Bioinspired metagel with broadband tunable impedance matching

Measurement of acoustic impedance distribution within the head of an Indo-Pacific humpback dolphin using computed tomography scanning and tissue experiments

A dead female Indo-Pacific humpback dolphin with a body length of 2.7 m and a weight of 280 kg was found stranded in Quanzhou waters on 14 March 2015 (24). Computed tomography scanning of the dolphin’s head with a slice width of 0.625 mm was carried out at the Radiology Department of Affiliated Zhongshan Hospital of Xiamen University. All images were collected with a power setting of 120 kV × 600 mA and with a matrix size of 512 × 512. Hounsfield unit (HU) in the computed tomography images was obtained by comparing the linear attenuation coefficient of a voxel with that of water. The tissues denser than water would have higher HU values. The high-resolution computed tomography imaging allowed us to extract the fine anatomical structure of the dolphin, and then the axial cross section, coronal cross section, and sagittal cross section were presented (fig. S1).

Furthermore, the acoustic impedance being the product of tissue density and sound speed was obtained from the tissue experiments. The forehead soft tissues were sectioned into 10 transverse slices, which were further cut into 42 small pieces. The sample weights m were measured by an electronic balance with an accuracy of 0.001 g. The volumes V were measured by a measurement cylinder with an accuracy of 1 ml. The density ρ = m/V of each tissue was determined. In addition, the sound speeds of these tissue samples were measured by using an ultrasound velocimeter with a pulse frequency of 3.5 MHz at room temperature. The tissue thickness d was measured with a vernier caliper. The acoustic travel time t = t₂ − t₁ within each tissue was measured with an oscilloscope, as shown in fig. S2, where t₁ represents the first reflected pulse at the upper surface of the tissue and t₂ represents the second reflected pulse at the lower tissue surface. On the basis of the measured m, V, d, and t, the acoustic impedance of the tissue sample was computed asγ=2×mdVt(4)

The acoustic impedance of each sample was measured five times to obtain the average value. HU and the acoustic impedance of one tissue sample were measured as 91.5 and 1.654 × 10⁶ Pa·s/m, respectively. Air has low acoustic impedance of 4.15 × 10² Pa·s/m, while the skull has high acoustic impedance of 6.241 × 10⁶ Pa·s/m. Recent study has found that the HU values of soft tissues were linearly related to the acoustic impedance (24); therefore, combing computed tomography imaging with tissue experiment, the gradient acoustic impedance distribution within the dolphin’s head was obtained (Fig. 1A).

Acoustic field simulation of the dolphin model and BMIT

Finite element simulation was applied to perform full-wave simulation of the acoustic transmission processes within the dolphin’s head. In the fluid media (such as air, water, melon, and other soft tissues), only longitudinal waves will propagate, which can be described by the wave equation1ρ0cs2∂2p∂t2+∇·(−1ρ∇p)=0(5)where p is the sound pressure, ρ₀ is the density, and c_s is the sound speed. A variable density ρ is included, since the forehead complex is inhomogeneous. In the skull structures, both shear and compressional waves should be considered asρ∂2v∂t2=(λ+μ)∇(∇•v)+μ∇2v(6)where v is the velocity vector and λ and μ are the two Lamé constants, characterizing compression and shear moduli, respectively. The acoustic fields in fig. S3A were numerically derived by solving these wave equations with proper boundary conditions using COMSOL Multiphysics Modeling software (Stockholm, Sweden) (28, 29). The sound speed and density of the tissues were obtained by computed tomography imaging and tissue experiments. The boundary conditions require that the sound pressure and normal velocity at the boundary of the fluid media are continuous, while normal velocity and mechanical stress at the skull-tissue boundary are continuous.

Furthermore, to numerically investigate the broadband performance, a short-duration pulse was used asps={eα1tsin(2πf0t)0≤t≤t0e[−α2t+2α2t0)]sin(2πf0t)t0≤t≤3t003t0≤t(7)where α₁ and α₂ are the attenuation factor, f₀ is the peak frequency, and t₀ = 1/f₀ is the point at which the signal reaches the highest value. For the short-duration pulse in fig. S3B, the transmitted signal by the acoustic channel shows the broadband spectrum in Fig. 1C, where α₁ = 4 × 10⁴ and α₂ = 2 × 10⁵. The central frequency f₀ was set as 100 kHz according to our field measurement of the Indo-Pacific humpback dolphin in Xiamen Bay. Using the finite element simulation, full-wave simulations of BMIT for the impulse signal were performed for comparison with QIT, where the gradient impedance function was realized by setting the density function ρ(x)=[a+de−ln(b−ad)x/L]/c kg/m³, and the following parameters were used as sound speed c = 2000 m/s, a = 0.01Z₁, d = 0.99Z₁, b = Z₀, Z₀ = 33.5 × 10⁶ Pa·s/m, and Z₁ = 1.48 × 10⁶ Pa·s/m, respectively. Differing from the narrowband of QIT, the more broadband transmitted signal of BMIT was then obtained using this method (Fig. 1D and fig. S3D).

Nonuniform transmission line theory of BMIT

For one-dimensional transverse electromagnetic and acoustic waves, the spatial distribution of the reflection coefficient within an impedance gradient medium could be described in a unified way, by a nonlinear differential equation (13, 18)∂R(x)∂x−2ikR(x)+12∂[ln γ(x)]∂x[1−R2(x)]=0(8)where i² = −1, R is the reflection coefficient, and γ(x) is the gradient acoustic characteristic impedance. To couple the transducer with water, the impedance of both sides of BMIT satisfy γ( − L) = Q and γ(0) = 1 and thus have T = 1 − ∣R( − L)∣².

For the uniform medium with impedance γ₀, Eq. 8 was reduced to∂R(x)∂x−2ikR(x)=0 (9)leading to the solution R₀e^2ikx, where R0=γ0−1γ0+1 was determined by the boundary impedance γ₀ at x = 0. QIT had γ0=Q, and then we had the following relationship for T = 1λ=4L2n−1,  (n=1,2,⋯) (10)We thus had Eq. 1 to describe the length-wavelength relationship.

However, for the impedance gradient medium of BMIT, Eq. 8 is a Riccati equation whose analytical solution is difficult to obtain except for quite few transformers, such as an exponential transformer (18). Substituting γ(x) = γ₀ + εe^ax into Eq. 8, we then obtained Eq. 3. Numerical solutions in Fig. 3 were obtained by applying a fourth-order Runge-Kutta method to numerically integrate Eq. 3, where the boundary condition was R(0) = R₀ and the integration step was L/10,000.

In addition, we also used the approximate theories of small reflections (18) as well as small impedance perturbations (19) to derive the approximate solutions. For small reflection approximation R << 1, Eq. 3 was reduced to∂R(x)∂x−2ikR(x)+εaeax2(γ0+εeax)=0 (11)and then R(x) was solved asR(x)=εa2e2ikx∫0−Le(a−2ik)x(γ0+εeax)dx+C (12)where C is the constant determined by the reflection coefficient R₀ at x = 0. For the matched impedance at x = 0, R₀ = 0 led to C = 0. L/λ >> 1 led to T → 1. It could be more clearly seen when γ₀ = 0. In this case, Eq. 12 was solved as R(−L)=a4ik[e−2ikL−1]. When γ₀≠0, Eq. 12 was solved as R(−L)=e−2ikLεaeaξ2(γ0+εeaξ)∫0−Le−2ikxdx, according to the integral mean value theorem, where εae^aξ/(γ₀ + εe^aξ) is a continuous function and ξ ∈ [ − L,0]. It can be further approximated as R(−L)∼e−2ikL1−2ik[e2ikL−1]. Therefore, the ratio R( − L) between the impedance gradient and uniform media can be obtained as∣R(−L) (gradient)R(−L) (uniform)∣ ∼1k(13)

When λ → 0 or k → ∞, L/λ = kL/2π → ∞ led to ∣R(−L) (nonuniform)R(−L) (uniform)∣→0. In comparison with uniform material, the impedance gradient material of BMIT significantly suppresses the reflection and then T → 1. Thus, the independent length-wavelength relationship could be demonstrated. The approximate solution at low frequencies (L/λ < 0.25) might be incorrect because of negative T (Fig. 2). This small reflection approximate theory was based on the assumption of R << 1.

For the small impedance perturbation approximation γ(x) = γ₀ + ε′e^ax (ε′is the perturbation coefficient), we assumed R = R⁽⁰⁾ + ε′R⁽¹⁾ and then derived Eq. 3 into the following leading order and perturbation equations∂R(0)(x)∂x−2ikR(0)(x)=0(14)∂R(1)(x)∂x−2ikR(1)(x)+aeax2(γ0+ε′eax)[1−R(0)2(x)]=0(15)

The reflection coefficients at x = −L of these two equations were further solved asR(0)(−L)=R0e−2ikL(16)R(1)(−L)=a2e−2ikL∫0−Le(a−2ik)x[1−R02e4ikx](γ0+ε′eax)dx(17)

Note that this approximate theory required that the impedance perturbation coefficient ε′ should be sufficiently smaller than 1, and thus, ε′= 0.2 was used in Fig. 2A. In addition, to investigate the effect of Q on the transmission performance of QIT and BMIT, we calculated their transmission coefficients where Q = 11.4 and 32.1 correspond to aluminum and steel, respectively, as shown in Fig. 2B. We also tested the transmission performances of copper (Q = 28.3), gold (Q = 42), and quartz (Q = 10.1), respectively, in fig. S4.

Hydrogel-based design of BMIT

Agarose hydrogels in 3% concentrations with a 60-mm sample thickness were built. Ultrapure agarose powder was measured as 6.48 g by the electronic balance and added to the 216-ml volume of deionized water in a beaker for complete mixing. To avoid water loss during heating, the mixture was covered by a filtering paper. The mixture in the beaker was heated 2 min to boil in a microwave oven. The heated liquid agarose solution was placed in a fabricated cooling tank and cooled down to about 60°C. A hexagonal array of steel cylinders was prepared in an acrylic mold with a dimension of 6 cm by 6 cm by 6 cm. The solution was put into the acrylic mold to compose the steel-hydrogel composite and was slightly overfilled to avoid the formation of bubbles. After solidified, gels were carefully removed from the mold and used for acoustic transmission experiments. Using the method in fig. S2, density, sound speed, and acoustic impedance of agarose hydrogel were measured as 1.017 × 10³kg/m³, 1517 m/s, and 1.54 × 10⁶ Pa·s/m (fig. S5). For comparison, density, sound speed, and acoustic impedance of steel sample were measured as 7.856 × 10³kg/m³, 5419 m/s, and 42.57 × 10⁶ Pa·s/m, which are consistent with the literature values ³ and thus verified our experimental procedures.

Determination of the acoustic impedance function of BMIT using metagel

The hexagonal array of steel cylinders with variant radii r was embedded in the agarose hydrogel to achieve the impedance function in Fig. 1B, where the hexagonal lattice constant is a = 2 mm and the filling ratio Φ of the steel cylinder satisfies Φ=2πr23a2. In experiments, the lattice constant of the cylinders was much smaller than the wavelength. The physical effect of effective density and sound speed of BMIT could be understood from the point of view of the effective medium theory (3, 30, 32). On the basis of the long-wavelength approximation, the effective parameters of the hydrogel-steel composite were described asρ*=Φρ1+(1−Φ)ρ0, cL*=λ*+2G*ρ*,and cT*=G*ρ*(18)where ρ^*, cL*, and cT* represent the effective density and sound speeds of longitudinal and transverse waves, respectively. The effective Lamé coefficient and shear modulus satisfy λ*=(M−1)G*+(M+1)G0+λ01−M and G*=G1(1+Φ)+G0(1−Φ)G1(1−Φ)+G0(1+Φ)G0, respectively, where M=[λ1+G1−(λ0+G0)]Φλ1+G1+G0. According to previous studies (29–32), the densities of hydrogel and steel were used as ρ₀ = 1070 kg/m³ and ρ₁ = 7800 kg/m³, respectively. The Lamé coefficient of hydrogel and steel were used as λ₀ = 2.3GPa and λ₁= 107.4 GPa, respectively. The shear modulus of hydrogel and steel were used as G₀ = 0.24 MPa and G₁ = 84.4 GPa, respectively. Figure S6 (A to D) shows the effects of the filling ratio Φ on the effective density, the effective sound speed of the longitudinal wave, the effective sound speed of the transverse wave, and the effective longitudinal impedance, respectively, where the parameters of the steel and agarose hydrogel were given for comparisons. With the increase of Φ, the effective density increased from 1070 to 7800 kg/m³. On the basis of Φ=2πr23a2, the effect of cylinder diameter on the acoustic impedance could be obtained. Therefore, the acoustic impedance profile in BMIT in Fig. 3B could be obtained.

Fabrication and acoustic transmission experiments of BMIT

We assembled BMIT using agarose hydrogel to embed a hexagonal matrix of steel cylinders. The agarose hydrogel layer (fig. S7) was sandwiched between two acrylic rectangular plates with a dimension of 6 cm by 6 cm by 0.5 cm. The plates were fabricated on the basis of a hexagonal honeycomb metamaterial design. The hexagonal lattice constant was 2 mm, and the steel cylinder diameter D of BMIT was progressively increased from 0.1 to 2 mm.

To tune the acoustic impedance, the steel-hydrogel composite without two acrylic plates was compressed by pushing the plastic plate at x = 0, and the movement was measured with a vernier caliper. Forty percent compression of the steel-hydrogel composite was performed by pushing the plastic plate from x = 0 to x = −0.4L using an external force. Using the experimental method in Fig. 2, the acoustic impedance profile in Fig. 1B could be approached for BMIT, and they show a significant linear correlation (P < 0.001, r² = 0.98).

Ultrasound transmission experiments were performed in an anechoic water tank. A tank with dimension of 1.2 m by 1.2 m by 1 m was covered by sound-absorbing material to approach the condition of the free acoustic field. A broadband acoustic transducer with a diameter of 0.03 m was used to produce the tone-burst signal with a five-cycle duration, where the applied frequency range was from 50 to 230 kHz. The signals were recorded by a broadband receiver at a distance of 0.1 m. The signal was then analog to digital (A/D) converted with the sampling rate of 1M kHz. The signals were measured 10 times to test the reproducibility. When the transducer frequency was increased from 60 to 120 kHz and the transformer length was decreased from 2.5 to 1.5 cm, both frequency and length dependencies for maximal transmission are found in QIT; however, BMIT was not influenced by increasing frequency and compression. Frequency shift between theoretical predictions and experimental measurements in Fig. 3C may be associated with the thin water layer between the acoustic transducer and the transformer.

Underwater ultrasound transmission experiments were performed using a single-beam echosounder (BioSonics, Seattle, WA, USA) with a transducer diameter of 26 cm and center frequency of 123 kHz. To fit the sizes of the transformers and reduce the effect of wave reflections, extra spaces of the transducer beyond the transformers were covered by sound-absorbing material. An iron cylinder with a length of 10 cm and a diameter of 9 cm was used as an acoustic ranging object. In Fig. 4, the time-distance acoustic backscatter intensity profiles for QIT and BMIT were given for L = 2.5 and 1.5 cm, where the horizontal axis is time, the vertical axis is the ranging distance, and the color bar gives the backscattered signal amplitude in decibel. The echo amplitude significantly increased when the steel wall was detected, while the object has a smaller backscattering area. The steel wall was shown in brown, the object was shown in yellow, and the water was shown in green. We tested the ultrasound detection performances of QIT and BMIT by including the following cases: (I) without object, (II) with an immobile object, and (III) with a swaying object. Decreasing the transformer length to L = 1.5 cm at the compression ratio of 40% might not significantly affect underwater ultrasound detection.