Assessment of Goldmann Tonometry Using Numerical Modelling

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Evaluation of Goldmann Applanation Tonometry Using a Nonlinear FiniteElement Ocular Model

AHMED ELSHEIKH ,1

DEFU WANG,1

AACHAL KOTECHA,2,3

MICHAEL BROWN,1

and DAVID GARWAY-HEATH2,3

1Division of Civil Engineering, Faculty of Engineering, University of Dundee, Dundee, DD1 4HN, UK; 2Glaucoma ResearchUnit, Moorfields Eye Hospital, London, EC1V 2PD, UK; and 3Department of Optometry and Visual Science, City University,

London, UK

(Received 22 February 2006; accepted 29 August 2006; published online: 28 September 2006)

AbstractGoldmann applanation tonometry (GAT) is theinternationally accepted standard for intra-ocular pressure(IOP) measurement, which is important for the diagnosis ofglaucoma. The technique does not consider the effect of thenatural variation in the corneal thickness, curvature and

material properties. As these parameters affect the structuralresistance of the cornea, their variation is expected to lead toinaccuracies in IOP determination. Numerical Analysis basedon the finite element method has been used to simulate theloading conditions experienced in GAT and hence assess theeffect of variation in corneal parameters on GAT IOPmeasurements. The analysis is highly nonlinear and considersthe hyper-elastic J-shaped stressstrain properties of cornealtissue observed in laboratory tests. The results reveal a clearassociation between both the corneal thickness and materialproperties, and the measured IOP. Corneal curvature has aconsiderably lower effect. Similar trends have been foundfrom analysis of clinical data involving 532 patients referredto the Glaucoma Unit at Moorfields Hospital, and fromearlier mathematical analyses. Nonlinear modelling is shownto trace the behaviour of the cornea under both IOP andtonometric pressure, and to be able to provide additional,and potentially useful, information on the distribution ofstress, strain, contact pressure and gap closure.

KeywordsTonometry, Intra-ocular pressure, Cornea,

Numerical modelling.

ABBREVIATIONS

CCT central corneal thickness

GAT Goldmann applanation tonometry

IOP intra-ocular pressureIOPG intra-ocular pressure as measured by GAT

IOPT true intra-ocular pressure

PCT peripheral corneal thickness

INTRODUCTION

Intra-ocular pressure (IOP) measurement in tonom-

etry is important for the diagnosis and management of a

number of conditions, most notably glaucoma, thesecond most common cause of irreversible blindness

in the world. Goldmann applanation tonometry (GAT)

developed in the mid-1950s, is still the internationally

accepted standard for IOP determination. It makes a

pseudo-static measurement of the force required to

flatten a fixed area of the central cornea and uses

this force to estimate the value of IOP. The natural

variation in the central corneal thickness (CCT),

corneal curvature and material properties, which have a

direct bearing on the corneal structural resistance,

can affect the accuracy of IOP measurement, and

since IOP is a major risk factor for glaucoma and

forms part of the classification of ocular hypertension

and normal tension glaucoma,13 errors in IOP

measurement can lead to the misdiagnosis of these

conditions.

The possible impact of corneal thickness on IOP

measurement using GAT was identified and discussed

briefly by Goldmann and Schmidt.16 Later studies by

Ehlers and co-workers9 drew attention to the effect of

CCT on IOP measurement, and the interest in this

effect grew further with the advent of refractive surgery

procedures involving an iatrogenic thinning of the

cornea, see Refs. [5 ,7,18,29,32]. The overall conclusion

that can be drawn from these studies is that IOPmeasurements using tonometry are affected by differ-

ences in CCT. While the vast majority of the studies

were based on statistical analyses of clinical data,

mathematical analysis was successfully used in a

number of studies including those by Orssengo and

Pye29 and by Liu and Roberts25 and produced results

with a similar trend. All found that high CCT led to

IOP overestimations while low CCT led to IOP

underestimations. However, there is no agreement yet

Address correspondence to Ahmed Elsheikh CEng MICE PhD,

Division of Civil Engineering, Faculty of Engineering, University of

Dundee, Dundee, DD1 4HN, UK. Electronic mail: a.i.h.elsheikh@

dundee.ac.uk

Annals of Biomedical Engineering, Vol. 34, No. 10, October 2006 ( 2006) pp. 16281640

DOI: 10.1007/s10439-006-9191-8

0090-6964/06/1000-1628/0 2006 Biomedical Engineering Society

1628


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on the pressure correction factors that should be used

to consider CCT variations.

Other sources of tonometry errors include the cor-

neal curvature as has been reported by Gunvant

et al.,18 Liu and Roberts25 and Kanngiesser et al.23

Furthermore, wound healing following surgical pro-

cedures is also believed to lead to changes in the bio-

mechanical properties of corneal tissue. These changes,

although not yet quantified, are expected to influence

the structural resistance of the cornea and might

therefore affect the accuracy of IOP measurements in

tonometry.25

The aim of this study is to use numerical modelling

based on nonlinear finite element analysis to create a

representative model of the GAT procedure. Numeri-

cal modelling is adopted as it has the potential to

represent real life conditions without having to adopt

the simplifications necessary with mathematical closed

form solutions. The development of the numerical

model has undergone a number of stages to optimise

its construction and improve its accuracy. The com-plexity of the ocular structure, both at the microscopic

and macroscopic levels, makes it essential to distin-

guish between the parameters that have a considerable

effect on behaviour (and which should therefore be

incorporated in model construction), and the param-

eters which can be ignored for their negligible effect.

The construction of the model was followed by a val-

idation process against experimental tests before it was

used in a parametric study in which GAT procedure

was simulated with different values of corneal thick-

ness, curvature and material properties. The results are

compared against the outcome of statistical analysis ofnew clinical data and the results of earlier mathemat-

ical analysis.

MODEL CONSTRUCTION

Nonlinear finite element modelling was used in this

research to enable a detailed representation of biome-

chanical behaviour and to provide a systematic

approach to determine the impact of variation in cor-

neal parameters on tonometry. The complexity of the

structure and form of the cornea at both the micro-

scopic and macroscopic levels presented a particular

challenge during the development of the numerical

models. On one hand, there was a desire to simulate

the real structure of the cornea in order to improve

accuracy, but on the other there was a practical

requirement to simplify the models and keep them at a

reasonable level of complexity to reduce computa-

tional cost. In order to strike the best balance between

computational cost and accuracy, a study10 was con-

ducted to identify the effect of individual parameters

on the models behaviour. The parameters that were

found to have a small or a negligible effect (with an

effect on results below 1%) were not considered in the

final construction of the model.

The parameters considered in the study10 were the

thickness variation between the limbus and the corneal

centre, representation of the boundary conditions

along the edge of the cornea, the material properties

and the corneal topography. The density of the finite

element mesh and the cornea discretisation method

were also considered. According to the findings of this

study, an optimum model construction involves the

following details:

The discretisation method shown in Fig. 1 is used toensure all element internal angles are kept within

practical limits (20 and 80). The model follows

the structural form of diamatic skeletal domes built

in structural engineering applications.27

Solid six-noded elements are used with six degrees of

freedom per node (u, v, w, hx, hy, hz). These elementshave been found to closely simulate the behaviour of

experimental test specimens while enabling a good

representation of corneal variable thickness.

The model has a number of element layers to enabletracing the stress and strain distributions across the

corneal thickness. The number of layers is at least

two under uniform pressures (e.g. IOP) and six

under concentrated effects such as point loads and

tonometry pressures. This feature is also important

for future development of the numerical model as it

FIGURE 1. Discretisation method adopted in construction ofcorneal modelmodel shown has 6 segments, 7 rings and294 elements per layer.

Assessment of Goldmann Tonometry Using Numerical Modelling 1629


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allows the use of different material properties for the

epithelium, endothelium and stroma when they

become available.

The minimum number of elements in each layer is294, arranged in six segments and seven rings.

Although the model predictions with this number of

elements are close to experimental observations,

more elements, and hence a finer mesh, would be

needed if smooth contour distributions are to be

obtained.

The thickness variation between a minimum at thecentre to a maximum along the limbus is modelled

to approximate natural topography.

The corneal model is provided with edge rollersupports oriented at 23 to the limbal plane. This

choice of boundary conditions and orientation angle

creates working conditions similar to those created

by the actual connection to the sclera, see Fig. 2. In

arriving at this angle, a whole eye model was

compared to a cornea-only model provided with

edge roller supports. The angle or orientation of theroller supports was varied until the behaviour

predictions of the two models under both distributed

pressure and concentrated load were in close agree-

ment (less than 2% on average). Details of this study

can be found in Refs. [1,10].

The corneo-scleral intersection is oval (elliptical)with the temporalnasal diameter (DTN) 10% larger

than the inferiorsuperior diameter (DIS). At the

same time, constant curvature is assumed along the

corneas two main meridian lines. This choice of

topography, which represents an approximation of

the actual aspheric form of the cornea, was neces-

sary because of (1) the lack of available corneal

topography maps and (2) the need to adopt an easy

to define and modify topography in the numerical

models.

The nonlinear material properties of corneal tissueobserved in laboratory tests1 are incorporated in the

model. The stressstrain relationship observed dem-

onstrates clear hyper-elastic behaviour with an

initial low stiffness phase followed by another with

much higher stiffness. The effect of using a simpli-

fied linear-elastic material model in Goldmann

tonometry modelling is illustrated in this paper.

The nonlinear material stressstrain property is

incorporated using a hyper-elastic material model

based on Ogdens strain energy function28:

UXN

i1

2ui

a2i

kai1 kai2 k

ai3 3

XN

i1

1

DiJel 12i

1

where Uis the strain energy per unit volume, ki are the

principal stretches, N is a material parameter defining

Angle

(a)

0

0.005

0.01

0.015

0.02

0.025

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Central corneal rise (mm)

Intra-ocularpr

essure(N/mm

2) Cornea model - 15 roller supports

Cornea model - 20 roller supports




Whole eye model

(b)

FIGURE 2. Corneal model with roller edge supports to simulate actual connection with the sclera. (a) Angle h found from com-parisons with a full ocular model to be 23 (Ref. [10]), (b) comparison of behaviour predictions between a whole eye model and acornea only model under IOP and with different angles h.

ELSHEIKH et al.1630


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the order of the equation, ui, ai, and Di are tempera-

ture-dependent material parameters and Jel is the

elastic volume ratio.

The Abaqus software package20 is used in this work.

The analyses consider both geometric nonlinearities due

to change of joint coordinates and material nonlinear-

ities. In tracing the nonlinear behaviour the Riks arc

method31 is adopted. In this method, load increments

vary according to the current stage of overall behaviour

and are controlled automatically such that a solution is

obtained even after the point of mechanical failure.

CORNEAL THICKNESS

Several recent studies measured corneal thickness in

recognition of its likely effect on tonometry measure-

ments. Most studies concentrated on the central

corneal thickness and used analysis of clinical obser-

vations to give its average value and range of variation.

The average values of CCT vary slightly betweenstudies, for instance: 0.548 mm is reported by Cho and

Cheung4 and 0.545 mm for the right eye and 0.547 mm

for the left eye by Lam and Chan.24 The natural var-

iation of CCT in the population was also estimated in

earlier studies including Feltgen et al.12 who reported

values between 0.448 and 0.713 mm. In the present

numerical study, a wider range of variation in CCT is

considered; between 0.320 and 0.720 mm to go beyond

the natural range.

The relationship between the CCT and the periph-

eral corneal thickness (PCT) is not yet established.

While it is known that PCT is always larger than CCT,

it is not clear whether PCT changes with CCT, and if

so by what degree. In this study, it is assumed that PCT

is always 0.150 mm larger than CCT. This is compat-

ible with the corneal dimensions of the Gullstrands

No. 1 schematic eye,1 in which CCT = 0.520 mm and

PCT = 0.670 mm.

CORNEAL CURVATURE

Corneal curvature variation has been considered as

another possible cause of error in tonometry mea-

surements.6 The radius of anterior curvature, R, theterm used to describe corneal curvature, is found by

Doughty and Zaman6 to change between 7.53 and

8.09 mm for children, 7.58 and 8.14 mm for adults,

and 7.36 and 7.86 mm for elderly adults. Similar val-

ues, between 7.60 and 8.14 mm, are reported by

Dubbelman et al.8 A wider range of variation in R

between 7.20 and 8.40 mm is considered in this study

to determine the effect of corneal curvature on IOP

measurements.

CORNEAL MATERIAL PROPERTIES

The material properties of corneal tissue clearly

have an effect on the structural resistance of the cor-

nea, and hence the accuracy of IOP measurement.

Despite this logical association, the material properties

have rarely been considered an important parameter in

earlier attempts to improve the accuracy of tonometry.

A recent exception is the work published by Liu andRoberts25 which found the material properties to have

a profound effect on GAT IOP.

There is a wide variation between the material

properties reported in earlier publications.33 The vari-

ation seems to be related to three main factors, the

pressure or stress at which Youngs modulus (E) is

determined, the test method and the test strain

rate. Hoeltzel et al.,21 for instance, reports two values

of E, 0.34 N/mm2 under IOP = 10 mmHg (0.0013 N/

mm2) and 4.1 N/mm2 under IOP = 400 mmHg

(0.053 N/mm2). Kampeier et al.22 also reports a lower

E = 0.4 N/mm2

at strain = 0.02 and a higher E =3.00 N/mm2 at strain = 0.08. Other researchers pres-

ent a hyper-elastic material model with E increasing

gradually with strain, yet still the values of initial E

vary considerably. For example, Bryant and McDon-

nell3 have an initial value of 0.08 N/mm2, Zeng et al.36

have E = 0.27 N/mm2 and Nash et al .26 have

E = 20.1 N/mm2 (all at strain = 0.01).

Another likely reason for the variation is the test

method. The strip testing adopted by Nash et al.26 has

a number of inherent geometric inefficiencies related to

the initial curved form of the corneal specimen, which

does not lend itself to strip testing.11 On the other

hand, inflation testing used for example by Bryant and

McDonnell,3 which maintains the cornea in its natural

working condition, is considered more accurate and its

load application speed more representative of the

normal state. For this reason, the test method is

expected to have a significant effect on the values of E

obtained experimentally.

Another factor that adds to the variation in material

models is the highly visco-elastic behaviour of corneal

tissue. This behaviour makes the material properties

dependent on the strain rate used in testing. For this

reason, it is significant that the studies discussed above

adopted different strain rates and in some cases, thestrain rates were not reported.

In this work, a stressstrain relationship of the form

shown in Fig. 3 with an initial E = 0.30 N/mm2 is

used. This relationship has been obtained from a lim-

ited experimental study conducted earlier by the

authors11 and is quite similar to the results reported by

Hoeltzel et al.,21 Kampeier et al.22 and Zeng et al.36

However, as will be seen below, the parametric studies

on effect of CCT and R have been repeated considering



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another material model with initial E = 0.08 N/mm2

to demonstrate the effect of variation in material

properties on GAT IOP. This additional model, which

was obtained from work by Bryant and McDonnell,3 is

also shown in Fig. 3.Note should be made that the material properties

obtained from inflation tests and reported in the lit-

erature were recorded after loading the cornea to the

inflation point. Before this point, the corneal behav-

iour was unstable and the readings remained incon-

sistent until the cornea had taken its natural inflated

form. For this reason and since the numerical models

described in this paper are based on these material

properties, these models can only describe the behav-

iour beyond the inflation point.

RESULTS

Analysis of a Case with Calibration Dimensions

GAT is based on the modified ImbertFick law,

which states that the tonometric pressure, and the effect

of surface tension along the edge of the tonometer

caused by the tear film, equal the IOP and the effect of

the bending resistance of the cornea, see Fig. 4.

Using experimental observations made by Gold-

mann and Schmidt,17 Ehlers et al.9 and Whitacre

et al.,34 it is found that for certain corneal dimensions,

the effect of surface tension cancels out the effect of

bending resistance. As a result, the external tonometry

pressure at applanation, (also called the intra-ocular

pressure measured using Goldmann tonometry, or

IOPG) equals the true intra-ocular pressure, denoted

IOPT. In this special case, the correction factor,

K = IOPG/IOPT equals 1. The corneal dimensions at

which this special case arises are called the calibration

dimensions, namely CCT = 0.520 mm and

R = 7.80 mm. Numerical modelling of GAT starts

with this special case and the value of the numerical

correction factor is compared with the expected value

of 1 found experimentally.

In this work, a corneal model with 17,424 six-noded

solid elements arranged in 6 layers and 22 rings is used.

This large number of elements has been necessary to

model the concentrated effect of tonometry and to

create a fine mesh at the contact area with the

tonometer. The tonometer model, which has a

3.06 mm diameter, uses similar solid elements. The

anterior surface of the cornea and the posterior surface

of the tonometer are described in the analysis as con-

tact surfaces to prevent over-closure of the gap

between them. The edge supports of the cornea are

roller supports set at 23 to the limbal plane in order to

represent the effect of connection with the sclera.10

The material properties used in the analysis have an

initial Youngs modulus, E = 0.30 N/mm2. When

analysed using the Ogden strain formula given in Eq.

(1) and assuming a forth order (N = 4) for improved

accuracy, the following values of u and a parameters

are obtained: u1 =)

110.3, u2 = 55.64, u3 = 108.2,u4 = )53.54, a1 = 14.97, a2 = 16.06, a3 = 12.93,

a4 = 11.99. These values provide a close fit with the

stressstrain relationship with initial E = 0.30 N/mm2

shown in Fig. 3. Note that since the extrafibrillar

matrix of the cornea is principally water, the corneal

tissue is characterized as a nearly incompressible

material.3,19,22,30,33 In Abaqus,22 materials defined as

incompressible are given an elastic volume ratio, Jel, of

1. Further, when the material response is incompress-

ible, the solution to the analysis problem cannot be

obtained in terms of the displacement history only

since a hydrostatic pressure can be added withoutchanging the displacements. This difficulty is removed

in Abaqus by treating the pressure as an independently

interpolated basic solution variable, coupled to the

displacement solution through the constitutive theory

and the compatibility condition, with this coupling

implemented by a Lagrange multiplier. This process is

implemented through the use of hybrid elements that

use a mixture of displacement and stress variables with

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Strain (mm/mm)

Stress(N/mm

2)

Model with initial E = 0.30 N/mm2

Model with initial E = 0.08 N/mm2

FIGURE 3. Material models considered in parametricstudies.

Intr

a-o

cula

rpres

sure

Tonometric pressure

Surface tensionBending resistance

FIGURE 4. Corneal deformation under IOP and tonometricpressure.

ELSHEIKH et al.1632


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an augmented variational principle to approximate the

equilibrium equations and compatibility conditions.

Further details of this process can be found in the

program documentation.20

The analysis starts by subjecting the corneal model

to an IOP with a predetermined value of 0.002 N/mm2

(15 mmHgapproximately at middle of normal IOP

range), acting as uniform pressure on the internal faces

of the internal layer of solid elements. After inflation

under full IOP, the model is subjected to contact

pressure from the tonometer, which is pushed gradu-

ally and concentrically against the cornea until com-

plete applanation is achieved. The stress distributions

recorded during this process are shown in Fig. 5. It can

be seen that increases in stress are limited to the

tonometer contact area with limited effect elsewhere.

The closure of gap between the tonometer and the

corneal anterior surface is continuously monitored

during the analysis to determine the point at which

applanation has occurred. Figure 6 shows the progress

of gap closure over five stages of the analysis, the lastof which represents the point of full applanation.

The contact stress distribution between the tonom-

eter and the corneal anterior surface is also monitored

during the progress of applanation. It is interesting to

observe how the maximum values of contact stress

change location with the progress of applanation, see

Fig. 7. At the start, the stress is highest at the centre of

the tonometer as this is where the contact is initiated.

Then as applanation progresses, the area of highest

contact stress shifts away from the centre and finally

locates at approximately half the tonometer radius at

full applanation.

At applanation, the force required to push the

tonometer model to this point is divided by the contact

area to obtain the external pressure. This pressure is

produced in actual tonometry by two effects, one is the

tonometry pressure (referred to as Goldmann IOP or

IOPG) and the other is the effect of tear film surface

tension. The value of surface tension is taken as

0.0455 N/m from work carried out at the University of

New South Wales and not yet published. This value is

slightly less than that for water, 0.0728 N/m.

The surface tension acts along the edge of the con-

tact area. Therefore, its effect on the IOPG calculations

is determined by multiplying the surface tension by the

tonometer perimeter (2 1.53p) and dividing it by thecontact area (1.532p). The resulting pressure is

subtracted from the previously calculated externalpressure to calculate IOPG. In the analysis, the true

intra-ocular pressure (IOPT) applied is 0.002 N/mm2

(15 mmHg) and the external pressure is determined as

0.002074 N/mm2. The small effect of tear film surface

tension is 0.0455 10)3 (2 1.53p)/(1.532p) =0.000059 N/mm2. Therefore IOPG = 0.002074

0.000059 = 0.002015 N/mm2 = 15.11 mmHg. This

FIGURE 5. Stress distribution during the analysis: (a) following application of IOP = 0.002 N/mm2 (15 mmHg), (b) following fullapplanationview without tonometer, (c) following full applanationview with tonometer. Contours are drawn on model withoriginal undeformed geometry. Stress range: red= 0.09 N/mm2, blue= 0.00 N/mm2.



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gives a correction factor = 15.11/15 = 1.007. This

value is close to 1.0, which is expected with the cali-

bration dimensions.

Effect of Material Modelling Approaches

The analysis of the above case with the calibration

dimensions has been repeated with the adoption of a

constant E equal to the initial E of the model ( E =

0.30 N/mm2). In this case, applanation is achieved

under an external pressure of 0.001714 N/mm2. After

removing the effect of surface tension, IOPG is found

as 0.001655 N/mm2 or 12.42 mmHgwith an error of

2.58 mmHg or 17.2%. Further study of the behaviour

of the numerical model with both the linear and non-

linear material definitions reveals significant differ-

ences in behaviour. For instance Fig. 8 shows the

anterior displacement of points along a corneal

meridian line starting from the centre. The model with

hyper-elastic material definition demonstrates a non-

linear displacement rate during both inflation and

applanation, giving a strong indication that the stress

regime within the model rose beyond the initial low

stiffness phase of the material. This behaviour is not

detected with linear material definition where the

displacement rate is almost linear and as a result,

applanation is achieved at a different pressure level.

It has therefore been concluded that the nonlinear

material model should be used in all further GAT

modelling since the stresses generated during the

procedure could exceed those associated with the first

low-stiffness phase of material behaviour.

Parametric Study 1Effect of Variation in CCT

Figure 9 shows numerical estimation of the effect of

CCT variation on IOP measurements using GAT

(IOPG). The figure shows the results obtained with

FIGURE 6. Closure of gap between the tonometer and the corneal anterior surface over five stages of analysisthe last stagemarks the full applanation pointFigure shows the tonometer and part of the corneal anterior surface with contours drawn onmodel with original undeformed geometry. Gap width range: red= 0.150 mm, blue= 0.00 mm.

FIGURE 7. Distribution of contact pressure on the tonometer surface with the progress of applanationfigure shows thetonometer and part of the corneal anterior surface with contours drawn on model with original undeformed geometry. Contactstress range: red= 0.0045 N/mm2, blue= 0.00 N/mm2.

ELSHEIKH et al.1634


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two material models, with initial Youngs modulus,

E = 0.30 and 0.08 N/mm2, respectively. Within the

CCT range considered (between 0.320 and 0.720 mm),

IOPG changes by 7.03 and 2.35 mmHg (or 1.75 and

0.60 mmHg per 0.100 mm variation in CCT) for the

two material models, respectively. In all cases, R is

kept constant at 7.8 mm. The effect of CCT on IOPGfor the first material model (0.0175 mmHg per 1 lm) is

comparable to values obtained earlier in population

based studies. Earlier values include 0.015 and 0.018 by

Foster et al.,15 0.018 and 0.024 by Foster et al.14 and

0.019 mmHg by Wolfs et al.35

The results indicate that IOP is overestimated with

CCT values larger than 0.520 mm and underesti-

mated with CCT values below it. There is also

evidence that the change in material stiffness has a

clear effect on the response to CCT variation. With astiffer material, the corneal resistance increases and

makes the CCT thickness effect on IOPG more

pronounced.

Parametric Study 2Effect of Variation in R

The numerical estimation of the effect of corneal

curvature, as described by the anterior radius, R, on

IOPG is depicted in Fig. 10. In this numerical study,

R is varied between 7.2 and 8.4 mm, while CCT is

kept unchanged at 0.520 mm. The study has been

conducted twice for material models with initial

Youngs modulus of 0.30 and 0.08 N/mm2.

As R increases, the corneal curvature decreases

leading to a reduction in the structural resistance and

hence an underestimation of IOP. With R increasing

from 7.2 to 8.4 mm, IOPG reduces by 1.63 and

1.58 mmHg (or 1.35 and 1.32 mmHg per 1.0 mm

variation in R) for the two material models, respec-

tively. The changes in IOPG are more pronounced

with R below the calibration value of 7.8 mm.

Beforeapplication

ofIOP

Afterapplication

ofIOP

Pointofapplanation

IOP = 0 to 15 mmHg Tonometric pressure

(exceeding point of applanation)

7.6

7.7

7.8

7.9

8.0

8.1

8.2

8.3

Centre point

Intermediate point

Intermediate point

Intermediate point

Intermediate point

Point under edge of tonometer

Point outside tonometer area

Centre point

Intermediate point

Intermediate point

Intermediate point

Intermediate point

Point under edge of tonometer

Point outside tonometer areaAnteriordisplacementofcornealsurfac

e(mm)

Beforeapplication

ofIOP

Afterapplication

ofIOP

Pointofapplanation

IOP = 0 to 15 mmHg Tonometric pressure

(exceeding point of applanation)

7.6

7.8

8.0

8.2

8.4

8.6

8.8

Anteriordisplacementofcornealsurfac

e(mm)

(a) (b)

FIGURE 8. Anterior displacement of the corneal surface under the tonometer during both inflation and applanation. (a) Modelwith nonlinear material properties, (b) model with linear material properties.

12

13

14

15

16

17

18

19

20

21

22

0.3 0.4 0.5 0.6 0.7 0.8

CCT (mm)

IOPG(m

mHg

)

Material model with initial E = 0.30 N/mm2


FIGURE 9. Numerical estimation of the influence of CCT onIOP measurement using GAT for two material models IOPT = 15 mmHg in all cases.



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Parametric Study 3Effect of Variation in Material

Constitutive Relationship

The third parametric study considers five materialmodels including the models with initial Youngs

modulus, E = 0.30 and 0.08 N/mm2. The other three

models have an initial E = 0.15, 0.60 and 1.20 N/

mm2. The results of considering different material

models on IOPG are illustrated in Fig. 11. The change

in IOPG caused by the maximum change in material

properties (from E = 0.08 to 1.20 N/mm2) is

17.0 mmHg, most of which is observed when the

material model is stiffer than 0.30 N/mm2.

Clinical Data: Methods

The results from the modelling study were com-

pared to those obtained from a clinical dataset from

the Glaucoma Research Unit, Moorfields Eye Hospi-

tal, London. GAT IOP, ultrasound measured CCT

and central corneal curvature measurements were ac-

quired from 532 eyes of 532 new referrals to the Unit

over a 14-month period. None of the patients were

taking topical IOP-lowering medication. Measurement

of CCT and corneal curvature were obtained by one of

four technicians, and GAT IOP measurements were

made by a single experienced clinician. CCT mea-

surements were performed using a contact ultrasound

pachymeter (20 MHz solid tip probe; Optikron 2000,

Roma, Italy). R measurements were made with a

noncontact keratometer (IOLMaster; Carl Zeiss

Meditec, AG, Germany). The device measures the

curvature of a 2.3 mm diameter circular area of the

central cornea. Only one eye per patient was used in

the analysis.

In an attempt to isolate the relative effects of R and

CCT on GAT IOP measurements, two analyses were

performed. The first investigated the effect of varying

CCT on GAT IOP measurements for three R ranges

(110 eyes with R = 6.87.5 mm, 317 eyes with

R = 7.517.9 mm, 105 eyes with R = 7.918.6 mm)

whilst the second assessed the effect ofR on GAT IOP

measurements for three CCT ranges (126 eyes with

CCT = 410530 lm, 290 eyes with CCT = 531

590 lm, 116 eyes with CCT = 591660 lm). The

results are presented graphically in Figs. 12 and 13.

Linear predictive models were chosen to assess the

relationships between corneal parameters and GAT

IOP measurement. The purpose of this exercise was to

assess how clinical findings related to our numerical

studies, and it was felt that a linear analysis would give

a good indication of agreement, if any.

GAT IOP measurements increased with increasing

CCT in all R groups, of the order between 1.6 and

3.3 mmHg per 0.1 mm increase in CCT. The trend was

most significant in the 7.58.6, or flat and middle

ranges of corneal curvature, where CCT accounted for

approximately 7.5% of the measurement variation.

This suggests that although the effect of CCT has astatistically significant effect on GAT IOP measure-

ments, its effect is small compared with other sources

of measurement variation. Other sources of variation

include the technique of IOP measurement in the

clinical setting (which is subject to both inter- and

intra-observer variation), variation in IOPT, variation

in the effect of CCT at different IOPT, and, poten-

tially, variation in corneal material properties, all of

which will add to the spread of data. This measure-

ment imprecision is a source of noise which may mask

the effect of corneal parameters on IOP measurement.

Therefore, it is possible that the effect of CCT may be agreater predictor of GAT IOP measurement than the

findings suggest. The composite graph showing

the linear trends for all three R ranges suggests that the

GAT IOP measurement is somewhat underestimated

12

13

14

15

16

17

18

7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6

Anterior radius, R (mm)

IOPG(mmHg)



FIGURE 10. Numerical estimation of the influence of R onIOP measurement using GAT for two material models.IOPT = 15 mmHg in all cases.

10

15

20

25

30

35

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Initial E (N/mm2)

IOPG(mm

Hg

)

FIGURE 11. Numerical estimation of the influence of mate-rial properties on IOP measurement using GAT.IOPT = 15 mmHg in all cases.

ELSHEIKH et al.1636


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in flat corneas (R range 7.58.6 mm) compared with

more curved corneas (R range 6.87.5 mm).

Changes in R produce similar effects, but, as in the

numerical model, are not as significant as changes in

CCT, and only account for up to 2% of the measure-

ment error. As R becomes progressively greater, the

result is an underestimation of IOP of the order of

2 mmHg per 1 mm change in R, although the effect is

considerably smaller in the middle range of CCT

between 531 and 590 lm.

Overall, the clinical data suggests that in eyes with

steep (R between 6.8 and 7.5 mm) and thick (591

660 lm) corneas, the GAT IOP measurement is over-

estimated.

DISCUSSION

There is strong evidence that nonlinear material

properties should be adopted in reproducing the behav-

iour of the human cornea under Goldmann tonometry.

The displacement change during both inflation and

applanation is nonlinear, giving an indication that the

stress regime within the model rose beyond the initial low

stiffness phase of the material. Simplifying the analysis

by considering only the initial Youngs modulus leads to

considerable results variation and significant underesti-

mation of intra-ocular pressure.

The numerical model has been successful in

obtaining a correction factor close to 1 for an eye with

the calibration dimensions (CCT = 0.520 mm,

R = 7.8 mm) and with a nonlinear material model

with an initial Eof 0.30 N/mm2. This case was studied

in detail to show that the GAT procedure did not lead

to notable changes in the stress distribution in the

cornea outside the applanated area. The analysis also

showed that the contact pressure between the tonom-

eter and the anterior cornea was not uniform, but

reached its maximum value along a ring with about

half the tonometer radius.

The model with the nonlinear material properties

predicts a clear effect of CCT and R variation on IOP as

measured using Goldmann tonometry. The effect is more

pronounced with CCT values above 0.520 mm and R

IOPG = 0.0162xCCT + 8.8808

R2 = 0.0246

0

5

10

15

20

25

30

35

400 500 600 700 400 500 600 700

400 500 600 700 400 500 600 700

CCT (m)

IOPG(mm

Hg

)

IOPG = 0.0334xCCT - 1.0115

R2 = 0.0746

0

5

10

15

20

25

30

35

CCT (m)

IOPG(mm

Hg

)

(a) (b)

IOPG = 0.0305xCCT + 0.3053

R2 = 0.0761

0

5

10

15

20

25

30

35

CCT (m)

IOPG(

mm

Hg

)

0

5

10

15

20

25

CCT (m)

IOPG(mm

Hg

)

R=6.8 to 7.50R=7.51 TO 7.9

R=7.91 to 8.6

(c) (d)

FIGURE 12. Analysis of clinical observation to determine the effect of CCT variations on intra-ocular pressure readings usingGAT. (a) Observations for patients with anterior radius between 6.8 and 7.5 mm, (b) observations for patients with anterior radiusbetween 7.51 and 7.9 mm, (c) observations for patients with anterior radius between 7.91 and 8.6 mm, (d) linear trend lines for theabove three cases.



11/13

below 7.8 mmcollectively termed the calibration

dimensions. With IOPT of 15 mmHg, a nonlinearmaterial model with initial Youngs modulus, E =

0.30 N/mm2, increasing CCT from 0.520 mm to 0.620

and 0.720 mm (i.e. increases by 19% and 38%) results in

IOP overestimations by 12.0% (16.96)15.11 =

1.85 mmHg) and 30.8% (19.84) 15.11 = 4.73 mmHg),

respectively. Reducing CCT by the same percentages

leads to smaller IOP underestimations of 9.0% (15.11)

13.73 = 1.38 mmHg) and 15.0% (15.11) 12.71 =

2.30 mmHg). The study with a less stiff material model

(initial E = 0.08 N/mm2) shows consistently smaller

effect of CCT variation on IOP measurement.

The effect of changes in R is less notable. Reducing R

from 7.8 mm to 7.5 and 7.2 mm results in IOP overes-

timations of 3.1% (15.59) 15.11 = 0.48 mmHg) and

6.4% (16.07)15.11 = 0.96 mmHg), respectively. On

the other hand, increasing R to 8.1 and 8.4 mm leads to

slightly smaller effects of 3.1% (15.11)14.63 =

0.48 mmHg) and 4.1% (15.11) 14.49 = 0.62 mmHg).

The effects on IOP measurement are again slightly

reduced when using the less stiff material model.

These results are compatible with the previously

reported effect of CCT and R on the structural resistance

of the cornea against applanation. Also as CCT is the

factor with the higher effect on the structural resistance,it is expected that its variation will lead to larger

discrepancies in IOP measurement.

The material model is also found to have a pro-

found effect on IOP measurements when studied sep-

arately. The range of variation in initial Youngs

modulus is varied within a wide range (0.081.2 N/

mm2). The effect of increasing the material stiffness by

a factor of 4 (from 0.30 to 1.2 N/mm2) is found to lead

to a large overestimation of IOP by 100.9%

(30.35) 15.11 = 15.24 mmHg) while reducing the

stiffness by the same factor (from 0.30 to 0.08 N/mm2)

only results in 11.9% underestimation of IOP

(15.11)

13.31 = 1.80 mmHg). It should be noted here

that clinical research is needed to determine the actual

range of variation in material stiffness to be expected

between different people, during the life span of the

same person and even during the different hours of

day.

The results of the numerical study have been com-

pared with the statistical results of the clinical data and

also the earlier mathematical predictions made by Liu

and Roberts25 and by Orssengo and Pye.29 The data

IOPG = -2.335xR + 34.102

R2 = 0.0192

0

5

10

15

20

25

30

35

6.6 7.1 7.6 8.1 8.6 6.6 7.1 7.6 8.1 8.6

6.6 7.1 7.6 8.1 8.6 6.6 7.1 7.6 8.1 8.6

R (mm)

IOPG = -0.2557xR + 19.718

R2 = 0.0002

0

5

10

15

20

25

30

35

R (mm)

IOPG(mm

Hg

)

IOPG

(mm

Hg

)

IOPG

(mm

Hg

)

IOPG(mm

Hg

)

(a) (b)

IOPG = -2.0711xR + 35.302

R2 = 0.0161

0

5

10

15

20

25

30

35

R (mm)

0

5

10

15

20

25

R (mm)

CCT = 410 - 530CCT = 531 - 590

CCT = 591 - 659

(c) (d)

FIGURE 13. Analysis of clinical observation to determine the effect of anterior radius variations on intra-ocular pressure readingsusing GAT. (a) Observations for patients with CCT between 410 and 530 lm, (b) observations for patients with CCT between 531and 590 lm, (c) observations for patients with CCT between 591 and 660 lm, (d) linear trend lines for the above three cases.

ELSHEIKH et al.1638


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available for comparison were provided by Liu and

Roberts who used CCT = 0.526 mm, R = 7.8 mm

and E = 0.19 N/mm2. Orssengo and Pye used

CCT = 0.520 mm, R = 7.8 mm and Ewas taken as a

function in IOP in the form E = 0.0229 IOPT. The

comparisons are shown in Figs. 1416. The overall

trends in all cases are similar and there is a notable

measure of agreement between the results, particularly

between our data and those obtained by Liu and

Roberts. The clinical data also shows a similar trend to

the numerical results in spite of three possible sources

of error. Firstly, referrals to the Glaucoma Unit are

usually on the basis of high IOP, unlike the numerical

models which have a moderate IOP of 15 mmHg.

Secondly, the clinical dataset is likely to have a higherthan usual percentage of individuals with stiffer and/or

thicker corneas, as these properties result in overesti-

mated IOP measurements. Thirdly, the assumption of

an almost spherical topography in the numerical

models represents an approximation of the actual

corneal topography. However, the match between the

numerical and clinical data remains encouraging and

should be indicative of the validity of the simulations

used.

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Assessment of Goldmann Tonometry Using Numerical Modelling