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Data-driven approximation of haemodynamics by combined reduced order modeling and deep neural networks

The use of numerical modeling and scientific computing in the fields of biomedical engineering has steadily grown in the last decade, providing novel means for the study of human physiological and pathological conditions. This is mainly due to the recent advancements in numerical methods for the solution of the underlying complex phenomena and the ever-increasing availability of high-performance computing resources. The study of the cardiovascular system is one of the many applications of these technologies, which can also be employed in the clinical setting for the diagnosis and treatment of numerous diseases. For example, when considering an arterial aneurysm, a reliable prediction of its risk of rupture is a key factor in support of medical decisions. Indicators of the risk of rupture are not only the size and other geometrical characteristics of the aneurysm, but also the blood flow patterns and induced stresses, which have an effect on the progressive weakening of the arterial wall [@piccinelli09; @sangalli09]. In this respect, the numerical simulation of blood flow in pathological patient-specific conditions represents a non-invasive approach for understanding the process and possibly improving the risk assessment of aneurysms rupture. Such simulations involve the approximation of the Navier-Stokes equations, or of more complex Fluid-Structure Interaction (FSI) models describing the mutual interaction of the blood flow with the deformable arterial wall. All these models are derived as systems of non-linear and time-dependent Partial Differential Equations (PDEs). An accurate approximation of the solution of the considered PDEs can be obtained by employing a high-fidelity model, as the finite element method, which involves the solution of a non-linear system, typically of large dimension. Through PDE models, a remarkably accurate approximation of the blood dynamics is computed, providing access to significant quantities which are of critical medical interest, such as the wall shear stress on the artery wall.

The procedure for numerically solving such kind of high-fidelity models typically involves several steps, namely the generation of a computational domain geometry, the (repeated) assembly and solution of a large linear system, and the post-processing of the results. Because of the low degree of automatization that, nowadays, is involved in the generation of computational meshes from medical images, and depending on the required level of accuracy, the first two steps of the pipeline are usually too time-consuming to make high-fidelity numerical simulations a viable option in many practical cases, since they do not allow for a timely evaluation which is often needed in clinical situations.

We are interested in numerical modeling and scientific computing in the fields of vascular flows, in particular for the computation of flow indicators for risk factors, for example for the evaluation of the risk of rupture of an anerusims, or the risk of heart-stroke. We aim at developing numerical methods to greatly improve the applicability of these methodologies in clinical settings. The project builds on the following main keypoints.* Reduced Basis and Deep Neural Networks for time-dependent parametrized PDEs: we plan to extend our work presented in [@dalsanto2019pdednn] to the time dependent case. The Reduced Basis (RB) method allows to efficiently solve parametrized PDEs by reducing the dimensionality (i.e. the number of variables involved) of the high-fidelity model. Through its coupling with Deep Neural Networks (DNNs), in [@dalsanto2019pdednn] we show that we are able to incorporate measured data in the simulation: this allows to recover the physical parameters of the problem and - as a result of the proven robustness of DNNs with respect to noisy input - to alleviate the effects of uncertainty. In heamodynamics applications the time dependence plays a central role and the extension will therefore require careful settings and analysis. * Reduced Basis and Domain Decomposition methods for modular approximation of arteries: the underlying idea is to approximate the geometry of arteries as a composition of building blocks, such as tubes and bifurcations which are obtained from parametrized deformations of reference geometries. The advantage with respect to employing high-quality meshes of arteries is the development of automatic algorithms to generate such approximated geometries directly from medical images and the fast solution of the equations deriving from the RB method. For this reason, this technique can be considered a valid alternative to the popular geometrical multiscale models (see e.g. [@malossi2013implicit; @blanco2013continuity; @malossi2011algorithms]) for the approximation of the flow in large portions of arteries. As we shall discuss in Section 2.1.2, the modular approach based on the RB method was originally proposed in [@Maday2002195] and further developments have been carried out in [@iapichino2012reduced; @iapichino2014reduced; @iapichino2016reduced]. All these efforts focus on two-dimensional problems and often on the (linear) Stokes equations; in our research, we plan instead to tackle the three-dimensional Navier-Stokes equations and reduced FSI models [@figueroa2006coupled; @figueroa2009computational].