Biomedical Engineering Reference
In-Depth Information
the ratio
V
c
,where
V
c
is the typical speed of the clotting front. Thus its determination
passes through the study of the motion of the front where
A
A
crit
, which is pre-
cisely the clotting front. The latter is preceded by an activation front, characterized
by the equality
(
t
)=
σ
=
σ
thr
(the shear stress becoming critical for platelet activation).
Thus, from the mathematical point of view, the clot growth model is a problem with
two free boundaries, governed by the flow equation and by the Fibrin production
differential system.
For lack of space we cannot deal with the fluid dynamical problem, nor with the
study of the free boundaries, which is performed in great detail in [10] and which
is considerably complicated. We just mention that the final conclusion is that, in
the framework of the selected parameters,
t
clot
can be estimated to be 120
s
, thus
concluding that the whole scheme is self consistent.
3.5 Conclusions
The paper is very concise, due to space limitations. In the first part of the paper
we have reviewed the complex mechanisms regulating the various stages of blood
coagulation (primary hemostasis, secondary hemostasis, fibrinolysis). After a brief
historical section illustrating the main discoveries, we have described the many ele-
ments entering the process and the bleeding disorders caused by their deficiency
or dysfunctions. The recently formulated cell-based coagulation model has been il-
lustrated in detail, also explaining why it replaced the previous 3-pathway cascade
model, that has been used for over forty years, since 1964. Some space has been
devoted to important illnesses such as Deep Vein Thrombosis, Thrombocytopenia,
Disseminated Intravascular Coagulation, for which mathematical models have been
proposed.
In the second part we have illustrated a selection of mathematical models, em-
phasizing different aspects. We could not point out which model is the best. First of
all we have to say that the shift from the cascade to the cell-based model is rather
recent and it poses a new challenge to mathematicians. Moreover, we remark that
even models listing a “complete” set of reaction-diffusion equations miss to men-
tion some basic feature, like e.g. the role of vWF or the volumetric contribution to
thrombus growth by cells entrapment. In general the coupling of chemistry with me-
chanics is not frequently addressed. If the scenario depicted in most of the models
may be incomplete, on the contrary there is sometimes a too scrupulous attention
to chemical details, which can possibly be avoided since many reactions are very
fast. Actually, opposite trends are identifiable: there are attempts to include as much
as possible of the biochemistry (e.g. [23, 24, 85], while other models are extremely
concise in order to emphasize specific aspects (e.g. [32, 33, 98]). Finally, there are
phenomena not yet accounted for in mathematical models, like for instance the re-
cently ascertained production of TF by platelets ([17, 63]) and the influence of blood
slip at the vessel wall, contributing additional platelets to the coagulation site. Thus
we conclude that, despite the considerable amount of literature on the subject, there
is still much work to do in search of a mathematical model consistent with the mod-
