Investigating platelet motion towards vessel
walls in the presence of red blood cells
(Complex Fluids in Biological Systems)
Lindsay Crowl and Aaron Fogelson
Department of Mathematics
University of Utah
SIAM Conference on the Life Sciences
August 5th, 2008
Blood is a Heterogeneous Medium
1
2
3
Plasma
water-based solution
contains electrolytes and proteins
incompressible Newtonian uid
40% of blood by volume
highly deformable cells
used for oxygen transport
0.3% of blood by volume
rigid elliptical cells
repair damaged vessel wall
Red blood cells
Platelets
Platelets
The main function of platelets is to repair damage to vessels...
... but platelet concentration in
blood is low.
Fortunately, platelets are not
uniformly distributed in blood
but are highly concentrated
near vessel walls.
How does this occur? What factors aect the distribution of
platelets in blood?
Experiments: Eckstein et al. 1994
This prole depends on hematocrit (% of red blood cells).
0%
15%
40%
(x-axis is distance from wall, y-axis is concentration of platelets)
Experiments: Eckstein et al. 1994
This prole also depends on shear rate.
250 sec
1
560 sec
1
800 sec
1
1220 sec
1
(x-axis is distance from wall, y-axis is concentration of platelets)
Modeling Blood
We want to simulate a large number of red blood cells and
platelets in a blood vessel over a considerable amount of time.
This is computationally expensive.
We use a parallel version of an immersed boundary-lattice
Boltzmann method to solve for the coupled system of
uid-lled cells immersed in a uid of similar density.
Lattice Boltzmann Discretization
The lattice Boltzmann equations govern the behavior of particle
distribution functions f (x; ei ; t ) that live at each lattice node x at
time t . ei are the discretized velocity vectors on a lattice dened
by:


(0; 0)
for i = 0
for i = 1; 3; 5; 7
(c 2cos((i 1)=4); c 2sin((i 1)=4)) for i = 2; 4; 6; 8;
ei =  (c pcos((i 1)=4); c sin(p(i 1)=4))
where c = ht is the particle speed.
Lattice Boltzmann Equations
The equations governing the particle distribution functions are:
fi (x + t ei ; t + t ) fi (x; ei ; t ) =
|
{z
Advection
}
1
| fi (x; t ) fieq ((x; t ); u(x; t ))
{z
Collision
where fi (x; t ) is shorthand for f (x; ei ; t ), is the relaxation
parameter and f eq is the equilibrium distribution.
}
Lattice Boltzmann Equilibrium Distribution
The exact form of fieq depends on lattice geometry. For our
nine-velocity model it is:
e u
(e u)2 u u
fieq (; u) = wi 1 + i 2 + i 4
cs
2cs
2cs2
where cs = pc3 is the speed of sound and the weights, wi , are
wi =
8
<
:
4=9 for i = 0
1=9 for i = 1; 3; 5; 7
1=36 for i = 2; 4; 6; 8:
The macroscopic quantities (x; t ) and u(x; t ) are the density and
macroscopic uid velocity.
Lattice Boltzmann Method
The macroscopic quantities can be obtained by evaluating the
hydrodynamic moments of fi (x; t ).
Fluid Density
(x; t )
=
X
i
fi (x; t )
Momentum Density
(x; t )u(x; t )
=
X
i
fi (x; t )ei
Relation to Navier Stokes
In the limit that
kuk ! 0 and h ! 0;
c
L
where L is the hydrodynamic length scale, the lattice Boltzmann
equations approximate the Navier Stokes equations. The
macroscopic quantities:
p (x; t ) = cs2 =
c2
3
and u(x; t )
from the lattice Boltzmann method are equivalent to the velocity
and pressure of the Navier Stokes equations. In addition the
viscosity is
= cs t
2
1
2
:
Modeling Cells: Immersed Boundary Method
How can we couple the background uid dynamics to a cellular
membrane submersed in the uid?
Fluid lies on Eulerian grid (x)
Boundary lives on Lagrangian
grid (X(q ; t ))
Immersed Boundary Method
Typically, the uid is governed by the incompressible Navier Stokes
equations:
r u
u + u ru
@t
@
= 0
=
rp + r2u + Ff ;
where the external force is imposed by the presence of the
immersed boundary object (in our case a cellular membrane).
Immersed Boundary Method
The force felt by uid is related to the force felt by the
boundary:
Ff
=
Z
FIB (x
X(q ; t ))d x:
Immersed Boundary Method
The force felt by uid is related to the force felt by the
boundary:
Ff
=
Z
FIB (x
X(q ; t ))d x:
The boundary moves with the uid velocity:
@X
@t
(q ; t ) = u(X; t ) =
Z
u(x; t ) (x
X(q ; t ))d x:
Immersed Boundary Method
The force felt by uid is related to the force felt by the
boundary:
Ff
X(q ; t ))d x:
FIB (x
The boundary moves with the uid velocity:
@X
@t
=
Z
(q ; t ) = u(X; t ) =
Z
u(x; t ) (x
X(q ; t ))d x:
We discretize the delta function:
(x)
where ^(xi ) =
1
4h
0
= ^(x1 ) ^(x2 );
1 + cos
xi
2h
for jxi j 2h
for jxi j > 2h
:
Lattice Boltzmann-Immersed Boundary Method
We can include an external force into the lattice Boltzmann
equations with the following additions:
fi
(x + ei t ; t + t )
fi
(x; t ) = 1 (fi (x; t )
"
1
w
t i
( ))+
( ) + A : (ei eTi
2cs4
eq ; v
1
Ff ei
2
cs2
fi
where v = u + t2Ff and the matrix A is
A=
t + (uFTf
t + Ff uT ):
cs2
I)
#
;
What is Ff ?
Recall that the force on the membrane is distributed to the uid.
Ff
=
Z
X(q ; t ))d x:
FIB (x
How do we model the behavior of cellular membranes that resist
stretching and have a non-circular equilibrium shape?
FIB
=
@T
@l
=
@
@l
(T t + q n):
What is Ff ?
FIB
=
@T
@l
=
@
@l
(T t + q n):
We use a one-dimensional version of the Skalak membrane law:
sk = G (2 1)(1 C 2 (2 + 1));
Tiso
ds is the principal stretch ratio.
where = dS
What is Ff ?
FIB
=
@T
@l
=
@
@l
(T t + q n):
We use a one-dimensional version of the Skalak membrane law:
sk = G (2 1)(1 C 2 (2 + 1));
Tiso
ds is the principal stretch ratio. Then we set the
where = dS
bending energy to be minimized at some equilibrium shape
q=
d
[E ((l )
dl B
0 (l ))] ;
where (l ) is the instantaneous curvature and 0 (l ) is the
curvature of minimum energy.
Whole Blood Simulation
Wall shear rate is 800 sec
1
and hematocrit is 40%.
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
50
40
30
20
.... now some movies!
10
0
0
10
20
30
40
50
60
70
80
90
Preliminary Results: Comparison to Eckstein's Data
Eckstein's experiment
LB-IB Simulation
30
Number of Platelets
25
20
15
10
5
0
0
5
10
15
Distance From Wall (µ m)
20
25
Platelet Movement
25
25
20
20
Distance from center (µ m)
Distance from center (µ m)
Lateral movement of platelets is highly sensitive to the presence of
red blood cells.
15
10
5
0
15
10
5
0
100
200
300
400
500
600
700
Time (msec)
without RBCs
800
900
1000
0
0
100
200
300
400
Time (msec)
with RBCs
500
600
Future Directions
Speed up code (OpenMP/MPI?)
Vary parameters
Hematocrit
Deformability of RBCs
Shear rate
Platelet shape and size
Vessel diameter
Understand why lateral platelet motion occurs
Find a way to incorporate the eect of red blood cells on
platelets into coagulation models