Impact of decentration of astigmatic intra-ocular lenses on the residual refraction after cataract surgery*

Linear optics of the eye and optical systems: a review of methods and applications

The purpose of this paper is to review the basic principles of linear optics. A paraxial optical system is represented by a symplectic matrix called the transference, with entries that represent the fundamental properties of a paraxial optical system. Such an optical system may have elements that are astigmatic and decentred or tilted. Nearly all the familiar optical properties of an optical system can be derived from the transference. The transference is readily obtainable, as shown, for Gaussian and astigmatic optical systems, including systems with elements that are decentred or tilted. Four special systems are described and used to obtain the commonly used optical properties including power, refractive compensation, vertex powers, neutralising powers, the generalised Prentice equation and change in vergence across an optical system. The use of linear optics in quantitative analysis and the consequences of symplecticity are discussed.

A systematic review produced 84 relevant papers for inclusion in this review on optical properties of linear systems. Topics reviewed include various magnifications (transverse, angular, spectacle, instrument, aniseikonia, retinal blur), cardinal points and axes of the eye, chromatic aberrations, positioning and design of intraocular lenses, flipped, reversed and catadioptric systems and gradient indices. The optical properties are discussed briefly, with emphasis placed on results and their implications. Many of these optical properties have applications for vision science and eye surgery and some examples of using linear optics for quantitative analyses are mentioned.

Dioptric power and refractive behaviour: a review of methods and applications

Myopia is a global healthcare concern and effective analyses of dioptric power are important in evaluating potential treatments involving surgery, orthokeratology, drugs such as low-dose (0.05%) atropine and gene therapy. This paper considers issues of concern when analysing refractive state such as data normality, transformations, outliers and anisometropia. A brief review of methods for analysing and representing dioptric power is included but the emphasis is on the optimal approach to understanding refractive state (and its variation) in addressing pertinent clinical and research questions.

Although there have been significant improvements in the analysis of refractive state, areas for critical consideration remain and the use of power matrices as opposed to power vectors is one such area. Another is effective identification of outliers in refractive data. The type of multivariate distribution present with samples of dioptric power is often not considered. Similarly, transformations of samples (of dioptric power) towards normality and the effects of such transformations are not thoroughly explored. These areas (outliers, normality and transformations) need further investigation for greater efficacy and proper inferences regarding refractive error. Although power vectors are better known, power matrices are accentuated herein due to potential advantages for statistical analyses of dioptric power such as greater simplicity, completeness, and improved facility for quantitative and graphical representation of refractive state.

The relationship between angle kappa and astigmatism after phacoemulsification with implanting of spherical and aspheric intraocular lens

Purpose:

To determine the significance of any association between either change in angle kappa (K°) or the rectilinear displacement (L, mm) of the first Purkinje image relative to the pupil center and unexpected changes in astigmatism after phacoemulsification.

Methods:

Orbscan II (Bausch and Lomb) measurements were taken at 1, 2, and 3 months after unremarkable phacoemulsification in patients implanted with spherical (group 1, SA60AT, Alcon) or aspheric (group 2, SN60WF, Alcon) nontoric IOLs. The outputs were used to calculate L. Astigmatism, measured by autorefractometry and subjective refraction, was subjected to vector analysis (polar and cartesian formats) to determine the actual change induced over the periods 1–2 and 2–3 months postop.

Results:

Chief findings were that the mean (n, ±SD, 95%CI) values for L over each period were as follows: Group 1, 0.407 (38, ±0.340, 0.299–0.521), 0.315 (23, ±0.184, 0.335–0.485); Group 2, 0.442 (45, ±0.423, 0.308–0.577), 0.372 (26, ±0.244, 0.335–0.485). Differences between groups were not significant. There was a significant linear relationship between (A) the change in K (ΔK = value at 1 month-value at 2 months) and K at 1 month (x), where ΔK =0.668-3.794X (r = 0.812, n = 38, P = <0.001) in group 1 and ΔK = 0.263x -1.462 (r = 0.494, n = 45, P = 0.002) in group 2, (B) L and the J₄₅ vector describing the actual change in astigmatism between 1 and 2 months in group 2, where J₄₅ (by autorefractometry) =0.287L-0.160 (r = 0.487, n = 38, P = 0.001) and J₄₅ (by subjective refraction) =0.281L-0.102 (r = 0.490, n = 38, P = 0.002), and (C) J₄₅ and ΔK between 2 and 3 months in group 2, where J₄₅ (by subjective refraction) =0.086ΔK-0.063 (r = 0.378, n = 26, P = 0.020).

Conclusion:

Changes in the location of the first Purkinje image relative to the pupil center after phacoemulsification contributes to changes in refractive astigmatism. However, the relationship between the induced change in astigmatism resulting from a change in L is not straightforward.

[Combination of lens decentration and tilt in phakic and pseudophakic eyes-Optical simulation of defocus, astigmatism and coma]

BACKGROUND AND PURPOSE

The effect of lens decentration and tilt on retinal image quality has been extensively studied in the past in simulations and clinical studies. The purpose of this study was to analyze the effect of combined lens decentration and tilt on the induction of defocus, astigmatism and coma in phakic and pseudophakic eyes.

METHODS

Simulations were performed with Zemax on the Liou-Brennan schematic model eye. Based on the position of the gradient lens the image plane was determined (best focus). The lens was decentered horizontally from -1.0 mm to 1.0 mm in steps of 0.2 mm and tilted with respect to the vertical axis from -10° to 10° in steps of 2° (in total 121 combinations of decentration and tilt). For each combination of decentration and tilt defocus, astigmatism (in 0/180°) and horizontal coma was extracted from wave front error and recorded for a pupil size of 4 mm. After replacement of the gradient lens with an aberration correcting artificial lens implant model with the equatorial plane of the artificial lens aligned to the equatorial plane of the gradient lens, the simulations were repeated for the pseudophakic eye model.

RESULTS

For the lens positioned according to the Liou-Brennan schematic model eye the simulation yielded a defocus of 0.026 dpt/-0.001 dpt, astigmatism of -0.045 dpt/-0.018 dpt, and a coma of -0.015 µm/0.047 µm for phakic/pseudophakic eyes. Maximum values were observed for a horizontal decentration of 1.0 mm and a tilt with respect to the vertical axis of 10° with 1.547 dpt/2.982 dpt for defocus, 0.971 dpt/1.871 dpt for astigmatism, and 0.441 µm/1.209 µm for coma. Maximum negative values occurred in phakic/pseudophakic eyes with -0.293 dpt/-1.224 dpt for defocus, for astigmatism -0.625 dpt/-0.663 dpt and for coma -0.491 µm /-0.559 µm, respectively.

CONCLUSION

In this simulation study the effect of a combination of lens decentration in horizontal direction and tilt with respect to the vertical axis on defocus, astigmatism and horizontal coma was analyzed. The results may help to describe in clinical routine if with a decentered or tilted artificial lens implant the postoperative refraction does not match the target refraction or the resulting astigmatism after cataract surgery is not fully explained by measurement of corneal astigmatism.

Relationship between effective lens position and axial position of a thick intraocular lens

PURPOSE

To discuss the impact of intraocular lens-(IOL)-power, IOL-thickness, IOL-shape, corneal power and effective lens position (ELP) on the distance between the anterior IOL vertex (ALP) of a thick IOL and the ELP of its thin lens equivalent.

METHODS

We calculated the ALP of a thick IOL in a model eye, which results in the same focal plane as a thin IOL placed at the ELP using paraxial approximation. The model eye included IOL-power (P), ELP, IOL-thickness (Th), IOL-shape-factor (X), and corneal power (DC). The initial values were P = 10 D (diopter: 1 D = 1 m-1), 20 D, 30 D, Th = 0.9 mm, ELP = 5 mm, X = 0, DC = 43 D. The difference between ALP and the ELP was illustrated as a function of each of the model parameters.

RESULTS

The ALP of a thick lens has to be placed in front of the ELP for P>0 IOLs to achieve the same optical effect as the thin lens equivalent. The difference ALP-ELP for the initial values is -0.57 mm. Minus power IOLs (ALP-ELP = -0.07 mm, for IOL-power = -5 D) and convex-concave IOLs (ALP-ELP = -0.16 mm, for X = 1) have to be placed further posterior. The corneal power and ELP have less influence, but corneal power cannot be neglected.

CONCLUSION

The distance between ELP and ALP primarily depends on IOL-power, IOL-thickness, and shape-factor.

Impact of intraocular lens displacement on the fixation axis

To investigate the impact of intraocular lens (IOL) decentration ≤±1  mm and IOL tilt ≤±10° on the fixation axis and spherical equivalent refraction (SE), 50 pseudo-phakic eyes were simulated using numerical ray-tracing. We computed the position of the object point whose image ends up at the virtual fovea for each scenario and estimated the corresponding change of fixation axis and SE. The eye turned opposite to the direction of IOL decentration or tilt to compensate for the associated prismatic effect (angle <1.2°). Decentration of the aspheric IOL resulted in a hyperopic shift (<0.57 D), and tilt in a myopic shift (<0.77 D).

Conditions in linear optics for sharp and undistorted retinal images, including Le Grand's conditions for distant objects

In 1945 Yves Le Grand published conditions, now largely forgotten, on the 4×4 matrix of an astigmatic eye for the eye to be emmetropic and an additional condition for retinal images to be undistorted. The conditions also applied to the combination of eye and the lens used to compensate for the refractive error. The conditions were presented with almost no justification. The purpose of this paper is to use linear optics to derive such conditions. It turns out that Le Grand's conditions are correct for sharp images but his condition such that the images are undistorted prove to be neither necessary nor sufficient in general although they are necessary but not sufficient in most situations of interest in optometry and vision science. A numerical example treats a model eye which satisfies Le Grand's condition of no distortion and yet forms elliptical and noncircular images of distant circles on the retina. The conditions for distant object are generalized to include the case of objects at finite distances, a case not examined by Le Grand.

[Determination of toric intraocular lenses]

BACKGROUND

In the last decades, toric posterior chamber lenses (TPCLs) for cataract surgery and phakic toric lenses (PTLs) for refractive surgery have become more and more popular for correcting high or excessive corneal astigmatism. The purpose of this article is to present a vergence-based calculation scheme for TPCLs and PTLs.

METHODS

In Gaussian optics (in the paraxial space), spherocylindrical optical surfaces can be described in a mathematically equivalent formulation as vergences. There are dual notations: The standard notation is used for transforming vergences through a homogeneous optical medium, and the component notation is applied to add up the power of a refractive surface to the vergence. Both notations can be used interchangeably. For calculating TPCLs, the vergences in front of and behind the predicted pseudophakic lens position are determined and subtracted. For calculating PTLs, the anterior vergence at the predicted lens position is estimated for the preoperative and postoperative states, and the difference between the two yields the desired lens power. WORKING EXAMPLES: In the 1(st) example, the power of a thin TPCL is determined step by step by applying the presented calculation scheme, which was designed to be transferred directly to a simple computer program (e.g., Microsoft Excel). In the 2(nd) example, the postoperative refraction is estimated for a simulation in which a TPCL similar to that in example 1 is implanted in a slightly misaligned orientation. In a 3(rd) example, the power of a PTL is determined step by step using the above-mentioned calculation scheme.

CONCLUSIONS

The presented calculation scheme allows determination of"thin" TPCLs or PTLs to achieve spherocylindrical target refraction with a cylinder axis at random or to predict the postoperative refraction for any toric lens implanted in any axis. The concept can be easily generalized to"thick" toric intraocular lenses if the geometric data and refraction index of the material are known.