Layer-specific ultrasound elastography using a multi-layered shear wave dispersion model for assessing the viscoelastic properties

Abstract

Mechanical properties of biological tissues are significant biomarkers for diagnosing various diseases. Assessing the viscoelastic properties of multi-layer tissues has remained challenging for a long time. Some shear wave models have been proposed to estimate thin-layer tissues’ viscoelasticity recently. However, the potential applications of these models are highly restricted since few biological tissues are single-layered. Here we proposed a multi-layer model for layer-specific viscoelasticity estimation of biological tissues. Integrating the theoretical model and ultrasonic micro-elastography imaging system, the viscoelasticity of both layers was assessed. Dual-layer phantoms and ex-vivo porcine eyes were used to verify the proposed model. Results obtained from the mechanical test and shear wave rheological model using bulk phantoms were provided as validation criteria. The representative phantom had two layers with elastic moduli of 1.6 ± 0.2 kPa and 18.3 ± 1.1 kPa, and viscosity moduli of 0.56 ± 0.16 Pa·s and 2.11 ± 0.28 Pa·s, respectively. The estimated moduli using the proposed model were 1.3 ± 0.2 kPa and 16.20 ± 1.8 kPa, and 0.80 ± 0.31 Pa·s and 1.87 ± 0.67 Pa·s, more consistent with the criteria (one-tailed t-test, p<0.1). By contrast, other methods, including the group velocity method and single-layer Rayleigh-Lamb model, generate significant errors in their estimates. For the ex-vivo porcine eye, the estimated viscoelasticity was 23.2 ± 8.3 kPa and 1.0 ± 0.4 Pa·s in the retina, and 158.0 ± 17.6 kPa and 1.2 ± 0.4 Pa·s in the sclera. This study demonstrated the potential of the proposed method to significantly improve accuracy and expand clinical applications of shear wave elastography.

1. Introduction

The mechanical properties of biological tissues are considered significant biomarkers for diagnosing various diseases. For example, breast cancer [1] and liver fibrosis [2] are clinically examined based on tissue stiffness. Metabolic syndrome [3], coronary artery disease [4], and hypertension [5] are associated with increased arterial stiffness. Assessment of cornea’s viscoelasticity can improve the diagnosis and monitoring of ophthalmic diseases, like keratoconus [6, 7]. The mechanical properties of posterior eye tissues, such as retina and sclera, can be altered by various eye diseases, including age-related macular degeneration (AMD) and glaucoma [8, 9]. Therefore, accurately assessing the viscoelasticity of biological tissues is important in disease monitoring and clinical diagnosis [10, 11]. As an imaging technique for non-invasively mapping the viscoelastic properties of biological tissues, shear wave elastography (SWE) has received substantial attention in recent years [8]-[23].

In elastography, an elastic wave is induced by acoustic radiation force (ARF) or mechanical shaker, and propagates in tissue. The induced wave propagation or vibration is detected by various imaging modalities, including ultrasonography [12–17], OCT [18–24] and MRI [25–27], to quantify the wave’s amplitude and velocity. Based on this information about the induced wave propagation, tissue viscoelasticity can be derived utilizing different biomechanical models. Many remarkable efforts have been devoted to investigating the relationship between wave propagation and tissue mechanical properties. However, none of the proposed models can be utilized for layer-specific assessments of multi-layered tissue viscoelasticity.

Group-velocity-based SWE was developed as the first model that can help determine the absolute Young’s modulus. It assumes that a shear wave propagates in a purely elastic, isotropic, homogeneous, infinitely large medium. Such methods have been verified and applied in large-scale tissues, such as breast [1, 14], liver [2, 12, 15], and kidney [28, 29]. Recent work by Rouze et al provided a new theoretical method to characterize the viscoelasticity using group shear wave velocity [30]. However, these group-velocity-based methods only work with bulky tissues and will result in biased estimation of elasticity in confined geometries, like skin, cornea, and artery. This is due to the fact that in tissue with thickness comparable to the wavelength, wave propagation switches from shear wave mode to Rayleigh-Lamb wave mode [16, 31].

Phase-velocity-based SWE was proposed to map both elasticity and viscosity in tissues [16, 24, 32]. By transforming the spatial-temporal domain displacement map to the k-space map, the phase velocity-frequency dispersion curve can be retrieved. The dispersion curve is then fit to the theoretical model to estimate tissue elasticity and viscosity. The scope of applications for this method can be expanded by adopting different wave propagation models. The Rayleigh-Lamb wave model is one of the most popular models to quantify the viscoelasticity of tissues with thin layered structures. The Rayleigh-Lamb wave model with different boundary conditions has been investigated theoretically or with finite-element methods [33, 34], and has been applied to assess the viscoelasticity of various thin-layer tissues [35, 36], including arteries [16, 31], bladder wall [37], skin [38] and corneas [16, 17, 39].

However, all models in these studies can only estimate the properties of single-layer tissues. This critical limitation forced them to consider multi-layer structures as one layer and ignore the viscoelasticity distribution among layers. For example, although skin has multi-layer structures, including epidermis, dermis, and a subcutaneous layer attached to muscles, some studies had to treat the skin as a single-layer structure and assumed that the subcutaneous layer has an infinite thickness [38, 40, 41]. Anatomical studies demonstrate that there are many biological tissues, like skin, arteries [42] and retina [43], that have multi-layer structures with different viscoelasticity in each layer. Layer-specific elastography of these multi-layered tissues can provide more detailed information and would be potentially beneficial in clinical diagnosis. Many efforts have been devoted to investigating the elastic waves in multi-layer tissues [44–49], while a multi-layer model in biomedical field is still an unmet need. Traditionally, to do layer-specific measurements, target tissues are excised and layered to test them separately to accurately measure the viscoelasticity of each layer. However, it is impossible to do in vivo measurements in this way. Due to the internal reflections at interfaces, shear wave propagation in such multi-layer tissues is different from and more complex than in single-layer tissues.

In this paper, we first derive the phase velocity-frequency dispersion relationship of a dual-layer model from wave equations. Then, dual-layer phantom experiments are conducted by integrating our theoretical model and ultrasonic micro-elastography imaging system [50]. Results obtained from the mechanical test and shear wave rheological model using bulk phantoms are provided as verification criteria. All results verify the validity and feasibility of our dual-layer model and the inaccuracy of the single-layer model. Ex vivo experiments conducted on porcine posterior eyes are reported. Further, a methodology for extending the given dual-layer model to multi-layer models is discussed.

2. Theory

To investigate wave propagation in multi-layers solids, we start from a dual-layer model with the basic wave equations [51] of the longitudinal (1) and shear (2) waves assuming the excitation is harmonic:

$$\rho \frac{{\partial }^2\varnothing }{{\partial }^{2}t}=(\lambda +2\mu ){\nabla }^2\varnothing$$ (1)

$$\rho \frac{{\partial }^2\mathit{\psi }}{{\partial }^{2}t}=\mu {\nabla }^2\mathit{\psi }$$ (2)

where λ and μ are Lame constants. ∅ and ψ are scalar and vector potentials of the viscoelastic wave field. ρ is the material density. Hence, the longitudinal and shear wave velocities are $c_L=\sqrt{\left(\lambda +2\mu \right)/\rho }$ and $c_s=\sqrt{\mu /\rho }$ Wave propagation in fluids can be seen as a special case of wave propagation in solids, where shear modulus μ equals zero, blocking the existence of shear waves in fluids. Soft tissue is considered to be a Voigt material, so its shear modulus μ = μ₁ + jωμ₂, where μ₁ and μ₂ are the shear elasticity and viscosity [52].

Because shear wave propagation is assumed to be cylindrical and each layer is isotropic, wave propagation is axisymmetric and independent of the angle θ, which is the angle of rotation of the r-axis around the z-axis in Fig. 1. Expressing the gradient and the curl operators in cylindrical coordinates and introducing this simplification into (1) & (2), the following equations are obtained:

$$\frac{1}{c_L^2}\frac{{\partial }^2\varnothing }{\partial t^2}=\frac{{\partial }^2\varnothing }{\partial r^2}+\frac{1}{r}\frac{\partial \varnothing }{\partial r}+\frac{{\partial }^2\varnothing }{\partial z^2}$$ (3)

$$\frac{1}{c_s^2}\frac{{\partial }^2\psi }{\partial t^2}=\frac{{\partial }^2\psi }{\partial r^2}+\frac{1}{r}\frac{\partial \psi }{\partial r}+\frac{{\partial }^2\psi }{\partial z^2}-\frac{\psi }{r^2}$$ (4)

Here ψ = ψ_ϑ is one of the scalar components of vector potential ψ. Since the wave propagation is independent of the angle θ and the velocity component ν_ϑ=0, we have ψ_r = ψ_z = 0. The boundary conditions between layer (i) and layer (i+1) are:

$${\sigma }_{zzi}={\sigma }_{zz\left(i+1\right)}$$ (5)

$${\sigma }_{rzi}={\sigma }_{rz\left(i+1\right)}$$ (6)

$${\epsilon }_{zi}={\epsilon }_{z\left(i+1\right)}$$ (7)

$${\epsilon }_{ri}={\epsilon }_{r\left(i+1\right)}$$ (8)

Fig. 1.

(a) The mechanical model of dual-layer biological tissue and (b) Schematic diagram of the ultrasonic micro-elastography imaging system, with ring-shape focused pushing transducer and high-frequency needle transducer.

Here ε is displacement and σ is stress, which have relationships with potential functions [31]:

$${\epsilon }_r=\frac{\partial \varnothing }{\partial r}-\frac{\partial \psi }{\partial z}$$ (9)

$${\epsilon }_z=\frac{\partial \varnothing }{\partial z}-\frac{1}{r}\frac{\partial \left(r\psi \right)}{\partial r}$$ (10)

$${\sigma }_{rz}=\mu \left(\frac{\partial {\epsilon }_z}{\partial r}+\frac{\partial {\epsilon }_r}{\partial z}\right)$$ (11)

$${\sigma }_{zz}=\left(\lambda +2\mu \right)\frac{\partial {\epsilon }_z}{\partial z}+\frac{\lambda }{r}\frac{\partial \left(r{\epsilon }_r\right)}{\partial r}$$ (12)

At the interface between solid and fluid, the boundary condition defined by (8) need not be satisfied because particle displacement in the r direction (ε_ri) is not continuous. Also, the shear stresses (6) here are zero. Considering wave reflection at the boundaries, the field in the z-direction should have the expression of potential functions in solids and can therefore be written as [53]

$${\varnothing }_i=\int \left[A_i\text{cosh}\left(q_{i}z\right)+B_i\text{sinh}\left(q_{i}z\right)\right]H_0^{\left(2\right)}\left(kr\right)e^{j\omega t}d\omega$$ (13)

$${\psi }_i=\int \left[C_i\text{cosh}\left(s_{i}z\right)+D_i\text{sinh}\left(s_{i}z\right)\right]H_1^{\left(2\right)}\left(kr\right)e^{j\omega t}d\omega$$ (14)

where the subscript i represents the layer number, $q_i^2=k^2-k_{Li}^2$, and $s_i^2=k^2-k_{si}^2$. k_Li and k_si are the wave vectors of longitudinal and shear wave propagation in layer (i), equal to $\frac{\omega }{c_{Li}}$ and $\frac{\omega }{c_{si}}$ individually. $k=\frac{\omega }{c}$ is the wave vector of the Lamb wave, and c is the phase velocity. $H_0^{\left(2\right)}$ and $H_1^{\left(2\right)}$ are the Hankel functions of the second kind and zero/first order.

For potential functions in fluids, a natural boundary condition exists: since waves can propagate to infinity, the potential function cannot be infinite in the region of interest, and therefore, the solution to (13)–(14) is a traveling waveform and the unstable solutions should be discarded. With this condition, we obtain:

$${\varnothing }_1=\int \left(A_1{e}^{-q_{1}z}\right)H_0^{\left(2\right)}\left(kr\right)e^{j\omega t}d\omega$$ (15)

$${\varnothing }_4=\int \left(A_4{e}^{-q_{4}z}\right)H_0^{\left(2\right)}\left(kr\right)e^{j\omega t}d\omega$$ (16)

Subscripts 1 and 4 indicate the up-and down-side fluid.

Biological tissues are modeled as dual-layer structures submerged in an incompressible fluid similar to water. As shown in Fig.1, there are three boundaries at z = 0, h₁, and h₂. Either interface between solid and fluid has three boundary conditions (5–7) and the boundary between solids has four boundary conditions (5–8). By introducing (9–12) and (13–16) into (5–8), we can express these ten boundary conditions in Matrix form, where the first three rows and last three rows represent solid-fluid boundaries at z=h₁, h₂ and the middle four rows represent the solid-solid boundary at z=0:

$$\begin{gathered} \text{X}=\left[{\text{A}}_1{\text{A}}_2{\text{B}}_2{\text{C}}_2{\text{D}}_2{\text{A}}_3{\text{B}}_3{\text{C}}_3{\text{D}}_3{\text{A}}_4\right] \\ \text{M}\cdot \text{X}=\widehat{0} \end{gathered}$$ (17)

$$\text{det}\left(\text{M}\right)=0$$ (18)

where u_i = μ_i1 + jωμ_i2 (i = 2,3) are the shear moduli of the second and third layer, and $m_i=2k^2-k_{si}^2\left(i=2,3\right)$ are used to simplify the expression of M.

$$\text{M}=\left[\begin{array}{cccccccccc}p{w}^2{e}^{-q_1{h}_1}& -u_2{m}_1\text{sinh}\left(q_2{h}_1\right)& -u_2{m}_1\text{cosh}\left(q_2{h}_1\right)& 2k{u}_2{s}_2\text{cosh}\left(s_2{h}_1\right)& -2k{u}_2{s}_2\text{sinh}\left(s_2{h}_1\right)& 0& 0& 0& 0& 0\\ 0& 2k{q}_2\text{cosh}\left(q_2{h}_1\right)& 2k{q}_2\text{sinh}\left(q_2{h}_1\right)& m_1\text{sinh}\left(s_2{h}_1\right)& m_1\text{cosh}\left(s_2{h}_1\right)& 0& 0& 0& 0& 0\\ q_1{e}^{-q_1{h}_1}& q_2\text{cosh}\left(q_2{h}_1\right)& q_2\text{sinh}\left(q_2{h}_1\right)& k\text{sinh}\left(s_2{h}_1\right)& k\text{cosh}\left(s_2{h}_1\right)& 0& 0& 0& 0& 0\\ 0& 0& u_2{m}_1& 2k{u}_2{s}_2& 0& 0& -u_3{m}_2& -2k{u}_3{s}_3& 0& 0\\ 0& 2k{u}_2{q}_2& 0& 0& u_2{m}_1& -2k{u}_3{q}_3& 0& 0& -u_3{m}_2& 0\\ 0& q_2& 0& 0& k& -q_3& 0& 0& -k& 0\\ 0& 0& k& s_2& 0& 0& -k& -s_3& 0& 0\\ 0& 0& 0& 0& 0& -u_3{m}_2\text{sinh}\left(q_3{h}_2\right)& -u_3{m}_2\text{cosh}\left(q_3{h}_2\right)& -2k{u}_3{s}_3\text{cosh}\left(s_3{h}_2\right)& -2k{u}_3{s}_3\text{sinh}\left(s_3{h}_2\right)& \rho {\omega }^2{e}^{q_4{h}_2}\\ 0& 0& 0& 0& 0& 2k{q}_3\text{cosh}\left(q_3{h}_2\right)& 2k{q}_3\text{sinh}\left(q_3{h}_2\right)& m_2\text{sinh}\left(s_3{h}_2\right)& m_2\text{cosh}\left(s_3{h}_2\right)& 0\\ 0& 0& 0& 0& 0& -q_3\text{cosh}\left(a_2{h}_2\right)& -q_3\text{sinh}\left(a_2{h}_2\right)& -k\text{sinh}\left(s_3{h}_2\right)& -k\text{cosh}\left(s_3{h}_2\right)& q_4{e}^{q_4{h}_2}\end{array}\right]$$

To ensure that the system of equations (17) has a nontrivial solution, the determinant of M must be zero (18). We note that there are only two variables in M, angular frequency ω and phase velocity c. Therefore, by solving (18), phase velocity-frequency dispersion curves can be derived. These curves are significant relative to the viscoelastic modulus of both layers. By fitting the experimentally measured phase velocity-frequency dispersion curves to the model results, elasticity and viscosity coefficients of both layers can be determined.

Now that we have derived the characteristic matrix for a dual-layer model, it can be extended to a multi-layer model easily. We note that whenever there is one more layer, four more equations (5)–(8) occur at the new interface and four more variables in (13) and (14) occur in the new layer. Therewith, the size of matrix M increases by four in both directions. The same calculations in (16) and (18) can be applied to the new expanded matrix for M, and the phase velocity-frequency dispersion curves of the new multi-layer structures can be determined.

3. Materials and Methods

To verify the validity and feasibility of the theoretical development described in Section II above, experiments with phantoms and ex vivo tissue were conducted.

A. Experimental setup

The schematic of the experimental setup is shown in Fig. 1(b). A focused 6.8-MHz ring transducer and a flat 48-MHz needle transducer were used in this elastography system. The two transducers were aligned carefully in a confocal manner along both axial (z) and lateral (r) directions, under the guidance of a hydrophone before the experiments. The 6.8-MHz ring transducer made of modified PZT was used for generating the ARF to induce tissue displacement. It has a diameter of 23 mm, and a central hole with a diameter of 10 mm. The focal length is 23 mm. The −6 dB lateral width of the focal beam at the focus is 0.2 mm. The axial depth of focus was 3.2 mm. A multi-layered sample was positioned in the focal zone of the pushing beam (radiation pressure beam), which is shown in Fig. 1(b). The axial depth of 3.2 mm covered both first and second layers of the phantom. The 48-MHz needle transducer was made of a flat PIN-PMN-PT piece with a size of 0.3 mm by 0.3 mm. It was used for detecting shear wave propagation inside dual-layer structures. In pulse-echo test, its −6 dB bandwidth was 83.09%. The needle transducer was attached to a motor platform (SGSP33–200, OptoSigma Corporation, Santa Ana, CA, USA) for mechanical scanning with an incremental motor step of 39 μm. The 6.8-MHz ring transducer was excited by a function generator (AFG3252C, Tektronix, Beaverton, OR, USA) connected to a radiofrequency power amplifier (100A250A, Amplifier Research, Souderton, PA, USA). Sinusoidal tone bursts of 200-μs duration (1360 cycles at 6.8 MHz) were produced using the function generator, then the output bursts were amplified to peak-to-peak amplitudes of 100 V by the power amplifier. A pulser-receiver (JSR500, Ultrasonics, NY, USA), triggered by the function generator, was used to drive the 48-MHz needle transducer for transmitting and receiving ultrasound signals with a pulse repetition frequency (PRF) of 10 kHz. The ultrasound signals backscattered from the tissue were filtered using a bandpass filter. The PRF trigger from the function generator was also used to synchronize the acquisition of backscattered signals at a maximum sampling frequency of 1.8 GHz by a 12-bit analog-to-digital converter (ADC) (ATS9360, Alazartech, Montreal, QC, Canada). The pulser-receiver, function generator, and ADC were all synchronized using a synchronization generator (33250A; Agilent, Santa Clara, CA, USA) while the needle and ring transducers were operated continuously. In this setup, the shear wave in the tissue was induced by the impulse method. The needle transducer received the backscatter signal before tissue motion (t<0.1 ms) for determining the initial tissue position. As opposed to an ultrasound array, it is almost impossible to use a single-element ultrasound transducer to implement a high imaging speed. To obtain images quick enough to track shear waves, a previously reported ultrasonic micro-elastography system was used in this study [54]. When the imaging transducer was at each lateral position, the pushing transducer was activated once for 0.2 ms and the imaging transducer was activated for 10 ms with a PRF of 10 kHz to acquire 100 A-lines (M-mode). After that, the imaging transducer was moved to the next lateral position and a 100 ms delay was introduced to let previous vibrations settle out. The pushing and imaging cycle were then repeated. Using this setup, although the imaging speed was relatively low, the interval between each frame was only 0.1 ms, which provided an equivalent frame rate of 10 kHz.

B. Phantom and biological tissue preparation

In the study, gelatin (Gelatin G8-500, Fisher Scientific, USA) based tissue-mimicking phantoms were fabricated, together with the same concentration of silicon carbide powder (S5631, Sigma-Aldrich, St. Louis, MO, USA) as the acoustic scatters. Phantoms comprising gelatin at 5%, 8.5%, and 15% concentrations were used in this study to represent materials with different elasticity and viscosity. Liquid gelatin was poured into a container with a cylindrical cup with a 10-mm diameter at its bottom. After the lower layer gelatin had gelled, the upper layer gelatin was poured on top of it. Then, the cap was removed, and the phantoms were considered to have a dual thin-layer structure containing two free boundaries with a 10-mm diameter. Detailed parameters of dual-layer phantoms were listed in Table I. Homogeneous bulk phantoms for shear wave rheological model had a size of 100-mm length, 100-mm width, and 50-mm depth and same gelatin concentrations as layered phantoms. All experiments were performed at a constant room temperature of 25 °C to avoid drift in the stiffness with temperature [55]. Repeated measurements were performed three times, rotating the sample by 120° each time. To check the accuracy of the estimated elasticity in this study, uniaxial quasi-static mechanical testing (Model 5942, Instron Corp., MA, USA) was performed on phantoms with the same concentration and 20-mm diameter and 10-mm height to measure the shear elasticity. In these experiments, the phase velocities were measured by the impulse method, and numerical fitting was performed on the velocity at frequencies in the range of 200–1000 Hz. Bar graph (1) in Fig. 2 illustrates the fitted results of the shear elasticity and shear viscosity of all phantoms. Biological tissue experiments were performed on fresh un-scalded porcine eyeballs ex vivo, which were collected from a local slaughterhouse (Sierra Medical Science, Inc., Whittier, CA, USA), with all experiments being performed within 12 h of collection. To avoid acoustic distortion resulting from the anterior eye, the cornea was removed. The prepared samples were immersed in a saline solution in a water tank when performing the experiments. All experiments were performed at room temperature (25 °C).

Table 1.

The designed parameters of phantoms used in this study.

	Thickness (mm)		Gelatin Concentration (%)
	Layer 1	Layer 2	Layer 1	Layer 2
SAMPLE. 0	1	3	5	15
SAMPLE. 1	1	3	5	15
SAMPLE. 2	1	3	8.5	15
SAMPLE. 3	1	3	15	8.5
SAMPLE. 4	0.5	3	8.5	15
SAMPLE. 5	1.5	3	8.5	15

Fig. 2.

(a-c) Post-processing results using the proposed multi-layer model. Maps of axial displacement (a), k-space (b), and a plot of phase velocity vs. frequency (c). In the axial displacements map, color represents the average displacement of the phantom in μm. (d)&(e) Axial displacement maps of either layer. Red lines are the linear regression using the group velocity method. (f) Phase velocity vs. frequency curves acquired using the single-layer Rayleigh-Lamb model. All estimation results from different methods are combined in bar graphs (g) & (h), which demonstrate the estimated elasticity and viscosity, respectively. Phantom results from bulk samples and the mechanical test are also given in the last two columns as validation criteria.

C. Post-Processing

The raw data acquired by our system was processed using MATLAB R2016b software (The MathWorks, Natick, MA, USA). Fig. 2 demonstrates the post-processing steps in all methods to retrieve the shear viscoelasticity of tissues from the raw data. The axial displacements over time along each lateral position and depth were computed using the normalized cross-correlation algorithm with a 1.5λ window size [16]. To obtain the axial displacement maps along the lateral position and time for the multi-layer model, the axial displacements were averaged over depth in all layers. In contrast, when the group velocity method and single-layer Rayleigh-Lamb model were used, the axial displacements were averaged in either layer separately. Distinguishing two layers was realized by analyzing B-mode imaging data. In single-layer Rayleigh-Lamb model, all boundary conditions were set to fluid-solid boundary. More details of single-layer model can be found in our previous work [16]. Using the 2-D Fourier transform, the k-space map along the wavenumber and frequency were then transformed from half of the spatiotemporal map. The main wavenumber k at each frequency f was identified as the maximum intensity at that frequency. Finally, the phase velocity-frequency curve was calculated using:

$$c_p=2\pi f/k$$ (19)

Curve fitting between the experimental phase velocity-frequency curve and the curve derived from the multi-layer model was performed using the linear least-squares method. In this method, the error is defined as E² = ∑(c_m(f_i) − c_e(f_i))², where c_m is velocity calculated from the curves and c_e is the velocity acquired from the experimental data. The shear elasticity (μ₁₁, μ₁₂) and viscosity (μ₂₁, μ₂₂) were varied in the model with a step size of 0.1 kPa and 0.1 Pa·s until the error E² was minimized. In this study, the error ratio (ER) was also calculated to describe the deviation between the fitted curve and the experimental data.

$$\text{ER}=100\%*\text{Average}(\sqrt{{\left(c_m\left(f_i\right)/c_e\left(f_i\right)-1\right)}^2})$$ (20)

In the group velocity method, linear regression was applied to the propagation time versus the propagation distance in the axial displacement maps of either layer, to obtain the shear wave’s group velocities. The wavefront of shear waves envelope was determined by the maximum derivative of displacement with respect to time at every lateral position. Then group velocity was determined by the slope of the line as c_g = Δx/Δt. Given the assumption that the shear wave propagates in a purely elastic, isotropic, homogeneous, infinitely large medium, the relationship between shear elasticity (μ) and group velocity (c_g) is given by:

$$\mu =\rho c_g^2$$ (21)

To verify the estimated viscosity using our dual-layer model, the shear wave rheological method was applied to bulk phantoms with the same gelatin concentrations. Dispersive phase velocity in bulk phantoms is related to the frequency as (22) [56, 57], where μ₁, and μ₂ are the shear elasticity and viscosity, respectively, and c_p is given by:

$$c_p=\sqrt{\frac{2\left({\mu }_1^2+{\omega }^2{\mu }_2^2\right)}{\rho ({\mu }_1+\sqrt{\left({\mu }_1^2+{\omega }^2{\mu }_2^2\right)})}}$$ (22)

4. Results

A. Estimated results using five different methods

In Fig. 2, representative results of each post-processing step are shown, which are obtained from the phantom experiment on the Sample 0 (1 mm and 3 mm, 5% and 15% gelatin). From top to bottom, Fig. 2(a–c) show the axial displacement map, k-space map, and phase velocity-frequency curve, respectively. The guided wave traveled axisymmetrically from the focal region of the pushing transducer. The axial displacements map in Fig. 2(a) shows the propagating wave in (r) directions. The area surrounded by the orange rectangular was transformed to k-space using a 2D-FFT. Since high frequency (>1000 Hz) components are attenuated quickly and low-frequency (<200 Hz) components are distorted by low-frequency noise, we selected the k-space region from 200 Hz to 1000 Hz for the dispersion map and rejected all frequency above and below this band [16]. The black circles in the k-space map (Fig. 2(b)), represent the main wavenumber at each frequency. The phase velocity-frequency curve was calculated using (19) and numerically fit to our model. Fitted elasticity and viscosity are shown as the first column in bar graphs Fig. 2(g) and (h), respectively, with a relatively low ER=0.85%. The estimated moduli using the proposed model are 1.3 ± 0.2 kPa and 16.2 ± 1.3 kPa, and 0.80 ± 0.31 Pa·s and 1.87 ± 0.67 Pa·s. The single-layer Rayleigh-Lamb model and the group velocity method were also used to process the same phantom data. Details of single-layer model can found in our previous work [16]. When the single-layer model was applied to fit this dual-layer phantom, two fitted curves with relatively high ERs=22.37% and 8.8% were obtained and shown in Fig. 2(f). Fitted elasticity and viscosity are shown as the second column in bar graphs (g) and (h) in Fig. 2. The estimated elastic moduli were 32.4 ± 4.2 kPa and 13.8 ± 1.8 kPa. Fig. 2(d)&(e) show the linear regression results in either layer and the elasticities estimated using the group velocity method are shown as the third column in bar graphs. The group velocity method estimated the elasticity as 4.92 ± 0.6 kPa and 8.39 ± 0.5 kPa. The fourth and fifth columns in bar graphs show estimated elasticity and viscosity using the bulk sample and mechanical tests, which are considered validation criteria, with the elastic moduli of 1.6 ± 0.2 kPa and 18.3 ± 1.1 kPa, and viscosity moduli of 0.56 ± 0.16 Pa·s and 2.11 ± 0.28 Pa·s, respectively.

One-tailed t test was conducted to compare the accuracy of estimated elasticity using the single-layer model and the multi-layer model. The results supported that the proposed multi-layer model significantly increased the accuracy in elasticity measurements in both layers (first layer: p<0.0001, second layer: p<0.1).

B. Phantoms with varied gelatin concentrations

Fig. 3(a) compares the phase velocity curves and fitted results of three types of phantoms (Samples 1–3), with gelatin concentrations of 5%, 8.5%, and 15%. The mechanically tested shear elasticity was 2.1 ± 0.2, 6.4 ± 0.2, and 20.3 ± 0.4 kPa, respectively. Phantom samples 1–3 had the same dual-layer structures but different gelatin concentrations. Detailed parameters of the sample can be found in Table I. In Fig. 3(b), bar graphs collect and compare the estimated elasticities and viscosities of three phantoms obtained from all methods.

Fig. 3.

Samples having different gelatin concentrations and distributions. Phantom results and fitted curves using the multi-layer model are shown in (a), with errors smaller than 1.5%. The bar graphs (b) collect the elasticities (first column) and viscosities (second column) of all three samples, estimated by different methods.

C. Phantoms with varied thickness

Fig. 4 demonstrate the results of phantoms with different thickness (Samples 2, 4 and 5). The first layer thicknesses of three phantoms were 1 mm, 0.5 mm, and 1.5 mm, while the second layer has a thickness of 3 mm. Detailed parameters can be found in Table I. All three samples had the same gelatin concentration and distribution with the first layer being 8.5% Gelatin and the second layer 15% Gelatin, so their estimated elasticity and viscosity was the same. Using the multi-layer model, the elasticity and viscosity of Sample 2 were estimated as 5.4 ± 0.3 kPa and 17.8 ± 1.4 kPa, 1.1 ± 0.36 Pa·s and 2.21 ± 0.64 Pa·s, for both layers respectively. Acquired parameters were input back to the dual-layer model, and the thicknesses of the first layers in the model were changed to 0.5 mm and 1.5 mm, to acquire new phase velocity-frequency curves. These new curves are compared with phantom results from Samples 4 and 5, whose first layer thicknesses were also 0.5 mm and 1.5 mm. Fig. 4(a) demonstrates acceptable matching between model and phantom results with altered thickness. The observed error in this case was less than 2%.

Fig. 4.

Phase velocity vs. frequency curves for three samples with different thicknesses compared with the model-fitted curves (a). Errors in estimated elasticity and viscosity caused by using the wrong thickness for the first layer (b)(d) and second layer(c)(e).

Figs. 4(b)–(e) demonstrate the thickness-dependent errors of the proposed model when Sample 2 was used as the representative sample. The real thickness was 1 mm for the first layer and 3 mm for the second layer. Over a range of thickness from −50% to 50% of the original in increments of 10% of the real thickness in the dual-layer model, the experimental phase velocity-frequency curves retained a good fit to the model. Errors caused by using incorrect thickness for the first layer are shown in Fig. 4(b) (elasticity) and Fig. 4(d) (viscosity). Errors caused by using incorrect thickness for the second layer are shown in Figs. 4(c) and (e) for elasticity and viscosity, respectively. When the first layer’s thickness was altered by 20%, the errors in elasticity were less than 19% for the first layer and 3% for the second layer. The errors in viscosity are less than 6% and 9%. When the second layer’s thickness was altered by 20%, the errors in elasticity were smaller than 14% and 13% for the first and second layers respectively. The errors in viscosity for both first and second layers were less than 22% and 16% respectively. Thickness bias in either layer affects the estimated viscoelasticity individually and the thickness-dependent errors were within an acceptable range.

D. Ex vivo experiments on porcine eye

Dual-layer structures of the porcine eye can be discriminated in B-mode images as illustrated by Fig. 5(d). This is due to the fact that the two layers have different signal amplitudes and strong reflections can be observed at boundary between layers. The thicknesses of the porcine retina and sclera were measured to be 0.45 ± 0.10 mm and 0.85 ± 0.19 mm. Fig. 5 shows the ex vivo results for the posterior part of a porcine eye. The wave propagations in retina and sclera are presented in Figs. 5(a)–(c) as spatial-temporal maps at three timepoints. The wave dispersion curve and the model-fitting curve are shown in Fig. 5(e). The experimental results were fit using the proposed dual-layer model with an ER of 8.51%. The elasticities and viscosities of both layers were measured to be 23.2 ± 8.3 and 153.5 ± 17.6 kPa, 1.0 ± 0.4 and 1.2 ± 0.4 Pa·s, respectively. For comparison, the group velocity method was also applied and estimated the elasticities of both layers as 78.4 ± 5.0 and 89.7 ± 12.9 kPa.

Fig. 5.

Ex vivo results for a posterior porcine eye. Spatial-temporal map of the wave propagation at different time points (a)-(c). The B-mode image (d) was used to discriminate the two layers. Calculated phase velocity and the fitted curve using the multi-layer model (e). Estimated elasticity (black bar) and viscosity (brown bar) (f).

5. Discussion

Taken together, the results presented in this paper demonstrate that the experimental phase-velocity data is best-described using the proposed multi-layer model, with the conventional group velocity method and single-layer Lamb wave model yielding less satisfactory results. When the subject tissue has a multi-layer structure, the estimated viscoelasticity data using the multi-layer model are in good agreement with the calibrated values under the assumption of Voigt model. However, large errors occurred in the single-layer Rayleigh-Lamb model, especially when it was used on the thinner layer. Also, the multi-layer model has smaller ERs, which implies that it fits the experiment results better, while the single-layer Rayleigh-Lamb model has much larger ERs, suggesting that the single-layer model cannot be used to fit the multi-layer tissues. This due to the fact that the internal reflections at interfaces were ignored in the single-layer model and strong interference between the multiple layers greatly affects wave dispersion. It is generally understood that the variance of estimates increases with the number of parameters. However, in our case, there were some instances for which the variance in the single-layer model was even larger than that for the dual-layer model. The reason for this discrepancy is that the single-layer model is not appropriate for a dual-layer phantom. For example, in Fig. 2(b), the single-layer model does not provide a decreasing dispersion curve, and in this case unacceptably large errors were observed, and the variance of estimates becomes large as well. Estimated elasticities using the group velocity method are between the validated elasticities of two layers, because the wavefront in two layers converges into one shear wave and neutralizes their group velocities.

To confirm the feasibility of the proposed method in various types of biological tissues, the multi-layer model was applied to phantoms with different gelatin concentrations, distributions, and thicknesses. Three different gelatin concentrations and three different thicknesses were used in the experiment. All estimated results were in good agreement with the validation criteria. The gelatin distribution was also reversed, and the proposed model still resulted in similarly accurate estimates. Furthermore, since it is difficult to accurately determine the true thickness of each layer in real tissue, the effect of thickness error on the model-based estimation was evaluated and noted in Figs. 4(b)–(e). The thickness error in each layer results in estimation bias in both layers. The bias is within an acceptable range when the thickness error is smaller than 30%. Another possible factor causing bias in estimation is that the proposed model assumed the thicknesses are spatially invariable, which can be inconsistent with the real conditions. This bias should be negligible in most practical applications. For example, the retina thickness only varies ~50 um [58], which is much shorter than the wavelength of shear waves (mm level) and does not affect wave propagation. In this study, the posterior eye was chosen as the respective multi-layer tissue for the ex vivo experiment. The posterior eye is normally understood to be a three-layer structure, consisting of retina, choroid, and sclera. Assessing the viscoelasticity of all three parts of the posterior eye is meaningful and important for clinical diagnosis and treatment. Various eye diseases like AMD, glaucoma and some retina diseases usually lead to a change in the viscoelasticity of the retina and sclera. There have been many efforts devoted to measuring the mechanical properties of posterior eyes and their results vary. Measured elasticity of retina ranged from 5 to 22 kPa [18, 19]. For the sclera, measured elasticity ranged from 300 kPa to 2.3 MPa [25, 59, 60]. However, none of the previous studies considered the posterior eye’s confined multi-layer geometry. The complex multi-layer structure of the posterior eye has prevented researchers from achieving accurate estimations in previous work. Significant misestimation of elasticity has occurred in past work using the group velocity method. Due to the fact that the retina is much softer than the other two tissue structures, and it is tightly attached to the choroid, shear waves propagating in the retina are strongly distorted by the choroid and sclera. Changes in the retina’s viscoelasticity are difficult to observe using the conventional group velocity method or Lamb wave model.

The estimation of viscosity is more challenging than estimating elasticity. Regardless of the methods or models used, viscosity usually have larger error bars/standard deviations compared with elasticity. [61] In general, the results reported in this manuscript were within the normal range. Viscosity of gelatin phantoms reported in different literatures varied a lot as well. Bo et al. [62] reported the viscosity of 10% and 15% gelatin phantoms were 1.65–2.15 Pa·s and 0.72–4.04 Pa·s, estimated by both impulse and continuous wave methods. Chih-Chung et al. [63] reported the viscosity of 3% and 7% gelatin phantoms were 0.12 ± 0.02 Pa·s and 0.86 ± 0.05 Pa·s. Ivan et al. [31] reported the viscosity of 15% gelatin phantoms varied from 0.3 to 1.1 Pa·s. In our previous work [16], the viscosity of 7% and 12% gelatin phantoms varied from 0.2 to 0.69 Pa·s and 0.68 to 1.7 Pa·s.

The proposed multi-layer model in this study has the potential to accurately estimate the viscoelasticity of a multi-layer structure. To simplify the model, the choroid is considered as part of the sclera in this study. This should not induce noticeable errors because the elasticity of choroid is normally around 1 MPa [64], which is in the same range as the elasticity of sclera. In our reported results, the estimated elasticities of both retina and sclera using the proposed multi-layer model are within the normal range. The estimated elasticity of sclera is in the same order of magnitude as other studies, although slightly reduced. This reduced elasticity is also reflected in a slower calculated group velocity. This effect may be attributed to removal of the anterior eye which was necessary for preparation of the experimental setup and results in almost zero intraocular pressure (IOP). Lower IOP normally decreases the elasticity of surrounding tissues [65]. Finally, the curvature of the posterior eye can be considered to improve the displacement measurement and thereby enhances the accuracy of viscoelasticity estimation.

In our ex vivo experiments, the cornea was removed for enhanced image quality. Although ultrasound with frequencies below 20 MHz can be used for posterior eyes clinically, the purpose of this manuscript is to introduce and verify the dual-layer model. Hence a high-frequency imaging transducer at 48 MHz was used for better resolution. It is difficult to accurately measure the viscosity of sclera due to the fact that in the Voigt model, the elasticity and viscosity are highly intertwined, appearing as the real part μ₁ and the imaginary part iωμ₂, respectively. However, the elasticity of sclera is an order of magnitude greater than the imaginary part, and therefore the small change in viscosity is hardly reflected in the model. This error in viscosity estimation occurs with all biological tissues that have a large difference between elasticity and viscosity [66].

The proposed model is only for horizontally layered structures (different layers in z- direction), not for orthogonally layered structures (different layers in r-direction). In the theoretical development above, a generalized multi-layer model was presented, however in practice it has only been applied to dual-layer structures in this study. As the number of layers increases, the complexity of the model grows, the variances of estimates are larger, and it becomes more likely that the model will overfit the experimental data. With each additional layer in the structure, four more variables are added in the model. However, the sample size of effective experimental data is constrained by the ultrasonic micro-elastography system and cannot be increased with the number of layers. To overcome this limitation, enhanced performance of the imaging system is required, since the amount of experimental data is determined by the imaging resolution, speed, and field of view. For instance, OCT, which provides better resolution and faster imaging speed, can be a good replacement for the current high-frequency ultrasound imaging system. It is possible to expand the high frequency range from 1 kHz to 5 kHz and therefore have more experimental data to fit using OCT. The higher-frequency range can also reduce the aforementioned errors in the viscosity estimation since the viscosity is tied to frequency in the Voigt model. Therefore, by improving the imaging system in future work, the proposed multi-layer model has the potential for in vivo layer-specific viscoelasticity measurement of various multi-layer biological tissues, such as the posterior of the eye, intestine, skin, and artery walls.

One limitation of this study is the assumption of the Voigt model. The Voigt model used in this study (μ = μ₁ + jωμ₂) is one of the most common models in shear wave elastography and has been extensively used to model the behavior of soft biological tissues. It has been verified to be valid and relatively accurate in various biological and medical applications because the elasticities of most biological tissues are almost independent of frequency in the low frequency range (f < 1000 Hz) [67, 68]. However, just as all models attempt to approximate real conditions, the Voigt model is no exception and is far from perfect. One of the most significant limitations of the Voigt model is that the elasticity is a constant and independent of the frequency, which is not consistent with real conditions especially in some diseased tissues [69–72]. This raises a significant problem which is that it is almost impossible to acquire the ground truth values of viscoelasticity under the assumption of the Voigt model. As an example, we only measured the static elasticity, but true elasticity is frequency dependent. Since the viscosity would change significantly with the testing frequency, it cannot be verified by the quasi-static mechanical test. The Voigt rheological model in a bulk phantom was used to verify the value of viscous modulus in this study [70–72].

Advanced models like fractional-derivative models [61], rheological models [73], and dynamic shear modulus G = G′(ω) + iG″(ω) were proposed and used for a more accurate estimation of viscoelasticity. For example, dynamic shear modulus benefits from the feature that both shear storage (elastic part G′) and shear loss modulus (viscous behavior G″) are frequency dependent [66]. By measuring phase velocity and attenuation, both frequency-dependent G′ and G″ can be obtained.

At present, simplified models like the Voigt model have unique advantages in applications as compared with other advanced models [61]. For example, in the current study, after investigating multi-layer shear wave propagation behavior, we note that frequency dependent phase velocity is the only parameter that needs to be measured for to obtain viscoelasticity using the Voigt model. However, if the dynamic shear modulus was used, an additional measurement of viscosity-induced attenuation or phase delay between stress and strain, would be required to derive material viscoelasticity. Another advantage of the proposed method is that the impulse pushing force was applied only once for a short time (~200 μs), which makes this method faster and safer. In contrast, the current dynamic shear modulus method based on harmonics transmit larger acoustic energy into tissues and requires longer imaging time.

6. Conclusions

In this study, we investigated shear wave propagation in confined multi-layer tissues and presented a theoretical multi-layer model to assess tissue elasticity and viscosity together with Voigt model. Combining our model with a previously developed ultrasound micro-elastography imaging system, we successfully measured viscoelasticity for each tissue layer. In phantom experiments, measured elasticities were in good agreement with results obtained from mechanical tests and a shear wave rheological model using bulk phantoms. The estimated viscosities also approximate results from the rheological model using bulk phantoms. In contrast, it was demonstrated that the group velocity method and single-layer Lamb wave model generated noticeable errors in viscoelasticity estimation when they were applied to the multi-layered phantoms. Furthermore, the feasibility and accuracy of the proposed model were confirmed using various types of phantoms with different gelatin concentration distributions and different thicknesses. The proposed model was also applied to assess the elasticity and viscosity of the porcine retina and sclera ex vivo. Assuming a Voigt model, all results support the validity of the proposed model and its feasibility to assess the viscoelasticity of multi-layer tissues. To the best of our knowledge, this is the first study that investigates shear wave dispersion in multi-layer tissues and utilizes it to estimate both elasticity and viscosity. The proposed methodology has the potential to greatly expand the application of current elastography techniques, with enhanced accuracy, which should lead to improved medical diagnoses in diverse clinical areas, including diagnosis of diseases of the eye.