Optimal speed in Thoroughbred horse racing.

The objective of this work is to provide a mathematical analysis on how a Thoroughbred horse should regulate its speed over the course of a race to optimize performance. 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. Our model relies on mechanics, energetics (both aerobic and anaerobic) and motor control. It is 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. In order to identify the parameters meaningful for Thoroughbred horses, we use velocity data on races in Chantilly (France) provided by France Galop, the French governing body of flat horse racing in France. Our numerical simulations of performance optimization then provide the optimal speed along the race, the oxygen uptake evolution in a race, as well as the energy or the propulsive force. It also predicts how the horse has to change its effort and velocity according to the topography (altitude and bending) of the track.

Introduction

Very little is known about the optimal strategy for a Thoroughbred horse to run and win a race. 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 ability according to the race distance or to optimize how to regulate its speed along the race can be crucial.

Due to limitations in the measurement of the mean oxygen uptake () for a horse at high exercise, no information is available on the full profile in a race, depending on the distance. Up to now, for Thoroughbreds, only measurements on treadmills have been obtained using masks [1-4]. Nevertheless, a portable mask technology has been developed [5], but to our knowledge no track tests have been yet performed on Thoroughbred racehorses. It is only for Standardbred and endurance horses that some study have been made, see [6] for instance. A review book on horse physiology is [7] and some information can also be found in the report [8]. What is known is that horses have a high aerobic capacity, about twice that of human beings, due to a high capacity for oxygen carriage and extractions, as well as a high stroke volume [9, 10]. They reach the maximal oxygen uptake () much quicker than humans, nevertheless no precise estimate of the time needed to reach a steady state is known. Similarly, no information is available about the distance or time at which the starts decreasing at the end of a race. Eventually, for long distances, it is not known whether the remains almost constant for the most part of the race or oscillates around a mid value, and then at which period and at which amplitude. Estimates on the proportion of energy derived from aerobic and anaerobic pathways during competitive events have been made according to breed and length of race [10] but the relationship that exists between performance and anaerobic capacity remains to be determined. This paper will provide pieces of information for all these issues.

Reference [11] is the only one that we know of where pacing strategy for horses is analyzed, together with the effect of drafting. Several directions of study have been investigated to better understand the effort or mechanical work developed by horses. One of them is to measure the propulsive force either on a force plate [12] or with an instrumented horse shoe [13, 14]. More on the biomechanics of athletic horses can be found in [15] for instance.

The aim of this paper is to provide a mathematical model able to predict how a Thoroughbred horse should regulate its speed over the course of a race in order to optimize performance. We will see how this depends on the distance to run, but also on the shape and topography of the track. It is based on the model developed in [16-18] for human races and it is adapted here to horses to fit the data.

Based on data provided by France Galop and the Mc Lloyd tracking device in French horse races, we are able to model the optimal horse efforts and velocity for a fixed distance, depending on the curvature and change of slopes and ramps. Our model yields in particular information on the profile.

Materials and methods

Data

The data consist of two dimensional position and speed sampled at 10 Hz for horses racing in Chantilly (France), at the end of the 2019 Thoroughbred horse racing season. The races were all run on a PSF track in a standard surface condition. The data are provided by France Galop the French operational body for flat horse racing, and are from roughly ten races. The tracking system is developed by Mc Lloyd. It is a miniaturized device which does not bring any discomfort to the horse or the jockey. It weighs 90g and is put by the jockey under the saddle. Reliable data is obtained thanks to a patented positioning technology and robust mobile network data transmission, even during crowded events. The accuracy of the system was validated on horses by France Galop by comparing 1ms-accurate photo finish data on more than one hundred races with gap data on the finish line obtained after processing latitude and longitude data: the device mean accuracy was confirmed to be 25cm or 2 hundredth of a second. It is now used by France Galop on all races to provide live position information to the audience. The tracking system provides the latitude and longitude data sampled at 10Hz, as well as the velocity. The latitude and longitude data given by the tracker are projected over a reference track leading to the position of horses in the race which is therefore given as live information to the audience on all races.

Model

Once we have these raw data, we have to smooth the speed using a third order Savitzky–Golay filter. Therefore, the raw data provide two curves for each horse sampled at 10Hz:

the curve of time vs distance from start, projected on the reference track, so that each horse runs the same distance,

the curve of velocity vs distance from start, projected on the reference track.

These curves will allow to determine all the parameters specific to horses.

The model of [16-18] yields an optimal control problem based on a system of coupled ordinary differential equations for the instantaneous velocity v(s), the propulsive force per unit of mass f(s), the anaerobic energy e(s), where s is the distance from start. The system relies on Newton’s second law of motion and the energy balance which takes into account the physiology of horses. The energy balance is between the aerobic contribution or σ(e), the anaerobic contribution e(s) and the power developed by the propulsive force. The mechanical part takes into account the positive slope or negative ramp α(s), the bending of the track and the control on the variations of the propulsive force. A crucial piece of information to be taken care of is the centrifugal force in the bends. For this purpose, the horse is identified with a bending rod. The centrifugal force does not act as such in the equation of movement but limits the propulsive force through a constraint which yields a decrease in the effective propulsive force in the bends.

The physiology of the horse is taken into account through a number of parameters:

the maximal propulsive force per unit of mass f,

the global friction coefficient τ which encompasses all kinds of friction, both from joint and track. In total, fτ is the maximal velocity,

the maximal decrease rate and increase rate of the propulsive force which is related to the motor control of the horse: a horse, like a human being, cannot stop or start its effort instantaneously, but needs some time or distance to do it. This is what our control parameters u− and u+ will provide,

the total anaerobic energy or maximal accumulated oxygen deficit e0,

the profile as a function of distance, namely the distance d1 at which the maximum of is reached, the distance d2 at which decreases, and the relative decrease with respect to the maximum value. This is a curve σ(s) where s is the relative distance from start, but in fact, in the model, it is a curve σ(e(s)) where e(s) is the remaining anaerobic energy. The profile of σ is to be identified from the data.

These parameters are not measured or given but they are computed numerically for each horse and each race: they are the parameters that allow the optimal control problem to best fit the data, as we will explain below.

Therefore, for a fixed distance, the model predicts the final time, the velocity curve and the effort developed by the horse to produce the optimal strategy. It depends on the geometry of the track, the ramps and slopes and the physiology of the horse.

For the ease of completeness, we provide below the full optimal control problem though the results of the paper do not require to understand it and the reader can skip this paragraph as a first reading. Instead of writing the equations of motion in the time variable, we write them using the distance from start s. This amounts to dividing by v the derivatives in time in order to get the derivatives in space. We also write the equations per unit of mass. Let d be the length of the race and g = 9.81 the gravity. Let c(s) denote the curvature at distance s from the start, which is provided for each track and α(s) be the slope coefficient. Let v0 be the initial velocity. Let e0, f, τ, u−, u+ and the function σ be given. They are identified for a specific race and horse. The optimal control problem (where minimizing the final time is equivalent to minimizing the integral of the inverse of the velocity) coming from the Newton law of motion and the energy conservation is and under the state constraint for s ∈ [0, d].

The optimal control problem for horse performance is solved using Bocop, an open licence software developed by Inria-Saclay France [19]. The discretisation used to solve the system of coupled ODE’s is set to one point every two meters which ensure the same accuracy whatever the race. The solution of the optimal control problem yields, for each race and each horse, the optimal effort, regulation of velocity and the profile depending on the length and topography of the race. Our aim is therefore to identify the physiological parameters f, τ, u+, u−, e0, σ(e) from the available data.

Identification process

The identification process is made through a bi-level optimization procedure looking for minimizing errors between the response of a Bocop simulation and the data through the following objective where the subscript simu (resp. data) refers to the variable extracted from the simulation (resp. data). The distance s1 = 0 is the beginning of the race and s = d is the length of the race, while s are intermediate distances. The parameter p refers to the vector containing all the variables to identify: where σ is the maximal value of σ and σ its final value. The objective is made up of two parts: first the difference in final time at the end of the race d, and then the mean square error over the speed measured at N points. For our identification process, N is taken equal to one thousand which ensures a good accuracy for the objective calculation (since it is one point every 2 meters for the longest race as the Bocop calculation) while keeping a relative low time cost. The points are evenly distributed between s1 and s.

The algorithm used here is a particle swarm optimization method [20] available in the pyswarm library in Python 3, part of the family of the heuristic optimization methods [21]. The main advantage of such a method is its good ability to explore the design space and its ease of use and implementation. A swarm of designs {p} is tested, that is the optimal control problem is solved for these parameters using Bocop, and its objective value is calculated. The performance influences the speed and the direction of the particles inside the design space for the next iteration. At the end of the process, the best score particle is kept as the result. The stopping criterion of the algorithm is set such that the algorithm is unable to find new particles for which the objective is at least 10−7 better than the best score observed until then. For all the examples treated in this paper, the swarm size is set to 50 particles and a maximum of 150 iterations. Each identification process has reached the stopping criterion described. This method is well suited to our problem since the objective space has a lot of local minimizers so that gradient based methods can get stuck in local minimizers. It insures overall robustness in the results as the design space is always well explored before converging toward a particular area of the design space.

Topography of the tracks

We have studied three types of races: a 1300 meters, a 1900 meters and a 2100 meters in Chantilly. The GPS track is shown in Fig 1. The 1300m starts with a straight, then there is a bend before the final straight. The 1900m starts earlier with an almost straight and follows the 1300m. The 2100m starts in the final straight of the other races, makes a closed loop before reaching the same final straight and finish line.

Fig 1

GPS track of the 1300m, 1900m and 2100m in Chantilly, France.

The elevation and curvature profiles are provided by France Galop and illustrated in Fig 2. The tracks are made up of straights (zero curvature), arcs of circles (constant curvature) and clothoids (curvature increasing linearly with distance). A clothoid is the usual way to match smoothly a straight and an arc of a circle since the curvature increases linearly. It is used for train tracks and roads as well. It allows smooth variations of velocities which are more comfortable for horses.

Fig 2

Altitude and curvature vs distance for different tracks.

(a) 1300m (first column), (b) 1900m (second column) and (c) 2100m (third column). The last 1300m are always the same.

The specificity of the track is that there is a bend of 500 meters before the final straight, where the track is going down in the first quarter (about 1.5%) and going up in the second quarter (about 2%). In the 2100 meters, there is also a bend of 400 meters just after the start, which is first going up and then down. We will see that curvature and altitude have a strong effect on the optimal velocity.

Let us point out that the track is banked but, because data correspond to horses close to the inner part of the track, the banking is not meaningful for the data and will not be taken into account here.

Results

Among all the available data, we have watched the videos of the races and chosen three races and three horses with the following criteria: no interactions (or at least very little) with other horses that modified the speed, no specific strategy from the jockey to regulate the speed. Therefore, we have chosen horses which seemed to be close to have run a race which would have been similar if they were alone, and could be qualified as their optimal race. We are going to present three significant races of 1300m, 1900m and 2100m. They are all races on a PSF track in a standard surface condition. The 1300m was for 2-year-old horses, the 1900m for 3-year-old and the 2100m for 4-year-old.

We describe below the results of our simulations.

1300 meters

The parameters identified for this race are in Table 1. The velocity data (raw and smoothed) and the velocity computed with our model are plotted in Fig 3 for the 1300m. We observe a very good match between the curves: there is a strong start with the maximal velocity being reached in 200 meters. Then the velocity decreases, and in particular in the bend, between 300 and 600 meters from start. Though the track is going down, the centrifugal force reduces the propulsive force as we see in Fig 4b (the black curve shows the limitation due to the centrifugal force). It is only when reaching the clothoid, 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 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.

Table 1

Identified parameters for the 1300m.

τ	e0	fM	d1	d2	σM	σf	u−	u+
3.911	2731	5.150	421.5	559.9	47.0	40.71	-1.693e-03	1.504e-03

Fig 3

Velocity data for the 1300 meters race.

Raw data, smoothed data, (t = 76.544s) and computed velocity for the identified paramaters (t = 76.544s).

Fig 4

and propulsive force vs distance for the 1300 meters race.

(left in green) and propulsive force (right): blue is the propulsive force f(s) in the direction of movement, black is the effective propulsive force taking into account the centrifugal force, where c is the curvature.

The curve vs distance and propulsive force vs distance are plotted in Fig 4. We measure σ in J/s/kg but we want to plot the results in terms of ml/mn/kg knowing that one liter of oxygen produces roughly 21 kJ (see [22] and also [23]), so we have to multiply our data for σ by 60/21. For a maximal value of σ equal to 47, this yields a of 133.6 ml/mn/kg. We see that the is increasing for about 400 meters, while the force is decreasing. Then when the 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. We point out that the value of the propulsive force is higher than the ones found in [13, 14] but the velocity is also much higher.

1900 meters

The parameters identified for this race are in Table 2. The velocity data (raw and smoothed) and the velocity computed with the model are plotted in Fig 5 for the 1900 meters. We observe that 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 is with a strong acceleration before the final slight slow down. The curve vs distance and propulsive force vs distance are plotted in Fig 6. We see that is increasing for about 400 meters, while the force starts at maximal value. Then, the is constant, the force and the velocity decrease to a mean value. At the end, the 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.

Table 2

Identification parameters for the 1900 meters.

τ	e0	fM	d1	σM	d2	σf	u−	u+
3.73	2702	4.74	379	54.6	1392	40.3	-3.69e-03	3.94e-03

Fig 5

Velocity data for the 1900 meters race.

Raw and smoothed data (t = 116.460s) and computed velocity (t = 116.460s) for the identified parameters.

Fig 6

and propulsive force vs distance for the 1900 meters race.

(left in green) and propulsive force (right): blue is the propulsive force f(s) in the direction of movement, black is the effective propulsive force taking into account the centrifugal force, where c is the curvature.

In Fig 7, we have plotted a zoom on the velocity curve for the identified parameters, and then have removed the effect of the slope (flat track), of the curvature (straight track) and of both (flat, straight track). This allows to notice the specific effects: a bend reduces strongly the velocity (pink curve); on the real track (brown curve), because the first part of the bend is going down, the reduction in propulsive force and velocity is not so strong. The end of the track is uphill and one notices that the velocity curves corresponding to going up (brown, red) cannot provide a speeding up as high as the two others. The combination of slopes and ramps of this track (red curve) reduce the velocity and final time in total though the profile is very similar. We also point out that though these are local effects, they have a global influence on the strategy since they change the mean velocity.

Fig 7

Effect of the slope and curvature on the velocity curve (zoom) for the 1900 meters race.

Brown is the velocity curve of the race, red with the slope only (straight track), pink with curvature only (flat track) and green is a flat, straight track.

2100 meters

The parameters identified for this race are in Table 3. The velocity data and the computed velocity are plotted in Fig 8 for the 2100 meters. The and propulsive force are plotted in Fig 9. Here, the is modified to match the behaviour observed in [24] for humans where the first reaches a peak value, before the mean race value. The first bend going up requires a rise of at the beginning of the race. We observe that there is 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 race in [25] and is related to a turnpike phenomenon. It is very likely that the horse optimal velocity for long races can be analyzed with this mathematical tool as well.

Table 3

Identification parameters for the 2100 meters.

τ	e0	fM	d1	σ1	d2	σM	d3	σf	u−	u+
3.637	3567	5.50	304	51.29	524	46.71	1613	41.67	-1.928e-03	9.922e-04

Fig 8

Velocity data for the 2100 meters race.

Raw and smoothed data t = 130.933s) and computed velocity (t = 130.933s) for the identified parameters.

Fig 9

and propulsive force vs distance for the 2100 meters race.

(left in green) and propulsive force (right): blue is the propulsive force f(s) in the direction of movement, black is the effective propulsive force taking into account the centrifugal force, where c is the curvature.

The first bend has a strong curvature and therefore reduces drastically the velocity as we can see in Fig 9b: 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.

Discussion

Results on

From experiments on human races from 400m to 1500m (that is of similar duration of the races we analyze here) [24, 26], it is expected that the curve vs distance is

increasing to a maximal value and then decreasing for short exercises,

increasing and reaching the maximal value , and then decreasing at the end of the race when the residual anaerobic energy is less than 30%,

reaching a peak value which is higher than the value along the race for moderate length exercises.

Our simulations and identifications yields that the behaviour is the same for horses. The results of our simulations even provide precise information on the curve all along the race. We see in Figs 4a, 6a and 9a, that

the maximal value of is reached in about 400m, that is about 20 to 30 seconds from start, which is indeed much quicker than humans, (this is consistent with [1]),

the 1300 meters is a short exercise where increases and decreases,

for the 1900 and 2100 meters race, the reaches a mean value during the race and decreases about 500 meters from the finish line, when, as for humans, the residual anaerobic energy is about a third of the initial value. The launch of the sprint is optimal when the starts decreasing at the end of the race, and this follows from optimal control theory. Numerically, we can observe that the decrease in and increase in velocity at the end of the race take place at the same time,

in a 2100 meters race, the first reaches a peak value before decreasing to a plateau value.

The longer the horse can maintain its maximal value , the better the performance is. Because the change of slope in is related not exactly to the distance but to the available stock in anaerobic capacity, a high anaerobic capacity is all the more important to maintain high velocities all along the race. A strong acceleration at the beginning of the race allows to reach the maximal value of quickly and is the best strategy. Jockeys often start slower than the optimal strategy being afraid that if the horse accelerates too strongly with respect to its capacity at the beginning of the race, then the drop in velocity at the end of the race will be bigger.

It is very likely that for horses, as for humans and explained in [24], results on treadmills are very different from measurements during a race. Indeed, as pointed out in [24], exercise at constant velocity yield contradictory results with exercises in a real race in terms of or pacing strategy. On a treadmill, the increases to reach a maximum value, whereas on a real race a decrease of is observed. Our numerical simulations obtained with varying parameters around the identified ones illustrate that in many cases, the best performance is achieved with a fast start, when the pace at the beginning of the race is higher than the pace at the end. While a fast start helps to speed up kinetics, and limits the participation of the anaerobic system in the intermediate part of the race, nevertheless, if it is too fast, it has the potential to cause fatigue and have an overall negative effect on the performance. Therefore, a departure which is too fast with respect to the horse’s capacity increases the participation of the anaerobic system at the beginning of the race and can damage the final performance. Eventually, a fast start does not necessarily induce a good performance. But a high value of v can allow a faster start velocity without increasing the O2 deficit.

As soon as there is a slope or ramp, the is also impacted. Here, in Chantilly, the slope coefficients are strong enough to change the optimal velocity but not strong enough to modify the overall profile. The only effect is on the optimization of performace. For stronger slopes, one would need to take into account additionally that increases with slopes both downhill and uphill as explained in [27] and [28].

Energy

Horses have two distinct types of energy supply, aerobic and anaerobic. In our model, it is e0 which estimates the anaerobic energy supply. For human races, recent research suggest that energy is derived from each of the energy-producing pathways during almost all exercise activities [29], which is what we also observe in our simulations.

In Table 4, we have computed the percentage of anaerobic energy to the total energy according to the length and duration of the race. The horse of the 1900m race has a very strong and therefore uses a lower anaerobic energy. We point out that the values estimated in [1] on a treadmill seem to be under estimated with respect to ours: for an exercise of duration 130 seconds, they find an anaerobic contribution of around 30%, which is smaller than our value. Indeed, in a race, for a similar duration of exercise, velocities and forces are much higher than on a treadmill, leading to a bigger contribution of the anaerobic supply [29].

Table 4

Percentage of anaerobic contribution in the total energy during the race.

Race length (m)	duration (s)	V˙O2 deficit
1300	76	47.32%
1900	116	32.09%
2100	131	38.01%

Fig 10 provides the evolution of the anaerobic energy of the three horses vs the distance to the finish line. We have highlighted the points where the changes slope. There is a strong anaerobic energy consumption both at the beginning and end of the race, that we can identify through strong slopes. The beginning of the race corresponds to the distance to reach .

Fig 10

Evolution of the anaerobic energy vs the distance to the finish line.

Green: horse of the 1300m, orange: horse of the 1900m, pink: horse of the 2100m. The distances are highlighted when the energy has a change of slope and the energy value at this point is indicated as well: it is at the beginning of the race when the is reached, and at the end of the race, when the starts decreasing.

The horse of the 1300m is a young horse and he needs more time (or a bigger distance) to reach (1214m vs 839 or 889 for the two others). The horse of the 1900m has a higher , therefore he needs less anaerobic energy when he is at to maintain his speed. Indeed, once the maximal oxygen uptake is attained, the extra energy to increase speed further is provided by anaerobic pathways. So if a horse has a higher , he will use less anaerobic energy to run at the same speed as a horse with a lower . In total, this explains why the horse of the 1900m has a lower contribution of the anaerobic energy to the total energy.

Effect of slopes, ramps and bends on a race

As evident from the data of [30], on a tight bend, horses slow down a lot. In Chantilly, the bends have a radius of at most 100m, which is not tight, but still has a strong impact on the optimal velocity.

To better illustrate this effect, we choose a race of 1900 meters and set an imaginary slope or ramp or bend for one third of the race at the beginning, middle or end of the race (that is roughly 630m) with the following configuration:

either a positive slope of +3% for 630m,

or a negative slope of −3% for 630m,

or an arc of circle with a curvature of 1/100m−1 for 630m.

Figs 11, 12 and 13 provide the optimal velocity vs distance for the 1900 meters parameters. The common feature is that a local change of elevation or bend does not only produce a local change in velocity but changes the whole velocity profile and mean value.

Fig 11

Effect of slope on the velocity profile for a 1900m race.

Flat straight track (blue), +3% slope for 630m at the beginning of the race (orange), +3% slope for 630m at the middle of the race (green), +3% slope for 630m at the end of the race (red).

Fig 12

Effect of ramp on the velocity profile for a 1900m race.

Flat straight track (blue), +3% ramp down for 630m at the beginning of the race (orange), +3% ramp for 630m at the middle of the race (green), +3% slope for 630m at the end of the race (red).

Fig 13

Effect of bend on the velocity profile for a 1900m race.

Flat straight track (blue), bend for 630m at the beginning of the race (orange), bend for 630m at the middle of the race (green), bend for 630m at the end of the race (red).

For a slope going up, as illustrated in Fig 11, the best time is obtained when the slope is at the end of the race. Indeed, a good horse can still provide a strong effort at the end of the race, even if he is tired. If the slope is at the beginning, it has a strong effect on the velocity which cannot reach its maximum value. In the middle of the race, the slope reduces the mean velocity and therefore the final time.

For a ramp going down, as illustrated in Fig 12, it is the opposite: the best time is obtained when the ramp is at the beginning of the race. Indeed the horse speeds up more easily and more quickly.

For the bend, as illustrated in Fig 13, the best time is obtained when the bend is at the middle of the race. This is where it has the smallest decrease on the velocity profile. At the beginning, it prevents the horse from reaching its maximal speed. Of course, here the effect is exaggerated with respect to a real race because the bend is long but it yields the general flavour. Similarly, at the end, it prevents the horse from sprinting.

Therefore, we have seen that the optimal velocity has to be analyzed with respect to the changes of slopes, ramps or bends in order to optimize the horse effort. For a given track, it is very likely that the turnpike theory of [25] should yield more precise and detailed analytical estimates of the increase of decrease of velocity.

Conclusion

Thanks to precise velocity data obtained on different races, we are able to set a mathematical model which provides information on how horses have to regulate their speed and effort on a given distance. It relies on both mechanical, energetic considerations and motor control. The process consists in identifying the physiological parameters of the horse from the data. Then the optimal control problem provides information on the velocity, the propulsive force, the and the anaerobic energy. 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. The horse that slows down the least at the end of the race is the one that wins the race. We understand from the optimal control problem that this slow down is related to the anaerobic supply, the and the ability to maintain maximal force at the end of the race. 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 profile, such as when steady state is reached, when the decay of starts. The ability to maintain a high for a long time is related to the ability to maintain velocity. The 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.

We also understand better the effects of altitude and the bends and find that they are not local effects producing a local perturbation on the velocity strategy, but on the contrary have a global 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.

Future works will be devoted to taking additionally into account drafting and the horse psychology [31] since an alternative strategy can be to stay behind to save energy and overtake in the last straight [11].

To maximize an individual horse’s potential for winning, it should be entered in races appropriate for its racing ability. Therefore information on a horse speed, endurance or running economy coupled with simulations can help to predict how a horse profile is adapted to some distances to run.

25 Aug 2020

PONE-D-20-17088

Pacing strategy in horse racing

PLOS ONE

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PLOS ONE

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Reviewer #1: Partly

Reviewer #2: Yes

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Reviewer #1: N/A

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #2: Yes

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Reviewer #1: This article addresses a completely new concept in the racehorse. It is particularly interesting to bring a bit of science to a discipline as traditional as gallop racing.

However, perhaps I missed it, but I have the impression that you define the optimal strategy, not from the analysis of a substantial amount of race data, but by having a priori chosen for each distance the profile that seemed optimal to you. It's intellectually embarrassing.

Introduction

Line 4 - « Up to now, only measurements on treadmills have been obtained using masks”. This is true for Thoroughbreds, but track measurements have been made for Standardbred and endurance horses.

Line 14 - Could you define the notion of "pacing strategy"? This notion can be ambiguous in the horse, a species in which there are naturally several paces and where the notion of rhythm or cadence has its own definition in traditional equitation.

Lines 28-29 – “Data are provided by France Galop and are from roughly ten races.”

This is very imprecise. How many races, of what level, how many different horses, of what age, male or female, same day or different day? The physiology of a 2 year old is not the same as that of a 3 or 4 year old. The quality of the terrain varies from one day to the next and influences the fatigue of the horse during the race. Good horses will not necessarily have the same behaviour as poor quality horses ...

Lines 32-34 – Has the accuracy of the GPS system been validated by you? Is it published?

Line 44 – “We use the model developed by [10-12] for the optimal strategy in running and adapt it for horses.” On what scientific basis can you state that the model defined in humans is also suitable for horses? How can you calculate physiological parameters such as strength, anaerobic energy or VO2 without including data obtained from the horse? These are two very different athletes. Can you explain?

Lines 85-86 – “This yields the optimal strategy depending on the length of the race, including the velocity profile and the VO2 profile.” The notion of “optimal strategy” troubles me since you have only analysed 3 races with 3 different horses.

Line 93 – Why did you chose N= one thousand points?

Line 94 – “the points are evenly distributed between s1 and sN.” This means that the distance between the points varies with the running distance, and therefore the accuracy?

Line 127 – “We have chosen three significant races of 1300m, 1900m and 2100m. For each one, we have taken the data of a horse which seems to have run an optimal race.”

What do you mean by significant?

How did you choose this horse? On what criteria? You try to model an optimal race and starting from a race that you judge optimal. It's contradictory. How do you know that this race is optimal?

Why didn't you take data from several horses, over several races of the same distance?

Lines 138-139 – « Nevertheless, a decrease in velocity at the end of such a race takes place even on a flat track.” Is this an observation from your model or a simple statement?

It seems to me that in the horse, the winners are often those who are capable of a final speed peak, or at least capable of maintaining a relatively constant speed while the losers suffocate and lose speed.

Moreover, if we compare the speed curves over 1300 and 1900m, we can see that the speed increases between 1300 and 1900m.

I'm very surprised at the VO2 values calculated in your model. They are relatively equivalent between the 1300 and 2100m races, but much higher for the 1900m race. Were the races and horses chosen really comparable? Doesn't this show that your model, if it is relevant to describe what happens during a race, is still very race-dependent? Can you discuss or propose an explanation?

Figure 7 and comments in the text. You use several terms to describe different conditions (e.g. pink/purple curve, curvature only) to describe the same things.

Line 180 – “The first bend has a strong curvature and therefore reduces drastically the velocity” – How can you say that? you have nearly the same velocities as for the 1900m race

Line 205 –What do you mean by “mean value of the race” ?

Lines 207-224 – I fully understand the idea and what intuitively sounds like an ideal. However, I don't understand how your results lead to this conclusion. That would require a comparison of the races of winners and losers. All you describe is that the best horses have a higher VO2 max, which is nothing new.

Line 232-233. Ref [22] is not really “recent”

Line 240 – Another explanation is that the forces and energy involved on a treadmill are much less than on a track at the same speed.

Table 4. It's very embarrassing that you don't have horses of equal quality or average data on comparable races.

General question: Why didn't you use data from other horses/races to test your assumptions?

Conclusion

Lines 271-272 – I disagree. You need to be more nuanced. You modeled 3 races of 3 different horses over 3 different distances on one track. This allows you to explain what happens and the physiological limits (= everything you develop afterwards). You cannot say that this provides you with the optimal pacing strategy. You would have to analyze data from many races, with horses of different categories.

For me, it would be more explicit to group together figures [3, 5, 8], [4, 6, 9] and [7, 10-12] to facilitate comparisons.

Reviewer #2: I have reviewed "Pacing strategy in horse racing" by Mercier and Aftalion.

Overall, the paper will contribute towards knowledge in the area of the mathematical model able to predict the pacing strategy and worthy of publication. The authors have documented the development of a mathematical model to predict the pacing strategy depending on the distance to run, the shape and topology of the track. A vital issue to address is the discussion of the results mainly concerning VO2max during the 1900 m race. The outcome is clear, but not all conclusions can be justified based on the study's data. I have a few comments and questions that I have detailed below.

Title

The title is clear and informative; however, including the horse breed would be advised since the study was performed in a single breed.

Abstract:

Overall I feel that the research question is not addressed adequately in the abstract. I would like to see a brief rationale as to why the optimal strategy for a horse to run a race is essential in Thoroughbred horses.

The abstract does not contain any data or indication of statistical analysis.

Introduction

Well written and clear. However, the authors do not state a hypothesis.

L5-6: In my opinion, this reference [5] is ancient. You can keep it, but please add a new reference to this information. And authors must specify the interesting breed, the Thoroughbred.

"What is known is that horses have a high aerobic capacity, about twice that of human beings".

This is true for Thoroughbred racehorses. Furthermore, the metabolic demand for Thoroughbred horses during a race is quite different. We have racing distances from 1000 to 3200 m. For example, for 3200 m (2-mile race), the aerobic contribution may be up to 90%.

Please, you must discuss and review the information in the introduction.

Materials and methods

A statement of ethics approval is required before the materials and methods information.

Results

Fig 4., Fig 6. and Fig 9.: "blue is propulsive force." Please note that VO2 (left) is blue too. Improve the caption text to increase readability.

Discussion

L188: "From experiments on human races," What kind of races? Please clarify it.

L236: "The horse of the 1900m race has a very strong VO2max and therefore uses lower anaerobic energy".

Here we must remember that to reach the velocity related to VO2max, and horses need of the anaerobic contribution. Anaerobic contribution (glycolysis) starts to supply ATP from 55% VOmax (see: DOI: 10.1152/japplphysiol.00909.2001). Usually, this velocity/intensity corresponds to the lactate threshold. Besides, Thoroughbred horse locomotor muscles usually contain high percentages of type 2A fibers. Type 2A fibers have a considerable number of capillaries and mitochondria and rely on glycolytic and oxidative metabolism. Also, there are fibers 2AX, a hybrid fibers type. Thus, I wonder how the Thoroughbred does to keep its VO2max during the 1900 m race without the anaerobic metabolism contributing. May be through of the high-energy phosphate system like phosphocreatine pathway? The discussion does not clarify the role of the non-mitochondrial metabolic pathways contribution (i. e., glycolytic, and phosphagen metabolic pathways) during a race. These aspects should be included when discussing and reviewing the results of the current study.

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Reviewer #2: Yes: Guilherme Camargo Ferraz

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23 Sep 2020

Reviewer 1 :

I have the impression that you define the optimal strategy, not from the analysis of a substantial amount of race data, but by having a priori chosen for each distance the profile that seemed optimal to you. It's intellectually embarrassing.

We have had the data for about 30 races of various distances. Because for the time being our model is for a single horse running alone, the model does not take into account interactions between horses : in particular how staying behind for some time saves energy and allows to overtake and win, or the selection of a strategy by the jockey such as starting slowly or on the opposite starting very strongly. Therefore, we have tried to select, among the data we had, horses that ran races close to a race they would have run alone : that is they didn’t seem to have interactions or at least very little, and their strategy didn’t seem to follow from a group strategy or the jockey strategy. Our definition of the optimal race is something close to a race alone. It is not that we have picked the races to match the model, it is just that if a horse runs a race leading it from the beginning to the end, it is easier to fit a model corresponding to a single horse.

Our aim is to extend the model to take into account the two effects of interactions and strategy. But our model as it is cannot predict the time run by the last horse : indeed, the last horse is likely to have run too strongly with respect to his capacity at the beginning of the race and not to be able to run properly till the end. We are working at including strategy and interactions but to start with we needed to have values for the physiological parameters which are consistent for a single horse.

Moreover, if one takes a large amount of races on the same track, on a similar ground, there is not an optimal strategy for all the horses or even for all the winners, because the strategy will depend on interactions: for instance if some horses start too strongly or if everyone starts slowly. In order to predict all types of strategies in the future, we need to fit our one horse model and have the profile for each horse running. Later, we hope to be able to deal with multi horses strategy or change of strategy with respect to the optimal regulation of speed for a single horse.

We have added a paragraph at the beginning of the section « Results » to try and clarify this.

Introduction

Line 4 - « Up to now, only measurements on treadmills have been obtained using masks”. This is true for Thoroughbreds, but track measurements have been made for Standardbred and endurance horses.

We didn’t know this, we have added « for Thoroughbreds » in the text and two references, but we are ready to add other references if the referee is willing to provide some.

Line 14 - Could you define the notion of "pacing strategy"? This notion can be ambiguous in the horse, a species in which there are naturally several paces and where the notion of rhythm or cadence has its own definition in traditional equitation.

We agree with the referee that the notion of pacing is ambiguous, it is used for humans but is probably not as meaningful for horses. We have tried to replace it with « regulating speed to optimize performance », but one of our references has this word in the title, so we could not completely remove the word from our paper.

Lines 28-29 – “Data are provided by France Galop and are from roughly ten races.”

This is very imprecise. How many races, of what level, how many different horses, of what age, male or female, same day or different day? The physiology of a 2 year old is not the same as that of a 3 or 4 year old. The quality of the terrain varies from one day to the next and influences the fatigue of the horse during the race. Good horses will not necessarily have the same behaviour as poor quality horses ...

As pointed out below, when we got the data, only 26 races had been tracked from october 2019 on. Then, there was winter, the covid, and it is only starting slowly again, so there are not many races available. Because it was the end of the season, the races did not involve the best horses. Among these 26 races, France Galop provided us with 10 races with « good » horses.

We have added information on terrain and age. The three races were not the same day. France Galop does not allow us to put the date.

Lines 32-34 – Has the accuracy of the GPS system been validated by you? Is it published?

No, we have not validated the GPS system ourselves. The system was bought by France Galop, the French operational body for flat horse racing in France and they have checked it on about a hundred races comparing the photo every 200m on the last 600m and got accuracy up to the 1000th. More info is on

https://mclloyd.com/wp-content/uploads/2020/09/HPV2-Data.pdf

The device is now used on every race and some data are available live. We have tried to make our paragraph clearer

Line 44 – “We use the model developed by [10-12] for the optimal strategy in running and adapt it for horses.” On what scientific basis can you state that the model defined in humans is also suitable for horses? How can you calculate physiological parameters such as strength, anaerobic energy or VO2 without including data obtained from the horse? These are two very different athletes. Can you explain?

The previous formulation was very bad. We did not mean that the model defined for humans is suitable for horses. What we meant is that the laws on which the model is based, Newton 2nd law of motion and energy conservation are the same principles that are used for horses. But of course, the model takes into account the physiology of horses. What it does not take into account is the detail of the stride, or on the fact that the horse is going up and down as it gallops. But it did not take into account the detail of the stride for humans either. Nevertheless, this captures the mean velocity on a stride, which is the essential feature leading to velocity computations and interesting information.

The strength of the model is that we do not need to measure all these physiological parameters (strength, VO2, anaerobic energy etc). The values are identified from the GPS data and velocity data and seem to match pretty well what is expected. So of course, we include the data from the horses to compute all the parameters of the problem.

We have tried to improve the explanations in the paper.

Lines 85-86 – “This yields the optimal strategy depending on the length of the race, including the velocity profile and the VO2 profile.” The notion of “optimal strategy” troubles me since you have only analysed 3 races with 3 different horses.

We are not performing a statistical analysis to determine the strategy depending on the distance. Given a horse, a distance to run and a shape of track, the issue is to determine how the horse should regulate its speed in the course of the race to make the best final time. It turns out that for each horse and race analyzed, we have found parameters that match the velocity/gps profile extremely well. If we had analyzed 10 races or 10 horses, it would have been different parameters. But they are always in the same range. It is not the number that is going to provide a global strategy.

But the next step is given the speed profile, to analyze the possible strategy to put when there are two different horses. Each horse and each race corresponds to an optimal strategy.

Line 93 – Why did you chose N= one thousand points?

Line 94 – “the points are evenly distributed between s1 and sN.” This means that the distance between the points varies with the running distance, and therefore the accuracy?

The discretisation for the optimisation problem is to take one point every two meters, we have added this element. Then for the longest race, 2100 meters, taking N=1000 corresponds to one point every two meters as well. So we could have taken N smaller for shorter races, which we have not done, but it does not raise an accuracy problem.

Line 127 – “We have chosen three significant races of 1300m, 1900m and 2100m. For each one, we have taken the data of a horse which seems to have run an optimal race.”

What do you mean by significant?

How did you choose this horse? On what criteria?

We refer to our paragraph at the very beginning to answer this part and we have rephrased the paragraph in the text.

You try to model an optimal race and starting from a race that you judge optimal. It's contradictory.

No, we choose a race that we think optimal and identify all the physiological parameters (VO2, anaerobic energy, tau, f_max) etc on it.

If the horse starts slowly because the jockey wants it like this, then there is no hope that the model will fit this race whatever the parameters.

Nevertheless, in the future, now that we have identified parameters for horses, we hope to be able to model any strategy.

How do you know that this race is optimal?

We assume so by watching the video and checking that it is not full of overtaking, so not too much interaction with other horses. If it was not, the model could not fit any parameter.

Why didn't you take data from several horses, over several races of the same distance?

Because for the moment, very few races have been tracked with the device, and we do not have so many races of the same horse for instance. We also wanted to show the difference on different distances, and not analyze one specific horse.

Lines 138-139 – « Nevertheless, a decrease in velocity at the end of such a race takes place even on a flat track.” Is this an observation from your model or a simple statement?

It is an observation from GPS data : though you have the visual impression that the horse is speeding up, in effect in the last 200m, even the best horses slow down (though slightly of course). This is the case in the data we present, but also in all the data we are aware of.

It seems to me that in the horse, the winners are often those who are capable of a final speed peak, or at least capable of maintaining a relatively constant speed while the losers suffocate and lose speed.

They speed up in the last 600m or so but in the last 200m, they do not maintain this speed peak and in the very end, the best ones are the ones who slow down least. Of course, you cannot see it with speed data every 200m, but you see it clearly with GPS data.

Moreover, if we compare the speed curves over 1300 and 1900m, we can see that the speed increases between 1300 and 1900m.

Not exactly. The maximal speed is higher for the 1300 than the 1900 and the mean speed (distance divided by time) is also higher for the 1300 though the 1900 is run with better horses (3 year old instead of 2).

I'm very surprised at the VO2 values calculated in your model. They are relatively equivalent between the 1300 and 2100m races, but much higher for the 1900m race. Were the races and horses chosen really comparable?

It is true that the horses were not comparable, now this is stated better, but we do not think that it is a problem for the paper. It shows the effect of the VO2 on the race and that our model is rather powerful.

Doesn't this show that your model, if it is relevant to describe what happens during a race, is still very race-dependent? Can you discuss or propose an explanation?

Of course, the model is race and horse dependent and hopefully because not all races are run the same. For instance, the 1900m is run in 116 seconds while the 2100 in 131, so it is obvious from the total time that the horses of the 2100 were not as strong and the interest of our model is that it determines the physiological parameters which are different.

Figure 7 and comments in the text. You use several terms to describe different conditions (e.g. pink/purple curve, curvature only) to describe the same things.

Everything is now pink

Line 180 – “The first bend has a strong curvature and therefore reduces drastically the velocity” – How can you say that? you have nearly the same velocities as for the 1900m race

No, maybe the scale is not easy to read but the data stay below 17 (the identification slightly above), for the 2100, while for the 1900, they top speed at 17.6 at the beginning of the race, and there is a difference in the acceleration profile. Nevertheless, we have removed « drastically ».

Line 205 –What do you mean by “mean value of the race” ?

We have changed the wording, we mean that the VO2 reaches a plateau, which is not the maximal value of the race, which is reached before.

Lines 207-224 – I fully understand the idea and what intuitively sounds like an ideal. However, I don't understand how your results lead to this conclusion. That would require a comparison of the races of winners and losers. All you describe is that the best horses have a higher VO2 max, which is nothing new.

Actually, we can vary slightly the physiological parameters in the model to see the evolution of the optimal velocity. Once we have identified parameters on a horse, we can thus simulate a stronger departure or other situations and see how the velocity evolves. We do not need extra data. This is what we comment. We do not show figures because there are already many figures in the paper. We include a figure in the pdf file for the answer for the 1300m, where the blue is the identified solution, the orange has a slow start (and therefore speeds up more at the end but in total is 4 hundredth slower ) and the green has a fast start, slows down more at the end and is also about 4 hundredth slower)

Line 232-233. Ref [22] is not really “recent”

We have removed the word recent.

Line 240 – Another explanation is that the forces and energy involved on a treadmill are much less than on a track at the same speed.

We have added forces, we are not sure of any treadmill at the same speed as the ones in the race we present.

Table 4. It's very embarrassing that you don't have horses of equal quality or average data on comparable races.

It is the very beginning of the tracking system in France. In 2019, only 26 races were officially tracked as we mentionned above and very little has been done since. It is starting again now.

Nevertheless, in our point of view, it would be a different work to compare a same horse on different distances. As for average data, it is not the same type of mathematics since it would be statistics, whereas we need data for a single horse on a single race for identification.

Our model, as it is, once we have identified on a race, we can predict the velocity for the same horse on another distance or with another topography.

General question: Why didn't you use data from other horses/races to test your assumptions?

As explained before, this was the very beginning of the tracking so that not so many data were available and we thought it was interesting to show the first results we got on these simulations.

Conclusion

Lines 271-272 – I disagree. You need to be more nuanced. You modeled 3 races of 3 different horses over 3 different distances on one track. This allows you to explain what happens and the physiological limits (= everything you develop afterwards). You cannot say that this provides you with the optimal pacing strategy. You would have to analyze data from many races, with horses of different categories.

We have changed the sentence accordingly.

For me, it would be more explicit to group together figures [3, 5, 8], [4, 6, 9] and [7, 10-12] to facilitate comparisons.

We understand that this would have advantages but if one wants to see the relationships between speed, force and VO2, it is better to have them grouped together for the same horse, so we prefer to keep our initial grouping of figures.

Reviewer 2 :

A vital issue to address is the discussion of the results mainly concerning VO2max during the 1900 m race. The outcome is clear, but not all conclusions can be justified based on the study's data.

Title

The title is clear and informative; however, including the horse breed would be advised since the study was performed in a single breed.

Thoroughbred has been added to the title

Abstract:

Overall I feel that the research question is not addressed adequately in the abstract. I would like to see a brief rationale as to why the optimal strategy for a horse to run a race is essential in Thoroughbred horses.

The abstract does not contain any data or indication of statistical analysis.

We have rewritten the asbtract and tried to take into account the referee’s comments about the research question. There maybe a misunderstanding about the maths we are using : we definitely do not do any statistics, what we use are deterministics mathematics, that is coupled ordinary differential equations. We have tried to clarify this.

Introduction

Well written and clear. However, the authors do not state a hypothesis.

Since we use a deterministic model of coupled ordinary differential equations leading to an optimal control problem, we perform simulations with adequate parameters and analyze the results of the simulations, but with these types of mathematics, we cannot make hypothesis, we just follow the simulations results. The only hypothesis that we can think of is that some horses run an « optimal » race and therefore there is a hope to fit the data with our model. We have added a paragraph related to this issue.

L5-6: In my opinion, this reference [5] is ancient. You can keep it, but please add a new reference to this information.

We have added a more recent reference, which is a book but if the referee has others to suggest, we are willing to add them.

And authors must specify the interesting breed, the Thoroughbred.

Done

"What is known is that horses have a high aerobic capacity, about twice that of human beings".

This is true for Thoroughbred racehorses. Furthermore, the metabolic demand for Thoroughbred horses during a race is quite different. We have racing distances from 1000 to 3200 m. For example, for 3200 m (2-mile race), the aerobic contribution may be up to 90%.

Please, you must discuss and review the information in the introduction.

We have added a reference and discussed this issue but it seems that it is not so well known in the literature. The relationship that exists between performance and anaerobic capacity remains to be determined and only estimates according to distances exist in the literature.

Materials and methods

A statement of ethics approval is required before the materials and methods information.

We have not performed any experiments on horses. The races are official French races operated by France Galop, the French governing body for French flat races. For these races, they have added a GPS device of 90g on the saddle. So the only effect is to slightly increase the weight by 90g. France Galop has signed an agreement about horse welfare in March 2016 at the annual Agricultural Fair in Paris. It can be found on

http://www.france-galop.com/en/our-responsibilities

but i do not see how we could write of statement of ethics, since our only task was to receive position and velocity data. In reference 11 where they received data from English races, there is not ethics statement.

Results

Fig 4., Fig 6. and Fig 9.: "blue is propulsive force." Please note that VO2 (left) is blue too. Improve the caption text to increase readability.

We have changed the color of VO2 to green.

Discussion

L188: "From experiments on human races," What kind of races? Please clarify it.

from 400m to 1500m (that is of similar duration of the races we analyze here), added to the text

L236: "The horse of the 1900m race has a very strong VO2max and therefore uses lower anaerobic energy".

Here we must remember that to reach the velocity related to VO2max, and horses need of the anaerobic contribution. Anaerobic contribution (glycolysis) starts to supply ATP from 55% VOmax (see: DOI: 10.1152/japplphysiol.00909.2001). Usually, this velocity/intensity corresponds to the lactate threshold. Besides, Thoroughbred horse locomotor muscles usually contain high percentages of type 2A fibers. Type 2A fibers have a considerable number of capillaries and mitochondria and rely on glycolytic and oxidative metabolism. Also, there are fibers 2AX, a hybrid fibers type. Thus, I wonder how the Thoroughbred does to keep its VO2max during the 1900 m race without the anaerobic metabolism contributing. May be through of the high-energy phosphate system like phosphocreatine pathway? The discussion does not clarify the role of the non-mitochondrial metabolic pathways contribution (i. e., glycolytic, and phosphagen metabolic pathways) during a race. These aspects should be included when discussing and reviewing the results of the current study.

We have tried to clarify these aspects in the text and we have added a figure which plots the anaerobic energy vs the distance to the finish line. From this, it is clear that in the middle of the race where the aerobic consumption is maximum, the horse of the 1900m uses less anaerobic energy than the others, since the slope is smaller. As explained in ref. 10, when the VO2max is higher, then less anaerobic energy is needed to maintain the same speed. So the anaerobic metabolism contributes for the 1900m horse, but less than the others. Our model treats the anaerobic energy as a global tank so that we are unable to be more precise about the pathway.

If we take into account the running economy (VO2max/speed), then the horse of the 1900m has a much higher RE than the others.

Submitted filename: Replytoreviewers.pdf

Click here for additional data file.

19 Oct 2020

Optimal speed in Thoroughbred horse racing

PONE-D-20-17088R1

Dear Dr. Aftalion,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Dalton Müller Pessôa Filho, Ph.D.

Academic Editor

PLOS ONE

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Reviewers are agreement about the considerable changes in the manuscript and considered it suitable to be published in Plos One.

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Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

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Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: N/A

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Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

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6. Review Comments to the Author

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Reviewer #1: Thank you for considering each of my remarks, responding to them in detail and modifying your manuscript accordingly.

The value of your model and the fact that it needs to be adapted to each situation may not yet be sufficiently reflected in the text. Perhaps more emphasis should be placed on the fact that this is a preliminary study, and that not only the racing distance varies, but also the age of the horses, their level of fitness and probably their aptitude, the racing conditions ...

Nevertheless, this study is original, and the analyses well carried out. It deserves to be published.

Reviewer #2: The reviewer would like to thank the authors for taking the time to provide additional analysis and discussion of their research data and for making extensive additions to the manuscript to address comments from the initial review. The authors have addressed all recommendations for revision, and therefore the reviewer recommends accepting the manuscript for publication.

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Reviewer #1: No

Reviewer #2: Yes: Guilherme C Ferraz

30 Oct 2020

PONE-D-20-17088R1

Optimal speed in Thoroughbred horse racing

Dear Dr. Aftalion:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

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on behalf of

Prof. Dr. Dalton Müller Pessôa Filho

Academic Editor

PLOS ONE