Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 532

7.7.2 Feature-Based Classification of Coagulation Models

Different coagulation models include different features, i.e., they take into account
different physical phenomena which results in more or less complete mathematical
models. Besides models for the biochemistry, additional models for the flow,
structure and other physical phenomena can be included. The reason for including
(resp. omitting) certain physical phenomena in a full model is mainly due to
economical reasons, i.e., based on the computational efficiency and accuracy of
the considered model. To some extent it also reflects the historical evolution of
coagulation models with visible trend to include more features and in the most
recent state-of-the-art models. In this section we will mainly, but not exclusively,
focus on the macroscopic continuum models, as many of the other biochemistry-
only sub-micro and microscale models were already described.

1. Biochemistry (Only) Models

Within this class, two separate subgroups of models can be recognized, based on
whether they allow just for temporal or also spatial variations of chemical fields.

SpatiallyHomogeneousModels

This class of models deals only with the chemical part of the coagulation process.
The stoichiometric equations used in classical chemistry are translated into the
mathematical language of ordinary differential equations describing the evolution in
time of concentrations C i .t/ of various components taking part in the coagulation
process.

dC i
dt D R i .C 1 ;C 2 ;:::;C N /iD 1;:::;N

(7.71)

These equations are called the ReactionRate Equations(RRE) . The corresponding
models are sometimes referred to as zero-dimensional 82 models, because all spatial
variations of concentrations are excluded and thus no diffusion (or convection)
processes are taken into account. The functions R i are in general nonlinear and can
depend on all considered concentrations. Typical forms are the first-order , second-
order ,or Michaelis-Menten kinetics terms:

ǛCi first order kinetics

ǛCi C j second order kinetics

ǛCi C j
LJ C C j

Michaelis-Menten kinetics

The rate constants Ǜ, LJ in all reaction terms have to be determined experimentally.

82 In contrast with spatially two- or three-dimensional models.