A Model for Fluid Drainage by the Lymphatic System

Abstract

This study investigates the fluid flow through tissues where lymphatic drainage occurs. Lymphatic drainage requires the use of two valve systems, primary and secondary. Primary valves are located in the initial lymphatics. Overlapping endothelial cells around the circumferential lining of lymphatic capillaries are presumed to act as a unidirectional valve system. Secondary valves are located in the lumen of the collecting lymphatics and act as another unidirectional valve system; these are well studied in contrast to primary valves. We propose a model for the drainage of fluid by the lymphatic system that includes the primary valve system. The analysis in this work incorporates the mechanics of the primary lymphatic valves as well as the fluid flow through the interstitium and that through the walls of the blood capillaries. The model predicts a piecewise linear relation between the drainage flux and the pressure difference between the blood and lymphatic capillaries. The model describes a permeable membrane around a blood capillary, an elastic primary lymphatic valve and the interstitium lying between the two.

A Model for Interstitial Drainage Through a Sliding Lymphatic Valve

“…This is particularly true of the Youngs' modulus E (tissue swelling is almost independent of E). It is interesting to speculate on the possible advantages of a sliding valve over those of a valve operated by pressure differences between the lymphatic lumen and the surrounding tissue (as described in [20]). An evolutionary advantage of the sliding valve system over the pressure operated valve would seem to be that it responds directly to tissue swelling.…”

“…In this work we formulate a mathematical model based on Rossi's 'sliding door' hypothesis of lymphatic valve opening and compare our results to previous models based on pressure operated valves [20,38]. This model makes use of a poroelastic description of the interstitium and assumes that the endothelial cells in the walls of the lymphatic capillary are firmly anchored to the surrounding interstitium, except in the overlap regions that form the primary lymphatic valves.…”

“…The usual hypothesis for primary lymphatic valve function is that valve opening occurs in response to pressure differences between the interstitium and the lymphatic lumen which, when they are sufficiently large, lead to the valve flaps bending into the lymphatic lumen and creating a narrow gap through which interstitial fluid can flow into the lymphatic capillary. Models based on this hypothesis include [20,38]. However here we will be primarily concerned with an alternative hypothesis, due to Rossi [46], which proposes instead that the overlapping primary lymphatic valves slide open in response to expansion of the intersititium about the lymphatic capillary, as fluid accumulates in the tissue.…”

Mathematical Modelling of the Structure and Function of the Lymphatic System

“…In contrast to models of secondary valves, models of the primary valves often take into account the geometry of the valve-the curvature of the cell [67,68].…”

“…The geometrical structure of the interstitial space with initial lymphatics and blood capillaries is considered in [68], and a mechanical model for primary valve dynamics concerning their curvatures is used to describe flow through the junctions between endothelial cells. As a result of this two-dimensional analysis, the formula for lymph flux per unit length q in terms of the pressure difference between bloodp and lymphatic capillaries p 0 was obtained:…”