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Numerical modelling of corneal biomechanicalbehaviourAhmed Elsheikh a & Defu Wang aa Division of Civil Engineering, University of Dundee, Dundee, DD1 4HN, UKVersion of record first published: 10 Mar 2011.
To cite this article: Ahmed Elsheikh & Defu Wang (2007): Numerical modelling of corneal biomechanical behaviour, ComputerMethods in Biomechanics and Biomedical Engineering, 10:2, 85-95
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Numerical modelling of corneal biomechanical behaviour†
AHMED ELSHEIKH* and DEFU WANG
Division of Civil Engineering, University of Dundee, Dundee DD1 4HN, UK
(Received 25 September 2005; in final form 15 August 2006)
Numerical modelling based on finite element analysis is used to represent the biomechanical behaviourof the cornea. The construction details of the model including the discretisation method, the meshdensity, the thickness distribution, the topography idealisation, the boundary conditions and thematerial properties, are optimised to improve efficiency. Factors which are found to have a considerableeffect on model accuracy are considered and those with effect below a certain low threshold are ignoredto reduce cost of analysis. The model is validated against laboratory tests involving pressure inflation ofcorneal trephinates while monitoring their behaviour. To illustrate the potential of the validated modelin studying corneal biomechanics, its use in modelling Goldmann applanation tonometry (GAT) isbriefly described. In studying GAT, the model is able to accurately trace the behaviour of the corneaunder tonometric pressure and monitor the gap closure and the progress of deformation to the point ofapplanation.
Keywords: Biomechanical behaviour; Intraocular pressure; Cornea; Numerical modelling
1. Introduction
The transparent cornea forms part of the outermost ocular
tunic. It provides a tough protective envelope for the
ocular contents and helps give the eye its general shape.
The anterior corneal surface accounts for over two-thirds
of the optical power of the eye. This important role can be
defined in terms of corneal shape, regularity and clarity,
and is a function of its refractive index. The cornea’s
contribution to ocular image formation can be degraded by
abnormalities in shape, by diseases and the effects of
surgery. The ability to predict the corneal response
brought on by disease and surgery, as well as intra-ocular
pressure (IOP) elevation, is of great clinical importance.
To this end, attempts have been made to numerically
model the mechanical behaviour of the cornea. Buzard
(1992) and Bryant and McDonnell (1996) developed a
finite element model in which corneas were modelled
using 2D axi-symmetric elements. Their work confirmed
the effectiveness of numerical modelling in corneal
biomechanics, but was unable to model asymmetrical
effects such as disease or injury. A more detailed 3D
model was used by Pinsky and Datye (1991) to predict the
immediate change in corneal topography following
refractive surgery. This model was based on a linear
elastic behaviour pattern despite the strong evidence
confirming the visco-elastic, nonlinear material behaviour.
Similar assumptions were adopted by Velinsky and Bryant
(1992) who produced a structural model of the whole eye
to determine the number and depth of cuts in surgical
operations to correct myopia. Their work showed how
finite element modelling could be customised so that the
process becomes patient specific using clinically
measured data.
The research presented in this paper is an attempt to
improve the accuracy and reliability of numerical
predictive modelling, which has clear potential in corneal
studies. The research attempts to improve accuracy by
approaching real life conditions and through validation
against laboratory test results. However, there is
appreciation of the complex structure of the cornea at
both the micro and macro levels, which potentially could
lead to prohibitively expensive models.
For this reason, a study has been conducted to evaluate
the importance of parameters, which could have an effect
on the model accuracy. These parameters include the
thickness distribution, connection with the sclera, material
properties, topography and the composite nature of the
cornea. Factors, which are found important, have been
considered in the model construction and those with an
Computer Methods in Biomechanics and Biomedical Engineering
ISSN 1025-5842 print/ISSN 1476-8259 online q 2007 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/10255840600976013
†Financial support—European Commission
*Corresponding author. Email: [email protected]
Computer Methods in Biomechanics and Biomedical Engineering,Vol. 10, No. 2, April 2007, 85–95
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effect below a certain low threshold are ignored. The
density of the finite element mesh and the discretisation
method are also considered.
The research starts with testing corneal trephinates
(buttons) under IOP elevations to evaluate the stress–
strain behaviour of the corneal tissue. The results are used
to validate the numerical model and guide the optimisation
process described in this paper. The research has so far
concentrated on porcine corneas because of their close
similarity with human corneas in structure and constitutive
relationship (Zeng et al. 2001), and because of their
availability. Once this step of the research is completed,
the main findings will be validated for human corneas.
2. Experimental programme
The experimental programme involved testing ten
trephinate porcine corneas under inflation conditions, i.e.
using a gradually elevated IOP. The specimens were
obtained from a local abattoir and were no more than 4 h
post mortem when tested. The ten trephinate specimens,
which included the cornea and a narrow ring of
surrounding scleral tissue, were mechanically separated
from the rest of the eye globe using a sharp cutting tool,
then mounted onto a specially designed test rig to provide
watertight edge fixity for the specimens along their ring of
scleral tissue. Care was taken to avoid damage to the
epithelium or endothelium in spite of their reported small
contribution to corneal biomechanics (Greene 1978).
The specimens were subjected to a gradually increasing
posterior pressure caused by a column of saline water to
simulate the effect of elevated IOP. In the meantime, a
laser (Keyence, CCD laser displacement sensor, LK
series) was used to continually monitor the displacement
at the apex of the cornea. The data related to the applied
pressure and the corresponding apical displacement was
automatically recorded for later analysis. Mechanical
clamps and cyano-acrylate glue were used to ensure a
good seal along the edge of the specimens. The specimens
were coated with mineral oil to prevent loss of hydration
and any subsequent changes in dimensions and mechan-
ical performance. All corneas were subjected to a
gradually increasing posterior pressure up to a maximum
value of 14 kPa (105mmHg). This pressure was above the
level at which the corneas entered a stage of stable
behaviour that was expected to continue until bursting
(Shin et al. 1997, Voorhies 2003).
The tests reported in this paper are part of an
experimental study conducted to evaluate the corneal
material properties, especially the stress–strain relation-
ship. This work was preceded by a review of available
literature in which it became evident that there was some
considerable disagreement regarding the form of s–1
curve that could be used in numerical modelling
(Anderson 2005). The testing programme was, therefore,
conducted to enable the development of reliable material
properties that could be counted on to produce accurate
numerical models.
The results in figure 1 show the pressure apical rise
relationships obtained for a representative selection of the
inflation tests. The results show a long phase of linear
behaviour followed by gradual stiffening at about
0.004Nmm22 (30mmHg). Based on the results of earlier
studies on the corneal microstructure (Woo et al. 1972,
Hjortdal 1993, 1998) and what has been found in current
laboratory testing, it is suggested to divide the stress–
strain relationship into two distinctive phases: a matrix
regulated phase with low stiffness followed by a collagen
regulated phase with a much higher stiffness.
Mathematical analysis based on shell theory has been
used to derive the material constitutive relationship from
the pressure-apical rise experimental results. The analysis
considers both in-plane and out-of-plane stiffness
components, and assumes that the cornea can be
approximated as a homogenous spherical structure (Vito
et al. 1989). Details of the analysis procedure can be found
in an earlier publication (Anderson et al. 2004). The final
equation relating the internal pressure, p, to the apical rise,
r, is:
r ¼p·R2
2Etð1 2 nÞ 1 2 e2bgcosðbgÞ
� �; ð1Þ
where R is the radius of the corneal median surface, t the
average thickness, n Poisson’s ratio taken as 0.49 based on
the assumption that the cornea behaves as an almost
incompressible body (Bryant and McDonnell 1996), g
half the central angle of curvature, g ¼ sin21ðRi=RÞ, Ri
the radius of the corneo-scleral intersection and
b ¼ffiffiffiffiffiffiffiR=t
p·
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1 2 n2Þ4
p.
Equation (1) is used for each set of p–r data, and the
corresponding (instantaneous) Young’s modulus, E, is
obtained. The strain at this level of loading is then
obtained using the relationship (Anderson et al. 2004):
1f ¼1
EtðNf 2 nNuÞ; ð2Þ
Figure 1. Pressure apical rise relationships of a selection of inflationtests.
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where Nf ¼ ðp·RÞ=2 and Nu ¼ ððp·RÞ=2Þ2 ðððp·RÞ=2Þ �
ð1 2 nÞe2bg{cosbg}Þ: Therefore,
1f ¼pR
2Etð1 2 nÞ 1 þ n e2bgcosbg
� �: ð3Þ
The corresponding stress is obtained using:
s ¼ 1E: ð4Þ
The results for each of the ten inflation tests have been
analysed using this procedure and the average stress–
strain curve obtained is shown in figure 2.
3. Model construction
The complexity of the structure and form of the cornea at
both the micro and macro 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 cost. In order to strike the best
balance between cost and accuracy, a study was conducted
to identify the effect of individual parameters on the
models’ results. 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 study are the
thickness distribution, 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 are also considered.
The following discussion considers the effect of these
parameters one by one.
3.1 Discretisation method
The corneal geometric form makes it reasonable to build
the numerical model using layers of shell and/or solid
elements with each layer containing elements arranged in
circular rings. Our modelling efforts started with the
discretisation technique adopted in earlier studies and
illustrated in figure 3(a). As this approach uses a constant
number of elements per ring, regardless of the size of the
ring, the elements included in the inner-most ring are
usually unacceptably shaped with the tip angles being too
small. For example, in the model shown in figure 3(a), 40
elements are used per ring and as a result, the angle at the
element tips at the corneal centre is 98—well below the 308
considered minimum for ensuring a high accuracy of
results (Hibbitt, Karlsson and Sorensen, Inc. 2005). For
this reason, another discretisation approach based on the
form of skeletal diamatic domes has been adopted
(Nooshin and Tomatsuri 1995), see figure 3(b). In this
case, the number of elements per ring is proportional to the
meridian distance between the ring and the corneal centre.
As a result, the minimum angles in all elements remain
within the 30–608 range.
3.2 Mesh density
Increasing the density of finite element meshes normally
means two effects: better accuracy and higher analysis
Figure 2. Stress strain relationship of corneal tissue as obtainedexperimentally.
Figure 3. Discretisation methods: (a) old meshing technique; (b) new meshing technique.
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cost due to the larger number of nodes and hence number
of equilibrium equations. The trend of improved accuracy
with denser meshes typically continues up to a certain
limit, beyond which the benefit of using denser meshes
diminishes. The study presented in this section is intended
to determine this limit. Two variables are considered in the
study; the number of elements per layer and the number of
layers. See figure 4 for a description of the terminology
used in model construction including ring, layer and
segment.
Figure 5 shows the load apical rise predictions for a
cornea under two separate loading conditions: an IOP and
an inward concentric point load. The predictions are
obtained using one-layer models employing 54, 150, 294,
486 and 2,400 six-noded solid elements. All models have
6 segments and have 3, 5, 7, 9 and 20 rings, respectively.
Figure 5 shows that models with 3 element rings were too
rough to produce reliable results. It also shows that the
predictions have not changed much with meshes using
more than 7 element rings. Increasing the mesh density
from 294 to 486 elements results in a maximum change in
predictions of 0.9% and would, therefore, be difficult to
justify considering the associated increase in analysis cost.
Varying the number of layers has also been attempted.
Models with 1, 2, 4, 6 and 8 layers, each containing 294
solid elements are used, and the results obtained are shown
in figure 6. The results show that under IOP, increasing the
number of layers beyond 2 results in a negligible change in
model predictions (,0.3% on average). Under a point
load, using more layers continues to produce notable
changes in results until the model with 6 layers. Increasing
the number of layers from 6 to 8 produces an average
change in predictions below 0.5%.
3.3 Boundary conditions along the corneo-scleralconnection
Awhole-eye model incorporating both the cornea and the
sclera would undoubtedly provide a better representation
of the actual state of the cornea than a model of the cornea
alone. However, a whole-eye model is clearly more
expensive to develop and run. A compromise could be
reached if the cornea-only model could be formed such
that the boundary conditions along its edge are made
representative of the effect of the connection with the
sclera. This is done in appreciation that the sclera,
although stiffer than the cornea, should not be expected to
provide the cornea with supports that are prevented from
both translation and rotation. In reality, the sclera will
deform under pressure and this deformation will affect the
corneal behaviour.
A study has been carried out to find a reasonable
approximation of the corneal edge supports. The approach
considers the edge nodes of the corneal model to be
attached to roller supports in an inclined direction as
shown in figure 7(a). This approach is similar to that
Figure 4. Numerical model with 2 layers of solid elements arranged in 6 segments and 12 rings—Total number of elements is 144 elements per segmentper layer £ 6 segments £ 2 layers ¼ 1728. Available in colour online.
Figure 5. Behaviour predictions using numerical models with different densities: (a) under IOP; (b) under concentric point load.
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adopted by Orssengo and Pye (1999). The angle of support
orientation, u, is changed between 15 and 608, and the
predictions of the model with different support orien-
tations are presented in figure 7(b) and compared with the
results of a whole-eye model. The comparisons show that
a support orientation of 238 provides the best match with
the whole-eye model with an average difference of 0.7%.
With these inclined supports, the cornea-only model
would be expected to approximate the whole-eye model
and be suitable for applications where the focus is only on
corneal behaviour.
3.4 Material properties
The stress strain behaviour of corneal tissue as obtained in
the laboratory tests and reported above is highly nonlinear.
However, attempts have been made before to consider
only the initial low stiffness of the material in order to
simplify the numerical model and reduce the cost of
analysis. The effect of this simplification has been
assessed by comparing the load-rise predictions of the
model once assuming a linear behaviour and another
considering the nonlinear behaviour observed in labora-
tory tests, see figure 8. The nonlinear behaviour is
incorporated using a hyper-elastic material model
available in Abaqus based on Ogden’s strain energy
function (Ogden 1984):
U ¼XN
i¼1
2ui
a2i
lai1 þ lai2 þ lai3 2 3� �
þXN
i¼1
1
D1
J el 2 1� �2i
ð5Þ
where li are the principal stretches, is a material parameter
and ui, ai, and Di are temperature dependent material
Figure 6. Behaviour predictions using numerical models with different numbers of element layers: (a) under IOP; (b) under concentric point load.
Figure 7. Behaviour predictions obtained using models with roller edge supports with a variable orientation: (a) schematic view; (b) behaviourunder IOP.
Study of corneal biomechanical behaviour 89
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parameters. As can be seen in figure 8, considering only
the initial material stiffness leads to a behaviour under IOP
that is quite different from that observed experimentally.
Similarly, the model experiences a significant change in
behaviour under point loads when the simplified material
model is used.
3.5 Thickness distribution
Corneal thickness varies from a minimum at the centre to a
maximum along the limbus. In modelling terms, it is
easier to assume a constant thickness than to consider the
actual thickness variation. The effect of this simplification
on the modelling accuracy is illustrated using the example
shown in figure 9. In this figure, the predictions using a
model with a variable thickness (0.670mm along the edge
and 0.520mm at the centre) are compared with those of
two constant-thickness models with t ¼ 0.520mm and
t ¼ 0.595mm, respectively. Note that while 0.520mm is
equal to the central corneal thickness (CCT) in the first
model, 0.595mm is the average thickness. All models are
built using 588 solid elements arranged in 2 layers and 7
element rings per layer. The results show that the model
with variable thickness produces intermediate results
between the two models with constant thickness. The
difference between the variable thickness model and the
other two is sufficiently high to justify the need for
considering the actual thickness variation in model
construction, especially in studying cases under point
loads.
Figure 8. Behaviour predictions obtained using two corneal representations employing a nonlinear material model and an elastic material model: (a)behaviour under IOP; (b) behaviour under a concentric point load.
Figure 9. Behaviour predictions obtained using three models with different thickness representations: (a) three models; (b) behaviour under IOP; (c)behaviour under a concentric point load.
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3.6 Corneal topography
The corneo-scleral intersection is oval (elliptical) with the
temporal–nasal diameter (DT – N) approximately 10%
larger than the inferior–superior diameter (DI – S). This
gives rise to a corneal topography which is not purely
spherical. The effect of assuming a spherical form on the
model predictions is assessed by building two models: one
with the topography described above where DT –
N ¼ 11.7 mm and DI – S ¼ 10.6 mm, and one with a purely
spherical topography with D ¼ 0.5 (DT – S þ DI – S) ¼
11.15 mm. The two models are subjected to IOP
elevations and concentric point loads. Both models have
588 solid elements arranged in 7 element rings and 2
layers. The behaviour comparisons are shown in figure 10.
The difference between the two model predictions is
below 1% in both cases suggesting that the cornea could
be approximated as a spherical object without any
significant effect on accuracy.
3.7 Optimised model construction
The study discussed above helped quantify the influence
of different parameters on the results of numerical
modelling. The results obtained suggest that certain
parameters have a notable effect on model predictions and,
therefore, need to be considered in model construction.
These parameters include the thickness distribution, the
boundary conditions along the limbus and the nonlinear
material properties. Approximating the corneal topogra-
phy as a pure spherical object produces a negligible effect
and could, therefore, be adopted in model construction.
These findings have been adopted in building the models
used in the remainder of this paper. Models subjected to
uniform pressure, such as IOP, have at least 588 solid
elements arranged in 2 layers and 7 rings per layer and are
meshed using the new discretisation method that
maintains element internal angles within acceptable
limits. In cases under non-uniform loads, such as point
loads and tonometry loads, the number of element layers is
increased to 6, with the subsequent increase in the
minimum number of elements to 1,764.
4. Results
The optimised numerical model has been validated against
the laboratory inflation test results, then used to provide
further insight into the behaviour of the tested specimens.
As an example of the potential applications of the model,
it has been used to simulate Goldmann applanation
tonometry (GAT). Further applications are currently
underway and will be published in future papers.
4.1 Validation of numerical model
The numerical model has been used to simulate the
inflation laboratory tests described above. For each
specimen, the dimensions measured experimentally and
the average material properties given in figure 2 are used.
The pressure rise predictions are then compared against
the laboratory observations, see for example figure 11. The
comparisons show that the model predictions closely
match the laboratory observations with average differ-
ences remaining below 4%.
Figure 10. Behaviour predictions obtained using two models employing different topography representations: (a) behaviour under IOP; (b) behaviourunder a concentric point load.
Figure 11. Behaviour of inflation test specimens as obtainednumerically and experimentally.
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4.2 Further insight into experimental behaviour
Having validated the model using pressure rise data, it can
be used to describe the performance of the trephinate
specimens tested in this research to a level of detail
impossible to achieve with laboratory tests. The
information obtained includes the distribution of displace-
ment, stress and strain over the surface area of the cornea.
Examples of the output of the model for a typical
trephinate specimen under two levels of IOP (0.002 and
0.014Nmm22 or 15 and 105mmHg) are given in
figure 12. Note for instance how the model displacements
are smallest near the supported edge nodes and highest at
the centre, as would be expected. On the other hand, the
peripheral corneal region experiences the highest values of
first principal strains and stresses, although the variation
between different zones remains within low limits.
4.3 Modelling of Goldmann applanation tonometry
Tonometry is a procedure to measure the IOP of the eye.
Ophthalmologists use IOP measurements to diagnose a
number of conditions including hyphema (trauma and
inflation of the iris) and glaucoma, the second most
common cause of irreversible blindness in the world
(Wilson and Kass 2002). The most common tonometry
technique is the GAT which makes a pseudo-static
measurement of the force required to flatten a fixed area of
the central cornea with 3.06mm diameter and uses this
force to estimate the value of IOP (Goldmann and Schmidt
1961, Feltgen et al. 2001). The procedure ignores the
natural variations in corneal dimensions, mainly the
thickness and radius, and as a result it can not accurately
eliminate the effects of corneal structural resistance from
its IOP estimations (Tonnu et al. 2005).
A study is being carried out by the authors to derive
correction factors that can eliminate the effect of
dimension variations. The study uses the numerical
model developed in this research in a parametric
investigation covering the practical range of thickness
and radius variation. As this study concerns the human
cornea, the modulus of elasticity is taken as 0.0229 £ IOP
according to Orssengo and Pye (1999). Only the case with
CCT ¼ 0.520mm, limbal thickness ¼ 0.670mm and
R ¼ 7.80 mm (collectively called the calibration dimen-
sions) is discussed here, mainly to demonstrate the
potential of numerical modelling in studying corneal
behaviour. The full parametric study will be covered in a
future publication.
In this work, a corneal model with 17,424 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, see figure 13. The
tonometer model, which has a 3.06mm diameter, uses
similar six-noded solid elements. The anterior surface of
the cornea and the posterior surface of the tonometer are
Figure 12. Displacement, stress and strain distributions of an inflationtest specimen: (a) displacement under IOP ¼ 0.002Nmm22; (b) stressunder IOP ¼ 0.002Nmm22; (c) strain under IOP ¼ 0.002Nmm22; (d)displacement under IOP ¼ 0.014 Nmm22; (e) stress underIOP ¼ 0.014 Nmm22; (f) strain under IOP ¼ 0.014 Nmm22—Displacement range: red ¼ 0.670mm, blue ¼ 0.00, stress range:red ¼ 0.266 Nmm22, blue ¼ 0.00, strain range: red ¼ 0.130,blue ¼ 0.00.
Figure 13. Stress distribution during the analysis: (a) followingapplication of IOP ¼ 0.002Nmm22; (b) following full applanation—view without tonometer; (c) following full applanation—view withtonometer—Contours are drawn on model with original undeformedgeometry—Stress range: red ¼ 0.09Nmm22, blue ¼ 0.00.
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described in the analysis as contact surfaces to prevent
over-closure of the gap between them. The edge supports
of the cornea are roller supports set at 238 to the limbal
plane in order to represent the effect of connection with
the sclera as discussed above.
The analysis starts with subjecting the corneal model to
an IOP with a predetermined value (0.002Nmm22 ¼ 15
mmHg), 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 gradually and concen-
trically against the cornea until complete applanation is
achieved. The stress distributions recorded during this
process are shown in figure 13. Notice how the tonometer
affects mainly the stress values at and around the contact
area, but has little effect elsewhere.
The closure of the 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 14 shows the progress of gap closure
over five stages of the analysis, the last of which represents
the point of full applanation.
The contact stress distribution between the tonometer
and the corneal anterior surface is also monitored during
the progress of applanation. It has been interesting to
observe how the maximum values of contact stress change
location with the progress of applanation, see figure 15. 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 surface tension. The value of surface tension
is taken as 0.0455Nm21 from work carried out by Mr
Figure 14. Closure of gap between the tonometer and the corneal anterior surface over five stages of analysis—the last stage marks the full applanationpoint—Contours are drawn on model with original undeformed geometry—Gap width range: red ¼ 0.150mm, blue ¼ 0.00.
Figure 15. Distribution of contact pressure on the tonometer surface with the progress of applanation—Contours are drawn on model with originalundeformed geometry—Contact stress range: red ¼ 0.0045Nmm22, blue ¼ 0.00.
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David Pye of the University of New South Wales. This
value, which is not yet published, is slightly less than that
for water, 0.0728Nm21.
The surface tension acts along the edgeof the contact 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 the contact area
(1.532p). The resulting pressure is subtracted from the
previously calculated external pressure to calculate IOPG. In
the analysis, the true intra-ocular pressure (IOPT) applied is
0.002 N mm22 (15mmHg) and the external pressure is
determined as 0.002074 N mm22. The effect of surface
tension is 0.0455 £ 1023 £ (2 £ 1.53p)/(1.532p) ¼
0.000059 Nm m22. Therefore, IOPG ¼ 0.002074
2 0.000059 ¼ 0.002015 Nm m22 ¼ 15.11 mmHg. This
gives a correction factor ¼ 15.11/15 ¼ 1.007. This value
is close to 1.0, which is expected with the calibration
dimensions. As stated in Goldmann and Schmidt (1961),
Ehlers et al. (1975) and Whitacre et al. (1993), at these
dimensions, the effects of surface tension and corneal
structural resistance cancel each other out, leading to the
tonometry estimate, IOPG, being equal to the true intra-
ocular pressure, IOPT.
5. Discussion
Numerical modelling of the cornea is used to determine its
biomechanical properties and predict its performance
when subjected to external effects such as disease, surgery
or injury, and internal effects including IOP elevation.
Modelling should approach real life conditions if it is to
produce accurate and reliable performance predictions.
However, there is a limit in sophistication beyond which
the model becomes difficult to build and expensive to run
and analyse. A parametric study has been conducted to
determine the parameters that have a notable effect on
model predictions and should therefore be incorporated in
model construction. Other parameters that have a
negligible influence on model predictions can be safely
ignored. Parameters that belong to the first group include
the thickness variation between the centre and the limbus,
the connection with the sclera and the nonlinear material
properties. On the other hand, the cornea’s topography can
be approximated as a pure spherical object without a
notable effect on results. Following the study, the model
has been successfully validated against the laboratory
inflation test results.
The numerical model has been used to provide further
insight into the behaviour of the inflation test specimens.
The distributions of deformation, stress and strain point at
the areas where the structural demands are high. This level
of detailed information would be useful in understanding
the corneal biomechanical response to various effects such
as corrective surgery, contact lenses or sports impact.
Numerical modelling can also be useful in simulating
clinical procedures. An example is provided on the
modelling of GAT and the results obtained in the specific
case with the calibration dimensions matches the
expectation with a correction factor close to 1.
Besides the ability to simulate real life conditions and to
provide detailed insight into behaviour, numerical
modelling has the potential to reduce need for animal
tests, which are becoming increasingly limited on both
ethical and legal grounds.
Acknowledgements
This research was conducted as part of a project on the use
of tissue engineering to construct an artificial cornea
funded by the European Commission.
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