Comprehensive characterization of nonlinear viscoelastic properties of arterial tissues using guided-wave optical coherence elastography

Abstract

The mechanical properties of arterial walls are critical for maintaining vascular function under pulsatile pressure and are closely linked to the development of cardiovascular diseases. Despite advances in imaging and elastography, comprehensive characterization of the complex mechanical behavior of arterial tissues remains challenging. Here, we present a broadband guided-wave optical coherence elastography (OCE) technique, grounded in viscoelasto-acoustic theory, for quantifying the nonlinear viscoelastic, anisotropic, and layer-specific properties of arterial walls with high spatial and temporal resolution. Our results reveal a strong stretch dependence of arterial viscoelasticity, with increasing prestress leading to a reduction in tissue viscosity. Under mechanical loading, the adventitia becomes significantly stiffer than the media, attributable to engagement of collagen fibers. Chemical degradation of collagen fibers highlighted their role in nonlinear viscoelasticity. This study demonstrates the potential of OCE as a powerful tool for detailed profiling of vascular biomechanics, with applications in basic research and future clinical diagnosis.

Introduction

The mechanical properties of arterial walls are fundamental to cardiovascular function. Alterations in these properties are associated with a range of vascular pathologies, including hypertension¹, coronary artery diseases², and aneurysm³. Arterial stiffening⁴ and weakening directly impact hemodynamics and can lead to rupture or bulging. Shear stress on arterial walls has been implicated in tortuosity^5,6, buckling⁷, dissection^8,9,10, vasa vasorum circulation¹¹, and atherosclerotic plaque development^12,13. Furthermore, changes in tissue nonlinearity and anisotropy reflect underlying structural and compositional remodeling. A non-destructive method capable of characterizing these sophisticated mechanical properties is therefore highly desirable for disease diagnosis and monitoring.

Arterial mechanics primarily derive from the fiber-reinforced structures of the media and adventitia. The adventitia contains a dense network of helically arranged, wavy collagen fibers, which confer tensile strength¹⁴. The media consists of concentric elastic lamellae composed of elastic fibers¹⁴, interspersed with transmural elastic fibers, collagen, proteoglycans, and smooth muscle cells^15,16. Elastin and collagen govern tensile responses in low- and high-strain regimes^17,18, respectively, while their orientation determines in-plane anisotropy^19,20. Non-fibrous components also contribute to shear resistance⁹. The heterogeneous organization of these structures underlies the tissue’s anisotropic mechanical response. Age-related stiffening is more prominent longitudinally^21,22, whereas aneurysm exhibit greater circumferential stiffening²³.

Conventional techniques to measure arterial mechanics include planar^21,22 and uniaxial tension tests²⁴, inflation-extension tests^25,26, shear^9,27, rotated-axes biaxial tests²⁸, and torsion^11,29. However, these bulky mechanical techniques are not amenable to in vivo application. Ultrasound can monitor arterial diameter and pressure to estimate circumferential modulus^30,31, but suffers from reduced accuracy in small vessels³². Pulse wave velocity (PWV), the gold clinical gold standard for stiffness assessment³³, fails to account for mechanical and geometric heterogeneities³⁴. Other indices based on pressure waveforms, such as augmentation index³⁵, central pulse pressure³⁶, back wave amplitude³⁷ and harmonic distortion³⁸, are often compounded by high heart rate, aging, pathology, and pharmacological interventions^32,39,40. These methods cannot directly or quantitatively assess the intrinsic mechanical properties of arterial walls.

Elastography based on ultrasound^41,42,43 and magnetic resonance imaging (MRI)⁴⁴ enables noninvasive stiffness mapping, but with limited resolution. Optical coherence elastography (OCE), an extension of optical coherence tomography (OCT), offers superior spatial resolution and sensitivity. OCE techniques include quasi-static compression OCE⁴⁵ and wave-based OCE⁴⁶. Compression OCE is capable of measuring Young’s modulus of tissues, whereas wave-based OCE has been used to measure shear modulus in tissues such as skin^47,48, cornea^49,50, sclera⁵¹, and artery⁵². Recently, we further advanced wave-based OCE method by developing a multi-wave approach capable of simultaneously quantifying both tensile and shear moduli in the cornea⁵³.

In the present study, we further advanced this OCE method and applied it to characterize the mechanical properties of porcine aortas ex vivo. Wave propagation velocities were measured in both circumferential and longitudinal directions over a 1–20 kHz frequency range. By analyzing dispersion under biaxial stretch, we extracted nonlinear and anisotropic shear and tensile modulus parameters. To capture viscoelastic behavior, we developed a viscoelastic two-layer guided wave model grounded in our newly proposed generalized acousto-viscoelastic theory⁵⁴, enabling quantification of layer-specific mechanical properties in the media and adventitia, including their stretch- and frequency-dependent viscous parameters. Finally, we used selective chemical treatments to remove collagen and investigated the distinct contributions of collagen and elastin to viscoelastic tensile and shear properties. This study demonstrates the utility of OCE for comprehensive mechanical characterization of arterial tissues, with relevance to both basic research and potential clinical applications.

Results

OCE detection of A0- and S0 wave modes

Porcine aorta samples were cut-open and mounted on a biaxial stretching device (Fig. 1a). Waves were excited using a PZT probe along either the axial or circumferential direction. During measurements, the intimal-media surface faced upward, while the adventitia remained in contact with PBS to prevent dehydration (Fig. 1b). Although the intimal surface was exposed to air, small drops of water were frequently applied around the probe excitation area to prevent local dehydration, and any excess water was wiped off before testing. A representative OCT cross-section of the arterial wall is shown in Fig. 1c, where the media and adventitia are distinguishable by their reflectivity and thickness. Based on OCT images, the average wall thickness was 1.53 ± 0.11 mm (N = 10), with a media-to-adventitia thickness ratio of ~1:1.4. Wave displacements were recorded at excitation frequencies ranging from 1 to 20 kHz. Representative displacement maps at 8 kHz and 16 kHz are shown in Fig. 1d, exhibiting sinusoidal oscillations with exponential decay (Fig. 1e). FFT analysis of the displacement profiles revealed spatial frequency components corresponding to the fundamental quasi-antisymmetric (A0) and quasi-symmetric (S0) Lamb wave modes, which represent the lowest-order elastic waves in plates – A0 involving flexural motion and S0 involving extensional motion (Fig. 1f).

Fig. 1: S0- and A0-waves in the artery.

a Schematic of a flattened artery tissue on a water bath to avoid dehydration. The elastic waves are excited by the contact probe and are measured by an OCT beam. b Photograph of the setup. c Typical OCT image of a sample. M: Media. A: Adventitia. d Representative wave motion profile measured in the artery for two different wave frequencies of 8 and 16 kHz. The displacement map (real part) is overlaid on the gray scale optical coherence tomography image. e Displacement extracted along the sample surface at 8 kHz. Solid and dashed lines denote the real and imagery parts of the displacement, respectively. f The displacement is Fourier transformed to wavenumber space, in which the primary A0 and S0 can be resolved. g Representative experimental data (circles) for phase velocities measured at different frequencies. Two modes are identified between 5 and 10 kHz, corresponding to the A0 and S0 modes. At high frequencies above 10 kHz, only a single mode is reliably detected, which is interpreted as the S0 mode in the limit of Rayleigh surface wave regime. The error bars represent the standard deviation of three replicate measurements on a single sample. h Schematics of a pure dilatational wave profile and a pure flexural wave displacement. i Finite element simulation results for the modal shapes of the A0 wave, S0 wave, and a combination of the two modes with equal amplitudes.

Figure 1g illustrates frequency-dependent velocities. At frequencies below 5 kHz, only the A0 mode was detectable. Between 5 and 10 kHz, both the A0 and S0 modes were observed, while above 10 kHz, the S0 mode dominated. This is because a wave is most efficiently excited when its half-wavelength approximately matches the contact length of the probe tip. According to Lamb wave theory, the low-frequency S0 mode corresponds to dilatational motion associated with tensile deformation, whereas the A0 mode reflects bending motion involving shear deformation (Fig. 1h)⁵³. The phase velocity of the A0 mode increases with frequency and asymptotically approaches to the Scholte wave velocity at the tissue-fluid interface. In contrast, the S0 mode velocity decreases toward the Rayleigh wave limit at the air-tissue interface. Finite element simulations (Fig. 1i) confirmed the presence of both A0 and S0 modes, with asymmetric mode profiles due to the differing boundary conditions on each surface.

Elastic wave analysis of biaxially stretched tissues

Wave velocity profiles were measured in both axial and circumferential directions across varying stretch ratios ($\lambda$ = 1.0–1.4). As shown in Fig. 2, phase velocities for both A0 and S0 modes increased with stretching. At stretch ratios above 1.2, the measurement of circumferential S0 mode velocities became unreliable due to low wave amplitudes—as the tissue stiffens at higher stretch ratios, the wave amplitude decreases with the same excitation force, which leads to less reliable velocity estimates.

Fig. 2: Phase velocities in axial and circumferential directions, and their fitting curves using the single-layer elastic model.

a Axial dispersion relations of A0 and S0 modes measured at varying stretch ratios. b Circumferential dispersion relations of A0 and S0 when stretch ratio $\lambda$ increases from 1 to 1.4. Markers: experiments. Lines: fitting curves using the single-layer elastic model. The error bars represent the standard deviation of three replicate measurements on a single sample.

To extract elastic moduli, we modeled the arterial wall as an incompressible elastic plate bordered by air and water. The incremental stress $\Sigma$ is related to the displacement $u$ as⁵⁵:

$${\Sigma }_{{ij}}={{\mathcal{A}}}_{{ijkl}}^{0}{u}_{l,k}-\hat{p}{\delta }_{{ij}}+p{u}_{i,j}$$

(1)

where ${{\mathcal{A}}}_{{ijkl}}^{0}$ is the Eulerian elasticity tensor, $p$ is the Lagrange multiplier enforcing incompressibility, and $\hat{p}$ its incremental term. Using a stream function $\psi$, the wave equation becomes (see details in Supplementary Note 1):

$$\alpha {\psi }_{,{xxxx}}+2\beta {\psi }_{,{xxyy}}+\gamma {\psi }_{,{yyyy}}=\rho ({\psi }_{,{xxtt}}+{\psi }_{,{yytt}})$$

(2)

Here, $\alpha ={{\mathcal{A}}}_{{xyxy}}^{0}$, $2\beta ={{\mathcal{A}}}_{{xxxx}}^{0}+{{\mathcal{A}}}_{{yyyy}}^{0}-2{{\mathcal{A}}}_{{xxyy}}^{0}-2{{\mathcal{A}}}_{{xyyx}}^{0}$, $\gamma ={{\mathcal{A}}}_{{yxyx}}^{0}$. These incremental elastic moduli characterize resistance to shear and in-plane tensile deformation⁴⁹. For $\psi \propto {e}^{{sky}}{e}^{i({kx}-\omega t)}$, the characteristic equation becomes:

$$\gamma {s}^{4}-\left(2\beta -\rho \frac{{\omega }^{2}}{{k}^{2}}\right){s}^{2}+\alpha -\rho \frac{{\omega }^{2}}{{k}^{2}}=0$$

(3)

For a bulk shear wave polarized in the y-direction ($s$ = 0), this yields $v=\omega /k=\sqrt{\alpha /\rho }$, indicating that $\alpha$ represents the shear modulus. A static plate analysis shows that $2\beta +2\gamma$ corresponds to the in-plane tensile modulus (see Supplementary Note 6). Applying boundary conditions at the air and water interfaces, we solved the resulting secular equation $\det \left({{\rm{M}}}_{5\times 5}^{{\rm{e}}}\right)=0$ (matrix components in Supplementary Note 1) to fit the dispersion data and extracted $\alpha$, $\beta$, and $\gamma$. The derived moduli are listed in Table 1, and corresponding fit curves are plotted in Fig. 2.

Table 1 Measured stretch-dependent elastic moduli from the single-layer elastic model (1–20 kHz)

The elastic model captured overall trends but showed discrepancies—particularly for the A0 mode below 5 kHz at $\lambda$ = 1–1.2 and for the S0 mode above 15 kHz. Notably, in unstretched samples ($\lambda$ = 1), the S0 velocity increased with frequency, whereas the elastic model predicted a monotonous decrease toward the Rayleigh surface wave limit. These deviations suggest viscoelastic contributions, addressed in the next section.

Viscoelastic single-layer wave model analysis

To account for frequency-dependent behavior of arterial tissues⁵⁶, we incorporated a Kelvin–Voigt fractional derivative (KVFD) viscoelastic model^57,58. In this formulation, a viscoelastic “spring-pot” element operates in parallel with an elastic spring (Fig. 3a). The spring-pot is defined by a complex, frequency-dependent parameter:

$$\Omega ={\eta (i\omega )}^{\delta }$$

(4)

where $\delta$ is the fractional order, and $\eta$ (unit: ${{\rm{s}}}^{\delta }$) denotes the relative strength of the spring-pot viscosity compared to the elasticity of the accompanying spring. When $\delta =1$, the model reduces to the classical Kelvin–Voigt model, where $\Omega =i\eta \omega$. In the linear regime under negligible pre-stress, the parallel combination of a spring-pot and a purely elastic spring with storage modulus $\mu$ yields a complex dynamic modulus of $\left(1+\Omega \right)\mu$. Note that $\Omega =0$ in response to static stress (since $\omega =0$). At equilibrium with static pre-stress, the viscous response of the spring-pot has fully relaxed. However, when additional dynamic strain is introduced by acoustic waves, the spring-pot can contribute significantly to the material response.

Fig. 3: Viscoelastic model analysis of experimental data.

a–d Single-layer viscoelastic model analysis. a Schematic of the KVFD model. b Circumferential dispersion relations of A0 and S0 modes in the stress-free state. Markers: experiments. Lines: fitting curves using the single-layer viscoelastic model. The error bars represent the standard deviation of three replicate measurements on a single sample. c Axial $\alpha$ parameter values derived with the pure elastic model (from Table 1). d Product of $\alpha$ times $G$ ($=1+{\eta (i\omega )}^{\delta }$) obtained from the axial values in Table 2. Solid curves: real values, Dashed curves: imaginary values. e, f Two-layer viscoelastic model analysis. e Axial dispersion relations of A0 and S0 modes. f Circumferential dispersion relations of A0 and S0 modes. Markers: experiments. Lines: fitting curves using the two-layer viscoelastic model.

Incorporating the KVFD model into the pre-stressed, dynamic-strain regime, the incremental stress tensor $\Sigma$ is modified to⁵⁴:

$${\Sigma }_{{ij}}=G{{\mathcal{A}}}_{{ijkl}}^{0}{u}_{l,k}-\hat{q}{\delta }_{{ij}}+q{u}_{i,j}-G\hat{Q}{\delta }_{{ij}}+{GQ}{u}_{i,j}-\Omega {\sigma }_{{Dik}}^{e}{u}_{j,k}$$

(5)

Here, $G=1+\Omega$, and $q$ is the Lagrange multiplier with its increment $\hat{q}$. $Q$ $={\sigma }_{{ii}}^{e}/3$ and $\hat{Q}$ is its increment. The elastic Cauchy stress-strain relation is ${\sigma }^{e}=(\partial W/\partial F){F}^{{\rm{T}}}$, and the deviatoric elastic stress is ${\sigma }_{D}^{e}={\sigma }^{e}-QI$. When $\Omega$ = 0, Eq. (5) reduces to the elastic form given by Eq. (1), as $q+Q$ corresponds to the original Lagrange multiplier $p$. Inserting this modified stress expression into the wave equation and applying the stream function $\psi$, we obtain (see details in Supplementary Note 2):

$$ \left(G\gamma -\Omega {\sigma }_{{Dyy}}^{e}\right){s}^{4}+\left[\rho \frac{{\omega }^{2}}{{k}^{2}}-2G\beta +\Omega ({\sigma }_{{Dxx}}^{e}+{\sigma }_{{Dyy}}^{e})\right]{s}^{2}\\ +\left(G\alpha -\Omega {\sigma }_{{Dxx}}^{e}-\rho \frac{{\omega }^{2}}{{k}^{2}}\right)=0$$

(6)

Applying the same boundary conditions used in the elastic model, we solved the corresponding secular equation $\det \left({{\rm{M}}}_{5\times 5}^{{\rm{v}}}\right)=0$, where the matrix components are detailed in Supplementary Note 2. By fitting this model to the experimentally measured dispersion curves, we extracted both the elastic parameters $\alpha$, $\beta$ and $\gamma$ and the viscoelastic parameters $\eta$ and $\delta$. The results are summarized in Table 2.

Table 2 Measured viscoelastic parameters from the single-layer viscoelastic model (1–20 kHz)

Figure 3b shows a representative set of fitting curves using the single-layer viscoelastic model (see Supplementary Fig. S1 for all fitting curves). Compared to the purely elastic model, the viscoelastic model provided a substantially improved fit, particularly in capturing the dispersion behavior of the A0 mode at low frequencies. The inclusion of viscosity resulted in a more gradual, yet continuous, increase in A0 phase velocity with frequency. However, some mismatch remained for the S0 mode at high frequencies. This discrepancy could not be resolved solely by increasing viscosity, as doing so introduced errors in other frequency regions. These limitations are further addressed in the next section.

From Eq. (6) for $s=0$, we find that the bulk shear modulus is equal to $G\alpha -\Omega {\sigma }_{{Dxx}}^{e}$, where ${\sigma }_{{Dxx}}^{e}$ is typically an order of magnitude smaller than $\alpha$ (see Supplementary Note 7). To better understand the role of viscoelasticity, we compared the shear modulus $\alpha$ obtained from the elastic model (Fig. 3c) with the real and imaginary parts of $\alpha G$ obtained from the viscoelastic model (Fig. 3d). The real-part of $\alpha G$ represents the storage modulus, reflecting elastic energy retention, and was in close agreement with the elastic model around 10 kHz. As expected for fiber-reinforced tissues, the storage modulus increased with stretch. The imaginary part of $\alpha G$, corresponding to the loss modulus, quantifies viscous energy dissipation and was found to decrease with increasing stretch ratio. This stretch-dependent reduction in loss modulus indicates that arterial tissues exhibit a transition toward more elastic and less viscous behavior as they are deformed.

Viscoelastic two-layer wave model analysis

The media and adventitia exhibit distinct structural compositions and mechanical properties¹⁴, with adventitia containing more collagen-rich, highly anisotropic fibers, and the media dominated by elastic lamellae. To account for this heterogeneity, we extended the viscoelastic wave model to a two-layer configuration. Each layer was assigned independent elastic moduli, denoted ${\alpha }_{1}$, ${\beta }_{1}$, and ${\gamma }_{1}$ for the media and ${\alpha }_{2}$, ${\beta }_{2}$, and ${\gamma }_{2}$ for the adventitia, while the viscous parameters $\eta$ and $\delta$ were assumed to be identical across layers to reduce the number of free parameters. Continuity conditions for displacement and stress were applied at the media-adventitia interface, in addition to the boundary conditions at the air-tissue and fluid-tissue interfaces. These interfacial conditions yielded a secular equation of the form: $\det \left({{\rm{M}}}_{9\times 9}^{{\rm{v}}}\right)=0$ with matrix components detailed in Supplementary Note 3.

A phantom experiment was first conducted on two PDMS films with different stiffness. Each film was independently tested using OCE to determine its viscoelastic parameters. The two films were then bonded together to form a bilayer system, and guided-wave dispersion was measured again using OCE. The results showed that the experimental dispersion data agreed well with the theoretical predictions derived from the two-layer viscoelastic model and the independently characterized material parameters (RMSE = 1.2 m/s, Supplementary Fig. S2), confirming the model’s validity for analyzing the dispersion behavior of layered soft materials.

Figure 3e, f present the fitted dispersion curves. The two-layer viscoelastic model successfully captured key trends in the high-frequency behavior of both A0 and S0 modes: specifically, the upward trend of the S0 mode at $\lambda$ = 1 and 1.1, and the downward trend of the A0 mode at higher stretches ratios ($\lambda$ = 1.3 and 1.4). These behaviors reflect the evolving mode shape with increasing frequency. Depending on the modulus ratio of the two layers, the asymptotic velocities of the S0 and A0 modes differ (see details in Supplementary Note 8). In the stress-free state ($\lambda =1$), where the media is slightly stiffer than the adventitia (${\alpha }_{2}/{\alpha }_{1}\approx 0.8$), the S0 mode approaches the shear wave velocity of the adventitia, and the A0 mode approaches the fluid–adventitia interface, resembling a Scholte wave limit. Under stretched conditions ($\lambda \ge 1.1$), the adventitia becomes significantly stiffer than the media (${\alpha }_{2}/{\alpha }_{1} > 1.4$); thus, the S0 mode approaches the shear wave velocity of the media, and the A0 mode tends toward the air–media interface, resembling the Rayleigh wave limit. Importantly, these trends could not be reproduced by a two-layer elastic model with $\eta$ = 0, which yielded dispersion curves similar to those of the single-layer elastic case (Supplementary Fig. S3 and Supplementary Note 4). This confirms that the observed high-frequency behaviors result from the combined effects of viscosity and spatially varying stiffness.

Fitted viscoelastic parameters are summarized in Table 3 and plotted in Fig. 4. Both the shear and tensile moduli increased with stretch. A particularly sharp increase in adventitial tensile modulus was observed with $\lambda$ > 1.1 (Fig. 4e), in agreement with previous biaxial tensile studies²². In the unstressed condition ($\lambda$ = 1), the media exhibited greater stiffness than the adventitia (Fig. 4c, f). However, under stretch, the adventitia stiffened more rapidly, eventually surpassing the media in stiffness. Additionally, circumferential elastic moduli were consistently greater than axial moduli under tension.

Fig. 4: Shear and tensile moduli of the media and the adventitia.

a Axial and circumferential shear moduli of the media as functions of the stretch ratio $\lambda$. b Axial and circumferential shear moduli of the adventitia. c The ratio of adventitial over medial shear moduli along both axial and circumferential directions. d Axial and circumferential tensile moduli of the media. e Axial and circumferential tensile moduli of the adventitia. f The ratio of adventitial over medial tensile moduli along both axial and circumferential directions. The error bars represent the standard deviation of the fitted parameters.

Table 3 Measured viscoelastic parameters from the two-layer viscoelastic model (1–20 kHz)

Stretch-dependent viscosity parameters of arterial tissues

Figure 5 summarizes the viscoelastic parameters $\eta$ and $\delta$ extracted from the two-layer model and listed in Table 3. The amplitude parameter $\eta$ increases with stretch up to $\lambda$ = 1.2, then decreases at higher stretch ratios. The underlying mechanistic basis for this non-monotonic trend remains unclear but may reflect microstructural changes in fiber alignment and fluid redistribution during deformation. In contrast, the fractional order $\delta$ decreases consistently with increasing stretch, remaining within a range of 0–0.5—comparable to prior reports (0.1–0.3) from uniaxial stress relaxation experiments⁵⁶.

Fig. 5: Viscoelasticity of the artery.

a Axial and circumferential viscous parameters $\eta$ (unit ${{\rm{s}}}^{\delta }$) with respect to the stretch ratio. b Axial and circumferential viscous parameters $\delta$ (fractional order, unitless) with respect to the stretch ratio. The error bars represent the standard deviation of the fitted parameters. c Wave attenuation in the axial direction, with $\lambda$ varying from 1 to 1.2. Markers: experiments. Dashed lines: two-layer model-predicted attenuation of the A0 mode using viscoelastic parameters in Table 3. The attenuation curves for the S0 mode are similar (Supplementary Fig. S4). The error bars represent the standard deviation of three replicate measurements on a single sample.

Representing the dynamic modulus of the spring-pot, both the real and imaginary components of $\alpha \Omega$ and $\left(2\beta +2\gamma \right)\Omega$ were found to decrease with increasing stretch. This trend indicates a reduction in both energy storage and dissipation contributed by the spring-pot component. The loss tangent, defined as the ratio of the loss modulus to the storage modulus, also decreases with stretch. Collectively, these findings demonstrate that arterial viscoelasticity is highly deformation-dependent with tissues exhibiting reduced viscosity under increasing tension.

The attenuation of acoustic waves, visible in the wave profile (Fig. 1f), also reflects this viscoelastic behavior. To quantify attenuation, we fit the displacement amplitude profiles to an exponential decay model of the form, ${e}^{-{k}_{{\rm{im}}}x}$, where ${k}_{{\rm{im}}}$ is the imaginary part of the wavenumber. The measured attenuation coefficients (Fig. 5c) increased with frequency, in agreement with the rising loss modulus shown in Fig. 3d. Notably, the attenuation decreased as the stretch ratio increased, further supporting the observation that arterial tissues become less dissipative and more elastically dominated as they are stretched.

Effects of removal of collagen fibrils

To investigate the roles of collagen and elastic fibers we treated arterial tissues with cyanogen bromide (CNBr), a process that degrades and removes collagen fibers, cellular components, and other extracellular matrix elements, while largely preserving the elastin fiber network. This treatment reduced the wall thickness from 1.53 ± 0.11 mm to 1.19 ± 0.19 (N = 5). Figure 6a shows representative circumferential velocity measurements before and after CNBr treatment. Both shear and tensile moduli were substantially reduced following treatment.

Fig. 6: Viscoelastic properties of arterial tissues after treating with CNBr.

a Representative dispersion relations for an arterial tissue before and after treatment. A 10% strain ($\lambda$ = 1.1) was applied in both cases. Curves: fitting with the viscoelastic single-layer model. The error bars represent the standard deviation of three replicate measurements on a single sample. b–e Viscoelastic parameters after treatment at different stretch ratios: Axial and circumferential shear moduli (b), tensile moduli (c), with respect to the stretch ratio, viscous parameters $\eta$ (d), and fractional orders ${\rm{\delta }}$ (e). The error bars represent the standard deviation of the fitted parameters. f Product of $\alpha$ times $G=1+{\eta (i\omega )}^{\delta }$ obtained from the axial values. g Product of axial $2(\beta +\gamma )$ times $G$. Solid curves: real values, Dashed curves: imaginary values.

Due to the loss of collagen, the samples could be stretched up to a stretch ratio of $\lambda$ = 1.1, beyond which mechanical failure occurred at the hooks. From the measured velocity data with the single-layer viscoelastic model (Supplementary Fig. S5), we derived the corresponding mechanical parameters. Table 4 and Fig. 6b–e summarize these results. The shear and tensile moduli of the treated tissues, now dominated by the elastin network, exhibited a substantial decrease compared to those of the intact sample. Circumferential moduli remained slightly higher than axial values, which reflects the anisotropy of the elastin network^17,20. The viscous parameters were generally comparable to those of the intact arteries. The frequency-dependent axial shear and tensile moduli, $\alpha G$ and $2(\beta +\gamma )G$ are plotted in Fig. 6f, g. Compared to intact arteries, the treated tissues exhibited considerable stiffening in storage modulus at relatively low strain levels (5–10%). In contrast, changes in loss moduli were modest, if not negligible.

Table 4 Viscoelastic parameters measured on collagen-degraded tissues (1–20 kHz)

We applied the GOH constitutive model (see Eq. (10)), which is widely used to describe fiber-reinforced cardiovascular tissues. The model parameters, estimated from the measured axial and circumferential elastic moduli (Tables 2–4), are listed in Supplementary Table S1. In intact samples, the parameter ${k}_{1}$ in the adventitia was higher than that in the media, consistent with the higher collagen content in the adventitia. Following collagen removal, both ${\mu }_{0}$ and ${k}_{1}$ decreased, and notably, the nonlinear exponent ${k}_{2}$ was reduced by more than half.

Discussion

We have presented a comprehensive characterization of the mechanical properties of arterial tissues using a broadband OCE technique. The high spatial resolution (~10 µm) and vibration sensitivity (~1 nm per A-line) of OCE enabled precise detection of mechanical wave propagation in the tissue. The wave velocities, measured over a broad frequency (1–20 kHz), exhibited rich spectral features, allowing us to extract various mechanical parameters, including shear and tensile elastic moduli as well as viscous coefficients. This acousto-elastic analysis is grounded in continuum mechanics theory and augmented by our recently developed viscoelastic model framework⁵⁴. Our analytic approach leveraged the layered, two-dimensional architecture of the vascular wall. The tissue supports two distinct types of guided acoustic waves: quasi-antisymmetric (A0) and quasi-symmetric (S0) modes. The OCE system was optimized for efficient excitation and detection of both modes. At frequencies below 5 kHz, the A0 mode was predominant, with dispersion profile providing information about the viscoelastic shear modulus. At higher frequencies (>10 kHz), the S0 mode dominated. The extrapolated dispersion of the S0 mode to lower frequencies yielded estimates of the tensile modulus, while its asymptotic behavior above 15 kHz enabled the estimation of layer-specific viscoelastic parameters.

Our measurements revealed key features of arterial wall mechanics, including anisotropy, nonlinearity, viscoelasticity, and layer-inhomogeneity across physiologically relevant stretch levels in the 1–20 kHz frequency range. Both shear and tensile moduli increased with stretch, with circumferential values consistently exceeding axial ones. The adventitia exhibited greater stiffness than the media under prestressed conditions, highlighting the load-bearing role of collagen fibers²². The experimentally observed stretch-dependent moduli (i.e. nonlinear elasticity) reflect, on one hand, the strain-stiffening behavior of collagen and elastin fibers and may be related to the porous interstitial microstructure of the arterial wall^59,60. We compared our experimentally derived moduli with literature values obtained from conventional mechanical testing methods^61,62 (see Supplementary Note 9). Overall, the trends are consistent: circumferential moduli are higher than axial moduli, and the adventitia-to-media modulus ratio increases markedly under stretch—from below unity in the unloaded state to values greater than one under physiological tension. A previous study used compression OCE to quantify the Young’s modulus ($E$) of human arterial plaque⁶⁰, reporting values on the order of MPa at ~10% strain, which is comparable to the tensile modulus measured in our study. The tensile modulus reported here, $2\beta +2\gamma$, reduces to $4E/3$ under linear isotropic elastic and incompressible conditions. This parameter corresponds to a plane-strain tensile response with constrained transverse deformation, while Young’s modulus describes uniaxial tension without such constraint. Therefore, caution should be exercised when directly comparing Young’s modulus with tensile modulus. In older individuals, aortic stiffness has been reported to be greater in the longitudinal direction than in the circumferential direction²¹. The current OCE technique holds potential for investigating age-related changes in the anisotropy of human arteries and merits further exploration.

An important finding of our study is the stretch-dependent modulation of arterial viscoelasticity. With increasing prestress, we observed a consistent decrease in both wave attenuation and the fractional orders of the viscoelastic model, indicating a shift toward more elastic behavior. Notably, the imaginary components of $\alpha G$ and $(2\beta +2\gamma )G$ —corresponding to the shear and tensile loss moduli, respectively, and thus indicative of viscous energy dissipation—decreased with increasing stretch ratio. This reduction in loss modulus suggests that arterial tissues transition toward a more elastic, energy-efficient state under physiological loading, potentially optimizing function during cyclic deformation. While nonlinear viscoelasticity in arteries and other biological tissues has been investigated previously^{63,64,65,66,67}, our quantitative findings offer new insight into this behavior. These results have important implications for constitutive modeling of arteries and may inform the design of bioinspired materials or therapeutic interventions for vascular disease. Further studies are warranted to elucidate the underlying biophysical mechanisms and to determine whether similar viscoelastic trends are observed in other tissue types.

Following collagen removal by CNBr treatment, the treated samples exhibited substantially reduced elastic moduli ($\alpha$, $\beta$, and $\gamma$), which is in agreement with prior studies using enzymatic digestion^68,69. These results are consistent with previous findings that elastin and collagen fibers predominantly govern tensile behavior in the low- and high-strain regimes, respectively^17,18. Interestingly, despite the loss of collagen, the complex viscoelastic parameter $\Omega$ (the ‘spring-pot’) remained comparable to that of intact tissues. This may reflect that collagen fibers contribute minimally to viscoelastic damping at low strains, but play a nonlinear role in viscous dissipation under larger deformations. It is also possible that the interstitial fluid within the arterial wall contributes to the observed significant viscoelastic behavior after collagen removal⁶⁰. A recent study on porcine sclera also found that collagen fibers are not the sole determinant of tissue nonlinearity⁷⁰.

In this study, using a full-field FFT algorithm, the lateral resolution in the measurement of wave speed is ~2–3 wavelengths. For example, the resolution is ~4 mm at 2 kHz and ~1 mm at 20 kHz. A higher lateral resolution was demonstrated in our previous work on the human cornea using a phase-gradient method⁴⁹. However, in the present case, strong wave attenuation due to arterial viscoelasticity makes phase-gradient methods less reliable. The FFT-based approach provides improved noise tolerance to yield more reliable wave speed estimates at the expense of spatial resolution. The two-layer model allows us to resolve and quantify the stiffness of the media and adventitia. By employing higher-frequency mechanical excitation—thereby reducing the elastic wavelength—and using a three-layer model, it may be possible to characterize the mechanical properties of the intima.

The demonstrated OCE technique and acousto-viscoelastic model have several limitations. First, it characterizes mechanical properties at frequencies ranging from 1 to 20 kHz, which are substantially higher than the physiologically relevant frequencies near 1 Hz. While the KVFD model allows extrapolation of viscoelastic moduli to lower frequencies, the accuracy of this extrapolation remains to be validated. Second, the acoustic wavelengths in our measurements ranged from ~2 mm (A0 mode) to 7 mm (S0 mode). Since these wavelengths are shorter than the radius of curvature of aortas, our measurements on flattened samples reasonably approximate those in intact cylindrical vessels. However, for lower-frequency waves or smaller-diameter vessels, curvature effects may significantly alter wave propagation, which should be considered in modeling⁷¹. Third, in our experimental setup, the intima surface was exposed to air to facilitate wave excitation, whereas physiologically it is in contact with blood, and the adventitia is surrounded by soft connective tissues. These in vivo boundary conditions differ from our experimental configuration and are expected to affect the observed mechanical responses, even if the intrinsic properties of the medial and adventitial layers remain the same. Fourth, the accuracy of velocity measurements is constrained by the efficiency of wave excitation and the optical signal-to-noise ratio. Incomplete dispersion curves and parameter interdependence in the fitting model contribute to uncertainties in some mechanical parameters, which exceeded 50%. Future system optimizations of the probe shape, contact angle, and excitation amplitude will help improve the velocity measurement.

Lastly, a major limitation of the current technique is the need for a contact probe to excite guided waves. The present contact PZT probe predominantly exerts mechanical surface forcing. Alternatively, focused ultrasound excitation, similar to that used in shear-wave ultrasound elastography⁷², can produce a more balanced excitation of A0 and S0 modes (Supplementary Note 10 and Supplementary Fig. S11). More importantly, a non-contact ultrasound-based system would enhance the translation of OCE for clinical applications. Both ultrasound and optical beams could be delivered through intravascular fiber-optic catheters^73,74,75,76, enabling simultaneous structural and mechanical assessment of arterial walls. In addition, handheld devices could be developed for noninvasive access to carotid arteries in the head and neck⁷⁷, further broadening the utility of OCE in vascular diagnostics.

In conclusion, this study presents a systematic OCE-based approach for characterizing arterial stretch-dependent anisotropy, layer-specific inhomogeneity, and viscoelasticity. The results reveal a decreasing trend in arterial viscoelasticity and an increasing trend of layer-inhomogeneity with increasing mechanical stretch. The elastin network was found to exhibit obvious viscoelasticity at low strain. These findings provide new insights into cardiovascular biomechanics and could open the way for early-stage cardiovascular disease diagnosis and intervention strategies.

Methods

Sample preparation

Fresh porcine descending thoracic aortas (N = 10) were obtained from a licensed commercial supplier and transported to lab on ice. Surrounding connective tissue was carefully removed. Square samples (2 × 2 cm) were cut with edges aligned along the circumferential and longitudinal directions of the arterial wall. Ten full-thickness aortic samples were prepared for measurements.

To investigate the specific contributions of collagen, five samples were further subjected to cyanogen bromide (CNBr) treatment, which effectively degrades collagen and other cellular and extracellular components while leaving elastin intact¹⁷. Briefly, samples were incubated in 50 mg/ml CNBr dissolved in 70% formic acid at room temperature for 19 h, followed by heating at 60 °C for 1 h and boiling for 5 min to deactivate the reagent. Treated samples were stored in 1× phosphate-buffered saline (PBS) until further experiments.

Optical coherence elastography (OCE)

The OCE system was based on a swept-source OCT platform^48,49, utilizing a 1300 nm wavelength-swept laser with an 80 nm bandwidth operating at 43.2 kHz. The axial and lateral resolutions of the optical beam were ~15 μm and 30 μm, respectively. Galvanometer mirrors (Cambridge Technology, 6210H) enabled lateral scanning. A piezoelectric (PZT) actuator, coupled with a custom 3D-printed probe tip (2 mm wide, with an ~1 mm contact length), was used to generate surface vibrations in the sample. Pure-tone stimuli (1–20 kHz, ${f}_{s}$) were used to generate harmonic waves in the tissue, with each frequency excited individually. The number of excitation wave cycles at each transverse position was proportional to the frequency – for example, 4 cycles at 1 kHz, 8 cycles at 2 kHz, and up to 80 cycles at 20 kHz. For each OCE measurement, M-B scan mode was employed: 96 transverse positions (B-scan) were sampled, with 172 A-lines (M scan) recorded per position at 43.2 kHz. Fast Fourier Transform (FFT) analysis of M-scan profiles yielded the local amplitude and phase of displacements. Subsequent Fourier transformation in space revealed wave modes and wavenumbers. Phase velocity was calculated as $v=\omega /\mathrm{Re}(k)$, where $\omega =2\pi {f}_{s}$ and $k$ is the wavenumber.

Tissue samples were mounted on a custom biaxial stretcher. Carbon particles (~200 µm diameter) served as fiducial markers for stretch ratio calculations. OCE measurements were conducted along the axial and circumferential directions of the arterial samples at equibiaxial stretch ratios of 1 (stress-free), 1.1, 1.2, 1.3, and 1.4. OCE measurements were performed on the media side (air exposed), while the adventitial side was submerged in saline to prevent dehydration. Five measurements were performed at a single location for each stretch condition. At each stretch ratio, the wall thicknesses of the media and adventitia for each sample were directly measured from the OCT images, and these values were used as input for the subsequent wave dispersion fitting.

Analytic modeling of guided acoustic waves

Acoustic waves guided along the arterial wall exhibit frequency-dependent dispersion characteristics^72,78. When the tissue is under prestress, the dispersion relation is modulated via the acoustoelastic effect. To model this, we used the incremental dynamic theory of elasticity⁵⁵, where the wave equation is expressed as:

$$\nabla \cdot \Sigma =\rho {u}_{,{tt}}$$

(7)

Here, $\Sigma$ is the incremental stress tensor induced by acoustic waves with displacement $u$, $\rho$ is the tissue mass density, and $t$ denotes time.

Given a strain energy function $W$ of deformation gradient tensor $F$, the fourth-order Eulerian elasticity tensor is defined as ${{\mathcal{A}}}_{{ijkl}}^{0}={F}_{{iI}}{F}_{{kJ}}\frac{{\partial }^{2}W}{\partial {F}_{{jI}}\partial {F}_{{lJ}}}$^55,79. We adopted a Cartesian coordinate system with x and z axes in the tissue plane and the y axis normal to the tissue surface. The equibiaxial in-plane stretching condition is (${\lambda }_{x}={\lambda }_{z}$; incompressibility yields ${\lambda }_{y}={\left({\lambda }_{x}{\lambda }_{z}\right)}^{-1}$; and $F={\rm{diag}}({\lambda }_{x},{\lambda }_{y},{\lambda }_{z})$.

To simplify the wave equation, we used a scalar stream function $\psi$, such that ${u}_{x}={\psi }_{,y}$ and ${u}_{y}={-\psi }_{,x}$. For harmonic guided waves propagating along the x-axis, the assumed form is:

$$\psi \propto {e}^{{sky}}{e}^{i({kx}-\omega t)}$$

(8)

where $s$ is a complex decay parameter. Inserting Eq. (8) into the wave equation Eq. (7) yield a biquadratic equation in ${s}^{2}$, with two solutions ${s}_{1}^{2}$ and ${s}_{2}^{2}$. The full stream function is then written as $\psi ={\sum }_{j=-2,-1,1,2} {\psi }_{j}{e}^{{sign}(j){s}_{|j|}{ky}}{e}^{i({kx}-\omega t)}$.

In this study, we sequentially proposed single-layer elastic, single-layer viscoelastic, and two-layer viscoelastic models to progressively investigate the intrinsic viscoelastic tissue properties and layer-specific nature of the arterial wall. Boundary conditions are applied at the tissue-air and tissue-fluid interfaces. For the single-layer model, the upper surface is stress-free, and the lower surface maintains stress and displacement continuity with the fluid. For the two-layer model, additional continuity conditions are imposed at the media-adventitia interface. These boundary conditions lead to a secular equation:

$$\det \left({\rm{M}}\right)=0$$

(9)

where ${\rm{M}}$ is a 5×5 matrix for single-layer and a 9×9 matrix for two-layer models. Explicit forms of ${\rm{M}}$ are detailed in Supplementary Notes 1–4. Solving this equation yields the phase velocities for the A0 and S0 Lamb wave modes. These correspond to the two lowest-order solutions without a frequency cutoff⁸⁰, exhibiting quasi-symmetric and quasi-antisymmetric displacement profiles due to asymmetric boundary conditions.

To extract mechanical parameters, dispersion curves were fitted using a least-squares error function: $\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}{\left({v}_{i}^{(\exp )}-{v}_{i}^{({\rm{model}})}\right)}^{2}}$, where ${v}_{i}^{(\exp )}$ and ${v}_{i}^{({\rm{model}})}$ are the experimental and model-predicted phase velocities, respectively. A genetic algorithm was employed for minimization. The parameter space used in the fitting was set to $1{\rm{kPa}} < \alpha < 500{\rm{kPa}}$, $0.1 < \gamma /\alpha \le 1$, $1\le \beta /\alpha < 10$, $0 < \delta < 0.5$, and $0 < \eta < 0.05$ based on literature data^56,64,81.

Gasser–Ogden–Holzapfel (GOH) constitutive model

To describe fiber-reinforced anisotropic elasticity in arteries, we employed the Gasser–Ogden–Holzapfel (GOH) model⁸². This model assumes two symmetrically distributed families of collagen fibers embedded in a non-fibrous matrix. The strain energy density function is:

$$W=\frac{{\mu }_{0}}{2}\left({I}_{1}-3\right)+\frac{{k}_{1}}{{k}_{2}}\left({e}^{{k}_{2}{\left(\kappa {I}_{1}+\left(1-3\kappa \right){I}^{{\prime} }-1\right)}^{2}}-1\right)$$

(10)

Here, ${\mu }_{0}$ is the ground matrix shear modulus, ${k}_{1}$ is the fiber stiffness coefficient, and ${k}_{2}$ is a dimensionless exponent indicating nonlinear stiffening. The fiber dispersion parameter $\kappa$ ranges from 0 (aligned fibers) to 1/3 (random orientation). ${I}_{1}$ and $I{\prime}$ are the strain invariants defined as ${I}_{1}={{\lambda }_{c}}^{2}+{{\lambda }_{r}}^{2}+{{\lambda }_{a}}^{2}$ and $I{\prime} ={{\lambda }_{c}}^{2}{\cos }^{2}\varphi +{{\lambda }_{a}}^{2}{\sin }^{2}\varphi$. The invariants of the two fiber families, ${I}_{4}$ and ${I}_{6}$, are equal and thus combined as $I{\prime}$ (see details in Supplementary Note 5). ${\lambda }_{r}$, ${\lambda }_{c}$, and ${\lambda }_{a}$ are stretch ratios in the radial, circumferential, and axial directions, and $\varphi$ is the mean fiber angle.

The GOH model parameters were fitted to experimental values of $\alpha$, $\beta$ and $\gamma$, determined at multiple stretch ratios in both axial and circumferential directions (see Supplementary Note 5).

Finite element simulation

Finite element simulation (Abaqus/CAE 6.14, Dassault Systèmes) was conducted to verify the mode mixture of A0 and S0 modes observed in the OCE experiments. The model included a thin plate (2 mm thickness) atop a fluid substrate with a plane-strain configuration. The plate’s shear modulus was set to 100 kPa based on experimental data. Approximately 8000 plane-strain elements (CPE8RH) and 1600 acoustic elements (AC2D8) were used to discretize the tissue and fluid domains, respectively. Frequency analysis step was adopted to determine the modal shapes of the layer structure. Mesh convergence was verified via refinement tests. The spatial displacement fields under the A0 and S0 modes were extracted from the model and compared with the experimental results.

Phantom preparation and experiment

Two polydimethylsiloxane (PDMS) film phantoms were prepared using Sylgard 184 (Dow Corning) with base-to-curing agent ratios of 17:1 and 40:1 to produce stiff and soft layers, respectively, mimicking the adventitia and intima-media under stretched conditions. The stiff and soft layer thicknesses were ~1.0 mm and 0.7 mm. To enhance optical scattering and improve OCT imaging contrast, titanium dioxide nanopowder (~0.1 wt%) was added to both mixtures before curing. After thorough mixing and degassing, the samples were poured into molds and cured at 70 °C for 12 h.

In the OCE experiments, each layer was first tested individually in air using pure-tone excitations from 1 to 20 kHz to measure phase velocities. The two layers were then bonded together (soft layer on top, stiff layer bottom, with the lower surface in contact with water) through the natural tackiness of PDMS. The probe excited the bilayer sample from the top surface and the phase velocities were measured subsequently.

Statistical analysis

All results are presented as mean ± standard deviation. Comparisons between axial and circumferential data, as well as between media and adventitia data, were performed using two-sided unpaired Student’s t tests. A p-values less than 0.05 was considered statistically significant.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.