Toric intraocular lens (IOL) implantation successfully treats corneal astigmatism and provides independence from distance spectacles in more than 70% of all cases compared to less than 50% in the case of non-toric IOLs.1 However, a moderate number of patients still have relevant remaining astigmatism after toric IOL implantation, resulting in patient dissatisfaction.
One way to evaluate and quantify sources of error in (toric) IOL power calculation is ray-tracing combined with Gaussian error propagation. This error quantification was performed for monofocal IOL power calculation in the past,2,3 but not for toric IOLs. Although a variety of publications focus on single or multiple errors in toric IOL power calculation,4–7 to our knowledge, no error quantification using Gaussian error propagation has been published so far.
The aim of this study was to detect and quantify the most relevant sources of error in toric IOL power calculation, including data from the literature and retrospectively analyzed data.
Patients and Methods
This retrospective ray-tracing and Gaussian error propagation analysis included parameters that were considered to have an influence on toric IOL power calculation from different studies using toric IOLs or related to toric IOL implantation. This included axial eye length, anterior chamber depth, central corneal thickness, keratometry (corneal radii and steep axis of anterior and posterior surface), diurnal changes of the cornea, inter-device differences of different corneal measurement techniques (including the posterior surface of the cornea), rotational misalignment of the IOL, tilt and decentration of the IOL, pupil size, angle kappa (angle between line of sight and pupillary axis), and surgically induced astigmatism.
The following concept was used: for those parameters that are usually included in toric IOL power calculation (axial eye length, anterior chamber depth, and corneal radii [anterior and posterior]), the standard deviation of measurement reproducibility was used. The reason is that the magnitude of error for these parameters depends mainly on the reproducibility of the measurement. For those parameters that are usually not included in toric IOL power calculation (diurnal changes of the cornea, inter-device differences [including posterior surface of the cornea], central corneal thickness, rotational misalignment of the IOL, tilt and decentration of the IOL, pupil size, angle kappa, and surgically induced astigmatism), the standard deviation between patients (inter-patient deviation) was used. The reason is that these factors either are not considered in toric IOL power calculation or a constant factor is used.
Methods of Evaluation
An eye model based on Liou and Brennan8 was developed using Zemax software (version 16) by one of the authors (NB) and a 3.00 diopters (D) astigmatic cornea was simulated.9 For the eye model, 42 studies2,10–51 (and 584 eyes of retrospectively analyzed data) were included in the analysis. Additionally, one study was used to assess the influence of the posterior surface of the cornea52 and three studies to evaluate the influence of surgically induced astigmatism.13,24,53
Ray-tracing was used to calculate the effect of one standard deviation of each parameter quantified as dioptric value. Vector analysis was performed as suggested by Thibos and Horner.54 In the next step, Gaussian error propagation was performed. Microsoft Excel 2011 software version 14.2.3 for Mac (Microsoft Corporation) with a Xlstat 2012 plug-in (Addinsoft) was used for statistical analysis.
In total, 4,949 eyes (4,365 eyes of 42 studies and 584 eyes of retrospectively analyzed data) were included in the analysis. Table 1 summarizes the included values for the eye model. Total error as a vector difference representing astigmatism caused by error with reference to a starting point of no astigmatism was 0.81 D.
Values for Each Eye Model
In the next step, the parameter difference between devices (including posterior surface), corneal radii, central corneal thickness, and diurnal changes were merged because they are part of the corneal measurement error. Merging was performed using a sum vector of all parameters. This corneal measurement error was larger compared to all other parameters (27.0% or 0.59 D) and it sums up 44.5% (0.26 D) of differences between devices (including posterior surface of the cornea), 24.0% (0.14 D) of corneal radii, 16.3% (0.10 D) of diurnal changes, and 15.2% (0.09 D) of central corneal thickness, respectively. Table 2 and Figure 1 represent the distribution of error.
Distribution of Error
Error distribution in toric intraocular lens (IOL) power calculation (percent) using a combined vector for all corneal measurement errors and other parameters. Calculations are based on ray-tracing and Gaussian error propagation. SICA = surgically induced corneal astigmatism; ACD = anterior chamber depth; AL = axial length
The posterior surface of the cornea was included in the model as part of the inter-device difference. The estimated average error deriving from neglecting the posterior surface of the cornea is 0.22 D. In other words, neither measuring nor estimating the posterior surface of the cornea results in an error of approximately 0.22 D, or 37.4% of the corneal measurement error. Additionally, the influence of the standard deviation of the posterior surface reproducibility measurements was assessed. Because this value is small (0.03 D), it was not mentioned in the table.
The measurement error of the cornea depends on the measurement technique. Therefore, a subanalysis using difference vectors from Hoffmann et al25 was used.
The corneal measurement error for swept-source optical coherence tomography, automated keratometry, and Scheimpflug imaging was 19.2% (0.42 D), 25.6% (0.56 D), and 32.0% (0.70 D), respectively (Figure 2). These values account for the simulation of a 3.00 D astigmatic cornea. In the case of low astigmatism, the corneal measurement error increases.
Double-angle plot for postoperative refractive astigmatism prediction error on the corneal plane using three different measurement techniques: triangle = swept-source optical coherence tomography technology, circle = autokeratometry, square = Scheimpflug measurement; D = diopters
In this eye model, the influence of the toric IOL would only be expressed by differences in rotational misalignment. Due to the fact that all modern toric IOLs show a mean absolute rotational misalignment below 5°, the difference was found to be negligible.
Simulating the surgically induced corneal astigmatism with a standard deviation of 0.31 D (104 eyes from the VIROS database) and 0.37 D resulted in a difference vector of 0.70 D in the ray-tracing model. However, if only the flattening effect of a 2.2-mm temporal incision was used (0.20 ± 0.24 D), the percentage of the total error would be 7.8%. This value was included in the Gaussian error propagation.
This ray-tracing and Gaussian error propagation model shows that more than one-quarter of the error in toric IOL power calculation derives from the corneal measurement. It should be mentioned that the calculation would have been slightly different if another toric IOL power had been used. Furthermore, it is difficult to take interactions between different parameters into account, and this is a potential bias of this study. However, the mean difference vector in the eye model used was similar to results from clinical studies.25,55,56 This similarity can be used for validation purposes of the eye model. Hoffmann et al25 showed similar results for conventional measurement techniques, but the difference vector for swept-source optical coherence tomography (OCT) measurements was lower.
Directly quantifying the error of the preoperative corneal measurement is not possible because the “true power” of the cornea is not known and changes over time and even during the day were observed. Therefore, a surrogate parameter was used: the standard deviation of the difference vector between corneal measurement devices. To aim for a real-life scenario, different measurement techniques (eg, keratometry, Placido disc-based topography, and Scheimpflug- and OCT-based tomography) were included in the model and the vector difference between devices was averaged. This means that this value also takes into account the posterior surface of the cornea. As shown by Hoffmann et al,25 modern measurement techniques such as swept-source OCT technology may reduce the prediction error, especially in combination with keratometry and ray-tracing. It should also be mentioned that the measurement error is relatively larger in low astigmatism, especially due to difficulties in meridian detection57 compared to moderate or high corneal astigmatism.24 Additionally, fluctuations of the tear film changes and eye drops58,59 significantly influence measurements of the cornea.
The influence of the posterior surface is relevant and the difference vector between anterior and total astigmatism was 0.30 D, with a maximum of 1.50 D.60
One problem in the ray-tracing and Gaussian error propagation model is that it is difficult to take interactions between parameters into account. Examples of such interactions are reproducibility-based error and diurnal change error. To improve this model, we used a sum vector of all explanatory variables summarized in “corneal measurement” to reduce this problem. In the second part of this analysis, we also calculated the influence of the posterior surface of the cornea on the corneal measurement error. Neglecting (not measuring and not estimating) the posterior surface of the cornea results in approximately one-third of the corneal measurement error.
Misalignment is the second largest source of error in toric IOL power calculation and it is the sum vector of preoperative/intraoperative marking error, implantation error, and postoperative rotation of a toric IOL. Preoperative marking was shown to result in a minor but relevant rotational error and there was a device-dependent difference.38,60 Intraoperative use of augmented reality was shown to reduce misalignment slightly.61 Postoperative rotation was a large source of error in the past, but modern toric IOLs have been shown to have good rotational stability.62 Visser et al63 calculated the combined vector for all steps of mis-alignment and found similar values to our study. Although mean absolute misalignment is usually small, there is a large deviation between patients, resulting in a relevant source of error in toric IOL power calculation. However, this study did not include a comparative analysis between different marking and aligning techniques. Therefore, it is possible that this amount of error differs between different centers.
Tilt and Angle Kappa
Tilt and angle kappa were shown to be relevant sources of error that are usually not considered in toric IOL power calculation. This finding was also confirmed for tilt in a previous study.64 A tilt prediction algorithm was recently introduced using preoperative tilt measurements with swept-source OCT. It was shown that the orientation of the postoperative tilt can be predicted with good precision.65 It is possible that taking angle kappa and tilt prediction into account results in a better toric IOL power calculation.
Pupil size was shown to be a relatively small source of error. It appears that different designs of toric IOLs are less dependent on pupil size compared to others.66 For the error propagation, a commonly available aspheric toric IOL was used for simulation purposes. Nevertheless, there are case reports suggesting that pupil size can be a relevant cause of remaining astigmatism.67
Surgically Induced Corneal Astigmatism
Surgically induced corneal astigmatism showed the limits of Gaussian error propagation. The reason is that surgically induced corneal astigmatism is on average small, but the standard deviation between patients is large. However, this standard deviation is influenced by a variety of factors, such as corneal measurement error, tear film problems, and diurnal changes. All of these interactions between parameters influence the standard deviation, resulting in an unreliable parameter. Therefore, the flattening effect of a 2.2-mm temporal incision was included in the model. In a previous study using partial least squares regression modelling, it was shown that the influence of surgically induced corneal astigmatism is small with a high amount of unpredictability.4
Anterior Chamber Depth and Axial Eye Length
Anterior chamber depth and axial eye length had a minor impact on toric IOL power calculation. Both parameters are important to predict the axial position of a toric IOL. Although this mainly influences the spherical equivalent, it also has an influence on residual astigmatism, especially in the case of high-powered toric IOLs.20
Contrary to tilt, decentration plays a minor role in the case of toric IOLs in terms of residual astigmatism. This was also shown in a previous study on the optic bench.66
There are several limitations of this study. First, the different parameters had to be simplified to be used for Gaussian error propagation. This is especially the case for the factor “corneal measurement.” Depending on the measurement method, the error distribution changes. We added a subanalysis in this study, but a more detailed comparison in a prospective trial including the influence of the posterior surface of the cornea in Gaussian error propagation would be useful. Another limitation is that the influence of each parameter changes depending on the astigmatic correction. In case of low corneal astigmatism, the influence of the corneal measurement increases, whereas misalignment is more relevant in correction of high corneal astigmatism.
A second minor limitation is the choice of the eye model. The Liou and Brennan eye model was chosen because it is a “finite” eye model instead of a paraxial model. Therefore, this model allows better simulation of optical imaging farther from the visual axis and simulation of a larger pupil size. This suggestion was taken from a publication from Atchison and Thibos.68 However, other eye models would have performed slightly differently.
The main source of error in toric IOL power calculation is the preoperative corneal measurement, followed by misalignment and tilt of the IOL. These findings suggest that measurement techniques should be improved to increase the predictability of toric IOL power calculation, which includes the measurement of the posterior surface of the cornea. Furthermore, angle kappa and tilt should be used in toric IOL power calculation in the future.
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Values for Each Eye Model
|Source of Error||SD|
|Difference between devices (including posterior surface of the cornea)||0.24 D12,25,29,42,48,+unpublished|
|Keratometry reading||0.11 D14,16,17,32,33,37,39,41,43,46,48,+unpublished|
|Pupil size||0.76 mmref2|
|Diurnal changes||0.04 D (unpublished)|
|SICA (temporal 2.2 mm)||0.20 D (SD: 0.24) (unpublished)|
|Anterior chamber depth||0.03 mm29,30,44|
|Central corneal thickness||0.00432 mm29,44|
|Axial length||0.02 mm21,29,30,31,40,44|
Distribution of Error
|Source of Error (3.00 D Corneal Astigmatism)||% of Total Error Deriving From Gaussian Error Propagation|
|Corneal measurement (including posterior surface of the cornea)||27.0 (of this, 37.4% deriving from the posterior surface of the cornea)|
|SICA (temporal incision)||7.8|
|Anterior chamber depth||7.5|