A Model of and Generation of Tractions on the Retina Amabile Tatone 1 Rodolfo Repetto 2 1 Department of Information Technology and Mathematics, University of L Aquila, Italy 2 Department of Civil, Chemical and Environmental Engineering, University of Genoa, Italy 8th European Solid Mechanics Conference, Graz, Austria, July 9-13, 2012 Minisymposium MS-41 Theoretical and Numerical Modelling of the Functions of the Eye
a) b) c) d) Types of (from Kakehashi et al.,1997)
Rhegmatogenous retinal detachment Synchysis (liquefaction) Syneresis (dehydration and shrinkage) Weakening of vitreoretinal adhesion Fast eye-rotation generated oscillations Quasi-static shrinkage of the vitreous How to best evaluate the presence or absence of vitreous traction and how to quantitate the degree of vitreous traction is presently not known [Sebag, 1989]
The mechanical model D s I D f Solid D s Fluid Df
Balance principle (solid & fluid) For any test velocity field w b w dv + t w da D D D T w dv τ [[w]] da = 0 I b t T τ bulk force per unit volume in D traction per unit area on D Cauchy stress tensor interface stress D Solid D s I Fluid Df
Balance principle (fluid) For any test velocity field w b f w f dv + D f t f w f da + D f T f w f dv = 0 D f I t f w f da D s b f t f t f T f bulk force per unit volume in D f traction per unit area on D f traction per unit area on I Cauchy stress tensor Solid D s I D f Fluid Df
Balance principle (solid & boundary membrane) For any test velocity field w b s w s dv + t s w s da + t s w s da D s D s I T s w s dv N m m w s da + fm w s dl = 0 D s I I b s bulk force per unit volume in D s D s t s t s T s N m fm traction per unit area on D s traction per unit area on I Cauchy stress tensor Membrane stress tensor traction on I Solid D s I D f Fluid Df
Material characterization Incompressibility det F = 1 div v f = 0 Viscoelastic solid T s = ˆT s (F) p s I + 2µ s sym v s Newtonian fluid T f = p f I + 2µ f sym v f Incompressible Mooney-Rivlin material (plane strain) response ˆT(F) = 2 c 0 dev (FF T )
Boundary conditions (solid & fluid) τ = t f = t s on I v s = v f on I u s = ū on D s v f = ū on D f D s I D f Solid D s Fluid Df
Moving grid γ : D D γ s = φ s u f = 0
Balance principle for the solid in the reference shape For any test velocity field w b s w s dv + t s w s da + D s D s S s w s dv N m m w s da + D s I I I t s w s da f m w s dl = 0 b s bulk force per unit volume in D s D s t s t s S s N m f m traction per unit area on D s traction per unit area on I Piola stress tensor Membrane stress tensor traction on I Solid D s I D f Fluid Df
Balance principle for the fluid in the reference shape For any test velocity field w b f w f dv + D f t f w f da + D f S f w f dv = 0 D f I t f w f da b f = ρ f ( vf + v f Γ 1 (v f u γ ) ) det Γ S f = ( p f I + 2µ sym( v f Γ 1 ) ) Γ T det Γ
Incompressibility conditions (weak form) For any regular test scalar field p s on D s D s (det F 1) p s dv = 0 For any regular test scalar field p f on D f D f tr( v f Γ 1 ) p f dv = 0
Boundary conditions (weak form) Saccadic rotation Perfect adhesion condition u γ (x) = (R I) (x x 0 ) D f (u s u γ ) t s da = 0 No-slip condition D f (v f u γ ) t f da = 0
Interface conditions (weak form) Velocity field continuity I (v f u s ) τ da = 0 From the interface integrals t s w s da + where Summarizing I I I t f = t s = τ t f w f da τ (w f w s ) + (v f u s ) τ da
Saccadic movement
Constitutive parameters Radius R = 0.012 m Fluid viscosity µ f = 10 3 Pa s Mass density ρ s = ρ f = 1000 kg/m 3
Deviatoric stress and velocity field [c3 case 008] Right m 0 = 20 Pa µ s = 0.5 Pa s Left
Deviatoric stress and velocity field [c3 case 008] Right m 0 = 20 Pa µ s = 0.5 Pa s Left
Deviatoric stress and velocity field [c3 case 008] Right m 0 = 20 Pa µ s = 0.5 Pa s Left
Deviatoric stress and velocity field [c3 case 008] Right m 0 = 20 Pa µ s = 0.5 Pa s Left
Deviatoric stress and velocity field [c3 case 008] Right m 0 = 20 Pa µ s = 0.5 Pa s Left
Different PVD shapes Left Right [a1 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [a2 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [a3 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [a4 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [b5 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [b6 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [b7 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [b8 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [c1 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [c2 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [c3 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes Left Right [c4 case 008] m 0 = 20 Pa µ s = 0.5 Pa s
Different PVD shapes (c) 0.09 0.08 Normal Tangential (d) 0.09 0.08 Normal Tangential Maximum traction - left [N/m] 0.07 0.06 0.05 0.04 0.03 0.02 Maximum traction - right [N/m] 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.01 0 0 b 1 b 2 b 3 a 2 b 4 b 1 b 2 b 3 a 2 b 4 Configuration Configuration c 0 = 5 m 0 = 20 Pa Pa µ s = 0.5 Pa s
Different PVD shapes (e) 0.09 0.08 Normal Tangential (f) 0.09 0.08 Maximum traction - left [N/m] 0.07 0.06 0.05 0.04 0.03 0.02 Maximum traction - right [N/m] 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.01 0 Normal Tangential c 1 c 2 c 3 c 4 c 1 c 2 c 3 c 4 Configuration Configuration c 0 = 5 m 0 = 20 Pa Pa µ s = 0.5 Pa s
Traction induced by saccadic eye movements Preceding results from: R. Repetto, A. Tatone, A. Testa, E. Colangeli, Biomech Model Mechanobiol, 2010
Vitreous remodeling A shrinkage and adhesion model With ageing the vitreous humor undergoes the processes of synchysis (liquefaction) and syneresis (dehydration and shrinkage)
Vitreous remodeling Age-related vitreous liquefaction and PVD. Pockets of liquid appear within the central vitreous that gradually coalesce. There is a concurrent weakening of postoral vitreoretinal adhesion. Eventually, this can progress to PVD, where the liquid vitreous dissects the residual cortical gel away from the ILL on the inner surface of the retina as far anteriorly as the posterior border of the vitreous base. [Le Goff and Bishop, Eye (22) 2008]
Vitreous remodeling Schematic representation of the cooperation between two networks responsible for the gel structure of the vitreous. A network of collagen fibrils maintains the gel state and provides the vitreous with tensile strength. A network of hyaluronan fills the spaces between these collagen fibrils and provides a swelling pressure to inflate the gel. [Le Goff and Bishop, Eye (22) 2008]
Vitreous remodeling (a) The collagen fibrils (thick grey lines) form an extended network of small bundles. Within each bundle, the collagen fibrils are both connected together and spaced apart by type IX collagen chains. (b) With ageing the loss of the type IX collagen from the fibril surfaces combined with an increased surface exposure of type II collagen results in collagen fibrillar aggregation. [Le Goff and Bishop, Eye (22) 2008]
Vitreous remodeling Diagram representing the postbasal vitreoretinal junction. Weakening of the adhesion at this interface predisposes to posterior vitreous detachment. Vitreoretinal adhesion may be dependent upon intermediary molecules acting as a molecular glue and linking the cortical vitreous collagen fibrils to components of ILL. It is possible that opticin, because it binds to both vitreous collagen fibrils and HS proteoglycans in the ILL, contributes towards this molecular glue. [Le Goff and Bishop, Eye (22) 2008]
a) b) c) d) Types of (from Kakehashi et al.,1997)
Kröner-Lee decomposition of the deformation gradient p p, p D D reference shape current shape G relaxed shape F
Balance Principle For any test velocity field (w, V) t w da S w dv + D D D (Q A) V dv = 0 t S A Q traction per unit reference area reference Piola stress tensor inner remodeling couple per unit reference volume outer remodeling couple per unit reference volume
Balance equations div S + b = 0 in D t = S n on D Q A = 0 in D
Material characterization Incompressibility det F = 1 Spherical shrinkage Viscoelastic solid G = gi T = ˆT(F) pi + 2µ sym v Incompressible Mooney-Rivlin material (plane strain) response ˆT(F) = 2 c 0 dev (FF T ) Eshelby tensor A = J(F T S ϕ(f)i)
Boundary conditions Diagram representing the postbasal vitreoretinal junction. Weakening of the adhesion at this interface predisposes to posterior vitreous detachment. Vitreoretinal adhesion may be dependent upon intermediary molecules acting as a molecular glue and linking the cortical vitreous collagen fibrils to components of ILL. It is possible that opticin, because it binds to both vitreous collagen fibrils and HS proteoglycans in the ILL, contributes towards this molecular glue. [Le Goff and Bishop, Eye (22) 2008]
Boundary conditions Boundary traction Adhesive force t = t ad + t rep t ad (x, t) = k ad d ad (x, t) u 0 Adhesion force potential d ad (x, t) = u(x, t) ( ) e dad (x,t) 2 u u(x, t) 0 d ad (x, t) φ ad (d ad (x, t)) = 1 ( ( ) 2 k ad u 0 e dad (x,t) 2 ) u 0 1
Boundary conditions Repulsive force ( ( d ) 0 2 ( t rep (x, t) = k rep d rep (x, t) d 0 d rep (x, t) ) 3 ) r(x, t) r(x, t) Repulsive force potential r(x, t) = p(x, t) x 0 d rep (x, t) = R r(x, t) + d 0 φ rep (d rep (x, t)) = 1 2 k rep d 0 (1 d ) 0 2 d rep (x, t)
Boundary conditions t ad k ad, trep k rep 0.5 0.4 0.3 0.2 0.1 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.1 d ad, d rep 0.2 Adhesive force t ad (red line) Repulsive force t rep (blue line)
Boundary conditions φ ad k ad, φrep k rep dad, d rep 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 2. 10 6 4. 10 6 6. 10 6 8. 10 6 0.00001 Adhesive force potentials φ ad (red line) Repulsive force potentials φ rep (blue line)
Case-02 (Strong adhesion) Slip pulses [case 08] m 0 = 10 Pa µ = 0.1 Pa s
Case-02 (Strong adhesion) [case 08] m 0 = 10 Pa µ = 0.1 Pa s
Case-02 (Strong adhesion) [case 08] m 0 = 10 Pa µ = 0.1 Pa s
Case-02 (Strong adhesion) [case 08] m 0 = 10 Pa µ = 0.1 Pa s
Case-02 (Strong adhesion) [case 08] m 0 = 10 Pa µ = 0.1 Pa s
Case-03 (Strong focal adhesion) [case 01] m 0 = 10 Pa µ = 0.1 Pa s
Case-03 (Strong focal adhesion) [case 01] m 0 = 10 Pa µ = 0.1 Pa s
Case-03 (Strong focal adhesion) [case 01] m 0 = 10 Pa µ = 0.1 Pa s
Case-03 (Strong focal adhesion) [case 01] m 0 = 10 Pa µ = 0.1 Pa s
Case-03 (Strong focal adhesion) [case 01] m 0 = 10 Pa µ = 0.1 Pa s
Case-04 (Strong focal adhesion) [case 02] m 0 = 10 Pa µ = 0.1 Pa s
Case-04 (Strong focal adhesion) [case 02] m 0 = 10 Pa µ = 0.1 Pa s
Case-04 (Strong focal adhesion) [case 02] m 0 = 10 Pa µ = 0.1 Pa s
Case-04 (Strong focal adhesion) [case 02] m 0 = 10 Pa µ = 0.1 Pa s
Case-04 (Strong focal adhesion) [case 02] m 0 = 10 Pa µ = 0.1 Pa s
Case-05 (Weak adhesion and stiff cortex) [case 05] m 0 = 10 Pa µ = 0.1 Pa s
Case-05 (Weak adhesion and stiff cortex) [case 05] m 0 = 10 Pa µ = 0.1 Pa s
Case-05 (Weak adhesion and stiff cortex) [case 05] m 0 = 10 Pa µ = 0.1 Pa s
Case-05 (Weak adhesion and stiff cortex) [case 05] m 0 = 10 Pa µ = 0.1 Pa s
Case-05 (Weak adhesion and stiff cortex) [case 05] m 0 = 10 Pa µ = 0.1 Pa s
Case-06 (Weak adhesion and stiffer cortex) [case 06] m 0 = 1000 Pa µ = 0.1 Pa s
Case-06 (Weak adhesion and stiffer cortex) [case 06] m 0 = 1000 Pa µ = 0.1 Pa s
Case-06 (Weak adhesion and stiffer cortex) [case 06] m 0 = 1000 Pa µ = 0.1 Pa s
Case-06 (Weak adhesion and stiffer cortex) [case 06] m 0 = 1000 Pa µ = 0.1 Pa s
Case-06 (Weak adhesion and stiffer cortex) [case 06] m 0 = 1000 Pa µ = 0.1 Pa s
Conclusions The shrinkage process induces an additional contribution to the traction, which is independent of the eye movements; The PVD shapes obtained through the simulations look close to the observed PVD shapes, even with a simple spherical shrinkage.