Numerical simulations of non-spherical bubble collapse

Ultrasound-responsive microbubbles and nanodroplets: A pathway to targeted drug delivery

Ultrasound-responsive agents have shown great potential as targeted drug delivery agents, effectively augmenting cell permeability and facilitating drug absorption. This review focuses on two specific agents, microbubbles and nanodroplets, and provides a sequential overview of their drug delivery process. Particular emphasis is given to the mechanical response of the agents under ultrasound, and the subsequent physical and biological effects on the cells. Finally, the state-of-the-art in their pre-clinical and clinical implementation are discussed. Throughout the review, major challenges that need to be overcome in order to accelerate their clinical translation are highlighted.

Ultra-high-speed dynamics of acoustic droplet vaporization in soft biomaterials: Effects of viscoelasticity, frequency, and bulk boiling point

Phase-shift droplets are a highly adaptable platform for biomedical applications of ultrasound. The spatiotemporal response of phase-shift droplets to focused ultrasound above a certain pressure threshold, termed acoustic droplet vaporization (ADV), is influenced by intrinsic features (e.g., bulk boiling point) and extrinsic factors (e.g., driving frequency and surrounding media). A deep understanding of ADV dynamics is critical to ensure the robustness and repeatability of an ADV-assisted application. Here, we integrated ultra-high-speed imaging, at 10 million frames per second, and confocal microscopy for a full-scale (i.e., from nanoseconds to seconds) characterization of ADV. Experiments were conducted in fibrin-based hydrogels to mimic soft tissue environments. Effects of fibrin concentration (0.2 to 8 % (w/v)), excitation frequency (1, 2.5, and 9.4 MHz), and perfluorocarbon core (perfluoropentane, perfluorohexane, and perfluorooctane) on ADV dynamics were studied. Several fundamental parameters related to ADV dynamics, such as expansion ratio, expansion velocity, collapse radius, collapse time, radius of secondary rebound, resting radius, and equilibrium radius of the generated bubbles were extracted from the radius vs time curves. Diffusion-driven ADV-bubble growth was fit to a modified Epstein-Plesset equation, adding a material stress term, to estimate the growth rate. Our results indicated that ADV dynamics were significantly impacted by fibrin concentration, frequency, and perfluorocarbon liquid core. This is the first study to combine ultra-high-speed and confocal microscopy techniques to provide insights into ADV bubble dynamics in tissue-mimicking hydrogels.

Numerical investigation of acoustic cavitation and viscoelastic tissue deformation

Acoustic cavitation and tissue deformation are studied by modifying a level-set method for compressible two-phase flows to consider viscoelastic tissue deformation. The numerical simulations performed using different shear moduli and bubble-tissue distances demonstrate various interactions between bubble and viscoelastic tissue, including inverted cone-shape bubbles, bubble migration, liquid jet formation, compressive and expansive tissue deformation, and tissue perforation. The bubble is observed to grow larger with increasing tissue bulk modulus and density. The maximum tissue deformation generally increases with decreasing initial bubble-tissue distance and with increasing tissue bulk modulus and density. The tissue shear modulus conditions that maximize tissue deformation are in the range of 1-10 MPa, unless the tissue density is very large.

Bio-inspired apparatus to produce luminescent cavitation in a rigid walled chamber

A mechanical device inspired by the rapid rotational motion of the pistol shrimp plunger has been developed to experimentally study the contraction/expansion dynamics of a gas bubble inside a confined liquid volume and in the vicinity of solid surfaces. The apparatus consists of a limb with a V-shaped end, which fits into a socket forming a cylindrical compression chamber. Air bubbles of different sizes and in different positions inside the chamber were seeded to study their shape evolution in liquids when subjected to pressure pulses induced by the limb closure. By changing the standoff and curvature parameters, as well as the closing power of the limb it was possible to control the dynamical behavior of the cavity. Four stages describing the dynamic behavior of the bubble were found: 1) A slight expansion-contraction stage accompanied by very weak volumetric oscillations. 2) First compression stage. The formation of gas and liquid micro-jets is observed when the vertical symmetry axis of the bubble is initially located outside of the chamber symmetry axis, on the other hand, when there is a coincidence between these axes, the bubble only contracts exhibiting non-spherical shapes, alternating between oblate and prolate spheroidal structures. 3) An expansion stage where the cavity reaches the walls of the chamber exhibiting irregular shapes on its surface. 4) Second compression stage. This process begins when the limb rebounds and stops sealing the chamber allowing a jet of liquid to enter from the fluid medium outside, inducing a very violent collapse accompanied by the emission of light. The proposed technique represents a novel alternative to study the dynamic evolution of bubbles near and on solid boundaries of various geometries. Other attractive features of the apparatus are its low manufacturing cost, simple design and compact size which makes it easily portable.

Numerical and experimental investigations on the jet and shock wave dynamics during the cavitation bubble collapsing near spherical particles based on OpenFOAM

The interaction between cavitation bubbles and particles is essential for the operational performance many kinds of fluid machineries. In the present paper, jet dynamics and shock waves induced by the cavitation bubble collapsing near two spherical particles are numerically investigated based on OpenFOAM. The numerical scheme is validated by the experimental data obtained based on our high-speed camera cavitation system. Our results reveal that bubble split induced by annular jet is the primary feature during bubble collapsing with four typical cases defined. For the jet formation, the localized high pressure produced at the bubble split point is the main reason and the split point also serves as the source of the shock waves shown by the numerical schlieren. Furthermore, the nondimensional bubble-particle distance is the most paramount parameter influencing the jet phenomenon (e.g. jet velocity).

Numerical investigation of shock-induced bubble collapse dynamics and fluid-solid interactions during shock-wave lithotripsy

In this paper we investigate the bubble collapse dynamics under shock-induced loading near soft and rigid bio-materials, during shock wave lithotripsy. A novel numerical framework was developed, that employs a Diffuse Interface Method (DIM) accounting for the interaction across fluid-solid-gas interfaces. For the resolution of the extended variety of length scales, due to the dynamic and fine interfacial structures, an Adaptive Mesh Refinement (AMR) framework for unstructured grids was incorporated. This multi-material multi-scale approach aims to reduce the numerical diffusion and preserve sharp interfaces. The presented numerical framework is validated for cases of bubble dynamics, under high and low ambient pressure ratios, shock-induced collapses, and wave transmission problems across a fluid-solid interface, against theoretical and numerical results. Three different configurations of shock-induced collapse applications near a kidney stone and soft tissue have been simulated for different stand-off distances and bubble attachment configurations. The obtained results reveal the detailed collapse dynamics, jet formation, solid deformation, rebound, primary and secondary shock wave emissions, and secondary collapse that govern the near-solid collapse and penetration mechanisms. Significant correlations of the problem configuration to the overall collapse mechanisms were found, stemming from the contact angle/attachment of the bubble and from the properties of solid material. In general, bubbles with their center closer to the kidney stone surface produce more violent collapses. For the soft tissue, the bubble movement prior to the collapse is of great importance as new structures can emerge which can trap the liquid jet into induced crevices. Finally, the tissue penetration is examined for these cases and a novel tension-driven tissue injury mechanism is elucidated, emanating from the complex interaction of the bubble/tissue interaction during the secondary collapse phase of an entrapped bubble in an induced crevice with the liquid jet.

Direct van der Waals simulation (DVS) of phase-transforming fluids

We present the method of direct van der Waals simulation (DVS) to study computationally flows with liquid-vapor phase transformations. Our approach is based on a discretization of the Navier-Stokes-Korteweg equations, which couple flow dynamics with van der Waals' nonequilibrium thermodynamic theory of phase transformations, and opens an opportunity for first-principles simulation of a wide range of boiling and cavitating flows. The proposed algorithm enables unprecedented simulations of the Navier-Stokes-Korteweg equations involving cavitating flows at strongly under-critical conditions and 𝒪(10⁵) Reynolds number. The proposed technique provides a pathway for a fundamental understanding of phase-transforming flows with multiple applications in science, engineering, and medicine.

Monitoring cavitation dynamics evolution in tissue mimicking hydrogels for repeated exposures via acoustic cavitation emissions

A 700 kHz histotripsy array is used to generate repeated cavitation events in agarose, gelatin, and polyacrylamide hydrogels. High-speed optical imaging, a broadband hydrophone, and the narrow-band receive elements of the histotripsy array are used to capture bubble dynamics and acoustic cavitation emissions. Bubble radii, lifespan, shockwave amplitudes are noted to be measured in close agreement between the different observation methods. These features also decrease with increasing hydrogel stiffness for all of the tested materials. However, the evolutions of these properties during the repeated irradiations vary significantly across the different material subjects. Bubble maximum radius initially increases, then plateaus, and finally decreases in agarose, but remains constant across exposures in gelatin and polyacrylamide. The bubble lifespan increases monotonically in agarose and gelatin but decreases in polyacrylamide. Collapse shockwave amplitudes were measured to have different-shaped evolutions between all three of the tested materials. Bubble maximum radii, lifespans, and collapse shockwave amplitudes were observed to express evolutions that are dependent on the structure and stiffness of the nucleation medium.

Cavitation erosion by shockwave self-focusing of a single bubble

The ability of cavitation bubbles to effectively focus energy is made responsible for cavitation erosion, traumatic brain injury, and even for catalyse chemical reactions. Yet, the mechanism through which material is eroded remains vague, and the extremely fast and localized dynamics that lead to material damage has not been resolved. Here, we reveal the decisive mechanism that leads to energy focusing during the non-spherical collapse of cavitation bubbles and eventually results to the erosion of hardened metals. We show that a single cavitation bubble at ambient pressure close to a metal surface causes erosion only if a non-axisymmetric energy self-focusing is at play. The bubble during its collapse emits shockwaves that under certain conditions converge to a single point where the remaining gas phase is driven to a shockwave-intensified collapse. We resolve the conditions under which this self-focusing enhances the collapse and damages the solid. High-speed imaging of bubble and shock wave dynamics at sub-picosecond exposure times is correlated to the shockwaves recorded with large bandwidth hydrophones. The material damage from several metallic materials is detected in situ and quantified ex-situ via scanning electron microscopy and confocal profilometry. With this knowledge, approaches to mitigate cavitation erosion or to even enhance the energy focusing are within reach.

Cavitation in a soft porous material

We study the collapse and expansion of a cavitation bubble in a deformable porous medium. We develop a continuum-scale model that couples compressible fluid flow in the pore network with the elastic response of a solid skeleton. Under the assumption of spherical symmetry, our model can be reduced to an ordinary differential equation that extends the Rayleigh-Plesset equation to bubbles in soft porous media. The extended Rayleigh-Plesset equation reveals that finite-size effects lead to the breakdown of the universal scaling relation between bubble radius and time that holds in the infinite-size limit. Our data indicate that the deformability of the porous medium slows down the collapse and expansion processes, a result with important consequences for wide-ranging phenomena, from drug delivery to spore dispersion.

Fundamentals, biomedical applications and future potential of micro-scale cavitation-a review

Thanks to the developments in the area of microfluidics, the cavitation-on-a-chip concept enabled researchers to control and closely monitor the cavitation phenomenon in micro-scale. In contrast to conventional scale, where cavitation bubbles are hard to be steered and manipulated, lab-on-a-chip devices provide suitable platforms to conduct smart experiments and design reliable devices to carefully harness the collapse energy of cavitation bubbles in different bio-related and industrial applications. However, bubble behavior deviates to some extent when confined to micro-scale geometries in comparison to macro-scale. Therefore, fundamentals of micro-scale cavitation deserve in-depth investigations. In this review, first we discussed the physics and fundamentals of cavitation induced by tension-based as well as energy deposition-based methods within microfluidic devices and discussed the similarities and differences in micro and macro-scale cavitation. We then covered and discussed recent developments in bio-related applications of micro-scale cavitation chips. Lastly, current challenges and future research directions towards the implementation of micro-scale cavitation phenomenon to emerging applications are presented.

Observation of the Formation of Multiple Shock Waves at the Collapse of Cavitation Bubbles for Improvement of Energy Convergence

The collapse of a cavitation bubble is always associated with the radiation of intense shock waves, which are highly relevant in a variety of applications. To radiate a strong shock wave, it is necessary to converge energy at the collapse, and understanding generation processes of multiple shock waves at the collapse is a key issue. In the present study, we investigated the formation of multiple shock waves generated by the collapse of a laser-induced bubble. We used a high-speed imaging system with unprecedented spatiotemporal resolution. We developed a triggering procedure of high precision and reproducibility based on the deflection of a laser beam by the shockwave passage. The high-speed videos clearly show that: (A) a first shockwave is emitted as the micro-jet hits the bottom of the bubble interface, followed by a second shock wave due to the collapse of the remaining toroidal bubble; (B) a sequential collapse of elongated bubbles, where the top part of the bubble collapses slightly before the bottom of the bubble; and (C) the formation of compression shock waves from multiple sites on a toroidal bubble.

Dynamics of a Laser-Induced Bubble above the Flat Top of a Solid Cylinder—Mushroom-Shaped Bubbles and the Fast Jet

The dynamics of a laser-induced bubble on top of a solid cylinder is studied both experimentally and numerically. When the bubble is generated close to the flat top along the axis of the cylinder and its maximum radius exceeds the one of the flat top surface, it collapses in the form of a mushroom with a footing on the cylinder, a long stem and a hat-like cap typical for a mushroom head. The head may collapse forming a thin, fast liquid jet into the stem, depending on bubble size and bubble distance to the top of the cylinder. Several experimental and numerical examples are given. The results represent a contribution to understand the behavior of bubbles collapsing close to structured surfaces and in particular, how thin, fast jets are generated.

Modeling cavitation bubble dynamics in an autoinjector and its implications on drug molecules

The collapse of cavitation bubbles induced by abrupt acceleration of the syringe in an autoinjector device can lead to protein aggregation. The details of bubble dynamics are investigated using an axisymmetric, three-dimensional simulation with passive tracers to illustrate the transport of protein molecules. When a bubble near the syringe wall collapses, protein molecules are concentrated in the re-entrant jet, pushed towards the syringe wall, and then spread across the wall, potentially leading to protein adsorption on the syringe wall and aggregation. This phenomenon is more prominent for bubbles positioned closer to the bottom wall, growing to a larger maximum radius. The bubble's maximum radius decreases with the bubble's distance from the syringe wall and air gap pressure, and increases with an increase in liquid column height and nucleus size. The strain rate induced by the bubble collapse is not large enough to unfold the proteins. When the re-entrant jet impacts the bubble surface or syringe wall, the bubble breaks up, generating smaller bubbles with high surface concentration of protein molecules, potentially inducing aggregation in the bulk. The bubble dynamics are influenced by dimensionless distance of the nucleus from the wall, normalized by maximum bubble radius (γ). The re-entrant jet velocity increases with γ, while the maximum liquid pressure, typically 100∼1000 bar, first decreases and then increases with γ. For a cloud of cavitation bubbles, i.e., closely clustered bubbles, coalescence of bubbles can occur, leading to a higher peak pressure at collapse.

A 3D Smoothed Particle Hydrodynamics Study of a Non-Symmetrical Rayleigh Collapse for an Empty Cavity

In this work the first 3D Smoothed Particle Hydrodynamics model of a Rayleigh collapse for an empty cavity is proposed with the aim of improving the hydrodynamic analysis of a non-symmetrical collapse. The hydrodynamics of the model is validated against the solution of the Rayleigh-Plesset equation for a symmetrical collapse. The model is then used to simulate a non-symmetrical collapse of an empty cavity attached to a solid surface with γ=0.6 induced by an external pressure of 50 [MPa]. The results shows that is possible to identify three regions where the hydrodynamics of the collapsing cavity shows different features. For all the stages of the collapse the simulation shows smooth pressure and velocity fields in the liquid and in the solid phase with the formation of a vortex ring in the final phase of the collapse. Finally, the model is compared to a previous 2D model to highlight strong, weak points and the key differences of both approaches in final phase of the collapse.

Inertial Cavitation Behaviors Induced by Nonlinear Focused Ultrasound Pulses

Inertial cavitation induced by pulsed high-intensity focused ultrasound (pHIFU) has previously been shown to successfully permeabilize tumor tissue and enhance chemotherapeutic drug uptake. In addition to HIFU frequency, peak rarefactional pressure ( p- ), and pulse duration, the threshold for cavitation-induced bioeffects has recently been correlated with asymmetric distortion caused by nonlinear propagation, diffraction and formation of shocks in the focal waveform, and therefore with the transducer F -number. To connect previously observed bioeffects with bubble dynamics and their attendant physical mechanisms, the dependence of inertial cavitation behavior on shock formation was investigated in transparent agarose gel phantoms using high-speed photography and passive cavitation detection (PCD). Agarose phantoms with concentrations ranging from 1.5% to 5% were exposed to 1-ms pulses using three transducers of the same aperture but different focal distances ( F -numbers of 0.77, 1.02, and 1.52). Pulses had central frequencies of 1, 1.5, or 1.9 MHz and a range of p- at the focus varying within 1-18 MPa. Three distinct categories of bubble behavior were observed as the acoustic power increased: stationary near-spherical oscillation of individual bubbles, proliferation of multiple bubbles along the pHIFU beam axis, and fanned-out proliferation toward the transducer. Proliferating bubbles were only observed under strongly nonlinear or shock-forming conditions regardless of frequency, and only where the bubbles reached a certain threshold size range. In stiffer gels with higher agarose concentrations, the same pattern of cavitation behavior was observed, but the dimensions of proliferating clouds were smaller. These observations suggest mechanisms that may be involved in bubble proliferation: enhanced growth of bubbles under shock-forming conditions, subsequent shock scattering from the gel-bubble interface, causing an increase in the repetitive tension created by the acoustic wave, and the appearance of a new growing bubble in the proximal direction. Different behaviors corresponded to specific spectral characteristics in the PCD signals: broadband noise in all cases, narrow peaks of backscattered harmonics in the case of stationary bubbles, and broadened, shifted harmonic peaks in the case of proliferating bubbles. The shift in harmonic peaks can be interpreted as a Doppler shift from targets moving at speeds of up to 2 m/s, which correspond to the observed bubble proliferation speeds.

Estimation of chemical and physical effects of cavitation by analysis of cavitating single bubble dynamics

Cavitation events create extreme conditions in a localized 'bubble collapse' region, leading to the formation of hydroxyl radicals, shockwaves and microscopic high-speed jets, which are useful for many chemical and physical transformation processes. Single bubble dynamics equations have been used previously to investigate the chemical and physical effects of cavitation. In the present study, the state of the art of the single bubble dynamics equations was reviewed and certain noteworthy modifications were implemented. Simulations reaffirmed previously reported collapse temperatures of the order ~5,000 K and collapse pressures well over ~1,000 bar under varying operating conditions. The chemical effects were assessed in terms of the hydroxyl radical generation rate (OHG), calculated by applying the minimization of the Gibb's Free Energy method using simulated collapse conditions. OHG values as high as 1x10¹²OH molecules per collapse event were found under certain operating conditions. A new equation was proposed to assess the physical effects, in terms of the impact pressure of the water jet - termed as the jet hammer pressure (JHP), formed due to the asymmetrical collapse of a bubble near a wall. The predicted JHP were found to be within a range of ~100 to 1000 bar under varying operating conditions. Important issues such as the onset of cavitation and chaotic solutions, for a cavitating single bubble dynamics were discussed. The Blake threshold pressure was found to be a sufficient criterion to capture the onset of cavitation. The impact of key operating parameters on the chemical and physical effects of cavitation were investigated exhaustively through simulations, over the parameter ranges relevant to acoustic and hydrodynamic cavitation processes. Presented methodology and results will be useful for optimisation and further investigations of a broad range of acoustic and hydrodynamic cavitation-based applications.

Shock-induced collapse of surface nanobubbles

The collapse of cavitation bubbles often releases high-speed liquid jets capable of surface damage, with applications in drug delivery, cancer treatment, and surface cleaning. Spherical cap-shaped surface nanobubbles have previously been found to exist on immersed substrates. Despite being known nucleation sites for cavitation, their collapsing dynamics are currently unexplored. Here, we use molecular dynamics simulations to model the shock-induced collapse of different surface nanobubble sizes and contact angles. Comparisons are made with additional collapsing spherical nanobubble simulations near a substrate, to investigate the differences in their jet formation and resulting substrate pitting damage. Our main finding is that the pitting damage in the surface nanobubble simulations is greatly reduced, when compared to the spherical nanobubbles, which is primarily caused by the weaker jets formed during their collapse. Furthermore, the pit depths for surface nanobubble collapse do not depend on bubble size, unlike in the spherical nanobubble cases, but instead depend only on their contact angle. We also find a linear scaling relationship for all bubble cases between the final substrate damage and the peak pressure impulse at the impact centre, which can now be exploited to assess the relative damage in other computational studies of collapsing bubbles. We anticipate the more controlled surface-damage features produced by surface nanobubble cavitation jets will open up new applications in advanced manufacturing, medicine, and precision cleaning.

Mechanically Induced Cavitation in Biological Systems

Cavitation bubbles form in soft biological systems when subjected to a negative pressure above a critical threshold, and dynamically change their size and shape in a violent manner. The critical threshold and dynamic response of these bubbles are known to be sensitive to the mechanical characteristics of highly compliant biological systems. Several recent studies have demonstrated different biological implications of cavitation events in biological systems, from therapeutic drug delivery and microsurgery to blunt injury mechanisms. Due to the rapidly increasing relevance of cavitation in biological and biomedical communities, it is necessary to review the current state-of-the-art theoretical framework, experimental techniques, and research trends with an emphasis on cavitation behavior in biologically relevant systems (e.g., tissue simulant and organs). In this review, we first introduce several theoretical models that predict bubble response in different types of biological systems and discuss the use of each model with physical interpretations. Then, we review the experimental techniques that allow the characterization of cavitation in biologically relevant systems with in-depth discussions of their unique advantages and disadvantages. Finally, we highlight key biological studies and findings, through the direct use of live cells or organs, for each experimental approach.

Non-Symmetrical Collapse of an Empty Cylindrical Cavity Studied with Smoothed Particle Hydrodynamics

The non-symmetrical collapse of an empty cylindrical cavity is modeled using Smoothed Particle Hydrodynamics. The presence of a nearby surface produces an anisotropic pressure field generating a high-velocity jet that hits the surface. The collapse follows a different dynamic based on the initial distance between the center of the cavity and the surface. When the distance is greater than the cavity radius (detached cavity) the surface is hit by traveling shock waves. When the distance is less than the cavity radius (attached cavity) the surface is directly hit by the jet and later by other shock waves generated in the last stages of the of the collapse. The results show that the surface is hit by a stronger shock when distance between the center of the cavity and the surface is zero while showing more complex double peaks behavior for other distances.

On the thermodynamic behaviors and interactions between bubble pairs: A numerical approach

Thermodynamic behaviors and interactions between bubble pairs are important to better understand the cavitation phenomena. In this study, a compressible two-phase model, accounting for thermal effects to investigate the thermodynamic behaviors and interactions between bubble pairs, is developed in OpenFOAM. The volume of fluid (VOF) method is adopted to capture the interface. Validations are performed by comparing the simulation results of a single bubble and bubble pairs with corresponding experimental data. The dynamical behaviors of bubble pairs and their thermodynamic effect at different relative distances γ are investigated and discussed, which help reveal the bubble cloud dynamics. The quantitative analysis of γ effects on the maximum temperature during bubble collapse is performed with three distinct stages identified. For a single bubble collapsing near the rigid surface, the thermodynamic characteristics at different relative distances are similar to that of the bubble pairs, but the maximum temperature is higher since the single bubble can collapse to a smaller volume.

Shape stability of a gas bubble in a soft solid

Predicting the onset of non-spherical oscillations of bubbles in soft matter is a fundamental cavitation problem with implications to sonoprocessing, polymeric materials synthesis, and biomedical ultrasound applications. The shape stability of a bubble in a Kelvin-Voigt viscoelastic medium with nonlinear elasticity, the simplest constitutive model for soft solids, is analytically investigated and compared to experiments. Using perturbation methods, we develop a model reducing the equations of motion to two sets of evolution equations: a Rayleigh-Plesset-type equation for the mean (volume-equivalent) bubble radius and an equation for the non-spherical mode amplitudes. Parametric instability is predicted by examining the natural frequency and the Mathieu equation for the non-spherical modes, which are obtained from our model. Our theoretical results show good agreement with published experiments of the shape oscillations of a bubble in a gelatin gel. We further examine the impact of viscoelasticity on the time evolution of non-spherical mode amplitudes. In particular, we find that viscosity increases the damping rate, thus suppressing the shape instability, while shear modulus increases the natural frequency, which changes the unstable mode. We also explain the contributions of rotational and irrotational fields to the viscoelastic stresses in the surroundings and at the bubble surface, as these contributions affect the damping rate and the unstable mode. Our analysis on the role of viscoelasticity is potentially useful to measure viscoelastic properties of soft materials by experimentally observing the shape oscillations of a bubble.

A smoothed particle hydrodynamics study of the collapse for a cylindrical cavity

In this study, we propose a mesh-free (particle-based) Smoothed Particle Hydrodynamics model for simulating a Rayleigh collapse. Both empty and gas cavities are investigates and the role of heat diffusion is also accounted for. The system behaves very differently according to the ratio between the characteristic time of collapse and the characteristic time of thermal diffusion. This study identifies five different possible behaviours that range from isothermal to adiabatic.

Optothermal generation, trapping, and manipulation of microbubbles

The most common approach to optically generate and manipulate bubbles in liquids involves temperature gradients induced by CW lasers. In this work, we present a method to accomplish both the generation of microbubbles and their 3D manipulation in ethanol through optothermal forces. These forces are triggered by light absorption from a nanosecond pulsed laser (λ = 532 nm) at silver nanoparticles photodeposited at the distal end of a multimode optical fiber. Light absorbed from each laser pulse quickly heats up the silver-ethanol interface beyond the ethanol critical-point (∼ 243 °C) before the heat diffuses through the liquid. Therefore, the liquid achieves a metastable state and owing to spontaneous nucleation converted to a vapor bubble attached to the optical fiber. The bubble grows with semi-spherical shape producing a counterjet in the final stage of the collapse. This jet reaches the hot nanoparticles vaporizing almost immediately and ejecting a microbubble. This microbubble-generation mechanism takes place with every laser pulse (10 kHz repetition rate) leading to the generation of a microbubbles stream. The microbubbles' velocities decrease as they move away from the optical fiber and eventually coalesce forming a larger bubble. The larger bubble is attracted to the optical fiber by the Marangoni force once it reaches a critical size while being continuously fed with each bubble of the microbubbles stream. The balance of the optothermal forces owing to the laser-pulse drives the 3D manipulation of the main bubble. A complete characterization of the trapping conditions is provided in this paper.

Collapse dynamics of spherical cavities in a solid under shock loading

Extraordinary states of highly localised pressure and temperature can be generated upon the collapse of impulsively driven cavities. Direct observation of this phenomenon in solids has proved challenging, but recent advances in high-speed synchrotron radiography now permit the study of highly transient, subsurface events in real time. We present a study on the shock-induced collapse of spherical cavities in a solid polymethyl methacrylate medium, driven to shock states between 0.49 and 16.60 GPa. Utilising multi-MHz phase contrast radiography, extended sequences of the collapse process have been captured, revealing new details of interface motion, material failure and jet instability formation. Results reveal a rich array of collapse characteristics dominated by strength effects at low shock pressures and leading to a hydrodynamic response at the highest loading conditions.

Bubble cloud dynamics in an ultrasound field

The dynamics of bubble clouds induced by high-intensity focused ultrasound are investigated in a regime where the cloud size is similar to the ultrasound wavelength. High-speed images show that the cloud is asymmetrical; the bubbles nearest the source grow to a larger radius than the distal ones. Similar structures of bubble clouds are observed in numerical simulations that mimic the laboratory experiment. To elucidate the structure, a parametric study is conducted for plane ultrasound waves with various amplitudes and diffuse clouds with different initial void fractions. Based on an analysis of the kinetic energy of liquid induced by bubble oscillations, a new scaling parameter is introduced to characterize the dynamics. The new parameter generalizes the cloud interaction parameter originally introduced by d'Agostino & Brennen (1989). The dynamic interaction parameter controls the energy localization and consequent anisotropy of the cloud. Moreover, the amplitude of the far-field, bubble-scattered acoustics is likewise correlated with the proposed parameter. Findings of the present study not only shed light on the physics of cloud cavitation, but may also be of use to quantification of the effects of cavitation on outcomes of ultrasound therapies including HIFU-based lithotripsy.

Numerical simulation of single bubble dynamics under acoustic standing waves

The objective of this paper is to apply numerical method to simulate the single bubble dynamics under the acoustic standing waves, which is an extensive research of our previous work (Ma et al. Ultrason. Sonochem., vol. 42, 2018, pp. 619-630). The Navier-Stokes equation, which considers the acoustic radiation force caused by acoustic standing wave, is used to capture the transient shape variation, pressure fluctuation, and the direction of the bubble motion, especially for the case of the bubble near the rigid boundary. Several normalized parameters, such as acoustic pressure amplitude, acoustic wave number, and bubble size, are investigated in temporal and spatial scales to actively influence the direction of the liquid jet caused by bubble collapse. The numerical results show that due to the strong interaction with the acoustic standing wave, the bubble loses the spherical shape and generates a high-speed liquid jet. It worth noting that a significantly high pressure and velocity peak is respectively founded at the boundary wall, which is caused by the toroidal bubble collapse. Furthermore, in the standing wave field, single bubble would have distinctly different behaviors with the change of its resonance radius size. The high-speed liquid jet is always directed towards the node of an acoustic standing wave when the radius of bubble is larger than the resonance size, while the liquid jet is directed to the antinode when the radius of bubble is much smaller than the resonance size, closely with the primary Bjerknes force. Finally, the investigation shows that the single bubble will collapse much earlier during the deformation process with the increase of the normalized pressure amplitude.

Energy shielding by cavitation bubble clouds in burst wave lithotripsy

Combined laboratory experiment and numerical simulation are conducted on bubble clouds nucleated on the surface of a model kidney stone to quantify the energy shielding of the stone caused by cavitation during burst wave lithotripsy (BWL). In the experiment, the bubble clouds are visualized and bubble-scattered acoustics are measured. In the simulation, a compressible, multi-component flow solver is used to capture complex interactions among cavitation bubbles, the stone, and the burst wave. Quantitative agreement is confirmed between results of the experiment and the simulation. In the simulation, a significant shielding of incident wave energy by the bubble clouds is quantified. The magnitude of shielding can reach up to 90% of the energy of the incoming burst wave that otherwise would be transmitted into the stone, suggesting a potential loss of efficacy of stone comminution. There is a strong correlation between the magnitude of the energy shielding and the amplitude of the bubble-scattered acoustics, independent of the initial size and the void fraction of the bubble cloud within a range addressed in the simulation. This correlation could provide for real-time monitoring of cavitation activity in BWL.

Dependence of inertial cavitation induced by high intensity focused ultrasound on transducer F-number and nonlinear waveform distortion

Pulsed high intensity focused ultrasound was shown to enhance chemotherapeutic drug uptake in tumor tissue through inertial cavitation, which is commonly assumed to require peak rarefactional pressures to exceed a certain threshold. However, recent studies have indicated that inertial cavitation activity also correlates with the presence of shocks at the focus. The shock front amplitude and corresponding peak negative pressure (p -) in the focal waveform are primarily determined by the transducer F-number: less focused transducers produce shocks at lower p -. Here, the dependence of inertial cavitation activity on the transducer F-number was investigated in agarose gel by monitoring broadband noise emissions with a coaxial passive cavitation detector (PCD) during pulsed exposures (pulse duration 1 ms, pulse repetition frequency 1 Hz) with p- varying within 1-15 MPa. Three 1.5 MHz transducers with the same aperture, but different focal distances (F-numbers 0.77, 1.02, 1.52) were used. PCD signals were processed to extract cavitation probability, persistence, and mean noise level. At the same p -, all metrics indicated enhanced cavitation activity at higher F-numbers; specifically, cavitation probability reached 100% when shocks formed at the focus. These results provide further evidence supporting the excitation of inertial cavitation at reduced p - by waveforms with nonlinear distortion and shocks.

Shock wave-induced permeabilization of mammalian cells

Controlled permeabilization of mammalian cell membranes is fundamental to develop gene and cell therapies based on macromolecular cargo delivery, a process that emerged against an increasing number of health afflictions, including genetic disorders, cancer and infections. Viral vectors have been successfully used for macromolecular delivery; however, they may have unpredictable side effects and have been limited to life-threatening cases. Thus, several chemical and physical methods have been explored to introduce drugs, vaccines, and nucleic acids into cells. One of the most appealing physical methods to deliver genes into cells is shock wave-induced poration. High-speed microjets of fluid, emitted due to the collapse of microbubbles after shock wave passage, represent the most significant mechanism that contributes to cell membrane poration by this technique. Herein, progress in shock wave-induced permeabilization of mammalian cells is presented. After covering the main concepts related to molecular strategies whose applications depend on safer drug delivery methods, the physics behind shock wave phenomena is described. Insights into the use of shock waves for cell membrane permeation are discussed, along with an overview of the two major biomedical applications thereof-i.e., genetic modification and anti-cancer shock wave-assisted chemotherapy. The aim of this review is to summarize 30 years of data showing underwater shock waves as a safe, noninvasive method for macromolecular delivery into mammalian cells, encouraging the development of further research, which is still required before the introduction of this promising tool into clinical practice.

Bubble-Induced Color Doppler Feedback Correlates with Histotripsy-Induced Destruction of Structural Components in Liver Tissue

Bubble-induced color Doppler (BCD) is a histotripsy-therapy monitoring technique that uses Doppler ultrasound to track the motion of residual cavitation nuclei that persist after the collapse of the histotripsy bubble cloud. In this study, BCD is used to monitor tissue fractionation during histotripsy tissue therapy, and the BCD signal is correlated with the destruction of structural and non-structural components identified histologically to further understand how BCD monitors the extent of treatment. A 500-kHz, 112-element phased histotripsy array is used to generate approximately 6- × 6- × 7-mm lesions within ex vivo bovine liver tissue by scanning more than 219 locations with 30-1000 pulses per location. A 128-element L7-4 imaging probe is used to acquire BCD signals during all treatments. The BCD signal is then quantitatively analyzed using the time-to-peak rebound velocity (tprv) metric. Using the Pearson correlation coefficient, the tprv is compared with histologic analytics of lesions generated by various numbers of pulses using a significance level of 0.001. Histologic analytics in this study include viable cell count, reticulin-stained type III collagen area and trichrome-stained type I collagen area. It is found that the tprv metric has a statistically significant correlation with the change in reticulin-stained type III collagen area with a Pearson correlation coefficient of -0.94 (p <0.001), indicating that changes in BCD are more likely because of destruction of the structural components of tissue.

Pressure and tension waves from bubble collapse near a solid boundary: A numerical approach

The acoustic waves being generated during the motion of a bubble in water near a solid boundary are calculated numerically. The open source package OpenFOAM is used for solving the Navier-Stokes equation and extended to include nonlinear acoustic wave effects via the Tait equation for water. A bubble model with a small amount of gas is chosen, the gas obeying an adiabatic law. A bubble starting from a small size with high internal pressure near a flat, solid boundary is studied. The sequence of events from bubble growth via axial microjet formation, jet impact, annular nanojet formation, torus-bubble collapse, and bubble rebound to second collapse is described. The different pressure and tension waves with their propagation properties are demonstrated.

Numerical studies of cavitation erosion on an elastic–plastic material caused by shock-induced bubble collapse

We present a study of shock-induced collapse of single bubbles near/attached to an elastic–plastic solid using the free-Lagrange method, which forms the latest part of our shock-induced collapse studies. We simulated the collapse of 40 μm radius single bubbles near/attached to rigid and aluminium walls by a 60 MPa lithotripter shock for various scenarios based on bubble–wall separations, and the collapse of a 255 μm radius bubble attached to aluminium foil with a 65 MPa lithotripter shock. The coupling of the multi-phases, compressibility, axisymmetric geometry and elastic–plastic material model within a single solver has enabled us to examine the impingement of high-speed liquid jets from the shock-induced collapsing bubbles, which imposes an extreme compression in the aluminium that leads to pitting and plastic deformation. For certain scenarios, instead of the high-speed jet, a radially inwards flow along the aluminium surface contracts the bubble to produce a ‘mushroom shape’. This work provides methods for quantifying which parameters (e.g. bubble sizes and separations from the solid) might promote or inhibit erosion on solid surfaces.

Effect of Shock-Induced Cavitation Bubble Collapse on the damage in the Simulated Perineuronal Net of the Brain

The purpose of this study is to conduct modeling and simulation to understand the effect of shock-induced mechanical loading, in the form of cavitation bubble collapse, on damage to the brain's perineuronal nets (PNNs). It is known that high-energy implosion due to cavitation collapse is responsible for corrosion or surface damage in many mechanical devices. In this case, cavitation refers to the bubble created by pressure drop. The presence of a similar damage mechanism in biophysical systems has long being suspected but not well-explored. In this paper, we use reactive molecular dynamics (MD) to simulate the scenario of a shock wave induced cavitation collapse within the perineuronal net (PNN), which is the near-neuron domain of a brain's extracellular matrix (ECM). Our model is focused on the damage in hyaluronan (HA), which is the main structural component of PNN. We have investigated the roles of cavitation bubble location, shockwave intensity and the size of a cavitation bubble on the structural evolution of PNN. Simulation results show that the localized supersonic water hammer created by an asymmetrical bubble collapse may break the hyaluronan. As such, the current study advances current knowledge and understanding of the connection between PNN damage and neurodegenerative disorders.

Review on Lithotripsy and Cavitation in Urinary Stone Therapy

Cavitation is the sudden formation of vapor bubbles or voids in liquid media and occurs after rapid changes in pressure as a consequence of mechanical forces. It is mostly an undesirable phenomenon. Although the elimination of cavitation is a major topic in the study of fluid dynamics, its destructive nature could be exploited for therapeutic applications. Ultrasonic and hydrodynamic sources are two main origins for generating cavitation. The purpose of this review is to give the reader a general idea about the formation of cavitation phenomenon and existing biomedical applications of ultrasonic and hydrodynamic cavitation. Because of the high number of the studies on ultrasound cavitation in the literature, the main focus of this review is placed on the lithotripsy techniques, which have been widely used for the treatment of urinary stones. Accordingly, cavitation phenomenon and its basic concepts are presented in Section II. The significance of the ultrasound cavitation in the urinary stone treatment is discussed in Section III in detail and hydrodynamic cavitation as an important alternative for the ultrasound cavitation is included in Section IV. Finally, side effects of using both ultrasound and hydrodynamic cavitation in biomedical applications are presented in Section V.

Surface waves on a soft viscoelastic layer produced by an oscillating microbubble

Ultrasound-driven bubbles can cause significant deformation of soft viscoelastic layers, for instance in surface cleaning and biomedical applications. The effect of the viscoelastic properties of a boundary on the bubble-boundary interaction has been explored only qualitatively, and remains poorly understood. We investigate the dynamic deformation of a viscoelastic layer induced by the volumetric oscillations of an ultrasound-driven microbubble. High-speed video microscopy is used to observe the deformation produced by a bubble oscillating at 17-20 kHz in contact with the surface of a hydrogel. The localised oscillating pressure applied by the bubble generates surface elastic (Rayleigh) waves on the gel, characterised by elliptical particle trajectories. The tilt angle of the elliptical trajectories varies with increasing distance from the bubble. Unexpectedly, the direction of rotation of the surface elements on the elliptical trajectories shifts from prograde to retrograde at a distance from the bubble that depends on the viscoelastic properties of the gel. To explain these behaviours, we develop a simple three-dimensional model for the deformation of a viscoelastic solid by a localised oscillating force. By using as input for the model the values of the shear modulus obtained from the propagation velocity of the Rayleigh waves, we find good qualitative agreement with the experimental observations.

Characterization of periodic cavitation in optical tweezers

Microscopic vapor explosions or cavitation bubbles can be generated repeatedly in optical tweezers with a microparticle that partially absorbs at the trapping laser wavelength. In this work we measure the size distribution and the production rate of cavitation bubbles for microparticles with a diameter of 3 μm using high-speed video recording and a fast photodiode. We find that there is a lower bound for the maximum bubble radius R(max)∼2  μm which can be explained in terms of the microparticle size. More than 94% of the measured R(max) are in the range between 2 and 6 μm, while the same percentage of the measured individual frequencies f(i) or production rates are between 10 and 200 Hz. The photodiode signal yields an upper bound for the lifetime of the bubbles, which is at most twice the value predicted by the Rayleigh equation. We also report empirical relations between R(max), f(i), and the bubble lifetimes.

Cell mechanics in biomedical cavitation

Studies on the deformation behaviours of cellular entities, such as coated microbubbles and liposomes subject to a cavitation flow, become increasingly important for the advancement of ultrasonic imaging and drug delivery. Numerical simulations for bubble dynamics of ultrasound contrast agents based on the boundary integral method are presented in this work. The effects of the encapsulating shell are estimated by adapting Hoff's model used for thin-shell contrast agents. The viscosity effects are estimated by including the normal viscous stress in the boundary condition. In parallel, mechanical models of cell membranes and liposomes as well as state-of-the-art techniques for quantitative measurement of viscoelasticity for a single cell or coated microbubbles are reviewed. The future developments regarding modelling and measurement of the material properties of the cellular entities for cutting-edge biomedical applications are also discussed.

Numerical modeling of bubble dynamics in viscoelastic media with relaxation

Cavitation occurs in a variety of non-Newtonian fluids and viscoelastic materials. The large-amplitude volumetric oscillations of cavitation bubbles give rise to high temperatures and pressures at collapse, as well as induce large and rapid deformation of the surroundings. In this work, we develop a comprehensive numerical framework for spherical bubble dynamics in isotropic media obeying a wide range of viscoelastic constitutive relationships. Our numerical approach solves the compressible Keller-Miksis equation with full thermal effects (inside and outside the bubble) when coupled to a highly generalized constitutive relationship (which allows Newtonian, Kelvin-Voigt, Zener, linear Maxwell, upper-convected Maxwell, Jeffreys, Oldroyd-B, Giesekus, and Phan-Thien-Tanner models). For the latter two models, partial differential equations (PDEs) must be solved in the surrounding medium; for the remaining models, we show that the PDEs can be reduced to ordinary differential equations. To solve the general constitutive PDEs, we present a Chebyshev spectral collocation method, which is robust even for violent collapse. Combining this numerical approach with theoretical analysis, we simulate bubble dynamics in various viscoelastic media to determine the impact of relaxation time, a constitutive parameter, on the associated physics. Relaxation time is found to increase bubble growth and permit rebounds driven purely by residual stresses in the surroundings. Different regimes of oscillations occur depending on the relaxation time.

Nanobubble collapse on a silica surface in water: billion-atom reactive molecular dynamics simulations

Cavitation bubbles occur in fluids subjected to rapid changes in pressure. We use billion-atom reactive molecular dynamics simulations on a 163,840-processor BlueGene/P supercomputer to investigate damage caused by shock-induced collapse of nanobubbles in water near an amorphous silica surface. Collapse of an empty bubble generates a high-speed nanojet, which causes pitting on the silica surface. We find pit radii are close to bubble radii, and experiments also indicate linear scaling between them. The gas-filled bubbles undergo partial collapse and, consequently, the damage on the silica surface is mitigated.

Shock-induced collapse of a bubble inside a deformable vessel

Shockwave lithotripsy repeatedly focuses shockwaves on kidney stones to induce their fracture, partially through cavitation erosion. A typical side effect of the procedure is hemorrhage, which is potentially the result of the growth and collapse of bubbles inside blood vessels. To identify the mechanisms by which shock-induced collapse could lead to the onset of injury, we study an idealized problem involving a preexisting bubble in a deformable vessel. We utilize a high-order accurate, shock- and interface-capturing, finite-volume scheme and simulate the three-dimensional shock-induced collapse of an air bubble immersed in a cylindrical water column which is embedded in a gelatin/water mixture. The mixture is a soft tissue simulant, 10% gelatin by weight, and is modeled by the stiffened gas equation of state. The bubble dynamics of this model configuration are characterized by the collapse of the bubble and its subsequent jetting in the direction of the propagation of the shockwave. The vessel wall, which is defined by the material interface between the water and gelatin/water mixture, is invaginated by the collapse and distended by the impact of the jet. The present results show that the highest measured pressures and deformations occur when the volumetric confinement of the bubble is strongest, the bubble is nearest the vessel wall and/or the angle of incidence of the shockwave reduces the distance between the jet tip and the nearest vessel surface. For a particular case considered, the 40 MPa shockwave utilized in this study to collapse the bubble generated a vessel wall pressure of almost 450 MPa and produced both an invagination and distention of nearly 50% of the initial vessel radius on a 𝒪(10) ns timescale. These results are indicative of the significant potential of shock-induced collapse to contribute to the injury of blood vessels in shockwave lithotripsy.

Theoretical microbubble dynamics in a viscoelastic medium at capillary breaching thresholds

In order to predict bioeffects in contrast-enhanced diagnostic and therapeutic ultrasound procedures, the dynamics of cavitation microbubbles in viscoelastic media must be determined. For this theoretical study, measured 1.5 to 7.5 MHz pulse pressure waveforms, which were used in experimental determinations of capillary breaching thresholds for contrast-enhanced diagnostic ultrasound in a rat kidney, were used to calculate cavitation nucleated from contrast agent microbubbles. A numerical model for cavitation in tissue was developed based on the Keller-Miksis equation (a compressible extension of the Rayleigh-Plesset equation for spherical bubble dynamics), with a Kelvin-Voigt constitutive relation. From this model, the bubble dynamics corresponding to the experimentally obtained capillary breaching thresholds were determined. Values of the maximum radius and temperature corresponding to previously determined bioeffect thresholds were computed for a range of ultrasound pulses and bubble sizes for comparison to inertial cavitation threshold criteria. The results were dependent on frequency, the gas contents, and the tissue elastic properties. The bioeffects thresholds were above previously determined inertial cavitation thresholds, even for the tissue models, suggesting the possibility of a more complex dosimetry for capillary injury in tissue.

Observations of translation and jetting of ultrasound-activated microbubbles in mesenteric microvessels

High-speed photomicrography was used to study the translational dynamics of single microbubbles in microvessels of ex vivo rat mesenteries. The microbubbles were insonated by a single 2 μs ultrasound pulse with a center frequency of 1 MHz and peak negative pressures spanning the range of 0.8-4 MPa. The microvessel diameters ranged from 10-80 μm. The high-speed image sequences show evidence of ultrasound-activated microbubble translation away from the nearest vessel wall; no microbubble showed a net translation toward the nearest vessel wall. Microbubble maximum translation displacements exceeded 20 μm. Microjets with the direction of the jets identifiable were also observed; all microjets appear to have been directed away from the nearest vessel wall. These observations appear to be characteristic of a strong coupling between ultrasound-driven microbubbles and compliant microvessels. Although limited to mesenteric tissues, these observations provide an important step in understanding the physical interactions between microbubbles and microvessels.

Observations of the collapses and rebounds of millimeter-sized lithotripsy bubbles

Bubbles excited by lithotripter shock waves undergo a prolonged growth followed by an inertial collapse and rebounds. In addition to the relevance for clinical lithotripsy treatments, such bubbles can be used to study the mechanics of inertial collapses. In particular, both phase change and diffusion among vapor and noncondensable gas molecules inside the bubble are known to alter the collapse dynamics of individual bubbles. Accordingly, the role of heat and mass transport during inertial collapses is explored by experimentally observing the collapses and rebounds of lithotripsy bubbles for water temperatures ranging from 20 to 60 °C and dissolved gas concentrations from 10 to 85% of saturation. Bubble responses were characterized through high-speed photography and acoustic measurements that identified the timing of individual bubble collapses. Maximum bubble diameters before and after collapse were estimated and the corresponding ratio of volumes was used to estimate the fraction of energy retained by the bubble through collapse. The rebounds demonstrated statistically significant dependencies on both dissolved gas concentration and temperature. In many observations, liquid jets indicating asymmetric bubble collapses were visible. Bubble rebounds were sensitive to these asymmetries primarily for water conditions corresponding to the most dissipative collapses.

Model for the dynamics of two interacting axisymmetric spherical bubbles undergoing small shape oscillations

Interaction between acoustically driven or laser-generated bubbles causes the bubble surfaces to deform. Dynamical equations describing the motion of two translating, nominally spherical bubbles undergoing small shape oscillations in a viscous liquid are derived using Lagrangian mechanics. Deformation of the bubble surfaces is taken into account by including quadrupole and octupole perturbations in the spherical-harmonic expansion of the boundary conditions on the bubbles. Quadratic terms in the quadrupole and octupole amplitudes are retained, and surface tension and shear viscosity are included in a consistent manner. A set of eight coupled second-order ordinary differential equations is obtained. Simulation results, obtained by numerical integration of the model equations, exhibit qualitative agreement with experimental observations by predicting the formation of liquid jets. Simulations also suggest that bubble-bubble interactions act to enhance surface mode instability.

Shock-induced bubble jetting into a viscous fluid with application to tissue injury in shock-wave lithotripsy

Shock waves in liquids are known to cause spherical gas bubbles to rapidly collapse and form strong re-entrant jets in the direction of the propagating shock. The interaction of these jets with an adjacent viscous liquid is investigated using finite-volume simulation methods. This configuration serves as a model for tissue injury during shock-wave lithotripsy, a medical procedure to remove kidney stones. In this case, the viscous fluid provides a crude model for the tissue. It is found that for viscosities comparable to what might be expected in tissue, the jet that forms upon collapse of a small bubble fails to penetrate deeply into the viscous fluid "tissue." A simple model reproduces the penetration distance versus viscosity observed in the simulations and leads to a phenomenological model for the spreading of injury with multiple shocks. For a reasonable selection of a single efficiency parameter, this model is able to reproduce in vivo observations of an apparent 1000-shock threshold before wide-spread tissue injury occurs in targeted kidneys and the approximate extent of this injury after a typical clinical dose of 2000 shock waves.