Optimal Control and Shape Optimization of Aorto-Coronaric Bypass Anastomoses

Abstract

In this paper we present a new approach in the study of Aorto-Coronaric bypass anastomoses configurations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary. The aim is to provide design indications in the perspective of future development for prosthetic bypasses. With a reduced model based on Stokes equations and a vorticity functional in the down field zone of bypass, a Taylor like patch is found. A feedback procedure with Navier-Stokes fluid model is proposed based on the analysis of wall shear stress and its related indexes such as OSI.

Shape optimization for viscous flows by reduced basis methods and free‐form deformation

“…Nevertheless, we can consider this shape optimal only for the Stokes model and just optimized for the Navier-Stokes model. Previous feedback results were provided in [33]. We report in Fig.…”

“…In this work we want to improve the results obtained in [33] by developing new tools of model order reduction (not considered in [1,33]), and formulating the problem as a suitable parametrized shape optimization problem (as will be explained in Sect. 4).…”

“…The reader interested in medical and bio-engineering aspects related to the aorto-coronaric bypass problem can refer to Appendix A. In the perspective of using low order methods for shape optimization featuring reasonable computational time, we adopt the steady Stokes equations (rather than the more appropriate NavierStokes equations, which are going to be used as "high fidelity" validation in feedback) for modelling low Reynolds blood flow in mid size arteries [13,33], like e.g. the coronary arteries.…”

“…Lipschitz domain} and D is a given fixed rectangle of area |D|; first order necessary conditions can be derived by either introducing a suitable adjoint problem [6,33,34] or using the approach based on Lagrange multipliers [16].…”

A reduced computational and geometrical framework for inverse problems in hemodynamics

“…On the other hand, whenever some parameters are uncertain, we aim at inferring their values (and/or distributions) from indirect observations (and/or measures) by solving an inverse problem: given an observed output, can we deduce the value of the parameters that resulted in this output? Such problems are often encountered in cardiovascular mathematics as problems of parameter identification [7,70], variational data assimilation [13,50,63,66], or shape optimization [1,44,51,64]. Computational inverse problems are characterized by two main difficulties:…”