# Determining the influence of endoskeleton friction on the damping of pulsating antibubbles

Nicole Anderton, Craig Stuart Carlson, Vered Aharonson, Michiel Postema

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Nicole Anderton, Craig Stuart Carlson, Vered Aharonson, Michiel Postema. Determining the influence of endoskeleton friction on the damping of pulsating antibubbles. 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.781-784. 10.1515/cdbme-2022-1199 . hal-03696318

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<figure><img src="image_1.png" /></figure>
Nicole Anderton*, Craig S. Carlson, Vered Aharonson, and Michiel Postema

# Determining the influence of endoskeleton friction on the damping of pulsating antibubbles

https://doi.org/10.1515/cdbme-2022-1199

**Abstract:** Recent *in-vivo* work showed the suitability of Pickering-stabilized antibubbles in harmonic imaging and ultrasound-guided drug delivery. To date, however, theoretical considerations of antibubble core properties and their effects on antibubble dynamics have been rather sparse. The purpose of this study was to investigate the influence of skeletal friction on the damping of a pulsating antibubble and the pulsation phase of an antibubble relative to the incident sound wave. Numerical simulations were performed to compute damping terms and pulsation phases of micron-sized antibubbles with thin elastic shells and 30% endoskeleton volume fraction. The simulations showed that the damping owing to skeleton presence dominates the damping mechanism for antibubbles of radii less than 2.5 µm, whilst it is negligible for greater radii. The pulsation phase of such small antibubbles was simulated to have a phase delay of up to $\frac{1}{6}\pi$ with respect to pulsating free gas bubbles. Our findings demonstrate that the presence of an endoskeleton inside a bubble influences pulsation phase and damping of small antibubbles. Antibubbles of radii less than 3 µm are of interest for the use as ultrasound contrast agents.

**Keywords:** Acoustic driving, ultrasound contrast agent, endoskeleton, antibubble damping modelling, harmonic oscillation.

## 1 Introduction

Endoskeletal antibubbles comprise gas bubbles with one or more liquid cores. These cores are suspended by a solid skele-

<figure><img src="image_2.png" /><figcaption>**Fig. 1:** Microscopic photograph of an endoskeletal antibubble and a schematic representation thereof: (a) shell; (b) liquid core; (c) skeleton; (d) gas.</figcaption></figure>

tal structure consisting of hydrophobic particles [1], as shown in Figure 1.

As free, unencapsulated, antibubbles, are short-lived [2–5], antibubbles are typically stabilized by adsorbing nanoparticles to the liquid–gas interfaces. This process has been referred to as Pickering-stabilizing [6].

When subjected to ultrasound, the presence of an incompressible core allows for asymmetric pulsation excursions, even at modest acoustic driving amplitudes [7]. Consequently, antibubbles have been proposed as ultrasound contrast agents for low-mechanical-index harmonic imaging [8, 9]. If the liquid cores are loaded with therapeutics, antibubbles act as vehicles for ultrasound-guided drug delivery, as shown recently *in vivo* [10].

For diagnostic as well as therapeutic applications, it is highly relevant to predict antibubble dynamics and to quantify the influence of the presence of liquid and solid cores on these dynamics. In this study, we investigated the influence of skeletal friction on the pulsation phase of an antibubble relative to the incident sound wave. The purpose of this investigation was to quantify the core presence from pulsation phase observations.

Pulsation phases of damped oscillators have been analyzed thoroughly for forced mass–spring–dashpot systems [11]. They had been simulated for pulsating ultrasound contrast agent microbubbles with various shell thicknesses [12, 13]. Pulsation phases of antibubbles have not been previously reported on.

Radial pulsations of shell-encapsulated microbubbles have been modeled using adaptations of the Rayleigh-Plesset equation. These have been modified to account for specific

<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>Craig S. Carlson, 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>

<footnote>Vered Aharonson, School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, Braamfontein, South Africa and School of Sciences, University of Central Lancashire — Cyprus, Pyla, Cyprus and Afeka Tel Aviv Academic College of Engineering, Tel Aviv, Israel</footnote>
properties of the surrounding medium, for high-amplitude acoustic driving [14], for changes in surface tension owing to buckling [15], for the presence of a solid elastic shell [16], a thin lipid shell [17], a viscous or viscoelastic shell [18, 19], for the presence of another oscillator nearby [20], and for teth- ering [21], just to name a few. Several review articles have been dedicated to comparing the many models [22–24]. In principle, the most basic Rayleigh-Plesset equation suffices for a precise estimation of the radial dynamics of a shell-encapsulated microbubble, provided that the pulsation veloc- ity does not approach the speed of sound, the pulsation ex- cursion amplitude is much less than the resting size, and only few pulsation cycles are taken into account, so that viscous or viscoelastic effects remain negligible. In this study, we were assuming low-amplitude very short-pulsed ultrasound, which justifies our choice for a rather simplistic model.

In previous simulation studies, the combined solid and liq- uid internal structure had been represented by an incompress- ible volume V_c = φV_0, where φ ∈ [0, 1) is the constant volume fraction and V_0 is the volume of the antibubble [7]. Empirical evidence to justify a constant size-independent volume frac- tion exists in the form of scanning electron microscopy and confocal microscopy images [1].

## 2 Theory

Let us consider an antibubble of resting radius R₀ with an in- finitesimal shell and surrounded by an infinite liquid of den- sity ρ. If the antibubble is forced by a short short pressure pulse p(ωt) of angular center frequency ω, whose amplitude is so small that that the pulsation amplitude of the antibubble is less than R₀, we may regard this system as a forced damped oscillator with an effective mass m = 4πρ(R₀ + x)³ [25], in- stantaneous excursion x, damping coefficient δ [25, 26], and angular resonance frequency ωᵣ.

If the amplitude of the driving function function is in- stantly lessened [27], the damping coefficient of the pulsating entity undergoing damping can then be determined by mea- suring two consecutive decaying excursion amplitudes xᵢ and xᵢ₊₁ (cf. Figure 2) and substituting them into [11]:

$$ \delta = \frac{2 \ln \frac{x_i}{x_{i+1}}}{(2\pi)^2 + \left(\ln \frac{x_i}{x_{i+1}}\right)^2}. \quad (1) $$

We propose that, for an endoskeletal antibubble, the damping coefficient consists of five components, four of which are identical to those of shell-encapsulated microbubbles:

$$ \delta = \delta_v + \delta_r + \delta_{\text{g}} + \delta_{\theta}, \quad (2) $$

<figure><img src="image_3.png" /><figcaption>Fig. 2: Antibubble excursion as a function of time, for an antibub- ble of resting radius R₀ = 5 µm, forced by a short pulse. The un- forced pulsation part is represented by a red line.</figcaption></figure>

where

$$ \delta_v = \frac{4\eta}{\omega\rho(R_0 + x)^2} \quad (3) $$

is the viscous damping [26], in which η is the dynamic viscos- ity of the surrounding medium,

$$ \delta_r = \frac{\omega(R_0 + x)}{c} \quad (4) $$

is the damping owing to reradiation [26], in which c is the speed of sound of the surrounding medium,

$$ \delta_{\text{g}} = \frac{S_{\text{g}}}{m\omega} \quad (5) $$

is the damping owing to friction in the shell, in which S_g is the outer shell friction parameter [28],

$$ \delta_{\text{g}} = \varphi^n \frac{S_{\text{g}}}{m\omega} \quad (6) $$

is the damping owing to friction in the endoskeleton, in which n is a noninteger power and S_g is a thus-far undefined skeleton friction parameter, and

$$ \delta_{\theta} = \frac{\frac{\sinh X + \sin X}{\cosh X - \cos X} - \frac{2}{X}}{\frac{X}{3(\gamma-1)} + \frac{\sinh X + \sin X}{\cosh X - \cos X}} \left(\frac{\omega_r}{\omega}\right)^2 \quad (7) $$

is the thermal damping, in which X = R₀ / l_D (1 - φ¹) > 1. The thermal boundary layer thickness is given by Eller [29]:

$$ l_D = \sqrt{\frac{K_g}{2\omega\rho_g C_p}}, \quad (8) $$

where C_p is the specific heat of the gas, K_g is the thermal con- ductivity of the gas, ρ_g is the density of the gas.

An expression for the linear angular resonance frequency of a shell-encapsulated antibubble has been stated by Kudo [1]. A derivation of the difference in pulsation phase of any base-forced damped oscillator was shown by Attenborough and Postema [11]. A solution for a single bubble structure was presented by Postema and Schmitz [13]:

$$ \alpha = \pi + \arctan \left( \frac{\left( \frac{\omega}{\omega_r} \right) \delta}{1 - \left( \frac{\omega}{\omega_r} \right)^2} \right), \quad (9) $$
where $\alpha$ is the phase difference between the antibubble pulsation and the incident sound field.

## 3 Methods

Numerical solutions of (2)–(9) were computed using MATLAB®. The input parameters were chosen such that they simulated experimental situations in antibubble literature:

$c = 1480 \text{ m s}^{-1}$, $C_p = 1000 \text{ J kg}^{-1} \text{ K}^{-1}$, $K_g = 0.025 \text{ W m}^{-1} \text{ K}^{-1}$, $\eta = 1.00 \text{ mPa}$, $\rho = 998 \text{ kg m}^{-3}$, $\rho_g = 1.00 \text{ kg m}^{-3}$, $\sigma = 0.072 \text{ N m}^{-1}$, $\omega = 2\pi \times 1.0 \times 10^6 \text{ s}^{-1}$. The outer shell was considered of negligible stiffness $\le 0.2 \text{ N m}^{-1}$.

During our simulations, the expression $(S_0 + \phi^n S_g)$ was treated as a single variable.

The resting radius was varied from 0.5 µm to 12 µm.

## 4 Results and discussion

<figure><img src="image_4.png" /><figcaption>**Fig. 3:** Dimensionless damping coefficient and its contributing components as a function of antibubble resting radius.</figcaption></figure>

Figure 3 shows the damping coefficient and its contributing components for an antibubble with an endoskeleton of 30% volume fraction. The shell friction and skeleton friction parameters had been chosen conservatively, with equal values of 0.27 µN s m⁻¹, similar to some lipids. For greater values, the damping was observed to be dominated by thermal damping and reradiation. For antibubbles of resting radii less than 2.5 µm, however, the damping owing to shell and endoskeleton presence was even greater than the viscous damping term. The trade-off size coincided with the resonant size at the driving frequency of 1 MHz. It is noted that antibubbles with greater volume fraction have an even stronger skele-

ton friction damping term (not shown). As ultrasound contrast agent particles need to have diameters less than those of capillaries, the findings of the smaller antibubbles are most relevant to medical imaging.

<figure><img src="image_5.png" /><figcaption>**Fig. 4:** Pulsation phase with respect to the incident sound field, as a function of resting radius, for an antibubble (black) and a free gas bubble (blue).</figcaption></figure>

Figure 4 shows the pulsation phase of an antibubble and a free gas bubble, with respect to an incident pulse of 1-MHz central driving frequency. At the parameters chosen, only the smaller antibubbles were found to have a substantially different pulsation phase compared to the free gas bubbles. A phase difference of up to $\frac{1}{6}\pi$ was computed. The finding is useful in optically or acoustically determining whether a bubble is a core-comprising antibubble or an empty gas bubble.

## 5 Conclusions

Pulsation phases of micron-sized antibubbles differed from those of free gas bubbles. These differences may be attributed to the friction of the antibubble shells and skeletons.

For smaller antibubbles, shell and skeleton friction were found to be the dominant damping mechanisms of pulsating antibubbles driven at frequencies less than their resonance frequencies.

### Author statement

Research funding: This work was supported by the National Research Foundation of South Africa, Grant Numbers 97742 and 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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