**Purpose.**:
To describe a quantitative ultrasonic spectroscopy technique for the noninvasive characterization of corneal biomechanical properties and to compare these measurements with established techniques in a porcine eye model.

**Methods.**:
An ultrasound system was constructed to accurately acquire acoustic reflections from corneas through a saline bath. Corneal properties (including thickness, density and aggregate modulus) were estimated from the measured reflection spectra based on wave propagation analysis. Twenty fresh porcine corneas were measured using the quantitative ultrasound method and other established techniques that can only be applied to dissected corneas.

**Results.**:
The ultrasonic measurements of corneal thickness and aggregate modulus were significantly correlated with the measurements of established techniques (Pearson's correlation = 0.99 and 0.61; *P* < 0.005), and good sample-to-sample consistency was achieved. The measurement of corneal density agreed well in terms of mean and range, but the correlation did not achieve statistical significance (Pearson's correlation = 0.41; *P* = 0.07).

**Conclusions.**:
The ultrasonically measured corneal biomechanical properties agreed well with the measurements obtained by using established techniques, validating the accuracy of the ultrasound method. Quantitative ultrasound spectroscopy may provide a noninvasive approach for in vivo characterization of corneal biomechanical properties.

^{ 1 }Measurement of the corneal elastic modulus has been an important goal. Stress–strain tests have been performed on corneal strips or buttons, and a fairly wide range of the Young's modulus has been reported in human corneas.

^{ 2–4 }It is believed that this large range may be caused in part by variations in the experimental conditions, including hydration, strain rate, donor age, and the use of different testing techniques (e.g., strip extensiometry or button inflation). The true variability of the human corneal modulus is yet to be determined.

^{ 5 }Because the cornea strongly attenuates high-frequency shear waves, it was challenging to obtain reliable shear-wave measurements. Surface-wave elastometry has been developed to measure the propagation speed of surface waves in the superficial stroma along the direction that is parallel to the cornea surface.

^{ 6 }The challenge of this approach is that water highly attenuates the surface waves, and so this technique requires an essentially dry corneal surface. Strain imaging has been developed to measure the distribution of the elastic modulus. Ultrasound elasticity microscopy

^{ 7 }and optical coherence tomography (Dupps WJ Jr, et al.

*IOVS 2*007;48:E-Abstract 3864) have been used to image strain distribution in corneas. In these methods, deformation of the cornea was induced by compression through an external flat plate or changes in the intraocular pressure. The subtle motions within the corneal tissue were computed from the ultrasonic or optical images to obtain the distribution of strain within the corneal stroma. The strain image reflects the relative distribution of the elastic modulus within the corneal stroma and may provide useful information to guide refractive surgery. However, it does not provide a direct measure of the elastic modulus that can be compared across subjects.

^{ 8,9 }Lame's constant λ, similar to Young's modulus, relates stress and strain; and it was introduced for convenience in expressing the normal stresses in terms of normal strains for general theoretical development.

^{ 10 }The second Lame's constant μ is also known as the shear modulus. For linear, elastic, and isotropic materials, aggregate modulus can be calculated from other elastic constants (e.g., Young's modulus and Poisson's ratio, see the 1). Since the cornea is neither linearly elastic nor isotropic, a single mathematical equation that defines the relationship between corneal aggregate modulus and Young's modulus may not exist. Nevertheless, aggregate modulus provides a measure of stiffness (in terms of relating stress and strain) and can be used for comparative characterization of the elastic properties of the cornea across subjects.

^{ 11 }The present study examines the validity of the measurements in enucleated porcine eyes comparing the quantitative ultrasound method with established techniques.

^{2}were propagated along the axial direction of the eye. The ultrasound beam was centered at the corneal apex by adjusting the position of the transducer by using precision linear stages (1 μm step size; Newport, Irvine CA). Potential errors introduced when the beam is not perfectly centered at apex were analyzed previously.

^{ 11 }The dissected corneas were mounted on an artificial anterior chamber during the measurements. The anterior chamber was pressurized to 10 mm Hg and maintained at that pressure during the measurements. The ultrasonic reflections from the corneas were displayed real-time on the PC monitor, and the signals were sampled by the digitizer and recorded on the hard-drive of the PC. The reference signal was obtained from a sample with known acoustic properties (Soflens-59; Bausch & Lomb, Rochester, NY).

^{ 12 }The model has been described in detail elsewhere.

^{ 11,13 }Briefly, continuity equations for stress and deformation at the boundaries in the path of the wave propagation (e.g., the anterior and posterior corneal surfaces) were solved for, to obtain the ultrasonic reflection spectra corresponding to any given combination of corneal thickness, density, and aggregate modulus. The model solves for the forward problem (i.e., the corneal properties are known and the reflection spectrum is the unknown). To estimate the corneal properties from the measured ultrasonic reflection spectra, the inverse problem must be solved (i.e., the reflection spectrum is known, and the corneal properties are unknown). To solve the inverse problem, a parameter estimation algorithm based on the Levenberg-Marquardt algorithm (MatLab, The MathWorks, Natick, MA) was implemented to search for the set of corneal parameters that minimizes the difference between the model-predicted and the experimentally measured spectra. The schematic of the wave propagation analysis is shown in Figure 1b.

^{ 14,15 }The Archimedes' method is a standard method that has been used to determine corneal density.

^{ 16 }Combining the information from the substitution method (speed of sound) and the Archimedes' method (density), the aggregate modulus can be calculated based on the mathematical relationship between these parameters

^{ 17 }: where

*E*is aggregate modulus, ρ is density, and

*V*is speed of sound.

*V*) and the thickness (

*h*) of the cornea were calculated from the known speed of sound of water and the measured times of flight: where

*V*is the speed of sound in cornea;

*V*

_{water}is the speed of sound in water;

*t*

_{1}is the time of flight of the ultrasonic pulse reflected from corneal anterior surface;

*t*

_{2}is the time of flight of the pulse reflected from corneal posterior surface;

*t*

_{3}is the time of flight of the ultrasonic pulse passing through the cornea and reflected from the plastic plate; and

*t*

_{4}is the time of flight of the ultrasonic pulse reflected from the plastic plate when the cornea was removed from the path. The definitions of

*t*

_{1},

*t*

_{2},

*t*

_{3}, and

*t*

_{4}are illustrated in Figure 2.

^{ 16 }Briefly, the mass of the cornea in air (

*m*

_{1}) and in distilled water (

*m*

_{2}) was measured by an analytical balance (accuracy: 0.1 mg; Denver Instrument, Denver, Colorado). The density of the cornea was calculated as:

^{ 18 }

*P*< 0.001, Fig. 4a). The ranges of these two measurements were 702.0 to 1008.0 μm (mean ± SD: 883.7 ± 76.6 μm) and 715.1 to 1011.3 μm (mean ± SD: 882.1 ± 76.7 μm), respectively. The mean difference of the two thickness measures was 1.6 μm (SD 11.2). The Bland-Altman plot for thickness measurements is shown in Figure 5a.

^{3}(mean ± SD: 1.074 ± 0.006 g/cm

^{3}) by the Archimedes' method and 1.064 to 1.087 g/cm

^{3}(mean ± SD: 1.075 ± 0.007 g/cm

^{3}) by the quantitative ultrasound method. The mean difference of the two density measures was 0.0004 g/cm

^{3}(SD 0.0074). The density measurements from the quantitative ultrasound method had a Pearson's correlation of 0.41 with the measurements using the Archimedes' method. The correlation did not achieve statistical significance (

*P*= 0.07, Fig. 4b). The Bland-Altman plot for density measurements is shown in Figure 5b.

*P*= 0.004; Fig. 4c). The ranges of these two measurements were 2.475 to 2.792 GPa (mean ± SD: 2.628 ± 0.072 GPa) and 2.485 to 2.839 GPa (2.629 ± 0.081 GPa), respectively. The mean difference between the two aggregate modulus measures was 0.0016 GPa (SD 0.0678). The Bland-Altman plot for aggregate modulus measurements is shown in Figure 5c.

*P*= 0.002). The substitution method gave a speed of sound reading of (mean ± SD) 1564 ± 18 m/s in the twenty fresh porcine corneas, whereas the quantitative ultrasound method generated a value of (mean ± SD) 1560 ± 23 m/s.

Thickness (μm) | Aggregate Modulus (GPa) | Density (g/cm^{3}) | |
---|---|---|---|

Established techniques | |||

Mean | 883.7 | 2.628* | 1.074 |

SD | 76.6 | 0.072 | 0.006 |

Range | 702.0–1008.0 | 2.475–2.792 | 1.054–1.083 |

Quantitative ultrasound method | |||

Mean | 882.1 | 2.629 | 1.075 |

SD | 76.7 | 0.081 | 0.007 |

Range | 715.1–1011.3 | 2.485–2.839 | 1.064–1.087 |

Mean difference | −1.6 | 0.002 | 0.0004 |

SD of difference | 11.2 | 0.068 | 0.0074 |

^{ 16 }The near constancy of this parameter may explain the weaker correlation (

*R*= 0.41,

*P*= 0.07) between the density measurements of the two different methods. When the true values are close from sample to sample, the correlation between different measurement methods could be low, because the experimental noise, which may account for a significant portion of the observed sample-to-sample difference, is not necessarily correlated. The speed of sound measurements in the present study also agreed well with the literature. Thijssen et al.

^{ 14 }reported a speed of sound of 1555 ± 2 m/s in porcine corneas, which is close to the values we obtained. Note that the speed of sound in porcine corneas is markedly lower than that in human corneas (the average speed of sound in human corneas is 1640 m/s, as used in ultrasound pachymetry).

^{ 19 }used Brillouin scattering to determine the longitudinal elastic modulus (similar to aggregate modulus as explained below) of crystalline lens and cornea. Brillouin scattering uses thermally excited hyperfrequency sound waves (1–10 GHz) to determine the longitudinal elastic modulus when the density is made available through other measurements. They reported a longitudinal elastic modulus of 2.56 GPa for bovine corneas, a value that is quite close to the aggregate modulus we measured in fresh porcine corneas (∼2.63 GPa). This agreement may be explained by the fact that both the aggregate modulus and the longitudinal elastic modulus are related to the speed of sound and the density of the material in the same way as described in equation 1. If the speed of sound does not change with frequency (i.e., is nondispersive), the high-frequency longitudinal elastic modulus measured by Brillouin scattering should be the same as the aggregate modulus measured by the quantitative ultrasound method. The results in the present study indicate that the cornea may be minimally dispersive so that the frequency of the acoustic waves does not significantly affect the speed of sound in cornea. The advantage of the quantitative ultrasound method compared with Brillouin scattering is that the ultrasound method provides a measure of both density and aggregate modulus (enabled by the wave propagation analysis), thus does not require the separate measurement of density.

^{ 20 }Corneal hysteresis, however, may not represent the elastic modulus of the cornea, because it is affected by both the viscous and the elastic properties of the tissue. According to a recent study, a low hysteresis could be associated with either high or low elastic modulus depending on viscosity.

^{ 21 }Therefore, although corneal hysteresis may provide independent and useful information about the cornea's biomechanical characteristics, other parameters such as aggregate modulus are needed for in vivo estimation of the elastic modulus (or stiffness) of the cornea.

^{ 22 }and the degree of collagen cross-linking in the cornea,

^{ 23 }and thus may provide useful comparative characterization of the biochemical factors that are associated with aggregate modulus. Future research calls for detailed characterization of the aggregate modulus of human corneas to determine its association with microstructure and its potential use in vision care and research. Conversely, aggregate modulus may not be a sensitive indicator of the anisotropic properties of the cornea. Young's modulus could be several times different in comparisons of the through-thickness and the in-plane directions of the cornea, but the aggregate modulus may differ much less (studies in myocardium have shown a 5% to 10% difference between the along-fiber and the through-fiber directions

^{ 24 }). Thus, for applications that emphasize the anisotropic elastic properties of the cornea, the measure of aggregate modulus may have limited use.

*I*

_{SPTA,3}= 3 mW/cm

^{2}) is within the U.S. Food and Drug Administration's safety guideline for ophthalmic ultrasound (

*I*

_{SPTA,3}≤ 17 mW/cm

^{2}). This is important for avoiding the potential thermal effects of the ultrasound application (i.e., the temperature rise of the tissue due to the absorption of acoustic energy). In terms of the measurement procedure and the subject experience, the quantitative ultrasound method is not substantially different from the current A-mode ophthalmic ultrasound. The quantitative ultrasound method is different from ophthalmic clinical ultrasound in two ways. First, the quantitative ultrasound method utilizes more information in the reflected wave forms. Clinical ultrasound utilizes only the information of the location and the strength of the reflected pulses. The quantitative ultrasound method utilizes the full-wave form (including phase information) and thus requires a high-resolution digitizer to capture the detailed information in the reflected waves. Second, the quantitative ultrasound method performs a wave propagation analysis that is not available in clinical ultrasound systems. The wave propagation analysis requires not only accurate measurements of the wave form but also accurate modeling of the interactions between the ultrasonic waves and the cornea. Because the cornea is directly accessible to acoustical excitations, no overlying tissues or structures are present to alter and complicate the ultrasonic beam before it interacts with the cornea. The cornea is also thin enough that the acoustic field for a weakly focused or nonfocused ultrasound beam is essentially homogeneous throughout its thickness. In addition, the cornea has very low attenuation to longitudinal ultrasound waves even at high frequencies.

^{ 15 }These factors make it possible to reliably measure the ultrasonic reflections and to accurately model the wave propagation for quantitative analysis. In essence, the unique anatomic position and the acoustically simple structure of the cornea make it advantageous to apply wave propagation analysis and to use the reflection spectra to noninvasively determine aggregate modulus. The present study has demonstrated the accuracy of noninvasive measurements obtained by the quantitative ultrasound method, as they were not different from the measurements made by using established techniques that can only be used on dissected corneas.

^{ 13,25 }and biomedicine.

^{ 26 }This study demonstrates the validity of a quantitative ultrasound spectroscopy method designed for noninvasive determination of corneal thickness, density, and aggregate modulus. The measurements of the quantitative ultrasound method agreed well with the measurements obtained from established techniques, suggesting that the quantitative ultrasound spectroscopy method may provide a noninvasive approach for in vivo characterization of corneal biomechanical properties.

^{27}For a linear, elastic, and isotropic material, only two independent elastic constants are needed to characterize the relationship between stress and strain. Young's modulus (E) and Poisson's ratio ν are usually used. For example, Hooke's law for isotropic material in stiffness matrix form can be expressed using E and ν as follows: The matrix can also be expressed by Lame's constants λ and μ as follows: The relationship between these constants is defined by the following equation: For linear, elastic, and isotropic material, the aggregate modulus (λ + 2μ) and Young's modulus (E) are related by a coefficient determined by Poisson's ratio (ν). The cornea is known to have anisotropic properties, and thus the relationship between the corneal Young's modulus and the corneal aggregate modulus may be complex.

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