** Introduction: **This project aims at the design and the analysis of innovative methods for reducing the computational complexity of mathematical models relevant for medical applications, by appropriate surrogating techniques. Although it is now evident the potential role of mathematical modeling in medical applications for many proofs of concept, the real penetration of clinical practice is prevented by many factors. One of those is the high computational costs that the solution of complex problems like medical ones may require and the need of remote high performance computing architectures. These requirements may impair the real impact of scientific computing in clinics. As an example, we mention the data assimilation techniques required to personalize numerical models to patient-specific settings. These problems almost invariably lead to the solution of inverse problems, with constrained minimization - computationally intensive - procedures. More specifically, we mention the identification of arterial compliance based on imaging, the solution of inverse fluid-structure interaction problems, the identification of cardiac conductivities/fibers from ventricular potential measures or the identification of sources in electroencephalography and magnetoencephalography. More in general, all the numerical models developed for medical applications - as for other fields - need to be equipped by Uncertainty Quantification procedures. These require additional computational costs. On the other hand, the concept of accuracy in clinics is different than in mathematics, as it relies on the correct stratification of patients into operative clusters more than on the number of decimal digits of a numerical simulation. Surrogate models, even if not completely correct from the physical point of view, may be an appropriate trade-off between accuracy and efficiency, leading to clinically correct conclusions in reasonable timelines. In this project we aim at developing and customizing model reduction techniques specifically for some medical applications with a clear clinical impact.

The assembled team shares relevant and complementary skills for the success of the project. We have a consolidated expertise on advanced discretisation schemes based on variational approaches, such as h-type finite elements (FEM), isogeometric analysis (IGA), and spectral elements (SEM). Several components of the team have competences in Reduced Order Models (ROM), including Reduced Basis (RB) methods, Proper Orthogonal Decomposition (POD), Proper Generalized Decomposition (PGD) and Hierarchical Model (HiMod) reduction, all representing effective strategies to reduce the computational costs demanded by standard high-fidelity models. Additionally, some members of the team have strict collaborations with medical doctors at hospitals on cardiac and circulatory diseases, so that they will provide the necessary support to transfer the methodological knowledge into the clinical practice.

** Description: **A growing amount of research has been devoted in the last decade to Reduced Order Methods in all the disciplines of applied sciences and engineering facing increasingly complex problems. In this project we focus on medical modeling, thanks to some already achieved expertise of some members of the team. We identify two research lines. (1) As the main purpose of reduced models is to gain efficiency by partially giving up the accuracy of a numerical simulation, we aim at identifying methods that retain the efficiency with an improved level of accuracy. Starting from well-established general purpose methods like Reduced Basis (RB), Proper Orthogonal Decomposition (POD), and other more specific techniques oriented to the solution of problems relevant to medical applications, we aim at exploring and analyzing their combinations for the efficient solution of complex direct problems. This specific aim will be called Direct Modeling (DM). (2) In many problems, the customization of a model to a specific patient as well as the optimization of a therapy/devise requires the solution of assimilation/optimization procedures solving iteratively constrained minimization problems, the constraint being a system of partial differential equations. The rationale of this research line is that the computational costs of these procedures can be affordable if we replace the high-fidelity models with reduced order models, without impairing the quality of the final result. This specific aim will be called Inverse Modeling (IM).

For both DM and IM, we plan to devise new methodologies and to verify them on standard benchmark problems to be meant as demonstrative proofs of concept; successively, we will move to software development in order to efficiently advance scientific computing on modern platforms; last possible step will be a validation of the proposed methodologies on real (patient-specific) data to make a first step towards reliable real-time simulations:

* Methodologies: *DM will be oriented the deep understanding, consolidation and refinement of computational reduction techniques recently employed in medical applications, with particular interest for (i) hemodynamics modeling in stenotic/aneurysmatic vessels and arteries and (ii) modeling cardiac electrophysiology.

For the former, we recall that for modeling blood flow in the arterial network a popular model, alternative to the full 3D Navier-Stokes equations with fluid-structure interaction (currently not tractable in real scenarios beyond limited vascular districts), is given by the Euler equations. In spite of their efficiency, these equations miss any local transversal dynamics that unfortunately may be relevant for clinical reasons (e.g., formation of aneurysms or stenoses). Among other techniques, we are here mainly concerned with the so called Hierarchical Model (HiMod) Reduction, stemming from reformulation of the problem inspired by separation of variables with a different numerical discretization of the different components. Classical finite elements are used for the axial part, while spectral techniques take care of the transverse one. Using spectral methods allows to define a hierarchy of models that may be easily or adaptively tuned to be locally more accurate. After several works on the general assessment of the methodology, we started exploring the combination of HiMod with IsoGeometric Analysis (IGA), a discretization technique based on Non-Uniform Rational B-Spline (NURBS), with very promising results when replacing finite elements for the axial dependence with an IGA discretization. We intend to expand the application of this approach to real vascular geometries, that are in fact reconstructed by image processing tools like Vasculat Modeling ToolKit with splines. The interplay between HiMod and IGA in real scenarios is expected to be very effective.

Another aspect that deserves to be investigated further is the management of nontrivial geometries like bifurcations. The mathematical workaround is to resort to domain decomposition techniques. In this project we plan to extend the basic HiMod approach for cylindrical domains to bifurcated geometries with either overlapping or nonoverlapping approaches. Interface conditions (basically inspired by optimized Schwarz approaches) for this numerical modeling will be properly investigated.

For cardiac electrophysiology modeling, a very well-known paradigm is represented by the bi-domain equations describing the polarization of the cardiac tissue cells at a macro-scale level. This model is often replaced in the literature by the mono-domain equations featuring a lower complexity, the extra- and intra-cellular potentials of the bi-domain model being replaced in the mono-domain model by the transmembrane potential. The rationale of this approximation has poor physical justification, then with a judicious tuning of the parameters it is possible to manage the Monodomain model with reduced costs and good accuracy. The correct identification of the conductivity parameters to tune is a critical step. We plan to use the Proper General Decomposition method for this. The latter is a technique for parametrized partial differential equations, where the parameters are regarded as independent variables and the solution of the resulting problem is achieved by smart techniques of separation of variables. As the identification of the optimal parameters for reproducing the Bidomain solution by the Monodomain model requires testing over several values of the conductivity tensor, we do think PGD is the method of choice. Using PGD in this field is a new idea we intend to explore.

Other research tasks in DM are related with the methodological developments of reduced order tools for the detection of loss of uniqueness of the solution for parametric problems and the related branching of the solutions. Applications of these studies in cardiovascular mathematics are related with the Coanda effect in mitral valves regurgitant flows, for example. A further task is related with the effort of reduced order modelling for parametric moderate turbulent patterns based on the variational multiscale method.

The basic task of IM will be to the assessment of effective Data Assimilation techniques for the parameter identification of patient-specific models, by using Reduced Models to surrogate the constrained minimization process. In particular, we will refer to the so called offline/online paradigm. Parameters might be both physical (material or flow properties, boundary conditions, forcing terms) and geometric (e.g., quantities characterizing the shape of the domain). In all these cases, the first step is the collection of high-resolution approximations for different (physical/geometric) scenarios during a so-called offline phase for different values of the parameters of interest. In the online phase, the collected solutions will be properly combined to guide the identification of the parameters of interest. This is in general pursued by selecting few basis functions, representing the most relevant features of the phenomenon at hand. Reduced order methods proposed in this context include RB, POD, and PGD. Also, a parametric version of the HiMod reduction has been proposed, with the aim of reducing the computational effort typical of a standard offline phase. The idea proposed consists in applying the POD to solutions already reduced via HiMod replacing standard high-resolution approximations. This smart combination of HiMod with POD techniques (HiPOD) features very promising results, yet it deserves deep investigations. Specifically, we are interested in properly combining the different techniques available for the approximation of parametric PDEs in order to exploit the good properties of every method. Preliminary results refer to the combination of RB and HiMod as well as PGD and HiMod. A generalization of these new methods to a nonlinear setting via empirical interpolation is a task of the project.

Efficient methods to tackle parametric models play a crucial role in multi-query contexts, such as optimization, control, uncertainty quantification, general inverse problems, since classical high-fidelity discretisation methods for each new query lead to unaffordable computational costs. We do expect that the ROM considered in the project will provide a benefit in such a direction, furnishing an effective strategy to contain the overall computational burden.

Other tasks are devoted to the development of parameter space reduction techniques, based on active subspace properties and their combinations with computational reduction techniques, like POD. Concerning IM other contributions are expected in reduced order methods for optimal flow control, as well as multiphysics like fluid-structure interaction.

*Software development:** *We plan to produce codes which implement the new ROM approaches of the project in software libraries already developed by the different components of the team. The basic library for Finite Elements will be LifeV (developed by a consortium of different groups worldwide) where HiMOD has been implemented too. Moreover the RBniCS library (open source), based on FEniCS, will be improved, as well as the PyGeM package for the geometrical parametrization and deformation (http://mathlab.sissa.it/cse-software).

*Validation:** *A critical part of the assessment of the new methodologies is the validation in practical contexts. As for the two main applications of the project, we plan to test the performances of ROM in computational hemodynamics and electrocardiology:

(a) In collaboration with the Hospital Giovanni XXIII in Bergamo (Dr. G. Guagliumi), we will collect images of coronary arteries and we will perform computational fluid dynamics analyses with traditional techniques and with HiMod combined with IGA. The computational advantages and the accuracy will be assessed over a number of different patients. We will start with mono-cylindrical domains and move to bifurcated coronary arteries. The clinical quality of the results will be measured in terms of predicted Fractional Flow Reserve, i.e., the pressure drop along the coronary branch.

(b) In collaboration with the Campus Biomedico in Rome (Dr. A. Gizzi), we will take advantage from potential measured in animal specimens to estimate cardiac conductivities to validate the PGD procedure. As data are already available, we do not envision any particular troubles in the data collection. We will start with PGD parameter estimation on slabs of cardiac tissue to move later on more realistic geometries.

(c) In collaboration with Sunnybrook Hospital (Dr. L. Jimenez) and University of Toronto (Prof. P. Triverio) on numerical simulation of blood flows in aorto-coronary bypass.

*Conclusive Statement:** *This project will open new scenarios, making possible the solution of complex problems of interest in medical modeling, with a significantly reduced computational effort, towards real-time computing practice. Nevertheless, at a more general level, we expect that the techniques devised here may be beneficial for other applications in fluid-dynamics, electromagnetism, structural mechanics, and, more in general, in multiphysics problems in computational science and engineering.

**Simona Perotto (P.I), MOX, Dipartimento di Matematica, Politecnico di Milano.**

__Responsabile:____Alessandro Reali (Universit`a degli Studi di Pavia), Simone Morganti (Universit`a degli Studi di Pavia), Alessandro Veneziani (Istituto Universitario di Studi Superiori di Pavia), Gianluigi Rozza (SISSA mathLab Trieste), Luca Heltai (SISSA mathLab Trieste), Ana Alonso Rodrıguez (Universit`a degli Studi di Trento), Alberto Valli (Universit`a degli Studi di Trento), Yves Antonio Brandes Costa Barbosa (Politecnico di Milano), Massimo Carraturo (Universit`a degli Studi di Pavia), Michele Giuliano Carlino (Istituto Universitario di Studi Superiori di Pavia), Francesco Ballarin (SISSA mathLab Trieste), Monica Nonino (SISSA mathLab Trieste), Federico Pichi (SISSA mathLab Trieste), Giovanni Stabile (SISSA mathLab Trieste), Maria Strazzullo (SISSA mathLab Trieste), Marco Tezzele (SISSA mathLab Trieste), Zakia Zainib (SISSA mathLab), Matteo Zancanaro (SISSA mathLab Trieste).__

*Participants:*