Optimally controlling nutrition and propulsion force in a long distance running race

Cameron Cook; Guoxun Chen; William W Hager; Suzanne Lenhart

doi:10.3389/fnut.2023.1096194

Optimally controlling nutrition and propulsion force in a long distance running race

Front Nutr. 2023 May 18:10:1096194. doi: 10.3389/fnut.2023.1096194. eCollection 2023.

Authors

Cameron Cook¹, Guoxun Chen², William W Hager³, Suzanne Lenhart⁴

Affiliations

¹ Research Triangle Institute (RTI) Health Solutions, Research Triangle Park, NC, United States.
² Department of Nutrition, University of Tennessee, Knoxville, Knoxville, TN, United States.
³ Department of Mathematics, University of Florida, Gainsville, FL, United States.
⁴ Department of Mathematics, University of Tennessee, Knoxville, Knoxville, TN, United States.

Abstract

Introduction: Runners competing in races are looking to optimize their performance. In this paper, a runner's performance in a race, such as a marathon, is formulated as an optimal control problem where the controls are: the nutrition intake throughout the race and the propulsion force of the runner. As nutrition is an integral part of successfully running long distance races, it needs to be included in models of running strategies.

Methods: We formulate a system of ordinary differential equations to represent the velocity, fat energy, glycogen energy, and nutrition for a runner competing in a long-distance race. The energy compartments represent the energy sources available in the runner's body. We allocate the energy source from which the runner draws, based on how fast the runner is moving. The food consumed during the race is a source term for the nutrition differential equation. With our model, we are investigating strategies to manage the nutrition and propulsion force in order to minimize the running time in a fixed distance race. This requires the solution of a nontrivial singular control problem.

Results: As the goal of an optimal control model is to determine the optimal strategy, comparing our results against real data presents a challenge; however, in comparing our results to the world record for the marathon, our results differed by 0.4%, 31 seconds. Per each additional gel consumed, the runner is able to run 0.5 to 0.7 kilometers further in the same amount of time, resulting in a 7.75% increase in taking five 100 calorie gels vs no nutrition.

Discussion: Our results confirm the belief that the most effective way to run a race is to run approximately the same pace the entire race without letting one's energies hit zero, by consuming in-race nutrition. While this model does not take all factors into account, we consider it a building block for future models, considering our novel energy representation, and in-race nutrition.

Keywords: bioenergetics; differential equation models; metabolism; nutrition; optimization; running.

Grants and funding

Support by the US National Science Foundation under grant 2031213 and by the US Office of Naval Research under grant N00014-22-1-2397 is gratefully acknowledged.