Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 517

LBM simulates the flow using the evolution of fictitious microscopic particles
living on a set of discrete lattice nodes. Their dynamics depend only on
interactions between particles in neighboring lattice points. The LBM tracks
particle distribution functions at points of a lattice, instead of tracking discrete
particles. The LBM can also be seen as a discrete approximate solution method
for a specific form of the continuous Boltzmann equation known in statistical
mechanics [ 54 ].

The fictitious particles in LBM only live in lattice nodes, thus only discrete
particle velocity (shift) vectors are allowed. These discrete velocities are denoted by
e i . The spatio-temporal distribution of the particles having velocity e i is described
by a scalar function f i .x;t/. The evolution in time of these particle distribution
functions is governed by:

f i .x;t/ f e i .; u /

1

f i .x C ıte i ;tC ıt/ D f i .x;t/ (7.34)

Here ıt is the time-step, is the relaxation time. The equilibrium distribution
f e i .; u / depends on the macroscopic fluid velocity u , on the density , and of
course on the structure of the computational lattice, resp. on the discrete velocities
e i , see, e.g., [ 54 , 258 ]. The macroscopic density .x;t/and the velocity u .x;t/at
point x and time t can be evaluated using the following simple relations:
.x;t/D X
i

f.x;e i ;t/ (7.35)

.x;t/ u .x;t/D X
i

f.x;e i ;t/e i (7.36)

A more detailed description of the LBM method can be found, e.g., in [ 197 ]. In most
cases the LBM is only used for (blood) flow simulations. Some applications also
include coagulation related phenomena. For example, the platelet motion induced
by RBCs was simulated in [ 54 ]. A study of fully resolved blood flow (including
blood cells representation with IB method) through aneurysmal vessels using LBM
was shown in [ 180 ]. Red blood cell aggregation and dissociation in shear flows is
simulated by the lattice Boltzmann method in [ 278 ]. An interesting extension of
LBM for deformable particles and flexible fibers was used in [ 257 , 258 ]. A simple
LBM-based model of thrombosis in intracranial aneurysms was used in [ 187 ]. A
clotting initiation mechanism based on particleresidencetime was adopted in [ 101 ]
and [ 27 ]. A blood damage model using the LBM approach was developed in [ 172 ].
Statistical methods for simulations in (bio)chemistry are much less common. An
important contribution in this field was provided in [ 89 , 90 ] where a model based
on the exact stochastic simulation of coupled chemical reactions was introduced for
well-stirred chemical systems. 69

A very comprehensive review of these stochastic

69 See [ 93 ] for the relation between this stochastic approach and continuous deterministic reaction
rate equations.