Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 513

Conservative force . This is the repulsive potential force acting between particles

i and j.

n a 1

r c

r ij

r ij r ij r c O

F ij .r ij / D

(7.19)

0 r ij >r c

Here a denotes the repulsive force coefficient corresponding to the maximum of
this force at distance r ij D r c .
Dissipative force .

Represents the friction force approximated by

F ij .r ij / D w D .r ij /.

r ij v ij / O r ij O (7.20)

where is the friction coefficient and w D stands for the dissipative force
amplitude weight function

n 1

r c 2k

r ij

r ij r c

w D .r ij / D (7.21)

0 r ij >r c

The exponentk D 1=2 is used in the standard DPD method, however other values
are possible (e.g., k D 0:25 see [ 222 ]) to modify the diffusivity of the method.
Random force . Based on the Brownian motion of particles, the random forces can
be expressed as

F ij .r ij / D w R .r ij / ij dt 1=2 r ij ; O (7.22)

where ij is a random variable with normal distribution, zero mean, and unit
variance satisfying the symmetry ij D ji . The random force coefficient is
linked to friction coefficient an d absolu te temperature T via the Boltzmann

constant k B by the relation D p 2k B T . The random force weight function is
in this case given by

n 1

r c k

r ij

r ij r c

w R .r ij / D (7.23)

0 r ij >r c

The random and dissipative forces form must satisfy the fluctuation-dissipation
theorem so that the DPD model preserves the equilibrium temperature. This leads
to the condition w D D w R which is satisfied in the above case. Parametrization for
external forces F i as well as for various additional structural (bonding, torsion,
adhesion, etc.) forces can be found in specialized literature.
The DPD method has been successfully used for simulations of blood coagula-
tion related phenomena. For details on the implementation of the DPD model of