Gamma detector dead time correction using Lambert W function

Abstract

Background: For therapeutic applications of several isotopes (e.g., 131I, 153Sm, 177Lu) in nuclear medicine, the high activities typically applied require accurate dead time correction in early time point imaging. We present a novel, straightforward dead time correction method using the Lambert W function, which is in principle exact for the paralyzable detector model with a single parameter τ (i.e., dead time). Results: As a proof of concept, the method is validated with a simple model: a commonly used isotope, 99mTc, with a single photopeak. We measured count rates of a gamma camera both intrinsically and extrinsically (i.e., with collimators) with point sources in air and in a scatter phantom (extrinsic only). τ was estimated for both open window (τOW) and a 99mTc photopeak window (τTc), using a “graphical” method for fitting the count rate of decaying sources. These values for τ were subsequently used for dead time correction. τ varied significantly between the different geometries for both energy windows, but τOW was more reproducible than τTc, particularly for the scatter phantom measurements. τOW measured from the phantom measurements was approximately 30% lower than τOW from the intrinsic measurement but corresponded within 15% with the extrinsic point source measurements. Accordingly, using the intrinsic τOW led to an overcorrection of 8% at high count rates; τOW from the extrinsic point source measurements corrected the phantom measurement to within 2%. However, significant differences were observed between τTc values. All measured τTc values underestimated dead time losses in a second independent phantom measurement, with even τTc from the first phantom measurement underestimating activity with 5–10% at the highest count rates. Based on measurements of the effect of energy window settings and geometry, we tentatively attribute the added dead time losses to pulse pile-up. Conclusions: Analytic dead time correction based on the Lambert W function is accurate for the range in which gamma detectors behave as paralyzable systems. However, further investigation indicated measured τ values to be variable with geometry as well as window fraction. We propose that dead time correction should be based on the open window value, τOW, corrected for window fraction.