application of structural analysis to the mechanical ... · kevin anderson, ahmed el-sheikh† and...

, published 22 November 2004, doi: 10.1098/rsif.2004.00021 2004 J. R. Soc. Interface Kevin Anderson, Ahmed El-Sheikh and Timothy Newson of the corneaApplication of structural analysis to the mechanical behaviour

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J. R. Soc. Interface (2004) 1, 3–15DOI: 10.1098/rsif.2004.0002

Application of structural analysis to themechanical behaviour of the cornea

Kevin Anderson, Ahmed El-Sheikh† and Timothy Newson

Division of Civil Engineering, University of Dundee, Dundee DD1 4HN, UK

Structural engineering analysis tools have been used to improve the understanding of thebiomechanical behaviour of the cornea. The research is a multi-disciplinary collaborationbetween structural engineers, mathematical and numerical analysts, ophthalmologists andclinicians. Mathematical shell analysis and nonlinear finite-element modelling have been usedin conjunction with laboratory experiments to study the behaviour of the cornea under differentloading states and to provide improved predictions of the mechanical response to disease andinjury. The initial study involved laboratory tests and mathematical back analysis to determinethe corneal material properties and topography. These data were then used to facilitate theconstruction of accurate finite-element models that are able to reliably trace the performanceof cornea upon exposure to disease, injury or elevated intra-ocular pressure. The models arebeing adapted to study the response to keratoconus (a disease causing loss of corneal tissue)and to tonometry procedures, which are used to measure the intra-ocular pressure. This paperintroduces these efforts as examples of the application of structural engineering analysis toolsand shows their potential in the field of corneal biomechanics.

Keywords: structural analysis; corneal biomechanics; mathematical analysis;numerical modelling

1. INTRODUCTION

The cornea performs three major functions: (a) it pro-tects the inner contents of the eye; (b) it maintains theshape of the eye; and (c) it refracts light. Disease andinjury can alter the shape and thickness of the cornea,leading to serious changes in the visual performance ofthe eye (Leibowitz & Waring 1998). For this reason,understanding the biomechanical response of the corneato agents such as disease, surgery and injury is ofgreat clinical importance. This task is made difficultby the layered structure of the cornea and its lackof homogeneity. Both factors mean that losing cornealtissue due to disease and surgery affects the cornea’smaterial properties and makes predicting the effects ofthese agents more difficult.

The research presented in this paper employs engi-neering, mathematical and numerical techniques tomodel the behaviour of the cornea under external effectssuch as disease and surgery and internal effects suchas intra-ocular pressure (IOP) change. Accuracy isachieved through incorporation of the actual materialproperties and topography of the cornea. Experimentaltesting of corneal trephinates (buttons) is conducted todetermine the material properties, and the specimenswere further subjected to laser scanning to determinetheir topography.

Mathematical analysis based on shell theory wasutilized to derive the material constitutive relationship

†Author for correspondence ([email protected]).

from the experimental results. The constitutive rela-tionship found was highly nonlinear, with two distinc-tive phases indicating significant stiffening beyond acertain level of loading. This knowledge of the materialproperties and corneal topography was essential forbuilding an accurate representative model based onfinite-element techniques. This paper presents a briefaccount of the research undertaken and the resultsobtained. Reference will be given to earlier publicationsand future directions for the work described.

The research has concentrated on porcine corneasbecause of their close similarity with human corneasin structure and constitutive relationship (Zeng et al.2001). This work has enabled the identification ofimportant aspects of the material properties and thedetermination of the significant factors that should beconsidered in the numerical modelling. This step isimportant because of the inherent complexity of thecorneal structure at both the micro and macro levels,and hence the need to simplify the model and allow it toincorporate only the significant factors. Once this stepof the research is completed, the main findings will bevalidated for human corneas.

2. CORNEAL STRUCTURE

The cornea consists of three distinct cell layers: theouter epithelium; the inner endothelium; and the cen-tral stroma, see figure 1. Between these layers arespecialized extracellular structures called Bowman’sand Descemet’s membranes. The constitution of thethree layers varies considerably, with the endothelium

Received 1 April 2004 3 c© 2004 The Royal SocietyAccepted 28 April 2004Published online 5 May 2004



4 Study of corneal biomechanics K. Anderson and others

epithelial cells(keratinocytes)

stromal cells(keratocytes)

endothelialcells

Bowman's membrane

extracellularmatrix ofcollagen fibrilsand proteoglycans

Descemet'smembrane

Figure 1. Cross-section through the cornea.

Figure 2. The interlacing of collagen lamellae in the cornealstroma.

and epithelium layers known to possess higher in-plane stiffness compared with the initial stiffness of thestroma. This variation poses the question of whetherthe cornea should be treated as a composite structure oras a homogeneous structure with equivalent properties.The answer to this question will undoubtedly haveimplications for the cost of modelling and analysis.

At the micro level, the stroma, which forms about90% of the corneal thickness, is made up of interlacinglayers of collagen fibrils embedded within a matrixof proteoglycans (Radner et al. 1998; Scott 1991), asdepicted in figure 2. This construction not only raisesquestions on whether the stroma can be considereda homogeneous structure from a macro viewpoint, italso leads to difficulties in understanding the materialbehaviour as explained below.

3. EXPERIMENTAL TESTING

Early experimental efforts were undertaken by Wooet al. (1972). In their experiments whole corneas werepressure tested and the results showed a nonlinearmaterial behaviour with significant stiffening at higherpressure levels. However, the method they used to moni-tor the displacement under pressure, which involvedcovering the corneas in ink, was thought to have affectedtheir response to pressure. Similar tests were morerecently undertaken by Bryant and McDonnell (1996)and they gave further evidence of the nonlinear natureof the corneal material. The material was observedto behave linearly to a range of IOPs between 2 and4 kPa. Beyond this pressure, the modulus of elasticitygrew suddenly. Bryant & McDonnell (1996) used a fibreoptic probe and a small dot of paint (1–2mm) onthe cornea’s centre to measure displacement—a methodwhich was reported to have produced some inaccuraciesin the results. The nonlinear behaviour of the corneawas also confirmed by Nyquist as early as 1968 whenhe investigated the biomechanical behaviour of porcinecorneas using strip testing (Nyquist 1968).

saline solutiontank

vert

ical

m

ovem

ent

0

0.5 m

1.0 m

wall-mountedpressure gauge

pressurechamber cornea

magneticsupport

triaxialmachine

verticalmovement

lasercentreline

Figure 3. Trephinate inflation test rig.

In the present research, 20 porcine corneal specimenswere subjected to internal pressure increases while theirperformance was monitored. The specimens (also calledbuttons or trephinates) included the cornea and a nar-row ring of surrounding scleral tissue and they were nomore than four hours post mortem when tested. Theywere mechanically separated from the rest of the eyeglobe using a sharp trephine cutting tool, then mountedonto a specially designed test rig that could providewatertight edge fixity for the specimens along theirring of scleral tissue. The specimens were subjected toa gradually increasing posterior pressure caused by acolumn of saline water to simulate the effect of elevatedIOP, as shown in figure 3. In the meantime, a laser(Keyence, CCD laser displacement sensor, LK series)was used to continually monitor the displacement atthe apex of the cornea. The data related to the appliedpressure and the corresponding apical displacementwere automatically recorded for later analysis. Care wastaken to consider the natural differences in the size ofthe specimens and in positioning the specimens relativeto the laser. To ensure a good seal along the edge of thespecimens, mechanical clamps and cyano-acrylate gluewere used and found to be effective. Loss of hydrationwas also prevented by coating the specimens in mineraloil. All corneas were subjected to a gradually increasingposterior pressure up to a maximum pressure of 14 kPa.This pressure was well above the level at which thecorneas entered a stage of stable behaviour that wasexpected to continue until bursting.

4. EXPERIMENTAL RESULTS

The results in figure 4 show the pressure–apical riserelationships obtained for a representative selection ofthe trephinate inflation tests. The results show a shortinitial inflating stage preceding a long phase of linearbehaviour, followed by sudden stiffening at about 4 kPa(0.004Nmm−2). Based on the results of earlier studiesby Woo et al. (1972) and Hjortdal (1993, 1998) on thecorneal microstructure and what has been found in cur-rent laboratory testing, it is suggested that the stress–strain relationship is divided into two distinctive phases:a matrix regulated phase with low stiffness followed by

J. R. Soc. Interface (2004)



Study of corneal biomechanics K. Anderson and others 5

trephinate 1trephinate 2trephinate 3

0.004

0.008

0.012

0.016

0 0.2 0.4 0.6 0.8corneal rise (mm)

pres

sure

(N

mm

−2)

Figure 4. Pressure–apical rise experimental results.

intr

a-oc

ular

pre

ssur

e (I

OP)

matrix-regulated phase collagen-regulated phase

collagen tensionpoint

corneal deflection

Figure 5. Corneal behaviour with an initial matrix regulatedphase followed by a higher stiffness collagen regulated phase.

a collagen regulated phase with much higher stiffness,as seen in figure 5. In the first phase, the behaviour isdominated by the corneal matrix, particularly withinthe stroma layer. As a result, the apical rise of thecornea increases almost linearly with pressure at a lowstiffness. The collagen fibril layers in this phase remainloose and unable to contribute notably to the overallperformance. Then, with the start of the second phase,it is expected that the fibril layers become taut and, dueto their much higher stiffness, they start to control theoverall behaviour and lead quickly to a much increasedcorneal stiffness. Note also that these two phases ofbehaviour are preceded by a short phase in whichthe corneal specimen is initially relaxed and graduallyinflated to assume its natural curved topography.

5. MATERIAL CONSTITUTIVEMODELLING

The nonlinear behaviour observed experimentally couldbe due to the curved profile of the cornea or due tononlinear material properties. Mathematical analysis isused in this work to assess the effect of these factors andto develop a material constitutive relationship, whichcould be incorporated into the finite-element model.For simplicity, the mathematical analysis assumes thatthe cornea can be approximated as a homogenous(Vito et al. 1989) spherical structure with a constantthickness.

N Rd

M Rd

XY

Z

Qr

d

Qr

Qr

d

d(d

)/d

]d+

[

+M r[d( d )/d ]d

Nr

+[d

(

d)/

d]d

d

d

A

B

D

C

CL

ofc

orne

a

r

r

R

R

parallelcircle

parallelcircle

mer

idia

nlin

e

m

eridia

n line

A

BC

D

d

A, B

C, D

Rφ φ γ

φ

φ

φθ

M rd

N rd

M Rdθ φN Rdθ φ

φφ

M Rdθ φ φ

N r

d

θ

θ

θφ

φφ

φφ

φ

φθ

φθ

φφ

θφ

θφ

θ

θ

(a)

(b)

(c)

φ

Figure 6. Mathematical analysis of corneal trephinate infla-tion tests: (a) element ABCD bounded by two meridian linesand two parallel circles; (b) section along a meridian line;(c) straining actions affecting element ABCD.

The mathematical analysis is based on shell theoryand is used to study the behaviour of corneal buttonsduring the inflation tests. The analysis considers bothin-plane and out-of-plane stiffness. It uses the pressure–apical rise data obtained experimentally in order toobtain the stress–strain relationship (or the constitutiverelationship) of the material.

The analysis starts with the general spherical ele-ment bounded by two meridian lines and two parallel





N R d dY

Z

Qr d

Qr

d(d

)/d]d

[

M rM r

d +[d( d )/d ]d

Nr

Nr

d+

[d(

d)/d

]d

d

r = Rsin

φ θ φ φφ θ

φ

θ

φθ

φφ φ

θφ

φ φφ

θ φ θ

φθ

φ

Mr d

φθ

N r dφθ

Qr dφ

θ

φ

Figure 7. Straining actions acting in the meridian direction.

circles as shown in figure 6. The element is assumedto have both bending and axial stiffness. It is analysedunder a uniform IOP, p, acting in the negative localZ-direction (for background reading see Timoshenko &Woinowsky-Krieger (1959) and Gibson (1980)).

Force equilibrium in the Y -direction (also called themeridian direction) yields

d(Nϕr dθ)dϕ

dϕ − NθR dϕ dθ cos ϕ − Qϕr dθ dϕ = 0

(5.1)or

d(Nϕr)dϕ

− NθR cos ϕ − Qϕr = 0, (5.2)

where r = R sin ϕ and R is the radius of the cornealmedian surface, see figure 7. Similarly, force equilibriumin the Z-direction yields

d(Qϕr dθ)dϕ

dϕ + NθR dϕ dθ sin ϕ

+ Nϕr dθ dϕ − pR dϕ r dθ = 0 (5.3)

or

d(Qϕr)dϕ

+ NθR sin ϕ + Nϕr − pRr = 0. (5.4)

Finally, moment equilibrium about the X-axis gives

d(Mϕr dθ)dϕ

dϕ − Qϕr dθ (R dϕ)

− Mθ(R dϕ) dθ cos ϕ = 0 (5.5)

ord(Mϕr)

dϕ− QϕrR − MθR cos ϕ = 0. (5.6)

Equations (5.2), (5.4) and (5.6) are not enough todetermine the values of the five unknowns Nθ, Nϕ, Qϕ,Mθ and Mϕ. More equations are needed and these can

r

r r w’= + sin + vcos

wv

A’ A

CC’

w

w+(d

/d)d

vv

+(d/d

)d

α

d

d +

−

befo

rede

form

ation

after

defor

mation

O

O’

φφ

φφφ

φ

φβ

α

φ

φ

β

Figure 8. Displacements and rotations in the Y –Z plane.

be obtained by considering the deformed shape of theelement ABCD in figure 8. Study of this figure revealsthat

εϕ =A′C′ − AC

AC=

(dv/dϕ) dϕ + (R + w) dϕ − R dϕ

R dϕ

=dv

R dϕ+

w

R, (5.7)

εθ =2πr′ − 2πr

2πr=

2π(r + w sin ϕ + v cos ϕ) − 2πr

2πr

=w sin ϕ + v cos ϕ

r=

w + v cot ϕ

R, (5.8)

where r = R sin ϕ.Now, with the rotation of the element, the centre

point O will move to O′ resulting in a change in thecentral angle from dϕ to dϕ + β − α. Therefore, whilelength AC is R dϕ, the length of the deformed arc A′C′will be R′(dϕ + β − α). The change in curvature in themeridian direction can then be determined as

χϕ =1R′ −

1R

=(dϕ + β − α)

A′C′ − dϕ

AC.

Substituting values for AC and A′C′ from equa-tion (5.7), χϕ is obtained as

χϕ∼= −1

R2

(w +

dv

dϕ

). (5.9)

Similarly, by considering the deformation in the paralleldirection, the corresponding change in curvature, χθ, isobtained as

χθ∼= cot ϕ

R2

(v +

dw

dϕ

). (5.10)





5.1. Overall equations

The meridian and parallel strains, εϕ and εθ, can berelated to Nϕ and Nθ according to Hooke’s Law,

εϕ =1Et

(Nϕ − νNθ), (5.11)

εθ =1

Et(Nθ − νNϕ), (5.12)

where E is the material’s modulus of elasticity andν is Poisson’s ratio. Poisson’s ratio is taken as 0.5based on the assumption that the cornea behaves asan incompressible body (Bryant & McDonnell 1996).Therefore,

Nϕ =Et

(1 − ν2)(εϕ + νεθ),

Nθ =Et

(1 − ν2)(εθ + νεϕ).

(5.13)

Furthermore, Mϕ and Mθ can be related to the changesin curvature in the following form:

Mϕ = −D(χϕ + νχθ) and Mθ = −D(χθ + νχϕ),(5.14)

where D is the flexural rigidity per unit width of thecorneal surface which is equal to Et3/12(1 − ν2).

The values of εϕ, εθ, χϕ and χθ in terms of v and w asgiven in equations (5.7)–(5.10) can now be substitutedinto equations (5.13) and (5.14) to obtain four newequations that, together with equations (5.2), (5.4)and (5.6), give seven equations in the seven unknowns:Nϕ, Nθ, Qϕ, Mϕ, Mθ, v and w.

These equations have been solved for the case witha pinned edge to represent laboratory boundary condi-tions, where at ϕ = γ (see figure 6b) both Mϕ = 0 andthe horizontal displacement equals zero. In this case,

Nϕ =pR

2, (5.15)

Nθ =pR

2− pR

2(1 − ν)eβη(cos βη), (5.16)

Mϕ =pR2

4β2(1 − ν)eβη(sin βη), (5.17)

Mθ = νMϕ, (5.18)

Rise (w at ϕ = 0)

=pR2

2Et(1 − ν) − νR

Et

pR

2(1 − ν)e−βγ(cos βγ), (5.19)

where β = (R/t)1/2[3(1 − ν2)]1/4 and η = ϕ − γ. Equa-tions (5.15)–(5.19) can be simplified for the case wherethe bending stiffness of the cornea can be ignored andthe cornea assumed to act as a membrane. In this case,

Nθ = Nϕ =pR

2, Mθ = Mϕ = 0

and

Rise =pR2

2Et(1 − ν).

5.2. Mathematical procedure

At the inflation point, the corneal height, H0, andthe diameter of the interface between the cornea andthe sclera, S, are measured experimentally. The initialradius is determined from

R20 =

(S

2

)2

+ (R0 − H0)2. (5.20)

After the first pressure application, p1, the cornealheight will increase by the apical rise recorded exper-imentally,

H1 = H0 + Rise1, (5.21)

and the corresponding radius is obtained from

R1 =

[(S

2

)2

+ (R1 − H1)2]1/2

. (5.22)

Assuming no change in volume of material in the corneaafter loading,

2πR0H0t0 = 2πR1H1t1. (5.23)

Therefore, the new thickness, t1, is

t1 =R0H0t0R1H1

. (5.24)

Also angle γ becomes

γ1 = sin−1

(S

2R1

). (5.25)

The values of R1, p1, t1 and γ1 are then substitutedinto equation (5.19) to obtain a value for the currentmodulus of elasticity, E1. The corresponding strain, ε1,can be determined using equation (5.11) after obtain-ing values for Nϕ and Nθ from equations (5.15) and(5.16), respectively. The corresponding stress is alsodetermined from

σ1 = ε1E1. (5.26)

This process is repeated for every pressure incrementand the experimental values of the pressure and rise areused, as explained, to obtain the corresponding valuesof stress and strain. The results for each of the 20tests were analysed and the average stress–strain curveobtained is shown in figure 9.

6. NUMERICAL MODELLING

Numerical modelling based on the finite-elementmethod is used to accurately predict the performanceof porcine corneas and to provide a detailed account oftheir response to various mechanical actions. Previousstudies attempted this approach with varying degrees ofsuccess. Buzard (1992) and Bryant & McDonnell (1996)developed a finite-element model in which corneaswere modelled using two-dimensional axisymmetricelements. Their work confirmed the effectiveness ofnumerical modelling in corneal biomechanics, but wasunable to model asymmetrical effects such as diseaseor injury. A more detailed three-dimensional model was





0 0.02 0.04 0.06

0.02

0.04

0.06

0.08

0.10

strain

stre

ss (

N m

m−2

)

Figure 9. The average stress–strain relationship obtainedfrom the trephinate inflation tests.

Figure 10. Three-dimensional numerical model.

used by Pinsky & Datye (1991) to predict the immedi-ate change in corneal topography following refractivesurgery. This model was based on a linear elasticbehaviour pattern, despite the strong evidence confirm-ing the visco-elastic, nonlinear material behaviour. Sim-ilar assumptions were adopted by Velinsky & Bryant(1992), who produced a structural model of the wholeeye to determine the number and depth of cuts in sur-gical operations to correct myopia. Their work showedhow finite-element modelling could be customized sothat the process becomes patient specific using clinicallymeasured data.

The numerical models developed in this researchattempt to improve accuracy by using:

(1) the corneal topography data determined experi-mentally;

(2) the material constitutive relationship obtainedfrom the trephinate inflation tests; and

(3) a three-dimensional modelling strategy able torepresent unsymmetrical effects, see figure 10.

The models are comprised of triangular thinshell elements with six degrees of freedom pernode (u, v, w, θx, θy, θz). The Abaqus software package(Hibbitt, Karlsson and Sorenson, Inc. 2001), knownfor its reliability and versatility, is used in this work.The nonlinear material stress–strain property is incor-porated using a hyperelastic material model available

in Abaqus based on Ogden’s strain energy function(Ogden 1984):

U =N∑

i=1

2ui

α2i

(λαi1 + λαi

2 + λαi3 − 3) +

N∑i=1

1D1

(Jel − 1)2i,

(6.1)where λi are the principal stretches, N is a materialparameter, Jel is the elastic volume ratio and ui, αi andDi are temperature-dependent material parameters.

The complexity of the structure and the form of thecornea at both the micro and macro levels presenteda particular challenge during the development of themodels. On the one hand, there was a desire to simulatethe real structure and form of the cornea in orderto improve accuracy and, on the other, there was apractical requirement to simplify the models and keepthem at a reasonable level of complexity to reducecost. In order to strike the best balance between costand accuracy, a study was conducted to identify theeffect of individual parameters on the models’ accuracy.The parameters that were found to have a small or anegligible effect (with an effect on accuracy below 2%)were not considered in the final construction of themodel.

The study considered several parameters, includingthe density of the finite-element mesh, the thicknessvariation between the corneal centre and edge, thecomposite structure of the cornea, the significance ofthe out-of-plane flexural and torsional resistance of thecornea and the boundary condition along the edge ofthe cornea as provided by the sclera. The accuracyof the model in each case was checked against thepressure–apical rise results of the trephinate tests. Thefollowing discussion considers the effect of these param-eters one by one.

6.1. Density of finite-element mesh

Increasing the density of finite-element meshes normallyhas two effects: (1) better accuracy; and (2) higheranalysis cost due to the larger number of nodes andhence number of equilibrium equations. The trend ofimproved accuracy with denser meshes commonly con-tinues up to a certain limit, beyond which the benefitof using denser meshes diminishes. The study presentedin this section is intended to determine this limit.

Figure 11 shows the pressure–apical rise predictionsfor a typical trephinate specimen tested under increas-ing pressure in the present experimental programme.The predictions are obtained using numerical modelsemploying 320, 640 and 960 triangular elements. Themodels have between 8 and 12 rings of elements andbetween 40 and 80 elements per ring. The figure showsthat the accuracy of predictions is not improved withmeshes using more than 640 elements. Increasing themesh density from 640 to 960 elements results in only0.80% improvement in accuracy and would therefore bedifficult to justify considering the associated increase inanalysis cost. This could be compared with the 3.5%improvement in accuracy when using 640 rather than320 elements.





0 0.2 0.4 0.6 0.8

0.010

0.020

pres

sure

(N

mm

−2)

mesh with 320 elements



experimental results

apical displacement (mm)

Figure 11. Effect of using finite-element meshes with differ-ent element densities.

pres

sure

(N

mm

−2)



0.010

0.020 mesh with membrane elementsmesh with shell elements

0 0.2 0.4 0.6 0.8

Figure 12. Effect of using finite-element meshes with shelland membrane elements.

6.2. Bending resistance of cornea

Two options are available in modelling the cornea.It could be modelled using shell elements consideringboth in-plane and out-of-plane effects (and even thecomposite nature of the cornea) or modelled withmembrane elements that ignore the corneal flexural andtorsional out-of-plane resistance. The second option isundoubtedly cheaper in analysis cost.

Both options have been attempted when modellingthe behaviour of a typical experimental trephinatespecimen and the predictions are presented in figure 12.Comparison of the results against experimental dataclearly confirms the significance of the out-of-planeeffects. Using membrane elements in this example leadsto an average loss in accuracy of 4.3% as compared tothe model with shell elements.

6.3. Thickness variation

The corneal thickness is smallest at the centre andgrows gradually towards the limbus—the intersectionwith the sclera. In modelling terms, it is easier toassume a constant thickness than to consider the actualthickness variation. The effect of this simplification onthe modelling accuracy is illustrated using the exampleshown in figure 13. In this figure, the predictions usinga model with a variable thickness (0.645mm along theedge and 0.545mm at the centre) and a model witha constant thickness (0.595mm, which is the average

pres

sure

(N

mm

−2)



0.010

0.020

0 0.2 0.4 0.6 0.8

mesh with constant thickness

mesh with variable thickness

Figure 13. Effect of using finite-element meshes with con-stant and variable corneal thickness.

(a)

(b)

Figure 14. Deformation of the cornea under IOP whileignoring or assuming scleral deformation: (a) without scleraldeformation; (b) with scleral deformation.

between the corneal central and edge thickness) areplotted against the experimental results. The resultsshow only a small change in predictions caused by thissimplification. Assuming a constant thickness results inan average change in predictions of 2%. This outcome,which was repeated for other trephinate tests, indicatesthat this modelling simplification could be adoptedwithout a significant loss in accuracy.

6.4. Corneo–scleral connection

A whole-eye model incorporating both the cornea andthe sclera, and the enclosed fluid, would undoubtedlyprovide a better representation of the actual state ofthe cornea, including the boundary conditions along itsedge, than a model of the cornea alone. However, awhole-eye model is clearly more expensive to developand run. A compromise could be reached if the cornea-only model could be formed such that the boundaryconditions along its edge are accurately represented.This can be done whilst appreciating that the sclera,although stiffer than the cornea, should not be expectedto provide the cornea with supports that are preventedfrom both translation and rotation. In reality, thesclera will deform under pressure or other effects andthis deformation will affect the corneal behaviour, seefigure 14.

A study has been carried out to find a reasonableapproximation of the corneal edge supports. The first





θ

(a)

(b)

θangle

Figure 15. Numerical models: (a) corneal model with edgeroller supports, the angle θ changed in the study between35◦ and 45◦; (b) full ocular model.

approach considers the edge nodes of the corneal modelto be attached to roller supports in an inclined directionas shown in figure 15. The angle of support orientation,θ, is changed between 30◦ and 60◦, and the predictionsof the model with different support orientations arepresented in figure 16, where they are compared withthe results of a whole-eye model. It is evident that asupport orientation of 40◦ provides the best match withthe whole-eye model with an average difference of 0.7%.This approach is similar to that adopted by Orssengo& Pye (1999).

With these inclined supports, the cornea-only modelwould be expected to approximate the whole-eye modeland be suitable for applications where the focus is onlyon the corneal behaviour. Applications that involvethe sclera or require the modelling of the internal eyefluid would still need a whole-eye model. Examplesinclude the modelling of dynamic tonometry, wherethe eye is subjected to an air impulse in a procedureused to measure the IOP. In this case, the pressurewaves inside the eye and the fluid structure interactionduring the movement of the pressure waves must be

pres

sure

(N

mm

−2)


0

0.02

0.04

0.4 0.8 1.2

whole-eye model

roller 35º

roller 45º

roller 37ºroller 40º

Figure 16. Comparison between the cornea model withdifferent support orientations and the whole-eye model.

considered. For these applications, a whole-eye modelmust be used. Further work will be undertaken to studyother approximations of the corneo–scleral connection,for instance using a roller support with a rotationalspring. This idealization would be expected to providea more accurate representation of the effect of the sclerain resisting corneal curvature.

7. RESULTS OF THE NUMERICAL MODEL

7.1. Further insight into experimental behaviour

The numerical model has been used to describe theperformance of the trephinate specimens tested in thisresearch to a level of detail impossible to achieve withlaboratory tests. The information obtained includes thedistributions of displacement, stress and strain over thesurface area of the cornea. Examples of the outputof the model for a typical trephinate specimen underIOP = 2kPa or 15.04mmHg are given in figure 17.Based on the initial study to identify the parameterswith significant influence, the model has 640 shellelements and constant thickness. It has pinned supportsalong the edge nodes to simulate the laboratory testconditions (note that in this case the sclera did notform part of the test specimens). The model alsoincorporates the nonlinear material constitutive rela-tionship obtained from the trephinate inflation testsand implemented using the hyperelastic material modelavailable in Abaqus. The load–apical rise results matchthe laboratory data closely, as shown in figure 17. Inaddition to these results, the deformation, stress andstrain distributions given in the figure provide a usefulinsight into the behaviour of the corneal specimen andthe changes encountered due to elevated IOP.

7.2. Practical applications

7.2.1. Modelling of keratoconic eyes. Keratoconus ischaracterized by a deterioration of the structure of thecornea mainly in the form of a localized loss of up to 75%of the corneal tissue (Bron 1988). As a result, the corneachanges shape under IOP, with serious implications onits refractive power (Andreassen 1980; Edmund 1989).The disease affects 55 individuals per 100 000 of thegeneral population. Research is currently underway to





(a)

8.400 × 10−3

9.474 × 10−3

1.055 × 10−2

1.162 × 10−2

1.269 × 10−2

1.751 × 10−2

1.995 × 10−2

2.239 × 10−2

2.483 × 10−2

2.727 × 10−2(b)

3.003 × 10−1

(c)

2.252 × 10−1

1.502 × 10−1

7.508 × 10−2

0

0.010

0.020

0 0.2 0.4 0.6 0.8

numerical results

experimental resultspr

essu

re (

N m

m−2

)


(d)

Figure 17. Example results of the numerical model under an IOP of 2 kPa (15.04 mmHg): (a) distribution of Mises strain(N mm−2); (b) distribution of maximum principal strain (Nmm−2); (c) distribution of displacement (mm); (d) comparisonbetween numerical and experimental IOP rise results.

study the effect of tissue loss on the corneal topographyand hence refractive power. The research assumes thedisease can affect the cornea either at or away fromits centre. The thickness loss and the percentage ofthe surface area affected by the disease are two moreparameters in the study.

The study is based on using the numerical modelto simulate the effects of keratoconus on the cornealbiomechanics. In particular, predictions of deformation,change of shape and stress distribution are important,as they may present useful information for the clinicalmanagement of the disease. Typical examples of theresults obtained from the initial parts of this work arepresented in figure 18 for two cases with keratoconusaffecting the centre of the cornea and an area half-waybetween the centre and the edge. A maximum tissueloss of 75% of the average thickness is assumed at thecentre of the disease, gradually reducing to no loss at theedge of the affected area. The affected area is assumedto have a diameter equal to a quarter of the corneal

meridian length. The predicted behaviour for a normalcornea is also shown in figure 18 for comparison.

The results presented in figure 18 show the increasedcorneal deformation as a result of the disease. Thisanalysis suggests an increase in the elevation of thecornea of approximately 0.080mm, which may have asignificant effect on the refractive power of the eye,although this may be partially countered by accom-modation causing changes to the lens and the shapeof the cornea (He et al. 2003). The distribution ofthe deformation is dependent on the location of theaffected area; when the tissue loss is at the centre, thedeformation distribution remains symmetrical, but thesymmetry is lost when the affected area is shifted awayfrom the centre. This type of approach can help provideinsights into the cause of this disease, the effects of lossof tissue and corneal stiffness and the progression of thedisease. Further comparison with patient specific datawill provide calibration and validation of the computermodels.





0.02

0.04

0.06

0.08

0.10

−6.0 −4.0 −2.0 0 2.0 6.0

normal cornea

disease centre L/8 away from corneal centre

disease centre 2L/8 away from corneal centre



X coordinate (mm)4.0

Z di

ffer

ence

(mm

)

Figure 18. Effect of keratoconus on the corneal profile under an IOP of 2 kPa; L is the meridian length of the cornea.

0

4.333 × 10−38.667 × 10−31.300 × 10−21.733 × 10−22.167 × 10−22.600 × 10−2

(a)

(c)

(b)

(d)

Figure 19. Modelling the GAT procedure using a cornea-only model (units are Nmm−2): (a) before contact; (b) initialcontact; (c) intermediate step; (d) full applanation.

7.2.2. Static tonometry. Tonometry is a procedure tomeasure IOP of the eye. Ophthalmologists use theIOP measurements to diagnose a number of condi-tions including hyphema (trauma and inflation of theiris) and glaucoma, the second most common causeof irreversible blindness in the world (Wilson & Kass2002). The two common tonometry techniques are

the Goldmann applanation tonometer (GAT) and thepneumo-tonometer. The GAT is based on makinga pseudo-static measurement of the force requiredto flatten a fixed central area of the cornea with3.06mm diameter (Goldmann & Schmidt 1961; Feltgenet al. 2001; Whitacre et al. 1993), while the pneumo-tonometer is a dynamic test where the eye is subjected





00.5 × 10−2

(a)

(c)

(b)

(d)

1.0 × 10−21.5 × 10−22.0 × 10−22.5 × 10−23.0 × 10−23.5 × 10−24.0 × 10−24.5 × 10−25.0 × 10−2

Figure 20. The pressure wave travelling through the eye in dynamic tonometry (units are N mm−2): (a) initial contact;(b) pressure wave; (c) dispersion of pressure wave; (d) pressure wave back at cornea following impact.

to an air impulse and the time taken to flatten thecentral cornea is used to determine the value of the IOP.

A study is underway to numerically model the GATprocedure with the view of determining the effect ofcorneal parameters such as thickness and diameter onthe IOP measurements. The study is parametric andinvolves corneal models with different dimensions andIOPs. The corneas are subjected to a static pressurefrom a solid object with 3.06mm diameter representingthe GAT. The pressure required to achieve completecontact between the tonometer and the cornea ismeasured and plotted to enable a comparison of theoutcome of the procedure with the actual IOP in theeye.

The model used in this study is a cornea-only model,as the connection with the sclera and the interactionbetween the eye and its internal fluid is expected tohave little effect on the results. The cornea modelhas 640 shell elements and is provided with rollersupports (with 40◦ inclination) along all edge nodes.The modulus of elasticity and Poisson’s ratio are takenas 0.0229× IOP and 0.5, respectively, according toOrssengo & Pye (1999). The tonometer is modelledusing 40 five-sided solid elements with two triangular

and three rectangular faces as shown in figure 19.The figure shows the progress of the procedure untilcomplete contact is established. The initial parts of thestudy have compared the results of fully explicit modelsof the indentor with simulations, where the indentor hasbeen represented by a uniformly distributed load (as hasbeen done previously, Orssengo & Pye (1999)), for bothlinear and nonlinear elastic corneal models. Furthercomparisons have been made with clinical data fromDoughty & Zaman (2000). Both data sets indicate thatthe IOP is over-predicted for thin corneas and under-predicted for thick corneas. Finite-element analysisshows a strong correlation with the clinical studies andthe prediction using explicit modelling of the indentorand material nonlinearity produces the best results.Further work will help provide more accurate correctionfactors for IOP measurements to accommodate varyingcorneal thickness.

7.2.3. Dynamic tonometry. A similar study is beingcarried out to determine the effect of corneal thick-ness and diameter on the IOP measurements usingthe pneumo-tonometer. This procedure is based onmeasuring the time taken for an air impulse to flatten





0 10 20 30 40 50

0

−0.40

0.40

corn

eal d

ispl

acem

ent (

mm

)

time (× 10−3)

Figure 21. The displacement at the corneal centre pointupon impact with the air impulse in pneumo-tonometry.

the central region of the cornea. As this procedure isa non-contact procedure and is quick to apply, it iscurrently finding increasing acceptance.

Pneumo-tonometry is a dynamic procedure and theinteraction between the cornea and the sclera andbetween the eye and its internal fluid is certainlyimportant for the accurate modelling of the procedure.A whole-eye model is used in this study with thecornea and sclera modelled using 640 and 3200 shellelements, respectively, see figure 20. The internal fluidis modelled as a set of 3840 internal fluid elementswith the initial pressure being equal to the eye’s IOP.The air impulse is modelled as a spherical formationof 150 four-sided membrane elements enclosing 150fluid elements with density and pressure equal to theair density and atmospheric pressure, respectively. Theair impulse model is triggered with an initial velocitytowards the centre of the cornea. The displacements ofthe cornea upon impact at several points at and awayfrom the centre are monitored and plotted to enable thedetermination of the time needed to flatten the centralcorneal region. The flow of pressure waves within theeye, as seen in figure 20, is monitored along with theassociated pressure changes at the region of impact, toimprove the understanding of the behaviour of the eyestructure and to enable more accurate correction factorsfor the IOP measurements to be obtained.

The dynamic response of the eye to the impact withthe air impulse is illustrated in figure 21, showing thedisplacement of the corneal centre point with time.A phenomenon that has been observed experimentallyand confirmed in this study is how the cornea deforms inonly the posterior direction in the first two deformationcycles. This finding is currently under study to establishthe reasons behind it. The possibilities include theinteraction between the eye and its internal fluid andthe peculiar geometry of the corneo–scleral structure.

8. CONCLUSIONS

The research presented in this paper is part of a long-term project to develop accurate predictive tools thatcan assist in the clinical management of the cornea.Accuracy is achieved through the use of laboratory teststhat are representative of in vivo conditions and theadoption of realistic corneal topography and materialconstitutive relationships. Laboratory inflation testsof corneal trephinates confirm the highly nonlinearbehaviour of the material. Mathematical analyses areperformed to determine the material constitutive rela-tionship and they confirm that the nonlinear behaviourobserved experimentally is primarily due to the natureof the material and its microstructure. An assumptionof a linear material behaviour might simplify the analy-sis, but will undoubtedly lead to serious underestima-tions of stiffness under high pressures.

Enhancements of the predictive models are underwayto consider the hysteretic behaviour observed undercyclic loading and to improve the stability of analysisat the stage where the modulus of elasticity increasesrapidly. Planned applications of the model includepredicting the effects of sports injuries and refractivesurgery. Research is also underway to study the loadingconditions in tonometry with a view to improving theaccuracy of existing techniques in measuring the IOP.

This research demonstrates the usefulness ofengineering analytical tools within multi-disciplinaryprojects in the field of biomechanics.

This work was partially supported by a Doctorate Training

award from the Engineering and Physical Sciences Research

Council. The authors are grateful for the valuable contribu-

tions made by Mr David Garway-Heath, the lead clinical

researcher of the Glaucoma Research Unit, Moorefields

Hospital, London.

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