  The effect of the intrinsic lutetium radioactivity on the detection
performances of a LYSO based in-beam PET prototype used for quality control of
hadron therapy treatments is studied. This radioactivity leads to a background
that degrades the measurement of the $\beta$ + signal. In particular, it
prevents the measurement of faint signals originating from low activity $\beta$
+ sources. This paper presents a method to estimate the minimum $\beta$ +
activity that can be measured for any acquisition time taking into account the
non-extensible dead time of the detector. This method is illustrated with
experimental data collected with the in-beam PET prototype. The results
presented in this paper are therefore specific to this detector. The method can
however be applied in other contexts, either to other lutetium based PET
detectors or even to non-PET detectors affected by lutetium radioactivity. The
dead time correction formalism can also be used generally to scale signal and
background yields in any non-paralizable detector, even those in which the
background is not due to the presence of intrinsic radioactivity.
