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 k_a) to, and a first-order elimination (rate constant k_e) from, a one-compartment body model where A″=(γDose)k_a/(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 ^A≠B )_∞ =B/λ ₂ ² -A/λ ₁ ² differs fromAUMC _corr ^A≠B ) =A″t _C=0 (1/λ₂-1/λ₁ +A″(1/λ ₂ ² -1/λ ₁ ² ). (AUMC _corr ^A″ ) =A″(1/λ ₂ ² -1/λ ₁ ² ) whenC-t plots start att=0.AUMC _uncorr ^{A ≠ B} is not valid. The (MRT _uncorr ^{A ≠ B} )_∞ 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 ^{A ≠ B} =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 _t ^p vs. timet have a slope of — CL_ME, (the negative of the clearance of the metabolite) and an intercept of the clearance of the precursor, CL_PM, 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=CL_PM/V _P) of the precursor to the metabolite.