Reduction strategies for PDE-constrained oprimization problems in Haemodynamics

In the last two decades several mathematical models and numerical methods have been proposed and progressively used as a tool for supporting medical research and patient treatment in the cardiovascular science. However, the main obstacle to making these models extensively useful and reliable in the clinical context, is that they have to be personalized, i.e. to be patient specific. Fortunately, at the same time in the last years the amount of (patient specific) quantitative data available to the clinician has grown significantly thanks to the advances in medical monitoring and imaging technology. Thanks to these tools, it is now possible to directly get many of the patient specific data needed by the numerical models, for example the patient anatomy. Yet, other quantities required by the numerical simulations, e.g. boundary conditions or model parameters, cannot be always obtained through direct measurements and thus need to be estimated using the available clinical measurements. We are thus faced to the problem of reconstructing unknown (or uncertain) data of a numerical model using partial observations of the model variables, i.e. to data assimilation and inverse problems. The numerical solution of this kind of problems, which are generally referred to as PDE-constrained optimization problems, usually features several computational complexities, since it requires the solution of a coupled system of (possibly non linear) PDEs arising from the optimality conditions. This task becomes even more challenging whenever the state system (or the cost functional to be minimized) depends on a set of parameters and we are interested to solve a PDE-constrained optimization problem for many different scenarios, corresponding to different set of parameter values. This is often the case in the cardiovascular contexts, where the parameter values identify the patient: they can be linked to the geometry of the patient anatomy or to some physiological parameters and clinical measurements. Solving these parametrized optimization problems with high-order discretization techniques, such as the finite element method, may yield an overwhelming computational complexity. Substantial computational saving becomes possible thanks to a reduced order model which relies on the reduced basis method [3]. In this talk, we apply this framework to the approximation of the whole PDEs system arising from the optimality conditions, thus following a optimize-then-discretize-then-reduce approach. Since the proposed reduction strategy is applied to some data reconstruction/assimilation problems arising in the haemodynamics context, we develop the methodology in order to deal with quadratic optimization problems constrained by either advection-diffusion or Navier-Stokes equations, discussing both the theoretical aspects (well-posedness, convergence and a posteriori error estimates) and the computational procedures. The first problem we consider deals with the reconstruction, from areal data provided by eco-doppler measurements, of the blood velocity field across a two dimensional section of a carotid artery [1]. The problem can be seen as a problem of surface reconstruction and it turns out that it can be modeled as a minimization problem for a suitable PDE-penalized least-square cost functional. We consider two sets of parameters: a set of geometrical parameters allowing to describe a wide variety of shape configurations of the vessel section, and a set of parameters related to the measured data. Then we consider some inverse boundary problems for the blood flow in carotid bifurcations/bypass configurations, inspired by the one proposed in [2]. Here we want to reconstruct some missing boundary conditions - in order to retrieve the velocity and pressure fields on the whole domain - starting from measured velocity profiles (which for instance could be given as the outputs of the previous problem) on a portion of the domain. The problem consists in finding the unknown Dirichlet/Neumann data on the boundary which minimize a cost functional penalizing the mismatch between the measured blood velocity and the one predicted by the state model, i.e. the Navier-Stokes equations. As in previous case, we consider both geometrical parameters, thus allowing to deal with different shape configurations, and parameters related to the measured velocity profile. [1] L. Azzimonti et al. Surface estimation via spatial spline models with PDE penalization. In Proceedings of S.Co.2011 Conference, Padova, 2011. [2] M. D'Elia et al. Applications of Variational Data Assimilation in Computational Hemodynamics. In Modelling of Physiological Flows, volume 5 of MS&A Series. Springer, 2011. [3] G. Rozza et al. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng., 2008. [4] G. Rozza et al. Reduction strategies for PDE-constrained optimization problems in haemodynamics. In Proceedings of ECCOMAS 2012, Vienna, 2012.