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

## TL;DR

This paper introduces a comprehensive one-dimensional mathematical model of collecting lymphatics coupled with an innovative electro-fluid-mechanical contraction model and valve dynamics, enabling detailed simulation of lymphatic function and pathologies.

## Contribution

The study presents a novel coupled PDE-ODE model for lymphatic contractions and valve behavior, including stability analysis and simulation of stenosis and regurgitation effects.

## Key findings

- The EFMC model replicates pressure and shear stress effects on contraction frequency.
- Stenotic valves significantly reduce pump flow at high contraction frequencies.
- Incompetent valves diminish net lymph flow as incompetence severity increases.

## 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 compliant lymphatic vessel is a set of hyperbolic Partial Differential Equations (PDEs). The EFMC model combines the electrical activity of lymphangions (action potentials) with fluid-mechanical feedback (stretch of the lymphatic wall and wall shear stress) and the mechanical variation of the lymphatic wall properties (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 complete mathematical analysis of the stability of the stationary state of the EFMC model. Overall, the EFMC model allows imitating the influence of pressure and wall shear stress on the frequency of contractions observed experimentally. Lymphatic valves are modelled using a well-established lumped-parameter model which allows simulating stenotic and regurgitant valves. We analysed several lymphodynamical indexes of a single lymphangion for a wide range of upstream and downstream pressure combinations. Stenotic and regurgitant valves were modelled, and their effects are here quantified. Results for stenotic valves showed in the downstream lymphangion that for low frequencies of contractions the Calculated Pump Flow (CPF) index remained almost unaltered, while for high frequencies the CPF dramatically decreased depending on the severity of the stenosis (up to 93% for a severe stenosis). Results for incompetent valves showed that the net flow during a lymphatic cycle tends to zero as the degree of incompetence increases.