VOLUME 15
`
`RLUk
`
`second Edition
`Revised and Expanded
`
` DRUGS AND THE PHARMACEUTICAL SCIENCES
`
`informa
`
`Milo Gibaldi
`
`Delile msleatae
`
`CELGENE 2148
`APOTEX v. CELGENE
`IPR2023-00512
`
`healthcare
`
`
`
`

`

`

`

`Pharmacokinetics
`
`

`

`DRUGS AND THE PHARMACEUTICAL SCIENCES
`
`A Series of Textbooks and Monographs
`
`Edited by
`dames Swarbrick
`School of Pharmacy
`University of North Carolina
`Chapel Hilti, North Carolina
`
`Volume 1,
`
`Volume 2.
`
`PHARMACOKINETICS, Milo Gibaldi and Donald Perrier
`fout ofprint}
`
`GOOD MANUFACTURING PRACTICES FOR
`PHARMACEUTICALS: APLAN FOR TOTAL QUALITY
`CONTROL,Sidney H. Willig, Murray M. Tuckerman, and
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`MICROENCAPSULATION, edited by J. A. Nixon {out of print}
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`Volume4.
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`DRUG METABOLISM: CHEMICAL AND BIOCHEMICAL
`ASPECTS, Bernard Testa and Peter Jenner
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`Volume 6.
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`NEW DRUGS: DISCOVERY AND DEVELOPMENT,
`edited by Alan A, Rubin
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`SUSTAINED AND CONTROLLED RELEASE DRUG DELIVERY
`SYSTEMS, édited by Joseph A. Robinson
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`MODERN PHARMACEUTICS,edited by Gilbert S.
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`Volume 9.
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`PRESCRIPTION DAUGS IN SHORT SUPPLY: CASE
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`ACTIVATED CHARCOAL: ANTIDOTAL AND OTHER
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`CONCEPTS IN DRUG METABOLISM {in two parts), edited
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`PHARMACEUTICAL ANALYSIS: MODERN METHODS
`(in two parts}, edited by Jarnes W. Munson
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`TECHNIQUES OF SOLUBILIZATION OF DRUGS,
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`NOVEL DRUG DELIVERY SYSTEMS: FUNDAMENTALS,
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`Volume 15.
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`PHARMACOKINETICS, Second Edition, Revised and Expanded,
`Milo Gibatdi and Donald Perrier
`
`Other Volumes in Preparation
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`

`

`Pharmacokinetics
`
`SECOND EDITION, REVISED AND EXPANDED
`
`Milo Gibaldi
`University of Washington
`School of Pharmacy
`Seattle, Washington
`
`Donald Perrier
`School of Pharmacy
`University of Arizona
`Tucson, Arizona
`
`informa
`
`healthcare
`
`New York London
`
`

`

`Informa Healthcare USA, Inc.
`52 Vanderbilt Avenue
`New York, NY 10017
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`© 2007 by Informa Healthcare USA, Inc.
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`

`

`Preface
`
`Pharmacokinetics is the study of the time course of drug absorption,
`distribution, metabolism, and excretion.
`It also concerns the relation-
`ship of these processes to the intensity and time course of pharma-
`cologic (therapeutic and toxicologic) effects of drugs and chemicals.
`Pharmacokinetics is a quantitative study that requires a preexisting
`competence in mathematics at least through calculus.
`It is also a
`biologic study and can be very useful to the biomedical scientist.
`At a fundamental level, pharmacokinetics is a tool to optimize the
`design of biological experiments with drugs and chemicals, All bio-
`logists would benefit from some knowledge of pharmacokinetics when-
`ever they engage in data analysis.
`It has become increasingly impor-
`tant in the design and development of new drugs and in the reass-
`essment of old drugs. Clinical applications of pharmacokinetics have
`resulted in improvements in drug utilization and direct benefits to
`patients,
`There is consensus that the origin of pharmacokinetics can be
`traced to two papers entitled "Kinetics of distribution of substances
`administered to the body" written by Torsten Teorell and published
`in the International Archives of Pharmacodynamics in 1937. Since
`this unheralded beginning, the study of pharmacokinetics has matured
`rapidly; undoubtedly growth has been stimulated by major break-
`throughs in analytical chemistry, which permit us to quantitatively
`detect minute concentrations of drugs and chemicals in exceedingly
`small volumes of biological fluids, in data processing, and by the bril-
`liant insights of many scientists. Dost, Kruger-Theimer, Nelson,
`Wagner, Riegelman, and Levy are among those scientists and must be
`reserved a special place in the history of the development of phar-
`macokinetics.
`Our goals in preparing this revision were similar to those that
`prompted us to undertake the initial effort. The need for revision
`was amply clear to us each time we looked at ourfiles, bulging with
`research papers and commentaries on pharmacokinetic methods and
`
`iii
`
`

`

`iv
`
`Preface
`
`applications published since 1975, The buzz words today are clearance
`concepts, noncompartmental models, and physiologic pharmacokinetics.
`Again, we strived to present the material in an explicit and detailed
`manner, We continue to believe that Pharmacokinetics can be used in
`formal courses, for self-study, or for reference purposes.
`We thank our colleagues for their work and publications, our
`staffs for their labors and support, and our families for their love
`and understanding.
`
`Milo Gibaldi
`Donald Perrier
`
`

`

`Contents
`
`Preface
`
`iil
`
`i.
`
`2.
`
`1
`
`One-Compartment Model
`Intravenous Injection
`Intravenous Infusion
`33
`First-Order Absorption
`Apparent Zero-Order Absorption
`References
`42
`
`2
`27
`
`40
`
`45
`
`Multicompartment Models
`Intravenous Injection
`Intravenous Infusion
`81
`First-Order Absorption
`Determination of Pharmacokinetic Parameters
`References
`109
`
`48
`63
`
`84
`
`113
`Multiple Dosing
`Intravenous Administration
`Intravenous Infusion
`128
`132
`First-Order Absorption
`Determination of Pharmacokinetic Parameters from
`Multiple-Dosing Data
`143
`References
`143
`
`113
`
`Absorption Kinetics and Bioavailability
`Absorption Rate
`146
`167
`Extent of Absorption
`Statistical Considerations in Comparative Bioavailability
`Studies
`185
`Sustained Release
`References
`195
`
`145
`
`188
`
`

`

`vi
`
`Contents
`
`199
`Apparent Volume of Distribution
`Relationship Between Volume of Distribution, Drug Binding
`and Elimination, and Anatomic Volume
`200
`Tissue Binding
`209
`Estimation of Apparent Volumes of Distribution
`References
`218
`
`211
`
`221
`Kinetics of Pharmacologic Response
`Kinetics of Directly Reversible Pharmacologic Response
`Kinetics of Indirect Pharmacologic Response
`245
`254
`Kinetics of Irreversible Pharmacologic Response
`Appendix: Solutions for Cs, Cp, and Cy for Cell Systems
`Sensitive to Drugs That are Cell Cycle Specific
`265
`References
`267
`
`221
`
`271
`Nonlinear Pharmacokinetics
`271
`Michaelis-Menten Kinetics
`Some Pharmacokinetic Characteristics of Michaelis-Menten
`Processes
`272
`277
`In Vivo Estimation of Km and Vy,
`Clearance, Half-Life, and Volume of Distribution
`Drug Concentration at Steady State
`289
`Time to Steady State
`290
`Area Under the Curve and Bioavailability
`Composition of Urinary Excretion Products
`Other Nonlinear Elimination Processes
`301
`Enzyme Induction
`303
`Nonlinear Binding
`307
`Some Problems in Quantifying Nonlinear Pharmacokinetics
`References
`315
`
`294
`297
`
`287
`
`313
`
`319
`
`Clearance Concepts
`319
`Organ Clearance
`321
`Total Clearance
`322
`Hepatic Clearance
`Hepatic Clearance and Drug Binding in Blood
`Drug Binding and Free Drug Concentration
`Half-Life, Intrinsic Clearance, and Binding
`First-Pass Effect
`332
`Gut Wali Clearance
`336
`338
`Lung Clearance
`341
`Renal Clearance
`Clearance Concepts Applied to Metabolites
`Physical Models of Organ Clearance
`347
`Blood Clearance Versus Plasma Clearance
`References
`351
`
`327
`330
`331
`
`344
`
`349
`
`

`

`Contents
`
`vii
`
`355
`
`9.
`
`364
`
`Physiological Pharmacokinetic Models
`Blood Flow Rate-Limited Models
`358
`Experimental Considerations
`Blood Clearance
`366
`Lung Clearance
`368
`Apparent Volume of Distribution
`Nonlinear Disposition
`370
`372
`Membrane-Limited Models
`Species Similarity and Scale-Up
`References
`382
`
`369
`
`375
`
`10. Application of Pharmacokinetic Principles
`Multiple Dosing
`385
`Dose Adjustments in Renal Failure
`Hemodialysis
`397
`Methods for Determination of Individual Patient Parameters
`References
`405
`
`385
`
`393
`
`401
`
`410
`
`11. Noncompartmental Analysis Based on Statistical Moment
`Theory
`409
`Statistical Moments
`Bioavailability
`411
`Clearance
`411
`Half-Life
`412
`413
`Absorption Kinetics
`Apparent Volume of Distribution
`Fraction Metabolized
`414
`Predicting Steady-State Concentrations
`Predicting Time to Steady State
`415
`Conclusions
`416
`References
`416
`
`413
`
`414
`
`Appendix A Method of Laplace Transforms
`References
`423
`
`419
`
`Appendix B Method for Solving Linear Mammillary Models
`References
`431
`
`425
`
`Appendix C Method of Residuals
`
`433
`
`Appendix D Estimation of Areas
`Reference
`449
`
`445
`
`Appendix E Prediction of Drug Concentrations on Multiple Dosing
`Using the Principle of Superposition
`451
`References
`457
`
`

`

`vil
`
`Contents
`
`Appendix F Estimation of Rates
`Reference
`463
`
`459
`
`Appendix G Selective Derivations
`465
`Michaelis-Menten Equation
`Time To Reach a Fraction of Steady State for a Drug Eliminated
`by Parallel First-Order and Capacity-Limited Processes
`467
`Reference
`473
`
`465
`
`Computer Programs
`Appendix H
`References
`476
`
`475
`
`Author Index
`Subject Index
`
`479
`489
`
`

`

`7 N
`
`onlinear Pharmacokinetics
`
`At therapeutic or nontoxic plasma concentrations, the pharmacokinetics
`of most drugs can be adequately described by first-order or linear
`processes. However, there are a small number of well-documented
`examples of drugs which have nonlinear absorption or distribution
`characteristics [e.g., ascorbic acid [1] and naproxen [2,3], re-
`spectively], and several examples drugs that are eliminated from
`the body in a nonlinear fashion.
`
`MICHAELIS-MENTEN KINETICS
`
`Drug biotransformation, renal tubular secretion, and biliary secretion
`usually require enzyme or carrier systems. These systems are rel-
`atively specific with respect to substrate and have finite capacities
`(i.e., they are said to be capacity Himited). Frequently, the kinetics
`of these capacity-limited processes can be described by the Michaelis-
`Menten equation:
`
`_ ac mo
`dt K,te
`
`(7.1)
`°
`
`where —dC/dt is the rate of decline of drug concentration at time t,
`Vm the theoretical maximum rate of the process, and Km the Michaelis
`constant.
`It is readily seen by determining C when —dC/dt = (1/2)Vm
`that Kp is in fact equal to the drug concentration at which the rate
`of the process is equal to one-half its theoretical maximum rate.
`Equation (7.1) can be derived based on the following scheme (see
`Appendix G for derivation):
`
`Ky
`k
`E+Cc => EC ye ESM
`Ky
`
`271
`
`

`

`272
`
`Pharmacokinetics
`
`In this scheme C is the concentration of drug, E the concentration of
`enzyme, EC the concentration of the enzyme-drug complex, and M the
`concentration of metabolite. The constants kg and k_, are first-order
`rate constants, and kj is a second-order rate constant. The Michaelis-
`Menten equation is of value for describing in vitro and in situ as well
`as certain in vivo rate processes. For in vivo systems the constants
`Vm and Kp are affected by distributional and other factors and there-
`fore must be viewed as functional, model-dependent constants.
`
`SOME PHARMACOKINETIC CHARACTERISTICS OF
`MICHAELIS-MENTEN PROCESSES
`
`There are two limiting cases of the Michaelis-Menten equation.
`is much larger than C, (7.1) reduces to
`
`If Ky
`
`dC _om
`-z Cc
`
`_
`
`(7.2)
`
`This equation has the same form as that describing first-order elimina-
`tion of a drug:
`(1) after intravenous administration in a one-compart-
`ment model, (2) in the postabsorptive phase after some other route of
`administration in a one-compartment model, or (3) in the postabsorp-
`tive, postdistributive phase in a multicompartment model. Assuming
`apparent first-order elimination of a drug which confers one-compart-
`ment characteristics to the body and which is eliminated by a single
`biotransformation process, the first-order rate constant K is actually
`Vm/Km. As shown in (7.2), if treatment with an enzyme inducer
`causes an increase in the amount of enzyme (and therefore of V,,), the
`apparent first-order rate constant of the process will also be increased.
`Given the fact that drug elimination is so frequently observed to follow
`apparent first-order kinetics, one must conclude that the drug con-
`centration in the body (or, more correctly, at the site of an active
`process) resulting from the usual therapeutic dosage regimens of
`most drugs is well below the Km of the processes involved in the dis-
`position of these drugs.
`There are some notable exceptions to this generalization and among
`them are ethanol [4], salicylate [5,6], and phenytoin [7]. The elim-
`ination kinetics of phenytoin [8] and ethanol [9] appear to be ade-
`quately described by a single Michaelis-Menten expression, while
`salicylate elimination [6] may be described by two capacity-limited
`and three linear processes. Marked deviations from apparent first-
`order drug elimination have also been noted frequently in cases of
`drug intoxications.
`In the latter situation there is often some ambigu-
`ity as to whether the deviations are due to capacity-limited biotrans-
`formation of the high drug levels in the body [described by (7.1)] or
`due to some toxicologic effect of the drug.
`
`

`

`7 / Nonlinear Pharmacokinetics
`
`273
`
`Another limiting case of the Michaelis-Menten equation is that
`which results when the drug concentration is considerably greater
`than Ky,. Equation (7,1) then reduces to
`
`dc
`dt
`
`m
`
`(7.3)
`
`Under these conditions, the rate is independent of drug concentra-
`tion, so that the process occurs at a constant rate Vy,. The kinetics
`of biotransformation of ethanol [4] have been observed to approach
`the condition described by (7.3) even at drug levels in the body
`that are appreciably lower than those considered to be toxic.
`Based on the discussion above, if ~dC/dt is plotted as a function
`of plasma concentration, —dC/dt would initially increase linearly
`with concentration, indicating first-order kinetics (Fig. 7.1). As the
`concentration increases further, —dC/dt would increase at a rate less
`
`than proportional to concentration, and eventually asymptote at a S
`
`
`
`
`
` 0 190 150 200
`
`
`
`CONCENTRATION (g/m?)
`
`Fig. 7.1 Relationship between drug elimination rate -dC/dt and drug
`concentration C for a Michaelis-Menten process.
`In this particular
`example the Michaelis constant K,, is equal to 10 ug/ml and the max-
`imum rate V,, is equal to 2ug/m))h-1,
`
`

`

`274
`
`Pharmacokinetics
`
`(ug/ml) 0
`Logscale—plasmaconcentrationofdiphenythydantoin
`
`
`50
`
`joo
`
`150
`
`Time measured from !2h after last dose (h)
`
`Fig. 7.2 Phenytoin (diphenylhydantoin) concentration in plasma 12h
`after the last dose of a 3 day regimen of the drug at three different
`daily doses. The data are described by Eq. (7.9). O:
`17.9 mg/kg;
`A: 4.7 mg/kg; O: 2.3 mg/kg.
`(From Ref. 10, © 1972 PJD Publica-
`tions Ltd., reprinted with permission.)
`
`rate equal to V,, which would be independentof concentration (i.e.,
`a zero-order rate).
`The time course of drug plasma concentration after intravenous
`injection of a drug that is eliminated only by a single capacity-limited
`process can be described for a one-compartment system by the inte-
`grated form of the Michaelis-Menten equation. Rearrangement of
`(7.1) yields
`
`_
`ac
`C (C+K) =v, dat
`
`or
`
`K dc
`
`-ac -—— =v at
`Cc
`m
`
`(7.4)
`
`(7.5)
`
`

`

`7 / Nonlinear Pharmacokinetics
`
`Integration of this equation gives the expression
`
`-C -K_ mnC=Vit+ti
`m
`m
`
`275
`
`(7.6)
`
`where iis an integration constant. Evaluating i at t = 0, where
`C=Cq, yields
`i=-C)- Ki. in Cy
`
`(7.7)
`
`Substituting this expression for iin (7.6) and rearranging terms gives
`
`(7.8)
`
`1.0
`
`a3-=:
`
`c.,-c¢
`- 8,
`t =
`7
`m
`
`Cc
`K
`_m,, 2
`+ 7 in c
`m
`
`100
`
`o
`Te) 250=500
`a
`TIME (min)
`
`
`
`°c
`
`o = _z>S
`
`o=<
`
`4
`
`Of
`
`o
`
`200
`
`400
`TIME (min)
`
`600
`
`800
`
`Fig. 7.3 Amount of drug in the body following intravenous adminis-
`tration of 1, 10, and 100 mg doses of a drug that is eliminated by a
`single Michaelis-Menten process. A one-compartment system is as-
`sumed; Km = 10 mg and V,, = 0.2 mg/min. The inset shows a plot of
`amount of drug in the body divided by administered dose versus time
`to show that the principle of superposition does not apply.
`
`

`

`276
`
`3000
`
`2000
`
`Pharmacokinetics
`
`
`
`Amountunexcreted(mgospirinequivolent)
`
`
`
`
`
`500
`
`200
`
`100
`
`re,°
`
`ore.‘as
`
`1000
`
`°
`
`"a,
`~e
`™
`
`e
`
`“a,
`ese
`“e
`Beginning of
`N\ ~
`LN Gem first-order
`*
`
`teo
`
`Time
`
`(h)
`
`Fig. 7.4 Elimination of salicylate after oral administration of 0.25,
`1.0, and 1,5 g doses of aspirin. Vertical arrows on the time axis
`indicate t5qg, the time to eliminate 50% of the dose.
`(From Ref. 5,
`reprinted with permission.)
`
`Unfortunately, it is not possible to solve this equation explicitly for
`Cc. Rather, one must determine the time t at which the initial concen-
`tration Cg has decreased to C. A modified form of (7.8), that is,
`
`c,-cC kK
`Cc
`0
`m
`0
`t-th=—y_ +-——In >
`m
`m
`
`(7.9)
`
`has been used to fit phenytoin levels in the plasma as a function of
`time 12 h after administration of the last of several oral doses to human
`subjects (see Fig. 7.2).
`In this case Cg represents the phenytoin
`
`plasma concentration at 12 h after the last dose, tg = 12h, andCis
`the phenytoin plasma concentration at time t, where t > ty.
`Conversion of (7.8) to common logarithms (In x = 2.303 log x)
`and solving for log C yields
`Via
`Cy 7
`log C = s-a59x «+ 1°06 Cg ~ Frg0gK t
`2.303K
`2.303K
`m
`m
`
`(7.10)
`
`

`

`7 / Nonlinear Pharmacokinetics
`
`277
`
`Figure 7.3 shows the time course of elimination, as described by (7.10),
`of three different doses of a drug that is eliminated by a process with
`Michaelis-Menten kinetics. The lowest dose represents the case where
`Ky, >> C. At this dose the decline in plasma concentrations is first
`order with a slope of —V,,/2.303K,,. On the other hand, the highest
`dose yields initial concentrations which are considerably above Ky,
`so that drug levels decline initially at an essentially constant rate (see
`inset to Fig. 7.3). The curves show that the time required for an
`initial drug concentration to decrease by 50% is not independent of
`dose, but, in fact, increases with increasing dose. This particular
`pharmacokinetic property may present considerable clinical difficulty
`in the treatment of drug intoxications. Figure 7.3 also shows that re-
`gardless of the initial dose, when the plasma concentration becomes
`significantly less than Kp, elimination is describable by first-order
`kinetics and the slope of this linear portion of the curve is inde-
`pendent of dose. Semilogarithmic plots of plasma concentration or
`amount unexcreted versus time after administration of phenytoin (Fig.
`7.2) or salicylate (Fig. 7.4) show characteristics that are remarkably
`similar to those described by the curves in Fig. 7.3.
`To assess whether or not a drug possesses nonlinear kinetic prop-
`erties, a series of single doses of varying size should be administered.
`If a plot of the resulting plasma concentrations divided by the ad-
`ministered dose are superimposable, the drug in question has linear
`kinetic properties over the concentration range examined.
`If, how-
`ever, the resulting curves are not superimposable (see inset to Fig.
`7.3), the drug behaves nonlinearly.
`
`IN VIVO ESTIMATION OF K,, AND Vy,
`
`For a drug that is eliminated by a single capacity-limited process,
`there are a number of general methods which permit the initial estima-
`tion of apparent in vivo Km and V,, values from plasma concentration-
`time data in the postabsorptive-postdistributive phase. Such estimates
`require the determination of the rate of change of the plasma concentra-
`tion from one sampling time to the next, AC/At, as a function of the
`plasma concentration Cy, at the midpoint of the sampling interval (see
`Appendix F). The data are usually plotted according to one of the
`linearized forms of the Michaelis-Menten equation, such as the Line-
`weaver-Burk expression,
`
`K
`
`1
`1
`m
`=
`+-~—
`AC/At
`Viaem Vin
`
`(7,11)
`
`so that a plot of the reciprocal of AC/At versus the reciprocal of Cm
`yields a straight line with intercept 1/V,, and slope Km/Vm. Two
`sometimes more reliable [11,12] plots are the Hanes-Woolf plot [13] and
`
`

`

`278
`
`Pharmacokinetics
`
`the Woolf-Augustinsson-Hofstee plot [13]. They are based on the
`relationships
`Cay
`Kn
` Cn
`AC/At VV
`m
`m
`
`(7,12)
`
`and
`
`SC iy _ OCHOKa(7,13)
`At
`m
`Cc.
`,
`
`respectively. Based on (7.12), a plot of C,,/(AC/At) versus Cy
`should yield a straight line with a slope of 1/V, and an intercept of
`Ky/Vm. Equation (7.13) indicates that a plot of AC/At versus
`(4C/At)/C,, gives a straight line with a slope of —Km and an inter-
`cept of Vy.
`A method for estimating V,, and K,, directly from log C versus time
`data, obtained following the intravenous administration of a drug that
`can be adequately described by a one-compartment system, is also
`available [14]. Extrapolation of the terminal log-linear portion of the
`log C versus time plot, where the plot is described by (7.10), would
`yield a zero-time intercept of log Ch (see Fig. 7.5). The resulting
`straight line can be described by
`
`=
`
`log C = log C,
`
`Vv
`‘mS
`
`ra0aK_*
`
`(7.14)
`
`At low plasma concentrations (7.10) and (7.14) are identical. By
`setting the right-hand sides of these two equations equal to each
`other, the following expression is obtained:
`
`c.-c
`
`Vv
`
`—9_siongcc —- —™_ t=log cc. - ——2-
`2.303K
`0
`2.303K
`0
`2.308K
`
`Vv
`
`(7.15)
`
`Cancellation of the common term, Vpt/2.303K,,, and rearrangement
`yields
`
`*
`
`Cc
`c.-c
`0 =loo—2
`
`7.303K.
`~ 8
`(7.16)
`
`m
`
`0
`
`Since the equality given by (7.15) is valid only at low concentrations,
`Cy can be assumed to be significantly greater than C, and therefore
`C,—-C Cg. Making this simplification in (7.16) and solving the re-
`sulting expression for K,, gives
`Cg
`K, ” 2,308 log (C#IE))
`
`(7.17)
`
`

`

`7 / Nonlinear Pharmacokinetics
`
`279
`
`SLOPE = “V2 -303K,,
`
`(ug/ml) 4
`CONCENTRATIONINPLASMA
`
`8
`
`le
`
`16
`
`20
`
`24
`
`TIME (h)
`
`Fig. 7.5 Graphical method for estimating Km and V,, after intravenous
`administration of a drug that is eliminated by a single Michaelis-
`Menten process. The solid line is described by Eq. (7.10). The
`terminal slope gives an estimate of the ratio of Vy, to K,,, and the
`ratio of Ch to Cg is used to estimate K,, [see Eq. (7.17)].
`
`Since C* and Cg can be estimated from a log C versus time plot, an
`estimate of K,, is possible employing (7.17). V,, can be calculated
`from the slope of the terminal log-linear segment of the concentration
`versus time curve. Since slope = —-V_y/2.303K,, Vin = —2.303(slope)Km.
`It is plausible to consider that drug elimination may involve a
`eapacity-limited process in parallel with one or more first-order proc-
`esses. Under these conditions, the foregoing methods for estimating
`Vm and Ky do not apply. When capacity-limited and first-order elim-
`ination occur in parallel, the rate of decline of drug levels in the
`plasma after intravenous administration in a one-compartment system
`is given by
`
`VC
`dC Liaw pm
`at
`RCT RSS
`
`(7.18)
`
`where K' is the rate constant characterizing the various parallel
`first-order processes. The time course of drug levels under these
`conditions may be determined by integration of (7.18) as follows.
`Expansion of (7.18) yields
`
`

`

`280
`
`Pharmacokinetics
`
`K'C(K +C)+V_C K'K C+vV C+K'C?
`-e =2. (7.19)
`K +C
`~
`K +C
`dt ~
`'
`m
`m
`
`Further simplification gives rise to
`
`CCK'K + KV_
`+ K'C)
`d
`t
`~d@ omms Ca + K'C)
`at
`K +C
`K +¢
`m
`m
`
`(7.20)
`
`where a = K'K,, + Vm.
`
`Inversion and rearrangement of (7.20) yields
`
`m
`m
`dat
`dC C(a+K'C)” Clat+K'C)
`
`1
`a+kK'C
`
`This equation is separable and can be rewritten as
`
`_ —Kde
`
`C(a + K'C)
`
`_
`
`dt
`
`dc
`
`a+ K'C
`
`(7.22)
`
`(1.22)
`
`.
`
`The two terms in this equation are of the form 1/x(a + bx) and
`1/(a + bx), respectively, the integrals of which are (—1/a) In [(a +
`bx)/x] and (1/b) In (a + bx) [15]. Therefore, integration of (7.22)
`gives
`
`.
`;
`atKiC 1
`_om,
`tes ins
`xin (a+ KIC) +3
`
`(7.23)
`
`Evaluating i at t = 0, where C = Cg, yields
`Ka
`at KiCy
`1
`eoMy '
`(7,24)
`a
`In c
`tim In (a+ K'C))
`i
`Substituting this value of iin (7.23) and simplifying the resulting
`expression yields
`
`at Kc)
`a
`Cy
`at=K In | + x” ¥, nMoyKC
`
`(7.25)
`
`Since a = K'Ky, + Vm;
`
`(K'K +Vo)t=Ko
`m
`m
`
`m
`
`{KK+V
`Cc,
`In—+| —S"-xk }]1
`Ki
`Cc
`m
`
`KK +V_ +K'C,
`m
`TKK +V +K'C
`m
`m
`
`(7.26)
`
`or
`
`
`
`cv (C_+K )K'+V
`
`
`_ _0,m1 0 m m
`
`m
`m
`™m
`m
`
`aa s,m e+ In aK OKT |
`
`
`
`(7,27)
`
`

`

`7 / Nonlinear Pharmacokinetics
`
`281
`
`{0
`
`100
`
`4K
`
`y
`
`|
`
`Dose
`
`12
`
`tso%,
`
`oa
`
`°
`
`001
`
`0.1
`
`Fig. 7.6 Comparison of dose dependence of t599 (time for elimination
`of 50% of the dose) after intravenous administration of a drug that is
`eliminated by a single Michaelis-Menten process (upper curve) or
`one that is eliminated by a single Michaelis-Menten process in parallel
`with a first-order process (lower curve).
`In each case, Kp = 1.0
`and Vm = 0.2. The rate constant K' for the first-order process is
`equal to 0.1.
`(Data from Ref, 16.)
`
`Equation (7.27), like (7.8), does not permit an explicit solution for
`C. Both (7.8) and (7.27) indicate that the time required to reduce
`an initial drug concentration by 50% is indeed dependent on the ad-
`ministered dose. Examples of this dependency are shown in Fig. 7.6.
`Expanding (7.27) and solving for In C gives
`Via
`Vv
`(Co +Kme + Via
`oOmm ! —_
`in C nC * igiK "CrkVV K *K t
`
`(7.28)
`
`=
`
`which in terms of common logarithms is
`Via
`(C,+K. )K! tv K' +V/K
`eC =e Cot eK ME (Ce KOK TV 2.308
`
`(7.29)
`
`

`

`282
`
`Pharmacokinetics
`
`At low concentrations (i.e., Ky, >> C), (7.29) becomes
`Via
`(Cy + KK + Vin
`K' + Vin Km
`log C= 108 Cy * ii~ log——"FetyVy-~— yon ~—t
`m
`m
`
`or
`
`log C = log Cy~ at
`
`K'+V_/K
`
`where
`
`(7.30)
`
`(7.31)
`
`Via
`(Cy + Ke + Va
`log Co =log Co*KKaeKKAV (7.32)
`
`As can be seen from (7.31), the slope of the terminal linear portion of
`a semilogrithmic plot of plasma concentration versus time at low plasma
`concentrations (i.e., K,, >> C) will yield an estimate of the first-order
`elimination rate constant of a drug, K' + Vm/K,, (see Fig. 7.7). The
`extrapolated intercept of this terminal linear phase wiil be log Cp.
`For certain drugs that exhibit parallel capacity-limited and first-
`order elimination, it may be possible to administer sufficiently high
`doses intravenously so that initial drug concentrations are substan-
`tially larger than K,. Under these conditions and where a one-com-
`partment model applies, the initial segment of a semilogarithmic plot
`of plasma concentration versus time will be linear (see Fig. 7.7).
`The slope of this linear segment will be —K'/2.303 [14]. This can be
`demonstrated by assuming C to be much greater than K,, in (7.18)
`and solving for C. Therefore, estimates of both K' and K' + Vm/Ky,
`ean be obtained directly from a semilogarithmic plot of plasma concen-
`tration versus time and the ratio V,,/K, can be calculated for the case
`where there is one capacity-limited process in parallel with one or
`more first-order processes. The following approach can then be used
`to obtain initial estimates of Ky, and Vp for this model. Expansion of
`the logarithmic term of (7.32) and rearrangement of this equation
`yields
`K'K|
`
`
`0
`
`m
`
`m
`
`Division of the numerator and denominator of this logarithmic term by
`K,, gives
`K'K
`
`m
`Vv,
`
`*
`
`Cc
`0
`Cy
`
`log —— = oe fe (7.34)
`
`Cc oh
`K(kh+V7K)
`
`*
`Cy
`
`Cok!
`
`log & = log (eg
`
`v
`
`m
`
`(7.33)
`
`

`

`7 / Nonlinear Pharmacokinetics
`
`283
`
`400
`
`200
`
`100
`
`10
`
`=
`ad
`oOo
`
`=<
`
`x 5<
`
`x
`amd
`a
`
`Go=
`o
`<=
`x=
`i
`wy
`
`So
`
`40
`
`20
`
`TIME (h)
`
`Fig. 7.7 4-Hydroxybutyric acid (4-HBA) concentration in plasma
`after intravenous administration. The compound appearsto be elim-
`inated by a Michaelis-Menten process in parallel with a first-order
`process. The initial slope gives an estimate of K' and the terminal
`slope provides an estimate of the ratio of V,, to K,,. K,, may be de-
`termined from Eq. (7.35). The deviation from theory for a short
`time after administration probably reflects drug distribution and the
`lack of strict adherence to a one-compartment model.
`(Data from
`Ref. 14.)
`
`A solution for K,, based on this equation is
`t
`'
`_— COKU(K' + V_/K.)
`m
`*
`K'K/V
`(Co/Co)
`-1
`
`(7.35)
`
`

`

`284
`
`Pharmacokinetics
`
`Since Cy and C4 can be obtained directly from a semilogarithmic plasma
`concentration-time plot, and K' and Vp/K,;, can be estimated as de-
`scribed above, an estimate of K,, is possible employing (7.35). Once
`K,, is known, V,, can be readily determined.
`Although this is an interesting approach for the estimation of K',
`Vm» and Km where there are parallel first-order and nonlinear elimina-
`tion pathways, caution must be exercised.
`Initial plasma concentrations
`have to be sufficiently high (i.e., C >> K,,) to yield a semilogarithmic
`plasma concentration-time curve which is truly linear.
`If these con-
`
`23.0
`
`21.0
`
`19.0
`
`4.0
`
`3.0
`
`2.0
`
`0.0
`
`(-dInCidt)x102
`
`J K+
`
`m= SLOPE = Vm
`
`0.1
`
`0.2
`
`Hatt
`10
`30
`50
`70
`90
`110 130
`0.3
`1/C x 107
`
`Fig. 7.8 Graphical method for estimating K' (designated K, in the
`plot), Vm and Km based on Eqs. (7.36) to (7.39). At high concentra-
`tions, a plot of -A In C/At versus 1/C will be linear with a slope of
`Vin and an intercept equal to K' [see Eq. (7.38)], whereas at low con-
`centrations the plot will asymptotically approach a limiting value equal
`to K' + (Vi,/Km) [see Eq. (7.39)].
`(From Ref. 17, © 1973 Plenum
`Publishing Corp.)
`
`

`

`7 / Nonlinear Pharmacokinetics
`
`285
`
`centrations are not attained, an overestimate of K' will result. This,
`in turn, will produce errors in the estimates of Vm and Ky.
`A different approach can also be used for the estimation of Ky
`and V,, where nonlinear and linear processes of drug elimination occur
`in parallel. At high plasma concentrations (i.e., C >> Km), (7.18)
`reduces to
`
`dt
`
`(7.36)
`- de = xtc tv,
`Division of both sides of (7.36) by C and recognition that (—dC/dt)/C
`equals —d In C/dt gives
`
`Vv
`_dinc _,,, m
`aC K' + C
`
`At low plasma concentrations (i.e., Ky >> C), (7.18) becomes
`
`Vv
`aC Lia,
`at “Ke +R
`and, therefore, the analogous expression to (7.37) is
`
`a ame =KI+ om
`
`Vv
`
`m
`
`(7.37)
`
`(7.38)
`
`(7.39)
`
`A plot of — A In C/At versus 1/C will consequently be linear with a
`slope of V,, and an intercept of K' at high plasma concentrations [Eq.
`(7.37)], but will reach an asymptotic value of K' + V,)/K,, at low con-
`centrations [Eq. (7.39)] from which K,, can be calculated (see
`Fig. 7.8).
`This method for estimating K', V,,, and K,, has limitations which
`are similar to those noted for the previous approach. Sufficiently
`high plasma concentrations are required to yield a straight line from
`the -A In C/At versus 1/C plot to permit accurate estimates of V,,
`and K'.
`Urine data can also be used to estimate V,, and K,,. Consider
`the following scheme:
`
`XE
`
`Kt
`
`M—>M
`mu
`
`

`

`286
`
`Pharmacokinetics
`
`where X is the amount of drug in the body, Xp the amount of drug
`eliminated by the linear or first-order processes, M the amount of
`metabolite in the body which is formed by a capacity-limited process,
`and M,, the amount of this metabolite present in the urine. All of
`these amounts are time dependent. The constants K' and ky are
`first-order rate constants, Vin is the maximum rate of metabolite forma-
`tion in units of amount per time, and Kin is the Michaelis constant in
`units of amount. Assuming that the urinary excretion rate of the
`metabolite (4M,,/At) is rate limited by its formation, and therefore
`reflects the rate of formation, the following relationship for AMy/dt
`‘ean be written:
`
`AM,
`
`Vin em
`
`m
`
`= 100
`
`980
`
`2<c z
`
`660
`
`2BR
`Lu
`co
`2
`peo
`Qga
`Ri
`a!
`°a
`Zz
`gs
`
`20
`
`0.5
`
`4.0
`
`1.5
`
`2.0
`
`2.5
`
`SA IN BODY (g}
`
`Fig. 7.9 Plot of linearized form of the Michaelis-Menten equation to
`describe the formation of salicy] phenolic glucuronide (SPG) after a
`single dose of salicylic acid (SA). According to Eq. (7.40) and the
`co