  In this paper, we are concerned with the simulation of blood flow in
microvascular networks and the surrounding tissue. To reduce the computational
complexity of this issue, the network structures are modeled by a
one-dimensional graph, whose location in space is determined by the centerlines
of the three-dimensional vessels. The surrounding tissue is considered as a
homogeneous porous medium. Darcy's equation is used to simulate flow in the
extra-vascular space, where the mass exchange with the blood vessels is
accounted for by means of line source terms. However, this model reduction
approach still causes high computational costs, in particular, when larger
parts of an organ have to be simulated. This observation motivates the
consideration of a further model reduction step. Thereby, we homogenize the
fine scale structures of the microvascular networks resulting in a new hybrid
approach modeling the fine scale structures as a heterogeneous porous medium
and the flow in the larger vessels by one-dimensional flow equations. Both
modeling approaches are compared with respect to mass fluxes and averaged
pressures. The simulations have been performed on a microvascular network that
has been extracted from the cortex of a rat brain.
