Thrombosis is a serious medical complication because excessive thrombus formation could lead to obstruction of blood vessels, resulting in tissue ischemia or death. This study is aimed to understand the cell-biology and biophysics, at the molecular level, of thrombosis in a blood vessel. We first review the biology underlying thrombosis and the previous mathematical models of thrombus formation (Chapter 1). In Chapter 2, mathematical modeling and sensitivity analysis were used to explore the mechanistic model of thrombus formation; the results support our working hypothesis that computationally derived points of fragility of human relevant cascades could be used as a rational basis for target selection despite model uncertainty. Chapter 3 focuses on the impact of parametric sampling strategies of Monte-Carlo sensitivity analysis and network structural uncertainty upon the assessment of the qualitative importance of molecular interactions in the coagulation network. While parametric uncertainty can be partially overcome by sampling feasible parameter regions using one of several strategies, structural uncertainty remains a critical determinant of our ability to classify mechanisms as fragile or robust in networks relevant to human health. The mathematical model was further extended to describe platelet activation by other agonists besides thrombin and the intrinsic pathway in Chapter 4. An ensemble of probable parameter sets were estimated using nine experimental data sets from a cell-based model and then used to predict the thrombin generation in experiments using patient-derived plasma for coronary artery and hemophilia A patients. In Chapter 5, analysis of the sensitivity results discloses that the intrinsic protease factor XI could be an excellent therapeutic target for thrombosis treatment with the advantage of not affecting hemostasis. Chapter 6 describes the effect of flow on the initiation of arterial clotting under conditions of exposed tissue factor (TF). Thresholds in shear rates or TF patch sizes were observed for the initiation of clot formation; the balance between the generation of active factors and the removal of those factors by flow was possibly the reason for the threshold phenomena.