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This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology

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The transport of platelets in blood is commonly assumed to obey an advection-diffusion equation with a diffusion constant given by the so-called Zydney-Colton theory. Here we reconsider this hypothesis based on experimental observations and numerical simulations including a fully resolved suspension of red blood cells and platelets subject to a shear. We observe that the transport of platelets perpendicular to the flow can be characterized by a non-trivial distribution of velocities with and exponential decreasing bulk, followed by a power law tail. We conclude that such distribution of velocities leads to diffusion of platelets about two orders of magnitude higher than predicted by Zydney-Colton theory. We tested this distribution with a minimal stochastic model of platelets deposition to cover space and time scales similar to our experimental results, and confirm that the exponential-powerlaw distribution of velocities results in a coefficient of diffusion significantly larger than predicted by the Zydney-Colton theory.

Platelets, or thrombocytes, are an essential blood constituent, from a physiological and heamodynamical point of view. Their motion is mainly a consequence of mechanical and hydrodynamic interactions with deformable red blood cells and the plasma, which makes an accurate description of their transport challenging.

Among the blood constituents, platelets are the second most numerous cells in blood, after red blood cells, with a concentration of 150–450 × 10^{9}/

From a physical point of view, platelets are small rigid suspensions interacting with other, larger, deformable suspensions, the red blood cells. It is well recognized that the physics of such systems is rich and complicated

Adhesion and aggregation of platelets depend not only on their affinity with the endothelium or the deposition surface but also on their flow towards this surface. Platelets movement in the blood is affected by their interactions with red blood cells (

In a blood flow subject to a shear rate, platelets experience an enhanced random motion in the direction perpendicular to the flow. The accepted description of this process, the so-called Zydney-Colton theory (_{
PRP
} is the diffusivity of platelets in a platelet-rich plasma (i.e. without red blood cells) with a typical value of _{
RBC
} is the typical diameter of a red blood cell. For _{
ZC
} is

The present study was motivated by the determination of platelets adhesion and aggregation rates from ^{–8} ^{2}
^{−1} is needed. This value is more than two orders of magnitude larger than the value predicted by

The Zydney-Colton model has been extensively validated by numerous numerical studies in which red blood cells and platelets were resolved (

In this context, we have developed a high-fidelity numerical blood flow solver, Palabos-npFEM, described and validated in

Palabos-npFEM is a computational framework for the simulation of blood flow with fully resolved constituents. The software computes the movement and deformation of red blood cells and platelets, and the complex interaction between them. The tool combines the lattice Boltzmann solver Palabos for the simulation of blood plasma (fluid phase), a finite element method (FEM) solver for the resolution of blood cells (solid phase), and an immersed boundary method (IBM) for the coupling of the two phases. Palabos-npFEM provides, on top of a CPU-only version, the option to simulate the deformable bodies on Graphic Processing Units (GPUs), thus the code is tailored for the fastest supercomputers

In more details, the framework resolves blood cells like red blood cells and platelets individually (both trajectories and deformed state), including their detailed non-linear viscoelastic behavior and the complex interaction between them.

The fluid solver is based on the lattice Boltzmann method (LBM) and solves indirectly the weakly compressible Navier-Stokes equations. The solid solver is based on the nodal projective finite elements method (npFEM)

Collisions between blood particles, whether red blood cells or platelets, are implemented through a repulsive force acting as a spring, when the surfaces delimiting two particles are getting too close to each other. In the current study, we employ the same parameters as reported in

From the fully resolved simulations we have recorded the position _{
i
}(^{–5} _{
moy
}:

We find that the probability distribution function of the platelets absolute velocities _{0}, _{min} and _{min}, a velocity threshold separating the exponentially decreasing bulk of the distribution to its heavy tail, and

The values of _{0} and

These two equations for _{0} and _{min} and _{
moy
}. The solution is _{0} = 1015.24 and

Here we assume that

From the expression of

The resulting velocity is:

Of course, when generating a velocity from

For each stochastic simulation we used the following procedure. First, we set the initial positions _{
i
} (0) by placing randomly N platelets on a segment of length L. Then at each iteration the velocities _{
i
} were drawn from the distribution described previously and the platelets positions were updated accordingly 3. A platelet _{
i
} ≤ 0 is removed from the system (deposited platelet), while the boundary _{
i
} >

We have performed simulations of the system depicted in ^{−1} and an hematocrit of 35% as in the Impact-R experiments. We have analyzed platelets trajectories along the vertical direction, that is perpendicular to the flow. We consider these simulations for 1_{
mw
} so as to produce the desired shear flow. Although these simulations are significantly smaller in size and time than the actual Impact-R experiment, they require supercomputing capabilities.

_{
mw
} (velocity of the top moving wall). The system has periodic boundary conditions along the

In what follows, we focus on shear rate 100^{−1} as in the Impact-R experiments, and on the smallest system of size

A typical trajectory is shown in

_{
y
} (blue) and the variance of the velocity

From the values of _{
i
}(_{
i
} indicates an average over the platelets, in our case those that are still in the domain at time _{
MSD
} = MSD(

We consider platelets that remain in the domain of interest during all the simulation time to compute a first quantity _{
in
}. Further, we compute _{
in&out
} from the trajectories of all the platelets until they leave the domain.

We obtain, for the present small system, _{
in&out
} = 1.71 × 10^{−10}
^{2}
^{−1} and _{
in
} = 1.99 × 10^{−10}
^{2}
^{−1} (_{
ZC
}. However, as shown below, the determination of

^{–5} ^{–3} ^{−b
}, with

Our goal with this stochastic model is to generate representative platelets trajectories, much faster than with the fully resolved blood flow simulation, and in any geometry, including scales and time span similar to the Impact-R experiment. To do so, we still need to determine the mean time between two random changes of ^{−3}

The fact that VACF(^{−3}

From the value of Δ

A typical trajectory is obtained as

_{
val
} < 10^{–64}) between the analytical expression and the distribution of velocity of the stochastic model, and a correlation coefficient 0.995 (_{
val
} < 10^{–35}) between the analytical expression and the distribution of velocity of the fully resolved model.

^{−2}
_{
zc
}. It is also about 4 times larger than the value obtained from a direct measurement of MSD(

One can further use our stochastic model to simulate the number

It is important to notice that our results are sensitive to the values _{min} and _{
moy
} measured from the velocity distribution given in ^{−9}
^{2}
^{−1} and approximately 1300 deposited platelets. Our fully resolved simulation does not show such values for

Velocity distribution, number of deposited platelets and finite size effect of the diffusion coefficient for different values of _{
moy
} and _{min} _{
moy
} and _{min}.

We then investigate the effect of changing _{
moy
} and _{min}. For a sake of simplicity, and also to keep the distribution of velocity smooth, we multiply concordantly _{
moy
} and _{min} by values between 0.5 and 3 (see ^{−7}
^{−5}
_{
moy
} and _{min}. Interestingly, for the highest _{
moy
} we tested, we find a diffusion coefficient similar to the one obtained with ^{–9} ^{2}
^{−1}), but a number

This paper proposes a detailed analysis of the statistics of platelets velocities when subject to an imposed shear flow of red blood cells and plasma. Our motivation was to better understand the deviation between the transport properties and deposition of platelets predicted by the Zydney-Colton relation 1) and those inferred from our Impact-R experiments.

Using fully resolved blood flow numerical simulations, in which deformable red blood cells and platelets are in suspension in a shear rate flow created between two walls, we were able to reconstruct the probability distribution

From the simulation data, we found that _{
ZC
}, predicted by the Zydney-Colton theory. However, for small system sizes, the diffusion coefficient inferred from the evaluation of the MSD within the systems boundaries gives a value of the same order than predicted by the ZC relation. More

Overall our results go in the direction of the experimental observation about the flux of platelets in the Impact-R device, which requires a much larger diffusion coefficient than _{
ZC
} but still a diffusion constant at least one order of magnitude larger than what is found here. More analysis is still needed to clarify this discrepancy.

In view of the importance of a right characterization of platelets transport in clinical devices to correctly test platelets functionality, we hope that this study will stimulate more experimental and numerical work.

The datasets generated and analyzed for this study can be found on Zenodo:

CK, FR and BC contributed equally to this work. CK developed the code for the fully resolved blood Palabos-npFEM model and performed the simulations. FR analyzed the data and developed the code to analyze them, including the stochastic model. He contributed to the interpretation of the results, produced figures and wrote parts of the manuscript. JL contributed to the development of the Palabos-npFEM code and the interpretation of the results. RD contributed to the interpretation of the results. FD contributed to the experimental data acquisition. KZ brought the biomedical questions and knowledge, collected experimental data, interpreted the results and contributed to the design of the research and the writing of the paper. BC designed the research project proposed the interpretation of the results, contributed to the data analysis and wrote the first draft of the paper. All authors contributed to the writing and reading of the manuscript.

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 823712 (CompBioMed2 project). Furthermore, this work was supported by grants from the CHU Charleroi (Fonds pour la Chirurgie Cardiaque; Fonds de la Recherche Medicale en Hainaut) and by the Swiss PASC project “Virtual Physiological Blood: an HPC framework for blood flow simulations in vasculature and in medical devices”. Open access funding was provided by the University of Geneva.

We acknowledge support from 1) the Swiss National Supercomputing Center, project ID 323, Digital blood: a study of platelet transport in blood, 2) the National Supercomputing Center in the Netherlands (Surfsara, Cartesius supercomputer), and 3) the HPC Facilities of the University of Geneva (Baobab cluster). This work also got support from EUREKA Eurostar project 3DPlts.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.