A one-dimensional mathematical model of collecting lymphatics coupled with an electro-fluid-mechanical contraction model and valve dynamics

Abstract

We propose a one-dimensional model for collecting lymphatics coupled with a novel Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions, based on a modified FitzHugh-Nagumo model for action potentials. The one-dimensional model for a deformable lymphatic vessel is a nonlinear system of hyperbolic Partial Differential Equations (PDEs). The EFMC model combines the electrical activity of lymphangions (action potentials) with fluid-mechanical feedback (circumferential stretch of the lymphatic wall and wall shear stress) and lymphatic vessel wall contractions. The EFMC model is governed by four Ordinary Differential Equations (ODEs) and phenomenologically relies on: (1) environmental calcium influx, (2) stretch-activated calcium influx, and (3) contraction inhibitions induced by wall shear stresses. We carried out a stability analysis of the stationary state of the EFMC model. Contractions turn out to be triggered by the instability of the stationary state. Overall, the EFMC model allows emulating the influence of pressure and wall shear stress on the frequency of contractions observed experimentally. Lymphatic valves are modelled by extending an existing lumped-parameter model for blood vessels. Modern numerical methods are employed for the one-dimensional model (PDEs), for the EFMC model and valve dynamics (ODEs). Adopting the geometrical structure of collecting lymphatics from rat mesentery, we apply the full mathematical model to a carefully selected suite of test problems inspired by experiments. We analysed several indices of a single lymphangion for a wide range of upstream and downstream pressure combinations which included both favourable and adverse pressure gradients. The most influential model parameters were identified by performing two sensitivity analyses for favourable and adverse pressure gradients.

Mathematical Modelling of the Structure and Function of the Lymphatic System

“…The majority of models of lymph flow in the lymphangion or series of lymphangions are based on the zero-dimensional (including lumped ones) [17,22,37,38] or one-dimensional (including quasi-one-dimensional ones) [18,[39][40][41] approaches. "Lumped" models here are models based on electrical circuit theory, where the network of vessels is substituted with the circuit and where the voltage, current, and charge are equivalent to pressure, lymph flux, and mass, respectively [38,42,43].…”

“…It is sufficiently simple and detailed to describe normal functioning of valves and their pathologies. This model is used in [39] to describe the valve influence on lymph flow. The equation for time-varying flux through valve q v has the following form:…”

“…The tonic response is a long-lasting variation of elastic properties of the vessel wall, and phase contractions are rapid and periodic. All these types of contractions can be modelled together [17] or separately [18,22,[37][38][39]41,54,55] In early works, contractions of lymphangions were modeled using the same approach as in the models of heart pumping with time-varying elasticity [17]:…”

“…Despite the existence of models concerning the cell structure of blood vessels, in the majority of investigations, empirical curves for the s-p relationship are used [57]. Similar representations are also used for all existing models of lymph flow on the basis of physiological experiments [18,22,37,39]. Despite the similarity of s-p curves for different pressure-area dependencies, mathematical solutions (in particular, the velocity of small perturbations) can differ significantly [57].…”

“…Two more refined forms of this relation are used by the same authors in [58]. A similar approach with more parameters is used in [39]. These parameters are determined by fitting the data from physiological experiments.…”