Statistics of the integrated backscatter estimate from a blood-mimicking fluid

This work evaluates the variance of the integrated backscatter (IBS) from moving blood [or blood-mimicking fluid (bmf)] as a way of determining the quality of the mean IBS estimate. The main motivation for this work comes from the fact that absolute IBS values from tissues adjacent to arterial blood can be found by normalizing the measured backscatter energy with the IBS of moving, deaggregated blood. The paper describes the parameters that control the statistics of the IBS estimate, which is calculated for the stochastic ultrasound backscatter signals from flowing blood. It further formulates how the measurement parameters should be specified so that an appropriately low blood IBS variance is ensured or, alternatively, a specified accuracy of the tissue IBS estimate is obtained. First, the paper provides an analytic formulation of the statistics of the IBS, based on a sequence of sampled echoes from a nonstationary Gaussian scattering medium. The analysis incorporates the correlation between the sample values as well as the correlation between the IBS of the individual echoes. The estimate of the mean IBS has been shown to be chi-squared distributed with a determinable order. With the degree of correlation between the samples and between the IBS of individual echoes specified, the number of measurements required to obtain an IBS estimate with a specified variance is readily calculated. Next, a sequence of synthetic echoes is produced and arranged as columns in a data matrix. The echoes are generated such that the second-order statistics along the rows and columns of the matrix match that of actually observed echoes. The actual variance of the mean IBS estimate for the synthetic echoes is calculated and compared with the variance determined from the analytic model, and a good agreement has been found. Finally, sequences of actual backscattered echoes from circulating blood-mimicking fluid are acquired and analyzed to determine the variance of their mean IBS estimate. Based on the measured second-order statistics of the rows and columns of the data matrix for the actual echoes, the observed variance of the mean IBS estimate was compared with the analytically determined variance and with good agreement. Thus, the paper has shown through modeling, simulations, and experiments how the variance of the IBS estimate of the blood backscatter signal can be quantified and reduced to a specified tolerable level.