To image strain, confocal images of the cornea were taken before and after deformation. Before measurement, the cornea was in slight contact with the deformation plate. Two surface deformation steps of 50 μm were then applied, capturing ultrasound data at no deformation and at each step. Speckle was tracked between these images with a correlation-based algorithm. 15–17 Tissue motion was computed from speckle tracking, and strains were determined by numerically differentiating displacement components.
To maintain high resolution throughout the cornea, we imaged with a 50-MHz center frequency, single element transducer. Its depth of field was 260 μm, but typical corneal thicknesses are greater than 1 mm. Speckle tracking outside the focal zone of a fixed focus transducer is poor because of speckle decorrelation, so confocal image processing was used. In this process, the transducer is scanned at multiple axial positions, and resulting signals in the depth of field are spliced together guided by correlation. 18,19 Without confocal imaging, determining strain through the entire thickness of a cornea would be difficult.
With an f/number of 1.4 and a quality factor of 2.25, the transducer has a theoretical resolution of 46 μm laterally and 37 μm axially for conventional B-scans. In previous experiments 12 with the same system, axial resolution was measured to be 52 μm for displacement and 71 μm for strain. Resolution is slightly reduced at each step in processing, because displacement is calculated from several points in a B-scan and strain is calculated from multiple displacements.
Direct Mechanical Elasticity and Finite Element Model
To corroborate strain images obtained with this instrument, we created a finite element simulation based on parameters obtained from direct mechanical measurements. A mechanical technique was used to measure elasticity distributions in large tissue samples. 20 This technique produces nearly direct measures of elasticity and is widely accepted. 21,22 Using a femtosecond corneal surgical laser, 4,23 five cuts were made at increasing corneal depths to produce a series of circular disks approximately 200 μm thick and 7 mm in diameter. In contrast to microkeratome cuts in which the slices have complex curvatures and variable thickness, femtosecond laser cutting can create custom geometries. To produce disks for mechanical measurements, a plane was cut parallel to the applanated anterior corneal surface. If these disks assume natural curvatures in the intact globe, resulting anterior and posterior surfaces would no longer be planar when the applanating contact lens is removed after resection. In practice, an unloaded tissue block has insufficient rigidity to revert to its loaded curvature and assumes a planar shape through surface tension when placed on a flat surface. The resulting disk sections have uniform thickness with parallel, planar surfaces. 24 Each sample was positioned horizontally on the rigid platform of a scale. Deformations were applied with an 8-mm diameter piston covering the entire surface of the sample. The measured Young's elastic modulus as a function of tissue depth is shown in Figure 3. Elasticity variations with depth are shown for 10%, 20%, 30%, and 40% deformations. Because of the difficulty in cutting into thick porcine corneas, we were only able to obtain samples for two thirds of the thickness. No measurements were obtained for the endothelium or Descemet's membrane. In direct mechanical measurements, spatial resolution is determined by the section depth and diameter. Unlike direct mechanical measurements, ultrasound based elasticity has better axial and lateral resolution, and it can measure strain in vivo.
Mechanical measurements produce a direct measure of elasticity, a material property independent of geometry. Axial strain depends on geometry, 13,25 so a finite element model is needed to convert elasticity from mechanical measurements into axial strain to compare with ultrasonic measurements. Finite element simulations describing static deformations of a linear elastic mechanical body induced by small externally applied deformation were performed using a commercially available finite element package (ABAQUS, HKS Inc., Pawtucket, RI). Corneal thickness was set to 1.30 mm as determined from the ultrasonic B-scan. Because exact elasticity (Young's modulus) distribution in the entire cornea was not known a priori, the analysis was performed for several cases. In case 1, the cornea was modeled as an object with two layers, where the anterior 160 μm layer was 2.2 times stiffer than the background tissue. In case 2, an additional 130 μm thick layer was introduced in the posterior. This layer was 1.8 times stiffer than the corneal stroma. Mechanical properties of all layers were guided by the direct mechanical measurements. In case 3, anterior and posterior layers in the cornea were 4.4 and 3.6 times stiffer than the background tissue. Although case 3 doubles anterior and posterior stiffness from case 2, mechanical measurements suggest these elastic stiffness values are plausible. In all cases, materials were assumed to be nearly incompressible, that is, having a Poisson ratio of 0.5.
A plane strain approximation of the deformation experiment was used in simulations. Such approximations are reasonable for our experimental conditions and sufficient for qualitative analysis and comparison. The model cornea was deformed from the top by downward motion of a rigid plate with a 2.3-mm gap located in the center of symmetry. The bottom boundary of the cornea was constrained in the vertical direction, and horizontal slippage was allowed at the bottom and top boundaries. Side boundaries of 25-mm long vertical segments were unconstrained. Also, the 2.3-mm gap of the top boundary was unconstrained. A Cartesian spatial grid with nonuniform sampling was used in simulations. Underneath the slit, in the imaging region, a 25-μm grid was used, and elsewhere a larger grid was sufficient. Further decrease of the grid spacing did not produce significant change and was unnecessary.
Results of the finite element models are shown in Figure 4. Positive strain represents expansional deformation, and negative strain designates compressional deformation. For case 1, the model predicts that the stroma, Descemet's membrane, and endothelial layer will expansionally deform and that the epithelium and Bowman's layer will generally compressionally deform under this load. For case 2, the endothelial and Descemet's layers will have significantly reduced deformation compared with case 1. Case 3 behaves similar to case 2, except there is greater deformation in the stroma. In all three cases, the model predicts bipolar strain (i.e., expansional and compressional deformation in the same sample under the conditions of our experiment). Such results are unusual, but reasonable, considering the layered structure of corneas and the mechanical response of the layers.
Results of ultrasonic measurements on a porcine cornea are shown in Figure 5. In the B-scan (Fig. 5A), the transducer is positioned at the top of the image, pointing toward page bottom. The top bright echo indicates the boundary between the coupling medium (water) and the epithelial surface. Below this, a second bright echo indicates the boundary of Bowman's layer. Lower yet, stromal tissue appears as uniform speckle. At the bottom of the image, Descemet's membrane and the endothelium appear as bright echoes. Commonly, fluid in the anterior chamber does not scatter ultrasound in this frequency range, so it appears as a region with no signal. For accurate strain imaging, a high fidelity conventional B-scan image is needed because strain imaging requires spatial differentiation of the B-scan. Our image compares favorably with published images of standard ultrasound techniques in corneas. 1,2,26
Figure 5B shows a strain image in which positive and negative strains are mapped. Mid-gray to white indicates positive strain (expansional deformation). Mid-gray to black indicates negative strain (compressional deformation). As we follow strain from the middle epithelium to the anterior stroma, it changes rapidly from compressional to expansional deformation in a short distance. This result showing bipolar strain is unusual but within reason and is predicted by the finite element model.
To emphasize the behavior of corneal layers, the strain image was split in two. Strain images were overlaid on top of the B-scan to reference strain to corneal structure. Figure 6A emphasizes stromal behavior. In this image, strain ranges from + 0.2% to + 3.5%. High strains are presented in yellow and white and lower strains in red. This figure indicates general bulging in the stroma, but it is not uniform. The anterior stroma bulges more than the posterior stroma. In Figure 6B, the epithelium, Bowman's layer, and endothelium are emphasized by the color map. Strain values range from −3.5% to + 0.2%. Lower strain magnitudes are presented in magenta, and higher strain in light and dark blue. The epithelium compressionally deforms, and the endothelium and Bowman's layer bulge slightly, if at all.
Figure 7 shows average nominal axial strain through the thickness of the cornea. As with the strain images, this figures shows an expanding stroma with a compressing epithelium, but in this figure, the bipolar nature of the strain is even more apparent. The depth dependence of corneal strain measured with ultrasound most closely follows case 3 (Fig. 4). Strain processing averages over depth, and so anterior portions of epithelial strain average with zero strain from the coupling medium. This blending can be seen in Figures 5B and 6, but it does not have physical significance. To emphasize only physically significant data, depth in Figure 7 begins slightly inside the epithelium. Variations in strain in Figures 5, 6, and 7 include physical variations in the specimen and measurement uncertainty. Similar variations appear in previous studies. 12–14 Previous measurements of elasticity in corneal tissue, including our mechanical measurements, provide only an average bulk measurement, not the high resolution of the ultrasonic elasticity microscope.
The mechanical behavior of corneal layers indicated by strain imaging is reasonable given our experience with strain in layered tissue mimicking phantoms 12,13 and what is known about corneal structure. 11,27,28 Previous studies have shown that Bowman's layer and the combination of Descemet's membrane and the endothelium are stiffer than stroma. Layered tissue phantoms are created by pouring gelatin into a square mold and letting it set. The mold constrains the phantom on all sides except the top. From the top, it is deformed with a slitted plate. Because it is incompressible and restrained everywhere except the slit, the tissue phantom bulges into the slit when deformed.
In our experiment, stroma behaves similar to a single layer phantom. Descemet's membrane, the endothelial layers, and fluid pressure in the eye globe limit movement of the stroma into the anterior chamber. Like a single-layer phantom, the only place left for the stroma to move is into the slit. In contrast to a phantom, Bowman's layer prevents the stroma from fully expanding into the slit. As it acts against stromal movement, Bowman's layer is compressed. If we did not already know the general stiffnesses of corneal structures from ex vivo data, we could deduce them from the strain map.
Strain measured with ultrasound is predicted well with the finite element model considering the simplicity of the model and the potentially complex nature of corneal elasticity. The model assumes layers of three distinct elastic moduli. In real biologic tissue, we might expect a transition from one area of stiffness to the next. Also, ultrasonic strain data hints at inhomogeneity and a depth distribution of stromal elasticity. Anisotropy in the corneal plane is known to exist 29 but is not accounted for herein. Changes in hydration may explain some differences between the model guided by mechanical measurements and the ultrasonic measurement of strain, although exposure to water for both measurement systems was similar. Hydration might affect the specific distribution of elasticity, but would not affect basic mechanisms enabling ultrasonic measurement of corneal deformation. To reduce effects from hydration changes during the experiment, measurement time was kept as minimal as allowed by equipment. In clinical settings, the corneal pump mechanisms would maintain eye hydration, and measurements would take a matter of seconds. The necessary speed for clinical measurement is fundamentally possible, but would require additional engineering and more investment in advanced equipment.
With a finite element model, we obtained strain from elasticity data, but this process can be inverted. An elasticity map can be reconstructed from the strain image. Elasticity reconstruction algorithms have been developed 25,30,31 and applied to tissue mimicking phantoms. 13 Reconstruction is more complicated with a multilayered structure such as the cornea and is a subject of current investigation. Elasticity reconstruction will produce material properties, independent of geometry and other experimental parameters, enabling specimen to specimen studies of corneal elasticity.
Equipment and measurement geometry were arranged to emphasize the normal axial strain component. The slit in the deformation plate is long in the out-of-plane dimension and narrow in the lateral dimension. This geometry greatly reduces out-of-plane displacement and its resulting strain. 12,25,30 Assuming a two-dimensional model, lateral strain should be equal to axial strain in magnitude, but lateral displacement is much smaller than its axial counterpart. We calculated lateral displacement from the same data set, but, because of its small magnitude, lateral displacement and strain had a poor signal-to-noise ratio. Shear strains are also present in this experiment, but they occur mostly underneath edges of the slit. Because of small lateral displacement, shear strain also would suffer from poor signal to noise. Calculating shear strain from axial displacement should be possible using tissue incompressibility. 16
Another subject of ongoing study is the integration of ultrasound elasticity measurements into femtosecond laser refractive surgical instrumentation. Although we have shown that strain can be measured in corneas, additional work is needed before clinical application. In one goal toward this end, eliminating the slit in the deformation plate would allow simultaneous application for laser surgery and deformation for strain imaging, but the plate material must meet a tight set of requirements. It must be acoustically and optically transparent, yet stiff enough to apply the required deformation. The FDA has already approved femtosecond lasers for cutting corneal flaps. As part of that procedure, 3,4 a glass plate deforms the cornea until it is flat. Pressure increase from this procedure is under 10 mm Hg, well under four times less than FDA-approved limits. This deformation is larger than what was applied in our experiment.
An important motivation for the development of strain imaging is elasticity mapping as input for models of the biomechanical response of the cornea. An analytic model has been developed, comparing favorably with clinical data from refractive surgery. 4,10 In this model, values of Young's modulus were chosen to fit clinical data. The model does not match unless Young's modulus varies with depth. With reconstruction of an elasticity image, ultrasound can determine depth dependent variation of mechanical strength in individual eyes.
With input from ultrasonic elasticity imaging, a biomechanical model could guide surgeons in determining how much tissue to remove to produce the desired corneal shape. These models can aid the refinement of existing nomograms (i.e., LASIK) and the development of new procedures (femtosecond laser correction and femtosecond laser keratotomy). In this way, ultrasound elasticity may improve the predictability of clinical outcomes. After surgery, ultrasonic elasticity can assess changes in biomechanical properties of tissue caused by refractive procedures and the wound healing response. With the noninvasive nature of ultrasound and the minimally invasive nature of femtosecond laser surgery, a second fine-tuning refractive surgery may be possible if needed. Such a system may be capable of refraction correction of greater accuracy than current LASIK procedures.
We have shown that ultrasound can produce high-resolution strain images of a porcine cornea. The ultimate goal of this research is in vivo measurement of elasticity in human corneas. Use of the ultrasound elasticity microscope on a whole intact eye globe shows the usefulness of this technique for in vivo measurements. Within the cornea, variation of strain with depth may indicate a nonuniform distribution of elasticity. The overall trend of strain values from ultrasound compare well with finite element modeling guided by direct mechanical measurements of Young's modulus. This independent measure gives confidence to ultrasonic strain data.
The authors thank Ramon Erkamp and Xunchang Chen in the Department of Biomedical Engineering at the University of Michigan for help with the direct mechanical measurements and strain image processing, respectively. We also thank Dr. Andrei Skovoroda of the Institute for Mathematical Problems in Biology of the Russian Academy of Sciences for useful discussions.
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Keywords:© 2002 Lippincott Williams & Wilkins, Inc.
Cornea; Corneal properties; Elasticity; Refractive surgery; Strain imaging; Ultrasonic microscopy; Ultrasound