Karel October Tůma 24, Simulation 2018 of a bovine eye 1/19 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
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 Healthy vitreous 1 Pathological vitreous 1 Sharif-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 ρ t + [ v]v = div T, T = T T, where the Cauchy stress tensor T = pi + 2µ 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) µ3 G2 µ2 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. G1 µ1 1D stress-strain relation ( 1 σ + + 1 ) τ 1 τ 2 σ + 1 τ 1 τ 2 σ = ( G1 + G ) 2 ε + (G 1 + G 2 ) ε τ 2 τ 1 "Generalization to 3D": Response can be described by ( 1 S + + 1 ) S + 1 ( G1 S = 2 + G ) 2 D + 2(G 1 + G 2 ) D τ 1 τ 2 τ 1 τ 2 τ 2 τ 1 S:= S + v S ( v)s S( v)t t S = S 2LṠ 2ṠLT LS S L T 2LSL T L 2 S S(L T ) 2 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. ( 1 S + + 1 ) S + 1 S = 2 τ 1 τ 2 τ 1 τ 2 can be written in the form B1 + 1 τ 1 (B 1 I) = 0, B2 + 1 τ 2 (B 2 I) = 0. T = pi + 2µ s D + S, ( G1 + G 2 τ 2 τ 1 ) D + 2(G 1 + G 2 ) D T = pi + 2µ s D + G 1 (B 1 I) + G 2 (B 2 I), 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, J = det F, C = F T F ( ) ρ 2 u W t 2 = Div F κ = 1000µ 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 ˆϕ from the physical domain in Ω x to computational domain Ω χ using arbitrary Langrangian-Eulerian description Ω χ ˆϕ : x = χ + û ˆF = I + χ û Ωx Ĵ = det ˆF new variable û arbitrary deformation of the domain and the mesh, for material points the relation dû/dt = 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 Displacement uz [mm] 0.0-0.5-1.0-1.5-2.0-2.5 Burgers Navier-Stokes -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 Applied deformation[mm] 12 10 8 6 4 2 Force[N] 20 15 10 5 Burgers Navier-Stokes 0 0 100 200 300 400 Time[s] 0 0 100 200 300 400 Time[s] Karel Tůma Simulation of a bovine eye 15/19
Dependence of force on time Navier-Stokes vs. Burgers 2.5 2.0 Burgers Navier-Stokes 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: Future: 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 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