Nonlinear Model for Athletic Training

This was research I conducted at Duke University to:

mathematically model the relationship between athletic training and performance and
use the model to optimize training protocols subject to constraints.

For example, the model can relate the time it takes you to cycle 10 miles (your performance metric) to the duration and intensity of your cycling training (your training stress). Then, using an optimization algorithm, you can create a training protocol that maximizes your performance on race day.

The model we developed accounts for important physiological effects, such as saturation and over-training, that existing linear models in the literature didn’t capture. Our model was

\begin{gather*} p = p_0 + f - u \\ \dot{f} + \frac{1}{\tau_1} f^\alpha = k_1 \sigma \\ \dot{u} + \frac{1}{\tau_2} u^\beta = k_2 \sigma \end{gather*}

where \(p\) is performance as a function of time, \(p_0\) is the individual’s performance in an untrained state, \(f\) is fitness as a function of time, \(u\) is fatigue as a function of time, \(σ\) is training stress impulse as a function of time, \(τ_1\) and \(τ_2\) are time constants, \(k_1\) and \(k_2\) are gain terms, \(α\) and \(β\) are exponents that represent the model’s nonlinearities, and \(t\) is time. An overdot indicates a time derivative. From a dynamical systems perspective, the independent variable is \(t\), the state variables are \(f\) and \(u\), the initial conditions are \(f_0\) and \(u_0\), and the parameters are \(p_0\), \(τ_1\), \(τ_2\), \(k_1\), \(k_2\), \(α\), and \(β\).

In addition to introducing the model, a core part of the research was using the model to optimize training protocols subject to various realistic constraints (such as limits on fatigue). The figure below is fig. 16 from the journal article. It shows the result of optimizing a training protocol for an example athlete to maximize performance after 12 weeks of training. The optimization was subject to constraints related to the individual’s fitness and fatigue over time. The prescribed training protocol in subfigure (c) is similar to the accepted practice for training in preparation for a competition (initial training progression, followed by a high-intensity phase, and then a taper).

A figure with three plots sharing the same horizontal axis, which is labeled “t [time]”. The bottom plot, with vertical axis “σ [stress]” shows stress increase gradually at the beginning, then remain relatively constant, then drop to almost zero at the start of the taper, and then remain near zero until the end. The same plot is overlaid with lines indicating two constraints; the fitness-based constraint is active during the gradual increase at the beginning, and the fatigue-based constraint is active during the middle. The top plot, with vertical axis “p [performance]”, shows the performance dip slightly initially, then increase at a decreasing rate for most of the time, then increase sharply at the onset of the taper, and then flatten out until the end. The middle plot, with vertical axis “u/f”, shows the ratio of fatigue to performance increase sharply at the beginning, followed by a relatively constant middle, and then a moderately rapid decrease starting at the onset of the taper. The same plot illustrates a constraint on the fatigue/fitness ratio of 0.8. — Optimized training schedule to maximize performance \(p\) after 12 weeks of training. Subfigure (a) shows the performance as a function of time \(t\) during training, (b) shows the ratio of fatigue \(u\) to fitness \(f\), and (c) shows the prescribed training stress \(σ\).

This research provides a new mathematical foundation for modeling and optimizing athletic training protocols subject to an individual athlete’s physiology, constraints, and performance goals. See the abstract and journal article (PDF) for more details.

Publication

Turner, J. D., M. J. Mazzoleni, J. A. Little, D. Sequeira, and B. P. Mann. “A nonlinear model for the characterization and optimization of athletic training and performance”. Biomedical Human Kinetics, 9.1 (Feb. 2017), pp. 82–93. (journal article (PDF) and journal webpage)

Abstract

This is the abstract from the journal article:

Study aim: Mathematical models of the relationship between training and performance facilitate the design of training protocols to achieve performance goals. However, current linear models do not account for nonlinear physiological effects such as saturation and over-training. This severely limits their practical applicability, especially for optimizing training strategies. This study describes, analyzes, and applies a new nonlinear model to account for these physiological effects.

Material and methods: This study considers the equilibria and step response of the nonlinear differential equation model to show its characteristics and trends, optimizes training protocols using genetic algorithms to maximize performance by applying the model under various realistic constraints, and presents a case study fitting the model to human performance data.

Results: The nonlinear model captures the saturation and over-training effects; produces realistic training protocols with training progression, a high-intensity phase, and a taper; and closely fits the experimental performance data. Fitting the model parameters to subsets of the data identifies which parameters have the largest variability but reveals that the performance predictions are relatively consistent.

Conclusions: These findings provide a new mathematical foundation for modeling and optimizing athletic training routines subject to an individual’s personal physiology, constraints, and performance goals.

Turner, J. D., et al., Biomedical Human Kinetics, 2017.