Deformation of bovine eye fluid structure interaction between - PowerPoint PPT Presentation

Deformation of bovine eye – fluid structure interaction between viscoelastic vitreous, non-linear elastic lens and sclera Karel Tůma 1 joint work with J. Stein 2 and V. Průša 1 1 Faculty of Mathematics and Physics, Charles University, Czechia 2 Faculty of Mathematics and Computer Sciences, Heidelberg University, Germany October 24, 2018 Karel Tůma Simulation of a bovine eye 1/19

Bovine eye transparent, colorless, gelatinous fluid 98% of water, NaCl, hyaluronan maintains the shape of the eye, keeps a clear path to the retina created during embryonic period, do not regenerate viscoelastic behavior (parameters by Sharif-Kashani et al. 2011) experiment with a bovine vitreous body (Shah et al. 2016) Karel Tůma Simulation of a bovine eye 2/19

Mathematical modeling: vitreous body Rheology of the vitreous Healthy (densely packed collagen fibers): elastic AND viscous behavior ⇒ viscoelastic fluid Pathological (liquefaction or complete vitrectomy): same properties like water ⇒ pure viscous fluid Collagen fibers in HA Elastic & viscous behavior Karel Tůma Simulation of a bovine eye 3/19

Mathematical modeling: vitreous body Motivation of the material specific equations for the stress: 1D mechanical analog: spring = elasticity, dashpot = viscosity Pathological vitreous Healthy vitreous 1 1Sharif-Kashani et al.: Rheology of the vitreous gel: Effects of macromolecule organization on the viscoelastic properties , J. Biomech. 44 (3): 419–423, 2011 Karel Tůma Simulation of a bovine eye 4/19

Incompressible viscoelastic fluid like models Incompressible rate-type fluid models div v = 0 , � ∂ v � T = T T , ρ ∂t + [ ∇ v ] v = div T , where the Cauchy stress tensor T = − p I + 2 µ s D + S , D = ( ∇ v + ( ∇ v ) T ) / 2 and S satisfies an evolutionary equation ∂ S ∂t + v · ∇ S + f ( ∇ v , S , ∇ S ) = 0 . Karel Tůma Simulation of a bovine eye 5/19

Burgers model Creep test by Sharif-Kashani et al. (2011) µ 3 G 2 µ 2 Corresponding material parameters: ρ = 1007 kg/m 3 , µ s = 2 . 37 Pa s , G 1 = 0 . 645 Pa , τ 1 = 1600 . 16 s , G 2 = 0 . 898 Pa , τ 2 = 25 . 06 s . µ 1 G 1 1D stress-strain relation � 1 + 1 � 1 � G 1 + G 2 � ¨ σ + σ + ˙ σ = ε + ( G 1 + G 2 )¨ ˙ ε τ 1 τ 2 τ 1 τ 2 τ 2 τ 1 "Generalization to 3D": Response can be described by � 1 + 1 � S + 1 � G 1 + G 2 � ▽▽ ▽ ▽ S + S = 2 D + 2( G 1 + G 2 ) D τ 1 τ 2 τ 1 τ 2 τ 2 τ 1 S := ∂ S ▽ ∂t + v · ∇ S − ( ∇ v ) S − S ( ∇ v ) T ▽▽ SL T − ˙ L T − 2 LSL T − L 2 S − S ( L T ) 2 S = ¨ S − 2 L ˙ S − 2 ˙ LS − S ˙ Karel Tůma Simulation of a bovine eye 6/19

Burgers model Thermodynamic derivation: second law thermodynamics, objectivity, physical meaning of B , symmetric equations, initial conditions Málek J., Rajagopal K.R., Tůma K.: Derivation of the Variants of the Burgers Model Using a Thermodynamic Approach and Appealing to the Concept of Evolving Natural Configurations , Fluids, Vol. 3, No. 4, pp. 1–18, 2018. T = − p I + 2 µ s D + S , � 1 + 1 � S + 1 � G 1 + G 2 � ▽▽ ▽ ▽ S + S = 2 D + 2( G 1 + G 2 ) D τ 1 τ 2 τ 1 τ 2 τ 2 τ 1 can be written in the form T = − p I + 2 µ s D + G 1 ( B 1 − I ) + G 2 ( B 2 − I ) , B 1 + 1 ▽ ( B 1 − I ) = 0 , τ 1 B 2 + 1 ▽ ( B 2 − I ) = 0 . τ 2 Karel Tůma Simulation of a bovine eye 6/19

Experiment (Shah et al. 2016) 2 cm thick plate cut out put into loading machine and glued on the sides let it relax and then 4 × prolongated in 3 mm increments tracking of markers on the top surface Karel Tůma Simulation of a bovine eye 7/19

Experiment (Shah et al. 2016) Karel Tůma Simulation of a bovine eye 8/19

a need to compute 3D problem first attempt: the deformation of vitreous only, deformation too different from what they did vitreous fluid can not hold by itself ⇒ need to compute a more complex problem Karel Tůma Simulation of a bovine eye 9/19

full model for vitreous, sclera and lens compressible elastic neo-Hookean solid with a strain energy density W ( F ) = 1 2 µ ( J − 2 / 3 tr C − 3) + 1 2 κ (ln J ) 2 , C = F T F J = det F , ρ∂ 2 u � ∂W � ∂t 2 = Div ∂ F κ = 1000 µ makes the material almost incompressible sclera: µ = 330 kPa (Grytz et al. 2014) lens: µ = 10 kPa (Fisher 1971) interaction: continuous displacement and stress Karel Tůma Simulation of a bovine eye 10/19

Full 3D simulation in deforming domains problem computed on a fixed mesh ⇒ the weak formulation transformed by ˆ ϕ from the physical domain in Ω x to computational domain Ω χ using arbitrary Langrangian-Eulerian description ϕ : x = χ + ˆ ˆ u Ω χ Ω x ˆ F = I + ∇ χ ˆ u J = det ˆ ˆ F new variable ˆ u – arbitrary deformation of the domain and the mesh, for material points the relation dˆ u / d t = v holds elastic sclera and lens in Lagrangian description, displacement u monolithic approach Karel Tůma Simulation of a bovine eye 11/19

FEM implementation weak ALE formulation implemented in AceFEM/AceGen system (J. Korelc Ljublan) non-linearities treated with the Newton method automatic differentiation provides exact tangent matrix implicit backward Euler method with a apriori given fixed time step (smaller during the deformation) hexahedral mesh: 12 896 elements (Q1-Q1-Q1-P0/Q1/Q1) 100 208 DOFs (Dirichlet BC excluded) MKL Pardiso solver used: iterative CGS solver with LU decomposition as a preconditioner CGS stopping criterion 10 − 4 , Newton solver 10 − 9 ∼ 20 CGS iterations during deformation ∼ 2 CGS iterations during relaxation Karel Tůma Simulation of a bovine eye 12/19

Bovine eye deformation (Burgers model) Karel Tůma Simulation of a bovine eye 13/19

Bovine eye deformation (Burgers model) Karel Tůma Simulation of a bovine eye 13/19

Dependence of u z of test particle on time 0.0 Burgers - 0.5 Displacement u z [ mm ] Navier - Stokes - 1.0 - 1.5 - 2.0 - 2.5 - 3.0 0 100 200 300 400 Time [ s ] Karel Tůma Simulation of a bovine eye 14/19

Dependence of force on time Navier-Stokes vs. Burgers 20 12 Burgers Applied deformation [ mm ] Navier - Stokes 10 15 8 Force [ N ] 10 6 4 5 2 0 0 0 100 200 300 400 0 100 200 300 400 Time [ s ] Time [ s ] Karel Tůma Simulation of a bovine eye 15/19

Dependence of force on time Navier-Stokes vs. Burgers 2.5 Burgers Navier - Stokes 2.0 Force [ mN ] 1.5 1.0 0.5 0.0 0 100 200 300 400 Time [ s ] Karel Tůma Simulation of a bovine eye 16/19

Results Quantification of the stress in a deforming eye The healthy vitreous shows twice as high stress as the pathological one. Magnitude of the stress inside the vitreous after a deformation Colors: magnitude of the stress (blue = 0 Pa, red = 8 Pa) Karel Tůma Simulation of a bovine eye 17/19

Results Quantification of the stress in a deforming eye Higher stresses in the healthy vitreous damp the external mechanical load due to the elastic collagen network compare to the pathological vitreous. Karel Tůma Simulation of a bovine eye 18/19

Conclusion Summary of the results: Realization of numerical experiments focused on simulation of flow of vitreous humour in a deforming eye ⇒ quantitative characterization of vitreous flow in different scenarios Rheological properties of the vitreous influence the mechanical stress distribution in the vitreous Difference in stresses between Navier-Stokes (pathological) and Burgers (healthy) Eye pathologies such as vitreous and retinal detachment are thought to be closely linked to mechanical processes with high stresses Future: cooperation with Heidelberg University Hospital Dept. of Ophthalmology (retinal detachment) Karel Tůma Simulation of a bovine eye 19/19

Q1-P0 elements aim: to decrease the overall cost of calculation problems on a regular mesh with Dirichlet BC: pure spurious pressure mode (checkerboard) h → 0 ill-conditioned numerical experiments: on a general distorted mesh the spurious pressure modes disappear, and the inf-sup constant is independent of the mesh size (Brezzi, Fortin (1991)) Karel Tůma Simulation of a bovine eye 20/19