Biomedical Engineering Reference
In-Depth Information
tors in the background state are extremely low and, therefore, experimental studies
become very difficult. In [57] a system of ODEs (no diffusion) was considered to
model the background state of blood coagulation. In these equations rates of enzy-
matic reactions were described by Michaelis-Menten approximations, whereas rates
of interaction of enzymes with inhibitors were second order kinetic equations. Be-
sides diffusion, the model also assumed the absence of platelets. Using this model the
effects of deficiencies of factors II, V, VII, VIII, IX and X on the background state
of hemostasis are also studied. Numerical simulations have shown that a decrease
in concentration of FVII inhibited the activation of factors II, IX, X and protein C,
whereas a decrease in the concentrations of factors VIII and IX had an insignificant
effect on activation of factors II and X or protein C. These results suggest that only
the extrinsic pathway of the background state is active, whereas FIX remains almost
inactive, which is consistent with the fact that, in patients with hemophilia B (FIX de-
ficiency) the level of FX activation is similar to that of healthy people. In our opinion
these conclusions based on numerical results should be treated with caution, since
the Hemophilia A and Hemophilia B are considered as the most important bleeding
disorders (Sect. 3.3.1).
We referred to the Hageman Factor (FXII) deficiency at the end of Sect. 3.3.1.
It is known that this deficiency has almost no effect on blood coagulation, see [58].
This paper deals with a first order nonlinear ODE system to study the conditions
of FXII deficiency. Unlike other models, here the kinetic rate constants are taken
from experimental studies only. Three positive feedbacks are considered: Throm-
bin activation of FV, FVIII and FXI. Solving the system by Runge-Kutta methods,
numerical simulations show that the amount of Thrombin first rises, after activa-
tion, and then declines to background level zero. This is because blood coagulation
is activated in the presence of background concentrations of initiation factors, pro-
vided that activated platelets are also present. Numerical simulations also show that
about 200-fold decrease in the concentrations of FXIIa has no effect on the maximal
Thrombin concentration but, on the contrary, a similar decrease on concentrations
of FVIII or FIX implies a significant decrease on Thrombin concentration. As a con-
sequence hemorrhage will take place.
Besides the conclusion that the absence of FXIIa is irrelevant for blood coa-
gulation, simulations also show that when this deficiency occurs, the time interval
required for Thrombin to reach its maximal concentration increases by a factor two
(in the case of a 200-fold decrease in the concentration of FXIIa) or even by a fac-
tor three in the absence of activated platelets which, therefore, can be considered
as responsible for the insensitivity of the blood coagulation system to the Hageman
deficiency. The same conclusion applies to the positive feedback activating FXI. As
already referred, this model does not take into account the effects of diffusion or
blood flow. It may simulate Thrombin production in a closed cavity (hematoma).
Within the context of models accounting for protein deficiencies and diseases we
refer to [90]. This paper addresses spatial aspects of clotting as a function of Factor
IX deficiency (hemophilia B). It has been experimentally observed that clot forma-
tion and growth in hemophilia B plasma is substantially affected by the different
levels of FIX deficiency. A detailed mathematical model consisting of a system of
