Indicator Transit Time Considered as a Gamma Variate

Abstract

Indicator dilution curves before the onset of the recirculation peak bear a striking resemblance to the frequency distribution curves of the class of random variables called "gamma variates." The mathematical expression for such distribution curves leads to a possible mathematical representation for indicator concentration as a function of time [image] where C(t) = indicator concentration at time, t, after injection; AT = appearance time; [alpha],[beta] = parameters of distribution; and K is a proportionality constant. With the help of a digital computer, the methods of moments and least squares were applied to each of 114 dye curves from 70 patients and 2 normals to fit the equation to the experimental curves. For all the curves, values for [alpha] and [beta] could be found which provided excellent fits. Observed and computed values for C(t) were compared by means of intraclass correlation coefficients ( R ). Closer fits were obtained by least squares (0.9805 < R < 0.9996) than by the less elaborate method of moments (0.9332 < R < 0.9990). Because of the very close correspondence between members of the family of curves in the former equation and experimental curves, indicator transit time may be considered, for practical purposes, to behave mathematically like a gamma variate. The gamma variates have well-known and convenient mathematical properties. It is anticipated that the availability of an analytical expression for indicator concentration as a function of time will facilitate theoretical analysis of arterial indicator dilution curves, characterization of normal and abnormal curves, and handling of experimental curves using high-speed computers.