A mathematical model to predict endothelial cell density following penetrating keratoplasty with selective dropout from graft failure

Abstract

Purpose: We constructed several mathematical models that predict endothelial cell density (ECD) for patients after penetrating keratoplasty (PK) for a moderate-risk condition (principally Fuchs' dystrophy or pseudophakic/aphakic corneal edema).

Methods: In a subset (n = 591) of Cornea Donor Study participants, postoperative ECD was determined by a central reading center. Various statistical models were considered to estimate the ECD trend longitudinally over 10 years of follow-up. A biexponential model with and without a logarithm transformation was fit using the Gauss-Newton nonlinear least squares algorithm. To account for correlated data, a log-polynomial model was fit using the restricted maximum likelihood method. A sensitivity analysis for the potential bias due to selective dropout was performed using Bayesian analysis techniques.

Results: The three models using a logarithm transformation yield similar trends, whereas the model without the transform predicts higher ECD values. The adjustment for selective dropout turns out to be negligible. However, this is possibly due to the relatively low rate of graft failure in this cohort (19% at 10 years). Fuchs' dystrophy and pseudophakic/aphakic corneal edema (PACE) patients had similar ECD decay curves, with the PACE group having slightly higher cell densities by 10 years.

Conclusions: Endothelial cell loss after PK can be modeled via a log-polynomial model, which accounts for the correlated data from repeated measures on the same subject. This model is not significantly affected by the selective dropout due to graft failure. Our findings warrant further study on how this may extend to ECD following endothelial keratoplasty.

Keywords: endothelial cell density; mathematical model; penetrating keratoplasty.

© ARVO.

Figures

Figure 1

Endothelial cell density over time for all subjects (n = 591) and restricted to subjects with a surviving graft and a gradable image at 10 years (n = 176). Vertical axis is on a logarithmic scale. In each box, the black dot indicates the geometric mean, horizontal lines inside the box indicate medians, and the bottom and top of the boxes indicate the 25th and 75th percentiles, respectively.

Figure 2

Endothelial cell density over time with modeled curves – untransformed scale (A) and a logarithmic scale (B) for the vertical axis. In each box, the black dot indicates the mean (A) or geometric mean (B), horizontal lines inside the box indicate medians, and the bottom and top of the boxes indicate the 25th and 75th percentiles, respectively.

Figure 3

Endothelial cell density over time with modeled curves – Fuchs' dystrophy diagnosis (A), PACE diagnosis (B), and the Bayesian MCMC models for Fuchs' dystrophy and PACE (C). In each box in (A) and (B), the black dot indicates the mean, horizontal lines inside the box indicate medians, and the bottom and top of the boxes indicate the 25th and 75th percentiles, respectively.