From Numerical Discretization to Real-Time Simulation: Reduced Order Models in Cardiovascular Hemodynamics

The increasing demand for accurate and efficient cardiovascular simulations in clinical practice highlights the need for the development of efficient computational tools. Although high-fidelity simulations can achieve excellent accuracy, the large cost associated with full order models, especially in complex patient-specific geometries and under varying physical and geometrical parameters, motivates the development of Reduced Order Models (ROMs). This thesis investigates several ROM strategies, including data-driven, equation-based and hybrid approaches, for incompressible fluid dynamics problems in realistic cardiovascular applications and biomedical benchmarks. Moreover, preliminary investigations of ongoing projects start to explore the use of surrogate modeling approaches combined with machine learning techniques for haemodynamics and cardiac electrophysiology. We first introduce a data-driven ROM framework that integrates linear reduction techniques with interpolation strategies, applied to blood flow in coronary arteries. This approach enables efficient parametrization of both physical and geometrical features and its effectiveness is demonstrated in the context of forward and optimal control problems. Next, we extend our investigation to a more challenging vascular domain, the left atrium, where we estimate the time of blood residence starting from forward haemodynamic simulations. Furthermore, we propose a hybrid reduced order modeling approach for blood flow in large vessels, specifically applied to the aortic arch. The method combines classical projection-based ROM techniques with a lifting strategy to incorporate nonhomogeneous, time-dependent pressure boundary conditions at the outlets, often simplified as homogeneous in many existing models. This enhancement allows for a more physiologically accurate representation of the system. In addition, we integrate machine learning techniques into the ROM framework to improve the ability of the model to generalize across different time values. Concerning biomedical benchmarks, we address challenges associated with transitional and turbulent flows in the food and drug administration benchmark nozzle, an idealized medical device that exhibits flow features similar to those of the cardiovascular system. Due to the convection-dominated nature of the problem, we evaluate and compare different stabilization strategies within the classical projection-based ROM, highlighting their strengths and limitations in accurately reproducing time-averaged flow quantities. Finally, we outline several early-stage investigations and exploratory works, including physics-informed modeling and strategies to decompose complex domains for cardiac electrophysiology and time-dependent optimal control formulations for drug delivery and non-Newtonian fluid flow. Overall, this thesis provides a comprehensive and critical perspective on reduced order modeling for cardiovascular applications, highlighting both the mathematical modeling challenges and the practical implications of different ROM techniques across various biomedical contexts