Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 547

negligible inertia a simplified flow model can be built starting from the Navier-
Stokes equations. The resulting approximate model is known as the Darcy's law .In
its generic form it leads to a velocity field inside the clot defined as

k
rp

u D (7.116)

The permeability k is related to the clot porosity and to the diameter of fibrin fibers
D f by

D f
64.1 / 3=2 .1:0 56.1 / 3 /

k D (7.117)

This model was used in [ 264 , 265 ] as a component of a multi-scale model of clot
formation. A similar model with only slightly different coefficients was also used in
[ 60 ]. The same kind of permeability formulae, but with Brinkmann model instead
of the Darcy model was used, e.g., in [ 130 ] to study fibrin networks regulating
protein transport during thrombus 96 development. The Brinkmann model adds an
extra Laplacian-like term to Darcy's law ( 7.116 ) in low porosity (permeable external
clot) regions where viscous dissipation of momentum is no more negligible.
Generalized-Newtonian models are the most common non-Newtonian models
used to describe the macroscopic behavior of blood flow. They capture its shear-
thinning behavior by introducing a variable (local shear rate dependent) apparent
viscosity instead of the classical constant viscosity used for Newtonian fluids. This
type of non-Newtonian models was used in [ 32 ] for both, blood and clot in a
complex coagulation model adapted from [ 8 , 11 ]. The basic principle is very simple.
The concentration of one of the coagulation products, e.g., fibrin in the described
case, is used as a tracer (indicator) to distinguish between blood and clot regions.
The fluid (blood/clot) viscosity is adjusted according to this tracer concentration.
The resulting model equations can be written in the form generalized Navier-Stokes
equations :
@ u

@t C u r u

h

Q .ŒIa/. P / r u Cr u T i

Drp Cr (7.118)

The shear-rate 97 dependent (shear-thinning) fluid viscosity .P / can be defined
using generalized Cross model :

1
1 C . P / b a . P / D 1 C . 0 1 / (7.119)

96 The thrombus model employs a composite structure with an impermeable core (activated
platelets and fibrin) and a permeable sh ell (fibr in cap).
97 The shear-rate is defined as

D 2 p D W D where D D . r u Cr u T /=2 is the symmetric part P

of velocity gradient.