Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 542

tion aspects are naturally embedded in small-scale coagulation models dealing with
separate blood cells within the blood flow. Several examples can be found in micro-
scale ([ 123 ] using the MPS method, [ 193 ] with DPD method) or multi-scale models
([ 262 , 265 ] using coupled continuous and CPM models) discussed in previous
sections of this text. These methods are capable to distinguish between blood,
thrombus, and vessel wall points or particles. This allows to use different rheological
and structural models for each of these distinct components of biological system.
Much less common are the macroscopic continuum-based flow-structure interac-
tion multiphase coagulation models. A simple model of this type was developed and
tested in [ 249 , 250 ]and[ 251 ] solving a free boundary problem modeling thrombus
growth using the level set method. The mathematical model is composed of the
incompressible Navier-Stokes equations for blood flow description, coupled with
a single scalar transport equation for platelet concentration. The flow-structure
(thrombus) interaction is provided by tracking in time the spatial evolution of
thrombus and adjusting the blood/thrombus boundary. This moving boundary
problem is solved using the Level-SetMethod(LSM) . The flow model can be written
down as:

r u D 0 (7.101)

@ u

@t C u r u

Drp C u (7.102)

This system is solved 91

simultaneously with the platelet transport (advection-

diffusion) equation

@c
@t C u rc Dr.D rc/

(7.103)

Here c.x;t/ is the platelet concentration and D is the corresponding diffusion
coefficient. Considering wall-bounded flow, the platelet boundary condition can be
of Neumann-type:

k.s/c on clotting (injured) surface

D @c
@ n D

(7.104)

O

0 on impermeable (healthy) wall

Here the surface adhesion of platelets is described as a first order chemical reaction
with rate k.s/ depending on the local surface shear rate s D S n tj= with O

t,

S being the stress tensor and n the clotting surface tangent, resp. normal unit

vectors. When the volume growth of the thrombus is non-negligible, a level set
function .x;t/can be introduced to distinguish between fluid and structure (blood
and thrombus) regions. The function is initialized as a signed-distance 92 from the

O

91 Subject to appropriate initial and boundary conditions.
92 Being positive, e.g., in the fluid (blood) and negative in the solid (thrombus).