IOL calculation using paraxial matrix optics

Artificial intelligence-based refractive error prediction and EVO-implantable collamer lens power calculation for myopia correction

BACKGROUND

Implantable collamer lens (ICL) has been widely accepted for its excellent visual outcomes for myopia correction. It is a new challenge in phakic IOL power calculation, especially for those with low and moderate myopia. This study aimed to establish a novel stacking machine learning (ML) model for predicting postoperative refraction errors and calculating EVO-ICL lens power.

METHODS

We enrolled 2767 eyes of 1678 patients (age: 27.5 ± 6.33 years, 18-54 years) who underwent non-toric (NT)-ICL or toric-ICL (TICL) implantation during 2014 to 2021. The postoperative spherical equivalent (SE) and sphere were predicted using stacking ML models [support vector regression (SVR), LASSO, random forest, and XGBoost] and training based on ocular dimensional parameters from NT-ICL and TICL cases, respectively. The accuracy of the stacking ML models was compared with that of the modified vergence formula (MVF) based on the mean absolute error (MAE), median absolute error (MedAE), and percentages of eyes within ± 0.25, ± 0.50, and ± 0.75 diopters (D) and Bland-Altman analyses. In addition, the recommended spheric lens power was calculated with 0.25 D intervals and targeting emmetropia.

RESULTS

After NT-ICL implantation, the random forest model demonstrated the lowest MAE (0.339 D) for predicting SE. Contrarily, the SVR model showed the lowest MAE (0.386 D) for predicting the sphere. After TICL implantation, the XGBoost model showed the lowest MAE for predicting both SE (0.325 D) and sphere (0.308 D). Compared with MVF, ML models had numerically lower values of standard deviation, MAE, and MedAE and comparable percentages of eyes within ± 0.25 D, ± 0.50 D, and ± 0.75 D prediction errors. The difference between MVF and ML models was larger in eyes with low-to-moderate myopia (preoperative SE >  - 6.00 D). Our final optimal stacking ML models showed strong agreement between the predictive values of MVF by Bland-Altman plots.

CONCLUSION

With various ocular dimensional parameters, ML models demonstrate comparable accuracy than existing MVF models and potential advantages in low-to-moderate myopia, and thus provide a novel nomogram for postoperative refractive error prediction and lens power calculation.

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.

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).

Magnifications of single and dual element accommodative intraocular lenses: paraxial optics analysis

PURPOSE

Using an analytical approach of paraxial optics, we evaluated the magnification of a model eye implanted with single-element (1E) and dual-element (2E) translating-optics accommodative intraocular lenses (AIOL) with an objective of understanding key control parameters relevant to their design. Potential clinical implications of the results arising from pseudophakic accommodation were also considered.

METHODS

Lateral and angular magnifications in a pseudophakic model eye were analyzed using the matrix method of paraxial optics. The effects of key control parameters such as direction (forward or backward) and distance (0 to 2 mm) of translation, power combinations of the 2E-AIOL elements (front element power range +20.0 D to +40.0 D), and amplitudes of accommodation (0 to 4 D) were tested. Relative magnification, defined as the ratio of the retinal image size of the accommodated eye to that of unaccommodated phakic (rLM(1)) or pseudophakic (rLM(2)) model eyes, was computed to determine how retinal image size changes with pseudophakic accommodation.

RESULTS

Both lateral and angular magnifications increased with increased power of the front element in 2E-AIOL and amplitude of accommodation. For a 2E-AIOL with front element power of +35 D, rLM(1) and rLM(2) increased by 17.0% and 16.3%, respectively, per millimetre of forward translation of the element, compared to the magnification at distance focus (unaccommodated). These changes correspond to a change of 9.4% and 6.5% per dioptre of accommodation, respectively. Angular magnification also increased with pseudophakic accommodation. 1E-AIOLs produced consistently less magnification than 2E-AIOLs. Relative retinal image size decreased at a rate of 0.25% with each dioptre of accommodation in the phakic model eye. The position of the image space nodal point shifted away from the retina (towards the cornea) with both phakic and pseudophakic accommodation.

CONCLUSION

Power of the mobile element, and amount and direction of the translation (or the achieved accommodative amplitude) are important parameters in determining the magnifications of the AIOLs. The results highlight the need for caution in the prescribing of AIOL. Aniso-accommodation or inter-ocular differences in AIOL designs (or relative to the natural lens of the contralateral eye) may introduce dynamic aniseikonia and consequent impaired binocular vision. Nevertheless, some designs, offering greater increases in magnification on accommodation, may provide enhanced near vision depending on patient needs.

[Correction of corneal astigmatism with toric lenses : Theory and clinical aspects]

In the last decades, the implantation of pseudophakic and phakic toric lenses has become widespread for correcting corneal astigmatism: in cataract surgery cases with implantation of a posterior chamber lens and in refractive surgery cases with implantation of phakic lenses. The purpose of this educational and training article is to familiarize the reader with the application of pseudophakic and phakic toric lenses, to show which parameters are necessary for calculating toric lenses, to present a matrix-based calculation scheme for pseudophakic and phakic toric lenses, to explicitly demonstrate the step-by-step calculations with clinical examples, and to show the impact of lens dislocation (especially rotation) on refractive outcome.