A mathematical analysis of capillary - tissue fluid exchange

Mathematical models for conventional transport of physiological fluids are explored analytically, including the characteristics and influences of the boundaries and media through which the flow occurs. The flow in fine capillaries with permeable walls was considered on the basis of some variants of the Krogh model for capillary tissue exchange and the Wiederhielm model for the extravascular circulation. Filtration from a cylindrical capillary into a concentrically surrounding tissue space; flow from a capillary into the tissue across a thin membrane; filtration from a rectangular, a cylindrical and a conical channel, bounded by a permeable material of uniform or regionally different permeability, and transcapillary fluid exchange were analyzed in some detail. For biological systems, which are characterized by low permeability, the calculations show that, independent of detailed geometry, any factor which produces a nonlinear distribution of pressure in the capillary will increase the filtration efficiency per unit of permeable area. Nonlinear pressure distributions will arise, for example, from an asymmetry of geometrical structure or as a result of interaction between red cell and wall during cell movement down the capillary. The filtration process is adequately described by a linear filtration law (the Starling relationship). Non linear laws are only of minor interest. In the systems considered the influence of velocity slip on filtration is negligible. The proposed models behave in such a way that the total amount of filtered fluid for a given capillary cannot exceed some limiting value. Thin membranes of low permeability on sufficiently thick layers of the tissue reduce the pressure gradient in the tissue to a value which is very small compared with the pressure gradient of blood flow in the capillary. The results obtained closely correspond to the microcirculation with respect to permeation rates and pressure distribution in the tissue. It was found that the system responds stably to changes in pressure, changes in rate of lymph flow etc. because of mutual compensation between the factors involved.