Biomedical Engineering Reference
In-Depth Information
27 ODEs derived in [64] has been adapted and numerical simulations have shown
good agreement with the experimental results. The initiation and propagation phases
of blood clotting is affected by severe deficiency in any of the components of the
intrinsic pathway, FVIII, FIX or FXI, corresponding to hemophilia A, B or C, re-
spectively. This study can also be found in [59]. We have already mentioned the
paper [12] modelling the thrombocytopenia, which in perspective can provide an
important tool to decide whether or not splenectomy is appropriate (see Sect. 3.3.1).
A model for Disseminated Intravascular Coagulation (DIC)
In the papers [32, 33] a space-dependent model has been formulated for the forma-
tion of mini thrombi in the blood flow. This process, known as Disseminated In-
travascular Coagulation (DIC), is the result of abnormal delivery of TF into blood,
following various pathological conditions. DIC may be lethal. Ultrasound inspection
[92] reveals clouds of mini thrombi transported along the bloodstream. Besides the
damage possibly caused by thrombi to various organs, DIC is harmful also because
it drains from the patient's blood platelets and other thrombogenic elements with the
possible consequence of spontaneous bleeding. References can be found in [32, 33].
The proposed model takes an oversimplified scheme of the coagulation cascade
and, on the contrary, focuses in detail on the fibrin polymer formation. It is an
ingenious combination of two fields for the first time put together in the mathe-
matical literature: morphogenesis (describing pattern formation) and coagulation-
fragmentation (describing the growth of polymer chains). From morphogenesis it
borrows a system of just two reaction-diffusion equations for the evolution of throm-
bin and of an anticoagulant (representing the class of anticoagulants), in which the
positive feedback is represented by the typical autocatalytic terms of morphogenesis.
The approach of [32] is based on ordinary differential equations since a spatially ho-
mogeneous case is considered, while in [33] diffusion and convection are added. The
development of the fibrin polymeric chain is described starting from the fibrinogen-
thrombin interaction by means of the classical coagulation-fragmentation infinite
differential system. The latter system is eventually replaced by the study of the first
few moments of polymer concentrations
M
0
,
M
1
,
M
2
,
M
3
,definedas
M
k
=
∑
n
=
1
n
k
F
n
,
where
F
n
is the concentration of the chains consisting of precisely
n
monomers. A
closure condition is necessary. The one adopted is
M
1
M
3
=
M
2
(see [32] for a jus-
tification). Numerical simulations show that the model does reproduce the micro
thombi cloud formation.
3.4.3 Models incorporating mechanical and biochemical actions
None of the previous models includes the direct effects of mechanical loading on the
process of thrombus formation and lysis, or the mechanical properties of blood itself
and the forming clots. Most of them are also based in ODEs, neglecting the spatial
distribution of reactants, catalysts and products, which is a simplifying assumption,
since different reactions of blood coagulation take place in different regions. More-
over, practically all models concerned with hemodynamic factors and hemostasis do
not capture all biochemical reactions that take place in flowing blood.
