The perfect horse race? It’s all in the math (and training, of course)

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The modeling provided insights into how horses run races. Image: Amandine Aftalion

Scientists have weighed in with mathematical analysis on the question of how Thoroughbreds can run the perfect race.

Quentin Mercier and Amandine Aftalion, with the Center for Social Analysis and Mathematics in Paris, gained insights into how a Thoroughbred should regulate its speed over the course of a race to optimize performance.

“Because the racing career of a Thoroughbred horse is not so long, and therefore the number of racing opportunities is limited, any information that can help to determine a horse’s ability according to the race distance or to optimize how to regulate its speed along the race can be crucial,” the pair said.

They noted that little is known about the optimal strategy for a Thoroughbred horse to run and win a race.

“Because Thoroughbred horses are not capable of running the whole race at top speed, determining what pace to set and when to unleash the burst of speed is essential,” Mercier and Aftalion reported in the open-access journal PLOS ONE.

The mathematical model they produced relies on mechanics, energy-related calculations centered around aerobic and anaerobic exertion, and motor control.

“It is,” they reported, “a system of coupled ordinary differential equations on the velocity, the propulsive force and the anaerobic energy, that leads to an optimal control problem that we solve.”

To identify the parameters meaningful to Thoroughbred performance, the pair used velocity data on races at Chantilly, France, provided by the country’s governing body of flat horse racing, France Galop, as well as information from tracking devices.

Mercier and Aftalion said their numerical simulations of performance optimization provided the optimal speed during the race, the oxygen uptake evolution in a race, as well as the energy or the propulsive force.

Cornering and altitude considered

Their modeling can even predict how the horse has to change its effort and velocity according to the altitude and tightness of the corners on the track.

The pair reported the findings of simulations over 1300m, 1900m and 2100m.

“We see that horses have to start strongly and reach a maximal velocity. The velocity decreases in the bends; when going out of the bend, the horse can speed up again and our model can quantify exactly how and when.”

Effectively, the horse that slows down the least at the end of the race is the one that wins.

“We understand from the optimal control problem that this slow-down is related to the anaerobic supply, the VO2 maximum, and the ability to maintain maximal force at the end of the race.” VO2 is an index of the body’s efficiency at producing work. It is expressed in milliliters of oxygen consumed per minute, and adjusted for body weight in kilograms.

“Therefore, horses that have a tendency to slow down too much at the end of the race should put less force at the beginning and slow down slightly through the whole race in order to have the ability to maintain velocity at the end.

“From our simulations, we are also able to get information on the VO2 profile, such as when steady state VO2 is reached, when the decay of VO2 starts.

The ability to maintain a high VO2 for a long time is related to the ability to maintain velocity, they said. The VO2 starts to decrease when the residual anaerobic energy is too low, and this corresponds to the optimal time to launch the sprint for a long race.

The pair said their work also provided a better understanding of the effects of altitude and the nature of corners, which can have an effect on the whole race.
The research provided a better understanding of the effects of altitude and the nature of corners, which can have an effect on the whole race. Image by H. Hach

The pair said their work also provided a better understanding of the effects of altitude and the nature of corners, which can have an effect on the whole race.

“Therefore, a good knowledge of the track and training are crucial to adapt the global pacing strategy rather than slowing down because of bends or slopes.”

The authors said future work will be devoted to taking into account drafting (tracking behind another horse to reduce wind resistance) and horse psychology, since an alternative strategy can be to stay behind to save energy and overtake in the last straight.

“To maximize an individual horse’s potential for winning, it should be entered in races appropriate for its racing ability,” they said.

“Therefore information on a horse’s speed, endurance or running economy coupled with simulations can help to predict how a horse profile is adapted to some distances to run.”

The authors provided simulations and modeling from three significant races over different distances at the Chantilly track, focusing on one horse for each race:

1300 metres

There is a strong start with the maximal velocity being reached in 200 meters. Then the velocity decreases, particularly in the bend, between 300 and 600 meters from start. Though the track is going down, the centrifugal force reduces the propulsive force. It is only when reaching a point before the final straight, after 600 meters, that the horse can speed up again. The end of the race is uphill and the velocity decreases even though the horse reaches the straight. Nevertheless, a decrease in velocity at the end of such a race takes place even on a flat track.

“We see that the VO2 is increasing for about 400 meters, while the force is decreasing. Then when the VO2 decreases, the force and thus the velocity increase until 900 meters when the slope and end of race lead to a decrease of force and velocity.”

1900 metres

There is a strong start with the maximal velocity being reached in 300 meters. Then the velocity decreases. Between 900 and 1400m, we see the effect of the bend: at the beginning of the bend, the track is going down and the horse slightly speeds up; then the centrifugal force reduces the velocity but the velocity increases again at the end of the bend. The end of the race involves strong acceleration before the final slight slow down.

“We see that VO2 is increasing for about 400 meters, while the force starts at maximal value. Then, the VO2 is constant, the force and the velocity decrease to a mean value. At the end, the VO2 decreases when the residual anaerobic energy reaches a third of its initial value. The effect of the bend and centrifugal force are obvious: it leads to a decrease in propulsive force and velocity. We see on the force profile that there is a very strong acceleration in the end. It can only take place after the bend where the centrifugal force reduces the available propulsive force.”

2100 metres

The first bend going up requires a rise of VO2 at the beginning of the race. They observed a strong start with the maximal velocity being reached in 200 meters. Then the velocity decreases and reaches a plateau. This plateau has been analyzed for human races in other studies and is related to what is known as a turnpike phenomenon. It is very likely that the horse’s optimal velocity for long races can be analyzed with this mathematical tool as well, they said.

“The first bend has a strong curvature and therefore drastically reduces the velocity. The propulsive force is reduced in the first bend. In the last bend, as in the previous race, the velocity decreases and increases again at the end of the bend.

“The end of the race is similar to the 1300 meters, with a strong acceleration before the final slow down. The horse in this race is not as good in terms of performance as the one in the 1900m and he cannot maintain his velocity similarly at the end of the race.”

Mercier Q, Aftalion A (2020) Optimal speed in Thoroughbred horse racing. PLoS ONE 15(12): e0235024. https://doi.org/10.1371/journal.pone.0235024

The study, published under a Creative Commons License, can be read here

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