Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 537

boundary conditions simulating healthy/injured vessel wall. According to [ 273 ], the
activation stimulus is provided via an extra diffusion equation for factor XIa:

@t D k 11 Cr D XIa rŒXIa

@ŒXIa

(7.90)

supplemented by boundary conditions

n

A on the clotting surface

@ŒIXa
@ n D

(7.91)

O

0 elsewhere

where n is the boundary surface outer normal unit vector and the number (or

function) A is the prescribed activation.
The diffusion-reaction (without advection) models are usually studied to clarify
the spatio-temporal dynamics of clotting and pattern formation in quiescent blood.
To do so, the coagulation models are often simplified to allow at least partial
analytical solution or mathematical analysis [ 14 , 275 ]. Some generalizations were
also introduced, e.g., to take into account spatially nonuniform diffusion coefficients
in a simple model of fibrin polymerization [ 153 ]. On the other hand, a much more
complicated model of this type was published in [ 189 ]. This model was developed
to describe the spatial propagation and localization of blood coagulation and their
regulation in intrinsic and protein C pathways. The whole model, however, consists
of almost 30 differential equations containing more than 100 parameters. This model
was used, e.g., in [ 230 ] to study the spatial dynamics of contact-activated fibrin clot
formation.
The diffusion coefficients D i enter the model as “tuning parameters” and
usually have to be determined from experiments, simplified analytical models or
microscopic numerical simulations. The diffusion coefficients for proteins were
estimated, e.g., in [ 272 ] using correlation based on Stokes-Einstein relation leading
to a simple expression

O

D D 8:34 10 8 T
M 1=3

.cm 2 s 1 / (7.92)

depending just on the absolute temperature T , the solvent dynamic viscosity ,and
the molecular weight M. This approximation was developed under the assumptions
of relatively large (with respect to solvent) protein molecules with globular shape.
Another estimate relaxing these shape restrictions was published in [ 42 ]. There are
many other methods and models developed for prediction of diffusion coefficients.
For a detailed study the above papers and references therein can be used as a starting
point.