Flow Optimization in Vascular Networks

The development of mathematical models for studying phenomena observed in vascular networks is very useful for its potential applications in medicine and physiology. Detailed 3D studies of flow in the arterial system based on the Navier-Stokes equations require high computational power, hence reduced models are often used, both for the constitutive laws and the spatial domain. In order to capture the major features of the phenomena under study, such as variations in arterial pressure and flow velocity, the resulting PDE models on networks require appropriate junction and boundary conditions. Instead of considering an entire network, we simulate portions of the latter and use inflow and outflow conditions which realistically mimic the behavior of the network that has not been included in the spatial domain. The resulting PDEs are solved numerically using a discontinuous Galerkin scheme for the spatial and Adam-Bashforth method for the temporal discretization. The aim is to study the e↵ect of truncation to the flow in the root edge of a fractal network, the e↵ect of adding or subtracting an edge to a given network, and optimal control strategies on a network in the event of a blockage or unblockage of an edge or of an entire subtree.