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Ophthalmologists rely on a device known as the Goldmann applanation tonometer to make intraocular pressure (IOP) measurements. It measures the force required to press a flat disc against the cornea to produce a flattened circular region of known area. The IOP is deduced from this force using the Imbert-Fick principle. However, there is scant analytical justification for this analysis. We present a mathematical model of tonometry to investigate the relationship between the pressure derived by tonometry and the IOP. An elementary equilibrium analysis suggests that there is no physical basis for traditional tonometric analysis. Tonometry is modelled using a hollow spherical shell of solid material enclosing an elastic liquid core, with the shell in tension and the core under pressure. The shell is pressed against a rigid flat plane. The solution is found using finite element analysis. The shell material is anisotropic. Values for its elastic constants are obtained from literature except where data are unavailable, when reasonable limits are explored. The results show that the force measured by the Goldmann tonometer depends on the elastic constant values. The relationship between the IOP and the tonometer readings is complex, showing potentially high levels of inaccuracy that depend on IOP.

Intraocular pressure (IOP)―the difference between atmospheric pressure and the pressure inside the eye―is a parameter of key importance in assessing the health of the eye. For the human eye IOP values in the range 1.6 - 3.3 kPa (12 - 25 mmHg) are considered normal. Higher values are always a cause of concern since they may indicate glaucoma, a serious optic nerve neuropathy that may lead to blindness.

A device called a tonometer is used to measure IOP. A large variety of tonometer designs are in use, which give fast and non-invasive measurements at varying levels of accuracy. The instrument considered to be the “gold standard” by the ophthalmic community, and the focus of this study, is the Goldmann Applanation Tonometer (“GAT”). When a GAT is used, a flat metal disc of a known, standardized diameter is brought into contact with the cornea and pressed against it until the whole area of the disc is in contact and “applanation” (observed via a slit lamp) is judged to have taken place. The force on the disc required to achieve full contact is measured, so that the mean pressure on the disc surface can be calculated. Routinely, the tonometer measurement is interpreted by equating this measured pressure with the IOP. The assumption embodied in this interpretation is known as the Imbert-Fick law, which asserts that every gm force required for applanation corresponds to 1 mmHg IOP (from which it may easily be calculated that the diameter of the applanation disc is 3.06 mm). This paper is an evaluation of the Imbert-Fick law, for which there appears to be no theoretical justification. We shall first demonstrate a simple equilibrium analysis that shows that the law is at best an estimation technique that may correlate with IOP. We then proceed to use a numerical technique to give a critical review of the relationship between tonometry measurements and IOP.

From a mathematical perspective, the tonometry problem may be modelled using solid mechanics. We shall adopt the finite element approach in which the model is three-dimensional. We perform calculations on a sphere with a linear elastic solid outer shell containing a material that is essentially an elastic fluid. Pressure is applied to the inner material by prestraining the outer shell. Then, boundary conditions are applied by introducing a plane surface touching the sphere’s outer surface, corresponding to the tonometer, and applying equilibrating displacements to at the opposite surface of the sphere to compress it against the plane.

The work is divided as follows: Section 1.1 outlines a brief introduction to the anatomical, physiological, clinical and pathological facts relevant to the study of the IOP. Section 2 presents the analysis, including details of the finite element models used and the material elastic properties. The results are presented and discussed in Section 3, where the tonometer performance is evaluated, and finally conclusions follow in Section 4.

Anatomical, Physiological, and Clinical ConsiderationsAccording to Fatt and Weissman [

The cornea is a transparent organ composed mostly of water and collagen. It possesses a high content of nerve fibres but almost entirely lacks blood vessels. This organ makes up almost one sixth of the outer surface area of the eye. The principal function of the cornea is optical since it is thought to be responsible for as much as 70.0% of the total refractive power of the eye due to its transparency; however its mechanical properties also make it capable of resisting the intraocular pressure and providing a protective coat for the eye. The cornea’s protective qualities are attributed to the collagen fibres within this organ [

The sclera is a tissue commonly known as “the white” of the eye and comprises the rest of the outer shell of the eye. This tissue is composed of approximately a third of interwoven elastic collagen fibres, which are responsible for its elasticity, and two thirds of water. The main function of the sclera is to protect the intraocular organs from injury and damage; it is stiff enough to resist the IOP. The thickness of the sclera varies with age as well as location. For an adult healthy human eye, Fatt and Weissman [

The vitreous body is a transparent gel-like substance comprising 80% of the total volume of the eye. Among the functions attributed to the vitreous body are its ability to both work as an optical medium, and to retain the retina. (The retina is attached to the choroid, which in turn is attached to the sclera.)

Though a scientific foundation for the Imbert-Fick law is largely lacking, the Goldmann-type applanation tonometer has nevertheless been designed under the assumption that this law is valid. The Imbert-Fick law seems to have been used for the first time for tonometer design in the last decade of the nineteenth century by Imbert [

When reflecting on the applanation process, it soon becomes clear that the Imbert-Fick law cannot perfectly describe it. During the applanation process, the cornea is flattened, and this in itself must require a force to create sufficient bending stress, even in the absence of any IOP. The effect of the surface tension of the tear liquid on the cornea surface must also be accounted for. This latter effect has been calculated to be equivalent to an added 0.06 kPa tension on the tonometer [

To fix ideas, in

For F = 0 this gives the standard solution for a pressurised thin sphere [

In contrast, the Imbert-Fick law is

Further insight can be gained by creating a cylindrical section with radius r < R, again with the tonometer contact area of radius a centred on the vertical symmetry axis, as shown in

F is assumed to be distributed about the contact area radius a, to give the mean contact pressure s_{c}:

This equation demonstrates the extreme case where p is small enough to be neglected. Then, s_{c} > p, t < 0. This value of s_{c} corresponds to that required to flatten the cornea. This corresponds to the observation of Orssengo and Pye [

With no applanation s_{c} = 0 and t > 0. On applanation, as s_{c} increases, t decreases. Considering the case r = a, as long as t remains positive, s_{c} < p. However, when the stress required to bend the cornea is the dominant factor as discussed above, s_{c} > p and t < 0. These findings are confirmed by the numerical studies reported below.

The above reasoning suggests that Equation (3) is not generally valid. Śródka [

Equations (2) and (5) indicate that s_{c} will be correlated positively with p, especially given the customary use of a standard value of area in applanation tonometry. The equations cannot be used to evaluate p directly, as s_{t} and t are unknown. Solutions to the problem obtained by numerical means are described below.

An early instance of the use of the finite element method to model the mechanical behaviour of the eye is due to Buzard [

Elsheikh et al. [

We have created in-house software that generates input files that, when operated upon by the finite element package ABAQUS [

As in

First, the sphere is pressurised. This is done by exploiting the thermal expansion capability of ABAQUS. A thermal expansion coefficient is assigned to the shell material and a fictitious temperature drop imposed. The temperature drop/expansion coefficient combination is adjusted to give an appropriate pressure level in the interior. We investigated three pressure levels: 1.2, 2.1 and 3.3 kPa (6, 15.75 and 24.75 mmHg).

Secondly, displacement boundary conditions are imposed on a circular region of nodes on the top nodes centred around the vertical axis. They are all displaced downwards by the same value. The lower surface of the sphere rests on a flat rigid analytical surface that is held stationary. As the displacement increases, the circular contact area increases and the value of the contact area is output.

The accuracy of the contact area calculation depends on the areas of the elements on the lower surface. Elements are concentrated in the contact zone for this reason. Two basic methods for this were used. In the first (type A) shown in

All the models possess fourfold rotational symmetry about the vertical axis, containing at the centre a cube of elements from which vectors are produced and on which nodes are positioned. The number of elements in the

cube controls the mesh density. In the meshes shown, the cube is an array of 20 ´ 40 ´ 40 elements, with a 40 ´ 40 face opposite the contact area. The model contains a total of 70,400 solid elements. Meshes with cube arrays 20 ´ 30 ´ 30 (45,200 elements) were also generated for comparison and to establish mesh convergence. Eight-noded linear hexahedral elements (C3D8) were used throughout. To correspond to typical human eye dimensions, the outer radius of the model was 12.5 mm and the inner radius of the shell was 11.98 mm. The displacement boundary condition was applied to nodes on the top surface within a circular area of radius 5 mm centred on the vertical symmetry axis.

For the (liquid) core material a compressible Gaussian model was used, as defined by the strain energy density function W:

J is the volume ratio defined by

where

In Equation (6) m and B are material parameters, with m governing shape change and B controlling volume change. With

which ensure that the stresses within the core material are essentially hydrostatic.

We treat the outer shell―the scleral and corneal regions―as linear elastic solids. In view of the observed nonlinear behaviour [

Outside the region of the cornea, the outer layer of the eye consists mainly of sclera of the order of 1mm thick, with an added thickness of ~0.08 mm of choroid. Modulus values of the sclera and choroid are given respectively as 2.3 and 0.6 MPa (Friberg and Lace [

There is evidence that the corneal modulus is smaller than that of the sclera [

The default assumption that elastic materials are isotropic is an attractive one, and this premise has been used before in the modelling of the eye (e.g. [

which may be rewritten as

where S is termed the compliance matrix. We shall further assume that the shell material is transversely isotropic with the isotropic planes normal to a radius. Then [

and for the isotropic plane the shear modulus is given by

There are now five independent elastic constants that must be specified. We identify

To summarise, the values may be expressed as the compliance matrix S of Equation (9) as follows. For the cornea:

where

For the sclera:

where

During applanation, the mechanical properties local to the deformation-those of the cornea-are significant to the process, whereas the scleral properties are of little significance. Therefore, our principal results will be for models for which the whole of the shell has been assigned cornea properties. However, we will present results for models with sclera properties in the whole of the shell also, to illustrate the importance of the mechanical properties to the tonometry results.

Here we compare the pressure on the contacting rigid plane with the internal pressure in the model. During each simulation downward displacements were applied incrementally to the top surface causing the contact area on the lower surface and the associated reaction force to increase. The analyses were continued until the contact area was approximately twice that of the tonometer, and pressure results taken at the tonometer area of 7.354 (=p ´ (3.06/2)) mm^{2}. In the results to be presented below, it is clear that there is consistently some slight nonlinearity, with the slope of the contact area-contact force curve increasing slightly with area. The force at the tonometer area was derived from a quadratic polynomial fit to this curve; this is more accurate than relying on a single point calculation, as there is some degree of scatter in the area evaluation. The IOP is calculated during the analysis, and increases slightly, but not significantly, by only up to less than 1% when applanated at the tonometer area.

The limits for

sclera material of Equation (13). This will provide useful insights into the pressure dependence of the tonometer response, while still leaving some uncertainty in the absolute values it provides. For

In

refinement is used. This model, in which the internal pressure is 3.3 kPa, shows typical behaviour. Other mechani- cal properties are given in Section 2, with the shell being assigned the cornea properties of Equation (12) and the value of ^{2} has been attained.

The response in terms of compression force versus contact area is shown in

quadratic fitting for each set of data we obtain the force on the tonometer at the contact area 7.354 mm^{2} and calculate the tonometer pressure. For the case of

We also made similar simulations using Type B mesh refinement at IOP values in the range 1.2 - 3.3 kPa and assuming

Tonometer pressure is observed to be an approximately linear function of IOP with a positive intercept. The intercept can be associated with the force required purely to flatten the cornea, increasing with increasing

A detailed appraisal of the stresses local to the applanation zone helps to understand the physical mechanisms involved, particularly in terms of the free body diagram of _{r} and s_{r}_{q} with vertical forces. A positive value of s_{r}_{q} is associated with a downward force.

In ^{2}.

case is accurately predicted by the Imbert-Fick law. In _{r} are not constant on the contact plane, being greater in magnitude nearer the centre. In this region the contact stress is greater than the IOP, at around 1.5 - 1.6 kPa, but at the outer regions of the zone they are less than the IOP. This is consistent with the mean compressive contact stress s_{c} (see Equation (5)) being at around the IOP, p. We also show shear stresses s_{r}_{q} equivalent to the stress t in Equation (5) in the right image of

change sign around the edges of the contact zone. This is consistent with IOP = p −s_{c} in Equation (5) when r = a.

Turning attention to cases that are not accurately predicted by Imbert-Fick, consider the case

and IOP = 3.3 kPa. The IOP is under-predicted by our tonometry simulation as shown in _{c} lower than IOP, corresponding to the low tonometer pressure shown in _{c} < p when r = a.

We now consider the case _{c} higher than IOP. The shear stresses in the contact area in the right image of _{c} > p and t < 0 when r = a.

The above detailed examination of the contact stresses explains how the Imbert-Fick law is capable of both under-predicting and over-predicting the IOP.

To evaluate the effect of varying the mechanical properties of the shell, we performed a similar exercise using the scleral properties of Equation (13). The stiffer material gives rise to a broader range of variation of tonometer pressure as a function of

The slopes of the data now lie in the range 0.4 - 0.5, similar to those in

for a cornea outside the normal human stiffness range, IOPs predicted by GAT readings could be highly misleading.

All of the results that have been discussed so far have been obtained with purely spherical models. This is a simplification as in reality (see

the radius of the eyeball. To explore the effects of the corneal radius we have created a series of smaller spherical models with radii in a range relevant to the human cornea. The applanation of these spheres resembles locally the applanation of a cornea of realistic size forming part of an eyeball. The shell thickness is kept the same as in the previous models, and the meshing used is type B, based on 20 ´ 40 ´ 40 element cube, as shown in

In

tional dependence on the radius for both large and small values of

drop in tonometer pressure of approximately 10% as the radius changes from 7.2 to 8.4 mm. This effect is very similar to that observed in the modelling of Elsheikh et al. [

While the effect of radius is clearly significant, we can conclude that it will not fundamentally affect the functional dependences of tonometry pressure on IOP as seen in

A mathematical model describing the working mechanism of Goldmann-type applanation tonometers has been developed. The models consist of a hollow elastic shell of orthotropic elastic material, representing the sclera and cornea, containing an elastic liquid core representing the vitreous body. The models, once pressurised, are compressed across a diameter against a rigid flat plane, with the contact area increasing as the compression continues. The finite element approach allows the IOP to be compared with the predictions of it deduced via tonometry and the Imbert-Fick law―the tonometer pressures.

The mechanical behaviours of the sclera and cornea are crucial to the analysis. Given the stiffness values taken from tensile data in the literature, it becomes clear that the assumption of isotropy is entirely inappropriate. The shell of the model eye is assumed to be orthotropic, and as far as possible its elastic constants are taken from published measurements.

The tonometer pressures are approximately a linear function of IOP with a positive intercept (

Results with the present modelling approach could be improved by the acquisition of more precise cornea and sclera elastic properties. The method can be trivially extended to more complex geometries, exploring variables such as corneal thickness and incorporating non-axisymmetric effects. Also, pressure rather than displacement boundary conditions could be applied to model alternative tonometry methods. Finally, the effects of scleral buckling could also be investigated using models of this kind.

Gabriela GonzálezCastro,Alastair D.Fitt,JohnSweeney, (2016) On the Validity of the Imbert-Fick Law: Mathematical Modelling of Eye Pressure Measurement. World Journal of Mechanics,06,35-51. doi: 10.4236/wjm.2016.63005