Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 512

to investigate platelet activation and blood damage in [ 259 ]. Platelets were also
investigated using an IB model in [ 215 ], where different versions of immersed
boundary representations of platelets are tested or as a part of a complex multiscale
model in [ 78 ]. Aggregation of RBCs under shear flow was studied using IB in [ 268 ]
or [ 16 ].
Discrete-ParticleMethods(DPM) . This subclass of Lagrangian tracking-based
methods is characterized by using both real and fictitious particles to describe
the whole fluid mechanics problem including the fluid, the immersed structure,
or domain boundaries. In these methods the mesh generation is unnecessary
for typical Eulerian fluid description. The motion of continuum is reconstructed
from the assembly behavior of discretized particles carrying the information
about physical quantities such as position, velocity, pressure, and density.
Many different implementations of DPM are used for specific applications
[ 37 , 66 , 135 , 269 ]. Only few examples will be mentioned here to demonstrate
the basic modeling principles and possible biomedical applications with special
focus on blood coagulation.
(a) Dissipative Particle Dynamics (DPD) . In this method the dissipative particles
represent mesoscopic portions (e.g., clusters of molecules) of a real fluid [ 108 ].
These fictitious coarse-grained particles interact with the surrounding particles
through elementary pair-wise forces. The particles motion is governed by the
Newton's second law. For a given set of N particles having mass m i , positions
r i and velocity v i , this can be written as:

dr i
dt D v i

i D 1;:::;N (7.17)

X

N

F ij C F ij C F ij C F i :

m i dv i

dt D

(7.18)

j D 1;j ¤ i

The three main interaction forces applied in DPD are the conservative force
F ij , the dissipative force F ij , and the random force F ij . The external force
F i represents, e.g., the contributions of the pressure gradient or gravity forces.
Let's denote by r ij D .r i r j / the vector connecting the particle j with the
particle i, r ij D .r ij r ij / 1=2 being the magnitude of this vector and

O

r ij D r ij =r ij
the corresponding unit vector. Similarly v ij D .v i v j / denotes the relative velocity
of particles i and j. The forces are assumed to act only within a spherical cut-off 65
region with the characteristic radius r c .

65 In a similar way as in Smoothed Particle Hydrodynamics (SPH) [ 146 , 175 ] where the interpolation
kernel is usually truncated to have a compact support. The SPH method differs significantly from
many other particle methods because the equations of motion for the fictitious particles in SPH are
derived directly from the partial differential equations of fluid mechanics by integration using an
interpolation kernel [ 66 ].