Numerical investigation of the effect of bubble properties on the linear resonance frequency shift due to inter-bubble interactions in ultrasonically excited lipid coated microbubbles

Highlights

• •Microbubble (MB) resonance frequency (f_r) rises with increasing concentration.
• •f_r of larger MBs increase more rapidly with concentration than smaller ones.
• •Shell properties impact the rate of fr changes with concentration.
• •MB size is more influential in fr changes with concentration than shell properties.

Abstract

Ultrasonically excited microbubbles (MBs) have numerous applications in various fields, such as drug delivery, and imaging. Ultrasonically excited MBs are known to be nonlinear oscillators that generate secondary acoustic emissions in the media when excited by a primary ultrasound wave. The propagation of acoustic waves in the liquid is limited to the speed of sound, resulting in each MB receiving the primary and secondary waves at different times depending on their distance from the ultrasound source and the distance between MBs. These delays are referred to as primary and secondary delays, respectively. A previous study demonstrated that the inclusion of secondary delays in a model describing the interactions between MBs exposed to ultrasound results in an increase in the linear resonance frequency of MBs as they approach each other. This work investigates the impact of various MB properties on the change in linear resonance frequency resulting from changes in inter-bubble distances. The effects of shell properties, including the initial surface tension, surface dilatational viscosity of the shell monolayer, elastic compression modulus of the shell, and the initial radius of the MBs, are examined. MB size is a significant factor influencing the rate of linear resonance frequency increase with increasing concentration. Moreover, it is found that the shell properties of MBs play a negligible role in the rate of change in linear resonance frequency of MBs as the inter-bubble distances change.The findings of this study have implications for various applications of MBs in the biomedical field. By understanding the impact of inter-bubble distances and shell properties on the linear resonance frequency of MBs, the utilization of MBs in applications reliant on their resonant behavior can be optimized.

1. Introduction

Microbubbles (MBs) are gas-filled bubbles with diameters ranging from 1 to 20 microns [1]. They have been widely investigated in various fields ranging from medicine to industry due to their unique dynamics and properties [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. One area of interest is the dynamics of ultrasonically excited MBs, where MBs are subjected to an ultrasound pressure wave leading to their radial oscillations. This phenomenon has garnered particular interest in the field as it has shown that ultrasonically excited MBs are capable of producing rich, complex nonlinear behaviour [12], [13], [14], [15], [16] which has led to the development of new applications and techniques [17], [18], [19], [20]. The fundamental understanding of the nonlinear dynamics of MBs under ultrasonic excitation is crucial for optimizing the performance of MB-based applications and predicting potential side effects. A significant body of research has been dedicated to studying the ultrasonically excited MBs [21], [22], [23], [24], [25], [26]. The nonlinear response of MBs to ultrasonic excitation has shown to benefit ultrasound imaging applications where MBs are used as contrast agents [27]. When exposed to specific ultrasound exposure parameters, MBs have been shown to be capable of violent oscillations capable of generating powerful shockwaves inside the media that has been utilized in histotripsy [28] and lithotripsy [29]. Moreover, the stable oscillations of MBs generating microstreaming in the media have been studied for pumping and stirring of liquids in miniature scales [30]. The same mechanism is also utilized in medicine for site specific drug delivery and gene delivery [31] in sonoporation [32] applications, reversible opening of blood–brain-barrier [33], [34], [35] increasing drug uptake near oscillating MBs [36]. Sonoluminescence [37], [38], another aspect of MB dynamics has been shown to be potentially beneficial to be utilized in sonochemical reactors [39].

Understanding the dynamics of MBs is the key to all these applications. The dynamical behaviour of MBs in response to ultrasound excitation is known to be nonlinear and complex and dependent on many parameters. One of the basic acoustic properties of MBs is their resonance frequency $f_r$ [40], [41]. The energy transfer from the ultrasound wave to the MB is maximized at the resonance frequency of the MBs. For instance, this will result in maximized scattering-cross section of MBs [42]. Previously, it has been shown that by knowing the resonance frequency of MBs the exposure parameters suitable for stable oscillations of MBs can be estimated [43], [44], [45].

The resonance frequency of a MB is dependent on several factors which can be broadly categorized as:

• •The Physical properties of the MB
• •The Physical properties of the host media and the gas/media interface
• •The Physical properties of the incoming ultrasound wave
• •The Bubble’s surrounding (e.g. presence of wall, other bubbles)

The physical properties of the MB that affect the resonance frequency of the MB include the initial radius of the MB [46], the properties of the gas inside [47], and the properties of the enclosing shell [48], [49], [50] or lack there of. It has also been shown that the properties of the host media can affect the resonance frequency of the MBs. These properties include the density, viscosity and sound speed of the media [51]. Moreover, changing the media also affects the surface tension at the gas/media interface, which can modify the shell properties and therefore the resonant characteristics of the MBs. Previous studies have also shown that the pressure amplitude of the incoming ultrasound wave can modify the resonance frequency of MBs where the higher pressure amplitudes lower the resonance frequency of MBs [52]. The MB’s surrounding can also play a role in modifying its resonance frequency. Previous works have demonstrated that a MB in proximity to a wall [48], inside a blood vessel [53] and near other MBs will exhibit different resonance frequencies [54], [55].

The radial oscillations of MBs in response to an ultrasound field generate secondary acoustic emissions in the media [56]. We refer to these as secondary waves [54]. Therefore, in a cluster of MBs, each MB oscillates in response to the primary wave and the summation of the secondary waves it receives from all of the MBs in its proximity. Hence, a MB cluster can be thought of as a system of coupled oscillators where the primary wave first triggers the oscillations and then the coupling is done through the secondary emissions in the media. The majority of MB applications employ clusters of MBs. A solid understanding of the effect of coupling on the dynamics of individual MBs is paramount to utilize their full potential in the applications.

Since the speed of sound in the medium is finite, the primary and secondary waves are received at different times by MBs in the cluster. The arrival time of these waves depend on the spatial locations of the MBs. This results in a delay between when the primary and secondary waves are emitted, and when they are incident upon an MB. In a previous work, we have defined these delays as primary and secondary delays [54]. In [54], we demonstrated that the theoretical predictions of the resonance frequency of uncoated MBs are influenced by the inclusion of the primary and secondary delays. The results indicate that the primary delays lead to the broadening of the resonance frequency range (up to 0.12 MHz for the examined cases) among identical uncoated MBs. Specifically, the MB closest to the acoustic source exhibits the lowest resonance frequency, while the furthest MB resonates at the highest frequency. In that work the impact of secondary delays were found to be significant. Without considering secondary delays, the model predicted a maximum decrease of up to 26% in the resonance frequency of four interacting identical uncoated MBs as the inter-bubble distances were reduced. However, it was revealed that incorporating secondary delays reversed the trend predicted by the model, and not including delays and ultimately led to an increase in the resonance frequency of closely situated MBs ($\approx 20-40\times$ initial MB radius). The increase in resonance frequency was found to be up to 58% observed for the case of four interacting identical uncoated MBs.

This study focuses on numerically examining the influence of secondary delays on the resonance frequency of lipid-coated MBs. To accomplish this, we extended the Marmottant model [57] to incorporate inter-bubble interactions, explicitly accounting for secondary delays. By configuring our model with two identical lipid-coated bubbles exposed to the ultrasound field simultaneously (excluding the primary delay, see Fig. 1), we aimed to isolate and investigate the impact of secondary delays, which we have previously determined to be more significant than primary delays [54].

Fig. 1.

Schematic of the used geometry, where the fist and second MBs are labelled as 1 and 2 with a center to center distance of d. MBs are positioned along a line parallel to the plane of the incoming ultrasound wave eliminating the effect of primary delays.

2. Methods

2.1. The lipid coated bubble model

The Marmottant model (Eq. 1) [57] has been widely used to mathematically model the dynamics of a single oscillating lipid coated MB in the media.

$${\rho }_L\left(R\ddot{R}+\frac{3\dot{R}}{2}\right)=\left[P_0+\frac{2\sigma (R_0)}{R_0}\right]{\left(\frac{R}{R_0}\right)}^{-3\gamma }\left(1-\frac{3\gamma }{c}\right)-P_0-\frac{2\sigma (R)}{R}-\frac{4{\mu }_L\dot{R}}{R}-\frac{4{\kappa }_s\dot{R}}{R^2}-P_a\sin \omega t$$ (1)

The terms $R,\dot{R},\ddot{R}$ and t in Eq. 1 respectively denote the MB radius, wall velocity, acceleration and time. Moreover, ${\rho }_L,{\mu }_L,\sigma (R),{\kappa }_s,\gamma$, and $P_0$ respectively denote liquid (water) density and viscosity, the effective surface tension of the MB coating as a function of MB radius and the surface dilatational viscosity of the MB coating, polytropic exponent of the gas (air) enclosed within the MB and ambient pressure. The acoustic exposure parameters of the primary ultrasound wave, pressure and angular frequency are denoted with $P_a$ and $\omega$ in Eq. 1.

The Marmottant model is based on the assumption that the surface tension of a lipid-coated MB is a function of its radius (Eq. 2) and varies between 0 N/m and the surface tension of the media (in this case water at 0.0725 N/m).

$$\sigma (R)=\left\{\begin{array}{cc}0\hfill & R⩽R_{\mathit{buckling}}\hfill \\ \chi \left(\frac{R^2}{R_{\mathit{buckling}}^2}-1\right)\hfill & R_{\mathit{buckling}}<R<R_{\mathit{break}-\mathit{up}}\hfill \\ {\sigma }_{\mathit{water}}\hfill & R⩾R_{\mathit{break}-\mathit{up}}\hfill \end{array}\right)$$ (2)

Oscillating MBs are shown to generate secondary pressure waves, amplitude ($P_{\mathit{rd}}$) of which at a distance of d from the center of the MB can be calculated as [56]

$$P_{\mathit{rd}}={\rho }_L\left(\frac{2R\dot{R}+\ddot{R}\dot{R}}{d}\right)$$ (3)

Using Eq. 3 and the Marmottant model presented in Eq. 1, we can write a set of equations describing the dynamics of acoustically coupled lipid-coated MBs where the coupling is done through the secondary pressure waves they receive from all of the MBs in the cluster. This can be written as:

$${\rho }_L\left(R_i\ddot{R_i}+\frac{3\dot{R_i}}{2}\right)=\left[P_0+\frac{2\sigma (R_{i0})}{R_{i0}}\right]{\left(\frac{R_i}{R_{i0}}\right)}^{-3\gamma }\left(1-\frac{3\gamma }{c}\right)-P_0-\frac{2\sigma (R_i)}{R_i}-\frac{4{\mu }_L\dot{R_i}}{R_i}-\frac{4{\kappa }_s\dot{R_i}}{R_i^2}-P_a\sin \omega t-{\displaystyle \sum _{j=1,j\ne i}^N}{\rho }_L\left(\frac{2R_j\dot{R_j}+\ddot{R_j}\dot{R_j}}{d_{\mathit{ij}}}\right)$$ (4)

The summation term on the right-hand side of Eq. 4 stands for the summation of secondary pressure waves received by the ith MB from all the other MBs in the cluster. $d_{\mathit{ij}}$ in Eq. 4 represents the distance between the centers of the ith and the jth MB in the N-MB cluster. In its presented form, Eq. 4 assumes that all of the MBs receive the primary acoustic wave ($P_a\sin \omega t$) and the secondary waves (${\rho }_L\frac{2R_j\dot{R_j}+\ddot{R_j}\dot{R_j}}{d_{\mathit{ij}}}$) at the same time. This forms a system of coupled ordinary differential equations to model the dynamics of N acoustically coupled lipid-coated MBs. However, in our previous work, we have shown that the assumption above stating that the primary and secondary waves are instantaneously received by all of the MBs is not physical, and by correcting it (taking the finite speed of sound in the media into account), the resonance behaviour of MBs becomes a function of the MB distance [54]. To correct Eq. 4, we assume that the primary acoustic wave originates at the origin of a 3D Cartesian coordinate system and that it is travelling along the $+x$ direction on the x-axis. Therefore, we can account for the finite speed of sound c we can add delays in the primary wave as:

$$P_a\sin (\omega t-\mathit{kx})$$ (5)

and to account for the delays in the secondary waves we can modify the summation term on the right-hand side of Eq. 4 into:

$${\displaystyle \sum _{j=1,j\ne i}^N}{\rho }_L\left(\frac{2R_j(t-d_{\mathit{ij}}/c)\dot{R_j}(t-d_{\mathit{ij}}/c)+\ddot{R_j}(t-d_{\mathit{ij}}/c)\dot{R_j}(t-d_{\mathit{ij}}/c)}{d_{\mathit{ij}}}\right)$$ (6)

Therefore by using Eq. 5 and Eq. 6 we can correct Eq. 4 into

$${\rho }_L\left(R_i\ddot{R_i}+\frac{3\dot{R_i}}{2}\right)=\left[P_0+\frac{2\sigma (R_{i0})}{R_{i0}}\right]{\left(\frac{R_i}{R_{i0}}\right)}^{-3\gamma }\left(1-\frac{3\gamma }{c}\right)-P_0-\frac{2\sigma (R_i)}{R_i}-\frac{4{\mu }_L\dot{R_i}}{R_i}-\frac{4{\kappa }_s\dot{R_i}}{R_i^2}-P_a\sin (\omega t-\mathit{kx})-{\displaystyle \sum _{j=1,j\ne i}^N}{\rho }_L\left(\frac{2R_j(t-d_{\mathit{ij}}/c)\dot{R_j}(t-d_{\mathit{ij}}/c)+\ddot{R_j}(t-d_{\mathit{ij}}/c)\dot{R_j}(t-d_{\mathit{ij}}/c)}{d_{\mathit{ij}}}\right)i=1,2,\dots ,N$$ (7)

Eq. 7 is a system of coupled neutral delay differential equations for the dynamics of N coupled lipid coated MBs based on the Marmottant model.

2.2. Parameters and numerical method

Previously, we demonstrated that identical MBs exhibit a range of resonance frequencies due to primary delays [54]. Moreover, we found the effect of the secondary delays to be more significant than the primary delays. For this study, we aim to investigate how the properties of MBs impact the rate of increase in resonance frequency when interacting MBs move closer to each other. To eliminate the effect of primary delays on our numerical investigation, we selected a geometry illustrated in Fig. 1 that involves two identical MBs positioned parallel to the wavefront plane, along a line, ensuring that they receive the primary wave simultaneously. By using this setup, we can ensure that the MBs exhibit the same resonance frequency, enabling us to analyze the impact of MB parameters on the changing resonance frequency more clearly.

Eq. 7 is a system of coupled neutral delay differential equations. A numerical solution starting at $t=0$ will require values for the summation terms on the right-hand side at $(t-d_{\mathit{ij}}/c)\le 0$. We define this as history function as follows

$$\left\{\begin{array}{c}{R}_i(t-d_{\mathit{ij}}/c\le 0)=R_{i0}\hfill \\ \dot{R_i}(t-d_{\mathit{ij}}/c\le 0)=0\hfill \\ \ddot{R_i}(t-d_{\mathit{ij}}/c\le 0)=0\hfill \end{array}\right)$$ (8)

The physical interpretation of Eq. 8 is that all MBs have a known radius of $R_{i0}$ at $t=0$ and they are at rest with zero radial wall velocity and acceleration before being exposed to the ultrasound wave. To solve Eq. 7 with the conditions detailed in Eq. 8 we used the DifferentialEquations.jl package of the Julia programming language [58]. We used the ten-stage, fourth order strong stability preserving method of Ketcheson (SSPRK104) which is a strong stability preserving Runge–Kutta solver. To ensure the accuracy of the solution we used time-steps of ${10}^{-10}s$ for the solver.

The output of the solver contains the radial oscillations of the MBs as a function of time (R(t)-t). For example, Fig. 2b shows the radial oscillations of a lipid-coated MB ($R_0=2\mu$m) interacting with an identical MB at $d_{i}j=50\mu$m. These oscillations are induced by 100 cycles of an ultrasound wave with a pressure amplitude of 1 kPa and a frequency of 8 MHz.

Fig. 2.

a) The frequency response graph of a MB with an initial radius of 2μm at a distance of d = 50μm away from an identical MB. The properties of the lipid coating are $\chi =1$ N/m, ${\kappa }_s=12\times {10}^{-9}$ kg/s, $\sigma (R_0)=0.02$ N/m, $R_{\mathit{break}}=1.02\times R_0$. The peak of the frequency response graph corresponds to the resonance frequency of the MB at the examined conditions. b) Normalized radius of lipid-coated MB as a function of ultrasound cycles (periods). The MB has an initial radius of 2μm and is d = 50μm away from an identical MB. The properties of the lipid coating are $\chi =1$ N/m, ${\kappa }_s=12\times {10}^{-9}$ kg/s, $\sigma (R_0)=0.02$ N/m, $R_{\mathit{break}}=1.02\times R_0$. The incoming ultrasound wave is 100 cycles long and has a frequency and pressure amplitude of 8 MHz and 1 kPa respectively.

In order to calculate the linear resonance frequency of the bubbles, we performed the following steps [59]:

• 1-Sonicate the MBs at a low pressure of 1 kPa for 100 cycles across for a range of frequencies.
• 2-Find the maximum amplitude of oscillations in the last 10 cycles (after the transient phase, see the first oscillation cycles in Fig. 2b for reference).
• 3-Map the maximum amplitude of the oscillatory radius to the corresponding frequency (Fig. 2a).
• 4-Resonance frequency of the MB is frequency of the peak of the generated frequency response curve (green arrow in Fig. 2a).

Therefore, the frequency response graph of the MB is generated by plotting the maximum radius of oscillations against the incoming ultrasound frequency (Fig. 2a). This graph illustrates the relationship between the maximum oscillation amplitude at varying ultrasound frequencies. The resonance frequency of the MB can be identified as the peak on the frequency response graph, where the oscillation amplitude is maximized.

We chose MBs with initial radii of 2μm, 4μm and 7μm in order to represent a wide range of applications using lipid-coated MBs. As a starting point a MB with the following shell properties was chosen [60] for all of the aforementioned initial radii.

$$\left\{\begin{array}{c}\sigma (R_0)=0.02N/m\hfill \\ {\kappa }_s=12\times {10}^{-9}\mathit{kg}/s\hfill \\ \chi =1N/m\hfill \end{array}\right)$$ (9)

3. Results

Fig. 3 illustrates the variation in resonance frequency of lipid-coated MBs with similar shell properties, except for shell elasticity. MBs with initial radii of 2μm, 4μm, and 7μm are examined at various separating distances, ranging from 1μm wall-to-wall distance ($d-2\times R_0$) to 150μm center-to-center distance (d). Fig. 3 depicts the relationship between shell elasticity and the change in resonance frequency for the three sizes. It demonstrates that lower values of shell elasticity result in a smaller change in resonance frequency for all three sizes. Moreover, the spread of the maximum change in resonance frequency across different $\chi$ values is more pronounced for larger MBs. Specifically, the spread is approximately 2.7%, 4.3%, and 6% for the 2μm, 4μm, and 7μm MBs, respectively (see Fig. 3).

Fig. 3.

Percent change in the resonance frequency of lipid-coated MBs with initial radius of 2μm, 4μm, 7μm as a function of d at $\chi$ values of 1 N/m (red), 1.5 N/m (green), 2.5 N/m (blue), 5 N/m (black), 10 N/m (cyan).

Additionally, Fig. 3 reveals that the rate of change in resonance frequency due to varying elasticity values is significantly smaller compared to the rate of change resulting from varying the MB radius. For instance, when examining the case of $\chi =1$ N/m (red lines) across the examined MBs in Fig. 3, we observe significant differences in the increase of resonance frequency. Specifically, at d = 15μm the resonance frequency increases by approximately 7.6%, 17.7%, and 29.6% for the 2μm, 4μm, and 7μm MBs, respectively. This implies that with the increase in the concentration of different monodisperse MB solutions, each characterized by different MB radii, the resonance frequency of smaller MBs demonstrates a comparatively slower rate of increase than larger MBs. To further elucidate this relationship, we can estimate the equivalent concentration of MBs based on their separation distance. By considering a neighbor-to-neighbor distance (first neighbor) of d, we can approximate the equivalent concentration as $n={(1/d)}^3$. We use the term equivalent concentration rather than concentration in this context. This choice is based on our extrapolation from the distance between two MBs to the concentration yielding the specified inter-bubble distance.

Fig. 4 illustrates the relationship between the change in resonance frequency as a function of equivalent concentration. Notably, the graph reveals that larger MBs display a higher rate of resonance increase compared to smaller MBs as the concentration increases. However, it is important to note that since it is possible to increase the concentration of smaller MBs further, it is possible to achieve similar resonance increases for smaller MBs by further increasing their concentration.

Fig. 4.

Percent change in the resonance frequency of lipid-coated MBs with initial radius of 2μm, 4μm, 7μm as a function of equivalent concentration at $\chi$ values of 1 N/m (red), 1.5 N/m (green), 2.5 N/m (blue), 5 N/m (black), 10 N/m (cyan).

Fig. 5 illustrates the variation in resonance frequency of lipid-coated MBs for three different sizes as a function of d. Three different sizes of MBs with initial radii of 2μm, 4μm and 7μm at five different values of ${\kappa }_s$ at $0.1\times {10}^{-9}$ kg/s, $1\times {10}^{-9}$ kg/s, $5\times {10}^{-9}$ kg/s, $12\times {10}^{-9}$ kg/s, $25\times {10}^{-9}$kg/s are examined. Fig. 5 shows a maximum variation of 7.3%, 7.6% and 4.6% in the resonance frequency of $R_0=2\mu$m, $R_0=4\mu$m and $R_0=7\mu$m respectively. Similar to the cases presented in Fig. 3, Fig. 5 shows that the effect of initial radius is much larger than the effect of ${\kappa }_s$ in the increasing resonance frequency of MBs with increasing concentration. For instance, at $d=15\mu$m and ${\kappa }_s=2.5$ N/m we see an resonance frequency increase of 31.6%, 17.7% and 7.2% for MBs of $R_0=2\mu$m, $R_0=4\mu$m and $R_0=7\mu$m respectively. Therefore, regardless of different values of ${\kappa }_s$, the resonance frequency of monodisperse MB solutions will rise more rapidly for solutions with larger MBs. Similar to Fig. 4, Fig. 6 plots the data in terms of equivalent concentration for the examined ${\kappa }_s$s and $R_0$s.

Fig. 5.

Percent change in the resonance frequency of lipid-coated MBs with initial radius of 2μm, 4μm, 7μm as a function of d at ${\kappa }_s$ values of $0.1\times {10}^{-9}$ kg/s (red), $1\times {10}^{-9}$ kg/s (green), $5\times {10}^{-9}$ kg/s (blue), $12\times {10}^{-9}$ kg/s, $25\times {10}^{-9}$ kg/s (cyan).

Fig. 6.

Percent change in the resonance frequency of lipid-coated MBs with initial radius of 2μm, 4μm, 7μm as a function of equivalent concentration at ${\kappa }_s$ values of $0.1\times {10}^{-9}$ kg/s (red), $1\times {10}^{-9}$ kg/s (green), $5\times {10}^{-9}$ kg/s (blue), $12\times {10}^{-9}$ kg/s, $25\times {10}^{-9}$ kg/s (cyan).

Fig. 7 presents the changes in the resonance frequency of lipid-coated MBs as a function of d for $R_0=2\mu$m, $R_0=4\mu$m and $R_0=7\mu$m for different $\sigma (R_0)$ values. For all of the examined MBs, the initial surface tension $\sigma (R_0)$ was set to 0.001 N/m, 0.02 N/m, 0.04 N/m and 0.07 N/m. Fig. 7 shows maximum variations of 5.8%, 4.1% and 2.1% for MBs with $R_0=2\mu$m, $R_0=4\mu$m and $R_0=7\mu$m respectively. Hence, Fig. 7 shows that the smaller MBs have a larger response to varying $\sigma (R_0)$. However, similar to the previous two cases, the initial MB sizes play a much larger role in the overall change of the resonance frequency of MBs with varying concentrations. For instance, Fig. 7 shows that at $d=15\mu$m, the change in the resonance frequency for $R_0=2\mu$m, $R_0=4\mu$m and $R_0=7\mu$m is 32.3%, 19.8% and 9.4%, respectively.

Fig. 7.

Percent change in the resonance frequency of lipid-coated MBs with initial radius of 2μm, 4μm, 7μm as a function of d at $\sigma (R_0)$ values of 0.001 N/m (red), 0.02 N/m (green), 0.04 N/m (blue), 0.07 N/m (black).

Equating the d in Fig. 7 into an equivalent concentration using the same method as the previous cases the change in the resonance frequency of MBs as a function of equivalent concentration is obtained and presented in Fig. 8. Fig. 8 shows that the resonance frequency of larger MBs rises more rapidly with increasing concentration.

Fig. 8.

Percent change in the resonance frequency of lipid-coated MBs with initial radius of 2μm, 4μm, 7μm as a function of equivalent concentration at $\sigma (R_0)$ values of 0.001 N/m (red), 0.02 N/m (green), 0.04 N/m (blue), 0.07 N/m (black).

4. Discussion

In this study, we aimed to investigate further the impact of secondary delays on the linear resonance frequency of acoustically coupled MBs, explicitly focusing on the influence of MB lipid coating and MB sizes. Our findings build upon our previous work [54], demonstrating that lipid-coated MBs, like their uncoated counterparts, exhibit higher linear resonance frequencies when positioned closer to each other due to secondary delays. Hence to focus on the effect of secondary delays alone, we conducted a detailed examination of this phenomenon using a pair of identical MBs.

Three different MB sizes of $2\mu m,4\mu m$, and $7\mu m$ were selected to encompass a wide range of MB sizes utilized in various applications. Starting from the parameters detailed in Eq. 9, each parameter was altered independently to investigate its effect on the rate of resonance frequency increase with increasing concentration. The lipid coating elasticity, $\chi$, was varied from $1N/m$ to $10N/m$. The surface dilatational viscosity, ${\kappa }_s$, of the lipid coating was examined over a range from $0.1\times {10}^{-9}\mathit{kg}/s$ to $25\times {10}^{-9}\mathit{kg}/s$. Furthermore, the initial surface tension, $\sigma (R_0)$, of the lipid coating was explored from $0.001N/m$ to $0.07N/m$. It is important to note that in practice, it is challenging and often impractical to isolate and modify only one of the physical properties of the MB shell while keeping the others constant.

Our results indicate that while the shell properties contribute to the overall change in resonance frequency as MB concentration increases, their effect is relatively minor compared to the impact of MB size. Our analysis notably emphasizes the crucial role of MB size in the rate at which the resonance frequency increases with concentration where larger MBs exhibited a more pronounced response to changes in MB concentration, showcasing a faster upward shift in their resonance frequencies compared to smaller MBs. This phenomenon can be attributed to the capacity of larger MBs to generate higher amplitude secondary pressure fields within the media during oscillation. Consequently, the impact of inter-bubble interactions, facilitated through their secondary emissions, becomes significantly more pronounced for larger MBs. Moreover, in a previous work [54] we have shown that increasing the number of MBs would also enhance the rate of increasing resonance frequency of the MBs with increasing concentration and that primary delays lead to a broadening of the resonance frequency range for identical MBs. This investigation serves as a theoretical foundation, providing insights into how changes in shell parameters and MB size affect the resonance frequency of MBs as the inter-bubble interaction strength between them varies with concentration.

It is important to note that in this study we have looked at the extent of the discussed effects in the case of two stationary MBs fixed the space. However, MBs exposed to ultrasound move in the space in response to the primary and secondary Bjerknes forces [61], [62]. This would modify the Eq. 7 into a neutral delay differential equations with variable delays where the delays become a function of time. [63] provides detailed insights into bubble–bubble interactions under varied conditions, focusing on the effects of driving frequency, MB size, and the mechanical properties of MB coating. Their findings reveal that the interaction between bubbles, including attraction and repulsion, is significantly influenced by the resonance frequencies relative to the driving frequency, and that bubble shell visco-elasticity plays a critical role in modulating these interactions. This complements our work by offering a mechanistic explanation of bubble dynamics, emphasizing the potential to tailor bubble interactions for enhanced performance in clinical applications of ultrasound contrast agents (UCAs). To study the linear resonance frequency (a fundamental property) of MBs, we limited the pressure amplitude of the primary ultrasound wave to 1 kPa. An investigation on the changing pressure-dependent resonance frequency of MBs at higher pressure as a function of MB concentration is beyond the scope of this study and merits a dedicated future study. In this study, we focused on examining the linear resonance frequency of MBs, a fundamental property for understanding their acoustic behavior and the interactions between the MBs. This necessitated maintaining the pressure amplitude of the primary ultrasound wave at a conservative level of 1 kPa, ensuring that we captured their fundamental resonance characteristics without introducing nonlinear, pressure-dependent effects [52]. However, exploring how resonance frequency evolves under higher pressure conditions and its correlation with MB concentration presents a complex and interesting avenue for research. An analysis of the pressure-dependent dynamics of MB resonance at elevated pressures falls outside the aim of our current study. With the potential for uncovering new insights into MB behavior, this topic warrants future studies to elucidate these intricate interactions to explore their implications for diagnostic and therapeutic ultrasound applications.

5. Conclusion

This study has provided critical insights into the interplay between MB size, shell properties, and their collective impact on the changing linear resonance frequency of MBs as a function of MB concentration, thereby advancing our understanding of lipid-coated MB dynamics in acoustical applications. Our findings reveal that the MB sizes significantly influence the resonance frequency shift, with larger MBs exhibiting a more pronounced response due to enhanced inter-bubble interactions. This phenomenon underscores the importance of considering MB size in designing and optimizing of MB-based technologies for medical imaging and drug delivery, where precision in controlling acoustic response is paramount.

Moreover, the investigation into the effects of shell properties, including elasticity, surface dilatational viscosity, and initial surface tension, has shown that these factors play a secondary role compared to the impact of MB size on resonance frequency changes with changing concentration. This nuanced understanding of MB behavior offers a foundational basis for tailoring MB formulations to specific application needs, optimizing their performance in diagnostic and therapeutic settings. The implications of this work extend beyond fundamental understanding, as these findings can inform the design and optimization of MB-based technologies. For instance, in medical imaging, knowledge of the relationship between MB size, density, and resonance frequency can aid in enhancing imaging contrast and spatial resolution. Similarly, in drug delivery applications, understanding the impact of MB properties on their interaction with targeted tissues can help improve therapeutic efficacy. Generally, in applications where the concentration of MBs may vary, the varying concentration of smaller MBs is likely to have a less substantial impact than that of larger ones.

In summary, our investigation highlights the effect of secondary delays and lipid coating on the changing resonance frequency of acoustically coupled MBs. By elucidating the contributions of shell properties and MB sizes, we provide valuable insights for developing of MB-based systems across a range of applications, where manipulating MB behavior and response is paramount.

CRediT authorship contribution statement

Hossein Haghi: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Visualization. Michael C. Kolios: Supervision, Project administration, Funding acquisition, Resources, Writing - review & editing.

Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Hossein Haghi reports financial support was provided by Natural Sciences and Engineering Research Council of Canada. Hossein Haghi reports financial support was provided by Terry Fox Research Institute. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.