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Cornea & External Disease  |   January 2025
Determinants of Human Corneal Mechanical Wave Dispersion for In Vivo Optical Coherence Elastography
Author Affiliations & Notes
  • Chaitanya Duvvuri
    College of Optometry, University of Houston, Houston, TX, USA
  • Manmohan Singh
    Department of Biomedical Engineering, University of Houston, Houston, TX, USA
  • Gongpu Lan
    School of Physics and Optoelectronic Engineering, Foshan University, Foshan, Guangdong, China
  • Salavat R. Aglyamov
    Department of Mechanical Engineering, University of Houston, Houston, TX, USA
  • Kirill V. Larin
    Department of Biomedical Engineering, University of Houston, Houston, TX, USA
  • Michael D. Twa
    College of Optometry, University of Houston, Houston, TX, USA
  • Correspondence: Michael D. Twa, College of Optometry, University of Houston, 4401 Martin Luther King Blvd., Houston, TX 77004, USA. e-mail: [email protected] 
Translational Vision Science & Technology January 2025, Vol.14, 26. doi:https://doi.org/10.1167/tvst.14.1.26
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      Chaitanya Duvvuri, Manmohan Singh, Gongpu Lan, Salavat R. Aglyamov, Kirill V. Larin, Michael D. Twa; Determinants of Human Corneal Mechanical Wave Dispersion for In Vivo Optical Coherence Elastography. Trans. Vis. Sci. Tech. 2025;14(1):26. https://doi.org/10.1167/tvst.14.1.26.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose: To characterize frequency-dependent wave speed dispersion in the human cornea using microliter air-pulse optical coherence elastography (OCE), and to evaluate the applicability of Lamb wave theory for determining corneal elastic modulus using high-frequency symmetric (S0) and anti-symmetric (A0) guided waves in cornea.

Methods: Wave speed dispersion analysis for transient (0.5 ms) microliter air-pulse stimulation was performed in four rabbit eyes ex vivo and compared to air-coupled ultrasound excitation. The effects of stimulation angle and sample geometry on the dispersion were evaluated in corneal phantoms. Corneal wave speed dispersion was measured in 36 healthy human eyes in vivo.

Results: Air-pulse-induced dispersion was comparable to ultrasound-induced dispersion between 0.7 and 5 kHz (mean-difference ± 1.96 × SD: 0.006 ± 0.5 m/s) in ex vivo rabbit corneas. Stimulation 0° relative to the surface normal generated A0 Lamb waves in corneal tissue phantoms, while oblique stimulation (35° and 65°) generated S0 waves. Stimulating normal to the human corneal apex in vivo (0°) induced A0 waves, plateauing at 10.87 to 13.63 m/s at 4 kHz, and when obliquely stimulated at the periphery (65°), produced S0 waves, plateauing at 13.10 to 15.98 m/s at 4 kHz.

Conclusions: Air-pulse OCE can be used to measure human corneal Lamb wave dispersion of A0 and S0 propagation modes in vivo. These modes are selectively excited by changing the stimulation angle. Accounting for wave speed dispersion enables reliable estimation of corneal elastic modulus in vivo.

Translational Relevance: This work demonstrates the feasibility of air-pulse stimulation for robust OCE measurements of corneal stiffness in vivo for disease detection and therapy evaluation.

Introduction
The cornea is the primary refractive element of the eye, providing about two-thirds of the total refractive power. To provide optimal optical performance, the cornea must be structurally and physiologically stable. Abnormalities in corneal biomechanical properties (e.g., as in keratoconus13) can alter corneal shape, causing poor optical performance and vision loss. Corneal mechanical properties are also intentionally modified to improve vision in corneal refractive surgery4,5 and in ultraviolet corneal cross-linking for keratoconus.68 There is longstanding interest in developing methods for noninvasive clinical measurements of corneal mechanical properties to diagnose and monitor ocular disease progression and to improve existing treatments that alter corneal tissue properties. 
Optical coherence elastography (OCE)9 has emerged as a powerful imaging method to quantify corneal biomechanical properties such as elasticity (Young's modulus) and viscosity.1013 A common method of dynamic OCE uses a mechanical loading system to induce tissue motion and a phase-sensitive optical coherence tomography imaging (OCT) system to detect the resulting tissue motion with sub-micron detection sensitivity.1416 The wave features and propagation dynamics (e.g., speed, frequency, amplitude, and damping) are then used to estimate tissue mechanical properties such as Young's modulus by using analytical models such as those based on shear waves,17,18 Rayleigh waves,19,20 or Lamb waves.10,2023 In previous work, we used microliter air-pulse stimulation (<1 kHz) as a mechanical loading system with OCE to characterize mechanical wave propagation and dispersion in the cornea.21,24 We observed that these high-frequency mechanical waves propagate in corneal tissue as guided waves, reflecting from both the top and bottom corneal surfaces. Although Lamb wave theory was developed for nondestructive mechanical testing applications and has been widely used to analyze guided waves in steel plates and shells to detect structural abnormalities,2529 there is a need to better understand the applicability of Lamb waves as a quantitative framework for modeling mechanical wave propagation dynamics in viscoelastic biological tissues such as the cornea. 
Lamb waves are classified into two basic types: symmetric and anti-symmetric, which are defined by the motion relative to the sample midplane (Fig. 1a).20,3032 Lamb waves exhibit velocity dispersion (Fig. 1b), meaning the wave speed depends on frequency. The fundamental modes—symmetric (S0) and anti-symmetric (A0)—carry the most energy and are the only modes that exist at low frequencies, whereas higher-order modes (A1, S1, A2, S2, etc.) exist at higher frequencies.20,23 The displacement profiles of the fundamental Lamb wave modes along the sample thickness also varies with frequency.31,33 At lower frequencies, the sample motion across the thickness linearly varies for symmetric and anti-symmetric modes, converging to the displacement characteristics of conventional axial and flexural plate waves, respectively.31,33 At intermediate frequencies, where the wavelength of the mechanical wave is similar to the thickness of the sample, the sample motion across the thickness resembles typical A0 and S0 mode shapes,31,33 as depicted in Figure 1a. At higher frequencies, mechanical wave reflections from the sample's bottom surface and the guided wave behavior dissipate, causing these fundamental modes to propagate as surface (Rayleigh) waves.31,33 Determining Lamb wave propagation mode by analyzing wave propagation dynamics and measuring the speed dispersion is critical in accurately reconstructing the mechanical properties of the sample. In cornea, mechanical wave propagation speed is a parameter commonly used for computing the elastic modulus. However, these corneal mechanical waves exhibit velocity dispersion, and the Lamb wave dispersion in Figure 1b indicates that multiple wave speeds can exist at the same frequency. Thus wave speed measurement at a single frequency is insufficient to characterize corneal mechanical properties and can result in large errors when estimating elastic modulus for the cornea. The multimodal and dispersive characteristics of corneal mechanical waves may explain some of the wide variability reported for corneal Young's modulus (ranging from kilopascals to tens of megapascals).11 Characterizing corneal mechanical wave velocity dispersion, identifying the propagation mode, and applying the Lamb wave model appropriate for the observed mode would improve the accuracy and clinical utility of dynamic OCE in quantifying corneal mechanical properties. 
Figure 1.
 
Illustration of Lamb wave propagation modes and their characteristic wave speed dispersion. (a) The relative motion of the top and the bottom surfaces are out of phase for antisymmetric Lamb mode and in phase for symmetric mode relative to the sample midplane (dashed line). Rayleigh waves propagate along the surface. (b) Wave speed dispersion for the fundamental Lamb modes (A0 and S0) and higher-order modes (A1, S1, A2, S2). The fundamental modes plateau at Rayleigh wave speed and higher-order modes plateau at shear wave speed. Adapted from Ryden et al.30
Figure 1.
 
Illustration of Lamb wave propagation modes and their characteristic wave speed dispersion. (a) The relative motion of the top and the bottom surfaces are out of phase for antisymmetric Lamb mode and in phase for symmetric mode relative to the sample midplane (dashed line). Rayleigh waves propagate along the surface. (b) Wave speed dispersion for the fundamental Lamb modes (A0 and S0) and higher-order modes (A1, S1, A2, S2). The fundamental modes plateau at Rayleigh wave speed and higher-order modes plateau at shear wave speed. Adapted from Ryden et al.30
A reported limitation of high-frequency microliter air-pulse stimulation is that the mechanical waves generated using this approach are only comprised of lower frequencies (<1 kHz), and therefore unsuitable to characterize the full spectrum of mechanical wave dispersion in corneal tissues.3437 In this study, we address this limitation by using air-pulse stimulation parameters capable of generating high-frequency and broadband (>5 kHz) mechanical waves in the cornea and quantify the resulting wave propagation dynamics and speed dispersion characteristics for the human cornea in vivo. 
The incident angle of stimulation is another parameter from Lamb wave theory that is known to influence mechanical wave dispersion when testing non-biological samples.3840 We evaluated the influence of stimulation angle, sample geometry, and lateral geometric sample boundaries on speed dispersion characteristics by using two stimulation paradigms (orthogonal [i.e., central] and oblique [i.e., peripheral, stimulation]) in corneal mimicking tissue phantoms and in the human cornea in vivo. Finally, we assessed the impact of wave speed dispersion on the repeatability of Young's modulus calculations using dispersion measurements from healthy human participants. 
Methods
We used a microliter air-pulse based OCE to capture wave propagation dynamics in the cornea. To evaluate the capability of air-pulse stimulation to induce high-frequency and broadband mechanical waves in the cornea, we evaluated corneal wave speed at multiple frequencies (0.7 kHz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz) in four whole rabbit eyes ex vivo (Pel-Freez Biologicals, Rogers, AR, USA). We compared corneal responses, mechanical wave wavelength, and dispersion induced by air-pulse stimulation to air-coupled ultrasound (ACUS) stimulation,37,41 which allows precise control over tissue stimulation frequency parameters. To determine the influence of stimulation angle and lateral geometric boundaries on Lamb wave propagation modes, we measured wave speed dispersion in a corneal phantom and in vivo human cornea at different stimulation angles and various geometrical locations. 
OCE System
A schematic of the OCE system is shown in Figure 2.42,43 The OCE system consisted of a mechanical stimulator (air-pulse or ACUS), which induced micromechanical waves that were detected with high sensitivity (∼6 nm) with a phase-sensitive spectral domain OCT imaging system.43 Two mechanical stimulators were used in the study—a low pressure (∼5 kPa), spatiotemporally discrete (driving pulse width: 520 µs) air-pulse stimulator,44 and a confocal air-coupled ultrasound stimulator.41 For mechanical stimulation using the air-pulse stimulator, pressurized medical-grade air was delivered to the sample using a cannula (∼900 µm lateral spot size), which was placed ∼2 mm away from the sample. The temporal duration of the air pulse was controlled using an electronic pulse generator (Model 577; Berkley Nucleonics Corp., San Rafael, CA, USA). The x-y position of the stimulator was controlled using motorized stages. For tissue stimulation using ACUS, a hemispherically shaped (∼34 mm diameter with a focal length of 20 mm) ACUS transducer was used, producing a focused acoustic beam (∼350 µm lateral spot size).41 A 10 mm diameter central opening in the transducer allowed for simultaneous tissue stimulation and OCT imaging.41 The ACUS transducer was used to stimulate the tissue at multiple discrete frequencies, as mentioned earlier. The generation and amplification of the transducer signal were achieved using a function generator (DG4062; Rigol Technologies, Beijing, China) and a power amplifier (A150; Electronics & Innovation, Rochester, NY, USA), respectively, as described in detail in our previous work.41 The OCT system was described in detail in our previous studies.42 Briefly, the OCT system consists of a superluminescent diode (cBLMD-D-840-HP-I, Superlum Diodes Ltd., Cork, Ireland) with a central wavelength of 845 nm and a bandwidth of 170 nm. The axial and lateral resolution of the OCT system, measured in air, were ∼2.7 µm and ∼7.8 µm, respectively. The OCT system also incorporates a fixation target and a camera to monitor the corneal position and to correct for lateral eye motion during imaging of human subjects. 
Figure 2.
 
Schematic of the clinical OCE system. Microliter air-pulse and ACUS were used as mechanical stimulation methods. The stimulators (green boxes) provided localized tissue excitation and were not used simultaneously. Phase-sensitive OCT (blue boxes) tracked the resulting dynamics of elastic wave propagation in the samples.
Figure 2.
 
Schematic of the clinical OCE system. Microliter air-pulse and ACUS were used as mechanical stimulation methods. The stimulators (green boxes) provided localized tissue excitation and were not used simultaneously. Phase-sensitive OCT (blue boxes) tracked the resulting dynamics of elastic wave propagation in the samples.
Wave Speed Calculations
The M-B-mode scan protocol was employed to capture wave propagation in corneas over 251 scanning positions.17,34,35 The OCT system operated at a 50 kHz A-scan rate. The OCE imaging protocol consisted of acquiring 1000 A-lines spanning 20 ms at a specific lateral position while synchronously triggering the excitation system with the OCT frame trigger. This procedure was repeated over 251 lateral positions. At a given lateral position, x, and specific sample depth, z, sub-micron tissue displacements, dx,z(t), induced by the mechanical stimulator were tracked and corrected at all lateral positions imaged.45 The axial particle velocities at these locations were computed by performing numerical differentiation in time (Δt = 20 µs) on these displacements, which were used to generate a two-dimensional (2D) space-time map (particle velocity as a function of distance from the stimulation). The slope of this 2D position-time map was used to compute the wave speed.46 
Dispersion Analysis
To measure wave speed frequency dispersion, a 2D Fast Fourier Transform (FFT) was applied to the 2D position-time map to obtain a map in the wavenumber (k) and frequency (f) domain. The maximum peaks in this wavenumber-frequency map for each frequency were used to compute the phase velocity (cp) using \({{c}_p} = \ \frac{{{{f}_n}}}{{{{k}_{peak}}}}\), where n was the total number of frequency bins obtained after FFT.47,48 After this computation, the phase velocities were plotted as a function of frequency to generate dispersion curves. 
Air-Pulse Frequency Spectrum Determination
Experiments were performed on ex vivo corneas of four whole rabbit eyes (Pel-Freez Biologicals) within 24 hours of enucleation. The eyes were cleaned of extraorbital connective tissues and frequently hydrated using a 1X phosphate-buffered saline solution during imaging. Before the experiments, the eyes were treated with 20% dextran49 to maintain physiological corneal thickness. The corneal thickness in these eyes was determined using OCT imaging one hour after the dextran treatment, and it was 356 ± 43 um (mean ± SD). After dextran treatment, the eyes were placed in a custom-designed holder, and the intraocular pressure (IOP) was artificially controlled using a closed-loop IOP control system, which consisted of a pressure transducer and a micro-infusion pump.6 Experiments were performed while maintaining the IOP at 10, 20, and 30 mm Hg. The cornea was stimulated using a microliter air pulse and ACUS in separate measurements. Tissue displacements and the wave propagation dynamics were captured using OCE across 25l lateral positions spanning 4.44 mm. 
Microliter air-pulse stimulation consisted of a single pulse with a pulse width of 520 µs. Air-pulse stimulation was performed perpendicular to the corneal surface, 2.7 mm away from the corneal apex. The tissue displacements induced by the temporally short air-pulse stimulation consisted of broad frequency bandwidth (Fig. 3a) with a full-width half maximum of 5 kHz. To obtain corneal wave speed at a specific frequency, tissue responses from the air-pulse stimulation were filtered using a finite impulse response filter with a bandwidth of 400 Hz. A zero-phase filtering method was performed to prevent any aberrant calculations of speed due to filtering-induced phase shifts.50 Corneal wave speeds at multiple frequencies (0.7 kHz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz) were obtained using these filtering methods. ACUS was used to stimulate the corneal apex at these same frequencies (0.7 kHz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz), and each frequency stimulation consisted of a single pulse. To achieve a comparable frequency spectrum between air-pulse and ACUS, tissue responses from ACUS stimulation were also band-pass filtered using frequency filtering methods. Figure 3 shows a comparison of the spectra and the temporal response in an ex vivo rabbit cornea at 1 kHz. 
Figure 3.
 
Frequency-filtering for wave propagation analysis. (a) Power spectrum for air-pulse excitation stimulus in ex vivo rabbit corneal tissue. (b) The filtered temporal response using the finite impulse response (FIR) filter. Red vertical lines indicate the bandwidth of the filter (1 kHz ± 200 Hz). (c) Power spectrum response for the 1 kHz ACUS in ex vivo rabbit corneal tissue and (d) filtered temporal displacement response.
Figure 3.
 
Frequency-filtering for wave propagation analysis. (a) Power spectrum for air-pulse excitation stimulus in ex vivo rabbit corneal tissue. (b) The filtered temporal response using the finite impulse response (FIR) filter. Red vertical lines indicate the bandwidth of the filter (1 kHz ± 200 Hz). (c) Power spectrum response for the 1 kHz ACUS in ex vivo rabbit corneal tissue and (d) filtered temporal displacement response.
The similarity of the frequency spectrum and equivalence of the sample response with air-pulse and ACUS stimulation was evaluated by comparing corneal wave speed propagation measured at specific frequencies. In each sample eye, the wave speed measurements were repeated three times at each frequency and for each IOP level. The mean and standard deviation was computed from these three wave speed measurements for both air-pulse and ACUS stimulations. The mean values were used to compute the difference in the wave speed between air-pulse and ACUS stimulation methods. Bland-Altman analysis51,52 was performed to assess the bias and agreement between the corneal wave speeds obtained from air-pulse and ACUS stimulation methods. The mean difference for corneal wave speeds between air-pulse and ACUS stimulations was defined as Δcp = cp,airpulsecp,ACUS. The speed difference was computed with the measurements paired by eye, IOP and at specific frequencies. 
Ex Vivo Lamb Wave Dispersion Using OCE
Tissue mimicking silicone phantoms were fabricated by mixing polydimethylsiloxane (PDMS) with a curing agent (room temperature vulcanizing silicone type A) and a crosslinker with ratios 10:1:10, respectively. Two test samples were made—a corneal phantom: 500 µm thick, 15 mm diameter, and a thick sample: 10 mm thick, 30 mm diameter. The 500 µm thick corneal phantom was placed on an artificial anterior chamber (Barron Precision Instruments, L.L.C, Grand Blanc, MI, USA). Mechanical stimulation was performed using a microliter air-pulse. 
To assess the impact of stimulation angle, lateral boundaries, and geometric constraints on wave dispersion, we stimulated the samples either orthogonal to the sample surface or at oblique angles (35° and 65°) in both central and peripheral locations. The stimulation angle was adjusted by rotating the samples using a precision optical goniometer stage (GOH-40A15; OptoSigma Corp., Santa Ana, CA, USA). Stimulation angle was defined as the relative angle between the air-pulse stimulation and the sample surface normal. 
Wave propagation dynamics were captured using OCE imaging across 251 lateral positions spanning 8.2 mm. Speed dispersion measurements for each stimulation protocol and for each stimulation angle were calculated using the dispersion analysis. The wave speed (cR) at 4.5 kHz was used to calculate Young's moduli of the 500 µm and the 10 mm thick samples using equation for Rayleigh surface wave in an incompressible medium: \(E = 3\rho {{( {\frac{{{{c}_R}}}{{0.955}}} )}^2}\), where ρ was the sample density. These computed Young's moduli were compared to Young's moduli measured with a uniaxial mechanical test frame (Model 5943, Instron, Norwood, MA, USA) using previously described methods.53 
In Vivo Human Corneal Dispersion Using OCE
This study was conducted under University of Houston institutional review board approval and adhered to the tenets of the Declaration of Helsinki. Informed consent was provided by all participants. This study was performed on the right eye of 36 healthy participants (age range 23–34 years). Clinical examinations were performed to confirm that participants had no ocular disease, previous ocular surgery, or any systemic condition or medication use that could have affected corneal health or sensory function. The IOP and central corneal thickness were obtained from a Scheimpflug imaging-based air-puff tonometer (Corvis ST; OCULUS Optikgeräte GmbH, Wetzlar, Germany). 
We conducted two sets of OCE measurements to evaluate the impact of stimulation angle on Lamb wave dispersion characteristics in the human cornea in vivo. In the first experiment, we stimulated the cornea perpendicularly at the corneal apex and measured wave propagation at distances ranging from 1 mm to 4.4 mm on both sides of the apex. In the second experiment, we stimulated the cornea at a distance of 6.5 mm from the corneal apex, using a stimulation angle of 65°, and measured wave propagation at distances ranging from 0.5 mm to 6 mm from the stimulation spot. In vivo human corneal wave speed dispersion for both normal and oblique stimulation angles was obtained using the dispersion analysis as described above for in vitro samples. In vivo corneal wave speed at the frequencies of 1 kHz, 2 kHz, 3 kHz, 4 kHz, and 5 kHz was used to compute Young's modulus using \(E = 3\rho {{( {\frac{{{{c}_R}}}{{0.955}}} )}^2}\), where ρ is the tissue density. 
Results
Air-Pulse Frequency Spectrum
Figure 4 shows a comparison of corneal mechanical wave propagation induced by air-pulse and ACUS stimulation in an ex vivo rabbit eye (IOP = 10 mm Hg). The data at 1 kHz, 3 kHz, and 5 kHz are shown. The wave propagation snapshots depicted in Figure 4 show that the wavelength of the mechanical wave induced by the air-pulse stimulation filtered at a specific frequency was similar to the wavelength produced by ACUS stimulation. The temporal profiles of the corneal displacements were also similar between these two stimulation techniques. Corneal displacement amplitudes in Figure 4 were computed on the surface, 0.25 mm away from the mechanical stimulation. The plots show that the displacement amplitudes from air pulse stimulation were greater compared to the ACUS stimulation. As expected, the corneal displacement amplitude decreased with increasing frequency in both stimulation methods. 
Figure 4.
 
Comparison of mechanical wave propagation features at 1 kHz (a, b), 3 kHz (c, d), and 5 kHz (e, f) in an ex vivo rabbit cornea (IOP = 10 mm Hg) induced by air-pulse and ACUS stimulation. Air-pulse and ACUS stimulation frequencies were bandpass filtered (±200 Hz) at each frequency. Similar displacement profiles and particle velocity fields were observed at each frequency (note lower amplitude for ACUS). White horizontal rule with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity in depth.
Figure 4.
 
Comparison of mechanical wave propagation features at 1 kHz (a, b), 3 kHz (c, d), and 5 kHz (e, f) in an ex vivo rabbit cornea (IOP = 10 mm Hg) induced by air-pulse and ACUS stimulation. Air-pulse and ACUS stimulation frequencies were bandpass filtered (±200 Hz) at each frequency. Similar displacement profiles and particle velocity fields were observed at each frequency (note lower amplitude for ACUS). White horizontal rule with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity in depth.
Figure 5 shows the Bland-Altman51,52 comparison between corneal wave speeds measured from air-pulse and air coupled ultrasound (ACUS) stimulation in four rabbit eyes at multiple frequencies (0.7 kHz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz) and at IOPs of 10, 20, and 30 mm Hg. The differences in wave propagation speed between air-pulse and ACUS stimulation were calculated from paired measurements matched by eye, IOP, and stimulation frequency. The results show good agreement overall and little bias between the wave speeds obtained from air-pulse and ACUS stimulation at 10 and 20 mm Hg. At 30 mm Hg, the wave speeds from air-pulse stimulation were lower (<1.1 m/s difference) compared to ACUS stimulation. The median [IQR] wave speeds measured with air-pulse at 4 kHz across all four eyes tested were 3.69 [0.44] m/s, 5.95 [0.50] m/s, and 6.76 [0.74] m/s at intraocular pressures of 10, 20, and 30 mm Hg, respectively (Supplementary Fig. S1). The variability in wave speeds from air-pulse stimulation was smaller compared to the differences observed in the Bland-Altman plot. The analysis revealed no consistent trend in wave speed differences as a function of frequency and there was good agreement across all tested frequencies. The average bias across all tested IOPs and frequencies was −0.05 m/s, and the upper and lower limits of agreement (mean difference ± 1.96 × SD of the differences) were 0.67 and −0.77 m/s, respectively. 
Figure 5.
 
Bland-Altman analysis shows good agreement and minimal bias between the corneal wave propagation speeds obtained using air-pulse and ACUS stimulations, measured in four ex vivo rabbit eyes at multiple frequencies (0.7 kHz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz) and at IOPs of 10, 20, and 30 mm Hg. The wave speed differences were color-coded by frequency, with distinct shapes representing the different IOP levels. Solid lines represent the upper and lower 95% limits of agreement, and the dashed lines represent the mean difference across all tested IOPs and frequencies.
Figure 5.
 
Bland-Altman analysis shows good agreement and minimal bias between the corneal wave propagation speeds obtained using air-pulse and ACUS stimulations, measured in four ex vivo rabbit eyes at multiple frequencies (0.7 kHz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz) and at IOPs of 10, 20, and 30 mm Hg. The wave speed differences were color-coded by frequency, with distinct shapes representing the different IOP levels. Solid lines represent the upper and lower 95% limits of agreement, and the dashed lines represent the mean difference across all tested IOPs and frequencies.
Lamb Wave Dispersion in Tissue-Mimicking Phantoms
Mechanical wave propagation and the corresponding speed dispersion profiles in tissue-mimicking silicone phantoms at a specific stimulation angle and location are depicted in Figure 6. The particle velocities on the top and bottom surfaces in the corneal phantom were in phase with each other during orthogonal stimulation (Figs. 6a, 6b), resembling the Lamb wave A0 mode. When the stimulation angle was oblique relative to the sample surface (Figs. 6c, 6d), the particle velocity on the top and bottom surfaces in the corneal phantom were out of phase with each other, resembling the Lamb wave S0 mode. Speed dispersion profiles for the orthogonal stimulation angle resembled the Lamb wave A0 mode, both for the center and the peripheral stimulation (Fig. 6e; 0°). Whereas for oblique stimulation angles, the speed dispersion profiles resembled S0 mode irrespective of the stimulation location (Fig. 6e; 35°, 65°). The wave dispersion profiles for the A0 and the S0 propagation modes gradually plateaued beyond 4 kHz, and the plateauing wave speeds for the A0 mode (8.0 m/s for center stimulation and 7.7 m/s for peripheral stimulation at 4.5 kHz) were similar to the plateau speeds of the S0 mode (7.9 m/s for center stimulation and 7.7 m/s for peripheral stimulation at 4.5 kHz). 
Figure 6.
 
Lamb wave mode for the 500 µm thick corneal phantom was determined by the stimulation angle, not the stimulation location. Particle velocity maps for normal stimulation angle (a, b) resembled A0 mode irrespective of stimulation location (center or periphery). (c, d) Particle velocity maps for oblique stimulation angles (35° or 65°) resembled S0 mode. The speed dispersion profiles (e) in corneal phantom also resembled A0 mode for normal stimulation angles and S0 mode for oblique stimulation angles. For the 10 mm thick sample, wave propagation was similar regardless of whether the phantom was stimulated at a normal stimulation angle (f, g) or at oblique angles (h, i) and whether the stimulation was at the center (f, h) or periphery (g, i). The thick sample dispersion plots (j) were similar across all stimulation conditions. The gray-shaded region in the particle velocity maps represents the air-pulse stimulator location— where stimulation was performed at the center of the gray-shaded region. The stimulation location in each panel is indicated as center or periphery. White horizontal rule (mm) with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity along depth axis.
Figure 6.
 
Lamb wave mode for the 500 µm thick corneal phantom was determined by the stimulation angle, not the stimulation location. Particle velocity maps for normal stimulation angle (a, b) resembled A0 mode irrespective of stimulation location (center or periphery). (c, d) Particle velocity maps for oblique stimulation angles (35° or 65°) resembled S0 mode. The speed dispersion profiles (e) in corneal phantom also resembled A0 mode for normal stimulation angles and S0 mode for oblique stimulation angles. For the 10 mm thick sample, wave propagation was similar regardless of whether the phantom was stimulated at a normal stimulation angle (f, g) or at oblique angles (h, i) and whether the stimulation was at the center (f, h) or periphery (g, i). The thick sample dispersion plots (j) were similar across all stimulation conditions. The gray-shaded region in the particle velocity maps represents the air-pulse stimulator location— where stimulation was performed at the center of the gray-shaded region. The stimulation location in each panel is indicated as center or periphery. White horizontal rule (mm) with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity along depth axis.
The wave propagation (Figs. 6f–i) and speed dispersion (Fig. 6j) in the 10 mm thick tissue-mimicking silicone phantom did not exhibit guided wave dynamics because the wavelength of the mechanical wave (1.5 mm) was shorter than the sample thickness. Unlike the corneal tissue phantom, there was no change in propagation mode from A0 to S0 with variations in stimulation angle (Figs. 6f vs. 6h; 6g vs. 6i) or stimulation location (Figs. 6f vs. 6g; 6h vs. 6i). The dispersion observed under these stimulation conditions in the thick sample is likely due to viscosity rather than guided wave propagation or sample geometry. The wave speed at 4.5 kHz was similar for all the stimulation conditions (7.8 ± 0.2 m/s; mean ± SD). Young's modulus of the 10 mm thick sample measured by mechanical testing (198 ± 10 kPa) was similar to the modulus calculated from the OCE propagation speed at 4.5 kHz (204 ± 10 kPa). Both the 10 mm thick phantom and the 500 µm thick corneal phantom were made from the same silicone solution, so their mechanical properties are expected to be comparable. Young's modulus computed from the OCE propagation speed at 4.5 kHz in the 500 µm thick corneal phantom (201 ± 14 kPa) closely matched the values obtained for the 10 mm thick phantom using both OCE and mechanical testing. 
In Vivo Human Corneal Dispersion Using OCE
In vivo human corneal wave propagation responses and subsequent dispersion profiles are shown in Figure 7. Corneal surface wave propagation responses and particle velocity maps filtered at a particular frequency for normal and oblique stimulation angles (∼650) are shown in Figures 7a–d. Corneal responses on the top and the bottom surfaces at a specific frequency when the stimulation was normal to the corneal surface (Figs. 7a, 7c) are in phase with each other, resembling the A0 mode. When the cornea was stimulated at oblique angles (Figs. 7b and 7d), particle velocity maps, as well as surface responses at 1 kHz are in phase with each other. Whereas at 5 kHz during oblique stimulation angles, the top and bottom tissue surfaces are out of phase with each other resembling S0 mode. Wave propagation snapshots for the normal and oblique stimulation filtered at 1 kHz and 5 kHz show that the wavelength and the propagation distance of the induced mechanical wave decreased at a higher frequency, as expected. It can be noted that the corneal wave dispersion in the sample shown resembled Lamb wave's A0 mode when stimulated perpendicular to the corneal surface (Fig. 7e) and resembled S0 mode when stimulated at oblique angles (Fig. 7f). The dispersion begins to plateau beyond 4 kHz for both the A0 and the S0 modes. 
Figure 7.
 
Stimulation angle determines Lamb wave mode and speed dispersion dynamics in the human cornea in vivo. (ad) Frequency filtered surface displacements (1 kHz and 5 kHz) and particle velocity fields at normal and oblique simulation angles. (a, c) Normal corneal stimulation produced (e) A0 mode dispersion (speed increases with frequency). (b, d) Oblique stimulation angles produced (f) S0 mode dispersion (speed decreases with frequency) and anti-symmetric propagation dynamics. Figure insets (e and f) show tissue stimulation location (*) relative to the OCE scan location (red line). White horizontal rule (mm) with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity along depth axis.
Figure 7.
 
Stimulation angle determines Lamb wave mode and speed dispersion dynamics in the human cornea in vivo. (ad) Frequency filtered surface displacements (1 kHz and 5 kHz) and particle velocity fields at normal and oblique simulation angles. (a, c) Normal corneal stimulation produced (e) A0 mode dispersion (speed increases with frequency). (b, d) Oblique stimulation angles produced (f) S0 mode dispersion (speed decreases with frequency) and anti-symmetric propagation dynamics. Figure insets (e and f) show tissue stimulation location (*) relative to the OCE scan location (red line). White horizontal rule (mm) with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity along depth axis.
Figure 8 shows the mean and central tendency for corneal wave speed dispersion measurements across all 36 participants. For perpendicular stimulation at the corneal apex, dispersion resembled the A0 Lamb wave mode, where speed increased with frequency before plateauing at 4 kHz. For oblique tissue stimulation angles, the dispersion resembled the S0 Lamb wave mode, where speed decreased with increasing frequency. With oblique corneal stimulation, there was greater variation in measured wave speed at lower frequencies (29.1 m/s to 15.9 m/s). This was observed for both within- and between-subject measures. Variation in the A0 and the S0 propagation modes for each subject are shown in Supplementary Figure S2. At 4 kHz and beyond, the wave speed propagation plateaued and had less variable range within- and between subjects (12.5 m/s to 15.9 m/s). The measured asymptotic wave speed from the peripheral stimulation was similar to the plateauing speed for the normal stimulation, which ranged from 11.1 m/s to 13.6 m/s. The plateauing speeds for these propagating modes were consistent across all participants. Figure 8b depicts Young's modulus estimated from the mean wave speed of these 36 patients at discrete frequencies (1-5 kHz at 1 kHz intervals). Young's modulus computed from the wave speeds at lower frequencies shows high variability across all participants, whereas similar Young's modulus estimations can be observed at higher frequencies (4 and 5 kHz). 
Figure 8.
 
(a) In vivo human corneal Lamb wave speed dispersion for all 36 subjects (mean ± SD of the intrasubject mean; n = 3) with stimulation normal to the corneal surface (blue) and at oblique angles (red). Solid line represents the mean and shaded region represents SD. (b) Per-subject Young's modulus computed from the mean wave speeds at specific frequencies (1–5 kHz at 1 kHz interval) where blue represents stimulation normal to the corneal surface and red represents oblique stimulation. The median value (50th percentile) of the dataset is represented by the center line in each box, whereas the box indicates the interquartile range (25th to 75th percentiles). The whiskers extend to the fifth and 95th percentiles and the individual data points are indicated by circles.
Figure 8.
 
(a) In vivo human corneal Lamb wave speed dispersion for all 36 subjects (mean ± SD of the intrasubject mean; n = 3) with stimulation normal to the corneal surface (blue) and at oblique angles (red). Solid line represents the mean and shaded region represents SD. (b) Per-subject Young's modulus computed from the mean wave speeds at specific frequencies (1–5 kHz at 1 kHz interval) where blue represents stimulation normal to the corneal surface and red represents oblique stimulation. The median value (50th percentile) of the dataset is represented by the center line in each box, whereas the box indicates the interquartile range (25th to 75th percentiles). The whiskers extend to the fifth and 95th percentiles and the individual data points are indicated by circles.
Discussion
The findings of this study further extend the clinical utility of microliter air-pulse stimulation in measuring corneal mechanical properties in vivo. This non-contact, low-force stimulation combined with 0.5 ms pulse duration generated broadband (>5 kHz) mechanical waves in corneal tissues with frequency content suitable for characterizing dispersion responses. Generating these broadband mechanical waves allowed for visualization and analysis of the guided wave behavior and improved speed dispersion measurements in corneal tissues in vivo. The wave propagation dynamics and the speed dispersion profiles measured in the human cornea in vivo, ex vivo rabbit cornea, and in tissue-mimicking phantoms were well predicted by Lamb wave theory. The results confirm previous work that shows that both A0 and S0 Lamb wave propagation modes are possible in the corneal tissues.54 These fundamental Lamb wave modes (A0 and S0) can be selectively excited in the in vivo human cornea by changing the angle of the mechanical stimulation relative to the corneal surface. The oblique stimulation angle was used to selectively excite S0 propagation mode, which was achieved by moving the air-pulse stimulator towards the peripheral cornea. The A0 and S0 Lamb wave modes have similar plateauing speeds at higher frequencies. Young's modulus computed from these speeds had lower variation compared to Young's modulus computed from wave speeds at lower frequencies. These findings emphasize the importance of correctly identifying the propagation mode and accurately characterizing Lamb wave dispersion characteristics in the human cornea in vivo, which can provide clinically reliable estimates of corneal biomechanical properties. 
The power spectrum (Fig. 3a) and particle velocity maps (Fig. 4) from microliter air-pulse stimulation illustrate the broadband (>5 kHz) frequency content of this non-contact technique. The broadband air-pulse power spectrum (Fig. 3a) shows reduced magnitude at high frequencies, which may contribute to the increased noise in particle velocity maps for the ex vivo rabbit cornea (Fig. 4). The decrease in amplitude and wave propagation distance at higher frequencies,55 could also contribute the increased noise observed at higher frequencies in air-pulse generated particle velocity maps. The broadband frequency content of the air-pulse stimulation, despite the noise at higher frequencies, offers several clinical advantages over traditional high-frequency stimulation techniques like vibrational contact probes or ACUS. First, the broadband spectrum enables wave speed computation at multiple frequencies, saving time and eliminating the need for repeated corneal stimulation at multiple frequencies. This single-scan approach is also helpful for minimizing wave speed variability caused by eye movement. Second, this wide frequency range allows for evaluating wave dynamics in highly stiff corneal tissue (∼0.5 MPa) at lower frequencies (<1 kHz), where ACUS often fails due to its low amplitude in this frequency range.56,57 Analyzing low-frequency propagation is particularly useful for differentiating Lamb wave modes in the cornea, and measuring low-frequency dispersion could help quantify corneal tensile modulus.58 
The primary effect on corneal wave propagation as IOP increases is a decrease in both displacement amplitude and wave propagation distance.5961 This reduction in propagation distance has been reported to introduce variability in wave speed calculations, primarily due to the limited number of pixels available for quantifying wave speed.60 With ACUS-generated waves, which are typically low in amplitude, this variability could become even more pronounced. The variability resulting from reduced propagation distance may explain the increased interquartile range of air-pulse wave speeds at higher IOPs (Supplementary Fig. S1) and could also account for the small differences in wave speed (<1.1 m/s) observed between air-pulse and ACUS stimulation methods in the Bland-Altman plot (Fig. 5) at 30 mm Hg. The Bland-Altman plot also shows fewer data points at lower frequencies (0.7 kHz and 1 kHz) at 30 mm Hg because ACUS stimulations did not generate measurable wave propagation in this frequency range. ACUS is known to generate low-amplitude mechanical waves at low frequencies, primarily because of the acoustic impedance mismatch between the transducer and air.56,57 This is consistent with our previous findings that show a decrease in the spectral magnitude from ACUS stimulation at 1kHz using a transient single pulse.41 It was also shown that increased corneal stiffness at high IOPs further reduced ACUS magnitude and mechanical wave detectability.41 
The speed dispersion results from both the in vivo human cornea and the corneal tissue phantoms show that the A0 mode was dominant when the stimulation was normal to the corneal surface, whereas during oblique stimulation (35° and 65° relative to the tissue surface normal), S0 mode was observed. This change in propagation mode with stimulation angle is characteristic of guided wave propagation,20,23,39 which is due to the sample geometry and should not be affected by stimulation type. These findings are consistent with previous studies evaluating Lamb wave modes in a steel plate.3840 These previous studies demonstrated that selective excitation of A0 mode is possible when the stimulation angle is between 0° and 20°, and S0 mode is dominant when the angle of stimulation is between 35° and 70°. Previous studies evaluating the structural integrity of metal plates have also indicated that the A0 propagation mode has higher sensitivity in detecting lateral abnormalities or sample discontinuities such as cracks. The S0 mode can improve detection sensitivity for axial features such as inclusions or delaminations.25,28,6264 This suggests that selectively exciting A0 and S0 Lamb wave modes in the human cornea in vivo by changing the stimulation angle can be useful in detecting the lateral and axial mechanical abnormalities in the human cornea during corneal pathologies such as keratoconus. Moreover, in a study using composite metal plates, Jiang and colleagues demonstrated that the S0 mode dispersion at low-frequencies depends on the tensile elastic modulus, whereas A0 dispersion at high frequencies depends predominantly on the shear elastic modulus.58 These findings indicate that selectively exciting S0 and A0 modes in the human cornea could be useful in computing tensile and shear elastic moduli respectively. 
The results from the human cornea and corneal phantom experiments show that the mechanical wave speed dispersion of the A0 and the S0 modes converged to Rayleigh speed at higher frequencies. These Rayleigh speeds were useful in improving the repeatability when estimating the elastic modulus. However, a single high-frequency wave speed measurement may not always correspond to the Rayleigh speed, as first-order A1 and S1 modes can also be generated at these frequencies. These first-order modes appear before fundamental modes converge to Rayleigh speeds (Fig. 1) and they can be selectively generated by adjusting the stimulation angle.65 This indicates that, depending on the stimulation paradigm, the mechanical waves propagating in cornea could exhibit the dispersion characteristics of either the fundamental or first-order Lamb modes. As a result, wave speeds measured at higher frequencies may correspond to the Rayleigh wave speed, shear wave speed, or the speed of a higher-order mode before it converges to the shear wave speed. Therefore identifying the propagation mode by analyzing the wave dynamics and measuring speed dispersion at multiple frequencies is essential for determining the Rayleigh wave speed and obtaining precise estimates of corneal elastic modulus. 
The particle velocity maps shown for the A0 mode in the human cornea (Fig. 7) are consistent with the characteristic tissue motion associated with anti-symmetric wave propagation. However, particle velocity maps for the S0 mode do not follow the expected propagation dynamics of the symmetric mode. These results are in agreement with Li and colleagues,54 where S0 and A0 modes were measured simultaneously, and they observed anti-symmetric displacement profiles in the low-frequency range. It is well established from Lamb wave theory that wave propagation dynamics and the displacements profiles for A0 and S0 modes vary with frequency.31,33 At lower frequencies, the sample motion across the thickness for A0 and S0 modes linearly varies and converges to the displacement characteristics of conventional axial and flexural plate waves, respectively.31,33 These displacement characteristics can be clearly observed from A0 and S0 corneal particle velocity maps (at 1 kHz) shown in Figure 7. Furthermore, the corneal particle velocity maps in Figure 7 also indicate that the wavelength of the mechanical wave for the A0 mode was shorter than the S0 mode. This is consistent with Lamb wave theory, which predicts that the A0 wavelength is always shorter than the S0 mode.20,31,33,66 The measured wave speeds for the A0 and the S0 modes are also consistent with the product of the mechanical wave wavelength and the frequency at which the mechanical wave was observed. Furthermore, in the low-frequency range, the S0 mode approximates a conventional axial plate wave, making the Lamb wave speed close to the axial plate wave speed \({{c}_a} = \sqrt {E/\rho } \) . This speed compares well with the Rayleigh wave speed in the high-frequency range, as illustrated in Figure 8. Young's moduli calculated from the speeds of the S0 and A0 modes in the high-frequency range (0.553 ± 0.1 MPa) show good agreement with the Young's modulus estimated from the axial plate wave speed ca in the low-frequency range (0.562 ± 0.15 MPa). This further confirms that the S0 propagation mode is generated in the in vivo human cornea during oblique stimulation. 
Lamb wave theory predicts that the characteristic propagation dynamics for symmetric mode can be observed at frequencies where the wavelength of mechanical wave is similar to the sample thickness.31,32 At these frequencies where the expected propagation dynamics of symmetric mode appear, speed of wave propagation matches closely with the surface (Rayleigh) wave speed.31,32 However, wavelength of corneal mechanical waves at 5 kHz for S0 mode was 2 mm (Fig. 7), which is greater than the corneal thickness (∼0.55 mm). Hence, particle velocity maps for S0 mode at 5 kHz (Fig. 7) do not show the expected propagation dynamics of the symmetric mode. To identify the Rayleigh wave speed in the human cornea, we determined the frequency at which the wavelength of the mechanical wave is similar to corneal thickness. This was performed by fitting the measured exponential decay of wavelength as a function of frequency to a second-order exponential function. This computational fit showed that at 18.6 kHz the wavelength of the corneal mechanical wave would be 0.55 mm, with the Rayleigh wave speed of 10.23 m/s. This frequency closely approximates the results of Li and colleagues,54 who demonstrated that the wave propagation in the human cornea at 16 to 18 kHz appears to have the characteristic propagation dynamics of the symmetric mode. This computed Rayleigh wave speed is comparable to the plateauing wave speeds shown in Figure 7 at 4 kHz (10.43 m/s) and 5 kHz (10.33 m/s). Although wave speeds at 4 and 5 kHz in the in vivo human cornea are close to the Rayleigh speeds, they are still in the Lamb domain and are not, strictly speaking, equivalent to the Rayleigh wave speed. Therefore we refer to the plateauing wave speeds observed at 4 and 5 kHz as the approximate Rayleigh Lamb wave (aRLW) speed hereafter. 
For a young cohort of normal healthy participants, we found that the human corneal aRLW speeds for the A0 mode ranged from 10.8 to 13.6 m/s (at 4 kHz) when measured in vivo with corneal apex stimulation. The human corneal speed dispersion profiles and aRLW speed range for the A0 mode reported here are consistent with those reported by Jin and colleagues.48 In this previous study, Jin and colleagues48 used air-pulse stimulation at the corneal apex and found Rayleigh wave speeds of 12.73 ± 1.46 m/s. Additionally, the in vivo corneal wave speeds measured at lower frequencies in the present work (2.4 to 5.2 m/s at 1 kHz) for the A0 mode matches with the corneal wave speeds (2.5 to 4 m/s) observed in our previous studies of the human cornea in vivo.42 
The human corneal A0 aRLW speeds in this study were also similar to the asymptotic speed measured at 4 kHz for the S0 mode, which ranged from 13.1 m/s to 15.9 m/s. The frequency-dependent decrease in corneal wave speed with increasing frequency (S0 mode) reported in our findings was also observed in a recent in vivo study by Ramier and colleagues,55 who used a vibrational contact probe to stimulate the peripheral cornea at discrete frequencies ranging from 2 to 16 kHz at 2 kHz intervals. The in vivo aRLW speeds for the S0 mode in our study (14.5 ± 1.4 m/s) were higher than those reported by Ramier et al.55 (7.86 ± 0.75 m/s). A key difference between our approach and Ramier's study is that we used broadband mechanical waves to compute phase velocity, whereas Ramier and colleagues55 used discrete frequencies to compute group velocities. This discrepancy in the observed Raleigh wave speeds could be due to the differences between group and phase velocities.20,67 As suggested by Matsuda and Biwa,67 phase and group velocity matching conditions must be applied to obtain similar aRLW speeds. To apply these matching conditions, transverse velocity (CT) should be computed from the measured phase (Cp) and the group (Cg) velocities using \({{C}_T}{\rm{\ }} = \ \frac{{{{C}_p}}}{{\sqrt 2 }}\) and \({{C}_T}{\rm{\ }} = \ \sqrt 2 {{C}_g}\). This matching case is applicable only when the observed Lamb modes are equivalent to Lamé modes,31 which may be true for the cornea, particularly for the S0 mode in x-y plane. Another plausible factor contributing to the difference in measured S0 aRLW speeds between these two studies is the high-frequency mechanical stimulation used by Ramier and colleagues. These high frequencies can also generate higher-order Lamb wave modes in the cornea. The dispersion profile for these higher-order modes asymptotically approaches the shear wave speed at higher frequencies, which is faster than the Rayleigh wave speed.20,23,62 The differences in observed wave speeds could also be due to variations in the spatiotemporal widths of the mechanical stimulators used or the inherent biomechanical variability of the human cornea in each study population. Optimization studies on stimulator geometry should be evaluated as another objective of future translational research. 
Accurately reconstructing the corneal elastic moduli from the measured wave dispersion requires the application of an appropriate mechanical model. To this end, different Lamb wave models describing the wave propagation in the cornea have been proposed. The most commonly used analytical models for the cornea include the modified Rayleigh-Lamb frequency equation (mRLFE) model,21,24 nearly incompressible transverse isotropy (NITI) model,68 and Holzapfel-Gasser-Ogden (HGO) model.54 Our previous work has shown that corneal thickness, corneal curvature and the axial geometrical boundaries (air on one side and fluid on the other) can affect wave speed dispersion, ultimately impacting the estimation of corneal elastic moduli.21,24,69,70 These factors affecting wave speed dispersion were incorporated in our previous mRLFE model and several ex vivo studies have used this model to measure both Young's modulus and the shear viscosity of the cornea.22,41,47,71 However, these previous studies have only observed the A0 propagation mode to compute corneal mechanical properties. Assessing corneal viscoelasticity with the S0 propagation mode with this modified Rayleigh-Lamb model while considering the effects of angle and location of mechanical stimulation continues to be a focus in our research. 
The NITI model proposed by Pitre et al.68 decouples the shear and tensile moduli of the cornea by utilizing a transverse isotropic model. In this model, the axial mechanical properties are anisotropic, while the transverse properties are isotropic. This assumption is important in modeling the mechanical properties of the cornea, particularly given the anatomical ultrastructural properties of corneal stroma. While this model was used to measure the tensile and shear moduli of the ex vivo porcine cornea maintained at low IOPs, there was an observed change in the mode shape at IOPs >20 mm Hg, and the model was unable to account for this change in mode shape. The authors suggested that this change in mode shape is due to the nonlinearity of corneal tissue. Moreover, the authors claimed that exciting A0 and S0 propagation modes in the cornea could provide more accurate estimates of corneal shear modulus, but the S0 mode cannot be effectively generated in corneal tissues due to corneal anisotropy. Our results from the corneal phantom and the in vivo human cornea using oblique sample stimulation, as well as results from Li et al.,54 show that S0 mode can be generated in the cornea. Including oblique sample stimulation (>35°) could generate the S0 mode during corneal wave propagation modeling, potentially leading to better characterization of corneal mechanical properties. 
In a recent study, Li et al.54 demonstrated A0 and S0 propagation modes in the in vivo cornea using a piezoelectric contact probe-based OCE. The authors applied the HGO model for the cornea, which has been used for modeling tissue mechanics in arterial walls.72 This model takes into account the nonlinear collagen stiffening effect that happens at larger strains and, hence, can be used to describe the nonlinear effects of increased IOP on corneal wave propagation. By fitting the A0 and S0 modes to this model and applying appropriate corneal boundary conditions, the authors computed the out-of-plane shear modulus and in-plane tensile modulus of the human cornea in vivo. The authors used a wide range of higher stimulation frequencies (4-16 kHz) to characterize the A0 mode, and the S0 mode was measured using a few discrete high frequencies (12, 14, 16 kHz). With this approach, the authors observed the characteristic propagation dynamics of the symmetric mode and measured decreasing speed for the S0 mode at these frequencies. This high-frequency corneal stimulation can improve spatial resolution in computing tissue mechanical properties73 and may be useful in detecting mechanically abnormal corneal regions in conditions such as keratoconus. As discussed earlier, at these frequencies, where the expected propagation dynamics of the symmetric mode appear, wave propagation speeds closely approximate the Rayleigh wave speed.31,33 Including additional lower frequencies when characterizing S0 mode could further improve reconstruction of the corneal dispersion response and improve the utility of the HGO model for the determination of corneal elastic moduli. 
Results from our experiments demonstrate that measuring speed dispersion and identifying the propagation mode is important when determining the appropriate mechanical model for reconstructing corneal biomechanical properties. The speed dispersion results from the human cornea in vivo indicate that identifying aRLW speeds can improve the repeatability of the corneal Young's modulus measurements. However, a single wave speed measurement at higher frequencies may not be sufficient to correctly identify aRLW speeds for all individuals because the convergence of S0 and A0 propagation modes to Rayleigh wave speeds can be influenced by factors such as corneal curvature, thickness, and IOP, all of which can vary across individuals. Additionally, depending on the bandwidth of the mechanical stimulator, higher-order modes may be generated in the cornea, meaning a single wave speed measurement at higher frequencies could correspond to the Rayleigh wave speed, shear wave speed, or the speed of a higher-order mode before converging to the shear wave speed. Therefore obtaining speed dispersion profiles for each individual is essential to correctly determine Rayleigh speeds and precisely estimate corneal elasticity. Furthermore, corneal biomechanical properties are viscoelastic and spatially heterogeneous, emphasizing that a single mechanical parameter may not adequately characterize the full complexity of corneal biomechanics. In future studies, we will include S0 propagation modes along with the A0 mode in our modified Rayleigh-Lamb model by adding symmetric wave equations from the Lamb wave theory. Because propagation dynamics and the dispersion profiles of the S0 mode at low frequencies are largely governed by tensile modulus and those of the A0 mode at high frequencies are governed by the shear modulus, selectively exciting these propagation modes and fitting the dispersion curves of these fundamental propagation modes with our analytical model in the cornea can be useful in measuring tensile and shear moduli respectively. Quantifying these corneal elastic moduli can provide a more comprehensive understanding of human corneal biomechanical properties in both healthy and diseased conditions, which continues to be the focus of our research. 
Conclusions
This work demonstrates the capability of microliter air-pulse–based OCE for measuring wave speed dispersion in the human cornea in vivo. The results show that using microliter air-pulse-based mechanical wave stimulation results in corneal wave propagation that is dispersive and multimodal, and these properties are well described using established Lamb wave theory. The stimulation angle relative to the corneal surface can selectively excite A0 and S0 Lamb wave propagation modes in the human cornea. Identifying the aRLW speeds of these A0 and S0 Lamb wave modes reduced the variability in computing corneal elastic modulus. However, because the frequency at which the wave speed plateaus is dependent on the stimulation parameters, determining the plateauing speed for a particular stimulation paradigm is important. Understanding the dispersive multimodal characteristics of mechanical waves is essential for high-resolution quantitative mapping of corneal properties in clinical applications. 
Acknowledgments
Supported by the US National Institutes of Health grants R01EY033978, R01EY022362, R01EY034114, and a core grant P30EY07551. 
Disclosure: C. Duvvuri, None; M. Singh, ElastEye LLC (I), US20220386862A1 (P), US11839483B2 (P), US10405740B2 (P); G. Lan, WO2018140703A1 (P); S.R. Aglyamov, US10405740B2 (P); K.V. Larin, ElastEye LLC (I), US20220386862A1 (P), US11839483B2 (P), US10405740B2 (P); M.D. Twa, US11839483B2 (P), WO2018140703A1 (P) 
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Figure 1.
 
Illustration of Lamb wave propagation modes and their characteristic wave speed dispersion. (a) The relative motion of the top and the bottom surfaces are out of phase for antisymmetric Lamb mode and in phase for symmetric mode relative to the sample midplane (dashed line). Rayleigh waves propagate along the surface. (b) Wave speed dispersion for the fundamental Lamb modes (A0 and S0) and higher-order modes (A1, S1, A2, S2). The fundamental modes plateau at Rayleigh wave speed and higher-order modes plateau at shear wave speed. Adapted from Ryden et al.30
Figure 1.
 
Illustration of Lamb wave propagation modes and their characteristic wave speed dispersion. (a) The relative motion of the top and the bottom surfaces are out of phase for antisymmetric Lamb mode and in phase for symmetric mode relative to the sample midplane (dashed line). Rayleigh waves propagate along the surface. (b) Wave speed dispersion for the fundamental Lamb modes (A0 and S0) and higher-order modes (A1, S1, A2, S2). The fundamental modes plateau at Rayleigh wave speed and higher-order modes plateau at shear wave speed. Adapted from Ryden et al.30
Figure 2.
 
Schematic of the clinical OCE system. Microliter air-pulse and ACUS were used as mechanical stimulation methods. The stimulators (green boxes) provided localized tissue excitation and were not used simultaneously. Phase-sensitive OCT (blue boxes) tracked the resulting dynamics of elastic wave propagation in the samples.
Figure 2.
 
Schematic of the clinical OCE system. Microliter air-pulse and ACUS were used as mechanical stimulation methods. The stimulators (green boxes) provided localized tissue excitation and were not used simultaneously. Phase-sensitive OCT (blue boxes) tracked the resulting dynamics of elastic wave propagation in the samples.
Figure 3.
 
Frequency-filtering for wave propagation analysis. (a) Power spectrum for air-pulse excitation stimulus in ex vivo rabbit corneal tissue. (b) The filtered temporal response using the finite impulse response (FIR) filter. Red vertical lines indicate the bandwidth of the filter (1 kHz ± 200 Hz). (c) Power spectrum response for the 1 kHz ACUS in ex vivo rabbit corneal tissue and (d) filtered temporal displacement response.
Figure 3.
 
Frequency-filtering for wave propagation analysis. (a) Power spectrum for air-pulse excitation stimulus in ex vivo rabbit corneal tissue. (b) The filtered temporal response using the finite impulse response (FIR) filter. Red vertical lines indicate the bandwidth of the filter (1 kHz ± 200 Hz). (c) Power spectrum response for the 1 kHz ACUS in ex vivo rabbit corneal tissue and (d) filtered temporal displacement response.
Figure 4.
 
Comparison of mechanical wave propagation features at 1 kHz (a, b), 3 kHz (c, d), and 5 kHz (e, f) in an ex vivo rabbit cornea (IOP = 10 mm Hg) induced by air-pulse and ACUS stimulation. Air-pulse and ACUS stimulation frequencies were bandpass filtered (±200 Hz) at each frequency. Similar displacement profiles and particle velocity fields were observed at each frequency (note lower amplitude for ACUS). White horizontal rule with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity in depth.
Figure 4.
 
Comparison of mechanical wave propagation features at 1 kHz (a, b), 3 kHz (c, d), and 5 kHz (e, f) in an ex vivo rabbit cornea (IOP = 10 mm Hg) induced by air-pulse and ACUS stimulation. Air-pulse and ACUS stimulation frequencies were bandpass filtered (±200 Hz) at each frequency. Similar displacement profiles and particle velocity fields were observed at each frequency (note lower amplitude for ACUS). White horizontal rule with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity in depth.
Figure 5.
 
Bland-Altman analysis shows good agreement and minimal bias between the corneal wave propagation speeds obtained using air-pulse and ACUS stimulations, measured in four ex vivo rabbit eyes at multiple frequencies (0.7 kHz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz) and at IOPs of 10, 20, and 30 mm Hg. The wave speed differences were color-coded by frequency, with distinct shapes representing the different IOP levels. Solid lines represent the upper and lower 95% limits of agreement, and the dashed lines represent the mean difference across all tested IOPs and frequencies.
Figure 5.
 
Bland-Altman analysis shows good agreement and minimal bias between the corneal wave propagation speeds obtained using air-pulse and ACUS stimulations, measured in four ex vivo rabbit eyes at multiple frequencies (0.7 kHz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz) and at IOPs of 10, 20, and 30 mm Hg. The wave speed differences were color-coded by frequency, with distinct shapes representing the different IOP levels. Solid lines represent the upper and lower 95% limits of agreement, and the dashed lines represent the mean difference across all tested IOPs and frequencies.
Figure 6.
 
Lamb wave mode for the 500 µm thick corneal phantom was determined by the stimulation angle, not the stimulation location. Particle velocity maps for normal stimulation angle (a, b) resembled A0 mode irrespective of stimulation location (center or periphery). (c, d) Particle velocity maps for oblique stimulation angles (35° or 65°) resembled S0 mode. The speed dispersion profiles (e) in corneal phantom also resembled A0 mode for normal stimulation angles and S0 mode for oblique stimulation angles. For the 10 mm thick sample, wave propagation was similar regardless of whether the phantom was stimulated at a normal stimulation angle (f, g) or at oblique angles (h, i) and whether the stimulation was at the center (f, h) or periphery (g, i). The thick sample dispersion plots (j) were similar across all stimulation conditions. The gray-shaded region in the particle velocity maps represents the air-pulse stimulator location— where stimulation was performed at the center of the gray-shaded region. The stimulation location in each panel is indicated as center or periphery. White horizontal rule (mm) with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity along depth axis.
Figure 6.
 
Lamb wave mode for the 500 µm thick corneal phantom was determined by the stimulation angle, not the stimulation location. Particle velocity maps for normal stimulation angle (a, b) resembled A0 mode irrespective of stimulation location (center or periphery). (c, d) Particle velocity maps for oblique stimulation angles (35° or 65°) resembled S0 mode. The speed dispersion profiles (e) in corneal phantom also resembled A0 mode for normal stimulation angles and S0 mode for oblique stimulation angles. For the 10 mm thick sample, wave propagation was similar regardless of whether the phantom was stimulated at a normal stimulation angle (f, g) or at oblique angles (h, i) and whether the stimulation was at the center (f, h) or periphery (g, i). The thick sample dispersion plots (j) were similar across all stimulation conditions. The gray-shaded region in the particle velocity maps represents the air-pulse stimulator location— where stimulation was performed at the center of the gray-shaded region. The stimulation location in each panel is indicated as center or periphery. White horizontal rule (mm) with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity along depth axis.
Figure 7.
 
Stimulation angle determines Lamb wave mode and speed dispersion dynamics in the human cornea in vivo. (ad) Frequency filtered surface displacements (1 kHz and 5 kHz) and particle velocity fields at normal and oblique simulation angles. (a, c) Normal corneal stimulation produced (e) A0 mode dispersion (speed increases with frequency). (b, d) Oblique stimulation angles produced (f) S0 mode dispersion (speed decreases with frequency) and anti-symmetric propagation dynamics. Figure insets (e and f) show tissue stimulation location (*) relative to the OCE scan location (red line). White horizontal rule (mm) with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity along depth axis.
Figure 7.
 
Stimulation angle determines Lamb wave mode and speed dispersion dynamics in the human cornea in vivo. (ad) Frequency filtered surface displacements (1 kHz and 5 kHz) and particle velocity fields at normal and oblique simulation angles. (a, c) Normal corneal stimulation produced (e) A0 mode dispersion (speed increases with frequency). (b, d) Oblique stimulation angles produced (f) S0 mode dispersion (speed decreases with frequency) and anti-symmetric propagation dynamics. Figure insets (e and f) show tissue stimulation location (*) relative to the OCE scan location (red line). White horizontal rule (mm) with 0.5 mm markings. Vertical scale bar: 0.5 mm. Color bar represents particle velocity along depth axis.
Figure 8.
 
(a) In vivo human corneal Lamb wave speed dispersion for all 36 subjects (mean ± SD of the intrasubject mean; n = 3) with stimulation normal to the corneal surface (blue) and at oblique angles (red). Solid line represents the mean and shaded region represents SD. (b) Per-subject Young's modulus computed from the mean wave speeds at specific frequencies (1–5 kHz at 1 kHz interval) where blue represents stimulation normal to the corneal surface and red represents oblique stimulation. The median value (50th percentile) of the dataset is represented by the center line in each box, whereas the box indicates the interquartile range (25th to 75th percentiles). The whiskers extend to the fifth and 95th percentiles and the individual data points are indicated by circles.
Figure 8.
 
(a) In vivo human corneal Lamb wave speed dispersion for all 36 subjects (mean ± SD of the intrasubject mean; n = 3) with stimulation normal to the corneal surface (blue) and at oblique angles (red). Solid line represents the mean and shaded region represents SD. (b) Per-subject Young's modulus computed from the mean wave speeds at specific frequencies (1–5 kHz at 1 kHz interval) where blue represents stimulation normal to the corneal surface and red represents oblique stimulation. The median value (50th percentile) of the dataset is represented by the center line in each box, whereas the box indicates the interquartile range (25th to 75th percentiles). The whiskers extend to the fifth and 95th percentiles and the individual data points are indicated by circles.
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