Translational dynamics of lipid-coated microbubbles driven by ultrasound

TL;DR

This study investigates the translational dynamics of phospholipid-coated microbubbles driven by the primary acoustic radiation force, using optical tweezers and ultra-high-speed imaging to capture time-resolved radial and translational responses in free space. It couples a modified Rayleigh–Plesset radial model with Marmottant shell stresses to a translational force balance that includes added-mass, quasi-steady drag, and a history drag term, showing that zero-Re history theory suffices up to Re \approx 2 while offering a consistent description with both zero-Re and finite-Re histories. A key finding is the linear scaling $d/R_0 \sim \Delta V / V_0$, enabling simple prediction of displacement from radial expansion and informing transport strategies that balance speed with bubble integrity; dissolution accelerates beyond $\Delta V / V_0 \approx 0.75$ when shape modes and jetting arise. The results suggest practical guidance for designing mild, short, and repeated ultrasound pulses to achieve targeted transport in biomedical settings, while noting limitations related to blood, confinement, and flow conditions in vivo.

Abstract

Ultrasound-driven microbubbles are increasingly being investigated for both molecular imaging and therapeutic applications. To be effective, these bubbles must be brought into close proximity or direct contact with the target site. Leveraging the acoustic radiation force provides a powerful strategy to direct their movement. In this study, we examine the translational dynamics of a microbubble with unprecedented accuracy by simultaneously time-resolving both its radial and translational responses and by using optical tweezers to study the bubble in free space. Our experimental results show excellent agreement with theoretical predictions for the bubble sizes considered, provided the history drag force is included in the force balance. For the Reynolds numbers considered (up to Re = 2), the zero-Reynolds-number history force performs as well as its finite-Reynolds-number extension. Although non-spherical modes may arise at larger bubble expansions, they do not appear to significantly influence the bubble translational motion. A major finding is that the normalised transport distance of the bubble scales linearly with the normalised volumetric expansion during its oscillation, greatly simplifying the design and analysis of transport strategies. We also investigated bubble stability during transport and observed a marked increase in dissolution rate once a threshold in bubble expansion is exceeded. These insights can be leveraged to develop optimal transport strategies that balance both transport speed and bubble stability for targeted delivery applications.

Figures (7)

• Figure 1: Experimental setup. (A) Agarose, (AOTF) Acousto-optic tunable filter, (C) Camera, (CO) Condenser, (DB) Dichroic beamsplitter, (DW) Deionised water, (GC) Glass coverslip, (L) Laser, (LS) Light source, (MB) Microbubble, (OL) Objective lens, (SLM) Spatial light modulator, (TC) Test chamber, (TL) Tube lens, (US) Ultrasound transducer. The inset provides a close-up view of the pre-test conditions, showing a single optically-trapped microbubble positioned more than 50µm from the test chamber's coverslip.
• Figure 2: Representative simulation of the radial and translational dynamics, and of the corresponding force balance in the direction of motion, for a free bubble with equilibrium radius $R_0 = 2.44µm$ driven by an ultrasound pulse at $f_{\rm d} = 1.5MHz$ and pressure amplitude $p_{\mathrm{a}} = 100kPa$. The time histories span two ultrasound cycles. (a) Time evolution of the bubble radius. (b) Time evolution of the bubble translation speed and displacement. (c) Forces acting on the bubble in the direction of translational motion: $F_{\rm I}$ inertial force, $F_{\rm US}$ primary acoustic radiation force, $F_{\rm AM}$ added mass force, $F_{\rm QS}$ quasi-steady drag force, $F_{\rm H}$ history drag force.
• Figure 3: Radial and translational motion of a bubble (equilibrium radius $R_0 = 2.48µm$) in free space, driven by a 20-cycle ultrasound pulse at a frequency $f_{\rm d} = 1.5MHz$ with a pressure amplitude of $p_{\mathrm{a}} = 50kPa$. (a) Image sequence showing the bubble response to the ultrasound pulse, extracted from Supplementary Movie 1. The bubble exhibits buckling during compression phases. No shape mode oscillations occur. (b) Comparison between the experimentally measured bubble radius over time and the theoretical prediction (Eq. \ref{['eq:RP']}). (c) Trajectory of the bubble translational motion in the horizontal $xy$-plane. (d) Comparison between the experimentally measured bubble displacement (measured along the straight line from its initial to final position) over time and three theoretical models: one without the history force (No HF), one incorporating the zero-Reynolds-number approximation of the history force (0Re HF, Eq. \ref{['eq:0Re']}), and one including its finite-Reynolds-number extension (fRe HF, Eq. \ref{['eq:fRe']}). (e) Comparison between the experimentally measured time evolution of the bubble averaged displacement velocity and the predictions from the same three models: No HF, 0Re HF, and fRe HF. The uncertainty in the experimental measurements corresponds to half the pixel size (80nm).
• Figure 4: Radial and translational motion of a bubble (equilibrium radius $R_0 = 2.44µm$) in free space, driven by a 20-cycle ultrasound pulse at a frequency $f_{\rm d} = 1.5MHz$ with a pressure amplitude of $p_{\mathrm{a}} = 100kPa$. (a) Image sequence showing the bubble response to the ultrasound pulse, extracted from Supplementary Movie 2. The bubble displays a distinct alternating rigid-body motion at half the ultrasound frequency, overlaid on its overall translational movement. This motion results from the excitation of the shape mode with wavenumber $l=1$, induced by the Faraday instability. (b) Comparison between the experimentally measured bubble radius over time and the theoretical prediction (Eq. \ref{['eq:RP']}). (c) Trajectory of the bubble translational motion in the horizontal $xy$-plane. (d) Comparison between the experimentally measured bubble displacement (measured along the straight line from its initial to final position) over time and three theoretical models: one without the history force (No HF), one incorporating the zero-Reynolds-number approximation of the history force (0Re HF, Eq. \ref{['eq:0Re']}), and one including its finite-Reynolds-number extension (fRe HF, Eq. \ref{['eq:fRe']}). (e) Comparison between the experimentally measured time evolution of the bubble averaged displacement velocity and the predictions from the same three models: No HF, 0Re HF, and fRe HF. The uncertainty in the experimental measurements corresponds to half the pixel size (80nm).
• Figure 5: Trajectories and displacements over time for four different cases of ultrasound-driven bubbles exhibiting the $l=1$ shape mode caused by the Faraday instability, illustrating the varying directions of the associated rigid-body motion: (a-b) Motion aligned with the ultrasound propagation direction. (c-d) Motion perpendicular to the ultrasound propagation direction. (e-f) Motion oriented obliquely relative to the ultrasound propagation direction. (g-h) Motion that transitions during the ultrasound pulse—from parallel to perpendicular relative to the propagation direction. The amplitude of the rigid-body motion increases with ultrasound cycles until saturation, a characteristic feature of a parametric instability, such as the Faraday instability. The uncertainty in the experimental measurements corresponds to half the pixel size (80nm).