Modeling of scintillation camera systems

Despite their widespread use, the satisfactory modeling of scintillation camera systems has remained difficult. Although the resolving time and deadtime T of a nonparalyzable counter are identical and also invariable, a distinction needs to be made between the fixed resolving time tau0 and the variable deadtime tau of a paralyzable counter. It is shown here that tau = tau0(e(n) - 1)/n, where n = Ntau0 = N/Nmax is the normalized input rate and N the absolute input rate. The normalized output rate, r = Rtau0, where R is the absolute output rate, has a maximum value r(max) = 1/e approximately 0.368 at the input rate n(max) = 1, where tau = tau0(e - 1) approximately 1.718tau0. It is also shown that the response of a system of nonparalyzable and paralyzable components at all input rates is determined by just the dominant nonparalyzable and paralyzable components in the system, the response at any particular input rate being that of the component with the higher of the two deadtimes T or tau. A system can be purely paralyzable (kT = T/tau0 < or = 1), combined paralyzable/nonparalyzable (1 < kT < or = 1.718), or essentially nonparalyzable (kT > 1.718), the combined paralyzable/nonparalyzable system having a lower nonparalyzable (T > tau) and an upper paralyzable (tau > T) operating range separated by a threshold input rate n(t) = ln(1 + kTn(t)) at which tau = T. A highly accurate and explicit expression for n(t) has also been derived. In the essentially nonparalyzable case, the system operates as nonparalyzable all the way up to the system's peak response point, which may occur at or above n(max) = 1. A two-component system with kT > 1 can also be described mathematically as nonparalyzable using r = n/(1 + k(tau)n), where k(tau) = tau/tau0 = kT for n < or = n(t), and k(tau) = (e(n) - 1)/n for n > or = n(t), or as paralyzable using r = ne(-nk0) with k0 = [ln(1 + kTn)]/n for n < or = n(t) and k0 = 1 for n > or = n(t). These alternative descriptions will be of considerable importance in the measurement of T and tau0 for such systems. The model described is able to account fully for the three different operating modes possible with scintillation camera systems.