Aurore Ecarnot
email: aurore.ecarnot@u-psud.fr

Giovanni Magno

Vy Yam

Yam And

Ultra efficient nanoparticles trapping by integrated plasmonic dimers

Ultra-efficient nanoparticle trapping by integrated plasmonic dimers

Optical forces, arising from light-matter interaction as the result of their momentum exchange, are the main characters acting in the theatre of optical trapping systems. The pioneer Arthur Ashkin in 1970 demonstrated that, thanks to the radiation pressure, it is possible to trap small particles released into a liquid when two counterpropagating laser beams were focused on these small objects [1]. In dielectric systems [2], the tweezing action undergoes diffraction limit which prevents subwavelength objects to be firmly trapped [3]. Furthermore, plasmonic structures are able to concentrate light in deep subwavelength volumes and represent an enticing alternative to overcome the limits of a dielectric system. This occurs thanks to surface plasmon polaritons (SPPs) and localized surface plasmons (LSPs) at the interface between a dielectric material and noble metal nanoparticles (MNPs) [4]. By taking advantage of plasmonic resonators inside tweezing systems, very deep and narrow potential energy wells can be tailored to achieve stable optical tweezing down to subwavelength particles.

Different plasmonic structures, such as gold disks [5], nanopillars [6] and double nanoholes [7], have been proposed to trap the smallest particles. However, in these structures, the excitation is supplied by internal total reflection resulting in a weak near field excitation. To overcome this issue, we recently proposed an integrated solution consisting in a gold MNPs chain composed by nanocylindrical particles placed on top of a silicon on insulator (SOI) waveguide. The strong coupling occuring between the SOI waveguide and the plasmonic structure [8] enables a giant excitation of the MNPs chain near field [9]. This efficient near field excitation leads to a very efficient tweezing of nearby nanobeads. In [10], we demonstrated the trapping of very low refractive index biological particles (n=1.38) by means of short plasmonic chains containing up to five MNPs. In particular, by optimising the chain geometry, it is possible to continuously change the trapping position along the chain as a function of the wavelength, paving the way for the conception of novel integrated optical linear nanopositioners.

A specific topology of plasmonic resonators that revealed to be very promising for optical tweezing is the dimer. A dimer consists of two juxtaposed MNPs, of an arbitrary shape and separated by a gap g, such as two disks [11], bowties [12] or two squares [13]. Dimers are well known because of the important enhancement of the near field in the proximity of the gap, which increases as the latter shrinks. As we will show further in this letter, this strong near field localization is essential to achieve a stable trapping of subwavelengh sized particles.

The performances of optical tweezers depend on both the size and the depth of the optomechanical potential energy well. The latter is a function of the indivisible system constituted by the optical fields, providing with a trapping centre, and the object we want to tweeze. The stiffness k models the intensity of the restoring action along a given direction, exerted by the trapping centre when the particle tries to escape from the trap. It corresponds to the local curvature when the potential energy is at its minimum: the narrower the energy well, the higher the stiffness. However, it does not suffice as an exhaustive figure of merit to quantify the performances of the tweezing system. In fact, particles dispersed into a liquid are affected by the thermal agitation of the environment which needs to be compensated. As a stability criterion, Ashkin et al [14] suggested that the Brownian dynamic is compensated when the potential energy is much deeper (~ at least 10 times) than the thermal energy kBT of the system. When this condition is fulfilled, the object is steadily trapped, independently of the random kicks the object will suffer because of the thermal agitation.

In this letter, we numerically conceive ultra-effective optical tweezers thanks to the coupling by exploiting a dimer-like plasmonic resonator integrated on top of a traditional SOI waveguide. As it will be shown, this structure takes advantage of a very high coupling and of a narrow localization of the electric field in the gap between the two gold particles constituting the dimer. In particular, we will demonstrate that, by decreasing the size of the dimer gap g from 70 down to 2.5 nm, it is possible to attain a stable trapping of a polystyrene bead having a radius r from 250 down to 50 nm.

As depicted in Fig. 1(a) the dimer is constituted of two gold ellipsoidal nanocylinders with radii rx = 40 nm, ry = 100 nm and a thickness h = 30 nm. The latter is placed on top of a monomode SOI waveguide, having a height hSi = 220 nm and a width wSi = 500 nm. The structure is submerged in water (n = 1.33) where polystyrene beads (nb = 1.59) are dispersed. Fig. 1(b) shows the transmittance (blue) and the reflectance (red) spectra of a mode propagating through the SOI waveguide, in the case of a dimer gap g = 20 nm and a bead radius r = 250 nm. The dimer's resonance occurs near 1.52 µm, where the absorbance exceeding 55% demonstrating the high coupling between the SOI waveguide and the plasmonic dimer, despite the compactness of the latter. The calculation of the total force F acting on the bead is performed via 3D Finite Difference Time Domain (FDTD)-based numerical simulations (Lumerical FDTD Solutions). Particularly, we calculate the total electromagnetic field for a set of discrete positions that the bead can occupy on top of the MNPs. For each of these positions, we calculate the three components of the total force arising from the interaction between the bead and the dimer near field. For sake of simplicity, we can most likely assume that the bead is located at the top surface of the dimer in z-direction because the z-component of the force Fz is always negative (i.e attractive) along this direction. In fact, a polystyrene bead with r = 250 nm (r = 50 nm) placed on the dimer's top surface and centred with respect to the same undergoes Fz = -4000 pN/W (Fz = -300 pN/W). Furthermore, because the ycomponent of the force has to be null for symmetry reasons (as we will show the dimer behave as a symmetric dipole), we assume that the bead is located along the centre of the waveguide with respect to the y-direction, which constitutes an equilibrium locus (stable or unstable depending either on the bead size and on the gap size) for the y-component of the optomechanical force. We modelled all the material involved by considering lossy and dispersive models [START_REF] Palik | Handbook of Optical Constants of Solids[END_REF]. We used ellipsometric data for the gold permittivity fitted by a Drude model with a plasma frequency ωp = 1.29x10 16 rad/s and a collision frequency γ = 6.478x10 13 rad/s. We enclosed the calculation region with perfectly matched layer (PML) boundary conditions and we used inside a variable mesh as described in [8]. In each calculation, a 10 mW broadband source, matching the fundamental TE mode, excites the SOI waveguide at telecom wavelengths (from 1.35 to 1.75 µm).

Fig. 2 shows the electric field squared modulus |E|² distribution for three different decreasing gap sizes (g = 70, 20 and 2.5 nm). For the larger gap (Fig. 2(a)), the electric field is mainly localized near each vertex of the MNPs, entailing that they are in a resonant state and behave like distinct dipoles. For gaps g ≤ 20 nm (Fig. 2(b) and 2(c)), we observe an increased localisation and an enhancement of the electric field located between the two MNPs. More precisely, the maximum value of |E| 2 shows an enhancement of about a factor of 3 when the gap size decreases from 70 to 2.5 nm. Because for a given excitation power deeper energy wells correspond to higher |E| 2 , smaller gap sizes can be exploited to maximize the optical forces intensities and to steadily trap smaller objects. As a necessary condition to trap a particle, the stiffness value needs to be negative, since this would entail the existence of a restoring action on the object. Fig. 3 shows the value of stiffness along the x-direction kx, as a function of the x-direction and of the wavelength, calculated for a gap g = 70 nm (Fig. 3(a,b)) and g = 20 nm (Fig. 3(c,d)) when the bead radii r = 250 nm (Fig. 3(a,c)) and r = 50 nm (Fig. 3(b,d)). The blue dotted curves, representing the locus where the stiffness is null, delimit the states where the trapping is possible. The two white solid lines correspond to the middle of each gold particles. Fig. 3(a) shows that the maximum stiffness value occurs at resonance and when a polystyrene bead with r =250 nm is located exactly in between the two MNPs, with kx = -3.77x10 3 fN•(nm•W) -1 (at λ =1.54 µm) when a gap of 70 nm is considered. However, by decreasing the bead radius to 50 nm (Fig. 3(b)), the stiffness between the two MNPs is positive (unstable equilibrium) and the trapping is no longer possible along y = 0 (we will show that two distinct symmetric wells form). Fig. 3(c) and (d) show that when g = 20 nm the bead undergoes a restoring force along the x-direction with kx = -12.98x10 3 fN•(nm•W) -1 (at λ =1.52 µm) and kx = -3.12x10 3 fN•(nm•W) -1 (at λ =1.51 µm) when r = 250 nm and r = 50 nm respectively.

Fig. 4(a) shows the stiffness kx as a function of the gap size for three different values of the bead radius (r = 50, 100 and 250 nm). We observe that the stiffness absolute value increases as the gap shrinks. In absence of a brownian dynamic, a bead with r = 250 nm (red curve) would be trapped regardless of the gap size within the observation range. For a bead radius r = 50 nm (blue curve), a negative value of the stiffness is achieved only for g < 40 nm. It is worth pointing out that, for a feasible gap size of 20 nm and a sized bead of 50 nm, the stiffness absolute value is still as high as 3.12x10 3 fN•nm -1 •W -1 . Fig. 4(a). Stiffness kx as a function of the gap g for bead radii r = 50 nm (blue curve), r = 100 nm (green curve) and r = 250 nm (red curve). (b) Poynting vector calculated on the dimer's top plane as a function of the x and y -coordinates, when g = 20 nm and at plasmonic resonance when λ = 1.53 µm. The intensity of the Poynting vector is given by the colorbar. The superposed black curves delimit the position of the MNPs.

However, to achieve a stable trapping in presence of thermal agitation, the Ashkin criterion needs to be satisfied. This can be accounted by calculating the potential energy distribution U, along a given direction, by searching for its minima and by evaluating the depth ΔU(xi) = U(∞)-U(xi) for each point xi where a minimum occurs. In practical situations, when only the total force is available, an approximation of U, up to a constant term, can be obtained by integrating the total force F over the space that can be occupied by the object (in our case we consider the hemispace z ≥ r). Even by limiting this calculation to only two sections for z = r (along the xdirection when y = 0 and along the y-direction when x corresponds to the gap centre) with a sufficient spatial resolution, the computational efforts are massive. To circumvent this hurdle, we calculate U by using the method based on the convolution between the gradient force potential density and the object (Gradient Force Density Convolution, GFDC) discussed in [10] (see Eq. ( 1) of this reference). Here, we calculated only the conservative force terms due to the unperturbed structure, neglecting both the perturbation due to the presence of the object on the plasmonic resonator and the contribution to non-conservative force terms arising from the field scattered by the object. This is enough to characterise the trapping operations either when the refractive index of the trapped bead is close to that of the environment or when the radiation pressure becomes negligible because of any compensation mechanism of the Poynting vector along the observation direction. And, indeed, it has been recently demonstrated in structures with an infinite chain of MNPs on top of SOI waveguide that, near the plasmonic resonance, the average value of the x-component of the Poynting vector is very small in the hemispace above the MNPs due to the formation of Poynting vector vortexes, associated to a slowed down light [8]. Although the dimer that we considered is composed of only two elements, a similar behaviour occurs near resonance, because of its steady-state nature. In fact, Fig. 4(b) shows the Poynting vector at the plasmonic resonance in the [x,y]-plane at the top facet of the dimer. For symmetry reason, along y-direction, the energy flow forms two symmetric vortexes near the first MNP vertices. In Fig. 4(b), in the proximity of the first MNP's vertices the Poynting vector is more intense than elsewhere (the intensity is given by the arrow colour).

Fig. 5 displays the trapping energy, normalized to the thermal energy (kBT), obtained by means of the FDTD method (solids lines) and of the GFDC method (dashes lines) for a bead radius equals to 250 nm (blue and red curves) and 50 nm (black and green curves) as a function of x (Fig. 5 The reported curves are comparable in terms of shape and amplitudes. However, in Fig. 5(a) when |x| > 0.5 µm, the FDTD curves appear somewhat tilted with respect to the corresponding GFDC one. Indeed, far from the MNPs and from the Poynting vector vortexes, the scattering force (taken into account only in FDTD simulation) no more cancels out over the bead volume. More precisely, the scattering term arises from the evanescent part of the waveguide propagating mode: on the left side of the dimer, the scattering force drives the particle toward the dimer, whereas on the right side it contributes to push the particle away from the dimer. Besides, far from the MNPs along the y-direction, the contribution of the waveguide to the scattering force is negligible. Near the dimer, the two counter-rotating vortexes formed in the MNPs vertices (as shown in Fig. 4(b)), play a positive role, enhancing the trapping along this direction. In fact, in Fig. 5(b), a difference between the two curves minimum values appears. Nevertheless, it is worth to notice that, for smaller particles, the potential well splits in two symmetric and distinct dips along the ydirection. Less evident differences appear between the two methods when r = 50 nm (black and green curves in Fig. 5), due to the less important impact of the scattering force on smaller particles. These results enable us to conclude that the GFCD method does not overestimate the optomechanical well depth and provides fast and reasonable approximation of the potential energy.

Fig. 6 shows the map of the potential energy U, normalized by the thermal energy, as a function of the x and y-positions for g = 20 nm and r = 50 nm, obtained by using the GFCD method. We observe the presence of two distinct potential wells (as in Fig. 5(b)) with a depth reaching the values of -26.6 when the bead is located at x = 15.9 nm and y = ± 76.5 nm. This confirms that the polystyrene bead can be efficiently trapped at this position. The stiffness along the ydirection, estimated by the derivation of the potential energy U, thanks to the GFDC method, reaches a value of -5.89x10 3 fN•(nm•W) -1 which is one order of magnitude smaller than that calculated along x-direction kx in the same case (the same behaviour have been observed experimentally in [START_REF] Dagens | CLEO/QELS[END_REF] for 5-elements MNP chains). Furthermore, FDTD calculations of the stiffness along the ydirection show a comparable value of -4.07 x10 3 fN•(nm•W) -1 . Fig. 6. Map of the normalized optomechanical energy as a function of the x and y-directions obtained with GFCD method for g = 20 nm and r = 50 nm. The black dashes curves correspond to the limit of the Ashkin criterion. Fig. 7 represents the normalized potential energy minimum value (accounting for the well depth) as a function of the trapped bead radius, for a gap varying from 2.5 to 70 nm and calculated by using the GFDC method. On one hand, when a constant bead radius is considered, the energy well becomes deeper as the gap size shrinks. When a small radius is considered, the energy depth changes only negligibly with the gap size, entailing a large fabrication tolerance for this parameter. On the other hand, when a constant gap size is considered, the energy depth decreases as radius shrinks: the Ashkin criterion is no more satisfied when r < 25 nm regardless of the dimer gap size. In this case, the trapping can still be achieved by increasing the injected power (i.e. 30 mW are required to trap a bead having r = 25 nm). As anticipated, for r = 50 nm and g = 70 nm the positive stiffness at y = 0 does not entail instability: FDTD calculation reveals that two symmetrical wells are found, with minima located at y = ±76.5 nm.

To conclude, we have numerically demonstrated the breakthrough provided by a simple structure easy to fabricate: an integrated plasmonic tweezers made of a SOI waveguide coupled with a gold dimer which enable to trap small bead with low injected power. By decreasing the gap size and thanks to light concentration in plasmonic dimers, the gradient of the electric field and then the optical forces are considerably enhanced. A high stiffness value, calculated by the FDTD method, of -6.73x10 3 fN•nm -1 •W -1 is obtained for a gap size of 2.5 nm and a bead radius of 50 nm, which represents an improvement of a factor of 1.5 with respect to literature [13]. Furthermore, we demonstrated that, near the dimer, the occurrence of two counter-rotating Poynting vector vortexes increases the trapping stability. Finally, thanks to highly efficient dimer excitation from the SOI waveguide, the Ashkin criterion is satisfied and stable trapping is ensured for a polystyrene bead having radius down to 50 nm and for a power injected in the waveguide as low as 10 mW. Additionally, a multiparameter analysis was required to identify dimer designs. We have firstly compared time-consuming complete 3D FDTD numerical simulation with a fast, semi-analytical Gradient Force Density Convolution (GFDC) method and shown the relevance and the limits of the latter for the large scan study. Main results shed light on the importance of having a very small dimer gap when dealing with the trapping of highly subwavelength beads, but also of having a large tolerance to gap size fluctuation which could occur during technological fabrication. This integrated plasmonic tweezers open the perspective to have efficient integrated sensor requiring particle immobilisation and is also suitable with low-consuming lab-on-chip design.

Funding. Paris-Sud University Chair (PSA).

Fig. 1 (

 1 Fig. 1(a) Sketch of the proposed device. (b) Transmittance (blue) and reflectance (red) spectra for g = 20 nm and r = 250 nm.

Fig. 2 .

 2 Fig. 2. Maps of |E|² calculated on the dimer's top facet for g equal to (a) 70 nm, (b) 20 nm and (c) 2.5 nm.

Fig. 3 .

 3 Fig. 3. Stiffness map kx [fN/nm] as a function of the x-direction and of the wavelength for g = 70 nm (a, b) and g = 20 nm (c, d) when r = 250 nm (a, c) and r = 50 nm (b, d). The blue dotted curves depict the locus where the stiffness is null. The white lines indicate the middle of each MNP.

  (a)) and y (Fig.5(b)) directions.

Fig. 5 .

 5 Fig. 5. Comparison between the FDTD (solid lines) and the GFDC (dashed lines) calculation of the trapping energy normalized to the thermal energy (kBT) as a function of the (a) x and (b) y coordinates. These results are obtained for g = 20 nm and at plasmonic resonance and for r = 250 nm (blue and red curves) when x = 50 nm (a) and y = 0 (b)) and r = 50 nm (black and green curves) when x = 15.9 nm (a) and y = 76.5 nm (b).

Fig. 7 .

 7 Fig. 7. Normalized trapping energy minimum as a function of the trapped bead radius for different gaps. The inset shows the same quantity as a function of the gap size when r = 50 nm at y = ±76.5 nm.


Adrien Krifa

Florian Spinelli

Stéphane Junca

On the convergence of the Elo rating system for a Bernoulli model and round-robin tournaments

Keywords: AMS Classification: 60J20, 91D30 Ranking, Elo, rate, Markov chain, law of large numbers, ergodicity, least squares

. The Bernoulli model is first presented for 2 players. Then we'll study the N > 2 players case by numerically simulating Round-robin tournaments between the players. In all the case the Elo rating system correctly sorts people's strength.

Introduction

The Elo rating system is a famous ranking system created by Physicist Arpad Elo around 1970 to rank chess players [START_REF] Elo | The rating of chessplayers, past and present[END_REF]. It is now widely used to classify participants, teams, objects or anything that can be compared in pairs. Many online games use this system to rank players and match players of similar strength in order to have more interesting games.

In this report we mainly study the properties of the Elo model for a fixed finite number of players of constant strengths. That means that player improvement or regression is not taken into account. In this simple case, a natural Bernoulli model suffices to classify the players. The Bernoulli model is used to simulate many games when "everyone plays everyone" and to study the convergence and the convergence rate of the Elo ratings. The Elo rating is then seen as a random variable updated after each tournament. The Elo rating system does rank players and is an easy tool to use as its formula is very basic. In this report, we have shown multiple results numerically.

For 2 players we obtain numerical and theoritical results.

the logistic function of the expected values of the Elo difference of two players D ij = the expected values of the logistic function of D ij which converges towards p ij the probability that player i wins against player j we found the function towards which the Elo seems to converge to when n → ∞ For 3 or more players, many numerical tests show the right convergence of the Elo.

The report is organized as follows. First, the Elo rating system is recalled for two players. Then the natural Bernoulli model is given and used in round-robin tournaments. The section 2 deals with 2 players, the section 3 with 3 players and the section 4 with more, typically 100 players. Some conclusions are listed at the end.

The Elo system for 2 players

Consider two players i and j, i = j with respective Elo rankings R n i and R n j at the discrete time n. The result or score of the match is S n ij . The update of the ranking proceeds as follows :

R n+1 i = R n i + K S n ij -b(R n i -R n j ) , (1)
R n+1 j = R n j + K S n ji -b(R n j -R n i ) . (2)
K is the K-factor, K = 20 is commonly used. It satisfies the contractivity property to insure some convergence of the Elo,

K sup x∈R b (x) << 1. (3)
The scores satisfy S n ij ∈ [0, 1]. We use S n ij = 0 when i loses nth and S n ij = 1 when i wins. Thus, there is a skew-symmetric relation,

S n ij + S n ji = 1. (4)
The bonus-malus function b(.) which predicts the mean score of the match :

b ∈ [0, 1]. ( 5
)
Usually it is a distribution function. This function is also increasing, ∀x ∈ R,

0 < b (x). (6)
The bigger difference in Elo ratings two players have, the more important is the mean score expected.

This function only depends on the difference in players' ratings, which is the main ingredient of the Elo.

The conservation of the total sum of the Elo rating is granted and a constant,

R n+1 i + R n+1 j = R n i + R n j , (7)
since b() satisfies a similar relation to the score (4),

b(x) + b(-x) = 1. ( 8
)
The function usually used is the logistic function,

b(x) = 1 1 + 10 -x/400 . (9)
This choice satisfies a transivity property for more than 2 players and will be discussed later. sup b = b (0) = ln(10)/1600 ≈ 0.0014 [START_REF] Junca | Contractions to update Elo ratings for round-robin tournaments[END_REF] Here is a plot of the b function on a [-500, 500] interval.

The Bernoulli model

The natural mathematical way, without using the Elo model to measure the strength of the players is to know p ij the probabibiity that the player i beats the player j, 1 ≤ i, j ≤ N . A natural convention is to set p ii = 1/2. These quantities are easy to estimate, a priori, using the Law of Large Numbers (LLN). This N × N matrix P satisfies a symmetry.

p ij + p ji = 1, P + P = U, (11)
where U ij = 1 (for every i,j) is the matrix of the orthogonal projector on the line x 1 = x 2 = . . . = x N . In other words P -(1/2)U is a skew-symmetric matrix.

There are only (N 2 -N )/2 free parameters for the matrix P. The other parameters can be deduced by symmetry.

The ranking of the players in a round-robin tournament (everyone meets everyone) can be predicted by knowing what is their power, the expected number of wins for each player, that is for player i,

p i. = k =i p ik . (12)
Notice that p i. /(N -1) represents a mean probability of win in a round-robin tournament (RRT). And we can assume that the label of any player corresponds to their ranking, (else the players are sorted accordind to their power)

p 1. ≤ p 2. ≤ . . . ≤ p N .
When player i meets player j for the nth encounter, the score is the random variable S n ij following the Bernoulli law B(p ij ). All the scores are assumed to be independent except for the symmetric relation S n ij + S n ji = 1. Thus, the matrix S n = (S n ij ) 1≤i,j≤N has the same symmetry as the matrix P, S n + (S n ) = U.

The round-robin tournament

In this report, we would like to study the Elo rating system with the most stable and fairest tournament type that can possibly represent the strengths of each player. The elimination tournament type is not considered here for multiple reasons (e.g. a lot of match-ups wouldn't be covered, the best player could very well lose early in the tournament, etc). The round-robin tournament is considered as the fairest tournament to rank all players without using any ranking system (such as the Elo rating system). Counting the number of wins of each player is enough as long as we have a big enough sample. Indeed the expected nimber of wins is exactly the power p i. as discussed with the Bernoulli model before.

Here everyone plays everyone else once. Then the Elo rating is updated,

R n+1 i = R n i + K j =i S n ij -b(R n i -R n j ) (13)
For online video games, the Elo is often updated after each match. For a large number of matches it is expected that it corresponds to updating the Elo after each tournament as long as we have as many matches [START_REF] Jabin | Stéphane: A continuous model for ratings[END_REF], we have also shown this graphically in section 3. Moreover, our numerical simulations show that the Elo is less fluctuating when it is updated after each match than after each tournament.

The report is organized as follows. In section 2 the Elo model is presented for 2 players. In particular the important property of mass conservation is presented. In section 3 the problem of whether the Elo has a transitivity property or not is pointed out for 3 players. In section 4 the case of many players ( N ≥ 4 ) is studied.

'2' players 2.1 Mathematical remarks for players

Here we mainly study the simplest case, that is when the 'strength' of the players is constant and the results of the matches are independent. Therefore, p ij , the probability that player i wins against player j is assumed to be constant. Of course, this quantity is enough to know who is the stronger player, that is player i if p ij > 1/2, player j otherwise. The point of the Elo ranking is, in practice, to predict the outcome of a given match. In this report, we will check this expectation on the Elo ranking.

That means that (S n ij ) n are independent Bernoulli random variables B(p n ij ) and (R n i , R n j ) is a Markov chain [START_REF] Meyn | Markov chains and stochastic stability[END_REF].

Reduction to a scalar discrete process (D n ij ) For two players, the 2-dimensional Elo process can be reduced to a scalar one. Let D n ij = R n i -R n j , due to the fact that the expected result of an encounter depends only on the strength difference, the process (D n ij ) satisfies the relation, using (1), ( 2), ( 4), [START_REF] Jabin | On the Elo convergence rate[END_REF],

D n+1 ij = D n ij + 2K(S n ij -b(D n ij )), D 0 ij = R 0 i -R 0 j . (14)
All of the information is contained in the process (D n ij ), through the conservation of the sum of the Elos (7),

2R n i = R 0 i + R 0 j + D n ij , (15)
2R n j = R 0 i + R 0 j -D n ij . ( 16
)
The set of the difference values D Let D be the set of all possible values of the process. This set can be built by recurrence. Let D n be the set of all values of the random variable D n ij . That means that all values in this set have a non-zero probability to be a value of D n ij ( if 0 < p = p ij < 1).

D := n≥0 D n . ( 17
)
Here D is a countable set as it is a countable union of countable sets since {D 0 ij } is a singleton. If D 0 ij is a continuous law (its distribution function is continuous) then D is not countable.

Let us define the two increasing functions

g 1 (x) = x + 2K(1 -b(x)), (18) g 0 (x) = x -2Kb(x). ( 19
)
under the contractivity assumption,

2K sup b < 1. ( 20
)
Now, the set D n is built by recurrence,

D 0 = D 0 ij , (21)
D 1 = D 0 ij + 2K(1 -b(D 0 ij )), D 0 ij + 2K(0 -b(D 0 ij )) = g 1 (D 0 ij ), g 0 (D 0 ij ) , (22)
D n+1 = g 1 (D n ) ∪ g 0 (D n ). ( 23
)
That means that the size of this set is (usually) multiplied by 2 every time,

cardinal D n ≤ 2 n
(and usually with an equality). As a consequence, the Elo ranking lives in an infinite, countable and unbounded set.

Proposition 2.1 The set D is infinite and unbounded.

Proof: Notice that it is possible that a new value in D n+1 can be an ancient value in D n . Indeed, D is infinite because it is unbounded Now, it comes down to proving that D is unbounded.

Let M n = max D n , M 0 = D 0 ij , M 1 = g 1 (M 0 ) and M n+1 = g 1 (M n ).
To have the maximum at time n, the player i has to win all of the n matches. Therefore, P (D n = M n ) = p n > 0 becomes a very rare event (with 0 < p < 1). Since b is increasing (6) and b ∈ [0, 1] then, ∀x, 0 < b(x) < 1. Hence, ∀x, g 1 (x) > x, g 1 has no fixed point, thus, (M n ) n is increasing and M n → +∞. Which means that D has no upper bound which conludes the proof. Notice that, in the same way, ∀x, g 0 (x) < x and D has no lower bound. Proof: We can estimate max D n ij . The maximal growth corresponds to player i winning all of their matches. This occurs with probability p n ij which is exponentially small except if p ij = 1. Thus, the deterministic sequence d n+1 = g 1 (d n ) is considered using the notations of the proof of the proposition 2.1. The sequence (d n ) diverges towards +∞, since

Proposition 2.2 (Logarithmic growth of R n i L ∞ (Ω) ) max ω∈Ω R n i (ω) ∼ ln n 2 c
g 1 (x) > x, ∀x. At +∞, g 1 (x) = x + 2K exp(-c x) + O(exp(-2c x)), so d n+1 ≤ d n + C exp(-c d n ), with a constant C > 2K. Notice that d n = o(n) since C exp(-c d n ) → 0.
It can be also proved that d n = o(n a ) for all a > 0 since (n + 1) a -n a ∼ a/n 1-a exp(-cn a ). Let r n be d n / ln n, n > 1, so r n = o(n a / ln n). We claim that (r n ) is bounded. The sequence is already positive for large n since d n → +∞, only a upper bound for (r n ) is necessary. The sequence r n satisfies the inequality ln(n+1) ln n r n ≤ r n + C exp(-c dn)

ln n which can be rewritten

r n+1 -r n ≤ 1 ln n C exp(-c d n ) - r n n + O r n + exp(-c d n ) n 2 ≤ 1 ln n C exp(-c d n ) - r n n + O n -3/2 ,
taking 0 < a = 1/2. Now, let A such that c A > 1 , assume that n is large enough and r n > A, then d n > A ln n and exp(-cd n ) < 1/n cA . That means that r n+1 -r n < 0 . Moreover the right side is equivalent to a term of a divergent sum since r n /n > A/n, thus in finite time r n becomes less than A. This is enough to conclude that r n = O(1). This asymptotic bound is valid for all A > 1/c which means that lim sup r n ≤ 1/c. Thus, estimating lim inf r n in a similar way, we can prove that r n → 1/c and max

D n ij ∼ ln(n) c .
The Factor 1/2 in the proposition comes from (15).

Ergodicity The ergodicity reduces the cost of numerical simulations. A first ergodic result is the following :

Proposition 2.3 (Ergodicity of b(R n i -R n j )) 1 N n 0≤n<N b(R n i -R n j ) = p ij + O ln ln N N a.s. (24)
The temporal mean of the expected scores defined by the rating difference of the players converges towards the real expected score.

This graph shows the Elo evolution over 1000 games of two players A and B, A has a p = b(ρ 1 -ρ 2 ) chance of winning against B. ρ is meant to be the "true" (theoretical) force of a player; b(ρ 1 -ρ 2 ) is what we will name the "natural" probability of A winning against B. Here we have ρ 1 = 1500, ρ 2 = 700 and the players start with 1000 and 1200 Elo respectively. Let's now see how the convergence rate evolves if ρ varies for both players, starting with the same initial condition : a 1500 Elo rating.

We notice that the bigger the difference in ρ 1 and ρ 2 is, the slower the convergence rate becomes. Some convergences in law were proven in [START_REF] Aldous | Elo ratings and the sports model: a neglected topic in applied probability?[END_REF][START_REF] Avdeev | Stationary distribution of the player rating in the Elo model with one adversary[END_REF][START_REF] Jabin | On the Elo convergence rate[END_REF][START_REF] Junca | Contractions to update Elo ratings for round-robin tournaments[END_REF]]. Let's assume that lim n E(R n i ) converges. Since the Elo formula of the difference D n ij = R n i -R n j only depends on K, p ij , b(.), D 0 ij . We guess that the limit does not depend on the initial data D 0 ij , but due to the total sum conservation (7) it does depend on M 0 = R 0 i + R 0 j . Thus, the limit only is determined by K, p ij and M 0 .For K = 20 fixed we conjecture formula (27). The other equalities are then a direct consequence of (27).

lim n E(D n ij ) = 2G(p ij ), ( 27
) lim n E(R n i ) = M 0 /2 + G(p ij ), ( 28
) lim n E(R n j ) = M 0 /2 -G(p ij ), . (29)
Notice that for the equality (29) the term +G(p ji ) is expected. However, it can be replaced by -G(p ij ) since, from the conservation of the total Elo ( 7) and p = p ij we have

G(p) + G(1 -p) = 0. ( 30
) This relation is valid for all p ∈ [0, 1]. Indeed, G(p ij ) is the limit of E(R n i ) when M 0 = 0. We guess that G(1/2) = 0, G(1) = +∞, G(0) = -∞
We want to estimate the unknown function G. From the symmetry (34), and the expected singular behavior the study is limited to ]1/2, 1[. A parameter N of discretisation of this interval ]1/2, 1[ is choosen and h = 2 -N -1 . p k = 1/2 + kh, k = 1, ..., 2 N . This choice is motivated to have nested grid. Now, ρ k i will be estimated for all p ij = p k .

Initial Elo

The convergence of this process is slow. To improve the convergence, it is important to cautiously choose the initial data. Taking the expectation in the Elo formula (1) yields,

E(R n+1 i ) -E(R n i ) = K p ij -E(b(R n i -R n j )) . (31)
Passing to the limit, the left side vanishes and

p ij = lim n E(b(D n ij )) (32) Var D n ij is related to K, indeed Kb (0) 1. A relatively 'small' variance is expected, a natural choice is to take b(D 0 ij ) = b(ρ i -ρ j ) = p ij . ( 33
)
Notice that for p ij = 1/2 we recover the optimal choice D 0 ij = 0 = G(p ij ) But for p ij not limited to 1/2, this choice may be worse.

Another good possibility is to start with D 0 = 0 for 1/2 and p 1 and then to proceed with the well known continuation method. p = 1/2, R 0 i = R 0 j is the right choice. After the computation of ρ k i , to compute ρ k+1 i for the initial data we use the final limit of the previous case.

R 0 i = ρ k i , R 0 j = ρ k j .
The critical non random case p ij = 1 This case is critical. For the Elo ranking we simply have R n i = R 0 i thus the convergence is effective. But for p ij 1 and p ij < 1, we expect that G(p ij )

1.

If the G function that we will discuss in a few paragraphs is the correct one, then G(1 -0) = +∞ Mean estimates Temporal mean I: A nice way to calculate lim n E(R n i ) is described here :

lim n E(R n i ) 1 m m n=1 R n i
Same approximation, with a smaller cost 2m. This is valid if ergodicity occurs for the Elo ranking [START_REF] Meyn | Markov chains and stochastic stability[END_REF].

The G function should predict the Elo a player gets after a big sample of matches, given their initial elo and their opponent's elo. To do that we first simulated games between two players A and B of 1200 and 1000 elo ratings respectively, those players would play each other over 1000 games 10 times with a probability of p = 0.01 being the chance of A beating B. We would then average those 10 final values to get a good idea of what player A and B's Elos would be. Then the probability would increase by 0.01 and the process would repeat itself until we reach p = 0.99. We tried approaching the G function in different ways but what seemed to work for us was first finding the reciprocal function of b. We saw that it almost matched the results that we had gotten, it turns out that by dividing the reciprocal of the logistic function b we would get a function that matched almost perfectly our results. Thus we concluded that

G(x) = -200 log 10 ((1 -x)/x) = b -1 (x)/2 (34) 2.3 E[b(D)] = b(E[D])
We noticed an equality between E(b(D)) and b(E(D)); in order to do find this result, we empirically simulated these (while also verifying that they converge towards p) by using the Law of Large Numbers.

In order to make sure that the equality is true we zoomed into a range of 100 games and we noticed that the graphs were very similar to each other. To be more rigorous we then plotted the difference between E(b(D)) and b(E(D)) and we noticed that the maximum error over that 750 games sample was of about 0.006 .

Variance and expected value of the difference D

We were interested in the expected value and the variance of D varying with K. To do so, we simulated results with the use of the LLN, and here are the plots we got :

For small values of K, the expected value seems to vary, this is due to the convergence rate which is too slow. Once we get a big enough K the expectancy doesn't seem to depend on K anymore. We conjecture that for an infinite number of games, the expected value of D stays the same no matter which K is chosen.

The variance seems to be increasing in a linear way while K varies. To check this hypothesis in a numerical way, we used the Least Squares Method and it does seem to be the case.

N = players

In this section the case of 3 players is considered. Now, we want to test the validity of the Elo rating system. After many encounters the Elo rating has to correspond to the real strengths of the players in order to be practical. Just like before we will test the Elo rating system with the use of round-robin tournaments. The Elo is initialized at time n = 0 so R 0 i ; i = 1, 2, 3 are fixed. Then players play a round robin tournament at each time n. The Elo ranking is updated after each round-robin tournament.

R n+1 i = R n i + K S n ij + S n ik -b(R n i -R n j ) -b(R n i -R n k ) , (35)
where i = j = k = i, i, j, k ∈ {1, 2, 3}. Due to the symmetrical relations the total sum of the ranking is conserved,

k R n k = k R 0 k . ( 36
)
With this P matrix we hope that players will end up with the same strength, as they all have the same ρ.

In this example p 12 = p 13 = p 23 = 2/3. This case is contradictory to the Arpad Elo assumption (37). This case is critical since, for a round robin tournament, all the players have the same power, p i. = 4/3, i = 1, 2, 3.

The next graphs show the Elo evolution with the use of 3 different methods over 400 games of three players A, B and C with 1200, 1100 and 1000 initial Elo respectively, probabilities are determined by the P matrix, here we have : In this section, the case for N players is studied.

The round-robin tournament is still studied.

Transitivity relation sastified by the matrices (p ij ) 1≤i,j≤N 100 players Now we generate 100 different ρ values, going from 600 to 1600, with a step of 10 between each value. Here the win probability of a given player is defined as p = b(ρ i -ρ j ) as defined earlier. 500 tournaments take place in which everyone plays everyone, Elos are updated after each tournament and we calculate the average Elo each player gets. Above is the graph obtained. We also plotted the ρ values to compare them with our results. We observe that the players are correctly ordered.

We notice that all of the players have their final elo close to their initial ρ, they fluctuate around this value. Also, the worse our initial condition, the more time we need in order to converge towards ρ.

No transitivity In reality, the transitivity is not satisfied.

In order to verify that Elo correctly ranks people, we tried using the P matrix previously defined 1.2 with the difference that its probabilities are generated by a uniform law. We used 200 different players all starting at 1500 initial Elo.

Just like previously, the theoretical strength of a player is defined by the sum of a player's probabilities to win against all of the other players. Thus, we ordered them as we did in 1.2 p 1. ≤ p 2. ≤ . . . ≤ p N . .

Then we make them play over 500 tournaments where everyone plays everyone and we calculate the average Elo every player gets after these tournaments. It turns out that players' order is very close to what they were expected to get and the graph has an increasing tendency which is expected. From this simulation we can conclude that Elo correctly orders players.

Let's now get interested in the Elo repartition with the use of a histogram.

Most of the players are close to the initial elo (1500), the further away we are from this elo the more extreme of a case we are in and the more we stand out compared to the other players.

Conclusion

In this paper we studied the Elo convergence from 2 to N players. This report was mainly made from R simulations to illustrate some properties of the Elo system. We numerically verified some known and expected Elo properties. Elo properly sorts people's strength also when the assumption p ij = b(ρ i -ρ j ) is not fulffilled. We also numerically found the limit of the mean Elo for two players: the G function ( b -1 /2 ) which determines a player's Elo's limit

E[b(D)] = b(E[D]) = p ij
The equality E[b(D)] = p ij is proven here in an ergodic context and it is easily seen when the expectations converge. The other identity is more difficult b(E[D]) = p ij and not proven anywhere. These results were expected by Arpad Elo himself [START_REF] Elo | The rating of chessplayers, past and present[END_REF] but such explicit limits are still an open mathematical problem. Another interesting study could be made with a different "bonus" function used instead of the logistic one [START_REF] Junca | Contractions to update Elo ratings for round-robin tournaments[END_REF], and an amazing result would also be to numerically find a G function for any given N players.

Remark 2 . 1

 21 The expansion of the set D n is very slow. Usually, b is concave on R + and b (x) → 0 when x → +∞, thus, for the logistic function (9) g 1 (x) -x ∼ exp(-c x), c = ln 10/400 1/200. It suggests that getting a very big Elo by chance requires an extraordinarily large number of games.

3. 1

 1 Matchmaking 3.2 Round-Robin tournament : updated after each tournament 3.3 Round-Robin tournament : updated after each match 4 Round-robin tournament for N 1

Proof: The recurrence relation ( 14) can be rewriiten as follows,

). Adding this relation for n = 0, ..., N -1 yields,

Using the law of iterated logarithm for n<N S n ij (or the central limit theorem) [START_REF] Feller | An introduction to probability theory and its applications[END_REF], the previous bound for D n ij and dividing by N yields the announced result,

With the Central Limit Theorem, the error has the order a little bit smaller O(N -1/2 ), but, with a confidence less than 100%.

We also expect to have ergodicity to compute the limit of E(R n i ), [START_REF] Meyn | Markov chains and stochastic stability[END_REF] . Conjecture: Let D ∞ ij be the limit in law of D n ij and

ij for the nonlinear function b(.) is surprising. The law limit of D ∞ ij has a symmetry with respect to the function b(.).

Simulations for 2 Players

This graph shows the Elo evolution over 500 games of two players A and B with 1200 and 1000 initial Elo respectively, player A has a 60% chance of winning against player B

In order to study Elo rankings for more than 2 players, we decided to use three kinds of simulations : Round-Robin tournament : updated after each tournament Round-Robin tournament : updated after each game Matchmaking : a random pair of players play against each other while all of the others don't.

In the end we mainly used the Round Robin tournament type in which rankings are updated after each tournament is over.

We now use the probability matrix defined in 1.2.

Out of convention, we will have p ii = 0.5, even though we won't make players face themselves.

To illustrate our different modelings for the N = 3 players case, we will use the matrix generated by the following graph :

P transitive

We consider that the P matrix satisfies Arpad Elo's assumption (37). The simplest case is considered. The real strengths of the 3 players are fixed and ordered. The Arpad Elo assumption means that the probability that player i beats player j only depends on the difference of the 'real' (theoritical) ratings, ρ 1 ≥ ρ 2 ≥ ρ 3 , fix the probabilities,

The Elo system is invariant by translation so, in our simulations, we always assume that

That means that there are only 2 free parameters (ρ 1 , ρ 2 , ρ 3 ) (2=3-1 due to the invariance by translation) and not 3 for the P matrix. For instance, we can fix only 2 differences δ 12 , δ 13 where δ ij = ρ i -ρ j , (then δ ji = -δ ij ) to know all the other possible differences of 'theoretical' ratings δ 23 = δ 21 + δ 13 = δ 13 -δ 12 . Therefore the Elo hypothesis (37) reduces the natural choice of the P matrix from a set of dimension 3 to a set of dimension 2.

For instance, due to (37) p 12 ≥ p 13 , which is not always the case in reality. The realest case will be considered later in section 3.5.

Numerical simulations

No constraint on P

We will now observe the Elo evolution without the transitivity relation. Here we generate the P probabilities by using a uniform law on [0, 1], with the symmetrical relationship p ij = 1 -(p ji ) and p ii = 0.5.

We have 3 players A, B and C who respectively start from 800, 1000 and 1200 elo. Then we observe the different simulations over 4 random selections of independent P matrices. We can observe tendencies varying with the results, and we get a good estimate of every player's ρ value.

Analogously the same results were obtained when we updated Elos after every match. Here is a small example of the Elo convergence for ten players of the aforementioned process.


Jocelyn Inamo

Endocrinologie Chu

Marie-Laure Lalanne-Mistrih

Moana Gelu-Simeon Gastroentérologie

Véronique Baccini

Fritz-Line VELAYOUDOM épse CEPHISE

Laurent BRUREAU

Urologie

REMERCIEMENTS :

A l'issue de la rédaction de ma thèse, je suis convaincu que c'est loin d'être un travail solitaire. Sophie HAJJAR (oui, toi Sophie !), je tiens à te remercier de tout mon coeur pour ton soutien malgré les épreuves, ta bonne humeur malgré les chutes, ta grandeur d'esprit malgré ta petite taille. On a été ensemble dès le premier semestre et depuis, je suis admiratif de ce que tu es devenue. Passionnée, méritante et forte, je suis fier de t'avoir rencontré et je suis fier de toi ! Cette thèse t'appartient doudou. Mille mercis à toi ! Je tiens à remercier ma directrice de thèse, Madame Laurence BERAL, pour la confiance qu'elle m'a accordée en acceptant d'encadrer cette thèse, pour ses très nombreux conseils et pour toutes les heures qu'elle a consacrées à diriger cette étude. J'aimerais également lui dire à quel point j'ai apprécié sa grande disponibilité et son respect sans faille des délais serrés de relecture des documents que je lui ai adressés. Enfin, j'ai été extrêmement sensible à ses qualités humaines d'écoute et de compréhension tout au long de ce travail.

Je remercie sincèrement PrThierry DAVID pour avoir été le Président de mon jury mais

RÉSUMÉ :

Objectifs :

Identifier les facteurs de risques génétiques, systémiques et biologiques de maculopathie dans la drépanocytose. Évaluer les anomalies microvasculaires maculaires dans la drépanocytose grâce à l'optical coherence tomography angiography (OCTA).

Méthodes

Résultats :

Notre étude a inclu 43 patients HbSS, 30 HbSC, 4 HbS/β+ et 1 HbSLepore. La maculopathie drépanocytaire est survenue chez 66 yeux (44%), en particulier chez les patients HbSS (71%, p = 0,004). L'analyse multivariée a identifié le génotype HbSS et l'allongement du temps de prothrombine (TP) comme les facteurs de risques les plus significatifs de maculopathie (p= 0 ,01). La rétinopathie drépanocytaire proliférative a aussi été associée à la maculopathie (p = 0,02). L'augmentation du taux de lactate déshydrogénase a été significativement associée à l'élargissement de la ZAC (p = 0,02). L'ischémie maculaire a été plus sévère chez les patients avec un taux bas de base en hémoglobine (p = 0,04) et un TP allongé (p = 0,01

ABSTRACT :

Purpose:

To identify genetic, systemic and biologic risk factors of the occurrence and severity of maculopathy in sickle cell disease (SCD). To evaluate microvascular macular alterations in SCD using optical coherence tomography angiography (OCTA).

Methods:

In this cross-sectional monocentric study, 151 eyes of 78 adult patients with SCD and 40 eyes of 20 race-matched healthy controls underwent spectral domain optical coherence tomography (SD-OCT) and OCTA using Spectralis (Heidelberg Engineering, Heidelberg, Germany) to evaluate 3 criteria : the occurrence of sickle cell maculopathy (SCM), the foveal avascular zone (FAZ) enlargement and the severity of macular ischemia. We also collected clinical, genetic, systemic and biologic data and used multivariate logistic regression analysis to evaluate association between each potential risk factor and SCM, FAZ area and macular ischemia.

Results:

Our study included 43 HbSS, 30 HbSC, 4 HbS/β+ and 1 HbSLepore patients. SCM occurred in 66 eyes (44%), especially in HbSS patients (71%, p = 0,004). Multivariate analysis identified HbSS genotype and lower prothrombin ratio (PR) as the most predictive risk factors of SCM (p = 0,01). Proliferative sickle cell retinopathy was also associated with SCM (p = 0,02). FAZ enlargement was statistically associated with higher lactate dehydrogenase (LDH) level (p = 0,02). Macular ischemia was significantly more severe in patients with lower hemoglobin level (p = 0,004) and lower PR (p = 0,01). Areas of no flow were identified with OCTA even in eyes with no macular thinning (36 eyes, 42%) and happened more frequently in the temporal superior subfield (36%). Irregularities of the FAZ borders and perifoveal venular loops were found respectively in 70 eyes (46%) and in 38 eyes (25%) using OCTA.

Conclusion:

HbSS genotype, abnormal coagulation and hemolysis increase the risk of SCM. OCTA

provides valuable criteria to identify potential risk factors of SCM. OCTA also proved to be useful tool in order to detect early microvascular changes before the onset of macular thinning.

Maculopathy as defined as macular thinning on OCT is now outdated and we need to find other valuable criteria in order to define when begins sickle cell maculopathy and determine its natural history.

Sickle Cell

INTRODUCTION :

Sickle cell disease (SCD) is the most common genetic disease in the world affecting around 120 millions of people and about 6000 to 7000 patients in France. [START_REF] Modell | Global epidemiology of haemoglobin disorders and derived service indicators[END_REF][START_REF] Elagouz | Sickle cell disease and the eye: old and new concepts[END_REF] SCD is caused by a mutation of the β-globin gene resulting in the production of an abnormal hemoglobin called hemoglobin S (HbS). Hemoglobin S polymerizes under hypoxic conditions, which causes a mechanical distortion of red blood cells into a sickle-like shape. These sickled red blood cells are fragile and rigid, resulting in chronic hemolytic anemia and vaso-occlusive like complications in multiple organs, respectively, in particular in the eyes. [START_REF] Rees | Sickle-cell disease[END_REF] SCD is a generic term encompassing different syndromes according to genotype: homozygotes HbSS or compound heterozygotes (HbSvariant).

Sickle cell retinopathy (SCR) has been well described, proliferative sickle cell retinopathy (PSR) being a major sight-threatening complication. [START_REF] Abdalla Elsayed Mea | Sickle cell retinopathy. A focused review[END_REF] Since the development of Spectral Domain Optical Coherence Tomography (SD-OCT), macular vascular abnormalities in SCD are now commonly detected. [START_REF] Mathew | Spectral domain optical coherence tomography in patients with sickle cell disease[END_REF] Sickle cell maculopathy (SCM) is defined as patchy areas of severe retinal thinning in the temporal macula. The exact etiology of macular thinning remains unclear but it may be related to the temporal macula ending along the horizontal raphe making it a watershed zone between the vascular arcades of the retinal circulation and thus more susceptible to vascular occlusions. [START_REF] Stevens | Sickling hemoglobinopathies: macular and perimacular vascular abnormalities[END_REF] By allowing depth-resolved visualization of macular vascular network with high resolution, Optical Coherence Tomography Angiography (OCTA) recently detected much more macular vascular changes than previously and provided additional features, including enlargement and irregularities of the foveal avascular zone (FAZ), hairpin venular loops and areas of flow loss responsible for irreversible macular thinning. [START_REF] Minvielle | Macular microangiopathy in sickle cell disease using optical coherence tomography angiography[END_REF][START_REF] Han | Evaluation of macular vascular abnormalities identified by optical coherence tomography angiography in sickle cell disease[END_REF] Previous studies showed that HbSC genotype, male gender, lower fetal hemoglobin (HbF), higher hemoglobin levels and higher blood viscosity (particularly in HbSS genotype) are associated with severe PSR. [START_REF] Fox | Risk factors for proliferative sickle retinopathy[END_REF][START_REF] Duan | Clinical and ophthalmic factors associated with the severity of sickle cell retinopathy[END_REF][START_REF] Lemaire | Severe proliferative retinopathy is associated with blood hyperviscosity in sickle cell hemoglobin-C disease but not in sickle cell anemia[END_REF] But the clinical staging for SCR does not correspond to macular involvement. Little is known about the impact of genetic, biologic and systemic factors on the occurrence and severity of SCM. The question is to know if the systemic severity of SCD can be correlated in any way to ocular damage and to findings reported on OCTA.

Identifying risk factors could lead to reliable prediction models for SCM management and to minimize the onset of irreversible macular scotoma, decreased retinal sensitivity or vision loss described in eyes with severe macular thinning. [START_REF] Martin | Visual function in asymptomatic patients with homozygous sickle cell disease and temporal macular atrophy[END_REF][START_REF] Chow | Structural and functional correlation in sickle cell retinopathy using spectral-domain optical coherence tomography and scanning laser ophthalmoscope microperimetry[END_REF] The primary purpose of this study was to identify the systemic, biologic and genetic risk factors of SCM on OCT. The secondary purpose was to analyze macular microvascular network using OCTA and test if FAZ area and macular ischemia could be useful to identify more specific risk factors of SCM.

MATERIALS AND METHODS :

Patients:

This study received the ethical approval of the Regional Ethics Committee (CPP Sud-Ouest Outre-Mer III, Bordeaux, France) and adhered to the principles set by the Declaration of Helsinki. Written informed consent was obtained from each participant. This cross-sectional monocentric study concerned 117 patients with confirmed SCD diagnosis using standard methods and undergoing regular ophthalmological evaluation at the Ophthalmology Department of the University Hospital of Guadeloupe, French West Indies, between January 1 st , 2019 and January 1 st , 2020. Exclusion criteria were age under 18, history of other confounding retinal vascular disease such as hypertension, diabetes or retinal vein occlusion, prior retinal surgery, spherical equivalent (SE) > ± 1D, retinal involvement such as vitreomacular traction syndrome, epiretinal membrane or druses, poor quality images or ocular media opacities preventing detailed imaging. Sixty-nine eyes of 37 patients met these criteria and were excluded, so that 151 eyes of 78 patients were analyzed. An ethnic-matched control group of 20 healthy African Caribbean subjects was also recruited and underwent the same ocular procedures.

Multimodal imaging:

Ophthalmic examination included best corrected visual acuity (BCVA) measured with Snellen chart, slit-lamp evaluation, dilated fundus examination. SCR was graded according to the Goldberg classification including peripheral arteriolar occlusions (stage I), peripheral arteriovenous anastomoses (stage II), preretinal neovascularization (stage III), vitreous hemorrhage (stage IV) and retinal detachment (stage V). [START_REF] Goldberg | Classification and pathogenesis of proliferative sickle retinopathy[END_REF] Eyes with no retinopathy were graded as Goldberg stage 0. Patients with previously performed focal laser photocoagulation for peripheral neovascularization were classified as stage III. Analyses were performed using three categories of retinopathy: no retinopathy (stage 0), non-proliferative retinopathy (stage I and II) and proliferative retinopathy (stage III, IV and V).

All patients and control group underwent a macular SD-OCT and OCTA using the Spectralis HRA+OCT (Heidelberg Engineering, Heidelberg, Germany). For each eye, the standard scanning protocol included an automated « posterior pole » 30x25° cuboid volume scan of 61 high resolution OCT scans and 10° and 20° OCTA scans centered on the fovea.

Multiple scans were acquired as needed to secure adequate quality criteria and the best quality scans were selected by the lead grader for analysis. Images were anonymized and analyzed by two independent retina fellowship trained graders in a blinded fashion. In case of disagreement, scans were reviewed and arbitrary graded together. Sickle cell maculopathy was identified as patchy retinal thinning areas on OCT (i.e., blue areas of markedly decreased thickness on retinal thickness color-coded map). The first step involved manual delineation of the full thickness retina FAZ border taken from 10° OCTA scans using the « area » function of the software (Figure 1). Scans were then qualitatively graded by the investigators for macular ischemia from no (0) to severe (3) macular ischemia (including extension of flow loss areas, vessel density, enlargement of the FAZ, presence of perifoveal venular loops and focal disruption of the perifoveal anastomotic capillary arcade). Figure 2 illustrates our macular ischemia graduation. Clinical data were retrospectively collected from patient's charts at the last on-site consultation to the Sickle cell disease center of Guadeloupe and included : SCD type (genotype, αthalassemia status), medical and surgery history (other systemic condition, spleen status, cholecystectomy), current treatment (hydroxyurea, chelation therapy, chronic transfusion, anticoagulants) and history of SCD systemic complications (acute chest syndrome, cerebrovascular accident, severe vaso-occlusive crisis frequency, osteonecrosis). Frequency of severe vaso-occlusive crises was defined using Duan et al's method, i.e. as rare if once a year or less, occasional if twice a year and frequent if three times a year or more, averaged over the two years prior to the index ocular examination. [START_REF] Duan | Clinical and ophthalmic factors associated with the severity of sickle cell retinopathy[END_REF] Severe vaso-occlusive crises were defined as requiring acute Emergency Department visit or inpatient admission.

We also collected laboratory results including fetal hemoglobin (HbF) level, baseline hemoglobin level, hematocrit, C-reactive protein (CRP), prothrombin ratio (PR), reticulocytes, leucocytes, neutrophils and platelets count, lactate dehydrogenase (LDH), total bilirubin, serum ferritin and fibrinogen levels and G6PD deficiency.

Statistical analysis:

To analyze demographic, clinical and ocular characteristics of the population and to compare with the occurrence of maculopathy and FAZ enlargement, a Mann-Whitney test was used. P-value < 0.05 was considered as significant. Three ocular criteria were included in the analysis: the occurrence of maculopathy, the FAZ enlargement and the severity of the macular ischemia. Because simple comparison of the distribution of systemic and biologic factors between patients can be misleading as certain factors are interdependent or age related, we used multiple logistic regression analysis, adjusted by genders, genotypes and factors showing a pvalue < 0.1 in univariate analysis to identify risk factors independently associated with sickle cell maculopathy, FAZ enlargement and macular ischemia in an ordinal Generalized Estimating Equations (GEE) model.

Complete data were not available for all patients, so we used a simple imputation with a predictive mean matching (PMM) method for the missing data. Matched controls were used to compare FAZ measures with SCD patients using a Mann-Whitney test. Statistical analyses were performed using R statistical software.

RESULTS:

Demographic results: HbSS patients are more hemolytic: lower hemoglobin and hematocrit (p < 0.001), higher LDH and total bilirubin levels (p < 0.001). Finally, HbSS patients seemed to exhibit a proinflammatory state: higher leucocytes and neutrophils levels (p < 0.001 et <0.02, respectively) and a trend through higher level of CRP (p=0.087).

Faz area:

On OCTA, FAZ was measured in 142 eyes (94%) of SCD patients and in all healthy control eyes. In 9 eyes, it was not detectable by the software related to extensive macular ischemia (Figure 3) due to central arterial occlusion (4 eyes) and extensive perifoveal capillary remodeling with a poorly defined FAZ (5 eyes). For these 9 eyes, the quantitative value of their surely enlarged FAZ could be misleading in the analysis. Accordingly, we used a censored statistical model that assigned 1.0 to larger FAZ area than 1.0. Univariate analysis (Table 3)

showed that FAZ enlargement was associated with HbSS genotype (p = 0.04), hydroxyurea use (p = 0.04), anticoagulant medication (p = 0.05), acute chest syndrome history (p = 0.02), occasional vaso-occlusive crises (p = 0.01), lower hematocrit and hemoglobin levels (p = 0.01) and higher LDH level (p = 0.02). Unexpectedly, we also found that male SCD patients had a smaller FAZ size (p = 0.01). In multivariate analysis only high LDH level was independently associated with FAZ enlargement (a rise of 0.03 µm 2 for every 100 units of LDH, p = 0.02).

Interestingly, positive history for acute chest syndrome was marginally associated with FAZ enlargement (p = 0.07). Even if not statistically significant, chronic transfusion and frequent vaso occlusive crisis also tended to enlarge the FAZ size (p = 0.06 and p = 0.08, respectively).

Comparison between SCD patients with or without SCM and age, ethnic and sex -matched controls revealed that the FAZ area was larger in our patients than controls (respectively 0.52 ± 0.19 vs 0.38 ± 0.17, p < 0.01).

Macular ischemia:

Macular ischemia was subjectively identified using SD-OCT and OCTA with quantitative and qualitative criteria: irregularity of the FAZ, the extension of no flow areas and the general vascular density. We graded ischemia from no (0) to severe (3) for all 151 eyes of the study. The size of the grade 3 (severe) group of macular ischemia was not large enough and was regrouped with the grade 2 (moderate) group for the analysis. Thus, we had 49 eyes (32.5%) with no ischemia, 52 with mild ischemia (34.4%) and 50 with moderate or severe ischemia (33.1%). Areas of no flow were more frequent in the temporal superior (36%) subfield then the temporal inferior (27%), nasal superior (9%) and nasal inferior (5%) subfields. Seventy eyes had irregularities of the FAZ borders (46%). Thirty-eight eyes had perifoveal venular loops (25%). Abnormal macular flow loss was visible even in SCD eyes with no macular thinning on OCT (36 eyes, 42%). Univariate analysis (Table 4) revealed that 80% of the eyes with moderate and severe macular ischemia are from HbSS patients (p < 0.001). We also showed a significant association between the severity of macular ischemia and the severity of anemia (p < 0.001), lower hematocrit and PR (p = 0.01 and < 0.001, respectively), higher platelet level (p = 0.01) and increased LDH (p = 0.003). In multivariate analysis, the severity of anemia (p = 0.004) and lower PR (p = 0.01) remained independently associated with macular ischemia severity.

DISCUSSION :

The current study is the largest study to date evaluating SCD patients with OCTA findings. The analysis revealed that HbSS genotype, lower PR and proliferative sickle cell retinopathy are independently associated with sickle cell maculopathy. Correlation between proliferative sickle cell retinopathy and maculopathy is an expected outcome. An association between genotype and sickle cell maculopathy is less apparent, especially since it is well accepted that sickle cell retinopathy is more common and more severe in SC patients than in HbSS. [START_REF] Goldberg | Classification and pathogenesis of proliferative sickle retinopathy[END_REF][START_REF] Downes | Incidence and natural history of proliferative sickle cell retinopathy[END_REF] Mathew et al reported a higher rate of retinal thinning in HbSS patients than in HbSC patients (48% vs 35%), [START_REF] Mathew | Spectral domain optical coherence tomography in patients with sickle cell disease[END_REF] whereas other studies found no statistical difference. [START_REF] Han | Evaluation of macular vascular abnormalities identified by optical coherence tomography angiography in sickle cell disease[END_REF][START_REF] Dell'arti | Sickle cell maculopathy: Identification of systemic risk factors, and microstructural analysis of individual retinal layers of the macula[END_REF] Our study demonstrated that retinal thinning occurs more frequently in HbSS eyes compared to the other form of SCD (HbSC, HbS/β+, HbSlepore). This result can easily be explained. Indeed, prior studies proposed that a higher rate of vaso-occlusive crisis in HbSS patients could lead to repeated ischemic episodes in the eyes and thus the development of extensive retinal atrophy. [START_REF] Fox | Risk factors for proliferative sickle retinopathy[END_REF][START_REF] Serjeant | Blood rheology and proliferative retinopathy in sickle cell-haemoglobin C disease[END_REF][START_REF] Serjeant | Blood rheology and proliferative retinopathy in homozygous sickle cell disease[END_REF] On the other hand, milder degree of retinal ischemia in HbSC genotype would be not enough to infarct the retina but could result in the production of angiogenic factors and a higher risk of peripheral neovascularization. [START_REF] Ballas | Beyond the definitions of the phenotypic complications of sickle cell disease: An update on management[END_REF][START_REF] Lopes | Key endothelial cell angiogenic mechanisms are stimulated by the circulating milieu in sickle cell disease and attenuated by hydroxyurea[END_REF] Natural history of sickle cell retinopathy is unique because peripheral neovessels can infarct itself over time. [START_REF] Downes | Incidence and natural history of proliferative sickle cell retinopathy[END_REF] In this study, we graded as PSR the few eyes with focal photocoagulation for a prior neovascularization, so PSR eyes may have been overrepresented in the analysis. However, our results suggest a strong relationship between maculopathy and retinopathy. This relationship may be explained by the fact that the temporal macular area and the retinal periphery are both supplied by small caliber terminal arterioles, therefore vascular occlusion can easily damage these areas. [START_REF] Mathew | Spectral domain optical coherence tomography in patients with sickle cell disease[END_REF] Interestingly, PR was found significantly lower in patients with maculopathy. Low PR means a prolonged prothrombin time which is common in SCD even if the reason is not well established. [START_REF] Raffini | Prolongation of the prothrombin time and activated partial thromboplastin time in children with sickle cell disease[END_REF][START_REF] Nilesh | Haemostatic alterations in patients of sickle cell trait and homozygous sickle cell disease -A hospital based case control study[END_REF] It is likely that prolonged prothrombin time is a marker of chronic vascular inflammation among others like platelets and adhesion molecules. Prolonged PR could be related to several conditions such as liver dysfunction that decreases the synthesis of clotting factors or dysfunctional coagulation factors on vitamin K deficiency due to cholestasis for example.

OCTA enabled us to determine valuable findings of maculopathy and to identify or confirm specific risk factors for macular damages in SCD. FAZ area and macular ischemia were correlated with several biologic factors. Indeed, high serum LDH level was significantly associated with FAZ area enlargement in multivariate analysis with a rise of 0.03 µm 2 for every 100 units of LDH. Also, lower hemoglobin level was significantly associated with more severe macular ischemia in multivariate analysis with a rise of 0.36 µm 2 for every unit of hemoglobin.

It is noteworthy that lower hemoglobin, lower hematocrit and higher serum LDH levels, which are good markers of chronic intravascular hemolytic state, [START_REF] Kato | Vasculopathy in sickle cell disease: Biology, pathophysiology, genetics, translational medicine, and new research directions[END_REF] were statistically associated with the occurrence of maculopathy even if they did not remain in multivariate regression analysis.

These results are consistent with hemolytic severity, so we can hypothesize on hemolysis as the onset factor of sickle cell maculopathy. Indeed, two mechanisms are involved in the pathophysiology of SCD. Kato et al suggests a model that associates clinical events (vaso occlusive crises, acute chest syndrome or osteonecrosis for example) reflecting hyperviscosity and vaso occlusion phenotype, versus other clinical events (leg ulcers, pulmonary hypertension, priapism) mainly associated with hemolytic phenotype. [START_REF] Kato | Vasculopathy in sickle cell disease: Biology, pathophysiology, genetics, translational medicine, and new research directions[END_REF] By analogy, few studies found that severe sickle cell retinopathy is more frequent in patients with higher hemoglobin and hematocrit levels, suggesting blood viscosity as a potential risk factor. 9,25 These studies comforted in the capacity of phlebotomy to treat blood viscosity and to prevent retinopathy and maybe even maculopathy. In our study, we failed to found any statistical association between SCM and evident complications of hyperviscosity, apart from acute chest syndrome. Yet, we can not formally conclude if SCM belongs to one phenotype or the other.

Another potential risk factor barely failed the multivariate analysis: chronic transfusion.

SCD patients have chronic anemia which is corrected by occasional or regular transfusion. By reducing the HbS level, chronic transfusion prevents hemolysis and hemolytic complications. [START_REF] Rees | Sickle-cell disease[END_REF] We could not conclude to the capacity of blood transfusion to prevent macular damages, especially since our study is transversal.

Concerning hydroxyurea use, it induces an increase of HbF preventing HbS polymerization and red blood cells deformation. [START_REF] Kato | Vasculopathy in sickle cell disease: Biology, pathophysiology, genetics, translational medicine, and new research directions[END_REF] We suggest that it is efficient against vascular occlusion causing macular and peripheral ischemia like it is reported to reduce the frequency of sickling crises in SCD patients. [START_REF] Lopes | Key endothelial cell angiogenic mechanisms are stimulated by the circulating milieu in sickle cell disease and attenuated by hydroxyurea[END_REF] Similar studies concurred in the same direction. [START_REF] Estepp | Protection from sickle cell retinopathy is associated with elevated HbF levels and hydroxycarbamide use in children[END_REF][START_REF] Dembélé | Prévalence et facteurs de risque de la rétinopathie drépanocytaire dans un centre de suivi drépanocytaire d'Afrique subsaharienne[END_REF] A previous study conducted on 123 HbSS children showed a protective effect of HbF on retinopathy when superior at 15%. [START_REF] Estepp | Protection from sickle cell retinopathy is associated with elevated HbF levels and hydroxycarbamide use in children[END_REF] Dell 'Arti showed a correlation between lower HbF and sickle cell maculopathy but their analyses did not reach significance. [START_REF] Dell'arti | Sickle cell maculopathy: Identification of systemic risk factors, and microstructural analysis of individual retinal layers of the macula[END_REF] The current study failed to show any association with hydroxyurea use. Analyzing hydroxyurea use as a binary variable was problematic because it did not account for age of initiation or cumulative dosage to really establish a correlation with the FAZ area, the occurrence or the severity of SCM.

On OCT, sickle cell maculopathy is defined as a macular thinning. In our study, the occurrence of maculopathy in patients with SCD was 43.7% which confirms prior studies where maculopathy frequency varied from 44% to 60%. [START_REF] Mathew | Spectral domain optical coherence tomography in patients with sickle cell disease[END_REF][START_REF] Hoang | central macular splaying and outer retinal thinning in asymptomatic sickle cell patients by spectral-domain optical coherence tomography[END_REF][START_REF] Guerra | Sickle cell retinopathy: What we now understand using optical coherence tomography angiography. A systematic review[END_REF] A great advantage of OCTA is to detect microvascular abnormalities before retinal thinning occurs. For instance, Minvielle et al already detected perifoveal abnormalities in fluorescence angiography in only 50% of eyes whereas OCTA detected in 100% of cases. [START_REF] Minvielle | Macular microangiopathy in sickle cell disease using optical coherence tomography angiography[END_REF] Our results comfort prior studies. [START_REF] Mathew | Spectral domain optical coherence tomography in patients with sickle cell disease[END_REF][START_REF] Hood | Temporal macular atrophy as a predictor of neovascularization in sickle cell retinopathy[END_REF] Areas of no flow on OCTA were present in 36/85 eyes without maculopathy on OCT. This data suggests that OCTA may detect early signs of macular changes in patients and can likely be considered as a clinically useful marker in the screening and the evaluation of SCD progression or even for preventive treatment.

As expected, FAZ area was found larger in SCD patients than controls. This finding should be read with caution because loss of capillaries may have been the result of blood flow velocity below the detection threshold of the OCTA rather than no perfusion. In addition, it is important acknowledge that FAZ area can vary with ethnies. [START_REF] Wagner-Schuman | Raceand sex-related differences in retinal thickness and foveal pit morphology[END_REF] An enlarged FAZ area is not specific of patients with SCD but also a physiological variation for healthy subjects from African origin. To avoid any bias related to ethnicity effect, the control group was selected in the same population. We also noticed a substantial variability in FAZ area and irregularity between patients and controls. Some studies in healthy eyes support that this parameter can be ambiguous and thus may be insufficient for discriminating occurrence and severity of maculopathy in SCD. [START_REF] Linderman | Variability of foveal avascular zone metrics derived from optical coherence tomography angiography images[END_REF][START_REF] Magrath | Variability in foveal avascular zone and capillary density using optical coherence tomography angiography machines in healthy eyes[END_REF] Because of the clinical implication of significant disruption of the FAZ extending beyond the borders of the OCTA scans and, even if a quantitative value could be misleading in the 9 concerned eyes, we did not exclude them from analysis but used a censored statistical model instead. As such, eyes with more severe macular vascular pathology could be represented in this study in order to establish association with potential risk factors. In our study, there was no statistical difference between male and female in the occurrence of maculopathy and the macular ischemia, but men had a lower FAZ area than women.

Interestingly, some studies have questioned a protective female effect in SCR. Indeed, Fox found that HbSS men had a 2.5-fold higher risk of developing severe SCR than women. 9 They attributed the female protective effect to estrogen levels through a vascular protective effect by increasing endothelial nitric oxide synthase activity. Further studies are warranted to address this issue.

Finally, our analysis also revealed that the occurrence of the maculopathy or the severity of macular ischemia can vary significantly between 2 eyes of the same patient. Future prospective studies may elucidate the ocular risk factors that may predispose the injury of one eye compared to the other.

Our study had several limitations. First, it was a cross-sectional study, so we could not estimate the incidence and the evolution of the sickle cell maculopathy. Further studies are warranted to evaluate macular microvascular changes with OCTA in the same patients over time in a longitudinal manner. Secondly, we had strict inclusion and exclusion criteria resulting in a modest cohort of patients, which limited the multivariate analysis for potential risk factors.

Yet, we did not exclude patients with prior photocoagulation because we think that it reflects a real-life situation and their exclusion can underestimate a number of severe patients in the analysis. Another limitation concerned image acquisition and the software used. OCTA has the advantage to be a non-invasive dye-free technique and to provide structural microvascular information. However, it requires a precise and stable ocular fixation from the patients for several seconds. Furthermore, the software we used could not provide any vascular density, so we could not have any quantitative criteria for ischemia. For this reason, we subjectively graded macular ischemia because we suspect that OCTA is a better tool to reveal maculopathy than OCT. However, the study may underestimate the frequency of macular abnormalities in patients with more severe systemic events.

In conclusion, the identification of systemic, genetic and biologic factors associated with the occurrence and the severity of maculopathy confirms the complexity of this disease.

Association between sickle cell maculopathy and other organ damages in SCD is a significant question. OCTA is a highly sensitive method to detect microvascular macular abnormalities before the onset of macular thinning that commonly define the maculopathy status. Avoiding the occurrence of these abnormalities is critical to minimize and to manage the functional implication of retinal thinning in these young patients. Prospective studies are needed to determine the natural history of SCM and its relevance regarding preventive strategies for SCD patients. Furthermore, the emergence of novel technology such as OCTA could provide valuable information to contribute in a public health screening and follow up of patients with SCD.

LISTE DES TABLEAUX ET FIGURES :

  CHU de POINTE-A-PITRE/ABYMES Tel : 05 90 89 10 10 Maître de Conférences des Universités -Praticiens Hospitaliers

  surtout pour votre pédagogie et votre droiture. Je remercie également Pr Maturin TABUE TEGUO d'avoir participé au jury. Mes remerciements vont évidemment à Pr Harold MERLE d'avoir accepté de participer à ce jury de thèse et d'avoir été mon premier Maître dans la discipline. Vous m'avez transmis la passion de la profession. Je souhaiterais aussi adresser ma gratitude à Maryse ETIENNE-JULAN de l'Unité Transversale de la Drépanocytose sans laquelle mon étude ne serait certainement pas aussi riche. Je suis particulièrement reconnaissant à Élise BOULANGER-SCEMAMA, ainsi que Marc ROMANA et Philippe CONNES, pour l'intérêt qu'ils ont manifesté à l'égard de cette thèse en s'engageant dans sa relecture. À Pr Françoise FROUSSART de l'HIA Percy pour sa rigueur médicale et son sangfroid à toute épreuve. Je vous remercie de m'avoir pris sous votre aile lors de mon premier stage parisien. Ce stage n'a été que plaisir jusqu'à la dernière minute ! A toute l'équipe Percy, en particulier aux docteurs Maxime DELBARRE, Marie MARECHAL et Marouen BERGUIGA, vous me manquez ! Merci au Pr Yannick LE MER et à sa formidable équipe : Sébastien BRUNEAU, Élise BOULANGER-SCEMAMA, Nadine MANASSEH, Claire SCEMAMA, Yona GEISMAR, Houyem JEGUIRIM, Allan BENAROUS, Céline GIRAUD et Anthony MANASSERO. Vous m'avez tant apporté aussi bien au niveau médical que chirurgical ! À Pr Isabelle COCHEREAU que je ne saurais assez remercier pour tout ce qu'elle m'a apporté. J'ai eu beaucoup de plaisir à travailler avec vous et votre équipe. Un remerciement particulier à mes Maîtres pendant ce stage : Dr Serge DOAN pour m'avoir supporté pendant de nombreux mois et à Dr Cherif TITAH de m'avoir transmis la passion de l'inflammation oculaire. Je souhaite aussi remercier toute l'équipe avec qui je me suis senti chez moi : Benjamin PENAUD, Éric GABISON, Charles SITBON, Pierre DUBRULLE, Sophie STEPHAN, Minh Hanh DUONG et Gilles MARTIN. Au terme de ce long périple d'étudiant en médecine, j'aimerais aussi remercier ceux qui me sont chers et que j'ai quelque peu délaissés ces derniers mois voire dernières années. Merci à tous pour leurs attentions et encouragements qui m'ont accompagné tout au long de ce parcours. Je leur en suis infiniment redevable. A ma famille (Mounir, Jocelyne, Mehdi et Sondos) pour m'avoir supporté pendant un quart de siècle. A Rémi, Stanislas, Alexandre et Laura, équipe P1 de choc. Réussite en grande partie grâce à votre soutien. A Malyza, Diane, MJ, Claire : que d'aventures et bons moments passés en votre compagnie aux Cordeliers. A tous mes cointernes antillais et tout particulièrement NicoPaul, Beverly, Karine et Thomas. Best team antillaise ! Ti-bo. Merci aux « Pisse froids » (Hoang Maï, Adèle, Amira et Céline) pour cette ambiance de stage exceptionnelle. Notre rap final me restera toujours en mémoire. Et enfin, mes remerciements n'iront pas à Microsoft Office et particulièrement Word, pour tous les plantages qui ont pimenté l'élaboration de cette thèse.

:

  Dans cette étude transversale observationnelle monocentrique, 151 yeux de 78 patients atteints de drépanocytose et 40 yeux de 20 sujets sains de même origine ethnique ont bénéficié des examens spectral domain optical coherence tomography (SD-OCT) et OCTA sur l'angiographe Spectralis (Heidelberg, Allemagne) pour évaluer 3 critères : la prévalence de maculopathie drépanocytaire, l'élargissement de la zone avasculaire centrale (ZAC) et la sévérité de l'ischémie maculaire. Nous avons aussi recueilli les données cliniques, génétiques, biologiques et avons fait une analyse multivariée en régression logistique pour identifier une association entre chaque facteur de risque potentiel et la maculopathie drépanocytaire, l'aire de la ZAC et l'ischémie maculaire.

Figure 1 :Figure 2 :

 12 Figure 1 : Manual delineation of FAZ area

Figure 3 :

 3 Figure 3 : Exemple of difficulty in FAZ delineation on OCTA scan from a patient with severe (3) macular ischemia.

Figure 3 :Figure 4 :

 34 Figure 3 : Manual delineation of FAZ area

Figure 3 :

 3 Figure 3 : Exemple of difficulty in FAZ delineation on OCTA scan from a patient with severe (3) macular ischemia.

Maculopathy: microstructural analysis using OCTA and identification of genetic, systemic and biologic risk factors Selim

  Fares a , Sophie Hajjar a , Marc Romana b,c,d , Philippe Connes d,e , Malik Acomat a , Coralie Zorobabel a , Kevin Zuber f , Elise Boulanger-Scemama f , Maryse Etienne-Julan g , Thierry David a and Laurence Beral a

	a Service d'ophtalmologie, CHU de Pointe-à-Pitre/Abymes, route de Chauvel,
	Pointe-à-Pitre, Guadeloupe, France
	b Université de Paris, UMR_S1134, BIGR, INSERM, F-75015, Paris, France;
	c Université des Antilles, UMR_S1134, BIGR, F-97157, Pointe-à-Pitre, France;
	d Laboratoire d'Excellence GR-Ex, Paris, France;
	e Laboratoire Interuniversitaire de Biologie de la Motricité (LIBM) EA7424, Team
	« Vascular Biology and Red Blood Cell », Université Claude Bernard Lyon 1,
	Université de Lyon, France
	f Fondation Ophtalmologique Adolphe de Rothschild, Paris, France
	g Unité Transversale de la Drépanocytose, CHU de Pointe-à-Pitre/Abymes, route
	de Chauvel, Pointe-à-Pitre, Guadeloupe, France
	Déclaration d'intérêt : aucune
	Statut de l'article : en cours de soumission à l'American Journal of Ophthalmology

Table 1

 1 summarizes patients' demographical and clinical characteristics. All participants were of African origin. 151 eyes of 78 patients (25 men and 53 women) with a mean age of 37 years (range 18 -61 years old) were enrolled in this study. Five eyes were excluded due to unilateral high myopia (1 eye), phthisis bulbi (1 eye), poor acquisition due to no fixation (2 eyes) and a large foveal vascular pigmentary epithelium drusen (1 eye). Several sickle cell syndromes were analyzed including homozygotes HbSS (43 patients, 55%) and compound heterozygotes HbSvariant (35 patients, 45%) including HbSC (30 patients), HbS/β+ (4 patients) and HbSLepore (1 patient). Visual acuity ranged from 20/25 to 20/20 for all patients. The most frequently systemic events in our population were painful vaso occlusive

crisis (49%), acute chest syndrome (42%) and osteonecrosis (31%). There was no statistical difference in age, gender, functional α-globin gene number, vaso-occlusive crisis, osteonecrosis, anticoagulant medication and chronic transfusion between HbSS and HbSvariant groups. However, HbSS patients used more hydroxyurea (p < 0.001), exhibited more ulcers (p = 0.019), more acute chest syndrome (p < 0.01) and more cerebrovascular accidents (p < 0.01).

Table 1 :

 1 Baseline demographic characteristics of the sickle cell disease population of this study.

	Total	Hb SS	Hb Svariant	P values

Table 2 :

 2 Univariate analysis for occurrence of sickle cell maculopathy

	* ACS = acute chest syndrome; VOC = vaso-occlusive crisis; HbF = hemoglobin F; PR =
	prothrombin ratio; FAZ = foveal avascular zone; CRP = C-reactive protein; LDH = lactate
	dehydrogenase; GGT = gamma-glutamyl transferase

Table 3 :

 3 Univariate analysis of FAZ area

			FAZ area
		Estimator Std error t	P value
	SYSTEMIC
	CHARACTERISTICS
	Male	-0.09	0.03	7.30	0.01
	HbSS Genotype	0.07	0.03	2.07	0.04
	Hydroxyurea use	0.07	0.03	2.05	0.04
	Anticoagulation	0.09	0.04	1.97	0.05
	Chronic transfusion	0.12	0.06	1.91	0.06
	Regular phlebotomy	-0.12	0.03	-3.53 <0.01
	Osteonecrosis	0.04	0.04	1.28	0.20
	CVA*	0.00	0.06	0.00	0.98
	Ulcers	0.05	0.06	0.69	0.41
	ACS* history	0.08	0.03	5.83	0.02
	VOC* frequency /2 years
	Rare	-0.01	0.03	0.10	0.75
	Occasional	0.09	0.03	6.75	0.01
	Frequent	0.06	0.04	3.14	0.08
	OCULAR
	CHARACTERISTICS
	Macular ischemia index
	1	0.00122	0.01761	0.00	0.94
	2	0.02777	0.02651	1.10	0.29
	3	0.03558	0.03679	0.94	0.33
	Proliferative retinopathy	-0.03	0.02	1.24	0.27
	BIOLOGICAL CRITERIA
	Baseline Hemoglobin	-0.03	0.01	-2.73 0.01
	HbF* < 15%	-0.03	0.03	0.99	0.32
	Hematocrit	-0.01	0.00	6.84	0.01
	Platelets	0.00	0.00	2.03	0.15
	Leucocytes	0.01	0.01	0.80	0.37
	Neutrophils	0.00	0.01	0.27	0.60
	PR*	-0.0004	0.00	0.24	0.62
	Fibrinogen	0.02	0.02	0.87	0.35
	LDH*	0.0003	0.00	5.85	0.02
	GGT*	0.00	0.00	2.11	0.15
	α globin gene number	-0.01	0.03	-0.21 0.83
	* ACS = acute chest syndrome; CVA = cerebrovascular accident; VOC = vaso-occlusive crisis;
	HbF = hemoglobin F; PR = prothrombin ratio; LDH = lactate dehydrogenase; GGT = gamma-
	glutamyl transferase

Table 4 :

 4 Univariate analysis of risk factors in macular ischemia

		No ischemia	Mild ischemia	Moderate or	P value
				severe ischemia
	Number of eyes (%)	49 (32.5)	52 (34.4)	50 (33.1)	-
	Female, eyes (%)	33 (67.3)	41 (78.8)	28 (56.0)	0.40
	HbSS genotype, eyes (%)	17 (34.7)	28 (53.9)	40 (80.0)	<0.001
	Age, mean ± SD	34.2 ± 11.2	41.0 ± 10.2	34.0 ± 9.0	0.912
	SYSTEMIC
	CHARACTERISTICS
	Hydroxyurea use, eyes (%)	11 (22.4)	21 (40.4)	22 (44.0)	0.055
	Regular phlebotomy, eyes	17 (34.7)	18 (34.6)	10 (20.0)	0.16
	(%)
	Chronic transfusion, eyes (%)	2 (4.1)	1 (1.9)	8 (16.0)	0.09
	Anticoagulant, eyes (%)	5 (10.2)	7 (13.5)	11 (22.0)	0.25
	Osteonecrosis, eyes (%)	11 (22.4)	22 (42.3)	13 (26.0)	0.66
	ACS* history, eyes (%)	17 (34.7)	21 (40.4)	26 (52.0)	0.16
	OCULAR
	CHARACTERISTICS
	Occurrence of maculopathy,
	eyes
	No maculopathy	49 (100)	34 (65)	2 (4)	< 0.001
	Maculopathy	0 (0)	18 (35)	48 (96)	-
	Proliferative retinopathy,	24 (49.0)	28 (53.9)	37 (74.0)	0.53
	eyes (%)
	Photocoagulation, eyes (%)	12 (24.5)	19 (36.5)	21 (42.0)	0.123
	BIOLOGICAL CRITERIA
	Baseline hemoglobin (g/dL),	10.2 ± 1.7	9.0 ± 1.4	8.6 ± 1.6	<0.001
	mean ± SD
	HbF* > 15%, eyes (%)	5 (10.2)	10 (19.2)	8 (16.0)	0.42
	Hematocrit, mean ± SD	30.2 ± 5.1	27.3 ± 4.2	25.6 ± 5.6	0.01
	Platelets (G/L), mean ± SD	298.8 ± 140.1	311.4 ± 141.4	395.3 ± 163.4	0.01
	Leucocytes (G/L), mean ± SD	7.7 ± 3.0	7.7 ± 2.6	8.8 ± 2.9	0.2077
	Neutrophils (G/L), mean ±	4.1 ± 2.1	4.4 ± 2.0	4.5 ± 2.4	0.471
	SD
	CRP* (mg/L), mean ± SD	5.9 ± 6.1	6.0 ±7.0	5.6 ±5.2	0.842
	PR* (%), mean ± SD	86.2 ± 8.9	82.7 ± 17.1	77.6 ± 13.5	<0.001
	Fibrinogen (g/L), mean ± SD	3.2 ± 0.7	3.3 ± 0.9	3.2 ± 0.7	0,847
	LDH* (U/L), mean ± SD	323.4 ± 140.5	399.8 ± 183.5	465.5 ± 176.9	0.003
	Total bilirubin (µmol/L),	30.7 ± 27.1	33.4 ± 25.0	47.1 ± 33.7	0.156
	mean ± SD
	GGT* (U/L), mean ± SD	50.8 ± 46.5	68.7 ± 60.8	55.4 ± 46.2	0.597
	Serum ferritin (ng/ml), mean	174.3 ±249.6	221.4 ± 245.7	275.5 ± 476.1	0.19
	± SD

* ACS = acute chest syndrome; HbF = hemoglobin F; PR = Prothrombin ratio; CRP = Creactive protein; LDH = lactate dehydrogenase; GGT = gamma-glutamyl transferase

Table 1 :

 1 Baseline demographic characteristics of the sickle cell disease population of this study.

Table 2 :

 2 Univariate analysis for occurrence of sickle cell maculopathy

Table 3 :

 3 Univariate analysis of FAZ area

Table 4 :

 4 Univariate analysis of risk factors in macular ischemia

 20 (25.6) 12 (32) 8 (26) -4

47 (60.3) 24 (63) 23 (74) -

Acknowledgments: We would like to thank Unity of Sickle Cell Disease -Pointe-à-Pitre, Guadeloupe, for providing the clinical, genetic and biological data.

SERMENT D'HIPPOCRATE

Au moment d'être admis à exercer la médecine, en présence des maîtres de cette école et de mes condisciples, je promets et je jure d'être fidèle aux lois de l'honneur et de la probité qui la régissent.

Mon premier souci sera, de rétablir, de préserver ou de promouvoir la santé dans tous les éléments physiques et mentaux, individuels collectifs et sociaux. Je respecterai toutes les personnes, leur autonomie et leur volonté, sans aucune discrimination selon leur état ou leurs convictions. J'interviendrai pour les protéger si elles sont affaiblies, vulnérables ou menacées dans leur intégrité ou dignité.

Même sous la contrainte, je ne ferai usage de mes connaissances contre les lois de l'humanité. J'informerai les patients de décisions envisagées, de leurs raisons et de leurs conséquences.

Je ne tromperai jamais leur confiance et n'exploiterai pas le pouvoir hérité des circonstances pour forcer leurs consciences.

Je donnerai mes soins à l'indigent et à quiconque me les demandera.

Je ne me laisserai influencer ni par la recherche du gain ni par la recherche de la gloire.

Admis dans l'intimité des personnes, je tairai les secrets qui me sont confiés.

Reçu à l'intérieur des maisons, je respecterai les secrets des foyers.

Et ma conduite ne servira pas à corrompre les moeurs.

Je ferai tout pour soulager les souffrances, sans acharnement.

Je ne provoquerai jamais la mort délibérément.

Je préserverai l'indépendance nécessaire à l'accomplissement de ma mission.

Que je sois modéré en tout, mais insatiable de mon amour de la science.

A Matthias, l'homme de ma vie. Je t'aime.

APPENDIX

RÉSUMÉ :

Objectifs : Identifier les facteurs de risques génétiques, systémiques et biologiques de maculopathie dans la drépanocytose. Évaluer les anomalies microvasculaires maculaires dans la drépanocytose grâce à l'optical coherence tomography angiography (OCTA). Méthodes : Dans cette étude transversale observationnelle monocentrique, 151 yeux de 78 patients atteints de drépanocytose et 40 yeux de 20 sujets sains de même origine ethnique ont bénéficiés des examens spectral domain optical coherence tomography (SD-OCT) et OCTA sur l'angiographe Spectralis (Heidelberg, Allemagne) pour évaluer 3 critères : la prévalence de maculopathie drépanocytaire, l'élargissement de la zone avasculaire centrale (ZAC) et la sévérité de l'ischémie maculaire. Nous avons aussi recueilli les données cliniques, génétiques, biologiques et avons fait une analyse multivariée en régression logistique pour identifier une association entre chaque facteurs de risques potentiels et la maculopathie drépanocytaire, l'aire de la ZAC et l'ischémie maculaire. Résultats : Notre étude a inclus 43 patients HbSS, 30 HbSC, 4 HbS/β+ et 1 HbSLepore. La maculopathie drépanocytaire est survenue chez 66 yeux (44%), en particulier chez les patients HbSS (71%, p = 0,004). L'analyse multivariée a identifié le génotype HbSS et l'allongement du temps de prothrombine (TP) comme les facteurs de risques les plus significatifs de maculopathie (p= 0 ,01). La rétinopathie drépanocytaire proliférative a aussi été associée à la maculopathie (p = 0,02). L'augmentation du taux de lactate déshydrogénase a été significativement associée à l'élargissement de la ZAC (p = 0,02). L'ischémie maculaire a été plus sévère chez les patients avec un taux bas de base en hémoglobine (p = 0,04) et un TP allongé (p = 0,01). Les aires de non perfusion en OCTA ont été identifiées chez des patients sans amincissement rétinien (36 yeux, 42%) et sont survenues plus fréquemment dans la partie temporale supérieure de la macula. (36%). Des irrégularités de la ZAC et des boucles vasculaires périfovéales ont été retrouvées respectivement chez 70 yeux (46%) et 38 yeux (25%) en OCTA. Conclusion : Le génotype HbSS, une coagulation anormale et l'hémolyse sont associés à la maculopathie drépanocytaire. L'OCTA apporte de nouveaux critères pour identifier les facteurs de risque de maculopathie dans la drépanocytose. L'OCTA est utile pour détecter des anomalies microvasculaires avant même l'apparition d'un amincissement maculaire. La définition de la maculopathie drépanocytaire comme un amincissement maculaire à l'OCT n'est plus d'actualité et d'autres critères sont à définir afin d'identifier l'apparition précoce de la maculopathie et évaluer son évolution. --------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------