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The two-compartment open-system kinetic model: A review of its clinical implications and applications
May 1976 The Journal o f P E D I A T R I C S
869
The two-compartment open-system kinetic model." A review of its clinical implications and applications The purpose of th& communication is to present an outline of the principles and methods of twocompartment kinetic analysis. Particular emphasis is given to the therapeutic significance of these concepts and to the magnitude of errors which can result from inappropriate use of simpler kinetic models. This outline shouM assist in the interpretation of the data reported in the following paper describing the pharmacokinetics of theophylline in young children.
P. M. Loughnan, M.B.B.S., M.R.A.C.P.,* D. S. Sitar, Ph.D., R. I. Ogilvie, M.D., F.R.C.P.(C), and A. H. Neims, M.D., Ph.D., Montreal, P.Q., Canada
IN RECENT YEARS detailed studies of drug disposition after intravenous administration have shown that the simple one-compartment model is often inadequate to describe observed plasma concentration versus time relationships. 1 Many drugs behave as though the body consisted of two separate compartments-central and peripheral. On occasion, drug disposition is more complex and three or more compartments must be postulated to explain observed data. Indeed, distinctive aspects o f the pharmacodynamics of certain drugs are predicted by such complex kinetic considerations. Examples include the delayed onset of drug action, persistence of effect long after cessation of administration, and varying duration and intensity of effect following repeated doses. ~ Increasingly, such pharmacokinetic data are being reported in the adult and pediatric literature and contribute to the design of more rational therapeutic regimens. :~-~ Application requires not only the computation of various kinetic factors, but also a demonstration of From the Roche Developmental Pharmacology Unit, Department o f Pharmacology and Therapeutics, McGill University," and the Division of Clinical Pharmacology, Montreal General Hospital. *Holder of a Fellowshipin Clinical Sciences (Clinical Pharmacology), National Health & Medical Research Council of Australia. Reprint address: Dr P. M. Loughnan, Department of Pharmacology & Therapeutics, Mclntyre Medical Sciences Bldg., Room 717, 3655 Drummond St., Montreal, Quebec H3G 1 Y6, Canada.
correlation between therapeutic effect and the concentration (or amount) of drug in one or the other compartment. TWO-COMPARTMENT KINETIC M O D E L
OPEN-SYSTEM
This model was initially described by TeorelP and has since been modified and extended by Riegelman and associates' and many others. It was proposed to explain the observation that log10 plasma concentration versus time curves for many drugs after intravenous administration are not linear from time zero, but conform to the curve depicted in Fig. 1. The graph shows an initial rapid
See related article, p. 874. decline in plasma concentration of drug and subsequently a slower decline. The terminal linear portion of this curve is termed the fi-phase of drug disposition. A first order rate constant, fi, can be defined and equals the slope of this line multiplied by -2.303. The plasma half-life (tV2) of a drug can be derived from fi: 0.693 --, B The fi-phase represents the net effect not only of drug elimination but also of movement of drug into and out of various tissues. The same is true of the earlier ral~id decline of plasma concentration, termed the a-phase; at these early times, however, the process of distribution Plasma t89 =
VoL 88, No. 5, pp. 869-873
870
Loughnan et al.
25- .t 20-
The Journal of Pediatrics May 1976
The disposition of a drug whose disappearance from plasma exhibits these characteristics can be interpreted as though the body consisted of two separate compartments, central and peripheral, as seen in Fig. 2. The first order rate constants--kl~, k~l, and ke~ apply to the processes depicted in Fig. 2. Each constant contributes to the value of a and fi defined above; for this reason a and fi are often referred to as hybrid disposition rate constants? In a first order process of elimination, the rate at which a drug leaves a compartment at any given time is directly proportional to the amount of drug in that compartment. This is reflected by an exponential decline with time (t) in the amount of drug, X, or its concentration, in that compartment (Fig. 3):
Cp o
A1_~5 -
Z ul U Z O
O
1
~ P H A S
E
_1
X = X~ k,
0
2
TIME
4
6
Fig. 1. Semilogarithmic plot of plasma drug concentration versus time with arbitrary units and hypothetical points to illustrate the graphical method of analyzing two-compartment kinetic data (see text).
where k is a first order rate constant and X ~ represents the amount of drug in the compartment at time zero. In the familiar one-compartment model, where there is no kinetically detectable peripheral compartment, the plasma concentration of drug, Cp, behaves similarly: Cp = C~ e -k~.
ABSORPTION i i i i
V CENTRAL
k12
COMPARTMENT
PERIPHERAL COMPARTMENT
k21 11
kel
METABOLISM & EXCRETION Fig. 2. The two-compartment open-system pharmacokinetic model. predominates. This phase, and its rate constant, a, is characterized by the derived line with steep slope shown in Fig. 1. This line is computed by subtracting from the observed data points the corresponding points on a linear extrapolation of the B-phase to time zero. These residuals are plotted on the same graph and a line of best fit drawn. The slope of this line multiplied by -2.303 defines the disposition rate constant, a. Linear extrapolation of this line to time zero yields the intercept, A, and similar extrapolation of the B-phase yields the intercept, B. The extrapolated plasma concentration at time zero, C~,, equals the sum of A and B.
In the two-compartment open-system, these first-order characteristics apply not only to the process of drug elimination from the central compartment, but also to the movement of drug out of central to peripheral compartment and vice versa. The following assumptions are made in the application of a two-compartment open-system kinetic model. 1. An initial dose of drug is delivered into the central compartment and is distributed instantaneously through this compartment. 2. Drug is eliminated only from the central compartment in a process characterized by the first order rate constant kel. 3. Drug passes into and out of the peripheral compartment in accord with processes characterized by the first order rate constants k,~ and k2,, respectively. 4. The volume of the central compartment and the rate constants do not vary with time. The central and peripheral compartments and their respective volumes are derived units and are not meant to correspond to any physiologic or anatomic area. Nevertheless, the concept of volume of distribution is essential to the Understanding and application of pharmacokinetics.~ If equal doses of different drugs are given intravenously, it is well known that the initial plasma concentrations achieved vary widely. This is not surprising since many factors such as molecular weight, lipid solubility, ionization, and plasma or tissue protein binding will influence the observed total plasma concen-
Volume 88 Number 5
Two-compartment open-system kinetic model
87 1
40-
30-
1-3 -
~.1o
o.Oa ~
i
i
i
TIME
i
i
i
TIME
Fig. 3. Linear (left) and semilogarithmic (right) plots of the amount of drug (X) in a compartment as a function of time. Units are arbitrary.
tration. The volume of the central compartment, V1, is defined as: dose g I -
C~ P
That is, V1 is the volume into which the administered dose would appear to distribute instantaneously in order to yield the extrapolated zero-time concentration (A + B). Hence, an observed concentration of drug, Cp, in the central compartment at any time, taken with V1, allows calculation of the amount of drug (X,) in the central compartment at that time, such that: X~ = V,C~,. The amount of drug eliminated per unit time, Q, is equal to ko~ multiplied by the amount of drug present. Since Q = kolX,, it follows that Q
distribute, if the initial instantaneous distribution had occurred into a larger, single compartment. Exclusive use of this volume term has serious limitations and might greatly underestimate "early" plasma concentrations after an intravenous infusion of drug. It is apparent that the actual plasma concentration at any time in the two-compartment model (Fig. 1) is the sum of two exponential processes, such that Cp = Ae -~t + Be fit With use of the terms defined above, algebraic manipulations allow derivation of the following relationships. 1 V~
dose A+B
a + f i = k,~ + k2, + k~, aft = ko,k21
= k e l V 1 C P.
kel = The plasma clearance of a drug, like the renal clearance of a substance, is an important determination which, in the context of this model, remains constant with time and can be shown to be equal to ke,V,. It is important to emphasize that the volume of the central compartment is simply a useful derived constant. This volume is likely to represent those areas into which a drug can distribute rapidly such as plasma and parts of highly perfused organs. Nevertheless, attempts to relate the volume terms to specific physiologic spaces such as extracellular fluid volume or total body water are inappropriate and potentially misleading. Another volume term, the apparent volume of distribution, AVD, is defined as dose/B where B is the intercept of the extrapolation of the fi-phase to time zero. It therefore represents the volume into which the drug would have appeared to
C~ A+B'
B Therefore, computation of the intercepts A and B, and the "hybrid" disposition rate constants a and fi enable all features of the two-compartment model to be derived. It should be noted, however, that the graphic method described is less accurate than nonlinear least squares curve fitting programs to derive the individual first order rate constants and V1. Certain aspects of the clinical applicability of this model deserve attention. The calculation of a maintenance dose rate from the above data is straightforward. It was shown above that for a given plasma concentration, Cp, the amount of drug eliminated from the system per unit time is Q = ko~'V,-Cp.
872
Loughnan et al.
The Journal of Pediatrics May 1976
maxo;'
100
Cp
/ Load= ~
9
s,ow,.)
b
PERIPHERAL
50-
u,i 0 a 20
i
0 I,z u,.I ~0
mir
Cp
w 0.
C
CENTRAL 10-
d
Fig. 4. Plasma concentration versus time curves predicted with use of an identical maintenance infusion rate after four different circumstances of drug loading (see text).
s~
,
;
,
~
,
TIME
Hence, if the amount eliminated per unit time, Q, is replaced by continuous infusion, then a plateau plasma concentration (C~) will be maintained. Q is thus the maintenance dose rate which will maintain C~. For many drugs, initiation of a continuous infusion would be expected to generate a plasma concentration of 97% of plateau value in approximately five B-half lives 7 (Fig. 4, d). One method proposed to rapidly achieve and maintain a given plasma concentration, Cp, is to administer a loading dose, R, where': R=
Q
ke~ ' V~ 9 C v
- = B B When this dose is given very rapidly, initial p!asma concentrations are as shown in Fig. 4, a. It is readily calculated that when this is done, the initial plasma concentratiofi is ~=
koj /~ " c ; ~
Since ko~ is always greater than B, this method of loading results in a C ~ that exceeds C~. The potential therapeutic or toxic effect produced by these high early plasma concentrations will depend upon whether such an effect is related to the drug concentration in the central or the peripheral compartment. In practice, it may be preferable to avoid these high plasma concentrations by administering the !oading dose more slowly (Fig. 4, b). Wagneff has described a method for calculation of an appropriate infusion rate of the loading dose, and has applied the
Fig. 5. Percent of dose in the central and peripheral compartments as a function of time. Units are arbitrary. method to theophylline in simulations. An alternative suggestion ~~ involves a loading dose, R = Q/kol (Fig. 4, e). This method avoids early high plasma concentrations but results in a subsequent decrease in plasma concentrations followed by a gradual increase to C~.8 The time course of this decrease relates to the half-life of the drug; in the case of lidocaine, plasma concentrations decreased to about half the initial value at 20 minutes and had risen to 90% of this value by four hours? Optimal suppression of cardiac arrhythmias might therefore not be obtained with use of this regimen. Finally, as alluded to above, with drugs whose disposition is best interpreted with use of the two-compartment model, one or another action of the drug might correlate better with concentration of drug in the central or in the peripheral compartment. Knowledge of the volume terms and relevant rate constants enables the calculation at any time of tlae amount of drug in the peripheral compartment (Fig. 5). It can also be shown that, throughout the Bphase, the amount of drug in the central compartment as a fraction oi" drug still in the body remains constant and equals B/ko, These relationships are important in the interpretation of plasma concentration data. It is not uncommon to have available in the literature plasma concentrations measured only during the B-phase. The use of these data to calculate maintenance dose rate
Volume 88 Number 5
Two-compartment open-system kinetic model
therefore necessitates the assumption of a one-compartment model: Q = fi.AVt~.Cp. Unless the logarithm of plasma concentration declines with time in a perfectly linear fashion from time zero, an error must be introduced. To evaluate the magnitude of this error, consider the equation:
k~
C~ A+B
fi
I f A l a is much smaller than Blfi, then o Cp
kel
kel *
dose o Cp
873
frequent plasma concentration measurements required for complex multicompartment analyses. It then becomes important to consider the magnitude of errors introduced by. using simpler models. Such errors cannot be ignored, but may be clinically acceptable depending upon the characteristics of the drug involved and the particular aims of the study. In conclusion, all kinetic models, no matter how sophisticated, contain many assumptions which may be more or less valid. Therefore, any predictions made from single dose kinetics should be tested in an appropriate clinical study before acceptance for general usage. Furthermore, such kinetic predictions are applicable only to a similar patient population with respect to age, condition, and disease.
B/fi dose rot.
- -
B
ke~ "V, ~ fi'AVD. Therefore, it is only when A / a is much less than B/fl, that plasma clearance and maintenance dose rate calculated using a one- or two-compartment analysis will be approximately equal? This outline of two compartment kinetics is necessarily limited and can be found in more detail elsewhere? ....... 3 However, it is designed to stress three important uses of a two-compartment analysis: (1) If an inappropriate model (e.g., one-compartment) is selected tO analyze data for a drug in which A / a cannot be ignored, then grossly erroneous loading and maintenance doses would be developed. (2) To define plasma concentrations shortly after rapid intravenous administration, a two-compartment analysis is essential, and is required .to determine appropriate infusion rates of intravenous loading doses) (3) Factors other than elimination by metabolism and/or excretion may have a major effect on drug disposition. For example, the rate of release of the drug from peripheral compartment may in itself largely determine the fi-phase half-life. 12 This factor cannot be ignored in designing dosage regimens in situations of impaired drug elimination such as renal failure, especially for those drugs which exhibit pronounced "two-compartment characteristics." These considerations will be of importance not only in the design of dosage regimens in different clinical situations, but also in defining the particular variables responsible for changing dosage requirements with age. In pediatric studies, ethical considerations may limit the
REFERENCES
1. Riegelman S, Loo JC, and Rowland M: Shortcomings in pharmacokinetic analysis by conceiving the body to exhibit properties of a single compartment, J Pharm Sci 57:117, 1968. 2. Gibaldi M, Levy G, and Weintraub H: Drug distribution and pharmacologic effects, Clin Pharmacol Ther 12:734, 1971. 3. Mitenko PA, and Ogilvie RI: Pharmacokinetics of intravenous theophylline, Clin Pharmacol Ther 14:509, 1973. 4. Koch-Weser J, and Klein SW: Procainamide dosage schedules, plasma concentrations, and clinical effects, JAMA 215:1454, 1971. 5. Teorell T: Kinetics of distribution of substances administered to the body. L The extravascular modes of administration, Arch Int Pharmacodyn 57:205, 1937. 6. BenetLZ, and Ronfeld RA: Volume terms in pharmacokinetics, J Pharm Sci 58:639, 1969. 7. Goldstein A, Aronow L, and Kalman SM: Principles of drug action, ed 2, New York, 1974, John Wiley & Sons, Inc, pp 311-318. 8. Mitenko PA, and Ogilvie RI: Rapidly achieved plasma concentration plateaus, with observations on theophylline kinetics, Clin Pharmacol Ther 13:329, 1972. 9. Wagner JG: A safe method for rapidly achieving plasma concentration plateaus, Clin Pharmacol Ther 16:691, 1974. 10. Boyes RN, Scott DB, Jebson PJ, Godman MJ, and Julian DG: Pharmacokinetics of lidocaine in man, Clin Pharmacol Ther 12:105, 1971. 11. Reigelman S, Loo J, and Rowland M: Concept of a volume of distribution and possible errors in calculation of this parameter, J Pharm Sci 57:128, 1968. 12. Jusko WJ, and Gibaldi M: Effects of change in elimination on various parameters of the two compartment open model, J Pharm Sci 61:1270, 1972. 13. Levy G, and Gibaldi M: Pharmacokinetics of drug action, Ann Rev Pharmacol 12:85, 1972.
869
The two-compartment open-system kinetic model." A review of its clinical implications and applications The purpose of th& communication is to present an outline of the principles and methods of twocompartment kinetic analysis. Particular emphasis is given to the therapeutic significance of these concepts and to the magnitude of errors which can result from inappropriate use of simpler kinetic models. This outline shouM assist in the interpretation of the data reported in the following paper describing the pharmacokinetics of theophylline in young children.
P. M. Loughnan, M.B.B.S., M.R.A.C.P.,* D. S. Sitar, Ph.D., R. I. Ogilvie, M.D., F.R.C.P.(C), and A. H. Neims, M.D., Ph.D., Montreal, P.Q., Canada
IN RECENT YEARS detailed studies of drug disposition after intravenous administration have shown that the simple one-compartment model is often inadequate to describe observed plasma concentration versus time relationships. 1 Many drugs behave as though the body consisted of two separate compartments-central and peripheral. On occasion, drug disposition is more complex and three or more compartments must be postulated to explain observed data. Indeed, distinctive aspects o f the pharmacodynamics of certain drugs are predicted by such complex kinetic considerations. Examples include the delayed onset of drug action, persistence of effect long after cessation of administration, and varying duration and intensity of effect following repeated doses. ~ Increasingly, such pharmacokinetic data are being reported in the adult and pediatric literature and contribute to the design of more rational therapeutic regimens. :~-~ Application requires not only the computation of various kinetic factors, but also a demonstration of From the Roche Developmental Pharmacology Unit, Department o f Pharmacology and Therapeutics, McGill University," and the Division of Clinical Pharmacology, Montreal General Hospital. *Holder of a Fellowshipin Clinical Sciences (Clinical Pharmacology), National Health & Medical Research Council of Australia. Reprint address: Dr P. M. Loughnan, Department of Pharmacology & Therapeutics, Mclntyre Medical Sciences Bldg., Room 717, 3655 Drummond St., Montreal, Quebec H3G 1 Y6, Canada.
correlation between therapeutic effect and the concentration (or amount) of drug in one or the other compartment. TWO-COMPARTMENT KINETIC M O D E L
OPEN-SYSTEM
This model was initially described by TeorelP and has since been modified and extended by Riegelman and associates' and many others. It was proposed to explain the observation that log10 plasma concentration versus time curves for many drugs after intravenous administration are not linear from time zero, but conform to the curve depicted in Fig. 1. The graph shows an initial rapid
See related article, p. 874. decline in plasma concentration of drug and subsequently a slower decline. The terminal linear portion of this curve is termed the fi-phase of drug disposition. A first order rate constant, fi, can be defined and equals the slope of this line multiplied by -2.303. The plasma half-life (tV2) of a drug can be derived from fi: 0.693 --, B The fi-phase represents the net effect not only of drug elimination but also of movement of drug into and out of various tissues. The same is true of the earlier ral~id decline of plasma concentration, termed the a-phase; at these early times, however, the process of distribution Plasma t89 =
VoL 88, No. 5, pp. 869-873
870
Loughnan et al.
25- .t 20-
The Journal of Pediatrics May 1976
The disposition of a drug whose disappearance from plasma exhibits these characteristics can be interpreted as though the body consisted of two separate compartments, central and peripheral, as seen in Fig. 2. The first order rate constants--kl~, k~l, and ke~ apply to the processes depicted in Fig. 2. Each constant contributes to the value of a and fi defined above; for this reason a and fi are often referred to as hybrid disposition rate constants? In a first order process of elimination, the rate at which a drug leaves a compartment at any given time is directly proportional to the amount of drug in that compartment. This is reflected by an exponential decline with time (t) in the amount of drug, X, or its concentration, in that compartment (Fig. 3):
Cp o
A1_~5 -
Z ul U Z O
O
1
~ P H A S
E
_1
X = X~ k,
0
2
TIME
4
6
Fig. 1. Semilogarithmic plot of plasma drug concentration versus time with arbitrary units and hypothetical points to illustrate the graphical method of analyzing two-compartment kinetic data (see text).
where k is a first order rate constant and X ~ represents the amount of drug in the compartment at time zero. In the familiar one-compartment model, where there is no kinetically detectable peripheral compartment, the plasma concentration of drug, Cp, behaves similarly: Cp = C~ e -k~.
ABSORPTION i i i i
V CENTRAL
k12
COMPARTMENT
PERIPHERAL COMPARTMENT
k21 11
kel
METABOLISM & EXCRETION Fig. 2. The two-compartment open-system pharmacokinetic model. predominates. This phase, and its rate constant, a, is characterized by the derived line with steep slope shown in Fig. 1. This line is computed by subtracting from the observed data points the corresponding points on a linear extrapolation of the B-phase to time zero. These residuals are plotted on the same graph and a line of best fit drawn. The slope of this line multiplied by -2.303 defines the disposition rate constant, a. Linear extrapolation of this line to time zero yields the intercept, A, and similar extrapolation of the B-phase yields the intercept, B. The extrapolated plasma concentration at time zero, C~,, equals the sum of A and B.
In the two-compartment open-system, these first-order characteristics apply not only to the process of drug elimination from the central compartment, but also to the movement of drug out of central to peripheral compartment and vice versa. The following assumptions are made in the application of a two-compartment open-system kinetic model. 1. An initial dose of drug is delivered into the central compartment and is distributed instantaneously through this compartment. 2. Drug is eliminated only from the central compartment in a process characterized by the first order rate constant kel. 3. Drug passes into and out of the peripheral compartment in accord with processes characterized by the first order rate constants k,~ and k2,, respectively. 4. The volume of the central compartment and the rate constants do not vary with time. The central and peripheral compartments and their respective volumes are derived units and are not meant to correspond to any physiologic or anatomic area. Nevertheless, the concept of volume of distribution is essential to the Understanding and application of pharmacokinetics.~ If equal doses of different drugs are given intravenously, it is well known that the initial plasma concentrations achieved vary widely. This is not surprising since many factors such as molecular weight, lipid solubility, ionization, and plasma or tissue protein binding will influence the observed total plasma concen-
Volume 88 Number 5
Two-compartment open-system kinetic model
87 1
40-
30-
1-3 -
~.1o
o.Oa ~
i
i
i
TIME
i
i
i
TIME
Fig. 3. Linear (left) and semilogarithmic (right) plots of the amount of drug (X) in a compartment as a function of time. Units are arbitrary.
tration. The volume of the central compartment, V1, is defined as: dose g I -
C~ P
That is, V1 is the volume into which the administered dose would appear to distribute instantaneously in order to yield the extrapolated zero-time concentration (A + B). Hence, an observed concentration of drug, Cp, in the central compartment at any time, taken with V1, allows calculation of the amount of drug (X,) in the central compartment at that time, such that: X~ = V,C~,. The amount of drug eliminated per unit time, Q, is equal to ko~ multiplied by the amount of drug present. Since Q = kolX,, it follows that Q
distribute, if the initial instantaneous distribution had occurred into a larger, single compartment. Exclusive use of this volume term has serious limitations and might greatly underestimate "early" plasma concentrations after an intravenous infusion of drug. It is apparent that the actual plasma concentration at any time in the two-compartment model (Fig. 1) is the sum of two exponential processes, such that Cp = Ae -~t + Be fit With use of the terms defined above, algebraic manipulations allow derivation of the following relationships. 1 V~
dose A+B
a + f i = k,~ + k2, + k~, aft = ko,k21
= k e l V 1 C P.
kel = The plasma clearance of a drug, like the renal clearance of a substance, is an important determination which, in the context of this model, remains constant with time and can be shown to be equal to ke,V,. It is important to emphasize that the volume of the central compartment is simply a useful derived constant. This volume is likely to represent those areas into which a drug can distribute rapidly such as plasma and parts of highly perfused organs. Nevertheless, attempts to relate the volume terms to specific physiologic spaces such as extracellular fluid volume or total body water are inappropriate and potentially misleading. Another volume term, the apparent volume of distribution, AVD, is defined as dose/B where B is the intercept of the extrapolation of the fi-phase to time zero. It therefore represents the volume into which the drug would have appeared to
C~ A+B'
B Therefore, computation of the intercepts A and B, and the "hybrid" disposition rate constants a and fi enable all features of the two-compartment model to be derived. It should be noted, however, that the graphic method described is less accurate than nonlinear least squares curve fitting programs to derive the individual first order rate constants and V1. Certain aspects of the clinical applicability of this model deserve attention. The calculation of a maintenance dose rate from the above data is straightforward. It was shown above that for a given plasma concentration, Cp, the amount of drug eliminated from the system per unit time is Q = ko~'V,-Cp.
872
Loughnan et al.
The Journal of Pediatrics May 1976
maxo;'
100
Cp
/ Load= ~
9
s,ow,.)
b
PERIPHERAL
50-
u,i 0 a 20
i
0 I,z u,.I ~0
mir
Cp
w 0.
C
CENTRAL 10-
d
Fig. 4. Plasma concentration versus time curves predicted with use of an identical maintenance infusion rate after four different circumstances of drug loading (see text).
s~
,
;
,
~
,
TIME
Hence, if the amount eliminated per unit time, Q, is replaced by continuous infusion, then a plateau plasma concentration (C~) will be maintained. Q is thus the maintenance dose rate which will maintain C~. For many drugs, initiation of a continuous infusion would be expected to generate a plasma concentration of 97% of plateau value in approximately five B-half lives 7 (Fig. 4, d). One method proposed to rapidly achieve and maintain a given plasma concentration, Cp, is to administer a loading dose, R, where': R=
Q
ke~ ' V~ 9 C v
- = B B When this dose is given very rapidly, initial p!asma concentrations are as shown in Fig. 4, a. It is readily calculated that when this is done, the initial plasma concentratiofi is ~=
koj /~ " c ; ~
Since ko~ is always greater than B, this method of loading results in a C ~ that exceeds C~. The potential therapeutic or toxic effect produced by these high early plasma concentrations will depend upon whether such an effect is related to the drug concentration in the central or the peripheral compartment. In practice, it may be preferable to avoid these high plasma concentrations by administering the !oading dose more slowly (Fig. 4, b). Wagneff has described a method for calculation of an appropriate infusion rate of the loading dose, and has applied the
Fig. 5. Percent of dose in the central and peripheral compartments as a function of time. Units are arbitrary. method to theophylline in simulations. An alternative suggestion ~~ involves a loading dose, R = Q/kol (Fig. 4, e). This method avoids early high plasma concentrations but results in a subsequent decrease in plasma concentrations followed by a gradual increase to C~.8 The time course of this decrease relates to the half-life of the drug; in the case of lidocaine, plasma concentrations decreased to about half the initial value at 20 minutes and had risen to 90% of this value by four hours? Optimal suppression of cardiac arrhythmias might therefore not be obtained with use of this regimen. Finally, as alluded to above, with drugs whose disposition is best interpreted with use of the two-compartment model, one or another action of the drug might correlate better with concentration of drug in the central or in the peripheral compartment. Knowledge of the volume terms and relevant rate constants enables the calculation at any time of tlae amount of drug in the peripheral compartment (Fig. 5). It can also be shown that, throughout the Bphase, the amount of drug in the central compartment as a fraction oi" drug still in the body remains constant and equals B/ko, These relationships are important in the interpretation of plasma concentration data. It is not uncommon to have available in the literature plasma concentrations measured only during the B-phase. The use of these data to calculate maintenance dose rate
Volume 88 Number 5
Two-compartment open-system kinetic model
therefore necessitates the assumption of a one-compartment model: Q = fi.AVt~.Cp. Unless the logarithm of plasma concentration declines with time in a perfectly linear fashion from time zero, an error must be introduced. To evaluate the magnitude of this error, consider the equation:
k~
C~ A+B
fi
I f A l a is much smaller than Blfi, then o Cp
kel
kel *
dose o Cp
873
frequent plasma concentration measurements required for complex multicompartment analyses. It then becomes important to consider the magnitude of errors introduced by. using simpler models. Such errors cannot be ignored, but may be clinically acceptable depending upon the characteristics of the drug involved and the particular aims of the study. In conclusion, all kinetic models, no matter how sophisticated, contain many assumptions which may be more or less valid. Therefore, any predictions made from single dose kinetics should be tested in an appropriate clinical study before acceptance for general usage. Furthermore, such kinetic predictions are applicable only to a similar patient population with respect to age, condition, and disease.
B/fi dose rot.
- -
B
ke~ "V, ~ fi'AVD. Therefore, it is only when A / a is much less than B/fl, that plasma clearance and maintenance dose rate calculated using a one- or two-compartment analysis will be approximately equal? This outline of two compartment kinetics is necessarily limited and can be found in more detail elsewhere? ....... 3 However, it is designed to stress three important uses of a two-compartment analysis: (1) If an inappropriate model (e.g., one-compartment) is selected tO analyze data for a drug in which A / a cannot be ignored, then grossly erroneous loading and maintenance doses would be developed. (2) To define plasma concentrations shortly after rapid intravenous administration, a two-compartment analysis is essential, and is required .to determine appropriate infusion rates of intravenous loading doses) (3) Factors other than elimination by metabolism and/or excretion may have a major effect on drug disposition. For example, the rate of release of the drug from peripheral compartment may in itself largely determine the fi-phase half-life. 12 This factor cannot be ignored in designing dosage regimens in situations of impaired drug elimination such as renal failure, especially for those drugs which exhibit pronounced "two-compartment characteristics." These considerations will be of importance not only in the design of dosage regimens in different clinical situations, but also in defining the particular variables responsible for changing dosage requirements with age. In pediatric studies, ethical considerations may limit the
REFERENCES
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