Kendall R. Waters

On the applicability of Kramers-Kronig relations for ultrasonic attenuation obeying a frequency power law

In the recent literature concern has been raised regarding the validity of Kramers-Kronig relations for media with ultrasonic attenuation obeying a frequency power law. It is demonstrated, however, that the Kramers-Kronig dispersion relations for application to these types of media are available. The developed dispersion relations are compared with measurements on several liquids, and agreement is found to better than 1 m/s over the experimentally available bandwidth. A discussion regarding the validity of these dispersion relations, in particular how the dispersion relations relate to the so-called Paley-Wiener conditions, forms the conclusion.

Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion

Causality imposes restrictions on both the time-domain and frequency-domain responses of a system. The Kramers-Kronig (K-K) relations relate the real and imaginary parts of the frequency-domain response. In ultrasonics, K-K relations often are used to link attenuation and dispersion. We review both integral and differential forms of the frequency-domain K-K relations that are relevant to theoretical models and laboratory measurements. We consider two methods for implementing integral K-K relations for the case of finite-bandwidth data, namely, extrapolation of data and restriction of integration limits. For the latter approach, we discuss the accuracy of K-K predictions for specific classes of system behavior and how the truncation of the integrals affects this accuracy. We demonstrate the accurate prediction of attenuation and dispersion using several forms of the K-K relations relevant to experimental measurements of media with attenuation coefficients obeying a frequency power law and media consisting of resonant scatterers. We also review the time-causal relations that describe the time-domain consequences of causality in the wave equation. These relations can be thought of as time-domain analogs of the (frequency-domain) K-K relations. Causality-imposed relations, such as the K-K and time-causal relations, provide useful tools for the analysis of measurements and models of acoustic systems.

Differential forms of the Kramers-Kronig dispersion relations

Differential forms of the Kramers-Kronig dispersion relations provide an alternative to the integral Kramers-Kronig dispersion relations for comparison with finite-bandwidth experimental data. The differential forms of the Kramers-Kronig relations are developed in the context of tempered distributions. Results are illustrated for media with attenuation obeying an arbitrary frequency power law (/spl alpha/(/spl omega/) = /spl alpha//sub 0/ + /spl alpha//sub 1/ |/spl omega/|/sup y/). Dispersion predictions using the differential dispersion relations are compared to the measured dispersion for a series of specimens (two polymers, an egg yolk, and two liquids) exhibiting attenuation obeying a frequency power law (1.00 /spl les/ y /spl les/ 1.99), with very good agreement found. For this form of ultrasonic attenuation, the differential Kramers-Kronig dispersion prediction is found to be identical to the (integral) Kramers-Kronig dispersion prediction.

On a time-domain representation of the Kramers–Krönig dispersion relations

The development of Kramers-Kronig dispersion relations is typically carried out in the frequency domain. An alternative approach known as the time-causal theory develops dispersion relations for media with attenuation obeying a frequency power law through analysis in the time domain [T. L. Szabo, J. Acoust. Soc. Am. 96, 491-500 (1994)]. Although both approaches predict identical dispersion relations, it is perceived that these two approaches are distinct from each other. It is shown, however, that the time-causal theory is in essence a time-domain formulation of the Kramers-Kronig dispersion relations for the special case of media with attenuation obeying a frequency power law. Additionally, it is shown that time-domain representations of the Kramers-Kronig dispersion relations are available for a broader class of media than simply those with power law attenuation. The time-causal theory and the Kramers-Kronig dispersion relations can be viewed as two complementary, yet equivalent, approaches to the study of dispersion.

Kramers–Kronig relations applied to finite bandwidth data from suspensions of encapsulated microbubbles

In this work, the Kramers-Kronig (K-K) relations are applied to experimental data of resonant nature by limiting the interval of integration to the measurement spectrum. The data are from suspensions of encapsulated microbubbles (Albunex) and have the characteristics of an ultrasonic notch filter. The goal is to test the consistency of this dispersion and attenuation data with the Kramers-Kronig relations in a strict manner, without any parameters from outside the experimental bandwidth entering in to the calculations. In the course of reaching the goal, the artifacts associated with the truncation of the integrals are identified and it is shown how their impacts on the results can be minimized. The problem is first approached analytically by performing the Kramers-Kronig calculations over a restricted spectral band on a specific Hilbert transform pair (Lorentzian curves). The resulting closed-form solutions illustrate the type of artifacts that can occur due to truncation and also show that accurate results can be achieved. Next, both twice-subtracted and lower-order Kramers-Kronig relations are applied directly to the attenuation and dispersion data from the encapsulated microbubbles. Only parameters from within the experimental attenuation coefficient and phase velocity data sets are used. The twice-subtracted K-K relations produced accurate estimates for both the attenuation coefficient and dispersion across all 12 data sets. Lower-order Kramers-Kronig relations also produced good results over the finite spectrum for most of the data. In 2 of the 12 cases, the twice-subtracted relations tracked the data markedly better than the lower-order predictions. These calculations demonstrate that truncation artifacts do not overwhelm the causal link between the phase velocity and the attenuation coefficient for finite bandwidth calculations. This work provides experimental evidence supporting the validity of the subtracted forms of the acoustic K-K relations between the phase velocity and attenuation coefficient.

Experimental determination of phase velocity of perfluorocarbons: Applications to targeted contrast agents

Targeted acoustic contrast agents are designed to enhance the sensitivity and specificity of ultrasonic diagnoses. We have previously developed a ligand targeted ultrasonic contrast system that is a lipid-encapsulated, liquid-perfluorocarbon emulsion. The emulsion particles are small (250 nm) and have inherently low echogenicity unless bound to a surface by a pretargeted ligand through avidin-biotin interactions. We have recently proposed a simple acoustic transmission line model that treats the emulsion particles as a thin layer over the targeted surface. In this model, the acoustic reflectivity of the sample increases for perfluorocarbons with smaller velocities of longitudinal sound or lower densities. In this study, we measure and report the velocity of longitudinal sound for 20 perfluorocarbons using a broadband phase spectroscopic approach for estimating phase velocities. Experimentally determined velocities ranged from 520/spl plusmn/2 m/sec (perfluorohexane) to 705/spl plusmn/5 m/s (perfluorodecalin). No measurable dispersion was observed over the useful bandwidth of 2 to 22 MHz. Increasing carbon backbone chain length and fluorine substitution with halogens of greater atomic weight increased the measured speed of sound. Our experimental data were consistent (R=0.87) with a published empirical model that predicts velocity as a function of molecular structure. These data provide a rational basis for optimizing targeted perfluorocarbon-based contrast agents and offer further insight into the physical mechanisms responsible for the observed enhancement of surface acoustic reflectivity.

Measurements and predictions of the phase velocity and attenuation coefficient in suspensions of elastic microspheres

The phase velocities and attenuation coefficients for suspensions of narrowly sized polymer microspheres are reported over a broadband spectrum from 3 to 30 MHz. The six suspensions used in this work contain microspheres with respective average diameters near 40 μm, 50 μm, 60 μm, 70 μm, 80 μm, and 100 μm. The results of these measurements are compared with theoretical expressions for the phase velocity and attenuation coefficient derived from the scattering properties of an elastic sphere in water using the weak scattering limit of the Waterman and Truell dispersion relation [J. Math. Phys. 2, 512–537 (1961)]. This single-scattering limit of the theory is found to be sufficient for predicting the ultrasonic transport properties of these suspensions to a considerable degree of accuracy.

Kramers-Kronig analysis of attenuation and dispersion in trabecular bone.

A restricted-bandwidth form of the Kramers-Kronig dispersion relations is applied to in vitro measurements of ultrasonic attenuation and dispersion properties of trabecular bone specimens from bovine tibia. The Kramers-Kronig analysis utilizes only experimentally measured properties and avoids extrapolation of ultrasonic properties beyond the known bandwidth. Compensation for the portions of the Kramers-Kronig integrals over the unknown bandwidth is partially achieved by the method of subtractions, where a subtraction frequency acts as an adjustable parameter. Good agreement is found between experimentally measured and Kramers-Kronig reconstructed dispersions. The restricted-bandwidth approach improves upon other forms of the Kramers-Kronig relations and may provide further insight into how ultrasound interacts with trabecular bone.

Acoustic measurements of scattering by objects of irregular shape

Although many scatterers in nature are irregular in shape, the data on the scattering of sound by asymmetrical bodies is relatively limited. Our understanding of the scattering of underwater sound by nonregular shaped targets is therefore significantly less developed than that due to scattering by scatterers of symmetry. In the present work, measurements are reported on the backscattering from irregularly shaped solid elastic particles in both the time and frequency domains. Backscatter data have been collected for a variety of irregularly shaped particles at varying orientations. These observations show a number of characteristics which are comparable with scattering by targets of symmetry, particularly the property of the scattered echo being significantly longer in duration than the insonifying pulse. This pulse elongation for symmetrical targets is normally interpreted using resonance scattering theory, RST, as arising from radiation due to surface eigen waves on the body of the scatterer. In the present work the interpretation of the scattering by irregular shaped targets has been guided by resonance scattering studies, and an understanding of the response formulated within this framework.

Backscatter imaging and myocardial tissue characterization

The goal of myocardial ultrasonic tissue characterization is to complement two-dimensional and Doppler echocardiography by providing information (such as assessment of regional viability based on localized values of backscatter) beyond that derived from an assessment of myocardial dimensions and motion. Quantitative backscatter imaging can be subdivided into three broad areas: (1) direct applications, in which specific pathologies are identified and monitored, (2) indirect applications, in which quantitative techniques designed for use in tissue characterization serve to expand the role of echocardiography, and (3) contributions to the understanding of cardiac structure and function.