**Purpose**:
To describe a formula to back-calculate the theoretical position of the principal object plane of an intraocular lens (IOL), as well as the theoretical anatomic position in a thick lens eye model. A study was conducted to ascertain the impact of variations in design and IOL power, on the refractive outcomes of cataract surgery.

**Methods**:
A schematic eye model was designed and manipulated to reflect changes in the anterior and posterior radii of an IOL, while keeping the central thickness and paraxial powers static. Modifications of the shape factor (X) of the IOL affects the thick lens estimated effective lens position (ELP). Corresponding postoperative spherical equivalent (SE) were computed for different IOL powers (–5 diopters [D], 5 D, 15 D, 25 D, and 35 D) with X ranging from –1 to +1 by 0.1.

**Results**:
The impact of the thick lens estimated effective lens position shift on postoperative refraction was highly dependent on the optical power of the IOL and its thickness. Design modifications could theoretically induce postoperative refraction variations between approximately 0.50 and 3.0 D, for implant powers ranging from 15 D to 35 D.

**Conclusions**:
This work could be of interest for researchers involved in the design of IOL power calculation formulas. The importance of IOL geometry in refractive outcomes, especially for short eyes, should challenge the fact that these data are not usually published by IOL manufacturers.

**Translational Relevance**:
The back-calculation of the estimated effective lens position is central to intraocular lens calculation formulas, especially for artificial intelligence–based optical formulas, where the algorithm can be trained to predict this value.

^{1}The quality of the patient‘s postoperative vision largely depends on predictable selection of the intraocular lens (IOL)’s optical power, which influences the postoperative refraction. Precise biometric measurements and accurate IOL power calculation methods are required. The adequate collection and use of data in modern healthcare provides an opportunity for significant improvements within a wide range of sub-specialties.

^{2}Recently, IOL calculation approaches using artificial intelligence have shown improving performances, even though the superiority of artificial intelligence to previous methods still remains controversial.

^{3}

^{–}

^{10}Although the inner principles of the most recent IOL formulas have not been published, most of them are based on optical formulas, with only a few described as purely artificial intelligence based.

^{11}Regardless of the algorithms that an optical formula uses, the back-calculation of the estimated effective position of the lens (ELP) is necessary to train the algorithms to predict this value.

^{12}Supervised learning allows algorithms to be trained in predicting an outcome from labelled feature recognition, with increasing dataset leading to increased accuracy. Although it is possible to know the anatomic position of the IOL by imaging the anterior segment,

^{13}

^{–}

^{15}this information is not routinely available within large training datasets. These generally include information limited to the preoperative ocular biometry, and final refractive outcome achieved (spherical equivalent [SE]). The back-calculation of the “matching” estimated ELP can be performed systematically; this value integrates by definition within the same parameter, all the errors induced by the underlying assumptions made in the eye model.

^{3}

^{,}

^{4}Datasets typically contain the preoperative ocular biometric parameters including anterior corneal curvature radius, corneal thickness, anterior chamber depth (measured from epithelium to lens), lens thickness, and corneal diameter, as well as the axial length. The postoperative refraction of each eye is provided as the value of the SE in the spectacle plane. Because the IOL surface curvature and thickness vary with power, the geometric characteristics of the inserted IOL such as anterior and posterior radii (R

_{ai}and R

_{pi}), thickness (d

_{i}) and refractive index (n

_{i}) should also ideally be available. The posterior corneal radius can be measured by many current biometers, but was previously usually inferred from the anterior corneal radius using the keratometric index.

_{s}. The total corneal power is denoted D

_{c}. It can be obtained from the value of the anterior and posterior radii of curvature (R

_{ca}and R

_{cp}), and from the refractive indices of air, stroma and aqueous humor (n

_{a}). The distances between the anterior and posterior vertices of the cornea with the main object planes and images of the cornea can be calculated from these values (see Appendix A1).

_{i}), the index variations between that of the media in contact with these surfaces as well as that of the lens itself.

_{i}. It can be calculated from the characteristics of the implant: curvatures of the anterior and posterior surfaces (R

_{ia}and R

_{ip}), central thickness d

_{i}, refractive index and the refractive indices of aqueous humor and vitreous. The main object planes and images of the implant can also be calculated from these values (see Appendix A2).

_{e}and H’

_{e}of the entire eye (cornea + IOL) from the paraxial thick lens formulas (see Appendix A3).

*De*is expressed by the Gullstrand formula (thick lenses):

_{c}) from the principal object plane of the IOL (H

_{i}). Note that all algebraic distances must be converted in meters for numerical applications using the formulas presented in this article.

_{T}and the ALP of the IOL defined as the anterior IOL vertex position (ALP = S1S3) is expressed as follows:

_{,}where F’

_{e}is the back focal point of the paraxial schematic pseudophakic eye.

_{T}is equal to the anatomic axial length of the emmetropic eye decreased by the distance between the principal planes of the implant\({\rm{\;}}(\overline {{H_i}{{H^{\prime}}_i}} > 0)\) and the distance between the anterior surface of the cornea and the secondary principal plane of the cornea (\(\overline {{S_1}{{H^{\prime}}_c}} < 0\)).

_{T}and Dc. Figure 1 provides a geometric representation of the relationships between ELP

_{T}and ALP, and between

*AL*and

_{A}*AL*of this emmetropic pseudophakic eye.

_{T}_{T}

_{T}leads to the following explicit formula:

_{i}> 0, and by + when D

_{i}< 0.

_{t}, which would be obtained in a thin lens model where the cornea and IOL have null thickness. It is generally not identical to the position of the anterior vertex, posterior vertex, or the principal planes. In this scenario, S

_{1}S

_{2}= S

_{3}S

_{4}= 0, AL

_{T}= AL

_{A}.

_{T}value can be computed in an ametropic eye (SE ≠ 0) after replacing D

_{c}by D

_{ce}in Equation 8, where D

_{ce}is the sum of Dc and the vergence in the corneal plane of a spectacle lens of power equal to SE, placed at distance d from the corneal vertex (ignoring the distance \(\overline {{S_1}{H_c}} \)).

_{T}

_{ia}→ ∞, planar posterior IOL) and 0 (R

_{ip}→ ∞, planar anterior IOL).

^{12}For the same ALP, optical power, and central thickness, a variation in the shape factor results in an axial displacement of the principal planes of the IOL (see in Appendix A2 Equations A6 and A7, which depend on Rip and Ria, respectively). This variation also affects the length of the segment \(\overline {{S_3}{H_i}} \) , which is involved in the calculation of ELP

_{T}(Equation 8).

_{T}on the Postoperative Refraction

_{c}provides the expected total corneal power D

_{ce}, for which the considered eye is emmetropic when an IOL of power D

_{i}is positioned so that its object principal plane H

_{i}is located at distance ELP

_{T}from the image principal plane of the cornea.

_{T}( \(\overline {H{{\rm{^{\prime}}}_c}{H_i}} \) ) and ALP (\(\overline {{S_1}{S_3}} \)) From a Known Dataset

_{ce}needed to achieve emmetropia, using Equation 9. The AL

_{T}is computed from Equation 6. Finally, we obtain the value of the ELP

_{T}using Equation 8. Here we provide a numerical example, for which a comprehensive description of the corneal and IOL design parameters is available.

- D
_{c}= 43.23 D (Equation A1), D_{i}= 20 D (Equation A5), D_{ce}= 42 D (Equation 9), \(\overline {{S_1}{{H^{\prime}}_c}} \) = 0.0522 mm (Equation A2), \(\overline {{H_i}{{H^{\prime}}_i}} \) = –0.0892 mm (Equation A8), AL_{T}= 23.513 mm (Equation 6) - Using Equation 8, we finally obtain: ELP
_{T}= 3.758 mm. Equation 3b allows to get the physical position of the IOL, defined as the distance between the corneal and IOL vertices: ALP = \(\overline {{S_1}{S_3}} \) = 3.210 mm

_{T}from its intended plane, caused by an IOL's design variation.

_{ia}and R

_{ip}(see Appendix C), but the anterior vertex and posterior vertex were kept at the same distance from the corneal vertex (\(\overline {{S_1}{S_3}} \) and IOL thickness were kept constant regardless of the value taken by the Coddington shape factor). The corneal parameters were identical for each of these theoretical examples. For each tested IOL configuration, the impact of a variation of the X shape factor on the ELP

_{T}was computed using Equation A6 and Equation 3. For each of the selected IOL powers, the axial length was adjusted to induce emmetropia for a biconvex symmetrical IOL (X = 0) using Equation 7b. For positive powers, the central thickness of the IOL was determined so that the thickness of the 6 mm diameter optic at the haptic junction was 250 ± 5 microns.

_{T}on the SE was computed for different IOL powers (–5 D, 5 D, 15 D, 25 D, and 35 D) with X varying from –1 to +1 by 0.1 steps using the method described in 1.5 of the Methods section. The theoretical locations of the ELP

_{T}for X = –1, X = 0, X = +1, and of the ELP

_{t}(corresponding with the position of an IOL of null thickness, making the considered eye emmetropic) were plotted for each of the computations (Figs. 3a through 3e).

^{16}

^{,}

^{17}

^{,}

^{18}The estimation of postoperative IOL position is essential to IOL power calculations for cataract surgery and is also a critical variable in ray tracing. Similarly, the prediction of the ELP is an important issue to improve the refractive precision of the calculation of the IOL power with machine learning. Holladay et al.

^{19}were the first to publish an explicit formula for calculating the effective position of the implant in a thin lens model. Later, they discussed the relationship between the ELP of a thin versus thick lens of equivalent power.

^{20}At the time of this pioneering work, it was not routine practice to measure the posterior surface of the cornea, which was considered as a single dioptric surface. The net optical power of the cornea was obtained using a fictitious refractive index of the cornea equal to 4/3.

^{21}The determination of the ELP is formula dependent and does not need to reflect the real postoperative IOL position in the anatomic sense. However, knowing the geometry of the IOL makes it possible using a thick lens paraxial model to relate the optically estimated position of an IOL to its estimated anatomic position. Fernández et al.

^{22}studied the relationship between the measured ALP and the back-calculated ELP and demonstrated the differences between both values, induced by assumptions made in theoretical eye models.

_{t}in our model) does not provide any direct information about the position of the actual thick lens within the eye. Conversely, in a thick lens eye model, the ELP corresponds to the distance between the principal image plane of the cornea and the principal object plane of the IOL. In our model, this distance was labeled “ELP

_{T}” and appears in the third term of the Gullstrand equation. For given corneal and IOL total powers, it suffices to know the ELP

_{T}value to determine the paraxial refractive properties of that dual optics refracting component. Once these properties are known, it is necessary to know the axial length to predict the refraction of the eye in consideration. In our thick lens schematic eye model, the axial length considered for the calculation using Equation 7 is altered as it is augmented by the segment connecting the corneal vertex to the principal corneal image plane \(\overline {{S_1}H{{\rm{^{\prime}}}_c}} \) and decreased by the interstice between the IOL's principal plane \(\overline {{H_i}H{{\rm{^{\prime}}}_i}} \). This transformation is expected in our paraxial model, because it corresponds with the suppression of the interstice between the principal planes of the system. Owing to the small dimensions of these algebraic segments and their opposite sign, this operation may have no clinically significant consequences on the results of numerical calculations involving the axial length. For a given pseudophakic eye, Equation 7 can be used to back-calculate an optimized axial length, which produces a refractive prediction error of zero, in the context of regression calculus such as that which has been used to improve the IOL power calculations in eyes with axial lengths of more than 25 mm.

^{23}

_{T}shift is highly dependent on the optical power of the IOL as suggested by the Gullstrand equation. As the thickness of the implants increases with their power, this tendency is increased, and we calculated that these design variations could induce postoperative refraction variations between approximately 0.50 and 3.0 D, for implant powers ranging from 15 D to 35 D. Therefore, the use of high index, thinner implants reduce the influence of variations in optical design. We did not explore the influence of the corneal geometry and power in the ELP

_{T}prediction. Using paraxial matrix optical calculations, Schröder and Langenbucher

^{24}have studied the relationship between the thin lens predicted ELP and axial position of a thick IOL achieving the same refraction (respectively referred to as ELP

_{t}and ALP in the present work). They found that the corneal power had less influence than the lens power and design, on the difference between the ELP and the ALP. In all scenarios, the ALP was shorter than the ELP, which was confirmed in the present work where we also found that the ALP of a thick lens must be placed in front of the thin and thick ELP locations for achieving the same refraction. We limited our analysis to biconvex lenses, although these authors included concave–convex minus powered IOLs. Although the latter did increase the discrepancy between ALP and ELP, the influence of variations in the ELP

_{T}distance on postoperative refraction appears to be relatively weak and not clinically significant for low power implants, even for negative implants, whose convex concave design could induce a more marked variation between the position of their vertices and the plane of the ELP

_{T}. The improvement of the calculation formulas for long eyes requires other adjustments than those related to the actual position of the implant, which rather concern the measurement of the posterior segment of the eye and the historical assumptions used by current biometers to infer the axial length from the measured optical path length.

^{25}

^{,}

^{26}This finding is reflected in a recent study

^{27}in which there was no significant difference in the accuracy of thick lens IOL power formula based on calculated versus manufacturer's IOL data for eyes with ALs of 22 mm and more. Fernández et al.

^{22}suggested modifying the refractive index of the cornea to correct errors beyond the ELP prediction, including assumptions from the biometers.

^{14}

^{,}

^{28}

^{–}

^{30}In a work evaluating the prediction accuracy of ELP after cataract surgery using a multiobjective evolutionary algorithm,

^{31}the ELP was defined as the distance from the cornea to the anterior surface of the IOL at 3 months after surgery plus the distance to the presumed principal object point of the IOLs. The predicted ELP was obtained by a multiobjective evolutionary algorithm and multiple regression analysis, including a dozen parameters, and the difference between the predicted value and measured value (prediction accuracy) was compared between the two methods. The study demonstrated that ELP prediction by a multiobjective evolutionary algorithm was more accurate and was a method with less fluctuation than that of stepwise multiple regression and conventional formulas such as SRK/T and Haigis formulas.

^{32}

^{,}

^{33}This might be even more important in eyes that have undergone corneal refractive surgery.

^{34}

^{35}This makes it possible to study the theoretical influence of factors linked to the design of the implant, on its effective position and the potential refractive variations that may result from it. This article aimed to provide specialists interested in the field of implant calculation and related problems, with a basis of explicit equations intended to solve numerically, problems related to dual thick lens optics with four refracting surfaces. It also underlines the critical importance of IOL geometry in IOL formulas training and calculation processes, especially for short eyes. This data is not usually released by the manufacturers; in an era where most of the optical parameters of the eye can be accurately measured or predicted, this consensus should be challenged.

^{36}to improve the accuracy of models useful for theoretical formulas and machine learning algorithms for IOL power calculation.

**D. Gatinel,**None;

**G. Debellemanière,**None;

**A. Saad,**None;

**M. Dubois,**None;

**R. Rampat,**None

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_{ca}is the radius of curvature of the anterior cornea, R

_{cp}is the radius of curvature of the posterior cornea, n

_{s}is the refractive index of stroma, n

_{a}is the refractive index of aqueous, and d

_{c}is the corneal thickness.

_{s}= 1.376, n

_{a}= 1.337, d

_{c}= 0.545 mm (0.00545 m) R

_{a}= 7.75 mm (0.00775 m), R

_{p}= 6.45 mm (0.00645 m) → D

_{c}= 42.59 D

_{ss}is given by:

_{i}is the labeled IOL power, n

_{i}is the refractive index of the IOL, d

_{i}is the central thickness of the IOL, R

_{ia}is the radius of curvature of the anterior surface, R

_{ip}is the radius of curvature of the posterior surface, n

_{i}is the refractive index of the IOL, n

_{a}is the refractive index of aqueous, and d

_{i}is the IOL thickness.

_{i}= 1.52, n

_{ss}= 1.335, d

_{i}= 0.7 mm, R

_{ia}= 23.97 mm, R

_{ip}= –12.91 mm, we get D

_{i}= 22 D

_{i}may change because of the changes in refractive index. Using the (previous) equation with n

_{a}= 1.337 and n

_{v}= 1.336 appropriately substituted to n

_{ss}we get D

_{i}= 21.84 D.

_{c}and the principal object plane of the IOL H

_{i}. This distance corresponds with the ELP in a thick lens paraxial pseudophakic eye model, denoted ELP

_{T}in the manuscript. Hence, the total power of the eye is given by:

_{T}= \(\overline {H{{\rm{^{\prime}}}_c}{H_i}} \) and \(\overline {H{{\rm{^{\prime}}}_i}H{{\rm{^{\prime}}}_e}} = - \overline {H{{\rm{^{\prime}}}_c}{H_i}} \frac{{{n_v}{D_c}}}{{{n_a}{D_e}}}\)

_{ip}by this expression in Equation A5 produces an expression which allows computation of the value of R

_{ia}for a specific IOL power D

_{i}and Coddington shape factor X. The corresponding value of R

_{ip}is obtained using Equation C2.