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″=(γDose)ka/(k a−k e) V. The rate constants (whenk a>3k e) are frequently determined mined by the “method of residuals” or “feathering”. The rate constantk a 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 whenk e>3k a. Whether −k a or −k e is estimable from the terminal lnC-t slope can be determined from which apparent volume of distribution,V, derived from the Baterman function is the most reasonable. The Bateman function and “feathering” fail when the rate constants are equal. The time course is then expressed byC=γDtk e −kt . The determination of such equalk values can be obtained by the nonlinear fitting of suchC-t data with random error to the Bateman function. Also, rate constant equality can be concluded when 1/t max and the k min (value ofk e at the minimum value of $$e^{ - k_e t_{max} } /k_e$$ plotted against variablek e values) are synonymous or whenk min t max approximates unity. Simpler methods exist to evaluateC-t data. When a drug has 100% bioavailability, regression ofDose/V/C onAUC/C in the nonabsorption phase givesk e no matter what is the ratio ofm=k a /k e . Sincek e t max=lnm/(m−1),m can be determined from the given table relatingm andk e t max. When γ is unknown,k e can be estimated from the abscissas of intersections of plots of $$C_{max} e^{k_e t_{max} }$$ andk e AUC, both plotted vs. arbitrary values ofk e, and γD/V values are estimable from the ordinate of the intersection. Also, when γ is unknown,k e can be estimated from the abscissas of intersections (or of closest approaches) of $$e^{k_e t_{max} } /k_e$$ andAUC/C max, both plotted vs. arbitrary values ofk e. TheC-t plot of the Modified Bateman function, $$C = Be^{ - \lambda _2 t} - Ae^{ - \lambda _1 t}$$ , does not commence at the origin (i.e., whent c=0=0 and when a lag time does not exist). However,T C=0 = ln(A/B/(λ1-λ2 whenA>B. AUC A″ without time lag is the same asAUC A≠B and $$A'' = Be^{ - \lambda _2 t} = Ae^{ - \lambda _1 t}$$ . Thet max of theC-t plot of the latter ist c=0 later than thet max of theC-t plot of the former which commences att=0. However, (AUMC uncorr )∞ =B/λ 2 2 -A/λ 1 2 differs fromAUMC corr ) =A″t C=0 (1/λ2-1/λ1 +A″(1/λ 2 2 -1/λ 1 2 ). (AUMC corr ) =A″(1/λ 2 2 -1/λ 1 2 ) whenC-t plots start att=0.AUMC uncorr is not valid. The (MRT uncorr )∞ is also an invalidMRT estimate, $$(B/\lambda _2^2 - A/\lambda _1^2 )/e^{t_{c = 0} } (B/\lambda _2 - A/\lambda _1 )$$ , but whenA>B, C-t curves which start at the origin,C t=0 , haveMRT values displaced byMRT corr =MRT [A′ or A' = A = B] ; +t C=0 . Thet max of the Bateman function is also displaced byt C=0 when theA exceeds theB of its modified form. Dose-dependent pharmacokinetics can be concluded fromC-t data generated by various firstorder invading nonintravenous doses if drug absorption is 100%. Thek e values can be determined if the apparent volume of distribution of the one-compartment body model is known. Plots ofm/AUC vs. timet have a slope of — CLME, (the negative of the clearance of the metabolite) and an intercept of the clearance of the precursor, CLPM, provided that all of the precursor had been absorbed. Similar studies could determine the appararent volume of distribution of the metabolite and the clearance (and thus the rate constant,k PM=CLPM/V P) of the precursor to the metabolite.