Transit compartments versus gamma distribution function to model signal transduction processes in pharmacodynamics

Delayed effects for pharmacodynamic responses can be observed for many signal transduction processes. Three approaches are summarized in this report to describe such effects caused by cascading steps: stochastic process model, gamma distribution function, and transit compartment model. The gamma distribution function, a probability density function of the waiting time for the final step in a stochastic process model, is a function of time with two variables: number of compartments N, and the expected number of compartments occurring per unit time k. The parameter k is equal to 1/tau, where tau is the mean transit time in the stochastic process model. Effects of N and k on the gamma distribution function were examined. The transit compartment model can link the pharmacokinetic profile of the tested compound, receptor occupancy, and cascade steps for the signal transduction process. Time delays are described by numbers of steps, the mean transit time tau, and the amplification or suppression of the process as characterized by a power coefficient gamma. The effects of N, tau, and gamma on signal transduction profiles are shown. The gamma distribution function can be utilized to estimate N and k values when the final response profile is available, but it is less flexible than transit compartments when dose-response relationships, receptor dynamics, and efficiency of the transduction process are of concern. The transit compartment model is useful in pharmacokinetic/pharmacodynamic modeling to describe precursor/product relationships in signal transduction process.