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

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.