The Bateman function, A''(e(-ket)) -e(-kat)), 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'' = (gamma Dose) k(a)/(k(a) - k(e))V. The rate constants (when k(a) > 3k(e)) are freguently determined by the ''method of residuals'' or ''feathering.'' The rate constant k(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 when k(e) > 3k(a). Whether -k(a) or -k(e) is estimable from the terminal In C-t slope can be determined from which apparent volume of distribution, V. derived from tire 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/t(max) and the k(min) (value of k(e) at the minimum value of e(ketmax)/k(e) plotted against variable k(e) values) are synonymous or when k(min)t(max) 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 k(e) no matter what is the ratio of m = k(a)/k(e). Since k(e)t(max)= ln m/(m-1), m can be determined from tire given table relating m and k(e)t(max). When gamma is unknown, k(e) can be estimated from the abscissas of intersections of plots of C-max e(ketmax) and k(e)AUC, both plotted us. arbitrary values of k(e), and gamma D/V values are estimable from the ordinate of the intersection. Also, when gamma is unknown, k(e) can be estimated from the abscissas of intersections (or of closest approaches) of e(ketmax)/k(e) and AUC/C-max, both plotted us. arbitrary values of k(e). The C-t plot of the Modified Bateman function, C = B e (-lambda 2t)-A e(-lambda 1t), does not commence at the origin (i.e.. when t(c=0)=0 and when a lag time does not exist). However, t(c)=ln(A/B)/ (lambda(1) lambda(2)) when A > B. AUC(A'') without time lag is the same as AUC(A not equal B) and A'' = B e(-lambda 2t) =A e(-lambda 1t). The t(max) of the C-t plot of tire latter is t(c=0) later than the t(max) of the C-t plot of the former which commences at t=0. However, (AUMC(uncorr)(A not equal B) infinity=B/lambda(2)(2)-A/lambda(1)(2) differs from (AUMC(corr)(A not equal B)) = A''t(C=0)(1/lambda(2) - 1/lambda(1)) + A'' (1/lambda)(2)(2)-1/lambda(1)(2)). (AUMC(corr) (A'')) = A''(1/lambda(2)(2) - 1/lambda(1)(2)) when C-t plots start at t=0. AUMC(uncorr)(A not equal B) is not valid. The (MRT(uncorr)(A not equal B)) infinity is also an invalid MRT estimate, (B/lambda(2)(2) - A/lambda(1)(2))/e(tc=0)(B/lambda(2) - A/lambda(1)), but when A > B, C-t curves which start at the origin, C-t=0 have MRT values displaced by t(c=0), i.e., MRT(corr)(A not equal B) = MRT ([A''or A'= A=B]) + t(C=0). The t(max) of the Bateman function is also displaced by t(c=0) when the A exceeds the B of its modified form. Dose-dependent pharmacokinetics can be concluded from C-t data generated by various first-order invading nonintravenous doses if drug absorption is 100% The k(e) values can be determined if the apparent volume of distribution of the one-compartment body model is known. Plots of m/AUC(t)(p) vs. time t have a slope of -CL(ME), (the negative of the clearance of the metabolite) and an intercept of the clearance of rite precursor, CL(PM), provided that all of tire precursor had been absorbed. Similar studies could determine the apparent 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.
