Joel Mobley

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.

Broadband through-transmission signal loss measurements of Albunex® suspensions at concentrations approaching in vivo doses

Broadband normalized signal loss of commercially available Albunex®, an ultrasonic contrast agent, was measured in vitro at concentrations approaching those which may be found in vivo for clinical doses. The measurements were made using a novel specimen chamber, careful material handling procedures, and a broadband square wave pulser system. Results were obtained over the full bandwidth of the experimental system (1 to 20 MHz) at concentrations up to 1.9×106 microspheres/mL. Further results were obtained over a partial bandwidth of the experimental system at concentrations up to 1.5×107 microspheres/mL. The frequency-dependent signal loss exhibited a peak for all concentrations investigated. In the meaningful bandwidth of the system, the signal loss (expressed in dB) was directly proportional to microsphere concentration. The experimental results for normalized signal loss were compared with predictions from a linear single-scattering model for encapsulated bubbles. The experimental data was used to estim...

Cyclic Variation of Integrated Backscatter: Dependence of Time Delay on the Echocardiographic View Used and the Myocardial Segment Analyzed

To determine the influence of myocardial anisotropy in ultrasonic tissue characterization, we measured the time delay (and magnitude) of the cyclic variation of myocardial integrated backscatter from specific segments visualized in the 4 standard transthoracic echocardiographic views. The cyclic variation data in 10 myocardial regions were obtained from analyses of 2-dimensional integrated backscatter images from 23 healthy subjects. Resultant values (mean +/- SD) for the time delay were as follows: parasternal long-axis view: 1.08 +/- 0.17 (septum) and 1.00 +/- 0.14 (posterior wall); parasternal short-axis view: 1.03 +/- 0.16 (anterior septum), 1.03 +/- 0.14 (posterior wall), 2.22 +/- 0.71 (lateral wall), and 1.65 +/- 0.66 (posterior septum); apical 4-chamber view: 1.08 +/- 0.31 (septum) and 2.20 +/- 0.79 (lateral wall); and apical 2-chamber view: 1.68 +/- 0.62 (inferior wall) and 2.04 +/- 0.72 (anterior wall). Hence, results of this study indicate that myocardial ultrasonic characterization that uses the cyclic variation is influenced by the echocardiographic view and the specific segment of the left ventricle.

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.

Broadband measurements of phase velocity in Albunex® suspensions

The phase velocities in suspensions of Albunex, an ultrasound contrast agent, are determined using the technique of broadband phase spectroscopy. In these suspensions, phase velocities were measured as a function of Albunex concentration in Isoton II (buffered saline). The dispersion over the measurement bandwidth (1-15 MHz) grew with increasing Albunex concentration, exhibiting a 221-m/s change (17% increase) in phase velocity between 1.0 and 3.8 MHz at the highest concentration reported (32 microL Albunex/8 mL Isoton). The dispersion behavior of the Albunex suspensions is described using a model of encapsulated gas bubbles. The influence of the dispersion in Albunex on broadband pulses is discussed, as well as the potential impact this dispersion may have on the formation of ultrasonic images of the heart.

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.

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.

Development of a compact, handheld Raman instrument with no moving parts for use in field analysis

This article describes a lightweight, self-contained, field portable Raman instrument that has been developed for rapid on-site determination of primary mixture components. The instrument consists of a helium neon (HeNe) laser for excitation, an acousto-optic tunable filter (AOTF) for wavelength discrimination, and an avalanche photodiode for detection. The AOTF is the primary component of this system and has been selected based on its spectral range (600–900 nm) along with its high resolution, ∼7.5 cm−1. Labview™ based software for controlling the AOTF frequency and the signal acquisition has also been developed. Several different samples were analyzed (both solids and liquids) using this instrument for the evaluation of parameters such as spectral resolution, sensitivity, and data acquisition speed for certain environmentally important compounds. The results from these measurements are compared with those using a conventional laboratory Raman system consisting of a grating-based spectrograph and a charg...

Broadband measurements of the attenuation coefficient and backscatter coefficient for suspensions: A potential calibration tool

Tissue characterization would be facilitated by improved methods of calibration with which to make absolute measurements of the backscatter and attenuation of tissue or contrast agents. In this paper, the use of polystyrene microspheres is examined as a potential broadband in vitro calibration tool by employing an experimental system designed to characterize ultrasonic contrast agents. The frequency-dependent attenuation coefficient and backscatter coefficient were measured for a series of microsphere size distributions with a broadband ultrasound system using a custom-built specimen chamber and a novel suspension mixing technique. The measurements were used in a broadband reduction method to yield the backscatter coefficient. The range of ka spanned in this study is from ka=0.5 to ka=12. The broadband nature of the pulses permitted simultaneous measurement of different regimes of scattering of the microspheres (specifically, the transition region from Rayleigh to short-wavelength asymptotic scattering). ...