A Mathematical Model for Marathon Running

Benjamin Rapoport had only six miles to go in the 2005 New York City Marathon when he felt his energy level drop dramatically — a sensation known to marathon runners as “hitting the wall.” “You feel like you’re not going anywhere,” Rapoport said of the experience, in a Massachusetts Institute of Technology (MIT) press release. “You can’t will yourself to run any faster.”

Rapoport was not alone. According to one recent study, more than 40 percent of marathon runners hit the wall, as Rapoport did. About 1 to 2 percent of race participants drop out before reaching the finish line, 26.2 miles from the starting line.

Now, however, thanks to Rapoport — who also happens to be an M.D.-Ph.D. student in the Harvard-MIT Division of Health Sciences and Technology — marathon runners can use a mathematical model to develop a personalized plan for avoiding the dreaded “wall.” Rapoport describes his mathematical model in the October 2010 issue of the journal PLoS Computational Biology.
Hitting the Wall

The experience of hitting the wall, or “bonking” as it is sometimes called, is not the result of simple fatigue or lack of motivation. Rather, it is a physiological response that occurs when the body’s carbohydrate supply runs out.

The human body uses both fats and carbohydrates as sources of energy. However, the process of metabolizing carbohydrates (that is, breaking them down for energy) is more efficient and requires less oxygen than metabolizing fat. Therefore, when the body has an immediate need for vast amounts of energy — during a marathon, for example — it relies heavily on carbohydrates. The main sources of carbohydrates available to a marathon runner include glycogen, a starch, stored in the leg muscles and the liver, and a small amount of glucose, a sugar, in the blood.

If the body’s carbohydrate stores run out during a marathon, fat becomes the body’s only available energy source. At this point, runners notice a significant drop in energy level: in short, they hit the wall. As they continue to run, their sense of physical exhaustion becomes even worse; fat metabolism produces by-products called ketones which accumulate in the body, resulting in increased pain and fatigue.
Modeling the Marathon

Some runners view hitting the wall as an inevitable part of running a marathon. Others try to avoid running out of carbohydrates by “carbohydrate loading” before the race — that is, eating a diet low in carbohydrates for a period of time, and then shifting to a high-carbohydrate diet in the days immediately before the race. However, as Jake Emmett, a professor of kinesiology and sports studies at Eastern Illinois University states in the MIT press release, “There is a lot of guesswork out there about carbo-loading and about carb intake during a marathon.”

Rapoport decided to take the guesswork out of the process by developing a mathematical model that determines how a runner can avoid hitting the wall. His model involves two key factors that influence metabolism: exercise intensity and the amount of glycogen that can be stored in the runner’s leg muscles.

Exercise intensity is essentially a measure of how hard the body is working while performing a certain exercise. It can be defined as the ratio of the athlete’s heart rate — measured while the athlete is engaging in a particular exercise — to the athlete’s maximum heart rate. Exercise intensity is a key factor in Rapoport’s model because it determines how much of the body’s energy is being drawn from carbohydrates versus how much is being drawn from fats. As Rapoport notes in his paper, “carbohydrates [account for] a greater proportion [of energy] at high intensities.”

If runners want to avoid hitting the wall during the marathon, they must run at a slow enough pace — maintaining a low-enough exercise intensity — to allow the body to achieve the right balance between fat and carbohydrate metabolism. Specifically, the proportion of fats to carbohydrates metabolized should be fine-tuned so that the body’s carbohydrate stores last until the very end of the race. Rapoport’s model allows runners to pinpoint the exact pace they should maintain to achieve this goal.

The second key component in Rapoport’s model is the leg muscles’ ability to store glycogen. Glycogen in the leg muscles constitutes a large portion of the carbohydrates metabolized during a marathon. However, the amount of glycogen in these muscles can vary significantly from one runner to the next, depending on leg muscle size and density. Using Rapoport’s model, runners can determine how their muscles’ capacity for storing glycogen can influence their ideal pace and the amount of carbohydrates they should consume, either before or during the race.
Trying It Out

Rapoport has tested his model using data for typical male and female marathon runners, and his results provide strong evidence that the model is accurate. According to the model, an 18-to-34-year-old man who has been training intensively for a marathon can expect to finish the race in 3 hours and 10 minutes without hitting the wall. This turns out to be the exact time that male runners in this age group must achieve to be eligible to participate in the Boston marathon. Similarly, Rapoport’s estimated time for a female marathon participant in this age group exactly matches the Boston Marathon’s qualifying time of 3 hours and 40 minutes.

Marathon runners interested in finding out their ideal pace, along with personalized guidelines for carbohydrate consumption before and during the race, can consult Rapoport’s paper. Rapoport’s model is also available in the form of a simple online calculator at endurancecalculator.com/ index.html. The calculator requires the user to enter a few basic pieces of information such as age, heart rate (which can be used to estimate exercise intensity) and weight (which can be used to estimate leg-muscle size and density).

For marathon runners, Rapoport’s model is a long-awaited tool that answers important questions about how to pace oneself during a marathon. For those of us who have never participated in a marathon, we can only hope that Rapoport’s next paper will answer yet another pressing question — how to run 26.2 miles without all of the hard work.

Discussion Questions

Do you think the calculator described here would work equally well for other endurance sports, such as long-distance cycling or swimming? Why or why not? The calculator uses age, heart rate, and weight as data; are there other types of health/exercise related data that you think would be relevant? How might you try to assess — in a controlled and ideally quantitative way — the relative importance of the factors used in calculating an ideal running pace?

How about running insoles? Can stopping shock and giving your feet more bounce really help with running? Many specialist recommend wearing special insoles (for example), however an equal amount of other specialists discourage their usage.

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