Use of parallel Erlang density functions to analyze first-pass pulmonary uptake of multiple indicators in dogs

The gamma and Erlang density functions describe a large class of lagged, right-skewed distributions. The Erlang distribution has been shown to be the analytic solution for a chain of compartments with identical rate constants. This relationship makes it useful for the analysis of first-pass pulmonary drug uptake data following intravenous bolus administration and the incorporation of this analysis into an overall systemic drug disposition model. However, others have shown that one Erlang density function characterizes the residence time distribution of solutes in single tissues with significant systematic error. We propose a model of two Erlang density functions in parallel that does characterize well the arterial appearance of indocyanine green, antipyrine, and alfentanil administered simultaneously by right atrial bolus injection. We derive the equations that permit calculation of the higher order moments of a system consisting of two parallel Erlang density functions and use the results of these calculations from the data for all three indicators to estimate pulmonary capillary blood volume and mean transit time in the dog.