We present a ray-tracing approach to calculate the optimum IOL power. This approach was previously used to model the optical performance in normal eyes [33
] and in eyes implanted with IOLs [7
]. Both studies showed a good correspondence between measured and computed eye’s aberrations in a customized eye model that took into consideration the IOL´s tilt and decentration. In both cases, all IOL parameters were known, including IOL power used in the surgery as well as its placement in the eye, and tilt and misalignments measured with Purkinje systems. These eye models could be used to model optical performance for new IOLs that correct for the higher order aberrations of the cornea. The impact of the correction of these higher order aberrations can be reduced if the IOL power is not properly chosen. Our aim was to further develop these customized models to be based on preoperative data so that they can be used as a predictive tool.
In addition to monochromatic aberration, polychromatic behavior is also considered in our model. To our knowledge, there is no other IOL power calculation method incorporating this aspect, which may be important for a realistic simulation of the optical quality of the eye. In this direction, recently, the visual impact of correcting chromatic aberrations in IOLs was evaluated [35
]. More particularly, it has been shown that the effect of chromatic aberration in pseudophakic eyes is mainly due to IOL material [36
]. In order to emphasize the importance of the combined effect between monochromatic and chromatic aberrations in IOL power calculations we had performed the IOL power prediction with our ray tracing procedure in different conditions. The impact of pure chromatic aberration can be seen in
, where the IOL power prediction has been performed correcting monochromatic corneal aberrations for the same subject with a virtual IOL model having the same geometrical properties but different Abbe numbers, that is, different chromatic aberration values, both in white light and monochromatic conditions (540nm). As expected, as higher is the Abbe number, there is less difference between both calculations, leading to the same calculated IOL power. In fact, chromatic aberration starts to play a role only for very dispersive, nonrealistic IOL materials, because available commercial IOLs present Abbe numbers between 35 and 60 [36
]. The scenario is different when monochromatic aberrations are added to the model and considered in the IOL power prediction (
). The polychromatic calculation resembles the effect of chromatic aberration for lower although commercially available IOL’s Abbe numbers with lower MTF values, due to the impact of monochromatic aberrations in comparison with the previous calculation where they were corrected (), affecting the selected IOL power. Then, we can conclude that the impact of chromatic aberration is modulated also by monochromatic aberrations. It is important to note that current IOL power calculations do not incorporate dispersion effects and for that reason cannot differentiate between IOL materials.
Fig. 10 Area under the radial MTF calculated with our customized procedure, correcting corneal aberrations, as a function of the IOL power with different IOL materials. Therefore, in this plot we show the pure chromatic aberration effect due to the IOL material (more ...)
Fig. 11 Area under the radial MTF calculated with our customized procedure, including corneal aberrations, as a function of the IOL power with different IOL materials. Therefore, in this plot we show the combined effect between chromatic aberration due to the (more ...)
Former formulas for IOL power calculations are based just in paraxial optics to predict IOL power, with no reference to aberrations that are more often present in the new aspheric IOLs. On the other hand, in older subjects the average pupil under photopic conditions is 3.5mm while in mesopic conditions is 5mm [37
]. In both cases, aberrations start to play a role in optical performance, so paraxial optics should not be used. In this study, we decided to use 4mm pupil since it’s realistic under the previous data. However, we didn’t find a correlation between the difference with standard regression formulas and our predictions. This may be due to the fact that we included normal eyes that were not highly aberrated. In order to show the importance of the aberrations in IOL power prediction, we chose one subject and we calculated the IOL power with our procedure considering his natural corneal aberrations and with two, three, four and five times the amount of these corneal aberrations (
). The IOL power calculated becomes different as higher is the amount of aberrations. Obviously, these changes would not affect the regression formulas due to their paraxial nature. This increase in aberrations might be high but not impossible to find in real eyes. For example, those eyes that had undergone LASIK surgery present increased amounts of corneal aberrations. For them, even the highest level of aberrations used in the example is perfectly possible.
Fig. 12 Area under the radial MTF calculated with our customized procedure as a function of the IOL power for different amounts of corneal aberrations (CWA) referred to 4mm pupil. Their impact on IOL power calculation is studied by increasing the original aberration (more ...)
In our approach, the only reference to not personalized data we use is the IOL placement based on a previous found high correlation between the natural lens position and the measured IOL position. The best scenario would be a complete theoretical model only depending on the IOL and particular eye’s characteristic. This will be the subject of further research to fully customize the model.
It is important to note that because we are using an exact ray tracing procedure, the effective lens position used in paraxial formulas cannot be introduced here. Those authors that have previously studied the IOL power calculation problem with either the thick lens theory or ray tracing have faced the same problem. Olsen [15
] or Norrby [38
] have developed elaborated actual lens position predictions based on multiple biometric parameters. These or another actual lens position can be introduced in the customized ray tracing procedure that we are presenting in this paper in order to evaluate their accuracy.
In order to evaluate the impact of a different IOL placement prediction on the ray tracing procedure, we repeat the calculations for all the population included in the study by using an ACD prediction recently presented by Norrby et al. [39
]. We decided to use this prediction because it has been developed for the same IOL model used here and it includes axial length and anterior chamber depth, which are parameters we measured.
shows the relationship between both ACD predictions for all the population of the study (r2
= 0.63). The average difference between both predictions was 0.11 ± 0.15mm, resulting our prediction in a deeper IOL placement on average than that developed by Norrby et al. It is also well accepted that the ACD prediction is highly correlated with axial length. However, we did not include this parameter in our prediction. In order to explore the impact of axial length in the IOL position prediction, we represent in the difference between ours and Norrby’s approach, which includes the AXL as predictor, as a function of eye’s axial length. There is, in fact, a linear correlation between the ACD difference and AXL, although there was some dispersion, leading to a weak correlation between both parameters (r2
= 0.27). Further studies will reveal the method achieving the highest accuracy for the IOL power procedure, although we still believe that a fully theoretical customized procedure for IOL placement is the best approach in order to complete the eye modeling for IOL power calculations.
Fig. 13 (a) ACD prediction presented in this paper described by Eq. (1) versus the Norrby prediction  for all the study population and (b) difference between both ACD predictions as a function of the axial length.
In order to explore the impact of these differences in the IOL power prediction, we applied the same ray tracing procedure, by generating three customized eye models corresponding to three different corneal topographies per subject, being the selected IOL power the mode of them. Then, the only difference between previous and these results is purely the IOL placement prediction, considered in this case as Norrby et al. [39
shows the IOL power difference between the ray tracing prediction considering our ACD prediction and Norrby’s as a function of the ACD difference. As expected, a deeper IOL placement will lead to a higher predicted IOL power, although there is a range where different ACD placement does not translate to a difference between both predictions. This is due to the fact that we are considering an IOL model with 0.5D steps. However, we did not find an ACD difference value wherein there is a clear step into the higher or lower IOL power. This might be related to the IOL power selection is based on the mode of the results coming from three different corneal topographies, making more complex the results interpretation.
Fig. 14 IOL power difference between ray tracing considering the ACD prediction described by Eq. (1) and that presented by Norrby et al.  as a function of the difference between ACD predictions. In both cases, the Norrby’s prediction is considered (more ...)
It is important to note that although there are differences in the ACD placement, the maximum discrepancy between ray tracing approaches is 0.5D, which is the IOL power step and the variability in the procedure, as we have shown in . In addition, the average IOL power with the Norrby’s ACD prediction was 19.44 ± 4.84D with a median of 21D, the lowest of all the methods considered in the study, including the ray tracing approach with the ACD prediction presented in this paper, as can be seen at . However, as we had pointed out through the paper, we do not have subject’s final refraction results, because they did not undergo cataract surgery. Therefore we can only establish differences between procedures. Further clinical studies will show the most accurate procedure for ACD placement to predict IOL power by ray tracing, although the point of this comparison was also to show the plasticity of the procedure to adopt different ACD predictions.
It is important to note the differences between formulas in a large range of powers for ammetropic eyes, establishing also their inaccuracy in those cases. Different solutions has been pointed out in order to increase the accuracy of IOL power calculation considering these formulas, from the modification of the A constants [2
], a myopic targeting for an emmetropic outcome or the transformation of axial length measurements to improve refractive outcome in extreme eyes without sacrificing the outcome in normal eyes [40
]. However, the regression and paraxial nature remains leading to outliers.
We found that the differences between our calculations and the empirical formulas were higher as greater was the subject’s refraction. From this result we only can state the difference but, because subjects didn’t undergo actual surgery. Future studies involving actual clinical surgeries could show the absolute differences.
Cataract surgery modifies the cornea by the incision, inducing additional aberrations [41
]. Due to the relative low value of these inductions and the tendency to be reduced with time, we believe that the pre-surgery corneal topography provides enough and valid information for the IOL power prediction, although the impact of these changes will be subject of future research.
The introduction of the different biometric parameters in the ray-tracing prediction can be a limit of its accuracy because all the errors involved to those measurements. Norrby [19
] quantified the sources of error in IOL power calculations, finding that the most limiting parameter is the IOL placement, followed by the actual determination of the postoperative refraction and the different biometric parameters considered in the calculation. In our case, the variability between topographies showed a maximum difference of 0.5D between power predictions. This variability could be avoided by a prior evaluation of those corneal topographies. Therefore, only those topographies within an astigmatism standard deviation smaller than 0.05 microns should be considered for the IOL power prediction.
Another possible limitation to the procedure is the introduction of a fixed equivalent refractive index to account the power of the posterior corneal. Although there are instruments measuring the posterior corneal surface, we believe that the current model incorporating anterior corneal aberrations provides enough important and valid information without increasing the amount of experimental error in the procedure. Then, the extension of the procedure to consider the posterior corneal is possible, however it would be subject of further research, as well as the modification of the calculated equivalent refractive index for post-LASIK patients.