Analytical model for pulse pileup spectra and count statistics in photon counting detectors with seminonparalyzable behavior.

Abstract

Background: Photon counting detectors (PCDs) with energy discriminating capabilities enable quantitative imaging of materials. However, the accuracy of estimates may be substantially degraded due to pulse pileup effects (PPEs) at high count rates. Accurate description of the output spectrum and count rate behavior of a PCD subject to pulse pileup is crucial to the development of photon counting computed tomography (PCCT).

Purpose: This study presents a fully analytical model to accurately predict the pulse pileup spectrum and count statistics (mean and covariance of energy-binned counts) for a non-paralyzable detector with nonzero pulse length and, therefore, seminonparalyzable behavior, that is, retriggering of dead time by pulses incident during the previous dead time.

Methods: We recursively computed the probability density function (PDF) of pulse pileup spectra at different pulse pileup orders. To do this, we considered the following factors: the unipolar pulse shape, the incident pulse spectrum, the distribution of time intervals between incident pulses, and the trigger threshold. We then derived the count rate and spectrum-dependent expression of total count statistics (mean and variance of total counts) based on renewal theory. We simulated a non-paralyzable PCD using Monte Carlo simulation to separately validate the spectrum and count statistics model outputs. Finally, we investigated the model accuracy in predicting material decomposition noise using the Cramér-Rao lower bound (CRLB) and a multibin system model. A comparison between predictions of the proposed model and Monte Carlo simulation is presented.

Results: The results show excellent agreement between the proposed model prediction of pulse pileup spectrum and count statistics and Monte Carlo simulation for relative count rates (the average number of counts detected during one dead time, $\lambda \tau$ ) of up to $2.5$ . The coefficient of variation (CV) values between the spectra from model prediction and Monte Carlo simulation are less than $9\%$ , and the coefficient of determination values, $R^2$ , between the count statistics from model prediction and Monte Carlo simulation are greater than $0.99$ . The proposed model also accurately predicts material decomposition noise for a non-paralyzable PCD for relative count rates of up to $1.0$ , with relative error (RE) less than $11\%$ .

Conclusions: We developed a fully analytical model of the pulse pileup spectrum and count statistics for a non-paralyzable detector model with nonzero pulse length. The model predictions agree with the Monte Carlo simulation outputs. This model could be used to correct and compensate for pulse pileup when imaging with PCDs.