# Valve-induced nonlinearity in the lymphatic junctions

Alaa Bou Orm

► To cite this version:

Alaa Bou Orm. Valve-induced nonlinearity in the lymphatic junctions. Multidisciplinary Biomechanics Journal, 2025, 49th congress of the Société de Biomécanique, 10.46298/mbj.14577. hal-04738070v2

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Submitted on 31 Jan 2025

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# Valve-induced nonlinearity in the lymphatic junctions

Alaa Bou Orm*

BMBI, UMR CNRS 7338, UTC, Compiègne, France

* Corresponding author: alaa.bou-orm@utc.fr

Received date: 07/04/2024

Accepted date: 28/06/2024

Publication date: 31/01/2025

**Keywords:** lymph, lymphatic valve, fluid-structure interaction, computational fluid dynamics

© 2025 The Authors
Licence CC-BY 4.0
Published by Société de Biomécanique

## 1. Introduction

The lymphatic system is little known and less studied compared to the cardiovascular system. It is only recently that it has attracted the attention of scientists because of its important link with cancer proliferation. The lymphatic system consists of initial lymphatic vessels, which absorb lymph liquid, and collecting lymphatic vessels, which drain the lymph through the lymphatic network and its nodes (Moore and Bertram 2018; Padera et al. 2016). Unlike the cardiovascular system, whose heart, acts as the central pump, that circulates blood in our body, the lymphatic network does not have a central pump. The pumping of the lymph is ensured thanks to synchronization between the contractions-relaxations of the walls of the lymphatic vessels and the opening-closing of the lymphatic valves, composed of two leaflets. These valves prevent backflow and ensure the unidirectional motion of the fluid. A malfunction of this system can lead to conditions such as lymphedema, which is a serious bad side effect in some cases of cancer treatment and currently has no cure (Scallan et al. 2016; Zawieja 2009).

We are developing a fully coupled fluid-structure computational model using the lattice Boltzmann method to compute the fluid and the immersed boundary method to accomplish the fluid-structure interaction. The results obtained from 2D simulations for the performance of a T-junction are presented in this article.

## 2. Methods

Several models and numerical methods are used to model and capture the correct mechanics of the valves under flow (Li et al. 2019; Ballard et al. 2018; Wolf et al. 2021; Li et al. 2022; Bertram et al. 2014). In addition to the techniques already proposed in the literature, we suggest the use of bicuspid valves, composed of two leaflets, to prevent reflux. This study is motivated by recovering properly the dynamics of the lymphatic valves and that involves in the pumping of the lymph liquid through the network of the lymphatic vessels. We are developing a fully coupled two-dimensional fluid-structure computational model to simulate the dynamics of a two-leaflet valve under channel flow. The lattice Boltzmann method (LBM) (Succi 2001) is used to compute the fluid flow using the lbm2d library (Kaoui 2024), and the immersed boundary method (IBM) (Peskin 2002) to couple the lymph flow with the mechanics of the valves. The valves are modeled with a spring network, where adjacent points are interconnected with a Hookean spring, with the stretching force and the restoring force.

## 3. Results and discussion

We focus our study on the fluid behaviour in a complex network of collecting lymphatic vessels. This study aims to understand the nonlinearity in pressure loss due to the presence of the valves and offers insights into their role in vessel mechanics. This helps to better understand the pumping of lymph in the collecting
lymphatic vessels, which involve bifurcations and junctions that converge at the drainage channel. To do this, we started with a simple T-junction geometry with two inlets and one outlet with a valve at each inlet and at the outlet. We consider the vessel walls to be rigid, as we are primarily interested in the mechanics of the valve subjected to flow. The valves are deformable structures. Flow rates $Q_1$, $Q_2$ and $Q_{out}$ and pressures $P_1$, $P_2$ and $P_{out}$ are measured respectively at the inlets and at the outlet. We define $\Delta P = P_1 - P_{out}$ for representing the reference pressure difference between the inlets and the outlet of the tested geometry. The Poiseuille law that relates the flow rate $Q$ directly to the pressure drop $\Delta P$ by $\Delta P = RQ$ where R is the hydraulic resistance. In figure 1, panel a) shows the pressure field and the velocity's vectors developed in such geometry in presence of the valves. The pressure increases at the inlets where the valves are located. The panel b) shows the variation of the rate flow as a function of the pressure gradient for the two cases: with and without valve in the T-junction geometry and in the channel. In the case of having no valve, the flow rate increases linearly with the pressure. Whereas for the case with the valves, the flow rate increases in a nonlinear way with the pressure gradient, which reflects the effect of the presence of valves. The panel c) shows the distance of the gap between the two leaflets of the valve at the inlets and at the outlet. The gap increases nonlinearly with pressure gradient, similar to the flow rate. Furthermore, the effect of the variation of the elasticity of the valve is also investigated in such geometry, as we can see in panel d). We notice that by increasing the elasticity of the valve, the pressure difference increases. More the valves are rigid, more their resistance is bigger.

## 4. Conclusions

We developed a numerical method in 2D that is capable of predicting the dynamics of lymphatic valves. These results suggest that the valve stiffness have an effect on overall lymph pumping performance. The outcome of this study will be used to further investigate the lymphatic valves dynamics encountered in the lymph pumping, and to elucidate the causes and emergence of lymphedema.

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 1.** a) pressure and velocity in the T-junction in presence of the valves, b) the flow rate as a function of the applied pressure gradient in the T-junction, and in the channel with and without valves, c) the gap between the two valve's leaflets at the inlets and the outlet, d) effect of the variation of the valve elasticity.</figcaption></figure>
## Acknowledgements

I would like to thank my PhD supervisor Dr. Badr Kaoui for his help.

## Funding

This work is financed and supported by the ANR (ANR-20-CE45-0008-01).

## Conflict of Interest Statement

None.

## References

Ballard M *et al*. 2018. Probing the effect of morphology on lymphatic valve dynamic function. *Biomechanics and modeling in mechanobiology*, 17:1343-1356.

Bertram C D *et al*. 2014. Development of a model of a multi lymphangion lymphatic vessel incorporating realistic and measured parameter values. *Biomechanics and modeling in mechanobiology*, 13:401-416.

Kaoui B. 2024. https://gitlab.utc.fr/kaouibad/lbm2d

Li H *et al*. 2019. The effects of valve leaflet mechanics on lymphatic pumping assessed using numerical simulations. *Scientific reports*, 9:10649.

Moore Jr J E, Bertran C D. 2018. Lymphatic System Flows. *Annual Review of Fluid Mechanics*, 50:459-82.

Padera T P, Meijer E F J, Munn L L. 2016. The lymphatic System in Disease Processes and Cancer Progression. *Annual Review of Biomedical Engineering*, 18:125-58.

Peskin C S. 2002. The immersed boundary method. *Acta Numerica*, 11:479-517.

Scallan J P *et al*. 2016. Lymphatic pumping: mechanics, mechanisms and malfunction. *The journal of physiology*, 5749-5768.

Succi S. 2001. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond (Oxford University Press, Oxford, UK).

Wolf K T, Dixon J B, Alexeev A. 2021. Fluid pumping of peristaltic vessel fitted with elastic valves. *Journal of fluid mechanics*, 918 A28-1.

Zawieja D C. 2009. Proceedings of a Mini-Symposium: Lymphedema: An Overview of the Biology, Diagnosis, and Treatment of the Disease Contractile Physiology of Lymphatics. *Lymphatic research and biology*, 7:87-96.