Article Text

AI-powered effective lens position prediction improves the accuracy of existing lens formulas
  1. Tingyang Li1,
  2. Joshua Stein2,3,4,
  3. Nambi Nallasamy1,2
  1. 1Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, Michigan, USA
  2. 2Kellogg Eye Center, Department of Ophthalmology and Visual Sciences, University of Michigan, Ann Arbor, Michigan, USA
  3. 3Department of Health Management and Policy, University of Michigan School of Public Health, Ann Arbor, Michigan, USA
  4. 4Center for Eye Policy and Innovation, University of Michigan, Ann Arbor, Michigan, USA
  1. Correspondence to Dr. Nambi Nallasamy, Kellogg Eye Center, University of Michigan, 1000 Wall St, Ann Arbor, MI 48105, USA; nnallasa{at}


Aims To assess whether incorporating a machine learning (ML) method for accurate prediction of postoperative anterior chamber depth (ACD) improves the refraction prediction performance of existing intraocular lens (IOL) calculation formulas.

Methods A dataset of 4806 patients with cataract was gathered at the Kellogg Eye Center, University of Michigan, and split into a training set (80% of patients, 5761 eyes) and a testing set (20% of patients, 961 eyes). A previously developed ML-based method was used to predict the postoperative ACD based on preoperative biometry. This ML-based postoperative ACD was integrated into new effective lens position (ELP) predictions using regression models to rescale the ML output for each of four existing formulas (Haigis, Hoffer Q, Holladay and SRK/T). The performance of the formulas with ML-modified ELP was compared using a testing dataset. Performance was measured by the mean absolute error (MAE) in refraction prediction.

Results When the ELP was replaced with a linear combination of the original ELP and the ML-predicted ELP, the MAEs±SD (in Diopters) in the testing set were: 0.356±0.329 for Haigis, 0.352±0.319 for Hoffer Q, 0.371±0.336 for Holladay, and 0.361±0.331 for SRK/T which were significantly lower (p<0.05) than those of the original formulas: 0.373±0.328 for Haigis, 0.408±0.337 for Hoffer Q, 0.384±0.341 for Holladay and 0.394±0.351 for SRK/T.

Conclusion Using a more accurately predicted postoperative ACD significantly improves the prediction accuracy of four existing IOL power formulas.

  • lens and zonules
  • treatment surgery

Data availability statement

Data may be obtained from the Sight Outcomes Research Collaborative (SOURCE) repository for participating institutions and are not publicly available.

This is an open access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited, appropriate credit is given, any changes made indicated, and the use is non-commercial. See:

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The estimation of postoperative intraocular lens (IOL) position is essential to IOL power calculations for cataract surgery. Norrby and Olsen have reported that inaccuracy in the prediction of the postoperative anterior chamber depth (ACD) is the number one source of error for postoperative refraction prediction.1 2 In addition to its vital role in IOL formulas, the postoperative ACD is also a critical variable in ray tracing, where the uncertainty in the postoperative ACD directly affects the accuracy of the results. Methods to improve the accuracy of the prediction of postoperative ACD have been studied for decades. In first-generation formulas, the lens position was represented by a constant. Later, more and more preoperative biometric variables such as the axial length (AL) and the corneal power were added to calculate the postoperative IOL position. In 1993, Holladay first proposed the term ‘expected lens position’ or ELP to indicate the location of the lens as it relates to a given optical model of the eye.3 The ELP estimates in SRK/T, Holladay1 and Hoffer Q are derived based on theoretical formulas. The ELP estimate in the Haigis formula is a simple linear combination of the AL and the preoperative ACD. Although ELP was initially intended to estimate the position of the IOL, ELPs in the aforementioned formulas were developed to account for different formula-specific assumptions and regression results.1 4 In order to reflect the use of ELP to account for these formula-specific assumptions and regression results, the term ELP today refers to ‘effective lens position’ rather than ‘expected lens position’. In view of the limitations of the ELP in existing formulas, recently, more efforts have been devoted to constructing ELPs that better reflect the true location of the IOL.5–9 New IOL power prediction methods have also been developed based on the new-generation ELP prediction methods, and they have shown that using a more accurately predicted IOL position helps to improve the IOL power prediction accuracy.5

It is so far largely unexplored whether inserting a more accurately predicted ELP into existing formulas improves refraction prediction accuracy. This is an important question because: (1) it provides a fast and efficient way to modify and improve on existing IOL formulas whose reliability has been tested extensively; (2) such research can provide supports for translating the continued improvements in accuracy in postoperative ACD prediction into better refraction predictions in published formulas. Several previous studies had modified the ELPs in existing formulas in order to achieve better refraction prediction results in certain cataract cases. Modification of ELP calculation in the Haigis formula for sulcus-implanted IOLs was reported to improve performance.10 Kim et al adjusted the ELP estimation in SRK/T formulas with the corneal height in postrefractive patients and achieved satisfactory accuracy.11 It remains to be explored whether improvement of ELP estimates for in-the-bag IOL placement can improve IOL power calculations of existing formulas for general cataract patients.

Since most recently published IOL formulas (eg, Barrett Universal II,12 13 Holladay 2, Olsen formula14) are either not disclosed to the public or do not have the option to customise the value of ELP during the prediction of postoperative refraction, here we applied our previously developed postoperative ACD prediction methods to a dataset of 4806 cataract surgery patients and replaced the ELP estimates in 4 existing IOL formulas: Haigis, Hoffer Q, Holladay and SRK/T. We combined our machine learning (ML) prediction of true postoperative ACD with the original ELP estimated by each formula and substituted this updated ELP prediction for each formula. We then compared the refraction prediction performance of each formula using its original and enhanced ELP estimates. The findings reported here demonstrate that existing formulas can benefit from improved methods for predicting true postoperative ACD.

Materials and methods

Postoperative ACD prediction ML model

In previous work,15 we developed an ML-based postoperative ACD prediction model, which predicts the postoperative ACD (in mm) based on preoperative biometry. Here, in the presented study, an ACD prediction ML model was trained using the method and dataset (847 patients, 1205 eyes, 4137 records) described in the previous research. The dataset was composed of the preoperative and postoperative biometry measured by the Lenstar LS900 optical biometers (Haag-Streit USA, EyeSuite software V.i9.1.0.0) at the University of Michigan’s Kellogg Eye Center. The postoperative ACD was defined as the distance from the front surface of the cornea to the front surface of the IOL. The postoperative ACD predicted by the ML model is referred to as Embedded Image in this manuscript.

Data collection

In this study, biometry records were collected using the same approach as for the development of the ML postoperative ACD prediction model at University of Michigan’s Kellogg Eye Center.15

The inclusion criteria were: (1) patients who had cataract surgery (Current Procedural Terminology (CPT) code=66 984 or 66 982) but no prior refractive surgery and no additional surgical procedures at the time of cataract surgery. (2) The implanted lens was an Alcon SN60WF single-piece acrylic monofocal lens (Alcon, USA). Each case in the dataset corresponds to one operation of a single eye with preoperative and postoperative information. The preoperative information includes the measurements of the AL, lens thickness (LT), ACD, flat keratometry (K1), steep keratometry (K2), and the average keratometry which was calculated as Embedded Image. The postoperative information includes the postoperative refraction (spherical component SC and cylindrical component CC) where the time when it was recorded was closest to 1 month (30 days) after surgery. Since the patients were measured in a lane of 10 feet long (3.048 m), which was shorter than the standard length of 20 feet (6 m), the SC was adjusted for the vergence distance by adding Embedded Image according to Simpson and Charman’s recommendation.16 The spherical equivalent (SE) refraction was therefore calculated as Embedded Image. Samples that were used to train the postoperative ACD prediction ML model were excluded from the dataset so that the dataset better simulates unseen samples.

The dataset in total consisted of 4806 patients (figure 1). The dataset was split into a training dataset used for the development of the methods and a testing dataset used for performance comparison. Eighty per cent of the patients were randomly assigned to the training set, and the rest of the patients (20%) were assigned to the testing set. For patients who had more than one associated case in the testing set (ie, patients who had both eyes operated on), one case was randomly selected to ensure each patient had the same weight when the prediction performance was evaluated. At the end of this process, the training set had 3845 patients (5761 eyes), and the testing set had 961 patients (961 eyes).

Figure 1

The analysis pipeline of the presented study. Embedded Image = the effective lens position (ELP) estimated by the existing formulas. Embedded Image = the postoperative anterior chamber depth (ACD) predicted by the machine learning method. Embedded Image = the back-calculated ELP (see main text). Embedded Image is a term that refers to a new ELP that is used to replace the Embedded Image in the existing formulas.

Linear regression model

We implemented four existing formulas (Haigis, Hoffer Q, Holladay, and SRK/T) in Python based on their publications.17–24 The existing formulas calculated the ELP (Embedded Image) as a function of the preoperative biometry (figure 1): Embedded Image. The predicted ELP (Embedded Image) was then used to predict the postoperative refraction: Embedded Image. Here, the goal was to reduce the refraction prediction error by replacing Embedded Image with a different value, Embedded Image. Our approach involves two steps: (1) finding the theoretically most optimal ELP values, (2) modelling the most optimal ELP with Embedded Image and the ML-predicted postoperative ACD, denoted Embedded Image.

In the first step, the most optimal ELP (denoted Embedded Image) was found by the standard method of back-calculating the ELP when the predicted refraction was set to equal the true refraction (ie, Embedded Image). In other words, when Embedded Image, the refraction prediction errors of all patients equal zero. More details on the computation of Embedded Image can be found in online supplemental materials.

After the computation of Embedded Image, Embedded Image, and Embedded Image we modelled Embedded Image using a linear function of Embedded Image and/or Embedded Image so as to obtain an approximation of the most optimal ELP using available variables. We compared four different approaches of approximating Embedded Image : (1) original, Embedded Image : using the original Embedded Image (2) Formula LR, Embedded Image : using linearly adjusted Embedded Image (3) ML LR, Embedded Image : using linearly adjusted Embedded Image (4) Formula & ML LR, Embedded Image using a linear combination of Embedded Image and Embedded Image. Here, Embedded Image Embedded Image, and Embedded Image are constants. Outliers with large refraction errors (ie, Embedded Image or Embedded Image) were excluded for each formula before establishing the linear regression model, in order to obtain better modelling results. The refraction prediction errors were calculated as Embedded Image. The linear regression was performed using scikit-learn 0.20.3.

On the testing set, Embedded Image was calculated based on the values of Embedded Image, Embedded Image, and Embedded Image obtained through linear regression. The predicted refraction was calculated as Embedded Image The mean absolute error (MAE), median absolute error (MedAE) and mean error (ME) were calculated for performance comparison.

A-constant optimisation

The A-constants for the formulas were optimised based on the training dataset so that the ME in refraction prediction was closest to zero. The A-constants were optimised separately for the unmodified formulas and formulas with a modified ELP estimate (see additional details in the A-constant optimisation section and online supplemental figure S1). The optimised A-constants for the original formulas were: a0=−0.733, a1=−0.234, a2=0.217 for Haigis, ACD constant=5.724 for Hoffer Q, surgeon factor=1.864 for Holladay, and A=119.089 for SRK/T (online supplemental table S1).

Statistical analysis

Linear regression analysis was used to assess the significance of the correlation between Embedded Image, Embedded Image, and Embedded Image. To test whether the MAE and ME of different methods were significantly different, a Friedman test followed by a post hoc paired Wilcoxon signed-rank test with Bonferroni correction was used. Statistical significance was defined as the p value<0.05. All the above analyses were performed with Python V.3.7.3.


Dataset overview

The cases in the training and testing datasets had a similar distribution according to the summary statistics shown in table 1. As elaborated in he Materials and methods section, we calculated Embedded Image, Embedded Image and Embedded Image based on the formulas and their optimised A-constants. The mean and SD of the ELPs calculated based on the original formulas were summarised in online supplemental table S2. Embedded Image and Embedded Image had similar mean values in contrast to Embedded Image.

Table 1

The summary statistics for the patient demographics for the training and testing dataset

The Pearson correlation coefficients (R) between Embedded Image Embedded Image and Embedded Image were shown in table 2. Three ELP-related variables were positively intercorrelated with each other. The correlation coefficients, R, between Embedded Image and Embedded Image were the weakest among the three pairs of variables across all formulas.

Table 2

The Pearson correlation coefficients (R) between Embedded Image, Embedded Image, andEmbedded Image

Linear regression results on the training set

Linear regression models were established based on the training set and the Embedded Image of alternative linear models were shown in table 3. The coefficients of the fitted linear regression line are shown in online supplemental table S3. The mean and SD of the Embedded Image resulting from different models are shown in online supplemental table S4. For ‘Formula LR’, the Embedded Image was larger than that of ‘ML LR’ for all four formulas. For ‘Formula & ML LR’, the Embedded Image was larger than that when one of Embedded Image or Embedded Image was excluded from the linear combination for all four formulas.

Table 3

The Embedded Image of alternative least-squares linear regression models in the training set

Refraction prediction performance comparison on the testing set

We tested the performance of four scenarios on the testing set and summarised the MAE and SD in table 4. The ME and MedAE were shown in online supplemental tables S5 and S6. Statistical tests were used to compare the difference in the MAEs of different models (see the Materials and methods section). Using a linear combination of Embedded Image and Embedded Image, the refraction prediction results of four existing formulas were significantly improved compared with original Embedded Image (statistical test results shown in online supplemental tables S7 and S8).

Table 4

Performance in the testing set

We further compared the MAEs of ‘Original’ and ‘Formula & ML LR’ among patients with short, medium and long AL (online supplemental table S9). It was observed that the short and medium AL groups had a higher percentage decrease in MAE than the long AL group for Hoffer Q and SRK/T. For Haigis, the medium AL group achieved higher decrease than the other two groups. And for Holladay, the long AL group achieved more decrease in MAE than the other two groups.


In this study, we applied a previously developed ML method for postoperative ACD prediction to an unseen dataset of 4806 cataract surgery patients to assess whether it was possible to improve the performance of existing IOL formulas (Haigis, Hoffer Q, Holladay, and SRK/T) by replacing each formula’s ELP estimate.

We computed three ELP-related quantities: the ML-predicted postoperative ACD (Embedded Image), formula-predicted ELP (Embedded Image), and a back-calculated ELP (Embedded Image) that minimised the refraction error for each eye in the dataset. They are strongly correlated with each other (table 2), which indicates that (1) Embedded Image and Embedded Image are both predictive of the most optimal ELP Embedded Image, (2) Embedded Image and Embedded Image contain partially overlapping information, which is consistent with our expectation. Embedded Image is an estimation of the value of the true postoperative ACD. On the other hand, Embedded Image was designed by the originators of each formula to serve a similar purpose but was based on the theoretical assumptions in each formula. Our findings are consistent with observations of previous studies that the ELP estimates made by IOL formulas were numerically different from the true postoperative ACD.9

Using a training dataset of 3845 patients, we sought to evaluate whether the machine-predicted postoperative ACD, Embedded Image, was able to provide information that could be used to refine each formula’s predicted ELP, Embedded Image. We established regression models between the Embedded Image, Embedded Image, and Embedded Image to evaluate whether a linear combination of Embedded Image and Embedded Image used in place of the original Embedded Image could lower the refraction prediction error. Using the modified ELPs, we obtained significantly lower MAEs in refraction prediction compared with the formulas with the original ELPs on the unseen testing set (table 4). Notably, the accurately predicted postoperative ACD (Embedded Image) alone did not outperform the original ELP (Embedded Image) when it was inserted into the formulas (table 4, row 3 compared with row 1). This is likely because the original method of calculating ELP in each formula compensates for its particular model of the eye and its associated assumptions. Our Embedded Image, however, does not have any components that compensate for the assumptions and constants in the formulas. On the other hand, Embedded Image has information about the true postoperative ACD, which it appears can beneficially alter the original ELP estimate.

In this study, the A-constants were optimised separately when Embedded Image was replaced with different Embedded Image. The means of Embedded Image, as shown in online supplemental table S4, were numerically close to those of Embedded Image as shown in online supplemental table S2. However, in our method, the similarity between Embedded Image and Embedded Image was not among the restrictions and goals of the optimisation. The reason that Embedded Image and the original Embedded Image had similar means might be that the other parts of each formula put restrictions on the values of ELP in order to obtain reasonable results. This could also be the reason why Embedded Image and Embedded Image had similar means as shown in online supplemental table S2.

Previous studies involving replacement of ELP in existing formulas have focused on special cases, such as sulcus implantation and postrefractive surgery eyes, where ELP estimates of traditional formulas would be expected to be inapplicable.10 11 However, the method for replacing ELP estimates presented here provides a simple way of improving the refraction prediction performance of existing formulas for the general cataract surgery population. While it would be ideal to evaluate this method on modern formulas such as Barrett Universal II or Holladay 2, the absence of published equations for these formulas prevents such a study. As such, we studied the application of the ML predicted postoperative ACD in four existing formulas whose mathematical equations were published. Although it awaits to be further validated, similar results can likely be transferred to other refraction prediction methods, since many modern IOL power formulas use predicted postoperative ACD as an intermediate step for predicting postoperative refraction. A limitation of the study was the absence of an external validation set, despite the use of a large unseen testing dataset (961 eyes). Accordingly, evaluation of the method at additional institutions and the extension to additional formulas will be future directions of this work.

In summary, the results of this study demonstrate that an ML method for postoperative ACD prediction based on postoperative optical biometry can be incorporated into a variety of existing IOL power formulas to improve their accuracy in refraction prediction.

Data availability statement

Data may be obtained from the Sight Outcomes Research Collaborative (SOURCE) repository for participating institutions and are not publicly available.

Ethics statements

Ethics approval

Institutional review board approval was obtained for the study, and it was determined that informed consent was not required because of its retrospective nature and the anonymised data used in this study. The study was carried out in accordance with the tenets of the Declaration of Helsinki.


Supplementary materials

  • Supplementary Data

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  • Contributors TL: data analysis, programming and writing of the manuscript. JS: data collection. NN: data collection, guidance on method development, and writing of the manuscript.

  • Funding This work was supported by the Lighthouse Guild, New York, NY (JDS) and National Eye Institute, Bethesda, MD, 1R01EY026641-01A1 (JDS).

  • Competing interests None declared.

  • Provenance and peer review Not commissioned; externally peer reviewed.

  • Supplemental material This content has been supplied by the author(s). It has not been vetted by BMJ Publishing Group Limited (BMJ) and may not have been peer-reviewed. Any opinions or recommendations discussed are solely those of the author(s) and are not endorsed by BMJ. BMJ disclaims all liability and responsibility arising from any reliance placed on the content. Where the content includes any translated material, BMJ does not warrant the accuracy and reliability of the translations (including but not limited to local regulations, clinical guidelines, terminology, drug names and drug dosages), and is not responsible for any error and/or omissions arising from translation and adaptation or otherwise.

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