Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 514

thrombosis, see, e.g., [ 75 , 233 ]. The influence of RBCs on platelets aggregation was
studied using DPD in [ 193 ] or more recently in [ 222 ].

(b) FluidParticleModel(FPM) . This method can be seen as an extension of the DPD
method, i.e., the interactions in the FPM are modeled by forces of a finite range.
In comparison with the DPD method the FPM allows the fluid particles to rotate
in space and also the interaction range for FPM is usually shorter due to a more
realistic interaction forcing model [ 65 ]. Some information about the theoretical
background of this method can be found, e.g., in [ 184 ]. The blood coagulation
related applications of this method can be found, e.g., in [ 36 ] simulating fibrin
aggregation and blood flow in capillaries. The RBCs aggregation in capillary
vessels was studied using FPM e.g., in [ 65 ].
(c) Moving ParticleSemi-implicit(MPS) method . This method has been introduced
in [ 135 ]. Its formulation is based on the Lagrangian form of the Navier-Stokes
equations for a viscous incompressible fluid

D
Dt D 0

(7.24)

D u

Dt D r

p C u C f: (7.25)

External forces (including inter- and intra-cellular bonding) are summed up in
f . The spatial gradients and Laplacians of quantities in MPS formalism are
approximated in a specific way. Let's consider a scalar quantity assigned to a
particle with index i. The gradient of this quantity is approximated by 66 :

X

N

dim
n 0

j min
r ij

r i D r ij w .r ij / O (7.26)

j D 1;j ¤ i

where dim is the spatial dimension of the considered model, i.e., dim D 2; 3 in
practical simulations. The initial (reference) particle number density is denoted
by n 0 .The min stands for the local discrete minimum of the quantity among
the particles surrounding the one with index i (within the radius of influence
r c ). The kernel (weighting) function w .r ij / is defined by the following formula

r c
r ij

1 r ij r c

n

w .r ij / D (7.27)

0 r ij >r c

with r c being the cut-off distance (radius of influence) assuring the compact sup-
port for particle interaction forces. Similarly, the Laplacian can be expressed as

66 We use the same notation of particles position vectors as in the description of the DPD method.