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
capsule
Finite element model
Schematic eye
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) 217-231
Anatomy of the human eye
Aqueous
Ciliary body
Zonules
Vitreous
Accommodation (Helmholtz 1909)
Cornea
Iris
Ciliary body
Zonule
Unaccommodated
Accommodated
Lens : geometric model
Axis of symmetry
Lens outline
MRI data on 29 and 45 year lenses.
Hermans et al. 2008
ref. Wolff’s Anatomy
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
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. 1999
Lens capsule: geometric model
Capsule
Lens
250 Microns
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
http://www.kumc.edu/instruction/medicine/anatomy/h
istoweb/eye_ear/eye_ear.htm
Anterior pole
Posterior pole
Mechanics of the lens capsule
(a) Uniaxial Test (Krag et al. 2003) Stress MPa
Sample cut from
lens capsule
Strain %
(b) Biaxial tests
(i) Isolated capsule inflation test (Fisher 1969)
P
Initial capsule geometry
(ii) In-situ capsule inflation (Pedrigi et al. 2007)
Linear elastic model; data on Young’s modulus
Young's modulus (MPa)
8
Fisher (1969)
Krag and Andreassen (2003)
Pedrigi et al. (2007)
6
uniaxial test
4
2
biaxial test
0
0
20
40
60
Age (years)
80
A structural model for the lens capsule
50 nm
(a) Structure of the lens capsule
Barnard et al. 1992
Filaments of collagen type IV
Barnard et al. 1992 J. Struct. Biol.
(b) Components of a structural model
Non-linear pin-jointed bars ( 2 parameters)
Neo-Hookean matrix ( 1 parameter)
a2
after Barnard et al. 1992
a1
(b) Components of a structural model
(i) Strain energy density
t
wn 1 , 2   wm 1 , 2 
W 
t ref
network
aa2
2
aa1
1
2
(ii) Neo-Hookean model for matrix
wm
1
matrix


1
2
2

  1  2  2 2  3
2
1 2 
(b) Components of a structural model
(iii) Strain energy density for bars
wb  c c1 b 1
b  1
1  b  L
b  
L


wb  c eQ  1
wb 



2
2

where

L 2
b 


  b  L  
dwb
d b
b  
L
b  1

1 2

Q  c1  b  1 
2

b
2
Implementation in multi-scale finite element model

1 
1
W
Li wb ( λbi )   Lj wb ( λbj )  wm ( λ1 , λ2 )


a1 a2 internal
2 edge

internal
bars
edge bars
Specify stretch ratios 1 and 2
Apply periodic boundary conditions
Constrain one joint to be fixed
a2
Compute updated joint coordinates (dW=0)
Compute derivatives
a1
Initial configuration
W
1
W
2
 2W
2
1
 2W
12
 2W
2
2
Generating the internal mesh
(b) Distorted hexagonal mesh
(a) Regular hexagonal mesh
2
1
a2
L0
a1
Boundaries
of periodic cell
Calibration tests
(a) Uniaxial test
(b) Biaxial test
W
Membrane traction =
1
W
Membrane traction =
1
44
3.5
3.5
33

c
 0.365 N/m
L0
c1
2.5
2.5
 9.67 N/m
 90 .49
 L  1.3
22
1.5
1.5
11
Biaxial calibration data
Membrane traction (N/m)
Uniaxial
Uniaxialcalibration
calibrationdata
data
Micronet
Micronet
Membrane traction (N/m)
Membrane traction (N/m)
W
2
30
55
4.5
4.5

Micronet
25
20
15
1
10
5
0.5
0.5
00
11
0
1.02
1.02 1.04
1.04 1.06
1.06 1.08
1.08 1.1
1.1 1.12
1.12
Stretch
Stretchratio
ratio((
1)1)
1
1.02
1.04
1.06
1.08
1.1
Stretch ratio(1 =2)
1.12
Simulation of isolated capsule inflation test
Fisher (1969)
2 mm
c
Polar
axis
Computed outline
of inflated capsule
Internal
pressure, p
z
r
Initial shape of capsule
Pinned boundary to
represent the clamped
edge of the capsule
Simulation of in-situ capsule inflation test
Pedrigi et al. (2007)
Simulation of in-situ capsule inflation test
Pedrigi et al. (2007)
Point D
Simulation of in-situ
capsule inflation test
Point X
circumferential
meridional
5
Point D
Simulation of in-situ
capsule inflation test
3
Point F
2
5
1
circumferential
and meridional
0
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
Green Strain
4
Pressure (kPa)
Pressure (kPa)
4
3
2
1
circumferential
meridional
0
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
Green Strain
Conclusions
•
3-parameter structural model for the lens capsule
•
Implementation in axisymmetric finite element
analysis
•
Comparison with previous capsule inflation data