Quadratic time-scale detection of microemboli in flowing blood with Doppler ultrasound

Small formed elements and gas bubbles in flowing blood, called microemboli, can be detected using pulse Doppler ultrasound. In this application, a pulsed constant-frequency ultrasound signal insonates a volume of blood in the middle cerebral artery, and microemboli moving through this sample volume produce a Doppler shifted transient reflection. Current detection methods include searching for these transients in a short-time Fourier transform (STFT) of the reflected signal. However, since the embolus transit time through the Doppler sample volume is inversely proportional to the embolus velocity (Doppler shift frequency), a matched-filter detector should in principle use a wavelet transform, rather than a short-time Fourier transform, for optimal results. Closer examination of the Doppler shift signals usually shows a chirping behavior apparently due to acceleration or deceleration of the emboli during their transit through the Doppler sample volume. These variations imply that a linear wavelet detector is not optimal. We argue from physical principles that quadratic time-scale detectors provide a robustness to variations that is nearly optimal for embolus detection. Using a theory for optimal quadratic time-scale detection, we develop a method for designing the optimal detection kernels from training data and derive efficient algorithms for implementation of the resulting quadratic time-scale detectors. The performance of these detectors is compared with optimized STFT and wavelet detectors. The performance for all of these methods is found to be very similar, and on average only about 2.5 dB less than an "oracle" detector which provides a theoretical upper bound for microembolus detection.