Our experimental results show that the distortion due to refraction at the anterior surface and within the gradient produces an error in the posterior radius of curvature that is within the experimental variability of the system. This finding suggests that accurate values of the in vitro
posterior radius can be obtained by simply rescaling the distances using Eq. (1)
with the appropriate value of the index. Experimentally, the average error in the posterior radius of curvature was found to 0.12mm with a 95% confidence interval of 0.73mm, which is very close to the error found with the simulations. The results of the simulations ( and
) show that most of the error can be corrected by using a ray-tracing procedure assuming a uniform refractive index equal to the average group refractive index of the lens. Interestingly, the optimal refractive index for the correction is closer to the average index than the equivalent index. The majority of previous studies have used the equivalent index for correction.
The effect of the distortions on asphericity was more variable. For the posterior surface, the distortion produces a mean difference of −0.182, with a 95% confidence interval of +/−1.9. The experimental results suggest that reliable measurements of asphericity of the distorted surface cannot be obtained with the simple correction of Eq. (1)
. The result of the simulations () suggest that a correction algorithm taking into account the GRIN is required to produce accurate values of the asphericity.
Overall, the simulations show that the GRIN makes a significant contribution to the distortion of the posterior surface, particularly in its estimated asphericity. When the GRIN is considered, the simulated posterior surface radius and asphericity are in very good agreement with the measurements obtained through the anterior surface (). Also, the best reconstruction of the posterior lens surface from OCT images in comparison to the nominal surface (obtained by direct imaging of the posterior surface in “posterior up” position) is obtained when the optical distortion correction algorithm considers the GRIN (). Interestingly, the results of suggest that correction of the optical refraction by the anterior surface using a GRIN model produces a relatively small improvement for the radius of curvature over a model that assumes a homogeneous index of refraction. The presence of GRIN has a larger impact in the reconstruction of the peripheral areas of the lens, as the largest improvements occur for the asphericity estimates.
The results have important implications in the OCT imaging of the crystalline lens, as the presence of GRIN will alter the estimated shape of the posterior lens surface. Our results suggest that for in vitro OCT measurements, where the posterior lens surface can be visualized directly by flipping the lens (and therefore having a measurement of the undistorted and distorted posterior lens surface), the calculated distortion could be used to reconstruct the GRIN of the lens. We have recently demonstrated a method that uses the optical path difference distortions in OCT to reconstruct in vitro the 3-D GRIN distribution in the porcine lens [23
], and the 2-D GRIN distribution in the human lens of various ages.
In the current study, we used the Goncharov 3-variable model to describe the GRIN. While it is not the only possible GRIN lens model (we have obtained similar results with a different 3-variable model), choosing an adequate GRIN model that is representative of the actual lens gradient is critical. Simpler models, such as Goncharov’s 2-variable models, failed to reproduce the distortion of the posterior surface.
In summary, we show that the GRIN produces significant distortions of the posterior shape of the lens, particularly in the lens periphery. However, when imaging in vitro lenses, accurate values of the central radius of curvature can be obtained using a simple correction that does not take into account refraction. The distortions can be predicted and corrected using a ray-tracing algorithm that incorporates an adequate model of the GRIN of the lens. Correction algorithms that assume a homogeneous index provide accurate values of the radius of curvature, but not of the asphericity. It is important to remember that these findings are applicable to in vitro studies. When imaging the lens in vivo, refraction by the cornea may induce significant additional distortions in both the radius and the asphericity.