Understanding the effect of red blood cells on lateral platelet motion
L INDSAY C ROWL E RICKSON AND A ARON F OGELSON
Department of Mathematics, University of Utah
Platelets play an essential role in blood clotting; they adhere to damaged tissue and release chemicals that activate other platelets. Yet in order to adhere, platelets, which account for only 0.3% of the blood’s volume, must first come into contact with the injured
vessel wall. Fortunately, fluid dynamics has solved this problem. Under arterial and
arteriolar flow conditions, platelets have an enhanced concentration near blood vessel
walls. This non-uniform cell distribution depends on the fluid dynamics of blood as a
heterogeneous medium; no such effect occurs in a Newtonian fluid.
Although lateral platelet motion has been well documented, the fluid dynamics are
poorly understood. We use a parallelized lattice Boltzmann-immersed boundary
method to solve the flow dynamics of red cells and platelets in a 2D vessel with noslip boundary conditions. Using this method we analyze the influence of hematocrit
and shear rate on lateral platelet motion. Insight gained from this work could lead to
significant improvements to current models for platelet adhesion where the presence of
red blood cells is neglected due to computational intensity.
Lateral platelet motion
Blood is a heterogenous medium composed primarily of red
blood cells, plasma and platelets. Platelets have a non-uniform
concentration that is highly localized near vessel walls where
they repair vessel damage. Experiments by Eckstein et al. [3]
show that this concentration profile is highly sensitive to both
shear rate and hematocrit.
Immersed boundary-lattice Boltzmann method
Results
t Ff
fi(x + eit ; t + t ) fi(x; t) = fi(x; t) f ; u + 2 +
#
"
1 (Ff ei ) + (vFT + Ff vT ) : (3ei eT c2 I)
1
i
f
2
3t wi
c2
2c4
1
eq
i
X
i
fi(x; t) ,
(x; t)u(x; t) =
X
i
fi (x; t)ei:
In the limit that the lattice spacing approaches zero and the particle speed c approaches
infinity, the lattice Boltzmann equations approximate the Navier Stokes equations
where the fluid pressure and viscosity are
p(x; t) = (c2 =3) ,
Z
= (c2=3)t (
FIB (x X(q; t))dx , @@tX (q; t) =
0.1
0.1
0.05
0.05
10
= :
1 2)
Z
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0
0
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10
20% − WSR 400
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40% − WSR 400
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
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40
0
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Above are histograms of our model’s concentration of platelets as a function of lateral
distance. We see platelet near wall excess occur for high wall shear rates (WSR). Below
are sample platelet trajectories in which we see that diffusion is a function of lateral
position.
20% − WSR 1200
Diffusion of platelets
−6
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3
x 10
2.5
40
u(x; t)(x X(q; t))dx:
2
30
20
1.5
1
0.5
10
0
0
numerical data
smooth function
0
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−0.5
0
1
2
3
Lateral Position
4
5
−3
x 10
To model the behavior of one-dimensional cellular membranes, we use the following
force equation
@ (T t + d [E ((l) )] n):
FIB = @@lT = @l
0
dl B
where (l) is the instantaneous curvature and 0 is the curvature of minimum energy.
sk
2
Tiso
= G(
1)(1
where is the principal stretch ratio.
However, this model is phenomenological and does not explain how this motion occurs. We would like to develop a model that can accurately predict how platelets
behave in the presence of red blood cells without have to replicate the experimental
conditions.
0.15
0
0
where is the relaxation parameter, c = ht is the particle speed, and f eq is the equilibrium distribution. The macroscopic quantities (x; t) and u(x; t) are the density and
macroscopic fluid velocity. These quantities can be calculated from the particle distribution functions by
Tension is both tangent to plane (due to stretching) and normal to plane (due to bending). For in-plane tension we use a one-dimensional reduction of the Skalak membrane
law
@c = D @ 2 c @A(y)c
@t 2 @y2
@y
p
Y (t + dt) = Y (t) + A(Y (t))dt + (D)dW
0.2
0.15
t
Ff =
To simulate this effect in models of platelet adhesion, Eckstein et al. [2] suggested continuum and stochastic drift-diffusion models for platelet motion with constant diffusion
and a drift defined by the equilibrium platelet concentration taken from experiments
that pushes platelets toward the wall.
0.25
0.2
t
(x; t) =
40% − WSR 1200
0.25
The equations governing the particle distribution functions in the presence of an immersed boundary force are
The immersed boundary method is a way of coupling fluid dynamics to the mechanics
of some elastic boundary location within the fluid. Fluid lies on Eulerian grid x and the
boundary lives on a Lagrangian grid X(q; t). An external force is imposed on the fluid
by the presence of the immersed boundary object. The boundary moves with the fluid
velocity, and the force felt by fluid is related to the force felt by the boundary.
The figure above demonstrates the effect of shear rate on lateral platelet motion (left to
right: 250, 560, 800, 1220 s 1 ). The figure below shows that hematocrit also influences
lateral platelet motion (left to right: 0% 15% 40%).
20% − WSR 1200
The lattice Boltzmann equations govern the behavior of the fluid in
our simulations. These equations track particle distribution functions
f (x; ei; t) that live at each Eulerian lattice node x at time t. Each PDF
is allowed to move along a set of discretized velocity vectors, ei .
D(y) cm2/sec
Abstract
C2(2 + 1));
Conclusions and Future directions
Our preliminary results show that lateral platelet motion is caused by interactions with
red blood cells. We also have reason to believe that enhanced diffusion alone is not
enough to generate this motion; that a barrier may exist at the edge of the red blood cell
layer and this barrier effectively causes a net “drift” toward the vessel wall.
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In the future, we hope to:
create
stochastic differential equation model of platelet transport
characterize platelet motion around
Numerical experiments
a clot
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5
0
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Below are simulations of red blood cells and platelets under pressure-driven flow for
(a) 20% hematocrit and (b) 40% hematocrit.
References
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10
0
0
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[1] L. C ROWL AND A. F OGELSON, Computational model of whole blood exhibiting lateral
platelet motion induced by red blood cells, Comm. Numer. Meth. Engng, (Accepted).
[2] E.C. E CKSTEIN AND F. B ELGACEM, Model of platelet transport in flowing blood with
drift and diffusion terms, Biophysical Journal, 60 (1991).
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60
(b)
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100
[3] C. Y EH AND E.C. E CKSTEIN, Transient lateral transport of platelet-sized particles in
flowing blood suspensions, Biophysical Journal, 66 (1994).