In order to predict the pulsed, nonlinear ultrasound field from medical phased array transducers, a numerical model has been developed that computes this field in a large-scale, three-dimensional configuration. The model is free of an assumed (quasi-plane) directionality in both the diffraction and nonlinear distortion, and it is solved with a Green's function method, which through proper filtering is efficiently and accurately evaluated with a discrization of two points per wavelength. In the method, the nonlinear terms in the wave equation are treated as constrast source terms, which are iteratively estimated by solving the associated linear wave problem. In this contribution we numerically study the behavior of these contrast sources. Firstly, we show from the computed nonlinear wavefield excited by a point source that the nonlinear distortion is indeed free of directionality. Secondly, we study the distribution of the contrast sources across the spatiotemporal domain, and we show that only the contrast sources between the primary source and the domain of observation have to be accounted for. Thirdly, we study the iterative estimation of the contrast sources for strong nonlinear distortion, and we show that for sufficient accuracy a spatiotemporal filtering of the contrast sources is necessary and may be performed straightforwardly.@en

In diagnostic medical ultrasound, it has become increasingly important to evaluate the nonlinear field of an acoustic beam that propagates in a weakly nonlinear, dissipative medium and that is steered off-axis up to very wide angles. In this case, computations cannot be based on the widely used KZK equation since it applies only to small angles. To benefit from successful computational schemes from elastodynamics and electromagnetics, we propose to use two first-order acoustic field equations, accompanied by two constitutive equations, as an alternative basis. This formulation quite naturally results in the contrast source formalism, makes a clear distinction between fundamental conservation laws and medium behavior, and allows for a straightforward inclusion of any medium inhomogenities. This paper is concerned with the derivation of relevant constitutive equations. We take a pragmatic approach and aim to find those constitutive equations that represent the same medium as implicitly described by the recognized, full wave, nonlinear equations such as the generalized Westervelt equation. We will show how this is achieved by considering the nonlinear case without attenuation, the linear case with attenuation, and the nonlinear case with attenuation. As a result we will obtain surprisingly simple constitutive equations for the full wave case.@en

A generalized theoretical framework for acoustic, electromagnetic and elastodynamic waves would give fruitful insights into equivalent phenomena in these physical domains and would form a basis to draw up general analytical or numerical solution methods. In this contribution we adopt the structure of Maxwell's equations for electromagnetic fields, encompassing the formulation of two first-order field equations and two first-order constitutive equations, and we apply it to the area of nonlinear acoustics. We derive the constitutive equations of a fluid directly from its thermodynamic equation of state (EOS). In the constitutive equations, the nonlinear medium behaviour of the fluid is described by a pressure-dependent density and compressibility. The resulting equations are general, making them valid for phenomena occuring in applications with finite amplitude waves of any magnitude, like waveform distortion or radiation pressure. This paper concerns with obtaining constitutive equations for fluids obeying an ideal gas law, a Tait-Kirkwood EOS or a 2-term Taylor approximation of the EOS employing the B/A nonlinearity parameter. The latter EOS is used in many of the classical model equations of nonlinear acoustics. We show that all three types result in simple expressions for the density and compressibility.@en

Nowadays, the design of phased array transducers for medical diagnostic ultrasound asks for an understanding of the nonlinear propagation of acoustic wavefields. In the last decade an imaging modality called Tissue Harmonic Imaging (THI) has become the standard for many echography investigations. THI specifically benefits from the nonlinear distortion of ultrasound propagation in tissue. Since most existing numerical models are based on a linear approximation of the underlying nonlinear physical reality, they cannot account for this kind of disortion. Several numerical models have been developed in recent years that incorporate weak nonlinear propagation. However, as yet no model has enabled the computation of large scale, full-wave nonlinear wavefields in the time domain. In this study, we present an approach that handles weak nonlinear propagation by means of an iterative Neumann scheme. This approach enables the successive use of the solution of a linear wave problem, where the nonlinearity is treated as a contrast source. Thus, we can employ well-known linear methods for large scale wave problems to obtain the desired nonlinear wavefield. The general formalism is outlined and applied to a one-dimensional nonlinear wave problem to obtain the desired nonlinear wavefield. The general formalism is outlined and applied to a one-dimensional nonlinear wave problem. The wavefield is evaluated, including harmonic frequencies up to the fifth harmonic. For each successive linear step a Green's function approach is employed. The results that already after a small number of iterations the results are in very good agreement with this exact result. The proposed method can easily be extended can easily be extended to more complex problems. The Green's function approach enables us to discretize the spatiotemporal domain very efficiently, which opens the road to solving time domain nonlinear wave problems in three dimensions. Furthermore, media with attenuation and inhomogeneity can be inlcuded straightforwardly in the algorithm.@en

Predictions of nonlinear ultrasound beams of a medical phased array transducer, steered at 0, 22.5 and 45 degrees, and generated by a forward-wave and a full-wave numerical model are presented. The results from both models show a good agreement in the unsteered situation and in all fundamental field profiles, but they deviate significantly in the harmonic field profiles for the steered situations.@en

The design of phased array transducers for medical diagnostic ultrasound asks for an understanding of the nonlinear propagation of acoustic wavefields. Most existing numerical models are based on the linearized model equations, but in the recent decades several numerical models have been developed that incorporate weak nonlinear propagation as well. In this study, we present an approach that enables the computation of large-scale, three dimensional, weakly nonlinear wavefields in the time domain. It is based on the Neumann iterative method. which enables the successive use of the solution of a linear wave problem, where the nonlinearity is treated as a contrast source. The linear wave problem is formulated as a convolution integral with a free space Green's function kernel. If the Green's function is adequately regularized, this method enables us to discretize the spatiotemporal domain with step sizes up to the limit given by the Nyquist-Shannon sampling theorem for a bandlimited signal. The remaining spatiotemporal derivatives are handled with high-order finite difference schemes. The proposed method is validated with aone-dimensional nonlinear wave problem. We evaluate the wavefield, including harmonic frequencies up to the fifth harmonic. The results are compared with a solution of the lossless Burger's equation. It is observed that already after a small number of iterations the computed result shows perfect agreement with those of the Burger's equation. In this paper, we make plausible that the proposed method can be straightforwardly extended to more complex problems. The volume integral equation approach opens the ros to solving time domain nonlinear wave problems in three dimensions. Furthermore, the method can be adapted to media with attenuation and inhomogeneity.@en

A full wave 3d time domain algorithm is presented that calculates the nonlinear acoustic wavefield of medical phased array transducers emitting an arbitratily steered, focused apodized beam. The algorithm can be used for computer simulation studies of the occuring second and higher harmonic wavefields in these configurations and for the optimization of the transducer design. It is based on the iterative employment of a Green's Function (or impulse response) method with the nonlinearity included as a contrast source. Several comparisons with validated algorithms are made, showing a perfect agreement.@en