Joint entropy of continuously differentiable ultrasonic waveforms

Ultrasound k-nearest neighbor entropy imaging: Theory, algorithm, and applications

Ultrasound information entropy is a flexible approach for analyzing ultrasound backscattering. Shannon entropy imaging based on probability distribution histograms (PDHs) has been implemented as a promising method for tissue characterization and diagnosis. However, the bin number affects the stability of entropy estimation. In this study, we introduced the k-nearest neighbor (KNN) algorithm to estimate entropy values and proposed ultrasound KNN entropy imaging. The proposed KNN estimator leveraged the Euclidean distance between data samples, rather than the histogram bins by conventional PDH estimators. We also proposed cumulative relative entropy (CRE) imaging to analyze time-series radiofrequency signals and applied it to monitor thermal lesions induced by microwave ablation (MWA). Computer simulation phantom experiments were conducted to validate and compare the performance of the proposed KNN entropy imaging, the conventional PDH entropy imaging, and Nakagami-m parametric imaging in detecting the variations of scatterer densities and visualizing inclusions. Clinical data of breast lesions were analyzed, and porcine liver MWA experiments ex vivo were conducted to validate the performance of KNN entropy imaging in classifying benign and malignant breast tumors and monitoring thermal lesions, respectively. Compared with PDH, the entropy estimation based on KNN was less affected by the tuning parameters. KNN entropy imaging was more sensitive to changes in scatterer densities and performed better visualizable capability than typical Shannon entropy (TSE) and Nakagami-m parametric imaging. Among different imaging methods, KNN-based Shannon entropy (KSE) imaging achieved the higher accuracy in classification of benign and malignant breast tumors and KNN-based CRE imaging had larger lesion-to-normal contrast when monitoring the ablated areas during MWA at different powers and treatment durations. Ultrasound KNN entropy imaging is a potential quantitative ultrasound approach for tissue characterization.

Information Entropy and Its Applications

Ultrasound is a first-line diagnostic tool for imaging many disease states. A number of statistical distributions have been proposed to describe ultrasound backscattering measured from tissues having different disease states. As an example, in this chapter we use nonalcoholic fatty liver disease (NAFLD), which is a critical health issue on a global scale, to demonstrate the capabilities of ultrasound to diagnose disease. Ultrasound interaction with the liver is typically characterized by scattering, which is quantified for the purpose of determining the degree of liver steatosis and fibrosis. Information entropy provides an insight into signal uncertainty. This concept allows for the analysis of backscattered statistics without considering the distribution of data or the statistical properties of ultrasound signals. In this chapter, we examined the background of NAFLD and the sources of scattering in the liver. The fundamentals of information entropy and an algorithmic scheme for ultrasound entropy imaging are then presented. Lastly, some examples of using ultrasound entropy imaging to grade hepatic steatosis and evaluate the risk of liver fibrosis in patients with significant hepatic steatosis are presented to illustrate future opportunities for clinical use.

Application of entropy and signal energy for ultrasound-based classification of three-dimensional printed polyetherketoneketone components

This paper describes a preliminary method for the classification of annealed and unannealed polyetherketoneketone (PEKK) components manufactured using a material extrusion three-dimensional (3D) printing process. PEKK is representative of a class of high-performance thermoplastics that are increasingly employed as feedstocks for use in 3D printing. PEKK components may be used continuously at elevated temperatures, are chemically resistant, and able to withstand large mechanical loads. These properties render PEKK suitable as a metal component replacement in aerospace applications, high-temperature industrial applications, and surgical implants. The structure of PEKK is semi-crystalline with the specific crystallinity correlating to the final properties during application, making determination of this property crucial. This study compares three different signal processing techniques intended to distinguish annealed (high crystallinity) from unannealed (low crystallinity) components using backscattered ultrasound. The first is energy-based and is unable to detect annealing. The second two are based on different entropies of the backscattered signal: a limiting form of Renyi's entropy and a limiting form of joint entropy. The joint entropy values for the annealed and unannealed specimens fall into two non-overlapping intervals and have a statistical separation of two standard deviations.

Resolution of Murine Toxic Hepatic Injury Quantified With Ultrasound Entropy Metrics

Image-based classification of liver disease generally lacks specificity for distinguishing between acute, resolvable injury and chronic irreversible injury. We propose that ultrasound radiofrequency data acquired in vivo from livers subjected to toxic drug injury can be analyzed with information theoretic detectors to derive entropy metrics, which classify a statistical distribution of pathologic scatterers that dissipate over time as livers heal. Here we exposed 38 C57BL/6 mice to carbon tetrachloride to cause liver damage, and imaged livers in vivo 1, 4, 8, 12 and 18 d after exposure with a broadband 15-MHz probe. Selected entropy metrics manifested monotonic recovery to normal values over time as livers healed, and were correlated directly with progressive restoration of liver architecture by histologic assessment (r² ≥ 0.95, p < 0.004). Thus, recovery of normal liver microarchitecture after toxic exposure can be delineated sensitively with entropy metrics.

Hepatic Steatosis Assessment with Ultrasound Small-Window Entropy Imaging

Nonalcoholic fatty liver disease is a type of hepatic steatosis that is not only associated with critical metabolic risk factors but can also result in advanced liver diseases. Ultrasound parametric imaging, which is based on statistical models, assesses fatty liver changes, using quantitative visualization of hepatic-steatosis-caused variations in the statistical properties of backscattered signals. One constraint with using statistical models in ultrasound imaging is that ultrasound data must conform to the distribution employed. Small-window entropy imaging was recently proposed as a non-model-based parametric imaging technique with physical meanings of backscattered statistics. In this study, we explored the feasibility of using small-window entropy imaging in the assessment of fatty liver disease and evaluated its performance through comparisons with parametric imaging based on the Nakagami distribution model (currently the most frequently used statistical model). Liver donors (n = 53) and patients (n = 142) were recruited to evaluate hepatic fat fractions (HFFs), using magnetic resonance spectroscopy and to evaluate the stages of fatty liver disease (normal, mild, moderate and severe), using liver biopsy with histopathology. Livers were scanned using a 3-MHz ultrasound to construct B-mode, small-window entropy and Nakagami images to correlate with HFF analyses and fatty liver stages. The diagnostic values of the imaging methods were evaluated using receiver operating characteristic curves. The results demonstrated that the entropy value obtained using small-window entropy imaging correlated well with log₁₀(HFF), with a correlation coefficient r = 0.74, which was higher than those obtained for the B-scan and Nakagami images. Moreover, small-window entropy imaging also resulted in the highest area under the receiver operating characteristic curve (0.80 for stages equal to or more severe than mild; 0.90 for equal to or more severe than moderate; 0.89 for severe), which indicated that non-model-based entropy imaging-using the small-window technique-performs more favorably than other techniques in fatty liver assessment.

Small-window parametric imaging based on information entropy for ultrasound tissue characterization

Constructing ultrasound statistical parametric images by using a sliding window is a widely adopted strategy for characterizing tissues. Deficiency in spatial resolution, the appearance of boundary artifacts, and the prerequisite data distribution limit the practicability of statistical parametric imaging. In this study, small-window entropy parametric imaging was proposed to overcome the above problems. Simulations and measurements of phantoms were executed to acquire backscattered radiofrequency (RF) signals, which were processed to explore the feasibility of small-window entropy imaging in detecting scatterer properties. To validate the ability of entropy imaging in tissue characterization, measurements of benign and malignant breast tumors were conducted (n = 63) to compare performances of conventional statistical parametric (based on Nakagami distribution) and entropy imaging by the receiver operating characteristic (ROC) curve analysis. The simulation and phantom results revealed that entropy images constructed using a small sliding window (side length = 1 pulse length) adequately describe changes in scatterer properties. The area under the ROC for using small-window entropy imaging to classify tumors was 0.89, which was higher than 0.79 obtained using statistical parametric imaging. In particular, boundary artifacts were largely suppressed in the proposed imaging technique. Entropy enables using a small window for implementing ultrasound parametric imaging.

Acoustic firearm discharge detection and classification in an enclosed environment

Two different signal processing algorithms are described for detection and classification of acoustic signals generated by firearm discharges in small enclosed spaces. The first is based on the logarithm of the signal energy. The second is a joint entropy. The current study indicates that a system using both signal energy and joint entropy would be able to both detect weapon discharges and classify weapon type, in small spaces, with high statistical certainty.

Entropy vs. Energy Waveform Processing: A Comparison Based on the Heat Equation

Virtually all modern imaging devices collect electromagnetic or acoustic waves and use the energy carried by these waves to determine pixel values to create what is basically an “energy” picture. However, waves also carry “information”, as quantified by some form of entropy, and this may also be used to produce an “information” image. Numerous published studies have demonstrated the advantages of entropy, or “information imaging”, over conventional methods. The most sensitive information measure appears to be the joint entropy of the collected wave and a reference signal. The sensitivity of repeated experimental observations of a slowly-changing quantity may be defined as the mean variation (i.e., observed change) divided by mean variance (i.e., noise). Wiener integration permits computation of the required mean values and variances as solutions to the heat equation, permitting estimation of their relative magnitudes. There always exists a reference, such that joint entropy has larger variation and smaller variance than the corresponding quantities for signal energy, matching observations of several studies. Moreover, a general prescription for finding an “optimal” reference for the joint entropy emerges, which also has been validated in several studies.

Additional results for "joint entropy of continuously differentiable ultrasonic waveforms" [J. Acoust. Soc. Am. 133(1), 283-300 (2013)]

Previous results on the use of joint entropy for detection of targeted nanoparticles accumulating in the neovasculature of MDA435 tumors [Fig. 7 of M. S. Hughes et al., J. Acoust. Soc. Am. 133, 283-300 (2013)] are extended, with sensitivity improving by nearly another factor of 2. This result is obtained using a "quasi-optimal" reference waveform in the computation of the joint entropy imaging technique used to image the accumulating nanoparticles.

Entropic imaging of cataract lens: an in vitro study

Phacoemulsification is a common surgical method for treating advanced cataracts. Determining the optimal phacoemulsification energy depends on the hardness of the lens involved. Previous studies have shown that it is possible to evaluate lens hardness via ultrasound parametric imaging based on statistical models that require data to follow a specific distribution. To make the method more system-adaptive, nonmodel-based imaging approach may be necessary in the visualization of lens hardness. This study investigated the feasibility of applying an information theory derived parameter - Shannon entropy from ultrasound backscatter to quantify lens hardness. To determine the physical significance of entropy, we performed computer simulations to investigate the relationship between the signal-to-noise ratio (SNR) based on the Rayleigh distribution and Shannon entropy. Young's modulus was measured in porcine lenses, in which cataracts had been artificially induced by the immersion in formalin solution in vitro. A 35-MHz ultrasound transducer was used to scan the cataract lenses for entropy imaging. The results showed that the entropy is 4.8 when the backscatter data form a Rayleigh distribution corresponding to an SNR of 1.91. The Young's modulus of the lens increased from approximately 8 to 100 kPa when we increased the immersion time from 40 to 160 min (correlation coefficient r = 0.99). Furthermore, the results indicated that entropy imaging seemed to facilitate visualizing different degrees of lens hardening. The mean entropy value increased from 2.7 to 4.0 as the Young's modulus increased from 8 to 100 kPa (r = 0.85), suggesting that entropy imaging may have greater potential than that of conventional statistical parametric imaging in determining the optimal energy to apply during phacoemulsification.