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We have presented some optimal design problems aimed at improving the shape of cardiovascular prostheses. Two different worst-case optimization formulations have been proposed to solve the problem of bypass design under uncertainty, in order to reduce the downfield vorticity: (i) a boundary control formulation, which simplifies the geometry and treats only the angle of the anastomosis as a boundary control variable, and (ii) a shape optimization formulation relying on a FFD shape parametrization to represent the bypass anastomosis. Thanks to computational reduction allowed by the reduced basis method, the numerical solution of these problems can be greatly enhanced, above all in the robust optimization case.

In the optimal control case (Sect. 5.3) we have considered a simplified two-dimensional bypass configuration, parametrized with respect to the anastomosis angle and the residual flow through the occluded artery. Four different cost functionals taken from literature and suited for the reduction of downstream vorticity were studied. The optimal anastomosis angle was found to depend strongly on the total residual flow from the occluded branch, but not as strongly on the particular shape of the flow profile. Some numerical tests confirm the robustness of the obtained anastomosis angle with respect to the unknown residual incoming flows. We validated the 2D model by comparing the results obtained against a 3D boundary control problem. Three dimensional effects were found not to have a large impact on the total downstream vorticity at moderate Reynolds numbers. The largest vorticity was observed for the case of total occlusion in the host artery, so that a robust bypass shape should be tailored for that particular situation.

Similar conclusions were drawn, for two different bypass configurations, by considering a shape optimization problem (Sect. 5.4). In this case, a FFD shape parametrization provides a strong reduction in geometrical complexity – in term of the number of parameters, reduction is of about

100 times with respect to traditional shape parametrization based on local boundary variation – yet allowing to describe a wide family of shapes. Moreover, active control points can be selected according to suitable ad hoc procedures. Also in the shape optimization case, different cost functionals proposed for the reduction of downstream vorticity were studied. The condition leading to the strongest development of vorticity cores is the presence of a complete occlusion, for which the flow through the bypass starts creating a strong vortex in the proximity of the anastomosis. Regarding the robust shape optimization, we find in both the aorto-coronary and the femoro-popliteal case that a bypass should be designed mainly for the case of total occlusion, in order to be robust over a range of possible residual flows.

The models we have presented are very simplified, since they do not take into account important features like the pulsatility of the flow and the periodic detachment/separation of the flow layer from the wall. Extension of the proposed applications to (i) more realistic three-dimensional configurations and (ii) unsteady flows, already taken into consideration in several problems solved by traditional discretization schemes (also in presence of uncertainties), represent the focus of future research work, parallel to the development of the reduced basis methodology to more complex unsteady and/or three-dimensional fluid dynamics problems.

6

Inverse parametrized problems

for blood flows

In this chapter we present a second group of applications dealing with inverse identification problems related to blood flows in parametrized geometrical configurations. Our goal is to investigate the dependence of simple blood flows by shape parameters using the reduced framework discussed in Parts I and II. After a brief survey on the interplay between blood flows and geometrical features of the vessels, we present some possible strategies to represent realistic vessel geometries based on suitable shape parametrization techniques. Thanks to shape parametrization, we can deal with both simple shape identification problems and more realistic parametric coupling problems for representing fluid-structure interactions. Then, three inverse identification problems are presented and discussed: (i) identification of arterial wall and flow properties aiming at atherosclerosis risk assessment in a deformable stenotic artery; (ii) identification of residual flows in a bypass model; (ii) identification of shape features related to pathologies. Both a deterministic and a statistical inverse framework are employed, fitting in both cases the many-query opportunity context which has driven the development of our reduced framework.

6.1

Parametrized problems of interest in haemodynamics

As already pointed out in chapter 6, a strong mutual interaction exists between haemodynamic factors and vessels geometry. On the one hand, flow adapts to the vessel shape; on the other hand, it also exerts forces affecting the vessel behavior and its morphology, resulting in a closed-loop mechanism. For instance, plaques occur preferentially in regions where arteries bend and in the vicinity of branches, but at the same time the changes in vessel shape due to this pathology clearly affect blood flows. Nevertheless, this interaction is far from being completely understood. Thus, improving the knowledge of the interplay between flows and geometries may be useful not only for the sake of design of better prosthetic devices, but also to characterize physical and geometrical properties of the flows which may be related for instance to pathological risks, such as in the case of narrowing or thickening of an arterial vessel.

It is therefore interesting to analyze the sensitivity of physical outputs related to viscous flows – such as wall shear stresses, wall shear stress gradients, vorticity, dissipated energy – with respect to shape morphology and possible variations. All these indices are influenced by lumen geometry, characterize local haemodynamics effects and may assess a risk of artery occlusion. Hence, rather than numerical simulations on a relatively small number of different configurations, it becomes crucial to explore a wide family of geometries – thus spanning a broad variety of shapes – in order to take into account their variability (for instance, among patients) and provide a more complete representation of blood flows and related outputs with respect to shape variation.

This may yield to the solution either of a forward problem – for any geometrical configuration, we wish to characterize the flow – or of a more complex inverse problem – where we wish to identify some properties, for instance related to the geometry, from data measurements. Thus, we need to face both with real-time problems (whenever, given a new geometry, we are interested to obtain a numerical approximation of the flow in a very small amount of time) and many-query contexts (because of the need to span a large set of configurations, or to solve an inverse problem), for which suitable model reduction techniques are requested, since full simulations may result very expensive if they have to be carried out from the beginning for each new geometry.

In general, whenever some parameters related to the flow can be isolated as representative features, parametrized frameworks and reduction techniques provide an excellent computational opportunity in view of sensitivity studies. Several works have focused on flow parameters such as Reynolds, Dean or Womersley numbers [58, 85], aiming at the description of three-dimensional characteristic flow patterns occurring where vessels bend, bifurcate or narrow. Other cases of significative interest can be the blood flow analysis with respect to inflow/outflow boundary conditions [20], or the sensitivity analysis of optimal shapes of artificial grafts with respect to flow parameters [323, 114, 252]. Since we basically deal with reduced basis simulations of two-dimensional flows, we do not take into account the effect of flow parameters such as Reynolds or Dean numbers. Nevertheless, the reduced framework can be easily extended in view of dealing with this kind of features [79, 168], provided that the truth approximation beneath can provide reliable numerical simulations of this kind of flows.

Besides parametric studies, inverse identification problems have been playing an ever increasing role in haemodynamics, mainly because of the huge amount of data (provided for instance through measures or images) that can be merged into a numerical model in order to get improved simulations. In this framework – the so-called data assimilation procedures [76] – several problems can be faced. However, our focus consists in exploiting the reduced framework analyzed in chapter 4 for the solution of inverse parametrized problems, where the goal is the identification of parameter values or combinations leading to some observed or measured flow conditions – for the time being, through the assimilation of some (surrogate) output of interest. Although restrictive (we will not deal neither with distributed observations of field variables, nor with the identification of physical or geometrical fields), this class of problems can be seen as a first step towards the integration of reduction strategies into more general inverse problems.

As motivated above, in the examples presented in this chapter we focus on parametrized flows with respect to inflow boundary conditions, to geometrical configuration of blood vessels and to physical properties related to the arterial wall. This choice is dictated by the capabilities of our current reduced approximation framework and motivated by the investigation of inverse problems related with shape variation fitting our parametrized context. In particular, we aim at:

1. providing a real-time simulator for blood flows in parametrized realistic geometries, un- der some modelling simplifications, possibly supported by some shape identification pre- processing stage;

2. performing parametric investigations of outputs of clinical interest over a range of possible configurations;

3. identifying flow and/or shape parametric features from surrogate measures of physical outputs through a suitable inverse identification process.

Since observations and measures are usually affected by experimental noise, we also want to deal with some basic issues arising in uncertainty quantification, for instance in order to quantify how these noise sources affect the predicted quantities of the simulations.

Results of the following sections are based on two works carried out in collaboration with Quarteroni, Rozza [205] and also with Lassila [180].