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

Adv Physiol Educ. 2013 Jun;37(2):134-52. doi: 10.1152/advan.00078.2011.


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.

Keywords: Banister impulse-response model; coaching education; critical power model; exercise physiology; physical fitness.

Publication types

  • Review

MeSH terms

  • Adaptation, Physiological
  • Athletic Performance*
  • Computer-Assisted Instruction
  • Curriculum
  • Exercise / physiology*
  • Exercise Therapy
  • Exercise Tolerance
  • Humans
  • Models, Biological*
  • Physiology / education*
  • Software
  • Teaching / methods*
  • Time Factors