Consistency equations in native detector coordinates and timing calibration for time-of-flight PET

Abstract

Fully 3-D time-of-flight (TOF) PET scanners offer the potential of previously unachievable image quality in clinical PET imaging. Timing calibration is critical to achieve the best possible timing resolution, to maximize the true counts and minimize the random counts. In this paper, we consider a data-driven timing calibration based on TOF data consistency. First, we derive two consistency equations for TOF data parameterized in the native detector coordinates where each line of response (LOR) is represented by the two azimuthal angles and axial coordinates of the two crystals detecting the LOR. By exploiting the consistency equations that must be satisfied by the time corrected TOF data, we derive the timing offset equations which only involve the zeroth and first moments with respect to the TOF variable. We also provide a numerical solution that unknown timing offsets can be directly estimated from the two moments by solving linear equations. The timing offsets can be determined up to a global constant which is immaterial for timing calibration. The data-driven timing calibration can be applied to data acquired with an arbitrary tracer distribution, which eliminates the need for a specialized data acquisition. We evaluate the timing calibration method with two-dimensional numerical simulations. The results show that the timing offsets can be determined with root mean square (RMS) errors of 1 ps and 15 ps from noise-free data and from noisy data with 16 million counts, respectively. Next generation TOF PET scanners have significantly improved timing resolution using silicon photomultiplier (SiPM) based detectors, which may require frequent timing calibration and close monitoring in performance compared to photomultiplier-tube based detectors. The proposed data-driven method allows the residual timing offsets be corrected automatically using clinical data sets whenever computing resources are available.