In this paper we shall review some recent mathematical models which have led to new conceptual views of the ultrastructural pathways by which water, solutes and large molecules cross the endothelial interface between tissue and blood. In particular, we shall show how a sequence of models for the endothelium and underlying tissue in large arteries have finally led to the experimental discovery of the large pore via which LDL and other large molecules enter the artery wall and how a new three-dimensional model for the interendothelial cleft in capillaries might reconcile the several long standing paradoxes relating to the measured filtration and solute permeability coefficients in the transcapillary exchange of water and hydrophilic solutes in the microcirculation.

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