [Maronski, 1996]
- Maronski, R. (1996). Minimum-time running and swimming: Bn optimal control approach. JOURNAL OF BIOMECHANICS, 29(2):245–249. [ bib ]
- [Behncke, 1993]
Behncke, H. (1993).
A mathematical-model for the force and energetics in competitive
JOURNAL OF MATHEMATICAL BIOLOGY, 31(8):853–878.
[ bib ]
A simple mathematical model for competitive running is developed. This model contains the force and energy reserves as key variables and it describes 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.
Keywords: ATHLETICS; RUNNING; FORCE; ENERGY; HYDRAULIC MODEL
- [Olds et al., 1993]
- Olds, T., Norton, K., and Craig, N. (1993). Mathematical-model of cycling performance. JOURNAL OF APPLIED PHYSIOLOGY, 75(2):730–737. [ bib ]
- [Jenkins and Quigley, 1992]
- Jenkins, D. and Quigley, B. (1992). Endurance training enhances critical power. MEDICINE AND SCIENCE IN SPORTS AND EXERCISE, 24(11):1283–1289. [ bib ]
- [MORTON, 1990]
- MORTON, R. (1990). Modeling human power and endurance. JOURNAL OF MATHEMATICAL BIOLOGY, 28(1):49–64. [ bib ]
- [Morton, 1986b]
- Morton, R. (1986b). A three component model of human bioenergetics. Journal of Mathematical Biology, 24(4):451–466. [ DOI | bib ]
- [Morton, 1986a]
- Morton, R. (1986a). On a model of human bioenergetics .2. maximal power and endurance. EUROPEAN JOURNAL OF APPLIED PHYSIOLOGY AND OCCUPATIONAL PHYSIOLOGY, 55(4):413–418. [ bib ]
- [Morton, 1985a]
- Morton, R. (1985a). A mathematical and computer-simulation model of the running athlete. BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 32(3):469–472. [ bib ]
- [Morton, 1985c]
- Morton, R. (1985c). Two-dimensional short-term model of oxygen-uptake kinetics. JOURNAL OF APPLIED PHYSIOLOGY, 58(5):1736–1740. [ bib ]
- [Morton, 1985b]
- Morton, R. (1985b). On a model of human bioenergetics. EUROPEAN JOURNAL OF APPLIED PHYSIOLOGY AND OCCUPATIONAL PHYSIOLOGY, 54(3):285–290. [ bib ]
- [Keller, 1974]
- Keller, J. (1974). Optimal velocity in a race. AMERICAN MATHEMATICAL MONTHLY, 81(5):474–480. [ bib ]
Xia, T., Frey, L. A. (2008). A theoretical ap- proach for modelling peripheral muscle fatigue and recovery. Journal of Biomechanics, 41, 3046- 3052. doi:10.1016/j.jbiomech.2008.07.013
This file was generated by bibtex2html 1.95.