Journal of Biomechanics 47 (2014) 723–728
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Viscoelastic shear properties of the corneal stroma Hamed Hatami-Marbini n School of Mechanical and Aerospace Engineering, Oklahoma State University, 218 Engineering North, Stillwater, OK 74078 5016, United States
art ic l e i nf o
a b s t r a c t
Article history: Accepted 18 November 2013
The cornea is a highly specialized transparent tissue which covers the front of the eye. It is a tough tissue responsible for refracting the light and protecting the sensitive internal contents of the eye. The biomechanical properties of the cornea are primarily derived from its extracellular matrix, the stroma. The majority of previous studies have used strip tensile and pressure inflation testing methods to determine material parameters of the corneal stroma. Since these techniques do not allow measurements of the shear properties, there is little information available on transverse shear modulus of the cornea. The primary objectives of the present study were to determine the viscoelastic behavior of the corneal stroma in shear and to investigate the effects of the compressive strain. A thorough knowledge of the shear properties is required for developing better material models for corneal biomechanics. In the present study, torsional shear experiments were conducted at different levels of compressive strain (0– 30%) on porcine corneal buttons. First, the range of linear viscoelasticity was determined from strain sweep experiments. Then, frequency sweep experiments with a shear strain amplitude of 0.2% (which was within the region of linear viscoelasticity) were performed. The corneal stroma exhibited viscoelastic properties in shear. The shear storage modulus, G′, and shear loss modulus, G″, were reported as a function of tissue compression. It was found that although both of these parameters were dependent on frequency, shear strain amplitude, and compressive strain, the average shear storage and loss moduli varied from 2 to 8 kPa, and 0.3 to 1.2 kPa, respectively. Therefore, it can be concluded that the transverse shear modulus is of the same order of magnitude as the out-of-plane Young's modulus and is about three orders of magnitude lower than the in-plane Young's modulus. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Corneal stromal mechanics Oscillatory torsional experiments Linear viscoelastic behavior Porcine cornea
1. Introduction The cornea is a highly specialized tissue which refracts and transmits light. It forms about one-sixth of the outer layer of the eye and is considered as its main refractive element. The normal cornea transmits over 90% of the light and causes 75% of the light to scatter at angles larger than 301. In addition to unique optical properties, the cornea has important structural roles in protecting internal contents of the eye and maintaining its shape. The tissue is constantly subjected to mechanical stresses caused by internal forces (e.g. intraocular pressure) and external forces (e.g. getting poked and eye rubbing). Over the past few decades, various computational models have been proposed to understand and predict the biomechanical behavior of the cornea (Anderson et al., 2004; Elsheikh et al., 2007; Fernandez et al., 2006; Pandolfi, 2010; Pandolfi and Holzapfel, 2008). Although these mathematical models have been successful in capturing certain aspects of the corneal mechanics, more studies are required to precisely predict the tissue behavior.
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0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.11.019
It is well-known that the accuracy of the predictions of computational models is highly dependent on the accuracy of constitutive (stress–strain) relation that has been presumed. The material parameters required for constitutive models are obtained from material characterization experiments. Under physiological conditions, the cornea is mainly subjected to intraocular pressure which puts the tissue under membrane tension. Therefore, extensometry and inflation tests have commonly been used to investigate the tensile properties of the cornea (Boyce et al., 2007; Elsheikh and Alhasso, 2009; Jue and Maurice, 1986; Kampmeier et al., 2000; Klyce et al., 1971; Olsen and Sperling, 1987). The cornea in these studies has often been considered as an isotropic material with two distinct material constants, i.e. the Young's modulus and Poisson's ratio. Although there are various studies in the literature promoting this simple material model, there still exists a large range of variation for the reported Young's modulus (Elsheikh et al., 2007; Jue and Maurice, 1986). This significant inconsistency could be due to the use of different testing conditions, experimental procedures, and samples. The biomechanical properties of the cornea are primarily due to the composition and properties of the stromal layer, a connective tissue forming the bulk of the corneal thickness. The stroma is a unique extracellular matrix in that it has a highly
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organized microstructure and is subsequently transparent. Despite the widespread use of the oversimplified isotropic material model, the microstructure of the corneal extracellular matrix is inhomogeneous and anisotropic (Maurice, 1957, 1984; Meek, 2008). The stroma is composed of many stacks of lamellae; each comprising a regular network of collagen fibrils and proteoglycans. The collagen fibrils of almost uniform diameter are distributed in a pseudohexagonal arrangement and proteoglycans fill the spaces between the fibrils (Lewis et al., 2010; Maurice, 1957, 1984). This particular ultrastructure of the stroma implies that in-plane and out-of-plane material properties should be significantly different from each other (Hatami-Marbini and Etebu, 2013a). Therefore, the biomechanical response of the cornea could be represented more accurately with anisotropic material models. Because of the lamellar structure of the stromal layer, an appealing choice is a transversely isotropic constitutive law in which the axis of material symmetry is perpendicular to the surface. The transversely isotropic models are generally more complex than isotropic material models. In particular, the linear transversely isotropic model is defined in terms of five material constants: the out-ofplane Young's modulus and Poisson's ratio, the in-plane Young's modulus and Poisson's ratio, and the transverse shear modulus. We have recently determined these material parameters by measuring the unconfined compression response of the stroma and analyzing the experimental data by a linear transversely isotropic biphasic theory (Hatami-Marbini and Etebu, 2013a, 2013b). Nevertheless, our study did not provide any information about the transverse shear modulus. This is because no shear deformation, similar to the standard inflation and strip testing methods, is introduced in the samples when tested under unconfined compression. The torsional shear experiment is a well-known testing procedure to determine the viscoelastic shear properties of soft tissues. In this experimental procedure, a rheometer is used to subject circular samples to oscillatory angular deformation (or torque) while measuring the applied torque (or angular deformation). Although this method has been widely used to investigate the shear properties of articular cartilage, meniscus, skin, and brain tissue (Bilston et al., 1997; Geerligs et al., 2011; Hayes and Bodine, 1978; Zhu et al., 1994; Zhu et al., 1993), there are only few studies on corneal shear properties. Following Nickerson (2005), Petsche et al. (2012) used the torsional rheometry to measure the transverse shear properties of four cornea pairs. The shear tests were performed at a single frequency and strain amplitude in order to show that there is a significant difference between the transverse shear moduli of anterior and posterior regions. With the exception of the above two limited studies and to the best of our knowledge, there are no previous reports in the literature on viscoelastic shear properties of the cornea using torsional rheometry. The primary objectives of the present study were to characterize the dynamic shear properties of the porcine corneal stroma and to investigate the effects of the compressive strain. The torsional oscillatory experiments, i.e. the strain sweep and frequency sweep dynamic shear tests, were conducted at different levels of axial compressive strain. These experiments were used to investigate, the effects of compressive strain, frequency, and shear strain amplitude on shear properties of the cornea.
2. Methods 2.1. Oscillatory shear deformation The oscillatory torsional experiment is an effective method to investigate the behavior of viscoelastic materials (Barnes et al., 1989). In this procedure, the
sinusoidal oscillatory shear strain given by γðtÞ ¼ γ 0 expðiωtÞ;
ð1Þ pffiffiffiffiffiffiffiffi is applied to the material. In this equation, i is the imaginary number 1, ω is the angular frequency, and γ 0 is the shear strain amplitude. Within the range of linear viscoelastic behavior, the corresponding shear stress is represented by τðtÞ ¼ τ0 expðiðωt þ δÞÞ;
ð2Þ
where τ0 is the shear stress amplitude, and δ is the phase shift angle between the applied shear strain and resulting shear stress. The complex shear modulus Gn ðωÞ is obtained from the shear stress–strain relation, i.e. τðtÞ ¼ Gn ðωÞγðtÞ. The magnitude of the complex shear modulus is a measure of shear stiffness. It is common to write the complex shear modulus as Gn ¼ G′ þ iG″, where G′ is referred to as the storage modulus and G″ as the loss modulus. The storage and loss moduli at any frequency represent the elastic properties (solid-like response) and viscous properties (fluidlike response) of the material, respectively. 2.2. Sample preparation Fresh enucleated porcine eye globes were obtained from an abattoir and transported to the laboratory on ice. After excising corneal scleral skirts from the eyes, a circular trephine was used to punch specimens of diameter 8 mm from the central region. The epithelial and endothelial layers were rubbed off with a dull scalpel blade and Kimwipe, respectively (Doughty, 2000; Kim et al., 1971). A DHR-2 rheometer (TA Instruments, Delaware) with a minimum torque oscillation of 2 nN m, torque resolution of 0.1 nN m, and displacement resolution of 10 nrad was used to perform the experiments. Sandpapers were glued to the loading platens in order to increase friction and prevent possible slippage. This method has commonly been used in previous studies for satisfying the necessary no slip boundary conditions (Bilston et al., 2001; Nickerson, 2005; Petsche et al., 2012). A series of pilot studies was conducted to assess the possible effects of coarseness of the sandpapers; no major difference was observed between 320 grit and 80 grit sandpapers. To be consistent with previous studies (Nickerson, 2005; Petsche et al., 2012), 320 grit sandpaper was used in all experimental measurements. 2.3. Experiments Prior to testing, the specimens were equilibrated in OBSS solution (ALCON laboratories, Inc., Fort Worth) for 30 min. Meanwhile, the submersion chamber of the rheometer was filled with OBSS solution. We have recently shown that corneal material properties strongly depend on the level of axial compressive strain (Hatami-Marbini and Etebu, 2013b). An initial constant axial load of 0.17 N (equivalent to swelling pressure of about 25 mmHg) was applied in all experiments. The application of this tare stress not only ensured firm clamping of specimens between the parallel platens but it also allowed conducting the experiments at relevant levels of swelling pressure (the average physiological swelling pressure of the porcine cornea is about 52 mmHg, Hatami-Marbini et al., 2013). The thickness of the specimens at this tare stress was taken as their initial thickness and the compressive strain ε was defined as the change in thickness divided by the initial thickness. Dynamic shear experiments were performed at four levels of compressive strain, ε ¼0%, 10%, 20%, and 30%. For each compressive strain step, an axial displacement rate of 1 mm/s and a relaxation time of about 30 min were used. In order to characterize the shear viscoelastic properties, two types of oscillatory tests, i.e. the strain and frequency sweep experiments, were conducted at each level of compressive strain, Fig. 1. For characterizing the range of linear viscoelasticity, the strain sweep experiments (8 samples) were done at frequency 1 Hz over shear strain amplitudes ranging from 0.01% to 10%. For frequency sweep experiments (8 samples), the frequencies of 0.01–2 Hz and a shear strain magnitude of 0.2% (which was within the region of linear viscoelasticity) were selected. The shear storage modulus G′ and shear loss modulus G″ of corneal disks at each compressive strain were calculated. Furthermore, the equilibrium axial stress at each compressive strain was obtained from dividing the equilibrium axial force by the initial cross-sectional area.
3. Results A typical stress–strain curve, from which the loss and storage moduli were calculated, is plotted in Fig. 1c. The data shown in this figure is for an experiment conducted at a frequency of 1 Hz and a shear strain amplitude of 1%. Fig. 2 shows the variation of the measured storage and loss moduli at different compressive strains and as a function of shear strain magnitude for strain sweep experiments conducted at frequency f ¼1 Hz. It is observed that the storage modulus and loss modulus both increased with increasing compressive strain. While the loss modulus at each
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Fig. 2. (a) Storage modulus (G′) and (b) loss modulus (G″) of the porcine corneal stromal specimens as a function of shear strain amplitudes at frequency 1 Hz. The results are shown for four different levels of compressive strain; it is seen that with increasing the compressive strain, both storage and loss moduli increased. The symbols denote the average of experimental measurements and the error bars indicate7 one standard deviation.
Fig. 1. (a) A schematic plot of the dynamic torsional experimental setup. (b) A schematic plot of stress–strain hysteresis loop for a viscoelastic material and the relevant material parameters. (c) An actual shear stress–strain curve obtained from an oscillatory experiment performed at frequency 1 Hz and shear strain amplitude 1%.
compressive strain increment was almost constant, there was a sudden drop in the storage modulus at shear strain larger than a critical shear strain of about 1.5%. The results from the small shear strain (γ¼ 0.2%) oscillatory experiments at various compressive axial strains (i.e. ε ¼0%, 10%, 20%, and 30%) are shown in Fig. 3; the storage modulus increased with increasing frequency at each compressive strain step. Furthermore, increasing the compressive strain increased both the loss and storage moduli.
4. Discussion The primary purpose of the present study was to investigate the dynamic shear properties of the porcine corneal stroma using oscillatory experiments. The mechanical properties of the corneal extracellular matrix have been commonly studied using the uniaxial tensile and inflation experiments (Boyce et al., 2007; Elsheikh
and Alhasso, 2009; Jue and Maurice, 1986; Kampmeier et al., 2000; Klyce et al., 1971; Olsen and Sperling, 1987). Although these experiments are quite successful in demonstrating the non-linear tensile properties, they suffer from a number of inherent drawbacks. For example, it is extremely hard to determine material properties while the tissue thickness (axial compressive strain) is fully controlled. Many of the previous studies used an isotropic material model with two material constants to interpret the experimental measurements. The stromal microstructure clearly indicates that an anisotropic material model should be used. Recently, we determined anisotropic material constants of the porcine corneal stroma from unconfined compression stressrelaxation experiments (Hatami-Marbini and Etebu, 2013a, 2013b). Similar to the strip tensile and pressure inflation experiments, this technique was unable to characterize the shear elastic modulus because of causing no shear deformation. A detailed knowledge of shear properties is required for understanding and modeling corneal biomechanics. The torsional experiments have been employed to determine the viscoelastic properties of soft tissues such as articular cartilage, brain, skin, and meniscus (Bilston et al., 2001; Geerligs et al., 2011; Zhu et al., 1994, 1993). In this experimental method, circular specimens are sheared between two parallel plates, Fig. 1. Despite popularity of oscillatory torsional
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Fig. 4. Normalized average storage modulus (G′) of the porcine corneal stromal specimens as a function of shear strain amplitudes at frequency 1 Hz and four different levels of compressive strain. The dashed lines are guides illustrating the linear range of viscoelasticity and the dotted line is the average of experimental data at various compressive strains. It is seen that a characteristic curve, independent of the compressive strain, resulted.
Fig. 3. (a) Storage modulus (G′) and (b) loss modulus (G″) of the porcine corneal stromal specimens as a function of frequency. The experiments were done using a strain amplitude of 0.2% and at four different levels of compressive strain. With increasing the compressive strain, both storage and loss moduli increased. The symbols denote the average of experimental measurements and the error bars indicate7 one standard deviation.
experiments in soft tissue biomechanics, they have only been used to examine corneal mechanics in few studies and limited data is available on viscoelastic shear properties of corneal stroma. The present study, to the best of our knowledge, is the first time that corneal dynamic shear property, the range of linear viscoelasticity, and effects of compressive strain have been investigated. Figs. 2 and 3 show the range of linear viscoelasticity and the frequency-dependent shear moduli of the porcine corneal stroma as a function of compressive strain, respectively. These plots confirm that the corneal stroma, similar to other soft tissues, shows viscoelastic behavior in shear. The strain sweep experiments showed that, at all levels of compressive strain, the measured storage modulus was fairly constant over small amplitude shear strain and decreased nonlinearly with increasing strain at a critical shear strain γcrt. Within the range of linear viscoelastic response, the viscoelastic parameters are independent of the shear strain amplitude. In order to determine the limit of linear viscoelasticity, the critical shear strain was defined as the shear strain amplitude at which a 95% decrease in the initial shear modulus (at γ ¼0.01%) was observed. In Fig. 4, the results of the strain sweep experiments at each compressive strain level were normalized and replotted. A characteristic curve independent of
Fig. 5. The average magnitude of the complex modulus of porcine corneal stroma at frequency 1 Hz over the range of linear viscoelasticity as a function of compressive strain. It is seen that shear stiffness of the samples increased nonlinearly with the compressive strain. The symbols denote the average and the error bars indicate7 one standard deviation.
the compressive strain and with a critical shear strain of γcrt 1.5% resulted. Additional strain sweep experiments were done to further probe the possible dependence of the limit of viscoelasticity on frequency (results not shown here). It was observed that the critical shear strain value did not vary significantly with frequency. The range of linear viscoelasticity for the porcine corneal stroma is consistent with that of other soft tissues. In general, soft materials, composed of loosely connected soft constituents, have a linear regime for strains below 1%. For example, the range of linear viscoelastic limit of the brain tissue lays between 0.2% and 1% (Bilston et al., 2001; Nicolle et al., 2005). Furthermore, a shear strain amplitude of 0.4% has been estimated to be within the linear viscoelastic limit of the articular cartilage (Zhu et al., 1993). In Fig. 5, the average corneal complex modulus over the range of linear viscoelasticity (0.01 oγo γcrt) was plotted as a function of compressive strain. For this purpose, the magnitude of the complex modulus, Gn ¼ ðG′2 þ G″2 Þ1=2 , at each level of compression was computed and averaged over the linear viscoelasticity limit. This figure shows that the shear stiffness of the
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corneal extracellular matrix increased with increasing compressive strain. It is noted that although the magnitude of the complex shear modulus is a measure of shear stiffness, it is higher than its equilibrium shear modulus. The corneal stroma depicted a decrease of stiffness for shear strains larger than γcrt, Fig. 2. It is first noted that a similar behavior was previously reported for the meniscus in the range of 0.5–5% (Zhu et al., 1994). The non-linear viscoelastic behavior in large amplitude oscillations can be quantified using the Fourier transform rheology which results in the introduction of more material constants (Lamers et al., 2013; Wilhelm, 2002). The transition from the linear to non-linear response can be attributed to the relative motion of collagen fibers and proteoglycans. This significant decrease in the measured storage modulus for strains larger than the critical shear strain is different from the results of the tensile strip and pressure inflation experiments. In these experiments, the corneal stroma exhibited a stiffer elasticity with increasing the applied deformation. A possible explanation for the difference in corneal behavior in shear and tensile modes of deformation might be given in terms of the microstructure. The corneal extracellular matrix is composed of stacks of collagen lamellae, each comprising bundles of thin collagen fibrils and proteoglycans. The gap between the collagen fibrils is filled with a network of the proteoglycans which are possibly responsible to maintain the uniform spacing of the fibrils (Lewis et al., 2010; Maurice, 1957, 1984). When subjected to uniaxial tensile strain in the strip testing method, the collagen lamellae are mainly loaded in tension. It is known that collagen fibrils have non-linear stress–strain behavior; therefore, a strain-stiffening response is expected with increasing deformation. Nevertheless, the shear stiffness is mainly provided by the proteoglycan matrix. When subjected to transverse shear deformation larger than critical shear strain, the matrix material possibly damages due to displacement of the adjacent lamellae and a lower shear stiffness is obtained. Despite the above discussion, future studies are required to fully understand the physical origins of this behavior. Once the range of linear viscoelastic response was determined, frequency sweep experiments were performed on corneal samples with a strain amplitude of 0.2% (which is well within the range of linear viscoelasticity). It was observed that corneal storage modulus increased with increasing both the frequency and the compressive strain. This behavior is similar to the response of the articular cartilage, meniscus, and intervertebral disk (Iatridis et al., 1999; Zhu et al., 1994, 1993). Depending on the compressive strain and frequency, the values of storage modulus for the articular cartilage, meniscus, and lumbar annulus fibrous were found to be approximately within the ranges of 180–2500 kPa, 22–198 kPa, and 50–225 kPa respectively. The present study showed that corneal storage modulus is much lower than that of the articular cartilage but closer to the shear stiffness of the meniscus and annulus fibrous. Figs. 2, 3 and 5 confirm that the corneal shear parameters were dependent on the compressive strain (tissue thickness). We observed a similar behavior for the corneal in-plane and out-of-plane Young's moduli (Hatami-Marbini and Etebu, 2013a). The dependence of the measured parameters on compressive strain may also be explained in terms of the stromal microstructure. As it was discussed earlier, the corneal extracellular matrix is primarily composed of collagen fibrils at regular distances. It has been suggested that side chains of proteoglycans (i.e. negatively charged glycosaminoglycan molecules) could form interfibrillar duplexes and tie collagen fibrils together (Lewis et al., 2010; Scott, 1991). The increase in the shear modulus with increasing compressive strain might be due to enhanced shortrange interactions between collagen fibrils and proteoglycans at high compressive strains (Cheng et al., 2013; Hatami-Marbini, 2013). In other words, the increase in density of the negative
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charges due to the tissue compression leads to higher shear stiffness. The complex shear modulus from the present study was computed at the frequency (f ¼0.03 Hz) and shear strain magnitude (γ ¼1%) in order to compare the static shear properties of porcine and human corneas (Petsche et al., 2012). The complex modulus of the porcine specimens varied from 2.5 kPa to 9 kPa while the equilibrium axial compressive stress varied from 3.4 kPa (25 mmHg) to 15 kPa (110 mmHg). Within the same range of compressive stress, the human corneal complex shear modulus varied from 3 kPa to 15 kPa (Petsche et al., 2012). Therefore, the porcine corneal stroma had only slightly lower shear modulus. Nonetheless, it should be noted that the protocols of these studies are different and future studies are required to fully characterize the differences between shear properties of the porcine and human corneas. In the present study, the specimens were allowed to relax for 30 min at each level of compressive strain but the Petsche et al. (2012) selected a short 3 min relaxation time. Previous studies showed that the axial force in stress-relaxation experiments on human corneal samples stabilizes only after a relaxation time of about 0.5–1 h (Olsen and Sperling, 1987). We observed a similar trend for the porcine axial stress-relaxation behavior (Hatami-Marbini and Etebu, 2013a). Finally, Fig. 5 shows while the average transverse shear modulus of the porcine corneal stroma is of the same order of magnitude as the out-of-plane Young's modulus, it is approximately three orders of magnitude lower than its in-plane Young's modulus (Hatami-Marbini and Etebu, 2013b). The present study provided important data on the effects of compressive strain, frequency, and shear strain amplitude on shear properties of the cornea. Nevertheless, future studies are required to fully characterize the mechanical behavior of the cornea in shear. The rheological nonlinearities in large strain response of the cornea require non-linear viscoelastic models which often include many material constants and are computationally expensive (Bilston et al., 2001; Lamers et al., 2013). The variations in the reported shear properties could be due to the non-uniformity of tissue specimens. For example, we considered neither the age nor the weight of the animals in the present investigation. Furthermore, the preparation procedure (e.g., excising and punching corneal buttons) might have damaged the internal microstructure of the specimens. It is well-known that the corneal stroma has an inhomogeneous microstructure through the thickness. In other words, the lamellae approximately lie on top of each other in the posterior thirds while they become slightly oblique and branched in the anterior thirds (Maurice, 1984). The present study did not consider these inhomogeneities and its findings should only be considered as the macroscale (effective) shear properties of the tissue. Similarly possible molecular scale effects of keratocytes were not considered. During the experiments, the natural curvature of the specimens was flattened. This potentially produced a non-uniform initial stress state and affected the experimental measurements. Although it is expected that amount of this prestress to be negligible (Hedbys and Dohlman, 1963; Olsen and Sperling, 1987; Petsche et al., 2012), this issue could be investigated by using corneal buttons of different sizes in future studies. Finally, the present study was not able to quantify the shear parameters of the samples at higher frequencies because of their very soft behavior. Although the frequency of shear deformation (e.g. due to eye rubbing) is expected to be relatively low, future studies using custom design shear testing apparatus (or advanced commercial machines) are required to probe corneal mechanics at high frequencies (Arbogast et al., 1997). It is also noted that previous studies have shown that the time–temperature superposition can be used to extend the range of shear rates for soft tissues such as brain tissue (Peters et al., 1997). The applicability of this principle to the linear viscoelastic response of corneal tissue is
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yet to be determined. Despite the above limitations, the results of the present study were in agreement with previous reports and characterized the linear viscoelastic properties of the porcine cornea for the first time. It is expected that a thorough knowledge of shear properties would facilitate developing more complex material models which are required for understanding and predicting the biomechanics of the cornea. Conflict of interest None. Acknowledgments This project has been funded in whole or in part with the startup fund from Oklahoma State University. The author would like to thank the members of computational biomechanics laboratory for helping him conduct the experiments. References Anderson, K., El-Sheikh, A., Newson, T., 2004. Application of structural analysis to the mechanical behaviour of the cornea. J. R. Soc. Lond. Interface 1, 3–15. Arbogast, K.B., Thibault, K.L., Pinheiro, B.S., Winey, K.I., Margulies, S.S., 1997. A highfrequency shear device for testing soft biological tissues. J. Biomech. 30, 757–759. Barnes, H.A., Hutton, J.F., Walters, K., 1989. An Introduction to Rheology. Elsevier, Amsterdam. Bilston, L.E., Liu, Z., Phan-Thien, N., 1997. Linear viscoelastic properties of bovine brain tissue in shear. Biorheology 34, 377–385. Bilston, L.E., Liu, Z., Phan-Thien, N., 2001. Large strain behaviour of brain tissue in shear: some experimental data and differential constitutive model. Biorheology 38, 335–345. Boyce, B.L., Jones, R.E., Nguyen, T.D., Grazier, J.M., 2007. Stress-controlled viscoelastic tensile response of bovine cornea. J. Biomech. 40, 2367–2376. Cheng, X., Hatami-Marbini, H., Pinsky, P.M., 2013. Modeling collagen–proteoglycan structural interactions in the human cornea. In: Holzapfel, G.A., Kuhl, E. (Eds.), Computer Models in Biomechanics. Springer, pp. 11–24. Doughty, M.J., 2000. Swelling of the collagen-keratocytematrix of the bovine corneal stroma ex vivo in various solutions and its relationship to tissue thickness. Tissue Cell 32, 478–493. Elsheikh, A., Alhasso, D., 2009. Mechanical anisotropy of porcine cornea and correlation with stromal microstructure. Exp. Eye Res. 88, 1084–1091. Elsheikh, A., Wang, D., Pye, D., 2007. Determination of the modulus of elasticity of the human cornea. J. Refract. Surg. 23, 808–818. Fernandez, D.C., Niazy, A.M., Kurtz, R.M., Djotyan, G.P., Juhasz, T., 2006. A finite element model for ultrafast laser-lamellar keratoplasty. Ann. Biomed. Eng. 34, 169–183. Geerligs, M., Oomens, C., Ackermans, P., Baaijens, F., Peters, G., 2011. Linear shear response of the upper skin layers. Biorheology 48, 229–245. Hatami-Marbini, H., 2013. Mechano-electrochemical mixture theories for the multiphase fluid infiltrated poroelastic media. In: Li, S., Gao, X.-L. (Eds.), Handbook on Micromechanics and Nanomechanics. Pan Stanford Publishing Pte. Ltd, pp. 273–302.
Hatami-Marbini, H., Etebu, E., 2013a. An experimental and theoretical analysis of unconfined compression of corneal stroma. J. Biomech. 46, 1752–1758. Hatami-Marbini, H., Etebu, E., 2013b. A new method to determine rate-dependent material parameters of corneal extracellular matrix. Ann. Biomed. Eng. 41, 2399–2408. Hatami-Marbini, H., Etebu, E., Rahimi, A., 2013. Swelling pressure and hydration behavior of porcine corneal stroma. Curr. Eye Res. 38, 1124–1132. Hayes, W.C., Bodine, A.J., 1978. Flow-independent viscoelastic properties of articular cartilage matrix. J. Biomech. 11, 407–419. Hedbys, B.O., Dohlman, C.H., 1963. A new method for the determination of the swelling pressure of the corneal stroma in vitro. Exp. Eye Res. 2, 122–129. Iatridis, J.C., Kumar, S., Foster, R.J., Weidenbaum, M., Mow, V.C., 1999. Shear mechanical properties of human lumbar annulus fibrosus. J. Orthop. Res. 17, 732–737. Jue, B., Maurice, D.M., 1986. The mechanical properties of the rabbit and human cornea. J. Biomechan. 19, 847–853. Kampmeier, J., Radt, B., Birngruber, R., Brinkmann, R., 2000. Thermal and biomechanical parameters of porcine cornea. Cornea 19, 355–363. Kim, J.H., Green, K., Martinez, M., Paton, D., 1971. Solute permeability of the corneal endothelium and Descemet's membrane. Exp. Eye Res. 12, 231–238. Klyce, S.D., Dohlman, C.H., Tolpin, D.W., 1971. In vivo determination of corneal swelling pressure. Exp. Eye Res. 11, 220–229. Lamers, E., vanKempen, T.H.S., Baaijens, F.P.T., Peters, G.W.M., Oomens, C.W.J., 2013. Large amplitude oscillatory shear properties of human skin. J. Mech. Behav. Biomed. Mater. 28, 462–470. Lewis, P.N., Pinali, C., Young, R.D., Meek, K.M., Quantock, A.J., Knupp, C., 2010. Structural interactions between collagen and proteoglycans are elucidated by three-dimensional electron tomography of bovine cornea. Structure 18, 239–245. Maurice, D.M., 1957. The structure and transparency of the cornea. J. Physiol. (London) 136, 263–286. Maurice, D.M., 1984. The cornea and sclera. In: Davason, H. (Ed.), The Eye. Academic Press, London, pp. 1–184. Meek, K.M., 2008. The cornea and sclera. In: Fratzl, P. (Ed.), Collagen: Structure and Mechanics. Springer Science, pp. 359–396. Nickerson, C.S., 2005. Engineering the Mechanical Properties of Ocular Tissue (Ph.D. Thesis), California Institute of Technology, California. Nicolle, S., Lounis, M., Willinger, R., Palierne, J.F., 2005. Shear linear behavior of brain tissue over a large frequency range. Biorheology 42, 209–223. Olsen, T., Sperling, S., 1987. The swelling pressure of the human corneal stroma as determined by a new method. Exp. Eye Res. 44, 481–490. Pandolfi, A., 2010. Computational biomechanics of the human cornea. In: De, S., GuilaK, F., Mofrad, M. (Eds.), Computational Modeling in Biomechanics. Springer, pp. 435–466. Pandolfi, A., Holzapfel, G.A., 2008. Three-dimensional modeling and computational analysis of the human cornea considering distributed collagen fibril orientations. J. Biomech. Eng. 130. (061006-061001-061006-061012). Peters, G.W., Meulman, J.H., Sauren, A.A., 1997. The applicability of the time/ temperature superposition principle to brain tissue. Biorheology 34, 127–138. Petsche, S.J., Chernyak, D., Martiz, J., Levenston, M.E., Pinsky, P.M., 2012. Depthdependent transverse shear properties of the human corneal stroma. Investig. Ophthalmol. Vis. Sci. 53, 873–880. Scott, J.E., 1991. Proteoglycan: collagen interactions and corneal ultrastructure. Biochem. Soc. Trans. 19, 877–881. Wilhelm, M., 2002. Fourier-transform rheology. Macromol. Mater. Eng. 287, 83–105. Zhu, W., Chern, K.Y., Mow, V.C., 1994. Anisotropic viscoelastic shear properties of bovine meniscus. Clin. Orthop. Related Res., 34–45. Zhu, W., Mow, V.C., Koob, T.J., Eyre, D.R., 1993. Viscoelastic shear properties of articular cartilage and the effects of glycosidase treatments. J. Orthop. Res. 11, 771–781.
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Viscoelastic shear properties of the corneal stroma Hamed Hatami-Marbini n School of Mechanical and Aerospace Engineering, Oklahoma State University, 218 Engineering North, Stillwater, OK 74078 5016, United States
art ic l e i nf o
a b s t r a c t
Article history: Accepted 18 November 2013
The cornea is a highly specialized transparent tissue which covers the front of the eye. It is a tough tissue responsible for refracting the light and protecting the sensitive internal contents of the eye. The biomechanical properties of the cornea are primarily derived from its extracellular matrix, the stroma. The majority of previous studies have used strip tensile and pressure inflation testing methods to determine material parameters of the corneal stroma. Since these techniques do not allow measurements of the shear properties, there is little information available on transverse shear modulus of the cornea. The primary objectives of the present study were to determine the viscoelastic behavior of the corneal stroma in shear and to investigate the effects of the compressive strain. A thorough knowledge of the shear properties is required for developing better material models for corneal biomechanics. In the present study, torsional shear experiments were conducted at different levels of compressive strain (0– 30%) on porcine corneal buttons. First, the range of linear viscoelasticity was determined from strain sweep experiments. Then, frequency sweep experiments with a shear strain amplitude of 0.2% (which was within the region of linear viscoelasticity) were performed. The corneal stroma exhibited viscoelastic properties in shear. The shear storage modulus, G′, and shear loss modulus, G″, were reported as a function of tissue compression. It was found that although both of these parameters were dependent on frequency, shear strain amplitude, and compressive strain, the average shear storage and loss moduli varied from 2 to 8 kPa, and 0.3 to 1.2 kPa, respectively. Therefore, it can be concluded that the transverse shear modulus is of the same order of magnitude as the out-of-plane Young's modulus and is about three orders of magnitude lower than the in-plane Young's modulus. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Corneal stromal mechanics Oscillatory torsional experiments Linear viscoelastic behavior Porcine cornea
1. Introduction The cornea is a highly specialized tissue which refracts and transmits light. It forms about one-sixth of the outer layer of the eye and is considered as its main refractive element. The normal cornea transmits over 90% of the light and causes 75% of the light to scatter at angles larger than 301. In addition to unique optical properties, the cornea has important structural roles in protecting internal contents of the eye and maintaining its shape. The tissue is constantly subjected to mechanical stresses caused by internal forces (e.g. intraocular pressure) and external forces (e.g. getting poked and eye rubbing). Over the past few decades, various computational models have been proposed to understand and predict the biomechanical behavior of the cornea (Anderson et al., 2004; Elsheikh et al., 2007; Fernandez et al., 2006; Pandolfi, 2010; Pandolfi and Holzapfel, 2008). Although these mathematical models have been successful in capturing certain aspects of the corneal mechanics, more studies are required to precisely predict the tissue behavior.
n
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0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.11.019
It is well-known that the accuracy of the predictions of computational models is highly dependent on the accuracy of constitutive (stress–strain) relation that has been presumed. The material parameters required for constitutive models are obtained from material characterization experiments. Under physiological conditions, the cornea is mainly subjected to intraocular pressure which puts the tissue under membrane tension. Therefore, extensometry and inflation tests have commonly been used to investigate the tensile properties of the cornea (Boyce et al., 2007; Elsheikh and Alhasso, 2009; Jue and Maurice, 1986; Kampmeier et al., 2000; Klyce et al., 1971; Olsen and Sperling, 1987). The cornea in these studies has often been considered as an isotropic material with two distinct material constants, i.e. the Young's modulus and Poisson's ratio. Although there are various studies in the literature promoting this simple material model, there still exists a large range of variation for the reported Young's modulus (Elsheikh et al., 2007; Jue and Maurice, 1986). This significant inconsistency could be due to the use of different testing conditions, experimental procedures, and samples. The biomechanical properties of the cornea are primarily due to the composition and properties of the stromal layer, a connective tissue forming the bulk of the corneal thickness. The stroma is a unique extracellular matrix in that it has a highly
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organized microstructure and is subsequently transparent. Despite the widespread use of the oversimplified isotropic material model, the microstructure of the corneal extracellular matrix is inhomogeneous and anisotropic (Maurice, 1957, 1984; Meek, 2008). The stroma is composed of many stacks of lamellae; each comprising a regular network of collagen fibrils and proteoglycans. The collagen fibrils of almost uniform diameter are distributed in a pseudohexagonal arrangement and proteoglycans fill the spaces between the fibrils (Lewis et al., 2010; Maurice, 1957, 1984). This particular ultrastructure of the stroma implies that in-plane and out-of-plane material properties should be significantly different from each other (Hatami-Marbini and Etebu, 2013a). Therefore, the biomechanical response of the cornea could be represented more accurately with anisotropic material models. Because of the lamellar structure of the stromal layer, an appealing choice is a transversely isotropic constitutive law in which the axis of material symmetry is perpendicular to the surface. The transversely isotropic models are generally more complex than isotropic material models. In particular, the linear transversely isotropic model is defined in terms of five material constants: the out-ofplane Young's modulus and Poisson's ratio, the in-plane Young's modulus and Poisson's ratio, and the transverse shear modulus. We have recently determined these material parameters by measuring the unconfined compression response of the stroma and analyzing the experimental data by a linear transversely isotropic biphasic theory (Hatami-Marbini and Etebu, 2013a, 2013b). Nevertheless, our study did not provide any information about the transverse shear modulus. This is because no shear deformation, similar to the standard inflation and strip testing methods, is introduced in the samples when tested under unconfined compression. The torsional shear experiment is a well-known testing procedure to determine the viscoelastic shear properties of soft tissues. In this experimental procedure, a rheometer is used to subject circular samples to oscillatory angular deformation (or torque) while measuring the applied torque (or angular deformation). Although this method has been widely used to investigate the shear properties of articular cartilage, meniscus, skin, and brain tissue (Bilston et al., 1997; Geerligs et al., 2011; Hayes and Bodine, 1978; Zhu et al., 1994; Zhu et al., 1993), there are only few studies on corneal shear properties. Following Nickerson (2005), Petsche et al. (2012) used the torsional rheometry to measure the transverse shear properties of four cornea pairs. The shear tests were performed at a single frequency and strain amplitude in order to show that there is a significant difference between the transverse shear moduli of anterior and posterior regions. With the exception of the above two limited studies and to the best of our knowledge, there are no previous reports in the literature on viscoelastic shear properties of the cornea using torsional rheometry. The primary objectives of the present study were to characterize the dynamic shear properties of the porcine corneal stroma and to investigate the effects of the compressive strain. The torsional oscillatory experiments, i.e. the strain sweep and frequency sweep dynamic shear tests, were conducted at different levels of axial compressive strain. These experiments were used to investigate, the effects of compressive strain, frequency, and shear strain amplitude on shear properties of the cornea.
2. Methods 2.1. Oscillatory shear deformation The oscillatory torsional experiment is an effective method to investigate the behavior of viscoelastic materials (Barnes et al., 1989). In this procedure, the
sinusoidal oscillatory shear strain given by γðtÞ ¼ γ 0 expðiωtÞ;
ð1Þ pffiffiffiffiffiffiffiffi is applied to the material. In this equation, i is the imaginary number 1, ω is the angular frequency, and γ 0 is the shear strain amplitude. Within the range of linear viscoelastic behavior, the corresponding shear stress is represented by τðtÞ ¼ τ0 expðiðωt þ δÞÞ;
ð2Þ
where τ0 is the shear stress amplitude, and δ is the phase shift angle between the applied shear strain and resulting shear stress. The complex shear modulus Gn ðωÞ is obtained from the shear stress–strain relation, i.e. τðtÞ ¼ Gn ðωÞγðtÞ. The magnitude of the complex shear modulus is a measure of shear stiffness. It is common to write the complex shear modulus as Gn ¼ G′ þ iG″, where G′ is referred to as the storage modulus and G″ as the loss modulus. The storage and loss moduli at any frequency represent the elastic properties (solid-like response) and viscous properties (fluidlike response) of the material, respectively. 2.2. Sample preparation Fresh enucleated porcine eye globes were obtained from an abattoir and transported to the laboratory on ice. After excising corneal scleral skirts from the eyes, a circular trephine was used to punch specimens of diameter 8 mm from the central region. The epithelial and endothelial layers were rubbed off with a dull scalpel blade and Kimwipe, respectively (Doughty, 2000; Kim et al., 1971). A DHR-2 rheometer (TA Instruments, Delaware) with a minimum torque oscillation of 2 nN m, torque resolution of 0.1 nN m, and displacement resolution of 10 nrad was used to perform the experiments. Sandpapers were glued to the loading platens in order to increase friction and prevent possible slippage. This method has commonly been used in previous studies for satisfying the necessary no slip boundary conditions (Bilston et al., 2001; Nickerson, 2005; Petsche et al., 2012). A series of pilot studies was conducted to assess the possible effects of coarseness of the sandpapers; no major difference was observed between 320 grit and 80 grit sandpapers. To be consistent with previous studies (Nickerson, 2005; Petsche et al., 2012), 320 grit sandpaper was used in all experimental measurements. 2.3. Experiments Prior to testing, the specimens were equilibrated in OBSS solution (ALCON laboratories, Inc., Fort Worth) for 30 min. Meanwhile, the submersion chamber of the rheometer was filled with OBSS solution. We have recently shown that corneal material properties strongly depend on the level of axial compressive strain (Hatami-Marbini and Etebu, 2013b). An initial constant axial load of 0.17 N (equivalent to swelling pressure of about 25 mmHg) was applied in all experiments. The application of this tare stress not only ensured firm clamping of specimens between the parallel platens but it also allowed conducting the experiments at relevant levels of swelling pressure (the average physiological swelling pressure of the porcine cornea is about 52 mmHg, Hatami-Marbini et al., 2013). The thickness of the specimens at this tare stress was taken as their initial thickness and the compressive strain ε was defined as the change in thickness divided by the initial thickness. Dynamic shear experiments were performed at four levels of compressive strain, ε ¼0%, 10%, 20%, and 30%. For each compressive strain step, an axial displacement rate of 1 mm/s and a relaxation time of about 30 min were used. In order to characterize the shear viscoelastic properties, two types of oscillatory tests, i.e. the strain and frequency sweep experiments, were conducted at each level of compressive strain, Fig. 1. For characterizing the range of linear viscoelasticity, the strain sweep experiments (8 samples) were done at frequency 1 Hz over shear strain amplitudes ranging from 0.01% to 10%. For frequency sweep experiments (8 samples), the frequencies of 0.01–2 Hz and a shear strain magnitude of 0.2% (which was within the region of linear viscoelasticity) were selected. The shear storage modulus G′ and shear loss modulus G″ of corneal disks at each compressive strain were calculated. Furthermore, the equilibrium axial stress at each compressive strain was obtained from dividing the equilibrium axial force by the initial cross-sectional area.
3. Results A typical stress–strain curve, from which the loss and storage moduli were calculated, is plotted in Fig. 1c. The data shown in this figure is for an experiment conducted at a frequency of 1 Hz and a shear strain amplitude of 1%. Fig. 2 shows the variation of the measured storage and loss moduli at different compressive strains and as a function of shear strain magnitude for strain sweep experiments conducted at frequency f ¼1 Hz. It is observed that the storage modulus and loss modulus both increased with increasing compressive strain. While the loss modulus at each
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Fig. 2. (a) Storage modulus (G′) and (b) loss modulus (G″) of the porcine corneal stromal specimens as a function of shear strain amplitudes at frequency 1 Hz. The results are shown for four different levels of compressive strain; it is seen that with increasing the compressive strain, both storage and loss moduli increased. The symbols denote the average of experimental measurements and the error bars indicate7 one standard deviation.
Fig. 1. (a) A schematic plot of the dynamic torsional experimental setup. (b) A schematic plot of stress–strain hysteresis loop for a viscoelastic material and the relevant material parameters. (c) An actual shear stress–strain curve obtained from an oscillatory experiment performed at frequency 1 Hz and shear strain amplitude 1%.
compressive strain increment was almost constant, there was a sudden drop in the storage modulus at shear strain larger than a critical shear strain of about 1.5%. The results from the small shear strain (γ¼ 0.2%) oscillatory experiments at various compressive axial strains (i.e. ε ¼0%, 10%, 20%, and 30%) are shown in Fig. 3; the storage modulus increased with increasing frequency at each compressive strain step. Furthermore, increasing the compressive strain increased both the loss and storage moduli.
4. Discussion The primary purpose of the present study was to investigate the dynamic shear properties of the porcine corneal stroma using oscillatory experiments. The mechanical properties of the corneal extracellular matrix have been commonly studied using the uniaxial tensile and inflation experiments (Boyce et al., 2007; Elsheikh
and Alhasso, 2009; Jue and Maurice, 1986; Kampmeier et al., 2000; Klyce et al., 1971; Olsen and Sperling, 1987). Although these experiments are quite successful in demonstrating the non-linear tensile properties, they suffer from a number of inherent drawbacks. For example, it is extremely hard to determine material properties while the tissue thickness (axial compressive strain) is fully controlled. Many of the previous studies used an isotropic material model with two material constants to interpret the experimental measurements. The stromal microstructure clearly indicates that an anisotropic material model should be used. Recently, we determined anisotropic material constants of the porcine corneal stroma from unconfined compression stressrelaxation experiments (Hatami-Marbini and Etebu, 2013a, 2013b). Similar to the strip tensile and pressure inflation experiments, this technique was unable to characterize the shear elastic modulus because of causing no shear deformation. A detailed knowledge of shear properties is required for understanding and modeling corneal biomechanics. The torsional experiments have been employed to determine the viscoelastic properties of soft tissues such as articular cartilage, brain, skin, and meniscus (Bilston et al., 2001; Geerligs et al., 2011; Zhu et al., 1994, 1993). In this experimental method, circular specimens are sheared between two parallel plates, Fig. 1. Despite popularity of oscillatory torsional
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Fig. 4. Normalized average storage modulus (G′) of the porcine corneal stromal specimens as a function of shear strain amplitudes at frequency 1 Hz and four different levels of compressive strain. The dashed lines are guides illustrating the linear range of viscoelasticity and the dotted line is the average of experimental data at various compressive strains. It is seen that a characteristic curve, independent of the compressive strain, resulted.
Fig. 3. (a) Storage modulus (G′) and (b) loss modulus (G″) of the porcine corneal stromal specimens as a function of frequency. The experiments were done using a strain amplitude of 0.2% and at four different levels of compressive strain. With increasing the compressive strain, both storage and loss moduli increased. The symbols denote the average of experimental measurements and the error bars indicate7 one standard deviation.
experiments in soft tissue biomechanics, they have only been used to examine corneal mechanics in few studies and limited data is available on viscoelastic shear properties of corneal stroma. The present study, to the best of our knowledge, is the first time that corneal dynamic shear property, the range of linear viscoelasticity, and effects of compressive strain have been investigated. Figs. 2 and 3 show the range of linear viscoelasticity and the frequency-dependent shear moduli of the porcine corneal stroma as a function of compressive strain, respectively. These plots confirm that the corneal stroma, similar to other soft tissues, shows viscoelastic behavior in shear. The strain sweep experiments showed that, at all levels of compressive strain, the measured storage modulus was fairly constant over small amplitude shear strain and decreased nonlinearly with increasing strain at a critical shear strain γcrt. Within the range of linear viscoelastic response, the viscoelastic parameters are independent of the shear strain amplitude. In order to determine the limit of linear viscoelasticity, the critical shear strain was defined as the shear strain amplitude at which a 95% decrease in the initial shear modulus (at γ ¼0.01%) was observed. In Fig. 4, the results of the strain sweep experiments at each compressive strain level were normalized and replotted. A characteristic curve independent of
Fig. 5. The average magnitude of the complex modulus of porcine corneal stroma at frequency 1 Hz over the range of linear viscoelasticity as a function of compressive strain. It is seen that shear stiffness of the samples increased nonlinearly with the compressive strain. The symbols denote the average and the error bars indicate7 one standard deviation.
the compressive strain and with a critical shear strain of γcrt 1.5% resulted. Additional strain sweep experiments were done to further probe the possible dependence of the limit of viscoelasticity on frequency (results not shown here). It was observed that the critical shear strain value did not vary significantly with frequency. The range of linear viscoelasticity for the porcine corneal stroma is consistent with that of other soft tissues. In general, soft materials, composed of loosely connected soft constituents, have a linear regime for strains below 1%. For example, the range of linear viscoelastic limit of the brain tissue lays between 0.2% and 1% (Bilston et al., 2001; Nicolle et al., 2005). Furthermore, a shear strain amplitude of 0.4% has been estimated to be within the linear viscoelastic limit of the articular cartilage (Zhu et al., 1993). In Fig. 5, the average corneal complex modulus over the range of linear viscoelasticity (0.01 oγo γcrt) was plotted as a function of compressive strain. For this purpose, the magnitude of the complex modulus, Gn ¼ ðG′2 þ G″2 Þ1=2 , at each level of compression was computed and averaged over the linear viscoelasticity limit. This figure shows that the shear stiffness of the
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corneal extracellular matrix increased with increasing compressive strain. It is noted that although the magnitude of the complex shear modulus is a measure of shear stiffness, it is higher than its equilibrium shear modulus. The corneal stroma depicted a decrease of stiffness for shear strains larger than γcrt, Fig. 2. It is first noted that a similar behavior was previously reported for the meniscus in the range of 0.5–5% (Zhu et al., 1994). The non-linear viscoelastic behavior in large amplitude oscillations can be quantified using the Fourier transform rheology which results in the introduction of more material constants (Lamers et al., 2013; Wilhelm, 2002). The transition from the linear to non-linear response can be attributed to the relative motion of collagen fibers and proteoglycans. This significant decrease in the measured storage modulus for strains larger than the critical shear strain is different from the results of the tensile strip and pressure inflation experiments. In these experiments, the corneal stroma exhibited a stiffer elasticity with increasing the applied deformation. A possible explanation for the difference in corneal behavior in shear and tensile modes of deformation might be given in terms of the microstructure. The corneal extracellular matrix is composed of stacks of collagen lamellae, each comprising bundles of thin collagen fibrils and proteoglycans. The gap between the collagen fibrils is filled with a network of the proteoglycans which are possibly responsible to maintain the uniform spacing of the fibrils (Lewis et al., 2010; Maurice, 1957, 1984). When subjected to uniaxial tensile strain in the strip testing method, the collagen lamellae are mainly loaded in tension. It is known that collagen fibrils have non-linear stress–strain behavior; therefore, a strain-stiffening response is expected with increasing deformation. Nevertheless, the shear stiffness is mainly provided by the proteoglycan matrix. When subjected to transverse shear deformation larger than critical shear strain, the matrix material possibly damages due to displacement of the adjacent lamellae and a lower shear stiffness is obtained. Despite the above discussion, future studies are required to fully understand the physical origins of this behavior. Once the range of linear viscoelastic response was determined, frequency sweep experiments were performed on corneal samples with a strain amplitude of 0.2% (which is well within the range of linear viscoelasticity). It was observed that corneal storage modulus increased with increasing both the frequency and the compressive strain. This behavior is similar to the response of the articular cartilage, meniscus, and intervertebral disk (Iatridis et al., 1999; Zhu et al., 1994, 1993). Depending on the compressive strain and frequency, the values of storage modulus for the articular cartilage, meniscus, and lumbar annulus fibrous were found to be approximately within the ranges of 180–2500 kPa, 22–198 kPa, and 50–225 kPa respectively. The present study showed that corneal storage modulus is much lower than that of the articular cartilage but closer to the shear stiffness of the meniscus and annulus fibrous. Figs. 2, 3 and 5 confirm that the corneal shear parameters were dependent on the compressive strain (tissue thickness). We observed a similar behavior for the corneal in-plane and out-of-plane Young's moduli (Hatami-Marbini and Etebu, 2013a). The dependence of the measured parameters on compressive strain may also be explained in terms of the stromal microstructure. As it was discussed earlier, the corneal extracellular matrix is primarily composed of collagen fibrils at regular distances. It has been suggested that side chains of proteoglycans (i.e. negatively charged glycosaminoglycan molecules) could form interfibrillar duplexes and tie collagen fibrils together (Lewis et al., 2010; Scott, 1991). The increase in the shear modulus with increasing compressive strain might be due to enhanced shortrange interactions between collagen fibrils and proteoglycans at high compressive strains (Cheng et al., 2013; Hatami-Marbini, 2013). In other words, the increase in density of the negative
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charges due to the tissue compression leads to higher shear stiffness. The complex shear modulus from the present study was computed at the frequency (f ¼0.03 Hz) and shear strain magnitude (γ ¼1%) in order to compare the static shear properties of porcine and human corneas (Petsche et al., 2012). The complex modulus of the porcine specimens varied from 2.5 kPa to 9 kPa while the equilibrium axial compressive stress varied from 3.4 kPa (25 mmHg) to 15 kPa (110 mmHg). Within the same range of compressive stress, the human corneal complex shear modulus varied from 3 kPa to 15 kPa (Petsche et al., 2012). Therefore, the porcine corneal stroma had only slightly lower shear modulus. Nonetheless, it should be noted that the protocols of these studies are different and future studies are required to fully characterize the differences between shear properties of the porcine and human corneas. In the present study, the specimens were allowed to relax for 30 min at each level of compressive strain but the Petsche et al. (2012) selected a short 3 min relaxation time. Previous studies showed that the axial force in stress-relaxation experiments on human corneal samples stabilizes only after a relaxation time of about 0.5–1 h (Olsen and Sperling, 1987). We observed a similar trend for the porcine axial stress-relaxation behavior (Hatami-Marbini and Etebu, 2013a). Finally, Fig. 5 shows while the average transverse shear modulus of the porcine corneal stroma is of the same order of magnitude as the out-of-plane Young's modulus, it is approximately three orders of magnitude lower than its in-plane Young's modulus (Hatami-Marbini and Etebu, 2013b). The present study provided important data on the effects of compressive strain, frequency, and shear strain amplitude on shear properties of the cornea. Nevertheless, future studies are required to fully characterize the mechanical behavior of the cornea in shear. The rheological nonlinearities in large strain response of the cornea require non-linear viscoelastic models which often include many material constants and are computationally expensive (Bilston et al., 2001; Lamers et al., 2013). The variations in the reported shear properties could be due to the non-uniformity of tissue specimens. For example, we considered neither the age nor the weight of the animals in the present investigation. Furthermore, the preparation procedure (e.g., excising and punching corneal buttons) might have damaged the internal microstructure of the specimens. It is well-known that the corneal stroma has an inhomogeneous microstructure through the thickness. In other words, the lamellae approximately lie on top of each other in the posterior thirds while they become slightly oblique and branched in the anterior thirds (Maurice, 1984). The present study did not consider these inhomogeneities and its findings should only be considered as the macroscale (effective) shear properties of the tissue. Similarly possible molecular scale effects of keratocytes were not considered. During the experiments, the natural curvature of the specimens was flattened. This potentially produced a non-uniform initial stress state and affected the experimental measurements. Although it is expected that amount of this prestress to be negligible (Hedbys and Dohlman, 1963; Olsen and Sperling, 1987; Petsche et al., 2012), this issue could be investigated by using corneal buttons of different sizes in future studies. Finally, the present study was not able to quantify the shear parameters of the samples at higher frequencies because of their very soft behavior. Although the frequency of shear deformation (e.g. due to eye rubbing) is expected to be relatively low, future studies using custom design shear testing apparatus (or advanced commercial machines) are required to probe corneal mechanics at high frequencies (Arbogast et al., 1997). It is also noted that previous studies have shown that the time–temperature superposition can be used to extend the range of shear rates for soft tissues such as brain tissue (Peters et al., 1997). The applicability of this principle to the linear viscoelastic response of corneal tissue is
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