Nonlinear ultrasound simulation in an axisymmetric coordinate system using a k-space pseudospectral method.

A full-wave model for nonlinear ultrasound propagation through a heterogeneous and absorbing medium in an axisymmetric coordinate system is developed. The model equations are solved using a nonstandard or k-space pseudospectral time domain method. Spatial gradients in the axial direction are calculated using the Fourier collocation spectral method, and spatial gradients in the radial direction are calculated using discrete trigonometric transforms. Time integration is performed using a k-space corrected finite difference scheme. This scheme is exact for plane waves propagating linearly in the axial direction in a homogeneous and lossless medium and significantly reduces numerical dispersion in the more general case. The implementation of the model is described, and performance benchmarks are given for a range of grid sizes. The model is validated by comparison with several analytical solutions. This includes one-dimensional absorption and nonlinearity, the pressure field generated by plane-piston and bowl transducers, and the scattering of a plane wave by a sphere. The general utility of the model is then demonstrated by simulating nonlinear transcranial ultrasound using a simplified head model.

Representing arbitrary acoustic source and sensor distributions in Fourier collocation methods.

Accurately representing acoustic source distributions is an important part of ultrasound simulation. This is challenging for grid-based collocation methods when such distributions do not coincide with the grid points, for instance when the source is a curved, two-dimensional surface embedded in a three-dimensional domain. Typically, grid points close to the source surface are defined as source points, but this can result in "staircasing" and substantial errors in the resulting acoustic fields. This paper describes a technique for accurately representing arbitrary source distributions within Fourier collocation methods. The method works by applying a discrete, band-limiting convolution operator to the continuous source distribution, after which source grid weights can be generated. This allows arbitrarily shaped sources, for example, focused bowls and circular pistons, to be defined on the grid without staircasing errors. The technique is examined through simulations of a range of ultrasound sources, and comparisons with analytical solutions show excellent accuracy and convergence rates. Extensions of the technique are also discussed, including application to initial value problems, distributed sensors, and moving sources.

Rapid calculation of acoustic fields from arbitrary continuous-wave sources.

A Green's function solution is derived for calculating the acoustic field generated by phased array transducers of arbitrary shape when driven by a single frequency continuous wave excitation with spatially varying amplitude and phase. The solution is based on the Green's function for the homogeneous wave equation expressed in the spatial frequency domain or k-space. The temporal convolution integral is solved analytically, and the remaining integrals are expressed in the form of the spatial Fourier transform. This allows the acoustic pressure for all spatial positions to be calculated in a single step using two fast Fourier transforms. The model is demonstrated through several numerical examples, including single element rectangular and spherically focused bowl transducers, and multi-element linear and hemispherical arrays.

Time domain reconstruction of sound speed and attenuation in ultrasound computed tomography using full wave inversion.

Ultrasound computed tomography (USCT) is a non-invasive imaging technique that provides information about the acoustic properties of soft tissues in the body, such as the speed of sound (SS) and acoustic attenuation (AA). Knowledge of these properties can improve the discrimination between benign and malignant masses, especially in breast cancer studies. Full wave inversion (FWI) methods for image reconstruction in USCT provide the best image quality compared to more approximate methods. Using FWI, the SS is usually recovered in the time domain, and the AA is usually recovered in the frequency domain. Nevertheless, as both properties can be obtained from the same data, it is desirable to have a common framework to reconstruct both distributions. In this work, an algorithm is proposed to reconstruct both the SS and AA distributions using a time domain FWI methodology based on the fractional Laplacian wave equation, an adjoint field formulation, and a gradient-descent method. The optimization code employs a Compute Unified Device Architecture version of the software k-Wave, which provides high computational efficiency. The performance of the method was evaluated using simulated noisy data from numerical breast phantoms. Errors were less than 0.5% in the recovered SS and 10% in the AA.

Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian.

The absorption of compressional and shear waves in many viscoelastic solids has been experimentally shown to follow a frequency power law. It is now well established that this type of loss behavior can be modeled using fractional derivatives. However, previous fractional constitutive equations for viscoelastic media are based on temporal fractional derivatives. These operators are non-local in time, which makes them difficult to compute in a memory efficient manner. Here, a fractional Kelvin-Voigt model is derived based on the fractional Laplacian. This is obtained by splitting the particle velocity into compressional and shear components using a dyadic wavenumber tensor. This allows the temporal fractional derivatives in the Kelvin-Voigt model to be replaced with spatial fractional derivatives using a lossless dispersion relation with the appropriate compressional or shear wave speed. The model is discretized using the Fourier collocation spectral method, which allows the fractional operators to be efficiently computed. The field splitting also allows the use of a k-space corrected finite difference scheme for time integration to minimize numerical dispersion. The absorption and dispersion behavior of the fractional Laplacian model is analyzed for both high and low loss materials. The accuracy and utility of the model is then demonstrated through several numerical experiments, including the transmission of focused ultrasound waves through the skull.

A first-order k-space model for elastic wave propagation in heterogeneous media.

A pseudospectral model of linear elastic wave propagation is described based on the first order stress-velocity equations of elastodynamics. k-space adjustments to the spectral gradient calculations are derived from the dyadic Green's function solution to the second-order elastic wave equation and used to (a) ensure the solution is exact for homogeneous wave propagation for timesteps of arbitrarily large size, and (b) also allows larger time steps without loss of accuracy in heterogeneous media. The formulation in k-space allows the wavefield to be split easily into compressional and shear parts. A perfectly matched layer (PML) absorbing boundary condition was developed to effectively impose a radiation condition on the wavefield. The staggered grid, which is essential for accurate simulations, is described, along with other practical details of the implementation. The model is verified through comparison with exact solutions for canonical examples and further examples are given to show the efficiency of the method for practical problems. The efficiency of the model is by virtue of the reduced point-per-wavelength requirement, the use of the fast Fourier transform (FFT) to calculate the gradients in k space, and larger time steps made possible by the k-space adjustments.

Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method.

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.

A k-space Green's function solution for acoustic initial value problems in homogeneous media with power law absorption.

An efficient Green's function solution for acoustic initial value problems in homogeneous media with power law absorption is derived. The solution is based on the homogeneous wave equation for lossless media with two additional terms. These terms are dependent on the fractional Laplacian and separately account for power law absorption and dispersion. Given initial conditions for the pressure and its temporal derivative, the solution allows the pressure field for any time t>0 to be calculated in a single step using the Fourier transform and an exact k-space time propagator. For regularly spaced Cartesian grids, the former can be computed efficiently using the fast Fourier transform. Because no time stepping is required, the solution facilitates the efficient computation of the pressure field in one, two, or three dimensions without stability constraints. Several computational aspects of the solution are discussed, including the effect of using a truncated Fourier series to represent discrete initial conditions, the use of smoothing, and the properties of the encapsulated absorption and dispersion.

Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian.

The efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics. Previous wave equations which use time-domain fractional operators require the storage of the complete pressure field at previous time steps (such operators are convolution based). This makes them unsuitable for many three-dimensional problems of interest. Here, a wave equation that utilizes two lossy derivative operators based on the fractional Laplacian is derived. These operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into Fourier based pseudospectral and k-space methods without the increase in memory required by their time-domain fractional counterparts. A framework for encoding the developed wave equation using three coupled first-order constitutive equations is discussed, and the model is demonstrated through several one-, two-, and three-dimensional simulations.

Photoacoustic tomography with a single detector in a reverberant cavity.

In conventional biomedical photoacoustic tomography (PAT), ultrasonic pulses generated through the absorption of nanosecond pulses of near-infrared light are recorded over an array of detectors and used to recover an image of the initial acoustic pressure distribution within soft tissue. This image is related to the tissue optical coefficients and therefore carries information about the tissue physiology. For high resolution imaging, a large-area detector array with a high density of small, sensitive elements is required. Such arrays can be expensive, so reverberant-field PAT has been suggested as a means of obtaining PAT images using arrays with a smaller number of detectors. By recording the reflections from an acoustically reverberant cavity surrounding the sample, in addition to the primary acoustic pulse, sufficient information may be captured to allow an image to be reconstructed without the need for a large-area array. An initial study using two-dimensional simulations was performed to assess the feasibility of using a single detector for PAT. It is shown that reverberant-field data recorded at a single detector are sufficient to reconstruct the initial pressure distribution accurately, so long as the shape of the reverberant cavity makes it ray-chaotic. The practicalities of such an approach to photoacoustic imaging are discussed.

Equivalent-Source Acoustic Holography for Projecting Measured Ultrasound Fields Through Complex Media.

Holographic projections of experimental ultrasound measurements generally use the angular spectrum method or Rayleigh integral, where the measured data are imposed as a Dirichlet boundary condition. In contrast, full-wave models, which can account for more complex wave behavior, often use interior mass or velocity sources to introduce acoustic energy into the simulation. Here, a method to generate an equivalent interior source that reproduces the measurement data is proposed based on gradient-based optimization. The equivalent-source can then be used with full-wave models (for example, the open-source k-Wave toolbox) to compute holographic projections through complex media including nonlinearity and heterogeneous material properties. Numerical and experimental results using both time-domain and continuous-wave sources are used to demonstrate the accuracy of the approach.

k-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics.

Biomedical applications of photoacoustics, in particular photoacoustic tomography, require efficient models of photoacoustic propagation that can incorporate realistic properties of soft tissue, such as acoustic inhomogeneities both for purposes of simulation and for use in model-based image reconstruction methods. k-space methods are well suited to modeling high-frequency acoustics applications as they require fewer mesh points per wavelength than conventional finite element and finite difference models, and larger time steps can be taken without a loss of stability or accuracy. They are also straightforward to encode numerically, making them appealing as a general tool. The rationale behind k-space methods and the k-space approach to the numerical modeling of photoacoustic waves in fluids are covered in this paper. Three existing k-space models are applied to photoacoustics and demonstrated with examples: an exact model for homogeneous media, a second-order model that can take into account heterogeneous media, and a first-order model that can incorporate absorbing boundary conditions.

Characterisation of a phantom for multiwavelength quantitative photoacoustic imaging.

Quantitative photoacoustic imaging (qPAI) has the potential to provide high- resolution in vivo images of chromophore concentration, which may be indicative of tissue function and pathology. Many strategies have been proposed recently for extracting quantitative information, but many have not been experimentally verified. Experimental phantom-based validation studies can be used to test the robustness and accuracy of such algorithms in order to ensure reliable in vivo application is possible. The phantoms used in such studies must have well-characterised optical and acoustic properties similar to tissue, and be versatile and stable. Polyvinyl chloride plastisol (PVCP) has been suggested as a phantom for quality control and system evaluation. By characterising its multiwavelength optical properties, broadband acoustic properties and thermoelastic behaviour, this paper examines its potential as a phantom for qPAI studies too. PVCP's acoustic properties were assessed for various formulations, as well as its intrinsic optical absorption, and scattering with added TiO2, over a range of wavelengths from 400-2000 nm. To change the absorption coefficient, pigment-based chromophores that are stable during the phantom fabrication process, were used. These yielded unique spectra analogous to tissue chromophores and linear with concentration. At the high peak powers typically used in photoacoustic imaging, nonlinear optical absorption was observed. The Grüneisen parameter was measured to be [Formula: see text] = 1.01 ± 0.05, larger than typically found in tissue, though useful for increased PA signal. Single and multiwavelength 3D PA imaging of various fabricated PVCP phantoms were demonstrated.

Time domain simulation of harmonic ultrasound images and beam patterns in 3D using the k-space pseudospectral method.

A k-space pseudospectral model is developed for the fast full-wave simulation of nonlinear ultrasound propagation through heterogeneous media. The model uses a novel equation of state to account for nonlinearity in addition to power law absorption. The spectral calculation of the spatial gradients enables a significant reduction in the number of required grid nodes compared to finite difference methods. The model is parallelized using a graphical processing unit (GPU) which allows the simulation of individual ultrasound scan lines using a 256 x 256 x 128 voxel grid in less than five minutes. Several numerical examples are given, including the simulation of harmonic ultrasound images and beam patterns using a linear phased array transducer.

k-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields.

A new, freely available third party MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields is described. The toolbox, named k-Wave, is designed to make realistic photoacoustic modeling simple and fast. The forward simulations are based on a k-space pseudo-spectral time domain solution to coupled first-order acoustic equations for homogeneous or heterogeneous media in one, two, and three dimensions. The simulation functions can additionally be used as a flexible time reversal image reconstruction algorithm for an arbitrarily shaped measurement surface. A one-step image reconstruction algorithm for a planar detector geometry based on the fast Fourier transform (FFT) is also included. The architecture and use of the toolbox are described, and several novel modeling examples are given. First, the use of data interpolation is shown to considerably improve time reversal reconstructions when the measurement surface has only a sparse array of detector points. Second, by comparison with one-step, FFT-based reconstruction, time reversal is shown to be sufficiently general that it can also be used for finite-sized planar measurement surfaces. Last, the optimization of computational speed is demonstrated through parallel execution using a graphics processing unit.

Quantitative Photoacoustic Image Reconstruction using Fluence Dependent Chromophores.

In biomedical photoacoustic imaging the images are proportional to the absorbed optical energy density, and not the optical absorption, which makes it difficult to obtain a quantitatively accurate image showing the concentration of a particular absorbing chromophore from photoacoustic measurements alone. Here it is shown that the spatially varying concentration of a chromophore whose absorption becomes zero above a threshold light fluence can be estimated from photoacoustic images obtained at increasing illumination strengths. This technique provides an alternative to model-based multiwavelength approaches to quantitative photoacoustic imaging, and a new approach to photoacoustic molecular and functional imaging.

Estimating chromophore distributions from multiwavelength photoacoustic images.

Biomedical photoacoustic tomography (PAT) can provide qualitative images of biomedical soft tissue with high spatial resolution. However, whether it is possible to give accurate quantitative estimates of the spatially varying concentrations of the sources of photoacoustic contrast-endogenous or exogenous chromophores-remains an open question. Even if the chromophores' absorption spectra are known, the problem is nonlinear and ill-posed. We describe a framework for obtaining such quantitative estimates. When the optical scattering distribution is known, adjoint and gradient-based optimization techniques can be used to recover the concentration distributions of the individual chromophores that contribute to the overall tissue absorption. When the scattering distribution is unknown, prior knowledge of the wavelength dependence of the scattering is shown to be sufficient to overcome the absorption-scattering nonuniqueness and allow both distributions of chromophore concentrations and scattering to be recovered from multiwavelength photoacoustic images.

Fast calculation of pulsed photoacoustic fields in fluids using k-space methods.

Two related numerical models that calculate the time-dependent pressure field radiated by an arbitrary photoacoustic source in a fluid, such as that generated by the absorption of a short laser pulse, are presented. Frequency-wavenumber (k-space) implementations have been used to produce fast and accurate predictions. Model I calculates the field everywhere at any instant of time, and is useful for visualizing the three-dimensional evolution of the wave field. Model II calculates pressure time series for points on a straight line or plane and is therefore useful for simulating array measurements. By mapping the vertical wavenumber spectrum directly to frequency, this model can calculate time series up to 50 times faster than current numerical models of photoacoustic propagation. As the propagating and evanescent parts of the field are calculated separately, model II can be used to calculate far- and near-field radiation patterns. Also, it can readily be adapted to calculate the velocity potential and thus particle velocity and acoustic intensity vectors. Both models exploit the efficiency of the fast Fourier transform, and can include the frequency-dependent directional response of an acoustic detector straightforwardly. The models were verified by comparison with a known analytic solution and a slower, but well-understood, numerical model.

Magnesium supplementation of lactating ewes and lamb growth.

Ewes rearing twin lambs on a restricted ration of irrigated pasture were either supplemented daily with 12.5 mg magnesium chloride or not. Serum magnesium levels measured 24 hours after the previous day's dose did not differ between treatments. They averaged 0.74 mmol/l during the period of supplementation with 36% of values below 0.7 mmol/l, the lower limit of the "normal" range. Milk consumption by and liveweight gains of lambs were not increased by magnesium supplementation indicating that ewe magnesium levels below 0.7 mmol/l serum did not limit production in this instance.

Serology of a Neospora abortion outbreak on a dairy farm in New Zealand: a case study.

AIM: To describe the kinetics of serological titres after an abortion outbreak in April-May 1995 due to Neospora caninum affected 17 dairy cows in a herd of 320. METHODS: Thirty-five cows, that had either aborted, carried mummified calves, were not pregnant or calved normally were: bled several times at regular intervals and the sera tested for Neospora antibodies in the indirect fluorescence antibody test (IFAT). RESULTS: Maximal IFAT titres of up to 1:4000 occurred within 6 weeks of the abortion outbreak, decreased over the next 2 months to < or = 1:200 and remained at this level until the next scheduled bleed a further 2 months later. A rise in titres was subsequently observed in the cows that had aborted or were not pregnant (at the time of the abortions) or had carried mummified foetuses. Seroconversion was also observed in some of the control cows, which had, up until then, remained seronegative. A dog and cat in contact with the cows in the herd investigated were, however, negative in the IFAT. CONCLUSIONS: Maximal serological titres in Neospora abortions are observed within weeks of the abortion event and then quickly return to very low levels. Subsequently, a recrudescence of titres can be observed in infected cows during the next pregnancy, without it being associated with repeat abortions.