Experimental Surface Strain Mapping of Porcine Peripapillary Sclera Due to Elevations of Intraocular Pressure

Experimental Setup and Testing Protocol

Four porcine eyes were harvested from a slaughterhouse and stored in a phosphate buffered saline solution (PBS) at 4°C for 48 hours until the experiments were conducted. A previous study [5] reported that donor eyes can be stored up to 72 hours at 4°C in PBS without risking deterioration of the uni-axial mechanical behavior of the sclera.

At the time of experiments, the anterior chamber, vitreous, retina and choroid of each eye were removed in order to isolate the posterior half of the sclera, which was then mounted and clamped at the equator on a custom-built pressurization apparatus (Fig. 1).

Pneumatic control of saline inflow and outflow was driven by a pressure regulator (Model 81, Fairchild, Winston-Salem, NC), which allowed each posterior scleral sample to be pressurized with a resolution of 0.15 mm Hg. Before the experiment, each pressurized scleral sample was quickly blotted with a gauze pad, and black dyed microspheres (10-µm-diameter Polybead Polystyrene Microspheres, Polysciences, Inc, Warrington, PA), dissolved in a 70% ethanol solution, were randomly scattered onto the tissue surface. The microspheres were allowed to adhere to the scleral surface for 5 minutes while the ethanol dried and were used as surface markers for the optical tracking method described below. All tissue samples were then kept moist with drops of PBS applied throughout the entire experiment, which only minimally affected the optical tracking of the surface marker position.

Each scleral shell was subjected to IOP preconditioning, consisting of 20 IOP cycles from 5 mm Hg to a maximum of 15 mm Hg at a rate of 5 mm Hg per second, and then allowed to recover for 360 seconds.

An optical measurement system, consisting of a high-resolution microscope (SZX-12, Olympus, Center Valley, PA) and a high-resolution digital camera (MicroFire, Optronics, Goleta, CA) was employed to capture images of the markers at a resolution of 2 µm/pixel within a 4 mm ×4 mm region of sclera. These images were acquired for each of the four quadrants of the peripapillary sclera around the ONH, in the following sequence: superior (S), nasal (N), inferior (I), and temporal (T) (Fig. 2-A). Pixel resolution was calibrated using a slide mounted micrometer scale, and the sclera was illuminated using a fiber optic ring light to ensure even illumination.

For each scleral shell, pressure was slowly increased by incremental steps of 1 mm Hg, from 5 mm Hg to 45 mm Hg, and images were generated for the inferior quadrant only. Each scleral shell was then depressurized from 45 to 5 mm Hg until equilibrium was reached, and the process of image acquisition was repeated three times for the three other (temporal, nasal, and superior) quadrants. Each image was post processed with a sharpening filter to increase the contrast of the surface marker image, then a contour was defined for each marker, and its centroid was calculated (Fig. 2-B).

Based on the set of images generated for each quadrant, the 2-D displacement field was extracted using the optical flow equation as described below, which allowed accurate tracking of the markers for each 1 mm Hg elevation of IOP.

Optical Flow Algorithm for Marker Tracking

Manually tracking the movements of hundreds of markers in each image is a daunting and time-consuming process. This study, therefore, utilized RGB digital images of markers and an optical flow algorithm to automate the tracking of marker movements, from which the pixel displacement of each marker could be determined.

It was assumed that scleral illumination was uniform and the intensity of the markers did not change with motion from one image to the next. Considering the pixel intensity I_k of an RGB image, the previous assumption can be mathematically transcribed to

Here, the index k ranges from 1 to 3 corresponding to the three components of an RGB image (i.e. red, green, and blue). The pixel intensity I_k of the 2-D image of the marker can be expressed as a function of space and time, I_k (x, y,t), in which x and y represent the pixel coordinates. Taylor’s first order approximation of Eq. (1) can be rewritten as

or

where v_x and v_y are defined as the pixel velocity components. Equation (3) is known as the color optical flow equation [11] and can be applied to two 2-D color pixel images taken at two consecutive time points.

A large body of work has been done to solve the optical flow equation using numerical approximations. Barron et al [12] summarized and evaluated the most commonly used optical flow algorithms and their performance. For the present study, we chose to adopt the Lucas-Kanade algorithm [13, 12] as described below because our preliminary comparative study found this reliable for RGB images [11].

If each image is captured at a physical equilibrium state, the time variable becomes secondary in Eq. (2). It is then reasonable to approximate Eq. (2) with dx = u_x and dy = u_y, where u_x and u_y are defined as the pixel displacements. In the Lucas-Kanade approximation [13, 12], the displacement field can be assumed to be smooth over a region of interest, R , such that for each pixel, the adjacent pixels are subjected to the same displacement field. As a result, if R is a square patch of 3 × 3 pixels with the pixel of interest in the center of that region, it requires a total of 9 equations to be solved for each color channel (Red, Green and Blue) for the center pixel of R. This results in an over-determined system of 27 equations with 2 unknowns to be solved for the center pixel, which can be rearranged in a matrix equation form:

where (x_i, y_i) for i ∈ [1,9] are the pixel-coordinates of the region of interest R , and I_k for k ∈ [1,3] are the corresponding RGB pixel intensities.

Using the least square minimization method, Eq. (4) can be left-multiplied by the matrix transpose of M

For each pixel, therefore, the displacements can be simply expressed as

The solution of Eq. (6) was computed for each pixel and repeated over the entire pixel domain in the two consecutive images taken before and after each 1 mm Hg IOP increment. The partial derivatives $\frac{\partial I_k}{\partial x},\frac{\partial I_k}{\partial y}$ in Eq. (4) were estimated using a 7-point rule, and the term $\frac{\partial I_k}{\partial t}\cdot dt$, with a 2-point rule.

Using the aforementioned optical flow algorithm, Eq. (3) – Eq. (6), the spatial coordinates X_C (i)and Y_C (i) of the centroid of each surface marker were processed to estimate its pixel displacement vector {u_Cx(i),u_Cy(i)} after the incremental IOP increase. Once the pixel displacement of a marker centroid was estimated by the aforementioned optical flow algorithm, a small circular area around the estimated centroid location {X_C (i) + u_Cx(i), Y_C (i) + u_Cy(i)} with a diameter of two-pixels was searched to locate any marker centroid in the second image, as shown in Fig. 3. This newly located marker centroid was then registered as the true marker centroid after the deformation. This step was repeated for each particle in the image, for each 1 mm Hg pressure increment, and summed over an IOP elevation from 5 to 45 mm Hg. Hence, the actual displacement vector was accurately determined for the IOP elevations from 5 to 10 mm Hg, from 10 to 30 mm Hg, and from 30 to 45 mm Hg.

Preliminary Validation of the Tracking Method using a Rubber Balloon

The experimental protocol and the optical flow algorithm method were validated using a half-spherical rubber balloon mounted and imaged as described previously, for incremental pressure increases from 70 to 80 mm Hg. The images were acquired within a 4 mm × 4 mm region located midway between the equator and the pole of the balloon. The displacement field vector was generated for each 1 mm Hg pressure increment using the optical flow algorithm method. Since rubber is an isotropic, homogeneous elastic material, and the balloon wall is of relatively constant thickness, the resulting 2-D displacement vector field as observed from the camera should be radial from the center of the image field.

Determination of Lagrangian Strain Field

After the displacement vector field was determined by the optical flow algorithm, the Lagrangian strain field on the scleral surface was calculated. To accomplish this, the 4 mm × 4 mm domain in each field image was first meshed with equal sized 4-noded quadrilateral finite elements, and the nodal displacements were smoothed from the above displacement vector field using the thin plate smoothing spline method (Matlab, The Mathworks, Nattick, MA). For each quadrilateral element, a continuous displacement field function was then approximated by a bilinear interpolation function as

in which the coefficients, a_i, could be determined from the nodal displacement values. The Lagrangian strain tensor E was calculated at Gaussian points in each quadrilateral element as

where F is the deformation gradient tensor expressed as

The magnitudes and directions of the corresponding principal Lagrangian strain were also obtained by calculating the eigenvalues and eigenvectors of the Lagrangian strain tensor shown in Eq. (8), respectively. It should be noted that the posterior scleral geometry pressurized to 5 mm Hg was chosen as the reference configuration. Therefore all Lagrangian strains mentioned in this study were computed from this reference configuration.

Validation Results with Rubber Balloon

The experimental displacement field obtained from a rubber balloon as a part of validation process of the optical flow method is shown in Fig. 4. The pattern of the displacement field was radial, and was qualitatively very similar to that of an isotropic, homogeneous spherical pressure vessel. For the entire 4 mm × 4 mm region, displacement components u and v were well fitted with two surface planes.

The Lagrangian strain tensor was calculated according to Eq. (8), and its maximum and minimum principal strains were found to be 0.90 % and 0.79 %, respectively. Although the maximum and minimum principal strains obtained for the rubber balloon were not exactly identical, they were close enough so that the rubber material could be assumed to be quasiisotropic.

Experimental Results with Porcine Specimen

Fig. 5. and Fig. 6. illustrate that the maximum principal strain was highest and primarily circumferential in trajectory in the peripapillary scleral region (immediately adjacent to the scleral canal). All four porcine eyes tested showed this specific pattern. However, the strain pattern was nonspecific in the peripheral scleral region further away from the scleral canal. Maximum increase of the principal Lagrangian strain for each IOP increase was averaged for all quadrants and for all specimen, and was found to be 0.013 ± 0.005 for an IOP elevation from 5 to 10 mm Hg (ΔIOP=5 mm Hg), 0.014 ± 0.004 for an IOP elevation from 10 to 30 mm Hg (ΔIOP=20 mm Hg), and 0.004 ± 0.001 for an IOP elevation from 30 to 45 mm Hg (ΔIOP=15 mm Hg). This result shows that the additional strain per unit increase in IOP was significantly lower at higher IOPs, demonstrating a high degree of nonlinearity in the IOP-maximum principal strain relationship. For each posterior scleral shell, this surface strain mapping also implied that the scleral stiffness was relatively low for low IOPs (<10 mm Hg), but dramatically increased at higher IOPs (>10 mm Hg), as shown in Fig. 7.