title: Spatio-energetic cross talk in photon counting detectors: Detector model and correlated Poisson data generator.
abstract: OBJECTIVE
  An x-ray photon interacts with photon counting detectors (PCDs) and generates an electron charge cloud or multiple clouds. The clouds (thus, the photon energy) may be split between two adjacent PCD pixels when the interaction occurs near pixel boundaries, producing a count at both of the pixels. This is called double-counting with charge sharing. (A photoelectric effect with K-shell fluorescence x-ray emission would result in double-counting as well). As a result, PCD data are spatially and energetically correlated, although the output of individual PCD pixels is Poisson distributed. Major problems include the lack of a detector noise model for the spatio-energetic cross talk and lack of a computationally efficient simulation tool for generating correlated Poisson data. A Monte Carlo (MC) simulation can accurately simulate these phenomena and produce noisy data; however, it is not computationally efficient.
  
  
  METHODS
  In this study, the authors developed a new detector model and implemented it in an efficient software simulator that uses a Poisson random number generator to produce correlated noisy integer counts. The detector model takes the following effects into account: (1) detection efficiency; (2) incomplete charge collection and ballistic effect; (3) interaction with PCDs via photoelectric effect (with or without K-shell fluorescence x-ray emission, which may escape from the PCDs or be reabsorbed); and (4) electronic noise. The correlation was modeled by using these two simplifying assumptions: energy conservation and mutual exclusiveness. The mutual exclusiveness is that no more than two pixels measure energy from one photon. The effect of model parameters has been studied and results were compared with MC simulations. The agreement, with respect to the spectrum, was evaluated using the reduced χ2statistics or a weighted sum of squared errors, χred2(≥1), where χred2=1 indicates a perfect fit.
  
  
  RESULTS
  The model produced spectra with flat field irradiation that qualitatively agree with previous studies. The spectra generated with different model and geometry parameters allowed for understanding the effect of the parameters on the spectrum and the correlation of data. The agreement between the model and MC data was very strong. The mean spectra with 90 keV and 140 kVp agreed exceptionally well: χred2values were 1.049 with 90 keV data and 1.007 with 140 kVp data. The degrees of cross talk (in terms of the relative increase from single pixel irradiation to flat field irradiation) were 22% with 90 keV and 19% with 140 kVp for MC simulations, while they were 21% and 17%, respectively, for the model. The covariance was in strong agreement qualitatively, although it was overestimated. The noisy data generation was very efficient, taking less than a CPU minute as opposed to CPU hours for MC simulators.
  
  
  CONCLUSIONS
  The authors have developed a novel, computationally efficient PCD model that takes into account double-counting and resulting spatio-energetic correlation between PCD pixels. The MC simulation validated the accuracy.
authors.affiliation: Division of Medical Imaging Physics, The Russell H. Morgan Department of Radiology and Radiological Science, Johns Hopkins University School of Medicine, Baltimore, Maryland 21287.
authors.forename: Katsuyuki
authors.initials: K
authors.lastname: Taguchi
authors.affiliation: Institute for Clinical Radiology, Ludwig-Maximilians-University Hospital, Munich 80539, Germany and Computed Tomography, Siemens Healthcare GmbH, Forchheim 91301, Germany.
authors.forename: Christoph
authors.initials: C
authors.lastname: Polster
authors.affiliation: Division of Medical Imaging Physics, The Russell H. Morgan Department of Radiology and Radiological Science, Johns Hopkins University School of Medicine, Baltimore, Maryland 21287.
authors.forename: Okkyun
authors.initials: O
authors.lastname: Lee
authors.affiliation: Computed Tomography, Siemens Healthcare GmbH, Forchheim 91301, Germany.
authors.forename: Karl
authors.initials: K
authors.lastname: Stierstorfer
authors.affiliation: Computed Tomography, Siemens Healthcare GmbH, Forchheim 91301, Germany.
authors.forename: Steffen
authors.initials: S
authors.lastname: Kappler
mesh_terms: D008962:Models, Theoretical; D017785:Photons; D016012:Poisson Distribution; D013057:Spectrum Analysis