Multicompartment Pharmacokinetic Models
`and Pharmacologie Effects
`
`GERHARD LEVY, MILO GIBALDI, and WILLIAM J. JUSKO
`
`The pharmacokinetic analysis of drug concentration in
`Abstract
`the plasma versus time data on the basis of multicompartment
`models makes it possible to examine not only the relationship be-
`tween drug concentration in the plasma (or serum) and the intensity
`of a pharmacologie effect, but permits also an assessment of the
`relationship between pharmacologic effect and the relative drug
`levels in other apparent compartments of the body. Experimental
`difficulties such as sampling and assay problems limit the precision
`of data obtainable in the early (distributive) phase of drug concen-
`tration decline, so that most available data tend to fit a two -
`compartment rather than a more complex model. The rate constants
`derived from a two -compartment analysis of drug concentration
`data are almost certainly not "pure" but still hybrid, though "purer"
`than rate constants obtained by assuming the still simpler single -
`compartment open model. Suitable pharmacologic effect data,
`obtained at frequent intervals after drug administration, can show
`whether the site of action can be considered as part of a homo-
`geneous tissue compartment (of the two -compartment system) or if
`the site of action must be considered as a distinct and separate
`pharmacokinetic compartment. This is illustrated by actual example,
`using previously published data of drug concentrations in the plasma
`and pharmacologie effects of lysergic acid diethylamide in man.
`
`Pharmacokinetic models - pharmacologic effects
`Keyphrases
`Action -site relation -model compartment drug concentration
`LSD -phar-
`Models, multicompartment - selection, data fitting
`macokinetic analysis
`
`Interest in the kinetics of drug absorption, distribu-
`tion, and elimination has resulted in the development of
`many analytical and mathematical techniques which
`permit relatively sophisticated pharmacokinetic analyses
`(1). It is now frequently possible to characterize the
`distribution and elimination of a drug on the basis of
`two- or even three -compartment open models, and to
`calculate the relative drug concentrations in each of
`these compartments as a function of time. Similarly,
`clinical pharmacologie techniques have advanced to the
`point where it has become possible to make quantitative
`correlations between the intensity of certain pharma-
`cologic effects and drug concentrations in plasma (or
`amounts of drug in the body). A number of these cor-
`relations have been shown to be consistent with basic
`pharmacokinetic principles (2, 3), and it has now be-
`come feasible to deal with even more complex systems
`such as the kinetics of apparently delayed pharmacologie
`effects (4). With a combination of precise and sufficiently
`frequent drug concentration and pharmacologic effect
`data, it should be possible to determine if the site of
`action of a drug is a pharmacokinetically indistinguish-
`able part of one of the hypothetical compartments of
`the body (as determined for the particular drug), or if
`this site is in fact (part of) a distinctly separate pharma-
`cokinetic compartment. Thus, suitable pharmacologic
`effect data may add another dimension to pharmaco-
`kinetic analyses by either indicating an association of
`
`422
`
`Journal of Pharmaceutical Sciences
`
`the site of action with one of the compartments evolved
`from the pharmacokinetic analysis of drug concentra-
`tion data, or by suggesting the presence of a distinctly
`separate compartment - embodying the site of action
`of a drug-which is not readily discernible from the
`drug concentration data alone. It is the purpose of this
`communication to consider some of the relationships
`between the time course of pharmacologic effects and
`drug concentrations in the plasma and in other pharma-
`cokinetically identifiable compartments of the body,
`and to show by actual example how a combination of
`drug concentration and pharmacologie activity data
`can be helpful (a) in the development of appropriate
`pharmacokinetic models; (b) in assessing the usefulness
`and limitations of pharmacokinetic analyses based only
`on drug concentration data; and (c) in obtaining a
`better understanding of the relationship of drug con-
`centration in the plasma or serum and the intensity of
`pharmacologie effects. The example to be used is a
`pharmacologic effect (impairment of ability to solve
`mathematical problems) of lysergic acid diethylamide
`(LSD) in man, the intensity of this effect having been
`determined at frequent intervals after intravenous ad-
`ministration of the drug, concurrently with determina-
`tions of L SD concentrations in the plasma.
`
`METHODS
`
`Plasma concentrations and pharmacologie effects of LSD follow-
`ing intravenous administration of 2 mcg. /kg. body weight of the
`drug to five human subjects were obtained from Aghajanian and
`Bing (5). The plasma concentrations (Cp) were given equal weight
`and were used as input data for the digital computer program of
`Marquardt (6) to provide a bi- exponential and tri -exponential
`least- squares regression fit to the data. The constants thus obtained
`were used as digital computer input along with the appropriate
`equations as described by Rescigno and Sege (7) to evaluate the
`rate constants and compartment drug levels of the two - and three -
`compartment open models (Models I and II).
`
`kit
`
`central
`tissue
`compartment k21 compartment
`,1. ksi
`
`Model I
`
`k14
`
`rapidly
`slowly
`ki.
`accessible `
`± accessible
`central
`compartment kit compartment ko, compartment
`k.i
`Model II
`
`RESULTS AND DISCUSSION
`
`A bi- exponential fit of the LSD -plasma- concentration data as a
`function of time (t) yielded the following expression:
`
`Cp = 5.469 é 7.Blbe + 6.924 e ° -2a"
`
`(Eq. 1)
`
`with a zero -time intercept of 12.39 ng. /ml. (Cp °). The rate constants
`
`EXHIBIT
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`20
`
`480,
`
`80-
`
`0.05
`
`0.20
`0.40
`0.10
`FRACTION OF DOSE
`Figure 2- Relationship between the fractional amount of LSD in
`the tissue compartment of Model I and the intensity of the pharma-
`cologic_effect. The number next to each symbol represents the time
`in minutes when the measurements were made.
`
`Figure 3 shows the relative amounts of LSD in the central, the
`rapidly accessible, and the slowly accessible compartments as a
`function of time, calculated according to Model II. Comparing
`Figs. 1 and 3 it appears that the experimental plasma level data fit
`very well to both Model I and Model II. The sum of the squared
`deviations of the observed from the calculated plasma concentra-
`tions is 0.003 on the basis of Model II, but 0.052 on the basis of
`Model I.
`A plot of the intensity of pharmacologie effect versus the loga-
`rithm of the fractional amount of LSD in the rapidly accessible
`compartment of Model II shows a relationship similar to that
`depicted in Fig. 2. On the other hand, a similar plot with respect to
`the drug level in the slowly accessible compartment of Model II
`yields a straight line (Fig. 4). Unlike the case shown in Fig. 2, the
`data obtained in the distribution period fit very well on the regression
`line for all of the data points and give no evidence of a systematic
`deviation The correlation coefficient of the data in Fig. 4 is 0.98.
`These observations suggest that the site of action of LSD is part of
`the slowly accessible compartment of Model II and show that
`Model I is insufficient to explain the total time course of LSD effects
`(9). The good correlation between the pharmacologie effect of LSD
`and the drug level in the slowly accessible compartment of Model II
`does not mean necessarily that the site of action of LSD and the
`slowly accessible compartment are identical. Rather, Model II is a
`better approximation of the biologic system than is Model I.
`Berman (10), in reviewing the application of multicompartmental
`analysis to pharmacokinetics, pointed out that compartmental
`models are frequently a consequence of limited resolution in the
`
`20
`
`40 cr
`0
`60 uVi
`
`80 w
`
`0
`
`100 0
`za
`2tro,L
`
`cew
`d
`
`0
`
`4 T
`
`IME, hr.
`
`Figure 3 -LSD in the central, rapidly accessible, and slowly accessible
`compartments as a function of time. Data analysis according to
`Model II. Key: , relative concentration of LSD in the plasma;
`, central compartment; - - -, rapidly
`O, performance test scores;
`accessible compartment; - -, slowly accessible compartment.
`
`Vol. 58, No. 4, April 1969 Q 423
`
`obtained according to Model I are:
`kei = 0.407
`k12 = 3.083
`k2, = 4.358 hr. -1
`
`Figure 1 shows the relative amounts of LSD in the central and
`tissue compartments as a function of time, calculated according to
`Model I. Shown also are the actual plasma concentrations divided
`by C,, °, and the intensities of the pharmacologic effect at various
`times after drug administration. Comparison of the time course of
`pharmacologic effect and of drug levels in the central compartment
`shows that the site of action of LSD is apparently not located in the
`central compartment. Otherwise, the earliest measurement of phar-
`macologic effect would have been expected to yield the highest in-
`hibition of normal performance (8). Can the site of action, therefore,
`be considered an indistinguishable part of the tissue compartment?
`This possibility can be tested by relating drug levels in the tissue
`compartment to the intensity of the pharmacologic effect. If a given
`tissue level yields essentially the same intensity of pharmacologic
`effect during the distributive phase (when tissue levels are rising) as
`during the period when tissue levels are declining, it can be con-
`cluded that the site of action is a pharmacokinetically indistinguish-
`able part of the tissue compartment. Figure 2 shows the relationship
`between the fractional amount of LSD in the tissue compartment
`and the pharmacologic effect. It is readily apparent that data ob-
`tained during the early (distributive) phase do not show the same
`relationship between drug level and effect as data obtained during
`the period when tissue levels of LSD were declining. It should be
`noted that the deviation of the early data is not random but sys-
`tematic in that the data converge with time on the response -log dose
`regression line obtained with the other data points. Thus, these data
`do not permit the conclusion that the site of action of LSD is an
`indistinguishable part of the tissue compartment. It appears that the
`site of action may in fact be (part of) a distinctly separate pharma-
`cokinetic compartment.
`A tri- exponential fit of the LSD -plasma -concentration data as a
`function of time yielded the expression:
`Cp = 14.64 e-21'231 + 2.016 a 2.4841 + 6.531 e- o.217e
`with a zero-time intercept of 23.19 ng. /ml. The rate constants ob-
`tained according to Model II are:
`kt = 0.738 hr.-1
`k18 = 14.54 hr.-1
`k31 = 10.21
`k14 = 1.565
`k41 = 1.879
`
`(Eq. 2)
`
`1.0
`
`0.5
`
`j O.
`
`O
`
`20
`
`40 ceo
`60 Ñ
`
`0
`
`80 F
`
`100 Z
`
`2e
`
`tOwc
`
`ew
`a_
`
`6
`
`2
`
`4
`TIME, hr.
`Figure 1-LSD in the central and tissue compartments as a function of
`time after intravenous administration of 2 mcg. /kg.; average of five
`normal human subjects. Data analysis according to Model I. The
`upper and lower curves represent the central and tissue compartments,
`respectively. Closed circles are the relative plasma concentrations of
`LSD (i.e., the actual concentrations divided by the calculated zero-
`time concentration); the open circles are performance test scores
`(expressed as percent of normal performance in solving mathematical
`problems). Data from Reference 5.
`
`Page 2
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`

`

`20
`
`40 -
`
`30
`
`,
`/60
`/ 120
`
`5
`
`2a0
`
`0.65
`
`0.15
`0.10
`0.075
`FRACTION OF DOSE
`Figure 4- Relationship between the fractional amount of LSD in
`the slowly accessible compartment of Model II and the intensity of
`a pharmacologic effect. The number next to each symbol represents
`the time in minutes when the measurements were made.
`
`u.20
`
`data. He then stated that "In model building one starts with the
`simplest model consistent with known information, with precon-
`ceptions of the investigator, and with the data. New experiments are
`then designed to test the model further and to reveal new features
`about the system that are not contained in the model. The model is
`then modified to include the new information, and the process
`is repeated."
`This is the approach that has been used here in that the pharma-
`cologic activity data were used to test the two -compartment model
`(Model I), caused it to be rejected, and led to the three -compartment
`model (Model II) which is consistent with the available data. How-
`ever, Model Il is not the only three -compartment model which fits
`the data. For example, the data also fit to a three -compartment
`model (11) in which drug elimination occurs from the rapidly
`accessible compartment.
`It is virtually impossible, in most instances, to distinguish between
`a two-compartment and more complex pharmacokinetic system on
`the basis of plasma concentrations alone. This difficulty is exempli-
`fied in the data shown in Fig. 5 which depicts the plasma concentra-
`tion of LSD during the first hour and the theoretical curves related
`to Model I and Model II. The experimental data appear to fit each
`curve equally well. A distinction between the two curves could have
`been made only on the basis of experimental data obtained during
`the first two or three minutes after injection, providing that blood
`mixing problems would not have interfered. However, Wichmann
`et al. (12) have observed considerable fluctuation in the serum con-
`centration of BSP during the first few minutes after intravenous in-
`jection of this drug, and attribute this to incomplete mixing of BSP
`in the blood. It is likely that this difficulty would be encountered
`also with other drugs.
`It should be apparent, upon reflection, that any pharmacokinetic
`model is, by definition, a simplification of the real biologic system.
`In this context, it is to be regretted that a distinction has sometimes
`been made between "true" and "hybrid" rate constants when refer-
`ring to the results of pharmacokinetic analyses based on two- and one-
`compartment models, respectively. The development of more de-
`tailed models simply results in increasingly closer approximation of
`the real biologic system. This process of refinement or purification
`may be thought of as being analogous to the successive processes of
`purification of an enzyme from its source.
`It should be recognized that the pharmacokinetic analysis of the
`LSD data presented here has led to the development of a model
`which is consistent with the experimental data; this does not mean
`that the model is the correct one. There may be certain unrecognized
`factors, such as a possible delay in the pharmacologie effect of LSD
`relative to the time course of its concentration at the site of action,
`which could affect a theoretical analysis of the data. If and when
`such factors become evident, the model has to be revised. Many of
`the complexities of a pharmacokinetic analysis of pharmacologie
`
`424
`
`Journal of Pharmaceutical Sciences
`
`<
`
`<
`
`0.8
`
`1.2
`
`0.0
`
`0.4
`T1ME, hr.
`Figure 5- Concentrations of LSD in the plasma during the first hour
`after intravenous injection and the theoretical curves which are
`). Note that a distinction
`related to Model 1(- - -) and Model 11(
`between these two curves can only be made in the first 2 to 3 min.
`
`effects can be evaded by restricting the analysis to that time period
`when drug levels decline mono -exponentially,) since, as is evident in
`Fig. 3 and as has been shown theoretically (11, 13), the ratio of drug
`levels in each hypothetical compartment of the body is constant
`during this period. Examples of analyses of this type have been
`presented previously (2). While this paper has dealt specifically with
`a pharmacokinetic analysis of LSD data, it has been the intention of
`the authors to use the specific example to present principles which
`should be applicable whenever suitable drug concentration and
`pharmacologie effect data are available.
`
`REFERENCES
`
`(1) E. R. Garrett, Antibiot. Chemotherapia, 12, 227(1964).
`(2) G. Levy, Clin. Pharmacol. Therap., 7, 362(1966).
`(3) E. R. Garrett, A. J. Agren, and H. J. Lambert, Intern J. Clin.
`Pharm., 1, 1(1967).
`(4) R. Nagashima, R. A. O'Reilly, and G. Levy, Clin. Pharmacol.
`Therap., 10, 22(1969).
`(5) G. K. Aghajanian and O. H. L. Bing, ibid. 5, 611(1964).
`(6) D. W. Marquardt, "DPE- NLIN," Share General Library
`Program No. 7 -1354.
`(7) A. Rescigno and G. Segre, "Drug and Tracer Kinetics,"
`Blaisdell, Waltham, Mass., 1966, pp. 27 -28, 93 -94.
`(8) J. G. Wagner, G. K. Aghajanian, and O. H. L. Bing, Clin.
`Pharmacol. Therap., 9, 635(1968).
`(9) G. Levy, ibid., 10, 134(1969).
`(10) M. Berman, Ann. Internal Med., 68, 423(1968).
`(11) R. Nagashima, G. Levy, and R. A. O'Reilly, J. Pharm. Sci.,
`57, 1888(1968),
`(12) H. M. Wichmann, H. Rind, and E. Gladtke, Zeit. Kinder -
`heilk., 103, 262(1968).
`(13) M. Gibaldi, R. Nagashima, and G. Levy, J. Pharm. Sci.,
`in press.
`
`ACKNOWLEDGMENTS AND ADDRESSES
`
`Received October 17, 1968, from the Department of Pharma-
`ceutics, School of Pharmacy, State University of New York at Buffalo,
`Buffalo, NY 14214
`Accepted for publication December 5, 1968.
`
`x Pharmacokinetic models in which drug levels in the central compart-
`ment appear to decline mono -exponentially, but do not, will be con-
`sidered in a subsequent communication.
`
`Page 3
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