The authors have declared that no competing interests exist.
Conceived and designed the experiments: FJR ACD JMD. Performed the experiments: FJR. Analyzed the data: FJR. Contributed reagents/materials/analysis tools: ACD JMD. Wrote the paper: FJR. Gave technical support and conceptual advice: ACD JMD. Supervised the project: ACD JMD.
To test a pseudophakic eye model that allows for intraocular lens power (IOL) calculation, both in normal eyes and in extreme conditions, such as postLASIK.
Participants: The model’s efficacy was tested in 54 participants (104 eyes) who underwent LASIK and were assessed before and after surgery, thus allowing to test the same method in the same eye after only changing corneal topography.
The LiouBrennan eye model was used as a starting point, and biometric values were replaced by individual measurements. Detailed corneal surface data were obtained from topography (Orbscan®) and a grid of elevation values was used to define corneal surfaces in an optical raytracing software (Zemax®). To determine IOL power, optimization criteria based on values of the modulation transfer function (MTF) weighted according to contrast sensitivity function (CSF), were applied.
Preoperative refractive assessment calculated by our eye model correlated very strongly with SRK/T (r = 0.959, p<0.001) with no difference of average values (16.9±2.9 vs 17.1±2.9 D, p>0.05). Comparison of postoperative refractive assessment obtained using our eye model with the average of currently used formulas showed a strong correlation (r = 0.778, p<0.001), with no difference of average values (21.5±1.7 vs 21.8±1.6 D, p>0.05).
Results suggest that personalized pseudophakic eye models and raytracing allow for the use of the same methodology, regardless of previous LASIK, independent of population averages and commonly used regression correction factors, which represents a clinical advantage.
Modelling the optics of an individual patient's eye, and predicting the resulting visual performance are major goals for visual optics and clinical researchers. Although generic eye models are of great use, they do not reflect individual anatomical characteristics, and are thus limited. Therefore, the development of personalized models, using individual biometry data and encompassing individual aberrations
Wavefront technology and ray tracing are very promising technologies that have been used to improve IOL power calculation error
In this paper, we report the design of a personalized pseudophakic model which overcomes those two needs by using a full 3D definition of the cornea based on detailed corneal elevation data, obtained from topography, and an optimization metric based on the MTF and CSF. The model was tested in 54 participants (104 eyes) who were assessed before and after LASIK, thus allowing to test the same method in the same eye after only changing corneal topography. In order to evaluate the efficacy of the model, results were compared to currently used methods of IOL power calculation.
54 participants (104 eyes), with average age of 33.8±8.0 years, with preLASIK refraction of –3.07±1.95 D, corneal anterior radii of 7.74±0.26 mm, ACD of 3.04±0.31 mm (distance from corneal endothelium to lens), lens thickness of 3.87±0.36 mm and vitreous chamber depth of 16.96±1.05 mm, scheduled to undergo LASIK refractive surgery, were assessed before and 1 month after LASIK. Topography data was obtained with an Orbscan II® (Bausch and Lomb Inc., Rochester, NY, USA) and contact biometry data without immersion with Ocuscan® (Ultrasound biometry Alcon RxP). Given Orbscan II® is based on slit scan beam imaging and uses mathematical calculations to recreate the posterior cornea, this strategy can cause false positive readings of posterior corneal elevation
The LiouBrennan eye model
In order to define individual corneal surfaces, a full 59×59 grid of elevation values from corneal anterior and posterior surface data were obtained from topography with an Orbscan II® and using a 10 mm diameter. Elevation data allow a full spatial morphological description of the corneal surface
Tridimensional corneal representation. Corneal elevation data generated from topography was reformatted and imported to Zemax®. Afterwards, a full definition of the surface shape was obtained through a bicubic spline interpolation of the imported data, thus allowing for raytracing. xx and yy axis represent value distribution of the grid over a corneal surface of 10 mm, zz axis represents elevation values.
Since one of the goals of our model is IOL calculation, we have used preoperative data to estimate postoperative anterior chamber depth (ACD_{post}). In order to define the IOL lens position, ACD_{post}, taken as the distance from the corneal endothelium to the anterior IOL surface, was calculated using the measured preoperative ACD and lens thickness (LT) and considering IOL position at the lens equator defined by the LiouBrennan model. Hence, the used formula was: ACD_{post} = ACD+0,395 LT. IOL was defined by its geometry  anterior and posterior curvature radius, thickness and refraction index, according to the catalogue of the AR40_{e} (AMO) ® IOL. It should be noted that ACD_{post} will always be an estimation, since it cannot be physically measured before IOL placement. Since our aim was that our eye model would be as independent as possible from regressive factors derived from population studies, ACD_{post} estimation was based solely on biometric values and the definition of equator lens.
The optical software Zemax® was used to construct a pseudophakic eye model. Once the virtual eye is defined, this software uses wavefront technology and exact raytracing to modulate light propagation through the optic system to the surface defined as image – the retina. The resulting amplitude distribution and phase of a ray beam allow the analysis of different optic phenomena.
This model takes into consideration the optical aberrations that limit the quality of the human eye retinal image, and an optimization procedure has to be adopted in order to choose the best corrective solution. In fact, the optimization procedure is the key process for the calculations of optical components in the virtual eye through the minimization of a predefined merit function. Although it is unknown which criteria the human eye actually uses for focusing, and as such the ideal optimization method is yet to be determined, wavefront rootmeansquare (RMS) minimization has been the most commonly used optimization criterion of best focus plane in raytracing. However, previous studies have shown that it does not correspond to subjective refraction, always retaining a significant amount of residual Zernike defocus
A schematic representation of the various inputs of the personalized model introduced in the optical analysis software described above is shown in
With the schematically represented algorithm, an individual pseudophakic model was obtained for each of the 104 assessed eyes, both before and after refractive surgery. Individual raytracing was then performed to allow IOL power calculation.
Correlations were assessed using the Pearson correlation coefficient. Linear regressions of the form y = Bx+A were performed and standard errors σ of all parameters were calculated. Means were compared using ttests. Tests were considered significant at p = 0.05 significance level (twotailed).
To test the efficacy of the representation of the cornea in our model, corneal anterior radii of the Zemax® representation of the 104 corneas were evaluated before refractive surgery and compared to values obtained by keratometry. In
Regression parameters y (σ = 0.084) = 0.948 (σ = 0.045)x+0.516 (σ = 0.347). Pearson correlation parameters: r = 0.949, p<0.001.
mean±sd  Mean absolute difference  Median  Median absolute difference  


keratometry  7.74±0.26 
0.12  7.72  0.12  
Our model  7.85±0.26 
7.80  


Olsen 2  4.87±0.24 
0.36  4.86  0.34  
Our model  5.22±0.23 
5.22  


SRK/T  17.2±2.9  0.6  17.4  0.5  
Our model  16.9±2.8  17.5  


Average IOL  21.8±1.6  0.9  22.0  0.5  
Our mode  21.5±1.7  21.5 
p<0.05 compared to our model, unpaired ttest.
The value of ACD_{post} is necessarily an estimation, needed for postoperative IOL power calculation. ACD_{post} calculation using our model used solely lens and anterior chamber biometric values, as previously described in the Methods Section. In order to validate the calculated values, we have tested the correlation between our ACD_{post} estimation and values obtained using the Olsen 2 formula, transformed to ACD prediction algorithm as described by Jin et al
Regression parameters y (σ = 0.164) = 0.657 (σ = 0.069)x+2.014 (σ = 0.335). Pearson correlation parameters: r = 0.688, p<0.001.
In order to validate the preoperative IOL power estimated using our model, we have analysed the correlation between the values obtained and the ones calculated using the SandersRetzlaffKraftTheoretical (SRKT) formula, as well as the differences between the mean values. The SRKT formula was used as a comparator since it is the most frequently used for IOL power calculations. There was a very strong correlation between preoperative IOL power estimation using our model and using the SRKT formula –
All values were rounded to 0.5 dioptres, in order to reflect currently available IOL powers. Regression parameters y (σ = 0.745) = 0.959 (σ = 0.026)x+0.409 (σ = 0.446). Pearson correlation parameters: r = 0.966, p<0.001.
The SRKT formula has been shown not to be accurate for post LASIK IOL power calculation, and of the several currently used methods, the ones using surgically induced changes in manifest refraction or using no prior data have been shown to have smaller IOL prediction errors and variances and greater percentages of eyes within ±0.50 and ±1.00 D of the refractive prediction errors
All values were rounded to 0.5 dioptres, in order to reflect currently available IOL powers. Regression parameters y (σ = 1.048) = 0.788 (σ = 0.063)x+4.340 (σ = 1.381). Pearson correlation parameters: r = 0.778, p<0.001.
Although wavefront technology and exact raytracing have been used for IOL power calculations
Comparison of corneal anterior radii values calculated by Zemax® and evaluated by keratometry not only showed a very strong correlation but also no statistically significant difference in mean values. The difference of 0.5% was below 1.5%, which is the percentage described by Preussner
Another extremely important parameter is ACD_{post} estimation. Its importance on refractive result is well established, since ACD_{post} prediction error accounts for 42% of all sources of error on IOL power calculation
Our eye model showed a strong correlation with ACD_{post} estimation using the Olsen 2 formula, with error associated with regression values estimation varying between 3.0% and 3.9%, when considering the ACD_{post} values of our population sample, corresponding to an error of 0.25 dioptres in refractive error
These results may be explained due to the fact that the Olsen 2 empirical formula, which also correlates with lens thickness and ACD, uses the Gaussian approximation of the “effective lens position” and not the physical position of the IOL.
Our eye model estimates ACD_{post} based solely on biometric values – ACD_{pre} and lens thickness –, enhancing the equator definition by using the derivative of population studies which are the base for the LiouBrennan model
The comparison of preLASIK IOL power using our eye model with the SRKT formula showed a very strong correlation, with no difference in mean values. The error associated with regression values estimation varied between 2.9% and 8.3%, when considering the preLASIK IOL values of our population sample, representing a standard deviation of 0.53 D. Considering the available IOLs vary in steps of 0.5 D, which we have taken into account on our linear fitting, and IOL manufacturers should apply an internal tolerance of ±0.25 D to all IOLs
Given the above, we have chosen as a comparator the average IOL power in the calculator available at the ASCRS website
The reality that a perfect solution for postLASIK IOL power estimation is yet to be attained underscores the importance of seeking new methods to determine IOL in pseudophakic eyes. Currently used formulas are only applicable to eyes that fall within population average values and if using the technologies they were developed for, and thus, whenever one of these variables change, new population studies are needed, leading to a timegap that leaves clinical practice without immediate solutions.
Results show that our eye model is applicable to IOL power calculation, both pre and postLASIK, in a personalized way, without the need for population averages, which in cases such as postLASIK may be very different from the general population.
The modelling of a human pseudophakic eye not only has multiple current clinical applications but may also be used for future diagnostic and correction challenges. Several relevant factors are still unknown, but the results presented in this paper suggest that the development of these eye models, considering individual aberrations, using wavefront technology and exact raytracing, enhanced by the image metric based on MTF and CSF, allowing for the prompt incorporation of parameters that are currently not measurable in clinical practice, in a personalized manner, if and when they become available, without the need for redefining population correction factors, can be used even in face of abnormal corneas or when clinical history is unknown.