Christian C. Anderson

Interference between wave modes may contribute to the apparent negative dispersion observed in cancellous bone

Previous work has shown that ultrasonic waves propagating through cancellous bone often exhibit a linear-with-frequency attenuation coefficient, but a decrease in phase velocity with frequency (negative dispersion) that is inconsistent with the causality-imposed Kramers-Kronig relations. In the current study, interfering wave modes similar to those observed in bone are shown to potentially contribute to the observed negative dispersion. Biot theory, the modified Biot-Attenborogh model, and experimental results are used to aid in simulating multiple-mode wave propagation through cancellous bone. Simulations entail constructing individual wave modes exhibiting a positive dispersion using plausible velocities and amplitudes, and then summing the individual modes to create mixed-mode output wave forms. Results of the simulations indicate that mixed-mode wave forms can exhibit negative dispersion when analyzed conventionally under the assumption that only one wave is present, even when the individual interfering waves exhibit positive dispersions in accordance with the Kramers-Kronig relations. Furthermore, negative dispersion is observed when little or no visual evidence of interference exists in the time-domain data. Understanding the mechanisms responsible for the observed negative dispersion could aid in determining the true material properties of cancellous bone, as opposed to the apparent properties measured using conventional data analysis techniques.

CALIBRATED QUANTITATIVE ULTRASOUND IMAGING OF SKELETAL MUSCLE USING BACKSCATTER ANALYSIS

We evaluated the ability of an ultrasound method, which can characterize cardiac muscle pathology and has reliability across different imaging systems, to obtain calibrated quantitative estimates of backscatter of skeletal muscle. Our procedure utilized a tissue‐mimicking phantom to establish a linear relationship between ultrasound grayscale and backscatter levels. We studied skeletal muscles of 82 adults: 45 controls and 37 patients with hereditary myopathies. We found that skeletal muscle ultrasound backscatter levels varied with probe orientation, age, and muscle contraction and pathology. Reliability was greater with the probe in longitudinal compared with transverse planes. Backscatter levels were higher in those >40 years of age, in muscle extension than flexion, and in myopathic patients than controls. Calibrated measurements of muscle backscatter have sensitivity and specificity in identifying and reliably measuring levels of skeletal muscle pathology. Muscle Nerve 38: 893–898, 2008

Inverse problems in cancellous bone: estimation of the ultrasonic properties of fast and slow waves using Bayesian probability theory.

Quantitative ultrasonic characterization of cancellous bone can be complicated by artifacts introduced by analyzing acquired data consisting of two propagating waves (a fast wave and a slow wave) as if only one wave were present. Recovering the ultrasonic properties of overlapping fast and slow waves could therefore lead to enhancement of bone quality assessment. The current study uses Bayesian probability theory to estimate phase velocity and normalized broadband ultrasonic attenuation (nBUA) parameters in a model of fast and slow wave propagation. Calculations are carried out using Markov chain Monte Carlo with simulated annealing to approximate the marginal posterior probability densities for parameters in the model. The technique is applied to simulated data, to data acquired on two phantoms capable of generating two waves in acquired signals, and to data acquired on a human femur condyle specimen. The models are in good agreement with both the simulated and experimental data, and the values of the estimated ultrasonic parameters fall within expected ranges.

Determining attenuation properties of interfering fast and slow ultrasonic waves in cancellous bone

Previous studies have shown that interference between fast waves and slow waves can lead to observed negative dispersion in cancellous bone. In this study, the effects of overlapping fast and slow waves on measurements of the apparent attenuation as a function of propagation distance are investigated along with methods of analysis used to determine the attenuation properties. Two methods are applied to simulated data that were generated based on experimentally acquired signals taken from a bovine specimen. The first method uses a time-domain approach that was dictated by constraints imposed by the partial overlap of fast and slow waves. The second method uses a frequency-domain log-spectral subtraction technique on the separated fast and slow waves. Applying the time-domain analysis to the broadband data yields apparent attenuation behavior that is larger in the early stages of propagation and decreases as the wave travels deeper. In contrast, performing frequency-domain analysis on the separated fast waves and slow waves results in attenuation coefficients that are independent of propagation distance. Results suggest that features arising from the analysis of overlapping two-mode data may represent an alternate explanation for the previously reported apparent dependence on propagation distance of the attenuation coefficient of cancellous bone.

Bone sonometry: Reducing phase aberration to improve estimates of broadband ultrasonic attenuation

Previous studies suggest that phase cancellation at the receiving transducer can result in the overestimation of the frequency dependent ultrasonic attenuation of bone, a quantity that has been shown to correlate with bone mineral density and ultimately with osteoporotic fracture risk. Evidence supporting this interpretation is provided by phase insensitive processing of the data, which appear to reduce the apparent overestimates of attenuation. The present study was designed to clarify the components underlying phase aberration artifacts in such through-transmission measurements by conducting systematic studies of the simplest possible test objects capable of introducing phase aberration. Experimental results are presented for a Lexan phantom over the frequency range 300-700 kHz and a Plexiglas phantom over the 3-7 MHz range. Both phantoms were flat and parallel plates featuring a step discontinuity milled into one of their initially flat sides. The through-transmitted signals were received by a 0.6 mm diameter membrane hydrophone that was raster scanned over a grid coaxial with the transmitting transducer. Signals received by the pseudoarray were processed offline to emulate phase sensitive and phase insensitive receivers with different aperture diameters. The data processed phase sensitively were focused to demonstrate the results of planar, geometrical, and correlation-based aberration correction methods. Results are presented illustrating the relative roles of interference in the ultrasonic field and phase cancellation at the receiving transducer in producing phase aberration artifacts. It was found that artifacts due to phase cancellation or interference can only be minimized with phase insensitive summation techniques by choosing an appropriately large receiving aperture. Data also suggest the potentially confounding role of time-and frequency-domain artifacts on ultrasonic measurements and illustrate the advantages of two-dimensional receiving arrays in determining the slope of attenuation (nBUA) for the clinical assessment of osteoporosis.

Phase Velocity of Cancellous Bone: Negative Dispersion Arising from Fast and Slow Waves, Interference, Diffraction, and Phase Cancellation at Piezoelectric Receiving Elements

Frequency-dependent phase velocity measurements may prove useful in bone quality assessment. However, the physical mechanisms of ultrasonic wave propagation in cancellous bone that govern phase velocity are not yet fully understood, particularly the phenomena that lead to the observed anomalous negative dispersion. This chapter provides an overview of phase velocity studies of cancellous bone, especially negative dispersion, and proposals for resolving the apparent conflict with the causality-imposed Kramers-Kronig relations.

PARAMETER ESTIMATION IN ULTRASONIC MEASUREMENTS ON TRABECULAR BONE

Ultrasonic tissue characterization has shown promise for clinical diagnosis of diseased bone (e.g., osteoporosis) by establishing correlations between bone ultrasonic characteristics and the state of disease. Porous (trabecular) bone supports propagation of two compressional modes, a fast wave and a slow wave, each of which is characterized by an approximately linear‐with‐frequency attenuation coefficient and monotonically increasing with frequency phase velocity. Only a single wave, however, is generally apparent in the received signals. The ultrasonic parameters that govern propagation of this single wave appear to be causally inconsistent [1]. Specifically, the attenuation coefficient rises approximately linearly with frequency, but the phase velocity exhibits a decrease with frequency. These inconsistent results are obtained when the data are analyzed under the assumption that the received signal is composed of one wave. The inconsistency disappears if the data are analyzed under the assumption that t...

DECOMPOSITION OF INTERFERING ULTRASONIC WAVES IN BONE AND BONE‐MIMICKING MATERIALS

Cancellous bone is a lattice‐like arrangement of solid trabeculae surrounded by soft bone marrow. When interrogated with ultrasonic waves, the complex architecture gives rise to two longitudinal modes known as a fast and a slow wave. Depending on experimental conditions and the ultrasonic characteristics of the bone sample under investigation, the two waves may strongly overlap in the ultrasounic data. Analyzing such data conventionally, as if only one wave were present, can potentially mask or alter bone quality parameters that are commonly used in clinical sonometry.In this study, ultrasonic data were acquired on a bovine femur condyle specimen and on a plastic bone‐mimicking phantom constructed from Lucite and Lexan blocks. The acquired data were used as inputs to a program that implements a Bayesian calculation to model the ultrasound signal as two interfering plane waves and then estimates the ultrasonic parameters of the fast and slow waves. The calculations were carried out using Markov chain Monte...

Measurement artifacts in sonometry of cancellous bone: The relative impact of phase cancellation and interference on measurements of phase-distorting phantoms

The goal of the present study is to demonstrate the relative effects of interference in the field and phase cancellation at the surface of a receiver on measurements of three increasingly more complex phantoms. Distinct from phase cancellation occurring at the surface of a phase-sensitive receiver, which results in an irrecoverable loss of energy, loss due to interference is only ostensible. Both apparent sources of loss have been shown by our lab to result in negative dispersion measured in cancellous bone. The transmitting transducer consisted of a single-element, planar, 2.9 cm diameter transducer centered at 500 kHz. The through-transmitted signals were received by a 0.6 mm diameter membrane hydrophone that was raster scanned in 0.04 mm steps over a 5.8 cm by 5.8 cm grid coaxial with the transmitting transducer. Signals received by the pseudo-array were processed offline to emulate a phase insensitive receiver of increasing aperture. Apparent attenuation coefficients were determined as functions of frequency for each configuration. Results suggest the potentially confounding role of time- and frequency-domain artifacts on measurements and illustrate the advantages of two-dimensional receiving arrays in determining the slope of attenuation (nBUA) for the clinical assessment of osteoporosis. This work was supported in part by NSF CBET 0717830 ldquoScholar in Residence at the FDArdquo and NIH HL 040302.

Model Selection in Ultrasonic Measurements on Trabecular Bone

Previous work from our laboratory showed that the widely reported decrease in phase velocity with frequency (negative dispersion) for ultrasonic waves propagating through trabecular bone can arise from the interference of two compressional waves, each of which exhibits a positive dispersion. Previous simulations suggest that Bayesian probability theory can be employed to recover the material properties linked to these two interfering waves, even when the waves overlap sufficiently that visual inspection cannot distinguish two modes. In the present study, Bayesian probability theory is applied first to simulated data and then to representative experimental bone data to determine whether one or two compressional wave modes are present. Model selection is implemented by evaluating the posterior probability for each model. The calculation is implemented by defining a model indicator and then using Markov chain Monte Carlo with simulated annealing to draw samples from the joint posterior probability for the ul...