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Determination of KM and Vmax
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Equation 9.1 relates the rate of drug biotransformation to the concentration of the drug in the body. The same equation may be applied to determine the rate of enzymatic reaction of a drug in vitro (Equation 9.7). When an experiment is performed with solutions of various concentration of drug C, a series of reaction rates (v) may be measured for each concentration. Special plots may then be used to determine KM and Vmax (see also Chapter 11).
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Equation 9.7 may be rearranged into Equation 9.8.
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Equation 9.8 is a linear equation when 1/v is plotted against 1/C. The y intercept for the line is 1/Vmax, and the slope is KM/Vmax. An example of a drug reacting enzymatically with rate (v) at various concentrations C is shown in Table 9-6 and Fig. 9-5. A plot of 1/v versus 1/C is shown in Fig. 9-6. A plot of 1/v versus 1/C is linear with an intercept of 0.33 mmol. Therefore,
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because the slope = 1.65 = KM/Vmax = KM/3 or KM = 3 × 1.65 μmol/mL = 5 μmol/mL. Alternatively, KM may be found from the x intercept, where −1/KM is equal to the x intercept. (This may be seen by extending the graph to intercept the x axis in the negative region.)
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With this plot (Fig. 9-6), the points are clustered. Other methods are available that may spread the points more evenly. These methods are derived from rearranging Equation 9.8 into Equations 9.9 and 9.10.
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A plot of C/v versus C would yield a straight line with 1/Vmax as slope and KM/Vmax as intercept (Equation 9.9). A plot of v versus v/C would yield a slope of −KM and an intercept of Vmax (Equation 9.10).
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The necessary calculations for making the above plots are shown in Table 9-7. The plots are shown in Figs. 9-7 and 9-8. It should be noted that the data are spread out better by the two latter plots. Calculations from the slope show that the same KM and Vmax are obtained as in Fig. 9-6. When the data are more scattered, one method may be more accurate than the other. A simple approach is to graph the data and examine the linearity of the graphs. The same basic type of plot is used in the clinical literature to determine KM and Vmax for individual patients for drugs that undergo capacity-limited kinetics.
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Determination of KM and Vmax in Patients
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Equation 9.7 shows that the rate of drug metabolism (v) is dependent on the concentration of the drug (C). This same basic concept may be applied to the rate of drug metabolism of a capacity-limited drug in the body (see Chapter 11). The body may be regarded as a single compartment in which the drug is dissolved. The rate of drug metabolism will vary depending on the concentration of drug Cp as well as on the metabolic rate constants KM and Vmax of the drug in each individual.
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An example for the determination of KM and Vmax is given for the drug phenytoin. Phenytoin undergoes capacity-limited kinetics at therapeutic drug concentrations in the body. To determine KM and Vmax, two different dose regimens are given at different times, until steady state is reached. The steady-state drug concentrations are then measured by assay. At steady state, the rate of drug metabolism (v) is assumed to be the same as the rate of drug input R (dose/day). Therefore Equation 9.11 may be written for drug metabolism in the body similar to the way drugs are metabolized in vitro (Eq. 9.7). However, steady state will not be reached if the drug input rate, R, is greater than the Vmax; instead, drug accumulation will continue to occur without reaching a steady-state plateau.
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where R = dose/day or dosing rate, CSS = steady-state plasma drug concentration, Vmax = maximum metabolic rate constant in the body, and KM = Michaelis–Menten constant of the drug in the body.
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Determination of KM and Vmax by Direct Method
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When steady-state concentrations of phenytoin are known at only two dose levels, there is no advantage in using the graphic method. KM and Vmax may be calculated by solving two simultaneous equations formed by substituting CSS and R (Equation 9.11) with C1, R1, C2, and R2. The equations contain two unknowns, KM and Vmax, and may be solved easily.
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Combining the two equations yields Equation 9.15.
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where C1 is steady-state plasma drug concentration after dose 1, C2 is steady-state plasma drug concentration after dose 2, R1 is the first dosing rate, and R2 is the second dosing rate. To calculate KM and Vmax, use Equation 9.15 with the values C1 = 8.6 mg/L, C2 = 25.1 mg/L, R1 = 150 mg/d, and R2 = 300 mg/d. The results are
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Substitute KM into either of the two simultaneous equations to solve for Vmax.
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Interpretation of KM and Vmax
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An understanding of Michaelis–Menten kinetics provides insight into the nonlinear kinetics and helps avoid dosing a drug at a concentration near enzyme saturation. For example, in the above phenytoin dosing example, since KM occurs at 0.5Vmax, KM = 27.3 mg/L, the implication is that at a plasma concentration of 27.3 mg/L, enzymes responsible for phenytoin metabolism are eliminating the drug at 50% Vmax, ie, 0.5 × 626 mg/day or 313 mg/day. When the subject is receiving 300 mg of phenytoin per day, the plasma drug concentration of phenytoin is 8.6 mg/L, which is considerably below the KM of 27.3 mg/L. In practice, the KM in patients can range from 1 to 15 mg/L, Vmax can range from 100 to 1000 mg/d. Patients with a low KM tend to have greater changes in plasma concentrations during dosing adjustments. Patients with a smaller KM (same Vmax) will show a greater change in the rate of elimination when plasma drug concentration changes compared to subjects with a higher KM. A subject with the same Vmax, but different KM, is shown in Fig. 9-12. (For another example, see the slopes of the two curves generated in Fig. 9-4.)
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Dependence of Elimination Half-Life on Dose
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For drugs that follow linear kinetics, the elimination half-life is constant and does not change with dose or drug concentration. For a drug that follows nonlinear kinetics, the elimination half-life and drug clearance both change with dose or drug concentration. Generally, the elimination half-life becomes longer, clearance becomes smaller, and the area under the curve becomes disproportionately larger with increasing dose. The relationship between elimination half-life and drug concentration is shown in Equation 9.16. The elimination half-life is dependent on the Michaelis–Menten parameters and concentration.
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Some pharmacokineticists prefer not to calculate the elimination half-life of a nonlinear drug because the elimination half-life is not constant. Clinically, if the half-life is increasing as plasma concentration increases, and there is no apparent change in metabolic or renal function, then there is a good possibility that the drug may be metabolized by nonlinear kinetics.
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Dependence of Clearance on Dose
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The total body clearance of a drug given by IV bolus injection that follows a one-compartment model with Michaelis–Menten elimination kinetics changes with respect to time and plasma drug concentration. Within a certain drug concentration range, an average or mean clearance (Clav) may be determined. Because the drug follows Michaelis–Menten kinetics, Clav is dose dependent. Clav may be estimated from the area under the curve and the dose given (Wagner et al, 1985).
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According to the Michaelis–Menten equation,
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Inverting Equation 9.17 and rearranging yields
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where V′max is the maximum velocity for metabolism. Units for V′max are mass/compartment volume per unit time. V′max = Vmax/VD; Wagner et al (1985) used Vmax in Equation 9.20 as mass/time to be consistent with biochemistry literature, which considers the initial mass of the substrate reacting with the enzyme.
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Integration of Equation 9.18 from time 0 to infinity gives Equation 9.20.
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where VD is the apparent volume of distribution.
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Because the dose Equation 9.20 may be expressed as
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To obtain mean body clearance, Clav is then calculated from the dose and the AUC.
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Alternatively, dividing Equation 9.17 by Cp gives Equation 9.24, which shows that the clearance of a drug that follows nonlinear pharmacokinetics is dependent on the plasma drug concentration Cp, KM, and Vmax.
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Equation 9.22 or 9.23 calculates the average clearance Clav for the drug after a single IV bolus dose over the entire time course of the drug in the body. For any time period, clearance may be calculated (see Chapters 6 and 11) as
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In Chapter 11, the physiologic model based on blood flow and intrinsic clearance is used to describe drug metabolism. The extraction ratios of many drugs are listed in the literature. Actually, extraction ratios are dependent on dose, enzymatic system, and blood flow, and for practical purposes, they are often assumed to be constant at normal doses.
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Except for phenytoin, there is a paucity of KM and Vmax data defining the nature of nonlinear drug elimination in patients. However, abundant information is available supporting variable metabolism due to genetic polymorphism (Chapter 11). The clearance (apparent) of many of these drugs in patients who are slow metabolizers changes with dose, although these drugs may exhibit linear kinetics in subjects with the “normal” phenotype. Metoprolol and many β-adrenergic antagonists are extensively metabolized. The plasma levels of metoprolol in slow metabolizers (Lennard et al, 1986) were much greater than other patients, and the AUC, after equal doses, is several times greater among slow metabolizers of metoprolol (Fig. 9-13). A similar picture is observed with another β-adrenergic antagonist, timolol. These drugs have smaller clearance than normal.
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