Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 516

where again k ad is the adhesion elastic (spring) force coefficient and r ad

ij is the natural

length of the spring.
Further details on the platelets and RBCs models within the MPS method
framework can be found, e.g., in [ 123 ]or[ 124 ], where the mechanical interaction
between a thrombus and red blood cells was studied. A more detailed MPS-based
model of platelets adhesion dynamics under shear flow taking into account various
receptors bonding forces was recently published in [ 125 ]. A specific problem of
malaria infected RBCs flow in capillaries was studied using an MPS model in [ 116 ]
or [ 134 ].

CellularPottsModel(CPM) . The CPM is a cell-level lattice model based on
energy-minimization 68 following the ideas of [ 95 ]. The effective energy E of
the system sums up the true energies, like the cell-to-cell adhesion associated
energy, with other energy-like contributions, e.g., the effects of blood flow on
cell or virtual energies arising from dimensions constraints and chemotaxis (see,
e.g., [ 262 , 263 ]).

E D E Adhesion C E Flow C E Dim C E Chem (7.33)

From an effective energy the resulting cell motion can be calculated using
algorithms based on the Monte-Carlo Boltzmann acceptance rule. The CPM uses
integer indices defined on a lattice to describe cells. The value of the index (kind
of marker) at a specific lattice site (in the position .i;j;k/ within the cartesian
lattice) is equal to C if the site lies in the cell C. The sets of lattice sites having the
same index represent cells. The cell is thus treated as a set of discrete sub-cells
that can rearrange to form a cell motion or shape changes.
The CPM is capable to predict, e.g., microscopic cell motion, aggregation and
interaction of cells, their adhesion, differentiation or division. The CPM was used in
[ 45 - 47 ] as a component of a multiscale model in three-dimensional morphogenesis
simulations. Thrombus development was studied using CPM in [ 262 , 263 ].

3. Mesoscale Models
The mesoscale in this context refers to models at the sub-continuum scale .They
are classified in this group mainly due to their common modeling principles based
on statistical methods, rather than to their spatial resolution. The particulate matter
in blood is treated in ensembles, described, e.g., using particle probability densities
and their spatial integrals. This statistical approach requires spatial control volumes
being much larger than typical particle sizes or inter-particular distances. The
mesoscale methods typically use information about microscopic particle interac-
tions to provide information about macroscopic quantities.

LatticeBoltzmannMethod(LBM) . This method is often used as a discretization
tool for numerical simulations of Navier-Stokes like models in fluid mechanics.

68 It uses similar principles as the Monte Carlo method in sub-microscale models.