Fully nonlinear three-dimensional modeling of parametric interactions in the field of a dual-frequency acoustic array.

An algorithm is developed for fully nonlinear three-dimensional (3D) simulation of a difference-frequency acoustic beam resulting from the interaction of two high-intensity pump waves. Simulations are performed in the frequency domain based on the Khokhlov-Zabolotskaya-Kuznetsov equation. A spectrum filtering method is used to enable accurate solutions for the difference-frequency fields in strongly nonlinear beams and with a high downshift frequency ratio using only dozens of spectral components retained in the algorithm. As an example, the dual-frequency operation of an underwater multi-element ellipsoidal array is considered, and numerical solutions describing parametric interactions in the array field are analyzed. It is shown that difference-frequency beams are more symmetric in transverse directions compared with the pump beams. The most efficient parametric generation of difference-frequency beams corresponded to close and beyond shock-forming conditions. Axial pressure amplitude of the difference frequency was shown to grow first quadratically with the source pressure following the quasi-linear solution and then linearly once shocks start to develop. The percentage of the total power converted to the difference frequency from pump waves increased at high power outputs without saturation. Up to twofold increase in directivity angles of difference-frequency beams under shock-forming conditions was observed compared with quasi-linear conditions.

Comparative Characterization of Nonlinear Ultrasound Fields Generated by Sonalleve V1 and V2 MR-HIFU Systems.

A Sonalleve magnetic resonance-guided high-intensity focused ultrasound (MR-HIFU) clinical system (Profound Medical, Mississauga, ON, Canada) has been shown to generate nonlinear ultrasound fields with shocks up to 100 MPa at the focus as required for HIFU applications such as boiling histotripsy of hepatic and renal tumors. The Sonalleve system has two versions V1 and V2 of the therapeutic array, with differences in focusing angle, focus depth, arrangement of elements, and the size of a central opening that is twice larger in the V2 system compared to the V1. The goal of this study was to compare the performance of the V1 and V2 transducers for generating high-amplitude shock-wave fields and to reveal the impact of different array geometries on shock amplitudes at the focus. Nonlinear modeling of the field in water using boundary conditions reconstructed from holography measurements shows that at the same power output, the V2 array generates 10-15-MPa lower shock amplitudes at the focus. Consequently, substantially higher power levels are required for the V2 system to reach the same shock-wave exposure conditions in histotripsy-type treatments. Although this difference is mainly caused by the smaller focusing angle of the V2 array, the larger central opening of the V2 array has a nontrivial impact. By excluding coherently interacting weakly focused waves coming from the central part of the source, the presence of the central opening results in a somewhat higher effective focusing angle and thus higher shock amplitudes at the focus. Axisymmetric equivalent source models were constructed for both arrays, and the importance of including the central opening was demonstrated. These models can be used in the "HIFU beam" software for simulating nonlinear fields of the Sonalleve V1 and V2 systems in water and flat-layered biological tissues.

Dual-Mode 1-D Linear Ultrasound Array for Image-Guided Drug Delivery Enhancement Without Ultrasound Contrast Agents.

Pulsed high-intensity focused ultrasound (pHIFU) uses nonlinearly distorted millisecond-long ultrasound pulses of moderate intensity to induce inertial cavitation in tissue without administration of contrast agents. The resulting mechanical disruption permeabilizes the tissue and enhances the diffusion of systemically administered drugs. This is especially beneficial for tissues with poor perfusion such as pancreatic tumors. Here, we characterize the performance of a dual-mode ultrasound array designed for image-guided pHIFU therapies in producing inertial cavitation and ultrasound imaging. The 64-element linear array (1.071 MHz, an aperture of 14.8×51.2 mm, and a pitch of 0.8 mm) with an elevational focal length of 50 mm was driven by the Verasonics V-1 ultrasound system with extended burst option. The attainable focal pressures and electronic steering range in linear and nonlinear operating regimes (relevant to pHIFU treatments) were characterized through hydrophone measurements, acoustic holography, and numerical simulations. The steering range at ±10% from the nominal focal pressure was found to be ±6 mm axially and ±11 mm azimuthally. Focal waveforms with shock fronts of up to 45 MPa and peak negative pressures up to 9 MPa were achieved at focusing distances of 38-75 mm from the array. Cavitation behaviors induced by isolated 1-ms pHIFU pulses in optically transparent agarose gel phantoms were observed by high-speed photography across a range of excitation amplitudes and focal distances. For all focusing configurations, the appearance of sparse, stationary cavitation bubbles occurred at the same P- threshold of 2 MPa. As the output level increased, a qualitative change in cavitation behavior occurred, to pairs and sets of proliferating bubbles. The pressure P- at which this transition was observed corresponded to substantial nonlinear distortion and shock formation in the focal region and was thus dependent on the focal distance of the beam ranging within 3-4 MPa for azimuthal F -numbers of 0.74-1.5. The array was capable of B-mode imaging at 1.5 MHz of centimeter-sized targets in phantoms and in vivo pig tissues at depths of 3-7 cm, relevant to pHIFU applications in abdominal targets.

Quantitative Assessment of Boiling Histotripsy Progression Based on Color Doppler Measurements.

Boiling histotripsy (BH) is a mechanical tissue liquefaction method that uses sequences of millisecond-long high intensity focused ultrasound (HIFU) pulses with shock fronts. The BH treatment generates bubbles that move within the sonicated volume due to acoustic radiation force. Since the velocity of the bubbles and tissue debris is expected to depend on the lesion size and liquefaction completeness, it could provide a quantitative metric of the treatment progression. In this study, the motion of bubble remnants and tissue debris immediately following BH pulses was investigated using high-pulse repetition frequency (PRF) plane-wave color Doppler ultrasound in ex vivo myocardium tissue. A 256-element 1.5 MHz spiral HIFU array with a coaxially integrated ultrasound imaging probe (ATL P4-2) produced 10 ms BH pulses to form volumetric lesions with electronic beam steering. Prior to performing volumetric BH treatments, the motion of intact myocardium tissue and anticoagulated bovine blood following isolated BH pulses was assessed as two limiting cases. In the liquid blood the velocity of BH-induced streaming at the focus reached over 200 cm/s, whereas the intact tissue was observed to move toward the HIFU array consistent with elastic rebound of tissue. Over the course of volumetric BH treatments tissue motion at the focus locations was dependent on the axial size of the forming lesion relative to the corresponding size of the HIFU focal area. For axially small lesions, the maximum velocity after the BH pulse was directed toward the HIFU transducer and monotonically increased over time from about 20-100 cm/s as liquefaction progressed, then saturated when tissue was fully liquefied. For larger lesions obtained by merging multiple smaller lesions in the axial direction, the high-speed streaming away from the HIFU transducer was observed at the point of full liquefaction. Based on these observations, the maximum directional velocity and its location along the HIFU propagation axis were proposed and evaluated as candidate metrics of BH treatment completeness.

Mechanical damage thresholds for hematomas near gas-containing bodies in pulsed HIFU fields.

Objective. Boiling histotripsy (BH) is a novel high intensity focused ultrasound (HIFU) application currently being developed for non-invasive mechanical fractionation of soft tissues and large hematomas. In the context of development of BH treatment planning approaches for ablating targets adjacent to gas-containing organs, this study aimed at investigation of the ultrasound pressure thresholds of atomization-induced damage to the tissue-air interface and correlation of the danger zone dimensions with spatial structure of nonlinear HIFU field parameters.Approach. A flat interface with air of freshly clotted bovine blood was used as anex vivomodel due to its homogenous structure and higher susceptibility to ultrasound-induced mechanical damage compared to soft tissues. Three 1.5 MHz transducers of differentF-numbers (0.77, 1 and 1.5) were focused at various distances before or beyond a flat clot surface, and a BH exposure was delivered either at constant, high-amplitude output level, or at gradually increasing level until a visible damage to the clot surface occurred. The HIFU pressure field parameters at the clot surface were determined through a combination of hydrophone measurements in water, forward wave propagation simulation using 'HIFU beam' software and an image source method to account for the wave reflection from the clot surface and formation of a standing wave. The iso-levels of peak negative pressure in the resulting HIFU field were correlated to the outlines of surface erosion to identify the danger zone around the BH focus.Main results. The outline of the danger zone was shown to differ from that of a typical BH lesion produced in a volume of clot material. In the prefocal area, the zone was confined within the 4 MPa contour of the incident peak-to-peak pressure; within the main focal lobe it was determined by the maximum BH lesion width, and in the postfocal area-by the transverse size of the focal lobe and position of the first postfocal pressure axial null.Significance. The incident HIFU pressure-based danger zone boundaries were outlined around the BH focus and can be superimposed onto in-treatment ultrasound image to avoid damage to adjacent gas-containing bodies.

Robust and durable aberrative and absorptive phantom for therapeutic ultrasound applications.

Phase aberration induced by soft tissue inhomogeneities often complicates high-intensity focused ultrasound (HIFU) therapies by distorting the field and, previously, we designed and fabricated a bilayer gel phantom to reproducibly mimic that effect. A surface pattern containing size scales relevant to inhomogeneities of a porcine body wall was introduced between gel materials with fat- and muscle-like acoustic properties-ballistic and polyvinyl alcohol gels. Here, the phantom design was refined to achieve relevant values of ultrasound absorption and scattering and make it more robust, facilitating frequent handling and use in various experimental arrangements. The fidelity of the interfacial surface of the fabricated phantom to the design was confirmed by three-dimensional ultrasound imaging. The HIFU field distortions-displacement of the focus, enlargement of the focal region, and reduction of focal pressure-produced by the phantom were characterized using hydrophone measurements with a 1.5 MHz 256-element HIFU array and found to be similar to those induced by an ex vivo porcine body wall. A phase correction approach was used to mitigate the aberration effect on nonlinear focal waveforms and enable boiling histotripsy treatments through the phantom or body wall. The refined phantom represents a practical tool to explore HIFU therapy systems capabilities.

Dual-Use Transducer for Ultrasound Imaging and Pulsed Focused Ultrasound Therapy.

Pulsed focused ultrasound (pFUS) uses short acoustic pulses delivered at low duty cycle and moderate intensity to noninvasively apply mechanical stress or introduce disruption to tissue. Ultrasound-guided pFUS has primarily been used for inducing cavitation at the focus, with or without contrast agents, to promote drug delivery to tumors. When applied in tandem with contrast agents, pFUS is often administered using an ultrasound imaging probe, which has a small footprint and does not require a large acoustic window. The use of nonlinear pFUS without contrast agents was recently shown to be beneficial for localized tissue disruption, but required higher ultrasound pressure levels than a conventional ultrasound imaging probe could produce. In this work, we present the design of a compact dual-use 1-MHz transducer for ultrasound-guided pFUS without contrast agents. Nonlinear pressure fields that could be generated by the probe, under realistic power input, were simulated using the Westervelt equation. In water, fully developed shocks of 42-MPa amplitude and peak negative pressure of 8 MPa were predicted to form at the focus at 458-W acoustic power or 35% of the maximum reachable power of the transducer. In absorptive soft tissue, fully developed shocks formed at higher power (760 W or 58% of the maximum reachable power) with the shock amplitude of 33 MPa and peak negative pressure of 7.5 MPa. The electronic focus-steering capabilities of the array were evaluated and found to be sufficient to cover a target with dimensions of 19 mm in axial direction and 44 mm in transversal direction.

"HIFU Beam:" A Simulator for Predicting Axially Symmetric Nonlinear Acoustic Fields Generated by Focused Transducers in a Layered Medium.

"HIFU beam" is a freely available software tool that comprises a MATLAB toolbox combined with a user-friendly interface and binary executable compiled from FORTRAN source code (HIFU beam. (2021). Available: http://limu.msu.ru/node/3555?language=en). It is designed for simulating high-intensity focused ultrasound (HIFU) fields generated by single-element transducers and annular arrays with propagation in flat-layered media that mimic biological tissues. Numerical models incorporated in the simulator include evolution-type equations, either the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation or one-way Westervelt equation, for radially symmetric ultrasound beams in homogeneous and layered media with thermoviscous or power-law acoustic absorption. The software uses shock-capturing methods that allow for simulating strongly nonlinear acoustic fields with high-amplitude shocks. In this article, a general description of the software is given along with three representative simulation cases of ultrasound transducers and focusing conditions typical for therapeutic applications. The examples illustrate major nonlinear wave effects in HIFU fields including shock formation. Two examples simulate propagation in water, involving a single-element source (1-MHz frequency, 100-mm diameter, 90-mm radius of curvature) and a 16-element annular array (3-MHz frequency, 48-mm diameter, and 35-mm radius of curvature). The third example mimics the scenario of a HIFU treatment in a "water-muscle-kidney" layered medium using a source typical for abdominal HIFU applications (1.2-MHz frequency, 120-mm diameter, and radius of curvature). Linear, quasi-linear, and shock-wave exposure protocols are considered. It is intended that "HIFU beam" can be useful in teaching nonlinear acoustics; designing and characterizing high-power transducers; and developing exposure protocols for a wide range of therapeutic applications such as shock-based HIFU, boiling histotripsy, drug delivery, immunotherapy, and others.

A Prototype Therapy System for Boiling Histotripsy in Abdominal Targets Based on a 256-Element Spiral Array.

Boiling histotripsy (BH) uses millisecond-long ultrasound (US) pulses with high-amplitude shocks to mechanically fractionate tissue with potential for real-time lesion monitoring by US imaging. For BH treatments of abdominal organs, a high-power multielement phased array system capable of electronic focus steering and aberration correction for body wall inhomogeneities is needed. In this work, a preclinical BH system was built comprising a custom 256-element 1.5-MHz phased array (Imasonic, Besançon, France) with a central opening for mounting an imaging probe. The array was electronically matched to a Verasonics research US system with a 1.2-kW external power source. Driving electronics and software of the system were modified to provide a pulse average acoustic power of 2.2 kW sustained for 10 ms with a 1-2-Hz repetition rate for delivering BH exposures. System performance was characterized by hydrophone measurements in water combined with nonlinear wave simulations based on the Westervelt equation. Fully developed shocks of 100-MPa amplitude are formed at the focus at 275-W acoustic power. Electronic steering capabilities of the array were evaluated for shock-producing conditions to determine power compensation strategies that equalize BH exposures at multiple focal locations across the planned treatment volume. The system was used to produce continuous volumetric BH lesions in ex vivo bovine liver with 1-mm focus spacing, 10-ms pulselength, five pulses/focus, and 1% duty cycle.

Bilayer aberration-inducing gel phantom for high intensity focused ultrasound applications.

Aberrations induced by soft tissue inhomogeneities often complicate high-intensity focused ultrasound (HIFU) therapies. In this work, a bilayer phantom made from polyvinyl alcohol hydrogel and ballistic gel was built to mimic alternating layers of water-based and lipid tissues characteristic of an abdominal body wall and to reproducibly distort HIFU fields. The density, sound speed, and attenuation coefficient of each material were measured using a homogeneous gel layer. A surface with random topographical features was designed as an interface between gel layers using a 2D Fourier spectrum approach and replicating different spatial scales of tissue inhomogeneities. Distortion of the field of a 256-element 1.5 MHz HIFU array by the phantom was characterized through hydrophone measurements for linear and nonlinear beam focusing and compared to the corresponding distortion induced by an ex vivo porcine body wall of the same thickness. Both spatial shift and widening of the focal lobe were observed, as well as dramatic reduction in focal pressures caused by aberrations. The results suggest that the phantom produced levels of aberration that are similar to a real body wall and can serve as a research tool for studying HIFU effects as well as for developing algorithms for aberration correction.

Effect of surface roughness on nonlinear reflection of weak shock waves.

The authors have recently shown that irregular reflections of spark-generated pressure weak shocks from a smooth rigid surface can be studied using an optical interferometer [Karzova, Lechat, Ollivier, Dragna, Yuldashev, Khokhlova, and Blanc-Benon, J. Acoust. Soc. Am. 145(1), 26-35 (2019)]. The current study extends these results to the reflection from rough surfaces. A Mach-Zehnder interferometer is used to measure pressure waveforms. Simulations are based on the solution of axisymmetric Euler equations. It is shown that roughness causes a decrease of the Mach stem height and the appearance of oscillations in the pressure waveforms. Close to rough surfaces, the pressure was higher compared to the smooth surface.

Simulation of nonlinear trans-skull focusing and formation of shocks in brain using a fully populated ultrasound array with aberration correction.

Multi-element high-intensity focused ultrasound phased arrays in the shape of hemispheres are currently used in clinics for thermal lesioning in deep brain structures. Certain side effects of overheating non-targeted tissues and skull bones have been revealed. Here, an approach is developed to mitigate these effects. A specific design of a fully populated 256-element 1-MHz array shaped as a spherical segment (F-number, F# = 1) and filled by randomly distributed equal-area polygonal elements is proposed. Capability of the array to generate high-amplitude shock fronts at the focus is tested in simulations by combining three numerical algorithms for linear and nonlinear field modeling and aberration correction. The algorithms are based on the combination of the Rayleigh integral, a linear pseudo-spectral time domain Kelvin-Voigt model, and nonlinear Westervelt model to account for the effects of inhomogeneities, aberrations, reflections, absorption, nonlinearity, and shear waves in the skull. It is shown that the proposed array can generate nonlinear waveforms with shock amplitudes >60 MPa at the focus deep inside the brain without exceeding the existing technical limitation on the intensity of 40 W/cm2 at the array elements. Such shock amplitudes are sufficient for mechanical ablation of brain tissues using the boiling histotripsy approach and implementation of other shock-based therapies.

Irregular reflection of spark-generated shock pulses from a rigid surface: Mach-Zehnder interferometry measurements in air.

The irregular reflection of weak acoustic shock waves, known as the von Neumann reflection, has been observed experimentally and numerically for spherically diverging waves generated by an electric spark source. Two optical measurement methods are used: a Mach-Zehnder interferometer for measuring pressure waveforms and a Schlieren system for visualizing shock fronts. Pressure waveforms are reconstructed from the light phase difference measured by the interferometer using the inverse Abel transform. In numerical simulations, the axisymmetric Euler equations are solved using finite-difference time-domain methods and the spark source is modeled as an instantaneous energy injection with a Gaussian shape. Waveforms and reflection patterns obtained from the simulations are in good agreement with those measured by the interferometer and the Schlieren methods. The Mach stem formation is observed close to the surface for incident pressures within the range of 800 to 4000 Pa. Similarly, as for strong shocks generated by blasts, it is found that for spherical weak shocks the Mach stem length increases with distance following a parabolic law. This study confirms the occurrence of irregular reflections at acoustic pressure levels and demonstrates the benefits of the Mach-Zehnder interferometer method when microphone measurements cannot be applied.

A Prototype Therapy System for Transcutaneous Application of Boiling Histotripsy.

Boiling histotripsy (BH) is a method of focused ultrasound surgery that noninvasively applies millisecond-length pulses with high-amplitude shock fronts to generate liquefied lesions in tissue. Such a technique requires unique outputs compared to a focused ultrasound thermal therapy apparatus, particularly to achieve high in situ pressure levels through intervening tissue. This paper describes the design and characterization of a system capable of producing the necessary pressure to transcutaneously administer BH therapy through clinically relevant overlying tissue paths using pulses with duration up to 10 ms. A high-voltage electronic pulser was constructed to drive a 1-MHz focused ultrasound transducer to produce shock waves with amplitude capable of generating boiling within the pulse duration in tissue. The system output was characterized by numerical modeling with the 3-D Westervelt equation using boundary conditions established by acoustic holography measurements of the source field. Such simulations were found to be in agreement with directly measured focal waveforms. An existing derating method for nonlinear therapeutic fields was used to estimate in situ pressure levels at different tissue depths. The system was tested in ex vivo bovine liver samples to create BH lesions at depths up to 7 cm. Lesions were also created through excised porcine body wall (skin, adipose, and muscle) with 3-5 cm thickness. These results indicate that the system is capable of producing the necessary output for transcutaneous ablation with BH.

Shock formation and nonlinear saturation effects in the ultrasound field of a diagnostic curvilinear probe.

Newer imaging and therapeutic ultrasound technologies may benefit from in situ pressure levels higher than conventional diagnostic ultrasound. One example is the recently developed use of 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 has been used to deliver the acoustic pushing pulses. The probe is a curvilinear array comprising 128 elements equally spaced along a convex cylindrical surface. The effectiveness of the treatment can be increased by using higher transducer output to provide a 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 three-dimensional Westervelt equation with the boundary condition set to match low power measurements of the acoustic pressure field. Nonlinear focal waveforms simulated for different numbers of operating elements of the array at several output power levels were compared to fiber-optic hydrophone measurements and were found to be in good agreement. It was shown that saturation effects do limit the acoustic pressure in the focal region of a diagnostic imaging probe.

Reconstruction of nonlinear ultrasound field of an annular therapeutic array from acoustic holograms of its individual elements.

Acoustic holography method has been shown to provide accurate reconstruction of 3D ultrasound fields generated by various medical transducers including multi-element arrays as well as to set a boundary condition for nonlinear field modeling at high pressure levels. Here an approach of measuring holograms of individual array elements for modeling of an entire array field is proposed and tested for a 3 MHz 16-element annular array (48 mm diameter and 35 mm radius of curvature). The array is a part of a high intensity focused ultrasound system with magnetic resonance guidance used for developing thermal and mechanical methods of tissue ablation in mouse tumors. The holograms measured separately for each array element were combined together to obtain a boundary condition for the array with all operating elements. Modeling results were compared to low-amplitude beam scans and good agreement was demonstrated. Then, nonlinear field simulations were performed at increasing power outputs based on the 3D Westervelt equation. It was shown that the transducer is capable to produce focal waveforms with 140 MPa shock amplitude at 110 W acoustic power and thus is well suited for evaluating shock-based ablation therapies in small animals.

Statistics of peak overpressure and shock steepness for linear and nonlinear N-wave propagation in a kinematic turbulence.

Linear and nonlinear propagation of high amplitude acoustic pulses through a turbulent layer in air is investigated using a two-dimensional KZK-type (Khokhlov-Zabolotskaya-Kuznetsov) equation. Initial waves are symmetrical N-waves with shock fronts of finite width. A modified von Kármán spectrum model is used to generate random wind velocity fluctuations associated with the turbulence. Physical parameters in simulations correspond to previous laboratory scale experiments where N-waves with 1.4 cm wavelength propagated through a turbulence layer with the outer scale of about 16 cm. Mean value and standard deviation of peak overpressure and shock steepness, as well as cumulative probabilities to observe amplified peak overpressure and shock steepness, are analyzed. Nonlinear propagation effects are shown to enhance pressure level in random foci for moderate initial amplitudes of N-waves thus increasing the probability to observe highly peaked waveforms. Saturation of the pressure level is observed for stronger nonlinear effects. It is shown that in the linear propagation regime, the turbulence mainly leads to the smearing of shock fronts, thus decreasing the probability to observe high values of steepness, whereas nonlinear effects dramatically increase the probability to observe steep shocks.

Design of HIFU Transducers for Generating Specified Nonlinear Ultrasound Fields.

Various clinical applications of high-intensity focused ultrasound have different requirements for the pressure levels and degree of nonlinear waveform distortion at the focus. The goal of this paper is to determine transducer design parameters that produce either a specified shock amplitude in the focal waveform or specified peak pressures while still maintaining quasi-linear conditions at the focus. Multiparametric nonlinear modeling based on the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation with an equivalent source boundary condition was employed. Peak pressures, shock amplitudes at the focus, and corresponding source outputs were determined for different transducer geometries and levels of nonlinear distortion. The results are presented in terms of the parameters of an equivalent single-element spherically shaped transducer. The accuracy of the method and its applicability to cases of strongly focused transducers were validated by comparing the KZK modeling data with measurements and nonlinear full diffraction simulations for a single-element source and arrays with 7 and 256 elements. The results provide look-up data for evaluating nonlinear distortions at the focus of existing therapeutic systems as well as for guiding the design of new transducers that generate specified nonlinear fields.

Design of HIFU transducers to generate specific nonlinear ultrasound fields.

Various clinical applications of high intensity focused ultrasound (HIFU) have different requirements on the pressure level and degree of nonlinear waveform distortion at the focus. Applications that utilize nonlinear waves with developed shocks are of growing interest, for example, for mechanical disintegration as well as for accelerated thermal ablation of tissue. In this work, an inverse problem of determining transducer parameters to enable formation of shocks with desired amplitude at the focus is solved. The solution was obtained by performing multiple direct simulations of the parabolic Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation for various parameters of the source. It is shown that results obtained within the parabolic approximation can be used to describe the focal region of single element spherical sources as well as complex transducer arrays. It is also demonstrated that the focal pressure level at which fully developed shocks are formed mainly depends on the focusing angle of the source and only slightly depends on its aperture and operating frequency. Using the simulation results, a 256-element HIFU array operating at 1.5 MHz frequency was designed for a specific application of boiling-histotripsy that relies on the presence of 90-100 MPa shocks at the focus. The size of the array elements and focusing angle of the array were chosen to satisfy technical limitations on the intensity at the array elements and desired shock amplitudes in the focal waveform. Focus steering capabilities of the array were analysed using an open-source T-Array software developed at Moscow State University.

Characterization of spark-generated N-waves in air using an optical schlieren method.

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.