Resonance behaviors of encapsulated microbubbles oscillating nonlinearly with ultrasonic excitation

Highlights

• •Resonance behaviors of microbubbles (MB) are studied via radius response analysis.
• •Nonlinear harmonic and sub-harmonic resonances of MBs are examined.
• •Microbubble resonance is radius-, pressure- and frequency-dependent.
• •Large MBs close to one MB strongly influence its resonance behaviors.
• •Lipid shell and surrounding medium dampen the resonant oscillations of MBs.

Abstract

The resonance behaviors of a few lipid-coated microbubbles acoustically activated in viscoelastic media were comprehensively examined via radius response analysis. The size polydispersity and random spatial distribution of the interacting microbubbles, the rheological properties of the lipid shell and the viscoelasticity of the surrounding medium were considered simultaneously. The obtained radius response curves present a successive occurrence of linear resonances, nonlinear harmonic and sub-harmonic resonances with the acoustic pressure increasing. The microbubble resonance is radius-, pressure- and frequency-dependent. Specifically, the maximum bubble expansion ratio at the main resonance peak increases but the resonant radius decreases as the ultrasound pressure increases, while both of them decrease with the ultrasound frequency increasing. Moreover, compared to an isolated microbubble case, it is found that large microbubbles in close proximity prominently suppress the resonant oscillations while slightly increase the resonant radii for both harmonic and subharmonic resonances, even leading to the disappearance of the subharmonic resonance with the influences increasing to a certain degree. In addition, the results also suggest that both the encapsulating shell and surrounding medium can substantially dampen the harmonic and subharmonic resonances while increase the resonant radii, which seem to be affected by the medium viscoelasticity to a greater degree rather than the shell properties. This work offers valuable insights into the resonance behaviors of microbubbles oscillating in viscoelastic biological media, greatly contributing to further optimizing their biomedical applications.

1. Introduction

Ultrasound contrast agent (UCA) microbubbles, typically composed of perfluorocarbon gas cores and stabilizing shells with proteins, lipids or polymers, have been attracting considerable interests in biomedicine [1], [2], [3], [4]. It is now well established that the UCA microbubbles will oscillate with rich and complex dynamic behaviors in response to the driving ultrasound, which is highly responsible for the contrast-enhanced ultrasound imaging and microbubble-mediated ultrasound therapies [5], [6], [7]. Subjected to ultrasonic excitation, the microbubbles have a damped resonance frequency, where the acoustic response of microbubbles will be maximal [8], [9], [10]. The damping effects arise from energy loss through radiation damping, thermal damping, and damping due to the viscosity of the surrounding medium and encapsulating shell [11], [12], [13]. To efficiently utilize the microbubbles and maximize the desired outcomes in their theranostic applications, it is necessary to match the driving ultrasound frequency to the resonance frequency of microbubbles as closely as possible. In the resonant state, the energy transfer from the acoustic wave into the microbubble oscillation is optimal and the oscillation amplitude achieves its maximum value [8], [9], [10]. Consequently, it can maximize scattered ultrasound signals for the contrast-enhanced imaging and can also result in powerful biological effects for highly effective microbubble-mediated ultrasound therapies.

The time-varying microbubble oscillations can be numerically modeled by some well-known equations, such as the Rayleigh-Plesset equation [14], the Keller-Miksis equation [15], the Gilmore equation [16], and so on. Based on these equations, the oscillation of one UCA microbubble with an encapsulating shell has also been proposed with modifications to consider the effects of the shell elasticity and viscosity via various viscoelastic models as reviewed previously [5], [17], [18], [19]. Solving the linearized equation which describes the bubble oscillation gives the damped resonance frequency. It has been demonstrated that the resonant frequency of an UCA microbubble depends on various parameters, including its size, core and shell composition [20], [21], [22], the exciting ultrasound pressure [8], [9], [23], the surrounding medium or boundary as well as its mechanical properties [24], [25], the ambient pressure [26], the fluid viscosity [27], and so on. Thus, understanding the complex resonance response of the UCA microbubbles in biological media is crucial to develop optimal strategies for microbubble-associated biomedical applications.

In practice, multiple UCA microbubbles always exist in the form of bubble cluster and they interact with each other all the time. Similar to the case of a single bubble, extensive numerical research has been devoted to the dynamics of the multi-bubble system, including the radial oscillation [28], [29], [30], translation motion [31], [32], bubble deformation [33], resonance frequency [34], and the secondary Bjerknes force [35], [36] for two oscillating bubbles in water, or the radial oscillation and/or translation motion of two interacting bubbles in viscoelastic media [37], [38], [39], [40], as well as the radial oscillation and resonance characteristics of multiple bubbles in water [41], [42], [43], [44], etc. The results showed that the inter-bubble interaction can significantly affect the bubble dynamic behaviors, such as enhancing/suppressing oscillation amplitude [28], [30], [37], [44], forcing translation motion of two bubbles close to/away from each other [35], [36], [37] and left shifting the resonance frequency [28], [34], etc. Moreover, it also demonstrated that the inter-bubble interaction highly depends on the ultrasound parameters, the initial bubble radii, the distances between bubbles, the number of bubbles and the medium viscoelasticity [28], [37], [39], [40], [44]. It should be pointed out that the interaction between multiple bubbles is relatively more complex than that of two bubbles, due to more governing equations and additional influencing factors. To simplify the problem, most of the current research assumes that: all bubbles have the same size and they are equally spaced [41], [42] or randomly distributed in space [43], or all bubbles have different sizes and are equidistant from each other [44]. These assumptions ignore the effects of the size polydispersity or spatial arrangement of bubbles, and even both of them on the inter-bubble interaction, thereby neglecting the resultant impacts on the dynamic behaviors of multiple bubbles.

Typical UCA microbubbles have a broad size distribution and are randomly distributed after injection into the human body. Under the same ultrasonic excitation, each bubble in the multi-bubble system with polydispersity in bubble size and random distribution in space will respond differently (e.g., some bubbles oscillate on– or off-resonance). Moreover, it can be expected that different mutual inter-bubble interaction will act on each other because of the size difference and unequal distance between these interacting bubbles [28], [37], [39], [40], [44]. More importantly, the bubble resonance and inter-bubble interaction can also be influenced by the viscoelastic properties of encapsulating shells and surrounding medium while applying the UCA microbubbles in biomedicine [37], [38], [39], [40], [45]. Consequently, investigating the dynamic behaviors of multiple interacting UCA microbubbles subjected to ultrasound in viscoelastic biological media would be more complicated due to fewer simplifying assumptions and additional influencing factors, and yet little attention has been paid to this issue. Therefore, understanding the acoustic resonant responses of UCA microbubbles and how they are affected by the key influencing factors (e.g., the acoustic settings, the initial bubble radii, the shell properties as well as the interactions with adjacent bubbles and surrounding viscoelastic media) is of vital importance for UCA microbubbles relevant applications.

The present study sought to investigate the dynamics of multiple interacting UCA microbubbles with lipid shells in viscoelastic biological media, with an emphasis on the nonlinear resonance response to the ultrasonic excitation. A comprehensive model was developed with simultaneously considering the impacts of the inter-bubble interaction, the shell properties and the medium viscoelasticity. Radius response analysis was utilized to describe the bubble oscillations, which directly display the oscillation amplitudes at different bubble radii and consequently the resonant radii can be easily obtained at the resonance peaks. Four representative models of a three-bubble system were first compared, and then the effects of the ultrasound amplitude, the ultrasound frequency, the initial radii of nearby bubbles as well as the inter-bubble distances were assessed in a more practical case. In addition, the influences of the shell properties and the medium viscoelasticity were also examined. This study might gain deeper insights into the acoustic resonant responses of lipid-shelled UCA microbubbles in viscoelastic media, further contributing to more tailored and more efficient microbubble-assisted ultrasound applications.

2. Theory and methods

2.1. Numerical bubble model

The radial oscillations of lipid-coated UCA microbubbles subjected to ultrasonic excitation were numerically simulated by solving the well-known Gilmore equation in the present study, since it has the largest applicability range with a high Mach number up to $\dot{R}/c$ = 2.2, as compared to the widely used Rayleigh-Plesset and Keller-Miksis equations [46]. The Gilmore model is more suitable for the situations involving large-amplitude oscillations at relatively high acoustic pressures, especially in the microbubble-mediated therapeutic applications [47], [48]. For a multi-bubble system, each bubble experiences additional driving pressures emitted by the nearby pulsating bubbles, hence the Gilmore equations were modified by considering the effects of the bubble–bubble interactions on the radial oscillations. For the bubble i, they are written as follows [39]:

$$\begin{array}{c}{R}_i{\ddot{R}}_i(1-\frac{{\dot{R}}_i}{C_i})+\frac{3}{2}{\dot{R}}_i^2(1-\frac{{\dot{R}}_i}{3C_i})=(1+\frac{{\dot{R}}_i}{C_i})(H_i-\frac{{\left({\tau }_{\mathit{rr}}\right|}_{R_i}}{\rho }+\frac{3q_i}{\rho })+\frac{R_i}{C_i}[{\dot{H}}_i(1-\frac{{\dot{R}}_i}{C_i})-\frac{{\left({\dot{\tau }}_{\mathit{rr}}\right|}_{R_i}}{\rho }+\frac{3{\dot{q}}_i}{\rho }]\\ -\sum _{j=1,j\ne i}^N\frac{2R_j{\dot{R}}_j^2+R_j^2{\ddot{R}}_j}{d_{\mathit{ij}}}\end{array}$$ (1)

$$H_i=\frac{1}{\rho }\frac{m}{m-1}(p_0+p_a+B)[{(\frac{p_{b,i}+B}{p_0+p_a+B})}^{\frac{m-1}{m}}-1]$$ (2)

$$C_i=c_0{(\frac{p_{b,i}+B}{p_0+p_a+B})}^{\frac{m-1}{2m}}$$ (3)

where R_i(t) and R_j(t) are the time-varying radii of the bubbles i and j, the dot denotes time derivative. H_i, C_i and ${{\tau }_{\mathit{rr}}|}_{R_i}$ are the enthalpy, the local speed of sound and the stress in the r direction at the wall of the bubble i, respectively. d_ij is the distance between the centers of bubbles i and j, N is the number of the interacting bubbles. ρ is the density of the surrounding medium, c₀ is the sound speed in the medium, B and m are specific constants in the Gilmore model. p₀ is the atmospheric pressure, p_a is the varying acoustic pressure, and p_b,i is the pressure at the wall of bubble i. The driving ultrasound is p_a(t) =  − p_Asin(2πft), where p_A and f are the ultrasound amplitude and frequency, respectively. q_i is given by $q_i={\int }_{R_i}^{\infty }\frac{{\tau }_{\mathit{rr}}}{r}dr$, which is introduced to couple the bubble model with the viscoelastic model of the surrounding medium.

2.2. Zener viscoelastic model

For the viscoelasticity of the surrounding medium, the Zener model was used since it has a distinct advantage in describing both creep recovery and stress relaxation of soft tissues simultaneously, which is given by [39], [46]:

$${\tau }_{\mathit{rr}}+{\lambda }_1{\dot{\tau }}_{\mathit{rr}}=2G{\gamma }_{\mathit{rr}}+2\mu {\dot{\gamma }}_{\mathit{rr}}$$ (4)

where ${\tau }_{\mathit{rr}}$ is the stress in the r direction, ${\gamma }_{\mathit{rr}}$ is the strain, ${\dot{\gamma }}_{\mathit{rr}}$ is the strain rate. G, μ and λ₁ are the elasticity, viscosity and relaxation time of the surrounding medium, respectively. According to the continuity equation, the ${\dot{\gamma }}_{\mathit{rr}}=-2R^2\dot{R}/r^3$ and ${\gamma }_{\mathit{rr}}=-2(R^3-R_0^3)/3r^3$ can be derived and then substituting them into Eq. (4) with r = R_i, thus one can obtain the stress at the wall of bubble i [46]:

$${\left({\tau }_{\mathit{rr}}\right|}_{R_i}+{\left({\lambda }_1{\dot{\tau }}_{\mathit{rr}}\right|}_{R_i}=-\frac{4G}{3}(1-\frac{R_{0i}^3}{R_i^3})-4\mu \frac{{\dot{R}}_i}{R_i}$$ (5)

where R_0i is the initial bubble radius of the bubble i. For the variable $q_i$, dividing Eq. (5) by r and integrating the resulting equation from R_i to ∞ can obtain [46]:

$$q_i+{\lambda }_1{\dot{q}}_i+{\lambda }_1\frac{{\dot{R}}_i{\left({\tau }_{\mathit{rr}}\right|}_{R_i}}{R_i}=\frac{1}{3}[-\frac{4G}{3}(1-\frac{R_{0i}^3}{R_i^3})-4\mu \frac{{\dot{R}}_i}{R_i}]$$ (6)

2.3. Nonlinear shell elasticity and viscosity of the encapsulating shell

In this study, nonlinear rheological modifications were made by adding elastic and viscous terms to the Gilmore equation for the effective elastic and viscous contributions of the lipid shell. The Marmottant model was applied to account for the shell elasticity by introducing the radius-dependent surface tension σ(R_i) [20]:

$$\sigma (R_i)=\left\{\begin{array}{c}0\\ \chi \left(R_i^2/R_{b,i}^2-1\right)\\ {\sigma }_t\end{array}\right)\begin{array}{c}\mathit{if}{R}_i\le R_{b,i}\\ \mathit{if}{R}_{b,i}<R_i<R_{r,i}\\ \mathit{if}rupturedand{R}_i\ge R_{r,i}\end{array}$$ (7)

where χ is the shell elasticity, and σ_t is the surface tension of the surrounding medium. R_b,i and R_r,i are the bucking and rupture radii of bubble i, defined respectively as $R_{b,i}=R_{0i}/\sqrt{1+{\sigma }_0(R_{0i})/\chi }$ and $R_{r,i}=R_{b,i}\sqrt{1+{\sigma }_t/\chi }$, where σ₀(R_0i) is the initial surface tension. Please refer to the previous literatures for more details [20], [23], [49], [50].

Furthermore, the nonlinear shell viscosity ${\kappa }_{s,i}$ was modified according to the Cross law [51]:

$${\kappa }_{s,i}=\frac{{\kappa }_0}{1+\alpha \left|{\dot{R}}_i\right|/R_i}$$ (8)

where ${\kappa }_0$ is the initial shell viscous parameter, α is a characteristic time constant.

2.4. Boundary definition at the bubble wall

The nonlinear elasticity and viscosity of the lipid shell as described by Eqs. (7), (8) respectively, as well as the viscoelasticity of the surrounding medium can strongly affect the pressure at the bubble wall. As a result, they were considered simultaneously as pressure boundary conditions at the wall of bubble i, which can be expressed as [46]:

$$p_{b,i}=p_{\mathit{in},i}-\frac{2\sigma (R_i)}{R_i}-\frac{4{\kappa }_{s,i}{\dot{R}}_i}{R_i^2}+{\left({\tau }_{\mathit{rr}}\right|}_{R_i}$$ (9)

where $p_{\mathit{in},i}$ is the pressure inside the bubble i. The gas in a lipid-coated bubble i is assumed to obey the van der Waals theorem and it can be determined by [40]:

$$p_{\mathit{in},i}=[p_0+\frac{2{\sigma }_0(R_{0i})}{R_{0i}}]{(\frac{R_{0i}^3-h_i^3}{R_i^3-h_i^3})}^n$$ (10)

where h_i = R_0i/5.6 is the van der Waals hard-core radius for the bubble i, n is the polytropic exponent of the gas within the bubble.

2.5. Initial conditions and numerical solution

Eqs. (1), (5), (6) are a system of ordinary differential equations. In conventional methods, Runge-Kutta numerical algorithms can be employed to obtain a solution with 4 N initial conditions (at t = 0) as follows:

$$R_i=R_{0i},{\dot{R}}_i=0,{\left({\dot{\tau }}_{\mathit{rr}}\right|}_{R_i}=0,{\dot{q}}_i=0i=1,2,...N$$ (11)

For the case of N interacting microbubbles, there are N coupled second-order differential equations as shown in Eq. (1). When using the Runge-Kutta algorithms to numerically solve the system of equations, the algebraic expressions for the highest order of differentiations are needed for each one equation. Consequently, the expressions would become very large, cumbersome and tedious to deal with for a large number of bubbles (N ≥ 3), limiting the number of interacting bubbles that can be simulated. This has been discussed in detail elsewhere and one can see more details in the cited references [44], [52], [53].

To simplify the numerical methods, one can re-write the Eq. (1) to have all the terms with ${\ddot{R}}_i$ on the left side as:

$$(1-\frac{{\dot{R}}_i}{C_i})R_i{\ddot{R}}_i+\frac{4{\kappa }_{s,i}}{\rho C_i{R}_i}(1-\frac{{\dot{R}}_i}{C_i}){(\frac{p_{b,i}+B}{p_0+p_a+B})}^{-\frac{1}{m}}{\ddot{R}}_i+\sum _{j=1,j\ne i}^N\frac{R_j^2}{d_{\mathit{ij}}}{\ddot{R}}_j=A_i$$ (12)

where

$$A_i=-\frac{3}{2}(1-\frac{{\dot{R}}_i}{3C_i}){\dot{R}}_i^2+(1+\frac{{\dot{R}}_i}{C_i})(H_i-\frac{{\left({\tau }_{\mathit{rr}}\right|}_{R_i}}{\rho }+\frac{3q_i}{\rho })+\frac{R_i}{C_i}[F_i(1-\frac{{\dot{R}}_i}{C_i})-\frac{{\left({\dot{\tau }}_{\mathit{rr}}\right|}_{R_i}}{\rho }+\frac{3{\dot{q}}_i}{\rho }]-\sum _{j=1,j\ne i}^N\frac{2R_j{\dot{R}}_j^2}{d_{\mathit{ij}}}$$ (13)

$$\begin{array}{c}{F}_i=\frac{1}{\rho }\frac{m}{m-1}[1-\frac{1}{m}{\left(\frac{p_{b,i}+B}{p_0+p_a+B}\right)}^{\frac{m-1}{m}}]p_A\cos \left(2\pi ft\right)2\pi f\\ +\frac{1}{\rho }{\left(\frac{p_{b,i}+B}{p_0+p_a+B}\right)}^{-\frac{1}{m}}[{\dot{p}}_{\mathit{in},i}-\frac{2\dot{\sigma }\left(R_i\right)}{R_i}+\frac{2\sigma \left(R_i\right){\dot{R}}_i}{R_i^2}+\frac{8{\kappa }_{s,i}{\dot{R}}_i^2}{R_i^3}+{\left({\dot{\tau }}_{\mathit{rr}}\right|}_{R_i}]\end{array}$$ (14)

$${\dot{p}}_{\mathit{in},i}=-3n[p_0+\frac{2{\sigma }_0(R_{0i})}{R_{0i}}]\frac{{(R_{0i}^3-h_i^3)}^n}{{(R_i^3-h_i^3)}^{n+1}}{R}_i^2{\dot{R}}_i$$ (15)

By defining D_i as$D_i=(1-\frac{{\dot{R}}_i}{C_i})[R_i+\frac{4{\kappa }_{s,i}}{\rho C_i{R}_i}{(\frac{p_{b,i}+B}{p_0+p_a+B})}^{-\frac{1}{m}}]$, Eq. (12) can be expressed in matrix format as:

$$\left[\begin{array}{cccc}{D}_1& \frac{R_2^2}{d_{12}}& \dots & \frac{R_N^2}{d_{1N}}\\ \frac{R_1^2}{d_{21}}& D_2& \dots & \frac{R_N^2}{d_{2N}}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{R_1^2}{d_{N1}}& \frac{R_2^2}{d_{N2}}& \dots & D_N\end{array}\right]\left[\begin{array}{c}{\ddot{R}}_1\\ {\ddot{R}}_2\\ \dots \\ {\ddot{R}}_N\end{array}\right]=\left[\begin{array}{c}{A}_1\\ A_2\\ \dots \\ A_N\end{array}\right]$$ (16)

The solution matrix can be obtained by an inverse matrix method, which is given by

$$\left[\begin{array}{c}{\ddot{R}}_1\\ {\ddot{R}}_2\\ \dots \\ {\ddot{R}}_N\end{array}\right]={\left[\begin{array}{cccc}{D}_1& \frac{R_2^2}{d_{12}}& \dots & \frac{R_N^2}{d_{1N}}\\ \frac{R_1^2}{d_{21}}& D_2& \dots & \frac{R_N^2}{d_{2N}}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{R_1^2}{d_{N1}}& \frac{R_2^2}{d_{N2}}& \dots & D_N\end{array}\right]}^{-1}\left[\begin{array}{c}{A}_1\\ A_2\\ \dots \\ A_N\end{array}\right]$$ (17)

The solution to the radial oscillations of N interacting microbubbles thus can be obtained by numerically solving Eq. (17) with using the Runge-Kutta algorithms.

2.6. Numerical analysis method

Fig. 1 illustrates a system of three interacting lipid-coated microbubbles (N = 3), which have different initial radii (i.e., R₀₁, R₀₂ and R₀₃) and are randomly distributed (determined by d₁₂, d₁₃ and θ) in viscoelastic media. To ensure that all bubbles do not collide during oscillations, a minimum inter-bubble distance is defined as 50 μm, and if the distances between any two bubbles are closer than the minimum allowed value, the spatial formations of a multiple bubble system will be reset. This process is repeated for every setting.

Fig. 1.

Schematic diagram for describing the oscillations of three-interacting bubbles in viscoelastic tissues.

Simulation was performed up to T = 100/f to ensure that a steady state solution has been reached. The relative expansion ratio of bubble i is defined as

$$X_i=\frac{R_{\max ,i}-R_{0i}}{R_{0i}}$$ (18)

where R_max,i is the maximum radius of the bubble i during its steady oscillations. The radius response curve, which plots the X_i versus the initial bubble radius R_0i at fixed ultrasound pressure and frequency, was used to describe the bubble oscillation, exhibiting a clear dependency of the bubble resonance on the initial bubble radius. Unless specified otherwise, the physical parameters and their default values were set according to Table 1.

Table 1.

Simulation conditions.

Parameters	Values	Units	Parameters	Values	Units
f	1	MHz	G	20	kPa
pA	300	kPa	μ	9	mPa∙s
ρ	1060	kg/m3	λ	3	ns
c0	1540	m/s	R01	0.1 ∼ 10	μm
p0	1.01 × 105	Pa	R02	3	μm
σ0	0.02	N/m	R03	10	μm
χ	0.44	N/m	d12	100	μm
κ0	5 × 10−8	kg/s	d13	50	μm
α	3	μs	θ	π/3	/
σt	0.056	N/m	n	1.07	/

3. Results and discussion

3.1. Comparisons of bubble dynamics in four representative cases

Based on different assumptions on the initial bubble radii and the inter-bubble distances (as shown in Table 2), four representative models for a three-bubble system (Models I-IV) can be developed, and the bubble oscillation characteristics subjected to ultrasonic excitation in the form of radius response curve were first compared between different models. Compared to typically used frequency response analysis, radius response curve has a great advantage for clearly giving the dependence of the bubble oscillation amplitude on its initial radius and the resonant radius can be readily obtained at certain acoustic settings. The radius response curves obtained with the four representative models are presented in Fig. 2(a) - (d), respectively. In the legends, the symbols “w” and “w/o” mean the results with and without considering the inter-bubble interaction, respectively.

Table 2.

Representative model assumptions, simulation cases as well as results in the maximum of relative expansion ratio X_max and the resonant radius R_res.

Model	Assumptions	Simulation cases	Xmax	Rres
I	Bubbles have the same initial radii and are equally spaced.	R01 = R02 = R03 = 0.1 ∼ 10 μm d12 = d13 = d23 = 100 μm	↓	↓
II	Bubbles have the same initial radii and are unequally spaced.	R01 = R02 = R03 = 0.1 ∼ 10 μm d12 = 100 μm, d13 = 50 μm,θ = π/3	↓	↓
III	Bubbles are polydisperse in size and equally spaced.	R01 = 0.1 ∼ 10 μm, R02 = 3 μm, R03 = 10 μm d12 = d13 = d23 = 100 μm	↓↓	↑
IV	Bubbles are polydisperse in size and unequally spaced.	R01 = 0.1 ∼ 10 μm, R02 = 3 μm, R03 = 10 μm d12 = 100 μm, d13 = 50 μm, θ = π/3	↓↓↓	↑↑

Fig. 2.

Radius response curves for acoustically excited microbubbles using (a-d) four representative models (I-IV) respectively in two cases with (w) or without (w/o) considering the inter-bubble interaction. Specifically, three lipid-coated microbubbles having (a, b) the same initial radii (R₀₁ = R₀₂ = R₀₃ = 0.1 ∼ 10 μm) or (c, d) different initial radii (R₀₁ = 0.1 ∼ 10 μm, R₀₂ = 3 μm and R₀₃ = 10 μm) are (a, c) equally (d₁₂ = d₁₃ = 100 μm and θ = π/3) or (b, d) unequally spaced (d₁₂ = 100 μm, d₁₃ = 50 μm and θ = π/3) in viscoelastic media.

It can be seen that the variation tendencies are similar in all cases, but there are still some noticeable differences. For the simplest Model I assuming that all microbubbles are identical and equidistant from each other, all microbubbles oscillate in the same manner, as displayed in Fig. 2(a). As compared to the case of an isolated microbubble, it is obvious that the maxima of relative expansion ratios at the main resonance peak of the radius response curves X_max decrease to a very small degree and the radius response curves slightly shift to left (i.e., a slight decrease in the resonant radius R_res) because of the bubble–bubble interaction. With an assumption of all microbubbles having the same size but the inter-bubble distances being unequal (Model II), Fig. 2(b) shows that the radius response curves of the three interacting microbubbles are considerably different between each other, indicating that distinct inter-bubble interactions exert on each microbubble. This can be attributed to the differences in the inter-bubble distance, which is a vital factor affecting the bubble–bubble interaction, as demonstrated in previous studies [28], [30], [37], [39], [44].

In contrast, for the cases of all microbubbles having different initial radii (Models III and IV), the radius response curves exhibit that the microbubble oscillation amplitudes are always reduced within the whole variation range of R₀₁ = 0.1 ∼ 10 μm, probably due to the prominent suppression effect from a large bubble (R₀₃ = 10 μm), as indicated by the previous research on the interacting polydisperse microbubble clusters in water [44]. Furthermore, an obvious evidence that the microbubble oscillation is more significantly reduced in the Model IV can also verify the dominant effect of the large bubble, since a stronger suppression effect exerts on the smaller bubble R₀₁ as the nearby large bubble R₀₃ becomes closer (i.e., d₁₃ = 100 μm in Model III versus d₁₃ = 50 μm in Model IV). Interestingly, it is worth noting that the resonance peak near R₀₁ = 6.5 μm disappears in the Model IV. Taken together, Table 2 summarizes the variations in both X_max and R_res for the cases with considering the bubble–bubble interaction, as compared to an isolated microbubble case. In this table, ‘↑’ stands for an increase, whereas ‘↓’ stands for a decrease. Additionally, the number of ‘↑’ or ‘↓’ means the degree of increase or decrease, and the more the number is, the greater the degree becomes. These results highlight that obvious differences in the microbubble oscillation characteristics, such as the oscillation amplitude and resonant radius, can be caused by different assumptions on the multi-bubble model. Moreover, these findings also raise intriguing questions regarding the reasonable model assumptions on the initial radius and spatial arrangement of multiple bubbles. Thus, developing a more realistic model simultaneously considering the size polydispersity of UCA microbubbles and their random spatial distribution in the viscoelastic biological medium, and meanwhile comprehensively investigating the oscillation characteristics of this multi-bubble system is highly necessary for optimizing UCA microbubble-mediated applications. In the following sections, Model IV is preferentially selected to investigate the dynamics of three interacting lipid-coated microbubbles in viscoelastic media with a focus on the resonance response via analyzing the radius response curve.

3.2. Pressure-dependent linear and nonlinear resonances

The dynamic behaviors of microbubbles are highly pressure-dependent, which will undergo a transform from linear response to nonlinear response with the acoustic pressure p_A increasing. As examples of a three-bubble system without and with taking into account the mutual interactions with nearby microbubbles, Fig. 3(a) and (b) display the radius response curves of the lipid-coated microbubbles at relatively low and high acoustic pressures, respectively. It shows just one peak (i.e., the linear resonance) at a resonant radius when the acoustic pressure is sufficiently small (Fig. 3(a)). There are small differences between the two cases of the isolated microbubble and interacting microbubbles (e.g., X_max-w/o - X_max-w ≈ 0.005 at p_A = 10 kPa), primarily due to the negligible inter-bubble interactions at such low acoustic pressures. In contrast, with the acoustic pressure increasing, the maximum of the relative expansion ratio X_max dramatically increases with the corresponding resonant radius R_res decreasing and additional resonance peaks (marked by arrows) appear near the main resonance peak, as displayed in Fig. 3(b). It can be explained that the bubble would undergo nonlinear resonances (i.e., harmonic and subharmonic resonances) at high enough acoustic pressures, which have been broadly investigated for the bubble oscillating in water [22], [23], [53], [54], [55]. Furthermore, the X_max and R_res at various acoustic pressures are also presented in Fig. 3(c) and (d), respectively. It is obvious that with the acoustic pressure increasing, the microbubbles will undergo more intense expansions, exhibiting that X_max increases monotonically at a near-linear trend. By comparison, R_res decreases continuously when the acoustic pressure increases, but with a larger decrease rate at lower acoustic pressures. In addition, comparing the two cases without and with considering the inter-bubble interaction exhibits obvious increases of the differences in both X_max and R_res as the ultrasound pressure increases. The presented evidence reveals that the influences of the inter-bubble interaction on both bubble oscillation amplitude and resonant radius are amplified at a higher acoustic pressure, due to a stronger inter-bubble interaction with the acoustic pressure increasing as demonstrated in previous studies [31], [34], [37], [40].

Fig. 3.

Radius response curves of lipid-coated microbubbles (R₀₁ = 0.1 ∼ 10 μm) subjected to relatively (a) low and (b) high acoustic pressures in two cases without (w/o) and with (w) considering the inter-bubble interaction. (c) and (d) represent the maximum of relative expansion ratio (X_max) and the resonant radius (R_res) as a function of the acoustic pressure (p_A), respectively.

For multiple microbubbles in realistic scenarios, ultrasound pressure as an important parameter can significantly change the microbubble dynamics, the inter-bubble interactions and therefore the resonance behaviors as well. At very low acoustic pressures, microbubbles will oscillate along with the frequency of the incident ultrasound wave, showing stable, symmetrical and relatively small radial oscillations. With an increase in the acoustic pressure, not only the oscillation amplitude increases, but also non-linear microbubble oscillation can be achieved. Moreover, it should be noticed that the main response peak seems to be much broader at a higher acoustic pressure than that for the case at a lower acoustic pressure. As a result, a microbubble can actively oscillate within a wide range of initial radius which does not include the linear resonant radius, since the radius response curve will shift left considerably with increasing the acoustic pressure. In other words, this means that a microbubble of linear resonant radius might be less active at sufficiently higher acoustic pressures due to the nonlinearity of bubble oscillations. Thus, limiting the size distribution of microbubbles to be monodisperse and meanwhile ensuring that the initial size is equal to/near the linear resonant radius might be not critical when utilizing microbubbles in therapeutic applications, especially for some cases where high ultrasound pressures are applied.

To further evaluate the linear and nonlinear acoustic responses of the differently sized microbubbles subjected to ultrasonic excitation, Fig. 4 exemplifies the normalized radii (R₁/R₀₁) over time and the acoustic emission spectra of some representative on-resonance microbubbles at various acoustic pressures. For an isolated microbubble case, at low acoustic pressures, the periodic bubble oscillations repeat their pattern once every-one acoustic driving period (marked by hollow circles at the end of each driving period similar to previous studies [23], [55]), and the circles have the same value at each period, therefore a period 1 (P1) resonance occurs as displayed in Fig. 4(a). In comparison, there are two distinct values for R₁/R₀₁ at the end of each driving period as displayed by purple circles (e.g., R₀₁ = 6.25 μm and p_A = 300 kPa shown in Fig. 4(b)), namely the bubble can undergo period 2 (P2) oscillations at a high enough acoustic pressure. At p_A = 1 kPa, the corresponding frequency spectrum of the acoustic emission shown in Fig. 4(c) has only one peak at the fundamental frequency, which is indicative of linear resonance. By contrast, as the acoustic pressure increases, the bubble will oscillate nonlinearly, concurrently generating the second and higher harmonics in the acoustic emission spectra, such as the cases of R₀₁ = 4.55 μm and p_A = 10 kPa, as well as R₀₁ = 2.9 μm and p_A = 300 kPa. Surprisingly, it can be found that upon the same acoustic excitation (p_A = 300 kPa), the microbubbles with initial radii of 2.9 μm and 6.25 μm undergo P1 and P2 oscillations respectively, as displayed by arrows in Fig. 4(b). With the occurrence of P2 oscillation, the subharmonic peaks appear in the frequency spectrum. The corresponding results with considering the bubble–bubble interaction are presented in Fig. 4(d) - (f), respectively. Comparatively, under the same initial conditions of R₀₁ = 6.25 μm and p_A = 300 kPa as the isolated microbubble case, there are no P2 oscillation occurrence and no subharmonics generation when the inter-bubble interaction is taken into account. Nevertheless, the P2 oscillation occurs at a relatively higher acoustic pressure (p_A = 400 kPa), and the subharmonic resonant radius increases up to 6.7 μm. This difference can be attributed to the inter-bubble interaction with nearby microbubbles. Overall, these results succeed in correlating the spectral components of acoustic emissions with the bubble oscillation behaviors directly. More importantly, it can be found that the bubble resonance behaviors, including the linear and nonlinear resonances (i.e., harmonic and subharmonic resonances), are highly dependent on the initial bubble size, the acoustic pressure as well as the presence of nearby bubbles. In addition, the effects of some key influencing factors on the resonance behaviors of three interacting microbubbles were also examined in the following sections.

Fig. 4.

(a, b) Radius-time curves of an isolated lipid-coated microbubble and (c) the frequency spectra of the scattered pressures under some representative resonance conditions. (d-f) represent the corresponding results for the cases with considering the inter-bubble interactions with two nearby microbubbles.

3.3. Effects of the ultrasound frequency

Fig. 5 examines the resonance response of a lipid-coated microbubble (R₀₁ = 1 ∼ 10 μm) as a function of ultrasound frequency f in two cases without and with taking its interactions with two nearby microbubbles into account. It is apparent in Fig. 5(a) that the X_max decreases as the ultrasound frequency increases, exhibiting a greater reduction in the lower frequency range. It can be explained that the low-frequency ultrasound can provide a longer time-period during which the bubble is continuously driven by the negative acoustic pressure, giving sufficient time for bubble expansion and hence resulting in a larger bubble expansion ratio. Moreover, the R_res also decreases with the ultrasound frequency increasing as shown in Fig. 5(b). Previous investigations on the resonance response of a single bubble have demonstrated that the resonance frequency of a microbubble is radius-dependent, which varies inversely with the bubble radius [9], [21]. In order to be more in line with the practical applications of bubbles, the interactions between bubbles are considered, our findings show that the X_max is reduced and the R_res is shifted toward larger values as compared to the single bubble case. In particular, it seems that the bubble resonance responses are influenced by the inter-bubble interaction to a greater degree at a lower ultrasound frequency. Together these results shown in Fig. 3, Fig. 4, Fig. 5 can provide important insights into the linear and nonlinear resonance behaviors of lipid-coated microbubbles under various acoustic settings, and can also highlight the need for comprehensively examining the effects of the initial bubble size, acoustic pressure, ultrasound frequency and inter-bubble interaction on the bubble resonances, especially for the cases at higher acoustic pressures and lower acoustic frequencies that are usually applied for ultrasound therapeutic applications.

Fig. 5.

(a) Maximum of relative expansion ratio X_max and (b) corresponding resonant radius R_res as a function of ultrasound frequency f, which are calculated by the radius response curve of a lipid-coated microbubble (R₀₁ = 0.1 ∼ 10 μm). Two cases without (w/o) and with (w) considering the inter-bubble interaction are compared.

3.4. Effects of the initial bubble radii

As shown in Fig. 2, Fig. 3, Fig. 4, although subjected to the same ultrasonic excitation, microbubbles will respond differently depending on their initial radii. Moreover, microbubbles of different initial radii can also exert influences on neighboring microbubbles to different degrees. For a system of three interacting microbubbles, the effects of the initial radii of the two nearby microbubbles (R₀₂ and R₀₃) on the resonance response of one microbubble were further examined via plotting the radius response curve (R₀₁ = 0.1 ∼ 10 μm). From the radius response curve, the X_max, R_res, as well as the relative expansion ratio at the 1/2 order subharmonic resonance peak X_max-1/2 and the subharmonic resonant radius R_res-1/2 were further calculated and mapped as a function of R₀₂ and R₀₃. As shown in Fig. 6. the increase of X_max, R_res, X_max-1/2 and R_res-1/2 is displayed in red while the decrease is displayed in blue as compared to the case of an isolated microbubble. It is apparent from Fig. 6(a) that the X_max is notably reduced with an increase in R₀₂, R₀₃ or both of them, indicating that the larger the nearby microbubbles are, the stronger suppression effects will act on owing to the stronger inter-bubble interaction. These results are in well agreement with those obtained by recent studies that focused on the inter-bubble interaction in water [28], [44]. For the R_res shown in Fig. 6(b), their variation trends as a function of R₀₂ and R₀₃ corresponds well to the strength of the inter-bubble interaction. Specifically, a stronger inter-bubble interaction at a larger R₀₂ and/or R₀₃ can give rise to a greater shift of the R_res. More importantly, it should also be worth noting that increasing R₀₃ appears to exert a stronger suppression effect than that caused by increasing R₀₂, indicating that the inter-bubble interaction seems to be more sensitive to the increase of R₀₃ rather than R₀₂. This finding might be explained by a smaller inter-bubble distance examined here (i.e., d₁₃ < d₁₂) which can result in a stronger inter-bubble interaction.

Fig. 6.

Effects of the nearby microbubbles with various initial radii (R₀₂ = 0.5 ∼ 10 μm and R₀₃ = 0.5 ∼ 10 μm) on the harmonic and subharmonic resonances of microbubbles (R₀₁ = 0.1 ∼ 10 μm), including (a) the maximum of relative expansion ratio X_max and (b) resonant radius R_res at the harmonic resonance peak, as well as (c) the relative expansion ratio X_max-1/2 and (d) resonant radius R_res-1/2 at the 1/2 order subharmonic resonance peak as a function of R₀₂ and R₀₃. An increase is displayed in red while a decrease is displayed in blue as compared to the case of an isolated microbubble. The gray shading in (c) and (d) represents that no 1/2 order subharmonic resonance occurs.

Furthermore, the influences of the inter-bubble interaction at various R₀₂ and R₀₃ on the 1/2 order subharmonic resonance (i.e., X_max-1/2 and R_res-1/2) are manifested in Fig. 6(c) and (d), respectively. If there were no subharmonic resonance peaks in the radius response curves, the X_max-1/2 and R_res-1/2 were set to zeros and plotted by gray shading in the mappings. It is obvious that no subharmonic resonance occurs at a larger R₀₃, whereas subharmonic resonance occurs all the time in the whole range of R₀₂ examined here. There are two possible explanations for this result. One is that the larger microbubbles exhibit a much stronger suppression effect on the subharmonic resonance of the smaller microbubble, according well with the previous research on the inter-bubble interactions of microbubble clusters in water [44]. Another one is that the inter-bubble distances d₁₃ was set to be smaller than d₁₂, and consequently the inter-bubble interaction between the microbubbles with a smaller inter-bubble distance of d₁₃ will become stronger, since the mutual interaction of bubbles is oppositely related to the distances between them [28], [30], [37]. When the subharmonic resonance occurs, the inter-bubble interaction exerts a much stronger effect on the X_max-1/2 and R_res-1/2 with the R₀₂ and/or R₀₃ increasing, similar to the effects on the harmonic resonance. Taken together, the results suggest that the resonance response of one microbubble will experience a very slight influence when the nearby microbubbles are of smaller sizes, whereas it will be prominently influenced by the presence of larger microbubbles in the neighborhood. In addition, the findings also highlight the importance of taking into account the size difference between bubbles in a polydisperse bubble cluster while investigating the bubble resonance response subjected to ultrasound, hence further contributing to the maximization of the strength of bubble oscillations and/or the generation of subharmonics.

3.5. Effects of the inter-bubble distance

As indicated in Fig. 6, the resonance response of one lipid-coated microbubble that is interacting with neighboring microbubbles can be noticeably affected by the inter-bubble distance between each other, since the bubble–bubble interaction is highly dependent on the inter-bubble distance [28], [30], [37]. Similarly, the effects of the inter-bubble distance on the X_max, R_res, X_max-1/2 and R_res-1/2 as a function of d₁₂ and d₁₃ are illustrated in Fig. 7(a-i) - (a-iv), respectively. The white region means that the inter-bubble distance is smaller than the minimum allowed distance which is set to make ensure all bubbles do not collide during oscillations. For a three-bubble system examined here, the inter-bubble distance is determined by three parameters (i.e., d₁₂, d₁₃ and θ) as defined in Fig. 1. At a fixed θ = π/3, the X_max increases but R_res decreases with d₁₂ and/or d₁₃ increasing. With respect to the 1/2 order subharmonic resonance, there is no subharmonic resonance occurring at a smaller d₁₃ (marked in gray), but the subharmonic resonance appears and the corresponding X_max-1/2 gradually increases with increasing d₁₂ and/or d₁₃. In contrast, the R_res-1/2 slightly decreases with the increase of d₁₂ and/or d₁₃. Moreover, the effects of the inter-bubble interaction on the bubble resonance at various d₁₃ and θ are presented in Fig. 7(b). Similar trends can be seen as the d₁₃ or θ increases, as compared to Fig. 7(a). These results indicate that the effects of the inter-bubble interaction on the bubble resonances are reduced as the interacting microbubbles stay away from each other, but the reductions appear to be quite different with the inter-bubble distances increasing. On the other hand, it can also be expected that the interaction effects become negligible and the interacting microbubbles oscillate as isolated microbubbles at sufficiently large distances. These results accord well with previous observations on the inter-bubble interactions of two or multiple bubbles oscillating in water, which demonstrated that the closer the interacting bubbles become, the stronger the inter-bubble interaction is [28], [30]. Additionally, it can also be clearly explained by the inspection of the Eq. (1), revealing that the magnitude of the secondary acoustic field is inversely proportional to the distance between the interacting bubbles.

Fig. 7.

Effects of the inter-bubble interaction on the bubble resonance at various inter-bubble distances, including (a) variation of the d₁₂ and d₁₃ as well as (b) variation of the d₁₃ and θ. In both figures, (i) the maximum of relative expansion ratio X_max and (ii) resonant radius R_res at the harmonic resonance peak, as well as (iii) the relative expansion ratio X_max-1/2 and (iv) resonant radius R_res-1/2 at the 1/2 order subharmonic resonance peak are plotted. An increase is displayed in red while a decrease is displayed in blue as compared to the case without considering the inter-bubble interactions. The white region means that the inter-bubble distance is smaller than 50 μm, and the gray shading represents that no 1/2 order subharmonic resonance occurs.

Considering the results shown in Fig. 7 together, in spite of the same variation tendencies with d₁₂, d₁₃ or θ increasing, there are evident differences in the degree of the influences. By contrast, it seems that increasing d₁₃ can lead to the variations of X_max, R_res, X_max-1/2 and R_res-1/2 to a larger degree as compared to the case with d₁₂ or θ increasing respectively. A possible explanation for this might be that a much larger microbubble whose initial radius is R₀₃ = 10 μm can impose a stronger interaction on the observed bubble than that impacted by a microbubble with a smaller initial radius (R₀₂ = 3 μm), as illustrated in Fig. 6. Overall, combining the results shown in Fig. 6, Fig. 7, there seems to be substantial evidence to indicate that the inter-bubble interaction will dramatically affect the microbubble resonance response, especially in the case of larger microbubbles being in close proximity. These findings further illustrate that ignoring the inter-bubble interaction for simplicity to predict and optimize dynamics of multiple microbubbles would inevitably lead to erroneous conclusions, thereby drawing our attention to consider the inter-bubble interaction in designing and optimizing microbubble-associated applications because of most applications using microbubble in polydisperse clusters.

3.6. Effects of the rheological properties of the encapsulating shell

The effects of the rheological properties of the lipid shell, including the initial surface tension σ₀, shell elasticity χ and viscosity κ₀, on the harmonic resonance (X_max and R_res) and 1/2 order subharmonic resonance (X_max-1/2 and R_res-1/2) were further examined, as shown in Fig. 8. It is obvious from Fig. 8(a) that increasing σ₀ will increase X_max but lead to a slight decrease in R_res, while enhance the 1/2 order subharmonic resonance. Specifically, there is no subharmonic resonance occurring at lower σ₀ (σ₀ < 16 mN/m), whereas it appears and the corresponding X_max-1/2 increases with an increase in σ₀. This can be attributed to the asymmetric variation of the effective surface tension between the buckled and the ruptured state of the lipid coating, according to the previous study investigating nonlinear dynamics of an isolated lipid-coated microbubble oscillating in water [23]. Nevertheless, it should be pointed out that subharmonic oscillations can occur at pressures as low as 1 kPa in the previous study [23], whereas in the present study no subharmonic resonance occurs as σ₀ < 16 mN/m at much larger acoustic pressures (e.g., 350 kPa) probably due to the inter-bubble interactions with the nearby larger microbubbles and the damping effects of the surrounding viscoelastic medium.

Fig. 8.

Resonance behaviors of lipid-coated microbubbles whose shells own various (a) initial surface tension σ₀, (b) elasticity χ and (c) viscosity κ₀ at p_A = 350 kPa, including (i) the maximum of relative expansion ratio X_max and resonant radius R_res at the harmonic resonance peak, as well as (ii) the relative expansion ratio X_max-1/2 and resonant radius R_res-1/2 at the 1/2 order subharmonic resonance peak.

The corresponding results at various shell elasticities χ and viscosities κ₀ are presented in Fig. 8(b) and (c), respectively. It can be found that there appear to exist similar variation trends in X_max and R_res with either χ or κ₀ increasing. The larger the χ or κ₀ is, the smaller the X_max is but the larger the R_res becomes. Nevertheless, it seems that increasing the shell viscosity exhibits a much greater effect than the shell elasticity. For the subharmonic resonance, it can be observed that the increase of χ slightly enhances the occurrence and strength of the subharmonic resonance, whereas the increase of κ₀ dramatically suppresses the subharmonic resonance. The results may further support previous research which has been quantified that for a constant σ₀, a lipid-shelled microbubble with a higher χ exhibits nonlinear behaviors at a lower pressure threshold [23].

To sum up, the lipid-coated microbubbles display a lower oscillation amplitude and a larger resonant radius for both harmonic and subharmonic resonances, as compared to the uncoated microbubble. In practical applications, to prolong the microbubble lifetime, the UCA microbubbles are typically stabilized by various shells, including lipids, polymers, and proteins [1], [2], [3], [4]. Fig. 8 exemplifies the dependence of the bubble resonance behaviors on the rheological properties of the encapsulating shell. The results indicate that changing the shell properties can lead to significant variations in their acoustic responses (e.g., bubble oscillation amplitudes and resonant radii at harmonic and subharmonic resonances), confirming high sensitivity to shell properties. Notably, there is a trend of shell stiffness impacting the bubble resonance behaviors to a much lesser degree while the initial surface tension and viscosity of shells affecting the bubble resonances to a greater degree within the range of variations examined in the present study. Together these results provide important insights into the effects of the rheological properties of the encapsulating shell on the bubble resonance responses, potentially giving assistance to an optimal outcome of desired applications associated with shell-encapsulated microbubbles.

3.7. Effects of the medium viscoelasticity

Fig. 9 exemplifies the radius response curves of the lipid-coated microbubbles embedded in various viscoelastic media. From Fig. 9(a) - (c), it can be seen that an increase of the medium viscosity μ leads to a remarkable decrease in the strength of both harmonic and subharmonic resonances (i.e., X_max and X_max-1/2), which is in accordance with previous studies investigating the dynamics of uncoated bubbles in viscoelastic media [37], [39]. However, there seems to exist an increase in the resonant radii (i.e., R_res and R_res-1/2) with μ increasing. This might be explained by the fact that suppressed bubble oscillations reduce the soft spring feature and hence increase the resonant radius [45]. A slight decrease in R_res at certain μ (approximately 20 mPa∙s) might be explained by the fact that the two resonance peaks become one broader peak as μ increases to a certain value, as displayed in Fig. 9(a). Moreover, from the perspective of subharmonic oscillation, the 1/2 order subharmonic resonance peak is reduced with μ increasing, and even disappears at sufficiently larger μ, since microbubbles experience a stronger viscous damping effect and consequently oscillate weakly. With regard to the influence of the medium elasticity G, similar trends can be observed in Fig. 9(d) - (f). Increasing G decreases the oscillation amplitude while increases the resonant radii at both harmonic and subharmonic resonances. Noticeably, it seems to have a smaller impact on the oscillation amplitude but exhibit a larger impact on the resonant radius as compared to the medium viscosity.

Fig. 9.

Radius response curves for lipid-coated microbubbles (R₀₁ = 0.1 ∼ 10 μm) in viscoelastic media of various (a) viscosities μ (d) and elasticities G. (b) and (e) plot the maximum of relative expansion ratio X_max (left y-axis) and resonant radius R_res (right y-axis) at the harmonic resonance peak as a function of μ and G respectively, while (c) and (f) represent the corresponding variations in the relative expansion ratio X_max-1/2 and resonant radius R_res-1/2 at 1/2 order subharmonic resonance.

Furthermore, comparing the two cases with (w) and without (w/o) considering the bubble–bubble interaction, the differences seem to become smaller as μ and/or G increases. It can be explained that the effects of the inter-bubble interaction are reduced at larger values of μ and/or G, in accord with recent studies demonstrating the bubble–bubble interaction of a two-bubble system in viscoelastic media [37], [39], [40]. In particular, the medium with larger viscoelasticity can prevent embedded microbubbles from undergoing large-amplitude oscillations and subharmonic oscillations. Note that at a lower G, no subharmonic resonance occurs in the case of considering the inter-bubble interaction, probably due to the intense oscillations of nearby large microbubbles acting on strong suppression effects on the observed microbubble. These results illustrate the effects of the medium viscoelasticity on the bubble resonance behaviors, which is of great importance for the microbubble-mediated biomedical applications due to the biological tissues individually displaying certain viscoelasticity.

4. Conclusion

This study set out to comprehensively examine the harmonic and subharmonic resonance behaviors of three lipid-coated microbubbles embedded in viscoelastic media under ultrasonic excitation. A representative three-interacting bubble model was developed to match the practical application scenarios better via simultaneously considering the size polydispersity and random spatial distribution of microbubbles, the rheological properties of the lipid shell as well as the viscoelasticity of the surrounding medium. The radius response curves for microbubbles present only one peak (i.e., the linear resonance occurs at a resonant radius) at lower acoustic pressures, whereas the resonance peak will gradually shift to left along with the successive occurrence of nonlinear harmonic and subharmonic resonances when the acoustic pressure increases. The results clearly indicate that the microbubble resonance is radius-, pressure- and frequency-dependent. Specifically, the maximum of relative expansion ratio at the main resonance peak increases as the ultrasound pressure increases but decreases with the ultrasound frequency increasing, while the corresponding resonant radius varies inversely with the ultrasound pressure and frequency. Moreover, the microbubble resonance behaviors are also highly dependent on the inter-bubble interaction. Compared to an isolated microbubble case, the resonant oscillations of microbubbles which are interacting with nearby microbubbles are prominently weakened as the adjacent microbubbles are larger and/or get closer, even giving rise to the subharmonic resonance disappearing with the suppression effect increasing to a certain degree. In addition, the microbubble resonance behaviors are prominently affected by the shell properties and medium viscoelasticity. Together, the results suggest that the viscosity and elasticity of both encapsulating shell and surrounding medium can substantially dampen the harmonic and subharmonic resonances while increase the corresponding resonant radii, which seem to be affected by the medium viscoelasticity to a greater degree rather than the shell properties. The present study may provide a deeper insight into the harmonic and subharmonic resonance behaviors of UCA microbubble clusters oscillating in viscoelastic biological media, and might be of great assistance to further optimize their applications in biomedical imaging and therapy. Nevertheless, due to so many different parameters involved in the multi-bubble model, further research should also be undertaken to explore how they affect the bubble dynamics, especially for a polydisperse bubble clouds consist of dozens or even hundreds of interacting UCA microbubbles. Moreover, the dynamics of UCA microbubbles in viscoelastic media excited by dual- or multi-frequency acoustic excitations would also be a fruitful area for further work.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.