[Paper Review] Mathematical modelling of the cardiovascular system

Alfio Quarteroni|ArXiv.org|May 1, 2003

Elasticity and Material Modeling15 references18 citations

TL;DR

This paper presents a mathematical framework for simulating blood flow in large and medium-sized arteries using fluid-structure interaction models based on the Arbitrary Lagrangian Eulerian (ALE) formulation of the Navier-Stokes equations. It introduces a monolithic, partitioned iterative algorithm for coupling fluid dynamics with vessel wall mechanics, and proposes a multiscale modeling approach integrating 3D, 1D, and lumped-parameter models to simulate the entire circulatory system with improved accuracy and computational efficiency.

ABSTRACT

In this paper we will address the problem of developing mathematical models for the numerical simulation of the human circulatory system. In particular, we will focus our attention on the problem of haemodynamics in large human arteries.

Motivation & Objective

• To develop a robust mathematical and numerical framework for simulating blood flow in compliant large arteries, accounting for dynamic vessel wall motion.
• To address the challenge of fluid-structure interaction in moving domains using the Arbitrary Lagrangian Eulerian (ALE) formulation for accurate domain mapping.
• To enable clinically relevant simulations of hemodynamic phenomena such as flow separation, shear stress, and post-surgical outcomes by coupling detailed 3D models with simplified circulatory models.
• To establish a hierarchical multiscale modeling strategy that integrates 3D fluid-structure interaction, 1D hyperbolic systems, and lumped-parameter models for whole-circulation simulation.
• To ensure numerical stability and accuracy in coupled simulations through a strongly implicit, iterative partitioned algorithm with sub-iteration and mesh deformation via harmonic extension.

Proposed method

• Formulates the Navier-Stokes equations in the ALE framework to handle time-dependent fluid domains with moving boundaries, particularly the compliant vessel wall.
• Uses a monolithic approach to couple the incompressible Navier-Stokes equations with a structural model of the vessel wall, where wall displacement and velocity are solved simultaneously with fluid velocity and pressure.
• Employs a partitioned iterative algorithm with sub-iterations to enforce kinematic and dynamic consistency between fluid and structure at the interface, using extrapolated wall motion and velocity to update the ALE mapping.
• Applies harmonic extension to compute the ALE mapping from the wall displacement, ensuring smooth mesh deformation in the fluid domain without excessive distortion.
• Integrates 3D fluid-structure interaction models with 1D reduced models (hyperbolic conservation laws) and lumped-parameter models (ODE systems analogous to electrical circuits) for global circulatory system representation.
• Implements a convergence check based on L2 norms of displacement and velocity differences between iterations, with under-relaxation to stabilize the iterative process.

Experimental results

Research questions

• RQ1 How can the fluid-structure interaction problem in large arteries be accurately modeled when the vessel wall deforms under pulsatile blood flow?
• RQ2 What is the most stable and accurate numerical strategy for coupling the Navier-Stokes equations with structural mechanics models in a moving domain?
• RQ3 How can multiscale modeling techniques be used to simulate the entire circulatory system while preserving local hemodynamic accuracy in regions of interest?
• RQ4 What role does the choice of coupling algorithm (e.g., partitioned vs. monolithic) play in the stability and convergence of fluid-structure simulations?
• RQ5 How can the effects of pathological conditions such as stenosis be quantitatively assessed through numerical simulation of altered hemodynamics?

Key findings

• The ALE formulation enables accurate and stable simulation of blood flow in time-varying domains, particularly when the vessel wall undergoes up to 10% radial deformation.
• The partitioned iterative algorithm with under-relaxation ensures convergence and stability, with the requirement of strong coupling at the discrete level to prevent numerical instabilities.
• The use of harmonic extension for mesh deformation maintains mesh quality and avoids excessive distortion, even over long simulation intervals.
• The proposed multiscale framework successfully couples 3D fluid-structure interaction models with 1D and lumped-parameter models, enabling whole-circulation simulations with reduced computational cost.
• The method allows for clinically relevant simulations, such as coronary bypass hemodynamics, by capturing the influence of geometry on flow patterns and wall shear stress.
• The algorithm is computationally expensive due to repeated solution of fluid, structure, and mesh deformation problems at each sub-iteration, but is necessary for stability in complex hemodynamic scenarios.