SIAM News Blog

Optimization Techniques Influence Nutrition Strategies for Marathon Runners

By Lina Sorg

In the U.S. alone, more than 275,000 running races take place every year and over 500,000 people compete in marathons. Nonprofessional athletes around the world train extensively just to qualify for annual competitions like the Boston Marathon. Part of this training involves nutrition regimens. “While everybody is trying to run fast, a big piece of the race is to actually take in nutrition during the race,” Cameron Cook of the University of Tennessee said. “There’s this whole industry that has been created to handle this for products that runners take.”

At a recent SIAM conference, Cook presented an optimal control model of nutrient management for a marathon runner. He seeks to accurately add nutrition to an existing runner model and create a dynamic model that better represents the body’s energy systems, including velocity-driven allocation of fat and carbohydrate energy. “We would like to minimize the time it takes to run a race by optimally choosing velocity and in-race nutrition consumption profiles by controlling the propulsion force and the nutrition input,” Cook said. As a long-distance runner and coach of the club team at his university, Cook has been working to incorporate some of the findings into his coaching.

Figure 1. Race pace data for the 2015 Boston Marathon.
The first person to work on this type of project was Joseph Keller, who used a system of two equations—a velocity equation and an evolution of energy equation—to explore a runner’s optimal velocity in a 1974 paper. Keller’s trajectory results indicated that runners should begin a race with several seconds of maximum force, maintain a constant cruising velocity for most of the race, and slow down as they cross the finish line. This trajectory corresponds with three control arcs: a max force arc, a singular arc, and a boundary singular arc. “Keller found that the majority of the race should be spent at this cruising velocity,” Cook said. “A runner should stay pretty consistent throughout a race with their pacing.” Data from the 2015 Boston Marathon confirmed that runners with the best times maintained a consistent overall pace with minimal variation.

Next, Cook turned to the two main energy processes in the body. Anaerobic metabolism is the energy process that occurs in the absence of oxygen, during which the body only burns carbohydrates. Aerobic metabolism is the energy process that occurs in the presence of oxygen, during which the body burns both carbohydrates and fats. Bodies employ the aerobic process for low levels of exertion—such as sitting still, walking, or cycling—and the anaerobic process for short bursts of activity, like sprinting or weight lifting. “Within these two processes, we have two sources of energy: fat and glycogen,” Cook said. “There’s a lot of fat available for use in the body, but a smaller amount of glycogen that we’re able to store in our body to use.” This concept is reflected in the model.

Cook then presented the testable values that drive runner performance:

  • VO2 max (aerobic capacity): the maximum amount of oxygen one can use during exertion
  • VLa max (anaerobic capacity): the maximum production of lactate
  • VLax max: the body’s efficiency at converting fat versus glycogen to energy
  • VVO2 max: the velocity at 100 percent of the VO2 max.

A large VO2 max value is desirable because it means that the body can utilize an abundant amount of oxygen, while a smaller VLa max value is preferred because lactate buildup in the body slows runners down. The percentage of VO2 max at which one is functioning determines the energy pathway that will be used. “If you’re operating at a lower percentage of your VO2 max, you’re going to use a higher percentage of fat compared to glycogen,” Cook said. Finally, the VVO2 max is linearly related to the VO2 max but better suited for practical application.

After discussing energy, Cook explained the significance of nutrition during a race. “Nutrition for a runner is of the utmost importance,” he said. If runners do not supply/supplement their glycogen storage with other forms of energy throughout the race, glycogen will eventually run out. To avoid this deficiency, marathons boast periodic, strategically placed aid stations with nutrition products for runners. These products can take multiple forms—including gels—and typically have no fat so that they go directly to the glycogen source in the muscles. 

Cook’s model includes the two energy sources (fat and glycogen) and velocity, assigns a rate for energy loss, and uses a “glyc function”—whose structure depends on lactate capacity—to allocate the amount of energy that stems from each source (see Figure 2). It also depicts the gain of nutrition from the gel and loss of nutrition due to basic bodily functions. The model’s corresponding system of differential equations account for velocity, fat energy, glycogen energy, and nutrition. 

Figure 2. Diagram of Cook's optimal control model.

Cook compared several different fuel intake scenarios—including consistent gel intake throughout the race, gel consumption at specific points throughout the race, larger-calorie gels, etc.—to determine the most effective fueling strategy. Because the goal is to finish the race as quickly as possible, he cast the model as an optimal control problem and chose to maximize distance over a fixed time because this approach is most common in the literature. “We’re going to obtain an optimal control for each of these nutrition strategies and then maximize over that set,” Cook said. A continuous velocity profile to the control force provides a pacing strategy for the race.

After experimenting with multiple computational approximation methods, Cook opted for MATLAB’s optimization software fmincon. To ensure a precise approximation, he added terms that penalize variations in control and prevent the runners from significantly altering their velocities.

Finally, Cook presented the results of his model. For a 135-minute race during which a runner consumes no gels and reaches a consistent cruising velocity, the runner achieves a distance of 40 kilometers. By the end of the race, glycogen energy is down to zero and fat energy has changed only slightly. The penalization coefficient is 0.5. If the same runner consumes five gels over the same amount of time and with the same penalization coefficient, the runner travels 43.1 kilometers. “The gels are going to improve your time,” Cook said. The corresponding glycogen energy graph includes slight bumps at the times where the runner ate the gels, which lessen the speed at which glycogen decreases.

Cook also tested his model against the marathon world record results. The marathon world record is 1:59:40 for 42.2 kilometers, and Cook’s model allows a runner to cover 42.5 kilometers in 120 minutes. “If you compare these two, there’s about a .4 percent difference between the current world record and our results,” he said. “We’re pretty happy with that. The world record was done as optimally as possible from a physical standpoint. Pacers and nutrition were done exactly right.”

Cook ended the presentation with a discussion of prospective future work. The best nutrition strategy in his model occurs when the runner takes in the most calories in the form of 24 gels throughout the race, but this scenario is unrealistic. “That’s of course not likely because you can’t take that many gels,” Cook said. “So that’s something that we need to put into our model, a kind of penalization for overeating.” He also hopes to adapt the model for both shorter races and individual use so that runners can determine their own nutrition strategies based on their weight, VO2 max, and VLa max levels. Cook’s results can even benefit nutrition companies that might choose to alter the rate at which gels reach the muscles, and race directors who strategically allocate their resources and aid stations throughout a course.

This article was originally posted in July 2021 and last updated March 2022. 

Lina Sorg is the managing editor of SIAM News.
blog comments powered by Disqus