Journal of Refractive Surgery

Original Article Supplemental Data

Application of a Partial Least Squares Regression Algorithm for Posterior Chamber Phakic Intraocular Lens Sizing and Postoperative Vault Prediction

Adam Oleszko, PhD; Jarosław Marek, MD, PhD; Maria Muzyka-Wozniak, MD, PhD

Abstract

PURPOSE:

To develop and validate a new algorithm for predicting the postoperative vault of the myopic EVO Visian Implantable Collamer Lens (ICL) V4c (STAAR Surgical AG).

METHODS:

This study included 81 eyes of 43 patients who had undergone ICL implantation. Preoperative data obtained by swept-source optical coherence tomography, Scheimpflug camera, and anterior segment optical coherence tomography were applied to develop a new partial least squares (PLS) regression algorithm. ICL sizing was performed using the standard white-to-white method with the online calculation and ordering system. The postoperative vault was assessed based on anterior segment optical coherence tomography. The PLS approach was applied to create the calibration model for predicting the postoperative vault. The new PLS model was cross-validated using the leave-one-out method and compared to a recently published linear regression model. Agreement between the actual and predicted vault values for the two methods was assessed by the Bland-Altman method.

RESULTS:

There was a statistically significant correlation (P < .001, r = 0.73) between the postoperative vault values and those predicted by the PLS algorithm. Validation of the PLS model yielded lower mean differences and limits of agreement (0 and 410 µm, respectively) than the linear regression method (400 and 750 µm, respectively).

CONCLUSIONS:

The PLS algorithm increases the precision of ICL vault prediction. However, it shows a tendency to overestimate small vault values and underestimate high vaults.

[J Refract Surg. 2020;36(9):606–612.]

Abstract

PURPOSE:

To develop and validate a new algorithm for predicting the postoperative vault of the myopic EVO Visian Implantable Collamer Lens (ICL) V4c (STAAR Surgical AG).

METHODS:

This study included 81 eyes of 43 patients who had undergone ICL implantation. Preoperative data obtained by swept-source optical coherence tomography, Scheimpflug camera, and anterior segment optical coherence tomography were applied to develop a new partial least squares (PLS) regression algorithm. ICL sizing was performed using the standard white-to-white method with the online calculation and ordering system. The postoperative vault was assessed based on anterior segment optical coherence tomography. The PLS approach was applied to create the calibration model for predicting the postoperative vault. The new PLS model was cross-validated using the leave-one-out method and compared to a recently published linear regression model. Agreement between the actual and predicted vault values for the two methods was assessed by the Bland-Altman method.

RESULTS:

There was a statistically significant correlation (P < .001, r = 0.73) between the postoperative vault values and those predicted by the PLS algorithm. Validation of the PLS model yielded lower mean differences and limits of agreement (0 and 410 µm, respectively) than the linear regression method (400 and 750 µm, respectively).

CONCLUSIONS:

The PLS algorithm increases the precision of ICL vault prediction. However, it shows a tendency to overestimate small vault values and underestimate high vaults.

[J Refract Surg. 2020;36(9):606–612.]

The wide range of refractive error correction and reversibility of phakic intraocular lens (IOL) implantation represents a major advantage over keratorefractive surgery. The EVO Implantable Collamer Lens (ICL) (STAAR Surgical AG) is a type of phakic IOL that is placed in the ciliary sulcus.1 An optimal ICL vault is important for long-term ICL safety, rotational stability, proper angle configuration, and undisturbed pupil function.2–5 The vault is the axial distance between the crystalline lens and the phakic IOL. An ideal ICL vault value is between 250 and 750 µm, but this depends mainly on the diameter of the implanted lens.2,6 There are four ICL sizes available: 12.1, 12.6, 13.2, and 13.7 mm.

The most common method of ICL sizing, as recommended by the manufacturer, is based on horizontal corneal diameter (white-to-white [WTW]) and anterior chamber depth (ACD). The major drawbacks of this method include the high variability of WTW measurements and the low correlation of WTW distance with the sulcus-to-sulcus (STS) distance.7–9

Direct measurement of STS distance is possible only with ultrasound biomicroscopy, which poses many difficulties in terms of measurement accuracy.10 The very high-frequency (VHF) digital ultrasound measurement of STS distance is a time-consuming and invasive procedure (water bath) that requires an experienced examiner and meticulous analysis of image alignment.11–13 In contrast to anterior segment optical coherence tomography (AS-OCT), access to VHF digital ultrasound is limited.

The AS-OCT technique allows for the quick, easy, and reproducible measurement of anterior chamber dimensions. The distance between the two opposing iridocorneal angles, angle-to-angle (ATA), shows a good correlation with the STS distance, but these two measurements cannot be used interchangeably.14 The difference between ICL diameter and ATA distance is known to be correlated with the postoperative vault after ICL implantation.15 This difference, as a single independent variable, can be applied with linear regression to predict the vault.15

Devices based on AS-OCT and the Scheimpflug principle are widely used in ophthalmology clinics, delivering multiple anterior segment parameters that can be used to better predict the ICL position. Partial least squares (PLS) regression analysis is used to model a response variable when there are many predictor variables. PLS analysis constructs new predictor variables, known as components, as linear combinations of the original predictor variables.16 PLS analysis is typically applied in spectroscopic quantitative analysis of chemical compounds.17,18 To date, the medical application of PLS analysis is limited to anatomy and geometric morphometrics of neurocranial growth.19

The aim of our study was to develop and validate a new PLS-based algorithm for predicting the postoperative ICL vault using anterior segment biometric parameters, axial length, ICL size, and refraction.

Patients and Methods

Study Population

The study was performed in compliance with the tenets of the Declaration of Helsinki and the research was reviewed and approved by the ethics committee of Wroclaw Medical University (no. KB-13/2019). Prior informed consent was obtained from all participants. This study enrolled patients who visited the hospital for pre-operative measurements prior to phakic IOL (EVO Visian ICL V4c) implantation in a private hospital.

Inclusion criteria for our study were as follows: age 21 to 45 years, presence of myopia with a manifest refraction spherical equivalent (SE) below −20.00 diopters (D), and astigmatism lower than 5.00 D. Patients were excluded from the analysis if they had previous ocular surgery, an ACD (defined as distance from the corneal endothelium to the lens) below 2.8 mm, corneal pathology, cataract, history of trauma, glaucoma, retinal detachment, diabetic retinopathy, macular degeneration, neuro-ophthalmic disease, or other contraindications according to the manufacturer's directions for use. Only individuals with good fixation were eligible.

Sample size calculation was performed as described elsewhere.20 To prove correlation between postoperative vault and PLS 10-factor simulated vault, 25 eyes were needed (at significance level α = 0.05 and test power β = 0.80). Vault standard deviation values (AS-OCT measured and PLS computed) were calculated based on data obtained from the first 15 eyes enrolled.

Examination and Calculation of ICL Size

Preoperative examination included subjective refractive error, corrected distance visual acuity, endothelial cell density, and biometry. The participants were examined using three devices: swept-source optical coherence topography (SS-OCT) (IOLMaster 700; Carl Zeiss Meditec), Scheimpflug camera (Pentacam AXL; Oculus Optikgeräte GmbH), and AS-OCT (Visante OCT; Carl Zeiss Meditec). Measurements for the three devices were taken no longer than 15 minutes apart for each participant. The ICL size was determined based on the manufacturer's recommended standard WTW method with the online calculation and ordering system ( https://ocos.staarag.ch). Emmetropia was selected as the target refraction for all patients to minimize the postoperative SE of refractive error. The postoperative subjective refractive error, corrected distance visual acuity, and vault, assessed by AS-OCT, were examined at least 8 weeks after surgery.

Vault was examined using AS-OCT and embedded software enabling distance measurement with a digital caliper. The vault was assessed as the distance from the posterior surface of the ICL to the anterior surface of the crystalline lens, aligned parallel to the visual axis.3 We provided the same photopic conditions15 during each vault examination with a white diode lamp with 1.2 W input power, placed 20 cm in front of the examined eye. After the light reflex was induced by shining a diode lamp into the contralateral eye, the AS-OCT image was acquired and photopic vault measurements were performed.15,21 To ensure a nonaccommodative state, the patient was asked to look at the far off distance during examination. Measurements were taken while the patient fixated on a collimated light-emitting diode (focus set at infinity).3,22

PLS Regression Model Development

Matlab R2009b software (MathWorks, Inc) was adapted to create and validate the PLS calibration model. Predefined PLS function (plsregress) using the SIMPLS algorithm was applied. The following biometric parameters were collected to develop the PLS calibration model: ATA distance, ACD, and lens elevation (LE) obtained with AS-OCT (Figure AA, available in the online version of this article) mean keratometry (Km), lens thickness (LT), and axial length (AL) obtained with the SS-OCT biometer (Figure AB); posterior cornea mean curvature (Rm) and anterior chamber volume (ACV) obtained with the Scheimpflug camera (Figure AC); and ICL size and SE of preoperative refraction error.

Biometric parameters used in the partial least squares (PLS) regression model. (A) Angle-to-angle (ATA) distance, anterior chamber depth (ACD), and lens elevation (LE) from anterior segment optical coherence tomography (AS-OCT). (B) Axial length (AL), lens thickness (LT), and mean keratometry (Km) from swept-source optical coherence tomography (SS-OCT). (C) Mean radius (Rm) of posterior cornea curvature and anterior chamber volume (ACV) from the Scheimpflug camera.

Figure A.

Biometric parameters used in the partial least squares (PLS) regression model. (A) Angle-to-angle (ATA) distance, anterior chamber depth (ACD), and lens elevation (LE) from anterior segment optical coherence tomography (AS-OCT). (B) Axial length (AL), lens thickness (LT), and mean keratometry (Km) from swept-source optical coherence tomography (SS-OCT). (C) Mean radius (Rm) of posterior cornea curvature and anterior chamber volume (ACV) from the Scheimpflug camera.

To measure ATA distance, ACD, and LE, we used AS-OCT and embedded software (anterior segment biometry tool), as shown in Figure A. The device was checked daily for measurement correctness with a test eye provided by the manufacturer. ACD was defined as the distance between the corneal endothelium and the anterior surface of the crystalline lens. These preoperative data (10 × 81 matrix) and postoperative vault results (1 × 81 vector) were the input data for the plsregress function to generate a calibration model.

After obtaining the calibration model, PLS regression coefficients were used to create an application (Figure B, available in the online version of this article) written in visual basic programming language and implemented in Excel software (Microsoft Corporation). The application enables easy vault calculation for every ICL size using patients' refractive error and biometric parameters.

Postoperative vault calculator screenshot. ICL = Visian Implantable Collamer Lens (STAAR Surgical AG); OCT = optical coherence tomography; ATA = angle-to-angle distance; ACD = anterior chamber depth; Rm = mean radius; SE = spherical equivalent

Figure B.

Postoperative vault calculator screenshot. ICL = Visian Implantable Collamer Lens (STAAR Surgical AG); OCT = optical coherence tomography; ATA = angle-to-angle distance; ACD = anterior chamber depth; Rm = mean radius; SE = spherical equivalent

Statistical Analysis

The same input data set was used for leave-one-out cross-validation.23 Of the 81 eyes, one was entered into the validation set and the rest (n – 1) were placed in the calibration set. The new calibration set was used to calculate and predict the postoperative vault of the eye from the validation set. Next, the eye in the validation set was replaced by an eye from the calibration set, creating another calibration model that was used for vault prediction of the eye in the validation set. This procedure was repeated until every eye was placed in the validation set. Calculated vault values of eyes from validation sets were compared to real postoperative vaults, and Spearman's rank correlation coefficient (r) was calculated to assess the correlation between predicted and actual values.

Our ATA and ICL size data were used to calculate and predict vaults using a recently published formula14: postoperative vault (µm) = 660.9 × (ICL size [mm] – ATA [mm]) + 86.6. Correlation with the actual postoperative vault was assessed by r and P values.

Calculated vault values were compared with postoperative vault data by the Bland-Altman method, as described elsewhere.17,24 The absolute error, defined as the difference between computed and real vault values, was calculated to assess the precision of PLS and the recently published model.

Data are presented as mean ± standard deviation. For statistical analyses, a P value less than .05 was considered significant. The 95% limits of agreement were defined as mean ±1.96 standard deviation of the difference between the vault predicted by PLS analysis and the real postoperative vault. Statistical calculations and plot creation were performed with Origin 8.0 software (OriginLab, Co).

Results

Descriptive statistics for the preoperative and postoperative data of our study population are presented in Table 1. Our study comprised 81 (41 right and 40 left) eyes of 43 patients, and 30 participants were female. The mean axial length was 26.81 ± 1.60 mm (range: 24.04 to 30.69 mm), as measured by SS-OCT. A myopic ICL was implanted in 29 eyes (36%) and a myopic toric ICL in 52 eyes (64%). No intraoperative or postoperative complications were reported and all ICL implantation surgeries were uneventful. Figure C (available in the online version of this article) shows detailed refractive outcomes. The vault was assessed 3 ± 1 months (range: 2 to 6 months) postoperatively, at least 8 weeks after surgery.

Characteristics of the Study Population

Table 1:

Characteristics of the Study Population

Outcomes of Visian Implantable Collamer Lens (ICL) (STAAR Surgical AG) implantation surgery for 81 eyes from 43 patients at 2 months postoperatively. (A) Uncorrected distance visual acuity (UDVA), (B) postoperative UDVA versus preoperative corrected distance visual acuity (CDVA), (C) spherical equivalent refractive accuracy, and (D) postoperative refractive cylinder. D = diopters

Figure C.

Outcomes of Visian Implantable Collamer Lens (ICL) (STAAR Surgical AG) implantation surgery for 81 eyes from 43 patients at 2 months postoperatively. (A) Uncorrected distance visual acuity (UDVA), (B) postoperative UDVA versus preoperative corrected distance visual acuity (CDVA), (C) spherical equivalent refractive accuracy, and (D) postoperative refractive cylinder. D = diopters

In the new PLS calibration model, 56% of the variance in postoperative vault was explained by 10 PLS components. A calculated correlation R2 between real and calculated vault values was 0.56. PLS regression coefficients were computed in the Matlab programming environment (Table 2). According to this model, the following formula should be applied to estimate the vault:

Vault[μm]=320×ICL size [mm]-9×SE[D]-197×ATA[mm]+233×ACD[mm]-8×LE[mm]+23×AL[mm]-235×LT[mm]+9×Km[D]+188×Rm[mm]-2×ACV[mm3]-3,192
Parameters of the PLS Model for Postoperative Vault Prediction and Calculation of PLS Regression Coefficients

Table 2:

Parameters of the PLS Model for Postoperative Vault Prediction and Calculation of PLS Regression Coefficients

Leave-one-out cross-validation results for the new PLS model (r = 0.73, P < .001) and vault values calculated with the linear regression model of Igarashi et al15 (r = 0.27, P = .02) showed significant correlation with postoperative vault values.

Table 3 and Figures 12 show the results of the Bland-Altman analysis of two vault prediction models compared to the real postoperative vault values. The mean difference between real vault values and PLS cross-validated values was 2 ± 110 µm (range: 250 to 270 µm). The mean difference between real vault values and those calculated by the Igarashi et al15 method was 400 ± 190 µm (range: −910 to 90 µm).

Bland-Altman Analysis of the Performance of Two Models for Postoperative Vault Prediction

Table 3:

Bland-Altman Analysis of the Performance of Two Models for Postoperative Vault Prediction

Bland-Altman plot of the performance of the partial least squares (PLS) regression model. Mean difference between the real and calculated vault is 0.00 mm, 95% limits of agreement (LoA) is 0.41 mm.

Figure 1.

Bland-Altman plot of the performance of the partial least squares (PLS) regression model. Mean difference between the real and calculated vault is 0.00 mm, 95% limits of agreement (LoA) is 0.41 mm.

Bland-Altman plot of the performance of the model recently published by Igarashi et al.15 Mean difference between the real and calculated vault is −0.40 mm, 95% limit of agreement is 0.75 mm.

Figure 2.

Bland-Altman plot of the performance of the model recently published by Igarashi et al.15 Mean difference between the real and calculated vault is −0.40 mm, 95% limit of agreement is 0.75 mm.

We calculated correlation coefficients between the average of simulated and obtained vault and their differences for both models (Figures 12). We have not found an association between the average of vault magnitude and vault difference for the Igarashi et al method15 (r2 = 0.01 and P = .461),15 but there was a significant association for the PLS model (r2 = 0.18 and P = .002).

Discussion

Our ICL vault prediction model is a novel application of PLS in ophthalmology. In our study, the PLS method showed high precision in predicting postoperative ICL vault (Table A, available in the online version of this article). The absolute difference between the calculated and real vault was within 100 µm in 90% of cases, and no higher than 300 µm in 100% of cases.

Precision of Vault Prediction in the Analyzed Group (n = 81), No. of Eyes (%)

Table A:

Precision of Vault Prediction in the Analyzed Group (n = 81), No. of Eyes (%)

An adequate vault remains the main prerequisite for long-term ICL safety. Direct measurement of the STS distance is considered to be the ideal parameter for choosing the size of the ICL.10–13 However, VHF digital ultrasound, which allows for such a measurement, has not gained popularity, mainly due to its complex and invasive image acquisition when compared to the no-touch, easy procedures for modern ophthalmic diagnostic devices.

Universally used AS-OCT devices yield fast and reproducible non-invasive measurements of ATA distance. The correlation between STS and ATA is stronger than the correlation between STS and WTW,13 and the reproducibility of ATA measurements is better than WTW measurements.13,15 A meta-analysis of the literature demonstrated that STS and WTW measurement-based sizing methods do not result in clinically meaningful or statistically significant differences in vault.2 Igarashi et al15 found that the most important determinants of postoperative ICL vault were the ATA distance and ICL size, and identified a statistically significant correlation between ICL size and ATA distance, but not between ICL size and WTW distance. Other studies have also shown a significant correlation between ICL vault and preoperative ATA and ACD values.25,26 Moreover, WTW measurements may vary depending on the device used for measurement.7–9 Therefore, we decided not to include WTW distance in our PLS calculation model. Despite this, our method yielded a 0 µm mean difference between the vault predicted by PLS analysis and the real vault (367 µm, 95% limits of agreement; Figure 1, Table 3). The impact of ATA distance on vault prediction in our cohort was 16.1%. Interestingly, the impact of LT and ACD on vault prediction was slightly higher than that for ATA (Table 2).

It seems that our new PLS model has a tendency to overestimate small vault values and underestimate high vaults (Figure 1). We will try to address this issue during further studies, when we complete a larger patient database and recompute PLS model parameters. On the other hand, the Igarashi et al model15 shows biased, overestimated values in full vault ranges (Figure 2).

The impact of Rm on vault prediction in the PLS model was 15.4%, whereas the impact of Km was only 0.7%. The posterior cornea has gained much interest over the past few years. It can be measured with Scheimpflug camera–based devices, as well as with newer OCT devices. This parameter is not influenced by tear film abnormalities; therefore, including Rm in the calculation model seems promising.

There were no cases of insufficient vaults (< 150 µm) and only 3 cases (4%) of slightly excessive vaults (> 750 µm) in our cohort of 81 eyes (values of 800, 820, and 860 µm). A previous study reported 9.3% insufficient and 9.3% excessive vaults in their cohort.27 The sizing method proposed by Malyugin et al,28 employing an algorithm with distance from iris pigment end to iris pigment end measured with AS-OCT, resulted in optimal vault values in only 16 of 29 eyes, with slightly excessive vault (710 to 840 µm) in 24% of eyes.

Variability in vault measurement is an important issue. Yan et al25 found no statistically significant differences in vault measurements performed 1 day, 1 week, and 2 years following surgery. This is in contrast with earlier research that showed significant differences between vault values measured 1 month after ICL implantation and 3, 6, and 12 months following surgery, with vault stabilization after the third month.26 This may be explained by different luminescence conditions and pupil diameter, which play an important role in vault measurement.15,21 In the current study, we measured the vault with AS-OCT under the same standardized photopic conditions at least 8 weeks after surgery.

The original calculation method described by the manufacturer includes WTW, ACD, Km, central corneal thickness, and preoperative SE, and does not predict the exact vault, only suggesting the size of ICL to use. We rejected the use of WTW and central corneal thickness values, and instead added ICL size, ATA distance, AL, LE, LT, Rm, and ACV. Our 10-factor PLS model explained 56% of variance in postoperative vault results. Therefore, more than 40% of variance can be attributed to other biometric parameters not considered in this study. Despite obtaining good results in vault prediction, there is still room for improvement. Some of the input data had a relatively low impact on the final calculation result (eg, LE, Km, preoperative SE, AL, and ACV [less than 2% of final calculated vault]), making the use of these variables optional. The limitation of our study is that our model simulates vault value after 2 to 6 months postoperatively, not the long term.

Our concept was to develop an algorithm resembling a regression IOL calculation formula, based on a statistical approach rather than physics (vergence IOL calculation formulas) and anatomy. We used specific devices to obtain the input data; therefore, the use of other equipment may affect the precision of the algorithm.

The implication of this novel PLS model seems to improve the precision of ICL vault prediction. However, it shows a tendency to overestimate small vault values and underestimate high vaults. The inclusion of more preoperative parameters in the calculation may increase the long-term safety of the ICL.

References

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Characteristics of the Study Population

ParameterMean ± SDRange
Age (years)30 ± 621 to 43
Preoperative CDVA (logMAR)0.06 ± 0.11−0.10 to 0.40
Preoperative manifest SE (D)−10.40 ± 3.64−3.50 to −20.00
Preoperative manifest cylinder (D)1.34 ± 1.120.00 to 4.50
SE of ICL power (D)−11.33 ± 3.22−4.50 to −18.00
Postoperative UDVA (logMAR)0.02 ± 0.12−0.30 to 0.40
Postoperative CDVA (logMAR)0.00 ± 0.10−0.30 to 0.40
Postoperative manifest SE (D)−0.07 ± 0.31−1.75 to +0.88
Postoperative manifest cylinder (D)0.15 ± 0.370.00 to 1.50
Vault (AS-OCT) (µm)440 ± 160170 to 860

Parameters of the PLS Model for Postoperative Vault Prediction and Calculation of PLS Regression Coefficients

No.ParameterPLS CoefficientImpact on Calculated Vault Value (%)
1ICL size (mm)32026.1
2Preoperative manifest SE (D)−90.7
3ATA (AS-OCT) (mm)−19716.1
4ACD (AS-OCT) (mm)23319.1
5LE (AS-OCT) (mm)−80.7
6AL (SS-OCT biometer) (mm)231.9
7LT (SS-OCT biometer) (D)−23519.2
8Km (SS-OCT biometer) (D)90.7
9Rm (Scheimplfug camera) (mm)18815.4
10ACV (Scheimplfug camera) (mm3)−20.2
11Constant−3,192

Bland-Altman Analysis of the Performance of Two Models for Postoperative Vault Prediction

MethodPLS ModelIgarashi et al,15 2019
Mean difference ± SD (µm)2 ± 110−400 ± 190
95% CI (µm)−20 to 20−440 to −360
Minimum–maximum difference (µm)−250 to 270−910 to 90
LoA 95% (range) (µm)(367)−774 to −30 (745)

Precision of Vault Prediction in the Analyzed Group (n = 81), No. of Eyes (%)

Vault DifferencePLS ModelIgarashi et al15 2019
Within ±100 µm59 (73)3 (4)
Within ±200 µm79 (90)12 (15)
Within ±300 µm81 (100)25 (31)
> 300 µm56 (69)
Authors

From Ophthalmology Clinical Centre SPEKTRUM, Wroclaw, Poland.

The authors have no financial or proprietary interest in the materials presented herein.

AUTHOR CONTRIBUTIONS

Study concept and design (AO, JM, MM-W); data collection (AO, JM, MM-W); analysis and interpretation of data (AO); writing the manuscript (AO, MM-W); critical revision of the manuscript (JM, MM-W); statistical expertise (AO); supervision (JM, MM-W)

Correspondence: Adam Oleszko, PhD, Drzewieckiego 16/8, 54-129 Wroclaw, Poland. Email: oleszko.a@gmail.com

Received: January 21, 2020
Accepted: June 24, 2020

10.3928/1081597X-20200630-01

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