Application of Renyi entropy for ultrasonic molecular imaging

Modeling of COVID-19 vaccination rate using odd Lomax inverted Nadarajah-Haghighi distribution

Since the spread of COVID-19 pandemic in early 2020, modeling the related factors became mandatory, requiring new families of statistical distributions to be formulated. In the present paper we are interested in modeling the vaccination rate in some African countries. The recorded data in these countries show less vaccination rate, which will affect the spread of new active cases and will increase the mortality rate. A new extension of the inverted Nadarajah-Haghighi distribution is considered, which has four parameters and is obtained by combining the inverted Nadarajah-Haghighi distribution and the odd Lomax-G family. The proposed distribution is called the odd Lomax inverted Nadarajah-Haghighi (OLINH) distribution. This distribution owns many virtuous characteristics and attractive statistical properties, such as, the simple linear representation of density function, the flexibility of the hazard rate curve and the odd ratio of failure, in addition to other properties related to quantile, the r^th-moment, moment generating function, Rényi entropy, and the function of ordered statistics. In this paper we address the problem of parameter estimation from frequentest and Bayesian approach, accordingly a comparison between the performance of the two estimation methods is implemented using simulation analysis and some numerical techniques. Finally different goodness of fit measures are used for modeling the COVID-19 vaccination rate, which proves the suitability of the OLINH distribution over other competitive distributions.

Rényi Entropy in Statistical Mechanics

Rényi entropy was originally introduced in the field of information theory as a parametric relaxation of Shannon (in physics, Boltzmann-Gibbs) entropy. This has also fuelled different attempts to generalise statistical mechanics, although mostly skipping the physical arguments behind this entropy and instead tending to introduce it artificially. However, as we will show, modifications to the theory of statistical mechanics are needless to see how Rényi entropy automatically arises as the average rate of change of free energy over an ensemble at different temperatures. Moreover, this notion is extended by considering distributions for isospectral, non-isothermal processes, resulting in relative versions of free energy, in which the Kullback-Leibler divergence or the relative version of Rényi entropy appear within the structure of the corrections to free energy. These generalisations of free energy recover the ordinary thermodynamic potential whenever isothermal processes are considered.

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.

An unmet clinical need: The history of thrombus imaging

Robust thrombus imaging is an unresolved clinical unmet need dating back to the mid 1970s. While early molecular imaging approaches began with nuclear SPECT imaging, contrast agents for virtually all biomedical imaging modalities have been demonstrated in vivo with unique strengths and common weaknesses. Two primary molecular imaging targets have been pursued for thrombus imaging: platelets and fibrin. Some common issues noted over 40 years ago persist today. Acute thrombus is readily imaged with all probes and modalities, but aged thrombus remains a challenge. Similarly, anti-coagulation continues to interfere with and often negate thrombus imaging efficacy, but heparin is clinically required in patients suspected of pulmonary embolism, deep venous thrombosis or coronary ruptured plaque prior to confirmatory diagnostic studies have been executed and interpreted. These fundamental issues can be overcome, but an innovative departure from the prior approaches will be needed.

Perfluorooctylbromide nanoparticles for ultrasound imaging and drug delivery

Perfluorooctylbromide nanoparticles (PFOB NPs) are a type of multifunctional nanotechnology that has been studied for various medical applications. Commercial ultrasound contrast agents (UCAs) suffer from the following limitations: short half-lives in vivo, high background signal and restricted distribution in the vascular circulation due to their micrometer dimensions. PFOB NPs are new potential UCAs that persist for long periods in the circulatory system, possess a relatively stable echogenic response without increasing the background signal and exhibit lower acoustic attenuation than commercial UCAs. Furthermore, PFOB NPs may also serve as drug delivery vehicles in which drugs are dissolved in the outer lipid or polymer layer for subsequent delivery to target sites in site-targeted therapy. The use of PFOB NPs as carriers has the potential advantage of selectively delivering payloads to the target site while improving visualization of the site using ultrasound (US) imaging. Unfortunately, the application of PFOB NPs to the field of ultrasonography has been limited because of the low intensity of US reflection. Numerous researchers have realized the potential use of PFOB NPs as UCAs and thus have developed alternative approaches to apply PFOB NPs in ultrasonography. In this article, we review the latest approaches for using PFOB NPs to enhance US imaging in vivo. In addition, this article emphasizes the application of PFOB NPs as promising drug delivery carriers for cancer and atherosclerosis treatments, as PFOB NPs can transport different drug payloads for various applications with good efficacy. We also note the challenges and future study directions for the application of PFOB NPs as both a delivery system for therapeutic agents and a diagnostic agent for ultrasonography.

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.

Perfluorocarbon nanoparticles for physiological and molecular imaging and therapy

Herein, we review the use of non-nephrotoxic perfluorocarbon nanoparticles (PFC NPs) for noninvasive detection and therapy of kidney diseases, and we provide a synopsis of other related literature pertinent to their anticipated clinical application. Recent reports indicate that PFC NPs allow for quantitative mapping of kidney perfusion and oxygenation after ischemia-reperfusion injury with the use of a novel multinuclear (1)H/(19)F magnetic resonance imaging approach. Furthermore, when conjugated with targeting ligands, the functionalized PFC NPs offer unique and quantitative capabilities for imaging inflammation in the kidney of atherosclerotic ApoE-null mice. In addition, PFC NPs can facilitate drug delivery for treatment of inflammation, thrombosis, and angiogenesis in selected conditions that are comorbidities for kidney failure. The excellent safety profile of PFC NPs with respect to kidney injury positions these nanomedicine approaches as promising diagnostic and therapeutic candidates for treating and following acute and chronic kidney diseases.

Joint entropy of continuously differentiable ultrasonic waveforms

This study is based on an extension of the concept of joint entropy of two random variables to continuous functions, such as backscattered ultrasound. For two continuous random variables, X and Y, the joint probability density p(x,y) is ordinarily a continuous function of x and y that takes on values in a two dimensional region of the real plane. However, in the case where X=f(t) and Y=g(t) are both continuously differentiable functions, X and Y are concentrated exclusively on a curve, γ(t)=(f(t),g(t)), in the x,y plane. This concentration can only be represented using a mathematically "singular" object such as a (Schwartz) distribution. Its use for imaging requires a coarse-graining operation, which is described in this study. Subsequently, removal of the coarse-graining parameter is accomplished using the ergodic theorem. The resulting expression for joint entropy is applied to several data sets, showing the utility of the concept for both materials characterization and detection of targeted liquid nanoparticle ultrasonic contrast agents. In all cases, the sensitivity of these techniques matches or exceeds, sometimes by a factor of two, that demonstrated in previous studies that employed signal energy or alternate entropic quantities.

Use of smoothing splines for analysis of backscattered ultrasonic waveforms: application to monitoring of steroid treatment of dystrophic mice

Duchenne muscular dystrophy (DMD) is an X-linked genetic disease characterized by progressive weakness and wasting of skeletal and cardiac muscle; boys present with weakness by the age of 5 years and, if left untreated, are unable to walk without assistance by the age of 10 years. Therapy for DMD has been primarily palliative, with oral steroids emerging as a first-line approach even though this treatment has serious side-effects. Consequently, low-cost imaging technology suitable for improved diagnosis and treatment monitoring of DMD would be of great value, especially in remote and underserved areas. Previously, we reported use of the logarithm of the signal energy, log [E(f)], and a new method for ultrasound signal characterization using entropy, H(f), to monitor prednisolone treatment of skeletal muscle in a dystrophin-deficient mouse model. Three groups were studied: mdx mice treated with prednisolone, a control group of mdx mice treated with saline, and a control group of wild-type mice treated with saline. It was found that both log [E(f)] and H(f) were required to statistically differentiate the three groups. In the current study, we show that preprocessing of the raw ultrasound using optimal smoothing splines before computation of either log [E(f)] or a rapidly computable variant of Hf, denoted I(f,∞), permits delineation of all three groups by either metric alone. This opens the way to the ultimate goal of this study, which is identification and implementation of new diagnostically sensitive algorithms on the new generation of low-cost hand-held clinical ultrasonic imaging systems.

Improved signal processing to detect cancer by ultrasonic molecular imaging of targeted nanoparticles

In several investigations of molecular imaging of angiogenic neovasculature using a targeted contrast agent, Renyi entropy [I(f)(r)] and a limiting form of Renyi entropy (I(f,∞)) exhibited significantly more sensitivity to subtle changes in scattering architecture than energy-based methods. Many of these studies required the fitting of a cubic spline to backscattered waveforms prior to calculation of entropy [either I(f)(r) or I(f,∞)]. In this study, it is shown that the robustness of I(f,∞) may be improved by using a smoothing spline. Results are presented showing the impact of different smoothing parameters. In addition, if smoothing is preceded by low-pass filtering of the waveforms, further improvements may be obtained.

Angiogenesis imaging with vascular-constrained particles: the why and how

Angiogenesis is a keystone in the treatment of cancer and potentially many other diseases. In cancer, first-generation antiangiogenic therapeutic approaches have demonstrated survival benefit in subsets of patients, but their high cost and notable adverse side effect risk have fueled alternative development efforts to personalize patient selection and reduce off-target effects. In parallel, rapid advances in cost-effective genomic profiling and sensitive early detection of high-risk biomarkers for cancer, atherosclerosis, and other angiogenesis-related pathologies will challenge the medical imaging community to identify, characterize, and risk stratify patients early in the natural history of these disease processes. Conventional diagnostic imaging techniques were not intended for such sensitive and specific detection, which has led to the emergence of novel noninvasive biomedical imaging approaches. The overall intent of molecular imaging is to achieve greater quantitative characterization of pathologies based on microanatomical, biochemical, or functional assessments; in many approaches, the capacity to deliver effective therapy, e.g., antiangiogenic therapy, can be combined. Agents with both diagnostic and therapy attributes have acquired the moniker "theranostics." This review will explore biomedical imaging options being pursued to better segment and treat patients with angiogenesis-influenced disease using vascular-constrained contrast platform technologies.

Application of a real-time, calculable limiting form of the Renyi entropy for molecular imaging of tumors

Previously, we reported new methods for ultrasound signal characterization using entropy, H(f); a generalized entropy, the Renyi entropy, I(f)(r); and a limiting form of Renyi entropy suitable for real-time calculation, I(f),(infinity). All of these quantities demonstrated significantly more sensitivity to subtle changes in scattering architecture than energy-based methods in certain settings. In this study, the real-time calculable limit of the Renyi entropy, I(f),(infinity), is applied for the imaging of angiogenic murine neovasculature in a breast cancer xenograft using a targeted contrast agent. It is shown that this approach may be used to reliably detect the accumulation of targeted nanoparticles at five minutes post-injection in this in vivo model.

Theragnostics for tumor and plaque angiogenesis with perfluorocarbon nanoemulsions

Molecular imaging agents are extending the potential of noninvasive medical diagnosis from basic gross anatomical descriptions to complicated phenotypic characterizations based upon the recognition of unique cell-surface biochemical signatures. Although originally the purview of nuclear medicine, "molecular imaging" is now studied in conjunction with all clinically relevant imaging modalities. Of the myriad of particles that have emerged as prospective candidates for clinical translation, perfluorocarbon nanoparticles offer great potential for combining targeted imaging with drug delivery, much like the "magic bullet" envisioned by Paul Ehrlich 100 years ago. Perfluorocarbon nanoparticles, once studied in Phase III clinical trials as blood substitutes, have found new life for molecular imaging and drug delivery. The particles have been adapted for use with all clinically relevant modalities and for targeted drug delivery. In particular, their intravascular constraint due to particle size provides a distinct advantage for angiogenesis imaging and antiangiogenesis therapy. As perfluorocarbon nanoparticles have recently entered Phase I clinical study, this review provides a timely focus on the development of this platform technology and its application for angiogenesis-related pathologies.

Liquid perfluorocarbons as contrast agents for ultrasonography and (19)F-MRI

Perfluorocarbons (PFCs) are fluorinated compounds that have been used for many years in clinics mainly as gas/oxygen carriers and for liquid ventilation. Besides this main application, PFCs have also been tested as contrast agents for ultrasonography and magnetic resonance imaging since the end of the 1970s. However, most of the PFCs applied as contrast agents for imaging were gaseous. This class of PFCs has been recently substituted by liquid PFCs as ultrasound contrast agents. Additionally, liquid PFCs are being tested as contrast agents for (19)F magnetic resonance imaging (MRI), to yield dual contrast agents for both ultrasonography and (19)F MRI. This review focuses on the development and applications of the different contrast agents containing liquid perfluorocarbons for ultrasonography and/or MRI: large and small size emulsions (i.e. nanoemulsions) and nanocapsules.

Real-time calculation of a limiting form of the Renyi entropy applied to detection of subtle changes in scattering architecture

Previously a new method for ultrasound signal characterization using entropy H(f) was reported, and it was demonstrated that in certain settings, further improvements in signal characterization could be obtained by generalizing to Renyi entropy-based signal characterization I(f)(r) with values of r near 2 (specifically r=1.99) [M. S. Hughes et al., J. Acoust. Soc. Am. 125, 3141-3145 (2009)]. It was speculated that further improvements in sensitivity might be realized at the limit r-->2. At that time, such investigation was not feasible due to excessive computational time required to calculate I(f)(r) near this limit. In this paper, an asymptotic expression for the limiting behavior of I(f)(r) as r-->2 is derived and used to present results analogous to those obtained with I(f)(1.99). Moreover, the limiting form I(f,infinity) is computable directly from the experimentally measured waveform f(t) by an algorithm that is suitable for real-time calculation and implementation.