O. V. Rudenko

Acoustic radiation force and streaming induced by focused nonlinear ultrasound in a dissipative medium

Based on asymptotic methods recently developed in nonlinear acoustics, analytical solutions of the equations for the radiation force induced by nonlinear focused ultrasound in a dissipative medium are considered. Equations describing spatial structure of the radiation force field in the paraxial region of the ultrasound beam and the spatial–temporal behavior of the induced nonlinear streaming are derived. The equations enable analytical investigation of dependencies of radiation pressure and resulting streaming on the acoustic field and medium parameters. Estimates have shown that nonlinearity of medium can significantly enhance radiation force in the focal region at the intensities lower than those used in ultrasound devices for medical imaging. The initiation of acoustical streaming by radiation force is considered, and the spatial and temporal characteristics of induced flow are discussed. Both acoustic and hydrodynamic nonlinearities are taken into account. The paper concludes by discussing possible m...

BIOMEDICAL APPLICATIONS OF RADIATION FORCE OF ULTRASOUND: HISTORICAL ROOTS AND PHYSICAL BASIS

Radiation force is a universal phenomenon in any wave motion, electromagnetic or acoustic. Although acoustic and electromagnetic waves are both characterized by time variation of basic quantities, they are also both capable of exerting a steady force called radiation force. In 1902, Lord Rayleigh published his classic work on the radiation force of sound, introducing the concept of acoustic radiation pressure, and some years later, further fundamental contributions to the radiation force phenomenon were made by L. Brillouin and P. Langevin. Many of the studies discussing radiation force published before 1990 were related to techniques for measuring acoustic power of therapeutic devices; also, radiation force was one of the factors considered in the search for noncavitational, nonthermal mechanisms of ultrasonic bioeffects. A major surge in various biomedical applications of acoustic radiation force started in the 1990s and continues today. Numerous new applications emerged including manipulation of cells in suspension, increasing the sensitivity of biosensors and immunochemical tests, assessing viscoelastic properties of fluids and biological tissues, elasticity imaging, monitoring ablation of lesions during ablation therapy, targeted drug and gene delivery, molecular imaging and acoustical tweezers. We briefly present in this review the major milestones in the history of radiation force and its biomedical applications. In discussing the physical basis of radiation force and its applications, we present basic equations describing the relationship of radiation stress with parameters of acoustical fields and with the induced motion in the biological media. Momentum and force associated with a plane-traveling wave, equations for nonlinear and nonsteady-state acoustic streams, radiation stress tensor for solids and biological tissues and radiation force acting on particles and microbubbles are considered.

The 40th anniversary of the Khokhlov-Zabolotskaya equation

The 40th anniversary of the Khokhlov-Zabolotskaya equation was marked by a special session of the 158th Meeting of the Acoustical Society of America (October 2009, San Antonio, Texas, United States). A response on the part of Acoustical Physics to this date is also quite appropriate, all the more so because Russian scientists were the main players involved in formulating and using this equation during the period of time between the middle 1960s and middle 1980s. In this article, the author—a participant and witness of those events—presents his view of the dramatic history of the formulation of this equation and related models in the context of earlier and independent work in aerodynamics and nonlinear wave theory. The main problems and physical phenomena described by these mathematical models are briefly considered.

Principle of an a priori use of symmetries in the theory of nonlinear waves.

The principle of an a priori use of symmetries is proposed as a new approach to solving nonlinear problems on the basis of a reasonable complication of mathematical models. Such a complication often causes an additional symmetry and, hence, opens up possibilities for finding new analytical solutions. The application of group analysis to the problems of nonlinear acoustics is outlined. The potentialities of the proposed approach are illustrated by exact solutions, which are of interest for wave theory.

Radiation force and shear motions in inhomogeneous media.

An action of radiation force induced by ultrasonic beam in waterlike media such as biological tissues (where the shear modulus is small as compared to the bulk compressibility) is considered. A new, nondissipative mechanism of generation of shear displacement due to a smooth (nonreflecting) medium inhomogeneity is suggested, and the corresponding medium displacement is evaluated. It is shown that a linear primary acoustic field in nondissipative, isotropic elastic medium cannot excite a nonpotential radiation force and, hence, a shear motion, whereas even smooth inhomogeneity makes this effect possible. An example is considered showing that the generated displacement pulse can be significantly longer than the primary ultrasound pulse. It is noted that, unlike the dissipative effect, the nondissipative action on a localized inhomogeneity (such as a lesion in a tissue) changes its sign along the beam axis, thus stretching or compressing the focus area.

Analytical method for describing the paraxial region of finite amplitude sound beams

A special analytical method, which combines the parabolic approximation (KZ equation) with nonlinear geometrical acoustics, is developed to model nonlinear and diffraction effects near the axis of a finite amplitude sound beam. The corresponding system of nonlinear equations describing waveform evolution is derived. For the case of an initially sinusoidal wave radiated by a Gaussian source, an analytic solution of the coupled equations is obtained for the paraxial region of the beam. The axial solution is expressed in both the time and frequency domains, and is analyzed in detail for both unfocused and focused beams in their preshock regions. Harmonic propagation curves are compared with finite difference solutions of the KZ equation, and good agreement is obtained for a variety of parameter values.

Wave Biomechanics of the Skeletal Muscle

Results of acoustic measurements in skeletal muscle are generalized. It is shown that assessment of the pathologies and functional condition of the muscular system is possible with the use of shear waves. The velocity of these waves in muscles is much smaller than the velocity of sound; therefore, a higher symmetry type is formed for them. In the presence of a preferential direction (along muscle fibers), it is characterized by only two rather than five (as in usual media with the same anisotropy) moduli of elasticity. A covariant form of the corresponding wave equation is presented. It is shown that dissipation properties of skeletal muscles can be controlled by contracting them isometrically. Pulsed loads (shocks) and vibrations are damped differently, depending on their frequency spectrum. Characteristic frequencies on the order of tens and hundreds of hertz are attenuated due to actin-myosin bridges association/dissociation dynamics in the contracted muscle. At higher (kilohertz) frequencies, when the muscle is tensed, viscosity of the tissue increases by a factor of several tens because of the increase in friction experienced by fibrillar structures as they move relative to the surrounding liquid; the tension of the fibers changes the hydrodynamic conditions of the flow around them. Finally, at higher frequencies, the attenuation is associated with the rheological properties of biological molecules, in particular, with their conformational dynamics in the wave field. Models that describe the controlled shock dissipation mechanisms are proposed. Corresponding solutions are found, including those that allow for nonlinear effects.

Nonlinear standing waves, resonance phenomena, and frequency characteristics of distributed systems

This review is dedicated to resonator oscillations under conditions of a strongly expressed nonlinearity under which steep shock fronts emerge in the wave profiles. Models and approximated methods for their analysis for quadratic and cubic nonlinear media are examined, as well as for nonlinearity when taking into account the mobility of boundaries. The forms of the profiles are calculated both for a steady-state oscillation regime and during the establishment of the profiles. Dissipative losses and selective losses at specially chosen frequencies are considered. An analysis of nonlinear Q-factor is given. The possibility of increasing the acoustic energy accumulated in the cavity of the resonator is discussed. Special attention is given to various physical phenomena that are exhibited only in nonlinear acoustic fields.

Nonlinear standing waves in a layer excited by the periodic motion of its boundary

A new analytical approach is developed for the description of standing waves caused by arbitrary periodic vibration of a boundary. The approach is based on the nonlinear evolution equation written for an auxiliary function. This equation offers the possibility to study not only the steady-state acoustic field, but also its evolution in time. One can take into account the dissipative properties of the medium and the difference between one of the resonant frequencies and the fundamental frequency of the driving motion of the wall. An exact nonsteady-state solution is derived corresponding to the sawtooth-like periodic vibration of the boundary. The maximal “amplitude” values of the particle velocity and the energy of a standing wave are calculated. The temporal profiles of standing waves at different points of the layer are presented. A new possibility of pumping a high acoustic energy into a resonator is indicated for the case of a special type of the wall motion having the form of an “inverse saw.” Theoretically, such a vibration leads to an “explosive instability” and an unlimited growth of the standing wave. For a harmonic excitation, the exact non-steady-state solution is derived as well. The standing wave profiles are described by Mathieu functions, and the energy characteristics by their eigenvalues.

Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response

Simplified nonlinear evolution equations describing non-steady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach based on a nonlinear functional equation is used. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically on the base of exact nonlinear solutions for different laws of periodic motion of the wall. For harmonic excitation the wave profiles are described by Mathieu functions, and their mean energy characteristics by the corresponding eigenvalues. The sawtooth-shaped motion of the boundary leads to a similar process of evolution of the profile, but the solution has a very simple form. Some possibilities to enhance the Q-factor of a nonlinear system by suppression of nonlinear energy losses are discussed.