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Extended one-dimensional reduced model for blood flow within a stenotic artery

Suncica Canic, Shihan Guo, Yifan Wang, Xiaohe Yue, Haibiao Zheng

TL;DR

An adapted one-dimensional (1D) reduced model aimed at analyzing blood flow within stenosed arteries is introduced and a method to extract radial velocity information from the 1D results during post-processing is proposed, enabling the generation of two-dimensional (2D) velocity data.

Abstract

In this paper, we introduce an adapted one-dimensional (1D) reduced model aimed at analyzing blood flow within stenosed arteries. Differing from the prevailing 1D model \cite{Formaggia2003, Sherwin2003_2, Sherwin2003, Quarteroni2004, 10.1007/978-3-642-56288-4_10}, our approach incorporates the variable radius of the blood vessel. Our methodology begins with the non-dimensionalization of the Navier-Stokes equations for axially symmetric flow in cylindrical coordinates and then derives the extended 1D reduced model, by making additional adjustments to accommodate the effects of variable radii of the vessel along the longitudinal direction. Additionally, we propose a method to extract radial velocity information from the 1D results during post-processing, enabling the generation of two-dimensional (2D) velocity data. We validate our model by conducting numerical simulations of blood flow through stenotic arteries with varying severities, ranging from 23% to 50%. The results were compared to those from the established 1D model and a full three-dimensional (3D) simulation, highlighting the potential and importance of this model for arteries with variable radius. All the code used to generate the results presented in the paper is available at https://github.com/qcutexu/Extended-1D-AQ-system.git.

Extended one-dimensional reduced model for blood flow within a stenotic artery

TL;DR

An adapted one-dimensional (1D) reduced model aimed at analyzing blood flow within stenosed arteries is introduced and a method to extract radial velocity information from the 1D results during post-processing is proposed, enabling the generation of two-dimensional (2D) velocity data.

Abstract

In this paper, we introduce an adapted one-dimensional (1D) reduced model aimed at analyzing blood flow within stenosed arteries. Differing from the prevailing 1D model \cite{Formaggia2003, Sherwin2003_2, Sherwin2003, Quarteroni2004, 10.1007/978-3-642-56288-4_10}, our approach incorporates the variable radius of the blood vessel. Our methodology begins with the non-dimensionalization of the Navier-Stokes equations for axially symmetric flow in cylindrical coordinates and then derives the extended 1D reduced model, by making additional adjustments to accommodate the effects of variable radii of the vessel along the longitudinal direction. Additionally, we propose a method to extract radial velocity information from the 1D results during post-processing, enabling the generation of two-dimensional (2D) velocity data. We validate our model by conducting numerical simulations of blood flow through stenotic arteries with varying severities, ranging from 23% to 50%. The results were compared to those from the established 1D model and a full three-dimensional (3D) simulation, highlighting the potential and importance of this model for arteries with variable radius. All the code used to generate the results presented in the paper is available at https://github.com/qcutexu/Extended-1D-AQ-system.git.
Paper Structure (27 sections, 125 equations, 7 figures, 1 table)

This paper contains 27 sections, 125 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Derivation of the extended $1D$ reduced model for blood flow with variable radius
  3. The Navier-Stokes equation in cylindrical coordinates assuming variable radius
  4. Derivation of the system in non-dimensional form
  5. The conservation of mass in nondimensional form
  6. The radial momentum equation in nondimensional form
  7. The axial momentum equation in nondimensional form
  8. The leading order system and its averaged equations
  9. The conservation of mass in reduced form
  10. The axial momentum equation in reduced form
  11. Reduced $(\widetilde{A}, \widetilde{Q})$ system and its reconstruction to the original dimensional form
  12. Recovery of the conservation of mass in dimensional form
  13. Recovery of the axial momentum equation in dimensional form
  14. (A, Q) system of Eqs. (\ref{['quick_summary']}) in the dimensional form
  15. The structure equation to close the system
  16. ...and 12 more sections

Figures (7)

  • Figure 1: The $3D$ mesh representing the domain of a stenotic artery with a 50% narrowing.
  • Figure 2: Angiography of the stenosis with a steep change Shirai2020. (A): Severe stenosis is seen in the mid-portion of the right coronary artery. (B) Total occlusion is observed in the left anterior descending coronary artery.
  • Figure 3: The $3D$ mesh showing the reference domain of stenotic arteries along with their corresponding geometric equations. The mesh size is around 100k tetrahedron elements.
  • Figure 4: The longitudinal velocity $u_z$ obtained from our extended $1D$ model (in green) is compared with the established $1D$ model (in red) and the full $3D$ model (in black) results across all cases. Our extended $1D$ model shows good agreement with the $3D$ results, while the established $1D$ model fails.
  • Figure 5: The pressure $p$ obtained from our extended $1D$ model (in green) is compared with the established $1D$ model (subfigure (d)) and the $3D$ full model (in black) results across all cases. Our extended $1D$ model again shows good agreement with the $3D$ results, while the established $1D$ model does not yield accurate solutions.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Remark 2.1