Amandine Aftalion
email: aman-dine.aftalion@ehess.fr

HOW TO RUN 100 METERS?

Keywords: Optimal control, running race, anaerobic energy, singular arc, state constraint

The aim of this paper is to bring a mathematical justification to the optimal way of organizing one's effort when running. It is well known from physiologists that all running exercises of duration less than 3mn are run with a strong initial acceleration and a decelerating end; on the contrary, long races are run with a final sprint. This can be explained using a mathematical model describing the evolution of the velocity, the anaerobic energy, and the propulsive force: a system of ordinary differential equations, based on Newton's second law and energy conservation, is coupled to the condition of optimizing the time to run a fixed distance. We show that the monotony of the velocity curve vs time is the opposite of that of the oxygen uptake ( V O2) vs time. Since the oxygen uptake is monotone increasing for a short run, we prove that the velocity is exponentially increasing to its maximum and then decreasing. For longer races, the oxygen uptake has an increasing start and a decreasing end and this accounts for the change of velocity profiles. Numerical simulations are compared to timesplits from real races in world championships for 100m, 400m and 800m and the curves match quite well.

1. Introduction. When watching a 100m race in a world championship or the Olympic games, one is not always aware that runners do not finish the race speeding up but slowing down. More precisely, they accelerate for the first 70m and then slow down in the last 30m. This can be checked by looking at the time splits every 10m for all athletes. Table 1.1 provides an example for the winner of the World Championship in 2011. One can notice that at 70m, the timesplit increases, which means that the velocity decreases. This way of running is not because they accelerate too strongly at Table 1.1 Blake's timesplits for the 2011 World Championships. Line 1 : distance in meter, line 2, cumulated time in seconds, line 3, time splits for 10 meters in seconds.

10m 20m 30m 40m 50m 60m 70m 80m 90m 100m 1.87 2.89 3.82 4.70 5.56 6.41 7.27 8.13 9.00 9.88 1.87 1.02 0.93 0.88 0.86 0.85 0.86 0.86 0.87 0.88 the beginning of the race, or are exhausted, but because this is the best way to run a 100m from the physiological point of view. One of the aims of this paper is to bring a mathematical justification and explanation to this phenomenon, and explain why, using coupled ordinary differential equations, the best use of one's ressources leads to a run with the last third in deceleration.

In fact all distances are not run the same way [START_REF] Hanon | Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the V O2 response[END_REF]: for distances up to 400m, the last part of the race is run slowing down, while for distances longer than 1500m, the first part of the race has an initial acceleration, the middle part is run at almost constant velocity and there is a final acceleration. The 800m is an intermediate race. This way of running is well known from physiologists but they do not have an explanation of why the human body does not optimize the same way according to the distance. This paper aims at understanding this optimization according to the distance with a mathematical justification using the model developed in [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF].

Up to now, very simple mathematical models have been used in the field of sport sciences to model the velocity evolution in a sprint. The recently published works [START_REF] Samozino | A simple method for measuring power, force, velocity properties, and mechanical effectiveness in sprint running[END_REF][START_REF] Slawinski | How 100-m event analyses improve our understanding of world-class men's and women's sprint performance[END_REF] model the velocity curve in a 100m as a double exponential v(t) = v max (e -t/τ1e -t/τ2 ), and they fit the parameters τ 1 and τ 2 to match real curves. This is indeed very close to the real velocity curves but the authors do not provide an explanation of their modelling. In this paper, we provide a model of coupled equations between velocity, energy and propulsive force which accounts for this double exponential approximation.

A pioneering mathematical work is that of Keller [START_REF] Keller | Optimal velocity in a race[END_REF] relying on Newton's law of motion and energy conservation. Though his analysis reproduces quite well the record times for distances up to 10km, it does not reproduce the champions' way of running. Indeed, Keller's model relies on the assumption of constant V O2, that is constant oxygen uptake and it is proved in [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF], that the race is made up of exactly three parts:

1. initial speed up phase at maximal force 2. the velocity is constant 3. the velocity decreases and the runner runs with zero energy. Therefore, in order to find non constant velocities as in real races, one has to take into account a realistic V O2 description. Several improvements of Keller's model have been introduced: the effect of fatigue [START_REF] Maroński | Minimum-time running and swimming: an optimal control approach[END_REF][START_REF] Woodside | The optimal strategy for running a race (a mathematical model for world records from 50 m to 275 km)[END_REF], the variation in maximal oxygen uptake [START_REF] Behncke | A mathematical model for the force and energetics in competitive running[END_REF][START_REF] Behncke | Optimization models for the force and energy in competitive running[END_REF], air resistance and altitude [START_REF] Behncke | Small effects in running[END_REF][START_REF] Quinn | The effects of wind and altitude in the 400-m sprint[END_REF], track curvature [START_REF] Behncke | Small effects in running[END_REF]. Other related works include [START_REF] Alvarez-Ramirez | An improved peronnet-thibault mathematical model of human running performance[END_REF][START_REF] Morton | A 3-parameter critical power model[END_REF][START_REF] Morton | The critical power and related whole-body bioenergetic models[END_REF][START_REF] Péronnet | Mathematical analysis of running performance and world running records[END_REF][START_REF] Ward-Smith | A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world-class athletes[END_REF].

In [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF], a new model is introduced relying on Keller's equations [START_REF] Keller | Optimal velocity in a race[END_REF] but improving them using a hydraulic analogy [START_REF] Morton | The critical power and related whole-body bioenergetic models[END_REF] and physiological indications [START_REF] Hanon | Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the V O2 response[END_REF], and taking into account a realistic model for V O2, the oxygen uptake. Numerical simulations are performed in [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF] for a 1500m. Let us introduce the model of [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF] that we adapt to short races. The first equation is the equation of motion, as in Keller's paper:

dv dt + v τ = f (t) (1.1)
where t is the time, v(t) is the instantaneous velocity, f (t) is the propulsive force and v/τ is a resistive force per unit mass. The resistive force can be modified to include another power of v or the influence of slopes, by adding to the right hand side a term of the form -g sin α(d(t)), where α(d(t)) is the slope at distance d(t). We can relate sin α(d) to A(d), the altitude of the center of mass of the runner at distance d, by

sin α(d(t)) = A (d(t)) 1 + (A (d(t))) 2 .
For most races, one can assume that sin α(d(t)) ∼ A (d(t)) and here, we assume for simplicity that A is constant along the race.

The second equation is an equation governing the energy. In fact, human energy can be split into aerobic energy called e ae (t), which is the energy provided by oxygen consumption, and anaerobic energy e(t), which is provided by glycogen and lactate. A very good review on different types of modeling can be found in [START_REF] Morton | The critical power and related whole-body bioenergetic models[END_REF]. In [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF], it is assumed that the anaerobic energy has finite capacity and is modeled by a container of finite height (set to 1) and section as illustrated in Figure 1 depleting by a height h, then what is called in physiology the accumulated oxygen deficit which is proportional to the height, is reduced to e 0 -e, where e 0 is the initial energy. It is assumed that the aerobic energy is of infinite capacity and flows at a rate of σ through a connecting tube R 1 into the anaerobic container. Note that σ is the energetic equivalent per unit time of V O2, the volume of oxygen used by unit of time. This equivalent can be determined thanks to the Respiratory Exchange Ratio and depends on the intensity of effort. Nevertheless, a reasonable average value is that 1l of oxygen produces 20kJ [START_REF] Peronnet | Table of nonprotein respiratory quotient: an update[END_REF]. The flow of the aerobic container into the anaerobic one depends on the anaerobic energy level, which is why we model σ as a function of e: if the level of the anaerobic container is above the connecting tube R 1 , as on the figure, then σ is proportional to h, the difference of fluid heights in the containers, that is e 0 -e, while if it is below, then σ is constant.

For a short race, R 1 is connected to the bottom of the anaerobic container as in Figure 1.1. Therefore, σ is proportional to e 0 -e, but its maximal value σ is not reached, so that σ can be modelled as

σ(e) = λσ(e 0 -e) (1.2)
where λ is such that λe 0 < 1. The available flow at the bottom of the anaerobic container is the work of the propulsive force f (t)v(t) and is equal to the creation of available energy through σ(e) (aerobic energy) and -de/dt (anaerobic energy). This leads to the energy equation:

de dt = σ(e(t)) -f (t)v(t). (1.3)
Let us point out that e(t) is a decreasing function of time, therefore σ(e(t)) is an increasing function of time, as expected. An example of σ(e(t)) for a 400m is illustrated in Figure 4.1.

When the race is longer, the dependence of σ on time is well known [START_REF] Hanon | Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the V O2 response[END_REF] and varies according to the length of the race: σ, as a function of time, is linear increasing until it reaches the maximal value σ and then is decreasing to its final value σ f , which corresponds to the rate of aerobic transfer when there is no anaerobic energy left. As a function of energy, we can take the following model, which has the opposite monotony since e(t) decreases from e 0 to 0 with time:

σ(e) =    σ e e 0 ecrit + σ f (1 -e e 0 ecrit
) if e e 0 < e crit σ if e e 0 ≥ e crit and λ(e 0 -e) ≥ 1 λσ(e 0 -e) if λ(e 0 -e) < 1 (1.4) where e 0 is the initial value of energy, σ is the maximal value of σ, σ f is the final value of σ at the end of the race, e crit the critical energy at which the flow of aerobic energy into the anaerobic container starts to depend on the residual anaerobic energy, because of an extra retroc control mechanism. The parameters λ, e 0 , e crit , σ, σ f depend on the runner and on the race. Then (1.3) also holds. An example of σ(e(t)) for a 800m are illustrated in Figure 4.1 or for a 1500m in Figure 4.4.

Constraints have to be imposed; the force is controlled by the runner but it cannot exceed a maximal value f M and the energy is nonnegative:

0 ≤ f (t) ≤ f M , and e(t) ≥ 0, (1.5)
with the initial conditions:

v(0) = 0, e(0) = e 0 . (1.6)
The aim is to minimize the time T , given the distance D = T 0 v(t) dt. The minimization problem depends on numerical parameters τ , f M , e 0 , σ, σ f , λ, e crit .

2. Numerical presentation of the models. Our numerical simulations are based on the Bocop toolbox for solving optimal control problems [START_REF] Bonnans | Bocop, the optimal control solver, open source toolbox for optimal control problems[END_REF]. This software combines a user friendly interface, general Runge-Kutta discretization schemes described in [START_REF] Hager | Runge-Kutta methods in optimal control and the transformed adjoint system[END_REF], and the numerical resolution of the discretized problem using the nonlinear programming problems solver IPOPT [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF].

Numerically, given the time splits of Table 1.1, Bocop identifies the parameters τ , f M , e 0 , σ, σ f , λ, e crit that match the time splits and provide the optimal velocity curve. Another protocol to identify the parameters has been described in [START_REF] Aftalion | How to identify the physiological parameters and run the optimal race[END_REF].

In Figure 2.1, the numerically computed velocity is plotted as a function of time on the left, while, on the right, a mean value every 10m is computed (star) to compare with the mean value from Table 1.1 (square). The matching is quite good. The initial velocity is taken to be 4, instead of 0, to better take into account the departure in the starting blocks [START_REF] Samozino | A simple method for measuring power, force, velocity properties, and mechanical effectiveness in sprint running[END_REF]. The left velocity curve indeed looks like a double exponential.

The rest of the paper is devoted to a mathematical justification of the race.

3. Mathematical analysis. We consider the following state equation

ḣ(t) = v(t), v(t) = f (t) -v(t)/τ, ė(t) = σ(e(t)) -f (t)v(t), (3.1)
We assume that the recreation function σ(e) satisfies σ(e) is C 2 and nonnegative.

(3.2)

Fig. 2.1.
Top: Plot of the velocity v(t) vs time for Blakes' 100m. Bottom: every 10m, the numerical velocity is averaged (star) and compared to the mean value obtained from Table 1.1, as the ratio of 10 to the timesplit (square). When they coincide, only the square is visible. The velocity is in m.s -1 and is plotted vs the distance in m.

We will apply it to σ(e) = α(e 0 -e) for the sprint and to a C 2 regularization of (1.4) for longer races. The initial conditions are

h(0) = 0, v(0) = 0, e(0) = e 0 > 0, (3.3)
and the constraints are

0 ≤ f (t) ≤ f M , e(t) ≥ 0, t ∈ (0, T ), h(T ) = D. (3.4)
The optimal control problem is to minimize the final time: Note that we could as well take the final constraint as -h(T ) ≤ -D. Writing an inequality in this way yields the sign of the Lagrange multiplier.

It is established in [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF] that the optimal solutions (for a given distance D) with corresponding time T are also solutions of the problem of maximizing the distance over a time interval T . So, in the following, we will rather use this second formulation for the proofs.

The optimal control problem is The Hamiltonian function is

H[p](f, v, e) := -v + p h v + p v (f -v/τ ) + p e (σ(e) -f v). (3.7)
The costate equation is therefore, omitting time arguments: ) is an arc with maximal force. A singular arc is one over which the bound constraints are not active. We recall a result from [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF]: Theorem 3.1. [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF] The above problem (3.6) has at least one optimal solution. An optimal trajectory starts with a maximal force arc, and is such that e(T ) = 0 and µ has no jump. Moreover, p e (t) < 0 and p v (t) < 0 for t ∈ [0, T ).

   -ṗh = 0 -ṗv = -1 -p v /τ -p e f -dp
Numerically, the zero energy arc is only a few points and cannot be seen on a 100m. Note that if we plug f (t) = f M in the second equation of (3.1), then we find as a solution v(t) = f M τ (1 -e -t/τ ).

Once we know that the race starts with an arc of maximal force and ends with a zero energy arc, we want to know more about the singular intermediate arc.

3.1. Justification of the velocity decrease in a 100m race. In this section, we will find properties of the singular arc. We will prove the following: Theorem 3.2. Let σ(e) = α(e 0 -e) with ατ < 1. Then an optimal trajectory of (3.6) starts with a maximal force arc, followed by a singular arc on which

v + αv = 0 (3.11)
and ends with a zero energy arc. Therefore, the velocity profile is

v(t) = f M τ (1 -e -t/τ ) for 0 ≤ t ≤ t 1 (3.12) v(t) = f M τ (1 -e -t1/τ )e -α(t-t1) for t 1 ≤ t ≤ t 2 . (3.13)
Note that for a 100m, τ is between 1 and 2 sec and α is of order 10 -2 sec -1 so that ατ < 1. We point out that t 2 is very close to T and the values of t 1 and t 2 are determined by the condition on the distance and the energy equation as we will see below. The final value of the propulsive force is limited by how much aerobic energy can be provided by the runner since on the zero energy arc f (t) = σ(0)/v(t).

Proof. We are going to use the properties of the switching function to determine the type of arcs. Indeed Theorem 3.1 states that the trajectory starts with an arc of maximal force. We want to investigate what follows, whether the constraints are active and what the shape of the singular arc (no active constraint) is.

We have that Pontryagin's principle holds in qualified form, i.e., each optimal trajectory (f, v, e) is associated with at least one multiplier (p, µ) such that the relaxed control minimizes the Hamiltonian.

The switching function is

Ψ(t) = ∂H ∂f (t) = p v (t) -p e (t)v(t). (3.14)
We can differentiate this with respect to t so that Ψ(t) = ṗv -ṗe v -p e v, and we use (3.8) to get,

Ψ = 1 + p v /τ + p e f + p e σ (e)v -p e (f -v/τ ) = 1 + Ψ(t)/τ + p e v(σ (e) + 2/τ ). (3.15)
We recall from Theorem 3.1 that at t = 0, the trajectory starts on a maximal force arc and that, since v = 0, Ψ(0) = p v (0) < 0. We want to investigate what can follow this arc of maximal force. Since we have the constraints 0 ≤ f ≤ f M and e ≥ 0, there could be arcs of zero force, of zero energy, other arcs of maximal force or a singular arc.

Step 1: Study of the singular arc. On a singular arc, Ψ(t) = 0. We can differentiate this and find that Ψ = 0, so that, from (3.15), p e v(σ (e) + 2/τ ) = -1. In particular, on a singular arc, σ (e) + 2/τ = 0. We differentiate again and obtain, after dividing by p e , and using the equation for ė and v:

v 1 - v 2 σ (e) σ (e) + 2/τ -v σ (e) + σ (e)(σ(e) -v 2 /τ ) σ (e) + 2/τ = 0. (3.16)
In a regime where σ(e) = α(e 0 -e), the previous equation simplifies to (3.11).

Step 2: There is no zero force arc and hence, Ψ(t) ≤ 0, for all t ∈ [0, T ]. Let (t a , t b ) be a zero force arc, over which necessarily Ψ is nonnegative. By what we have seen t a > 0, and so Ψ(t a ) = 0, Ψ(t a+ ) ≥ 0. On a zero force arc, the equations yield that p e v = λe (α-1/τ )t , hence is increasing since ατ < 1 and p e (t) < 0. By (3.15), we have that

Ψ(t b-) -Ψ(t a+ ) = 1 τ Ψ(t b-) + ( 2 τ -α)((p e v)(t b-) -(p e v)(t a+ ) ≥ 0
meaning that the zero force arc cannot end before time T , contradicting the final condition e(T ) = 0 recalled in Theorem 3.1.

Step 3: The only maximal force arc is the one starting at t = 0. On a maximal force arc (t a , t b ) with t a > 0, the speed increases and ṗe = αp e , thus p e decreases and is negative. Therefore, p e v decreases. Equation (3.15) implies Ψ(t b-) < Ψ(t a+ ) ≤ 0, and since Ψ ≤ 0 along the maximal force arc, it follows that Ψ(t b ) < 0, meaning that the maximal force arc ends at time T , which is in contradiction with e(T ) = 0 in Theorem 3.1.

Step 4: end of the proof. The existence of a maximal force arc starting at time 0 is established in Theorem 3.1. Let t a ∈ (0, T ) be its exit point. Let t b ∈ (0, T ) be the first time at which the energy vanishes.

If t a < t b , over (t a , t b ), Ψ is equal to zero and hence, (t a , t b ) is a singular arc on which (3.11) is satisfied. Finally let us show that the energy is zero on (t b , T ). Otherwise there would exist t c , t d with t b ≤ t c < t d ≤ T such that e(t c ) = e(t d ) = 0, and e(t) > 0, for all t ∈ (t c , t d ). Then (t c , t d ) is a singular arc, over which ė = α(e 0 -e) -f v. We differentiate again and recall that ḟ = -αf and v = -αv on a singular arc, so that ë = -α ė + 2αf v.

Therefore, the function e cannot have a positive maximum which gives a contradiction since the energy is positive. The result follows.

Equation (3.11) is the main result of the paper. It allows us to make the difference between short races and long races. Indeed, when the duration of exercise is less than 3mn, the maximal value of V O2 is not reached, therefore σ is a decreasing, almost linear function of energy and an increasing function of time, so that (3.11) provides that v is a decreasing function of time once the runner can no longer run at maximal force. Therefors, the race starts at maximal force and v is increasing. When on the singular arc, the velocity becomes decreasing and its evolution is governed by (3.11). This matches the model of [START_REF] Samozino | A simple method for measuring power, force, velocity properties, and mechanical effectiveness in sprint running[END_REF] of double exponential but provides a mathematical justification.

We use (3.12)-(3.13) to integrate the energy equation de dt = α(e 0 -e) -f v from 0 to t 2 recalling that f = f M for t < t 1 and that on the singular arc, we derive from (3.11) and (1.1) that f = v(1/τ -α). We find

e 0 f 2 M τ e αt2 = e αt1 -1 α - e (α-1 τ )t1 -1 α -1 τ +(1-e -t1/τ ) 2 e 2αt1 (1-ατ ) e -αt1 -e -αt2 α . (3.17)
We also recall D = t2 0 v(t) dt. In fact, there should be the last arc at zero energy, but since it is very short, we can make the approximation that the race is run in t 2 . We find .18) This equation allows to solve for e -αt2 , and replace it in (3.17) to solve for t 1 . The expression of t 1 is not analytic in general, but to get an idea on the dependance of the parameters, we can make assumptions. For instance, if we assume that α is small (of order 10 -2 sec -1 ), then we can expand the exponentials in α and find that (3.17)- (3.18) becomes

D f M τ = t 1 -τ (1 -e -t1/τ ) + (1 -e -t1/τ ) 1 -e -α(t2-t1) α . ( 3
D f M τ = t 1 + (1 -e -t1/τ )(t 2 -t 1 -τ ), (3.19) D f M τ (1 -e -t1/τ ) - e 0 f 2 M τ = (τ -t 1 -τ e -t1/τ )e -t1/τ . (3.20)
Given the order of magnitude, the right hand side of (3.20) can be neglected at leading order, which yields

D f M τ (1 -e -t1/τ ) - e 0 f 2 M τ = 0. (3.21)
If e 0 = f M D, then t 1 is of order t 2 , that is the race is run at maximal force and the e -t1/τ term can be in fact neglected, and one has to go further in the expansion in α of the exponentials. In general, e 0 is not so high, and t 1 is given by (3.21), that is

t 1 = -τ ln(1 - e 0 f M D ). (3.22)
Therefore, t 1 is an increasing function of e 0 : the bigger the initial energy, the longer f M is maintained. Moreover, t 1 is a decreasing function of f M : the bigger f M , the shorter it can be maintained.

We have found that the optimal race is to start at maximal force, and then decrease the propulsive force, because the runner does not have enough energy to maintain his maximal propulsive force for the whole duration of the race. The decrease in propulsive force leads to a singular arc and a decrease in velocity. It turns out that the decrease in the propulsive force predicted by our model is too strong compared to what the runner can produce. Therefore, in a real race, the runner keeps his maximal force for a shorter time and has a decrease which is weaker. This implies that in our model we have to take into account another constraint, which is the bound on the variation of the propulsive force. This implies also that in order to improve one's performance, improving the ability to vary the propulsive force is an important criterion. This is why interval training is often used.

Bounding variations of the force.

It seems desirable to avoid strong variations of the force which occur with the previous model, and to introduce bounds on ḟ . The force becomes then a state and the new control ḟ is denoted by g. So the state equation is

ḣ = v; v = f -v/τ ; ė = σ(e) -f v; ḟ = g, (3.23)
with constraints

0 ≤ f ≤ f M ; e ≥ 0; g m ≤ g ≤ g M . (3.24)
Note that g m is negative. We minimize as before -T 0 v(t)dt. The Hamiltonian is The costate equation are now

H = -v + p v (f -v/τ ) + p e (σ(e) -f v) + p f g. (3.25)
   -ṗv = -1 -p v /τ -p e f, -dp e = σ (e)p e dt -dµ, -ṗf = p v -p e v. (3.26)
The state constraint e ≥ 0 is of second order, and we may expect a jump of the measure µ at time T . The final condition for the costate are therefore

p v (T ) = 0; p e (T ) = 0; p f (T ) = 0. (3.27)
We may expect that the optimal trajectory is such that g is bang-bang (i.e., always on its bounds), except if a state constraint is active (the state constraints now include bound constraints on the force), as is confirmed in our numerical experiments: for a 100m, the race starts with an arc of maximal force followed by an arc where g = g m and ends with an arc of zero energy.

The model with a bound on the derivative of the force matches the real race of world champions better than the previous system. This can be explained from the fact that the propulsive force cannot be varied too quickly from muscular and motor control reasons [START_REF] Bouc | Computational dissection of dopamine motor and motivational functions in humans[END_REF]. This is the model which is simulated to get Figure 2

.1. Analytically, if ḟ = -β, that is f (t) = f M -β(t -t 0 )
, then the velocity equation can be solved explicitly:

v(t) = v 0 e -(t-t0)/τ + f M τ (1 -e -(t-t0)/τ ) + βτ 2 (1 -e -(t-t0)/τ ) -βτ (t -t 0 ).

So an approximation of the velocity close to

t 0 is f M τ -β(t -t 0 ) 2 /2.
This formula can lead to an easier identification of the parameters: f M τ is the peak velocity, β is related to the decrease of velocity and τ can be identified by matching the exponential at the beginning of the race. Top: Plot of the velocity v(t) vs time for a 400m. Bottom: every 50m, the numerical velocity is averaged (star) and compared to the mean value obtained from Table 4.1, as the ratio of 50 to the timesplit (square). The velocity is in m.s -1 and is plotted vs the distance in m. 4. Longer races. In order to describe mathematically a race, one needs to know the V O2 curve. We get information from [START_REF] Hanon | Effects of optimal pacing strategies for 400-, 800-, and 1500-m races on the V O2 response[END_REF] to construct the curve σ(e). In Figure 4.1, we have plotted the V O2 curve (that is σ(e(t))) vs time for a 400m and for an 800m.

We have obtained data for Tables 4.1 and 4.2 from Christine Hanon [9] as well 4.2, as the ratio of 100 to the timesplit (square). The velocity is in m.s -1 and is plotted vs the distance in m.

as estimated data for σ: for the 400m, it corresponds to French world level athletes, while for the 800m, it is the Olympic and World record of Rudisha (London 2012), where he was in front for the whole run, so that no strategy related to overtaking modified his way of running.

In the 400m case, the V O2 curve is increasing which leads to a velocity profile (Figure 4.2 left) similar to a 100m, increasing and then decreasing though the decrease is stronger and the increase is only during the first quarter of the race, as can be confirmed in Table 4.1. The first part of the race is at maximal force, then the force is decreased and so is the velocity due to the fact that V O2 is increasing. The description of the race of Theorem 3.2 holds, as well as (3.22). This is consistent with previous analyses in the literature leading to double exponentials for short races, but additionally provides a relation between the time where the runner starts slowing down and his total anaerobic energy.

In the case of an 800m, the curve σ(e(t)) is increasing and then decreasing, which leads to a complicated race strategy: the beginning of the race is at maximal force and the velocity is increasing. Then, the V O2 is still increasing, and on the singular arc, the velocity is decreasing: indeed equation (3.11) holds but α = σ (e) > 0. At t = 40s, the curve σ(e(t)) changes monotony and α changes sign and is negative again. Therefore, from (3.11), the curve v(t) is increasing again until the runner reaches a situation where he can no longer increase his velocity because he does not have enough energy left and he moves onto a constrained arc where the derivative of the force is constant, before finishing on a zero energy arc. We have no mathematical proof that this is the best strategy in this order, but it is consistent with different possible arcs and the curve σ(e(t)). More precisely, the possibility of an arc of maximal force is allowed when σ is an increasing function of energy, that is a decreasing function of time. In Theorem 3.2, Step 3, it was ruled out because σ had the opposite monotony. We see from this part of the proof of Theorem 3.2 that the decrease of velocity at the end of the race is related to the monotony of σ. Therefore, if σ changes monotony, as it is the case for longer races, then another maximal force arc is not excluded at the end of the race. This is why for longer races, at the end of the race, σ is a decreasing function of time, and the velocity increases again: this is the final sprint.

Let us point out that this complex behaviour of the 800m race, had not been captured by other mathematical modeling: after the initial acceleration, Keller's model leads to a constant velocity and Ward Smith [START_REF] Ward-Smith | A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world-class athletes[END_REF] or Reardon [START_REF] Reardon | Optimal pacing for running 400-and 800-m track races[END_REF] to a decreasing velocity, while the reality of the race is more involved.

When the duration of the race is longer than 3mn, the maximal value σ of V O2 is reached, σ is a decreasing function of time at the end of the race, or an increasing linear function of energy, and therefore, (3.16) implies that v is increasing again.

Varying the parameters, we can notice that a high V O2 (high σ) improves performance as well as a high ability to vary the propulsive force. Hence interval training exercises are favored to improve this force variation.

The 1500m has been studied in [START_REF] Aftalion | Optimization of running strategies based on anaerobic energy and variations of velocity[END_REF] and [START_REF] Aftalion | How to identify the physiological parameters and run the optimal race[END_REF], in particular the effects of the various parameters. Here, we use the data of [START_REF] Hanon | Pacing strategy and V O2 kinetics during a 1500-m race[END_REF] to produce the velocity curve and V O2 curve in Figure 4.4. Again, we can derive information from (3.16): when in the part where V O2 is linearly increasing, on the singular arc, we have (3.11) with α > 0 and the velocity is decreasing; on the part where σ is constant, the velocity is constant, and when σ is linearly decreasing, we have (3.11) with α < 0 and the velocity is increasing so that there is a final sprint. The very last points where the velocity decreases again is due to the zero energy part at the end of the optimization.

For longer races, we do not have detailed timesplits but we believe that the model remains valid, and could provide interesting indications in the case of hilly and windy races.

Conclusion.

Given a distance D to run, our model relies on the knowledge of the V O2 curve, that is the data of σ, σ f , λ and e crit and on 3 parameters: the maximal propulsive force f M , the initial anaerobic energy e 0 and the friction coefficient τ . On the basis of these parameters, the energy conservation and Newton's second law, we can determine the optimal way of running, that is the instantaneous velocity, propulsive force and anaerobic energy. The same model holds for all distances but provides very different shapes of velocities according to distances.

The structure of the 100m or 400m race is that v increases exponentially quickly when f is at its maximal value f M . Then v decreases exponentially slowly until the energy reaches the zero value for a few points of computation. Though the double exponential profile structure for the velocity was already known in the literature, a precise dependance on the different parameters is clearly identified. One sees that in order to reach a very high velocity quickly, one needs a very strong propulsive force (f M ) and small internal friction dues to joints or running economy (high τ ). But on the other hand, the limiting value becomes the available anaerobic energy because reaching a high velocity quickly requires a lot of energy, and if little is left, the runner cannot maintain his velocity, which falls down exponentially. Nevertheless, the best strategy is to put as high force and acceleration at the beginning to reach strong velocity, even if this strong velocity cannot be maintained, rather than accelerating more slowly and all along the race.

If the race is long enough so that σ reaches its maximal value and decreases on the last third of the race, then the runner can speed up again. We provide a precise structure of the 800m race, for which the velocity curve has 4 pieces: increasing, decreasing, increasing and decreasing again. For a 1500m or more, there is a long part of the race at almost constant velocity, which gets close to Keller's model. Numerically, our model matches 100m, 400m, 800m and 1500m races for olympic or world championships.

The interest of the mathematical model is that it allows to play on the parameters variations. For instance, one can identify how the variation of one parameter or another has an influence on the deceleration on the second part of the race or how to better improve the final time since our model is strong enough to match all type of distances and human optimization.

. 1 .Fig. 1 . 1 .

 111 Fig. 1.1. Scheme of the container modeling.

  Min T ; s.t. (3.1) and (3.3)-(3.4).(3.5)

  )dt; s.t. (3.1) and (3.3)-(3.4) hold. (3.6)

  e = p e σ (e)dt -dµ, (3.8) with final conditions p v (T ) = p e (T ) = 0. (3.9) We can fix therefore p h to 0. The variables p v and p e are called costate variables. Here dµ, identified to the bounded variation function µ on [0, T ], is a Borel measure (that can be interpreted as a Lagrange multiplier) associated to the state constraint -e ≤ 0 which satisifes dµ ≥ 0; supp(dµ) ⊂ {t ∈ [0, T ]; e(t) = 0}. (3.10) If 0 ≤ a < b ≤ T is such that e(t) = 0 for t ∈ [a, b], but e(•) does not vanish over an interval in which [a, b] is strictly included, then we say that (a, b) is an arc with zero energy. Similarly, if f (t) = f M a.e. over (a, b) but not over an open interval strictly containing (a, b), we say that (a, b

Fig. 4 .

 4 Fig. 4.1. V O2 curve vs time, that is σ(e(t)) vs time for a 400m (left) and 800m (right).

Fig. 4 .

 4 Fig.4.2. Top: Plot of the velocity v(t) vs time for a 400m. Bottom: every 50m, the numerical velocity is averaged (star) and compared to the mean value obtained from Table4.1, as the ratio of 50 to the timesplit (square). The velocity is in m.s -1 and is plotted vs the distance in m.

Fig. 4 . 3 .

 43 Fig. 4.3. Top: Plot of the velocity v(t) vs time for an 800m. Bottom: every 100m, the numerical velocity is averaged (star) and compared to the mean value obtained from Table4.2, as the ratio of 100 to the timesplit (square). The velocity is in m.s -1 and is plotted vs the distance in m.

Fig. 4 . 4 .

 44 Fig. 4.4. Velocity curve vs time for a 1500m (left) and V O2 curve, that is σ(e(t)) vs time (right).

Table 4 .

 4 1 400m, French world level. Line 1 : distance in meter, line 2, time splits for 50 meters in seconds. Total time 44.43 seconds.

	50m 100m 150m 200m 250m 300m 350m 400m
	6.10	4.94	5.0	5.17	5.37	5.52	5.86	6.45

Table 4 .

 4 2 800m, Rudisha's timesplits for the 2012 Olympic games, World record and olympic record. Line 1 : distance in meter, line 2, time splits for 100 meters in seconds. Total time 100.91 seconds.

	100m 200m 300m 400m 500m 600m 700m 800m
	12,3	11,2	12,55 13,23 12,74 12,28 13,02 13,59

Acknowledgments. The author would like to thank her colleague Martin Andler, who is both a talented mathematician and a former 800m runner, and whose remarks and advice all along this work were crucial. Martin Andler introduced the author to a physiologist and former French champion Christine Hanon. Christine Hanon provided all real race data for 400, 800 and 1500m [9]. Though she participated strongly to this work, she did not want to be involved in the authorship. Her involvement into making physiologists and sport scientists believe in mathematics is strongly acknowledged here, as well as her terrific ability to understand mathematics. The author is also very grateful to Frédéric Bonnans and Pierre Martinon on Bocop and to people from the sprint project at Insep, namely Antoine Couturier, Gael Guilhem and Giuseppe Rabita who provided the timesplits of Table 1.1.


E Alphandéry

 S Faure

L Raison

E Duguet

P A Howse

D A Bazylinski

Heat Production by Bacterial Magnetosomes Exposed to an Oscillating Magnetic Field

In this work, we examine the mechanisms of heat production by whole intact cells of magnetotactic bacteria, Magnetospirillum magneticum strain AMB-1, as well as by their extracted chains of magnetosomes or extracted individual magnetosomes when they are exposed to an oscillating magnetic field of frequency 108 kHz and field amplitudes varied between 23 mT and 88 mT. For intact bacterial cells that contain chains of magnetosomes, heat is generated through hysteresis losses yielding specific absorption rates (SAR) of 110 W/gFe at 23 mT and 860 W/gFe at 88 mT. When the chains of magnetosomes are extracted from the bacterial cells and exposed to the same magnetic field, the heatproducing mechanism includes an additional contribution, one that is due to their rotation in the magnetic field. This results in the production of higher SAR of 860 W/gFe at 23 mT and 1240 W/gFe at 88 mT. Lower SAR values of 540 W/gFe at 23 mT and 970 W/gFe at 88 mT are obtained with the individual magnetosomes. This appears to be due to aggregation of the individual magnetosomes in the liquid, which prevents them from rotating as efficiently as the chains of magnetosomes.

I. Introduction:

Magnetotactic bacteria are a unique group of bacteria that biomineralize intracellular, membranebounded, nanometer-sized single-magnetic-domain crystals of the iron oxide magnetite (Fe3O4), maghemite (Fe2O3) and/or iron sulfide greigite (FeS4) [START_REF] Bazylinski | For a comprehensive review on magnetotactic bacteria[END_REF]. These structures, called magnetosomes, are most often arranged as a chain within the cells. They are used by the bacteria to navigate in the direction of the Earth's magnetic field and help them to find optimal conditions for their growth and survival [START_REF] Bazylinski | For a comprehensive review on magnetotactic bacteria[END_REF].

Magnetosomes have been shown to be useful in a number of scientific, commercial and health applications (2). Recently, great efforts have been devoted to synthesizing nanoparticles, which are able to induce the production of heat when an oscillating magnetic field is applied to them (3). The aim of this effort is to improve the treatments of tumors by hyperthermia or thermoablation or to enable the release of drugs within a localized region of the body (3)(4)(5)(6)(7)(8)(9). Although rapid progresses have been made in this field, concerns have been raised regarding the toxicity induced by the presence of the nanoparticles in the body (10). In order to minimize the potential side effects of nanoparticle based clinical treatments, the nanoparticles need to be administered in a quantity as small as possible. For that, the latter have to generate a sufficiently large amount of heat (specific absorption rates SAR). This can be achieved by using nanoparticles with either a large volumes or a high magnetocrystalline anisotropy (11). The magnetosomes belong to the largest single domain ferrite nanoparticles, making them good candidates for the applications mentioned above. Although the magnetosomes have been shown to produce a large amount of heat when they are subjected to an oscillating magnetic field (11)(12)(13), this property has not been fully explained.

In this paper, we provide a detailed study of the mechanisms of heat production by magnetosomes biomineralized by the magnetotactic bacteria. The magnetosomes studied in this paper are iron oxide nanoparticles of mean sizes 40 nm, which are oxidized in maghemite and are arranged in chains 26/06/17 containing ~ 6 magnetosomes (14,15). Since maghemite and magnetite have similar magnetic properties at room temperature, the fact that the magnetosomes are oxidized in maghemite does not change the conclusions drawn in this paper. We compare the heat-producing properties of three different types of magnetosome arrangements (14,15). We examine the properties of intact cells that contain chains of magnetosomes and those of the magnetosomes that have been extracted from cells and are either arranged in chains or form individual nanocrystals. It is known that for large ferromagnetic nanoparticles, there are two main heat-producing mechanisms. The first one is due to the physical rotation of magnetic nanoparticles in a magnetic field and the second one is a result of hysteresis losses (16). In order to determine which one of these two mechanisms is responsible for heat production by the three different types of magnetosome arrangements mentioned above, we compare the heating rates of the samples in water, in which rotation of the cells and extracted magnetosomes is possible, with those in a gel, where the rotation is inhibited. In this way, the amount of heat generated by the rotation of the bacteria or extracted magnetosomes and that arising from hysteresis losses can be determined. In order to verify that the heat produced in the gel is due to hysteresis losses, we measure the hysteresis losses independently using magnetic measurements. hysteresis loops of the whole intact cells (Figure 1(d)) also increase when the magnetic field amplitude increases from 23 mT up to 88 mT. Using the method of Hergt et al (10), we deduce from the areas of the minor hysteresis loops shown in Figure 1(d) that the hysteresis losses of the intact cells increase from 110 W/gFe at 23 mT up to 810 W/gFe at 88 mT (Figure 1(e)). In order to determine if the rotation of the intact cells contributes to the observed increase in temperature, we estimate the SAR of the intact cells suspended in liquid. The SAR of the cell suspension is estimated at 22 0 C from the slopes of the variation with time of the temperature, T/t, using the formula (1) below (12):

II. Results and discussion:

m water x 1 δt ΔT C SAR        (1)
, where Cwater is the specific heat capacity of water (Cwater = 4.184 J/g.K) and xm is weight of iron per mL of liquid (water). As shown in Figure 1(e), the SAR of the bacterial suspension increase from 110 W/gFe up to 860 W/gFe when the magnetic field amplitude is increased from 23 mT up to 88 mT. These values of the SAR are similar to those measured from the areas of the minor hysteresis loops shown in Figure 1(e). Unexpectedly, the SAR determined for cells fixed in the agarose gel is significantly smaller than the areas of the minor hysteresis loops, which correspond to the hysteresis losses (Figure 1(e)). This could be due to poor diffusion of heat within the bacterial cells when they are contained in the agarose gel. From these results, we hypothesize that the rotation of the whole bacterial cells does not contribute to the production of heat. This hypothesis is supported by the fact that the magnetosomes occupy a small volume within the whole bacteria. This small volume induces insufficient rotation of the cells to generate heat. Furthermore, the absence of contribution of the rotation can be confirmed by estimating the SAR due to the rotation of the bacterial cells, SARrot. The latter is estimated using the formula (2) below (16,[START_REF]we have assumed that the Brownian relaxation time, b, is much smaller than the Néel relaxation time n, where b = 3V[END_REF]:

26/06/17     b b b 2 o s rot ωτ 1 ωτ τ 1 ρV 1 T K V H M 2 1 SAR   (2)
In ( 2),  = 2f, where f = 108 kHz is the frequency of the oscillating magnetic field, Ms is the saturating magnetization of maghemite (Ms = 390 emu/cm 3 ), H0 is the amplitude of the applied magnetic field (23 mT < H0 < 88 mT), V ~ 20 10 -17 cm 3 is the volume of a typical chain of magnetosomes ( 13),  ~ 5 g/cm 3 is the specific weight of maghemite, Kb ~ 1.38 10 -23 J/K is the Boltzmann constant and b ~ 10 seconds is the Brownian relaxation time of intact bacterial cell [START_REF]3Vh/KbT, where Vh is the hydrodynamic radius. For the whole magnetotactic bacteria[END_REF]. Using these values, we find that SARrot lies between 5.10 -2 W/gFe and 7.10 -1 W/gFe for 23 mT < H0 < 88 mT. These values are much smaller than the measured hysteresis losses, which lie between ~ 50 W/gFe at 23 mT and ~ 810 W/gFe at 88 mT (Fig.

1(e))

. Thus, the rotation of the whole bacterial cells does not appear to contribute to heat production. As indicated in the column bar plots of Figure 1(f), the SAR arises at 100 % from hysteresis losses and becomes much more significant at a magnetic field amplitude of 88 mT (SAR ~ 860 W/gFe) than at a field amplitude of 23 mT (SAR ~ 110 W/gFe). This is due to the increase of hysteresis losses with increasing magnetic field amplitude as previously observed with chemically synthesized magnetite nanoparticles (16). The SAR per cycle of the whole cells in suspension, which is defined as the SAR divided by the frequency of the oscillating magnetic field, lies between 1 J/kgFe and 8 J/kgFe. These values are higher than those observed with chemically synthesized nanoparticles, which typically lie between 0.001 J/kgFe and 1.2 J/kgFe for a wide range of nanoparticle sizes and compositions as well as for a large choice of magnetic field frequencies and amplitudes (16,(20)(21)(22)[START_REF] Chan | In Scientific and clinical applications of magnetic carriers[END_REF]. We conclude that the suspensions of whole magnetotactic bacteria produce a larger amount of heat than most of the chemically synthesized nanoparticles.

In order to increase even more the amount of heat produced by the magnetosomes, chains of magnetosomes are extracted from bacterial cells to presumably enhance their rotation in the magnetic . This suggests either that the extracted chains of magnetosomes produce larger hysteresis losses than intact bacteria cells or that their rotation contributes to the heat production or both. In order to discern which, if any, of these explanations is plausible, hysteresis losses of the extracted chains of magnetosomes are determined. . This decrease could be explained by the stronger enhancement with increasing magnetic field amplitude of the hysteresis losses than of the SAR due to the rotation of the chains of magnetosomes in the magnetic field (16).

The third sample we test is a suspension of extracted magnetosomes whose membranes have been disrupted by a combination of heat and sodium deodecyl sulfate (SDS). Disruption of the magnetosome membrane results in a suspension of individual magnetosomes, which are not arranged in chains. Figure 3(a) shows a typical TEM micrograph of these magnetosomes and clearly shows that magnetosomes are not organized in chains. These nanocrystals interact and organize within compact assemblies of individual nanocrystals (14,15,[START_REF] Kobayashi | [END_REF], forming significantly more compact and aggregated assemblies than those observed in Figure 2 Therefore, the difference in SAR observed between the extracted chains of magnetosomes and the individual magnetosomes suspended in liquid must result from a difference in the contribution of the rotation to the heating mechanism. Equation 2 predicts that the SAR due to the rotation of the individual magnetosomes suspended in liquid should be the same as that deduced for the chains of magnetosomes, SARrot ~ 3600 W/gFe at 23 mT. Therefore the lower heating rate observed for the extracted individual magnetosomes is likely due to the fact that the latter are more prone to aggregation than the chains of magnetosomes. Aggregation of the individual magnetosomes is clearly evident using electron microscopy (Figure 3(a)) and can also be observed visually in liquid suspension. It prevents these magnetosomes from rotating as easily as the extracted chains of magnetosomes. As can be deduced from Figure 3(f), the contribution of the rotation to the heating mechanism lies between 44 % and 60 % of the total SAR. These percentages are lower than those obtained with the extracted chains of magnetosomes.

III Conclusion

In this study, we have examined the heat-production by intact whole magnetotactic bacterial cells, extracted chains of magnetosomes and extracted individual magnetosomes exposed to an oscillating magnetic field. The SAR of each of these three magnetic samples is larger than that reported for smaller superparamagnetic nanoparticles. The predominant contribution to heat production by the intact bacterial cells appears to be hysteresis losses while physical rotation and hysteresis losses are both responsible for the generation of heat for the extracted chains of magnetosomes and individual magnetosomes. Magnetosomes organized in chains produce a greater amount of heat than individual 26/06/17 magnetosomes (suspended in liquid) probably due to the latter's strong tendency to form aggregates in liquid.

IV. Experimental section:

IV.1 Preparation of the samples:

Magnetospirillum magneticum strain AMB-1 magnetotactic bacteria are purchased from the ATCC (ATCC 700274). The bacteria are cultivated using the growth medium recommended by the ATCC, which contains in 1 L of distilled water, 5 mL of ATCC trace mineral supplement, 10 mL of ATCC vitamin supplement, 2 mL of a ferric quinate solution, 0.45 mL of 0.1 % resazurin, 0.68 g of KH2PO4, 0.12 g of NaNO3, 0.035 g of ascorbic acid, 0.37 g of tartaric acid, 0.37 g of succinic acid and 0.05 g of sodium acetate. The ATCC vitamin supplement solution (ATCC MD-VS) is prepared by adding to 1 L of distilled water 2 mg of folic acid, 10 mg of pyridoxine hydrochloride, 5 mg of riboflavin, 2 mg of biotin, 5 mg of thiamine, 5 mg of nicotinic acid, 5 mg of pantothenic acid, 0.1 mg of vitamin B12, 5 mg of p-aminobenzoic acid, 5 mg of thioctic acid and 900 mg of monopotassium phosphate. The solution of mineral supplement (ATCC MD-TMS) is prepared by adding to 1 L of distilled water 0.5 g of EDTA, 3 g of MgSO4.7H2O, 0.5 g of MnSO4.H2O, 1 g of NaCl, 0.1 g of FeSO4.7H2O, 0.1 g of Co(NO3)2.6H2O, 0.1 g of CaCl2 (anhydrous), 0.1 g of ZnSO4.7H2O, 0.01 g of Na2WO4.2H2O and 0.02 g of NiCl2.6H2O.

The ferric quinate solution is prepared by adding to 100 mL of distilled water 0.27 g of FeCl3 and 0.19 g of quinic acid. After having added the chemicals indicated above, the pH of the growth media is adjusted to 6.75 using a 1 M NaOH solution. The growth media is then autoclaved at 121 C for 15 min.

The bacteria provided by the ATCC are warmed up to room temperature and introduced in the three different growth media. The bacteria are kept at room temperature for one week until a change in the coloration of the growth medium is observed (from pink to white). 10 26/06/17

Three different types of samples are prepared from intact whole cells of M. magneticum. Cells are harvested at stationary phase and by centrifugation at 8000 rpm for 15 minutes. The supernatant (spent growth medium) is discarded and cells are resuspended in 3 mL of deionized water. For suspensions of whole intact cells, this sample is not treated further. To extract the chains of magnetosomes, 1 ml of the cell suspension is recentrifuged and resuspended in a 10 mM Tris.HCl buffer (pH ~ 7.4) and then sonicated for 60 min. at 30 W to lyse releasing the chains of magnetosomes. After sonication, the suspension of extracted chains of magnetosomes is placed next to a strong magnet and the supernate containing cells debris and organic material is removed. The magnetosome chains are washed 10 times in this way and are finally resuspended in dionized water. Individual magnetosomes (not in chains) are obtained by heating the suspension of magnetosome chains for five hours at 90 0 C in the presence of 1 % sodium dodecyl sulfate (SDS) in dionized water to disrupt the magnetosome membranes. Individual magnetosomes are washed as described for magnetosome chains and resuspended in dionized water.

IV.2 Characterization of the samples:

Samples are examined using a JEOL model JEM 1011 transmission electron microscope (JEOL Ltd, Tokyo) operating at 100 kV. Five l of a solution containing 2 10 -4 % in weight of magnetosomes are deposited on a carbon-coated copper grid and the grids are allowed to dry before examination. The same relative quantity of magnetosomes are used to prepare all samples, thus aggregation in a particular sample does not arise from a difference in the concentration of the magnetosomes.

Magnetic measurements are carried out using a vibrating sample magnetometer (VSM, Quantum design, San Diego). For magnetic measurements, 25 l of a liquid suspension of magnetotactic bacterial cells, extracted chains of magnetosomes or extracted individual magnetosomes containing 2 10 -3 % in weight of magnetosomes, are deposited on top of a silica substrate. The samples are then positioned inside a capsule made of hard gelatin in a direction parallel to that of the magnetic field.

26/06/17

Three types of magnetic measurements are performed, those of the saturating isothermal remanent magnetization (SIRM) and major or minor hysteresis loops. The SIRM measurements are used to determine the composition of the magnetosomes following a method similar to that previously described (15) and show that the magnetosomes have been almost completely oxidized in maghemite.

This result is expected since we don't use fresh suspensions of magnetotactic bacteria and magnetosomes are known to oxidize in maghemite over time [START_REF] Chen | [END_REF]. The fact that the magnetosomes are made of maghemite instead of magnetite does not change the conclusions drawn in this paper since maghemite and magnetite have very similar magnetic properties at room temperature (15). Major hysteresis loop measurements are carried out at 300 K in order to determine the amount of maghemite contained within sample. The latter is determined by dividing the saturating magnetization of the samples by the saturating magnetization of maghemite. For nanoparticles as large as the magnetosomes, the saturating magnetization is that of the bulk material (in this case bulk maghemite) (ref). Finally, measurements of minor hysteresis loops are also carried out by recording the magnetization of the samples as a function of a continuous magnetic field, which is applied between -H0 and H0 where H0 is 23 mT, 36 mT, 66 mT or 88 mT.

IV.3 Heating experiments:

These experiments are carried out with the whole bacteria, extracted chains of magnetosomes and extracted individual magnetosomes either suspended in ultrapure water (18,6 M) or in aqueous agarose gel (2 % by weight). The concentration of maghemite is 457 µg mL -1 for the liquid suspension containing the whole cells, 435 µg mL -1 for that containing the extracted chains of magnetosomes and 380 µg mL -1 for that containing the extracted individual magnetosomes. 250 µl of each of these three suspensions are poured inside polypropylene tubes and positioned at the center of a coil producing an oscillating magnetic field of frequency 108 kHz, the field amplitude being fixed at 23 mT, 36 mT, 66

Figure 1 (

 1 Figure 1(a) depicts a transmission electron micrograph (TEM) of cells of Magnetospirillum

6 26/06/ 17 field.

 617 To verify that the magnetosomes are actually extracted from the bacteria and remain in chains, we use electron microscopy. Figure2(a) shows typical assemblies of chains of magnetosomes(14,15), which are not aggregated in clumps but are sufficiently close one to another as chains to be interacting in several regions. Heating rates of the extracted chains of magnetosomes are shown in Figures2(b) and 2(c) for the magnetic field amplitudes of 23 mT and 88 mT respectively. In the gel, they are similar to those determined for suspensions of intact whole bacterial cells while in solution, they are between 5 and 13 times larger than those of the whole cells (Figs. 1(b), 1(c), 2(b) and 2(c))

Figure 2 (

 2 Figure 2(d) shows the minor hysteresis loops of the chains at 23 mT and 88 mT. Compared with the

  (a). The heating rates of the liquid suspension containing these magnetosomes are shown in Figures 3(b) and 3(c) for the magnetic field amplitude of 23 mT and 88 mT. They are lower than those observed for the extracted chains of magnetosomes suspended in liquid both at 23 mT and 88 mT (Figures 2(b), 2(c), 3(b) and 3(c)). The difference between the heating rates of the chains and individual magnetosomes can either be due to a difference in the contribution of the magnetosome rotation or hysteresis losses to the SAR. The hysteresis losses measured from the areas of 8 26/06/17 the minor hysteresis loops (Figure 3(d)) or from the heating rates of the individual magnetosomes in the gel (Figures 3(b) and 3(c)) lie between ~ 220 W/gFe and ~ 430 W/gFe (Figure 3(e)), which are similar values than the hysteresis losses estimated for the extracted chains of magnetosomes (Figure 2(e)).

Figure 1 .26/06/ 17 Figure 2 . 2 %

 11722 Figure 1. Magnetic properties of intact cells of the magnetotactic bacterium Magnetospirillum

Figure 3 .

 3 Figure 3. Properties of extracted individual magnetosomes from cells of Magnetospirillumm

Figure 2

 2 Figure 2.

Figure 3 .

 3 Figure 3.

Acknowledgment:

We would like to thank the French biotech company Nanobacterie, SARL, for participating in this study.

26/06/17


C Keilani

An Uncommon Case of Symptomatic Multiple Meningiomas with Bilateral Compressive Optic Neuropathy Rapidly Induced Under Cyproterone Acetate Treatment

Keywords: Papilledema, bilateral optic nerve compression, multiple meningiomas, cyproterone acetate, pharmacovigilance, exposure time

Multiple meningiomas growth in patients under cyproterone acetate (CPA) is now well known. However, time between initial CPA intake and diagnosis remains unclear. The exposure time differs in each reported case: from 2 to 10 years. We present the case of an old man with acute visual impairment caused by an unusual bilateral optic nerve compression by three likely planum sphenoidale meningiomas rapidly induced by the admistration of CPA for prostatic adenocarinoma. This case is the first reported with a short exposure time (7 months) to CPA treatment before diagnosis of multiple meningiomas and stabilization on clinical follow-up after CPA treatment discontinuation.

INTRODUCTION

Cyproterone acetate (CPA) has an anti-androgenic, progestagenic and antigonadotropic effect. CPA can induce meningioma growth. In fact, meningiomas are known to be hormonesensitive tumors expressing progesterone receptors [START_REF] Marosi | Meningioma[END_REF][START_REF] Pravdenkova | Progesterone and estrogen receptors: opposing prognostic indicators in meningiomas[END_REF]. Meningiomas are mostly benign tumors with meningothelial cells originating from the arachnoid tissue. Few cases have been reported and showed an increased risk of meningioma growth in patients treated with high doses of CPA [START_REF] Gazzeri | Growth of a meningioma in a transsexual patient after estrogen-progestin therapy[END_REF][START_REF] Froelich | Endocrine abstracts[END_REF][START_REF] Bergoglio | Symptomatic meningioma induced by cross-sex hormone treatment in a male-to-female transsexual[END_REF][START_REF] Sys | Unilateral proptosis and blindness caused by meningioma in a patient treated with cyproterone acetate[END_REF]. The most recent case in the relevant field was in 2016 [START_REF] Alderman | Probable drug-related meningioma detected during the course of medication review services[END_REF].

The review by the European Pharmacovigilance Working Party in 2009 concluded to a causal relationship between high dose CPA (≥ 25 mg) and the occurrence of multiple meningiomas.

It has therefore resulted in formal contraindication of CPA treatment in patients with meningioma or a history of meningioma. Recently, in 2011 a population-based cohort study among 2474 patients showed that the exposure to high dose of CPA increases the risk of meningioma occurrence : 2474 users of high-dose cyproterone (6663 person-years), among whom four cases of meningioma were identified, with an incidence rate of 11 (95% CI: 16, 154) per 100,000 person-years after adjusting for age and sex, which was significantly higher than that observed among non-users (IR: 6.6; 95% CI: 6.0, 7.3) and among female users of low-dose cyproterone (IR: 0.0; 95% CI upper limit 5.5) [START_REF] Gil | Risk of meningioma among users of high doses of cyproterone acetate as compared with the general population: evidence from a population-based cohort study[END_REF]. Nevertheless the most detailed study of this question was a much larger cohort study in a population of 2,171,287 (14,059,934 person-years), among whom 745 patients with meningioma were identified and compared with 10,000 age-and sex-matched controls [START_REF] Cea-Soriano | Hormonal therapies and meningioma: is there a link?[END_REF]. There was a significantly increased risk of meningioma among men who used androgen analogues (OR: 19.09; CI: 2.81-129.74) and among users of high-dose cyproterone acetate (>50 mg/day; OR: 6.30; CI: 1.37-29) compared with non-users. It was the first cohort-study studying the causal relationship between CPA treatment and multiple meningiomas.

However, time between initial CPA intake and diagnosis is long. According to most studies, this period is at least two years.

Here, we present the case of an 83-year old man with acute visual impairment caused by three likely planum sphenoidale meningiomas with bilateral optic nerve compression induced after 7 months of CPA treatment. As far as we know, this case is the first with a short CPA exposure time before diagnosis. In addition, bilateral cases of optic nerve compression like this case are rare.

CASE PRESENTATION

An 83-year old man presented to the emergency department of François Quesnay hospital (Mantes-la-Jolie, France) for sudden bilateral vision loss. The patient was referred to our department of ophthalmology for examination. His family history was unremarkable. His past medical history was significant for high blood pressure treated by angiotensin converting enzyme inhibitors (enalapril 20 mg per day) and thiazide diuretic (hydrochlorothiazide 12,5 mg per day) since two years, arthrosis treated by analgesic (paracetamol 1 g three times per day) since 10 years and prostate cancer treated by cyproterone acetate (100 mg twice a day) for seven months. Ocular examination revealed light perception acuity in his right eye and 6/10 on Monoyer scale in his left eye. There was a right relative afferent pupillary defect. Fundus examination and optical coherence tomography showed generalized, marked bilateral papiledema with peripapillary hemorrhages (figure 1 and2). Goldmann visual field exam showed an inferior altitudinal scotoma (figure 3). Systemic examination was within normal limits with no evidence of neurofibromatosis type 2. A brain and orbit magnetic resonance imaging (MRI) was performed. We have highlighted three voluminous expansive tumoral processes of the planum of the sphenoid respectively weighted 18, 11, 9 mm. T1 and T2-weighted showed three iso-intense tumoral processes. T1-weighted with gadolinium injection showed diffuse enlargement and contrast enhancement (figure 4). One of the tumoral processes seemed to fill the right optical hole and compress the right optic nerve without infiltrating it. All these findings were consistent with meningiomas. The main hypothesis was that the recent administration of cyproterone acetate induced multiple meningiomas. These meningiomas caused bilateral optic nerve compression explaining acute visual impairment. The patient was referred to neurosurgeons. Neurosurgeons decided a therapeutic abstention considering the advanced age and the past medical history. In the context of prostate cancer a cerebral biopsy should have been performed to formally eliminate metastases. However, MRI findings were typically consistent with meningiomas and neurosurgeons decided not to perform biopsy because the patient was bedridden and because of the old age. In addition, a work-up for spread of prostate cancer had been carried out by a cerebral MRI before CPA treatment and was normal. This finding provided an additional argument for diagnosis of multiples meningiomas. Cyproterone acetate was discontinued. The patient had a prescription of corticoid (Predinsone 60 mg per day) and carbonic anhydrase inhibitor (Acetazolamid 500 mg per day) for the bilateral papiledema. These treatments were introduced on the advice of the neurosurgeons. Corticoid and carbonic anhydrase inhibitor were withdrawn after ten days because effectiveness of these treatments on optic nerve compression had never been proved by studies and no clinical improvement was noticed at ten days. The follow-up period after treatment withdrawal was more than twelve months with no clinical progression. After twelve months of CPA treatment withdrawal left visual acuity was improved at 7/10 on Monoyer scale and right eye visual acuity was unchanged.

Left eye fundus examination was within the limits of normal. Radiological follow-up by MRI has not been performed because one month later the patient had cardiac rhythm disorders and he needed a pacemaker. Therefore, MRI was contraindicated. We decided an only clinical follow-up every 6 months.

DISCUSSION

After informing the pharmacology department of the hospital, we tried to establish a statement with the Pharmacovigilance causality assessment. The causality assessment can be defined as an estimation of the putative causal relationship between a drug treatment and the occurrence of an adverse event, for a given person at a given time. There were indicators to consider a direct accountability of CPA to explain the multiple meningiomas growth. Causality was assessed using the Naranjo Algorithm. Our case had a score of 5 (probable adverse drug reaction).

In our case, cerebral MRI findings are consistent with meningiomas despite we have no histological certainty and metastases of prostate cancer could not be formally eliminated.

Cerebral MRI before CPA treatment was normal and this finding provided an additional argument for diagnosis of multiple meningiomas under CPA treatment.

To our knowledge, this case is the first reporting rapidly induced multiple meningiomas by CPA treatment. There have not been case studies with such a short exposure time. Usually, this period ranges from 2 to 10 years before diagnosis (Table 1). We have no explanation to this rapidly induced phenomenon. A hypothesis could be the higher exposure dose to CPA treatment (200 mg per day) than other past cases (50 mg to 100 mg per day) (table 1).

Another hypothesis could be a more pronounced progesterone and estrogen receptors. We suppose that these hormones may influence the growth in tumor size by increasing blood volume and extracellular and intracellular fluid as well as by increasing proliferation of the tumor tissue [START_REF] Blankenstein | Occurrence, regulation, and significance of progesterone receptors in human meningioma[END_REF][START_REF] Adams | Hormonal dependency of cerebral meningiomas. Part 2: In vitro effect of steroids, bromocriptine, and epidermal growth factor on growth of meningiomas[END_REF].

After 6 months of CPA treatment discontinuation, clinical follow-up showed an improvement in visual acuity and fundus examination. In absence of radiological follow-up we could not observe a tumor regression but we can talk at least of stabilization. Given the clinical improvement of this case, it is unlikely that corticoid and carbonic anhydrase inhibitor had any influence. These treatments are subject to controverse. Effectiveness of corticoid and anhydrase inhibitor on optic nerve compression had never been proved by studies. Few reviews showed that CPA discontinuation leads to a rapid regression of meningiomas [START_REF] Cebula | Regression of meningiomas after discontinuation of cyproterone acetate in a transsexual patient[END_REF][START_REF] Gonçalves | Abrupt regression of a meningioma after discontinuation of cyproterone treatment[END_REF][START_REF] Bernat | Growth stabilization and regression of meningiomas after discontinuation ofcyproterone ace tate: a case series of 12 patients[END_REF][START_REF] Botella | Méningiomes intracrâniens et utilisation prolongée d'acétate de cyprotérone à dose conventionnelle chez la femme: à propos de deux cas de régression tumorale après arrêt du traitement[END_REF][START_REF] Zairi | Close follow-up after discontinuation of cyproterone acetate: a possible option to defer surgery in patients with voluminous intracranial meningioma[END_REF] (table 2). In these studies, only one reported case had menigioma stabilization like in our patient. These indicators seem to be evidences of the link between meningiomas growth and cyproterone acetate. Therefore, in front of induced CPA meningiomas, waiting before a surgery could be suitable because spontaneous regression or stabilization can occur.

Meningioma induced by CPA therapy is now a well-known phenomenon that has been put forward by five clinical cases and one observational study (Table 1). Our clinical presentation case was similar to last reported cases: visual impairment.

Our review of literature showed that most reported cases were younger than our patient.

We have no explanation for this difference. However, one cohort study has reported 4 cases older than 60 years with multiple meningiomas induced under CPA treatment (Table 1). These cases had short exposure time: around 2 years (table 1). We suspect a link between CPA exposure time and patient age. It seems that in elderly patients, time between initial CPA intake and diagnosis of multiple meningiomas is short.

Multiple meningiomas represent only 5 % of all reported cases. Moreover, meningiomas account for 5-10 % of all orbital tumors [START_REF] Saeed | Optic nerve sheath meningiomas[END_REF] and bilateral cases of optic nerve compression like our case are rare and represent just 5 % of all reported cases [START_REF] Miller | The evolving management of optic nerve sheath meningiomas[END_REF]. In our case, the unchanged visual acuity on right eye is probably due to irreversible damage caused by the compression of the right optic nerve.

Most reported cases undergoing CPA treatment have skull base location meningiomas like in our case. Therefore, we suspect a particular histopathologic entity considering the preferential skull base location of this type of meningiomas (table 2). We suppose that these meningiomas present overexpressed progesterone and estrogen receptors. Post mortem biopsy for histopathologic analyze could be interesting. We present this report to show that meningiomas can occur rapidly under CPA treatment.

CONCLUSION

Thus, we suggest an ophthalmogic examination including a fundus examination and a visual field every 6 months for patients undergoing CPA treatment to prevent a delay in diagnosis (figure 5).

Finally, if a meningioma is found with a patient under CPA medication, conservative tumor management with clinical and imaging follow-up could be an option as stabilization may occur after treatment discontinuation (figure 5). Tables.

Figures.Figure 1 .

 1 Figures.

Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 1 .

 23451 Figure 2. OCT of left eye (right) and right eye (left)

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.

Figure 4 .

 4 Figure 4.

Figure 5 .

 5 Figure 5.

Table 1 . Reported clinical cases of multiple meningiomas induced by cyproterone acetateTable 2 . Reported cases of multiple meningiomas regression and stabilization after cyproterone acetate withdrawal

 12

ACKNOWLEDGMENTS

Dr. K. KEILANI, E. ROBBEN, S. EVEILLARD, N. JOUZIER, C. REGNAT.

Declaration of interests

The authors declare that they have no conflicts of interest related to this article.

Anterior temporal lobe extending into the left orbit with optic nerve compression

Visual impairment ° The more or less correspond to mean values +/-standard deviation