Mathematical formulation and numerical simulation of a 1D synthetic blood coagulation model

Blood coagulation is a biological process of fundamental importance and extreme complexity. It consists on the formation of blood clots at the site of vascular injury, preventing the blood loss. This process involves complex interactions among multiple molecular and cellular components in the blood and vessel wall, and it is also influenced by the flow of blood.

Mathematical modeling of the blood coagulation and fibrinolysis processes is a way of conceptualizing and understanding this complicated system, helping to optimize design of artificial devices and also to identify the regions of the arterial tree susceptible to the formation of thrombotic plaques and possible rupture in stenosed arteries. A good model should be simple enough in order to be applied in numerical simulations, and at the same time should be able to capture the process complexity, so to allow its better understanding.

The blood coagulation model we are working on consists of a system of convection-reaction-diffusion equations, describing the cascade of biochemical reactions, coupled with rheological models for the blood flow (Newtonian, shear-thinning and viscoelastic models). We introduce the effect of blood slip at the vessel wall emphasizing an extra supply of activated platelets to the clotting site. We expect that such contribution could be dominant, resulting in the acceleration of thrombin production and eventually of the whole clot progression. Such model will have the capacity to predict effects of specific perturbations in the hemostatic system that can't be done by laboratory tests, and will assist in clinical diagnosis and therapies of blood coagulation diseases.

Numerical results for 1D case will be presented, based on the solution of a system of reaction-diffusion equations, using the Finite Element Method. Evolution of concentration of biochemical species and clot formation and growth will be investigated in the injury site of the vessel wall.