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`Basic Pharmacokinetics
`
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`

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`Basic Pharmacokinetics
`
`Sunil S Jambhekar
`MS, PhD, Professor
`Department of Pharmaceutical Sciences
`LECOM-Bradenton, School of Pharmacy
`Bradenton, Florida, USA
`
`and
`
`Philip J Breen
`PhD, Associate Professor
`College of Pharmacy
`University of Arkansas for Medical Sciences
`Little Rock, Arkansas, USA
`
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`Published by the Pharmaceutical Press
`An imprint of RPS Publishing
`
`1 Lambeth High Street, London SE1 7JN, UK
`100 South Atkinson Road, Suite 200, Grayslake, IL 60030-7820, USA
`
`ÓPharmaceutical Press 2009
`
`is a trade mark of RPS Publishing
`
`RPS Publishing is the publishing organisation of the Royal
`Pharmaceutical Society of Great Britain
`
`First published 2009
`
`Typeset by Thomson Digital, Noida, India
`
`Printed in Great Britain by J International, Padstow, Cornwall
`
`ISBN 978 0 85369 772 5
`
`All rights reserved. No part of this publication may be reproduced,
`stored in a retrieval system, or transmitted in any form or by any
`means, without the prior written permission of the copyright
`holder.
`The publisher makes no representation, express or implied,
`with regard to the accuracy of the information contained in this
`book and cannot accept any legal responsibility or liability for
`any errors or omissions that may be made.
`The right of Sunil Jambhekar and Philip Breen to be identified
`as the authors of this work has been asserted by them in
`accordance with the Copyright, Designs and Patents Act, 1988.
`
`A catalogue record for this book is available from the British Library.
`
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`Contents
`
`Preface xiii
`About the authors
`
`xv
`
`1
`
`17
`
`1
`
`Introduction and overview
`
`1.1 Use of drugs in disease states 1
`1.2 Important definitions and descriptions 2
`1.3 Sites of drug administration 4
`1.4 Review of ADME processes 6
`1.5 Pharmacokinetic models 7
`1.6 Rate processes 12
`
`2
`
`Mathematical review
`
`2.1 Introduction 17
`2.2 A brief history of pharmacokinetics 17
`2.3 Hierarchy of algebraic operations 18
`2.4 Exponents and logarithms 18
`2.5 Variables, constants and parameters 19
`2.6 Significant figures 21
`2.7 Units and their manipulation 21
`2.8 Slopes, rates and derivatives 21
`2.9 Time expressions 23
`2.10 Construction of pharmacokinetic sketches (profiles) 23
`
`3
`
`Intravenous bolus administration (one-compartment model)
`
`29
`
`3.1 Introduction 29
`3.2 Useful pharmacokinetic parameters 30
`3.3 The apparent volume of distribution (V ) 32
`3.4 The elimination half life (t1/2) 36
`3.5 The elimination rate constant (K or Kel) 38
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`Contents
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`3.6 Plotting drug concentration versus time 40
`3.7 Intravenous bolus administration of drugs: summary 42
`3.8 Intravenous bolus administration: monitoring drug in urine 42
`3.9 Use of urinary excretion data 44
`
`4
`
`Clearance concepts
`
`4.1 Introduction 53
`4.2 Clearance definitions 55
`4.3 Clearance: rate and concentration 56
`4.4 Clearance: tank and faucet analogy 56
`4.5 Organ clearance 58
`4.6 Physiological approach to clearance 59
`4.7 Estimation of systemic clearance 64
`4.8 Calculating renal clearance (Clr) and metabolic
`clearance (Clm) 64
`4.9 Determination of the area under the plasma concentration
`versus time curve: application of the trapezoidal rule 65
`4.10 Elimination mechanism 67
`4.11 Use of creatinine clearance to determine renal function 68
`
`Problem set 1
`
`5
`
`Drug absorption from the gastrointestinal tract
`
`5.1 Gastrointestinal tract 87
`5.2 Mechanism of drug absorption 89
`5.3 Factors affecting passive drug absorption 92
`5.4 pH–partition theory of drug absorption 93
`
`6
`
`Extravascular routes of drug administration
`
`6.1 Introduction 97
`6.2 Drug remaining to be absorbed, or drug remaining
`at the site of administration 99
`6.3 Determination of elimination half life (t1/2) and
`elimination rate constant (K or Kel) 101
`6.4 Absorption rate constant (Ka) 102
`6.5 Lag time (t0) 103
`6.6 Some important comments on the absorption rate constant 104
`6.7 The apparent volume of distribution (V ) 105
`
`53
`
`77
`
`87
`
`97
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`v i i
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`6.8 Time of maximum drug concentration, peak time (tmax) 105
`6.9 Maximum (peak) plasma concentration (Cp)max 107
`6.10 Some general comments 109
`6.11 Example for extravascular route of drug administration 110
`6.12 Flip-flop kinetics 114
`
`Problem set 2
`
`7
`
`Bioavailability/bioequivalence
`
`7.1 Introduction 125
`7.2 Important definitions 126
`7.3 Types of bioavailability 126
`7.4 Bioequivalence 129
`7.5 Factors affecting bioavailability 130
`7.6 The first-pass effect (presystemic clearance) 130
`7.7 Determination of the area under the plasma concentration–time
`curve and the cumulative amount of drug eliminated in urine 131
`7.8 Methods and criteria for bioavailability testing 135
`7.9 Characterizing drug absorption from plasma concentration
`versus time and urinary data following the administration of
`a drug via different extravascular routes and/or dosage forms 143
`7.10 Equivalency terms 145
`7.11 Food and Drug Administration codes 145
`7.12 Fallacies on bioequivalence 147
`7.13 Evidence of generic bioinequivalence or of therapeutic
`inequivalence for certain formulations approved
`by the Food and Drug Administration 148
`
`Problem set 3
`
`8
`
`Factors affecting drug absorption: physicochemical factors
`
`8.1 Dissolution rate 159
`8.2 Dissolution process 159
`8.3 Noyes–Whitney equation and drug dissolution 160
`8.4 Factors affecting the dissolution rate 161
`
`117
`
`125
`
`149
`
`159
`
`9
`
`Gastrointestinal absorption: role of the dosage form
`
`171
`
`9.1 Introduction 171
`9.2 Solution (elixir, syrup and solution) as a dosage form 172
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`Contents
`
`9.3
`9.4
`9.5
`9.6
`9.7
`9.8
`
`Suspension as a dosage form 172
`Capsule as a dosage form 173
`Tablet as a dosage form 173
`Dissolution methods 175
`Formulation and processing factors 175
`Correlation of in vivo data with in vitro dissolution data 178
`
`10
`
`Continuous intravenous infusion (one-compartment model)
`
`185
`
`10.1 Introduction 185
`10.2 Monitoring drug in the body or blood (plasma/serum) 188
`10.3 Sampling drug in body or blood during infusion 189
`10.4 Sampling blood following cessation of infusion 203
`10.5 Use of post-infusion plasma concentration data to
`obtain half life, elimination rate constant and the
`apparent volume of distribution 204
`10.6 Rowland and Tozer method 208
`
`Problem set 4
`
`11
`
`Multiple dosing: intravenous bolus administration
`
`211
`
`221
`
`11.1 Introduction 221
`11.2 Useful pharmacokinetic parameters in multiple dosing 225
`11.3 Designing or establishing the dosage regimen for a drug 233
`11.4 Concept of drug accumulation in the body (R) 233
`11.5 Determination of fluctuation (F): intravenous
`bolus administration 236
`11.6 Number of doses required to reach a fraction of the
`steady-state condition 239
`11.7 Calculation of loading and maintenance doses 239
`11.8 Maximum and minimum drug concentration at steady state 240
`
`12
`
`Multiple dosing: extravascular routes of drug administration
`
`243
`
`12.1 Introduction 243
`12.2 The peak time in multiple dosing to steady state (t0
`max) 245
`12.3 Maximum plasma concentration at steady state 246
`12.4 Minimum plasma concentration at steady state 247
`12.5 ‘‘Average’’ plasma concentration at steady state:
`extravascular route 248
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`i x
`
`12.6 Determination of drug accumulation: extravascular route 249
`12.7 Calculation of fluctuation factor (F) for multiple
`extravascular dosing 250
`12.8 Number of doses required reaching a fraction of steady state:
`extravascular route 251
`12.9 Determination of loading and maintenance dose:
`extravascular route 252
`12.10 Interconversion between loading, maintenance,
`oral and intravenous bolus doses 253
`
`Problem set 5
`
`13
`
`Two-compartment model
`
`13.1 Introduction 269
`13.2 Intravenous bolus administration: two-compartment model 272
`13.3 Determination of the post-distribution rate constant
`(b) and the coefficient (B) 276
`13.4 Determination of the distribution rate constant
`(a) and the coefficient (A) 277
`13.5 Determination of micro rate constants: the inter-compartmental
`rate constants (K21 and K12) and the pure elimination
`rate constant (K10) 278
`13.6 Determination of volumes of distribution (V) 280
`13.7 How to obtain the area under the plasma concentration–time
`curve from time zero to time t and time ¥ 282
`13.8 General comments 282
`13.9 Example 283
`13.10 Futher calculations to perform and determine the answers 286
`
`Problem set 6
`
`14
`
`Multiple intermittent infusions
`
`14.1 Introduction 289
`14.2 Drug concentration guidelines 291
`14.3 Example: determination of a multiple intermittent infusion
`dosing regimen for an aminoglycoside antibiotic 292
`14.4 Dose to the patient from a multiple intermittent infusion 293
`14.5 Multiple intermittent infusion of a two-compartment
`drug: vancomycin ‘‘peak’’ at 1 h post-infusion 294
`
`257
`
`269
`
`287
`
`289
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`x
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`Contents
`
`14.6 Vancomycin dosing regimen problem 295
`14.7 Adjustment for early or late drug concentrations 296
`
`Problem set 7
`
`15
`
`Non-linear pharmacokinetics
`
`15.1 Introduction 301
`15.2 Capacity-limited metabolism 304
`15.3 Estimation of Michaelis–Menten
`parameters (Vmax and Km) 305
`15.4 Relationship between the area under the plasma
`concentration versus time curve and the administered dose 309
`15.5 Time to reach a given fraction of steady state 311
`15.6 Example: calculation of parameters
`for phenytoin 313
`
`Problem set 8
`
`16
`
`Drug interactions
`
`16.1 Introduction 319
`16.2 The effect of protein-binding interactions 320
`16.3 The effect of tissue-binding interactions 327
`16.4 Cytochrome P450-based drug interactions 328
`16.5 Drug interactions linked to transporters 336
`
`299
`
`301
`
`317
`
`319
`
`17
`
`Pharmacokinetic and pharmacodynamic relationships
`
`337
`
`17.1 Introduction 337
`17.2 Generation of a pharmacokinetic–pharmacodynamic
`(PKPD) equation 338
`17.3 Pharmacokinetic and pharmacodynamic
`drug interactions 342
`
`18
`
`Pharmacokinetics and pharmacodynamics of biotechnology drugs
`
`345
`
`18.1 Introduction 345
`18.2 Proteins and peptides 345
`18.3 Monoclonal antibodies 351
`18.4 Oligonucleotides 355
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`18.5 Vaccines (immunotherapy) 356
`18.6 Gene therapies 357
`
`Appendix: Statistical moment theory in pharmacokinetics
`
`A.1 Introduction 361
`A.2 Statistical moment theory 362
`A.3 Applications 374
`
`Glossary
`
`References
`
`Index
`
`Contents
`
`x i
`
`361
`
`377
`
`383
`
`391
`
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`6
`
`Extravascular routes of drug administration
`
`Objectives
`
`Upon completion of this chapter, you will have the ability to:
`
`· calculate plasma drug concentration at any given time after the administration of an extravascular
`dose of a drug, based on known or estimated pharmacokinetic parameters
`· interpret the plasma drug concentration versus time curve of a drug administered extravascularly as
`the sum of an absorption curve and an elimination curve
`· employ extrapolation techniques to characterize the absorption phase
`· calculate the absorption rate constant and explain factors that influence this constant
`· explain possible reasons for the presence of lag time in a drug’s absorption
`· calculate peak plasma drug concentration, (Cp)max, and the time, tmax, at which this occurs
`· explain the factors that influence peak plasma concentration and peak time
`· decide when flip-flop kinetics may be a factor in the plasma drug concentration versus time curve of
`a drug administered extravascularly.
`
`6.1 Introduction
`
`Drugs, through dosage forms, are most frequently
`administered extravascularly and the majority of
`them are intended to act systemically; for this
`reason, absorption is a prerequisite for pharma-
`cological effects. Delays or drug loss during
`absorption may contribute to variability in drug
`response and, occasionally, may result in a failure
`of drug therapy.
`The gastrointestinal membrane separates the
`absorption site from the blood. Therefore, passage
`of drug across the membrane is a prerequisite for
`absorption. For this reason, drug must be in a
`solution form and dissolution becomes very crit-
`ical for the absorption of a drug. The passage of
`
`drug molecules from the gastrointestinal tract to
`the general circulation and factors affecting this
`are shown in Figs 6.1 and 6.2. Any factor influ-
`encing dissolution of the drug is likely to affect
`the absorption of a drug. These factors will be
`discussed, in detail, later in the text.
`Drug, once in solution, must pass through
`membranes before reaching the general circula-
`tion. Hence, the physicochemical properties of
`the drug molecule (pKa of the drug, partition coef-
`ficient of the drug, drug solubility, etc.), pH at the
`site of drug administration, nature of the mem-
`brane and physiological factors will also influence
`the absorption of a drug.
`The present discussion will deal with general
`principles that determine the rate and extent of
`
`9 7
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`10 mols
`of drug
`dissolved
`in GI tract
`
`Biliary
`excretion of
`1 mol of
`drug (to
`feces)
`
`1 mol
`of drug
`metabolized
`in gut wall
`
`9 8
`
`Basic Pharmacokinetics
`
`10 mols
`of drug
`ingested
`
`8 mols of drug
`carried by portal
`circulation to
`liver
`
`2 mols
`of drug
`metabolized
`in liver
`
`6 mols
`of drug escaping
`metabolism go on
`to systemic
`circulation
`(F po = 6/10 = 0.6)
`
`Figure 6.1
`
`Barriers to gastrointestinal absorption.
`
`drug absorption and the methods used to assess
`these and other pharmacokinetic parameters,
`from plasma concentration versus time data fol-
`lowing oral administration of drugs. Emphasis is
`placed upon absorption of drugs following oral
`administration because it illustrates all sources of
`variability encountered during drug absorption.
`Please note that a similar approach may be
`applied to determine pharmacokinetic parameters
`of drugs when any other extravascular route is
`used.
`The following assumptions are made:
`
`· drug exhibits
`the characteristics of one-
`compartment model
`· absorption and elimination of a drug follow
`the first-order process and passive diffusion is
`operative at all the time
`
`· drug is eliminated in unchanged form (i.e. no
`metabolism occurs)
`· drug is monitored in the blood
`
`Useful pharmacokinetic parameters
`
`Figure 6.3 outlines the absorption of a drug that
`fits a one-compartment model with first-order
`elimination. The following information is
`useful.
`
`1. Equation for determining the plasma concen-
`tration at any time, t
`2. Determination of the elimination half life
`(t1/2) and rate constant (K or Kel)
`3. Determination of the absorption half
`(t1/2)abs and absorption rate constant (Ka)
`
`life
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`Extravascular routes of drug administration
`
`9 9
`
`Tablet
`
`Gastric emptying
`
`Intestinal transit time
`
`Disintegration
`time
`
`Dissolution
`time
`
`pH of lumen fluid
`
`Surface
`area
`
`Transport across
`columnar cell
`
`Metabolism
`
`Figure 6.2
`
`Passage of drug in the gastrointestinal tract until transport across the membrane.
`
`Mesenteric
`blood flow
`
`6.2 Drug remaining to be absorbed,
`or drug remaining at the site
`of administration
`
`Equation 6.1 describes the changes with drug over
`time at the site of administration.
`
` dXa
`
`¼ KaðXaÞ
`
`t
`
`ð6:1Þ
`
`4. Lag time (t0), if any
`5. Determination of the apparent volume of
`distribution (V or Vd) and fraction of drug
`absorbed (F)
`6. Determination of the peak time (tmax.)
`7. Determination of the peak plasma or serum
`concentration, (Cp)max.
`
`dt
`
`SCHEME:
`
`Xa
`(absorbable
`drug at absorption
`site)
`
`Ka (h−1)
`absorption
`
`X
`(drug in
`body or
`blood)
`
`K (h−1)
`elimination
`
`Xu
`
`SETUP:
`
`Xu
`Xa
`Figure 6.3 Absorption of a one-compartment drug with first-order elimination. where Xa is the mass or amount of absorbable
`drug remaining in the gut, or at the site of administration, at time t (i.e. drug available for absorption at time t); X is the mass or
`amount of drug in the blood at time, t; Xu is the mass or amount of drug excreted unchanged in the urine at time, t; Ka is the first-
`
`
`1).1 or min1 or min1); and K (or Kel) is the first-order elimination rate constant (h
`
`order absorption rate constant (h
`
`Ka
`
`K
`
`X
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`

`rate constant; and, therefore, we go to other alter-
`natives such as monitoring drug in the blood and/
`or urine to determine the absorption rate con-
`stant and the absorption characteristics.
`
`Monitoring drug in the blood
`(plasma/serum) or site of measurement
`
`The differential equation that follows relates
`changes in drug concentration in the blood with
`time to the absorption and the elimination rates
`
`¼ KaXa KX
`
`dX
`dt
`
`ð6:4Þ
`
`1) of change of
`where dX/dt is the rate (mg h
`amount of drug in the blood; X is the mass or
`amount of drug in the blood or body at time, t;
`Xa is the mass or amount of absorbable drug at the
`absorption site at time t; Ka and K are the first-
`order absorption and elimination rate constants,
`1); KaXa is the first-order rate
`respectively (e.g. h
`1; mg h
`1, etc); and KX is the
`of absorption (mg h
`1).
`first-order rate of elimination (e.g. mg h
`Equation 6.4 clearly indicates that rate of
`change in drug in the blood reflects the difference
`between the absorption and the elimination rates
`(i.e. KaXa and KX, respectively). Following the
`administration of a dose of drug, the difference
`between the absorption and elimination rates (i.e.
`
`KaXa KX) becomes smaller as time increases; at
`
`peak time, the difference becomes zero.
`Please note that, most of the time, the absorp-
`tion rate constant is greater than the elimination
`
`Intercept = (Xa)0 or FX0
`
`Slope =
`
`–Ka
`2.303
`
`(b)
`
`Xa (mg)
`
`1 0 0
`
`Basic Pharmacokinetics
`
`where dX/dt is the decrease in the amount of
`absorbable drug present at the site of administra-
`1); Ka is the first-
`tion per unit time (e.g. mg h
`1; min
`1); and
`order absorption rate constant (h
`(Xa)t is the mass or amount of absorbable drug at
`the site of administration (e.g. the gastrointesti-
`nal tract) at time t.
`Upon integration of Eq. 6.1, we obtain the
`following:
`
`ðXaÞ
`
`t
`
`¼ ðXaÞ
`
`t¼0e
`
` Kat ¼ FX0e
`
` Kat
`
`ð6:2Þ
`
`where (Xa)t=0 is the mass or amount of absorbable
`drug at the site of administration at time t¼ 0 (for
`(Xa)t=0
`extravascular administration of drug,
`equals FX0); and F is the fraction or percentage
`of the administered dose that is available to reach
`the general circulation; X0 is the administered
`dose of drug.
`If F¼ 1.0, that is, if the drug is completely
`(100%) absorbed, then
`
`ðXaÞ
`
`t
`
`¼ X0e
`
` Kat
`
`ð6:3Þ
`
`Both Eqs 6.2 and 6.3 and Fig. 6.4 clearly indi-
`cate that the mass, or amount, of drug that
`remains at the absorption site or site of adminis-
`tration (or remains to be absorbed) declines
`monoexponentially with time.
`However,
`since we cannot measure the
`amount of drug remaining to be absorbed (Xa)
`directly, because of practical difficulty, Eqs 6.2
`and 6.3, for the time being, become virtually use-
`less for the purpose of determining the absorption
`
`(a)
`
`Xa (mg)
`
`t = 0
`
`Time (h)
`
`t = 0
`
`Time (h)
`
`Figure 6.4 Amount of drug remaining at the site of administration against time in a rectilinear plot (a) and a semilogarithmic
`plot (b). Xa, amount of absorbable drug at the site of administration; (Xa)0, amount of absorbable drug at the site of
`administration at time t¼ 0; F, fraction of administered dose that is available to reach the general circulation.
`
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`Extravascular routes of drug administration
`
`1 0 1
`
`to reach the general circulation, which is the
`same as the bioavailable fraction times the
`administered dose.
`Equation 6.5 and Fig. 6.5 show that the mass or
`amount of drug in the body or blood follows a
`biexponential profile,
`first
`rising and then
`declining.
`For orally or extravascularly administered
`
`drugs, generally Ka K; therefore, the rising por-
`If K Ka (perhaps indicating a dissolution-
`
`tion of the graph denotes the absorption phase.
`
`rate-limited absorption) the exact opposite will
`hold true. (Please see the discussion of the flip-
`flop model at the end of this chapter.)
`
`6.3 Determination of elimination
`half life (t1/2) and elimination rate
`constant (K or Kel)
`
`Equation 6.5, when written in concentration (Cp)
`terms, takes the following form:
` Kt e
`
` KatŠ
`
`ð6:6Þ
`
`ðCpÞ
`
`t
`
`rate constant. (The exceptional situation when
`K > Ka,
`termed “flip-flop kinetics,” will be
`addressed in the last section of this chapter.)
`Furthermore, immediately following the admin-
`istration of a dose of drug, the amount of (absorb-
`able) drug present at the site of administration
`will be greater than the amount of drug in the
`blood. Consequently, the rate of absorption will
`be greater than the rate of elimination up to a
`certain time (prior to peak time); then, exactly
`at peak time, the rate of absorption will become
`equal to the rate of elimination. Finally, the rate
`of absorption will become smaller than the rate of
`elimination (post peak time). This is simply the
`result of a continuous change in the amount of
`absorbable drug remaining at the site of adminis-
`tration and the amount of drug in the blood. Also,
`please note that rate of absorption and the rate of
`elimination change with time (consistent with
`the salient feature of the first-order process),
`whereas the absorption and the elimination rate
`constants do not change.
`Integration of Eq. 6.4 gives:
`
`½e
`
` Kt e
`
` KatŠ
`
`ðXÞ
`
`t
`
`¼ KaðXaÞ
`t¼0
`Ka K
`½e
`¼ KaFX0
`Ka K
`
` Kt e
`
` KatŠ
`
`ð6:5Þ
`
`where (X)t is the mass (amount) of drug in the
`body at time t; X0 is the mass of drug at the site
`of administration at t¼ 0 (the administered dose);
`F is the fraction of drug absorbed; (Xa)0¼ FD0 and
`
`is the mass of administered dose that is available
`
`VðKa KÞ½e
`¼ KaFX0
`KaFX0
`VðKa KÞ is the intercept of plasma drug
`concentration versus time plot (Fig. 6.6).
`When time is large, because of the fact that
` Kat approaches zero, and Eq. 6.6
`reduces to:
`
`where
`
`Ka  K, e
`
`ðCpÞ
`
`t
`
`¼ KaFX0
`VðKa KÞ½e
`
` KtŠ
`
`ð6:7Þ
`
`KaXa = KX
`
`Elimination phase
`(KX » KaXa)
`
`X (mg)
`
`Absorption phase
`(KaXa » KX)
`
`Time (h)
`Figure 6.5 A typical rectilinear profile illustrating amount of drug (X) in blood or body against time. Xa, amount of absorbable
`drug at the absorption site at time t; Ka and K, first-order absorption and elimination rate constants, respectively; KaXa and KX,
`first-order rates of absorption and elimination, respectively.
`
`t = 0
`
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`

`1 0 2
`
`Basic Pharmacokinetics
`
`Intercept =
`
`KaFX0
`V (Ka – K )
`
`Slope =
`
`–K
`2.303
`
`(b)
`
`Cp (ng mL–1)
`
`(a)
`
`Cp (ng mL–1)
`
`t = 0
`
`Time (h)
`Time (h)
`Figure 6.6 A plot of plasma concentration (Cp) against time on rectilinear (a) and semilogarithmic (b) paper.(Xa)0, amount of
`absorbable drug at the site of administration at time t¼ 0; F, fraction of administered dose that is available to reach the general
`circulation; Ka and K, first-order absorption and elimination rate constants, respectively; V, apparent volume of distribution.
`
`t = 0
`
`versus time data obtained or provided to you and
`the plot of the data (as shown in Fig. 6.8) we can
`construct a table with headings and columns as in
`Table 6.1 for the purpose of determining the
`absorption rate constant.
`In column 1 of the table, the time values are
`recorded that correspond to the observed plasma
`concentrations. This is done only for the absorp-
`tion phase. In column 2, the observed plasma con-
`centration values provided only from the
`absorption phase are recorded (i.e. all values prior
`to reaching maximum or highest plasma concen-
`tration value). In column 3, the plasma concentra-
`tion values obtained only from the extrapolated
`portion of the plasma concentration versus time
`plot are recorded (these values are read from the
`plasma concentration–time plot); and, in column
`4, the differences in the plasma concentrations
`(Cp)diff between the extrapolated and observed
`values for each time in the absorption phase are
`recorded.
`in plasma concentrations
`The differences
`between the extrapolated and observed values
`(in column 4 of Table 6.1) should decline mono-
`exponentially
`according
`to the
`following
`equation:
`
`ðCpÞ
`
`diff
`
`¼ KaFX0
`VðKa KÞ½e
`
` KatŠ
`
`ð6:8Þ
`
`The elimination half life and elimination rate
`constant can be obtained by methods described
`earlier and illustrated in Figure 6.7.
`
`6.4 Absorption rate constant (Ka)
`
`The absorption rate constant is determined by a
`method known as “feathering,” “method of
`residuals” or “curve stripping.” The method allows
`the separation of the monoexponential constitu-
`ents of a biexponential plot of plasma concentra-
`tion against time. From the plasma concentration
`
`Intercept =
`
`KaFX0
`V (Ka – K )
`
`Slope =
`
`–K
`2.303
`
`t
`
`½
`
`Time (h)
`
`Cp (mg L–1)
`
`Semilogarithmic plot of plasma drug concentra-
`Figure 6.7
`tion (Cp) versus time of an extravascular dosage form: visual-
`ization of elimination half life (t1/2). Other abbreviations as in
`Fig. 6.6.
`
`where KaFX0
`VðKa KÞ is the intercept of plasma drug con-
`centration versus time plot. A plot of this differ-
`ence between extrapolated and observed plasma
`concentrations against time, on semilogarithmic
`
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`

`Extravascular routes of drug administration
`
`1 0 3
`
`Intercept =
`
`KaFX0
`V (Ka – K )
`Extrapolated concentration
`values
`
`Feathered or residual line
`
`Elimination
`phase
`
`Cp (mg mL–1)
`
`Absorption
`phase
`
`Time (h)
`Semilogarithmic plot of plasma concentration (Cp) versus time of an extravascular dosage form, showing the
`Figure 6.8
`method of residuals. Other abbreviations as in Fig. 6.6.
`
`t = 0
`
`Table 6.1
`
`Time (h)
`
`Illustration of the table created for determination of the first-order absorption rate constant Ka
`(Cp)diff = (Cp)extrap (Cp)obs
`
`Observed plasma
`concentration (Cp)obs
`
`Extrapolated plasma
`concentration (Cp)extrap
`
`Time values corresponding
`to observed plasma
`concentrations for
`absorption phase only
`
`Values only from the
`absorption phase (i.e. all
`values prior to reaching
`maximum or highest plasma
`concentration) (units, e.g.
`1)
`mg mL
`
`Values only from the
`extrapolated portion of the
`plot of plasma
`concentration–time (units,
`1)
`e.g. (mg mL
`
`Differences between
`extrapolated and observed
`values for each time in the
`absorption phase (units,
`1)
`e.g. mg mL
`
`paper (Fig. 6.9), should yield a straight line,
`which, in turn, should allow determination of:
`· the half life of the feathered or residual line
`(i.e. the t1/2 of absorption phase)
`· the first-order absorption rate constants,
`using the equation Ka¼ 0.693/(t1/2)abs, or
`Ka¼ (slope) 2.303.
`
`6.5 Lag time (t0)
`
`Theoretically, intercepts of the terminal linear
`portion and the feathered line in Fig. 6.8 should
`be the same; however, sometimes, these two
`lines do not have the same intercepts, as seen
`in Fig. 6.10.
`A plot showing a lag time (t0) indicates that
`absorption did not start immediately following
`the administration of drug by the oral or other
`
`extravascular route. This delay in absorption
`may be attributed to some formulation-related
`problems, such as:
`· slow tablet disintegration
`· slow and/or poor drug dissolution from the
`dosage form
`· incomplete wetting of drug particles (large
`contact angle may result in a smaller effective
`surface area) owing to the hydrophobic nature
`of the drug or the agglomeration of smaller
`insoluble drug particles
`· poor formulation, affecting any of the above
`· a delayed release formulation.
`
`Negative lag time (t0)
`
`Figure 6.11 shows a plot with an apparent nega-
`tive lag time.
`
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`

`1 0 4
`
`Basic Pharmacokinetics
`
`Intercept =
`
`KaFX0
`V (Ka – K )
`
`Slope =
`
`–Ka
`2.303
`
`or
`
`Ka = 0.693
`(t ½) abs
`
`t ½
`absorption
`
`(Cp)diff (mg mL–1)
`
`Time (h)
`Semilogarithmic plots of plasma concentration (Cp)diff. between calculated residual concentrations and mea-
`Figure 6.9
`sured ones versus time, allowing the calculation of the absorption rate constant. (t1/2)abs, absorption half life; other abbrevia-
`tions as in Fig. 6.6.
`
`What does negative lag time mean? Does it
`mean that absorption has begun prior to the
`administration of a drug? That cannot be possi-
`ble unless the body is producing the drug! The
`presence of a negative lag time may be attrib-
`uted to a paucity of data points in the absorp-
`tion as well as in the elimination phase. Another
`possible reason may be that the absorption rate
`constant is not much greater than the elimina-
`tion rate constant.
`The absorption rate constant obtained by the
`feathering, or residual, method could be errone-
`ous under the conditions stated above. Should
`that be the case, it is advisable to employ some
`other methods (Wagner and Nelson method, sta-
`tistical moment analysis, Loo–Rigelman method
`for a two-compartment model, just to mention a
`few) of determining the absorption rate constant.
`Though these methods
`tend to be highly
`
`mathematical and rather complex, they do pro-
`vide an accurate estimate of the absorption rate
`constant, which, in turn, permits accurate estima-
`tion of other pharmacokinetic parameters such as
`peak time, peak plasma concentration, as well as
`the assessment of bioequivalence and compara-
`tive and/or relative bioavailability.
`
`6.6 Some important comments on
`the absorption rate constant
`
`Figure 6.12 indicates that the greater the differ-
`ence between the absorption and the elimina-
`
`tion rate constants (i.e. Ka  K), the faster is drug
`
`absorption and the quicker is the onset of action
`(in Fig. 6.12, apply the definition of onset of
`action). Please note the shift in the peak time
`
`Intercept of
`feathered line and
`extrapolated line
`
`Cp (mg L–1)
`
`Theoretical
`Intercept
`
`Cp (mg L–1)
`
`t = 0
`
`Time (h)
`
`Time (h)
`
`Log time (t0)
`Semilogarithmic plots of the extrapolated plasma concentration (Cp) versus time showing the lag time (t0).
`
`Figure 6.10
`
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`

`Extravascular routes of drug administration
`
`1 0 5
`
`Intercept of
`feathered and
`extrapolated lines
`
`6.7 The apparent volume
`of distribution (V)
`
`For a drug administered by the oral, or any other
`extravascular, route of administration, the appar-
`ent volume of distribution cannot be calculated
`from plasma drug concentration data alone. The
`reason is that the value of F (the fraction of
`administered dose that reaches the general circu-
`lation) is not known. From Eqs 6.7 and 6.8:
`
`Intercept ¼ KaFX0
`VðKa KÞ
`
`ð6:9Þ
`
`Feathered line
`
`Cp (mg L–1)
`
`(–t0)
`
`Time (h)
`
`Semilogarithmic plot of plasma concentra-
`Figure 6.11
`tion (Cp) versus time showing a negative value for the lag
`time (t0).
`
`and peak plasma concentration values as the
`difference between absorption rate constant
`(Ka) and elimination rate constant (K) becomes
`smaller, as you go from left to right of the figure.
`If the absorption rate constant (Ka) is equal to
`the elimination rate constant (K), we need
`to employ a different pharmacokinetic model to
`fit the data.
`Please note that the absorption rate constant
`for a given drug can change as a result of changing
`the formulation, the dosage form (tablet, suspen-
`sion and capsule) or the extravascular route of
`drug administration (oral, intramuscular, subcu-
`taneous, etc.). Administration of a drug with or
`without food will also influence the absorption
`rate constant for the same drug administered
`orally through the same formulation of the same
`dosage form.
`
`If we can reasonably assume, or if it has been
`reported in the scientific literature, that F¼ 1.0
`(i.e. the entire administered dose has reached
`the general circulation), only then can we calcu-
`late the apparent volume of distribution follow-
`ing the administration of a drug by the oral or any
`other extravascular route.
`In the absence of data for the fraction of admin-
`istered dose that reaches the general circulation,
`the best one can do is to obtain the ratio of V/F:
`
`
`
`ð6:10Þ
`
`
`
`1
`Intercept
`
`¼ KaX0
`ðKa KÞ
`
`V F
`
`6.8 Time of maximum drug
`concentration, peak time (tmax)
`
`The peak time (tmax)
`is the time at which
`the body displays
`the maximum plasma
`
`Ka K
`
`Ka» K
`
`Ka K
`
`MTC
`Therapeutic
`range
`MEC
`
`Ka @ K
`Problems ?
`
`Cp (µg L–1)
`
`Time (h)
`Rectilinear plot of plasma concentration (Cp) versus time for various magnitudes of absorption (Ka) and
`Figure 6.12
`elimination (K) rate constants. MTC, minimum toxic concentration; MEC, minimum effective concentration.
`
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`

`1 0 6
`
`Basic Pharmacokinetics
`
`Cp (mg L–1)
`
`At this time (t max)
`rate of absorption =
`rate of elimination
`(KaXa = KX )
`
`Cp (mg L–1)
`
`t max
`
`t max estimated
`from graph
`
`R