Corneal biomechanical properties have received significant attention in recent years due to their hypothesized role in predicting subclinical keratoconus and ectasia after refractive surgery.1–4 This prediction is possible only if in vivo quantification of properties exists. However, the human cornea exhibits both viscous and elastic responses to induced stress,5–7 and in vivo quantifications of both features in patients are challenging. The viscoelastic material assumption may be necessary to understand corneal deformation due to air-puff applanation.8 Porcine corneas are naturally thicker and differ from human corneas in terms of in situ fiber distribution.8 Thus, differences between the deformation of porcine and human corneas in response to the same air-puff applanation could be expected. Hence, we investigated whether a viscoelastic response can be determined from in vivo deformation of patient corneas in this study.
The Ocular Response Analyzer (Reichert Technologies, Depew, NY) and Corvis ST (Oculus Optikgeräte GmbH, Wetzlar, Germany) are the only current clinical devices for in vivo assessment of corneal biomechanical properties with air-puff applanation. Only the Corvis ST segregates the corneal deformation and whole globe movement during the applanation.7 The Corvis ST can quantify the entire applanation process due to the capture of 140 image frames at a sampling rate of 4,330 frames per second with an ultra–high-speed Scheimpflug camera.2 Currently, two classes of mathematical approaches are available for the calculation of biomechanical properties from the deformation amplitude: rheological closed-form analytical2,9,10 and inverse finite element methods.6,7,11 The rheological models are simpler, computationally non-intensive, and fast.2,9,10
The current study used the rheological model to assess the in vivo corneal deformation. The method was applied to three groups of patients: normal eyes, topographically normal fellow eyes of patients with asymmetric keratoconus, and patients with clinical keratoconus.
Patients and Methods
This study was a retrospective analysis of Corvis ST data (Asian-Indian eyes) collected at the Narayana Nethralaya Multi-specialty Eye Hospital, Bangalore, India. The study was conducted in accordance with the tenets of the Declaration of Helsinki, after obtaining institutional ethics committee approval. The study included 312 normal eyes (only one eye was chosen at random with a coin toss from 312 patients), 107 fellow eyes of patients with asymmetric keratoconus (a subset of 107 of 289 patients with keratoconus), and 289 keratoconic eyes (289 patients). These eyes formed the study population of an earlier study.2 The corneal tomographic features were derived from Pentacam (Oculus Optikgeräte GmbH; software version 1.21b26) and Corvis ST (software version 1.5r1902). The Corvis ST data were exported in the form of a comma separated value file, which included the deformation amplitude (total), deflection amplitude (corneal only), whole eye movement (extracorneal only), and air-puff pressure waveform.2,9,10 Here, deformation amplitude was the arithmetic sum of deflection amplitude and whole eye movement.
Data were analyzed using two spring-dashpot models. The first one was a standard linear solid model (SLM), assuming the cornea was an elastic material and extracorneal tissue was a viscoelastic material (Figure AA, available in the online version of this article). This model has already been used in interpreting the corneal and extracorneal material properties in a variety of conditions, such as aging, myopia, autoimmune response, keratoconus, after refractive surgery, and after corneal cross-linking.2,9,10,12–15 The second model was a two-compartment Kelvin-Voigt model in which both the corneal and extracorneal tissue were modeled as viscoelastic materials (Figure AB). Both models were solved for every eye with a non-linear least square technique in MATLAB R2013a (Math-Works, Inc., Natick, MA).2,9,10,12–15 The following parameters were derived from the above models, as described in our previous publications2,9,10,12–15:
kc (constant): constant corneal stiffness (N/m), a measure of the linear elastic response of the cornea.
kc (mean): mean corneal stiffness (N/m), a measure of the non-linear elastic response of the cornea as a function of the applied air-puff pressure.
µc: corneal viscosity (Pa.sec), calculated only for the second model.
kg: extracorneal stiffness (N/m).
µg: extracorneal viscosity (Pa.sec).
Deformation hysteresis area: the area enclosed by the applied air-puff force and deformation amplitude (N.m) (calculated directly from measured patient data). This area is a theoretical measure of the degree of viscous response of the whole globe. The greater the area, the greater will be the viscosity (µg) of the whole globe.
Deflection hysteresis area: the area enclosed by the corneal force to deflection amplitude plot (N.m) (using the two spring-dashpot model only). This area is a theoretical measure of the degree of viscous response of the cornea only. The greater the area, the greater will be the viscosity (µc) of the cornea.
From fundamental mechanics, it is well known that the measured deformation of a viscoelastic material will lag in phase relative to the applied force (eg, the peak deformation will be reached after the peak force has passed). To assess whether the second model was able to sense any viscoelastic feature of the cornea, the deflection amplitude derived from the in vivo measurement was advanced in time by one frame (up to 6 frames) at a time and then added to the in vivo measurement of the whole eye movement to obtain a modified deformation amplitude. The shift was performed in the direction of increasing time (Figure B, available in the online version of this article). Each shift by one frame led to an additional time difference of 0.231 msec between peak air-puff pressure and peak modified deformation amplitude. Physically, this would imply that µc would increase in magnitude with every frame shift. Figure B shows the schematic of the deformation amplitude waveform after shifts by 2, 4, and 6 frames for a normal eye (Figure BA), fellow eye (Figure BB), and keratoconic eye (Figure BC). The changes in the visco-elastic parameters after frame shift from the baseline were analyzed for each group.
Statistical analyses were performed using Med-Calc software (version 19.0.3; MedCalc Inc., Ostend, Belgium). The statistical analyses were based on non-parametric assumptions as determined by the Shapiro– Wilk test. All reported values were median and 95% confidence interval. The Kruskal–Wallis test was used to determine the statistical differences between the groups. The statistical difference between parameters calculated before and after frame shift was done using the Friedman test. The Conover post-hoc test was used to account for multigroup comparisons. The agreement between parameters calculated from the two models was calculated using the concordance correlation coefficient (Cor). The Cor was defined without the assumptions of normal distribution.16 Cor values of greater than 0.99, between 0.95 and 0.99, between 0.90 and 0.95, and less than 0.90 were considered as almost perfect, substantial, moderate, and poor, respectively.17 A P value less than .05 was considered statistically significant.
Results
Table 1 shows the demographics of eyes included in the study. Age (P = .003) and intraocular pressure (P = .001) showed significant differences, but the magnitude of the observed difference was clinically insignificant (less than 1 mm Hg between the groups). All other parameters, such as central corneal thickness, thinnest corneal thickness, flat keratometry (K1), steep keratometry (K2), mean keratometry (Kmean), maximum keratometry (Kmax), Corvis Biomechanical Index (CBI), Tomographic Biomechanical Index (TBI), root mean square of lower order (LORMS), and higher order (HORMS) anterior aberrations, showed significant differences between the groups (P = .001 for all). The groupwise significances of the above parameters are described in Table 1. The K1, K2, and Kmean were similar between normal eyes and fellow eyes, but differed from keratoconic eyes (Table 1). However, other parameters such as CBI, TBI, LORMS, and HORMS showed a significant difference between all groups when evaluated pairwise.
Table 2 shows the parameters calculated from the two models using the Corvis ST deformation amplitude. Both models segregate the deflection amplitude and whole eye movement to derive corneal and extracorneal biomechanical parameters. In both the SLM and KVM, kc (constant) and kc (mean) differed significantly between all groups (P < .001). This indicated significant differences between the corneal stiffnesses of the three groups. In SLM, kg of normal eyes differed significantly from fellow eyes and keratoconic eyes (P = .001). Further, µg of keratoconic eyes differed significantly from the normal eyes and fellow eyes (P = .001). The same was observed with the KVM model (Table 2). In the KVM, µc (corneal viscosity) of normal eyes differed significantly from that of fellow eyes and keratoconic eyes (P = .001). However, µc was extremely small in magnitude (order of 10−13) and this indicated no detectable viscosity of the cornea with air-puff applanation. This affected the measurement of both the deflection hysteresis and deformation hysteresis area, which were also small in magnitude (order of 10−6). All variables were highly correlated between the SLM and KVM (Cor > 0.99). This indicated virtually no difference between the magnitudes of the variables from the SLM and KVM models, and was also confirmed statistically (P > .05 for all variables in Table 2).
Figure 1 shows the change in corneal and extracorneal parameters using the KVM when the deflection amplitude waveform was artificially frame shifted. Both kc (constant) (Figure 1A) and kc (mean) (Figure 1B) initially increased and then decreased with the increasing frame shift. However, µc increased nearly linearly with increasing frame shift (Figure 1C). A trend opposing the change in corneal stiffness was noted with the change in kg (Figure 1D) and µg (Figure 1E). Interestingly, the finite magnitude of µc with increasing frame shift was several orders of magnitude (order of 10−2) greater than the in vivo measurement (Table 2). Thus, true viscous response of the cornea was lacking under air-puff applanation because even a single frame shift, if present in vivo, changed the order of magnitude of corneal viscosity from 10−13 to 10−2. Figure 1 also presented a visual description of changes in the deformation amplitude, if corneal viscous effects truly existed during in vivo air-puff applanation.
Discussion
In viscoelastic tissues, the measured deformation lags in phase relative to the applied force.18 This phenomenon is well known among soft tissues. Further, the magnitude of tissue viscosity also depends on the time rate of change in the applied stress (ie, the loading rate).18 In this study, no significant difference between the calculated parameters from the two models was noted (high magnitude of Cor). Further, the order of magnitude of the calculated µc from the KVM was insignificant and virtually zero (Table 2). We also investigated whether the KVM was capable of sensing viscous effects during the deformation of the cornea and whole globe. Hence, we conducted simulated perturbations where the deflection amplitude was artificially delayed in time to induce a lag in phase relative to the air-puff pressure pulse (Figure 1). This effect was analogous to inducing a “virtual” viscous effect.18 Then, the KVM distinctly reported higher magnitudes of µc (Figure 1C). This indicated that with the existing loading magnitude and loading rate (applied air-puff pressure profile), the cornea may not be able to undergo any viscous deformation in vivo. Another recent study used the Burgers viscoelastic model to study corneal viscosity in normal and keratoconic corneas and concluded that that viscosity of the cornea does not contribute substantially to the deformation and deflection amplitude.19
Another study compared normal and keratoconic eyes using a stiffness model.20 However, that study did not segregate the effect of deflection amplitude from the deformation amplitude in their stiffness model. Therefore, the study erroneously concluded that viscous properties of the cornea were detectable with air-puff applanation.20 The study reported a mean deformation hysteresis area of 6.06 × 10−6 and 7.78 × 10−6 Nm in normal and keratoconic eyes, respectively.20 Our results were smaller in magnitude due to the use of a patient-specific raw air-puff pressure profile instead of a smoothed average pressure curve in the earlier study.20 In Figures 1A–1B, the change in corneal stiffness first increased and then decreased with increasing frame shift. This was because with the increasing “virtual” frame shift, the time difference between the peak air-puff pressure and peak deformation amplitude was increasing while the time difference between the peak deformation amplitude and the peak whole eye movement was decreasing. The interaction between these time differences for the solution of the SLM and KVM led to the trends shown in Figures 1A–1B. Likewise, the trends in Figures 1D–1E were opposite to Figures 1A–1B.
Keratoconus reduced the elastic modulus of the cornea.21 However, the viscous properties of keratoconic corneas are unknown. The in vivo imaging of keratoconic corneas suggested local degeneration of biomechanical properties.22 Therefore, this degeneration most likely was a combination of both elastic and viscous properties, although exact magnitudes of each property are unknown.22,23 Nonetheless, the air-puff applanation technique appeared inadequate to assess the viscous properties with the analytical models. An increase in the duration of the air-puff pulse and decrease in the magnitude of the air-puff pressure may assist in vivo quantification of viscous properties of the normal and diseased human cornea. The former will allow a greater time scale for viscous lag to occur and the latter will minimize the effects of whole globe movement. Figure 1 showed the changes expected in the stiffness and viscous parameters, if corneal viscous response truly existed during air-puff applanation. These data could serve as an important reference, if modifications were made to the device (eg, increase in duration of air-puff or decreased in magnitude of air-puff pressure).
A limitation of this study was that the model did not segregate the sclera contribution to the deformation amplitude. A recent finite element study concluded that the sclera could play a role in assessing corneal biomechanics.24 The study showed that the corneal deformation differed under different ex vivo boundary conditions of fixed versus flexible limbus.24 Despite this limitation, inverse finite element models of air-puff applanation predicted accurately the postoperative deformation of patients' corneas after refractive surgery.7,12 Another limitation of the study was the lack of anisotropic properties in the biomechanical model. The SLM model formed the basis of our earlier studies using inverse finite elements.7,12 Hence, the findings of this study with respect to corneal viscoelastic properties were also applicable to the inverse finite element models. This study was limited to keratoconic eyes and fellow eyes. Future studies need to investigate whether other corneal degenerations enable detection of corneal viscosity with air-puff applanation. The viscous properties of the cornea may be below a measureable range with air-puff applanation under in vivo conditions. The conditions that could enable finite measurements of µc were simulated artificially by delaying the deflection amplitude in phase with respect to the applied air-puff pressure. Further evaluation in eyes from other regional populations should be conducted in future studies.
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Median [95% Confidence Interval] of Clinical Parameters of the Study Eyes
| Parameter | NE | FE | KE | P | Conover Post-hoc Test of Significance |
|---|
| Sample size (eyes) | 300 | 102 | 293 | – | NA |
| Age (years) | 25 [24 to 25] | 24 [22 to 25] | 24 [23 to 25] | .003 | NE to FE and KE |
| IOP (mm Hg) | 16.7 [16.5 to 17] | 16.1 [15.78 to 16.4] | 16 [15.7 to 16.3] | .001 | NE to FE and KE |
| CCT (µm) | 528.5 [525 to 533.71] | 505 [498 to 515] | 464 [458 to 471] | .001 | All |
| TCT (µm) | 525 [521 to 530.41] | 502 [491.97 to 506.01] | 453 [447 to 459] | .001 | All |
| K1 (D) | 43.20 [42.96 to 43.50] | 43.40 [43.20 to 43.88] | 45.20 [45.00 to 45.70] | .001 | KE to NE and FE |
| K2 (D) | 44.4 [44.3 to 44.7] | 44.7 [44.3 to 45] | 49.2 [48.7 to 49.5] | .001 | KE to NE and FE |
| Kmean (D) | 43.8 [43.5 to 44.1] | 44.05 [43.7 to 44.4] | 47.1 [46.6 to 47.53] | .001 | KE to NE and FE |
| Kmax (D) | 44.9 [44.7 to 45.14] | 45.6 [45.3 to 46.3] | 53.5 [52.77 to 54.8] | .001 | All |
| CBI | 0.02 [0.01 to 0.03] | 0.79 [0.56 to 0.89] | 1 [1 to 1] | .001 | All |
| TBI | 0.29 [0.23 to 0.36] | 0.99 [0.96 to 1] | 1 [1 to 1] | .001 | All |
| LORMS aberration (µm) | 1.58 [1.5 to 1.66] | 1.93 [1.77 to 2.07] | 8.05 [7.26 to 8.58] | .001 | All |
| HORMS aberration (µm) | 0.37 [0.37 to 0.39] | 0.51 [0.46 to 0.55] | 2.03 [1.72 to 2.14] | .001 | All |
Groupwise Median [95% Confidence Interval] Obtained From the 2 Models
| Parameters | NE | FE | KE | Pa | Conover Post-hoc Test of Significance |
|---|
| SLM | | | | | |
| kc (constant) (N/m) | 107.37 [105.57 to 108.85] | 100.87 [98.79 to 104.78] | 94.85 [93.77 to 96.34] | .001 | All |
| kc (mean) (N/m) | 103.05 [101.34 to 104.71] | 92.56 [90.5 to 95.62] | 85.8 [84.04 to 88.77] | .001 | All |
| kg (N/m) | 71.46 [69.19 to 73.32] | 59.28 [53.99 to 62.76] | 59.38 [56.42 to 61.75] | .001 | NE to FE and KE |
| µg (Pasec) | 0.3 [0.29 to 0.31] | 0.28 [0.26 to 0.31] | 0.25 [0.24 to 0.26] | .001 | KE to NE and FE |
| Deflection hysteresis area (Nm) | 4.26 × 10−6 [3.99 × 10−6 to 4.66 × 10−6] | 4.63 × 10−6 [4.23 × 10−6 to 5.04 × 10−6] | 5.82 × 10−6 [5.48 × 10−6 to 6.22 × 10−6] | .001 | KE to NE and FE |
| KVM | | | | | |
| kc (constant) (N/m) | 106.99 [105.34 to 108.7] | 100.87 [98.79 to 104.78] | 94.85 [93.77 to 96.34] | .001 | All |
| kc (mean) (N/m) | 102.59 [100.64 to 104.15] | 93.13 [89.99 to 95.6] | 85.54 [83.97 to 89.17] | .001 | All |
| µc (Pasec) | 4.36 × 10−9 [1.33 × 10−10 to 1.8 × 10−8] | 9.48 × 10−12 [1.25 × 10−13 to 1.03 × 10−10] | 4.29 × 10−12 [1.04 × 10−12 to 1.49 × 10−11] | .001 | NE to FE and KE |
| kg (N/m) | 72.14 [70.26 to 75.86] | 59.51 [55.17 to 63.26] | 59.82 [56.63 to 62.12] | .001 | NE to FE and KE |
| µg (Pasec) | 0.31 [0.3 to 0.33] | 0.29 [0.26 to 0.32] | 0.25 [0.24 to 0.27] | .001 | KE to NE and FE |
| Deflection hysteresis area (Nm) | 4.69 × 10−6 [4.26 × 10−6 to 4.93 × 10−6] | 4.62 × 10−6 [4.15 × 10−6 to 4.87 × 10−6] | 5.80 × 10−6 [5.47 × 10−6 to 6.14 × 10−6] | .001 | KE to NE and FE |
| Deformation hysteresis area (Nm) | 2.07 × 10−5 [1.94 × 10−5 to 2.16 × 10−5] | 2.22 × 10−5 [2.02 × 10−5 to 2.37 × 10−5] | 2.14 × 10−5 [2.06 × 10−5 to 2.2 × 10−5] | .04 | NE to FE |