The Bateman function revisited: a critical reevaluation of the quantitative expressions to characterize concentrations in the one compartment body model as a function of time with first-order invasion and first-order elimination.

The Bateman function, A"(e-k(e)t--e-k(a)t), quantifies the time course of a first-order invasion (rate constant ka) to, and a first-order elimination (rate constant ke) from, a one-compartment body model where A" = (gamma Dose)ka/(ka-ke)V. The rate constants (when ka > 3ke) are frequently determined by the "method of residuals" or "feathering." The rate constant ka is actually the sum of rate constants for the removal of drug from the invading compartment. "Flip-flop," the interchange of the values of the evaluated rate constants, occurs when ke > 3ka. Whether -ka or -ke is estimable from the terminal ln C-t slope can be determined from which apparent volume of distribution, V, derived from the Bateman function is the most reasonable. The Bateman function and "feathering" fail when the rate constants are equal. The time course is then expressed by C = gamma Dtk e-kt. The determination of such equal k values can be obtained by the nonlinear fitting of such C-t data with random error to the Bateman function. Also, rate constant equality can be concluded when 1/tmax and the kmin (value of ke at the minimum value of ek(e)tmax/ke plotted against variable ke values) are synonymous or when kmintmax approximates unity. Simpler methods exist to evaluate C-t data. When a drug has 100% bioavailability, regression of Dose/V/C on AUC/C in the nonabsorption phase gives ke no matter what is the ratio of m = ka/ke. Since k(e)tmax = ln m/(m-1), m can be determined from the given table relating m and k(e)tmax. When gamma is unknown, ke can be estimated from the abscissas of intersections of plots of Cmax ek(e)tmax and keAUC, both plotted vs. arbitrary values of ke, and gamma D/V values are estimable from the ordinate of the intersection. Also, when gamma is unknown, ke can be estimated from the abscissas of intersections (or of closest approaches) of ek(e)tmax/ke and AUC/Cmax, both plotted vs. arbitrary values of ke. The C-t plot of the Modified Bateman function, C = Be-lambda 2t-A e-lambda 1t, does not commence at the origin (i.e., when tc = 0 = 0 and when a lag time does not exist).(ABSTRACT TRUNCATED AT 400 WORDS)