A Unified Model for Blood and Lymph Flow with Coupled Nonsmooth Biochemical Dynamics

TL;DR

This paper develops a comprehensive mathematical model combining fluid dynamics and biochemical regulation to describe lymph and blood flow, capturing valve behavior and oscillatory pumping consistent with experimental data.

Contribution

It introduces a coupled PDE-ODE framework with non-smooth dynamics for lymphatic valve regulation, providing new insights into lymph flow oscillations and stability.

Findings

Model predicts stable limit cycles matching lymphatic pumping

Coupled biochemical and fluid dynamics elucidate valve control mechanisms

Analysis identifies parameter regimes for oscillatory lymph flow

Abstract

We present a unified mathematical framework for modeling blood and lymph flow in biological vessels, with a particular focus on lymph transport through lymphangions. Starting from first principles, we rigorously derive a system of partial differential equations (PDEs) that govern the fluid dynamics using perturbative methods. To capture the active regulation of lymphangion valves, we couple these PDEs with a system of two nonlinear ordinary non-smooth differential equations (ODEs) describing the chemical kinetics of calcium ions and nitric oxide. These biochemical species play a critical role in valve opening and closing, influencing lymph propulsion. We further analyze a reduced model consisting of two non-smooth ODEs, identifying parameter regimes that guarantee the existence of a stable limit cycle. This oscillatory behavior aligns with experimental observations of lymphatic pumping, providing theoretical validation and new insights into lymphatic physiology. Our results offer a comprehensive mathematical description of lymph flow regulation and open possibilities for future studies on pathological conditions and therapeutic interventions.