Investigating platelet motion towards vessel walls in the presence of red blood cells

Transcription

1 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

2 Blood is a Heterogeneous Medium 1 Plasma water-based solution contains electrolytes and proteins incompressible Newtonian uid 2 Red blood cells 40% of blood by volume highly deformable cells used for oxygen transport 3 Platelets 0.3% of blood by volume rigid elliptical cells repair damaged vessel wall

3 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?

4 Experiments: Eckstein et al This prole depends on hematocrit (% of red blood cells). 0% 15% 40% (x-axis is distance from wall, y-axis is concentration of platelets)

5 Experiments: Eckstein et al This prole also depends on shear rate. 250 sec sec sec sec 1 (x-axis is distance from wall, y-axis is concentration of platelets)

6 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.

7 Lattice Boltzmann Discretization The lattice Boltzmann equations govern the behavior of particle distribution functions f (x; e i ; t) that live at each lattice node x at time t. e i are the discretized velocity vectors on a lattice dened by: e i = (0; 0) for i = 0 (c cos((i 1)=4); c sin((i 1)=4)) for i = 1; 3; 5; 7 (cp 2 cos((i 1)=4); cp 2 sin((i 1)=4)) for i = 2; 4; 6; 8; where c = h t is the particle speed.

8 Lattice Boltzmann Equations The equations governing the particle distribution functions are: f i (x + t e i ; t + t ) f i (x; e i ; t) = {z } Advection 1 f i (x; t) f eq ((x; t); u(x; t)) i {z } Collision where f i (x; t) is shorthand for f (x; e i ; t), is the relaxation parameter and f eq is the equilibrium distribution.

9 Lattice Boltzmann Equilibrium Distribution The exact form of f eq i nine-velocity model it is: depends on lattice geometry. For our f eq (; u) = w i i 1 + e i u c 2 + (e i u) 2 s 2c 4 s u u where c s = c p 3 is the speed of sound and the weights, w i, are w i = 8 < : 4=9 for i = 0 1=9 for i = 1; 3; 5; 7 1=36 for i = 2; 4; 6; 8: 2c 2 s The macroscopic quantities (x; t) and u(x; t) are the density and macroscopic uid velocity.

10 Lattice Boltzmann Method The macroscopic quantities can be obtained by evaluating the hydrodynamic moments of f i (x; t). Fluid Density (x; t) = X f i (x; t) Momentum Density (x; t)u(x; t) = i X i f i (x; t)e i

11 Relation to Navier Stokes In the limit that kuk! 0 and h! 0; L c where L is the hydrodynamic length scale, the lattice Boltzmann equations approximate the Navier Stokes equations. The macroscopic quantities: p(x; t) = c 2 s = c2 and u(x; t) 3 from the lattice Boltzmann method are equivalent to the velocity and pressure of the Navier Stokes equations. In addition the viscosity is = c 2 1 s t : 2

12 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))

13 Immersed Boundary Method Typically, the uid is governed by the incompressible Navier Stokes equations: r u = u + u ru = rp + r 2 u + F f where the external force is imposed by the presence of the immersed boundary object (in our case a cellular membrane).

14 Immersed Boundary Method The force felt by uid is related to the force felt by the boundary: F f = Z F IB (x X(q; t))dx:

15 Immersed Boundary Method The force felt by uid is related to the force felt by the boundary: F f = Z F IB (x X(q; t))dx: The boundary moves with the uid (q; t) = u(x; t) Z u(x; t)(x X(q; t))d x:

16 Immersed Boundary Method The force felt by uid is related to the force felt by the boundary: F f = Z F IB (x X(q; t))dx: The boundary moves with the uid (q; t) = u(x; t) Z u(x; t)(x X(q; t))d x: We discretize the delta function: (x) = ^(x1) ^(x2); 1 where ^(x 4h i ) = 1 + cos x i 2h for jxi j 2h 0 for jx i j > : 2h

17 Lattice Boltzmann-Immersed Boundary Method We can include an external force into the lattice Boltzmann equations with the following additions: f i (x + e i t ; t + t) f i (x; t) = 1 (f i (x; t) f eq i (; v)) + t w i [ ( ) (Ff e i ) c 2 s + A : (e i e T i c 2 s I) 2c 4 s ] ; where v = u + tf f 2 and the matrix A is A = t + t (uf T f + F f u T ):

18 What is F f? Recall that the force on the membrane is distributed to the uid. F f = Z F IB (x X(q; t))dx: How do we model the behavior of cellular membranes that resist stretching and have a non-circular equilibrium shape? F (T t +

19 What is F f? F (T t + We use a one-dimensional version of the Skalak membrane law: where = ds ds T sk iso = G (2 1)(1 C 2 ( 2 + 1)); is the principal stretch ratio.

20 What is F f? F (T t + We use a one-dimensional version of the Skalak membrane law: T sk iso = G (2 1)(1 C 2 ( 2 + 1)); where = ds is the principal stretch ratio. Then we set the ds bending energy to be minimized at some equilibrium shape q = d dl [E B ((l) 0 (l))] ; where (l) is the instantaneous curvature and 0 (l) is the curvature of minimum energy.

21 Whole Blood Simulation Wall shear rate is 800 sec 1 and hematocrit is 40% now some movies!

22 Preliminary Results: Comparison to Eckstein's Data LB-IB Simulation Eckstein's experiment Number of Platelets Distance From Wall (µ m)

23 Platelet Movement Lateral movement of platelets is highly sensitive to the presence of red blood cells Distance from center (µ m) Distance from center (µ m) Time (msec) Time (msec) without RBCs with RBCs

24 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