Biomedical Engineering Reference
In-Depth Information
The platelets activation by shear stress was described introducing a stress thresh-
old
σ
thr
and supposing that activation depends on the exposure time to stress beyond
that threshold. This was the motivation for defining the activation number as follows
t
0
H
(
e
k
II
T
1
d
A
(
t
)=
II
T
−
σ
thr
)
σ
thr
−
τ
H
([
FI
a
]
−
[
FI
a
]
thr
)
(3.5)
1
3
is the Heaviside function,
T
where
H
=
T
−
(
tr
T
)
1
,and
k
is a positive number.
H
([
]
−
[
]
thr
)
The factor
keeps into account that a sufficient Fibrin concen-
tration is needed to produce clotting. Following [2], clot formation was supposed to
occur instantaneously when
A
FI
a
FI
a
(
t
)
exceeds a critical value
A
crit
.
is the consequence of the known chemical cas-
cade involving numerous steps. A key simplification introduced in [10] was to
single out Prothrombinase as pivotal element in the cascade. If the Prothrombi-
nase concentration
Of course the evolution of
[
FIa
]
is supposed to be known as a function of time, then the
only surviving section of the reaction-diffusion system involves just the concen-
trations
[
W
]
[
FI
]
,
[
FIa
]
,
[
FII
]
,
[
FIIa
]
, together with those of the fibrinolysis factors
[
tPA
]
,
[
PLA
]
,
[
L
2
AP
]
. The Prothrombinase concentration
[
W
]
just enters the two bal-
ance equations for
[
FII
]
,
[
FIIa
]
, describing the Thrombin production, with a reaction
k
2
[
W
][
FII
]
K
2
m
+[
term of the form
.
In [10] Prothrombinase is supposed to be formed in a scenario of
mild inflamma-
tion
of the blood vessel producing an increase of
FII
]
[
W
]
from 0 to some
[
W
]
∞
, which is
selected so to control the Thrombin production rate.
The following time scales are recognized to have an important role in the process:
t
f
=
ρ
R
2
R
2
D
(largest diffusive time),
t
chem
(the time scale of
Fibrin production),
t
clot
(the time scale of clot growth). The determination of these
time scales relies on Table 3.1.
With those data and with the rate constants taken from [2] we have
t
f
(fluid dynamics),
t
di f f
=
ˆ
η
=
33
s
,
10
−
3
nM
,then
t
chem
=
10
3
s
, and, assuming
10
4
s
. Thus we
t
di f f
=
1
.
3
×
[
W
]
∞
=
5
×
got the separation
t
f
1
s
) is compara-
tively short and plays no role. We remark that this is an abnormal situation, since
when clotting has the purpose of wound sealing it has to be fast. Nevertheless, as we
said, these conditions are imposed with the aim of keeping a simple geometry and can
be released if the latter is not required (clearly leading to a much more complicated
problem). The above chain of inequalities allows to justify the following approxi-
mation:
all concentrations in the biochemical system depend on time only, reducing
it to an ODE system
. If in addition we want that
the blood flow is quasi steady during
the whole process
, then we also need
t
f
t
di f f
t
chem
. The heart beat time scale (
≈
t
clot
. The time scale
t
clot
can be defined as
Table 3.1.
Parameters for determining time scales
ˆ
R
D
ρ
η
10
−
2
cm
10
−
7
cm
2
sec
−
1
10
3
gcm
−
3
10
−
2
2
×
3
×
1
.
2
×
Poise
