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`Pharmacokinetics
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`DRUGS AND THE PHARMACEUTICAL SCIENCES
`
`A SeriH of TfJxtbook, and Monograph,
`
`Edittldby
`J.m•• Sw.rbrick
`School of PharmllCY
`University of North Carolina
`Chapel Hill, North Carolina
`
`Volume 1. PHARMACOKINETICS, Milo Gibaldi and Donald Perrier
`(out of print)
`
`Volume 2. GOOD MANUFACTURING PRACTICES FOR
`PHARMACEUTICALS: A PLAN FOR TOTAL OUALITY
`CONTROL, Sidney H. Willig, Murray M. Tuckerman, and
`William S. Hitchings IV (out ofprint)
`
`Volume 3. MICROENCAPSULATION, edited by J. R. Nixon (out of print)
`
`Volume 4. DRUG METABOLISM: CHEMICAL AND BIOCHEMICAL
`ASPECTS, Bernard Testaand Pettlr Jenner
`
`Volume 5. NEW DRUGS: DISCOVERY AND DEVELOPMENT,
`edited by Alan A. Rubin
`
`Volume 6. SUSTAINED AND CONTROLLED RELEASE DRUG DELIVERY
`SYSTEMS,edited by Joseph R. Robinson
`
`Volume 7. MODERN PHARMACEUTICS, edited by Gilbert S.
`Banker and Christopher T. Rhodes
`
`Volume 8. PRESCRIPTION DRUGS IN SHORT SUPPLY: CASE
`HISTORIES, Michael A. Schwartz
`
`Volume 9. ACTIVATED CHARCOAL: ANTIDOTAL AND OTHER
`MEDICAL USES,David O. Cooney
`
`Volume 10. CONCEPTS IN DRUG METABOLISM (in two parts). edited
`by Peter Jenner and Bernard Testa
`
`Volume 11. PHARMACEUTICAL ANAL YSIS: MODERN METHODS
`(in two parts), edited by James W. Munson
`
`Volume 12. TECHNIOUES OF SOLUBILIZATION OF DRUGS,
`edited by Samuel H. Yalkowsky
`
`Volume 13. ORPHAN DRUGS,edited by Fred E. Karch
`
`Volume 14. NOVEL DRUG DELIVERY SYSTeMS: FUNDAMENTALS,
`DEVELOPMENTAL CONCEPTS, BIOMEDICAL ASSESSMENTS,
`Yie W. Chien
`
`Volume 15. PHARMACOKINETICS, Second Edition, Revised and Expanded,
`Milo Gibaldi 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
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`Informa Healthcare USA, Inc.
`52 Vanderbilt Avenue
`New York, NY 10017
`
`© 2007 by Informa Healthcare USA, Inc.
`Informa Healthcare is an Informa business
`
`No claim to original U.S. Government works
`Printed in the United States of America on acid-free paper
`30 29 28 27 26 25 24 23 22 21
`
`International Standard Book Number-10: 0-8247-1042-8 (Hardcover)
`International Standard Book Number-13: 978-0-8247-1042-2 (Hardcover)
`
`This book contains information obtained from authentic and highly regarded sources. Reprinted material
`is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
`efforts have been made to publish reliable data and information, but the author and the publisher cannot
`assume responsibility for the validity of all materials or for the consequences of their use.
`
`No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic,
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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(cid:173)
`ship of these processes to the intensity and time course of pharma(cid:173)
`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(cid:173)
`logists would benefit from some knowledge of pharmacokinetics when(cid:173)
`ever they engage in data analysis.
`It has become increasingly impor(cid:173)
`tant in the design and development of new drugs and in the reass(cid:173)
`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(cid:173)
`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(cid:173)
`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(cid:173)
`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 our files, bulging with
`research papers and commentaries on pharmacokinetic methods and
`
`iii
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`iv
`
`Preface
`
`applications publiahed 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 publicationa , our
`staffs for their labors and support, and our families for their love
`and understanding.
`
`Milo Gibaldi
`Donald Perrier
`
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`Contents
`
`Preface
`
`iii
`
`1. One-Compartment Model
`Intravenous Injection
`Intravenous Infusion
`33
`First-Order Absorption
`Apparent Zero-Order Absorption
`References
`42
`
`2
`27
`
`40
`
`45
`
`2. Multicompartment Models
`Intravenous Injection
`Intravenous Infusion
`First-Order Absorption
`81
`Determination of Pharmacokinetic Parameters
`References
`109
`
`48
`63
`
`84
`
`113
`3. Multiple Dosing
`Intravenous Administration
`Intravenous Infusion
`128
`First-Order Absorption
`132
`Determination of Pharmacokinetic Parameters from
`Multiple-Dosing Data
`143
`References
`143
`
`113
`
`4. Absorption Kinetics and Bioavailability
`Absorption Rate
`146
`Extent of Absorption
`167
`Statistical Considerations in Comparative Bioavailability
`Studies
`185
`Sustained Release
`References
`195
`
`145
`
`188
`
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`vi
`
`Contents
`
`199
`5. Apparent Volume of Distribution
`Relationship Between Volume of Distribution, Drug Binding
`200
`and Elimination, and Anatomic Volume
`Tissue Binding
`209
`Estimation of Apparent Volumes of Distribution
`References
`218
`
`211
`
`221
`6. Kinetics of Pharmacologic Response
`Kinetics of Directly Reversible Pharmacologic Response
`Kinetics of Indirect Pharmacologic Response
`245
`Kinetics of Irreversible Pharmacologic Response
`254
`Appendix: Solutions for Cs' Cr, and CT for Cell Systems
`Sensitive to Drugs That are Cell Cycle Specific
`265
`References
`267
`
`221
`
`271
`7. Nonlinear Pharmacokinetics
`Michaelis-Menten Kinetics
`271
`Some Pharmacokinetic Characteristics of Michaelis-Menten
`Processes
`272
`In Vivo Estimation of Km and Vm
`277
`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
`8. Clearance Concepts
`Organ Clearance
`319
`Total Clearance
`321
`Hepatic Clearance
`322
`Hepatic Clearance and Drug Binding in Blood
`Drug Binding and Free Drug Concentration
`Half-Life, Intrinsic Clearance, and Binding
`First-Pass Effect
`332
`Gut Wall Clearance
`336
`Lung Clearance
`338
`Renal Clearance
`341
`Clearance Concepts Applied to Metabolites
`Physical Models of Organ Clearance
`347
`Blood Clearance Versus Plasma Clearance
`References
`351
`
`327
`330
`331
`
`344
`
`349
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`Contents
`
`vii
`
`355
`
`9.
`
`364
`
`Physiological Pharmacokinetlc 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
`Absorption Kinetics
`413
`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
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`viii
`
`Appendix F Estimation of Rates
`Reference
`463
`
`1159
`
`Contents
`
`Appendix C 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
`
`1165
`
`Computer Programs
`Appendix H
`References
`476
`
`1175
`
`Author Index
`Subject Index
`
`1179
`489
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`ne-Compartment Model
`
`1 O
`
`The most commonly employed approach to the pharmacokinetic char(cid:173)
`acterization of a drug is to represent the body as a system of com(cid:173)
`partments, even though these compartments usually have no physio(cid:173)
`logic or anatomic reality, and to assume that the rate of transfer be(cid:173)
`tween compartments and the rate of drug elimination from compart(cid:173)
`ments follow first-order or linear kinetics. The one-compartment
`model, the simplest model, depicts the body as a single, kinetically
`homogeneous unit. This model is particularly useful for the pharma(cid:173)
`cokinetic analysis of drugs that distribute relatively rapidly through(cid:173)
`out the body. Almost invariably, the plasma or serum is the anatomical
`reference compartment for the one-compartment model, but we do not
`assume that the drug concentration in plasma is equal to the concen(cid:173)
`tration of drug in other body fluids or in tissues, for this is rarely
`the case. Rather, we assume that the rate of change of drug concen(cid:173)
`tration in plasma reflects quantitatively the change in drug concen(cid:173)
`trations throughout the body.
`In other words, if we see a 20% de(cid:173)
`crease in drug concentration in plasma over a certain period of time,
`we assume that the drug concentrations in kidney, liver, cerebro(cid:173)
`spinal fluid, and all other fluids and tissues also decrease by 20%
`during this time.
`Drug elimination from the body can and often does occur by
`several pathways, including urinary and biliary excretion, excretion
`in expired air, and biotransformation in the liver or other fluids or
`tissues. Glomerular filtration in the kidneys is clearly a diffusional
`process, the rate of which can be characterized by first-order kinetics,
`but tubular secretion in the kidneys, biliary secretion, and biotrans(cid:173)
`formation usually involves enzymatic (active) processes that are ca(cid:173)
`pacity limited. However, as demonstrated in subsequent sections of
`the text dealing with capacity-limited and nonlinear processes (Chap.
`7), at
`low concentrations of drug (Le . , concentrations typically as(cid:173)
`sociated with therapeutic doses) the rate of these enzymatic processes
`can be approximated very well by first -order kinetics. Hence we find
`
`1
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`2
`
`Pharmacokinetics
`
`that the elimination of most drugs in humans and animals following
`therapeutic or nontoxic doses can be characterized as an apparent
`first-order process (Le., the rate of elimination of drug from the
`body at any time is proportional to the amount of drug in the body
`at that time). The proportionality constant relating the rate and
`amount is the first-order elimination rate constant.
`Its units are
`reciprocal time (I.e., min- 1 or h- 1). The first-order elimination rate
`constant characterizing the overall elimination of a drug from a one(cid:173)
`compartment model is usually written as K and usually represents the
`sum of two or more rate constants characterizing individual elimination
`processes:
`K = k + k
`+ k' + k
`m
`m
`e
`b
`where ke and kb are apparent first-order elimination rate constants for
`renal and biliary excretion, respectively, and km and kin are apparent
`first-order rate constants for two different biotransformation (metabo(cid:173)
`lism) processes. These constants are usually referred to as apparent
`first-order rate constants to convey the fact that the kinetics only
`approximate first-order.
`
`( 1.1)
`
`+ •••
`
`INTRAVENOUS INJECTION
`Drug Concentrations In the Plasma
`
`FOllowing rapid intravenous injection of a drug that distributes in the
`body according to a one-compartment model and is eliminated by ap(cid:173)
`parent first-order kinetics, the rate of loss of drug from the body is
`given by
`
`~; = -KX
`
`(1.2)
`
`time t after injection.
`where X is the amount of drug in the body at
`K, as defined above, is the apparent first -order elimination rate con(cid:173)
`stant for the drug. The negative sign indicates that drug is being
`lost from the body.
`To describe the time course of the amount of drug in the body after
`injection, Eq. (1. 2) must be integrated. The method of Laplace trans(cid:173)
`forms in Appendix A will be employed. The transform of (1. 2) is
`sX - X = -KX
`o
`where Xo is the amount injected (I.e., the dose) and s is the Laplace
`operator. Rearrangement of (1. 3) yields
`
`
`(1. 3)
`
`XoX= - (cid:173)_
`
`s+K
`
`(1.4)
`
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`1 lOne-Compartment Model
`
`3
`
`(1. 5)
`
`which when solved using a table of Laplace transforms (Appendix A)
`gives
`X = X e-Kt
`o
`where e represents the base of the natural logarithm. Taking the
`natural logarithm of both sides of (1. 5) gives
`In X = In Xo - Kt
`Then, based on the relationship
`2.303 log a =In a
`Eq. (1. 6) can be converted to common logarithms (base 10, log):
`Kt
`log X =log Xo - 2.303
`The body is obviously not homogeneous even if plasma concentra(cid:173)
`tion and urinary excretion data can be described by representing the
`body as a one-compartment model. Drug concentrations in the liver,
`kidneys, heart, muscle, fat, and other tissues usually differ from one
`another as well as from the concentration in the plasma.
`If the rela(cid:173)
`tive binding of a drug to components of these tissues and fluids is
`essentially independent of drug concentration,
`the ratio of drug con(cid:173)
`centrations in the various tissues and fluids is constant. Conse(cid:173)
`quently, there will exist a constant relationship between drug con(cid:173)
`centration in the plasma C and the amount of drug in the body:
`
`(1. 6)
`
`(1. 7)
`
`(1.8)
`
`X = VC
`The proportionality constant V in this equation has the units of
`volume and is known as the apparent volume of distribution. Despite
`its name, this constant usually has no direct physiologic meaning and
`does not refer to a real volume. For example, the apparent volume of
`distribution of a drug in a 70 kg human can be several hundred liters.
`The relationship between plasma concentration and the amount of
`drug in the body, as expressed by Eq , (1. 9), enables the conversion
`of Eq , (1. 8) from an amount -time to a concentration-time relationship:
`
`(1. 9)
`
`Kt
`log C = log Co - 2. 303
`
`( 1.10)
`
`where Co is the drug concentration in plasma immediately after injec(cid:173)
`tion. Equation (1.10) indicates that a plot of log C versus t will be
`linear under the conditions stated (Fig. 1.1). Co may be obtained by
`extrapolation of the log C versus t plot to time zero. This intercept,
`CO, may be used in the calculation of the apparent volume of distribu(cid:173)
`tion. Since Xo equals the amount of drug injected intravenously (Le.,
`the intravenous dose), V may be estimated from the relationship
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`4
`
`Pharmacokinetics
`
`500
`
`200
`
`100
`
`50
`
`20
`
`2
`
`4
`
`6
`
`8
`
`10
`
`12
`
`Time (h)
`
`Fig. 1. 1 Prednisolone concentration in plasma following an intra(cid:173)
`venous dose equivalent to 20 mg prednisone to a kidney transplant
`patient. The data show monoexponential decline that can be described
`by Eq. (1.10). Co = intravenous dose/V; slope = -K/2.303.
`(Data
`from Ref. 1.)
`
`V = intravenous dose
`Co
`
`( 1.11)
`
`Equation (1.11) is theoretically correct only for a one-compartment
`model where instantaneous distribution of drug between plasma and
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`1 lOne-Compartment Model
`
`5
`
`tissues takes place. Since this is rarely true, a calculation based on
`Eq. (1.11) will almost always overestimate the apparent volume of
`distribution. Sometimes the error is trivial, but often the overestimate
`is substantial and the calculation may be misleading. More accurate and
`more general methods of estimating V will be discussed subsequently.
`The slope of the line resulting from a plot of log C versus time is
`equal to -K/2.303 and K may be estimated directly from this slope.
`It is easier, however,
`to estimate K from the relationship
`
`K = 0.693
`t
`1/ 2
`
`( 1.12)
`
`where t1/2 is the biologic or elimination half-life of the drug. This
`parameter is readily determined from a semilogarithmic plot of plasma
`drug concentration (on logarithmic scale) versus time (on linear
`scale), as illustrated in Fig. 1.1. The time required for the drug
`concentration at any point on the straight line to decrease by one-
`half is the biologic half-life. An important characteristic of first-
`order processes is that the time required for a given concentration to
`decrease by a given percentage is independent of concentration. Equa(cid:173)
`tion (1.12) is easily derived by setting C equal to COl 2 and t equal
`to t1/2 in Eq. (1.10).
`In principle. a plot of the logarithm of tissue drug concentration
`versus time should also be linear and give exactly the same slope as
`the plasma concentration-time curve. This is illustrated in Fig. 1. 2.
`Estimates of CO. t1l2. and K are often obtained from the best
`straight-line fit (by eye) to the log C versus time data. However,
`a more objective method is to convert all concentration values to log(cid:173)
`arithms, and then to determine the best-fitting line by the method of
`least squares, described in elementary textbooks of statistics [3].
`Computer programs are available (see Appendix H) that do not require
`logarithmic conversions for nonlinear least-squares fitting of data.
`
`Urinary Excretion Data
`
`It is sometimes possible to determine the elimination kinetics of a drug
`from urinary excretion data. This requires that at least some of the
`drug be excreted unchanged. Consider a drug eliminated from the
`body partly by renal excretion and partly by nonrenal processes such
`as biotransformation and biliary excretion. as shown in Scheme 1,
`
`Scheme 1
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`6
`
`Pharmacokinetics
`2,.--------------------,
`
`I -
`
`•
`• 0
`
`0
`
`0
`
`•
`
`u;
`~
`:>
`on
`on
`:;::
`
`c
`
`~
`
`e
`J
`s
`~
`uc
`0
`u
`CJ'
`"
`00.11-
`
`0
`
`•
`
`0
`
`0
`
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`
`•
`
`~
`
`~
`
`~
`
`o
`
`5
`
`~
`TIme (h)
`Fig. 1.2 Dipyridamole concentrations in serum (0) and heart tissue
`(e) after a single oral dose of the drug to guinea pigs. Drug con(cid:173)
`centrations in serum and heart decline in a parallel manner.
`(Data
`from Ref. 2.)
`
`where Xu and Xnr are the cumulative amounts of drug eliminated un(cid:173)
`changed in the urine and eliminated by all nonrenal pathways, re(cid:173)
`spectively. The elimination rate constant K is the sum of the individ(cid:173)
`ual rate constants that characterize the parallel elimination processes.
`Thus
`K = k + k
`e
`nr
`where ke is the apparent first-order rate constant for renal excretion
`and k nr is the sum of all other apparent first-order rate constants
`for drug elimination by nonrenal pathways. Since in first-order
`kinetics, the rate of appearance of intact drug in the urine is propor(cid:173)
`tional to the amount of drug in the body, the excretion rate of un(cid:173)
`changed drug, dXu/dt, can be defined as
`
`( 1.13)
`
`dX
`--!:!.= k X
`dt
`e
`
`where X is the amount of drug in the body at time t ,
`Substitution for X according to Eq , (1.5) yields
`
`dXu
`ill = keXOe
`
`-Kt
`
`( 1.14)
`
`( 1.15)
`
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`
`Therefore,
`
`7
`
`( 1.16)
`
`Kt
`2.303
`
`dXu
`log d't= log keX O -
`Equation (1.16) states that a semilogarithmic plot of excretion rate
`of unmetabolized drug versus time is linear, with a slope of -K/2. 303.
`This slope is the same as that Obtained from a semilogarithmic plot of
`drug concentration in plasma versus time. Thus the elimination rate
`constant of a drug can be obtained from either plasma concentration or
`urinary excretion data.
`It must be emphasized that the slope of the
`log excretion rate versus time plot is related to the elimination rate
`constant K, not to the excretion rate constant ke.
`Urinary excretion rates are estimated by collecting all urine for
`a fixed period of time, determining the concentration of drug in the
`urine, multiplying the concentration by the volume of urine collected
`to determine the amount excreted, and dividing the amount excreted
`by the collection time. These experimentally determined excretion
`rates are obviously not instantaneous rates (Le ,; dXu/dt) but are
`f:, Xu I f:,t ) , However, we
`average rates over a finite time period (i. e.,
`often find that the average excretion rate closely approximates the
`
`Table 1.1 Calculation of Excretion Rate Versus Time Data for
`Estimating Half-Life
`
`t (h)
`
`Xu (mg)
`
`0
`1
`2
`
`3
`6
`12
`24
`36
`48
`
`0.0
`4.0
`7.8
`
`11. 3
`
`20.4
`33.9
`48.6
`55.0
`57.8
`
`lit
`
`1
`
`1
`
`1
`
`3
`
`6
`12
`12
`12
`
`lIXu
`
`4.0
`3.8
`3.5
`
`9.1
`13.5
`14.7
`6.4
`2.8
`
`liXu/lit (mg/h)
`
`4.0
`3.8
`3.5
`3.0
`
`2.2
`1.2
`0.53
`0.23
`
`t m
`
`0.5
`1.5
`2.5
`
`4.5
`9.0
`18.0
`30.0
`42.0
`
`t , cumulative time after intra(cid:173)
`Note: The symbols are as follows:
`venous administration; Xu' cumulative amount of unmetabolized drug
`excreted in the urine;
`!:> t , urine collection interval;
`II Xu' amount of
`drug excreted during each interval;
`f:;. Xul f:;. t, experimentally de(cid:173)
`termined excretion rate; t m, midpoint of the collection interval.
`
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`8
`
`Pharmacokinetics
`
`instantaneous excretion rate at the midpoint of the urine collection
`period. The validity of this approximation depends on the collection
`period relative to the half-life of the drug. An individual collection
`period should not exceed one biologic half-life and, ideally, should
`be considerably less. These considerations are discussed in Appendix
`F.
`It is important to remember that urinary excretion rates must be
`plotted against the midpoints of the urine collection periods and not at
`the beginning or end of these periods (see Table 1.1 and Figs. 1. 3
`and 1. 4).
`Fluctuations in the rate of drug elimination are reflected to a high
`degree in excretion rate plots. At times the data are so scattered that
`an estimate of the half-life is difficult. To overcome this problem an
`
`4.0
`
`~-.
`
`2.0 o~
`
`0
`
`1.0
`
`0.5
`
`0.2
`
`:c
`"''"
`E.....e
`'....b
`
`)(
`
`w
`
`e0
`
`6
`
`12
`
`18
`
`24
`
`30
`
`36
`
`40
`
`Time (h)
`
`Fig. 1. 3 Semilogarithmic plot of excretion rate versus time after in(cid:173)
`travenous administration of a drug. Data taken from Table 1.1. Each
`excretion rate is plotted at the midpoint of the urine collection interval.
`The data are described by Eq. (1.16). Slope = - K/ 2.303.
`
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`
`9
`
`100
`
`10
`
`o
`
`5
`
`10
`Time (h)
`Fig. 1. 4 Urinary excretion rate of norephedrine after oral administra(cid:173)
`tion of a single dose of the drug to a healthy adult subjeet ,
`[From
`Ref. 4. © 1968 American Society for Pharmacology and Experimental
`Therapeutics, The Williams and Wilkins Company (agent).]
`
`15
`
`20
`
`alternative approach, termed the sigma-minus method, is available.
`This method is considered less sensitive to fluctuations in drug elim(cid:173)
`ination rate. The Laplace transform of Eq. (1.14) is
`
`sX = k X
`u
`e
`
`( 1.17)
`
`Substitution for X from Eq. (1. 4) and rearrangement yields
`keX O
`s(s + K)
`which when solved gives the following relationship between amount of
`drug in the urine and time:
`
`X =
`u
`
`( 1.18)
`
`kX
`X =~ (1- e- Kt )
`K
`u
`where Xu is the cumulative amount of unchanged drug excreted to
`time t . The amount of unmetabolized drug ultimately eliminated in
`the urine, X~, can be determined by setting time in (1.19) equal to
`infinity; it is given by
`
`( 1.19)
`
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`Pharmacokinetics
`
`(1. 20)
`
`For a drug eliminated solely by renal excretion, K = ke and the amount
`In
`ultimately excreted, X~, will be equal to the intravenous dose, XO'
`all cases the ratio of Xu to X0 equals the ratio of ke to K. This re(cid:173)
`lationship is commonly employed to estimate ke from urinary excretion
`data once the half-life of the drug is determined.
`Substitution of X~ for keXO/K in (1.19) and rearrangement yields
`00 -Kt
`X -X =X e
`u
`u
`u
`which in logarithmic form is
`
`(1. 21)
`
`00
`
`00
`
`00
`
`(1. 22)
`
`Kt
`log (X - X ) = log X - - -
`2.303
`u
`u
`u
`The term (X~ - Xu) is commonly called the amount of unchanged drug
`remaining to be excreted, or A.R.E. A plot of log A.R.E. versus time
`is linear (Fig. 1.5) with a slope equal to -K/2.303. Hence the elimina(cid:173)
`tion rate constant may be estimated from plots of log drug concentra(cid:173)
`tion in plasma versus time,
`log excretion rate versus time (the rate
`method), and log A.R.E. versus time (the sigma-minus method). To
`determine X00, total urine collection must be carried out until no un(cid:173)
`changed dru~ can be detected in the urine.
`It is incorrect to plot
`log (dose - Xu) rather than log (Xu - Xu) versus time.
`total urine collection should be continued for a
`When possible,
`period of time equal to about seven half-lives of the drug to accurately
`estimate X~. This can be very difficult if the drug has a long half(cid:173)
`life. The problem does not arise if the log excretion rate versus time
`plots are used since urine need be collected for only three or four
`half-lives to obtain an accurate estimate of the elimination rate constant.
`The rate method also obviates the need to collect all urine (Le., urine
`samples may be lost or intentionally discarded to minimize the number
`of assays) since the determination of a single point on a rate plot simply
`requires the collection of two consecutive urine samples.
`
`Renal Clearance
`
`The kinetics of renal excretion of a drug may be characterized not only
`by a renal excretion rate constant ke, but also by a renal clearance Clr.
`The concept of drug clearance is discussed in Chap. 8. At this point
`it suffices to state that the renal clearance of drug is equal to the
`volume of blood flowing through the kidneys per unit time from which
`all drug is extracted and excreted.
`The renal clearance of a drug cannot exceed the renal blood flow.
`In pharmaeo-
`Clearance has units of flow (I.e., ml/min or liters/h).
`
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`-,•,~
`
`•
`
`1 / One-Compartment Model
`
`100
`
`10
`
`11
`
`•"-.
`"".'\.
`
`20
`
`40
`
`60
`
`80
`
`100
`
`120
`
`140
`
`Time (h)
`Fig. 1.5 Semilogarithmic plot of the average percentage unmetabolized
`drug remaining to be excreted versus time after oral administration
`t1/2 =36 h.
`of 250 mg of chlorpropamide to six healthy subjects.
`(Data from Ref. 5.)
`
`CI =
`r
`
`kinetic terms renal clearance is simply the ratio of urinary excretion
`rate to drug concentration in the blood or plasma:
`dX /dt
`u
`C
`In practice, renal clearance is estimated by dividing the average
`t.Xu / t.t , by the drug concentration in plasma
`urinary excretion rate,
`at the time corresponding to the midpoint of the urine collection
`period.
`Since excretion rate is the product of the urinary excretion rate
`constant and the amount of drug in the body [Eq. (1.14)], we can write
`
`(1. 23)
`
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`Pharmacokinetics
`
`k X
`Cl = _e_
`C
`r
`Recognizing that X/C is simply the apparent volume of distribution
`[Eq. (1.9)], we can shown that renal clearance is the product of the
`urinary excretion rate constant and the apparent volume of distribu(cid:173)
`tion:
`
`(1, 24)
`
`CI = k V
`r
`e
`
`(1, 25)
`
`All clearance terms can be expressed in terms of a rate constant and
`a volume.
`An estimation of renal clearance by means of Eq , (1, 23) may be
`misleading because like all rate processes in the body, renal excre(cid:173)
`tion is subject to biologic variability. A more satisfactory approach is
`to plot urinary excretion rate versus drug concentration in plasma at
`the times corresponding to the midpoints of the urine collection periods
`(see Fig. 1. 6) . Since rearrangement of Eq. (1, 23) yields
`
`dX
`~ = Cl C
`dt
`r
`
`(1.26)
`
`the slope of an excretion rate-plasma concentration plot is equal to
`renal clearance.
`A second method for calculating renal clearance requires simul(cid:173)
`taneous collection of plasma and urine.
`Integrating Eq. (1.26) from
`t1 to t2 yields
`
`t
`(X ) 2 = Cl
`u t 1
`
`r
`
`It 2 C dt ,
`t1
`
`(1. 27)
`
`where (Xu)~~ is the amount of unmetabolized drug excreted in the urine
`t
`during the time interval from t1 to t 2 and It: e dt is the area under the
`drug concentration in plasma versus time curve during the same time
`interval (see Fig. 1. 7). Terms for area have units of concentration-
`
`time. A plot of (Xu)~~ versus Ji; C dt yields a straight line with a
`
`slope equal to renal clearance.
`Integration of Eq , (1.26) from time zero to time infinity, and re(cid:173)
`arrangement, gives an expression for the average renal clearance over
`the entire time course of drug in the body after a single dose:
`
`Cl =
`r
`
`OO
`
`X u
`J; e dt
`
`OO
`
`X
`u
`= Aue
`
`(1, 28)
`
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`1 / One-Compartment Model
`
`13
`
`•
`•
`
`•
`•
`
`•
`
`•
`
`•
`
`200
`
`e
`E
`....
`Cl':t
`
`see
`
`.s 100
`"§
`
`• •
`
`•
`
`50
`
`u"G
`
`Oe0
`
`5:::;
`
`2.0
`1.0
`Serum concentration (j.Lg/ml)
`
`Fig. 1.6 Relationship between urinary excretion rates of tetracycline
`and serum concentrations of the drug determined at the midpoints of
`each uri