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Deformation of bovine eye – fluid structure interaction between viscoelastic vitreous, non-linear elastic lens and sclera

Karel Tůma1 joint work with J. Stein2 and V. Průša1

1Faculty of Mathematics and Physics, Charles University, Czechia 2Faculty of Mathematics and Computer Sciences, Heidelberg University, Germany

October 24, 2018

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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)

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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

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Mathematical modeling: vitreous body

Motivation of the material specific equations for the stress: 1D mechanical analog: spring = elasticity, dashpot = viscosity Healthy vitreous1 Pathological vitreous

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, ρ ∂v ∂t + [∇v]v

• = div T,

T = TT, where the Cauchy stress tensor T = −pI + 2µsD + S, D = (∇v + (∇v)T)/2 and S satisfies an evolutionary equation ∂S ∂t + v · ∇S + f(∇v, S, ∇S) = 0.

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Burgers model Creep test by Sharif-Kashani et al. (2011)

µ3 µ1 µ2 G2 G1

Corresponding material parameters: ρ = 1007 kg/m3, µs = 2.37 Pa s, G1 = 0.645 Pa, τ1 = 1600.16 s, G2 = 0.898 Pa, τ2 = 25.06 s.

1D stress-strain relation ¨ σ + 1 τ1 + 1 τ2

• ˙

σ + 1 τ1τ2 σ = G1 τ2 + G2 τ1

• ˙

ε + (G1 + G2)¨ ε "Generalization to 3D": Response can be described by

▽▽

S + 1 τ1 + 1 τ2

• ▽

S + 1 τ1τ2 S = 2 G1 τ2 + G2 τ1

• D + 2(G1 + G2)

▽

D

▽

S:= ∂S ∂t + v · ∇S − (∇v)S − S(∇v)T

▽▽

S = ¨ S − 2L ˙ S − 2 ˙ SLT − ˙ LS − S ˙ LT − 2LSLT − L2S − S(LT)2

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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 = −pI + 2µsD + S,

▽▽

S + 1 τ1 + 1 τ2

• ▽

S + 1 τ1τ2 S = 2 G1 τ2 + G2 τ1

• D + 2(G1 + G2)

▽

D can be written in the form T = −pI + 2µsD + G1(B1 − I) + G2(B2 − I),

▽

B1 + 1 τ1 (B1 − I) = 0,

▽

B2 + 1 τ2 (B2 − I) = 0.

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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

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Experiment (Shah et al. 2016)

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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

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full model for vitreous, sclera and lens compressible elastic neo-Hookean solid with a strain energy density W(F) = 1 2µ(J−2/3 tr C − 3) + 1 2κ(ln J)2, J = det F, C = FTF ρ∂2u ∂t2 = Div ∂W ∂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

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Full 3D simulation in deforming domains

problem computed on a fixed mesh ⇒ the weak formulation transformed by ˆ ϕ from the physical domain in Ωx to computational domain Ωχ using arbitrary Langrangian-Eulerian description

ˆ ϕ : x = χ + ˆ u ˆ F = I + ∇χˆ u ˆ J = det ˆ F Ωχ Ωx

new variable ˆ u – arbitrary deformation of the domain and the mesh, for material points the relation dˆ u/dt = v holds elastic sclera and lens in Lagrangian description, displacement u monolithic approach

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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

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Bovine eye deformation (Burgers model)

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Bovine eye deformation (Burgers model)

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Dependence of uz of test particle on time

Burgers Navier-Stokes

100 200 300 400

• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 Time [s] Displacement uz [mm]

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Dependence of force on time

Navier-Stokes vs. Burgers

100 200 300 400 2 4 6 8 10 12 Time [s] Applied deformation [mm]

Burgers Navier-Stokes

100 200 300 400 5 10 15 20 Time [s] Force [N] Karel Tůma Simulation of a bovine eye 15/19

Dependence of force on time

Navier-Stokes vs. Burgers

Burgers Navier-Stokes

100 200 300 400 0.0 0.5 1.0 1.5 2.0 2.5 Time [s] Force [mN] 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

• ne.

Magnitude of the stress inside the vitreous after a deformation Colors: magnitude of the stress (blue = 0 Pa, red = 8 Pa)

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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.

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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)

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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))

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