  We present a new method for modeling tissue perfusion on the capillary scale.
The microvasculature is represented by a network of one-dimensional vessel
segments embedded in the extra-vascular space. Vascular and extra-vascular
space exchange fluid over the vessel walls. This exchange is modeled by
distributed sources using smooth kernel functions for the extra-vascular
domain. It is shown that the proposed method may significantly improve the
approximation of the exchange flux, in comparison with existing methods for
mixed-dimension embedded problems. Furthermore, the method exhibits better
convergence rates of the relevant quantities due to the increased regularity of
the extra-vascular pressure solution. Numerical experiments with a vascular
network from the rat cortex show that the error in the approximation of the
exchange flux for coarse grid resolution may be decreased by a factor of $3$.
This may open the way for computing on larger network domains, where a fine
grid resolution cannot be achieved in practical simulations due to constraints
in computational resources, for example in the context of uncertainty
quantification.
