  In the context of micro-circulation, the coexistence of two distinct length
scales - the vascular radius and the tissue/organ scale - with a substantial
difference in magnitude, poses significant challenges. To handle slender
inclusions and simplify the geometry involved, a technique called topological
dimensionality reduction is employed, which suppresses manifold dimensions
associated with the smaller characteristic length. However, the resulting
discretized system's algebraic structure presents a challenge in constructing
efficient solution algorithms. This chapter addresses this challenge by
developing a robust preconditioner for the 3d-1d problem using the operator
preconditioning technique. Robustness of the preconditioner is demonstrated
with respect to problem parameters, except for the vascular radius. The
vascular radius, as demonstrated, plays a fundamental role in mathematical
well-posedness of the problem and the preconditioner's effectiveness.
