Theoretical studies of steady-state transcapillary exchange in countercurrent systems

As a first step in modeling microvascular exchange in the renal medulla, we developed mathematical models to explore the effects of blood flow, permeability, and anatomical arrangement of microvessels on the steady-state distribution of solute in the blood and the interstitial fluid (ISF).

Single capillaries and countercurrent capillary loops were used to model microvessels that were surrounded by a secretory epithelium over either the whole or part of the capillary length. Solute concentration in the vessels and the ISF were derived analytically. We also derived approximate solutions that ignored axial diffusion of solute.

The full and approximate solutions were in good agreement with data based on measurements in the renal medulla. Model results revealed that concentration in the ISF falls rapidly with distance beyond the region of solute secretion and equilibrates with the concentration in capillaries, even with countercurrent exchange between the two limbs of the capillary loop. The ratio of the product of the permeability and area to the flow of the afferent limb, gamma 1, is an important parameter. When gamma 1 > 4, countercurrent exchange in a capillary loop facilitates a greater ISF concentration gradient than with a single capillary. Changes in flow also have a greater effect on this gradient.

The model of countercurrent exchange presented here not only demonstrates the sensitivity of interstitial concentration gradients of solute to flow through capillary loops but also reveals the importance of the absolute value of gamma 1 in determining the magnitude and direction of these gradients.