Air-pulse optical coherence elastography: how excitation angle affects mechanical wave propagation

Abstract

We evaluate the effect of excitation angles on the observation and characterization of surface wave propagations used to derive tissue’s mechanical properties in optical coherence tomography (OCT)-based elastography (OCE). Air-pulse stimulation was performed at the center of the sample with excitation angles ranging from oblique (e.g., 70° or 45°) to perpendicular (0°). OCT scanning was conducted radially to record en face mechanical wave propagations in 360°, and the wave features (amplitude, attenuation, group and phase velocities) were calculated in the spatiotemporal or wavenumber-frequency domains. We conducted measurements on isotropic, homogeneous samples (1–1.6% agar phantoms), anisotropic samples (chicken breast), and samples with complex boundaries, coupling media, and stress conditions (ex vivo porcine cornea, intraocular pressure (IOP): 5–20 mmHg). Our findings indicate that mechanical wave velocities are less affected by excitation angles compared to displacement features, demonstrating the robustness of using mechanical waves for elasticity estimations. Agar and chicken breast sample measurements showed that all these metrics (particularly wave velocities) are relatively consistent when excitation angles are smaller than 45°. However, significant disparities were observed in the porcine cornea measurements across different excitation angles (even between 15° and 0°), particularly at high IOP levels (e.g., 20 mmHg). Our findings provide valuable insights for enhancing the accuracy of biomechanical assessments using air-pulse-based or other dynamic OCE approaches. This facilitates the refinement and clinical translation of the OCE technique and could ultimately improve diagnostic and therapeutic applications across various biomedical fields.

1. Introduction

Soft tissue biomechanics, such as stiffness, elasticity, and viscosity, are vital indicators of tissue health, aging, and function. Pathological changes, including inflammation, swelling, and tumors, often trigger changes in the biomechanical properties at the cellular, tissue, or organ level [1,2]. In clinical settings, changes in tissue biomechanics have become a hallmark for diagnosing various diseases. For example, liver fibrosis is associated with cirrhosis [3], while arterial wall stiffness is connected to systemic inflammation and atherosclerosis in hypertension [4]. Breast tissue fibrosis and tumor formations have different stiffness compared to normal or benign tissue formations [5]. In ophthalmology, there is a growing interest to understand ocular biomechanics as it has implications for various ocular conditions including keratoconus [6,7], glaucoma [8], and myopia [9,10], as well as a potential risk of iatrogenic corneal ectasia [11], which is a severe complication caused by the reduction of the tensile strength of the cornea post-refractive surgery [12]. Elastography methods utilizing tissue stimulation and non-invasive high-resolution imaging techniques are actively being developed to objectively and quantitively detect soft tissue biomechanics caused by diseases, making it a dynamic and evolving field of study in clinics.

Optical coherence elastography (OCE) [13] is an innovative elasticity imaging technique combining a stimulation sub-system to generate displacements (strain) or elastic waves in tissues and a high-resolution optical coherence tomography (OCT) sub-system to record the tissue's mechanical response. Tissue properties (e.g., elastic modulus [14], viscoelasticity [15,16], and natural frequency [17–19]) can be then derived or estimated from the applied deformation force and observed tissue response. The development of phase-sensitive OCT technique [20–23] has enhanced displacement detection sensitivity from the micrometer (structure/intensity detection) to nanometer or sub-nanometer scale (phase detection). Previous research demonstrated sub-nanometer displacement detection sensitivities [24], enabling high-resolution quantification of tissue displacement [17] and the visualization and analysis of laterally propagating elastic waves in dynamic OCE [25,26]. As OCE has been developed for various applications, numerous stimulation strategies have also been introduced, such as (quasi-)static or dynamic bulk compression [27], needle probe compression [28], magnetomotive [29,30], nano-bomb [31], audio sound [32], pulsed laser (photothermal excitation) [33], ultrasound [34,35] and air-coupled ultrasound [36–38]-induced acoustic radiation force (ARF), air puff/pulse indentation [25,39–41], and passive sources like heartbeat-induced pulsation signals [42,43]. These loading strategies can be classified into various categories, including active or passive loading, contact or non-contact loading, (quasi-)static or dynamic loading, internal or external loading, and local or global loading methods [44]. Among them, the microliter air-pulse stimulation method [39] is a non-contact, highly spatialized and low-force technique capable of providing transient (e.g., millisecond scale for the pulse duration) and broadband frequency (e.g., ∼0–1 kHz for a 3-ms duration pulse) excitation, inducing micrometer to sub-nanometer range displacement magnitudes within tissue. Thereby, it has been considered as an appropriate method for biomechanical property detection of delicate tissue such as the cornea [44]. Ex vivo measurements using the microliter air-pulse OCE technique in rabbit and porcine corneas have revealed that corneal stiffness increases following corneal collagen crosslinking [45,46] at higher intraocular pressure [47] and with older age. Furthermore, the technique has been employed in recent in vivo studies of human corneal biomechanical properties, resulting in successful shear wave propagation tracking [25] and natural frequency distribution characterization [48].

Given the physical constraints imposed by the bulk of the loading system and limited space between the OCT scan lens and the sample, most reported OCE systems have to stimulate the tissue obliquely, even though perpendicular stimulation is considered optimal for providing evenly distributed mechanical waves and simplifying the modeling methods used to derive mechanical properties from the observed response [49]. This raises several critical questions: How does oblique or perpendicular tissue excitation affect mechanical wave propagation in bulk or layered tissues, or those with complex interfaces and boundaries? What are the effects on isotropic, homogeneous samples versus anisotropic, heterogeneous samples? How would these excitations influence the observation and calculation of group or phase velocities used to derive tissue’s mechanical properties? These questions are crucial for accurately interpreting mechanical properties from observed mechanical propagation features, especially under oblique tissue excitation. Unfortunately, we still lack definitive answers to these questions [49].

For air-pulse OCE applications, researchers typically used straight-type air cannulas [39,49,50] to stimulate samples at an oblique angle, typically 45°, as shown in Fig. 1(a). Recent studies have also employed 45° [17,25,48] or 90°-angled [15,51–53] curved air cannulas to enable perpendicular sample stimulation, as illustrated in Figs. 1(b) and (c). In fact, the air-pulse OCE system with a curved air cannula is one of the few systems capable of achieving perpendicular tissue excitation without significantly obstructing OCT scanning beams. In addition, by rotating the 90°-angled curved cannula, we also gain the flexibility to achieve tissue excitation at any desired angle, making the air-pulse OCE system suitable for evaluating the effects of excitation angle (the angle between the stimulation direction and the sample's normal) on mechanical wave propagation. In this paper, we conducted a comprehensive study to evaluate the effect of excitation angle on mechanical wave propagation features in various types of samples. Utilizing a pattern of central air-pulse stimulation and radial OCT scanning, we performed OCE measurements of en face surface wave propagations in 360° directions, quantifying features such as magnitudes and attenuation of wave fronts, group and phase velocities, in both spatial-temporal and wavenumber-frequency domains. Additionally, we evaluated the roundness, ellipticity, and fractional anisotropy of these features in 360° directions. For the measurement experiments, we set up 12 groups of excitation angles (70°–30° with 5° reductions and 30°–0° with 10° reductions) to assess directional wave propagation features in isotropic, homogeneous samples (1%, 1.3%, and 1.6% agar phantoms). This allowed us to study the effect of excitation angles on samples with different stiffness. We then analyzed the impact of excitation angles (from 45° to 0°) on different types of anisotropic, heterogeneous samples, including chicken breast to represent samples with fibrous structures and preferred alignment directions, and ex vivo porcine corneas to represent tissues with layered structures, complex air-water boundaries, and different intraocular pressure (IOP) levels. This study systematically examines the influence of excitation angles on mechanical wave propagation features across a wide range of sample types and conditions. The insights gained from this research could enhance the accuracy of biomechanical assessments using air-pulse-based or other dynamic OCE approaches. Our goal is to continually refine dynamic OCE techniques to facilitate clinical translations and improve diagnostic and therapeutic methods across various applications.

Fig. 1.

Three types of air cannula utilized for air-pulse stimulation in OCE. (a) straight-type air cannula; (b) 45°-angled curved air cannula; (c) 90°-angled curved air cannula.

2. Methods

2.1. OCE system

Figure 2(a) shows the layout of the air-pulse OCE system. Detailed explanations of the OCT/OCE system are available in our previous publications [52]. Briefly, a transient, low-pressure air pulse was delivered to the sample via an air cannula controlled by a solenoid valve, inducing mechanical wave propagations. The OCT system was synchronized with the air-pulse stimulation system to record the dynamic response. The OCT system [54] was equipped with a superluminescent diode (SLD, IPSDS1307C-1311, Inphenix Inc., Livermore, CA, USA) light source with a wavelength of 1290 ± 40 nm. The light emitted from the SLD was split into a sample arm and a reference arm by 50:50. In the sample arm, the light beam was collimated to a diameter of 4 mm, scanned by two-dimensional galvo mirrors, and focused by a telecentric scan objective (LSM54-1310, Thorlabs Inc., Newton, NJ, USA) with a focal length of 54 mm and an 18.8 × 18.8 mm² field of view. The maximum output power at the sample surface was 1.8 mW. Interference signals from the sample and reference arms were recorded by a linear-wavenumber (k) spectrometer (PSLKS1300-001-GL, Pharostek, Rochester, Minnesota, USA) equipped with an InGaAs line scan camera (GL2048L-10A-ENC-STD-210, Sensors Unlimited, Inc., Princeton, NJ, USA) at a maximum line rate of 76 kHz. The linear-k spectrometer disperses the spectrum optically in the k domain using an optimized combination of a grating and a prism, ensuring an improved signal-to-noise ratio for deeper tissue and reducing the computing time to generate OCT images [55]. The interference signals were sent through a frame grabber (PCIE-1433, National Instruments Corp., Austin, Texas, USA) into a computer and processed directly via Fourier transform to acquire depth profiles (A-scans) using code written in LabVIEW, without any digital interpolation.

Fig. 2.

Experiment set-up. (a) Schematic of the air-pulse OCE system that comprised of microliter air-pulse stimulation and high-resolution phase-sensitive OCT detection. SLD: superluminescent laser diode with a waveband of 1290 ± 40 nm. A linear-wavenumber (k) spectrometer disperses the interference spectrum in the k-domain prior to Fourier transform processing for OCT. (b) Geometry of angled stimulation and radial OCT scanning. Air pulse stimulation was performed at the center of the sample, with the excitation angle ( $\gamma$ ) defined as the angle between the air pulse and the sample's normal, typically aligned with the optical axis. An angular space of ±18° was reserved for locating the canula tip. The OCT scan was performed in 20 radial directions, from 18° to 342°, with an interval angle of 18°.

2.2. Angled stimulation, radial scanning, and M-B mode measurement protocols

To assess the impact of excitation angles on 360° wave propagation, we employed a central stimulation and radial scanning OCE measurement geometry (Fig. 2(b)). A 90°-angled curved air cannula delivered air pulses to the sample’s center and was rotated to vary the excitation angle ( $\gamma$ ), defined as the angle between the stimulation direction and the sample’s normal (aligned with the OCT optical axis). Air pressure (100–2000Pa) was adjusted to ensure uniform displacement magnitudes across samples of varying stiffness and excitation angles. OCT imaging covered 20 radial directions (0°–360°, spaced at 18° intervals), with 17 spatial sampling points per direction. The cannula tip was carefully aligned to the sample center during rotation. As the cannula rotated, it blocked several measurement points from the OCT imaging (Fig. 2(b), bottom). These blocked data points, varying with rotation angle, were later interpolated from neighboring data during data processing to preserve spatial continuity. Elastic wave propagation was tracked using M-B mode imaging (a hybrid of M-mode depth-resolved and B-mode cross-sectional imaging) to capture time-resolved surface displacements. Air pulses were applied every 0.5 s, synchronized with OCT acquisition to record 800 A-scans over 10.5 ms per measurement point, enabling high-temporal-resolution displacement profiling.

The temporal profile of the induced tissue response was recorded at each measurement point using the phase-sensitive detection technique. The surface displacement $\mathrm{\Delta }Z(t_j-t_0)$ can be derived from the unwrapped phase change $\mathrm{\Delta }\phi (t_j-t_0)$ at the time $t_j$ relative to the time $t_0$ in the successive A-scan signals, as

$$\mathrm{\Delta }Z(t_j-t_0)=\frac{{\lambda }_0}{4\pi }\mathrm{\Delta }\phi (t_j-t_0)$$ (1)

where ${\lambda }_0$ is the center wavelength of OCT detection. Since we only access surface wave propagation, we do not need to consider the refractive index of the tissue.

2.3. Samples and measurement parameter setting

We performed OCE measurements on three sample types: (1) pure agar phantoms (1.0%, 1.3%, and 1.6% concentrations) to represent homogeneous soft tissues with different elasticities/stiffnesses, (2) chicken breast sample to quantify anisotropic mechanical behavior, and (3) ex vivo porcine cornea to evaluate the IOP effects (5–20 mmHg). OCE parameters were optimized per sample group (Table 1). For agar phantoms, excitation angles were systematically reduced from 70° to 0° (5° increments between 70°–30°, 10° increments between 30°–0°), enabling comprehensive characterization of directional wave propagation in isotropic media. For biological tissues (chicken breast and cornea), excitation angles were restricted to 45° to 0°, focusing on more practical ranges as applied in most studies [39,49,50], reducing total data acquisition time and minimize dehydration-induced mechanical artifacts, a critical consideration for corneal measurement. Radial scanning spanned 0.93–2.80 mm (17 points per direction) for flat samples (agar, chicken breast) and 0.37–1.87 mm for the curved corneal surface.

Table 1. OCE parameter settings and sample grouping.

Sample	Grouping Parameters	Excitation Angle Range	Radial Distance Range
Agar Phantom	Concentration: 1%, 1.3%, 1.6%	70° → 0°	0.93–2.80 mm
Chicken Breast	Fiber orientation: vertical, tilted, parallel	45° → 0°	0.93–2.80 mm
Ex vivo Cornea	IOP: 5 mmHg, 10 mmHg, 15 mmHg, 20 mmHg	45° → 0°	0.37–1.87 mm

The agar samples were freshly prepared before measurement, following the procedure described previously [17,19]. We mixed agar powder (Biowest agarose 111860, XHLY company Ltd., Beijing, China) and distilled water by weight according to the desired concentrations. The mixture was sealed to prevent evaporation, heated to 220°C, and stirred at 450 rpm until the agar powder was fully dissolved. After cooling the solution to 80°C, it was poured into a Petri dish with outer and inner diameters of 35 mm and 32 mm, respectively, with an inner height of 12.5 mm, and with a net weight of 8.54 g. all the phantoms were made with their height equal to the Petri dish’s height. The agar phantom was refrigerated for 2 hours until fully solidified, then left at room temperature for 30 minutes before OCE measurement. The densities of these pure agar phantoms were 1.113 kg/m³ (1.0%), 1.115 kg/m³ (1.3%), and 1.118 kg/m³ (1.6%), respectively, and the pure weight of the phantoms was ∼11.2 g. The agar phantom remained inside the Petri dish during OCE measurement.

We obtained fresh chicken breast from live chickens at the local market. A section with clear fiber orientation was cut into an 8 mm thick, ∼30 mm diameter piece to fit the Petri dish. This piece was placed atop a nearly solidified 1% agar phantom within the dish. Agar solution filled peripheral gaps, stabilizing the sample. After solidification, measurements were performed at the central region (1.86–5.6 mm diameter) to ensure characterization of the chicken breast alone, avoiding agar interference. The entire OCE measurement procedure was completed within 4 hours post-mortem.

Fresh porcine eyes were obtained from the local market. We carefully removed any remaining eye muscles from each specimen and secured the eye globe in a 3D-printed holder. An automatic infusion syringe pump (CTN-W100, Chengyitong Co., Ltd., Beijing, China) was used to inject saline solution into the vitreous chamber of the porcine eye. A digital pressure gauge (YB-80A, XSENR, Suzhou, China) was employed to monitor IOP (5–20 mmHg) during OCE measurement [19]. The entire OCE measurement procedure was completed within 12 hours post-mortem.

2.4. Group and phase velocities

In wave-based OCE measurements (see Ref. [56] for more details), the precise measurement of mechanical wave propagation speed is fundamental before utilizing a specific analytical model (e.g., shear, Rayleigh, or Lamb waves, depending on the coupling media and boundary conditions) to interpret mechanical properties from the measured wave speeds. Typically, mechanical waves travel faster in stiffer media and slower in softer media. In this section, we begin with the shear wave model as an example to demonstrate the characterization of wave propagation speed in OCE.

The elasticity of a material, Young's modulus ( $E$ ), is determined by the ratio of stress to strain. Soft tissues are commonly classified as incompressible media (Poisson's ratio $\upsilon \approx 0.499$ ). In this case, Young’s modulus E can be represented by the shear modulus $\mu$ as [57]

$$E=2\mu (1+\upsilon )\approx 3\mu .$$ (2)

In an incompressible, homogeneous, and isotropic medium, the shear wave velocity $V_{shear}$ is directly correlated with the mechanical properties of the tissue. The relation between the shear modulus $\mu$ and the shear wave velocity $V_{shear}$ is

$$\mu =\rho V_{shear}^2,$$ (3)

where $\rho$ is the density. Therefore, the elastic properties ( $\mu$ and $E$ ) of the soft tissues can be determined by generating a shear wave within the tissue and measuring its propagation speed, as proposed in shear wave elasticity imaging modalities.

Group velocity refers to the speed at which the envelope of a wave packet or group of waves (i.e., wavefront) travels through a medium. It is essentially the velocity at which the overall shape of the waves’ amplitudes propagates. Thereby, the first and most common method to measure group velocity is to directly calculate it from the wave propagation distance ( $\mathrm{\Delta }X$ ) and time ( $\mathrm{\Delta }T$ ) in the spatial-temporal domain, as [25]

$$V_{group\mathrm{\_}1}=\frac{\mathrm{\Delta }X}{\mathrm{\Delta }T}.$$ (4)

This method is known as the time of flight (TOF) method. For multiple measured points at different distances and their corresponding measurement times, we can simply use linear fitting or cross-correlation methods to calculate $V_{group\mathrm{\_}1}$ .

Due to the viscoelasticity of the soft-tissue, the shear and elastic moduli can be frequency-dependent and written in complex forms. The shear modulus can be represented as

$$\tilde{\mu }(\omega )={\mu }_s(\omega )+i{\mu }_l(\omega ),$$ (5)

where ${\mu }_s(\omega )$ and ${\mu }_l(\omega )$ are the storage and loss moduli, respectively; and $\omega$ is the angular frequency ( $\omega =2\pi f,f:$ linear frequency). Thereby, the shear wave speed and shear modulus in Eq. (3) can be represented using a complex form

$${\tilde{V}}_{shear}(\omega )=\sqrt{\frac{\tilde{\mu }(\omega )}{\rho }}.$$ (6)

The wave number $(k=\omega /V_{shear})$ can also be written in a complex form

$$\tilde{k}(\omega )=\frac{\omega }{{\tilde{V}}_{shear}(\omega )}=\frac{\omega }{\sqrt{\tilde{\mu }(\omega )/\rho }}=\beta (\omega )-i\alpha (\omega )=\frac{\omega }{V_{Phase}(\omega )}-i\alpha (\omega ),$$ (7)

The real part $\beta (\omega )$ can be used to calculate the frequency-dependent phase velocity $V_{Phase}(\omega )$

$$V_{Phase}(\omega )=\frac{\omega }{\beta (\omega )}.$$ (8)

The second method to calibrate group velocity can be represented using a differential form [58]

$$V_{group\mathrm{\_}2}=\frac{d\omega }{d\beta (\omega )}.$$ (9)

The imaginary part $\alpha (\omega )$ of Eq. (7) represents the attenuation factor of the shear wave with distance. Ignoring the influence of the excitation profile, a shear wave traveling in the + x direction can be represented in complex exponential form as follows:

$$u(t,x)=\underset{Attenuation}{\underbrace{A{e}^{-\alpha (\omega )x}}}\underset{Velocity}{\underbrace{e^{j[\omega t-\beta (\omega )x]}}},$$ (10)

where $u(t,x)$ is displacement, A is the amplitude, and j is the imaginary unite. Equation (10) illustrates both the attenuation and the wave propagation speed features. Similar to the calculations of phase and group velocities, we can either calculate the attenuation feature in the frequency domain or consider the attenuation of the wavefront amplitudes for the group waves in the temporal domain.

Shear wave theory provides clear insights into the mechanical properties of homogeneous organs or tissues, such as the liver or kidney, because the induced waves are less affected by boundaries. However, in complex tissues with layered structures like the cornea or arterial walls, Lamb wave theory is more appropriate for evaluating dispersive, guided waves that propagate along the surface and through the thickness of thin, plate-like structures. Recent studies have demonstrated the use of OCE systems to measure the two fundamental guided waves of Lamb waves: symmetric (S₀) and antisymmetric (A₀) modes [14,59]. The S₀ wave, or symmetric zero-order mode, involves particle motion that is symmetric about the mid-plane of the structure, creating compressional and extensional movements. In contrast, the A₀ wave, or antisymmetric zero-order mode, features particle motion that is asymmetric, resulting in flexural or bending movements. For thin elastic plates in a vacuum, a low-frequency analytical approximation for the phase velocity of the A₀ mode is expressed as $V_{A_0}(\omega )=\sqrt{\omega h{V}_{shear}}/\sqrt[4]{3}$ , where h represents thickness. In the case of soft tissues surrounded by liquid media, such as the arterial wall or cornea, an empirical analytical formula that approximates the phase velocity of a leaky Lamb wave can be expressed as $V_{A_0}(\omega )=\sqrt{0.5\omega h{V}_{shear}}/\sqrt[4]{3}$ [60]. Regardless of the analytical models used, the primary task of wave-based OCE imaging is to measure the mechanical waves precisely with high temporal, spatial, and frequency resolutions. This paper focuses on measuring mechanical waves and evaluating the effect of excitation angles on measurement results without converting the waves into the elastic modulus using any analytical models.

2.5. Mechanical wave quantification methods

Using central stimulation and a radial scanning pattern (Fig. 2(b)), we can measure 360° mechanical wave propagations and quantify wave propagation metrics (i.e., group velocity, phase velocity, amplitude, and attenuation) in both temporal-spatial and wavenumber-frequency domains. The quantification method is demonstrated in Fig. 3, using perpendicular air-pulse stimulation ( $\gamma$  = 0°) to measure a 1.3% agar phantom sample. Measurements were performed in 360° with radii ranging from 0.93 to 2.80 mm.

Fig. 3.

Demonstration of en face surface wave quantification methods in spatial-temporal and wavenumber-frequency domains. Sample: 1.3% agar phantom. (a) Spatial-temporal profile of surface wave propagation in the 180° direction, with peak displacement marked as the wavefront. A window with the width of FWHM (full width at half maximum) for the average wave envelope was used for 2D-FFT for (d). (b, c) Spatial-temporal analysis method: (b) wavefront amplitude decay over distance, and (c) group velocity calculation using linear fitting of the peak displacements ( $V_{group\mathrm{\_}1}$ , Eq. (4)). (d–f) Wave propagation analysis in the wavenumber-frequency domain: (d) Normalized real component ( $\beta (\omega )$ ) of the wavenumber ( $\tilde{k}(\omega )$ ), acquired using 2D-FFT of the windowed wave envelope in (a). Red and green curves indicate the slops of $\beta (\omega )$ for A₀ and S₀ modes, respectively. (e, f) Dispersion curves (A₀, S₀) for phase and group velocities, respectively. The A₀ curves were used to represent phase velocity ( $V_{Phase}(\omega )$ , Eq. (8)) and group velocity ( $V_{group\mathrm{\_}2}$ , Eq. (9)). (g) 360° en face wave propagation profile over time. See Visualization 1 ^{(45.1MB, avi)} in the Supplementary Material. (h) Wavefront map. (i–k) Spatial-temporal analysis results: (i) Amplitude, (j) attenuation, and (k) group velocity 1 in 360° directions. (l–n) Wave propagation analysis in the wavenumber-frequency domain: (l) Frequency-dependent phase velocity for A₀ mode, (m) averaged phase velocity for A₀ mode, and (n) Averaged group velocity 2 for A₀ mode.

Figure 3(a) shows typical displacement profiles along a wave propagation direction (180° as illustrated in Fig. 2(b)). Primary deformation was observed as the displacement increased from baseline to the maximum negative displacement and then recovered. The primary deformation peaks, denoted as wavefronts, were used for surface wave quantification in the spatial-temporal domain (Figs. 3(b, c)). In Fig. 3(b), the displacement amplitude decreased from -1.88 µm to -0.54 µm. We modified the first component of Eq. (10) to $u(t,x)=A{e}^{-\alpha (x-0.93mm)}$ , simplifying $\alpha$ as a constant for this frequency range. The fitted values were -1.80 μm ( $A$ ) and 0.70 ( $\alpha$ ) with a R² of 0.98. Figure 3(c) shows the linear fitting between the time delay and peak magnitudes of the wavefronts, where the slope represents the group velocity of the surface wave along the 180° direction of the sample, measured as 4.47 m/s (95% confidence interval: 4.24–4.70 m/s, R² = 0.99).

Figures 3(d–f) illustrate the surface wave quantification methods in the wavenumber-frequency domain. Figure 3(d) displays the normalized real component $\beta (\omega )$ of the wavenumber $\tilde{k}(\omega )$ , obtained using the 2D-FFT of the spatial-temporal profile from the windowed wave envelopes in Fig. 3(a). We defined the window's width as the average full width at half maximum (FWHM) of the primary wave envelopes across the measurement distance, to maintain mechanical contrast and reduce noise and artifacts. A frequency range of 0–1000 Hz and a wavenumber range of 0–600 m⁻¹ were selected to capture the maximum negative and positive values for characterizing the dispersive wave features. We set a threshold of 0.3 times the maximum positive/negative amplitudes to isolated regions of interest. The negative and positive regions were then segmented using contour lines, followed by primary component analysis (PCA) to define the directions of each enclosed region. Subsequently, these primary components were connected to calculate the phase velocities using Eq. (8). The characterization results of phase velocity, shown in Fig. 3(e), indicate that the negative and positive values correspond well to the antisymmetric mode (A₀) and the symmetric mode (S₀) in the Lamb wave model, respectively. Since the wavenumber ( $k$ ) is much smaller than 600 m⁻¹, we can estimate that the wavelength of the surface waves is greater than 10.4 mm. This fits the Lamb wave condition, where the sample's thickness is comparable to or smaller than the wavelengths of the mechanical waves, causing the waves to be guided by the boundaries and exhibit dispersion in frequency. Figure 3(f) presents the characterized group velocities ( $V_{group\mathrm{\_}2}$ ) from the A₀ and S₀ dispersion curves, determined using the differential equation in Eq. (9). Notably the group velocities remained more consistent than phase velocities for both the A₀ and S₀ curves. In comparison (Figs. 3(e, f)), the A₀ mode predominantly governs wave propagation in lower-frequency ranges—characterized by a relatively uniform dispersion profile—whereas the S⁰ mode governs higher-frequency components and exhibits significant dispersion variability (Fig. 3(e)). To ensure analytical consistency, we selected the relatively flat portion of the A₀ mode curve (frequency range: 30–383 Hz) to calculate the average values of ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ , then compared these metrics across all directions.

Figure 3(g) displays the en face surface wave propagation profiles in a 4-dimensional form. The color map indicates displacement values (µm), while the horizontal plane represents a polar coordinate for wave propagation angles and distances, and the vertical axis represents time. Due to the perpendicular stimulation and the homogeneity of the testing sample, a highly symmetric wave propagation pattern was observed. The corresponding wave propagation video is provided in the Supplementary Material (Visualization 1 ^{(45.1MB, avi)} ). Notably, the wavefronts of the mechanical waves were located in the time frame of 4.97 ms to 5.44 ms. Figure 3(h) highlights the wavefronts with red dots (directly measured values) and green dots (interpolated values) and plots them in a 3-dimensional form. The analysis results in the spatiotemporal domain are presented in Figs. 3(i–k), showing the amplitudes (Fig. 3(i)), attenuations (Fig. 3(j)), and group velocities ( $V_{group\mathrm{\_}1}$ , Fig. 3(k)) in 360° directions. Similarly, Figs. 3(l–n) display the 360° quantification results in the wavenumber-frequency domain, with the frequency-dependent phase velocity of the A₀ mode (Fig. 3(l)), the averaged phase velocity ( ${\overline{V}}_{phase}$ ) of the A₀ mode in the frequency range of 30–383 Hz (Fig. 3(m)), and the corresponding averaged group velocity for the A₀ mode ( ${\overline{V}}_{group\mathrm{\_}2}$ , Fig. 3(n)).

2.6. 360° wave shape characterization

We used the parameters of roundness, ellipticity aspect ratio, and fractional anisotropy to quantify the 360° wave propagation parameters, including amplitudes, attenuations, and phase/group velocities. This serves two purposes: First, to evaluate the directional distribution of wave propagations influenced by the excitation angle ( $\gamma$ ). Second, to assess the anisotropic mechanical properties of anisotropic media, such as skeletal muscle and corneal tissues. Figure 4 illustrates the distribution of the mechanical wave parameters with the values of ${\rho }_i$ ( $i$  = 1 to 20) in different directions. If the wave shape approximates an ellipse, we fit the shape into an ellipse with an angle $\theta$ between the long axis and the x-axis.

Fig. 4.

Demonstration of en face wave shape quantifications, including roundness, ellipticity, and fractional anisotropy. ${\rho }_i$ : polar distance for the measurement data ( $i$  = 1, …, 20); $\theta$ , a, and $b$ : angle, and lengths of the long and short axes for elliptical fitting.

Roundness is the measure of how closely the shape approaches that of a mathematically perfect circle. It can be mathematically described using the formula:

$$Roundness=\frac{4\pi \times Area}{Perimete{r}^2}.$$ (11)

For a perfect circle, the roundness is equal to 1. As the shape becomes less circular and more irregular, the roundness value decreases, approaching 0 for highly non-circular shapes.

We can also use the ratio of the length of the long axis ( $a$ ) to the short axis ( $b$ ) to describe the ellipticity:

$$\text{Aspect}\text{ratio}=\frac{a}{b}.$$ (12)

For a perfect circle, the aspect ratio is 1, and it increases towards infinity as the ellipse becomes more elongated.

The degree of anisotropy can be estimated by the fractional anisotropy ( $FA$ ) as [61]

$$FA=\sqrt{2\frac{{(a-\overline{\rho })}^2+{(b-\overline{\rho })}^2}{a^2+b^2}},$$ (13)

where $FA$ denotes the quadratic norm of the deviation compared to the mean value ( $\overline{\rho }$ ). An increased variation leads to a higher degree of anisotropy. A $FA$ value of 0 indicates an isotropic medium, while a value approaching 1 signifies a highly anisotropic medium.

3. Results

3.1. Measurement results of agar phantoms

For agar phantoms (1.0–1.6% concentration), we analyzed 360° radial wave propagation trends induced by excitation angles spanning a broad range from 70° (oblique) to 0° (perpendicular). We adjusted the air-pulse pressures to maintain a similar maximum displacement amplitude (∼2 µm) for all these agar phantoms during perpendicular stimulation. The measurement results are shown in Figs. 5 and 6. Figure 5(a) shows an example of the waveforms of the 1.0% agar phantom for these 12 excitation angle groups. Notably, the displacement waveforms transitioned from oblique to circular as the excitation angle decreased from 70° to 0°. In fact, once the excitation angle was below 40°, the displacement waveform shape approximated a circular form in this data set. Figure 5(b) presents the quantification results of wave propagation metrics in 360° directions, including amplitude ( $A$ ), attenuation ( $\alpha$ ), group velocity 1 ( $V_{group\mathrm{\_}1}$ ), averaged phase velocity ( ${\overline{V}}_{phase}$ ), and averaged group velocity 2 ( ${\overline{V}}_{group\mathrm{\_}2}$ ), respectively. It was noted that the amplitude and attenuation metrics exhibited elliptical forms when the excitation angle was large (e.g., 70°), transitioning to a near-circular shape as the excitation angle decreased (e.g., < 40°). An off-center effect in amplitude ( $A$ ) and attenuation ( $\alpha$ ) also emerged at excitation angles ≥45°, stemming from a spatial offset between the nominal stimulation center (air cannula tip) and the actual surface stimulation point. We can recalibrate the actual stimulation center using elliptic fitting of the displacement distribution, leveraging the rapid exponential decay of displacement amplitude with propagation distance. In contrast, group and phase velocities ( $V_{group\mathrm{\_}1}$ , ${\overline{V}}_{phase}$ , and ${\overline{V}}_{group\mathrm{\_}2}$ ) maintained near-circular distributions across all angles, demonstrating their robustness to tilt-induced geometric offsets.

Fig. 5.

En face surface wave propagation on 1% agar phantoms with excitation angles decreasing from oblique (70°) to perpendicular (0°). Measurements were performed in 360° with radii ranging from 0.93 to 2.80 mm. (a) Selected waveforms for 12 different excitation angles. (b) 360° surface wave feature quantification.

Fig. 6.

Quantification results of 360° surface wave propagation for 1%, 1.3%, and 1.6% agar phantoms, with excitation angles ranging from 70° to 0°. (a) Attenuation ( $\alpha$ ) and mechanical wave velocities ( $V_{group\mathrm{\_}1}$ , ${\overline{V}}_{phase}$ , and ${\overline{V}}_{group\mathrm{\_}2}$ ) for different excitation angle groups. (b–d) Quantification of roundness, aspect ratio, and fractional anisotropy for the angular distribution shapes of different mechanical wave metrics, including amplitude ( $A$ ), attenuation ( $\alpha$ ), group velocity 1 ( $V_{group\mathrm{\_}1}$ ), averaged phase velocity ( ${\overline{V}}_{phase}$ ), and averaged group velocity 2 ( ${\overline{V}}_{group\mathrm{\_}2}$ ). Shaded areas represent the mean ± STD of these metrics for the three agar phantoms.

Figure 6 displays the quantification results of the measurement metrics for 1%, 1.3%, and 1.6% agar phantoms. Since different air-pulse pressures were applied to each agar phantom to achieve comparable displacements, the amplitude ( $A$ ) values alone cannot reflect differences in sample stiffness. Therefore, we only compared the distributions of the amplitude ( $A$ ) in 360° directions for these three agar phantoms. Figure 6(a) shows the means and standard deviations (mean ± STD) of attenuation ( $\alpha$ ), group velocity 1 ( $V_{group\mathrm{\_}1}$ ), averaged phase velocity ( ${\overline{V}}_{phase}$ ) of A₀ mode, and averaged group velocity 2 ( ${\overline{V}}_{group\mathrm{\_}2}$ ) of A₀ mode across all measurement directions for 1–1.6% agar phantoms. As anticipated, mechanical waves decay faster in stiffer samples (e.g., higher concentration agar phantoms). In this dataset, $V_{group\mathrm{\_}1}$ is more effective than ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ at distinguishing agar phantoms of different concentrations (1%, 1.3%, and 1.6%) across all excitation angles.

Figures 6(b–d) present the evaluation results of roundness (Eq. (11)), aspect ratio (Eq. (12)), and fractional anisotropy (Eq. (13)) for the angular distribution shapes of mechanical wave metrics, including amplitude ( $A$ ), attenuation ( $\alpha$ ), group velocity 1 ( $V_{group\mathrm{\_}1}$ ), averaged phase velocity ( ${\overline{V}}_{phase}$ , A₀ mode), and averaged group velocity 2 ( ${\overline{V}}_{group\mathrm{\_}2}$ , A₀ mode). For homogeneous, isotropic samples (1%–1.6% agar phantoms), larger excitation angles (70°–45°) exhibit a stronger influence on amplitude ( $A$ ) and attenuation ( $\alpha$ ) compared to the group/phase velocities. Notably, $V_{group\mathrm{\_}1}$ demonstrates minimal sensitivity to larger excitation angles, outperforming ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ in preserving near-circular spatial distributions and mitigating pseudo-anisotropy artifacts caused by geometric offsets. This distinction likely stems from spatial variations in dispersion effects—such as shifts between A₀/S₀ modes or higher-order modes—introduced by oblique excitation. While $V_{group\mathrm{\_}1}$ integrates contributions from all modal and frequency components, yielding isotropic velocity profiles, ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ rely solely on the A₀ mode. The latter’s dependence on a single mode amplifies minor spatial dispersion variations, resulting in slight pseudo-anisotropic distributions.

3.2. Measurement results of chicken breast sample and ex vivo porcine cornea

In anisotropic media such as skeletal muscle and corneal tissues, the presence of a fibrous structure induces anisotropy, meaning that the material's properties vary depending on the direction of measurement. Previous ultrasonic studies have indicated a significant difference in shear wave velocities along the fiber direction compared to those perpendicular to the fiber direction [62]. These findings highlight the importance of considering directional dependencies when evaluating tissue properties. Here, we evaluate the OCE measurement of anisotropic samples under excitation angles of 45°–0°.

Figure 7 shows the 360° measurement results for the chicken breast sample with fiber orientations in vertical (a), oblique (b), and horizontal (c) directions. Generally, the wave amplitude ( $A$ ) was larger and the attenuation coefficient ( $\alpha$ ) was smaller along the direction of the muscle fibers, indicating that the mechanical waves travel more efficiently and lose less energy in this direction. Specifically, the mechanical wave velocities were much higher along the muscle fiber direction compared to those measured perpendicular to the fibers. However, the directional distribution of $V_{group\mathrm{\_}1}$ was more elliptical, with a higher value of fractional anisotropy ( $\overline{FA}$  = 0.63) compared to the distributions of ${\overline{V}}_{phase}$ , and ${\overline{V}}_{group\mathrm{\_}2}$ ( $\overline{FA}$  = 0.23). In addition, the measurement results for $V_{group\mathrm{\_}1}$ showed marginally higher variability across different excitation angle groups (45°, 30°, and 0°) compared to those for ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ (quantitative details are provided in Fig. 9). This discrepancy arises because $V_{group\mathrm{\_}1}$ was calculated using wavefronts (i.e., peak displacements), whereas ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ were derived from wave packets within a FWHM window (Fig. 3). Consequently, wavefront tracking enhances sensitivity to directional mechanical contrasts (e.g., anisotropy in chicken breast fibers) by isolating peak energy propagation; however, it amplifies noise due to limited spatial sampling (i.e., 17 points per direction). In contrast, time-window averaging (FWHM) reduces noise by integrating wave energy over a broader temporal range, thereby improving robustness in noisy or sparsely sampled conditions within the 45°–0° excitation range. Nevertheless, this averaging diminishes detection sensitivity to subtle directional variations in the chicken breast sample. Considering the effect of the excitation angles, the chicken breast measurements showed that although relatively small excitation angles ( $\gamma$  = 45° or 30°) still affect the measurements of amplitude and attenuation compared to normal excitation ( $\gamma$  = 0°), they have a lesser effect on the measurements of mechanical wave propagation velocities.

Fig. 7.

Surface wave propagation in anisotropic chicken breast samples with fiber orientations: vertical (a), tilted (b), and horizontal (c). Excitation angles were 45°, 30°, and 0°, respectively; radial distance: 0.93–2.80 mm. The selected waveforms and quantification metrics are organized into groups for clarity. $\overline{FA}$ : Averaged fractional anisotropy.

Fig. 9.

Comparison of ${\overline{V}}_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ , and the residual error analysis for oblique to perpendicular tissue excitations for ${\overline{V}}_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ . Data are derived from ${\overline{V}}_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ in Figs. 5–8. (a) Direct comparison between ${\overline{V}}_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ . (b, c) Residual errors (mean vs. difference) for oblique to perpendicular tissue excitations for ${\overline{V}}_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ , respectively.

Figure 8 presents the 360° measurement results for an ex vivo porcine cornea of the right eye under varying IOPs ranging from 5 mmHg to 20 mmHg. The long axis from the white-to-white region of the porcine eye was defined as the 0°–180° axis, extending from the nasal (N = 0°) to temporal (T = 180°) directions, with 90°–270° representing the inferior (I = 270°) and superior (S = 90°) directions. The air cannula was positioned on the nasal side with the excitation angle $\gamma$ at the corneal apex (Fig. 8(a)). To maintain optimal mechanical contrast across different IOP and excitation angle groups, we fine-tuned the air-pulse pressures for each experimental set to approximately 100-1000 Pascals. In Fig. 8(b), we observed faster wave propagation speeds in the 20 mmHg IOP group compared to the 10 mmHg group, as indicated by the wavefronts in the time series (marked by red and green dots). Figure 8(c) shows the 360° quantification metrics ( $A$ , $\alpha$ , $V_{group\mathrm{\_}1}$ , ${\overline{V}}_{phase}$ , and ${\overline{V}}_{group\mathrm{\_}2}$ ) for the ex vivo porcine cornea measurement. Several phenomena emerged from the observation and quantification results. First, anisotropy values for both group and phase velocities exhibited a biphasic response to IOP, increasing from 5–15 mmHg before declining slightly at 15–20 mmHg (Figs. 8(d) and (e)). This trend likely reflects IOP-mediated shifts in collagen fiber alignment and stress-dependent corneal stiffening (see Discussion). Secondly, the directional distribution of $V_{group\mathrm{\_}1}$ was more elliptical and exhibited higher $FA$ values compared to ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ . Additionally, we observed larger disparities in $V_{group\mathrm{\_}1}$ measurements across different excitation angles (even between 15° and 0°), especially at high IOP levels (e.g., 20 mmHg). We also noticed different dispersion curves of A₀ and S₀ modes for $V_{phase}$ and $V_{group\mathrm{\_}2}$ in the frequency domain across different excitation angles (data not illustrated). However, since we only compared the averaged values ( ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ ) from the relatively flat curves of the A₀ mode, these differences were greatly reduced in these results. The highly excitation angle-dependent measurement results of the porcine cornea highlight the necessity for meticulous attention to excitation angles during corneal OCE mechanical property measurements.

Fig. 8.

En face surface wave quantification on ex vivo porcine cornea. IOP: 5–20 mmHg; excitation angles: 45°–0°; Radial distance: 0.37-1.87 mm. (a) Set-up for porcine cornea measurement. S: superior; I: inferior; N: nasal; T: temporal. (b) Demonstration of the 360° waveforms for 10 mmHg and 20 mmHg groups, respectively. (c) 360° quantification metrics. (d, e) Fractional anisotropy (FA) analysis for $V_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ .

From Figs. 5–8, we observed that ${\overline{V}}_{group\mathrm{\_}2}$ exhibited similar values and trends to ${\overline{V}}_{phase}$ . This similarity arises because both values were derived from the same time window (Fig. 3(a)) and calculated from the A₀ mode (Figs. 3(e, f)). Therefore, in Fig. 9(a), we only compared the correlation between $V_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ , which were derived from different time windows (peak displacement vs. FWHM, Fig. 3(a)) and using different analytical methods (Eq. (4) vs. Equation (8)). The results indicated a positive correlation between ${\overline{V}}_{phase}$ and $V_{group\mathrm{\_}1}$ (Pearson correlation coefficient, r = 0.51 for all samples). However, both the correlation coefficients and the linear regression slopes varied significantly across different sample types. For 1–1.6% agar phantoms, the regression equation was y = 0.53x + 4.38 (R² = 0.42) with r = 0.65, attributed to the wide range of excitation groups. Limiting the excitation angle range to ≤ 45° improved the correlation coefficient to r = 0.76, with the regression equation y = 0.56x + 4.37 (R² = 0.58). For the chicken breast sample, the regression was y = 0.27x + 3.31 (R² = 0.75) with r = 0.86, while for the porcine cornea, it was y = 0.36x + 3.39 (R² = 0.77) with r = 0.88. Figures 9(b, c) compare the residual errors (mean vs. difference) for oblique (up to 70° to 10°) to perpendicular tissue excitations for $V_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ , respectively. Notably, the measurement of $V_{group\mathrm{\_}1}$ exhibited significant residual errors in porcine cornea measurements, especially at high IOP levels; whereas the measurement of ${\overline{V}}_{phase}$ displayed substantial residual errors when the excitation angles were large.

4. Discussion

In this study, we systematically evaluated the effect of excitation angles—from oblique to perpendicular—on key mechanical wave properties (amplitude, attenuation, group velocity, and phase velocity) by adjusting the tilt of an air cannula relative to the tissue surface. OCT scanning was performed radially to record en face wave propagations in 360°. Wave propagation features were characterized in multiple domains: magnitude ( $A$ ), attenuation ( $\alpha$ ), and group velocity 1 ( $V_{group\mathrm{\_}1}$ ) were calculated in the spatio-temporal domain, whereas the phase-dependent velocity ( $V_{phase}$ ), group velocity 2 ( $V_{group\mathrm{\_}2}$ ) and their averaged values ( ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ for A₀ mode) were calculated in the wavenumber-frequency domain. The distributions of these metrics were quantified using roundness, aspect ratio, and fractional anisotropy in polar coordinates.

We conducted measurements on various types of samples, including isotropic, homogeneous samples (1–1.6% agar phantoms), anisotropic samples (chicken breast), and samples with complex boundaries, coupling media, and stress conditions (ex vivo porcine cornea). Our findings indicate that mechanical wave velocities ( $V_{group\mathrm{\_}1}$ , ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ ) are less affected by excitation angles compared to displacement measurements ( $A$ and $\alpha$ ), indicating the robustness of using mechanical waves as indicators for elasticity measurement. In general, the tilting of the excitation angle has a more pronounced effect on the wave propagation features for tissues with greater complexity, such as those that are more anisotropic, have more complex boundary conditions, or are under greater tension. The agar measurement results (Figs. 5 and 6) showed that all these measurement metrics are relatively consistent when $\gamma$  < 40°, but more or less affected by the excitation angles when $\gamma$  ≥ 45°. We further evaluated the effects of smaller excitation angles ( $\gamma$  ≤ 45°) on anisotropic samples. The chicken breast measurements agreed with the agar phantom results, showing that wave velocity metrics remain more consistent compared to displacement metrics when $\gamma$  ≤ 45° (Fig. 7). However, significant disparities were observed in the ex vivo porcine cornea measurements (Fig. 8) across different excitation angles ( $\gamma$ : 45°, 30°, 15°, and 0°), particularly at IOP levels (e.g., 20 mmHg). Notably, group and phase velocities exhibited a biphasic anisotropy response to IOP, increasing from 5–15 mmHg before declining at 15–20 mmHg. These phenomena can be attributed to several factors. First, the cornea's layered structure and air-water boundaries can cause reflections, superposition, and mode conversions of the mechanical waves. High IOP may exacerbate these interactions, making wave propagation more angle-dependent. Second, elevated IOP increases the stress within the corneal tissue, altering its mechanical properties. Under high stress, the tissue may exhibit different elastic and viscoelastic behaviors, which affects how mechanical waves propagate. In stressed tissues, guided waves can be more sensitive to the excitation angle, leading to variations in wave speed and attenuation. Third, the porcine cornea has a complex structure with collagen fibers aligned in preferred directions. High IOP can alter the tension states of these fibers. As IOP increases from low to high, the corneal mechanical properties may shift from being predominantly driven by axial mechanical properties to predominantly lateral mechanical properties at the extremes. Excitations at different angles can selectively highlight or accentuate different tissue properties. Our study highlights the importance of using perpendicular stimulation for corneal biomechanics measurements, especially at high IOP levels. This approach is likely crucial for the clinical evaluation of diseased eyes, such as those with keratoconus, and post-refractive surgery corneas, which exhibit more complex structures and guided wave patterns under excitation. Furthermore, our findings suggest that perpendicular excitation may improve the accuracy of biomechanical assessments and help in the development of better diagnostic tools and therapeutic strategies for various corneal conditions. Future research should focus on refining these measurement techniques and exploring their clinical applications, particularly in cases involving complex tissue structures and varying stress conditions. This could lead to significant advancements in the field of corneal biomechanics and enhance our understanding of ocular health and disease.

Across all these samples (agar phantoms, chicken breast, and porcine cornea), ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ exhibit similar values and trends in every measurement direction and under different excitation angles. This consistency arises because both metrics were derived from the same time window, processed via identical 2D-FFT methods, and focused exclusively on the A₀ mode. However, as shown in Fig. 9, discrepancies exist between $V_{group\mathrm{\_}1}$ and ${\overline{V}}_{phase}$ . These differences stem from methodological distinctions: $V_{group\mathrm{\_}1}$ captures contributions from all frequency/mode components via wavefront tracking (peak displacements), while ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ isolate the A₀ mode within an FWHM window (Fig. 3). Oblique excitation angles (e.g., 70°–45°) induce spatial dispersion variations, such as shifts between A₀/S₀ modes or excitation of higher-order modes. These effects amplify differences between $V_{group\mathrm{\_}1}$ (integrating all modes) and ${\overline{V}}_{phase}$ / ${\overline{V}}_{group\mathrm{\_}2}$ (A₀ mode only), particularly at larger angles (Figs. 6 and 9). Wavefront-based $V_{group\mathrm{\_}1}$ provides superior mechanical contrast, ideal for resolving directional anisotropies (e.g., muscle fibers in chicken breast), but is more susceptible to noise due to sparse spatial sampling (17 points/direction). Conversely, FWHM-windowed metrics ( ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ ) prioritize noise robustness by averaging across time, albeit at the cost of reduced sensitivity to subtle directional variations (Fig. 7). Different window sizes (full window for the wave envelope, FWHM, or a narrower window that focuses on peak displacement) can achieve varying levels of mechanical contrast. Additionally, selecting an appropriate windowing function (e.g., Hanning, Hamming, Blackman-Harris, flat top, etc.) can further optimize the 2D-FFT results in the wavenumber-frequency domain (Fig. 3(d)) for better analysis of wave dispersion characteristics (Fig. 3(e, f)). Optimizing the windowing size and function is one of our priorities for future research.

Another interesting phenomenon is that the attenuation value ( $\alpha$ ) exhibits positive correlations with velocities ( $V_{group\mathrm{\_}1},{\overline{V}}_{phase}$ , and ${\overline{V}}_{group\mathrm{\_}2}$ ) in isotropic samples, but has negative correlations in anisotropic materials. In the 1–1.6% agar phantom measurements (Fig. 5), $\alpha$ followed the same trend as the wave speeds, with both increasing as the elasticity/stiffness increased. However, in the measurements of chicken breast and porcine cornea (Figs. 7 and 8), $\alpha$ exhibited an opposite trend to the wave speeds. This was particularly evident in the chicken breast sample, where the mechanical wave velocities were significantly higher along the muscle fiber direction compared to those measured perpendicular to the fibers. Conversely, $\alpha$ was markedly higher perpendicular to the fiber direction than along the fiber directions. Understanding these distinct behaviors among different metrics in various tissue types is crucial for increasing the accuracy of interpreting mechanical wave data and for characterizing the biomechanical properties of soft tissues.

This study has several limitations that need to be addressed in future developments. Firstly, we used a 3 ms duration air-pulse stimulation, which resulted in a limited excitation spectrum of less than 1 kHz (see Refs. [19,53] for more details). Thereby, we could only observe a portion of the A₀ mode and an even smaller portion of the S₀ mode, as shown in Figs. 3(e, f), hindering our capability to quantify the dispersive Lamb waves. Consequently, our work focused solely on analyzing the A₀ mode within a limited low frequency range. Specifically, we calculated the average values of ${\overline{V}}_{phase}$ and ${\overline{V}}_{group\mathrm{\_}2}$ within 0–400 Hz for the A₀ mode. Expanding the analysis to include the dispersive curves of both A₀ and S₀ modes across a broader frequency spectrum would offer a more comprehensive understanding of the material's properties. Such an approach could potentially enable the separation of lateral and axial tissue properties [14], as well as provide insights into the material's viscosity. It also worth emphasize that all the results and conclusions derived from the OCE measurements (Figs. 5–9) were all from the low excitation spectrum. The behavior of the material under higher frequency excitation could differ, potentially revealing additional insights into its mechanical properties as well as different excitation-dependent mechanical behaviors. Recently, Duvvuri C. et al [59] utilized a 0.5 ms duration air-pulse system to generate an excitation spectrum of up to 8 kHz, suggesting that a shorter duration air-pulse system could improve the quantification of Lamb wave dispersive properties across a wider frequency range. In their analysis within the spectrum of up to 5 kHz, they found that a larger excitation angle (e.g., > 30°) tends to excite S₀ modes in human corneas, which exhibit greater dispersion and more variable wave speeds compared to A₀ modes at closer to 0° excitation angles. Additionally, they observed an interesting phenomenon where increasing IOP from 10 mmHg to 30 mmHg caused ex vivo rabbit corneas to demonstrate mode-switching from A₀ to S₀ after about 20 mmHg. Unfortunately, we lack the capability to observe the mode-switching phenomenon, primarily due to our limited excitation spectrum. Secondly, we constantly adjusted the air pressures within a range of approximately 100–2000 Pascals to maintain comparable displacement magnitudes across samples of different stiffnesses and for various excitation angle groups. However, we were unable to use a dual-channel air-pulse stimulation system [53] for real-time monitoring of air-pulse parameters (e.g., duration, pressure magnitude) during OCE measurements, because the pressure sensor could not be conveniently rotated with the air cannula for different excitation angle setups (Fig. 2(b)). With the unknown air pressure magnitude, we could not precisely predict the trend of displacement amplitude under different excitation angles and in samples of varying stiffness. For example, we did not report the absolute values of A in Fig. 6(a), but reported the distribution shape of A instead, in Figs. 6(b–d). Additionally, even if we set a constant duration time (i.e., 3 ms) for the air pulse generator, the actual pulse duration varied with changes in pressure amplitude (see Fig. 6 in Ref. [15]). This variation in pulse duration can alter the excitation spectrum, possibly impacting the characterization of the frequency-dependent $V_{phase}$ and $V_{group\mathrm{\_}2}$ in the wavenumber-frequency domain. Furthermore, the changes in magnitude and duration of the air-pulse stimulation profile may introduce larger measurement variations due to the nonlinear elastic properties of soft tissues, particularly the cornea [44]. In the future, we plan to employ shorter duration air-pulse systems to expand the excitation spectrum for better dispersive wave analysis and to utilize dual-channel OCE systems for more precise quantification of mechanical properties.

In summary, this study systematically examines the influence of excitation angles on mechanical wave propagation across diverse sample types. Our findings provide valuable insights for enhancing the accuracy of biomechanical assessments using air-pulse-based or other dynamic OCE approaches. Specifically, we underscore the importance of accounting for excitation angles when interpreting mechanical wave data, particularly for tissues with complex geometry and stress conditions, such as the cornea. This comprehensive understanding lays the groundwork for the continuous refinement of the OCE technique, facilitating its clinical translation and ultimately improving diagnostic and therapeutic applications across various biomedical fields.

Funding

National Natural Science Foundation of China10.13039/501100001809 (61975030, 62305120); Basic and Applied Basic Research Foundation of Guangdong Province10.13039/501100021171 (2024A1515011344); Guangdong Provincial Pearl River Talents Program10.13039/100016691 (2019ZT08Y105); Guangdong Eye Intelligent Medical Imaging Equipment Engineering Technology Research Center (2022E076); Guangdong-Hong Kong-Macao Intelligent Micro-Nano Optoelectronic Technology Joint Laboratory (2020B1212030010); National Eye Institute10.13039/100000053 (P30EY003039, P30EY07551, R01EY022362).

Disclosures

GL, JX and YH: Weiren Meditech Co., Ltd. (C); JQ. and LA: Weiren Meditech Co., Ltd. (E).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.