Statistical properties of cerebral CT perfusion imaging systems. Part II. Deconvolution-based systems

Abstract

Purpose: The purpose of this work was to develop a theoretical framework to pinpoint the quantitative relationship between input parameters of deconvolution-based cerebral computed tomography perfusion (CTP) imaging systems and statistical properties of the output perfusion maps.

Methods: Deconvolution-based CTP systems assume that the arterial input function, tissue enhancement curve, and flow-scaled residue function k(t) are related to each other through a convolution model, and thus by reversing the convolution operation, k(t) and the associated perfusion parameters can be estimated. The theoretical analysis started by deriving analytical formulas for the expected value and autocovariance of the residue function estimated using the singular value decomposition-based deconvolution method. Next, it analyzed statistical properties of the "max" and "arg max" operators, based on which the signal and noise properties of cerebral blood flow (CBF) and time-to-max ( $t_{\max }$ ) are quantitatively related to the statistical model of the estimated residue function [ $k^{*}\left(t\right)$ ] and system parameters. To validate the theory, CTP images of a digital head phantom were simulated, from which signal and noise of each perfusion parameter were measured and compared with values calculated using the theoretical model. In addition, an in vivo canine experiment was performed to validate the noise model of cerebral blood volume (CBV).

Results: For the numerical study, the relative root mean squared error between the measured and theoretically calculated value is ≤0.21% for the autocovariance matrix of $k^{*}\left(t\right)$ , and is ≤0.13% for the expected form of $k^{*}\left(t\right)$ . A Bland-Altman analysis demonstrated no significant difference between measured and theoretical values for the mean or noise of each perfusion parameter. For the animal study, the theoretical CBV noise fell within the 25th and 75th percentiles of the experimental values. To provide an example of the theory's utility, an expansion of the CBV noise formula was performed to unveil the dominant role of the baseline image noise in deconvolution-based CBV. Correspondingly, data of the three canine subjects used in the Part I paper were retrospectively processed to confirm that preferentially partitioning dose to the baseline frames benefits both nondeconvolution- and deconvolution-based CBV maps.

Conclusions: Quantitative relationships between the statistical properties of deconvolution-based CTP maps, source image acquisition and reconstruction parameters, contrast injection protocol, and deconvolution parameters are established.

Keywords: CT perfusion; autocovariance; bias; cascaded systems analysis; deconvolution; linear systems theory; quantitative imaging; singular value decomposition; stroke imaging.