A quantitative two-dimensional theoretical model is developed to describe the movement of water and salt along the long narrow extracellular channels which appear to be a common structural feature of all epithelial membranes. This study examines the transport behaviour of both open and closed membrane systems as a function of the geometric specialization of the channel and the active transport site location under the influence of three driving forces: trans-membrane osmotic and hydrodynamic pressure differentials and active transport. The previous one-dimensional hydrodynamic model of Diamond & Bossert (1967) and Segel(1970) was confined to closed channel systems such as the gall bladder in which the only mechanism for water movement is local osmosis due to active transport.
Approximate analytical solutions are presented for long constant-area open channels in which the active transport sites have been idealized as point solute sources. A streamwise co-ordinate straining technique has been used in these solutions to describe the nonlinear effects of convection over long distances. Closed-form solutions are also presented for the pressure and solute concentration distributions within simplified models of channel constrictions with varying degrees of occlusion.
Numerical results of the model have been compared with Cole's (1961, 1962) in vivo and in vitro experiments on the rabbit ciliary body. Satisfying agreement with the measured values of the solute and water fluxes has been obtained for both the living eye and the excised ciliary body. These results strongly suggest that the formation of aqueous humour in the rabbit is a pressure-dependent process in which local osmosis due to active transport accounts for only one-third of the total aqueous flow. The model has also been applied to the gall bladder epithelium using more general boundary conditions than allowed for in the model of Diamond & Bossert. New solutions yielding a vanishing diffusional flux at the channel exit were obtained. However, the model, like that of Diamond & Bossert, does not provide a rational explanation as to how the water in the cell interior is replenished.