Frequency Power Law Attenuation Research Articles

Acoustic Radiation Forces at the Crossroads of Ultrasound Exposimetry, HIFU, and Elastography.

The applications of the acoustic radiation force (ARF) continue to multiply and extend from elastography into high-intensity focused ultrasound (HIFU), diagnostic imaging, lithotripsy, sonochemistry, levitation, and microsonics yet fundamental principles remain shrouded in mystery. A well-known and popular equation often used for calculating ARF in elastography is in conflict with the equation commonly employed for calculating ARF for determining acoustic power in radiation force balances (RFBs). Controversies have sparked debate for over a century concerning the physical mechanisms underlying ARFs. For over four decades, the science of ultrasound exposimetry has steadily improved and has provided clues in terms of accumulated data about the characteristics of transmitted ultrasound fields. Concurrently, the availability and capability of predicting these fields have improved significantly. The author draws on these sources to re-examine the physical principles behind ARFs. Conflicts are shown to stem from idealized configurations and simplistic assumptions. By more fully accounting for the pulse shape and spectrum, the effect of frequency power law attenuation, diffraction, and nonlinearity, more accurate equations are developed for ARFs for practical applications which are more consistent with exposimetry observations. Simulations compare well to corrected 1.5 MHz RFB data. While some questions await resolution, the approach presented here settles several longstanding conflicts and provides a new broadband framework for future ARF work.

Acoustic Propagation in a Medium with Spatially Distributed Relaxation Processes and a Possible Explanation of a Frequency Power Law Attenuation

A possible medium is considered with a large number of diverse but possibly similar relaxation processes. The equations for single spatially located relaxation processes of general type are developed and extended to a large spatially extended system. Each process is characterized by a relaxation time and a relaxation strength and is excited by the local pressure in an incident acoustic wave. A constant frequency model is initially assumed with complex amplitudes associated with acoustic pressure and relaxation responses. A smearing process results in the collective relaxation responses being regarded as a continuous function of relaxation times. An expression for entropy perturbation is developed in terms of the relaxation responses. The equations of fluid mechanics then lead to an expression for the pressure perturbation in terms of the dilatation and the relaxation processes. The extended result yields a time-domain wave equation for propagation through an inhomogeneous medium with distributed relaxation processes. Expressions for frequency-dependent attenuation and phase velocity are derived in which relaxation is characterized by a single function giving strength as a continuous function of relaxation time. Possible choices for this function are discussed, and it is shown that some choices lead to an attenuation varying nearly as a power of the frequency over any fixed and possibly large range of frequencies. A model set of parameters leads to good agreement with attenuation and phase velocity measurements for a suspension of human red blood cells in saline solution, as communicated to the authors by Treeby and analogous to those reported by Treeby, Zhang, Thomas, and Cox.

Determination of Phase Jumps in the Measurement of Phase Velocity of Samples Obeying a Frequency Power-Law Attenuation Coefficient Using Kramers-Kronig Relations.

Ultrasonic phase velocity spectroscopy is a very sensitive technique used in the measurement of material properties. In a phase velocity calculation, ambiguities can arise in the spectral phases, in the form of integer multiples of 2π rad, which, if not corrected, results in large errors. In this work, we propose a method for determining these ambiguities, more specifically, the number of 2π rad phase jumps, using the Kramers-Kronig relations, for samples exhibiting a frequency power-law attenuation coefficient. The method is based on a first estimate of the phase velocity from group velocity and attenuation coefficient that are not affected by phase jumps. This estimated phase velocity is used to obtain the number of 2π rad phase jumps, which in turn is used to calculate the corrected phase velocity. The method was tested with samples of liquids with a frequency power-law attenuation coefficient (exponent y varying from 1.5 to 2) and a solid [polymethyl methacrylate (PMMA)] with y ∼ 1 , and velocity dispersions ranging from 0 to 34 (cm/s)/MHz.

Time-domain simulation of ultrasound propagation with fractional Laplacians for lossy-medium biological tissues with complicated geometries.

Simulations of ultrasound wave propagation inside biological tissues have a wide range of practical applications. In previous studies, wave propagation equations in lossy biological media are solved either with convolutions, which consume a large amount of memory, or with pseudo-spectral methods, which cannot handle complicated geometries effectively. The approach described in the paper employed a fractional central difference method (FCD), combined with the immersed boundary (IB) method for the finite-difference, time-domain simulation. The FCD method can solve the fractional Laplace terms in Chen and Holm's lossy-medium equations directly in the physical domain without integral transforms. It also works naturally with the IB method, which enables a simple Cartesian-type grid mesh to be used to solve problems with complicated geometries. The numerical results agree very well with the analytical solutions for frequency power-law attenuation lossy media.

Time-Domain Simulation of Ultrasound Propagation in a Tissue-Like Medium Based on the Resolution of the Nonlinear Acoustic Constitutive Relations

A time-domain numerical code based on the constitutive relations of nonlinear acoustics for simulating ultrasound propagation is presented. To model frequency power law attenuation, such as observed in biological media, multiple relaxation processes are included and relaxation parameters are fitted to both exact frequency power law attenuation and empirically measured attenuation of a variety of tissues that does not fit an exact power law. A computational technique based on artificial relaxation is included to correct the non-negligible numerical dispersion of the numerical method and to improve stability when shock waves are present. This technique avoids the use of high order finite difference schemes, leading to fast calculations. The numerical code is especially suitable to study high intensity and focused axisymmetric acoustic beams in tissue-like medium, as it is based on the full constitutive relations that overcomes the limitations of the parabolic approximations, while some specific effects not contemplated by the Westervelt equation can be also studied. The accuracy of the method is discussed by comparing the proposed simulation solutions to one-dimensional analytical ones, to $k$-space numerical solutions and also to experimental data from a focused beam propagating in a frequency power law attenuation media.

Frequency power-law attenuation and dispersion in marine sediments

Acoustic waves in marine sediments, including the fine-grained muds and clays, often exhibit an attenuation in the form of a frequency power law in which the exponent is close to unity. The frequency dependence of the wave speed, or dispersion, associated with such a power law has long been a subject of debate. Causality arguments, often in the form of the Kramers-Kronig dispersion relations, are usually applied to the problem, but these are not sufficient to characterize the dispersion fully. Following an alternative approach, a wave equation will be introduced which predicts the dispersion when the attenuation is of the frequency power law type. Based on this wave equation, it will be shown that only certain values of the frequency exponent are allowed, otherwise causality is violated, indicating that the associated attenuation and dispersion are not physically realizable. In effect, for these prohibited values of the frequency exponent, the Fourier components in the dispersion and attenuation cannot combine to give a zero response for negative times. Emerging from the theory is an expression for the complex wavenumber that is complete and exact, and from which all the properties of the dispersion and attenuation may be derived.

On the Nonlinear Effects in Focused Ultrasound Beams with Frequency Power Law Attenuation

When finite amplitude ultrasound propagation is considered, changes in spatial features of focused ultrasound beams can be observed. These nonlinear effects typically appear in thermoviscous fluids as focal displacements, beam-width variations or gain changes. However, in soft-tissue media, the frequency dependence of the attenuation doesn’t obey a squared law. In this way, these complex media response leads to weak dispersion that prevents the cumulative processes of energy transfer to higher harmonics. In this work we explore the influence of different frequency power law attenuation responses and its influence on the self-defocusing effects in focused ultrasound beams. Thus, we numerically explore the spatial field distributions produced by low-Fresnel number devices and High Intensity Focused Ultrasound (HIFU) radiating trough different soft-tissue media.

Acoustic precursor wave propagation in viscoelastic media

Precursor field theory has been developed to describe the dynamics of electromagnetic field evolution in causally attenuative and dispersive media. In Debye dielectrics, the so-called Brillouin precursor exhibits an algebraic attenuation rate that makes it an ideal pulse waveform for communication, sensing, and imaging applications. Inspired by these studies in the electromagnetic domain, the present paper explores the propagation of acoustic precursors in dispersive media, with emphasis on biological media. To this end, a recently proposed causal dispersive model is employed, based on its interpretation as the acoustic counterpart of the Cole¿Cole model for dielectrics. The model stems from the fractional stress¿strain relation, which is consistent with the empirically known frequency power-law attenuation in viscoelastic media. It is shown that viscoelastic media described by this model, including human blood, support the formation and propagation of Brillouin precursors. The amplitude of these precursors exhibits a sub-exponential attenuation rate as a function of distance, actually being proportional to z(-p), where z is the distance traveled within the medium and 0.5 <p < 1. The precursors identified in this work facilitate the design of optimal waveforms for propagation in complex media, creating new possibilities for acoustic-pulse-based communication and imaging systems.

On a fractional Zener elastic wave equation

Abstract This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a Maxwell-Wiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.

Rationale behind the Iterative Nonlinear Contrast Source method

Modern medical echoscopy increasingly relies on imaging modalities that exploit the features of nonlinear ultrasound. The development of these modalities and corresponding dedicated transducers requires accurate simulations of pulsed nonlinear acoustic wave fields in realistic biomedical situations. This involves nonlinear media with frequency power law attenuation and spatially dependent acoustic properties. Simulations frequently concern strongly steered beams, hundreds of wavelengths long, and their grating lobes. The Iterative Nonlinear Contrast Source (INCS) method is a full-wave method that has been developed for this purpose. It treats the nonlinear term in the Westervelt equation as a contrast source that operates, alongside other source terms, in a homogeneous linear ‘background’ medium. The background Green’s function is known analytically, and convolution with the source terms yields an integral equation. This is solved iteratively to obtain the nonlinear pressure field. The convolution over the four-dimensional computational domain is performed with FFT’s, and a grid with only two points per wavelength suffices due to prior filtering of the involved quantities. The present talk elaborates on the characteristic steps of the INCS method, i.e. the contrast source formulation and the filtered convolution. Comparisons with other methods will be made, and recent developments will be presented.

Full wave modeling of therapeutic ultrasound: Efficient time-domain implementation of the frequency power-law attenuation

For the simulation of therapeutic ultrasound applications, a method including frequency-dependent attenuation effects directly in the time domain is highly desirable. This paper describes an efficient numerical time-domain implementation of the power-law attenuation model presented by Szabo [Szabo, J. Acoust. Soc. Am. 96, 491-500 (1994)]. Simulations of therapeutic ultrasound applications are feasible in conjunction with a previously presented finite differences time-domain (FDTD) algorithm for nonlinear ultrasound propagation [Ginter et al., J. Acoust. Soc. Am. 111, 2049-2059 (2002)]. Szabo implemented the empirical frequency power-law attenuation using a causal convolutional operator directly in the time-domain equation. Though a variety of time-domain models has been published in recent years, no efficient numerical implementation has been presented so far for frequency power-law attenuation models. Solving a convolutional integral with standard time-domain techniques requires enormous computational effort and therefore often limits the application of such models to 1D problems. In contrast, the presented method is based on a recursive algorithm and requires only three time levels and a few auxiliary data to approximate the convolutional integral with high accuracy. The simulation results are validated by comparison with analytical solutions and measurements.

An explicit numerical time domain formulation to simulate pulsed pressure waves in viscous fluids exhibiting arbitrary frequency power law attenuation

An explicit time domain algorithm is developed which is capable of numerically simulating pulsed pressure waves propagating through media whose attenuation increases with frequency according to a power law dependency. Because of possible noninteger exponents in the power law formulation, standard temporal differential operators cannot be defined and traditional finite difference approximations are therefore inappropriate. We derive a method that is consistent with the fact that the complex wavenumber often must include a nonlinear phase if system causality is to be ensured. This phase is derived from the power law attenuation and, due to their interdependency, the two terms in the wave equation corresponding to attenuation and phase can be combined into a single factor. This so-called dispersion wave equation is mapped into complex discrete-time frequency. In this domain, noninteger exponents can be eliminated via a power series expansion, and the resulting equations transform naturally to discrete time operators. The algorithm is tested by comparing numerically evaluated attenuation with the exact power law form, and the issues of stability in relation to the convergence of the power series and the accuracy of the mapping are investigated.