The art of running: a mathematical model to optimize performance

Antoine Le Hyaric, the deputy director of the Mathematics of modeling specialty from the Jacques-Louis Lions laboratory¹  and Amandine Aftalion, a research director from the Center for Anaysis and Social Mathematics², have developed a model to optimize running performance. This new model takes into account physiological and psychological parameters to advise athletes on the best training strategies.

What are the objectives of your mathematical model?

Amandine Aftalion: The main objective of this model is to determine the best running strategies based on the physiological and psychological capabilities of each athlete. We use optimization techniques to integrate equations relating to movement, energy management—whether aerobic or anaerobic—and motor control by the brain. The goal is to optimize effort, energy and time to maximize racing performance.

Antoine Le Hyaric: This model is a powerful tool because it enables you to simulate different strategies and predict their effectiveness based on the real conditions of a race.

What data did you use to develop this model?

A. A.: The innovative aspect of this study was the ability to calculate all the runner's physiological parameters simply from the extremely precise data collected during the European athletics championships, where the runners' speeds were measured ten times per second. The model that we have developed provides access to all the variables of the problem, such as energy expenditure, oxygen consumption (or VO2³), force, and more. To do this, we analyzed data from the performances of Matthew Hudson-Smith (400 m), Femke Bol (400 m) and Jakob Ingebrigtsen (1,500 m) at the 2022 European Athletics Championships in Munich, and Gaia Sabbatini (1,500 m) at the 2021 European U23 Athletics Championships in Tallinn.

A. L. H: We wrote the code that solves the equations based on the optimal control work carried out by Emmanuel Trélat, director of the Jacques-Louis Lions laboratory.

What are the main physiological and psychological parameters taken into account in your model and how do they influence performance?

A. A.: Our model integrates various parameters. On a physiological level, we analyze in particular the propulsion force, anaerobic energy (which does not come from oxygen) and aerobic energy (from the transformation of oxygen). Aerobic energy, for example, is not constant and reaches its maximum variably depending on the athlete's respiratory capacities and the duration of the race.

We also study psychological parameters, notably motivation. The latter is crucial because it directly impacts the intensity and speed of the athlete's motor response. For this, we relied on an equation developed by the group of Mathias Pessiglione, neuroscientist at the Brain Institute⁴

Once we have modeled the race, we can then vary these parameters to understand how they impact performance. Quantifying costs and benefits in the model provides immediate access to the best strategy to achieve optimal runner performance.

A. L. H: The influence of these parameters also depends on the length of the race and the physiology of the runner because you do not run a sprint like you run a marathon.

How is motor control variability important for running performance?

A. A.: This is the first time that a model takes into account the variability of motor control and the role of the brain in the movement production process. Motivation impacts performance in two ways. On one hand, it affects the intensity of an action: if you are motivated, you can start more quickly in a sprint or accelerate at the end of an endurance race because you are ready to make a greater effort. On the other hand, it affects the reaction speed and the effort. If I am motivated, I will produce the effort more quickly.

We introduced these two parameters into the model, knowing that a repeated variation of the effort during acceleration or deceleration during a race is very costly and that it must be limited as much as possible in a race.

What are the most important findings from your study regarding optimizing running performance?

A. A.: One of the most interesting discoveries concerns the optimal strategy for the 400m. Accelerating throughout the race is not the strategy that allows you to achieve the best time. We have found that starting fast, accelerating over the first 70 meters, using mainly anaerobic energy, then reaching your maximum speed by limiting your fall over the remaining 330 meters with a high VO2 max, is the most effective.

For the 1500m, improving aerobic metabolism (oxygen absorption) and the ability to maintain a high VO2 to accelerate at the end of the race are crucial. It's about not starting too hard to be able to accelerate on the last two laps.

More generally, this model showed that it was not enough to train athletes with advice based solely on mechanics, but that it was important to take into account energy expenditure and mentality.

A. L. H.: We also observed that indoor corridors are unfavorable for performance due to the physical constraints they impose. You have to try to have good results during qualifying to avoid lane number 1.

How do you think athletes and coaches could use your model to improve their performance at the Paris Olympics?

A. A.: If it is now too late to change the situation for the next Olympics, coaches can take advantage of the information from our model by targeting the physiological parameters to work on according to the targeted performance.

Generally speaking, there are two things to develop in athletes: a cardiac output which allows a lot of oxygen to be transformed per unit of time into energy and a significant anaerobic energy stock. But each athlete is different and must have targeted training according to the abilities to be worked on.

How could your model be extended to other sporting disciplines?

A. A.: To be validated, the model requires very precise and good quality GPS data. If this exists, then the mechanical part can be adapted to other sprint or endurance sports such as cross-country skiing, rowing, cycling, swimming and speed skating.

In the long term, this type of model could also help in the rehabilitation of a convalescent patient who resumes exercise on the treadmill, for example.

A. L. H.: In the coming months, we want to improve our model for 200m indoor tracks where the turns are tighter and the centrifugal force is greater, but also take into account the interaction between runners.

 ^{1 The Jacques-Louis Lions laboratory, co-supervised by Sorbonne University, CNRS, Université Paris Cité
 2 The Center for Analysis and Social Mathematics co-supervised by CNRS and EHESS
3 Flow of oxygen transformed into energy in the lungs. It varies during a race and depends on respiratory capacity, which is not the same for someone who is trained or for someone who is not.
4 The Brain Institute, co-supervised by Sorbonne University, CNRS, Inserm, APHP)}

For more information:

• Be a Champion, 40 facts you didnt know about sports and science, Amandine Aftalion, Springer, 2024
• Pourquoi est-on penché dans les virages ? Le sport expliqué par les sciences en 40 questions, Amandine Aftalion, CNRS Editions 2023
• Publication scientifique du modèle dans Frontiers