Custom speckle interferometer for the study of eye diseases

Mechanical stress and strain are thought to be significant risk factors for many ocular diseases. Indeed, significant correlations between the mechanical load imposed by intraocular pressure (IOP) and the etiology of ocular pathologies—such as glaucoma,¹ myopia,² and keratoconus³—have been found in many studies. Furthermore, IOP is thought to be the most relevant risk factor for the onset and development of permanent visual damage caused by glaucoma (a worldwide ocular pathology and a leading cause of permanent visual loss). The results of several numerical studies have been used to explain the relationship between ocular tissue biomechanics and the development of ocular diseases.^4–8 However, the significance and reliability of these numerically based studies (based mainly around finite element methods) are strongly dependent on the underlying assumptions needed to build the numerical models and simulations. For instance, assumptions on the boundary conditions and material constitutive equations can severely affect the outcome of the numerical simulations, and in many cases the validity of the assumptions is hard to confirm experimentally.

In addition to the numerical studies of the relationship between mechanical deformation of ocular tissue and the occurrence of ocular diseases, several experimental investigations have been conducted on ex vivo animal tissue samples,^{9, 10} and human donor eyes.¹¹ From these studies, it has been found that the most accurate full-field non-contact methods for predicting the mechanical response of ocular tissues to IOP were based on digital image correlation¹² and electronic speckle interferometry^{11, 13} (ESPI) techniques. Moreover, recently developed non-invasive imaging techniques (e.g., optical coherence tomography) mean that in vivo mechanical characterizations of ocular tissues may be possible.^{14, 15}

As part of this growing area of research, we have formed a collaboration between the University of Alabama at Birmingham's Department of Ophthalmology and the Department of Mechanical, Energy and Management Engineering of the University of Calabria (Italy). Our specific aim for this collaboration is to develop mechanical testing techniques that are customized for the analysis of ocular tissue biomechanics. In previous work, we have thus investigated—with the use of the ESPI technique and a commercial interferometer—the local mechanical response of the human sclera to IOP.^{11, 13} Our results suggested that it could be possible to produce an imaging device for measuring the 3D displacement field under time-dependent loading conditions (i.e., rapid IOP increases) for eyes with different sizes and shapes, and for variable imaging conditions. With the use of commercial apparatus, however, we found that it was not possible to carry out dynamic tests (because the single displacement components must be measured separately, at different times). In addition, the flexibility of the interferometer—in terms of light-intensity control and the sensitivity of vector adjustment—was very poor.

To try and overcome many of the limitations that we encountered with the commercial interferometer device, we have therefore more recently designed and built a customized speckle interferometer (see Figure 1).¹⁶ The optical layout of this device is based on the replication of four identical image acquisition modules. We can thus measure four linearly independent displacement components with our interferometer. To optimize the geometry of the experimental equipment, we use suitably oriented mirrors to redirect the image of the specimen toward the center of the cameras at four different points and thus break the observation directions into two segments. Although three independent displacement components are sufficient to determine the 3D displacement vector for each point of the specimen being tested, our use of a fourth imaging module allows us to make a more accurate computation of the three unknowns (i.e., the three components of the displacement vector), by means of an optimization operation.

Figure 1. Schematic illustration of the optical setup of the customized speckle interferometer. The achromat inserted in the four identical acquisition modules (A–D) creates the spherical wavefront that is used as the reference beam. Beam steering is used to redirect the object beam toward the specimen along the z-direction. The piezoelectric transducer (PZT) drives the prism, which is used to apply the phase-shifting procedure (up to a maximum frequency that is not less than one third of the mechanical resonant frequency of the PZT/holder/prism system). BS: Beamsplitter. PBS: Polarizing BS. λ/2, λ/4: Half-wave and quarter-wave retardation plate. RP: Rotating polarizer. FP: Fixed polarizer.

We have also designed our experimental equipment to be as operationally flexible as possible, and we thus included a number of specific features. First, we combine a half-wave retardation plate with a polarizing beamsplitter so that we can optimize the ratio between the object and the intensity of the reference beams. Second, our combination of a rotating polarizer and a fixed polarizer allows independent optimization of the light intensity for each of the four reference beams. We can also easily change the camera lenses to work at different magnification ratios. In addition, we can adjust the sensitivity vector directions—by varying the distance between the cameras and the specimen—and these can be measured precisely with a fast calibration procedure. Lastly, our cameras can acquire images at up to 150 frames per second (in a 2 × 2 binning, synchronized acquisition mode).

We have also developed a software procedure (using LabView and Mathematica) for acquiring and processing our experimental data. The shape of a specimen that we retrieved by applying a 3D multiview stereo technique¹⁷ to 64 images (900 × 900 pixels) is shown in Figure 2. These images were acquired with four cameras and at 16 different orientations of the specimen (as it was rotated around its vertical axis). The region of interest was a 27mm-diameter circle and was represented by 40,000 3D points—see Figure 2(a)—in our approach. By applying our custom fitting procedure,¹⁸ we obtained a mathematical representation of the shape—Figure 2(b)—from which we could estimate the local strain tensors.

Figure 2. Shape of a specimen, retrieved by using a 3D multiview stereo technique. (a) The 3D coordinates of 40,000 points were determined from 64 (900 ×900 pixel) images. (b) Fitted shape of the specimen, and the local coordinate reference system (i.e., r, n, and t axes) used to determine the strain components.

The three components of the displacement vector that we obtained from an inflation test (during which the internal pressure was increased by 10mmHg) on a rubber specimen are illustrated in Figure 3. We thermally hardened the central region of this specimen so that we could induce local deformation gradients. Such gradients mimic the differential mechanical response of the central and peripheral regions of the cornea. We can also easily compute the strain components of this specimen by coupling the mathematical representation of the shape, with the mathematical differentiation of the displacement field (see Figure 4). In addition, we can determine the local coordinate system—as shown in Figure 2(b)—from the analytical expression of the specimen's shape.

Figure 3. Components of the displacement vector (u_x, u_y, and u_z) represented as contour bands on the fitted shape of the specimen, as shown in Figure 2(b).

Figure 4. Different strain components calculated from the local shape orientation and displacement gradients. ε_r, ε_t, and γ_rt: Strains calculated in the local coordinate system shown in Figure 2(b). ε₁and ε₂: Principal strains. ε_VM: Von Mises strain.

In summary, we have presented our customized electronic pattern speckle interferometer with which we can measure the shape (with micrometric accuracy) and the full displacement field (with nanometric accuracy, up to 150Hz) of a specimen. We use this custom device in a number of projects, to investigate the association between ocular tissue biomechanics and the development of ocular diseases (e.g., glaucoma and myopia). In our future work, we plan to use our device to study the fundamentals of mechanical characterization of ocular tissue. For example, we will characterize the mechanical response of ocular tissues to IOP (e.g., the determination of elastic, plastic, viscous material properties and the identification of anisotropic behavior), the effects of intervention procedures that are aimed at altering the mechanical response of ocular tissues (e.g., caused by cross-in keratoconus), and the correlation between local mechanical properties of the sclera and the response to IOP in optic nerve head tissues.