A Multi-Scale Model for the Mechanics of the Human Lens Capsule Harvey Burd Civil Engineering Research Group Department of Engineering Science, Oxford University, UK Finite element model Schematic eye capsule

Scope Background  Accommodation mechanism  Finite element analysis of the human lens Mechanics of the lens capsule  Uniaxial and biaxial test data.  Structural constitutive model (Micronet) 1 Multi-scale finite element analysis  Implementation of the Micronet model in an axisymmetric hyperelastic finite element program  Example analyses 1. Burd (2009) Biomech Model Mechanobiol 8(3)

Anatomy of the human eye Aqueous Vitreous Ciliary body Zonules

Accommodation (Helmholtz 1909) Zonule Ciliary body Iris Cornea AccommodatedUnaccommodated

Lens : geometric model ref. Wolff’s Anatomy Axis of symmetry Lens outline MRI data on 29 and 45 year lenses. Hermans et al Zonule geometry: Age-related model for the geometry of the intersection of zonules with capsule. Canals et al. 1996; Farnsworth and Shyne 1979 Ciliary body radius: MRI data. Strenk et al Nucleus outline Brown, 1973; Dubbelman et al., 2003; Hermans et al., 2007; Kasthurirangan et al., 2008; Sweeney and Truscott, 1998; Ayaki et al., 1993; Gullapalli et al., 1995

Lens capsule: geometric model istoweb/eye_ear/eye_ear.htm 250 Microns Capsule Lens Data from: Barraquer et al. (2006). “Human lens capsule thickness as a function of age and location along the sagittal lens perimeter.” IOVS Capsule thickness Anterior pole Posterior pole

(a) Uniaxial Test (Krag et al. 2003) Sample cut from lens capsule Stress MPa Strain % Mechanics of the lens capsule

(b) Biaxial tests (i) Isolated capsule inflation test (Fisher 1969) Initial capsule geometry P

(ii) In-situ capsule inflation (Pedrigi et al. 2007)

uniaxial test biaxial test Linear elastic model; data on Young’s modulus

A structural model for the lens capsule (a) Structure of the lens capsule Barnard et al nm Filaments of collagen type IV Barnard et al J. Struct. Biol.

(b) Components of a structural model Non-linear pin-jointed bars ( 2 parameters) Neo-Hookean matrix ( 1 parameter) after Barnard et al. 1992

(b) Components of a structural model (i) Strain energy density ( ii) Neo-Hookean model for matrix networkmatrix a1a1 a2a2

(b) Components of a structural model (iii) Strain energy density for bars where

Implementation in multi-scale finite element model a1a1 a2a2 internal bars edge bars Specify stretch ratios 1 and  2 Apply periodic boundary conditions Constrain one joint to be fixed Compute updated joint coordinates (  W=0) Compute derivatives Initial configuration

Generating the internal mesh

Calibration tests Membrane traction = (a) Uniaxial test (b) Biaxial test Membrane traction =

Simulation of isolated capsule inflation test Fisher (1969)

Simulation of in-situ capsule inflation test Pedrigi et al. (2007)

Simulation of in-situ capsule inflation test Pedrigi et al. (2007)

Simulation of in-situ capsule inflation test circumferential meridional Point D Point X

Simulation of in-situ capsule inflation test Point D Point F

Conclusions 3-parameter structural model for the lens capsule Implementation in axisymmetric finite element analysis Comparison with previous capsule inflation data