Adjoint-based shape optimization for the minimization of flow-induced hemolysis in biomedical applications

Abstract

This paper reports on the derivation and implementation of a shape optimization procedure for the minimization of hemolysis induction in blood flows through biomedical devices.Despite the significant progress in relevant experimental studies, the ever-growing advances in computational science have made computational fluid dynamics an indispensable tool for the design of biomedical devices. However, even the latter can lead to a restrictive cost when the model requires an extensive number of computational elements or when the simulation needs to be overly repeated. This work aims at the formulation of a continuous adjoint complement to a power-law hemolysis prediction model dedicated to efficiently identifying the shape sensitivity to hemolysis. The proposed approach can accompany any gradient-based optimization method at the cost of approximately one additional flow solution per shape update. The approach is verified against analytical solutions of a benchmark problem and computed sensitivity derivatives are validated by a finite differences study on a generic 2D stenosed geometry. The included application addresses a 3D ducted geometry which features typical characteristics of blood-carrying devices. An optimized shape, leading to a potential improvement up to 22%, is identified. It is shown that the improvement persists for different hemolysis-evaluation parameters.

Parameter-free shape optimization: various shape updates for engineering applications

“…[77]. The primal and adjoint pressure-velocity coupling, which has been extensively verified and validated [92,47,52,50,15], follows the SIMPLE method, and possible parallelization is realized using a domain decomposition approach [101,102]. Convective fluxes for the primal [adjoint] momentum are approximated using the Quadratic Upwind [Downwind] Interpolation of Convective Kinematics (QUICK) [QDICK] scheme [92] and the self-adjoint diffusive fluxes follow a central difference approach.…”

“…We include Sobolev gradients into our studies, which can be seen as a well-established concept that is applied in many studies to obtain a so-called descent direction (which leads to the shape update) from a shape derivative, see e.g. [48,15] for engineering and [82,98,99] for mathematical studies. We also look at more recently-developed approaches like the Steklov-Poincaré approach developed in [81] and further investigated in [83,99] and the p-harmonic descent approach, which was proposed in [19] and further investigated in [69].…”

Adjoint shape sensitivities of blood flows considering non‐Newtonian properties

“…Here, we follow the methodology introduced in Reference 36 and frequently employed thereafter 21,26,28 and approximate 𝛿u i on the design wall through a first order Taylor series expansion of the no-slip condition, namely,…”

“…Excessively high stresses induced by peculiarities of the blood flow may damage the red blood cells or even lead to thrombus formation. 21,23 Furthermore, the objective functional is constructed in such a way that the observation region is detached from the design. This allows us to study the impact of upstream shape changes of the graft-where the surgeon can act before or during the operation-on downstream hemodynamics.…”

“…17 Due to its computational efficiency, therefore, the use of the adjoint method has expanded from traditional CFD-based optimization problems, such as optimization of transportation vehicles, [18][19][20] to applications of biomedical engineering. 21,22 However, in the majority of biomedical cases, in which adjoint-based shape optimization is considered, blood is modeled exclusively as a Newtonian fluid in either primal and adjoint or viscosity derivatives are dropped from the adjoint equations. 23 The latter is similar to the simplifying assumption of frozen viscosity that one might find in the adjoint (dual) problem of turbulent flows, for example, see This article is concerned with the derivation of a consistent dual problem to the Navier-Stokes (NS) equations with a generalized Newtonian constitutive model.…”