MODELING OF BLOOD FLOW IN CAPILLARY VESSELS USING DISCRETE PARTICLESKrzysztof Boryczko1,2, Witold Dzwinel1, David A.Yuen2,

1Institute of Computer Science, AGH University of Technology, Mickiewicza 30, 30-059 Kraków, Poland, 2Minnesota Supercomputing Institute, University of Minnesota, MN 55455, Minneapolis, USA dzwinel@uci.agh.edu.pl, boryczko@uci.agh.edu.pl, davey@msi.umn.edu

Fluid particle with fibrin monomers

coarse graining of many molecules

Fibrin monomer

m= 1.02×10-20 kg

Fluid particle 150 nm

coarse graining of many plasma molecules m= 3.13×10-17 kg

Wall and red blood cells particles

F=k(r-r0)+FD+FB

Fluid particles

F=FC+FD+FB FC -repulsive

Fluid with fibrins

F=FC+FD+FB FC -repulsive

Fluid with fibrins

F=kf(r-r0f)+FD+FB FC

45 nm

MW 340 kD

Discrete-particle model of the microscopic blood regime. We show the principal components of the model made of particles, such ascapillary walls, plasma, red blood cells and fibrins. We display the character of interparticle forces acting between particles of various types.

DISCRETE PARTICLE MODEL AGGREGATION AND CLOTTING

PHYSICAL CONDITIONS MODELED

AbstractWe have devised a numerical model based on discrete particles for simulating blood

flow and clotting in capillaries with diameters comparable or less than the red blood cell (RBC) size [1,2]. The mesoscopic dynamics of blood is simulated in the microscopic capillary vessels about 100µm long and with diameters on order of 7-10µm. We assume that the plasma consists of fluid particles containing fibrin monomers, while the red blood cells and capillary walls are represented by elastic mesh of “solid” particles. The fluid particles are modeled by using fluid particle model (FPM) in which particles interact with each other with a short-ranged, repulsive dissipative force [3].

Erythrocyte deformability is a critical determinant of blood flow in the microcirculation. Therefore, we model short-time dynamics of RBCs in capillaries assuming different stiffness and shapes of the cells. Both biconcave discs -mimicking normal RBCs - and sickle cells – the cells characteristic for rare anemia– are considered. We show that both the red blood cells and walls undergo high stresses and strong deformations during the flow. The rheological properties of flow assuming various stiffness and shapes of RBCs differ considerably. Harder RBCs and sickle cells impede the flow more than normal biconcave cells.

We show also that the discrete-particles model of blood reflects clearly the aggregation properties of the red blood cells flowing in the capillary channels. There is a strong tendency to produce RBC clusters in capillaries. The choking points and other irregularities in geometry considerably increase the clotting effect. We discuss also other clotting factors coming from the fibrinogen. The polymerization of fibrin monomers into hydrated fibrins is modeled by the change of the interactions between fluid particles from repulsive to attractive forces. This process occurs with a probability being an increasing function of the local density. In [4] we show that due to the density fluctuations caused by high acceleration, the fibrin chains are produced within a very short time (0.5 ms). Fibrin aggregation modifies the rheological properties of blood, slows down the incipient flow, and entraps the red blood cells, thus forming dangerous clots. Modeling has been carried out with adequate resolution by using 1 to 10 million particles on multiprocessor systems [5,6]. Discrete particle simulations open a new pathway for modeling the dynamics of complex, viscoelastic fluids at the microscale where both liquid and solid phases are treated with discrete particles.

References

1. Dzwinel W, Boryczko K, Yuen DA, A Discrete-Particle Model of Blood Dynamics in Capillary Vessels , J Colloid Int Sci, 258/1, 163-173, 2003

2. Boryczko K, Dzwinel W, Yuen DA, Dynamical clustering of red blood cells in capillary vessels, J Mol. Modeling, 9, 16-33, 2003 (e-paper)

3. Español P. 1998. Fluid particle model. Physical Review E 57(3):2930-2948 1998

4. Boryczko K, Dzwinel W, Yuen DA, Modeling Fibrin Polymerization in Blood Flow with Discrete-Particles, Computer Models and Programs in Biomedicine, in press, March 2004.

5. Boryczko K, Dzwinel W, Yuen DA, Modeling mesoscopic heterogeneous fluids in irregular geometries on shared memory systems, Concurrency and Computation: Practice and Experience, submitted in February 2004

6. Boryczko K, Dzwinel W, Yuen DA, Clustering Revealed in High-Resolution Simulations and Visualization of Multi-Resolution Features in Fluid-Particle Models, Concurrency and Computation: Practice and Experience, 15, 101-116, 2003

The capillary walls are made of particles on springs. The endothelial layer of particles covering the internal surface of the capillary is modeled by using modified Lennard-Jones potential. The picture on the right shows deformations caused by blood flow.

A microscopic picture of red blood cell (left) and its 3-D discrete particle model (right). Only surface particles are visible

The bent deflection for the three RBC models: a) flat b) dual concavity and c) discs with holes under the same load. The lines on the surface of RBC reveal the nature of vector field associated with the maximum curvature.

The value of deflection response for the bending and stretching of flat and biconcave discs.

Tracing motion of every particle on the Power Wall at the University of Minnesota

Aggregation of RBC for free flow in the tube of 25µm diameter. More than 2 million particles were employed. Approximate shear is 100s-1. On the right, the image of real blood under much smaller shear conditions (5s-1) with a gap of 0.15 mm. This image was captured on the Linkam CSS450 by using whole blood of 40% haematocrit. Figure courtesy of R de Roeck and M.R.Mackley, Department of Chemical Engineering, University of Cambridge http://www.cheng.cam.ac.uk/~rmdr2/Nice_Pics.html.

This snapshot shows the formation of a clot consisting of the red blood cells (red plates) and fibrins close to the capillary neck placed on the right hand side of the figure. The fibrinogen, which is responsible for clot formation concentrates mainly in the stagnation points of the flow, producing fibrins entangling with the red blood cells. The fluid and wall particles are not shown in this figure. The figure on the left shows the realistic image from the scanning microscope.

Red blood cells flowing past the choking point for different viscosity of plasma suspension and the elasticity of blood cells. In Figs.a,b,c we present the magnitude of the velocity fields, the cross-sections through the vector velocity fields and the stream lines for the same moment of time. The figure on the left shows the realistic image from the scanning microscope.

Red blood cells flowing past the choking point for different shapes of blood cells. In the upper figure we present the flow of normal biconcave RBCs while in the bottom figure the RBCs has a sickle shape. The figure on the left shows the realistic image from the scanning microscope.