Biomedical Engineering Reference
In-Depth Information
⎧
⎨
d
[
FVII
a
−
TF
]
dt
=[
FVII
a
−
TF
]
0
−
h
1
[
FVII
a
−
TF
][
TFPI
][
FX
a
]
k
14
[
FVIII
a
]
d
2
+[
FVIII
a
]
c
0
+
d
[
FX
a
]
dt
f
([
FII
a
])
1
+
f
([
FII
a
])
=
k
2
[
FVII
a
−
TF
]+
k
21
[
FIX
a
]
−
h
21
[
TFPI
][
FX
a
]
−
h
22
[
ATIII
][
FX
a
]
d
[
FIX
a
]
dt
h
3
[
=
k
3
[
FVII
a
−
TF
]
−
ATIII
][
FIX
a
]
(3.3)
⎩
[
FV
a
]
d
1
+[
FV
a
]
c
0
+
d
[
FII
a
]
dt
f
([
FII
a
])
1
+
f
([
FII
a
])
h
4
[
=
k
4
[
FX
a
]
−
ATIII
][
FII
a
]
d
[
FV
a
]
dt
=
k
5
[
FII
a
][
FV
]
−
h
5
[
FV
a
]
d
[
FVIII
a
]
dt
k
6
[
=
−
FII
a
][
FVIII
]
−
h
6
+[
FVIII
a
]
.
The model shows that the formation of FXa is the most complex. In fact, during
the initiation stage, FX is activated by the FVIIa-TF complex, and by the complex
FIXa-FVIIIa during the propagation stage. Moreover, FXa is inhibited both by An-
tithrombin III (ATIII) and by Tissue Factor Inhibitor (TFPI). By inhibiting FXa-TF
complex, FXa-TFPI can inhibit the formation of more FXa. Tissue Factor (TF) is
not formed but assumed to be exposed initially. The amount of TF is supposed to be
limited and forming complex FVII-TF, which quickly activates FVIIa. That is why
the equation for FVIIa-TF only shows the decrease of FVIIa-TF and is inhibited by
FXa-TFPI.
A stability linear analysis, similar to the one in [44] is considered. Asymptotic sta-
bility for a zero equilibrium point is shown. Zero is the only equilibrium point when
a certain quantity, “the strength of pro-coagulant stimulus”,
, is equal to zero. Oth-
erwise, there exists also a non zero equilibrium point, which is acceptable (positive)
if
β
β
>
β
0
, a threshold value.
The asymptotic stability leads to the conclusion that
must exceed the threshold
value for the coagulation cascade to be initiated. This value is related to the rate of
activation of platelets
c
and the binding constant
d
2
.For
β
β
<
β
0
numerical calcula-
tions show concentrations of FVa and FIIa approaching zero in time, leading to the
conclusion that, in this case, the system does not amplify and therefore coagulation
does not take place.
For the non-zero equilibrium point numerical simulations are performed. They
suggest that FVIII and FIX can contribute to the amplification stage of the cascade
only when
reaches the threshold level. Simulations also show that if FVIII is lack-
ing, blood coagulation does not occur since only small amounts of Thrombin are
produced. On the other hand, when the level of FIX is below 1 % of normal levels,
the rate of Thrombin generation reduces to a point that severe bleeding tendency
may occur, which confirms the great dependency of coagulation on FVIII and FIX.
β
Protein deficiencies and diseases
The hemostatic system is maintained in a state of permanent background activity.
This fact is confirmed by the presence of fragments of clotting factors split, as a
result of their activation. It should be noted that the concentrations of activation fac-
