Biomedical Engineering Reference
In-Depth Information
As previously mentioned, blood coagulation is a very complex process. It is there-
fore not reasonable to expect to have a mathematical model that “fits all” the features
of clot formation and lysis. Indeed, the important influence of shear stress (both on
platelets activation and on the mechanical interaction of the blood flow with the clot)
should be captured by the model which is therefore strongly dependent on the en-
vironmental conditions in which coagulation takes place. The path for an overall
understanding of the whole process seems to be constructing mathematical models
which look upon a particular phase or aspects of the process, along with numerical
simulations. For example, the role of Tissue Factor in the initiation of blood coagu-
lation, or that of other activators, or inhibitors, from a threshold point of view; or, the
main step leading to clot formation, and how much Thrombin is formed; existence
and control of the several feedbacks in the chain of chemical reactions (positive and
negative); existence of different time scales that, in some cases, allow for a reduc-
tion in the size or in the nature of the differential system; or the important role of
platelets in the whole process, how they aggregate and their behaviour under shear
flow conditions. Bleeding disorders or deficiencies leading to diseases can also be
taken into account in a mathematical model. Moreover, blood rheology and the spe-
cific flow regime for blood clotting cannot be neglected. These are the main issues
that will now be considered, while giving a short overview of some relevant math-
ematical models of blood coagulation proposed in the literature. It is important to
notice that there remains a pressing need for further experimental data to validate
these models and to develop new ones in order to better understand the interplay
between biochemical and mechanical factors under different flow conditions found
in the human vasculature.
3.4.1 Early models of blood coagulation
On the wake of the celebrated
cascade
[48], or
waterfall
[18] models, already in 1966
Levine [47] proposed one of the first mathematical models for blood coagulation
based on a series of enzymatic reactions leading to the formation of Fibrin, but where
no feedbacks were considered. The process was modelled by a linear system or first
order ODEs taking the form
⎧
⎨
dy
1
a
dt
=
k
1
y
1
[
U
(
t
)
−
U
(
t
−
a
)]
−
K
1
y
1
a
dy
2
a
dt
=
k
2
y
2
y
1
a
−
K
2
y
2
a
(3.1)
⎩
.
dy
N
a
dt
=
−
k
N
y
N
y
(
N
−
1
)
a
−
K
N
y
N
a
where
y
i
a
denotes the concentration of the activated form of
y
i
,and
k
i
and
K
i
are
rate reaction constants. It was assumed that the first reactant
y
1
was activated over a
time interval of amplitude
a
, after which the stimulus was terminated. The delay was
modelled through the unit step function
U
(
t
)
so that
U
(
t
)
−
U
(
t
−
a
)
was considered
