The lymphatic system plays a crucial role in the human body. It is widely know for its immune function, including production of lymphocytes and filtering pathogens in lymphatic nodes.
However, lymphatic system is also responsible for keeping the fluid homeostasis. The lymphatic vessels collect the extensive fluid (called lymph) from the tissues and transport it back to the blood vessels. Contrary to cardiovascular system, lymphatic system does not have a central organ responsible for pumping the fluid. Instead, lymphatic vessels propel lymph by regular contractions of muscle cells, located in the vessel walls. This process is supported by the valves, that prevent the retrograde motion. The lymphatic vessel segment located between a pair of valves is called lymphangion.
In our work, we focused on modelling the lymph flow through a single lymphangion. The model consists of two main components: fluid dynamics and biochemical dynamics.
Fluid dynamics
The fluid dynamics can be described by Navier-Stokes equations, as a viscous flow through an axisymmetric tube with dynamic boundary. We rigorously derived a system of partial differential equations (PDEs) that govern the fluid dynamics using perturbative methods.
Biochemical dynamics
The lymphangions contractions are regulated by chemical reactions. Among the key regulators of this process are calcium ions (Ca2+) and nitric oxide (NO). Similarly as in blood vessels, Ca2+ influx initiates the contraction of lymphatic muscle cells. Nitric oxide, on the other hand, is the vasodilator – which means it leads to muscle cell relaxation. The interaction of these two substances leads to the regular contraction-relaxation cycle, resulting in lymph propagation.
Some of the Ca2+ and NO production mechanisms are activated when calcium ions level exceeds some threshold. Due to this fact, we described the biochemical dynamics as a non-smooth dynamical system. We managed to find the physiological parameters for which the system exhibits a periodic limit cycle. The existence of these oscillations in the non-smooth formulation shows the robustness of the physiological mechanism and provides a deterministic explanation for the rhythmic contractions observed in real life.
In our future work we will focus on developing an analysis of the coupled flow-biochemistry model along with designing efficient numerical methods to solve the main nonlinear (and possibly) nonlocal system of differential equations.
By Bogna Jaszczak-Dyka and Łukasz Płociniczak (Wrocław University of Science and Technology)