2.1.

Experimental Setup

For the measurements, we used a clinical spectral-domain OCT system (SD-OCT) (Spectralis OCT, Heidelberg Engineering, Heidelberg, Germany) with a central wavelength of $\lambda =870\mathrm{nm}$. To allow imaging of the cornea, it was equipped with an add-on objective lens (Anterior Segment Module, Heidelberg Engineering, Heidelberg, Germany). The rectangular polystyrene tissue cultivation flasks (Primaria $25{\mathrm{cm}}^2$ Canted-Neck Cell Culture Flask, Corning Inc., New York) containing the corneoscleral grafts [Fig. 2(a)] were mounted in front of the OCT system [Fig. 2(b)].

The corneal grafts were gathered from the LIONS cornea bank Saar-Lor-Lux, Trier/Westpfalz in Homburg/Saar and measured inside the sterile culture flask. The graft is fixated onto a polyethylene cornea holder with four brackets [Bausch + Lomb, Heidelberg, Germany; Fig. 2(a)]. The measurements were conducted at room temperature of $\sim 23°\mathrm{C}$ and within 1 day after reimmersement because corneal metabolism particles obfuscate the culture medium, and the folding distorts the posterior surface topography. After measurement, the culture flasks were returned to the storing environment at 36°C. The storage and transportation were done according to the German regulations for cornea banking. According to these the allowed temperature range in the second medium is between 10°C and 40°C.¹⁰

The data were collected on the basis of a volume reconstruction of cross-sectional images through the posterior surface of the cornea (Fig. 3). The OCT was manually aligned to capture the central part of the cornea. The average penetration depth was 1.927 mm and the axial resolution $3.893\mu \mathrm{m}$ across the corneal diameter of 8.3 mm [Fig. 4(a)]. The horizontal resolution was $10.821\mu \mathrm{m}$. In total, the measured volume consisted of a vertical sequence of 81 cross-sectional images. With a distance of $69\mu \mathrm{m}$ between each cross-section, the area had a vertical expanse of 5.52 mm. After acquisition of the vertical array (scan 1), the scan window of the OCT was rotated by 90 deg with the sample in place, and the acquisition was repeated (scan 2). The two volume scans were exported as video files (image stacks) and combined in one coordinate system.

Fig. 3

OCT measurement software (Heidelberg Eye Explorer, Heidelberg Engineering, Heidelberg, Germany). The flask reflection is visible in the lower region of the left image. By tilting the culture flask, the reflected light portion is directed out of the scan area (blue grid in the IR reflectance image on the left side).

Fig. 4

Measuring scheme. (a) Scan area. The measured space was spanned of two overlapping measurements, each containing a $5.52\mathrm{mm}\times 8.3\mathrm{mm}$ array of 81 cross-sections (scan 1 and scan 2). The green and blue areas depict the resulting (ROI, $5.037\mathrm{mm}\times 5.66\mathrm{mm}$). (b) Sample image slice. Mirror artifacts are prevalent in the peripheral regions.

2.2.

Data Analysis

The data analysis was implemented in MATLAB (The MathWorks Inc., Massachusetts). The distinct steps of the procedure are shown in Fig. 5. At first, the region of interest (ROI) was applied to cutoff mirror artifacts in the periphery arising from the Fourier transform of the OCT fringe signal.³⁴ The central 5.66 mm from the horizontal dimension of the corneal diameter ($x$-axis) and the central 5.037 mm from the vertical dimension ($y$-axis) were selected as ROI from the exported image files [Fig. 4(b)]. The cropped images had a resolution of $568\times 496\text{pixels}$ with a scale of $10.821\mu \mathrm{m}/\text{pixel}$ laterally and $3.893\mu \mathrm{m}/\text{pixel}$ axially. Afterward, the cropped images were denoised, and the corneal volume was reconstructed as described in the following section. The two orthogonal measurements were placed in a common coordinate system [Fig. 4(a)] following an edge detection of the corneal surfaces. With the aid of a biconic surface fit, the two orthogonal radii of curvature and conic constants were determined.

Fig. 5

Data analysis procedure.

2.2.1.

Denoising and edge detection

Prior to the edge detection, the image data had to be refined from the remaining artifacts of the flask wall, reflections, and particles while maintaining the anterior and posterior corneal surfaces. For that reason, the background noise was eliminated with a ($5\times 5$) median filter. The number of particularly bright spots, such as reflections from the central corneal surfaces or mirrored artifacts from the back wall of the flask varied along the corneal diameter. The contrast range was adjusted by remapping the gray values of each image linearly to the interval [0, 1]. Clustered background noise components and mirror artifacts were selectively removed by a morphological opening. The size of the structuring element was chosen to be large enough to remove salty noise while being as small as possible avoiding corneal edge distortion.

Variations in image brightness toward the periphery were corrected by adjusting the contrast in relation to the distance from the center. Remaining particles were removed by deleting small objects [Fig. 6(c)]. Missing parts of the corneal tissue due to the shadowing were filled up by a top-hat-transformation. This morphological operation performs another opening with a disk-shaped structuring element with a radius of 45 pixels. It was applied to the inverted image. As a result, the shadowings were opened as bright spots. The resulting image was subtracted from the original image. In this manner, the shadowed areas were closed. Finally, the edges were detected using the Canny method,³⁵ starting with a low gradient threshold of 0.25 and including connected edge pixels up to a high gradient threshold of 0. The resulting edges were assembled as surface data.

Fig. 6

Denoising routine, part 2. (a) Original image. (b) Morphologically opened image. (c) Image after removal of small particles. (d) Top-hat transform of the complementary image of (c). (e) image (c)–(d). (f) Detected edge. Gray values were scaled and do not reflect actual image gray values. Axes are in pixels.

2.2.2.

Biconic surface fit

To determine two orthogonal radii of curvature $R_x$ and $R_y$ as well as the related conic constants ${\kappa }_x$ and ${\kappa }_y$, we expanded the well-known equation for the height data $z(x,y)$ of a biconic surface.

At first, the detected edges of the anterior and posterior surfaces from each scan were transformed into spatial coordinates. The penetration depth was defined as the $z$-axis, the horizontal expanse of each cross-section as the $x$-axis, and the vertical sequence of cross-sections as the $y$-axis. Then the corneal volume data of scan 1 and scan 2 were combined in one coordinate system $z(x,y)$ with $x=x_{\text{measurement}1}\vee y_{\text{measurement}2}$ and $y=y_{\text{measurement}1}\vee x_{\text{measurement}2}$.

To extract the required parameters, the biconic surface fit had to be independent from the spatial alignment of the corneoscleral button within the culture flask in relation to the measuring system.

For that reason, the spatial coordinates $x$ and $y$ were transformed to $x^{\prime }$ and $y^{\prime }$. The azimuthal rotation angle $\alpha$ of the orthogonal meridians around the $z$-axis was incorporated by multiplication of the coordinate vector $X={[xy]}^T$ with a rotation matrix $R$. Translational shifts $x_0$, $y_0$ of the anticipatory apex $z=0$ from the origin of the coordinate system were addressed by subsequent multiplication with a translation matrix T and yielded the new coordinate vector $X_{\text{new}}=XRT$. The slant of the surface plane due to the varying fixation of the culture flask in relation to the OCT device, and the floating of the corneal graft was included by adding two tilted planes with the tilt angles $\beta$ and $\gamma$. Furthermore, the depth offset $z_0$ of the apex was subtracted from the resulting biconic equation. The final equation $z_{\text{model}}(x^{\prime },y^{\prime },ϵ)$ yields 10 degrees of freedom in terms of the corneal parameters $R_x$, $R_y$, ${\kappa }_x$, and ${\kappa }_y$ and the spatial parameters $x_0$, $y_0$, $z_0$, $\alpha$, $\beta$, and $\gamma$, merged in the parameter vector $ϵ=[R_x,R_y,{\kappa }_x,{\kappa }_y{x}_0,y_0,z_{0,}\alpha ,\beta ,\gamma ]$:

Eq. (1)

$$z_{\text{model}}(x^{\prime },y^{\prime },ϵ)=\frac{\frac{x^{\prime 2}}{R_x}+\frac{y^{\prime 2}}{R_y}}{1+\sqrt{1-\frac{x^{\prime 2}}{R_x^2}(1+{\kappa }_x)-\frac{y^{\prime 2}}{R_y^2}(1+{\kappa }_y)}}-z_0+\tan (\beta )·(x-x_0)+\tan (\gamma )·(y-y_0).$$

The values of the corneal parameters for a given spatial setting were derived with an iterative nonlinear least squares fit with trust region algorithm. In the first step A of the fitting procedure (Fig. 7), the spatial parameters were estimated by fitting a rotationally symmetrical anterior and posterior surfaces with average radii of curvature ($R_x=R_y=7.8\mathrm{mm}$ for the anterior corneal surface and 6.4 mm for the posterior corneal surface) and conic constants (${\kappa }_x={\kappa }_y=-0.2$ anterior and $-0.4$ posterior)^36,37 to the measurement data. The initial spatial parameters of ${ϵ}_0$ were set to 0. The root-mean-square error (RMSE) was calculated, and outliers were excluded from the data set.

Fig. 7

Fitting procedure.

In the second step B, $z_0$ was assessed for the average cornea using the RMSE from step A as an initial parameter for ${ϵ}_1$. All the data points $z_i$ with $\text{residual}(i)>\mathrm{RMSE}·1.5$ were excluded. Afterward, the tilt angles $\beta$ and $\gamma$ were calculated for the average cornea with the $z_0$ value from step B. Data points $z_i$ with $\text{residual}(i)>\mathrm{RMSE}·1.2$ were discarded. The resulting data set $z_{i,\text{reduced}}(x_{\text{reduced}},y_{\text{reduced}})$ was used as the initial data set for steps D and E of the fitting procedure.

The previously probed values for $\beta$, $\gamma$, $x_0$, $y_0$, and $z_0$ were set as initial parameters for step D, and the limits of their 95% confidence interval were set as lower and upper constraints. Then $R_x$ and $R_y$ as well as $\alpha$ were calculated iteratively until the fit converged, i.e., $\frac{|{ϵ}_{\text{iteration}-1}^n-{ϵ}_{\text{iteration}}^n|}{{ϵ}_{\text{iteration}-1}^n}<{10}^{-6}$ for each parameter ${ϵ}^n$ of $ϵ=({ϵ}^1,\dots ,{ϵ}^m)$. Following each iteration, data points with $\text{residual}(i)>Q_{0.0125}$ or $\text{residual}(i)>Q_{0.9875}$ with $Q$ as the corresponding quantiles were ruled out. In the last fitting step, E, ${\kappa }_x$ and ${\kappa }_y$ were estimated while again using ${ϵ}_{\text{iteration}-1}$ as initial parameters and the limits of their 95% confidence interval as constraints until the fit converged. Once more, the data points of the upper and lower 1.25% quantiles of the residual were excluded from the initial data set $z_{i,\text{reduced}}$ after each iteration.

4.1.

Corneal Parameters

In this study, the geometrical properties of corneal grafts were assessed by means of an OCT-based screening method. The donor corneas were maintained in their sterile cultivation flasks within the eye bank’s storage procedure. Using a biconic surface fit, two orthogonal radii of curvature and the corresponding conic constants $\kappa$ were elicited from the anterior and posterior corneal surfaces, respectively. Refractive power and astigmatism were calculated in a way that allows direct comparison with in vivo measured corneas. The corneal thickness, particularly at the anterior apex (CCT), was derived from the determined corneal volume model. The median radii of curvature of $R_{\mathrm{f},\text{anterior}}=7.890\mathrm{mm}$, $R_{\mathrm{s},\text{anterior}}=7.705\mathrm{mm}$, $R_{\mathrm{f},\text{posterior}}=6.538\mathrm{mm}$, and $R_{\mathrm{s},\text{posterior}}=6.354\mathrm{mm}$ correspond to average horizontal and vertical corneal curvatures found in previous studies in vivo.^33,37,38 The mean CCT of $616\mu \mathrm{m}$ is slightly higher than the average value found in normal human corneas ($534\mu \mathrm{m}$).³⁹ This effect can be attributed to the cultivation process of the graft as the culture medium increases stromal hydration.⁴⁰ We conducted our measurements within 1 day after the grafts had been reimmersed in the deswelling medium, and in some cases, this time span might have been insufficient. The deswelling of organ-cultured corneal tissue was investigated by Schnitzler et al. who found that the condition of the endothelium might have a significant effect on the deswelling aside from the osmotic effect of the culture medium.²⁷ Thus, measuring corneal thickness before reimmersement and at a standardized time after reimmersement might be one indicative factor for endothelium activity.

Schnitzler et al. reported a minimal corneal thickness after 24 h of deswelling.²⁷ We chose a time span of 1 day as a good compromise between the time for deswelling and residual OCT image contrast.

One cornea was anamnestically reported to have undergone refractive surgery (cornea 1). The type of refractive surgery, however, was unknown. However, the corneal parameters, refractive power, and the CCT ($539.9\mu \mathrm{m}$) did not differ significantly from the median values. The visual inspection of cornea 1 did not reveal any signs for refractive surgery. Previous studies found that LASIK interfaces can be seen in the short term in the stromal reflectivity data as bright spots. For longer periods of time, the interface intensity fades and manifests in a lower internal reflectivity.²⁶ Lin et al. found the LASIK corneas to have flatter anterior curvature radii, which could not be detected in cornea 1 of this study.

Cornea 2 with the flattest anterior curvature radius showed a large discrepancy between the two curvature radii $R_{\mathrm{f}}=9.214\mathrm{mm}$ and $R_{\mathrm{s}}=6.099\mu \mathrm{m}$ and is, therefore, causing the extreme outlier within the astigmatism values (20.84 D). The corresponding conic constants of the anterior surface also show extreme values (${\kappa }_{\mathrm{f}}=-1.074$ and ${\kappa }_{\mathrm{s}}=-0.844$). Although the posterior surface is within the average range, this cornea exhibits the highest anterior refractive power ($D_{\mathrm{f}}=61.65\mathrm{dpt}$).

With cornea 3, all corneal parameters except for the CCT ($844.28\mu \mathrm{m}$) were close to the median values. However, the ratios of the anterior to posterior curvature radius (0.794 and 0.784 for the flatter and the steeper meridian, respectively) are lower than the average ratio (0.82).

4.2.

Fitting Procedure

Due to the manual mounting of the cultivation flask in front of the objective of the OCT, the exact position of the corneal graft was unknown and not reproducible. By involving rotation, tilt angles, and translation parameters, the automatic fitting procedure was made independent of the alignment of the donor cornea to the corneal holder and its positioning within the culture flask.

Because of the rotationally nonsymmetrical toroidal surface of the cornea, a general biconic representation was elected as a mathematical model for the nonlinear least squares fitting procedure. In contrast to the ellipsoidal model, the individual parameters are independent from each other.⁴¹ The denoising routine yielded a smooth edge that eliminates particles and reflections and preserves the corneal surface. During the fit, we used two different refinement criteria to exclude outliers in the residual data. The first restricted the data to 1.5·RMSE, the second one to 1.2·RMSE. These criteria were arbitrarily chosen as a compromise between quality of the fit and convergence of the fit while still considering a sufficient amount of data. On average, $29\pm 20$ iterations were necessary until the fit converged (minimum: 6, maximum: 95). The average RMSE was $15.13\mu \mathrm{m}\pm 13.30\mu \mathrm{m}$ (minimum: $3.72\mu \mathrm{m}$ and maximum: $69.32\mu \mathrm{m}$), and the highest values occurred when the number of iterations culminated.

Calculating the biconic surface with the average anterior radii of curvature ($R_{\mathrm{f}}=8.030\mathrm{mm}$ and $R_{\mathrm{s}}=7.624\mathrm{mm}$) and conic constants (${\kappa }_{\mathrm{f}}=-0.286$ and ${\kappa }_{\mathrm{s}}=-0.252$), a difference of $\mathrm{\Delta }\kappa =-0.1$ manifests in the peripheral regions of the measurement area ($-2.83\mathrm{mm}$ and 2.83 mm) in an elevation difference of $1.7\mu \mathrm{m}$. Considering the depth resolution of $3.893\mu \mathrm{m}$ and the mean RMSE of the fitting procedure, this calculation of $R$ and $\kappa$ is rather inaccurate compared with clinical measurement methods. The average human corneal diameter (horizontal limbus-to-limbus distance) is 11 mm.³⁶ Screening a larger, circular-shaped area would significantly improve the calculation of the conic constant. Preliminarily estimating the unknown location of the corneal graft in relation to the OCT device by fitting an average cornea reduces the number of fitting parameters. Especially in the peripheral regions, a difference in the elevation data can almost equally be assigned to either different corneal parameters or varying tilt and rotation angles. This may lead to an inconclusive fitting result. To eliminate this drawback, the spatial parameters have been estimated beforehand in steps A–D of the fitting procedure (Fig. 6). The accuracy of this anticipatory approach has been verified with the PMMA model lens. Tilt, translation (especially relative to the optical axis of the OCT’s objective lens), and asymmetry of the cornea will affect the quality of the measurement data and the surface-fitting procedure, as they might induce significant refraction or displacement of the OCT beam. In the fitting procedure tilt, translation and corneal asymmetry will lead to errors in the biconic fit due to fitting the data to different data points. Furthermore, keratometric values will be compromised as well. Therefore, some care should be taken for a properly placed specimen during the image acquisition procedure. This is, in fact, one of the limitations in the study, the upright measurement procedure using a clinical OCT and the fact that the cornea might most probably not be in its natural geometric state, will affect our results (compare cornea 2). Alternative fixation techniques for corneal grafts are available that might help to reduce this problem. In addition, a marker on the corneal graft may assist proper alignment to improve the results. However, regulatory decrees do not allow an easy change of the fixation procedure. Another limitation arises from the OCT technique. One should take care of correcting the refraction of the rays at the first refractive surface of the cornea, which is especially important when measuring the cornea in air due to the high difference in the refractive indices between cornea and air. Exact measurement of geometrical parameters also requires correction of refraction at the interface between culture medium and cornea. However, we assumed that this effect would be rather small due to the small difference between the refractive indices of these media. The resulting error in thickness measurement due to neglecting the optical path differences between the surrounding medium and the corneal tissue was estimated to be $<3\%$.

The use of Zernike polynomials for the reconstruction of corneal surfaces either as a global approach or as an additional decomposition of the residual error from a basic surface fit has been widely discussed in the literature.^41–43 While optical aberrations can be sufficiently modeled by Zernike decomposition and might be beneficial in the detection of keratoconus,^44,45 the coefficients yield no direct information about geometrical properties such as curvature or asphericity. Moreover, the number of coefficients necessary to minimize the RMSE when representing abnormal corneas increases rapidly.

4.3.

Optical Coherence Tomography Technique

In comparison with other measurement methods like x-ray or magnetic resonance techniques, OCT offers the best spatial resolution and does not harm the examined tissue.²⁰ Unlike confocal microscopy, it is independent from the aperture of the system and provides a higher penetration depth. For that reason, OCT has become a widespread measurement method in ophthalmology,⁴⁶ and it has previously been applied for screening corneal grafts aside from imaging the cornea in vivo.^{22,24,26,28,40,47} Considering the necessity of a sterile environment for the graft, the use of a contact-free measurement technique is obligatory. Because of the reflective behavior of the flask walls, other noncontact, high-resolution measurement methods, such as slit-projection or Placido topography would require the exposure of the tissue to air. Ultrasound techniques necessitate the application of an immersion medium, which is less comfortable.

The imaging modality of the OCT device exerts an influence on the signal-to-noise ratio and duration of the measurement. SD-OCT imaging offers several advantages over time-domain OCT (TD-OCT) devices. The Spectralis OCT is a Fourier-domain OCT (FD-OCT) and has a broadband light source, which simultaneously collects the measurement signal of multiple wavelengths. Therefore, it works much faster than the sequential image acquisition of TD-OCT devices. Although FD-OCT devices show difficulties in eliminating speckle and electron noise, several studies have shown that they offer a higher sensitivity and better spatial resolution, especially for the transparent tissue of the eye, but not in highly scattering tissue.^48–50