Projection-based Reduced Order Modelling for Unsteady Parametrized Optimal Control Problems in 3D Cardiovascular Flows

TL;DR

A projection-based reduced order modelling (ROM) framework for unsteady parametrized optimal control problems (OCP(μ)s) arising from CV applications that provides an efficient and accurate approach for simulating parametrized CV flows.

Abstract

This paper presents a projection-based reduced order modelling (ROM) framework for unsteady parametrized optimal control problems (OCP$_{(μ)}$s) arising from cardiovascular (CV) applications. In real-life scenarios, accurately defining outflow boundary conditions in patient-specific models poses significant challenges due to complex vascular morphologies, physiological conditions, and high computational demands. These challenges make it difficult to compute realistic and reliable CV hemodynamics by incorporating clinical data such as 4D magnetic resonance imaging. To address these challenges, we focus on controlling the outflow boundary conditions to optimize CV flow dynamics and minimize the discrepancy between target and computed flow velocity profiles. The fluid flow is governed by unsteady Navier--Stokes equations with physical parametric dependence, i.e. the Reynolds number. Numerical solutions of OCP$_{(μ)}$s require substantial computational resources, highlighting the need for robust and efficient ROMs to perform real-time and many-query simulations. Here, we aim at investigating the performance of a projection-based reduction technique that relies on the offline-online paradigm, enabling significant computational cost savings. The Galerkin finite element method is used to compute the high-fidelity solutions in the offline phase. We implemented a nested-proper orthogonal decomposition (nested-POD) for fast simulation of OCP$_{(μ)}$s that encompasses two stages: temporal compression for reducing dimensionality in time, followed by parametric-space compression on the precomputed POD modes. We tested the efficacy of the methodology on vascular models, namely an idealized bifurcation geometry and a patient-specific coronary artery bypass graft, incorporating stress control at the outflow boundary, observing consistent speed-up with respect to high-fidelity strategies.

Figures (15)

• Figure 1: Three-dimensional vascular flow models.
• Figure 2: Streamlines of target, and comparison of uncontrolled and controlled high-fidelity profiles with $Re=50$ at $t=0.25$ s.
• Figure 3: Comparison of uncontrolled and controlled high-fidelity profiles with $\left(Re_1, Re_2 \right) = \left(50, 50 \right)$ at the maximum inlet.
• Figure 4: Normalized POD singular values for velocity, pressure, and supremizers for $Re = 50$.
• Figure 5: Comparison of velocity profiles at $Re = 50$ and $t = 0.5$ s. (a) High-fidelity solution, (b) Reduced-order solution, and (c) Absolute error.