TEXTBUUK 0F
`BIUPHAHMACEUTICS
`AND CLINICAL
`PHARMACUKINETICS
`
`
`
`Sarfaraz Niazi, Ph.D.
`Associate Professor of Pharmacy
`College of Pharmacy
`University oi‘Iilinois at the Medical Center
`Chicago, Illinois
`
`Appleton-Cenmry-Crofis
`New York
`
`i
`i
`
`Hospira v. Genentech
`Hospira v. Genentech
`IPR2017-00805
`IPR2017—00805
`Genentech Exhibit 2006
`Genentech Exhibit 2006
`
`

`

`Copyright © 1979 by APPLETON-CENTUBY—CBOFTS
`A Publishing Division of Prentice-Hall, Inc.
`
`All rights reserved. This book, or any parts thereof, may not be used
`or reproduced in any manner without written permission. For infor-
`mation, address App]eton-Century-Crofts, 292 Madison Avenue, New
`York, N.Y. 10017.
`
`7980818283/1098765432
`
`Prentice-Hall International, Inc., London
`Prentice-Hall of Australia, Pty. Ltd, Sydney
`Prentice-Hall of India Private Limited, New Delhi
`Prentice-Hall of Japan, 1110., Tokyo
`Prentice-Hall of Southeast Asia (Pte.) Ltd, Singapore
`Whitehall Books Ltd., Wellington, New Zealand
`
`Library of Congress Cataloging in Publication Data
`
`Niazi, Sarfaraz, 1949-
`
`Textbook ofbiophannaceuties and clinical pharmacokinetics.
`
`Includes index.
`
`2. Pharmacokinetics.
`1. Biopharmaceuties.
`I. Title.
`[DNLM:
`1. Biopharmaceutics.
`2. Kinetics.
`3. Pharmacology. QV38.3 N577t]
`BM301.4.N52
`615’.7
`79-10869
`ISBN 0-8385-8868-9
`
`Design: Meryl Sussman Levavi
`
`PRINTED IN THE UNITED STATES OF AMERICA
`
`ii
`ii
`
`

`

`This material may be protected by Copyright law (Title 17 U.S. Code)
`
`

`

`142
`
`TEXTBOOK OF BIOPHARMACEUTICS AND CLINICAL PHARMACOKINETICS
`
`Figure 7.1. Single compartment vis—
`
`ualization.
`
`One compartment One compartment
`model before
`model
`immediately
`administration
`after administration
`
`minute, it is reasonable to expect that by the time the injection is completed
`all those tissues which are highly perfused will have equilibrated with the
`concentration in the plasma (Fig. 7.1). Not Surprisingly, a large number of
`drugs do show almost instantaneous equilibration with the body tissues and
`are distributed homogeneously throughout the body. Note that the word
`homogeneous does not mean equal concentration in this discussion. It simply
`means that an equilibration is reached between the plasma and the various
`tissues and fluids in the body and that any change in the plasma concentration
`can be attributed to the elimination of drug from the body rather than uptake
`by the tissues. Of course, as the drug is removed from the plasma, body
`tissues de—equilibrate and achieve a new equilibrium. All of these processes
`are very quick in nature, so the plasma concentration can be used to predict
`the amount of drug remaining in the body:
`
`X = V C
`
`(Eq. 7.3)
`
`where V 2 volume of distribution or a proportionality constant between the
`amount of drug, X, in the body and the plasma concentration, C. A substitution
`of Equation 7.3 into Equation 7.2 and conversion to log10 base yields (see
`Appendix A):
`
`log C = log Co —
`
`
`Kt
`
`2.303
`
`(Eq. 7.4)
`
`Where C 0 is the plasma concentration immediately after intravenous injection
`or the concentration extrapolated to time zero on a plasma concentration : time
`profile (Fig. 7.2). Thus if a known amount of dose, X0, is introduced into the
`body and the plasma concentration is measured periodically the unknown
`pharmacokinetic parameters can be easily obtained:
`
`

`

`PHARMACOKINETIC PRINCIPLES
`
`143
`
`V = Xufcu
`
`K = - slope X 2.303
`
`(Eq.
`
`(Eq. 7.6)
`
`Once we have calculated these two parameters by sampling the plasma at
`various times, the complete profile can be predicted using Equation 7.2. The
`calculated pharmacokinetic parameter can be used to describe the general
`nature of the drug elimination, i.e., biotransiormation and excretion. A quick
`understanding of the rate of drug removal from the body can be obtained by
`calcrilating the time needed to decrease the body level by one-half, or the
`HALF-LIFE, which is calculated from:
`
`log X 2 loan -‘
`
`Kt
`
`Thus
`
`= *Xa, t = tuvfi, O" the half-ller
`
`ktflvfi
`2.303
`
`_
`log an — logXo
`
`_ __ Kt“:
`log it —
`2.303
`
`m = 0.693lK
`
`(Eq.
`
`(Eq. 7.8)
`
`(Eq. 7.9)
`
`(Eq. 7.10)
`
`It should be noted from Equation 7.10 that the half-life is independent of the
`dose administered, a characteristic of first order processes. ‘The half-life can
`also be calculated directly from a plasma concentration profile (Fig. 7.2) by
`reading the time needed for the concentration to decrease by one-half from
`any arbitrary point on a log concentrationztime plot. Since the calculation of
`
`70 i 0
`50
`30
`I 20'105
`
`10 '
`
`Sim‘”K’2v303
`
`concentration
`7.2. Plasma
`Figure
`profile following intravenous adminis~
`[ration in a singie compartment open
`model. Data from Example 7.1. See
`text for detaits.
`
` C
`
`Co=64mgfml
`X°=600mg
`v =9.3?5
`liters
`
`K=o.533hr"
`
`5
`
`|
`
`‘E
`“a,
`3,
`C
`.2
`E
`E
`t:
`8
`
`*
`‘
`
`4
`3
`I+ 2
`Time (hours)
`
`5
`
`

`

`H4
`
`TEXTBOOK OF BIOPHARMACEUTICS AND CLINICAL PHARMACOKINETICS
`
`half-life is simpler than caiculating the rate constant, it is suggested that the
`rate constant be calculated From the half-life values.
`
`The model presented in Figure 7.1 is called a single compartment model
`since the body can be assumed to have homogeneous characteristics as far as
`the elimination of the drug is concerned. The term COMPARTMENT refers to
`the homogeneity of the rate process rather than a physical part of the body
`within which the drug has distributed, though the latter use is always pos~
`sible.l
`
`The following example illustrates calculations of the pharmacokinetic paw
`rameter in a single compartment open model.
`
`Example 7.1:
`A single 600 mg dose oi‘ampicillin was administered intravenously to an adult
`and the following plasma cancentration profile was obtained:
`
`l (hr)
`
`C (ugfml)
`
`1
`2
`3
`5
`
`3?
`21.5
`12.5
`4.5
`
`The pharmacokinetic parameters can be calculated by first plotting the plasma
`concentration data on a semilogarithmic scale (Fig. 7.2) to obtain a straight line
`drawn through these points (best fit line).
`C a (64 agiml) is directly read as the intercept of the extrapolation on the
`ordinate.
`
`V {9.375 liters) is obtained by diViding CD into the dose administered.
`t“, (1.3 hours) is obtained from the time needed for the plasma concentration
`to decrease from 60 [item] to 30 uglml. You should try using other ranges as
`well.
`
`K (0.533 hr”) is obtained by dividing the half-life into 0.693 or by determining
`the slepe oi" the best fit line and multiplying that by —2.303. Note that the slope
`can be calculated by:
`
`slope =
`
`logA — log 8
`(Eq
`gifts
`where A and B are any two concentrations on the profile and t4 and t 3 are the
`corresponding times at which these concentrations occur. A common error made
`in the calculation of slopes is the failure to convert the concentration on the
`ordinate scale to log scale. Remember that the values read from the plasma
`concentration axis are not log C but simply C.
`
`.Iu
`
`)
`
`All of these parameters can be summed up in a compartmental represen-
`tation of the model, as shown'in the inset of Fig. 7.2.
`The pharmacokinetic parameters have great physiologic and clinical mean-
`ing and the following discussion is hoped to shed some light on it. The full
`potential of the pharn'iacokinetic modeling excercises can only become ap-
`parent atter all applications are diseussed (as presented in later chapters).
`
`

`

`PHARMACOKINETIC PRINCIPLES
`
`145
`
`The Half-life
`
`As defined above, the half-life is the time required for the amount of drug in
`the body to decrease to one-half of its initial value. Thus ifthe initial amount
`is 100 mg and if the lialfwlife is one-hour, only 50 mg will remain in the body
`at the end of a one hour period:
`
`Time (hr)
`
`Amount {mg}
`
`0
`1
`2
`3
`4
`
`100 {dose}
`50
`25
`1 2.5
`6.25
`
`Note that after each hour the amount is decreased to one-half of the amount
`
`in the previous heur. Thus the half-life can be calculated at any time on the
`amounticoncentration profile. The half-life denotes how quickly a drug is
`removed from the body by biotransformation or excretion. Since most drugs
`require a minimum effective concentration in the plasma, a drug which is
`eliminated quickly requires more frequent dosing than a drug with a long
`half-life. Half-lives vary greatly between drugs. For instance, penicillin has
`a half-life of about 30 minutes, phenobarbital has a half-life of about 5 days
`and, as shown in Figure 5.5, a drug can persist in the blood for years. The
`half-lives can also vary between structurally similar drugs. For example,
`sulfarnethylthiazide has a half—life of about two hours, whereas sulfarnethoxy—
`pyridazine requires 34 hours for 50 percent elimination.
`'
`The half-lives also vary between species and even within a given species,
`depending on various physiologic and pathologic functions. In normal sub-
`jects the half-lives remain fairly constant, but subtle variations due to such
`factors as diurnal functions are always possible.
`In bioavailability studies it is recommended that the urinary excretion be
`monitored for at least seven half-lives to assure complete elimination. The
`rationale for this comes from the following calculations:
`
`Cumulative
`Fraction
`Percent
`
`1M
`Remaining
`Remaining
`
`1 00
`1
`0
`50
`HQ
`1
`25
`1 i4
`2
`12. 5
`1i8
`3
`6. 25
`1 i 1 5
`4
`3. 1 2
`1i32
`5
`1 . 56
`I
`1 i64
`5
`
`
`1 i 1 287 0. 78
`
`Thus at the end of seven half-lives less than 1 percent of the administered
`
`

`

`145
`
`TEXTBOOK OF BIOPHAFIMACEUTICS AND CLINICAL PHARMACOKINETICS
`
`dose remains in the body. Note that at least in theory infinite half-lives will
`he required to reach a 0 percent value in the body.
`Since the half-life is an indication of the efficiency of the elimination
`processes of the body, any change in the half-life will reflect changes in these
`elimination organ functions, such as liver biotransformation or excretion in
`the kidneys. Thus such reference compounds as creatinine or inulin can be
`used to estimate the kidney function or bromosulfonphthalein can be uSed
`to study the liver function in terms of the half-lives of these compounds.
`The changes in the half-lives of drugs are. used as a prime measure for
`dosage adjustment in disease states. Table 7.1 lists half-lives of some com-
`monly used dnigs. Note that most commonly used drugs can be classified
`into four broad categories based on their half-lives. The group of drugs
`undergoing ultra-fast disposition (UFD) must be administered more fre-
`quently than other drugs in order to maintain a desirable plasma concentra-
`tion in the blood. In some instances a continuous intravenous infusion is
`
`needed to provide effective drug concentration due to these drugs’ rapid
`disposition. On the other hand, with the drugs which are slowly or very
`slowly eliminated from the body (SD and VSD), administration of a dose
`even once a day is often sufficient, since the fluctuation in the plasma con-
`centration during the dosing interval is not quite as large as that observed for
`UFDs or FDs.
`
`In clinical pharmacokinetic studies, the half-lives of drugs in the patients
`are individually determined rather than simply assuming the reported values
`in normal individuals and the dosage regimens are adjusted accordingly.
`
`
`
`Table 7.1. DRUG CLASSIFICATION BASED ON THE HALF-LIVES OF DISPOSITION
`
`
`
`
`
` DRUG HALFviJFE DRUG
`
`HAL F-UFE
`
`Uiira—Fasi Disposition rimEi 5 I how}
`Acetylsalicyhc acid2
`0.25
`Para‘aminosaricylic acid3
`0.90
`Amoxaciilin‘
`1.00
`Carbenicillin"
`1.00
`Cephalexin"
`1.00
`Cephalolhin‘
`0.50
`'Ch'ixacillinfl
`0.40
`Corlisone”
`0.50
`Dicioxacillin‘“
`0.90
`Furosemide”
`0.50
`Heiacillinl2
`0.30
`Insulin"
`0.10
`Melhicillin'”
`0.40
`Nafcillin”
`0.50
`Naiidixic acid“
`1.00
`Oxacillin"
`0.40
`
`‘
`
`
`
`Moderate Disposilion (I‘M, A to 8 hours)
`Amiloride‘"
`6.0
`Cl'nlortetracyc!inc“1
`5.5
`Lincomycin‘”
`2.5w1 1.5
`Sulfisoxazolem
`6.0
`Tetracycline“
`7.0—9.0
`Theophyllinei’
`4.0-7.0
`Toibuiamide“3
`5.0—9.0
`Trimelhoprimé‘
`0.0
`
`Siow Disposition (IM 8 to 24 hours]
`'
`Amphetamine“
`Anlipyrinei’"
`ChlordiazepOxide”
`Cycloserine“
`Dapsone“
`
`70—140
`70—350
`6.0—150
`3.0—1.5.0
`1?.0—210
`
`

`

`PHARMACOKINETIC PRINCIPLES
`
`147
`
`
`
`Table 7.1.
`DRUG
`
`{CONTINUED}
`HALF-LIFE
`
`DRUG
`
`HALF-LIFE
`
`UHra~FasI Disposition {cont}
`Penicillin G'“
`Pivampicitlin"
`Propylthi ouraoil ‘5
`
`0.?0
`0.10
`1.00
`
`Fast Disposition ff“ 1‘ to 4 hours)
`
`Acetaminophen “‘
`Alprenolol‘z”
`Amikacin’n
`Ampicillin’
`Bupivacaine”
`Cetazolin’
`Cueph.aIcnridiI-se1r
`Chloramphenicol 1‘
`Clindamioin“
`Colistimethale”
`Cyctophosphamide“
`Cytarabine”
`Disodium cromogiycate“
`Ethambulol”
`Gentamicin“
`Heparin”ll
`Hydroconisone”
`lndomethacin”
`Isoniazid"
`Kanamycin“
`Lidooalineas
`Meperidine“
`Methyltestosterone"
`Morphine“
`Phena'cerinl-
`Prednisolone"
`Procainamide‘"
`Propranolol"
`Fliiampioin‘”
`Salicylamide"
`Salécyfic acid“
`Streptomycin“
`Testosterone“
`Tobramycin“
`Warfari r1“i
`
`1.0-3.0
`2.0
`2.5
`1.0—1.5
`2. 5
`2.0
`1.5
`1.?—2.8
`3.0
`3.0
`3.0—6.0
`0.4—3.5
`1.0—1.8
`4.0
`2. 0
`0.7-2.5
`2.0
`2.0
`3.5
`2.0
`2.0
`3.0
`3. 5
`2.0
`ova—1.5
`3.5
`3.0
`4.5
`3.0
`1.2
`4.0
`2.5
`1 .8
`2.0
`2.0
`
`
`
`Slow Disposition (cont)
`Daunorut’tioin‘“I
`DemeolooyclinoIn
`Desi rnipran'li n2"
`121::mycycIine"l
`Giulethimide“
`Griseofulvin“
`
`Iodochlorhydroxyqui n“
`Lithium"
`Meprobamate
`Methaoycline"
`Methadone“
`Minocycline“
`Practolol '“
`Suifadiazi ne”
`
`12.0—27.0
`15.0
`140—25. 0
`12.0
`5.0—22.0
`9.0—220
`
`1 1 .0-140
`14.0—24.0
`15.0—16.0
`12.0
`15.0
`16.0
`12.0
`13.0~25.0
`
`Very 810w Disposition (In ) 24 hours)
`
`Acetylsulfisoxazole"
`Amobarbital“
`Apobarbital"
`Almpine“
`Barbital'2
`Carbamezapine"
`Chlorpromazine"
`Chlorpropamide"
`Diazepam"
`Di oumarol"
`Digitoxin"
`Digoxi n”
`Elhosuximide“
`Haloperidol“
`Methaquamne“
`I'Hiortriptyfineu
`Pentobarbital“
`Phenobarbital"
`Phenylbutazone“
`Sulfadimethoxine"
`Sulfamerazine"
`
`9.0420
`14.0—42.0
`12.0-36.0
`12.0—38.0
`60.0—78.0
`36.0
`30.0
`25.0—42.0
`55.0
`8.0—740
`200.0
`120—1320
`52.0
`13.0—35.0
`10.0—40.0
`18.0-35.0
`48.0
`48.0—1 20.0
`84.0
`40.0
`15.0—45.0
`
`
`
`

`

`148
`
`TEXTBOOK OF BIOPHARMACEUTICS AND CLINICAL PHARMACOKJNETICS
`
`The Elimination Rate Constant
`
`The first order rate constant of elimination, K, has units of reciprocal time
`and represents the fraction of drug removed per unit time. Thus if K = 0.1
`min", it means that 10 percent of the remaining amount is removed per
`minute. Note the term REMAINING AMOUNT in the definition. Since the drug
`is continuously removed from the body, the remaining amount is continu-
`oust changing and thus accurate calculation of the amount remaining in the
`body can only be made by using Equation 7.2. However, ifthe time interval
`for which the calculation is made is very small (about one-Seventh or one-
`eighth of the half-life) compared to the half-life of the drug, a direct calcu-
`lation can be made with little error without using Equation 7.2. For example,
`ifK is equal to 0.1 day’1 (half-life = 6.93 days) and the amount remaining in
`the body at the end of each day is calcalated using the three approaches
`described above, then there will be little error ifthe half-life is much longer
`than the duration of calculation. However, as the half-life decreases, signifi-
`cant errors can be introduced into the direct calculation of the amounts
`
`remaining in the body, based on the rate constant (Table 7.2).
`
`
`
`Table 7.2.
`
`COMPARISON OF ELIMINATION HATE CALCULATIONS
`
`. AMOUNT REMAINING
`IF 10% 0F INITIAL
`DOSE REMOVED PER
`DAY (my)
`100
`90
`30
`70
`60
`
`AMOUNT REMAINING
`IE 50% OF INITIAL
`DOSE REMOVED PER
`DAY (mg)
`
`TIME
`
`(day)
`
`hum—l0
`
`TIME
`(day)
`
`100
`50
`0
`_
`
`—
`
`boom—-
`
`AMOIINI REMAINING
`IF i0% OF REMAINING
`DOSE REMOVED PER
`DAY Ing
`100
`so
`81
`72.9
`55.5
`
`AMOUNT REMAINING
`IF 50% OF REMAINING
`DOSE REMOVED PER
`DAY (mg)
`
`100
`50
`25
`12.5
`6.25
`
`'
`
`x = x, e‘”
`(ng
`100
`90.43
`81.87
`74.08
`67.03
`
`x = x, e"“--"
`(mg)
`
`100
`60.65
`35.79
`22.31
`13.53
`
`Thus direct calculation of the-amount remaining as a function of time is
`possible for MDS, SDs, and VSDs on an hourly basis, or on a daily basis if
`the half-lives are six or seven times longer than the time interval chosen.
`The elimination rate constant represents the overall drug elimination from
`the body, which includes urinary excretion, biliary secretion, biotransi'or—
`mation, and all other mechanisms possible for the removal of the drug from
`
`

`

`PHARMACOKINETIC PRINCIPLES
`
`149
`
`the body. All of these individual processes are described by individual first
`order reaction constants and K is simply the sum of all of these rate constants:
`
`K =ke ‘l' kn + kn! + km + kg“ + ”'
`
`Where it? is the urinary excretion constant, in, and kb' are the biotransformation
`constants for two routes of structural modification, km is the biliary excretion
`constant, and km is the. fraction of drug removed in the lungs. These are
`apparent first order constants describing the apparent nature of the order of
`reaction. It is possible that as the drug concentration changes in the body
`these constants might also change, as in the saturation of enzymes reSponsible
`for the biotransformation of a drug to a specific product.
`The additive property of the rate constants is of great importance since it
`allows calculation of unknown rate constants and the total fractions of the
`
`drugs removed from the body by a specific route. For example, the total
`fraction of drug hiotranstbrmed in the body is given by:
`
`p,
`
`_ k, + k,’
`K
`
`(Eq. 7.13)
`
`where the numerator represents the sum of all rate constants representing
`the biotransfonnation of drug. One can similarly calculate the fraction of the
`available close excreted in the kidneys or eliminated through the bile.
`
`Example 7.2:
`The half-life of oxacillin is 0.5 hours and 30 percent of the available close is
`excreted unchanged in the urine; the rest undergoes biotransformation. What is
`the overall rate constant for biotransformation?
`
`Thus
`
`kt = K ‘ke
`
`= (0.693105) — in,
`
`kit
`E = 0.3
`
`k, = (0.69305) e 0.3{0.693l0.5)
`
`= 0.97 hr"1
`
`Example 7.3:
`In the example given above calculate the disposition half-life if the renal func-
`tion decreases by one-half.
`
`K = k, + it,
`
`= k, + 0.51:,
`
`= 0.97 + 0.5 x 0.3 x (0.69305)
`
`= 1.178
`
`m = 0.6932r 1.178 = 0.59 hours
`
`

`

`150
`
`TEXTBOOK OF BIOPHARMACEUTICS AND CLINICAL PHARMACOKINETICS
`
`The calculation above can be repeated with a 50 percent reduction in the
`biotransformation function to yield a half-life of 0.77 hours, a much greater
`increase since biotransfimnation is a much more important factor for this drug
`than renal excretion.
`
`The Volume of Distribution
`
`As discussed earlier, the volume of distribution is given by:
`
`v
`
`gfi
`Co
`
`quM)
`
`This is simply a proportionality constant between the amount of drug in the
`body and the plasma concentration, but when the drug distributes to specific
`fluids of the body, as occurs with Evans blue dye (which remains restricted
`to the blood compartment), the volume of distribution represents that pool of
`fluid. If an instantaneous equilibration is reached, then Equation 7.14 de—
`scribes the volumes of distribution throughout the course of drug disposition.
`But when tissues which might equilibrate slowly are involved the volume of
`distribution changes with time.‘ This concept will be discussed later.
`An important application of the volume of distribution comes in the cal—
`culation of clearance of the drug from the body. The clearance, Q, is defined
`as:
`
`Q=KV
`
`maiw)
`
`The units for clearance are volume/time and represent the part of the volume
`of distribution which is cleared of the drug per unit of time. The total body
`clearance is the sum of individual clearances:
`
`KV=krV+ka+kb’V+...
`
`(Eq. 7.16)
`
`Thus each of the component clearance values represents the fraction of
`volume of distribution cleared by the given elimination mechanism of excre-
`tion, biotransformation, biliary excretion, etc. The concept of clearance can
`be easily understood from the examples of the excretion of inulin and peni—
`cillin in the kidneys. Inulin distributes only in the blood and in each cycle
`through the kidneys about 125 ml of plasma is filtered per minute as ultra-
`tiltl'ate containing inulin. If all of the drug contained in the filtered volume
`is removed, the renal clearance is equal to the volume of filtrate. In the case
`of penicillin almost 100 percent of the total'clrug circulating through the
`kidneys is removed in each cycle and thus the renal clearance of penicillin
`is equal to the total blood flow rate.
`
`Example 7.4:
`Calculate the half-lite ofa drug ii'it is completely removed in each cycle through
`the kidneys and this represents the only route of drug elimination. The volume
`of distribution is 50 liters.
`
`

`

`PHARMACOKINETIC PRINCIPLES
`
`151
`
`V lg. = O = K V = 650 mllmin
`
`K = 65050000 = 0.013 min“1
`
`I.” = 069310.013 = 53.3 min
`
`Example 7.5:
`What is the shortest halt-life possible for a drug if it is cornpietely removed in
`each cycle through the kidneys and this represents the only mechanism of
`elimination?
`
`The smallest volume of distribution is about 3 liters, or the plasma volume:
`
`K = 650.8,000 = 0.22 min"
`
`t” = 0.693l0.22 = 3.15 min
`
`Example 7.6:
`What is the half-life ofa drug if its excretion ratio is 0.45, and 50 percent of the
`blood flowing through the liver is cleared of the drug in each cycle? The volume
`of distribution is 50 liters and the normal hepatic flow rate is 1.25 literslmin.
`
`V k, = Excretion ratio >< inulin clearance
`
`= 0.45 X 125 = 56.25 mlhnin
`
`k? = 562550.000 = 1.125 X 10—5 min"l
`
`V kb = 0.5 x hepatic flow rate
`
`= 0.5 X 1.25 = 625 mlfrnin
`
`k5 = 625J'50,000 = 1.25 X 10-3 min"
`
`K = k, + k, = 0.01363 min"1
`
`t.” = 0593;001:563 = 50.9 mill
`
`It is quite obvious that these calculations are required for dosage calculations
`in the event of kidney or liver malfunction, when the half-life in the patient
`can be calculated based on the component clearance values. However, in
`calculating the half-life in disease states the clearance component due to
`biotransforlnation in the blood and excretion in the breath, saliva, lungs, and
`other locations should also be considered if it contributes significantly to the
`Overall disposition of the drug.
`The discussion presented above is applied to the pharmacokinetic analysis
`following intravenous administration, when the plasma concentration is mon-
`itored. However, other concentrations (such as urinary levels) can also be
`monitored for similar pharmacokinetic analyses.
`
`Urinary Excretion Rate Data
`
`One advantage in using urinary excretion data to analyze a pharmacokinetic
`system is the noninvasive nature ofsuch data. It is much more convenient to
`
`

`

`152
`
`TEXTBOOK 0F BIOPHARMACEUTICS AND CLINICAL PHARMACOKINETICS
`
`collect a urine sample than to draw the blood periodically. Another advantage
`in using urinary excretion data is that this allows direct measurements of
`bioavailability, both absolute and relative, without the necessity of fitting the
`data to a mathematical model.
`
`The rate of urinary drug excretion is proportional to the amount of drug in
`the body:
`
`
`dX V X
`dt
`0C
`
`E . 7.17
`( q
`
`l
`
`The proportionality constant is simply the rate constant for the renal excretion
`of drug:
`
`qu
`E—=k¢,X :ng0640
`
`(Eq.
`
`= hive“: 'K'=Q,.Cae"“
`
`(Eq.7.19)
`
`The urinary excretion rate is therefore the product of renal clearance and the
`plasma concentration with units of amountftime. It is experimentally deter-
`mined by collecting the urine during a specified time interval and analyzing
`the drug concentration in an aliquot of the collected volume:
`
`an A (urinary concentration) (urine volume)
`cit
`(time interval for collection)
`
`(Eq 7 20)
`
`For example, if after an intravenous dose of 100 mg of a drug the volume of
`urine collected during the first three hours is 2.50 ml and the concentration
`of chug in this volume is 20 ug/ml, the urinary excretion rate is:
`
`20 X 250
`
`3
`
`= 1655.67 jig/hr
`
`(Eq. 7.20A)
`
`Analogous to the treatment of plasma level data, the urinary excretion rate
`data can be used to calculate the half—life and excretion rate constant of the
`
`drug. Equation 7.18 can be converted to:
`
`log (dXHIdt) = log kEX‘, — [Ct/2.303
`
`(Eq. 7.21)
`
`Therefore, a plot of the log of the excretion rate of unchanged drug in the
`urine against time (the midpoint time of the duration of urine collection) will
`yield a straight line (Fig. 7.3) from which the following information can he
`obtained:
`'
`
`l. Half—life of the drug is simply the time required for the excretion rate to
`decrease to one-half on the descending part of the plot.
`2. The elimination rate constant, K, is determined by: 0.693/to,5
`3. The urinary excretion rate constant is determined by: intercept on Y-axis!
`X0
`
`

`

`inie rcept = he X0
`
`
`
`PHARMACOKINETIC PRINCIPLES
`
`153
`
`Figure 7.3. Calculation of pharmacokinetic
`parameters from urinary excretion rate data.
`See text for details of the data points.
`
` '42-22 :3": '
`
`
`I
`
`il I
`
`I1 l
`
`7
`6
`4 5
`3
`2
`Time {hours}. Midpoint
`
`8
`
`
`
`
`
` l
`
`
`
`
`
`UrinaryExcretionRate{pg/hr}
`
`4. The extrarenal excretion rate constant (mainly the biotransformation) is
`obtained by: K — kg
`5. If the volume of distribution is known—as may be obtained from blood
`level data—the total body clearance, renal clearance, extrarena] clearance,
`and the corresponding plasma concentration for a given urinary excretion
`rate can be easily obtained.
`
`Another approach to handling urinary excretion data is by integrating Eq.
`7.18 (Appendix C) to give the total urinary excretion as a Function of time:
`
`
`Xu
`
`= k?" {1 — 9—1“)
`
`‘
`
`(Eq. 7.22)
`
`Figure 7.4 shows a characteristic plot for Equation 7.22. Note that as time
`approaches infinity (for our purpose, 6 to 7 half-lives):
`
`X.,°° = %Xn
`
`(Eq. 7.23)
`
`k9
`u _ _ _ _ _ _ _ _ _ _.
`
`intercepFT x0
`
`X M
`
`._
`
`.4.
`
`'
`
`Figure 7
`
`'
`
`r‘n
`
`r
`
`'
`
`.
`
`Cumulative u I ary exc etion
`
` Xu
`
`

`

`154
`
`TEXTBOOK OF BIOPHARMACEUTICS AND CLINICAL PHAHMACOKINETICS
`
`If the drug is eliminated only by renal excretion. K will be equal to in. and
`X,,°° will become equal to the dose administered (intravenously). X...
`
`Example 7.7:
`The following urinary excretion data are obtained ibllowing intravenous admin-
`istration ol‘a 100 mg (lose:
`
`Time
`(hr)
`
`0— 1
`i - 3
`3- 5
`5‘9
`
`Urine Voiume
`(mi)
`
`Drug Concentration
`(Mimi)
`
`1.85
`200
`2.86
`1 50
`0. 70
`300
`
`700 0.20
`
`If the volume of distribution is 100 liters, calculate the half-life, K; k2, renal
`clearance; the halfwlife in case 0? anuria, or in the case of 50 percent renal
`function; plasma concentration at! —- 3; and percent lJiotranslormation in normal
`subjects.
`
`Midpoint (hr)
`0.5
`2.0
`4.0
`7.0
`
`Excretion Rate (,uginr)
`370
`214.5
`1 05
`35
`
`From a plot of the excretion rate against time on a send-logarithmic scale, the
`half-life is calculated to be armmd two hours:
`
`K = 0.693i2 = 0.347 hr“
`
`kg = interceptan = (450 lugihrMUOO t‘l]g>< 1000 ugimg)
`= 0.0045 hr"
`Qe = keV = 0.0045 x 100 = 0.45 litersihr
`k9 = K — ice = 0.347 - 0.0045 = 0.3425 hr“
`In anuria, ice = 0: it, 3 K; t“, = 0.693i0.3425 = 2.02 hr
`In 50% renal function: K = 0.51:0 + k3. = 0.34475 hr“
`tog. = 0693034475 = 2.01 hr
`Plasma concentration att = 3: Cu = XJV = l ugiml
`C = Cue—m = l e"“‘3‘7"3 = 0.353 agiml
`Percent biotransformation = (kbiK) 100 = 98.7%
`
`First Order Input Data
`
`The discussion presented above pertains to instantaneous input or intrave-
`nous administration, wliich shows an immediate distribution in the body.
`However, a large number of drugs are administered orally or by other routes
`from which the input or absorption is not instantaneous and instead follows
`a first order input:
`
`

`

`PHARMACOKINETIC PRINCIPLES
`
`155
`
`dx
`5 = kaXa -H KX
`
`(Eq. 7.24)
`
`where k“ is the absorption rate constant and Xa is the amount of drug re
`maining at the site of absorption, such as the stomach or intestine. Equation
`7.24 can be integrated (Appendix C) to yield:
`
`or
`
`X = g%(e-M — e-w)
`
`—m
`kflFX"
`C = Wfi? — t?
`
`—s
`
`“0
`
`(Eq. 7.25)
`
`(Eq.
`
`where F is the fraction of the administered dose, X“, which is absorbed
`following administration by the oral or other routes.
`In most instances the absorption rate constant, kg, is larger than the elim-
`ination rate constant, K, and the plasma concentration profiles (such as those
`shown in Figure 3.7) are obtained. The peak plasma concentration represents
`the time at which the absorption rate becomes equal to the rate of elimination:
`
`kaXfl = KX
`
`(Eq. 7.27)
`
`As the amount of drug remaining at the site of absorption decreases, the rate
`of absorption also decreases until:
`
`aw ~ 0
`
`_
`
`(Eq. 7.28)
`
`and the plasma concentration is described only by the elimination constant
`of the drug:
`
`knFXu
`C =(k,—i<)V" "m
`
`(Eq. 7.29)
`
`This is referred to as the post-absorptive phase, in which the half-life of the
`drug can be easily determined from the slope of the plasma concentration : time
`profile (Fig. 7.5). The absorption rate constant, kc, is determined by a tech-
`nique which is appropriately termed the METHOD OF RESIDUALS. This involves
`the subtraction of two rate processes,
`in this case the subtraction of the
`elimination rate process from the overall disposition of the drug, to yield the
`rate of absorption. Recalling Equation 7.26:
`
`C = A3 "‘7 — Ae‘ks‘
`
`(Eq. 7.30)
`
`where
`
`lanXa
`A =(k,—K)V
`
`(Eq. 7.31)
`
`

`

`156
`
`TEXTBODK OF BIOPHAHMACEUTICS AND CLINICAL PHARMACOKINETICS
`
`7.5. Semilogarithmic
`Figure
`of
`plot
`plasma concentration against
`time. Note
`that
`it
`is composed of
`two exponential
`terms; one describing the absorption and
`the other the elimination. During the post-
`absorptive phase the ptasma concentration
`is described by a single exponent.
`
`
`
`Post Absorptive
`Phase
`
`
`
`a
`he
`g
`:
`g
`
`8 5
`
`0
`
`U E 3a
`
`Time
`
`Since the terminal portion of the plasma concentration (log scale):time plot
`represents only the elimination phase, or Equation 7.29, it can be extrapolated
`back to time zero on the plasma concentration axis and the absorption rate
`constant can be determined by the following subtraction:
`
`Ae 4““ = A8 ‘K’ — C
`
`(Eq. 7.32)
`
`
`
`Table 7.3.
`
`1,, VALUES OF SOME COMMONLY USED DRUGS IN HUMANS
`
`DRUG
`
`1,,
`
`DRUG
`
`Acetaminophen
`Acetazolamide
`Acetohexamide
`Amitriptyline
`5-Azacytidine
`Cytosine
`Cromolyn sodium
`Chlorothiazide
`Chlorodiazepoxide
`Chlorproparnide
`Chlcrphenssin carbamate
`Clofibrate
`Chlorpromazine
`Diazepam
`Ethchlorvinyl
`Flcxuridine
`Furosemide
`Glutethimide
`Hexamethyiamine
`
`10 min to 1 hr
`2 to 8 hr
`1.5 to 2 hr
`2 to 4 hr
`30 min
`510 15 min
`15 min
`1.5 to 2.5 hr
`2 hr
`2 to 4 hr
`2 hr
`5 to 12 hr
`3 to 4 hr
`1 hr
`1 hr
`15 to 20 min
`1 to 2 hr
`2.2 hr
`2 to 3 hr
`
`Hydroxyurea
`Lidocaine
`Lithium carbonate
`Methadone
`Morphine
`B-Mercaplopurine
`Methaqualone
`Nortriptytine
`Pentazocine
`Propoxyphene
`Practolol
`Procainarnide
`Propranolol
`Procarbazine
`F’hentormin
`Protryptyline
`Ouinidine
`Salicyiate
`Wariarin
`
`t,
`
`0.5 to 2 hr
`45 to 60 min
`1.33 hr
`4 hr
`45 min
`2 hr
`2 hr
`5.5 hr
`2 hr
`2 hr
`2 to 4 hr
`1 hr
`2 hr
`1 hr
`2 to 4 hr
`24 to 30 hr
`0.5 to 4.5 hr
`2 hr
`2 to 9 hr
`
`

`

`PHARMACOKINETIC PRINCIPLES
`
`157
`
`Equation 7.32 describes the residual line on the plot from which the absorp—
`tion half-life can be calculated simply by reading the time required for a 50
`percent decrease in the values on the plot. To obtain good estimates, at least
`three arbitrary points should be marked on the plasma concentration plot and
`the corresponding values on the extrapolated elimination phase line read
`from the graph. These values are then subtracted and the residual values are
`plotted at the same time values to obtain the line described by Equation 7.32
`(see also Fig. 7.5). Note that the intercept for both lines on the graph is
`always the same and from this the volume of distribution can be calculated
`if the bioavailability is knewn:
`
`V =
`
`knFXo
`
`(kn H K) (intercept)
`
`(Eq. 7.33)
`
`The time at which the peak plasma concentration occurs can be determined
`easily since at this time:
`
`or
`
`kflxa = xx
`
`dC
`RT _ 0
`
`wax
`
`a Wes.» _ Effie-m»
`
`k
`
`X
`
`(Eq. 7.34)
`
`(Eq. 7.35)
`
`(Eq. 7.36)
`
`where t,, = time at which C = Cm“, peak concentration. Equation 7.36
`yields:
`
`ts:
`
`
`2.303
`kn
`(k _ K) log?
`
`(Eq. 7.37)
`
`C...ax can he obtained simply by substituting the value of t, in Equation 7.26.
`Table 7.3 reports the t, values of some commonly used drugs.
`It should be noted that as it“ becomes larger than K, t, becomes smaller
`since the term (it, — K) increases much faster than log (k afK) in Equation
`7.37. In some instances where the absorption is extremely fast, it is possible
`to’ miss the peak entirely and the plasma concentration profile resembles that
`obtained after intravenous or instantaneous input.
`In many instances it is possible to calculate the absorption half-life (0.693!
`kg) from the t, and the elimination half-life:
`
`t” (absorption) =
`
`0.693 e "‘39"
`K 8—!“
`
`(Eq. 7.38)
`
`It should be noted that Equation 7.38 is mathematically insoluble. How-
`ever, with the aid of an iterating computer program, using a calculator, it can
`
`
`
`

`

`158
`
`TEXTBOOK OF BIOPHARMACEUTICS AND CLINICAL PHARMACOKINETICS
`
`be easily solved for a given elimination half-life and a time for peak concen-
`tration.
`
`The discussion presented above applies only to those situations where the
`absorption rate constant is larger than the elimination rate constant. However,
`in some situations these two constants may be of comparable dimensions, as
`with drugs listed as undergoing very fast or fast disposition (Table 7.1). The
`plasma concentration under these conditions is described as:3