- Email: [email protected]
Ultrasonic model and system for measurement of corneal biomechanical properties and validation on phantoms
ARTICLE IN PRESS
Journal of Biomechanics 40 (2007) 1177–1182
Short communication
www.elsevier.com/locate/jbiomech www.JBiomech.com
Ultrasonic model and system for measurement of corneal biomechanical properties and validation on phantoms Jun Liua,b,, Xiaoyin Heb, Xueliang Panc, Cynthia J. Robertsa,b a
Biomedical Engineering Department, The Ohio State University, 270 Bevis Hall, 1080 Carmack Rd., Columbus, OH 43210, USA b Department of Ophthalmology, The Ohio State University, USA c Department of Statistics, The Ohio State University, USA Accepted 28 April 2006
Abstract Non-invasive measurement of biomechanical properties of corneas may provide important information for ocular disease management and therapeutic procedures. An ultrasonic non-destructive evaluation method with a wave propagation model was developed to determine corneal biomechanical properties in vivo. In this study, we tested the feasibility of the approach in differentiating the mechanical properties of soft contact lenses as corneal phantoms. Three material types of soft contact lenses (six samples in each group) were measured using a broadband ultrasound transducer. The ultrasonic reflections from the contact lenses were recorded by a 500 MHz/8-bit digitizer, and displayed and processed by a PC. A reference signal was recorded to compute the normalized power spectra using Fast Fourier Transformation. An inverse algorithm based on least-squares minimization was used to reconstruct three parameters of the contact lenses: density, thickness, and elastic constants l+2m. The thickness of each sample was verified using an electronic thickness gauge, and the averaged density for each type of lenses was verified using Archimedes’ principle and manufacturer’s report. Our results demonstrated that the ultrasonic system was able to differentiate the elastic properties of the three types of the soft contact lenses with statistical significance (P-valueo0.001). The reconstructed thicknesses and densities agreed well with the independent measurements. Our studies on corneal phantoms indicated that the ultrasonic system was sensitive and accurate in measuring the material properties of cornea-like structures. It is important to optimize the system for in vivo measurements. r 2006 Elsevier Ltd. All rights reserved. Keywords: Cornea biomechanics; Ultrasound; Non-destructive evaluation; Elastic wave propagation; Thin layer
1. Introduction The biomechanical properties of corneal tissue are essential for the eye’s normal physiological function, i.e., maintaining a spherical shape for visual acuity (Ethier et al., 2004). These properties may be altered by either disease or surgical operations. Studies have shown that, keratoconus, a corneal disease that manifests as buckling Corresponding author. Tel.: +1 614 247 8904; fax: +1 614 292 7301. E-mail address: [email protected] (J. Liu).
0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2006.04.017
of corneal tissue around apex, is correlated with corneal thinning and softening (Edmund, 1988). Ablative refractive surgery, a therapeutic procedure in which corneal tissue is removed in a specific pattern to correct myopia or hyperopia, may introduce changes in biomechanical properties (Hjortdal et al., 1996). Variations in corneal biomechanical properties may be a significant confounding factor for tonometry measurement of intraocular pressure, a routine practice for glaucoma screening (Liu and Roberts, 2005). Non-invasive determination of corneal mechanical properties is therefore important for detection and monitoring of ocular diseases.
ARTICLE IN PRESS 1178
J. Liu et al. / Journal of Biomechanics 40 (2007) 1177–1182
The elastic modulus of ex vivo cornea tissue has been studied in the past (Nyquist, 1968; Woo et al., 1972; Nash et al., 1982; Jue and Maurice, 1986; Hoeltzel et al., 1992), yet non-invasive methods for in vivo measurements are not yet available. Human corneas present a unique opportunity for the application of ultrasonic techniques due to their direct accessibility and structural simplicity. The cornea can be treated as a homogenous layer for ultrasonic modeling and property reconstruction. Ultrasound spectroscopy has been used to characterize a thin layer embedded between two substrates (Brekhovskikh, 1996; Kinra and Iyer, 1995a, b; Lavrentyev and Rokhlin, 1997). The spectra of the ultrasonic reflections from the thin layer are dependent on a set of layer material properties such as density, thickness, and elastic moduli. In this study, we developed a model and system for measurements of human corneas using ultrasound spectroscopic methods. To validate this approach in terms of the accuracy of the estimated properties, we first performed measurements on soft contact lenses to avoid unknown confounding factors associated with biological tissue samples (e.g., hydration status of corneas). The ultrasonic estimation of the properties was then compared with those obtained by standard methods.
2. Method 2.1. Experimental samples Soft contact lenses made of three different materials were obtained: 1. Hydrogel lenses: Biomedics 55 (Ocular Sciences), 2. Silicone–Hydrogel lenses: Night & Day (CIBA Vision Corp), and 3. Silicone lenses: DuraSoft2 (Wesley Jessen Visioncare). Six lenses, with identical specifications, of each type were used. The lenses were stored in 0.9% saline for more than 24 h before measurements.
x2
saline
(O)
(R1) x1
(T1)
h
(R2) cornea or contact lens (T2)
aqueous humor or saline Fig. 1. Plane wave propagation in a thin layer structure of cornea or contact lens: the incident wave (O), the reflected wave in water bath (R1), the transmitted wave in cornea or contact lens (T1), the reflected wave in cornea or contact lens (R2), and the transmitted wave in aqueous humor (T2). The thickness of the layer is h.
was chosen as 7–16 MHz, corresponding to the bandwidth of the transducer used in the experiments. 2.3. Measurement system and data acquisition All lenses were immersed in 0.9% saline during ultrasonic measurement. A broadband ultrasound transducer (XMS, Panametrics-NDT) was excited by a pulser-receiver (5900PR, Panametrics-NDT). The X, Y, and Z positions of the transducer were adjusted using precision linear stages (1 mm step size, Newport), to center the ultrasonic beam to the center of lens apex. The distance from the transducer surface to lens apex was kept the same for all samples. The ultrasonic reflections were recorded using a digitizer (DP105, Acqiris, 500 MHz/8-bit), and displayed and stored on a PC. All measurements were performed under the same pulser/receiver and digitizer settings. The ultrasonic reflections from the contact lens layer were converted into experimental reflection spectra using Fast Fourier Transformation.
2.2. Wave propagation model A mathematical model of elastic wave propagation was constructed to simulate ultrasound propagation in contact lenses immersed in liquid bath. Mechanically, the system is composed of a thin layer (i.e., contact lens) embedded between two continuous subspaces (i.e., saline) as shown in Fig. 1. The details of the model derivation are presented in online Supplementary material. Briefly, the reflection coefficient of the thin layer (defined as the ratio between the magnitude of the reflected wave AR1 and that of the original wave AO) was solved by enforcing continuity conditions at the interfaces between the layer and the substrates. By calculating reflection coefficients at a range of frequencies, a reflection spectrum was obtained. The frequency range
2.4. Inverse algorithm The physical properties (thickness h, density r, and modulus l+2m) of the lens layer were reconstructed using an inverse algorithm. The inverse algorithm searched the multidimensional space to minimize the following function: ðh; r; kÞ ¼
fm X
ðjRe ðf Þj jRt ðf ÞjÞ2 ,
(1)
f ¼f 1
where m is the number of the data points at different frequencies; Re and Rt are the experimental and theoretical reflection coefficients, which are functions of the layer properties and frequency f; e is the error term
ARTICLE IN PRESS J. Liu et al. / Journal of Biomechanics 40 (2007) 1177–1182
that represents the discrepancy between the reconstructed reflection spectra Rt and the experimental reflection spectra Re. A MatLab non-linear least-square optimization algorithm was adopted for the minimization search. 2.5. Thickness validation The thickness of each lens was measured separately using an electronic thickness gauge ET-1 (Rehder). This device is a standard for measuring the thickness of soft contact lenses (Fatt, 1997). Three readings were taken from each lens, and the average was used to compare with the reconstructed thickness obtained through the ultrasonic method. 2.6. Density validation Density of the lenses was measured by comparing the mass of the samples in air and in water (Archimedes’ principle). Due to their small mass, all six lenses of the same type were scaled together using an analytical balance in air and in water. The density was calculated as: r ¼ ðmair rwater =mair mwater Þ, where mair and mwater were the lens mass measured in air and water, and rwater ¼ 1.0 g/cm3. Manufacturers’ reports on the density of the lenses were also obtained.
3. Results
Normalized Reflection Coefficients
The ultrasonic spectra from all contact lenses measured in this study are presented in Fig. 2. The
spectral signals were consistent within the same type lenses in terms of the magnitude and distance between maxima or minima. The spectral curves appeared ‘‘shifted’’ along the frequency axis for some lenses compared to others of the same type (‘‘1’’ in Night & Day, and ‘‘2’’, ‘‘3’’ and ‘‘4’’ in DuraSoft2). The spectra of different material types differed in terms of the height of the maxima and the distance between adjacent maxima or minima. Table 1 presents the mean and standard deviation of the reconstructed properties for each type of the contact lenses. Night & Day had the lowest reconstructed longitudinal modulus, while Biomedics had the intermediate, and DuraSoft2, the highest. Pair-wise Student t-test showed that the modulus l+2m was significantly different for different types of lenses (Po0:001). Fig. 3 compares the reconstructed and experimental spectra for one lens of each group. The reconstructed spectra were calculated theoretically using the reconstructed properties obtained from the ultrasonic measurements of the respective lens. Fig. 4 is a comparison of the reconstructed and the directly measured thickness for each lens. The two measurements were highly correlated (R ¼ 92:5%), and all data points were close to the ‘‘equivalent’’ line. It is observed that although close to equivalent, the thicknesses reconstructed by the ultrasonic method were higher than those obtained using the thickness gauge for Night & Day lenses, while lower for DuraSoft2 lenses. It is also observed that sample 1 had a very different thickness compared to other Night and Day samples; and so were samples 2, 3 and 4 of DuraSoft2.
0.6 0.5
3
4
2
0.4 0.3 0.2 0.1
1 0 7
9
1179
13 11 Frequency (MHz)
15
BM 1 BM 2 BM 3 BM 4 BM 5 BM 6 ND1 ND2 ND3 ND4 ND5 ND6 DS 1 DS 2 DS 3 DS 4 DS 5 DS 6
Fig. 2. Measured ultrasonic reflection spectra from three types of soft contact lenses: BM (dotted line, Biomedics), ND (solid line, Night & Day), and DS (open circles, DuraSoft2). The lenses of the same material type shared similar magnitudes of the minima and maxima, although frequency positions may vary due to variations in thickness (sample 1 in ND, samples 2, 3, and 4 in DS). Lenses of different material types had distinct spectral characteristics.
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Table 1 Reconstructed material properties of soft contact lenses Elastic constantsa (GPa)
Thickness (mm)
Biomedics Night & day Durasoft2 a
Mean
SD
Mean
SD
Mean
SD
108.64 86.14 76.18
3.87 9.72 15.03
2.74 2.14 2.91
0.05 0.23 0.07
1.11 1.09 1.17
0.02 0.01 0.03
Difference was statistically significant. P value o0.001, pair-wise Student t-test.
0.25 Reflection Coefficients
Density (g/cm3)
Reconstructed
Table 2 Comparison of contact lens density values (g/cm3) obtained from different methods
Experimental
0.2 0.15
Biomedics Night & day DuraSoft2
0.1 0.05
Direct measurement
Ultrasound reconstruction
Manufacturers’ report
1.108 1.097 1.203
1.107 1.094 1.174
1.060 1.080 1.176
0 7
9
11 13 Frequency (MHz)
three measurements were small, and not statistically significant.
15
Fig. 3. Comparison of experimental and reconstructed reflection spectra. The reconstructed reflection spectra were computed using lens thickness, density and elastic constants measured through the ultrasonic method.
120 Biomedics Night & Day DuraSoft 2
Ultrasound Reconstruction (um)
110 100 90 80
2
70
1
3
60 4 50 50
60
70
80
90
100
110
120
Direct Measurement (um) Fig. 4. Comparison of thicknesses measured directly by an electronic thickness gauge and reconstructed from the ultrasonic method. The dots on the diagonal line indicate perfect agreement between the two methods. The samples 1, 2, 3, and 4 correspond to those marked in Fig. 4. The Pearson’s correlation coefficient for these two measurements was 0.925.
The comparison of the reconstructed, the directly measured, and reported density of each group of lenses is presented in Table 2. The differences among these
4. Discussion Our study showed that the ultrasonic approach was able to differentiate the mechanical properties of the three types of soft contact lenses made of different polymers. The standard deviation of the reconstructed modulus within each type of lenses was small, indicating consistency in measuring samples made of the same material. The thicknesses obtained from the ultrasonic method and direct readings from the thickness gauge agreed from sample to sample. The averaged density readings from ultrasonic reconstruction were consistent with the direct measurements, as well as the manufacturers’ reports. Therefore, the ultrasonic measurements of two out of the three unknown properties (thickness, density, and modulus) were validated through standard methods. We also demonstrated that the theoretical spectra calculated from ultrasonic measurements of the properties agreed well with the experimental spectra (Fig. 4). Since the theoretical spectra are uniquely determined by the three properties (see online Supplementary information), it could be inferred that the reconstructed modulus were accurate. This approach can potentially apply to in vivo measurements of human corneas. Safety is paramount for in vivo applications. Ultrasound exposure may produce thermal, mechanical or cavitation effects in biological tissues (Dalecki, 2004). According to their distinct mechanisms, only thermal effects may be of
ARTICLE IN PRESS J. Liu et al. / Journal of Biomechanics 40 (2007) 1177–1182
potential concern in our approach. Thermal effect, which is the absorption of acoustic energy to cause temperature rise in tissue, is determined by the spatially and temporally averaged intensity, ISPTA,3, of the transducer output. FDA 510(k) guideline for ultrasound exposure of ocular tissue is ISPTA,3p17 mW/cm2. We expect the ISPTA,3 of the transducer used in this study is far below the threshold due to its unfocused nature. Theoretic estimation was performed based on extrapolation of the reported values in literature (Silverman et al., 2001), and we found the ISPTA,3 in this study may be as low as 0.2 mW/cm2. Experimental measurements are needed to obtain the actual output characteristics. Ultrasonic exposure could be further reduced by minimizing the dwelling time of sound waves on corneas. For example, the transducer could be grossly positioned before it is turned on. It can then be turned on to facilitate fine tune of the positioning, which takes about 1 or 2 min in our experiments. After the transducer is positioned, the actual data acquisition requires only a few seconds to complete. The positioning stages we adopted allow a resolution of 1 mm in each direction of adjustment to ensure good positioning of the transducer with respect to the sample. To measure the error introduced when the ultrasonic beam was not perfectly centered at cornea apex, we compared the reconstructed values for centered and offcentered (by 40 mm in one direction) measurements on one contact lens. The measured values deviated from the original values by about 1%. This may be explained by the fairly large ultrasonic beam size (2 mm) with respect to the dimensions of the contact lens. In addition, when the transducer is significantly offcentered, the ultrasonic reflections become much smaller in amplitude which alerts the operator as mal-positioning. A water bath for transduction of sound waves is necessary for using immersion-type ultrasonic transducers. The water bath can be applied by using an eye cup in the same fashion as in ophthalmic ultrasound imaging (Pavlin and Foster, 1995). Sound waves may be transmitted into the cornea through tear film; however, the near field effect of the transducer could make amplitude-dependent measurements extremely difficult due to the variations of acoustic intensity in this region. Contact-mode transducers may be considered. The design and manufacturing of such transducers with high transmitting efficiency to biological tissue may be challenging. IOP loading will likely change the elastic modulus of corneas due to the intrinsic nonlinearity of corneal tissue (Bryant and McDonnell, 1996). The nonlinearity of intact corneas can be studied by performing ultrasonic measurements on enucleated eyes with IOPs maintained and monitored at various levels to provide useful information for constitutive modeling of cornea tissue.
1181
In conclusion, we have demonstrated the feasibility of an ultrasonic approach for non-destructive evaluation of the mechanical properties of corneal phantoms. This approach can be potentially applied to in vivo noninvasive measurement of human corneas.
Acknowledgements The authors would like to thank Michael Twa, O.D., M.S., for obtaining contact lens samples. This work is partially funded by the Ann Ellis Fund from the Columbus Foundation.
Appendix A. Supplementary materials Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech. 2006.04.017
References Brekhovskikh, L.M., 1996. Waves in Layered Media. Academic, New York. Bryant, M.R., McDonnell, P.J., 1996. Constitutive laws for biomechanical modeling of refractive surgery. Journal of Biomechanical Engineering-Transactions of the ASME 118 (4), 473–481. Dalecki, D., 2004. Mechanical bioeffects of ultrasound. Annual Review of Biomedical Engineering 6, 229–248. Edmund, C., 1988. Corneal elasticity and ocular rigidity in normal and keratoconic eyes. Acta Ophthalmologica 66 (2), 134–140. Ethier, R.C., Johnson, M., Ruberti, J., 2004. Ocular biomechanics and biotransport. Annual Review of Biomedical Engineering 6, 249–273. Fatt, I., 1997. Comparative study of some physiologically important properties of six brands of disposable hydrogel contact lenses. Journal of Contact Lens Association of Ophthalmologists (CLAO) 23 (1), 49–54. Hjortdal, J.O., Bohm, A., Kohlhaas, M., Olsen, H., Lerche, R., Ehlers, N., Draeger, J., 1996. Mechanical stability of the cornea after radial keratotomy and photorefractive keratectomy. Journal of Refractive Surgery 12 (4), 459–466. Hoeltzel, D.A., Altman, P., Buzard, K., Choe, K., 1992. Strip extensiometry for comparison of the mechanical response of bovine, rabbit and human corneas. Journal of Biomechanical Engineering 114, 202–215. Jue, B., Maurice, D.M., 1986. The mechanical properties of rabbit and human cornea. Journal of Biomechanics 19, 847–853. Kinra, V.K., Iyer, V.R., 1995a. Ultrasonic measurement of the thickness, phase velocity, density or attenuation of a thinviscoelastic plate. Part I: The forward problem. Ultrasonics 33 (2), 95–109. Kinra, V.K., Iyer, V.R., 1995b. Ultrasonic measurement of the thickness, phase velocity, density or attenuation of a thinviscoelastic plate. Part II: the inverse problem. Ultrasonics 33 (2), 111–122. Lavrentyev, A.I., Rokhlin, S.I., 1997. Determination of elastic moduli, density, attenuation, and thickness of a layer using ultrasonic
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spectroscopy at two angles. Journal of the Acoustical Society of the America 102 (6), 3467–3477. Liu, J., Roberts, C., 2005. Quantitative analysis of effects of biomechanics on IOP measurement. Journal of Cataract and Refractive Surgery 31 (1), 146–155. Nash, I.S., Greene, P.R., Foster, C.S., 1982. Comparison of mechanical properties of keratoconus and normal cornea. Experimental Eye Research 35, 413–424. Nyquist, G.W., 1968. Rheology of the cornea: experimental techniques and results. Experimental Eye Research 7, 183–188.
Pavlin, C.J., Foster, F.S., 1995. Ultrasound Biomicroscopy of the Eye. Springer, Berlin. Silverman, R.H., Lizzi, F.L., Ursea, B.G., Cozzarelli, L., Ketterling, J.A., Deng, C.X., Folberg, R., Coleman, D.J., 2001. Safety levels for exposure of cornea and lens to very highfrequency ultrasound. Journal of Ultrasound in Medicine 20 (9), 979–986. Woo, S.L.-Y., Kobayashi, A.S., Shlegel, W.A., Lawrence, C., 1972. Nonlinear material properties of intact cornea and sclera. Experimental Eye Research 14, 29–39.
Journal of Biomechanics 40 (2007) 1177–1182
Short communication
www.elsevier.com/locate/jbiomech www.JBiomech.com
Ultrasonic model and system for measurement of corneal biomechanical properties and validation on phantoms Jun Liua,b,, Xiaoyin Heb, Xueliang Panc, Cynthia J. Robertsa,b a
Biomedical Engineering Department, The Ohio State University, 270 Bevis Hall, 1080 Carmack Rd., Columbus, OH 43210, USA b Department of Ophthalmology, The Ohio State University, USA c Department of Statistics, The Ohio State University, USA Accepted 28 April 2006
Abstract Non-invasive measurement of biomechanical properties of corneas may provide important information for ocular disease management and therapeutic procedures. An ultrasonic non-destructive evaluation method with a wave propagation model was developed to determine corneal biomechanical properties in vivo. In this study, we tested the feasibility of the approach in differentiating the mechanical properties of soft contact lenses as corneal phantoms. Three material types of soft contact lenses (six samples in each group) were measured using a broadband ultrasound transducer. The ultrasonic reflections from the contact lenses were recorded by a 500 MHz/8-bit digitizer, and displayed and processed by a PC. A reference signal was recorded to compute the normalized power spectra using Fast Fourier Transformation. An inverse algorithm based on least-squares minimization was used to reconstruct three parameters of the contact lenses: density, thickness, and elastic constants l+2m. The thickness of each sample was verified using an electronic thickness gauge, and the averaged density for each type of lenses was verified using Archimedes’ principle and manufacturer’s report. Our results demonstrated that the ultrasonic system was able to differentiate the elastic properties of the three types of the soft contact lenses with statistical significance (P-valueo0.001). The reconstructed thicknesses and densities agreed well with the independent measurements. Our studies on corneal phantoms indicated that the ultrasonic system was sensitive and accurate in measuring the material properties of cornea-like structures. It is important to optimize the system for in vivo measurements. r 2006 Elsevier Ltd. All rights reserved. Keywords: Cornea biomechanics; Ultrasound; Non-destructive evaluation; Elastic wave propagation; Thin layer
1. Introduction The biomechanical properties of corneal tissue are essential for the eye’s normal physiological function, i.e., maintaining a spherical shape for visual acuity (Ethier et al., 2004). These properties may be altered by either disease or surgical operations. Studies have shown that, keratoconus, a corneal disease that manifests as buckling Corresponding author. Tel.: +1 614 247 8904; fax: +1 614 292 7301. E-mail address: [email protected] (J. Liu).
0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2006.04.017
of corneal tissue around apex, is correlated with corneal thinning and softening (Edmund, 1988). Ablative refractive surgery, a therapeutic procedure in which corneal tissue is removed in a specific pattern to correct myopia or hyperopia, may introduce changes in biomechanical properties (Hjortdal et al., 1996). Variations in corneal biomechanical properties may be a significant confounding factor for tonometry measurement of intraocular pressure, a routine practice for glaucoma screening (Liu and Roberts, 2005). Non-invasive determination of corneal mechanical properties is therefore important for detection and monitoring of ocular diseases.
ARTICLE IN PRESS 1178
J. Liu et al. / Journal of Biomechanics 40 (2007) 1177–1182
The elastic modulus of ex vivo cornea tissue has been studied in the past (Nyquist, 1968; Woo et al., 1972; Nash et al., 1982; Jue and Maurice, 1986; Hoeltzel et al., 1992), yet non-invasive methods for in vivo measurements are not yet available. Human corneas present a unique opportunity for the application of ultrasonic techniques due to their direct accessibility and structural simplicity. The cornea can be treated as a homogenous layer for ultrasonic modeling and property reconstruction. Ultrasound spectroscopy has been used to characterize a thin layer embedded between two substrates (Brekhovskikh, 1996; Kinra and Iyer, 1995a, b; Lavrentyev and Rokhlin, 1997). The spectra of the ultrasonic reflections from the thin layer are dependent on a set of layer material properties such as density, thickness, and elastic moduli. In this study, we developed a model and system for measurements of human corneas using ultrasound spectroscopic methods. To validate this approach in terms of the accuracy of the estimated properties, we first performed measurements on soft contact lenses to avoid unknown confounding factors associated with biological tissue samples (e.g., hydration status of corneas). The ultrasonic estimation of the properties was then compared with those obtained by standard methods.
2. Method 2.1. Experimental samples Soft contact lenses made of three different materials were obtained: 1. Hydrogel lenses: Biomedics 55 (Ocular Sciences), 2. Silicone–Hydrogel lenses: Night & Day (CIBA Vision Corp), and 3. Silicone lenses: DuraSoft2 (Wesley Jessen Visioncare). Six lenses, with identical specifications, of each type were used. The lenses were stored in 0.9% saline for more than 24 h before measurements.
x2
saline
(O)
(R1) x1
(T1)
h
(R2) cornea or contact lens (T2)
aqueous humor or saline Fig. 1. Plane wave propagation in a thin layer structure of cornea or contact lens: the incident wave (O), the reflected wave in water bath (R1), the transmitted wave in cornea or contact lens (T1), the reflected wave in cornea or contact lens (R2), and the transmitted wave in aqueous humor (T2). The thickness of the layer is h.
was chosen as 7–16 MHz, corresponding to the bandwidth of the transducer used in the experiments. 2.3. Measurement system and data acquisition All lenses were immersed in 0.9% saline during ultrasonic measurement. A broadband ultrasound transducer (XMS, Panametrics-NDT) was excited by a pulser-receiver (5900PR, Panametrics-NDT). The X, Y, and Z positions of the transducer were adjusted using precision linear stages (1 mm step size, Newport), to center the ultrasonic beam to the center of lens apex. The distance from the transducer surface to lens apex was kept the same for all samples. The ultrasonic reflections were recorded using a digitizer (DP105, Acqiris, 500 MHz/8-bit), and displayed and stored on a PC. All measurements were performed under the same pulser/receiver and digitizer settings. The ultrasonic reflections from the contact lens layer were converted into experimental reflection spectra using Fast Fourier Transformation.
2.2. Wave propagation model A mathematical model of elastic wave propagation was constructed to simulate ultrasound propagation in contact lenses immersed in liquid bath. Mechanically, the system is composed of a thin layer (i.e., contact lens) embedded between two continuous subspaces (i.e., saline) as shown in Fig. 1. The details of the model derivation are presented in online Supplementary material. Briefly, the reflection coefficient of the thin layer (defined as the ratio between the magnitude of the reflected wave AR1 and that of the original wave AO) was solved by enforcing continuity conditions at the interfaces between the layer and the substrates. By calculating reflection coefficients at a range of frequencies, a reflection spectrum was obtained. The frequency range
2.4. Inverse algorithm The physical properties (thickness h, density r, and modulus l+2m) of the lens layer were reconstructed using an inverse algorithm. The inverse algorithm searched the multidimensional space to minimize the following function: ðh; r; kÞ ¼
fm X
ðjRe ðf Þj jRt ðf ÞjÞ2 ,
(1)
f ¼f 1
where m is the number of the data points at different frequencies; Re and Rt are the experimental and theoretical reflection coefficients, which are functions of the layer properties and frequency f; e is the error term
ARTICLE IN PRESS J. Liu et al. / Journal of Biomechanics 40 (2007) 1177–1182
that represents the discrepancy between the reconstructed reflection spectra Rt and the experimental reflection spectra Re. A MatLab non-linear least-square optimization algorithm was adopted for the minimization search. 2.5. Thickness validation The thickness of each lens was measured separately using an electronic thickness gauge ET-1 (Rehder). This device is a standard for measuring the thickness of soft contact lenses (Fatt, 1997). Three readings were taken from each lens, and the average was used to compare with the reconstructed thickness obtained through the ultrasonic method. 2.6. Density validation Density of the lenses was measured by comparing the mass of the samples in air and in water (Archimedes’ principle). Due to their small mass, all six lenses of the same type were scaled together using an analytical balance in air and in water. The density was calculated as: r ¼ ðmair rwater =mair mwater Þ, where mair and mwater were the lens mass measured in air and water, and rwater ¼ 1.0 g/cm3. Manufacturers’ reports on the density of the lenses were also obtained.
3. Results
Normalized Reflection Coefficients
The ultrasonic spectra from all contact lenses measured in this study are presented in Fig. 2. The
spectral signals were consistent within the same type lenses in terms of the magnitude and distance between maxima or minima. The spectral curves appeared ‘‘shifted’’ along the frequency axis for some lenses compared to others of the same type (‘‘1’’ in Night & Day, and ‘‘2’’, ‘‘3’’ and ‘‘4’’ in DuraSoft2). The spectra of different material types differed in terms of the height of the maxima and the distance between adjacent maxima or minima. Table 1 presents the mean and standard deviation of the reconstructed properties for each type of the contact lenses. Night & Day had the lowest reconstructed longitudinal modulus, while Biomedics had the intermediate, and DuraSoft2, the highest. Pair-wise Student t-test showed that the modulus l+2m was significantly different for different types of lenses (Po0:001). Fig. 3 compares the reconstructed and experimental spectra for one lens of each group. The reconstructed spectra were calculated theoretically using the reconstructed properties obtained from the ultrasonic measurements of the respective lens. Fig. 4 is a comparison of the reconstructed and the directly measured thickness for each lens. The two measurements were highly correlated (R ¼ 92:5%), and all data points were close to the ‘‘equivalent’’ line. It is observed that although close to equivalent, the thicknesses reconstructed by the ultrasonic method were higher than those obtained using the thickness gauge for Night & Day lenses, while lower for DuraSoft2 lenses. It is also observed that sample 1 had a very different thickness compared to other Night and Day samples; and so were samples 2, 3 and 4 of DuraSoft2.
0.6 0.5
3
4
2
0.4 0.3 0.2 0.1
1 0 7
9
1179
13 11 Frequency (MHz)
15
BM 1 BM 2 BM 3 BM 4 BM 5 BM 6 ND1 ND2 ND3 ND4 ND5 ND6 DS 1 DS 2 DS 3 DS 4 DS 5 DS 6
Fig. 2. Measured ultrasonic reflection spectra from three types of soft contact lenses: BM (dotted line, Biomedics), ND (solid line, Night & Day), and DS (open circles, DuraSoft2). The lenses of the same material type shared similar magnitudes of the minima and maxima, although frequency positions may vary due to variations in thickness (sample 1 in ND, samples 2, 3, and 4 in DS). Lenses of different material types had distinct spectral characteristics.
ARTICLE IN PRESS J. Liu et al. / Journal of Biomechanics 40 (2007) 1177–1182
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Table 1 Reconstructed material properties of soft contact lenses Elastic constantsa (GPa)
Thickness (mm)
Biomedics Night & day Durasoft2 a
Mean
SD
Mean
SD
Mean
SD
108.64 86.14 76.18
3.87 9.72 15.03
2.74 2.14 2.91
0.05 0.23 0.07
1.11 1.09 1.17
0.02 0.01 0.03
Difference was statistically significant. P value o0.001, pair-wise Student t-test.
0.25 Reflection Coefficients
Density (g/cm3)
Reconstructed
Table 2 Comparison of contact lens density values (g/cm3) obtained from different methods
Experimental
0.2 0.15
Biomedics Night & day DuraSoft2
0.1 0.05
Direct measurement
Ultrasound reconstruction
Manufacturers’ report
1.108 1.097 1.203
1.107 1.094 1.174
1.060 1.080 1.176
0 7
9
11 13 Frequency (MHz)
three measurements were small, and not statistically significant.
15
Fig. 3. Comparison of experimental and reconstructed reflection spectra. The reconstructed reflection spectra were computed using lens thickness, density and elastic constants measured through the ultrasonic method.
120 Biomedics Night & Day DuraSoft 2
Ultrasound Reconstruction (um)
110 100 90 80
2
70
1
3
60 4 50 50
60
70
80
90
100
110
120
Direct Measurement (um) Fig. 4. Comparison of thicknesses measured directly by an electronic thickness gauge and reconstructed from the ultrasonic method. The dots on the diagonal line indicate perfect agreement between the two methods. The samples 1, 2, 3, and 4 correspond to those marked in Fig. 4. The Pearson’s correlation coefficient for these two measurements was 0.925.
The comparison of the reconstructed, the directly measured, and reported density of each group of lenses is presented in Table 2. The differences among these
4. Discussion Our study showed that the ultrasonic approach was able to differentiate the mechanical properties of the three types of soft contact lenses made of different polymers. The standard deviation of the reconstructed modulus within each type of lenses was small, indicating consistency in measuring samples made of the same material. The thicknesses obtained from the ultrasonic method and direct readings from the thickness gauge agreed from sample to sample. The averaged density readings from ultrasonic reconstruction were consistent with the direct measurements, as well as the manufacturers’ reports. Therefore, the ultrasonic measurements of two out of the three unknown properties (thickness, density, and modulus) were validated through standard methods. We also demonstrated that the theoretical spectra calculated from ultrasonic measurements of the properties agreed well with the experimental spectra (Fig. 4). Since the theoretical spectra are uniquely determined by the three properties (see online Supplementary information), it could be inferred that the reconstructed modulus were accurate. This approach can potentially apply to in vivo measurements of human corneas. Safety is paramount for in vivo applications. Ultrasound exposure may produce thermal, mechanical or cavitation effects in biological tissues (Dalecki, 2004). According to their distinct mechanisms, only thermal effects may be of
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potential concern in our approach. Thermal effect, which is the absorption of acoustic energy to cause temperature rise in tissue, is determined by the spatially and temporally averaged intensity, ISPTA,3, of the transducer output. FDA 510(k) guideline for ultrasound exposure of ocular tissue is ISPTA,3p17 mW/cm2. We expect the ISPTA,3 of the transducer used in this study is far below the threshold due to its unfocused nature. Theoretic estimation was performed based on extrapolation of the reported values in literature (Silverman et al., 2001), and we found the ISPTA,3 in this study may be as low as 0.2 mW/cm2. Experimental measurements are needed to obtain the actual output characteristics. Ultrasonic exposure could be further reduced by minimizing the dwelling time of sound waves on corneas. For example, the transducer could be grossly positioned before it is turned on. It can then be turned on to facilitate fine tune of the positioning, which takes about 1 or 2 min in our experiments. After the transducer is positioned, the actual data acquisition requires only a few seconds to complete. The positioning stages we adopted allow a resolution of 1 mm in each direction of adjustment to ensure good positioning of the transducer with respect to the sample. To measure the error introduced when the ultrasonic beam was not perfectly centered at cornea apex, we compared the reconstructed values for centered and offcentered (by 40 mm in one direction) measurements on one contact lens. The measured values deviated from the original values by about 1%. This may be explained by the fairly large ultrasonic beam size (2 mm) with respect to the dimensions of the contact lens. In addition, when the transducer is significantly offcentered, the ultrasonic reflections become much smaller in amplitude which alerts the operator as mal-positioning. A water bath for transduction of sound waves is necessary for using immersion-type ultrasonic transducers. The water bath can be applied by using an eye cup in the same fashion as in ophthalmic ultrasound imaging (Pavlin and Foster, 1995). Sound waves may be transmitted into the cornea through tear film; however, the near field effect of the transducer could make amplitude-dependent measurements extremely difficult due to the variations of acoustic intensity in this region. Contact-mode transducers may be considered. The design and manufacturing of such transducers with high transmitting efficiency to biological tissue may be challenging. IOP loading will likely change the elastic modulus of corneas due to the intrinsic nonlinearity of corneal tissue (Bryant and McDonnell, 1996). The nonlinearity of intact corneas can be studied by performing ultrasonic measurements on enucleated eyes with IOPs maintained and monitored at various levels to provide useful information for constitutive modeling of cornea tissue.
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In conclusion, we have demonstrated the feasibility of an ultrasonic approach for non-destructive evaluation of the mechanical properties of corneal phantoms. This approach can be potentially applied to in vivo noninvasive measurement of human corneas.
Acknowledgements The authors would like to thank Michael Twa, O.D., M.S., for obtaining contact lens samples. This work is partially funded by the Ann Ellis Fund from the Columbus Foundation.
Appendix A. Supplementary materials Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech. 2006.04.017
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