Multidimensional Analysis of Physiological Entropy during Self-Paced Marathon Running

Abstract

The pacing of a marathon is arguably the most challenging aspect for runners, particularly in avoiding a sudden decline in speed, or what is colloquially termed a "wall", occurring at approximately the 30 km mark. To gain further insight into the potential for optimizing self-paced marathon performance through the coding of comprehensive physiological data, this study investigates the complex physiological responses and pacing strategies during a marathon, with a focus on the application of Shannon entropy and principal component analysis (PCA) to quantify the variability and unpredictability of key cardiorespiratory measures. Nine recreational marathon runners were monitored throughout the marathon race, with continuous measurements of oxygen uptake (V˙O₂), carbon dioxide output (V˙CO₂), tidal volume (Vt), heart rate, respiratory frequency (Rf), and running speed. The PCA revealed that the entropy variance of V˙O₂, V˙CO₂, and Vt were captured along the F1 axis, while cadence and heart rate variances were primarily captured along the F2 axis. Notably, when distance and physiological responses were projected simultaneously on the PCA correlation circle, the first 26 km of the race were positioned on the same side of the F1 axis as the metabolic responses, whereas the final kilometers were distributed on the opposite side, indicating a shift in physiological state as fatigue set in. The separation of heart rate and cadence entropy variances from the metabolic parameters suggests that these responses are independent of distance, contrasting with the linear increase in heart rate and decrease in cadence typically observed. Additionally, Agglomerative Hierarchical Clustering further categorized runners' physiological responses, revealing distinct clusters of entropy profiles. The analysis identified two to four classes of responses, representing different phases of the marathon for individual runners, with some clusters clearly distinguishing the beginning, middle, and end of the race. This variability emphasizes the personalized nature of physiological responses and pacing strategies, reinforcing the need for individualized approaches. These findings offer practical applications for optimizing pacing strategies, suggesting that real-time monitoring of entropy could enhance marathon performance by providing insights into a runner's physiological state and helping to prevent the onset of hitting the wall.

Keywords: entropy; fatigue; hitting the wall; marathon running; pacing; performance; physiological responses.

Figure 1

Biplot from the principal component analysis (PCA) illustrating the relationship between physiological and cadence responses during the marathon, with Runner 3 serving as a representative example for all nine marathoners. The red vectors represent the variables, including heart rate (HR), cadence (Cad), oxygen uptake ($\dot{\mathrm{V}}$O₂), carbon dioxide output ($\dot{\mathrm{V}}$CO₂), respiratory frequency (Rf), tidal volume (Vt), respiratory exchange ratio (RER), ventilation ($\dot{\mathrm{V}}$E), and speed. The blue points represent the observations corresponding to the different kilometers of the marathon. The F1 axis explains 47.13% of the variance, while the F2 axis accounts for 17.19%, together capturing a total of 64.32% of the variance in the data.

Figure 2

The dendrogram for the classification of entropy profiles during the marathon using Agglomerative Hierarchical Clustering. The y-axis represents the dissimilarity between clusters, while the x-axis shows the kilometers run by each participant. (a) The dendrogram for Runner 6, showing four distinct clusters, representative of four other runners with a four-class profile. (b) The dendrogram for Runner 2, showing three clusters, typical of three runners with a three-class profile. (c) The dendrogram for Runner 8, displaying two clusters, indicating a two-class profile unique to this runner.