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Higher order Abberations

Besides Short sight, LongSight, Astigmatism and Presbyopia called as Lower order Abberations (LOA's), another group of refractive errors exist called as Higher Order Aberations (HOA's). Unlike LOA's which predominantly  affect the quantity of vision, HOA's predominantly affect the quality of vision which was previously undefined. As Ophthalmology made rapid advances & as science progressed, it was seen that many patients who were tested to be normal using the usual methods were still complaining about visual deficiencies that could not be 'Measured'. This led investigators to explore the Quality aspect of vision which unravelled an entire new range of defects together called as Higher order abberations (HOA's).

For a layman to understand the concept of HOA's, Night sky photography will make the concept clear. Images of Solar eclipses taken on a normal camera, or as seen through normal spectacle lenses are vastly different in terms of quality  when seen through telescopes. The reason being that these telescopes are corrected for HOA's or Wavefront errors as they are known in optics to enhance optical perfection.  For an illustration on these images click here.  For those of you interested in more details about Wavefront errors and corneal abberation, read on what investigators already know about HOA's...

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 necessaryto 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 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 topography. Geometrical path lengths 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 2_ 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 whole. 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.

The Following table indicates the type of error( Zer nicke coefficient), followed by the Radial order( the row in which it is classified) followed by the Angular Frequency( the number of times the error can be repeated within the optic zone for ex- Spherical Abberation has a angular freq of 0, while coma/tilt has a ang freq of 1, Astigmatism has a angular freq of 2 ,trefoil /3fold has AF of 3, tetra foil/4fold -4, Pentafoil/5 fold -5) and the common name given to the error.

 

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 discussion 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 Howland. From the Zernike coefficients the root-mean-square (RMS)

Table 1: Double-index Zernike polynomials describing optical aberrations up to the sixth order coefficients

(Zfn, n means radial order and f angular frequency).

wavefront errors for coma like aberrations (third order components Z3 i and fifth-order components Z5 i) and

 spherical-like aberrations (fourth-order component Z4 0 and sixth-order component Z6 0) 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 (Z3 i +

 Z4 i + Z5   + Z6 i)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

 

 

aberrations. The magnitude of the induced aberration for a large pupil diameter appears to be a function of corneal irregularity in the periphery. Obviously, an increase in spherical aberration is expected with pupillary dilation, which is highly pupil size dependant.

If Abnormal Q values are obtained, normally it would correlate with high WF errors in larger Pupils/Mesopic conditions.  With normal Q values, WF errors may not be significant in Mesopic conditions.  Any Deviations from this and the cause has to be elaborated. For Ex, Abn Q values and low WF errors, See if the Individual Zernicke Coefficients are cancelling out Each other producing a Constructive WF. If normal Q Values and having abnormal WF errors, other Causes like Tear Film abnormalities, Media Opacities, Pupillary abnormalities that may increase optical path lengths to produce destructive interference may have to be looked into. If low WF errors are producing symptoms , see if the zernicke modes are in the same radial order, different angular frequency or different sign. As mentioned before , if the zernicke modes are in the centre of the pyramid they tend to have a greater effect on vision than the peripheral modes even if the total WF errors are minimal.

Comparison of the wavefront analysis and derived conclusions between techniques or treatments need to be done inside of severe research protocols. Therefore, videokeratography, which produces an analysis of corneal surface curvature based on the radii of corneal curvature at points interpolated from reflected mires, allows the wavefront aberrations on the cornea to be calculated and separated into different optical aberrations. These corneal optical aberrations give us the opportunity to study the optical quality of the cornea and to relate it with the expected visual function in any patient.

CONCLUSIONS

Corneal topography provides the needed data to determine the optical quality of the cornea. By using Fourier analysis, which allows to quantify the irregularity level of the cornea by dividing its regular and irregular power components, and by means of the Zernike polynomial expansion, which inform us about the different corneal aberrations; we have a powerful instrument to explore the visual function in order to improve unaided vision created in some cases of irregular astigmatism.

 

New technologies applications as Fourier and Zernike analysis in Ophthalmology is still a young field. With this technology and new ideas about how to apply them in refractive surgery, the future of irregular astigmatism and the visual consequences derived from it would be solved in a short period of time.

Send mail to drsaravana@srivenkateshwaranethralaya.com with questions or comments about this web site.
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