Thrombus formation in flowing blood is a complex time- and space-dependent process of cell adhesion and fibrin gel formation controlled by huge intricate networks of biochemical reactions. This combination of complex biochemistry, non-Newtonian hydrodynamics, and transport processes makes thrombosis difficult to understand. That is why numerous attempts to use mathematical modeling for this purpose were undertaken during the last decade. In particular, recent years witnessed something of a transition from the “systems biology” to the “systems pharmacology/systems medicine” stage: computational modeling is being increasingly applied to practical problems such as drug development, investigation of particular events underlying disease, analysis of the mechanism(s) of drug’s action, determining an optimal dosing protocols, etc. Here we review recent advances and challenges in our understanding of thrombus formation.

We perform a sensitivity analysis for a thus far unstudied mathematical model for the formation, growth and lysis of clots in vitro. The sensitivity analysis procedure uses an ensemble standard deviation for species concentrations, and is equivalent to a variance decomposition procedure also available in the literature. Our analysis shows that fibrin production is most sensitive to the rate constant governing activation of prothrombin to thrombin. Further, the time-averaged sum of all species’ concentrations is most sensitive to the rate constants governing the inactivation of VIIIa (intrinsic as well as by APC). We therefore conclude that the rate constants for VIIIa inactivation affect the model the greatest: this conclusion must be experimentally verified to determine if such is indeed the case for hemostasis.

This paper presents numerical results based on a macroscopic blood coagulation model coupled with a non-linear viscoelastic model for blood flow. The system of governing equations is solved using a central finite-volume scheme for space discretization and an explicit Runge-Kutta time-integration. An artificial compressibility method is used to resolve pressure and a non-linear TVD filter is applied for stabilization. A simple test case of flowing blood over a clotting surface in a straight 3D vessel is solved. This work presents a significant extension of the previous studies [10] and [9].

A stenosis is the narrowing of the artery, this narrowing is usually the result of the formation of an atheromatous plaque infiltrating gradually the artery wall, forming a bump in the ductus arteriosus. This arterial lesion falls within the general context of atherosclerotic arterial disease that can affect the carotid arteries, but also the arteries of the heart (coronary), arteries of the legs (PAD), the renal arteries... It can cause a stroke (hemiplegia, transient paralysis of a limb, speech disorder, sailing before the eye). In this paper we study the blood-plaque and blood-wall interactions using a fluid-structure interaction model. We first propose a 2D analytical study of the generalized Navier-Stokes equations to prove the existence of a weak solution for incompressible non-Newtonian fluids with non standard boundary conditions. Then, coupled, based on the results of the theoretical study approach is given. And to form a realistic model, with high accuracy, additional conditions due to fluid-structure coupling are proposed on the border undergoing inetraction. This coupled model includes (a) a fluid model, where blood is modeled as an incompressible non-Newtonian viscous fluid, (b) a solid model, where the arterial wall and atherosclerotic plaque will be treated as non linear hyperelastic solids, and (c) a fluid-structure interaction (FSI) model where interactions between the fluid (blood) and structures (the arterial wall and atheromatous plaque) are conducted by an Arbitrary Lagrangian Eulerian (ALE) method that allows accurate fluid-structure coupling.

Properties of blood cells and their interaction determine their distribution in flow. It is observed experimentally that erythrocytes migrate to the flow axis, platelets to the vessel wall, and leucocytes roll along the vessel wall. In this work, a three-dimensional model based on Dissipative Particle Dynamics method and a new hybrid (discrete-continuous) model for blood cells is used to study the interaction of erythrocytes with platelets and leucocytes in flow. Erythrocytes are modelled as elastic highly deformable membranes, while platelets and leucocytes as elastic membranes with their shape close to a sphere. Separation of erythrocytes and platelets in flow is shown for different values of hematocrit. Erythrocyte and platelet distributions are in a good qualitative agreement with the existing experimental results. Migration of leucocyte to the vessel wall and its rolling along the wall is observed.

In this work we propose a method for analysis of postsurgical haemodynamics after femoral artery treatment of occlusive vascular disease. Patient specific reconstruction algorithm of 1D core network based on MRI data is proposed as a tool for such analysis. Along with presurgical ultrasound data fitting it provides effective personalizing predictive method that is validated with clinical observations.

We present a fully automatic approach to recover boundary conditions and locations of the vessel wall, given a crude initial guess and some velocity cross-sections, which can be corrupted by noise. This paper contributes to the body of work regarding patient-specific numerical simulations of blood flow, where the computational domain and boundary conditions have an implicit uncertainty and error, that derives from acquiring and processing clinical data in the form of medical images. The tools described in this paper fit well in the current approach of performing patient-specific simulations, where a reasonable segmentation of the medical images is used to form the computational domain, and boundary conditions are obtained as velocity cross-sections from phase-contrast magnetic resonance imaging. The only additional requirement in the proposed methods is to obtain additional velocity cross-section measurements throughout the domain. The tools developed around optimal control theory, would then minimize a user defined cost function to fit the observations, while solving the incompressible Navier-Stokes equations. Examples include two-dimensional idealized geometries and an anatomically realistic saccular geometry description.

A traceless variant of the Johnson-Segalman viscoelastic model is presented and applied to blood flow simulations. The viscoelastic extra stress tensor is decomposed into its traceless (deviatoric) and spherical parts, leading to a reformulation of the classical Johnson-Segalman model. The equivalence of the two models is established comparing model predictions for simple test cases. The new model is validated using several 2D benchmark problems, designed to reproduce difficulties that arise in the simulation of blood flow in blood vessels or medical devices. The structure and behaviour of the new model are discussed and the future use of the new model in envisioned, both on the theoretical and numerical perspectives.