Point spread function of photon-counting detectors under pile-up conditions: a proposed framework

X-ray detectors with photon-counting capabilities promise to revolutionise medical imaging. For an efficient comparison of detectors of various materials and with different setup choices, reliable detector performance measures are needed. The detector point spread function (PSF) is a commonly used measure, which describes the spatial response of an X-ray detector to the irradiation of a single pixel, given the energy spectrum of the source. In the case of an energy-resolving PCD, the detector PSF is typically derived for each energy bin and characterises its resolution. Moreover, it is commonly determined under low count rate conditions, to avoid dead time and pile-up related distortions. Under these assumptions, the PSF can be determined in a straightforward manner, but does not fully characterise the detector under all conditions encountered in clinical practice. This is especially true since the number of registered counts per energy bin depends on both the incident spectrum and the fluence rate, due to pile-up and dead time. We therefore propose a new metric, the differential point spread function (dPSF), which describes the change in the output count rate due to a small change in the input spectrum, for a given combination of incident spectrum and fluence rate. The dPSF can be used to characterize the spectral and spatial performance of a PCD under high-fluence conditions, i.e. when its response becomes non-linear. We illustrate the use of the dPSF by performing a Monte-Carlo study in which we compare the response of direct-conversion and scintillationbased PCDs at different fluence rates.