# Fragmentation thresholds simulated for antibubbles with various infinitesimal elastic shells

Nicole Anderton, Michiel Postema

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Nicole Anderton, Michiel Postema. Fragmentation thresholds simulated for antibubbles with various infinitesimal elastic shells. Current Directions in Biomedical Engineering, 2022, Proceedings of the 2022 Joint Annual Conference of the Austrian (ÖGBMT), German (VDE DGBMT) and Swiss (SSBE) Societies for Biomedical Engineering, 8 (2), pp.73-76. 10.1515/cdbme-2022-1020. hal-03696317

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Nicole Anderton* and Michiel Postema

# Fragmentation thresholds simulated for antibubbles with various infinitesimal elastic shells

https://doi.org/10.1515/cdbme-2022-1020

**Abstract:** Antibubbles are small gas bubbles comprising one or multiple liquid or solid cores, typically surrounded by stabilising shells. Acoustically active microscopic antibubbles have been proposed for use as theranostic agents. For clinical applications such as ultrasound-guided drug delivery and flash-echo, it is relevant to know the fragmentation threshold of antibubbles and the influence of the stabilising shells thereon. For antibubbles with an infinitesimal frictionless elastic shell of constant surface tension, we simulated ultrasound-assisted fragmentation by computing radial pulsation as a function of time using an adapted Rayleigh-Plesset equation, and converting the solutions to time-variant kinetic energy of the shell and time-variant surface energy deficit. By repetition over a range of pressure amplitudes, fragmentation thresholds were found for antibubbles of varying size, core volume, shell stiffness, and driving frequency. As backscattering increases with scatterer size, and as drug delivery would require vehicles just small enough to pass through capillaries with a relatively large payload, we chose to present typical results for antibubbles of resting diameter 6 µm with a 90% incompressible core. At a driving frequency of 13 MHz, the fragmentation threshold was found to correspond to a mechanical indices less than 0.4, irrespective of shell stiffness. This mechanical index is not considered unsafe in diagnosis. That means that antibubbles acting as drug-carrying vehicles could release their payload under diagnostic conditions.

**Keywords:** Acoustic fragmentation, ultrasound contrast agent, shell stiffness, antibubble oscillation modelling, Rayleigh-Plesset equation.

## 1 Introduction

Ultrasound contrast agents are used both in diagnosis and therapy, and are therefore referred to as theranostic agents [1]. They comprise microscopic gas bubbles surrounded by stabilising shells [2]. The pulsation dynamics of a spherically symmetric microbubble surrounded by an elastic shell has been modelled with a Rayleigh-Plesset equation, adjusted for the presence of a shell by introducing a shell stiffness parameter [2–5]. Knowing under which conditions ultrasound contrast agent microbubbles fragment might be of interest in echography and ultrasound-guided drug delivery [6, 7]. Shell-encapsulated microbubbles, called parents, subjected to pressures below the inertial cavitation threshold have been observed to typically fragment into eight or more so-called daughter microbubbles [8]. The number of fragments depends on the energy subjected to the parent [8]. The presence of a shell has been found to be of influence on the fragmentation threshold of such microbubbles [7].

Antibubbles are small gas bubbles comprising one or multiple liquid or solid cores [9]. If a surrounding shell is absent, an antibubble is very short-lived [10–12]. The presence of an encapsulating shell or an endoskeleton drastically increases the antibubble lifetime [13–16]. Fluids comprising antibubbles have been proposed for the use as theranostic agents [11, 15, 17, 18].

The dynamic response of microscopic antibubbles by stabilising shells subjected to ultrasound has been studied in *silico* [19, 20], *in vitro* [16, 18, 21], and, more recently, *in vivo* [22]. The simulations and experiments of most of these preliminary studies concentrated on the radial pulsations of antibubbles and the accompanying generation of harmonics. These studies were highly relevant for potential applications of antibubbles in diagnostic harmonic imaging. For potential therapeutic applications of antibubbles, however, it is more relevant to know under which acoustic conditions the core material is released [22]. Ultrasound-assisted disruption of microscopic antibubbles had been demonstrated by high-speed camera footage [21, 22]. From such footage, the fragment size distribution could be predicted [23].

The purpose of this study was to simulate the fragmentation threshold of microscopic antibubbles with infinitesimal

<footnote>*Corresponding author: Nicole Anderton, BioMediTech, Faculty of Medicine and Health Technology, Tampere University, Korkeakoulunkatu 3, 33720 Tampere, Finland, e-mail: nicole.anderton@tuni.fi</footnote>

<footnote>Michiel Postema, BioMediTech, Faculty of Medicine and Health Technology, Tampere University, Tampere, Finland and School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, Braamfontein, South Africa</footnote>
shells of different biomaterial composition. The outcomes of this study may aid in the development of novel theranostic antibubble agents.

## 2 Methods

An infinite viscous fluid was assumed to surround a perfectly spherical shell-encapsulated antibubble containing one or multiple incompressible cores, subjected to a sound pulse whose wavelength is much greater than the antibubble size. The shell was considered homogeneous, of infinitesimal thickness, elastic, frictionless, and of constant surface tension. For modelling purposes, we replaced the total core volume inside the antibubble by an equivalent core radius. Following a prior derivation [24] but incorporating damping and shell stiffness terms [25], the resulting fundamental pulsation equation used in this study is then given by

$$R\ddot{R} + \frac{3}{2}\dot{R}^2 = \frac{1}{\rho} \left[ \left( p_0 - p_v + \frac{2\sigma R_0}{R_0^2 - R_c^2} \right) \left( \frac{R_0^3 - R_c^3}{R^3 - R_c^3} \right)^\gamma \right. \\ \left. + p_v - \frac{2\sigma R}{R^2 - R_c^2} - 2\chi \left( \frac{1}{R_0} - \frac{1}{R} \right) - \frac{4\eta\dot{R}}{R} - \delta\omega\rho R\dot{R} - p_0 - p(t) \right], \quad (1)$$

where $p(t)$ is the time-dependent acoustic driving function, $p_0$ is the ambient pressure, $p_v$ is the vapour pressure, $R$ is the instantaneous radius, $R_0$ is the initial radius, $R_c$ is the equivalent core radius, $\gamma$ is the ratio of specific heats, $\delta$ is the damping coefficient, $\eta$ is the liquid viscosity, $\rho$ is the liquid density, $\sigma$ is the surface tension, $\chi$ is the shell stiffness, and $\omega$ is the angular driving frequency. The viscous damping had been directly included in (1). Therefore, the damping coefficient $\delta$ only comprised the damping owing to reradiation and the thermal damping, $\delta \approx \frac{\omega R}{c} + \frac{3}{5}(\gamma - 1)$, where $c$ is the speed of sound of the medium. The angular resonance frequency of a shell-encapsulated antibubble was found by adjusting the resonance frequency of a free-surface antibubble [21] for the surface pressure components in (1) and for the presence of an infinitesimal elastic shell [7]:

$$\omega_r = \frac{1}{R_0 \sqrt{\rho}} \sqrt{\frac{3\gamma \left( p_0 - p_v + \frac{2\sigma R_0}{R_0^3 - R_c^3} \right)}{1 - \frac{R_c^3}{R_0^3}} - \frac{2\sigma R_0}{R_0^2 - R_c^2} + \frac{4\eta^2}{R_0^2 \rho} + \frac{2\chi}{R_0}} \quad (2)$$

The fragmentation threshold pressure was defined as the acoustic pressure amplitude at which the kinetic energy of the parent surface [26]

$$E_k \approx 2\pi\rho R^3 \dot{R}^2 \quad (3)$$

<figure><img src="image_3.png" /><figcaption>Fig. 1: Radius and instantaneous energies as a function of time simulated for an antibubble of initial radius $R_0 = 3 \mu m$ with a 90% core radius and shell stiffness $\chi = 7.6 N m^{-1}$ driven with a 0.6-MI pulse of centre frequency 1 MHz (a,c) and 13 MHz (b,d). Instantaneous kinetic energies are indicated by blue lines and surface energy deficits by red lines.</figcaption></figure>

surpassed the difference in surface energy between the parent entity and at least eight daughters [6, 7],

$$\Delta E_s \approx 4\pi R^2 \sigma, \quad (4)$$

here referred to as energy deficit.

Numerical solutions of (1) were computed using the ○de45 differential equation solver of MATLAB® (The MathWorks, Inc., Natick, MA, USA). The three-cycle sinoidal acoustic driving function was defined by $p(\omega t) = A \sin \omega t \quad \forall \omega t \in [0, 6\pi] \wedge p(\omega t) = 0 \quad \forall \omega t \notin [0, 6\pi]$, in which A was varied to find the fragmentation pressure threshold. The following parameters were chosen in the simulations: $c = 1568 m s^{-1}$, representing saline [27], $p_0 = 1.00$ atm, $p_v = 2.33$ kPa, $\gamma = 1.4$, $\eta = 1.00$ mPa s, $\rho = 998$ kg m$^{-3}$, $\sigma = 0.072 N m^{-1}$. Values of $R_0, R_c, \chi$, and $\omega$ were variables. Throughout this paper, $R_c$ is expressed as a percentage of $R_0$. The $R(t)$ curves computed and their time derivatives were converted to kinetic energy and surface energy deficit vectors. Fragmentation pressure thresholds were expressed in mechanical index,

$$MI = \frac{A[\text{MPa}]}{\sqrt{\frac{\omega}{2\pi}[\text{MHz}]}} \quad (5)$$

for interpreting the clinical relevance of the findings. An $MI \le 0.7$ is not considered unsafe [28]. Through iteration in MI steps of 0.005, the fragmentation threshold was determined automatically as a function of the variables of choice. The size range was limited to radii small enough to pass through capillaries. The range of shell stiffnesses was limited to those com-
<figure><img src="image_4.png" /><figcaption>**Fig. 2:** Fragmentation threshold in MI as a function of shell stiffness $\chi$, simulated for a 3-µm radius antibubble with a 90% core radius, subjected to a pulse of centre frequency 1 MHz (red) and 13 MHz (blue).</figcaption></figure>

mon in biomaterials. The frequency range was limited to those common in commercial probes.

# 3 Results and discussion

Two representative $R(t)$ curves and their corresponding instantaneous energies are shown in Figure 1, for an antibubble of 3-µm initial radius with a 90% core radius, subjected to a 0.6-MI pulse. At a 1-MHz driving frequency, the kinetic energy simulated was too low to cause fragmentation. At a 13-MHz driving frequency, however, the kinetic energy clearly surpassed the surface energy deficit to cause fragmentation.

Figure 2 shows the fragmentation threshold for 3-µm radius antibubbles as a function of shell stiffness at two different sonication frequencies. At a 1-MHz driving frequency, the fragmentation threshold was found to increase with shell stiffness. Here, antibubbles with a shell stiffness less than $7 \text{ N m}^{-1}$ were simulated to fragment at an $MI \le 1$. At a 13-MHz driving frequency, however, the fragmentation threshold was found to decrease with shell stiffness. Here, for each stiffness simulated, antibubbles fragmented at an $MI < 0.4$. A straightforward explanation of this counterintuitive result is that the resonance frequency of an antibubble increases with its shell stiffness. Consequently, the difference between antibubble resonance and driving frequency may increase or decrease with

<figure><img src="image_5.png" /><figcaption>**Fig. 3:** Fragmentation threshold in MI as a function of driving frequency, simulated for a 3-µm radius antibubble with a 90% core radius and a $7.6 \text{ N m}^{-1}$ shell stiffness.</figcaption></figure>

shell stiffness. The fragmentation threshold should be lowest at resonance. The size of the antibubble core was observed to be only of minor influence on the fragmentation threshold (data not shown).

Figure 3 shows the fragmentation threshold for 3-µm radius antibubbles with shells of $7.6 \text{ N m}^{-1}$ stiffness and a 90% equivalent core radius as a function of driving frequency. For driving frequencies greater than 1.5 MHz, the fragmentation thresholds corresponded to $MI < 0.7$. The fragmentation threshold had a minimum of $MI = 0.1$ at a driving frequency of 4.5 MHz. From (2), it followed that the resonance frequency of such an antibubble is 4.54 MHz. Hence, the driving frequency at the simulated minimum corresponded to the resonance frequency.

Even at a core radius of 90% of the antibubble radius, the fragmentation threshold at 13-MHz driving corresponded to an $MI$ of less than 0.4, which is not considered unsafe in diagnosis. As a consequence, antibubbles acting as drug-carrying vehicles would release their payload under diagnostic conditions.

# 4 Conclusions

Our simulations show that at lower driving frequencies, the shell stiffness is of major influence on antibubble fragmentation, whilst at higher driving frequencies, the shell mate-
rial is hardly of influence. At 13-MHz driving, stiff-shell-
encapsulated antibubbles were simulated to fragment at acous-
tic amplitudes that are not considered unsafe in diagnosis.
These findings imply that drug-loaded antibubbles, stabilised
by rigid shells, could be forced to release their contents using
diagnostic ultrasound. This research is of interest in flash-echo
and ultrasound-guided drug delivery.

Author statement

Research funding: This work was supported by the National Research Foundation of South Africa, Grant Number 127102, and by the Academy of Finland, Grant Number 340026. Conflict of interest: Authors state no conflict of interest. Informed consent: Authors state that informed consent is not applicable. Ethical approval: Authors state that no ethical approval was required for this research as no human or animal samples or data were used.

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