Computer-based analysis of rhegmatogenous retinal detachment Damiano Natali 1, Rodolfo Repetto 1, Jennifer H. Siggers 2, Tom H. Williamson 3,4, Jan O. Pralits 1 1 Dept of Civil, Chemical and Environmental Engineering, University of Genova, Italy, 2 Dept of Bioengineering, Imperial College London, London, UK, 3 Retina Surgery, 50-52 New Cavendish Street, London, UK 4 NHS, St Thomas Hospital, London, UK September, 17 th 2015 Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 1 / 24 September 17, 2015 1 / 24
Anatomy of the eye Background Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 2 / 24 September 17, 2015 2 / 24
Retinal detachment Background Posterior vitreous detachment (PVD) and vitreous degeneration: more common in myopic eyes; preceded by changes in vitreous macromolecular structure and in vitreoretinal interface possibly mechanical reasons. If the retina detaches from the underlying layers loss of vision; Rhegmatogeneous retinal detachment: fluid enters through a retinal break into the sub retinal space and peels off the retina. Risk factors: myopia; posterior vitreous detachment (PVD); lattice degeneration;... Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 3 / 24 September 17, 2015 3 / 24
Background Scleral buckling and vitrectomy Scleral bluckling Scleral buckling is the application of a rubber band around the eyeball at the site of a retinal tear in order to promote reachtachment of the retina. Vitrectomy The vitreous may be completely replaced with tamponade fluids: silicon oils, water, gas,..., usually immiscible with the eye s own aqueous humor Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 4 / 24 September 17, 2015 4 / 24
Background The retinal detachment: cases considered here A) horseshoe tear (when large, >90, GRT), B) retinal hole Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 5 / 24 September 17, 2015 5 / 24
Background The retinal detachment: cases considered here Retinal tear Retinal hole A A A A retinal tear retinal hole Retinal surface Retinal surface Retinal horse shoe tear Hole Vitreous chamber Section A-A Vitreous chamber Section A-A Eye wall Eye wall Retinal ap Liqui ed vitreous Detached retina Liqui ed vitreous Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 6 / 24 September 17, 2015 6 / 24
Motivation Motivation Retinal tear and retinal holes are treated using surgery but it is not always clear what type of retinal break and under what conditions is more prone to further detach. It would therefore be useful to parametrize (size, attachment angles, size of retinal hole,...) different retinal breaks during eye motion and evaluate a measure of the tendency to further detach. Retinal tear Retinal hole retinal tear retinal hole A A A A Retinal surface Retinal surface Retinal horse shoe tear Hole Vitreous chamber Section A-A Vitreous chamber Section A-A Eye wall Eye wall Retinal ap Liqui ed vitreous Detached retina Liqui ed vitreous Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 7 / 24 September 17, 2015 7 / 24
The model The model - tear configuration y Ω Ω top ρ, ν Γ 1 Ω left Ω right X(s, t) θ Π x L f X p(t) Ω bottom Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 8 / 24 September 17, 2015 8 / 24
The model The model - hole configuration y Ω top Ω ρ, ν Ω left Γ 1 τ Σ Γ 2 Ω right X(s, t) n θ θ Π x L f X p(t) Ω bottom Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 9 / 24 September 17, 2015 9 / 24
Governing equations The model y Ω Ω top ρ, ν Ω left X(s, t) Γ 1 θ L f X p(t) Ω bottom For the viscous incompressible fluid u 1 + u u = p + t Re 2 u + f u = 0 Π Ω right Periodicity is imposed at Ω left and Ω right, and symmetry at Ω top and Ω bottom. Non slip boundary conditions are imposed on solid surfaces. x, For the slender 1D structure ρ 1 2 X t 2 = s ( T X ) ( 2 2 ) X s s 2 K b s 2 + ρ 1 g F The structure is clamped at a certain angle θ at the wall, which moves according to X p(t). Incompressibility of the structure is imposed and non-slip/no penetration of the fluid is enforced. Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 10 / 24 September 17, 2015 10 / 24
Dimensionless parameters The model The governing equations can be non-dimensionalized with the following characteristic scales: x = x L, u = u U, f = Doing so, several dimensionless parameters arises: fl ρ 0 U 2, F = FL ρ 1 U 2 Re = U L, Fr = gl ν U 2, ρ = ρ 1 ρ 0 L, γ = K b ρ 1 U L 2 2 Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 11 / 24 September 17, 2015 11 / 24
Plate imposed motion The model We model isolated rotations using the analytical relationship proposed by Repetto et al. (2005). 1 0.9 Xp(t/D) up(t/d) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t/d The angle is 8 The maximum velocity is 0.061 m/s The duration is 0.045 s Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 12 / 24 September 17, 2015 12 / 24
Parameters Parameters used in the computations Quantity Value Reference Properties of the retinal flap Density ρ S 1300 kg/m 3 Length L 1.5 2.5 mm Thickness 70 µm Alamouti and Funk (2003), Foster et al. (2010), Ethier et al. (2004), Bowd et al. (2000), Wollensak and Eberhard (2004), Dogramaci and Williamson (2013) Bending stiffness K b 2.98 10 11 Nm 2 Eh 3 /12 Young s modulus E 1.21 10 3 N/m2 Jones et al. (1992), Wollensak and Eberhard (2004), Reichenbach et al. (1991), Sigal et al. (2005) Properties of the fluid Density ρ F 1000 kg/m 3 Foster et al. (2010) Dynamic viscosity µ 1.065 10 3 kg/ms Foster et al. (2010) Table: Parameter values used for the simulations and corresponding references when available. Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 13 / 24 September 17, 2015 13 / 24
Dynamics for retinal tear Results L=2 mm, θ = 33.6 Movie 1 Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 14 / 24 September 17, 2015 14 / 24
Dynamics for retinal hole Results L=2 mm, θ = 33.6, = 0.17 mm Movie 2 Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 15 / 24 September 17, 2015 15 / 24
Results Clamping force and torque evaluation 40 35 30 25 20 15 10 5 0 5 F c,n(t/d) M c(t/d) u p(t/d) 3.5 3 2.5 2 1.5 1 0.5 0 10 0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t/d We evaluate the wall-normal force (F c,n) and torque (M c) at the clamping point as a function of time. These values are then used to model the tendency to further detach. Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 16 / 24 September 17, 2015 16 / 24
Winkler theory Results v(s) F c,n s M c k T r(s) Semi-infinite foundation (in green) subject to a punctual force F c,n and torque M c at the finite end, and supported by elastic spring of stiffness k T (in red). The soil reaction r(s) (in blue) is proportional to the foundation displacement v(s). v(s) = e αs 2α 3 {αmc [cos (αs) sin (αs)] + Fc,ncos (αs)}, γ d = max(v s=0, 0) = max( αmc + Fc,n 2α 3, 0), γ where α is the ratio between the soil spring rigidity k T and the foundation beam stiffness γ. d is the tendency to detach Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 17 / 24 September 17, 2015 17 / 24
Tendency to detach Results 140 120 100 80 60 40 20 0 F c,n(t/d) αm c(t/d) u p(t/d) d/d max (t/d) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 20 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t/d d attains a maximum value for a finite value of t/d Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 18 / 24 September 17, 2015 18 / 24
Results Different filament lengths L: maximum tendency to detach clamping angle θ = 33.56, = 0.17mm (retinal hole) 1.25 1.25 dmax /dmax,l=2 1 dmax /dmax,l=2 1 0.75 1.5 1.75 2 2.25 2.5 0.75 1.5 1.75 2 2.25 2.5 L [mm] L [mm] Tear Hole Increasing L increases the maximum value of d Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 19 / 24 September 17, 2015 19 / 24
Results Different clamping angles θ: maximum tendency to detach length L = 2 mm, = 0.17mm (retinal hole) 1 1 0.95 0.9 0.85 0.75 dmax /dmax,θ=25 0.8 0.75 0.7 0.65 dmax /dmax,θ=33.56 0.5 0.6 0.55 0.5 15 17.5 20 22.5 25 27 30 33.56 0.25 15 25 33.56 45 55 θ [degrees] θ [degrees] Tear Hole A maximum value of d is found Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 20 / 24 September 17, 2015 20 / 24
Results Comparison horseshoe tear & hole: maximum tendency to detach clamping angle θ = 33.56, = 0.17mm (retinal hole) 3.5 3.0 dhole/dflap 2.5 2.0 1.5 1.25 2 2.25 2.5 L [mm] The retinal hole is more prone to detach compared to horseshoe tear Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 21 / 24 September 17, 2015 21 / 24
Conclusions Conclusion Numerical investigation of the tendency to further detach two types of retinal breaks (horseshoe tear & hole) Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 22 / 24 September 17, 2015 22 / 24
Conclusions Conclusion Numerical investigation of the tendency to further detach two types of retinal breaks (horseshoe tear & hole) Achieved solving a fluid-structure interaction problem using a finite-volume code developed in Matlab c with an immersed boundary approach Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 22 / 24 September 17, 2015 22 / 24
Conclusions Conclusion Numerical investigation of the tendency to further detach two types of retinal breaks (horseshoe tear & hole) Achieved solving a fluid-structure interaction problem using a finite-volume code developed in Matlab c with an immersed boundary approach The parameters used are realistic for the human eye (according to the literature) Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 22 / 24 September 17, 2015 22 / 24
Conclusions Conclusion Numerical investigation of the tendency to further detach two types of retinal breaks (horseshoe tear & hole) Achieved solving a fluid-structure interaction problem using a finite-volume code developed in Matlab c with an immersed boundary approach The parameters used are realistic for the human eye (according to the literature) The main results show: Increasing the length L increases the tendency to detach (both hole & tear). The maximum tendency to detach is found for a clamping angle of 25 and 34 for tear and hole, respectively. The inter tip distance (hole size) has little effect on the tendency to detach. The tendency to detach of a retinal hole, compared to a tear, is 2-3 times larger for retinal filaments of 1.5-2.5 mm, with increasing values of d for increasing values of the filament length. Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 22 / 24 September 17, 2015 22 / 24
Conclusions Conclusion Numerical investigation of the tendency to further detach two types of retinal breaks (horseshoe tear & hole) Achieved solving a fluid-structure interaction problem using a finite-volume code developed in Matlab c with an immersed boundary approach The parameters used are realistic for the human eye (according to the literature) The main results show: Increasing the length L increases the tendency to detach (both hole & tear). The maximum tendency to detach is found for a clamping angle of 25 and 34 for tear and hole, respectively. The inter tip distance (hole size) has little effect on the tendency to detach. The tendency to detach of a retinal hole, compared to a tear, is 2-3 times larger for retinal filaments of 1.5-2.5 mm, with increasing values of d for increasing values of the filament length. Collaborations with a surgeon confirms that these results will give useful guidelines for treatment of retinal breaks. Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 22 / 24 September 17, 2015 22 / 24
References I Conclusion B. Alamouti and J. Funk. Retinal thickness decreases with age: an oct study. British Journal of Ophthalmology, 87(7):899 901, 2003. C. Bowd, R. N. Weinreb, B. Lee, A. Emdadi, and L. M. Zangwill. Optic disk topography after medical treatment to reduce intraocular pressure. American Journal of Ophthalmology, 130(3): 280 286, 2000. ISSN 0002-9394. doi: http://dx.doi.org/10.1016/s0002-9394(00)00488-8. URL http://www.sciencedirect.com/science/article/pii/s0002939400004888. M. Dogramaci and T. H. Williamson. Dynamics of epiretinal membrane removal off the retinal surface: a computer simulation project. Br. J. Ophthalmol., 97 (9):1202 1207, 2013. doi: 10.1136/bjophthalmol-2013-303598. C. R. Ethier, M. Johnson, and J. Ruberti. Ocular biomechanics and biotransport. Annu. Rev. Biomed. Eng., 6:249 273, 2004. W. J. Foster, N. Dowla, S. Y. Joshi, and M. Nikolaou. The fluid mechanics of scleral buckling surgery for the repair of retinal detachment. Graefe s Archive for Clinical and Experimental Ophthalmology, 248(1):31 36, 2010. I. L. Jones, M. Warner, and J. D. Stevens. Mathematical modelling of the elastic properties of retina: a determination of young s modulus. Eye, (6):556 559, 1992. A. Reichenbach, W. Eberhardt, R. Scheibe, C. Deich, B. Seifert, W. Reichelt, K. Dähnert, and M. Rödenbeck. Development of the rabbit retina. iv. tissue tensility and elasticity in dependence on topographic specializations. Experimental Eye Research, 53(2):241 251, 1991. ISSN 0014-4835. doi: http://dx.doi.org/10.1016/0014-4835(91)90080-x. URL http://www.sciencedirect.com/science/article/pii/001448359190080x. Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 23 / 24 September 17, 2015 23 / 24
References II Conclusion R. Repetto, A. Stocchino, and C. Cafferata. Experimental investigation of vitreous humour motion within a human eye model. Phys. Med. Biol., 50:4729 4743, 2005. I. A. Sigal, J. G. Flanagan, and C. R. Ethier. Factors influencing optic nerve head biomechanics. Investigative Ophthalmology & Visual Science, 46(11):4189, 2005. doi: 10.1167/iovs.05-0541. URL +http://dx.doi.org/10.1167/iovs.05-0541. G. Wollensak and S. Eberhard. Biomechanical characteristics of retina. Retina, (24):967 970, 2004. Natali, Repetto, Siggers, Williamson, Pralits () AIMETA 2015 24 / 24 September 17, 2015 24 / 24