Biomedical Engineering Reference
In-Depth Information
2.
Biochemistry and Flow models
It is evident that the biochemical reactions of blood coagulation are in physiological
situations strongly affected by the flow of blood. Most of the chemicals and
particulate matter necessary to build up the clots (or thrombus) are supplied by the
flowing blood. On the other hand, the flow itself is finally affected by the presence
and growth of the clot. Flow-induced shear stress (to which von Willebrand factor
is sensitive) is also known to affect the initiation of clotting as well as the structure
and mechanical properties of the resulting clot. The flow variables or fields can be
incorporated into the coagulation models in several ways.
(a)
Parametric
. A characteristic flow velocity or shear rate is used in some of the
model parameters to represent the effects of flow on blood coagulation. This
is typically only one way coupling, i.e., the coagulation has no direct effect
on the flow field that is usually considered as fixed (i.e., a priori prescribed).
This simplest approach was used, e.g., in the model published in [
137
]. This
is a rather complex, spatially homogeneous model of surface-mediated control
of blood coagulation described by reaction rate equations. It consists of 59
coupled ODEs and besides the chemistry it also includes the role of binding
site densities and platelet deposition. This model is coupled to flow via variable
(flow dependent) chemistry kinetics-like coefficients
89
:
V
2
D
R
2
L
2
1=3
3
4
k
flow
c
D
(7.93)
This coefficient depends on flow velocity V , molecular diffusion coefficient
D, vessel size (radius) R, and injury length L. It is derived from steady state
advection-diffusion problem assuming a parabolic velocity profile and chemical
boundary layer over a finite-length surface source. Using this coefficient a first
order kinetics source/sink term is built
Ǚ k
flo
c
.C C
out
/
where C is the local concentration and C
out
is the concentration far from the
injury. This extra term is used to take into account the combined effect of
advection and diffusion on the chemical transport to/from the coagulation site.
The flow parameters play also an important role in the diffusion processes
during coagulation. Quite common is to consider shear-dependent diffusion
coefficients. A correlation for the enhancement of diffusion depending on the
characteristic shear rate and hematocrit in concentrated erythrocyte suspensions
was described, e.g., in [
245
]. Another such shear rate-dependent diffusion
89
See [
80
] for details of derivation and use.
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