Matrix-based calculation scheme for toric intraocular lenses

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.

Ratio of torus and equivalent power to refractive cylinder and spherical equivalent in phakic lenses - a Monte-Carlo simulation study

BACKGROUND

Spherical and astigmatic powers for phakic intraocular lenses are frequently calculated using fixed ratios of phakic lens refractive power to refractive spherical equivalent, and of phakic lens astigmatism to refractive cylinder. In this study, a Monte-Carlo simulation based on biometric data was used to investigate how variations in biometrics affect these ratios, in order to improve the calculation of implantable lens parameters.

METHODS

A data set of over sixteen thousand biometric measurements including axial length, phakic anterior chamber depth, and corneal equivalent and astigmatic power was used to construct a multidimensional probability density distribution. From this, we determined the axial position of the implanted lens and estimated the refractive spherical equivalent and refractive cylinder. A generic data model resampled the density distributions and interactions between variables, and the implantable lens power was determined using vergence propagation.

RESULTS

50 000 artificial data sets were used to calculate the phakic lens spherical equivalent and astigmatism required for emmetropization, and to determine the corresponding ratios for these two values. The spherical ratio ranged from 1.0640 to 1.3723 and the astigmatic ratio from 1.0501 to 1.4340. Both ratios are unaffected by the corneal spherical / astigmatic powers, or the refractive cylinder, but show strong correlation with the refractive spherical equivalent, mild correlation with the lens axial position, and moderate negative correlation with axial length. As a simplification, these ratios could be modelled using a bi-variable linear regression based on the first two of these factors.

CONCLUSION

Fixed spherical and astigmatic ratios should not be used when selecting high refractive power phakic IOLs as their variation can result in refractive errors of up to ±0.3 D for a 8 D lens. Both ratios can be estimated with clinically acceptable precision using a linear regression based on the refractive spherical equivalent and the axial position.

Calculation of equivalent and toric power in AddOn lenses based on a Monte Carlo simulation

BACKGROUND

Additional lenses implanted in the ciliary sulcus (AddOn) are one option for permanent correction of refractive error or generate pseudoaccommodation in the pseudophakic eye. The purpose of this paper was to model the power and magnification behaviour of toric AddOn and to show the effect sizes with a Monte-Carlo simulation.

METHODS

Anonymized data of a cataractous population uploaded for formula constant optimization were extracted from the IOLCon platform. After filtering out data with refractive spherical equivalent (RSEQ) between -0.75 to 0.25 dpt and refractive cylinder (RCYL) lower than 0.75, for each of the N=6588 redcords a toric AddOn was calculated which transfers the refraction error from spectacle plane to AddOn plane using a matrix-based calculation strategy based on linear Gaussian optics. The equivalent (AddOnEQ) and toric (AddOnCYL) power of the AddOn and the overall lateral magnification change and meridional magnification was derived for the situations before and after AddOn implantation, and a linear modelling was fitted for all 4 parameters.

RESULTS

RSEQ is the dominant effect size in the prediction of AddOnEQ and overall change in magnification (ΔM) , whereas the lens position (LP), corneal thickness (CCT) and mean corneal radius (CPa) play a minor role. In a simplified model AddOnEQ can be estimated by 0.0179 + 1.4104 · RSEQ. RCYL and corneal radius difference (CPad) are the dominant effect sizes in the prediction of AddOnCYL and the change in meridional magnification (ΔMmer) , whereas LP, CCT, CPa and RSEQ play a minor role. In a simplified model AddOnCYL can be predicted by -0.0005+ 0.0328 · CPad + 1.4087 · RCYL. Myopic eyes gain in overall magnification whereas in hyperopic eyes we observe a loss. Meridional distortion could be in general reduced to 35% on average with a toric AddOn.

CONCLUSION

Our simulation shows that with a linear model the equivalent and toric AddOn power as well as overall change in magnification, meridional distortion before and after AddOn implantation as well as the reduction in meridional distortion can be easily predicted from the biometric data in pseudophakic eyes with moderate refractive error.

The Castrop formula for calculation of toric intraocular lenses

Purpose

To explain the concept behind the Castrop toric lens (tIOL) power calculation formula and demonstrate its application in clinical examples.

Methods

The Castrop vergence formula is based on a pseudophakic model eye with four refractive surfaces and three formula constants. All four surfaces (spectacle correction, corneal front and back surface, and toric lens implant) are expressed as spherocylindrical vergences. With tomographic data for the corneal front and back surface, these data are considered to define the thick lens model for the cornea exactly. With front surface data only, the back surface is defined from the front surface and a fixed ratio of radii and corneal thickness as preset. Spectacle correction can be predicted with an inverse calculation.

Results

Three clinical examples are presented to show the applicability of this calculation concept. In the 1st example, we derived the tIOL power for a spherocylindrical target refraction and corneal tomography data of corneal front and back surface. In the 2nd example, we calculated the tIOL power with keratometric data from corneal front surface measurements, and considered a surgically induced astigmatism and a correction for the corneal back surface astigmatism. In the 3rd example, we predicted the spherocylindrical power of spectacle refraction after implantation of any toric lens with an inverse calculation.

Conclusions

The Castrop formula for toric lenses is a generalization of the Castrop formula based on spherocylindrical vergences. The application in clinical studies is needed to prove the potential of this new concept.

[Monte Carlo simulation of biometric effect sizes and their influence on the translational ratio of corneal astigmatism in the cylinders of toric intraocular lenses]

Hintergrund und Zielsetzung

Torische Kapselsacklinsen bieten heutzutage eine zuverlässige Option der permanenten Korrektur eines Hornhautastigmatismus. Zur Ermittlung der für den gewünschten Ausgleich erforderlichen Linsenstärke kann der Operateur entweder auf die in seinem Biometriegerät implementierten Berechnungsmodi oder auf den vom Linsenhersteller angebotenen Kalkulationsservice zurückgreifen. In vielen Fällen wird dabei allerdings keine klassische Linsenberechnung aus biometrischen Daten durchgeführt, sondern nur mit einer vereinfachten Abschätzung gearbeitet, die den Hornhautastigmatismus in den Torus der tIOL übersetzt. Dieses dann zumeist als durchschnittlicher Standardwert genutzte Übersetzungsverhältnis kann jedoch eine erhebliche Schwankungsbreite aufweisen, sodass im ungünstigsten Fall eine Unterkorrektur des refraktiven Zylinders um bis zu 12,5 % oder eine Überkorrektur um bis zu 17 % resultieren kann. Ziel dieser Studie war es aufzuzeigen, welche biometrischen Einflussgrößen das Verhältnis zwischen dem zu korrigierenden Hornhautastigmatismus und dem für dessen Vollkorrektur notwendigen Torus einer Kapselsacklinse bestimmen.

Methoden

Aus der WEB-Plattform IOLCon wurden 16.744 Datensätze extrahiert, und anhand der präoperativen biometrischen Größen und dem postoperativen sphärischen Äquivalent wurde zunächst die axiale Position der Kapselsacklinse formelunabhängig abgeleitet. Anschließend wurde, basierend auf der Propagation sphärozylindrischer Vergenzen, der entsprechende Brechwert einer emmetropisierenden Kapselsacklinse ermittelt. Das Übersetzungsverhältnis als Quotient aus dem Torus der Linse und dem Hornhautastigmatismus wurde mit einer Monte-Carlo-Simulation auf seine potenziellen Einflussgrößen hin untersucht.

Ergebnisse

Die Monte-Carlo-Simulation zeigt, dass nicht von einem konstanten Übersetzungsverhältnis ausgegangen werden kann. Für die hier zugrunde gelegten klinischen Fälle ergibt sich ein mittleres Übersetzungsverhältnis von 1,3938 ± 0,0595 (Median 1,3921) mit einer Spannweite von 1,2131 bis 1,5974. Den größten Einfluss hat hierbei die axiale Position der Kapselsacklinse – je weiter posterior sich diese befindet, desto höher ist das Übersetzungsverhältnis. Aufgrund der Korrelation der axialen Linsenposition mit der Augenlänge kann die Augenlänge als indirekte Einflussgröße gewertet werden. Der Äquivalentbrechwert sowie der Astigmatismus der Hornhaut besitzen keinen nennenswerten Effekt auf das Übersetzungsverhältnis.

Diskussion

In einer ganzen Reihe von Berechnungsmodulen wird die Kalkulation des Torus der Kapselsacklinse dahingehend vereinfacht, dass dieser mittels eines einfachen konstanten Umrechnungsfaktors aus dem gemessenen Hornhautastigmatismus abgeleitet wird. Die vorliegende Studie zeigt jedoch, dass diese Vereinfachung zu deutlich fehlerhaften Ergebnissen führen kann. Dementsprechend wird eine individuelle Berechnung des Torus der IOL aus gemessenen biometrischen Größen (z. B. mittels Vergenzpropagation, Matrizen oder mittels Full-aperture-Raytracing) empfohlen.

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.

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.

[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.

Assessment and statistics of surgically induced astigmatism

The aim of the thesis was to develop methods for assessment of surgically induced astigmatism (SIA) in individual eyes, and in groups of eyes. The thesis is based on 12 peer-reviewed publications, published over a period of 16 years. In these publications older and contemporary literature was reviewed(1). A new method (the polar system) for analysis of SIA was developed. Multivariate statistical analysis of refractive data was described(2-4). Clinical validation studies were performed. The description of a cylinder surface with polar values and differential geometry was compared. The main results were: refractive data in the form of sphere, cylinder and axis may define an individual patient or data set, but are unsuited for mathematical and statistical analyses(1). The polar value system converts net astigmatisms to orthonormal components in dioptric space. A polar value is the difference in meridional power between two orthogonal meridians(5,6). Any pair of polar values, separated by an arch of 45 degrees, characterizes a net astigmatism completely(7). The two polar values represent the net curvital and net torsional power over the chosen meridian(8). The spherical component is described by the spherical equivalent power. Several clinical studies demonstrated the efficiency of multivariate statistical analysis of refractive data(4,9-11). Polar values and formal differential geometry describe astigmatic surfaces with similar concepts and mathematical functions(8). Other contemporary methods, such as Long's power matrix, Holladay's and Alpins' methods, Zernike(12) and Fourier analyses(8), are correlated to the polar value system. In conclusion, analysis of SIA should be performed with polar values or other contemporary component systems. The study was supported by Statens Sundhedsvidenskabeligt Forskningsråd, Cykelhandler P. Th. Rasmussen og Hustrus Mindelegat, Hotelejer Carl Larsen og Hustru Nicoline Larsens Mindelegat, Landsforeningen til Vaern om Synet, Forskningsinitiativet for Arhus Amt, Alcon Denmark, and Desirée and Niels Ydes Fond.

Modeling of lateral magnification changes due to changes in corneal shape or refraction

BACKGROUND AND PURPOSE

Especially after corneal surgery the lateral magnification of the eye providing the retinal image size of an object is a crucial factor influencing visual acuity and binocularity. The purpose of this study is to describe a paraxial computing scheme calculating lateral magnification changes (ratio of the image sizes before and after surgery) due to variation in corneal shape and spectacle refraction.

CALCULATION STRATEGY

From the 4 x 4 refraction and translation matrices the system matrix representing the entire 'optical system eye' and the pupil matrix describing the sub-system from the spectacle correction to the aperture stop were defined for the state before and after surgery. As the chief ray is assumed to pass through the centre of the aperture stop, the 2 x 2 matrix of the lateral magnification ratio from preoperative to postoperative is described by the 2 x 2 sub-matrices of the respective pupil matrices. The cardinal meridians can be extracted by calculating the eigenvalues and eigenvectors.

WORKING EXAMPLE

Vertex distance 14 mm, measured distance between corneal apex and aperture stop 3.6mm, keratometry 39 D+6D/0 degrees to 47D+3D/30 degrees and refraction 3.5D-5-5D/5 degrees to -4.0 D-3.5D/25 degrees preoperatively to postoperatively. The matrix of magnification ratio from preop to postop yields (0.8960 -0.0085;0.0074 0.9371) and the eigenvalues decomposition provided a 10.7% minified image at 170.1 degrees and a minified image of 6.1% at 78.7 degrees , which both are clinically relevant.

CONCLUSION

We presented a straight-forward computer-based strategy for calculation of retinal image size changes using 4 x 4 matrix notation. With this model the meridional changes in lateral magnification from the preoperative to the postoperative stage or between follow-up stages can be estimated from keratometry, refraction, vertex distance and anterior chamber depth, which might be important for binocularity and vision tests in corneal surgery.

Calculating the power of toric phakic intraocular lenses

BACKGROUND AND PURPOSE

A toric phakic intraocular lens (IOL) implanted in the anterior or posterior chamber of the eye has the potential to correct high or excessive ametropia and astigmatism with high predictability of the postoperative refraction and preservation of phakic accommodation. The calculation of spherical phakic lenses has been described previously, but a formalism for estimating the power of toric phakic lenses has not yet been published. The purpose of this study is to describe a mathematical strategy for calculating toric phakic IOLs.

METHODS

The method presented in this paper is based on vergence transformation in the paraxial Gaussian space. Parameters used for the calculations are the spherocylindrical spectacle refraction before implantation, corneal power (sphere and astigmatism) and (spherocylindrical) target refraction, together with the vertex distance and the predicted position of the phakic IOL. The lens power is determined as the difference in vergences between the spectacle-corrected eye and the uncorrected eye at the reference plane of the predicted lens position. The axes of the preoperative refraction, the target refraction and the corneal astigmatism are at random (not necessarily aligned).

RESULTS

The method was applied to two clinical examples. In example 1 we calculate the power of a phakic lens for the simple case, when the target refraction is plano and the axis of the preoperative refraction is aligned to the axis of the corneal astigmatism. In example 2, the cylindrical axis of the preoperative refraction is not aligned to the corneal astigmatism and the target refraction is spherocylindrical (and the axis of the target refraction is not aligned to the preoperative refractive cylinder or the corneal astigmatism). The calculations for both examples are described step-by-step and illustrated in a table.

CONCLUSIONS

The calculation scheme can be generalized to an unlimited number of crossed cylinders in the optical pathway. Based on paraxial raytracing, the spherical and cylindrical power as well as the orientation of the cylinder are determined from the preoperative refraction (including vertex distance), the corneal power, the intended target refraction (including vertex distance) and the predicted position of the phakic lens implant provided by the lens manufacturer. This calculation scheme can be easily implemented in a simple computer program (i.e. in Microsoft excel or matlab).

Evaluation of a toric intraocular lens with a Z-haptic

PURPOSE

To evaluate the efficacy and rotational stability of the MicroSil 6116TU foldable 3-piece silicone toric intraocular lens (IOL) (HumanOptics).

SETTING

Department of Ophthalmology, Hillingdon Hospital, Uxbridge, Middlesex, United Kingdom.

METHODS

This prospective observational study included 21 eyes of 14 consecutive patients with more than 1.50 diopters (D) of preexisting corneal astigmatism having cataract surgery. Phacoemulsification was performed, and a MicroSil 6116TU toric IOL was inserted through a 3.4 mm temporal corneal incision. LogMAR uncorrected visual acuity (UCVA), best corrected visual acuity, refraction, keratometry, and cylinder axis of the toric IOL were measured.

RESULTS

The mean preoperative refractive and keratometric astigmatism was 3.52 D +/- 1.11 (SD) and 3.08 +/- 0.76 D, respectively. Six months postoperatively, the logMAR UCVA in eyes without ocular comorbidity (n = 14) was 0.20 +/- 0.15 (Snellen 20/32). Seventy-nine percent (11 eyes) had a visual acuity of 0.24 (Snellen 20/35) or better. The mean refractive astigmatism at 6 months was 1.23 +/- 0.90 D. Vector analysis using the Holladay-Cravy-Koch method showed a mean reduction in refractive astigmatism of 2.16 +/- 2.33 D. The mean difference between intended and achieved cylinder axis at 6 months was 5.2 degrees (range 0 to 15 degrees). No IOL rotated more than 5 degrees during the follow-up period.

CONCLUSIONS

The MicroSil 6116TU toric IOL reduced visually significant keratometric astigmatism and increased spectacle independence. The IOL was stable in the capsular bag, showing no significant rotation up to 6 months postoperatively.