Vascular Endothelium - Tissue Functioning and Remodeling in the Circulatory and Ventilatory Systems - page 561

9.8.5 Mathematical Modeling and Numerical Simulations

Mathematical multiscale 143 models of either clotting on a breach of the vessel wall
or thrombosis after a rupture of an atherosclerotic plaque has been developed in the
presence of a flow of an incompressible viscous fluid [ 1037 , 1038 ]. In the explored
fluid domain, the model set incorporates the platelets and coagulation factors,
that are involved in the chemical reaction chain, each step being triggered by the
activation of the corresponding zymogen into enzyme, and coagulation inhibitors.
Compounds and platelet transport by convection and diffusion, are assumed to take
place in a near-wall thin plasma layer. Competition occurs between the activation
of the coagulation stages and removal by the flowing fluid of the clotting factors
and cells away from the reaction site. 144 Numerical simulations use the immersed
boundary method (Vol. 7 - Chap. 8. Numerical Simulations). Adhesion of platelets
to the injured wall and cohesion between activated platelets are modeled by
distributed elastic links. 145 Strain-dependent breaking of cohesive bonds between
platelets and adhesive links between the platelet and the vessel wall are treated
by a closure approximation of the evolution equations. The probability of platelet
aggregation increases quickly after activation, remains nearly maximum for a
significant time interval, and then declines [ 1039 ]. The fibrinogen concentration and
density in surface receptors strongly affect the time constant of platelet aggregation.
The thrombin production depends on available binding sites. Thrombus growth,
with possible vessel occlusion, and embolus shedding from the thrombus can be
predicted by the stress field exerted by the moving fluid on the thrombus.
Interactions between mechanical factors introduced by the flowing blood and
the biochemical agents with their multiple positive and negative feedbacks and
regulators have also been mathematically modeled, not only during clot formation,
but also during degradation [ 1040 , 1041 ]. Clot reaction kinetics are affected by shear

platelet shape changes are associated with: (1) expression of pro-inflammatory molecules, such as
P-selectin, soluble CD40 ligand (also expressed by platelets and endothelial and smooth muscle
cells); (2) stimulation of platelet procoagulant activity; (3) potentiation of aggregation by other
prothrombotic factors; and (4) activation of α 2B β 3 -integrins. Activation of endothelial cells by

TNFSF5 from activated platelets promotes the production of reactive oxygen species, chemokines,
and cytokines as well as expression of adhesion molecules (selectin, ICAM1, and VCAM1).
In addition, platelet integrins bind to fibrinogen and von Willebrand factor, thereby causing platelet
aggregation and thrombus formation. Serotonin released from activated platelets: (1) accelerates
aggregation of platelets; (2) promotes proliferation, migration, and contraction of vascular smooth
myocytes; and (3) heightens and reduces procoagulant and fibrinolytic activities, respectively, of
endothelial cells via 5HT 2 A receptors, as it provokes expression of tissue factor and plasminogen
activator inhibitor-1 [ 1034 ]. Thromboxane-A2 causes platelet aggregation in synergy with other
secreted agonists [ 1035 ].
143 The model requires several spatial scales: the nanoscale of the coagulation factors, the
microscale of the platelets, and the macroscale of the blood vessel.
144 The flux of activated coagulation factors and platelets is flow dependent.
145 Platelets aggregate assuming that bridges form isotropically, linear springs modeling the clusters
of binding sites.