Acoustic Characteristics of Biosynthetic Bubbles for Ultrasound

May 23, 2019 - Acoustic Characteristics of Biosynthetic Bubbles for Ultrasound Contrast Imaging ... Our findings provide physical insight into the bio...
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Article Cite This: Langmuir 2019, 35, 10213−10222

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Acoustic Characteristics of Biosynthetic Bubbles for Ultrasound Contrast Imaging Fei Li,†,‡,# Yu Wang,†,§,∥,# Xinghai Mo,⊥ Zhiting Deng,†,‡ and Fei Yan*,†,‡

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†

Paul C. Lauterbur Research Center for Biomedical Imaging, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China ‡ Shenzhen Key Laboratory of Ultrasound Imaging and Therapy, Shenzhen 518055, China § The Second School of Clinical Medicine, Southern Medical University, Guangzhou 510515, China ∥ Guangdong Second Provincial General Hospital, Guangzhou 510317, China ⊥ Department of Ultrasound in Medicine, Shanghai East Hospital, School of Medicine, Tongji University, Shanghai 200120, China ABSTRACT: Biosynthetic bubbles produced by floating microorganisms, such as bacteria and algae, have recently attracted wide attention as novel ultrasound contrast agents owing to their significant potential in ultrasound imaging and acoustic reporter gene-based imaging. However, the acoustics properties of these bubbles are unclear. In this study, we developed a finite-element model to describe the oscillation of nonspherical biosynthetic bubbles composed of a gas core encapsulated in a protein shell. In this model, the elastic properties of the bubble shells were characterized in terms of the density, thickness, Young’s modulus, and Poisson’s ratio. Theoretical calculations were performed for a single bubble and an assembly of randomly oriented bubbles. Our results demonstrate that (1) there are many types of surface oscillation modes for nonspherical biosynthetic bubbles, and a systematic relationship exists between the surface modes and the resonance frequencies; (2) the bubble shell shape has a significant effect on the acoustic behavior; (3) the resonance frequency of an ellipsoidal bubble decreases with the decrease in its polar radius-to-equatorial axis ratio; and (4) the acoustic scattering of a randomly oriented suspension is isotropic at and below the first resonance frequency. Our findings provide physical insight into the biomedical applications of biosynthetic bubbles and can be used to optimize the acoustics properties of such bubbles.

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INTRODUCTION

Although microbubbles have been successfully utilized in blood pool imaging, there are several drawbacks limiting their further application, particularly for the molecular imaging of extravascular tissues: (1) Microbubbles have a micrometer particle size, typically with a diameter range of 1−8 μm; therefore, the imaging is limited to within the vasculature. (2) Microbubbles have a short half-life when injected into circulation (typically 5−10 min) because the gas core escapes from the bubbles by diffusing into the surrounding biological fluids.10,20,21 To date, solid, liquid, hollow, and phase-change contrast agents have been developed as microbubble alternatives; however, they have not been widely used because of their limitations when it comes to echogenicity, stability, and synthesis procedures.22−25 Recently, research has been ongoing for nanobubbles,26,27 which have potential applications in ultrasonic imaging, floatation in water treatment and the mining industry, aeration and disinfection of water, and cleaning.28−30 Notably,

Over the past decades, ultrasound contrast agents (UCAs), such as microbubbles, comprising a lipid or a protein spherical shell have gradually become the focus of a widely and rapidly expanding field of research,1 with their benefits being repeatedly demonstrated in ultrasound imaging,2−5 drug delivery,6−11 and the noninvasive estimation of blood pressure.12−16 These applications mainly depend on the acoustics properties of the coated microbubbles. For example, at a particular driving frequency f, such as the resonance frequency, shelled bubbles have been predicted and observed to scatter energy at harmonic frequencies. These harmonic signals originate from the nonlinear oscillation of these bubbles, not only resulting in second- (2f) and higher-order harmonics (3f, 4f, 5f,...) but also producing subharmonics (1/ 2f) and ultraharmonics (3/2f, 5/2f,...) in the backscattered signals. On the basis of the nonlinear scattering signals, many types of ultrasound contrast imaging techniques, including second-harmonic imaging, subharmonic imaging, pulse inversion imaging, targeted molecular imaging, and ultrafast superresolution imaging,3,17−19 have been developed to enhance the contrast-to-tissue ratio (CTR) and thus improve the image quality. © 2019 American Chemical Society

Special Issue: Microbubbles: Exploring Gas-Liquid Interfaces for Biomedical Applications Received: April 26, 2019 Revised: May 22, 2019 Published: May 23, 2019 10213

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Figure 1. Biosynthetic bubbles: (a) TEM image of a biosynthetic bubble. (b) Three-dimensional ellipsoid geometry model of a biosynthetic bubble and its elliptical cross section and circular cross section. (c) Geometry of an arbitrarily oriented ellipsoidal bubble in the xyz Cartesian coordinate system. The orientation of the ellipsoid is characterized by the polar angle θ and the azimuthal angle φ. (d) Finite-element model of a vibrating biosynthetic bubble (θ = 90°, φ = 0°) in an incident acoustic field.

biosynthetic gas vesicles generated by microorganisms, such as bacteria and archaea, have been introduced as ultrasonic molecular reporters.31 Nanobubbles exhibit typical nonspherical nanostructures (with a width range of 45−250 nm and a length range of 100−600 nm)32 and a stiff protein shell with a Young’s modulus of several GPa.33 Compared to microbubbles with a soft shell, nanobubbles have a resonance frequency >80 MHz,34 thus exhibiting a low echogenicity in the frequency range of 1−20 MHz, which is the widely used range for clinical diagnosis and small animal imaging. Fortunately, the size and shape of nanobubbles can be controlled by genetic manupulation,31 making it possible to optimize the acoustic properties of gas vesicles for imaging. Previous studies on gas vesicles have mainly focused on preparation and biomedical applications. Studies, particularly theoretical ones, on the acoustic scattering properties of nanobubbles are few. This could be because the various dynamic ordinary differential equations derived for UCA bubbles are suitable only for describing the spherical and symmetrical oscillation of microbubbles.35−39 These models fail to predict the vibrations of biosynthetic bubbles with nonspherical shapes. Finite-element methods have been used to model acoustic scattering and surface modes of UCA microbubbles,40−44 demonstrating the ability to simulate the oscillation and acoustic scattering properties of nonspherical nanobubbles. Herein, we proposed a finite-element model to study the interaction between an incident plane wave and a nonspherical biosynthetic bubble. The numerical results show the significant effect of the shell shape on the acoustic scattering properties of

the bubbles, and a relationship is established between the driving frequencies and far-field backscattering.

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NUMERICAL METHODS

Halo gas vesicles were prepared on the basis of a previous report by Shapiro et al.31 From the TEM images of halo gas vesicles, as shown in Figure 1a, it is reasonable to simplify a single biosynthetic bubble as an ellipsoid with the pole semiaxis Ra at the elliptical cross section in the xz plane and the equatorial semiaxis Rb at the circular cross section in the yz plane, as shown in Figure 1b. Furthermore, the ratio R Ra/Rb can be used to define the shape of the bubble: (1) R a > 1, b

ellipsoid; (2)

Ra Rb

= 1, sphere; and (3)

Ra Rb

< 1, oblate spheroid or disk.

Figure 1b shows the ellipsoidal bubble represented in a Cartesian coordinate system. The origin o coincides with the geometrical center of the ellipsoid. The symmetrical axis of the bubble is the z axis, and the incident acoustic plane wave propagates along the z direction. To describe the orientation of the ellipsoidal bubble and calculate the scattering pressure at a distance r = 10 mm from the origin (in the far field), we introduced a spherical coordinate system over the Cartesian coordinate system, as shown in Figure 1c. The orientation of the ellipsoidal bubble (i.e., the direction of the pole axis of the ellipsoid) is defined by the polar angle θ and the azimuthal angle φ in the xyz system. We simulated the oscillation and acoustic scattering of the bubble based on the following assumptions:45 (1) the bubble is surrounded by an infinite and ideal fluid; (2) the gas inside the bubble is encapsulated in a solid, incompressible, linear elastic solid shell; (3) the shell replaces the surface tensions at the shell−liquid and shell−gas interfaces; (4) the shell thickness is the same for bubbles of different sizes; and 10214

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wave propagation with the assumptions of small acoustic disturbances, an inviscid fluid, and adiabatic processes46

(5) there is no gas or liquid transport across the interface during the insonation. Because the nonlinear dynamic equations describing the oscillation of a bubble have been derived for a microsphere and its spherically symmetrical vibration, we used the finite-element method to investigate the oscillation modes and acoustic properties of a nonspherical bubble. The simulation procedure is implemented in the finite-element package COMSOL Multiphysics 5.2a. A linear acoustic model (Helmholtz equation) coupled with a linear elastic solid mechanics module was used to determine the first-order resonant acoustic pressure, displacement, and acoustic particle velocity fields. The surface modes and acoustic scattering properties of the bubbles were then obtained. Figure 1d shows the finite-element model of an ellipsoidal bubble immersed in a fluid. The bubble is located at the center of the computational domain and is excited by a plane wave traveling in the positive z direction. A spherical wave radiation boundary is applied to the computational domain boundary to simulate the infinite medium. Table 1 lists the model parameters used in the simulation.31,33,41

∇2 pt = −

pt = psc + pbk

3 1100 177.6 60 0.48 11.21 100

nm kg/m3 MPa MPa

shell thickness shell density shell’s Young’s modulus shell’s Poisson’s ratio inner gas density inner gas speed of sound

ds ρs Es vs ρg cg water

2 1100 2800 0.33 1.29 340

nm kg/m3 MPa

density speed of sound

ρl cl

1000 1500

(2)

This will be solved using the finite-element method. The background pressure field is defined as follows pbk = pin e−ixk

(3)

where pin is the incident acoustic pressure amplitude and k is the wave vector representing the wave propagation direction with magnitude |k| = ω/cl (i.e., the wavenumber). Solid−Fluid Interactions. The solid−fluid (liquid−shell and gas−shell) interactions are described by the pressure load Fp on the solid−fluid interfaces.41 The surrounding fluid applies an external load to the solid, where n is the unit normal vector pointing outward from the solid shell.

BR14 ds ρs Es Gs vs ρg cg Halo GV

(1)

where pt is the total acoustic pressure field, cl is the speed of the longitudinal wave in the surrounding fluid, and ω is the driving angular frequency. The total acoustic pressure field pt is divided into the scattering pressure field psc and the background pressure field pbk.

Table 1. Model Parameters shell thickness shell density shell’s Young’s modulus shell’s shear modulus shell’s Poisson’s ratio inner gas density inner gas speed of sound

ω2 p cl 2 t

Fp = npt

(4)

The normal acceleration a for the acoustic pressure on the interface is equivalent to the second derivative of the shell displacement u with respect to time.

kg/m3 m/s

a = n·ü

(5)

Wave Propagation in Solids. The elastic wave propagating in the shell can be described using the displacement field u44 2 − ρω u − ∇· σ = Fp s

kg/m3 m/s

(6)

where σ is the total stress tensor. The constitutive relation of linear elastic solid materials is utilized to characterize the solid shell’s stress−strain relationship (σ − ε) that is described by Hook’s law for isotropic materials as follows

kg/m3 m/s

σ = [D]ε

Wave Propagation in Fluids. The acoustic waves in a fluid medium (surrounding liquid or inner gas of a bubble) can be characterized using the Helmholtz wave equation for linear elastic

(7)

where the elastic matrix [D] is determined by Young’s modulus E and Poisson’s ratio v as follows.

Figure 2. Acoustic properties of microbubbles. (a) Normalized scattering power spectra calculated using a finite-element method for BR14 microbubbles with diameters of 3, 5, and 6 μm, respectively. (b) Resonance frequency as a function of the radius of the BR14 microbubbles. The results obtained using the Church−Hoff model (blue solid line), finite-element method (inverted red triangles), and measurements (cross) are compared. The incident acoustic pressure is 80 kPa. 10215

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Figure 3. Calculated, normalized scattering power of ellipsoidal bubbles (θ = 90°, φ = 0°) with different sizes: (a) Ra = 150 nm, Rb = 50 nm; (b) Ra = 150 nm, Rb = 100 nm; (c) Ra = Rb = 150 nm; (d) Ra = 150 nm, Rb = 250 nm; and (e) Ra = 100 nm, Rb = 300 nm. (f) Linear resonance frequencies of the ellipsoidal bubbles with respect to Ra/Rb. The incident acoustic pressure is 80 kPa. [D] =

E (1 + ν)(1 − 2ν) ÄÅ ÅÅ1 − ν ν ν 0 ÅÅ ÅÅ 1−ν ν 0 ÅÅ ν ÅÅ ÅÅ ν ν 1−ν 0 ÅÅ ÅÅ ÅÅ 1 − 2ν ÅÅ 0 0 0 ÅÅÅ 2 ÅÅ ÅÅ ÅÅ ÅÅ 0 0 0 0 ÅÅ ÅÅ ÅÅ ÅÅ ÅÅ ÅÅ 0 0 0 0 ÅÅ ÅÇ

ÉÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ 0 0 ÑÑ ÑÑ ÑÑ ÑÑ ÑÑ 1 − 2ν ÑÑ 0 ÑÑ 2 ÑÑ ÑÑ 1 − 2ν ÑÑÑÑ 0 Ñ 2 ÑÑÑÖ

0 0 0

scattering calculation for equally probable orientations. The averaged scattering pressures psc(θ, φ) for a sample of randomly oriented biosynthetic bubbles can be determined by integrating the scattering pressure in a given direction over all of the orientations in the threedimensional space

0 0 0

k

psc (θ , φ) =

l

∑ ∑ psc (θi , φj)wi ,j i=1 j=1

(9)

where k and l are the numbers of polar angles θ ∈ [0°, 360°] and azimuth angles φ ∈ [0°, 360°], respectively, and wi,j = 1/(k × l) is the number weight of biosynthetic bubbles with an orientation of (θi, φj). To estimate the total computation time, the averaged far-field acoustic pressure distributions in the xz and yz plane were preliminarily calculated for both θ and φ in the [0°, 90°] range with the same step (Δθ = Δφ). The obtained far-field patterns are largely similar for the three steps of Δθ = Δφ = 1°, 5°, and 10°. Because we are interested only in the directivity of the averaged acoustic field and considering the amount of calculation time, a step of 5° (Δθ = Δφ) was used to sweep the polar and azimuthal angles during the numerical simulation in this study.

(8)

Acoustic Scattering by Randomly Oriented Biosynthetic Bubbles. Furthermore, we numerically investigated the acoustic scattering properties of an assembly of randomly oriented, identical ellipsoidal bubbles. The far-field distributions of the scattering pressures psc(θ, φ) at the xz and yz cross sections were computed for an arbitrarily oriented bubble.47 Assuming that the biosynthetic bubble orientation exhibits a uniform distribution, we estimated the angular average of the scattering pressures by performing the 10216

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Figure 4. Calculated surface amplitude displacement (vibration mode) of a single ellipsoidal bubble (Ra = 150 nm, Rb = 100 nm, θ = 90°, and φ = 0°) at driving frequencies of 21, 467, 617, 779, 938, 1091, 1111, 1253, and 1308 MHz. The incident acoustic pressure is 80 kPa.

■

where ae is the equilibrium radius of the microbubble, ρl is the density of the surrounding medium, κ is the polythropic exponent of the gas core, Gs is the shear modulus of the shell, and ds is the shell thickness. The resonance frequency predicted using the finite-element model is in good agreement with the analytical solution calculated using the equation.45 The obtained curve of the resonance frequency as a function of the bubble size has the same tendency as that plotted using the experimental data.48 Acoustic Scattering Power Spectra of Biosynthetic Bubbles. To investigate the acoustic properties of the biosynthetic bubbles, we computed the acoustic scattering powers of biosynthetic bubbles with different shapes, as shown in Figure 3: (a) Ra = 150 nm, Rb = 50 nm; (b) Ra = 150 nm, Rb = 100 nm; (c) Ra = Rb = 150 nm; (d) Ra = 150 nm, Rb = 250 nm; and (e) Ra = 100 nm, Rb = 300 nm. The linear resonance frequencies are 982, 467, and 352 MHz for cases a−c, respectively. The calculated resonance frequencies are significantly greater than the usual working frequency range of 1−20 MHz for clinical diagnosis and small animal ultrasound imaging. The scattering powers at the resonance frequencies are approximately 6−10 orders of magnitude higher than those in the 1−20 MHz frequency range (the resonance frequency range of 1−10 μm lipid-coated microbubbles), indicating that biosynthetic bubbles undergo small-amplitude oscillations with weak nonlinear harmonic scattering signals (2f, 3f, 4f, 5f...) at driving frequencies in the range of 1−20 MHz. In this context, it is difficult to excite the subharmonic and ultraharmonic signals, which provide a higher CTR than harmonic signals for ultrasound contrast imaging. In addition, there is only one resonance peak in the

RESULTS AND DISCUSSION

Acoustics Properties of Microbubbles. To validate the finite-element model, we calculated the acoustic responses from a lipid microbubble (BR14) and compared them with the results obtained using the Church−Hoff model and measurements.45 The finite-element model was first used to predict the normalized acoustic scattering power of the BR14 microbubbles with a radius ranging from 1 to 5 μm. Figure 2a shows the scattering power spectra originating from the linear oscillations of the BR14 microbubbles with diameters of 3, 5, and 6 μm. There are obvious peaks in the scattering spectra, which form the first peaks occurring at the linear resonance frequencies of the microbubbles. The resonance frequency is plotted against the bubble size, as shown in Figure 2b. The resonance frequency obtained using the finite-element method decreases from 7.9 to 0.9 MHz with the increase in the microbubble radius from 1 to 5 μm. This frequency is compared to the analytical solution formulated by Hoff et al.45 In their work, after neglecting the surface tensions at the shell− liquid and shell−gas interfaces, a simplified version of the Church equation, namely, the Church−Hoff equation, was derived to describe the nonlinear oscillation of a microbubble with a thin shell. With the linearization of the Church−Hoff equation, the linear resonance frequency f 0 of a shellencapsulated microbubble can be given by45 f0 =

1 2πae

d yz 1 ijj jj3κp0 + 12Gs s zzz ρl jk ae z{

(10) 10217

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Figure 5. Finite-element solutions for angular scattering in the xz plane from a biosynthetic bubble (Ra = 150 nm, Rb = 100 nm, θ = 90°, and φ = 0°) at driving frequencies of 21, 467, 617, 779, 938, 1091, 1111, 1253, and 1308 MHz. The plots are of the scattering acoustic pressure amplitude (Pa) vs angle (degrees). Backscattering occurs at 270°. The incident acoustic pressure is 80 kPa.

scattering spectra in the spherical case (Figure 3c), whereas multiple significant resonance peaks can be observed in the ellipsoid spectra, as shown in Figure 3a,b, suggesting the existence of various surface modes for vibrating ellipsoidal nanobubbles. Therefore, the frequencies of the surface modes can be estimated. For cases d and e, the resonance frequency decreases to 90 and 36 MHz, respectively. Ra/Rb is a shape factor used to characterize the flattening of the ellipsoid. Clearly, the lower the value of Ra/Rb, the flatter the ellipsoid. The linear resonance frequency decreases with the decrease in the value of Ra/Rb, as shown in Figure 3f. When Ra/Rb decreases to 0.33, a linear resonance frequency of 20 MHz can be obtained, which falls into the widely used frequency range (1−20 MHz) for ultrasound imaging. The results indicate that the shell shape has a significant effect on the acoustic scattering properties of biosynthetic bubbles. It would be more advantageous to produce oblate biosynthetic bubbles to enhance the imaging contrast. Surface Modes of a Single Biosynthetic Bubble. The surface modes of bubbles may play a role in exciting the acoustic scattering energy for harmonic contrast imaging. They may also reveal physical insights into the shell characteristics

for modeling. For therapy using ultrasound combined with microbubbles, the surface modes will be of aid in comprehending the rupture mechanism of microbubbles41 because the stress and strain distributions in the shell can be determined by investigating the surface mode of a bubble. Because of the spherical symmetry, the surface mode of a spherical bubble is usually characterized by a single integer n representing the number of undulations along the circumference of the bubble.49,50 For an ellipsoidal bubble, the oscillations can be categorized using two indices n and m, where n is the number of nodes along the direction from pole to pole and m the number of nodes in the equatorial direction.51 The surface modes or deformations of an ellipsoidal bubble (Ra = 150 nm, Rb = 100 nm, θ = 90°, and φ = 0°) are demonstrated at different driving frequencies, as shown in Figure 4. Figure 3b shows that the linear resonance frequency of the bubble is 467 MHz, which is much higher than the attainable highest frequency (21 MHz) for the B-mode ultrasound imaging scanner in our group. Other driving frequencies above 467 MHz correspond to higher-order resonance modes. The surface modes can be observed through 10218

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Figure 6. Finite-element solutions for angular scattering in the yz plane from a biosynthetic bubble (Ra = 150 nm, Rb = 100 nm, θ = 90°, and φ = 0°) at driving frequencies of 21, 467, 617, 779, 938, 1091, 1111, 1253, and 1308 MHz. The plots are of the scattering acoustic pressure amplitude (Pa) vs angle (degrees). Backscattering occurs at 270°. The incident acoustic pressure is 80 kPa.

systematic relationship between the resonance frequencies and the surface modes. To further understand the backscattering properties of the bubbles, we calculated the far-field scattering pressure at a distance of r = 10 mm from a bubble (Ra = 150 nm, Rb = 100 nm, θ = 90°, and φ = 0°) insonified by an incident plane wave in the z direction. Figures 5 and 6 show the angular scattering pressures in the xz and yz planes containing the ellipsoidal and circular cross sections, respectively. At 21 MHz, the angular scattering pressure distributions in the xz and yz planes are nearly isotropic. The far-field patterns in the two planes correspond to the surface mode (n = 0, m = 0), known as the monopole mode or the breathing mode. Because the bubble size is significantly lower than the acoustic wavelength of 21 MHz (λ = 71.4 μm), Rayleigh scattering when kd ≪ 1 (k, number of wavelengths; d, particle size) dominates. Thus, the scattering pressure distribution is independent of the shapes of the cross sections in the xz and yz planes, and there is more backscattering (270°) than forward scattering (90°), as shown in Figures 5 and 6. Below the first resonance frequency of 467 MHz, the far-field pressure distributions have largely the same pattern at the different driving frequencies.

a three-dimensional displacement plot of the bubble shell, as shown in Figure 4. At 21 MHz or other frequencies below the first resonance frequency (467 MHz), the bubble with mode (n = 0, m = 0) maintains its ellipsoidal shape, contracting and expanding periodically. At the first resonance frequency of 467 MHz, the deformed bubble with mode (n = 2, m = 0) looks like a capsule, the shape of which is slightly different from that of an ellipsoid and maximum displacement appears at the equator in the cross section. It is noted that there also exists an oscillation mode (n = 2, m = 0) at 1308 MHz, while the polar displacement dominates and can be used to distinguish the two different shape oscillations with the same mode number. The reason may be: compared to the direction parallel to the equator plane, the bubble’s structure has a stronger stiffness along the polar direction, thus resulting in a higher resonance frequency. At other frequencies, the bubble would undergo a significant shape deformation. Compared to a spherical microbubble exhibiting a combination of various surface modes above the monopole resonance,41 the ellipsoidal biosynthetic bubble analyzed in this study exhibits a clear 10219

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Figure 7. Finite-element solutions for averaged angular scattering by randomly oriented biosynthetic bubbles (Ra = 150 nm; Rb = 100 nm; θ, φ ∈ [0°, 360°]; and Δθ = Δφ = 5°) at driving frequencies of 21, 467, 617, 779, 938, 1091, 1111, 1253, and 1308 MHz. The plots are of the angularly averaged scattering acoustic pressure amplitude (Pa) vs angle (degrees). Backscattering occurs at 270°. The incident acoustic pressure is 80 kPa.

shows the angular averaging of the scattering pressures in the xz plane. Because of the equally probable orientation, the acoustic scattering properties in an arbitrary plane containing the acoustic axis remain the same for enough samples with orientation angles (θ, φ). Therefore, the radiation pattern, shown in Figure 7, also represents the acoustic scattering map in an arbitrary plane containing the z axis. At 21, 467, 617, 779, and 938 MHz, the angular average of the total scattering pressures shows a nearly isotropic radiation pattern. This indicates that the incident angle of an acoustic wave from a transducer does not significantly affect the ultrasound image quality in the widely used 1−20 MHz frequency range. Because of the weak Rayleigh scattering, the scattering pressure at 21 MHz is much lower than those at other resonance frequencies. At 1091, 1111, and 1253 MHz, forward scattering is slightly more than backscattering. The averaged scattering amplitude at 1308 MHz is directional and is distinct from the quadrupole mode, shown in Figure 5, for a single biosynthetic bubble. We also calculate the angular averaging of scattering pressures from a flattened ellipsoid (Ra = 150 nm, Rb = 250 nm) at 21 MHz and its first resonance frequency at 90 MHz corresponding to the same oscillation mode (n = 2, m = 0). The averaged scattering also has the characteristics of isotropy.

At higher frequencies, the far-field pattern is frequencydependent. Figure 6 shows the acoustic radiation pattern in the yz plane. The pressure distributions behave as in the monopole mode at driving frequencies of 21, 467, 617, 779, and 938 MHz. The radiation patterns are directional at other resonance frequencies and are similar to those of the calculated angular scattering pressure for a spherical bubble with a diameter of 4 μm and a shell thickness of 100 nm.41 Because of the various values of n, the patterns of the acoustic scattering pressure distribution are different in the xz plane, as shown in Figure 5. The forward scattering pressure is equal to the backscattering pressure at 467 MHz, and the radiation pattern is the same as that of a monopole source. The pressure magnitude of forward scattering is largely the same as that of backscattering at 617, 779, and 938 MHz, and the pressure distributions excluding the minimum values seem to indicate a dipole mode. At 1091, 1111, and 1253 MHz, the scattering amplitude is directional and is lower in the backward direction than in the forward direction. The far-field radiation pressure distribution at 1308 MHz is similar to that in the quadrupole mode, and the magnitude of forward scattering is lower than that of backscattering. Angular Averaging of Scattering Pressures by Randomly Oriented Biosynthetic Bubbles. Figure 7 10220

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S h e n z h e n B a s i c R e s e a r c h P r o g r a m (g r a n t n o s . JCYJ20170413100222613 and JCYJ20170307165254568).

Compared to the UCA microbubbles with a spherically symmetric shape, this study considered the effect of the orientation of an ellipsoidal biosynthetic bubble on the acoustic scattering to calculate the average scattering pressure by randomly oriented biosynthetic bubbles. Besides the assumption of equally probable orientations for a population of randomly oriented and identical biosynthetic bubbles, it is assumed that the waves scattered from the biosynthetic bubbles are uncorrelated because a small concentration of bubbles is used in ultrasound imaging to avoid artifacts originating from the front bubbles barring the signals scattered by the rear bubbles. Therefore, the total scattering pressure can be calculated by the superposition of the scattering pressures from all of the bubbles. This method has been successfully used in predicting the acoustic spectra from UCA bubbles with different sizes.14,52 Additionally, we did not consider the bubble concentration during the calculation because the concentration will determine the magnitude of scattering pressure rather than the far-field pattern. In the future, we will further develop the calculation method for the acoustic attenuation of a group of biosynthetic bubbles with a low concentration and a certain size distribution and compare the theoretical values with the experimental results.

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CONCLUSIONS In this study, we proposed a finite-element model to simulate the acoustic scattering properties of nonspherical biosynthetic bubbles. This model was developed through acoustic wave propagation equations in a fluid and a solid. Various surface oscillation modes were found for the nonspherical bubbles, and a relationship between the surface modes and the backscattering pattern was established. The effect of shell shape on the resonance frequencies of the bubbles was found to be significant, providing physical insights into their backscattering properties. In addition, we developed a numerical method to determine the acoustic scattering by randomly oriented biosynthetic bubbles, which may be used to determine the attenuation of a randomly oriented suspension. The established method may be helpful in improving the acoustic scattering of bubbles for imaging applications. In conclusion, our study provides a tool to understand the scattering and shape oscillations of nonspherical gas nanostructures.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Fei Yan: 0000-0003-4874-9582 Author Contributions #

These authors contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (grant nos. 11774369, 81871376, 81530056, 11534013, 81527901, 81727805, and 11574342), the Shenzhen Key Laboratory of Ultrasound Imaging and Therapy (grant no. ZDSYS20180206180631473), and the Key Laboratory for Magnetic Resonance and Multimodality Imaging of Guangdong Province (2014B030301013) and the 10221

DOI: 10.1021/acs.langmuir.9b01225 Langmuir 2019, 35, 10213−10222

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NOTE ADDED AFTER ASAP PUBLICATION This paper published ASAP on June 5, 2019 with an error in equation 9. The corrected paper reposted to the Web on June 7, 2019.

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DOI: 10.1021/acs.langmuir.9b01225 Langmuir 2019, 35, 10213−10222