Transport and deposition problems in blood flow
Keywords: asymptotic analysis
mathematical modelling
blood flow
fluid mechanics

In this thesis, we examine blood flow dynamics in single vessels and the deposition of transported species to vessel walls. We develop continuum mathematical models to explore the role of fluid mechanics in two applications. First, we examine magnetic targeting applied to stem cell delivery for regenerative medicine. Second, we model the initial stages of arterial blood clot formation (thrombosis). In both applications, we use asymptotic analysis to develop reduced models. This allows us to i) design and test parameter estimation schemes which can effectively calibrate the mathematical models using in vitro data and ii) determine optimal therapeutic parameters by exploring regimes not tested in vitro. Finally, we use large scale numerical simulations to examine regimes which are not amenable to asymptotic reduction: larger vessels and high Reynolds number flows. Firstly, to examine magnetically controlled stem cell delivery we present a 2D continuum model of the flow of blood and magnetically tagged stem cells in a single channel. Cell capture leads to the growth of a cell mass on the wall of the vessel closest to the magnet. For a safe and effective therapy, the stem cells must be delivered in large numbers under physiological flow conditions, but critically, the aggregation of cells at the target site must be controlled. Using numerical simulations we analyse the model in the regime of in vitro experiments: where the size of the magnet is comparable to the channel height. The model qualitatively agrees with experimental results showing better capture with stronger magnetic fields and at lower levels of red blood cells. Using lubrication theory we explore cell delivery regimes beyond the in vitro setup when the channel height is much smaller than the size of the magnet. This allows us to identify parameter regimes where finite sized aggregates form and those in which potentially dangerous vessel blockage is predicted. Secondly, we develop a 3D continuum model for thrombus formation in a diseased artery model. Blood protein Von Willebrand Factor (VWF) is critical in facilitating arterial thrombosis. At pathologically high shear rates the protein unfolds and rapidly captures platelets from the flow. Our model extends existing continuum models for thrombosis by explicitly modelling the VWF unfolding dynamics using a modified viscoelastic fluid model. Since thrombus initiation occurs over several minutes we use the initial fluid mechanics in the stenosis to predict VWF unfolding and platelet deposition. We examine clot formation in an arterial-scale stenosis at high Reynolds numbers numerically. We demonstrate that the initial clot location and deposition rate depends on the ratio of VWF length to Reynolds number. Finally, we examine the thrombosis model in a rectangular microfluidic geometry motivated by in vitro data gathered by our collaborators. We exploit the small aspect ratio of the geometry to reduce the model using lubrication theory. We then explore methods of quantifying unknown model parameters using existing data for VWF dynamics and the thrombosis data obtained in this microfluidic device.