Justine S Pujos

Mathilde Reyssat

Anne Le Goff
email: anne.le-goff@utc.fr

Experimental and numerical study of platelets rolling on a von Willebrand factor-coated surface

Keywords: Microfluidics, Platelet, Transport, Rolling, Method of characteristics paraformaldehyde SID: shear-induced diffusion vWF: von Willebrand factor

published or not. The documents may come

Experimental and numerical study of platelets rolling on a von Willebrand factor-coated surface

Justine Pujos, Mathilde Reyssat, Anne

Introduction

Platelets are small anucleated blood cells that have an essential role in the process of hemostasis. In a wounded vessel, platelets adhere to the wall through ligand-receptor interactions and initiate coagulation. The large multimeric protein von Willebrand factor (vWF) is such a ligand [1]. At high shear rates, vWF unfolds and exposes platelet-binding domains [2,3]. The resulting platelet-vWF bonds are transient, causing platelets to roll along a vWF-expressing wall [4,[START_REF] Doggett | Selectin-Like Kinetics and Biomechanics Promote Rapid Platelet Adhesion in Flow: The GPIbα-vWF Tether Bond[END_REF]. vWF can also mediate firm adhesion through interaction with activated integrin α IIb β 3 [START_REF] Jackson | The growing complexity of platelet aggregation[END_REF]. Cell rolling is well documented for platelets [START_REF] Kumar | Kinetics of GPIbα-vWF-A1 Tether Bond under Flow: Effect of GPIbα Mutations on the Association and Dissociation Rates[END_REF] but also leukocytes [START_REF] Dong | Biomechanics of cell rolling: shear flow, cell-surface adhesion, and cell deformability[END_REF] and tumour cells [START_REF] Tremblay | Mechanisms by which E-Selectin Regulates Diapedesis of Colon Cancer Cells under Flow Conditions[END_REF]. Clinicians need reliable platelet function assays, either for the diagnosis of bleeding disorders [START_REF] Yago | Platelet glycoprotein Ibα forms catch bonds with human WT vWF but not with type 2B von Willebrand disease vWF[END_REF] or the monitoring of anticoagulant treatments. Microfluidics is a convenient way to manipulate small blood samples [START_REF] Toner | BLOOD-ON-A-CHIP[END_REF] and devices have been developed to assess the effects of anti-platelet treatments with measurements of the concentration platelets bound to the surface of a micro-channel [START_REF] Colace | Microfluidics and Coagulation Biology[END_REF][START_REF] Maloney | P2Y12 or P2Y1inhibitors reduce platelet deposition in a microfluidic model of thrombosis while apyrase lacks efficacy under flow conditions[END_REF][START_REF] Lincoln | Integrated system investigating shearmediated platelet interactions with von Willebrand factor using microliters of whole blood[END_REF].

In this article, we study the flow of fixed platelets in a vWF-coated microfluidic chamber. The local concentration of platelets adhering to the wall, C s , is measured and found to be a decreasing function of the distance x between the channel entrance and the position of observation. This phenomenon, called axial dependency, has been observed in experiments involving whole blood [START_REF] Sakariassen | Axial Dependence of Platelet-Collagen Interactions in Flowing Blood[END_REF]. To the best of our knowledge, although several theoretical and numerical studies focused on the diffusivity of platelets in the transverse direction [START_REF] Eckstein | Model of platelet transport in flowing blood with drift and diffusion terms[END_REF][START_REF] Zhao | Shear-induced particle migration and margination in a cellular suspension[END_REF][START_REF] Lopez | Shear-induced diffusion in dilute suspensions of spherical or nonspherical particles: Effects of irreversibility and symmetry breaking[END_REF] or on the adhesion mechanism [START_REF] Yago | Platelet glycoprotein Ibα forms catch bonds with human WT vWF but not with type 2B von Willebrand disease vWF[END_REF], no predictions have been made regarding the axial dependency of platelet concentration.

We present a basic model for the transport of blood platelets in the microchannel. This model involves specific cell-wall interactions and can include advection, diffusion in the transverse direction and cell rolling. The relative importance of the different features of this model is explored through comparison with experimental observations. Physical parameters, such as platelet attachment and detachment rates K on and K of f , are used as fitting parameters and we compare their values with experimental measurements or simulated estimations when available.

Experimental Methods

Microfluidics

Microchannels are fabricated using standard soft lithography techniques [19] and have a rectangular cross-section of fixed width W = 400 µm and length L = 4 cm, and variable height H ∈ [START_REF] Lincoln | Integrated system investigating shearmediated platelet interactions with von Willebrand factor using microliters of whole blood[END_REF]63] µm. After plasma treatment, channels are sealed on a glass slide, filled with a phosphate buffer saline (PBS, Lonza) solution containing 20 µg.mL -1 vWF (Wilfactin, LFB Biomedicaments, Les Ulis, France) and incubated overnight at 4 • C. Prior to the experiment, the channel is rinsed with PBS in order to remove unbound vWF.

The cell reservoir is connected to the channel with capillary tubing (internal diameter 228 µm). The homogeneity of the suspension in the entrance reservoir is maintained by gentle agitation. The flow is driven by a pressure control system (Fluigent MFCS-4C). Platelet perfusions are then performed at a wall shear rate compatible with the unfolding of immobilized vWF ( γ = 1400 -1800 s -1 ).

Biological Material

Blood platelets isolated from whole blood are provided by Etablissement Français du Sang (EFS) as the following agreement (CPSL C UNT-06/EFS/029), prepared as described by Dunois-Lardé and co-workers [START_REF] Dunois-Lardé | Exposure of human megakaryocytes to high shear rates accelerates platelet production[END_REF], then fixed with paraformaldehyde (PFA), rinsed and diluted in PBS to reach a concentration of 1.4 10 8 mL -1 . The goal of fixation is to preserve as much as possible the structure of platelets but to avoid further metabolic reactions, thus preventing effects such as on-chip activation [START_REF] Corum | The effect of upstream platelet-fibrinogen interactions on downstream adhesion and activation[END_REF].

Videomicroscopy

We use a Leica DMI 6000 B inverted microscope, equipped with a 40x magnification dry lens focused on the glass surface of the chip, and a camera (Photron Fastcam SA3) at an acquisition rate of 0.5 fps with a 20 ms shutter to measure the rolling velocity of adherent cells. During the kinetics measurements, only snapshots are acquired. As the motorized stage navigates between several pre-recorded positions along the whole channel length in less than 2 minutes, the experimental error associated with time is ∆t = 1 min. We define channel entrance, denoted x = 0, as the center of the 750 µm-wide hole where inlet tubing is inserted. The precision for position is therefore of the order of 300 µm.

Images are analysed with ImageJ software, using a routine to automatically count the number of adherent cells in a microscope field. In the case of dense surface coverage, the program fails to separate entangled platelets. Such images are analysed independently by two experimentalists. The discrepancy between their two counts was used to estimate the measurement error on platelet surface concentration ∆C s ∼ 20%C s .

Experimental observations

Figure 1 shows photographs of the channel surface after 95 minutes of perfusion, showing evidence of the axial dependency of platelet adsorption. The exposure time is long, so that only platelets bound to vWF appear sharp. Flowing platelets are not visible. The colors from blue to red corresponds to the time, the scale is 20µm. Both stationary platelets (frame 1) and rolling platelets (frame 2) can thus be observed. One adhesion event is identified in (frame 3) as the first time this platelet is measured is in the middle of the experiment. Likewise a desorption event is seen (frame 4) as the platelet disappears from the surface before the end of the experiment.

Model

Based on the above observations, we develop a model which describes the transport of blood platelets in the channel. Our goal is here to understand the physical phenomena governing the repartition of platelets between volume and surface along the channel. Three types of dynamical events are involved in the model: adhesion, desorption and rolling. As shown in Figure 3, we define a small control volume and consider the number of platelets in this volume and on the surface, respectively noted n v and n s . The mass balance sketched in Figure 4 involves their time variations:

Equations

         n v (x, t + dt) = n v (x, t) + n v,in -n v,out -n exch + n dif f usion n s (x, t + dt) = n s (x, t) + n s,in -n s,out + n exch (1)
where n v,in and n v,out (resp. n s,in and n s,out ) are the number of cells that enter and exit the volume (resp. surface) due to advection (resp. rolling) during dt. n exch represents the number of cells exchanged between surface and volume.

The additional source term n dif f usion will be discussed later.

The exchange term n exch is composed of two parts. The formation of bonds between vWF and platelets on the one hand is described by the adhesion rate K on , depending linearly on the number n v of available platelets, and limited by the presence of platelets already on the surface. On the other hand, for the detachment of platelets, with a rate K of f , from the surface, only a linear dependency with the number of adsorbed platelets is considered, as platelet suspensions are dilute enough in our experiments:

n exch = K on dt n v (1 -n s /n s,max ) -K of f dt n s
with n s,max the maximum number of platelets that can adhere to the control surface.

We rewrite equation (1) using the volume and surface concentration of platelets:

n v,in = C v (x, t) V v dt h dy n v,out = C v (x + dx, t) V v dt h dy n s,in = C s (x, t) V s dt dy n s,out = C s (x + dx, t) V s dt dy (2)
Here, h represents the thickness of the fluid layer interacting with the surface.

In an extremely shallow channel, all platelets in the bulk interact with the walls and the concentration C v in the exchange layer is equal to the average bulk concentration in the channel. In some situations however, the influence of the wall may not reach the core of the flow. The region involved in the exchanges with the wall is more shallow than the channel. There, new platelet adhesion events can induce a significant decrease of the volume concentration C v . This depletion can be compensated by a diffusive term. Calling D the diffusion coefficient, n ∞ v the quantity of platelets in the bulk, above the exchange layer, and l D the typical distance over which diffusion occurs, we get:

n dif f usion = D dt /l 2 D (n ∞ v -n v ).
Introducing the partial derivatives of the concentration with respect to time and space, we get:

               ∂C v ∂t + V v ∂C v ∂x = -J + D l 2 D (C ∞ v -C v ) ∂C s ∂t + V s ∂C s ∂x = hJ J = K on C v 1 - C s C s,max -K of f C s h (3)
C s , C v and C ∞ v are respectively the surface concentration, the volume concentration in the exchange layer and the volume concentration in the bulk; C s,max the maximal concentration on the surface at the packing limit.

Initially, both the surface and the volume of the channel are empty:

C v (x, t = 0) = 0 C s (x, t = 0) = 0 (4)
At the channel entrance, new platelets are injected in the volume only:

C v (x = 0, t) = C ∞ v C s (x = 0, t) = 0 (5)

Dimensionless equations

From equation (3), we can apply a series of basic transformations described in the Appendix A to reach the following dimensionless equations:

     v t + v x = -j + d(v ∞ -v) s t + s x = j j = v (1 -s) -αs (6)
and boundary conditions:

         v(x, t = 0) = 0 s(x, t = 0) = 0 v(x = 0, t) = v ∞ s(x = 0, t) = 0 (7)
where v, s and v ∞ are the dimensionless concentrations C v , C s and C ∞ v respectively, and the subscripts represent partial derivation with respect to dimensionless time and space. The parameters d, and α are defined as a non-

dimensional diffusion coefficient d = D l 2 D Kon , the ratio of longitudinal velocities = Vs
Vv and the ratio of exchange rates α =

K of f
Kon .

Here the system is composed of two half-spaces v and s, where a same object has different speed 1 and and an exchange value j between the half-spaces.

The diffusion can be easily added or neglected. These dimensionless coupled equations thus offer the opportunity to generalise this study to any similar system.

Analytical approach

The equations (6) represent the transport of a population in two interacting spaces with different characteristic speeds. We apply here the method of characteristics to these coupled equations. Briefly, three subspaces are defined in the (t, x) plane, as shown in Figure 5: a yellow domain where x > t, a grey domain where t < x < t and a blue domain where x < t. The boundaries of these subspaces are lines of constant speed, as 1 and are the non-dimensional speeds of the cells in the volume and at the surface respectively. We can define two sets of characteristic curves in the (t, x) space, one for each concentration v and s. The set of curves K 1 is defined by dx dt = 1 (blue lines in Figure 5) and

the equation dv dt = -j(s, v) + d(v ∞ -v
) is true on them; the set of curves K is defined by dx dt = (red lines in Figure 5) and associated with ds dt = j(s, v).

Let us consider the blue sub-space defined by x < t and the point M 1 (t 1 , x 1 ) as shown in Figure 6. Since the evolution of v is known on the characteristic lines of slope 1 and that of s on lines of slope , the values v 1 and s 1 at M 1 only depend on the initial values of v and s at M 0 and M 0 and on the distance on each characteristic, that is to say on x 1 .

If M1 is also in the sub-space and at the same height x 1 as M 1 , then the solutions v and s are identical to those of M 1 , since they derive from the values Three sub-spaces are presented here: in yellow the sub-space defined by x > t, in blue the sub-space x < t and grey the sub-space between the two and where most of the interesting physics of the system resides.

measured in M0 and M 0 , that are the same as in M 0 and M 0 . Thus all points at a position (t, x 1 ) with t > x 1 / share similar values of the concentrations s and v since they originate from a similar boundary condition v = v ∞ and s = 0.

This means that, for each position x, the system reaches a stationary state for t > x/ . Experimentally, this fact is confirmed for small positions. For large positions, no stationary state is observed, because the condition t > x/ is not achieved during the course of the experiment.

A similar approach shows that the sub-space defined by t < x has solutions that do not vary with position. If diffusion is neglected (d = 0), then the concentrations in this subspace are zero, as no object can move faster than the highest speed in the system and the only source of platelets is the injection at the origin of the channel (v(t,

x = 0) = v ∞ ). When diffusion is considered, it
is the only source of cells in this subspace. The solution still does not change with respect to the position, because the diffusion coefficient is a constant and the boundary conditions are the same all along the channel.

Figure 7 shows how, having explored these asymptotic regimes, we can sketch the shape of kinetics and axial dependency curves. The region explored in experiments corresponds to the grey zone and not to the asymptotic regimes described above. The temporal resolution of our experiments does not allow us to capture the delay between starting times at different positions. For example, for two positions x 1 and x 2 , we expect kinetics curves to start with a delay

τ = x2-x1 Vv . If V v is 1 mm•s -1
, which is the order of magnitude of the average bulk velocity, then even for the largest possible separation x 2 -x 1 = 4 cm, we

Figure 6: Details of the sub-space defined by x < t where M 0 and M 0 (respectively M0 and M 0 ) are the boundary conditions relative to M 1 (respectively M1 ). On the horizontal axis, values of v and s are fixed to v ∞ and 0. From these values, the solution at the point M 1 (t 1 , x 1 ) can be calculated. This solution is similar to the one at the point M1 ( t1 , x 1 ) as they originate from the same set of equations and previous states. Thus, all points on the line x = x 1 at times larger than x 1 / share the same values of v and s, and the solution is stationary in this subspace. find τ < 1 min. In spite of this, the overall shape of kinetics curves resembles that of experimental ones.

Numerical resolution

In order to explore the intermediate zone, we use a time-step numerical method to solve equation 6. We compute volume and surface platelet concentrations for all values of x and t, and not only in asymptotic cases. The programs are written in Matlab and run on a recent Intel-based desktop computer.

The equations involve seven parameters. d, and α describe the physics of the system, while the other four render time t, position x and the two concentrations C v and C s dimensionless. We look for the set of parameters which yields the simulation closest to the experimental points. We can thus describe this as minimizing a score function in a seven-dimensional space. The score value r is a simple distance between simulated and experimental points. Since the parameters do not appear explicitly in the score function, we can not run a classical optimisation search. Indeed, a whole simulation must be re-run for each set tested. We thus wrote a simplified search process, sketched in Figure 8.

We start the search with a limited number of different values for each parameters and test all possible combinations. All parameters are varied on a logarithmic scale with the exception of C s,max which is tested on a linear scale.

We then select the two sets which lead to the best fits between simulation and experiments but differ by more than one parameter. All combinations are tested for the two previously selected sets. At each iteration, we keep the two best fits and use them as input for the next refinement. Every three iterations, the dif- ference between the input values and the tested parameters is reduced so we can reach a greater precision for the result.

Stationary v = 0 s = 0 x = x 1 x = x 2 C s x = x 1 x = x 2 S ta ti o n a ry C s Stationary solution t = t 1 t = t 2 t = t 1 t = t 2 t = x v = 0 s = 0 v = v ∞ s = 0 t = x/
The selection of the two best fits aims to avoid finding a local minimum of our score function instead of the global minimum we are looking for. This search pattern, and in particular the fact that we limit our search to nine iterations, determines the maximum values each parameters can reach. An a-posteriori verification is performed: the values found at the end of the search should not be at the border of the search range.

C s,max V v V s K on K of f D h (mm -2 ) (mm•s -1 ) (mm•s -1 ) (s -1 ) (s -1 ) (mm 2 •s -1 ) (mm) min 5

Discussion

We start by using a simplified model without diffusion. Table 1 contains the range of variation for all the other parameters. Figure 9 shows that the model provides an overall shape consistent with experimental measurements, consisting of an initial increase followed by saturation, close to the channel entrance. The consistent with the rate constant associated with the transition of free platelets from the volume to a surface bounded state, that was estimated by Fitzgibbon et al. [START_REF] Fitzgibbon | Examining platelet adhesion via Stokes flow simulations and microfluidic experiments[END_REF]: r 2-3 0.07 -0.4 s -1 . When modeling the adhesion of platelets to subendothelial matrix, Leiderman and Fogelson [START_REF] Leiderman | Grow with the flow: a spatial-temporal model of platelet deposition and blood coagulation under flow[END_REF] find a value of 1 s -1 . In this situation, platelet adhesion is mediated by a larger number of molecules than vWF alone, which can explain this larger adhesion rate.

The desorption rate K of f is found to be much smaller than K on : K of f 6 10 -5 s -1 . This small value is due to the dense coating of vWF which forms several bonds with each platelet, that have to be simultaneously broken for the platelet to be released. Experimental measurements with platelet-rich plasma on a non-adhesive surface yield a value k of f 5 • 10 -2 s -1 [START_REF] Taylor | Development of a platelet adhesion transport equation for a computational thrombosis model[END_REF]. It is reasonable that the value we measure on an adhesive surface much smaller.

The length h is the height in the volume where platelets can interact with the surface and form bonds. We find h 0.1 µm which is in agreement with the size used by Fitzgibbon et al. [START_REF] Fitzgibbon | Examining platelet adhesion via Stokes flow simulations and microfluidic experiments[END_REF]: L = 0.2 µm. This also corresponds to the diameter of the repeating units constituting the vWF protein [2]. Because h is smaller than the size of a single platelet, the volume concentration in the layer of thickness h should rather be considered as a probability for a platelet from the bulk to reach the surface and adhere to it.

The advection velocity V v is found to be 0.36 mm • s -1 . The average fluid velocity in the channel is much larger V ∼ 6 mm•s -1 . But in the exchange layer located close to the channel bottom the velocity can be estimated as

V v = γh/2.
This yields a prediction for V v ∼ 0.6 mms • s -1 , that is in fair agreement with the value provided by the fit.

The rolling velocity V s is the only parameter whose fitted value seems inconsistent. The typical velocity of rolling platelets is 1 to 10 µm•s -1 , which is 5 orders of magnitude beyond the numerical result. The decrease in average velocity could be explained by the presence of platelets activated prior to fixation and incapable of rolling. This is consistent with experimental observations of a few immobile platelets. Far from the entrance, our model fails to capture the curvature of kinetics curves. This underestimation of platelet adsorption suggests that a source term is missing from the initial model. A possible explanation is that platelets diffuse from the core of the channel towards the walls. Brownian diffusion is parametrized by the coefficient D B = k B T /(6πηR) ∼ 2 10 -13 m 2 s -1 , with the water viscosity η water ∼ 10 -3 Pa • s and the particle radius R ∼ 10 -6 m. Shear-induced diffusion (SID) is defined as an effective diffusive motion of particles in a shear flow. It increases with shear rate and depends on the shape of diffusing objects [START_REF] Rusconi | Shear-Induced Diffusion of Platelike Particles in Microchannels[END_REF]. The SID coefficient is written D SID = γR 2 f (φ), with γ = 1400 s -1 the local shear rate, and φ ∼ 0.5 10 -3 the volume fraction of the particles. For anisotropic particles, the function f is different than for spherical particles and the diffusivity is more important [START_REF] Lopez | Shear-induced diffusion in dilute suspensions of spherical or nonspherical particles: Effects of irreversibility and symmetry breaking[END_REF]. In the dilute limit the diffusivity is proportional to φ and we can estimate the diffusion coefficient for plate-like objects [START_REF] Rusconi | Shear-Induced Diffusion of Platelike Particles in Microchannels[END_REF]: D SID = γR 2 6.9 φ ∼ 6.2 10 -12 m 2 s -1 . It thus appears that shear-induced diffusion should predominate over Brownian diffusion in our system.

Influence of diffusion

The Peclet number P e = measurements. Here we consider that l D is of the order of the channel thickness H ∼ 10 -5 m [START_REF] Squires | Making it stick: convection, reaction and diffusion in surface-based biosensors[END_REF]. We get P e ∼ 2, indicating that diffusion cannot be neglected.

When diffusion is taken into account, numerical curves lose their convexity, as illustrated in Figure 10. The best fit is obtained for D = 5.9 10 -12 m 2 s -1 , which is in fair agreement with theoretical predictions. The value of parameters obtained with this full model, presented in the second line of Table 2, remain consistent with the estimations presented in the previous paragraph.

Influence of the rolling behaviour

Even when diffusion is taken into account, the value of V s deduced from the model remains extremely low. In the equation expressing transport at the surface, the term

V s ∂Cs ∂x ∼ V s Cs,max L can balance the desorption term K of f C s if V s ∼ K of f L, that is of the order of 0.1 µm•s -1 .
The adjusted values of V s are several orders of magnitude smaller. This seems to indicate that rolling can be neglected in our problem. In supplementary Figure 12, we adjust experimental data with new numerical simulations in which rolling velocity is set to 0. Of course, in a different system where the velocity contrast between surface and volume is smaller, we could expect a significant contribution from rolling to surface coverage. Again, the values of all parameters stay in the same order of magnitude, although they may vary by a factor 2 or 3. Thus the rolling behaviour of platelets seems negligible in this case. Nevertheless, it may very well be fundamental for the formation of blood clots, such as in microfluidic models of stenosis and coagulation [START_REF] Colace | Microfluidics and Coagulation Biology[END_REF][START_REF] Combariza | Nonlinear Dynamic Modelling of Platelet Aggregation via Microfluidic Devices[END_REF].

Influence of the non-linearity

Experimental values of C s never exceed 15000 mm -2 , a much smaller value than those obtained for C s,max in the best fits. We further simplify the model by suppressing the non-linear surface saturation term:

1 - C s C s,max 1
in equation (3). Far from the channel entrance, the linear model yields results that are very similar to the ones obtained with the full model. As seen in supplementary Figure 13, results are less satisfying close to the channel entrance, where the surface coverage is maximal. As suppressing the non-linear term represents a significant gain of time, this linear model could be interesting for applications where a prediction of surface coverage far from the channel entrance is desired.

Influence of desorption

Since the desorption rate K of f is much smaller than that of the adsorption rate K on , we also attempt to adjust experimental curves while neglecting desorption. The results are displayed in Figure 11: the obtained numerical curves increase fast and clearly fail to grasp the trend towards saturation exhibited by experimental data. In fact, although both rates share the same dimension,

C s,max V v V s K on K of f D h (mm -2 ) (mm• s -1 ) (mm• s -1 ) (s -1 ) (s -1 ) (mm 2 s -1 ) (mm) no diffusion 1.7 •

Variability

The weak contribution of rolling to the transport of platelets along the channel may be explained by the fact that some of the adherent platelets are activated and do not translocate. This feature is likely to vary from one sample to another. In supplementary Figure 14, we show that the shape of adhesion kinetics changes when experimental conditions are varied, but the main features, such as axial dependency and saturation close to the entrance, are conserved. Parameters such as channel height, platelet sample, platelet concentration and wall shear rate are varied (see supplementary Table 3). While the model without rolling remains in fair agreement with experimental data for other samples, the adjusted diffusion coefficient can be as much as three orders of magnitude lower than the one measured for sample A. In shear-induced diffusion, the apparent diffusion coefficient strongly depends on the particules' size and aspect ratio.

Platelets have a roughly discoidal shape, but for some patients, blood also contains incompletely formed, larger platelets that have a dumbbell shape. It can therefore be understood that shear-induced diffusion plays a significant role in platelet transport for some, but not all patients.

We also notice that the adjusted value of K on is small for this particular sample (sample B). This is consistent with the measured values of platelet concentration that remain low even at long times, indicating that the affinity of platelets from this sample for vWF is smaller than usual.

For yet another example (sample C), the shape of kinetics curves and results of fitting are more similar to the ones obtained with sample A, as well as the values of the fitting parameters. The shape of curves at intermediate positions however is not well reproduced by the simulations. The axial dependency is very marked in this experiment, and the values of C s measured at positions far from the entrance are almost uniformly equal to 0. These data with extremely low values bias the fitting process, that becomes therefore less efficient in the regions of finite C s . The choice of the observation positions for this experiment is not optimal. A more accurate fitting could be performed by selecting only the experimental data containing significant information.

Conclusions

We presented in this article an experimental set-up and an associated model to study the transport of fixed blood platelets in a micro channel with coated walls. In spite of the variations that we observe between models or between experiments, some of the conclusions are repeatable. In all three examples, we find that the thickness of the exchange layer h is of the order of a few hundred nanometers. The adsorption and desorption constants K on and K of f extracted from this work characterize the behavior of whole cells and not of single bonds. The desorption rate K of f is always much smaller than K on but desorption is not negligible and accounts for the saturation observed close to the channel entrance. In some samples, shear-induced diffusion needs to be taken into account and the diffusion coefficient D found by adjusting experimental data is in agreement with theoretical predictions. To further improve the fitting, we would need to take into account the fact that diffusion occurs across the cellfree layer, whose thickness is not uniform. By choosing a constant characteristic length for diffusion, we underestimated the diffusive flux towards the surface close to the channel entrance, where the cell-free layer is extremely thin and the concentration gradient large.

When whole blood is used instead of a platelet suspension, the surface is covered much more rapidly [START_REF] Sakariassen | Axial Dependence of Platelet-Collagen Interactions in Flowing Blood[END_REF]. This is due to margination effects : red blood cells present in the core of the vessel flow push platelets towards the surface, increasing their availability in the vicinity of the walls [START_REF] Zhao | Shear-induced particle migration and margination in a cellular suspension[END_REF]. We assume that our model could remain valid in such a situation, if an effective diffusion coefficient depending on red blood cell concentration and deformation replaced the shearinduced diffusion coefficient.

We also quantified axial dependency and showed that concentration profiles exhibit extremely sharp decreases close to the channel entrance. Measurements of the surface concentration in adsorbed platelets is therefore highly sensitive to a small error in position. We therefore think that experiments where observations are made on a single location in the channel should be handled with care, and that measurements made at several positions should be preferred.

Although sophisticated multiscale models exist, practical applications could make use of less costly micro scale simulations [START_REF] Rugonyi | Effect of Blood Flow on Near-the-Wall Mass Transport of Drugs and Other Bioactive Agents: A Simple Formula to Estimate Boundary Layer Concentrations[END_REF]. This paves the way towards more efficient simulations of cell suspension dynamics.

D (C ∞ v -C v ) ∂C s ∂t + V s ∂C s ∂x = hJ J = K on C v 1 - C s C s,max -K of f C s h
We first focus on the two concentrations and apply the following transformation:

C v ←- C v h C s,max = v C s ←- C s C s,max = s
The transversal dimension h is naturally used to bridge the gap between the surface and volume concentrations as it is characteristic of the dimension that differ between the two concentrations. The equations thus transform into:

               ∂v ∂t + V v ∂v ∂x = -J + D l 2 D (v ∞ -v) ∂s ∂t + V s ∂s ∂x = J J = Jh C s,max = K on v (1 -s) -K of

Exp

C s,max

V v K on K of f D h (mm -2 ) (mm•s -1 ) (s -1 ) (s -1 ) (mm 2 s -1 ) (mm)

Figure 1 :

 1 Figure 1: Photographs of the surface of the micro channel after 95 minutes experiment. Positions are equally spaced and separated by 1mm. The first picture is acquired x = 0.6 mm away from the channel entrance. Scale bar represents 20 µm.

Figure 2

 2 Figure 2 illustrates the diversity of behaviours observed at the surface. Positions of surface-adsorbed platelets are shown, with time colour-coded from purple to red. Some platelets roll along the surface, creating rainbow-like patterns, while others are stationary and appear white. Events of platelet adhesion and desorption from the surface also occur, generating incomplete rainbows.

Figure 2 :

 2 Figure 2: Superposition of pictures from a video focused on the surface of the micro-channel.The colors from blue to red corresponds to the time, the scale is 20µm. Both stationary platelets (frame 1) and rolling platelets (frame 2) can thus be observed. One adhesion event is identified in (frame 3) as the first time this platelet is measured is in the middle of the experiment. Likewise a desorption event is seen (frame 4) as the platelet disappears from the surface before the end of the experiment.

Figure 3 :

 3 Figure 3: Interactions of platelets with the vWF-coated wall. Platelets in the volume are advected with a speed Vv and can bind with the vWF with a coefficient Kon. Adherent platelets can roll with a speed Vs and detach with a coefficient K of f .

Figure 4 :

 4 Figure4: The wall surface in green and the volume in orange respectively contain ns(x, t)and nv(x, t) platelets at the time t in the space between x and x + dx. The platelets in the lighter region enter these spaces between t and t + dt.

Figure 5 :

 5 Figure 5: Characteristic curves defined from the coupled equations (6) in the (t, x) space. Three sub-spaces are presented here: in yellow the sub-space defined by x > t, in blue the sub-space x < t and grey the sub-space between the two and where most of the interesting physics of the system resides.

Figure 7 :

 7 Figure7: A kinetics (resp. axial dependency) curve is built by intersecting a horizontal line with the (t, x) (resp. (x, t)) diagram. Significant variations occur when this line cross the grey region, connecting with the two asymptotic solutions at times (reap. positions) defined by the dashed lines.

)Figure 9 :

 9 Figure 9: Temporal evolution of the surface concentration of platelets at different positions in the channel: x = 0.6 mm (•) x = 1.6 mm ( ), x = 2.6 mm ( ), x = 3.6 mm ( ), x = 4.6 mm ( ), x = 5.6 mm (•), x = 6.6 mm ( ), x = 7.6 mm ( ). Solid lines represent numerical predictions obtained by neglecting diffusion.

)Figure 10 :

 10 Figure 10: Adsorption kinetics 7.6 mm away from the channel entrance. Experimental data are compared with numerical simulations obtained with a simple model neglecting (in red) or taking into account diffusion (in black).

  τ dif f τconv compares diffusion and convection time scales. Here, P e = Vvl 2 D DL , with l D the typical distance over which diffusion occurs in the transverse direction and L ∼ 10 -3 m the distance between two

)Figure 11 :

 11 Figure 11: Temporal evolution of platelet surface concentration at different positions in the channel: x = 0.6 mm (•) x = 1.6 mm ( ), x = 2.6 mm ( ), x = 3.6 mm ( ), x = 4.6 mm ( ), x = 5.6 mm (•), x = 6.6 mm ( ), x = 7.6 mm ( ). The experimental points with their error bar are compared with the model obtained when no desorption is considered (lines).

)Figure 12 :

 12 Figure12: Temporal evolution of platelet surface concentration at different positions in the channel: x = 0.6 mm (•) x = 1.6 mm ( ), x = 2.6 mm ( ), x = 3.6 mm ( ), x = 4.6 mm ( ), x = 5.6 mm (•), x = 6.6 mm ( ), x = 7.6 mm ( ). The experimental points with their error bar are compared with the model obtained when no rolling is considered (lines).

)Figure 13 :

 13 Figure 13: Temporal evolution of platelet surface concentration at different positions in the channel: x = 0.6 mm (•) x = 1.6 mm ( ), x = 2.6 mm ( ), x = 3.6 mm ( ), x = 4.6 mm ( ), x = 5.6 mm (•), x = 6.6 mm ( ), x = 7.6 mm ( ). The experimental points with their error bar are compared with the linear model (lines).

B 3. 8

 8 • 10 4 0.01 7.2 • 10 -3 1.4 • 10 -4 1.8 • 10 -10 1.1 • 10 -• 10 -4 2.5 • 10 -6 4.9 • 10 -4Table4: Parameters obtained by fitting data shown in Figure14with a model neglecting rolling.

Figure 14 :

 14 Figure 14: Adjustments of two extra experimental kinetics with the model neglecting rolling. The experimental parameters are listed in Table 3. (a) Curves obtained for sample B at positions x = 0.5 mm (•) x = 1 mm ( ), x = 1.5 mm ( ), x = 2 mm ( ), x = 2.5 mm ( ), x = 4 mm (•), x = 7 mm ( ), x = 38 mm ( ). (b) Curves obtained for sample C at positions x = 0.5 mm (•) x = 1 mm ( ), x = 1.5 mm ( ), x = 2 mm ( ), x = 3 mm ( ), x = 5 mm (•), x = 10 mm ( ), x = 30 mm ( ).

Table 1 :

 1 Extreme values used in the search process.

	10 3	1.7 10 -5	0	1.7 10 -6 10 -12	0	10 -7
	max 13.6 10 4	373	373	37.3	37.3	10 2	1

Table 2 :

 2 Parameters value for the different models tested they should not be directly compared with one another. The exchange rate J involves two terms, K on C v and K of f Cs h that are actually of the same order of magnitude when computed with the fitting parameters. It is thus necessary to keep the desorption term in the analysis.

Table 3 :

 3 f s Conditions corresponding to the experimental results. Three different batches of platelets were tested. We varied channel height H, platelet concentration in the suspension [P LT ] as well as wall shear rate γ. Sample A corresponds to data shown in Figures 9 to 13

	Exp	H	PLT source	[P LT ]	γ
		(µm) batch number (mL -1 ) (s -1 )
	A	23	1	1.4•10 8	1400
	B	23	2	2.8•10 8	1800
	C	63	3	1.4•10 8	1800

Acknowledgements

Work financed in part by ANR RPIB 2011-009 grant and Fondation Pierre-Gilles de Gennes. We wish to thank Dominique Baruch and Aurélie Magniez who prepared the platelets; Antoine Blin and Hyacinthe Buisson who performed preliminary experiments; and Olivier Dauchot for inspiring discussions.

saturation is no longer observed further away from the entrance (x > 4 mm).

For distant positions, experimental curves are concave while numerical ones are convex. Before trying to refine the model to better adjust the shape of experimental curves, let us discuss the order of magnitude obtained for the model parameters, presented in Table 2.

C s,max is defined as the threshold beyond which no new platelets can adsorb to the surface. This situation may occur when all the vWF binding sites are used, or if the platelet close packing limit is reached. As the surface is densely coated in vWF, we expect the second situation to be the limiting factor. Considering platelets as discs of average diameter 3.1 µm [START_REF] Frojmovic | Human platelet size, shape and related functions in health and disease[END_REF], the maximal 2D packing fraction is reached when C s,max 12 • 10 4 mm -2 . The value found with our algorithm 1.7 • 10 4 mm -2 is smaller than this estimation. This can be easily explained, as many platelets in our sample have a shape more complex than a discoid, and the packing of such objects cannot be as compact as that of discs. Also, in the vicinity of adherent platelets, perturbations of the local flow may prevent new platelets from being captured by the surface.

The adsorption rate of the model is taken as K on = 0.24 s -1 . This value is

Appendix A: Dimensionless equations

The goal is to render the following coupled equations dimensionless:

We then render time and position dimensionless using the characteristic speed V s and characteristic time 1/K on .

t ←-t K on x ←-x K on / V v which leads to the following coupled equations:

The dimensionless physical parameters appear directly in the above expression and we identify them:

This leads to the following system:


Differential use of Transitional Probabilities and Frequency in Statistical Learning of Pseudowords

The ability to learn transitional probabilities (probability of a stimulus given a preceding stimulus) and frequency of the sequence is central to language processing (Saffran, Newport, & Aslin, 1996;[START_REF] Pacton | An attention-based associative account of adjacent and nonadjacent dependency learning[END_REF][START_REF] Gebhart | Statistical learning of adjacent and nonadjacent dependencies among nonlinguistic sounds[END_REF]Thiessen, Kronstein, & Hufnagle, 2013).

These two factors (Transitional Probability and Frequency) have generally been studied separately in statistical learning of sequences, although they could interplay. For instance, a few studies that have addressed the combined effect of both transitional probabilities (TPs) and frequency have suggested a prevalence of TPs over frequency [START_REF] Endress | Transitional probabilities count more than frequency, but might not be used for memorization[END_REF], Mirman, Graf Estes & Magnusson, 2010, Perruchet & Poulin-Charronnat, 2012[START_REF] Lazartigues | Statistical learning of unbalanced exclusive-or temporal sequences in humans[END_REF]. Nevertheless, the role of the type of task on the use of these factors was not studied yet.

The present study investigated the role of transitional probabilities and frequency in pseudoword learning through two tasks.

Our hypothesis was that one or the other statistic can be used depending on the task, with a use of TPs in a task involving prediction and a use of frequency in a task involving recognition.

We selected only pseudowords allowing a dissociation of TPs and frequency.

Detection task

The results showed that a low-frequency pseudoword resulted in lower accuracy (resulting in an absence of drop in performance during the switch phase). Furthermore, for equal values of frequency high enough to benefit learning, the results showed an effect of first-order TPs on accuracy, with higher performance for higher TPs (but with no significant effect on RTs).

2AFC

A too low frequency appeared to hinder the learning of pseudowords.

Model comparison

In order to test the involvement of TPs and frequency in each task, an Akaike information criterion (AIC, [START_REF] Akaike | Factor analysis and aic[END_REF] was computed for three models for each task, including TPs or Frequency or TPs × Frequency. The AIC allowed us to determine which model best explained the results, with a lower AIC corresponding to a better model (more parsimonious).

For the correct answer rate in the detection task, the best model corresponded to the model including only TPs (AIC = -1098 versus AIC = -1113 for the models including frequency and no lower score for the model including both frequency and TPs).

Conversely, for the correct recognition rate in the 2AFC task, the best model corresponded to the one including only frequency (AIC = 179 versus AIC = 183 for the two other models).

Two tasks were used (detection task and 2AFC task) focusing on either online prediction or memorization and showed stronger effects of TPs or frequency, respectively.

The results suggested that both TP and frequency were learned during the experimental sessions but that participants did not use the same statistic in the two tasks. Thus, the information used by participants depends on the task at stake. The present study highlights the relationship between task properties and information used :

-The detection task allowed prediction of a stimulus from the preceding one. In that case, the TP was the information used by participants. When prediction can help online processing of sequentially presented stimuli, the TP appeared to be the relevant information about which stimulus will come next in the sequence.

-Conversely, the 2AFC task required offline recognition of pseudowords (triplets of syllables presented at the same time) and hence did not require predicting a stimulus from the preceding stimuli. In that case, frequency had a stronger effect on recognition than TP. When learning is tested offline, the frequency becomes more relevant than TP and influences recognition, as is the case in word reading [START_REF] Albrengues | Linguistic processes do not beat visuo-motor constraints, but they modulate where the eyes move regardless of word boundaries: Evidence against top-down word-based eye-movement control during reading[END_REF]. This differential use of TP or frequency is in line with Thiessen et al. ( 2013) view that frequency and TP are involved in different types of statistical learning mechanisms. Further, our results show that TPS and frequency are both learned, based on a single set of stimuli. But, the use of one or the other depends on the task, depending on whether it requires recognition or prediction. 30 participants completed the two tasks.

Detection task 16 pseudowords composed with three syllables were constructed with a manipulation of TPs and frequency. Participants had to detect two vowels appearing only in the last syllable (fully predictable based on TPs) of each pseudoword. The target vowels could be, for example, either "e" or "u", each associated with a specific key. Participants had to press the "q" key if they saw the "e" vowel and press the "m" key if they saw the "u" vowel (note that the target vowels were not the same for all participants). Participants were asked to press the key as fast as possible when the vowel appeared. Correct answer rates and response times were recorded. All pseudowords were presented in random order in each block. A block consisted of 80 trials and each participant completed five sessions of 10 blocks, one session per day for five days. A switch phase of 80 trials was added at the end of the last session (corresponding to a switch of the last stimuli to reset the TPs to zero in order to test participants' learning).

2AFC task

After the first task, a two-alternative forced-choice recognition task was conducted. Participants were asked to choose between the learned pseudowords, and lures constructed using the same syllables as in the learned pseudowords but in different combinations.

Detection task

2AFC

Method Figures Introduction

Results