A mathematical analysis of capillary-tissue fluid exchange¹

Article type: Research Article

Authors: Apelblat, A.^{; *} | Katzir-Katchalsky, A.^{; †} | Silberberg, A.

Affiliations: Weizmann Institute of Science, Rehovot, Israel

Note: [1] Represents part of a Ph.D. thesis submitted by A.A. to the Feinberg Graduate School, Weizmann Institute of Science.

Note: [*] Present address: Israel Atomic Energy Commission, Nuclear Research Center-Negev, Beersheva.

Note: [†] Professor Aharon Katzir-Katchalsky fell victim to the attack on innocent passengers by terrorists on 30 May 1972 at Lod Airport, Israel. “It was one of his last papers. He wished that in spite of its length it should be published as a single whole. One part of this paper was cited by Professor A. Silberberg in his Memorial Address at the First International Congress of Biorheology, Lyon, France on the 5th of September 1972 (Biorheology 10, 109 1973). A.L.C. and G.W.S.B.”

Abstract: 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 [Krogh Physiol. 52 (1919), 391] for capillary-tissue exchange and the Wiederhielm [In: Physical Bases of Circulatory Transport: Regulation and Exchange (Edited by Reeve and Guyton) p. 313. Saunder, 1967, J. Gen. Physiol. 52 (1968), Suppl. Pt. 2, 295] 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, due to 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 very small as 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.

DOI: 10.3233/BIR-1974-11101

Journal: Biorheology, vol. 11, no. 1, pp. 1-49, 1974