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.

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.

A parametric evaluation of shear wave speeds estimated with time-of-flight calculations in viscoelastic media.

Shear wave elasticity imaging (SWEI) uses an acoustic radiation force to generate shear waves, and then soft tissue mechanical properties are obtained by analyzing the shear wave data. In SWEI, the shear wave speed is often estimated with time-of-flight (TOF) calculations. To characterize the errors produced by TOF calculations, three-dimensional (3D) simulated shear waves are described by time-domain Green's functions for a Kelvin-Voigt model evaluated for multiple combinations of the shear elasticity and the shear viscosity.

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)≈α|ω|. 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.

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.

Linear and nonlinear ultrasound simulations using the discontinuous Galerkin method.

A nodal discontinuous Galerkin (DG) code based on the nonlinear wave equation is developed to simulate transient ultrasound propagation. The DG method has high-order accuracy, geometric flexibility, low dispersion error, and excellent scalability, so DG is an ideal choice for solving this problem. A nonlinear acoustic wave equation is written in a first-order flux form and discretized using nodal DG.

GPU-based Green's function simulations of shear waves generated by an applied acoustic radiation force in elastic and viscoelastic models.

Shear wave calculations induced by an acoustic radiation force are very time-consuming on desktop computers, and high-performance graphics processing units (GPUs) achieve dramatic reductions in the computation time for these simulations. The acoustic radiation force is calculated using the fast near field method and the angular spectrum approach, and then the shear waves are calculated in parallel with Green's functions on a GPU. This combination enables rapid evaluation of shear waves for push beams with different spatial samplings and for apertures with different f/#.

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

The Caputo fractional wave equation [Geophys. J. R.

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.

Mapped Chebyshev pseudo-spectral method for simulating the shear wave propagation in the plane of symmetry of a transversely isotropic viscoelastic medium.

Shear wave elastography is a versatile technique that is being applied to many organs. However, in tissues that exhibit anisotropic material properties, special care must be taken to estimate shear wave propagation accurately and efficiently. A two-dimensional simulation method is implemented to simulate the shear wave propagation in the plane of symmetry in transversely isotropic viscoelastic media.

Attenuated Fractional Wave Equations With Anisotropy.

This paper develops new fractional calculus models for wave propagation. These models permit a different attenuation index in each coordinate to fully capture the anisotropic nature of wave propagation in complex media. Analytical expressions that describe power law attenuation and anomalous dispersion in each direction are derived for these fractional calculus models.

Synthesis of monopolar ultrasound pulses for therapy: the frequency-compounding transducer.

In diagnostic ultrasound, broadband transducers capable of short acoustic pulse emission and reception can improve axial resolution and provide sufficient bandwidth for harmonic imaging and multi-frequency excitation techniques. In histotripsy, a cavitation-based ultrasound therapy, short acoustic pulses (<2 cycles) can produce precise tissue ablation wherein lesion formation only occurs when the applied peak negative pressure exceeds an intrinsic threshold of the medium. This paper investigates a frequency compounding technique to synthesize nearly monopolar (half-cycle) ultrasound pulses.

Components of a hyperthermia clinic: recommendations for staffing, equipment, and treatment monitoring.

Like other technically sophisticated medical endeavours, a hyperthermia clinic relies on skilled staffing. Physicians, physicists and technologists perform multiple tasks to ensure properly functioning equipment, appropriate patient selection, and to plan and administer this treatment. This paper reviews the competencies and tasks that are used in a hyperthermia clinic.

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.

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.

Fast ultrasound beam prediction for linear and regular two-dimensional arrays.

Real-time beam predictions are highly desirable for the patient-specific computations required in ultrasound therapy guidance and treatment planning. To address the longstanding issue of the computational burden associated with calculating the acoustic field in large volumes, we use graphics processing unit (GPU) computing to accelerate the computation of monochromatic pressure fields for therapeutic ultrasound arrays. In our strategy, we start with acceleration of field computations for single rectangular pistons, and then we explore fast calculations for arrays of rectangular pistons.

Improving conformal tumour heating by adaptively removing control points from waveform diversity beamforming calculations: a simulation study.

Waveform diversity is a phased array beamforming strategy that determines an optimal sequence of excitation signals to maximise power at specified tumour control points while simultaneously minimising power delivered to sensitive normal tissues. Waveform diversity is combined with mode scanning, a deterministic excitation signal synthesis algorithm, and an adaptive control point removal algorithm in an effort to achieve higher, more uniform tumour temperatures. Simulations were evaluated for a 1444 element spherical section ultrasound phased array that delivers therapeutic heat to a 3 cm spherical tumour model located 12 cm from the array.

Automated PET/CT brain registration for accurate attenuation correction.

Computed Tomography (CT) is used for the attenuation correction of Positron Emission Tomography (PET) to enhance the efficiency of data acquisition process and to improve the quality of the reconstructed PET data in the brain. Due to the use of two different modalities, chances of misalignment between PET and CT images are quite significant. The main cause of this misregistration is the motion of the patient during the PET scan and between the PET and CT scans.

Fractal ladder models and power law wave equations.

The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100 MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots.

A waveform diversity method for optimizing 3-d power depositions generated by ultrasound phased arrays.

A waveform-diversity-based approach for 3-D tumor heating is compared to spot scanning for hyperthermia applications. The waveform diversity method determines the excitation signals applied to the phased array elements and produces a beam pattern that closely matches the desired power distribution. The optimization algorithm solves the covariance matrix of the excitation signals through semidefinite programming subject to a series of quadratic cost functions and constraints on the control points.

Optimal simulations of ultrasonic fields produced by large thermal therapy arrays using the angular spectrum approach.

The angular spectrum approach is evaluated for the simulation of focused ultrasound fields produced by large thermal therapy arrays. For an input pressure or normal particle velocity distribution in a plane, the angular spectrum approach rapidly computes the output pressure field in a three dimensional volume. To determine the optimal combination of simulation parameters for angular spectrum calculations, the effect of the size, location, and the numerical accuracy of the input plane on the computed output pressure is evaluated.

Analytical time-domain Green's functions for power-law media.

Frequency-dependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation ["Time domain wave-equations for lossy media obeying a frequency power-law," J. Acoust.

A 2D fast near-field method for calculating near-field pressures generated by apodized rectangular pistons.

Analytical two-dimensional (2D) integral expressions are derived for fast calculations of time-harmonic and transient near-field pressures generated by apodized rectangular pistons. These 2D expressions represent an extension of the fast near-field method (FNM) for uniformly excited pistons. After subdividing the rectangular piston into smaller rectangles, the pressure produced by each of the smaller rectangles is calculated using the uniformly excited FNM expression for a rectangular piston, and the total pressure generated by an apodized rectangular piston is the superposition of the pressures produced by all of the subdivided rectangles.

Automated cardiac motion compensation in PET/CT for accurate reconstruction of PET myocardial perfusion images.

Error-free reconstruction of PET data with a registered CT attenuation map is essential for accurate quantification and interpretation of cardiac perfusion. Misalignment of the CT and PET data can produce an erroneous attenuation map that projects lung attenuation parameters onto the heart wall, thereby underestimating the attenuation and creating artifactual areas of hypoperfusion that can be misinterpreted as myocardial ischemia or infarction. The major causes of misregistration between CT and PET images are the respiratory motion, cardiac motion and gross physical motion of the patient.

Causal impulse response for circular sources in viscous media.

The causal impulse response of the velocity potential for the Stokes wave equation is derived for calculations of transient velocity potential fields generated by circular pistons in viscous media. The causal Green's function is numerically verified using the material impulse response function approach. The causal, lossy impulse response for a baffled circular piston is then calculated within the near field and the far field regions using expressions previously derived for the fast near field method.