**Purpose.**:
To develop a mathematical model that can predict refractive changes after Descemet stripping endothelial keratoplasty (DSEK).

**Methods.**:
A mathematical formula based on the Gullstrand eye model was generated to estimate the change in refractive power of the eye after DSEK. This model was applied to four DSEK cases retrospectively, to compare measured and predicted refractive changes after DSEK.

**Results.**:
The refractive change after DSEK is determined by calculating the difference in the power of the eye before and after DSEK surgery. The power of the eye post-DSEK surgery can be calculated with modified Gullstrand eye model equations that incorporate the change in the posterior radius of curvature and change in the distance between the principal planes of the cornea and lens after DSEK. Analysis of this model suggests that the ratio of central to peripheral graft thickness (CP ratio) and central thickness can have significant effect on refractive change where smaller CP ratios and larger graft thicknesses result in larger hyperopic shifts. This model was applied to four patients, and the average predicted hyperopic shift in the overall power of the eye was calculated to be 0.83 D. This change reflected in a mean of 93% (range, 75%–110%) of patients' measured refractive shifts.

**Conclusions.**:
This simplified DSEK mathematical model can be used as a first step for estimating the hyperopic shift after DSEK. Further studies are necessary to refine the validity of this model.

^{ 1 –3 }and Descemet stripping automated endothelial keratoplasty (DSAEK)

^{ 4 }are lamellar corneal surgical techniques used to replace abnormal corneal endothelium in patients with endothelial diseases, such as Fuchs' corneal endothelial dystrophy and pseudophakic bullous keratopathy. The abnormal corneal endothelium is removed and replaced with donor posterior lamellae of cornea containing healthy endothelium. DSEK offers several advantages over traditional penetrating keratoplasty, including faster visual recovery and less postoperative astigmatism, because the structural integrity of the eye is largely maintained.

^{ 4 –6 }This shift should be considered during preoperative planning, in particular when choosing the power of the intraocular lens in combination surgery of DSEK along with phacoemulsification and intraocular lens placement. However, no quantitative model exists to explain or predict the hyperopic shift that results after placement of the donor graft. Because DSEK is largely a posterior corneal procedure, studies have suggested that the hyperopic shift occurs because of a change in the posterior corneal curvature.

^{ 7 }In a previous study, Yoo et al.

^{ 8 }demonstrated a linear correlation between the ratio of central to peripheral thickness (CP ratio) of the donor graft and the resulting postsurgical hyperopic shift. However, the mathematical reason for such a correlation is not well understood.

Variable | Description | Average Value/Figure | Standard Gullstrand Eye Model Variable |
---|---|---|---|

F _{eye} | Power of the eye | 58.6 D | Yes |

F _{cornea} | Power of the cornea | 43.0 D | Yes |

F _{lens} | Power of the lens | 19.1 D | Yes |

d | Distance between second principal plane of the cornea and the first principal plane of the lens | 0.0057 m | Yes |

F _{ac} | Power of the anterior cornea | 48.8 D | Yes |

F _{pc} | Power of the posterior cornea | −5.88 D | Yes |

t | Thickness of recipient cornea | 0.0005 m | Yes |

t _{transplant} | Thickness of corneal transplant graft | Figures 2, 3, 6 7–8 | No |

n _{3} | Index of refraction of aqueous humor | 1.336 | Yes |

n _{2} | Index of refraction of cornea | 1.376 | Yes |

n _{1} | Index of refraction of air | 1.00 | Yes |

r _{ac} | Radius of curvature of anterior cornea | 0.0077 m | Yes |

r _{pc} | Radius of curvature of posterior cornea | 0.0068 m | Yes |

r _{pc}′ | Radius of curvature of posterior cornea with transplant of even width throughout | Figures 2, 3, A1 | No |

r _{pc}″ | Radius of curvature of posterior cornea with peripherally thicker donor cornea | Figures 3, 6, 7, A1 | No |

h′ | Shortest distance between a point on the periphery of a uniform width corneal graft and the visual axis | 0.0015 m was used in this paper, Figures 6, 7, A1 | No |

h″ | Shortest distance between a point on the periphery of a nonuniform width corneal graft and the visual axis | 0.0015 m used in this paper, Figures 3, 6, 7, A1 | No |

- The shape of the anterior curvature of the recipient cornea does not change after surgery.
- The shape of the posterior curvature (excluding the corneal graft) of the recipient cornea does not change after surgery.
- The thickness of the recipient cornea does not change after endothelial stripping (i.e., removal of endothelium does not affect recipient corneal thickness).
- The corneal graft's radius of curvature can be estimated at a distance
*h*′ = 1.5 mm from the visual axis. The visual axis is defined as the axis that passes through the geometric center or vertex of the cornea. - Calculations using the radius of curvature, sagittal depth, and chord length (sag equation) assume that the sag is much less than the radius of curvature.
- The recipient cornea and donor graft are symmetric around the visual axis.
- The variation in thickness of the transplant is symmetrical around the visual axis.
- The donor cornea, when attached to the recipient cornea, behaves as one uniform refractive medium.
- The refractive index of the donor cornea is the same as that of the recipient cornea.
- The index of refraction of cornea and air does not change after surgery.
- The angle q (the angle between the radius of curvature and the visual axis) in Figure A1 is much less than 1 so that sin θ ∼ 0 and cos θ ∼ 1.
- The posterior corneal radius of curvature measured from postsurgical anterior segment optical coherence tomography (AS-OCT) readings approximates presurgical corneal posterior radius of curvatures.
- Intraocular lens power calculations are precise and accurate.

^{ 8,9 }This theory is supported by the observation that the corneal posterior radius of curvature changes after DSEK surgery

^{ 10 }and the CP ratio of transplanted corneal tissue correlates with refractive change after DSEK surgery.

^{ 8 }

*F*

_{pc}, in diopters), which is given by

^{ 11 }where

*n*

_{3}is the index of refraction of the aqueous humor,

*n*

_{2}is the index of refraction of the cornea, and

*r*

_{pc}is the radius of curvature of the posterior cornea (in meters). Reference values are listed in Table 1.

^{ 11 }

*r*

_{pc}′). Any deviation from parallelism will cause additional change, the value of which we wish to predict, in the power of the eye. By mathematically understanding how the donor CP ratio (Fig. 3) alters the geometry of the recipient cornea after surgery, the resulting change in the posterior radius of curvature can be determined.

*r*

_{pc}can be visualized by drawing a circle with radius

*r*

_{pc}(Fig. 2). Because a uniformly wide donor corneal surface would lie parallel to the recipient posterior surface, the post-DSEK radius of curvature with a uniform width graft is given by

*r*

_{pc}′ =

*r*

_{pc}−

*t*

_{transplant}, where

*t*

_{transplant}is the thickness of the donor graft (Fig. 2).

^{ 6,8 }(Fig. 3). As the thickness along the periphery of the posterior cornea increases with the graft's peripheral thickness, the posterior radius of curvature decreases (Fig. 3). To help understand this change in radius of curvature, two other quantities can be determined: the chord length and the sagittal depth. One half of the chord length (

*h*″ in Fig. 3) represents the shortest vertical distance between a peripheral point on the posterior post-DSEK corneal surface and the visual axis. The sagittal depth (

*s*

_{i}″ in Fig. 3) represents the shortest horizontal distance along the visual axis between the chord length and the central region of the posterior post-DSEK cornea.

^{ 12 }):

*r*

_{pc}= (0.5 · chord length)

^{2}/(2 · sagittal depth). Because presurgical AS-OCT and anterior eye segment measurements (Pentacam; Oculus, Wetzlar, Germany) were not available for these patients, their presurgical posterior radii of curvatures were estimated using the sag equation and post-DSEK AS-OCT measurements of sagittal depth and chord length, excluding donor graft thickness. The recipient's posterior radius of curvature was assumed to be unchanged before and after surgery and to be unaffected by endothelial stripping (assumptions 2 and 3).

*r*

_{pc}″ when compared with the curvature of an ideal donor cornea with even thickness throughout

*r*

_{pc}′. After the difference in the sag between a donor cornea of even thickness (CP ratio = 1) and a nonuniform width donor cornea (with a CP ratio not equal to 1) is calculated, the nonuniform post-DSEK radius of curvature,

*r*

_{pc}″, can be estimated by the equation (derivation shown in the 1) Because the peripheral thicknesses of each graft used in the graft CP ratio calculations were taken using pachymetry 1.5 mm from the vertex, we used

*h*′ = 1.5 mm throughout this study. The calculated post-DSEK radius of curvature can then be used to calculate the power of the post-DSEK posterior cornea (equation 1).

- Measure corneal thickness (e.g., pachymetry).
- Calculate the presurgical corneal posterior radius of curvature using the equation (0.5 · chord length)
^{2}/(2 · sagittal depth), which is derived from the sagittal depth (sag) equation (Figs. 4, 6). The chord length and sagittal depth in the sag equation are measured from a presurgical AS-OCT image. Alternatively, the posterior radius of curvature can be directly measured by using an anterior segment evaluation (Pentacam; Oculus). - Obtain presurgical graft thickness and CP ratios using pachymetry and data provided by the eye bank that provides the corneal transplants.
- Use the equations given below to estimate the refractive changes in the power of the eye after DSEK surgery.To estimate the refractive changes in the power of the eye after DSEK, first determine the corneal power, which depends on the central graft thickness and is given by where
*F*_{cornea+DSEK}is the power of the cornea after DSEK,*F*_{ac}is the power of the anterior cornea,*F*_{pc+DSEK}is the power of the posterior cornea after DSEK, and*t*is the distance between the anterior and posterior surfaces of the recipient cornea (corneal thickness). The transplant thickness is added to the host corneal thickness to obtain the new combined corneal thickness (*t*+*t*_{transplant}). In equation 3, where*n*_{1}is the index of refraction of air and*r*_{ac}is the radius of curvature of the anterior cornea (average*r*_{ac}is 0.0077 m^{11}), where Finally, the overall power of the eye with the graft is given by where*F*_{eye+DSEK}equals the power of the eye after DSEK,*F*_{cornea+DSEK}equals the power of the cornea after DSEK,*F*_{lens}equals the power of the lens (the average lens is estimated to be 19 D^{11}or it can be replaced with the pseudophakic IOL power), and The derivation will be shown below in step A1. A summary of the above equations and sample calculations are shown in Figures 6 and 7. - Next, the power of the eye
*without*the graft is calculated by using the same equations, where*t*_{transplant}equals 0, the CP ratio is 1, and*r*_{pc}″ equals the measured presurgical corneal posterior radius of curvature. - The difference between the two powers is calculated
*F*_{eye+DSEK}−*F*_{eye without DSEK}. This difference is the predicted refractive change after DSEK surgery.

*d*equals the distance between the second principal plane of the cornea and the first principal plane of the lens (1st principal plane of lens − 2nd principal plane of cornea). The first principal plane of a normal lens is approximately 0.005678 m posterior to the cornea.

^{ 11 }The location of the second principal plane of the cornea is given by The point of reference is from the vertex of the anterior cornea. Negative values represent positions in front of the anterior cornea (outside the eye), and positive values represent positions behind the anterior cornea (inside the eye).

Pt | Procedure | Pre-op Central Corneal Thickness (μm) | Pre-op Central Graft Thickness (μm) | Pre-op Graft Peripheral Thickness (μm) | Pre-op Graft C:P Ratio | Posterior Radius of Curvature without Graft (mm) | Predicted Posterior Radius of Curvature with Graft (mm) | Measured Refractive Change (D) | Predicted Hyperopic Corneal Refractive Change (D) | Predicted Hyperopic Shift of Eye (D) |
---|---|---|---|---|---|---|---|---|---|---|

1 | DSEK | 650 | 142 | 159, 167, 163, 151 | 0.89 | 5.27 | 4.74 | 1.0 | 0.79 | 0.75 |

2 | DSEK | 816 | 129 | 148, 156, 169, 132 | 0.85 | 5.95 | 5.22 | 0.75 | 0.88 | 0.83 |

3 | DSEK | 573 | 147 | 129, 163, 199, 186 | 0.87 | 5.80 | 5.09 | 1.0 | 0.91 | 0.86 |

4 | Triple | 764 | 95 | 112, 86, 141, 140 | 0.79 | 6.19 | 5.37 | 0.82 | 0.93 | 0.88 |

Pt | Procedure | Post-op Corneal Thickness without Graft (μm) | Measured Post-op Radius of Curvature with Graft (mm) | Post-op Graft Thickness (μm) | Post-op Graft C:P Ratio | Measured Refractive Change (D) | Age | Time Post-op |
---|---|---|---|---|---|---|---|---|

1 | DSEK | 600 | 5.35 | 150 | 0.88 | 1.0 | 85 | 2 years (23 months) |

2 | DSEK | 490 | 5.71 | 150 | 0.83 | 0.75 | 56 | 2 years (22 months) |

3 | DSEK | 580 | 5.00 | 200 | 0.89 | 1.0 | 73 | 5 months |

4 | Triple | 630 | 5.34 | 110 | 0.88 | 0.82 | 72 | 3 months |

^{ 5 –8,13,14 }with averages close to 1.0 D. In particular, the posterior radius of curvature has been suggested to be the main refractive component that changes after surgery.

^{ 7,10 }This change can be modeled by calculating the change in sag of the presurgical posterior corneal surface to estimate the change in posterior corneal power after DSEK surgery. We developed a mathematical model to estimate the hyperopic shift after DSEK surgery. The variables for the model include preoperative corneal thickness, graft thickness, graft CP ratio, and recipient posterior corneal curvature.

Corneal Thickness (μm) | C:P Ratio | Transplant Thickness (μm) | Predicted Posterior Radius of Curvature (mm) | Predicted Change in Corneal Power (D) | Predicted Change in Eye Power (D) | Predicted Hyperopic Refractive Shift (D) |
---|---|---|---|---|---|---|

500 | — | 0 | 6.80 | 0.00 | 0.00 | 0.00 |

500 | 0.7 | 100 | 5.34 | −1.56 | −1.45 | +1.45 |

500 | 0.7 | 105 | 5.28 | −1.63 | −1.52 | +1.52 |

500 | 0.7 | 110 | 5.23 | −1.71 | −1.59 | +1.59 |

500 | 0.7 | 115 | 5.17 | −1.79 | −1.67 | +1.67 |

500 | 0.7 | 120 | 5.12 | −1.87 | −1.74 | +1.74 |

500 | 0.7 | 125 | 5.07 | −1.94 | −1.81 | +1.81 |

500 | 0.7 | 130 | 5.01 | −2.02 | −1.88 | +1.88 |

500 | 0.7 | 135 | 4.96 | −2.10 | −1.96 | +1.96 |

500 | 0.7 | 140 | 4.91 | −2.18 | −2.03 | +2.03 |

500 | 0.7 | 145 | 4.87 | −2.25 | −2.10 | +2.10 |

500 | 0.7 | 150 | 4.82 | −2.33 | −2.17 | +2.17 |

500 | 0.7 | 155 | 4.77 | −2.41 | −2.25 | +2.25 |

500 | 0.7 | 160 | 4.73 | −2.49 | −2.32 | +2.32 |

500 | 0.7 | 165 | 4.68 | −2.56 | −2.39 | +2.39 |

500 | 0.7 | 170 | 4.64 | −2.64 | −2.46 | +2.46 |

500 | 0.7 | 175 | 4.60 | −2.72 | −2.53 | +2.53 |

500 | 0.7 | 180 | 4.55 | −2.80 | −2.61 | +2.61 |

500 | 0.7 | 185 | 4.51 | −2.87 | −2.68 | +2.68 |

500 | 0.7 | 190 | 4.47 | −2.95 | −2.75 | +2.75 |

500 | 0.7 | 195 | 4.43 | −3.03 | −2.82 | +2.82 |

500 | 0.7 | 200 | 4.39 | −3.10 | −2.90 | +2.90 |

Corneal Thickness (μm) | C:P Ratio | Transplant Thickness (μm) | Predicted Posterior Radius of Curvature (mm) | Predicted Change in Corneal Power (D) | Predicted Change in Eye Power (D) | Predicted Hyperopic Refractive Shift (D) |
---|---|---|---|---|---|---|

500 | — | 0 | 6.8 | 0.00 | 0.00 | 0.00 |

500 | 1.0 | 100 | 6.70 | −0.07 | −0.07 | +0.07 |

500 | 0.985 | 100 | 6.64 | −0.12 | −0.12 | +0.12 |

500 | 0.97 | 100 | 6.58 | −0.17 | −0.17 | +0.17 |

500 | 0.955 | 100 | 6.52 | −0.23 | −0.22 | +0.22 |

500 | 0.94 | 100 | 6.45 | −0.29 | −0.27 | +0.27 |

500 | 0.925 | 100 | 6.39 | −0.35 | −0.33 | +0.33 |

500 | 0.91 | 100 | 6.33 | −0.41 | −0.39 | +0.39 |

500 | 0.895 | 100 | 6.26 | −0.47 | −0.45 | +0.45 |

500 | 0.88 | 100 | 6.20 | −0.54 | −0.51 | +0.51 |

500 | 0.865 | 100 | 6.13 | −0.61 | −0.57 | +0.57 |

500 | 0.85 | 100 | 6.06 | −0.68 | −0.64 | +0.64 |

500 | 0.835 | 100 | 5.99 | −0.75 | −0.70 | +0.70 |

500 | 0.82 | 100 | 5.93 | −0.83 | −0.77 | +0.77 |

500 | 0.805 | 100 | 5.86 | −0.91 | −0.85 | +0.85 |

500 | 0.79 | 100 | 5.78 | −0.99 | −0.92 | +0.92 |

500 | 0.775 | 100 | 5.71 | −1.08 | −1.00 | +1.00 |

500 | 0.76 | 100 | 5.64 | −1.16 | −1.09 | +1.09 |

500 | 0.745 | 100 | 5.57 | −1.26 | −1.17 | +1.17 |

500 | 0.73 | 100 | 5.49 | −1.35 | −1.26 | +1.26 |

500 | 0.715 | 100 | 5.41 | −1.45 | −1.35 | +1.35 |

500 | 0.7 | 100 | 5.34 | −1.56 | −1.45 | +1.45 |

500 | 0.685 | 100 | 5.26 | −1.67 | −1.55 | +1.55 |

500 | 0.67 | 100 | 5.18 | −1.78 | −1.66 | +1.66 |

^{ 6 }and that graft thinning stabilizes after ∼6 months.

^{ 9 }Furthermore, a prospective study showed that the central pachymetry was significantly decreased from 0.70 to 0.66 mm

^{ 15 }6 months after DSEK surgery. Additional studies are needed to determine how much graft CP ratios and recipient corneal thicknesses change 6 months after surgery, when compared to their preoperative thicknesses, and whether they stabilize or continue to change beyond 6 months. The four cases studied, with a range of postoperative measurements ranging from 3 months to 2 years, suggest that the CP ratio does not change dramatically over time. Finally, a larger prospective study would be helpful in determining the true predictive accuracy of the mathematical model.

^{ 4 –6,10 }The model demonstrates that the hyperopic shift depends on at least four variables, and that can explain why correlations between two variables (e.g., graft thickness with hyperopic shift) may not appear significant if other variables are not held constant. Furthermore, this is also the first model to predict refractive changes based on an individual's presurgical measurements.

*r*

_{pc}′ and

*r*

_{pc}″, we find where the approximation in equation 13 uses equation 10. Taking the difference of the two sags (

*s*

_{i}″ −

*s*

_{i}′) results in Equation 11 can be rewritten as Note that the difference at the periphery of the donor graft (between the graft of nonuniform thickness and the idealized graft with even thickness throughout) can be defined as where CP is the central to peripheral ratio of the donor cornea at height

*h*″ away from the central axis of the donor cornea and

*t*

_{transplant}is the transplant thickness at the vertex or central region. A value of

*w*= 0 means that the graft is of uniform width.

*s*

_{i}″ −

*s*

_{i}′) replaced with

*t*

_{transplant}[(1/CP) − 1], because both are equal to

*w*, and the equation can be rewritten to determine the new radius of curvature which is given by Since

*r*

_{pc}′ =

*r*

_{pc}−

*t*

_{transplant},