Projection-based reduced order modelling for unsteady parametrized optimal control problems in 3D cardiovascular flows

Background and Objective: 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 hinder the computation of realistic and reliable cardiovascular (CV) haemodynamics by incorporating clinical data such as 4D magnetic resonance imaging. The objective is to control the outflow boundary conditions to optimize CV haemodynamics and minimize the discrepancy between target and computed flow velocity profiles. This paper presents a projection-based reduced order modelling (ROM) framework for unsteady parametrized optimal control problems (OCP(μ)s) arising from CV applications. Methods: 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. We investigate the performance of a projection-based reduction technique that relies on the offline-online paradigm, enabling significant computational cost savings. In this study, the fluid flow is governed by unsteady Navier–Stokes equations with physical parametric dependence, i.e. the Reynolds number. 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. Results: We tested the efficacy of the proposed methodology on vascular models, namely an idealized bifurcation geometry and a patient-specific coronary artery bypass graft, incorporating stress control at the outflow boundary and observing consistent speed-up with respect to high-fidelity strategies. We observed the inter-dependency between the state, adjoint, and control solutions and presented detailed flow field characteristics, providing valuable insights into factors such as atherosclerosis risk. Conclusion: The projection-based ROM framework provides an efficient and accurate approach for simulating parametrized CV flows. By enabling real-time, patient-specific modelling, this advancement supports personalized medical interventions and improves the predictions of disease progression in vascular regions.