Ординатура / Офтальмология / Английские материалы / Wavefront Analysis Aberrometers and Corneal Topography_Boyd_2003

VISION

Jorge L. Alió, MD, PhD

Robert Montés-Micó, OD, MPhil

INTRODUCTION TO IRREGULAR ASTIGMATISM

It is well known that there exists two types of refractive errors, one named spherical, related with the axial length and the dioptric power of the different refracting structures of the eye, and another named astigmatic, related with the asphericity of the cornea (most commonly the anterior cornea) and the crystalline lens of the eye1. The last type of refractive error is dependent upon meridian.

Astigmatism is usually regular, which means that the principal meridians are perpendicular to each other, and astigmatism is correctable with conventional sphero-cylindrical lenses. However, when the cornea have an irregular shape that cannot be described by spherical, toric, or conic section aspheric geometry we call that this patient shows irregular astigmatism2. Irregular astigmatism occurs when the principal meridians are not perpendicular to each other or there are other rotational asymmetries that are not correctable with the conventional spherocylindrical lenses. This denotes a condition in which poor focusing results from asymmetrical or local

variations in the curvature of one or more of the eye’s refracting surfaces, notably the corneal. Irregular astigmatism can be devastating to vision when it occurs centrally within the pupillary area; it can be present even on a surface that clinically appears relatively smooth. Common causes of irregular astigmatism include: dry eye, corneal scars, ectatic corneal degenerations, pterygium, trauma, surgery including cataract surgery, penetrating keratoplasty and refractive surgery3.

CORNEAL TOPOGRAPHY PATTERNS OF IRREGULAR ASTIGMATISM

Irregular astigmatism has a variety of causes and appearances on topography. Generally, every deviation from a pure ellipsoidal shape is called irregular astigmatism. Computer assisted videokeratography provides theoretical and practical advantages in the assessment of the corneal topography, over keratometry and photokeratoscopy together. Based on the topography, the following classification for irregular astigmatism is proposed3:

A.- Irregular Astigmatism with Defined Pattern

We define irregular astigmatism with defined pattern when there is a steep or flat area of at least 2 mm of diameter, at any location of the corneal topography, which is the main cause of the irregular astigmatism. It is divided into five groups:

A.1.- Decentered Ablation: Shows a corneal topographic pattern with decentered myopic ablation in more than 1.5 mm in relation to the center of the cornea. The flat area is not centered in the center of the cornea; the optical zone of the cornea has one flat and one steep area (Figure 1).

A.2.- Decentered Steep: Shows a decentered corneal hyperopic treatment in more than 1.5 mm in relation to the center of the cornea (Figure 2).

A.3.- Central Island: Shows an image with an increase in the central power of the ablation zone for myopic treatment ablation at least 3.00D and 1.5 mm in diameter, surrounded by areas of lesser curvature (Figure 3).

Section III: Clinical Applications of Topography

A.4.- Central Irregularity: Shows an irregular pattern with more than one area not larger than 1.0mm and no more than 1.50D in relationship with the flattest radius, located into the area of the myopic ablation treatment (Figure 4).

A.5.- Peripheral Irregularity: It is a corneal topographic pattern, similar to central island, extending to the periphery. The myopic ablation is not homogeneous, there is a central zone measuring 1.5 mm in diameter and 3.00 D in relation to the flattest radius, connected with the periphery of the ablation zone in one meridian (Figure 5).

B.- Irregular Astigmatism with Undefined Pattern

We consider irregular astigmatism with undefined pattern when the image shows a surface with multiples irregularities; big and small steep and flat areas, defined as more than one area measuring more than 3 mm in diameter in the central 6 mm (Figure 6). The differential between flat and steep areas were not possible to calculate in the Profile

and the flat k, given in diopters at the cross of the profile map. A plane line is produced when the k cannot recognize the difference between the steep k and the flat k in severe corneal surface irregularities. This irregularly irregular astigmatism difficulties most of the therapeutic approaches to improve the quality of the corneal surface, mainly those based on selective or diametral zonal ablations, or those directly guided by topography (Topolink®).

QUANTITATIVE ANALYSIS OF IRREGULAR ASTIGMATISM FROM TOPOGRAPHY

In the simplest mathematical model describing an astigmatic error in the eye, the power P of the eye can be thought of as varying with azimuth angle q in the pupil according to the following equation:

P(θ) = Psph + Pcyl sin2(θ - α)

According to this, the correction is composed of a spherical component of power Psph

Psph Pcyl x α. If we consider a patient’s treatment with regular astigmatism this would be an easy way to quantify and analyze it. When a refractive procedure needs to be evaluated the change using this representation is considered. Several investigators have developed elaborate methods of further describing and characterizing astigmatic changes4. However, when irregular astigmatism is present powerful descriptors are needed to evaluate it, for example5:

Simulated Keratometry (SimK), that provides the powers and axes of the steepest and flattest meridians; Corneal Eccentricity Index (CEI), that indicates the eccentricity of the central cornea, Average Corneal Power (ACP), that corresponds to an areacorrected average of the corneal power before the entrance pupil; Surface Regularity Index (SRI), that measures the meridional mire-to-mire changes in power to the cornea over the entrance pupil of the eye and Ray-tracing (RT) that uses a ray-tracing to estimate predicted corneal visual acuity, superficial corneal surface quality and distortion index.

Fourier Analysis Applied to Corneal Topography

In addition to previous topographic indexes the study of irregular astigmatism could be expanded by means of the introduction of the Fourier analysis. Fourier analysis6 is a mathematical procedure that breaks any function into a sum of sine wave components with different frequencies, amplitudes and phases. For example, if we have a periodic function f(x), we can break it into a sum of sine components by applying the Fourier series formulation:

f (x)=ao + x an ?cos(2πnx / p) + bn ?sin (2πnx / p) n=1

where p is the period and a0, an and bn the coefficients of the series, which can be easily calculated. But for a general function, which can be periodic or not, the Fourier analysis is performed by means of

Section III: Clinical Applications of Topography

the Fast Fourier Transform (FFT), which is essentially the same:

FFT [f(x)]=F(w)= f(x)?exp(-i2πwx)dx

With the FFT the frequency spectrum of the function f(x) is obtained. That is, the output of the FFT is a sum of components with increasing frequencies and different amplitudes and phases, which added together allow to obtain the original function f(x). The main advantage of this procedure is that each of this frequencies gives information about different attributes of the function f(x). We can apply the FFT to f(x), isolate one of the output frequencies and rebuild the original function only with this frequency by applying the inverse FFT. Thus, one isolated attribute of the whole function can be studied.

f(x)=FFT-1[F(w)]= F(w)?exp(i2πxw)dw

Fourier analysis applied to videokeratographic data has been proposed in several studies7-10. When applied to videokeratographic data, Fourier analysis has the great advantage that the cycle-com- ponents of F(w) represent real physic corneal attributes8,9. Thus, the zero frequency represents the average spherical power of the cornea; the first frequency component represents the decentration of the videokeratography from the corneal apex; the third frequency component represents regular astigmatism, and the rest of higher order frequency components gives information about corneal irregularities.

Since Fourier analysis allows corneal irregularities isolation, it will be possible to quantify the level of corneal irregularity in normal and irregular corneas, establishing thus standard irregularity parameters for any cornea12. An illustration of the process can be observed in Figure 7. Figure 7A shows a corneal videokeratography. Figure 7B represents the three first frequencies for one ring of that videokeratography: 0, in green, representing the average spherical power of the ring; 1, in red, representing the decentration; and 2, in blue, representing the regular astigmatism (both the cylinder, given by the difference between the maximum and minimum power/radius values, and the axis, given by the first

minimum). Figure 7C shows regular astigmatism for all the rings obtained from topography (regular component, first three frequencies) and Figure 7D shows the irregular parts of the Fourier decomposition (irregular component, higher frequencies). Figures 7E and 7F show the regular and irregular corneal 3Dsurface, respectively. If a quantitative analysis of

irregular astigmatism is required, from the irregular part of the cornea (figure 7D) an irregular parameter may be defined as the mean value of the standard deviations for corneal radii data of all rings. Thus defined, for a given cornea, the higher parameter is, the more irregular the cornea is.

Fourier analysis of the videokeratography data allows to separate the regular and irregular components of the cornea, then, this applied-tool is an useful index for the quantitative study of corneas with irregular astigmatism. Assessing the irregular component in patients submitted to any refractive surgery procedure or contact lens fitting (i.e. keratoconus) will give us quantitative information of the successful in the irregular astigmatism treatment.

Corneal Aberration Identification from Topography

As it is well know, the cornea is the major refractive component of the human eye, contributing approximately two thirds of the eye’s optical power. The ideal optical shape of the anterior surface of the cornea is a prolate ellipsoid. Notwithstanding, there are wide variations in shape producing common aberrations such as astigmatism. Deviations from this ideal optical shape provoke significant amounts of asymmetric aberrations that cannot be corrected with traditional spectacles. To know exactly the shape of the corneal surface is important for several reasons: that is, corneal refractive surgery and contact lens design or fitting require accurate modeling of this surface. In both cases, characterization of corneal shape by means of topography is necessary to ensure good optical and visual outcomes after these procedures.

To establish the contribution that the corneal surface makes to vision, one can take the measurement of the anterior corneal surface using noninvasive instruments such as the videokeratoscope and apply geometrical and wave optics to determine the wavefront aberration error. In order to estimate optical aberrations induced by anterior surface of the cornea it is necessary to determine corneal contour. This can be defined in terms of sagittal depths measured from a plane surface tangential to the most prominent part of the corneal (generally the apex). All measurements from the tangential plane will be defined as positive if measured from the plane

Section III: Clinical Applications of Topography

towards the retina. Assuming that the corneal apex coincides with the center of the Placido rings, sagittal depths may be estimated from the radius of curvature recorded by topography13. Geometrical pathlengths can then be calculated starting from an arbitrary axial object location, through each point over the corneal surface to an axial image point, and converted to actual optical pathlengths by multiplying pathlengths by the refractive index of the medium. Due to that the quality of an image is dependent on the phase of light rays, after passing through different parts of the optical system, it is necessary to divide the optical pathlength by wavelength (to multiply by 2π to obtain phase angle in radians). If we have equal pathlengths over the pupil this will result in an aberration-free image. However, if the phases differ the image will be aberrated. These phase differences can be resolved into corneal aberration terms by expressing the distribution of pathlengths over the pupil as a combination of functions described mathematically as Zernike polynomials14. These polynomials may be divided into components or terms, each of which has a coefficient which describes the contribution of that element to the image as a whole15. Then, for a single surface optical system, the polynomial describing the perfect wavefront would have coefficients which were all equal to zero, and for an aberrated optical system the magnitude of each coefficient is a measure of that term’s contribution to the total wavefront error. In the Zernike polynomial expansion, different optical aberrations are described by terms which are raised to different orders (Table 1). First and second order terms, which describe tilt, astigmatism and spherical refractive error are easily corrected with ophthalmic lenses. Third, fourth and higher-orders which describe spherical aberration, coma and the rest of aberrations, are less amenable to correction by ophthalmic lenses and can contribute substantially to the total aberration.

Obtaining corneal data from the videokeratoscope and reducing them into Zernike polynomials, will give us a magnitude of corneal aberrations

evaluating each term of the polynomial expansion. This analysis has been applied to determine changes in the optical quality of the cornea after some refractive surgery treatments16-22. Since this section shows clinical applications of topography and this particular chapter is related with irregular astigmatism, we are going to apply the previous methodology of Zernike polynomial expansion to a case of irregular astigmatism.

Videokeratographic data were obtained by computerized videokeratography (Orbscan II, Orbtek Inc, Salt Lake City, UT) in patient with irregular

astigmatism (Figure 8). Measurements in each eye were repeated until a well focussed and aligned image was obtained. Corneal videokeratographic data were downloaded onto floppy disks in ASCII files which contained information about corneal elevation, curvature, power and position of the pupil. The corneal data were fitted with Zernike polynomials up to the sixth order to determine aberration coefficients, from which the corneal aberration function was reconstructed using the descriptive polynomial method of Howland and Howland23. From the Zernike coefficients the root-mean-square (RMS)

wavefront errors for comalike aberrations (thirdorder components Z3i and fifth-order components Z5i) and spherical-like aberrations (fourth-order component Z40 and sixth-order component Z60) were calculated. RMS is the square root of the average squared difference between the observed measurements and their expected values, as estimated from the curve that best fits the actual data. Because of the linear independence of the Zernike terms, the total wavefront error may be computed by summing all components (Z3i + Z4i + Z5i + Z6i)24. To derive aberration coefficients for different pupil diameters, the raw data images were masked to include only the data inside the required pupil diameter before proceeding with Zernike analysis. In this case, 3.5-mm

and 6.5-mm pupil diameters were studied. Different pupil diameters were assessed to allow evaluation of the central-peripheral irregularity of the cornea.

Table 2 shows the Zernike coefficients obtained for both pupil diameters and the RMS wavefront error of the different optical aberrations in this patient. From this table, we are able to asses each coefficient separately and its contribution to total wavefront error. RMS gives us the magnitude of each type of aberration (i.e. spherical-like, coma-like) and how contributes to total wavefront error. If an additional parameter is evaluated, such as pupillary dilation, these descriptors allow a direct comparison between coefficient and/or orders. For example, as Table 2 shows, pupillary dilation from 3.5 to 6.5-mm in the eye increases the amount and character of all