[Paper Review] Analysis of blood flow in one dimensional elastic artery using Navier-Stokes conservation laws

Cristóbal Rodero|arXiv (Cornell University)|Oct 15, 2017

Navier-Stokes equation solutions68 references3 citations

TL;DR

This paper develops a one-dimensional blood flow model in elastic arteries using the Navier-Stokes equations and tube laws, applying the method of characteristics to analyze shock wave formation. It presents an original analytical result identifying precise conditions for shock wave development and validates the model via discontinuous Galerkin finite element simulations with biologically realistic parameters.

ABSTRACT

In the last years, medical computer simulation has seen a great growth in several scientific branches, from modelling to numerical methods, going through computer science. The main goals of this incipient discipline are testing hypotheses before an intervention, or see what effect could have a drug in the system before actually taking it, among others. In this work we deduce from the most basic physical principles a one dimensional model for the simulation of blood flow in elastic arteries. We will provide some historical background, as well as a brief state of the art of these models. We will also study from a calculus point of view the equations of the model obtained, achieving an original result for the formation of shock waves in compliant vessels. Afterwards we will make some numerical simulations using Galerkin Discontinuous Finite Element Method. Since this is actually a family of methods, we will motivate and detail the elections and the implementation strategies.

Motivation & Objective

• To develop a one-dimensional mathematical model of blood flow in compliant arteries based on fundamental physical principles.
• To analyze the theoretical behavior of the model, particularly the formation of shock waves in elastic vessels.
• To implement and validate the model numerically using the discontinuous Galerkin finite element method with realistic hemodynamic parameters.
• To explore the feasibility and stability of the model under varying physiological conditions and parameter sensitivity.
• To provide a foundation for future personalized hemodynamic simulations and disease characterization.

Proposed method

• Derives the one-dimensional Navier-Stokes equations for blood flow in elastic tubes using conservation of mass and momentum.
• Incorporates tube laws relating vessel radius to pressure and flow, based on vessel elasticity and wall mechanics.
• Applies the method of characteristics to analyze wave propagation and derive conditions for shock wave formation.
• Uses the discontinuous Galerkin (DG) finite element method for spatial and temporal discretization, with numerical flux and basis function selection.
• Implements a semi-discrete DG formulation with Lebesgue measure notation and local coordinate transformations.
• Performs numerical simulations using biologically realistic parameters from the literature, including blood density, viscosity, and vessel elasticity.

Experimental results

Research questions

• RQ1 Under what physical and geometric conditions does a shock wave form in a compliant arterial vessel according to the 1D model?
• RQ2 How do variations in hemodynamic parameters (e.g., pulse rate, vessel stiffness) affect blood flow velocity and pressure waveforms?
• RQ3 What is the convergence and stability behavior of the discontinuous Galerkin method when applied to this 1D blood flow system?
• RQ4 How do different choices of test and basis functions, and numerical fluxes, affect the accuracy and robustness of the simulation?
• RQ5 To what extent can the model predict hemodynamic responses under personalized or pathological conditions?

Key findings

• The study derives an original analytical condition for shock wave formation in compliant arteries, identifying a precise time and location where wave steepening occurs.
• Numerical simulations using the discontinuous Galerkin method show stable and convergent behavior across varying mesh sizes and time steps.
• The condition number of the mass matrix grows linearly with polynomial degree when using Legendre polynomials, indicating good conditioning for high-order methods.
• Maximum arterial amplitude and blood flow velocity vary significantly with pulse rate (BPM) and vessel stiffness parameter β, as shown in parametric simulations.
• The model successfully captures wave propagation and discontinuity evolution, with discontinuity measures decreasing under refined spatial and temporal discretizations.
• The framework enables biomedical simulations sensitive to key parameters, supporting future applications in personalized hemodynamics and disease modeling.