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Shape Optimization of hemolysis for shear thinning flows in moving domains

Valentin Calisti, Šárka Nečasová

TL;DR

This paper shows that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain of the blood flow, namely the minimization of red blood cells damage.

Abstract

We consider the $3$D problem of shape optimization of blood flows in moving domains. Such a geometry is adopted to take into account the modeling of rotating systems and blood pumps for instance. The blood flow is described by generalized Navier-Stokes equations, in the particular case of shear-thinning flows. For a sequence of converging moving domains, we show that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain. Thus, we extended the result given in (Sokołowski, Stebel, 2014, in \textit{Evol. Eq. Control Theory}) for $q \geq 11/5$, to the range $6/5< q < 11/5$, where $q$ is the exponent of the rheological law. This shape continuity property allows us to show the existence of minimal shapes for a class of functionals depending on the blood velocity field and its gradient. This allows to consider in particular the problem of hemolysis minimization in blood flows, namely the minimization of red blood cells damage.

Shape Optimization of hemolysis for shear thinning flows in moving domains

TL;DR

This paper shows that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain of the blood flow, namely the minimization of red blood cells damage.

Abstract

We consider the D problem of shape optimization of blood flows in moving domains. Such a geometry is adopted to take into account the modeling of rotating systems and blood pumps for instance. The blood flow is described by generalized Navier-Stokes equations, in the particular case of shear-thinning flows. For a sequence of converging moving domains, we show that a sequence of associated solutions to blood equations converges to a solution of the problem written on the limit moving domain. Thus, we extended the result given in (Sokołowski, Stebel, 2014, in \textit{Evol. Eq. Control Theory}) for , to the range , where is the exponent of the rheological law. This shape continuity property allows us to show the existence of minimal shapes for a class of functionals depending on the blood velocity field and its gradient. This allows to consider in particular the problem of hemolysis minimization in blood flows, namely the minimization of red blood cells damage.
Paper Structure (21 sections, 14 theorems, 81 equations, 1 figure)

This paper contains 21 sections, 14 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A blood model in moving domains
  3. Shape optimization problem and main result
  4. Shape optimization settings
  5. Optimization settings: moving domains
  6. Functional settings
  7. Blood flows equations in moving domains
  8. Shape continuity of blood flows
  9. Existence of solutions and uniform energy estimate
  10. Equivalent homogeneous Dirichlet problem
  11. Approximation
  12. Uniform energy estimate
  13. Continuity of blood flows solutions
  14. Identification of $\mathbf{H}_{\mathbf{V}}$
  15. Identification of the stress
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\mathcal{J} ( \Omega , \mathbf{V} )$ be the shape functional defined in eq:def:SHPfun, where $j$ satisfies assumptions item:assump:j:1-item:assump:j:2. Then, the shape optimization problem eq:pbm:minimization:abstract admits at least one solution $( \Omega^* , \mathbf{V}^* )$ in the class

Figures (1)

  • Figure 1: Moving domain $Q$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Example 1.2: Hemolysis minimization
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 13 more