Blood vessels are flow-induced diffusive molecular channels equipped with transport mechanisms across their walls for conveying substances between the organs in the body. Mathematical modeling of the blood vessel as a molecular transport channel can be used for the characterization of the underlying processes and higher-level functions in the circulatory system. Besides, the mathematical model can be utilized for designing and realizing nano-scale molecular communication systems for healthcare applications including drug delivery systems. In this paper, a continuous-time Markov chain framework is proposed to simply model active transport mechanisms e.g. transcytosis, across the single-layered endothelial cells building the inner vessel wall. Correspondingly, a general homogeneous boundary condition over the vessel wall is introduced. Coupled with the derived boundary condition, the flow-induced diffusion problem in an ideal vessel structure with a cylindrical shape is accurately formulated which takes into account variation in all three dimensions. The corresponding concentration Green's function is analytically derived in terms of a convergent infinite series. Particle-based simulation results confirm the proposed analysis. Also, the effects of system parameters on the concentration Green's function are examined.