Traditional and contemporary approaches to mathematical fitness-fatigue models in exercise science: A practical guide with resources. Part I.

Abstract

The fitness-fatigue model (FFM) has been around for more than 40 years and is one of most prominent conceptual models within exercise science. Translation from a purely conceptual form into a mathematical structure reveals there is no single model, but instead a collective of models with common properties. The greatest potential use of FFMs is to predict future performance of athletes with sufficient accuracy to assist with training program design. However, despite a long history and consistent study, there has been limited uptake in practice. This is most likely due to a lack of accessible resources explaining key concepts and processes required to fit models, and insufficient knowledge of predictive validity. The current review provides a comprehensive overview of FFMs and discusses three key aspects of fitting models in practice: 1) training load quantification; 2) criterion performance selection; and 3) parameter estimation. As the majority of athletes engage in sports where performance is complex and determined by a range of physical, psychological and technical factors, it is argued that greater focus should be placed on FFMs being used to quantify components of fitness (e.g. strength, power) targeted by an athletes training rather than competitive outcomes. Additionally, contemporary approaches to training monitoring (e.g. barbell velocity and repetitions in reserve) are recommended as tools to generate high frequency “performance” measurements to fit FFMs. Necessary further developments require collaboration between researchers and practitioners with larger data sets to establish conditions where suitable predictions to future training can be obtained.

The suitability of a quasi-Newton algorithm for estimating fitness-fatigue models: Sensitivity, troublesome local optima, and implications for future research (An in silico experimental design)

“…Although the assumption of a completely deterministic model of athlete response is unrealistic due to simplification by design within the modelling process, this experimental approach is believed to be reasonable in a research context to enable lower-bound study of the fitting process in a manner not possible with real data. Furthermore, the simulated performance data were not unreasonable with regard to change in the performance profile, and true parameters (such as the decay constants) were chosen as to be interpretable with regard to model dynamics (Stephens Hemingway et al, 2021). In the real world, fitting FFMs involves non-zero (possibly large) residual solutions that make it impossible to be sure that the fitted estimates represent a unique global minimum.…”

“…Therefore, fitting an FFM constitutes a nonlinear optimisation problem in its model parameters. FFMs dependent on time-invariant parameters (Banister et al, 1975), or in special cases time-varying parameters (Busso et al, 1997;Kolossa et al, 2017) that cannot be inferred from observation and must instead be estimated from quantified training load and measured performance data (Stephens Hemingway et al, 2021). The fitting process takes as input a time-series of measured performances (denoted ) and training load values (denoted ) and provides as output model parameter estimates ( ∈ ℝ ! )…”

Validity and Accuracy of Impulse-Response Models for Modeling and Predicting Training Effects on Performance of Swimmers

“…T here is renewed interest in the mathematical models of training effects on performance to guide training planning (1)(2)(3)(4)(5). The most widely used models are impulse-response models, which were especially suited to acquiring knowledge on taper for performance peaking (6).…”