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

Bogna Jaszczak-Dyka, Łukasz Płociniczak

TL;DR

A unified mathematical framework for modeling blood and lymph flow in biological vessels, with a particular focus on lymph transport through lymphangions, is presented, identifying parameter regimes that guarantee the existence of a stable limit cycle.

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.

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

TL;DR

A unified mathematical framework for modeling blood and lymph flow in biological vessels, with a particular focus on lymph transport through lymphangions, is presented, identifying parameter regimes that guarantee the existence of a stable limit cycle.

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.
Paper Structure (13 sections, 4 theorems, 89 equations, 12 figures, 3 tables)

This paper contains 13 sections, 4 theorems, 89 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Derivation of the flow model
  3. Navier-Stokes equations and the constitutive law
  4. Nondimensionalization and the lubrication approximation
  5. The leading-order model
  6. Averaging the Navier-Stokes equations
  7. Valve boundary conditions for the lymphangion
  8. Biochemical dynamics model
  9. Derivation
  10. Analysis of the nonsmooth system
  11. Existence of limit cycles
  12. Bifurcations
  13. Conclusion and future work

Key Result

Theorem 1

Assume eqn:ConditionsOsc. Then, the system eq:alternative_scaling_system exhibits a periodic limit cycle.

Figures (12)

  • Figure 1: A diagrammatic cross-section of the tube.
  • Figure 2: Curves \ref{['eqn:RahbarPressure-Radius']}, \ref{['eqn:ReciprocalPressure-Radius']} fitted to data describing pressure-radius relation.
  • Figure 3: Example realization of $F_{\Sigma^+}.H(x)$ (solid line) and $F_{\Sigma^-}.H(x)$ (dashed line).
  • Figure 4: Three main cases of boundary classification.
  • Figure 5: Three main cases of boundary classification - phase planes.
  • ...and 7 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Definition 4
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 1 more