Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 522

The nonlinear chemical reaction terms R i are mainly based on second order or
Michaelis-Menten kinetics. As an example we mention the reaction term R Ia in
the equation for fibrin 75 :

k 1 ŒIIaŒI
K 1M C ŒI

h 1 ŒPLAŒIa
H 1M C ŒIa

R Ia D (7.48)

The concentrations of thrombin (ŒIIa), fibrinogen (ŒI), fibrin (ŒIa), and
plasminogen ([PLA]) are used to evaluate the reaction term R Ia . The chemical
kinetics rates k 1 , h 1 and constants K 1M , H 1M are known (taken from [ 11 ]). The
values of the diffusion parameters D i and the exact form of the reaction terms
R i are given in [ 32 ], where the model has been for the first time implemented
and used in 3D simulations. The clot formation is initiated by flux boundary
conditions imposed within a small region on the internal surface of the vessel.
No-flux, homogeneous Neumann conditions are imposed elsewhere.
Coupling Strategy . The coupling between blood flow and biochemistry is based
on the fibrin concentration. It is described in detail elsewhere in this chapter. 76
The fluid viscosity 1 is multiplied by a non-dimensional factor Q 1 that locally

depends on fibrin concentration ŒIa.
Q 1 .ŒIa/ D min 1 C

C clot ŒIa;

1

(7.49)

In this case the values D 100 and C clot D 1;000nM were used. In practice,
when the fibrin concentration is sufficiently large such a high viscosity is
generated to immobilize the clot.
Numerical Methods . The system of governing equations is rather complex and
highly nonlinear. A semi-discretization approach is adopted to first discretize the
PDEs in space and then integrate in time the resulting system of ODEs. The
same discretization is employed for flow variables, viscoelastic stress tensor, and
concentrations in the biochemistry model.

We adopted a space discretization based on a simple central finite-volume
discretization on a structured grid with hexahedral cells. The multiblock grid
topology with wall-fitted cells was used. The viscous fluxes were also discretized
using finite volumes over diamond-shaped cells adjoint to primary control volumes
faces. See, e.g., [ 31 , 33 , 35 ] for more details.
The time integration was performed using a Runge-Kutta (RK) multistage
scheme. A specific advection-diffusion optimized RK method has been used to
reduce the computational cost. The basic idea behind this subclass of RK methods
is to split the space-discretization operator into its inviscid and viscous parts. The

75 The subscript Ia refers to the chemical notation for fibrin.
76 See p. 554 for details.