Systems Modelling of the Relationship Between Training and Performance


Mathematical models may provide a method of describing and predicting the effect of training on performance. The current models attempt to describe the effects of single or multiple bouts of exercise on the performance of a specific task on a given day. These models suggest that any training session increases fitness and provokes a fatigue response. Various methods of quantifying the training stimulus (training impulse, absolute work, psychophysiological rating) and physical performance (criterion scale, arbitrary units) are employed in these models. The models are empirical descriptions and do not use current knowledge regarding the specificity of training adaptations. Tests of these models with published data indicate discrepancies between the predicted and measured time course of physiological adaptations, and between the predicted and measured performance responses to training. The relationship between these models and the underlying physiology requires clarification. New functional models that incorporate specificity of training and known physiology are required to enhance our ability to guide athletic training, rehabilitation and research.

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Table I
Table II


  1. 1.

    Keul J, Konig D, Huonker M, et al. Adaptation to training and performance in elite athletes. Res Q Exerc Sport 1996; 67(3): S29–36

    PubMed  CAS  Google Scholar 

  2. 2.

    Coyle EF. Integration of the physiological factors determining endurance performance ability. Exerc Sport Sci Rev 1995; 23: 25–63

    PubMed  Article  CAS  Google Scholar 

  3. 3.

    Swanson GD. Assembling control models from pulmonary gas exchange dynamics. Med Sci Sports Exerc 1990; 22(1): 80–7

    PubMed  CAS  Google Scholar 

  4. 4.

    Bouchard C, Rankinen T. Individual differences in response to regular physical activity. Med Sci Sports Exerc 2001; 33 (6 Suppl.): S446–51

    PubMed  CAS  Google Scholar 

  5. 5.

    Haskell WL. What to look for in assessing responsiveness to exercise in a health context. Med Sci Sports Exerc 2001; 33 (6 Suppl.): S454–8

    PubMed  CAS  Google Scholar 

  6. 6.

    Keener J, Sneyd J. Mathematical physiology. New York: Springer, 1998

    Google Scholar 

  7. 7.

    Banister EW, Calvert TW. A systems model of training for athletic performance. Aust J Sports Med 1975; 7: 57–61

    Google Scholar 

  8. 8.

    Banister EW, Hamilton CL. Variations in iron status with fatigue modelled from training in female distance runners. Eur J Appl Physiol 1985; 54(1): 16–23

    Article  CAS  Google Scholar 

  9. 9.

    Calvert TW, Banister EW, Savage MV, et al. A systems model of the effects of training on physical performance. IEEE Trans Syst Man Cybern 1976; 6(2): 94–102

    Article  Google Scholar 

  10. 10.

    Sonna LA, Sharp MA, Knapik JJ, et al. Angiotensin-converting enzyme genotype and physical performance during US Army basic training. J Appl Physiol 2001; 91: 1355–63

    PubMed  CAS  Google Scholar 

  11. 11.

    Thomas SG, Cunningham DA, Rechnitzer PA, et al. Determinants of the training response in elderly men. Med Sci Sports Exerc 1985; 17: 667–72

    PubMed  Article  CAS  Google Scholar 

  12. 12.

    Morton RH, Fitz-Clarke JR, Banister EW. Modeling human performance in running. J Appl Physiol 1990; 69(3): 1171–7

    PubMed  CAS  Google Scholar 

  13. 13.

    Busso T, Carasso C, Lacour JR. Adequacy of a systems structure in the modeling of training effects on performance. J Appl Physiol 1991; 71(5): 2044–9

    PubMed  CAS  Google Scholar 

  14. 14.

    Busso T, Hakkinen K, Pakarinen A, et al. A systems model of training responses and its relationship to hormonal responses in elite weight-lifters. Eur J Appl Physiol 1990; 61(1–2): 48–54

    Article  CAS  Google Scholar 

  15. 15.

    Fitz-Clarke JR, Morton RH, Banister EW. Optimizing athletic performance by influence curves. J Appl Physiol 1991; 71(3): 1151–8

    PubMed  CAS  Google Scholar 

  16. 16.

    Busso T, Candau R, Lacour JR. Fatigue and fitness modelled from the effects of training on performance. Eur J Appl Physiol 1994; 69(1): 50–4

    Article  CAS  Google Scholar 

  17. 17.

    Neufer PD, Costill DL, Fielding RA, et al. Effect of reduced training on muscular strength and endurance in competitive swimmers. Med Sci Sports Exerc 1987; 19(5): 486–90

    PubMed  CAS  Google Scholar 

  18. 18.

    Houmard JA, Costill DL, Mitchell JB, et al. Reduced training maintains performance in distance runners. Int J Sports Med 1990; 11(1): 46–52

    PubMed  Article  CAS  Google Scholar 

  19. 19.

    Busso T, Denis C, Bonnefoy R, et al. Modeling of adaptations to physical training by using a recursive least squares algorithm. J Appl Physiol 1997; 82(5): 1685–93

    PubMed  CAS  Google Scholar 

  20. 20.

    Busso T, Benoit H, Bonnefoy R, et al. Effects of training frequency on the dynamics of performance response to a single training bout. J Appl Physiol 2002; 92(2): 572–80

    PubMed  Google Scholar 

  21. 21.

    Martin D, Carl K, Lehnertz K. Handbuch Trainingslehre. Schorndorf: Verlag Hofmann Schorndorf, 1993

    Google Scholar 

  22. 22.

    Mujika I, Busso T, Lacoste L, et al. Modeled responses to training and taper in competitive swimmers. Med Sci Sports Exerc 1996; 28(2): 251–8

    PubMed  Article  CAS  Google Scholar 

  23. 23.

    Rowbottom DG, Keast D, Garcia-Webb P, et al. Training adaptation and biological changes among well-trained male triathletes. Med Sci Sports Exerc 1997; 29(9): 1233–9

    PubMed  Article  CAS  Google Scholar 

  24. 24.

    Banister EW, Calvert TW. Planning for future performance: implications for long term training. Can J Appl Sport Sci 1980; 5(3): 170–6

    PubMed  CAS  Google Scholar 

  25. 25.

    Busso T, Hakkinen K, Pakarinen A, et al. Hormonal adaptations and modelled responses in elite weightlifters during 6 weeks of training. Eur J Appl Physiol 1992; 64(4): 381–6

    Article  CAS  Google Scholar 

  26. 26.

    Millet GP, Candau RB, Barbier B, et al. Modelling the transfers of training effects on performance in elite triathletes. Int J Sports Med 2002; 23(1): 55–63

    PubMed  Article  CAS  Google Scholar 

  27. 27.

    Banister EW. Modeling elite athletic performance. In: MacDougall JD, Wenger HA, Green HJ, Canadian Association of Sports Sciences, editors. Physiological testing of the high-performance athlete. 2nd ed. Champaign (IL): Human Kinetics Books, 1991: 403–24

    Google Scholar 

  28. 28.

    Matveev LP. Fundamentals of sports training. Moscow: Progress Publishers, 1981

    Google Scholar 

  29. 29.

    Fry RW, Morton AR, Keast D. Periodisation of training stress: a review. Can J Sport Sci 1992; 17(3): 234–40

    PubMed  CAS  Google Scholar 

  30. 30.

    McArdle WD, Katch FI, Katch VL. Exercise physiology: energy, nutrition and human performance. Philadelphia (PA): Lea and Febiger, 1991

    Google Scholar 

  31. 31.

    Gaesser GA, Poole DC. Blood lactate during exercise: time course of training adaptation in humans. Int J Sports Med 1988; 9(4): 284–8

    PubMed  Article  CAS  Google Scholar 

  32. 32.

    Hurley BF, Hagberg JM, Allen WK, et al. Effect of training on blood lactate levels during submaximal exercise. J Appl Physiol 1984; 56(5): 1260–4

    PubMed  CAS  Google Scholar 

  33. 33.

    Morton RH. Modeling training and overtraining. J Sports Sci 1997; 15(3): 335–40

    PubMed  Article  CAS  Google Scholar 

  34. 34.

    MacIntosh BR, Rassier DE. What is fatigue? Can J Appl Physiol 2002; 27(1): 42–55

    PubMed  Article  CAS  Google Scholar 

  35. 35.

    Gandevia SC. Spinal and supraspinal factors in human muscle fatigue. Physiol Rev 2001; 81(4): 1725–89

    PubMed  CAS  Google Scholar 

  36. 36.

    Raastad T, Hallen J. Recovery of skeletal muscle contractility after high- and moderate-intensity strength exercise. Eur J Appl Physiol 2000; 82(3): 206–14

    PubMed  Article  CAS  Google Scholar 

  37. 37.

    Banister EW, Morton RH, Fitz-Clarke J. Dose/response effects of exercise modeled from training: physical and biochemical measures. Ann Physiol Anthropol 1992; 11(3): 345–56

    PubMed  Article  CAS  Google Scholar 

  38. 38.

    James DVB, Doust JH. Time to exhaustion during severe intensity running: response following a single bout of interval training. Eur J Appl Physiol 2000; 81(4): 337–45

    PubMed  Article  CAS  Google Scholar 

  39. 39.

    Green HJ, Coates G, Sutton JR, et al. Early adaptations in gas exchange, cardiac function and haematology to prolonged exercise training in man. Eur J Appl Physiol 1991; 63(1): 17–23

    Article  CAS  Google Scholar 

  40. 40.

    Govindasamy D, Paterson DH, Poulin MJ, et al. Cardiorespiratory adaptation with short term training in older men. Eur J Appl Physiol 1992; 65(3): 203–8

    Article  CAS  Google Scholar 

  41. 41.

    Hickson RC, Bomze HA, Holloszy JO. Linear increase in aerobic power induced by a strenuous program of endurance exercise. J Appl Physiol 1977; 42(3): 372–6

    PubMed  CAS  Google Scholar 

  42. 42.

    Green HJ, Jones LL, Painter DC. Effects of short-term training on cardiac function during prolonged exercise. Med Sci Sports Exerc 1990; 22(4): 488–93

    PubMed  CAS  Google Scholar 

  43. 43.

    Swaine IL, Linden RJ, Mary DA. Loss of exercise traininginduced bradycardia with continued improvement in fitness. J Sports Sci 1994; 12(5): 477–81

    PubMed  Article  CAS  Google Scholar 

  44. 44.

    Hickson RC, Hagberg JM, Ehsani AA, et al. Time course of the adaptive responses of aerobic power and heart rate to training. Med Sci Sports Exerc 1981; 13(1): 17–20

    PubMed  CAS  Google Scholar 

  45. 45.

    Banister EW, Carter JB, Zarkadas PC. Training theory and taper: validation in triathlon athletes. Eur J Appl Physiol 1999; 79(2): 182–91

    Article  CAS  Google Scholar 

  46. 46.

    Hooper SL, Mackinnon LT, Ginn EM. Effects of three tapering techniques on the performance, forces and psychometric measures of competitive swimmers. Eur J Appl Physiol Occup Physiol 1998; 78(3): 258–63

    PubMed  Article  CAS  Google Scholar 

  47. 47.

    Shepley B, MacDougall JD, Cipriano N, et al. Physiological effects of tapering in highly trained athletes. J Appl Physiol 1992; 72(2): 706–11

    PubMed  CAS  Google Scholar 

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This work was supported through a research contract with the Canadian Forces. The authors have no conflicts of interest that are directly relevant to the content of this manuscript.

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Correspondence to Scott G. Thomas.

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Taha, T., Thomas, S.G. Systems Modelling of the Relationship Between Training and Performance. Sports Med 33, 1061–1073 (2003).

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  • Training Session
  • Training Stimulus
  • Training Load
  • Fatigue Effect
  • Cycle Training