Biomedical Engineering Reference
In-Depth Information
introduced. Moreover, a criterion to quantify platelets activation due to prolonged
exposure to shear stress was defined. Finally, the model also included convection-
reaction-diffusion equations to account for fibrinolysis.
Numerical results for a 3D simplified version of this model, where the viscoelas-
ticity of blood was not considered, can be found in [9]. Preliminary stability results
have been obtained for this clot model in quiescent plasma (see [84]). In particu-
lar, a continuum of equilibria has been found for the kinetics of some of the most
relevant Michaelis-Menten chemical reactions involved in the model and, using a
nontangency-based Lyapunov criterion, semistability has been proved.
This model undergoes some oversimplifications and is far from being complete.
In fact, biochemical factors have been partially included and, on the other hand,
both the blood and the clot were treated as homogeneized continua with a very sim-
ilar constitutive structure, which does not allow to account for the rheology of their
constituents. Nevertheless, the model provides a preliminary framework for the de-
velopment of more comprehensive mathematical models that could better capture
the relevant biochemical and mechanical processes of hemostasis and its regulation.
In Weller [98] a model has been proposed in which thrombus growth is described as
the result of the aggregation of activated platelets driven to the coagulation site by
diffusion.
A model with a platelets activation front and a clotting front
In [10] a clotting model was presented based on the idea, suggested in [2], that
platelets mechanical activation can be monitored by an activation number. Though
the general principles adopted in [10] can be applied to any geometry, the paper was
dealing with the special case of a clot growing in an arteriole of radius R in con-
ditions of cylindrical symmetry, allowing a thoroughly mathematical investigation.
The possibility of having an axisymmetric thrombus requires a neat separation of
the time scales of the various concurrent processes, and this is a delicate aspect ad-
dressed in [10]. Hemodynamics was assumed to be described by a shear-thinning
model in which the Cauchy stress tensor
T
is defined as
T
=
η
(
D
,
[
FI
a
])
D
where
D
is the shear rate tensor and
1
δ
ˆ
η
(
D
,
[
FI
a
])
D
=
ηη
∞
([
FI
a
])
+
(3.4)
(
+
α
II
D
)
n
1
is the shear dependent viscosity. In (3.4) ˆ
η
is a reference viscosity,
η
∞
is a dimen-
sionless increasing function of the Fibrin concentration
[
FIa
]
(see [72]),
δ
is a di-
2
√
tr
D
2
1
mensionless positive constant,
II
D
is a
positive constant (measured in
s
2
), and
n
is a positive exponent. A nontrivial pro-
perty of the constitutive law (3.4) is that the dependence on the second invariant may
not be monotone, with interesting consequences on the fluid dynamics, that will not
be dealt with here. The blood density
=
is the second invariant of
D
,
α
ρ
was taken constant.
