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

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## 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.

## Keywords

One-dimensional model for lymphatics FitzHugh–Nagumo Collecting lymphatics Lymphangions Lymphatic action potential## Notes

### Acknowledgements

The authors gratefully acknowledge the suggestions given by Prof. Christian Vergara from the Department of Mathematics, Politecnico di Milano, Italy. The authors also acknowledge the excellent work done by anonymous referees that greatly contributed to improve this paper.

### Compliance with ethical standards

### Conflict of Interest

The authors declare that they have no conflict of interests.

## Supplementary material

## References

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