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A hybrid Reduced Order Model to enforce outflow pressure boundary conditions in computational haemodynamics

Pierfrancesco Siena, Pasquale Claudio Africa, Michele Girfoglio, Gianluigi Rozza

TL;DR

This work presents a hybrid POD–Galerkin reduced-order model for cardiovascular haemodynamics that enforces nonhomogeneous outlet pressure via a lifting-function approach and uses a Windkessel model to represent downstream pressures. A neural network interpolates the outflow pressure for times outside the training data, enabling accurate time reconstruction in a time-dependent setting. The method is validated on a 2D idealized vessel and a 3D patient-specific aortic arch, achieving substantial online speedups (up to ~10^5) while maintaining high fidelity to the full-order model. The approach combines physics-based reduced bases with data-driven time-interval interpolation, offering a practical framework for rapid parametric studies in cardiovascular simulations.

Abstract

This paper deals with the development of a Reduced-Order Model (ROM) to investigate haemodynamics in cardiovascular applications. It employs the use of Proper Orthogonal Decomposition (POD) for the computation of the basis functions and the Galerkin projection for the computation of the reduced coefficients. The main novelty of this work lies in the extension of the lifting function method, which typically is adopted for treating nonhomogeneous inlet velocity boundary conditions, to the handling of nonhomogeneous outlet boundary conditions for the pressure, representing a very delicate point in the numerical simulations of the cardiovascular system. Moreover, we incorporate a properly trained neural network in the ROM framework to approximate the mapping from the time parameter to the outflow pressure, which in the most general case is not available in closed form. We define our approach as "hybrid", because it merges physics-based elements with data-driven ones. At full order level, a Finite Volume method is employed for the discretization of the unsteady Navier-Stokes equations while a two-element Windkessel model is adopted to enforce a valuable estimation of the outflow pressure. Numerical results, firstly related to a 2D idealized blood vessel and then to a 3D patient-specific aortic arch, demonstrate that our ROM is able to accurately approximate the FOM with a significant reduction in the computational cost.

A hybrid Reduced Order Model to enforce outflow pressure boundary conditions in computational haemodynamics

TL;DR

This work presents a hybrid POD–Galerkin reduced-order model for cardiovascular haemodynamics that enforces nonhomogeneous outlet pressure via a lifting-function approach and uses a Windkessel model to represent downstream pressures. A neural network interpolates the outflow pressure for times outside the training data, enabling accurate time reconstruction in a time-dependent setting. The method is validated on a 2D idealized vessel and a 3D patient-specific aortic arch, achieving substantial online speedups (up to ~10^5) while maintaining high fidelity to the full-order model. The approach combines physics-based reduced bases with data-driven time-interval interpolation, offering a practical framework for rapid parametric studies in cardiovascular simulations.

Abstract

This paper deals with the development of a Reduced-Order Model (ROM) to investigate haemodynamics in cardiovascular applications. It employs the use of Proper Orthogonal Decomposition (POD) for the computation of the basis functions and the Galerkin projection for the computation of the reduced coefficients. The main novelty of this work lies in the extension of the lifting function method, which typically is adopted for treating nonhomogeneous inlet velocity boundary conditions, to the handling of nonhomogeneous outlet boundary conditions for the pressure, representing a very delicate point in the numerical simulations of the cardiovascular system. Moreover, we incorporate a properly trained neural network in the ROM framework to approximate the mapping from the time parameter to the outflow pressure, which in the most general case is not available in closed form. We define our approach as "hybrid", because it merges physics-based elements with data-driven ones. At full order level, a Finite Volume method is employed for the discretization of the unsteady Navier-Stokes equations while a two-element Windkessel model is adopted to enforce a valuable estimation of the outflow pressure. Numerical results, firstly related to a 2D idealized blood vessel and then to a 3D patient-specific aortic arch, demonstrate that our ROM is able to accurately approximate the FOM with a significant reduction in the computational cost.
Paper Structure (13 sections, 51 equations, 22 figures, 5 tables)

This paper contains 13 sections, 51 equations, 22 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The full order model
  3. Problem formulation
  4. The discrete formulation
  5. The reduced order model
  6. Lifting function method
  7. Proper orthogonal decomposition
  8. Galerkin projection
  9. Neural network interpolation
  10. Numerical results
  11. Case 1: Idealized blood vessel
  12. Case 2: Aortic arch
  13. Conclusions and perspectives

Figures (22)

  • Figure 1: Sketch of the three-element Windkessel model.
  • Figure 2: Schematic flowchart of the ROM framework.
  • Figure 3: Sketch of a feedforward neural network.
  • Figure 4: Case 1: sketch of the domain and of the computational grid.
  • Figure 5: Case 1: Time evolution of the boundary conditions for the velocity (a) and the pressure (b).
  • ...and 17 more figures