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Metabolic Kinetics of Nonproductive Binding Inhibition
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Metabolic Kinetics of Nonproductive Binding Inhibition To the Editor: Frequently, nonlinear biotransformation can be described by the Michaelis-Menten equation.' However, simple MichaelisMenten kinetics cannot be applied in cases such as product inhibition.2 Recently, the kinetics of competitive drug metabolism inhibition in uiuo have been described with the use of computer simulation? in which the metabolite concentrationtime profiles offer a distinct advantage as a sensitive index of inhibition when parallel routes of metabolism and a selective inhibitory action exist. Therefore, the kinetic profile of metabolite is important in such an inhibition study. In general, the metabolite concentration-time profile declines more slowly than that of drug when the elimination rate constant of themetabolite is smaller than that of the parent drug, and conversely, the metabolite concentration-time profile declines in parallel with that of drug when the elimination rate constant of the drug is smaller than that of the m e t a b ~ l i t e . ~ A plateau concentration-time profile of metabolite when the elimination rate constant is larger than that of the drug has not been reported so far. Such concentration-time profiles cannot be described by Michaelis-Menten kinetics or product inhibition kinetics. In the present study, a simple model of nonproductive binding simulating the pIateau concentration-time profile of the metabolite, in which the elimination rate constant of metabolite is larger than that of drug, is investigated. The assumption that the elimination of metabolite is faster than that of the drug may be true when polar compounds such as glucuronide, sulfate, or glycine conjugates are the major biotransmation product^.^ The kinetic scheme for a nonproductive binding has been defined as outlined in Scheme 1: where E, S,P, and S-Erepresent enzyme, substrate, product, and enzyme-substrate complex, respectively, K , is the Michaelis-Menten constant, Ki is the inhibition constant, and kcat is a rate constant. As shown in Scheme 1,the nonproductive binding inhibition is different from a suicide substrate inhibition. In the inactivation of an enzyme by a suicide substrate, the enzyme is inactivated irreversively, thus the time-course of the remaining enzyme activity is decreased according to pseudo-first-order kinetics.7.8 In order to describe the plateau concentration-time profile of metabolite, the model depicted in Figure 1 is considered. In the model, metabolism inhibition by nonproductive binding6 is incorporated with another linear elimination process such as renal excretion, and the metabolite formation is linear with respect to concentration and time at basal conditions, i.e., in the absence of inhibition. The model is almost compatible with that of by Shaw and Houston3 except for such modification. The blood concentration-time profile of drug (C)and metabolite (C,) after intravenous bolus administration of drug can be described by the following equations:
where V,*
= Vm,,/V,
kother,
E-S'
Scheme 1 V
J
other
Figure 1-The pharmacokineticmodel for metabolism inhibition by nonproductivebinding. In the model, metabolism inhibition by nonproductive binding is incorporated with another linear elimination process. The metabolite formation is considered linear with respect to concentration and time in the absence of inhibition. Table 1-Pharmacoklnetlc Parameters Descrlblng the Disposition of Drug and Its Metaboltte In Metabolism Inhlbltlon by Nonproductlve Binding fm
Parameters
0.25
0.5
0.75
elimination rate constants, V,, is the maximum rate of the metabolite formation, and V and V , are the volumes of distribution of the parent drug and metabolite. In the abesence of inhibition by nonproductive binding, i.e., at basal condition, the disposition of both drug and metabolite is linear with respect to concentration and time. No sequential metabolism9 was presumed in the present study. Drug and metabolite concentration-time profiles were computer-generated usings eqs 1 and 2 by the Runge-Kutta-Gill method. The dose was 50 mg and the value of each concentration was generated at every 1h from t = 0 to t = 24 h with a computing interval of 0.1 h. The parameters describing the disposition of drug and its metabolite under basal condition are summarized in Table 1. For the purpose of the present study, the elimination rate constant of the metabolite was fixed to be 5 times larger than that of the parent drug. Inhibition constants over a range of 0.01-0.5 mg/L were investigated. The fractional intrinsic clearance for the formation of the given metabolite by the capacity-limited process was calculated at the basal conditions as follows:
and k(ml are the first-order
0 1994, American Chemical Society and American pharmaceutical Association
I
rnax
0022-3549/94/1200- 1 18 1$04.50/0
(3) Journal of Pharmaceutical Sciences / 1181 Vol. 83, No. 8. August 1994
0.0005
0.0001‘
i
\
0.0001
1
5
10
15
20
\
5
25
Time
10
15
20
25
(h)
Figure 2-The effect of various 4s on the concentration-time profiles for a drug and its metabolite at f, = 0.25 (A), 0.5 (B), and 0.75 (C). A dose of 50 mg is administered intravenously and the pharmacokinetic parameters used are shown in Table 1. Curve A represents the basal state and the remaining curves refer to particular K,s, i.e., K, = 0.5 for curve B, K, = 0.1 for curve C, 6 = 0.05 for curve D, and K, = 0.01 for curve E.
The area under the blood concentration-time (AUC) and the first moment-time (AUMC)curves were calculated for drug and metabolite by the trapezoidal method and extrapolated to infinity using the slope of the terminal log-linear phase. The slope of the terminal log-linear phase for each blood concentration was calculated using the four last succesive data points. The total body clearance (CL) and mean residence time (MRT) of the drug were calculated by eqs 4 and 5:
CL = D/AUC
(4)
MRT = AUMC/AUC
(5)
in which D is the dose of drug administered. The mean residence time of the metabolite (MRT,) was calculated by eq 6:1°
MRT, = AUMC,/AUC,
- MRT
(6)
in which AUMC, and AUC, are AUMC and AUC of metabolite following administration of the parent drug. The percentage of apparent inhibition at the calculated time was estimated by eq 7.
I .
inhibiton ( % ) = (1- CL,,JCL,)
AUC,,;CL, = ( l - AUC,CL
X
100
) x 100
(7)
in which CL, and CL,,i are the formation clearance of the given metabolite by the capacity-limited process with and without of inhibition, and CLi and AUC,,i are CL and AUC, in the presence of inhibition. In the present study, we assumed that the disposition of both drug and metabolite are linear with respect to concentration and time in the absence of inhibition by nonproductive binding. 1182 / Journal of Pharmaceutical Sciences Vol. 83, No. 8, August 1994
Figure 2 illustrates the effect of variousKis on the concentrationtime profiles for a drug and its metabolite at f, = 0.25,0.5, and 0.75. In the absence of inhibition by nonproductive binding, the drug concentration-time profiles were linear at thesef, values and the metabolite concentration-time course declines in parallel to the drug because the elimination rate constant of the metabolite is larger than that of the drug. The maximum metabolite concentration and “the time to reach a maximum Concentration” change progressively as Ki decreases. The duration of the plateau metabolite level a t the same Ki was prolonged with an increase of f, and the plateau concentrations of metabolite can be simulated at Ki = 0.01 for all fm values, although the drug concentration-time profiles are slightlyconvex. A typical case of the plateau concentration of metabolite can be seen a t Ki = 0.01 in f, = 0.75. The plateau metabolite level could not be observed by the product inhibition model using the same parameters shown in Table 1 (data not shown). The plateau metabolite level may be obtainable when the elimination rate of the metabolite is slower than that of the parent drug; however, the time to the peak of the metabolite would be delayed in a such case. In the case of f, = 0.75, the change in MRT, was enormous in comparison with the modest change in CL and MRT of the parent drug (Table2);this was due to the plateau concentration of metabolite. In addition, the percent inhibition at Ki = 0.01 and 0.05 seemed to be underestimated when using f, = 0.75 because of miscalculation of the terminal log-linear slope, i.e., the elimination rate constant, due to the plateau metabolite level. Although MRT, extrapolated to infinity may be overestimated due to such miscalculation of the elimination rate constant, MRT,s calculated up to 24 h in the presence of inhibition were also larger than those in the absence of inhibition (data not shown). Even taking into account the above inheritant error, we could conclude that a much larger increase of MRT, than that of the
Table 2-Effect of The Size of Inhlbltion Rate Constant In Nonproductive Blndlng on Pharmacoklnetic Parameters of Drug and Its Metabolite at Various Formation Rates of Metabolite
4
c 0.25 Inhibition
CL MRT MRT,,, 0.5
Inhibition CL MRT MRT,,, 0.75
Inhibition CL MRT MRT,
0.01
0.05
0.1
0.5
m
95.0 9.48 5.13 12.0
85.1 9.80 4.85 5.63
76.5 10.1 4.68 4.3 1
45.2 11.0 4.29 2.26
0 12.4 3.96 0.99
92.0 6.46 7.46 42.3
85.2 7.08 6.42 7.79
77.5 7.60 5.87 5.01
46.1 9.54 4.72 2.33
0 12.4 3.96 0.99
76.9 3.35 14.5 314
78.9 4.16 10.6 31.6
76.9 4.96 8.43 9.02
47.3 7.96 5.34 2.44
0 12.4 3.96 0.99
parent drug when Ki is decreased might be the typical characteristic of drug metabolism inhibition in nonproductive binding. In conclusion, the plateau concentration of the metabolite can be simulated with a nonproductive binding inhibition model when the elimination rate constant of the metabolite is larger than that of drug. That kind of blood-level profile cannot be simulated by a simple Michaelis-Menten and/or product inhibition model when the elimination rate constant of metabolite is larger than that of drug.
References and Notes 1. Gibaldi, M.; Perrier, D. Pharmacokinetics, 2nd ed.; Marcel Dekker: New York, 1982; pp 271. 2. Perrier, D.; Ashley, J. J.; Levy, G. J. Pharmacokinet. Biopharm. 1973,1, 231-242. 3. Shaw, P. N.; Houston, J. B. J . Pharmacokinet. Biopharm. 1987, 15,497-510. 4. Rowland, M.; Tozer, T. N. Clinical Pharmacokinetics: Concepts and Applications, 2nd ed.; Lea & Febiger: Philadelphia, 1989; pp 351. 5. Balant, L. P.; Doelker, E.; Buri, P. In New Trends in Pharmacokinetics; Rescigno, A.; Thakker, A., Eds.; Plenum Press: New York, 1991; pp 281-299. 6. Cornish-Bowden,A. Principle of Enzyme Kinetics; Butterworth & Co: London, 1976; pp 64-66. 7. Funaki, T.; Takanohashi,Y.; Fukazawa, H.; Kuruma, I. Biochim. Biophys. Acta 1991, 1078, 43-46. 8. Funaki, T.; Ichihara, S.; Fukazawa, H.; Kuruma, I. Biochim. Biophys. Acta, 1991, 1118, 21-24. 9. Xu, X.; Pang, K. S. J. Pharmacokinet. Biopharm. 1989,17,645672. 10. Veng-Pedersen,P.; Gillepie, W. R. Biopharm. Drug Dispos. 1987, 8, 395-401.
TOMOOFUNAKI~, HIDEOFUKAZAWA, AND ISAMI KURUMA Nippon Roche Research Center 200 Kajiwara, Kamakura-shi Kanagawa 247, Japan Received June 4, 1993. Accepted for publication April 15, 1994.
Journal of Pharmaceutical Sciences / 1183 Vol. 83, No. 8, August 1994
Metabolic Kinetics of Nonproductive Binding Inhibition To the Editor: Frequently, nonlinear biotransformation can be described by the Michaelis-Menten equation.' However, simple MichaelisMenten kinetics cannot be applied in cases such as product inhibition.2 Recently, the kinetics of competitive drug metabolism inhibition in uiuo have been described with the use of computer simulation? in which the metabolite concentrationtime profiles offer a distinct advantage as a sensitive index of inhibition when parallel routes of metabolism and a selective inhibitory action exist. Therefore, the kinetic profile of metabolite is important in such an inhibition study. In general, the metabolite concentration-time profile declines more slowly than that of drug when the elimination rate constant of themetabolite is smaller than that of the parent drug, and conversely, the metabolite concentration-time profile declines in parallel with that of drug when the elimination rate constant of the drug is smaller than that of the m e t a b ~ l i t e . ~ A plateau concentration-time profile of metabolite when the elimination rate constant is larger than that of the drug has not been reported so far. Such concentration-time profiles cannot be described by Michaelis-Menten kinetics or product inhibition kinetics. In the present study, a simple model of nonproductive binding simulating the pIateau concentration-time profile of the metabolite, in which the elimination rate constant of metabolite is larger than that of drug, is investigated. The assumption that the elimination of metabolite is faster than that of the drug may be true when polar compounds such as glucuronide, sulfate, or glycine conjugates are the major biotransmation product^.^ The kinetic scheme for a nonproductive binding has been defined as outlined in Scheme 1: where E, S,P, and S-Erepresent enzyme, substrate, product, and enzyme-substrate complex, respectively, K , is the Michaelis-Menten constant, Ki is the inhibition constant, and kcat is a rate constant. As shown in Scheme 1,the nonproductive binding inhibition is different from a suicide substrate inhibition. In the inactivation of an enzyme by a suicide substrate, the enzyme is inactivated irreversively, thus the time-course of the remaining enzyme activity is decreased according to pseudo-first-order kinetics.7.8 In order to describe the plateau concentration-time profile of metabolite, the model depicted in Figure 1 is considered. In the model, metabolism inhibition by nonproductive binding6 is incorporated with another linear elimination process such as renal excretion, and the metabolite formation is linear with respect to concentration and time at basal conditions, i.e., in the absence of inhibition. The model is almost compatible with that of by Shaw and Houston3 except for such modification. The blood concentration-time profile of drug (C)and metabolite (C,) after intravenous bolus administration of drug can be described by the following equations:
where V,*
= Vm,,/V,
kother,
E-S'
Scheme 1 V
J
other
Figure 1-The pharmacokineticmodel for metabolism inhibition by nonproductivebinding. In the model, metabolism inhibition by nonproductive binding is incorporated with another linear elimination process. The metabolite formation is considered linear with respect to concentration and time in the absence of inhibition. Table 1-Pharmacoklnetlc Parameters Descrlblng the Disposition of Drug and Its Metaboltte In Metabolism Inhlbltlon by Nonproductlve Binding fm
Parameters
0.25
0.5
0.75
elimination rate constants, V,, is the maximum rate of the metabolite formation, and V and V , are the volumes of distribution of the parent drug and metabolite. In the abesence of inhibition by nonproductive binding, i.e., at basal condition, the disposition of both drug and metabolite is linear with respect to concentration and time. No sequential metabolism9 was presumed in the present study. Drug and metabolite concentration-time profiles were computer-generated usings eqs 1 and 2 by the Runge-Kutta-Gill method. The dose was 50 mg and the value of each concentration was generated at every 1h from t = 0 to t = 24 h with a computing interval of 0.1 h. The parameters describing the disposition of drug and its metabolite under basal condition are summarized in Table 1. For the purpose of the present study, the elimination rate constant of the metabolite was fixed to be 5 times larger than that of the parent drug. Inhibition constants over a range of 0.01-0.5 mg/L were investigated. The fractional intrinsic clearance for the formation of the given metabolite by the capacity-limited process was calculated at the basal conditions as follows:
and k(ml are the first-order
0 1994, American Chemical Society and American pharmaceutical Association
I
rnax
0022-3549/94/1200- 1 18 1$04.50/0
(3) Journal of Pharmaceutical Sciences / 1181 Vol. 83, No. 8. August 1994
0.0005
0.0001‘
i
\
0.0001
1
5
10
15
20
\
5
25
Time
10
15
20
25
(h)
Figure 2-The effect of various 4s on the concentration-time profiles for a drug and its metabolite at f, = 0.25 (A), 0.5 (B), and 0.75 (C). A dose of 50 mg is administered intravenously and the pharmacokinetic parameters used are shown in Table 1. Curve A represents the basal state and the remaining curves refer to particular K,s, i.e., K, = 0.5 for curve B, K, = 0.1 for curve C, 6 = 0.05 for curve D, and K, = 0.01 for curve E.
The area under the blood concentration-time (AUC) and the first moment-time (AUMC)curves were calculated for drug and metabolite by the trapezoidal method and extrapolated to infinity using the slope of the terminal log-linear phase. The slope of the terminal log-linear phase for each blood concentration was calculated using the four last succesive data points. The total body clearance (CL) and mean residence time (MRT) of the drug were calculated by eqs 4 and 5:
CL = D/AUC
(4)
MRT = AUMC/AUC
(5)
in which D is the dose of drug administered. The mean residence time of the metabolite (MRT,) was calculated by eq 6:1°
MRT, = AUMC,/AUC,
- MRT
(6)
in which AUMC, and AUC, are AUMC and AUC of metabolite following administration of the parent drug. The percentage of apparent inhibition at the calculated time was estimated by eq 7.
I .
inhibiton ( % ) = (1- CL,,JCL,)
AUC,,;CL, = ( l - AUC,CL
X
100
) x 100
(7)
in which CL, and CL,,i are the formation clearance of the given metabolite by the capacity-limited process with and without of inhibition, and CLi and AUC,,i are CL and AUC, in the presence of inhibition. In the present study, we assumed that the disposition of both drug and metabolite are linear with respect to concentration and time in the absence of inhibition by nonproductive binding. 1182 / Journal of Pharmaceutical Sciences Vol. 83, No. 8, August 1994
Figure 2 illustrates the effect of variousKis on the concentrationtime profiles for a drug and its metabolite at f, = 0.25,0.5, and 0.75. In the absence of inhibition by nonproductive binding, the drug concentration-time profiles were linear at thesef, values and the metabolite concentration-time course declines in parallel to the drug because the elimination rate constant of the metabolite is larger than that of the drug. The maximum metabolite concentration and “the time to reach a maximum Concentration” change progressively as Ki decreases. The duration of the plateau metabolite level a t the same Ki was prolonged with an increase of f, and the plateau concentrations of metabolite can be simulated at Ki = 0.01 for all fm values, although the drug concentration-time profiles are slightlyconvex. A typical case of the plateau concentration of metabolite can be seen a t Ki = 0.01 in f, = 0.75. The plateau metabolite level could not be observed by the product inhibition model using the same parameters shown in Table 1 (data not shown). The plateau metabolite level may be obtainable when the elimination rate of the metabolite is slower than that of the parent drug; however, the time to the peak of the metabolite would be delayed in a such case. In the case of f, = 0.75, the change in MRT, was enormous in comparison with the modest change in CL and MRT of the parent drug (Table2);this was due to the plateau concentration of metabolite. In addition, the percent inhibition at Ki = 0.01 and 0.05 seemed to be underestimated when using f, = 0.75 because of miscalculation of the terminal log-linear slope, i.e., the elimination rate constant, due to the plateau metabolite level. Although MRT, extrapolated to infinity may be overestimated due to such miscalculation of the elimination rate constant, MRT,s calculated up to 24 h in the presence of inhibition were also larger than those in the absence of inhibition (data not shown). Even taking into account the above inheritant error, we could conclude that a much larger increase of MRT, than that of the
Table 2-Effect of The Size of Inhlbltion Rate Constant In Nonproductive Blndlng on Pharmacoklnetic Parameters of Drug and Its Metabolite at Various Formation Rates of Metabolite
4
c 0.25 Inhibition
CL MRT MRT,,, 0.5
Inhibition CL MRT MRT,,, 0.75
Inhibition CL MRT MRT,
0.01
0.05
0.1
0.5
m
95.0 9.48 5.13 12.0
85.1 9.80 4.85 5.63
76.5 10.1 4.68 4.3 1
45.2 11.0 4.29 2.26
0 12.4 3.96 0.99
92.0 6.46 7.46 42.3
85.2 7.08 6.42 7.79
77.5 7.60 5.87 5.01
46.1 9.54 4.72 2.33
0 12.4 3.96 0.99
76.9 3.35 14.5 314
78.9 4.16 10.6 31.6
76.9 4.96 8.43 9.02
47.3 7.96 5.34 2.44
0 12.4 3.96 0.99
parent drug when Ki is decreased might be the typical characteristic of drug metabolism inhibition in nonproductive binding. In conclusion, the plateau concentration of the metabolite can be simulated with a nonproductive binding inhibition model when the elimination rate constant of the metabolite is larger than that of drug. That kind of blood-level profile cannot be simulated by a simple Michaelis-Menten and/or product inhibition model when the elimination rate constant of metabolite is larger than that of drug.
References and Notes 1. Gibaldi, M.; Perrier, D. Pharmacokinetics, 2nd ed.; Marcel Dekker: New York, 1982; pp 271. 2. Perrier, D.; Ashley, J. J.; Levy, G. J. Pharmacokinet. Biopharm. 1973,1, 231-242. 3. Shaw, P. N.; Houston, J. B. J . Pharmacokinet. Biopharm. 1987, 15,497-510. 4. Rowland, M.; Tozer, T. N. Clinical Pharmacokinetics: Concepts and Applications, 2nd ed.; Lea & Febiger: Philadelphia, 1989; pp 351. 5. Balant, L. P.; Doelker, E.; Buri, P. In New Trends in Pharmacokinetics; Rescigno, A.; Thakker, A., Eds.; Plenum Press: New York, 1991; pp 281-299. 6. Cornish-Bowden,A. Principle of Enzyme Kinetics; Butterworth & Co: London, 1976; pp 64-66. 7. Funaki, T.; Takanohashi,Y.; Fukazawa, H.; Kuruma, I. Biochim. Biophys. Acta 1991, 1078, 43-46. 8. Funaki, T.; Ichihara, S.; Fukazawa, H.; Kuruma, I. Biochim. Biophys. Acta, 1991, 1118, 21-24. 9. Xu, X.; Pang, K. S. J. Pharmacokinet. Biopharm. 1989,17,645672. 10. Veng-Pedersen,P.; Gillepie, W. R. Biopharm. Drug Dispos. 1987, 8, 395-401.
TOMOOFUNAKI~, HIDEOFUKAZAWA, AND ISAMI KURUMA Nippon Roche Research Center 200 Kajiwara, Kamakura-shi Kanagawa 247, Japan Received June 4, 1993. Accepted for publication April 15, 1994.
Journal of Pharmaceutical Sciences / 1183 Vol. 83, No. 8, August 1994