A mathematical model for filtration and macromolecule transport across capillary walls

Abstract

Metabolic substrates, such as oxygen and glucose, are rapidly delivered to the cells of large organisms through filtration across microvessels walls. Modelling this important process is complicated by the strong coupling between flow and transport equations, which are linked through the osmotic pressure induced by the colloidal plasma proteins. The microvessel wall is a composite media with the internal glycocalyx layer exerting a strong sieving effect on macromolecules, with respect to the external layer composed by the endothelial cells. The physiological structure of the microvessel is represented as the superimposition of two membranes with different properties; the inner membrane represents the glycocalyx, while the outer membrane represents the surrounding endothelial cells. Application of the mass conservation principle and thermodynamic considerations lead to a model composed of two coupled second-order ordinary differential equations for the hydrostatic and osmotic pressures, one, expressing volumetric mass conservation and the other, which is non-linear in the unknown osmotic pressure, expressing macromolecules mass conservation. Despite the complexity of the system, the assumption that the properties of the layers are piece-wise constant allows us to obtain analytical solutions for the two pressures. This solution is in agreement with experimental observations, which contrary to common belief, show that flow reversal cannot occur in steady-state conditions unless the hydrostatic pressure in the lumen drops below physiologically plausible values. The observed variations of the volumetric flux and the solute mass flux in case of a significant reduction of the hydrostatic pressure at the lumen are in qualitative agreement with observed variations during detailed experiments reported in the literature. On the other hand, homogenising the microvessel wall into a single-layer membrane with equivalent properties leads to a very different distribution of pressure across the microvessel walls, not consistent with observations.

Keywords: Analytical solution; Blood-brain barrier; Capillary wall; Glycocalyx damage; Mathematical model; Nonlinear transport of macromolecules; Solute extravasation; Starling's law; Ultrafiltration; Venous hypertension.