Unsteady Model of Blood Coagulation in Aneurysms of Blood Vessels

Introduction. A two-dimensional hydrodynamic problem is numerically solved in the “stream function-vorticity” formulation for an open rectangular cavity simulating blood flow and its coagulation within a vascular aneurysm. The model accounts for a simplified nonlinear mathematical description of the first phase of blood coagulation (30 seconds).
Materials and Methods. To accelerate the numerical solution of the unsteady problem with an explicit finite-difference scheme for the vorticity dynamics equation, an n-fold splitting method of the explicit scheme (n = 100, 200) was employed, along with the use of a symmetry plane in the rectangular aneurysm domain. The splitting method was also applied to solve the dynamic system of advection-diffusion equations with nonlinear source terms for the activator and inhibitor blood factors (N = 70). The maximum time step τ0 was synchronized across both splitting cycles. The computation was performed on half of the rectangular aneurysm using a uniform 100×50 grid with equal spacing h₁ = h₂ = 0.01. The inverse matrix required for solving the Poisson equation in the “stream function-vorticity” formulation with a finite number of elementary operations was computed using the Msimsl library.
Results. The numerical solution demonstrated that, in arterioles (Re = 3.6), advection and diffusion of fibrin occur according to the nonlinear dynamics of activator and inhibitor factors, as if fibrin were moving counter to the blood flow. The maximum fibrin density forms in the central region of the vessel in the shape of a “fibrin horseshoe”. For higher Reynolds numbers (Re = 3000) corresponding to arteries, fibrin motion occurs along the main flow, and the central part of the vessel is separated from the aneurysm by a “fibrin foot” along its geometric boundary. In arterioles, a layered fibrin growth effect was also observed, with periodic variations in fibrin density near the aneurysm wall, consistent with other authors’ findings. In arteries, the fibrin film within the aneurysm forms in approximately one second — significantly shorter than the first coagulation phase (30 seconds).
Discussion. The finite-difference approximation achieves sixth-order accuracy at interior nodes and fourth-order accuracy at boundary nodes. The model was applied to simulate blood flow in arterial aneurysms at high Reynolds numbers (Re = 3000) and in arteriole aneurysms (Re = 3.6). The dimensionless range of fibrin density variation is consistent with data reported by other researchers.
Conclusions. The study proposes a system of equations representing a simplified unsteady model of blood motion and fibrin (thrombus) formation in vascular aneurysms. The proposed model provides a qualitative understanding of thrombus formation mechanisms in aneurysms of arteries and arterioles, as well as in elements of medical equipment.

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