Modeling, Analysis, and Simulation of Thrombosis and Hemostasis

Abstract

This thesis investigates the influences of shear stress, saturation-dependent changes in surface reactivity, and thrombus growth on platelet deposition to reactive materials, which is of paramount interest in bioengineering and clinical practice. For this purpose, two mathematical models based on the Navier-Stokes equations and on particle conservation are developed. The first model is formulated on a fixed domain (“FD-model”) and describes the initial phase of platelet adhesion, whereas the second one is a free boundary problem capturing long-term thrombus growth. Several vessel geometries are considered: Stagnation point flow, tubular expansion, and t-junction. Model parameters are optimized to fit the data and their so obtained values are justified on the basis of experimental observations. The FD-model does not match the experimental data at all, when platelet adhesion is assumed independent of shear stress. In contrast, when adhesion is assumed shear-dependent, at least qualitative agreement is achieved. Solely by consideration of both shear stress and saturation-dependent changes in surface reactivity, good quantitative agreement of FD-model and data is obtainable. Such changes in surface reactivity are taken into account by coupling platelet flux conditions to ordinary differential equations (ODEs) for the evolution of surface-bound platelets. The free boundary problem is simulated by the level set method. Like the FD-model, it shows good qualitative agreement with the experimental evidence when shear stress is taken into account, whereas negligence of shear leads to completely false predictions. Regarding mathematical well-posedness of the FD-model, existence of weak solutions is shown for generalized parabolic systems having ODE-coupled flux conditions. Uniqueness and positivity of solutions are also investigated. Regarding the free boundary problem, a detailed proof of classical solvability in terms of Holder spaces is presented.