Dynamic changes in corneal topography and its influenceon the point-spread function of the eye

Damian Siedlecki, Henryk Kasprzak, and Barbara K. Pierscionek

The dynamic changes of the anterior surface of the eye are investigated. A Twyman–Green interferometeris used to record topographic images at 40 ms intervals. A method of analysis of the dynamic changes intopography by use of Zernike polynomials enables a general distinction to be made between dynamicalterations in the shape of the cornea itself and the changes in the layer of the tears. The influence ofdeviations in the shape of the anterior surface of the eye on the retinal image is estimated. © 2007Optical Society of America

OCIS codes: 120.3180, 330.4460, 330.5370, 110.3000, 170.4470.

1. Introduction

The cornea, as the main refracting element of the eye,is not engaged in refractive changes on the scale re-quired for altering focus to meet visual demand. It has,however, been shown to be amenable to some changesin shape, albeit it small ones, when the tear filmspreads on the cornea,1–3 as the eye accommodates4–6

(this is not consistent; some studies7,8 report that thecornea as an optical medium is stable during accom-modation), or as a result of gravitational forces on theintraocular pressure (IOP).9 Long-term changes in cor-neal toricity have been attributed to extraocular mus-cle (EOM) forces.10 IOP is affected by both the tonicand the phasic responses of the extraocular muscles.11

If the tonicity of the EOM has a measureable effect onthe IOP, and both affect corneal shape, it is possiblethat the corneal topography exhibits dynamic changesin response to muscle tonus and IOP. Such shape al-terations would be very small and require very sensi-tive measurement.

This study investigates whether dynamic changesin the front surface of the cornea could be observedusing a well-described1,2,12–14 interferometric tech-

nique. The method that we used was, like most ofthe commercial videokeratometric devices and cor-neal topographers (i.e., incorporating the Placido disctechnique15), based on the reflectance from the firstoptical surface of the eye, i.e., the tear film. Becauseit covers the anterior surface of the cornea, measure-ment of the shape of the tear film can provide someinformation about the shape of the anterior cornealsurface. The breakup time of tears is substantiallylonger16 than the 200 ms duration of observation inthis study. Hence any changes in the tear layer overthe measurement period would have been insufficientto mask alterations in the interferograms that mayhave arisen from changes in corneal topography.Changes in anterior surface shape were incorporatedinto a model eye to determine the effect of shape onthe point-spread function (PSF) of the eye.

2. Methods

Six interferograms were used in the analysis of thecorneal topography of a single eye. The subject (male)was an emmetropic presbyope who did not sufferfrom any ocular pathology that may affect measure-ments (dry eye syndromes or conditions that causeexcessive tearing). A Twyman–Green interferometerwas used to record the six images at 40 ms intervals,and recorded images were sent to a monitor. Theexperimental design is shown in Fig. 1. The head ofthe subject was supported by a frame with a chin andforehead rest and fixation was maintained on a pointdirectly in front of the measured eye. The illuminatedarea of the cornea had a diameter of 5 mm. The sub-ject did not blink between images; the first image wastaken immediately after a blink. All images remainedclear and undisturbed, indicating that there was no

D. Siedlecki (damian.siedlecki@pwr.wroc.pl) and H. Kasprzakare with the Institute of Physics, Wroclaw University of Technol-ogy, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland. B. K.Pierscionek is with the School of Biomedical Sciences, University ofUlster at Coleraine County Londonderry, Northern Ireland BT521SA, UK.

Received 2 May 2006; revised 28 November 2006; accepted 5December 2006; posted 14 December 2006 (Doc. ID 70532); pub-lished 20 February 2007.

0003-6935/07/081361-06$15.00/0© 2007 Optical Society of America

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localized thinning of the tear film during measure-ment.

The interferograms were analyzed by superimpos-ing 40 vertical scan lines on each image and a Car-tesian coordinate system to indicate the points ofintersection of scan lines and interferometric fringes.The coordinate points underwent corrections for anyeye movements that may have occurred.14 On thebasis of the corrected coordinates, and using interpo-lation techniques, the deviation of the actual shape ofthe anterior corneal surface from an ideal sphere wasdetermined for each of the interferograms. The shapeof each surface was approximated with 36 Zernikepolynomials using the method of least squares. Thenumbering of the polynomials was in accordance withthe standards set out by VSIA standards taskforcemembers.17 To describe the dynamics of corneal shapechange within a 200 ms time duration, the quantitieswere defined as the difference

fi � �max�Zi�k � min�Zi�k

�Zi�k�, (1)

where Zi is the value of the coefficient (given by i)of the Zernike polynomial (i � 1–36), min�Zi�k andmax�Zi�k indicate the minimum and maximum valuesof the coefficient (given by i) for the surface (given byk), k is the coefficient that identifies the surface (inthis case k � 1–6), and �Zi�k is the average value ofthe coefficient (given by i) for the surface (given by k).

To normalize the values, Fi are defined as

Fi �fi

�i

fi� 100%, (2)

where Fi describes the range of changes in the coef-ficient given by i in relation to the average value ofthese coefficients for all surfaces. The smaller thevalue of Fi, the smaller the contribution of the corre-

sponding coefficient to changes in the shape of thesurface.

The effects of changes in corneal topography on theoptical image (PSF) were calculated using parame-ters for an ideal, diffraction-limited model of the eyewith an axial length of 24 mm. Aberrations of theanterior corneal surface, as calculated using Zernikepolynomials from interferometric measurements ofthe cornea, were represented by a phase plate withvarying thickness (Fig. 2). The refractive index of theplate was made equal to that of the cornea. Zemaxoptical design and analysis software was used forcalculating the PSF.

3. Results

The six images, shown in Fig. 3, illustrate the devi-ation of the actual shape of the first ocular surfacefrom a model spherical surface. The pattern of fringeschanges from image to image. A common and fixedfeature of all images is the astigmatism of the corneawith the major axes at about 10° and 100°.

Along the 10° axis, the fringes change from dark tolight and the number of fringes changes by 1 or 2, butthe basic shape remains the same. Along the 100°axis, there are changes in both shape and number offringes. The greater separation of fringes along the10° axis compared to that along the 100° axis indi-cates the greater curvature in the former direction.This is in agreement with the keratometric measure-ments of this subject, which showed that the cornealastigmatism was at an axis of 100°.

Figure 4 shows the deviation of the anterior cornealsurface topography from the model spherical surfacefor each of the Zernike coefficients calculated for thearea of 5 mm. The results confirm the existence ofastigmatism (coefficients Z3, Z5, Z11, and Z13), whichis clearly seen in the interferograms. The values ofFi for the first 15 Zernike coefficients are shown inFig. 5.

The values of Fi for the individual Zernike coeffi-cients range from 0.09% (Z12) to almost 12% (Z14). Adistinction can be made between the Zernike coeffi-cients that are relatively steady in time and thosethat show large changes with time. The least chang-ing coefficients are analagous to the following opticalaberrations: astigmatism of first (Z3 and Z5) and sec-ond order (Z11 and Z13) and spherical aberration offirst order (Z12). The most highly varying coefficientsare analagous to vertical tilt (Z1), vertical coma (Z7),and quatrefoil (Z14). These coefficients have thegreatest effect on the changes in the images andtherefore also contribute most to changes in the PSF.

Fig. 1. Diagrammatic representation of the experimental design(L, HeNe laser; Sh, shutter; M1, M2, mirrors; P, polarizers; BE,beam expander; BS, beam splitter; L1, L2, lenses; CCD, CCDcamera; DVR, digital video recorder; PC, computer).

Fig. 2. Diagrammatic representation of the system used to cal-culate the PSF.

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The PSFs calculated for the various states of cor-neal topography that were recorded in the interfero-grams are shown in Fig. 6. The images in this figureshow that the intensity of the PSF decreases sub-stantially in relation to the intensity of the PSF for

the diffraction-limited model. A significant amountof energy goes into off-axis local maxima. The max-imum intensity is significantly smaller than in thediffraction-limited model—the Strehl ratio is about0.2–0.25 and stays almost steady in time. In Fig. 6

Fig. 3. Consecutive interferograms of corneal topography recorded using a Twyman–Green interferometer at intervals of 40 ms.

Fig. 4. Deviation of the anterior corneal surface from the model spherical surface (�) for each of the Zernike expansion coefficientscorresponding to the six interferograms.

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there are two characteristic maxima of the intensity(marked A and B) that are present in all calculatedimages of the PSF. Figure 7 presents the changes inthe positions of the A (diamonds) and B (squares)maxima in relation to the position of the maximum ofthe diffraction-limited PSF. The influence of the cor-neal astigmatism is noticeable.

Figure 8 shows tangential and sagittal cross sec-tions of the modulation transfer function (MTF) cal-culated for all interferograms analyzed with a 5 mmpupil, compared with the diffraction-limited MTF.There is a significant loss of the MTF in relation to thediffraction-limited MTF, especially in the sagittal di-rection. The changes in the MTF, calculated for sev-eral interferograms, are also noticeable—10%–20%in the range of 20–130 cycles�degree. The sagittal

MTF [Fig. 8(b)] has two additional local minima,which are the results of first- and second-ordermaxima in the intensity of the image (Fig. 6). Thefirst minimum takes place between 35 and 40 cycles�degree, which is especially significant because it fallsunder the average contrast sensitivity curve of theeye. It should be noted that, although informationcarried by such frequencies is partially suppressed ina normal eye, the resolution limit for foveally pre-sented gratings is about 60 cycles�degree.18 The pres-ence of this minimum may have some effect on thefrequencies in the optical system of the eye and berelated to the presence of astigmatism. In the tan-gential direction, there are no such minima and the

Fig. 5. Values of Fi for each of the Zernike coefficients.

Fig. 6. Intensity of the PSF for a schematic model eye with a 5 mm aperture and corneal topography corresponding to images capturedin the interferograms shown in Fig. 3. Each image is 43.6 �m wide. Intensity of images is not normalized. A and B are intensity maxima.

Fig. 7. Traces of the two characteristic maxima marked with Aand B in Fig. 6(a) calculated for several interferograms from Fig. 3.

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contrast sensitivity function, for the pupil size used,is normal.

4. Discussion

Utilization of the Twyman–Green interferometerfor measuring corneal topography was describedpreviously.1,2,12–14 In these studies, high levels of ac-curacy and repeatability were found from measure-ments tested on a set of glass objects of knowncurvatures. Testing biological tissues in vivo is moredifficult. With a living eye, errors could be introducedby rapid eye movements and, if measurement dura-tion was sufficiently long, as a result of evaporation ofthe tear layer. The measurements therefore had to beconducted on a trained observer who could maintainfixation and be completed within a very short periodof time.

The results show that the front surface of the eyeundergoes subtle but measurable changes within avery short time period. The method of analysis ofthese dynamic changes is described using Zernikepolynomials. Certain Zernike coefficients, those thatcorrespond to astigmatism and spherical aberration,were hardly altered, and this is because of the rela-tively high magnitudes of these aberrations. The great-est changes were found to occur in the coefficients thatcorrespond to the aberrations of coma and trefoil.

The main difficulty in the study was to distinguishfluctuations in the shape of the cornea from any pos-sible fluctuations in the thickness of the tear layer. Themeasurements were taken over a very short time pe-riod to minimize any changes in the latter. However,these cannot be discounted. Measurable changes incorneal surface may have been expected to inducegreater changes in astigmatism, spherical aberration,and defocus,19 but given the relative magnitude ofthese aberrations, substantial changes would havebeen required to register a measurable difference. Is-kander and Collins,19 who examined dynamic changesin corneal topography, have also noted that defocus isrelatively stable for a period up to 17 s after a blink,whereas vertical coma shows significant variations.Coma, as well as trefoil, was found to vary in thesubject observed in this study. Although there may besome contribution to this variation from the tear layer,this is unlikely to be predominant. It has been shownthat the surface asymmetry index, which is used tocharacterize tear film behavior, had a 4 s period ofdecay followed by a 10 s period of constancy in a subjectwho exhibited tear film instability.19 The observationtime in this study was more than an order of magni-tude less than the observation time in the Iskanderand Collins study.19 Moreover, the subject in this studydid not suffer from any instability in the tear film. Ifthe interferograms were changing purely as a result oftear layer changes, greater variations and more ex-pressive changes in the fringe patterns would beexpected20 and this would be reflected in the magni-tude of the Fi factor for all of the Zernike coefficients.In contrast to earlier studies of Buehren et al.7 andSchachar,8 who claimed that the cornea is stable dur-ing accommodation, this work suggests that someslight shape alterations can occur, and this is consis-tent with observations made using high-speed video-keratoscopy.19 Changes in the shape of the cornealsurface have been linked with respiration and withpulse rate.21–24

Any change in the shape of the first refractingsurface of the eye will have an effect on the retinalimage. Analysis of the PSF, relative to a diffraction-limited model, shows that the sharp point imageof the diffraction-limited model becomes a line ofpoints when the real cornea is used, indicating thepresence of astigmatism. This alters with time intoa spread of points with noticeable maxima of thefirst and second order, which occur in each intensitydistribution calculated for several interferograms.Although it is recognized that the eye for which thePSFs were calculated was based on a diffraction-limited model eye and that the PSF may be differ-ent in a real eye, the results suggest that over a veryshort duration, the fixating eye will undergo sub-tle but measurable changes to the general cornealshape. These may alter the retinal imagery. Artalet al.25,26 have suggested that internal elements ofthe eye compensate for total aberrations of the corneato prevent a noticeable deterioration in the qualityof the retinal image. It remains to be seen whetherchanges to the retinal image caused by dynamic alter-

Fig. 8. (a) Tangential and (b) sagittal cross sections of MTFscalculated for the analyzed series of interferograms in comparisonwith the diffraction-limited MTF (the pupil diameter is 5 mm).

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ations in the corneal shape are likewise compensatedby fine changes in the internal components of the eye.The most significant changes would be expected fromthe lens, but, unless the dynamic changes in the cor-neal shape are part of a synchronized alteration thataffects the entire eye, it is difficult to envisage how thelens could dynamically change its shape to compensatefor that of the cornea. Alternatively, there may be aneural mechanism that maintains image quality andstabilizes any variations. Micromovements of the eyeare known to occur20 and are advantageous to the vi-sual process; the Troxler phenomenon is evidence ofthe detrimental effect on vision when the eye main-tains a completely steady gaze. The changes in cornealshape, over a short duration of time, may indeed be amanifestation of micromovements of the eye. Whetheror not they are, it is clear that the cornea is sufficientlyflexible to undergo relatively rapid changes in shape.Such changes have been found with accommodation4,5

and pupil fluctuations.24 Although the magnitude ofthe rapid changes is small, it indicates that the cornealshape is not rigid and hence can, potentially, be al-tered. Whether or not such alterations could be sus-tainable over a long-term period and so exploited tobenefit the refractive status of the eye requires furtherinvestigation.

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