Abstract
purpose. To evaluate a biexponential decay model for describing the loss of corneal endothelial cells with age as well as the increased loss of cells after cataract surgery and penetrating keratoplasty.
methods. Data from previous studies were identified and the sum of two exponentials, d = p · exp(−a t) + q · exp(−b t) (where d is cell density at time t, p and q are constants the sum of which is equal to the initial cell density, and a and b are exponential rate constants), fitted to each data set by a nonlinear least-squares algorithm. Goodness of fit was indicated by the residual standard deviation. Half times were calculated from the exponential rate constants.
results. The model identified in each instance a rapid and a slow component to the cell loss. The half time for the slow component of the loss with age was 224 years, underlining the excess endothelial capacity in normal eyes. After surgery, the rapid component of the cell loss was probably due to surgical trauma and, after penetrating keratoplasty, cell-mediated rejection and other complications. The half times of the slow component were only 26 years after cataract surgery and 21 years after penetrating keratoplasty.
discussion. The loss of endothelial cells followed a biexponential decay and could thus be described by a single equation. The half times of the slow component of the cell loss after surgery were substantially less than for the loss with age, indicating a markedly increased rate of cell loss that persisted for many years after surgery. A mechanism for this accelerated cell loss is suggested that involves a nonspecific, innate response initiated by the breakdown of the blood–ocular barrier. The model was used to calculate endothelial cell loss in the long term after penetrating keratoplasty and to predict when cell density would reach levels that are incompatible with maintenance of transparency and graft function. Thus, a rationale is presented for the setting of minimum donor cell densities by eye banks.
Human corneal endothelial cells undergo mitotic division only rarely in situ, and there is a well-documented, gradual decline in endothelial cell density with increasing age.
1 2 3 As cell density decreases, there is a corresponding increase in cell surface area as the remaining cells spread to maintain the barrier properties of the endothelial mosaic. After penetrating keratoplasty, the loss of endothelial cells is greatly exacerbated. Although especially marked in the short term, this abnormal cell loss persists, albeit at a much reduced rate, for many years after transplantation.
4 This potentially jeopardizes the long-term survival of grafts as the cell density declines toward levels, variously reported to range from 250 to 500 cells/mm
2 5 6 that are inconsistent with the control of stromal hydration and maintenance of corneal transparency. It has been suggested that a low donor endothelial cell density is a risk factor for long-term endothelial failure.
4 If that is indeed the case, then a mathematical description of the postoperative cell loss in corneal grafts may allow long-term survival to be predicted and provide a rational basis for setting a minimum donor endothelial cell density compatible with long-term survival.
Because many biological processes are first-order, changing with time in an exponential manner, Redmond et al.
7 modeled endothelial cell loss in grafts with up to 4 years’ follow-up, with the equation,
D t =
D 0.
e −λt, where
D 0 and
D t are, respectively, the initial cell density and cell density at postoperative time
t, and λ is the exponential rate constant. The half-time for cell loss, calculated from the exponential rate constant, was only 3.4 years. This suggests that cell density could approach critical levels in only 7 to 10 years, which is somewhat sooner than longer term observations indicate.
4 Use of a single phenomenologic exponential coefficient therefore appears unable to account for the likelihood that different mechanisms of cell loss predominate in the early and late postoperative phases. Intuitively, the initially high loss of endothelium could be attributed to adverse events such as surgical trauma, cell-mediated rejection episodes, and other postoperative complications. With time, these early mechanisms of cell loss would have a diminishing influence, and the underlying slow attrition of cells, albeit at a higher than normal rate, would predominate. Similarly, after cataract surgery, surgical trauma would cause and early rapid loss of cells, but an accelerated cell loss also persists for many years after surgery.
8 Thus, two components to the cell loss may be defined after surgery: a rapid component that dominates the early postoperative period and a slow component that persists for many years.
Even with the decline in cell density with age in nondiseased corneas, there is evidence of more than one mechanism for the decrease. Møller-Pedersen
3 found a higher rate of loss in the younger age groups. His data suggest an annual loss of 2.9% up to 14 years of age but only 0.3% after 14 years. Clearly, the mechanisms are different from those responsible for the cell loss after surgery; but, if it is assumed that there are indeed two components to the cell loss, both after surgery and with increasing age, then a biexponential decay model may be considered more appropriate than the monoexponential equation used by Redmond et al.
7 Our purpose, therefore, was to evaluate a biexponential decay model for describing the normal endothelial cell loss with age, as well as the postoperative cell loss observed after both penetrating keratoplasty and cataract surgery.
The biexponential model we have presented is not put forth as definitive, rather the goal was to present an approach that may, as more data become available, lead to a robust mathematical description of postoperative cell loss. Other mathematical models could have been chosen. However, when the monoexponential used by Redmond et al.
7 was applied to long-term follow-up data, it underestimated early cell loss and significantly overestimated cell loss in the long term (data not presented). Alternatively, Møller-Pedersen
3 used a two-phase linear model that required dividing the data into two groups to which separate lines were fitted by linear regression. The advantage of using the biexponential model is that a single curve is fitted to the entire data set without the need arbitrarily to divide the data.
Clearly, there are deficiencies in the fitting procedure for the corneal graft data. Each time point does not represent a series of independent measurements, nor, given that these are repeated measures, are the same grafts represented at every time point. These factors could have two adverse effects on the reliability of the model. First, the early phase of cell loss may be exaggerated by cell counts in grafts that undergo rapid cell loss and early failure. Second, the data for the longest postoperative time are the least reliable (i.e., lowest number of observations), yet are likely to have a disproportionately large influence on the fitted curve and, hence, on the estimates for the rate constants. This means that extrapolation much beyond the final measured time point could become increasingly unreliable. It is also uncertain how the undoubted variations in the distributions of both patient age and diagnosis at each time point would influence the overall pattern of decreasing cell density.
Earlier data from Bourne et al.
10 and Ing et al.,
11 who reported measurements on cell densities in grafts for which the same patients were measured at every time point, were therefore compared with the cell densities predicted by the model
(Table 1) . Notably, both sets of data included only grafts with no reported rejection episodes and no reoperations affecting the corneal endothelium. These two cohorts are subsets of the cohort used to create the model. Despite this overlap, the agreement appears to be good, with the largest differences evident at two months after surgery. At all other time points, the deviations from the predicted values varied by only a few percentage points and both sets of data were well within the 95% prediction limits.
A Rationale for a Minimum Donor Endothelial Cell Density for Penetrating Keratoplasty
The loss of endothelial cells and indeed graft survival will initially be dominated by complications such as vascularization, inflammation and cell-mediated rejection. The indication for the graft will itself strongly influence graft survival, especially in the first few years after transplantation. In the longer term, however, Bourne
4 found that grafts that had endothelial failure had not lost cells any faster than grafts that did not fail; rather, the failed grafts had lower initial donor endothelial cell densities and simply reached cell densities incompatible with graft function sooner. The time taken to reach a putative critical cell density may be estimated by extrapolation of the current model
(Fig. 3) . If the critical density is taken to be 500 cells/mm
2, it can be seen from
Figure 4 that corneas with initial cell densities lower than 2000 cells/mm
2 could reach the critical density in less than 20 years. With initial densities above 2500 cells/mm
2, however, the grafts should remain functioning for at least 30 years. These extrapolations are based on the upper limit of critical cell density reported in the literature
5 6 : assuming a lower critical density would increase graft survival times accordingly.
In those European eye banks that use organ culture, the suitability of corneas for penetrating keratoplasty is determined by endothelial cell density rather than by donor age, which allows corneas from older donors (>70 years) to be made available for transplantation.
18 Minimum donor cell densities are typically 2000 to 2500 cells/mm
2. Provided that grafts survive beyond the early complications, the model suggests that donor cell densities within this range should provide sufficient endothelial capacity for at least 20 years of graft survival before the upper limit (i.e., 500 cells/mm
2) of the critical range of cell density is reached
(Fig. 4) .
In summary, a biexponential decay model was used to describe the change in endothelial cell density with increasing age in nondiseased eyes, and the postoperative decline in cell density after cataract surgery and penetrating keratoplasty. The intention was to suggest an approach that may in time produce a robust mathematical description of endothelial cell loss. Only with the gathering of more extensive data can the central assumptions of the model be properly tested, principally that the underlying slow rate of attrition is independent of the initial cell density. The impact of other factors, such as donor age and storage method of corneal grafts, also should be evaluated. Even in its present form, however, insights may be gained and questions posed about the mechanisms of cell loss and the similarity between the loss after cataract surgery and penetrating keratoplasty. Moreover, the 10-fold decrease suggested by the model in the half time of the slow component of cell loss after surgery compared with that of nondiseased eyes emphasizes that this accelerated cell loss is evident for many years after the surgical intervention. Finally, although acknowledging the constraints of the current model and the uncertainty of extrapolation, it provides a rationale for setting a minimum donor endothelial cell density for use in eye banks.
Supported in part by National Eye Institute Grant EY02037 (WMB).
Submitted for publication December 9, 2002; revised February 21, 2003; accepted February 26, 2003.
Disclosure:
W.J. Armitage, None;
A.D. Dick, None;
W.M. Bourne, None
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “
advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Corresponding author: W. John Armitage, Division of Ophthalmology, Bristol Eye Hospital, Bristol BS1 2LX, UK;
w.j.armitage@bristol.ac.uk.
Table 1. Postoperative Endothelial Cell Density after Penetrating Keratoplasty
Table 1. Postoperative Endothelial Cell Density after Penetrating Keratoplasty
Postoperative Time (mo) | Endothelial Cell Density (cells/mm2) | | | | |
| Calculated (95% PI)* | Study A, † | Δ (%), ‡ | Study B, § | Δ (%) |
0 | 2854 (2499–3210) | 2956 (546) | +3 | 3019 (594) | +6 |
2 | 2630 (2276–2985) | 2417 (756) | −8 | 2366 (801) | −10 |
12 | 1909 (1557–2260) | 1998 (705) | +5 | 1969 (735) | +3 |
36 | 1348 (1003–1693) | 1418 (600) | +5 | 1373 (551) | +2 |
60 | 1199 (858–1540) | 1214 (533) | +1 | 1171 (489) | −2 |
120 | 1010 (666–1353) | — | | 958 (471) | −5 |
Table 2. Summary of Half Times for the Rapid and Slow Components of Exponential Cell Loss with Age, after Cataract Surgery and after Penetrating Keratoplasty
Table 2. Summary of Half Times for the Rapid and Slow Components of Exponential Cell Loss with Age, after Cataract Surgery and after Penetrating Keratoplasty
| Half Times (years) | |
| Rapid | Slow |
Age | | |
Møller-Pedersen 3 | 3.00 | 224 |
Yee et al. 2 | 4.00 | 277 |
Cataract* | 0.04 | 26 |
Penetrating keratoplasty, † | 0.70 | 21 |
The authors thank Stephen Kaye, Monica Berry, and Andrew Tullo for helpful comments and stimulating discussion.
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