Previous work has demonstrated that a signal receiver based on a limiting form of the Shannon entropy is, in certain settings, more sensitive to subtle changes in scattering architecture than conventional energy-based signal receivers ͓M. S. Hughes et al., J. Acoust. Soc. Am. 121, 3542-3557 ͑2007͔͒. In this paper new results are presented demonstrating further improvements in sensitivity using a signal receiver based on the Renyi entropy.
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 ͑r͒ with values of r near 2 ͑specifically r = 1.99͒ ͓M. S. Hughes et al., J. Acoust. Soc. Am. 125, 3141-3145 ͑2009͔͒. 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 ͑r͒ near this limit. In this paper, an asymptotic expression for the limiting behavior of I f ͑r͒ as r → 2 is derived and used to present results analogous to those obtained with I f ͑1.99͒. Moreover, the limiting form I f,ϱ is computable directly from the experimentally measured waveform f͑t͒ by an algorithm that is suitable for real-time calculation and implementation.
Duchenne Muscular Dystrophy 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 age 10 years. Therapy for DMD has been primarily palliative, with oral steroids are 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[Ef], and a new method for ultrasound signal characterization using entropy, Hf, 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[Ef] and Hf were required to statistically differentiate the three groups. In the current study we show that preprocessing of the raw ultrasound using optimal smoothing splines prior to computation of either log[Ef] or a rapidly computable variant of Hf, denoted If,∞, permits delineation of all three groups by either metric alone. This opens the way for the ultimate aim 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.
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