November 2009
Volume 50, Issue 11
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Cornea  |   November 2009
A Quantitative Ultrasonic Spectroscopy Method for Noninvasive Determination of Corneal Biomechanical Properties
Author Affiliations & Notes
  • Xiaoyin He
    From the Departments of Biomedical Engineering and
  • Jun Liu
    From the Departments of Biomedical Engineering and
    Ophthalmology, The Ohio State University, Columbus, Ohio.
  • Corresponding author: Jun Liu, 270 Bevis Hall, 1080 Carmack Road, Columbus, OH 43210; liu.314@osu.edu
Investigative Ophthalmology & Visual Science November 2009, Vol.50, 5148-5154. doi:10.1167/iovs.09-3439
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      Xiaoyin He, Jun Liu; A Quantitative Ultrasonic Spectroscopy Method for Noninvasive Determination of Corneal Biomechanical Properties. Invest. Ophthalmol. Vis. Sci. 2009;50(11):5148-5154. doi: 10.1167/iovs.09-3439.

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      © 2017 Association for Research in Vision and Ophthalmology.

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Abstract

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.

Corneal biomechanics may play an important role in the normal function and the pathophysiology of the eye. 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. 24 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. 
Noninvasive methods of determining the elastic properties of the cornea have been investigated. Ultrasonic shear waves have been applied to corneas by using a fused silica stage to measure the shear wave speed in the through-thickness direction of the corneal stroma. 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 2007;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. 
We have developed a quantitative ultrasound spectroscopy method to noninvasively determine an elastic constant, the aggregate modulus, of the cornea. Aggregate modulus is the sum of the Lame's constants λ and 2μ (aggregate modulus = λ + 2μ). This modulus was also used previously in the modeling of corneal swelling stress. 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. 
We previously applied the quantitative ultrasound method in determining the properties of soft contact lenses and validated the measurements. 11 The present study examines the validity of the measurements in enucleated porcine eyes comparing the quantitative ultrasound method with established techniques. 
Materials and Methods
Porcine Globe Preparation
Twenty enucleated porcine globes were obtained and measured within 24 hours postmortem (Sioux-Preme Packing Co., Sioux City, IA). Before all measurements, corneal epithelia were carefully removed, and the globes were immersed in a 20% dextran-saline solution to reduce the postmortem swelling of the corneas. Corneas were dissected for the measurements of the established techniques. Although the quantitative ultrasound method can be applied without corneal dissection, the measurements were performed after corneal dissection so that the measurements of different methods could be implemented close in time, to minimize the influence of uncontrollable alterations of ex vivo corneas between measurements. 
Quantitative Ultrasound Spectroscopy Method
The quantitative ultrasound spectroscopy method has two major components: the measurement system and the wave propagation analysis. 
Ultrasonic Measurement System.
The ultrasonic measurement system is composed of a transducer (11 MHz center frequency, XMS, Panametrics-NDT, Waltham, MA), a pulser-receiver (5900PR; Panametrics-NDT), a digitizer (500 MHz/8-bit; DP105; Acqiris, Monroe, NY), and a computer (Fig. 1a). Short-duration ultrasonic pulses with an acoustic energy less than 3 mW/cm2 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). 
Figure 1.
 
The two components of the quantitative ultrasound method: the measurement system and the wave propagation analysis. (a) The ultrasound system measures the reflection signals from the corneas. A saline bath is used to couple the acoustic waves between the cornea and the transducer. (b) A wave-propagation model generates a theoretical reflection spectrum to be compared with the measured spectrum. An iterative process (dashed line) implemented in the parameter estimation algorithm ensures optimal fitting of the theoretical spectrum to the measured spectrum.
Figure 1.
 
The two components of the quantitative ultrasound method: the measurement system and the wave propagation analysis. (a) The ultrasound system measures the reflection signals from the corneas. A saline bath is used to couple the acoustic waves between the cornea and the transducer. (b) A wave-propagation model generates a theoretical reflection spectrum to be compared with the measured spectrum. An iterative process (dashed line) implemented in the parameter estimation algorithm ensures optimal fitting of the theoretical spectrum to the measured spectrum.
Wave Propagation Analysis.
The ultrasonic reflection spectra were analyzed to generate estimates of corneal properties. The acquired, time-domain, ultrasonic waveforms were converted into power spectra with a Fourier transform, and the spectra within the usable frequency range of the transducer (i.e., 10–12 MHz) were obtained. The power spectra from the corneas were calibrated against that of the reference signal to obtain normalized spectra. 
A mathematical model based on the theories of wave propagation was developed to relate the reflection spectrum to the properties of the cornea. 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. 
Measurement of Corneal Properties with Established Techniques
For 20 porcine corneas, the measurements using the quantitative ultrasound method were compared with the measurements using other established techniques. The substitution method (detailed below) is an established acoustic method to determine corneal thickness and speed of sound. 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. 
Substitution Method.
The substitution method requires the placement of an optically flat reflector at the posterior side of the measured sample: the cornea. A flat plastic plate was used for this purpose in the present study. The same ultrasonic measurement system described in Figure 1a was used for acquiring the acoustic signals in the substitution method. Two sets of acoustic signals were acquired with (Fig. 2a) or without (Fig. 2b) the cornea intervening. The 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; Vwater is the speed of sound in water; t1 is the time of flight of the ultrasonic pulse reflected from corneal anterior surface; t2 is the time of flight of the pulse reflected from corneal posterior surface; t3 is the time of flight of the ultrasonic pulse passing through the cornea and reflected from the plastic plate; and t4 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 t1, t2, t3, and t4 are illustrated in Figure 2
Figure 2.
 
The substitution method for measuring corneal speed of sound and thickness.(a) Ultrasound measurements with the cornea intervening; (b) ultrasound measurements without corneal intervening.
Figure 2.
 
The substitution method for measuring corneal speed of sound and thickness.(a) Ultrasound measurements with the cornea intervening; (b) ultrasound measurements without corneal intervening.
Archimedes' Method.
The use of Archimedes' principle for the direct measurement of corneal density has been described previously. 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:    
Statistical Analysis
Pearson's correlations between the measurements of corneal thickness, density and aggregate modulus obtained from the quantitative ultrasound method and the established techniques were calculated (SAS software, ver. 9.1; SAS Institute Inc., Cary, NC). The Bland-Altman plots were used to examine the agreement between the methods. 18  
Results
The thickness, density, and aggregate modulus of 20 porcine corneas were determined by using the quantitative ultrasound method. For each cornea, the reflection spectra calculated from the estimated corneal properties fitted well with the experimentally measured spectra. One example of the spectral fitting is presented in Figure 3
Figure 3.
 
The ultrasound reflection spectra from two porcine corneas are shown, to demonstrate the goodness of fit of between the resultant spectra based on wave propagation analysis and the experimentally measured spectra.
Figure 3.
 
The ultrasound reflection spectra from two porcine corneas are shown, to demonstrate the goodness of fit of between the resultant spectra based on wave propagation analysis and the experimentally measured spectra.
The measurements from the quantitative ultrasound method were compared with those obtained by using the established techniques. 
The thickness measurements from the quantitative ultrasound method were significantly correlated with the measurements from the substitution method (Pearson's correlation = 0.99, 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. 
Figure 4.
 
The correlation between the measurements of the quantitative ultrasound method and the established techniques. (a) CCT measurements (Pearson's correlation = 0.99, P < 0.001). (b) Density measurements (Pearson's correlation = 0.41, P = 0.07). (c) Aggregate modulus measurements (Pearson's correlation = 0.61, P = 0.004). CCT, central corneal thickness; AM, aggregate modulus.
Figure 4.
 
The correlation between the measurements of the quantitative ultrasound method and the established techniques. (a) CCT measurements (Pearson's correlation = 0.99, P < 0.001). (b) Density measurements (Pearson's correlation = 0.41, P = 0.07). (c) Aggregate modulus measurements (Pearson's correlation = 0.61, P = 0.004). CCT, central corneal thickness; AM, aggregate modulus.
Figure 5.
 
Bland-Altman plots showing the mean of the two measurements (quantitative ultrasound method and established techniques) against the difference between the measurements: (a) for CCT, (b) for density and (c) for aggregate modulus. CCT, central corneal thickness; AM, aggregate modulus.
Figure 5.
 
Bland-Altman plots showing the mean of the two measurements (quantitative ultrasound method and established techniques) against the difference between the measurements: (a) for CCT, (b) for density and (c) for aggregate modulus. CCT, central corneal thickness; AM, aggregate modulus.
The ranges of the densities in the 20 fresh porcine corneas were fairly narrow: 1.054 to 1.083 g/cm3 (mean ± SD: 1.074 ± 0.006 g/cm3) by the Archimedes' method and 1.064 to 1.087 g/cm3 (mean ± SD: 1.075 ± 0.007 g/cm3) by the quantitative ultrasound method. The mean difference of the two density measures was 0.0004 g/cm3 (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. 
The aggregate modulus measurements from the quantitative ultrasound method correlated significantly with the values calculated from the speed of sound (substitution method) and density (Archimedes' method). The Pearson's correlation of these two measures was 0.61 (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. 
The mean difference was close to 0 for all parameters, and most of the data points were within ±2 SD (for each parameter, there was only one sample that fell beyond the ±2 SD range). 
As described before, the corneal speed of sound was determined by the substitution method. The speed of sound can also be calculated by applying equation 1 using the aggregate modulus and density obtained by the quantitative ultrasound method. These two speed-of-sound measurements were also compared. The Pearson's correlation of these two measures was 0.66 (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. 
The comparison of the measurements is summarized in Table 1
Table 1.
 
Comparison of the Measurements Using the Quantitative Ultrasound Method and Established Techniques
Table 1.
 
Comparison of the Measurements Using the Quantitative Ultrasound Method and Established Techniques
Thickness (μm) Aggregate Modulus (GPa) Density (g/cm3)
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
Discussion
The experimentally measured reflection spectra from fresh porcine corneas agreed well with the calculated spectra based on the estimated corneal properties (Fig. 3), demonstrating excellent goodness of fit between the wave propagation analysis and the ultrasound measurements. More important, the corneal properties measured by the quantitative ultrasound method and the established methods showed satisfactory agreement, as will be discussed. 
For thickness and aggregate modulus, sample-to-sample agreement was obtained between the measurement methods (high correlation and small difference), indicating that the quantitative ultrasound method was essentially equivalent to the established methods in terms of measuring these two corneal properties. The density measurements also agreed well according to Bland-Altman analysis; however, the correlation did not achieve statistical significance, probably due to the small variance of this parameter in the measured samples. Both the direct measurement and the ultrasound measurement revealed a small variance of density in fresh porcine corneas (the SD, was approximately 0.5% of the mean value). Similar small variance of density was also reported in previous studies. 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). 
Few studies have reported the measurement of corneal aggregate modulus. Vaughan and Randall 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. 
In vivo determination of corneal stiffness is an important goal for several ophthalmic applications. Corneal hysteresis (as measured by the Ocular Response Analyzer [ORA]; Reichert, Depew, NY), is the one of the few biomechanical measures that are currently available for clinical use. 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. 
The corneal aggregate modulus measured by the quantitative ultrasound method may provide useful information for some ophthalmic applications but may have limited use for others. For example, the aggregate modulus may be related to the collagen content 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. 
The advantage of the quantitative ultrasound method lies in its noninvasiveness and its application to in vivo measurements. The energy level of the acoustic excitation used in the quantitative ultrasound method (spatially and temporally averaged intensity, I SPTA,3 = 3 mW/cm2) is within the U.S. Food and Drug Administration's safety guideline for ophthalmic ultrasound (I SPTA,3 ≤ 17 mW/cm2). 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. 
Quantitative ultrasound methods in general have been used in engineering 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. 
Footnotes
 Supported by the Columbus Foundation Ann Ellis Fund.
Footnotes
 Disclosure: X. He, None; J. Liu, None
Footnotes
 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.
The authors thank Xueliang (Jeff) Pan, PhD (Center for Biostatistics, The Ohio State University), for assistance with statistical analysis, and Mark A. Bullimore, MCOptom, PhD, Karla Zadnik, PhD, (College of Optometry, The Ohio State University), and J. Crawford Downs, PhD (Devers Eye Institute, Portland, OR) for reviewing the manuscript. 
Appendix
Brief Review of Elastic Constants
For linear and elastic materials, Hooke's law states that the stress is a linear function of all the components of strain.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. 
References
Ethier CR Johnson M Ruberti J . Ocular biomechanics and biotransport. Ann Rev Biomed Eng. 2004;6:249–273. [CrossRef]
Bryant MR McDonnell PJ . Constitutive laws for biomechanical modeling of refractive surgery. J Biomech Eng. 1996;118:473–481. [CrossRef] [PubMed]
Hjortdal JO . Regional elastic performance of the human cornea. J Biomech. 1996;29:931–942. [CrossRef] [PubMed]
Woo SLY Schlegel WA Kobayashi AS Lawrence C . Nonlinear material properties of intact cornea and sclera. Exp Eye Res. 1972;14:29–39. [CrossRef] [PubMed]
Wang HC Prendiville PL McDonnell PJ Chang WV . An ultrasonic technique for the measurement of the elastic moduli of human cornea. J Biomechan. 1996;29:1633–1636. [CrossRef]
Dupps WJ Netto MV Herekar S Krueger RR . Surface wave elastometry of the cornea in porcine and human donor eyes. J Refract Surg. 2007;23:66–75. [PubMed]
Hollman KW Emelianov SY Neiss JH . Strain imaging of corneal tissue with an ultrasound elasticity microscope. Cornea. 2002;21:68–73. [CrossRef] [PubMed]
Eisenberg SR Grodzinsky AJ . Swelling of articular-cartilage and other connective tissues: electromechanochemical forces. J Orthopaed Res. 1985;3:148–159. [CrossRef]
Eisenberg SR Grodzinsky AJ . The of chemically-induced nonequilibrium swelling of articular-cartilage and corneal stroma. J Biomechan Eng 1987;109:79–89. [CrossRef]
Davis R Selvadurai APS . Elasticity and Geomechanics. Cambridge, UK: Cambridge University Press; 1996.
Liu J He X Pan X Roberts CJ . Ultrasonic model and system for measurement of corneal biomechanical properties and validation on phantoms. J Biomechan. 2007;40:1177–1182. [CrossRef]
Brekhovskikh LM . Waves in Layered Media. New York: Academic Press; 1960.
Lavrentyev AI Rokhlin SI . Determination of elastic moduli, density, attenuation, and thickness of a layer using ultrasonic spectroscopy at two angles. J Acoust Soc Am. 1997;102:3467–3477. [CrossRef]
Thijssen JM Mol HJM Timmer MR . Acoustic parameters of ocular-tissues. Ultrasound Med Biol. 1985;11:157–161. [CrossRef] [PubMed]
Ye SG Harasiewicz KA Pavlin CJ Foster FS . Ultrasound characterization of normal ocular tissue in the frequency-range from 50 Mhz to 100 Mhz. IEEE Trans Ultrason Ferroelectr Freq Cont. 1995;42:8–14. [CrossRef]
Kampmeier J Radt B Birngruber R Brinkmann R . Thermal and biomechanical parameters of porcine cornea. Cornea. 2000;19:355–363. [CrossRef] [PubMed]
Chtistensen D . Ultrasonic Bioinstrumentation. New York: John Wiley & Sons; 1988.
Bland JM Altman DG . Statistical methods for assessing agreement between 2 methods of clinical measurement. Lancet. 1986;1:307–310. [CrossRef] [PubMed]
Vaughan JM Randall JT . Brillouin-scattering, density and elastic properties of the lens and cornea of the eye. Nature. 1980;284:489–491. [CrossRef] [PubMed]
Luce DA . Determining in vivo biomechanical properties of the cornea with an ocular response analyzer. J Cataract Refract Surg. 2005;31:156–162. [CrossRef] [PubMed]
Glass DH Roberts CJ Litsky AS Weber PA . A viscoelastic biomechanical model of the cornea describing the effect of viscosity and elasticity on hysteresis. Invest Ophthalmol Vis Sci. 2008;49:3919–3926. [CrossRef] [PubMed]
Harley R James D Miller A White JW . Phonons and elastic-moduli of collagen and muscle. Nature. 1977;267:285–287. [CrossRef] [PubMed]
Scarcelli G Yun SH . Confocal Brillouin microscopy for three-dimensional mechanical imaging. Nat Photon. 2008;2:39–43. [CrossRef]
Baldwin SL Yang M Marutyan KR Wallace KD Holland MR Miller JG . Measurements of the anisotropy of ultrasonic velocity in freshly excised and formalin-fixed myocardial tissue. J Acoust Soc Am. 2005;118:505–513. [CrossRef] [PubMed]
Chang FH Flynn PL Gordon DE Bell JR . Principles and application of ultrasonic spectroscopy in NDE of adhesive bonds. IEEE Trans Son Ultrason. 1976;23:334–338. [CrossRef]
Gluer CC . Quantitative ultrasound techniques for the assessment of osteoporosis: expert agreement on current status. J Bone Min Res. 1997;12:1280–1288. [CrossRef]
Auld BA . Acoustic Fields and Waves in Solids. Malabar, FL: RE Krieger; 1990.
Figure 1.
 
The two components of the quantitative ultrasound method: the measurement system and the wave propagation analysis. (a) The ultrasound system measures the reflection signals from the corneas. A saline bath is used to couple the acoustic waves between the cornea and the transducer. (b) A wave-propagation model generates a theoretical reflection spectrum to be compared with the measured spectrum. An iterative process (dashed line) implemented in the parameter estimation algorithm ensures optimal fitting of the theoretical spectrum to the measured spectrum.
Figure 1.
 
The two components of the quantitative ultrasound method: the measurement system and the wave propagation analysis. (a) The ultrasound system measures the reflection signals from the corneas. A saline bath is used to couple the acoustic waves between the cornea and the transducer. (b) A wave-propagation model generates a theoretical reflection spectrum to be compared with the measured spectrum. An iterative process (dashed line) implemented in the parameter estimation algorithm ensures optimal fitting of the theoretical spectrum to the measured spectrum.
Figure 2.
 
The substitution method for measuring corneal speed of sound and thickness.(a) Ultrasound measurements with the cornea intervening; (b) ultrasound measurements without corneal intervening.
Figure 2.
 
The substitution method for measuring corneal speed of sound and thickness.(a) Ultrasound measurements with the cornea intervening; (b) ultrasound measurements without corneal intervening.
Figure 3.
 
The ultrasound reflection spectra from two porcine corneas are shown, to demonstrate the goodness of fit of between the resultant spectra based on wave propagation analysis and the experimentally measured spectra.
Figure 3.
 
The ultrasound reflection spectra from two porcine corneas are shown, to demonstrate the goodness of fit of between the resultant spectra based on wave propagation analysis and the experimentally measured spectra.
Figure 4.
 
The correlation between the measurements of the quantitative ultrasound method and the established techniques. (a) CCT measurements (Pearson's correlation = 0.99, P < 0.001). (b) Density measurements (Pearson's correlation = 0.41, P = 0.07). (c) Aggregate modulus measurements (Pearson's correlation = 0.61, P = 0.004). CCT, central corneal thickness; AM, aggregate modulus.
Figure 4.
 
The correlation between the measurements of the quantitative ultrasound method and the established techniques. (a) CCT measurements (Pearson's correlation = 0.99, P < 0.001). (b) Density measurements (Pearson's correlation = 0.41, P = 0.07). (c) Aggregate modulus measurements (Pearson's correlation = 0.61, P = 0.004). CCT, central corneal thickness; AM, aggregate modulus.
Figure 5.
 
Bland-Altman plots showing the mean of the two measurements (quantitative ultrasound method and established techniques) against the difference between the measurements: (a) for CCT, (b) for density and (c) for aggregate modulus. CCT, central corneal thickness; AM, aggregate modulus.
Figure 5.
 
Bland-Altman plots showing the mean of the two measurements (quantitative ultrasound method and established techniques) against the difference between the measurements: (a) for CCT, (b) for density and (c) for aggregate modulus. CCT, central corneal thickness; AM, aggregate modulus.
Table 1.
 
Comparison of the Measurements Using the Quantitative Ultrasound Method and Established Techniques
Table 1.
 
Comparison of the Measurements Using the Quantitative Ultrasound Method and Established Techniques
Thickness (μm) Aggregate Modulus (GPa) Density (g/cm3)
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
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