Nonlinear pulsed pressure field from focused rectangular apertures: experimental and numerical simulation results

The popular KZK equation has been widely used to model the nonlinear acoustic wave field. However, since it is based on a parabolic approximation, it has various limitations, both at close distances from the transducer and at large radial extents from the beam axis or, in case of phased arrays, at steering angles larger than, say, 20/spl deg/. The purpose of this work is to model the nonlinear acoustic wave field without using the parabolic approximation. The model is based on the full nonlinear wave equation known as the Westervelt equation. This equation is solved numerically using an explicit finite difference scheme with fourth order centred space differences and second order centered or second order backward time differences. To check the validity of our model, numerical results for the lossless linear case were compared to analytical results. Moreover, the nonlinear beam profiles were compared to the results of a numerical solution of the KZK equation and to hydrophone measurements performed with a single element, unfocused transducer with a radius of 6.35 mm and a centre frequency of 1 MHz. We used a pulsed excitation of 6 cycles with pressure levels of 40, 220 and 500 kPa. The axial beam profiles and the lateral beam profiles at several distances were measured. For a low drive level, our numerical predictions show a good agreement with the analytic solution as well as with the measured data for all fundamental frequency beam profiles. At higher excitations, our simulations give a significant second harmonic component at 15 db below the fundamental level, which is in agreement with the measurements. The numerical data reproduces all the features at the fundamental and harmonic frequency beam profiles. For regions where the KZK equation is known to be less accurate, our model performs better. The sidelobes created at larger observation angles (above 20/spl deg/) and the nearfield variations are accurately reproduced. The performance of our model is good since it predicts accurately the harmonic field at any position with respect to the transducer. It can be used for the whole scanning sector that is required for harmonic beam profile simulations for phased arrays.

The Khokhlov - Zabolotskaya - Kuznetsov (KZK) equation is a parabolic approximation to the Westervelt equation that models the effects of diffraction, attenuation, and nonlinearity. Although the KZK equation is only valid in the far field of the paraxial region for mildly focused or unfocused transducers, the KZK equation is widely applied in medical ultrasound simulations. For a continuous wave input, the KZK equation is effectively modeled by the Bergen Code [J. Berntsen, Numerical Calculations of Finite Amplitude Sound Beams, in M. F. Hamilton and D. T. Blackstock, editors, Frontiers of Nonlinear Acoustics: Proceedings of 12th ISNA, Elsevier, 1990], which is a finite difference model that utilizes operator splitting. Similar C++ routines have been developed for FOCUS, the ‘Fast Object-Oriented C++ Ultrasound Simulator’ (http://www.egr.msu.edu/∼fultras-web) to calculate nonlinear pressure fields generated by axisymmetric flat circular and spherically focused ultrasound transducers. This new routine complements an existing FOCUS program that models nonlinear ultrasound propagation with the angular spectrum approach [P. T. Christopher and K. J. Parker, J. Acoust. Soc. Am. 90, 488–499 (1991)]. Results obtained from these two nonlinear ultrasound simulation approaches are evaluated and compared for continuous wave linear simulations. The simulation results match closely in the farfield of the paraxial region, but the results differ in the nearfield. The nonlinear pressure field generated by a spherically focused transducer with a peak surface pressure of 0.2MPa radiating in a lossy medium with β = 3.5 is simulated, and the computation times are also evaluated. The nonlinear simulation results demonstrate acceptable agreement in the focal zone. These two related nonlinear simulation approaches are now included with FOCUS to enable convenient simulations of nonlinear pressure fields on desktop and laptop computers.

Many bio-medical applications entail the problems of spatially manipulating microbubbles by means of high-intensity focused ultrasound (HIFU) pressure field. This paper investigates the nonlinear coupling between radial pulsations, axisymmetric modes of shape oscillations and translational motion of a single spherical gas bubble in water domain when it is subjected to a focused nonlinear acoustic pressure. A multiphysics mathematical model is investigated to obtain nonlinear acoustic field from a focused transducer and to account for bubble oscillations in the resultant HIFU pressure field. The effects of acoustic nonlinearity on the bubble dynamics are examined to determine the instability of the mode shapes of a bubble, which is contributing to form the translational instability, known as dancing motion. Furthermore, acoustic streaming effects caused by radial pulsations of a microbubble in the surrounding water domain are reported.

Eighth international symposium on nonlinear acoustics

High intensity focused ultrasound (HIFU) has been proven to be promising in non-invasive therapies, in which precise prediction of the focused ultrasound field is crucial for its accurate and safe application. Although the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation has been widely used in the calculation of the nonlinear acoustic field of HIFU, some deviations still exist when it comes to dispersive medium. This problem also exists as an obstacle to the Westervelt model and the Spherical Beam Equation. Considering that the KZK equation is the most prevalent model in HIFU applications due to its accurate and simple simulation algorithms, there is an urgent need to improve its performance in dispersive medium. In this work, a modified KZK (mKZK) equation derived from a fractional order derivative is proposed to calculate the nonlinear acoustic field in a dispersive medium. By correcting the power index in the attenuation term, this model is capable of providing improved prediction accuracy, especially in the axial position of the focal area. Simulation results using the obtained model were further compared with the experimental results from a gel phantom. Good agreements were found, indicating the applicability of the proposed model. The findings of this work will be helpful in making more accurate treatment plans for HIFU therapies, as well as facilitating the application of ultrasound in acoustic hyperthermia therapy.

Numerical modeling of nonlinear (NL) ultrasound (US) beam propagation plays a key role in designing state-of-art medical US systems due to the improvements in image quality coming from the use of second harmonic overtones. Describing the combined effects of diffraction, absorption and nonlinearity, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation is the most commonly used model to address this simulation problem. This work introduces an algorithm capable of providing good estimations of pressure profiles at arbitrary depth without requiring stepping techniques. Our proposal consists in a perturbative approach to the solution of the KZK equation cast in a discrete Fourier-Bessel domain applicable to axisymmetric propagation from circular transducers. The resulting algorithm is capable of simulating NL US fields with significant computational savings and an accuracy at least comparable to other standard approaches to the same problem. Comparing our algorithm to the output of a publicly available NL KZK solver, significant cost reductions and performance improvements were recorded.

Biomedical tissues usually show inhomogeneity in their acoustic medium parameters. These inhomogeneities cause refraction and scattering of diagnostic and therapeutic ultrasound waves. A method that is able to model the effects of inhomogeneity in the attenuation and in the nonlinearity is essential for the design of transducers for new ultrasound modalities and the development of novel ultrasound applications. The Iterative Nonlinear Contrast Source (INCS) method has originally been designed for the accurate modeling of nonlinear acoustic wave fields in homogeneous media. It considers the nonlinear term from the Westervelt equation as a distributed contrast source, and the corresponding integral equation is solved using an iterative Neumann scheme. This paper presents an extension of the INCS method that can handle inhomogeneity in the attenuation and in the coefficient of nonlinearity. Results are presented for the one-dimensional case. These show that in this case the presented method correctly predicts the effects related to nonlinear propagation and scattering by inhomogeneities in the attenuation and the coefficient of nonlinearity.

In order to study the pressure pulsation characteristics of a centrifugal pump with a guide vane, a pump with specific speed ns= 190 was chosen as the research object. The numerical simulation was carried out using the standard k-ε turbulence model. The pressure pulsation characteristic was researched at a set monitoring point. The reliability of the numerical simulation was verified with experiment results. The pressure pulsation characteristic curves of each calculation domain were obtained by dimensionless data processing. The results elucidate that the rotor/stator interaction at the outlet of the impeller is the main source for the pressure pulsation. With increasing distance, the pressure pulsation gradually decreases; the pressure pulsation amplitude of the pressure surface is larger than the suction surface; the pressure pulsation in the flow channel of the guide vane remains the same; the pressure fluctuation near the outlet of the annular volute chamber is larger, and the pressure pulsation near the upstream of the flow channel is quite complicated. The pressure pulsation at the annular volute and axial symmetry is the same.

We have numerically investigated nonlinear nonstationary velocity and pressure fields in a viscous incompressible fluid near an oscillating absolutely rigid plate of an infinitely small thickness and a limited length in the direction of oscillations. It is shown that the limitation imposed on the plate size along the direction of its oscillations leads to nonlinearity of the problem even for a small amplitude of surface oscillations of the plate. We have analyzed the velocity and pressure fields for large amplitudes of plate oscillations. The emergence of displacement flows during oscillations of an infinitely thin plate of a finite length is explained. The mechanism of formation of instantaneous vortex motions in the viscous fluid is investigated.

The advantages of nonlinear ultrasound propagation are exploited in an ever increasing number of applications. As a consequence, much research is devoted to the development of numerical methods capable of modeling nonlinear pressure wave fields propagating through (inhomogeneous) biomedical tissue. These tools are essential to design and optimize ultrasound transducers and to investigate novel ultrasound modalities or devices. The iterative nonlinear contrast source (INCS) method is an accurate full-wave method for modeling three-dimensional nonlinear acoustic wave fields propagating through soft biomedical tissue containing arbitrary inhomogeneities in the coefficient of nonlinearity, attenuation and speed of sound. This method is directionally independent and can deal with large three-dimensional domains, measuring hundreds of wavelengths and periods, and scattering. Here, we present the first synthetic harmonic ultrasound images generated with INCS. The images are created using a delay-and-sum aperture focusing scheme. Results show the applicability of the INCS method as a simulator for ultrasound harmonic imaging systems.

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.

As the unique rotating component, the impeller is the core component of a nuclear reactor coolant pump (RCP), the dynamic properties of the impeller are critical for the safe operation of the whole reactor. The purpose of this study was to shed comprehensive light on the pressure pulsation and modal properties of a scaled RCP impeller via experimental and numerical methods. The numerical model was validated by an experiment connecting the pressure pulsation signals at the diffuser inlet, and a good agreement was obtained between the numerical and experimental results. Pressure pulsation acting on the impeller's blade is mainly dominated by the impeller rotating frequency, the vane passing frequency, and the double blade‐passing frequency, and the pressure pulsation acting on the blade's pressure surface is more intense than on the suction surface. The modal properties were obtained via the modal test and numerical methods with the impeller suspended as a free body in the air and submerged inside water. The reduction in the impeller natural frequencies was between 31.63% and 37.77% for the corresponding mode shape due to the added mass effect of the fluid. Based on the pressure pulsation characteristics acting on the impeller and the natural frequency of the impeller, it is considered there is no risk of resonance in the impeller. Finally, it is expected that the present work can provide scientific guidance to avoid hydraulic resonance in nuclear reactor coolant pumps.

The reachability problem is one of the most important issues in the verification of hybrid systems. But unfortunately the reachable sets for most of hybrid systems are not computable. In the literature, only some special families of linear vector fields are proved with decidable reachability problem, let alone nonlinear ones. In this paper, we investigate the reachability problem of nonlinear vector fields by identifying three families of nonlinear vector fields with solvability and prove that their reachability problems are decidable. An $n$ -dimension dynamical system is called solvable if its state variables can be partitioned into $m$ groups such that the derivatives of the variables in the $i$ th group are linear in themselves, but possibly nonlinear in the variables from the 1st to $i-1$ th groups. The three families of nonlinear solvable vector fields under consideration are: the matrices corresponding to the linear parts of any vector field in the first family are nilpotent; the matrices corresponding to the linear parts of any vector in the second family are only with real eigenvalues; the matrices corresponding to the linear parts of any vector field in the third family are only with pure imaginary eigenvalues. The experimental results indicate the efficiency of our approach.

Frequency response of pressure pulsations and source identification in a suction manifold

Focused ultrasound therapy has been widely used for the treatment of various diseases, employing different types of transducers. The focused ultrasound pressure fields inevitably exhibit nonlinear effects, which can influence the ablation region. However, the nonlinear effects exhibit noticeable variations across different applications. The characterization of the nonlinear pressure fields of ultrasound is important for the effective implementation of focused ultrasound therapy. The traditional angular spectrum method (ASM) was extended to accurately and efficiently simulate the propagation of weakly nonlinear ultrasound in heterogeneous mediums of clinical model. The nonlinear effects were further analyzed in relationship to the transducer parameters that are different in various applications. The pressure fields were simulated using the extended ASM, incorporating calculations for phase aberration in the frequency domain and magnitude compensation in the spatial domain to account for heterogeneous acoustic impedance mismatch. Validation was performed by comparison to k-Wave simulation results using two simplified clinical models, an abdominal soft tissue and a transcranial skull model. The nonlinear effects were then analyzed in relation to the transducer parameters of f-number and effective source area based on the same acoustic output power. The analysis of nonlinear effects was conducted under both homogeneous medium and the clinical models. The simulation results demonstrated a maximum error of 3.93% in the calculated harmonic pressure of the abdominal model, and a maximum error of 4.89% within the transcranial model when comparing the extended ASM simulation results to those obtained from k-Wave simulations. The characterization of the nonlinear effects reveals a strong correlation with the transducer parameters. Specifically, the results indicate that the nonlinear effects intensify with an increase in the effective source area and f-number, under the same acoustic output power of the transducer. However, the clinical model also showed an influence on the nonlinear effects in relation to the f-number. The extended ASM was demonstrated as an accurate and efficient simulation tool, and the simulation results provide a reference for evaluating the intensity of nonlinear effects in various transducer designs.