Mean First-Passage Time in a Sphere Traversed by a Cylinder

We study Mean First-Passage Times (MFPTs) for diffusive migration in finite three-dimensional domains motivated by organ-vessel architectures and intracellular transport. Focusing on a sphere traversed by a thin cylinder (an idealised organ surrounding a blood vessel), and on a deformed hollow sphere with an inner trap, we formulate MFPT via boundary integral equations that encode Dirichlet, Neumann, and Robin conditions. Two complementary strategies emerge: (i) a deformation-from-solvable-geometry approach that yields accurate approximations for mixed boundaries, and (ii) a “conductor-term” augmentation that restores solvability when standard integral equations fail. These frameworks expose narrow escape/capture regimes and quantify how trap size and interface semi-permeability control MFPT. We complement theory with numerical solutions of the adjoint Poisson problem, covering cases not reached analytically: first, a sphere with Dirichlet boundary condition, traversed by a cylinder with Neumann one (complementing the integral equations approach); second, a sphere with Neumann boundary condition, traversed by a cylinder with Dirichlet one (uncovered by the integral equations approach). Beyond methodological value, our results provide interpretable MFPT maps relevant to cell migration and particle delivery in organ-like geometries.