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

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.

A Multi-Grid Iterative Method for Photoacoustic Tomography

Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multi-grid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction.

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.

Sensitivity of simulated transcranial ultrasound fields to acoustic medium property maps

High intensity transcranial focused ultrasound is an FDA approved treatment for essential tremor, while low-intensity applications such as neurostimulation and opening the blood brain barrier are under active research. Simulations of transcranial ultrasound propagation are used both for focusing through the skull, and predicting intracranial fields. Maps of the skull acoustic properties are necessary for accurate simulations, and can be derived from medical images using a variety of methods. The skull maps range from segmented, homogeneous models, to fully heterogeneous models derived from medical image intensity. In the present work, the impact of uncertainties in the skull properties is examined using a model of transcranial propagation from a single element focused transducer. The impact of changes in bone layer geometry and the sound speed, density, and acoustic absorption values is quantified through a numerical sensitivity analysis. Sound speed is shown to be the most influential acoustic property, and must be defined with less than 4% error to obtain acceptable accuracy in simulated focus pressure, position, and volume. Changes in the skull thickness of as little as 0.1 mm can cause an error in peak intracranial pressure of greater than 5%, while smoothing with a 1 [Formula: see text] kernel to imitate the effect of obtaining skull maps from low resolution images causes an increase of over 50% in peak pressure. The numerical results are confirmed experimentally through comparison with sonications made through 3D printed and resin cast skull bone phantoms.

Fractional biharmonic operator equation model for arbitrary frequency-dependent scattering attenuation in acoustic wave propagation

This paper proposes a fractional biharmonic operator equation model in the time-space domain to describe scattering attenuation of acoustic waves in heterogeneous media. Compared with the existing models, the proposed fractional model is able to describe arbitrary frequency-dependent scattering attenuation, which typically obeys an empirical power law with an exponent ranging from 0 to 4. In stark contrast to an extensive and rapidly increasing application of the fractional derivative models for wave absorption attenuation in the literature, little has been reported on frequency-dependent scattering attenuation. This is largely because the order of the fractional Laplacian is from 0 to 2 and is infeasible for scattering attenuation. In this study, the definition of the fractional biharmonic operator in space with an order varying from 0 to 4 is proposed, as well as a fractional biharmonic operator equation model of scattering attenuation which is consistent with arbitrary frequency power-law dependency and obeys the causal relation under the smallness approximation. Finally, the correlation between the fractional order and the ratio of wavelength to the diameter of the scattering heterogeneity is investigated and an expression on exponential form is also provided.

Uniform scattering and attenuation of acoustically sorted ultrasound contrast agents: Modeling and experiments

The sensitivity and efficiency in contrast-enhanced ultrasound imaging and therapy can potentially be increased by the use of resonant monodisperse bubbles. However, bubbles of the same size may respond differently to ultrasound due to differences in their phospholipid shell. In an acoustic bubble sorting chip, resonant bubbles can be separated from the polydisperse agent. Here, a sample of acoustically sorted bubbles is characterized by measuring scattering and attenuation simultaneously using narrowband acoustic pulses at peak negative pressures of 10, 25, and 50 kPa over a 0.7-5.5 MHz frequency range. A second sample is characterized by attenuation measurements at acoustic pressures ranging from 5 to 75 kPa in steps of 2.5 kPa. Scattering and attenuation coefficients were modeled by integration over the pressure and frequency dependent response of all bubbles located within the non-uniform acoustic characterization beam. For all driving pressures and frequencies employed here, the coefficients could be modeled using a single and unique set of shell parameters confirming that acoustically sorted bubbles provide a uniform acoustic response. Moreover, it is shown that it is crucial to include the pressure distribution of the acoustic characterization beam in the modeling to accurately determine shell parameters of non-linearly oscillating bubbles.

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.

Nonlinear acoustic pulse propagation in dispersive sediments using fractional loss operators

The nonlinear progressive wave equation (NPE) is a time-domain formulation of the Euler fluid equations designed to model low-angle wave propagation using a wave-following computational domain. The wave-following frame of reference permits the simulation of long-range propagation and is useful in modeling blast wave effects in the ocean waveguide. Existing models do not take into account frequency-dependent sediment attenuation, a feature necessary for accurately describing sound propagation over, into, and out of the ocean sediment. Sediment attenuation is addressed in this work by applying lossy operators to the governing equation that are based on a fractional Laplacian. These operators accurately describe frequency-dependent attenuation and dispersion in typical ocean sediments. However, dispersion within the sediment is found to be a secondary process to absorption and effectively negligible for ranges of interest. The resulting fractional NPE is benchmarked against a Fourier-transformed parabolic equation solution for a linear case, and against the analytical Mendousse solution to Burgers' equation for the nonlinear case. The fractional NPE is then used to investigate the effects of attenuation on shock wave propagation.

Estimation of shear modulus in media with power law characteristics

Shear wave propagation in tissue generated by the radiation force is usually modeled by either a lossless or a classical viscoelastic equation. However, experimental data shows power law behavior which is not consistent with those approaches. It is well known that fractional derivatives results in power laws, therefore a time fractional wave equation, the Caputo equation, which can be derived from the fractional Kelvin-Voigt stress and strain relation is tested. This equation is solved using the finite difference method with experimental parameters obtained from the existing literature. The equation is characterized by a fractional order which is also the power law exponent of the frequency dependent shear modulus. It is shown that for fractional order between 0 and 1, the equation gives smaller shear modulus than the classical model. The opposite situation applies for fractional order greater than 1. The numerical simulation also shows that the shear wave velocity method is only reliable for small losses. In our case, this is only for a small fractional order. Based on the published values of fractional order from other studies, there is therefore a chance for biased estimation of the shear modulus.

Modeling of wave propagation for medical ultrasound: a review

Numerical modeling of medical ultrasound has advanced tremendously in the past two decades. This opens up a great number of opportunities for medical ultrasound and associated technologies. Numerous new governing equations and algorithms have emerged and been applied to studying various medical ultrasound applications, including ultrasound imaging, photo-acoustic imaging, and therapeutic ultrasound. In addition, thanks to the rapid development of computers, modeling acoustic wave propagation in three-dimensional, large-scale domains has become a reality. This article will provide an indepth literature and technical review of recent progress on numerical modeling of medical ultrasound. Future challenges will also be discussed.

STOCHASTIC SOLUTIONS FOR FRACTIONAL WAVE EQUATIONS

A fractional wave equation replaces the second time derivative by a Caputo derivative of order between one and two. In this paper, we show that the fractional wave equation governs a stochastic model for wave propagation, with deterministic time replaced by the inverse of a stable subordinator whose index is one half the order of the fractional time derivative.

Ultrasonic broadband characterization of a viscous liquid: methods and perturbation factors

The perturbation factors involved in ultrasonic broadband characterization of viscous fluids are analyzed. Precisely, the normal incidence error and the thermal sensitivity of the properties have been identified as dominant parameters. Thus, the sensitivity of the ultrasonic parameters of attenuation and phase velocity were measured at room temperature in the MHz frequency range for two reference silicone oils, namely 47V50 and 47V350 (Rhodorsil). Several methods of characterization were carried out: time of flight, cross-correlation and spectral method. These ultrasonic parameters are measured at room temperature. For this family of silicone oil, the dispersion of the attenuation spectrum is modeled by a power law. The velocity dispersion is modeled by two dispersion models: the quasi-local and the temporal causal. The impact of the experimental reproducibility of the phase velocity and acoustic attenuation was measured in the MHz frequency range, using a set of ultrasonic transducers with different center frequencies. These measurements are used to identify the dispersion of the ultrasonic parameters as a function of the frequency. A first experimental and descriptive approach is developed to assess the reproducibility of the normal incidence between the acoustic beam and the viscoelastic material. Thus, the relative error on the measurements of velocity and attenuation are directly related to the angular deviation of the ultrasonic wave, as well as the sampling and signal-to-noise ratio. A second experimental and phenomenological approach deals with the effect of a temperature change, typical of a polymerization reaction. As a result, the sensitivity of the phase velocity of silicone oil 47V50 was evaluated around -2 ms(-1) K(-1).

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.

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.

Inverse problems of ultrasound tomography in models with attenuation

We develop efficient methods for solving inverse problems of ultrasound tomography in models with attenuation. We treat the inverse problem as a coefficient inverse problem for unknown coordinate-dependent functions that characterize both the speed cross section and the coefficients of the wave equation describing attenuation in the diagnosed region. We derive exact formulas for the gradient of the residual functional in models with attenuation, and develop efficient algorithms for minimizing the gradient of the residual by solving the conjugate problem. These algorithms are easy to parallelize when implemented on supercomputers, allowing the computation time to be reduced by a factor of several hundred compared to a PC. The numerical analysis of model problems shows that it is possible to reconstruct not only the speed cross section, but also the properties of the attenuating medium. We investigate the choice of the initial approximation for iterative algorithms used to solve inverse problems. The algorithms considered are primarily meant for the development of ultrasound tomographs for differential diagnosis of breast cancer.

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.

Probe beam deflection technique as acoustic emission directionality sensor with photoacoustic emission source

The goal of this paper is to demonstrate the unique capability of measuring the vector or angular information of propagating acoustic waves using an optical sensor. Acoustic waves were generated using photoacoustic interaction and detected by the probe beam deflection technique. Experiments and simulations were performed to study the interaction of acoustic emissions with an optical sensor in a coupling medium. The simulated results predict the probe beam and wavefront interaction and produced simulated signals that are verified by experiment.

Modeling nonlinear wave propagation on nonuniform grids using a mapped k-space pseudospectral method

Simulating the propagation of nonlinear ultrasound waves is computationally difficult because of the dense grids needed to capture high-frequency harmonics. Here, a mapped k-space pseudospectral method is presented which allows the use of nonuniform grid spacings. This enables grid points to be clustered around steep regions of the wave field. Compared with using a uniform grid, this significantly reduces the total number of grid points needed for accurate simulations. Two methods for selecting a suitable nonuniform grid mapping are discussed.

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.

A k-space method for moderately nonlinear wave propagation

A k-space method for moderately nonlinear wave propagation in absorptive media is presented. The Westervelt equation is first transferred into k-space via Fourier transformation, and is solved by a modified wave-vector time-domain scheme. The present approach is not limited to forward propagation or parabolic approximation. One- and two-dimensional problems are investigated to verify the method by comparing results to analytic solutions and finite-difference time-domain (FDTD) method. It is found that to obtain accurate results in homogeneous media, the grid size can be as little as two points per wavelength, and for a moderately nonlinear problem, the Courant-Friedrichs-Lewy number can be as large as 0.4. Through comparisons with the conventional FDTD method, the k-space method for nonlinear wave propagation is shown here to be computationally more efficient and accurate. The k-space method is then employed to study three-dimensional nonlinear wave propagation through the skull, which shows that a relatively accurate focusing can be achieved in the brain at a high frequency by sending a low frequency from the transducer. Finally, implementations of the k-space method using a single graphics processing unit shows that it required about one-seventh the computation time of a single-core CPU calculation.

Photoacoustic image reconstruction in an attenuating medium using singular-value decomposition

Attenuation effects can be significant in photoacoustic tomography since the generated pressure signals are broadband, and ignoring them may lead to image artifacts and blurring. La Rivière et al. [Opt. Lett. 31(6), pp. 781-783, (2006)] had previously derived a method for modeling the attenuation effect and correcting for it in the image reconstruction. This was done by relating the ideal, unattenuated pressure signals to the attenuated pressure signals via an integral operator. We derive an integral operator relating the attenuated pressure signals to the absorbed optical energy for a planar measurement geometry. The matrix operator relating the two quantities is a function of the temporal frequency, attenuation coefficient and the two-dimensional spatial frequency. We perform singular-value decomposition (SVD) of this integral operator to study the problem further. We find that the smallest singular values correspond to wavelet-like eigenvectors in which most of the energy is concentrated at times corresponding to greater depths in tissue. This allows us to characterize the ill-posedness of recovering the absorbed optical energy distribution at different depths in an attenuating medium. This integral equation can be inverted using standard SVD methods, and the initial pressure distribution can be recovered. We conduct simulations and derive an algorithm for image reconstruction using SVD for a planar measurement geometry. We also study the noise and resolution properties of this image-reconstruction method.

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.

Experimental evaluation of time domain models for ultrasound attenuation losses in photoacoustic imaging

Understanding and compensating ultrasound attenuation losses is an important issue in photoacoustic imaging. To contribute to this effort, simulated attenuated time domain waveforms are compared to experimental waveforms. The experimental waveforms are acquired by transmitting broadband ultrasound pulses through distilled water and porcine fat tissue. Three well-known modeling approaches are examined in detail with regard to accuracy and computation time. Furthermore, the influence of attenuation on imaging resolution is addressed. In the present paper, the focus lies on the calculation of attenuated detector signals. The results, however, also provide clues about the quality of image reconstruction.

Time-reversal transcranial ultrasound beam focusing using a k-space method

This paper proposes the use of a k-space method to obtain the correction for transcranial ultrasound beam focusing. Mirroring past approaches, a synthetic point source at the focal point is numerically excited, and propagated through the skull, using acoustic properties acquired from registered computed tomography of the skull being studied. The received data outside the skull contain the correction information and can be phase conjugated (time reversed) and then physically generated to achieve a tight focusing inside the skull, by assuming quasi-plane transmission where shear waves are not present or their contribution can be neglected. Compared with the conventional finite-difference time-domain method for wave propagation simulation, it will be shown that the k-space method is significantly more accurate even for a relatively coarse spatial resolution, leading to a dramatically reduced computation time. Both numerical simulations and experiments conducted on an ex vivo human skull demonstrate that precise focusing can be realized using the k-space method with a spatial resolution as low as only 2.56 grid points per wavelength, thus allowing treatment planning computation on the order of minutes.

Linking multiple relaxation, power-law attenuation, and fractional wave equations

The acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media, e.g., biological tissue, polymers, rocks, and rubber. Recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power-law attenuation. On the other hand, a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by Nachman et al. [J. Acoust. Soc. Am. 88, 1584-1595 (1990)]. Through assumption of a continuum of relaxation mechanisms, each with an effective compressibility described by a distribution related to the Mittag-Leffler function, this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation. This work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling.

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.

A causal and fractional all-frequency wave equation for lossy media

This work presents a lossy partial differential acoustic wave equation including fractional derivative terms. It is derived from first principles of physics (mass and momentum conservation) and an equation of state given by the fractional Zener stress-strain constitutive relation. For a derivative order α in the fractional Zener relation, the resulting absorption α(k) obeys frequency power-laws as α(k) ∝ ω(1+α) in a low-frequency regime, α(k) ∝ ω(1-α/2) in an intermediate-frequency regime, and α(k) ∝ ω(1-α) in a high-frequency regime. The value α=1 corresponds to the case of a single relaxation process. The wave equation is causal for all frequencies. In addition the sound speed does not diverge as the frequency approaches infinity. This is an improvement over a previously published wave equation building on the fractional Kelvin-Voigt constitutive relation.

Nonlinear acoustic wave equations with fractional loss operators

Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlinear wave equation which describes attenuation and dispersion laws that match observations. This wave equation is a generalization of the Westervelt equation, and also leads to a fractional version of the Khokhlov-Zabolotskaya-Kuznetsov and Burgers' equations.

Newtonian viscous effects in ultrasonic emboli removal from blood

We have modeled the removal of emboli from cardiopulmonary bypass circuits via acoustic radiation force. Unless removed, emboli can result in cognitive deficit for those undergoing heart surgery with the use of extracorporeal circuits. There are a variety of mathematical formulations in the literature describing acoustic radiation force, but a lingering question that remains is how important viscosity of the blood and/or embolus is to the process. We implemented both inviscid and viscous models for acoustic radiation force on a sphere immersed in a fluid. We found that for this specific application, the inviscid model seems to be sufficient for predicting acoustic force upon emboli when compared with the chosen viscous model. Thus, the much simpler inviscid model could be used to optimize experimental techniques for ultrasonic emboli removal.

Wave simulation in biologic media based on the Kelvin-Voigt fractional-derivative stress-strain relation

The acoustic behavior of biologic media can be described more realistically using a stress-strain relation based on fractional time derivatives of the strain, since the fractional exponent is an additional fitting parameter. We consider a generalization of the Kelvin-Voigt rheology to the case of rational orders of differentiation, the so-called Kelvin-Voigt fractional-derivative (KVFD) constitutive equation, and introduce a novel modeling method to solve the wave equation by means of the Grünwald-Letnikov approximation and the staggered Fourier pseudospectral method to compute the spatial derivatives. The algorithm can handle complex geometries and general material-property variability. We verify the results by comparison with the analytical solution obtained for wave propagation in homogeneous media. Moreover, we illustrate the use of the algorithm by simulation of wave propagation in normal and cancerous breast tissue.

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.

Measurement of the ultrasound attenuation and dispersion in whole human blood and its components from 0-70 MHz

The ultrasound attenuation coefficient and dispersion from 0-70 MHz in whole human blood and its components (red blood cells and plasma) at 37°C is reported. The measurements are made using a fixed path substitution technique that exploits optical mechanisms for the generation and detection of ultrasound. This allows the measurements to cover a broad frequency range with a single source and receiver. The measured attenuation coefficient and dispersion in solutions of red blood cells and physiological saline for total haemoglobin concentrations of 10, 15 and 20 g/dL are presented. The attenuation coefficient and dispersion in whole human blood taken from four healthy volunteers by venipuncture is also reported. The power law dependence of the attenuation coefficient is shown to vary across the measured frequency range. This is due to the varying frequency dependence of the different mechanisms responsible for the attenuation. The attenuation coefficient measured at high frequencies is found to be significantly higher than that predicted by historical power law parameters. A review of the attenuation mechanisms in blood along with previously reported experimental measurements is given. Values for the sound speed and density in the tested samples are also presented.