Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 528

120

0.95

0.9

0.85

90

0.75

0.6

0.4

60

0.2

-0.1

-0.3

30

-0.5

-0.7

-0.9

0

0 2 4 6 8 10

Axial distance

Fig. 7.18 Flow velocity reduction ı in the near-wall layer along the axial direction

coagulation as well as of the clot lysis.

@ŒC i
@t C u r

ŒC i D r .D i r ŒC i / C R i i D 1;:::;13 (7.51)

The specific form of the corresponding reaction terms R i is summarized below
in Eqs. ( 7.52 )-( 7.64 ). These equations can be grouped according to which coag-
ulation phase they describe. The initiation and amplification phases are modeled
explicitly using an equivalent chemical reaction ( 7.52 ). This equation for the
prothrombinase 78 concentration ŒW does not represent a chemical reaction
in the classical sense, but rather a virtual reaction synthesizing the positive
feedback loop prothrombinase-thrombin-prothrombinase in combination with
the thrombin production described below in ( 7.53 ).
R W D k W C P ŒIIa 1

ŒIIa
ŒIIa max

h 1W ŒAPCŒW h 2W ŒATIIIŒW
(7.52)
The rate constants in ( 7.52 ) have to be carefully adjusted to reproduce results
comparable with the experimentally observed behavior of the realistic biochem-
ical cascade.

78 We should clarify that the simplification introduced assigning a pivotal role to prothrombinase
and summarizing in one equation the complex process leading to its production makes sense only
in the framework of the normal physiological process. If we have to consider any type of pathology
referring, e.g., to a defective or missing factor, the model has to incorporate the equations involving
the dynamics related to that specific factor. In other words, the model is conceived in an elastic way,
adapting the number of equations to the complexity that needs to be taken into account.