Power Law Wave Equation Research Articles

Numerical spatial impulse response calculations for a circular piston radiating in a lossy medium.

Exact analytical expressions for the spatial impulse response are available for certain transducer geometries. These exact expressions for the spatial impulse response, which are only available for lossless media, analytically evaluate the Rayleigh integral to describe the effect of diffraction in the time domain. To extend the concept of the spatial impulse response by including the effect of power law attenuation in a lossy medium, time-domain Green's functions for the Power Law Wave Equation, which are expressed in terms of stable probability density functions, are computed numerically and superposed. Numerical validations demonstrate that the lossy spatial impulse for a circular piston converges to the analytical lossless spatial impulse response as the value of the attenuation constant grows small. The lossy spatial impulse response is then evaluated in different spatial locations for four specific values of the power law exponent using several different values for the attenuation constant. As the attenuation constant or the distance from the source increases, the amplitude decreases while an increase in temporal broadening is observed. The sharp edges that appear in the time-limited lossless impulse response are replaced by increasingly smooth curves in the lossy impulse response, which decays slowly as a function of time.

Numerical evaluations of the lossy spatial impulse response for a circular piston in the far field

Numerical evaluations of the lossy spatial impulse response are achieved by substituting Green's functions from the Power Law Wave Equation into the Rayleigh integral. The addition of loss introduces noticeable attenuation, dispersion, and delay into the impulse response, which become more pronounced in the far field.

Numerical evaluation of a fractional calculus model for the spatial impulse response of a circular piston

In lossless media, exact analytical expressions for the spatial impulse response describe the effect of diffraction in the time domain for several specific transducer shapes. At present, no exact analytical expressions are available for lossy media, so numerical evaluation of the spatial impulse response is required. For attenuation that follows a power law, the spatial impulse response is numerically evaluated by superposing contributions from time-domain Green's functions for the Power Law Wave Equation, which is a fractional calculus model for power law attenuation. To demonstrate examples of lossy spatial impulse responses obtained with these time-domain Green's functions, which are expressed in terms of maximally skewed stable densities, numerical results are computed for a circular transducer using different attenuation values and compared with the analytical result for a lossless medium. The results show that the numerically computed spatial impulse response for a lossy medium converges to the analytical result evaluated in a lossless medium as the value of the attenuation constant decreases. As the attenuation constant grows larger, the temporal extent increases, and the sharp edges are replaced by increasingly smooth curves, as observed in numerical evaluations of the lossy spatial impulse response evaluated in multiple locations.

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 continuous power law wave equation for biomedical ultrasound

The attenuation coefficient in most biological media is a power law with respect to frequency. Measurements indicate that the power law exponent y is near or exactly one for many classes of tissue. In J. Acous. Soc. Am. [124 2861–2872 (2008)], a power law wave equation (PLWE) was proposed to model this power law dependence using Riemann-Liouville fractional derivatives. The PLWE, like most other fractional calculus models for attenuation in the time-domain, is invalid for y =1 due to a discontinuity in the phase velocity. To address this problem, a continuous power law wave equation (CPLWE) is proposed that is valid for all power law exponents between zero and two. The CPLWE utilizes the recently developed Zolotarev fractional derivative of order y, which is a nonlocal operator even for y = 1. The 3D Green's function for the CPLWE that is valid for homogeneous media is derived using stable probability density functions in the Zolotarev M-parameterization. Solutions to the CPLWE that are valid for inhomoge...

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

The attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency that is modeled by several different causal and noncausal fractional partial differential equations. To demonstrate some of the similarities and differences that are observed in three related time-fractional partial differential equations, time-domain Green's functions are calculated numerically for the power law wave equation, the Szabo wave equation, and for the Caputo wave equation. These Green's functions are evaluated for water with a power law exponent of y = 2, breast with a power law exponent of y = 1.5, and liver with a power law exponent of y = 1.139. Simulation results show that the noncausal features of the numerically calculated time-domain response are only evident very close to the source and that these causal and noncausal time-domain Green's functions converge to the same result away from the source. When noncausal time-domain Green's functions are convolved with a short pulse, no evidence of noncausal behavior remains in the time-domain, which suggests that these causal and noncausal time-fractional models are equally effective for these numerical calculations.

Numerical evaluation of absorbing boundary layers for the transient Khokhlov–Zabolotskaya–Kuznetsov Equation

FOCUS, the “Fast Object-Oriented C + + Ultrasound Simulator” (http://www.egr.msu.edu/~fultras-web), simulates nonlinear ultrasound propagation in the time-domain by numerically evaluating the transient Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation. KZK simulations in FOCUS previously required a computational grid with a large radial distance relative to the aperture radius to reduce the effect of reflections from the boundary. To decrease the size of the grid required for these calculations, an absorbing boundary layer derived from the power law wave equation provides an alternative to the stretched coordinate system in a perfectly matched layer. Simulations of the linear pressure fields generated by a spherically focused transducer with an aperture radius of 1.5 cm and a radius of curvature of 6 cm are evaluated for a short pulse with a center frequency of 1 MHz and a peak surface pressure of 0.5 MPa. Numerical results for linear KZK simulations with and without the absorbing boundary layer are compared to an on-axis analytical solution. Results of nonlinear KZK simulations with and without the PML are also evaluated, and all of these show that the absorbing layer effectively attenuates the wavefronts that reach the boundary of the computational grid. [This work was supported in part by NIH Grant R01 EB012079.]

Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography

A set of wave equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law wave equation and the causal fractional Laplacian wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional constitutive equations, whereas the former wave equations have been derived from the desire to model power law attenuation in applications like medical ultrasound. This has consequences for use in modeling and simulation, especially for applications that do not satisfy the low-frequency approximation, such as shear wave elastography. In such applications, the wave equations based on constitutive equations are the viable ones.

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.

FRACTIONAL WAVE EQUATIONS WITH ATTENUATION.

Fractional wave equations with attenuation have been proposed by Caputo [5], Szabo [27], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

Analytical and numerical approximations for the lossy on-axis impulse response of a circular piston

In biological tissues, the frequency dependence of attenuation and speed of sound for ultrasound propagation is described by the power law wave equation, which is a partial differential equation with fractional time derivatives. As demonstrated previously, the time domain Green's function for the power law wave equation combined with the Rayleigh-Sommerfeld integral is an effective reference for calculations of the lossy on-axis impulse response of a circular piston. Using the result obtained from this reference, two different approximations to the lossy on-axis impulse response are evaluated. The first approximation is an analytical expression that is proportional to the difference between two cumulative distribution functions for maximally skewed stable probability densities. The second approximation numerically convolves the lossless impulse response with a maximally skewed stable probability density function. The results show that both approximations achieve relatively small errors. Furthermore, the analytical approximation provides an excellent estimate for the arrival time of the lossy impulse response, whereas the departure time of the lossy impulse response is more difficult to characterize due to the heavy tail of the maximally skewed stable probability density function. Both approximations are rapidly calculated with the STABLE toolbox. [supported in part by NIH Grant R01 EB012079.]

Stochastic solution to a time-fractional attenuated wave equation.

The power law wave equation uses two different fractional derivative terms to model wave propagation with power law attenuation. This equation averages complex nonlinear dynamics into a convenient, tractable form with an explicit analytical solution. This paper develops a random walk model to explain the appearance and meaning of the fractional derivative terms in that equation, and discusses an application to medical ultrasound. In the process, a new strictly causal solution to this fractional wave equation is developed.

Evaluation of the power-law impulse response for liver in the time- and frequency-domains

Ultrasound attenuation in biological media often follows a frequency-dependent power-law relationship. Power-law attenuation is accompanied by dispersion in which phase velocity also changes as a function of frequency. The power-law wave equation has exact time- and frequency-domain Green's function solutions that are used in numerical evaluations of the Rayleigh–Sommerfeld diffraction integral. These Green's functions contain stable probability density functions that are evaluated in the time-domain using the STABLE toolbox and in the frequency-domain by evaluating the characteristic function of a stable distribution. The impulse response for a circular piston is evaluated on-axis in the time- and frequency-domain using values for human liver. The results show the accuracy of the time-domain impulse response calculation is dependent upon adequate spatial sampling of the piston face. The frequency-domain impulse response exhibits aliasing caused by wrap around of the heavy tail of the stable distribution. Frequency-domain accuracy is dependent upon the spatial sampling of the piston face and the density of the frequency samples. Numerical evaluations show that the frequency- and time-domain calculations converge to the same result. [This work was supported in part by NIH Grant R01 EB012079.]