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A new parabolic equation is derived to describe the propagation of nonlinear sound waves in inhomogeneous moving media. The equation accounts for diffraction, nonlinearity, absorption, scalar inhomogeneities (density and sound speed), and vectorial inhomogeneities (flow). A numerical algorithm employed earlier to solve the KZK equation is adapted to this more general case. A two-dimensional version of the algorithm is used to investigate the propagation of nonlinear periodic waves in media with random inhomogeneities. For the case of scalar inhomogeneities, including the case of a flow parallel to the wave propagation direction, a complex acoustic field structure with multiple caustics is obtained. Inclusion of the transverse component of vectorial random inhomogeneities has little effect on the acoustic field. However, when a uniform transverse flow is present, the field structure is shifted without changing its morphology. The impact of nonlinearity is twofold: it produces strong shock waves in focal regions, while, outside the caustics, it produces higher harmonics without any shocks. When the intensity is averaged across the beam propagating through a random medium, it evolves similarly to the intensity of a plane nonlinear wave, indicating that the transverse redistribution of acoustic energy gives no considerable contribution to nonlinear absorption. © Nauka/Interperiodica 2006.

Newer imaging and therapeutic ultrasound technologies require higher in situ pressure levels compared to conventional diagnostic values. One example is the recently developed use of focused ultrasonic radiation force to move kidney stones and residual fragments out of the urinary collecting system. A commercial diagnostic 2.3 MHz C5-2 array probe is used to deliver the acoustic pushing pulses. The probe comprises 128 elements equally spaced at the 55 mm long convex cylindrical surface with 38 mm radius of curvature. The efficacy of the treatment can be increased by using higher intensity at the focus to provide stronger pushing force; however, nonlinear acoustic saturation can be a limiting factor. In this work nonlinear propagation effects were analyzed for the C5-2 transducer using a combined measurement and modeling approach. Simulations were based on the 3D Westervelt equation; the boundary condition was set to match the focal geometry of the beam as measured at a low power output. Focal waveforms simulated for increased output power levels were compared with the fiber-optic hydrophone measurements and were found in good agreement. It was shown that saturation effects do limit the acoustic pressure in the focal region of the transducer. This work has application to standard diagnostic probes and imaging.

Acoustic fields of powerful ultrasound sources with Gaussian spatial apodization and initial excitation in the form of a periodic wave or single pulse are examined based on the numerical solution of the Khokhlov-Zabolotskaya-Kuznetsov equation. The influence of nonlinear effects on the spatial structure of focused beams, as well as on the limiting values of the acoustic field parameters is compared. It is demonstrated that pressure saturation in periodic fields is mainly due to the effect of nonlinear absorption at a shock front, while in pulsed fields is due to the effect of nonlinear refraction. The limiting attainable values for the peak positive pressure in periodic fields turned out to be higher than the analogous values in pulsed acoustic fields. The total energy in a beam of periodic waves decreases with the distance from the source faster than in the case of a pulsed field, but it becomes concentrated within much smaller spatial region in the vicinity of the focus. These special features of nonlinear effect manifestation provide an opportunity to use pulsed beams for more efficient delivery of wave energy to the focus and to use periodic beams for attaining higher values of pressure in the focal region.

Nonlinear propagation of short N-waves (wavelength of 1.4 cm) through a turbulent layer with outer scale of about 16 cm was simulated based on the 2D parabolic KZK-type equation. A modified von Karman spectrum was used to generate random fluctuations of wind velocity associated with the presence of turbulence. Statistics of peak overpressure and shock front steepness were analyzed for linear and various degrees of nonlinear effects in N-wave propagation. It was shown that in case of linear propagation, the turbulence mainly led to smearing the shock fronts and resulted in low probabilities of observing steepened shocks. At higher pressure level, nonlinear effects resulted in steepening of the shock fronts and hence counteracted the effects introduced by the turbulence. Due to nonlinear enhancement of focusing gain in random foci, up to twofold increase of the cumulative probability of observing highly peaked waveforms was shown for nonlinear wave propagation comparing to the linear case. However, it was a...

Therapeutic ultrasound sources, which are typically piezoelectric transducers, are intended to deliver known acoustic pressures to targeted tissue sites. Each transducer vibrates in a unique way and radiates a corresponding 3D ultrasound field. Accordingly, transducer vibrations should be known accurately in order to characterize the pressures delivered to the patient. Acoustic holography is a technique that relies on hydrophone measurements to reconstruct a source hologram that characterizes transducer vibrations [Sapozhnikov et al., JASA, 138(3), 1515–1532 (2015)]. In this way, a hologram is a signature of each transducer that can be monitored over time for quality assurance. Using holography-defined source boundary conditions, numerical forward projection of the ultrasound field based on the nonlinear wave equation can be used to accurately predict in situ temperatures and pressures in heterogeneous media for treatment planning. As such, acoustic holography goes beyond simple hydrophone scans and is uniquely suited to meet clinical needs for quantifying therapeutic ultrasound fields. In this paper, several examples of acoustic holography implementation are presented, including the characterization of single-element and multi-element flat and spherically curved sources working in linear and nonlinear regimes and in continuous and pulsed modes. [Work supported by NIH 1R01EB025187, R01EB007643, and R21CA219793; RFBR 17-02-00261 and 17-54-33034.]

One-way wave propagation models based on the parabolic approximation or its wide-angle extensions are often used for describing bounded acoustic beams. Such models are highly demanded when solving nonlinear problems that are computationally intensive and thus technically difficult to solve using full-wave approaches. When describing the propagation of a wave beam in a homogeneous medium, the one-way assumption is fulfilled exactly, and therefore, the inaccuracy is caused only by the limitations of the parabolic approximation. Such an error is significantly reduced within the wide-angle approach and completely disappears when using the exact propagator in the framework of the angular spectrum method. The situation is less obvious in the case of a heterogeneous environment, when a part of the wave energy is inevitably reflected becoming a counter-propagating wave and thus is not taken into account in the one-way approximation. The degree this phenomenon affects the accuracy of the one-way approach is still under discussion. In the current paper, a one-way propagator based on the pseudo-differential wide-angle equation is proposed. The propagator is tested for the homogeneous medium and for several configurations of media with regular and random inhomogeneities. The corresponding solutions are compared with those obtained using the k-Wave toolbox. Results of comparison show how the one-way propagator accuracy depends on the contrast and smoothness of the inhomogeneities. [Work supported by RSF No. 18-72-00196.] One-way wave propagation models based on the parabolic approximation or its wide-angle extensions are often used for describing bounded acoustic beams. Such models are highly demanded when solving nonlinear problems that are computationally intensive and thus technically difficult to solve using full-wave approaches. When describing the propagation of a wave beam in a homogeneous medium, the one-way assumption is fulfilled exactly, and therefore, the inaccuracy is caused only by the limitations of the parabolic approximation. Such an error is significantly reduced within the wide-angle approach and completely disappears when using the exact propagator in the framework of the angular spectrum method. The situation is less obvious in the case of a heterogeneous environment, when a part of the wave energy is inevitably reflected becoming a counter-propagating wave and thus is not taken into account in the one-way approximation. The degree this phenomenon affects the accuracy of the one-way approach is still ...

The rapid development of nonlinear acoustics, which began in the second half of the 20th century, had several distinct centers in different countries, and one of them was M. V. Lomonosov Moscow State University (MSU) in Russia. From MSU came one of the basic equations in nonlinear acoustics—the Khokhlov-Zabolotskaya (KZ) equation, the 50th anniversary of which is celebrated this year [Sov. Phys. Acoust. 15(1), 35–40 (1969)]. By the 1960–1970s, the work conducted by Academician Rem Khokhlov and his colleagues made MSU a leading international center for nonlinear acoustics and, even more influentially, for nonlinear optics. Nonlinear optics at that time underwent rapid growth related to the invention of lasers, and many phenomena in optics had analogs in acoustics. One of them was a parabolic approximation for wave beams, which led to formulation of the KZ equation after translation to acoustics with consideration of the details of nonlinear effects in weakly dispersive media. Along with the development of theoretical foundations, initial experiments on nonlinear acoustic phenomena were carried out at MSU. The current paper highlights these studies, as well as contemporary work and achievements on nonlinear acoustics at Moscow State University.

Thermal effect of high-intensity focused ultrasound on biological tissue behind the rib cage phantom has been numerically simulated. A phased array was used as a source of ultrasound. Temperature and thermal dose distributions in the biological tissue were calculated and used to find the form of thermal lesions. The results of the calculations are in good agreement with experiment and show that the model is applicable for predicting the effect produced by ultrasound in tissue.

High-intensity focused ultrasound (HIFU) therapies are often performed using multi-element phased arrays. Independent variation of the amplitudes and phases at the array elements allows electronic steering of the focus and compensation for aberrations. To suppress the formation of grating lobes, the arrangement of the array elements should be non-periodic. Three approaches that have been recently employed in our studies to solve this problem are presented and discussed: a random element arrangement [L.R. Gavrilov and J.W. Hand, IEEE Trans. UFFC 2000], a multi-armed spiral arrangement [V.A. Khokhlova et al., Physics Procedia 2016], and recently proposed array design with tight packing of the elements based on the capacity-constrained tessellation [P.B. Rosnitskiy et al., IEEE UFFC 2018]. The efficiency of two arrays with the same geometric and physical parameters is compared: a 256-element array with a compact 16-spirals layout of circular elements and a fully populated array comprising polygonal elements of equal area. It is shown that for the same intensity at the elements, the fully-populated array provides twofold higher total power while maintaining the same electronic focusing capabilities as compared to the spiral one which can be beneficial for high-power applications such as histotripsy. [The work was supported by NIH R01EB7643 and R01EB025187, RFBR 17-02-00261, and Ph.D. student stipends from the “Basis” Foundation and of the President of Russia SP-2644.2018.4.]

Enhanced chemotherapeutic drug delivery has been shown using pulsed high intensity focused ultrasound (pHIFU) without contrast agents. The threshold of these inertial cavitation effects has recently been correlated to the formation of shocks at the focus. The shock amplitude and corresponding peak negative pressure (p-) are primarily determined by the transducers F-number with less focused (higher F-number) transducers producing shocks at lower p-. Here, the dependence of inertial cavitation activity on F-number was investigated in gel phantoms using passive cavitation detection (PCD) and high-speed photography. HIFU transducers with the same aperture but different F-numbers (0.77, 1.02, and 1.52), operable at three driving frequencies (1 MHz, 1.5 MHz, and 1.9 MHz), were utilized with driving conditions consisting of 1 ms pulses delivered every second and p- from 1 to 15 MPa. Broadband noise emissions recorded by PCD were batch-processed to extract cavitation probability and persistence while concurrent imaging was performed in the focal pHIFU region. At the same p-, both PCD metrics and the imaging revealed enhanced cavitation activity at higher F-numbers. These results support the use of less focused, smaller-footprint transducers for achieving desired cavitation-aided drug delivery. [Work supported by NIH R01EB023910, K01DK104854, R01EB007643, and RFBR 17-54-33034.]

Acoustic holography measurements and corresponding numerical projection methods have recently gained acceptance for characterizing the fields generated by high intensity therapeutic ultrasound (HITU) applications. To facilitate the standardization of such measurement-based simulation methods, it is important to understand how practical measurement challenges and limitations impact the results. Toward this end, holography measurements were acquired and analyzed for characterization of two therapeutic array transducers, each with 256 elements. For different focusing configurations, measurements comprised 2D holography scans as well as 1D scans used to independently quantify the focal lobe. The fields projected from holograms were compared to independent 1D scans to evaluate the impacts of hydrophone directivity, non-orthogonality of the axes of the positioner used to move the hydrophone, and overall mechanical stability of measurements. Results demonstrate that directivity effects exceeded those expected based on hydrophone size and altered focal pressures on the order of 10%. Non-orthogonality of the positioner shifted apparent focal locations by measurable distances, which could be confused with small shifts in device fixturing. However, holographic reconstruction of the ultrasound field structure near the focal lobe was robust with respect to directivity and non-orthogonality, which enables compensation by relative calibration of source output levels. [Funding support by NIH R01-EB025187, R01-EB007643, and P01-DK043881.]Acoustic holography measurements and corresponding numerical projection methods have recently gained acceptance for characterizing the fields generated by high intensity therapeutic ultrasound (HITU) applications. To facilitate the standardization of such measurement-based simulation methods, it is important to understand how practical measurement challenges and limitations impact the results. Toward this end, holography measurements were acquired and analyzed for characterization of two therapeutic array transducers, each with 256 elements. For different focusing configurations, measurements comprised 2D holography scans as well as 1D scans used to independently quantify the focal lobe. The fields projected from holograms were compared to independent 1D scans to evaluate the impacts of hydrophone directivity, non-orthogonality of the axes of the positioner used to move the hydrophone, and overall mechanical stability of measurements. Results demonstrate that directivity effects exceeded those expected ba...

This study is aimed at developing a method to controllably manipulate kidney stones in the human body using a single ultrasound source. Calculation and measurement of the lateral force on a sphere using various acoustic beams (vortex and radially-varying phased beams) were compared. A 256-element, 1.5-MHz array was used to synthesize the beams. Spheres of 1–6 mm (6 < ka < 20) diameter made of glass, ceramic, or brass were positioned on an acoustically matched platform at the focus rigidly attached to the transducer by a frame. The transducer and platform were rotated to the angle at which the trapped sphere fell. The acoustic power was

Boiling histotripsy (BH) is a HIFU approach that uses millisecond-long ultrasound bursts with high-amplitude shocks to mechanically fractionate tissue. In this work, the performance of a BH pre-clinical system operating at 1.5 MHz was characterized by hydrophone measurements in water and lesion formation capabilities were tested in ex vivo tissue. The system comprises a 256 element array driven by Verasonics Ultrasound Engine with a 1.2 kW external power source enhanced by an additional capacitor bank. The maximum pulse average acoustic power of the system was 2.2 kW (as measured by radiation force balance), sustained for 10 ms with

Acoustic radiation force (ARF) is successfully used in a recently proposed ultrasonic technology for kidney stone propulsion. The planning of the treatment requires calibration of ARF for stones of different dimensions and locations. However, such calibration remains a problem. Here, a method of measuring ARF acting on a mm-sized spherical object positioned on the axis of a focused ultrasound beam is proposed and tested. Acoustic field was generated by a single-element 1.072 MHz transducer of 100 mm aperture and 70 mm focal length positioned at the bottom of the water tank. Measurements were performed for nylon and glass spherical scatterers with diameters from 2 to 4 mm located at different distances from the source along the vertical beam axis. For each scatterer, the source power was gradually decreased until the scatterer started to fall down from the trap. At this threshold power of the source, the value of ARF was determined as a difference between the gravity and buoyancy forces acting on the scatterer. At other source power outputs, the value of ARF was linearly scaled. ARF was also calculated from pressure distributions reconstructed from acoustic hologram of the source and physical parameters of the scatterers. Experimental and theoretical results were found in a good agreement. It was shown that the most effective pushing was observed at distances where the beam was slightly wider than the scatterer. [Work supported by the stipend of the President of Russia (SP-2621.2016.4), RFBR №18-32-00659, and NIH P01 DK43881.]

Irregular reflection of weak acoustic shock waves occurs under the framework of the von Neumann paradox. In this study, the influence of the surface roughness on the reflection pattern was studied experimentally using spark-generated spherically divergent N-waves of 1.4 cm length reflecting from rigid rough surfaces in air. Dimensions of the roughness were varied from 20 up to 500 μm for different surfaces. A Mach-Zehnder interferometry method was used to reconstruct the pressure waveforms near the surface. The reconstruction was performed by applying the inverse Abel transform to the phase of the signal measured by the interferometer. It was shown that the height of the Mach stem became shorter for surfaces with larger dimensions of the roughness and disappeared when the surface roughness was large enough. Such tendency was also observed in simulations based on the Euler equations where the acoustic source was introduced as a Gaussian-envelope energy injection and the roughness was either sinusoidal or random and described by a Gaussian correlation function. [Work supported by RSF-17-72-10277 and by the Labex CeLyA of Universite de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ ANR-11-IDEX-0007).]Irregular reflection of weak acoustic shock waves occurs under the framework of the von Neumann paradox. In this study, the influence of the surface roughness on the reflection pattern was studied experimentally using spark-generated spherically divergent N-waves of 1.4 cm length reflecting from rigid rough surfaces in air. Dimensions of the roughness were varied from 20 up to 500 μm for different surfaces. A Mach-Zehnder interferometry method was used to reconstruct the pressure waveforms near the surface. The reconstruction was performed by applying the inverse Abel transform to the phase of the signal measured by the interferometer. It was shown that the height of the Mach stem became shorter for surfaces with larger dimensions of the roughness and disappeared when the surface roughness was large enough. Such tendency was also observed in simulations based on the Euler equations where the acoustic source was introduced as a Gaussian-envelope energy injection and the roughness was either sinusoidal or r...

The effect of focus splitting after propagation of focused ultrasound through a rib cage is investigated theoretically. It is shown that the mechanism of this effect is caused by the interference of waves from two or more spatially separated sources, such as intercostal spaces. Analytical estimates of the parameters of splitting are obtained, i.e., the number of foci, their amplitudes, diameter, and the distance between them, depending on the transducer parameters, as well as the dimensions of the rib cage and position of ribs relative to the radiator. Various configurations of the relative positioning of ribs and radiator are considered; it is shown which of them are the most effective for real surgical operations.

The field of spectral components of a focused finite-amplitude ultrasound beam in water behind the layer that introduces a constant shift randomly distributed over its plane is studied experimentally and numerically. Based on the focal field distributions obtained and the criterion proposed, the degree of maintenance of the focusing condition for the first six harmonics of the beam radiated at 1.1 MHz is evaluated. Several layer positions at a distance from the radiator are considered. It is shown that focusing of the higher-order harmonics may be less subjected to destruction by the phase layer than that of the wave at the fundamental frequency. The theory and experiment are compared for the 90° and 180° phase layers. For the latter case, selective destruction of focusing of the odd harmonics is demonstrated.

Nonlinear propagation causes high-intensity ultrasound waves to distort and generate higher harmonics, which are more readily absorbed and converted to heat than the fundamental frequency. Although such nonlinear effects have been investigated previously and found to not significantly alter high-intensity focused ultrasound (HIFU) treatments, two results reported here change this paradigm. One is that at clinically relevant intensity levels, HIFU waves not only become distorted but form shock waves in tissue. The other is that the generated shock waves heat the tissue to boiling in much less time than predicted for undistorted or weakly distorted waves. In this study, a 2-MHz HIFU source operating at peak intensities up to 25,000 W/cm(2) was used to heat transparent tissue-mimicking phantoms and ex vivo bovine liver samples. Initiation of boiling was detected using high-speed photography, a 20-MHz passive cavitation detector and fluctuation of the drive voltage at the HIFU source. The time to boil obtained experimentally was used to quantify heating rates and was compared with calculations using weak shock theory and the shock amplitudes obtained from nonlinear modeling and measurements with a fiber optic hydrophone. As observed experimentally and predicted by calculations, shocked focal waveforms produced boiling in as little as 3 ms and the time to initiate boiling was sensitive to small changes in HIFU output. Nonlinear heating as a result of shock waves is therefore important to HIFU, and clinicians should be aware of the potential for very rapid boiling because it alters treatments.

Propagation of a probing pulse through a thermal inhomogeneity produced in the medium as a result of heating by a focused ultrasonic beam is considered. The wave equation is used to numerically calculate the waveform of the pulse arriving at the receiver whose focus coincides with that of the radiator. Calculations are performed for a pulse focused at the center of the inhomogeneity and crossing it in the transverse direction. The results of numerical simulation are compared with experimental data. Conditions under which the diffraction effects should be taken into account in calculating the pulse delay at the receiver are formulated.

In this work, the influence of nonlinear and diffraction effects on amplification factors of focused ultrasound systems is investigated. The limiting values of acoustic field parameters obtained by focusing of high power ultrasound are studied. The Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation was used for the numerical modeling. Solutions for the nonlinear acoustic field were obtained at output levels corresponding to both pre- and post- shock formation conditions in the focal area of the beam in a weakly dissipative medium. Numerical solutions were compared with experimental data as well as with known analytic predictions.

Both mechanically induced acoustic cavitation and thermally induced boiling can occur during high intensity focused ultrasound (HIFU) medical therapy. The goal was to monitor the temperature as boiling was approached using magnetic resonance imaging (MRI). Tissue phantoms were heated for 20 s in a 4.7-T magnet using a 2-MHz HIFU source with an aperture and radius of curvature of 44 mm. The peak focal pressure was 27.5 MPa with corresponding beam width of 0.5 mm. The temperature measured in a single MRI voxel by water proton resonance frequency shift attained a maximum value of only 73 degrees C after 7 s of continuous HIFU exposure when boiling started. Boiling was detected by visual observation, by appearance on the MR images, and by a marked change in the HIFU source power. Nonlinear modeling of the acoustic field combined with a heat transfer equation predicted 100 degrees C after 7 s of exposure. Averaging of the calculated temperature field over the volume of the MRI voxel (0.3 x 0.5 x 2 mm(3)) yielded a maximum of 73 degrees C that agreed with the MR thermometry measurement. These results have implications for the use of MRI-determined temperature values to guide treatments with clinical HIFU systems.

The influence of nonlinear and diffraction effects on distortion of the spatial structure of peak positive and negative pressures in focused acoustic beams was studied for a weakly dissipative propagation medium. The problem was solved numerically based on the Khokhlov-Zabolotskaya-Kuznetsov equation for beams with uniform and Gaussian distributions of the harmonic signal amplitude at the source.

Acoustic characterization of high intensity focused ultrasound (HIFU) fields is important both for the accurate prediction of ultrasound induced bioeffects in tissues and for the development of regulatory standards for clinical HIFU devices. In this paper, a method to determine HIFU field parameters at and around the focus is proposed. Nonlinear pressure waveforms were measured and modeled in water and in a tissue-mimicking gel phantom for a 2 MHz transducer with an aperture and focal length of 4.4 cm. Measurements were performed with a fiber optic probe hydrophone at intensity levels up to 24,000 W/cm(2). The inputs to a Khokhlov-Zabolotskaya-Kuznetsov-type numerical model were determined based on experimental low amplitude beam plots. Strongly asymmetric waveforms with peak positive pressures up to 80 MPa and peak negative pressures up to 15 MPa were obtained both numerically and experimentally. Numerical simulations and experimental measurements agreed well; however, when steep shocks were present in the waveform at focal intensity levels higher than 6000 W/cm(2), lower values of the peak positive pressure were observed in the measured waveforms. This underrepresentation was attributed mainly to the limited hydrophone bandwidth of 100 MHz. It is shown that a combination of measurements and modeling is necessary to enable accurate characterization of HIFU fields.

Recent advances in the fields of physical acoustics, imaging technologies, piezoelectric materials, and ultrasonic transducer design have lead to emerging of novel methods and apparatus for ultrasonic diagnostics, therapy, and anesthetics as well as an expansion of traditional application fields. The chapter presents the results of theoretical modeling and experimental study of different High Intensity Focused Ultrasound (HIFU) transducers. Numerical solutions of parabolic Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation were obtained for nonlinear focused fields. Technological peculiarities of the HIFU transducer design as well as theoretical and numerical models of such transducers and the corresponding HIFU fields are discussed. Several HIFU transducers of different design have been fabricated using different advanced piezoelectric materials. Acoustic field measurements for those transducers have been performed using a calibrated fiber optic hydrophone and an ultrasonic measurement system. The results of theoretical modeling and ex vivo experiments with different tissues (fresh porcine mussels, adipose tissue and bovine liver), as well as in vivo experiments with blood vessels are present that prove the efficacy, safety and selectivity of the developed HIFU transducers and methods.

A Mach-Zehnder interferometer is used to measure spherically diverging N-waves in homogeneous air. An electrical spark source is used to generate high-amplitude (1800 Pa at 15 cm from the source) and short duration (50 μs) N-waves. Pressure waveforms are reconstructed from optical phase signals using an Abel-type inversion. It is shown that the interferometric method allows one to reach 0.4 μs of time resolution, which is 6 times better than the time resolution of a 1/8-in. condenser microphone (2.5 μs). Numerical modeling is used to validate the waveform reconstruction method. The waveform reconstruction method provides an error of less than 2% with respect to amplitude in the given experimental conditions. Optical measurement is used as a reference to calibrate a 1/8-in. condenser microphone. The frequency response function of the microphone is obtained by comparing the spectra of the waveforms resulting from optical and acoustical measurements. The optically measured pressure waveforms filtered with the microphone frequency response are in good agreement with the microphone output voltage. Therefore, an optical measurement method based on the Mach-Zehnder interferometer is a reliable tool to accurately characterize evolution of weak shock waves in air and to calibrate broadband acoustical microphones.

The importance of nonlinear acoustic wave propagation and ultrasound-induced cavitation in the acceleration of thermal lesion production by high intensity focused ultrasound was investigated experimentally and theoretically in a transparent protein-containing gel. A numerical model that accounted for nonlinear acoustic propagation was used to simulate experimental conditions. Various exposure regimes with equal total ultrasound energy but variable peak acoustic pressure were studied for single lesions and lesion stripes obtained by moving the transducer. Static overpressure was applied to suppress cavitation. Strong enhancement of lesion production was observed for high amplitude waves and was supported by modeling. Through overpressure experiments it was shown that both nonlinear propagation and cavitation mechanisms participate in accelerating lesion inception and growth. Using B-mode ultrasound, cavitation was observed at normal ambient pressure as weakly enhanced echogenicity in the focal region, but was not detected with overpressure. Formation of tadpole-shaped lesions, shifted toward the transducer, was always observed to be due to boiling. Boiling bubbles were visible in the gel and were evident as strongly echogenic regions in B-mode images. These experiments indicate that nonlinear propagation and cavitation accelerate heating, but no lesion displacement or distortion was observed in the absence of boiling.

A method for remote monitoring of temperature in the focal region of a high-intensity ultrasonic transducer is described. Results of measurements and theoretical simulation are presented. The measurements were conducted on a polymer sample with thermophysical and acoustic parameters close to the properties of a soft biological tissue. The sample was heated by a focused piezoelectric transducer with different values of radiation power. The delay of a probe pulse transmitted through the heated region perpendicularly to the axis of the intense ultrasonic beam was detected. The local character of temperature measurements was provided by focusing the probe pulse at the heated region. The application of an additional transducer installed confocally with the probing one provided an opportunity to enhance the precision of measurements. An analysis was conducted on the basis of a numerical solution of the heat conduction equation. The function of thermal sources in the heat conduction equation was calculated according to the results of measuring the pressure distribution in the focal region of the heating transducer. The experimental data obtained agree well with the results of simulation and demonstrate a fundamental possibility of using the proposed ultrasonic technique for remote temperature measurements.

A classical effect of nonlinear acoustics is that a plane sinusoidal acoustic wave propagating in a nonlinear medium transforms to a sawtooth wave with one shock per cycle. However, the waveform evolution can be quite different in the near field of a plane source due to diffraction. Previous numerical simulations of nonlinear acoustic waves in the near field of a circular piston source predict the development of two shocks per wave cycle [Khokhlova et al., J. Acoust. Soc. Am. 110, 95–108 (2001)]. Moreover, at some locations the peak pressure may be up to 4 times the source amplitude. The motivation of this work was to experimentally verify and further explain the phenomena of the nonlinear waveform distortion. Measurements were conducted in water with a 47-mm-diameter unfocused transducer, working at 1-MHz frequency. For pressure amplitudes higher than 0.5 MPa, two shocks per cycle were observed in the waveform beyond the last minimum of the fundamental harmonic amplitude. With the increase of the observation distance, these two shocks collided and formed one shock (per cycle), i.e., the waveform developed into the classical sawtooth wave. The experimental results were in a very good agreement with the modeling based on the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation.

Mechanical erosion of soft tissues into subcellular debris has been demonstrated with pulsed high-intensity focused ultrasound, facilitated by either boiling or cavitation bubbles. In this work, we propose acoustic streaming as a primary cause of tissue erosion at a tissue-fluid interface. Bovine liver tissue and polyacrylamide gels were sonicated in a degassed water bath, with the focus positioned at the tissue/fluid interface. Pulses with duration between 1 to 104 μs and constant duty cycle of 0.5% were applied from a 1 MHz transducer generating focal pressures |p- | ≤17 MPa and p+ ≤90 MPa. Results showed a strongly nonlinear change in erosion with pulse duration, being greatest for pulse lengths between 50-500 μs. For longer pulses (>1 ms), high-speed videos showed streaming velocities >10 m/s. Moreover, lesions >1cm in depth were produced in tissue phantoms even with a single pulse and tissue disruption was evident where no bubbles were observed at the tissue-fluid interface. Enhancement of erosion wa...

High intensity focused ultrasound (HIFU) can destroy tumors or stop internal bleeding. The primary physical mechanism in HIFU is the conversion of acoustic energy to heat, which as HIFU amplitude increases is enhanced by nonlinear acoustic propagation and nonlinear scattering from bubbles. The goal of this work is to study and separate the effects of nonlinear propagation and cavitation during HIFU heating of tissue. Transparent polyacrylamide gel was used as a tissue-mimicking phantom to visualize HIFU lesion growth. Lesion size was also measured in excised turkey breast. Lesions were produced by the same time-averaged intensity, but with different peak acoustic pressure amplitudes compensated by different duty cycles. In order to separate cavitation and nonlinear wave effects, experiments were performed under static pressure (10.34MPa) greater than the peak negative pressure amplitude of the sound waves (8.96MPa). Suppression of cavitation by overpressure was measured by reduced acoustic scattering and transmission loss in the treatment region. We found that, with the same time-averaged intensity, a shorter, higher amplitude wave created a larger lesion than a longer, lower amplitude wave with or without overpressure.

Therapeutic ultrasound is an emerging field with many medical applications. High intensity focused ultrasound (HIFU) provides the ability to localize the deposition of acoustic energy within the body, which can cause tissue necrosis and hemostasis. Similarly, shock waves from a lithotripter penetrate the body to comminute kidney stones, and transcutaneous ultrasound enhances the transport of chemotherapy agents. New medical applications have required advances in transducer design and advances in numerical and experimental studies of the interaction of sound with biological tissues and fluids. The primary physical mechanism in HIFU is the conversion of acoustic energy into heat, which is often enhanced by nonlinear acoustic propagation and nonlinear scattering from bubbles. Other mechanical effects from ultrasound appear to stimulate an immune response, and bubble dynamics play an important role in lithotripsy and ultrasound-enhanced drug delivery. A dramatic shift to understand and exploit these nonlinear and mechanical mechanisms has occurred over the last few years. Specific challenges remain, such as treatment protocol planning and real-time treatment monitoring. An improved understanding of the physical mechanisms is essential to meet these challenges and to further advance therapeutic ultrasound.

In high intensity focused ultrasound (HIFU) therapy, an ultrasound beam is focused within the body to locally affect the targeted site without damaging intervening tissues. The most common HIFU regime is thermal ablation. Recently there has been increasing interest in generating purely mechanical lesions in tissue (histotripsy). This paper provides an overview of several studies on the development of histotripsy methods toward clinical applications. Two histotripsy approaches and examples of their applications are presented. In one approach, sequences of high-amplitude, short (microsecond-long), focused ultrasound pulses periodically produce dense, energetic bubble clouds that mechanically disintegrate tissue. In an alternative approach, longer (millisecond-long) pulses with shock fronts generate boiling bubbles and the interaction of shock fronts with the resulting vapour cavity causes tissue disintegration. Recent preclinical studies on histotripsy are reviewed for treating benign prostatic hyperplasia (BPH), liver and kidney tumours, kidney stone fragmentation, enhancing anti-tumour immune response, and tissue decellularisation for regenerative medicine applications. Potential clinical advantages of the histotripsy methods are discussed. Histotripsy methods can be used to mechanically ablate a wide variety of tissues, whilst selectivity sparing structures such as large vessels. Both ultrasound and MR imaging can be used for targeting and monitoring the treatment in real time. Although the two approaches utilise different mechanisms for tissue disintegration, both have many of the same advantages and offer a promising alternative method of non-invasive surgery.

When focused ultrasound waves of moderate intensity in liquid encounter an air interface, a chain of drops emerges from the liquid surface to form what is known as a drop-chain fountain. Atomization, or the emission of micro-droplets, occurs when the acoustic intensity exceeds a liquid-dependent threshold. While the cavitation-wave hypothesis, which states that atomization arises from a combination of capillary-wave instabilities and cavitation bubble oscillations, is currently the most accepted theory of atomization, more data on the roles of cavitation, capillary waves, and even heat deposition or boiling would be valuable. In this paper, we experimentally test whether bubbles are a significant mechanism of atomization in drop-chain fountains. High-speed photography was used to observe the formation and atomization of drop-chain fountains composed of water and other liquids. For a range of ultrasonic frequencies and liquid sound speeds, it was found that the drop diameters approximately equalled the ultrasonic wavelengths. When water was exchanged for other liquids, it was observed that the atomization threshold increased with shear viscosity. Upon heating water, it was found that the time to commence atomization decreased with increasing temperature. Finally, water was atomized in an overpressure chamber where it was found that atomization was significantly diminished when the static pressure was increased. These results indicate that bubbles, generated by either acoustic cavitation or boiling, contribute significantly to atomization in the drop-chain fountain.

The aim of this study is to show the evidence of Mach stem formation for very weak shock waves with acoustic Mach numbers on the order of 10(-3) to 10(-2). Two representative cases are considered: reflection of shock pulses from a rigid surface and focusing of nonlinear acoustic beams. Reflection experiments are performed in air using spark-generated shock pulses. Shock fronts are visualized using a schlieren system. Both regular and irregular types of reflection are observed. Numerical simulations are performed to demonstrate the Mach stem formation in the focal region of periodic and pulsed nonlinear beams in water.

An equivalent source model was proposed as a boundary condition to the nonlinear parabolic Khokhlov-Zabolotskaya (KZ) equation to simulate high intensity focused ultrasound (HIFU) fields generated by medical ultrasound transducers with the shape of a spherical shell. The boundary condition was set in the initial plane; the aperture, the focal distance, and the initial pressure of the source were chosen based on the best match of the axial pressure amplitude and phase distributions in the Rayleigh integral analytic solution for a spherical transducer and the linear parabolic approximation solution for the equivalent source. Analytic expressions for the equivalent source parameters were derived. It was shown that the proposed approach allowed us to transfer the boundary condition from the spherical surface to the plane and to achieve a very good match between the linear field solutions of the parabolic and full diffraction models even for highly focused sources with F-number less than unity. The proposed method can be further used to expand the capabilities of the KZ nonlinear parabolic equation for efficient modeling of HIFU fields generated by strongly focused sources.

Simulation of enhanced heating of clinically relevant tissue volumes using nonlinear ultrasound waves generated by a multi-element HIFU phased array were conducted based on the combined Westervelt and bio-heat equations. A spatial spectral approach using the fast Fourier transform algorithm and a corresponding analytic solution to the bioheat equation were used to optimize temperature modeling in tissue. Localized shock-wave heating within a much larger treated tissue volume and short, single HIFU pulses within a much longer overall exposure time were accounted for in the algorithm. Separation of processes with different time and spatial scales made the calculations faster and more accurate. With the proposed method it was shown that for the same time-average power, the use of high peak power pulsing schemes that produce high-amplitude shocks at the focus result in faster tissue heating compared to harmonic, continuous-wave sonications. Nonlinear effects can significantly accelerate volumetric heating while also permitting greater spatial control to reduce the impact on surrounding tissues. Such studies can be further used to test and optimize various steering trajectories of shock-wave sonications for faster and more controlled treatment of tissue volumes.

Accurate measurement of high-amplitude, broadband shock pulses in air is an important part of laboratory-scale experiments in atmospheric acoustics. Although various methods have been developed, specific drawbacks still exist and need to be addressed. Here, a schlieren optical method was used to reconstruct the pressure signatures of nonlinear spherically diverging short acoustic pulses generated using an electric spark source (2.5 kPa, 33 μs at 10 cm from the source) in homogeneous air. A high-speed camera was used to capture light rays deflected by refractive index inhomogeneities, caused by the acoustic wave. Pressure waveforms were reconstructed from the light intensity patterns in the recorded images using an Abel-type inversion method. Absolute pressure levels were determined by analyzing at different propagation distances the duration of the compression phase of pulses, which changed due to nonlinear propagation effects. Numerical modeling base on the generalized Burgers equation was used to evaluate the smearing of the waveform caused by finite exposure time of the high-speed camera and corresponding limitations in resolution of the schlieren technique. The proposed method allows the study of the evolution of spark-generated shock waves in air starting from the very short distances from the spark, 30 mm, up to 600 mm.

Steering abilities of a typical HIFU therapeutic array operated in linear and nonlinear regimes were compared using numerical simulation with the 3D Westervelt equation. The array included 256 elements of 1.2 MHz frequency and 6.6 mm diameter distributed in a quasi-random pattern over a spherical shell with a 130 mm aperture and a focal length of 120 mm. In the case of linear focusing, thermal effects are proportional to the intensity level and the criterion for safe array operation is that the intensity in the grating lobes should be less than 10% of the intensity in the main focus. In the case of nonlinear focusing, the heating effect is no longer proportional to intensity; therefore the heat deposition rate was chosen as the relevant metric, using the same 10% threshold for the secondary lobe in comparison with the focal maximum. When steering the focus, the same linearly predicted intensity level at the main focus was maintained by increasing the array power. Numerical simulations of the acoustic field were performed for nonlinear propagation both in water and in tissue. It was shown that for shock-forming conditions in the main focus, the steering range of safe electronic focusing is larger than that for linear propagation conditions. Nonlinear sonication regimes therefore can be used to enlarge tissue volumes that can be sonicated using electronic steering of the focus of HIFU arrays.

Mach stem is a well-known structure typically observed in the process of strong (acoustic Mach numbers greater than 0.4) step-shock waves reflection from a rigid boundary. However, this phenomenon has been much less studied for weak shocks in nonlinear acoustic fields where Mach numbers are in the range from 0.001 to 0.01 and pressure waveforms have more complicated waveforms than step shocks. The goal of this work was to demonstrate experimentally how nonlinear reflection occurs in air for very weak spherically divergent acoustic spark-generated pulses resembling an N-wave. Measurements of reflection patterns were performed using a Mach-Zehnder interferometer. A thin laser beam with sub-millimeter cross-section was used to obtain the time resolution of 0.4 µs, which is 6 times higher than the time resolution of the condenser microphones. Pressure waveforms were reconstructed using the inverse Abel transform applied to the phase of the signal measured by the interferometer. The Mach stem formation was observed experimentally as a result of collision of the incident and reflected shock pulses. It was shown that irregular reflection of the pulse occurred in a dynamic way and the length of the Mach stem increased linearly while the pulse propagated along the surface. Since the front shock of the spark-generated pulse was steeper than the rear shock, irregular type of reflection was observed only for the front shock of the pulse while the rear shock reflection occurred in a regular regime.

In the paper the use of receiving and radiating system, which allows to determine the parameters of bone by nonlinear pulse-echo technique and to image of brain structures through the skull bones, was proposed. Accuracy of the skull bone characterization is due to higher measured harmonic and is significantly better than in linear case. In the experimental part focused piezoelectric transducer with diameter 100 mm, focal distance 100 mm, the frequency of 1.092 MHz was used. It was shown that skull bone profiling can be performed with the use of 3rd harmonic since 1st harmonic can be used for visualization of the underlying objects. The use of wideband systems for both skull profiling and brain visualization is restricted by skull attenuation and resulting low effective sensitivity.

The development of medical ultrasound devices requires characterization of the ultrasound fields they radiate. Recently, acoustic holography methods have gained increased acceptance for field characterization. Because such an approach allows reconstruction of the full 3D field, the beam axis, its orientation relative to the positioner, and any field quantities of interest can be readily determined. Inherently, the accuracy of such projection methods can be sensitive to uncertainties in the positions at which field measurements are recorded. Although commonly used industrial positioning systems have linear axes with repeatability specifications much less than an ultrasound wavelength, the assembled orthogonality of three linear axes is not guaranteed. Here we analyze a typical raster scanning holography experiment by considering two distinct coordinate systems: ideal rectilinear coordinates aligned with the transducer (1.2 MHz, aperture 124 mm, F-number 1) and non-orthogonal coordinates aligned with the po...

Trans-skull ultrasound imaging of brain structures is a challenging problem because of strong attenuation of the skull and reflections from its boundaries. In addition, because the speed of sound in skull is much higher than in soft tissues, nonuniform thickness and heterogeneous bone structures cause strong refraction and aberration effects. In this work, transcranial ultrasound imaging of a 3D volume was simulated numerically. A linear wave equation in an inhomogeneous medium was modeled using the k-space method. A phased array comprising 10000 identically shaped square elements distributed over 70 x 70 mm2 area was used to generate a quasi-plane 2-cycle pulsed wave at 1 MHz. A spherical 3-mm diameter scatterer was placed 30 mm behind a cranial bone phantom with mass density 1900 kg/m3, sound speed 2500 m/s, and an irregular thickness varying from 5 to 8 mm. First, two reflections from the face and back sides of the phantom were used to determine its thickness. Then, a delay and sum algorithm was applie...

Varying ethanol concentration with depth, Hobaek et al. [AIP Proc. 838, 2006)] simulated the atmospheric sound speed profile scaled in a water tank and mimicked sonic boom propagation. Some limitations of their groundbreaking design included complexity, fragility, size, safety, one-dimensionality, tonal bursts instead of impulsive shocks, and the inability to scale density, absorption, turbulence, or other nonlinearities. Recognizing the time and cost savings of scaled overflight experiments in a laboratory environment, this work is meant to address certain limitations and investigate the Mach cut-off phenomena. Our primary advances are to use gel layers instead of fluid mixtures and lithotripter shock pulses. We defocused and modified a Dornier Compact S lithotripter to vary source angle in a 500-liter polycarbonate tank, measured shocks with a fiber optic hydrophone, and developed fluid-like gels with negligible shear-wave generation and robustness against cavitation. The sound speed, attenuation, inhomogeneity, and nonlinearity are adjustable within each gel layer. The result is a more flexible, controllable, and durable atmospheric analogue. [Work supported by the FAA. The opinions, findings, conclusions, and recommendations expressed in this material are those of the authors and do not necessarily reflect the views of ASCENT FAA Center of Excellence sponsor organizations.]Varying ethanol concentration with depth, Hobaek et al. [AIP Proc. 838, 2006)] simulated the atmospheric sound speed profile scaled in a water tank and mimicked sonic boom propagation. Some limitations of their groundbreaking design included complexity, fragility, size, safety, one-dimensionality, tonal bursts instead of impulsive shocks, and the inability to scale density, absorption, turbulence, or other nonlinearities. Recognizing the time and cost savings of scaled overflight experiments in a laboratory environment, this work is meant to address certain limitations and investigate the Mach cut-off phenomena. Our primary advances are to use gel layers instead of fluid mixtures and lithotripter shock pulses. We defocused and modified a Dornier Compact S lithotripter to vary source angle in a 500-liter polycarbonate tank, measured shocks with a fiber optic hydrophone, and developed fluid-like gels with negligible shear-wave generation and robustness against cavitation. The sound speed, attenuation, inhom...

Recent pre-clinical studies on boiling histotripsy (BH) of kidney in vivo have shown that the presence of soft tissue inhomogeneities such as skin, fat, and muscle layers do not prevent shock formation required for treatment. However, the increase in source power required to compensate for tissue aberrations may be impractically high and result in nearfield damage. Simulations can provide deeper insight into the impact of aberrations on shock formation in tissue and mitigation methods. A previously developed numerical solver of the Westervelt equation was refined to account for tissue inhomogeneities assuming that backward scattering effects can be neglected. Irradiation of human kidney using a single-element 1-MHz transducer with 5-cm radius and 9-cm focal distance was considered. Spatial distributions of the sound speed and mass density in tissue were reconstructed directly from a 3D CT scan. Values of nonlinear and absorption coefficients at each spatial location were assigned by CT image segmentation,...

Most progress in acoustic trapping and levitation has been achieved with the use of multiple sound sources, standing waves, and low density or very small objects. Repositioning kidney stones in the human body is an important clinical application where acoustic levitation could be employed; however, it requires manipulation of larger and heavier objects. The goal of this work was to calibrate the acoustic output of a 1.5-MHz, 256-element array designed in our laboratory for HIFU research, which is also capable of generating vortex beams to manipulate mm-sized objects in water and to move them in any direction without moving the source. Electrical circuits were developed for matching each element of the array to an output channel of a Verasonics ultrasound engine to allow for efficient power transfer to the transducer. Acoustic holography was used to calibrate and equalize outputs across all channels. Manipulation of artificial kidney stone targets made of plastic, glass, or cement (2—8 mm) and comparable i...

An approach that combines numerical modeling with measurements is gaining acceptance for field characterization in medical ultrasound. The general method is suitable for accurate simulation of the fields radiated by therapeutic transducers in both water and tissue. Here, three characterization methods are compared. Simulations based on the 3D Westervelt equation with a boundary condition determined from acoustic holography measurements are used as the most accurate benchmark method. Two simplified methods are based on an axially symmetric nonlinear parabolic formulation, either the KZK equation or its wide-angle extension. Various approaches for setting a boundary condition to the parabolic models are presented and discussed. Simulation results obtained with the proposed methods are compared for a typical therapeutic array and a strongly focused single-element transducer, with validation measurements recorded by a fiber optic hydrophone at the focus at increasing acoustic outputs. It is shown that the wid...

Ultrasound (US) imaging of brain structures is a challenging, but highly promising diagnostic technology in medical ultrasound. Recent advances in transcranial US therapy suggest the potential to implement diagnostic US at higher frequencies, ideally for full brain imaging. In this work. we present experimental results of ultrasound imaging of spherical and tubular scatterers placed behind a skull phantom. The phantom was produced from a casting compound with acoustic properties matching those of skull. Phantom shape was defined from CT data of a human skull and 3D printing of a mold. A two-dimensional ultrasound array was simulated by mechanical translation of the focal spot of a broadband single-element 2 MHz transducer over the phantom surface. This synthesized array mimicked a 2D flexible phased array placed on the top of the patient's head. A pulse-echo technique was used for reconstructing the thickness of the skull phantom and detecting backscattered signals from the test objects. Transcranial imag...

In the 1980s shockwave lithotripsy emerged as a revolutionary advancement for the treatment of kidney stones. Initial studies with patients showed SWL to be highly effective. The technology was elegant, outcomes exceptionally positive and early tests suggested treatment was safe. As experience with SWL grew, limitations surfaced. A key finding was that SWs have the potential to induce significant trauma to the kidney. Our group convinced the NIH it was time to conduct a rigorous assessment to characterize the adverse effects of SWL and determine the mechanisms of SW action in stone breakage and tissue injury. The NIH Program Project Grant mechanism mandated we establish a panel of external advisors to help guide our work. We needed expertise in physical acoustics, cavitation and animal models of ultrasound exposure. We wanted a leading expert. We were extremely fortunate to land Ed Carstensen. Ed worked with us for nearly 15 years, well into our third renewal cycle. He was a brilliant scientist, a man ded...