# Mathematical Model for Evaluating Soft Contact Lens Fit

Purpose To evaluate the effect of varying lens and ocular topography parameters on soft contact lens (SCL) fit, using a novel computer spreadsheet model. Although SCLs are worn by more than 100 million ametropes, the factors governing their fitting characteristics are poorly understood.

Methods A spreadsheet-based computer model used a novel ellipto-conical corneal model coupled with population data on corneoscleral topography obtained in a previous clinical study. The model calculated lens edge strain (circumferential elongation) as a predictor of lens tightness. The following parameters were systematically varied: corneal curvature, corneal diameter, corneal shape factor, corneoscleral junction angle, lens base curve (BC), and diameter.

Results The ellipto-conical corneal model showed closer concordance with actual measurements of corneal sagittal height than a simple elliptical model (limits of agreement, ±0.20 vs. ±0.25 mm; p = 0.0015). For an eye with average ocular parameters wearing a typical SCL design (BC, 8.60; diameter, 14.2 mm), the model calculated an edge strain of 2.7%. For the same SCL, the tightest fit (8.5% strain) was found with the eye showing the combination of smallest, flattest, most aspheric cornea. Conversely, the loosest fitting (−2.6%) was found with the eye showing the combination of largest, steepest, least aspheric cornea. A change in BC of 0.4 mm typically resulted in changes in edge strain of less than 2.5%, whereas a change in diameter of 0.5 mm resulted in a change of less than 2%. Using the typical SCL design and average corneal model, wide variations in corneoscleral junction angle did not critically affect lens fit. More extreme combinations of SCL and ocular parameters resulted in edge strain likely to result in a tight (>6%) or loose fit (<0%).

Conclusions A novel ellipto-conical corneal model in conjunction with spreadsheet mathematical modeling proved to be a useful tool for attempting to understand the factors governing SCL fit.

*MPhil, PhD, FCOptom, FAAO

Visioncare Research Ltd, Farnham, United Kingdom.

**Graeme Young**, Visioncare Research Ltd Craven House, W Street Farnham, Surrey, GU9 7EN United Kingdom e-mail: g.young@visioncare.co.uk

Soft contact lenses (SCLs) have been used for more than 40 years and are worn by more than 100 million ametropes; however, the factors governing SCL fit are poorly understood. The traditional practice of selecting soft lens base curve (BC) radius based on central curvature (K-reading) has long been discredited.^{1} Some workers have attempted to relate soft lens fit to ocular sagittal height^{1–4} and, in fact, several clinical studies have noted a correlation between lens centration and corneal sagittal height, with greater sagittal height leading to greater decentration.^{5,6} Palpebral aperture has been positively correlated with the amount of lens postblink movement, presumably because of greater interaction between the lid and lens during the blink.^{6} However, the principal defining characteristic of soft lens fit is tightness as assessed by the push-up test.^{7} Martin et al.^{8} found a correlation between lens tightness and the pressure generated behind the lens when on the eye. In turn, they noted a correlation between this postlens pressure and the squeeze pressure at the periphery of the lens.^{9} Thus, tightness is governed by the amount of deformation necessary to fit the lens to the eye as well as the physical characteristics of the lens, such as thickness, modulus, and BC.^{9} A recent clinical study found a relationship between tightness and the amount of flexing required in the lens periphery.^{6}

Most of the potential strain energy generated when an SCL is pressed onto the eye arises from circumferential stretching.^{10} An SCL can therefore be envisaged as a series of concentric elastic bands, which flex and stretch to align with the peripheral ocular shape.^{11} When a large amount of peripheral stretching is required, this results in relatively high squeeze pressure and a tight lens fit. Conversely, when the lens is relatively large and no stretching is required to align with the eye, the lens is relatively loose and may even show edge standoff. The optimum condition probably involves a small amount of stretching in the lens periphery, enough to encourage centration but not so much as to cause excess pressure on the conjunctiva.

Various models have been used to describe ocular topography.^{12–16} The cornea is most commonly described as an elliptical conic section. Measurements suggest that most central corneas approximate a prolate (i.e., flattening) ellipse, although corneas closer to an oblate ellipse or sphere have also been measured.^{12} Elliptical geometry, however, is a poor estimator of the peripheral cornea.^{16} Anterior segment ocular coherence tomography (OCT) has allowed the imaging of cross sections of the cornea and sclera. These images show that the transition between the cornea and sclera is relatively smooth, with the junction angle averaging 177 degrees.^{6,17} The images also suggest some flattening of both the corneal and scleral profiles toward the limbus to affect a smoother transition between the two.

Mathematical models have been used to design rigid contact lenses and to investigate factors affecting their fit.^{18–22} A previous study used a spreadsheet-based fluorescein pattern simulator to assess the effect on rigid lens alignment of various rigid lens back surface design parameters.^{22} However, a simple elliptical model is less suited to assessing soft lens fit as this tends to overestimate corneal sagittal height.^{16} Fig. 1 shows unpublished data from a previous clinical study^{17} of corneal topography in which theoretical corneal sagittal height has been plotted against actual OCT measurements of height.

The purpose of this study has been to develop a spreadsheet-based mathematical model using a novel corneoscleral model and to evaluate factors affecting SCL fit, specifically tightness of fit.

## METHODS

A spreadsheet-based mathematical model was constructed to evaluate the fit of various spherical SCL shapes on a range of corneal topographies. The sensitivity of soft lens fit to various changes in ocular and lens design variables was systematically evaluated. The key lens fit variable of peripheral circumferential elongation, or “edge strain,” was utilized as an indicator of lens tightness.

### Corneal Model

The corneal model used for this evaluation comprises a central ellipse and separate conical periphery; the perilimbal sclera is also modeled using a cone (Fig. 2). The corneal model is rotationally symmetric and therefore makes no allowance for astigmatism or other asymmetries. Four aspects of ocular topography can be varied: apical corneal radius, corneal shape factor, corneal diameter, and corneoscleral junction (CSJ) angle.

The central elliptical corneal zone was set at 84% of the corneal diameter for most calculations. This value was determined using the data set in Fig. 1; the horizontal and vertical central zone diameters were varied until the mean corneal sagittal heights were closest to the mean horizontal and vertical corneal sagittal heights measured by OCT (Fig. 3). The optimal elliptical corneal diameters (i.e., those corresponding to zero mean difference between the calculated and actual sagittal heights) were found to be approximately 88 and 79% for the horizontal and vertical meridians, respectively.

The *x*-*y* coordinates of the elliptical zone (where *x* represents sagittal height) are calculated from the following formula:

where *R* is the corneal apical radius and SF is the corneal shape factor (1 − *e* ^{2}).

The outer conical corneal zone is tangential to the elliptical zone. The slope of the corneal conical zone is set by estimating the slope of the periphery (outer 0.3 mm) of the elliptical zone. The slope of the conical scleral zone is calculated from the CSJ angle and slope of the peripheral corneal zone.

Clinical ocular topography data from a previous study^{17} were used to test the accuracy of the corneal model. This study collected videokeratoscopy (VK) and OCT measurements from more than 200 subjects. The OCT images allowed measurement of true corneal diameter and sagittal height, whereas the VK measurements provided corresponding corneal curvature and shape factor data. The VK and OCT corneal diameter measurements were used in the model to calculate theoretical corneal sagittal heights for the same eyes.

### Lens Fit Model

The contact lens model uses a spherical back surface and is, therefore, defined by the lens BC and diameter. To calculate the on-eye diameter of the lens, the model assumes that the lens perfectly aligns the ocular surface and that there is no change in the arc length (AL) of the lens. The AL of circular shapes, such as the lens cross section, are easily calculated from the following formula:

where *R* is the radius, *&thetas;* is the angular subtense of the arc in radians, orwhere *D* is the chord diameter.

where *D* is the chord diameter.

However, calculating the AL of elliptical shapes is mathematically more complex and requires integral calculus. For simplicity, the model therefore uses an iterative calculation in which the ellipse is divided into 0.05-mm sections and an approximation of each AL calculated using the Pythagoras theorem (i.e., square root of 0.05^{2} + square of corresponding sagittal height). The total AL of the ellipse is calculated by summing the iterative results. Because the corneal diameter value is entered to the nearest 0.1 mm, using 0.05-mm iterations ensures that the full extent of the central cornea is covered and that there are no rounding errors. The accuracy of this method can be tested by specifying the cornea to be the arc of a circle (i.e., SF = 1.0) and comparing the result with the true value calculated from the above formula. The two results (in millimeters) are invariably in agreement to four decimal places.

The overlap of the lens beyond the limbus is calculated by subtracting the total AL of the cornea from the AL of the lens and projecting this onto the *x*-axis. The on-eye lens diameter equates to corneal diameter plus overlap and is usually larger than the off-eye diameter.

The stretching of the lens edge (edge strain) is calculated by comparing the off-eye diameter with the on-eye diameter once the lens has aligned with the shape of the eye. It is assumed that when the lens is forced to stretch beyond a certain extent, the lens fit will be tight. Conversely, it is assumed that a lens whose on-eye diameter is larger than the corresponding ocular diameter will be loose.

Although the key outcome variable was strain at the edge of the lens, it was also possible to plot the strain across the entire lens diameter to show other points of significant strain. This was also achieved by comparing the *x*-component (chord diameter) at various points across the eye with that of a given spherical lens at the point corresponding to similar AL.

### Lens Fit Evaluations

Using the ellipto-conical lens fit model, the effect on edge strain was calculated for variations in lens and ocular variables. Data from a previous ocular topography study^{17} were used to provide typical ocular parameters along with an indication of how these varied within a typical population (Table 1). These data were collected in a UK-based study involving more than 200 subjects (65% female) in the age range 18 to 65 years and mean spherical equivalent refraction ranging from −10.2 to +3.5.

Using the population average ocular parameters shown in Table 1, lens BC and diameter were systematically varied to evaluate the effect of edge strain. In a similar fashion, using a representative soft lens design (BC, 8.60 mm; diameter, 14.2 mm), each of the ocular parameters was systematically varied. Corneoscleral junction angle was systematically varied for a wide range of corneal diameters.

For the purposes of this exercise, the threshold between an acceptable and tight lens fit was taken to be 6.0% edge strain, whereas the threshold for an unacceptably loose lens fit was taken to be 0%. These thresholds were estimated from the aforementioned calculations to optimize the elliptical diameter. Using the ocular topography data from the previous study,^{17} the range of edge strains was calculated for a typical SCL design. The expected proportion of tight and loose fittings was estimated and the thresholds were accordingly selected. In practice, the true threshold would also be dependent on the modulus of lens material and the thickness of lens edge.

## RESULTS

### Corneal Sagittal Height

To check the accuracy of the ellipto-conical corneal model, calculations of horizontal corneal sagittal height were compared with equivalent measurements from OCT (Fig. 4). These showed closer concordance than the simple elliptical model (Fig. 2). A Bland-Altman plot (Fig. 5) shows the limits of agreement, which were significantly narrower than those for the elliptical model: ±0.20 mm versus +0.25 mm (F = 1.52, p = 0.0015). With the simple elliptical corneal model, there was increasing overestimation of corneal sagittal height with deeper corneas.

### Effect of Lens Parameters

For a typical lens design (BC, 8.60 mm; diameter, 14.2 mm) worn on an eye with average ocular parameters (Table 1), the model calculated an edge strain value of 2.7%. This corresponds to an increase in lens diameter of 0.4 mm and edge circumference of 1.2 mm, which is consistent with on-eye measurements.^{23}

Fig. 6 shows a three-dimensional representation of the variation in strain for a range of BCs and diameters on an eye with average ocular topography. The tightest fitting (i.e., greatest strain) was with the largest, steepest lens. With a smaller cornea, the same fitting would be even tighter; for instance, with a 12.6-mm cornea, the tightest lens in this range would be 17.2% rather than 15.6%. The loosest fitting was with the flattest and smallest lens, particularly on a larger diameter cornea. With a larger cornea, the same fitting would be even looser; for instance, with a 14.0-mm cornea, the loosest lens in this range would be −3.2% rather than −2.2%. Several of the lenses at this end of the range were significantly loose and could be expected to show edge standoff.

The analysis gives some indication of the effect of changing BC and diameter (Fig. 6). For instance, with commonly used BCs (8.40 to 8.80 mm), a change in BC of 0.4 mm results in a change in edge strain of less than 2.5%. However, with most lens diameters, a change in BC of more than 1.0 mm is required to change from a loose to a tight fitting. Similarly, for a given BC, a change in lens diameter of 0.5 mm results in a small change in edge strain: less than 2%.

### Effect of Ocular Parameters

The range of edge strain was calculated when a single ocular topography parameter was varied across a population range equivalent to ±2 SDs (Table 1). Corneal curvature showed the greatest effect on edge strain (4.4%) across the population range followed by corneal shape factor (3.2%); CSJ angle showed the least (1.1%).

More detailed results for variation in ocular topography are summarized in Figs. 7 to 9. The variation in strain for a range of corneal radii and diameters using an average corneal asphericity (SF = 0.68) is shown in Fig. 7. As expected, the tightest lens fit (8.5%) was with the smallest, flattest cornea and the loosest (−2.6%) was with the largest, steepest cornea.

Fig. 8 shows the effect of varying corneal shape factor as well as corneal curvature for an average corneal diameter (CD, 13.3 mm). Corneas with greater peripheral flattening (i.e., lower SF) showed tighter lens fittings. A large proportion of the plot lie in the “loose” zone; however, a high proportion of corneas in the population lie within the zone indicated by the circle.

Using the typical soft lens design and average corneal curvature model, wide variations in CSJ angle did not critically affect lens fit across a wide range of corneal diameters (Fig. 9). The tightest lens fit was with the smallest corneal diameter (11.6 mm) combined with sharpest CSJ angle (160 degrees), which gave an edge strain of 8.9%. The greatest variation in strain with CSJ angle was with smaller corneas. In Fig. 9, the variation is 4.4% (4.3 to 7.7%) with the smallest cornea, but close to zero with the largest cornea (0.8 to 1.0%).

### Strain Distribution

Fig. 10 shows the distribution of circumferential strain across a standard lens on a typical eye. Because the lens BC is flatter than the corneal radius, there is no strain across most of the cornea, and, in fact, it shows negative strain toward the peripheral cornea (i.e., circumferential shrinkage). The strain gradually increases toward the peripheral cornea, reaching a maximum (2.8%) at the lens edge. Fig. 11 shows a plot of the same lens on an eye combining extreme ocular parameters disposed toward lens tightness, (i.e., AR, 8.40 mm; CSF, 0.4; CD, 12.2 mm; CSJ, 177 degrees). Again, there is little or no strain across most of the cornea, with strain beginning approximately 2 mm from the limbus and rising to 7.5% at the lens edge.

## DISCUSSION

Computer modeling is a useful tool for understanding aspects of SCL fit. Systematically varying a single parameter gives some insight into which variables have the most impact on lens fit. Computer modeling also facilitates the design of more effective and versatile lenses. Its usage is common in the contact lens industry but not widely published for reasons of commercial sensitivity.

Although relatively simplistic, the concept of edge strain can be used to better understand several aspects of SCL performance, including tight-fitting lenses, edge standoff, decentration, clinically equivalent fittings, and superior epithelial arcuate lesion.

The ocular topography variable, which appeared to have the greatest effect on strain, and therefore tightness, was corneal radius followed by corneal shape factor. The variation in strain across a typical population range of corneal radii (4.4%, Table 1) is likely to be enough to cause a significant variation in lens fit. The range for corneal shape factor was slightly less (3.2%) and also theoretically capable of effecting a significant change in fit. As noted in a previous paper,^{1} because there is a positive correlation in the population between corneal diameter and corneal radius, any variations due to corneal radius are likely to be mitigated by corresponding variations in corneal diameter. Because flatter corneas tend to be larger, few corneas correspond to the “tight” and “loose” zones shown in Fig. 7.

It is notable that, for an eye with average ocular topography, typical soft lens designs provide edge strain corresponding to optimal fitting. Although Fig. 6 shows a wide range of lens designs, most contemporary soft lenses fall within a narrow range of BCs and diameters; these are indicated by the square in Fig. 6, which lies midway between the tight and loose thresholds.

The concept of clinically equivalent fit is common to scleral, rigid, and SCL fitting and specifies the amount of change in BC required to compensate for a change in diameter while maintaining similar fitting characteristics. Based on the data from this study, a 0.1-mm change in BC is required to compensate for a 0.2-mm change in lens diameter. For example, on an average eye, the following lenses (BC/diameter) experience similar edge strain: 8.40/14.0 and 8.60/14.4 mm.

Soft contact lens decentration can also be explained in terms of edge strain.^{10} When the ocular topography is asymmetrical, the edge strain for a centered lens will be different on opposing sides. The lens will therefore decenter until the edge strain energy is balanced. Because the temporal CSJ angle tends to be flatter than the nasal, this would encourage temporal decentration, which, indeed, is the most common direction of decentration.^{17} In cases where there is negative edge strain (i.e., loose fitting), there may be no centering force from the lens and, consequently, centration will be primarily influenced by lid forces.

During lens insertion, SCLs are often placed so as to only partially cover the cornea but quickly center themselves with little intervention. This may be partly attributed to greater stretching in that part of the lens located on the relatively flat sclera; this generates greater strain energy on the scleral side, which reduces as the lens edge moves toward the limbus.

An important characteristic of soft lens fit is blink-induced movement, which is thought to help maintain postlens lubrication. However, this computer model does not attempt to predict lens movement. Martin et al.^{8} have shown that there does not appear to be a simple correlation between lens movement and tightness, although when the lens’ squeeze pressure is above a certain threshold, movement is greatly reduced. Other factors, such as edge design, postlens tear film, and lid configuration, have a significant influence on soft lens movement. The geometry of the lens edge is often governed by the manufacturing process and therefore varies between lens brands and this is thought to have a significant effect on lens movement.^{24} Another important factor is the thickness of the postlens tear film.^{25,26}

Superior epithelial arcuate lesions are occasionally seen as a complication of SCL wear. These derive from excessive pressure on the peripheral cornea resulting in a mechanical abrasion corresponding to the arcuate zone of pressure.^{27} Two possible causes of the excess pressure are either a large mismatch between the shape of the lens and cornea or, alternatively, resistance to flexure (stiffness) of the lens as it bends to align with the perilimbal sclera. The strain profiles generated by this spreadsheet show relatively little strain in the peripheral cornea (Fig. 11), suggesting that the key factor is resistance to flexure resulting in a pressure point. The fact that the corneal staining tends to be located away from the limbus further supports this assumption.

As with all computer models, several limitations and inaccuracies are likely to affect the outcome variable. The most obvious approximation is that the model assumes rotational symmetry and, thus, overlooks the effect of astigmatism and other lateral differences in topography. It is known, for instance, that the temporal CSJ angle tends to be flatter than the nasal.^{17} Another major limitation of the model is that it assumes complete lens alignment, whereas OCT work has shown some lens clearance at the limbus, particularly where the CSJ angle is sharp.^{28} The degree of clearance is likely to be greatest in lenses of higher elastic modulus.

An often overlooked aspect of soft lens fit is the effect of temperature on lens diameter. Soft contact lenses shrink to a greater or lesser extent when raised from room to eye temperature.^{29} This can be accounted for in the model by adjusting the diameter to account for the known shrinkage characteristics of a given lens material.

Several factors, other than those considered in the model, can affect SCL tightness of fit. Lens thickness and elastic modulus affect lens stiffness, which, in turn, affects tightness of fit.^{9} However, the output variable considered in the model (edge strain) is unaffected by these. This can be reconciled by considering the critical strain thresholds at which the lens becomes critically tight or loose; these will vary according to lens stiffness. For instance, the tightness threshold for a high-modulus lens will be lower because the resulting stress for a given elongation is greater for a high- than for a low-modulus material.

The main output of the model is strain at the edge of the lens whereas soft lens fit is likely to depend on the relationship of the whole lens profile. However, the strain at the lens edge is essentially a summation of the stretching across the entire lens profile and is, therefore, invariably greatest at this location. Nevertheless, because the spreadsheet is able to calculate strain across the lens, a possible future refinement would be to explore outputs based on total strain. Another obvious refinement would be to incorporate more sophisticated, asymmetric ocular topographies.

In conclusion, a novel ellipto-conical corneal model provides a more accurate representation of ocular topography. In conjunction with spreadsheet mathematical modeling, this is a useful tool for attempting to understand the factors governing SCL fit.

Graeme Young

Visioncare Research Ltd

Craven House, W Street

Farnham, Surrey, GU9 7EN

United Kingdom

e-mail: g.young@visioncare.co.uk

### ACKNOWLEDGMENTS

*I am grateful to Lee Hall and James Wolffsohn for allowing the use of the clinical database. I am also grateful to Philipe Jubin and Pierre-Yves Gerligand for their helpful comments.*

*Received January 3, 2014; accepted May 7, 2014.*

## REFERENCES

**Keywords:**

soft contact lens; corneal model; tightness; lens fit