Numerical Simulation of Viscoelastic Fluid‐Structure Interaction Problems and Drug Therapy in the Eye

Alexander Drobny,Elfriede Friedmann

doi:10.1002/pamm.202000260

Abstract

AbstractThe vitreous is a fluid‐like viscoelastic transparent medium located in the center of the human eye and is surrounded by hyperelastic structures like the sclera, lens and iris. This naturally gives rise to a fluid‐structure interaction (FSI) problem. While the healthy vitreous is viscoelastic and described by a viscoelastic Burgers‐type equation, the aging vitreous liquefies and is therefore modeled by the Navier‐Stokes equations. We derive a monolithic variational formulation employing the arbitrary Lagrangian Eulerian framework which is solved using the finite element method. To allow large 3D simulations the implementation is parallelized. Furthermore we study the vascular endothelial growth factor (VEGF) therapy in the vitreous which is modeled by four coupled convection‐diffusion‐reaction equations with an additional coupling to the underlying flow.

Highlights

1 Introduction In this work we perform three dimensional finite element simulations of viscoelastic fluid-structure interaction (FSI) problems in the eye and of a drug therapy model in the vitreous based on four coupled convection-diffusion-reaction equations
The model for the drug therapy consists of four coupled convection-diffusion-reaction equations and the Burgers or Navier-Stokes equations as the underlying flow
The numerical simulations are realized in deal.ii [6] and based on the FSI implementation in [7]

Summary

Introduction

1 Introduction In this work we perform three dimensional finite element simulations of viscoelastic FSI problems in the eye and of a drug therapy model in the vitreous based on four coupled convection-diffusion-reaction equations. The variational formulation for the viscoelastic FSI models (see [1] for details on the models and parameters) employing the arbitrary Lagrangian Eulerian (ALE) framework [2] using appropriate function spaces reads: Definition 2.1 Find {vf , vs, uf , us, pf , B1, B2} ∈ {vfD + Vf0,v} × Ls × {uDf + Vf0,u} × {uDs + Vs0} × L0f × Vf × Vf such that vf (0) = vf0, vs(0) = vs0, uf (0) = u0f , us(0) = u0s , B1(0) = I, B2(0) = I and for almost all time steps t ∈ I it holds: (Jρf ∂tvf , ψv)Ωf + (ρf J(F−1(vf − ∂tuf ) · ∇ˆ )vf , ψv)Ωf + (Jσf F−T , ∇ˆ ψv)Ωf − (ρf Jff , ψv)Ωf = 0 ∀ψv ∈ Vf0,Γi ,

Results

Conclusion