Biomedical Engineering Reference
In-Depth Information
then system (3.2) has two steady state solutions. It also shows that if the stimula-
tion intensity does not exceed its threshold, then there is only one zero stationary
point which is stable. In this case coagulation does not take place. On the contrary,
if
α
>
α
0
, there exists a stable non-zero stationary point, while the zero point be-
comes unstable. In this case there is Fibrin production. Note that a decrease of
α
0
favors hypercoagulation, while hypocoagulation might be possible in the opposite
case.
Other mathematical models are concerned with certain important issues related
to the coagulation response. It is for instance the case of [6] were the threshold re-
sponse of a hierarchy of enzyme-catalyzed positive feedback loops and inhibition,
acting in a cascade, has been investigated and supported by numerical simulations.
More recently, we can refer to [42], where the authors have characterized the thresh-
old response of initiation of blood clotting to the size of a patch of stimulus, based on
experiments and numerical simulations. The threshold response follows a scaling re-
lationship based on the Damk oler number, which describes the competition between
reaction and diffusion of molecules. The influence of the flow on the threshold patch
size has not been considered in this work.
As already mentioned, blood flow has a major role in the regulation of hemostasis
and thrombosis, being one of the components of the famous Virchow's triad [93].
Ataullakhanov and co-workers developed in [85] a detailed quantitative mechanism-
driven mathematical model of (TF)-initiated thrombus formation in flowing plasma
(adapted from a previous model without flow presented in [64]). The model includes
28 partial differential equations describing biochemical reactions, diffusion and con-
vection of the reactants, with all initial concentrations and kinetic constants obtained
from experiments. It was demonstrated that blood flow can regulate clotting onset in
the model in a threshold-like manner, in agreement with existing experimental evi-
dence. This is due to a combination of the positive feedback of FVII activation with
chemical inhibition of extrinsic Tenase, and effective removal of FXa by flow from
the activating patch, depriving the feedback of “ignition”. This mechanism is con-
trolled by the activity of Tissue Factor Pathway Inhibitor (TFPI) (see also
Remark 2
,
Sect. 3.2.4).
The role of activated platelets
Models characterized by a large number of reaction-diffusion equations with the in-
clusion of a greater number of factors have been developed by several authors. A
typical example is the biochemical model proposed by Kuharsky and Fogelson [45]
that also incorporates the effects of hemodynamic forces on the transport of reac-
tants, catalysts and products. More precisely, it includes plasma-phase and surface-
bound enzymes and zymogens, coagulation inhibitors and activated and non acti-
vated platelets, as well as membrane-phase reactions, and in a simplified way it ac-
counts for chemical and cellular transport by flow and diffusion. The model assumes
that FVII and FVIIa compete for the TF binding sites on the subendothelium, that
FIX and FX compete for the FVIIa-TF complex on the subendothelium, and that
each pair FII/FIIa, FV/FVa, FVIII/FVIIIa and FX/FXa has distinct binding sites on
activated platelets for which each zymogen and enzyme compete. The kinetic re-
