Mechanical characterization of functionally graded soft materials with ultrasound elastography

Abstract

Functionally graded soft materials (FGSMs) with microstructures and mechanical properties exhibiting gradients across a spatial volume to satisfy specific functions have received interests in recent years. How to characterize the mechanical properties of these FGSMs in vivo/in situ and/or in a non-destructive manner is a great challenge. This paper investigates the use of ultrasound elastography in the mechanical characterization of FGSMs. An efficient finite-element model was built to calculate the dispersion relation for surface waves in FGSMs. For FGSMs with large elastic gradients, the measured dispersion relation can be used to identify mechanical parameters. In the case where the elastic gradient is smaller than a certain critical value calculated here, our analysis on transient wave motion in FGSMs shows that the group velocities measured at different depths can infer the local mechanical properties. Experiments have been performed on polyvinyl alcohol (PVA) cryogel to demonstrate the usefulness of the method. Our analysis and the results may not only find broad applications in mechanical characterization of FGSMs but also facilitate the use of shear wave elastography in clinics because many diseases change the local elastic properties of soft tissues and lead to different material gradients.

This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.

1. Introduction

Functional gradient soft materials (FGSMs) are ubiquitous in both nature and engineering. In nature, basically, all soft tissues are FGSMs with their microstructures varying in space, leading to different spatial gradients in mechanical properties to meet specific functions [1–3]. Additionally, some diseases may alter the local mechanical properties of soft tissues and lead to gradients in elasticity. In engineering, the advance of modern technologies enables the fabrication of FGSMs with well-controlled microstructures. Microstructures with controlled spatial gradient endow FGSMs with unique physical properties, which find broad applications in a variety of fields, including biology, medicine, soft robotics and tissue engineering [1,4–6]. Understanding the mechanical properties of FGSMs plays an essential role for their practical applications and clinical diagnosis; however, this is by no means trivial when bearing the following challenges in mind. First, it is well known that the mechanical properties of FGSMs in vivo or under service conditions are sensitive to environments, including pH, temperature, humidity and so on [7–9]. Therefore, in situ testing methods to measure the mechanical properties of FGSMs in different environmental conditions are required. Second, the spatial variation in the mechanical properties of FGSMs brings great challenges to commonly used testing methods, e.g. indentation and pipette aspiration methods. Indeed, it is impossible or very difficult to probe the in situ mechanical properties of FGSMs in an interior region with the existing testing methods.

Bearing the above challenges in mind, this paper investigates the use of ultrasound elastography to measure the mechanical properties of FGSMs under different conditions in a non-destructive and non-invasive manner. FGSMs of interest here include not only biological soft tissues but also many artificial soft materials (such as functional hydrogels), which have received considerable attention in soft matter mechanics and soft matter physics [7–9]. Ultrasound shear wave elastography (SWE) is a quantitative testing method which relies on the use of mechanical vibrators or acoustic radiation force (ARF) to induce shear waves, and on ultrafast ultrasound imaging techniques to visualize the shear wave propagation [10–12]. From the displacement field corresponding to the wave propagation, we can measure the shear wave velocity (SWV) and further infer the mechanical properties of a soft material. The development and use of ultrasound elastography depend largely on the understanding of wave motion in complex soft materials. For example, in SWE of soft matter, an important issue is to understand how the finite pre-deformation in soft materials will affect the propagation speed of the small-amplitude shear wave. For this issue, the classic studies over the past five decades (e.g. Biot [13], Hayes & Rivlin [14,15], Truesdell [16], Chadwick [17] and Ogden [18]) provide much valuable information. Indeed, these theoretical studies lay the foundation for developing inverse approaches to characterize both linear and nonlinear elastic properties of the soft biological tissues with SWE [19,20]. Presently, SWE has been successfully used to characterize in vivo some homogeneous soft tissues including liver and skeletal muscles [21,22] and to assess breast tumours [23,24]. However, the use of SWE to characterize the mechanical properties of FGSMs is more challenging and has not been systematically explored yet.

The existence of a material gradient leads to dispersive Rayleigh surface waves and analytical dispersion relations can only be achieved in limited cases. Destrade [25] studied Rayleigh surface waves in elastic-graded materials with the elastic moduli and densities varying in the same manner and obtained the dispersion relation in analytical form. However, for most soft materials and biological tissues/organs, the mass density is close to that of water (approx. 1000 kg m⁻³) and usually assumed to be a constant throughout. The assumption that the mass density varies with depth may not be justified in this case. When elastic moduli and mass densities of FGSMs vary in different manners, Balogun & Achenbach presented an approximate solution for the dispersion relation in the high-frequency range [26,27]. However, due to the strong attenuation of the elastic wave at a high frequency, only the low frequency shear waves can be tracked experimentally in soft materials. Therefore, their approximate expression of the dispersion relation in the high-frequency range cannot be used to understand the SWE of FGSMs. Besides analytical analysis, numerical methods (e.g. propagator matrix method [28,29]) have been frequently used to study wave propagation in mechanically graded materials, usually modelled as layered structures. Each layer is homogeneous but different layers have different mechanical properties. If the thicknesses of the layers are thin enough, the layered structure can well approximate a functionally graded material. However, such a numerical method is time-consuming and difficult to be used in an inverse analysis to infer the material parameters of FGSMs using ultrasound elastography.

Based on the premise above, this paper investigates the use of ultrasound elastography for the mechanical characterization of FGSMs. The paper is organized as follows: in §2, we establish a finite-element (FE) model to calculate the dispersion relation of the Rayleigh surface waves in FGSMs. Floquet–Bloch type boundary conditions are adopted. Based on our analysis, an inverse approach is proposed to characterize the graded mechanical properties from the measured dispersion relation when the material gradient is large. In §3, we study the transient wave in the FGSMs induced by the moving ARF. Our results show that the group velocities of the shear waves at different depths depend on the local material properties only when the material gradient is smaller than a critical value. This finding indicates that when the dispersion of the elastic wave caused by the spatial variation of material parameters is weak, local elastic properties of FGSMs can be measured with SWE. To demonstrate the usefulness of the results given by the theoretical analysis, experiments are performed in §4 to characterize the graded mechanical properties of a polyvinyl alcohol (PVA) cryogel. Discussions are conducted in §5 on the potential use of our results. Section 6 gives the concluding remarks.

2. Dispersion relation of surface wave in a functionally graded soft material and its potential application in ultrasound elastography of FGSMs

In this section, the dispersion relation of a surface wave in FGSMs is studied. The shear modulus of the semi-infinite material (located in the region y ≥ 0) is supposed to vary with the depth as

$$\mu (y)={\mu }_0+({\mu }_1-{\mu }_0)\exp \left(\frac{-y}{\alpha }\right),$$ 2.1

where μ₀ and μ₁ denote the shear moduli at y → +∞ and y = 0, respectively. The material is assumed to be incompressible and the mass density ρ is a constant throughout. The parameter α determines how fast the shear modulus varies from μ₁ to μ₀ with depth. Here we focus on the case μ₁ ≤ μ₀, whereas the case in which μ₁ ≥ μ₀ is discussed subsequently.

To reveal the dispersion feature of the surface wave, an FE model (Abaqus 6.13, Dassault Systèmes^®) using Floquet–Bloch type boundary conditions is built to calculate the dispersion relation. The method used here is frequently used in the study of phononic crystals [30–32]. For the surface wave propagating along the x-axis, the displacements on the left and right sides of the model (denoted by u^l and u^r, respectively) are related by

$${\mathbf{\text{u}}}^r={\mathbf{\text{u}}}^l{\text{e}}^{ikL},$$ 2.2

according to Floquet–Bloch's theorem [30,31], where k and L denote the wavenumber and length of the model, respectively. Note that in equation (2.2), the displacement is a complex number, which cannot be directly dealt with in Abaqus. Therefore, two models with the same meshes are built (figure 1a); one denotes the real part of the displacement (${\mathbf{\text{u}}}_{\mathrm{Re}}$) and the other models its imaginary part (${\mathbf{\text{u}}}_{\mathrm{Im}}$) [33]. The displacement induced by wave motion can then be expressed as $\mathbf{\text{u}}={\mathbf{\text{u}}}_{\mathrm{Re}}+i{\mathbf{\text{u}}}_{\mathrm{Im}}$. Equation (2.2) can be rewritten as

$$\begin{array}{r}{\mathbf{\text{u}}}_{\mathrm{Re}}^r={\mathbf{\text{u}}}_{\mathrm{Re}}^l\cos (kL)-{\mathbf{\text{u}}}_{\mathrm{Im}}^l\sin (kL)\\ {\mathbf{\text{u}}}_{\mathrm{Im}}^r={\mathbf{\text{u}}}_{\mathrm{Re}}^l\sin (kL)+{\mathbf{\text{u}}}_{\mathrm{Im}}^l\cos (kL).\end{array}$$ 2.3

Figure 1.

(a) The two same meshes used in the FE analysis, which stand for the real and imaginary parts of the displacement field. (b) The displacement field of the surface wave mode obtained from the FE analysis. The wavenumber in this case is π/L. (c) The dispersion relation obtained from the FE analysis. The frequency of the surface wave mode is the lowest; therefore, the surface wave mode can be easily distinguished from the other modes. The parameters used in this model are μ₁ = 6.87 kPa, μ₀ = 60 kPa and α = 6 mm. (d) The dimensionless dispersion relations for different stiffness ratios μ₀/μ₁ = 1.0, 2.0, 4.0, 7.0, 10.0. (Online version in colour.)

The height H of the model is taken as 100L ($100L\gg \alpha$) to model the half space. To describe the graded mechanical properties, the material parameters are defined as temperature-dependent, and a temperature distribution which has the same form as equation (2.1) is defined in our model. In theory, arbitrary material distributions can be described with this definition.

For a certain wavenumber, conducting the natural frequency analysis (NFA) of the models enables the determination of the natural frequencies (i.e. ω) of different modes. Among those modes, the surface wave mode has the lowest frequency because the free surface is the softest [34,35]. In this case, one can easily distinguish the surface wave mode from all other modes. Figure 1b shows the lowest mode shape when k = π/L. By varying the wavenumber from 0 to π/L, the dispersion relation can be obtained, as shown in figure 1c.

Dimensional analysis on this problem suggests that only the following three independent dimensionless parameters are involved, i.e. c/c₀, fα/c₀ and μ₀/μ₁, where f = ω/2π and c = ω/k denote the frequency and the phase velocity, respectively, and $c_0=\sqrt{{\mu }_0/\rho }$. Here we define the first two dimensionless parameters as the dimensionless phase velocity and the dimensionless frequency, i.e.

$$\{\begin{array}{l}\overline{c}={\displaystyle \frac{c}{c_0}}\\ \overline{f}={\displaystyle \frac{f\alpha }{c_0}}\end{array}.$$ 2.4

Then for a given stiffness ratio μ₀/μ₁, the dimensionless phase velocity $\overline{c}$ is only dependent on the dimensionless frequency $\overline{f}$.

For illustration, we plot the dimensionless dispersion relations in figure 1d. Obviously, all the curves start from $\overline{c}=0.955$, which is the dimensionless Rayleigh surface wave velocity of a homogeneous isotropic, incompressible material [36]. This can be understood as follows: the elastic wave with a low frequency has a relatively long wavelength, which largely depends on the elastic properties in the interior region of the half space instead of its surface properties. When the frequency of the wave is high enough, the phase velocity tends to a constant, which is determined by the elastic properties of the surface (i.e. μ₁).

Since the dispersion relation is significantly affected by the material properties, we may in theory identify the material parameters by plotting the dispersion relation. If the dispersion relation is readily measured in a wide range of frequencies, the elastic modulus at the surface (i.e. μ₁) can be inferred from the dispersion curve at the high-frequency range, whereas the elastic modulus of the FGSMs deep in the half space (i.e. μ₀) can be measured using the data at low frequency range. The parameter α can be determined by fitting the dispersion curve with the theoretical solution obtained here.

In practice, the main challenge is to measure the dispersion curve in a sufficiently wide range of frequency with existing techniques. For example, with the ultrasound-based guided wave elastography (GWE) method, which has been successfully applied to measure the dispersion relation of thin-walled soft tissues, we can only measure the dispersion relation of FSGMs in the range of approximately 500–1500 Hz [37–40]. As shown in figure 2, the inverse problem to identify all the three parameters (i.e. μ₁, μ₀ and α) from the dispersion curve in the frequency range of 500–1500 Hz may be ill-posed. In this case, the determination of the elastic moduli μ₁ and μ₀ from other independent measurements may be necessary. For example, the surface elastic modulus μ₁ can be measured using indentation tests, while the deep elastic modulus μ₀ can be measured with SWE. Then the parameter α can be inferred from the dispersion curve with the known μ₁ and μ₀.

Figure 2.

Dispersion curves for different α. The material parameters used here are μ₁ = 6.87 kPa, μ₀ = 60 kPa.

The discussions above show that the dispersion relation of Rayleigh surface wave can be used to characterize the graded mechanical properties of FGSMs. This dispersion relation is particularly useful for characterizing the FGSMs with a large mechanical gradient (e.g. α = 1 mm). By contrast, in the case where the material gradient is small, the dispersion relation is less helpful. To characterize the mechanical properties of FGSMs with small material gradients, we study the transient wave motion in FGSMs in the next section.

3. Transient shear waves induced by focused acoustic radiation force in functionally graded soft materials

In this section, we investigate the effects of the parameter α on the SWE of the FGSMs. When α is small, dispersion of the elastic wave is strong, and SWE is inapplicable for assessing the mechanical properties of FGSMs, whereas when α is larger than a critical value the dispersion induced by the material gradient becomes weak and SWE may be directly used to measure the local properties of FGSMs. To determine this critical value for the parameter α, an axisymmetric FE model was built to study transient wave motion in FGSMs. The elastic wave was induced by the simulated ARF. The ARF was supposed to have a Gaussian distribution [41,42] and its centre moved with a speed of v_arf to generate elastic waves [10]. The specified expression for the ARF is $\mathbf{\text{f}}=f_2(x,y,t){\mathbf{\text{e}}}_y$ (${\mathbf{\text{e}}}_y$ is the unit vector along y axis)

$$f_2(x,y,t)=-F_0\exp \left[-\frac{(x^2/a^2+{(y-v_{\mathrm{arf}}t)}^2)}{b^2}\right],$$ 3.1

where the parameters a and b determine the distribution of ARF, here equal to 0.5 mm and 1.5 mm, respectively. v_arf denotes the moving speed of the ARF (here $v_{\mathrm{arf}}\approx 100\sqrt{{\mu }_0/\rho }$). Here F₀ determines the amplitude of the ARF and it is taken as a small value (10⁻⁶) to generate a small-amplitude wave. At t = 0, the centre of the ARF is located on the surface, and when t > 0, the centre moves into the half space. Therefore, in this FE model, the surface wave and shear body wave will be simultaneously generated.

The length and width of the model were 120 mm and 60 mm, respectively. A total of 17 451 CAX8RH elements were used in the simulations. The wave propagation process is sampled at a frame rate of 10 000 Hz. A total of 35 frames are recorded for subsequent analysis. Such frame rate and frame number are consistent with the experimental set-up presented in the next section.

Figure 3a shows an example for the wave propagation in an FGSM with a relatively small material gradient (α = 6 mm). We can observe that the wavefront at a deeper position propagates with larger velocity, leading to an oblique wavefront. To further study wave propagation at different depths y, figure 3b shows the spatio-temporal plot of v₂ (the vertical component of the particle velocity field). It can be clearly seen that the wave propagates faster at a deeper position from the surface (i.e. larger y) and that the wave is weakly dispersive. We then conduct the two-dimensional Fourier transformation (2DFT) of the spatio-temporal plot at y = 0 to study the dispersion relation of the surface wave [37,43], i.e.

$${\overline{v}}_2(k,f)=\iint v_2(x,t){\text{e}}^{-ikx}{\text{e}}^{-i2\pi ft}\text{d}x\text{d}t,$$ 3.2

where ${\overline{v}}_2(k,f)$ is the transformed result. As shown in figure 3c, in the Fourier space each frequency f corresponds to a wavenumber k/2π at which the amplitude of ${\overline{v}}_2(k,f)$ (i.e. $|{\overline{v}}_2(k,f)|$) is maximal. In this way, we can determine a relationship between f and k, which is the dispersion relation. The dispersion relation in the frequency range of 500–1500 Hz is shown in figure 3d, which agrees well with the theoretical results obtained in the previous section, suggesting that the phase velocity in this case is dominated by the surface elastic modulus μ₁. The peak of $|{\overline{v}}_2(k,f)|$ in figure 3c denotes the most favourable mode. The frequency of this mode is approximately 800 Hz. This is the central frequency involved in SWE [44]. Here we use f₀ to denote this characteristic frequency, enabling us to determine a characteristic wavelength λ₀, i.e. ${\lambda }_0=\sqrt{{\mu }_0/\rho }/f_0$. This wavelength and the parameter α determine a dimensionless parameter which can be used to evaluate the effect of the graded mechanical property on the wave propagation. In theory, if ${\lambda }_0\ll \alpha$, the surface wave speed will be dominated by the surface elastic modulus μ₁, whereas if ${\lambda }_0\gg \alpha$, the wave speed largely depends on the deep modulus μ₀.

Figure 3.

FE analysis of the transient wave propagation in FGSMs. (a) Snapshots of the wave propagation. (b) Spatio-temporal plot of v₂ at different depths. It can be observed that the waves propagate faster at deeper sites. (c) Fourier space of the spatio-temporal plot at y = 0. The favourable mode, i.e. the peak in the figure, appears at f = 800 Hz. (d) Dispersion relation obtained from (c), which agrees well with the theoretical solution. (e) Measurement of the wavefront propagating velocity at y = 0. In this case, the wave is dispersive but the wavefront can be clearly identified. (f) Comparison of the group velocity distribution and $\sqrt{\mu (y)/\rho }$. The parameters used in this model are μ₀ = 20 kPa, μ₁ = 3 kPa, α = 6 mm. (Online version in colour.)

Moreover, the group velocity of the wavefront can be determined via the time-of-flight method [45]. As shown in figure 3e, at t = 3 ms, we can clearly see the wavefront even though the wave is weakly dispersive. Linear fitting (r > 0.99) of the results gives the group velocity of the wavefront. The relative error between the theoretical solution of $c=\sqrt{{\mu }_1/\rho }$ and the measured velocity is less than 15%. Furthermore, group velocities of the wavefronts are measured at positions with an interval of 2 mm along the depth direction, and the results are plotted in figure 3f. The measured velocity agrees well with the distribution of $\sqrt{\mu (y)/\rho }$, suggesting in this case that we may use the propagating velocity of the wavefront to measure the spatial variation of the elastic modulus. Briefly, we can measure the group velocities at y = 0 (denoted by c|_y=0) and at y → ∞ (far enough from the surface that the group velocity no longer varies with the depth, denoted by c|_y=∞), to determine the two elastic moduli according to μ₁ ∼ ρc²|_y=0 and μ₀ = ρc²|_y→∞, respectively. Then we can determine the parameter α to minimize the differences between the group velocities and $\sqrt{\mu (y)/\rho }$.

It should be pointed out that the results and conclusions above are only valid for FGSMs with small material gradients, i.e. the elastic modulus μ₀ and μ₁ are close to each other or the parameter α is large. Therefore, it is important and necessary to determine the critical α beyond which the method suggested above is valid. To this end, we calculate the group velocity of the surface wave when α varies from 1 to 10 mm. As shown in figure 4a, where μ₀/μ₁ ≈ 7, when α decreases, the group velocity becomes significantly larger than $\sqrt{{\mu }_1/\rho }$. When α < 4 mm, the relative error between the group velocity and $\sqrt{{\mu }_1/\rho }$ is larger than 20%. For the example shown in figure 4b, the stiffness ratio is μ₀/μ₁ = 2. We can see that the relative error between group velocity and $\sqrt{{\mu }_1/\rho }$ is less than 10% even when α is smaller than 2 mm. This indicates that both the stiffness ratio and the magnitude of α determine whether SWE can be used to characterize the mechanical properties of FGSMs.

Figure 4.

Group velocities measured at y = 0 for different parameters α: (a) μ₁ = 6.87 kPa, μ₀ = 60 kPa; (b) μ₁ = 30 kPa, μ₀ = 60 kPa. When α is large (i.e. the gradient becomes small), the group velocity becomes close to $\sqrt{{\mu }_1/\rho }$. (c) The critical value α_c beyond which the relative error between the group velocity and $\sqrt{{\mu }_1/\rho }$ is less than 10%. (Online version in colour.)

We then determine an explicit expression to predict the critical α beyond which SWE can be used to characterize FGSMs. For FGSMs with different stiffness ratio μ₀/μ₁, we increase the parameter α until the relative error between the measured group velocity of the surface wave and $\sqrt{{\mu }_1/\rho }$ is less than 10%. This α is taken to be the critical value beyond which SWE works well, and denoted as α_c. In figure 4c, we show the relation between α_c/λ₀ and μ₀/μ₁, which can be approximately expressed as (r ≈ 0.99)

$$\frac{{\alpha }_c}{{\lambda }_0}\approx 0.26\hspace{0.17em}\frac{{\mu }_0}{{\mu }_1}-\mathrm{0.17.}$$ 3.3

In practical measurements, we can measure the graded mechanical properties (i.e. μ₀, μ₁ and α) with SWE. Then equation (3.3) can be used to check whether the measured material parameters are reliable.

4. Experiments

(a). Equipment and system

Experiments were performed in this section to demonstrate the usefulness of our results and conclusions above. The Verasonics Vantage 64LE System (Verasonics Inc., Kirkland, WA, USA) equipped with an L9-4 transducer (JiaRui Electronics Technology Co., Ltd., Shenzhen, China) was used. The central frequency of the transducer was approximately 6.5 MHz. The ARF induced by a long push (1000 cycles) was used to generate the elastic shear waves. The transducer acquired the in-phase and quadrature (IQ) data at a frame rate of 10 000 Hz. Based on the acquired IQ data, the Loupas' estimator [46] was then used to calculate the vertical component of the particle velocities.

(b). Preparation of the PVA cryogel

The FGSM was prepared from PVA cryogel, which is widely used as phantom material in ultrasound elastography. PVA powders (Sigma-Aldrich, Shanghai, China) were added into water and the solution was stirred until all the PVA was dissolved. During this process, the temperature of the solution was kept to be approximately 80°C. Then Sigmacell cellulose (20 µm, Sigma-Aldrich, Shanghai, China) was homogeneously dispersed into the solution to serve as scatters for ultrasound imaging. The weight ratios for PVA, pure water and cellulose were 10%, 87% and 3%, respectively. The solution was poured into a plastic box (150 mm × 100 mm × 100 mm, length ×  width × height), in which the solution underwent freeze/thaw (F/T) cycles. In each F/T cycle, the solution underwent 12 h of freezing (−20°C) and 12 h of thawing (20°C). Undergoing more F/T cycles makes the PVA cryogel stiffer [47]. In this study, the PVA cryogel underwent either 2 or 3 F/T cycles.

(c). Characterization of the PVA cryogel

As shown in figure 5a, the PVA cryogel was immersed in water during the experiments, and the ultrasound transducer was vertically placed on the surface of the PVA. Figure 5b shows the B-mode image, from which we can see the homogeneous speckles.

Figure 5.

(a) The PVA cryogel used in the experiments. The PVA is immersed in water during the experiments. (b) The B-mode image of the PVA cryogel. (Online version in colour.)

For the PVA with 2 F/T cycles, the experimental results are shown in figures 6 and 7. As shown in figure 6a, the wave propagation process (see also electronic supplementary material, videos) observed in the experiment is similar to that given by FE simulations. The waves at deeper positions propagate faster due to the graded mechanical properties of the PVA cryogel. Because the surface of the PVA cryogel was not flat, we started the measurement from y = 0.4 mm. Figure 6b,c shows the group velocities at surface and at y = 14.5 mm, respectively. The site of y = 14.5 mm is deep enough, as the group velocity no longer varies. The variation of the group velocity is presented in figure 6d (the measurements were repeated three times at each site). Then following the method suggested in the previous section, the graded mechanical properties can be measured. The results are listed in table 1. As mentioned, it is important to check whether the condition that α should be larger than α_c is satisfied. Inserting μ₀/μ₁ ≈ 5.7 and λ₀ ≈ 5.4 mm into the relation of α_c/λ₀ = 0.26μ₀/μ₁ − 0.17, we obtain α_c ≈ 7.1 mm, which is indeed smaller than α.

Figure 6.

(a) Comparison of the experiments with FE analysis. The experiments clearly show that the PVA cryogel is mechanically graded. (b,c) The measurements of the group velocities at y = 0.4 mm and y = 14.5 mm. According to the two measurements, the two elastic moduli μ₁ and μ₀ can be determined. (d) The distribution of the group velocity obtained from the experiments and its fitting curve. The parameter α measured for the 2 F/T cycles PVA cryogel is 7.9 mm. (Online version in colour.)

Figure 7.

FE analysis for the case μ₁ > μ₀. In this case, the shear moduli measured near the surface are significantly underestimated. (Online version in colour.)

Table 1.

The elastic parameters identified from experiments for 2 and 3 F/T cycles PVA cryogel.

F/T cycles	μ1 (kPa)	μ0 (kPa)	α (mm)	αc (mm)
2	3.2	18.5	7.9	7.1
3	9.0	35.5	7.1	5.7

Similarly, for the PVA with 3 F/T cycles, the gradient of the mechanical properties is also significant. According to table 1, the PVA becomes stiffer, and the elastic modulus agrees well with the previously reported data [48]. However, the stiffness ratio μ₀/μ₁ becomes smaller, which corresponds to a smaller α_c. The identified parameter α is 7.1 mm, which is close to the one obtained from the 2 F/T cycles PVA. In summary, we found that F/T cycles lead to graded mechanical properties in PVA cryogel, and the characteristic length of the gradient denoted by α here is approximately 7–8 mm.

5. Discussions

In this study, we mainly focused on FGSMs with elastic moduli increasing with depth. Such graded mechanical properties can model many soft materials such as the PVA cryogel used in our experimental study. However, in some cases, FGSMs may have a stiffer surface, i.e. μ₁ > μ₀ in equation (2.1). For illustration, we consider an example in which α = 6 mm, μ₁ = 60 kPa which is larger than μ₀ = 6.87 kPa. In comparison with the example shown in figure 3, it can be seen from figure 7 that the measured group velocities in the vicinity of the free surface show larger relative errors. The relative difference between group velocity at y = 0 and $\sqrt{{\mu }_1/\rho }$ is approximately 18%. However, when the measurement sites are not in the vicinity of the surface, the relative errors are less than 10%. Therefore, the critical α_c predicted in this study may also be applicable to the case of μ₁ > μ₀ when the regions of interest are not very close to the surface. In practical measurements, an extrapolation of the experimental data to the surface of the sample may enable the evaluation of the surface elastic properties, e.g. μ₁.

The experiments performed here are limited to phantom experiments. However, our results should provide considerable insight into the use of SWE in clinics. It is well known that many diseases may alter the local elastic properties of soft tissues, e.g. liver fibrosis at an early stage or the occurrence and development of cancer, leading to different material gradients. Our method and results show that SWE may be inapplicable when the dimension of the diseased region is smaller than a characteristic length that depends on the material gradient. This interesting issue deserves further investigations.

6. Conclusion and future prospects

In this study, ultrasound elastography of FGSMs is investigated. To this end, both direct and inverse problems are considered. In summary, the following contributions have been made.

A finite-element model based on a unit cell and the Floquet–Bloch type boundary conditions has been built to calculate the dispersion relation of the Rayleigh surface waves in FGSMs. The model in theory enables the investigation of arbitrary material gradients. The analysis of the dispersion relation allowed us to establish an inverse approach to measure the graded mechanical properties of FGSMs when the material gradient is large (e.g. α = 1 mm).

Transient wave motion in FGSMs induced by the moving ARF is studied to characterize the mechanical properties of FGSMs with small material gradients. Our results show that when the gradient of the material is smaller than a critical value, SWE can be directly used to measure the distributed mechanical properties. An explicit solution to predict the critical parameter α_c has been proposed.

PVA cryogel has been widely used in the phantom study with its acoustic properties similar to human tissues, including toughness and tuneable stiffness [47–50]. In this study, phantom experiments on PVA cryogels have been performed to demonstrate the usefulness of the results reported here. The bulk PVA cryogel cannot be homogeneously cooled/heated because of its large size, leading to graded elastic properties along the depth. The spatial-dependent material properties have been identified from our experiments.

Graded mechanical properties of FGSMs lead to complex wave motion problems and bring a great challenge for the inverse analysis to infer the material parameters when using dynamic elastography. This study represents the first attempt to address this challenging issue. Although some insights may be gained from our analysis, many open issues remain that deserve further efforts.

Data accessibility

The electronic supplementary material contains the movies of shear wave propagation in PVA cryogel.

Authors' contributions

G.Y.L., Z.Y.Z. and Y.Z. carried out theoretical and numerical analysis, participated in the experiments; J.Q., W.L., H.W. participated in the experiments; Y.P.C. conceived the study, designed the study and coordinated the study. G.Y.L. and Y.P.C. drafted the manuscript. All authors gave final approval for publication.

Competing interests

We declare that we have no competing interests.

Funding

This work is supported by the National Natural Science Foundation of China (grant nos. 11572179 and 11432008).