A numerical model can't help you finish a marathon, that will take a lot of running, but math can at least help you be the most efficient your body can be.

The best part about the night before a marathon is 'carb'ing up' the night before with a healthy pasta dinner but marathoners in training long before that make many mistakes with their diet, assuming "I'm in training" means they can eat twice as much of anything.

The best marathoners know the right fueling strategy is crucial and one dedicated marathoner,  an MD/PhD student in the Harvard-MIT Division of Health Sciences and Technology, has taken a more rigorous approach to calculating just how much carbohydrate a runner needs to fuel him or herself through 26.2 miles, and what pace that runner can reasonably expect to sustain.

The result allows runners to calculate personalized targets using an estimate of their aerobic capacity.

The Harvard-MIT scientist, Benjamin Rapoport, was inspired by his experience in the 2005 New York Marathon. As he entered Manhattan for the last several miles of the race, his legs just didn't want to keep up the pace. He was experiencing a common phenomenon among marathoners, known as "hitting the wall." Essentially, the body runs out of fuel, forcing the runner to slow down dramatically.

"You feel like you're not going anywhere," says Rapoport. "It's a big psychological letdown, because you feel powerless. You can't will yourself to run any faster."

Of the hundreds of thousands of people who run a marathon each year, more than 40 percent hit the figurative wall, and 1 to 2 percent drop out before finishing.

Computed distance athletes can run before completely depleting glycogen reserves (‘hitting the wall’), as a function of running intensity (expressed as a percent of VO2max ), relative leg muscle mass (leg muscle mass as a fraction of total body mass), and muscle glycogen density. The distance an athlete can run before ‘hitting the wall’ decreases with increasing levels of exertion, and as the shaded rectangular region labeled ‘Will “Hit the Wall”’ indicates, if this distance is less than 42.195 kilometers (26 miles and 385 yards) then the runner will not be able to complete a marathon without experiencing complete glycogen depletion (at least not without refueling midrace). Each colored curve corresponds to a particular density of muscle glycogen, as labeled (Red, 40; Orange, 60; Yellow, 80; Green, 100; Blue, 120; Purple, 140 kcal kg-1 leg muscle). The colored curves each correspond to athletes whose leg muscles constitute 21.4% of total body mass and are loaded with glycogen at a particular density (relative liver mass has been assumed constant at 2.5% of total body mass, and liver glycogen density has been assumed maximized at 360 kcal kg-1 liver mass; a shaded region around each colored curve fills the region corresponding to relative leg muscle masses of 15% to 25% total body mass. Darker shading indicates overlapping regions and identical failure distances for different sets of physical parameters: The area of densest shading straddles the 21-mile line for athletes running at intensities of 80% to 95% VO2max, indicating that many different athletic builds and levels of glycogen loading are subject to failure at these intensities around mile 21, which has been empirically identified as the distance at which marathon runners most commonly ‘hit the wall.’ By contrast, few runners will ‘hit the wall’ before mile 11, or when running a marathon at less than 55% VO2max.   See citation below for credits.

During strenuous exercise such as running, the body relies on carbohydrates for most of its energy, even though fat stores are usually much larger. Most of those carbohydrates come from glycogen stored in the liver and in the leg muscles. A small amount of glucose is also present in the blood.

Hitting the wall occurs when those stored carbohydrates are completely depleted, forcing the body to start burning fat. When that happens, the runner's pace can drop about 30 percent, and ketones, the byproducts of fat metabolism, start building up in the body, causing pain and fatigue.

"People think hitting the wall is inevitable, but it's not," says Rapoport, who has run 18 marathons, including a personal best of 2:55 at this year's Boston Marathon. "In order to avoid it, you need to know what your capabilities are. You need to set a target pace that will get you to the finish without hitting the wall. Once you do that, you need to make sure you appropriately carbo-load."

To create his new model, Rapoport identified two fundamental physiological factors that limit performance in endurance runners: aerobic capacity and the ability of the leg muscles to store carbohydrates as glycogen. Aerobic capacity, also known as VO2max, is a measure of how much oxygen the body can transport to the muscles and consume during aerobic exercise. Oxygen is critical to muscle performance because glucose can only be broken down completely in the presence of oxygen.

The average untrained male's VO2max of 45 ml/kg/min can be boosted with training, and elite marathoners often have VO2max in excess of 75 ml/kg/min. Measuring exact VO2max requires a treadmill stress test at maximum effort, but it can be estimated by measuring heart rate while running at a constant pace on a treadmill.

Using Rapoport's model, any runner training for a marathon who estimates his or her VO2max can figure out a range of paces, including the fastest safe pace he or she can sustain without hitting the wall. For example, a man with a VO2max of 60 ml/kg/min could run the race in 3:10 without consuming any carbs during the race.

A VO2max of 60 ml/kg/min is about the highest that most men can attain through training, and 3:10 happens to be a gold standard in marathoning: It's the time men age 18 to 34 must achieve to qualify for the Boston Marathon. For women of the same age, the qualifying time is 3:40, which is also the time that Rapoport's model predicts for a runner with a VO2max of 52 ml/kg/min, about the highest level the average woman can attain through training.

The model's predictions also depend on the runner's leg muscle mass, because larger muscles can store more glycogen. In the examples above, those finishing times assume that the runner's leg muscles make up at least 7.5 percent of his or her body mass, which is true of most people. For men, the values typically range from 14 to 27.5 percent, and in women, they range from 18 to 22.5 percent.

Rapoport's model also allows runners to calculate how much carbohydrate they need to consume during the race if they want to run a faster pace without hitting the wall. For example, a runner with VO2max of 50 ml/kg/min who wanted to achieve the 3:10 Boston Marathon qualifying time would need to consume 10 calories of carbohydrate per kilogram of body weight (about 700 calories for a 154-lb. runner), assuming that his legs make up at least 15 percent of his body mass.

While physiological models like Rapoport's can help runners plan for their races, Rapoport says that other factors such as mental toughness and course terrain also play important roles in how a runner will perform in any given marathon. One of the most important things a runner should do during a marathon is stick to his or her target pace, Rapoport advises. When runners start out too fast, they burn a higher percentage of carbohydrates, increasing the risk of hitting the wall.

"Once you figure out your target pace, you have to stay at it," he says. "People sometimes get too excited or change their game plan on the day of the race, and that's a tactical mistake."

Citation: Rapoport BI (2010) 'Metabolic Factors Limiting Performance in Marathon Runners', PLoS Comput Biol 6(10): e1000960. doi:10.1371/journal.pcbi.1000960