  We consider a multiscale approach based on immersed methods for the efficient
computational modeling of tissues composed of an elastic matrix (in two or
three-dimensions) and a thin vascular structure (treated as a co-dimension two
manifold) at a given pressure. We derive different variational formulations of
the coupled problem, in which the effect of the vasculature can be surrogated
in the elasticity equations via singular or hyper-singular forcing terms. These
terms only depend on information defined on co-dimension two manifolds (such as
vessel center line, cross sectional area, and mean pressure over cross
section), thus drastically reducing the complexity of the computational model.
We perform several numerical tests, ranging from simple cases with known exact
solutions to the modeling of materials with random distributions of vessels. In
the latter case, we use our immersed method to perform an in silico
characterization of the mechanical properties of the effective biphasic
material tissue via statistical simulations.
