Perfect Your Running (with Math!)

Any runner gunning for a PR knows that pacing strategy is important. And until now, most of those strategies look like this: Figure out what pace you can hold for the distance, and hold it. But two French researchers say a varied pace is better, and they’ve created the ultimate running algorithm to determine your optimal pace for every race condition. 

“No wearable device out there is based on real scientific formula, and large companies are cheating runners into thinking they are giving them more than just simple Excel-sheet calculations that measure heart rate,” says Amandine Aftalion, director at Centre de la Recherche Scientifique in Paris. “We managed to write mathematical forms based on the numerical velocity at each distance marker of a run within the unique physical parameters of a runner.”

In other words, Aftalion’s formulas will tell you what pace you should be running at any given moment during a run to reach your best time for the distance. Previous models assume the body should move and recover at a constant speed. Think: You should run 8:10s for this 10K. Aftalion and Bonnan found, however, that runners are more efficient when they vary their speed, rather than trying to keep a constant pace. If you are attempting a 3:30 marathon, it’s more efficient to bounce between a 7:30 and an 8:30 mile, they say, rather than maintain the 8:00-per-mile pace.

“A runner can know exactly when they should move back and forth between speed surges to optimize every bit of energy, instead of just guessing all of this information by how fast their heart is beating,” Aftalion says. It may be more complicated than calculating target heart rate zones, but the French equation promises to be more helpful in improving running efficiency.

The catch: There’s not yet an easy way to apply the complicated equation. Having access to such specific information would be a game changer in terms of fitness technology, but Aftalion and Bonnan have not secured funding to create a software product for consumers. 

“We have analyzed the mathematical and numerical properties to check that it can roughly reproduce reality,” says Bonnan. But until a product can be funded and created, a real-life test model is out of reach. In the future, they hope to create a watch or other wearable that would constantly track and update the best speed to run at any moment so you can nab that 3:30 finishing time based on course location, heart rate, speed, and energy output.

(For the math geeks among us, here’s a brief rundown of how the French equation works: A runner’s speed is the known variable. The muscular force being used and anaerobic function of the runner act as the unknown variables. Depending on your physical traits and your velocity at any certain point of a run, the equation can solve for output force, VO2 max, and muscular capacity.)

While Aftalion works on issue of practical use, she’s forging ahead on the math. She hopes to adapt the equation to triathlon, cycling, and skiing training in the future. “It’s not easy to convince sportsmen that science can be the answer when training,” she says. “My work is to apply science to the places it can help.”