Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 505

Two main experimental observations clash with the former 3-pathway cascade
model:

• The TF-pathway bypasses FVIII , but is nevertheless sufficient to produce

clotting,

• FXII deficiency, as we have seen in the previous section, produces no significant

bleeding disorder.

So the troubles about the 3-pathway cascade model come from the position
occupied by FVIII, whose deficiency we know to be associated with Hemophilia
A. If the TF-pathway alone would be enough for clotting, FVIII deficiency would
be inessential. Moreover, FXII deficiency would exclude FVIII from the process,
causing the same effects as FVIII deficiency. None of these facts are true, hence the
necessity of a model, like the cell-based model, which can produce the clot without
FXII and recognizes the crucial role of FVIII.
Nevertheless, it would be wrong to dismiss the role of FXII altogether. Its ability
of becoming activated on artificial surfaces can enhance clotting on implanted
bodies. A paper recently devoted to FXII and its possible alternative functions
is [ 212 ].

7.7 Mathematical Description of Coagulation Models

Mathematical models of blood coagulation are at the same time the main tools as
well as outputs to study the coagulation processes. Solutions of these models are
used for comparison with experimental observations to test our understanding of the
underlying processes. The wide range of physical phenomena to be considered and
the complexity of their interactions are responsible for a high number of different
models being developed for specific use.
There is a large number of coagulation models that can be applied for various
specific purposes [ 266 ]. They differ in many aspects like physical features to be
considered, scales of the phenomena, or the biochemistry model to be coupled to
biomechanics. Important differences are also in the mathematical formulation of
the coagulation models and in the numerical methods to solve them.
In order to give a brief overview of some of the existing mathematical models
of blood coagulation, we will classify the models in two ways. First, the models
will be grouped together according to the (spatial) scales they are able to describe
and resolve. Second, the models can be classified according to the physical features
they are including. These two classification schemes are not independent, as some
physical phenomena can only be described at appropriate scales. Nevertheless, these
classification schemes will be useful to demonstrate some of the main trends in the
past and current evolution of mathematical models of blood coagulation.