Pharmacokinetic Modeling of M6G Formation after Oral Administration of Morphine in Healthy Volunteers 

The study was conducted according to the Declaration of Helsinki on Biomedical Research Involving Human Subjects (Somerset West amendment). The University of Erlangen-Nurnberg Ethics Review Committee approved the protocol. Each participant gave written informed consent. Five healthy male volunteers (ages 25 to 34 yr; mean body weight, 75.2 +/− 6.9 kg) received 90 mg morphine orally (sustained-release tablets MST 30 and 60 [Mundipharma GmbH, Limburg/Lahn, Germany] containing 30 or 60 mg morphine sulfate 5H²O, equivalent to 22.6 mg and 45.2 mg morphine, respectively) together with 100 ml water. Only those persons were included who had already participated in a previous investigation of intravenous pharmacokinetics and pharmacodynamics of morphine and M6G; therefore, individual pharmacokinetic data after intravenous administration of morphine and preformed M6G were available for all participants. [7]The participants fasted overnight before drug administration. Eight hours after oral administration of morphine, the participants received a standard meal. At the beginning and at the end of the study, general clinical examination and routine clinical laboratory tests were performed, with special attention given to hepatic and renal function.

Blood samples (4 ml) were collected in potassium EDTA tubes before drug administration (baseline) and then every 30 min for 8 h, and then at 9, 10, 11, 12, 13, 14, 24, and 36 h after drug administration. Plasma samples were obtained within 15 min of blood collection (centrifugation, 10 min at 3,500g) and immediately stored together with quality control samples at -25 [degree sign]C until analysis. Urine was sampled fractionally over 36 h. After the urine volume was measured, 20-ml aliquots were stored at -25 [degree sign]C until analysis.

Morphine, M6G, and morphine-3-glucuronide (M3G) concentrations were assayed using a high-performance liquid chromatography method previously described. [10,11]The reliable limit of quantification was 10 ng/ml for all analytes (35.05 nM and 22.45 nM for morphine and M6G, respectively). The coefficient of variation over the calibration range of 10 to 500 ng/ml was less than 10%.

Participants were supervised continuously during the study. Specifically, blood oxygen saturation and heart rate were monitored continuously using a pulse oximeter (Nellcor N-200 pulse oximeter, Nellcor, Haywood, CA). The presence or absence of physical and psychologic effects was recorded. In addition, in accordance with previous studies in our laboratory, the participants rated the intensity of “tiredness,”“sickness,”“vertigo,” and “drowsiness” at the time of each blood sampling using visual analog scales (length, 100 mm), ranging from 0 ("no such symptom") to 100 ("symptom experienced at maximum").

Plasma Concentration-over-Time Profiles of Morphine and M6G. The kinetics of metabolite (M6G) formation after oral administration of morphine were analyzed using data from the current oral administration of the parent compound (morphine) and the individual pharmacokinetic parameters obtained from previously published intravenous data. [12]The pharmacokinetic model is presented in detail in appendix 1 of this article. This section focuses on the principal ideas rather than the exact mathematical equations.

The pharmacokinetic model was developed to provide a flexible tool to predict plasma concentrations of M6G in various clinical situations and dosing regimens of morphine. The main principle of the modeling was as follows: Imagine that we give morphine in different ways, such as an intravenous bolus, an intravenous infusion, oral administration of fast-release tablets, or oral administration of slow-release tablets. Then the plasma concentration-over-time curves are determined by how the drug enters the systemic circulation, the input function I(t)(i.e., intravenous infusion, absorption, and so on), and by how the body handles the drug, the disposition function f^D(t), whereby the latter is given by the concentration-time curve after intravenous bolus injection, f^D(t)= C (^iv)(t)/D^iv(Figure 1). Mathematically, this is a convolution of functions, which is noted with an asterisk:Equation 1To describe the plasma concentration-over-time curves of M6G after oral administration of morphine, we need to know the input functions I(t) and disposition function f (^D)(t) of M6G. Because with intravenous data the input function usually is known, the disposition function can be obtained from intravenous data using standard pharmacokinetic analysis tools (see also Glass et al. [13]). The disposition function is usually described as a sum of exponential decays f (^D)(t)= Sigmaⁱⁿ=¹[Greek small letter alpha]ⁱe^-[Greek small letter lambda]ⁱt where n is the number of exponentials (i.e., compartments; in the present case n = 2, Table 1). Each exponential term is associated with a coefficient [Greek small letter alpha] and an exponent [Greek small letter lambda], which can be used to calculate the elimination rate constant (half-life = ln(2)/elimination rate constant), and the transfer constants between compartments (for details, see Wagner [14]and appendix 2 of this article).

After having determined the disposition functions f^D(t) of morphine and M6G from the intravenous data (Table 1), [12]the input function, I(t), of M6G after oral morphine was to be found. Because M6G is a metabolite of morphine, the first step is to model the absorption of morphine. We selected an inverse Gaussian density distribution to describe the absorption of morphine (i.e., its input I(t)). The details of this function are described elsewhere [15]and in appendix 1 of this article (Equation 16). Briefly, this function provides the right bell shape and asymptotic behavior for the time course of absorption. In addition, this function permits direct estimation of two parameters of interest: MAT, the mean absorption time, and its normalized variance CV^A,p,²the shape factor of the absorption profile. The relation between morphine input, morphine plasma concentration, and the disposition of morphine can be expressed as a convolution of the same general form as Equation 1:Equation 2where

C^p,or (t)= plasma morphine concentrations over time

D^or= oral dose of morphine

F^p= oral bioavailability of morphine

f^A(t)= absorption function of morphine (the inverse Gaussian distribution function)

f^D,p (t)= disposition function of morphine (known from previous intravenous studies [12]).

Only a fraction F^mpof the total clearance of morphine CL^paccounts for the formation of M6G, the rest being metabolized to other metabolites or excreted unchanged. Furthermore, the formation of the metabolite and its appearance in the plasma takes time, which can be seen as a delay between the plasma concentration-over-time curve of morphine and M6G. This delay was accounted for by introducing the metabolic transit time function of morphine to M6G, f^M,pm (t)(for details, see Equation 24of appendix 1). Thus, after intravenous administration, the M6G input consists of the plasma concentration-over-time profile of morphine, the metabolic transit time, and the fraction of morphine that is metabolized to M6G. The relation between M6G input, M6G plasma concentration, and its disposition again can be expressed as a convolution of the same general form as Equation 1:Equation 3where

C^m,sys (t)= the plasma concentrations over time of the M6G formed from systemic morphine; that is, from the morphine in the circulation

F^mp= the fraction of the total clearance of morphine that accounts for the formation of M6G (known from previous intravenous studies [12])

CL^p= the total clearance of morphine (known from previous intravenous studies [12])

f^M,pm (t)= the metabolic transit time function, with time constant [Greek small letter lambda]^M(known from intravenous studies [12])

f^D,p (t)= the disposition function of M6G (known from previous intravenous studies [12]).

When morphine is administered parentally, Equation 3suffices as the description of the plasma concentration-over-time profile of M6G (Equation 23of appendix 1). However, when morphine is administered orally, the M6G formed during the first-pass liver metabolism of morphine adds to the M6G formed from the systemically available morphine. Only absorbed morphine is subjected to first-pass metabolism. The bioavailability, F, is the product of the fraction absorbed F^Aand the fraction that passes unmetabolized through the liver F^H,p. [16]Thus, Equation 4

The fraction extracted by the liver, E^H,p, is 1 minus the fraction of the drug that passes unmetabolized through the liver:Equation 5

The relation between M6G input from the first-pass metabolism of morphine, the M6G plasma concentration, and the disposition of M6G again can be expressed as a convolution of the same general form as Equation 1:Equation 6where

C^m,fp (t)= the plasma concentrations over time of M6G from first-pass metabolism

D^or= the oral dose of morphine

F^A= the fraction absorbed of morphine

F^H,p = the fraction of morphine that passes unmetabolized through the liver

h^mp= the fraction of hepatic morphine clearance CL^H,p that forms M6G

f^A(t)= the absorption function of morphine (the inverse Gaussian distribution function)

f^M,pm (t)= the metabolic transit time function, with time constant [Greek small letter lambda]^M(known from intravenous studies [12])

f^D,m (t)= the disposition function of M6G (known from previous intravenous studies [12]).

Using the relation h^mpCL^H,p = h^mp(1 - f^eh)CL (^p)= F^mpCL^p, where f^ehdenotes the fraction of morphine eliminated extrahepatically, the unknown h^mpfrom Equation 6can be substituted by F^mp/(1-f^eh), and Equation 6can be rewritten as Equation 7

The fraction of extrahepatic morphine elimination, f^eh, is the sum of the fraction of extrahepatically metabolized morphine and the fraction of morphine excreted unchanged in urine. The fraction of extrahepatically metabolized morphine was taken from the literature (38%). [17]The fraction of morphine excreted unchanged in urine, F^p,excr, was calculated from the amount of morphine excreted in urine, A^p,e, and the amount of morphine that had entered the systemic circulation, given by F (^p)[middle dot] D^or. Together, the measured plasma concentration over time of M6G after oral administration of morphine could be obtained easily by adding C^m,sys (t) to C^m,fp (t), as given in Equation 3and Equation 7:Equation 8where

C^m(t)= the plasma concentrations over time of total M6G

C^m,sys (t)= the plasma concentrations over time of the M6G formed from systemic morphine (i.e., from the morphine in the circulation)

C^m,fp (t)= the plasma concentrations over time of M6G from first-pass metabolism.

The pharmacokinetic analysis was performed using the Scientist 2.01 computer software (MicroMath Inc., Salt Lake City, UT). Analysis of oral data began with fitting of the C^p,or (t) curves observed after oral administration of morphine (p)(Equation 21of appendix 1) with fixed disposition parameters [Greek small letter alpha]^pand [Greek small letter lambda]^pof morphine (from intravenous data [12];Table 1). After the morphine absorption parameters MAT^p, CV^2A,p, and F (^p) were determined, they were used together with the disposition parameters of M6G ([Greek small letter alpha]^mand [Greek small letter lambda]^m;Table 1) and with the values of [Greek small letter lambda](^M) and F^mpto fit the plasma concentration-over-time profile of M6G after oral administration of morphine (Equation 27of appendix 1).

Urine Concentrations of Morphine and M6G. From the urine volumes and the urine concentrations of morphine and M6G, their cumulative amounts excreted in urine, A^e,p and A^e,m, respectively, were calculated (Equation 30, Equation 31, Equation 32of appendix 1). The fraction of morphine eliminated unchanged in urine was obtained as the quotient of the amount of morphine excreted in urine and the amount of morphine that had entered the system:Equation 9where

A^e,p = the amount of morphine excreted unchanged in urine

F^p= the oral bioavailability of morphine

D^or= the oral dose of morphine.

The fraction of M6G excreted in urine was calculated analogously; that is, from the amount of M6G excreted in urine divided by the total amount of M6G formed from morphine. The latter was given by the sum of the M6G formed from systemically available morphine and the amount of M6G formed during the first-pass liver metabolism of morphine (Equation 30and Equation 31of appendix 1, respectively).

Simulation of Plasma and Effect-Compartment Concentration-Time Profiles after Multiple Oral Dosing of Morphine. Considering the reported increase in the relative potency of oral morphine compared with parenteral morphine after multiple dosing, [9]we used the pharmacokinetic model to simulate the plasma concentration-over-time profiles of morphine and M6G after multiple oral dosing of morphine. Because we were interested in the effects rather than the plasma concentrations, we also simulated the concentrations at effect site. The concentration-over-time profile of a drug at effect site C^eff(t) can be described using the principle of Equation 1Equation 10as a convolution of the input function C(t)(plasma concentration-over-time profile) of the drug and the transfer function f^D,eff (t), which accounts for the time delay between C^eff(t) and C(t). As in the case of the metabolic transit time, we assume a simple exponential density Equation 11where k^e0is the rate constant for the transfer process. [18,19]Thus, by substituting Equation 11into Equation 10, concentration-over-time profiles at effect site C^eff(t) after oral drug administration were obtained by Equation 12where

C^eff(t)= the drug concentration-over-time profiles at effect site

K^e0= the rate constant for the transfer from plasma to the site of drug effect

C^p,or = the morphine plasma concentrations over time.

The effects site concentrations of M6G were calculated analogously. The k^e0values were taken from the literature [20](t (^1/2ke0)= 16.7 min for morphine, and 20.3 h for M6G; k^e0= 1n(2)/t (^1/2),ke0). The simulation was performed with N = 10 doses of 90 mg MST at an interval of [Greek small letter tau]= 12 h mimicking the clinical situation.

Because high M6G plasma concentrations have been related to side effects after morphine in patients with renal failure, [21]we also simulated M6G plasma and effect-compartment concentrations using a reduced plasma clearance of M6G of 10.6 ml/min as described for renal insufficiency [22](compared with a clearance found in healthy persons of 162 ml/min¹²or 187 ml/min²³). Reduced clearance translates into an altered disposition function f^D(t). To obtain the altered disposition parameters ([Greek small letter lambda]^1b;[Greek small letter lambda](^2b), [Greek small letter alpha]^1b, [Greek small letter alpha]^2b), the disposition function was reparameterized as a two-compartment model in terms of volumes, clearances, and rate constants rather than [Greek small letter alpha] and [Greek small letter lambda], using standard equations (for details, see Wagner [14]and appendix 2 of this article). Then the compartmental parameters were replaced by values available in the literature from persons with renal failure, [22]and the disposition function was reparameterized back to [Greek small letter alpha] and [Greek small letter lambda], again using standard equations (for details, see Wagner [14]and appendix 2 of this article). The new values of [Greek small letter lambda](^1b)= 3.554 h^-1, [Greek small letter lambda]^2b= 0.037 h^-1, [Greek small letter alpha]^1b= 0.063 1^-1(for comparison with the original values from healthy persons, see Table 1) were used to simulate plasma concentrations of M6G in renal failure. The simulated plasma concentration-over-time profiles of morphine and M6G were used to predict the concentration-time profile at the effect site, using Equation 12(Equation 33in appendix 1).

All participants completed the study. Side effects were generally mild to moderate and did not require medical assistance. During the first 8 h after dosing, the ratings of tiredness and drowsiness were elevated compared with baseline values; sickness and vertigo were mostly rated as “zero”(detailed data not given).

Fourteen hours after drug intake, plasma concentrations of both morphine and M6G were less than the lower limit of reliable quantification, and therefore sampling points of 24 and 36 h were discarded. M3G was the main metabolite of morphine that exceeded the concentrations of M6G by five to six times.

The plasma concentration-time data of morphine and M6G were well described by the pharmacokinetic model. Figure 2shows individual plasma concentrations over time. The estimated bioavailability of morphine was 34.5 +/− 8.7%, and the mean absorption time was 3.3 +/− 0.9 h for this commercially available morphine formulation. The fact that in the case of the metabolite model a reasonable fit of the M6G data could be achieved using only one free parameter in the equation (i.e., F^H,p) indicates the validity of the assumptions made. The total amount A^mof M6G formed from morphine was 25,532.7 +/− 4,091 nmol, which is equivalent to 13 +/− 3% of the amount of morphine absorbed. Of the total amount of M6G, 71 +/− 7% were formed during the first-pass metabolism of morphine, and 29 +/− 7% from systemically available morphine. The modeling of the metabolite kinetics of M6G also revealed a fraction of morphine absorbed F^Aof 82.3 +/− 13.7%, and a hepatosplanchnic availability of morphine (F^H,p) of 42 +/− 8%. Thus, a first-pass liver extraction ratio of 58 +/− 8%(E^H,p = 1 -F^H,p) was estimated. Table 2shows individual and mean pharmacokinetic parameters.

Most of the M6G was excreted in urine (92 +/− 17%). In contrast, only 9 +/− 3% of morphine was found as an unchanged substance in urine. Figure 3shows the cumulative renal excretion of morphine and M6G.

(Figure 4) shows a prediction of plasma and effect-compartment concentrations of morphine and M6G after 10 dose of 90 mg MST at a 12-h interval, based on k^eOvalues published by Kramer et al. [20]According to this simulation, neither morphine nor M6G is expected to accumulate in the plasma of healthy persons (i.e., in persons with normal renal function with this dosing regimen). In contrast, although there is no accumulation of morphine in the effect compartment under multiple dosing, the M6G concentration at the effect site increases progressively, reaching a steady state after approximately 80 to 100 h. Then concentrations of M6G were approximately two times higher than those of morphine. The simulation of M6G plasma and effect-compartment concentrations in patients with renal failure (Figure 4) shows that M6G is expected to accumulate in the plasma of those patients and, as a consequence, it may reach high and sustained concentrations at the effect site.

This study characterized the pharmacokinetics of M6G formed from orally administered morphine in healthy volunteers. The pharmacokinetic model analysis was based on (1) the currently obtained data after oral administration of the parent compound (morphine), and (2) the individual data obtained after the intravenous administration of the parent compound and the preformed metabolite (M6G). [12]Thus, the current approach is an extension of previous results on M6G kinetics after intravenous administration of morphine. The effect of first-pass formation of M6G from morphine also has been described. The results can be regarded as an experimental validation of the underlying noncompartmental model of metabolite kinetics. [24]As shown earlier on the basis of the areas under the curves, [16]the assessment of metabolite kinetics enables us to distinguish between the fraction absorbed (F^A) and the first-pass extraction (1 - F^H,p) of the drug as determinants of its bioavailability. Thus, in addition to specific information on the time profile of the generated metabolite, useful information on the pharmacokinetics of the precursor drug is obtained; this clearly exceeds information available from an analysis of the precursor kinetics alone. For drugs such as morphine in which a potential active metabolite is formed, this integrated mathematical model may enhance a differentiated approach to the pharmacokinetics and pharmacodynamics of the drug and its metabolite in specific clinical situations (e.g., renal failure).

The pharmacokinetic model applied here provides a general approach to the metabolite kinetics of M6G. The only parameters specific to the administered formulation (i.e., to MST) are the mean absorption time MAT (3.3 +/− 0.9 h) and the normalized variance of its distribution CV^A2(1.1 +/− 0.7). The estimated absolute bioavailability of 34 +/− 8% corresponds with that of 29 +/− 7% or 32 +/− 7% reported from other commercial oral morphine formulations. [25,26]In this respect, the results show the utility of the inverse Gaussian density (Equation 16) as a flexible input function describing the absorption of MST sustained-release tablets. It is noteworthy that the observed relative dispersion CV^A2of about 1 is characteristic of a nearly well-mixed system [27]and similar to the value estimated previously for a controlled-release formulation of another drug. [15]Furthermore, the liver extraction of 58% estimated in our five participants corresponds with that of 52% measured directly in eight healthy persons. [28] 

The model that describes the kinetics of a metabolite formed from a parent compound by first-pass and systemic metabolism was developed on the basis of morphine and M6G; however, its application is not limited to these specific substances. It may even serve as the basis for appropriate experiments involving other substances with comparable pharmacokinetics, provided that intravenous data for both the parent compound and the performed metabolite can be obtained.

The simulation of the time profiles of both plasma and effect-compartment concentrations indicated that in healthy volunteers neither morphine nor M6G are likely to accumulate in plasma after multiple administrations of morphine at a common dosing regimen. In contrast, the concentration of M6G but not that of morphine increases slowly at the effect site, reaching its steady state after four or five half-lives t^1/2,keO of the transfer process (t^1/2,keO = 0.693/k^eO). This corresponds with the observation of Hanks et al. [9]that after acute dosing, the relative analgesic potency of oral to parenteral morphine is 1:8, whereas after long-term dosing this relation increases to 1:2. Thus, a fourfold increase in potency is observed after single compared with multiple dosing of oral morphine. Hanks et al. [9]explained their observation by the contribution of M6G to the analgesic effects. Our simulation shows that steady state concentrations of M6G at the effect site are approximately two times higher than that of morphine when renal function is not compromised. Thus, the twofold higher M6G levels produce a fourfold increase in potency. This gives a relative potency of M6G to morphine of 2:1, which is not far from the value of 2.6:1 that Hanna et al. [4]obtained after intrathecal administration of morphine and M6G in humans. According to Aasmundstad et al., [29]the relative potency of M6G to morphine might be species specific. Thus, data obtained in animals showing a far higher relative potency of morphine (up to 650:1 [1]) may not reflect the human situation.

Because M6G is excreted almost completely by the kidneys, it is expected to accumulate in plasma and the effect-compartment in patients with renal failure. The high steady state plasma concentrations that we predicted are in the range of those seen by Tiseo et al. [2](approximately 4500 nM) in patients with renal dysfunction receiving long-term morphine treatment, indicating the relevance of our simulation. Patients with such elevated M6G concentrations had side effects after morphine therapy, specifically respiratory depression. Similar observations of an association of high M6G levels with opioid side effects have been reported by McQuay et al. [30]and Portenoy et al. [31] 

The simulation of effect-compartment concentrations of M6G may provide an explanation for the lack of analgesic activity after short-term administration of M6G, which we reported recently. [7]The failure of M6G to produce analgesia in that study probably reflects a pharmacokinetic problem. Specifically, M6G seems to reach pharmacodynamically relevant concentrations at the effect site only after long-term administration. Thus, in the time window of 4 h that was evaluated in that previous study, M6G probably did not reach effect-site concentrations high enough to produce clinical or analgesic effects. In contrast, under the conditions of long-term treatment, M6G might become an effective analgesic. Its possible therapeutic use should be evaluated under those long-term conditions. However, when considering the reports of increased toxicity of morphine in patients with renal impairment that has been attributed to increased levels of the predominantly renally eliminated M6G, the outcome of those studies regarding drug safety and toxicity, and thus the therapeutic index of M6G, is unpredictable.

The study focused on M6G because the clinical interest centered so far on this metabolite. The clinical importance of M3G, the primary metabolite of morphine, is difficult to estimate. There are several reports that M3G antagonizes the analgesic activity of both morphine and M6G and thus contributes to the development of tolerance to morphine. [32-35]This antagonism appears not to be mediated by opioid receptors. [36-40]Furthermore, a hyperglycemic effect of M3G by a nonopiate and nonhormonal mechanism has been shown. [41]On the other hand, several reports have questioned the hyperalgesic action of M3G. [42-46]Thus, the role of M3G in the effects of morphine remains unclear.

In conclusion, we have provided a detailed model of M6G metabolite kinetics that may serve as a pharmacokinetic basis for experiments evaluating the analgesic activity of M6G and may help to interpret the clinical effects of morphine in different patient populations.

The approach was based on a general model of metabolite kinetics [24]that was developed as an extension of a previously described steady state (or area-under-the curve-based) model. [16]Application of the approach to the plasma concentration-over-time profiles of drug and metabolite as a consecutive representation of subsystems was simplified using the Laplace transform f(s) of a time function f(t)(analogous to the modeling of pharmacokinetics after oral administration [15]). The advantage of the Laplace transform results from the reduction of the mathematically relatively complicated convolution of functions to a simple multiplication. Thus, the plasma concentration time curve of a drug was considered as a result of the independent and consecutive input and disposition processes described by an input function I(s) and disposition function f^D(s):Equation 13

In the following outline of model equations applied in this study, the precursor morphine and its metabolite M6G are characterized by indexes p and m, respectively.

Whereas for intravenous administration of a bolus dose D^iv, the concentration time curve of morphine is completely determined by its disposition function Equation 14for oral administration the input function I^p,or (s), which characterizes the absorption of a specific formulation (MST sustained-release tablets, Mundipharma GmbH, Limburg/Lahn, Germany), plays an important role Equation 15

The input function I^p,or (s) is determined by the processes of absorption and primary liver passage of the drug. The inverse Gaussian density was used as a model of the input time distribution Equation 16where MAT denotes the mean absorption time, and the normalized variance of the density function CV^A2is the shape parameter of the input function. Note that MAT includes the dissolution time of the pharmaceutical preparation. As shown recently, the inverse Gaussian density is a flexible input function with appropriate asymptotic behavior (Figure 5), with the advantage of reducing the effect of model mis-specification in parameter estimation. [15] 

Thus, the input can be written for a single oral dose D^oras Equation 17where F denotes bioavailability and Equation 18is the Laplace transform of the inverse Gaussian density.

Because inspection of the plasma concentration-over-time curves of morphine suggested in some participants a bisegmental absorption (i.e., two succeeding peaks of plasma morphine), the input model of morphine kinetics was extended to multiple-segment absorption. Specifically, to describe an absorption occurring in k segments, Equation 18was extended to Equation 19where aⁱis a weighting factor for each segment. However, given the small number of persons tested, the subsequent calculations and predictions were based on a one-segment absorption to maintain their generality. Furthermore, introduction of a second segment of absorption improved the fit only slightly (data not given), and the model selection criterion [14]increased from 1.4 +/− 0.4 to 1.5 +/− 0.6.

Based on the general assumption that the disposition curves of morphine and the preformed metabolite can be described by a sum of exponentials, the Laplace transform of the unit impulse response (i.e., the disposition functions) are given by Equation 20for the precursor, and the same Equation holdstrue for the metabolite with index m. Then it follows from Equation 13, Equation 14, Equation 15, Equation 16, Equation 17, Equation 18, Equation 19, Equation 20that the concentration-time curve of morphine after an oral dose can be described by Equation 21

Note that F = F^AF^H,p where F^Ais the fraction absorbed and F^H,p is the hepatosplanchnic availability of the drug (extraction ratio across the liver: E^H,p = 1 - F^H,p).

In the case of oral administration, the hepatic first-pass metabolism described by I^m,sys (s) must be considered, in addition to the metabolite formation I^m,fp (s) from plasma concentration C^p,or (s)(Equation 21):Equation 22where D^orF^Af^A(s) describes the drug input to the liver, and (1 - F^H,p) h^mpis the product of the extraction ratio and h^mp, the fraction of hepatic clearance CL^H,p that forms the metabolite m. The input rate I^m,fp (s) of the metabolite (m) generated from the systemically available parent drug (p) is given by (analogous to intravenous administration)Equation 23where CL^pis the total clearance of morphine, F^mpdenotes the fraction of drug p metabolized to the primary metabolite m, and f^M,pm (s) denotes the transit time density across the site of metabolism corresponding to the systemic formation of m from drug p (analogous to intravenous administration). The transit time density is based on the well-stirred model as the simplest liver model:Equation 24implying that the corresponding mean transit time across the site of metabolism is given by MTT^M= 1/[Greek small letter lambda]^M. In the case of first-pass metabolism, this transit time is included implicitly in the parameter MAT. Note that Equation 25where f^ehis the fraction of drug eliminated extrahepatically, because h^mpCL^H,p = h^mp(1 - f^eh)CL^p= F^mpCL^p. The model takes into consideration that after oral administration of morphine, there are two inputs of M6G: One part of M6G is formed from systemically available morphine analogous to intravenous administration, and another part is formed during the first-pass metabolism of morphine. Because the total input function of metabolite (i.e., the time course of metabolite generation) is the sum of I^m,sys (s)(Equation 22) and I^m,fp (s)(Equation 23), it follows analogously to Equation 13:Equation 26which finally leads to Equation 27

As shown in a previous application of this approach (intravenous data [12]), the disposition parameters of morphine ([Greek small letter alpha]^p,i and [Greek small letter lambda]^p,i, i = 1,2) and metabolite ([Greek small letter alpha]^m,i and [Greek small letter lambda]^m,i, i = 1,2) and the derived pharmacokinetic parameter, CL^p= 1/Sigmaⁱ²= 1 sup [Greek small letter alpha]^p,i/[Greek small letter lambda]^p,i, can be estimated from C(t) data after intravenous administration of morphine and M6G, respectively, using Equation 28

The parameters F^mpand [Greek small letter lambda]^Mare then obtained by fitting Equation 29to the time course of M6G concentration generated after intravenous administration of morphine (e.g., after a bolus dose D^iv). Because the parameters F, MAT, and CV^A2are estimated from the C^p,or (t) data of morphine (Equation 21), and F^A= F/F^H,p, the only parameter that remains to be estimated in Equation 27is F^H,p, the hepatosplanchnic availability of morphine.

The amount A^m,fp of M6G formed during the first pass through the liver can be derived from the Equation 22:Equation 30and the amount A^m,sys of M6G formed from systemically available morphine is given by Equation 31

The total amount of M6G A^mformed from morphine is then the sum of the amount of M6G formed during the first-pass metabolism, and the amount of M6G formed from systemically available morphine:Equation 32

The equations for multiple dosing can be formulated readily in the Laplace domain by substituting D^orwith D^orSigmaⁱ= 1^Ne (^-)(i - 1)s [Greek small letter tau]. To simulate concentrations at the effect site, an effect compartment with an equilibration time constant k^eObetween plasma concentrations and effect was added to the pharmacokinetic model. [18,19]Thus, concentrations at effect site C^eff(s) can be obtained easily by Equation 33

([14]) Disposition parameters of M6G were used to derive pharmacokinetic parameters Vⁱ, k²¹, k¹⁰, and k¹², where V (¹) denotes the volume of distribution of the central compartment, k¹⁰denotes the elimination rate constant, and k¹²and k²¹denote the transfer constants between compartments [14]:Equation 34

Clearance is given by CL = k¹⁰[middle dot] V¹. To calculate the disposition parameters for reduced clearance, CL^reduced, k (¹⁰) was recalculated as k¹⁰,reduced =(CL^reduced/V1), leaving V¹, k¹², and k²¹unchanged and taking as CL^reducedthe value of 10.6 ml/min published by Hanna et al. [22]from patients with renal failure. Then new disposition parameters [Greek small letter lambda]¹,reduced, [Greek small letter lambda]²,reduced, [Greek small letter alpha]¹,reduced, and [Greek small letter alpha]²,reduced were calculated as described by Wagner [14]Equation 35

The new values of [Greek small letter lambda]¹,reduced = 3.554 h^-1, [Greek small letter lambda]²,reduced = 0.037 h^-1, [Greek small letter alpha]¹,reduced = 0.063 l^-1, and [Greek small letter alpha]²,reduced = 0.058 l^-1(for comparison with the original values from healthy persons, see Table 2) were introduced in the equations instead of the original values to simulate plasma and effect-compartment concentrations of M6G in renal failure.

The authors thank Dr. J. Liefhold, Mundipharma GmbH, Limburg/Lahn, Germany, for supplying the study medication and the M6G as analytical standard; Dr. Steven L. Shafer, Palo Alto, California, for helpful discussions; Professor Dr. A. Barocka, Department of Psychiatry, University of Erlangen-Nurnberg, for his contribution; and Doris Schreiner for technical assistance.