Ultrasound monitoring of cartilaginous matrix evolution in degradable PEG hydrogels.

Ultrasound has potential as a non-destructive analytical technique to provide real-time online assessments of matrix evolution in cell-hydrogel constructs used in tissue engineering. In these studies, chondrocytes were encapsulated in poly(ethylene glycol) hydrogels, and gel degradation was manipulated to provide conditions with varying distribution of the large cartilage extracellular matrix molecule, collagen. Mechanical properties and matrix accumulations were simultaneously measured for each condition during 9.5 weeks of in vitro culture. Ultrasound data were used to construct cross-sectional B-scan images for qualitative observations of evolving constructs. Ultrasound data were also analyzed to calculate the speed of sound (SoS) and slope of attenuation (SoA) in developing constructs and a non-degrading hydrogel control without encapsulated chondrocytes. SoS and SoA were calculated from 50 and 100MHz ultrasound data, and sample correlation coefficients were calculated to identify important relationships between these ultrasound parameters and mechanical/biochemical properties of the evolving matrix. SoA appears to be more sensitive to the density of accumulated matrix molecules than SoS, while SoS appeared to be more sensitive to mechanical modulus than SoA in measurements performed at 100MHz. Correlation analysis revealed that ultrasound measurements at 100MHz are more likely to be good predictors of matrix evolution and neotissue function than measurements at 50MHz. Rigorous studies of the relationships identified in this study will lead to non-destructive real-time monitoring that will be useful in combination with bioreactors used to promote and study cartilage regeneration.

Introduction to the special issue on diagnostic and therapeutic applications of ultrasound in bone--Part I.

The 13 papers and one item of correspondence in this special issue focus on diagnostic and therapeutic applications of ultrasound in bone. In Part I manuscripts discuss instrumentation, numerical modeling, applications, and guided waves.

Causal determination of acoustic group velocity and frequency derivative of attenuation with finite-bandwidth Kramers-Kronig relations.

Kramers-Kronig (KK) analyses of experimental data are complicated by the extrapolation problem, that is, how the unexamined spectral bands impact KK calculations. This work demonstrates the causal linkages in resonant-type data provided by acoustic KK relations for the group velocity (c(g)) and the derivative of the attenuation coefficient (alpha') (components of the derivative of the acoustic complex wave number) without extrapolation or unmeasured parameters. These relations provide stricter tests of causal consistency relative to previously established KK relations for the phase velocity (c(p)) and attenuation coefficient (alpha) (components of the undifferentiated acoustic wave number) due to their shape invariance with respect to subtraction constants. For both the group velocity and attenuation derivative, three forms of the relations are derived. These relations are equivalent for bandwidths covering the entire infinite spectrum, but differ when restricted to bandlimited spectra. Using experimental data from suspensions of elastic spheres in saline, the accuracy of finite-bandwidth KK predictions for c(g) and alpha' is demonstrated. Of the multiple methods, the most accurate were found to be those whose integrals were expressed only in terms of the phase velocity and attenuation coefficient themselves, requiring no differentiated quantities.

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.

Tomographic imaging of an ultrasonic field in a plane by use of a linear array: theory and experiment.

Quantitative ultrasonic characterization of in-homogeneous and anisotropic materials is often difficult due to undesired phenomena such as beam steering and phase aberration of the insonifying field. We introduce a method based on tomographic reconstruction techniques for the visualization of an ultrasonic field using a linear array rotated in a plane. Tomographic reconstruction of the ultrasonic field is made possible through the phase-sensitive nature of the tall, narrow piezoelectric elements of a linear array that act as parallel line integrators of the pressure field. We validate the proposed imaging method through numerical simulations of propagated ultrasonic fields based upon the angular spectrum decomposition technique. We then demonstrate the technique with experimental measurements of two textile composites and a reference water path. We reconstruct images of the real and imaginary parts of a transmitted 2 MHz ultrasonic field that are then combined to reconstruct images of the power and unwrapped phase. We also construct images of the attenuation and phase shift for several regions of the composites. Our results demonstrate that tomographic imaging of an ultrasonic field in a plane using a rotated linear array can potentially improve ultrasonic characterization of complex materials.

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.

Temperature estimation using ultrasonic spatial compound imaging.

The feasibility of temperature estimation during high-intensity focused ultrasound therapy using pulse-echo diagnostic ultrasound data has been demonstrated. This method is based upon the measurement of thermally-induced modifications in backscattered RF echoes due to thermal expansion and local changes in the speed of sound. It has been shown that strong ripple artifacts due to the thermo-acoustic lens effect severely corrupt the temperature estimates behind the heated region. We propose here a new imaging technique that improves the temperature estimation behind the heated region and reduces the variance of the temperature estimates in the entire image. We replaced the conventional beamforming on transmit with multiple steered plane wave insonifications using several subapertures. A two-dimensional temperature map is estimated from axial displacement maps between consecutive RF images of identically steered plane wave insonifications. Temperature estimation is then improved by averaging the two-dimensional maps from the multiple steered plane wave insonifications. Experiments were conducted in a tissue-mimicking gelatin-based phantom and in fresh bovine liver.

Parametric analysis of carotid plaque using a clinical ultrasound imaging system.

We evaluated quantitative ultrasonic methods for assessment of carotid plaque content. In vitro measurements of fixed, carotid plaque specimens obtained by surgical endarterectomy were performed using a clinical Philips HDI 5000 imaging system connected to a radiofrequency (RF) signal-acquisition system. We acquired RF signals and grey-scale images from carotid specimens (n = 17) and a tissue-mimicking reference phantom. Imaged plaque sections were then classified according to histology. Parametric images were constructed from the integrated backscatter (IBS), and the midband, slope and intercept values of a straight-line fit to the apparent backscatter transfer function. Analysis was performed on 82 regions-of-interest (ROIs). The IBS values for collagen, lipid and hemorrhage plaques were 5.8 +/- 5.4, 3.9 +/- 3.7, 2.8 +/- 2.2 dB, respectively. Midband and IBS parameter images exhibited good agreement in morphology with histology, whereas the slope and intercept parameter images were noisy. Mean IBS, midband, and grey-scale values of complex plaques were found to be statistically different (p < 0.05) from lipid, hemorrhage and fibrolipid plaques. The bias and limits of agreement (1.3 +/- 4.9 dB) between the grey-scale and IBS methods, however, indicated that the two methods were not interchangeable. Results indicate necessary improvements, such as reduction of large measurement variances and identification of robust parameters, that will permit multiparametric characterization of carotid plaque under in vivo conditions.

Finite-bandwidth effects on the causal prediction of ultrasonic attenuation of the power-law form.

Kramers-Kronig (K-K) relations exist as a consequence of causality, placing nonlocal constraints on the relationship between dispersion and absorption. The finite-bandwidth method of applying these relations is examined where the K-K integrals are restricted to the spectrum of the experimental data. These finite-bandwidth K-K relations are known to work with resonant-type data and here are applied to dispersion data consistent with a power-law attenuation coefficient (exponent from 1 to 2). Bandwidth-restricted forms of the zero and once-subtracted K-K relations are used to determine the attenuation coefficient from phase velocity. Analytically, it is shown that these transforms produce the proper power-law form of the attenuation coefficient as a stand-alone term summed with artifacts that are dependent on the integration limits. Calculations are performed to demonstrate how these finite-bandwidth artifacts affect the K-K predictions under a variety of conditions. The predictions are studied in a local context as a function of subtraction frequency, bandwidth, and power-law exponent. The K-K predictions of the power-law exponent within various decades of the spectrum are also examined. In general, the agreement between finite-bandwidth K-K predictions and exact values grows as the power-law exponent approaches 1 and with increasing bandwidth.

Differential forms of the Kramers-Krönig dispersion relations.

Differential forms of the Kramers-Krönig dispersion relations provide an alternative to the integral Kramers-Krönig dispersion relations for comparison with finite-bandwidth experimental data. The differential forms of the Kramers-Krönig relations are developed in the context of tempered distributions. Results are illustrated for media with attenuation obeying an arbitrary frequency power law (alpha(omega) = alpha0 + alpha1(absolute value of omega)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 < or = y < or = 1.99), with very good agreement found. For this form of ultrasonic attenuation, the differential Kramers-Krönig dispersion prediction is found to be identical to the (integral) Kramers-Krönig dispersion prediction.