Abstract
Models describing the plasma concentrationtime curves of generated metabolite after iv and oral drug administration are presented. Utilizing numerical inverse Laplace transformation, the method can readily be used for parameter estimation and model simulation in conjunction with appropriate curvefitting software. The approach is not limited to compartment modeling and can be applied to any linear pharmacokinetic system exhibiting hepatic and renal elimination of the parent drug. The model is formulated for single and multiple dosing of the precursor, including bolus doses and/or infusions for iv administration and sustainedrelease dosage forms for oral administration.
To date, two different methods have been used for modeling of metabolite kinetics, i.e. the mathematical analysis of concentrationtime curves of a parent drug and the generated metabolite: classical compartmental models and socalled model independent approaches, which are not restricted to the assumption of wellmixed compartments and simple firstorder transfer processes. One reason why noncompartmental methods are applied relatively seldomly to the prediction (analysis) of concentrationtime curves of a formed metabolite is the need to apply convolution (deconvolution) methods. Although the discussion of deconvolution methods is out of the scope of this paper, it should be noted that these methods have practical limitations (instability in the case of noisy data and no direct estimation of metabolite parameters). We propose here a simple and practicable method that is based on a general model of metabolite kinetics after iv and oral administration of the parent drug (Weiss, 1988). This approach, which was originally used to estimate curve moments—i.e. pharmacokinetic parameters that are dependent on theAUCs^{1} or mean residence times—is now extended to the analysis of the timecourse of metabolite concentration. Since the dispositions of drug and metabolite can be regarded as consecutive subsystems, modeling is greatly simplified in the Laplace domain. While the model equations can be readily formulated, the crucial point for the application in practice is the availability of methods of the numerical inverse Laplace transformation to obtain C(t) fromĈ(s), the corresponding model in the Laplace domain.
In recent years, appropriate curvefitting software became available in which such numerical inversion methods are implemented: for example, SCIENTIST (MicroMath Scientific Software, Salt Lake City, UT) for DOS, and MINIM (Purves, 1995) for Macintosh computers. We have also applied this method in combination with the widely used ADAPT II program. For the evaluation of metabolite concentrationtime curve after oral administration of the parent drug, a flexible oral input model has been utilized (Weiss, 1996).
Materials and Methods
Pharmacokinetic Model.
As shown in fig. 1, the complete model for oral administration of drug and iv administration of precursor and preformed metabolite can be decomposed into subsystems describing drug input and the disposition processes of drug and metabolite, respectively.
Input models: drug dosing.
The input function in the Laplace domain for administration of a bolus dose D_{iv} (at t = 0)
Disposition models.
Disposition curves of the drug and the preformed metabolite are described by a sum of exponentials
ConcentrationTime Profiles of the Drug.
The plasma concentrationtime curveĈ_{p}(s) of the drug after an input or dosing function, Î_{p}(s) can be written as
ConcentrationTime Profiles of the Formed Metabolite.
For iv administration, the time course of C_{p}(t) determines the input rate Î_{mp}(s) of the metabolite (m) generated from the parent drug (p):
Results and Discussion
Model Simulation.
The effect of different dosing schedules on the time course of the parent drug and the formed metabolite has been simulated according to eqs. 9, 13, 17, and 21, respectively, using the software package SCIENTIST (fig. 2). The underlying model parameters of a highclearance drug (80% hepatic extraction) with a fraction metabolized to the primary metabolite of 10% are characteristic for the metabolism of morphine to morphine6glucuronide in humans (Hasselström and Säwe, 1993; Lötschet al., 1998). The dosing interval for the four intravenous bolus doses is 6 hr. For oral administration, the response to a slowrelease formulation (MAT = 5 hr, CV_{A}^{2} = 0.8) is compared with that of a normal tablet (MAT = 1 hr, CV_{A}^{2} = 0.8), assuming that the dose is divided into four doses administered in intervals of 6 hr, as in the case of intravenous administration.
Data Analysis.
Analogously to the situation in the analysis of pharmacokinetic systems after oral administration, in which intravenous data are necessary to identify the system, a complete identification of the metabolite formation kinetics (i.e. estimation of parameters in eq. 21) is only possible if disposition data of the preformed metabolite are available. Data analysis starts with fitting of the C_{p,iv}(t) and C_{m,iv}(t) curves observed after iv administration of precursor (p) and metabolite (m), respectively. In this case, the disposition curves in the time domain (eq. 4) can be used for parameter estimation, or for infusion with rate k_{0} over a time period T,
The number of exponential terms (np and nm) can be determined using a model selection criterion (e.g. the MSC criterion of SCIENTIST, which is a modified Akaike information criterion). The estimated disposition parameters of parent drug (α_{p,i} and λ_{p,i}, i = 1..np) and metabolite (α_{m,i} and λ_{m,i}, i = 1..nm) and the derived pharmacokinetic parameter, CL_{p} = 1/∑_{i=1}^{np} α_{p,i}/λ_{p,i}, are then substituted into eq. 17 in the case of iv drug administration. These parameters are held fixed in the fitting of the formed metabolite [C_{mp,iv}(t)] data to estimate λ_{H} (= 1/MTT_{H}) and F_{mp}. Since numerical inverse Laplace transformation C(t) = L^{−1}{Ĉ(s)} is part of the fitting procedure, all model equations can be used as given in the Laplace domain. For oral administration, the concentrationtime curve of the precursor (C_{p,or}) is not available in the time domain, and eq. 13 has to be fitted to the data to estimate F, MAT and CV_{A}^{2} (Weiss, 1996). Again the estimated parameters are held fixed in fitting eq. 21 to the curve of the metabolite generated after oral administration of the drug. (The only parameter that remains to be estimated in this fit is F_{H,p}, since F_{A} = F/F_{H,p}.)
Conclusion.
In this paper, we presented an approach to parameter estimation and model simulation in metabolite kinetics formulated in terms of Laplace transforms. Numerical inversion of the equations by a curvefitting software makes the method userfriendly. The SCIENTIST program files of the described solutions can be found in the . Despite some simplifying assumptions, the model is more general than previous concepts (Karol and Goodrich, 1988; Chan and Gibaldi, 1990;Nigrovic and Banoub, 1992). The basic structure of the noncompartmental model (fig. 1) was originally used in applying statistical moment theory to metabolite kinetics (Weiss, 1988). Apart from the intrinsic disposition parameters of the metabolite, its formation kinetics after intravenous administration of the parent drug are described by only two parameters, the fraction of drug metabolized to the metabolite and the time constant of the hepatic formation process, i.e. the mean transit time across the liver for drug input and metabolite output. The evaluation of metabolite kinetics after oral administration of the precursor is based on a flexible input function that allows an application to immediate and controlled release tablets (Weiss, 1996). Note that in the case of firstpass metabolite generation the liver transit time is part of MAT, the input time of the precursor. As shown earlier on the basis of the AUCs or steadystate concentrations (Weiss, 1990), the assessment of metabolite kinetics enables us to distinguish between the fraction absorbed (F_{A}) and firstpass extraction (1 − F_{H,p}) of the drug as determinants of bioavailability. The utilization of the present approach can be useful in the design and pharmacokinetic analysis of novel controlledrelease dosage forms. The flexibility of the input mode (multiple dosing) might also be of interest for toxicokinetic applications. Part of this theory has been applied successfully to analyze metabolite kinetics after iv administration of morphine to healthy volunteers (Lötsch et al., 1998).
Appendix
SCIENTIST program source listings for the case of twoexponential disposition of the precursor and preformed metabolite (np = nm = 2) following a bolus dose D_{iv}.
 // Equation 5
 IndVars: T
 DepVars: Civ
 Params: Div, a1, a2, l1, l2
 Civ = Div*(al*exp(−11*T) + a2*exp(−12*T))
 // Div constant
 ***
 C(t) of drug (precursor) after an oral dose D_{or}
 // Equation 13
 IndVars: T
 LaplaceVar: s
 DepVars: Co
 Params: Dor, F, CVs, MAT, 1h, a1, a2, l1, l2
 fp = a1/(s+11)+a2/(s+12)
 m = 2*MAT/CVs
 n = 1/(2*MAT*CVs)
 CoL = Dor*F*exp(−sqrt(m*(s+n))+1/CVs)
 Co = LAPLACEINVERSE(T, CoL, s)
 // Dor constant
 // a1, a2, l1, l2 fixed parameters if estimated in eq. 5
 ***
 C(t) of metabolite after an iv dose of precursor D_{iv}
 // Equation 17
 IndVars: T
 LaplaceVar: s
 DepVars: Cm
 Params: Div, Fmp, CL, 1h, a1, a2, l1, l2, am1, am2, lm1, lm2
 fp = a1/(s+11)+a2/(s+12)
 fm = am1/(s+lm1)+am2/(s+lm2)
 CmL = Div*Fmp*CL*(lh/(s+lh))*fp*fm
 Cm = LAPLACEINVERSE(T, CmL, s)
 // Div constant
 // a1, a2, l1, l2, am1, am2, lm1, lm2, CL fixed parameters
 ***
 C(t) of metabolite after an oral dose of precursor D_{or}
 // Equation 21
 IndVars: T
 LaplaceVar: s
 DepVars: Cmo
 Params: Dor, F, Fh, Fe, Fmp, CVs, MAT, CL, 1h, a1, a2, l1, l2, am1, am2, lm1, lm2
 fp = al/(s+11)+a2/(s+12)
 fm = am1/(s+lm1)+a2/(s+l2)
 m = 2*MAT/CVs
 n = 1/(2*MAT*CVs)
 x = exp(−sqrt(m*(s+n))+1/CVs)
 CmoL = Dor*(F/Fh)*Fmp*x*((1−Fh)/(1−Fe) + Fh*CL*(1h/(s+lh)) *fp)*fm
 Cmo = LAPLACEINVERSE(T, CmoL, s)
 // Dor constant
 // a1, a2, l1, l2, am1, am2, lm1, lm2, CL, 1h, Fmp, F, CVs, MAT fixed parameters
 ***
Footnotes

Send reprint requests to: Dr. Michael Weiss, Section of Pharmacokinetics, Department of Pharmacology, Martin Luther University HalleWittenberg, 06097 Halle, Germany.
 Abbreviations used are::
 AUC
 area under the curve
 C(t)
 concentrationtime curve
 MAT
 mean absorption time
 MTT
 mean transit time
 Received November 17, 1997.
 Accepted February 20, 1998.
 The American Society for Pharmacology and Experimental Therapeutics