Simulating ultrasonic pulse echo registration including multiple scattering, attenuation and nonlinearity

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 contrast-enhanced echography, the simulation of nonlinear propagation of ultrasound through a population of oscillating microbubbles imposes a computational challenge. Also, the numerical complexity increases because each scatterer has individual properties. To address these problems, the Iterative Nonlinear Contrast Source (INCS) method has been extended to include a large population of nonlinearly responding microbubbles. The original INCS method solves the Westervelt equation in a four-dimensional spatiotemporal domain by generating increasingly accurate field corrections to iteratively update the acoustic pressure. The field corrections are computed by the convolution of a nonlinear contrast source with the Green's function of the linear background medium. Because the convolution integral allows a coarse discretization, INCS can efficiently deal with large-scale problems. To include a population of microbubbles, these are considered as individual contrast point sources with their own nonlinear response. The field corrections are computed as before, but now, in each iteration, the temporal signature of each contrast point source is computed by solving the bubble's Marmottant equation. Physically, each iteration adds an order of multiple scattering. Here, the performance of the extended INCS method and the significance of multiple scattering is demonstrated through various results from different configurations.

The refraction and scattering of nonlinear acoustic waves play an important role in the realistic application of medical ultrasound. One cause of these effects is the tissue dependence of the nonlinear medium behavior. A method that is able to model those effects is essential for the design of transducers for novel ultrasound modalities. Starting from the Westervelt equation, nonlinear pressure wave fields can be modeled via a contrast source formulation, as has been done with the INCS method. An extension of this method will be presented that can handle inhomogeneities in the coefficient of nonlinearity. The contrast source formulation results in an integral equation, which is solved iteratively using a Neumann scheme. The convergence of this scheme has been investigatedfor relevant media (e.g., blood, brain, and liver). Further, as an example, the method has been applied to compute the 1D nonlinear acoustic wave field in an inhomogeneous medium insonified by a 1 MHz Gaussian pulse propagatingup to 100 mm. The results show that the method is able to predict the propagation and the scattering effects of nonlinear acoustic waves in media with inhomogeneities in the coefficient of nonlinearity. This motivates a similar extension ofthe 3D INCS method.

The refraction and scattering of nonlinear acoustic waves play an important role in the realistic application of medical ultrasound. One cause of these effects is the tissue dependence of the nonlinear medium behavior. A method that is able to model those effects is essential for the design of transducers for novel ultrasound modalities. Starting from the Westervelt equation, nonlinear pressure wave fields can be modeled via a contrast source formulation, as has been done with the INCS method. An extension of this method will be presented that can handle inhomogeneities in the coefficient of nonlinearity. The contrast source formulation results in an integral equation, which is solved iteratively using a Neumann scheme. The convergence of this scheme has been investigated for relevant media (e.g., blood, brain, and liver). Further, as an example, the method has been applied to compute the 1-D nonlinear acoustic wave field in an inhomogeneous medium insonified by a 1 MHz Gaussian pulse propagating up to 100 mm. The results show that the method is able to predict the propagation and the scattering effects of nonlinear acoustic waves in media with inhomogeneities in the coefficient of nonlinearity. This motivates a similar extension of the 3-D INCS method.

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.

Abstract. A numerical model for long-term simulation of gravity surface waves is described. The model is designed as a component of a coupled Wave Boundary Layer/Sea Waves model, for investigation of small-scale dynamic and thermodynamic interactions between the ocean and atmosphere. Statistical properties of nonlinear wave fields are investigated on a basis of direct hydrodynamical modeling of 1-D potential periodic surface waves. The method is based on a nonstationary conformal surface-following coordinate transformation; this approach reduces the principal equations of potential waves to two simple evolutionary equations for the elevation and the velocity potential on the surface. The numerical scheme is based on a Fourier transform method. High accuracy was confirmed by validation of the nonstationary model against known solutions, and by comparison between the results obtained with different resolutions in the horizontal. The scheme allows reproduction of the propagation of steep Stokes waves for thousands of periods with very high accuracy. The method here developed is applied to simulation of the evolution of wave fields with large number of modes for many periods of dominant waves. The statistical characteristics of nonlinear wave fields for waves of different steepness were investigated: spectra, curtosis and skewness, dispersion relation, life time. The prime result is that wave field may be presented as a superposition of linear waves is valid only for small amplitudes. It is shown as well, that nonlinear wave fields are rather a superposition of Stokes waves not linear waves. Potential flow, free surface, conformal mapping, numerical modeling of waves, gravity waves, Stokes waves, breaking waves, freak waves, wind-wave interaction.

In this paper the nonlinear pulsed acoustic pressure fields from the focused square and rectangular apertures are considered. Experimental results in water of the 4D nonlinear sound field radiated from a 2.8 MHz focused (focal distance F = 80 mm) transducer of a square (20 x 20 mm) and rectangular (10 x 25 mm) geometry for various excitation levels (producing the average pressure P0 at the source equal to 0.045, 0.225 and 0.45 MPa) are presented. The measurement results are compared with the numerical calculations obtained for the same boundary conditions. The novel, developed by the second author, numerical algorithm was used to simulate the 4D nonlinear pressure field propagation from nonaxisymmetric focused acoustic sources radiating pulsed waves in the attenuating media. Our theoretical model is based on the Time-Averaged Pressure Envelope (TAPE) method, recently developed in our lab. This method employs the representation of the propagated pulsed pressure waveform as a quasi-Fourier series, i.e. as a superposition of sinusoidal wavelet- like packets with carrier frequencies equal to harmonics of the initial tone burst, and with envelopes (being functions of space coordinates and time) determined by the TAPE method. The spectrum of the propagating pressure waveform is considered to be a sum of contributions of each wavelet spectrum. Our model uses the second-order operator splitting approach with an incremental propagation scheme whereby the effects of combined diffraction and absorption are computed separately from the effects of nonlinear harmonic interactions over incremental steps. The proposed model is free from paraxial approximation, is computationally efficient and capable of predicting the 4D ultrasound field in nonlinear and attenuating media with the arbitrary absorption from pulsed, arbitrary shaped, plane and focused sources (including phased arrays with an angular beam deflection). Using a computational power of a standard PC a computation time required for the full 4D nonlinear field simulation depends on the source dimensions, radiated frequency and excitation level as well as on the medium absorption and nonlinearity strength and can vary from few minutes to few hours. Comparison between experimental and simulation results clearly show that our numerical model predicts well the structure of the nonlinear ultrasound field for all boundary conditions considered. I. INTRODUCTION The theoretical and experimental studies of the finite amplitude acoustic waves propagation in attenuating media from nonaxisymmetric sources rather rarely can be found in literature in spite of the fact that probes of the rectangular geometry (such as linear phased arrays) are commonly used in clinical practice for medical ultrasonic imaging purposes. The main reason of such situation is a lack in simpler theoretical models and in computationally efficient numerical algorithms that are able to predict accurately the nonlinear effects in 4D ultrasound fields from pulsed, arbitrarily shaped sources (plane and focused) in biological media with arbitrary frequency- dependent absorption. In recent years the only study describing the computationally efficient numerical model that is able to simulate accurately the 4D nonlinear ultrasound field in water and in biological tissues from pulsed nonaxisymmetric sources was developed by Zemp et al. (1). Their model is based on the second order operator-splitting method, proposed by Tavakkoli et al., with the modified fractional step scheme whereby the combined effects of diffraction and absorption are accounted for over half-steps and the effects of nonlinear harmonic interactions over full incremental steps. The computation of diffraction and absorption sub-steps was based on the angular spectrum technique with modified sampling method (to obtain computational savings due to larger axial propagation steps) while the computation of nonlinear steps was based on the time-domain solution to Burgers' equation. There are not reports yet describing an experimental confirmation of an agreement between the simulated nonlinear acoustic pulsed fields in water or in soft tissues from nonaxisymmetric focused sources (obtained by using the numerical model proposed) and nonlinear field from realistic probes. In this work the experimental measurement results of the 4D nonlinear field from the both square and rectangular ultrasonic focused transducer radiating the pulsed pressure wave in water are presented. The realistic beam patterns are compared with the simulation results obtained by using our novel numerical algorithm.

Use of the Acoustic Shadow Width to Determine Kidney Stone Size with Ultrasound

On the KZK-type equation for modulated ultrasound fields

In the field of nonlinear waves, there exist two kinds of waves, i.e., one is Non-Coupled Simple Wave (nonlinear P wave) and Coupled Simple Wave (nonlinear S wave). In this paper, the problem is limitted to the two-dimensional case. After numerical computations, we have then found that the behavior of waves in the nonlinear wave field is governed by the theory of Simple Waves. Even complex wave form can be analyzed as composite Simple Waves.

In this work, the three-component coupled nonlinear Schrödinger (tc-CNLS) equation is systemically investigated. By using the Darboux transformation, the new breather wave and rogue wave solutions of the tc-CNLS equation are constructed. These solutions exhibit breather waves and rogue waves on a multi-soliton background. Furthermore, the dynamic behaviors of these solutions are analyzed with some graphics. Our results can be of much importance in enriching and predicting rogue wave phenomena arising in nonlinear wave fields.

The dispersionless Whitham modulation equations in one space dimension and time are generically hyperbolic or elliptic, and breakdown at the transition, which is a curve in the frequency-wavenumber plane. In this paper, the modulation theory is reformulated with a slow phase and different scalings resulting in a phase modulation equation near the singular curves which is a geometric form of the two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multiperiodic, quasiperiodic and multi-pulse localized solutions. This theory shows that the elliptic-hyperbolic transition is a rich source of complex behaviour in nonlinear wave fields. There are several examples of these transition curves in the literature to which the theory applies. For illustration the theory is applied to the complex nonlinear Klein-Gordon equation which has two singular curves in the manifold of periodic travelling waves.

Solitary wave solutions to the nonlinear evolution equations have recently attracted widespread interest in engineering and physical sciences. In this work, we investigate the fractional generalised nonlinear Pochhammer–Chree equation under the power-law of nonlinearity with order m. This equation is used to describe longitudinal deformation wave propagation in an elastic rod. In this study, we have secured a variety of exact solitary wave solutions by the assistance of the recently developed technique known as modified generalized exponential rational function method. Exact solutions of various categories, such as bright-dark, bright, mixed, singular, dark, complex, and combined solitons, are extracted. The applied approach is highly efficient and has a significant computational capability to efficiently tackle the solutions with a high degree of accuracy in nonlinear systems. To analyze the governing system, the equation under investigation is converted to an ordinary differential equation through the application of a suitable wave transformation with a \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-derivative. In addition to illustrate the behavior of the solution at various parameter values, we generate 2D and 3D graphs that incorporate pertinent parameters. Moreover, the Galilean transformation is employed to investigate the sensitivity analysis. This research’s results have the potential to enhance comprehension of the nonlinear dynamic characteristics displayed by the defined system and to verify the efficacy of the strategies that have been implemented. The results obtained are a substantial contribution to the comprehension of nonlinear science and nonlinear wave fields that are associated with higher dimensions.

Phase-resolved wave simulation and direct numerical simulation of turbulence are performed to investigate the surface wave effects on the energy transfer in overlying turbulent flow. The JONSWAP spectrum is used to initialize a broadband wave field. The nonlinear wave field is simulated using a high-order spectral method, and the resultant wave surface provides the bottom boundary conditions for direct numerical simulation of the overlying turbulent flow. Two wave ages of and 25 are considered, corresponding to slow and fast wave fields, respectively, where denotes the celerity of the peak wave and denotes the friction velocity. The energy transfer of turbulent motions in the presence of surface waves is investigated through the spectral analysis of the two-point correlation transport equation. It is found that the production term has an extra peak at the dominant wavelength scale in the vicinity of the surface, and the energy transported to the surface via viscous and spatial turbulent transport is enhanced in the region of . The presence of surface waves results in an inverse turbulent energy cascade in the near-surface region, where small-scale wave-related motions transfer energy back to the dominant wavelength scale. Pressure-related terms reflecting the spatial and inter-component energy transfer are strongly dependent on the wave age. Furthermore, triadic interaction analysis reveals that the energy influx at the dominant wavelength scale is due to the contribution of the neighbouring streamwise turbulent motions, and those at the harmonic wavelength scales contribute the most.

The modulation instability (MI) is a universal mechanism that is responsible for the disintegration of weakly nonlinear narrow-banded wave fields and the emergence of localized extreme events in dispersive media. The instability dynamics is naturally triggered, when unstable energy sidebands located around the main energy peak are excited and then follow an exponential growth law. As a consequence of four wave mixing effect, these primary sidebands generate an infinite number of additional sidebands, forming a triangular sideband cascade. After saturation, it is expected that the system experiences a return to initial conditions followed by a spectral recurrence dynamics. Much complex nonlinear wave field motion is expected, when the secondary or successive sideband pair that is created is also located in the finite instability gain range around the main carrier frequency peak. This latter process is referred to as higher-order MI. We report a numerical and experimental study that confirms observation of higher-order MI dynamics in water waves. Furthermore, we show that the presence of weak dissipation may counterintuitively enhance wave focusing in the second recurrent cycle of wave amplification. The interdisciplinary weakly nonlinear approach in addressing the evolution of unstable nonlinear waves dynamics may find significant resonance in other nonlinear dispersive media in physics, such as optics, solids, superfluids, and plasma.