Acoustic Radiation Stress Research Articles

Deformation of nucleated cells driven by ultrasonic standing waves

The deformation of nucleated cells driven by ultrasonic standing waves in fluid environment is investigated by theoretical analysis and numerical simulation. The nucleated cells are considered to consist of cell membrane, cytoplasm and nucleus The cell membrane is assumed to have in-plane deformation and bending resistances, in which the in-plane deformation is modeled by the generalized Hooke's law, and the bending resistance is modeled by the Helfrich bending energy formula. Due to the acoustic mismatch between the media inside and outside the cell, the generated acoustic radiation stress on the cell membrane causes the nucleated cell to deform. Considering the acoustic scattering of the nucleus, the acoustic radiation stress is formulated by applying the acoustic radiation stress tensor theory. Based on the proposed theoretical model, a finite element model is established to solve the coupling problem of ultrasonic propagation and cell deformation. The analytical solution for the acoustic induced deformation of nucleated cells in the limits of long wavelength and small deformation is given and successfully verifies the finite element model. The results show that the deformation of nucleated cells is much larger than that of enucleated cells due to the effect of the nucleus on wave propagation. The deformation of nucleated cells is significantly affected by nuclear size and nuclear density. This study is helpful to accurately extract the mechanical properties of nucleated cells and further promote the detection of cell-related diseases by acoustic deformation technology.

Mechanics of biconcave red blood cells deformed by acoustic tweezers

The deformation of biconcave red blood cells (RBCs) in a fluid environment driven by ultrasonic standing waves is studied theoretically and numerically. The cell contains homogeneous cytosol enclosed by a thin biological membrane. The cell membrane is modeled as a hyperelastic thin shell whose enclosed volume is assumed to be preserved to ensure osmotic equilibrium. Regarding acoustic deformation, the acoustic induced force acting on the cell membrane is calculated based on the acoustic radiation stress theory. The equations governing the interaction of the cell deformation and acoustic wave propagation are then numerically implemented in an axisymmetric finite element framework. The simulation results show that when the biconcave red blood cells are trapped at the pressure node of the acoustic standing wave, they gradually transform into an oblate disc shape with the increase of the acoustic input. Parametric studies are performed to systematically investigate the effects of cell mechanical properties and acoustic wavelength on cell deformation. Given that the mechanical properties of RBCs play an important role in disease development and mechanotransduction, this work is helpful to diagnose RBC-related diseases and study mechanotransduction using acoustic deformation technique.

Time-dependent deformation of biological tissue under ultrasonic irradiation

This work proposes a theoretical model for analyzing the time-dependent deformation of poroelastic materials, as a tissue model, under the action of an ultrasonic field, which considers the diffusion fluid transport that obeys Darcy's law. An acoustomechanical constitutive law is developed, which takes into account the mechanical stress generated by poroelastic deformation based on the classical poroelastic theory and the acoustic radiation stress generated by propagating ultrasonic waves. As an example, the time-dependent deformation of a poroelastic sheet subjected to two counter-propagating waves is calculated using the theoretical model, which is also verified by our finite element model. The results show that the sheet can be stretched or compressed, which depends on the ratio of the sheet thickness to the acoustic wavelength. Additionally, by exploring the relaxation process of the acoustic deformation, the role of the sheet thickness and porosity in determining the time of poroelastic relaxation is examined. This work is very helpful for understanding the deformation of tissues under ultrasonic fields, and also provides a method to extract the mechanical properties of tissues.

Nonlinear large deformation of a spherical red blood cell induced by ultrasonic standing wave.

A computational model is developed to investigate the nonlinear static deformation of a spherical (osmotically swollen) red blood cell (RBC) induced by ultrasonic standing wave. The ultrasonic standing wave can generate steady acoustic radiation stress to deform the cell, and in turn, the deformed cell reshapes the acoustic field. This is a real-time coupling problem between the acoustic field and the mechanical field. In the computational model, the acoustic radiation stress acting on the RBC membrane is modeled by adopting the nonviscous momentum flux theory. The RBC membrane is modeled as a hyperelastic shell considering the in-plane elasticity, bending elasticity, and surface tension of the membrane. The volume conservation constraint of the membrane sealing fluid is applied to ensure the osmotic balance of the membrane. To address this real-time coupling problem, the computational model is implemented by a finite element method algorithm. The numerical results are compared with the existing theoretical model and experimental data, and the strain hardening trend of the experimental data is successfully predicted, which verifies the accuracy and effectiveness of the computational model. The computational model can accurately extract the mechanical properties of cells from acoustic deformation experiments, which is helpful for the diagnosis of some human diseases.

Cavitation-induced damage model of soft materials in exposure to high-intensity focused ultrasound

High-intensity focused ultrasound can drive cavitation bubbles that damage soft materials. A cavitation-induced damage model is developed to study the mechanical damage of soft materials by high-intensity focused ultrasound. Based on the thermodynamics framework with the acoustic radiation stress theory and the nonlinear elasticity theory, an internal variable which reflects the mechanical damage caused by cavitation bubbles is incorporated into the Helmholtz free energy. Thus, the evolution of material damages for tissue-mimicking gel is established, which is related to the cavitation process and the mechanical dissipation from a macroscopic viewpoint. The theoretical model is then applied to analyze the material damaged area observed in the experiment and the theoretical prediction is found in agreement with the existing experimental results, which shows that the damage induced by cavitation bubbles develops rapidly, then slow down and finally reaches a plateau.

Extensional and flexural modes of Rayleigh–Lamb wave in an orthotropic thermoelastic layer lying over a viscoelastic half-space

In this article the characteristics of the extensional and flexural modes, propagating in a thermoelastic orthotropic layer lying over a viscoelastic half-space, are analyzed. The complete analysis is carried out in the framework of a thermodynamically consistent hyperbolic type heat conduction model without energy dissipation. The normal-mode-analysis is adopted and a general form of dispersive equation is derived for an anisotropic thermoelastic layered medium. A prominent distinction with the isotropic elastic solids is observed in the symmetric as well as anti-symmetric modes of dispersion curves. In turn, such deformation reshapes the wave propagation while the deformation stiffening changes significantly the phase velocities of the wave till the acoustic radiation stresses are balanced by elastic stresses in the current configuration of the hyperelastic medium.

Comment on "Acoustic deformation for the extraction of mechanical properties of lipid vesicle populations".

In a recent paper, Silva etal. [Phys. Rev. E 99, 063002 (2019)10.1103/PhysRevE.99.063002] consider the deformation of vesicles in response to the acoustic radiation stresses of ultrasonic standing waves. An important part of the analysis concerns the quadrupole projection of the radial component of radiation stress on a spherical vesicle having a radius much smaller than the acoustic wavelength evaluated with a software-aided expansion approach. Here we note that their projection agrees with results previously found for compressible spheres in the corresponding limit [J. Acoust. Soc. Am. 69, 1499 (1981)10.1121/1.385785] provided an error in Eq. (B5) of Silva etal. is corrected.

Acoustic deformation for the extraction of mechanical properties of lipid vesicle populations.

We use an ultrasonic standing wave to simultaneously trap and deform thousands of soft lipid vesicles immersed in a liquid solution. In our device, acoustic radiation stresses comparable in magnitude to those generated in optical stretching devices are achieved over a spatial extent of more than ten acoustic wavelengths. We solve the acoustic scattering problem in the long-wavelength limit to obtain the radiation stress. The result is then combined with thin-shell elasticity theory to form expressions that relate the deformed geometry to the applied acoustic field intensity. Using observation of the deformed geometry and this model, we rapidly extract mechanical properties, such as the membrane Young's modulus, from populations of lipid vesicles.

Acousto-thermo-mechanical deformation of hydrogels coupled with chemical diffusion.

We develop an acousto-thermo-mechanical theory for nonlinear (large) deformation of temperature-sensitive hydrogels subjected to temperature and ultrasonic inputs, with diffusion mass transport driven by osmotic pressure accounted for. On the basis of the strain energy due to network stretching, the mixing energy of polymers and small molecules, the Cauchy stress of the deformed hydrogel can be obtained. The acoustic radiation stress generated by the ultrasonic inputs is incorporated into the Cauchy stress to give the constitutive equations of the acousto-thermal-mechanical hydrogel. The mixing energy contains an interaction parameter as a function of temperature and polymer concentration so that hydrogel deformation is temperature dependent. By employing the incompressible condition of polymers and molecules, both the temperature and acoustic radiation stress contribute to osmotic pressure, inducing hydrogel swelling (or shrinking). Specifically, for a temperature-sensitive hydrogel layer immersed in solvent, its acoustic-triggered large deformation is comprehensively analysed under different boundary conditions (e.g. free swelling, uniaxial constraint and biaxial constraint).

An Acoustomechanical Constitutive Model of Gel Considering Cavitation Effect in Exposure to Ultrasound

In this paper, an acoustomechanical constitutive model is developed to describe the heating effect of a tissue-mimicking gel by cavitation in exposure to high-intensity focused ultrasound (HIFU). An internal variable, representing the evolution of cavitation process, is introduced into the Helmholtz free energy under the framework of thermodynamics that combines the acoustic radiation stress theory and the nonlinear elasticity theory together. Thus, the internal variable is related to the cavitation process and the mechanical energy dissipation of a tissue-mimicking gel from a macroscopic viewpoint. Since the temperature rise of cavitation phenomenon is more remarkable than that of heating waves, the temperature inside the tissue-mimicking gel rises rapidly mainly due to large amounts of cavitation bubbles. This phenomenon can be quantitatively described by the present model, which fits the existing experimental data well.

Self-controlled wave propagation in hyperelastic media

We demonstrate theoretically that an ultrasonic wave propagating in a hyperelastic medium can self-control its phase velocities. This phenomenon occurs because the propagation of the ultrasonic wave generates acoustic radiation stresses in the medium, which can induce large deformation of the medium with significant stiffening effect. In turn, such deformation reshapes the wave propagation while the deformation stiffening changes significantly the phase velocities of the wave till the acoustic radiation stresses are balanced by elastic stresses in the current configuration of the hyperelastic medium. As a result of deformation stiffening, an initially isotropic medium becomes anisotropic, thus enabling self-control or self-bending of the wave propagation. We further reveal that, due to snap-through instability of acoustomechanical deformation in the hyperelastic medium, the ultrasonic wave can discontinuously switch its phase velocities from one state to another by jumping over a large unstable regime. This self-control and switchable mechanism of ultrasonic wave propagation in homogenous hyperelastic media offers innovative design opportunities for phononic, thermal and acoustic materials and devices.

A nonlinear acoustomechanical field theory of polymeric gels

A nonlinear acoustomechanical field theory is presented for polymeric gels undergoing large deformation coupled with diffusion mass transport of solvent molecules in and out of the gel. The theory is developed by combining the acoustic radiation stress theory with the nonlinear elasticity theory of polymeric gels. Explicit velocity and acoustic fields are determined by solving the elastodynamical equations of wave propagation in Eulerian coordinates, which are then employed to determine the distribution of acoustic radiation stresses inside the gel. The nonlinear elasticity of gels is modeled by adopting the Flory–Rehner free energy functions for network stretching and molecules-polymer mixing. For illustration, the developed theory is applied to a layer of polymeric gel immersed in external solvent subjected to two counterpropagating acoustic waves. The acoustically actuated large deformation of the gel is analyzed under three different constraint conditions. Unique acoustomechanical behaviors of polymeric gels are revealed, such as periodical response and nonlinear chaos. This work is expected to enable novel design of ultrasound-triggered sensors and actuators made of polymeric gels, and can also enlighten the application of ultrasonic waves in biomedical engineering.

Nonlinear large deformation of acoustomechanical soft materials

An acoustomechanical theory of soft materials is proposed to account for the nonlinear large deformation of soft materials triggered by both the ultrasound waves and mechanical forces. This theory is formulated by employing the nonlinear elasticity theory of the soft material and the theory of acoustic radiation force, which takes into consideration of the combination loading of mechanical forces and acoustic inputs. While the propagation of acoustic wave depends on material configuration, the radiation force generated by wave propagation deforms the material configuration. This complex interaction reaches a steady state when the mechanical stress and the acoustic radiation stress are able to balance with the elastic deformation stress. The acoustomechanical theory is employed to characterize the acoustomechanical behaviors of thin soft material layers under different boundary conditions (e.g., equal-biaxial forces, uniaxial force, and uniaxial constraint). Prestretches arising from these boundary conditions are shown to play significant roles in affecting the acoustomechanical repsonse of soft material: the same material actuated from different prestretches and boundary conditions exhibits different stretch-stress relations. This novel functionality enables innovative design of acoustic sensors and actuators based on soft materials.

Acoustomechanical constitutive theory for soft materials

Acoustic wave propagation from surrounding medium into a soft material can generate acoustic radiation stress due to acoustic momentum transfer inside the medium and material, as well as at the interface between the two. To analyze acoustic-induced deformation of soft materials, we establish an acoustomechanical constitutive theory by combining the acoustic radiation stress theory and the nonlinear elasticity theory for soft materials. The acoustic radiation stress tensor is formulated by time averaging the momentum equation of particle motion, which is then introduced into the nonlinear elasticity constitutive relation to construct the acoustomechanical constitutive theory for soft materials. Considering a specified case of soft material sheet subjected to two counter-propagating acoustic waves, we demonstrate the nonlinear large deformation of the soft material and analyze the interaction between acoustic waves and material deformation under the conditions of total reflection, acoustic transparency, and acoustic mismatch.

Acoustomechanical giant deformation of soft elastomers with interpenetrating networks

We demonstrate giant deformation caused by ultrasound waves in soft elastomers with interpenetrating networks and reveal the physical mechanisms underlying the snap-through instability and phase transition. The snap-through instability can be harnessed to generate large deformation when the elastomer is subjected to combined mechanical and acoustical loading. We further demonstrate that the preserved stresses can enhance not only the mechanical tangential stiffness but also the acoustical tangential stiffness of the elastomer. However, with fixed acoustical loads, the preserved stresses reduce the mechanical tangential stiffness because the dependence of acoustic radiation stress on the stretch state overturns the effect of the preserved stresses. Our findings enable new strategies of device designs based on acoustomechanical soft elastomers having interpenetrating networks.

Nature of acoustic nonlinear radiation stress

When a fluid is insonified with ultrasound, a flow consequence of a net stress becomes observable, which has been described as acoustic streaming, quartz wind, acoustic radiation force, or acoustic fountain. Following Sir James Lighthill's formulation of the Reynold's streaming, these phenomena have been attributed to a cumulative viscous effect. Instead, a multiscale effect, whereby the constitutive elastic nonlinearity scales from the ultrasonic to the macroscopic time, is here proposed and formulated to explain its origin. This raises an additional term in the Navier-Stokes equation, which ultimately stems from the anharmonicity of the atomic potential. In our experimental validation, this theory is consistent in water and for a range of ultrasonic configurations, whereas the formerly established viscous theory fails by an order of magnitude. This ultrasonic-fluid interaction, called nonlinear mechanical radiation since it is able to remotely exert a stress field, correctly explains a wide range of industrial and biomedical active ultrasonic uses including jet engines, acoustic tweezers, cyanobacteria propulsion mechanisms, nanofluidics, or acoustic radiation force elastography.

Active damping of capillary oscillations on liquid columns

Active control of acoustic radiation pressure and of electrostatic stresses on liquid columns has been demonstrated to overcome the Rayleigh–Plateau instability that normally causes long liquid columns to break [M. J. Marr-Lyon et al., J. Fluid Mech. 351, 345 (1997); Phys. Fluids 12, 986–995 (2000)]. Though originally demonstrated for liquid–liquid systems in plateau tanks, the electrostatic method also works on columns in air in reduced gravity [D. B. Thiessen, M. J. Marr-Lyon, and P. L. Marston, ‘‘Active electrostatic stabilization of liquid bridges in low gravity,’’ J. Fluid Mech. (in press)]. In new research, the electrostatic stresses are applied in proportion to the velocity of the surface of the column so as to actively dampen capillary oscillations of the surface. The mode amplitude is optically sensed and the rate-of-change is electronically determined. Plateau tank measurements and theory both show that the change in damping rate is proportional to the feedback gain. The results suggest that either active control of electrostatic stresses or of acoustic radiation stresses can be used to suppress the response of interfaces to vibration. [Work supported by NASA.]

The acoustic radiation force in lossless fluids in Eulerian and Lagrangian coordinates

A general theory of the (Langevin) acoustic radiation force in three-dimensional fields in lossless fluids in Eulerian and Lagrangian coordinates is given. It is based on Brillouin’s approach of an acoustic radiation stress identified in Eulerian coordinates with the negative time average of the momentum flux density and in Lagrangian coordinates with the time average of the Piola–Kirchhoff–Boussinesq stress. It is shown in comparison to other statements from the literature that these stresses are not in general identical but that in the linear-acoustics approximation, the difference between the results in the two representations vanishes if the radiation force acting on an object entirely surrounded by the sound-propagating fluid is considered. It is just this situation which is of experimental relevance.

Acoustic Streaming and other Radiation Stress Effects

Acoustic Streaming and other Radiation Stress Effects

Acoustic energy quantities and radiation force in higher approximation

Extending the familiar linear acoustics treatment, the acoustic energy density, the acoustic energy flux density (acoustic intensity), and the acoustic radiation stress tensor in a three-dimensional acoustic field in a lossless and nondispersive fluid are calculated up to any order of approximation. The formulas obtained are used to discuss the question of the validity limitations of the familiar second-order result for the relation between radiation pressure and intensity. This question is important in connection with the widespread use of the radiation force method for ultrasonic power measurements. An answer is given by calculating the third- and fourth-order expressions for the specific case of a plane-progressive wave impinging on a perfect absorber. The result is that in the whole amplitude range covered by current ultrasonic medical equipment (therapeutic and diagnostic), the third- and fourth-order correction to the second-order formula is less than 1% and can thus be neglected.