Title: 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 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.