Time-domain comparisons of power law attenuation in causal and noncausal time-fractional wave equations

Three-dimensional time-domain Green's functions and spatial impulse responses for the van Wijngaarden wave equation

An exact analytical three-dimensional time-domain Green's function is introduced for the van Wijngaarden wave equation when the coefficients of the two loss terms satisfy a specific relationship. This analytical Green's function, which describes frequency-squared attenuation in acoustic media such as water, enables the subsequent derivation of new expressions that describe the lossy spatial impulse response for a circular piston. Initial time-domain assessments, which compare the Green's functions for the van Wijngaarden, Stokes, and power law wave equations using the attenuation and sound speed for water, indicate that these three lossy wave equations yield nearly identical results at distances greater than or equal to 10 μm. Lossy spatial impulse responses are also evaluated with increasing distance in and near the paraxial region of a circular piston radiating in water to reveal some interesting time-domain interactions between frequency-squared attenuation and diffraction. Similar behaviors are also demonstrated for the lossy far-field spatial impulse. In addition, the convergence is demonstrated for two analytically equivalent expressions applied to numerical computations of the lossy spatial impulse response. The results show that these new expressions are ideal for describing and explaining fundamental interactions between frequency-squared attenuation and diffraction in the time-domain.

mSOUND: An Open Source Toolbox for Modeling Acoustic Wave Propagation in Heterogeneous Media

mSOUND is an open-source toolbox written in MATLAB. This toolbox is intended for modeling linear/ nonlinear acoustic wave propagation in media (primarily biological tissues) with arbitrary heterogeneities, in which, the speed of sound, density, attenuation coefficient, power-law exponent, and nonlinear coefficient are all spatially varying functions. The computational model is an iterative one-way model based on a mixed domain method. In this article, a general guideline is given along with three representative examples to illustrate how to set up simulations using mSOUND. The first example uses the transient mixed-domain method (TMDM) forward projection to compute the transient acoustic field for a given source defined on a plane. The second example uses the frequency-specific mixed-domain method (FSMDM) forward projection to rapidly obtain the pressure distribution directly at the frequencies of interest, assuming linear or weakly nonlinear wave propagation. The third example demonstrates how to use TMDM backward projection to reconstruct the initial acoustic pressure field to facilitate photoacoustic tomography (PAT). mSOUND (https://m-sound.github.io/mSOUND/home) is designed to be complementary to existing ultrasound modeling toolboxes and is expected to be useful for a wide range of applications in medical ultrasound including treatment planning, PAT, transducer design, and characterization.

A modified mixed domain method for modeling acoustic wave propagation in strongly heterogeneous media

In this paper, phase correction and amplitude compensation are introduced to a previously developed mixed domain method (MDM), which is only accurate for modeling wave propagation in weakly heterogeneous media. Multiple reflections are also incorporated with the one-way model to improve the accuracy. The resulting model is denoted as the modified mixed domain method (MMDM) and is numerically evaluated for its accuracy and efficiency using four distinct cases. It is found that the MMDM is significantly more accurate than the MDM for strongly heterogeneous media, especially when the phase aberrating layer is approximately perpendicular to the acoustic beam. Additionally, a convergence study suggests that the second-order reflection could be sufficient for cases involving high contrast inhomogeneities and large loss values (e.g., skulls). The method developed in this work could facilitate therapeutic ultrasound for treating brain-related diseases and disorders.

Exact and approximate analytical time-domain Green's functions for space-fractional wave equations

The Chen-Holm and Treeby-Cox wave equations are space-fractional partial differential equations that describe power law attenuation of the form α(ω)≈α₀|ω|^y. Both of these space-fractional wave equations are causal, but the phase velocities differ, which impacts the shapes of the time-domain Green's functions. Exact and approximate closed-form time-domain Green's functions are derived for these space-fractional wave equations, and the resulting expressions contain symmetric and maximally skewed stable probability distribution functions. Numerical results are evaluated with ultrasound parameters for breast and liver at different times as a function of space and at different distances as a function of time, where the reference calculations are computed with the Pantis method. The results show that the exact and approximate time-domain Green's functions contain both outbound and inbound propagating terms and that the inbound component is negligible a short distance from the origin. Exact and approximate analytical time-domain Green's functions are also evaluated for the Chen-Holm wave equation with power law exponent y = 1. These comparisons demonstrate that single term analytical expressions containing stable probability densities provide excellent approximations to the time-domain Green's functions for the Chen-Holm and Treeby-Cox wave equations.

Time-domain analysis of power law attenuation in space-fractional wave equations

Ultrasound attenuation in soft tissue follows a power law as a function of the ultrasound frequency, and in medical ultrasound, power law attenuation is often described by fractional calculus models that contain one or more time- or space-fractional derivatives. For certain time-fractional models, exact and approximate time-domain Green's functions are known, but similar expressions are not available for the space-fractional models that describe power law attenuation. To address this deficiency, a numerical approach for calculating time-domain Green's functions for the Chen-Holm space-fractional wave equation and Treeby-Cox space-fractional wave equation is introduced, where challenges associated with the numerical evaluation of a highly oscillatory improper integral are addressed with the Filon integration formula combined with the Pantis method. Numerical results are computed for both of these space-fractional wave equations at different distances in breast and liver with power law exponents of 1.5 and 1.139, respectively. The results show that these two space-fractional wave equations are causal and that away from the origin, the time-domain Green's function for the Treeby-Cox space-fractional wave equation is very similar to the time-domain Green's function for the time-fractional power law wave equation.

A time-domain model and experimental validation of the acoustic field radiated by air-coupled transducers

This paper presents a time-domain model for the prediction of an acoustic field in an air-coupled, non-contact, ultrasonic setup, which includes an air-coupled Emitter, the Propagation space and an air-coupled Receiver (EPR). The model takes into account the finite size of the aperture receiver, attenuation in air, and the electric response of the emitter-receiver set h_e. The attenuation is characterized by a causal time-domain Green's function, allowing the wideband attenuation of a lossy medium obeying the power law α(ω)=α₀ω^η,1⩽η⩽2 to be included. The electrical response is recovered experimentally using a procedure which includes the deconvolution of air absorption effects. The model is implemented numerically using a discrete representation approach. In order to study the influence of receiver size and attenuation, five different computational approaches are proposed; each of these is evaluated quantitatively, by comparing the predicted acoustic field with the experimentally measured signal. The prediction error is studied in both the near and far fields, for three typical field features: the system's impulse response, the on-axis field distribution, and the directivity pattern, for the case of air-coupled transducers operating at two different central frequencies, namely 50kHz and 350kHz, with a 10mm diameter wideband receiver. It is shown that when the attenuation in air, the receiver size, and the accurately recovered electric response h_e, are correctly taken into account, the model allows the system's impulse response to be accurately predicted, with on-axis errors ranging between 0.2% in the far field and 1% in the near field. In the near-field area and within the far field -3dB beam spread width, the error is generally greater than on the axis, but globally remains smaller than 1%. Inclusion of the size of the receiver dimension in the model appears to be crucial to the accuracy of the near field predictions, and an approximate criterion is proposed for the evaluation of the influence of receiver. The procedure used to recover the electric response h_e is also presented in detail. The results obtained from this study are used to formulate various recommendations related to EPR modelling.

Approximate analytical time-domain Green's functions for the Caputo fractional wave equation

The Caputo fractional wave equation [Geophys. J. R. Astron. Soc. 13, 529-539 (1967)] models power-law attenuation and dispersion for both viscoelastic and ultrasound wave propagation. The Caputo model can be derived from an underlying fractional constitutive equation and is causal. In this study, an approximate analytical time-domain Green's function is derived for the Caputo equation in three dimensions (3D) for power law exponents greater than one. The Green's function consists of a shifted and scaled maximally skewed stable distribution multiplied by a spherical spreading factor 1/(4πR). The approximate one dimensional (1D) and two dimensional (2D) Green's functions are also computed in terms of stable distributions. Finally, this Green's function is decomposed into a loss component and a diffraction component, revealing that the Caputo wave equation may be approximated by a coupled lossless wave equation and a fractional diffusion equation.