In recent years, numerical simulations of hemodynamics have gained significant attention within the medical community, thanks to their ability of non-invasively estimating the blood flow conditions. However, high-fidelity simulations require extensive computing resources, which limits their use in routine clinical practice. In this regard, Reduced Order Models (ROMs) represent valuable tools, since they deliver sufficiently precise approximations of vascular dynamics at drastically lower computational costs. In medicine, these models proved to be especially useful in parameter estimation and uncertainty quantification tasks, which are crucial for developing accurate personalized computational models. Despite their advantages, ROMs also come with their own set of challenges. This thesis addresses two specific issues: temporal complexity and geometrical variability.
Efficiently dealing with temporal complexity is crucial in cardiovascular applications that entail blood flow simulations over several heartbeats. In such cases, the RB method does not allow to realize important computational gains, because it solely reduces the problem dimensionality in space. Therefore, we introduce space-time RB (ST-RB) methods, that compress the problem dynamics by extending the RB reduction paradigm to the temporal dimension. In particular, we formalize the application of ST-RB methods to the incompressible Navier-Stokes equations and we investigate well-posedness for saddle point problems. Furthermore, we combine ST-RB with a physics-based ROM of fluid-structure interaction, called Coupled Momentum, to conveniently account for vessel wall compliance. The numerical results, conducted on five different test cases, confirm that this approach outperforms the standard RB method in contexts characterized by moderately complex temporal dynamics.
Accurately reconstructing and effectively handling the unique shapes of blood vessels is essential for the development of faithful digital twins. While RB methods can yield precise results under elementary geometry variations, their performances deteriorate in more complex scenarios. Therefore, it emerges the need for model order reduction techniques, specifically designed to deal with the heterogeneity of vessel anatomies. To this aim, we propose a meshless surrogate model of blood flow, named USM4Hemo, based on deep learning techniques. Fundamentally, USM4Hemo solves an operator learning task, approximating the map between the space of geometries and the manifold of hemodynamics solutions. To handle geometric variations, we employ a pre-trained auto-decoder model, which simultaneously delivers two key ingredients: low-dimensional shape encodings and diffeomorphic maps to a universal coordinate system. The former serve as conditioning variables, while the latter enhance the generalization power of the model. USM4Hemo attains promising results on a cohort of synthetic healthy aortic geometries; however, data scarcity currently limits its generalization capabilities to unseen anatomies.