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.

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.

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. Estimated shear wave speeds are obtained from cross correlations and time-to-peak (TTP) calculations applied to shear wave particle velocities and shear wave particle displacements. The results obtained from these 3D shear wave simulations indicate that TTP calculations applied to shear wave particle displacements yield effective estimates of the shear wave speed if noise is absent, but cross correlations applied to shear wave particle displacements are more robust when the effects of noise and shear viscosity are included. The results also show that shear wave speeds estimated with TTP methods and cross correlations using shear wave particle velocities are more sensitive to increases in shear viscosity and noise, which suggests that superior estimates of the shear wave speed are obtained from noiseless or noisy shear wave particle displacements.

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.

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. A dynamic sub-grid scale stabilization method for reducing Gibbs oscillations in acoustic shock waves is then established. Linear and nonlinear numerical results from a two-dimensional axisymmetric DG code are presented and compared to numerical solutions obtained from linear and Khokhlov-Zabolotskaya-Kuznetsov-based simulations in FOCUS. The numerical results indicate that these nodal DG simulations capture nonlinearity, thermoviscous absorption, and diffraction for both flat and focused pistons in homogeneous media.

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/#. Relative to shear wave simulations that evaluate the same algorithm on an Intel i7 desktop computer, a high performance nVidia GPU reduces the time required for these calculations by a factor of 45 and 700 when applied to elastic and viscoelastic shear wave simulation models, respectively. These GPU-accelerated simulations also compared to measurements in different viscoelastic phantoms, and the results are similar. For parametric evaluations and for comparisons with measured shear wave data, shear wave simulations with the Green's function approach are ideally suited for high-performance GPUs.

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. The method uses a mapped Chebyshev pseudo-spectral method to calculate the spatial derivatives and an Adams-Bashforth-Moulton integrator with variable step sizes for time marching. The boundaries of the two-dimensional domain are surrounded by perfectly matched layers to approximate an infinite domain and minimize reflection errors. In an earlier work, we proposed a solution for estimating the apparent shear wave elasticity and viscosity of the spatial group velocity as a function of rotation angle through a low-frequency approximation by a Taylor expansion. With the solver implemented in MATLAB, the simulated results in this paper match well with the theory. Compared to the finite element method simulations we used before, the pseudo-spectral solver consumes less memory and is faster and achieves better accuracy.

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.

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.

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. More specifically, these pulses were generated using a custom transducer composed of 23 individual relatively-broadband piezoceramic elements with various resonant frequencies (0.5, 1, 1.5, 2, and 3 MHz). Each frequency component of the transducer was capable of generating 1.5-cycle pulses with only one high-amplitude negative half-cycle using a custom 23-channel high-voltage pulser. By varying time delays of individual frequency components to allow their principal peak negative peaks to arrive at the focus of the transducer constructively, destructive interference occurs elsewhere in time and space, resulting in a monopolar pulse approximation with a dominant negative phase (with measured peak negative pressure [P-]: peak positive pressure [P+] = 4.68: 1). By inverting the excitation pulses to individual elements, monopolar pulses with a dominant positive phase can also be generated (with measured P+: P- = 4.74: 1). Experiments in RBC phantoms indicated that monopolar pulses with a dominant negative phase were able to produce very precise histotripsy-type lesions using the intrinsic threshold mechanism. Monopolar pulses with a dominant negative phase can inhibit shock scattering during histotripsy, leading to more predictable lesion formation using the intrinsic threshold mechanism, while greatly reducing any constructive interference, and potential hot-spots elsewhere. Moreover, these monopolar pulses could have many potential benefits in ultrasound imaging, including axial resolution improvement, speckle reduction, and contrast enhancement in pulse inversion imaging.

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. 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].

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.

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. For single-piston calculations, we employ the fast near-field method (FNM) to accurately and efficiently estimate the complex near-field wave patterns for rectangular pistons in homogeneous media. The FNM is compared with the Rayleigh-Sommerfeld method (RSM) for the number of abscissas required in the respective numerical integrations to achieve 1%, 0.1%, and 0.01% accuracy in the field calculations. Next, algorithms are described for accelerated computation of beam patterns for two different ultrasound transducer arrays: regular 1-D linear arrays and regular 2-D linear arrays. For the array types considered, the algorithm is split into two parts: 1) the computation of the field from one piston, and 2) the computation of a piston-array beam pattern based on a pre-computed field from one piston. It is shown that the process of calculating an array beam pattern is equivalent to the convolution of the single-piston field with the complex weights associated with an array of pistons. Our results show that the algorithms for computing monochromatic fields from linear and regularly spaced arrays can benefit greatly from GPU computing hardware, exceeding the performance of an expensive CPU by more than 100 times using an inexpensive GPU board. For a single rectangular piston, the FNM method facilitates volumetric computations with 0.01% accuracy at rates better than 30 ns per field point. Furthermore, we demonstrate array calculation speeds of up to 11.5 X 10(9) field-points per piston per second (0.087 ns per field point per piston) for a 512-piston linear array. Beam volumes containing 256(3) field points are calculated within 1 s for 1-D and 2-D arrays containing 512 and 20(2) pistons, respectively, thus facilitating future real-time thermal dose predictions.

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. By selectively deleting tumour control points, the tumour volume heated above 42°C increased from 2.28 cm3 to 11.22 cm3. At the expense of a slight increase in the normal tissue volume heated above the target temperature of 42°C, the size of the tumour volume heated above 42°C after tumour points were deleted was almost five times larger than the size of the original heated tumour volume. Several other configurations were also simulated, and the largest heated tumour volumes, subject to a 43°C peak temperature constraint, were achieved when the tumour control points were located along the back edge of the tumour and laterally around the tumour periphery. The simulated power depositions obtained from the results of the adaptive control point removal algorithm, when optimised for waveform diversity combined with mode scanning, consistently increased the penetration depth and the size of the heated tumour volume while increasing the heated normal tissue volume by a small amount.

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. A numerical example simulates a 1444-element spherical-section phased array that delivers heat to a 3-cm-diameter spherical tumor located 12 cm from the array aperture, and the results show that waveform diversity combined with mode scanning increases the heated volume within the tumor while simultaneously decreasing normal tissue heating. Whereas standard single focus and multiple focus methods are often associated with unwanted intervening tissue heating, the waveform diversity method combined with mode scanning shifts energy away from intervening tissues where hotspots otherwise accumulate to improve temperature localization in deep-seated tumors.

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. This misalignment produces an erroneous CT attenuation map that can project the bone and water attenuation parameters onto the brain, thereby under- or over-estimating the attenuation. To avoid the misregistration artifact and potential diagnostic misinterpretation, automated software for PET/CT brain registration has been developed. This software extracts the brain surface information from the CT and PET images and compensates for the translational and rotational misalignment between the two scans. This procedure has been applied to the dataset of a patient with visible perfusion defect in the brain, and the results show that the CTAC produced after the image registration eliminates that hypoperfusion artifact caused by the erroneous attenuation of the PET images.

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. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the Kramers-Kronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters.